CINXE.COM

<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <meta content=" Adaptive control, composite learning, exponential stability, mismatched uncertainty, parameter convergence. " lang="en" name="keywords"/> <base href="/html/2401.10785v2/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S1" title="In Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">I </span><span class="ltx_text ltx_font_smallcaps">Introduction</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S2" title="In Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II </span><span class="ltx_text ltx_font_smallcaps">Problem Formulation</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3" title="In Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">III </span><span class="ltx_text ltx_font_smallcaps">Modular Backstepping Control Design</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4" title="In Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">IV </span><span class="ltx_text ltx_font_smallcaps">Composite Learning Design</span></span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.SS1" title="In IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-A</span> </span><span class="ltx_text ltx_font_italic">Composite Learning High-Order Tuner</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.SS2" title="In IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-B</span> </span><span class="ltx_text ltx_font_italic">Some Discussions</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S5" title="In Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">V </span><span class="ltx_text ltx_font_smallcaps">Theoretical Guarantees</span></span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S5.SS1" title="In V Theoretical Guarantees ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">V-A</span> </span><span class="ltx_text ltx_font_italic">Parameter Convergence Results</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S5.SS2" title="In V Theoretical Guarantees ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">V-B</span> </span><span class="ltx_text ltx_font_italic">Closed-Loop Stability Results</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S5.SS3" title="In V Theoretical Guarantees ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">V-C</span> </span><span class="ltx_text ltx_font_italic">Robustness Results</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S6" title="In Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">VI </span><span class="ltx_text ltx_font_smallcaps">Simulation Studies</span></span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S6.SS1" title="In VI Simulation Studies ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">VI-A</span> </span><span class="ltx_text ltx_font_italic">Stability and Convergence Comparison</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S6.SS2" title="In VI Simulation Studies ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">VI-B</span> </span><span class="ltx_text ltx_font_italic">Transient Performance Comparisons</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S7" title="In Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">VII </span><span class="ltx_text ltx_font_smallcaps">Conclusions</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A1" title="In Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A </span><span class="ltx_text" style="color:#000099;">The derivation of (<span class="ltx_text ltx_ref_tag">12</span>)</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A2" title="In Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B </span>The proof of Theorem 1</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A3" title="In Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">C </span>The proof of Theorem 2</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A4" title="In Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">D </span>The proof of Theorem 3</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Tian Shi, , Changyun Wen, , and Yongping Pan </span><span class="ltx_author_notes">This work was supported in part by the Major Key Project of PCL, China, under Grant No. PCL2024A04 (<span class="ltx_text ltx_font_italic" id="id1.1.id1">Corresponding author: Yongping Pan</span>).Yongping Pan is with the Peng Cheng Laboratory, Shenzhen 518057, China, and also the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail: eee-yppan@ntu.edu.sg).Tian Shi is with the School of Computer Science and Engineering, Sun Yat-sen University, Guangzhou 510006, China (e-mail: shit23@mail2.sysu.edu.cn).Changyun Wen is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail: ecywen@ntu.edu.sg).</span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id2.id1">Adaptive backstepping control provides a feasible solution to achieve asymptotic tracking for mismatched uncertain nonlinear systems. However, the closed-loop stability depends on high-gain feedback generated by nonlinear damping terms, and closed-loop exponential stability with parameter convergence involves a stringent condition named persistent excitation (PE). This paper proposes a composite learning backstepping control (CLBC) strategy based on modular backstepping and high-order tuners to compensate for the transient process of parameter estimation and achieve closed-loop exponential stability without the nonlinear damping terms and the PE condition. A novel composite learning mechanism that maximizes the staged exciting strength is designed for parameter estimation, such that parameter convergence can be achieved under a condition of interval excitation (IE) or even partial IE that is strictly weaker than PE. An extra prediction error is employed in the adaptive law to ensure the transient performance without nonlinear damping terms. The exponential stability of the closed-loop system is proved rigorously under the partial IE or IE condition. Simulations have demonstrated the effectiveness and superiority of the proposed method in both parameter estimation and control compared to state-of-the-art methods.</p> </div> <div class="ltx_keywords"> <h6 class="ltx_title ltx_title_keywords">Index Terms: </h6> Adaptive control, composite learning, exponential stability, mismatched uncertainty, parameter convergence. </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">I </span><span class="ltx_text ltx_font_smallcaps" id="S1.1.1">Introduction</span> </h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">Adaptive control is desirable due to its unique capacity to accommodate uncertain and time-varying properties of nonlinear systems, where recent survey papers can be referred to <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib1" title="">1</a>, <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib2" title="">2</a>, <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib3" title="">3</a>, <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib4" title="">4</a>, <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib5" title="">5</a>, <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib6" title="">6</a>]</cite>. The presence of mismatched uncertainties is a major obstacle to adaptive control of nonlinear systems. Adaptive integral backstepping with overparameterization, which combines integral backstepping and direct adaptive control, is a precursor to relax the above obstacle by designing an adaptive law to adjust a virtual control input at each backstepping step <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib7" title="">7</a>]</cite>. A tuning function approach is a direct adaptive backstepping approach that avoids overparameterization <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib8" title="">8</a>, Ch. 4]</cite>, where an adaptive law termed as a tuning function is constructed iteratively at each backstepping step, while an actual adaptive law is generated at the last step by all previous tuning functions. It is revealed that adaptive backstepping control driven by tuning functions has a higher-order tracking property <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib9" title="">9</a>]</cite>. There are two common drawbacks for the above adaptive backstepping approaches: 1) The “explosion of complexity” exists due to the repeated differentiation of virtual control inputs; 2) the exponential stability of the closed-loop system (implying parameter convergence and robustness) relies on a strict condition termed persistent excitation (PE), which requires system states contain sufficiently rich spectral information all the time.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">Adaptive dynamic surface control (DSC) applies a first-order linear filter to estimate the time derivative of each virtual control input at its backstepping step for addressing the complexity problem of the classical backstepping control <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib10" title="">10</a>]</cite>. The performance and robustness of adaptive DSC have been enhanced by integrating neural network (NN) approximation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib11" title="">11</a>]</cite>, nonlinear filtering <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib12" title="">12</a>]</cite>, power integration <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib13" title="">13</a>]</cite>, coordinate transformation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib14" title="">14</a>]</cite>, prescribed performance control <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib15" title="">15</a>]</cite>, etc. An approach similar to DSC named command-filtered backstepping control (CFBC) employs a second-order linear filter to estimate the time derivatives of virtual control inputs and introduces compensation terms for stability guarantees <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib16" title="">16</a>]</cite>. The performance and robustness of adaptive CFBC have been improved by some techniques, such as NN approximation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib17" title="">17</a>]</cite>, exact differentiation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib18" title="">18</a>]</cite>, and immersion and invariance <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib19" title="">19</a>]</cite>. But adaptive DSC and CFBC suffer from the explosion of the dynamic order and the loss of global stability and asymptotic tracking due to the employed filtering operations <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib20" title="">20</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">A modular backstepping approach follows the certainty equivalence principle that separates control and estimation designs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib8" title="">8</a>, Ch. 5]</cite>. A key feature of this approach is that the time derivatives of virtual control inputs are replaced by their partial derivatives with respect to system states and reference signals, while the resulting high-order time derivatives of parameter estimates are treated as additive disturbances. Therefore, the modular backstepping approach does not involve tuning functions and overparameterization and has a lower complexity. This approach ensures the closed-loop stability with strong robustness owing to introducing a nonlinear damping term in a stabilizing function at each backstepping step <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib21" title="">21</a>]</cite>. A standard gradient-descent identifier derived from a swapping scheme can be combined with modular backstepping to achieve asymptotic tracking <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib8" title="">8</a>, Ch. 6]</cite>. Nevertheless, due to the existence of the high-order time derivatives of parameter estimates in the closed-loop system, the modular backstepping approach can degrade the transient tracking performance and prevent exact parameter estimation even in the presence of PE.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">A high-order tuner (HOT) approach in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib22" title="">22</a>]</cite> can efficiently remove the negative influence caused by the time derivatives of parameter estimates in modular backstepping. The key idea of the HOT is to apply a linear filter with a sufficiently high relative degree to the adaptive law, such that the exact implementation of the high-order time derivatives of parameter estimates becomes feasible. In <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib23" title="">23</a>]</cite>, a direct adaptive control scheme was combined with the HOT to counteract the transient process of parameter estimates caused by their high-order time derivatives, improving the transient and steady-state tracking performance. In <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib24" title="">24</a>]</cite>, a memory regressor extension (MRE) identifier, which utilizes regressor extension with filtering, was combined with the HOT to design an indirect adaptive control law, where the HOT is applied to an extended regression model such that the high-order time derivatives of parameter estimates can be calculated exactly without filtering delay. However, the above two methods still need nonlinear damping terms to ensure closed-loop stability and transient performance and resort to the stringent PE condition for exponential stability guarantees.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.1">From the above discussions, existing modular backstepping methods have the following limitations:</p> <ol class="ltx_enumerate" id="S1.I1"> <li class="ltx_item" id="S1.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S1.I1.i1.p1"> <p class="ltx_p" id="S1.I1.i1.p1.1">The transient and steady-state tracking performances rely on nonlinear damping terms;</p> </div> </li> <li class="ltx_item" id="S1.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S1.I1.i2.p1"> <p class="ltx_p" id="S1.I1.i2.p1.1">The transient process of parameter estimates due to their high-order time derivatives can destroy the tracking performance and parameter convergence;</p> </div> </li> <li class="ltx_item" id="S1.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">3.</span> <div class="ltx_para" id="S1.I1.i3.p1"> <p class="ltx_p" id="S1.I1.i3.p1.1">The stringent PE condition must be fulfilled to realize the exponential stability of the closed-loop system.</p> </div> </li> </ol> <p class="ltx_p" id="S1.p5.2">Motivated by the above facts, this paper proposes a composite learning backstepping control (CLBC) strategy that ensures the exponential stability of the closed-loop system under relaxed excitation conditions for a class of strict-feedback uncertain nonlinear systems. The design procedure is as follows: First, the modular backstepping scheme without nonlinear damping terms is given to facilitate the control design; second, a generalized regression equation is constructed by the swapping technique and an interval integration; third, a linear filter is applied to the generalized regression equation to generate a generalized linearly parameterized model; fourth, a generalized prediction error is designed to exploit online data memory; fifth, a general prediction error is introduced to counteract a modeling error term; finally, a composite learning HOT is constructed by mixing the two prediction errors to implement the high-order time derivatives of parameter estimates exactly. The contributions of this study lie in three folds:</p> <ol class="ltx_enumerate" id="S1.I2"> <li class="ltx_item" id="S1.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S1.I2.i1.p1"> <p class="ltx_p" id="S1.I2.i1.p1.1">A feasible modular backstepping strategy termed CLBC is proposed to guarantee transient and steady-state tracking without nonlinear damping terms or high control gains;</p> </div> </li> <li class="ltx_item" id="S1.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S1.I2.i2.p1"> <p class="ltx_p" id="S1.I2.i2.p1.1">An algorithm of staged exciting strength maximization is designed to enhance the online data memory of composite learning in different partial excitation stages;</p> </div> </li> <li class="ltx_item" id="S1.I2.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">3.</span> <div class="ltx_para" id="S1.I2.i3.p1"> <p class="ltx_p" id="S1.I2.i3.p1.1">The exponential stability and robustness of the closed-loop system with parameter convergence are proven under the condition of interval excitation (IE) or even partial IE.</p> </div> </li> </ol> </div> <div class="ltx_para" id="S1.p6"> <p class="ltx_p" id="S1.p6.42"><span class="ltx_text ltx_font_italic" id="S1.p6.42.1">Notations</span>: <math alttext="\mathbb{R}" class="ltx_Math" display="inline" id="S1.p6.1.m1.1"><semantics id="S1.p6.1.m1.1a"><mi id="S1.p6.1.m1.1.1" xref="S1.p6.1.m1.1.1.cmml">ℝ</mi><annotation-xml encoding="MathML-Content" id="S1.p6.1.m1.1b"><ci id="S1.p6.1.m1.1.1.cmml" xref="S1.p6.1.m1.1.1">ℝ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.1.m1.1c">\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.1.m1.1d">blackboard_R</annotation></semantics></math>, <math alttext="\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S1.p6.2.m2.1"><semantics id="S1.p6.2.m2.1a"><msup id="S1.p6.2.m2.1.1" xref="S1.p6.2.m2.1.1.cmml"><mi id="S1.p6.2.m2.1.1.2" xref="S1.p6.2.m2.1.1.2.cmml">ℝ</mi><mo id="S1.p6.2.m2.1.1.3" xref="S1.p6.2.m2.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S1.p6.2.m2.1b"><apply id="S1.p6.2.m2.1.1.cmml" xref="S1.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S1.p6.2.m2.1.1.1.cmml" xref="S1.p6.2.m2.1.1">superscript</csymbol><ci id="S1.p6.2.m2.1.1.2.cmml" xref="S1.p6.2.m2.1.1.2">ℝ</ci><plus id="S1.p6.2.m2.1.1.3.cmml" xref="S1.p6.2.m2.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.2.m2.1c">\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\mathbb{R}^{n}" class="ltx_Math" display="inline" id="S1.p6.3.m3.1"><semantics id="S1.p6.3.m3.1a"><msup id="S1.p6.3.m3.1.1" xref="S1.p6.3.m3.1.1.cmml"><mi id="S1.p6.3.m3.1.1.2" xref="S1.p6.3.m3.1.1.2.cmml">ℝ</mi><mi id="S1.p6.3.m3.1.1.3" xref="S1.p6.3.m3.1.1.3.cmml">n</mi></msup><annotation-xml encoding="MathML-Content" id="S1.p6.3.m3.1b"><apply id="S1.p6.3.m3.1.1.cmml" xref="S1.p6.3.m3.1.1"><csymbol cd="ambiguous" id="S1.p6.3.m3.1.1.1.cmml" xref="S1.p6.3.m3.1.1">superscript</csymbol><ci id="S1.p6.3.m3.1.1.2.cmml" xref="S1.p6.3.m3.1.1.2">ℝ</ci><ci id="S1.p6.3.m3.1.1.3.cmml" xref="S1.p6.3.m3.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.3.m3.1c">\mathbb{R}^{n}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.3.m3.1d">blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\mathbb{R}^{m\times n}" class="ltx_Math" display="inline" id="S1.p6.4.m4.1"><semantics id="S1.p6.4.m4.1a"><msup id="S1.p6.4.m4.1.1" xref="S1.p6.4.m4.1.1.cmml"><mi id="S1.p6.4.m4.1.1.2" xref="S1.p6.4.m4.1.1.2.cmml">ℝ</mi><mrow id="S1.p6.4.m4.1.1.3" xref="S1.p6.4.m4.1.1.3.cmml"><mi id="S1.p6.4.m4.1.1.3.2" xref="S1.p6.4.m4.1.1.3.2.cmml">m</mi><mo id="S1.p6.4.m4.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S1.p6.4.m4.1.1.3.1.cmml">×</mo><mi id="S1.p6.4.m4.1.1.3.3" xref="S1.p6.4.m4.1.1.3.3.cmml">n</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S1.p6.4.m4.1b"><apply id="S1.p6.4.m4.1.1.cmml" xref="S1.p6.4.m4.1.1"><csymbol cd="ambiguous" id="S1.p6.4.m4.1.1.1.cmml" xref="S1.p6.4.m4.1.1">superscript</csymbol><ci id="S1.p6.4.m4.1.1.2.cmml" xref="S1.p6.4.m4.1.1.2">ℝ</ci><apply id="S1.p6.4.m4.1.1.3.cmml" xref="S1.p6.4.m4.1.1.3"><times id="S1.p6.4.m4.1.1.3.1.cmml" xref="S1.p6.4.m4.1.1.3.1"></times><ci id="S1.p6.4.m4.1.1.3.2.cmml" xref="S1.p6.4.m4.1.1.3.2">𝑚</ci><ci id="S1.p6.4.m4.1.1.3.3.cmml" xref="S1.p6.4.m4.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.4.m4.1c">\mathbb{R}^{m\times n}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.4.m4.1d">blackboard_R start_POSTSUPERSCRIPT italic_m × italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> denote the spaces of real numbers, positive real numbers, real <math alttext="n" class="ltx_Math" display="inline" id="S1.p6.5.m5.1"><semantics id="S1.p6.5.m5.1a"><mi id="S1.p6.5.m5.1.1" xref="S1.p6.5.m5.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S1.p6.5.m5.1b"><ci id="S1.p6.5.m5.1.1.cmml" xref="S1.p6.5.m5.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.5.m5.1c">n</annotation><annotation encoding="application/x-llamapun" id="S1.p6.5.m5.1d">italic_n</annotation></semantics></math>-vectors and real <math alttext="m\times n" class="ltx_Math" display="inline" id="S1.p6.6.m6.1"><semantics id="S1.p6.6.m6.1a"><mrow id="S1.p6.6.m6.1.1" xref="S1.p6.6.m6.1.1.cmml"><mi id="S1.p6.6.m6.1.1.2" xref="S1.p6.6.m6.1.1.2.cmml">m</mi><mo id="S1.p6.6.m6.1.1.1" lspace="0.222em" rspace="0.222em" xref="S1.p6.6.m6.1.1.1.cmml">×</mo><mi id="S1.p6.6.m6.1.1.3" xref="S1.p6.6.m6.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.6.m6.1b"><apply id="S1.p6.6.m6.1.1.cmml" xref="S1.p6.6.m6.1.1"><times id="S1.p6.6.m6.1.1.1.cmml" xref="S1.p6.6.m6.1.1.1"></times><ci id="S1.p6.6.m6.1.1.2.cmml" xref="S1.p6.6.m6.1.1.2">𝑚</ci><ci id="S1.p6.6.m6.1.1.3.cmml" xref="S1.p6.6.m6.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.6.m6.1c">m\times n</annotation><annotation encoding="application/x-llamapun" id="S1.p6.6.m6.1d">italic_m × italic_n</annotation></semantics></math>-matrices, respectively, <math alttext="\min\{\cdot\}" class="ltx_Math" display="inline" id="S1.p6.7.m7.2"><semantics id="S1.p6.7.m7.2a"><mrow id="S1.p6.7.m7.2.3.2" xref="S1.p6.7.m7.2.3.1.cmml"><mi id="S1.p6.7.m7.1.1" xref="S1.p6.7.m7.1.1.cmml">min</mi><mo id="S1.p6.7.m7.2.3.2a" xref="S1.p6.7.m7.2.3.1.cmml">⁡</mo><mrow id="S1.p6.7.m7.2.3.2.1" xref="S1.p6.7.m7.2.3.1.cmml"><mo id="S1.p6.7.m7.2.3.2.1.1" stretchy="false" xref="S1.p6.7.m7.2.3.1.cmml">{</mo><mo id="S1.p6.7.m7.2.2" lspace="0em" rspace="0em" xref="S1.p6.7.m7.2.2.cmml">⋅</mo><mo id="S1.p6.7.m7.2.3.2.1.2" stretchy="false" xref="S1.p6.7.m7.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.7.m7.2b"><apply id="S1.p6.7.m7.2.3.1.cmml" xref="S1.p6.7.m7.2.3.2"><min id="S1.p6.7.m7.1.1.cmml" xref="S1.p6.7.m7.1.1"></min><ci id="S1.p6.7.m7.2.2.cmml" xref="S1.p6.7.m7.2.2">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.7.m7.2c">\min\{\cdot\}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.7.m7.2d">roman_min { ⋅ }</annotation></semantics></math> represents the minimum operator, <math alttext="\lambda_{\min}(A)" class="ltx_Math" display="inline" id="S1.p6.8.m8.1"><semantics id="S1.p6.8.m8.1a"><mrow id="S1.p6.8.m8.1.2" xref="S1.p6.8.m8.1.2.cmml"><msub id="S1.p6.8.m8.1.2.2" xref="S1.p6.8.m8.1.2.2.cmml"><mi id="S1.p6.8.m8.1.2.2.2" xref="S1.p6.8.m8.1.2.2.2.cmml">λ</mi><mi id="S1.p6.8.m8.1.2.2.3" xref="S1.p6.8.m8.1.2.2.3.cmml">min</mi></msub><mo id="S1.p6.8.m8.1.2.1" xref="S1.p6.8.m8.1.2.1.cmml">⁢</mo><mrow id="S1.p6.8.m8.1.2.3.2" xref="S1.p6.8.m8.1.2.cmml"><mo id="S1.p6.8.m8.1.2.3.2.1" stretchy="false" xref="S1.p6.8.m8.1.2.cmml">(</mo><mi id="S1.p6.8.m8.1.1" xref="S1.p6.8.m8.1.1.cmml">A</mi><mo id="S1.p6.8.m8.1.2.3.2.2" stretchy="false" xref="S1.p6.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.8.m8.1b"><apply id="S1.p6.8.m8.1.2.cmml" xref="S1.p6.8.m8.1.2"><times id="S1.p6.8.m8.1.2.1.cmml" xref="S1.p6.8.m8.1.2.1"></times><apply id="S1.p6.8.m8.1.2.2.cmml" xref="S1.p6.8.m8.1.2.2"><csymbol cd="ambiguous" id="S1.p6.8.m8.1.2.2.1.cmml" xref="S1.p6.8.m8.1.2.2">subscript</csymbol><ci id="S1.p6.8.m8.1.2.2.2.cmml" xref="S1.p6.8.m8.1.2.2.2">𝜆</ci><min id="S1.p6.8.m8.1.2.2.3.cmml" xref="S1.p6.8.m8.1.2.2.3"></min></apply><ci id="S1.p6.8.m8.1.1.cmml" xref="S1.p6.8.m8.1.1">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.8.m8.1c">\lambda_{\min}(A)</annotation><annotation encoding="application/x-llamapun" id="S1.p6.8.m8.1d">italic_λ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_A )</annotation></semantics></math> and <math alttext="\lambda_{\max}(A)" class="ltx_Math" display="inline" id="S1.p6.9.m9.1"><semantics id="S1.p6.9.m9.1a"><mrow id="S1.p6.9.m9.1.2" xref="S1.p6.9.m9.1.2.cmml"><msub id="S1.p6.9.m9.1.2.2" xref="S1.p6.9.m9.1.2.2.cmml"><mi id="S1.p6.9.m9.1.2.2.2" xref="S1.p6.9.m9.1.2.2.2.cmml">λ</mi><mi id="S1.p6.9.m9.1.2.2.3" xref="S1.p6.9.m9.1.2.2.3.cmml">max</mi></msub><mo id="S1.p6.9.m9.1.2.1" xref="S1.p6.9.m9.1.2.1.cmml">⁢</mo><mrow id="S1.p6.9.m9.1.2.3.2" xref="S1.p6.9.m9.1.2.cmml"><mo id="S1.p6.9.m9.1.2.3.2.1" stretchy="false" xref="S1.p6.9.m9.1.2.cmml">(</mo><mi id="S1.p6.9.m9.1.1" xref="S1.p6.9.m9.1.1.cmml">A</mi><mo id="S1.p6.9.m9.1.2.3.2.2" stretchy="false" xref="S1.p6.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.9.m9.1b"><apply id="S1.p6.9.m9.1.2.cmml" xref="S1.p6.9.m9.1.2"><times id="S1.p6.9.m9.1.2.1.cmml" xref="S1.p6.9.m9.1.2.1"></times><apply id="S1.p6.9.m9.1.2.2.cmml" xref="S1.p6.9.m9.1.2.2"><csymbol cd="ambiguous" id="S1.p6.9.m9.1.2.2.1.cmml" xref="S1.p6.9.m9.1.2.2">subscript</csymbol><ci id="S1.p6.9.m9.1.2.2.2.cmml" xref="S1.p6.9.m9.1.2.2.2">𝜆</ci><max id="S1.p6.9.m9.1.2.2.3.cmml" xref="S1.p6.9.m9.1.2.2.3"></max></apply><ci id="S1.p6.9.m9.1.1.cmml" xref="S1.p6.9.m9.1.1">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.9.m9.1c">\lambda_{\max}(A)</annotation><annotation encoding="application/x-llamapun" id="S1.p6.9.m9.1d">italic_λ start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ( italic_A )</annotation></semantics></math> represent the minimal and maximum eigenvalues of <math alttext="A" class="ltx_Math" display="inline" id="S1.p6.10.m10.1"><semantics id="S1.p6.10.m10.1a"><mi id="S1.p6.10.m10.1.1" xref="S1.p6.10.m10.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.p6.10.m10.1b"><ci id="S1.p6.10.m10.1.1.cmml" xref="S1.p6.10.m10.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.10.m10.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.p6.10.m10.1d">italic_A</annotation></semantics></math>, respectively, <math alttext="\sigma_{\min}(A)" class="ltx_Math" display="inline" id="S1.p6.11.m11.1"><semantics id="S1.p6.11.m11.1a"><mrow id="S1.p6.11.m11.1.2" xref="S1.p6.11.m11.1.2.cmml"><msub id="S1.p6.11.m11.1.2.2" xref="S1.p6.11.m11.1.2.2.cmml"><mi id="S1.p6.11.m11.1.2.2.2" xref="S1.p6.11.m11.1.2.2.2.cmml">σ</mi><mi id="S1.p6.11.m11.1.2.2.3" xref="S1.p6.11.m11.1.2.2.3.cmml">min</mi></msub><mo id="S1.p6.11.m11.1.2.1" xref="S1.p6.11.m11.1.2.1.cmml">⁢</mo><mrow id="S1.p6.11.m11.1.2.3.2" xref="S1.p6.11.m11.1.2.cmml"><mo id="S1.p6.11.m11.1.2.3.2.1" stretchy="false" xref="S1.p6.11.m11.1.2.cmml">(</mo><mi id="S1.p6.11.m11.1.1" xref="S1.p6.11.m11.1.1.cmml">A</mi><mo id="S1.p6.11.m11.1.2.3.2.2" stretchy="false" xref="S1.p6.11.m11.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.11.m11.1b"><apply id="S1.p6.11.m11.1.2.cmml" xref="S1.p6.11.m11.1.2"><times id="S1.p6.11.m11.1.2.1.cmml" xref="S1.p6.11.m11.1.2.1"></times><apply id="S1.p6.11.m11.1.2.2.cmml" xref="S1.p6.11.m11.1.2.2"><csymbol cd="ambiguous" id="S1.p6.11.m11.1.2.2.1.cmml" xref="S1.p6.11.m11.1.2.2">subscript</csymbol><ci id="S1.p6.11.m11.1.2.2.2.cmml" xref="S1.p6.11.m11.1.2.2.2">𝜎</ci><min id="S1.p6.11.m11.1.2.2.3.cmml" xref="S1.p6.11.m11.1.2.2.3"></min></apply><ci id="S1.p6.11.m11.1.1.cmml" xref="S1.p6.11.m11.1.1">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.11.m11.1c">\sigma_{\min}(A)</annotation><annotation encoding="application/x-llamapun" id="S1.p6.11.m11.1d">italic_σ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_A )</annotation></semantics></math> is the minimum singular value of <math alttext="A" class="ltx_Math" display="inline" id="S1.p6.12.m12.1"><semantics id="S1.p6.12.m12.1a"><mi id="S1.p6.12.m12.1.1" xref="S1.p6.12.m12.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.p6.12.m12.1b"><ci id="S1.p6.12.m12.1.1.cmml" xref="S1.p6.12.m12.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.12.m12.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.p6.12.m12.1d">italic_A</annotation></semantics></math>, <math alttext="\|\bm{x}\|" class="ltx_Math" display="inline" id="S1.p6.13.m13.1"><semantics id="S1.p6.13.m13.1a"><mrow id="S1.p6.13.m13.1.2.2" xref="S1.p6.13.m13.1.2.1.cmml"><mo id="S1.p6.13.m13.1.2.2.1" stretchy="false" xref="S1.p6.13.m13.1.2.1.1.cmml">‖</mo><mi id="S1.p6.13.m13.1.1" xref="S1.p6.13.m13.1.1.cmml">𝒙</mi><mo id="S1.p6.13.m13.1.2.2.2" stretchy="false" xref="S1.p6.13.m13.1.2.1.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.13.m13.1b"><apply id="S1.p6.13.m13.1.2.1.cmml" xref="S1.p6.13.m13.1.2.2"><csymbol cd="latexml" id="S1.p6.13.m13.1.2.1.1.cmml" xref="S1.p6.13.m13.1.2.2.1">norm</csymbol><ci id="S1.p6.13.m13.1.1.cmml" xref="S1.p6.13.m13.1.1">𝒙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.13.m13.1c">\|\bm{x}\|</annotation><annotation encoding="application/x-llamapun" id="S1.p6.13.m13.1d">∥ bold_italic_x ∥</annotation></semantics></math> is the Euclidean norm of <math alttext="\bm{x}" class="ltx_Math" display="inline" id="S1.p6.14.m14.1"><semantics id="S1.p6.14.m14.1a"><mi id="S1.p6.14.m14.1.1" xref="S1.p6.14.m14.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S1.p6.14.m14.1b"><ci id="S1.p6.14.m14.1.1.cmml" xref="S1.p6.14.m14.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.14.m14.1c">\bm{x}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.14.m14.1d">bold_italic_x</annotation></semantics></math>, <math alttext="L_{\infty}" class="ltx_Math" display="inline" id="S1.p6.15.m15.1"><semantics id="S1.p6.15.m15.1a"><msub id="S1.p6.15.m15.1.1" xref="S1.p6.15.m15.1.1.cmml"><mi id="S1.p6.15.m15.1.1.2" xref="S1.p6.15.m15.1.1.2.cmml">L</mi><mi id="S1.p6.15.m15.1.1.3" mathvariant="normal" xref="S1.p6.15.m15.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p6.15.m15.1b"><apply id="S1.p6.15.m15.1.1.cmml" xref="S1.p6.15.m15.1.1"><csymbol cd="ambiguous" id="S1.p6.15.m15.1.1.1.cmml" xref="S1.p6.15.m15.1.1">subscript</csymbol><ci id="S1.p6.15.m15.1.1.2.cmml" xref="S1.p6.15.m15.1.1.2">𝐿</ci><infinity id="S1.p6.15.m15.1.1.3.cmml" xref="S1.p6.15.m15.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.15.m15.1c">L_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.15.m15.1d">italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math> is the space of bounded signals, <math alttext="I" class="ltx_Math" display="inline" id="S1.p6.16.m16.1"><semantics id="S1.p6.16.m16.1a"><mi id="S1.p6.16.m16.1.1" xref="S1.p6.16.m16.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S1.p6.16.m16.1b"><ci id="S1.p6.16.m16.1.1.cmml" xref="S1.p6.16.m16.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.16.m16.1c">I</annotation><annotation encoding="application/x-llamapun" id="S1.p6.16.m16.1d">italic_I</annotation></semantics></math> is an identity matrix, <math alttext="\bm{0}" class="ltx_Math" display="inline" id="S1.p6.17.m17.1"><semantics id="S1.p6.17.m17.1a"><mn id="S1.p6.17.m17.1.1" xref="S1.p6.17.m17.1.1.cmml">𝟎</mn><annotation-xml encoding="MathML-Content" id="S1.p6.17.m17.1b"><cn id="S1.p6.17.m17.1.1.cmml" type="integer" xref="S1.p6.17.m17.1.1">0</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.17.m17.1c">\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.17.m17.1d">bold_0</annotation></semantics></math> is a zero matrix, <math alttext="\Omega_{c}" class="ltx_Math" display="inline" id="S1.p6.18.m18.1"><semantics id="S1.p6.18.m18.1a"><msub id="S1.p6.18.m18.1.1" xref="S1.p6.18.m18.1.1.cmml"><mi id="S1.p6.18.m18.1.1.2" mathvariant="normal" xref="S1.p6.18.m18.1.1.2.cmml">Ω</mi><mi id="S1.p6.18.m18.1.1.3" xref="S1.p6.18.m18.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p6.18.m18.1b"><apply id="S1.p6.18.m18.1.1.cmml" xref="S1.p6.18.m18.1.1"><csymbol cd="ambiguous" id="S1.p6.18.m18.1.1.1.cmml" xref="S1.p6.18.m18.1.1">subscript</csymbol><ci id="S1.p6.18.m18.1.1.2.cmml" xref="S1.p6.18.m18.1.1.2">Ω</ci><ci id="S1.p6.18.m18.1.1.3.cmml" xref="S1.p6.18.m18.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.18.m18.1c">\Omega_{c}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.18.m18.1d">roman_Ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S1.p6.19.m19.1"><semantics id="S1.p6.19.m19.1a"><mo id="S1.p6.19.m19.1.1" xref="S1.p6.19.m19.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S1.p6.19.m19.1b"><csymbol cd="latexml" id="S1.p6.19.m19.1.1.cmml" xref="S1.p6.19.m19.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.19.m19.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S1.p6.19.m19.1d">:=</annotation></semantics></math> <math alttext="\{\bm{x}|\|\bm{x}\|\leq c\}" class="ltx_Math" display="inline" id="S1.p6.20.m20.3"><semantics id="S1.p6.20.m20.3a"><mrow id="S1.p6.20.m20.3.3.1" xref="S1.p6.20.m20.3.3.2.cmml"><mo id="S1.p6.20.m20.3.3.1.2" stretchy="false" xref="S1.p6.20.m20.3.3.2.1.cmml">{</mo><mi id="S1.p6.20.m20.2.2" xref="S1.p6.20.m20.2.2.cmml">𝒙</mi><mo id="S1.p6.20.m20.3.3.1.3" lspace="0em" rspace="0em" xref="S1.p6.20.m20.3.3.2.1.cmml">|</mo><mrow id="S1.p6.20.m20.3.3.1.1" xref="S1.p6.20.m20.3.3.1.1.cmml"><mrow id="S1.p6.20.m20.3.3.1.1.2.2" xref="S1.p6.20.m20.3.3.1.1.2.1.cmml"><mo id="S1.p6.20.m20.3.3.1.1.2.2.1" stretchy="false" xref="S1.p6.20.m20.3.3.1.1.2.1.1.cmml">‖</mo><mi id="S1.p6.20.m20.1.1" xref="S1.p6.20.m20.1.1.cmml">𝒙</mi><mo id="S1.p6.20.m20.3.3.1.1.2.2.2" stretchy="false" xref="S1.p6.20.m20.3.3.1.1.2.1.1.cmml">‖</mo></mrow><mo id="S1.p6.20.m20.3.3.1.1.1" xref="S1.p6.20.m20.3.3.1.1.1.cmml">≤</mo><mi id="S1.p6.20.m20.3.3.1.1.3" xref="S1.p6.20.m20.3.3.1.1.3.cmml">c</mi></mrow><mo id="S1.p6.20.m20.3.3.1.4" stretchy="false" xref="S1.p6.20.m20.3.3.2.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.20.m20.3b"><apply id="S1.p6.20.m20.3.3.2.cmml" xref="S1.p6.20.m20.3.3.1"><csymbol cd="latexml" id="S1.p6.20.m20.3.3.2.1.cmml" xref="S1.p6.20.m20.3.3.1.2">conditional-set</csymbol><ci id="S1.p6.20.m20.2.2.cmml" xref="S1.p6.20.m20.2.2">𝒙</ci><apply id="S1.p6.20.m20.3.3.1.1.cmml" xref="S1.p6.20.m20.3.3.1.1"><leq id="S1.p6.20.m20.3.3.1.1.1.cmml" xref="S1.p6.20.m20.3.3.1.1.1"></leq><apply id="S1.p6.20.m20.3.3.1.1.2.1.cmml" xref="S1.p6.20.m20.3.3.1.1.2.2"><csymbol cd="latexml" id="S1.p6.20.m20.3.3.1.1.2.1.1.cmml" xref="S1.p6.20.m20.3.3.1.1.2.2.1">norm</csymbol><ci id="S1.p6.20.m20.1.1.cmml" xref="S1.p6.20.m20.1.1">𝒙</ci></apply><ci id="S1.p6.20.m20.3.3.1.1.3.cmml" xref="S1.p6.20.m20.3.3.1.1.3">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.20.m20.3c">\{\bm{x}|\|\bm{x}\|\leq c\}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.20.m20.3d">{ bold_italic_x | ∥ bold_italic_x ∥ ≤ italic_c }</annotation></semantics></math> is the ball of radius <math alttext="c" class="ltx_Math" display="inline" id="S1.p6.21.m21.1"><semantics id="S1.p6.21.m21.1a"><mi id="S1.p6.21.m21.1.1" xref="S1.p6.21.m21.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S1.p6.21.m21.1b"><ci id="S1.p6.21.m21.1.1.cmml" xref="S1.p6.21.m21.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.21.m21.1c">c</annotation><annotation encoding="application/x-llamapun" id="S1.p6.21.m21.1d">italic_c</annotation></semantics></math>, <math alttext="\arg\max_{x\in S}f(x)" class="ltx_Math" display="inline" id="S1.p6.22.m22.1"><semantics id="S1.p6.22.m22.1a"><mrow id="S1.p6.22.m22.1.2" xref="S1.p6.22.m22.1.2.cmml"><mrow id="S1.p6.22.m22.1.2.2" xref="S1.p6.22.m22.1.2.2.cmml"><mi id="S1.p6.22.m22.1.2.2.1" xref="S1.p6.22.m22.1.2.2.1.cmml">arg</mi><mo id="S1.p6.22.m22.1.2.2a" lspace="0.167em" xref="S1.p6.22.m22.1.2.2.cmml">⁡</mo><mrow id="S1.p6.22.m22.1.2.2.2" xref="S1.p6.22.m22.1.2.2.2.cmml"><msub id="S1.p6.22.m22.1.2.2.2.1" xref="S1.p6.22.m22.1.2.2.2.1.cmml"><mi id="S1.p6.22.m22.1.2.2.2.1.2" xref="S1.p6.22.m22.1.2.2.2.1.2.cmml">max</mi><mrow id="S1.p6.22.m22.1.2.2.2.1.3" xref="S1.p6.22.m22.1.2.2.2.1.3.cmml"><mi id="S1.p6.22.m22.1.2.2.2.1.3.2" xref="S1.p6.22.m22.1.2.2.2.1.3.2.cmml">x</mi><mo id="S1.p6.22.m22.1.2.2.2.1.3.1" xref="S1.p6.22.m22.1.2.2.2.1.3.1.cmml">∈</mo><mi id="S1.p6.22.m22.1.2.2.2.1.3.3" xref="S1.p6.22.m22.1.2.2.2.1.3.3.cmml">S</mi></mrow></msub><mo id="S1.p6.22.m22.1.2.2.2a" lspace="0.167em" xref="S1.p6.22.m22.1.2.2.2.cmml">⁡</mo><mi id="S1.p6.22.m22.1.2.2.2.2" xref="S1.p6.22.m22.1.2.2.2.2.cmml">f</mi></mrow></mrow><mo id="S1.p6.22.m22.1.2.1" xref="S1.p6.22.m22.1.2.1.cmml">⁢</mo><mrow id="S1.p6.22.m22.1.2.3.2" xref="S1.p6.22.m22.1.2.cmml"><mo id="S1.p6.22.m22.1.2.3.2.1" stretchy="false" xref="S1.p6.22.m22.1.2.cmml">(</mo><mi id="S1.p6.22.m22.1.1" xref="S1.p6.22.m22.1.1.cmml">x</mi><mo id="S1.p6.22.m22.1.2.3.2.2" stretchy="false" xref="S1.p6.22.m22.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.22.m22.1b"><apply id="S1.p6.22.m22.1.2.cmml" xref="S1.p6.22.m22.1.2"><times id="S1.p6.22.m22.1.2.1.cmml" xref="S1.p6.22.m22.1.2.1"></times><apply id="S1.p6.22.m22.1.2.2.cmml" xref="S1.p6.22.m22.1.2.2"><arg id="S1.p6.22.m22.1.2.2.1.cmml" xref="S1.p6.22.m22.1.2.2.1"></arg><apply id="S1.p6.22.m22.1.2.2.2.cmml" xref="S1.p6.22.m22.1.2.2.2"><apply id="S1.p6.22.m22.1.2.2.2.1.cmml" xref="S1.p6.22.m22.1.2.2.2.1"><csymbol cd="ambiguous" id="S1.p6.22.m22.1.2.2.2.1.1.cmml" xref="S1.p6.22.m22.1.2.2.2.1">subscript</csymbol><max id="S1.p6.22.m22.1.2.2.2.1.2.cmml" xref="S1.p6.22.m22.1.2.2.2.1.2"></max><apply id="S1.p6.22.m22.1.2.2.2.1.3.cmml" xref="S1.p6.22.m22.1.2.2.2.1.3"><in id="S1.p6.22.m22.1.2.2.2.1.3.1.cmml" xref="S1.p6.22.m22.1.2.2.2.1.3.1"></in><ci id="S1.p6.22.m22.1.2.2.2.1.3.2.cmml" xref="S1.p6.22.m22.1.2.2.2.1.3.2">𝑥</ci><ci id="S1.p6.22.m22.1.2.2.2.1.3.3.cmml" xref="S1.p6.22.m22.1.2.2.2.1.3.3">𝑆</ci></apply></apply><ci id="S1.p6.22.m22.1.2.2.2.2.cmml" xref="S1.p6.22.m22.1.2.2.2.2">𝑓</ci></apply></apply><ci id="S1.p6.22.m22.1.1.cmml" xref="S1.p6.22.m22.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.22.m22.1c">\arg\max_{x\in S}f(x)</annotation><annotation encoding="application/x-llamapun" id="S1.p6.22.m22.1d">roman_arg roman_max start_POSTSUBSCRIPT italic_x ∈ italic_S end_POSTSUBSCRIPT italic_f ( italic_x )</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S1.p6.23.m23.1"><semantics id="S1.p6.23.m23.1a"><mo id="S1.p6.23.m23.1.1" xref="S1.p6.23.m23.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S1.p6.23.m23.1b"><csymbol cd="latexml" id="S1.p6.23.m23.1.1.cmml" xref="S1.p6.23.m23.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.23.m23.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S1.p6.23.m23.1d">:=</annotation></semantics></math> <math alttext="\{x\in S|f(y)" class="ltx_math_unparsed" display="inline" id="S1.p6.24.m24.1"><semantics id="S1.p6.24.m24.1a"><mrow id="S1.p6.24.m24.1b"><mo id="S1.p6.24.m24.1.2" stretchy="false">{</mo><mi id="S1.p6.24.m24.1.3">x</mi><mo id="S1.p6.24.m24.1.4">∈</mo><mi id="S1.p6.24.m24.1.5">S</mi><mo fence="false" id="S1.p6.24.m24.1.6" rspace="0.167em" stretchy="false">|</mo><mi id="S1.p6.24.m24.1.7">f</mi><mrow id="S1.p6.24.m24.1.8"><mo id="S1.p6.24.m24.1.8.1" stretchy="false">(</mo><mi id="S1.p6.24.m24.1.1">y</mi><mo id="S1.p6.24.m24.1.8.2" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S1.p6.24.m24.1c">\{x\in S|f(y)</annotation><annotation encoding="application/x-llamapun" id="S1.p6.24.m24.1d">{ italic_x ∈ italic_S | italic_f ( italic_y )</annotation></semantics></math> <math alttext="\leq" class="ltx_Math" display="inline" id="S1.p6.25.m25.1"><semantics id="S1.p6.25.m25.1a"><mo id="S1.p6.25.m25.1.1" xref="S1.p6.25.m25.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="S1.p6.25.m25.1b"><leq id="S1.p6.25.m25.1.1.cmml" xref="S1.p6.25.m25.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.25.m25.1c">\leq</annotation><annotation encoding="application/x-llamapun" id="S1.p6.25.m25.1d">≤</annotation></semantics></math> <math alttext="f(x)" class="ltx_Math" display="inline" id="S1.p6.26.m26.1"><semantics id="S1.p6.26.m26.1a"><mrow id="S1.p6.26.m26.1.2" xref="S1.p6.26.m26.1.2.cmml"><mi id="S1.p6.26.m26.1.2.2" xref="S1.p6.26.m26.1.2.2.cmml">f</mi><mo id="S1.p6.26.m26.1.2.1" xref="S1.p6.26.m26.1.2.1.cmml">⁢</mo><mrow id="S1.p6.26.m26.1.2.3.2" xref="S1.p6.26.m26.1.2.cmml"><mo id="S1.p6.26.m26.1.2.3.2.1" stretchy="false" xref="S1.p6.26.m26.1.2.cmml">(</mo><mi id="S1.p6.26.m26.1.1" xref="S1.p6.26.m26.1.1.cmml">x</mi><mo id="S1.p6.26.m26.1.2.3.2.2" stretchy="false" xref="S1.p6.26.m26.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.26.m26.1b"><apply id="S1.p6.26.m26.1.2.cmml" xref="S1.p6.26.m26.1.2"><times id="S1.p6.26.m26.1.2.1.cmml" xref="S1.p6.26.m26.1.2.1"></times><ci id="S1.p6.26.m26.1.2.2.cmml" xref="S1.p6.26.m26.1.2.2">𝑓</ci><ci id="S1.p6.26.m26.1.1.cmml" xref="S1.p6.26.m26.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.26.m26.1c">f(x)</annotation><annotation encoding="application/x-llamapun" id="S1.p6.26.m26.1d">italic_f ( italic_x )</annotation></semantics></math>, <math alttext="\forall y" class="ltx_Math" display="inline" id="S1.p6.27.m27.1"><semantics id="S1.p6.27.m27.1a"><mrow id="S1.p6.27.m27.1.1" xref="S1.p6.27.m27.1.1.cmml"><mo id="S1.p6.27.m27.1.1.1" rspace="0.167em" xref="S1.p6.27.m27.1.1.1.cmml">∀</mo><mi id="S1.p6.27.m27.1.1.2" xref="S1.p6.27.m27.1.1.2.cmml">y</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.27.m27.1b"><apply id="S1.p6.27.m27.1.1.cmml" xref="S1.p6.27.m27.1.1"><csymbol cd="latexml" id="S1.p6.27.m27.1.1.1.cmml" xref="S1.p6.27.m27.1.1.1">for-all</csymbol><ci id="S1.p6.27.m27.1.1.2.cmml" xref="S1.p6.27.m27.1.1.2">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.27.m27.1c">\forall y</annotation><annotation encoding="application/x-llamapun" id="S1.p6.27.m27.1d">∀ italic_y</annotation></semantics></math> <math alttext="\in" class="ltx_Math" display="inline" id="S1.p6.28.m28.1"><semantics id="S1.p6.28.m28.1a"><mo id="S1.p6.28.m28.1.1" xref="S1.p6.28.m28.1.1.cmml">∈</mo><annotation-xml encoding="MathML-Content" id="S1.p6.28.m28.1b"><in id="S1.p6.28.m28.1.1.cmml" xref="S1.p6.28.m28.1.1"></in></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.28.m28.1c">\in</annotation><annotation encoding="application/x-llamapun" id="S1.p6.28.m28.1d">∈</annotation></semantics></math> <math alttext="S\}" class="ltx_math_unparsed" display="inline" id="S1.p6.29.m29.1"><semantics id="S1.p6.29.m29.1a"><mrow id="S1.p6.29.m29.1b"><mi id="S1.p6.29.m29.1.1">S</mi><mo id="S1.p6.29.m29.1.2" stretchy="false">}</mo></mrow><annotation encoding="application/x-tex" id="S1.p6.29.m29.1c">S\}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.29.m29.1d">italic_S }</annotation></semantics></math>, <math alttext="\bm{g}\in\mathcal{C}^{k}" class="ltx_Math" display="inline" id="S1.p6.30.m30.1"><semantics id="S1.p6.30.m30.1a"><mrow id="S1.p6.30.m30.1.1" xref="S1.p6.30.m30.1.1.cmml"><mi id="S1.p6.30.m30.1.1.2" xref="S1.p6.30.m30.1.1.2.cmml">𝒈</mi><mo id="S1.p6.30.m30.1.1.1" xref="S1.p6.30.m30.1.1.1.cmml">∈</mo><msup id="S1.p6.30.m30.1.1.3" xref="S1.p6.30.m30.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p6.30.m30.1.1.3.2" xref="S1.p6.30.m30.1.1.3.2.cmml">𝒞</mi><mi id="S1.p6.30.m30.1.1.3.3" xref="S1.p6.30.m30.1.1.3.3.cmml">k</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.30.m30.1b"><apply id="S1.p6.30.m30.1.1.cmml" xref="S1.p6.30.m30.1.1"><in id="S1.p6.30.m30.1.1.1.cmml" xref="S1.p6.30.m30.1.1.1"></in><ci id="S1.p6.30.m30.1.1.2.cmml" xref="S1.p6.30.m30.1.1.2">𝒈</ci><apply id="S1.p6.30.m30.1.1.3.cmml" xref="S1.p6.30.m30.1.1.3"><csymbol cd="ambiguous" id="S1.p6.30.m30.1.1.3.1.cmml" xref="S1.p6.30.m30.1.1.3">superscript</csymbol><ci id="S1.p6.30.m30.1.1.3.2.cmml" xref="S1.p6.30.m30.1.1.3.2">𝒞</ci><ci id="S1.p6.30.m30.1.1.3.3.cmml" xref="S1.p6.30.m30.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.30.m30.1c">\bm{g}\in\mathcal{C}^{k}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.30.m30.1d">bold_italic_g ∈ caligraphic_C start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math> indicates that <math alttext="\bm{g}" class="ltx_Math" display="inline" id="S1.p6.31.m31.1"><semantics id="S1.p6.31.m31.1a"><mi id="S1.p6.31.m31.1.1" xref="S1.p6.31.m31.1.1.cmml">𝒈</mi><annotation-xml encoding="MathML-Content" id="S1.p6.31.m31.1b"><ci id="S1.p6.31.m31.1.1.cmml" xref="S1.p6.31.m31.1.1">𝒈</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.31.m31.1c">\bm{g}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.31.m31.1d">bold_italic_g</annotation></semantics></math> has continuous partial derivatives up to the order <math alttext="k" class="ltx_Math" display="inline" id="S1.p6.32.m32.1"><semantics id="S1.p6.32.m32.1a"><mi id="S1.p6.32.m32.1.1" xref="S1.p6.32.m32.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.p6.32.m32.1b"><ci id="S1.p6.32.m32.1.1.cmml" xref="S1.p6.32.m32.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.32.m32.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.p6.32.m32.1d">italic_k</annotation></semantics></math>, where <math alttext="A\in\mathbb{R}^{n\times n}" class="ltx_Math" display="inline" id="S1.p6.33.m33.1"><semantics id="S1.p6.33.m33.1a"><mrow id="S1.p6.33.m33.1.1" xref="S1.p6.33.m33.1.1.cmml"><mi id="S1.p6.33.m33.1.1.2" xref="S1.p6.33.m33.1.1.2.cmml">A</mi><mo id="S1.p6.33.m33.1.1.1" xref="S1.p6.33.m33.1.1.1.cmml">∈</mo><msup id="S1.p6.33.m33.1.1.3" xref="S1.p6.33.m33.1.1.3.cmml"><mi id="S1.p6.33.m33.1.1.3.2" xref="S1.p6.33.m33.1.1.3.2.cmml">ℝ</mi><mrow id="S1.p6.33.m33.1.1.3.3" xref="S1.p6.33.m33.1.1.3.3.cmml"><mi id="S1.p6.33.m33.1.1.3.3.2" xref="S1.p6.33.m33.1.1.3.3.2.cmml">n</mi><mo id="S1.p6.33.m33.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S1.p6.33.m33.1.1.3.3.1.cmml">×</mo><mi id="S1.p6.33.m33.1.1.3.3.3" xref="S1.p6.33.m33.1.1.3.3.3.cmml">n</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.33.m33.1b"><apply id="S1.p6.33.m33.1.1.cmml" xref="S1.p6.33.m33.1.1"><in id="S1.p6.33.m33.1.1.1.cmml" xref="S1.p6.33.m33.1.1.1"></in><ci id="S1.p6.33.m33.1.1.2.cmml" xref="S1.p6.33.m33.1.1.2">𝐴</ci><apply id="S1.p6.33.m33.1.1.3.cmml" xref="S1.p6.33.m33.1.1.3"><csymbol cd="ambiguous" id="S1.p6.33.m33.1.1.3.1.cmml" xref="S1.p6.33.m33.1.1.3">superscript</csymbol><ci id="S1.p6.33.m33.1.1.3.2.cmml" xref="S1.p6.33.m33.1.1.3.2">ℝ</ci><apply id="S1.p6.33.m33.1.1.3.3.cmml" xref="S1.p6.33.m33.1.1.3.3"><times id="S1.p6.33.m33.1.1.3.3.1.cmml" xref="S1.p6.33.m33.1.1.3.3.1"></times><ci id="S1.p6.33.m33.1.1.3.3.2.cmml" xref="S1.p6.33.m33.1.1.3.3.2">𝑛</ci><ci id="S1.p6.33.m33.1.1.3.3.3.cmml" xref="S1.p6.33.m33.1.1.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.33.m33.1c">A\in\mathbb{R}^{n\times n}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.33.m33.1d">italic_A ∈ blackboard_R start_POSTSUPERSCRIPT italic_n × italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\bm{x}\in\mathbb{R}^{n}" class="ltx_Math" display="inline" id="S1.p6.34.m34.1"><semantics id="S1.p6.34.m34.1a"><mrow id="S1.p6.34.m34.1.1" xref="S1.p6.34.m34.1.1.cmml"><mi id="S1.p6.34.m34.1.1.2" xref="S1.p6.34.m34.1.1.2.cmml">𝒙</mi><mo id="S1.p6.34.m34.1.1.1" xref="S1.p6.34.m34.1.1.1.cmml">∈</mo><msup id="S1.p6.34.m34.1.1.3" xref="S1.p6.34.m34.1.1.3.cmml"><mi id="S1.p6.34.m34.1.1.3.2" xref="S1.p6.34.m34.1.1.3.2.cmml">ℝ</mi><mi id="S1.p6.34.m34.1.1.3.3" xref="S1.p6.34.m34.1.1.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.34.m34.1b"><apply id="S1.p6.34.m34.1.1.cmml" xref="S1.p6.34.m34.1.1"><in id="S1.p6.34.m34.1.1.1.cmml" xref="S1.p6.34.m34.1.1.1"></in><ci id="S1.p6.34.m34.1.1.2.cmml" xref="S1.p6.34.m34.1.1.2">𝒙</ci><apply id="S1.p6.34.m34.1.1.3.cmml" xref="S1.p6.34.m34.1.1.3"><csymbol cd="ambiguous" id="S1.p6.34.m34.1.1.3.1.cmml" xref="S1.p6.34.m34.1.1.3">superscript</csymbol><ci id="S1.p6.34.m34.1.1.3.2.cmml" xref="S1.p6.34.m34.1.1.3.2">ℝ</ci><ci id="S1.p6.34.m34.1.1.3.3.cmml" xref="S1.p6.34.m34.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.34.m34.1c">\bm{x}\in\mathbb{R}^{n}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.34.m34.1d">bold_italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="c\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S1.p6.35.m35.1"><semantics id="S1.p6.35.m35.1a"><mrow id="S1.p6.35.m35.1.1" xref="S1.p6.35.m35.1.1.cmml"><mi id="S1.p6.35.m35.1.1.2" xref="S1.p6.35.m35.1.1.2.cmml">c</mi><mo id="S1.p6.35.m35.1.1.1" xref="S1.p6.35.m35.1.1.1.cmml">∈</mo><msup id="S1.p6.35.m35.1.1.3" xref="S1.p6.35.m35.1.1.3.cmml"><mi id="S1.p6.35.m35.1.1.3.2" xref="S1.p6.35.m35.1.1.3.2.cmml">ℝ</mi><mo id="S1.p6.35.m35.1.1.3.3" xref="S1.p6.35.m35.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.35.m35.1b"><apply id="S1.p6.35.m35.1.1.cmml" xref="S1.p6.35.m35.1.1"><in id="S1.p6.35.m35.1.1.1.cmml" xref="S1.p6.35.m35.1.1.1"></in><ci id="S1.p6.35.m35.1.1.2.cmml" xref="S1.p6.35.m35.1.1.2">𝑐</ci><apply id="S1.p6.35.m35.1.1.3.cmml" xref="S1.p6.35.m35.1.1.3"><csymbol cd="ambiguous" id="S1.p6.35.m35.1.1.3.1.cmml" xref="S1.p6.35.m35.1.1.3">superscript</csymbol><ci id="S1.p6.35.m35.1.1.3.2.cmml" xref="S1.p6.35.m35.1.1.3.2">ℝ</ci><plus id="S1.p6.35.m35.1.1.3.3.cmml" xref="S1.p6.35.m35.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.35.m35.1c">c\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.35.m35.1d">italic_c ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="f:\mathbb{R}\mapsto\mathbb{R}" class="ltx_Math" display="inline" id="S1.p6.36.m36.1"><semantics id="S1.p6.36.m36.1a"><mrow id="S1.p6.36.m36.1.1" xref="S1.p6.36.m36.1.1.cmml"><mi id="S1.p6.36.m36.1.1.2" xref="S1.p6.36.m36.1.1.2.cmml">f</mi><mo id="S1.p6.36.m36.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.p6.36.m36.1.1.1.cmml">:</mo><mrow id="S1.p6.36.m36.1.1.3" xref="S1.p6.36.m36.1.1.3.cmml"><mi id="S1.p6.36.m36.1.1.3.2" xref="S1.p6.36.m36.1.1.3.2.cmml">ℝ</mi><mo id="S1.p6.36.m36.1.1.3.1" stretchy="false" xref="S1.p6.36.m36.1.1.3.1.cmml">↦</mo><mi id="S1.p6.36.m36.1.1.3.3" xref="S1.p6.36.m36.1.1.3.3.cmml">ℝ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.36.m36.1b"><apply id="S1.p6.36.m36.1.1.cmml" xref="S1.p6.36.m36.1.1"><ci id="S1.p6.36.m36.1.1.1.cmml" xref="S1.p6.36.m36.1.1.1">:</ci><ci id="S1.p6.36.m36.1.1.2.cmml" xref="S1.p6.36.m36.1.1.2">𝑓</ci><apply id="S1.p6.36.m36.1.1.3.cmml" xref="S1.p6.36.m36.1.1.3"><csymbol cd="latexml" id="S1.p6.36.m36.1.1.3.1.cmml" xref="S1.p6.36.m36.1.1.3.1">maps-to</csymbol><ci id="S1.p6.36.m36.1.1.3.2.cmml" xref="S1.p6.36.m36.1.1.3.2">ℝ</ci><ci id="S1.p6.36.m36.1.1.3.3.cmml" xref="S1.p6.36.m36.1.1.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.36.m36.1c">f:\mathbb{R}\mapsto\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.36.m36.1d">italic_f : blackboard_R ↦ blackboard_R</annotation></semantics></math>, <math alttext="\bm{g}" class="ltx_Math" display="inline" id="S1.p6.37.m37.1"><semantics id="S1.p6.37.m37.1a"><mi id="S1.p6.37.m37.1.1" xref="S1.p6.37.m37.1.1.cmml">𝒈</mi><annotation-xml encoding="MathML-Content" id="S1.p6.37.m37.1b"><ci id="S1.p6.37.m37.1.1.cmml" xref="S1.p6.37.m37.1.1">𝒈</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.37.m37.1c">\bm{g}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.37.m37.1d">bold_italic_g</annotation></semantics></math> <math alttext=":" class="ltx_Math" display="inline" id="S1.p6.38.m38.1"><semantics id="S1.p6.38.m38.1a"><mo id="S1.p6.38.m38.1.1" xref="S1.p6.38.m38.1.1.cmml">:</mo><annotation-xml encoding="MathML-Content" id="S1.p6.38.m38.1b"><ci id="S1.p6.38.m38.1.1.cmml" xref="S1.p6.38.m38.1.1">:</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.38.m38.1c">:</annotation><annotation encoding="application/x-llamapun" id="S1.p6.38.m38.1d">:</annotation></semantics></math> <math alttext="\mathbb{R}^{n}\mapsto\mathbb{R}^{m}," class="ltx_Math" display="inline" id="S1.p6.39.m39.1"><semantics id="S1.p6.39.m39.1a"><mrow id="S1.p6.39.m39.1.1.1" xref="S1.p6.39.m39.1.1.1.1.cmml"><mrow id="S1.p6.39.m39.1.1.1.1" xref="S1.p6.39.m39.1.1.1.1.cmml"><msup id="S1.p6.39.m39.1.1.1.1.2" xref="S1.p6.39.m39.1.1.1.1.2.cmml"><mi id="S1.p6.39.m39.1.1.1.1.2.2" xref="S1.p6.39.m39.1.1.1.1.2.2.cmml">ℝ</mi><mi id="S1.p6.39.m39.1.1.1.1.2.3" xref="S1.p6.39.m39.1.1.1.1.2.3.cmml">n</mi></msup><mo id="S1.p6.39.m39.1.1.1.1.1" stretchy="false" xref="S1.p6.39.m39.1.1.1.1.1.cmml">↦</mo><msup id="S1.p6.39.m39.1.1.1.1.3" xref="S1.p6.39.m39.1.1.1.1.3.cmml"><mi id="S1.p6.39.m39.1.1.1.1.3.2" xref="S1.p6.39.m39.1.1.1.1.3.2.cmml">ℝ</mi><mi id="S1.p6.39.m39.1.1.1.1.3.3" xref="S1.p6.39.m39.1.1.1.1.3.3.cmml">m</mi></msup></mrow><mo id="S1.p6.39.m39.1.1.1.2" xref="S1.p6.39.m39.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.39.m39.1b"><apply id="S1.p6.39.m39.1.1.1.1.cmml" xref="S1.p6.39.m39.1.1.1"><csymbol cd="latexml" id="S1.p6.39.m39.1.1.1.1.1.cmml" xref="S1.p6.39.m39.1.1.1.1.1">maps-to</csymbol><apply id="S1.p6.39.m39.1.1.1.1.2.cmml" xref="S1.p6.39.m39.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.p6.39.m39.1.1.1.1.2.1.cmml" xref="S1.p6.39.m39.1.1.1.1.2">superscript</csymbol><ci id="S1.p6.39.m39.1.1.1.1.2.2.cmml" xref="S1.p6.39.m39.1.1.1.1.2.2">ℝ</ci><ci id="S1.p6.39.m39.1.1.1.1.2.3.cmml" xref="S1.p6.39.m39.1.1.1.1.2.3">𝑛</ci></apply><apply id="S1.p6.39.m39.1.1.1.1.3.cmml" xref="S1.p6.39.m39.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.p6.39.m39.1.1.1.1.3.1.cmml" xref="S1.p6.39.m39.1.1.1.1.3">superscript</csymbol><ci id="S1.p6.39.m39.1.1.1.1.3.2.cmml" xref="S1.p6.39.m39.1.1.1.1.3.2">ℝ</ci><ci id="S1.p6.39.m39.1.1.1.1.3.3.cmml" xref="S1.p6.39.m39.1.1.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.39.m39.1c">\mathbb{R}^{n}\mapsto\mathbb{R}^{m},</annotation><annotation encoding="application/x-llamapun" id="S1.p6.39.m39.1d">blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ↦ blackboard_R start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ,</annotation></semantics></math> <math alttext="S\subset\mathbb{R}" class="ltx_Math" display="inline" id="S1.p6.40.m40.1"><semantics id="S1.p6.40.m40.1a"><mrow id="S1.p6.40.m40.1.1" xref="S1.p6.40.m40.1.1.cmml"><mi id="S1.p6.40.m40.1.1.2" xref="S1.p6.40.m40.1.1.2.cmml">S</mi><mo id="S1.p6.40.m40.1.1.1" xref="S1.p6.40.m40.1.1.1.cmml">⊂</mo><mi id="S1.p6.40.m40.1.1.3" xref="S1.p6.40.m40.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.40.m40.1b"><apply id="S1.p6.40.m40.1.1.cmml" xref="S1.p6.40.m40.1.1"><subset id="S1.p6.40.m40.1.1.1.cmml" xref="S1.p6.40.m40.1.1.1"></subset><ci id="S1.p6.40.m40.1.1.2.cmml" xref="S1.p6.40.m40.1.1.2">𝑆</ci><ci id="S1.p6.40.m40.1.1.3.cmml" xref="S1.p6.40.m40.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.40.m40.1c">S\subset\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.40.m40.1d">italic_S ⊂ blackboard_R</annotation></semantics></math>, <math alttext="n,m" class="ltx_Math" display="inline" id="S1.p6.41.m41.2"><semantics id="S1.p6.41.m41.2a"><mrow id="S1.p6.41.m41.2.3.2" xref="S1.p6.41.m41.2.3.1.cmml"><mi id="S1.p6.41.m41.1.1" xref="S1.p6.41.m41.1.1.cmml">n</mi><mo id="S1.p6.41.m41.2.3.2.1" xref="S1.p6.41.m41.2.3.1.cmml">,</mo><mi id="S1.p6.41.m41.2.2" xref="S1.p6.41.m41.2.2.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.41.m41.2b"><list id="S1.p6.41.m41.2.3.1.cmml" xref="S1.p6.41.m41.2.3.2"><ci id="S1.p6.41.m41.1.1.cmml" xref="S1.p6.41.m41.1.1">𝑛</ci><ci id="S1.p6.41.m41.2.2.cmml" xref="S1.p6.41.m41.2.2">𝑚</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.41.m41.2c">n,m</annotation><annotation encoding="application/x-llamapun" id="S1.p6.41.m41.2d">italic_n , italic_m</annotation></semantics></math> are positive integers, and <math alttext="k" class="ltx_Math" display="inline" id="S1.p6.42.m42.1"><semantics id="S1.p6.42.m42.1a"><mi id="S1.p6.42.m42.1.1" xref="S1.p6.42.m42.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.p6.42.m42.1b"><ci id="S1.p6.42.m42.1.1.cmml" xref="S1.p6.42.m42.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.42.m42.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.p6.42.m42.1d">italic_k</annotation></semantics></math> is an non-negative integer.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">II </span><span class="ltx_text ltx_font_smallcaps" id="S2.1.1">Problem Formulation</span> </h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.1">Consider a class of <math alttext="n" class="ltx_Math" display="inline" id="S2.p1.1.m1.1"><semantics id="S2.p1.1.m1.1a"><mi id="S2.p1.1.m1.1.1" xref="S2.p1.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.p1.1.m1.1b"><ci id="S2.p1.1.m1.1.1.cmml" xref="S2.p1.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.p1.1.m1.1d">italic_n</annotation></semantics></math>th-order strict-feedback uncertain nonlinear systems as follows <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib8" title="">8</a>]</cite>:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx1"> <tbody id="S2.E4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\left\{\begin{array}[]{l}\dot{x}_{i}={\bm{\varphi}}_{i}^{T}({\bm{% x}}_{i})\bm{\theta}+x_{i+1},\\ \dot{x}_{n}={\bm{\varphi}}_{n}^{T}(\bm{x})\bm{\theta}+\beta(\bm{x})u,\\ y=x_{1}\end{array}\right." class="ltx_Math" display="inline" id="S2.E4.m1.1"><semantics id="S2.E4.m1.1a"><mrow id="S2.E4.m1.1.2.2" xref="S2.E4.m1.1.2.1.cmml"><mo id="S2.E4.m1.1.2.2.1" xref="S2.E4.m1.1.2.1.1.cmml">{</mo><mtable id="S2.E4.m1.1.1" rowspacing="0pt" xref="S2.E4.m1.1.1.cmml"><mtr id="S2.E4.m1.1.1a" xref="S2.E4.m1.1.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S2.E4.m1.1.1b" xref="S2.E4.m1.1.1.cmml"><mrow id="S2.E1.1.1.1" xref="S2.E1.1.1.1.1.cmml"><mrow id="S2.E1.1.1.1.1" xref="S2.E1.1.1.1.1.cmml"><msub id="S2.E1.1.1.1.1.3" xref="S2.E1.1.1.1.1.3.cmml"><mover accent="true" id="S2.E1.1.1.1.1.3.2" xref="S2.E1.1.1.1.1.3.2.cmml"><mi id="S2.E1.1.1.1.1.3.2.2" xref="S2.E1.1.1.1.1.3.2.2.cmml">x</mi><mo id="S2.E1.1.1.1.1.3.2.1" xref="S2.E1.1.1.1.1.3.2.1.cmml">˙</mo></mover><mi id="S2.E1.1.1.1.1.3.3" xref="S2.E1.1.1.1.1.3.3.cmml">i</mi></msub><mo id="S2.E1.1.1.1.1.2" xref="S2.E1.1.1.1.1.2.cmml">=</mo><mrow id="S2.E1.1.1.1.1.1" xref="S2.E1.1.1.1.1.1.cmml"><mrow id="S2.E1.1.1.1.1.1.1" xref="S2.E1.1.1.1.1.1.1.cmml"><msubsup id="S2.E1.1.1.1.1.1.1.3" xref="S2.E1.1.1.1.1.1.1.3.cmml"><mi id="S2.E1.1.1.1.1.1.1.3.2.2" xref="S2.E1.1.1.1.1.1.1.3.2.2.cmml">𝝋</mi><mi id="S2.E1.1.1.1.1.1.1.3.2.3" xref="S2.E1.1.1.1.1.1.1.3.2.3.cmml">i</mi><mi id="S2.E1.1.1.1.1.1.1.3.3" xref="S2.E1.1.1.1.1.1.1.3.3.cmml">T</mi></msubsup><mo id="S2.E1.1.1.1.1.1.1.2" xref="S2.E1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E1.1.1.1.1.1.1.1.1" xref="S2.E1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.E1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.E1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.E1.1.1.1.1.1.1.1.1.1" xref="S2.E1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.E1.1.1.1.1.1.1.1.1.1.2" xref="S2.E1.1.1.1.1.1.1.1.1.1.2.cmml">𝒙</mi><mi id="S2.E1.1.1.1.1.1.1.1.1.1.3" xref="S2.E1.1.1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.E1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.E1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.E1.1.1.1.1.1.1.2a" xref="S2.E1.1.1.1.1.1.1.2.cmml">⁢</mo><mi id="S2.E1.1.1.1.1.1.1.4" xref="S2.E1.1.1.1.1.1.1.4.cmml">𝜽</mi></mrow><mo id="S2.E1.1.1.1.1.1.2" xref="S2.E1.1.1.1.1.1.2.cmml">+</mo><msub id="S2.E1.1.1.1.1.1.3" xref="S2.E1.1.1.1.1.1.3.cmml"><mi id="S2.E1.1.1.1.1.1.3.2" xref="S2.E1.1.1.1.1.1.3.2.cmml">x</mi><mrow id="S2.E1.1.1.1.1.1.3.3" xref="S2.E1.1.1.1.1.1.3.3.cmml"><mi id="S2.E1.1.1.1.1.1.3.3.2" xref="S2.E1.1.1.1.1.1.3.3.2.cmml">i</mi><mo id="S2.E1.1.1.1.1.1.3.3.1" xref="S2.E1.1.1.1.1.1.3.3.1.cmml">+</mo><mn id="S2.E1.1.1.1.1.1.3.3.3" xref="S2.E1.1.1.1.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow></mrow><mo id="S2.E1.1.1.1.2" xref="S2.E1.1.1.1.1.cmml">,</mo></mrow></mtd></mtr><mtr id="S2.E4.m1.1.1c" xref="S2.E4.m1.1.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S2.E4.m1.1.1d" xref="S2.E4.m1.1.1.cmml"><mrow id="S2.E2.3.3.3" xref="S2.E2.3.3.3.1.cmml"><mrow id="S2.E2.3.3.3.1" xref="S2.E2.3.3.3.1.cmml"><msub id="S2.E2.3.3.3.1.2" xref="S2.E2.3.3.3.1.2.cmml"><mover accent="true" id="S2.E2.3.3.3.1.2.2" xref="S2.E2.3.3.3.1.2.2.cmml"><mi id="S2.E2.3.3.3.1.2.2.2" xref="S2.E2.3.3.3.1.2.2.2.cmml">x</mi><mo id="S2.E2.3.3.3.1.2.2.1" xref="S2.E2.3.3.3.1.2.2.1.cmml">˙</mo></mover><mi id="S2.E2.3.3.3.1.2.3" xref="S2.E2.3.3.3.1.2.3.cmml">n</mi></msub><mo id="S2.E2.3.3.3.1.1" xref="S2.E2.3.3.3.1.1.cmml">=</mo><mrow id="S2.E2.3.3.3.1.3" xref="S2.E2.3.3.3.1.3.cmml"><mrow id="S2.E2.3.3.3.1.3.2" xref="S2.E2.3.3.3.1.3.2.cmml"><msubsup id="S2.E2.3.3.3.1.3.2.2" xref="S2.E2.3.3.3.1.3.2.2.cmml"><mi id="S2.E2.3.3.3.1.3.2.2.2.2" xref="S2.E2.3.3.3.1.3.2.2.2.2.cmml">𝝋</mi><mi id="S2.E2.3.3.3.1.3.2.2.2.3" xref="S2.E2.3.3.3.1.3.2.2.2.3.cmml">n</mi><mi id="S2.E2.3.3.3.1.3.2.2.3" xref="S2.E2.3.3.3.1.3.2.2.3.cmml">T</mi></msubsup><mo id="S2.E2.3.3.3.1.3.2.1" xref="S2.E2.3.3.3.1.3.2.1.cmml">⁢</mo><mrow id="S2.E2.3.3.3.1.3.2.3.2" xref="S2.E2.3.3.3.1.3.2.cmml"><mo id="S2.E2.3.3.3.1.3.2.3.2.1" stretchy="false" xref="S2.E2.3.3.3.1.3.2.cmml">(</mo><mi id="S2.E2.1.1.1" xref="S2.E2.1.1.1.cmml">𝒙</mi><mo id="S2.E2.3.3.3.1.3.2.3.2.2" stretchy="false" xref="S2.E2.3.3.3.1.3.2.cmml">)</mo></mrow><mo id="S2.E2.3.3.3.1.3.2.1a" xref="S2.E2.3.3.3.1.3.2.1.cmml">⁢</mo><mi id="S2.E2.3.3.3.1.3.2.4" xref="S2.E2.3.3.3.1.3.2.4.cmml">𝜽</mi></mrow><mo id="S2.E2.3.3.3.1.3.1" xref="S2.E2.3.3.3.1.3.1.cmml">+</mo><mrow id="S2.E2.3.3.3.1.3.3" xref="S2.E2.3.3.3.1.3.3.cmml"><mi id="S2.E2.3.3.3.1.3.3.2" xref="S2.E2.3.3.3.1.3.3.2.cmml">β</mi><mo id="S2.E2.3.3.3.1.3.3.1" xref="S2.E2.3.3.3.1.3.3.1.cmml">⁢</mo><mrow id="S2.E2.3.3.3.1.3.3.3.2" xref="S2.E2.3.3.3.1.3.3.cmml"><mo id="S2.E2.3.3.3.1.3.3.3.2.1" stretchy="false" xref="S2.E2.3.3.3.1.3.3.cmml">(</mo><mi id="S2.E2.2.2.2" xref="S2.E2.2.2.2.cmml">𝒙</mi><mo id="S2.E2.3.3.3.1.3.3.3.2.2" stretchy="false" xref="S2.E2.3.3.3.1.3.3.cmml">)</mo></mrow><mo id="S2.E2.3.3.3.1.3.3.1a" xref="S2.E2.3.3.3.1.3.3.1.cmml">⁢</mo><mi id="S2.E2.3.3.3.1.3.3.4" xref="S2.E2.3.3.3.1.3.3.4.cmml">u</mi></mrow></mrow></mrow><mo id="S2.E2.3.3.3.2" xref="S2.E2.3.3.3.1.cmml">,</mo></mrow></mtd></mtr><mtr id="S2.E4.m1.1.1e" xref="S2.E4.m1.1.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S2.E4.m1.1.1f" xref="S2.E4.m1.1.1.cmml"><mrow id="S2.E3.1.1" xref="S2.E3.1.1.cmml"><mi id="S2.E3.1.1.2" xref="S2.E3.1.1.2.cmml">y</mi><mo id="S2.E3.1.1.1" xref="S2.E3.1.1.1.cmml">=</mo><msub id="S2.E3.1.1.3" xref="S2.E3.1.1.3.cmml"><mi id="S2.E3.1.1.3.2" xref="S2.E3.1.1.3.2.cmml">x</mi><mn id="S2.E3.1.1.3.3" xref="S2.E3.1.1.3.3.cmml">1</mn></msub></mrow></mtd></mtr></mtable><mi id="S2.E4.m1.1.2.2.2" xref="S2.E4.m1.1.2.1.1.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="S2.E4.m1.1b"><apply id="S2.E4.m1.1.2.1.cmml" xref="S2.E4.m1.1.2.2"><csymbol cd="latexml" id="S2.E4.m1.1.2.1.1.cmml" xref="S2.E4.m1.1.2.2.1">cases</csymbol><matrix id="S2.E4.m1.1.1.cmml" xref="S2.E4.m1.1.1"><matrixrow id="S2.E4.m1.1.1a.cmml" xref="S2.E4.m1.1.1"><apply id="S2.E1.1.1.1.1.cmml" xref="S2.E1.1.1.1"><eq id="S2.E1.1.1.1.1.2.cmml" xref="S2.E1.1.1.1.1.2"></eq><apply id="S2.E1.1.1.1.1.3.cmml" xref="S2.E1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.E1.1.1.1.1.3.1.cmml" xref="S2.E1.1.1.1.1.3">subscript</csymbol><apply id="S2.E1.1.1.1.1.3.2.cmml" xref="S2.E1.1.1.1.1.3.2"><ci id="S2.E1.1.1.1.1.3.2.1.cmml" xref="S2.E1.1.1.1.1.3.2.1">˙</ci><ci id="S2.E1.1.1.1.1.3.2.2.cmml" xref="S2.E1.1.1.1.1.3.2.2">𝑥</ci></apply><ci id="S2.E1.1.1.1.1.3.3.cmml" xref="S2.E1.1.1.1.1.3.3">𝑖</ci></apply><apply id="S2.E1.1.1.1.1.1.cmml" xref="S2.E1.1.1.1.1.1"><plus id="S2.E1.1.1.1.1.1.2.cmml" xref="S2.E1.1.1.1.1.1.2"></plus><apply id="S2.E1.1.1.1.1.1.1.cmml" xref="S2.E1.1.1.1.1.1.1"><times id="S2.E1.1.1.1.1.1.1.2.cmml" xref="S2.E1.1.1.1.1.1.1.2"></times><apply id="S2.E1.1.1.1.1.1.1.3.cmml" xref="S2.E1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.E1.1.1.1.1.1.1.3.1.cmml" xref="S2.E1.1.1.1.1.1.1.3">superscript</csymbol><apply id="S2.E1.1.1.1.1.1.1.3.2.cmml" xref="S2.E1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.E1.1.1.1.1.1.1.3.2.1.cmml" xref="S2.E1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.E1.1.1.1.1.1.1.3.2.2.cmml" xref="S2.E1.1.1.1.1.1.1.3.2.2">𝝋</ci><ci id="S2.E1.1.1.1.1.1.1.3.2.3.cmml" xref="S2.E1.1.1.1.1.1.1.3.2.3">𝑖</ci></apply><ci id="S2.E1.1.1.1.1.1.1.3.3.cmml" xref="S2.E1.1.1.1.1.1.1.3.3">𝑇</ci></apply><apply id="S2.E1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.E1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.E1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.E1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.E1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.E1.1.1.1.1.1.1.1.1.1.2">𝒙</ci><ci id="S2.E1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S2.E1.1.1.1.1.1.1.1.1.1.3">𝑖</ci></apply><ci id="S2.E1.1.1.1.1.1.1.4.cmml" xref="S2.E1.1.1.1.1.1.1.4">𝜽</ci></apply><apply id="S2.E1.1.1.1.1.1.3.cmml" xref="S2.E1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.E1.1.1.1.1.1.3.1.cmml" xref="S2.E1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.E1.1.1.1.1.1.3.2.cmml" xref="S2.E1.1.1.1.1.1.3.2">𝑥</ci><apply id="S2.E1.1.1.1.1.1.3.3.cmml" xref="S2.E1.1.1.1.1.1.3.3"><plus id="S2.E1.1.1.1.1.1.3.3.1.cmml" xref="S2.E1.1.1.1.1.1.3.3.1"></plus><ci id="S2.E1.1.1.1.1.1.3.3.2.cmml" xref="S2.E1.1.1.1.1.1.3.3.2">𝑖</ci><cn id="S2.E1.1.1.1.1.1.3.3.3.cmml" type="integer" xref="S2.E1.1.1.1.1.1.3.3.3">1</cn></apply></apply></apply></apply></matrixrow><matrixrow id="S2.E4.m1.1.1b.cmml" xref="S2.E4.m1.1.1"><apply id="S2.E2.3.3.3.1.cmml" xref="S2.E2.3.3.3"><eq id="S2.E2.3.3.3.1.1.cmml" xref="S2.E2.3.3.3.1.1"></eq><apply id="S2.E2.3.3.3.1.2.cmml" xref="S2.E2.3.3.3.1.2"><csymbol cd="ambiguous" id="S2.E2.3.3.3.1.2.1.cmml" xref="S2.E2.3.3.3.1.2">subscript</csymbol><apply id="S2.E2.3.3.3.1.2.2.cmml" xref="S2.E2.3.3.3.1.2.2"><ci id="S2.E2.3.3.3.1.2.2.1.cmml" xref="S2.E2.3.3.3.1.2.2.1">˙</ci><ci id="S2.E2.3.3.3.1.2.2.2.cmml" xref="S2.E2.3.3.3.1.2.2.2">𝑥</ci></apply><ci id="S2.E2.3.3.3.1.2.3.cmml" xref="S2.E2.3.3.3.1.2.3">𝑛</ci></apply><apply id="S2.E2.3.3.3.1.3.cmml" xref="S2.E2.3.3.3.1.3"><plus id="S2.E2.3.3.3.1.3.1.cmml" xref="S2.E2.3.3.3.1.3.1"></plus><apply id="S2.E2.3.3.3.1.3.2.cmml" xref="S2.E2.3.3.3.1.3.2"><times id="S2.E2.3.3.3.1.3.2.1.cmml" xref="S2.E2.3.3.3.1.3.2.1"></times><apply id="S2.E2.3.3.3.1.3.2.2.cmml" xref="S2.E2.3.3.3.1.3.2.2"><csymbol cd="ambiguous" id="S2.E2.3.3.3.1.3.2.2.1.cmml" xref="S2.E2.3.3.3.1.3.2.2">superscript</csymbol><apply id="S2.E2.3.3.3.1.3.2.2.2.cmml" xref="S2.E2.3.3.3.1.3.2.2"><csymbol cd="ambiguous" id="S2.E2.3.3.3.1.3.2.2.2.1.cmml" xref="S2.E2.3.3.3.1.3.2.2">subscript</csymbol><ci id="S2.E2.3.3.3.1.3.2.2.2.2.cmml" xref="S2.E2.3.3.3.1.3.2.2.2.2">𝝋</ci><ci id="S2.E2.3.3.3.1.3.2.2.2.3.cmml" xref="S2.E2.3.3.3.1.3.2.2.2.3">𝑛</ci></apply><ci id="S2.E2.3.3.3.1.3.2.2.3.cmml" xref="S2.E2.3.3.3.1.3.2.2.3">𝑇</ci></apply><ci id="S2.E2.1.1.1.cmml" xref="S2.E2.1.1.1">𝒙</ci><ci id="S2.E2.3.3.3.1.3.2.4.cmml" xref="S2.E2.3.3.3.1.3.2.4">𝜽</ci></apply><apply id="S2.E2.3.3.3.1.3.3.cmml" xref="S2.E2.3.3.3.1.3.3"><times id="S2.E2.3.3.3.1.3.3.1.cmml" xref="S2.E2.3.3.3.1.3.3.1"></times><ci id="S2.E2.3.3.3.1.3.3.2.cmml" xref="S2.E2.3.3.3.1.3.3.2">𝛽</ci><ci id="S2.E2.2.2.2.cmml" xref="S2.E2.2.2.2">𝒙</ci><ci id="S2.E2.3.3.3.1.3.3.4.cmml" xref="S2.E2.3.3.3.1.3.3.4">𝑢</ci></apply></apply></apply></matrixrow><matrixrow id="S2.E4.m1.1.1c.cmml" xref="S2.E4.m1.1.1"><apply id="S2.E3.1.1.cmml" xref="S2.E3.1.1"><eq id="S2.E3.1.1.1.cmml" xref="S2.E3.1.1.1"></eq><ci id="S2.E3.1.1.2.cmml" xref="S2.E3.1.1.2">𝑦</ci><apply id="S2.E3.1.1.3.cmml" xref="S2.E3.1.1.3"><csymbol cd="ambiguous" id="S2.E3.1.1.3.1.cmml" xref="S2.E3.1.1.3">subscript</csymbol><ci id="S2.E3.1.1.3.2.cmml" xref="S2.E3.1.1.3.2">𝑥</ci><cn id="S2.E3.1.1.3.3.cmml" type="integer" xref="S2.E3.1.1.3.3">1</cn></apply></apply></matrixrow></matrix></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E4.m1.1c">\displaystyle\left\{\begin{array}[]{l}\dot{x}_{i}={\bm{\varphi}}_{i}^{T}({\bm{% x}}_{i})\bm{\theta}+x_{i+1},\\ \dot{x}_{n}={\bm{\varphi}}_{n}^{T}(\bm{x})\bm{\theta}+\beta(\bm{x})u,\\ y=x_{1}\end{array}\right.</annotation><annotation encoding="application/x-llamapun" id="S2.E4.m1.1d">{ start_ARRAY start_ROW start_CELL over˙ start_ARG italic_x end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_italic_φ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) bold_italic_θ + italic_x start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL over˙ start_ARG italic_x end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = bold_italic_φ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_italic_x ) bold_italic_θ + italic_β ( bold_italic_x ) italic_u , end_CELL end_ROW start_ROW start_CELL italic_y = italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p1.16">with <math alttext="i=" class="ltx_Math" display="inline" id="S2.p1.2.m1.1"><semantics id="S2.p1.2.m1.1a"><mrow id="S2.p1.2.m1.1.1" xref="S2.p1.2.m1.1.1.cmml"><mi id="S2.p1.2.m1.1.1.2" xref="S2.p1.2.m1.1.1.2.cmml">i</mi><mo id="S2.p1.2.m1.1.1.1" xref="S2.p1.2.m1.1.1.1.cmml">=</mo><mi id="S2.p1.2.m1.1.1.3" xref="S2.p1.2.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.2.m1.1b"><apply id="S2.p1.2.m1.1.1.cmml" xref="S2.p1.2.m1.1.1"><eq id="S2.p1.2.m1.1.1.1.cmml" xref="S2.p1.2.m1.1.1.1"></eq><ci id="S2.p1.2.m1.1.1.2.cmml" xref="S2.p1.2.m1.1.1.2">𝑖</ci><csymbol cd="latexml" id="S2.p1.2.m1.1.1.3.cmml" xref="S2.p1.2.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.2.m1.1c">i=</annotation><annotation encoding="application/x-llamapun" id="S2.p1.2.m1.1d">italic_i =</annotation></semantics></math> 1 to <math alttext="n-1" class="ltx_Math" display="inline" id="S2.p1.3.m2.1"><semantics id="S2.p1.3.m2.1a"><mrow id="S2.p1.3.m2.1.1" xref="S2.p1.3.m2.1.1.cmml"><mi id="S2.p1.3.m2.1.1.2" xref="S2.p1.3.m2.1.1.2.cmml">n</mi><mo id="S2.p1.3.m2.1.1.1" xref="S2.p1.3.m2.1.1.1.cmml">−</mo><mn id="S2.p1.3.m2.1.1.3" xref="S2.p1.3.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.3.m2.1b"><apply id="S2.p1.3.m2.1.1.cmml" xref="S2.p1.3.m2.1.1"><minus id="S2.p1.3.m2.1.1.1.cmml" xref="S2.p1.3.m2.1.1.1"></minus><ci id="S2.p1.3.m2.1.1.2.cmml" xref="S2.p1.3.m2.1.1.2">𝑛</ci><cn id="S2.p1.3.m2.1.1.3.cmml" type="integer" xref="S2.p1.3.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.3.m2.1c">n-1</annotation><annotation encoding="application/x-llamapun" id="S2.p1.3.m2.1d">italic_n - 1</annotation></semantics></math> and <math alttext="{\bm{x}}_{i}(t):=[x_{1}(t),x_{2}(t),\cdots,x_{i}(t)]^{T}\in\mathbb{R}^{i}" class="ltx_Math" display="inline" id="S2.p1.4.m3.8"><semantics id="S2.p1.4.m3.8a"><mrow id="S2.p1.4.m3.8.8" xref="S2.p1.4.m3.8.8.cmml"><mrow id="S2.p1.4.m3.8.8.5" xref="S2.p1.4.m3.8.8.5.cmml"><msub id="S2.p1.4.m3.8.8.5.2" xref="S2.p1.4.m3.8.8.5.2.cmml"><mi id="S2.p1.4.m3.8.8.5.2.2" xref="S2.p1.4.m3.8.8.5.2.2.cmml">𝒙</mi><mi id="S2.p1.4.m3.8.8.5.2.3" xref="S2.p1.4.m3.8.8.5.2.3.cmml">i</mi></msub><mo id="S2.p1.4.m3.8.8.5.1" xref="S2.p1.4.m3.8.8.5.1.cmml">⁢</mo><mrow id="S2.p1.4.m3.8.8.5.3.2" xref="S2.p1.4.m3.8.8.5.cmml"><mo id="S2.p1.4.m3.8.8.5.3.2.1" stretchy="false" xref="S2.p1.4.m3.8.8.5.cmml">(</mo><mi id="S2.p1.4.m3.1.1" xref="S2.p1.4.m3.1.1.cmml">t</mi><mo id="S2.p1.4.m3.8.8.5.3.2.2" rspace="0.278em" stretchy="false" xref="S2.p1.4.m3.8.8.5.cmml">)</mo></mrow></mrow><mo id="S2.p1.4.m3.8.8.6" rspace="0.278em" xref="S2.p1.4.m3.8.8.6.cmml">:=</mo><msup id="S2.p1.4.m3.8.8.3" xref="S2.p1.4.m3.8.8.3.cmml"><mrow id="S2.p1.4.m3.8.8.3.3.3" xref="S2.p1.4.m3.8.8.3.3.4.cmml"><mo id="S2.p1.4.m3.8.8.3.3.3.4" stretchy="false" xref="S2.p1.4.m3.8.8.3.3.4.cmml">[</mo><mrow id="S2.p1.4.m3.6.6.1.1.1.1" xref="S2.p1.4.m3.6.6.1.1.1.1.cmml"><msub id="S2.p1.4.m3.6.6.1.1.1.1.2" xref="S2.p1.4.m3.6.6.1.1.1.1.2.cmml"><mi id="S2.p1.4.m3.6.6.1.1.1.1.2.2" xref="S2.p1.4.m3.6.6.1.1.1.1.2.2.cmml">x</mi><mn id="S2.p1.4.m3.6.6.1.1.1.1.2.3" xref="S2.p1.4.m3.6.6.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S2.p1.4.m3.6.6.1.1.1.1.1" xref="S2.p1.4.m3.6.6.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.p1.4.m3.6.6.1.1.1.1.3.2" xref="S2.p1.4.m3.6.6.1.1.1.1.cmml"><mo id="S2.p1.4.m3.6.6.1.1.1.1.3.2.1" stretchy="false" xref="S2.p1.4.m3.6.6.1.1.1.1.cmml">(</mo><mi id="S2.p1.4.m3.2.2" xref="S2.p1.4.m3.2.2.cmml">t</mi><mo id="S2.p1.4.m3.6.6.1.1.1.1.3.2.2" stretchy="false" xref="S2.p1.4.m3.6.6.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p1.4.m3.8.8.3.3.3.5" xref="S2.p1.4.m3.8.8.3.3.4.cmml">,</mo><mrow id="S2.p1.4.m3.7.7.2.2.2.2" xref="S2.p1.4.m3.7.7.2.2.2.2.cmml"><msub id="S2.p1.4.m3.7.7.2.2.2.2.2" xref="S2.p1.4.m3.7.7.2.2.2.2.2.cmml"><mi id="S2.p1.4.m3.7.7.2.2.2.2.2.2" xref="S2.p1.4.m3.7.7.2.2.2.2.2.2.cmml">x</mi><mn id="S2.p1.4.m3.7.7.2.2.2.2.2.3" xref="S2.p1.4.m3.7.7.2.2.2.2.2.3.cmml">2</mn></msub><mo id="S2.p1.4.m3.7.7.2.2.2.2.1" xref="S2.p1.4.m3.7.7.2.2.2.2.1.cmml">⁢</mo><mrow id="S2.p1.4.m3.7.7.2.2.2.2.3.2" xref="S2.p1.4.m3.7.7.2.2.2.2.cmml"><mo id="S2.p1.4.m3.7.7.2.2.2.2.3.2.1" stretchy="false" xref="S2.p1.4.m3.7.7.2.2.2.2.cmml">(</mo><mi id="S2.p1.4.m3.3.3" xref="S2.p1.4.m3.3.3.cmml">t</mi><mo id="S2.p1.4.m3.7.7.2.2.2.2.3.2.2" stretchy="false" xref="S2.p1.4.m3.7.7.2.2.2.2.cmml">)</mo></mrow></mrow><mo id="S2.p1.4.m3.8.8.3.3.3.6" xref="S2.p1.4.m3.8.8.3.3.4.cmml">,</mo><mi id="S2.p1.4.m3.5.5" mathvariant="normal" xref="S2.p1.4.m3.5.5.cmml">⋯</mi><mo id="S2.p1.4.m3.8.8.3.3.3.7" xref="S2.p1.4.m3.8.8.3.3.4.cmml">,</mo><mrow id="S2.p1.4.m3.8.8.3.3.3.3" xref="S2.p1.4.m3.8.8.3.3.3.3.cmml"><msub id="S2.p1.4.m3.8.8.3.3.3.3.2" xref="S2.p1.4.m3.8.8.3.3.3.3.2.cmml"><mi id="S2.p1.4.m3.8.8.3.3.3.3.2.2" xref="S2.p1.4.m3.8.8.3.3.3.3.2.2.cmml">x</mi><mi id="S2.p1.4.m3.8.8.3.3.3.3.2.3" xref="S2.p1.4.m3.8.8.3.3.3.3.2.3.cmml">i</mi></msub><mo id="S2.p1.4.m3.8.8.3.3.3.3.1" xref="S2.p1.4.m3.8.8.3.3.3.3.1.cmml">⁢</mo><mrow id="S2.p1.4.m3.8.8.3.3.3.3.3.2" xref="S2.p1.4.m3.8.8.3.3.3.3.cmml"><mo id="S2.p1.4.m3.8.8.3.3.3.3.3.2.1" stretchy="false" xref="S2.p1.4.m3.8.8.3.3.3.3.cmml">(</mo><mi id="S2.p1.4.m3.4.4" xref="S2.p1.4.m3.4.4.cmml">t</mi><mo id="S2.p1.4.m3.8.8.3.3.3.3.3.2.2" stretchy="false" xref="S2.p1.4.m3.8.8.3.3.3.3.cmml">)</mo></mrow></mrow><mo id="S2.p1.4.m3.8.8.3.3.3.8" stretchy="false" xref="S2.p1.4.m3.8.8.3.3.4.cmml">]</mo></mrow><mi id="S2.p1.4.m3.8.8.3.5" xref="S2.p1.4.m3.8.8.3.5.cmml">T</mi></msup><mo id="S2.p1.4.m3.8.8.7" xref="S2.p1.4.m3.8.8.7.cmml">∈</mo><msup id="S2.p1.4.m3.8.8.8" xref="S2.p1.4.m3.8.8.8.cmml"><mi id="S2.p1.4.m3.8.8.8.2" xref="S2.p1.4.m3.8.8.8.2.cmml">ℝ</mi><mi id="S2.p1.4.m3.8.8.8.3" xref="S2.p1.4.m3.8.8.8.3.cmml">i</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.4.m3.8b"><apply id="S2.p1.4.m3.8.8.cmml" xref="S2.p1.4.m3.8.8"><and id="S2.p1.4.m3.8.8a.cmml" xref="S2.p1.4.m3.8.8"></and><apply id="S2.p1.4.m3.8.8b.cmml" xref="S2.p1.4.m3.8.8"><csymbol cd="latexml" id="S2.p1.4.m3.8.8.6.cmml" xref="S2.p1.4.m3.8.8.6">assign</csymbol><apply id="S2.p1.4.m3.8.8.5.cmml" xref="S2.p1.4.m3.8.8.5"><times id="S2.p1.4.m3.8.8.5.1.cmml" xref="S2.p1.4.m3.8.8.5.1"></times><apply id="S2.p1.4.m3.8.8.5.2.cmml" xref="S2.p1.4.m3.8.8.5.2"><csymbol cd="ambiguous" id="S2.p1.4.m3.8.8.5.2.1.cmml" xref="S2.p1.4.m3.8.8.5.2">subscript</csymbol><ci id="S2.p1.4.m3.8.8.5.2.2.cmml" xref="S2.p1.4.m3.8.8.5.2.2">𝒙</ci><ci id="S2.p1.4.m3.8.8.5.2.3.cmml" xref="S2.p1.4.m3.8.8.5.2.3">𝑖</ci></apply><ci id="S2.p1.4.m3.1.1.cmml" xref="S2.p1.4.m3.1.1">𝑡</ci></apply><apply id="S2.p1.4.m3.8.8.3.cmml" xref="S2.p1.4.m3.8.8.3"><csymbol cd="ambiguous" id="S2.p1.4.m3.8.8.3.4.cmml" xref="S2.p1.4.m3.8.8.3">superscript</csymbol><list id="S2.p1.4.m3.8.8.3.3.4.cmml" xref="S2.p1.4.m3.8.8.3.3.3"><apply id="S2.p1.4.m3.6.6.1.1.1.1.cmml" xref="S2.p1.4.m3.6.6.1.1.1.1"><times id="S2.p1.4.m3.6.6.1.1.1.1.1.cmml" xref="S2.p1.4.m3.6.6.1.1.1.1.1"></times><apply id="S2.p1.4.m3.6.6.1.1.1.1.2.cmml" xref="S2.p1.4.m3.6.6.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.p1.4.m3.6.6.1.1.1.1.2.1.cmml" xref="S2.p1.4.m3.6.6.1.1.1.1.2">subscript</csymbol><ci id="S2.p1.4.m3.6.6.1.1.1.1.2.2.cmml" xref="S2.p1.4.m3.6.6.1.1.1.1.2.2">𝑥</ci><cn id="S2.p1.4.m3.6.6.1.1.1.1.2.3.cmml" type="integer" xref="S2.p1.4.m3.6.6.1.1.1.1.2.3">1</cn></apply><ci id="S2.p1.4.m3.2.2.cmml" xref="S2.p1.4.m3.2.2">𝑡</ci></apply><apply id="S2.p1.4.m3.7.7.2.2.2.2.cmml" xref="S2.p1.4.m3.7.7.2.2.2.2"><times id="S2.p1.4.m3.7.7.2.2.2.2.1.cmml" xref="S2.p1.4.m3.7.7.2.2.2.2.1"></times><apply id="S2.p1.4.m3.7.7.2.2.2.2.2.cmml" xref="S2.p1.4.m3.7.7.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.p1.4.m3.7.7.2.2.2.2.2.1.cmml" xref="S2.p1.4.m3.7.7.2.2.2.2.2">subscript</csymbol><ci id="S2.p1.4.m3.7.7.2.2.2.2.2.2.cmml" xref="S2.p1.4.m3.7.7.2.2.2.2.2.2">𝑥</ci><cn id="S2.p1.4.m3.7.7.2.2.2.2.2.3.cmml" type="integer" xref="S2.p1.4.m3.7.7.2.2.2.2.2.3">2</cn></apply><ci id="S2.p1.4.m3.3.3.cmml" xref="S2.p1.4.m3.3.3">𝑡</ci></apply><ci id="S2.p1.4.m3.5.5.cmml" xref="S2.p1.4.m3.5.5">⋯</ci><apply id="S2.p1.4.m3.8.8.3.3.3.3.cmml" xref="S2.p1.4.m3.8.8.3.3.3.3"><times id="S2.p1.4.m3.8.8.3.3.3.3.1.cmml" xref="S2.p1.4.m3.8.8.3.3.3.3.1"></times><apply id="S2.p1.4.m3.8.8.3.3.3.3.2.cmml" xref="S2.p1.4.m3.8.8.3.3.3.3.2"><csymbol cd="ambiguous" id="S2.p1.4.m3.8.8.3.3.3.3.2.1.cmml" xref="S2.p1.4.m3.8.8.3.3.3.3.2">subscript</csymbol><ci id="S2.p1.4.m3.8.8.3.3.3.3.2.2.cmml" xref="S2.p1.4.m3.8.8.3.3.3.3.2.2">𝑥</ci><ci id="S2.p1.4.m3.8.8.3.3.3.3.2.3.cmml" xref="S2.p1.4.m3.8.8.3.3.3.3.2.3">𝑖</ci></apply><ci id="S2.p1.4.m3.4.4.cmml" xref="S2.p1.4.m3.4.4">𝑡</ci></apply></list><ci id="S2.p1.4.m3.8.8.3.5.cmml" xref="S2.p1.4.m3.8.8.3.5">𝑇</ci></apply></apply><apply id="S2.p1.4.m3.8.8c.cmml" xref="S2.p1.4.m3.8.8"><in id="S2.p1.4.m3.8.8.7.cmml" xref="S2.p1.4.m3.8.8.7"></in><share href="https://arxiv.org/html/2401.10785v2#S2.p1.4.m3.8.8.3.cmml" id="S2.p1.4.m3.8.8d.cmml" xref="S2.p1.4.m3.8.8"></share><apply id="S2.p1.4.m3.8.8.8.cmml" xref="S2.p1.4.m3.8.8.8"><csymbol cd="ambiguous" id="S2.p1.4.m3.8.8.8.1.cmml" xref="S2.p1.4.m3.8.8.8">superscript</csymbol><ci id="S2.p1.4.m3.8.8.8.2.cmml" xref="S2.p1.4.m3.8.8.8.2">ℝ</ci><ci id="S2.p1.4.m3.8.8.8.3.cmml" xref="S2.p1.4.m3.8.8.8.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.4.m3.8c">{\bm{x}}_{i}(t):=[x_{1}(t),x_{2}(t),\cdots,x_{i}(t)]^{T}\in\mathbb{R}^{i}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.4.m3.8d">bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) := [ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) , ⋯ , italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="\bm{x}(t):=[x_{1}(t),x_{2}(t),\cdots,x_{n}(t)]^{T}" class="ltx_Math" display="inline" id="S2.p1.5.m4.8"><semantics id="S2.p1.5.m4.8a"><mrow id="S2.p1.5.m4.8.8" xref="S2.p1.5.m4.8.8.cmml"><mrow id="S2.p1.5.m4.8.8.5" xref="S2.p1.5.m4.8.8.5.cmml"><mi id="S2.p1.5.m4.8.8.5.2" xref="S2.p1.5.m4.8.8.5.2.cmml">𝒙</mi><mo id="S2.p1.5.m4.8.8.5.1" xref="S2.p1.5.m4.8.8.5.1.cmml">⁢</mo><mrow id="S2.p1.5.m4.8.8.5.3.2" xref="S2.p1.5.m4.8.8.5.cmml"><mo id="S2.p1.5.m4.8.8.5.3.2.1" stretchy="false" xref="S2.p1.5.m4.8.8.5.cmml">(</mo><mi id="S2.p1.5.m4.1.1" xref="S2.p1.5.m4.1.1.cmml">t</mi><mo id="S2.p1.5.m4.8.8.5.3.2.2" rspace="0.278em" stretchy="false" xref="S2.p1.5.m4.8.8.5.cmml">)</mo></mrow></mrow><mo id="S2.p1.5.m4.8.8.4" rspace="0.278em" xref="S2.p1.5.m4.8.8.4.cmml">:=</mo><msup id="S2.p1.5.m4.8.8.3" xref="S2.p1.5.m4.8.8.3.cmml"><mrow id="S2.p1.5.m4.8.8.3.3.3" xref="S2.p1.5.m4.8.8.3.3.4.cmml"><mo id="S2.p1.5.m4.8.8.3.3.3.4" stretchy="false" xref="S2.p1.5.m4.8.8.3.3.4.cmml">[</mo><mrow id="S2.p1.5.m4.6.6.1.1.1.1" xref="S2.p1.5.m4.6.6.1.1.1.1.cmml"><msub id="S2.p1.5.m4.6.6.1.1.1.1.2" xref="S2.p1.5.m4.6.6.1.1.1.1.2.cmml"><mi id="S2.p1.5.m4.6.6.1.1.1.1.2.2" xref="S2.p1.5.m4.6.6.1.1.1.1.2.2.cmml">x</mi><mn id="S2.p1.5.m4.6.6.1.1.1.1.2.3" xref="S2.p1.5.m4.6.6.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S2.p1.5.m4.6.6.1.1.1.1.1" xref="S2.p1.5.m4.6.6.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.p1.5.m4.6.6.1.1.1.1.3.2" xref="S2.p1.5.m4.6.6.1.1.1.1.cmml"><mo id="S2.p1.5.m4.6.6.1.1.1.1.3.2.1" stretchy="false" xref="S2.p1.5.m4.6.6.1.1.1.1.cmml">(</mo><mi id="S2.p1.5.m4.2.2" xref="S2.p1.5.m4.2.2.cmml">t</mi><mo id="S2.p1.5.m4.6.6.1.1.1.1.3.2.2" stretchy="false" xref="S2.p1.5.m4.6.6.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p1.5.m4.8.8.3.3.3.5" xref="S2.p1.5.m4.8.8.3.3.4.cmml">,</mo><mrow id="S2.p1.5.m4.7.7.2.2.2.2" xref="S2.p1.5.m4.7.7.2.2.2.2.cmml"><msub id="S2.p1.5.m4.7.7.2.2.2.2.2" xref="S2.p1.5.m4.7.7.2.2.2.2.2.cmml"><mi id="S2.p1.5.m4.7.7.2.2.2.2.2.2" xref="S2.p1.5.m4.7.7.2.2.2.2.2.2.cmml">x</mi><mn id="S2.p1.5.m4.7.7.2.2.2.2.2.3" xref="S2.p1.5.m4.7.7.2.2.2.2.2.3.cmml">2</mn></msub><mo id="S2.p1.5.m4.7.7.2.2.2.2.1" xref="S2.p1.5.m4.7.7.2.2.2.2.1.cmml">⁢</mo><mrow id="S2.p1.5.m4.7.7.2.2.2.2.3.2" xref="S2.p1.5.m4.7.7.2.2.2.2.cmml"><mo id="S2.p1.5.m4.7.7.2.2.2.2.3.2.1" stretchy="false" xref="S2.p1.5.m4.7.7.2.2.2.2.cmml">(</mo><mi id="S2.p1.5.m4.3.3" xref="S2.p1.5.m4.3.3.cmml">t</mi><mo id="S2.p1.5.m4.7.7.2.2.2.2.3.2.2" stretchy="false" xref="S2.p1.5.m4.7.7.2.2.2.2.cmml">)</mo></mrow></mrow><mo id="S2.p1.5.m4.8.8.3.3.3.6" xref="S2.p1.5.m4.8.8.3.3.4.cmml">,</mo><mi id="S2.p1.5.m4.5.5" mathvariant="normal" xref="S2.p1.5.m4.5.5.cmml">⋯</mi><mo id="S2.p1.5.m4.8.8.3.3.3.7" xref="S2.p1.5.m4.8.8.3.3.4.cmml">,</mo><mrow id="S2.p1.5.m4.8.8.3.3.3.3" xref="S2.p1.5.m4.8.8.3.3.3.3.cmml"><msub id="S2.p1.5.m4.8.8.3.3.3.3.2" xref="S2.p1.5.m4.8.8.3.3.3.3.2.cmml"><mi id="S2.p1.5.m4.8.8.3.3.3.3.2.2" xref="S2.p1.5.m4.8.8.3.3.3.3.2.2.cmml">x</mi><mi id="S2.p1.5.m4.8.8.3.3.3.3.2.3" xref="S2.p1.5.m4.8.8.3.3.3.3.2.3.cmml">n</mi></msub><mo id="S2.p1.5.m4.8.8.3.3.3.3.1" xref="S2.p1.5.m4.8.8.3.3.3.3.1.cmml">⁢</mo><mrow id="S2.p1.5.m4.8.8.3.3.3.3.3.2" xref="S2.p1.5.m4.8.8.3.3.3.3.cmml"><mo id="S2.p1.5.m4.8.8.3.3.3.3.3.2.1" stretchy="false" xref="S2.p1.5.m4.8.8.3.3.3.3.cmml">(</mo><mi id="S2.p1.5.m4.4.4" xref="S2.p1.5.m4.4.4.cmml">t</mi><mo id="S2.p1.5.m4.8.8.3.3.3.3.3.2.2" stretchy="false" xref="S2.p1.5.m4.8.8.3.3.3.3.cmml">)</mo></mrow></mrow><mo id="S2.p1.5.m4.8.8.3.3.3.8" stretchy="false" xref="S2.p1.5.m4.8.8.3.3.4.cmml">]</mo></mrow><mi id="S2.p1.5.m4.8.8.3.5" xref="S2.p1.5.m4.8.8.3.5.cmml">T</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.5.m4.8b"><apply id="S2.p1.5.m4.8.8.cmml" xref="S2.p1.5.m4.8.8"><csymbol cd="latexml" id="S2.p1.5.m4.8.8.4.cmml" xref="S2.p1.5.m4.8.8.4">assign</csymbol><apply id="S2.p1.5.m4.8.8.5.cmml" xref="S2.p1.5.m4.8.8.5"><times id="S2.p1.5.m4.8.8.5.1.cmml" xref="S2.p1.5.m4.8.8.5.1"></times><ci id="S2.p1.5.m4.8.8.5.2.cmml" xref="S2.p1.5.m4.8.8.5.2">𝒙</ci><ci id="S2.p1.5.m4.1.1.cmml" xref="S2.p1.5.m4.1.1">𝑡</ci></apply><apply id="S2.p1.5.m4.8.8.3.cmml" xref="S2.p1.5.m4.8.8.3"><csymbol cd="ambiguous" id="S2.p1.5.m4.8.8.3.4.cmml" xref="S2.p1.5.m4.8.8.3">superscript</csymbol><list id="S2.p1.5.m4.8.8.3.3.4.cmml" xref="S2.p1.5.m4.8.8.3.3.3"><apply id="S2.p1.5.m4.6.6.1.1.1.1.cmml" xref="S2.p1.5.m4.6.6.1.1.1.1"><times id="S2.p1.5.m4.6.6.1.1.1.1.1.cmml" xref="S2.p1.5.m4.6.6.1.1.1.1.1"></times><apply id="S2.p1.5.m4.6.6.1.1.1.1.2.cmml" xref="S2.p1.5.m4.6.6.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.p1.5.m4.6.6.1.1.1.1.2.1.cmml" xref="S2.p1.5.m4.6.6.1.1.1.1.2">subscript</csymbol><ci id="S2.p1.5.m4.6.6.1.1.1.1.2.2.cmml" xref="S2.p1.5.m4.6.6.1.1.1.1.2.2">𝑥</ci><cn id="S2.p1.5.m4.6.6.1.1.1.1.2.3.cmml" type="integer" xref="S2.p1.5.m4.6.6.1.1.1.1.2.3">1</cn></apply><ci id="S2.p1.5.m4.2.2.cmml" xref="S2.p1.5.m4.2.2">𝑡</ci></apply><apply id="S2.p1.5.m4.7.7.2.2.2.2.cmml" xref="S2.p1.5.m4.7.7.2.2.2.2"><times id="S2.p1.5.m4.7.7.2.2.2.2.1.cmml" xref="S2.p1.5.m4.7.7.2.2.2.2.1"></times><apply id="S2.p1.5.m4.7.7.2.2.2.2.2.cmml" xref="S2.p1.5.m4.7.7.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.p1.5.m4.7.7.2.2.2.2.2.1.cmml" xref="S2.p1.5.m4.7.7.2.2.2.2.2">subscript</csymbol><ci id="S2.p1.5.m4.7.7.2.2.2.2.2.2.cmml" xref="S2.p1.5.m4.7.7.2.2.2.2.2.2">𝑥</ci><cn id="S2.p1.5.m4.7.7.2.2.2.2.2.3.cmml" type="integer" xref="S2.p1.5.m4.7.7.2.2.2.2.2.3">2</cn></apply><ci id="S2.p1.5.m4.3.3.cmml" xref="S2.p1.5.m4.3.3">𝑡</ci></apply><ci id="S2.p1.5.m4.5.5.cmml" xref="S2.p1.5.m4.5.5">⋯</ci><apply id="S2.p1.5.m4.8.8.3.3.3.3.cmml" xref="S2.p1.5.m4.8.8.3.3.3.3"><times id="S2.p1.5.m4.8.8.3.3.3.3.1.cmml" xref="S2.p1.5.m4.8.8.3.3.3.3.1"></times><apply id="S2.p1.5.m4.8.8.3.3.3.3.2.cmml" xref="S2.p1.5.m4.8.8.3.3.3.3.2"><csymbol cd="ambiguous" id="S2.p1.5.m4.8.8.3.3.3.3.2.1.cmml" xref="S2.p1.5.m4.8.8.3.3.3.3.2">subscript</csymbol><ci id="S2.p1.5.m4.8.8.3.3.3.3.2.2.cmml" xref="S2.p1.5.m4.8.8.3.3.3.3.2.2">𝑥</ci><ci id="S2.p1.5.m4.8.8.3.3.3.3.2.3.cmml" xref="S2.p1.5.m4.8.8.3.3.3.3.2.3">𝑛</ci></apply><ci id="S2.p1.5.m4.4.4.cmml" xref="S2.p1.5.m4.4.4">𝑡</ci></apply></list><ci id="S2.p1.5.m4.8.8.3.5.cmml" xref="S2.p1.5.m4.8.8.3.5">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.5.m4.8c">\bm{x}(t):=[x_{1}(t),x_{2}(t),\cdots,x_{n}(t)]^{T}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.5.m4.8d">bold_italic_x ( italic_t ) := [ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) , ⋯ , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> <math alttext="\in\mathbb{R}^{n}" class="ltx_Math" display="inline" id="S2.p1.6.m5.1"><semantics id="S2.p1.6.m5.1a"><mrow id="S2.p1.6.m5.1.1" xref="S2.p1.6.m5.1.1.cmml"><mi id="S2.p1.6.m5.1.1.2" xref="S2.p1.6.m5.1.1.2.cmml"></mi><mo id="S2.p1.6.m5.1.1.1" xref="S2.p1.6.m5.1.1.1.cmml">∈</mo><msup id="S2.p1.6.m5.1.1.3" xref="S2.p1.6.m5.1.1.3.cmml"><mi id="S2.p1.6.m5.1.1.3.2" xref="S2.p1.6.m5.1.1.3.2.cmml">ℝ</mi><mi id="S2.p1.6.m5.1.1.3.3" xref="S2.p1.6.m5.1.1.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.6.m5.1b"><apply id="S2.p1.6.m5.1.1.cmml" xref="S2.p1.6.m5.1.1"><in id="S2.p1.6.m5.1.1.1.cmml" xref="S2.p1.6.m5.1.1.1"></in><csymbol cd="latexml" id="S2.p1.6.m5.1.1.2.cmml" xref="S2.p1.6.m5.1.1.2">absent</csymbol><apply id="S2.p1.6.m5.1.1.3.cmml" xref="S2.p1.6.m5.1.1.3"><csymbol cd="ambiguous" id="S2.p1.6.m5.1.1.3.1.cmml" xref="S2.p1.6.m5.1.1.3">superscript</csymbol><ci id="S2.p1.6.m5.1.1.3.2.cmml" xref="S2.p1.6.m5.1.1.3.2">ℝ</ci><ci id="S2.p1.6.m5.1.1.3.3.cmml" xref="S2.p1.6.m5.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.6.m5.1c">\in\mathbb{R}^{n}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.6.m5.1d">∈ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> is a measurable state, <math alttext="u(t)\in\mathbb{R}^{n}" class="ltx_Math" display="inline" id="S2.p1.7.m6.1"><semantics id="S2.p1.7.m6.1a"><mrow id="S2.p1.7.m6.1.2" xref="S2.p1.7.m6.1.2.cmml"><mrow id="S2.p1.7.m6.1.2.2" xref="S2.p1.7.m6.1.2.2.cmml"><mi id="S2.p1.7.m6.1.2.2.2" xref="S2.p1.7.m6.1.2.2.2.cmml">u</mi><mo id="S2.p1.7.m6.1.2.2.1" xref="S2.p1.7.m6.1.2.2.1.cmml">⁢</mo><mrow id="S2.p1.7.m6.1.2.2.3.2" xref="S2.p1.7.m6.1.2.2.cmml"><mo id="S2.p1.7.m6.1.2.2.3.2.1" stretchy="false" xref="S2.p1.7.m6.1.2.2.cmml">(</mo><mi id="S2.p1.7.m6.1.1" xref="S2.p1.7.m6.1.1.cmml">t</mi><mo id="S2.p1.7.m6.1.2.2.3.2.2" stretchy="false" xref="S2.p1.7.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.p1.7.m6.1.2.1" xref="S2.p1.7.m6.1.2.1.cmml">∈</mo><msup id="S2.p1.7.m6.1.2.3" xref="S2.p1.7.m6.1.2.3.cmml"><mi id="S2.p1.7.m6.1.2.3.2" xref="S2.p1.7.m6.1.2.3.2.cmml">ℝ</mi><mi id="S2.p1.7.m6.1.2.3.3" xref="S2.p1.7.m6.1.2.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.7.m6.1b"><apply id="S2.p1.7.m6.1.2.cmml" xref="S2.p1.7.m6.1.2"><in id="S2.p1.7.m6.1.2.1.cmml" xref="S2.p1.7.m6.1.2.1"></in><apply id="S2.p1.7.m6.1.2.2.cmml" xref="S2.p1.7.m6.1.2.2"><times id="S2.p1.7.m6.1.2.2.1.cmml" xref="S2.p1.7.m6.1.2.2.1"></times><ci id="S2.p1.7.m6.1.2.2.2.cmml" xref="S2.p1.7.m6.1.2.2.2">𝑢</ci><ci id="S2.p1.7.m6.1.1.cmml" xref="S2.p1.7.m6.1.1">𝑡</ci></apply><apply id="S2.p1.7.m6.1.2.3.cmml" xref="S2.p1.7.m6.1.2.3"><csymbol cd="ambiguous" id="S2.p1.7.m6.1.2.3.1.cmml" xref="S2.p1.7.m6.1.2.3">superscript</csymbol><ci id="S2.p1.7.m6.1.2.3.2.cmml" xref="S2.p1.7.m6.1.2.3.2">ℝ</ci><ci id="S2.p1.7.m6.1.2.3.3.cmml" xref="S2.p1.7.m6.1.2.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.7.m6.1c">u(t)\in\mathbb{R}^{n}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.7.m6.1d">italic_u ( italic_t ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> is a control input, <math alttext="y(t)\in\mathbb{R}" class="ltx_Math" display="inline" id="S2.p1.8.m7.1"><semantics id="S2.p1.8.m7.1a"><mrow id="S2.p1.8.m7.1.2" xref="S2.p1.8.m7.1.2.cmml"><mrow id="S2.p1.8.m7.1.2.2" xref="S2.p1.8.m7.1.2.2.cmml"><mi id="S2.p1.8.m7.1.2.2.2" xref="S2.p1.8.m7.1.2.2.2.cmml">y</mi><mo id="S2.p1.8.m7.1.2.2.1" xref="S2.p1.8.m7.1.2.2.1.cmml">⁢</mo><mrow id="S2.p1.8.m7.1.2.2.3.2" xref="S2.p1.8.m7.1.2.2.cmml"><mo id="S2.p1.8.m7.1.2.2.3.2.1" stretchy="false" xref="S2.p1.8.m7.1.2.2.cmml">(</mo><mi id="S2.p1.8.m7.1.1" xref="S2.p1.8.m7.1.1.cmml">t</mi><mo id="S2.p1.8.m7.1.2.2.3.2.2" stretchy="false" xref="S2.p1.8.m7.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.p1.8.m7.1.2.1" xref="S2.p1.8.m7.1.2.1.cmml">∈</mo><mi id="S2.p1.8.m7.1.2.3" xref="S2.p1.8.m7.1.2.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.8.m7.1b"><apply id="S2.p1.8.m7.1.2.cmml" xref="S2.p1.8.m7.1.2"><in id="S2.p1.8.m7.1.2.1.cmml" xref="S2.p1.8.m7.1.2.1"></in><apply id="S2.p1.8.m7.1.2.2.cmml" xref="S2.p1.8.m7.1.2.2"><times id="S2.p1.8.m7.1.2.2.1.cmml" xref="S2.p1.8.m7.1.2.2.1"></times><ci id="S2.p1.8.m7.1.2.2.2.cmml" xref="S2.p1.8.m7.1.2.2.2">𝑦</ci><ci id="S2.p1.8.m7.1.1.cmml" xref="S2.p1.8.m7.1.1">𝑡</ci></apply><ci id="S2.p1.8.m7.1.2.3.cmml" xref="S2.p1.8.m7.1.2.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.8.m7.1c">y(t)\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.8.m7.1d">italic_y ( italic_t ) ∈ blackboard_R</annotation></semantics></math> is a system output, <math alttext="\bm{\theta}\in\Omega_{{\rm c}_{\theta}}\subset\mathbb{R}^{N}" class="ltx_Math" display="inline" id="S2.p1.9.m8.1"><semantics id="S2.p1.9.m8.1a"><mrow id="S2.p1.9.m8.1.1" xref="S2.p1.9.m8.1.1.cmml"><mi id="S2.p1.9.m8.1.1.2" mathcolor="#000099" xref="S2.p1.9.m8.1.1.2.cmml">𝜽</mi><mo id="S2.p1.9.m8.1.1.3" mathcolor="#000099" xref="S2.p1.9.m8.1.1.3.cmml">∈</mo><msub id="S2.p1.9.m8.1.1.4" xref="S2.p1.9.m8.1.1.4.cmml"><mi id="S2.p1.9.m8.1.1.4.2" mathcolor="#000099" mathvariant="normal" xref="S2.p1.9.m8.1.1.4.2.cmml">Ω</mi><msub id="S2.p1.9.m8.1.1.4.3" xref="S2.p1.9.m8.1.1.4.3.cmml"><mi id="S2.p1.9.m8.1.1.4.3.2" mathcolor="#000099" mathvariant="normal" xref="S2.p1.9.m8.1.1.4.3.2.cmml">c</mi><mi id="S2.p1.9.m8.1.1.4.3.3" mathcolor="#000099" xref="S2.p1.9.m8.1.1.4.3.3.cmml">θ</mi></msub></msub><mo id="S2.p1.9.m8.1.1.5" mathcolor="#000099" xref="S2.p1.9.m8.1.1.5.cmml">⊂</mo><msup id="S2.p1.9.m8.1.1.6" xref="S2.p1.9.m8.1.1.6.cmml"><mi id="S2.p1.9.m8.1.1.6.2" mathcolor="#000099" xref="S2.p1.9.m8.1.1.6.2.cmml">ℝ</mi><mi id="S2.p1.9.m8.1.1.6.3" mathcolor="#000099" xref="S2.p1.9.m8.1.1.6.3.cmml">N</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.9.m8.1b"><apply id="S2.p1.9.m8.1.1.cmml" xref="S2.p1.9.m8.1.1"><and id="S2.p1.9.m8.1.1a.cmml" xref="S2.p1.9.m8.1.1"></and><apply id="S2.p1.9.m8.1.1b.cmml" xref="S2.p1.9.m8.1.1"><in id="S2.p1.9.m8.1.1.3.cmml" xref="S2.p1.9.m8.1.1.3"></in><ci id="S2.p1.9.m8.1.1.2.cmml" xref="S2.p1.9.m8.1.1.2">𝜽</ci><apply id="S2.p1.9.m8.1.1.4.cmml" xref="S2.p1.9.m8.1.1.4"><csymbol cd="ambiguous" id="S2.p1.9.m8.1.1.4.1.cmml" xref="S2.p1.9.m8.1.1.4">subscript</csymbol><ci id="S2.p1.9.m8.1.1.4.2.cmml" xref="S2.p1.9.m8.1.1.4.2">Ω</ci><apply id="S2.p1.9.m8.1.1.4.3.cmml" xref="S2.p1.9.m8.1.1.4.3"><csymbol cd="ambiguous" id="S2.p1.9.m8.1.1.4.3.1.cmml" xref="S2.p1.9.m8.1.1.4.3">subscript</csymbol><ci id="S2.p1.9.m8.1.1.4.3.2.cmml" xref="S2.p1.9.m8.1.1.4.3.2">c</ci><ci id="S2.p1.9.m8.1.1.4.3.3.cmml" xref="S2.p1.9.m8.1.1.4.3.3">𝜃</ci></apply></apply></apply><apply id="S2.p1.9.m8.1.1c.cmml" xref="S2.p1.9.m8.1.1"><subset id="S2.p1.9.m8.1.1.5.cmml" xref="S2.p1.9.m8.1.1.5"></subset><share href="https://arxiv.org/html/2401.10785v2#S2.p1.9.m8.1.1.4.cmml" id="S2.p1.9.m8.1.1d.cmml" xref="S2.p1.9.m8.1.1"></share><apply id="S2.p1.9.m8.1.1.6.cmml" xref="S2.p1.9.m8.1.1.6"><csymbol cd="ambiguous" id="S2.p1.9.m8.1.1.6.1.cmml" xref="S2.p1.9.m8.1.1.6">superscript</csymbol><ci id="S2.p1.9.m8.1.1.6.2.cmml" xref="S2.p1.9.m8.1.1.6.2">ℝ</ci><ci id="S2.p1.9.m8.1.1.6.3.cmml" xref="S2.p1.9.m8.1.1.6.3">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.9.m8.1c">\bm{\theta}\in\Omega_{{\rm c}_{\theta}}\subset\mathbb{R}^{N}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.9.m8.1d">bold_italic_θ ∈ roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊂ blackboard_R start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text" id="S2.p1.10.1" style="color:#000099;"> with <math alttext="{\rm c}_{\theta}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S2.p1.10.1.m1.1"><semantics id="S2.p1.10.1.m1.1a"><mrow id="S2.p1.10.1.m1.1.1" xref="S2.p1.10.1.m1.1.1.cmml"><msub id="S2.p1.10.1.m1.1.1.2" xref="S2.p1.10.1.m1.1.1.2.cmml"><mi id="S2.p1.10.1.m1.1.1.2.2" mathcolor="#000099" mathvariant="normal" xref="S2.p1.10.1.m1.1.1.2.2.cmml">c</mi><mi id="S2.p1.10.1.m1.1.1.2.3" mathcolor="#000099" xref="S2.p1.10.1.m1.1.1.2.3.cmml">θ</mi></msub><mo id="S2.p1.10.1.m1.1.1.1" mathcolor="#000099" xref="S2.p1.10.1.m1.1.1.1.cmml">∈</mo><msup id="S2.p1.10.1.m1.1.1.3" xref="S2.p1.10.1.m1.1.1.3.cmml"><mi id="S2.p1.10.1.m1.1.1.3.2" mathcolor="#000099" xref="S2.p1.10.1.m1.1.1.3.2.cmml">ℝ</mi><mo id="S2.p1.10.1.m1.1.1.3.3" mathcolor="#000099" xref="S2.p1.10.1.m1.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.10.1.m1.1b"><apply id="S2.p1.10.1.m1.1.1.cmml" xref="S2.p1.10.1.m1.1.1"><in id="S2.p1.10.1.m1.1.1.1.cmml" xref="S2.p1.10.1.m1.1.1.1"></in><apply id="S2.p1.10.1.m1.1.1.2.cmml" xref="S2.p1.10.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.p1.10.1.m1.1.1.2.1.cmml" xref="S2.p1.10.1.m1.1.1.2">subscript</csymbol><ci id="S2.p1.10.1.m1.1.1.2.2.cmml" xref="S2.p1.10.1.m1.1.1.2.2">c</ci><ci id="S2.p1.10.1.m1.1.1.2.3.cmml" xref="S2.p1.10.1.m1.1.1.2.3">𝜃</ci></apply><apply id="S2.p1.10.1.m1.1.1.3.cmml" xref="S2.p1.10.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.p1.10.1.m1.1.1.3.1.cmml" xref="S2.p1.10.1.m1.1.1.3">superscript</csymbol><ci id="S2.p1.10.1.m1.1.1.3.2.cmml" xref="S2.p1.10.1.m1.1.1.3.2">ℝ</ci><plus id="S2.p1.10.1.m1.1.1.3.3.cmml" xref="S2.p1.10.1.m1.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.10.1.m1.1c">{\rm c}_{\theta}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.10.1.m1.1d">roman_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> is an unknown constant parameter</span>, <math alttext="{\bm{\varphi}}_{i}:\mathbb{R}^{i}" class="ltx_Math" display="inline" id="S2.p1.11.m9.1"><semantics id="S2.p1.11.m9.1a"><mrow id="S2.p1.11.m9.1.1" xref="S2.p1.11.m9.1.1.cmml"><msub id="S2.p1.11.m9.1.1.2" xref="S2.p1.11.m9.1.1.2.cmml"><mi id="S2.p1.11.m9.1.1.2.2" xref="S2.p1.11.m9.1.1.2.2.cmml">𝝋</mi><mi id="S2.p1.11.m9.1.1.2.3" xref="S2.p1.11.m9.1.1.2.3.cmml">i</mi></msub><mo id="S2.p1.11.m9.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.p1.11.m9.1.1.1.cmml">:</mo><msup id="S2.p1.11.m9.1.1.3" xref="S2.p1.11.m9.1.1.3.cmml"><mi id="S2.p1.11.m9.1.1.3.2" xref="S2.p1.11.m9.1.1.3.2.cmml">ℝ</mi><mi id="S2.p1.11.m9.1.1.3.3" xref="S2.p1.11.m9.1.1.3.3.cmml">i</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.11.m9.1b"><apply id="S2.p1.11.m9.1.1.cmml" xref="S2.p1.11.m9.1.1"><ci id="S2.p1.11.m9.1.1.1.cmml" xref="S2.p1.11.m9.1.1.1">:</ci><apply id="S2.p1.11.m9.1.1.2.cmml" xref="S2.p1.11.m9.1.1.2"><csymbol cd="ambiguous" id="S2.p1.11.m9.1.1.2.1.cmml" xref="S2.p1.11.m9.1.1.2">subscript</csymbol><ci id="S2.p1.11.m9.1.1.2.2.cmml" xref="S2.p1.11.m9.1.1.2.2">𝝋</ci><ci id="S2.p1.11.m9.1.1.2.3.cmml" xref="S2.p1.11.m9.1.1.2.3">𝑖</ci></apply><apply id="S2.p1.11.m9.1.1.3.cmml" xref="S2.p1.11.m9.1.1.3"><csymbol cd="ambiguous" id="S2.p1.11.m9.1.1.3.1.cmml" xref="S2.p1.11.m9.1.1.3">superscript</csymbol><ci id="S2.p1.11.m9.1.1.3.2.cmml" xref="S2.p1.11.m9.1.1.3.2">ℝ</ci><ci id="S2.p1.11.m9.1.1.3.3.cmml" xref="S2.p1.11.m9.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.11.m9.1c">{\bm{\varphi}}_{i}:\mathbb{R}^{i}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.11.m9.1d">bold_italic_φ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : blackboard_R start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> <math alttext="\rightarrow\mathbb{R}^{N}" class="ltx_Math" display="inline" id="S2.p1.12.m10.1"><semantics id="S2.p1.12.m10.1a"><mrow id="S2.p1.12.m10.1.1" xref="S2.p1.12.m10.1.1.cmml"><mi id="S2.p1.12.m10.1.1.2" xref="S2.p1.12.m10.1.1.2.cmml"></mi><mo id="S2.p1.12.m10.1.1.1" stretchy="false" xref="S2.p1.12.m10.1.1.1.cmml">→</mo><msup id="S2.p1.12.m10.1.1.3" xref="S2.p1.12.m10.1.1.3.cmml"><mi id="S2.p1.12.m10.1.1.3.2" xref="S2.p1.12.m10.1.1.3.2.cmml">ℝ</mi><mi id="S2.p1.12.m10.1.1.3.3" xref="S2.p1.12.m10.1.1.3.3.cmml">N</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.12.m10.1b"><apply id="S2.p1.12.m10.1.1.cmml" xref="S2.p1.12.m10.1.1"><ci id="S2.p1.12.m10.1.1.1.cmml" xref="S2.p1.12.m10.1.1.1">→</ci><csymbol cd="latexml" id="S2.p1.12.m10.1.1.2.cmml" xref="S2.p1.12.m10.1.1.2">absent</csymbol><apply id="S2.p1.12.m10.1.1.3.cmml" xref="S2.p1.12.m10.1.1.3"><csymbol cd="ambiguous" id="S2.p1.12.m10.1.1.3.1.cmml" xref="S2.p1.12.m10.1.1.3">superscript</csymbol><ci id="S2.p1.12.m10.1.1.3.2.cmml" xref="S2.p1.12.m10.1.1.3.2">ℝ</ci><ci id="S2.p1.12.m10.1.1.3.3.cmml" xref="S2.p1.12.m10.1.1.3.3">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.12.m10.1c">\rightarrow\mathbb{R}^{N}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.12.m10.1d">→ blackboard_R start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT</annotation></semantics></math> is a known regressor, <math alttext="\beta:" class="ltx_Math" display="inline" id="S2.p1.13.m11.1"><semantics id="S2.p1.13.m11.1a"><mrow id="S2.p1.13.m11.1.1" xref="S2.p1.13.m11.1.1.cmml"><mi id="S2.p1.13.m11.1.1.2" xref="S2.p1.13.m11.1.1.2.cmml">β</mi><mo id="S2.p1.13.m11.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.p1.13.m11.1.1.1.cmml">:</mo><mi id="S2.p1.13.m11.1.1.3" xref="S2.p1.13.m11.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.13.m11.1b"><apply id="S2.p1.13.m11.1.1.cmml" xref="S2.p1.13.m11.1.1"><ci id="S2.p1.13.m11.1.1.1.cmml" xref="S2.p1.13.m11.1.1.1">:</ci><ci id="S2.p1.13.m11.1.1.2.cmml" xref="S2.p1.13.m11.1.1.2">𝛽</ci><csymbol cd="latexml" id="S2.p1.13.m11.1.1.3.cmml" xref="S2.p1.13.m11.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.13.m11.1c">\beta:</annotation><annotation encoding="application/x-llamapun" id="S2.p1.13.m11.1d">italic_β :</annotation></semantics></math> <math alttext="\mathbb{R}^{n}\rightarrow\mathbb{R}" class="ltx_Math" display="inline" id="S2.p1.14.m12.1"><semantics id="S2.p1.14.m12.1a"><mrow id="S2.p1.14.m12.1.1" xref="S2.p1.14.m12.1.1.cmml"><msup id="S2.p1.14.m12.1.1.2" xref="S2.p1.14.m12.1.1.2.cmml"><mi id="S2.p1.14.m12.1.1.2.2" xref="S2.p1.14.m12.1.1.2.2.cmml">ℝ</mi><mi id="S2.p1.14.m12.1.1.2.3" xref="S2.p1.14.m12.1.1.2.3.cmml">n</mi></msup><mo id="S2.p1.14.m12.1.1.1" stretchy="false" xref="S2.p1.14.m12.1.1.1.cmml">→</mo><mi id="S2.p1.14.m12.1.1.3" xref="S2.p1.14.m12.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.14.m12.1b"><apply id="S2.p1.14.m12.1.1.cmml" xref="S2.p1.14.m12.1.1"><ci id="S2.p1.14.m12.1.1.1.cmml" xref="S2.p1.14.m12.1.1.1">→</ci><apply id="S2.p1.14.m12.1.1.2.cmml" xref="S2.p1.14.m12.1.1.2"><csymbol cd="ambiguous" id="S2.p1.14.m12.1.1.2.1.cmml" xref="S2.p1.14.m12.1.1.2">superscript</csymbol><ci id="S2.p1.14.m12.1.1.2.2.cmml" xref="S2.p1.14.m12.1.1.2.2">ℝ</ci><ci id="S2.p1.14.m12.1.1.2.3.cmml" xref="S2.p1.14.m12.1.1.2.3">𝑛</ci></apply><ci id="S2.p1.14.m12.1.1.3.cmml" xref="S2.p1.14.m12.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.14.m12.1c">\mathbb{R}^{n}\rightarrow\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.14.m12.1d">blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT → blackboard_R</annotation></semantics></math> is a known gain function, and <math alttext="N" class="ltx_Math" display="inline" id="S2.p1.15.m13.1"><semantics id="S2.p1.15.m13.1a"><mi id="S2.p1.15.m13.1.1" xref="S2.p1.15.m13.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S2.p1.15.m13.1b"><ci id="S2.p1.15.m13.1.1.cmml" xref="S2.p1.15.m13.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.15.m13.1c">N</annotation><annotation encoding="application/x-llamapun" id="S2.p1.15.m13.1d">italic_N</annotation></semantics></math> is the number of parameter elements. Let <math alttext="y_{\rm r}(t)\in\mathbb{R}" class="ltx_Math" display="inline" id="S2.p1.16.m14.1"><semantics id="S2.p1.16.m14.1a"><mrow id="S2.p1.16.m14.1.2" xref="S2.p1.16.m14.1.2.cmml"><mrow id="S2.p1.16.m14.1.2.2" xref="S2.p1.16.m14.1.2.2.cmml"><msub id="S2.p1.16.m14.1.2.2.2" xref="S2.p1.16.m14.1.2.2.2.cmml"><mi id="S2.p1.16.m14.1.2.2.2.2" xref="S2.p1.16.m14.1.2.2.2.2.cmml">y</mi><mi id="S2.p1.16.m14.1.2.2.2.3" mathvariant="normal" xref="S2.p1.16.m14.1.2.2.2.3.cmml">r</mi></msub><mo id="S2.p1.16.m14.1.2.2.1" xref="S2.p1.16.m14.1.2.2.1.cmml">⁢</mo><mrow id="S2.p1.16.m14.1.2.2.3.2" xref="S2.p1.16.m14.1.2.2.cmml"><mo id="S2.p1.16.m14.1.2.2.3.2.1" stretchy="false" xref="S2.p1.16.m14.1.2.2.cmml">(</mo><mi id="S2.p1.16.m14.1.1" xref="S2.p1.16.m14.1.1.cmml">t</mi><mo id="S2.p1.16.m14.1.2.2.3.2.2" stretchy="false" xref="S2.p1.16.m14.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.p1.16.m14.1.2.1" xref="S2.p1.16.m14.1.2.1.cmml">∈</mo><mi id="S2.p1.16.m14.1.2.3" xref="S2.p1.16.m14.1.2.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.16.m14.1b"><apply id="S2.p1.16.m14.1.2.cmml" xref="S2.p1.16.m14.1.2"><in id="S2.p1.16.m14.1.2.1.cmml" xref="S2.p1.16.m14.1.2.1"></in><apply id="S2.p1.16.m14.1.2.2.cmml" xref="S2.p1.16.m14.1.2.2"><times id="S2.p1.16.m14.1.2.2.1.cmml" xref="S2.p1.16.m14.1.2.2.1"></times><apply id="S2.p1.16.m14.1.2.2.2.cmml" xref="S2.p1.16.m14.1.2.2.2"><csymbol cd="ambiguous" id="S2.p1.16.m14.1.2.2.2.1.cmml" xref="S2.p1.16.m14.1.2.2.2">subscript</csymbol><ci id="S2.p1.16.m14.1.2.2.2.2.cmml" xref="S2.p1.16.m14.1.2.2.2.2">𝑦</ci><ci id="S2.p1.16.m14.1.2.2.2.3.cmml" xref="S2.p1.16.m14.1.2.2.2.3">r</ci></apply><ci id="S2.p1.16.m14.1.1.cmml" xref="S2.p1.16.m14.1.1">𝑡</ci></apply><ci id="S2.p1.16.m14.1.2.3.cmml" xref="S2.p1.16.m14.1.2.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.16.m14.1c">y_{\rm r}(t)\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.16.m14.1d">italic_y start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT ( italic_t ) ∈ blackboard_R</annotation></semantics></math> be a reference signal. The following definitions are given for the subsequent analysis.</p> </div> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.4"><span class="ltx_text ltx_font_italic" id="S2.p2.4.5">Definition 1</span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib25" title="">25</a>]</cite>: <span class="ltx_text" id="S2.p2.4.4" style="color:#000099;">A bounded regressor <math alttext="\Phi(t)\in\mathbb{R}^{N\times n}" class="ltx_Math" display="inline" id="S2.p2.1.1.m1.1"><semantics id="S2.p2.1.1.m1.1a"><mrow id="S2.p2.1.1.m1.1.2" xref="S2.p2.1.1.m1.1.2.cmml"><mrow id="S2.p2.1.1.m1.1.2.2" xref="S2.p2.1.1.m1.1.2.2.cmml"><mi id="S2.p2.1.1.m1.1.2.2.2" mathcolor="#000099" mathvariant="normal" xref="S2.p2.1.1.m1.1.2.2.2.cmml">Φ</mi><mo id="S2.p2.1.1.m1.1.2.2.1" xref="S2.p2.1.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S2.p2.1.1.m1.1.2.2.3.2" xref="S2.p2.1.1.m1.1.2.2.cmml"><mo id="S2.p2.1.1.m1.1.2.2.3.2.1" mathcolor="#000099" stretchy="false" xref="S2.p2.1.1.m1.1.2.2.cmml">(</mo><mi id="S2.p2.1.1.m1.1.1" mathcolor="#000099" xref="S2.p2.1.1.m1.1.1.cmml">t</mi><mo id="S2.p2.1.1.m1.1.2.2.3.2.2" mathcolor="#000099" stretchy="false" xref="S2.p2.1.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.p2.1.1.m1.1.2.1" mathcolor="#000099" xref="S2.p2.1.1.m1.1.2.1.cmml">∈</mo><msup id="S2.p2.1.1.m1.1.2.3" xref="S2.p2.1.1.m1.1.2.3.cmml"><mi id="S2.p2.1.1.m1.1.2.3.2" mathcolor="#000099" xref="S2.p2.1.1.m1.1.2.3.2.cmml">ℝ</mi><mrow id="S2.p2.1.1.m1.1.2.3.3" xref="S2.p2.1.1.m1.1.2.3.3.cmml"><mi id="S2.p2.1.1.m1.1.2.3.3.2" mathcolor="#000099" xref="S2.p2.1.1.m1.1.2.3.3.2.cmml">N</mi><mo id="S2.p2.1.1.m1.1.2.3.3.1" lspace="0.222em" mathcolor="#000099" rspace="0.222em" xref="S2.p2.1.1.m1.1.2.3.3.1.cmml">×</mo><mi id="S2.p2.1.1.m1.1.2.3.3.3" mathcolor="#000099" xref="S2.p2.1.1.m1.1.2.3.3.3.cmml">n</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.1.1.m1.1b"><apply id="S2.p2.1.1.m1.1.2.cmml" xref="S2.p2.1.1.m1.1.2"><in id="S2.p2.1.1.m1.1.2.1.cmml" xref="S2.p2.1.1.m1.1.2.1"></in><apply id="S2.p2.1.1.m1.1.2.2.cmml" xref="S2.p2.1.1.m1.1.2.2"><times id="S2.p2.1.1.m1.1.2.2.1.cmml" xref="S2.p2.1.1.m1.1.2.2.1"></times><ci id="S2.p2.1.1.m1.1.2.2.2.cmml" xref="S2.p2.1.1.m1.1.2.2.2">Φ</ci><ci id="S2.p2.1.1.m1.1.1.cmml" xref="S2.p2.1.1.m1.1.1">𝑡</ci></apply><apply id="S2.p2.1.1.m1.1.2.3.cmml" xref="S2.p2.1.1.m1.1.2.3"><csymbol cd="ambiguous" id="S2.p2.1.1.m1.1.2.3.1.cmml" xref="S2.p2.1.1.m1.1.2.3">superscript</csymbol><ci id="S2.p2.1.1.m1.1.2.3.2.cmml" xref="S2.p2.1.1.m1.1.2.3.2">ℝ</ci><apply id="S2.p2.1.1.m1.1.2.3.3.cmml" xref="S2.p2.1.1.m1.1.2.3.3"><times id="S2.p2.1.1.m1.1.2.3.3.1.cmml" xref="S2.p2.1.1.m1.1.2.3.3.1"></times><ci id="S2.p2.1.1.m1.1.2.3.3.2.cmml" xref="S2.p2.1.1.m1.1.2.3.3.2">𝑁</ci><ci id="S2.p2.1.1.m1.1.2.3.3.3.cmml" xref="S2.p2.1.1.m1.1.2.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.1.1.m1.1c">\Phi(t)\in\mathbb{R}^{N\times n}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.1.1.m1.1d">roman_Φ ( italic_t ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_N × italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> is of PE if there exist constants <math alttext="t_{0}" class="ltx_Math" display="inline" id="S2.p2.2.2.m2.1"><semantics id="S2.p2.2.2.m2.1a"><msub id="S2.p2.2.2.m2.1.1" xref="S2.p2.2.2.m2.1.1.cmml"><mi id="S2.p2.2.2.m2.1.1.2" mathcolor="#000099" xref="S2.p2.2.2.m2.1.1.2.cmml">t</mi><mn id="S2.p2.2.2.m2.1.1.3" mathcolor="#000099" xref="S2.p2.2.2.m2.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p2.2.2.m2.1b"><apply id="S2.p2.2.2.m2.1.1.cmml" xref="S2.p2.2.2.m2.1.1"><csymbol cd="ambiguous" id="S2.p2.2.2.m2.1.1.1.cmml" xref="S2.p2.2.2.m2.1.1">subscript</csymbol><ci id="S2.p2.2.2.m2.1.1.2.cmml" xref="S2.p2.2.2.m2.1.1.2">𝑡</ci><cn id="S2.p2.2.2.m2.1.1.3.cmml" type="integer" xref="S2.p2.2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.2.2.m2.1c">t_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.2.2.m2.1d">italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\sigma" class="ltx_Math" display="inline" id="S2.p2.3.3.m3.1"><semantics id="S2.p2.3.3.m3.1a"><mi id="S2.p2.3.3.m3.1.1" mathcolor="#000099" xref="S2.p2.3.3.m3.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S2.p2.3.3.m3.1b"><ci id="S2.p2.3.3.m3.1.1.cmml" xref="S2.p2.3.3.m3.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.3.3.m3.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S2.p2.3.3.m3.1d">italic_σ</annotation></semantics></math>, and <math alttext="\tau_{\rm d}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S2.p2.4.4.m4.1"><semantics id="S2.p2.4.4.m4.1a"><mrow id="S2.p2.4.4.m4.1.1" xref="S2.p2.4.4.m4.1.1.cmml"><msub id="S2.p2.4.4.m4.1.1.2" xref="S2.p2.4.4.m4.1.1.2.cmml"><mi id="S2.p2.4.4.m4.1.1.2.2" mathcolor="#000099" xref="S2.p2.4.4.m4.1.1.2.2.cmml">τ</mi><mi id="S2.p2.4.4.m4.1.1.2.3" mathcolor="#000099" mathvariant="normal" xref="S2.p2.4.4.m4.1.1.2.3.cmml">d</mi></msub><mo id="S2.p2.4.4.m4.1.1.1" mathcolor="#000099" xref="S2.p2.4.4.m4.1.1.1.cmml">∈</mo><msup id="S2.p2.4.4.m4.1.1.3" xref="S2.p2.4.4.m4.1.1.3.cmml"><mi id="S2.p2.4.4.m4.1.1.3.2" mathcolor="#000099" xref="S2.p2.4.4.m4.1.1.3.2.cmml">ℝ</mi><mo id="S2.p2.4.4.m4.1.1.3.3" mathcolor="#000099" xref="S2.p2.4.4.m4.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.4.4.m4.1b"><apply id="S2.p2.4.4.m4.1.1.cmml" xref="S2.p2.4.4.m4.1.1"><in id="S2.p2.4.4.m4.1.1.1.cmml" xref="S2.p2.4.4.m4.1.1.1"></in><apply id="S2.p2.4.4.m4.1.1.2.cmml" xref="S2.p2.4.4.m4.1.1.2"><csymbol cd="ambiguous" id="S2.p2.4.4.m4.1.1.2.1.cmml" xref="S2.p2.4.4.m4.1.1.2">subscript</csymbol><ci id="S2.p2.4.4.m4.1.1.2.2.cmml" xref="S2.p2.4.4.m4.1.1.2.2">𝜏</ci><ci id="S2.p2.4.4.m4.1.1.2.3.cmml" xref="S2.p2.4.4.m4.1.1.2.3">d</ci></apply><apply id="S2.p2.4.4.m4.1.1.3.cmml" xref="S2.p2.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.p2.4.4.m4.1.1.3.1.cmml" xref="S2.p2.4.4.m4.1.1.3">superscript</csymbol><ci id="S2.p2.4.4.m4.1.1.3.2.cmml" xref="S2.p2.4.4.m4.1.1.3.2">ℝ</ci><plus id="S2.p2.4.4.m4.1.1.3.3.cmml" xref="S2.p2.4.4.m4.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.4.4.m4.1c">\tau_{\rm d}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.4.4.m4.1d">italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> such that<span class="ltx_note ltx_role_footnote" id="footnote1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note"><span class="ltx_text" id="footnote1.1.1.1" style="color:#000000;">1</span></span><span class="ltx_text" id="footnote1.4" style="color:#000000;">If one has </span><math alttext="0<t<\tau_{\rm d}" class="ltx_Math" display="inline" id="footnote1.m1.1"><semantics id="footnote1.m1.1b"><mrow id="footnote1.m1.1.1" xref="footnote1.m1.1.1.cmml"><mn id="footnote1.m1.1.1.2" mathcolor="#000000" xref="footnote1.m1.1.1.2.cmml">0</mn><mo id="footnote1.m1.1.1.3" mathcolor="#000000" xref="footnote1.m1.1.1.3.cmml"><</mo><mi id="footnote1.m1.1.1.4" mathcolor="#000000" xref="footnote1.m1.1.1.4.cmml">t</mi><mo id="footnote1.m1.1.1.5" mathcolor="#000000" xref="footnote1.m1.1.1.5.cmml"><</mo><msub id="footnote1.m1.1.1.6" xref="footnote1.m1.1.1.6.cmml"><mi id="footnote1.m1.1.1.6.2" mathcolor="#000000" xref="footnote1.m1.1.1.6.2.cmml">τ</mi><mi id="footnote1.m1.1.1.6.3" mathcolor="#000000" mathvariant="normal" xref="footnote1.m1.1.1.6.3.cmml">d</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="footnote1.m1.1c"><apply id="footnote1.m1.1.1.cmml" xref="footnote1.m1.1.1"><and id="footnote1.m1.1.1a.cmml" xref="footnote1.m1.1.1"></and><apply id="footnote1.m1.1.1b.cmml" xref="footnote1.m1.1.1"><lt id="footnote1.m1.1.1.3.cmml" xref="footnote1.m1.1.1.3"></lt><cn id="footnote1.m1.1.1.2.cmml" type="integer" xref="footnote1.m1.1.1.2">0</cn><ci id="footnote1.m1.1.1.4.cmml" xref="footnote1.m1.1.1.4">𝑡</ci></apply><apply id="footnote1.m1.1.1c.cmml" xref="footnote1.m1.1.1"><lt id="footnote1.m1.1.1.5.cmml" xref="footnote1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2401.10785v2#footnote1.m1.1.1.4.cmml" id="footnote1.m1.1.1d.cmml" xref="footnote1.m1.1.1"></share><apply id="footnote1.m1.1.1.6.cmml" xref="footnote1.m1.1.1.6"><csymbol cd="ambiguous" id="footnote1.m1.1.1.6.1.cmml" xref="footnote1.m1.1.1.6">subscript</csymbol><ci id="footnote1.m1.1.1.6.2.cmml" xref="footnote1.m1.1.1.6.2">𝜏</ci><ci id="footnote1.m1.1.1.6.3.cmml" xref="footnote1.m1.1.1.6.3">d</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m1.1d">0<t<\tau_{\rm d}</annotation><annotation encoding="application/x-llamapun" id="footnote1.m1.1e">0 < italic_t < italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text" id="footnote1.5" style="color:#000000;"> in Definitions 1–3, the integral interval is reset to be </span><math alttext="[0,t]" class="ltx_Math" display="inline" id="footnote1.m2.2"><semantics id="footnote1.m2.2b"><mrow id="footnote1.m2.2.3.2" xref="footnote1.m2.2.3.1.cmml"><mo id="footnote1.m2.2.3.2.1" mathcolor="#000000" stretchy="false" xref="footnote1.m2.2.3.1.cmml">[</mo><mn id="footnote1.m2.1.1" mathcolor="#000000" xref="footnote1.m2.1.1.cmml">0</mn><mo id="footnote1.m2.2.3.2.2" mathcolor="#000000" xref="footnote1.m2.2.3.1.cmml">,</mo><mi id="footnote1.m2.2.2" mathcolor="#000000" xref="footnote1.m2.2.2.cmml">t</mi><mo id="footnote1.m2.2.3.2.3" mathcolor="#000000" stretchy="false" xref="footnote1.m2.2.3.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="footnote1.m2.2c"><interval closure="closed" id="footnote1.m2.2.3.1.cmml" xref="footnote1.m2.2.3.2"><cn id="footnote1.m2.1.1.cmml" type="integer" xref="footnote1.m2.1.1">0</cn><ci id="footnote1.m2.2.2.cmml" xref="footnote1.m2.2.2">𝑡</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m2.2d">[0,t]</annotation><annotation encoding="application/x-llamapun" id="footnote1.m2.2e">[ 0 , italic_t ]</annotation></semantics></math><span class="ltx_text" id="footnote1.6" style="color:#000000;"> because the interval length must be greater than 0.</span></span></span></span></span></p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\int_{t-\tau_{\rm d}}^{t}\Phi(\tau)\Phi^{T}(\tau)d\tau\geq\sigma I,\forall t% \geq t_{0}." class="ltx_Math" display="block" id="S2.Ex1.m1.3"><semantics id="S2.Ex1.m1.3a"><mrow id="S2.Ex1.m1.3.3.1"><mrow id="S2.Ex1.m1.3.3.1.1.2" xref="S2.Ex1.m1.3.3.1.1.3.cmml"><mrow id="S2.Ex1.m1.3.3.1.1.1.1" xref="S2.Ex1.m1.3.3.1.1.1.1.cmml"><mrow id="S2.Ex1.m1.3.3.1.1.1.1.2" xref="S2.Ex1.m1.3.3.1.1.1.1.2.cmml"><msubsup id="S2.Ex1.m1.3.3.1.1.1.1.2.1" xref="S2.Ex1.m1.3.3.1.1.1.1.2.1.cmml"><mo id="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.2" mathcolor="#000099" xref="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.2.cmml">∫</mo><mrow id="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3" xref="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3.cmml"><mi id="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3.2" mathcolor="#000099" xref="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3.2.cmml">t</mi><mo id="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3.1" mathcolor="#000099" xref="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3.1.cmml">−</mo><msub id="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3.3" xref="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3.3.cmml"><mi id="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3.3.2" mathcolor="#000099" xref="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3.3.2.cmml">τ</mi><mi id="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3.3.3" mathcolor="#000099" mathvariant="normal" xref="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3.3.3.cmml">d</mi></msub></mrow><mi id="S2.Ex1.m1.3.3.1.1.1.1.2.1.3" mathcolor="#000099" xref="S2.Ex1.m1.3.3.1.1.1.1.2.1.3.cmml">t</mi></msubsup><mrow id="S2.Ex1.m1.3.3.1.1.1.1.2.2" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.cmml"><mi id="S2.Ex1.m1.3.3.1.1.1.1.2.2.2" mathcolor="#000099" mathvariant="normal" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.2.cmml">Φ</mi><mo id="S2.Ex1.m1.3.3.1.1.1.1.2.2.1" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.1.cmml">⁢</mo><mrow id="S2.Ex1.m1.3.3.1.1.1.1.2.2.3.2" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.cmml"><mo id="S2.Ex1.m1.3.3.1.1.1.1.2.2.3.2.1" mathcolor="#000099" stretchy="false" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.cmml">(</mo><mi id="S2.Ex1.m1.1.1" mathcolor="#000099" xref="S2.Ex1.m1.1.1.cmml">τ</mi><mo id="S2.Ex1.m1.3.3.1.1.1.1.2.2.3.2.2" mathcolor="#000099" stretchy="false" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.cmml">)</mo></mrow><mo id="S2.Ex1.m1.3.3.1.1.1.1.2.2.1a" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.1.cmml">⁢</mo><msup id="S2.Ex1.m1.3.3.1.1.1.1.2.2.4" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.4.cmml"><mi id="S2.Ex1.m1.3.3.1.1.1.1.2.2.4.2" mathcolor="#000099" mathvariant="normal" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.4.2.cmml">Φ</mi><mi id="S2.Ex1.m1.3.3.1.1.1.1.2.2.4.3" mathcolor="#000099" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.4.3.cmml">T</mi></msup><mo id="S2.Ex1.m1.3.3.1.1.1.1.2.2.1b" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.1.cmml">⁢</mo><mrow id="S2.Ex1.m1.3.3.1.1.1.1.2.2.5.2" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.cmml"><mo id="S2.Ex1.m1.3.3.1.1.1.1.2.2.5.2.1" mathcolor="#000099" stretchy="false" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.cmml">(</mo><mi id="S2.Ex1.m1.2.2" mathcolor="#000099" xref="S2.Ex1.m1.2.2.cmml">τ</mi><mo id="S2.Ex1.m1.3.3.1.1.1.1.2.2.5.2.2" mathcolor="#000099" stretchy="false" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.cmml">)</mo></mrow><mo id="S2.Ex1.m1.3.3.1.1.1.1.2.2.1c" lspace="0em" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.1.cmml">⁢</mo><mrow id="S2.Ex1.m1.3.3.1.1.1.1.2.2.6" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.6.cmml"><mo id="S2.Ex1.m1.3.3.1.1.1.1.2.2.6.1" mathcolor="#000099" rspace="0em" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.6.1.cmml">𝑑</mo><mi id="S2.Ex1.m1.3.3.1.1.1.1.2.2.6.2" mathcolor="#000099" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.6.2.cmml">τ</mi></mrow></mrow></mrow><mo id="S2.Ex1.m1.3.3.1.1.1.1.1" mathcolor="#000099" xref="S2.Ex1.m1.3.3.1.1.1.1.1.cmml">≥</mo><mrow id="S2.Ex1.m1.3.3.1.1.1.1.3" xref="S2.Ex1.m1.3.3.1.1.1.1.3.cmml"><mi id="S2.Ex1.m1.3.3.1.1.1.1.3.2" mathcolor="#000099" xref="S2.Ex1.m1.3.3.1.1.1.1.3.2.cmml">σ</mi><mo id="S2.Ex1.m1.3.3.1.1.1.1.3.1" xref="S2.Ex1.m1.3.3.1.1.1.1.3.1.cmml">⁢</mo><mi id="S2.Ex1.m1.3.3.1.1.1.1.3.3" mathcolor="#000099" xref="S2.Ex1.m1.3.3.1.1.1.1.3.3.cmml">I</mi></mrow></mrow><mo id="S2.Ex1.m1.3.3.1.1.2.3" mathcolor="#000099" xref="S2.Ex1.m1.3.3.1.1.3a.cmml">,</mo><mrow id="S2.Ex1.m1.3.3.1.1.2.2" xref="S2.Ex1.m1.3.3.1.1.2.2.cmml"><mrow id="S2.Ex1.m1.3.3.1.1.2.2.2" xref="S2.Ex1.m1.3.3.1.1.2.2.2.cmml"><mo id="S2.Ex1.m1.3.3.1.1.2.2.2.1" mathcolor="#000099" rspace="0.167em" xref="S2.Ex1.m1.3.3.1.1.2.2.2.1.cmml">∀</mo><mi id="S2.Ex1.m1.3.3.1.1.2.2.2.2" mathcolor="#000099" xref="S2.Ex1.m1.3.3.1.1.2.2.2.2.cmml">t</mi></mrow><mo id="S2.Ex1.m1.3.3.1.1.2.2.1" mathcolor="#000099" xref="S2.Ex1.m1.3.3.1.1.2.2.1.cmml">≥</mo><msub id="S2.Ex1.m1.3.3.1.1.2.2.3" xref="S2.Ex1.m1.3.3.1.1.2.2.3.cmml"><mi id="S2.Ex1.m1.3.3.1.1.2.2.3.2" mathcolor="#000099" xref="S2.Ex1.m1.3.3.1.1.2.2.3.2.cmml">t</mi><mn id="S2.Ex1.m1.3.3.1.1.2.2.3.3" mathcolor="#000099" xref="S2.Ex1.m1.3.3.1.1.2.2.3.3.cmml">0</mn></msub></mrow></mrow><mo id="S2.Ex1.m1.3.3.1.2" lspace="0em" mathcolor="#000099">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex1.m1.3b"><apply id="S2.Ex1.m1.3.3.1.1.3.cmml" xref="S2.Ex1.m1.3.3.1.1.2"><csymbol cd="ambiguous" id="S2.Ex1.m1.3.3.1.1.3a.cmml" xref="S2.Ex1.m1.3.3.1.1.2.3">formulae-sequence</csymbol><apply id="S2.Ex1.m1.3.3.1.1.1.1.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1"><geq id="S2.Ex1.m1.3.3.1.1.1.1.1.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.1"></geq><apply id="S2.Ex1.m1.3.3.1.1.1.1.2.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.2"><apply id="S2.Ex1.m1.3.3.1.1.1.1.2.1.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.2.1"><csymbol cd="ambiguous" id="S2.Ex1.m1.3.3.1.1.1.1.2.1.1.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.2.1">superscript</csymbol><apply id="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.2.1"><csymbol cd="ambiguous" id="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.1.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.2.1">subscript</csymbol><int id="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.2.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.2"></int><apply id="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3"><minus id="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3.1.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3.1"></minus><ci id="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3.2.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3.2">𝑡</ci><apply id="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3.3.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3.3"><csymbol cd="ambiguous" id="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3.3.1.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3.3">subscript</csymbol><ci id="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3.3.2.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3.3.2">𝜏</ci><ci id="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3.3.3.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.2.1.2.3.3.3">d</ci></apply></apply></apply><ci id="S2.Ex1.m1.3.3.1.1.1.1.2.1.3.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.2.1.3">𝑡</ci></apply><apply id="S2.Ex1.m1.3.3.1.1.1.1.2.2.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2"><times id="S2.Ex1.m1.3.3.1.1.1.1.2.2.1.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.1"></times><ci id="S2.Ex1.m1.3.3.1.1.1.1.2.2.2.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.2">Φ</ci><ci id="S2.Ex1.m1.1.1.cmml" xref="S2.Ex1.m1.1.1">𝜏</ci><apply id="S2.Ex1.m1.3.3.1.1.1.1.2.2.4.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.4"><csymbol cd="ambiguous" id="S2.Ex1.m1.3.3.1.1.1.1.2.2.4.1.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.4">superscript</csymbol><ci id="S2.Ex1.m1.3.3.1.1.1.1.2.2.4.2.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.4.2">Φ</ci><ci id="S2.Ex1.m1.3.3.1.1.1.1.2.2.4.3.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.4.3">𝑇</ci></apply><ci id="S2.Ex1.m1.2.2.cmml" xref="S2.Ex1.m1.2.2">𝜏</ci><apply id="S2.Ex1.m1.3.3.1.1.1.1.2.2.6.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.6"><csymbol cd="latexml" id="S2.Ex1.m1.3.3.1.1.1.1.2.2.6.1.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.6.1">differential-d</csymbol><ci id="S2.Ex1.m1.3.3.1.1.1.1.2.2.6.2.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.2.2.6.2">𝜏</ci></apply></apply></apply><apply id="S2.Ex1.m1.3.3.1.1.1.1.3.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.3"><times id="S2.Ex1.m1.3.3.1.1.1.1.3.1.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.3.1"></times><ci id="S2.Ex1.m1.3.3.1.1.1.1.3.2.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.3.2">𝜎</ci><ci id="S2.Ex1.m1.3.3.1.1.1.1.3.3.cmml" xref="S2.Ex1.m1.3.3.1.1.1.1.3.3">𝐼</ci></apply></apply><apply id="S2.Ex1.m1.3.3.1.1.2.2.cmml" xref="S2.Ex1.m1.3.3.1.1.2.2"><geq id="S2.Ex1.m1.3.3.1.1.2.2.1.cmml" xref="S2.Ex1.m1.3.3.1.1.2.2.1"></geq><apply id="S2.Ex1.m1.3.3.1.1.2.2.2.cmml" xref="S2.Ex1.m1.3.3.1.1.2.2.2"><csymbol cd="latexml" id="S2.Ex1.m1.3.3.1.1.2.2.2.1.cmml" xref="S2.Ex1.m1.3.3.1.1.2.2.2.1">for-all</csymbol><ci id="S2.Ex1.m1.3.3.1.1.2.2.2.2.cmml" xref="S2.Ex1.m1.3.3.1.1.2.2.2.2">𝑡</ci></apply><apply id="S2.Ex1.m1.3.3.1.1.2.2.3.cmml" xref="S2.Ex1.m1.3.3.1.1.2.2.3"><csymbol cd="ambiguous" id="S2.Ex1.m1.3.3.1.1.2.2.3.1.cmml" xref="S2.Ex1.m1.3.3.1.1.2.2.3">subscript</csymbol><ci id="S2.Ex1.m1.3.3.1.1.2.2.3.2.cmml" xref="S2.Ex1.m1.3.3.1.1.2.2.3.2">𝑡</ci><cn id="S2.Ex1.m1.3.3.1.1.2.2.3.3.cmml" type="integer" xref="S2.Ex1.m1.3.3.1.1.2.2.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex1.m1.3c">\int_{t-\tau_{\rm d}}^{t}\Phi(\tau)\Phi^{T}(\tau)d\tau\geq\sigma I,\forall t% \geq t_{0}.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1.m1.3d">∫ start_POSTSUBSCRIPT italic_t - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT roman_Φ ( italic_τ ) roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_τ ) italic_d italic_τ ≥ italic_σ italic_I , ∀ italic_t ≥ italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.p3"> <p class="ltx_p" id="S2.p3.4"><span class="ltx_text ltx_font_italic" id="S2.p3.4.1">Definition 2</span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib26" title="">26</a>]</cite>: A bounded regressor <math alttext="\Phi(t)\in\mathbb{R}^{N\times n}" class="ltx_Math" display="inline" id="S2.p3.1.m1.1"><semantics id="S2.p3.1.m1.1a"><mrow id="S2.p3.1.m1.1.2" xref="S2.p3.1.m1.1.2.cmml"><mrow id="S2.p3.1.m1.1.2.2" xref="S2.p3.1.m1.1.2.2.cmml"><mi id="S2.p3.1.m1.1.2.2.2" mathvariant="normal" xref="S2.p3.1.m1.1.2.2.2.cmml">Φ</mi><mo id="S2.p3.1.m1.1.2.2.1" xref="S2.p3.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S2.p3.1.m1.1.2.2.3.2" xref="S2.p3.1.m1.1.2.2.cmml"><mo id="S2.p3.1.m1.1.2.2.3.2.1" stretchy="false" xref="S2.p3.1.m1.1.2.2.cmml">(</mo><mi id="S2.p3.1.m1.1.1" xref="S2.p3.1.m1.1.1.cmml">t</mi><mo id="S2.p3.1.m1.1.2.2.3.2.2" stretchy="false" xref="S2.p3.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.p3.1.m1.1.2.1" xref="S2.p3.1.m1.1.2.1.cmml">∈</mo><msup id="S2.p3.1.m1.1.2.3" xref="S2.p3.1.m1.1.2.3.cmml"><mi id="S2.p3.1.m1.1.2.3.2" xref="S2.p3.1.m1.1.2.3.2.cmml">ℝ</mi><mrow id="S2.p3.1.m1.1.2.3.3" xref="S2.p3.1.m1.1.2.3.3.cmml"><mi id="S2.p3.1.m1.1.2.3.3.2" xref="S2.p3.1.m1.1.2.3.3.2.cmml">N</mi><mo id="S2.p3.1.m1.1.2.3.3.1" lspace="0.222em" rspace="0.222em" xref="S2.p3.1.m1.1.2.3.3.1.cmml">×</mo><mi id="S2.p3.1.m1.1.2.3.3.3" xref="S2.p3.1.m1.1.2.3.3.3.cmml">n</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.1.m1.1b"><apply id="S2.p3.1.m1.1.2.cmml" xref="S2.p3.1.m1.1.2"><in id="S2.p3.1.m1.1.2.1.cmml" xref="S2.p3.1.m1.1.2.1"></in><apply id="S2.p3.1.m1.1.2.2.cmml" xref="S2.p3.1.m1.1.2.2"><times id="S2.p3.1.m1.1.2.2.1.cmml" xref="S2.p3.1.m1.1.2.2.1"></times><ci id="S2.p3.1.m1.1.2.2.2.cmml" xref="S2.p3.1.m1.1.2.2.2">Φ</ci><ci id="S2.p3.1.m1.1.1.cmml" xref="S2.p3.1.m1.1.1">𝑡</ci></apply><apply id="S2.p3.1.m1.1.2.3.cmml" xref="S2.p3.1.m1.1.2.3"><csymbol cd="ambiguous" id="S2.p3.1.m1.1.2.3.1.cmml" xref="S2.p3.1.m1.1.2.3">superscript</csymbol><ci id="S2.p3.1.m1.1.2.3.2.cmml" xref="S2.p3.1.m1.1.2.3.2">ℝ</ci><apply id="S2.p3.1.m1.1.2.3.3.cmml" xref="S2.p3.1.m1.1.2.3.3"><times id="S2.p3.1.m1.1.2.3.3.1.cmml" xref="S2.p3.1.m1.1.2.3.3.1"></times><ci id="S2.p3.1.m1.1.2.3.3.2.cmml" xref="S2.p3.1.m1.1.2.3.3.2">𝑁</ci><ci id="S2.p3.1.m1.1.2.3.3.3.cmml" xref="S2.p3.1.m1.1.2.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.1.m1.1c">\Phi(t)\in\mathbb{R}^{N\times n}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.1.m1.1d">roman_Φ ( italic_t ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_N × italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> is of IE if there exist constants <math alttext="\sigma" class="ltx_Math" display="inline" id="S2.p3.2.m2.1"><semantics id="S2.p3.2.m2.1a"><mi id="S2.p3.2.m2.1.1" xref="S2.p3.2.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S2.p3.2.m2.1b"><ci id="S2.p3.2.m2.1.1.cmml" xref="S2.p3.2.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.2.m2.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S2.p3.2.m2.1d">italic_σ</annotation></semantics></math>, <math alttext="\tau_{\rm d}" class="ltx_Math" display="inline" id="S2.p3.3.m3.1"><semantics id="S2.p3.3.m3.1a"><msub id="S2.p3.3.m3.1.1" xref="S2.p3.3.m3.1.1.cmml"><mi id="S2.p3.3.m3.1.1.2" xref="S2.p3.3.m3.1.1.2.cmml">τ</mi><mi id="S2.p3.3.m3.1.1.3" mathvariant="normal" xref="S2.p3.3.m3.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p3.3.m3.1b"><apply id="S2.p3.3.m3.1.1.cmml" xref="S2.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S2.p3.3.m3.1.1.1.cmml" xref="S2.p3.3.m3.1.1">subscript</csymbol><ci id="S2.p3.3.m3.1.1.2.cmml" xref="S2.p3.3.m3.1.1.2">𝜏</ci><ci id="S2.p3.3.m3.1.1.3.cmml" xref="S2.p3.3.m3.1.1.3">d</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.3.m3.1c">\tau_{\rm d}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.3.m3.1d">italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="T_{\rm e}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S2.p3.4.m4.1"><semantics id="S2.p3.4.m4.1a"><mrow id="S2.p3.4.m4.1.1" xref="S2.p3.4.m4.1.1.cmml"><msub id="S2.p3.4.m4.1.1.2" xref="S2.p3.4.m4.1.1.2.cmml"><mi id="S2.p3.4.m4.1.1.2.2" xref="S2.p3.4.m4.1.1.2.2.cmml">T</mi><mi id="S2.p3.4.m4.1.1.2.3" mathvariant="normal" xref="S2.p3.4.m4.1.1.2.3.cmml">e</mi></msub><mo id="S2.p3.4.m4.1.1.1" xref="S2.p3.4.m4.1.1.1.cmml">∈</mo><msup id="S2.p3.4.m4.1.1.3" xref="S2.p3.4.m4.1.1.3.cmml"><mi id="S2.p3.4.m4.1.1.3.2" xref="S2.p3.4.m4.1.1.3.2.cmml">ℝ</mi><mo id="S2.p3.4.m4.1.1.3.3" xref="S2.p3.4.m4.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.4.m4.1b"><apply id="S2.p3.4.m4.1.1.cmml" xref="S2.p3.4.m4.1.1"><in id="S2.p3.4.m4.1.1.1.cmml" xref="S2.p3.4.m4.1.1.1"></in><apply id="S2.p3.4.m4.1.1.2.cmml" xref="S2.p3.4.m4.1.1.2"><csymbol cd="ambiguous" id="S2.p3.4.m4.1.1.2.1.cmml" xref="S2.p3.4.m4.1.1.2">subscript</csymbol><ci id="S2.p3.4.m4.1.1.2.2.cmml" xref="S2.p3.4.m4.1.1.2.2">𝑇</ci><ci id="S2.p3.4.m4.1.1.2.3.cmml" xref="S2.p3.4.m4.1.1.2.3">e</ci></apply><apply id="S2.p3.4.m4.1.1.3.cmml" xref="S2.p3.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.p3.4.m4.1.1.3.1.cmml" xref="S2.p3.4.m4.1.1.3">superscript</csymbol><ci id="S2.p3.4.m4.1.1.3.2.cmml" xref="S2.p3.4.m4.1.1.3.2">ℝ</ci><plus id="S2.p3.4.m4.1.1.3.3.cmml" xref="S2.p3.4.m4.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.4.m4.1c">T_{\rm e}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.4.m4.1d">italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> such that</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\int_{T_{\rm e}-\tau_{\rm d}}^{T_{\rm e}}\Phi(\tau)\Phi^{T}(\tau)d\tau\geq% \sigma I." class="ltx_Math" display="block" id="S2.Ex2.m1.3"><semantics id="S2.Ex2.m1.3a"><mrow id="S2.Ex2.m1.3.3.1" xref="S2.Ex2.m1.3.3.1.1.cmml"><mrow id="S2.Ex2.m1.3.3.1.1" xref="S2.Ex2.m1.3.3.1.1.cmml"><mrow id="S2.Ex2.m1.3.3.1.1.2" xref="S2.Ex2.m1.3.3.1.1.2.cmml"><msubsup id="S2.Ex2.m1.3.3.1.1.2.1" xref="S2.Ex2.m1.3.3.1.1.2.1.cmml"><mo id="S2.Ex2.m1.3.3.1.1.2.1.2.2" xref="S2.Ex2.m1.3.3.1.1.2.1.2.2.cmml">∫</mo><mrow id="S2.Ex2.m1.3.3.1.1.2.1.2.3" xref="S2.Ex2.m1.3.3.1.1.2.1.2.3.cmml"><msub id="S2.Ex2.m1.3.3.1.1.2.1.2.3.2" xref="S2.Ex2.m1.3.3.1.1.2.1.2.3.2.cmml"><mi id="S2.Ex2.m1.3.3.1.1.2.1.2.3.2.2" xref="S2.Ex2.m1.3.3.1.1.2.1.2.3.2.2.cmml">T</mi><mi id="S2.Ex2.m1.3.3.1.1.2.1.2.3.2.3" mathvariant="normal" xref="S2.Ex2.m1.3.3.1.1.2.1.2.3.2.3.cmml">e</mi></msub><mo id="S2.Ex2.m1.3.3.1.1.2.1.2.3.1" xref="S2.Ex2.m1.3.3.1.1.2.1.2.3.1.cmml">−</mo><msub id="S2.Ex2.m1.3.3.1.1.2.1.2.3.3" xref="S2.Ex2.m1.3.3.1.1.2.1.2.3.3.cmml"><mi id="S2.Ex2.m1.3.3.1.1.2.1.2.3.3.2" xref="S2.Ex2.m1.3.3.1.1.2.1.2.3.3.2.cmml">τ</mi><mi id="S2.Ex2.m1.3.3.1.1.2.1.2.3.3.3" mathvariant="normal" xref="S2.Ex2.m1.3.3.1.1.2.1.2.3.3.3.cmml">d</mi></msub></mrow><msub id="S2.Ex2.m1.3.3.1.1.2.1.3" xref="S2.Ex2.m1.3.3.1.1.2.1.3.cmml"><mi id="S2.Ex2.m1.3.3.1.1.2.1.3.2" xref="S2.Ex2.m1.3.3.1.1.2.1.3.2.cmml">T</mi><mi id="S2.Ex2.m1.3.3.1.1.2.1.3.3" mathvariant="normal" xref="S2.Ex2.m1.3.3.1.1.2.1.3.3.cmml">e</mi></msub></msubsup><mrow id="S2.Ex2.m1.3.3.1.1.2.2" xref="S2.Ex2.m1.3.3.1.1.2.2.cmml"><mi id="S2.Ex2.m1.3.3.1.1.2.2.2" mathvariant="normal" xref="S2.Ex2.m1.3.3.1.1.2.2.2.cmml">Φ</mi><mo id="S2.Ex2.m1.3.3.1.1.2.2.1" xref="S2.Ex2.m1.3.3.1.1.2.2.1.cmml">⁢</mo><mrow id="S2.Ex2.m1.3.3.1.1.2.2.3.2" xref="S2.Ex2.m1.3.3.1.1.2.2.cmml"><mo id="S2.Ex2.m1.3.3.1.1.2.2.3.2.1" stretchy="false" xref="S2.Ex2.m1.3.3.1.1.2.2.cmml">(</mo><mi id="S2.Ex2.m1.1.1" xref="S2.Ex2.m1.1.1.cmml">τ</mi><mo id="S2.Ex2.m1.3.3.1.1.2.2.3.2.2" stretchy="false" xref="S2.Ex2.m1.3.3.1.1.2.2.cmml">)</mo></mrow><mo id="S2.Ex2.m1.3.3.1.1.2.2.1a" xref="S2.Ex2.m1.3.3.1.1.2.2.1.cmml">⁢</mo><msup id="S2.Ex2.m1.3.3.1.1.2.2.4" xref="S2.Ex2.m1.3.3.1.1.2.2.4.cmml"><mi id="S2.Ex2.m1.3.3.1.1.2.2.4.2" mathvariant="normal" xref="S2.Ex2.m1.3.3.1.1.2.2.4.2.cmml">Φ</mi><mi id="S2.Ex2.m1.3.3.1.1.2.2.4.3" xref="S2.Ex2.m1.3.3.1.1.2.2.4.3.cmml">T</mi></msup><mo id="S2.Ex2.m1.3.3.1.1.2.2.1b" xref="S2.Ex2.m1.3.3.1.1.2.2.1.cmml">⁢</mo><mrow id="S2.Ex2.m1.3.3.1.1.2.2.5.2" xref="S2.Ex2.m1.3.3.1.1.2.2.cmml"><mo id="S2.Ex2.m1.3.3.1.1.2.2.5.2.1" stretchy="false" xref="S2.Ex2.m1.3.3.1.1.2.2.cmml">(</mo><mi id="S2.Ex2.m1.2.2" xref="S2.Ex2.m1.2.2.cmml">τ</mi><mo id="S2.Ex2.m1.3.3.1.1.2.2.5.2.2" stretchy="false" xref="S2.Ex2.m1.3.3.1.1.2.2.cmml">)</mo></mrow><mo id="S2.Ex2.m1.3.3.1.1.2.2.1c" lspace="0em" xref="S2.Ex2.m1.3.3.1.1.2.2.1.cmml">⁢</mo><mrow id="S2.Ex2.m1.3.3.1.1.2.2.6" xref="S2.Ex2.m1.3.3.1.1.2.2.6.cmml"><mo id="S2.Ex2.m1.3.3.1.1.2.2.6.1" rspace="0em" xref="S2.Ex2.m1.3.3.1.1.2.2.6.1.cmml">𝑑</mo><mi id="S2.Ex2.m1.3.3.1.1.2.2.6.2" xref="S2.Ex2.m1.3.3.1.1.2.2.6.2.cmml">τ</mi></mrow></mrow></mrow><mo id="S2.Ex2.m1.3.3.1.1.1" xref="S2.Ex2.m1.3.3.1.1.1.cmml">≥</mo><mrow id="S2.Ex2.m1.3.3.1.1.3" xref="S2.Ex2.m1.3.3.1.1.3.cmml"><mi id="S2.Ex2.m1.3.3.1.1.3.2" xref="S2.Ex2.m1.3.3.1.1.3.2.cmml">σ</mi><mo id="S2.Ex2.m1.3.3.1.1.3.1" xref="S2.Ex2.m1.3.3.1.1.3.1.cmml">⁢</mo><mi id="S2.Ex2.m1.3.3.1.1.3.3" xref="S2.Ex2.m1.3.3.1.1.3.3.cmml">I</mi></mrow></mrow><mo id="S2.Ex2.m1.3.3.1.2" lspace="0em" xref="S2.Ex2.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex2.m1.3b"><apply id="S2.Ex2.m1.3.3.1.1.cmml" xref="S2.Ex2.m1.3.3.1"><geq id="S2.Ex2.m1.3.3.1.1.1.cmml" xref="S2.Ex2.m1.3.3.1.1.1"></geq><apply id="S2.Ex2.m1.3.3.1.1.2.cmml" xref="S2.Ex2.m1.3.3.1.1.2"><apply id="S2.Ex2.m1.3.3.1.1.2.1.cmml" xref="S2.Ex2.m1.3.3.1.1.2.1"><csymbol cd="ambiguous" id="S2.Ex2.m1.3.3.1.1.2.1.1.cmml" xref="S2.Ex2.m1.3.3.1.1.2.1">superscript</csymbol><apply id="S2.Ex2.m1.3.3.1.1.2.1.2.cmml" xref="S2.Ex2.m1.3.3.1.1.2.1"><csymbol cd="ambiguous" id="S2.Ex2.m1.3.3.1.1.2.1.2.1.cmml" xref="S2.Ex2.m1.3.3.1.1.2.1">subscript</csymbol><int id="S2.Ex2.m1.3.3.1.1.2.1.2.2.cmml" xref="S2.Ex2.m1.3.3.1.1.2.1.2.2"></int><apply id="S2.Ex2.m1.3.3.1.1.2.1.2.3.cmml" xref="S2.Ex2.m1.3.3.1.1.2.1.2.3"><minus id="S2.Ex2.m1.3.3.1.1.2.1.2.3.1.cmml" xref="S2.Ex2.m1.3.3.1.1.2.1.2.3.1"></minus><apply id="S2.Ex2.m1.3.3.1.1.2.1.2.3.2.cmml" xref="S2.Ex2.m1.3.3.1.1.2.1.2.3.2"><csymbol cd="ambiguous" id="S2.Ex2.m1.3.3.1.1.2.1.2.3.2.1.cmml" xref="S2.Ex2.m1.3.3.1.1.2.1.2.3.2">subscript</csymbol><ci id="S2.Ex2.m1.3.3.1.1.2.1.2.3.2.2.cmml" xref="S2.Ex2.m1.3.3.1.1.2.1.2.3.2.2">𝑇</ci><ci id="S2.Ex2.m1.3.3.1.1.2.1.2.3.2.3.cmml" xref="S2.Ex2.m1.3.3.1.1.2.1.2.3.2.3">e</ci></apply><apply id="S2.Ex2.m1.3.3.1.1.2.1.2.3.3.cmml" xref="S2.Ex2.m1.3.3.1.1.2.1.2.3.3"><csymbol cd="ambiguous" id="S2.Ex2.m1.3.3.1.1.2.1.2.3.3.1.cmml" xref="S2.Ex2.m1.3.3.1.1.2.1.2.3.3">subscript</csymbol><ci id="S2.Ex2.m1.3.3.1.1.2.1.2.3.3.2.cmml" xref="S2.Ex2.m1.3.3.1.1.2.1.2.3.3.2">𝜏</ci><ci id="S2.Ex2.m1.3.3.1.1.2.1.2.3.3.3.cmml" xref="S2.Ex2.m1.3.3.1.1.2.1.2.3.3.3">d</ci></apply></apply></apply><apply id="S2.Ex2.m1.3.3.1.1.2.1.3.cmml" xref="S2.Ex2.m1.3.3.1.1.2.1.3"><csymbol cd="ambiguous" id="S2.Ex2.m1.3.3.1.1.2.1.3.1.cmml" xref="S2.Ex2.m1.3.3.1.1.2.1.3">subscript</csymbol><ci id="S2.Ex2.m1.3.3.1.1.2.1.3.2.cmml" xref="S2.Ex2.m1.3.3.1.1.2.1.3.2">𝑇</ci><ci id="S2.Ex2.m1.3.3.1.1.2.1.3.3.cmml" xref="S2.Ex2.m1.3.3.1.1.2.1.3.3">e</ci></apply></apply><apply id="S2.Ex2.m1.3.3.1.1.2.2.cmml" xref="S2.Ex2.m1.3.3.1.1.2.2"><times id="S2.Ex2.m1.3.3.1.1.2.2.1.cmml" xref="S2.Ex2.m1.3.3.1.1.2.2.1"></times><ci id="S2.Ex2.m1.3.3.1.1.2.2.2.cmml" xref="S2.Ex2.m1.3.3.1.1.2.2.2">Φ</ci><ci id="S2.Ex2.m1.1.1.cmml" xref="S2.Ex2.m1.1.1">𝜏</ci><apply id="S2.Ex2.m1.3.3.1.1.2.2.4.cmml" xref="S2.Ex2.m1.3.3.1.1.2.2.4"><csymbol cd="ambiguous" id="S2.Ex2.m1.3.3.1.1.2.2.4.1.cmml" xref="S2.Ex2.m1.3.3.1.1.2.2.4">superscript</csymbol><ci id="S2.Ex2.m1.3.3.1.1.2.2.4.2.cmml" xref="S2.Ex2.m1.3.3.1.1.2.2.4.2">Φ</ci><ci id="S2.Ex2.m1.3.3.1.1.2.2.4.3.cmml" xref="S2.Ex2.m1.3.3.1.1.2.2.4.3">𝑇</ci></apply><ci id="S2.Ex2.m1.2.2.cmml" xref="S2.Ex2.m1.2.2">𝜏</ci><apply id="S2.Ex2.m1.3.3.1.1.2.2.6.cmml" xref="S2.Ex2.m1.3.3.1.1.2.2.6"><csymbol cd="latexml" id="S2.Ex2.m1.3.3.1.1.2.2.6.1.cmml" xref="S2.Ex2.m1.3.3.1.1.2.2.6.1">differential-d</csymbol><ci id="S2.Ex2.m1.3.3.1.1.2.2.6.2.cmml" xref="S2.Ex2.m1.3.3.1.1.2.2.6.2">𝜏</ci></apply></apply></apply><apply id="S2.Ex2.m1.3.3.1.1.3.cmml" xref="S2.Ex2.m1.3.3.1.1.3"><times id="S2.Ex2.m1.3.3.1.1.3.1.cmml" xref="S2.Ex2.m1.3.3.1.1.3.1"></times><ci id="S2.Ex2.m1.3.3.1.1.3.2.cmml" xref="S2.Ex2.m1.3.3.1.1.3.2">𝜎</ci><ci id="S2.Ex2.m1.3.3.1.1.3.3.cmml" xref="S2.Ex2.m1.3.3.1.1.3.3">𝐼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex2.m1.3c">\int_{T_{\rm e}-\tau_{\rm d}}^{T_{\rm e}}\Phi(\tau)\Phi^{T}(\tau)d\tau\geq% \sigma I.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex2.m1.3d">∫ start_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_Φ ( italic_τ ) roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_τ ) italic_d italic_τ ≥ italic_σ italic_I .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.p4"> <p class="ltx_p" id="S2.p4.6"><span class="ltx_text ltx_font_italic" id="S2.p4.6.1">Definition 3:</span> A bounded regressor <math alttext="\Phi(t)\in\mathbb{R}^{N\times n}" class="ltx_Math" display="inline" id="S2.p4.1.m1.1"><semantics id="S2.p4.1.m1.1a"><mrow id="S2.p4.1.m1.1.2" xref="S2.p4.1.m1.1.2.cmml"><mrow id="S2.p4.1.m1.1.2.2" xref="S2.p4.1.m1.1.2.2.cmml"><mi id="S2.p4.1.m1.1.2.2.2" mathvariant="normal" xref="S2.p4.1.m1.1.2.2.2.cmml">Φ</mi><mo id="S2.p4.1.m1.1.2.2.1" xref="S2.p4.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S2.p4.1.m1.1.2.2.3.2" xref="S2.p4.1.m1.1.2.2.cmml"><mo id="S2.p4.1.m1.1.2.2.3.2.1" stretchy="false" xref="S2.p4.1.m1.1.2.2.cmml">(</mo><mi id="S2.p4.1.m1.1.1" xref="S2.p4.1.m1.1.1.cmml">t</mi><mo id="S2.p4.1.m1.1.2.2.3.2.2" stretchy="false" xref="S2.p4.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.p4.1.m1.1.2.1" xref="S2.p4.1.m1.1.2.1.cmml">∈</mo><msup id="S2.p4.1.m1.1.2.3" xref="S2.p4.1.m1.1.2.3.cmml"><mi id="S2.p4.1.m1.1.2.3.2" xref="S2.p4.1.m1.1.2.3.2.cmml">ℝ</mi><mrow id="S2.p4.1.m1.1.2.3.3" xref="S2.p4.1.m1.1.2.3.3.cmml"><mi id="S2.p4.1.m1.1.2.3.3.2" xref="S2.p4.1.m1.1.2.3.3.2.cmml">N</mi><mo id="S2.p4.1.m1.1.2.3.3.1" lspace="0.222em" rspace="0.222em" xref="S2.p4.1.m1.1.2.3.3.1.cmml">×</mo><mi id="S2.p4.1.m1.1.2.3.3.3" xref="S2.p4.1.m1.1.2.3.3.3.cmml">n</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p4.1.m1.1b"><apply id="S2.p4.1.m1.1.2.cmml" xref="S2.p4.1.m1.1.2"><in id="S2.p4.1.m1.1.2.1.cmml" xref="S2.p4.1.m1.1.2.1"></in><apply id="S2.p4.1.m1.1.2.2.cmml" xref="S2.p4.1.m1.1.2.2"><times id="S2.p4.1.m1.1.2.2.1.cmml" xref="S2.p4.1.m1.1.2.2.1"></times><ci id="S2.p4.1.m1.1.2.2.2.cmml" xref="S2.p4.1.m1.1.2.2.2">Φ</ci><ci id="S2.p4.1.m1.1.1.cmml" xref="S2.p4.1.m1.1.1">𝑡</ci></apply><apply id="S2.p4.1.m1.1.2.3.cmml" xref="S2.p4.1.m1.1.2.3"><csymbol cd="ambiguous" id="S2.p4.1.m1.1.2.3.1.cmml" xref="S2.p4.1.m1.1.2.3">superscript</csymbol><ci id="S2.p4.1.m1.1.2.3.2.cmml" xref="S2.p4.1.m1.1.2.3.2">ℝ</ci><apply id="S2.p4.1.m1.1.2.3.3.cmml" xref="S2.p4.1.m1.1.2.3.3"><times id="S2.p4.1.m1.1.2.3.3.1.cmml" xref="S2.p4.1.m1.1.2.3.3.1"></times><ci id="S2.p4.1.m1.1.2.3.3.2.cmml" xref="S2.p4.1.m1.1.2.3.3.2">𝑁</ci><ci id="S2.p4.1.m1.1.2.3.3.3.cmml" xref="S2.p4.1.m1.1.2.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.1.m1.1c">\Phi(t)\in\mathbb{R}^{N\times n}</annotation><annotation encoding="application/x-llamapun" id="S2.p4.1.m1.1d">roman_Φ ( italic_t ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_N × italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> is of partial IE, if there exist constants <math alttext="\sigma" class="ltx_Math" display="inline" id="S2.p4.2.m2.1"><semantics id="S2.p4.2.m2.1a"><mi id="S2.p4.2.m2.1.1" xref="S2.p4.2.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S2.p4.2.m2.1b"><ci id="S2.p4.2.m2.1.1.cmml" xref="S2.p4.2.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.2.m2.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S2.p4.2.m2.1d">italic_σ</annotation></semantics></math>, <math alttext="\tau_{\rm d}" class="ltx_Math" display="inline" id="S2.p4.3.m3.1"><semantics id="S2.p4.3.m3.1a"><msub id="S2.p4.3.m3.1.1" xref="S2.p4.3.m3.1.1.cmml"><mi id="S2.p4.3.m3.1.1.2" xref="S2.p4.3.m3.1.1.2.cmml">τ</mi><mi id="S2.p4.3.m3.1.1.3" mathvariant="normal" xref="S2.p4.3.m3.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p4.3.m3.1b"><apply id="S2.p4.3.m3.1.1.cmml" xref="S2.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S2.p4.3.m3.1.1.1.cmml" xref="S2.p4.3.m3.1.1">subscript</csymbol><ci id="S2.p4.3.m3.1.1.2.cmml" xref="S2.p4.3.m3.1.1.2">𝜏</ci><ci id="S2.p4.3.m3.1.1.3.cmml" xref="S2.p4.3.m3.1.1.3">d</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.3.m3.1c">\tau_{\rm d}</annotation><annotation encoding="application/x-llamapun" id="S2.p4.3.m3.1d">italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="T_{\rm a}" class="ltx_Math" display="inline" id="S2.p4.4.m4.1"><semantics id="S2.p4.4.m4.1a"><msub id="S2.p4.4.m4.1.1" xref="S2.p4.4.m4.1.1.cmml"><mi id="S2.p4.4.m4.1.1.2" xref="S2.p4.4.m4.1.1.2.cmml">T</mi><mi id="S2.p4.4.m4.1.1.3" mathvariant="normal" xref="S2.p4.4.m4.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p4.4.m4.1b"><apply id="S2.p4.4.m4.1.1.cmml" xref="S2.p4.4.m4.1.1"><csymbol cd="ambiguous" id="S2.p4.4.m4.1.1.1.cmml" xref="S2.p4.4.m4.1.1">subscript</csymbol><ci id="S2.p4.4.m4.1.1.2.cmml" xref="S2.p4.4.m4.1.1.2">𝑇</ci><ci id="S2.p4.4.m4.1.1.3.cmml" xref="S2.p4.4.m4.1.1.3">a</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.4.m4.1c">T_{\rm a}</annotation><annotation encoding="application/x-llamapun" id="S2.p4.4.m4.1d">italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext="\in" class="ltx_Math" display="inline" id="S2.p4.5.m5.1"><semantics id="S2.p4.5.m5.1a"><mo id="S2.p4.5.m5.1.1" xref="S2.p4.5.m5.1.1.cmml">∈</mo><annotation-xml encoding="MathML-Content" id="S2.p4.5.m5.1b"><in id="S2.p4.5.m5.1.1.cmml" xref="S2.p4.5.m5.1.1"></in></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.5.m5.1c">\in</annotation><annotation encoding="application/x-llamapun" id="S2.p4.5.m5.1d">∈</annotation></semantics></math> <math alttext="\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S2.p4.6.m6.1"><semantics id="S2.p4.6.m6.1a"><msup id="S2.p4.6.m6.1.1" xref="S2.p4.6.m6.1.1.cmml"><mi id="S2.p4.6.m6.1.1.2" xref="S2.p4.6.m6.1.1.2.cmml">ℝ</mi><mo id="S2.p4.6.m6.1.1.3" xref="S2.p4.6.m6.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S2.p4.6.m6.1b"><apply id="S2.p4.6.m6.1.1.cmml" xref="S2.p4.6.m6.1.1"><csymbol cd="ambiguous" id="S2.p4.6.m6.1.1.1.cmml" xref="S2.p4.6.m6.1.1">superscript</csymbol><ci id="S2.p4.6.m6.1.1.2.cmml" xref="S2.p4.6.m6.1.1.2">ℝ</ci><plus id="S2.p4.6.m6.1.1.3.cmml" xref="S2.p4.6.m6.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.6.m6.1c">\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S2.p4.6.m6.1d">blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> such that</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\int_{T_{\rm a}-\tau_{\rm d}}^{T_{\rm a}}\Phi_{\zeta}(\tau)\Phi_{\zeta}^{T}(% \tau)d\tau\geq\sigma I" class="ltx_Math" display="block" id="S2.Ex3.m1.2"><semantics id="S2.Ex3.m1.2a"><mrow id="S2.Ex3.m1.2.3" xref="S2.Ex3.m1.2.3.cmml"><mrow id="S2.Ex3.m1.2.3.2" xref="S2.Ex3.m1.2.3.2.cmml"><msubsup id="S2.Ex3.m1.2.3.2.1" xref="S2.Ex3.m1.2.3.2.1.cmml"><mo id="S2.Ex3.m1.2.3.2.1.2.2" xref="S2.Ex3.m1.2.3.2.1.2.2.cmml">∫</mo><mrow id="S2.Ex3.m1.2.3.2.1.2.3" xref="S2.Ex3.m1.2.3.2.1.2.3.cmml"><msub id="S2.Ex3.m1.2.3.2.1.2.3.2" xref="S2.Ex3.m1.2.3.2.1.2.3.2.cmml"><mi id="S2.Ex3.m1.2.3.2.1.2.3.2.2" xref="S2.Ex3.m1.2.3.2.1.2.3.2.2.cmml">T</mi><mi id="S2.Ex3.m1.2.3.2.1.2.3.2.3" mathvariant="normal" xref="S2.Ex3.m1.2.3.2.1.2.3.2.3.cmml">a</mi></msub><mo id="S2.Ex3.m1.2.3.2.1.2.3.1" xref="S2.Ex3.m1.2.3.2.1.2.3.1.cmml">−</mo><msub id="S2.Ex3.m1.2.3.2.1.2.3.3" xref="S2.Ex3.m1.2.3.2.1.2.3.3.cmml"><mi id="S2.Ex3.m1.2.3.2.1.2.3.3.2" xref="S2.Ex3.m1.2.3.2.1.2.3.3.2.cmml">τ</mi><mi id="S2.Ex3.m1.2.3.2.1.2.3.3.3" mathvariant="normal" xref="S2.Ex3.m1.2.3.2.1.2.3.3.3.cmml">d</mi></msub></mrow><msub id="S2.Ex3.m1.2.3.2.1.3" xref="S2.Ex3.m1.2.3.2.1.3.cmml"><mi id="S2.Ex3.m1.2.3.2.1.3.2" xref="S2.Ex3.m1.2.3.2.1.3.2.cmml">T</mi><mi id="S2.Ex3.m1.2.3.2.1.3.3" mathvariant="normal" xref="S2.Ex3.m1.2.3.2.1.3.3.cmml">a</mi></msub></msubsup><mrow id="S2.Ex3.m1.2.3.2.2" xref="S2.Ex3.m1.2.3.2.2.cmml"><msub id="S2.Ex3.m1.2.3.2.2.2" xref="S2.Ex3.m1.2.3.2.2.2.cmml"><mi id="S2.Ex3.m1.2.3.2.2.2.2" mathvariant="normal" xref="S2.Ex3.m1.2.3.2.2.2.2.cmml">Φ</mi><mi id="S2.Ex3.m1.2.3.2.2.2.3" xref="S2.Ex3.m1.2.3.2.2.2.3.cmml">ζ</mi></msub><mo id="S2.Ex3.m1.2.3.2.2.1" xref="S2.Ex3.m1.2.3.2.2.1.cmml">⁢</mo><mrow id="S2.Ex3.m1.2.3.2.2.3.2" xref="S2.Ex3.m1.2.3.2.2.cmml"><mo id="S2.Ex3.m1.2.3.2.2.3.2.1" stretchy="false" xref="S2.Ex3.m1.2.3.2.2.cmml">(</mo><mi id="S2.Ex3.m1.1.1" xref="S2.Ex3.m1.1.1.cmml">τ</mi><mo id="S2.Ex3.m1.2.3.2.2.3.2.2" stretchy="false" xref="S2.Ex3.m1.2.3.2.2.cmml">)</mo></mrow><mo id="S2.Ex3.m1.2.3.2.2.1a" xref="S2.Ex3.m1.2.3.2.2.1.cmml">⁢</mo><msubsup id="S2.Ex3.m1.2.3.2.2.4" xref="S2.Ex3.m1.2.3.2.2.4.cmml"><mi id="S2.Ex3.m1.2.3.2.2.4.2.2" mathvariant="normal" xref="S2.Ex3.m1.2.3.2.2.4.2.2.cmml">Φ</mi><mi id="S2.Ex3.m1.2.3.2.2.4.2.3" xref="S2.Ex3.m1.2.3.2.2.4.2.3.cmml">ζ</mi><mi id="S2.Ex3.m1.2.3.2.2.4.3" xref="S2.Ex3.m1.2.3.2.2.4.3.cmml">T</mi></msubsup><mo id="S2.Ex3.m1.2.3.2.2.1b" xref="S2.Ex3.m1.2.3.2.2.1.cmml">⁢</mo><mrow id="S2.Ex3.m1.2.3.2.2.5.2" xref="S2.Ex3.m1.2.3.2.2.cmml"><mo id="S2.Ex3.m1.2.3.2.2.5.2.1" stretchy="false" xref="S2.Ex3.m1.2.3.2.2.cmml">(</mo><mi id="S2.Ex3.m1.2.2" xref="S2.Ex3.m1.2.2.cmml">τ</mi><mo id="S2.Ex3.m1.2.3.2.2.5.2.2" stretchy="false" xref="S2.Ex3.m1.2.3.2.2.cmml">)</mo></mrow><mo id="S2.Ex3.m1.2.3.2.2.1c" lspace="0em" xref="S2.Ex3.m1.2.3.2.2.1.cmml">⁢</mo><mrow id="S2.Ex3.m1.2.3.2.2.6" xref="S2.Ex3.m1.2.3.2.2.6.cmml"><mo id="S2.Ex3.m1.2.3.2.2.6.1" rspace="0em" xref="S2.Ex3.m1.2.3.2.2.6.1.cmml">𝑑</mo><mi id="S2.Ex3.m1.2.3.2.2.6.2" xref="S2.Ex3.m1.2.3.2.2.6.2.cmml">τ</mi></mrow></mrow></mrow><mo id="S2.Ex3.m1.2.3.1" xref="S2.Ex3.m1.2.3.1.cmml">≥</mo><mrow id="S2.Ex3.m1.2.3.3" xref="S2.Ex3.m1.2.3.3.cmml"><mi id="S2.Ex3.m1.2.3.3.2" xref="S2.Ex3.m1.2.3.3.2.cmml">σ</mi><mo id="S2.Ex3.m1.2.3.3.1" xref="S2.Ex3.m1.2.3.3.1.cmml">⁢</mo><mi id="S2.Ex3.m1.2.3.3.3" xref="S2.Ex3.m1.2.3.3.3.cmml">I</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex3.m1.2b"><apply id="S2.Ex3.m1.2.3.cmml" xref="S2.Ex3.m1.2.3"><geq id="S2.Ex3.m1.2.3.1.cmml" xref="S2.Ex3.m1.2.3.1"></geq><apply id="S2.Ex3.m1.2.3.2.cmml" xref="S2.Ex3.m1.2.3.2"><apply id="S2.Ex3.m1.2.3.2.1.cmml" xref="S2.Ex3.m1.2.3.2.1"><csymbol cd="ambiguous" id="S2.Ex3.m1.2.3.2.1.1.cmml" xref="S2.Ex3.m1.2.3.2.1">superscript</csymbol><apply id="S2.Ex3.m1.2.3.2.1.2.cmml" xref="S2.Ex3.m1.2.3.2.1"><csymbol cd="ambiguous" id="S2.Ex3.m1.2.3.2.1.2.1.cmml" xref="S2.Ex3.m1.2.3.2.1">subscript</csymbol><int id="S2.Ex3.m1.2.3.2.1.2.2.cmml" xref="S2.Ex3.m1.2.3.2.1.2.2"></int><apply id="S2.Ex3.m1.2.3.2.1.2.3.cmml" xref="S2.Ex3.m1.2.3.2.1.2.3"><minus id="S2.Ex3.m1.2.3.2.1.2.3.1.cmml" xref="S2.Ex3.m1.2.3.2.1.2.3.1"></minus><apply id="S2.Ex3.m1.2.3.2.1.2.3.2.cmml" xref="S2.Ex3.m1.2.3.2.1.2.3.2"><csymbol cd="ambiguous" id="S2.Ex3.m1.2.3.2.1.2.3.2.1.cmml" xref="S2.Ex3.m1.2.3.2.1.2.3.2">subscript</csymbol><ci id="S2.Ex3.m1.2.3.2.1.2.3.2.2.cmml" xref="S2.Ex3.m1.2.3.2.1.2.3.2.2">𝑇</ci><ci id="S2.Ex3.m1.2.3.2.1.2.3.2.3.cmml" xref="S2.Ex3.m1.2.3.2.1.2.3.2.3">a</ci></apply><apply id="S2.Ex3.m1.2.3.2.1.2.3.3.cmml" xref="S2.Ex3.m1.2.3.2.1.2.3.3"><csymbol cd="ambiguous" id="S2.Ex3.m1.2.3.2.1.2.3.3.1.cmml" xref="S2.Ex3.m1.2.3.2.1.2.3.3">subscript</csymbol><ci id="S2.Ex3.m1.2.3.2.1.2.3.3.2.cmml" xref="S2.Ex3.m1.2.3.2.1.2.3.3.2">𝜏</ci><ci id="S2.Ex3.m1.2.3.2.1.2.3.3.3.cmml" xref="S2.Ex3.m1.2.3.2.1.2.3.3.3">d</ci></apply></apply></apply><apply id="S2.Ex3.m1.2.3.2.1.3.cmml" xref="S2.Ex3.m1.2.3.2.1.3"><csymbol cd="ambiguous" id="S2.Ex3.m1.2.3.2.1.3.1.cmml" xref="S2.Ex3.m1.2.3.2.1.3">subscript</csymbol><ci id="S2.Ex3.m1.2.3.2.1.3.2.cmml" xref="S2.Ex3.m1.2.3.2.1.3.2">𝑇</ci><ci id="S2.Ex3.m1.2.3.2.1.3.3.cmml" xref="S2.Ex3.m1.2.3.2.1.3.3">a</ci></apply></apply><apply id="S2.Ex3.m1.2.3.2.2.cmml" xref="S2.Ex3.m1.2.3.2.2"><times id="S2.Ex3.m1.2.3.2.2.1.cmml" xref="S2.Ex3.m1.2.3.2.2.1"></times><apply id="S2.Ex3.m1.2.3.2.2.2.cmml" xref="S2.Ex3.m1.2.3.2.2.2"><csymbol cd="ambiguous" id="S2.Ex3.m1.2.3.2.2.2.1.cmml" xref="S2.Ex3.m1.2.3.2.2.2">subscript</csymbol><ci id="S2.Ex3.m1.2.3.2.2.2.2.cmml" xref="S2.Ex3.m1.2.3.2.2.2.2">Φ</ci><ci id="S2.Ex3.m1.2.3.2.2.2.3.cmml" xref="S2.Ex3.m1.2.3.2.2.2.3">𝜁</ci></apply><ci id="S2.Ex3.m1.1.1.cmml" xref="S2.Ex3.m1.1.1">𝜏</ci><apply id="S2.Ex3.m1.2.3.2.2.4.cmml" xref="S2.Ex3.m1.2.3.2.2.4"><csymbol cd="ambiguous" id="S2.Ex3.m1.2.3.2.2.4.1.cmml" xref="S2.Ex3.m1.2.3.2.2.4">superscript</csymbol><apply id="S2.Ex3.m1.2.3.2.2.4.2.cmml" xref="S2.Ex3.m1.2.3.2.2.4"><csymbol cd="ambiguous" id="S2.Ex3.m1.2.3.2.2.4.2.1.cmml" xref="S2.Ex3.m1.2.3.2.2.4">subscript</csymbol><ci id="S2.Ex3.m1.2.3.2.2.4.2.2.cmml" xref="S2.Ex3.m1.2.3.2.2.4.2.2">Φ</ci><ci id="S2.Ex3.m1.2.3.2.2.4.2.3.cmml" xref="S2.Ex3.m1.2.3.2.2.4.2.3">𝜁</ci></apply><ci id="S2.Ex3.m1.2.3.2.2.4.3.cmml" xref="S2.Ex3.m1.2.3.2.2.4.3">𝑇</ci></apply><ci id="S2.Ex3.m1.2.2.cmml" xref="S2.Ex3.m1.2.2">𝜏</ci><apply id="S2.Ex3.m1.2.3.2.2.6.cmml" xref="S2.Ex3.m1.2.3.2.2.6"><csymbol cd="latexml" id="S2.Ex3.m1.2.3.2.2.6.1.cmml" xref="S2.Ex3.m1.2.3.2.2.6.1">differential-d</csymbol><ci id="S2.Ex3.m1.2.3.2.2.6.2.cmml" xref="S2.Ex3.m1.2.3.2.2.6.2">𝜏</ci></apply></apply></apply><apply id="S2.Ex3.m1.2.3.3.cmml" xref="S2.Ex3.m1.2.3.3"><times id="S2.Ex3.m1.2.3.3.1.cmml" xref="S2.Ex3.m1.2.3.3.1"></times><ci id="S2.Ex3.m1.2.3.3.2.cmml" xref="S2.Ex3.m1.2.3.3.2">𝜎</ci><ci id="S2.Ex3.m1.2.3.3.3.cmml" xref="S2.Ex3.m1.2.3.3.3">𝐼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex3.m1.2c">\int_{T_{\rm a}-\tau_{\rm d}}^{T_{\rm a}}\Phi_{\zeta}(\tau)\Phi_{\zeta}^{T}(% \tau)d\tau\geq\sigma I</annotation><annotation encoding="application/x-llamapun" id="S2.Ex3.m1.2d">∫ start_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_τ ) roman_Φ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_τ ) italic_d italic_τ ≥ italic_σ italic_I</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p4.9">in which <math alttext="\Phi_{\zeta}\in\mathbb{R}^{m\times n}" class="ltx_Math" display="inline" id="S2.p4.7.m1.1"><semantics id="S2.p4.7.m1.1a"><mrow id="S2.p4.7.m1.1.1" xref="S2.p4.7.m1.1.1.cmml"><msub id="S2.p4.7.m1.1.1.2" xref="S2.p4.7.m1.1.1.2.cmml"><mi id="S2.p4.7.m1.1.1.2.2" mathvariant="normal" xref="S2.p4.7.m1.1.1.2.2.cmml">Φ</mi><mi id="S2.p4.7.m1.1.1.2.3" xref="S2.p4.7.m1.1.1.2.3.cmml">ζ</mi></msub><mo id="S2.p4.7.m1.1.1.1" xref="S2.p4.7.m1.1.1.1.cmml">∈</mo><msup id="S2.p4.7.m1.1.1.3" xref="S2.p4.7.m1.1.1.3.cmml"><mi id="S2.p4.7.m1.1.1.3.2" xref="S2.p4.7.m1.1.1.3.2.cmml">ℝ</mi><mrow id="S2.p4.7.m1.1.1.3.3" xref="S2.p4.7.m1.1.1.3.3.cmml"><mi id="S2.p4.7.m1.1.1.3.3.2" xref="S2.p4.7.m1.1.1.3.3.2.cmml">m</mi><mo id="S2.p4.7.m1.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S2.p4.7.m1.1.1.3.3.1.cmml">×</mo><mi id="S2.p4.7.m1.1.1.3.3.3" xref="S2.p4.7.m1.1.1.3.3.3.cmml">n</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p4.7.m1.1b"><apply id="S2.p4.7.m1.1.1.cmml" xref="S2.p4.7.m1.1.1"><in id="S2.p4.7.m1.1.1.1.cmml" xref="S2.p4.7.m1.1.1.1"></in><apply id="S2.p4.7.m1.1.1.2.cmml" xref="S2.p4.7.m1.1.1.2"><csymbol cd="ambiguous" id="S2.p4.7.m1.1.1.2.1.cmml" xref="S2.p4.7.m1.1.1.2">subscript</csymbol><ci id="S2.p4.7.m1.1.1.2.2.cmml" xref="S2.p4.7.m1.1.1.2.2">Φ</ci><ci id="S2.p4.7.m1.1.1.2.3.cmml" xref="S2.p4.7.m1.1.1.2.3">𝜁</ci></apply><apply id="S2.p4.7.m1.1.1.3.cmml" xref="S2.p4.7.m1.1.1.3"><csymbol cd="ambiguous" id="S2.p4.7.m1.1.1.3.1.cmml" xref="S2.p4.7.m1.1.1.3">superscript</csymbol><ci id="S2.p4.7.m1.1.1.3.2.cmml" xref="S2.p4.7.m1.1.1.3.2">ℝ</ci><apply id="S2.p4.7.m1.1.1.3.3.cmml" xref="S2.p4.7.m1.1.1.3.3"><times id="S2.p4.7.m1.1.1.3.3.1.cmml" xref="S2.p4.7.m1.1.1.3.3.1"></times><ci id="S2.p4.7.m1.1.1.3.3.2.cmml" xref="S2.p4.7.m1.1.1.3.3.2">𝑚</ci><ci id="S2.p4.7.m1.1.1.3.3.3.cmml" xref="S2.p4.7.m1.1.1.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.7.m1.1c">\Phi_{\zeta}\in\mathbb{R}^{m\times n}</annotation><annotation encoding="application/x-llamapun" id="S2.p4.7.m1.1d">roman_Φ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_m × italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> is a sub-regressor obtained by eliminating some rows of <math alttext="\Phi" class="ltx_Math" display="inline" id="S2.p4.8.m2.1"><semantics id="S2.p4.8.m2.1a"><mi id="S2.p4.8.m2.1.1" mathvariant="normal" xref="S2.p4.8.m2.1.1.cmml">Φ</mi><annotation-xml encoding="MathML-Content" id="S2.p4.8.m2.1b"><ci id="S2.p4.8.m2.1.1.cmml" xref="S2.p4.8.m2.1.1">Φ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.8.m2.1c">\Phi</annotation><annotation encoding="application/x-llamapun" id="S2.p4.8.m2.1d">roman_Φ</annotation></semantics></math> with <math alttext="1\leq m<N" class="ltx_Math" display="inline" id="S2.p4.9.m3.1"><semantics id="S2.p4.9.m3.1a"><mrow id="S2.p4.9.m3.1.1" xref="S2.p4.9.m3.1.1.cmml"><mn id="S2.p4.9.m3.1.1.2" xref="S2.p4.9.m3.1.1.2.cmml">1</mn><mo id="S2.p4.9.m3.1.1.3" xref="S2.p4.9.m3.1.1.3.cmml">≤</mo><mi id="S2.p4.9.m3.1.1.4" xref="S2.p4.9.m3.1.1.4.cmml">m</mi><mo id="S2.p4.9.m3.1.1.5" xref="S2.p4.9.m3.1.1.5.cmml"><</mo><mi id="S2.p4.9.m3.1.1.6" xref="S2.p4.9.m3.1.1.6.cmml">N</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p4.9.m3.1b"><apply id="S2.p4.9.m3.1.1.cmml" xref="S2.p4.9.m3.1.1"><and id="S2.p4.9.m3.1.1a.cmml" xref="S2.p4.9.m3.1.1"></and><apply id="S2.p4.9.m3.1.1b.cmml" xref="S2.p4.9.m3.1.1"><leq id="S2.p4.9.m3.1.1.3.cmml" xref="S2.p4.9.m3.1.1.3"></leq><cn id="S2.p4.9.m3.1.1.2.cmml" type="integer" xref="S2.p4.9.m3.1.1.2">1</cn><ci id="S2.p4.9.m3.1.1.4.cmml" xref="S2.p4.9.m3.1.1.4">𝑚</ci></apply><apply id="S2.p4.9.m3.1.1c.cmml" xref="S2.p4.9.m3.1.1"><lt id="S2.p4.9.m3.1.1.5.cmml" xref="S2.p4.9.m3.1.1.5"></lt><share href="https://arxiv.org/html/2401.10785v2#S2.p4.9.m3.1.1.4.cmml" id="S2.p4.9.m3.1.1d.cmml" xref="S2.p4.9.m3.1.1"></share><ci id="S2.p4.9.m3.1.1.6.cmml" xref="S2.p4.9.m3.1.1.6">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.9.m3.1c">1\leq m<N</annotation><annotation encoding="application/x-llamapun" id="S2.p4.9.m3.1d">1 ≤ italic_m < italic_N</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.p5"> <p class="ltx_p" id="S2.p5.11">For convenience, a column <math alttext="{\bm{\phi}}_{j}(t)\in\mathbb{R}^{n}" class="ltx_Math" display="inline" id="S2.p5.1.m1.1"><semantics id="S2.p5.1.m1.1a"><mrow id="S2.p5.1.m1.1.2" xref="S2.p5.1.m1.1.2.cmml"><mrow id="S2.p5.1.m1.1.2.2" xref="S2.p5.1.m1.1.2.2.cmml"><msub id="S2.p5.1.m1.1.2.2.2" xref="S2.p5.1.m1.1.2.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S2.p5.1.m1.1.2.2.2.2" mathvariant="bold-italic" xref="S2.p5.1.m1.1.2.2.2.2.cmml">ϕ</mi><mi id="S2.p5.1.m1.1.2.2.2.3" xref="S2.p5.1.m1.1.2.2.2.3.cmml">j</mi></msub><mo id="S2.p5.1.m1.1.2.2.1" xref="S2.p5.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S2.p5.1.m1.1.2.2.3.2" xref="S2.p5.1.m1.1.2.2.cmml"><mo id="S2.p5.1.m1.1.2.2.3.2.1" stretchy="false" xref="S2.p5.1.m1.1.2.2.cmml">(</mo><mi id="S2.p5.1.m1.1.1" xref="S2.p5.1.m1.1.1.cmml">t</mi><mo id="S2.p5.1.m1.1.2.2.3.2.2" stretchy="false" xref="S2.p5.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.p5.1.m1.1.2.1" xref="S2.p5.1.m1.1.2.1.cmml">∈</mo><msup id="S2.p5.1.m1.1.2.3" xref="S2.p5.1.m1.1.2.3.cmml"><mi id="S2.p5.1.m1.1.2.3.2" xref="S2.p5.1.m1.1.2.3.2.cmml">ℝ</mi><mi id="S2.p5.1.m1.1.2.3.3" xref="S2.p5.1.m1.1.2.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.1.m1.1b"><apply id="S2.p5.1.m1.1.2.cmml" xref="S2.p5.1.m1.1.2"><in id="S2.p5.1.m1.1.2.1.cmml" xref="S2.p5.1.m1.1.2.1"></in><apply id="S2.p5.1.m1.1.2.2.cmml" xref="S2.p5.1.m1.1.2.2"><times id="S2.p5.1.m1.1.2.2.1.cmml" xref="S2.p5.1.m1.1.2.2.1"></times><apply id="S2.p5.1.m1.1.2.2.2.cmml" xref="S2.p5.1.m1.1.2.2.2"><csymbol cd="ambiguous" id="S2.p5.1.m1.1.2.2.2.1.cmml" xref="S2.p5.1.m1.1.2.2.2">subscript</csymbol><ci id="S2.p5.1.m1.1.2.2.2.2.cmml" xref="S2.p5.1.m1.1.2.2.2.2">bold-italic-ϕ</ci><ci id="S2.p5.1.m1.1.2.2.2.3.cmml" xref="S2.p5.1.m1.1.2.2.2.3">𝑗</ci></apply><ci id="S2.p5.1.m1.1.1.cmml" xref="S2.p5.1.m1.1.1">𝑡</ci></apply><apply id="S2.p5.1.m1.1.2.3.cmml" xref="S2.p5.1.m1.1.2.3"><csymbol cd="ambiguous" id="S2.p5.1.m1.1.2.3.1.cmml" xref="S2.p5.1.m1.1.2.3">superscript</csymbol><ci id="S2.p5.1.m1.1.2.3.2.cmml" xref="S2.p5.1.m1.1.2.3.2">ℝ</ci><ci id="S2.p5.1.m1.1.2.3.3.cmml" xref="S2.p5.1.m1.1.2.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.1.m1.1c">{\bm{\phi}}_{j}(t)\in\mathbb{R}^{n}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.1.m1.1d">bold_italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> (<math alttext="j=1" class="ltx_Math" display="inline" id="S2.p5.2.m2.1"><semantics id="S2.p5.2.m2.1a"><mrow id="S2.p5.2.m2.1.1" xref="S2.p5.2.m2.1.1.cmml"><mi id="S2.p5.2.m2.1.1.2" xref="S2.p5.2.m2.1.1.2.cmml">j</mi><mo id="S2.p5.2.m2.1.1.1" xref="S2.p5.2.m2.1.1.1.cmml">=</mo><mn id="S2.p5.2.m2.1.1.3" xref="S2.p5.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.2.m2.1b"><apply id="S2.p5.2.m2.1.1.cmml" xref="S2.p5.2.m2.1.1"><eq id="S2.p5.2.m2.1.1.1.cmml" xref="S2.p5.2.m2.1.1.1"></eq><ci id="S2.p5.2.m2.1.1.2.cmml" xref="S2.p5.2.m2.1.1.2">𝑗</ci><cn id="S2.p5.2.m2.1.1.3.cmml" type="integer" xref="S2.p5.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.2.m2.1c">j=1</annotation><annotation encoding="application/x-llamapun" id="S2.p5.2.m2.1d">italic_j = 1</annotation></semantics></math> to <math alttext="N" class="ltx_Math" display="inline" id="S2.p5.3.m3.1"><semantics id="S2.p5.3.m3.1a"><mi id="S2.p5.3.m3.1.1" xref="S2.p5.3.m3.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S2.p5.3.m3.1b"><ci id="S2.p5.3.m3.1.1.cmml" xref="S2.p5.3.m3.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.3.m3.1c">N</annotation><annotation encoding="application/x-llamapun" id="S2.p5.3.m3.1d">italic_N</annotation></semantics></math>) of a regressor <math alttext="\Phi^{T}(t)\in\mathbb{R}^{n\times N}" class="ltx_Math" display="inline" id="S2.p5.4.m4.1"><semantics id="S2.p5.4.m4.1a"><mrow id="S2.p5.4.m4.1.2" xref="S2.p5.4.m4.1.2.cmml"><mrow id="S2.p5.4.m4.1.2.2" xref="S2.p5.4.m4.1.2.2.cmml"><msup id="S2.p5.4.m4.1.2.2.2" xref="S2.p5.4.m4.1.2.2.2.cmml"><mi id="S2.p5.4.m4.1.2.2.2.2" mathvariant="normal" xref="S2.p5.4.m4.1.2.2.2.2.cmml">Φ</mi><mi id="S2.p5.4.m4.1.2.2.2.3" xref="S2.p5.4.m4.1.2.2.2.3.cmml">T</mi></msup><mo id="S2.p5.4.m4.1.2.2.1" xref="S2.p5.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S2.p5.4.m4.1.2.2.3.2" xref="S2.p5.4.m4.1.2.2.cmml"><mo id="S2.p5.4.m4.1.2.2.3.2.1" stretchy="false" xref="S2.p5.4.m4.1.2.2.cmml">(</mo><mi id="S2.p5.4.m4.1.1" xref="S2.p5.4.m4.1.1.cmml">t</mi><mo id="S2.p5.4.m4.1.2.2.3.2.2" stretchy="false" xref="S2.p5.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.p5.4.m4.1.2.1" xref="S2.p5.4.m4.1.2.1.cmml">∈</mo><msup id="S2.p5.4.m4.1.2.3" xref="S2.p5.4.m4.1.2.3.cmml"><mi id="S2.p5.4.m4.1.2.3.2" xref="S2.p5.4.m4.1.2.3.2.cmml">ℝ</mi><mrow id="S2.p5.4.m4.1.2.3.3" xref="S2.p5.4.m4.1.2.3.3.cmml"><mi id="S2.p5.4.m4.1.2.3.3.2" xref="S2.p5.4.m4.1.2.3.3.2.cmml">n</mi><mo id="S2.p5.4.m4.1.2.3.3.1" lspace="0.222em" rspace="0.222em" xref="S2.p5.4.m4.1.2.3.3.1.cmml">×</mo><mi id="S2.p5.4.m4.1.2.3.3.3" xref="S2.p5.4.m4.1.2.3.3.3.cmml">N</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.4.m4.1b"><apply id="S2.p5.4.m4.1.2.cmml" xref="S2.p5.4.m4.1.2"><in id="S2.p5.4.m4.1.2.1.cmml" xref="S2.p5.4.m4.1.2.1"></in><apply id="S2.p5.4.m4.1.2.2.cmml" xref="S2.p5.4.m4.1.2.2"><times id="S2.p5.4.m4.1.2.2.1.cmml" xref="S2.p5.4.m4.1.2.2.1"></times><apply id="S2.p5.4.m4.1.2.2.2.cmml" xref="S2.p5.4.m4.1.2.2.2"><csymbol cd="ambiguous" id="S2.p5.4.m4.1.2.2.2.1.cmml" xref="S2.p5.4.m4.1.2.2.2">superscript</csymbol><ci id="S2.p5.4.m4.1.2.2.2.2.cmml" xref="S2.p5.4.m4.1.2.2.2.2">Φ</ci><ci id="S2.p5.4.m4.1.2.2.2.3.cmml" xref="S2.p5.4.m4.1.2.2.2.3">𝑇</ci></apply><ci id="S2.p5.4.m4.1.1.cmml" xref="S2.p5.4.m4.1.1">𝑡</ci></apply><apply id="S2.p5.4.m4.1.2.3.cmml" xref="S2.p5.4.m4.1.2.3"><csymbol cd="ambiguous" id="S2.p5.4.m4.1.2.3.1.cmml" xref="S2.p5.4.m4.1.2.3">superscript</csymbol><ci id="S2.p5.4.m4.1.2.3.2.cmml" xref="S2.p5.4.m4.1.2.3.2">ℝ</ci><apply id="S2.p5.4.m4.1.2.3.3.cmml" xref="S2.p5.4.m4.1.2.3.3"><times id="S2.p5.4.m4.1.2.3.3.1.cmml" xref="S2.p5.4.m4.1.2.3.3.1"></times><ci id="S2.p5.4.m4.1.2.3.3.2.cmml" xref="S2.p5.4.m4.1.2.3.3.2">𝑛</ci><ci id="S2.p5.4.m4.1.2.3.3.3.cmml" xref="S2.p5.4.m4.1.2.3.3.3">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.4.m4.1c">\Phi^{T}(t)\in\mathbb{R}^{n\times N}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.4.m4.1d">roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_t ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_n × italic_N end_POSTSUPERSCRIPT</annotation></semantics></math> is named as a channel. Therefore, one has <math alttext="\Phi(t)" class="ltx_Math" display="inline" id="S2.p5.5.m5.1"><semantics id="S2.p5.5.m5.1a"><mrow id="S2.p5.5.m5.1.2" xref="S2.p5.5.m5.1.2.cmml"><mi id="S2.p5.5.m5.1.2.2" mathvariant="normal" xref="S2.p5.5.m5.1.2.2.cmml">Φ</mi><mo id="S2.p5.5.m5.1.2.1" xref="S2.p5.5.m5.1.2.1.cmml">⁢</mo><mrow id="S2.p5.5.m5.1.2.3.2" xref="S2.p5.5.m5.1.2.cmml"><mo id="S2.p5.5.m5.1.2.3.2.1" stretchy="false" xref="S2.p5.5.m5.1.2.cmml">(</mo><mi id="S2.p5.5.m5.1.1" xref="S2.p5.5.m5.1.1.cmml">t</mi><mo id="S2.p5.5.m5.1.2.3.2.2" stretchy="false" xref="S2.p5.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.5.m5.1b"><apply id="S2.p5.5.m5.1.2.cmml" xref="S2.p5.5.m5.1.2"><times id="S2.p5.5.m5.1.2.1.cmml" xref="S2.p5.5.m5.1.2.1"></times><ci id="S2.p5.5.m5.1.2.2.cmml" xref="S2.p5.5.m5.1.2.2">Φ</ci><ci id="S2.p5.5.m5.1.1.cmml" xref="S2.p5.5.m5.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.5.m5.1c">\Phi(t)</annotation><annotation encoding="application/x-llamapun" id="S2.p5.5.m5.1d">roman_Φ ( italic_t )</annotation></semantics></math> <math alttext="=" class="ltx_Math" display="inline" id="S2.p5.6.m6.1"><semantics id="S2.p5.6.m6.1a"><mo id="S2.p5.6.m6.1.1" xref="S2.p5.6.m6.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S2.p5.6.m6.1b"><eq id="S2.p5.6.m6.1.1.cmml" xref="S2.p5.6.m6.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.6.m6.1c">=</annotation><annotation encoding="application/x-llamapun" id="S2.p5.6.m6.1d">=</annotation></semantics></math> <math alttext="[{\bm{\phi}}_{1}(t),{\bm{\phi}}_{2}(t),\cdots,{\bm{\phi}}_{N}(t)]^{T}" class="ltx_Math" display="inline" id="S2.p5.7.m7.7"><semantics id="S2.p5.7.m7.7a"><msup id="S2.p5.7.m7.7.7" xref="S2.p5.7.m7.7.7.cmml"><mrow id="S2.p5.7.m7.7.7.3.3" xref="S2.p5.7.m7.7.7.3.4.cmml"><mo id="S2.p5.7.m7.7.7.3.3.4" stretchy="false" xref="S2.p5.7.m7.7.7.3.4.cmml">[</mo><mrow id="S2.p5.7.m7.5.5.1.1.1" xref="S2.p5.7.m7.5.5.1.1.1.cmml"><msub id="S2.p5.7.m7.5.5.1.1.1.2" xref="S2.p5.7.m7.5.5.1.1.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S2.p5.7.m7.5.5.1.1.1.2.2" mathvariant="bold-italic" xref="S2.p5.7.m7.5.5.1.1.1.2.2.cmml">ϕ</mi><mn id="S2.p5.7.m7.5.5.1.1.1.2.3" xref="S2.p5.7.m7.5.5.1.1.1.2.3.cmml">1</mn></msub><mo id="S2.p5.7.m7.5.5.1.1.1.1" xref="S2.p5.7.m7.5.5.1.1.1.1.cmml">⁢</mo><mrow id="S2.p5.7.m7.5.5.1.1.1.3.2" xref="S2.p5.7.m7.5.5.1.1.1.cmml"><mo id="S2.p5.7.m7.5.5.1.1.1.3.2.1" stretchy="false" xref="S2.p5.7.m7.5.5.1.1.1.cmml">(</mo><mi id="S2.p5.7.m7.1.1" xref="S2.p5.7.m7.1.1.cmml">t</mi><mo id="S2.p5.7.m7.5.5.1.1.1.3.2.2" stretchy="false" xref="S2.p5.7.m7.5.5.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p5.7.m7.7.7.3.3.5" xref="S2.p5.7.m7.7.7.3.4.cmml">,</mo><mrow id="S2.p5.7.m7.6.6.2.2.2" xref="S2.p5.7.m7.6.6.2.2.2.cmml"><msub id="S2.p5.7.m7.6.6.2.2.2.2" xref="S2.p5.7.m7.6.6.2.2.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S2.p5.7.m7.6.6.2.2.2.2.2" mathvariant="bold-italic" xref="S2.p5.7.m7.6.6.2.2.2.2.2.cmml">ϕ</mi><mn id="S2.p5.7.m7.6.6.2.2.2.2.3" xref="S2.p5.7.m7.6.6.2.2.2.2.3.cmml">2</mn></msub><mo id="S2.p5.7.m7.6.6.2.2.2.1" xref="S2.p5.7.m7.6.6.2.2.2.1.cmml">⁢</mo><mrow id="S2.p5.7.m7.6.6.2.2.2.3.2" xref="S2.p5.7.m7.6.6.2.2.2.cmml"><mo id="S2.p5.7.m7.6.6.2.2.2.3.2.1" stretchy="false" xref="S2.p5.7.m7.6.6.2.2.2.cmml">(</mo><mi id="S2.p5.7.m7.2.2" xref="S2.p5.7.m7.2.2.cmml">t</mi><mo id="S2.p5.7.m7.6.6.2.2.2.3.2.2" stretchy="false" xref="S2.p5.7.m7.6.6.2.2.2.cmml">)</mo></mrow></mrow><mo id="S2.p5.7.m7.7.7.3.3.6" xref="S2.p5.7.m7.7.7.3.4.cmml">,</mo><mi id="S2.p5.7.m7.4.4" mathvariant="normal" xref="S2.p5.7.m7.4.4.cmml">⋯</mi><mo id="S2.p5.7.m7.7.7.3.3.7" xref="S2.p5.7.m7.7.7.3.4.cmml">,</mo><mrow id="S2.p5.7.m7.7.7.3.3.3" xref="S2.p5.7.m7.7.7.3.3.3.cmml"><msub id="S2.p5.7.m7.7.7.3.3.3.2" xref="S2.p5.7.m7.7.7.3.3.3.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S2.p5.7.m7.7.7.3.3.3.2.2" mathvariant="bold-italic" xref="S2.p5.7.m7.7.7.3.3.3.2.2.cmml">ϕ</mi><mi id="S2.p5.7.m7.7.7.3.3.3.2.3" xref="S2.p5.7.m7.7.7.3.3.3.2.3.cmml">N</mi></msub><mo id="S2.p5.7.m7.7.7.3.3.3.1" xref="S2.p5.7.m7.7.7.3.3.3.1.cmml">⁢</mo><mrow id="S2.p5.7.m7.7.7.3.3.3.3.2" xref="S2.p5.7.m7.7.7.3.3.3.cmml"><mo id="S2.p5.7.m7.7.7.3.3.3.3.2.1" stretchy="false" xref="S2.p5.7.m7.7.7.3.3.3.cmml">(</mo><mi id="S2.p5.7.m7.3.3" xref="S2.p5.7.m7.3.3.cmml">t</mi><mo id="S2.p5.7.m7.7.7.3.3.3.3.2.2" stretchy="false" xref="S2.p5.7.m7.7.7.3.3.3.cmml">)</mo></mrow></mrow><mo id="S2.p5.7.m7.7.7.3.3.8" stretchy="false" xref="S2.p5.7.m7.7.7.3.4.cmml">]</mo></mrow><mi id="S2.p5.7.m7.7.7.5" xref="S2.p5.7.m7.7.7.5.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="S2.p5.7.m7.7b"><apply id="S2.p5.7.m7.7.7.cmml" xref="S2.p5.7.m7.7.7"><csymbol cd="ambiguous" id="S2.p5.7.m7.7.7.4.cmml" xref="S2.p5.7.m7.7.7">superscript</csymbol><list id="S2.p5.7.m7.7.7.3.4.cmml" xref="S2.p5.7.m7.7.7.3.3"><apply id="S2.p5.7.m7.5.5.1.1.1.cmml" xref="S2.p5.7.m7.5.5.1.1.1"><times id="S2.p5.7.m7.5.5.1.1.1.1.cmml" xref="S2.p5.7.m7.5.5.1.1.1.1"></times><apply id="S2.p5.7.m7.5.5.1.1.1.2.cmml" xref="S2.p5.7.m7.5.5.1.1.1.2"><csymbol cd="ambiguous" id="S2.p5.7.m7.5.5.1.1.1.2.1.cmml" xref="S2.p5.7.m7.5.5.1.1.1.2">subscript</csymbol><ci id="S2.p5.7.m7.5.5.1.1.1.2.2.cmml" xref="S2.p5.7.m7.5.5.1.1.1.2.2">bold-italic-ϕ</ci><cn id="S2.p5.7.m7.5.5.1.1.1.2.3.cmml" type="integer" xref="S2.p5.7.m7.5.5.1.1.1.2.3">1</cn></apply><ci id="S2.p5.7.m7.1.1.cmml" xref="S2.p5.7.m7.1.1">𝑡</ci></apply><apply id="S2.p5.7.m7.6.6.2.2.2.cmml" xref="S2.p5.7.m7.6.6.2.2.2"><times id="S2.p5.7.m7.6.6.2.2.2.1.cmml" xref="S2.p5.7.m7.6.6.2.2.2.1"></times><apply id="S2.p5.7.m7.6.6.2.2.2.2.cmml" xref="S2.p5.7.m7.6.6.2.2.2.2"><csymbol cd="ambiguous" id="S2.p5.7.m7.6.6.2.2.2.2.1.cmml" xref="S2.p5.7.m7.6.6.2.2.2.2">subscript</csymbol><ci id="S2.p5.7.m7.6.6.2.2.2.2.2.cmml" xref="S2.p5.7.m7.6.6.2.2.2.2.2">bold-italic-ϕ</ci><cn id="S2.p5.7.m7.6.6.2.2.2.2.3.cmml" type="integer" xref="S2.p5.7.m7.6.6.2.2.2.2.3">2</cn></apply><ci id="S2.p5.7.m7.2.2.cmml" xref="S2.p5.7.m7.2.2">𝑡</ci></apply><ci id="S2.p5.7.m7.4.4.cmml" xref="S2.p5.7.m7.4.4">⋯</ci><apply id="S2.p5.7.m7.7.7.3.3.3.cmml" xref="S2.p5.7.m7.7.7.3.3.3"><times id="S2.p5.7.m7.7.7.3.3.3.1.cmml" xref="S2.p5.7.m7.7.7.3.3.3.1"></times><apply id="S2.p5.7.m7.7.7.3.3.3.2.cmml" xref="S2.p5.7.m7.7.7.3.3.3.2"><csymbol cd="ambiguous" id="S2.p5.7.m7.7.7.3.3.3.2.1.cmml" xref="S2.p5.7.m7.7.7.3.3.3.2">subscript</csymbol><ci id="S2.p5.7.m7.7.7.3.3.3.2.2.cmml" xref="S2.p5.7.m7.7.7.3.3.3.2.2">bold-italic-ϕ</ci><ci id="S2.p5.7.m7.7.7.3.3.3.2.3.cmml" xref="S2.p5.7.m7.7.7.3.3.3.2.3">𝑁</ci></apply><ci id="S2.p5.7.m7.3.3.cmml" xref="S2.p5.7.m7.3.3">𝑡</ci></apply></list><ci id="S2.p5.7.m7.7.7.5.cmml" xref="S2.p5.7.m7.7.7.5">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.7.m7.7c">[{\bm{\phi}}_{1}(t),{\bm{\phi}}_{2}(t),\cdots,{\bm{\phi}}_{N}(t)]^{T}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.7.m7.7d">[ bold_italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) , bold_italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) , ⋯ , bold_italic_ϕ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_t ) ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math>. A channel <math alttext="{\bm{\phi}}_{j}(t)" class="ltx_Math" display="inline" id="S2.p5.8.m8.1"><semantics id="S2.p5.8.m8.1a"><mrow id="S2.p5.8.m8.1.2" xref="S2.p5.8.m8.1.2.cmml"><msub id="S2.p5.8.m8.1.2.2" xref="S2.p5.8.m8.1.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S2.p5.8.m8.1.2.2.2" mathvariant="bold-italic" xref="S2.p5.8.m8.1.2.2.2.cmml">ϕ</mi><mi id="S2.p5.8.m8.1.2.2.3" xref="S2.p5.8.m8.1.2.2.3.cmml">j</mi></msub><mo id="S2.p5.8.m8.1.2.1" xref="S2.p5.8.m8.1.2.1.cmml">⁢</mo><mrow id="S2.p5.8.m8.1.2.3.2" xref="S2.p5.8.m8.1.2.cmml"><mo id="S2.p5.8.m8.1.2.3.2.1" stretchy="false" xref="S2.p5.8.m8.1.2.cmml">(</mo><mi id="S2.p5.8.m8.1.1" xref="S2.p5.8.m8.1.1.cmml">t</mi><mo id="S2.p5.8.m8.1.2.3.2.2" stretchy="false" xref="S2.p5.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.8.m8.1b"><apply id="S2.p5.8.m8.1.2.cmml" xref="S2.p5.8.m8.1.2"><times id="S2.p5.8.m8.1.2.1.cmml" xref="S2.p5.8.m8.1.2.1"></times><apply id="S2.p5.8.m8.1.2.2.cmml" xref="S2.p5.8.m8.1.2.2"><csymbol cd="ambiguous" id="S2.p5.8.m8.1.2.2.1.cmml" xref="S2.p5.8.m8.1.2.2">subscript</csymbol><ci id="S2.p5.8.m8.1.2.2.2.cmml" xref="S2.p5.8.m8.1.2.2.2">bold-italic-ϕ</ci><ci id="S2.p5.8.m8.1.2.2.3.cmml" xref="S2.p5.8.m8.1.2.2.3">𝑗</ci></apply><ci id="S2.p5.8.m8.1.1.cmml" xref="S2.p5.8.m8.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.8.m8.1c">{\bm{\phi}}_{j}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.p5.8.m8.1d">bold_italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> is named as an <span class="ltx_text ltx_font_italic" id="S2.p5.11.1">active channel</span> if <math alttext="\|{\bm{\phi}}_{j}(t)\|\neq 0" class="ltx_Math" display="inline" id="S2.p5.9.m9.2"><semantics id="S2.p5.9.m9.2a"><mrow id="S2.p5.9.m9.2.2" xref="S2.p5.9.m9.2.2.cmml"><mrow id="S2.p5.9.m9.2.2.1.1" xref="S2.p5.9.m9.2.2.1.2.cmml"><mo id="S2.p5.9.m9.2.2.1.1.2" stretchy="false" xref="S2.p5.9.m9.2.2.1.2.1.cmml">‖</mo><mrow id="S2.p5.9.m9.2.2.1.1.1" xref="S2.p5.9.m9.2.2.1.1.1.cmml"><msub id="S2.p5.9.m9.2.2.1.1.1.2" xref="S2.p5.9.m9.2.2.1.1.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S2.p5.9.m9.2.2.1.1.1.2.2" mathvariant="bold-italic" xref="S2.p5.9.m9.2.2.1.1.1.2.2.cmml">ϕ</mi><mi id="S2.p5.9.m9.2.2.1.1.1.2.3" xref="S2.p5.9.m9.2.2.1.1.1.2.3.cmml">j</mi></msub><mo id="S2.p5.9.m9.2.2.1.1.1.1" xref="S2.p5.9.m9.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S2.p5.9.m9.2.2.1.1.1.3.2" xref="S2.p5.9.m9.2.2.1.1.1.cmml"><mo id="S2.p5.9.m9.2.2.1.1.1.3.2.1" stretchy="false" xref="S2.p5.9.m9.2.2.1.1.1.cmml">(</mo><mi id="S2.p5.9.m9.1.1" xref="S2.p5.9.m9.1.1.cmml">t</mi><mo id="S2.p5.9.m9.2.2.1.1.1.3.2.2" stretchy="false" xref="S2.p5.9.m9.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p5.9.m9.2.2.1.1.3" stretchy="false" xref="S2.p5.9.m9.2.2.1.2.1.cmml">‖</mo></mrow><mo id="S2.p5.9.m9.2.2.2" xref="S2.p5.9.m9.2.2.2.cmml">≠</mo><mn id="S2.p5.9.m9.2.2.3" xref="S2.p5.9.m9.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.9.m9.2b"><apply id="S2.p5.9.m9.2.2.cmml" xref="S2.p5.9.m9.2.2"><neq id="S2.p5.9.m9.2.2.2.cmml" xref="S2.p5.9.m9.2.2.2"></neq><apply id="S2.p5.9.m9.2.2.1.2.cmml" xref="S2.p5.9.m9.2.2.1.1"><csymbol cd="latexml" id="S2.p5.9.m9.2.2.1.2.1.cmml" xref="S2.p5.9.m9.2.2.1.1.2">norm</csymbol><apply id="S2.p5.9.m9.2.2.1.1.1.cmml" xref="S2.p5.9.m9.2.2.1.1.1"><times id="S2.p5.9.m9.2.2.1.1.1.1.cmml" xref="S2.p5.9.m9.2.2.1.1.1.1"></times><apply id="S2.p5.9.m9.2.2.1.1.1.2.cmml" xref="S2.p5.9.m9.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S2.p5.9.m9.2.2.1.1.1.2.1.cmml" xref="S2.p5.9.m9.2.2.1.1.1.2">subscript</csymbol><ci id="S2.p5.9.m9.2.2.1.1.1.2.2.cmml" xref="S2.p5.9.m9.2.2.1.1.1.2.2">bold-italic-ϕ</ci><ci id="S2.p5.9.m9.2.2.1.1.1.2.3.cmml" xref="S2.p5.9.m9.2.2.1.1.1.2.3">𝑗</ci></apply><ci id="S2.p5.9.m9.1.1.cmml" xref="S2.p5.9.m9.1.1">𝑡</ci></apply></apply><cn id="S2.p5.9.m9.2.2.3.cmml" type="integer" xref="S2.p5.9.m9.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.9.m9.2c">\|{\bm{\phi}}_{j}(t)\|\neq 0</annotation><annotation encoding="application/x-llamapun" id="S2.p5.9.m9.2d">∥ bold_italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) ∥ ≠ 0</annotation></semantics></math>, conversely termed as an <span class="ltx_text ltx_font_italic" id="S2.p5.11.2">inactive channel</span>. Without loss of generality, assume that there exists a proper time window <math alttext="\tau_{\rm d}" class="ltx_Math" display="inline" id="S2.p5.10.m10.1"><semantics id="S2.p5.10.m10.1a"><msub id="S2.p5.10.m10.1.1" xref="S2.p5.10.m10.1.1.cmml"><mi id="S2.p5.10.m10.1.1.2" xref="S2.p5.10.m10.1.1.2.cmml">τ</mi><mi id="S2.p5.10.m10.1.1.3" mathvariant="normal" xref="S2.p5.10.m10.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p5.10.m10.1b"><apply id="S2.p5.10.m10.1.1.cmml" xref="S2.p5.10.m10.1.1"><csymbol cd="ambiguous" id="S2.p5.10.m10.1.1.1.cmml" xref="S2.p5.10.m10.1.1">subscript</csymbol><ci id="S2.p5.10.m10.1.1.2.cmml" xref="S2.p5.10.m10.1.1.2">𝜏</ci><ci id="S2.p5.10.m10.1.1.3.cmml" xref="S2.p5.10.m10.1.1.3">d</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.10.m10.1c">\tau_{\rm d}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.10.m10.1d">italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT</annotation></semantics></math> satisfying either Definition 2 or Definition 3, and consider the case where the IE condition may not hold, <math alttext="\forall t\geq 0" class="ltx_Math" display="inline" id="S2.p5.11.m11.1"><semantics id="S2.p5.11.m11.1a"><mrow id="S2.p5.11.m11.1.1" xref="S2.p5.11.m11.1.1.cmml"><mrow id="S2.p5.11.m11.1.1.2" xref="S2.p5.11.m11.1.1.2.cmml"><mo id="S2.p5.11.m11.1.1.2.1" rspace="0.167em" xref="S2.p5.11.m11.1.1.2.1.cmml">∀</mo><mi id="S2.p5.11.m11.1.1.2.2" xref="S2.p5.11.m11.1.1.2.2.cmml">t</mi></mrow><mo id="S2.p5.11.m11.1.1.1" xref="S2.p5.11.m11.1.1.1.cmml">≥</mo><mn id="S2.p5.11.m11.1.1.3" xref="S2.p5.11.m11.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.11.m11.1b"><apply id="S2.p5.11.m11.1.1.cmml" xref="S2.p5.11.m11.1.1"><geq id="S2.p5.11.m11.1.1.1.cmml" xref="S2.p5.11.m11.1.1.1"></geq><apply id="S2.p5.11.m11.1.1.2.cmml" xref="S2.p5.11.m11.1.1.2"><csymbol cd="latexml" id="S2.p5.11.m11.1.1.2.1.cmml" xref="S2.p5.11.m11.1.1.2.1">for-all</csymbol><ci id="S2.p5.11.m11.1.1.2.2.cmml" xref="S2.p5.11.m11.1.1.2.2">𝑡</ci></apply><cn id="S2.p5.11.m11.1.1.3.cmml" type="integer" xref="S2.p5.11.m11.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.11.m11.1c">\forall t\geq 0</annotation><annotation encoding="application/x-llamapun" id="S2.p5.11.m11.1d">∀ italic_t ≥ 0</annotation></semantics></math>, but the partial IE condition holds at the beginning and some moments later. We aim to design a suitable adaptive control strategy for the system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S2.E4" title="In II Problem Formulation ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">4</span></a>) such that closed-loop stability with parameter convergence can be guaranteed in the absence of the PE condition.</p> </div> <div class="ltx_para" id="S2.p6"> <p class="ltx_p" id="S2.p6.13"><span class="ltx_text ltx_font_italic" id="S2.p6.13.1">Remark 1:</span> The PE and IE conditions require that all channels <math alttext="\bm{\phi}_{j}(t)" class="ltx_Math" display="inline" id="S2.p6.1.m1.1"><semantics id="S2.p6.1.m1.1a"><mrow id="S2.p6.1.m1.1.2" xref="S2.p6.1.m1.1.2.cmml"><msub id="S2.p6.1.m1.1.2.2" xref="S2.p6.1.m1.1.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S2.p6.1.m1.1.2.2.2" mathvariant="bold-italic" xref="S2.p6.1.m1.1.2.2.2.cmml">ϕ</mi><mi id="S2.p6.1.m1.1.2.2.3" xref="S2.p6.1.m1.1.2.2.3.cmml">j</mi></msub><mo id="S2.p6.1.m1.1.2.1" xref="S2.p6.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.p6.1.m1.1.2.3.2" xref="S2.p6.1.m1.1.2.cmml"><mo id="S2.p6.1.m1.1.2.3.2.1" stretchy="false" xref="S2.p6.1.m1.1.2.cmml">(</mo><mi id="S2.p6.1.m1.1.1" xref="S2.p6.1.m1.1.1.cmml">t</mi><mo id="S2.p6.1.m1.1.2.3.2.2" stretchy="false" xref="S2.p6.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.1.m1.1b"><apply id="S2.p6.1.m1.1.2.cmml" xref="S2.p6.1.m1.1.2"><times id="S2.p6.1.m1.1.2.1.cmml" xref="S2.p6.1.m1.1.2.1"></times><apply id="S2.p6.1.m1.1.2.2.cmml" xref="S2.p6.1.m1.1.2.2"><csymbol cd="ambiguous" id="S2.p6.1.m1.1.2.2.1.cmml" xref="S2.p6.1.m1.1.2.2">subscript</csymbol><ci id="S2.p6.1.m1.1.2.2.2.cmml" xref="S2.p6.1.m1.1.2.2.2">bold-italic-ϕ</ci><ci id="S2.p6.1.m1.1.2.2.3.cmml" xref="S2.p6.1.m1.1.2.2.3">𝑗</ci></apply><ci id="S2.p6.1.m1.1.1.cmml" xref="S2.p6.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.1.m1.1c">\bm{\phi}_{j}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.p6.1.m1.1d">bold_italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> (<math alttext="j=1" class="ltx_Math" display="inline" id="S2.p6.2.m2.1"><semantics id="S2.p6.2.m2.1a"><mrow id="S2.p6.2.m2.1.1" xref="S2.p6.2.m2.1.1.cmml"><mi id="S2.p6.2.m2.1.1.2" xref="S2.p6.2.m2.1.1.2.cmml">j</mi><mo id="S2.p6.2.m2.1.1.1" xref="S2.p6.2.m2.1.1.1.cmml">=</mo><mn id="S2.p6.2.m2.1.1.3" xref="S2.p6.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.2.m2.1b"><apply id="S2.p6.2.m2.1.1.cmml" xref="S2.p6.2.m2.1.1"><eq id="S2.p6.2.m2.1.1.1.cmml" xref="S2.p6.2.m2.1.1.1"></eq><ci id="S2.p6.2.m2.1.1.2.cmml" xref="S2.p6.2.m2.1.1.2">𝑗</ci><cn id="S2.p6.2.m2.1.1.3.cmml" type="integer" xref="S2.p6.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.2.m2.1c">j=1</annotation><annotation encoding="application/x-llamapun" id="S2.p6.2.m2.1d">italic_j = 1</annotation></semantics></math> to <math alttext="N" class="ltx_Math" display="inline" id="S2.p6.3.m3.1"><semantics id="S2.p6.3.m3.1a"><mi id="S2.p6.3.m3.1.1" xref="S2.p6.3.m3.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S2.p6.3.m3.1b"><ci id="S2.p6.3.m3.1.1.cmml" xref="S2.p6.3.m3.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.3.m3.1c">N</annotation><annotation encoding="application/x-llamapun" id="S2.p6.3.m3.1d">italic_N</annotation></semantics></math>) are activated in an uncorrelated manner within a time window <math alttext="[t-\tau_{\rm d},t]" class="ltx_Math" display="inline" id="S2.p6.4.m4.2"><semantics id="S2.p6.4.m4.2a"><mrow id="S2.p6.4.m4.2.2.1" xref="S2.p6.4.m4.2.2.2.cmml"><mo id="S2.p6.4.m4.2.2.1.2" stretchy="false" xref="S2.p6.4.m4.2.2.2.cmml">[</mo><mrow id="S2.p6.4.m4.2.2.1.1" xref="S2.p6.4.m4.2.2.1.1.cmml"><mi id="S2.p6.4.m4.2.2.1.1.2" xref="S2.p6.4.m4.2.2.1.1.2.cmml">t</mi><mo id="S2.p6.4.m4.2.2.1.1.1" xref="S2.p6.4.m4.2.2.1.1.1.cmml">−</mo><msub id="S2.p6.4.m4.2.2.1.1.3" xref="S2.p6.4.m4.2.2.1.1.3.cmml"><mi id="S2.p6.4.m4.2.2.1.1.3.2" xref="S2.p6.4.m4.2.2.1.1.3.2.cmml">τ</mi><mi id="S2.p6.4.m4.2.2.1.1.3.3" mathvariant="normal" xref="S2.p6.4.m4.2.2.1.1.3.3.cmml">d</mi></msub></mrow><mo id="S2.p6.4.m4.2.2.1.3" xref="S2.p6.4.m4.2.2.2.cmml">,</mo><mi id="S2.p6.4.m4.1.1" xref="S2.p6.4.m4.1.1.cmml">t</mi><mo id="S2.p6.4.m4.2.2.1.4" stretchy="false" xref="S2.p6.4.m4.2.2.2.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.4.m4.2b"><interval closure="closed" id="S2.p6.4.m4.2.2.2.cmml" xref="S2.p6.4.m4.2.2.1"><apply id="S2.p6.4.m4.2.2.1.1.cmml" xref="S2.p6.4.m4.2.2.1.1"><minus id="S2.p6.4.m4.2.2.1.1.1.cmml" xref="S2.p6.4.m4.2.2.1.1.1"></minus><ci id="S2.p6.4.m4.2.2.1.1.2.cmml" xref="S2.p6.4.m4.2.2.1.1.2">𝑡</ci><apply id="S2.p6.4.m4.2.2.1.1.3.cmml" xref="S2.p6.4.m4.2.2.1.1.3"><csymbol cd="ambiguous" id="S2.p6.4.m4.2.2.1.1.3.1.cmml" xref="S2.p6.4.m4.2.2.1.1.3">subscript</csymbol><ci id="S2.p6.4.m4.2.2.1.1.3.2.cmml" xref="S2.p6.4.m4.2.2.1.1.3.2">𝜏</ci><ci id="S2.p6.4.m4.2.2.1.1.3.3.cmml" xref="S2.p6.4.m4.2.2.1.1.3.3">d</ci></apply></apply><ci id="S2.p6.4.m4.1.1.cmml" xref="S2.p6.4.m4.1.1">𝑡</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.4.m4.2c">[t-\tau_{\rm d},t]</annotation><annotation encoding="application/x-llamapun" id="S2.p6.4.m4.2d">[ italic_t - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT , italic_t ]</annotation></semantics></math> (sliding in the case of PE), which is difficult to satisfy in many practical scenarios due to the presence of inactive channels (i.e., there exists <math alttext="j\in\{1,2,\cdots,N\}" class="ltx_Math" display="inline" id="S2.p6.5.m5.4"><semantics id="S2.p6.5.m5.4a"><mrow id="S2.p6.5.m5.4.5" xref="S2.p6.5.m5.4.5.cmml"><mi id="S2.p6.5.m5.4.5.2" xref="S2.p6.5.m5.4.5.2.cmml">j</mi><mo id="S2.p6.5.m5.4.5.1" xref="S2.p6.5.m5.4.5.1.cmml">∈</mo><mrow id="S2.p6.5.m5.4.5.3.2" xref="S2.p6.5.m5.4.5.3.1.cmml"><mo id="S2.p6.5.m5.4.5.3.2.1" stretchy="false" xref="S2.p6.5.m5.4.5.3.1.cmml">{</mo><mn id="S2.p6.5.m5.1.1" xref="S2.p6.5.m5.1.1.cmml">1</mn><mo id="S2.p6.5.m5.4.5.3.2.2" xref="S2.p6.5.m5.4.5.3.1.cmml">,</mo><mn id="S2.p6.5.m5.2.2" xref="S2.p6.5.m5.2.2.cmml">2</mn><mo id="S2.p6.5.m5.4.5.3.2.3" xref="S2.p6.5.m5.4.5.3.1.cmml">,</mo><mi id="S2.p6.5.m5.3.3" mathvariant="normal" xref="S2.p6.5.m5.3.3.cmml">⋯</mi><mo id="S2.p6.5.m5.4.5.3.2.4" xref="S2.p6.5.m5.4.5.3.1.cmml">,</mo><mi id="S2.p6.5.m5.4.4" xref="S2.p6.5.m5.4.4.cmml">N</mi><mo id="S2.p6.5.m5.4.5.3.2.5" stretchy="false" xref="S2.p6.5.m5.4.5.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.5.m5.4b"><apply id="S2.p6.5.m5.4.5.cmml" xref="S2.p6.5.m5.4.5"><in id="S2.p6.5.m5.4.5.1.cmml" xref="S2.p6.5.m5.4.5.1"></in><ci id="S2.p6.5.m5.4.5.2.cmml" xref="S2.p6.5.m5.4.5.2">𝑗</ci><set id="S2.p6.5.m5.4.5.3.1.cmml" xref="S2.p6.5.m5.4.5.3.2"><cn id="S2.p6.5.m5.1.1.cmml" type="integer" xref="S2.p6.5.m5.1.1">1</cn><cn id="S2.p6.5.m5.2.2.cmml" type="integer" xref="S2.p6.5.m5.2.2">2</cn><ci id="S2.p6.5.m5.3.3.cmml" xref="S2.p6.5.m5.3.3">⋯</ci><ci id="S2.p6.5.m5.4.4.cmml" xref="S2.p6.5.m5.4.4">𝑁</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.5.m5.4c">j\in\{1,2,\cdots,N\}</annotation><annotation encoding="application/x-llamapun" id="S2.p6.5.m5.4d">italic_j ∈ { 1 , 2 , ⋯ , italic_N }</annotation></semantics></math> such that <math alttext="\|\bm{\phi}_{j}(\tau)\|=0" class="ltx_Math" display="inline" id="S2.p6.6.m6.2"><semantics id="S2.p6.6.m6.2a"><mrow id="S2.p6.6.m6.2.2" xref="S2.p6.6.m6.2.2.cmml"><mrow id="S2.p6.6.m6.2.2.1.1" xref="S2.p6.6.m6.2.2.1.2.cmml"><mo id="S2.p6.6.m6.2.2.1.1.2" stretchy="false" xref="S2.p6.6.m6.2.2.1.2.1.cmml">‖</mo><mrow id="S2.p6.6.m6.2.2.1.1.1" xref="S2.p6.6.m6.2.2.1.1.1.cmml"><msub id="S2.p6.6.m6.2.2.1.1.1.2" xref="S2.p6.6.m6.2.2.1.1.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S2.p6.6.m6.2.2.1.1.1.2.2" mathvariant="bold-italic" xref="S2.p6.6.m6.2.2.1.1.1.2.2.cmml">ϕ</mi><mi id="S2.p6.6.m6.2.2.1.1.1.2.3" xref="S2.p6.6.m6.2.2.1.1.1.2.3.cmml">j</mi></msub><mo id="S2.p6.6.m6.2.2.1.1.1.1" xref="S2.p6.6.m6.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S2.p6.6.m6.2.2.1.1.1.3.2" xref="S2.p6.6.m6.2.2.1.1.1.cmml"><mo id="S2.p6.6.m6.2.2.1.1.1.3.2.1" stretchy="false" xref="S2.p6.6.m6.2.2.1.1.1.cmml">(</mo><mi id="S2.p6.6.m6.1.1" xref="S2.p6.6.m6.1.1.cmml">τ</mi><mo id="S2.p6.6.m6.2.2.1.1.1.3.2.2" stretchy="false" xref="S2.p6.6.m6.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p6.6.m6.2.2.1.1.3" stretchy="false" xref="S2.p6.6.m6.2.2.1.2.1.cmml">‖</mo></mrow><mo id="S2.p6.6.m6.2.2.2" xref="S2.p6.6.m6.2.2.2.cmml">=</mo><mn id="S2.p6.6.m6.2.2.3" xref="S2.p6.6.m6.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.6.m6.2b"><apply id="S2.p6.6.m6.2.2.cmml" xref="S2.p6.6.m6.2.2"><eq id="S2.p6.6.m6.2.2.2.cmml" xref="S2.p6.6.m6.2.2.2"></eq><apply id="S2.p6.6.m6.2.2.1.2.cmml" xref="S2.p6.6.m6.2.2.1.1"><csymbol cd="latexml" id="S2.p6.6.m6.2.2.1.2.1.cmml" xref="S2.p6.6.m6.2.2.1.1.2">norm</csymbol><apply id="S2.p6.6.m6.2.2.1.1.1.cmml" xref="S2.p6.6.m6.2.2.1.1.1"><times id="S2.p6.6.m6.2.2.1.1.1.1.cmml" xref="S2.p6.6.m6.2.2.1.1.1.1"></times><apply id="S2.p6.6.m6.2.2.1.1.1.2.cmml" xref="S2.p6.6.m6.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S2.p6.6.m6.2.2.1.1.1.2.1.cmml" xref="S2.p6.6.m6.2.2.1.1.1.2">subscript</csymbol><ci id="S2.p6.6.m6.2.2.1.1.1.2.2.cmml" xref="S2.p6.6.m6.2.2.1.1.1.2.2">bold-italic-ϕ</ci><ci id="S2.p6.6.m6.2.2.1.1.1.2.3.cmml" xref="S2.p6.6.m6.2.2.1.1.1.2.3">𝑗</ci></apply><ci id="S2.p6.6.m6.1.1.cmml" xref="S2.p6.6.m6.1.1">𝜏</ci></apply></apply><cn id="S2.p6.6.m6.2.2.3.cmml" type="integer" xref="S2.p6.6.m6.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.6.m6.2c">\|\bm{\phi}_{j}(\tau)\|=0</annotation><annotation encoding="application/x-llamapun" id="S2.p6.6.m6.2d">∥ bold_italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_τ ) ∥ = 0</annotation></semantics></math>, <math alttext="\forall\tau\in[t-\tau_{\rm d},t]" class="ltx_Math" display="inline" id="S2.p6.7.m7.2"><semantics id="S2.p6.7.m7.2a"><mrow id="S2.p6.7.m7.2.2" xref="S2.p6.7.m7.2.2.cmml"><mrow id="S2.p6.7.m7.2.2.3" xref="S2.p6.7.m7.2.2.3.cmml"><mo id="S2.p6.7.m7.2.2.3.1" rspace="0.167em" xref="S2.p6.7.m7.2.2.3.1.cmml">∀</mo><mi id="S2.p6.7.m7.2.2.3.2" xref="S2.p6.7.m7.2.2.3.2.cmml">τ</mi></mrow><mo id="S2.p6.7.m7.2.2.2" xref="S2.p6.7.m7.2.2.2.cmml">∈</mo><mrow id="S2.p6.7.m7.2.2.1.1" xref="S2.p6.7.m7.2.2.1.2.cmml"><mo id="S2.p6.7.m7.2.2.1.1.2" stretchy="false" xref="S2.p6.7.m7.2.2.1.2.cmml">[</mo><mrow id="S2.p6.7.m7.2.2.1.1.1" xref="S2.p6.7.m7.2.2.1.1.1.cmml"><mi id="S2.p6.7.m7.2.2.1.1.1.2" xref="S2.p6.7.m7.2.2.1.1.1.2.cmml">t</mi><mo id="S2.p6.7.m7.2.2.1.1.1.1" xref="S2.p6.7.m7.2.2.1.1.1.1.cmml">−</mo><msub id="S2.p6.7.m7.2.2.1.1.1.3" xref="S2.p6.7.m7.2.2.1.1.1.3.cmml"><mi id="S2.p6.7.m7.2.2.1.1.1.3.2" xref="S2.p6.7.m7.2.2.1.1.1.3.2.cmml">τ</mi><mi id="S2.p6.7.m7.2.2.1.1.1.3.3" mathvariant="normal" xref="S2.p6.7.m7.2.2.1.1.1.3.3.cmml">d</mi></msub></mrow><mo id="S2.p6.7.m7.2.2.1.1.3" xref="S2.p6.7.m7.2.2.1.2.cmml">,</mo><mi id="S2.p6.7.m7.1.1" xref="S2.p6.7.m7.1.1.cmml">t</mi><mo id="S2.p6.7.m7.2.2.1.1.4" stretchy="false" xref="S2.p6.7.m7.2.2.1.2.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.7.m7.2b"><apply id="S2.p6.7.m7.2.2.cmml" xref="S2.p6.7.m7.2.2"><in id="S2.p6.7.m7.2.2.2.cmml" xref="S2.p6.7.m7.2.2.2"></in><apply id="S2.p6.7.m7.2.2.3.cmml" xref="S2.p6.7.m7.2.2.3"><csymbol cd="latexml" id="S2.p6.7.m7.2.2.3.1.cmml" xref="S2.p6.7.m7.2.2.3.1">for-all</csymbol><ci id="S2.p6.7.m7.2.2.3.2.cmml" xref="S2.p6.7.m7.2.2.3.2">𝜏</ci></apply><interval closure="closed" id="S2.p6.7.m7.2.2.1.2.cmml" xref="S2.p6.7.m7.2.2.1.1"><apply id="S2.p6.7.m7.2.2.1.1.1.cmml" xref="S2.p6.7.m7.2.2.1.1.1"><minus id="S2.p6.7.m7.2.2.1.1.1.1.cmml" xref="S2.p6.7.m7.2.2.1.1.1.1"></minus><ci id="S2.p6.7.m7.2.2.1.1.1.2.cmml" xref="S2.p6.7.m7.2.2.1.1.1.2">𝑡</ci><apply id="S2.p6.7.m7.2.2.1.1.1.3.cmml" xref="S2.p6.7.m7.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S2.p6.7.m7.2.2.1.1.1.3.1.cmml" xref="S2.p6.7.m7.2.2.1.1.1.3">subscript</csymbol><ci id="S2.p6.7.m7.2.2.1.1.1.3.2.cmml" xref="S2.p6.7.m7.2.2.1.1.1.3.2">𝜏</ci><ci id="S2.p6.7.m7.2.2.1.1.1.3.3.cmml" xref="S2.p6.7.m7.2.2.1.1.1.3.3">d</ci></apply></apply><ci id="S2.p6.7.m7.1.1.cmml" xref="S2.p6.7.m7.1.1">𝑡</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.7.m7.2c">\forall\tau\in[t-\tau_{\rm d},t]</annotation><annotation encoding="application/x-llamapun" id="S2.p6.7.m7.2d">∀ italic_τ ∈ [ italic_t - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT , italic_t ]</annotation></semantics></math>). The partial IE condition relaxes the requirement by ignoring all inactive channels during <math alttext="[t-\tau_{\rm d},t]" class="ltx_Math" display="inline" id="S2.p6.8.m8.2"><semantics id="S2.p6.8.m8.2a"><mrow id="S2.p6.8.m8.2.2.1" xref="S2.p6.8.m8.2.2.2.cmml"><mo id="S2.p6.8.m8.2.2.1.2" stretchy="false" xref="S2.p6.8.m8.2.2.2.cmml">[</mo><mrow id="S2.p6.8.m8.2.2.1.1" xref="S2.p6.8.m8.2.2.1.1.cmml"><mi id="S2.p6.8.m8.2.2.1.1.2" xref="S2.p6.8.m8.2.2.1.1.2.cmml">t</mi><mo id="S2.p6.8.m8.2.2.1.1.1" xref="S2.p6.8.m8.2.2.1.1.1.cmml">−</mo><msub id="S2.p6.8.m8.2.2.1.1.3" xref="S2.p6.8.m8.2.2.1.1.3.cmml"><mi id="S2.p6.8.m8.2.2.1.1.3.2" xref="S2.p6.8.m8.2.2.1.1.3.2.cmml">τ</mi><mi id="S2.p6.8.m8.2.2.1.1.3.3" mathvariant="normal" xref="S2.p6.8.m8.2.2.1.1.3.3.cmml">d</mi></msub></mrow><mo id="S2.p6.8.m8.2.2.1.3" xref="S2.p6.8.m8.2.2.2.cmml">,</mo><mi id="S2.p6.8.m8.1.1" xref="S2.p6.8.m8.1.1.cmml">t</mi><mo id="S2.p6.8.m8.2.2.1.4" stretchy="false" xref="S2.p6.8.m8.2.2.2.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.8.m8.2b"><interval closure="closed" id="S2.p6.8.m8.2.2.2.cmml" xref="S2.p6.8.m8.2.2.1"><apply id="S2.p6.8.m8.2.2.1.1.cmml" xref="S2.p6.8.m8.2.2.1.1"><minus id="S2.p6.8.m8.2.2.1.1.1.cmml" xref="S2.p6.8.m8.2.2.1.1.1"></minus><ci id="S2.p6.8.m8.2.2.1.1.2.cmml" xref="S2.p6.8.m8.2.2.1.1.2">𝑡</ci><apply id="S2.p6.8.m8.2.2.1.1.3.cmml" xref="S2.p6.8.m8.2.2.1.1.3"><csymbol cd="ambiguous" id="S2.p6.8.m8.2.2.1.1.3.1.cmml" xref="S2.p6.8.m8.2.2.1.1.3">subscript</csymbol><ci id="S2.p6.8.m8.2.2.1.1.3.2.cmml" xref="S2.p6.8.m8.2.2.1.1.3.2">𝜏</ci><ci id="S2.p6.8.m8.2.2.1.1.3.3.cmml" xref="S2.p6.8.m8.2.2.1.1.3.3">d</ci></apply></apply><ci id="S2.p6.8.m8.1.1.cmml" xref="S2.p6.8.m8.1.1">𝑡</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.8.m8.2c">[t-\tau_{\rm d},t]</annotation><annotation encoding="application/x-llamapun" id="S2.p6.8.m8.2d">[ italic_t - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT , italic_t ]</annotation></semantics></math>, which allows it to satisfy at the beginning and some moments later due to the changing state <math alttext="\bm{x}(t)" class="ltx_Math" display="inline" id="S2.p6.9.m9.1"><semantics id="S2.p6.9.m9.1a"><mrow id="S2.p6.9.m9.1.2" xref="S2.p6.9.m9.1.2.cmml"><mi id="S2.p6.9.m9.1.2.2" xref="S2.p6.9.m9.1.2.2.cmml">𝒙</mi><mo id="S2.p6.9.m9.1.2.1" xref="S2.p6.9.m9.1.2.1.cmml">⁢</mo><mrow id="S2.p6.9.m9.1.2.3.2" xref="S2.p6.9.m9.1.2.cmml"><mo id="S2.p6.9.m9.1.2.3.2.1" stretchy="false" xref="S2.p6.9.m9.1.2.cmml">(</mo><mi id="S2.p6.9.m9.1.1" xref="S2.p6.9.m9.1.1.cmml">t</mi><mo id="S2.p6.9.m9.1.2.3.2.2" stretchy="false" xref="S2.p6.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.9.m9.1b"><apply id="S2.p6.9.m9.1.2.cmml" xref="S2.p6.9.m9.1.2"><times id="S2.p6.9.m9.1.2.1.cmml" xref="S2.p6.9.m9.1.2.1"></times><ci id="S2.p6.9.m9.1.2.2.cmml" xref="S2.p6.9.m9.1.2.2">𝒙</ci><ci id="S2.p6.9.m9.1.1.cmml" xref="S2.p6.9.m9.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.9.m9.1c">\bm{x}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.p6.9.m9.1d">bold_italic_x ( italic_t )</annotation></semantics></math> over time in general cases. This implies the inexistence of the case with all channels <math alttext="\|\bm{\phi}_{j}(t)\|\equiv 0" class="ltx_Math" display="inline" id="S2.p6.10.m10.2"><semantics id="S2.p6.10.m10.2a"><mrow id="S2.p6.10.m10.2.2" xref="S2.p6.10.m10.2.2.cmml"><mrow id="S2.p6.10.m10.2.2.1.1" xref="S2.p6.10.m10.2.2.1.2.cmml"><mo id="S2.p6.10.m10.2.2.1.1.2" stretchy="false" xref="S2.p6.10.m10.2.2.1.2.1.cmml">‖</mo><mrow id="S2.p6.10.m10.2.2.1.1.1" xref="S2.p6.10.m10.2.2.1.1.1.cmml"><msub id="S2.p6.10.m10.2.2.1.1.1.2" xref="S2.p6.10.m10.2.2.1.1.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S2.p6.10.m10.2.2.1.1.1.2.2" mathvariant="bold-italic" xref="S2.p6.10.m10.2.2.1.1.1.2.2.cmml">ϕ</mi><mi id="S2.p6.10.m10.2.2.1.1.1.2.3" xref="S2.p6.10.m10.2.2.1.1.1.2.3.cmml">j</mi></msub><mo id="S2.p6.10.m10.2.2.1.1.1.1" xref="S2.p6.10.m10.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S2.p6.10.m10.2.2.1.1.1.3.2" xref="S2.p6.10.m10.2.2.1.1.1.cmml"><mo id="S2.p6.10.m10.2.2.1.1.1.3.2.1" stretchy="false" xref="S2.p6.10.m10.2.2.1.1.1.cmml">(</mo><mi id="S2.p6.10.m10.1.1" xref="S2.p6.10.m10.1.1.cmml">t</mi><mo id="S2.p6.10.m10.2.2.1.1.1.3.2.2" stretchy="false" xref="S2.p6.10.m10.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p6.10.m10.2.2.1.1.3" stretchy="false" xref="S2.p6.10.m10.2.2.1.2.1.cmml">‖</mo></mrow><mo id="S2.p6.10.m10.2.2.2" xref="S2.p6.10.m10.2.2.2.cmml">≡</mo><mn id="S2.p6.10.m10.2.2.3" xref="S2.p6.10.m10.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.10.m10.2b"><apply id="S2.p6.10.m10.2.2.cmml" xref="S2.p6.10.m10.2.2"><equivalent id="S2.p6.10.m10.2.2.2.cmml" xref="S2.p6.10.m10.2.2.2"></equivalent><apply id="S2.p6.10.m10.2.2.1.2.cmml" xref="S2.p6.10.m10.2.2.1.1"><csymbol cd="latexml" id="S2.p6.10.m10.2.2.1.2.1.cmml" xref="S2.p6.10.m10.2.2.1.1.2">norm</csymbol><apply id="S2.p6.10.m10.2.2.1.1.1.cmml" xref="S2.p6.10.m10.2.2.1.1.1"><times id="S2.p6.10.m10.2.2.1.1.1.1.cmml" xref="S2.p6.10.m10.2.2.1.1.1.1"></times><apply id="S2.p6.10.m10.2.2.1.1.1.2.cmml" xref="S2.p6.10.m10.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S2.p6.10.m10.2.2.1.1.1.2.1.cmml" xref="S2.p6.10.m10.2.2.1.1.1.2">subscript</csymbol><ci id="S2.p6.10.m10.2.2.1.1.1.2.2.cmml" xref="S2.p6.10.m10.2.2.1.1.1.2.2">bold-italic-ϕ</ci><ci id="S2.p6.10.m10.2.2.1.1.1.2.3.cmml" xref="S2.p6.10.m10.2.2.1.1.1.2.3">𝑗</ci></apply><ci id="S2.p6.10.m10.1.1.cmml" xref="S2.p6.10.m10.1.1">𝑡</ci></apply></apply><cn id="S2.p6.10.m10.2.2.3.cmml" type="integer" xref="S2.p6.10.m10.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.10.m10.2c">\|\bm{\phi}_{j}(t)\|\equiv 0</annotation><annotation encoding="application/x-llamapun" id="S2.p6.10.m10.2d">∥ bold_italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) ∥ ≡ 0</annotation></semantics></math> (<math alttext="j=1" class="ltx_Math" display="inline" id="S2.p6.11.m11.1"><semantics id="S2.p6.11.m11.1a"><mrow id="S2.p6.11.m11.1.1" xref="S2.p6.11.m11.1.1.cmml"><mi id="S2.p6.11.m11.1.1.2" xref="S2.p6.11.m11.1.1.2.cmml">j</mi><mo id="S2.p6.11.m11.1.1.1" xref="S2.p6.11.m11.1.1.1.cmml">=</mo><mn id="S2.p6.11.m11.1.1.3" xref="S2.p6.11.m11.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.11.m11.1b"><apply id="S2.p6.11.m11.1.1.cmml" xref="S2.p6.11.m11.1.1"><eq id="S2.p6.11.m11.1.1.1.cmml" xref="S2.p6.11.m11.1.1.1"></eq><ci id="S2.p6.11.m11.1.1.2.cmml" xref="S2.p6.11.m11.1.1.2">𝑗</ci><cn id="S2.p6.11.m11.1.1.3.cmml" type="integer" xref="S2.p6.11.m11.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.11.m11.1c">j=1</annotation><annotation encoding="application/x-llamapun" id="S2.p6.11.m11.1d">italic_j = 1</annotation></semantics></math> to <math alttext="N" class="ltx_Math" display="inline" id="S2.p6.12.m12.1"><semantics id="S2.p6.12.m12.1a"><mi id="S2.p6.12.m12.1.1" xref="S2.p6.12.m12.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S2.p6.12.m12.1b"><ci id="S2.p6.12.m12.1.1.cmml" xref="S2.p6.12.m12.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.12.m12.1c">N</annotation><annotation encoding="application/x-llamapun" id="S2.p6.12.m12.1d">italic_N</annotation></semantics></math>), <math alttext="\forall t\geq 0" class="ltx_Math" display="inline" id="S2.p6.13.m13.1"><semantics id="S2.p6.13.m13.1a"><mrow id="S2.p6.13.m13.1.1" xref="S2.p6.13.m13.1.1.cmml"><mrow id="S2.p6.13.m13.1.1.2" xref="S2.p6.13.m13.1.1.2.cmml"><mo id="S2.p6.13.m13.1.1.2.1" rspace="0.167em" xref="S2.p6.13.m13.1.1.2.1.cmml">∀</mo><mi id="S2.p6.13.m13.1.1.2.2" xref="S2.p6.13.m13.1.1.2.2.cmml">t</mi></mrow><mo id="S2.p6.13.m13.1.1.1" xref="S2.p6.13.m13.1.1.1.cmml">≥</mo><mn id="S2.p6.13.m13.1.1.3" xref="S2.p6.13.m13.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.13.m13.1b"><apply id="S2.p6.13.m13.1.1.cmml" xref="S2.p6.13.m13.1.1"><geq id="S2.p6.13.m13.1.1.1.cmml" xref="S2.p6.13.m13.1.1.1"></geq><apply id="S2.p6.13.m13.1.1.2.cmml" xref="S2.p6.13.m13.1.1.2"><csymbol cd="latexml" id="S2.p6.13.m13.1.1.2.1.cmml" xref="S2.p6.13.m13.1.1.2.1">for-all</csymbol><ci id="S2.p6.13.m13.1.1.2.2.cmml" xref="S2.p6.13.m13.1.1.2.2">𝑡</ci></apply><cn id="S2.p6.13.m13.1.1.3.cmml" type="integer" xref="S2.p6.13.m13.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.13.m13.1c">\forall t\geq 0</annotation><annotation encoding="application/x-llamapun" id="S2.p6.13.m13.1d">∀ italic_t ≥ 0</annotation></semantics></math>.</p> </div> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">III </span><span class="ltx_text ltx_font_smallcaps" id="S3.1.1">Modular Backstepping Control Design</span> </h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.1">The following assumptions presented in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib8" title="">8</a>]</cite> are given for the modular backstepping control design of the system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S2.E4" title="In II Problem Formulation ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">4</span></a>).</p> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.5"><span class="ltx_text ltx_font_italic" id="S3.p2.5.6">Assumption 1:</span> <span class="ltx_text" id="S3.p2.5.5" style="color:#000099;">There exists a certain constant <math alttext="{\rm c}_{\rm r}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S3.p2.1.1.m1.1"><semantics id="S3.p2.1.1.m1.1a"><mrow id="S3.p2.1.1.m1.1.1" xref="S3.p2.1.1.m1.1.1.cmml"><msub id="S3.p2.1.1.m1.1.1.2" xref="S3.p2.1.1.m1.1.1.2.cmml"><mi id="S3.p2.1.1.m1.1.1.2.2" mathcolor="#000099" mathvariant="normal" xref="S3.p2.1.1.m1.1.1.2.2.cmml">c</mi><mi id="S3.p2.1.1.m1.1.1.2.3" mathcolor="#000099" mathvariant="normal" xref="S3.p2.1.1.m1.1.1.2.3.cmml">r</mi></msub><mo id="S3.p2.1.1.m1.1.1.1" mathcolor="#000099" xref="S3.p2.1.1.m1.1.1.1.cmml">∈</mo><msup id="S3.p2.1.1.m1.1.1.3" xref="S3.p2.1.1.m1.1.1.3.cmml"><mi id="S3.p2.1.1.m1.1.1.3.2" mathcolor="#000099" xref="S3.p2.1.1.m1.1.1.3.2.cmml">ℝ</mi><mo id="S3.p2.1.1.m1.1.1.3.3" mathcolor="#000099" xref="S3.p2.1.1.m1.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.1.1.m1.1b"><apply id="S3.p2.1.1.m1.1.1.cmml" xref="S3.p2.1.1.m1.1.1"><in id="S3.p2.1.1.m1.1.1.1.cmml" xref="S3.p2.1.1.m1.1.1.1"></in><apply id="S3.p2.1.1.m1.1.1.2.cmml" xref="S3.p2.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.p2.1.1.m1.1.1.2.1.cmml" xref="S3.p2.1.1.m1.1.1.2">subscript</csymbol><ci id="S3.p2.1.1.m1.1.1.2.2.cmml" xref="S3.p2.1.1.m1.1.1.2.2">c</ci><ci id="S3.p2.1.1.m1.1.1.2.3.cmml" xref="S3.p2.1.1.m1.1.1.2.3">r</ci></apply><apply id="S3.p2.1.1.m1.1.1.3.cmml" xref="S3.p2.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.p2.1.1.m1.1.1.3.1.cmml" xref="S3.p2.1.1.m1.1.1.3">superscript</csymbol><ci id="S3.p2.1.1.m1.1.1.3.2.cmml" xref="S3.p2.1.1.m1.1.1.3.2">ℝ</ci><plus id="S3.p2.1.1.m1.1.1.3.3.cmml" xref="S3.p2.1.1.m1.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.1.1.m1.1c">{\rm c}_{\rm r}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.1.1.m1.1d">roman_c start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="y_{\rm r}" class="ltx_Math" display="inline" id="S3.p2.2.2.m2.1"><semantics id="S3.p2.2.2.m2.1a"><msub id="S3.p2.2.2.m2.1.1" xref="S3.p2.2.2.m2.1.1.cmml"><mi id="S3.p2.2.2.m2.1.1.2" mathcolor="#000099" xref="S3.p2.2.2.m2.1.1.2.cmml">y</mi><mi id="S3.p2.2.2.m2.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S3.p2.2.2.m2.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.2.2.m2.1b"><apply id="S3.p2.2.2.m2.1.1.cmml" xref="S3.p2.2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p2.2.2.m2.1.1.1.cmml" xref="S3.p2.2.2.m2.1.1">subscript</csymbol><ci id="S3.p2.2.2.m2.1.1.2.cmml" xref="S3.p2.2.2.m2.1.1.2">𝑦</ci><ci id="S3.p2.2.2.m2.1.1.3.cmml" xref="S3.p2.2.2.m2.1.1.3">r</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.2.2.m2.1c">y_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.2.2.m2.1d">italic_y start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\dot{y}_{\rm r}" class="ltx_Math" display="inline" id="S3.p2.3.3.m3.1"><semantics id="S3.p2.3.3.m3.1a"><msub id="S3.p2.3.3.m3.1.1" xref="S3.p2.3.3.m3.1.1.cmml"><mover accent="true" id="S3.p2.3.3.m3.1.1.2" xref="S3.p2.3.3.m3.1.1.2.cmml"><mi id="S3.p2.3.3.m3.1.1.2.2" mathcolor="#000099" xref="S3.p2.3.3.m3.1.1.2.2.cmml">y</mi><mo id="S3.p2.3.3.m3.1.1.2.1" mathcolor="#000099" xref="S3.p2.3.3.m3.1.1.2.1.cmml">˙</mo></mover><mi id="S3.p2.3.3.m3.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S3.p2.3.3.m3.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.3.3.m3.1b"><apply id="S3.p2.3.3.m3.1.1.cmml" xref="S3.p2.3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.p2.3.3.m3.1.1.1.cmml" xref="S3.p2.3.3.m3.1.1">subscript</csymbol><apply id="S3.p2.3.3.m3.1.1.2.cmml" xref="S3.p2.3.3.m3.1.1.2"><ci id="S3.p2.3.3.m3.1.1.2.1.cmml" xref="S3.p2.3.3.m3.1.1.2.1">˙</ci><ci id="S3.p2.3.3.m3.1.1.2.2.cmml" xref="S3.p2.3.3.m3.1.1.2.2">𝑦</ci></apply><ci id="S3.p2.3.3.m3.1.1.3.cmml" xref="S3.p2.3.3.m3.1.1.3">r</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.3.3.m3.1c">\dot{y}_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.3.3.m3.1d">over˙ start_ARG italic_y end_ARG start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\cdots" class="ltx_Math" display="inline" id="S3.p2.4.4.m4.1"><semantics id="S3.p2.4.4.m4.1a"><mi id="S3.p2.4.4.m4.1.1" mathcolor="#000099" mathvariant="normal" xref="S3.p2.4.4.m4.1.1.cmml">⋯</mi><annotation-xml encoding="MathML-Content" id="S3.p2.4.4.m4.1b"><ci id="S3.p2.4.4.m4.1.1.cmml" xref="S3.p2.4.4.m4.1.1">⋯</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.4.4.m4.1c">\cdots</annotation><annotation encoding="application/x-llamapun" id="S3.p2.4.4.m4.1d">⋯</annotation></semantics></math>, <math alttext="y^{(n-1)}_{\rm r}\in\Omega_{{\rm c}_{\rm r}}\subset\mathbb{R}" class="ltx_Math" display="inline" id="S3.p2.5.5.m5.1"><semantics id="S3.p2.5.5.m5.1a"><mrow id="S3.p2.5.5.m5.1.2" xref="S3.p2.5.5.m5.1.2.cmml"><msubsup id="S3.p2.5.5.m5.1.2.2" xref="S3.p2.5.5.m5.1.2.2.cmml"><mi id="S3.p2.5.5.m5.1.2.2.2.2" mathcolor="#000099" xref="S3.p2.5.5.m5.1.2.2.2.2.cmml">y</mi><mi id="S3.p2.5.5.m5.1.2.2.3" mathcolor="#000099" mathvariant="normal" xref="S3.p2.5.5.m5.1.2.2.3.cmml">r</mi><mrow id="S3.p2.5.5.m5.1.1.1.1" xref="S3.p2.5.5.m5.1.1.1.1.1.cmml"><mo id="S3.p2.5.5.m5.1.1.1.1.2" mathcolor="#000099" stretchy="false" xref="S3.p2.5.5.m5.1.1.1.1.1.cmml">(</mo><mrow id="S3.p2.5.5.m5.1.1.1.1.1" xref="S3.p2.5.5.m5.1.1.1.1.1.cmml"><mi id="S3.p2.5.5.m5.1.1.1.1.1.2" mathcolor="#000099" xref="S3.p2.5.5.m5.1.1.1.1.1.2.cmml">n</mi><mo id="S3.p2.5.5.m5.1.1.1.1.1.1" mathcolor="#000099" xref="S3.p2.5.5.m5.1.1.1.1.1.1.cmml">−</mo><mn id="S3.p2.5.5.m5.1.1.1.1.1.3" mathcolor="#000099" xref="S3.p2.5.5.m5.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.p2.5.5.m5.1.1.1.1.3" mathcolor="#000099" stretchy="false" xref="S3.p2.5.5.m5.1.1.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S3.p2.5.5.m5.1.2.3" mathcolor="#000099" xref="S3.p2.5.5.m5.1.2.3.cmml">∈</mo><msub id="S3.p2.5.5.m5.1.2.4" xref="S3.p2.5.5.m5.1.2.4.cmml"><mi id="S3.p2.5.5.m5.1.2.4.2" mathcolor="#000099" mathvariant="normal" xref="S3.p2.5.5.m5.1.2.4.2.cmml">Ω</mi><msub id="S3.p2.5.5.m5.1.2.4.3" xref="S3.p2.5.5.m5.1.2.4.3.cmml"><mi id="S3.p2.5.5.m5.1.2.4.3.2" mathcolor="#000099" mathvariant="normal" xref="S3.p2.5.5.m5.1.2.4.3.2.cmml">c</mi><mi id="S3.p2.5.5.m5.1.2.4.3.3" mathcolor="#000099" mathvariant="normal" xref="S3.p2.5.5.m5.1.2.4.3.3.cmml">r</mi></msub></msub><mo id="S3.p2.5.5.m5.1.2.5" mathcolor="#000099" xref="S3.p2.5.5.m5.1.2.5.cmml">⊂</mo><mi id="S3.p2.5.5.m5.1.2.6" mathcolor="#000099" xref="S3.p2.5.5.m5.1.2.6.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.5.5.m5.1b"><apply id="S3.p2.5.5.m5.1.2.cmml" xref="S3.p2.5.5.m5.1.2"><and id="S3.p2.5.5.m5.1.2a.cmml" xref="S3.p2.5.5.m5.1.2"></and><apply id="S3.p2.5.5.m5.1.2b.cmml" xref="S3.p2.5.5.m5.1.2"><in id="S3.p2.5.5.m5.1.2.3.cmml" xref="S3.p2.5.5.m5.1.2.3"></in><apply id="S3.p2.5.5.m5.1.2.2.cmml" xref="S3.p2.5.5.m5.1.2.2"><csymbol cd="ambiguous" id="S3.p2.5.5.m5.1.2.2.1.cmml" xref="S3.p2.5.5.m5.1.2.2">subscript</csymbol><apply id="S3.p2.5.5.m5.1.2.2.2.cmml" xref="S3.p2.5.5.m5.1.2.2"><csymbol cd="ambiguous" id="S3.p2.5.5.m5.1.2.2.2.1.cmml" xref="S3.p2.5.5.m5.1.2.2">superscript</csymbol><ci id="S3.p2.5.5.m5.1.2.2.2.2.cmml" xref="S3.p2.5.5.m5.1.2.2.2.2">𝑦</ci><apply id="S3.p2.5.5.m5.1.1.1.1.1.cmml" xref="S3.p2.5.5.m5.1.1.1.1"><minus id="S3.p2.5.5.m5.1.1.1.1.1.1.cmml" xref="S3.p2.5.5.m5.1.1.1.1.1.1"></minus><ci id="S3.p2.5.5.m5.1.1.1.1.1.2.cmml" xref="S3.p2.5.5.m5.1.1.1.1.1.2">𝑛</ci><cn id="S3.p2.5.5.m5.1.1.1.1.1.3.cmml" type="integer" xref="S3.p2.5.5.m5.1.1.1.1.1.3">1</cn></apply></apply><ci id="S3.p2.5.5.m5.1.2.2.3.cmml" xref="S3.p2.5.5.m5.1.2.2.3">r</ci></apply><apply id="S3.p2.5.5.m5.1.2.4.cmml" xref="S3.p2.5.5.m5.1.2.4"><csymbol cd="ambiguous" id="S3.p2.5.5.m5.1.2.4.1.cmml" xref="S3.p2.5.5.m5.1.2.4">subscript</csymbol><ci id="S3.p2.5.5.m5.1.2.4.2.cmml" xref="S3.p2.5.5.m5.1.2.4.2">Ω</ci><apply id="S3.p2.5.5.m5.1.2.4.3.cmml" xref="S3.p2.5.5.m5.1.2.4.3"><csymbol cd="ambiguous" id="S3.p2.5.5.m5.1.2.4.3.1.cmml" xref="S3.p2.5.5.m5.1.2.4.3">subscript</csymbol><ci id="S3.p2.5.5.m5.1.2.4.3.2.cmml" xref="S3.p2.5.5.m5.1.2.4.3.2">c</ci><ci id="S3.p2.5.5.m5.1.2.4.3.3.cmml" xref="S3.p2.5.5.m5.1.2.4.3.3">r</ci></apply></apply></apply><apply id="S3.p2.5.5.m5.1.2c.cmml" xref="S3.p2.5.5.m5.1.2"><subset id="S3.p2.5.5.m5.1.2.5.cmml" xref="S3.p2.5.5.m5.1.2.5"></subset><share href="https://arxiv.org/html/2401.10785v2#S3.p2.5.5.m5.1.2.4.cmml" id="S3.p2.5.5.m5.1.2d.cmml" xref="S3.p2.5.5.m5.1.2"></share><ci id="S3.p2.5.5.m5.1.2.6.cmml" xref="S3.p2.5.5.m5.1.2.6">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.5.5.m5.1c">y^{(n-1)}_{\rm r}\in\Omega_{{\rm c}_{\rm r}}\subset\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.5.5.m5.1d">italic_y start_POSTSUPERSCRIPT ( italic_n - 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT ∈ roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊂ blackboard_R</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S3.p3"> <p class="ltx_p" id="S3.p3.5"><span class="ltx_text ltx_font_italic" id="S3.p3.5.5">Assumption 2:</span> <math alttext="\beta(\bm{x})\neq 0" class="ltx_Math" display="inline" id="S3.p3.1.m1.1"><semantics id="S3.p3.1.m1.1a"><mrow id="S3.p3.1.m1.1.2" xref="S3.p3.1.m1.1.2.cmml"><mrow id="S3.p3.1.m1.1.2.2" xref="S3.p3.1.m1.1.2.2.cmml"><mi id="S3.p3.1.m1.1.2.2.2" mathcolor="#000099" xref="S3.p3.1.m1.1.2.2.2.cmml">β</mi><mo id="S3.p3.1.m1.1.2.2.1" xref="S3.p3.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S3.p3.1.m1.1.2.2.3.2" xref="S3.p3.1.m1.1.2.2.cmml"><mo id="S3.p3.1.m1.1.2.2.3.2.1" mathcolor="#000099" stretchy="false" xref="S3.p3.1.m1.1.2.2.cmml">(</mo><mi id="S3.p3.1.m1.1.1" mathcolor="#000099" xref="S3.p3.1.m1.1.1.cmml">𝒙</mi><mo id="S3.p3.1.m1.1.2.2.3.2.2" mathcolor="#000099" stretchy="false" xref="S3.p3.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.p3.1.m1.1.2.1" mathcolor="#000099" xref="S3.p3.1.m1.1.2.1.cmml">≠</mo><mn id="S3.p3.1.m1.1.2.3" mathcolor="#000099" xref="S3.p3.1.m1.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.1.m1.1b"><apply id="S3.p3.1.m1.1.2.cmml" xref="S3.p3.1.m1.1.2"><neq id="S3.p3.1.m1.1.2.1.cmml" xref="S3.p3.1.m1.1.2.1"></neq><apply id="S3.p3.1.m1.1.2.2.cmml" xref="S3.p3.1.m1.1.2.2"><times id="S3.p3.1.m1.1.2.2.1.cmml" xref="S3.p3.1.m1.1.2.2.1"></times><ci id="S3.p3.1.m1.1.2.2.2.cmml" xref="S3.p3.1.m1.1.2.2.2">𝛽</ci><ci id="S3.p3.1.m1.1.1.cmml" xref="S3.p3.1.m1.1.1">𝒙</ci></apply><cn id="S3.p3.1.m1.1.2.3.cmml" type="integer" xref="S3.p3.1.m1.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.1.m1.1c">\beta(\bm{x})\neq 0</annotation><annotation encoding="application/x-llamapun" id="S3.p3.1.m1.1d">italic_β ( bold_italic_x ) ≠ 0</annotation></semantics></math><span class="ltx_text" id="S3.p3.5.4" style="color:#000099;">, <math alttext="\forall\bm{x}\in\mathbb{R}^{n}" class="ltx_Math" display="inline" id="S3.p3.2.1.m1.1"><semantics id="S3.p3.2.1.m1.1a"><mrow id="S3.p3.2.1.m1.1.1" xref="S3.p3.2.1.m1.1.1.cmml"><mrow id="S3.p3.2.1.m1.1.1.2" xref="S3.p3.2.1.m1.1.1.2.cmml"><mo id="S3.p3.2.1.m1.1.1.2.1" mathcolor="#000099" rspace="0.167em" xref="S3.p3.2.1.m1.1.1.2.1.cmml">∀</mo><mi id="S3.p3.2.1.m1.1.1.2.2" mathcolor="#000099" xref="S3.p3.2.1.m1.1.1.2.2.cmml">𝒙</mi></mrow><mo id="S3.p3.2.1.m1.1.1.1" mathcolor="#000099" xref="S3.p3.2.1.m1.1.1.1.cmml">∈</mo><msup id="S3.p3.2.1.m1.1.1.3" xref="S3.p3.2.1.m1.1.1.3.cmml"><mi id="S3.p3.2.1.m1.1.1.3.2" mathcolor="#000099" xref="S3.p3.2.1.m1.1.1.3.2.cmml">ℝ</mi><mi id="S3.p3.2.1.m1.1.1.3.3" mathcolor="#000099" xref="S3.p3.2.1.m1.1.1.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.2.1.m1.1b"><apply id="S3.p3.2.1.m1.1.1.cmml" xref="S3.p3.2.1.m1.1.1"><in id="S3.p3.2.1.m1.1.1.1.cmml" xref="S3.p3.2.1.m1.1.1.1"></in><apply id="S3.p3.2.1.m1.1.1.2.cmml" xref="S3.p3.2.1.m1.1.1.2"><csymbol cd="latexml" id="S3.p3.2.1.m1.1.1.2.1.cmml" xref="S3.p3.2.1.m1.1.1.2.1">for-all</csymbol><ci id="S3.p3.2.1.m1.1.1.2.2.cmml" xref="S3.p3.2.1.m1.1.1.2.2">𝒙</ci></apply><apply id="S3.p3.2.1.m1.1.1.3.cmml" xref="S3.p3.2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.p3.2.1.m1.1.1.3.1.cmml" xref="S3.p3.2.1.m1.1.1.3">superscript</csymbol><ci id="S3.p3.2.1.m1.1.1.3.2.cmml" xref="S3.p3.2.1.m1.1.1.3.2">ℝ</ci><ci id="S3.p3.2.1.m1.1.1.3.3.cmml" xref="S3.p3.2.1.m1.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.2.1.m1.1c">\forall\bm{x}\in\mathbb{R}^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.2.1.m1.1d">∀ bold_italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\bm{\varphi}_{i}\in\mathcal{C}^{n-i}" class="ltx_Math" display="inline" id="S3.p3.3.2.m2.1"><semantics id="S3.p3.3.2.m2.1a"><mrow id="S3.p3.3.2.m2.1.1" xref="S3.p3.3.2.m2.1.1.cmml"><msub id="S3.p3.3.2.m2.1.1.2" xref="S3.p3.3.2.m2.1.1.2.cmml"><mi id="S3.p3.3.2.m2.1.1.2.2" mathcolor="#000099" xref="S3.p3.3.2.m2.1.1.2.2.cmml">𝝋</mi><mi id="S3.p3.3.2.m2.1.1.2.3" mathcolor="#000099" xref="S3.p3.3.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S3.p3.3.2.m2.1.1.1" mathcolor="#000099" xref="S3.p3.3.2.m2.1.1.1.cmml">∈</mo><msup id="S3.p3.3.2.m2.1.1.3" xref="S3.p3.3.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p3.3.2.m2.1.1.3.2" mathcolor="#000099" xref="S3.p3.3.2.m2.1.1.3.2.cmml">𝒞</mi><mrow id="S3.p3.3.2.m2.1.1.3.3" xref="S3.p3.3.2.m2.1.1.3.3.cmml"><mi id="S3.p3.3.2.m2.1.1.3.3.2" mathcolor="#000099" xref="S3.p3.3.2.m2.1.1.3.3.2.cmml">n</mi><mo id="S3.p3.3.2.m2.1.1.3.3.1" mathcolor="#000099" xref="S3.p3.3.2.m2.1.1.3.3.1.cmml">−</mo><mi id="S3.p3.3.2.m2.1.1.3.3.3" mathcolor="#000099" xref="S3.p3.3.2.m2.1.1.3.3.3.cmml">i</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.3.2.m2.1b"><apply id="S3.p3.3.2.m2.1.1.cmml" xref="S3.p3.3.2.m2.1.1"><in id="S3.p3.3.2.m2.1.1.1.cmml" xref="S3.p3.3.2.m2.1.1.1"></in><apply id="S3.p3.3.2.m2.1.1.2.cmml" xref="S3.p3.3.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.p3.3.2.m2.1.1.2.1.cmml" xref="S3.p3.3.2.m2.1.1.2">subscript</csymbol><ci id="S3.p3.3.2.m2.1.1.2.2.cmml" xref="S3.p3.3.2.m2.1.1.2.2">𝝋</ci><ci id="S3.p3.3.2.m2.1.1.2.3.cmml" xref="S3.p3.3.2.m2.1.1.2.3">𝑖</ci></apply><apply id="S3.p3.3.2.m2.1.1.3.cmml" xref="S3.p3.3.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.p3.3.2.m2.1.1.3.1.cmml" xref="S3.p3.3.2.m2.1.1.3">superscript</csymbol><ci id="S3.p3.3.2.m2.1.1.3.2.cmml" xref="S3.p3.3.2.m2.1.1.3.2">𝒞</ci><apply id="S3.p3.3.2.m2.1.1.3.3.cmml" xref="S3.p3.3.2.m2.1.1.3.3"><minus id="S3.p3.3.2.m2.1.1.3.3.1.cmml" xref="S3.p3.3.2.m2.1.1.3.3.1"></minus><ci id="S3.p3.3.2.m2.1.1.3.3.2.cmml" xref="S3.p3.3.2.m2.1.1.3.3.2">𝑛</ci><ci id="S3.p3.3.2.m2.1.1.3.3.3.cmml" xref="S3.p3.3.2.m2.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.3.2.m2.1c">\bm{\varphi}_{i}\in\mathcal{C}^{n-i}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.3.2.m2.1d">bold_italic_φ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_C start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> for <math alttext="i=1" class="ltx_Math" display="inline" id="S3.p3.4.3.m3.1"><semantics id="S3.p3.4.3.m3.1a"><mrow id="S3.p3.4.3.m3.1.1" xref="S3.p3.4.3.m3.1.1.cmml"><mi id="S3.p3.4.3.m3.1.1.2" mathcolor="#000099" xref="S3.p3.4.3.m3.1.1.2.cmml">i</mi><mo id="S3.p3.4.3.m3.1.1.1" mathcolor="#000099" xref="S3.p3.4.3.m3.1.1.1.cmml">=</mo><mn id="S3.p3.4.3.m3.1.1.3" mathcolor="#000099" xref="S3.p3.4.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.4.3.m3.1b"><apply id="S3.p3.4.3.m3.1.1.cmml" xref="S3.p3.4.3.m3.1.1"><eq id="S3.p3.4.3.m3.1.1.1.cmml" xref="S3.p3.4.3.m3.1.1.1"></eq><ci id="S3.p3.4.3.m3.1.1.2.cmml" xref="S3.p3.4.3.m3.1.1.2">𝑖</ci><cn id="S3.p3.4.3.m3.1.1.3.cmml" type="integer" xref="S3.p3.4.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.4.3.m3.1c">i=1</annotation><annotation encoding="application/x-llamapun" id="S3.p3.4.3.m3.1d">italic_i = 1</annotation></semantics></math> to <math alttext="n" class="ltx_Math" display="inline" id="S3.p3.5.4.m4.1"><semantics id="S3.p3.5.4.m4.1a"><mi id="S3.p3.5.4.m4.1.1" mathcolor="#000099" xref="S3.p3.5.4.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.p3.5.4.m4.1b"><ci id="S3.p3.5.4.m4.1.1.cmml" xref="S3.p3.5.4.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.5.4.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.p3.5.4.m4.1d">italic_n</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S3.p4"> <p class="ltx_p" id="S3.p4.1">The virtual control inputs <math alttext="v_{1}(t),v_{i}(t)\in\mathbb{R}" class="ltx_Math" display="inline" id="S3.p4.1.m1.4"><semantics id="S3.p4.1.m1.4a"><mrow id="S3.p4.1.m1.4.4" xref="S3.p4.1.m1.4.4.cmml"><mrow id="S3.p4.1.m1.4.4.2.2" xref="S3.p4.1.m1.4.4.2.3.cmml"><mrow id="S3.p4.1.m1.3.3.1.1.1" xref="S3.p4.1.m1.3.3.1.1.1.cmml"><msub id="S3.p4.1.m1.3.3.1.1.1.2" xref="S3.p4.1.m1.3.3.1.1.1.2.cmml"><mi id="S3.p4.1.m1.3.3.1.1.1.2.2" xref="S3.p4.1.m1.3.3.1.1.1.2.2.cmml">v</mi><mn id="S3.p4.1.m1.3.3.1.1.1.2.3" xref="S3.p4.1.m1.3.3.1.1.1.2.3.cmml">1</mn></msub><mo id="S3.p4.1.m1.3.3.1.1.1.1" xref="S3.p4.1.m1.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S3.p4.1.m1.3.3.1.1.1.3.2" xref="S3.p4.1.m1.3.3.1.1.1.cmml"><mo id="S3.p4.1.m1.3.3.1.1.1.3.2.1" stretchy="false" xref="S3.p4.1.m1.3.3.1.1.1.cmml">(</mo><mi id="S3.p4.1.m1.1.1" xref="S3.p4.1.m1.1.1.cmml">t</mi><mo id="S3.p4.1.m1.3.3.1.1.1.3.2.2" stretchy="false" xref="S3.p4.1.m1.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.p4.1.m1.4.4.2.2.3" xref="S3.p4.1.m1.4.4.2.3.cmml">,</mo><mrow id="S3.p4.1.m1.4.4.2.2.2" xref="S3.p4.1.m1.4.4.2.2.2.cmml"><msub id="S3.p4.1.m1.4.4.2.2.2.2" xref="S3.p4.1.m1.4.4.2.2.2.2.cmml"><mi id="S3.p4.1.m1.4.4.2.2.2.2.2" xref="S3.p4.1.m1.4.4.2.2.2.2.2.cmml">v</mi><mi id="S3.p4.1.m1.4.4.2.2.2.2.3" xref="S3.p4.1.m1.4.4.2.2.2.2.3.cmml">i</mi></msub><mo id="S3.p4.1.m1.4.4.2.2.2.1" xref="S3.p4.1.m1.4.4.2.2.2.1.cmml">⁢</mo><mrow id="S3.p4.1.m1.4.4.2.2.2.3.2" xref="S3.p4.1.m1.4.4.2.2.2.cmml"><mo id="S3.p4.1.m1.4.4.2.2.2.3.2.1" stretchy="false" xref="S3.p4.1.m1.4.4.2.2.2.cmml">(</mo><mi id="S3.p4.1.m1.2.2" xref="S3.p4.1.m1.2.2.cmml">t</mi><mo id="S3.p4.1.m1.4.4.2.2.2.3.2.2" stretchy="false" xref="S3.p4.1.m1.4.4.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S3.p4.1.m1.4.4.3" xref="S3.p4.1.m1.4.4.3.cmml">∈</mo><mi id="S3.p4.1.m1.4.4.4" xref="S3.p4.1.m1.4.4.4.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.1.m1.4b"><apply id="S3.p4.1.m1.4.4.cmml" xref="S3.p4.1.m1.4.4"><in id="S3.p4.1.m1.4.4.3.cmml" xref="S3.p4.1.m1.4.4.3"></in><list id="S3.p4.1.m1.4.4.2.3.cmml" xref="S3.p4.1.m1.4.4.2.2"><apply id="S3.p4.1.m1.3.3.1.1.1.cmml" xref="S3.p4.1.m1.3.3.1.1.1"><times id="S3.p4.1.m1.3.3.1.1.1.1.cmml" xref="S3.p4.1.m1.3.3.1.1.1.1"></times><apply id="S3.p4.1.m1.3.3.1.1.1.2.cmml" xref="S3.p4.1.m1.3.3.1.1.1.2"><csymbol cd="ambiguous" id="S3.p4.1.m1.3.3.1.1.1.2.1.cmml" xref="S3.p4.1.m1.3.3.1.1.1.2">subscript</csymbol><ci id="S3.p4.1.m1.3.3.1.1.1.2.2.cmml" xref="S3.p4.1.m1.3.3.1.1.1.2.2">𝑣</ci><cn id="S3.p4.1.m1.3.3.1.1.1.2.3.cmml" type="integer" xref="S3.p4.1.m1.3.3.1.1.1.2.3">1</cn></apply><ci id="S3.p4.1.m1.1.1.cmml" xref="S3.p4.1.m1.1.1">𝑡</ci></apply><apply id="S3.p4.1.m1.4.4.2.2.2.cmml" xref="S3.p4.1.m1.4.4.2.2.2"><times id="S3.p4.1.m1.4.4.2.2.2.1.cmml" xref="S3.p4.1.m1.4.4.2.2.2.1"></times><apply id="S3.p4.1.m1.4.4.2.2.2.2.cmml" xref="S3.p4.1.m1.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S3.p4.1.m1.4.4.2.2.2.2.1.cmml" xref="S3.p4.1.m1.4.4.2.2.2.2">subscript</csymbol><ci id="S3.p4.1.m1.4.4.2.2.2.2.2.cmml" xref="S3.p4.1.m1.4.4.2.2.2.2.2">𝑣</ci><ci id="S3.p4.1.m1.4.4.2.2.2.2.3.cmml" xref="S3.p4.1.m1.4.4.2.2.2.2.3">𝑖</ci></apply><ci id="S3.p4.1.m1.2.2.cmml" xref="S3.p4.1.m1.2.2">𝑡</ci></apply></list><ci id="S3.p4.1.m1.4.4.4.cmml" xref="S3.p4.1.m1.4.4.4">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.1.m1.4c">v_{1}(t),v_{i}(t)\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.1.m1.4d">italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) , italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) ∈ blackboard_R</annotation></semantics></math> in the modular backstepping approach are recursively given by <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib8" title="">8</a>]</cite></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx2"> <tbody id="S3.Ex4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle v_{1}(x_{1},\hat{\bm{\theta}},y_{\rm r})=-k_{{\rm c}1}e_{1}-\bm{% \psi}_{1}^{T}\hat{\bm{\theta}}," class="ltx_Math" display="inline" id="S3.Ex4.m2.2"><semantics id="S3.Ex4.m2.2a"><mrow id="S3.Ex4.m2.2.2.1" xref="S3.Ex4.m2.2.2.1.1.cmml"><mrow id="S3.Ex4.m2.2.2.1.1" xref="S3.Ex4.m2.2.2.1.1.cmml"><mrow id="S3.Ex4.m2.2.2.1.1.2" xref="S3.Ex4.m2.2.2.1.1.2.cmml"><msub id="S3.Ex4.m2.2.2.1.1.2.4" xref="S3.Ex4.m2.2.2.1.1.2.4.cmml"><mi id="S3.Ex4.m2.2.2.1.1.2.4.2" xref="S3.Ex4.m2.2.2.1.1.2.4.2.cmml">v</mi><mn id="S3.Ex4.m2.2.2.1.1.2.4.3" xref="S3.Ex4.m2.2.2.1.1.2.4.3.cmml">1</mn></msub><mo id="S3.Ex4.m2.2.2.1.1.2.3" xref="S3.Ex4.m2.2.2.1.1.2.3.cmml">⁢</mo><mrow id="S3.Ex4.m2.2.2.1.1.2.2.2" xref="S3.Ex4.m2.2.2.1.1.2.2.3.cmml"><mo id="S3.Ex4.m2.2.2.1.1.2.2.2.3" stretchy="false" xref="S3.Ex4.m2.2.2.1.1.2.2.3.cmml">(</mo><msub id="S3.Ex4.m2.2.2.1.1.1.1.1.1" xref="S3.Ex4.m2.2.2.1.1.1.1.1.1.cmml"><mi id="S3.Ex4.m2.2.2.1.1.1.1.1.1.2" xref="S3.Ex4.m2.2.2.1.1.1.1.1.1.2.cmml">x</mi><mn id="S3.Ex4.m2.2.2.1.1.1.1.1.1.3" xref="S3.Ex4.m2.2.2.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.Ex4.m2.2.2.1.1.2.2.2.4" xref="S3.Ex4.m2.2.2.1.1.2.2.3.cmml">,</mo><mover accent="true" id="S3.Ex4.m2.1.1" xref="S3.Ex4.m2.1.1.cmml"><mi id="S3.Ex4.m2.1.1.2" xref="S3.Ex4.m2.1.1.2.cmml">𝜽</mi><mo id="S3.Ex4.m2.1.1.1" xref="S3.Ex4.m2.1.1.1.cmml">^</mo></mover><mo id="S3.Ex4.m2.2.2.1.1.2.2.2.5" xref="S3.Ex4.m2.2.2.1.1.2.2.3.cmml">,</mo><msub id="S3.Ex4.m2.2.2.1.1.2.2.2.2" xref="S3.Ex4.m2.2.2.1.1.2.2.2.2.cmml"><mi id="S3.Ex4.m2.2.2.1.1.2.2.2.2.2" xref="S3.Ex4.m2.2.2.1.1.2.2.2.2.2.cmml">y</mi><mi id="S3.Ex4.m2.2.2.1.1.2.2.2.2.3" mathvariant="normal" xref="S3.Ex4.m2.2.2.1.1.2.2.2.2.3.cmml">r</mi></msub><mo id="S3.Ex4.m2.2.2.1.1.2.2.2.6" stretchy="false" xref="S3.Ex4.m2.2.2.1.1.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.Ex4.m2.2.2.1.1.3" xref="S3.Ex4.m2.2.2.1.1.3.cmml">=</mo><mrow id="S3.Ex4.m2.2.2.1.1.4" xref="S3.Ex4.m2.2.2.1.1.4.cmml"><mrow id="S3.Ex4.m2.2.2.1.1.4.2" xref="S3.Ex4.m2.2.2.1.1.4.2.cmml"><mo id="S3.Ex4.m2.2.2.1.1.4.2a" xref="S3.Ex4.m2.2.2.1.1.4.2.cmml">−</mo><mrow id="S3.Ex4.m2.2.2.1.1.4.2.2" xref="S3.Ex4.m2.2.2.1.1.4.2.2.cmml"><msub id="S3.Ex4.m2.2.2.1.1.4.2.2.2" xref="S3.Ex4.m2.2.2.1.1.4.2.2.2.cmml"><mi id="S3.Ex4.m2.2.2.1.1.4.2.2.2.2" xref="S3.Ex4.m2.2.2.1.1.4.2.2.2.2.cmml">k</mi><mi id="S3.Ex4.m2.2.2.1.1.4.2.2.2.3" xref="S3.Ex4.m2.2.2.1.1.4.2.2.2.3.cmml">c1</mi></msub><mo id="S3.Ex4.m2.2.2.1.1.4.2.2.1" xref="S3.Ex4.m2.2.2.1.1.4.2.2.1.cmml">⁢</mo><msub id="S3.Ex4.m2.2.2.1.1.4.2.2.3" xref="S3.Ex4.m2.2.2.1.1.4.2.2.3.cmml"><mi id="S3.Ex4.m2.2.2.1.1.4.2.2.3.2" xref="S3.Ex4.m2.2.2.1.1.4.2.2.3.2.cmml">e</mi><mn id="S3.Ex4.m2.2.2.1.1.4.2.2.3.3" xref="S3.Ex4.m2.2.2.1.1.4.2.2.3.3.cmml">1</mn></msub></mrow></mrow><mo id="S3.Ex4.m2.2.2.1.1.4.1" xref="S3.Ex4.m2.2.2.1.1.4.1.cmml">−</mo><mrow id="S3.Ex4.m2.2.2.1.1.4.3" xref="S3.Ex4.m2.2.2.1.1.4.3.cmml"><msubsup id="S3.Ex4.m2.2.2.1.1.4.3.2" xref="S3.Ex4.m2.2.2.1.1.4.3.2.cmml"><mi id="S3.Ex4.m2.2.2.1.1.4.3.2.2.2" xref="S3.Ex4.m2.2.2.1.1.4.3.2.2.2.cmml">𝝍</mi><mn id="S3.Ex4.m2.2.2.1.1.4.3.2.2.3" xref="S3.Ex4.m2.2.2.1.1.4.3.2.2.3.cmml">1</mn><mi id="S3.Ex4.m2.2.2.1.1.4.3.2.3" xref="S3.Ex4.m2.2.2.1.1.4.3.2.3.cmml">T</mi></msubsup><mo id="S3.Ex4.m2.2.2.1.1.4.3.1" xref="S3.Ex4.m2.2.2.1.1.4.3.1.cmml">⁢</mo><mover accent="true" id="S3.Ex4.m2.2.2.1.1.4.3.3" xref="S3.Ex4.m2.2.2.1.1.4.3.3.cmml"><mi id="S3.Ex4.m2.2.2.1.1.4.3.3.2" xref="S3.Ex4.m2.2.2.1.1.4.3.3.2.cmml">𝜽</mi><mo id="S3.Ex4.m2.2.2.1.1.4.3.3.1" xref="S3.Ex4.m2.2.2.1.1.4.3.3.1.cmml">^</mo></mover></mrow></mrow></mrow><mo id="S3.Ex4.m2.2.2.1.2" xref="S3.Ex4.m2.2.2.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex4.m2.2b"><apply id="S3.Ex4.m2.2.2.1.1.cmml" xref="S3.Ex4.m2.2.2.1"><eq id="S3.Ex4.m2.2.2.1.1.3.cmml" xref="S3.Ex4.m2.2.2.1.1.3"></eq><apply id="S3.Ex4.m2.2.2.1.1.2.cmml" xref="S3.Ex4.m2.2.2.1.1.2"><times id="S3.Ex4.m2.2.2.1.1.2.3.cmml" xref="S3.Ex4.m2.2.2.1.1.2.3"></times><apply id="S3.Ex4.m2.2.2.1.1.2.4.cmml" xref="S3.Ex4.m2.2.2.1.1.2.4"><csymbol cd="ambiguous" id="S3.Ex4.m2.2.2.1.1.2.4.1.cmml" xref="S3.Ex4.m2.2.2.1.1.2.4">subscript</csymbol><ci id="S3.Ex4.m2.2.2.1.1.2.4.2.cmml" xref="S3.Ex4.m2.2.2.1.1.2.4.2">𝑣</ci><cn id="S3.Ex4.m2.2.2.1.1.2.4.3.cmml" type="integer" xref="S3.Ex4.m2.2.2.1.1.2.4.3">1</cn></apply><vector id="S3.Ex4.m2.2.2.1.1.2.2.3.cmml" xref="S3.Ex4.m2.2.2.1.1.2.2.2"><apply id="S3.Ex4.m2.2.2.1.1.1.1.1.1.cmml" xref="S3.Ex4.m2.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex4.m2.2.2.1.1.1.1.1.1.1.cmml" xref="S3.Ex4.m2.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S3.Ex4.m2.2.2.1.1.1.1.1.1.2.cmml" xref="S3.Ex4.m2.2.2.1.1.1.1.1.1.2">𝑥</ci><cn id="S3.Ex4.m2.2.2.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.Ex4.m2.2.2.1.1.1.1.1.1.3">1</cn></apply><apply id="S3.Ex4.m2.1.1.cmml" xref="S3.Ex4.m2.1.1"><ci id="S3.Ex4.m2.1.1.1.cmml" xref="S3.Ex4.m2.1.1.1">^</ci><ci id="S3.Ex4.m2.1.1.2.cmml" xref="S3.Ex4.m2.1.1.2">𝜽</ci></apply><apply id="S3.Ex4.m2.2.2.1.1.2.2.2.2.cmml" xref="S3.Ex4.m2.2.2.1.1.2.2.2.2"><csymbol cd="ambiguous" id="S3.Ex4.m2.2.2.1.1.2.2.2.2.1.cmml" xref="S3.Ex4.m2.2.2.1.1.2.2.2.2">subscript</csymbol><ci id="S3.Ex4.m2.2.2.1.1.2.2.2.2.2.cmml" xref="S3.Ex4.m2.2.2.1.1.2.2.2.2.2">𝑦</ci><ci id="S3.Ex4.m2.2.2.1.1.2.2.2.2.3.cmml" xref="S3.Ex4.m2.2.2.1.1.2.2.2.2.3">r</ci></apply></vector></apply><apply id="S3.Ex4.m2.2.2.1.1.4.cmml" xref="S3.Ex4.m2.2.2.1.1.4"><minus id="S3.Ex4.m2.2.2.1.1.4.1.cmml" xref="S3.Ex4.m2.2.2.1.1.4.1"></minus><apply id="S3.Ex4.m2.2.2.1.1.4.2.cmml" xref="S3.Ex4.m2.2.2.1.1.4.2"><minus id="S3.Ex4.m2.2.2.1.1.4.2.1.cmml" xref="S3.Ex4.m2.2.2.1.1.4.2"></minus><apply id="S3.Ex4.m2.2.2.1.1.4.2.2.cmml" xref="S3.Ex4.m2.2.2.1.1.4.2.2"><times id="S3.Ex4.m2.2.2.1.1.4.2.2.1.cmml" xref="S3.Ex4.m2.2.2.1.1.4.2.2.1"></times><apply id="S3.Ex4.m2.2.2.1.1.4.2.2.2.cmml" xref="S3.Ex4.m2.2.2.1.1.4.2.2.2"><csymbol cd="ambiguous" id="S3.Ex4.m2.2.2.1.1.4.2.2.2.1.cmml" xref="S3.Ex4.m2.2.2.1.1.4.2.2.2">subscript</csymbol><ci id="S3.Ex4.m2.2.2.1.1.4.2.2.2.2.cmml" xref="S3.Ex4.m2.2.2.1.1.4.2.2.2.2">𝑘</ci><ci id="S3.Ex4.m2.2.2.1.1.4.2.2.2.3.cmml" xref="S3.Ex4.m2.2.2.1.1.4.2.2.2.3">c1</ci></apply><apply id="S3.Ex4.m2.2.2.1.1.4.2.2.3.cmml" xref="S3.Ex4.m2.2.2.1.1.4.2.2.3"><csymbol cd="ambiguous" id="S3.Ex4.m2.2.2.1.1.4.2.2.3.1.cmml" xref="S3.Ex4.m2.2.2.1.1.4.2.2.3">subscript</csymbol><ci id="S3.Ex4.m2.2.2.1.1.4.2.2.3.2.cmml" xref="S3.Ex4.m2.2.2.1.1.4.2.2.3.2">𝑒</ci><cn id="S3.Ex4.m2.2.2.1.1.4.2.2.3.3.cmml" type="integer" xref="S3.Ex4.m2.2.2.1.1.4.2.2.3.3">1</cn></apply></apply></apply><apply id="S3.Ex4.m2.2.2.1.1.4.3.cmml" xref="S3.Ex4.m2.2.2.1.1.4.3"><times id="S3.Ex4.m2.2.2.1.1.4.3.1.cmml" xref="S3.Ex4.m2.2.2.1.1.4.3.1"></times><apply id="S3.Ex4.m2.2.2.1.1.4.3.2.cmml" xref="S3.Ex4.m2.2.2.1.1.4.3.2"><csymbol cd="ambiguous" id="S3.Ex4.m2.2.2.1.1.4.3.2.1.cmml" xref="S3.Ex4.m2.2.2.1.1.4.3.2">superscript</csymbol><apply id="S3.Ex4.m2.2.2.1.1.4.3.2.2.cmml" xref="S3.Ex4.m2.2.2.1.1.4.3.2"><csymbol cd="ambiguous" id="S3.Ex4.m2.2.2.1.1.4.3.2.2.1.cmml" xref="S3.Ex4.m2.2.2.1.1.4.3.2">subscript</csymbol><ci id="S3.Ex4.m2.2.2.1.1.4.3.2.2.2.cmml" xref="S3.Ex4.m2.2.2.1.1.4.3.2.2.2">𝝍</ci><cn id="S3.Ex4.m2.2.2.1.1.4.3.2.2.3.cmml" type="integer" xref="S3.Ex4.m2.2.2.1.1.4.3.2.2.3">1</cn></apply><ci id="S3.Ex4.m2.2.2.1.1.4.3.2.3.cmml" xref="S3.Ex4.m2.2.2.1.1.4.3.2.3">𝑇</ci></apply><apply id="S3.Ex4.m2.2.2.1.1.4.3.3.cmml" xref="S3.Ex4.m2.2.2.1.1.4.3.3"><ci id="S3.Ex4.m2.2.2.1.1.4.3.3.1.cmml" xref="S3.Ex4.m2.2.2.1.1.4.3.3.1">^</ci><ci id="S3.Ex4.m2.2.2.1.1.4.3.3.2.cmml" xref="S3.Ex4.m2.2.2.1.1.4.3.3.2">𝜽</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex4.m2.2c">\displaystyle v_{1}(x_{1},\hat{\bm{\theta}},y_{\rm r})=-k_{{\rm c}1}e_{1}-\bm{% \psi}_{1}^{T}\hat{\bm{\theta}},</annotation><annotation encoding="application/x-llamapun" id="S3.Ex4.m2.2d">italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , over^ start_ARG bold_italic_θ end_ARG , italic_y start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT ) = - italic_k start_POSTSUBSCRIPT c1 end_POSTSUBSCRIPT italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - bold_italic_ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over^ start_ARG bold_italic_θ end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle v_{i}({\bm{x}}_{i},\Theta_{i-1},{\bm{y}}_{{\rm r}i})=-k_{{\rm c}% i}e_{i}-e_{i-1}-\bm{\psi}_{i}^{T}\hat{\bm{\theta}}" class="ltx_Math" display="inline" id="S3.Ex5.m1.3"><semantics id="S3.Ex5.m1.3a"><mrow id="S3.Ex5.m1.3.3" xref="S3.Ex5.m1.3.3.cmml"><mrow id="S3.Ex5.m1.3.3.3" xref="S3.Ex5.m1.3.3.3.cmml"><msub id="S3.Ex5.m1.3.3.3.5" xref="S3.Ex5.m1.3.3.3.5.cmml"><mi id="S3.Ex5.m1.3.3.3.5.2" xref="S3.Ex5.m1.3.3.3.5.2.cmml">v</mi><mi id="S3.Ex5.m1.3.3.3.5.3" xref="S3.Ex5.m1.3.3.3.5.3.cmml">i</mi></msub><mo id="S3.Ex5.m1.3.3.3.4" xref="S3.Ex5.m1.3.3.3.4.cmml">⁢</mo><mrow id="S3.Ex5.m1.3.3.3.3.3" xref="S3.Ex5.m1.3.3.3.3.4.cmml"><mo id="S3.Ex5.m1.3.3.3.3.3.4" stretchy="false" xref="S3.Ex5.m1.3.3.3.3.4.cmml">(</mo><msub id="S3.Ex5.m1.1.1.1.1.1.1" xref="S3.Ex5.m1.1.1.1.1.1.1.cmml"><mi id="S3.Ex5.m1.1.1.1.1.1.1.2" xref="S3.Ex5.m1.1.1.1.1.1.1.2.cmml">𝒙</mi><mi id="S3.Ex5.m1.1.1.1.1.1.1.3" xref="S3.Ex5.m1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.Ex5.m1.3.3.3.3.3.5" xref="S3.Ex5.m1.3.3.3.3.4.cmml">,</mo><msub id="S3.Ex5.m1.2.2.2.2.2.2" xref="S3.Ex5.m1.2.2.2.2.2.2.cmml"><mi id="S3.Ex5.m1.2.2.2.2.2.2.2" mathvariant="normal" xref="S3.Ex5.m1.2.2.2.2.2.2.2.cmml">Θ</mi><mrow id="S3.Ex5.m1.2.2.2.2.2.2.3" xref="S3.Ex5.m1.2.2.2.2.2.2.3.cmml"><mi id="S3.Ex5.m1.2.2.2.2.2.2.3.2" xref="S3.Ex5.m1.2.2.2.2.2.2.3.2.cmml">i</mi><mo id="S3.Ex5.m1.2.2.2.2.2.2.3.1" xref="S3.Ex5.m1.2.2.2.2.2.2.3.1.cmml">−</mo><mn id="S3.Ex5.m1.2.2.2.2.2.2.3.3" xref="S3.Ex5.m1.2.2.2.2.2.2.3.3.cmml">1</mn></mrow></msub><mo id="S3.Ex5.m1.3.3.3.3.3.6" xref="S3.Ex5.m1.3.3.3.3.4.cmml">,</mo><msub id="S3.Ex5.m1.3.3.3.3.3.3" xref="S3.Ex5.m1.3.3.3.3.3.3.cmml"><mi id="S3.Ex5.m1.3.3.3.3.3.3.2" xref="S3.Ex5.m1.3.3.3.3.3.3.2.cmml">𝒚</mi><mrow id="S3.Ex5.m1.3.3.3.3.3.3.3" xref="S3.Ex5.m1.3.3.3.3.3.3.3.cmml"><mi id="S3.Ex5.m1.3.3.3.3.3.3.3.2" mathvariant="normal" xref="S3.Ex5.m1.3.3.3.3.3.3.3.2.cmml">r</mi><mo id="S3.Ex5.m1.3.3.3.3.3.3.3.1" xref="S3.Ex5.m1.3.3.3.3.3.3.3.1.cmml">⁢</mo><mi id="S3.Ex5.m1.3.3.3.3.3.3.3.3" xref="S3.Ex5.m1.3.3.3.3.3.3.3.3.cmml">i</mi></mrow></msub><mo id="S3.Ex5.m1.3.3.3.3.3.7" stretchy="false" xref="S3.Ex5.m1.3.3.3.3.4.cmml">)</mo></mrow></mrow><mo id="S3.Ex5.m1.3.3.4" xref="S3.Ex5.m1.3.3.4.cmml">=</mo><mrow id="S3.Ex5.m1.3.3.5" xref="S3.Ex5.m1.3.3.5.cmml"><mrow id="S3.Ex5.m1.3.3.5.2" xref="S3.Ex5.m1.3.3.5.2.cmml"><mo id="S3.Ex5.m1.3.3.5.2a" xref="S3.Ex5.m1.3.3.5.2.cmml">−</mo><mrow id="S3.Ex5.m1.3.3.5.2.2" xref="S3.Ex5.m1.3.3.5.2.2.cmml"><msub id="S3.Ex5.m1.3.3.5.2.2.2" xref="S3.Ex5.m1.3.3.5.2.2.2.cmml"><mi id="S3.Ex5.m1.3.3.5.2.2.2.2" xref="S3.Ex5.m1.3.3.5.2.2.2.2.cmml">k</mi><mrow id="S3.Ex5.m1.3.3.5.2.2.2.3" xref="S3.Ex5.m1.3.3.5.2.2.2.3.cmml"><mi id="S3.Ex5.m1.3.3.5.2.2.2.3.2" mathvariant="normal" xref="S3.Ex5.m1.3.3.5.2.2.2.3.2.cmml">c</mi><mo id="S3.Ex5.m1.3.3.5.2.2.2.3.1" xref="S3.Ex5.m1.3.3.5.2.2.2.3.1.cmml">⁢</mo><mi id="S3.Ex5.m1.3.3.5.2.2.2.3.3" xref="S3.Ex5.m1.3.3.5.2.2.2.3.3.cmml">i</mi></mrow></msub><mo id="S3.Ex5.m1.3.3.5.2.2.1" xref="S3.Ex5.m1.3.3.5.2.2.1.cmml">⁢</mo><msub id="S3.Ex5.m1.3.3.5.2.2.3" xref="S3.Ex5.m1.3.3.5.2.2.3.cmml"><mi id="S3.Ex5.m1.3.3.5.2.2.3.2" xref="S3.Ex5.m1.3.3.5.2.2.3.2.cmml">e</mi><mi id="S3.Ex5.m1.3.3.5.2.2.3.3" xref="S3.Ex5.m1.3.3.5.2.2.3.3.cmml">i</mi></msub></mrow></mrow><mo id="S3.Ex5.m1.3.3.5.1" xref="S3.Ex5.m1.3.3.5.1.cmml">−</mo><msub id="S3.Ex5.m1.3.3.5.3" xref="S3.Ex5.m1.3.3.5.3.cmml"><mi id="S3.Ex5.m1.3.3.5.3.2" xref="S3.Ex5.m1.3.3.5.3.2.cmml">e</mi><mrow id="S3.Ex5.m1.3.3.5.3.3" xref="S3.Ex5.m1.3.3.5.3.3.cmml"><mi id="S3.Ex5.m1.3.3.5.3.3.2" xref="S3.Ex5.m1.3.3.5.3.3.2.cmml">i</mi><mo id="S3.Ex5.m1.3.3.5.3.3.1" xref="S3.Ex5.m1.3.3.5.3.3.1.cmml">−</mo><mn id="S3.Ex5.m1.3.3.5.3.3.3" xref="S3.Ex5.m1.3.3.5.3.3.3.cmml">1</mn></mrow></msub><mo id="S3.Ex5.m1.3.3.5.1a" xref="S3.Ex5.m1.3.3.5.1.cmml">−</mo><mrow id="S3.Ex5.m1.3.3.5.4" xref="S3.Ex5.m1.3.3.5.4.cmml"><msubsup id="S3.Ex5.m1.3.3.5.4.2" xref="S3.Ex5.m1.3.3.5.4.2.cmml"><mi id="S3.Ex5.m1.3.3.5.4.2.2.2" xref="S3.Ex5.m1.3.3.5.4.2.2.2.cmml">𝝍</mi><mi id="S3.Ex5.m1.3.3.5.4.2.2.3" xref="S3.Ex5.m1.3.3.5.4.2.2.3.cmml">i</mi><mi id="S3.Ex5.m1.3.3.5.4.2.3" xref="S3.Ex5.m1.3.3.5.4.2.3.cmml">T</mi></msubsup><mo id="S3.Ex5.m1.3.3.5.4.1" xref="S3.Ex5.m1.3.3.5.4.1.cmml">⁢</mo><mover accent="true" id="S3.Ex5.m1.3.3.5.4.3" xref="S3.Ex5.m1.3.3.5.4.3.cmml"><mi id="S3.Ex5.m1.3.3.5.4.3.2" xref="S3.Ex5.m1.3.3.5.4.3.2.cmml">𝜽</mi><mo id="S3.Ex5.m1.3.3.5.4.3.1" xref="S3.Ex5.m1.3.3.5.4.3.1.cmml">^</mo></mover></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex5.m1.3b"><apply id="S3.Ex5.m1.3.3.cmml" xref="S3.Ex5.m1.3.3"><eq id="S3.Ex5.m1.3.3.4.cmml" xref="S3.Ex5.m1.3.3.4"></eq><apply id="S3.Ex5.m1.3.3.3.cmml" xref="S3.Ex5.m1.3.3.3"><times id="S3.Ex5.m1.3.3.3.4.cmml" xref="S3.Ex5.m1.3.3.3.4"></times><apply id="S3.Ex5.m1.3.3.3.5.cmml" xref="S3.Ex5.m1.3.3.3.5"><csymbol cd="ambiguous" id="S3.Ex5.m1.3.3.3.5.1.cmml" xref="S3.Ex5.m1.3.3.3.5">subscript</csymbol><ci id="S3.Ex5.m1.3.3.3.5.2.cmml" xref="S3.Ex5.m1.3.3.3.5.2">𝑣</ci><ci id="S3.Ex5.m1.3.3.3.5.3.cmml" xref="S3.Ex5.m1.3.3.3.5.3">𝑖</ci></apply><vector id="S3.Ex5.m1.3.3.3.3.4.cmml" xref="S3.Ex5.m1.3.3.3.3.3"><apply id="S3.Ex5.m1.1.1.1.1.1.1.cmml" xref="S3.Ex5.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex5.m1.1.1.1.1.1.1.1.cmml" xref="S3.Ex5.m1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.Ex5.m1.1.1.1.1.1.1.2.cmml" xref="S3.Ex5.m1.1.1.1.1.1.1.2">𝒙</ci><ci id="S3.Ex5.m1.1.1.1.1.1.1.3.cmml" xref="S3.Ex5.m1.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S3.Ex5.m1.2.2.2.2.2.2.cmml" xref="S3.Ex5.m1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.Ex5.m1.2.2.2.2.2.2.1.cmml" xref="S3.Ex5.m1.2.2.2.2.2.2">subscript</csymbol><ci id="S3.Ex5.m1.2.2.2.2.2.2.2.cmml" xref="S3.Ex5.m1.2.2.2.2.2.2.2">Θ</ci><apply id="S3.Ex5.m1.2.2.2.2.2.2.3.cmml" xref="S3.Ex5.m1.2.2.2.2.2.2.3"><minus id="S3.Ex5.m1.2.2.2.2.2.2.3.1.cmml" xref="S3.Ex5.m1.2.2.2.2.2.2.3.1"></minus><ci id="S3.Ex5.m1.2.2.2.2.2.2.3.2.cmml" xref="S3.Ex5.m1.2.2.2.2.2.2.3.2">𝑖</ci><cn id="S3.Ex5.m1.2.2.2.2.2.2.3.3.cmml" type="integer" xref="S3.Ex5.m1.2.2.2.2.2.2.3.3">1</cn></apply></apply><apply id="S3.Ex5.m1.3.3.3.3.3.3.cmml" xref="S3.Ex5.m1.3.3.3.3.3.3"><csymbol cd="ambiguous" id="S3.Ex5.m1.3.3.3.3.3.3.1.cmml" xref="S3.Ex5.m1.3.3.3.3.3.3">subscript</csymbol><ci id="S3.Ex5.m1.3.3.3.3.3.3.2.cmml" xref="S3.Ex5.m1.3.3.3.3.3.3.2">𝒚</ci><apply id="S3.Ex5.m1.3.3.3.3.3.3.3.cmml" xref="S3.Ex5.m1.3.3.3.3.3.3.3"><times id="S3.Ex5.m1.3.3.3.3.3.3.3.1.cmml" xref="S3.Ex5.m1.3.3.3.3.3.3.3.1"></times><ci id="S3.Ex5.m1.3.3.3.3.3.3.3.2.cmml" xref="S3.Ex5.m1.3.3.3.3.3.3.3.2">r</ci><ci id="S3.Ex5.m1.3.3.3.3.3.3.3.3.cmml" xref="S3.Ex5.m1.3.3.3.3.3.3.3.3">𝑖</ci></apply></apply></vector></apply><apply id="S3.Ex5.m1.3.3.5.cmml" xref="S3.Ex5.m1.3.3.5"><minus id="S3.Ex5.m1.3.3.5.1.cmml" xref="S3.Ex5.m1.3.3.5.1"></minus><apply id="S3.Ex5.m1.3.3.5.2.cmml" xref="S3.Ex5.m1.3.3.5.2"><minus id="S3.Ex5.m1.3.3.5.2.1.cmml" xref="S3.Ex5.m1.3.3.5.2"></minus><apply id="S3.Ex5.m1.3.3.5.2.2.cmml" xref="S3.Ex5.m1.3.3.5.2.2"><times id="S3.Ex5.m1.3.3.5.2.2.1.cmml" xref="S3.Ex5.m1.3.3.5.2.2.1"></times><apply id="S3.Ex5.m1.3.3.5.2.2.2.cmml" xref="S3.Ex5.m1.3.3.5.2.2.2"><csymbol cd="ambiguous" id="S3.Ex5.m1.3.3.5.2.2.2.1.cmml" xref="S3.Ex5.m1.3.3.5.2.2.2">subscript</csymbol><ci id="S3.Ex5.m1.3.3.5.2.2.2.2.cmml" xref="S3.Ex5.m1.3.3.5.2.2.2.2">𝑘</ci><apply id="S3.Ex5.m1.3.3.5.2.2.2.3.cmml" xref="S3.Ex5.m1.3.3.5.2.2.2.3"><times id="S3.Ex5.m1.3.3.5.2.2.2.3.1.cmml" xref="S3.Ex5.m1.3.3.5.2.2.2.3.1"></times><ci id="S3.Ex5.m1.3.3.5.2.2.2.3.2.cmml" xref="S3.Ex5.m1.3.3.5.2.2.2.3.2">c</ci><ci id="S3.Ex5.m1.3.3.5.2.2.2.3.3.cmml" xref="S3.Ex5.m1.3.3.5.2.2.2.3.3">𝑖</ci></apply></apply><apply id="S3.Ex5.m1.3.3.5.2.2.3.cmml" xref="S3.Ex5.m1.3.3.5.2.2.3"><csymbol cd="ambiguous" id="S3.Ex5.m1.3.3.5.2.2.3.1.cmml" xref="S3.Ex5.m1.3.3.5.2.2.3">subscript</csymbol><ci id="S3.Ex5.m1.3.3.5.2.2.3.2.cmml" xref="S3.Ex5.m1.3.3.5.2.2.3.2">𝑒</ci><ci id="S3.Ex5.m1.3.3.5.2.2.3.3.cmml" xref="S3.Ex5.m1.3.3.5.2.2.3.3">𝑖</ci></apply></apply></apply><apply id="S3.Ex5.m1.3.3.5.3.cmml" xref="S3.Ex5.m1.3.3.5.3"><csymbol cd="ambiguous" id="S3.Ex5.m1.3.3.5.3.1.cmml" xref="S3.Ex5.m1.3.3.5.3">subscript</csymbol><ci id="S3.Ex5.m1.3.3.5.3.2.cmml" xref="S3.Ex5.m1.3.3.5.3.2">𝑒</ci><apply id="S3.Ex5.m1.3.3.5.3.3.cmml" xref="S3.Ex5.m1.3.3.5.3.3"><minus id="S3.Ex5.m1.3.3.5.3.3.1.cmml" xref="S3.Ex5.m1.3.3.5.3.3.1"></minus><ci id="S3.Ex5.m1.3.3.5.3.3.2.cmml" xref="S3.Ex5.m1.3.3.5.3.3.2">𝑖</ci><cn id="S3.Ex5.m1.3.3.5.3.3.3.cmml" type="integer" xref="S3.Ex5.m1.3.3.5.3.3.3">1</cn></apply></apply><apply id="S3.Ex5.m1.3.3.5.4.cmml" xref="S3.Ex5.m1.3.3.5.4"><times id="S3.Ex5.m1.3.3.5.4.1.cmml" xref="S3.Ex5.m1.3.3.5.4.1"></times><apply id="S3.Ex5.m1.3.3.5.4.2.cmml" xref="S3.Ex5.m1.3.3.5.4.2"><csymbol cd="ambiguous" id="S3.Ex5.m1.3.3.5.4.2.1.cmml" xref="S3.Ex5.m1.3.3.5.4.2">superscript</csymbol><apply id="S3.Ex5.m1.3.3.5.4.2.2.cmml" xref="S3.Ex5.m1.3.3.5.4.2"><csymbol cd="ambiguous" id="S3.Ex5.m1.3.3.5.4.2.2.1.cmml" xref="S3.Ex5.m1.3.3.5.4.2">subscript</csymbol><ci id="S3.Ex5.m1.3.3.5.4.2.2.2.cmml" xref="S3.Ex5.m1.3.3.5.4.2.2.2">𝝍</ci><ci id="S3.Ex5.m1.3.3.5.4.2.2.3.cmml" xref="S3.Ex5.m1.3.3.5.4.2.2.3">𝑖</ci></apply><ci id="S3.Ex5.m1.3.3.5.4.2.3.cmml" xref="S3.Ex5.m1.3.3.5.4.2.3">𝑇</ci></apply><apply id="S3.Ex5.m1.3.3.5.4.3.cmml" xref="S3.Ex5.m1.3.3.5.4.3"><ci id="S3.Ex5.m1.3.3.5.4.3.1.cmml" xref="S3.Ex5.m1.3.3.5.4.3.1">^</ci><ci id="S3.Ex5.m1.3.3.5.4.3.2.cmml" xref="S3.Ex5.m1.3.3.5.4.3.2">𝜽</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex5.m1.3c">\displaystyle v_{i}({\bm{x}}_{i},\Theta_{i-1},{\bm{y}}_{{\rm r}i})=-k_{{\rm c}% i}e_{i}-e_{i-1}-\bm{\psi}_{i}^{T}\hat{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex5.m1.3d">italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , roman_Θ start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT roman_r italic_i end_POSTSUBSCRIPT ) = - italic_k start_POSTSUBSCRIPT roman_c italic_i end_POSTSUBSCRIPT italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_e start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT - bold_italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over^ start_ARG bold_italic_θ end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.E5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle+\sum_{k=1}^{i-1}\left(\frac{\partial v_{i-1}}{\partial x_{k}}x_{% k+1}+\frac{\partial v_{i-1}}{\partial\hat{\bm{\theta}}^{(k-1)}}\hat{\bm{\theta% }}^{(k)}+\frac{\partial v_{i-1}}{\partial y_{\rm r}^{(k-1)}}y_{\rm r}^{(k)}\right)" class="ltx_Math" display="inline" id="S3.E5.m1.5"><semantics id="S3.E5.m1.5a"><mrow id="S3.E5.m1.5.5" xref="S3.E5.m1.5.5.cmml"><mo id="S3.E5.m1.5.5a" xref="S3.E5.m1.5.5.cmml">+</mo><mrow id="S3.E5.m1.5.5.1" xref="S3.E5.m1.5.5.1.cmml"><mstyle displaystyle="true" id="S3.E5.m1.5.5.1.2" xref="S3.E5.m1.5.5.1.2.cmml"><munderover id="S3.E5.m1.5.5.1.2a" xref="S3.E5.m1.5.5.1.2.cmml"><mo id="S3.E5.m1.5.5.1.2.2.2" movablelimits="false" xref="S3.E5.m1.5.5.1.2.2.2.cmml">∑</mo><mrow id="S3.E5.m1.5.5.1.2.2.3" xref="S3.E5.m1.5.5.1.2.2.3.cmml"><mi id="S3.E5.m1.5.5.1.2.2.3.2" xref="S3.E5.m1.5.5.1.2.2.3.2.cmml">k</mi><mo id="S3.E5.m1.5.5.1.2.2.3.1" xref="S3.E5.m1.5.5.1.2.2.3.1.cmml">=</mo><mn id="S3.E5.m1.5.5.1.2.2.3.3" xref="S3.E5.m1.5.5.1.2.2.3.3.cmml">1</mn></mrow><mrow id="S3.E5.m1.5.5.1.2.3" xref="S3.E5.m1.5.5.1.2.3.cmml"><mi id="S3.E5.m1.5.5.1.2.3.2" xref="S3.E5.m1.5.5.1.2.3.2.cmml">i</mi><mo id="S3.E5.m1.5.5.1.2.3.1" xref="S3.E5.m1.5.5.1.2.3.1.cmml">−</mo><mn id="S3.E5.m1.5.5.1.2.3.3" xref="S3.E5.m1.5.5.1.2.3.3.cmml">1</mn></mrow></munderover></mstyle><mrow id="S3.E5.m1.5.5.1.1.1" xref="S3.E5.m1.5.5.1.1.1.1.cmml"><mo id="S3.E5.m1.5.5.1.1.1.2" xref="S3.E5.m1.5.5.1.1.1.1.cmml">(</mo><mrow id="S3.E5.m1.5.5.1.1.1.1" xref="S3.E5.m1.5.5.1.1.1.1.cmml"><mrow id="S3.E5.m1.5.5.1.1.1.1.2" xref="S3.E5.m1.5.5.1.1.1.1.2.cmml"><mstyle displaystyle="true" id="S3.E5.m1.5.5.1.1.1.1.2.2" xref="S3.E5.m1.5.5.1.1.1.1.2.2.cmml"><mfrac id="S3.E5.m1.5.5.1.1.1.1.2.2a" xref="S3.E5.m1.5.5.1.1.1.1.2.2.cmml"><mrow id="S3.E5.m1.5.5.1.1.1.1.2.2.2" xref="S3.E5.m1.5.5.1.1.1.1.2.2.2.cmml"><mo id="S3.E5.m1.5.5.1.1.1.1.2.2.2.1" rspace="0em" xref="S3.E5.m1.5.5.1.1.1.1.2.2.2.1.cmml">∂</mo><msub id="S3.E5.m1.5.5.1.1.1.1.2.2.2.2" xref="S3.E5.m1.5.5.1.1.1.1.2.2.2.2.cmml"><mi id="S3.E5.m1.5.5.1.1.1.1.2.2.2.2.2" xref="S3.E5.m1.5.5.1.1.1.1.2.2.2.2.2.cmml">v</mi><mrow id="S3.E5.m1.5.5.1.1.1.1.2.2.2.2.3" xref="S3.E5.m1.5.5.1.1.1.1.2.2.2.2.3.cmml"><mi id="S3.E5.m1.5.5.1.1.1.1.2.2.2.2.3.2" xref="S3.E5.m1.5.5.1.1.1.1.2.2.2.2.3.2.cmml">i</mi><mo id="S3.E5.m1.5.5.1.1.1.1.2.2.2.2.3.1" xref="S3.E5.m1.5.5.1.1.1.1.2.2.2.2.3.1.cmml">−</mo><mn id="S3.E5.m1.5.5.1.1.1.1.2.2.2.2.3.3" xref="S3.E5.m1.5.5.1.1.1.1.2.2.2.2.3.3.cmml">1</mn></mrow></msub></mrow><mrow id="S3.E5.m1.5.5.1.1.1.1.2.2.3" xref="S3.E5.m1.5.5.1.1.1.1.2.2.3.cmml"><mo id="S3.E5.m1.5.5.1.1.1.1.2.2.3.1" rspace="0em" xref="S3.E5.m1.5.5.1.1.1.1.2.2.3.1.cmml">∂</mo><msub id="S3.E5.m1.5.5.1.1.1.1.2.2.3.2" xref="S3.E5.m1.5.5.1.1.1.1.2.2.3.2.cmml"><mi id="S3.E5.m1.5.5.1.1.1.1.2.2.3.2.2" xref="S3.E5.m1.5.5.1.1.1.1.2.2.3.2.2.cmml">x</mi><mi id="S3.E5.m1.5.5.1.1.1.1.2.2.3.2.3" xref="S3.E5.m1.5.5.1.1.1.1.2.2.3.2.3.cmml">k</mi></msub></mrow></mfrac></mstyle><mo id="S3.E5.m1.5.5.1.1.1.1.2.1" xref="S3.E5.m1.5.5.1.1.1.1.2.1.cmml">⁢</mo><msub id="S3.E5.m1.5.5.1.1.1.1.2.3" xref="S3.E5.m1.5.5.1.1.1.1.2.3.cmml"><mi id="S3.E5.m1.5.5.1.1.1.1.2.3.2" xref="S3.E5.m1.5.5.1.1.1.1.2.3.2.cmml">x</mi><mrow id="S3.E5.m1.5.5.1.1.1.1.2.3.3" xref="S3.E5.m1.5.5.1.1.1.1.2.3.3.cmml"><mi id="S3.E5.m1.5.5.1.1.1.1.2.3.3.2" xref="S3.E5.m1.5.5.1.1.1.1.2.3.3.2.cmml">k</mi><mo id="S3.E5.m1.5.5.1.1.1.1.2.3.3.1" xref="S3.E5.m1.5.5.1.1.1.1.2.3.3.1.cmml">+</mo><mn id="S3.E5.m1.5.5.1.1.1.1.2.3.3.3" xref="S3.E5.m1.5.5.1.1.1.1.2.3.3.3.cmml">1</mn></mrow></msub></mrow><mo id="S3.E5.m1.5.5.1.1.1.1.1" xref="S3.E5.m1.5.5.1.1.1.1.1.cmml">+</mo><mrow id="S3.E5.m1.5.5.1.1.1.1.3" xref="S3.E5.m1.5.5.1.1.1.1.3.cmml"><mstyle displaystyle="true" id="S3.E5.m1.1.1" xref="S3.E5.m1.1.1.cmml"><mfrac id="S3.E5.m1.1.1a" xref="S3.E5.m1.1.1.cmml"><mrow id="S3.E5.m1.1.1.3" xref="S3.E5.m1.1.1.3.cmml"><mo id="S3.E5.m1.1.1.3.1" rspace="0em" xref="S3.E5.m1.1.1.3.1.cmml">∂</mo><msub id="S3.E5.m1.1.1.3.2" xref="S3.E5.m1.1.1.3.2.cmml"><mi id="S3.E5.m1.1.1.3.2.2" xref="S3.E5.m1.1.1.3.2.2.cmml">v</mi><mrow id="S3.E5.m1.1.1.3.2.3" xref="S3.E5.m1.1.1.3.2.3.cmml"><mi id="S3.E5.m1.1.1.3.2.3.2" xref="S3.E5.m1.1.1.3.2.3.2.cmml">i</mi><mo id="S3.E5.m1.1.1.3.2.3.1" xref="S3.E5.m1.1.1.3.2.3.1.cmml">−</mo><mn id="S3.E5.m1.1.1.3.2.3.3" xref="S3.E5.m1.1.1.3.2.3.3.cmml">1</mn></mrow></msub></mrow><mrow id="S3.E5.m1.1.1.1" xref="S3.E5.m1.1.1.1.cmml"><mo id="S3.E5.m1.1.1.1.2" rspace="0em" xref="S3.E5.m1.1.1.1.2.cmml">∂</mo><msup id="S3.E5.m1.1.1.1.3" xref="S3.E5.m1.1.1.1.3.cmml"><mover accent="true" id="S3.E5.m1.1.1.1.3.2" xref="S3.E5.m1.1.1.1.3.2.cmml"><mi id="S3.E5.m1.1.1.1.3.2.2" xref="S3.E5.m1.1.1.1.3.2.2.cmml">𝜽</mi><mo id="S3.E5.m1.1.1.1.3.2.1" xref="S3.E5.m1.1.1.1.3.2.1.cmml">^</mo></mover><mrow id="S3.E5.m1.1.1.1.1.1.1" xref="S3.E5.m1.1.1.1.1.1.1.1.cmml"><mo id="S3.E5.m1.1.1.1.1.1.1.2" stretchy="false" xref="S3.E5.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.E5.m1.1.1.1.1.1.1.1" xref="S3.E5.m1.1.1.1.1.1.1.1.cmml"><mi id="S3.E5.m1.1.1.1.1.1.1.1.2" xref="S3.E5.m1.1.1.1.1.1.1.1.2.cmml">k</mi><mo id="S3.E5.m1.1.1.1.1.1.1.1.1" xref="S3.E5.m1.1.1.1.1.1.1.1.1.cmml">−</mo><mn id="S3.E5.m1.1.1.1.1.1.1.1.3" xref="S3.E5.m1.1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.E5.m1.1.1.1.1.1.1.3" stretchy="false" xref="S3.E5.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></msup></mrow></mfrac></mstyle><mo id="S3.E5.m1.5.5.1.1.1.1.3.1" xref="S3.E5.m1.5.5.1.1.1.1.3.1.cmml">⁢</mo><msup id="S3.E5.m1.5.5.1.1.1.1.3.2" xref="S3.E5.m1.5.5.1.1.1.1.3.2.cmml"><mover accent="true" id="S3.E5.m1.5.5.1.1.1.1.3.2.2" xref="S3.E5.m1.5.5.1.1.1.1.3.2.2.cmml"><mi id="S3.E5.m1.5.5.1.1.1.1.3.2.2.2" xref="S3.E5.m1.5.5.1.1.1.1.3.2.2.2.cmml">𝜽</mi><mo id="S3.E5.m1.5.5.1.1.1.1.3.2.2.1" xref="S3.E5.m1.5.5.1.1.1.1.3.2.2.1.cmml">^</mo></mover><mrow id="S3.E5.m1.2.2.1.3" xref="S3.E5.m1.5.5.1.1.1.1.3.2.cmml"><mo id="S3.E5.m1.2.2.1.3.1" stretchy="false" xref="S3.E5.m1.5.5.1.1.1.1.3.2.cmml">(</mo><mi id="S3.E5.m1.2.2.1.1" xref="S3.E5.m1.2.2.1.1.cmml">k</mi><mo id="S3.E5.m1.2.2.1.3.2" stretchy="false" xref="S3.E5.m1.5.5.1.1.1.1.3.2.cmml">)</mo></mrow></msup></mrow><mo id="S3.E5.m1.5.5.1.1.1.1.1a" xref="S3.E5.m1.5.5.1.1.1.1.1.cmml">+</mo><mrow id="S3.E5.m1.5.5.1.1.1.1.4" xref="S3.E5.m1.5.5.1.1.1.1.4.cmml"><mstyle displaystyle="true" id="S3.E5.m1.3.3" xref="S3.E5.m1.3.3.cmml"><mfrac id="S3.E5.m1.3.3a" xref="S3.E5.m1.3.3.cmml"><mrow id="S3.E5.m1.3.3.3" xref="S3.E5.m1.3.3.3.cmml"><mo id="S3.E5.m1.3.3.3.1" rspace="0em" xref="S3.E5.m1.3.3.3.1.cmml">∂</mo><msub id="S3.E5.m1.3.3.3.2" xref="S3.E5.m1.3.3.3.2.cmml"><mi id="S3.E5.m1.3.3.3.2.2" xref="S3.E5.m1.3.3.3.2.2.cmml">v</mi><mrow id="S3.E5.m1.3.3.3.2.3" xref="S3.E5.m1.3.3.3.2.3.cmml"><mi id="S3.E5.m1.3.3.3.2.3.2" xref="S3.E5.m1.3.3.3.2.3.2.cmml">i</mi><mo id="S3.E5.m1.3.3.3.2.3.1" xref="S3.E5.m1.3.3.3.2.3.1.cmml">−</mo><mn id="S3.E5.m1.3.3.3.2.3.3" xref="S3.E5.m1.3.3.3.2.3.3.cmml">1</mn></mrow></msub></mrow><mrow id="S3.E5.m1.3.3.1" xref="S3.E5.m1.3.3.1.cmml"><mo id="S3.E5.m1.3.3.1.2" rspace="0em" xref="S3.E5.m1.3.3.1.2.cmml">∂</mo><msubsup id="S3.E5.m1.3.3.1.3" xref="S3.E5.m1.3.3.1.3.cmml"><mi id="S3.E5.m1.3.3.1.3.2.2" xref="S3.E5.m1.3.3.1.3.2.2.cmml">y</mi><mi id="S3.E5.m1.3.3.1.3.2.3" mathvariant="normal" xref="S3.E5.m1.3.3.1.3.2.3.cmml">r</mi><mrow id="S3.E5.m1.3.3.1.1.1.1" xref="S3.E5.m1.3.3.1.1.1.1.1.cmml"><mo id="S3.E5.m1.3.3.1.1.1.1.2" stretchy="false" xref="S3.E5.m1.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S3.E5.m1.3.3.1.1.1.1.1" xref="S3.E5.m1.3.3.1.1.1.1.1.cmml"><mi id="S3.E5.m1.3.3.1.1.1.1.1.2" xref="S3.E5.m1.3.3.1.1.1.1.1.2.cmml">k</mi><mo id="S3.E5.m1.3.3.1.1.1.1.1.1" xref="S3.E5.m1.3.3.1.1.1.1.1.1.cmml">−</mo><mn id="S3.E5.m1.3.3.1.1.1.1.1.3" xref="S3.E5.m1.3.3.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.E5.m1.3.3.1.1.1.1.3" stretchy="false" xref="S3.E5.m1.3.3.1.1.1.1.1.cmml">)</mo></mrow></msubsup></mrow></mfrac></mstyle><mo id="S3.E5.m1.5.5.1.1.1.1.4.1" xref="S3.E5.m1.5.5.1.1.1.1.4.1.cmml">⁢</mo><msubsup id="S3.E5.m1.5.5.1.1.1.1.4.2" xref="S3.E5.m1.5.5.1.1.1.1.4.2.cmml"><mi id="S3.E5.m1.5.5.1.1.1.1.4.2.2.2" xref="S3.E5.m1.5.5.1.1.1.1.4.2.2.2.cmml">y</mi><mi id="S3.E5.m1.5.5.1.1.1.1.4.2.2.3" mathvariant="normal" xref="S3.E5.m1.5.5.1.1.1.1.4.2.2.3.cmml">r</mi><mrow id="S3.E5.m1.4.4.1.3" xref="S3.E5.m1.5.5.1.1.1.1.4.2.cmml"><mo id="S3.E5.m1.4.4.1.3.1" stretchy="false" xref="S3.E5.m1.5.5.1.1.1.1.4.2.cmml">(</mo><mi id="S3.E5.m1.4.4.1.1" xref="S3.E5.m1.4.4.1.1.cmml">k</mi><mo id="S3.E5.m1.4.4.1.3.2" stretchy="false" xref="S3.E5.m1.5.5.1.1.1.1.4.2.cmml">)</mo></mrow></msubsup></mrow></mrow><mo id="S3.E5.m1.5.5.1.1.1.3" xref="S3.E5.m1.5.5.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.E5.m1.5b"><apply id="S3.E5.m1.5.5.cmml" xref="S3.E5.m1.5.5"><plus id="S3.E5.m1.5.5.2.cmml" xref="S3.E5.m1.5.5"></plus><apply id="S3.E5.m1.5.5.1.cmml" xref="S3.E5.m1.5.5.1"><apply id="S3.E5.m1.5.5.1.2.cmml" xref="S3.E5.m1.5.5.1.2"><csymbol cd="ambiguous" id="S3.E5.m1.5.5.1.2.1.cmml" xref="S3.E5.m1.5.5.1.2">superscript</csymbol><apply id="S3.E5.m1.5.5.1.2.2.cmml" xref="S3.E5.m1.5.5.1.2"><csymbol cd="ambiguous" id="S3.E5.m1.5.5.1.2.2.1.cmml" xref="S3.E5.m1.5.5.1.2">subscript</csymbol><sum id="S3.E5.m1.5.5.1.2.2.2.cmml" xref="S3.E5.m1.5.5.1.2.2.2"></sum><apply id="S3.E5.m1.5.5.1.2.2.3.cmml" xref="S3.E5.m1.5.5.1.2.2.3"><eq id="S3.E5.m1.5.5.1.2.2.3.1.cmml" xref="S3.E5.m1.5.5.1.2.2.3.1"></eq><ci id="S3.E5.m1.5.5.1.2.2.3.2.cmml" xref="S3.E5.m1.5.5.1.2.2.3.2">𝑘</ci><cn id="S3.E5.m1.5.5.1.2.2.3.3.cmml" type="integer" xref="S3.E5.m1.5.5.1.2.2.3.3">1</cn></apply></apply><apply id="S3.E5.m1.5.5.1.2.3.cmml" xref="S3.E5.m1.5.5.1.2.3"><minus id="S3.E5.m1.5.5.1.2.3.1.cmml" xref="S3.E5.m1.5.5.1.2.3.1"></minus><ci id="S3.E5.m1.5.5.1.2.3.2.cmml" xref="S3.E5.m1.5.5.1.2.3.2">𝑖</ci><cn id="S3.E5.m1.5.5.1.2.3.3.cmml" type="integer" xref="S3.E5.m1.5.5.1.2.3.3">1</cn></apply></apply><apply id="S3.E5.m1.5.5.1.1.1.1.cmml" xref="S3.E5.m1.5.5.1.1.1"><plus id="S3.E5.m1.5.5.1.1.1.1.1.cmml" xref="S3.E5.m1.5.5.1.1.1.1.1"></plus><apply id="S3.E5.m1.5.5.1.1.1.1.2.cmml" xref="S3.E5.m1.5.5.1.1.1.1.2"><times id="S3.E5.m1.5.5.1.1.1.1.2.1.cmml" xref="S3.E5.m1.5.5.1.1.1.1.2.1"></times><apply id="S3.E5.m1.5.5.1.1.1.1.2.2.cmml" xref="S3.E5.m1.5.5.1.1.1.1.2.2"><divide id="S3.E5.m1.5.5.1.1.1.1.2.2.1.cmml" xref="S3.E5.m1.5.5.1.1.1.1.2.2"></divide><apply id="S3.E5.m1.5.5.1.1.1.1.2.2.2.cmml" xref="S3.E5.m1.5.5.1.1.1.1.2.2.2"><partialdiff id="S3.E5.m1.5.5.1.1.1.1.2.2.2.1.cmml" xref="S3.E5.m1.5.5.1.1.1.1.2.2.2.1"></partialdiff><apply id="S3.E5.m1.5.5.1.1.1.1.2.2.2.2.cmml" xref="S3.E5.m1.5.5.1.1.1.1.2.2.2.2"><csymbol cd="ambiguous" id="S3.E5.m1.5.5.1.1.1.1.2.2.2.2.1.cmml" xref="S3.E5.m1.5.5.1.1.1.1.2.2.2.2">subscript</csymbol><ci id="S3.E5.m1.5.5.1.1.1.1.2.2.2.2.2.cmml" xref="S3.E5.m1.5.5.1.1.1.1.2.2.2.2.2">𝑣</ci><apply id="S3.E5.m1.5.5.1.1.1.1.2.2.2.2.3.cmml" xref="S3.E5.m1.5.5.1.1.1.1.2.2.2.2.3"><minus id="S3.E5.m1.5.5.1.1.1.1.2.2.2.2.3.1.cmml" xref="S3.E5.m1.5.5.1.1.1.1.2.2.2.2.3.1"></minus><ci id="S3.E5.m1.5.5.1.1.1.1.2.2.2.2.3.2.cmml" xref="S3.E5.m1.5.5.1.1.1.1.2.2.2.2.3.2">𝑖</ci><cn id="S3.E5.m1.5.5.1.1.1.1.2.2.2.2.3.3.cmml" type="integer" xref="S3.E5.m1.5.5.1.1.1.1.2.2.2.2.3.3">1</cn></apply></apply></apply><apply id="S3.E5.m1.5.5.1.1.1.1.2.2.3.cmml" xref="S3.E5.m1.5.5.1.1.1.1.2.2.3"><partialdiff id="S3.E5.m1.5.5.1.1.1.1.2.2.3.1.cmml" xref="S3.E5.m1.5.5.1.1.1.1.2.2.3.1"></partialdiff><apply id="S3.E5.m1.5.5.1.1.1.1.2.2.3.2.cmml" xref="S3.E5.m1.5.5.1.1.1.1.2.2.3.2"><csymbol cd="ambiguous" id="S3.E5.m1.5.5.1.1.1.1.2.2.3.2.1.cmml" xref="S3.E5.m1.5.5.1.1.1.1.2.2.3.2">subscript</csymbol><ci id="S3.E5.m1.5.5.1.1.1.1.2.2.3.2.2.cmml" xref="S3.E5.m1.5.5.1.1.1.1.2.2.3.2.2">𝑥</ci><ci id="S3.E5.m1.5.5.1.1.1.1.2.2.3.2.3.cmml" xref="S3.E5.m1.5.5.1.1.1.1.2.2.3.2.3">𝑘</ci></apply></apply></apply><apply id="S3.E5.m1.5.5.1.1.1.1.2.3.cmml" xref="S3.E5.m1.5.5.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S3.E5.m1.5.5.1.1.1.1.2.3.1.cmml" xref="S3.E5.m1.5.5.1.1.1.1.2.3">subscript</csymbol><ci id="S3.E5.m1.5.5.1.1.1.1.2.3.2.cmml" xref="S3.E5.m1.5.5.1.1.1.1.2.3.2">𝑥</ci><apply id="S3.E5.m1.5.5.1.1.1.1.2.3.3.cmml" xref="S3.E5.m1.5.5.1.1.1.1.2.3.3"><plus id="S3.E5.m1.5.5.1.1.1.1.2.3.3.1.cmml" xref="S3.E5.m1.5.5.1.1.1.1.2.3.3.1"></plus><ci id="S3.E5.m1.5.5.1.1.1.1.2.3.3.2.cmml" xref="S3.E5.m1.5.5.1.1.1.1.2.3.3.2">𝑘</ci><cn id="S3.E5.m1.5.5.1.1.1.1.2.3.3.3.cmml" type="integer" xref="S3.E5.m1.5.5.1.1.1.1.2.3.3.3">1</cn></apply></apply></apply><apply id="S3.E5.m1.5.5.1.1.1.1.3.cmml" xref="S3.E5.m1.5.5.1.1.1.1.3"><times id="S3.E5.m1.5.5.1.1.1.1.3.1.cmml" xref="S3.E5.m1.5.5.1.1.1.1.3.1"></times><apply id="S3.E5.m1.1.1.cmml" xref="S3.E5.m1.1.1"><divide id="S3.E5.m1.1.1.2.cmml" xref="S3.E5.m1.1.1"></divide><apply id="S3.E5.m1.1.1.3.cmml" xref="S3.E5.m1.1.1.3"><partialdiff id="S3.E5.m1.1.1.3.1.cmml" xref="S3.E5.m1.1.1.3.1"></partialdiff><apply id="S3.E5.m1.1.1.3.2.cmml" xref="S3.E5.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.E5.m1.1.1.3.2.1.cmml" xref="S3.E5.m1.1.1.3.2">subscript</csymbol><ci id="S3.E5.m1.1.1.3.2.2.cmml" xref="S3.E5.m1.1.1.3.2.2">𝑣</ci><apply id="S3.E5.m1.1.1.3.2.3.cmml" xref="S3.E5.m1.1.1.3.2.3"><minus id="S3.E5.m1.1.1.3.2.3.1.cmml" xref="S3.E5.m1.1.1.3.2.3.1"></minus><ci id="S3.E5.m1.1.1.3.2.3.2.cmml" xref="S3.E5.m1.1.1.3.2.3.2">𝑖</ci><cn id="S3.E5.m1.1.1.3.2.3.3.cmml" type="integer" xref="S3.E5.m1.1.1.3.2.3.3">1</cn></apply></apply></apply><apply id="S3.E5.m1.1.1.1.cmml" xref="S3.E5.m1.1.1.1"><partialdiff id="S3.E5.m1.1.1.1.2.cmml" xref="S3.E5.m1.1.1.1.2"></partialdiff><apply id="S3.E5.m1.1.1.1.3.cmml" xref="S3.E5.m1.1.1.1.3"><csymbol cd="ambiguous" id="S3.E5.m1.1.1.1.3.1.cmml" xref="S3.E5.m1.1.1.1.3">superscript</csymbol><apply id="S3.E5.m1.1.1.1.3.2.cmml" xref="S3.E5.m1.1.1.1.3.2"><ci id="S3.E5.m1.1.1.1.3.2.1.cmml" xref="S3.E5.m1.1.1.1.3.2.1">^</ci><ci id="S3.E5.m1.1.1.1.3.2.2.cmml" xref="S3.E5.m1.1.1.1.3.2.2">𝜽</ci></apply><apply id="S3.E5.m1.1.1.1.1.1.1.1.cmml" xref="S3.E5.m1.1.1.1.1.1.1"><minus id="S3.E5.m1.1.1.1.1.1.1.1.1.cmml" xref="S3.E5.m1.1.1.1.1.1.1.1.1"></minus><ci id="S3.E5.m1.1.1.1.1.1.1.1.2.cmml" xref="S3.E5.m1.1.1.1.1.1.1.1.2">𝑘</ci><cn id="S3.E5.m1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.E5.m1.1.1.1.1.1.1.1.3">1</cn></apply></apply></apply></apply><apply id="S3.E5.m1.5.5.1.1.1.1.3.2.cmml" xref="S3.E5.m1.5.5.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S3.E5.m1.5.5.1.1.1.1.3.2.1.cmml" xref="S3.E5.m1.5.5.1.1.1.1.3.2">superscript</csymbol><apply id="S3.E5.m1.5.5.1.1.1.1.3.2.2.cmml" xref="S3.E5.m1.5.5.1.1.1.1.3.2.2"><ci id="S3.E5.m1.5.5.1.1.1.1.3.2.2.1.cmml" xref="S3.E5.m1.5.5.1.1.1.1.3.2.2.1">^</ci><ci id="S3.E5.m1.5.5.1.1.1.1.3.2.2.2.cmml" xref="S3.E5.m1.5.5.1.1.1.1.3.2.2.2">𝜽</ci></apply><ci id="S3.E5.m1.2.2.1.1.cmml" xref="S3.E5.m1.2.2.1.1">𝑘</ci></apply></apply><apply id="S3.E5.m1.5.5.1.1.1.1.4.cmml" xref="S3.E5.m1.5.5.1.1.1.1.4"><times id="S3.E5.m1.5.5.1.1.1.1.4.1.cmml" xref="S3.E5.m1.5.5.1.1.1.1.4.1"></times><apply id="S3.E5.m1.3.3.cmml" xref="S3.E5.m1.3.3"><divide id="S3.E5.m1.3.3.2.cmml" xref="S3.E5.m1.3.3"></divide><apply id="S3.E5.m1.3.3.3.cmml" xref="S3.E5.m1.3.3.3"><partialdiff id="S3.E5.m1.3.3.3.1.cmml" xref="S3.E5.m1.3.3.3.1"></partialdiff><apply id="S3.E5.m1.3.3.3.2.cmml" xref="S3.E5.m1.3.3.3.2"><csymbol cd="ambiguous" id="S3.E5.m1.3.3.3.2.1.cmml" xref="S3.E5.m1.3.3.3.2">subscript</csymbol><ci id="S3.E5.m1.3.3.3.2.2.cmml" xref="S3.E5.m1.3.3.3.2.2">𝑣</ci><apply id="S3.E5.m1.3.3.3.2.3.cmml" xref="S3.E5.m1.3.3.3.2.3"><minus id="S3.E5.m1.3.3.3.2.3.1.cmml" xref="S3.E5.m1.3.3.3.2.3.1"></minus><ci id="S3.E5.m1.3.3.3.2.3.2.cmml" xref="S3.E5.m1.3.3.3.2.3.2">𝑖</ci><cn id="S3.E5.m1.3.3.3.2.3.3.cmml" type="integer" xref="S3.E5.m1.3.3.3.2.3.3">1</cn></apply></apply></apply><apply id="S3.E5.m1.3.3.1.cmml" xref="S3.E5.m1.3.3.1"><partialdiff id="S3.E5.m1.3.3.1.2.cmml" xref="S3.E5.m1.3.3.1.2"></partialdiff><apply id="S3.E5.m1.3.3.1.3.cmml" xref="S3.E5.m1.3.3.1.3"><csymbol cd="ambiguous" id="S3.E5.m1.3.3.1.3.1.cmml" xref="S3.E5.m1.3.3.1.3">superscript</csymbol><apply id="S3.E5.m1.3.3.1.3.2.cmml" xref="S3.E5.m1.3.3.1.3"><csymbol cd="ambiguous" id="S3.E5.m1.3.3.1.3.2.1.cmml" xref="S3.E5.m1.3.3.1.3">subscript</csymbol><ci id="S3.E5.m1.3.3.1.3.2.2.cmml" xref="S3.E5.m1.3.3.1.3.2.2">𝑦</ci><ci id="S3.E5.m1.3.3.1.3.2.3.cmml" xref="S3.E5.m1.3.3.1.3.2.3">r</ci></apply><apply id="S3.E5.m1.3.3.1.1.1.1.1.cmml" xref="S3.E5.m1.3.3.1.1.1.1"><minus id="S3.E5.m1.3.3.1.1.1.1.1.1.cmml" xref="S3.E5.m1.3.3.1.1.1.1.1.1"></minus><ci id="S3.E5.m1.3.3.1.1.1.1.1.2.cmml" xref="S3.E5.m1.3.3.1.1.1.1.1.2">𝑘</ci><cn id="S3.E5.m1.3.3.1.1.1.1.1.3.cmml" type="integer" xref="S3.E5.m1.3.3.1.1.1.1.1.3">1</cn></apply></apply></apply></apply><apply id="S3.E5.m1.5.5.1.1.1.1.4.2.cmml" xref="S3.E5.m1.5.5.1.1.1.1.4.2"><csymbol cd="ambiguous" id="S3.E5.m1.5.5.1.1.1.1.4.2.1.cmml" xref="S3.E5.m1.5.5.1.1.1.1.4.2">superscript</csymbol><apply id="S3.E5.m1.5.5.1.1.1.1.4.2.2.cmml" xref="S3.E5.m1.5.5.1.1.1.1.4.2"><csymbol cd="ambiguous" id="S3.E5.m1.5.5.1.1.1.1.4.2.2.1.cmml" xref="S3.E5.m1.5.5.1.1.1.1.4.2">subscript</csymbol><ci id="S3.E5.m1.5.5.1.1.1.1.4.2.2.2.cmml" xref="S3.E5.m1.5.5.1.1.1.1.4.2.2.2">𝑦</ci><ci id="S3.E5.m1.5.5.1.1.1.1.4.2.2.3.cmml" xref="S3.E5.m1.5.5.1.1.1.1.4.2.2.3">r</ci></apply><ci id="S3.E5.m1.4.4.1.1.cmml" xref="S3.E5.m1.4.4.1.1">𝑘</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E5.m1.5c">\displaystyle+\sum_{k=1}^{i-1}\left(\frac{\partial v_{i-1}}{\partial x_{k}}x_{% k+1}+\frac{\partial v_{i-1}}{\partial\hat{\bm{\theta}}^{(k-1)}}\hat{\bm{\theta% }}^{(k)}+\frac{\partial v_{i-1}}{\partial y_{\rm r}^{(k-1)}}y_{\rm r}^{(k)}\right)</annotation><annotation encoding="application/x-llamapun" id="S3.E5.m1.5d">+ ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i - 1 end_POSTSUPERSCRIPT ( divide start_ARG ∂ italic_v start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG italic_x start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT + divide start_ARG ∂ italic_v start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT end_ARG start_ARG ∂ over^ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT ( italic_k - 1 ) end_POSTSUPERSCRIPT end_ARG over^ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT + divide start_ARG ∂ italic_v start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_y start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k - 1 ) end_POSTSUPERSCRIPT end_ARG italic_y start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p4.23">with <math alttext="\Theta_{i-1}(t)" class="ltx_Math" display="inline" id="S3.p4.2.m1.1"><semantics id="S3.p4.2.m1.1a"><mrow id="S3.p4.2.m1.1.2" xref="S3.p4.2.m1.1.2.cmml"><msub id="S3.p4.2.m1.1.2.2" xref="S3.p4.2.m1.1.2.2.cmml"><mi id="S3.p4.2.m1.1.2.2.2" mathvariant="normal" xref="S3.p4.2.m1.1.2.2.2.cmml">Θ</mi><mrow id="S3.p4.2.m1.1.2.2.3" xref="S3.p4.2.m1.1.2.2.3.cmml"><mi id="S3.p4.2.m1.1.2.2.3.2" xref="S3.p4.2.m1.1.2.2.3.2.cmml">i</mi><mo id="S3.p4.2.m1.1.2.2.3.1" xref="S3.p4.2.m1.1.2.2.3.1.cmml">−</mo><mn id="S3.p4.2.m1.1.2.2.3.3" xref="S3.p4.2.m1.1.2.2.3.3.cmml">1</mn></mrow></msub><mo id="S3.p4.2.m1.1.2.1" xref="S3.p4.2.m1.1.2.1.cmml">⁢</mo><mrow id="S3.p4.2.m1.1.2.3.2" xref="S3.p4.2.m1.1.2.cmml"><mo id="S3.p4.2.m1.1.2.3.2.1" stretchy="false" xref="S3.p4.2.m1.1.2.cmml">(</mo><mi id="S3.p4.2.m1.1.1" xref="S3.p4.2.m1.1.1.cmml">t</mi><mo id="S3.p4.2.m1.1.2.3.2.2" stretchy="false" xref="S3.p4.2.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.2.m1.1b"><apply id="S3.p4.2.m1.1.2.cmml" xref="S3.p4.2.m1.1.2"><times id="S3.p4.2.m1.1.2.1.cmml" xref="S3.p4.2.m1.1.2.1"></times><apply id="S3.p4.2.m1.1.2.2.cmml" xref="S3.p4.2.m1.1.2.2"><csymbol cd="ambiguous" id="S3.p4.2.m1.1.2.2.1.cmml" xref="S3.p4.2.m1.1.2.2">subscript</csymbol><ci id="S3.p4.2.m1.1.2.2.2.cmml" xref="S3.p4.2.m1.1.2.2.2">Θ</ci><apply id="S3.p4.2.m1.1.2.2.3.cmml" xref="S3.p4.2.m1.1.2.2.3"><minus id="S3.p4.2.m1.1.2.2.3.1.cmml" xref="S3.p4.2.m1.1.2.2.3.1"></minus><ci id="S3.p4.2.m1.1.2.2.3.2.cmml" xref="S3.p4.2.m1.1.2.2.3.2">𝑖</ci><cn id="S3.p4.2.m1.1.2.2.3.3.cmml" type="integer" xref="S3.p4.2.m1.1.2.2.3.3">1</cn></apply></apply><ci id="S3.p4.2.m1.1.1.cmml" xref="S3.p4.2.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.2.m1.1c">\Theta_{i-1}(t)</annotation><annotation encoding="application/x-llamapun" id="S3.p4.2.m1.1d">roman_Θ start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S3.p4.3.m2.1"><semantics id="S3.p4.3.m2.1a"><mo id="S3.p4.3.m2.1.1" xref="S3.p4.3.m2.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S3.p4.3.m2.1b"><csymbol cd="latexml" id="S3.p4.3.m2.1.1.cmml" xref="S3.p4.3.m2.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.3.m2.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S3.p4.3.m2.1d">:=</annotation></semantics></math> <math alttext="\big{[}\hat{\bm{\theta}}(t),\dot{\hat{\bm{\theta}}}(t),\cdots,\hat{\bm{\theta}% }^{(i-1)}(t)\big{]}" class="ltx_Math" display="inline" id="S3.p4.4.m3.8"><semantics id="S3.p4.4.m3.8a"><mrow id="S3.p4.4.m3.8.8.3" xref="S3.p4.4.m3.8.8.4.cmml"><mo id="S3.p4.4.m3.8.8.3.4" maxsize="120%" minsize="120%" xref="S3.p4.4.m3.8.8.4.cmml">[</mo><mrow id="S3.p4.4.m3.6.6.1.1" xref="S3.p4.4.m3.6.6.1.1.cmml"><mover accent="true" id="S3.p4.4.m3.6.6.1.1.2" xref="S3.p4.4.m3.6.6.1.1.2.cmml"><mi id="S3.p4.4.m3.6.6.1.1.2.2" xref="S3.p4.4.m3.6.6.1.1.2.2.cmml">𝜽</mi><mo id="S3.p4.4.m3.6.6.1.1.2.1" xref="S3.p4.4.m3.6.6.1.1.2.1.cmml">^</mo></mover><mo id="S3.p4.4.m3.6.6.1.1.1" xref="S3.p4.4.m3.6.6.1.1.1.cmml">⁢</mo><mrow id="S3.p4.4.m3.6.6.1.1.3.2" xref="S3.p4.4.m3.6.6.1.1.cmml"><mo id="S3.p4.4.m3.6.6.1.1.3.2.1" stretchy="false" xref="S3.p4.4.m3.6.6.1.1.cmml">(</mo><mi id="S3.p4.4.m3.2.2" xref="S3.p4.4.m3.2.2.cmml">t</mi><mo id="S3.p4.4.m3.6.6.1.1.3.2.2" stretchy="false" xref="S3.p4.4.m3.6.6.1.1.cmml">)</mo></mrow></mrow><mo id="S3.p4.4.m3.8.8.3.5" xref="S3.p4.4.m3.8.8.4.cmml">,</mo><mrow id="S3.p4.4.m3.7.7.2.2" xref="S3.p4.4.m3.7.7.2.2.cmml"><mover accent="true" id="S3.p4.4.m3.7.7.2.2.2" xref="S3.p4.4.m3.7.7.2.2.2.cmml"><mover accent="true" id="S3.p4.4.m3.7.7.2.2.2.2" xref="S3.p4.4.m3.7.7.2.2.2.2.cmml"><mi id="S3.p4.4.m3.7.7.2.2.2.2.2" xref="S3.p4.4.m3.7.7.2.2.2.2.2.cmml">𝜽</mi><mo id="S3.p4.4.m3.7.7.2.2.2.2.1" xref="S3.p4.4.m3.7.7.2.2.2.2.1.cmml">^</mo></mover><mo id="S3.p4.4.m3.7.7.2.2.2.1" xref="S3.p4.4.m3.7.7.2.2.2.1.cmml">˙</mo></mover><mo id="S3.p4.4.m3.7.7.2.2.1" xref="S3.p4.4.m3.7.7.2.2.1.cmml">⁢</mo><mrow id="S3.p4.4.m3.7.7.2.2.3.2" xref="S3.p4.4.m3.7.7.2.2.cmml"><mo id="S3.p4.4.m3.7.7.2.2.3.2.1" stretchy="false" xref="S3.p4.4.m3.7.7.2.2.cmml">(</mo><mi id="S3.p4.4.m3.3.3" xref="S3.p4.4.m3.3.3.cmml">t</mi><mo id="S3.p4.4.m3.7.7.2.2.3.2.2" stretchy="false" xref="S3.p4.4.m3.7.7.2.2.cmml">)</mo></mrow></mrow><mo id="S3.p4.4.m3.8.8.3.6" xref="S3.p4.4.m3.8.8.4.cmml">,</mo><mi id="S3.p4.4.m3.5.5" mathvariant="normal" xref="S3.p4.4.m3.5.5.cmml">⋯</mi><mo id="S3.p4.4.m3.8.8.3.7" xref="S3.p4.4.m3.8.8.4.cmml">,</mo><mrow id="S3.p4.4.m3.8.8.3.3" xref="S3.p4.4.m3.8.8.3.3.cmml"><msup id="S3.p4.4.m3.8.8.3.3.2" xref="S3.p4.4.m3.8.8.3.3.2.cmml"><mover accent="true" id="S3.p4.4.m3.8.8.3.3.2.2" xref="S3.p4.4.m3.8.8.3.3.2.2.cmml"><mi id="S3.p4.4.m3.8.8.3.3.2.2.2" xref="S3.p4.4.m3.8.8.3.3.2.2.2.cmml">𝜽</mi><mo id="S3.p4.4.m3.8.8.3.3.2.2.1" xref="S3.p4.4.m3.8.8.3.3.2.2.1.cmml">^</mo></mover><mrow id="S3.p4.4.m3.1.1.1.1" xref="S3.p4.4.m3.1.1.1.1.1.cmml"><mo id="S3.p4.4.m3.1.1.1.1.2" stretchy="false" xref="S3.p4.4.m3.1.1.1.1.1.cmml">(</mo><mrow id="S3.p4.4.m3.1.1.1.1.1" xref="S3.p4.4.m3.1.1.1.1.1.cmml"><mi id="S3.p4.4.m3.1.1.1.1.1.2" xref="S3.p4.4.m3.1.1.1.1.1.2.cmml">i</mi><mo id="S3.p4.4.m3.1.1.1.1.1.1" xref="S3.p4.4.m3.1.1.1.1.1.1.cmml">−</mo><mn id="S3.p4.4.m3.1.1.1.1.1.3" xref="S3.p4.4.m3.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.p4.4.m3.1.1.1.1.3" stretchy="false" xref="S3.p4.4.m3.1.1.1.1.1.cmml">)</mo></mrow></msup><mo id="S3.p4.4.m3.8.8.3.3.1" xref="S3.p4.4.m3.8.8.3.3.1.cmml">⁢</mo><mrow id="S3.p4.4.m3.8.8.3.3.3.2" xref="S3.p4.4.m3.8.8.3.3.cmml"><mo id="S3.p4.4.m3.8.8.3.3.3.2.1" stretchy="false" xref="S3.p4.4.m3.8.8.3.3.cmml">(</mo><mi id="S3.p4.4.m3.4.4" xref="S3.p4.4.m3.4.4.cmml">t</mi><mo id="S3.p4.4.m3.8.8.3.3.3.2.2" stretchy="false" xref="S3.p4.4.m3.8.8.3.3.cmml">)</mo></mrow></mrow><mo id="S3.p4.4.m3.8.8.3.8" maxsize="120%" minsize="120%" xref="S3.p4.4.m3.8.8.4.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.4.m3.8b"><list id="S3.p4.4.m3.8.8.4.cmml" xref="S3.p4.4.m3.8.8.3"><apply id="S3.p4.4.m3.6.6.1.1.cmml" xref="S3.p4.4.m3.6.6.1.1"><times id="S3.p4.4.m3.6.6.1.1.1.cmml" xref="S3.p4.4.m3.6.6.1.1.1"></times><apply id="S3.p4.4.m3.6.6.1.1.2.cmml" xref="S3.p4.4.m3.6.6.1.1.2"><ci id="S3.p4.4.m3.6.6.1.1.2.1.cmml" xref="S3.p4.4.m3.6.6.1.1.2.1">^</ci><ci id="S3.p4.4.m3.6.6.1.1.2.2.cmml" xref="S3.p4.4.m3.6.6.1.1.2.2">𝜽</ci></apply><ci id="S3.p4.4.m3.2.2.cmml" xref="S3.p4.4.m3.2.2">𝑡</ci></apply><apply id="S3.p4.4.m3.7.7.2.2.cmml" xref="S3.p4.4.m3.7.7.2.2"><times id="S3.p4.4.m3.7.7.2.2.1.cmml" xref="S3.p4.4.m3.7.7.2.2.1"></times><apply id="S3.p4.4.m3.7.7.2.2.2.cmml" xref="S3.p4.4.m3.7.7.2.2.2"><ci id="S3.p4.4.m3.7.7.2.2.2.1.cmml" xref="S3.p4.4.m3.7.7.2.2.2.1">˙</ci><apply id="S3.p4.4.m3.7.7.2.2.2.2.cmml" xref="S3.p4.4.m3.7.7.2.2.2.2"><ci id="S3.p4.4.m3.7.7.2.2.2.2.1.cmml" xref="S3.p4.4.m3.7.7.2.2.2.2.1">^</ci><ci id="S3.p4.4.m3.7.7.2.2.2.2.2.cmml" xref="S3.p4.4.m3.7.7.2.2.2.2.2">𝜽</ci></apply></apply><ci id="S3.p4.4.m3.3.3.cmml" xref="S3.p4.4.m3.3.3">𝑡</ci></apply><ci id="S3.p4.4.m3.5.5.cmml" xref="S3.p4.4.m3.5.5">⋯</ci><apply id="S3.p4.4.m3.8.8.3.3.cmml" xref="S3.p4.4.m3.8.8.3.3"><times id="S3.p4.4.m3.8.8.3.3.1.cmml" xref="S3.p4.4.m3.8.8.3.3.1"></times><apply id="S3.p4.4.m3.8.8.3.3.2.cmml" xref="S3.p4.4.m3.8.8.3.3.2"><csymbol cd="ambiguous" id="S3.p4.4.m3.8.8.3.3.2.1.cmml" xref="S3.p4.4.m3.8.8.3.3.2">superscript</csymbol><apply id="S3.p4.4.m3.8.8.3.3.2.2.cmml" xref="S3.p4.4.m3.8.8.3.3.2.2"><ci id="S3.p4.4.m3.8.8.3.3.2.2.1.cmml" xref="S3.p4.4.m3.8.8.3.3.2.2.1">^</ci><ci id="S3.p4.4.m3.8.8.3.3.2.2.2.cmml" xref="S3.p4.4.m3.8.8.3.3.2.2.2">𝜽</ci></apply><apply id="S3.p4.4.m3.1.1.1.1.1.cmml" xref="S3.p4.4.m3.1.1.1.1"><minus id="S3.p4.4.m3.1.1.1.1.1.1.cmml" xref="S3.p4.4.m3.1.1.1.1.1.1"></minus><ci id="S3.p4.4.m3.1.1.1.1.1.2.cmml" xref="S3.p4.4.m3.1.1.1.1.1.2">𝑖</ci><cn id="S3.p4.4.m3.1.1.1.1.1.3.cmml" type="integer" xref="S3.p4.4.m3.1.1.1.1.1.3">1</cn></apply></apply><ci id="S3.p4.4.m3.4.4.cmml" xref="S3.p4.4.m3.4.4">𝑡</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.4.m3.8c">\big{[}\hat{\bm{\theta}}(t),\dot{\hat{\bm{\theta}}}(t),\cdots,\hat{\bm{\theta}% }^{(i-1)}(t)\big{]}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.4.m3.8d">[ over^ start_ARG bold_italic_θ end_ARG ( italic_t ) , over˙ start_ARG over^ start_ARG bold_italic_θ end_ARG end_ARG ( italic_t ) , ⋯ , over^ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT ( italic_i - 1 ) end_POSTSUPERSCRIPT ( italic_t ) ]</annotation></semantics></math> <math alttext="\in\mathbb{R}^{N\times i}" class="ltx_Math" display="inline" id="S3.p4.5.m4.1"><semantics id="S3.p4.5.m4.1a"><mrow id="S3.p4.5.m4.1.1" xref="S3.p4.5.m4.1.1.cmml"><mi id="S3.p4.5.m4.1.1.2" xref="S3.p4.5.m4.1.1.2.cmml"></mi><mo id="S3.p4.5.m4.1.1.1" xref="S3.p4.5.m4.1.1.1.cmml">∈</mo><msup id="S3.p4.5.m4.1.1.3" xref="S3.p4.5.m4.1.1.3.cmml"><mi id="S3.p4.5.m4.1.1.3.2" xref="S3.p4.5.m4.1.1.3.2.cmml">ℝ</mi><mrow id="S3.p4.5.m4.1.1.3.3" xref="S3.p4.5.m4.1.1.3.3.cmml"><mi id="S3.p4.5.m4.1.1.3.3.2" xref="S3.p4.5.m4.1.1.3.3.2.cmml">N</mi><mo id="S3.p4.5.m4.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S3.p4.5.m4.1.1.3.3.1.cmml">×</mo><mi id="S3.p4.5.m4.1.1.3.3.3" xref="S3.p4.5.m4.1.1.3.3.3.cmml">i</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.5.m4.1b"><apply id="S3.p4.5.m4.1.1.cmml" xref="S3.p4.5.m4.1.1"><in id="S3.p4.5.m4.1.1.1.cmml" xref="S3.p4.5.m4.1.1.1"></in><csymbol cd="latexml" id="S3.p4.5.m4.1.1.2.cmml" xref="S3.p4.5.m4.1.1.2">absent</csymbol><apply id="S3.p4.5.m4.1.1.3.cmml" xref="S3.p4.5.m4.1.1.3"><csymbol cd="ambiguous" id="S3.p4.5.m4.1.1.3.1.cmml" xref="S3.p4.5.m4.1.1.3">superscript</csymbol><ci id="S3.p4.5.m4.1.1.3.2.cmml" xref="S3.p4.5.m4.1.1.3.2">ℝ</ci><apply id="S3.p4.5.m4.1.1.3.3.cmml" xref="S3.p4.5.m4.1.1.3.3"><times id="S3.p4.5.m4.1.1.3.3.1.cmml" xref="S3.p4.5.m4.1.1.3.3.1"></times><ci id="S3.p4.5.m4.1.1.3.3.2.cmml" xref="S3.p4.5.m4.1.1.3.3.2">𝑁</ci><ci id="S3.p4.5.m4.1.1.3.3.3.cmml" xref="S3.p4.5.m4.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.5.m4.1c">\in\mathbb{R}^{N\times i}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.5.m4.1d">∈ blackboard_R start_POSTSUPERSCRIPT italic_N × italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="{\bm{y}}_{{\rm r}i}(t)" class="ltx_Math" display="inline" id="S3.p4.6.m5.1"><semantics id="S3.p4.6.m5.1a"><mrow id="S3.p4.6.m5.1.2" xref="S3.p4.6.m5.1.2.cmml"><msub id="S3.p4.6.m5.1.2.2" xref="S3.p4.6.m5.1.2.2.cmml"><mi id="S3.p4.6.m5.1.2.2.2" xref="S3.p4.6.m5.1.2.2.2.cmml">𝒚</mi><mrow id="S3.p4.6.m5.1.2.2.3" xref="S3.p4.6.m5.1.2.2.3.cmml"><mi id="S3.p4.6.m5.1.2.2.3.2" mathvariant="normal" xref="S3.p4.6.m5.1.2.2.3.2.cmml">r</mi><mo id="S3.p4.6.m5.1.2.2.3.1" xref="S3.p4.6.m5.1.2.2.3.1.cmml">⁢</mo><mi id="S3.p4.6.m5.1.2.2.3.3" xref="S3.p4.6.m5.1.2.2.3.3.cmml">i</mi></mrow></msub><mo id="S3.p4.6.m5.1.2.1" xref="S3.p4.6.m5.1.2.1.cmml">⁢</mo><mrow id="S3.p4.6.m5.1.2.3.2" xref="S3.p4.6.m5.1.2.cmml"><mo id="S3.p4.6.m5.1.2.3.2.1" stretchy="false" xref="S3.p4.6.m5.1.2.cmml">(</mo><mi id="S3.p4.6.m5.1.1" xref="S3.p4.6.m5.1.1.cmml">t</mi><mo id="S3.p4.6.m5.1.2.3.2.2" stretchy="false" xref="S3.p4.6.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.6.m5.1b"><apply id="S3.p4.6.m5.1.2.cmml" xref="S3.p4.6.m5.1.2"><times id="S3.p4.6.m5.1.2.1.cmml" xref="S3.p4.6.m5.1.2.1"></times><apply id="S3.p4.6.m5.1.2.2.cmml" xref="S3.p4.6.m5.1.2.2"><csymbol cd="ambiguous" id="S3.p4.6.m5.1.2.2.1.cmml" xref="S3.p4.6.m5.1.2.2">subscript</csymbol><ci id="S3.p4.6.m5.1.2.2.2.cmml" xref="S3.p4.6.m5.1.2.2.2">𝒚</ci><apply id="S3.p4.6.m5.1.2.2.3.cmml" xref="S3.p4.6.m5.1.2.2.3"><times id="S3.p4.6.m5.1.2.2.3.1.cmml" xref="S3.p4.6.m5.1.2.2.3.1"></times><ci id="S3.p4.6.m5.1.2.2.3.2.cmml" xref="S3.p4.6.m5.1.2.2.3.2">r</ci><ci id="S3.p4.6.m5.1.2.2.3.3.cmml" xref="S3.p4.6.m5.1.2.2.3.3">𝑖</ci></apply></apply><ci id="S3.p4.6.m5.1.1.cmml" xref="S3.p4.6.m5.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.6.m5.1c">{\bm{y}}_{{\rm r}i}(t)</annotation><annotation encoding="application/x-llamapun" id="S3.p4.6.m5.1d">bold_italic_y start_POSTSUBSCRIPT roman_r italic_i end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S3.p4.7.m6.1"><semantics id="S3.p4.7.m6.1a"><mo id="S3.p4.7.m6.1.1" xref="S3.p4.7.m6.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S3.p4.7.m6.1b"><csymbol cd="latexml" id="S3.p4.7.m6.1.1.cmml" xref="S3.p4.7.m6.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.7.m6.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S3.p4.7.m6.1d">:=</annotation></semantics></math> <math alttext="\big{[}y_{\rm r}(t)" class="ltx_math_unparsed" display="inline" id="S3.p4.8.m7.1"><semantics id="S3.p4.8.m7.1a"><mrow id="S3.p4.8.m7.1b"><mo id="S3.p4.8.m7.1.2" maxsize="120%" minsize="120%">[</mo><msub id="S3.p4.8.m7.1.3"><mi id="S3.p4.8.m7.1.3.2">y</mi><mi id="S3.p4.8.m7.1.3.3" mathvariant="normal">r</mi></msub><mrow id="S3.p4.8.m7.1.4"><mo id="S3.p4.8.m7.1.4.1" stretchy="false">(</mo><mi id="S3.p4.8.m7.1.1">t</mi><mo id="S3.p4.8.m7.1.4.2" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.p4.8.m7.1c">\big{[}y_{\rm r}(t)</annotation><annotation encoding="application/x-llamapun" id="S3.p4.8.m7.1d">[ italic_y start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math>, <math alttext="\dot{y}_{\rm r}(t)" class="ltx_Math" display="inline" id="S3.p4.9.m8.1"><semantics id="S3.p4.9.m8.1a"><mrow id="S3.p4.9.m8.1.2" xref="S3.p4.9.m8.1.2.cmml"><msub id="S3.p4.9.m8.1.2.2" xref="S3.p4.9.m8.1.2.2.cmml"><mover accent="true" id="S3.p4.9.m8.1.2.2.2" xref="S3.p4.9.m8.1.2.2.2.cmml"><mi id="S3.p4.9.m8.1.2.2.2.2" xref="S3.p4.9.m8.1.2.2.2.2.cmml">y</mi><mo id="S3.p4.9.m8.1.2.2.2.1" xref="S3.p4.9.m8.1.2.2.2.1.cmml">˙</mo></mover><mi id="S3.p4.9.m8.1.2.2.3" mathvariant="normal" xref="S3.p4.9.m8.1.2.2.3.cmml">r</mi></msub><mo id="S3.p4.9.m8.1.2.1" xref="S3.p4.9.m8.1.2.1.cmml">⁢</mo><mrow id="S3.p4.9.m8.1.2.3.2" xref="S3.p4.9.m8.1.2.cmml"><mo id="S3.p4.9.m8.1.2.3.2.1" stretchy="false" xref="S3.p4.9.m8.1.2.cmml">(</mo><mi id="S3.p4.9.m8.1.1" xref="S3.p4.9.m8.1.1.cmml">t</mi><mo id="S3.p4.9.m8.1.2.3.2.2" stretchy="false" xref="S3.p4.9.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.9.m8.1b"><apply id="S3.p4.9.m8.1.2.cmml" xref="S3.p4.9.m8.1.2"><times id="S3.p4.9.m8.1.2.1.cmml" xref="S3.p4.9.m8.1.2.1"></times><apply id="S3.p4.9.m8.1.2.2.cmml" xref="S3.p4.9.m8.1.2.2"><csymbol cd="ambiguous" id="S3.p4.9.m8.1.2.2.1.cmml" xref="S3.p4.9.m8.1.2.2">subscript</csymbol><apply id="S3.p4.9.m8.1.2.2.2.cmml" xref="S3.p4.9.m8.1.2.2.2"><ci id="S3.p4.9.m8.1.2.2.2.1.cmml" xref="S3.p4.9.m8.1.2.2.2.1">˙</ci><ci id="S3.p4.9.m8.1.2.2.2.2.cmml" xref="S3.p4.9.m8.1.2.2.2.2">𝑦</ci></apply><ci id="S3.p4.9.m8.1.2.2.3.cmml" xref="S3.p4.9.m8.1.2.2.3">r</ci></apply><ci id="S3.p4.9.m8.1.1.cmml" xref="S3.p4.9.m8.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.9.m8.1c">\dot{y}_{\rm r}(t)</annotation><annotation encoding="application/x-llamapun" id="S3.p4.9.m8.1d">over˙ start_ARG italic_y end_ARG start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math>, <math alttext="\cdots" class="ltx_Math" display="inline" id="S3.p4.10.m9.1"><semantics id="S3.p4.10.m9.1a"><mi id="S3.p4.10.m9.1.1" mathvariant="normal" xref="S3.p4.10.m9.1.1.cmml">⋯</mi><annotation-xml encoding="MathML-Content" id="S3.p4.10.m9.1b"><ci id="S3.p4.10.m9.1.1.cmml" xref="S3.p4.10.m9.1.1">⋯</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.10.m9.1c">\cdots</annotation><annotation encoding="application/x-llamapun" id="S3.p4.10.m9.1d">⋯</annotation></semantics></math>, <math alttext="y^{(i-1)}_{\rm r}(t)\big{]}^{T}" class="ltx_math_unparsed" display="inline" id="S3.p4.11.m10.2"><semantics id="S3.p4.11.m10.2a"><mrow id="S3.p4.11.m10.2b"><msubsup id="S3.p4.11.m10.2.3"><mi id="S3.p4.11.m10.2.3.2.2">y</mi><mi id="S3.p4.11.m10.2.3.3" mathvariant="normal">r</mi><mrow id="S3.p4.11.m10.1.1.1.1"><mo id="S3.p4.11.m10.1.1.1.1.2" stretchy="false">(</mo><mrow id="S3.p4.11.m10.1.1.1.1.1"><mi id="S3.p4.11.m10.1.1.1.1.1.2">i</mi><mo id="S3.p4.11.m10.1.1.1.1.1.1">−</mo><mn id="S3.p4.11.m10.1.1.1.1.1.3">1</mn></mrow><mo id="S3.p4.11.m10.1.1.1.1.3" stretchy="false">)</mo></mrow></msubsup><mrow id="S3.p4.11.m10.2.4"><mo id="S3.p4.11.m10.2.4.1" stretchy="false">(</mo><mi id="S3.p4.11.m10.2.2">t</mi><mo id="S3.p4.11.m10.2.4.2" stretchy="false">)</mo></mrow><mo id="S3.p4.11.m10.2.5" maxsize="120%" minsize="120%">]</mo><msup id="S3.p4.11.m10.2.6"><mi id="S3.p4.11.m10.2.6a"></mi><mi id="S3.p4.11.m10.2.6.1">T</mi></msup></mrow><annotation encoding="application/x-tex" id="S3.p4.11.m10.2c">y^{(i-1)}_{\rm r}(t)\big{]}^{T}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.11.m10.2d">italic_y start_POSTSUPERSCRIPT ( italic_i - 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT ( italic_t ) ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> <math alttext="\in\mathbb{R}^{i}" class="ltx_Math" display="inline" id="S3.p4.12.m11.1"><semantics id="S3.p4.12.m11.1a"><mrow id="S3.p4.12.m11.1.1" xref="S3.p4.12.m11.1.1.cmml"><mi id="S3.p4.12.m11.1.1.2" xref="S3.p4.12.m11.1.1.2.cmml"></mi><mo id="S3.p4.12.m11.1.1.1" xref="S3.p4.12.m11.1.1.1.cmml">∈</mo><msup id="S3.p4.12.m11.1.1.3" xref="S3.p4.12.m11.1.1.3.cmml"><mi id="S3.p4.12.m11.1.1.3.2" xref="S3.p4.12.m11.1.1.3.2.cmml">ℝ</mi><mi id="S3.p4.12.m11.1.1.3.3" xref="S3.p4.12.m11.1.1.3.3.cmml">i</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.12.m11.1b"><apply id="S3.p4.12.m11.1.1.cmml" xref="S3.p4.12.m11.1.1"><in id="S3.p4.12.m11.1.1.1.cmml" xref="S3.p4.12.m11.1.1.1"></in><csymbol cd="latexml" id="S3.p4.12.m11.1.1.2.cmml" xref="S3.p4.12.m11.1.1.2">absent</csymbol><apply id="S3.p4.12.m11.1.1.3.cmml" xref="S3.p4.12.m11.1.1.3"><csymbol cd="ambiguous" id="S3.p4.12.m11.1.1.3.1.cmml" xref="S3.p4.12.m11.1.1.3">superscript</csymbol><ci id="S3.p4.12.m11.1.1.3.2.cmml" xref="S3.p4.12.m11.1.1.3.2">ℝ</ci><ci id="S3.p4.12.m11.1.1.3.3.cmml" xref="S3.p4.12.m11.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.12.m11.1c">\in\mathbb{R}^{i}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.12.m11.1d">∈ blackboard_R start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="e_{1}(t)" class="ltx_Math" display="inline" id="S3.p4.13.m12.1"><semantics id="S3.p4.13.m12.1a"><mrow id="S3.p4.13.m12.1.2" xref="S3.p4.13.m12.1.2.cmml"><msub id="S3.p4.13.m12.1.2.2" xref="S3.p4.13.m12.1.2.2.cmml"><mi id="S3.p4.13.m12.1.2.2.2" xref="S3.p4.13.m12.1.2.2.2.cmml">e</mi><mn id="S3.p4.13.m12.1.2.2.3" xref="S3.p4.13.m12.1.2.2.3.cmml">1</mn></msub><mo id="S3.p4.13.m12.1.2.1" xref="S3.p4.13.m12.1.2.1.cmml">⁢</mo><mrow id="S3.p4.13.m12.1.2.3.2" xref="S3.p4.13.m12.1.2.cmml"><mo id="S3.p4.13.m12.1.2.3.2.1" stretchy="false" xref="S3.p4.13.m12.1.2.cmml">(</mo><mi id="S3.p4.13.m12.1.1" xref="S3.p4.13.m12.1.1.cmml">t</mi><mo id="S3.p4.13.m12.1.2.3.2.2" stretchy="false" xref="S3.p4.13.m12.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.13.m12.1b"><apply id="S3.p4.13.m12.1.2.cmml" xref="S3.p4.13.m12.1.2"><times id="S3.p4.13.m12.1.2.1.cmml" xref="S3.p4.13.m12.1.2.1"></times><apply id="S3.p4.13.m12.1.2.2.cmml" xref="S3.p4.13.m12.1.2.2"><csymbol cd="ambiguous" id="S3.p4.13.m12.1.2.2.1.cmml" xref="S3.p4.13.m12.1.2.2">subscript</csymbol><ci id="S3.p4.13.m12.1.2.2.2.cmml" xref="S3.p4.13.m12.1.2.2.2">𝑒</ci><cn id="S3.p4.13.m12.1.2.2.3.cmml" type="integer" xref="S3.p4.13.m12.1.2.2.3">1</cn></apply><ci id="S3.p4.13.m12.1.1.cmml" xref="S3.p4.13.m12.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.13.m12.1c">e_{1}(t)</annotation><annotation encoding="application/x-llamapun" id="S3.p4.13.m12.1d">italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S3.p4.14.m13.1"><semantics id="S3.p4.14.m13.1a"><mo id="S3.p4.14.m13.1.1" xref="S3.p4.14.m13.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S3.p4.14.m13.1b"><csymbol cd="latexml" id="S3.p4.14.m13.1.1.cmml" xref="S3.p4.14.m13.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.14.m13.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S3.p4.14.m13.1d">:=</annotation></semantics></math> <math alttext="x_{1}(t)-y_{\rm r}(t)\in\mathbb{R}" class="ltx_Math" display="inline" id="S3.p4.15.m14.2"><semantics id="S3.p4.15.m14.2a"><mrow id="S3.p4.15.m14.2.3" xref="S3.p4.15.m14.2.3.cmml"><mrow id="S3.p4.15.m14.2.3.2" xref="S3.p4.15.m14.2.3.2.cmml"><mrow id="S3.p4.15.m14.2.3.2.2" xref="S3.p4.15.m14.2.3.2.2.cmml"><msub id="S3.p4.15.m14.2.3.2.2.2" xref="S3.p4.15.m14.2.3.2.2.2.cmml"><mi id="S3.p4.15.m14.2.3.2.2.2.2" xref="S3.p4.15.m14.2.3.2.2.2.2.cmml">x</mi><mn id="S3.p4.15.m14.2.3.2.2.2.3" xref="S3.p4.15.m14.2.3.2.2.2.3.cmml">1</mn></msub><mo id="S3.p4.15.m14.2.3.2.2.1" xref="S3.p4.15.m14.2.3.2.2.1.cmml">⁢</mo><mrow id="S3.p4.15.m14.2.3.2.2.3.2" xref="S3.p4.15.m14.2.3.2.2.cmml"><mo id="S3.p4.15.m14.2.3.2.2.3.2.1" stretchy="false" xref="S3.p4.15.m14.2.3.2.2.cmml">(</mo><mi id="S3.p4.15.m14.1.1" xref="S3.p4.15.m14.1.1.cmml">t</mi><mo id="S3.p4.15.m14.2.3.2.2.3.2.2" stretchy="false" xref="S3.p4.15.m14.2.3.2.2.cmml">)</mo></mrow></mrow><mo id="S3.p4.15.m14.2.3.2.1" xref="S3.p4.15.m14.2.3.2.1.cmml">−</mo><mrow id="S3.p4.15.m14.2.3.2.3" xref="S3.p4.15.m14.2.3.2.3.cmml"><msub id="S3.p4.15.m14.2.3.2.3.2" xref="S3.p4.15.m14.2.3.2.3.2.cmml"><mi id="S3.p4.15.m14.2.3.2.3.2.2" xref="S3.p4.15.m14.2.3.2.3.2.2.cmml">y</mi><mi id="S3.p4.15.m14.2.3.2.3.2.3" mathvariant="normal" xref="S3.p4.15.m14.2.3.2.3.2.3.cmml">r</mi></msub><mo id="S3.p4.15.m14.2.3.2.3.1" xref="S3.p4.15.m14.2.3.2.3.1.cmml">⁢</mo><mrow id="S3.p4.15.m14.2.3.2.3.3.2" xref="S3.p4.15.m14.2.3.2.3.cmml"><mo id="S3.p4.15.m14.2.3.2.3.3.2.1" stretchy="false" xref="S3.p4.15.m14.2.3.2.3.cmml">(</mo><mi id="S3.p4.15.m14.2.2" xref="S3.p4.15.m14.2.2.cmml">t</mi><mo id="S3.p4.15.m14.2.3.2.3.3.2.2" stretchy="false" xref="S3.p4.15.m14.2.3.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.p4.15.m14.2.3.1" xref="S3.p4.15.m14.2.3.1.cmml">∈</mo><mi id="S3.p4.15.m14.2.3.3" xref="S3.p4.15.m14.2.3.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.15.m14.2b"><apply id="S3.p4.15.m14.2.3.cmml" xref="S3.p4.15.m14.2.3"><in id="S3.p4.15.m14.2.3.1.cmml" xref="S3.p4.15.m14.2.3.1"></in><apply id="S3.p4.15.m14.2.3.2.cmml" xref="S3.p4.15.m14.2.3.2"><minus id="S3.p4.15.m14.2.3.2.1.cmml" xref="S3.p4.15.m14.2.3.2.1"></minus><apply id="S3.p4.15.m14.2.3.2.2.cmml" xref="S3.p4.15.m14.2.3.2.2"><times id="S3.p4.15.m14.2.3.2.2.1.cmml" xref="S3.p4.15.m14.2.3.2.2.1"></times><apply id="S3.p4.15.m14.2.3.2.2.2.cmml" xref="S3.p4.15.m14.2.3.2.2.2"><csymbol cd="ambiguous" id="S3.p4.15.m14.2.3.2.2.2.1.cmml" xref="S3.p4.15.m14.2.3.2.2.2">subscript</csymbol><ci id="S3.p4.15.m14.2.3.2.2.2.2.cmml" xref="S3.p4.15.m14.2.3.2.2.2.2">𝑥</ci><cn id="S3.p4.15.m14.2.3.2.2.2.3.cmml" type="integer" xref="S3.p4.15.m14.2.3.2.2.2.3">1</cn></apply><ci id="S3.p4.15.m14.1.1.cmml" xref="S3.p4.15.m14.1.1">𝑡</ci></apply><apply id="S3.p4.15.m14.2.3.2.3.cmml" xref="S3.p4.15.m14.2.3.2.3"><times id="S3.p4.15.m14.2.3.2.3.1.cmml" xref="S3.p4.15.m14.2.3.2.3.1"></times><apply id="S3.p4.15.m14.2.3.2.3.2.cmml" xref="S3.p4.15.m14.2.3.2.3.2"><csymbol cd="ambiguous" id="S3.p4.15.m14.2.3.2.3.2.1.cmml" xref="S3.p4.15.m14.2.3.2.3.2">subscript</csymbol><ci id="S3.p4.15.m14.2.3.2.3.2.2.cmml" xref="S3.p4.15.m14.2.3.2.3.2.2">𝑦</ci><ci id="S3.p4.15.m14.2.3.2.3.2.3.cmml" xref="S3.p4.15.m14.2.3.2.3.2.3">r</ci></apply><ci id="S3.p4.15.m14.2.2.cmml" xref="S3.p4.15.m14.2.2">𝑡</ci></apply></apply><ci id="S3.p4.15.m14.2.3.3.cmml" xref="S3.p4.15.m14.2.3.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.15.m14.2c">x_{1}(t)-y_{\rm r}(t)\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.15.m14.2d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) - italic_y start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT ( italic_t ) ∈ blackboard_R</annotation></semantics></math> and <math alttext="e_{i}(t)" class="ltx_Math" display="inline" id="S3.p4.16.m15.1"><semantics id="S3.p4.16.m15.1a"><mrow id="S3.p4.16.m15.1.2" xref="S3.p4.16.m15.1.2.cmml"><msub id="S3.p4.16.m15.1.2.2" xref="S3.p4.16.m15.1.2.2.cmml"><mi id="S3.p4.16.m15.1.2.2.2" xref="S3.p4.16.m15.1.2.2.2.cmml">e</mi><mi id="S3.p4.16.m15.1.2.2.3" xref="S3.p4.16.m15.1.2.2.3.cmml">i</mi></msub><mo id="S3.p4.16.m15.1.2.1" xref="S3.p4.16.m15.1.2.1.cmml">⁢</mo><mrow id="S3.p4.16.m15.1.2.3.2" xref="S3.p4.16.m15.1.2.cmml"><mo id="S3.p4.16.m15.1.2.3.2.1" stretchy="false" xref="S3.p4.16.m15.1.2.cmml">(</mo><mi id="S3.p4.16.m15.1.1" xref="S3.p4.16.m15.1.1.cmml">t</mi><mo id="S3.p4.16.m15.1.2.3.2.2" stretchy="false" xref="S3.p4.16.m15.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.16.m15.1b"><apply id="S3.p4.16.m15.1.2.cmml" xref="S3.p4.16.m15.1.2"><times id="S3.p4.16.m15.1.2.1.cmml" xref="S3.p4.16.m15.1.2.1"></times><apply id="S3.p4.16.m15.1.2.2.cmml" xref="S3.p4.16.m15.1.2.2"><csymbol cd="ambiguous" id="S3.p4.16.m15.1.2.2.1.cmml" xref="S3.p4.16.m15.1.2.2">subscript</csymbol><ci id="S3.p4.16.m15.1.2.2.2.cmml" xref="S3.p4.16.m15.1.2.2.2">𝑒</ci><ci id="S3.p4.16.m15.1.2.2.3.cmml" xref="S3.p4.16.m15.1.2.2.3">𝑖</ci></apply><ci id="S3.p4.16.m15.1.1.cmml" xref="S3.p4.16.m15.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.16.m15.1c">e_{i}(t)</annotation><annotation encoding="application/x-llamapun" id="S3.p4.16.m15.1d">italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S3.p4.17.m16.1"><semantics id="S3.p4.17.m16.1a"><mo id="S3.p4.17.m16.1.1" xref="S3.p4.17.m16.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S3.p4.17.m16.1b"><csymbol cd="latexml" id="S3.p4.17.m16.1.1.cmml" xref="S3.p4.17.m16.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.17.m16.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S3.p4.17.m16.1d">:=</annotation></semantics></math> <math alttext="x_{i}(t)-v_{i-1}(t)-y_{\rm r}^{(i-1)}(t)\in\mathbb{R}" class="ltx_Math" display="inline" id="S3.p4.18.m17.4"><semantics id="S3.p4.18.m17.4a"><mrow id="S3.p4.18.m17.4.5" xref="S3.p4.18.m17.4.5.cmml"><mrow id="S3.p4.18.m17.4.5.2" xref="S3.p4.18.m17.4.5.2.cmml"><mrow id="S3.p4.18.m17.4.5.2.2" xref="S3.p4.18.m17.4.5.2.2.cmml"><msub id="S3.p4.18.m17.4.5.2.2.2" xref="S3.p4.18.m17.4.5.2.2.2.cmml"><mi id="S3.p4.18.m17.4.5.2.2.2.2" xref="S3.p4.18.m17.4.5.2.2.2.2.cmml">x</mi><mi id="S3.p4.18.m17.4.5.2.2.2.3" xref="S3.p4.18.m17.4.5.2.2.2.3.cmml">i</mi></msub><mo id="S3.p4.18.m17.4.5.2.2.1" xref="S3.p4.18.m17.4.5.2.2.1.cmml">⁢</mo><mrow id="S3.p4.18.m17.4.5.2.2.3.2" xref="S3.p4.18.m17.4.5.2.2.cmml"><mo id="S3.p4.18.m17.4.5.2.2.3.2.1" stretchy="false" xref="S3.p4.18.m17.4.5.2.2.cmml">(</mo><mi id="S3.p4.18.m17.2.2" xref="S3.p4.18.m17.2.2.cmml">t</mi><mo id="S3.p4.18.m17.4.5.2.2.3.2.2" stretchy="false" xref="S3.p4.18.m17.4.5.2.2.cmml">)</mo></mrow></mrow><mo id="S3.p4.18.m17.4.5.2.1" xref="S3.p4.18.m17.4.5.2.1.cmml">−</mo><mrow id="S3.p4.18.m17.4.5.2.3" xref="S3.p4.18.m17.4.5.2.3.cmml"><msub id="S3.p4.18.m17.4.5.2.3.2" xref="S3.p4.18.m17.4.5.2.3.2.cmml"><mi id="S3.p4.18.m17.4.5.2.3.2.2" xref="S3.p4.18.m17.4.5.2.3.2.2.cmml">v</mi><mrow id="S3.p4.18.m17.4.5.2.3.2.3" xref="S3.p4.18.m17.4.5.2.3.2.3.cmml"><mi id="S3.p4.18.m17.4.5.2.3.2.3.2" xref="S3.p4.18.m17.4.5.2.3.2.3.2.cmml">i</mi><mo id="S3.p4.18.m17.4.5.2.3.2.3.1" xref="S3.p4.18.m17.4.5.2.3.2.3.1.cmml">−</mo><mn id="S3.p4.18.m17.4.5.2.3.2.3.3" xref="S3.p4.18.m17.4.5.2.3.2.3.3.cmml">1</mn></mrow></msub><mo id="S3.p4.18.m17.4.5.2.3.1" xref="S3.p4.18.m17.4.5.2.3.1.cmml">⁢</mo><mrow id="S3.p4.18.m17.4.5.2.3.3.2" xref="S3.p4.18.m17.4.5.2.3.cmml"><mo id="S3.p4.18.m17.4.5.2.3.3.2.1" stretchy="false" xref="S3.p4.18.m17.4.5.2.3.cmml">(</mo><mi id="S3.p4.18.m17.3.3" xref="S3.p4.18.m17.3.3.cmml">t</mi><mo id="S3.p4.18.m17.4.5.2.3.3.2.2" stretchy="false" xref="S3.p4.18.m17.4.5.2.3.cmml">)</mo></mrow></mrow><mo id="S3.p4.18.m17.4.5.2.1a" xref="S3.p4.18.m17.4.5.2.1.cmml">−</mo><mrow id="S3.p4.18.m17.4.5.2.4" xref="S3.p4.18.m17.4.5.2.4.cmml"><msubsup id="S3.p4.18.m17.4.5.2.4.2" xref="S3.p4.18.m17.4.5.2.4.2.cmml"><mi id="S3.p4.18.m17.4.5.2.4.2.2.2" xref="S3.p4.18.m17.4.5.2.4.2.2.2.cmml">y</mi><mi id="S3.p4.18.m17.4.5.2.4.2.2.3" mathvariant="normal" xref="S3.p4.18.m17.4.5.2.4.2.2.3.cmml">r</mi><mrow id="S3.p4.18.m17.1.1.1.1" xref="S3.p4.18.m17.1.1.1.1.1.cmml"><mo id="S3.p4.18.m17.1.1.1.1.2" stretchy="false" xref="S3.p4.18.m17.1.1.1.1.1.cmml">(</mo><mrow id="S3.p4.18.m17.1.1.1.1.1" xref="S3.p4.18.m17.1.1.1.1.1.cmml"><mi id="S3.p4.18.m17.1.1.1.1.1.2" xref="S3.p4.18.m17.1.1.1.1.1.2.cmml">i</mi><mo id="S3.p4.18.m17.1.1.1.1.1.1" xref="S3.p4.18.m17.1.1.1.1.1.1.cmml">−</mo><mn id="S3.p4.18.m17.1.1.1.1.1.3" xref="S3.p4.18.m17.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.p4.18.m17.1.1.1.1.3" stretchy="false" xref="S3.p4.18.m17.1.1.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S3.p4.18.m17.4.5.2.4.1" xref="S3.p4.18.m17.4.5.2.4.1.cmml">⁢</mo><mrow id="S3.p4.18.m17.4.5.2.4.3.2" xref="S3.p4.18.m17.4.5.2.4.cmml"><mo id="S3.p4.18.m17.4.5.2.4.3.2.1" stretchy="false" xref="S3.p4.18.m17.4.5.2.4.cmml">(</mo><mi id="S3.p4.18.m17.4.4" xref="S3.p4.18.m17.4.4.cmml">t</mi><mo id="S3.p4.18.m17.4.5.2.4.3.2.2" stretchy="false" xref="S3.p4.18.m17.4.5.2.4.cmml">)</mo></mrow></mrow></mrow><mo id="S3.p4.18.m17.4.5.1" xref="S3.p4.18.m17.4.5.1.cmml">∈</mo><mi id="S3.p4.18.m17.4.5.3" xref="S3.p4.18.m17.4.5.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.18.m17.4b"><apply id="S3.p4.18.m17.4.5.cmml" xref="S3.p4.18.m17.4.5"><in id="S3.p4.18.m17.4.5.1.cmml" xref="S3.p4.18.m17.4.5.1"></in><apply id="S3.p4.18.m17.4.5.2.cmml" xref="S3.p4.18.m17.4.5.2"><minus id="S3.p4.18.m17.4.5.2.1.cmml" xref="S3.p4.18.m17.4.5.2.1"></minus><apply id="S3.p4.18.m17.4.5.2.2.cmml" xref="S3.p4.18.m17.4.5.2.2"><times id="S3.p4.18.m17.4.5.2.2.1.cmml" xref="S3.p4.18.m17.4.5.2.2.1"></times><apply id="S3.p4.18.m17.4.5.2.2.2.cmml" xref="S3.p4.18.m17.4.5.2.2.2"><csymbol cd="ambiguous" id="S3.p4.18.m17.4.5.2.2.2.1.cmml" xref="S3.p4.18.m17.4.5.2.2.2">subscript</csymbol><ci id="S3.p4.18.m17.4.5.2.2.2.2.cmml" xref="S3.p4.18.m17.4.5.2.2.2.2">𝑥</ci><ci id="S3.p4.18.m17.4.5.2.2.2.3.cmml" xref="S3.p4.18.m17.4.5.2.2.2.3">𝑖</ci></apply><ci id="S3.p4.18.m17.2.2.cmml" xref="S3.p4.18.m17.2.2">𝑡</ci></apply><apply id="S3.p4.18.m17.4.5.2.3.cmml" xref="S3.p4.18.m17.4.5.2.3"><times id="S3.p4.18.m17.4.5.2.3.1.cmml" xref="S3.p4.18.m17.4.5.2.3.1"></times><apply id="S3.p4.18.m17.4.5.2.3.2.cmml" xref="S3.p4.18.m17.4.5.2.3.2"><csymbol cd="ambiguous" id="S3.p4.18.m17.4.5.2.3.2.1.cmml" xref="S3.p4.18.m17.4.5.2.3.2">subscript</csymbol><ci id="S3.p4.18.m17.4.5.2.3.2.2.cmml" xref="S3.p4.18.m17.4.5.2.3.2.2">𝑣</ci><apply id="S3.p4.18.m17.4.5.2.3.2.3.cmml" xref="S3.p4.18.m17.4.5.2.3.2.3"><minus id="S3.p4.18.m17.4.5.2.3.2.3.1.cmml" xref="S3.p4.18.m17.4.5.2.3.2.3.1"></minus><ci id="S3.p4.18.m17.4.5.2.3.2.3.2.cmml" xref="S3.p4.18.m17.4.5.2.3.2.3.2">𝑖</ci><cn id="S3.p4.18.m17.4.5.2.3.2.3.3.cmml" type="integer" xref="S3.p4.18.m17.4.5.2.3.2.3.3">1</cn></apply></apply><ci id="S3.p4.18.m17.3.3.cmml" xref="S3.p4.18.m17.3.3">𝑡</ci></apply><apply id="S3.p4.18.m17.4.5.2.4.cmml" xref="S3.p4.18.m17.4.5.2.4"><times id="S3.p4.18.m17.4.5.2.4.1.cmml" xref="S3.p4.18.m17.4.5.2.4.1"></times><apply id="S3.p4.18.m17.4.5.2.4.2.cmml" xref="S3.p4.18.m17.4.5.2.4.2"><csymbol cd="ambiguous" id="S3.p4.18.m17.4.5.2.4.2.1.cmml" xref="S3.p4.18.m17.4.5.2.4.2">superscript</csymbol><apply id="S3.p4.18.m17.4.5.2.4.2.2.cmml" xref="S3.p4.18.m17.4.5.2.4.2"><csymbol cd="ambiguous" id="S3.p4.18.m17.4.5.2.4.2.2.1.cmml" xref="S3.p4.18.m17.4.5.2.4.2">subscript</csymbol><ci id="S3.p4.18.m17.4.5.2.4.2.2.2.cmml" xref="S3.p4.18.m17.4.5.2.4.2.2.2">𝑦</ci><ci id="S3.p4.18.m17.4.5.2.4.2.2.3.cmml" xref="S3.p4.18.m17.4.5.2.4.2.2.3">r</ci></apply><apply id="S3.p4.18.m17.1.1.1.1.1.cmml" xref="S3.p4.18.m17.1.1.1.1"><minus id="S3.p4.18.m17.1.1.1.1.1.1.cmml" xref="S3.p4.18.m17.1.1.1.1.1.1"></minus><ci id="S3.p4.18.m17.1.1.1.1.1.2.cmml" xref="S3.p4.18.m17.1.1.1.1.1.2">𝑖</ci><cn id="S3.p4.18.m17.1.1.1.1.1.3.cmml" type="integer" xref="S3.p4.18.m17.1.1.1.1.1.3">1</cn></apply></apply><ci id="S3.p4.18.m17.4.4.cmml" xref="S3.p4.18.m17.4.4">𝑡</ci></apply></apply><ci id="S3.p4.18.m17.4.5.3.cmml" xref="S3.p4.18.m17.4.5.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.18.m17.4c">x_{i}(t)-v_{i-1}(t)-y_{\rm r}^{(i-1)}(t)\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.18.m17.4d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) - italic_v start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ( italic_t ) - italic_y start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i - 1 ) end_POSTSUPERSCRIPT ( italic_t ) ∈ blackboard_R</annotation></semantics></math> are tracking errors, <math alttext="\bm{\psi}_{1}:=\bm{\varphi}_{1}" class="ltx_Math" display="inline" id="S3.p4.19.m18.1"><semantics id="S3.p4.19.m18.1a"><mrow id="S3.p4.19.m18.1.1" xref="S3.p4.19.m18.1.1.cmml"><msub id="S3.p4.19.m18.1.1.2" xref="S3.p4.19.m18.1.1.2.cmml"><mi id="S3.p4.19.m18.1.1.2.2" xref="S3.p4.19.m18.1.1.2.2.cmml">𝝍</mi><mn id="S3.p4.19.m18.1.1.2.3" xref="S3.p4.19.m18.1.1.2.3.cmml">1</mn></msub><mo id="S3.p4.19.m18.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.p4.19.m18.1.1.1.cmml">:=</mo><msub id="S3.p4.19.m18.1.1.3" xref="S3.p4.19.m18.1.1.3.cmml"><mi id="S3.p4.19.m18.1.1.3.2" xref="S3.p4.19.m18.1.1.3.2.cmml">𝝋</mi><mn id="S3.p4.19.m18.1.1.3.3" xref="S3.p4.19.m18.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.19.m18.1b"><apply id="S3.p4.19.m18.1.1.cmml" xref="S3.p4.19.m18.1.1"><csymbol cd="latexml" id="S3.p4.19.m18.1.1.1.cmml" xref="S3.p4.19.m18.1.1.1">assign</csymbol><apply id="S3.p4.19.m18.1.1.2.cmml" xref="S3.p4.19.m18.1.1.2"><csymbol cd="ambiguous" id="S3.p4.19.m18.1.1.2.1.cmml" xref="S3.p4.19.m18.1.1.2">subscript</csymbol><ci id="S3.p4.19.m18.1.1.2.2.cmml" xref="S3.p4.19.m18.1.1.2.2">𝝍</ci><cn id="S3.p4.19.m18.1.1.2.3.cmml" type="integer" xref="S3.p4.19.m18.1.1.2.3">1</cn></apply><apply id="S3.p4.19.m18.1.1.3.cmml" xref="S3.p4.19.m18.1.1.3"><csymbol cd="ambiguous" id="S3.p4.19.m18.1.1.3.1.cmml" xref="S3.p4.19.m18.1.1.3">subscript</csymbol><ci id="S3.p4.19.m18.1.1.3.2.cmml" xref="S3.p4.19.m18.1.1.3.2">𝝋</ci><cn id="S3.p4.19.m18.1.1.3.3.cmml" type="integer" xref="S3.p4.19.m18.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.19.m18.1c">\bm{\psi}_{1}:=\bm{\varphi}_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.19.m18.1d">bold_italic_ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT := bold_italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\bm{\psi}_{i}:=\bm{\varphi}_{i}-\sum_{k=1}^{i-1}\frac{\partial v_{i-1}}{% \partial x_{k}}\bm{\varphi}_{k}\in\mathbb{R}^{N}" class="ltx_Math" display="inline" id="S3.p4.20.m19.1"><semantics id="S3.p4.20.m19.1a"><mrow id="S3.p4.20.m19.1.1" xref="S3.p4.20.m19.1.1.cmml"><msub id="S3.p4.20.m19.1.1.2" xref="S3.p4.20.m19.1.1.2.cmml"><mi id="S3.p4.20.m19.1.1.2.2" xref="S3.p4.20.m19.1.1.2.2.cmml">𝝍</mi><mi id="S3.p4.20.m19.1.1.2.3" xref="S3.p4.20.m19.1.1.2.3.cmml">i</mi></msub><mo id="S3.p4.20.m19.1.1.3" lspace="0.278em" rspace="0.278em" xref="S3.p4.20.m19.1.1.3.cmml">:=</mo><mrow id="S3.p4.20.m19.1.1.4" xref="S3.p4.20.m19.1.1.4.cmml"><msub id="S3.p4.20.m19.1.1.4.2" xref="S3.p4.20.m19.1.1.4.2.cmml"><mi id="S3.p4.20.m19.1.1.4.2.2" xref="S3.p4.20.m19.1.1.4.2.2.cmml">𝝋</mi><mi id="S3.p4.20.m19.1.1.4.2.3" xref="S3.p4.20.m19.1.1.4.2.3.cmml">i</mi></msub><mo id="S3.p4.20.m19.1.1.4.1" rspace="0.055em" xref="S3.p4.20.m19.1.1.4.1.cmml">−</mo><mrow id="S3.p4.20.m19.1.1.4.3" xref="S3.p4.20.m19.1.1.4.3.cmml"><msubsup id="S3.p4.20.m19.1.1.4.3.1" xref="S3.p4.20.m19.1.1.4.3.1.cmml"><mo id="S3.p4.20.m19.1.1.4.3.1.2.2" xref="S3.p4.20.m19.1.1.4.3.1.2.2.cmml">∑</mo><mrow id="S3.p4.20.m19.1.1.4.3.1.2.3" xref="S3.p4.20.m19.1.1.4.3.1.2.3.cmml"><mi id="S3.p4.20.m19.1.1.4.3.1.2.3.2" xref="S3.p4.20.m19.1.1.4.3.1.2.3.2.cmml">k</mi><mo id="S3.p4.20.m19.1.1.4.3.1.2.3.1" xref="S3.p4.20.m19.1.1.4.3.1.2.3.1.cmml">=</mo><mn id="S3.p4.20.m19.1.1.4.3.1.2.3.3" xref="S3.p4.20.m19.1.1.4.3.1.2.3.3.cmml">1</mn></mrow><mrow id="S3.p4.20.m19.1.1.4.3.1.3" xref="S3.p4.20.m19.1.1.4.3.1.3.cmml"><mi id="S3.p4.20.m19.1.1.4.3.1.3.2" xref="S3.p4.20.m19.1.1.4.3.1.3.2.cmml">i</mi><mo id="S3.p4.20.m19.1.1.4.3.1.3.1" xref="S3.p4.20.m19.1.1.4.3.1.3.1.cmml">−</mo><mn id="S3.p4.20.m19.1.1.4.3.1.3.3" xref="S3.p4.20.m19.1.1.4.3.1.3.3.cmml">1</mn></mrow></msubsup><mrow id="S3.p4.20.m19.1.1.4.3.2" xref="S3.p4.20.m19.1.1.4.3.2.cmml"><mfrac id="S3.p4.20.m19.1.1.4.3.2.2" xref="S3.p4.20.m19.1.1.4.3.2.2.cmml"><mrow id="S3.p4.20.m19.1.1.4.3.2.2.2" xref="S3.p4.20.m19.1.1.4.3.2.2.2.cmml"><mo id="S3.p4.20.m19.1.1.4.3.2.2.2.1" rspace="0em" xref="S3.p4.20.m19.1.1.4.3.2.2.2.1.cmml">∂</mo><msub id="S3.p4.20.m19.1.1.4.3.2.2.2.2" xref="S3.p4.20.m19.1.1.4.3.2.2.2.2.cmml"><mi id="S3.p4.20.m19.1.1.4.3.2.2.2.2.2" xref="S3.p4.20.m19.1.1.4.3.2.2.2.2.2.cmml">v</mi><mrow id="S3.p4.20.m19.1.1.4.3.2.2.2.2.3" xref="S3.p4.20.m19.1.1.4.3.2.2.2.2.3.cmml"><mi id="S3.p4.20.m19.1.1.4.3.2.2.2.2.3.2" xref="S3.p4.20.m19.1.1.4.3.2.2.2.2.3.2.cmml">i</mi><mo id="S3.p4.20.m19.1.1.4.3.2.2.2.2.3.1" xref="S3.p4.20.m19.1.1.4.3.2.2.2.2.3.1.cmml">−</mo><mn id="S3.p4.20.m19.1.1.4.3.2.2.2.2.3.3" xref="S3.p4.20.m19.1.1.4.3.2.2.2.2.3.3.cmml">1</mn></mrow></msub></mrow><mrow id="S3.p4.20.m19.1.1.4.3.2.2.3" xref="S3.p4.20.m19.1.1.4.3.2.2.3.cmml"><mo id="S3.p4.20.m19.1.1.4.3.2.2.3.1" rspace="0em" xref="S3.p4.20.m19.1.1.4.3.2.2.3.1.cmml">∂</mo><msub id="S3.p4.20.m19.1.1.4.3.2.2.3.2" xref="S3.p4.20.m19.1.1.4.3.2.2.3.2.cmml"><mi id="S3.p4.20.m19.1.1.4.3.2.2.3.2.2" xref="S3.p4.20.m19.1.1.4.3.2.2.3.2.2.cmml">x</mi><mi id="S3.p4.20.m19.1.1.4.3.2.2.3.2.3" xref="S3.p4.20.m19.1.1.4.3.2.2.3.2.3.cmml">k</mi></msub></mrow></mfrac><mo id="S3.p4.20.m19.1.1.4.3.2.1" xref="S3.p4.20.m19.1.1.4.3.2.1.cmml">⁢</mo><msub id="S3.p4.20.m19.1.1.4.3.2.3" xref="S3.p4.20.m19.1.1.4.3.2.3.cmml"><mi id="S3.p4.20.m19.1.1.4.3.2.3.2" xref="S3.p4.20.m19.1.1.4.3.2.3.2.cmml">𝝋</mi><mi id="S3.p4.20.m19.1.1.4.3.2.3.3" xref="S3.p4.20.m19.1.1.4.3.2.3.3.cmml">k</mi></msub></mrow></mrow></mrow><mo id="S3.p4.20.m19.1.1.5" xref="S3.p4.20.m19.1.1.5.cmml">∈</mo><msup id="S3.p4.20.m19.1.1.6" xref="S3.p4.20.m19.1.1.6.cmml"><mi id="S3.p4.20.m19.1.1.6.2" xref="S3.p4.20.m19.1.1.6.2.cmml">ℝ</mi><mi id="S3.p4.20.m19.1.1.6.3" xref="S3.p4.20.m19.1.1.6.3.cmml">N</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.20.m19.1b"><apply id="S3.p4.20.m19.1.1.cmml" xref="S3.p4.20.m19.1.1"><and id="S3.p4.20.m19.1.1a.cmml" xref="S3.p4.20.m19.1.1"></and><apply id="S3.p4.20.m19.1.1b.cmml" xref="S3.p4.20.m19.1.1"><csymbol cd="latexml" id="S3.p4.20.m19.1.1.3.cmml" xref="S3.p4.20.m19.1.1.3">assign</csymbol><apply id="S3.p4.20.m19.1.1.2.cmml" xref="S3.p4.20.m19.1.1.2"><csymbol cd="ambiguous" id="S3.p4.20.m19.1.1.2.1.cmml" xref="S3.p4.20.m19.1.1.2">subscript</csymbol><ci id="S3.p4.20.m19.1.1.2.2.cmml" xref="S3.p4.20.m19.1.1.2.2">𝝍</ci><ci id="S3.p4.20.m19.1.1.2.3.cmml" xref="S3.p4.20.m19.1.1.2.3">𝑖</ci></apply><apply id="S3.p4.20.m19.1.1.4.cmml" xref="S3.p4.20.m19.1.1.4"><minus id="S3.p4.20.m19.1.1.4.1.cmml" xref="S3.p4.20.m19.1.1.4.1"></minus><apply id="S3.p4.20.m19.1.1.4.2.cmml" xref="S3.p4.20.m19.1.1.4.2"><csymbol cd="ambiguous" id="S3.p4.20.m19.1.1.4.2.1.cmml" xref="S3.p4.20.m19.1.1.4.2">subscript</csymbol><ci id="S3.p4.20.m19.1.1.4.2.2.cmml" xref="S3.p4.20.m19.1.1.4.2.2">𝝋</ci><ci id="S3.p4.20.m19.1.1.4.2.3.cmml" xref="S3.p4.20.m19.1.1.4.2.3">𝑖</ci></apply><apply id="S3.p4.20.m19.1.1.4.3.cmml" xref="S3.p4.20.m19.1.1.4.3"><apply id="S3.p4.20.m19.1.1.4.3.1.cmml" xref="S3.p4.20.m19.1.1.4.3.1"><csymbol cd="ambiguous" id="S3.p4.20.m19.1.1.4.3.1.1.cmml" xref="S3.p4.20.m19.1.1.4.3.1">superscript</csymbol><apply id="S3.p4.20.m19.1.1.4.3.1.2.cmml" xref="S3.p4.20.m19.1.1.4.3.1"><csymbol cd="ambiguous" id="S3.p4.20.m19.1.1.4.3.1.2.1.cmml" xref="S3.p4.20.m19.1.1.4.3.1">subscript</csymbol><sum id="S3.p4.20.m19.1.1.4.3.1.2.2.cmml" xref="S3.p4.20.m19.1.1.4.3.1.2.2"></sum><apply id="S3.p4.20.m19.1.1.4.3.1.2.3.cmml" xref="S3.p4.20.m19.1.1.4.3.1.2.3"><eq id="S3.p4.20.m19.1.1.4.3.1.2.3.1.cmml" xref="S3.p4.20.m19.1.1.4.3.1.2.3.1"></eq><ci id="S3.p4.20.m19.1.1.4.3.1.2.3.2.cmml" xref="S3.p4.20.m19.1.1.4.3.1.2.3.2">𝑘</ci><cn id="S3.p4.20.m19.1.1.4.3.1.2.3.3.cmml" type="integer" xref="S3.p4.20.m19.1.1.4.3.1.2.3.3">1</cn></apply></apply><apply id="S3.p4.20.m19.1.1.4.3.1.3.cmml" xref="S3.p4.20.m19.1.1.4.3.1.3"><minus id="S3.p4.20.m19.1.1.4.3.1.3.1.cmml" xref="S3.p4.20.m19.1.1.4.3.1.3.1"></minus><ci id="S3.p4.20.m19.1.1.4.3.1.3.2.cmml" xref="S3.p4.20.m19.1.1.4.3.1.3.2">𝑖</ci><cn id="S3.p4.20.m19.1.1.4.3.1.3.3.cmml" type="integer" xref="S3.p4.20.m19.1.1.4.3.1.3.3">1</cn></apply></apply><apply id="S3.p4.20.m19.1.1.4.3.2.cmml" xref="S3.p4.20.m19.1.1.4.3.2"><times id="S3.p4.20.m19.1.1.4.3.2.1.cmml" xref="S3.p4.20.m19.1.1.4.3.2.1"></times><apply id="S3.p4.20.m19.1.1.4.3.2.2.cmml" xref="S3.p4.20.m19.1.1.4.3.2.2"><divide id="S3.p4.20.m19.1.1.4.3.2.2.1.cmml" xref="S3.p4.20.m19.1.1.4.3.2.2"></divide><apply id="S3.p4.20.m19.1.1.4.3.2.2.2.cmml" xref="S3.p4.20.m19.1.1.4.3.2.2.2"><partialdiff id="S3.p4.20.m19.1.1.4.3.2.2.2.1.cmml" xref="S3.p4.20.m19.1.1.4.3.2.2.2.1"></partialdiff><apply id="S3.p4.20.m19.1.1.4.3.2.2.2.2.cmml" xref="S3.p4.20.m19.1.1.4.3.2.2.2.2"><csymbol cd="ambiguous" id="S3.p4.20.m19.1.1.4.3.2.2.2.2.1.cmml" xref="S3.p4.20.m19.1.1.4.3.2.2.2.2">subscript</csymbol><ci id="S3.p4.20.m19.1.1.4.3.2.2.2.2.2.cmml" xref="S3.p4.20.m19.1.1.4.3.2.2.2.2.2">𝑣</ci><apply id="S3.p4.20.m19.1.1.4.3.2.2.2.2.3.cmml" xref="S3.p4.20.m19.1.1.4.3.2.2.2.2.3"><minus id="S3.p4.20.m19.1.1.4.3.2.2.2.2.3.1.cmml" xref="S3.p4.20.m19.1.1.4.3.2.2.2.2.3.1"></minus><ci id="S3.p4.20.m19.1.1.4.3.2.2.2.2.3.2.cmml" xref="S3.p4.20.m19.1.1.4.3.2.2.2.2.3.2">𝑖</ci><cn id="S3.p4.20.m19.1.1.4.3.2.2.2.2.3.3.cmml" type="integer" xref="S3.p4.20.m19.1.1.4.3.2.2.2.2.3.3">1</cn></apply></apply></apply><apply id="S3.p4.20.m19.1.1.4.3.2.2.3.cmml" xref="S3.p4.20.m19.1.1.4.3.2.2.3"><partialdiff id="S3.p4.20.m19.1.1.4.3.2.2.3.1.cmml" xref="S3.p4.20.m19.1.1.4.3.2.2.3.1"></partialdiff><apply id="S3.p4.20.m19.1.1.4.3.2.2.3.2.cmml" xref="S3.p4.20.m19.1.1.4.3.2.2.3.2"><csymbol cd="ambiguous" id="S3.p4.20.m19.1.1.4.3.2.2.3.2.1.cmml" xref="S3.p4.20.m19.1.1.4.3.2.2.3.2">subscript</csymbol><ci id="S3.p4.20.m19.1.1.4.3.2.2.3.2.2.cmml" xref="S3.p4.20.m19.1.1.4.3.2.2.3.2.2">𝑥</ci><ci id="S3.p4.20.m19.1.1.4.3.2.2.3.2.3.cmml" xref="S3.p4.20.m19.1.1.4.3.2.2.3.2.3">𝑘</ci></apply></apply></apply><apply id="S3.p4.20.m19.1.1.4.3.2.3.cmml" xref="S3.p4.20.m19.1.1.4.3.2.3"><csymbol cd="ambiguous" id="S3.p4.20.m19.1.1.4.3.2.3.1.cmml" xref="S3.p4.20.m19.1.1.4.3.2.3">subscript</csymbol><ci id="S3.p4.20.m19.1.1.4.3.2.3.2.cmml" xref="S3.p4.20.m19.1.1.4.3.2.3.2">𝝋</ci><ci id="S3.p4.20.m19.1.1.4.3.2.3.3.cmml" xref="S3.p4.20.m19.1.1.4.3.2.3.3">𝑘</ci></apply></apply></apply></apply></apply><apply id="S3.p4.20.m19.1.1c.cmml" xref="S3.p4.20.m19.1.1"><in id="S3.p4.20.m19.1.1.5.cmml" xref="S3.p4.20.m19.1.1.5"></in><share href="https://arxiv.org/html/2401.10785v2#S3.p4.20.m19.1.1.4.cmml" id="S3.p4.20.m19.1.1d.cmml" xref="S3.p4.20.m19.1.1"></share><apply id="S3.p4.20.m19.1.1.6.cmml" xref="S3.p4.20.m19.1.1.6"><csymbol cd="ambiguous" id="S3.p4.20.m19.1.1.6.1.cmml" xref="S3.p4.20.m19.1.1.6">superscript</csymbol><ci id="S3.p4.20.m19.1.1.6.2.cmml" xref="S3.p4.20.m19.1.1.6.2">ℝ</ci><ci id="S3.p4.20.m19.1.1.6.3.cmml" xref="S3.p4.20.m19.1.1.6.3">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.20.m19.1c">\bm{\psi}_{i}:=\bm{\varphi}_{i}-\sum_{k=1}^{i-1}\frac{\partial v_{i-1}}{% \partial x_{k}}\bm{\varphi}_{k}\in\mathbb{R}^{N}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.20.m19.1d">bold_italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT := bold_italic_φ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i - 1 end_POSTSUPERSCRIPT divide start_ARG ∂ italic_v start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG bold_italic_φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT</annotation></semantics></math> are regressors, <math alttext="k_{{\rm c}1},k_{{\rm c}i}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S3.p4.21.m20.2"><semantics id="S3.p4.21.m20.2a"><mrow id="S3.p4.21.m20.2.2" xref="S3.p4.21.m20.2.2.cmml"><mrow id="S3.p4.21.m20.2.2.2.2" xref="S3.p4.21.m20.2.2.2.3.cmml"><msub id="S3.p4.21.m20.1.1.1.1.1" xref="S3.p4.21.m20.1.1.1.1.1.cmml"><mi id="S3.p4.21.m20.1.1.1.1.1.2" xref="S3.p4.21.m20.1.1.1.1.1.2.cmml">k</mi><mi id="S3.p4.21.m20.1.1.1.1.1.3" xref="S3.p4.21.m20.1.1.1.1.1.3.cmml">c1</mi></msub><mo id="S3.p4.21.m20.2.2.2.2.3" xref="S3.p4.21.m20.2.2.2.3.cmml">,</mo><msub id="S3.p4.21.m20.2.2.2.2.2" xref="S3.p4.21.m20.2.2.2.2.2.cmml"><mi id="S3.p4.21.m20.2.2.2.2.2.2" xref="S3.p4.21.m20.2.2.2.2.2.2.cmml">k</mi><mrow id="S3.p4.21.m20.2.2.2.2.2.3" xref="S3.p4.21.m20.2.2.2.2.2.3.cmml"><mi id="S3.p4.21.m20.2.2.2.2.2.3.2" mathvariant="normal" xref="S3.p4.21.m20.2.2.2.2.2.3.2.cmml">c</mi><mo id="S3.p4.21.m20.2.2.2.2.2.3.1" xref="S3.p4.21.m20.2.2.2.2.2.3.1.cmml">⁢</mo><mi id="S3.p4.21.m20.2.2.2.2.2.3.3" xref="S3.p4.21.m20.2.2.2.2.2.3.3.cmml">i</mi></mrow></msub></mrow><mo id="S3.p4.21.m20.2.2.3" xref="S3.p4.21.m20.2.2.3.cmml">∈</mo><msup id="S3.p4.21.m20.2.2.4" xref="S3.p4.21.m20.2.2.4.cmml"><mi id="S3.p4.21.m20.2.2.4.2" xref="S3.p4.21.m20.2.2.4.2.cmml">ℝ</mi><mo id="S3.p4.21.m20.2.2.4.3" xref="S3.p4.21.m20.2.2.4.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.21.m20.2b"><apply id="S3.p4.21.m20.2.2.cmml" xref="S3.p4.21.m20.2.2"><in id="S3.p4.21.m20.2.2.3.cmml" xref="S3.p4.21.m20.2.2.3"></in><list id="S3.p4.21.m20.2.2.2.3.cmml" xref="S3.p4.21.m20.2.2.2.2"><apply id="S3.p4.21.m20.1.1.1.1.1.cmml" xref="S3.p4.21.m20.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.p4.21.m20.1.1.1.1.1.1.cmml" xref="S3.p4.21.m20.1.1.1.1.1">subscript</csymbol><ci id="S3.p4.21.m20.1.1.1.1.1.2.cmml" xref="S3.p4.21.m20.1.1.1.1.1.2">𝑘</ci><ci id="S3.p4.21.m20.1.1.1.1.1.3.cmml" xref="S3.p4.21.m20.1.1.1.1.1.3">c1</ci></apply><apply id="S3.p4.21.m20.2.2.2.2.2.cmml" xref="S3.p4.21.m20.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.p4.21.m20.2.2.2.2.2.1.cmml" xref="S3.p4.21.m20.2.2.2.2.2">subscript</csymbol><ci id="S3.p4.21.m20.2.2.2.2.2.2.cmml" xref="S3.p4.21.m20.2.2.2.2.2.2">𝑘</ci><apply id="S3.p4.21.m20.2.2.2.2.2.3.cmml" xref="S3.p4.21.m20.2.2.2.2.2.3"><times id="S3.p4.21.m20.2.2.2.2.2.3.1.cmml" xref="S3.p4.21.m20.2.2.2.2.2.3.1"></times><ci id="S3.p4.21.m20.2.2.2.2.2.3.2.cmml" xref="S3.p4.21.m20.2.2.2.2.2.3.2">c</ci><ci id="S3.p4.21.m20.2.2.2.2.2.3.3.cmml" xref="S3.p4.21.m20.2.2.2.2.2.3.3">𝑖</ci></apply></apply></list><apply id="S3.p4.21.m20.2.2.4.cmml" xref="S3.p4.21.m20.2.2.4"><csymbol cd="ambiguous" id="S3.p4.21.m20.2.2.4.1.cmml" xref="S3.p4.21.m20.2.2.4">superscript</csymbol><ci id="S3.p4.21.m20.2.2.4.2.cmml" xref="S3.p4.21.m20.2.2.4.2">ℝ</ci><plus id="S3.p4.21.m20.2.2.4.3.cmml" xref="S3.p4.21.m20.2.2.4.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.21.m20.2c">k_{{\rm c}1},k_{{\rm c}i}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.21.m20.2d">italic_k start_POSTSUBSCRIPT c1 end_POSTSUBSCRIPT , italic_k start_POSTSUBSCRIPT roman_c italic_i end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> are control gain parameters, and <math alttext="i=2,\cdots,n" class="ltx_Math" display="inline" id="S3.p4.22.m21.3"><semantics id="S3.p4.22.m21.3a"><mrow id="S3.p4.22.m21.3.4" xref="S3.p4.22.m21.3.4.cmml"><mi id="S3.p4.22.m21.3.4.2" xref="S3.p4.22.m21.3.4.2.cmml">i</mi><mo id="S3.p4.22.m21.3.4.1" xref="S3.p4.22.m21.3.4.1.cmml">=</mo><mrow id="S3.p4.22.m21.3.4.3.2" xref="S3.p4.22.m21.3.4.3.1.cmml"><mn id="S3.p4.22.m21.1.1" xref="S3.p4.22.m21.1.1.cmml">2</mn><mo id="S3.p4.22.m21.3.4.3.2.1" xref="S3.p4.22.m21.3.4.3.1.cmml">,</mo><mi id="S3.p4.22.m21.2.2" mathvariant="normal" xref="S3.p4.22.m21.2.2.cmml">⋯</mi><mo id="S3.p4.22.m21.3.4.3.2.2" xref="S3.p4.22.m21.3.4.3.1.cmml">,</mo><mi id="S3.p4.22.m21.3.3" xref="S3.p4.22.m21.3.3.cmml">n</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.22.m21.3b"><apply id="S3.p4.22.m21.3.4.cmml" xref="S3.p4.22.m21.3.4"><eq id="S3.p4.22.m21.3.4.1.cmml" xref="S3.p4.22.m21.3.4.1"></eq><ci id="S3.p4.22.m21.3.4.2.cmml" xref="S3.p4.22.m21.3.4.2">𝑖</ci><list id="S3.p4.22.m21.3.4.3.1.cmml" xref="S3.p4.22.m21.3.4.3.2"><cn id="S3.p4.22.m21.1.1.cmml" type="integer" xref="S3.p4.22.m21.1.1">2</cn><ci id="S3.p4.22.m21.2.2.cmml" xref="S3.p4.22.m21.2.2">⋯</ci><ci id="S3.p4.22.m21.3.3.cmml" xref="S3.p4.22.m21.3.3">𝑛</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.22.m21.3c">i=2,\cdots,n</annotation><annotation encoding="application/x-llamapun" id="S3.p4.22.m21.3d">italic_i = 2 , ⋯ , italic_n</annotation></semantics></math>. The control law <math alttext="u" class="ltx_Math" display="inline" id="S3.p4.23.m22.1"><semantics id="S3.p4.23.m22.1a"><mi id="S3.p4.23.m22.1.1" xref="S3.p4.23.m22.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S3.p4.23.m22.1b"><ci id="S3.p4.23.m22.1.1.cmml" xref="S3.p4.23.m22.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.23.m22.1c">u</annotation><annotation encoding="application/x-llamapun" id="S3.p4.23.m22.1d">italic_u</annotation></semantics></math> is derived in the final step of backstepping as follows:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx3"> <tbody id="S3.E6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle u=\frac{1}{\beta(\bm{x})}\left(v_{n}(\bm{x},\Theta_{n-1},{\bm{y}% }_{{\rm r}n})+y_{\rm r}^{(n)}\right)." class="ltx_Math" display="inline" id="S3.E6.m1.4"><semantics id="S3.E6.m1.4a"><mrow id="S3.E6.m1.4.4.1" xref="S3.E6.m1.4.4.1.1.cmml"><mrow id="S3.E6.m1.4.4.1.1" xref="S3.E6.m1.4.4.1.1.cmml"><mi id="S3.E6.m1.4.4.1.1.3" xref="S3.E6.m1.4.4.1.1.3.cmml">u</mi><mo id="S3.E6.m1.4.4.1.1.2" xref="S3.E6.m1.4.4.1.1.2.cmml">=</mo><mrow id="S3.E6.m1.4.4.1.1.1" xref="S3.E6.m1.4.4.1.1.1.cmml"><mstyle displaystyle="true" id="S3.E6.m1.1.1" xref="S3.E6.m1.1.1.cmml"><mfrac id="S3.E6.m1.1.1a" xref="S3.E6.m1.1.1.cmml"><mn id="S3.E6.m1.1.1.3" xref="S3.E6.m1.1.1.3.cmml">1</mn><mrow id="S3.E6.m1.1.1.1" xref="S3.E6.m1.1.1.1.cmml"><mi id="S3.E6.m1.1.1.1.3" xref="S3.E6.m1.1.1.1.3.cmml">β</mi><mo id="S3.E6.m1.1.1.1.2" xref="S3.E6.m1.1.1.1.2.cmml">⁢</mo><mrow id="S3.E6.m1.1.1.1.4.2" xref="S3.E6.m1.1.1.1.cmml"><mo id="S3.E6.m1.1.1.1.4.2.1" stretchy="false" xref="S3.E6.m1.1.1.1.cmml">(</mo><mi id="S3.E6.m1.1.1.1.1" xref="S3.E6.m1.1.1.1.1.cmml">𝒙</mi><mo id="S3.E6.m1.1.1.1.4.2.2" stretchy="false" xref="S3.E6.m1.1.1.1.cmml">)</mo></mrow></mrow></mfrac></mstyle><mo id="S3.E6.m1.4.4.1.1.1.2" xref="S3.E6.m1.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S3.E6.m1.4.4.1.1.1.1.1" xref="S3.E6.m1.4.4.1.1.1.1.1.1.cmml"><mo id="S3.E6.m1.4.4.1.1.1.1.1.2" xref="S3.E6.m1.4.4.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.E6.m1.4.4.1.1.1.1.1.1" xref="S3.E6.m1.4.4.1.1.1.1.1.1.cmml"><mrow id="S3.E6.m1.4.4.1.1.1.1.1.1.2" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.cmml"><msub id="S3.E6.m1.4.4.1.1.1.1.1.1.2.4" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.4.cmml"><mi id="S3.E6.m1.4.4.1.1.1.1.1.1.2.4.2" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.4.2.cmml">v</mi><mi id="S3.E6.m1.4.4.1.1.1.1.1.1.2.4.3" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.4.3.cmml">n</mi></msub><mo id="S3.E6.m1.4.4.1.1.1.1.1.1.2.3" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.3.cmml">⁢</mo><mrow id="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.3.cmml"><mo id="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.3" stretchy="false" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.3.cmml">(</mo><mi id="S3.E6.m1.3.3" xref="S3.E6.m1.3.3.cmml">𝒙</mi><mo id="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.4" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.3.cmml">,</mo><msub id="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1" xref="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1.2" mathvariant="normal" xref="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1.2.cmml">Θ</mi><mrow id="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1.3" xref="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1.3.2" xref="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1.3.2.cmml">n</mi><mo id="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1.3.1" xref="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1.3.1.cmml">−</mo><mn id="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1.3.3" xref="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.5" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.3.cmml">,</mo><msub id="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2.cmml"><mi id="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2.2" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2.2.cmml">𝒚</mi><mrow id="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2.3" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2.3.cmml"><mi id="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2.3.2" mathvariant="normal" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2.3.2.cmml">r</mi><mo id="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2.3.1" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2.3.1.cmml">⁢</mo><mi id="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2.3.3" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2.3.3.cmml">n</mi></mrow></msub><mo id="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.6" stretchy="false" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.E6.m1.4.4.1.1.1.1.1.1.3" xref="S3.E6.m1.4.4.1.1.1.1.1.1.3.cmml">+</mo><msubsup id="S3.E6.m1.4.4.1.1.1.1.1.1.4" xref="S3.E6.m1.4.4.1.1.1.1.1.1.4.cmml"><mi id="S3.E6.m1.4.4.1.1.1.1.1.1.4.2.2" xref="S3.E6.m1.4.4.1.1.1.1.1.1.4.2.2.cmml">y</mi><mi id="S3.E6.m1.4.4.1.1.1.1.1.1.4.2.3" mathvariant="normal" xref="S3.E6.m1.4.4.1.1.1.1.1.1.4.2.3.cmml">r</mi><mrow id="S3.E6.m1.2.2.1.3" xref="S3.E6.m1.4.4.1.1.1.1.1.1.4.cmml"><mo id="S3.E6.m1.2.2.1.3.1" stretchy="false" xref="S3.E6.m1.4.4.1.1.1.1.1.1.4.cmml">(</mo><mi id="S3.E6.m1.2.2.1.1" xref="S3.E6.m1.2.2.1.1.cmml">n</mi><mo id="S3.E6.m1.2.2.1.3.2" stretchy="false" xref="S3.E6.m1.4.4.1.1.1.1.1.1.4.cmml">)</mo></mrow></msubsup></mrow><mo id="S3.E6.m1.4.4.1.1.1.1.1.3" xref="S3.E6.m1.4.4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S3.E6.m1.4.4.1.2" lspace="0em" xref="S3.E6.m1.4.4.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E6.m1.4b"><apply id="S3.E6.m1.4.4.1.1.cmml" xref="S3.E6.m1.4.4.1"><eq id="S3.E6.m1.4.4.1.1.2.cmml" xref="S3.E6.m1.4.4.1.1.2"></eq><ci id="S3.E6.m1.4.4.1.1.3.cmml" xref="S3.E6.m1.4.4.1.1.3">𝑢</ci><apply id="S3.E6.m1.4.4.1.1.1.cmml" xref="S3.E6.m1.4.4.1.1.1"><times id="S3.E6.m1.4.4.1.1.1.2.cmml" xref="S3.E6.m1.4.4.1.1.1.2"></times><apply id="S3.E6.m1.1.1.cmml" xref="S3.E6.m1.1.1"><divide id="S3.E6.m1.1.1.2.cmml" xref="S3.E6.m1.1.1"></divide><cn id="S3.E6.m1.1.1.3.cmml" type="integer" xref="S3.E6.m1.1.1.3">1</cn><apply id="S3.E6.m1.1.1.1.cmml" xref="S3.E6.m1.1.1.1"><times id="S3.E6.m1.1.1.1.2.cmml" xref="S3.E6.m1.1.1.1.2"></times><ci id="S3.E6.m1.1.1.1.3.cmml" xref="S3.E6.m1.1.1.1.3">𝛽</ci><ci id="S3.E6.m1.1.1.1.1.cmml" xref="S3.E6.m1.1.1.1.1">𝒙</ci></apply></apply><apply id="S3.E6.m1.4.4.1.1.1.1.1.1.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1"><plus id="S3.E6.m1.4.4.1.1.1.1.1.1.3.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.3"></plus><apply id="S3.E6.m1.4.4.1.1.1.1.1.1.2.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2"><times id="S3.E6.m1.4.4.1.1.1.1.1.1.2.3.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.3"></times><apply id="S3.E6.m1.4.4.1.1.1.1.1.1.2.4.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.4"><csymbol cd="ambiguous" id="S3.E6.m1.4.4.1.1.1.1.1.1.2.4.1.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.4">subscript</csymbol><ci id="S3.E6.m1.4.4.1.1.1.1.1.1.2.4.2.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.4.2">𝑣</ci><ci id="S3.E6.m1.4.4.1.1.1.1.1.1.2.4.3.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.4.3">𝑛</ci></apply><vector id="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.3.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2"><ci id="S3.E6.m1.3.3.cmml" xref="S3.E6.m1.3.3">𝒙</ci><apply id="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1.2">Θ</ci><apply id="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1.3"><minus id="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1.3.1"></minus><ci id="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1.3.2">𝑛</ci><cn id="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2"><csymbol cd="ambiguous" id="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2.1.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2">subscript</csymbol><ci id="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2.2.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2.2">𝒚</ci><apply id="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2.3.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2.3"><times id="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2.3.1.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2.3.1"></times><ci id="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2.3.2.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2.3.2">r</ci><ci id="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2.3.3.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.2.2.2.3.3">𝑛</ci></apply></apply></vector></apply><apply id="S3.E6.m1.4.4.1.1.1.1.1.1.4.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.4"><csymbol cd="ambiguous" id="S3.E6.m1.4.4.1.1.1.1.1.1.4.1.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.4">superscript</csymbol><apply id="S3.E6.m1.4.4.1.1.1.1.1.1.4.2.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.4"><csymbol cd="ambiguous" id="S3.E6.m1.4.4.1.1.1.1.1.1.4.2.1.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.4">subscript</csymbol><ci id="S3.E6.m1.4.4.1.1.1.1.1.1.4.2.2.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.4.2.2">𝑦</ci><ci id="S3.E6.m1.4.4.1.1.1.1.1.1.4.2.3.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.4.2.3">r</ci></apply><ci id="S3.E6.m1.2.2.1.1.cmml" xref="S3.E6.m1.2.2.1.1">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E6.m1.4c">\displaystyle u=\frac{1}{\beta(\bm{x})}\left(v_{n}(\bm{x},\Theta_{n-1},{\bm{y}% }_{{\rm r}n})+y_{\rm r}^{(n)}\right).</annotation><annotation encoding="application/x-llamapun" id="S3.E6.m1.4d">italic_u = divide start_ARG 1 end_ARG start_ARG italic_β ( bold_italic_x ) end_ARG ( italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( bold_italic_x , roman_Θ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT roman_r italic_n end_POSTSUBSCRIPT ) + italic_y start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p4.26">Applying (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.Ex4" title="III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">III</span></a>), (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E6" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">6</span></a>), and <math alttext="e_{i}" class="ltx_Math" display="inline" id="S3.p4.24.m1.1"><semantics id="S3.p4.24.m1.1a"><msub id="S3.p4.24.m1.1.1" xref="S3.p4.24.m1.1.1.cmml"><mi id="S3.p4.24.m1.1.1.2" xref="S3.p4.24.m1.1.1.2.cmml">e</mi><mi id="S3.p4.24.m1.1.1.3" xref="S3.p4.24.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p4.24.m1.1b"><apply id="S3.p4.24.m1.1.1.cmml" xref="S3.p4.24.m1.1.1"><csymbol cd="ambiguous" id="S3.p4.24.m1.1.1.1.cmml" xref="S3.p4.24.m1.1.1">subscript</csymbol><ci id="S3.p4.24.m1.1.1.2.cmml" xref="S3.p4.24.m1.1.1.2">𝑒</ci><ci id="S3.p4.24.m1.1.1.3.cmml" xref="S3.p4.24.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.24.m1.1c">e_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.24.m1.1d">italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext="=" class="ltx_Math" display="inline" id="S3.p4.25.m2.1"><semantics id="S3.p4.25.m2.1a"><mo id="S3.p4.25.m2.1.1" xref="S3.p4.25.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S3.p4.25.m2.1b"><eq id="S3.p4.25.m2.1.1.cmml" xref="S3.p4.25.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.25.m2.1c">=</annotation><annotation encoding="application/x-llamapun" id="S3.p4.25.m2.1d">=</annotation></semantics></math> <math alttext="x_{i}-v_{i-1}-y_{\rm r}^{(i-1)}" class="ltx_Math" display="inline" id="S3.p4.26.m3.1"><semantics id="S3.p4.26.m3.1a"><mrow id="S3.p4.26.m3.1.2" xref="S3.p4.26.m3.1.2.cmml"><msub id="S3.p4.26.m3.1.2.2" xref="S3.p4.26.m3.1.2.2.cmml"><mi id="S3.p4.26.m3.1.2.2.2" xref="S3.p4.26.m3.1.2.2.2.cmml">x</mi><mi id="S3.p4.26.m3.1.2.2.3" xref="S3.p4.26.m3.1.2.2.3.cmml">i</mi></msub><mo id="S3.p4.26.m3.1.2.1" xref="S3.p4.26.m3.1.2.1.cmml">−</mo><msub id="S3.p4.26.m3.1.2.3" xref="S3.p4.26.m3.1.2.3.cmml"><mi id="S3.p4.26.m3.1.2.3.2" xref="S3.p4.26.m3.1.2.3.2.cmml">v</mi><mrow id="S3.p4.26.m3.1.2.3.3" xref="S3.p4.26.m3.1.2.3.3.cmml"><mi id="S3.p4.26.m3.1.2.3.3.2" xref="S3.p4.26.m3.1.2.3.3.2.cmml">i</mi><mo id="S3.p4.26.m3.1.2.3.3.1" xref="S3.p4.26.m3.1.2.3.3.1.cmml">−</mo><mn id="S3.p4.26.m3.1.2.3.3.3" xref="S3.p4.26.m3.1.2.3.3.3.cmml">1</mn></mrow></msub><mo id="S3.p4.26.m3.1.2.1a" xref="S3.p4.26.m3.1.2.1.cmml">−</mo><msubsup id="S3.p4.26.m3.1.2.4" xref="S3.p4.26.m3.1.2.4.cmml"><mi id="S3.p4.26.m3.1.2.4.2.2" xref="S3.p4.26.m3.1.2.4.2.2.cmml">y</mi><mi id="S3.p4.26.m3.1.2.4.2.3" mathvariant="normal" xref="S3.p4.26.m3.1.2.4.2.3.cmml">r</mi><mrow id="S3.p4.26.m3.1.1.1.1" xref="S3.p4.26.m3.1.1.1.1.1.cmml"><mo id="S3.p4.26.m3.1.1.1.1.2" stretchy="false" xref="S3.p4.26.m3.1.1.1.1.1.cmml">(</mo><mrow id="S3.p4.26.m3.1.1.1.1.1" xref="S3.p4.26.m3.1.1.1.1.1.cmml"><mi id="S3.p4.26.m3.1.1.1.1.1.2" xref="S3.p4.26.m3.1.1.1.1.1.2.cmml">i</mi><mo id="S3.p4.26.m3.1.1.1.1.1.1" xref="S3.p4.26.m3.1.1.1.1.1.1.cmml">−</mo><mn id="S3.p4.26.m3.1.1.1.1.1.3" xref="S3.p4.26.m3.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.p4.26.m3.1.1.1.1.3" stretchy="false" xref="S3.p4.26.m3.1.1.1.1.1.cmml">)</mo></mrow></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.26.m3.1b"><apply id="S3.p4.26.m3.1.2.cmml" xref="S3.p4.26.m3.1.2"><minus id="S3.p4.26.m3.1.2.1.cmml" xref="S3.p4.26.m3.1.2.1"></minus><apply id="S3.p4.26.m3.1.2.2.cmml" xref="S3.p4.26.m3.1.2.2"><csymbol cd="ambiguous" id="S3.p4.26.m3.1.2.2.1.cmml" xref="S3.p4.26.m3.1.2.2">subscript</csymbol><ci id="S3.p4.26.m3.1.2.2.2.cmml" xref="S3.p4.26.m3.1.2.2.2">𝑥</ci><ci id="S3.p4.26.m3.1.2.2.3.cmml" xref="S3.p4.26.m3.1.2.2.3">𝑖</ci></apply><apply id="S3.p4.26.m3.1.2.3.cmml" xref="S3.p4.26.m3.1.2.3"><csymbol cd="ambiguous" id="S3.p4.26.m3.1.2.3.1.cmml" xref="S3.p4.26.m3.1.2.3">subscript</csymbol><ci id="S3.p4.26.m3.1.2.3.2.cmml" xref="S3.p4.26.m3.1.2.3.2">𝑣</ci><apply id="S3.p4.26.m3.1.2.3.3.cmml" xref="S3.p4.26.m3.1.2.3.3"><minus id="S3.p4.26.m3.1.2.3.3.1.cmml" xref="S3.p4.26.m3.1.2.3.3.1"></minus><ci id="S3.p4.26.m3.1.2.3.3.2.cmml" xref="S3.p4.26.m3.1.2.3.3.2">𝑖</ci><cn id="S3.p4.26.m3.1.2.3.3.3.cmml" type="integer" xref="S3.p4.26.m3.1.2.3.3.3">1</cn></apply></apply><apply id="S3.p4.26.m3.1.2.4.cmml" xref="S3.p4.26.m3.1.2.4"><csymbol cd="ambiguous" id="S3.p4.26.m3.1.2.4.1.cmml" xref="S3.p4.26.m3.1.2.4">superscript</csymbol><apply id="S3.p4.26.m3.1.2.4.2.cmml" xref="S3.p4.26.m3.1.2.4"><csymbol cd="ambiguous" id="S3.p4.26.m3.1.2.4.2.1.cmml" xref="S3.p4.26.m3.1.2.4">subscript</csymbol><ci id="S3.p4.26.m3.1.2.4.2.2.cmml" xref="S3.p4.26.m3.1.2.4.2.2">𝑦</ci><ci id="S3.p4.26.m3.1.2.4.2.3.cmml" xref="S3.p4.26.m3.1.2.4.2.3">r</ci></apply><apply id="S3.p4.26.m3.1.1.1.1.1.cmml" xref="S3.p4.26.m3.1.1.1.1"><minus id="S3.p4.26.m3.1.1.1.1.1.1.cmml" xref="S3.p4.26.m3.1.1.1.1.1.1"></minus><ci id="S3.p4.26.m3.1.1.1.1.1.2.cmml" xref="S3.p4.26.m3.1.1.1.1.1.2">𝑖</ci><cn id="S3.p4.26.m3.1.1.1.1.1.3.cmml" type="integer" xref="S3.p4.26.m3.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.26.m3.1c">x_{i}-v_{i-1}-y_{\rm r}^{(i-1)}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.26.m3.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_v start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT - italic_y start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i - 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> to (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S2.E4" title="In II Problem Formulation ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">4</span></a>) results in a closed-loop tracking error system</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx4"> <tbody id="S3.E7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{\bm{e}}=\Lambda\bm{e}+\Phi^{T}(\bm{x},\Theta_{n-1},{\bm{y}}_% {{\rm r}n})\tilde{\bm{\theta}}" class="ltx_Math" display="inline" id="S3.E7.m1.3"><semantics id="S3.E7.m1.3a"><mrow id="S3.E7.m1.3.3" xref="S3.E7.m1.3.3.cmml"><mover accent="true" id="S3.E7.m1.3.3.4" xref="S3.E7.m1.3.3.4.cmml"><mi id="S3.E7.m1.3.3.4.2" xref="S3.E7.m1.3.3.4.2.cmml">𝒆</mi><mo id="S3.E7.m1.3.3.4.1" xref="S3.E7.m1.3.3.4.1.cmml">˙</mo></mover><mo id="S3.E7.m1.3.3.3" xref="S3.E7.m1.3.3.3.cmml">=</mo><mrow id="S3.E7.m1.3.3.2" xref="S3.E7.m1.3.3.2.cmml"><mrow id="S3.E7.m1.3.3.2.4" xref="S3.E7.m1.3.3.2.4.cmml"><mi id="S3.E7.m1.3.3.2.4.2" mathvariant="normal" xref="S3.E7.m1.3.3.2.4.2.cmml">Λ</mi><mo id="S3.E7.m1.3.3.2.4.1" xref="S3.E7.m1.3.3.2.4.1.cmml">⁢</mo><mi id="S3.E7.m1.3.3.2.4.3" xref="S3.E7.m1.3.3.2.4.3.cmml">𝒆</mi></mrow><mo id="S3.E7.m1.3.3.2.3" xref="S3.E7.m1.3.3.2.3.cmml">+</mo><mrow id="S3.E7.m1.3.3.2.2" xref="S3.E7.m1.3.3.2.2.cmml"><msup id="S3.E7.m1.3.3.2.2.4" xref="S3.E7.m1.3.3.2.2.4.cmml"><mi id="S3.E7.m1.3.3.2.2.4.2" mathvariant="normal" xref="S3.E7.m1.3.3.2.2.4.2.cmml">Φ</mi><mi id="S3.E7.m1.3.3.2.2.4.3" xref="S3.E7.m1.3.3.2.2.4.3.cmml">T</mi></msup><mo id="S3.E7.m1.3.3.2.2.3" xref="S3.E7.m1.3.3.2.2.3.cmml">⁢</mo><mrow id="S3.E7.m1.3.3.2.2.2.2" xref="S3.E7.m1.3.3.2.2.2.3.cmml"><mo id="S3.E7.m1.3.3.2.2.2.2.3" stretchy="false" xref="S3.E7.m1.3.3.2.2.2.3.cmml">(</mo><mi id="S3.E7.m1.1.1" xref="S3.E7.m1.1.1.cmml">𝒙</mi><mo id="S3.E7.m1.3.3.2.2.2.2.4" xref="S3.E7.m1.3.3.2.2.2.3.cmml">,</mo><msub id="S3.E7.m1.2.2.1.1.1.1.1" xref="S3.E7.m1.2.2.1.1.1.1.1.cmml"><mi id="S3.E7.m1.2.2.1.1.1.1.1.2" mathvariant="normal" xref="S3.E7.m1.2.2.1.1.1.1.1.2.cmml">Θ</mi><mrow id="S3.E7.m1.2.2.1.1.1.1.1.3" xref="S3.E7.m1.2.2.1.1.1.1.1.3.cmml"><mi id="S3.E7.m1.2.2.1.1.1.1.1.3.2" xref="S3.E7.m1.2.2.1.1.1.1.1.3.2.cmml">n</mi><mo id="S3.E7.m1.2.2.1.1.1.1.1.3.1" xref="S3.E7.m1.2.2.1.1.1.1.1.3.1.cmml">−</mo><mn id="S3.E7.m1.2.2.1.1.1.1.1.3.3" xref="S3.E7.m1.2.2.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S3.E7.m1.3.3.2.2.2.2.5" xref="S3.E7.m1.3.3.2.2.2.3.cmml">,</mo><msub id="S3.E7.m1.3.3.2.2.2.2.2" xref="S3.E7.m1.3.3.2.2.2.2.2.cmml"><mi id="S3.E7.m1.3.3.2.2.2.2.2.2" xref="S3.E7.m1.3.3.2.2.2.2.2.2.cmml">𝒚</mi><mrow id="S3.E7.m1.3.3.2.2.2.2.2.3" xref="S3.E7.m1.3.3.2.2.2.2.2.3.cmml"><mi id="S3.E7.m1.3.3.2.2.2.2.2.3.2" mathvariant="normal" xref="S3.E7.m1.3.3.2.2.2.2.2.3.2.cmml">r</mi><mo id="S3.E7.m1.3.3.2.2.2.2.2.3.1" xref="S3.E7.m1.3.3.2.2.2.2.2.3.1.cmml">⁢</mo><mi id="S3.E7.m1.3.3.2.2.2.2.2.3.3" xref="S3.E7.m1.3.3.2.2.2.2.2.3.3.cmml">n</mi></mrow></msub><mo id="S3.E7.m1.3.3.2.2.2.2.6" stretchy="false" xref="S3.E7.m1.3.3.2.2.2.3.cmml">)</mo></mrow><mo id="S3.E7.m1.3.3.2.2.3a" xref="S3.E7.m1.3.3.2.2.3.cmml">⁢</mo><mover accent="true" id="S3.E7.m1.3.3.2.2.5" xref="S3.E7.m1.3.3.2.2.5.cmml"><mi id="S3.E7.m1.3.3.2.2.5.2" xref="S3.E7.m1.3.3.2.2.5.2.cmml">𝜽</mi><mo id="S3.E7.m1.3.3.2.2.5.1" xref="S3.E7.m1.3.3.2.2.5.1.cmml">~</mo></mover></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.E7.m1.3b"><apply id="S3.E7.m1.3.3.cmml" xref="S3.E7.m1.3.3"><eq id="S3.E7.m1.3.3.3.cmml" xref="S3.E7.m1.3.3.3"></eq><apply id="S3.E7.m1.3.3.4.cmml" xref="S3.E7.m1.3.3.4"><ci id="S3.E7.m1.3.3.4.1.cmml" xref="S3.E7.m1.3.3.4.1">˙</ci><ci id="S3.E7.m1.3.3.4.2.cmml" xref="S3.E7.m1.3.3.4.2">𝒆</ci></apply><apply id="S3.E7.m1.3.3.2.cmml" xref="S3.E7.m1.3.3.2"><plus id="S3.E7.m1.3.3.2.3.cmml" xref="S3.E7.m1.3.3.2.3"></plus><apply id="S3.E7.m1.3.3.2.4.cmml" xref="S3.E7.m1.3.3.2.4"><times id="S3.E7.m1.3.3.2.4.1.cmml" xref="S3.E7.m1.3.3.2.4.1"></times><ci id="S3.E7.m1.3.3.2.4.2.cmml" xref="S3.E7.m1.3.3.2.4.2">Λ</ci><ci id="S3.E7.m1.3.3.2.4.3.cmml" xref="S3.E7.m1.3.3.2.4.3">𝒆</ci></apply><apply id="S3.E7.m1.3.3.2.2.cmml" xref="S3.E7.m1.3.3.2.2"><times id="S3.E7.m1.3.3.2.2.3.cmml" xref="S3.E7.m1.3.3.2.2.3"></times><apply id="S3.E7.m1.3.3.2.2.4.cmml" xref="S3.E7.m1.3.3.2.2.4"><csymbol cd="ambiguous" id="S3.E7.m1.3.3.2.2.4.1.cmml" xref="S3.E7.m1.3.3.2.2.4">superscript</csymbol><ci id="S3.E7.m1.3.3.2.2.4.2.cmml" xref="S3.E7.m1.3.3.2.2.4.2">Φ</ci><ci id="S3.E7.m1.3.3.2.2.4.3.cmml" xref="S3.E7.m1.3.3.2.2.4.3">𝑇</ci></apply><vector id="S3.E7.m1.3.3.2.2.2.3.cmml" xref="S3.E7.m1.3.3.2.2.2.2"><ci id="S3.E7.m1.1.1.cmml" xref="S3.E7.m1.1.1">𝒙</ci><apply id="S3.E7.m1.2.2.1.1.1.1.1.cmml" xref="S3.E7.m1.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.E7.m1.2.2.1.1.1.1.1.1.cmml" xref="S3.E7.m1.2.2.1.1.1.1.1">subscript</csymbol><ci id="S3.E7.m1.2.2.1.1.1.1.1.2.cmml" xref="S3.E7.m1.2.2.1.1.1.1.1.2">Θ</ci><apply id="S3.E7.m1.2.2.1.1.1.1.1.3.cmml" xref="S3.E7.m1.2.2.1.1.1.1.1.3"><minus id="S3.E7.m1.2.2.1.1.1.1.1.3.1.cmml" xref="S3.E7.m1.2.2.1.1.1.1.1.3.1"></minus><ci id="S3.E7.m1.2.2.1.1.1.1.1.3.2.cmml" xref="S3.E7.m1.2.2.1.1.1.1.1.3.2">𝑛</ci><cn id="S3.E7.m1.2.2.1.1.1.1.1.3.3.cmml" type="integer" xref="S3.E7.m1.2.2.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="S3.E7.m1.3.3.2.2.2.2.2.cmml" xref="S3.E7.m1.3.3.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.E7.m1.3.3.2.2.2.2.2.1.cmml" xref="S3.E7.m1.3.3.2.2.2.2.2">subscript</csymbol><ci id="S3.E7.m1.3.3.2.2.2.2.2.2.cmml" xref="S3.E7.m1.3.3.2.2.2.2.2.2">𝒚</ci><apply id="S3.E7.m1.3.3.2.2.2.2.2.3.cmml" xref="S3.E7.m1.3.3.2.2.2.2.2.3"><times id="S3.E7.m1.3.3.2.2.2.2.2.3.1.cmml" xref="S3.E7.m1.3.3.2.2.2.2.2.3.1"></times><ci id="S3.E7.m1.3.3.2.2.2.2.2.3.2.cmml" xref="S3.E7.m1.3.3.2.2.2.2.2.3.2">r</ci><ci id="S3.E7.m1.3.3.2.2.2.2.2.3.3.cmml" xref="S3.E7.m1.3.3.2.2.2.2.2.3.3">𝑛</ci></apply></apply></vector><apply id="S3.E7.m1.3.3.2.2.5.cmml" xref="S3.E7.m1.3.3.2.2.5"><ci id="S3.E7.m1.3.3.2.2.5.1.cmml" xref="S3.E7.m1.3.3.2.2.5.1">~</ci><ci id="S3.E7.m1.3.3.2.2.5.2.cmml" xref="S3.E7.m1.3.3.2.2.5.2">𝜽</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E7.m1.3c">\displaystyle\dot{\bm{e}}=\Lambda\bm{e}+\Phi^{T}(\bm{x},\Theta_{n-1},{\bm{y}}_% {{\rm r}n})\tilde{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S3.E7.m1.3d">over˙ start_ARG bold_italic_e end_ARG = roman_Λ bold_italic_e + roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_italic_x , roman_Θ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT roman_r italic_n end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p4.40">in which <math alttext="\bm{e}(t)" class="ltx_Math" display="inline" id="S3.p4.27.m1.1"><semantics id="S3.p4.27.m1.1a"><mrow id="S3.p4.27.m1.1.2" xref="S3.p4.27.m1.1.2.cmml"><mi id="S3.p4.27.m1.1.2.2" xref="S3.p4.27.m1.1.2.2.cmml">𝒆</mi><mo id="S3.p4.27.m1.1.2.1" xref="S3.p4.27.m1.1.2.1.cmml">⁢</mo><mrow id="S3.p4.27.m1.1.2.3.2" xref="S3.p4.27.m1.1.2.cmml"><mo id="S3.p4.27.m1.1.2.3.2.1" stretchy="false" xref="S3.p4.27.m1.1.2.cmml">(</mo><mi id="S3.p4.27.m1.1.1" xref="S3.p4.27.m1.1.1.cmml">t</mi><mo id="S3.p4.27.m1.1.2.3.2.2" stretchy="false" xref="S3.p4.27.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.27.m1.1b"><apply id="S3.p4.27.m1.1.2.cmml" xref="S3.p4.27.m1.1.2"><times id="S3.p4.27.m1.1.2.1.cmml" xref="S3.p4.27.m1.1.2.1"></times><ci id="S3.p4.27.m1.1.2.2.cmml" xref="S3.p4.27.m1.1.2.2">𝒆</ci><ci id="S3.p4.27.m1.1.1.cmml" xref="S3.p4.27.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.27.m1.1c">\bm{e}(t)</annotation><annotation encoding="application/x-llamapun" id="S3.p4.27.m1.1d">bold_italic_e ( italic_t )</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S3.p4.28.m2.1"><semantics id="S3.p4.28.m2.1a"><mo id="S3.p4.28.m2.1.1" xref="S3.p4.28.m2.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S3.p4.28.m2.1b"><csymbol cd="latexml" id="S3.p4.28.m2.1.1.cmml" xref="S3.p4.28.m2.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.28.m2.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S3.p4.28.m2.1d">:=</annotation></semantics></math> <math alttext="[e_{1}(t),e_{2}(t),\cdots,e_{n}(t)]^{T}\in\mathbb{R}^{n}" class="ltx_Math" display="inline" id="S3.p4.29.m3.7"><semantics id="S3.p4.29.m3.7a"><mrow id="S3.p4.29.m3.7.7" xref="S3.p4.29.m3.7.7.cmml"><msup id="S3.p4.29.m3.7.7.3" xref="S3.p4.29.m3.7.7.3.cmml"><mrow id="S3.p4.29.m3.7.7.3.3.3" xref="S3.p4.29.m3.7.7.3.3.4.cmml"><mo id="S3.p4.29.m3.7.7.3.3.3.4" stretchy="false" xref="S3.p4.29.m3.7.7.3.3.4.cmml">[</mo><mrow id="S3.p4.29.m3.5.5.1.1.1.1" xref="S3.p4.29.m3.5.5.1.1.1.1.cmml"><msub id="S3.p4.29.m3.5.5.1.1.1.1.2" xref="S3.p4.29.m3.5.5.1.1.1.1.2.cmml"><mi id="S3.p4.29.m3.5.5.1.1.1.1.2.2" xref="S3.p4.29.m3.5.5.1.1.1.1.2.2.cmml">e</mi><mn id="S3.p4.29.m3.5.5.1.1.1.1.2.3" xref="S3.p4.29.m3.5.5.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S3.p4.29.m3.5.5.1.1.1.1.1" xref="S3.p4.29.m3.5.5.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.p4.29.m3.5.5.1.1.1.1.3.2" xref="S3.p4.29.m3.5.5.1.1.1.1.cmml"><mo id="S3.p4.29.m3.5.5.1.1.1.1.3.2.1" stretchy="false" xref="S3.p4.29.m3.5.5.1.1.1.1.cmml">(</mo><mi id="S3.p4.29.m3.1.1" xref="S3.p4.29.m3.1.1.cmml">t</mi><mo id="S3.p4.29.m3.5.5.1.1.1.1.3.2.2" stretchy="false" xref="S3.p4.29.m3.5.5.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.p4.29.m3.7.7.3.3.3.5" xref="S3.p4.29.m3.7.7.3.3.4.cmml">,</mo><mrow id="S3.p4.29.m3.6.6.2.2.2.2" xref="S3.p4.29.m3.6.6.2.2.2.2.cmml"><msub id="S3.p4.29.m3.6.6.2.2.2.2.2" xref="S3.p4.29.m3.6.6.2.2.2.2.2.cmml"><mi id="S3.p4.29.m3.6.6.2.2.2.2.2.2" xref="S3.p4.29.m3.6.6.2.2.2.2.2.2.cmml">e</mi><mn id="S3.p4.29.m3.6.6.2.2.2.2.2.3" xref="S3.p4.29.m3.6.6.2.2.2.2.2.3.cmml">2</mn></msub><mo id="S3.p4.29.m3.6.6.2.2.2.2.1" xref="S3.p4.29.m3.6.6.2.2.2.2.1.cmml">⁢</mo><mrow id="S3.p4.29.m3.6.6.2.2.2.2.3.2" xref="S3.p4.29.m3.6.6.2.2.2.2.cmml"><mo id="S3.p4.29.m3.6.6.2.2.2.2.3.2.1" stretchy="false" xref="S3.p4.29.m3.6.6.2.2.2.2.cmml">(</mo><mi id="S3.p4.29.m3.2.2" xref="S3.p4.29.m3.2.2.cmml">t</mi><mo id="S3.p4.29.m3.6.6.2.2.2.2.3.2.2" stretchy="false" xref="S3.p4.29.m3.6.6.2.2.2.2.cmml">)</mo></mrow></mrow><mo id="S3.p4.29.m3.7.7.3.3.3.6" xref="S3.p4.29.m3.7.7.3.3.4.cmml">,</mo><mi id="S3.p4.29.m3.4.4" mathvariant="normal" xref="S3.p4.29.m3.4.4.cmml">⋯</mi><mo id="S3.p4.29.m3.7.7.3.3.3.7" xref="S3.p4.29.m3.7.7.3.3.4.cmml">,</mo><mrow id="S3.p4.29.m3.7.7.3.3.3.3" xref="S3.p4.29.m3.7.7.3.3.3.3.cmml"><msub id="S3.p4.29.m3.7.7.3.3.3.3.2" xref="S3.p4.29.m3.7.7.3.3.3.3.2.cmml"><mi id="S3.p4.29.m3.7.7.3.3.3.3.2.2" xref="S3.p4.29.m3.7.7.3.3.3.3.2.2.cmml">e</mi><mi id="S3.p4.29.m3.7.7.3.3.3.3.2.3" xref="S3.p4.29.m3.7.7.3.3.3.3.2.3.cmml">n</mi></msub><mo id="S3.p4.29.m3.7.7.3.3.3.3.1" xref="S3.p4.29.m3.7.7.3.3.3.3.1.cmml">⁢</mo><mrow id="S3.p4.29.m3.7.7.3.3.3.3.3.2" xref="S3.p4.29.m3.7.7.3.3.3.3.cmml"><mo id="S3.p4.29.m3.7.7.3.3.3.3.3.2.1" stretchy="false" xref="S3.p4.29.m3.7.7.3.3.3.3.cmml">(</mo><mi id="S3.p4.29.m3.3.3" xref="S3.p4.29.m3.3.3.cmml">t</mi><mo id="S3.p4.29.m3.7.7.3.3.3.3.3.2.2" stretchy="false" xref="S3.p4.29.m3.7.7.3.3.3.3.cmml">)</mo></mrow></mrow><mo id="S3.p4.29.m3.7.7.3.3.3.8" stretchy="false" xref="S3.p4.29.m3.7.7.3.3.4.cmml">]</mo></mrow><mi id="S3.p4.29.m3.7.7.3.5" xref="S3.p4.29.m3.7.7.3.5.cmml">T</mi></msup><mo id="S3.p4.29.m3.7.7.4" xref="S3.p4.29.m3.7.7.4.cmml">∈</mo><msup id="S3.p4.29.m3.7.7.5" xref="S3.p4.29.m3.7.7.5.cmml"><mi id="S3.p4.29.m3.7.7.5.2" xref="S3.p4.29.m3.7.7.5.2.cmml">ℝ</mi><mi id="S3.p4.29.m3.7.7.5.3" xref="S3.p4.29.m3.7.7.5.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.29.m3.7b"><apply id="S3.p4.29.m3.7.7.cmml" xref="S3.p4.29.m3.7.7"><in id="S3.p4.29.m3.7.7.4.cmml" xref="S3.p4.29.m3.7.7.4"></in><apply id="S3.p4.29.m3.7.7.3.cmml" xref="S3.p4.29.m3.7.7.3"><csymbol cd="ambiguous" id="S3.p4.29.m3.7.7.3.4.cmml" xref="S3.p4.29.m3.7.7.3">superscript</csymbol><list id="S3.p4.29.m3.7.7.3.3.4.cmml" xref="S3.p4.29.m3.7.7.3.3.3"><apply id="S3.p4.29.m3.5.5.1.1.1.1.cmml" xref="S3.p4.29.m3.5.5.1.1.1.1"><times id="S3.p4.29.m3.5.5.1.1.1.1.1.cmml" xref="S3.p4.29.m3.5.5.1.1.1.1.1"></times><apply id="S3.p4.29.m3.5.5.1.1.1.1.2.cmml" xref="S3.p4.29.m3.5.5.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.p4.29.m3.5.5.1.1.1.1.2.1.cmml" xref="S3.p4.29.m3.5.5.1.1.1.1.2">subscript</csymbol><ci id="S3.p4.29.m3.5.5.1.1.1.1.2.2.cmml" xref="S3.p4.29.m3.5.5.1.1.1.1.2.2">𝑒</ci><cn id="S3.p4.29.m3.5.5.1.1.1.1.2.3.cmml" type="integer" xref="S3.p4.29.m3.5.5.1.1.1.1.2.3">1</cn></apply><ci id="S3.p4.29.m3.1.1.cmml" xref="S3.p4.29.m3.1.1">𝑡</ci></apply><apply id="S3.p4.29.m3.6.6.2.2.2.2.cmml" xref="S3.p4.29.m3.6.6.2.2.2.2"><times id="S3.p4.29.m3.6.6.2.2.2.2.1.cmml" xref="S3.p4.29.m3.6.6.2.2.2.2.1"></times><apply id="S3.p4.29.m3.6.6.2.2.2.2.2.cmml" xref="S3.p4.29.m3.6.6.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.p4.29.m3.6.6.2.2.2.2.2.1.cmml" xref="S3.p4.29.m3.6.6.2.2.2.2.2">subscript</csymbol><ci id="S3.p4.29.m3.6.6.2.2.2.2.2.2.cmml" xref="S3.p4.29.m3.6.6.2.2.2.2.2.2">𝑒</ci><cn id="S3.p4.29.m3.6.6.2.2.2.2.2.3.cmml" type="integer" xref="S3.p4.29.m3.6.6.2.2.2.2.2.3">2</cn></apply><ci id="S3.p4.29.m3.2.2.cmml" xref="S3.p4.29.m3.2.2">𝑡</ci></apply><ci id="S3.p4.29.m3.4.4.cmml" xref="S3.p4.29.m3.4.4">⋯</ci><apply id="S3.p4.29.m3.7.7.3.3.3.3.cmml" xref="S3.p4.29.m3.7.7.3.3.3.3"><times id="S3.p4.29.m3.7.7.3.3.3.3.1.cmml" xref="S3.p4.29.m3.7.7.3.3.3.3.1"></times><apply id="S3.p4.29.m3.7.7.3.3.3.3.2.cmml" xref="S3.p4.29.m3.7.7.3.3.3.3.2"><csymbol cd="ambiguous" id="S3.p4.29.m3.7.7.3.3.3.3.2.1.cmml" xref="S3.p4.29.m3.7.7.3.3.3.3.2">subscript</csymbol><ci id="S3.p4.29.m3.7.7.3.3.3.3.2.2.cmml" xref="S3.p4.29.m3.7.7.3.3.3.3.2.2">𝑒</ci><ci id="S3.p4.29.m3.7.7.3.3.3.3.2.3.cmml" xref="S3.p4.29.m3.7.7.3.3.3.3.2.3">𝑛</ci></apply><ci id="S3.p4.29.m3.3.3.cmml" xref="S3.p4.29.m3.3.3">𝑡</ci></apply></list><ci id="S3.p4.29.m3.7.7.3.5.cmml" xref="S3.p4.29.m3.7.7.3.5">𝑇</ci></apply><apply id="S3.p4.29.m3.7.7.5.cmml" xref="S3.p4.29.m3.7.7.5"><csymbol cd="ambiguous" id="S3.p4.29.m3.7.7.5.1.cmml" xref="S3.p4.29.m3.7.7.5">superscript</csymbol><ci id="S3.p4.29.m3.7.7.5.2.cmml" xref="S3.p4.29.m3.7.7.5.2">ℝ</ci><ci id="S3.p4.29.m3.7.7.5.3.cmml" xref="S3.p4.29.m3.7.7.5.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.29.m3.7c">[e_{1}(t),e_{2}(t),\cdots,e_{n}(t)]^{T}\in\mathbb{R}^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.29.m3.7d">[ italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) , italic_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) , ⋯ , italic_e start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> is a tracking error vector, <math alttext="\Phi" class="ltx_Math" display="inline" id="S3.p4.30.m4.1"><semantics id="S3.p4.30.m4.1a"><mi id="S3.p4.30.m4.1.1" mathvariant="normal" xref="S3.p4.30.m4.1.1.cmml">Φ</mi><annotation-xml encoding="MathML-Content" id="S3.p4.30.m4.1b"><ci id="S3.p4.30.m4.1.1.cmml" xref="S3.p4.30.m4.1.1">Φ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.30.m4.1c">\Phi</annotation><annotation encoding="application/x-llamapun" id="S3.p4.30.m4.1d">roman_Φ</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S3.p4.31.m5.1"><semantics id="S3.p4.31.m5.1a"><mo id="S3.p4.31.m5.1.1" xref="S3.p4.31.m5.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S3.p4.31.m5.1b"><csymbol cd="latexml" id="S3.p4.31.m5.1.1.cmml" xref="S3.p4.31.m5.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.31.m5.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S3.p4.31.m5.1d">:=</annotation></semantics></math> <math alttext="[\bm{\psi}_{1}" class="ltx_math_unparsed" display="inline" id="S3.p4.32.m6.1"><semantics id="S3.p4.32.m6.1a"><mrow id="S3.p4.32.m6.1b"><mo id="S3.p4.32.m6.1.1" stretchy="false">[</mo><msub id="S3.p4.32.m6.1.2"><mi id="S3.p4.32.m6.1.2.2">𝝍</mi><mn id="S3.p4.32.m6.1.2.3">1</mn></msub></mrow><annotation encoding="application/x-tex" id="S3.p4.32.m6.1c">[\bm{\psi}_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.32.m6.1d">[ bold_italic_ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\bm{\psi}_{2}" class="ltx_Math" display="inline" id="S3.p4.33.m7.1"><semantics id="S3.p4.33.m7.1a"><msub id="S3.p4.33.m7.1.1" xref="S3.p4.33.m7.1.1.cmml"><mi id="S3.p4.33.m7.1.1.2" xref="S3.p4.33.m7.1.1.2.cmml">𝝍</mi><mn id="S3.p4.33.m7.1.1.3" xref="S3.p4.33.m7.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p4.33.m7.1b"><apply id="S3.p4.33.m7.1.1.cmml" xref="S3.p4.33.m7.1.1"><csymbol cd="ambiguous" id="S3.p4.33.m7.1.1.1.cmml" xref="S3.p4.33.m7.1.1">subscript</csymbol><ci id="S3.p4.33.m7.1.1.2.cmml" xref="S3.p4.33.m7.1.1.2">𝝍</ci><cn id="S3.p4.33.m7.1.1.3.cmml" type="integer" xref="S3.p4.33.m7.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.33.m7.1c">\bm{\psi}_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.33.m7.1d">bold_italic_ψ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\cdots" class="ltx_Math" display="inline" id="S3.p4.34.m8.1"><semantics id="S3.p4.34.m8.1a"><mi id="S3.p4.34.m8.1.1" mathvariant="normal" xref="S3.p4.34.m8.1.1.cmml">⋯</mi><annotation-xml encoding="MathML-Content" id="S3.p4.34.m8.1b"><ci id="S3.p4.34.m8.1.1.cmml" xref="S3.p4.34.m8.1.1">⋯</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.34.m8.1c">\cdots</annotation><annotation encoding="application/x-llamapun" id="S3.p4.34.m8.1d">⋯</annotation></semantics></math>, <math alttext="\bm{\psi}_{n}]" class="ltx_math_unparsed" display="inline" id="S3.p4.35.m9.1"><semantics id="S3.p4.35.m9.1a"><mrow id="S3.p4.35.m9.1b"><msub id="S3.p4.35.m9.1.1"><mi id="S3.p4.35.m9.1.1.2">𝝍</mi><mi id="S3.p4.35.m9.1.1.3">n</mi></msub><mo id="S3.p4.35.m9.1.2" stretchy="false">]</mo></mrow><annotation encoding="application/x-tex" id="S3.p4.35.m9.1c">\bm{\psi}_{n}]</annotation><annotation encoding="application/x-llamapun" id="S3.p4.35.m9.1d">bold_italic_ψ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ]</annotation></semantics></math> <math alttext="\in\mathbb{R}^{N\times n}" class="ltx_Math" display="inline" id="S3.p4.36.m10.1"><semantics id="S3.p4.36.m10.1a"><mrow id="S3.p4.36.m10.1.1" xref="S3.p4.36.m10.1.1.cmml"><mi id="S3.p4.36.m10.1.1.2" xref="S3.p4.36.m10.1.1.2.cmml"></mi><mo id="S3.p4.36.m10.1.1.1" xref="S3.p4.36.m10.1.1.1.cmml">∈</mo><msup id="S3.p4.36.m10.1.1.3" xref="S3.p4.36.m10.1.1.3.cmml"><mi id="S3.p4.36.m10.1.1.3.2" xref="S3.p4.36.m10.1.1.3.2.cmml">ℝ</mi><mrow id="S3.p4.36.m10.1.1.3.3" xref="S3.p4.36.m10.1.1.3.3.cmml"><mi id="S3.p4.36.m10.1.1.3.3.2" xref="S3.p4.36.m10.1.1.3.3.2.cmml">N</mi><mo id="S3.p4.36.m10.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S3.p4.36.m10.1.1.3.3.1.cmml">×</mo><mi id="S3.p4.36.m10.1.1.3.3.3" xref="S3.p4.36.m10.1.1.3.3.3.cmml">n</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.36.m10.1b"><apply id="S3.p4.36.m10.1.1.cmml" xref="S3.p4.36.m10.1.1"><in id="S3.p4.36.m10.1.1.1.cmml" xref="S3.p4.36.m10.1.1.1"></in><csymbol cd="latexml" id="S3.p4.36.m10.1.1.2.cmml" xref="S3.p4.36.m10.1.1.2">absent</csymbol><apply id="S3.p4.36.m10.1.1.3.cmml" xref="S3.p4.36.m10.1.1.3"><csymbol cd="ambiguous" id="S3.p4.36.m10.1.1.3.1.cmml" xref="S3.p4.36.m10.1.1.3">superscript</csymbol><ci id="S3.p4.36.m10.1.1.3.2.cmml" xref="S3.p4.36.m10.1.1.3.2">ℝ</ci><apply id="S3.p4.36.m10.1.1.3.3.cmml" xref="S3.p4.36.m10.1.1.3.3"><times id="S3.p4.36.m10.1.1.3.3.1.cmml" xref="S3.p4.36.m10.1.1.3.3.1"></times><ci id="S3.p4.36.m10.1.1.3.3.2.cmml" xref="S3.p4.36.m10.1.1.3.3.2">𝑁</ci><ci id="S3.p4.36.m10.1.1.3.3.3.cmml" xref="S3.p4.36.m10.1.1.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.36.m10.1c">\in\mathbb{R}^{N\times n}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.36.m10.1d">∈ blackboard_R start_POSTSUPERSCRIPT italic_N × italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> is a new regressor, <math alttext="\tilde{\bm{\theta}}(t)" class="ltx_Math" display="inline" id="S3.p4.37.m11.1"><semantics id="S3.p4.37.m11.1a"><mrow id="S3.p4.37.m11.1.2" xref="S3.p4.37.m11.1.2.cmml"><mover accent="true" id="S3.p4.37.m11.1.2.2" xref="S3.p4.37.m11.1.2.2.cmml"><mi id="S3.p4.37.m11.1.2.2.2" xref="S3.p4.37.m11.1.2.2.2.cmml">𝜽</mi><mo id="S3.p4.37.m11.1.2.2.1" xref="S3.p4.37.m11.1.2.2.1.cmml">~</mo></mover><mo id="S3.p4.37.m11.1.2.1" xref="S3.p4.37.m11.1.2.1.cmml">⁢</mo><mrow id="S3.p4.37.m11.1.2.3.2" xref="S3.p4.37.m11.1.2.cmml"><mo id="S3.p4.37.m11.1.2.3.2.1" stretchy="false" xref="S3.p4.37.m11.1.2.cmml">(</mo><mi id="S3.p4.37.m11.1.1" xref="S3.p4.37.m11.1.1.cmml">t</mi><mo id="S3.p4.37.m11.1.2.3.2.2" stretchy="false" xref="S3.p4.37.m11.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.37.m11.1b"><apply id="S3.p4.37.m11.1.2.cmml" xref="S3.p4.37.m11.1.2"><times id="S3.p4.37.m11.1.2.1.cmml" xref="S3.p4.37.m11.1.2.1"></times><apply id="S3.p4.37.m11.1.2.2.cmml" xref="S3.p4.37.m11.1.2.2"><ci id="S3.p4.37.m11.1.2.2.1.cmml" xref="S3.p4.37.m11.1.2.2.1">~</ci><ci id="S3.p4.37.m11.1.2.2.2.cmml" xref="S3.p4.37.m11.1.2.2.2">𝜽</ci></apply><ci id="S3.p4.37.m11.1.1.cmml" xref="S3.p4.37.m11.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.37.m11.1c">\tilde{\bm{\theta}}(t)</annotation><annotation encoding="application/x-llamapun" id="S3.p4.37.m11.1d">over~ start_ARG bold_italic_θ end_ARG ( italic_t )</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S3.p4.38.m12.1"><semantics id="S3.p4.38.m12.1a"><mo id="S3.p4.38.m12.1.1" xref="S3.p4.38.m12.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S3.p4.38.m12.1b"><csymbol cd="latexml" id="S3.p4.38.m12.1.1.cmml" xref="S3.p4.38.m12.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.38.m12.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S3.p4.38.m12.1d">:=</annotation></semantics></math> <math alttext="\bm{\theta}-\hat{\bm{\theta}}(t)" class="ltx_Math" display="inline" id="S3.p4.39.m13.1"><semantics id="S3.p4.39.m13.1a"><mrow id="S3.p4.39.m13.1.2" xref="S3.p4.39.m13.1.2.cmml"><mi id="S3.p4.39.m13.1.2.2" xref="S3.p4.39.m13.1.2.2.cmml">𝜽</mi><mo id="S3.p4.39.m13.1.2.1" xref="S3.p4.39.m13.1.2.1.cmml">−</mo><mrow id="S3.p4.39.m13.1.2.3" xref="S3.p4.39.m13.1.2.3.cmml"><mover accent="true" id="S3.p4.39.m13.1.2.3.2" xref="S3.p4.39.m13.1.2.3.2.cmml"><mi id="S3.p4.39.m13.1.2.3.2.2" xref="S3.p4.39.m13.1.2.3.2.2.cmml">𝜽</mi><mo id="S3.p4.39.m13.1.2.3.2.1" xref="S3.p4.39.m13.1.2.3.2.1.cmml">^</mo></mover><mo id="S3.p4.39.m13.1.2.3.1" xref="S3.p4.39.m13.1.2.3.1.cmml">⁢</mo><mrow id="S3.p4.39.m13.1.2.3.3.2" xref="S3.p4.39.m13.1.2.3.cmml"><mo id="S3.p4.39.m13.1.2.3.3.2.1" stretchy="false" xref="S3.p4.39.m13.1.2.3.cmml">(</mo><mi id="S3.p4.39.m13.1.1" xref="S3.p4.39.m13.1.1.cmml">t</mi><mo id="S3.p4.39.m13.1.2.3.3.2.2" stretchy="false" xref="S3.p4.39.m13.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.39.m13.1b"><apply id="S3.p4.39.m13.1.2.cmml" xref="S3.p4.39.m13.1.2"><minus id="S3.p4.39.m13.1.2.1.cmml" xref="S3.p4.39.m13.1.2.1"></minus><ci id="S3.p4.39.m13.1.2.2.cmml" xref="S3.p4.39.m13.1.2.2">𝜽</ci><apply id="S3.p4.39.m13.1.2.3.cmml" xref="S3.p4.39.m13.1.2.3"><times id="S3.p4.39.m13.1.2.3.1.cmml" xref="S3.p4.39.m13.1.2.3.1"></times><apply id="S3.p4.39.m13.1.2.3.2.cmml" xref="S3.p4.39.m13.1.2.3.2"><ci id="S3.p4.39.m13.1.2.3.2.1.cmml" xref="S3.p4.39.m13.1.2.3.2.1">^</ci><ci id="S3.p4.39.m13.1.2.3.2.2.cmml" xref="S3.p4.39.m13.1.2.3.2.2">𝜽</ci></apply><ci id="S3.p4.39.m13.1.1.cmml" xref="S3.p4.39.m13.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.39.m13.1c">\bm{\theta}-\hat{\bm{\theta}}(t)</annotation><annotation encoding="application/x-llamapun" id="S3.p4.39.m13.1d">bold_italic_θ - over^ start_ARG bold_italic_θ end_ARG ( italic_t )</annotation></semantics></math> <math alttext="\in\mathbb{R}^{N}" class="ltx_Math" display="inline" id="S3.p4.40.m14.1"><semantics id="S3.p4.40.m14.1a"><mrow id="S3.p4.40.m14.1.1" xref="S3.p4.40.m14.1.1.cmml"><mi id="S3.p4.40.m14.1.1.2" xref="S3.p4.40.m14.1.1.2.cmml"></mi><mo id="S3.p4.40.m14.1.1.1" xref="S3.p4.40.m14.1.1.1.cmml">∈</mo><msup id="S3.p4.40.m14.1.1.3" xref="S3.p4.40.m14.1.1.3.cmml"><mi id="S3.p4.40.m14.1.1.3.2" xref="S3.p4.40.m14.1.1.3.2.cmml">ℝ</mi><mi id="S3.p4.40.m14.1.1.3.3" xref="S3.p4.40.m14.1.1.3.3.cmml">N</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.40.m14.1b"><apply id="S3.p4.40.m14.1.1.cmml" xref="S3.p4.40.m14.1.1"><in id="S3.p4.40.m14.1.1.1.cmml" xref="S3.p4.40.m14.1.1.1"></in><csymbol cd="latexml" id="S3.p4.40.m14.1.1.2.cmml" xref="S3.p4.40.m14.1.1.2">absent</csymbol><apply id="S3.p4.40.m14.1.1.3.cmml" xref="S3.p4.40.m14.1.1.3"><csymbol cd="ambiguous" id="S3.p4.40.m14.1.1.3.1.cmml" xref="S3.p4.40.m14.1.1.3">superscript</csymbol><ci id="S3.p4.40.m14.1.1.3.2.cmml" xref="S3.p4.40.m14.1.1.3.2">ℝ</ci><ci id="S3.p4.40.m14.1.1.3.3.cmml" xref="S3.p4.40.m14.1.1.3.3">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.40.m14.1c">\in\mathbb{R}^{N}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.40.m14.1d">∈ blackboard_R start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT</annotation></semantics></math> is a parameter estimation error, and</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx5"> <tbody id="S3.E8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\Lambda=\left[\begin{matrix}-k_{{\rm c}1}&1&0&\cdots&0\\ -1&-k_{{\rm c}2}&1&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&1\\ 0&0&0&\cdots&-k_{{\rm c}n}\\ \end{matrix}\right]\in\mathbb{R}^{n\times n}." class="ltx_Math" display="inline" id="S3.E8.m1.2"><semantics id="S3.E8.m1.2a"><mrow id="S3.E8.m1.2.2.1" xref="S3.E8.m1.2.2.1.1.cmml"><mrow id="S3.E8.m1.2.2.1.1" xref="S3.E8.m1.2.2.1.1.cmml"><mi id="S3.E8.m1.2.2.1.1.2" mathvariant="normal" xref="S3.E8.m1.2.2.1.1.2.cmml">Λ</mi><mo id="S3.E8.m1.2.2.1.1.3" xref="S3.E8.m1.2.2.1.1.3.cmml">=</mo><mrow id="S3.E8.m1.2.2.1.1.4.2" xref="S3.E8.m1.2.2.1.1.4.1.cmml"><mo id="S3.E8.m1.2.2.1.1.4.2.1" xref="S3.E8.m1.2.2.1.1.4.1.1.cmml">[</mo><mtable columnspacing="5pt" id="S3.E8.m1.1.1.1.1" rowspacing="0pt" xref="S3.E8.m1.1.1a.2.cmml"><mtr id="S3.E8.m1.1.1.1.1a" xref="S3.E8.m1.1.1a.2.cmml"><mtd id="S3.E8.m1.1.1.1.1b" xref="S3.E8.m1.1.1a.2.cmml"><mrow id="S3.E8.m1.1.1.1.1.1.1.1" xref="S3.E8.m1.1.1.1.1.1.1.1.cmml"><mo id="S3.E8.m1.1.1.1.1.1.1.1a" xref="S3.E8.m1.1.1.1.1.1.1.1.cmml">−</mo><msub id="S3.E8.m1.1.1.1.1.1.1.1.2" xref="S3.E8.m1.1.1.1.1.1.1.1.2.cmml"><mi id="S3.E8.m1.1.1.1.1.1.1.1.2.2" xref="S3.E8.m1.1.1.1.1.1.1.1.2.2.cmml">k</mi><mi id="S3.E8.m1.1.1.1.1.1.1.1.2.3" xref="S3.E8.m1.1.1.1.1.1.1.1.2.3.cmml">c1</mi></msub></mrow></mtd><mtd id="S3.E8.m1.1.1.1.1c" xref="S3.E8.m1.1.1a.2.cmml"><mn id="S3.E8.m1.1.1.1.1.1.2.1" xref="S3.E8.m1.1.1.1.1.1.2.1.cmml">1</mn></mtd><mtd id="S3.E8.m1.1.1.1.1d" xref="S3.E8.m1.1.1a.2.cmml"><mn id="S3.E8.m1.1.1.1.1.1.3.1" xref="S3.E8.m1.1.1.1.1.1.3.1.cmml">0</mn></mtd><mtd id="S3.E8.m1.1.1.1.1e" xref="S3.E8.m1.1.1a.2.cmml"><mi id="S3.E8.m1.1.1.1.1.1.4.1" mathvariant="normal" xref="S3.E8.m1.1.1.1.1.1.4.1.cmml">⋯</mi></mtd><mtd id="S3.E8.m1.1.1.1.1f" xref="S3.E8.m1.1.1a.2.cmml"><mn id="S3.E8.m1.1.1.1.1.1.5.1" xref="S3.E8.m1.1.1.1.1.1.5.1.cmml">0</mn></mtd></mtr><mtr id="S3.E8.m1.1.1.1.1g" xref="S3.E8.m1.1.1a.2.cmml"><mtd id="S3.E8.m1.1.1.1.1h" xref="S3.E8.m1.1.1a.2.cmml"><mrow id="S3.E8.m1.1.1.1.1.2.1.1" xref="S3.E8.m1.1.1.1.1.2.1.1.cmml"><mo id="S3.E8.m1.1.1.1.1.2.1.1a" xref="S3.E8.m1.1.1.1.1.2.1.1.cmml">−</mo><mn id="S3.E8.m1.1.1.1.1.2.1.1.2" xref="S3.E8.m1.1.1.1.1.2.1.1.2.cmml">1</mn></mrow></mtd><mtd id="S3.E8.m1.1.1.1.1i" xref="S3.E8.m1.1.1a.2.cmml"><mrow id="S3.E8.m1.1.1.1.1.2.2.1" xref="S3.E8.m1.1.1.1.1.2.2.1.cmml"><mo id="S3.E8.m1.1.1.1.1.2.2.1a" xref="S3.E8.m1.1.1.1.1.2.2.1.cmml">−</mo><msub id="S3.E8.m1.1.1.1.1.2.2.1.2" xref="S3.E8.m1.1.1.1.1.2.2.1.2.cmml"><mi id="S3.E8.m1.1.1.1.1.2.2.1.2.2" xref="S3.E8.m1.1.1.1.1.2.2.1.2.2.cmml">k</mi><mi id="S3.E8.m1.1.1.1.1.2.2.1.2.3" xref="S3.E8.m1.1.1.1.1.2.2.1.2.3.cmml">c2</mi></msub></mrow></mtd><mtd id="S3.E8.m1.1.1.1.1j" xref="S3.E8.m1.1.1a.2.cmml"><mn id="S3.E8.m1.1.1.1.1.2.3.1" xref="S3.E8.m1.1.1.1.1.2.3.1.cmml">1</mn></mtd><mtd id="S3.E8.m1.1.1.1.1k" xref="S3.E8.m1.1.1a.2.cmml"><mi id="S3.E8.m1.1.1.1.1.2.4.1" mathvariant="normal" xref="S3.E8.m1.1.1.1.1.2.4.1.cmml">⋯</mi></mtd><mtd id="S3.E8.m1.1.1.1.1l" xref="S3.E8.m1.1.1a.2.cmml"><mn id="S3.E8.m1.1.1.1.1.2.5.1" xref="S3.E8.m1.1.1.1.1.2.5.1.cmml">0</mn></mtd></mtr><mtr id="S3.E8.m1.1.1.1.1m" xref="S3.E8.m1.1.1a.2.cmml"><mtd id="S3.E8.m1.1.1.1.1n" xref="S3.E8.m1.1.1a.2.cmml"><mi id="S3.E8.m1.1.1.1.1.3.1.1" mathvariant="normal" xref="S3.E8.m1.1.1.1.1.3.1.1.cmml">⋮</mi></mtd><mtd id="S3.E8.m1.1.1.1.1o" xref="S3.E8.m1.1.1a.2.cmml"><mi id="S3.E8.m1.1.1.1.1.3.2.1" mathvariant="normal" xref="S3.E8.m1.1.1.1.1.3.2.1.cmml">⋮</mi></mtd><mtd id="S3.E8.m1.1.1.1.1p" xref="S3.E8.m1.1.1a.2.cmml"><mi id="S3.E8.m1.1.1.1.1.3.3.1" mathvariant="normal" xref="S3.E8.m1.1.1.1.1.3.3.1.cmml">⋮</mi></mtd><mtd id="S3.E8.m1.1.1.1.1q" xref="S3.E8.m1.1.1a.2.cmml"><mi id="S3.E8.m1.1.1.1.1.3.4.1" mathvariant="normal" xref="S3.E8.m1.1.1.1.1.3.4.1.cmml">⋱</mi></mtd><mtd id="S3.E8.m1.1.1.1.1r" xref="S3.E8.m1.1.1a.2.cmml"><mi id="S3.E8.m1.1.1.1.1.3.5.1" mathvariant="normal" xref="S3.E8.m1.1.1.1.1.3.5.1.cmml">⋮</mi></mtd></mtr><mtr id="S3.E8.m1.1.1.1.1s" xref="S3.E8.m1.1.1a.2.cmml"><mtd id="S3.E8.m1.1.1.1.1t" xref="S3.E8.m1.1.1a.2.cmml"><mn id="S3.E8.m1.1.1.1.1.4.1.1" xref="S3.E8.m1.1.1.1.1.4.1.1.cmml">0</mn></mtd><mtd id="S3.E8.m1.1.1.1.1u" xref="S3.E8.m1.1.1a.2.cmml"><mn id="S3.E8.m1.1.1.1.1.4.2.1" xref="S3.E8.m1.1.1.1.1.4.2.1.cmml">0</mn></mtd><mtd id="S3.E8.m1.1.1.1.1v" xref="S3.E8.m1.1.1a.2.cmml"><mn id="S3.E8.m1.1.1.1.1.4.3.1" xref="S3.E8.m1.1.1.1.1.4.3.1.cmml">0</mn></mtd><mtd id="S3.E8.m1.1.1.1.1w" xref="S3.E8.m1.1.1a.2.cmml"><mi id="S3.E8.m1.1.1.1.1.4.4.1" mathvariant="normal" xref="S3.E8.m1.1.1.1.1.4.4.1.cmml">⋯</mi></mtd><mtd id="S3.E8.m1.1.1.1.1x" xref="S3.E8.m1.1.1a.2.cmml"><mn id="S3.E8.m1.1.1.1.1.4.5.1" xref="S3.E8.m1.1.1.1.1.4.5.1.cmml">1</mn></mtd></mtr><mtr id="S3.E8.m1.1.1.1.1y" xref="S3.E8.m1.1.1a.2.cmml"><mtd id="S3.E8.m1.1.1.1.1z" xref="S3.E8.m1.1.1a.2.cmml"><mn id="S3.E8.m1.1.1.1.1.5.1.1" xref="S3.E8.m1.1.1.1.1.5.1.1.cmml">0</mn></mtd><mtd id="S3.E8.m1.1.1.1.1aa" xref="S3.E8.m1.1.1a.2.cmml"><mn id="S3.E8.m1.1.1.1.1.5.2.1" xref="S3.E8.m1.1.1.1.1.5.2.1.cmml">0</mn></mtd><mtd id="S3.E8.m1.1.1.1.1ab" xref="S3.E8.m1.1.1a.2.cmml"><mn id="S3.E8.m1.1.1.1.1.5.3.1" xref="S3.E8.m1.1.1.1.1.5.3.1.cmml">0</mn></mtd><mtd id="S3.E8.m1.1.1.1.1ac" xref="S3.E8.m1.1.1a.2.cmml"><mi id="S3.E8.m1.1.1.1.1.5.4.1" mathvariant="normal" xref="S3.E8.m1.1.1.1.1.5.4.1.cmml">⋯</mi></mtd><mtd id="S3.E8.m1.1.1.1.1ad" xref="S3.E8.m1.1.1a.2.cmml"><mrow id="S3.E8.m1.1.1.1.1.5.5.1" xref="S3.E8.m1.1.1.1.1.5.5.1.cmml"><mo id="S3.E8.m1.1.1.1.1.5.5.1a" xref="S3.E8.m1.1.1.1.1.5.5.1.cmml">−</mo><msub id="S3.E8.m1.1.1.1.1.5.5.1.2" xref="S3.E8.m1.1.1.1.1.5.5.1.2.cmml"><mi id="S3.E8.m1.1.1.1.1.5.5.1.2.2" xref="S3.E8.m1.1.1.1.1.5.5.1.2.2.cmml">k</mi><mrow id="S3.E8.m1.1.1.1.1.5.5.1.2.3" xref="S3.E8.m1.1.1.1.1.5.5.1.2.3.cmml"><mi id="S3.E8.m1.1.1.1.1.5.5.1.2.3.2" mathvariant="normal" xref="S3.E8.m1.1.1.1.1.5.5.1.2.3.2.cmml">c</mi><mo id="S3.E8.m1.1.1.1.1.5.5.1.2.3.1" xref="S3.E8.m1.1.1.1.1.5.5.1.2.3.1.cmml">⁢</mo><mi id="S3.E8.m1.1.1.1.1.5.5.1.2.3.3" xref="S3.E8.m1.1.1.1.1.5.5.1.2.3.3.cmml">n</mi></mrow></msub></mrow></mtd></mtr></mtable><mo id="S3.E8.m1.2.2.1.1.4.2.2" xref="S3.E8.m1.2.2.1.1.4.1.1.cmml">]</mo></mrow><mo id="S3.E8.m1.2.2.1.1.5" xref="S3.E8.m1.2.2.1.1.5.cmml">∈</mo><msup id="S3.E8.m1.2.2.1.1.6" xref="S3.E8.m1.2.2.1.1.6.cmml"><mi id="S3.E8.m1.2.2.1.1.6.2" xref="S3.E8.m1.2.2.1.1.6.2.cmml">ℝ</mi><mrow id="S3.E8.m1.2.2.1.1.6.3" xref="S3.E8.m1.2.2.1.1.6.3.cmml"><mi id="S3.E8.m1.2.2.1.1.6.3.2" xref="S3.E8.m1.2.2.1.1.6.3.2.cmml">n</mi><mo id="S3.E8.m1.2.2.1.1.6.3.1" lspace="0.222em" rspace="0.222em" xref="S3.E8.m1.2.2.1.1.6.3.1.cmml">×</mo><mi id="S3.E8.m1.2.2.1.1.6.3.3" xref="S3.E8.m1.2.2.1.1.6.3.3.cmml">n</mi></mrow></msup></mrow><mo id="S3.E8.m1.2.2.1.2" lspace="0em" xref="S3.E8.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E8.m1.2b"><apply id="S3.E8.m1.2.2.1.1.cmml" xref="S3.E8.m1.2.2.1"><and id="S3.E8.m1.2.2.1.1a.cmml" xref="S3.E8.m1.2.2.1"></and><apply id="S3.E8.m1.2.2.1.1b.cmml" xref="S3.E8.m1.2.2.1"><eq id="S3.E8.m1.2.2.1.1.3.cmml" xref="S3.E8.m1.2.2.1.1.3"></eq><ci id="S3.E8.m1.2.2.1.1.2.cmml" xref="S3.E8.m1.2.2.1.1.2">Λ</ci><apply id="S3.E8.m1.2.2.1.1.4.1.cmml" xref="S3.E8.m1.2.2.1.1.4.2"><csymbol cd="latexml" id="S3.E8.m1.2.2.1.1.4.1.1.cmml" xref="S3.E8.m1.2.2.1.1.4.2.1">delimited-[]</csymbol><apply id="S3.E8.m1.1.1a.2.cmml" xref="S3.E8.m1.1.1.1.1"><csymbol cd="latexml" id="S3.E8.m1.1.1a.2.1.cmml" xref="S3.E8.m1.1.1.1.1">matrix</csymbol><matrix id="S3.E8.m1.1.1.1.1.cmml" xref="S3.E8.m1.1.1.1.1"><matrixrow id="S3.E8.m1.1.1.1.1a.cmml" xref="S3.E8.m1.1.1.1.1"><apply id="S3.E8.m1.1.1.1.1.1.1.1.cmml" xref="S3.E8.m1.1.1.1.1.1.1.1"><minus id="S3.E8.m1.1.1.1.1.1.1.1.1.cmml" xref="S3.E8.m1.1.1.1.1.1.1.1"></minus><apply id="S3.E8.m1.1.1.1.1.1.1.1.2.cmml" xref="S3.E8.m1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.E8.m1.1.1.1.1.1.1.1.2.1.cmml" xref="S3.E8.m1.1.1.1.1.1.1.1.2">subscript</csymbol><ci id="S3.E8.m1.1.1.1.1.1.1.1.2.2.cmml" xref="S3.E8.m1.1.1.1.1.1.1.1.2.2">𝑘</ci><ci id="S3.E8.m1.1.1.1.1.1.1.1.2.3.cmml" xref="S3.E8.m1.1.1.1.1.1.1.1.2.3">c1</ci></apply></apply><cn id="S3.E8.m1.1.1.1.1.1.2.1.cmml" type="integer" xref="S3.E8.m1.1.1.1.1.1.2.1">1</cn><cn id="S3.E8.m1.1.1.1.1.1.3.1.cmml" type="integer" xref="S3.E8.m1.1.1.1.1.1.3.1">0</cn><ci id="S3.E8.m1.1.1.1.1.1.4.1.cmml" xref="S3.E8.m1.1.1.1.1.1.4.1">⋯</ci><cn id="S3.E8.m1.1.1.1.1.1.5.1.cmml" type="integer" xref="S3.E8.m1.1.1.1.1.1.5.1">0</cn></matrixrow><matrixrow id="S3.E8.m1.1.1.1.1b.cmml" xref="S3.E8.m1.1.1.1.1"><apply id="S3.E8.m1.1.1.1.1.2.1.1.cmml" xref="S3.E8.m1.1.1.1.1.2.1.1"><minus id="S3.E8.m1.1.1.1.1.2.1.1.1.cmml" xref="S3.E8.m1.1.1.1.1.2.1.1"></minus><cn id="S3.E8.m1.1.1.1.1.2.1.1.2.cmml" type="integer" xref="S3.E8.m1.1.1.1.1.2.1.1.2">1</cn></apply><apply id="S3.E8.m1.1.1.1.1.2.2.1.cmml" xref="S3.E8.m1.1.1.1.1.2.2.1"><minus id="S3.E8.m1.1.1.1.1.2.2.1.1.cmml" xref="S3.E8.m1.1.1.1.1.2.2.1"></minus><apply id="S3.E8.m1.1.1.1.1.2.2.1.2.cmml" xref="S3.E8.m1.1.1.1.1.2.2.1.2"><csymbol cd="ambiguous" id="S3.E8.m1.1.1.1.1.2.2.1.2.1.cmml" xref="S3.E8.m1.1.1.1.1.2.2.1.2">subscript</csymbol><ci id="S3.E8.m1.1.1.1.1.2.2.1.2.2.cmml" xref="S3.E8.m1.1.1.1.1.2.2.1.2.2">𝑘</ci><ci id="S3.E8.m1.1.1.1.1.2.2.1.2.3.cmml" xref="S3.E8.m1.1.1.1.1.2.2.1.2.3">c2</ci></apply></apply><cn id="S3.E8.m1.1.1.1.1.2.3.1.cmml" type="integer" xref="S3.E8.m1.1.1.1.1.2.3.1">1</cn><ci id="S3.E8.m1.1.1.1.1.2.4.1.cmml" xref="S3.E8.m1.1.1.1.1.2.4.1">⋯</ci><cn id="S3.E8.m1.1.1.1.1.2.5.1.cmml" type="integer" xref="S3.E8.m1.1.1.1.1.2.5.1">0</cn></matrixrow><matrixrow id="S3.E8.m1.1.1.1.1c.cmml" xref="S3.E8.m1.1.1.1.1"><ci id="S3.E8.m1.1.1.1.1.3.1.1.cmml" xref="S3.E8.m1.1.1.1.1.3.1.1">⋮</ci><ci id="S3.E8.m1.1.1.1.1.3.2.1.cmml" xref="S3.E8.m1.1.1.1.1.3.2.1">⋮</ci><ci id="S3.E8.m1.1.1.1.1.3.3.1.cmml" xref="S3.E8.m1.1.1.1.1.3.3.1">⋮</ci><ci id="S3.E8.m1.1.1.1.1.3.4.1.cmml" xref="S3.E8.m1.1.1.1.1.3.4.1">⋱</ci><ci id="S3.E8.m1.1.1.1.1.3.5.1.cmml" xref="S3.E8.m1.1.1.1.1.3.5.1">⋮</ci></matrixrow><matrixrow id="S3.E8.m1.1.1.1.1d.cmml" xref="S3.E8.m1.1.1.1.1"><cn id="S3.E8.m1.1.1.1.1.4.1.1.cmml" type="integer" xref="S3.E8.m1.1.1.1.1.4.1.1">0</cn><cn id="S3.E8.m1.1.1.1.1.4.2.1.cmml" type="integer" xref="S3.E8.m1.1.1.1.1.4.2.1">0</cn><cn id="S3.E8.m1.1.1.1.1.4.3.1.cmml" type="integer" xref="S3.E8.m1.1.1.1.1.4.3.1">0</cn><ci id="S3.E8.m1.1.1.1.1.4.4.1.cmml" xref="S3.E8.m1.1.1.1.1.4.4.1">⋯</ci><cn id="S3.E8.m1.1.1.1.1.4.5.1.cmml" type="integer" xref="S3.E8.m1.1.1.1.1.4.5.1">1</cn></matrixrow><matrixrow id="S3.E8.m1.1.1.1.1e.cmml" xref="S3.E8.m1.1.1.1.1"><cn id="S3.E8.m1.1.1.1.1.5.1.1.cmml" type="integer" xref="S3.E8.m1.1.1.1.1.5.1.1">0</cn><cn id="S3.E8.m1.1.1.1.1.5.2.1.cmml" type="integer" xref="S3.E8.m1.1.1.1.1.5.2.1">0</cn><cn id="S3.E8.m1.1.1.1.1.5.3.1.cmml" type="integer" xref="S3.E8.m1.1.1.1.1.5.3.1">0</cn><ci id="S3.E8.m1.1.1.1.1.5.4.1.cmml" xref="S3.E8.m1.1.1.1.1.5.4.1">⋯</ci><apply id="S3.E8.m1.1.1.1.1.5.5.1.cmml" xref="S3.E8.m1.1.1.1.1.5.5.1"><minus id="S3.E8.m1.1.1.1.1.5.5.1.1.cmml" xref="S3.E8.m1.1.1.1.1.5.5.1"></minus><apply id="S3.E8.m1.1.1.1.1.5.5.1.2.cmml" xref="S3.E8.m1.1.1.1.1.5.5.1.2"><csymbol cd="ambiguous" id="S3.E8.m1.1.1.1.1.5.5.1.2.1.cmml" xref="S3.E8.m1.1.1.1.1.5.5.1.2">subscript</csymbol><ci id="S3.E8.m1.1.1.1.1.5.5.1.2.2.cmml" xref="S3.E8.m1.1.1.1.1.5.5.1.2.2">𝑘</ci><apply id="S3.E8.m1.1.1.1.1.5.5.1.2.3.cmml" xref="S3.E8.m1.1.1.1.1.5.5.1.2.3"><times id="S3.E8.m1.1.1.1.1.5.5.1.2.3.1.cmml" xref="S3.E8.m1.1.1.1.1.5.5.1.2.3.1"></times><ci id="S3.E8.m1.1.1.1.1.5.5.1.2.3.2.cmml" xref="S3.E8.m1.1.1.1.1.5.5.1.2.3.2">c</ci><ci id="S3.E8.m1.1.1.1.1.5.5.1.2.3.3.cmml" xref="S3.E8.m1.1.1.1.1.5.5.1.2.3.3">𝑛</ci></apply></apply></apply></matrixrow></matrix></apply></apply></apply><apply id="S3.E8.m1.2.2.1.1c.cmml" xref="S3.E8.m1.2.2.1"><in id="S3.E8.m1.2.2.1.1.5.cmml" xref="S3.E8.m1.2.2.1.1.5"></in><share href="https://arxiv.org/html/2401.10785v2#S3.E8.m1.2.2.1.1.4.cmml" id="S3.E8.m1.2.2.1.1d.cmml" xref="S3.E8.m1.2.2.1"></share><apply id="S3.E8.m1.2.2.1.1.6.cmml" xref="S3.E8.m1.2.2.1.1.6"><csymbol cd="ambiguous" id="S3.E8.m1.2.2.1.1.6.1.cmml" xref="S3.E8.m1.2.2.1.1.6">superscript</csymbol><ci id="S3.E8.m1.2.2.1.1.6.2.cmml" xref="S3.E8.m1.2.2.1.1.6.2">ℝ</ci><apply id="S3.E8.m1.2.2.1.1.6.3.cmml" xref="S3.E8.m1.2.2.1.1.6.3"><times id="S3.E8.m1.2.2.1.1.6.3.1.cmml" xref="S3.E8.m1.2.2.1.1.6.3.1"></times><ci id="S3.E8.m1.2.2.1.1.6.3.2.cmml" xref="S3.E8.m1.2.2.1.1.6.3.2">𝑛</ci><ci id="S3.E8.m1.2.2.1.1.6.3.3.cmml" xref="S3.E8.m1.2.2.1.1.6.3.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E8.m1.2c">\displaystyle\Lambda=\left[\begin{matrix}-k_{{\rm c}1}&1&0&\cdots&0\\ -1&-k_{{\rm c}2}&1&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&1\\ 0&0&0&\cdots&-k_{{\rm c}n}\\ \end{matrix}\right]\in\mathbb{R}^{n\times n}.</annotation><annotation encoding="application/x-llamapun" id="S3.E8.m1.2d">roman_Λ = [ start_ARG start_ROW start_CELL - italic_k start_POSTSUBSCRIPT c1 end_POSTSUBSCRIPT end_CELL start_CELL 1 end_CELL start_CELL 0 end_CELL start_CELL ⋯ end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL - 1 end_CELL start_CELL - italic_k start_POSTSUBSCRIPT c2 end_POSTSUBSCRIPT end_CELL start_CELL 1 end_CELL start_CELL ⋯ end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL start_CELL ⋮ end_CELL start_CELL ⋮ end_CELL start_CELL ⋱ end_CELL start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL ⋯ end_CELL start_CELL 1 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL ⋯ end_CELL start_CELL - italic_k start_POSTSUBSCRIPT roman_c italic_n end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] ∈ blackboard_R start_POSTSUPERSCRIPT italic_n × italic_n end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(8)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.p5"> <p class="ltx_p" id="S3.p5.18">Consider linear filtering operations</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx6"> <tbody id="S3.E11"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\left\{\begin{array}[]{l}\dot{\bm{\zeta}}(t)=\Lambda\bm{\zeta}(t)% +\Phi^{T}(t)\hat{\bm{\theta}}(t)\\ \dot{\Phi}_{\rm s}^{T}(t)=\Lambda{\Phi}_{\rm s}^{T}(t)+\Phi^{T}(t)\end{array}\right." class="ltx_Math" display="inline" id="S3.E11.m1.1"><semantics id="S3.E11.m1.1a"><mrow id="S3.E11.m1.1.2.2" xref="S3.E11.m1.1.2.1.cmml"><mo id="S3.E11.m1.1.2.2.1" xref="S3.E11.m1.1.2.1.1.cmml">{</mo><mtable id="S3.E11.m1.1.1" rowspacing="0pt" xref="S3.E11.m1.1.1.cmml"><mtr id="S3.E11.m1.1.1a" xref="S3.E11.m1.1.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S3.E11.m1.1.1b" xref="S3.E11.m1.1.1.cmml"><mrow id="S3.E9.4.4" xref="S3.E9.4.4.cmml"><mrow id="S3.E9.4.4.6" xref="S3.E9.4.4.6.cmml"><mover accent="true" id="S3.E9.4.4.6.2" xref="S3.E9.4.4.6.2.cmml"><mi id="S3.E9.4.4.6.2.2" xref="S3.E9.4.4.6.2.2.cmml">𝜻</mi><mo id="S3.E9.4.4.6.2.1" xref="S3.E9.4.4.6.2.1.cmml">˙</mo></mover><mo id="S3.E9.4.4.6.1" xref="S3.E9.4.4.6.1.cmml">⁢</mo><mrow id="S3.E9.4.4.6.3.2" xref="S3.E9.4.4.6.cmml"><mo id="S3.E9.4.4.6.3.2.1" stretchy="false" xref="S3.E9.4.4.6.cmml">(</mo><mi id="S3.E9.1.1.1" xref="S3.E9.1.1.1.cmml">t</mi><mo id="S3.E9.4.4.6.3.2.2" stretchy="false" xref="S3.E9.4.4.6.cmml">)</mo></mrow></mrow><mo id="S3.E9.4.4.5" xref="S3.E9.4.4.5.cmml">=</mo><mrow id="S3.E9.4.4.7" xref="S3.E9.4.4.7.cmml"><mrow id="S3.E9.4.4.7.2" xref="S3.E9.4.4.7.2.cmml"><mi id="S3.E9.4.4.7.2.2" mathvariant="normal" xref="S3.E9.4.4.7.2.2.cmml">Λ</mi><mo id="S3.E9.4.4.7.2.1" xref="S3.E9.4.4.7.2.1.cmml">⁢</mo><mi id="S3.E9.4.4.7.2.3" xref="S3.E9.4.4.7.2.3.cmml">𝜻</mi><mo id="S3.E9.4.4.7.2.1a" xref="S3.E9.4.4.7.2.1.cmml">⁢</mo><mrow id="S3.E9.4.4.7.2.4.2" xref="S3.E9.4.4.7.2.cmml"><mo id="S3.E9.4.4.7.2.4.2.1" stretchy="false" xref="S3.E9.4.4.7.2.cmml">(</mo><mi id="S3.E9.2.2.2" xref="S3.E9.2.2.2.cmml">t</mi><mo id="S3.E9.4.4.7.2.4.2.2" stretchy="false" xref="S3.E9.4.4.7.2.cmml">)</mo></mrow></mrow><mo id="S3.E9.4.4.7.1" xref="S3.E9.4.4.7.1.cmml">+</mo><mrow id="S3.E9.4.4.7.3" xref="S3.E9.4.4.7.3.cmml"><msup id="S3.E9.4.4.7.3.2" xref="S3.E9.4.4.7.3.2.cmml"><mi id="S3.E9.4.4.7.3.2.2" mathvariant="normal" xref="S3.E9.4.4.7.3.2.2.cmml">Φ</mi><mi id="S3.E9.4.4.7.3.2.3" xref="S3.E9.4.4.7.3.2.3.cmml">T</mi></msup><mo id="S3.E9.4.4.7.3.1" xref="S3.E9.4.4.7.3.1.cmml">⁢</mo><mrow id="S3.E9.4.4.7.3.3.2" xref="S3.E9.4.4.7.3.cmml"><mo id="S3.E9.4.4.7.3.3.2.1" stretchy="false" xref="S3.E9.4.4.7.3.cmml">(</mo><mi id="S3.E9.3.3.3" xref="S3.E9.3.3.3.cmml">t</mi><mo id="S3.E9.4.4.7.3.3.2.2" stretchy="false" xref="S3.E9.4.4.7.3.cmml">)</mo></mrow><mo id="S3.E9.4.4.7.3.1a" xref="S3.E9.4.4.7.3.1.cmml">⁢</mo><mover accent="true" id="S3.E9.4.4.7.3.4" xref="S3.E9.4.4.7.3.4.cmml"><mi id="S3.E9.4.4.7.3.4.2" xref="S3.E9.4.4.7.3.4.2.cmml">𝜽</mi><mo id="S3.E9.4.4.7.3.4.1" xref="S3.E9.4.4.7.3.4.1.cmml">^</mo></mover><mo id="S3.E9.4.4.7.3.1b" xref="S3.E9.4.4.7.3.1.cmml">⁢</mo><mrow id="S3.E9.4.4.7.3.5.2" xref="S3.E9.4.4.7.3.cmml"><mo id="S3.E9.4.4.7.3.5.2.1" stretchy="false" xref="S3.E9.4.4.7.3.cmml">(</mo><mi id="S3.E9.4.4.4" xref="S3.E9.4.4.4.cmml">t</mi><mo id="S3.E9.4.4.7.3.5.2.2" stretchy="false" xref="S3.E9.4.4.7.3.cmml">)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr id="S3.E11.m1.1.1c" xref="S3.E11.m1.1.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S3.E11.m1.1.1d" xref="S3.E11.m1.1.1.cmml"><mrow id="S3.E10.3.3" xref="S3.E10.3.3.cmml"><mrow id="S3.E10.3.3.5" xref="S3.E10.3.3.5.cmml"><msubsup id="S3.E10.3.3.5.2" xref="S3.E10.3.3.5.2.cmml"><mover accent="true" id="S3.E10.3.3.5.2.2.2" xref="S3.E10.3.3.5.2.2.2.cmml"><mi id="S3.E10.3.3.5.2.2.2.2" mathvariant="normal" xref="S3.E10.3.3.5.2.2.2.2.cmml">Φ</mi><mo id="S3.E10.3.3.5.2.2.2.1" xref="S3.E10.3.3.5.2.2.2.1.cmml">˙</mo></mover><mi id="S3.E10.3.3.5.2.2.3" mathvariant="normal" xref="S3.E10.3.3.5.2.2.3.cmml">s</mi><mi id="S3.E10.3.3.5.2.3" xref="S3.E10.3.3.5.2.3.cmml">T</mi></msubsup><mo id="S3.E10.3.3.5.1" xref="S3.E10.3.3.5.1.cmml">⁢</mo><mrow id="S3.E10.3.3.5.3.2" xref="S3.E10.3.3.5.cmml"><mo id="S3.E10.3.3.5.3.2.1" stretchy="false" xref="S3.E10.3.3.5.cmml">(</mo><mi id="S3.E10.1.1.1" xref="S3.E10.1.1.1.cmml">t</mi><mo id="S3.E10.3.3.5.3.2.2" stretchy="false" xref="S3.E10.3.3.5.cmml">)</mo></mrow></mrow><mo id="S3.E10.3.3.4" xref="S3.E10.3.3.4.cmml">=</mo><mrow id="S3.E10.3.3.6" xref="S3.E10.3.3.6.cmml"><mrow id="S3.E10.3.3.6.2" xref="S3.E10.3.3.6.2.cmml"><mi id="S3.E10.3.3.6.2.2" mathvariant="normal" xref="S3.E10.3.3.6.2.2.cmml">Λ</mi><mo id="S3.E10.3.3.6.2.1" xref="S3.E10.3.3.6.2.1.cmml">⁢</mo><msubsup id="S3.E10.3.3.6.2.3" xref="S3.E10.3.3.6.2.3.cmml"><mi id="S3.E10.3.3.6.2.3.2.2" mathvariant="normal" xref="S3.E10.3.3.6.2.3.2.2.cmml">Φ</mi><mi id="S3.E10.3.3.6.2.3.2.3" mathvariant="normal" xref="S3.E10.3.3.6.2.3.2.3.cmml">s</mi><mi id="S3.E10.3.3.6.2.3.3" xref="S3.E10.3.3.6.2.3.3.cmml">T</mi></msubsup><mo id="S3.E10.3.3.6.2.1a" xref="S3.E10.3.3.6.2.1.cmml">⁢</mo><mrow id="S3.E10.3.3.6.2.4.2" xref="S3.E10.3.3.6.2.cmml"><mo id="S3.E10.3.3.6.2.4.2.1" stretchy="false" xref="S3.E10.3.3.6.2.cmml">(</mo><mi id="S3.E10.2.2.2" xref="S3.E10.2.2.2.cmml">t</mi><mo id="S3.E10.3.3.6.2.4.2.2" stretchy="false" xref="S3.E10.3.3.6.2.cmml">)</mo></mrow></mrow><mo id="S3.E10.3.3.6.1" xref="S3.E10.3.3.6.1.cmml">+</mo><mrow id="S3.E10.3.3.6.3" xref="S3.E10.3.3.6.3.cmml"><msup id="S3.E10.3.3.6.3.2" xref="S3.E10.3.3.6.3.2.cmml"><mi id="S3.E10.3.3.6.3.2.2" mathvariant="normal" xref="S3.E10.3.3.6.3.2.2.cmml">Φ</mi><mi id="S3.E10.3.3.6.3.2.3" xref="S3.E10.3.3.6.3.2.3.cmml">T</mi></msup><mo id="S3.E10.3.3.6.3.1" xref="S3.E10.3.3.6.3.1.cmml">⁢</mo><mrow id="S3.E10.3.3.6.3.3.2" xref="S3.E10.3.3.6.3.cmml"><mo id="S3.E10.3.3.6.3.3.2.1" stretchy="false" xref="S3.E10.3.3.6.3.cmml">(</mo><mi id="S3.E10.3.3.3" xref="S3.E10.3.3.3.cmml">t</mi><mo id="S3.E10.3.3.6.3.3.2.2" stretchy="false" xref="S3.E10.3.3.6.3.cmml">)</mo></mrow></mrow></mrow></mrow></mtd></mtr></mtable><mi id="S3.E11.m1.1.2.2.2" xref="S3.E11.m1.1.2.1.1.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="S3.E11.m1.1b"><apply id="S3.E11.m1.1.2.1.cmml" xref="S3.E11.m1.1.2.2"><csymbol cd="latexml" id="S3.E11.m1.1.2.1.1.cmml" xref="S3.E11.m1.1.2.2.1">cases</csymbol><matrix id="S3.E11.m1.1.1.cmml" xref="S3.E11.m1.1.1"><matrixrow id="S3.E11.m1.1.1a.cmml" xref="S3.E11.m1.1.1"><apply id="S3.E9.4.4.cmml" xref="S3.E9.4.4"><eq id="S3.E9.4.4.5.cmml" xref="S3.E9.4.4.5"></eq><apply id="S3.E9.4.4.6.cmml" xref="S3.E9.4.4.6"><times id="S3.E9.4.4.6.1.cmml" xref="S3.E9.4.4.6.1"></times><apply id="S3.E9.4.4.6.2.cmml" xref="S3.E9.4.4.6.2"><ci id="S3.E9.4.4.6.2.1.cmml" xref="S3.E9.4.4.6.2.1">˙</ci><ci id="S3.E9.4.4.6.2.2.cmml" xref="S3.E9.4.4.6.2.2">𝜻</ci></apply><ci id="S3.E9.1.1.1.cmml" xref="S3.E9.1.1.1">𝑡</ci></apply><apply id="S3.E9.4.4.7.cmml" xref="S3.E9.4.4.7"><plus id="S3.E9.4.4.7.1.cmml" xref="S3.E9.4.4.7.1"></plus><apply id="S3.E9.4.4.7.2.cmml" xref="S3.E9.4.4.7.2"><times id="S3.E9.4.4.7.2.1.cmml" xref="S3.E9.4.4.7.2.1"></times><ci id="S3.E9.4.4.7.2.2.cmml" xref="S3.E9.4.4.7.2.2">Λ</ci><ci id="S3.E9.4.4.7.2.3.cmml" xref="S3.E9.4.4.7.2.3">𝜻</ci><ci id="S3.E9.2.2.2.cmml" xref="S3.E9.2.2.2">𝑡</ci></apply><apply id="S3.E9.4.4.7.3.cmml" xref="S3.E9.4.4.7.3"><times id="S3.E9.4.4.7.3.1.cmml" xref="S3.E9.4.4.7.3.1"></times><apply id="S3.E9.4.4.7.3.2.cmml" xref="S3.E9.4.4.7.3.2"><csymbol cd="ambiguous" id="S3.E9.4.4.7.3.2.1.cmml" xref="S3.E9.4.4.7.3.2">superscript</csymbol><ci id="S3.E9.4.4.7.3.2.2.cmml" xref="S3.E9.4.4.7.3.2.2">Φ</ci><ci id="S3.E9.4.4.7.3.2.3.cmml" xref="S3.E9.4.4.7.3.2.3">𝑇</ci></apply><ci id="S3.E9.3.3.3.cmml" xref="S3.E9.3.3.3">𝑡</ci><apply id="S3.E9.4.4.7.3.4.cmml" xref="S3.E9.4.4.7.3.4"><ci id="S3.E9.4.4.7.3.4.1.cmml" xref="S3.E9.4.4.7.3.4.1">^</ci><ci id="S3.E9.4.4.7.3.4.2.cmml" xref="S3.E9.4.4.7.3.4.2">𝜽</ci></apply><ci id="S3.E9.4.4.4.cmml" xref="S3.E9.4.4.4">𝑡</ci></apply></apply></apply></matrixrow><matrixrow id="S3.E11.m1.1.1b.cmml" xref="S3.E11.m1.1.1"><apply id="S3.E10.3.3.cmml" xref="S3.E10.3.3"><eq id="S3.E10.3.3.4.cmml" xref="S3.E10.3.3.4"></eq><apply id="S3.E10.3.3.5.cmml" xref="S3.E10.3.3.5"><times id="S3.E10.3.3.5.1.cmml" xref="S3.E10.3.3.5.1"></times><apply id="S3.E10.3.3.5.2.cmml" xref="S3.E10.3.3.5.2"><csymbol cd="ambiguous" id="S3.E10.3.3.5.2.1.cmml" xref="S3.E10.3.3.5.2">superscript</csymbol><apply id="S3.E10.3.3.5.2.2.cmml" xref="S3.E10.3.3.5.2"><csymbol cd="ambiguous" id="S3.E10.3.3.5.2.2.1.cmml" xref="S3.E10.3.3.5.2">subscript</csymbol><apply id="S3.E10.3.3.5.2.2.2.cmml" xref="S3.E10.3.3.5.2.2.2"><ci id="S3.E10.3.3.5.2.2.2.1.cmml" xref="S3.E10.3.3.5.2.2.2.1">˙</ci><ci id="S3.E10.3.3.5.2.2.2.2.cmml" xref="S3.E10.3.3.5.2.2.2.2">Φ</ci></apply><ci id="S3.E10.3.3.5.2.2.3.cmml" xref="S3.E10.3.3.5.2.2.3">s</ci></apply><ci id="S3.E10.3.3.5.2.3.cmml" xref="S3.E10.3.3.5.2.3">𝑇</ci></apply><ci id="S3.E10.1.1.1.cmml" xref="S3.E10.1.1.1">𝑡</ci></apply><apply id="S3.E10.3.3.6.cmml" xref="S3.E10.3.3.6"><plus id="S3.E10.3.3.6.1.cmml" xref="S3.E10.3.3.6.1"></plus><apply id="S3.E10.3.3.6.2.cmml" xref="S3.E10.3.3.6.2"><times id="S3.E10.3.3.6.2.1.cmml" xref="S3.E10.3.3.6.2.1"></times><ci id="S3.E10.3.3.6.2.2.cmml" xref="S3.E10.3.3.6.2.2">Λ</ci><apply id="S3.E10.3.3.6.2.3.cmml" xref="S3.E10.3.3.6.2.3"><csymbol cd="ambiguous" id="S3.E10.3.3.6.2.3.1.cmml" xref="S3.E10.3.3.6.2.3">superscript</csymbol><apply id="S3.E10.3.3.6.2.3.2.cmml" xref="S3.E10.3.3.6.2.3"><csymbol cd="ambiguous" id="S3.E10.3.3.6.2.3.2.1.cmml" xref="S3.E10.3.3.6.2.3">subscript</csymbol><ci id="S3.E10.3.3.6.2.3.2.2.cmml" xref="S3.E10.3.3.6.2.3.2.2">Φ</ci><ci id="S3.E10.3.3.6.2.3.2.3.cmml" xref="S3.E10.3.3.6.2.3.2.3">s</ci></apply><ci id="S3.E10.3.3.6.2.3.3.cmml" xref="S3.E10.3.3.6.2.3.3">𝑇</ci></apply><ci id="S3.E10.2.2.2.cmml" xref="S3.E10.2.2.2">𝑡</ci></apply><apply id="S3.E10.3.3.6.3.cmml" xref="S3.E10.3.3.6.3"><times id="S3.E10.3.3.6.3.1.cmml" xref="S3.E10.3.3.6.3.1"></times><apply id="S3.E10.3.3.6.3.2.cmml" xref="S3.E10.3.3.6.3.2"><csymbol cd="ambiguous" id="S3.E10.3.3.6.3.2.1.cmml" xref="S3.E10.3.3.6.3.2">superscript</csymbol><ci id="S3.E10.3.3.6.3.2.2.cmml" xref="S3.E10.3.3.6.3.2.2">Φ</ci><ci id="S3.E10.3.3.6.3.2.3.cmml" xref="S3.E10.3.3.6.3.2.3">𝑇</ci></apply><ci id="S3.E10.3.3.3.cmml" xref="S3.E10.3.3.3">𝑡</ci></apply></apply></apply></matrixrow></matrix></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E11.m1.1c">\displaystyle\left\{\begin{array}[]{l}\dot{\bm{\zeta}}(t)=\Lambda\bm{\zeta}(t)% +\Phi^{T}(t)\hat{\bm{\theta}}(t)\\ \dot{\Phi}_{\rm s}^{T}(t)=\Lambda{\Phi}_{\rm s}^{T}(t)+\Phi^{T}(t)\end{array}\right.</annotation><annotation encoding="application/x-llamapun" id="S3.E11.m1.1d">{ start_ARRAY start_ROW start_CELL over˙ start_ARG bold_italic_ζ end_ARG ( italic_t ) = roman_Λ bold_italic_ζ ( italic_t ) + roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_t ) over^ start_ARG bold_italic_θ end_ARG ( italic_t ) end_CELL end_ROW start_ROW start_CELL over˙ start_ARG roman_Φ end_ARG start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_t ) = roman_Λ roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_t ) + roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_t ) end_CELL end_ROW end_ARRAY</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(11)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p5.5">and define an output vector <math alttext="\bm{p}(t)" class="ltx_Math" display="inline" id="S3.p5.1.m1.1"><semantics id="S3.p5.1.m1.1a"><mrow id="S3.p5.1.m1.1.2" xref="S3.p5.1.m1.1.2.cmml"><mi id="S3.p5.1.m1.1.2.2" xref="S3.p5.1.m1.1.2.2.cmml">𝒑</mi><mo id="S3.p5.1.m1.1.2.1" xref="S3.p5.1.m1.1.2.1.cmml">⁢</mo><mrow id="S3.p5.1.m1.1.2.3.2" xref="S3.p5.1.m1.1.2.cmml"><mo id="S3.p5.1.m1.1.2.3.2.1" stretchy="false" xref="S3.p5.1.m1.1.2.cmml">(</mo><mi id="S3.p5.1.m1.1.1" xref="S3.p5.1.m1.1.1.cmml">t</mi><mo id="S3.p5.1.m1.1.2.3.2.2" stretchy="false" xref="S3.p5.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.1.m1.1b"><apply id="S3.p5.1.m1.1.2.cmml" xref="S3.p5.1.m1.1.2"><times id="S3.p5.1.m1.1.2.1.cmml" xref="S3.p5.1.m1.1.2.1"></times><ci id="S3.p5.1.m1.1.2.2.cmml" xref="S3.p5.1.m1.1.2.2">𝒑</ci><ci id="S3.p5.1.m1.1.1.cmml" xref="S3.p5.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.1.m1.1c">\bm{p}(t)</annotation><annotation encoding="application/x-llamapun" id="S3.p5.1.m1.1d">bold_italic_p ( italic_t )</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S3.p5.2.m2.1"><semantics id="S3.p5.2.m2.1a"><mo id="S3.p5.2.m2.1.1" xref="S3.p5.2.m2.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S3.p5.2.m2.1b"><csymbol cd="latexml" id="S3.p5.2.m2.1.1.cmml" xref="S3.p5.2.m2.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.2.m2.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S3.p5.2.m2.1d">:=</annotation></semantics></math> <math alttext="\bm{e}(t)+\bm{\zeta}(t)\in\mathbb{R}^{n}" class="ltx_Math" display="inline" id="S3.p5.3.m3.2"><semantics id="S3.p5.3.m3.2a"><mrow id="S3.p5.3.m3.2.3" xref="S3.p5.3.m3.2.3.cmml"><mrow id="S3.p5.3.m3.2.3.2" xref="S3.p5.3.m3.2.3.2.cmml"><mrow id="S3.p5.3.m3.2.3.2.2" xref="S3.p5.3.m3.2.3.2.2.cmml"><mi id="S3.p5.3.m3.2.3.2.2.2" xref="S3.p5.3.m3.2.3.2.2.2.cmml">𝒆</mi><mo id="S3.p5.3.m3.2.3.2.2.1" xref="S3.p5.3.m3.2.3.2.2.1.cmml">⁢</mo><mrow id="S3.p5.3.m3.2.3.2.2.3.2" xref="S3.p5.3.m3.2.3.2.2.cmml"><mo id="S3.p5.3.m3.2.3.2.2.3.2.1" stretchy="false" xref="S3.p5.3.m3.2.3.2.2.cmml">(</mo><mi id="S3.p5.3.m3.1.1" xref="S3.p5.3.m3.1.1.cmml">t</mi><mo id="S3.p5.3.m3.2.3.2.2.3.2.2" stretchy="false" xref="S3.p5.3.m3.2.3.2.2.cmml">)</mo></mrow></mrow><mo id="S3.p5.3.m3.2.3.2.1" xref="S3.p5.3.m3.2.3.2.1.cmml">+</mo><mrow id="S3.p5.3.m3.2.3.2.3" xref="S3.p5.3.m3.2.3.2.3.cmml"><mi id="S3.p5.3.m3.2.3.2.3.2" xref="S3.p5.3.m3.2.3.2.3.2.cmml">𝜻</mi><mo id="S3.p5.3.m3.2.3.2.3.1" xref="S3.p5.3.m3.2.3.2.3.1.cmml">⁢</mo><mrow id="S3.p5.3.m3.2.3.2.3.3.2" xref="S3.p5.3.m3.2.3.2.3.cmml"><mo id="S3.p5.3.m3.2.3.2.3.3.2.1" stretchy="false" xref="S3.p5.3.m3.2.3.2.3.cmml">(</mo><mi id="S3.p5.3.m3.2.2" xref="S3.p5.3.m3.2.2.cmml">t</mi><mo id="S3.p5.3.m3.2.3.2.3.3.2.2" stretchy="false" xref="S3.p5.3.m3.2.3.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.p5.3.m3.2.3.1" xref="S3.p5.3.m3.2.3.1.cmml">∈</mo><msup id="S3.p5.3.m3.2.3.3" xref="S3.p5.3.m3.2.3.3.cmml"><mi id="S3.p5.3.m3.2.3.3.2" xref="S3.p5.3.m3.2.3.3.2.cmml">ℝ</mi><mi id="S3.p5.3.m3.2.3.3.3" xref="S3.p5.3.m3.2.3.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.3.m3.2b"><apply id="S3.p5.3.m3.2.3.cmml" xref="S3.p5.3.m3.2.3"><in id="S3.p5.3.m3.2.3.1.cmml" xref="S3.p5.3.m3.2.3.1"></in><apply id="S3.p5.3.m3.2.3.2.cmml" xref="S3.p5.3.m3.2.3.2"><plus id="S3.p5.3.m3.2.3.2.1.cmml" xref="S3.p5.3.m3.2.3.2.1"></plus><apply id="S3.p5.3.m3.2.3.2.2.cmml" xref="S3.p5.3.m3.2.3.2.2"><times id="S3.p5.3.m3.2.3.2.2.1.cmml" xref="S3.p5.3.m3.2.3.2.2.1"></times><ci id="S3.p5.3.m3.2.3.2.2.2.cmml" xref="S3.p5.3.m3.2.3.2.2.2">𝒆</ci><ci id="S3.p5.3.m3.1.1.cmml" xref="S3.p5.3.m3.1.1">𝑡</ci></apply><apply id="S3.p5.3.m3.2.3.2.3.cmml" xref="S3.p5.3.m3.2.3.2.3"><times id="S3.p5.3.m3.2.3.2.3.1.cmml" xref="S3.p5.3.m3.2.3.2.3.1"></times><ci id="S3.p5.3.m3.2.3.2.3.2.cmml" xref="S3.p5.3.m3.2.3.2.3.2">𝜻</ci><ci id="S3.p5.3.m3.2.2.cmml" xref="S3.p5.3.m3.2.2">𝑡</ci></apply></apply><apply id="S3.p5.3.m3.2.3.3.cmml" xref="S3.p5.3.m3.2.3.3"><csymbol cd="ambiguous" id="S3.p5.3.m3.2.3.3.1.cmml" xref="S3.p5.3.m3.2.3.3">superscript</csymbol><ci id="S3.p5.3.m3.2.3.3.2.cmml" xref="S3.p5.3.m3.2.3.3.2">ℝ</ci><ci id="S3.p5.3.m3.2.3.3.3.cmml" xref="S3.p5.3.m3.2.3.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.3.m3.2c">\bm{e}(t)+\bm{\zeta}(t)\in\mathbb{R}^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.3.m3.2d">bold_italic_e ( italic_t ) + bold_italic_ζ ( italic_t ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="\bm{\zeta}(t)\in\mathbb{R}^{n}" class="ltx_Math" display="inline" id="S3.p5.4.m4.1"><semantics id="S3.p5.4.m4.1a"><mrow id="S3.p5.4.m4.1.2" xref="S3.p5.4.m4.1.2.cmml"><mrow id="S3.p5.4.m4.1.2.2" xref="S3.p5.4.m4.1.2.2.cmml"><mi id="S3.p5.4.m4.1.2.2.2" xref="S3.p5.4.m4.1.2.2.2.cmml">𝜻</mi><mo id="S3.p5.4.m4.1.2.2.1" xref="S3.p5.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S3.p5.4.m4.1.2.2.3.2" xref="S3.p5.4.m4.1.2.2.cmml"><mo id="S3.p5.4.m4.1.2.2.3.2.1" stretchy="false" xref="S3.p5.4.m4.1.2.2.cmml">(</mo><mi id="S3.p5.4.m4.1.1" xref="S3.p5.4.m4.1.1.cmml">t</mi><mo id="S3.p5.4.m4.1.2.2.3.2.2" stretchy="false" xref="S3.p5.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.p5.4.m4.1.2.1" xref="S3.p5.4.m4.1.2.1.cmml">∈</mo><msup id="S3.p5.4.m4.1.2.3" xref="S3.p5.4.m4.1.2.3.cmml"><mi id="S3.p5.4.m4.1.2.3.2" xref="S3.p5.4.m4.1.2.3.2.cmml">ℝ</mi><mi id="S3.p5.4.m4.1.2.3.3" xref="S3.p5.4.m4.1.2.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.4.m4.1b"><apply id="S3.p5.4.m4.1.2.cmml" xref="S3.p5.4.m4.1.2"><in id="S3.p5.4.m4.1.2.1.cmml" xref="S3.p5.4.m4.1.2.1"></in><apply id="S3.p5.4.m4.1.2.2.cmml" xref="S3.p5.4.m4.1.2.2"><times id="S3.p5.4.m4.1.2.2.1.cmml" xref="S3.p5.4.m4.1.2.2.1"></times><ci id="S3.p5.4.m4.1.2.2.2.cmml" xref="S3.p5.4.m4.1.2.2.2">𝜻</ci><ci id="S3.p5.4.m4.1.1.cmml" xref="S3.p5.4.m4.1.1">𝑡</ci></apply><apply id="S3.p5.4.m4.1.2.3.cmml" xref="S3.p5.4.m4.1.2.3"><csymbol cd="ambiguous" id="S3.p5.4.m4.1.2.3.1.cmml" xref="S3.p5.4.m4.1.2.3">superscript</csymbol><ci id="S3.p5.4.m4.1.2.3.2.cmml" xref="S3.p5.4.m4.1.2.3.2">ℝ</ci><ci id="S3.p5.4.m4.1.2.3.3.cmml" xref="S3.p5.4.m4.1.2.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.4.m4.1c">\bm{\zeta}(t)\in\mathbb{R}^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.4.m4.1d">bold_italic_ζ ( italic_t ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="{\Phi}_{\rm s}(t)\in\mathbb{R}^{N\times n}" class="ltx_Math" display="inline" id="S3.p5.5.m5.1"><semantics id="S3.p5.5.m5.1a"><mrow id="S3.p5.5.m5.1.2" xref="S3.p5.5.m5.1.2.cmml"><mrow id="S3.p5.5.m5.1.2.2" xref="S3.p5.5.m5.1.2.2.cmml"><msub id="S3.p5.5.m5.1.2.2.2" xref="S3.p5.5.m5.1.2.2.2.cmml"><mi id="S3.p5.5.m5.1.2.2.2.2" mathvariant="normal" xref="S3.p5.5.m5.1.2.2.2.2.cmml">Φ</mi><mi id="S3.p5.5.m5.1.2.2.2.3" mathvariant="normal" xref="S3.p5.5.m5.1.2.2.2.3.cmml">s</mi></msub><mo id="S3.p5.5.m5.1.2.2.1" xref="S3.p5.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S3.p5.5.m5.1.2.2.3.2" xref="S3.p5.5.m5.1.2.2.cmml"><mo id="S3.p5.5.m5.1.2.2.3.2.1" stretchy="false" xref="S3.p5.5.m5.1.2.2.cmml">(</mo><mi id="S3.p5.5.m5.1.1" xref="S3.p5.5.m5.1.1.cmml">t</mi><mo id="S3.p5.5.m5.1.2.2.3.2.2" stretchy="false" xref="S3.p5.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.p5.5.m5.1.2.1" xref="S3.p5.5.m5.1.2.1.cmml">∈</mo><msup id="S3.p5.5.m5.1.2.3" xref="S3.p5.5.m5.1.2.3.cmml"><mi id="S3.p5.5.m5.1.2.3.2" xref="S3.p5.5.m5.1.2.3.2.cmml">ℝ</mi><mrow id="S3.p5.5.m5.1.2.3.3" xref="S3.p5.5.m5.1.2.3.3.cmml"><mi id="S3.p5.5.m5.1.2.3.3.2" xref="S3.p5.5.m5.1.2.3.3.2.cmml">N</mi><mo id="S3.p5.5.m5.1.2.3.3.1" lspace="0.222em" rspace="0.222em" xref="S3.p5.5.m5.1.2.3.3.1.cmml">×</mo><mi id="S3.p5.5.m5.1.2.3.3.3" xref="S3.p5.5.m5.1.2.3.3.3.cmml">n</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.5.m5.1b"><apply id="S3.p5.5.m5.1.2.cmml" xref="S3.p5.5.m5.1.2"><in id="S3.p5.5.m5.1.2.1.cmml" xref="S3.p5.5.m5.1.2.1"></in><apply id="S3.p5.5.m5.1.2.2.cmml" xref="S3.p5.5.m5.1.2.2"><times id="S3.p5.5.m5.1.2.2.1.cmml" xref="S3.p5.5.m5.1.2.2.1"></times><apply id="S3.p5.5.m5.1.2.2.2.cmml" xref="S3.p5.5.m5.1.2.2.2"><csymbol cd="ambiguous" id="S3.p5.5.m5.1.2.2.2.1.cmml" xref="S3.p5.5.m5.1.2.2.2">subscript</csymbol><ci id="S3.p5.5.m5.1.2.2.2.2.cmml" xref="S3.p5.5.m5.1.2.2.2.2">Φ</ci><ci id="S3.p5.5.m5.1.2.2.2.3.cmml" xref="S3.p5.5.m5.1.2.2.2.3">s</ci></apply><ci id="S3.p5.5.m5.1.1.cmml" xref="S3.p5.5.m5.1.1">𝑡</ci></apply><apply id="S3.p5.5.m5.1.2.3.cmml" xref="S3.p5.5.m5.1.2.3"><csymbol cd="ambiguous" id="S3.p5.5.m5.1.2.3.1.cmml" xref="S3.p5.5.m5.1.2.3">superscript</csymbol><ci id="S3.p5.5.m5.1.2.3.2.cmml" xref="S3.p5.5.m5.1.2.3.2">ℝ</ci><apply id="S3.p5.5.m5.1.2.3.3.cmml" xref="S3.p5.5.m5.1.2.3.3"><times id="S3.p5.5.m5.1.2.3.3.1.cmml" xref="S3.p5.5.m5.1.2.3.3.1"></times><ci id="S3.p5.5.m5.1.2.3.3.2.cmml" xref="S3.p5.5.m5.1.2.3.3.2">𝑁</ci><ci id="S3.p5.5.m5.1.2.3.3.3.cmml" xref="S3.p5.5.m5.1.2.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.5.m5.1c">{\Phi}_{\rm s}(t)\in\mathbb{R}^{N\times n}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.5.m5.1d">roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_t ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_N × italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> are filtered outputs. Following the swapping technique <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib8" title="">8</a>, Ch. 6]</cite> and (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E11" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">11</span></a>), one gets a static parametrized model</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx7"> <tbody id="S3.E12"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\bm{p}(t)=\Phi^{T}_{\rm s}(t)\bm{\theta}+\bm{\epsilon}_{\rm f}(t)" class="ltx_Math" display="inline" id="S3.E12.m1.3"><semantics id="S3.E12.m1.3a"><mrow id="S3.E12.m1.3.4" xref="S3.E12.m1.3.4.cmml"><mrow id="S3.E12.m1.3.4.2" xref="S3.E12.m1.3.4.2.cmml"><mi id="S3.E12.m1.3.4.2.2" xref="S3.E12.m1.3.4.2.2.cmml">𝒑</mi><mo id="S3.E12.m1.3.4.2.1" xref="S3.E12.m1.3.4.2.1.cmml">⁢</mo><mrow id="S3.E12.m1.3.4.2.3.2" xref="S3.E12.m1.3.4.2.cmml"><mo id="S3.E12.m1.3.4.2.3.2.1" stretchy="false" xref="S3.E12.m1.3.4.2.cmml">(</mo><mi id="S3.E12.m1.1.1" xref="S3.E12.m1.1.1.cmml">t</mi><mo id="S3.E12.m1.3.4.2.3.2.2" stretchy="false" xref="S3.E12.m1.3.4.2.cmml">)</mo></mrow></mrow><mo id="S3.E12.m1.3.4.1" xref="S3.E12.m1.3.4.1.cmml">=</mo><mrow id="S3.E12.m1.3.4.3" xref="S3.E12.m1.3.4.3.cmml"><mrow id="S3.E12.m1.3.4.3.2" xref="S3.E12.m1.3.4.3.2.cmml"><msubsup id="S3.E12.m1.3.4.3.2.2" xref="S3.E12.m1.3.4.3.2.2.cmml"><mi id="S3.E12.m1.3.4.3.2.2.2.2" mathvariant="normal" xref="S3.E12.m1.3.4.3.2.2.2.2.cmml">Φ</mi><mi id="S3.E12.m1.3.4.3.2.2.3" mathvariant="normal" xref="S3.E12.m1.3.4.3.2.2.3.cmml">s</mi><mi id="S3.E12.m1.3.4.3.2.2.2.3" xref="S3.E12.m1.3.4.3.2.2.2.3.cmml">T</mi></msubsup><mo id="S3.E12.m1.3.4.3.2.1" xref="S3.E12.m1.3.4.3.2.1.cmml">⁢</mo><mrow id="S3.E12.m1.3.4.3.2.3.2" xref="S3.E12.m1.3.4.3.2.cmml"><mo id="S3.E12.m1.3.4.3.2.3.2.1" stretchy="false" xref="S3.E12.m1.3.4.3.2.cmml">(</mo><mi id="S3.E12.m1.2.2" xref="S3.E12.m1.2.2.cmml">t</mi><mo id="S3.E12.m1.3.4.3.2.3.2.2" stretchy="false" xref="S3.E12.m1.3.4.3.2.cmml">)</mo></mrow><mo id="S3.E12.m1.3.4.3.2.1a" xref="S3.E12.m1.3.4.3.2.1.cmml">⁢</mo><mi id="S3.E12.m1.3.4.3.2.4" xref="S3.E12.m1.3.4.3.2.4.cmml">𝜽</mi></mrow><mo id="S3.E12.m1.3.4.3.1" xref="S3.E12.m1.3.4.3.1.cmml">+</mo><mrow id="S3.E12.m1.3.4.3.3" xref="S3.E12.m1.3.4.3.3.cmml"><msub id="S3.E12.m1.3.4.3.3.2" xref="S3.E12.m1.3.4.3.3.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S3.E12.m1.3.4.3.3.2.2" mathvariant="bold-italic" xref="S3.E12.m1.3.4.3.3.2.2.cmml">ϵ</mi><mi id="S3.E12.m1.3.4.3.3.2.3" mathvariant="normal" xref="S3.E12.m1.3.4.3.3.2.3.cmml">f</mi></msub><mo id="S3.E12.m1.3.4.3.3.1" xref="S3.E12.m1.3.4.3.3.1.cmml">⁢</mo><mrow id="S3.E12.m1.3.4.3.3.3.2" xref="S3.E12.m1.3.4.3.3.cmml"><mo id="S3.E12.m1.3.4.3.3.3.2.1" stretchy="false" xref="S3.E12.m1.3.4.3.3.cmml">(</mo><mi id="S3.E12.m1.3.3" xref="S3.E12.m1.3.3.cmml">t</mi><mo id="S3.E12.m1.3.4.3.3.3.2.2" stretchy="false" xref="S3.E12.m1.3.4.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.E12.m1.3b"><apply id="S3.E12.m1.3.4.cmml" xref="S3.E12.m1.3.4"><eq id="S3.E12.m1.3.4.1.cmml" xref="S3.E12.m1.3.4.1"></eq><apply id="S3.E12.m1.3.4.2.cmml" xref="S3.E12.m1.3.4.2"><times id="S3.E12.m1.3.4.2.1.cmml" xref="S3.E12.m1.3.4.2.1"></times><ci id="S3.E12.m1.3.4.2.2.cmml" xref="S3.E12.m1.3.4.2.2">𝒑</ci><ci id="S3.E12.m1.1.1.cmml" xref="S3.E12.m1.1.1">𝑡</ci></apply><apply id="S3.E12.m1.3.4.3.cmml" xref="S3.E12.m1.3.4.3"><plus id="S3.E12.m1.3.4.3.1.cmml" xref="S3.E12.m1.3.4.3.1"></plus><apply id="S3.E12.m1.3.4.3.2.cmml" xref="S3.E12.m1.3.4.3.2"><times id="S3.E12.m1.3.4.3.2.1.cmml" xref="S3.E12.m1.3.4.3.2.1"></times><apply id="S3.E12.m1.3.4.3.2.2.cmml" xref="S3.E12.m1.3.4.3.2.2"><csymbol cd="ambiguous" id="S3.E12.m1.3.4.3.2.2.1.cmml" xref="S3.E12.m1.3.4.3.2.2">subscript</csymbol><apply id="S3.E12.m1.3.4.3.2.2.2.cmml" xref="S3.E12.m1.3.4.3.2.2"><csymbol cd="ambiguous" id="S3.E12.m1.3.4.3.2.2.2.1.cmml" xref="S3.E12.m1.3.4.3.2.2">superscript</csymbol><ci id="S3.E12.m1.3.4.3.2.2.2.2.cmml" xref="S3.E12.m1.3.4.3.2.2.2.2">Φ</ci><ci id="S3.E12.m1.3.4.3.2.2.2.3.cmml" xref="S3.E12.m1.3.4.3.2.2.2.3">𝑇</ci></apply><ci id="S3.E12.m1.3.4.3.2.2.3.cmml" xref="S3.E12.m1.3.4.3.2.2.3">s</ci></apply><ci id="S3.E12.m1.2.2.cmml" xref="S3.E12.m1.2.2">𝑡</ci><ci id="S3.E12.m1.3.4.3.2.4.cmml" xref="S3.E12.m1.3.4.3.2.4">𝜽</ci></apply><apply id="S3.E12.m1.3.4.3.3.cmml" xref="S3.E12.m1.3.4.3.3"><times id="S3.E12.m1.3.4.3.3.1.cmml" xref="S3.E12.m1.3.4.3.3.1"></times><apply id="S3.E12.m1.3.4.3.3.2.cmml" xref="S3.E12.m1.3.4.3.3.2"><csymbol cd="ambiguous" id="S3.E12.m1.3.4.3.3.2.1.cmml" xref="S3.E12.m1.3.4.3.3.2">subscript</csymbol><ci id="S3.E12.m1.3.4.3.3.2.2.cmml" xref="S3.E12.m1.3.4.3.3.2.2">bold-italic-ϵ</ci><ci id="S3.E12.m1.3.4.3.3.2.3.cmml" xref="S3.E12.m1.3.4.3.3.2.3">f</ci></apply><ci id="S3.E12.m1.3.3.cmml" xref="S3.E12.m1.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E12.m1.3c">\displaystyle\bm{p}(t)=\Phi^{T}_{\rm s}(t)\bm{\theta}+\bm{\epsilon}_{\rm f}(t)</annotation><annotation encoding="application/x-llamapun" id="S3.E12.m1.3d">bold_italic_p ( italic_t ) = roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_t ) bold_italic_θ + bold_italic_ϵ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(12)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p5.17">where <math alttext="\bm{\epsilon}_{\rm f}(t)\in\mathbb{R}^{n}" class="ltx_Math" display="inline" id="S3.p5.6.m1.1"><semantics id="S3.p5.6.m1.1a"><mrow id="S3.p5.6.m1.1.2" xref="S3.p5.6.m1.1.2.cmml"><mrow id="S3.p5.6.m1.1.2.2" xref="S3.p5.6.m1.1.2.2.cmml"><msub id="S3.p5.6.m1.1.2.2.2" xref="S3.p5.6.m1.1.2.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S3.p5.6.m1.1.2.2.2.2" mathvariant="bold-italic" xref="S3.p5.6.m1.1.2.2.2.2.cmml">ϵ</mi><mi id="S3.p5.6.m1.1.2.2.2.3" mathvariant="normal" xref="S3.p5.6.m1.1.2.2.2.3.cmml">f</mi></msub><mo id="S3.p5.6.m1.1.2.2.1" xref="S3.p5.6.m1.1.2.2.1.cmml">⁢</mo><mrow id="S3.p5.6.m1.1.2.2.3.2" xref="S3.p5.6.m1.1.2.2.cmml"><mo id="S3.p5.6.m1.1.2.2.3.2.1" stretchy="false" xref="S3.p5.6.m1.1.2.2.cmml">(</mo><mi id="S3.p5.6.m1.1.1" xref="S3.p5.6.m1.1.1.cmml">t</mi><mo id="S3.p5.6.m1.1.2.2.3.2.2" stretchy="false" xref="S3.p5.6.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.p5.6.m1.1.2.1" xref="S3.p5.6.m1.1.2.1.cmml">∈</mo><msup id="S3.p5.6.m1.1.2.3" xref="S3.p5.6.m1.1.2.3.cmml"><mi id="S3.p5.6.m1.1.2.3.2" xref="S3.p5.6.m1.1.2.3.2.cmml">ℝ</mi><mi id="S3.p5.6.m1.1.2.3.3" xref="S3.p5.6.m1.1.2.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.6.m1.1b"><apply id="S3.p5.6.m1.1.2.cmml" xref="S3.p5.6.m1.1.2"><in id="S3.p5.6.m1.1.2.1.cmml" xref="S3.p5.6.m1.1.2.1"></in><apply id="S3.p5.6.m1.1.2.2.cmml" xref="S3.p5.6.m1.1.2.2"><times id="S3.p5.6.m1.1.2.2.1.cmml" xref="S3.p5.6.m1.1.2.2.1"></times><apply id="S3.p5.6.m1.1.2.2.2.cmml" xref="S3.p5.6.m1.1.2.2.2"><csymbol cd="ambiguous" id="S3.p5.6.m1.1.2.2.2.1.cmml" xref="S3.p5.6.m1.1.2.2.2">subscript</csymbol><ci id="S3.p5.6.m1.1.2.2.2.2.cmml" xref="S3.p5.6.m1.1.2.2.2.2">bold-italic-ϵ</ci><ci id="S3.p5.6.m1.1.2.2.2.3.cmml" xref="S3.p5.6.m1.1.2.2.2.3">f</ci></apply><ci id="S3.p5.6.m1.1.1.cmml" xref="S3.p5.6.m1.1.1">𝑡</ci></apply><apply id="S3.p5.6.m1.1.2.3.cmml" xref="S3.p5.6.m1.1.2.3"><csymbol cd="ambiguous" id="S3.p5.6.m1.1.2.3.1.cmml" xref="S3.p5.6.m1.1.2.3">superscript</csymbol><ci id="S3.p5.6.m1.1.2.3.2.cmml" xref="S3.p5.6.m1.1.2.3.2">ℝ</ci><ci id="S3.p5.6.m1.1.2.3.3.cmml" xref="S3.p5.6.m1.1.2.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.6.m1.1c">\bm{\epsilon}_{\rm f}(t)\in\mathbb{R}^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.6.m1.1d">bold_italic_ϵ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ( italic_t ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> is an exponential decay term that satisfies <math alttext="\bm{\epsilon}_{\rm f}(t)\equiv\bm{0}" class="ltx_Math" display="inline" id="S3.p5.7.m2.1"><semantics id="S3.p5.7.m2.1a"><mrow id="S3.p5.7.m2.1.2" xref="S3.p5.7.m2.1.2.cmml"><mrow id="S3.p5.7.m2.1.2.2" xref="S3.p5.7.m2.1.2.2.cmml"><msub id="S3.p5.7.m2.1.2.2.2" xref="S3.p5.7.m2.1.2.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S3.p5.7.m2.1.2.2.2.2" mathvariant="bold-italic" xref="S3.p5.7.m2.1.2.2.2.2.cmml">ϵ</mi><mi id="S3.p5.7.m2.1.2.2.2.3" mathvariant="normal" xref="S3.p5.7.m2.1.2.2.2.3.cmml">f</mi></msub><mo id="S3.p5.7.m2.1.2.2.1" xref="S3.p5.7.m2.1.2.2.1.cmml">⁢</mo><mrow id="S3.p5.7.m2.1.2.2.3.2" xref="S3.p5.7.m2.1.2.2.cmml"><mo id="S3.p5.7.m2.1.2.2.3.2.1" stretchy="false" xref="S3.p5.7.m2.1.2.2.cmml">(</mo><mi id="S3.p5.7.m2.1.1" xref="S3.p5.7.m2.1.1.cmml">t</mi><mo id="S3.p5.7.m2.1.2.2.3.2.2" stretchy="false" xref="S3.p5.7.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.p5.7.m2.1.2.1" xref="S3.p5.7.m2.1.2.1.cmml">≡</mo><mn id="S3.p5.7.m2.1.2.3" xref="S3.p5.7.m2.1.2.3.cmml">𝟎</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.7.m2.1b"><apply id="S3.p5.7.m2.1.2.cmml" xref="S3.p5.7.m2.1.2"><equivalent id="S3.p5.7.m2.1.2.1.cmml" xref="S3.p5.7.m2.1.2.1"></equivalent><apply id="S3.p5.7.m2.1.2.2.cmml" xref="S3.p5.7.m2.1.2.2"><times id="S3.p5.7.m2.1.2.2.1.cmml" xref="S3.p5.7.m2.1.2.2.1"></times><apply id="S3.p5.7.m2.1.2.2.2.cmml" xref="S3.p5.7.m2.1.2.2.2"><csymbol cd="ambiguous" id="S3.p5.7.m2.1.2.2.2.1.cmml" xref="S3.p5.7.m2.1.2.2.2">subscript</csymbol><ci id="S3.p5.7.m2.1.2.2.2.2.cmml" xref="S3.p5.7.m2.1.2.2.2.2">bold-italic-ϵ</ci><ci id="S3.p5.7.m2.1.2.2.2.3.cmml" xref="S3.p5.7.m2.1.2.2.2.3">f</ci></apply><ci id="S3.p5.7.m2.1.1.cmml" xref="S3.p5.7.m2.1.1">𝑡</ci></apply><cn id="S3.p5.7.m2.1.2.3.cmml" type="integer" xref="S3.p5.7.m2.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.7.m2.1c">\bm{\epsilon}_{\rm f}(t)\equiv\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.7.m2.1d">bold_italic_ϵ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ( italic_t ) ≡ bold_0</annotation></semantics></math>, <math alttext="\forall t\geq 0" class="ltx_Math" display="inline" id="S3.p5.8.m3.1"><semantics id="S3.p5.8.m3.1a"><mrow id="S3.p5.8.m3.1.1" xref="S3.p5.8.m3.1.1.cmml"><mrow id="S3.p5.8.m3.1.1.2" xref="S3.p5.8.m3.1.1.2.cmml"><mo id="S3.p5.8.m3.1.1.2.1" rspace="0.167em" xref="S3.p5.8.m3.1.1.2.1.cmml">∀</mo><mi id="S3.p5.8.m3.1.1.2.2" xref="S3.p5.8.m3.1.1.2.2.cmml">t</mi></mrow><mo id="S3.p5.8.m3.1.1.1" xref="S3.p5.8.m3.1.1.1.cmml">≥</mo><mn id="S3.p5.8.m3.1.1.3" xref="S3.p5.8.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.8.m3.1b"><apply id="S3.p5.8.m3.1.1.cmml" xref="S3.p5.8.m3.1.1"><geq id="S3.p5.8.m3.1.1.1.cmml" xref="S3.p5.8.m3.1.1.1"></geq><apply id="S3.p5.8.m3.1.1.2.cmml" xref="S3.p5.8.m3.1.1.2"><csymbol cd="latexml" id="S3.p5.8.m3.1.1.2.1.cmml" xref="S3.p5.8.m3.1.1.2.1">for-all</csymbol><ci id="S3.p5.8.m3.1.1.2.2.cmml" xref="S3.p5.8.m3.1.1.2.2">𝑡</ci></apply><cn id="S3.p5.8.m3.1.1.3.cmml" type="integer" xref="S3.p5.8.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.8.m3.1c">\forall t\geq 0</annotation><annotation encoding="application/x-llamapun" id="S3.p5.8.m3.1d">∀ italic_t ≥ 0</annotation></semantics></math> if <math alttext="\bm{\zeta}(0)=-\bm{e}(0)" class="ltx_Math" display="inline" id="S3.p5.9.m4.2"><semantics id="S3.p5.9.m4.2a"><mrow id="S3.p5.9.m4.2.3" xref="S3.p5.9.m4.2.3.cmml"><mrow id="S3.p5.9.m4.2.3.2" xref="S3.p5.9.m4.2.3.2.cmml"><mi id="S3.p5.9.m4.2.3.2.2" xref="S3.p5.9.m4.2.3.2.2.cmml">𝜻</mi><mo id="S3.p5.9.m4.2.3.2.1" xref="S3.p5.9.m4.2.3.2.1.cmml">⁢</mo><mrow id="S3.p5.9.m4.2.3.2.3.2" xref="S3.p5.9.m4.2.3.2.cmml"><mo id="S3.p5.9.m4.2.3.2.3.2.1" stretchy="false" xref="S3.p5.9.m4.2.3.2.cmml">(</mo><mn id="S3.p5.9.m4.1.1" xref="S3.p5.9.m4.1.1.cmml">0</mn><mo id="S3.p5.9.m4.2.3.2.3.2.2" stretchy="false" xref="S3.p5.9.m4.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.p5.9.m4.2.3.1" xref="S3.p5.9.m4.2.3.1.cmml">=</mo><mrow id="S3.p5.9.m4.2.3.3" xref="S3.p5.9.m4.2.3.3.cmml"><mo id="S3.p5.9.m4.2.3.3a" xref="S3.p5.9.m4.2.3.3.cmml">−</mo><mrow id="S3.p5.9.m4.2.3.3.2" xref="S3.p5.9.m4.2.3.3.2.cmml"><mi id="S3.p5.9.m4.2.3.3.2.2" xref="S3.p5.9.m4.2.3.3.2.2.cmml">𝒆</mi><mo id="S3.p5.9.m4.2.3.3.2.1" xref="S3.p5.9.m4.2.3.3.2.1.cmml">⁢</mo><mrow id="S3.p5.9.m4.2.3.3.2.3.2" xref="S3.p5.9.m4.2.3.3.2.cmml"><mo id="S3.p5.9.m4.2.3.3.2.3.2.1" stretchy="false" xref="S3.p5.9.m4.2.3.3.2.cmml">(</mo><mn id="S3.p5.9.m4.2.2" xref="S3.p5.9.m4.2.2.cmml">0</mn><mo id="S3.p5.9.m4.2.3.3.2.3.2.2" stretchy="false" xref="S3.p5.9.m4.2.3.3.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.9.m4.2b"><apply id="S3.p5.9.m4.2.3.cmml" xref="S3.p5.9.m4.2.3"><eq id="S3.p5.9.m4.2.3.1.cmml" xref="S3.p5.9.m4.2.3.1"></eq><apply id="S3.p5.9.m4.2.3.2.cmml" xref="S3.p5.9.m4.2.3.2"><times id="S3.p5.9.m4.2.3.2.1.cmml" xref="S3.p5.9.m4.2.3.2.1"></times><ci id="S3.p5.9.m4.2.3.2.2.cmml" xref="S3.p5.9.m4.2.3.2.2">𝜻</ci><cn id="S3.p5.9.m4.1.1.cmml" type="integer" xref="S3.p5.9.m4.1.1">0</cn></apply><apply id="S3.p5.9.m4.2.3.3.cmml" xref="S3.p5.9.m4.2.3.3"><minus id="S3.p5.9.m4.2.3.3.1.cmml" xref="S3.p5.9.m4.2.3.3"></minus><apply id="S3.p5.9.m4.2.3.3.2.cmml" xref="S3.p5.9.m4.2.3.3.2"><times id="S3.p5.9.m4.2.3.3.2.1.cmml" xref="S3.p5.9.m4.2.3.3.2.1"></times><ci id="S3.p5.9.m4.2.3.3.2.2.cmml" xref="S3.p5.9.m4.2.3.3.2.2">𝒆</ci><cn id="S3.p5.9.m4.2.2.cmml" type="integer" xref="S3.p5.9.m4.2.2">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.9.m4.2c">\bm{\zeta}(0)=-\bm{e}(0)</annotation><annotation encoding="application/x-llamapun" id="S3.p5.9.m4.2d">bold_italic_ζ ( 0 ) = - bold_italic_e ( 0 )</annotation></semantics></math> and <math alttext="{\Phi}_{\rm s}(0)=\bm{0}" class="ltx_Math" display="inline" id="S3.p5.10.m5.1"><semantics id="S3.p5.10.m5.1a"><mrow id="S3.p5.10.m5.1.2" xref="S3.p5.10.m5.1.2.cmml"><mrow id="S3.p5.10.m5.1.2.2" xref="S3.p5.10.m5.1.2.2.cmml"><msub id="S3.p5.10.m5.1.2.2.2" xref="S3.p5.10.m5.1.2.2.2.cmml"><mi id="S3.p5.10.m5.1.2.2.2.2" mathvariant="normal" xref="S3.p5.10.m5.1.2.2.2.2.cmml">Φ</mi><mi id="S3.p5.10.m5.1.2.2.2.3" mathvariant="normal" xref="S3.p5.10.m5.1.2.2.2.3.cmml">s</mi></msub><mo id="S3.p5.10.m5.1.2.2.1" xref="S3.p5.10.m5.1.2.2.1.cmml">⁢</mo><mrow id="S3.p5.10.m5.1.2.2.3.2" xref="S3.p5.10.m5.1.2.2.cmml"><mo id="S3.p5.10.m5.1.2.2.3.2.1" stretchy="false" xref="S3.p5.10.m5.1.2.2.cmml">(</mo><mn id="S3.p5.10.m5.1.1" xref="S3.p5.10.m5.1.1.cmml">0</mn><mo id="S3.p5.10.m5.1.2.2.3.2.2" stretchy="false" xref="S3.p5.10.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.p5.10.m5.1.2.1" xref="S3.p5.10.m5.1.2.1.cmml">=</mo><mn id="S3.p5.10.m5.1.2.3" xref="S3.p5.10.m5.1.2.3.cmml">𝟎</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.10.m5.1b"><apply id="S3.p5.10.m5.1.2.cmml" xref="S3.p5.10.m5.1.2"><eq id="S3.p5.10.m5.1.2.1.cmml" xref="S3.p5.10.m5.1.2.1"></eq><apply id="S3.p5.10.m5.1.2.2.cmml" xref="S3.p5.10.m5.1.2.2"><times id="S3.p5.10.m5.1.2.2.1.cmml" xref="S3.p5.10.m5.1.2.2.1"></times><apply id="S3.p5.10.m5.1.2.2.2.cmml" xref="S3.p5.10.m5.1.2.2.2"><csymbol cd="ambiguous" id="S3.p5.10.m5.1.2.2.2.1.cmml" xref="S3.p5.10.m5.1.2.2.2">subscript</csymbol><ci id="S3.p5.10.m5.1.2.2.2.2.cmml" xref="S3.p5.10.m5.1.2.2.2.2">Φ</ci><ci id="S3.p5.10.m5.1.2.2.2.3.cmml" xref="S3.p5.10.m5.1.2.2.2.3">s</ci></apply><cn id="S3.p5.10.m5.1.1.cmml" type="integer" xref="S3.p5.10.m5.1.1">0</cn></apply><cn id="S3.p5.10.m5.1.2.3.cmml" type="integer" xref="S3.p5.10.m5.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.10.m5.1c">{\Phi}_{\rm s}(0)=\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.10.m5.1d">roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( 0 ) = bold_0</annotation></semantics></math>. However, the parameter vector <math alttext="\bm{\theta}" class="ltx_Math" display="inline" id="S3.p5.11.m6.1"><semantics id="S3.p5.11.m6.1a"><mi id="S3.p5.11.m6.1.1" xref="S3.p5.11.m6.1.1.cmml">𝜽</mi><annotation-xml encoding="MathML-Content" id="S3.p5.11.m6.1b"><ci id="S3.p5.11.m6.1.1.cmml" xref="S3.p5.11.m6.1.1">𝜽</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.11.m6.1c">\bm{\theta}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.11.m6.1d">bold_italic_θ</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E12" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">12</span></a>) can not be estimated by classical adaptation schemes because the regressor <math alttext="\Phi_{\rm s}" class="ltx_Math" display="inline" id="S3.p5.12.m7.1"><semantics id="S3.p5.12.m7.1a"><msub id="S3.p5.12.m7.1.1" xref="S3.p5.12.m7.1.1.cmml"><mi id="S3.p5.12.m7.1.1.2" mathvariant="normal" xref="S3.p5.12.m7.1.1.2.cmml">Φ</mi><mi id="S3.p5.12.m7.1.1.3" mathvariant="normal" xref="S3.p5.12.m7.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p5.12.m7.1b"><apply id="S3.p5.12.m7.1.1.cmml" xref="S3.p5.12.m7.1.1"><csymbol cd="ambiguous" id="S3.p5.12.m7.1.1.1.cmml" xref="S3.p5.12.m7.1.1">subscript</csymbol><ci id="S3.p5.12.m7.1.1.2.cmml" xref="S3.p5.12.m7.1.1.2">Φ</ci><ci id="S3.p5.12.m7.1.1.3.cmml" xref="S3.p5.12.m7.1.1.3">s</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.12.m7.1c">\Phi_{\rm s}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.12.m7.1d">roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT</annotation></semantics></math> relies on the inaccessible high-order time derivatives <math alttext="\hat{\bm{\theta}}^{(k)}" class="ltx_Math" display="inline" id="S3.p5.13.m8.1"><semantics id="S3.p5.13.m8.1a"><msup id="S3.p5.13.m8.1.2" xref="S3.p5.13.m8.1.2.cmml"><mover accent="true" id="S3.p5.13.m8.1.2.2" xref="S3.p5.13.m8.1.2.2.cmml"><mi id="S3.p5.13.m8.1.2.2.2" xref="S3.p5.13.m8.1.2.2.2.cmml">𝜽</mi><mo id="S3.p5.13.m8.1.2.2.1" xref="S3.p5.13.m8.1.2.2.1.cmml">^</mo></mover><mrow id="S3.p5.13.m8.1.1.1.3" xref="S3.p5.13.m8.1.2.cmml"><mo id="S3.p5.13.m8.1.1.1.3.1" stretchy="false" xref="S3.p5.13.m8.1.2.cmml">(</mo><mi id="S3.p5.13.m8.1.1.1.1" xref="S3.p5.13.m8.1.1.1.1.cmml">k</mi><mo id="S3.p5.13.m8.1.1.1.3.2" stretchy="false" xref="S3.p5.13.m8.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.p5.13.m8.1b"><apply id="S3.p5.13.m8.1.2.cmml" xref="S3.p5.13.m8.1.2"><csymbol cd="ambiguous" id="S3.p5.13.m8.1.2.1.cmml" xref="S3.p5.13.m8.1.2">superscript</csymbol><apply id="S3.p5.13.m8.1.2.2.cmml" xref="S3.p5.13.m8.1.2.2"><ci id="S3.p5.13.m8.1.2.2.1.cmml" xref="S3.p5.13.m8.1.2.2.1">^</ci><ci id="S3.p5.13.m8.1.2.2.2.cmml" xref="S3.p5.13.m8.1.2.2.2">𝜽</ci></apply><ci id="S3.p5.13.m8.1.1.1.1.cmml" xref="S3.p5.13.m8.1.1.1.1">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.13.m8.1c">\hat{\bm{\theta}}^{(k)}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.13.m8.1d">over^ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT</annotation></semantics></math> (<math alttext="k=1" class="ltx_Math" display="inline" id="S3.p5.14.m9.1"><semantics id="S3.p5.14.m9.1a"><mrow id="S3.p5.14.m9.1.1" xref="S3.p5.14.m9.1.1.cmml"><mi id="S3.p5.14.m9.1.1.2" xref="S3.p5.14.m9.1.1.2.cmml">k</mi><mo id="S3.p5.14.m9.1.1.1" xref="S3.p5.14.m9.1.1.1.cmml">=</mo><mn id="S3.p5.14.m9.1.1.3" xref="S3.p5.14.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.14.m9.1b"><apply id="S3.p5.14.m9.1.1.cmml" xref="S3.p5.14.m9.1.1"><eq id="S3.p5.14.m9.1.1.1.cmml" xref="S3.p5.14.m9.1.1.1"></eq><ci id="S3.p5.14.m9.1.1.2.cmml" xref="S3.p5.14.m9.1.1.2">𝑘</ci><cn id="S3.p5.14.m9.1.1.3.cmml" type="integer" xref="S3.p5.14.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.14.m9.1c">k=1</annotation><annotation encoding="application/x-llamapun" id="S3.p5.14.m9.1d">italic_k = 1</annotation></semantics></math> to <math alttext="n-1" class="ltx_Math" display="inline" id="S3.p5.15.m10.1"><semantics id="S3.p5.15.m10.1a"><mrow id="S3.p5.15.m10.1.1" xref="S3.p5.15.m10.1.1.cmml"><mi id="S3.p5.15.m10.1.1.2" xref="S3.p5.15.m10.1.1.2.cmml">n</mi><mo id="S3.p5.15.m10.1.1.1" xref="S3.p5.15.m10.1.1.1.cmml">−</mo><mn id="S3.p5.15.m10.1.1.3" xref="S3.p5.15.m10.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.15.m10.1b"><apply id="S3.p5.15.m10.1.1.cmml" xref="S3.p5.15.m10.1.1"><minus id="S3.p5.15.m10.1.1.1.cmml" xref="S3.p5.15.m10.1.1.1"></minus><ci id="S3.p5.15.m10.1.1.2.cmml" xref="S3.p5.15.m10.1.1.2">𝑛</ci><cn id="S3.p5.15.m10.1.1.3.cmml" type="integer" xref="S3.p5.15.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.15.m10.1c">n-1</annotation><annotation encoding="application/x-llamapun" id="S3.p5.15.m10.1d">italic_n - 1</annotation></semantics></math>). Even when <math alttext="\hat{\bm{\theta}}^{(k)}" class="ltx_Math" display="inline" id="S3.p5.16.m11.1"><semantics id="S3.p5.16.m11.1a"><msup id="S3.p5.16.m11.1.2" xref="S3.p5.16.m11.1.2.cmml"><mover accent="true" id="S3.p5.16.m11.1.2.2" xref="S3.p5.16.m11.1.2.2.cmml"><mi id="S3.p5.16.m11.1.2.2.2" xref="S3.p5.16.m11.1.2.2.2.cmml">𝜽</mi><mo id="S3.p5.16.m11.1.2.2.1" xref="S3.p5.16.m11.1.2.2.1.cmml">^</mo></mover><mrow id="S3.p5.16.m11.1.1.1.3" xref="S3.p5.16.m11.1.2.cmml"><mo id="S3.p5.16.m11.1.1.1.3.1" stretchy="false" xref="S3.p5.16.m11.1.2.cmml">(</mo><mi id="S3.p5.16.m11.1.1.1.1" xref="S3.p5.16.m11.1.1.1.1.cmml">k</mi><mo id="S3.p5.16.m11.1.1.1.3.2" stretchy="false" xref="S3.p5.16.m11.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.p5.16.m11.1b"><apply id="S3.p5.16.m11.1.2.cmml" xref="S3.p5.16.m11.1.2"><csymbol cd="ambiguous" id="S3.p5.16.m11.1.2.1.cmml" xref="S3.p5.16.m11.1.2">superscript</csymbol><apply id="S3.p5.16.m11.1.2.2.cmml" xref="S3.p5.16.m11.1.2.2"><ci id="S3.p5.16.m11.1.2.2.1.cmml" xref="S3.p5.16.m11.1.2.2.1">^</ci><ci id="S3.p5.16.m11.1.2.2.2.cmml" xref="S3.p5.16.m11.1.2.2.2">𝜽</ci></apply><ci id="S3.p5.16.m11.1.1.1.1.cmml" xref="S3.p5.16.m11.1.1.1.1">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.16.m11.1c">\hat{\bm{\theta}}^{(k)}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.16.m11.1d">over^ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT</annotation></semantics></math> are available, the parameter convergence of classical adaptation schemes relies on the stringent PE condition, which requires that the reference trajectory <math alttext="y_{\rm r}" class="ltx_Math" display="inline" id="S3.p5.17.m12.1"><semantics id="S3.p5.17.m12.1a"><msub id="S3.p5.17.m12.1.1" xref="S3.p5.17.m12.1.1.cmml"><mi id="S3.p5.17.m12.1.1.2" xref="S3.p5.17.m12.1.1.2.cmml">y</mi><mi id="S3.p5.17.m12.1.1.3" mathvariant="normal" xref="S3.p5.17.m12.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p5.17.m12.1b"><apply id="S3.p5.17.m12.1.1.cmml" xref="S3.p5.17.m12.1.1"><csymbol cd="ambiguous" id="S3.p5.17.m12.1.1.1.cmml" xref="S3.p5.17.m12.1.1">subscript</csymbol><ci id="S3.p5.17.m12.1.1.2.cmml" xref="S3.p5.17.m12.1.1.2">𝑦</ci><ci id="S3.p5.17.m12.1.1.3.cmml" xref="S3.p5.17.m12.1.1.3">r</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.17.m12.1c">y_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.17.m12.1d">italic_y start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math> includes sufficiently rich spectral information all the time.</p> </div> <div class="ltx_para" id="S3.p6"> <p class="ltx_p" id="S3.p6.15"><span class="ltx_text ltx_font_italic" id="S3.p6.15.1">Remark 2:</span> The reference signals <math alttext="y_{\rm r}" class="ltx_Math" display="inline" id="S3.p6.1.m1.1"><semantics id="S3.p6.1.m1.1a"><msub id="S3.p6.1.m1.1.1" xref="S3.p6.1.m1.1.1.cmml"><mi id="S3.p6.1.m1.1.1.2" xref="S3.p6.1.m1.1.1.2.cmml">y</mi><mi id="S3.p6.1.m1.1.1.3" mathvariant="normal" xref="S3.p6.1.m1.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p6.1.m1.1b"><apply id="S3.p6.1.m1.1.1.cmml" xref="S3.p6.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p6.1.m1.1.1.1.cmml" xref="S3.p6.1.m1.1.1">subscript</csymbol><ci id="S3.p6.1.m1.1.1.2.cmml" xref="S3.p6.1.m1.1.1.2">𝑦</ci><ci id="S3.p6.1.m1.1.1.3.cmml" xref="S3.p6.1.m1.1.1.3">r</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.1.m1.1c">y_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.1.m1.1d">italic_y start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\dot{y}_{\rm r}" class="ltx_Math" display="inline" id="S3.p6.2.m2.1"><semantics id="S3.p6.2.m2.1a"><msub id="S3.p6.2.m2.1.1" xref="S3.p6.2.m2.1.1.cmml"><mover accent="true" id="S3.p6.2.m2.1.1.2" xref="S3.p6.2.m2.1.1.2.cmml"><mi id="S3.p6.2.m2.1.1.2.2" xref="S3.p6.2.m2.1.1.2.2.cmml">y</mi><mo id="S3.p6.2.m2.1.1.2.1" xref="S3.p6.2.m2.1.1.2.1.cmml">˙</mo></mover><mi id="S3.p6.2.m2.1.1.3" mathvariant="normal" xref="S3.p6.2.m2.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p6.2.m2.1b"><apply id="S3.p6.2.m2.1.1.cmml" xref="S3.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p6.2.m2.1.1.1.cmml" xref="S3.p6.2.m2.1.1">subscript</csymbol><apply id="S3.p6.2.m2.1.1.2.cmml" xref="S3.p6.2.m2.1.1.2"><ci id="S3.p6.2.m2.1.1.2.1.cmml" xref="S3.p6.2.m2.1.1.2.1">˙</ci><ci id="S3.p6.2.m2.1.1.2.2.cmml" xref="S3.p6.2.m2.1.1.2.2">𝑦</ci></apply><ci id="S3.p6.2.m2.1.1.3.cmml" xref="S3.p6.2.m2.1.1.3">r</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.2.m2.1c">\dot{y}_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.2.m2.1d">over˙ start_ARG italic_y end_ARG start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\cdots" class="ltx_Math" display="inline" id="S3.p6.3.m3.1"><semantics id="S3.p6.3.m3.1a"><mi id="S3.p6.3.m3.1.1" mathvariant="normal" xref="S3.p6.3.m3.1.1.cmml">⋯</mi><annotation-xml encoding="MathML-Content" id="S3.p6.3.m3.1b"><ci id="S3.p6.3.m3.1.1.cmml" xref="S3.p6.3.m3.1.1">⋯</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.3.m3.1c">\cdots</annotation><annotation encoding="application/x-llamapun" id="S3.p6.3.m3.1d">⋯</annotation></semantics></math>, <math alttext="y^{(n-1)}_{\rm r}\in\Omega_{{\rm c}_{\rm r}}" class="ltx_Math" display="inline" id="S3.p6.4.m4.1"><semantics id="S3.p6.4.m4.1a"><mrow id="S3.p6.4.m4.1.2" xref="S3.p6.4.m4.1.2.cmml"><msubsup id="S3.p6.4.m4.1.2.2" xref="S3.p6.4.m4.1.2.2.cmml"><mi id="S3.p6.4.m4.1.2.2.2.2" xref="S3.p6.4.m4.1.2.2.2.2.cmml">y</mi><mi id="S3.p6.4.m4.1.2.2.3" mathvariant="normal" xref="S3.p6.4.m4.1.2.2.3.cmml">r</mi><mrow id="S3.p6.4.m4.1.1.1.1" xref="S3.p6.4.m4.1.1.1.1.1.cmml"><mo id="S3.p6.4.m4.1.1.1.1.2" stretchy="false" xref="S3.p6.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S3.p6.4.m4.1.1.1.1.1" xref="S3.p6.4.m4.1.1.1.1.1.cmml"><mi id="S3.p6.4.m4.1.1.1.1.1.2" xref="S3.p6.4.m4.1.1.1.1.1.2.cmml">n</mi><mo id="S3.p6.4.m4.1.1.1.1.1.1" xref="S3.p6.4.m4.1.1.1.1.1.1.cmml">−</mo><mn id="S3.p6.4.m4.1.1.1.1.1.3" xref="S3.p6.4.m4.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.p6.4.m4.1.1.1.1.3" stretchy="false" xref="S3.p6.4.m4.1.1.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S3.p6.4.m4.1.2.1" xref="S3.p6.4.m4.1.2.1.cmml">∈</mo><msub id="S3.p6.4.m4.1.2.3" xref="S3.p6.4.m4.1.2.3.cmml"><mi id="S3.p6.4.m4.1.2.3.2" mathvariant="normal" xref="S3.p6.4.m4.1.2.3.2.cmml">Ω</mi><msub id="S3.p6.4.m4.1.2.3.3" xref="S3.p6.4.m4.1.2.3.3.cmml"><mi id="S3.p6.4.m4.1.2.3.3.2" mathvariant="normal" xref="S3.p6.4.m4.1.2.3.3.2.cmml">c</mi><mi id="S3.p6.4.m4.1.2.3.3.3" mathvariant="normal" xref="S3.p6.4.m4.1.2.3.3.3.cmml">r</mi></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.4.m4.1b"><apply id="S3.p6.4.m4.1.2.cmml" xref="S3.p6.4.m4.1.2"><in id="S3.p6.4.m4.1.2.1.cmml" xref="S3.p6.4.m4.1.2.1"></in><apply id="S3.p6.4.m4.1.2.2.cmml" xref="S3.p6.4.m4.1.2.2"><csymbol cd="ambiguous" id="S3.p6.4.m4.1.2.2.1.cmml" xref="S3.p6.4.m4.1.2.2">subscript</csymbol><apply id="S3.p6.4.m4.1.2.2.2.cmml" xref="S3.p6.4.m4.1.2.2"><csymbol cd="ambiguous" id="S3.p6.4.m4.1.2.2.2.1.cmml" xref="S3.p6.4.m4.1.2.2">superscript</csymbol><ci id="S3.p6.4.m4.1.2.2.2.2.cmml" xref="S3.p6.4.m4.1.2.2.2.2">𝑦</ci><apply id="S3.p6.4.m4.1.1.1.1.1.cmml" xref="S3.p6.4.m4.1.1.1.1"><minus id="S3.p6.4.m4.1.1.1.1.1.1.cmml" xref="S3.p6.4.m4.1.1.1.1.1.1"></minus><ci id="S3.p6.4.m4.1.1.1.1.1.2.cmml" xref="S3.p6.4.m4.1.1.1.1.1.2">𝑛</ci><cn id="S3.p6.4.m4.1.1.1.1.1.3.cmml" type="integer" xref="S3.p6.4.m4.1.1.1.1.1.3">1</cn></apply></apply><ci id="S3.p6.4.m4.1.2.2.3.cmml" xref="S3.p6.4.m4.1.2.2.3">r</ci></apply><apply id="S3.p6.4.m4.1.2.3.cmml" xref="S3.p6.4.m4.1.2.3"><csymbol cd="ambiguous" id="S3.p6.4.m4.1.2.3.1.cmml" xref="S3.p6.4.m4.1.2.3">subscript</csymbol><ci id="S3.p6.4.m4.1.2.3.2.cmml" xref="S3.p6.4.m4.1.2.3.2">Ω</ci><apply id="S3.p6.4.m4.1.2.3.3.cmml" xref="S3.p6.4.m4.1.2.3.3"><csymbol cd="ambiguous" id="S3.p6.4.m4.1.2.3.3.1.cmml" xref="S3.p6.4.m4.1.2.3.3">subscript</csymbol><ci id="S3.p6.4.m4.1.2.3.3.2.cmml" xref="S3.p6.4.m4.1.2.3.3.2">c</ci><ci id="S3.p6.4.m4.1.2.3.3.3.cmml" xref="S3.p6.4.m4.1.2.3.3.3">r</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.4.m4.1c">y^{(n-1)}_{\rm r}\in\Omega_{{\rm c}_{\rm r}}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.4.m4.1d">italic_y start_POSTSUPERSCRIPT ( italic_n - 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT ∈ roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> in Assumption 1 is only for ensuring the possibility of input-to-state stability from these signals to the state variable <math alttext="\bm{x}" class="ltx_Math" display="inline" id="S3.p6.5.m5.1"><semantics id="S3.p6.5.m5.1a"><mi id="S3.p6.5.m5.1.1" xref="S3.p6.5.m5.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S3.p6.5.m5.1b"><ci id="S3.p6.5.m5.1.1.cmml" xref="S3.p6.5.m5.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.5.m5.1c">\bm{x}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.5.m5.1d">bold_italic_x</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib27" title="">27</a>]</cite>. In Assumption 2, the control gain <math alttext="\beta(\bm{x})\neq 0" class="ltx_Math" display="inline" id="S3.p6.6.m6.1"><semantics id="S3.p6.6.m6.1a"><mrow id="S3.p6.6.m6.1.2" xref="S3.p6.6.m6.1.2.cmml"><mrow id="S3.p6.6.m6.1.2.2" xref="S3.p6.6.m6.1.2.2.cmml"><mi id="S3.p6.6.m6.1.2.2.2" xref="S3.p6.6.m6.1.2.2.2.cmml">β</mi><mo id="S3.p6.6.m6.1.2.2.1" xref="S3.p6.6.m6.1.2.2.1.cmml">⁢</mo><mrow id="S3.p6.6.m6.1.2.2.3.2" xref="S3.p6.6.m6.1.2.2.cmml"><mo id="S3.p6.6.m6.1.2.2.3.2.1" stretchy="false" xref="S3.p6.6.m6.1.2.2.cmml">(</mo><mi id="S3.p6.6.m6.1.1" xref="S3.p6.6.m6.1.1.cmml">𝒙</mi><mo id="S3.p6.6.m6.1.2.2.3.2.2" stretchy="false" xref="S3.p6.6.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.p6.6.m6.1.2.1" xref="S3.p6.6.m6.1.2.1.cmml">≠</mo><mn id="S3.p6.6.m6.1.2.3" xref="S3.p6.6.m6.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.6.m6.1b"><apply id="S3.p6.6.m6.1.2.cmml" xref="S3.p6.6.m6.1.2"><neq id="S3.p6.6.m6.1.2.1.cmml" xref="S3.p6.6.m6.1.2.1"></neq><apply id="S3.p6.6.m6.1.2.2.cmml" xref="S3.p6.6.m6.1.2.2"><times id="S3.p6.6.m6.1.2.2.1.cmml" xref="S3.p6.6.m6.1.2.2.1"></times><ci id="S3.p6.6.m6.1.2.2.2.cmml" xref="S3.p6.6.m6.1.2.2.2">𝛽</ci><ci id="S3.p6.6.m6.1.1.cmml" xref="S3.p6.6.m6.1.1">𝒙</ci></apply><cn id="S3.p6.6.m6.1.2.3.cmml" type="integer" xref="S3.p6.6.m6.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.6.m6.1c">\beta(\bm{x})\neq 0</annotation><annotation encoding="application/x-llamapun" id="S3.p6.6.m6.1d">italic_β ( bold_italic_x ) ≠ 0</annotation></semantics></math> is only for ensuring controllability of the system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S2.E4" title="In II Problem Formulation ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">4</span></a>); the regressors <math alttext="\bm{\varphi}_{i}\in\mathcal{C}^{n-i}" class="ltx_Math" display="inline" id="S3.p6.7.m7.1"><semantics id="S3.p6.7.m7.1a"><mrow id="S3.p6.7.m7.1.1" xref="S3.p6.7.m7.1.1.cmml"><msub id="S3.p6.7.m7.1.1.2" xref="S3.p6.7.m7.1.1.2.cmml"><mi id="S3.p6.7.m7.1.1.2.2" mathcolor="#000099" xref="S3.p6.7.m7.1.1.2.2.cmml">𝝋</mi><mi id="S3.p6.7.m7.1.1.2.3" mathcolor="#000099" xref="S3.p6.7.m7.1.1.2.3.cmml">i</mi></msub><mo id="S3.p6.7.m7.1.1.1" mathcolor="#000099" xref="S3.p6.7.m7.1.1.1.cmml">∈</mo><msup id="S3.p6.7.m7.1.1.3" xref="S3.p6.7.m7.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p6.7.m7.1.1.3.2" mathcolor="#000099" xref="S3.p6.7.m7.1.1.3.2.cmml">𝒞</mi><mrow id="S3.p6.7.m7.1.1.3.3" xref="S3.p6.7.m7.1.1.3.3.cmml"><mi id="S3.p6.7.m7.1.1.3.3.2" mathcolor="#000099" xref="S3.p6.7.m7.1.1.3.3.2.cmml">n</mi><mo id="S3.p6.7.m7.1.1.3.3.1" mathcolor="#000099" xref="S3.p6.7.m7.1.1.3.3.1.cmml">−</mo><mi id="S3.p6.7.m7.1.1.3.3.3" mathcolor="#000099" xref="S3.p6.7.m7.1.1.3.3.3.cmml">i</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.7.m7.1b"><apply id="S3.p6.7.m7.1.1.cmml" xref="S3.p6.7.m7.1.1"><in id="S3.p6.7.m7.1.1.1.cmml" xref="S3.p6.7.m7.1.1.1"></in><apply id="S3.p6.7.m7.1.1.2.cmml" xref="S3.p6.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.p6.7.m7.1.1.2.1.cmml" xref="S3.p6.7.m7.1.1.2">subscript</csymbol><ci id="S3.p6.7.m7.1.1.2.2.cmml" xref="S3.p6.7.m7.1.1.2.2">𝝋</ci><ci id="S3.p6.7.m7.1.1.2.3.cmml" xref="S3.p6.7.m7.1.1.2.3">𝑖</ci></apply><apply id="S3.p6.7.m7.1.1.3.cmml" xref="S3.p6.7.m7.1.1.3"><csymbol cd="ambiguous" id="S3.p6.7.m7.1.1.3.1.cmml" xref="S3.p6.7.m7.1.1.3">superscript</csymbol><ci id="S3.p6.7.m7.1.1.3.2.cmml" xref="S3.p6.7.m7.1.1.3.2">𝒞</ci><apply id="S3.p6.7.m7.1.1.3.3.cmml" xref="S3.p6.7.m7.1.1.3.3"><minus id="S3.p6.7.m7.1.1.3.3.1.cmml" xref="S3.p6.7.m7.1.1.3.3.1"></minus><ci id="S3.p6.7.m7.1.1.3.3.2.cmml" xref="S3.p6.7.m7.1.1.3.3.2">𝑛</ci><ci id="S3.p6.7.m7.1.1.3.3.3.cmml" xref="S3.p6.7.m7.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.7.m7.1c">\bm{\varphi}_{i}\in\mathcal{C}^{n-i}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.7.m7.1d">bold_italic_φ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_C start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="i=1" class="ltx_Math" display="inline" id="S3.p6.8.m8.1"><semantics id="S3.p6.8.m8.1a"><mrow id="S3.p6.8.m8.1.1" xref="S3.p6.8.m8.1.1.cmml"><mi id="S3.p6.8.m8.1.1.2" xref="S3.p6.8.m8.1.1.2.cmml">i</mi><mo id="S3.p6.8.m8.1.1.1" xref="S3.p6.8.m8.1.1.1.cmml">=</mo><mn id="S3.p6.8.m8.1.1.3" xref="S3.p6.8.m8.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.8.m8.1b"><apply id="S3.p6.8.m8.1.1.cmml" xref="S3.p6.8.m8.1.1"><eq id="S3.p6.8.m8.1.1.1.cmml" xref="S3.p6.8.m8.1.1.1"></eq><ci id="S3.p6.8.m8.1.1.2.cmml" xref="S3.p6.8.m8.1.1.2">𝑖</ci><cn id="S3.p6.8.m8.1.1.3.cmml" type="integer" xref="S3.p6.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.8.m8.1c">i=1</annotation><annotation encoding="application/x-llamapun" id="S3.p6.8.m8.1d">italic_i = 1</annotation></semantics></math> to <math alttext="n" class="ltx_Math" display="inline" id="S3.p6.9.m9.1"><semantics id="S3.p6.9.m9.1a"><mi id="S3.p6.9.m9.1.1" xref="S3.p6.9.m9.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.p6.9.m9.1b"><ci id="S3.p6.9.m9.1.1.cmml" xref="S3.p6.9.m9.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.9.m9.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.p6.9.m9.1d">italic_n</annotation></semantics></math> implies that they have continuous partial derivatives up to the order <math alttext="n-i" class="ltx_Math" display="inline" id="S3.p6.10.m10.1"><semantics id="S3.p6.10.m10.1a"><mrow id="S3.p6.10.m10.1.1" xref="S3.p6.10.m10.1.1.cmml"><mi id="S3.p6.10.m10.1.1.2" xref="S3.p6.10.m10.1.1.2.cmml">n</mi><mo id="S3.p6.10.m10.1.1.1" xref="S3.p6.10.m10.1.1.1.cmml">−</mo><mi id="S3.p6.10.m10.1.1.3" xref="S3.p6.10.m10.1.1.3.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.10.m10.1b"><apply id="S3.p6.10.m10.1.1.cmml" xref="S3.p6.10.m10.1.1"><minus id="S3.p6.10.m10.1.1.1.cmml" xref="S3.p6.10.m10.1.1.1"></minus><ci id="S3.p6.10.m10.1.1.2.cmml" xref="S3.p6.10.m10.1.1.2">𝑛</ci><ci id="S3.p6.10.m10.1.1.3.cmml" xref="S3.p6.10.m10.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.10.m10.1c">n-i</annotation><annotation encoding="application/x-llamapun" id="S3.p6.10.m10.1d">italic_n - italic_i</annotation></semantics></math> (i.e., <math alttext="\frac{\partial^{k}\bm{\varphi}_{i}}{\partial x_{j}^{k}}" class="ltx_Math" display="inline" id="S3.p6.11.m11.1"><semantics id="S3.p6.11.m11.1a"><mfrac id="S3.p6.11.m11.1.1" xref="S3.p6.11.m11.1.1.cmml"><mrow id="S3.p6.11.m11.1.1.2" xref="S3.p6.11.m11.1.1.2.cmml"><msup id="S3.p6.11.m11.1.1.2.1" xref="S3.p6.11.m11.1.1.2.1.cmml"><mo id="S3.p6.11.m11.1.1.2.1.2" xref="S3.p6.11.m11.1.1.2.1.2.cmml">∂</mo><mi id="S3.p6.11.m11.1.1.2.1.3" xref="S3.p6.11.m11.1.1.2.1.3.cmml">k</mi></msup><msub id="S3.p6.11.m11.1.1.2.2" xref="S3.p6.11.m11.1.1.2.2.cmml"><mi id="S3.p6.11.m11.1.1.2.2.2" xref="S3.p6.11.m11.1.1.2.2.2.cmml">𝝋</mi><mi id="S3.p6.11.m11.1.1.2.2.3" xref="S3.p6.11.m11.1.1.2.2.3.cmml">i</mi></msub></mrow><mrow id="S3.p6.11.m11.1.1.3" xref="S3.p6.11.m11.1.1.3.cmml"><mo id="S3.p6.11.m11.1.1.3.1" rspace="0em" xref="S3.p6.11.m11.1.1.3.1.cmml">∂</mo><msubsup id="S3.p6.11.m11.1.1.3.2" xref="S3.p6.11.m11.1.1.3.2.cmml"><mi id="S3.p6.11.m11.1.1.3.2.2.2" xref="S3.p6.11.m11.1.1.3.2.2.2.cmml">x</mi><mi id="S3.p6.11.m11.1.1.3.2.2.3" xref="S3.p6.11.m11.1.1.3.2.2.3.cmml">j</mi><mi id="S3.p6.11.m11.1.1.3.2.3" xref="S3.p6.11.m11.1.1.3.2.3.cmml">k</mi></msubsup></mrow></mfrac><annotation-xml encoding="MathML-Content" id="S3.p6.11.m11.1b"><apply id="S3.p6.11.m11.1.1.cmml" xref="S3.p6.11.m11.1.1"><divide id="S3.p6.11.m11.1.1.1.cmml" xref="S3.p6.11.m11.1.1"></divide><apply id="S3.p6.11.m11.1.1.2.cmml" xref="S3.p6.11.m11.1.1.2"><apply id="S3.p6.11.m11.1.1.2.1.cmml" xref="S3.p6.11.m11.1.1.2.1"><csymbol cd="ambiguous" id="S3.p6.11.m11.1.1.2.1.1.cmml" xref="S3.p6.11.m11.1.1.2.1">superscript</csymbol><partialdiff id="S3.p6.11.m11.1.1.2.1.2.cmml" xref="S3.p6.11.m11.1.1.2.1.2"></partialdiff><ci id="S3.p6.11.m11.1.1.2.1.3.cmml" xref="S3.p6.11.m11.1.1.2.1.3">𝑘</ci></apply><apply id="S3.p6.11.m11.1.1.2.2.cmml" xref="S3.p6.11.m11.1.1.2.2"><csymbol cd="ambiguous" id="S3.p6.11.m11.1.1.2.2.1.cmml" xref="S3.p6.11.m11.1.1.2.2">subscript</csymbol><ci id="S3.p6.11.m11.1.1.2.2.2.cmml" xref="S3.p6.11.m11.1.1.2.2.2">𝝋</ci><ci id="S3.p6.11.m11.1.1.2.2.3.cmml" xref="S3.p6.11.m11.1.1.2.2.3">𝑖</ci></apply></apply><apply id="S3.p6.11.m11.1.1.3.cmml" xref="S3.p6.11.m11.1.1.3"><partialdiff id="S3.p6.11.m11.1.1.3.1.cmml" xref="S3.p6.11.m11.1.1.3.1"></partialdiff><apply id="S3.p6.11.m11.1.1.3.2.cmml" xref="S3.p6.11.m11.1.1.3.2"><csymbol cd="ambiguous" id="S3.p6.11.m11.1.1.3.2.1.cmml" xref="S3.p6.11.m11.1.1.3.2">superscript</csymbol><apply id="S3.p6.11.m11.1.1.3.2.2.cmml" xref="S3.p6.11.m11.1.1.3.2"><csymbol cd="ambiguous" id="S3.p6.11.m11.1.1.3.2.2.1.cmml" xref="S3.p6.11.m11.1.1.3.2">subscript</csymbol><ci id="S3.p6.11.m11.1.1.3.2.2.2.cmml" xref="S3.p6.11.m11.1.1.3.2.2.2">𝑥</ci><ci id="S3.p6.11.m11.1.1.3.2.2.3.cmml" xref="S3.p6.11.m11.1.1.3.2.2.3">𝑗</ci></apply><ci id="S3.p6.11.m11.1.1.3.2.3.cmml" xref="S3.p6.11.m11.1.1.3.2.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.11.m11.1c">\frac{\partial^{k}\bm{\varphi}_{i}}{\partial x_{j}^{k}}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.11.m11.1d">divide start_ARG ∂ start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT bold_italic_φ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math> are continuous functions for <math alttext="k=0" class="ltx_Math" display="inline" id="S3.p6.12.m12.1"><semantics id="S3.p6.12.m12.1a"><mrow id="S3.p6.12.m12.1.1" xref="S3.p6.12.m12.1.1.cmml"><mi id="S3.p6.12.m12.1.1.2" xref="S3.p6.12.m12.1.1.2.cmml">k</mi><mo id="S3.p6.12.m12.1.1.1" xref="S3.p6.12.m12.1.1.1.cmml">=</mo><mn id="S3.p6.12.m12.1.1.3" xref="S3.p6.12.m12.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.12.m12.1b"><apply id="S3.p6.12.m12.1.1.cmml" xref="S3.p6.12.m12.1.1"><eq id="S3.p6.12.m12.1.1.1.cmml" xref="S3.p6.12.m12.1.1.1"></eq><ci id="S3.p6.12.m12.1.1.2.cmml" xref="S3.p6.12.m12.1.1.2">𝑘</ci><cn id="S3.p6.12.m12.1.1.3.cmml" type="integer" xref="S3.p6.12.m12.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.12.m12.1c">k=0</annotation><annotation encoding="application/x-llamapun" id="S3.p6.12.m12.1d">italic_k = 0</annotation></semantics></math> to <math alttext="n-i" class="ltx_Math" display="inline" id="S3.p6.13.m13.1"><semantics id="S3.p6.13.m13.1a"><mrow id="S3.p6.13.m13.1.1" xref="S3.p6.13.m13.1.1.cmml"><mi id="S3.p6.13.m13.1.1.2" xref="S3.p6.13.m13.1.1.2.cmml">n</mi><mo id="S3.p6.13.m13.1.1.1" xref="S3.p6.13.m13.1.1.1.cmml">−</mo><mi id="S3.p6.13.m13.1.1.3" xref="S3.p6.13.m13.1.1.3.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.13.m13.1b"><apply id="S3.p6.13.m13.1.1.cmml" xref="S3.p6.13.m13.1.1"><minus id="S3.p6.13.m13.1.1.1.cmml" xref="S3.p6.13.m13.1.1.1"></minus><ci id="S3.p6.13.m13.1.1.2.cmml" xref="S3.p6.13.m13.1.1.2">𝑛</ci><ci id="S3.p6.13.m13.1.1.3.cmml" xref="S3.p6.13.m13.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.13.m13.1c">n-i</annotation><annotation encoding="application/x-llamapun" id="S3.p6.13.m13.1d">italic_n - italic_i</annotation></semantics></math> and <math alttext="j=1" class="ltx_Math" display="inline" id="S3.p6.14.m14.1"><semantics id="S3.p6.14.m14.1a"><mrow id="S3.p6.14.m14.1.1" xref="S3.p6.14.m14.1.1.cmml"><mi id="S3.p6.14.m14.1.1.2" xref="S3.p6.14.m14.1.1.2.cmml">j</mi><mo id="S3.p6.14.m14.1.1.1" xref="S3.p6.14.m14.1.1.1.cmml">=</mo><mn id="S3.p6.14.m14.1.1.3" xref="S3.p6.14.m14.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.14.m14.1b"><apply id="S3.p6.14.m14.1.1.cmml" xref="S3.p6.14.m14.1.1"><eq id="S3.p6.14.m14.1.1.1.cmml" xref="S3.p6.14.m14.1.1.1"></eq><ci id="S3.p6.14.m14.1.1.2.cmml" xref="S3.p6.14.m14.1.1.2">𝑗</ci><cn id="S3.p6.14.m14.1.1.3.cmml" type="integer" xref="S3.p6.14.m14.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.14.m14.1c">j=1</annotation><annotation encoding="application/x-llamapun" id="S3.p6.14.m14.1d">italic_j = 1</annotation></semantics></math> to <math alttext="i" class="ltx_Math" display="inline" id="S3.p6.15.m15.1"><semantics id="S3.p6.15.m15.1a"><mi id="S3.p6.15.m15.1.1" xref="S3.p6.15.m15.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.p6.15.m15.1b"><ci id="S3.p6.15.m15.1.1.cmml" xref="S3.p6.15.m15.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.15.m15.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.p6.15.m15.1d">italic_i</annotation></semantics></math>), which ensures the realizability of the modular backstepping approach <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib8" title="">8</a>]</cite>.</p> </div> <div class="ltx_para" id="S3.p7"> <p class="ltx_p" id="S3.p7.12"><span class="ltx_text ltx_font_italic" id="S3.p7.12.1">Remark 3:</span> In all existing modular backstepping control methods, nonlinear damping terms like <math alttext="k_{\rm{d}\it i}\|\bm{\psi}_{i}\|^{2}" class="ltx_Math" display="inline" id="S3.p7.1.m1.1"><semantics id="S3.p7.1.m1.1a"><mrow id="S3.p7.1.m1.1.1" xref="S3.p7.1.m1.1.1.cmml"><msub id="S3.p7.1.m1.1.1.3" xref="S3.p7.1.m1.1.1.3.cmml"><mi id="S3.p7.1.m1.1.1.3.2" xref="S3.p7.1.m1.1.1.3.2.cmml">k</mi><mrow id="S3.p7.1.m1.1.1.3.3" xref="S3.p7.1.m1.1.1.3.3.cmml"><mi id="S3.p7.1.m1.1.1.3.3.2" mathvariant="normal" xref="S3.p7.1.m1.1.1.3.3.2.cmml">d</mi><mo id="S3.p7.1.m1.1.1.3.3.1" xref="S3.p7.1.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S3.p7.1.m1.1.1.3.3.3" xref="S3.p7.1.m1.1.1.3.3.3.cmml">i</mi></mrow></msub><mo id="S3.p7.1.m1.1.1.2" xref="S3.p7.1.m1.1.1.2.cmml">⁢</mo><msup id="S3.p7.1.m1.1.1.1" xref="S3.p7.1.m1.1.1.1.cmml"><mrow id="S3.p7.1.m1.1.1.1.1.1" xref="S3.p7.1.m1.1.1.1.1.2.cmml"><mo id="S3.p7.1.m1.1.1.1.1.1.2" stretchy="false" xref="S3.p7.1.m1.1.1.1.1.2.1.cmml">‖</mo><msub id="S3.p7.1.m1.1.1.1.1.1.1" xref="S3.p7.1.m1.1.1.1.1.1.1.cmml"><mi id="S3.p7.1.m1.1.1.1.1.1.1.2" xref="S3.p7.1.m1.1.1.1.1.1.1.2.cmml">𝝍</mi><mi id="S3.p7.1.m1.1.1.1.1.1.1.3" xref="S3.p7.1.m1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.p7.1.m1.1.1.1.1.1.3" stretchy="false" xref="S3.p7.1.m1.1.1.1.1.2.1.cmml">‖</mo></mrow><mn id="S3.p7.1.m1.1.1.1.3" xref="S3.p7.1.m1.1.1.1.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p7.1.m1.1b"><apply id="S3.p7.1.m1.1.1.cmml" xref="S3.p7.1.m1.1.1"><times id="S3.p7.1.m1.1.1.2.cmml" xref="S3.p7.1.m1.1.1.2"></times><apply id="S3.p7.1.m1.1.1.3.cmml" xref="S3.p7.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.p7.1.m1.1.1.3.1.cmml" xref="S3.p7.1.m1.1.1.3">subscript</csymbol><ci id="S3.p7.1.m1.1.1.3.2.cmml" xref="S3.p7.1.m1.1.1.3.2">𝑘</ci><apply id="S3.p7.1.m1.1.1.3.3.cmml" xref="S3.p7.1.m1.1.1.3.3"><times id="S3.p7.1.m1.1.1.3.3.1.cmml" xref="S3.p7.1.m1.1.1.3.3.1"></times><ci id="S3.p7.1.m1.1.1.3.3.2.cmml" xref="S3.p7.1.m1.1.1.3.3.2">d</ci><ci id="S3.p7.1.m1.1.1.3.3.3.cmml" xref="S3.p7.1.m1.1.1.3.3.3">𝑖</ci></apply></apply><apply id="S3.p7.1.m1.1.1.1.cmml" xref="S3.p7.1.m1.1.1.1"><csymbol cd="ambiguous" id="S3.p7.1.m1.1.1.1.2.cmml" xref="S3.p7.1.m1.1.1.1">superscript</csymbol><apply id="S3.p7.1.m1.1.1.1.1.2.cmml" xref="S3.p7.1.m1.1.1.1.1.1"><csymbol cd="latexml" id="S3.p7.1.m1.1.1.1.1.2.1.cmml" xref="S3.p7.1.m1.1.1.1.1.1.2">norm</csymbol><apply id="S3.p7.1.m1.1.1.1.1.1.1.cmml" xref="S3.p7.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.p7.1.m1.1.1.1.1.1.1.1.cmml" xref="S3.p7.1.m1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.p7.1.m1.1.1.1.1.1.1.2.cmml" xref="S3.p7.1.m1.1.1.1.1.1.1.2">𝝍</ci><ci id="S3.p7.1.m1.1.1.1.1.1.1.3.cmml" xref="S3.p7.1.m1.1.1.1.1.1.1.3">𝑖</ci></apply></apply><cn id="S3.p7.1.m1.1.1.1.3.cmml" type="integer" xref="S3.p7.1.m1.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.1.m1.1c">k_{\rm{d}\it i}\|\bm{\psi}_{i}\|^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p7.1.m1.1d">italic_k start_POSTSUBSCRIPT roman_d italic_i end_POSTSUBSCRIPT ∥ bold_italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="k_{\rm{d}\it i}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S3.p7.2.m2.1"><semantics id="S3.p7.2.m2.1a"><mrow id="S3.p7.2.m2.1.1" xref="S3.p7.2.m2.1.1.cmml"><msub id="S3.p7.2.m2.1.1.2" xref="S3.p7.2.m2.1.1.2.cmml"><mi id="S3.p7.2.m2.1.1.2.2" xref="S3.p7.2.m2.1.1.2.2.cmml">k</mi><mrow id="S3.p7.2.m2.1.1.2.3" xref="S3.p7.2.m2.1.1.2.3.cmml"><mi id="S3.p7.2.m2.1.1.2.3.2" mathvariant="normal" xref="S3.p7.2.m2.1.1.2.3.2.cmml">d</mi><mo id="S3.p7.2.m2.1.1.2.3.1" xref="S3.p7.2.m2.1.1.2.3.1.cmml">⁢</mo><mi id="S3.p7.2.m2.1.1.2.3.3" xref="S3.p7.2.m2.1.1.2.3.3.cmml">i</mi></mrow></msub><mo id="S3.p7.2.m2.1.1.1" xref="S3.p7.2.m2.1.1.1.cmml">∈</mo><msup id="S3.p7.2.m2.1.1.3" xref="S3.p7.2.m2.1.1.3.cmml"><mi id="S3.p7.2.m2.1.1.3.2" xref="S3.p7.2.m2.1.1.3.2.cmml">ℝ</mi><mo id="S3.p7.2.m2.1.1.3.3" xref="S3.p7.2.m2.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p7.2.m2.1b"><apply id="S3.p7.2.m2.1.1.cmml" xref="S3.p7.2.m2.1.1"><in id="S3.p7.2.m2.1.1.1.cmml" xref="S3.p7.2.m2.1.1.1"></in><apply id="S3.p7.2.m2.1.1.2.cmml" xref="S3.p7.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.p7.2.m2.1.1.2.1.cmml" xref="S3.p7.2.m2.1.1.2">subscript</csymbol><ci id="S3.p7.2.m2.1.1.2.2.cmml" xref="S3.p7.2.m2.1.1.2.2">𝑘</ci><apply id="S3.p7.2.m2.1.1.2.3.cmml" xref="S3.p7.2.m2.1.1.2.3"><times id="S3.p7.2.m2.1.1.2.3.1.cmml" xref="S3.p7.2.m2.1.1.2.3.1"></times><ci id="S3.p7.2.m2.1.1.2.3.2.cmml" xref="S3.p7.2.m2.1.1.2.3.2">d</ci><ci id="S3.p7.2.m2.1.1.2.3.3.cmml" xref="S3.p7.2.m2.1.1.2.3.3">𝑖</ci></apply></apply><apply id="S3.p7.2.m2.1.1.3.cmml" xref="S3.p7.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.p7.2.m2.1.1.3.1.cmml" xref="S3.p7.2.m2.1.1.3">superscript</csymbol><ci id="S3.p7.2.m2.1.1.3.2.cmml" xref="S3.p7.2.m2.1.1.3.2">ℝ</ci><plus id="S3.p7.2.m2.1.1.3.3.cmml" xref="S3.p7.2.m2.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.2.m2.1c">k_{\rm{d}\it i}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S3.p7.2.m2.1d">italic_k start_POSTSUBSCRIPT roman_d italic_i end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> being damping parameters (<math alttext="i=1" class="ltx_Math" display="inline" id="S3.p7.3.m3.1"><semantics id="S3.p7.3.m3.1a"><mrow id="S3.p7.3.m3.1.1" xref="S3.p7.3.m3.1.1.cmml"><mi id="S3.p7.3.m3.1.1.2" xref="S3.p7.3.m3.1.1.2.cmml">i</mi><mo id="S3.p7.3.m3.1.1.1" xref="S3.p7.3.m3.1.1.1.cmml">=</mo><mn id="S3.p7.3.m3.1.1.3" xref="S3.p7.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p7.3.m3.1b"><apply id="S3.p7.3.m3.1.1.cmml" xref="S3.p7.3.m3.1.1"><eq id="S3.p7.3.m3.1.1.1.cmml" xref="S3.p7.3.m3.1.1.1"></eq><ci id="S3.p7.3.m3.1.1.2.cmml" xref="S3.p7.3.m3.1.1.2">𝑖</ci><cn id="S3.p7.3.m3.1.1.3.cmml" type="integer" xref="S3.p7.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.3.m3.1c">i=1</annotation><annotation encoding="application/x-llamapun" id="S3.p7.3.m3.1d">italic_i = 1</annotation></semantics></math> to <math alttext="n" class="ltx_Math" display="inline" id="S3.p7.4.m4.1"><semantics id="S3.p7.4.m4.1a"><mi id="S3.p7.4.m4.1.1" xref="S3.p7.4.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.p7.4.m4.1b"><ci id="S3.p7.4.m4.1.1.cmml" xref="S3.p7.4.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.4.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.p7.4.m4.1d">italic_n</annotation></semantics></math>) must be incorporated into the virtual control inputs <math alttext="v_{i}" class="ltx_Math" display="inline" id="S3.p7.5.m5.1"><semantics id="S3.p7.5.m5.1a"><msub id="S3.p7.5.m5.1.1" xref="S3.p7.5.m5.1.1.cmml"><mi id="S3.p7.5.m5.1.1.2" xref="S3.p7.5.m5.1.1.2.cmml">v</mi><mi id="S3.p7.5.m5.1.1.3" xref="S3.p7.5.m5.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p7.5.m5.1b"><apply id="S3.p7.5.m5.1.1.cmml" xref="S3.p7.5.m5.1.1"><csymbol cd="ambiguous" id="S3.p7.5.m5.1.1.1.cmml" xref="S3.p7.5.m5.1.1">subscript</csymbol><ci id="S3.p7.5.m5.1.1.2.cmml" xref="S3.p7.5.m5.1.1.2">𝑣</ci><ci id="S3.p7.5.m5.1.1.3.cmml" xref="S3.p7.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.5.m5.1c">v_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.p7.5.m5.1d">italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.Ex4" title="III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">III</span></a>) to counteract the transient process arising from the modeling error term <math alttext="\Phi^{T}\tilde{\bm{\theta}}" class="ltx_Math" display="inline" id="S3.p7.6.m6.1"><semantics id="S3.p7.6.m6.1a"><mrow id="S3.p7.6.m6.1.1" xref="S3.p7.6.m6.1.1.cmml"><msup id="S3.p7.6.m6.1.1.2" xref="S3.p7.6.m6.1.1.2.cmml"><mi id="S3.p7.6.m6.1.1.2.2" mathvariant="normal" xref="S3.p7.6.m6.1.1.2.2.cmml">Φ</mi><mi id="S3.p7.6.m6.1.1.2.3" xref="S3.p7.6.m6.1.1.2.3.cmml">T</mi></msup><mo id="S3.p7.6.m6.1.1.1" xref="S3.p7.6.m6.1.1.1.cmml">⁢</mo><mover accent="true" id="S3.p7.6.m6.1.1.3" xref="S3.p7.6.m6.1.1.3.cmml"><mi id="S3.p7.6.m6.1.1.3.2" xref="S3.p7.6.m6.1.1.3.2.cmml">𝜽</mi><mo id="S3.p7.6.m6.1.1.3.1" xref="S3.p7.6.m6.1.1.3.1.cmml">~</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.p7.6.m6.1b"><apply id="S3.p7.6.m6.1.1.cmml" xref="S3.p7.6.m6.1.1"><times id="S3.p7.6.m6.1.1.1.cmml" xref="S3.p7.6.m6.1.1.1"></times><apply id="S3.p7.6.m6.1.1.2.cmml" xref="S3.p7.6.m6.1.1.2"><csymbol cd="ambiguous" id="S3.p7.6.m6.1.1.2.1.cmml" xref="S3.p7.6.m6.1.1.2">superscript</csymbol><ci id="S3.p7.6.m6.1.1.2.2.cmml" xref="S3.p7.6.m6.1.1.2.2">Φ</ci><ci id="S3.p7.6.m6.1.1.2.3.cmml" xref="S3.p7.6.m6.1.1.2.3">𝑇</ci></apply><apply id="S3.p7.6.m6.1.1.3.cmml" xref="S3.p7.6.m6.1.1.3"><ci id="S3.p7.6.m6.1.1.3.1.cmml" xref="S3.p7.6.m6.1.1.3.1">~</ci><ci id="S3.p7.6.m6.1.1.3.2.cmml" xref="S3.p7.6.m6.1.1.3.2">𝜽</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.6.m6.1c">\Phi^{T}\tilde{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S3.p7.6.m6.1d">roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math> and the high-order time derivatives <math alttext="\hat{\bm{\theta}}^{(k)}" class="ltx_Math" display="inline" id="S3.p7.7.m7.1"><semantics id="S3.p7.7.m7.1a"><msup id="S3.p7.7.m7.1.2" xref="S3.p7.7.m7.1.2.cmml"><mover accent="true" id="S3.p7.7.m7.1.2.2" xref="S3.p7.7.m7.1.2.2.cmml"><mi id="S3.p7.7.m7.1.2.2.2" xref="S3.p7.7.m7.1.2.2.2.cmml">𝜽</mi><mo id="S3.p7.7.m7.1.2.2.1" xref="S3.p7.7.m7.1.2.2.1.cmml">^</mo></mover><mrow id="S3.p7.7.m7.1.1.1.3" xref="S3.p7.7.m7.1.2.cmml"><mo id="S3.p7.7.m7.1.1.1.3.1" stretchy="false" xref="S3.p7.7.m7.1.2.cmml">(</mo><mi id="S3.p7.7.m7.1.1.1.1" xref="S3.p7.7.m7.1.1.1.1.cmml">k</mi><mo id="S3.p7.7.m7.1.1.1.3.2" stretchy="false" xref="S3.p7.7.m7.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.p7.7.m7.1b"><apply id="S3.p7.7.m7.1.2.cmml" xref="S3.p7.7.m7.1.2"><csymbol cd="ambiguous" id="S3.p7.7.m7.1.2.1.cmml" xref="S3.p7.7.m7.1.2">superscript</csymbol><apply id="S3.p7.7.m7.1.2.2.cmml" xref="S3.p7.7.m7.1.2.2"><ci id="S3.p7.7.m7.1.2.2.1.cmml" xref="S3.p7.7.m7.1.2.2.1">^</ci><ci id="S3.p7.7.m7.1.2.2.2.cmml" xref="S3.p7.7.m7.1.2.2.2">𝜽</ci></apply><ci id="S3.p7.7.m7.1.1.1.1.cmml" xref="S3.p7.7.m7.1.1.1.1">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.7.m7.1c">\hat{\bm{\theta}}^{(k)}</annotation><annotation encoding="application/x-llamapun" id="S3.p7.7.m7.1d">over^ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT</annotation></semantics></math> (<math alttext="k=1" class="ltx_Math" display="inline" id="S3.p7.8.m8.1"><semantics id="S3.p7.8.m8.1a"><mrow id="S3.p7.8.m8.1.1" xref="S3.p7.8.m8.1.1.cmml"><mi id="S3.p7.8.m8.1.1.2" xref="S3.p7.8.m8.1.1.2.cmml">k</mi><mo id="S3.p7.8.m8.1.1.1" xref="S3.p7.8.m8.1.1.1.cmml">=</mo><mn id="S3.p7.8.m8.1.1.3" xref="S3.p7.8.m8.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p7.8.m8.1b"><apply id="S3.p7.8.m8.1.1.cmml" xref="S3.p7.8.m8.1.1"><eq id="S3.p7.8.m8.1.1.1.cmml" xref="S3.p7.8.m8.1.1.1"></eq><ci id="S3.p7.8.m8.1.1.2.cmml" xref="S3.p7.8.m8.1.1.2">𝑘</ci><cn id="S3.p7.8.m8.1.1.3.cmml" type="integer" xref="S3.p7.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.8.m8.1c">k=1</annotation><annotation encoding="application/x-llamapun" id="S3.p7.8.m8.1d">italic_k = 1</annotation></semantics></math> to <math alttext="n-1" class="ltx_Math" display="inline" id="S3.p7.9.m9.1"><semantics id="S3.p7.9.m9.1a"><mrow id="S3.p7.9.m9.1.1" xref="S3.p7.9.m9.1.1.cmml"><mi id="S3.p7.9.m9.1.1.2" xref="S3.p7.9.m9.1.1.2.cmml">n</mi><mo id="S3.p7.9.m9.1.1.1" xref="S3.p7.9.m9.1.1.1.cmml">−</mo><mn id="S3.p7.9.m9.1.1.3" xref="S3.p7.9.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p7.9.m9.1b"><apply id="S3.p7.9.m9.1.1.cmml" xref="S3.p7.9.m9.1.1"><minus id="S3.p7.9.m9.1.1.1.cmml" xref="S3.p7.9.m9.1.1.1"></minus><ci id="S3.p7.9.m9.1.1.2.cmml" xref="S3.p7.9.m9.1.1.2">𝑛</ci><cn id="S3.p7.9.m9.1.1.3.cmml" type="integer" xref="S3.p7.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.9.m9.1c">n-1</annotation><annotation encoding="application/x-llamapun" id="S3.p7.9.m9.1d">italic_n - 1</annotation></semantics></math>), such that the boundedness of the closed-loop system can always be established even in the absence of adaptation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib8" title="">8</a>, <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib24" title="">24</a>]</cite>. However, the square terms <math alttext="\|\bm{\psi}_{i}\|^{2}" class="ltx_Math" display="inline" id="S3.p7.10.m10.1"><semantics id="S3.p7.10.m10.1a"><msup id="S3.p7.10.m10.1.1" xref="S3.p7.10.m10.1.1.cmml"><mrow id="S3.p7.10.m10.1.1.1.1" xref="S3.p7.10.m10.1.1.1.2.cmml"><mo id="S3.p7.10.m10.1.1.1.1.2" stretchy="false" xref="S3.p7.10.m10.1.1.1.2.1.cmml">‖</mo><msub id="S3.p7.10.m10.1.1.1.1.1" xref="S3.p7.10.m10.1.1.1.1.1.cmml"><mi id="S3.p7.10.m10.1.1.1.1.1.2" xref="S3.p7.10.m10.1.1.1.1.1.2.cmml">𝝍</mi><mi id="S3.p7.10.m10.1.1.1.1.1.3" xref="S3.p7.10.m10.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.p7.10.m10.1.1.1.1.3" stretchy="false" xref="S3.p7.10.m10.1.1.1.2.1.cmml">‖</mo></mrow><mn id="S3.p7.10.m10.1.1.3" xref="S3.p7.10.m10.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S3.p7.10.m10.1b"><apply id="S3.p7.10.m10.1.1.cmml" xref="S3.p7.10.m10.1.1"><csymbol cd="ambiguous" id="S3.p7.10.m10.1.1.2.cmml" xref="S3.p7.10.m10.1.1">superscript</csymbol><apply id="S3.p7.10.m10.1.1.1.2.cmml" xref="S3.p7.10.m10.1.1.1.1"><csymbol cd="latexml" id="S3.p7.10.m10.1.1.1.2.1.cmml" xref="S3.p7.10.m10.1.1.1.1.2">norm</csymbol><apply id="S3.p7.10.m10.1.1.1.1.1.cmml" xref="S3.p7.10.m10.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.p7.10.m10.1.1.1.1.1.1.cmml" xref="S3.p7.10.m10.1.1.1.1.1">subscript</csymbol><ci id="S3.p7.10.m10.1.1.1.1.1.2.cmml" xref="S3.p7.10.m10.1.1.1.1.1.2">𝝍</ci><ci id="S3.p7.10.m10.1.1.1.1.1.3.cmml" xref="S3.p7.10.m10.1.1.1.1.1.3">𝑖</ci></apply></apply><cn id="S3.p7.10.m10.1.1.3.cmml" type="integer" xref="S3.p7.10.m10.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.10.m10.1c">\|\bm{\psi}_{i}\|^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p7.10.m10.1d">∥ bold_italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> can lead to high-gain control, regardless of whether the estimation error <math alttext="\tilde{\bm{\theta}}" class="ltx_Math" display="inline" id="S3.p7.11.m11.1"><semantics id="S3.p7.11.m11.1a"><mover accent="true" id="S3.p7.11.m11.1.1" xref="S3.p7.11.m11.1.1.cmml"><mi id="S3.p7.11.m11.1.1.2" xref="S3.p7.11.m11.1.1.2.cmml">𝜽</mi><mo id="S3.p7.11.m11.1.1.1" xref="S3.p7.11.m11.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S3.p7.11.m11.1b"><apply id="S3.p7.11.m11.1.1.cmml" xref="S3.p7.11.m11.1.1"><ci id="S3.p7.11.m11.1.1.1.cmml" xref="S3.p7.11.m11.1.1.1">~</ci><ci id="S3.p7.11.m11.1.1.2.cmml" xref="S3.p7.11.m11.1.1.2">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.11.m11.1c">\tilde{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S3.p7.11.m11.1d">over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math> converges to <math alttext="\bm{0}" class="ltx_Math" display="inline" id="S3.p7.12.m12.1"><semantics id="S3.p7.12.m12.1a"><mn id="S3.p7.12.m12.1.1" xref="S3.p7.12.m12.1.1.cmml">𝟎</mn><annotation-xml encoding="MathML-Content" id="S3.p7.12.m12.1b"><cn id="S3.p7.12.m12.1.1.cmml" type="integer" xref="S3.p7.12.m12.1.1">0</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.12.m12.1c">\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S3.p7.12.m12.1d">bold_0</annotation></semantics></math> or not, which results in noise amplification and control saturation in practice. In this study, we aim to establish closed-loop stability without resorting to nonlinear damping terms as detailed in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4" title="IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">IV</span></a>.</p> </div> <div class="ltx_para" id="S3.p8"> <p class="ltx_p" id="S3.p8.17"><span class="ltx_text ltx_font_italic" id="S3.p8.17.17" style="color:#000099;">Remark 4:<span class="ltx_text ltx_font_upright" id="S3.p8.17.17.17" style="color:#000099;"> The transient and steady-state performance of the closed-loop system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E7" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">7</span></a>) is directly influenced by the modeling error <math alttext="\Phi^{T}\tilde{\bm{\theta}}" class="ltx_Math" display="inline" id="S3.p8.1.1.1.m1.1"><semantics id="S3.p8.1.1.1.m1.1a"><mrow id="S3.p8.1.1.1.m1.1.1" xref="S3.p8.1.1.1.m1.1.1.cmml"><msup id="S3.p8.1.1.1.m1.1.1.2" xref="S3.p8.1.1.1.m1.1.1.2.cmml"><mi id="S3.p8.1.1.1.m1.1.1.2.2" mathcolor="#000099" mathvariant="normal" xref="S3.p8.1.1.1.m1.1.1.2.2.cmml">Φ</mi><mi id="S3.p8.1.1.1.m1.1.1.2.3" mathcolor="#000099" xref="S3.p8.1.1.1.m1.1.1.2.3.cmml">T</mi></msup><mo id="S3.p8.1.1.1.m1.1.1.1" xref="S3.p8.1.1.1.m1.1.1.1.cmml">⁢</mo><mover accent="true" id="S3.p8.1.1.1.m1.1.1.3" xref="S3.p8.1.1.1.m1.1.1.3.cmml"><mi id="S3.p8.1.1.1.m1.1.1.3.2" mathcolor="#000099" xref="S3.p8.1.1.1.m1.1.1.3.2.cmml">𝜽</mi><mo id="S3.p8.1.1.1.m1.1.1.3.1" mathcolor="#000099" xref="S3.p8.1.1.1.m1.1.1.3.1.cmml">~</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.p8.1.1.1.m1.1b"><apply id="S3.p8.1.1.1.m1.1.1.cmml" xref="S3.p8.1.1.1.m1.1.1"><times id="S3.p8.1.1.1.m1.1.1.1.cmml" xref="S3.p8.1.1.1.m1.1.1.1"></times><apply id="S3.p8.1.1.1.m1.1.1.2.cmml" xref="S3.p8.1.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.p8.1.1.1.m1.1.1.2.1.cmml" xref="S3.p8.1.1.1.m1.1.1.2">superscript</csymbol><ci id="S3.p8.1.1.1.m1.1.1.2.2.cmml" xref="S3.p8.1.1.1.m1.1.1.2.2">Φ</ci><ci id="S3.p8.1.1.1.m1.1.1.2.3.cmml" xref="S3.p8.1.1.1.m1.1.1.2.3">𝑇</ci></apply><apply id="S3.p8.1.1.1.m1.1.1.3.cmml" xref="S3.p8.1.1.1.m1.1.1.3"><ci id="S3.p8.1.1.1.m1.1.1.3.1.cmml" xref="S3.p8.1.1.1.m1.1.1.3.1">~</ci><ci id="S3.p8.1.1.1.m1.1.1.3.2.cmml" xref="S3.p8.1.1.1.m1.1.1.3.2">𝜽</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.1.1.1.m1.1c">\Phi^{T}\tilde{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S3.p8.1.1.1.m1.1d">roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math>. The exponential decay term <math alttext="\bm{\epsilon}_{\rm f}" class="ltx_Math" display="inline" id="S3.p8.2.2.2.m2.1"><semantics id="S3.p8.2.2.2.m2.1a"><msub id="S3.p8.2.2.2.m2.1.1" xref="S3.p8.2.2.2.m2.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S3.p8.2.2.2.m2.1.1.2" mathcolor="#000099" mathvariant="bold-italic" xref="S3.p8.2.2.2.m2.1.1.2.cmml">ϵ</mi><mi id="S3.p8.2.2.2.m2.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S3.p8.2.2.2.m2.1.1.3.cmml">f</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p8.2.2.2.m2.1b"><apply id="S3.p8.2.2.2.m2.1.1.cmml" xref="S3.p8.2.2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p8.2.2.2.m2.1.1.1.cmml" xref="S3.p8.2.2.2.m2.1.1">subscript</csymbol><ci id="S3.p8.2.2.2.m2.1.1.2.cmml" xref="S3.p8.2.2.2.m2.1.1.2">bold-italic-ϵ</ci><ci id="S3.p8.2.2.2.m2.1.1.3.cmml" xref="S3.p8.2.2.2.m2.1.1.3">f</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.2.2.2.m2.1c">\bm{\epsilon}_{\rm f}</annotation><annotation encoding="application/x-llamapun" id="S3.p8.2.2.2.m2.1d">bold_italic_ϵ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT</annotation></semantics></math> appears as a disturbance in the regression equation (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E12" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">12</span></a>), which affects the accuracy of parameter estimation. Thus, <math alttext="{\bm{\epsilon}}_{\rm f}" class="ltx_Math" display="inline" id="S3.p8.3.3.3.m3.1"><semantics id="S3.p8.3.3.3.m3.1a"><msub id="S3.p8.3.3.3.m3.1.1" xref="S3.p8.3.3.3.m3.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S3.p8.3.3.3.m3.1.1.2" mathcolor="#000099" mathvariant="bold-italic" xref="S3.p8.3.3.3.m3.1.1.2.cmml">ϵ</mi><mi id="S3.p8.3.3.3.m3.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S3.p8.3.3.3.m3.1.1.3.cmml">f</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p8.3.3.3.m3.1b"><apply id="S3.p8.3.3.3.m3.1.1.cmml" xref="S3.p8.3.3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.p8.3.3.3.m3.1.1.1.cmml" xref="S3.p8.3.3.3.m3.1.1">subscript</csymbol><ci id="S3.p8.3.3.3.m3.1.1.2.cmml" xref="S3.p8.3.3.3.m3.1.1.2">bold-italic-ϵ</ci><ci id="S3.p8.3.3.3.m3.1.1.3.cmml" xref="S3.p8.3.3.3.m3.1.1.3">f</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.3.3.3.m3.1c">{\bm{\epsilon}}_{\rm f}</annotation><annotation encoding="application/x-llamapun" id="S3.p8.3.3.3.m3.1d">bold_italic_ϵ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT</annotation></semantics></math> must be compensated for in adaptive laws or eliminated by properly initializing <math alttext="\bm{\zeta}" class="ltx_Math" display="inline" id="S3.p8.4.4.4.m4.1"><semantics id="S3.p8.4.4.4.m4.1a"><mi id="S3.p8.4.4.4.m4.1.1" mathcolor="#000099" xref="S3.p8.4.4.4.m4.1.1.cmml">𝜻</mi><annotation-xml encoding="MathML-Content" id="S3.p8.4.4.4.m4.1b"><ci id="S3.p8.4.4.4.m4.1.1.cmml" xref="S3.p8.4.4.4.m4.1.1">𝜻</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.4.4.4.m4.1c">\bm{\zeta}</annotation><annotation encoding="application/x-llamapun" id="S3.p8.4.4.4.m4.1d">bold_italic_ζ</annotation></semantics></math> and <math alttext="{\Phi}_{\rm s}" class="ltx_Math" display="inline" id="S3.p8.5.5.5.m5.1"><semantics id="S3.p8.5.5.5.m5.1a"><msub id="S3.p8.5.5.5.m5.1.1" xref="S3.p8.5.5.5.m5.1.1.cmml"><mi id="S3.p8.5.5.5.m5.1.1.2" mathcolor="#000099" mathvariant="normal" xref="S3.p8.5.5.5.m5.1.1.2.cmml">Φ</mi><mi id="S3.p8.5.5.5.m5.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S3.p8.5.5.5.m5.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p8.5.5.5.m5.1b"><apply id="S3.p8.5.5.5.m5.1.1.cmml" xref="S3.p8.5.5.5.m5.1.1"><csymbol cd="ambiguous" id="S3.p8.5.5.5.m5.1.1.1.cmml" xref="S3.p8.5.5.5.m5.1.1">subscript</csymbol><ci id="S3.p8.5.5.5.m5.1.1.2.cmml" xref="S3.p8.5.5.5.m5.1.1.2">Φ</ci><ci id="S3.p8.5.5.5.m5.1.1.3.cmml" xref="S3.p8.5.5.5.m5.1.1.3">s</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.5.5.5.m5.1c">{\Phi}_{\rm s}</annotation><annotation encoding="application/x-llamapun" id="S3.p8.5.5.5.m5.1d">roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT</annotation></semantics></math> to obtain <math alttext="\bm{\epsilon}_{\rm f}(0)=\bm{0}" class="ltx_Math" display="inline" id="S3.p8.6.6.6.m6.1"><semantics id="S3.p8.6.6.6.m6.1a"><mrow id="S3.p8.6.6.6.m6.1.2" xref="S3.p8.6.6.6.m6.1.2.cmml"><mrow id="S3.p8.6.6.6.m6.1.2.2" xref="S3.p8.6.6.6.m6.1.2.2.cmml"><msub id="S3.p8.6.6.6.m6.1.2.2.2" xref="S3.p8.6.6.6.m6.1.2.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S3.p8.6.6.6.m6.1.2.2.2.2" mathcolor="#000099" mathvariant="bold-italic" xref="S3.p8.6.6.6.m6.1.2.2.2.2.cmml">ϵ</mi><mi id="S3.p8.6.6.6.m6.1.2.2.2.3" mathcolor="#000099" mathvariant="normal" xref="S3.p8.6.6.6.m6.1.2.2.2.3.cmml">f</mi></msub><mo id="S3.p8.6.6.6.m6.1.2.2.1" xref="S3.p8.6.6.6.m6.1.2.2.1.cmml">⁢</mo><mrow id="S3.p8.6.6.6.m6.1.2.2.3.2" xref="S3.p8.6.6.6.m6.1.2.2.cmml"><mo id="S3.p8.6.6.6.m6.1.2.2.3.2.1" mathcolor="#000099" stretchy="false" xref="S3.p8.6.6.6.m6.1.2.2.cmml">(</mo><mn id="S3.p8.6.6.6.m6.1.1" mathcolor="#000099" xref="S3.p8.6.6.6.m6.1.1.cmml">0</mn><mo id="S3.p8.6.6.6.m6.1.2.2.3.2.2" mathcolor="#000099" stretchy="false" xref="S3.p8.6.6.6.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.p8.6.6.6.m6.1.2.1" mathcolor="#000099" xref="S3.p8.6.6.6.m6.1.2.1.cmml">=</mo><mn id="S3.p8.6.6.6.m6.1.2.3" mathcolor="#000099" xref="S3.p8.6.6.6.m6.1.2.3.cmml">𝟎</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p8.6.6.6.m6.1b"><apply id="S3.p8.6.6.6.m6.1.2.cmml" xref="S3.p8.6.6.6.m6.1.2"><eq id="S3.p8.6.6.6.m6.1.2.1.cmml" xref="S3.p8.6.6.6.m6.1.2.1"></eq><apply id="S3.p8.6.6.6.m6.1.2.2.cmml" xref="S3.p8.6.6.6.m6.1.2.2"><times id="S3.p8.6.6.6.m6.1.2.2.1.cmml" xref="S3.p8.6.6.6.m6.1.2.2.1"></times><apply id="S3.p8.6.6.6.m6.1.2.2.2.cmml" xref="S3.p8.6.6.6.m6.1.2.2.2"><csymbol cd="ambiguous" id="S3.p8.6.6.6.m6.1.2.2.2.1.cmml" xref="S3.p8.6.6.6.m6.1.2.2.2">subscript</csymbol><ci id="S3.p8.6.6.6.m6.1.2.2.2.2.cmml" xref="S3.p8.6.6.6.m6.1.2.2.2.2">bold-italic-ϵ</ci><ci id="S3.p8.6.6.6.m6.1.2.2.2.3.cmml" xref="S3.p8.6.6.6.m6.1.2.2.2.3">f</ci></apply><cn id="S3.p8.6.6.6.m6.1.1.cmml" type="integer" xref="S3.p8.6.6.6.m6.1.1">0</cn></apply><cn id="S3.p8.6.6.6.m6.1.2.3.cmml" type="integer" xref="S3.p8.6.6.6.m6.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.6.6.6.m6.1c">\bm{\epsilon}_{\rm f}(0)=\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S3.p8.6.6.6.m6.1d">bold_italic_ϵ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ( 0 ) = bold_0</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib8" title="">8</a>, <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib24" title="">24</a>]</cite>. Here, the filter initializations <math alttext="\bm{\zeta}(0)=-\bm{e}(0)" class="ltx_Math" display="inline" id="S3.p8.7.7.7.m7.2"><semantics id="S3.p8.7.7.7.m7.2a"><mrow id="S3.p8.7.7.7.m7.2.3" xref="S3.p8.7.7.7.m7.2.3.cmml"><mrow id="S3.p8.7.7.7.m7.2.3.2" xref="S3.p8.7.7.7.m7.2.3.2.cmml"><mi id="S3.p8.7.7.7.m7.2.3.2.2" mathcolor="#000099" xref="S3.p8.7.7.7.m7.2.3.2.2.cmml">𝜻</mi><mo id="S3.p8.7.7.7.m7.2.3.2.1" xref="S3.p8.7.7.7.m7.2.3.2.1.cmml">⁢</mo><mrow id="S3.p8.7.7.7.m7.2.3.2.3.2" xref="S3.p8.7.7.7.m7.2.3.2.cmml"><mo id="S3.p8.7.7.7.m7.2.3.2.3.2.1" mathcolor="#000099" stretchy="false" xref="S3.p8.7.7.7.m7.2.3.2.cmml">(</mo><mn id="S3.p8.7.7.7.m7.1.1" mathcolor="#000099" xref="S3.p8.7.7.7.m7.1.1.cmml">0</mn><mo id="S3.p8.7.7.7.m7.2.3.2.3.2.2" mathcolor="#000099" stretchy="false" xref="S3.p8.7.7.7.m7.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.p8.7.7.7.m7.2.3.1" mathcolor="#000099" xref="S3.p8.7.7.7.m7.2.3.1.cmml">=</mo><mrow id="S3.p8.7.7.7.m7.2.3.3" xref="S3.p8.7.7.7.m7.2.3.3.cmml"><mo id="S3.p8.7.7.7.m7.2.3.3a" mathcolor="#000099" xref="S3.p8.7.7.7.m7.2.3.3.cmml">−</mo><mrow id="S3.p8.7.7.7.m7.2.3.3.2" xref="S3.p8.7.7.7.m7.2.3.3.2.cmml"><mi id="S3.p8.7.7.7.m7.2.3.3.2.2" mathcolor="#000099" xref="S3.p8.7.7.7.m7.2.3.3.2.2.cmml">𝒆</mi><mo id="S3.p8.7.7.7.m7.2.3.3.2.1" xref="S3.p8.7.7.7.m7.2.3.3.2.1.cmml">⁢</mo><mrow id="S3.p8.7.7.7.m7.2.3.3.2.3.2" xref="S3.p8.7.7.7.m7.2.3.3.2.cmml"><mo id="S3.p8.7.7.7.m7.2.3.3.2.3.2.1" mathcolor="#000099" stretchy="false" xref="S3.p8.7.7.7.m7.2.3.3.2.cmml">(</mo><mn id="S3.p8.7.7.7.m7.2.2" mathcolor="#000099" xref="S3.p8.7.7.7.m7.2.2.cmml">0</mn><mo id="S3.p8.7.7.7.m7.2.3.3.2.3.2.2" mathcolor="#000099" stretchy="false" xref="S3.p8.7.7.7.m7.2.3.3.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p8.7.7.7.m7.2b"><apply id="S3.p8.7.7.7.m7.2.3.cmml" xref="S3.p8.7.7.7.m7.2.3"><eq id="S3.p8.7.7.7.m7.2.3.1.cmml" xref="S3.p8.7.7.7.m7.2.3.1"></eq><apply id="S3.p8.7.7.7.m7.2.3.2.cmml" xref="S3.p8.7.7.7.m7.2.3.2"><times id="S3.p8.7.7.7.m7.2.3.2.1.cmml" xref="S3.p8.7.7.7.m7.2.3.2.1"></times><ci id="S3.p8.7.7.7.m7.2.3.2.2.cmml" xref="S3.p8.7.7.7.m7.2.3.2.2">𝜻</ci><cn id="S3.p8.7.7.7.m7.1.1.cmml" type="integer" xref="S3.p8.7.7.7.m7.1.1">0</cn></apply><apply id="S3.p8.7.7.7.m7.2.3.3.cmml" xref="S3.p8.7.7.7.m7.2.3.3"><minus id="S3.p8.7.7.7.m7.2.3.3.1.cmml" xref="S3.p8.7.7.7.m7.2.3.3"></minus><apply id="S3.p8.7.7.7.m7.2.3.3.2.cmml" xref="S3.p8.7.7.7.m7.2.3.3.2"><times id="S3.p8.7.7.7.m7.2.3.3.2.1.cmml" xref="S3.p8.7.7.7.m7.2.3.3.2.1"></times><ci id="S3.p8.7.7.7.m7.2.3.3.2.2.cmml" xref="S3.p8.7.7.7.m7.2.3.3.2.2">𝒆</ci><cn id="S3.p8.7.7.7.m7.2.2.cmml" type="integer" xref="S3.p8.7.7.7.m7.2.2">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.7.7.7.m7.2c">\bm{\zeta}(0)=-\bm{e}(0)</annotation><annotation encoding="application/x-llamapun" id="S3.p8.7.7.7.m7.2d">bold_italic_ζ ( 0 ) = - bold_italic_e ( 0 )</annotation></semantics></math> and <math alttext="{\Phi}_{\rm s}(0)=\bm{0}" class="ltx_Math" display="inline" id="S3.p8.8.8.8.m8.1"><semantics id="S3.p8.8.8.8.m8.1a"><mrow id="S3.p8.8.8.8.m8.1.2" xref="S3.p8.8.8.8.m8.1.2.cmml"><mrow id="S3.p8.8.8.8.m8.1.2.2" xref="S3.p8.8.8.8.m8.1.2.2.cmml"><msub id="S3.p8.8.8.8.m8.1.2.2.2" xref="S3.p8.8.8.8.m8.1.2.2.2.cmml"><mi id="S3.p8.8.8.8.m8.1.2.2.2.2" mathcolor="#000099" mathvariant="normal" xref="S3.p8.8.8.8.m8.1.2.2.2.2.cmml">Φ</mi><mi id="S3.p8.8.8.8.m8.1.2.2.2.3" mathcolor="#000099" mathvariant="normal" xref="S3.p8.8.8.8.m8.1.2.2.2.3.cmml">s</mi></msub><mo id="S3.p8.8.8.8.m8.1.2.2.1" xref="S3.p8.8.8.8.m8.1.2.2.1.cmml">⁢</mo><mrow id="S3.p8.8.8.8.m8.1.2.2.3.2" xref="S3.p8.8.8.8.m8.1.2.2.cmml"><mo id="S3.p8.8.8.8.m8.1.2.2.3.2.1" mathcolor="#000099" stretchy="false" xref="S3.p8.8.8.8.m8.1.2.2.cmml">(</mo><mn id="S3.p8.8.8.8.m8.1.1" mathcolor="#000099" xref="S3.p8.8.8.8.m8.1.1.cmml">0</mn><mo id="S3.p8.8.8.8.m8.1.2.2.3.2.2" mathcolor="#000099" stretchy="false" xref="S3.p8.8.8.8.m8.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.p8.8.8.8.m8.1.2.1" mathcolor="#000099" xref="S3.p8.8.8.8.m8.1.2.1.cmml">=</mo><mn id="S3.p8.8.8.8.m8.1.2.3" mathcolor="#000099" xref="S3.p8.8.8.8.m8.1.2.3.cmml">𝟎</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p8.8.8.8.m8.1b"><apply id="S3.p8.8.8.8.m8.1.2.cmml" xref="S3.p8.8.8.8.m8.1.2"><eq id="S3.p8.8.8.8.m8.1.2.1.cmml" xref="S3.p8.8.8.8.m8.1.2.1"></eq><apply id="S3.p8.8.8.8.m8.1.2.2.cmml" xref="S3.p8.8.8.8.m8.1.2.2"><times id="S3.p8.8.8.8.m8.1.2.2.1.cmml" xref="S3.p8.8.8.8.m8.1.2.2.1"></times><apply id="S3.p8.8.8.8.m8.1.2.2.2.cmml" xref="S3.p8.8.8.8.m8.1.2.2.2"><csymbol cd="ambiguous" id="S3.p8.8.8.8.m8.1.2.2.2.1.cmml" xref="S3.p8.8.8.8.m8.1.2.2.2">subscript</csymbol><ci id="S3.p8.8.8.8.m8.1.2.2.2.2.cmml" xref="S3.p8.8.8.8.m8.1.2.2.2.2">Φ</ci><ci id="S3.p8.8.8.8.m8.1.2.2.2.3.cmml" xref="S3.p8.8.8.8.m8.1.2.2.2.3">s</ci></apply><cn id="S3.p8.8.8.8.m8.1.1.cmml" type="integer" xref="S3.p8.8.8.8.m8.1.1">0</cn></apply><cn id="S3.p8.8.8.8.m8.1.2.3.cmml" type="integer" xref="S3.p8.8.8.8.m8.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.8.8.8.m8.1c">{\Phi}_{\rm s}(0)=\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S3.p8.8.8.8.m8.1d">roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( 0 ) = bold_0</annotation></semantics></math> are applied to simplify the derivations without loss of generality, which ensures <math alttext="\bm{\epsilon}_{\rm f}(t)\equiv\bm{0}" class="ltx_Math" display="inline" id="S3.p8.9.9.9.m9.1"><semantics id="S3.p8.9.9.9.m9.1a"><mrow id="S3.p8.9.9.9.m9.1.2" xref="S3.p8.9.9.9.m9.1.2.cmml"><mrow id="S3.p8.9.9.9.m9.1.2.2" xref="S3.p8.9.9.9.m9.1.2.2.cmml"><msub id="S3.p8.9.9.9.m9.1.2.2.2" xref="S3.p8.9.9.9.m9.1.2.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S3.p8.9.9.9.m9.1.2.2.2.2" mathcolor="#000099" mathvariant="bold-italic" xref="S3.p8.9.9.9.m9.1.2.2.2.2.cmml">ϵ</mi><mi id="S3.p8.9.9.9.m9.1.2.2.2.3" mathcolor="#000099" mathvariant="normal" xref="S3.p8.9.9.9.m9.1.2.2.2.3.cmml">f</mi></msub><mo id="S3.p8.9.9.9.m9.1.2.2.1" xref="S3.p8.9.9.9.m9.1.2.2.1.cmml">⁢</mo><mrow id="S3.p8.9.9.9.m9.1.2.2.3.2" xref="S3.p8.9.9.9.m9.1.2.2.cmml"><mo id="S3.p8.9.9.9.m9.1.2.2.3.2.1" mathcolor="#000099" stretchy="false" xref="S3.p8.9.9.9.m9.1.2.2.cmml">(</mo><mi id="S3.p8.9.9.9.m9.1.1" mathcolor="#000099" xref="S3.p8.9.9.9.m9.1.1.cmml">t</mi><mo id="S3.p8.9.9.9.m9.1.2.2.3.2.2" mathcolor="#000099" stretchy="false" xref="S3.p8.9.9.9.m9.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.p8.9.9.9.m9.1.2.1" mathcolor="#000099" xref="S3.p8.9.9.9.m9.1.2.1.cmml">≡</mo><mn id="S3.p8.9.9.9.m9.1.2.3" mathcolor="#000099" xref="S3.p8.9.9.9.m9.1.2.3.cmml">𝟎</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p8.9.9.9.m9.1b"><apply id="S3.p8.9.9.9.m9.1.2.cmml" xref="S3.p8.9.9.9.m9.1.2"><equivalent id="S3.p8.9.9.9.m9.1.2.1.cmml" xref="S3.p8.9.9.9.m9.1.2.1"></equivalent><apply id="S3.p8.9.9.9.m9.1.2.2.cmml" xref="S3.p8.9.9.9.m9.1.2.2"><times id="S3.p8.9.9.9.m9.1.2.2.1.cmml" xref="S3.p8.9.9.9.m9.1.2.2.1"></times><apply id="S3.p8.9.9.9.m9.1.2.2.2.cmml" xref="S3.p8.9.9.9.m9.1.2.2.2"><csymbol cd="ambiguous" id="S3.p8.9.9.9.m9.1.2.2.2.1.cmml" xref="S3.p8.9.9.9.m9.1.2.2.2">subscript</csymbol><ci id="S3.p8.9.9.9.m9.1.2.2.2.2.cmml" xref="S3.p8.9.9.9.m9.1.2.2.2.2">bold-italic-ϵ</ci><ci id="S3.p8.9.9.9.m9.1.2.2.2.3.cmml" xref="S3.p8.9.9.9.m9.1.2.2.2.3">f</ci></apply><ci id="S3.p8.9.9.9.m9.1.1.cmml" xref="S3.p8.9.9.9.m9.1.1">𝑡</ci></apply><cn id="S3.p8.9.9.9.m9.1.2.3.cmml" type="integer" xref="S3.p8.9.9.9.m9.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.9.9.9.m9.1c">\bm{\epsilon}_{\rm f}(t)\equiv\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S3.p8.9.9.9.m9.1d">bold_italic_ϵ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ( italic_t ) ≡ bold_0</annotation></semantics></math> to eliminate the disturbing effect in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E12" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">12</span></a>) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib8" title="">8</a>, Sec. 6.5]</cite>. It is worth noting that the regressor <math alttext="\Phi" class="ltx_Math" display="inline" id="S3.p8.10.10.10.m10.1"><semantics id="S3.p8.10.10.10.m10.1a"><mi id="S3.p8.10.10.10.m10.1.1" mathcolor="#000099" mathvariant="normal" xref="S3.p8.10.10.10.m10.1.1.cmml">Φ</mi><annotation-xml encoding="MathML-Content" id="S3.p8.10.10.10.m10.1b"><ci id="S3.p8.10.10.10.m10.1.1.cmml" xref="S3.p8.10.10.10.m10.1.1">Φ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.10.10.10.m10.1c">\Phi</annotation><annotation encoding="application/x-llamapun" id="S3.p8.10.10.10.m10.1d">roman_Φ</annotation></semantics></math> is explicitly a function of <math alttext="\bm{x}" class="ltx_Math" display="inline" id="S3.p8.11.11.11.m11.1"><semantics id="S3.p8.11.11.11.m11.1a"><mi id="S3.p8.11.11.11.m11.1.1" mathcolor="#000099" xref="S3.p8.11.11.11.m11.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S3.p8.11.11.11.m11.1b"><ci id="S3.p8.11.11.11.m11.1.1.cmml" xref="S3.p8.11.11.11.m11.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.11.11.11.m11.1c">\bm{x}</annotation><annotation encoding="application/x-llamapun" id="S3.p8.11.11.11.m11.1d">bold_italic_x</annotation></semantics></math>, <math alttext="\hat{\bm{\theta}}" class="ltx_Math" display="inline" id="S3.p8.12.12.12.m12.1"><semantics id="S3.p8.12.12.12.m12.1a"><mover accent="true" id="S3.p8.12.12.12.m12.1.1" xref="S3.p8.12.12.12.m12.1.1.cmml"><mi id="S3.p8.12.12.12.m12.1.1.2" mathcolor="#000099" xref="S3.p8.12.12.12.m12.1.1.2.cmml">𝜽</mi><mo id="S3.p8.12.12.12.m12.1.1.1" mathcolor="#000099" xref="S3.p8.12.12.12.m12.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S3.p8.12.12.12.m12.1b"><apply id="S3.p8.12.12.12.m12.1.1.cmml" xref="S3.p8.12.12.12.m12.1.1"><ci id="S3.p8.12.12.12.m12.1.1.1.cmml" xref="S3.p8.12.12.12.m12.1.1.1">^</ci><ci id="S3.p8.12.12.12.m12.1.1.2.cmml" xref="S3.p8.12.12.12.m12.1.1.2">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.12.12.12.m12.1c">\hat{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S3.p8.12.12.12.m12.1d">over^ start_ARG bold_italic_θ end_ARG</annotation></semantics></math>, and <math alttext="{\bm{y}}_{{\rm r}n}" class="ltx_Math" display="inline" id="S3.p8.13.13.13.m13.1"><semantics id="S3.p8.13.13.13.m13.1a"><msub id="S3.p8.13.13.13.m13.1.1" xref="S3.p8.13.13.13.m13.1.1.cmml"><mi id="S3.p8.13.13.13.m13.1.1.2" mathcolor="#000099" xref="S3.p8.13.13.13.m13.1.1.2.cmml">𝒚</mi><mrow id="S3.p8.13.13.13.m13.1.1.3" xref="S3.p8.13.13.13.m13.1.1.3.cmml"><mi id="S3.p8.13.13.13.m13.1.1.3.2" mathcolor="#000099" mathvariant="normal" xref="S3.p8.13.13.13.m13.1.1.3.2.cmml">r</mi><mo id="S3.p8.13.13.13.m13.1.1.3.1" xref="S3.p8.13.13.13.m13.1.1.3.1.cmml">⁢</mo><mi id="S3.p8.13.13.13.m13.1.1.3.3" mathcolor="#000099" xref="S3.p8.13.13.13.m13.1.1.3.3.cmml">n</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.p8.13.13.13.m13.1b"><apply id="S3.p8.13.13.13.m13.1.1.cmml" xref="S3.p8.13.13.13.m13.1.1"><csymbol cd="ambiguous" id="S3.p8.13.13.13.m13.1.1.1.cmml" xref="S3.p8.13.13.13.m13.1.1">subscript</csymbol><ci id="S3.p8.13.13.13.m13.1.1.2.cmml" xref="S3.p8.13.13.13.m13.1.1.2">𝒚</ci><apply id="S3.p8.13.13.13.m13.1.1.3.cmml" xref="S3.p8.13.13.13.m13.1.1.3"><times id="S3.p8.13.13.13.m13.1.1.3.1.cmml" xref="S3.p8.13.13.13.m13.1.1.3.1"></times><ci id="S3.p8.13.13.13.m13.1.1.3.2.cmml" xref="S3.p8.13.13.13.m13.1.1.3.2">r</ci><ci id="S3.p8.13.13.13.m13.1.1.3.3.cmml" xref="S3.p8.13.13.13.m13.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.13.13.13.m13.1c">{\bm{y}}_{{\rm r}n}</annotation><annotation encoding="application/x-llamapun" id="S3.p8.13.13.13.m13.1d">bold_italic_y start_POSTSUBSCRIPT roman_r italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, and similarly, <math alttext="\Phi_{\rm s}" class="ltx_Math" display="inline" id="S3.p8.14.14.14.m14.1"><semantics id="S3.p8.14.14.14.m14.1a"><msub id="S3.p8.14.14.14.m14.1.1" xref="S3.p8.14.14.14.m14.1.1.cmml"><mi id="S3.p8.14.14.14.m14.1.1.2" mathcolor="#000099" mathvariant="normal" xref="S3.p8.14.14.14.m14.1.1.2.cmml">Φ</mi><mi id="S3.p8.14.14.14.m14.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S3.p8.14.14.14.m14.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p8.14.14.14.m14.1b"><apply id="S3.p8.14.14.14.m14.1.1.cmml" xref="S3.p8.14.14.14.m14.1.1"><csymbol cd="ambiguous" id="S3.p8.14.14.14.m14.1.1.1.cmml" xref="S3.p8.14.14.14.m14.1.1">subscript</csymbol><ci id="S3.p8.14.14.14.m14.1.1.2.cmml" xref="S3.p8.14.14.14.m14.1.1.2">Φ</ci><ci id="S3.p8.14.14.14.m14.1.1.3.cmml" xref="S3.p8.14.14.14.m14.1.1.3">s</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.14.14.14.m14.1c">\Phi_{\rm s}</annotation><annotation encoding="application/x-llamapun" id="S3.p8.14.14.14.m14.1d">roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E12" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">12</span></a>) also depends on these variables. Thus, excitation is primarily provided by the transients of the tracking error <math alttext="\bm{e}" class="ltx_Math" display="inline" id="S3.p8.15.15.15.m15.1"><semantics id="S3.p8.15.15.15.m15.1a"><mi id="S3.p8.15.15.15.m15.1.1" mathcolor="#000099" xref="S3.p8.15.15.15.m15.1.1.cmml">𝒆</mi><annotation-xml encoding="MathML-Content" id="S3.p8.15.15.15.m15.1b"><ci id="S3.p8.15.15.15.m15.1.1.cmml" xref="S3.p8.15.15.15.m15.1.1">𝒆</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.15.15.15.m15.1c">\bm{e}</annotation><annotation encoding="application/x-llamapun" id="S3.p8.15.15.15.m15.1d">bold_italic_e</annotation></semantics></math> and the estimation error <math alttext="\tilde{\bm{\theta}}" class="ltx_Math" display="inline" id="S3.p8.16.16.16.m16.1"><semantics id="S3.p8.16.16.16.m16.1a"><mover accent="true" id="S3.p8.16.16.16.m16.1.1" xref="S3.p8.16.16.16.m16.1.1.cmml"><mi id="S3.p8.16.16.16.m16.1.1.2" mathcolor="#000099" xref="S3.p8.16.16.16.m16.1.1.2.cmml">𝜽</mi><mo id="S3.p8.16.16.16.m16.1.1.1" mathcolor="#000099" xref="S3.p8.16.16.16.m16.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S3.p8.16.16.16.m16.1b"><apply id="S3.p8.16.16.16.m16.1.1.cmml" xref="S3.p8.16.16.16.m16.1.1"><ci id="S3.p8.16.16.16.m16.1.1.1.cmml" xref="S3.p8.16.16.16.m16.1.1.1">~</ci><ci id="S3.p8.16.16.16.m16.1.1.2.cmml" xref="S3.p8.16.16.16.m16.1.1.2">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.16.16.16.m16.1c">\tilde{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S3.p8.16.16.16.m16.1d">over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math>, rather than by the exponential decay term <math alttext="{\bm{\epsilon}}_{\rm f}(t)" class="ltx_Math" display="inline" id="S3.p8.17.17.17.m17.1"><semantics id="S3.p8.17.17.17.m17.1a"><mrow id="S3.p8.17.17.17.m17.1.2" xref="S3.p8.17.17.17.m17.1.2.cmml"><msub id="S3.p8.17.17.17.m17.1.2.2" xref="S3.p8.17.17.17.m17.1.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S3.p8.17.17.17.m17.1.2.2.2" mathcolor="#000099" mathvariant="bold-italic" xref="S3.p8.17.17.17.m17.1.2.2.2.cmml">ϵ</mi><mi id="S3.p8.17.17.17.m17.1.2.2.3" mathcolor="#000099" mathvariant="normal" xref="S3.p8.17.17.17.m17.1.2.2.3.cmml">f</mi></msub><mo id="S3.p8.17.17.17.m17.1.2.1" xref="S3.p8.17.17.17.m17.1.2.1.cmml">⁢</mo><mrow id="S3.p8.17.17.17.m17.1.2.3.2" xref="S3.p8.17.17.17.m17.1.2.cmml"><mo id="S3.p8.17.17.17.m17.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="S3.p8.17.17.17.m17.1.2.cmml">(</mo><mi id="S3.p8.17.17.17.m17.1.1" mathcolor="#000099" xref="S3.p8.17.17.17.m17.1.1.cmml">t</mi><mo id="S3.p8.17.17.17.m17.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="S3.p8.17.17.17.m17.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p8.17.17.17.m17.1b"><apply id="S3.p8.17.17.17.m17.1.2.cmml" xref="S3.p8.17.17.17.m17.1.2"><times id="S3.p8.17.17.17.m17.1.2.1.cmml" xref="S3.p8.17.17.17.m17.1.2.1"></times><apply id="S3.p8.17.17.17.m17.1.2.2.cmml" xref="S3.p8.17.17.17.m17.1.2.2"><csymbol cd="ambiguous" id="S3.p8.17.17.17.m17.1.2.2.1.cmml" xref="S3.p8.17.17.17.m17.1.2.2">subscript</csymbol><ci id="S3.p8.17.17.17.m17.1.2.2.2.cmml" xref="S3.p8.17.17.17.m17.1.2.2.2">bold-italic-ϵ</ci><ci id="S3.p8.17.17.17.m17.1.2.2.3.cmml" xref="S3.p8.17.17.17.m17.1.2.2.3">f</ci></apply><ci id="S3.p8.17.17.17.m17.1.1.cmml" xref="S3.p8.17.17.17.m17.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.17.17.17.m17.1c">{\bm{\epsilon}}_{\rm f}(t)</annotation><annotation encoding="application/x-llamapun" id="S3.p8.17.17.17.m17.1d">bold_italic_ϵ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math>.</span></span></p> </div> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">IV </span><span class="ltx_text ltx_font_smallcaps" id="S4.1.1">Composite Learning Design</span> </h2> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S4.SS1.4.1.1">IV-A</span> </span><span class="ltx_text ltx_font_italic" id="S4.SS1.5.2">Composite Learning High-Order Tuner</span> </h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.16">For convenience, let <math alttext="\Phi_{{\rm s},\zeta}" class="ltx_Math" display="inline" id="S4.SS1.p1.1.m1.2"><semantics id="S4.SS1.p1.1.m1.2a"><msub id="S4.SS1.p1.1.m1.2.3" xref="S4.SS1.p1.1.m1.2.3.cmml"><mi id="S4.SS1.p1.1.m1.2.3.2" mathvariant="normal" xref="S4.SS1.p1.1.m1.2.3.2.cmml">Φ</mi><mrow id="S4.SS1.p1.1.m1.2.2.2.4" xref="S4.SS1.p1.1.m1.2.2.2.3.cmml"><mi id="S4.SS1.p1.1.m1.1.1.1.1" mathvariant="normal" xref="S4.SS1.p1.1.m1.1.1.1.1.cmml">s</mi><mo id="S4.SS1.p1.1.m1.2.2.2.4.1" xref="S4.SS1.p1.1.m1.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p1.1.m1.2.2.2.2" xref="S4.SS1.p1.1.m1.2.2.2.2.cmml">ζ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.1.m1.2b"><apply id="S4.SS1.p1.1.m1.2.3.cmml" xref="S4.SS1.p1.1.m1.2.3"><csymbol cd="ambiguous" id="S4.SS1.p1.1.m1.2.3.1.cmml" xref="S4.SS1.p1.1.m1.2.3">subscript</csymbol><ci id="S4.SS1.p1.1.m1.2.3.2.cmml" xref="S4.SS1.p1.1.m1.2.3.2">Φ</ci><list id="S4.SS1.p1.1.m1.2.2.2.3.cmml" xref="S4.SS1.p1.1.m1.2.2.2.4"><ci id="S4.SS1.p1.1.m1.1.1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1.1.1">s</ci><ci id="S4.SS1.p1.1.m1.2.2.2.2.cmml" xref="S4.SS1.p1.1.m1.2.2.2.2">𝜁</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.1.m1.2c">\Phi_{{\rm s},\zeta}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.1.m1.2d">roman_Φ start_POSTSUBSCRIPT roman_s , italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S4.SS1.p1.2.m2.1"><semantics id="S4.SS1.p1.2.m2.1a"><mo id="S4.SS1.p1.2.m2.1.1" xref="S4.SS1.p1.2.m2.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.2.m2.1b"><csymbol cd="latexml" id="S4.SS1.p1.2.m2.1.1.cmml" xref="S4.SS1.p1.2.m2.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.2.m2.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.2.m2.1d">:=</annotation></semantics></math> <math alttext="[\bm{\phi}_{{\rm s},k_{1}},\bm{\phi}_{{\rm s},k_{2}},\cdots,\bm{\phi}_{{\rm s}% ,k_{N_{\zeta}}}]^{T}" class="ltx_Math" display="inline" id="S4.SS1.p1.3.m3.10"><semantics id="S4.SS1.p1.3.m3.10a"><msup id="S4.SS1.p1.3.m3.10.10" xref="S4.SS1.p1.3.m3.10.10.cmml"><mrow id="S4.SS1.p1.3.m3.10.10.3.3" xref="S4.SS1.p1.3.m3.10.10.3.4.cmml"><mo id="S4.SS1.p1.3.m3.10.10.3.3.4" stretchy="false" xref="S4.SS1.p1.3.m3.10.10.3.4.cmml">[</mo><msub id="S4.SS1.p1.3.m3.8.8.1.1.1" xref="S4.SS1.p1.3.m3.8.8.1.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS1.p1.3.m3.8.8.1.1.1.2" mathvariant="bold-italic" xref="S4.SS1.p1.3.m3.8.8.1.1.1.2.cmml">ϕ</mi><mrow id="S4.SS1.p1.3.m3.2.2.2.2" xref="S4.SS1.p1.3.m3.2.2.2.3.cmml"><mi id="S4.SS1.p1.3.m3.1.1.1.1" mathvariant="normal" xref="S4.SS1.p1.3.m3.1.1.1.1.cmml">s</mi><mo id="S4.SS1.p1.3.m3.2.2.2.2.2" xref="S4.SS1.p1.3.m3.2.2.2.3.cmml">,</mo><msub id="S4.SS1.p1.3.m3.2.2.2.2.1" xref="S4.SS1.p1.3.m3.2.2.2.2.1.cmml"><mi id="S4.SS1.p1.3.m3.2.2.2.2.1.2" xref="S4.SS1.p1.3.m3.2.2.2.2.1.2.cmml">k</mi><mn id="S4.SS1.p1.3.m3.2.2.2.2.1.3" xref="S4.SS1.p1.3.m3.2.2.2.2.1.3.cmml">1</mn></msub></mrow></msub><mo id="S4.SS1.p1.3.m3.10.10.3.3.5" xref="S4.SS1.p1.3.m3.10.10.3.4.cmml">,</mo><msub id="S4.SS1.p1.3.m3.9.9.2.2.2" xref="S4.SS1.p1.3.m3.9.9.2.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS1.p1.3.m3.9.9.2.2.2.2" mathvariant="bold-italic" xref="S4.SS1.p1.3.m3.9.9.2.2.2.2.cmml">ϕ</mi><mrow id="S4.SS1.p1.3.m3.4.4.2.2" xref="S4.SS1.p1.3.m3.4.4.2.3.cmml"><mi id="S4.SS1.p1.3.m3.3.3.1.1" mathvariant="normal" xref="S4.SS1.p1.3.m3.3.3.1.1.cmml">s</mi><mo id="S4.SS1.p1.3.m3.4.4.2.2.2" xref="S4.SS1.p1.3.m3.4.4.2.3.cmml">,</mo><msub id="S4.SS1.p1.3.m3.4.4.2.2.1" xref="S4.SS1.p1.3.m3.4.4.2.2.1.cmml"><mi id="S4.SS1.p1.3.m3.4.4.2.2.1.2" xref="S4.SS1.p1.3.m3.4.4.2.2.1.2.cmml">k</mi><mn id="S4.SS1.p1.3.m3.4.4.2.2.1.3" xref="S4.SS1.p1.3.m3.4.4.2.2.1.3.cmml">2</mn></msub></mrow></msub><mo id="S4.SS1.p1.3.m3.10.10.3.3.6" xref="S4.SS1.p1.3.m3.10.10.3.4.cmml">,</mo><mi id="S4.SS1.p1.3.m3.7.7" mathvariant="normal" xref="S4.SS1.p1.3.m3.7.7.cmml">⋯</mi><mo id="S4.SS1.p1.3.m3.10.10.3.3.7" xref="S4.SS1.p1.3.m3.10.10.3.4.cmml">,</mo><msub id="S4.SS1.p1.3.m3.10.10.3.3.3" xref="S4.SS1.p1.3.m3.10.10.3.3.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS1.p1.3.m3.10.10.3.3.3.2" mathvariant="bold-italic" xref="S4.SS1.p1.3.m3.10.10.3.3.3.2.cmml">ϕ</mi><mrow id="S4.SS1.p1.3.m3.6.6.2.2" xref="S4.SS1.p1.3.m3.6.6.2.3.cmml"><mi id="S4.SS1.p1.3.m3.5.5.1.1" mathvariant="normal" xref="S4.SS1.p1.3.m3.5.5.1.1.cmml">s</mi><mo id="S4.SS1.p1.3.m3.6.6.2.2.2" xref="S4.SS1.p1.3.m3.6.6.2.3.cmml">,</mo><msub id="S4.SS1.p1.3.m3.6.6.2.2.1" xref="S4.SS1.p1.3.m3.6.6.2.2.1.cmml"><mi id="S4.SS1.p1.3.m3.6.6.2.2.1.2" xref="S4.SS1.p1.3.m3.6.6.2.2.1.2.cmml">k</mi><msub id="S4.SS1.p1.3.m3.6.6.2.2.1.3" xref="S4.SS1.p1.3.m3.6.6.2.2.1.3.cmml"><mi id="S4.SS1.p1.3.m3.6.6.2.2.1.3.2" xref="S4.SS1.p1.3.m3.6.6.2.2.1.3.2.cmml">N</mi><mi id="S4.SS1.p1.3.m3.6.6.2.2.1.3.3" xref="S4.SS1.p1.3.m3.6.6.2.2.1.3.3.cmml">ζ</mi></msub></msub></mrow></msub><mo id="S4.SS1.p1.3.m3.10.10.3.3.8" stretchy="false" xref="S4.SS1.p1.3.m3.10.10.3.4.cmml">]</mo></mrow><mi id="S4.SS1.p1.3.m3.10.10.5" xref="S4.SS1.p1.3.m3.10.10.5.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.3.m3.10b"><apply id="S4.SS1.p1.3.m3.10.10.cmml" xref="S4.SS1.p1.3.m3.10.10"><csymbol cd="ambiguous" id="S4.SS1.p1.3.m3.10.10.4.cmml" xref="S4.SS1.p1.3.m3.10.10">superscript</csymbol><list id="S4.SS1.p1.3.m3.10.10.3.4.cmml" xref="S4.SS1.p1.3.m3.10.10.3.3"><apply id="S4.SS1.p1.3.m3.8.8.1.1.1.cmml" xref="S4.SS1.p1.3.m3.8.8.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.3.m3.8.8.1.1.1.1.cmml" xref="S4.SS1.p1.3.m3.8.8.1.1.1">subscript</csymbol><ci id="S4.SS1.p1.3.m3.8.8.1.1.1.2.cmml" xref="S4.SS1.p1.3.m3.8.8.1.1.1.2">bold-italic-ϕ</ci><list id="S4.SS1.p1.3.m3.2.2.2.3.cmml" xref="S4.SS1.p1.3.m3.2.2.2.2"><ci id="S4.SS1.p1.3.m3.1.1.1.1.cmml" xref="S4.SS1.p1.3.m3.1.1.1.1">s</ci><apply id="S4.SS1.p1.3.m3.2.2.2.2.1.cmml" xref="S4.SS1.p1.3.m3.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.SS1.p1.3.m3.2.2.2.2.1.1.cmml" xref="S4.SS1.p1.3.m3.2.2.2.2.1">subscript</csymbol><ci id="S4.SS1.p1.3.m3.2.2.2.2.1.2.cmml" xref="S4.SS1.p1.3.m3.2.2.2.2.1.2">𝑘</ci><cn id="S4.SS1.p1.3.m3.2.2.2.2.1.3.cmml" type="integer" xref="S4.SS1.p1.3.m3.2.2.2.2.1.3">1</cn></apply></list></apply><apply id="S4.SS1.p1.3.m3.9.9.2.2.2.cmml" xref="S4.SS1.p1.3.m3.9.9.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.p1.3.m3.9.9.2.2.2.1.cmml" xref="S4.SS1.p1.3.m3.9.9.2.2.2">subscript</csymbol><ci id="S4.SS1.p1.3.m3.9.9.2.2.2.2.cmml" xref="S4.SS1.p1.3.m3.9.9.2.2.2.2">bold-italic-ϕ</ci><list id="S4.SS1.p1.3.m3.4.4.2.3.cmml" xref="S4.SS1.p1.3.m3.4.4.2.2"><ci id="S4.SS1.p1.3.m3.3.3.1.1.cmml" xref="S4.SS1.p1.3.m3.3.3.1.1">s</ci><apply id="S4.SS1.p1.3.m3.4.4.2.2.1.cmml" xref="S4.SS1.p1.3.m3.4.4.2.2.1"><csymbol cd="ambiguous" id="S4.SS1.p1.3.m3.4.4.2.2.1.1.cmml" xref="S4.SS1.p1.3.m3.4.4.2.2.1">subscript</csymbol><ci id="S4.SS1.p1.3.m3.4.4.2.2.1.2.cmml" xref="S4.SS1.p1.3.m3.4.4.2.2.1.2">𝑘</ci><cn id="S4.SS1.p1.3.m3.4.4.2.2.1.3.cmml" type="integer" xref="S4.SS1.p1.3.m3.4.4.2.2.1.3">2</cn></apply></list></apply><ci id="S4.SS1.p1.3.m3.7.7.cmml" xref="S4.SS1.p1.3.m3.7.7">⋯</ci><apply id="S4.SS1.p1.3.m3.10.10.3.3.3.cmml" xref="S4.SS1.p1.3.m3.10.10.3.3.3"><csymbol cd="ambiguous" id="S4.SS1.p1.3.m3.10.10.3.3.3.1.cmml" xref="S4.SS1.p1.3.m3.10.10.3.3.3">subscript</csymbol><ci id="S4.SS1.p1.3.m3.10.10.3.3.3.2.cmml" xref="S4.SS1.p1.3.m3.10.10.3.3.3.2">bold-italic-ϕ</ci><list id="S4.SS1.p1.3.m3.6.6.2.3.cmml" xref="S4.SS1.p1.3.m3.6.6.2.2"><ci id="S4.SS1.p1.3.m3.5.5.1.1.cmml" xref="S4.SS1.p1.3.m3.5.5.1.1">s</ci><apply id="S4.SS1.p1.3.m3.6.6.2.2.1.cmml" xref="S4.SS1.p1.3.m3.6.6.2.2.1"><csymbol cd="ambiguous" id="S4.SS1.p1.3.m3.6.6.2.2.1.1.cmml" xref="S4.SS1.p1.3.m3.6.6.2.2.1">subscript</csymbol><ci id="S4.SS1.p1.3.m3.6.6.2.2.1.2.cmml" xref="S4.SS1.p1.3.m3.6.6.2.2.1.2">𝑘</ci><apply id="S4.SS1.p1.3.m3.6.6.2.2.1.3.cmml" xref="S4.SS1.p1.3.m3.6.6.2.2.1.3"><csymbol cd="ambiguous" id="S4.SS1.p1.3.m3.6.6.2.2.1.3.1.cmml" xref="S4.SS1.p1.3.m3.6.6.2.2.1.3">subscript</csymbol><ci id="S4.SS1.p1.3.m3.6.6.2.2.1.3.2.cmml" xref="S4.SS1.p1.3.m3.6.6.2.2.1.3.2">𝑁</ci><ci id="S4.SS1.p1.3.m3.6.6.2.2.1.3.3.cmml" xref="S4.SS1.p1.3.m3.6.6.2.2.1.3.3">𝜁</ci></apply></apply></list></apply></list><ci id="S4.SS1.p1.3.m3.10.10.5.cmml" xref="S4.SS1.p1.3.m3.10.10.5">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.3.m3.10c">[\bm{\phi}_{{\rm s},k_{1}},\bm{\phi}_{{\rm s},k_{2}},\cdots,\bm{\phi}_{{\rm s}% ,k_{N_{\zeta}}}]^{T}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.3.m3.10d">[ bold_italic_ϕ start_POSTSUBSCRIPT roman_s , italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , bold_italic_ϕ start_POSTSUBSCRIPT roman_s , italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , ⋯ , bold_italic_ϕ start_POSTSUBSCRIPT roman_s , italic_k start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> <math alttext="\in\mathbb{R}^{N_{\zeta}\times n}" class="ltx_Math" display="inline" id="S4.SS1.p1.4.m4.1"><semantics id="S4.SS1.p1.4.m4.1a"><mrow id="S4.SS1.p1.4.m4.1.1" xref="S4.SS1.p1.4.m4.1.1.cmml"><mi id="S4.SS1.p1.4.m4.1.1.2" xref="S4.SS1.p1.4.m4.1.1.2.cmml"></mi><mo id="S4.SS1.p1.4.m4.1.1.1" xref="S4.SS1.p1.4.m4.1.1.1.cmml">∈</mo><msup id="S4.SS1.p1.4.m4.1.1.3" xref="S4.SS1.p1.4.m4.1.1.3.cmml"><mi id="S4.SS1.p1.4.m4.1.1.3.2" xref="S4.SS1.p1.4.m4.1.1.3.2.cmml">ℝ</mi><mrow id="S4.SS1.p1.4.m4.1.1.3.3" xref="S4.SS1.p1.4.m4.1.1.3.3.cmml"><msub id="S4.SS1.p1.4.m4.1.1.3.3.2" xref="S4.SS1.p1.4.m4.1.1.3.3.2.cmml"><mi id="S4.SS1.p1.4.m4.1.1.3.3.2.2" xref="S4.SS1.p1.4.m4.1.1.3.3.2.2.cmml">N</mi><mi id="S4.SS1.p1.4.m4.1.1.3.3.2.3" xref="S4.SS1.p1.4.m4.1.1.3.3.2.3.cmml">ζ</mi></msub><mo id="S4.SS1.p1.4.m4.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S4.SS1.p1.4.m4.1.1.3.3.1.cmml">×</mo><mi id="S4.SS1.p1.4.m4.1.1.3.3.3" xref="S4.SS1.p1.4.m4.1.1.3.3.3.cmml">n</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.4.m4.1b"><apply id="S4.SS1.p1.4.m4.1.1.cmml" xref="S4.SS1.p1.4.m4.1.1"><in id="S4.SS1.p1.4.m4.1.1.1.cmml" xref="S4.SS1.p1.4.m4.1.1.1"></in><csymbol cd="latexml" id="S4.SS1.p1.4.m4.1.1.2.cmml" xref="S4.SS1.p1.4.m4.1.1.2">absent</csymbol><apply id="S4.SS1.p1.4.m4.1.1.3.cmml" xref="S4.SS1.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p1.4.m4.1.1.3.1.cmml" xref="S4.SS1.p1.4.m4.1.1.3">superscript</csymbol><ci id="S4.SS1.p1.4.m4.1.1.3.2.cmml" xref="S4.SS1.p1.4.m4.1.1.3.2">ℝ</ci><apply id="S4.SS1.p1.4.m4.1.1.3.3.cmml" xref="S4.SS1.p1.4.m4.1.1.3.3"><times id="S4.SS1.p1.4.m4.1.1.3.3.1.cmml" xref="S4.SS1.p1.4.m4.1.1.3.3.1"></times><apply id="S4.SS1.p1.4.m4.1.1.3.3.2.cmml" xref="S4.SS1.p1.4.m4.1.1.3.3.2"><csymbol cd="ambiguous" id="S4.SS1.p1.4.m4.1.1.3.3.2.1.cmml" xref="S4.SS1.p1.4.m4.1.1.3.3.2">subscript</csymbol><ci id="S4.SS1.p1.4.m4.1.1.3.3.2.2.cmml" xref="S4.SS1.p1.4.m4.1.1.3.3.2.2">𝑁</ci><ci id="S4.SS1.p1.4.m4.1.1.3.3.2.3.cmml" xref="S4.SS1.p1.4.m4.1.1.3.3.2.3">𝜁</ci></apply><ci id="S4.SS1.p1.4.m4.1.1.3.3.3.cmml" xref="S4.SS1.p1.4.m4.1.1.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.4.m4.1c">\in\mathbb{R}^{N_{\zeta}\times n}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.4.m4.1d">∈ blackboard_R start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT × italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="1\leq k_{j}\leq N" class="ltx_Math" display="inline" id="S4.SS1.p1.5.m5.1"><semantics id="S4.SS1.p1.5.m5.1a"><mrow id="S4.SS1.p1.5.m5.1.1" xref="S4.SS1.p1.5.m5.1.1.cmml"><mn id="S4.SS1.p1.5.m5.1.1.2" xref="S4.SS1.p1.5.m5.1.1.2.cmml">1</mn><mo id="S4.SS1.p1.5.m5.1.1.3" xref="S4.SS1.p1.5.m5.1.1.3.cmml">≤</mo><msub id="S4.SS1.p1.5.m5.1.1.4" xref="S4.SS1.p1.5.m5.1.1.4.cmml"><mi id="S4.SS1.p1.5.m5.1.1.4.2" xref="S4.SS1.p1.5.m5.1.1.4.2.cmml">k</mi><mi id="S4.SS1.p1.5.m5.1.1.4.3" xref="S4.SS1.p1.5.m5.1.1.4.3.cmml">j</mi></msub><mo id="S4.SS1.p1.5.m5.1.1.5" xref="S4.SS1.p1.5.m5.1.1.5.cmml">≤</mo><mi id="S4.SS1.p1.5.m5.1.1.6" xref="S4.SS1.p1.5.m5.1.1.6.cmml">N</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.5.m5.1b"><apply id="S4.SS1.p1.5.m5.1.1.cmml" xref="S4.SS1.p1.5.m5.1.1"><and id="S4.SS1.p1.5.m5.1.1a.cmml" xref="S4.SS1.p1.5.m5.1.1"></and><apply id="S4.SS1.p1.5.m5.1.1b.cmml" xref="S4.SS1.p1.5.m5.1.1"><leq id="S4.SS1.p1.5.m5.1.1.3.cmml" xref="S4.SS1.p1.5.m5.1.1.3"></leq><cn id="S4.SS1.p1.5.m5.1.1.2.cmml" type="integer" xref="S4.SS1.p1.5.m5.1.1.2">1</cn><apply id="S4.SS1.p1.5.m5.1.1.4.cmml" xref="S4.SS1.p1.5.m5.1.1.4"><csymbol cd="ambiguous" id="S4.SS1.p1.5.m5.1.1.4.1.cmml" xref="S4.SS1.p1.5.m5.1.1.4">subscript</csymbol><ci id="S4.SS1.p1.5.m5.1.1.4.2.cmml" xref="S4.SS1.p1.5.m5.1.1.4.2">𝑘</ci><ci id="S4.SS1.p1.5.m5.1.1.4.3.cmml" xref="S4.SS1.p1.5.m5.1.1.4.3">𝑗</ci></apply></apply><apply id="S4.SS1.p1.5.m5.1.1c.cmml" xref="S4.SS1.p1.5.m5.1.1"><leq id="S4.SS1.p1.5.m5.1.1.5.cmml" xref="S4.SS1.p1.5.m5.1.1.5"></leq><share href="https://arxiv.org/html/2401.10785v2#S4.SS1.p1.5.m5.1.1.4.cmml" id="S4.SS1.p1.5.m5.1.1d.cmml" xref="S4.SS1.p1.5.m5.1.1"></share><ci id="S4.SS1.p1.5.m5.1.1.6.cmml" xref="S4.SS1.p1.5.m5.1.1.6">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.5.m5.1c">1\leq k_{j}\leq N</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.5.m5.1d">1 ≤ italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ≤ italic_N</annotation></semantics></math> and <math alttext="j=1" class="ltx_Math" display="inline" id="S4.SS1.p1.6.m6.1"><semantics id="S4.SS1.p1.6.m6.1a"><mrow id="S4.SS1.p1.6.m6.1.1" xref="S4.SS1.p1.6.m6.1.1.cmml"><mi id="S4.SS1.p1.6.m6.1.1.2" xref="S4.SS1.p1.6.m6.1.1.2.cmml">j</mi><mo id="S4.SS1.p1.6.m6.1.1.1" xref="S4.SS1.p1.6.m6.1.1.1.cmml">=</mo><mn id="S4.SS1.p1.6.m6.1.1.3" xref="S4.SS1.p1.6.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.6.m6.1b"><apply id="S4.SS1.p1.6.m6.1.1.cmml" xref="S4.SS1.p1.6.m6.1.1"><eq id="S4.SS1.p1.6.m6.1.1.1.cmml" xref="S4.SS1.p1.6.m6.1.1.1"></eq><ci id="S4.SS1.p1.6.m6.1.1.2.cmml" xref="S4.SS1.p1.6.m6.1.1.2">𝑗</ci><cn id="S4.SS1.p1.6.m6.1.1.3.cmml" type="integer" xref="S4.SS1.p1.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.6.m6.1c">j=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.6.m6.1d">italic_j = 1</annotation></semantics></math> to <math alttext="N_{\zeta}" class="ltx_Math" display="inline" id="S4.SS1.p1.7.m7.1"><semantics id="S4.SS1.p1.7.m7.1a"><msub id="S4.SS1.p1.7.m7.1.1" xref="S4.SS1.p1.7.m7.1.1.cmml"><mi id="S4.SS1.p1.7.m7.1.1.2" xref="S4.SS1.p1.7.m7.1.1.2.cmml">N</mi><mi id="S4.SS1.p1.7.m7.1.1.3" xref="S4.SS1.p1.7.m7.1.1.3.cmml">ζ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.7.m7.1b"><apply id="S4.SS1.p1.7.m7.1.1.cmml" xref="S4.SS1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.7.m7.1.1.1.cmml" xref="S4.SS1.p1.7.m7.1.1">subscript</csymbol><ci id="S4.SS1.p1.7.m7.1.1.2.cmml" xref="S4.SS1.p1.7.m7.1.1.2">𝑁</ci><ci id="S4.SS1.p1.7.m7.1.1.3.cmml" xref="S4.SS1.p1.7.m7.1.1.3">𝜁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.7.m7.1c">N_{\zeta}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.7.m7.1d">italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math> be an active sub-regressor of <math alttext="\Phi_{\rm s}" class="ltx_Math" display="inline" id="S4.SS1.p1.8.m8.1"><semantics id="S4.SS1.p1.8.m8.1a"><msub id="S4.SS1.p1.8.m8.1.1" xref="S4.SS1.p1.8.m8.1.1.cmml"><mi id="S4.SS1.p1.8.m8.1.1.2" mathvariant="normal" xref="S4.SS1.p1.8.m8.1.1.2.cmml">Φ</mi><mi id="S4.SS1.p1.8.m8.1.1.3" mathvariant="normal" xref="S4.SS1.p1.8.m8.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.8.m8.1b"><apply id="S4.SS1.p1.8.m8.1.1.cmml" xref="S4.SS1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.8.m8.1.1.1.cmml" xref="S4.SS1.p1.8.m8.1.1">subscript</csymbol><ci id="S4.SS1.p1.8.m8.1.1.2.cmml" xref="S4.SS1.p1.8.m8.1.1.2">Φ</ci><ci id="S4.SS1.p1.8.m8.1.1.3.cmml" xref="S4.SS1.p1.8.m8.1.1.3">s</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.8.m8.1c">\Phi_{\rm s}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.8.m8.1d">roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E12" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">12</span></a>), <math alttext="N_{\zeta}<N" class="ltx_Math" display="inline" id="S4.SS1.p1.9.m9.1"><semantics id="S4.SS1.p1.9.m9.1a"><mrow id="S4.SS1.p1.9.m9.1.1" xref="S4.SS1.p1.9.m9.1.1.cmml"><msub id="S4.SS1.p1.9.m9.1.1.2" xref="S4.SS1.p1.9.m9.1.1.2.cmml"><mi id="S4.SS1.p1.9.m9.1.1.2.2" xref="S4.SS1.p1.9.m9.1.1.2.2.cmml">N</mi><mi id="S4.SS1.p1.9.m9.1.1.2.3" xref="S4.SS1.p1.9.m9.1.1.2.3.cmml">ζ</mi></msub><mo id="S4.SS1.p1.9.m9.1.1.1" xref="S4.SS1.p1.9.m9.1.1.1.cmml"><</mo><mi id="S4.SS1.p1.9.m9.1.1.3" xref="S4.SS1.p1.9.m9.1.1.3.cmml">N</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.9.m9.1b"><apply id="S4.SS1.p1.9.m9.1.1.cmml" xref="S4.SS1.p1.9.m9.1.1"><lt id="S4.SS1.p1.9.m9.1.1.1.cmml" xref="S4.SS1.p1.9.m9.1.1.1"></lt><apply id="S4.SS1.p1.9.m9.1.1.2.cmml" xref="S4.SS1.p1.9.m9.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p1.9.m9.1.1.2.1.cmml" xref="S4.SS1.p1.9.m9.1.1.2">subscript</csymbol><ci id="S4.SS1.p1.9.m9.1.1.2.2.cmml" xref="S4.SS1.p1.9.m9.1.1.2.2">𝑁</ci><ci id="S4.SS1.p1.9.m9.1.1.2.3.cmml" xref="S4.SS1.p1.9.m9.1.1.2.3">𝜁</ci></apply><ci id="S4.SS1.p1.9.m9.1.1.3.cmml" xref="S4.SS1.p1.9.m9.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.9.m9.1c">N_{\zeta}<N</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.9.m9.1d">italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT < italic_N</annotation></semantics></math> be the number of active channels, and <math alttext="\bm{\psi}_{\zeta,i}" class="ltx_Math" display="inline" id="S4.SS1.p1.10.m10.2"><semantics id="S4.SS1.p1.10.m10.2a"><msub id="S4.SS1.p1.10.m10.2.3" xref="S4.SS1.p1.10.m10.2.3.cmml"><mi id="S4.SS1.p1.10.m10.2.3.2" xref="S4.SS1.p1.10.m10.2.3.2.cmml">𝝍</mi><mrow id="S4.SS1.p1.10.m10.2.2.2.4" xref="S4.SS1.p1.10.m10.2.2.2.3.cmml"><mi id="S4.SS1.p1.10.m10.1.1.1.1" xref="S4.SS1.p1.10.m10.1.1.1.1.cmml">ζ</mi><mo id="S4.SS1.p1.10.m10.2.2.2.4.1" xref="S4.SS1.p1.10.m10.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p1.10.m10.2.2.2.2" xref="S4.SS1.p1.10.m10.2.2.2.2.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.10.m10.2b"><apply id="S4.SS1.p1.10.m10.2.3.cmml" xref="S4.SS1.p1.10.m10.2.3"><csymbol cd="ambiguous" id="S4.SS1.p1.10.m10.2.3.1.cmml" xref="S4.SS1.p1.10.m10.2.3">subscript</csymbol><ci id="S4.SS1.p1.10.m10.2.3.2.cmml" xref="S4.SS1.p1.10.m10.2.3.2">𝝍</ci><list id="S4.SS1.p1.10.m10.2.2.2.3.cmml" xref="S4.SS1.p1.10.m10.2.2.2.4"><ci id="S4.SS1.p1.10.m10.1.1.1.1.cmml" xref="S4.SS1.p1.10.m10.1.1.1.1">𝜁</ci><ci id="S4.SS1.p1.10.m10.2.2.2.2.cmml" xref="S4.SS1.p1.10.m10.2.2.2.2">𝑖</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.10.m10.2c">\bm{\psi}_{\zeta,i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.10.m10.2d">bold_italic_ψ start_POSTSUBSCRIPT italic_ζ , italic_i end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext="\in\mathbb{R}^{N_{\zeta}}" class="ltx_Math" display="inline" id="S4.SS1.p1.11.m11.1"><semantics id="S4.SS1.p1.11.m11.1a"><mrow id="S4.SS1.p1.11.m11.1.1" xref="S4.SS1.p1.11.m11.1.1.cmml"><mi id="S4.SS1.p1.11.m11.1.1.2" xref="S4.SS1.p1.11.m11.1.1.2.cmml"></mi><mo id="S4.SS1.p1.11.m11.1.1.1" xref="S4.SS1.p1.11.m11.1.1.1.cmml">∈</mo><msup id="S4.SS1.p1.11.m11.1.1.3" xref="S4.SS1.p1.11.m11.1.1.3.cmml"><mi id="S4.SS1.p1.11.m11.1.1.3.2" xref="S4.SS1.p1.11.m11.1.1.3.2.cmml">ℝ</mi><msub id="S4.SS1.p1.11.m11.1.1.3.3" xref="S4.SS1.p1.11.m11.1.1.3.3.cmml"><mi id="S4.SS1.p1.11.m11.1.1.3.3.2" xref="S4.SS1.p1.11.m11.1.1.3.3.2.cmml">N</mi><mi id="S4.SS1.p1.11.m11.1.1.3.3.3" xref="S4.SS1.p1.11.m11.1.1.3.3.3.cmml">ζ</mi></msub></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.11.m11.1b"><apply id="S4.SS1.p1.11.m11.1.1.cmml" xref="S4.SS1.p1.11.m11.1.1"><in id="S4.SS1.p1.11.m11.1.1.1.cmml" xref="S4.SS1.p1.11.m11.1.1.1"></in><csymbol cd="latexml" id="S4.SS1.p1.11.m11.1.1.2.cmml" xref="S4.SS1.p1.11.m11.1.1.2">absent</csymbol><apply id="S4.SS1.p1.11.m11.1.1.3.cmml" xref="S4.SS1.p1.11.m11.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p1.11.m11.1.1.3.1.cmml" xref="S4.SS1.p1.11.m11.1.1.3">superscript</csymbol><ci id="S4.SS1.p1.11.m11.1.1.3.2.cmml" xref="S4.SS1.p1.11.m11.1.1.3.2">ℝ</ci><apply id="S4.SS1.p1.11.m11.1.1.3.3.cmml" xref="S4.SS1.p1.11.m11.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS1.p1.11.m11.1.1.3.3.1.cmml" xref="S4.SS1.p1.11.m11.1.1.3.3">subscript</csymbol><ci id="S4.SS1.p1.11.m11.1.1.3.3.2.cmml" xref="S4.SS1.p1.11.m11.1.1.3.3.2">𝑁</ci><ci id="S4.SS1.p1.11.m11.1.1.3.3.3.cmml" xref="S4.SS1.p1.11.m11.1.1.3.3.3">𝜁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.11.m11.1c">\in\mathbb{R}^{N_{\zeta}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.11.m11.1d">∈ blackboard_R start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="i=1" class="ltx_Math" display="inline" id="S4.SS1.p1.12.m12.1"><semantics id="S4.SS1.p1.12.m12.1a"><mrow id="S4.SS1.p1.12.m12.1.1" xref="S4.SS1.p1.12.m12.1.1.cmml"><mi id="S4.SS1.p1.12.m12.1.1.2" xref="S4.SS1.p1.12.m12.1.1.2.cmml">i</mi><mo id="S4.SS1.p1.12.m12.1.1.1" xref="S4.SS1.p1.12.m12.1.1.1.cmml">=</mo><mn id="S4.SS1.p1.12.m12.1.1.3" xref="S4.SS1.p1.12.m12.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.12.m12.1b"><apply id="S4.SS1.p1.12.m12.1.1.cmml" xref="S4.SS1.p1.12.m12.1.1"><eq id="S4.SS1.p1.12.m12.1.1.1.cmml" xref="S4.SS1.p1.12.m12.1.1.1"></eq><ci id="S4.SS1.p1.12.m12.1.1.2.cmml" xref="S4.SS1.p1.12.m12.1.1.2">𝑖</ci><cn id="S4.SS1.p1.12.m12.1.1.3.cmml" type="integer" xref="S4.SS1.p1.12.m12.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.12.m12.1c">i=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.12.m12.1d">italic_i = 1</annotation></semantics></math> to <math alttext="n" class="ltx_Math" display="inline" id="S4.SS1.p1.13.m13.1"><semantics id="S4.SS1.p1.13.m13.1a"><mi id="S4.SS1.p1.13.m13.1.1" xref="S4.SS1.p1.13.m13.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.13.m13.1b"><ci id="S4.SS1.p1.13.m13.1.1.cmml" xref="S4.SS1.p1.13.m13.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.13.m13.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.13.m13.1d">italic_n</annotation></semantics></math> be the <math alttext="i" class="ltx_Math" display="inline" id="S4.SS1.p1.14.m14.1"><semantics id="S4.SS1.p1.14.m14.1a"><mi id="S4.SS1.p1.14.m14.1.1" xref="S4.SS1.p1.14.m14.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.14.m14.1b"><ci id="S4.SS1.p1.14.m14.1.1.cmml" xref="S4.SS1.p1.14.m14.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.14.m14.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.14.m14.1d">italic_i</annotation></semantics></math>th column of <math alttext="\Phi_{{\rm s},\zeta}" class="ltx_Math" display="inline" id="S4.SS1.p1.15.m15.2"><semantics id="S4.SS1.p1.15.m15.2a"><msub id="S4.SS1.p1.15.m15.2.3" xref="S4.SS1.p1.15.m15.2.3.cmml"><mi id="S4.SS1.p1.15.m15.2.3.2" mathvariant="normal" xref="S4.SS1.p1.15.m15.2.3.2.cmml">Φ</mi><mrow id="S4.SS1.p1.15.m15.2.2.2.4" xref="S4.SS1.p1.15.m15.2.2.2.3.cmml"><mi id="S4.SS1.p1.15.m15.1.1.1.1" mathvariant="normal" xref="S4.SS1.p1.15.m15.1.1.1.1.cmml">s</mi><mo id="S4.SS1.p1.15.m15.2.2.2.4.1" xref="S4.SS1.p1.15.m15.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p1.15.m15.2.2.2.2" xref="S4.SS1.p1.15.m15.2.2.2.2.cmml">ζ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.15.m15.2b"><apply id="S4.SS1.p1.15.m15.2.3.cmml" xref="S4.SS1.p1.15.m15.2.3"><csymbol cd="ambiguous" id="S4.SS1.p1.15.m15.2.3.1.cmml" xref="S4.SS1.p1.15.m15.2.3">subscript</csymbol><ci id="S4.SS1.p1.15.m15.2.3.2.cmml" xref="S4.SS1.p1.15.m15.2.3.2">Φ</ci><list id="S4.SS1.p1.15.m15.2.2.2.3.cmml" xref="S4.SS1.p1.15.m15.2.2.2.4"><ci id="S4.SS1.p1.15.m15.1.1.1.1.cmml" xref="S4.SS1.p1.15.m15.1.1.1.1">s</ci><ci id="S4.SS1.p1.15.m15.2.2.2.2.cmml" xref="S4.SS1.p1.15.m15.2.2.2.2">𝜁</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.15.m15.2c">\Phi_{{\rm s},\zeta}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.15.m15.2d">roman_Φ start_POSTSUBSCRIPT roman_s , italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math>. <span class="ltx_text" id="S4.SS1.p1.16.1" style="color:#000099;">The following partial identifiability assumption is given for the composite learning HOT design, such that the parameter <math alttext="\bm{\theta}" class="ltx_Math" display="inline" id="S4.SS1.p1.16.1.m1.1"><semantics id="S4.SS1.p1.16.1.m1.1a"><mi id="S4.SS1.p1.16.1.m1.1.1" mathcolor="#000099" xref="S4.SS1.p1.16.1.m1.1.1.cmml">𝜽</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.16.1.m1.1b"><ci id="S4.SS1.p1.16.1.m1.1.1.cmml" xref="S4.SS1.p1.16.1.m1.1.1">𝜽</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.16.1.m1.1c">\bm{\theta}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.16.1.m1.1d">bold_italic_θ</annotation></semantics></math> in the parametrized model (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E12" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">12</span></a>) is identifiable.</span></p> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.6"><span class="ltx_text ltx_font_italic" id="S4.SS1.p2.6.1">Assumption 3:</span> There exists at least one regressor vector <math alttext="\bm{\psi}_{\zeta,i}\in\mathbb{R}^{N_{\zeta}}" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.2"><semantics id="S4.SS1.p2.1.m1.2a"><mrow id="S4.SS1.p2.1.m1.2.3" xref="S4.SS1.p2.1.m1.2.3.cmml"><msub id="S4.SS1.p2.1.m1.2.3.2" xref="S4.SS1.p2.1.m1.2.3.2.cmml"><mi id="S4.SS1.p2.1.m1.2.3.2.2" xref="S4.SS1.p2.1.m1.2.3.2.2.cmml">𝝍</mi><mrow id="S4.SS1.p2.1.m1.2.2.2.4" xref="S4.SS1.p2.1.m1.2.2.2.3.cmml"><mi id="S4.SS1.p2.1.m1.1.1.1.1" xref="S4.SS1.p2.1.m1.1.1.1.1.cmml">ζ</mi><mo id="S4.SS1.p2.1.m1.2.2.2.4.1" xref="S4.SS1.p2.1.m1.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p2.1.m1.2.2.2.2" xref="S4.SS1.p2.1.m1.2.2.2.2.cmml">i</mi></mrow></msub><mo id="S4.SS1.p2.1.m1.2.3.1" xref="S4.SS1.p2.1.m1.2.3.1.cmml">∈</mo><msup id="S4.SS1.p2.1.m1.2.3.3" xref="S4.SS1.p2.1.m1.2.3.3.cmml"><mi id="S4.SS1.p2.1.m1.2.3.3.2" xref="S4.SS1.p2.1.m1.2.3.3.2.cmml">ℝ</mi><msub id="S4.SS1.p2.1.m1.2.3.3.3" xref="S4.SS1.p2.1.m1.2.3.3.3.cmml"><mi id="S4.SS1.p2.1.m1.2.3.3.3.2" xref="S4.SS1.p2.1.m1.2.3.3.3.2.cmml">N</mi><mi id="S4.SS1.p2.1.m1.2.3.3.3.3" xref="S4.SS1.p2.1.m1.2.3.3.3.3.cmml">ζ</mi></msub></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.2b"><apply id="S4.SS1.p2.1.m1.2.3.cmml" xref="S4.SS1.p2.1.m1.2.3"><in id="S4.SS1.p2.1.m1.2.3.1.cmml" xref="S4.SS1.p2.1.m1.2.3.1"></in><apply id="S4.SS1.p2.1.m1.2.3.2.cmml" xref="S4.SS1.p2.1.m1.2.3.2"><csymbol cd="ambiguous" id="S4.SS1.p2.1.m1.2.3.2.1.cmml" xref="S4.SS1.p2.1.m1.2.3.2">subscript</csymbol><ci id="S4.SS1.p2.1.m1.2.3.2.2.cmml" xref="S4.SS1.p2.1.m1.2.3.2.2">𝝍</ci><list id="S4.SS1.p2.1.m1.2.2.2.3.cmml" xref="S4.SS1.p2.1.m1.2.2.2.4"><ci id="S4.SS1.p2.1.m1.1.1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1.1.1">𝜁</ci><ci id="S4.SS1.p2.1.m1.2.2.2.2.cmml" xref="S4.SS1.p2.1.m1.2.2.2.2">𝑖</ci></list></apply><apply id="S4.SS1.p2.1.m1.2.3.3.cmml" xref="S4.SS1.p2.1.m1.2.3.3"><csymbol cd="ambiguous" id="S4.SS1.p2.1.m1.2.3.3.1.cmml" xref="S4.SS1.p2.1.m1.2.3.3">superscript</csymbol><ci id="S4.SS1.p2.1.m1.2.3.3.2.cmml" xref="S4.SS1.p2.1.m1.2.3.3.2">ℝ</ci><apply id="S4.SS1.p2.1.m1.2.3.3.3.cmml" xref="S4.SS1.p2.1.m1.2.3.3.3"><csymbol cd="ambiguous" id="S4.SS1.p2.1.m1.2.3.3.3.1.cmml" xref="S4.SS1.p2.1.m1.2.3.3.3">subscript</csymbol><ci id="S4.SS1.p2.1.m1.2.3.3.3.2.cmml" xref="S4.SS1.p2.1.m1.2.3.3.3.2">𝑁</ci><ci id="S4.SS1.p2.1.m1.2.3.3.3.3.cmml" xref="S4.SS1.p2.1.m1.2.3.3.3.3">𝜁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.2c">\bm{\psi}_{\zeta,i}\in\mathbb{R}^{N_{\zeta}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.2d">bold_italic_ψ start_POSTSUBSCRIPT italic_ζ , italic_i end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> and a set of time instants <math alttext="\{t_{j}\}" class="ltx_Math" display="inline" id="S4.SS1.p2.2.m2.1"><semantics id="S4.SS1.p2.2.m2.1a"><mrow id="S4.SS1.p2.2.m2.1.1.1" xref="S4.SS1.p2.2.m2.1.1.2.cmml"><mo id="S4.SS1.p2.2.m2.1.1.1.2" stretchy="false" xref="S4.SS1.p2.2.m2.1.1.2.cmml">{</mo><msub id="S4.SS1.p2.2.m2.1.1.1.1" xref="S4.SS1.p2.2.m2.1.1.1.1.cmml"><mi id="S4.SS1.p2.2.m2.1.1.1.1.2" xref="S4.SS1.p2.2.m2.1.1.1.1.2.cmml">t</mi><mi id="S4.SS1.p2.2.m2.1.1.1.1.3" xref="S4.SS1.p2.2.m2.1.1.1.1.3.cmml">j</mi></msub><mo id="S4.SS1.p2.2.m2.1.1.1.3" stretchy="false" xref="S4.SS1.p2.2.m2.1.1.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.2.m2.1b"><set id="S4.SS1.p2.2.m2.1.1.2.cmml" xref="S4.SS1.p2.2.m2.1.1.1"><apply id="S4.SS1.p2.2.m2.1.1.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.2.m2.1.1.1.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1.1.1">subscript</csymbol><ci id="S4.SS1.p2.2.m2.1.1.1.1.2.cmml" xref="S4.SS1.p2.2.m2.1.1.1.1.2">𝑡</ci><ci id="S4.SS1.p2.2.m2.1.1.1.1.3.cmml" xref="S4.SS1.p2.2.m2.1.1.1.1.3">𝑗</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.2.m2.1c">\{t_{j}\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.2.m2.1d">{ italic_t start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT }</annotation></semantics></math> with <math alttext="t_{j}\in[T_{\rm a}-\tau_{\rm d},T_{\rm a}]\subset\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S4.SS1.p2.3.m3.2"><semantics id="S4.SS1.p2.3.m3.2a"><mrow id="S4.SS1.p2.3.m3.2.2" xref="S4.SS1.p2.3.m3.2.2.cmml"><msub id="S4.SS1.p2.3.m3.2.2.4" xref="S4.SS1.p2.3.m3.2.2.4.cmml"><mi id="S4.SS1.p2.3.m3.2.2.4.2" xref="S4.SS1.p2.3.m3.2.2.4.2.cmml">t</mi><mi id="S4.SS1.p2.3.m3.2.2.4.3" xref="S4.SS1.p2.3.m3.2.2.4.3.cmml">j</mi></msub><mo id="S4.SS1.p2.3.m3.2.2.5" xref="S4.SS1.p2.3.m3.2.2.5.cmml">∈</mo><mrow id="S4.SS1.p2.3.m3.2.2.2.2" xref="S4.SS1.p2.3.m3.2.2.2.3.cmml"><mo id="S4.SS1.p2.3.m3.2.2.2.2.3" stretchy="false" xref="S4.SS1.p2.3.m3.2.2.2.3.cmml">[</mo><mrow id="S4.SS1.p2.3.m3.1.1.1.1.1" xref="S4.SS1.p2.3.m3.1.1.1.1.1.cmml"><msub id="S4.SS1.p2.3.m3.1.1.1.1.1.2" xref="S4.SS1.p2.3.m3.1.1.1.1.1.2.cmml"><mi id="S4.SS1.p2.3.m3.1.1.1.1.1.2.2" xref="S4.SS1.p2.3.m3.1.1.1.1.1.2.2.cmml">T</mi><mi id="S4.SS1.p2.3.m3.1.1.1.1.1.2.3" mathvariant="normal" xref="S4.SS1.p2.3.m3.1.1.1.1.1.2.3.cmml">a</mi></msub><mo id="S4.SS1.p2.3.m3.1.1.1.1.1.1" xref="S4.SS1.p2.3.m3.1.1.1.1.1.1.cmml">−</mo><msub id="S4.SS1.p2.3.m3.1.1.1.1.1.3" xref="S4.SS1.p2.3.m3.1.1.1.1.1.3.cmml"><mi id="S4.SS1.p2.3.m3.1.1.1.1.1.3.2" xref="S4.SS1.p2.3.m3.1.1.1.1.1.3.2.cmml">τ</mi><mi id="S4.SS1.p2.3.m3.1.1.1.1.1.3.3" mathvariant="normal" xref="S4.SS1.p2.3.m3.1.1.1.1.1.3.3.cmml">d</mi></msub></mrow><mo id="S4.SS1.p2.3.m3.2.2.2.2.4" xref="S4.SS1.p2.3.m3.2.2.2.3.cmml">,</mo><msub id="S4.SS1.p2.3.m3.2.2.2.2.2" xref="S4.SS1.p2.3.m3.2.2.2.2.2.cmml"><mi id="S4.SS1.p2.3.m3.2.2.2.2.2.2" xref="S4.SS1.p2.3.m3.2.2.2.2.2.2.cmml">T</mi><mi id="S4.SS1.p2.3.m3.2.2.2.2.2.3" mathvariant="normal" xref="S4.SS1.p2.3.m3.2.2.2.2.2.3.cmml">a</mi></msub><mo id="S4.SS1.p2.3.m3.2.2.2.2.5" stretchy="false" xref="S4.SS1.p2.3.m3.2.2.2.3.cmml">]</mo></mrow><mo id="S4.SS1.p2.3.m3.2.2.6" xref="S4.SS1.p2.3.m3.2.2.6.cmml">⊂</mo><msup id="S4.SS1.p2.3.m3.2.2.7" xref="S4.SS1.p2.3.m3.2.2.7.cmml"><mi id="S4.SS1.p2.3.m3.2.2.7.2" xref="S4.SS1.p2.3.m3.2.2.7.2.cmml">ℝ</mi><mo id="S4.SS1.p2.3.m3.2.2.7.3" xref="S4.SS1.p2.3.m3.2.2.7.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.3.m3.2b"><apply id="S4.SS1.p2.3.m3.2.2.cmml" xref="S4.SS1.p2.3.m3.2.2"><and id="S4.SS1.p2.3.m3.2.2a.cmml" xref="S4.SS1.p2.3.m3.2.2"></and><apply id="S4.SS1.p2.3.m3.2.2b.cmml" xref="S4.SS1.p2.3.m3.2.2"><in id="S4.SS1.p2.3.m3.2.2.5.cmml" xref="S4.SS1.p2.3.m3.2.2.5"></in><apply id="S4.SS1.p2.3.m3.2.2.4.cmml" xref="S4.SS1.p2.3.m3.2.2.4"><csymbol cd="ambiguous" id="S4.SS1.p2.3.m3.2.2.4.1.cmml" xref="S4.SS1.p2.3.m3.2.2.4">subscript</csymbol><ci id="S4.SS1.p2.3.m3.2.2.4.2.cmml" xref="S4.SS1.p2.3.m3.2.2.4.2">𝑡</ci><ci id="S4.SS1.p2.3.m3.2.2.4.3.cmml" xref="S4.SS1.p2.3.m3.2.2.4.3">𝑗</ci></apply><interval closure="closed" id="S4.SS1.p2.3.m3.2.2.2.3.cmml" xref="S4.SS1.p2.3.m3.2.2.2.2"><apply id="S4.SS1.p2.3.m3.1.1.1.1.1.cmml" xref="S4.SS1.p2.3.m3.1.1.1.1.1"><minus id="S4.SS1.p2.3.m3.1.1.1.1.1.1.cmml" xref="S4.SS1.p2.3.m3.1.1.1.1.1.1"></minus><apply id="S4.SS1.p2.3.m3.1.1.1.1.1.2.cmml" xref="S4.SS1.p2.3.m3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p2.3.m3.1.1.1.1.1.2.1.cmml" xref="S4.SS1.p2.3.m3.1.1.1.1.1.2">subscript</csymbol><ci id="S4.SS1.p2.3.m3.1.1.1.1.1.2.2.cmml" xref="S4.SS1.p2.3.m3.1.1.1.1.1.2.2">𝑇</ci><ci id="S4.SS1.p2.3.m3.1.1.1.1.1.2.3.cmml" xref="S4.SS1.p2.3.m3.1.1.1.1.1.2.3">a</ci></apply><apply id="S4.SS1.p2.3.m3.1.1.1.1.1.3.cmml" xref="S4.SS1.p2.3.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p2.3.m3.1.1.1.1.1.3.1.cmml" xref="S4.SS1.p2.3.m3.1.1.1.1.1.3">subscript</csymbol><ci id="S4.SS1.p2.3.m3.1.1.1.1.1.3.2.cmml" xref="S4.SS1.p2.3.m3.1.1.1.1.1.3.2">𝜏</ci><ci id="S4.SS1.p2.3.m3.1.1.1.1.1.3.3.cmml" xref="S4.SS1.p2.3.m3.1.1.1.1.1.3.3">d</ci></apply></apply><apply id="S4.SS1.p2.3.m3.2.2.2.2.2.cmml" xref="S4.SS1.p2.3.m3.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.p2.3.m3.2.2.2.2.2.1.cmml" xref="S4.SS1.p2.3.m3.2.2.2.2.2">subscript</csymbol><ci id="S4.SS1.p2.3.m3.2.2.2.2.2.2.cmml" xref="S4.SS1.p2.3.m3.2.2.2.2.2.2">𝑇</ci><ci id="S4.SS1.p2.3.m3.2.2.2.2.2.3.cmml" xref="S4.SS1.p2.3.m3.2.2.2.2.2.3">a</ci></apply></interval></apply><apply id="S4.SS1.p2.3.m3.2.2c.cmml" xref="S4.SS1.p2.3.m3.2.2"><subset id="S4.SS1.p2.3.m3.2.2.6.cmml" xref="S4.SS1.p2.3.m3.2.2.6"></subset><share href="https://arxiv.org/html/2401.10785v2#S4.SS1.p2.3.m3.2.2.2.cmml" id="S4.SS1.p2.3.m3.2.2d.cmml" xref="S4.SS1.p2.3.m3.2.2"></share><apply id="S4.SS1.p2.3.m3.2.2.7.cmml" xref="S4.SS1.p2.3.m3.2.2.7"><csymbol cd="ambiguous" id="S4.SS1.p2.3.m3.2.2.7.1.cmml" xref="S4.SS1.p2.3.m3.2.2.7">superscript</csymbol><ci id="S4.SS1.p2.3.m3.2.2.7.2.cmml" xref="S4.SS1.p2.3.m3.2.2.7.2">ℝ</ci><plus id="S4.SS1.p2.3.m3.2.2.7.3.cmml" xref="S4.SS1.p2.3.m3.2.2.7.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.3.m3.2c">t_{j}\in[T_{\rm a}-\tau_{\rm d},T_{\rm a}]\subset\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.3.m3.2d">italic_t start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ [ italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT ] ⊂ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> to get rank<math alttext="\{\bm{\psi}_{\zeta,i}(t_{1}),\bm{\psi}_{\zeta,i}(t_{2}),\cdots,\bm{\psi}_{% \zeta,i}(t_{N_{\zeta}})\}" class="ltx_Math" display="inline" id="S4.SS1.p2.4.m4.10"><semantics id="S4.SS1.p2.4.m4.10a"><mrow id="S4.SS1.p2.4.m4.10.10.3" xref="S4.SS1.p2.4.m4.10.10.4.cmml"><mo id="S4.SS1.p2.4.m4.10.10.3.4" stretchy="false" xref="S4.SS1.p2.4.m4.10.10.4.cmml">{</mo><mrow id="S4.SS1.p2.4.m4.8.8.1.1" xref="S4.SS1.p2.4.m4.8.8.1.1.cmml"><msub id="S4.SS1.p2.4.m4.8.8.1.1.3" xref="S4.SS1.p2.4.m4.8.8.1.1.3.cmml"><mi id="S4.SS1.p2.4.m4.8.8.1.1.3.2" xref="S4.SS1.p2.4.m4.8.8.1.1.3.2.cmml">𝝍</mi><mrow id="S4.SS1.p2.4.m4.2.2.2.4" xref="S4.SS1.p2.4.m4.2.2.2.3.cmml"><mi id="S4.SS1.p2.4.m4.1.1.1.1" xref="S4.SS1.p2.4.m4.1.1.1.1.cmml">ζ</mi><mo id="S4.SS1.p2.4.m4.2.2.2.4.1" xref="S4.SS1.p2.4.m4.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p2.4.m4.2.2.2.2" xref="S4.SS1.p2.4.m4.2.2.2.2.cmml">i</mi></mrow></msub><mo id="S4.SS1.p2.4.m4.8.8.1.1.2" xref="S4.SS1.p2.4.m4.8.8.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.p2.4.m4.8.8.1.1.1.1" xref="S4.SS1.p2.4.m4.8.8.1.1.1.1.1.cmml"><mo id="S4.SS1.p2.4.m4.8.8.1.1.1.1.2" stretchy="false" xref="S4.SS1.p2.4.m4.8.8.1.1.1.1.1.cmml">(</mo><msub id="S4.SS1.p2.4.m4.8.8.1.1.1.1.1" xref="S4.SS1.p2.4.m4.8.8.1.1.1.1.1.cmml"><mi id="S4.SS1.p2.4.m4.8.8.1.1.1.1.1.2" xref="S4.SS1.p2.4.m4.8.8.1.1.1.1.1.2.cmml">t</mi><mn id="S4.SS1.p2.4.m4.8.8.1.1.1.1.1.3" xref="S4.SS1.p2.4.m4.8.8.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.SS1.p2.4.m4.8.8.1.1.1.1.3" stretchy="false" xref="S4.SS1.p2.4.m4.8.8.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p2.4.m4.10.10.3.5" xref="S4.SS1.p2.4.m4.10.10.4.cmml">,</mo><mrow id="S4.SS1.p2.4.m4.9.9.2.2" xref="S4.SS1.p2.4.m4.9.9.2.2.cmml"><msub id="S4.SS1.p2.4.m4.9.9.2.2.3" xref="S4.SS1.p2.4.m4.9.9.2.2.3.cmml"><mi id="S4.SS1.p2.4.m4.9.9.2.2.3.2" xref="S4.SS1.p2.4.m4.9.9.2.2.3.2.cmml">𝝍</mi><mrow id="S4.SS1.p2.4.m4.4.4.2.4" xref="S4.SS1.p2.4.m4.4.4.2.3.cmml"><mi id="S4.SS1.p2.4.m4.3.3.1.1" xref="S4.SS1.p2.4.m4.3.3.1.1.cmml">ζ</mi><mo id="S4.SS1.p2.4.m4.4.4.2.4.1" xref="S4.SS1.p2.4.m4.4.4.2.3.cmml">,</mo><mi id="S4.SS1.p2.4.m4.4.4.2.2" xref="S4.SS1.p2.4.m4.4.4.2.2.cmml">i</mi></mrow></msub><mo id="S4.SS1.p2.4.m4.9.9.2.2.2" xref="S4.SS1.p2.4.m4.9.9.2.2.2.cmml">⁢</mo><mrow id="S4.SS1.p2.4.m4.9.9.2.2.1.1" xref="S4.SS1.p2.4.m4.9.9.2.2.1.1.1.cmml"><mo id="S4.SS1.p2.4.m4.9.9.2.2.1.1.2" stretchy="false" xref="S4.SS1.p2.4.m4.9.9.2.2.1.1.1.cmml">(</mo><msub id="S4.SS1.p2.4.m4.9.9.2.2.1.1.1" xref="S4.SS1.p2.4.m4.9.9.2.2.1.1.1.cmml"><mi id="S4.SS1.p2.4.m4.9.9.2.2.1.1.1.2" xref="S4.SS1.p2.4.m4.9.9.2.2.1.1.1.2.cmml">t</mi><mn id="S4.SS1.p2.4.m4.9.9.2.2.1.1.1.3" xref="S4.SS1.p2.4.m4.9.9.2.2.1.1.1.3.cmml">2</mn></msub><mo id="S4.SS1.p2.4.m4.9.9.2.2.1.1.3" stretchy="false" xref="S4.SS1.p2.4.m4.9.9.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p2.4.m4.10.10.3.6" xref="S4.SS1.p2.4.m4.10.10.4.cmml">,</mo><mi id="S4.SS1.p2.4.m4.7.7" mathvariant="normal" xref="S4.SS1.p2.4.m4.7.7.cmml">⋯</mi><mo id="S4.SS1.p2.4.m4.10.10.3.7" xref="S4.SS1.p2.4.m4.10.10.4.cmml">,</mo><mrow id="S4.SS1.p2.4.m4.10.10.3.3" xref="S4.SS1.p2.4.m4.10.10.3.3.cmml"><msub id="S4.SS1.p2.4.m4.10.10.3.3.3" xref="S4.SS1.p2.4.m4.10.10.3.3.3.cmml"><mi id="S4.SS1.p2.4.m4.10.10.3.3.3.2" xref="S4.SS1.p2.4.m4.10.10.3.3.3.2.cmml">𝝍</mi><mrow id="S4.SS1.p2.4.m4.6.6.2.4" xref="S4.SS1.p2.4.m4.6.6.2.3.cmml"><mi id="S4.SS1.p2.4.m4.5.5.1.1" xref="S4.SS1.p2.4.m4.5.5.1.1.cmml">ζ</mi><mo id="S4.SS1.p2.4.m4.6.6.2.4.1" xref="S4.SS1.p2.4.m4.6.6.2.3.cmml">,</mo><mi id="S4.SS1.p2.4.m4.6.6.2.2" xref="S4.SS1.p2.4.m4.6.6.2.2.cmml">i</mi></mrow></msub><mo id="S4.SS1.p2.4.m4.10.10.3.3.2" xref="S4.SS1.p2.4.m4.10.10.3.3.2.cmml">⁢</mo><mrow id="S4.SS1.p2.4.m4.10.10.3.3.1.1" xref="S4.SS1.p2.4.m4.10.10.3.3.1.1.1.cmml"><mo id="S4.SS1.p2.4.m4.10.10.3.3.1.1.2" stretchy="false" xref="S4.SS1.p2.4.m4.10.10.3.3.1.1.1.cmml">(</mo><msub id="S4.SS1.p2.4.m4.10.10.3.3.1.1.1" xref="S4.SS1.p2.4.m4.10.10.3.3.1.1.1.cmml"><mi id="S4.SS1.p2.4.m4.10.10.3.3.1.1.1.2" xref="S4.SS1.p2.4.m4.10.10.3.3.1.1.1.2.cmml">t</mi><msub id="S4.SS1.p2.4.m4.10.10.3.3.1.1.1.3" xref="S4.SS1.p2.4.m4.10.10.3.3.1.1.1.3.cmml"><mi id="S4.SS1.p2.4.m4.10.10.3.3.1.1.1.3.2" xref="S4.SS1.p2.4.m4.10.10.3.3.1.1.1.3.2.cmml">N</mi><mi id="S4.SS1.p2.4.m4.10.10.3.3.1.1.1.3.3" xref="S4.SS1.p2.4.m4.10.10.3.3.1.1.1.3.3.cmml">ζ</mi></msub></msub><mo id="S4.SS1.p2.4.m4.10.10.3.3.1.1.3" stretchy="false" xref="S4.SS1.p2.4.m4.10.10.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p2.4.m4.10.10.3.8" stretchy="false" xref="S4.SS1.p2.4.m4.10.10.4.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.4.m4.10b"><set id="S4.SS1.p2.4.m4.10.10.4.cmml" xref="S4.SS1.p2.4.m4.10.10.3"><apply id="S4.SS1.p2.4.m4.8.8.1.1.cmml" xref="S4.SS1.p2.4.m4.8.8.1.1"><times id="S4.SS1.p2.4.m4.8.8.1.1.2.cmml" xref="S4.SS1.p2.4.m4.8.8.1.1.2"></times><apply id="S4.SS1.p2.4.m4.8.8.1.1.3.cmml" xref="S4.SS1.p2.4.m4.8.8.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p2.4.m4.8.8.1.1.3.1.cmml" xref="S4.SS1.p2.4.m4.8.8.1.1.3">subscript</csymbol><ci id="S4.SS1.p2.4.m4.8.8.1.1.3.2.cmml" xref="S4.SS1.p2.4.m4.8.8.1.1.3.2">𝝍</ci><list id="S4.SS1.p2.4.m4.2.2.2.3.cmml" xref="S4.SS1.p2.4.m4.2.2.2.4"><ci id="S4.SS1.p2.4.m4.1.1.1.1.cmml" xref="S4.SS1.p2.4.m4.1.1.1.1">𝜁</ci><ci id="S4.SS1.p2.4.m4.2.2.2.2.cmml" xref="S4.SS1.p2.4.m4.2.2.2.2">𝑖</ci></list></apply><apply id="S4.SS1.p2.4.m4.8.8.1.1.1.1.1.cmml" xref="S4.SS1.p2.4.m4.8.8.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.4.m4.8.8.1.1.1.1.1.1.cmml" xref="S4.SS1.p2.4.m4.8.8.1.1.1.1">subscript</csymbol><ci id="S4.SS1.p2.4.m4.8.8.1.1.1.1.1.2.cmml" xref="S4.SS1.p2.4.m4.8.8.1.1.1.1.1.2">𝑡</ci><cn id="S4.SS1.p2.4.m4.8.8.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS1.p2.4.m4.8.8.1.1.1.1.1.3">1</cn></apply></apply><apply id="S4.SS1.p2.4.m4.9.9.2.2.cmml" xref="S4.SS1.p2.4.m4.9.9.2.2"><times id="S4.SS1.p2.4.m4.9.9.2.2.2.cmml" xref="S4.SS1.p2.4.m4.9.9.2.2.2"></times><apply id="S4.SS1.p2.4.m4.9.9.2.2.3.cmml" xref="S4.SS1.p2.4.m4.9.9.2.2.3"><csymbol cd="ambiguous" id="S4.SS1.p2.4.m4.9.9.2.2.3.1.cmml" xref="S4.SS1.p2.4.m4.9.9.2.2.3">subscript</csymbol><ci id="S4.SS1.p2.4.m4.9.9.2.2.3.2.cmml" xref="S4.SS1.p2.4.m4.9.9.2.2.3.2">𝝍</ci><list id="S4.SS1.p2.4.m4.4.4.2.3.cmml" xref="S4.SS1.p2.4.m4.4.4.2.4"><ci id="S4.SS1.p2.4.m4.3.3.1.1.cmml" xref="S4.SS1.p2.4.m4.3.3.1.1">𝜁</ci><ci id="S4.SS1.p2.4.m4.4.4.2.2.cmml" xref="S4.SS1.p2.4.m4.4.4.2.2">𝑖</ci></list></apply><apply id="S4.SS1.p2.4.m4.9.9.2.2.1.1.1.cmml" xref="S4.SS1.p2.4.m4.9.9.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.4.m4.9.9.2.2.1.1.1.1.cmml" xref="S4.SS1.p2.4.m4.9.9.2.2.1.1">subscript</csymbol><ci id="S4.SS1.p2.4.m4.9.9.2.2.1.1.1.2.cmml" xref="S4.SS1.p2.4.m4.9.9.2.2.1.1.1.2">𝑡</ci><cn id="S4.SS1.p2.4.m4.9.9.2.2.1.1.1.3.cmml" type="integer" xref="S4.SS1.p2.4.m4.9.9.2.2.1.1.1.3">2</cn></apply></apply><ci id="S4.SS1.p2.4.m4.7.7.cmml" xref="S4.SS1.p2.4.m4.7.7">⋯</ci><apply id="S4.SS1.p2.4.m4.10.10.3.3.cmml" xref="S4.SS1.p2.4.m4.10.10.3.3"><times id="S4.SS1.p2.4.m4.10.10.3.3.2.cmml" xref="S4.SS1.p2.4.m4.10.10.3.3.2"></times><apply id="S4.SS1.p2.4.m4.10.10.3.3.3.cmml" xref="S4.SS1.p2.4.m4.10.10.3.3.3"><csymbol cd="ambiguous" id="S4.SS1.p2.4.m4.10.10.3.3.3.1.cmml" xref="S4.SS1.p2.4.m4.10.10.3.3.3">subscript</csymbol><ci id="S4.SS1.p2.4.m4.10.10.3.3.3.2.cmml" xref="S4.SS1.p2.4.m4.10.10.3.3.3.2">𝝍</ci><list id="S4.SS1.p2.4.m4.6.6.2.3.cmml" xref="S4.SS1.p2.4.m4.6.6.2.4"><ci id="S4.SS1.p2.4.m4.5.5.1.1.cmml" xref="S4.SS1.p2.4.m4.5.5.1.1">𝜁</ci><ci id="S4.SS1.p2.4.m4.6.6.2.2.cmml" xref="S4.SS1.p2.4.m4.6.6.2.2">𝑖</ci></list></apply><apply id="S4.SS1.p2.4.m4.10.10.3.3.1.1.1.cmml" xref="S4.SS1.p2.4.m4.10.10.3.3.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.4.m4.10.10.3.3.1.1.1.1.cmml" xref="S4.SS1.p2.4.m4.10.10.3.3.1.1">subscript</csymbol><ci id="S4.SS1.p2.4.m4.10.10.3.3.1.1.1.2.cmml" xref="S4.SS1.p2.4.m4.10.10.3.3.1.1.1.2">𝑡</ci><apply id="S4.SS1.p2.4.m4.10.10.3.3.1.1.1.3.cmml" xref="S4.SS1.p2.4.m4.10.10.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p2.4.m4.10.10.3.3.1.1.1.3.1.cmml" xref="S4.SS1.p2.4.m4.10.10.3.3.1.1.1.3">subscript</csymbol><ci id="S4.SS1.p2.4.m4.10.10.3.3.1.1.1.3.2.cmml" xref="S4.SS1.p2.4.m4.10.10.3.3.1.1.1.3.2">𝑁</ci><ci id="S4.SS1.p2.4.m4.10.10.3.3.1.1.1.3.3.cmml" xref="S4.SS1.p2.4.m4.10.10.3.3.1.1.1.3.3">𝜁</ci></apply></apply></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.4.m4.10c">\{\bm{\psi}_{\zeta,i}(t_{1}),\bm{\psi}_{\zeta,i}(t_{2}),\cdots,\bm{\psi}_{% \zeta,i}(t_{N_{\zeta}})\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.4.m4.10d">{ bold_italic_ψ start_POSTSUBSCRIPT italic_ζ , italic_i end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , bold_italic_ψ start_POSTSUBSCRIPT italic_ζ , italic_i end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , ⋯ , bold_italic_ψ start_POSTSUBSCRIPT italic_ζ , italic_i end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) }</annotation></semantics></math> <math alttext="=" class="ltx_Math" display="inline" id="S4.SS1.p2.5.m5.1"><semantics id="S4.SS1.p2.5.m5.1a"><mo id="S4.SS1.p2.5.m5.1.1" xref="S4.SS1.p2.5.m5.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.5.m5.1b"><eq id="S4.SS1.p2.5.m5.1.1.cmml" xref="S4.SS1.p2.5.m5.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.5.m5.1c">=</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.5.m5.1d">=</annotation></semantics></math> <math alttext="N_{\zeta}" class="ltx_Math" display="inline" id="S4.SS1.p2.6.m6.1"><semantics id="S4.SS1.p2.6.m6.1a"><msub id="S4.SS1.p2.6.m6.1.1" xref="S4.SS1.p2.6.m6.1.1.cmml"><mi id="S4.SS1.p2.6.m6.1.1.2" xref="S4.SS1.p2.6.m6.1.1.2.cmml">N</mi><mi id="S4.SS1.p2.6.m6.1.1.3" xref="S4.SS1.p2.6.m6.1.1.3.cmml">ζ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.6.m6.1b"><apply id="S4.SS1.p2.6.m6.1.1.cmml" xref="S4.SS1.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.6.m6.1.1.1.cmml" xref="S4.SS1.p2.6.m6.1.1">subscript</csymbol><ci id="S4.SS1.p2.6.m6.1.1.2.cmml" xref="S4.SS1.p2.6.m6.1.1.2">𝑁</ci><ci id="S4.SS1.p2.6.m6.1.1.3.cmml" xref="S4.SS1.p2.6.m6.1.1.3">𝜁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.6.m6.1c">N_{\zeta}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.6.m6.1d">italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS1.p3"> <p class="ltx_p" id="S4.SS1.p3.3">Multiplying (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E12" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">12</span></a>) by <math alttext="\Phi_{\rm s}" class="ltx_Math" display="inline" id="S4.SS1.p3.1.m1.1"><semantics id="S4.SS1.p3.1.m1.1a"><msub id="S4.SS1.p3.1.m1.1.1" xref="S4.SS1.p3.1.m1.1.1.cmml"><mi id="S4.SS1.p3.1.m1.1.1.2" mathvariant="normal" xref="S4.SS1.p3.1.m1.1.1.2.cmml">Φ</mi><mi id="S4.SS1.p3.1.m1.1.1.3" mathvariant="normal" xref="S4.SS1.p3.1.m1.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.1.m1.1b"><apply id="S4.SS1.p3.1.m1.1.1.cmml" xref="S4.SS1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.1.m1.1.1.1.cmml" xref="S4.SS1.p3.1.m1.1.1">subscript</csymbol><ci id="S4.SS1.p3.1.m1.1.1.2.cmml" xref="S4.SS1.p3.1.m1.1.1.2">Φ</ci><ci id="S4.SS1.p3.1.m1.1.1.3.cmml" xref="S4.SS1.p3.1.m1.1.1.3">s</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.1.m1.1c">\Phi_{\rm s}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.1.m1.1d">roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT</annotation></semantics></math> and letting <math alttext="\bm{\zeta}(0)=-\bm{e}(0)" class="ltx_Math" display="inline" id="S4.SS1.p3.2.m2.2"><semantics id="S4.SS1.p3.2.m2.2a"><mrow id="S4.SS1.p3.2.m2.2.3" xref="S4.SS1.p3.2.m2.2.3.cmml"><mrow id="S4.SS1.p3.2.m2.2.3.2" xref="S4.SS1.p3.2.m2.2.3.2.cmml"><mi id="S4.SS1.p3.2.m2.2.3.2.2" xref="S4.SS1.p3.2.m2.2.3.2.2.cmml">𝜻</mi><mo id="S4.SS1.p3.2.m2.2.3.2.1" xref="S4.SS1.p3.2.m2.2.3.2.1.cmml">⁢</mo><mrow id="S4.SS1.p3.2.m2.2.3.2.3.2" xref="S4.SS1.p3.2.m2.2.3.2.cmml"><mo id="S4.SS1.p3.2.m2.2.3.2.3.2.1" stretchy="false" xref="S4.SS1.p3.2.m2.2.3.2.cmml">(</mo><mn id="S4.SS1.p3.2.m2.1.1" xref="S4.SS1.p3.2.m2.1.1.cmml">0</mn><mo id="S4.SS1.p3.2.m2.2.3.2.3.2.2" stretchy="false" xref="S4.SS1.p3.2.m2.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p3.2.m2.2.3.1" xref="S4.SS1.p3.2.m2.2.3.1.cmml">=</mo><mrow id="S4.SS1.p3.2.m2.2.3.3" xref="S4.SS1.p3.2.m2.2.3.3.cmml"><mo id="S4.SS1.p3.2.m2.2.3.3a" xref="S4.SS1.p3.2.m2.2.3.3.cmml">−</mo><mrow id="S4.SS1.p3.2.m2.2.3.3.2" xref="S4.SS1.p3.2.m2.2.3.3.2.cmml"><mi id="S4.SS1.p3.2.m2.2.3.3.2.2" xref="S4.SS1.p3.2.m2.2.3.3.2.2.cmml">𝒆</mi><mo id="S4.SS1.p3.2.m2.2.3.3.2.1" xref="S4.SS1.p3.2.m2.2.3.3.2.1.cmml">⁢</mo><mrow id="S4.SS1.p3.2.m2.2.3.3.2.3.2" xref="S4.SS1.p3.2.m2.2.3.3.2.cmml"><mo id="S4.SS1.p3.2.m2.2.3.3.2.3.2.1" stretchy="false" xref="S4.SS1.p3.2.m2.2.3.3.2.cmml">(</mo><mn id="S4.SS1.p3.2.m2.2.2" xref="S4.SS1.p3.2.m2.2.2.cmml">0</mn><mo id="S4.SS1.p3.2.m2.2.3.3.2.3.2.2" stretchy="false" xref="S4.SS1.p3.2.m2.2.3.3.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.2.m2.2b"><apply id="S4.SS1.p3.2.m2.2.3.cmml" xref="S4.SS1.p3.2.m2.2.3"><eq id="S4.SS1.p3.2.m2.2.3.1.cmml" xref="S4.SS1.p3.2.m2.2.3.1"></eq><apply id="S4.SS1.p3.2.m2.2.3.2.cmml" xref="S4.SS1.p3.2.m2.2.3.2"><times id="S4.SS1.p3.2.m2.2.3.2.1.cmml" xref="S4.SS1.p3.2.m2.2.3.2.1"></times><ci id="S4.SS1.p3.2.m2.2.3.2.2.cmml" xref="S4.SS1.p3.2.m2.2.3.2.2">𝜻</ci><cn id="S4.SS1.p3.2.m2.1.1.cmml" type="integer" xref="S4.SS1.p3.2.m2.1.1">0</cn></apply><apply id="S4.SS1.p3.2.m2.2.3.3.cmml" xref="S4.SS1.p3.2.m2.2.3.3"><minus id="S4.SS1.p3.2.m2.2.3.3.1.cmml" xref="S4.SS1.p3.2.m2.2.3.3"></minus><apply id="S4.SS1.p3.2.m2.2.3.3.2.cmml" xref="S4.SS1.p3.2.m2.2.3.3.2"><times id="S4.SS1.p3.2.m2.2.3.3.2.1.cmml" xref="S4.SS1.p3.2.m2.2.3.3.2.1"></times><ci id="S4.SS1.p3.2.m2.2.3.3.2.2.cmml" xref="S4.SS1.p3.2.m2.2.3.3.2.2">𝒆</ci><cn id="S4.SS1.p3.2.m2.2.2.cmml" type="integer" xref="S4.SS1.p3.2.m2.2.2">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.2.m2.2c">\bm{\zeta}(0)=-\bm{e}(0)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.2.m2.2d">bold_italic_ζ ( 0 ) = - bold_italic_e ( 0 )</annotation></semantics></math> and <math alttext="{\Phi}_{\rm s}(0)=\bm{0}" class="ltx_Math" display="inline" id="S4.SS1.p3.3.m3.1"><semantics id="S4.SS1.p3.3.m3.1a"><mrow id="S4.SS1.p3.3.m3.1.2" xref="S4.SS1.p3.3.m3.1.2.cmml"><mrow id="S4.SS1.p3.3.m3.1.2.2" xref="S4.SS1.p3.3.m3.1.2.2.cmml"><msub id="S4.SS1.p3.3.m3.1.2.2.2" xref="S4.SS1.p3.3.m3.1.2.2.2.cmml"><mi id="S4.SS1.p3.3.m3.1.2.2.2.2" mathvariant="normal" xref="S4.SS1.p3.3.m3.1.2.2.2.2.cmml">Φ</mi><mi id="S4.SS1.p3.3.m3.1.2.2.2.3" mathvariant="normal" xref="S4.SS1.p3.3.m3.1.2.2.2.3.cmml">s</mi></msub><mo id="S4.SS1.p3.3.m3.1.2.2.1" xref="S4.SS1.p3.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S4.SS1.p3.3.m3.1.2.2.3.2" xref="S4.SS1.p3.3.m3.1.2.2.cmml"><mo id="S4.SS1.p3.3.m3.1.2.2.3.2.1" stretchy="false" xref="S4.SS1.p3.3.m3.1.2.2.cmml">(</mo><mn id="S4.SS1.p3.3.m3.1.1" xref="S4.SS1.p3.3.m3.1.1.cmml">0</mn><mo id="S4.SS1.p3.3.m3.1.2.2.3.2.2" stretchy="false" xref="S4.SS1.p3.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p3.3.m3.1.2.1" xref="S4.SS1.p3.3.m3.1.2.1.cmml">=</mo><mn id="S4.SS1.p3.3.m3.1.2.3" xref="S4.SS1.p3.3.m3.1.2.3.cmml">𝟎</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.3.m3.1b"><apply id="S4.SS1.p3.3.m3.1.2.cmml" xref="S4.SS1.p3.3.m3.1.2"><eq id="S4.SS1.p3.3.m3.1.2.1.cmml" xref="S4.SS1.p3.3.m3.1.2.1"></eq><apply id="S4.SS1.p3.3.m3.1.2.2.cmml" xref="S4.SS1.p3.3.m3.1.2.2"><times id="S4.SS1.p3.3.m3.1.2.2.1.cmml" xref="S4.SS1.p3.3.m3.1.2.2.1"></times><apply id="S4.SS1.p3.3.m3.1.2.2.2.cmml" xref="S4.SS1.p3.3.m3.1.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.p3.3.m3.1.2.2.2.1.cmml" xref="S4.SS1.p3.3.m3.1.2.2.2">subscript</csymbol><ci id="S4.SS1.p3.3.m3.1.2.2.2.2.cmml" xref="S4.SS1.p3.3.m3.1.2.2.2.2">Φ</ci><ci id="S4.SS1.p3.3.m3.1.2.2.2.3.cmml" xref="S4.SS1.p3.3.m3.1.2.2.2.3">s</ci></apply><cn id="S4.SS1.p3.3.m3.1.1.cmml" type="integer" xref="S4.SS1.p3.3.m3.1.1">0</cn></apply><cn id="S4.SS1.p3.3.m3.1.2.3.cmml" type="integer" xref="S4.SS1.p3.3.m3.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.3.m3.1c">{\Phi}_{\rm s}(0)=\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.3.m3.1d">roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( 0 ) = bold_0</annotation></semantics></math>, one obtains an extended regression equation</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx8"> <tbody id="S4.E13"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\Phi_{\rm s}(t)\Phi_{\rm s}^{T}(t)\bm{\theta}=\Phi_{\rm s}(t)\bm{% p}(t)." class="ltx_Math" display="inline" id="S4.E13.m1.5"><semantics id="S4.E13.m1.5a"><mrow id="S4.E13.m1.5.5.1" xref="S4.E13.m1.5.5.1.1.cmml"><mrow id="S4.E13.m1.5.5.1.1" xref="S4.E13.m1.5.5.1.1.cmml"><mrow id="S4.E13.m1.5.5.1.1.2" xref="S4.E13.m1.5.5.1.1.2.cmml"><msub id="S4.E13.m1.5.5.1.1.2.2" xref="S4.E13.m1.5.5.1.1.2.2.cmml"><mi id="S4.E13.m1.5.5.1.1.2.2.2" mathvariant="normal" xref="S4.E13.m1.5.5.1.1.2.2.2.cmml">Φ</mi><mi id="S4.E13.m1.5.5.1.1.2.2.3" mathvariant="normal" xref="S4.E13.m1.5.5.1.1.2.2.3.cmml">s</mi></msub><mo id="S4.E13.m1.5.5.1.1.2.1" xref="S4.E13.m1.5.5.1.1.2.1.cmml">⁢</mo><mrow id="S4.E13.m1.5.5.1.1.2.3.2" xref="S4.E13.m1.5.5.1.1.2.cmml"><mo id="S4.E13.m1.5.5.1.1.2.3.2.1" stretchy="false" xref="S4.E13.m1.5.5.1.1.2.cmml">(</mo><mi id="S4.E13.m1.1.1" xref="S4.E13.m1.1.1.cmml">t</mi><mo id="S4.E13.m1.5.5.1.1.2.3.2.2" stretchy="false" xref="S4.E13.m1.5.5.1.1.2.cmml">)</mo></mrow><mo id="S4.E13.m1.5.5.1.1.2.1a" xref="S4.E13.m1.5.5.1.1.2.1.cmml">⁢</mo><msubsup id="S4.E13.m1.5.5.1.1.2.4" xref="S4.E13.m1.5.5.1.1.2.4.cmml"><mi id="S4.E13.m1.5.5.1.1.2.4.2.2" mathvariant="normal" xref="S4.E13.m1.5.5.1.1.2.4.2.2.cmml">Φ</mi><mi id="S4.E13.m1.5.5.1.1.2.4.2.3" mathvariant="normal" xref="S4.E13.m1.5.5.1.1.2.4.2.3.cmml">s</mi><mi id="S4.E13.m1.5.5.1.1.2.4.3" xref="S4.E13.m1.5.5.1.1.2.4.3.cmml">T</mi></msubsup><mo id="S4.E13.m1.5.5.1.1.2.1b" xref="S4.E13.m1.5.5.1.1.2.1.cmml">⁢</mo><mrow id="S4.E13.m1.5.5.1.1.2.5.2" xref="S4.E13.m1.5.5.1.1.2.cmml"><mo id="S4.E13.m1.5.5.1.1.2.5.2.1" stretchy="false" xref="S4.E13.m1.5.5.1.1.2.cmml">(</mo><mi id="S4.E13.m1.2.2" xref="S4.E13.m1.2.2.cmml">t</mi><mo id="S4.E13.m1.5.5.1.1.2.5.2.2" stretchy="false" xref="S4.E13.m1.5.5.1.1.2.cmml">)</mo></mrow><mo id="S4.E13.m1.5.5.1.1.2.1c" xref="S4.E13.m1.5.5.1.1.2.1.cmml">⁢</mo><mi id="S4.E13.m1.5.5.1.1.2.6" xref="S4.E13.m1.5.5.1.1.2.6.cmml">𝜽</mi></mrow><mo id="S4.E13.m1.5.5.1.1.1" xref="S4.E13.m1.5.5.1.1.1.cmml">=</mo><mrow id="S4.E13.m1.5.5.1.1.3" xref="S4.E13.m1.5.5.1.1.3.cmml"><msub id="S4.E13.m1.5.5.1.1.3.2" xref="S4.E13.m1.5.5.1.1.3.2.cmml"><mi id="S4.E13.m1.5.5.1.1.3.2.2" mathvariant="normal" xref="S4.E13.m1.5.5.1.1.3.2.2.cmml">Φ</mi><mi id="S4.E13.m1.5.5.1.1.3.2.3" mathvariant="normal" xref="S4.E13.m1.5.5.1.1.3.2.3.cmml">s</mi></msub><mo id="S4.E13.m1.5.5.1.1.3.1" xref="S4.E13.m1.5.5.1.1.3.1.cmml">⁢</mo><mrow id="S4.E13.m1.5.5.1.1.3.3.2" xref="S4.E13.m1.5.5.1.1.3.cmml"><mo id="S4.E13.m1.5.5.1.1.3.3.2.1" stretchy="false" xref="S4.E13.m1.5.5.1.1.3.cmml">(</mo><mi id="S4.E13.m1.3.3" xref="S4.E13.m1.3.3.cmml">t</mi><mo id="S4.E13.m1.5.5.1.1.3.3.2.2" stretchy="false" xref="S4.E13.m1.5.5.1.1.3.cmml">)</mo></mrow><mo id="S4.E13.m1.5.5.1.1.3.1a" xref="S4.E13.m1.5.5.1.1.3.1.cmml">⁢</mo><mi id="S4.E13.m1.5.5.1.1.3.4" xref="S4.E13.m1.5.5.1.1.3.4.cmml">𝒑</mi><mo id="S4.E13.m1.5.5.1.1.3.1b" xref="S4.E13.m1.5.5.1.1.3.1.cmml">⁢</mo><mrow id="S4.E13.m1.5.5.1.1.3.5.2" xref="S4.E13.m1.5.5.1.1.3.cmml"><mo id="S4.E13.m1.5.5.1.1.3.5.2.1" stretchy="false" xref="S4.E13.m1.5.5.1.1.3.cmml">(</mo><mi id="S4.E13.m1.4.4" xref="S4.E13.m1.4.4.cmml">t</mi><mo id="S4.E13.m1.5.5.1.1.3.5.2.2" stretchy="false" xref="S4.E13.m1.5.5.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E13.m1.5.5.1.2" lspace="0em" xref="S4.E13.m1.5.5.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E13.m1.5b"><apply id="S4.E13.m1.5.5.1.1.cmml" xref="S4.E13.m1.5.5.1"><eq id="S4.E13.m1.5.5.1.1.1.cmml" xref="S4.E13.m1.5.5.1.1.1"></eq><apply id="S4.E13.m1.5.5.1.1.2.cmml" xref="S4.E13.m1.5.5.1.1.2"><times id="S4.E13.m1.5.5.1.1.2.1.cmml" xref="S4.E13.m1.5.5.1.1.2.1"></times><apply id="S4.E13.m1.5.5.1.1.2.2.cmml" xref="S4.E13.m1.5.5.1.1.2.2"><csymbol cd="ambiguous" id="S4.E13.m1.5.5.1.1.2.2.1.cmml" xref="S4.E13.m1.5.5.1.1.2.2">subscript</csymbol><ci id="S4.E13.m1.5.5.1.1.2.2.2.cmml" xref="S4.E13.m1.5.5.1.1.2.2.2">Φ</ci><ci id="S4.E13.m1.5.5.1.1.2.2.3.cmml" xref="S4.E13.m1.5.5.1.1.2.2.3">s</ci></apply><ci id="S4.E13.m1.1.1.cmml" xref="S4.E13.m1.1.1">𝑡</ci><apply id="S4.E13.m1.5.5.1.1.2.4.cmml" xref="S4.E13.m1.5.5.1.1.2.4"><csymbol cd="ambiguous" id="S4.E13.m1.5.5.1.1.2.4.1.cmml" xref="S4.E13.m1.5.5.1.1.2.4">superscript</csymbol><apply id="S4.E13.m1.5.5.1.1.2.4.2.cmml" xref="S4.E13.m1.5.5.1.1.2.4"><csymbol cd="ambiguous" id="S4.E13.m1.5.5.1.1.2.4.2.1.cmml" xref="S4.E13.m1.5.5.1.1.2.4">subscript</csymbol><ci id="S4.E13.m1.5.5.1.1.2.4.2.2.cmml" xref="S4.E13.m1.5.5.1.1.2.4.2.2">Φ</ci><ci id="S4.E13.m1.5.5.1.1.2.4.2.3.cmml" xref="S4.E13.m1.5.5.1.1.2.4.2.3">s</ci></apply><ci id="S4.E13.m1.5.5.1.1.2.4.3.cmml" xref="S4.E13.m1.5.5.1.1.2.4.3">𝑇</ci></apply><ci id="S4.E13.m1.2.2.cmml" xref="S4.E13.m1.2.2">𝑡</ci><ci id="S4.E13.m1.5.5.1.1.2.6.cmml" xref="S4.E13.m1.5.5.1.1.2.6">𝜽</ci></apply><apply id="S4.E13.m1.5.5.1.1.3.cmml" xref="S4.E13.m1.5.5.1.1.3"><times id="S4.E13.m1.5.5.1.1.3.1.cmml" xref="S4.E13.m1.5.5.1.1.3.1"></times><apply id="S4.E13.m1.5.5.1.1.3.2.cmml" xref="S4.E13.m1.5.5.1.1.3.2"><csymbol cd="ambiguous" id="S4.E13.m1.5.5.1.1.3.2.1.cmml" xref="S4.E13.m1.5.5.1.1.3.2">subscript</csymbol><ci id="S4.E13.m1.5.5.1.1.3.2.2.cmml" xref="S4.E13.m1.5.5.1.1.3.2.2">Φ</ci><ci id="S4.E13.m1.5.5.1.1.3.2.3.cmml" xref="S4.E13.m1.5.5.1.1.3.2.3">s</ci></apply><ci id="S4.E13.m1.3.3.cmml" xref="S4.E13.m1.3.3">𝑡</ci><ci id="S4.E13.m1.5.5.1.1.3.4.cmml" xref="S4.E13.m1.5.5.1.1.3.4">𝒑</ci><ci id="S4.E13.m1.4.4.cmml" xref="S4.E13.m1.4.4">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E13.m1.5c">\displaystyle\Phi_{\rm s}(t)\Phi_{\rm s}^{T}(t)\bm{\theta}=\Phi_{\rm s}(t)\bm{% p}(t).</annotation><annotation encoding="application/x-llamapun" id="S4.E13.m1.5d">roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_t ) roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_t ) bold_italic_θ = roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_t ) bold_italic_p ( italic_t ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(13)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p3.4">Integrating (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E13" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">13</span></a>) over a moving time window <math alttext="[t-\tau_{\rm d},t]" class="ltx_Math" display="inline" id="S4.SS1.p3.4.m1.2"><semantics id="S4.SS1.p3.4.m1.2a"><mrow id="S4.SS1.p3.4.m1.2.2.1" xref="S4.SS1.p3.4.m1.2.2.2.cmml"><mo id="S4.SS1.p3.4.m1.2.2.1.2" stretchy="false" xref="S4.SS1.p3.4.m1.2.2.2.cmml">[</mo><mrow id="S4.SS1.p3.4.m1.2.2.1.1" xref="S4.SS1.p3.4.m1.2.2.1.1.cmml"><mi id="S4.SS1.p3.4.m1.2.2.1.1.2" xref="S4.SS1.p3.4.m1.2.2.1.1.2.cmml">t</mi><mo id="S4.SS1.p3.4.m1.2.2.1.1.1" xref="S4.SS1.p3.4.m1.2.2.1.1.1.cmml">−</mo><msub id="S4.SS1.p3.4.m1.2.2.1.1.3" xref="S4.SS1.p3.4.m1.2.2.1.1.3.cmml"><mi id="S4.SS1.p3.4.m1.2.2.1.1.3.2" xref="S4.SS1.p3.4.m1.2.2.1.1.3.2.cmml">τ</mi><mi id="S4.SS1.p3.4.m1.2.2.1.1.3.3" mathvariant="normal" xref="S4.SS1.p3.4.m1.2.2.1.1.3.3.cmml">d</mi></msub></mrow><mo id="S4.SS1.p3.4.m1.2.2.1.3" xref="S4.SS1.p3.4.m1.2.2.2.cmml">,</mo><mi id="S4.SS1.p3.4.m1.1.1" xref="S4.SS1.p3.4.m1.1.1.cmml">t</mi><mo id="S4.SS1.p3.4.m1.2.2.1.4" stretchy="false" xref="S4.SS1.p3.4.m1.2.2.2.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.4.m1.2b"><interval closure="closed" id="S4.SS1.p3.4.m1.2.2.2.cmml" xref="S4.SS1.p3.4.m1.2.2.1"><apply id="S4.SS1.p3.4.m1.2.2.1.1.cmml" xref="S4.SS1.p3.4.m1.2.2.1.1"><minus id="S4.SS1.p3.4.m1.2.2.1.1.1.cmml" xref="S4.SS1.p3.4.m1.2.2.1.1.1"></minus><ci id="S4.SS1.p3.4.m1.2.2.1.1.2.cmml" xref="S4.SS1.p3.4.m1.2.2.1.1.2">𝑡</ci><apply id="S4.SS1.p3.4.m1.2.2.1.1.3.cmml" xref="S4.SS1.p3.4.m1.2.2.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p3.4.m1.2.2.1.1.3.1.cmml" xref="S4.SS1.p3.4.m1.2.2.1.1.3">subscript</csymbol><ci id="S4.SS1.p3.4.m1.2.2.1.1.3.2.cmml" xref="S4.SS1.p3.4.m1.2.2.1.1.3.2">𝜏</ci><ci id="S4.SS1.p3.4.m1.2.2.1.1.3.3.cmml" xref="S4.SS1.p3.4.m1.2.2.1.1.3.3">d</ci></apply></apply><ci id="S4.SS1.p3.4.m1.1.1.cmml" xref="S4.SS1.p3.4.m1.1.1">𝑡</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.4.m1.2c">[t-\tau_{\rm d},t]</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.4.m1.2d">[ italic_t - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT , italic_t ]</annotation></semantics></math>, one obtains a generalized regression equation</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx9"> <tbody id="S4.E14"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\Psi(t)\bm{\theta}=\bm{q}(t)" class="ltx_Math" display="inline" id="S4.E14.m1.2"><semantics id="S4.E14.m1.2a"><mrow id="S4.E14.m1.2.3" xref="S4.E14.m1.2.3.cmml"><mrow id="S4.E14.m1.2.3.2" xref="S4.E14.m1.2.3.2.cmml"><mi id="S4.E14.m1.2.3.2.2" mathvariant="normal" xref="S4.E14.m1.2.3.2.2.cmml">Ψ</mi><mo id="S4.E14.m1.2.3.2.1" xref="S4.E14.m1.2.3.2.1.cmml">⁢</mo><mrow id="S4.E14.m1.2.3.2.3.2" xref="S4.E14.m1.2.3.2.cmml"><mo id="S4.E14.m1.2.3.2.3.2.1" stretchy="false" xref="S4.E14.m1.2.3.2.cmml">(</mo><mi id="S4.E14.m1.1.1" xref="S4.E14.m1.1.1.cmml">t</mi><mo id="S4.E14.m1.2.3.2.3.2.2" stretchy="false" xref="S4.E14.m1.2.3.2.cmml">)</mo></mrow><mo id="S4.E14.m1.2.3.2.1a" xref="S4.E14.m1.2.3.2.1.cmml">⁢</mo><mi id="S4.E14.m1.2.3.2.4" xref="S4.E14.m1.2.3.2.4.cmml">𝜽</mi></mrow><mo id="S4.E14.m1.2.3.1" xref="S4.E14.m1.2.3.1.cmml">=</mo><mrow id="S4.E14.m1.2.3.3" xref="S4.E14.m1.2.3.3.cmml"><mi id="S4.E14.m1.2.3.3.2" xref="S4.E14.m1.2.3.3.2.cmml">𝒒</mi><mo id="S4.E14.m1.2.3.3.1" xref="S4.E14.m1.2.3.3.1.cmml">⁢</mo><mrow id="S4.E14.m1.2.3.3.3.2" xref="S4.E14.m1.2.3.3.cmml"><mo id="S4.E14.m1.2.3.3.3.2.1" stretchy="false" xref="S4.E14.m1.2.3.3.cmml">(</mo><mi id="S4.E14.m1.2.2" xref="S4.E14.m1.2.2.cmml">t</mi><mo id="S4.E14.m1.2.3.3.3.2.2" stretchy="false" xref="S4.E14.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E14.m1.2b"><apply id="S4.E14.m1.2.3.cmml" xref="S4.E14.m1.2.3"><eq id="S4.E14.m1.2.3.1.cmml" xref="S4.E14.m1.2.3.1"></eq><apply id="S4.E14.m1.2.3.2.cmml" xref="S4.E14.m1.2.3.2"><times id="S4.E14.m1.2.3.2.1.cmml" xref="S4.E14.m1.2.3.2.1"></times><ci id="S4.E14.m1.2.3.2.2.cmml" xref="S4.E14.m1.2.3.2.2">Ψ</ci><ci id="S4.E14.m1.1.1.cmml" xref="S4.E14.m1.1.1">𝑡</ci><ci id="S4.E14.m1.2.3.2.4.cmml" xref="S4.E14.m1.2.3.2.4">𝜽</ci></apply><apply id="S4.E14.m1.2.3.3.cmml" xref="S4.E14.m1.2.3.3"><times id="S4.E14.m1.2.3.3.1.cmml" xref="S4.E14.m1.2.3.3.1"></times><ci id="S4.E14.m1.2.3.3.2.cmml" xref="S4.E14.m1.2.3.3.2">𝒒</ci><ci id="S4.E14.m1.2.2.cmml" xref="S4.E14.m1.2.2">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E14.m1.2c">\displaystyle\Psi(t)\bm{\theta}=\bm{q}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.E14.m1.2d">roman_Ψ ( italic_t ) bold_italic_θ = bold_italic_q ( italic_t )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(14)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p3.5">where <math alttext="\Psi(t)\in\mathbb{R}^{N\times N}" class="ltx_Math" display="inline" id="S4.SS1.p3.5.m1.1"><semantics id="S4.SS1.p3.5.m1.1a"><mrow id="S4.SS1.p3.5.m1.1.2" xref="S4.SS1.p3.5.m1.1.2.cmml"><mrow id="S4.SS1.p3.5.m1.1.2.2" xref="S4.SS1.p3.5.m1.1.2.2.cmml"><mi id="S4.SS1.p3.5.m1.1.2.2.2" mathvariant="normal" xref="S4.SS1.p3.5.m1.1.2.2.2.cmml">Ψ</mi><mo id="S4.SS1.p3.5.m1.1.2.2.1" xref="S4.SS1.p3.5.m1.1.2.2.1.cmml">⁢</mo><mrow id="S4.SS1.p3.5.m1.1.2.2.3.2" xref="S4.SS1.p3.5.m1.1.2.2.cmml"><mo id="S4.SS1.p3.5.m1.1.2.2.3.2.1" stretchy="false" xref="S4.SS1.p3.5.m1.1.2.2.cmml">(</mo><mi id="S4.SS1.p3.5.m1.1.1" xref="S4.SS1.p3.5.m1.1.1.cmml">t</mi><mo id="S4.SS1.p3.5.m1.1.2.2.3.2.2" stretchy="false" xref="S4.SS1.p3.5.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p3.5.m1.1.2.1" xref="S4.SS1.p3.5.m1.1.2.1.cmml">∈</mo><msup id="S4.SS1.p3.5.m1.1.2.3" xref="S4.SS1.p3.5.m1.1.2.3.cmml"><mi id="S4.SS1.p3.5.m1.1.2.3.2" xref="S4.SS1.p3.5.m1.1.2.3.2.cmml">ℝ</mi><mrow id="S4.SS1.p3.5.m1.1.2.3.3" xref="S4.SS1.p3.5.m1.1.2.3.3.cmml"><mi id="S4.SS1.p3.5.m1.1.2.3.3.2" xref="S4.SS1.p3.5.m1.1.2.3.3.2.cmml">N</mi><mo id="S4.SS1.p3.5.m1.1.2.3.3.1" lspace="0.222em" rspace="0.222em" xref="S4.SS1.p3.5.m1.1.2.3.3.1.cmml">×</mo><mi id="S4.SS1.p3.5.m1.1.2.3.3.3" xref="S4.SS1.p3.5.m1.1.2.3.3.3.cmml">N</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.5.m1.1b"><apply id="S4.SS1.p3.5.m1.1.2.cmml" xref="S4.SS1.p3.5.m1.1.2"><in id="S4.SS1.p3.5.m1.1.2.1.cmml" xref="S4.SS1.p3.5.m1.1.2.1"></in><apply id="S4.SS1.p3.5.m1.1.2.2.cmml" xref="S4.SS1.p3.5.m1.1.2.2"><times id="S4.SS1.p3.5.m1.1.2.2.1.cmml" xref="S4.SS1.p3.5.m1.1.2.2.1"></times><ci id="S4.SS1.p3.5.m1.1.2.2.2.cmml" xref="S4.SS1.p3.5.m1.1.2.2.2">Ψ</ci><ci id="S4.SS1.p3.5.m1.1.1.cmml" xref="S4.SS1.p3.5.m1.1.1">𝑡</ci></apply><apply id="S4.SS1.p3.5.m1.1.2.3.cmml" xref="S4.SS1.p3.5.m1.1.2.3"><csymbol cd="ambiguous" id="S4.SS1.p3.5.m1.1.2.3.1.cmml" xref="S4.SS1.p3.5.m1.1.2.3">superscript</csymbol><ci id="S4.SS1.p3.5.m1.1.2.3.2.cmml" xref="S4.SS1.p3.5.m1.1.2.3.2">ℝ</ci><apply id="S4.SS1.p3.5.m1.1.2.3.3.cmml" xref="S4.SS1.p3.5.m1.1.2.3.3"><times id="S4.SS1.p3.5.m1.1.2.3.3.1.cmml" xref="S4.SS1.p3.5.m1.1.2.3.3.1"></times><ci id="S4.SS1.p3.5.m1.1.2.3.3.2.cmml" xref="S4.SS1.p3.5.m1.1.2.3.3.2">𝑁</ci><ci id="S4.SS1.p3.5.m1.1.2.3.3.3.cmml" xref="S4.SS1.p3.5.m1.1.2.3.3.3">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.5.m1.1c">\Psi(t)\in\mathbb{R}^{N\times N}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.5.m1.1d">roman_Ψ ( italic_t ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_N × italic_N end_POSTSUPERSCRIPT</annotation></semantics></math> is an excitation matrix given by</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx10"> <tbody id="S4.E15"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\Psi(t):=\int_{t-\tau_{\rm d}}^{t}\Phi_{\rm s}(\tau)\Phi_{\rm s}^% {T}(\tau)d\tau" class="ltx_Math" display="inline" id="S4.E15.m1.3"><semantics id="S4.E15.m1.3a"><mrow id="S4.E15.m1.3.4" xref="S4.E15.m1.3.4.cmml"><mrow id="S4.E15.m1.3.4.2" xref="S4.E15.m1.3.4.2.cmml"><mi id="S4.E15.m1.3.4.2.2" mathvariant="normal" xref="S4.E15.m1.3.4.2.2.cmml">Ψ</mi><mo id="S4.E15.m1.3.4.2.1" xref="S4.E15.m1.3.4.2.1.cmml">⁢</mo><mrow id="S4.E15.m1.3.4.2.3.2" xref="S4.E15.m1.3.4.2.cmml"><mo id="S4.E15.m1.3.4.2.3.2.1" stretchy="false" xref="S4.E15.m1.3.4.2.cmml">(</mo><mi id="S4.E15.m1.1.1" xref="S4.E15.m1.1.1.cmml">t</mi><mo id="S4.E15.m1.3.4.2.3.2.2" rspace="0.278em" stretchy="false" xref="S4.E15.m1.3.4.2.cmml">)</mo></mrow></mrow><mo id="S4.E15.m1.3.4.1" rspace="0.278em" xref="S4.E15.m1.3.4.1.cmml">:=</mo><mrow id="S4.E15.m1.3.4.3" xref="S4.E15.m1.3.4.3.cmml"><mstyle displaystyle="true" id="S4.E15.m1.3.4.3.1" xref="S4.E15.m1.3.4.3.1.cmml"><msubsup id="S4.E15.m1.3.4.3.1a" xref="S4.E15.m1.3.4.3.1.cmml"><mo id="S4.E15.m1.3.4.3.1.2.2" xref="S4.E15.m1.3.4.3.1.2.2.cmml">∫</mo><mrow id="S4.E15.m1.3.4.3.1.2.3" xref="S4.E15.m1.3.4.3.1.2.3.cmml"><mi id="S4.E15.m1.3.4.3.1.2.3.2" xref="S4.E15.m1.3.4.3.1.2.3.2.cmml">t</mi><mo id="S4.E15.m1.3.4.3.1.2.3.1" xref="S4.E15.m1.3.4.3.1.2.3.1.cmml">−</mo><msub id="S4.E15.m1.3.4.3.1.2.3.3" xref="S4.E15.m1.3.4.3.1.2.3.3.cmml"><mi id="S4.E15.m1.3.4.3.1.2.3.3.2" xref="S4.E15.m1.3.4.3.1.2.3.3.2.cmml">τ</mi><mi id="S4.E15.m1.3.4.3.1.2.3.3.3" mathvariant="normal" xref="S4.E15.m1.3.4.3.1.2.3.3.3.cmml">d</mi></msub></mrow><mi id="S4.E15.m1.3.4.3.1.3" xref="S4.E15.m1.3.4.3.1.3.cmml">t</mi></msubsup></mstyle><mrow id="S4.E15.m1.3.4.3.2" xref="S4.E15.m1.3.4.3.2.cmml"><msub id="S4.E15.m1.3.4.3.2.2" xref="S4.E15.m1.3.4.3.2.2.cmml"><mi id="S4.E15.m1.3.4.3.2.2.2" mathvariant="normal" xref="S4.E15.m1.3.4.3.2.2.2.cmml">Φ</mi><mi id="S4.E15.m1.3.4.3.2.2.3" mathvariant="normal" xref="S4.E15.m1.3.4.3.2.2.3.cmml">s</mi></msub><mo id="S4.E15.m1.3.4.3.2.1" xref="S4.E15.m1.3.4.3.2.1.cmml">⁢</mo><mrow id="S4.E15.m1.3.4.3.2.3.2" xref="S4.E15.m1.3.4.3.2.cmml"><mo id="S4.E15.m1.3.4.3.2.3.2.1" stretchy="false" xref="S4.E15.m1.3.4.3.2.cmml">(</mo><mi id="S4.E15.m1.2.2" xref="S4.E15.m1.2.2.cmml">τ</mi><mo id="S4.E15.m1.3.4.3.2.3.2.2" stretchy="false" xref="S4.E15.m1.3.4.3.2.cmml">)</mo></mrow><mo id="S4.E15.m1.3.4.3.2.1a" xref="S4.E15.m1.3.4.3.2.1.cmml">⁢</mo><msubsup id="S4.E15.m1.3.4.3.2.4" xref="S4.E15.m1.3.4.3.2.4.cmml"><mi id="S4.E15.m1.3.4.3.2.4.2.2" mathvariant="normal" xref="S4.E15.m1.3.4.3.2.4.2.2.cmml">Φ</mi><mi id="S4.E15.m1.3.4.3.2.4.2.3" mathvariant="normal" xref="S4.E15.m1.3.4.3.2.4.2.3.cmml">s</mi><mi id="S4.E15.m1.3.4.3.2.4.3" xref="S4.E15.m1.3.4.3.2.4.3.cmml">T</mi></msubsup><mo id="S4.E15.m1.3.4.3.2.1b" xref="S4.E15.m1.3.4.3.2.1.cmml">⁢</mo><mrow id="S4.E15.m1.3.4.3.2.5.2" xref="S4.E15.m1.3.4.3.2.cmml"><mo id="S4.E15.m1.3.4.3.2.5.2.1" stretchy="false" xref="S4.E15.m1.3.4.3.2.cmml">(</mo><mi id="S4.E15.m1.3.3" xref="S4.E15.m1.3.3.cmml">τ</mi><mo id="S4.E15.m1.3.4.3.2.5.2.2" stretchy="false" xref="S4.E15.m1.3.4.3.2.cmml">)</mo></mrow><mo id="S4.E15.m1.3.4.3.2.1c" lspace="0em" xref="S4.E15.m1.3.4.3.2.1.cmml">⁢</mo><mrow id="S4.E15.m1.3.4.3.2.6" xref="S4.E15.m1.3.4.3.2.6.cmml"><mo id="S4.E15.m1.3.4.3.2.6.1" rspace="0em" xref="S4.E15.m1.3.4.3.2.6.1.cmml">𝑑</mo><mi id="S4.E15.m1.3.4.3.2.6.2" xref="S4.E15.m1.3.4.3.2.6.2.cmml">τ</mi></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E15.m1.3b"><apply id="S4.E15.m1.3.4.cmml" xref="S4.E15.m1.3.4"><csymbol cd="latexml" id="S4.E15.m1.3.4.1.cmml" xref="S4.E15.m1.3.4.1">assign</csymbol><apply id="S4.E15.m1.3.4.2.cmml" xref="S4.E15.m1.3.4.2"><times id="S4.E15.m1.3.4.2.1.cmml" xref="S4.E15.m1.3.4.2.1"></times><ci id="S4.E15.m1.3.4.2.2.cmml" xref="S4.E15.m1.3.4.2.2">Ψ</ci><ci id="S4.E15.m1.1.1.cmml" xref="S4.E15.m1.1.1">𝑡</ci></apply><apply id="S4.E15.m1.3.4.3.cmml" xref="S4.E15.m1.3.4.3"><apply id="S4.E15.m1.3.4.3.1.cmml" xref="S4.E15.m1.3.4.3.1"><csymbol cd="ambiguous" id="S4.E15.m1.3.4.3.1.1.cmml" xref="S4.E15.m1.3.4.3.1">superscript</csymbol><apply id="S4.E15.m1.3.4.3.1.2.cmml" xref="S4.E15.m1.3.4.3.1"><csymbol cd="ambiguous" id="S4.E15.m1.3.4.3.1.2.1.cmml" xref="S4.E15.m1.3.4.3.1">subscript</csymbol><int id="S4.E15.m1.3.4.3.1.2.2.cmml" xref="S4.E15.m1.3.4.3.1.2.2"></int><apply id="S4.E15.m1.3.4.3.1.2.3.cmml" xref="S4.E15.m1.3.4.3.1.2.3"><minus id="S4.E15.m1.3.4.3.1.2.3.1.cmml" xref="S4.E15.m1.3.4.3.1.2.3.1"></minus><ci id="S4.E15.m1.3.4.3.1.2.3.2.cmml" xref="S4.E15.m1.3.4.3.1.2.3.2">𝑡</ci><apply id="S4.E15.m1.3.4.3.1.2.3.3.cmml" xref="S4.E15.m1.3.4.3.1.2.3.3"><csymbol cd="ambiguous" id="S4.E15.m1.3.4.3.1.2.3.3.1.cmml" xref="S4.E15.m1.3.4.3.1.2.3.3">subscript</csymbol><ci id="S4.E15.m1.3.4.3.1.2.3.3.2.cmml" xref="S4.E15.m1.3.4.3.1.2.3.3.2">𝜏</ci><ci id="S4.E15.m1.3.4.3.1.2.3.3.3.cmml" xref="S4.E15.m1.3.4.3.1.2.3.3.3">d</ci></apply></apply></apply><ci id="S4.E15.m1.3.4.3.1.3.cmml" xref="S4.E15.m1.3.4.3.1.3">𝑡</ci></apply><apply id="S4.E15.m1.3.4.3.2.cmml" xref="S4.E15.m1.3.4.3.2"><times id="S4.E15.m1.3.4.3.2.1.cmml" xref="S4.E15.m1.3.4.3.2.1"></times><apply id="S4.E15.m1.3.4.3.2.2.cmml" xref="S4.E15.m1.3.4.3.2.2"><csymbol cd="ambiguous" id="S4.E15.m1.3.4.3.2.2.1.cmml" xref="S4.E15.m1.3.4.3.2.2">subscript</csymbol><ci id="S4.E15.m1.3.4.3.2.2.2.cmml" xref="S4.E15.m1.3.4.3.2.2.2">Φ</ci><ci id="S4.E15.m1.3.4.3.2.2.3.cmml" xref="S4.E15.m1.3.4.3.2.2.3">s</ci></apply><ci id="S4.E15.m1.2.2.cmml" xref="S4.E15.m1.2.2">𝜏</ci><apply id="S4.E15.m1.3.4.3.2.4.cmml" xref="S4.E15.m1.3.4.3.2.4"><csymbol cd="ambiguous" id="S4.E15.m1.3.4.3.2.4.1.cmml" xref="S4.E15.m1.3.4.3.2.4">superscript</csymbol><apply id="S4.E15.m1.3.4.3.2.4.2.cmml" xref="S4.E15.m1.3.4.3.2.4"><csymbol cd="ambiguous" id="S4.E15.m1.3.4.3.2.4.2.1.cmml" xref="S4.E15.m1.3.4.3.2.4">subscript</csymbol><ci id="S4.E15.m1.3.4.3.2.4.2.2.cmml" xref="S4.E15.m1.3.4.3.2.4.2.2">Φ</ci><ci id="S4.E15.m1.3.4.3.2.4.2.3.cmml" xref="S4.E15.m1.3.4.3.2.4.2.3">s</ci></apply><ci id="S4.E15.m1.3.4.3.2.4.3.cmml" xref="S4.E15.m1.3.4.3.2.4.3">𝑇</ci></apply><ci id="S4.E15.m1.3.3.cmml" xref="S4.E15.m1.3.3">𝜏</ci><apply id="S4.E15.m1.3.4.3.2.6.cmml" xref="S4.E15.m1.3.4.3.2.6"><csymbol cd="latexml" id="S4.E15.m1.3.4.3.2.6.1.cmml" xref="S4.E15.m1.3.4.3.2.6.1">differential-d</csymbol><ci id="S4.E15.m1.3.4.3.2.6.2.cmml" xref="S4.E15.m1.3.4.3.2.6.2">𝜏</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E15.m1.3c">\displaystyle\Psi(t):=\int_{t-\tau_{\rm d}}^{t}\Phi_{\rm s}(\tau)\Phi_{\rm s}^% {T}(\tau)d\tau</annotation><annotation encoding="application/x-llamapun" id="S4.E15.m1.3d">roman_Ψ ( italic_t ) := ∫ start_POSTSUBSCRIPT italic_t - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_τ ) roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_τ ) italic_d italic_τ</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(15)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p3.6">and <math alttext="\bm{q}(t)\in\mathbb{R}^{N}" class="ltx_Math" display="inline" id="S4.SS1.p3.6.m1.1"><semantics id="S4.SS1.p3.6.m1.1a"><mrow id="S4.SS1.p3.6.m1.1.2" xref="S4.SS1.p3.6.m1.1.2.cmml"><mrow id="S4.SS1.p3.6.m1.1.2.2" xref="S4.SS1.p3.6.m1.1.2.2.cmml"><mi id="S4.SS1.p3.6.m1.1.2.2.2" xref="S4.SS1.p3.6.m1.1.2.2.2.cmml">𝒒</mi><mo id="S4.SS1.p3.6.m1.1.2.2.1" xref="S4.SS1.p3.6.m1.1.2.2.1.cmml">⁢</mo><mrow id="S4.SS1.p3.6.m1.1.2.2.3.2" xref="S4.SS1.p3.6.m1.1.2.2.cmml"><mo id="S4.SS1.p3.6.m1.1.2.2.3.2.1" stretchy="false" xref="S4.SS1.p3.6.m1.1.2.2.cmml">(</mo><mi id="S4.SS1.p3.6.m1.1.1" xref="S4.SS1.p3.6.m1.1.1.cmml">t</mi><mo id="S4.SS1.p3.6.m1.1.2.2.3.2.2" stretchy="false" xref="S4.SS1.p3.6.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p3.6.m1.1.2.1" xref="S4.SS1.p3.6.m1.1.2.1.cmml">∈</mo><msup id="S4.SS1.p3.6.m1.1.2.3" xref="S4.SS1.p3.6.m1.1.2.3.cmml"><mi id="S4.SS1.p3.6.m1.1.2.3.2" xref="S4.SS1.p3.6.m1.1.2.3.2.cmml">ℝ</mi><mi id="S4.SS1.p3.6.m1.1.2.3.3" xref="S4.SS1.p3.6.m1.1.2.3.3.cmml">N</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.6.m1.1b"><apply id="S4.SS1.p3.6.m1.1.2.cmml" xref="S4.SS1.p3.6.m1.1.2"><in id="S4.SS1.p3.6.m1.1.2.1.cmml" xref="S4.SS1.p3.6.m1.1.2.1"></in><apply id="S4.SS1.p3.6.m1.1.2.2.cmml" xref="S4.SS1.p3.6.m1.1.2.2"><times id="S4.SS1.p3.6.m1.1.2.2.1.cmml" xref="S4.SS1.p3.6.m1.1.2.2.1"></times><ci id="S4.SS1.p3.6.m1.1.2.2.2.cmml" xref="S4.SS1.p3.6.m1.1.2.2.2">𝒒</ci><ci id="S4.SS1.p3.6.m1.1.1.cmml" xref="S4.SS1.p3.6.m1.1.1">𝑡</ci></apply><apply id="S4.SS1.p3.6.m1.1.2.3.cmml" xref="S4.SS1.p3.6.m1.1.2.3"><csymbol cd="ambiguous" id="S4.SS1.p3.6.m1.1.2.3.1.cmml" xref="S4.SS1.p3.6.m1.1.2.3">superscript</csymbol><ci id="S4.SS1.p3.6.m1.1.2.3.2.cmml" xref="S4.SS1.p3.6.m1.1.2.3.2">ℝ</ci><ci id="S4.SS1.p3.6.m1.1.2.3.3.cmml" xref="S4.SS1.p3.6.m1.1.2.3.3">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.6.m1.1c">\bm{q}(t)\in\mathbb{R}^{N}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.6.m1.1d">bold_italic_q ( italic_t ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT</annotation></semantics></math> is an auxiliary variable given by</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx11"> <tbody id="S4.E16"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\bm{q}(t):=\int_{t-\tau_{\rm d}}^{t}\Phi_{\rm s}(\tau)\bm{p}(\tau% )d\tau." class="ltx_Math" display="inline" id="S4.E16.m1.4"><semantics id="S4.E16.m1.4a"><mrow id="S4.E16.m1.4.4.1" xref="S4.E16.m1.4.4.1.1.cmml"><mrow id="S4.E16.m1.4.4.1.1" xref="S4.E16.m1.4.4.1.1.cmml"><mrow id="S4.E16.m1.4.4.1.1.2" xref="S4.E16.m1.4.4.1.1.2.cmml"><mi id="S4.E16.m1.4.4.1.1.2.2" xref="S4.E16.m1.4.4.1.1.2.2.cmml">𝒒</mi><mo id="S4.E16.m1.4.4.1.1.2.1" xref="S4.E16.m1.4.4.1.1.2.1.cmml">⁢</mo><mrow id="S4.E16.m1.4.4.1.1.2.3.2" xref="S4.E16.m1.4.4.1.1.2.cmml"><mo id="S4.E16.m1.4.4.1.1.2.3.2.1" stretchy="false" xref="S4.E16.m1.4.4.1.1.2.cmml">(</mo><mi id="S4.E16.m1.1.1" xref="S4.E16.m1.1.1.cmml">t</mi><mo id="S4.E16.m1.4.4.1.1.2.3.2.2" rspace="0.278em" stretchy="false" xref="S4.E16.m1.4.4.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.E16.m1.4.4.1.1.1" rspace="0.278em" xref="S4.E16.m1.4.4.1.1.1.cmml">:=</mo><mrow id="S4.E16.m1.4.4.1.1.3" xref="S4.E16.m1.4.4.1.1.3.cmml"><mstyle displaystyle="true" id="S4.E16.m1.4.4.1.1.3.1" xref="S4.E16.m1.4.4.1.1.3.1.cmml"><msubsup id="S4.E16.m1.4.4.1.1.3.1a" xref="S4.E16.m1.4.4.1.1.3.1.cmml"><mo id="S4.E16.m1.4.4.1.1.3.1.2.2" xref="S4.E16.m1.4.4.1.1.3.1.2.2.cmml">∫</mo><mrow id="S4.E16.m1.4.4.1.1.3.1.2.3" xref="S4.E16.m1.4.4.1.1.3.1.2.3.cmml"><mi id="S4.E16.m1.4.4.1.1.3.1.2.3.2" xref="S4.E16.m1.4.4.1.1.3.1.2.3.2.cmml">t</mi><mo id="S4.E16.m1.4.4.1.1.3.1.2.3.1" xref="S4.E16.m1.4.4.1.1.3.1.2.3.1.cmml">−</mo><msub id="S4.E16.m1.4.4.1.1.3.1.2.3.3" xref="S4.E16.m1.4.4.1.1.3.1.2.3.3.cmml"><mi id="S4.E16.m1.4.4.1.1.3.1.2.3.3.2" xref="S4.E16.m1.4.4.1.1.3.1.2.3.3.2.cmml">τ</mi><mi id="S4.E16.m1.4.4.1.1.3.1.2.3.3.3" mathvariant="normal" xref="S4.E16.m1.4.4.1.1.3.1.2.3.3.3.cmml">d</mi></msub></mrow><mi id="S4.E16.m1.4.4.1.1.3.1.3" xref="S4.E16.m1.4.4.1.1.3.1.3.cmml">t</mi></msubsup></mstyle><mrow id="S4.E16.m1.4.4.1.1.3.2" xref="S4.E16.m1.4.4.1.1.3.2.cmml"><msub id="S4.E16.m1.4.4.1.1.3.2.2" xref="S4.E16.m1.4.4.1.1.3.2.2.cmml"><mi id="S4.E16.m1.4.4.1.1.3.2.2.2" mathvariant="normal" xref="S4.E16.m1.4.4.1.1.3.2.2.2.cmml">Φ</mi><mi id="S4.E16.m1.4.4.1.1.3.2.2.3" mathvariant="normal" xref="S4.E16.m1.4.4.1.1.3.2.2.3.cmml">s</mi></msub><mo id="S4.E16.m1.4.4.1.1.3.2.1" xref="S4.E16.m1.4.4.1.1.3.2.1.cmml">⁢</mo><mrow id="S4.E16.m1.4.4.1.1.3.2.3.2" xref="S4.E16.m1.4.4.1.1.3.2.cmml"><mo id="S4.E16.m1.4.4.1.1.3.2.3.2.1" stretchy="false" xref="S4.E16.m1.4.4.1.1.3.2.cmml">(</mo><mi id="S4.E16.m1.2.2" xref="S4.E16.m1.2.2.cmml">τ</mi><mo id="S4.E16.m1.4.4.1.1.3.2.3.2.2" stretchy="false" xref="S4.E16.m1.4.4.1.1.3.2.cmml">)</mo></mrow><mo id="S4.E16.m1.4.4.1.1.3.2.1a" xref="S4.E16.m1.4.4.1.1.3.2.1.cmml">⁢</mo><mi id="S4.E16.m1.4.4.1.1.3.2.4" xref="S4.E16.m1.4.4.1.1.3.2.4.cmml">𝒑</mi><mo id="S4.E16.m1.4.4.1.1.3.2.1b" xref="S4.E16.m1.4.4.1.1.3.2.1.cmml">⁢</mo><mrow id="S4.E16.m1.4.4.1.1.3.2.5.2" xref="S4.E16.m1.4.4.1.1.3.2.cmml"><mo id="S4.E16.m1.4.4.1.1.3.2.5.2.1" stretchy="false" xref="S4.E16.m1.4.4.1.1.3.2.cmml">(</mo><mi id="S4.E16.m1.3.3" xref="S4.E16.m1.3.3.cmml">τ</mi><mo id="S4.E16.m1.4.4.1.1.3.2.5.2.2" stretchy="false" xref="S4.E16.m1.4.4.1.1.3.2.cmml">)</mo></mrow><mo id="S4.E16.m1.4.4.1.1.3.2.1c" lspace="0em" xref="S4.E16.m1.4.4.1.1.3.2.1.cmml">⁢</mo><mrow id="S4.E16.m1.4.4.1.1.3.2.6" xref="S4.E16.m1.4.4.1.1.3.2.6.cmml"><mo id="S4.E16.m1.4.4.1.1.3.2.6.1" rspace="0em" xref="S4.E16.m1.4.4.1.1.3.2.6.1.cmml">𝑑</mo><mi id="S4.E16.m1.4.4.1.1.3.2.6.2" xref="S4.E16.m1.4.4.1.1.3.2.6.2.cmml">τ</mi></mrow></mrow></mrow></mrow><mo id="S4.E16.m1.4.4.1.2" lspace="0em" xref="S4.E16.m1.4.4.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E16.m1.4b"><apply id="S4.E16.m1.4.4.1.1.cmml" xref="S4.E16.m1.4.4.1"><csymbol cd="latexml" id="S4.E16.m1.4.4.1.1.1.cmml" xref="S4.E16.m1.4.4.1.1.1">assign</csymbol><apply id="S4.E16.m1.4.4.1.1.2.cmml" xref="S4.E16.m1.4.4.1.1.2"><times id="S4.E16.m1.4.4.1.1.2.1.cmml" xref="S4.E16.m1.4.4.1.1.2.1"></times><ci id="S4.E16.m1.4.4.1.1.2.2.cmml" xref="S4.E16.m1.4.4.1.1.2.2">𝒒</ci><ci id="S4.E16.m1.1.1.cmml" xref="S4.E16.m1.1.1">𝑡</ci></apply><apply id="S4.E16.m1.4.4.1.1.3.cmml" xref="S4.E16.m1.4.4.1.1.3"><apply id="S4.E16.m1.4.4.1.1.3.1.cmml" xref="S4.E16.m1.4.4.1.1.3.1"><csymbol cd="ambiguous" id="S4.E16.m1.4.4.1.1.3.1.1.cmml" xref="S4.E16.m1.4.4.1.1.3.1">superscript</csymbol><apply id="S4.E16.m1.4.4.1.1.3.1.2.cmml" xref="S4.E16.m1.4.4.1.1.3.1"><csymbol cd="ambiguous" id="S4.E16.m1.4.4.1.1.3.1.2.1.cmml" xref="S4.E16.m1.4.4.1.1.3.1">subscript</csymbol><int id="S4.E16.m1.4.4.1.1.3.1.2.2.cmml" xref="S4.E16.m1.4.4.1.1.3.1.2.2"></int><apply id="S4.E16.m1.4.4.1.1.3.1.2.3.cmml" xref="S4.E16.m1.4.4.1.1.3.1.2.3"><minus id="S4.E16.m1.4.4.1.1.3.1.2.3.1.cmml" xref="S4.E16.m1.4.4.1.1.3.1.2.3.1"></minus><ci id="S4.E16.m1.4.4.1.1.3.1.2.3.2.cmml" xref="S4.E16.m1.4.4.1.1.3.1.2.3.2">𝑡</ci><apply id="S4.E16.m1.4.4.1.1.3.1.2.3.3.cmml" xref="S4.E16.m1.4.4.1.1.3.1.2.3.3"><csymbol cd="ambiguous" id="S4.E16.m1.4.4.1.1.3.1.2.3.3.1.cmml" xref="S4.E16.m1.4.4.1.1.3.1.2.3.3">subscript</csymbol><ci id="S4.E16.m1.4.4.1.1.3.1.2.3.3.2.cmml" xref="S4.E16.m1.4.4.1.1.3.1.2.3.3.2">𝜏</ci><ci id="S4.E16.m1.4.4.1.1.3.1.2.3.3.3.cmml" xref="S4.E16.m1.4.4.1.1.3.1.2.3.3.3">d</ci></apply></apply></apply><ci id="S4.E16.m1.4.4.1.1.3.1.3.cmml" xref="S4.E16.m1.4.4.1.1.3.1.3">𝑡</ci></apply><apply id="S4.E16.m1.4.4.1.1.3.2.cmml" xref="S4.E16.m1.4.4.1.1.3.2"><times id="S4.E16.m1.4.4.1.1.3.2.1.cmml" xref="S4.E16.m1.4.4.1.1.3.2.1"></times><apply id="S4.E16.m1.4.4.1.1.3.2.2.cmml" xref="S4.E16.m1.4.4.1.1.3.2.2"><csymbol cd="ambiguous" id="S4.E16.m1.4.4.1.1.3.2.2.1.cmml" xref="S4.E16.m1.4.4.1.1.3.2.2">subscript</csymbol><ci id="S4.E16.m1.4.4.1.1.3.2.2.2.cmml" xref="S4.E16.m1.4.4.1.1.3.2.2.2">Φ</ci><ci id="S4.E16.m1.4.4.1.1.3.2.2.3.cmml" xref="S4.E16.m1.4.4.1.1.3.2.2.3">s</ci></apply><ci id="S4.E16.m1.2.2.cmml" xref="S4.E16.m1.2.2">𝜏</ci><ci id="S4.E16.m1.4.4.1.1.3.2.4.cmml" xref="S4.E16.m1.4.4.1.1.3.2.4">𝒑</ci><ci id="S4.E16.m1.3.3.cmml" xref="S4.E16.m1.3.3">𝜏</ci><apply id="S4.E16.m1.4.4.1.1.3.2.6.cmml" xref="S4.E16.m1.4.4.1.1.3.2.6"><csymbol cd="latexml" id="S4.E16.m1.4.4.1.1.3.2.6.1.cmml" xref="S4.E16.m1.4.4.1.1.3.2.6.1">differential-d</csymbol><ci id="S4.E16.m1.4.4.1.1.3.2.6.2.cmml" xref="S4.E16.m1.4.4.1.1.3.2.6.2">𝜏</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E16.m1.4c">\displaystyle\bm{q}(t):=\int_{t-\tau_{\rm d}}^{t}\Phi_{\rm s}(\tau)\bm{p}(\tau% )d\tau.</annotation><annotation encoding="application/x-llamapun" id="S4.E16.m1.4d">bold_italic_q ( italic_t ) := ∫ start_POSTSUBSCRIPT italic_t - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_τ ) bold_italic_p ( italic_τ ) italic_d italic_τ .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(16)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p3.12">To obtain the high-order time derivatives <math alttext="\hat{\bm{\theta}}^{(k)}" class="ltx_Math" display="inline" id="S4.SS1.p3.7.m1.1"><semantics id="S4.SS1.p3.7.m1.1a"><msup id="S4.SS1.p3.7.m1.1.2" xref="S4.SS1.p3.7.m1.1.2.cmml"><mover accent="true" id="S4.SS1.p3.7.m1.1.2.2" xref="S4.SS1.p3.7.m1.1.2.2.cmml"><mi id="S4.SS1.p3.7.m1.1.2.2.2" xref="S4.SS1.p3.7.m1.1.2.2.2.cmml">𝜽</mi><mo id="S4.SS1.p3.7.m1.1.2.2.1" xref="S4.SS1.p3.7.m1.1.2.2.1.cmml">^</mo></mover><mrow id="S4.SS1.p3.7.m1.1.1.1.3" xref="S4.SS1.p3.7.m1.1.2.cmml"><mo id="S4.SS1.p3.7.m1.1.1.1.3.1" stretchy="false" xref="S4.SS1.p3.7.m1.1.2.cmml">(</mo><mi id="S4.SS1.p3.7.m1.1.1.1.1" xref="S4.SS1.p3.7.m1.1.1.1.1.cmml">k</mi><mo id="S4.SS1.p3.7.m1.1.1.1.3.2" stretchy="false" xref="S4.SS1.p3.7.m1.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.7.m1.1b"><apply id="S4.SS1.p3.7.m1.1.2.cmml" xref="S4.SS1.p3.7.m1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p3.7.m1.1.2.1.cmml" xref="S4.SS1.p3.7.m1.1.2">superscript</csymbol><apply id="S4.SS1.p3.7.m1.1.2.2.cmml" xref="S4.SS1.p3.7.m1.1.2.2"><ci id="S4.SS1.p3.7.m1.1.2.2.1.cmml" xref="S4.SS1.p3.7.m1.1.2.2.1">^</ci><ci id="S4.SS1.p3.7.m1.1.2.2.2.cmml" xref="S4.SS1.p3.7.m1.1.2.2.2">𝜽</ci></apply><ci id="S4.SS1.p3.7.m1.1.1.1.1.cmml" xref="S4.SS1.p3.7.m1.1.1.1.1">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.7.m1.1c">\hat{\bm{\theta}}^{(k)}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.7.m1.1d">over^ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT</annotation></semantics></math>, a feasible solution is to apply a linear filter with sufficiently high relative degrees to (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E14" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">14</span></a>) and then design an adaptive law of <math alttext="\hat{\bm{\theta}}" class="ltx_Math" display="inline" id="S4.SS1.p3.8.m2.1"><semantics id="S4.SS1.p3.8.m2.1a"><mover accent="true" id="S4.SS1.p3.8.m2.1.1" xref="S4.SS1.p3.8.m2.1.1.cmml"><mi id="S4.SS1.p3.8.m2.1.1.2" xref="S4.SS1.p3.8.m2.1.1.2.cmml">𝜽</mi><mo id="S4.SS1.p3.8.m2.1.1.1" xref="S4.SS1.p3.8.m2.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.8.m2.1b"><apply id="S4.SS1.p3.8.m2.1.1.cmml" xref="S4.SS1.p3.8.m2.1.1"><ci id="S4.SS1.p3.8.m2.1.1.1.cmml" xref="S4.SS1.p3.8.m2.1.1.1">^</ci><ci id="S4.SS1.p3.8.m2.1.1.2.cmml" xref="S4.SS1.p3.8.m2.1.1.2">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.8.m2.1c">\hat{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.8.m2.1d">over^ start_ARG bold_italic_θ end_ARG</annotation></semantics></math>, such that <math alttext="\hat{\bm{\theta}}^{(k)}" class="ltx_Math" display="inline" id="S4.SS1.p3.9.m3.1"><semantics id="S4.SS1.p3.9.m3.1a"><msup id="S4.SS1.p3.9.m3.1.2" xref="S4.SS1.p3.9.m3.1.2.cmml"><mover accent="true" id="S4.SS1.p3.9.m3.1.2.2" xref="S4.SS1.p3.9.m3.1.2.2.cmml"><mi id="S4.SS1.p3.9.m3.1.2.2.2" xref="S4.SS1.p3.9.m3.1.2.2.2.cmml">𝜽</mi><mo id="S4.SS1.p3.9.m3.1.2.2.1" xref="S4.SS1.p3.9.m3.1.2.2.1.cmml">^</mo></mover><mrow id="S4.SS1.p3.9.m3.1.1.1.3" xref="S4.SS1.p3.9.m3.1.2.cmml"><mo id="S4.SS1.p3.9.m3.1.1.1.3.1" stretchy="false" xref="S4.SS1.p3.9.m3.1.2.cmml">(</mo><mi id="S4.SS1.p3.9.m3.1.1.1.1" xref="S4.SS1.p3.9.m3.1.1.1.1.cmml">k</mi><mo id="S4.SS1.p3.9.m3.1.1.1.3.2" stretchy="false" xref="S4.SS1.p3.9.m3.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.9.m3.1b"><apply id="S4.SS1.p3.9.m3.1.2.cmml" xref="S4.SS1.p3.9.m3.1.2"><csymbol cd="ambiguous" id="S4.SS1.p3.9.m3.1.2.1.cmml" xref="S4.SS1.p3.9.m3.1.2">superscript</csymbol><apply id="S4.SS1.p3.9.m3.1.2.2.cmml" xref="S4.SS1.p3.9.m3.1.2.2"><ci id="S4.SS1.p3.9.m3.1.2.2.1.cmml" xref="S4.SS1.p3.9.m3.1.2.2.1">^</ci><ci id="S4.SS1.p3.9.m3.1.2.2.2.cmml" xref="S4.SS1.p3.9.m3.1.2.2.2">𝜽</ci></apply><ci id="S4.SS1.p3.9.m3.1.1.1.1.cmml" xref="S4.SS1.p3.9.m3.1.1.1.1">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.9.m3.1c">\hat{\bm{\theta}}^{(k)}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.9.m3.1d">over^ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT</annotation></semantics></math> can be obtained by the direct differentiation of filtered elements regarding <math alttext="\Psi" class="ltx_Math" display="inline" id="S4.SS1.p3.10.m4.1"><semantics id="S4.SS1.p3.10.m4.1a"><mi id="S4.SS1.p3.10.m4.1.1" mathvariant="normal" xref="S4.SS1.p3.10.m4.1.1.cmml">Ψ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.10.m4.1b"><ci id="S4.SS1.p3.10.m4.1.1.cmml" xref="S4.SS1.p3.10.m4.1.1">Ψ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.10.m4.1c">\Psi</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.10.m4.1d">roman_Ψ</annotation></semantics></math> and <math alttext="\bm{q}" class="ltx_Math" display="inline" id="S4.SS1.p3.11.m5.1"><semantics id="S4.SS1.p3.11.m5.1a"><mi id="S4.SS1.p3.11.m5.1.1" xref="S4.SS1.p3.11.m5.1.1.cmml">𝒒</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.11.m5.1b"><ci id="S4.SS1.p3.11.m5.1.1.cmml" xref="S4.SS1.p3.11.m5.1.1">𝒒</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.11.m5.1c">\bm{q}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.11.m5.1d">bold_italic_q</annotation></semantics></math>. Thus, one introduces a linear filter with <math alttext="n-1" class="ltx_Math" display="inline" id="S4.SS1.p3.12.m6.1"><semantics id="S4.SS1.p3.12.m6.1a"><mrow id="S4.SS1.p3.12.m6.1.1" xref="S4.SS1.p3.12.m6.1.1.cmml"><mi id="S4.SS1.p3.12.m6.1.1.2" xref="S4.SS1.p3.12.m6.1.1.2.cmml">n</mi><mo id="S4.SS1.p3.12.m6.1.1.1" xref="S4.SS1.p3.12.m6.1.1.1.cmml">−</mo><mn id="S4.SS1.p3.12.m6.1.1.3" xref="S4.SS1.p3.12.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.12.m6.1b"><apply id="S4.SS1.p3.12.m6.1.1.cmml" xref="S4.SS1.p3.12.m6.1.1"><minus id="S4.SS1.p3.12.m6.1.1.1.cmml" xref="S4.SS1.p3.12.m6.1.1.1"></minus><ci id="S4.SS1.p3.12.m6.1.1.2.cmml" xref="S4.SS1.p3.12.m6.1.1.2">𝑛</ci><cn id="S4.SS1.p3.12.m6.1.1.3.cmml" type="integer" xref="S4.SS1.p3.12.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.12.m6.1c">n-1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.12.m6.1d">italic_n - 1</annotation></semantics></math> relative degrees</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx12"> <tbody id="S4.E17"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle H(s):=\prod_{i=1}^{n-1}\frac{\alpha_{i}}{s+\alpha_{i}}" class="ltx_Math" display="inline" id="S4.E17.m1.1"><semantics id="S4.E17.m1.1a"><mrow id="S4.E17.m1.1.2" xref="S4.E17.m1.1.2.cmml"><mrow id="S4.E17.m1.1.2.2" xref="S4.E17.m1.1.2.2.cmml"><mi id="S4.E17.m1.1.2.2.2" xref="S4.E17.m1.1.2.2.2.cmml">H</mi><mo id="S4.E17.m1.1.2.2.1" xref="S4.E17.m1.1.2.2.1.cmml">⁢</mo><mrow id="S4.E17.m1.1.2.2.3.2" xref="S4.E17.m1.1.2.2.cmml"><mo id="S4.E17.m1.1.2.2.3.2.1" stretchy="false" xref="S4.E17.m1.1.2.2.cmml">(</mo><mi id="S4.E17.m1.1.1" xref="S4.E17.m1.1.1.cmml">s</mi><mo id="S4.E17.m1.1.2.2.3.2.2" rspace="0.278em" stretchy="false" xref="S4.E17.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.E17.m1.1.2.1" rspace="0.278em" xref="S4.E17.m1.1.2.1.cmml">:=</mo><mrow id="S4.E17.m1.1.2.3" xref="S4.E17.m1.1.2.3.cmml"><mstyle displaystyle="true" id="S4.E17.m1.1.2.3.1" xref="S4.E17.m1.1.2.3.1.cmml"><munderover id="S4.E17.m1.1.2.3.1a" xref="S4.E17.m1.1.2.3.1.cmml"><mo id="S4.E17.m1.1.2.3.1.2.2" movablelimits="false" xref="S4.E17.m1.1.2.3.1.2.2.cmml">∏</mo><mrow id="S4.E17.m1.1.2.3.1.2.3" xref="S4.E17.m1.1.2.3.1.2.3.cmml"><mi id="S4.E17.m1.1.2.3.1.2.3.2" xref="S4.E17.m1.1.2.3.1.2.3.2.cmml">i</mi><mo id="S4.E17.m1.1.2.3.1.2.3.1" xref="S4.E17.m1.1.2.3.1.2.3.1.cmml">=</mo><mn id="S4.E17.m1.1.2.3.1.2.3.3" xref="S4.E17.m1.1.2.3.1.2.3.3.cmml">1</mn></mrow><mrow id="S4.E17.m1.1.2.3.1.3" xref="S4.E17.m1.1.2.3.1.3.cmml"><mi id="S4.E17.m1.1.2.3.1.3.2" xref="S4.E17.m1.1.2.3.1.3.2.cmml">n</mi><mo id="S4.E17.m1.1.2.3.1.3.1" xref="S4.E17.m1.1.2.3.1.3.1.cmml">−</mo><mn id="S4.E17.m1.1.2.3.1.3.3" xref="S4.E17.m1.1.2.3.1.3.3.cmml">1</mn></mrow></munderover></mstyle><mstyle displaystyle="true" id="S4.E17.m1.1.2.3.2" xref="S4.E17.m1.1.2.3.2.cmml"><mfrac id="S4.E17.m1.1.2.3.2a" xref="S4.E17.m1.1.2.3.2.cmml"><msub id="S4.E17.m1.1.2.3.2.2" xref="S4.E17.m1.1.2.3.2.2.cmml"><mi id="S4.E17.m1.1.2.3.2.2.2" xref="S4.E17.m1.1.2.3.2.2.2.cmml">α</mi><mi id="S4.E17.m1.1.2.3.2.2.3" xref="S4.E17.m1.1.2.3.2.2.3.cmml">i</mi></msub><mrow id="S4.E17.m1.1.2.3.2.3" xref="S4.E17.m1.1.2.3.2.3.cmml"><mi id="S4.E17.m1.1.2.3.2.3.2" xref="S4.E17.m1.1.2.3.2.3.2.cmml">s</mi><mo id="S4.E17.m1.1.2.3.2.3.1" xref="S4.E17.m1.1.2.3.2.3.1.cmml">+</mo><msub id="S4.E17.m1.1.2.3.2.3.3" xref="S4.E17.m1.1.2.3.2.3.3.cmml"><mi id="S4.E17.m1.1.2.3.2.3.3.2" xref="S4.E17.m1.1.2.3.2.3.3.2.cmml">α</mi><mi id="S4.E17.m1.1.2.3.2.3.3.3" xref="S4.E17.m1.1.2.3.2.3.3.3.cmml">i</mi></msub></mrow></mfrac></mstyle></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E17.m1.1b"><apply id="S4.E17.m1.1.2.cmml" xref="S4.E17.m1.1.2"><csymbol cd="latexml" id="S4.E17.m1.1.2.1.cmml" xref="S4.E17.m1.1.2.1">assign</csymbol><apply id="S4.E17.m1.1.2.2.cmml" xref="S4.E17.m1.1.2.2"><times id="S4.E17.m1.1.2.2.1.cmml" xref="S4.E17.m1.1.2.2.1"></times><ci id="S4.E17.m1.1.2.2.2.cmml" xref="S4.E17.m1.1.2.2.2">𝐻</ci><ci id="S4.E17.m1.1.1.cmml" xref="S4.E17.m1.1.1">𝑠</ci></apply><apply id="S4.E17.m1.1.2.3.cmml" xref="S4.E17.m1.1.2.3"><apply id="S4.E17.m1.1.2.3.1.cmml" xref="S4.E17.m1.1.2.3.1"><csymbol cd="ambiguous" id="S4.E17.m1.1.2.3.1.1.cmml" xref="S4.E17.m1.1.2.3.1">superscript</csymbol><apply id="S4.E17.m1.1.2.3.1.2.cmml" xref="S4.E17.m1.1.2.3.1"><csymbol cd="ambiguous" id="S4.E17.m1.1.2.3.1.2.1.cmml" xref="S4.E17.m1.1.2.3.1">subscript</csymbol><csymbol cd="latexml" id="S4.E17.m1.1.2.3.1.2.2.cmml" xref="S4.E17.m1.1.2.3.1.2.2">product</csymbol><apply id="S4.E17.m1.1.2.3.1.2.3.cmml" xref="S4.E17.m1.1.2.3.1.2.3"><eq id="S4.E17.m1.1.2.3.1.2.3.1.cmml" xref="S4.E17.m1.1.2.3.1.2.3.1"></eq><ci id="S4.E17.m1.1.2.3.1.2.3.2.cmml" xref="S4.E17.m1.1.2.3.1.2.3.2">𝑖</ci><cn id="S4.E17.m1.1.2.3.1.2.3.3.cmml" type="integer" xref="S4.E17.m1.1.2.3.1.2.3.3">1</cn></apply></apply><apply id="S4.E17.m1.1.2.3.1.3.cmml" xref="S4.E17.m1.1.2.3.1.3"><minus id="S4.E17.m1.1.2.3.1.3.1.cmml" xref="S4.E17.m1.1.2.3.1.3.1"></minus><ci id="S4.E17.m1.1.2.3.1.3.2.cmml" xref="S4.E17.m1.1.2.3.1.3.2">𝑛</ci><cn id="S4.E17.m1.1.2.3.1.3.3.cmml" type="integer" xref="S4.E17.m1.1.2.3.1.3.3">1</cn></apply></apply><apply id="S4.E17.m1.1.2.3.2.cmml" xref="S4.E17.m1.1.2.3.2"><divide id="S4.E17.m1.1.2.3.2.1.cmml" xref="S4.E17.m1.1.2.3.2"></divide><apply id="S4.E17.m1.1.2.3.2.2.cmml" xref="S4.E17.m1.1.2.3.2.2"><csymbol cd="ambiguous" id="S4.E17.m1.1.2.3.2.2.1.cmml" xref="S4.E17.m1.1.2.3.2.2">subscript</csymbol><ci id="S4.E17.m1.1.2.3.2.2.2.cmml" xref="S4.E17.m1.1.2.3.2.2.2">𝛼</ci><ci id="S4.E17.m1.1.2.3.2.2.3.cmml" xref="S4.E17.m1.1.2.3.2.2.3">𝑖</ci></apply><apply id="S4.E17.m1.1.2.3.2.3.cmml" xref="S4.E17.m1.1.2.3.2.3"><plus id="S4.E17.m1.1.2.3.2.3.1.cmml" xref="S4.E17.m1.1.2.3.2.3.1"></plus><ci id="S4.E17.m1.1.2.3.2.3.2.cmml" xref="S4.E17.m1.1.2.3.2.3.2">𝑠</ci><apply id="S4.E17.m1.1.2.3.2.3.3.cmml" xref="S4.E17.m1.1.2.3.2.3.3"><csymbol cd="ambiguous" id="S4.E17.m1.1.2.3.2.3.3.1.cmml" xref="S4.E17.m1.1.2.3.2.3.3">subscript</csymbol><ci id="S4.E17.m1.1.2.3.2.3.3.2.cmml" xref="S4.E17.m1.1.2.3.2.3.3.2">𝛼</ci><ci id="S4.E17.m1.1.2.3.2.3.3.3.cmml" xref="S4.E17.m1.1.2.3.2.3.3.3">𝑖</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E17.m1.1c">\displaystyle H(s):=\prod_{i=1}^{n-1}\frac{\alpha_{i}}{s+\alpha_{i}}</annotation><annotation encoding="application/x-llamapun" id="S4.E17.m1.1d">italic_H ( italic_s ) := ∏ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT divide start_ARG italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_s + italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(17)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p3.16">where <math alttext="\alpha_{i}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S4.SS1.p3.13.m1.1"><semantics id="S4.SS1.p3.13.m1.1a"><mrow id="S4.SS1.p3.13.m1.1.1" xref="S4.SS1.p3.13.m1.1.1.cmml"><msub id="S4.SS1.p3.13.m1.1.1.2" xref="S4.SS1.p3.13.m1.1.1.2.cmml"><mi id="S4.SS1.p3.13.m1.1.1.2.2" xref="S4.SS1.p3.13.m1.1.1.2.2.cmml">α</mi><mi id="S4.SS1.p3.13.m1.1.1.2.3" xref="S4.SS1.p3.13.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.SS1.p3.13.m1.1.1.1" xref="S4.SS1.p3.13.m1.1.1.1.cmml">∈</mo><msup id="S4.SS1.p3.13.m1.1.1.3" xref="S4.SS1.p3.13.m1.1.1.3.cmml"><mi id="S4.SS1.p3.13.m1.1.1.3.2" xref="S4.SS1.p3.13.m1.1.1.3.2.cmml">ℝ</mi><mo id="S4.SS1.p3.13.m1.1.1.3.3" xref="S4.SS1.p3.13.m1.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.13.m1.1b"><apply id="S4.SS1.p3.13.m1.1.1.cmml" xref="S4.SS1.p3.13.m1.1.1"><in id="S4.SS1.p3.13.m1.1.1.1.cmml" xref="S4.SS1.p3.13.m1.1.1.1"></in><apply id="S4.SS1.p3.13.m1.1.1.2.cmml" xref="S4.SS1.p3.13.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p3.13.m1.1.1.2.1.cmml" xref="S4.SS1.p3.13.m1.1.1.2">subscript</csymbol><ci id="S4.SS1.p3.13.m1.1.1.2.2.cmml" xref="S4.SS1.p3.13.m1.1.1.2.2">𝛼</ci><ci id="S4.SS1.p3.13.m1.1.1.2.3.cmml" xref="S4.SS1.p3.13.m1.1.1.2.3">𝑖</ci></apply><apply id="S4.SS1.p3.13.m1.1.1.3.cmml" xref="S4.SS1.p3.13.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p3.13.m1.1.1.3.1.cmml" xref="S4.SS1.p3.13.m1.1.1.3">superscript</csymbol><ci id="S4.SS1.p3.13.m1.1.1.3.2.cmml" xref="S4.SS1.p3.13.m1.1.1.3.2">ℝ</ci><plus id="S4.SS1.p3.13.m1.1.1.3.3.cmml" xref="S4.SS1.p3.13.m1.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.13.m1.1c">\alpha_{i}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.13.m1.1d">italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> (<math alttext="i=" class="ltx_Math" display="inline" id="S4.SS1.p3.14.m2.1"><semantics id="S4.SS1.p3.14.m2.1a"><mrow id="S4.SS1.p3.14.m2.1.1" xref="S4.SS1.p3.14.m2.1.1.cmml"><mi id="S4.SS1.p3.14.m2.1.1.2" xref="S4.SS1.p3.14.m2.1.1.2.cmml">i</mi><mo id="S4.SS1.p3.14.m2.1.1.1" xref="S4.SS1.p3.14.m2.1.1.1.cmml">=</mo><mi id="S4.SS1.p3.14.m2.1.1.3" xref="S4.SS1.p3.14.m2.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.14.m2.1b"><apply id="S4.SS1.p3.14.m2.1.1.cmml" xref="S4.SS1.p3.14.m2.1.1"><eq id="S4.SS1.p3.14.m2.1.1.1.cmml" xref="S4.SS1.p3.14.m2.1.1.1"></eq><ci id="S4.SS1.p3.14.m2.1.1.2.cmml" xref="S4.SS1.p3.14.m2.1.1.2">𝑖</ci><csymbol cd="latexml" id="S4.SS1.p3.14.m2.1.1.3.cmml" xref="S4.SS1.p3.14.m2.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.14.m2.1c">i=</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.14.m2.1d">italic_i =</annotation></semantics></math> 1 to <math alttext="n" class="ltx_Math" display="inline" id="S4.SS1.p3.15.m3.1"><semantics id="S4.SS1.p3.15.m3.1a"><mi id="S4.SS1.p3.15.m3.1.1" xref="S4.SS1.p3.15.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.15.m3.1b"><ci id="S4.SS1.p3.15.m3.1.1.cmml" xref="S4.SS1.p3.15.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.15.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.15.m3.1d">italic_n</annotation></semantics></math>) are filtering constants. Applying <math alttext="H(s)" class="ltx_Math" display="inline" id="S4.SS1.p3.16.m4.1"><semantics id="S4.SS1.p3.16.m4.1a"><mrow id="S4.SS1.p3.16.m4.1.2" xref="S4.SS1.p3.16.m4.1.2.cmml"><mi id="S4.SS1.p3.16.m4.1.2.2" xref="S4.SS1.p3.16.m4.1.2.2.cmml">H</mi><mo id="S4.SS1.p3.16.m4.1.2.1" xref="S4.SS1.p3.16.m4.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p3.16.m4.1.2.3.2" xref="S4.SS1.p3.16.m4.1.2.cmml"><mo id="S4.SS1.p3.16.m4.1.2.3.2.1" stretchy="false" xref="S4.SS1.p3.16.m4.1.2.cmml">(</mo><mi id="S4.SS1.p3.16.m4.1.1" xref="S4.SS1.p3.16.m4.1.1.cmml">s</mi><mo id="S4.SS1.p3.16.m4.1.2.3.2.2" stretchy="false" xref="S4.SS1.p3.16.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.16.m4.1b"><apply id="S4.SS1.p3.16.m4.1.2.cmml" xref="S4.SS1.p3.16.m4.1.2"><times id="S4.SS1.p3.16.m4.1.2.1.cmml" xref="S4.SS1.p3.16.m4.1.2.1"></times><ci id="S4.SS1.p3.16.m4.1.2.2.cmml" xref="S4.SS1.p3.16.m4.1.2.2">𝐻</ci><ci id="S4.SS1.p3.16.m4.1.1.cmml" xref="S4.SS1.p3.16.m4.1.1">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.16.m4.1c">H(s)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.16.m4.1d">italic_H ( italic_s )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E17" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">17</span></a>) to each side of (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E14" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">14</span></a>), one gets a generalized parameterized model</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx13"> <tbody id="S4.E18"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle Q(t)\bm{\theta}={\bm{q}}_{\rm f}(t)" class="ltx_Math" display="inline" id="S4.E18.m1.2"><semantics id="S4.E18.m1.2a"><mrow id="S4.E18.m1.2.3" xref="S4.E18.m1.2.3.cmml"><mrow id="S4.E18.m1.2.3.2" xref="S4.E18.m1.2.3.2.cmml"><mi id="S4.E18.m1.2.3.2.2" xref="S4.E18.m1.2.3.2.2.cmml">Q</mi><mo id="S4.E18.m1.2.3.2.1" xref="S4.E18.m1.2.3.2.1.cmml">⁢</mo><mrow id="S4.E18.m1.2.3.2.3.2" xref="S4.E18.m1.2.3.2.cmml"><mo id="S4.E18.m1.2.3.2.3.2.1" stretchy="false" xref="S4.E18.m1.2.3.2.cmml">(</mo><mi id="S4.E18.m1.1.1" xref="S4.E18.m1.1.1.cmml">t</mi><mo id="S4.E18.m1.2.3.2.3.2.2" stretchy="false" xref="S4.E18.m1.2.3.2.cmml">)</mo></mrow><mo id="S4.E18.m1.2.3.2.1a" xref="S4.E18.m1.2.3.2.1.cmml">⁢</mo><mi id="S4.E18.m1.2.3.2.4" xref="S4.E18.m1.2.3.2.4.cmml">𝜽</mi></mrow><mo id="S4.E18.m1.2.3.1" xref="S4.E18.m1.2.3.1.cmml">=</mo><mrow id="S4.E18.m1.2.3.3" xref="S4.E18.m1.2.3.3.cmml"><msub id="S4.E18.m1.2.3.3.2" xref="S4.E18.m1.2.3.3.2.cmml"><mi id="S4.E18.m1.2.3.3.2.2" xref="S4.E18.m1.2.3.3.2.2.cmml">𝒒</mi><mi id="S4.E18.m1.2.3.3.2.3" mathvariant="normal" xref="S4.E18.m1.2.3.3.2.3.cmml">f</mi></msub><mo id="S4.E18.m1.2.3.3.1" xref="S4.E18.m1.2.3.3.1.cmml">⁢</mo><mrow id="S4.E18.m1.2.3.3.3.2" xref="S4.E18.m1.2.3.3.cmml"><mo id="S4.E18.m1.2.3.3.3.2.1" stretchy="false" xref="S4.E18.m1.2.3.3.cmml">(</mo><mi id="S4.E18.m1.2.2" xref="S4.E18.m1.2.2.cmml">t</mi><mo id="S4.E18.m1.2.3.3.3.2.2" stretchy="false" xref="S4.E18.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E18.m1.2b"><apply id="S4.E18.m1.2.3.cmml" xref="S4.E18.m1.2.3"><eq id="S4.E18.m1.2.3.1.cmml" xref="S4.E18.m1.2.3.1"></eq><apply id="S4.E18.m1.2.3.2.cmml" xref="S4.E18.m1.2.3.2"><times id="S4.E18.m1.2.3.2.1.cmml" xref="S4.E18.m1.2.3.2.1"></times><ci id="S4.E18.m1.2.3.2.2.cmml" xref="S4.E18.m1.2.3.2.2">𝑄</ci><ci id="S4.E18.m1.1.1.cmml" xref="S4.E18.m1.1.1">𝑡</ci><ci id="S4.E18.m1.2.3.2.4.cmml" xref="S4.E18.m1.2.3.2.4">𝜽</ci></apply><apply id="S4.E18.m1.2.3.3.cmml" xref="S4.E18.m1.2.3.3"><times id="S4.E18.m1.2.3.3.1.cmml" xref="S4.E18.m1.2.3.3.1"></times><apply id="S4.E18.m1.2.3.3.2.cmml" xref="S4.E18.m1.2.3.3.2"><csymbol cd="ambiguous" id="S4.E18.m1.2.3.3.2.1.cmml" xref="S4.E18.m1.2.3.3.2">subscript</csymbol><ci id="S4.E18.m1.2.3.3.2.2.cmml" xref="S4.E18.m1.2.3.3.2.2">𝒒</ci><ci id="S4.E18.m1.2.3.3.2.3.cmml" xref="S4.E18.m1.2.3.3.2.3">f</ci></apply><ci id="S4.E18.m1.2.2.cmml" xref="S4.E18.m1.2.2">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E18.m1.2c">\displaystyle Q(t)\bm{\theta}={\bm{q}}_{\rm f}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.E18.m1.2d">italic_Q ( italic_t ) bold_italic_θ = bold_italic_q start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(18)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p3.18">with <math alttext="Q(t):=H(s)[\Psi(t)]" class="ltx_Math" display="inline" id="S4.SS1.p3.17.m1.4"><semantics id="S4.SS1.p3.17.m1.4a"><mrow id="S4.SS1.p3.17.m1.4.4" xref="S4.SS1.p3.17.m1.4.4.cmml"><mrow id="S4.SS1.p3.17.m1.4.4.3" xref="S4.SS1.p3.17.m1.4.4.3.cmml"><mi id="S4.SS1.p3.17.m1.4.4.3.2" xref="S4.SS1.p3.17.m1.4.4.3.2.cmml">Q</mi><mo id="S4.SS1.p3.17.m1.4.4.3.1" xref="S4.SS1.p3.17.m1.4.4.3.1.cmml">⁢</mo><mrow id="S4.SS1.p3.17.m1.4.4.3.3.2" xref="S4.SS1.p3.17.m1.4.4.3.cmml"><mo id="S4.SS1.p3.17.m1.4.4.3.3.2.1" stretchy="false" xref="S4.SS1.p3.17.m1.4.4.3.cmml">(</mo><mi id="S4.SS1.p3.17.m1.1.1" xref="S4.SS1.p3.17.m1.1.1.cmml">t</mi><mo id="S4.SS1.p3.17.m1.4.4.3.3.2.2" rspace="0.278em" stretchy="false" xref="S4.SS1.p3.17.m1.4.4.3.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p3.17.m1.4.4.2" rspace="0.278em" xref="S4.SS1.p3.17.m1.4.4.2.cmml">:=</mo><mrow id="S4.SS1.p3.17.m1.4.4.1" xref="S4.SS1.p3.17.m1.4.4.1.cmml"><mi id="S4.SS1.p3.17.m1.4.4.1.3" xref="S4.SS1.p3.17.m1.4.4.1.3.cmml">H</mi><mo id="S4.SS1.p3.17.m1.4.4.1.2" xref="S4.SS1.p3.17.m1.4.4.1.2.cmml">⁢</mo><mrow id="S4.SS1.p3.17.m1.4.4.1.4.2" xref="S4.SS1.p3.17.m1.4.4.1.cmml"><mo id="S4.SS1.p3.17.m1.4.4.1.4.2.1" stretchy="false" xref="S4.SS1.p3.17.m1.4.4.1.cmml">(</mo><mi id="S4.SS1.p3.17.m1.2.2" xref="S4.SS1.p3.17.m1.2.2.cmml">s</mi><mo id="S4.SS1.p3.17.m1.4.4.1.4.2.2" stretchy="false" xref="S4.SS1.p3.17.m1.4.4.1.cmml">)</mo></mrow><mo id="S4.SS1.p3.17.m1.4.4.1.2a" xref="S4.SS1.p3.17.m1.4.4.1.2.cmml">⁢</mo><mrow id="S4.SS1.p3.17.m1.4.4.1.1.1" xref="S4.SS1.p3.17.m1.4.4.1.1.2.cmml"><mo id="S4.SS1.p3.17.m1.4.4.1.1.1.2" stretchy="false" xref="S4.SS1.p3.17.m1.4.4.1.1.2.1.cmml">[</mo><mrow id="S4.SS1.p3.17.m1.4.4.1.1.1.1" xref="S4.SS1.p3.17.m1.4.4.1.1.1.1.cmml"><mi id="S4.SS1.p3.17.m1.4.4.1.1.1.1.2" mathvariant="normal" xref="S4.SS1.p3.17.m1.4.4.1.1.1.1.2.cmml">Ψ</mi><mo id="S4.SS1.p3.17.m1.4.4.1.1.1.1.1" xref="S4.SS1.p3.17.m1.4.4.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS1.p3.17.m1.4.4.1.1.1.1.3.2" xref="S4.SS1.p3.17.m1.4.4.1.1.1.1.cmml"><mo id="S4.SS1.p3.17.m1.4.4.1.1.1.1.3.2.1" stretchy="false" xref="S4.SS1.p3.17.m1.4.4.1.1.1.1.cmml">(</mo><mi id="S4.SS1.p3.17.m1.3.3" xref="S4.SS1.p3.17.m1.3.3.cmml">t</mi><mo id="S4.SS1.p3.17.m1.4.4.1.1.1.1.3.2.2" stretchy="false" xref="S4.SS1.p3.17.m1.4.4.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p3.17.m1.4.4.1.1.1.3" stretchy="false" xref="S4.SS1.p3.17.m1.4.4.1.1.2.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.17.m1.4b"><apply id="S4.SS1.p3.17.m1.4.4.cmml" xref="S4.SS1.p3.17.m1.4.4"><csymbol cd="latexml" id="S4.SS1.p3.17.m1.4.4.2.cmml" xref="S4.SS1.p3.17.m1.4.4.2">assign</csymbol><apply id="S4.SS1.p3.17.m1.4.4.3.cmml" xref="S4.SS1.p3.17.m1.4.4.3"><times id="S4.SS1.p3.17.m1.4.4.3.1.cmml" xref="S4.SS1.p3.17.m1.4.4.3.1"></times><ci id="S4.SS1.p3.17.m1.4.4.3.2.cmml" xref="S4.SS1.p3.17.m1.4.4.3.2">𝑄</ci><ci id="S4.SS1.p3.17.m1.1.1.cmml" xref="S4.SS1.p3.17.m1.1.1">𝑡</ci></apply><apply id="S4.SS1.p3.17.m1.4.4.1.cmml" xref="S4.SS1.p3.17.m1.4.4.1"><times id="S4.SS1.p3.17.m1.4.4.1.2.cmml" xref="S4.SS1.p3.17.m1.4.4.1.2"></times><ci id="S4.SS1.p3.17.m1.4.4.1.3.cmml" xref="S4.SS1.p3.17.m1.4.4.1.3">𝐻</ci><ci id="S4.SS1.p3.17.m1.2.2.cmml" xref="S4.SS1.p3.17.m1.2.2">𝑠</ci><apply id="S4.SS1.p3.17.m1.4.4.1.1.2.cmml" xref="S4.SS1.p3.17.m1.4.4.1.1.1"><csymbol cd="latexml" id="S4.SS1.p3.17.m1.4.4.1.1.2.1.cmml" xref="S4.SS1.p3.17.m1.4.4.1.1.1.2">delimited-[]</csymbol><apply id="S4.SS1.p3.17.m1.4.4.1.1.1.1.cmml" xref="S4.SS1.p3.17.m1.4.4.1.1.1.1"><times id="S4.SS1.p3.17.m1.4.4.1.1.1.1.1.cmml" xref="S4.SS1.p3.17.m1.4.4.1.1.1.1.1"></times><ci id="S4.SS1.p3.17.m1.4.4.1.1.1.1.2.cmml" xref="S4.SS1.p3.17.m1.4.4.1.1.1.1.2">Ψ</ci><ci id="S4.SS1.p3.17.m1.3.3.cmml" xref="S4.SS1.p3.17.m1.3.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.17.m1.4c">Q(t):=H(s)[\Psi(t)]</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.17.m1.4d">italic_Q ( italic_t ) := italic_H ( italic_s ) [ roman_Ψ ( italic_t ) ]</annotation></semantics></math> and <math alttext="{\bm{q}}_{\rm f}(t):=H(s)[\bm{q}(t)]" class="ltx_Math" display="inline" id="S4.SS1.p3.18.m2.4"><semantics id="S4.SS1.p3.18.m2.4a"><mrow id="S4.SS1.p3.18.m2.4.4" xref="S4.SS1.p3.18.m2.4.4.cmml"><mrow id="S4.SS1.p3.18.m2.4.4.3" xref="S4.SS1.p3.18.m2.4.4.3.cmml"><msub id="S4.SS1.p3.18.m2.4.4.3.2" xref="S4.SS1.p3.18.m2.4.4.3.2.cmml"><mi id="S4.SS1.p3.18.m2.4.4.3.2.2" xref="S4.SS1.p3.18.m2.4.4.3.2.2.cmml">𝒒</mi><mi id="S4.SS1.p3.18.m2.4.4.3.2.3" mathvariant="normal" xref="S4.SS1.p3.18.m2.4.4.3.2.3.cmml">f</mi></msub><mo id="S4.SS1.p3.18.m2.4.4.3.1" xref="S4.SS1.p3.18.m2.4.4.3.1.cmml">⁢</mo><mrow id="S4.SS1.p3.18.m2.4.4.3.3.2" xref="S4.SS1.p3.18.m2.4.4.3.cmml"><mo id="S4.SS1.p3.18.m2.4.4.3.3.2.1" stretchy="false" xref="S4.SS1.p3.18.m2.4.4.3.cmml">(</mo><mi id="S4.SS1.p3.18.m2.1.1" xref="S4.SS1.p3.18.m2.1.1.cmml">t</mi><mo id="S4.SS1.p3.18.m2.4.4.3.3.2.2" rspace="0.278em" stretchy="false" xref="S4.SS1.p3.18.m2.4.4.3.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p3.18.m2.4.4.2" rspace="0.278em" xref="S4.SS1.p3.18.m2.4.4.2.cmml">:=</mo><mrow id="S4.SS1.p3.18.m2.4.4.1" xref="S4.SS1.p3.18.m2.4.4.1.cmml"><mi id="S4.SS1.p3.18.m2.4.4.1.3" xref="S4.SS1.p3.18.m2.4.4.1.3.cmml">H</mi><mo id="S4.SS1.p3.18.m2.4.4.1.2" xref="S4.SS1.p3.18.m2.4.4.1.2.cmml">⁢</mo><mrow id="S4.SS1.p3.18.m2.4.4.1.4.2" xref="S4.SS1.p3.18.m2.4.4.1.cmml"><mo id="S4.SS1.p3.18.m2.4.4.1.4.2.1" stretchy="false" xref="S4.SS1.p3.18.m2.4.4.1.cmml">(</mo><mi id="S4.SS1.p3.18.m2.2.2" xref="S4.SS1.p3.18.m2.2.2.cmml">s</mi><mo id="S4.SS1.p3.18.m2.4.4.1.4.2.2" stretchy="false" xref="S4.SS1.p3.18.m2.4.4.1.cmml">)</mo></mrow><mo id="S4.SS1.p3.18.m2.4.4.1.2a" xref="S4.SS1.p3.18.m2.4.4.1.2.cmml">⁢</mo><mrow id="S4.SS1.p3.18.m2.4.4.1.1.1" xref="S4.SS1.p3.18.m2.4.4.1.1.2.cmml"><mo id="S4.SS1.p3.18.m2.4.4.1.1.1.2" stretchy="false" xref="S4.SS1.p3.18.m2.4.4.1.1.2.1.cmml">[</mo><mrow id="S4.SS1.p3.18.m2.4.4.1.1.1.1" xref="S4.SS1.p3.18.m2.4.4.1.1.1.1.cmml"><mi id="S4.SS1.p3.18.m2.4.4.1.1.1.1.2" xref="S4.SS1.p3.18.m2.4.4.1.1.1.1.2.cmml">𝒒</mi><mo id="S4.SS1.p3.18.m2.4.4.1.1.1.1.1" xref="S4.SS1.p3.18.m2.4.4.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS1.p3.18.m2.4.4.1.1.1.1.3.2" xref="S4.SS1.p3.18.m2.4.4.1.1.1.1.cmml"><mo id="S4.SS1.p3.18.m2.4.4.1.1.1.1.3.2.1" stretchy="false" xref="S4.SS1.p3.18.m2.4.4.1.1.1.1.cmml">(</mo><mi id="S4.SS1.p3.18.m2.3.3" xref="S4.SS1.p3.18.m2.3.3.cmml">t</mi><mo id="S4.SS1.p3.18.m2.4.4.1.1.1.1.3.2.2" stretchy="false" xref="S4.SS1.p3.18.m2.4.4.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p3.18.m2.4.4.1.1.1.3" stretchy="false" xref="S4.SS1.p3.18.m2.4.4.1.1.2.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.18.m2.4b"><apply id="S4.SS1.p3.18.m2.4.4.cmml" xref="S4.SS1.p3.18.m2.4.4"><csymbol cd="latexml" id="S4.SS1.p3.18.m2.4.4.2.cmml" xref="S4.SS1.p3.18.m2.4.4.2">assign</csymbol><apply id="S4.SS1.p3.18.m2.4.4.3.cmml" xref="S4.SS1.p3.18.m2.4.4.3"><times id="S4.SS1.p3.18.m2.4.4.3.1.cmml" xref="S4.SS1.p3.18.m2.4.4.3.1"></times><apply id="S4.SS1.p3.18.m2.4.4.3.2.cmml" xref="S4.SS1.p3.18.m2.4.4.3.2"><csymbol cd="ambiguous" id="S4.SS1.p3.18.m2.4.4.3.2.1.cmml" xref="S4.SS1.p3.18.m2.4.4.3.2">subscript</csymbol><ci id="S4.SS1.p3.18.m2.4.4.3.2.2.cmml" xref="S4.SS1.p3.18.m2.4.4.3.2.2">𝒒</ci><ci id="S4.SS1.p3.18.m2.4.4.3.2.3.cmml" xref="S4.SS1.p3.18.m2.4.4.3.2.3">f</ci></apply><ci id="S4.SS1.p3.18.m2.1.1.cmml" xref="S4.SS1.p3.18.m2.1.1">𝑡</ci></apply><apply id="S4.SS1.p3.18.m2.4.4.1.cmml" xref="S4.SS1.p3.18.m2.4.4.1"><times id="S4.SS1.p3.18.m2.4.4.1.2.cmml" xref="S4.SS1.p3.18.m2.4.4.1.2"></times><ci id="S4.SS1.p3.18.m2.4.4.1.3.cmml" xref="S4.SS1.p3.18.m2.4.4.1.3">𝐻</ci><ci id="S4.SS1.p3.18.m2.2.2.cmml" xref="S4.SS1.p3.18.m2.2.2">𝑠</ci><apply id="S4.SS1.p3.18.m2.4.4.1.1.2.cmml" xref="S4.SS1.p3.18.m2.4.4.1.1.1"><csymbol cd="latexml" id="S4.SS1.p3.18.m2.4.4.1.1.2.1.cmml" xref="S4.SS1.p3.18.m2.4.4.1.1.1.2">delimited-[]</csymbol><apply id="S4.SS1.p3.18.m2.4.4.1.1.1.1.cmml" xref="S4.SS1.p3.18.m2.4.4.1.1.1.1"><times id="S4.SS1.p3.18.m2.4.4.1.1.1.1.1.cmml" xref="S4.SS1.p3.18.m2.4.4.1.1.1.1.1"></times><ci id="S4.SS1.p3.18.m2.4.4.1.1.1.1.2.cmml" xref="S4.SS1.p3.18.m2.4.4.1.1.1.1.2">𝒒</ci><ci id="S4.SS1.p3.18.m2.3.3.cmml" xref="S4.SS1.p3.18.m2.3.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.18.m2.4c">{\bm{q}}_{\rm f}(t):=H(s)[\bm{q}(t)]</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.18.m2.4d">bold_italic_q start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ( italic_t ) := italic_H ( italic_s ) [ bold_italic_q ( italic_t ) ]</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS1.p4"> <p class="ltx_p" id="S4.SS1.p4.2">From Assumption 3, partial IE exists at the beginning and some moments later, and there exist certain constants <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS1.p4.1.m1.1"><semantics id="S4.SS1.p4.1.m1.1a"><mi id="S4.SS1.p4.1.m1.1.1" xref="S4.SS1.p4.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.1.m1.1b"><ci id="S4.SS1.p4.1.m1.1.1.cmml" xref="S4.SS1.p4.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.1.m1.1d">italic_σ</annotation></semantics></math>, <math alttext="\tau_{\rm d}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S4.SS1.p4.2.m2.1"><semantics id="S4.SS1.p4.2.m2.1a"><mrow id="S4.SS1.p4.2.m2.1.1" xref="S4.SS1.p4.2.m2.1.1.cmml"><msub id="S4.SS1.p4.2.m2.1.1.2" xref="S4.SS1.p4.2.m2.1.1.2.cmml"><mi id="S4.SS1.p4.2.m2.1.1.2.2" xref="S4.SS1.p4.2.m2.1.1.2.2.cmml">τ</mi><mi id="S4.SS1.p4.2.m2.1.1.2.3" mathvariant="normal" xref="S4.SS1.p4.2.m2.1.1.2.3.cmml">d</mi></msub><mo id="S4.SS1.p4.2.m2.1.1.1" xref="S4.SS1.p4.2.m2.1.1.1.cmml">∈</mo><msup id="S4.SS1.p4.2.m2.1.1.3" xref="S4.SS1.p4.2.m2.1.1.3.cmml"><mi id="S4.SS1.p4.2.m2.1.1.3.2" xref="S4.SS1.p4.2.m2.1.1.3.2.cmml">ℝ</mi><mo id="S4.SS1.p4.2.m2.1.1.3.3" xref="S4.SS1.p4.2.m2.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.2.m2.1b"><apply id="S4.SS1.p4.2.m2.1.1.cmml" xref="S4.SS1.p4.2.m2.1.1"><in id="S4.SS1.p4.2.m2.1.1.1.cmml" xref="S4.SS1.p4.2.m2.1.1.1"></in><apply id="S4.SS1.p4.2.m2.1.1.2.cmml" xref="S4.SS1.p4.2.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p4.2.m2.1.1.2.1.cmml" xref="S4.SS1.p4.2.m2.1.1.2">subscript</csymbol><ci id="S4.SS1.p4.2.m2.1.1.2.2.cmml" xref="S4.SS1.p4.2.m2.1.1.2.2">𝜏</ci><ci id="S4.SS1.p4.2.m2.1.1.2.3.cmml" xref="S4.SS1.p4.2.m2.1.1.2.3">d</ci></apply><apply id="S4.SS1.p4.2.m2.1.1.3.cmml" xref="S4.SS1.p4.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p4.2.m2.1.1.3.1.cmml" xref="S4.SS1.p4.2.m2.1.1.3">superscript</csymbol><ci id="S4.SS1.p4.2.m2.1.1.3.2.cmml" xref="S4.SS1.p4.2.m2.1.1.3.2">ℝ</ci><plus id="S4.SS1.p4.2.m2.1.1.3.3.cmml" xref="S4.SS1.p4.2.m2.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.2.m2.1c">\tau_{\rm d}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.2.m2.1d">italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> such that</p> <table class="ltx_equation ltx_eqn_table" id="S4.E19"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\Psi_{\zeta}(t):=\int_{t-\tau_{\rm d}}^{t}\Phi_{\rm s,\zeta}(\tau)\Phi_{\rm s,% \zeta}^{T}(\tau)d\tau\geq\sigma I" class="ltx_Math" display="block" id="S4.E19.m1.7"><semantics id="S4.E19.m1.7a"><mrow id="S4.E19.m1.7.8" xref="S4.E19.m1.7.8.cmml"><mrow id="S4.E19.m1.7.8.2" xref="S4.E19.m1.7.8.2.cmml"><msub id="S4.E19.m1.7.8.2.2" xref="S4.E19.m1.7.8.2.2.cmml"><mi id="S4.E19.m1.7.8.2.2.2" mathvariant="normal" xref="S4.E19.m1.7.8.2.2.2.cmml">Ψ</mi><mi id="S4.E19.m1.7.8.2.2.3" xref="S4.E19.m1.7.8.2.2.3.cmml">ζ</mi></msub><mo id="S4.E19.m1.7.8.2.1" xref="S4.E19.m1.7.8.2.1.cmml">⁢</mo><mrow id="S4.E19.m1.7.8.2.3.2" xref="S4.E19.m1.7.8.2.cmml"><mo id="S4.E19.m1.7.8.2.3.2.1" stretchy="false" xref="S4.E19.m1.7.8.2.cmml">(</mo><mi id="S4.E19.m1.5.5" xref="S4.E19.m1.5.5.cmml">t</mi><mo id="S4.E19.m1.7.8.2.3.2.2" rspace="0.278em" stretchy="false" xref="S4.E19.m1.7.8.2.cmml">)</mo></mrow></mrow><mo id="S4.E19.m1.7.8.3" rspace="0.111em" xref="S4.E19.m1.7.8.3.cmml">:=</mo><mrow id="S4.E19.m1.7.8.4" xref="S4.E19.m1.7.8.4.cmml"><msubsup id="S4.E19.m1.7.8.4.1" xref="S4.E19.m1.7.8.4.1.cmml"><mo id="S4.E19.m1.7.8.4.1.2.2" xref="S4.E19.m1.7.8.4.1.2.2.cmml">∫</mo><mrow id="S4.E19.m1.7.8.4.1.2.3" xref="S4.E19.m1.7.8.4.1.2.3.cmml"><mi id="S4.E19.m1.7.8.4.1.2.3.2" xref="S4.E19.m1.7.8.4.1.2.3.2.cmml">t</mi><mo id="S4.E19.m1.7.8.4.1.2.3.1" xref="S4.E19.m1.7.8.4.1.2.3.1.cmml">−</mo><msub id="S4.E19.m1.7.8.4.1.2.3.3" xref="S4.E19.m1.7.8.4.1.2.3.3.cmml"><mi id="S4.E19.m1.7.8.4.1.2.3.3.2" xref="S4.E19.m1.7.8.4.1.2.3.3.2.cmml">τ</mi><mi id="S4.E19.m1.7.8.4.1.2.3.3.3" mathvariant="normal" xref="S4.E19.m1.7.8.4.1.2.3.3.3.cmml">d</mi></msub></mrow><mi id="S4.E19.m1.7.8.4.1.3" xref="S4.E19.m1.7.8.4.1.3.cmml">t</mi></msubsup><mrow id="S4.E19.m1.7.8.4.2" xref="S4.E19.m1.7.8.4.2.cmml"><msub id="S4.E19.m1.7.8.4.2.2" xref="S4.E19.m1.7.8.4.2.2.cmml"><mi id="S4.E19.m1.7.8.4.2.2.2" mathvariant="normal" xref="S4.E19.m1.7.8.4.2.2.2.cmml">Φ</mi><mrow id="S4.E19.m1.2.2.2.4" xref="S4.E19.m1.2.2.2.3.cmml"><mi id="S4.E19.m1.1.1.1.1" mathvariant="normal" xref="S4.E19.m1.1.1.1.1.cmml">s</mi><mo id="S4.E19.m1.2.2.2.4.1" xref="S4.E19.m1.2.2.2.3.cmml">,</mo><mi id="S4.E19.m1.2.2.2.2" xref="S4.E19.m1.2.2.2.2.cmml">ζ</mi></mrow></msub><mo id="S4.E19.m1.7.8.4.2.1" xref="S4.E19.m1.7.8.4.2.1.cmml">⁢</mo><mrow id="S4.E19.m1.7.8.4.2.3.2" xref="S4.E19.m1.7.8.4.2.cmml"><mo id="S4.E19.m1.7.8.4.2.3.2.1" stretchy="false" xref="S4.E19.m1.7.8.4.2.cmml">(</mo><mi id="S4.E19.m1.6.6" xref="S4.E19.m1.6.6.cmml">τ</mi><mo id="S4.E19.m1.7.8.4.2.3.2.2" stretchy="false" xref="S4.E19.m1.7.8.4.2.cmml">)</mo></mrow><mo id="S4.E19.m1.7.8.4.2.1a" xref="S4.E19.m1.7.8.4.2.1.cmml">⁢</mo><msubsup id="S4.E19.m1.7.8.4.2.4" xref="S4.E19.m1.7.8.4.2.4.cmml"><mi id="S4.E19.m1.7.8.4.2.4.2.2" mathvariant="normal" xref="S4.E19.m1.7.8.4.2.4.2.2.cmml">Φ</mi><mrow id="S4.E19.m1.4.4.2.4" xref="S4.E19.m1.4.4.2.3.cmml"><mi id="S4.E19.m1.3.3.1.1" mathvariant="normal" xref="S4.E19.m1.3.3.1.1.cmml">s</mi><mo id="S4.E19.m1.4.4.2.4.1" xref="S4.E19.m1.4.4.2.3.cmml">,</mo><mi id="S4.E19.m1.4.4.2.2" xref="S4.E19.m1.4.4.2.2.cmml">ζ</mi></mrow><mi id="S4.E19.m1.7.8.4.2.4.3" xref="S4.E19.m1.7.8.4.2.4.3.cmml">T</mi></msubsup><mo id="S4.E19.m1.7.8.4.2.1b" xref="S4.E19.m1.7.8.4.2.1.cmml">⁢</mo><mrow id="S4.E19.m1.7.8.4.2.5.2" xref="S4.E19.m1.7.8.4.2.cmml"><mo id="S4.E19.m1.7.8.4.2.5.2.1" stretchy="false" xref="S4.E19.m1.7.8.4.2.cmml">(</mo><mi id="S4.E19.m1.7.7" xref="S4.E19.m1.7.7.cmml">τ</mi><mo id="S4.E19.m1.7.8.4.2.5.2.2" stretchy="false" xref="S4.E19.m1.7.8.4.2.cmml">)</mo></mrow><mo id="S4.E19.m1.7.8.4.2.1c" lspace="0em" xref="S4.E19.m1.7.8.4.2.1.cmml">⁢</mo><mrow id="S4.E19.m1.7.8.4.2.6" xref="S4.E19.m1.7.8.4.2.6.cmml"><mo id="S4.E19.m1.7.8.4.2.6.1" rspace="0em" xref="S4.E19.m1.7.8.4.2.6.1.cmml">𝑑</mo><mi id="S4.E19.m1.7.8.4.2.6.2" xref="S4.E19.m1.7.8.4.2.6.2.cmml">τ</mi></mrow></mrow></mrow><mo id="S4.E19.m1.7.8.5" xref="S4.E19.m1.7.8.5.cmml">≥</mo><mrow id="S4.E19.m1.7.8.6" xref="S4.E19.m1.7.8.6.cmml"><mi id="S4.E19.m1.7.8.6.2" xref="S4.E19.m1.7.8.6.2.cmml">σ</mi><mo id="S4.E19.m1.7.8.6.1" xref="S4.E19.m1.7.8.6.1.cmml">⁢</mo><mi id="S4.E19.m1.7.8.6.3" xref="S4.E19.m1.7.8.6.3.cmml">I</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E19.m1.7b"><apply id="S4.E19.m1.7.8.cmml" xref="S4.E19.m1.7.8"><and id="S4.E19.m1.7.8a.cmml" xref="S4.E19.m1.7.8"></and><apply id="S4.E19.m1.7.8b.cmml" xref="S4.E19.m1.7.8"><csymbol cd="latexml" id="S4.E19.m1.7.8.3.cmml" xref="S4.E19.m1.7.8.3">assign</csymbol><apply id="S4.E19.m1.7.8.2.cmml" xref="S4.E19.m1.7.8.2"><times id="S4.E19.m1.7.8.2.1.cmml" xref="S4.E19.m1.7.8.2.1"></times><apply id="S4.E19.m1.7.8.2.2.cmml" xref="S4.E19.m1.7.8.2.2"><csymbol cd="ambiguous" id="S4.E19.m1.7.8.2.2.1.cmml" xref="S4.E19.m1.7.8.2.2">subscript</csymbol><ci id="S4.E19.m1.7.8.2.2.2.cmml" xref="S4.E19.m1.7.8.2.2.2">Ψ</ci><ci id="S4.E19.m1.7.8.2.2.3.cmml" xref="S4.E19.m1.7.8.2.2.3">𝜁</ci></apply><ci id="S4.E19.m1.5.5.cmml" xref="S4.E19.m1.5.5">𝑡</ci></apply><apply id="S4.E19.m1.7.8.4.cmml" xref="S4.E19.m1.7.8.4"><apply id="S4.E19.m1.7.8.4.1.cmml" xref="S4.E19.m1.7.8.4.1"><csymbol cd="ambiguous" id="S4.E19.m1.7.8.4.1.1.cmml" xref="S4.E19.m1.7.8.4.1">superscript</csymbol><apply id="S4.E19.m1.7.8.4.1.2.cmml" xref="S4.E19.m1.7.8.4.1"><csymbol cd="ambiguous" id="S4.E19.m1.7.8.4.1.2.1.cmml" xref="S4.E19.m1.7.8.4.1">subscript</csymbol><int id="S4.E19.m1.7.8.4.1.2.2.cmml" xref="S4.E19.m1.7.8.4.1.2.2"></int><apply id="S4.E19.m1.7.8.4.1.2.3.cmml" xref="S4.E19.m1.7.8.4.1.2.3"><minus id="S4.E19.m1.7.8.4.1.2.3.1.cmml" xref="S4.E19.m1.7.8.4.1.2.3.1"></minus><ci id="S4.E19.m1.7.8.4.1.2.3.2.cmml" xref="S4.E19.m1.7.8.4.1.2.3.2">𝑡</ci><apply id="S4.E19.m1.7.8.4.1.2.3.3.cmml" xref="S4.E19.m1.7.8.4.1.2.3.3"><csymbol cd="ambiguous" id="S4.E19.m1.7.8.4.1.2.3.3.1.cmml" xref="S4.E19.m1.7.8.4.1.2.3.3">subscript</csymbol><ci id="S4.E19.m1.7.8.4.1.2.3.3.2.cmml" xref="S4.E19.m1.7.8.4.1.2.3.3.2">𝜏</ci><ci id="S4.E19.m1.7.8.4.1.2.3.3.3.cmml" xref="S4.E19.m1.7.8.4.1.2.3.3.3">d</ci></apply></apply></apply><ci id="S4.E19.m1.7.8.4.1.3.cmml" xref="S4.E19.m1.7.8.4.1.3">𝑡</ci></apply><apply id="S4.E19.m1.7.8.4.2.cmml" xref="S4.E19.m1.7.8.4.2"><times id="S4.E19.m1.7.8.4.2.1.cmml" xref="S4.E19.m1.7.8.4.2.1"></times><apply id="S4.E19.m1.7.8.4.2.2.cmml" xref="S4.E19.m1.7.8.4.2.2"><csymbol cd="ambiguous" id="S4.E19.m1.7.8.4.2.2.1.cmml" xref="S4.E19.m1.7.8.4.2.2">subscript</csymbol><ci id="S4.E19.m1.7.8.4.2.2.2.cmml" xref="S4.E19.m1.7.8.4.2.2.2">Φ</ci><list id="S4.E19.m1.2.2.2.3.cmml" xref="S4.E19.m1.2.2.2.4"><ci id="S4.E19.m1.1.1.1.1.cmml" xref="S4.E19.m1.1.1.1.1">s</ci><ci id="S4.E19.m1.2.2.2.2.cmml" xref="S4.E19.m1.2.2.2.2">𝜁</ci></list></apply><ci id="S4.E19.m1.6.6.cmml" xref="S4.E19.m1.6.6">𝜏</ci><apply id="S4.E19.m1.7.8.4.2.4.cmml" xref="S4.E19.m1.7.8.4.2.4"><csymbol cd="ambiguous" id="S4.E19.m1.7.8.4.2.4.1.cmml" xref="S4.E19.m1.7.8.4.2.4">superscript</csymbol><apply id="S4.E19.m1.7.8.4.2.4.2.cmml" xref="S4.E19.m1.7.8.4.2.4"><csymbol cd="ambiguous" id="S4.E19.m1.7.8.4.2.4.2.1.cmml" xref="S4.E19.m1.7.8.4.2.4">subscript</csymbol><ci id="S4.E19.m1.7.8.4.2.4.2.2.cmml" xref="S4.E19.m1.7.8.4.2.4.2.2">Φ</ci><list id="S4.E19.m1.4.4.2.3.cmml" xref="S4.E19.m1.4.4.2.4"><ci id="S4.E19.m1.3.3.1.1.cmml" xref="S4.E19.m1.3.3.1.1">s</ci><ci id="S4.E19.m1.4.4.2.2.cmml" xref="S4.E19.m1.4.4.2.2">𝜁</ci></list></apply><ci id="S4.E19.m1.7.8.4.2.4.3.cmml" xref="S4.E19.m1.7.8.4.2.4.3">𝑇</ci></apply><ci id="S4.E19.m1.7.7.cmml" xref="S4.E19.m1.7.7">𝜏</ci><apply id="S4.E19.m1.7.8.4.2.6.cmml" xref="S4.E19.m1.7.8.4.2.6"><csymbol cd="latexml" id="S4.E19.m1.7.8.4.2.6.1.cmml" xref="S4.E19.m1.7.8.4.2.6.1">differential-d</csymbol><ci id="S4.E19.m1.7.8.4.2.6.2.cmml" xref="S4.E19.m1.7.8.4.2.6.2">𝜏</ci></apply></apply></apply></apply><apply id="S4.E19.m1.7.8c.cmml" xref="S4.E19.m1.7.8"><geq id="S4.E19.m1.7.8.5.cmml" xref="S4.E19.m1.7.8.5"></geq><share href="https://arxiv.org/html/2401.10785v2#S4.E19.m1.7.8.4.cmml" id="S4.E19.m1.7.8d.cmml" xref="S4.E19.m1.7.8"></share><apply id="S4.E19.m1.7.8.6.cmml" xref="S4.E19.m1.7.8.6"><times id="S4.E19.m1.7.8.6.1.cmml" xref="S4.E19.m1.7.8.6.1"></times><ci id="S4.E19.m1.7.8.6.2.cmml" xref="S4.E19.m1.7.8.6.2">𝜎</ci><ci id="S4.E19.m1.7.8.6.3.cmml" xref="S4.E19.m1.7.8.6.3">𝐼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E19.m1.7c">\Psi_{\zeta}(t):=\int_{t-\tau_{\rm d}}^{t}\Phi_{\rm s,\zeta}(\tau)\Phi_{\rm s,% \zeta}^{T}(\tau)d\tau\geq\sigma I</annotation><annotation encoding="application/x-llamapun" id="S4.E19.m1.7d">roman_Ψ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t ) := ∫ start_POSTSUBSCRIPT italic_t - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_s , italic_ζ end_POSTSUBSCRIPT ( italic_τ ) roman_Φ start_POSTSUBSCRIPT roman_s , italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_τ ) italic_d italic_τ ≥ italic_σ italic_I</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(19)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p4.14">in which <math alttext="\Phi_{\rm s,\zeta}\in\mathbb{R}^{N_{\zeta}\times n}" class="ltx_Math" display="inline" id="S4.SS1.p4.3.m1.2"><semantics id="S4.SS1.p4.3.m1.2a"><mrow id="S4.SS1.p4.3.m1.2.3" xref="S4.SS1.p4.3.m1.2.3.cmml"><msub id="S4.SS1.p4.3.m1.2.3.2" xref="S4.SS1.p4.3.m1.2.3.2.cmml"><mi id="S4.SS1.p4.3.m1.2.3.2.2" mathvariant="normal" xref="S4.SS1.p4.3.m1.2.3.2.2.cmml">Φ</mi><mrow id="S4.SS1.p4.3.m1.2.2.2.4" xref="S4.SS1.p4.3.m1.2.2.2.3.cmml"><mi id="S4.SS1.p4.3.m1.1.1.1.1" mathvariant="normal" xref="S4.SS1.p4.3.m1.1.1.1.1.cmml">s</mi><mo id="S4.SS1.p4.3.m1.2.2.2.4.1" xref="S4.SS1.p4.3.m1.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p4.3.m1.2.2.2.2" xref="S4.SS1.p4.3.m1.2.2.2.2.cmml">ζ</mi></mrow></msub><mo id="S4.SS1.p4.3.m1.2.3.1" xref="S4.SS1.p4.3.m1.2.3.1.cmml">∈</mo><msup id="S4.SS1.p4.3.m1.2.3.3" xref="S4.SS1.p4.3.m1.2.3.3.cmml"><mi id="S4.SS1.p4.3.m1.2.3.3.2" xref="S4.SS1.p4.3.m1.2.3.3.2.cmml">ℝ</mi><mrow id="S4.SS1.p4.3.m1.2.3.3.3" xref="S4.SS1.p4.3.m1.2.3.3.3.cmml"><msub id="S4.SS1.p4.3.m1.2.3.3.3.2" xref="S4.SS1.p4.3.m1.2.3.3.3.2.cmml"><mi id="S4.SS1.p4.3.m1.2.3.3.3.2.2" xref="S4.SS1.p4.3.m1.2.3.3.3.2.2.cmml">N</mi><mi id="S4.SS1.p4.3.m1.2.3.3.3.2.3" xref="S4.SS1.p4.3.m1.2.3.3.3.2.3.cmml">ζ</mi></msub><mo id="S4.SS1.p4.3.m1.2.3.3.3.1" lspace="0.222em" rspace="0.222em" xref="S4.SS1.p4.3.m1.2.3.3.3.1.cmml">×</mo><mi id="S4.SS1.p4.3.m1.2.3.3.3.3" xref="S4.SS1.p4.3.m1.2.3.3.3.3.cmml">n</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.3.m1.2b"><apply id="S4.SS1.p4.3.m1.2.3.cmml" xref="S4.SS1.p4.3.m1.2.3"><in id="S4.SS1.p4.3.m1.2.3.1.cmml" xref="S4.SS1.p4.3.m1.2.3.1"></in><apply id="S4.SS1.p4.3.m1.2.3.2.cmml" xref="S4.SS1.p4.3.m1.2.3.2"><csymbol cd="ambiguous" id="S4.SS1.p4.3.m1.2.3.2.1.cmml" xref="S4.SS1.p4.3.m1.2.3.2">subscript</csymbol><ci id="S4.SS1.p4.3.m1.2.3.2.2.cmml" xref="S4.SS1.p4.3.m1.2.3.2.2">Φ</ci><list id="S4.SS1.p4.3.m1.2.2.2.3.cmml" xref="S4.SS1.p4.3.m1.2.2.2.4"><ci id="S4.SS1.p4.3.m1.1.1.1.1.cmml" xref="S4.SS1.p4.3.m1.1.1.1.1">s</ci><ci id="S4.SS1.p4.3.m1.2.2.2.2.cmml" xref="S4.SS1.p4.3.m1.2.2.2.2">𝜁</ci></list></apply><apply id="S4.SS1.p4.3.m1.2.3.3.cmml" xref="S4.SS1.p4.3.m1.2.3.3"><csymbol cd="ambiguous" id="S4.SS1.p4.3.m1.2.3.3.1.cmml" xref="S4.SS1.p4.3.m1.2.3.3">superscript</csymbol><ci id="S4.SS1.p4.3.m1.2.3.3.2.cmml" xref="S4.SS1.p4.3.m1.2.3.3.2">ℝ</ci><apply id="S4.SS1.p4.3.m1.2.3.3.3.cmml" xref="S4.SS1.p4.3.m1.2.3.3.3"><times id="S4.SS1.p4.3.m1.2.3.3.3.1.cmml" xref="S4.SS1.p4.3.m1.2.3.3.3.1"></times><apply id="S4.SS1.p4.3.m1.2.3.3.3.2.cmml" xref="S4.SS1.p4.3.m1.2.3.3.3.2"><csymbol cd="ambiguous" id="S4.SS1.p4.3.m1.2.3.3.3.2.1.cmml" xref="S4.SS1.p4.3.m1.2.3.3.3.2">subscript</csymbol><ci id="S4.SS1.p4.3.m1.2.3.3.3.2.2.cmml" xref="S4.SS1.p4.3.m1.2.3.3.3.2.2">𝑁</ci><ci id="S4.SS1.p4.3.m1.2.3.3.3.2.3.cmml" xref="S4.SS1.p4.3.m1.2.3.3.3.2.3">𝜁</ci></apply><ci id="S4.SS1.p4.3.m1.2.3.3.3.3.cmml" xref="S4.SS1.p4.3.m1.2.3.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.3.m1.2c">\Phi_{\rm s,\zeta}\in\mathbb{R}^{N_{\zeta}\times n}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.3.m1.2d">roman_Φ start_POSTSUBSCRIPT roman_s , italic_ζ end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT × italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> denotes a sub-regressor composed of all active channels <math alttext="{\bm{\phi}}_{{\rm s},k_{j}}" class="ltx_Math" display="inline" id="S4.SS1.p4.4.m2.2"><semantics id="S4.SS1.p4.4.m2.2a"><msub id="S4.SS1.p4.4.m2.2.3" xref="S4.SS1.p4.4.m2.2.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS1.p4.4.m2.2.3.2" mathvariant="bold-italic" xref="S4.SS1.p4.4.m2.2.3.2.cmml">ϕ</mi><mrow id="S4.SS1.p4.4.m2.2.2.2.2" xref="S4.SS1.p4.4.m2.2.2.2.3.cmml"><mi id="S4.SS1.p4.4.m2.1.1.1.1" mathvariant="normal" xref="S4.SS1.p4.4.m2.1.1.1.1.cmml">s</mi><mo id="S4.SS1.p4.4.m2.2.2.2.2.2" xref="S4.SS1.p4.4.m2.2.2.2.3.cmml">,</mo><msub id="S4.SS1.p4.4.m2.2.2.2.2.1" xref="S4.SS1.p4.4.m2.2.2.2.2.1.cmml"><mi id="S4.SS1.p4.4.m2.2.2.2.2.1.2" xref="S4.SS1.p4.4.m2.2.2.2.2.1.2.cmml">k</mi><mi id="S4.SS1.p4.4.m2.2.2.2.2.1.3" xref="S4.SS1.p4.4.m2.2.2.2.2.1.3.cmml">j</mi></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.4.m2.2b"><apply id="S4.SS1.p4.4.m2.2.3.cmml" xref="S4.SS1.p4.4.m2.2.3"><csymbol cd="ambiguous" id="S4.SS1.p4.4.m2.2.3.1.cmml" xref="S4.SS1.p4.4.m2.2.3">subscript</csymbol><ci id="S4.SS1.p4.4.m2.2.3.2.cmml" xref="S4.SS1.p4.4.m2.2.3.2">bold-italic-ϕ</ci><list id="S4.SS1.p4.4.m2.2.2.2.3.cmml" xref="S4.SS1.p4.4.m2.2.2.2.2"><ci id="S4.SS1.p4.4.m2.1.1.1.1.cmml" xref="S4.SS1.p4.4.m2.1.1.1.1">s</ci><apply id="S4.SS1.p4.4.m2.2.2.2.2.1.cmml" xref="S4.SS1.p4.4.m2.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.SS1.p4.4.m2.2.2.2.2.1.1.cmml" xref="S4.SS1.p4.4.m2.2.2.2.2.1">subscript</csymbol><ci id="S4.SS1.p4.4.m2.2.2.2.2.1.2.cmml" xref="S4.SS1.p4.4.m2.2.2.2.2.1.2">𝑘</ci><ci id="S4.SS1.p4.4.m2.2.2.2.2.1.3.cmml" xref="S4.SS1.p4.4.m2.2.2.2.2.1.3">𝑗</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.4.m2.2c">{\bm{\phi}}_{{\rm s},k_{j}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.4.m2.2d">bold_italic_ϕ start_POSTSUBSCRIPT roman_s , italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="\Phi_{\rm s}" class="ltx_Math" display="inline" id="S4.SS1.p4.5.m3.1"><semantics id="S4.SS1.p4.5.m3.1a"><msub id="S4.SS1.p4.5.m3.1.1" xref="S4.SS1.p4.5.m3.1.1.cmml"><mi id="S4.SS1.p4.5.m3.1.1.2" mathvariant="normal" xref="S4.SS1.p4.5.m3.1.1.2.cmml">Φ</mi><mi id="S4.SS1.p4.5.m3.1.1.3" mathvariant="normal" xref="S4.SS1.p4.5.m3.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.5.m3.1b"><apply id="S4.SS1.p4.5.m3.1.1.cmml" xref="S4.SS1.p4.5.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.p4.5.m3.1.1.1.cmml" xref="S4.SS1.p4.5.m3.1.1">subscript</csymbol><ci id="S4.SS1.p4.5.m3.1.1.2.cmml" xref="S4.SS1.p4.5.m3.1.1.2">Φ</ci><ci id="S4.SS1.p4.5.m3.1.1.3.cmml" xref="S4.SS1.p4.5.m3.1.1.3">s</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.5.m3.1c">\Phi_{\rm s}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.5.m3.1d">roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT</annotation></semantics></math>, i.e., <math alttext="\Phi_{\rm s,\zeta}:=[{\bm{\phi}}_{{\rm s},k_{1}},{\bm{\phi}}_{{\rm s},k_{2}},% \cdots,{\bm{\phi}}_{{\rm s},k_{N_{\zeta}}}]^{T}" class="ltx_Math" display="inline" id="S4.SS1.p4.6.m4.12"><semantics id="S4.SS1.p4.6.m4.12a"><mrow id="S4.SS1.p4.6.m4.12.12" xref="S4.SS1.p4.6.m4.12.12.cmml"><msub id="S4.SS1.p4.6.m4.12.12.5" xref="S4.SS1.p4.6.m4.12.12.5.cmml"><mi id="S4.SS1.p4.6.m4.12.12.5.2" mathvariant="normal" xref="S4.SS1.p4.6.m4.12.12.5.2.cmml">Φ</mi><mrow id="S4.SS1.p4.6.m4.2.2.2.4" xref="S4.SS1.p4.6.m4.2.2.2.3.cmml"><mi id="S4.SS1.p4.6.m4.1.1.1.1" mathvariant="normal" xref="S4.SS1.p4.6.m4.1.1.1.1.cmml">s</mi><mo id="S4.SS1.p4.6.m4.2.2.2.4.1" xref="S4.SS1.p4.6.m4.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p4.6.m4.2.2.2.2" xref="S4.SS1.p4.6.m4.2.2.2.2.cmml">ζ</mi></mrow></msub><mo id="S4.SS1.p4.6.m4.12.12.4" lspace="0.278em" rspace="0.278em" xref="S4.SS1.p4.6.m4.12.12.4.cmml">:=</mo><msup id="S4.SS1.p4.6.m4.12.12.3" xref="S4.SS1.p4.6.m4.12.12.3.cmml"><mrow id="S4.SS1.p4.6.m4.12.12.3.3.3" xref="S4.SS1.p4.6.m4.12.12.3.3.4.cmml"><mo id="S4.SS1.p4.6.m4.12.12.3.3.3.4" stretchy="false" xref="S4.SS1.p4.6.m4.12.12.3.3.4.cmml">[</mo><msub id="S4.SS1.p4.6.m4.10.10.1.1.1.1" xref="S4.SS1.p4.6.m4.10.10.1.1.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS1.p4.6.m4.10.10.1.1.1.1.2" mathvariant="bold-italic" xref="S4.SS1.p4.6.m4.10.10.1.1.1.1.2.cmml">ϕ</mi><mrow id="S4.SS1.p4.6.m4.4.4.2.2" xref="S4.SS1.p4.6.m4.4.4.2.3.cmml"><mi id="S4.SS1.p4.6.m4.3.3.1.1" mathvariant="normal" xref="S4.SS1.p4.6.m4.3.3.1.1.cmml">s</mi><mo id="S4.SS1.p4.6.m4.4.4.2.2.2" xref="S4.SS1.p4.6.m4.4.4.2.3.cmml">,</mo><msub id="S4.SS1.p4.6.m4.4.4.2.2.1" xref="S4.SS1.p4.6.m4.4.4.2.2.1.cmml"><mi id="S4.SS1.p4.6.m4.4.4.2.2.1.2" xref="S4.SS1.p4.6.m4.4.4.2.2.1.2.cmml">k</mi><mn id="S4.SS1.p4.6.m4.4.4.2.2.1.3" xref="S4.SS1.p4.6.m4.4.4.2.2.1.3.cmml">1</mn></msub></mrow></msub><mo id="S4.SS1.p4.6.m4.12.12.3.3.3.5" xref="S4.SS1.p4.6.m4.12.12.3.3.4.cmml">,</mo><msub id="S4.SS1.p4.6.m4.11.11.2.2.2.2" xref="S4.SS1.p4.6.m4.11.11.2.2.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS1.p4.6.m4.11.11.2.2.2.2.2" mathvariant="bold-italic" xref="S4.SS1.p4.6.m4.11.11.2.2.2.2.2.cmml">ϕ</mi><mrow id="S4.SS1.p4.6.m4.6.6.2.2" xref="S4.SS1.p4.6.m4.6.6.2.3.cmml"><mi id="S4.SS1.p4.6.m4.5.5.1.1" mathvariant="normal" xref="S4.SS1.p4.6.m4.5.5.1.1.cmml">s</mi><mo id="S4.SS1.p4.6.m4.6.6.2.2.2" xref="S4.SS1.p4.6.m4.6.6.2.3.cmml">,</mo><msub id="S4.SS1.p4.6.m4.6.6.2.2.1" xref="S4.SS1.p4.6.m4.6.6.2.2.1.cmml"><mi id="S4.SS1.p4.6.m4.6.6.2.2.1.2" xref="S4.SS1.p4.6.m4.6.6.2.2.1.2.cmml">k</mi><mn id="S4.SS1.p4.6.m4.6.6.2.2.1.3" xref="S4.SS1.p4.6.m4.6.6.2.2.1.3.cmml">2</mn></msub></mrow></msub><mo id="S4.SS1.p4.6.m4.12.12.3.3.3.6" xref="S4.SS1.p4.6.m4.12.12.3.3.4.cmml">,</mo><mi id="S4.SS1.p4.6.m4.9.9" mathvariant="normal" xref="S4.SS1.p4.6.m4.9.9.cmml">⋯</mi><mo id="S4.SS1.p4.6.m4.12.12.3.3.3.7" xref="S4.SS1.p4.6.m4.12.12.3.3.4.cmml">,</mo><msub id="S4.SS1.p4.6.m4.12.12.3.3.3.3" xref="S4.SS1.p4.6.m4.12.12.3.3.3.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS1.p4.6.m4.12.12.3.3.3.3.2" mathvariant="bold-italic" xref="S4.SS1.p4.6.m4.12.12.3.3.3.3.2.cmml">ϕ</mi><mrow id="S4.SS1.p4.6.m4.8.8.2.2" xref="S4.SS1.p4.6.m4.8.8.2.3.cmml"><mi id="S4.SS1.p4.6.m4.7.7.1.1" mathvariant="normal" xref="S4.SS1.p4.6.m4.7.7.1.1.cmml">s</mi><mo id="S4.SS1.p4.6.m4.8.8.2.2.2" xref="S4.SS1.p4.6.m4.8.8.2.3.cmml">,</mo><msub id="S4.SS1.p4.6.m4.8.8.2.2.1" xref="S4.SS1.p4.6.m4.8.8.2.2.1.cmml"><mi id="S4.SS1.p4.6.m4.8.8.2.2.1.2" xref="S4.SS1.p4.6.m4.8.8.2.2.1.2.cmml">k</mi><msub id="S4.SS1.p4.6.m4.8.8.2.2.1.3" xref="S4.SS1.p4.6.m4.8.8.2.2.1.3.cmml"><mi id="S4.SS1.p4.6.m4.8.8.2.2.1.3.2" xref="S4.SS1.p4.6.m4.8.8.2.2.1.3.2.cmml">N</mi><mi id="S4.SS1.p4.6.m4.8.8.2.2.1.3.3" xref="S4.SS1.p4.6.m4.8.8.2.2.1.3.3.cmml">ζ</mi></msub></msub></mrow></msub><mo id="S4.SS1.p4.6.m4.12.12.3.3.3.8" stretchy="false" xref="S4.SS1.p4.6.m4.12.12.3.3.4.cmml">]</mo></mrow><mi id="S4.SS1.p4.6.m4.12.12.3.5" xref="S4.SS1.p4.6.m4.12.12.3.5.cmml">T</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.6.m4.12b"><apply id="S4.SS1.p4.6.m4.12.12.cmml" xref="S4.SS1.p4.6.m4.12.12"><csymbol cd="latexml" id="S4.SS1.p4.6.m4.12.12.4.cmml" xref="S4.SS1.p4.6.m4.12.12.4">assign</csymbol><apply id="S4.SS1.p4.6.m4.12.12.5.cmml" xref="S4.SS1.p4.6.m4.12.12.5"><csymbol cd="ambiguous" id="S4.SS1.p4.6.m4.12.12.5.1.cmml" xref="S4.SS1.p4.6.m4.12.12.5">subscript</csymbol><ci id="S4.SS1.p4.6.m4.12.12.5.2.cmml" xref="S4.SS1.p4.6.m4.12.12.5.2">Φ</ci><list id="S4.SS1.p4.6.m4.2.2.2.3.cmml" xref="S4.SS1.p4.6.m4.2.2.2.4"><ci id="S4.SS1.p4.6.m4.1.1.1.1.cmml" xref="S4.SS1.p4.6.m4.1.1.1.1">s</ci><ci id="S4.SS1.p4.6.m4.2.2.2.2.cmml" xref="S4.SS1.p4.6.m4.2.2.2.2">𝜁</ci></list></apply><apply id="S4.SS1.p4.6.m4.12.12.3.cmml" xref="S4.SS1.p4.6.m4.12.12.3"><csymbol cd="ambiguous" id="S4.SS1.p4.6.m4.12.12.3.4.cmml" xref="S4.SS1.p4.6.m4.12.12.3">superscript</csymbol><list id="S4.SS1.p4.6.m4.12.12.3.3.4.cmml" xref="S4.SS1.p4.6.m4.12.12.3.3.3"><apply id="S4.SS1.p4.6.m4.10.10.1.1.1.1.cmml" xref="S4.SS1.p4.6.m4.10.10.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p4.6.m4.10.10.1.1.1.1.1.cmml" xref="S4.SS1.p4.6.m4.10.10.1.1.1.1">subscript</csymbol><ci id="S4.SS1.p4.6.m4.10.10.1.1.1.1.2.cmml" xref="S4.SS1.p4.6.m4.10.10.1.1.1.1.2">bold-italic-ϕ</ci><list id="S4.SS1.p4.6.m4.4.4.2.3.cmml" xref="S4.SS1.p4.6.m4.4.4.2.2"><ci id="S4.SS1.p4.6.m4.3.3.1.1.cmml" xref="S4.SS1.p4.6.m4.3.3.1.1">s</ci><apply id="S4.SS1.p4.6.m4.4.4.2.2.1.cmml" xref="S4.SS1.p4.6.m4.4.4.2.2.1"><csymbol cd="ambiguous" id="S4.SS1.p4.6.m4.4.4.2.2.1.1.cmml" xref="S4.SS1.p4.6.m4.4.4.2.2.1">subscript</csymbol><ci id="S4.SS1.p4.6.m4.4.4.2.2.1.2.cmml" xref="S4.SS1.p4.6.m4.4.4.2.2.1.2">𝑘</ci><cn id="S4.SS1.p4.6.m4.4.4.2.2.1.3.cmml" type="integer" xref="S4.SS1.p4.6.m4.4.4.2.2.1.3">1</cn></apply></list></apply><apply id="S4.SS1.p4.6.m4.11.11.2.2.2.2.cmml" xref="S4.SS1.p4.6.m4.11.11.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.p4.6.m4.11.11.2.2.2.2.1.cmml" xref="S4.SS1.p4.6.m4.11.11.2.2.2.2">subscript</csymbol><ci id="S4.SS1.p4.6.m4.11.11.2.2.2.2.2.cmml" xref="S4.SS1.p4.6.m4.11.11.2.2.2.2.2">bold-italic-ϕ</ci><list id="S4.SS1.p4.6.m4.6.6.2.3.cmml" xref="S4.SS1.p4.6.m4.6.6.2.2"><ci id="S4.SS1.p4.6.m4.5.5.1.1.cmml" xref="S4.SS1.p4.6.m4.5.5.1.1">s</ci><apply id="S4.SS1.p4.6.m4.6.6.2.2.1.cmml" xref="S4.SS1.p4.6.m4.6.6.2.2.1"><csymbol cd="ambiguous" id="S4.SS1.p4.6.m4.6.6.2.2.1.1.cmml" xref="S4.SS1.p4.6.m4.6.6.2.2.1">subscript</csymbol><ci id="S4.SS1.p4.6.m4.6.6.2.2.1.2.cmml" xref="S4.SS1.p4.6.m4.6.6.2.2.1.2">𝑘</ci><cn id="S4.SS1.p4.6.m4.6.6.2.2.1.3.cmml" type="integer" xref="S4.SS1.p4.6.m4.6.6.2.2.1.3">2</cn></apply></list></apply><ci id="S4.SS1.p4.6.m4.9.9.cmml" xref="S4.SS1.p4.6.m4.9.9">⋯</ci><apply id="S4.SS1.p4.6.m4.12.12.3.3.3.3.cmml" xref="S4.SS1.p4.6.m4.12.12.3.3.3.3"><csymbol cd="ambiguous" id="S4.SS1.p4.6.m4.12.12.3.3.3.3.1.cmml" xref="S4.SS1.p4.6.m4.12.12.3.3.3.3">subscript</csymbol><ci id="S4.SS1.p4.6.m4.12.12.3.3.3.3.2.cmml" xref="S4.SS1.p4.6.m4.12.12.3.3.3.3.2">bold-italic-ϕ</ci><list id="S4.SS1.p4.6.m4.8.8.2.3.cmml" xref="S4.SS1.p4.6.m4.8.8.2.2"><ci id="S4.SS1.p4.6.m4.7.7.1.1.cmml" xref="S4.SS1.p4.6.m4.7.7.1.1">s</ci><apply id="S4.SS1.p4.6.m4.8.8.2.2.1.cmml" xref="S4.SS1.p4.6.m4.8.8.2.2.1"><csymbol cd="ambiguous" id="S4.SS1.p4.6.m4.8.8.2.2.1.1.cmml" xref="S4.SS1.p4.6.m4.8.8.2.2.1">subscript</csymbol><ci id="S4.SS1.p4.6.m4.8.8.2.2.1.2.cmml" xref="S4.SS1.p4.6.m4.8.8.2.2.1.2">𝑘</ci><apply id="S4.SS1.p4.6.m4.8.8.2.2.1.3.cmml" xref="S4.SS1.p4.6.m4.8.8.2.2.1.3"><csymbol cd="ambiguous" id="S4.SS1.p4.6.m4.8.8.2.2.1.3.1.cmml" xref="S4.SS1.p4.6.m4.8.8.2.2.1.3">subscript</csymbol><ci id="S4.SS1.p4.6.m4.8.8.2.2.1.3.2.cmml" xref="S4.SS1.p4.6.m4.8.8.2.2.1.3.2">𝑁</ci><ci id="S4.SS1.p4.6.m4.8.8.2.2.1.3.3.cmml" xref="S4.SS1.p4.6.m4.8.8.2.2.1.3.3">𝜁</ci></apply></apply></list></apply></list><ci id="S4.SS1.p4.6.m4.12.12.3.5.cmml" xref="S4.SS1.p4.6.m4.12.12.3.5">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.6.m4.12c">\Phi_{\rm s,\zeta}:=[{\bm{\phi}}_{{\rm s},k_{1}},{\bm{\phi}}_{{\rm s},k_{2}},% \cdots,{\bm{\phi}}_{{\rm s},k_{N_{\zeta}}}]^{T}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.6.m4.12d">roman_Φ start_POSTSUBSCRIPT roman_s , italic_ζ end_POSTSUBSCRIPT := [ bold_italic_ϕ start_POSTSUBSCRIPT roman_s , italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , bold_italic_ϕ start_POSTSUBSCRIPT roman_s , italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , ⋯ , bold_italic_ϕ start_POSTSUBSCRIPT roman_s , italic_k start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="\|{\bm{\phi}}_{{\rm s},k_{j}}(\tau_{j})\|>0" class="ltx_Math" display="inline" id="S4.SS1.p4.7.m5.3"><semantics id="S4.SS1.p4.7.m5.3a"><mrow id="S4.SS1.p4.7.m5.3.3" xref="S4.SS1.p4.7.m5.3.3.cmml"><mrow id="S4.SS1.p4.7.m5.3.3.1.1" xref="S4.SS1.p4.7.m5.3.3.1.2.cmml"><mo id="S4.SS1.p4.7.m5.3.3.1.1.2" stretchy="false" xref="S4.SS1.p4.7.m5.3.3.1.2.1.cmml">‖</mo><mrow id="S4.SS1.p4.7.m5.3.3.1.1.1" xref="S4.SS1.p4.7.m5.3.3.1.1.1.cmml"><msub id="S4.SS1.p4.7.m5.3.3.1.1.1.3" xref="S4.SS1.p4.7.m5.3.3.1.1.1.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS1.p4.7.m5.3.3.1.1.1.3.2" mathvariant="bold-italic" xref="S4.SS1.p4.7.m5.3.3.1.1.1.3.2.cmml">ϕ</mi><mrow id="S4.SS1.p4.7.m5.2.2.2.2" xref="S4.SS1.p4.7.m5.2.2.2.3.cmml"><mi id="S4.SS1.p4.7.m5.1.1.1.1" mathvariant="normal" xref="S4.SS1.p4.7.m5.1.1.1.1.cmml">s</mi><mo id="S4.SS1.p4.7.m5.2.2.2.2.2" xref="S4.SS1.p4.7.m5.2.2.2.3.cmml">,</mo><msub id="S4.SS1.p4.7.m5.2.2.2.2.1" xref="S4.SS1.p4.7.m5.2.2.2.2.1.cmml"><mi id="S4.SS1.p4.7.m5.2.2.2.2.1.2" xref="S4.SS1.p4.7.m5.2.2.2.2.1.2.cmml">k</mi><mi id="S4.SS1.p4.7.m5.2.2.2.2.1.3" xref="S4.SS1.p4.7.m5.2.2.2.2.1.3.cmml">j</mi></msub></mrow></msub><mo id="S4.SS1.p4.7.m5.3.3.1.1.1.2" xref="S4.SS1.p4.7.m5.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.p4.7.m5.3.3.1.1.1.1.1" xref="S4.SS1.p4.7.m5.3.3.1.1.1.1.1.1.cmml"><mo id="S4.SS1.p4.7.m5.3.3.1.1.1.1.1.2" stretchy="false" xref="S4.SS1.p4.7.m5.3.3.1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS1.p4.7.m5.3.3.1.1.1.1.1.1" xref="S4.SS1.p4.7.m5.3.3.1.1.1.1.1.1.cmml"><mi id="S4.SS1.p4.7.m5.3.3.1.1.1.1.1.1.2" xref="S4.SS1.p4.7.m5.3.3.1.1.1.1.1.1.2.cmml">τ</mi><mi id="S4.SS1.p4.7.m5.3.3.1.1.1.1.1.1.3" xref="S4.SS1.p4.7.m5.3.3.1.1.1.1.1.1.3.cmml">j</mi></msub><mo id="S4.SS1.p4.7.m5.3.3.1.1.1.1.1.3" stretchy="false" xref="S4.SS1.p4.7.m5.3.3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p4.7.m5.3.3.1.1.3" stretchy="false" xref="S4.SS1.p4.7.m5.3.3.1.2.1.cmml">‖</mo></mrow><mo id="S4.SS1.p4.7.m5.3.3.2" xref="S4.SS1.p4.7.m5.3.3.2.cmml">></mo><mn id="S4.SS1.p4.7.m5.3.3.3" xref="S4.SS1.p4.7.m5.3.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.7.m5.3b"><apply id="S4.SS1.p4.7.m5.3.3.cmml" xref="S4.SS1.p4.7.m5.3.3"><gt id="S4.SS1.p4.7.m5.3.3.2.cmml" xref="S4.SS1.p4.7.m5.3.3.2"></gt><apply id="S4.SS1.p4.7.m5.3.3.1.2.cmml" xref="S4.SS1.p4.7.m5.3.3.1.1"><csymbol cd="latexml" id="S4.SS1.p4.7.m5.3.3.1.2.1.cmml" xref="S4.SS1.p4.7.m5.3.3.1.1.2">norm</csymbol><apply id="S4.SS1.p4.7.m5.3.3.1.1.1.cmml" xref="S4.SS1.p4.7.m5.3.3.1.1.1"><times id="S4.SS1.p4.7.m5.3.3.1.1.1.2.cmml" xref="S4.SS1.p4.7.m5.3.3.1.1.1.2"></times><apply id="S4.SS1.p4.7.m5.3.3.1.1.1.3.cmml" xref="S4.SS1.p4.7.m5.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p4.7.m5.3.3.1.1.1.3.1.cmml" xref="S4.SS1.p4.7.m5.3.3.1.1.1.3">subscript</csymbol><ci id="S4.SS1.p4.7.m5.3.3.1.1.1.3.2.cmml" xref="S4.SS1.p4.7.m5.3.3.1.1.1.3.2">bold-italic-ϕ</ci><list id="S4.SS1.p4.7.m5.2.2.2.3.cmml" xref="S4.SS1.p4.7.m5.2.2.2.2"><ci id="S4.SS1.p4.7.m5.1.1.1.1.cmml" xref="S4.SS1.p4.7.m5.1.1.1.1">s</ci><apply id="S4.SS1.p4.7.m5.2.2.2.2.1.cmml" xref="S4.SS1.p4.7.m5.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.SS1.p4.7.m5.2.2.2.2.1.1.cmml" xref="S4.SS1.p4.7.m5.2.2.2.2.1">subscript</csymbol><ci id="S4.SS1.p4.7.m5.2.2.2.2.1.2.cmml" xref="S4.SS1.p4.7.m5.2.2.2.2.1.2">𝑘</ci><ci id="S4.SS1.p4.7.m5.2.2.2.2.1.3.cmml" xref="S4.SS1.p4.7.m5.2.2.2.2.1.3">𝑗</ci></apply></list></apply><apply id="S4.SS1.p4.7.m5.3.3.1.1.1.1.1.1.cmml" xref="S4.SS1.p4.7.m5.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p4.7.m5.3.3.1.1.1.1.1.1.1.cmml" xref="S4.SS1.p4.7.m5.3.3.1.1.1.1.1">subscript</csymbol><ci id="S4.SS1.p4.7.m5.3.3.1.1.1.1.1.1.2.cmml" xref="S4.SS1.p4.7.m5.3.3.1.1.1.1.1.1.2">𝜏</ci><ci id="S4.SS1.p4.7.m5.3.3.1.1.1.1.1.1.3.cmml" xref="S4.SS1.p4.7.m5.3.3.1.1.1.1.1.1.3">𝑗</ci></apply></apply></apply><cn id="S4.SS1.p4.7.m5.3.3.3.cmml" type="integer" xref="S4.SS1.p4.7.m5.3.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.7.m5.3c">\|{\bm{\phi}}_{{\rm s},k_{j}}(\tau_{j})\|>0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.7.m5.3d">∥ bold_italic_ϕ start_POSTSUBSCRIPT roman_s , italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_τ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ∥ > 0</annotation></semantics></math>, <math alttext="\exists\tau_{j}\in[t-\tau_{\rm d},t]" class="ltx_Math" display="inline" id="S4.SS1.p4.8.m6.2"><semantics id="S4.SS1.p4.8.m6.2a"><mrow id="S4.SS1.p4.8.m6.2.2" xref="S4.SS1.p4.8.m6.2.2.cmml"><mrow id="S4.SS1.p4.8.m6.2.2.3" xref="S4.SS1.p4.8.m6.2.2.3.cmml"><mo id="S4.SS1.p4.8.m6.2.2.3.1" rspace="0.167em" xref="S4.SS1.p4.8.m6.2.2.3.1.cmml">∃</mo><msub id="S4.SS1.p4.8.m6.2.2.3.2" xref="S4.SS1.p4.8.m6.2.2.3.2.cmml"><mi id="S4.SS1.p4.8.m6.2.2.3.2.2" xref="S4.SS1.p4.8.m6.2.2.3.2.2.cmml">τ</mi><mi id="S4.SS1.p4.8.m6.2.2.3.2.3" xref="S4.SS1.p4.8.m6.2.2.3.2.3.cmml">j</mi></msub></mrow><mo id="S4.SS1.p4.8.m6.2.2.2" xref="S4.SS1.p4.8.m6.2.2.2.cmml">∈</mo><mrow id="S4.SS1.p4.8.m6.2.2.1.1" xref="S4.SS1.p4.8.m6.2.2.1.2.cmml"><mo id="S4.SS1.p4.8.m6.2.2.1.1.2" stretchy="false" xref="S4.SS1.p4.8.m6.2.2.1.2.cmml">[</mo><mrow id="S4.SS1.p4.8.m6.2.2.1.1.1" xref="S4.SS1.p4.8.m6.2.2.1.1.1.cmml"><mi id="S4.SS1.p4.8.m6.2.2.1.1.1.2" xref="S4.SS1.p4.8.m6.2.2.1.1.1.2.cmml">t</mi><mo id="S4.SS1.p4.8.m6.2.2.1.1.1.1" xref="S4.SS1.p4.8.m6.2.2.1.1.1.1.cmml">−</mo><msub id="S4.SS1.p4.8.m6.2.2.1.1.1.3" xref="S4.SS1.p4.8.m6.2.2.1.1.1.3.cmml"><mi id="S4.SS1.p4.8.m6.2.2.1.1.1.3.2" xref="S4.SS1.p4.8.m6.2.2.1.1.1.3.2.cmml">τ</mi><mi id="S4.SS1.p4.8.m6.2.2.1.1.1.3.3" mathvariant="normal" xref="S4.SS1.p4.8.m6.2.2.1.1.1.3.3.cmml">d</mi></msub></mrow><mo id="S4.SS1.p4.8.m6.2.2.1.1.3" xref="S4.SS1.p4.8.m6.2.2.1.2.cmml">,</mo><mi id="S4.SS1.p4.8.m6.1.1" xref="S4.SS1.p4.8.m6.1.1.cmml">t</mi><mo id="S4.SS1.p4.8.m6.2.2.1.1.4" stretchy="false" xref="S4.SS1.p4.8.m6.2.2.1.2.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.8.m6.2b"><apply id="S4.SS1.p4.8.m6.2.2.cmml" xref="S4.SS1.p4.8.m6.2.2"><in id="S4.SS1.p4.8.m6.2.2.2.cmml" xref="S4.SS1.p4.8.m6.2.2.2"></in><apply id="S4.SS1.p4.8.m6.2.2.3.cmml" xref="S4.SS1.p4.8.m6.2.2.3"><exists id="S4.SS1.p4.8.m6.2.2.3.1.cmml" xref="S4.SS1.p4.8.m6.2.2.3.1"></exists><apply id="S4.SS1.p4.8.m6.2.2.3.2.cmml" xref="S4.SS1.p4.8.m6.2.2.3.2"><csymbol cd="ambiguous" id="S4.SS1.p4.8.m6.2.2.3.2.1.cmml" xref="S4.SS1.p4.8.m6.2.2.3.2">subscript</csymbol><ci id="S4.SS1.p4.8.m6.2.2.3.2.2.cmml" xref="S4.SS1.p4.8.m6.2.2.3.2.2">𝜏</ci><ci id="S4.SS1.p4.8.m6.2.2.3.2.3.cmml" xref="S4.SS1.p4.8.m6.2.2.3.2.3">𝑗</ci></apply></apply><interval closure="closed" id="S4.SS1.p4.8.m6.2.2.1.2.cmml" xref="S4.SS1.p4.8.m6.2.2.1.1"><apply id="S4.SS1.p4.8.m6.2.2.1.1.1.cmml" xref="S4.SS1.p4.8.m6.2.2.1.1.1"><minus id="S4.SS1.p4.8.m6.2.2.1.1.1.1.cmml" xref="S4.SS1.p4.8.m6.2.2.1.1.1.1"></minus><ci id="S4.SS1.p4.8.m6.2.2.1.1.1.2.cmml" xref="S4.SS1.p4.8.m6.2.2.1.1.1.2">𝑡</ci><apply id="S4.SS1.p4.8.m6.2.2.1.1.1.3.cmml" xref="S4.SS1.p4.8.m6.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p4.8.m6.2.2.1.1.1.3.1.cmml" xref="S4.SS1.p4.8.m6.2.2.1.1.1.3">subscript</csymbol><ci id="S4.SS1.p4.8.m6.2.2.1.1.1.3.2.cmml" xref="S4.SS1.p4.8.m6.2.2.1.1.1.3.2">𝜏</ci><ci id="S4.SS1.p4.8.m6.2.2.1.1.1.3.3.cmml" xref="S4.SS1.p4.8.m6.2.2.1.1.1.3.3">d</ci></apply></apply><ci id="S4.SS1.p4.8.m6.1.1.cmml" xref="S4.SS1.p4.8.m6.1.1">𝑡</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.8.m6.2c">\exists\tau_{j}\in[t-\tau_{\rm d},t]</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.8.m6.2d">∃ italic_τ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ [ italic_t - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT , italic_t ]</annotation></semantics></math>, <math alttext="1\leq k_{j}\leq N" class="ltx_Math" display="inline" id="S4.SS1.p4.9.m7.1"><semantics id="S4.SS1.p4.9.m7.1a"><mrow id="S4.SS1.p4.9.m7.1.1" xref="S4.SS1.p4.9.m7.1.1.cmml"><mn id="S4.SS1.p4.9.m7.1.1.2" xref="S4.SS1.p4.9.m7.1.1.2.cmml">1</mn><mo id="S4.SS1.p4.9.m7.1.1.3" xref="S4.SS1.p4.9.m7.1.1.3.cmml">≤</mo><msub id="S4.SS1.p4.9.m7.1.1.4" xref="S4.SS1.p4.9.m7.1.1.4.cmml"><mi id="S4.SS1.p4.9.m7.1.1.4.2" xref="S4.SS1.p4.9.m7.1.1.4.2.cmml">k</mi><mi id="S4.SS1.p4.9.m7.1.1.4.3" xref="S4.SS1.p4.9.m7.1.1.4.3.cmml">j</mi></msub><mo id="S4.SS1.p4.9.m7.1.1.5" xref="S4.SS1.p4.9.m7.1.1.5.cmml">≤</mo><mi id="S4.SS1.p4.9.m7.1.1.6" xref="S4.SS1.p4.9.m7.1.1.6.cmml">N</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.9.m7.1b"><apply id="S4.SS1.p4.9.m7.1.1.cmml" xref="S4.SS1.p4.9.m7.1.1"><and id="S4.SS1.p4.9.m7.1.1a.cmml" xref="S4.SS1.p4.9.m7.1.1"></and><apply id="S4.SS1.p4.9.m7.1.1b.cmml" xref="S4.SS1.p4.9.m7.1.1"><leq id="S4.SS1.p4.9.m7.1.1.3.cmml" xref="S4.SS1.p4.9.m7.1.1.3"></leq><cn id="S4.SS1.p4.9.m7.1.1.2.cmml" type="integer" xref="S4.SS1.p4.9.m7.1.1.2">1</cn><apply id="S4.SS1.p4.9.m7.1.1.4.cmml" xref="S4.SS1.p4.9.m7.1.1.4"><csymbol cd="ambiguous" id="S4.SS1.p4.9.m7.1.1.4.1.cmml" xref="S4.SS1.p4.9.m7.1.1.4">subscript</csymbol><ci id="S4.SS1.p4.9.m7.1.1.4.2.cmml" xref="S4.SS1.p4.9.m7.1.1.4.2">𝑘</ci><ci id="S4.SS1.p4.9.m7.1.1.4.3.cmml" xref="S4.SS1.p4.9.m7.1.1.4.3">𝑗</ci></apply></apply><apply id="S4.SS1.p4.9.m7.1.1c.cmml" xref="S4.SS1.p4.9.m7.1.1"><leq id="S4.SS1.p4.9.m7.1.1.5.cmml" xref="S4.SS1.p4.9.m7.1.1.5"></leq><share href="https://arxiv.org/html/2401.10785v2#S4.SS1.p4.9.m7.1.1.4.cmml" id="S4.SS1.p4.9.m7.1.1d.cmml" xref="S4.SS1.p4.9.m7.1.1"></share><ci id="S4.SS1.p4.9.m7.1.1.6.cmml" xref="S4.SS1.p4.9.m7.1.1.6">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.9.m7.1c">1\leq k_{j}\leq N</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.9.m7.1d">1 ≤ italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ≤ italic_N</annotation></semantics></math>, and <math alttext="j=1" class="ltx_Math" display="inline" id="S4.SS1.p4.10.m8.1"><semantics id="S4.SS1.p4.10.m8.1a"><mrow id="S4.SS1.p4.10.m8.1.1" xref="S4.SS1.p4.10.m8.1.1.cmml"><mi id="S4.SS1.p4.10.m8.1.1.2" xref="S4.SS1.p4.10.m8.1.1.2.cmml">j</mi><mo id="S4.SS1.p4.10.m8.1.1.1" xref="S4.SS1.p4.10.m8.1.1.1.cmml">=</mo><mn id="S4.SS1.p4.10.m8.1.1.3" xref="S4.SS1.p4.10.m8.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.10.m8.1b"><apply id="S4.SS1.p4.10.m8.1.1.cmml" xref="S4.SS1.p4.10.m8.1.1"><eq id="S4.SS1.p4.10.m8.1.1.1.cmml" xref="S4.SS1.p4.10.m8.1.1.1"></eq><ci id="S4.SS1.p4.10.m8.1.1.2.cmml" xref="S4.SS1.p4.10.m8.1.1.2">𝑗</ci><cn id="S4.SS1.p4.10.m8.1.1.3.cmml" type="integer" xref="S4.SS1.p4.10.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.10.m8.1c">j=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.10.m8.1d">italic_j = 1</annotation></semantics></math> to <math alttext="N_{\zeta}" class="ltx_Math" display="inline" id="S4.SS1.p4.11.m9.1"><semantics id="S4.SS1.p4.11.m9.1a"><msub id="S4.SS1.p4.11.m9.1.1" xref="S4.SS1.p4.11.m9.1.1.cmml"><mi id="S4.SS1.p4.11.m9.1.1.2" xref="S4.SS1.p4.11.m9.1.1.2.cmml">N</mi><mi id="S4.SS1.p4.11.m9.1.1.3" xref="S4.SS1.p4.11.m9.1.1.3.cmml">ζ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.11.m9.1b"><apply id="S4.SS1.p4.11.m9.1.1.cmml" xref="S4.SS1.p4.11.m9.1.1"><csymbol cd="ambiguous" id="S4.SS1.p4.11.m9.1.1.1.cmml" xref="S4.SS1.p4.11.m9.1.1">subscript</csymbol><ci id="S4.SS1.p4.11.m9.1.1.2.cmml" xref="S4.SS1.p4.11.m9.1.1.2">𝑁</ci><ci id="S4.SS1.p4.11.m9.1.1.3.cmml" xref="S4.SS1.p4.11.m9.1.1.3">𝜁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.11.m9.1c">N_{\zeta}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.11.m9.1d">italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math>. If the IE condition holds, then there exists a finite time <math alttext="T_{\rm e}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S4.SS1.p4.12.m10.1"><semantics id="S4.SS1.p4.12.m10.1a"><mrow id="S4.SS1.p4.12.m10.1.1" xref="S4.SS1.p4.12.m10.1.1.cmml"><msub id="S4.SS1.p4.12.m10.1.1.2" xref="S4.SS1.p4.12.m10.1.1.2.cmml"><mi id="S4.SS1.p4.12.m10.1.1.2.2" xref="S4.SS1.p4.12.m10.1.1.2.2.cmml">T</mi><mi id="S4.SS1.p4.12.m10.1.1.2.3" mathvariant="normal" xref="S4.SS1.p4.12.m10.1.1.2.3.cmml">e</mi></msub><mo id="S4.SS1.p4.12.m10.1.1.1" xref="S4.SS1.p4.12.m10.1.1.1.cmml">∈</mo><msup id="S4.SS1.p4.12.m10.1.1.3" xref="S4.SS1.p4.12.m10.1.1.3.cmml"><mi id="S4.SS1.p4.12.m10.1.1.3.2" xref="S4.SS1.p4.12.m10.1.1.3.2.cmml">ℝ</mi><mo id="S4.SS1.p4.12.m10.1.1.3.3" xref="S4.SS1.p4.12.m10.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.12.m10.1b"><apply id="S4.SS1.p4.12.m10.1.1.cmml" xref="S4.SS1.p4.12.m10.1.1"><in id="S4.SS1.p4.12.m10.1.1.1.cmml" xref="S4.SS1.p4.12.m10.1.1.1"></in><apply id="S4.SS1.p4.12.m10.1.1.2.cmml" xref="S4.SS1.p4.12.m10.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p4.12.m10.1.1.2.1.cmml" xref="S4.SS1.p4.12.m10.1.1.2">subscript</csymbol><ci id="S4.SS1.p4.12.m10.1.1.2.2.cmml" xref="S4.SS1.p4.12.m10.1.1.2.2">𝑇</ci><ci id="S4.SS1.p4.12.m10.1.1.2.3.cmml" xref="S4.SS1.p4.12.m10.1.1.2.3">e</ci></apply><apply id="S4.SS1.p4.12.m10.1.1.3.cmml" xref="S4.SS1.p4.12.m10.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p4.12.m10.1.1.3.1.cmml" xref="S4.SS1.p4.12.m10.1.1.3">superscript</csymbol><ci id="S4.SS1.p4.12.m10.1.1.3.2.cmml" xref="S4.SS1.p4.12.m10.1.1.3.2">ℝ</ci><plus id="S4.SS1.p4.12.m10.1.1.3.3.cmml" xref="S4.SS1.p4.12.m10.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.12.m10.1c">T_{\rm e}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.12.m10.1d">italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="\Psi(T_{\rm e})\geq\sigma I" class="ltx_Math" display="inline" id="S4.SS1.p4.13.m11.1"><semantics id="S4.SS1.p4.13.m11.1a"><mrow id="S4.SS1.p4.13.m11.1.1" xref="S4.SS1.p4.13.m11.1.1.cmml"><mrow id="S4.SS1.p4.13.m11.1.1.1" xref="S4.SS1.p4.13.m11.1.1.1.cmml"><mi id="S4.SS1.p4.13.m11.1.1.1.3" mathvariant="normal" xref="S4.SS1.p4.13.m11.1.1.1.3.cmml">Ψ</mi><mo id="S4.SS1.p4.13.m11.1.1.1.2" xref="S4.SS1.p4.13.m11.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.p4.13.m11.1.1.1.1.1" xref="S4.SS1.p4.13.m11.1.1.1.1.1.1.cmml"><mo id="S4.SS1.p4.13.m11.1.1.1.1.1.2" stretchy="false" xref="S4.SS1.p4.13.m11.1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS1.p4.13.m11.1.1.1.1.1.1" xref="S4.SS1.p4.13.m11.1.1.1.1.1.1.cmml"><mi id="S4.SS1.p4.13.m11.1.1.1.1.1.1.2" xref="S4.SS1.p4.13.m11.1.1.1.1.1.1.2.cmml">T</mi><mi id="S4.SS1.p4.13.m11.1.1.1.1.1.1.3" mathvariant="normal" xref="S4.SS1.p4.13.m11.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="S4.SS1.p4.13.m11.1.1.1.1.1.3" stretchy="false" xref="S4.SS1.p4.13.m11.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p4.13.m11.1.1.2" xref="S4.SS1.p4.13.m11.1.1.2.cmml">≥</mo><mrow id="S4.SS1.p4.13.m11.1.1.3" xref="S4.SS1.p4.13.m11.1.1.3.cmml"><mi id="S4.SS1.p4.13.m11.1.1.3.2" xref="S4.SS1.p4.13.m11.1.1.3.2.cmml">σ</mi><mo id="S4.SS1.p4.13.m11.1.1.3.1" xref="S4.SS1.p4.13.m11.1.1.3.1.cmml">⁢</mo><mi id="S4.SS1.p4.13.m11.1.1.3.3" xref="S4.SS1.p4.13.m11.1.1.3.3.cmml">I</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.13.m11.1b"><apply id="S4.SS1.p4.13.m11.1.1.cmml" xref="S4.SS1.p4.13.m11.1.1"><geq id="S4.SS1.p4.13.m11.1.1.2.cmml" xref="S4.SS1.p4.13.m11.1.1.2"></geq><apply id="S4.SS1.p4.13.m11.1.1.1.cmml" xref="S4.SS1.p4.13.m11.1.1.1"><times id="S4.SS1.p4.13.m11.1.1.1.2.cmml" xref="S4.SS1.p4.13.m11.1.1.1.2"></times><ci id="S4.SS1.p4.13.m11.1.1.1.3.cmml" xref="S4.SS1.p4.13.m11.1.1.1.3">Ψ</ci><apply id="S4.SS1.p4.13.m11.1.1.1.1.1.1.cmml" xref="S4.SS1.p4.13.m11.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p4.13.m11.1.1.1.1.1.1.1.cmml" xref="S4.SS1.p4.13.m11.1.1.1.1.1">subscript</csymbol><ci id="S4.SS1.p4.13.m11.1.1.1.1.1.1.2.cmml" xref="S4.SS1.p4.13.m11.1.1.1.1.1.1.2">𝑇</ci><ci id="S4.SS1.p4.13.m11.1.1.1.1.1.1.3.cmml" xref="S4.SS1.p4.13.m11.1.1.1.1.1.1.3">e</ci></apply></apply><apply id="S4.SS1.p4.13.m11.1.1.3.cmml" xref="S4.SS1.p4.13.m11.1.1.3"><times id="S4.SS1.p4.13.m11.1.1.3.1.cmml" xref="S4.SS1.p4.13.m11.1.1.3.1"></times><ci id="S4.SS1.p4.13.m11.1.1.3.2.cmml" xref="S4.SS1.p4.13.m11.1.1.3.2">𝜎</ci><ci id="S4.SS1.p4.13.m11.1.1.3.3.cmml" xref="S4.SS1.p4.13.m11.1.1.3.3">𝐼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.13.m11.1c">\Psi(T_{\rm e})\geq\sigma I</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.13.m11.1d">roman_Ψ ( italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) ≥ italic_σ italic_I</annotation></semantics></math>; otherwise, <math alttext="T_{\rm e}=\infty" class="ltx_Math" display="inline" id="S4.SS1.p4.14.m12.1"><semantics id="S4.SS1.p4.14.m12.1a"><mrow id="S4.SS1.p4.14.m12.1.1" xref="S4.SS1.p4.14.m12.1.1.cmml"><msub id="S4.SS1.p4.14.m12.1.1.2" xref="S4.SS1.p4.14.m12.1.1.2.cmml"><mi id="S4.SS1.p4.14.m12.1.1.2.2" xref="S4.SS1.p4.14.m12.1.1.2.2.cmml">T</mi><mi id="S4.SS1.p4.14.m12.1.1.2.3" mathvariant="normal" xref="S4.SS1.p4.14.m12.1.1.2.3.cmml">e</mi></msub><mo id="S4.SS1.p4.14.m12.1.1.1" xref="S4.SS1.p4.14.m12.1.1.1.cmml">=</mo><mi id="S4.SS1.p4.14.m12.1.1.3" mathvariant="normal" xref="S4.SS1.p4.14.m12.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.14.m12.1b"><apply id="S4.SS1.p4.14.m12.1.1.cmml" xref="S4.SS1.p4.14.m12.1.1"><eq id="S4.SS1.p4.14.m12.1.1.1.cmml" xref="S4.SS1.p4.14.m12.1.1.1"></eq><apply id="S4.SS1.p4.14.m12.1.1.2.cmml" xref="S4.SS1.p4.14.m12.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p4.14.m12.1.1.2.1.cmml" xref="S4.SS1.p4.14.m12.1.1.2">subscript</csymbol><ci id="S4.SS1.p4.14.m12.1.1.2.2.cmml" xref="S4.SS1.p4.14.m12.1.1.2.2">𝑇</ci><ci id="S4.SS1.p4.14.m12.1.1.2.3.cmml" xref="S4.SS1.p4.14.m12.1.1.2.3">e</ci></apply><infinity id="S4.SS1.p4.14.m12.1.1.3.cmml" xref="S4.SS1.p4.14.m12.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.14.m12.1c">T_{\rm e}=\infty</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.14.m12.1d">italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT = ∞</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS1.p5"> <p class="ltx_p" id="S4.SS1.p5.24">The index <math alttext="k_{j}" class="ltx_Math" display="inline" id="S4.SS1.p5.1.m1.1"><semantics id="S4.SS1.p5.1.m1.1a"><msub id="S4.SS1.p5.1.m1.1.1" xref="S4.SS1.p5.1.m1.1.1.cmml"><mi id="S4.SS1.p5.1.m1.1.1.2" xref="S4.SS1.p5.1.m1.1.1.2.cmml">k</mi><mi id="S4.SS1.p5.1.m1.1.1.3" xref="S4.SS1.p5.1.m1.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.1.m1.1b"><apply id="S4.SS1.p5.1.m1.1.1.cmml" xref="S4.SS1.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p5.1.m1.1.1.1.cmml" xref="S4.SS1.p5.1.m1.1.1">subscript</csymbol><ci id="S4.SS1.p5.1.m1.1.1.2.cmml" xref="S4.SS1.p5.1.m1.1.1.2">𝑘</ci><ci id="S4.SS1.p5.1.m1.1.1.3.cmml" xref="S4.SS1.p5.1.m1.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.1.m1.1c">k_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.1.m1.1d">italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> of active channels <math alttext="\bm{\phi}_{{\rm s},k_{j}}" class="ltx_Math" display="inline" id="S4.SS1.p5.2.m2.2"><semantics id="S4.SS1.p5.2.m2.2a"><msub id="S4.SS1.p5.2.m2.2.3" xref="S4.SS1.p5.2.m2.2.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS1.p5.2.m2.2.3.2" mathvariant="bold-italic" xref="S4.SS1.p5.2.m2.2.3.2.cmml">ϕ</mi><mrow id="S4.SS1.p5.2.m2.2.2.2.2" xref="S4.SS1.p5.2.m2.2.2.2.3.cmml"><mi id="S4.SS1.p5.2.m2.1.1.1.1" mathvariant="normal" xref="S4.SS1.p5.2.m2.1.1.1.1.cmml">s</mi><mo id="S4.SS1.p5.2.m2.2.2.2.2.2" xref="S4.SS1.p5.2.m2.2.2.2.3.cmml">,</mo><msub id="S4.SS1.p5.2.m2.2.2.2.2.1" xref="S4.SS1.p5.2.m2.2.2.2.2.1.cmml"><mi id="S4.SS1.p5.2.m2.2.2.2.2.1.2" xref="S4.SS1.p5.2.m2.2.2.2.2.1.2.cmml">k</mi><mi id="S4.SS1.p5.2.m2.2.2.2.2.1.3" xref="S4.SS1.p5.2.m2.2.2.2.2.1.3.cmml">j</mi></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.2.m2.2b"><apply id="S4.SS1.p5.2.m2.2.3.cmml" xref="S4.SS1.p5.2.m2.2.3"><csymbol cd="ambiguous" id="S4.SS1.p5.2.m2.2.3.1.cmml" xref="S4.SS1.p5.2.m2.2.3">subscript</csymbol><ci id="S4.SS1.p5.2.m2.2.3.2.cmml" xref="S4.SS1.p5.2.m2.2.3.2">bold-italic-ϕ</ci><list id="S4.SS1.p5.2.m2.2.2.2.3.cmml" xref="S4.SS1.p5.2.m2.2.2.2.2"><ci id="S4.SS1.p5.2.m2.1.1.1.1.cmml" xref="S4.SS1.p5.2.m2.1.1.1.1">s</ci><apply id="S4.SS1.p5.2.m2.2.2.2.2.1.cmml" xref="S4.SS1.p5.2.m2.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.SS1.p5.2.m2.2.2.2.2.1.1.cmml" xref="S4.SS1.p5.2.m2.2.2.2.2.1">subscript</csymbol><ci id="S4.SS1.p5.2.m2.2.2.2.2.1.2.cmml" xref="S4.SS1.p5.2.m2.2.2.2.2.1.2">𝑘</ci><ci id="S4.SS1.p5.2.m2.2.2.2.2.1.3.cmml" xref="S4.SS1.p5.2.m2.2.2.2.2.1.3">𝑗</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.2.m2.2c">\bm{\phi}_{{\rm s},k_{j}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.2.m2.2d">bold_italic_ϕ start_POSTSUBSCRIPT roman_s , italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> may be changed under partial IE, which leads to the existence of multiple partial IE stages. To consider the changes of the sub-regressor <math alttext="\Phi_{{\rm s},\zeta}" class="ltx_Math" display="inline" id="S4.SS1.p5.3.m3.2"><semantics id="S4.SS1.p5.3.m3.2a"><msub id="S4.SS1.p5.3.m3.2.3" xref="S4.SS1.p5.3.m3.2.3.cmml"><mi id="S4.SS1.p5.3.m3.2.3.2" mathvariant="normal" xref="S4.SS1.p5.3.m3.2.3.2.cmml">Φ</mi><mrow id="S4.SS1.p5.3.m3.2.2.2.4" xref="S4.SS1.p5.3.m3.2.2.2.3.cmml"><mi id="S4.SS1.p5.3.m3.1.1.1.1" mathvariant="normal" xref="S4.SS1.p5.3.m3.1.1.1.1.cmml">s</mi><mo id="S4.SS1.p5.3.m3.2.2.2.4.1" xref="S4.SS1.p5.3.m3.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p5.3.m3.2.2.2.2" xref="S4.SS1.p5.3.m3.2.2.2.2.cmml">ζ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.3.m3.2b"><apply id="S4.SS1.p5.3.m3.2.3.cmml" xref="S4.SS1.p5.3.m3.2.3"><csymbol cd="ambiguous" id="S4.SS1.p5.3.m3.2.3.1.cmml" xref="S4.SS1.p5.3.m3.2.3">subscript</csymbol><ci id="S4.SS1.p5.3.m3.2.3.2.cmml" xref="S4.SS1.p5.3.m3.2.3.2">Φ</ci><list id="S4.SS1.p5.3.m3.2.2.2.3.cmml" xref="S4.SS1.p5.3.m3.2.2.2.4"><ci id="S4.SS1.p5.3.m3.1.1.1.1.cmml" xref="S4.SS1.p5.3.m3.1.1.1.1">s</ci><ci id="S4.SS1.p5.3.m3.2.2.2.2.cmml" xref="S4.SS1.p5.3.m3.2.2.2.2">𝜁</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.3.m3.2c">\Phi_{{\rm s},\zeta}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.3.m3.2d">roman_Φ start_POSTSUBSCRIPT roman_s , italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math> under different partial IE stages, let <math alttext="\mathcal{I}" class="ltx_Math" display="inline" id="S4.SS1.p5.4.m4.1"><semantics id="S4.SS1.p5.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p5.4.m4.1.1" xref="S4.SS1.p5.4.m4.1.1.cmml">ℐ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.4.m4.1b"><ci id="S4.SS1.p5.4.m4.1.1.cmml" xref="S4.SS1.p5.4.m4.1.1">ℐ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.4.m4.1c">\mathcal{I}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.4.m4.1d">caligraphic_I</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S4.SS1.p5.5.m5.1"><semantics id="S4.SS1.p5.5.m5.1a"><mo id="S4.SS1.p5.5.m5.1.1" xref="S4.SS1.p5.5.m5.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.5.m5.1b"><csymbol cd="latexml" id="S4.SS1.p5.5.m5.1.1.cmml" xref="S4.SS1.p5.5.m5.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.5.m5.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.5.m5.1d">:=</annotation></semantics></math> <math alttext="\{k_{1},k_{2},\cdots,k_{N_{\zeta}}\}" class="ltx_Math" display="inline" id="S4.SS1.p5.6.m6.4"><semantics id="S4.SS1.p5.6.m6.4a"><mrow id="S4.SS1.p5.6.m6.4.4.3" xref="S4.SS1.p5.6.m6.4.4.4.cmml"><mo id="S4.SS1.p5.6.m6.4.4.3.4" stretchy="false" xref="S4.SS1.p5.6.m6.4.4.4.cmml">{</mo><msub id="S4.SS1.p5.6.m6.2.2.1.1" xref="S4.SS1.p5.6.m6.2.2.1.1.cmml"><mi id="S4.SS1.p5.6.m6.2.2.1.1.2" xref="S4.SS1.p5.6.m6.2.2.1.1.2.cmml">k</mi><mn id="S4.SS1.p5.6.m6.2.2.1.1.3" xref="S4.SS1.p5.6.m6.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.SS1.p5.6.m6.4.4.3.5" xref="S4.SS1.p5.6.m6.4.4.4.cmml">,</mo><msub id="S4.SS1.p5.6.m6.3.3.2.2" xref="S4.SS1.p5.6.m6.3.3.2.2.cmml"><mi id="S4.SS1.p5.6.m6.3.3.2.2.2" xref="S4.SS1.p5.6.m6.3.3.2.2.2.cmml">k</mi><mn id="S4.SS1.p5.6.m6.3.3.2.2.3" xref="S4.SS1.p5.6.m6.3.3.2.2.3.cmml">2</mn></msub><mo id="S4.SS1.p5.6.m6.4.4.3.6" xref="S4.SS1.p5.6.m6.4.4.4.cmml">,</mo><mi id="S4.SS1.p5.6.m6.1.1" mathvariant="normal" xref="S4.SS1.p5.6.m6.1.1.cmml">⋯</mi><mo id="S4.SS1.p5.6.m6.4.4.3.7" xref="S4.SS1.p5.6.m6.4.4.4.cmml">,</mo><msub id="S4.SS1.p5.6.m6.4.4.3.3" xref="S4.SS1.p5.6.m6.4.4.3.3.cmml"><mi id="S4.SS1.p5.6.m6.4.4.3.3.2" xref="S4.SS1.p5.6.m6.4.4.3.3.2.cmml">k</mi><msub id="S4.SS1.p5.6.m6.4.4.3.3.3" xref="S4.SS1.p5.6.m6.4.4.3.3.3.cmml"><mi id="S4.SS1.p5.6.m6.4.4.3.3.3.2" xref="S4.SS1.p5.6.m6.4.4.3.3.3.2.cmml">N</mi><mi id="S4.SS1.p5.6.m6.4.4.3.3.3.3" xref="S4.SS1.p5.6.m6.4.4.3.3.3.3.cmml">ζ</mi></msub></msub><mo id="S4.SS1.p5.6.m6.4.4.3.8" stretchy="false" xref="S4.SS1.p5.6.m6.4.4.4.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.6.m6.4b"><set id="S4.SS1.p5.6.m6.4.4.4.cmml" xref="S4.SS1.p5.6.m6.4.4.3"><apply id="S4.SS1.p5.6.m6.2.2.1.1.cmml" xref="S4.SS1.p5.6.m6.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p5.6.m6.2.2.1.1.1.cmml" xref="S4.SS1.p5.6.m6.2.2.1.1">subscript</csymbol><ci id="S4.SS1.p5.6.m6.2.2.1.1.2.cmml" xref="S4.SS1.p5.6.m6.2.2.1.1.2">𝑘</ci><cn id="S4.SS1.p5.6.m6.2.2.1.1.3.cmml" type="integer" xref="S4.SS1.p5.6.m6.2.2.1.1.3">1</cn></apply><apply id="S4.SS1.p5.6.m6.3.3.2.2.cmml" xref="S4.SS1.p5.6.m6.3.3.2.2"><csymbol cd="ambiguous" id="S4.SS1.p5.6.m6.3.3.2.2.1.cmml" xref="S4.SS1.p5.6.m6.3.3.2.2">subscript</csymbol><ci id="S4.SS1.p5.6.m6.3.3.2.2.2.cmml" xref="S4.SS1.p5.6.m6.3.3.2.2.2">𝑘</ci><cn id="S4.SS1.p5.6.m6.3.3.2.2.3.cmml" type="integer" xref="S4.SS1.p5.6.m6.3.3.2.2.3">2</cn></apply><ci id="S4.SS1.p5.6.m6.1.1.cmml" xref="S4.SS1.p5.6.m6.1.1">⋯</ci><apply id="S4.SS1.p5.6.m6.4.4.3.3.cmml" xref="S4.SS1.p5.6.m6.4.4.3.3"><csymbol cd="ambiguous" id="S4.SS1.p5.6.m6.4.4.3.3.1.cmml" xref="S4.SS1.p5.6.m6.4.4.3.3">subscript</csymbol><ci id="S4.SS1.p5.6.m6.4.4.3.3.2.cmml" xref="S4.SS1.p5.6.m6.4.4.3.3.2">𝑘</ci><apply id="S4.SS1.p5.6.m6.4.4.3.3.3.cmml" xref="S4.SS1.p5.6.m6.4.4.3.3.3"><csymbol cd="ambiguous" id="S4.SS1.p5.6.m6.4.4.3.3.3.1.cmml" xref="S4.SS1.p5.6.m6.4.4.3.3.3">subscript</csymbol><ci id="S4.SS1.p5.6.m6.4.4.3.3.3.2.cmml" xref="S4.SS1.p5.6.m6.4.4.3.3.3.2">𝑁</ci><ci id="S4.SS1.p5.6.m6.4.4.3.3.3.3.cmml" xref="S4.SS1.p5.6.m6.4.4.3.3.3.3">𝜁</ci></apply></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.6.m6.4c">\{k_{1},k_{2},\cdots,k_{N_{\zeta}}\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.6.m6.4d">{ italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , ⋯ , italic_k start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT end_POSTSUBSCRIPT }</annotation></semantics></math> and <math alttext="\mathcal{I}^{\prime}" class="ltx_Math" display="inline" id="S4.SS1.p5.7.m7.1"><semantics id="S4.SS1.p5.7.m7.1a"><msup id="S4.SS1.p5.7.m7.1.1" xref="S4.SS1.p5.7.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p5.7.m7.1.1.2" xref="S4.SS1.p5.7.m7.1.1.2.cmml">ℐ</mi><mo id="S4.SS1.p5.7.m7.1.1.3" xref="S4.SS1.p5.7.m7.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.7.m7.1b"><apply id="S4.SS1.p5.7.m7.1.1.cmml" xref="S4.SS1.p5.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS1.p5.7.m7.1.1.1.cmml" xref="S4.SS1.p5.7.m7.1.1">superscript</csymbol><ci id="S4.SS1.p5.7.m7.1.1.2.cmml" xref="S4.SS1.p5.7.m7.1.1.2">ℐ</ci><ci id="S4.SS1.p5.7.m7.1.1.3.cmml" xref="S4.SS1.p5.7.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.7.m7.1c">\mathcal{I}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.7.m7.1d">caligraphic_I start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S4.SS1.p5.8.m8.1"><semantics id="S4.SS1.p5.8.m8.1a"><mo id="S4.SS1.p5.8.m8.1.1" xref="S4.SS1.p5.8.m8.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.8.m8.1b"><csymbol cd="latexml" id="S4.SS1.p5.8.m8.1.1.cmml" xref="S4.SS1.p5.8.m8.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.8.m8.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.8.m8.1d">:=</annotation></semantics></math> <math alttext="\{k_{1}^{\prime},k_{2}^{\prime},\cdots,k_{N_{\zeta}^{\prime}}^{\prime}\}" class="ltx_Math" display="inline" id="S4.SS1.p5.9.m9.4"><semantics id="S4.SS1.p5.9.m9.4a"><mrow id="S4.SS1.p5.9.m9.4.4.3" xref="S4.SS1.p5.9.m9.4.4.4.cmml"><mo id="S4.SS1.p5.9.m9.4.4.3.4" stretchy="false" xref="S4.SS1.p5.9.m9.4.4.4.cmml">{</mo><msubsup id="S4.SS1.p5.9.m9.2.2.1.1" xref="S4.SS1.p5.9.m9.2.2.1.1.cmml"><mi id="S4.SS1.p5.9.m9.2.2.1.1.2.2" xref="S4.SS1.p5.9.m9.2.2.1.1.2.2.cmml">k</mi><mn id="S4.SS1.p5.9.m9.2.2.1.1.2.3" xref="S4.SS1.p5.9.m9.2.2.1.1.2.3.cmml">1</mn><mo id="S4.SS1.p5.9.m9.2.2.1.1.3" xref="S4.SS1.p5.9.m9.2.2.1.1.3.cmml">′</mo></msubsup><mo id="S4.SS1.p5.9.m9.4.4.3.5" xref="S4.SS1.p5.9.m9.4.4.4.cmml">,</mo><msubsup id="S4.SS1.p5.9.m9.3.3.2.2" xref="S4.SS1.p5.9.m9.3.3.2.2.cmml"><mi id="S4.SS1.p5.9.m9.3.3.2.2.2.2" xref="S4.SS1.p5.9.m9.3.3.2.2.2.2.cmml">k</mi><mn id="S4.SS1.p5.9.m9.3.3.2.2.2.3" xref="S4.SS1.p5.9.m9.3.3.2.2.2.3.cmml">2</mn><mo id="S4.SS1.p5.9.m9.3.3.2.2.3" xref="S4.SS1.p5.9.m9.3.3.2.2.3.cmml">′</mo></msubsup><mo id="S4.SS1.p5.9.m9.4.4.3.6" xref="S4.SS1.p5.9.m9.4.4.4.cmml">,</mo><mi id="S4.SS1.p5.9.m9.1.1" mathvariant="normal" xref="S4.SS1.p5.9.m9.1.1.cmml">⋯</mi><mo id="S4.SS1.p5.9.m9.4.4.3.7" xref="S4.SS1.p5.9.m9.4.4.4.cmml">,</mo><msubsup id="S4.SS1.p5.9.m9.4.4.3.3" xref="S4.SS1.p5.9.m9.4.4.3.3.cmml"><mi id="S4.SS1.p5.9.m9.4.4.3.3.2.2" xref="S4.SS1.p5.9.m9.4.4.3.3.2.2.cmml">k</mi><msubsup id="S4.SS1.p5.9.m9.4.4.3.3.2.3" xref="S4.SS1.p5.9.m9.4.4.3.3.2.3.cmml"><mi id="S4.SS1.p5.9.m9.4.4.3.3.2.3.2.2" xref="S4.SS1.p5.9.m9.4.4.3.3.2.3.2.2.cmml">N</mi><mi id="S4.SS1.p5.9.m9.4.4.3.3.2.3.2.3" xref="S4.SS1.p5.9.m9.4.4.3.3.2.3.2.3.cmml">ζ</mi><mo id="S4.SS1.p5.9.m9.4.4.3.3.2.3.3" xref="S4.SS1.p5.9.m9.4.4.3.3.2.3.3.cmml">′</mo></msubsup><mo id="S4.SS1.p5.9.m9.4.4.3.3.3" xref="S4.SS1.p5.9.m9.4.4.3.3.3.cmml">′</mo></msubsup><mo id="S4.SS1.p5.9.m9.4.4.3.8" stretchy="false" xref="S4.SS1.p5.9.m9.4.4.4.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.9.m9.4b"><set id="S4.SS1.p5.9.m9.4.4.4.cmml" xref="S4.SS1.p5.9.m9.4.4.3"><apply id="S4.SS1.p5.9.m9.2.2.1.1.cmml" xref="S4.SS1.p5.9.m9.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p5.9.m9.2.2.1.1.1.cmml" xref="S4.SS1.p5.9.m9.2.2.1.1">superscript</csymbol><apply id="S4.SS1.p5.9.m9.2.2.1.1.2.cmml" xref="S4.SS1.p5.9.m9.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p5.9.m9.2.2.1.1.2.1.cmml" xref="S4.SS1.p5.9.m9.2.2.1.1">subscript</csymbol><ci id="S4.SS1.p5.9.m9.2.2.1.1.2.2.cmml" xref="S4.SS1.p5.9.m9.2.2.1.1.2.2">𝑘</ci><cn id="S4.SS1.p5.9.m9.2.2.1.1.2.3.cmml" type="integer" xref="S4.SS1.p5.9.m9.2.2.1.1.2.3">1</cn></apply><ci id="S4.SS1.p5.9.m9.2.2.1.1.3.cmml" xref="S4.SS1.p5.9.m9.2.2.1.1.3">′</ci></apply><apply id="S4.SS1.p5.9.m9.3.3.2.2.cmml" xref="S4.SS1.p5.9.m9.3.3.2.2"><csymbol cd="ambiguous" id="S4.SS1.p5.9.m9.3.3.2.2.1.cmml" xref="S4.SS1.p5.9.m9.3.3.2.2">superscript</csymbol><apply id="S4.SS1.p5.9.m9.3.3.2.2.2.cmml" xref="S4.SS1.p5.9.m9.3.3.2.2"><csymbol cd="ambiguous" id="S4.SS1.p5.9.m9.3.3.2.2.2.1.cmml" xref="S4.SS1.p5.9.m9.3.3.2.2">subscript</csymbol><ci id="S4.SS1.p5.9.m9.3.3.2.2.2.2.cmml" xref="S4.SS1.p5.9.m9.3.3.2.2.2.2">𝑘</ci><cn id="S4.SS1.p5.9.m9.3.3.2.2.2.3.cmml" type="integer" xref="S4.SS1.p5.9.m9.3.3.2.2.2.3">2</cn></apply><ci id="S4.SS1.p5.9.m9.3.3.2.2.3.cmml" xref="S4.SS1.p5.9.m9.3.3.2.2.3">′</ci></apply><ci id="S4.SS1.p5.9.m9.1.1.cmml" xref="S4.SS1.p5.9.m9.1.1">⋯</ci><apply id="S4.SS1.p5.9.m9.4.4.3.3.cmml" xref="S4.SS1.p5.9.m9.4.4.3.3"><csymbol cd="ambiguous" id="S4.SS1.p5.9.m9.4.4.3.3.1.cmml" xref="S4.SS1.p5.9.m9.4.4.3.3">superscript</csymbol><apply id="S4.SS1.p5.9.m9.4.4.3.3.2.cmml" xref="S4.SS1.p5.9.m9.4.4.3.3"><csymbol cd="ambiguous" id="S4.SS1.p5.9.m9.4.4.3.3.2.1.cmml" xref="S4.SS1.p5.9.m9.4.4.3.3">subscript</csymbol><ci id="S4.SS1.p5.9.m9.4.4.3.3.2.2.cmml" xref="S4.SS1.p5.9.m9.4.4.3.3.2.2">𝑘</ci><apply id="S4.SS1.p5.9.m9.4.4.3.3.2.3.cmml" xref="S4.SS1.p5.9.m9.4.4.3.3.2.3"><csymbol cd="ambiguous" id="S4.SS1.p5.9.m9.4.4.3.3.2.3.1.cmml" xref="S4.SS1.p5.9.m9.4.4.3.3.2.3">superscript</csymbol><apply id="S4.SS1.p5.9.m9.4.4.3.3.2.3.2.cmml" xref="S4.SS1.p5.9.m9.4.4.3.3.2.3"><csymbol cd="ambiguous" id="S4.SS1.p5.9.m9.4.4.3.3.2.3.2.1.cmml" xref="S4.SS1.p5.9.m9.4.4.3.3.2.3">subscript</csymbol><ci id="S4.SS1.p5.9.m9.4.4.3.3.2.3.2.2.cmml" xref="S4.SS1.p5.9.m9.4.4.3.3.2.3.2.2">𝑁</ci><ci id="S4.SS1.p5.9.m9.4.4.3.3.2.3.2.3.cmml" xref="S4.SS1.p5.9.m9.4.4.3.3.2.3.2.3">𝜁</ci></apply><ci id="S4.SS1.p5.9.m9.4.4.3.3.2.3.3.cmml" xref="S4.SS1.p5.9.m9.4.4.3.3.2.3.3">′</ci></apply></apply><ci id="S4.SS1.p5.9.m9.4.4.3.3.3.cmml" xref="S4.SS1.p5.9.m9.4.4.3.3.3">′</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.9.m9.4c">\{k_{1}^{\prime},k_{2}^{\prime},\cdots,k_{N_{\zeta}^{\prime}}^{\prime}\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.9.m9.4d">{ italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , ⋯ , italic_k start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT }</annotation></semantics></math> denote index sets of active channels in the current and previous partial IE stages, respectively, where <math alttext="1\leq k_{j}^{\prime}\leq N" class="ltx_Math" display="inline" id="S4.SS1.p5.10.m10.1"><semantics id="S4.SS1.p5.10.m10.1a"><mrow id="S4.SS1.p5.10.m10.1.1" xref="S4.SS1.p5.10.m10.1.1.cmml"><mn id="S4.SS1.p5.10.m10.1.1.2" xref="S4.SS1.p5.10.m10.1.1.2.cmml">1</mn><mo id="S4.SS1.p5.10.m10.1.1.3" xref="S4.SS1.p5.10.m10.1.1.3.cmml">≤</mo><msubsup id="S4.SS1.p5.10.m10.1.1.4" xref="S4.SS1.p5.10.m10.1.1.4.cmml"><mi id="S4.SS1.p5.10.m10.1.1.4.2.2" xref="S4.SS1.p5.10.m10.1.1.4.2.2.cmml">k</mi><mi id="S4.SS1.p5.10.m10.1.1.4.2.3" xref="S4.SS1.p5.10.m10.1.1.4.2.3.cmml">j</mi><mo id="S4.SS1.p5.10.m10.1.1.4.3" xref="S4.SS1.p5.10.m10.1.1.4.3.cmml">′</mo></msubsup><mo id="S4.SS1.p5.10.m10.1.1.5" xref="S4.SS1.p5.10.m10.1.1.5.cmml">≤</mo><mi id="S4.SS1.p5.10.m10.1.1.6" xref="S4.SS1.p5.10.m10.1.1.6.cmml">N</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.10.m10.1b"><apply id="S4.SS1.p5.10.m10.1.1.cmml" xref="S4.SS1.p5.10.m10.1.1"><and id="S4.SS1.p5.10.m10.1.1a.cmml" xref="S4.SS1.p5.10.m10.1.1"></and><apply id="S4.SS1.p5.10.m10.1.1b.cmml" xref="S4.SS1.p5.10.m10.1.1"><leq id="S4.SS1.p5.10.m10.1.1.3.cmml" xref="S4.SS1.p5.10.m10.1.1.3"></leq><cn id="S4.SS1.p5.10.m10.1.1.2.cmml" type="integer" xref="S4.SS1.p5.10.m10.1.1.2">1</cn><apply id="S4.SS1.p5.10.m10.1.1.4.cmml" xref="S4.SS1.p5.10.m10.1.1.4"><csymbol cd="ambiguous" id="S4.SS1.p5.10.m10.1.1.4.1.cmml" xref="S4.SS1.p5.10.m10.1.1.4">superscript</csymbol><apply id="S4.SS1.p5.10.m10.1.1.4.2.cmml" xref="S4.SS1.p5.10.m10.1.1.4"><csymbol cd="ambiguous" id="S4.SS1.p5.10.m10.1.1.4.2.1.cmml" xref="S4.SS1.p5.10.m10.1.1.4">subscript</csymbol><ci id="S4.SS1.p5.10.m10.1.1.4.2.2.cmml" xref="S4.SS1.p5.10.m10.1.1.4.2.2">𝑘</ci><ci id="S4.SS1.p5.10.m10.1.1.4.2.3.cmml" xref="S4.SS1.p5.10.m10.1.1.4.2.3">𝑗</ci></apply><ci id="S4.SS1.p5.10.m10.1.1.4.3.cmml" xref="S4.SS1.p5.10.m10.1.1.4.3">′</ci></apply></apply><apply id="S4.SS1.p5.10.m10.1.1c.cmml" xref="S4.SS1.p5.10.m10.1.1"><leq id="S4.SS1.p5.10.m10.1.1.5.cmml" xref="S4.SS1.p5.10.m10.1.1.5"></leq><share href="https://arxiv.org/html/2401.10785v2#S4.SS1.p5.10.m10.1.1.4.cmml" id="S4.SS1.p5.10.m10.1.1d.cmml" xref="S4.SS1.p5.10.m10.1.1"></share><ci id="S4.SS1.p5.10.m10.1.1.6.cmml" xref="S4.SS1.p5.10.m10.1.1.6">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.10.m10.1c">1\leq k_{j}^{\prime}\leq N</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.10.m10.1d">1 ≤ italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_N</annotation></semantics></math> with <math alttext="j=1" class="ltx_Math" display="inline" id="S4.SS1.p5.11.m11.1"><semantics id="S4.SS1.p5.11.m11.1a"><mrow id="S4.SS1.p5.11.m11.1.1" xref="S4.SS1.p5.11.m11.1.1.cmml"><mi id="S4.SS1.p5.11.m11.1.1.2" xref="S4.SS1.p5.11.m11.1.1.2.cmml">j</mi><mo id="S4.SS1.p5.11.m11.1.1.1" xref="S4.SS1.p5.11.m11.1.1.1.cmml">=</mo><mn id="S4.SS1.p5.11.m11.1.1.3" xref="S4.SS1.p5.11.m11.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.11.m11.1b"><apply id="S4.SS1.p5.11.m11.1.1.cmml" xref="S4.SS1.p5.11.m11.1.1"><eq id="S4.SS1.p5.11.m11.1.1.1.cmml" xref="S4.SS1.p5.11.m11.1.1.1"></eq><ci id="S4.SS1.p5.11.m11.1.1.2.cmml" xref="S4.SS1.p5.11.m11.1.1.2">𝑗</ci><cn id="S4.SS1.p5.11.m11.1.1.3.cmml" type="integer" xref="S4.SS1.p5.11.m11.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.11.m11.1c">j=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.11.m11.1d">italic_j = 1</annotation></semantics></math> to <math alttext="N_{\zeta}^{\prime}" class="ltx_Math" display="inline" id="S4.SS1.p5.12.m12.1"><semantics id="S4.SS1.p5.12.m12.1a"><msubsup id="S4.SS1.p5.12.m12.1.1" xref="S4.SS1.p5.12.m12.1.1.cmml"><mi id="S4.SS1.p5.12.m12.1.1.2.2" xref="S4.SS1.p5.12.m12.1.1.2.2.cmml">N</mi><mi id="S4.SS1.p5.12.m12.1.1.2.3" xref="S4.SS1.p5.12.m12.1.1.2.3.cmml">ζ</mi><mo id="S4.SS1.p5.12.m12.1.1.3" xref="S4.SS1.p5.12.m12.1.1.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.12.m12.1b"><apply id="S4.SS1.p5.12.m12.1.1.cmml" xref="S4.SS1.p5.12.m12.1.1"><csymbol cd="ambiguous" id="S4.SS1.p5.12.m12.1.1.1.cmml" xref="S4.SS1.p5.12.m12.1.1">superscript</csymbol><apply id="S4.SS1.p5.12.m12.1.1.2.cmml" xref="S4.SS1.p5.12.m12.1.1"><csymbol cd="ambiguous" id="S4.SS1.p5.12.m12.1.1.2.1.cmml" xref="S4.SS1.p5.12.m12.1.1">subscript</csymbol><ci id="S4.SS1.p5.12.m12.1.1.2.2.cmml" xref="S4.SS1.p5.12.m12.1.1.2.2">𝑁</ci><ci id="S4.SS1.p5.12.m12.1.1.2.3.cmml" xref="S4.SS1.p5.12.m12.1.1.2.3">𝜁</ci></apply><ci id="S4.SS1.p5.12.m12.1.1.3.cmml" xref="S4.SS1.p5.12.m12.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.12.m12.1c">N_{\zeta}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.12.m12.1d">italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="N_{\zeta}^{\prime}<N" class="ltx_Math" display="inline" id="S4.SS1.p5.13.m13.1"><semantics id="S4.SS1.p5.13.m13.1a"><mrow id="S4.SS1.p5.13.m13.1.1" xref="S4.SS1.p5.13.m13.1.1.cmml"><msubsup id="S4.SS1.p5.13.m13.1.1.2" xref="S4.SS1.p5.13.m13.1.1.2.cmml"><mi id="S4.SS1.p5.13.m13.1.1.2.2.2" xref="S4.SS1.p5.13.m13.1.1.2.2.2.cmml">N</mi><mi id="S4.SS1.p5.13.m13.1.1.2.2.3" xref="S4.SS1.p5.13.m13.1.1.2.2.3.cmml">ζ</mi><mo id="S4.SS1.p5.13.m13.1.1.2.3" xref="S4.SS1.p5.13.m13.1.1.2.3.cmml">′</mo></msubsup><mo id="S4.SS1.p5.13.m13.1.1.1" xref="S4.SS1.p5.13.m13.1.1.1.cmml"><</mo><mi id="S4.SS1.p5.13.m13.1.1.3" xref="S4.SS1.p5.13.m13.1.1.3.cmml">N</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.13.m13.1b"><apply id="S4.SS1.p5.13.m13.1.1.cmml" xref="S4.SS1.p5.13.m13.1.1"><lt id="S4.SS1.p5.13.m13.1.1.1.cmml" xref="S4.SS1.p5.13.m13.1.1.1"></lt><apply id="S4.SS1.p5.13.m13.1.1.2.cmml" xref="S4.SS1.p5.13.m13.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p5.13.m13.1.1.2.1.cmml" xref="S4.SS1.p5.13.m13.1.1.2">superscript</csymbol><apply id="S4.SS1.p5.13.m13.1.1.2.2.cmml" xref="S4.SS1.p5.13.m13.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p5.13.m13.1.1.2.2.1.cmml" xref="S4.SS1.p5.13.m13.1.1.2">subscript</csymbol><ci id="S4.SS1.p5.13.m13.1.1.2.2.2.cmml" xref="S4.SS1.p5.13.m13.1.1.2.2.2">𝑁</ci><ci id="S4.SS1.p5.13.m13.1.1.2.2.3.cmml" xref="S4.SS1.p5.13.m13.1.1.2.2.3">𝜁</ci></apply><ci id="S4.SS1.p5.13.m13.1.1.2.3.cmml" xref="S4.SS1.p5.13.m13.1.1.2.3">′</ci></apply><ci id="S4.SS1.p5.13.m13.1.1.3.cmml" xref="S4.SS1.p5.13.m13.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.13.m13.1c">N_{\zeta}^{\prime}<N</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.13.m13.1d">italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_N</annotation></semantics></math> is the number of previous active channels. Then, a staged exciting strength maximization algorithm for reconstructing the sub-regressor <math alttext="\Phi_{{\rm s},\zeta}" class="ltx_Math" display="inline" id="S4.SS1.p5.14.m14.2"><semantics id="S4.SS1.p5.14.m14.2a"><msub id="S4.SS1.p5.14.m14.2.3" xref="S4.SS1.p5.14.m14.2.3.cmml"><mi id="S4.SS1.p5.14.m14.2.3.2" mathvariant="normal" xref="S4.SS1.p5.14.m14.2.3.2.cmml">Φ</mi><mrow id="S4.SS1.p5.14.m14.2.2.2.4" xref="S4.SS1.p5.14.m14.2.2.2.3.cmml"><mi id="S4.SS1.p5.14.m14.1.1.1.1" mathvariant="normal" xref="S4.SS1.p5.14.m14.1.1.1.1.cmml">s</mi><mo id="S4.SS1.p5.14.m14.2.2.2.4.1" xref="S4.SS1.p5.14.m14.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p5.14.m14.2.2.2.2" xref="S4.SS1.p5.14.m14.2.2.2.2.cmml">ζ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.14.m14.2b"><apply id="S4.SS1.p5.14.m14.2.3.cmml" xref="S4.SS1.p5.14.m14.2.3"><csymbol cd="ambiguous" id="S4.SS1.p5.14.m14.2.3.1.cmml" xref="S4.SS1.p5.14.m14.2.3">subscript</csymbol><ci id="S4.SS1.p5.14.m14.2.3.2.cmml" xref="S4.SS1.p5.14.m14.2.3.2">Φ</ci><list id="S4.SS1.p5.14.m14.2.2.2.3.cmml" xref="S4.SS1.p5.14.m14.2.2.2.4"><ci id="S4.SS1.p5.14.m14.1.1.1.1.cmml" xref="S4.SS1.p5.14.m14.1.1.1.1">s</ci><ci id="S4.SS1.p5.14.m14.2.2.2.2.cmml" xref="S4.SS1.p5.14.m14.2.2.2.2">𝜁</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.14.m14.2c">\Phi_{{\rm s},\zeta}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.14.m14.2d">roman_Φ start_POSTSUBSCRIPT roman_s , italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math> and maximizing the exciting strength <math alttext="\sigma_{\min}(\Psi_{\zeta}(t))" class="ltx_Math" display="inline" id="S4.SS1.p5.15.m15.2"><semantics id="S4.SS1.p5.15.m15.2a"><mrow id="S4.SS1.p5.15.m15.2.2" xref="S4.SS1.p5.15.m15.2.2.cmml"><msub id="S4.SS1.p5.15.m15.2.2.3" xref="S4.SS1.p5.15.m15.2.2.3.cmml"><mi id="S4.SS1.p5.15.m15.2.2.3.2" xref="S4.SS1.p5.15.m15.2.2.3.2.cmml">σ</mi><mi id="S4.SS1.p5.15.m15.2.2.3.3" xref="S4.SS1.p5.15.m15.2.2.3.3.cmml">min</mi></msub><mo id="S4.SS1.p5.15.m15.2.2.2" xref="S4.SS1.p5.15.m15.2.2.2.cmml">⁢</mo><mrow id="S4.SS1.p5.15.m15.2.2.1.1" xref="S4.SS1.p5.15.m15.2.2.1.1.1.cmml"><mo id="S4.SS1.p5.15.m15.2.2.1.1.2" stretchy="false" xref="S4.SS1.p5.15.m15.2.2.1.1.1.cmml">(</mo><mrow id="S4.SS1.p5.15.m15.2.2.1.1.1" xref="S4.SS1.p5.15.m15.2.2.1.1.1.cmml"><msub id="S4.SS1.p5.15.m15.2.2.1.1.1.2" xref="S4.SS1.p5.15.m15.2.2.1.1.1.2.cmml"><mi id="S4.SS1.p5.15.m15.2.2.1.1.1.2.2" mathvariant="normal" xref="S4.SS1.p5.15.m15.2.2.1.1.1.2.2.cmml">Ψ</mi><mi id="S4.SS1.p5.15.m15.2.2.1.1.1.2.3" xref="S4.SS1.p5.15.m15.2.2.1.1.1.2.3.cmml">ζ</mi></msub><mo id="S4.SS1.p5.15.m15.2.2.1.1.1.1" xref="S4.SS1.p5.15.m15.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS1.p5.15.m15.2.2.1.1.1.3.2" xref="S4.SS1.p5.15.m15.2.2.1.1.1.cmml"><mo id="S4.SS1.p5.15.m15.2.2.1.1.1.3.2.1" stretchy="false" xref="S4.SS1.p5.15.m15.2.2.1.1.1.cmml">(</mo><mi id="S4.SS1.p5.15.m15.1.1" xref="S4.SS1.p5.15.m15.1.1.cmml">t</mi><mo id="S4.SS1.p5.15.m15.2.2.1.1.1.3.2.2" stretchy="false" xref="S4.SS1.p5.15.m15.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p5.15.m15.2.2.1.1.3" stretchy="false" xref="S4.SS1.p5.15.m15.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.15.m15.2b"><apply id="S4.SS1.p5.15.m15.2.2.cmml" xref="S4.SS1.p5.15.m15.2.2"><times id="S4.SS1.p5.15.m15.2.2.2.cmml" xref="S4.SS1.p5.15.m15.2.2.2"></times><apply id="S4.SS1.p5.15.m15.2.2.3.cmml" xref="S4.SS1.p5.15.m15.2.2.3"><csymbol cd="ambiguous" id="S4.SS1.p5.15.m15.2.2.3.1.cmml" xref="S4.SS1.p5.15.m15.2.2.3">subscript</csymbol><ci id="S4.SS1.p5.15.m15.2.2.3.2.cmml" xref="S4.SS1.p5.15.m15.2.2.3.2">𝜎</ci><min id="S4.SS1.p5.15.m15.2.2.3.3.cmml" xref="S4.SS1.p5.15.m15.2.2.3.3"></min></apply><apply id="S4.SS1.p5.15.m15.2.2.1.1.1.cmml" xref="S4.SS1.p5.15.m15.2.2.1.1"><times id="S4.SS1.p5.15.m15.2.2.1.1.1.1.cmml" xref="S4.SS1.p5.15.m15.2.2.1.1.1.1"></times><apply id="S4.SS1.p5.15.m15.2.2.1.1.1.2.cmml" xref="S4.SS1.p5.15.m15.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p5.15.m15.2.2.1.1.1.2.1.cmml" xref="S4.SS1.p5.15.m15.2.2.1.1.1.2">subscript</csymbol><ci id="S4.SS1.p5.15.m15.2.2.1.1.1.2.2.cmml" xref="S4.SS1.p5.15.m15.2.2.1.1.1.2.2">Ψ</ci><ci id="S4.SS1.p5.15.m15.2.2.1.1.1.2.3.cmml" xref="S4.SS1.p5.15.m15.2.2.1.1.1.2.3">𝜁</ci></apply><ci id="S4.SS1.p5.15.m15.1.1.cmml" xref="S4.SS1.p5.15.m15.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.15.m15.2c">\sigma_{\min}(\Psi_{\zeta}(t))</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.15.m15.2d">italic_σ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Ψ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t ) )</annotation></semantics></math> in each partial IE stage is given in Algorithm 1, where <math alttext="T_{\rm s}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S4.SS1.p5.16.m16.1"><semantics id="S4.SS1.p5.16.m16.1a"><mrow id="S4.SS1.p5.16.m16.1.1" xref="S4.SS1.p5.16.m16.1.1.cmml"><msub id="S4.SS1.p5.16.m16.1.1.2" xref="S4.SS1.p5.16.m16.1.1.2.cmml"><mi id="S4.SS1.p5.16.m16.1.1.2.2" xref="S4.SS1.p5.16.m16.1.1.2.2.cmml">T</mi><mi id="S4.SS1.p5.16.m16.1.1.2.3" mathvariant="normal" xref="S4.SS1.p5.16.m16.1.1.2.3.cmml">s</mi></msub><mo id="S4.SS1.p5.16.m16.1.1.1" xref="S4.SS1.p5.16.m16.1.1.1.cmml">∈</mo><msup id="S4.SS1.p5.16.m16.1.1.3" xref="S4.SS1.p5.16.m16.1.1.3.cmml"><mi id="S4.SS1.p5.16.m16.1.1.3.2" xref="S4.SS1.p5.16.m16.1.1.3.2.cmml">ℝ</mi><mo id="S4.SS1.p5.16.m16.1.1.3.3" xref="S4.SS1.p5.16.m16.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.16.m16.1b"><apply id="S4.SS1.p5.16.m16.1.1.cmml" xref="S4.SS1.p5.16.m16.1.1"><in id="S4.SS1.p5.16.m16.1.1.1.cmml" xref="S4.SS1.p5.16.m16.1.1.1"></in><apply id="S4.SS1.p5.16.m16.1.1.2.cmml" xref="S4.SS1.p5.16.m16.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p5.16.m16.1.1.2.1.cmml" xref="S4.SS1.p5.16.m16.1.1.2">subscript</csymbol><ci id="S4.SS1.p5.16.m16.1.1.2.2.cmml" xref="S4.SS1.p5.16.m16.1.1.2.2">𝑇</ci><ci id="S4.SS1.p5.16.m16.1.1.2.3.cmml" xref="S4.SS1.p5.16.m16.1.1.2.3">s</ci></apply><apply id="S4.SS1.p5.16.m16.1.1.3.cmml" xref="S4.SS1.p5.16.m16.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p5.16.m16.1.1.3.1.cmml" xref="S4.SS1.p5.16.m16.1.1.3">superscript</csymbol><ci id="S4.SS1.p5.16.m16.1.1.3.2.cmml" xref="S4.SS1.p5.16.m16.1.1.3.2">ℝ</ci><plus id="S4.SS1.p5.16.m16.1.1.3.3.cmml" xref="S4.SS1.p5.16.m16.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.16.m16.1c">T_{\rm s}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.16.m16.1d">italic_T start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> is a sampling time, <math alttext="T_{\rm a}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S4.SS1.p5.17.m17.1"><semantics id="S4.SS1.p5.17.m17.1a"><mrow id="S4.SS1.p5.17.m17.1.1" xref="S4.SS1.p5.17.m17.1.1.cmml"><msub id="S4.SS1.p5.17.m17.1.1.2" xref="S4.SS1.p5.17.m17.1.1.2.cmml"><mi id="S4.SS1.p5.17.m17.1.1.2.2" xref="S4.SS1.p5.17.m17.1.1.2.2.cmml">T</mi><mi id="S4.SS1.p5.17.m17.1.1.2.3" mathvariant="normal" xref="S4.SS1.p5.17.m17.1.1.2.3.cmml">a</mi></msub><mo id="S4.SS1.p5.17.m17.1.1.1" xref="S4.SS1.p5.17.m17.1.1.1.cmml">∈</mo><msup id="S4.SS1.p5.17.m17.1.1.3" xref="S4.SS1.p5.17.m17.1.1.3.cmml"><mi id="S4.SS1.p5.17.m17.1.1.3.2" xref="S4.SS1.p5.17.m17.1.1.3.2.cmml">ℝ</mi><mo id="S4.SS1.p5.17.m17.1.1.3.3" xref="S4.SS1.p5.17.m17.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.17.m17.1b"><apply id="S4.SS1.p5.17.m17.1.1.cmml" xref="S4.SS1.p5.17.m17.1.1"><in id="S4.SS1.p5.17.m17.1.1.1.cmml" xref="S4.SS1.p5.17.m17.1.1.1"></in><apply id="S4.SS1.p5.17.m17.1.1.2.cmml" xref="S4.SS1.p5.17.m17.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p5.17.m17.1.1.2.1.cmml" xref="S4.SS1.p5.17.m17.1.1.2">subscript</csymbol><ci id="S4.SS1.p5.17.m17.1.1.2.2.cmml" xref="S4.SS1.p5.17.m17.1.1.2.2">𝑇</ci><ci id="S4.SS1.p5.17.m17.1.1.2.3.cmml" xref="S4.SS1.p5.17.m17.1.1.2.3">a</ci></apply><apply id="S4.SS1.p5.17.m17.1.1.3.cmml" xref="S4.SS1.p5.17.m17.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p5.17.m17.1.1.3.1.cmml" xref="S4.SS1.p5.17.m17.1.1.3">superscript</csymbol><ci id="S4.SS1.p5.17.m17.1.1.3.2.cmml" xref="S4.SS1.p5.17.m17.1.1.3.2">ℝ</ci><plus id="S4.SS1.p5.17.m17.1.1.3.3.cmml" xref="S4.SS1.p5.17.m17.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.17.m17.1c">T_{\rm a}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.17.m17.1d">italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> is the first epoch in each partial IE stage, <math alttext="\Psi_{k_{j},k_{j}}(t)" class="ltx_Math" display="inline" id="S4.SS1.p5.18.m18.3"><semantics id="S4.SS1.p5.18.m18.3a"><mrow id="S4.SS1.p5.18.m18.3.4" xref="S4.SS1.p5.18.m18.3.4.cmml"><msub id="S4.SS1.p5.18.m18.3.4.2" xref="S4.SS1.p5.18.m18.3.4.2.cmml"><mi id="S4.SS1.p5.18.m18.3.4.2.2" mathvariant="normal" xref="S4.SS1.p5.18.m18.3.4.2.2.cmml">Ψ</mi><mrow id="S4.SS1.p5.18.m18.2.2.2.2" xref="S4.SS1.p5.18.m18.2.2.2.3.cmml"><msub id="S4.SS1.p5.18.m18.1.1.1.1.1" xref="S4.SS1.p5.18.m18.1.1.1.1.1.cmml"><mi id="S4.SS1.p5.18.m18.1.1.1.1.1.2" xref="S4.SS1.p5.18.m18.1.1.1.1.1.2.cmml">k</mi><mi id="S4.SS1.p5.18.m18.1.1.1.1.1.3" xref="S4.SS1.p5.18.m18.1.1.1.1.1.3.cmml">j</mi></msub><mo id="S4.SS1.p5.18.m18.2.2.2.2.3" xref="S4.SS1.p5.18.m18.2.2.2.3.cmml">,</mo><msub id="S4.SS1.p5.18.m18.2.2.2.2.2" xref="S4.SS1.p5.18.m18.2.2.2.2.2.cmml"><mi id="S4.SS1.p5.18.m18.2.2.2.2.2.2" xref="S4.SS1.p5.18.m18.2.2.2.2.2.2.cmml">k</mi><mi id="S4.SS1.p5.18.m18.2.2.2.2.2.3" xref="S4.SS1.p5.18.m18.2.2.2.2.2.3.cmml">j</mi></msub></mrow></msub><mo id="S4.SS1.p5.18.m18.3.4.1" xref="S4.SS1.p5.18.m18.3.4.1.cmml">⁢</mo><mrow id="S4.SS1.p5.18.m18.3.4.3.2" xref="S4.SS1.p5.18.m18.3.4.cmml"><mo id="S4.SS1.p5.18.m18.3.4.3.2.1" stretchy="false" xref="S4.SS1.p5.18.m18.3.4.cmml">(</mo><mi id="S4.SS1.p5.18.m18.3.3" xref="S4.SS1.p5.18.m18.3.3.cmml">t</mi><mo id="S4.SS1.p5.18.m18.3.4.3.2.2" stretchy="false" xref="S4.SS1.p5.18.m18.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.18.m18.3b"><apply id="S4.SS1.p5.18.m18.3.4.cmml" xref="S4.SS1.p5.18.m18.3.4"><times id="S4.SS1.p5.18.m18.3.4.1.cmml" xref="S4.SS1.p5.18.m18.3.4.1"></times><apply id="S4.SS1.p5.18.m18.3.4.2.cmml" xref="S4.SS1.p5.18.m18.3.4.2"><csymbol cd="ambiguous" id="S4.SS1.p5.18.m18.3.4.2.1.cmml" xref="S4.SS1.p5.18.m18.3.4.2">subscript</csymbol><ci id="S4.SS1.p5.18.m18.3.4.2.2.cmml" xref="S4.SS1.p5.18.m18.3.4.2.2">Ψ</ci><list id="S4.SS1.p5.18.m18.2.2.2.3.cmml" xref="S4.SS1.p5.18.m18.2.2.2.2"><apply id="S4.SS1.p5.18.m18.1.1.1.1.1.cmml" xref="S4.SS1.p5.18.m18.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p5.18.m18.1.1.1.1.1.1.cmml" xref="S4.SS1.p5.18.m18.1.1.1.1.1">subscript</csymbol><ci id="S4.SS1.p5.18.m18.1.1.1.1.1.2.cmml" xref="S4.SS1.p5.18.m18.1.1.1.1.1.2">𝑘</ci><ci id="S4.SS1.p5.18.m18.1.1.1.1.1.3.cmml" xref="S4.SS1.p5.18.m18.1.1.1.1.1.3">𝑗</ci></apply><apply id="S4.SS1.p5.18.m18.2.2.2.2.2.cmml" xref="S4.SS1.p5.18.m18.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.p5.18.m18.2.2.2.2.2.1.cmml" xref="S4.SS1.p5.18.m18.2.2.2.2.2">subscript</csymbol><ci id="S4.SS1.p5.18.m18.2.2.2.2.2.2.cmml" xref="S4.SS1.p5.18.m18.2.2.2.2.2.2">𝑘</ci><ci id="S4.SS1.p5.18.m18.2.2.2.2.2.3.cmml" xref="S4.SS1.p5.18.m18.2.2.2.2.2.3">𝑗</ci></apply></list></apply><ci id="S4.SS1.p5.18.m18.3.3.cmml" xref="S4.SS1.p5.18.m18.3.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.18.m18.3c">\Psi_{k_{j},k_{j}}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.18.m18.3d">roman_Ψ start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S4.SS1.p5.19.m19.1"><semantics id="S4.SS1.p5.19.m19.1a"><mo id="S4.SS1.p5.19.m19.1.1" xref="S4.SS1.p5.19.m19.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.19.m19.1b"><csymbol cd="latexml" id="S4.SS1.p5.19.m19.1.1.cmml" xref="S4.SS1.p5.19.m19.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.19.m19.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.19.m19.1d">:=</annotation></semantics></math> <math alttext="\int_{t-\tau_{\rm d}}^{t}\|\bm{\phi}_{{\rm s},k_{j}}(\tau)\|^{2}d\tau" class="ltx_Math" display="inline" id="S4.SS1.p5.20.m20.4"><semantics id="S4.SS1.p5.20.m20.4a"><mrow id="S4.SS1.p5.20.m20.4.4" xref="S4.SS1.p5.20.m20.4.4.cmml"><msubsup id="S4.SS1.p5.20.m20.4.4.2" xref="S4.SS1.p5.20.m20.4.4.2.cmml"><mo id="S4.SS1.p5.20.m20.4.4.2.2.2" xref="S4.SS1.p5.20.m20.4.4.2.2.2.cmml">∫</mo><mrow id="S4.SS1.p5.20.m20.4.4.2.2.3" xref="S4.SS1.p5.20.m20.4.4.2.2.3.cmml"><mi id="S4.SS1.p5.20.m20.4.4.2.2.3.2" xref="S4.SS1.p5.20.m20.4.4.2.2.3.2.cmml">t</mi><mo id="S4.SS1.p5.20.m20.4.4.2.2.3.1" xref="S4.SS1.p5.20.m20.4.4.2.2.3.1.cmml">−</mo><msub id="S4.SS1.p5.20.m20.4.4.2.2.3.3" xref="S4.SS1.p5.20.m20.4.4.2.2.3.3.cmml"><mi id="S4.SS1.p5.20.m20.4.4.2.2.3.3.2" xref="S4.SS1.p5.20.m20.4.4.2.2.3.3.2.cmml">τ</mi><mi id="S4.SS1.p5.20.m20.4.4.2.2.3.3.3" mathvariant="normal" xref="S4.SS1.p5.20.m20.4.4.2.2.3.3.3.cmml">d</mi></msub></mrow><mi id="S4.SS1.p5.20.m20.4.4.2.3" xref="S4.SS1.p5.20.m20.4.4.2.3.cmml">t</mi></msubsup><mrow id="S4.SS1.p5.20.m20.4.4.1" xref="S4.SS1.p5.20.m20.4.4.1.cmml"><msup id="S4.SS1.p5.20.m20.4.4.1.1" xref="S4.SS1.p5.20.m20.4.4.1.1.cmml"><mrow id="S4.SS1.p5.20.m20.4.4.1.1.1.1" xref="S4.SS1.p5.20.m20.4.4.1.1.1.2.cmml"><mo id="S4.SS1.p5.20.m20.4.4.1.1.1.1.2" lspace="0em" stretchy="false" xref="S4.SS1.p5.20.m20.4.4.1.1.1.2.1.cmml">‖</mo><mrow id="S4.SS1.p5.20.m20.4.4.1.1.1.1.1" xref="S4.SS1.p5.20.m20.4.4.1.1.1.1.1.cmml"><msub id="S4.SS1.p5.20.m20.4.4.1.1.1.1.1.2" xref="S4.SS1.p5.20.m20.4.4.1.1.1.1.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS1.p5.20.m20.4.4.1.1.1.1.1.2.2" mathvariant="bold-italic" xref="S4.SS1.p5.20.m20.4.4.1.1.1.1.1.2.2.cmml">ϕ</mi><mrow id="S4.SS1.p5.20.m20.2.2.2.2" xref="S4.SS1.p5.20.m20.2.2.2.3.cmml"><mi id="S4.SS1.p5.20.m20.1.1.1.1" mathvariant="normal" xref="S4.SS1.p5.20.m20.1.1.1.1.cmml">s</mi><mo id="S4.SS1.p5.20.m20.2.2.2.2.2" xref="S4.SS1.p5.20.m20.2.2.2.3.cmml">,</mo><msub id="S4.SS1.p5.20.m20.2.2.2.2.1" xref="S4.SS1.p5.20.m20.2.2.2.2.1.cmml"><mi id="S4.SS1.p5.20.m20.2.2.2.2.1.2" xref="S4.SS1.p5.20.m20.2.2.2.2.1.2.cmml">k</mi><mi id="S4.SS1.p5.20.m20.2.2.2.2.1.3" xref="S4.SS1.p5.20.m20.2.2.2.2.1.3.cmml">j</mi></msub></mrow></msub><mo id="S4.SS1.p5.20.m20.4.4.1.1.1.1.1.1" xref="S4.SS1.p5.20.m20.4.4.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS1.p5.20.m20.4.4.1.1.1.1.1.3.2" xref="S4.SS1.p5.20.m20.4.4.1.1.1.1.1.cmml"><mo id="S4.SS1.p5.20.m20.4.4.1.1.1.1.1.3.2.1" stretchy="false" xref="S4.SS1.p5.20.m20.4.4.1.1.1.1.1.cmml">(</mo><mi id="S4.SS1.p5.20.m20.3.3" xref="S4.SS1.p5.20.m20.3.3.cmml">τ</mi><mo id="S4.SS1.p5.20.m20.4.4.1.1.1.1.1.3.2.2" stretchy="false" xref="S4.SS1.p5.20.m20.4.4.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p5.20.m20.4.4.1.1.1.1.3" stretchy="false" xref="S4.SS1.p5.20.m20.4.4.1.1.1.2.1.cmml">‖</mo></mrow><mn id="S4.SS1.p5.20.m20.4.4.1.1.3" xref="S4.SS1.p5.20.m20.4.4.1.1.3.cmml">2</mn></msup><mo id="S4.SS1.p5.20.m20.4.4.1.2" lspace="0em" xref="S4.SS1.p5.20.m20.4.4.1.2.cmml">⁢</mo><mrow id="S4.SS1.p5.20.m20.4.4.1.3" xref="S4.SS1.p5.20.m20.4.4.1.3.cmml"><mo id="S4.SS1.p5.20.m20.4.4.1.3.1" rspace="0em" xref="S4.SS1.p5.20.m20.4.4.1.3.1.cmml">𝑑</mo><mi id="S4.SS1.p5.20.m20.4.4.1.3.2" xref="S4.SS1.p5.20.m20.4.4.1.3.2.cmml">τ</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.20.m20.4b"><apply id="S4.SS1.p5.20.m20.4.4.cmml" xref="S4.SS1.p5.20.m20.4.4"><apply id="S4.SS1.p5.20.m20.4.4.2.cmml" xref="S4.SS1.p5.20.m20.4.4.2"><csymbol cd="ambiguous" id="S4.SS1.p5.20.m20.4.4.2.1.cmml" xref="S4.SS1.p5.20.m20.4.4.2">superscript</csymbol><apply id="S4.SS1.p5.20.m20.4.4.2.2.cmml" xref="S4.SS1.p5.20.m20.4.4.2"><csymbol cd="ambiguous" id="S4.SS1.p5.20.m20.4.4.2.2.1.cmml" xref="S4.SS1.p5.20.m20.4.4.2">subscript</csymbol><int id="S4.SS1.p5.20.m20.4.4.2.2.2.cmml" xref="S4.SS1.p5.20.m20.4.4.2.2.2"></int><apply id="S4.SS1.p5.20.m20.4.4.2.2.3.cmml" xref="S4.SS1.p5.20.m20.4.4.2.2.3"><minus id="S4.SS1.p5.20.m20.4.4.2.2.3.1.cmml" xref="S4.SS1.p5.20.m20.4.4.2.2.3.1"></minus><ci id="S4.SS1.p5.20.m20.4.4.2.2.3.2.cmml" xref="S4.SS1.p5.20.m20.4.4.2.2.3.2">𝑡</ci><apply id="S4.SS1.p5.20.m20.4.4.2.2.3.3.cmml" xref="S4.SS1.p5.20.m20.4.4.2.2.3.3"><csymbol cd="ambiguous" id="S4.SS1.p5.20.m20.4.4.2.2.3.3.1.cmml" xref="S4.SS1.p5.20.m20.4.4.2.2.3.3">subscript</csymbol><ci id="S4.SS1.p5.20.m20.4.4.2.2.3.3.2.cmml" xref="S4.SS1.p5.20.m20.4.4.2.2.3.3.2">𝜏</ci><ci id="S4.SS1.p5.20.m20.4.4.2.2.3.3.3.cmml" xref="S4.SS1.p5.20.m20.4.4.2.2.3.3.3">d</ci></apply></apply></apply><ci id="S4.SS1.p5.20.m20.4.4.2.3.cmml" xref="S4.SS1.p5.20.m20.4.4.2.3">𝑡</ci></apply><apply id="S4.SS1.p5.20.m20.4.4.1.cmml" xref="S4.SS1.p5.20.m20.4.4.1"><times id="S4.SS1.p5.20.m20.4.4.1.2.cmml" xref="S4.SS1.p5.20.m20.4.4.1.2"></times><apply id="S4.SS1.p5.20.m20.4.4.1.1.cmml" xref="S4.SS1.p5.20.m20.4.4.1.1"><csymbol cd="ambiguous" id="S4.SS1.p5.20.m20.4.4.1.1.2.cmml" xref="S4.SS1.p5.20.m20.4.4.1.1">superscript</csymbol><apply id="S4.SS1.p5.20.m20.4.4.1.1.1.2.cmml" xref="S4.SS1.p5.20.m20.4.4.1.1.1.1"><csymbol cd="latexml" id="S4.SS1.p5.20.m20.4.4.1.1.1.2.1.cmml" xref="S4.SS1.p5.20.m20.4.4.1.1.1.1.2">norm</csymbol><apply id="S4.SS1.p5.20.m20.4.4.1.1.1.1.1.cmml" xref="S4.SS1.p5.20.m20.4.4.1.1.1.1.1"><times id="S4.SS1.p5.20.m20.4.4.1.1.1.1.1.1.cmml" xref="S4.SS1.p5.20.m20.4.4.1.1.1.1.1.1"></times><apply id="S4.SS1.p5.20.m20.4.4.1.1.1.1.1.2.cmml" xref="S4.SS1.p5.20.m20.4.4.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p5.20.m20.4.4.1.1.1.1.1.2.1.cmml" xref="S4.SS1.p5.20.m20.4.4.1.1.1.1.1.2">subscript</csymbol><ci id="S4.SS1.p5.20.m20.4.4.1.1.1.1.1.2.2.cmml" xref="S4.SS1.p5.20.m20.4.4.1.1.1.1.1.2.2">bold-italic-ϕ</ci><list id="S4.SS1.p5.20.m20.2.2.2.3.cmml" xref="S4.SS1.p5.20.m20.2.2.2.2"><ci id="S4.SS1.p5.20.m20.1.1.1.1.cmml" xref="S4.SS1.p5.20.m20.1.1.1.1">s</ci><apply id="S4.SS1.p5.20.m20.2.2.2.2.1.cmml" xref="S4.SS1.p5.20.m20.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.SS1.p5.20.m20.2.2.2.2.1.1.cmml" xref="S4.SS1.p5.20.m20.2.2.2.2.1">subscript</csymbol><ci id="S4.SS1.p5.20.m20.2.2.2.2.1.2.cmml" xref="S4.SS1.p5.20.m20.2.2.2.2.1.2">𝑘</ci><ci id="S4.SS1.p5.20.m20.2.2.2.2.1.3.cmml" xref="S4.SS1.p5.20.m20.2.2.2.2.1.3">𝑗</ci></apply></list></apply><ci id="S4.SS1.p5.20.m20.3.3.cmml" xref="S4.SS1.p5.20.m20.3.3">𝜏</ci></apply></apply><cn id="S4.SS1.p5.20.m20.4.4.1.1.3.cmml" type="integer" xref="S4.SS1.p5.20.m20.4.4.1.1.3">2</cn></apply><apply id="S4.SS1.p5.20.m20.4.4.1.3.cmml" xref="S4.SS1.p5.20.m20.4.4.1.3"><csymbol cd="latexml" id="S4.SS1.p5.20.m20.4.4.1.3.1.cmml" xref="S4.SS1.p5.20.m20.4.4.1.3.1">differential-d</csymbol><ci id="S4.SS1.p5.20.m20.4.4.1.3.2.cmml" xref="S4.SS1.p5.20.m20.4.4.1.3.2">𝜏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.20.m20.4c">\int_{t-\tau_{\rm d}}^{t}\|\bm{\phi}_{{\rm s},k_{j}}(\tau)\|^{2}d\tau</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.20.m20.4d">∫ start_POSTSUBSCRIPT italic_t - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ∥ bold_italic_ϕ start_POSTSUBSCRIPT roman_s , italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_τ ) ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d italic_τ</annotation></semantics></math> is the <math alttext="k_{j}" class="ltx_Math" display="inline" id="S4.SS1.p5.21.m21.1"><semantics id="S4.SS1.p5.21.m21.1a"><msub id="S4.SS1.p5.21.m21.1.1" xref="S4.SS1.p5.21.m21.1.1.cmml"><mi id="S4.SS1.p5.21.m21.1.1.2" xref="S4.SS1.p5.21.m21.1.1.2.cmml">k</mi><mi id="S4.SS1.p5.21.m21.1.1.3" xref="S4.SS1.p5.21.m21.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.21.m21.1b"><apply id="S4.SS1.p5.21.m21.1.1.cmml" xref="S4.SS1.p5.21.m21.1.1"><csymbol cd="ambiguous" id="S4.SS1.p5.21.m21.1.1.1.cmml" xref="S4.SS1.p5.21.m21.1.1">subscript</csymbol><ci id="S4.SS1.p5.21.m21.1.1.2.cmml" xref="S4.SS1.p5.21.m21.1.1.2">𝑘</ci><ci id="S4.SS1.p5.21.m21.1.1.3.cmml" xref="S4.SS1.p5.21.m21.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.21.m21.1c">k_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.21.m21.1d">italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>th diagonal element of <math alttext="\Psi(t)" class="ltx_Math" display="inline" id="S4.SS1.p5.22.m22.1"><semantics id="S4.SS1.p5.22.m22.1a"><mrow id="S4.SS1.p5.22.m22.1.2" xref="S4.SS1.p5.22.m22.1.2.cmml"><mi id="S4.SS1.p5.22.m22.1.2.2" mathvariant="normal" xref="S4.SS1.p5.22.m22.1.2.2.cmml">Ψ</mi><mo id="S4.SS1.p5.22.m22.1.2.1" xref="S4.SS1.p5.22.m22.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p5.22.m22.1.2.3.2" xref="S4.SS1.p5.22.m22.1.2.cmml"><mo id="S4.SS1.p5.22.m22.1.2.3.2.1" stretchy="false" xref="S4.SS1.p5.22.m22.1.2.cmml">(</mo><mi id="S4.SS1.p5.22.m22.1.1" xref="S4.SS1.p5.22.m22.1.1.cmml">t</mi><mo id="S4.SS1.p5.22.m22.1.2.3.2.2" stretchy="false" xref="S4.SS1.p5.22.m22.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.22.m22.1b"><apply id="S4.SS1.p5.22.m22.1.2.cmml" xref="S4.SS1.p5.22.m22.1.2"><times id="S4.SS1.p5.22.m22.1.2.1.cmml" xref="S4.SS1.p5.22.m22.1.2.1"></times><ci id="S4.SS1.p5.22.m22.1.2.2.cmml" xref="S4.SS1.p5.22.m22.1.2.2">Ψ</ci><ci id="S4.SS1.p5.22.m22.1.1.cmml" xref="S4.SS1.p5.22.m22.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.22.m22.1c">\Psi(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.22.m22.1d">roman_Ψ ( italic_t )</annotation></semantics></math>, <math alttext="\sigma_{\rm c}(t)\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S4.SS1.p5.23.m23.1"><semantics id="S4.SS1.p5.23.m23.1a"><mrow id="S4.SS1.p5.23.m23.1.2" xref="S4.SS1.p5.23.m23.1.2.cmml"><mrow id="S4.SS1.p5.23.m23.1.2.2" xref="S4.SS1.p5.23.m23.1.2.2.cmml"><msub id="S4.SS1.p5.23.m23.1.2.2.2" xref="S4.SS1.p5.23.m23.1.2.2.2.cmml"><mi id="S4.SS1.p5.23.m23.1.2.2.2.2" xref="S4.SS1.p5.23.m23.1.2.2.2.2.cmml">σ</mi><mi id="S4.SS1.p5.23.m23.1.2.2.2.3" mathvariant="normal" xref="S4.SS1.p5.23.m23.1.2.2.2.3.cmml">c</mi></msub><mo id="S4.SS1.p5.23.m23.1.2.2.1" xref="S4.SS1.p5.23.m23.1.2.2.1.cmml">⁢</mo><mrow id="S4.SS1.p5.23.m23.1.2.2.3.2" xref="S4.SS1.p5.23.m23.1.2.2.cmml"><mo id="S4.SS1.p5.23.m23.1.2.2.3.2.1" stretchy="false" xref="S4.SS1.p5.23.m23.1.2.2.cmml">(</mo><mi id="S4.SS1.p5.23.m23.1.1" xref="S4.SS1.p5.23.m23.1.1.cmml">t</mi><mo id="S4.SS1.p5.23.m23.1.2.2.3.2.2" stretchy="false" xref="S4.SS1.p5.23.m23.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p5.23.m23.1.2.1" xref="S4.SS1.p5.23.m23.1.2.1.cmml">∈</mo><msup id="S4.SS1.p5.23.m23.1.2.3" xref="S4.SS1.p5.23.m23.1.2.3.cmml"><mi id="S4.SS1.p5.23.m23.1.2.3.2" xref="S4.SS1.p5.23.m23.1.2.3.2.cmml">ℝ</mi><mo id="S4.SS1.p5.23.m23.1.2.3.3" xref="S4.SS1.p5.23.m23.1.2.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.23.m23.1b"><apply id="S4.SS1.p5.23.m23.1.2.cmml" xref="S4.SS1.p5.23.m23.1.2"><in id="S4.SS1.p5.23.m23.1.2.1.cmml" xref="S4.SS1.p5.23.m23.1.2.1"></in><apply id="S4.SS1.p5.23.m23.1.2.2.cmml" xref="S4.SS1.p5.23.m23.1.2.2"><times id="S4.SS1.p5.23.m23.1.2.2.1.cmml" xref="S4.SS1.p5.23.m23.1.2.2.1"></times><apply id="S4.SS1.p5.23.m23.1.2.2.2.cmml" xref="S4.SS1.p5.23.m23.1.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.p5.23.m23.1.2.2.2.1.cmml" xref="S4.SS1.p5.23.m23.1.2.2.2">subscript</csymbol><ci id="S4.SS1.p5.23.m23.1.2.2.2.2.cmml" xref="S4.SS1.p5.23.m23.1.2.2.2.2">𝜎</ci><ci id="S4.SS1.p5.23.m23.1.2.2.2.3.cmml" xref="S4.SS1.p5.23.m23.1.2.2.2.3">c</ci></apply><ci id="S4.SS1.p5.23.m23.1.1.cmml" xref="S4.SS1.p5.23.m23.1.1">𝑡</ci></apply><apply id="S4.SS1.p5.23.m23.1.2.3.cmml" xref="S4.SS1.p5.23.m23.1.2.3"><csymbol cd="ambiguous" id="S4.SS1.p5.23.m23.1.2.3.1.cmml" xref="S4.SS1.p5.23.m23.1.2.3">superscript</csymbol><ci id="S4.SS1.p5.23.m23.1.2.3.2.cmml" xref="S4.SS1.p5.23.m23.1.2.3.2">ℝ</ci><plus id="S4.SS1.p5.23.m23.1.2.3.3.cmml" xref="S4.SS1.p5.23.m23.1.2.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.23.m23.1c">\sigma_{\rm c}(t)\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.23.m23.1d">italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ( italic_t ) ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> is the current maximal exciting strength, and <math alttext="t_{\rm e}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S4.SS1.p5.24.m24.1"><semantics id="S4.SS1.p5.24.m24.1a"><mrow id="S4.SS1.p5.24.m24.1.1" xref="S4.SS1.p5.24.m24.1.1.cmml"><msub id="S4.SS1.p5.24.m24.1.1.2" xref="S4.SS1.p5.24.m24.1.1.2.cmml"><mi id="S4.SS1.p5.24.m24.1.1.2.2" xref="S4.SS1.p5.24.m24.1.1.2.2.cmml">t</mi><mi id="S4.SS1.p5.24.m24.1.1.2.3" mathvariant="normal" xref="S4.SS1.p5.24.m24.1.1.2.3.cmml">e</mi></msub><mo id="S4.SS1.p5.24.m24.1.1.1" xref="S4.SS1.p5.24.m24.1.1.1.cmml">∈</mo><msup id="S4.SS1.p5.24.m24.1.1.3" xref="S4.SS1.p5.24.m24.1.1.3.cmml"><mi id="S4.SS1.p5.24.m24.1.1.3.2" xref="S4.SS1.p5.24.m24.1.1.3.2.cmml">ℝ</mi><mo id="S4.SS1.p5.24.m24.1.1.3.3" xref="S4.SS1.p5.24.m24.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.24.m24.1b"><apply id="S4.SS1.p5.24.m24.1.1.cmml" xref="S4.SS1.p5.24.m24.1.1"><in id="S4.SS1.p5.24.m24.1.1.1.cmml" xref="S4.SS1.p5.24.m24.1.1.1"></in><apply id="S4.SS1.p5.24.m24.1.1.2.cmml" xref="S4.SS1.p5.24.m24.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p5.24.m24.1.1.2.1.cmml" xref="S4.SS1.p5.24.m24.1.1.2">subscript</csymbol><ci id="S4.SS1.p5.24.m24.1.1.2.2.cmml" xref="S4.SS1.p5.24.m24.1.1.2.2">𝑡</ci><ci id="S4.SS1.p5.24.m24.1.1.2.3.cmml" xref="S4.SS1.p5.24.m24.1.1.2.3">e</ci></apply><apply id="S4.SS1.p5.24.m24.1.1.3.cmml" xref="S4.SS1.p5.24.m24.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p5.24.m24.1.1.3.1.cmml" xref="S4.SS1.p5.24.m24.1.1.3">superscript</csymbol><ci id="S4.SS1.p5.24.m24.1.1.3.2.cmml" xref="S4.SS1.p5.24.m24.1.1.3.2">ℝ</ci><plus id="S4.SS1.p5.24.m24.1.1.3.3.cmml" xref="S4.SS1.p5.24.m24.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.24.m24.1c">t_{\rm e}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.24.m24.1d">italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> is the corresponding exciting time. Based on the above argument, define a generalized prediction error</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx14"> <tbody id="S4.E20"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle{\color[rgb]{0.00,0.00,0.60}\bm{\xi}(t):={\bm{q}}_{\rm f}(t,t_{% \rm e})-Q(t,t_{\rm e})\hat{\bm{\theta}}(t)}" class="ltx_Math" display="inline" id="S4.E20.m1.6"><semantics id="S4.E20.m1.6a"><mrow id="S4.E20.m1.6.6" xref="S4.E20.m1.6.6.cmml"><mrow id="S4.E20.m1.6.6.4" xref="S4.E20.m1.6.6.4.cmml"><mi id="S4.E20.m1.6.6.4.2" mathcolor="#000099" xref="S4.E20.m1.6.6.4.2.cmml">𝝃</mi><mo id="S4.E20.m1.6.6.4.1" xref="S4.E20.m1.6.6.4.1.cmml">⁢</mo><mrow id="S4.E20.m1.6.6.4.3.2" xref="S4.E20.m1.6.6.4.cmml"><mo id="S4.E20.m1.6.6.4.3.2.1" mathcolor="#000099" stretchy="false" xref="S4.E20.m1.6.6.4.cmml">(</mo><mi id="S4.E20.m1.1.1" mathcolor="#000099" xref="S4.E20.m1.1.1.cmml">t</mi><mo id="S4.E20.m1.6.6.4.3.2.2" mathcolor="#000099" rspace="0.278em" stretchy="false" xref="S4.E20.m1.6.6.4.cmml">)</mo></mrow></mrow><mo id="S4.E20.m1.6.6.3" mathcolor="#000099" rspace="0.278em" xref="S4.E20.m1.6.6.3.cmml">:=</mo><mrow id="S4.E20.m1.6.6.2" xref="S4.E20.m1.6.6.2.cmml"><mrow id="S4.E20.m1.5.5.1.1" xref="S4.E20.m1.5.5.1.1.cmml"><msub id="S4.E20.m1.5.5.1.1.3" xref="S4.E20.m1.5.5.1.1.3.cmml"><mi id="S4.E20.m1.5.5.1.1.3.2" mathcolor="#000099" xref="S4.E20.m1.5.5.1.1.3.2.cmml">𝒒</mi><mi id="S4.E20.m1.5.5.1.1.3.3" mathcolor="#000099" mathvariant="normal" xref="S4.E20.m1.5.5.1.1.3.3.cmml">f</mi></msub><mo id="S4.E20.m1.5.5.1.1.2" xref="S4.E20.m1.5.5.1.1.2.cmml">⁢</mo><mrow id="S4.E20.m1.5.5.1.1.1.1" xref="S4.E20.m1.5.5.1.1.1.2.cmml"><mo id="S4.E20.m1.5.5.1.1.1.1.2" mathcolor="#000099" stretchy="false" xref="S4.E20.m1.5.5.1.1.1.2.cmml">(</mo><mi id="S4.E20.m1.2.2" mathcolor="#000099" xref="S4.E20.m1.2.2.cmml">t</mi><mo id="S4.E20.m1.5.5.1.1.1.1.3" mathcolor="#000099" xref="S4.E20.m1.5.5.1.1.1.2.cmml">,</mo><msub id="S4.E20.m1.5.5.1.1.1.1.1" xref="S4.E20.m1.5.5.1.1.1.1.1.cmml"><mi id="S4.E20.m1.5.5.1.1.1.1.1.2" mathcolor="#000099" xref="S4.E20.m1.5.5.1.1.1.1.1.2.cmml">t</mi><mi id="S4.E20.m1.5.5.1.1.1.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S4.E20.m1.5.5.1.1.1.1.1.3.cmml">e</mi></msub><mo id="S4.E20.m1.5.5.1.1.1.1.4" mathcolor="#000099" stretchy="false" xref="S4.E20.m1.5.5.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.E20.m1.6.6.2.3" mathcolor="#000099" xref="S4.E20.m1.6.6.2.3.cmml">−</mo><mrow id="S4.E20.m1.6.6.2.2" xref="S4.E20.m1.6.6.2.2.cmml"><mi id="S4.E20.m1.6.6.2.2.3" mathcolor="#000099" xref="S4.E20.m1.6.6.2.2.3.cmml">Q</mi><mo id="S4.E20.m1.6.6.2.2.2" xref="S4.E20.m1.6.6.2.2.2.cmml">⁢</mo><mrow id="S4.E20.m1.6.6.2.2.1.1" xref="S4.E20.m1.6.6.2.2.1.2.cmml"><mo id="S4.E20.m1.6.6.2.2.1.1.2" mathcolor="#000099" stretchy="false" xref="S4.E20.m1.6.6.2.2.1.2.cmml">(</mo><mi id="S4.E20.m1.3.3" mathcolor="#000099" xref="S4.E20.m1.3.3.cmml">t</mi><mo id="S4.E20.m1.6.6.2.2.1.1.3" mathcolor="#000099" xref="S4.E20.m1.6.6.2.2.1.2.cmml">,</mo><msub id="S4.E20.m1.6.6.2.2.1.1.1" xref="S4.E20.m1.6.6.2.2.1.1.1.cmml"><mi id="S4.E20.m1.6.6.2.2.1.1.1.2" mathcolor="#000099" xref="S4.E20.m1.6.6.2.2.1.1.1.2.cmml">t</mi><mi id="S4.E20.m1.6.6.2.2.1.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S4.E20.m1.6.6.2.2.1.1.1.3.cmml">e</mi></msub><mo id="S4.E20.m1.6.6.2.2.1.1.4" mathcolor="#000099" stretchy="false" xref="S4.E20.m1.6.6.2.2.1.2.cmml">)</mo></mrow><mo id="S4.E20.m1.6.6.2.2.2a" xref="S4.E20.m1.6.6.2.2.2.cmml">⁢</mo><mover accent="true" id="S4.E20.m1.6.6.2.2.4" xref="S4.E20.m1.6.6.2.2.4.cmml"><mi id="S4.E20.m1.6.6.2.2.4.2" mathcolor="#000099" xref="S4.E20.m1.6.6.2.2.4.2.cmml">𝜽</mi><mo id="S4.E20.m1.6.6.2.2.4.1" mathcolor="#000099" xref="S4.E20.m1.6.6.2.2.4.1.cmml">^</mo></mover><mo id="S4.E20.m1.6.6.2.2.2b" xref="S4.E20.m1.6.6.2.2.2.cmml">⁢</mo><mrow id="S4.E20.m1.6.6.2.2.5.2" xref="S4.E20.m1.6.6.2.2.cmml"><mo id="S4.E20.m1.6.6.2.2.5.2.1" mathcolor="#000099" stretchy="false" xref="S4.E20.m1.6.6.2.2.cmml">(</mo><mi id="S4.E20.m1.4.4" mathcolor="#000099" xref="S4.E20.m1.4.4.cmml">t</mi><mo id="S4.E20.m1.6.6.2.2.5.2.2" mathcolor="#000099" stretchy="false" xref="S4.E20.m1.6.6.2.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E20.m1.6b"><apply id="S4.E20.m1.6.6.cmml" xref="S4.E20.m1.6.6"><csymbol cd="latexml" id="S4.E20.m1.6.6.3.cmml" xref="S4.E20.m1.6.6.3">assign</csymbol><apply id="S4.E20.m1.6.6.4.cmml" xref="S4.E20.m1.6.6.4"><times id="S4.E20.m1.6.6.4.1.cmml" xref="S4.E20.m1.6.6.4.1"></times><ci id="S4.E20.m1.6.6.4.2.cmml" xref="S4.E20.m1.6.6.4.2">𝝃</ci><ci id="S4.E20.m1.1.1.cmml" xref="S4.E20.m1.1.1">𝑡</ci></apply><apply id="S4.E20.m1.6.6.2.cmml" xref="S4.E20.m1.6.6.2"><minus id="S4.E20.m1.6.6.2.3.cmml" xref="S4.E20.m1.6.6.2.3"></minus><apply id="S4.E20.m1.5.5.1.1.cmml" xref="S4.E20.m1.5.5.1.1"><times id="S4.E20.m1.5.5.1.1.2.cmml" xref="S4.E20.m1.5.5.1.1.2"></times><apply id="S4.E20.m1.5.5.1.1.3.cmml" xref="S4.E20.m1.5.5.1.1.3"><csymbol cd="ambiguous" id="S4.E20.m1.5.5.1.1.3.1.cmml" xref="S4.E20.m1.5.5.1.1.3">subscript</csymbol><ci id="S4.E20.m1.5.5.1.1.3.2.cmml" xref="S4.E20.m1.5.5.1.1.3.2">𝒒</ci><ci id="S4.E20.m1.5.5.1.1.3.3.cmml" xref="S4.E20.m1.5.5.1.1.3.3">f</ci></apply><interval closure="open" id="S4.E20.m1.5.5.1.1.1.2.cmml" xref="S4.E20.m1.5.5.1.1.1.1"><ci id="S4.E20.m1.2.2.cmml" xref="S4.E20.m1.2.2">𝑡</ci><apply id="S4.E20.m1.5.5.1.1.1.1.1.cmml" xref="S4.E20.m1.5.5.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E20.m1.5.5.1.1.1.1.1.1.cmml" xref="S4.E20.m1.5.5.1.1.1.1.1">subscript</csymbol><ci id="S4.E20.m1.5.5.1.1.1.1.1.2.cmml" xref="S4.E20.m1.5.5.1.1.1.1.1.2">𝑡</ci><ci id="S4.E20.m1.5.5.1.1.1.1.1.3.cmml" xref="S4.E20.m1.5.5.1.1.1.1.1.3">e</ci></apply></interval></apply><apply id="S4.E20.m1.6.6.2.2.cmml" xref="S4.E20.m1.6.6.2.2"><times id="S4.E20.m1.6.6.2.2.2.cmml" xref="S4.E20.m1.6.6.2.2.2"></times><ci id="S4.E20.m1.6.6.2.2.3.cmml" xref="S4.E20.m1.6.6.2.2.3">𝑄</ci><interval closure="open" id="S4.E20.m1.6.6.2.2.1.2.cmml" xref="S4.E20.m1.6.6.2.2.1.1"><ci id="S4.E20.m1.3.3.cmml" xref="S4.E20.m1.3.3">𝑡</ci><apply id="S4.E20.m1.6.6.2.2.1.1.1.cmml" xref="S4.E20.m1.6.6.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.E20.m1.6.6.2.2.1.1.1.1.cmml" xref="S4.E20.m1.6.6.2.2.1.1.1">subscript</csymbol><ci id="S4.E20.m1.6.6.2.2.1.1.1.2.cmml" xref="S4.E20.m1.6.6.2.2.1.1.1.2">𝑡</ci><ci id="S4.E20.m1.6.6.2.2.1.1.1.3.cmml" xref="S4.E20.m1.6.6.2.2.1.1.1.3">e</ci></apply></interval><apply id="S4.E20.m1.6.6.2.2.4.cmml" xref="S4.E20.m1.6.6.2.2.4"><ci id="S4.E20.m1.6.6.2.2.4.1.cmml" xref="S4.E20.m1.6.6.2.2.4.1">^</ci><ci id="S4.E20.m1.6.6.2.2.4.2.cmml" xref="S4.E20.m1.6.6.2.2.4.2">𝜽</ci></apply><ci id="S4.E20.m1.4.4.cmml" xref="S4.E20.m1.4.4">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E20.m1.6c">\displaystyle{\color[rgb]{0.00,0.00,0.60}\bm{\xi}(t):={\bm{q}}_{\rm f}(t,t_{% \rm e})-Q(t,t_{\rm e})\hat{\bm{\theta}}(t)}</annotation><annotation encoding="application/x-llamapun" id="S4.E20.m1.6d">bold_italic_ξ ( italic_t ) := bold_italic_q start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) - italic_Q ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) over^ start_ARG bold_italic_θ end_ARG ( italic_t )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(20)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p5.30"><span class="ltx_text" id="S4.SS1.p5.30.6" style="color:#000099;">with <math alttext="Q(t,t_{\rm e})" class="ltx_Math" display="inline" id="S4.SS1.p5.25.1.m1.2"><semantics id="S4.SS1.p5.25.1.m1.2a"><mrow id="S4.SS1.p5.25.1.m1.2.2" xref="S4.SS1.p5.25.1.m1.2.2.cmml"><mi id="S4.SS1.p5.25.1.m1.2.2.3" mathcolor="#000099" xref="S4.SS1.p5.25.1.m1.2.2.3.cmml">Q</mi><mo id="S4.SS1.p5.25.1.m1.2.2.2" xref="S4.SS1.p5.25.1.m1.2.2.2.cmml">⁢</mo><mrow id="S4.SS1.p5.25.1.m1.2.2.1.1" xref="S4.SS1.p5.25.1.m1.2.2.1.2.cmml"><mo id="S4.SS1.p5.25.1.m1.2.2.1.1.2" mathcolor="#000099" stretchy="false" xref="S4.SS1.p5.25.1.m1.2.2.1.2.cmml">(</mo><mi id="S4.SS1.p5.25.1.m1.1.1" mathcolor="#000099" xref="S4.SS1.p5.25.1.m1.1.1.cmml">t</mi><mo id="S4.SS1.p5.25.1.m1.2.2.1.1.3" mathcolor="#000099" xref="S4.SS1.p5.25.1.m1.2.2.1.2.cmml">,</mo><msub id="S4.SS1.p5.25.1.m1.2.2.1.1.1" xref="S4.SS1.p5.25.1.m1.2.2.1.1.1.cmml"><mi id="S4.SS1.p5.25.1.m1.2.2.1.1.1.2" mathcolor="#000099" xref="S4.SS1.p5.25.1.m1.2.2.1.1.1.2.cmml">t</mi><mi id="S4.SS1.p5.25.1.m1.2.2.1.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS1.p5.25.1.m1.2.2.1.1.1.3.cmml">e</mi></msub><mo id="S4.SS1.p5.25.1.m1.2.2.1.1.4" mathcolor="#000099" stretchy="false" xref="S4.SS1.p5.25.1.m1.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.25.1.m1.2b"><apply id="S4.SS1.p5.25.1.m1.2.2.cmml" xref="S4.SS1.p5.25.1.m1.2.2"><times id="S4.SS1.p5.25.1.m1.2.2.2.cmml" xref="S4.SS1.p5.25.1.m1.2.2.2"></times><ci id="S4.SS1.p5.25.1.m1.2.2.3.cmml" xref="S4.SS1.p5.25.1.m1.2.2.3">𝑄</ci><interval closure="open" id="S4.SS1.p5.25.1.m1.2.2.1.2.cmml" xref="S4.SS1.p5.25.1.m1.2.2.1.1"><ci id="S4.SS1.p5.25.1.m1.1.1.cmml" xref="S4.SS1.p5.25.1.m1.1.1">𝑡</ci><apply id="S4.SS1.p5.25.1.m1.2.2.1.1.1.cmml" xref="S4.SS1.p5.25.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p5.25.1.m1.2.2.1.1.1.1.cmml" xref="S4.SS1.p5.25.1.m1.2.2.1.1.1">subscript</csymbol><ci id="S4.SS1.p5.25.1.m1.2.2.1.1.1.2.cmml" xref="S4.SS1.p5.25.1.m1.2.2.1.1.1.2">𝑡</ci><ci id="S4.SS1.p5.25.1.m1.2.2.1.1.1.3.cmml" xref="S4.SS1.p5.25.1.m1.2.2.1.1.1.3">e</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.25.1.m1.2c">Q(t,t_{\rm e})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.25.1.m1.2d">italic_Q ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT )</annotation></semantics></math> <math alttext="=" class="ltx_Math" display="inline" id="S4.SS1.p5.26.2.m2.1"><semantics id="S4.SS1.p5.26.2.m2.1a"><mo id="S4.SS1.p5.26.2.m2.1.1" mathcolor="#000099" xref="S4.SS1.p5.26.2.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.26.2.m2.1b"><eq id="S4.SS1.p5.26.2.m2.1.1.cmml" xref="S4.SS1.p5.26.2.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.26.2.m2.1c">=</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.26.2.m2.1d">=</annotation></semantics></math> <math alttext="H(s)[\Psi(t_{\rm e})]" class="ltx_Math" display="inline" id="S4.SS1.p5.27.3.m3.2"><semantics id="S4.SS1.p5.27.3.m3.2a"><mrow id="S4.SS1.p5.27.3.m3.2.2" xref="S4.SS1.p5.27.3.m3.2.2.cmml"><mi id="S4.SS1.p5.27.3.m3.2.2.3" mathcolor="#000099" xref="S4.SS1.p5.27.3.m3.2.2.3.cmml">H</mi><mo id="S4.SS1.p5.27.3.m3.2.2.2" xref="S4.SS1.p5.27.3.m3.2.2.2.cmml">⁢</mo><mrow id="S4.SS1.p5.27.3.m3.2.2.4.2" xref="S4.SS1.p5.27.3.m3.2.2.cmml"><mo id="S4.SS1.p5.27.3.m3.2.2.4.2.1" mathcolor="#000099" stretchy="false" xref="S4.SS1.p5.27.3.m3.2.2.cmml">(</mo><mi id="S4.SS1.p5.27.3.m3.1.1" mathcolor="#000099" xref="S4.SS1.p5.27.3.m3.1.1.cmml">s</mi><mo id="S4.SS1.p5.27.3.m3.2.2.4.2.2" mathcolor="#000099" stretchy="false" xref="S4.SS1.p5.27.3.m3.2.2.cmml">)</mo></mrow><mo id="S4.SS1.p5.27.3.m3.2.2.2a" xref="S4.SS1.p5.27.3.m3.2.2.2.cmml">⁢</mo><mrow id="S4.SS1.p5.27.3.m3.2.2.1.1" xref="S4.SS1.p5.27.3.m3.2.2.1.2.cmml"><mo id="S4.SS1.p5.27.3.m3.2.2.1.1.2" mathcolor="#000099" stretchy="false" xref="S4.SS1.p5.27.3.m3.2.2.1.2.1.cmml">[</mo><mrow id="S4.SS1.p5.27.3.m3.2.2.1.1.1" xref="S4.SS1.p5.27.3.m3.2.2.1.1.1.cmml"><mi id="S4.SS1.p5.27.3.m3.2.2.1.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS1.p5.27.3.m3.2.2.1.1.1.3.cmml">Ψ</mi><mo id="S4.SS1.p5.27.3.m3.2.2.1.1.1.2" xref="S4.SS1.p5.27.3.m3.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.p5.27.3.m3.2.2.1.1.1.1.1" xref="S4.SS1.p5.27.3.m3.2.2.1.1.1.1.1.1.cmml"><mo id="S4.SS1.p5.27.3.m3.2.2.1.1.1.1.1.2" mathcolor="#000099" stretchy="false" xref="S4.SS1.p5.27.3.m3.2.2.1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS1.p5.27.3.m3.2.2.1.1.1.1.1.1" xref="S4.SS1.p5.27.3.m3.2.2.1.1.1.1.1.1.cmml"><mi id="S4.SS1.p5.27.3.m3.2.2.1.1.1.1.1.1.2" mathcolor="#000099" xref="S4.SS1.p5.27.3.m3.2.2.1.1.1.1.1.1.2.cmml">t</mi><mi id="S4.SS1.p5.27.3.m3.2.2.1.1.1.1.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS1.p5.27.3.m3.2.2.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="S4.SS1.p5.27.3.m3.2.2.1.1.1.1.1.3" mathcolor="#000099" stretchy="false" xref="S4.SS1.p5.27.3.m3.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p5.27.3.m3.2.2.1.1.3" mathcolor="#000099" stretchy="false" xref="S4.SS1.p5.27.3.m3.2.2.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.27.3.m3.2b"><apply id="S4.SS1.p5.27.3.m3.2.2.cmml" xref="S4.SS1.p5.27.3.m3.2.2"><times id="S4.SS1.p5.27.3.m3.2.2.2.cmml" xref="S4.SS1.p5.27.3.m3.2.2.2"></times><ci id="S4.SS1.p5.27.3.m3.2.2.3.cmml" xref="S4.SS1.p5.27.3.m3.2.2.3">𝐻</ci><ci id="S4.SS1.p5.27.3.m3.1.1.cmml" xref="S4.SS1.p5.27.3.m3.1.1">𝑠</ci><apply id="S4.SS1.p5.27.3.m3.2.2.1.2.cmml" xref="S4.SS1.p5.27.3.m3.2.2.1.1"><csymbol cd="latexml" id="S4.SS1.p5.27.3.m3.2.2.1.2.1.cmml" xref="S4.SS1.p5.27.3.m3.2.2.1.1.2">delimited-[]</csymbol><apply id="S4.SS1.p5.27.3.m3.2.2.1.1.1.cmml" xref="S4.SS1.p5.27.3.m3.2.2.1.1.1"><times id="S4.SS1.p5.27.3.m3.2.2.1.1.1.2.cmml" xref="S4.SS1.p5.27.3.m3.2.2.1.1.1.2"></times><ci id="S4.SS1.p5.27.3.m3.2.2.1.1.1.3.cmml" xref="S4.SS1.p5.27.3.m3.2.2.1.1.1.3">Ψ</ci><apply id="S4.SS1.p5.27.3.m3.2.2.1.1.1.1.1.1.cmml" xref="S4.SS1.p5.27.3.m3.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p5.27.3.m3.2.2.1.1.1.1.1.1.1.cmml" xref="S4.SS1.p5.27.3.m3.2.2.1.1.1.1.1">subscript</csymbol><ci id="S4.SS1.p5.27.3.m3.2.2.1.1.1.1.1.1.2.cmml" xref="S4.SS1.p5.27.3.m3.2.2.1.1.1.1.1.1.2">𝑡</ci><ci id="S4.SS1.p5.27.3.m3.2.2.1.1.1.1.1.1.3.cmml" xref="S4.SS1.p5.27.3.m3.2.2.1.1.1.1.1.1.3">e</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.27.3.m3.2c">H(s)[\Psi(t_{\rm e})]</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.27.3.m3.2d">italic_H ( italic_s ) [ roman_Ψ ( italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) ]</annotation></semantics></math> and <math alttext="{\bm{q}}_{\rm f}(t,t_{\rm e})" class="ltx_Math" display="inline" id="S4.SS1.p5.28.4.m4.2"><semantics id="S4.SS1.p5.28.4.m4.2a"><mrow id="S4.SS1.p5.28.4.m4.2.2" xref="S4.SS1.p5.28.4.m4.2.2.cmml"><msub id="S4.SS1.p5.28.4.m4.2.2.3" xref="S4.SS1.p5.28.4.m4.2.2.3.cmml"><mi id="S4.SS1.p5.28.4.m4.2.2.3.2" mathcolor="#000099" xref="S4.SS1.p5.28.4.m4.2.2.3.2.cmml">𝒒</mi><mi id="S4.SS1.p5.28.4.m4.2.2.3.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS1.p5.28.4.m4.2.2.3.3.cmml">f</mi></msub><mo id="S4.SS1.p5.28.4.m4.2.2.2" xref="S4.SS1.p5.28.4.m4.2.2.2.cmml">⁢</mo><mrow id="S4.SS1.p5.28.4.m4.2.2.1.1" xref="S4.SS1.p5.28.4.m4.2.2.1.2.cmml"><mo id="S4.SS1.p5.28.4.m4.2.2.1.1.2" mathcolor="#000099" stretchy="false" xref="S4.SS1.p5.28.4.m4.2.2.1.2.cmml">(</mo><mi id="S4.SS1.p5.28.4.m4.1.1" mathcolor="#000099" xref="S4.SS1.p5.28.4.m4.1.1.cmml">t</mi><mo id="S4.SS1.p5.28.4.m4.2.2.1.1.3" mathcolor="#000099" xref="S4.SS1.p5.28.4.m4.2.2.1.2.cmml">,</mo><msub id="S4.SS1.p5.28.4.m4.2.2.1.1.1" xref="S4.SS1.p5.28.4.m4.2.2.1.1.1.cmml"><mi id="S4.SS1.p5.28.4.m4.2.2.1.1.1.2" mathcolor="#000099" xref="S4.SS1.p5.28.4.m4.2.2.1.1.1.2.cmml">t</mi><mi id="S4.SS1.p5.28.4.m4.2.2.1.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS1.p5.28.4.m4.2.2.1.1.1.3.cmml">e</mi></msub><mo id="S4.SS1.p5.28.4.m4.2.2.1.1.4" mathcolor="#000099" stretchy="false" xref="S4.SS1.p5.28.4.m4.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.28.4.m4.2b"><apply id="S4.SS1.p5.28.4.m4.2.2.cmml" xref="S4.SS1.p5.28.4.m4.2.2"><times id="S4.SS1.p5.28.4.m4.2.2.2.cmml" xref="S4.SS1.p5.28.4.m4.2.2.2"></times><apply id="S4.SS1.p5.28.4.m4.2.2.3.cmml" xref="S4.SS1.p5.28.4.m4.2.2.3"><csymbol cd="ambiguous" id="S4.SS1.p5.28.4.m4.2.2.3.1.cmml" xref="S4.SS1.p5.28.4.m4.2.2.3">subscript</csymbol><ci id="S4.SS1.p5.28.4.m4.2.2.3.2.cmml" xref="S4.SS1.p5.28.4.m4.2.2.3.2">𝒒</ci><ci id="S4.SS1.p5.28.4.m4.2.2.3.3.cmml" xref="S4.SS1.p5.28.4.m4.2.2.3.3">f</ci></apply><interval closure="open" id="S4.SS1.p5.28.4.m4.2.2.1.2.cmml" xref="S4.SS1.p5.28.4.m4.2.2.1.1"><ci id="S4.SS1.p5.28.4.m4.1.1.cmml" xref="S4.SS1.p5.28.4.m4.1.1">𝑡</ci><apply id="S4.SS1.p5.28.4.m4.2.2.1.1.1.cmml" xref="S4.SS1.p5.28.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p5.28.4.m4.2.2.1.1.1.1.cmml" xref="S4.SS1.p5.28.4.m4.2.2.1.1.1">subscript</csymbol><ci id="S4.SS1.p5.28.4.m4.2.2.1.1.1.2.cmml" xref="S4.SS1.p5.28.4.m4.2.2.1.1.1.2">𝑡</ci><ci id="S4.SS1.p5.28.4.m4.2.2.1.1.1.3.cmml" xref="S4.SS1.p5.28.4.m4.2.2.1.1.1.3">e</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.28.4.m4.2c">{\bm{q}}_{\rm f}(t,t_{\rm e})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.28.4.m4.2d">bold_italic_q start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT )</annotation></semantics></math> <math alttext="=" class="ltx_Math" display="inline" id="S4.SS1.p5.29.5.m5.1"><semantics id="S4.SS1.p5.29.5.m5.1a"><mo id="S4.SS1.p5.29.5.m5.1.1" mathcolor="#000099" xref="S4.SS1.p5.29.5.m5.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.29.5.m5.1b"><eq id="S4.SS1.p5.29.5.m5.1.1.cmml" xref="S4.SS1.p5.29.5.m5.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.29.5.m5.1c">=</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.29.5.m5.1d">=</annotation></semantics></math> <math alttext="H(s)[{\bm{q}}(t_{\rm e})]" class="ltx_Math" display="inline" id="S4.SS1.p5.30.6.m6.2"><semantics id="S4.SS1.p5.30.6.m6.2a"><mrow id="S4.SS1.p5.30.6.m6.2.2" xref="S4.SS1.p5.30.6.m6.2.2.cmml"><mi id="S4.SS1.p5.30.6.m6.2.2.3" mathcolor="#000099" xref="S4.SS1.p5.30.6.m6.2.2.3.cmml">H</mi><mo id="S4.SS1.p5.30.6.m6.2.2.2" xref="S4.SS1.p5.30.6.m6.2.2.2.cmml">⁢</mo><mrow id="S4.SS1.p5.30.6.m6.2.2.4.2" xref="S4.SS1.p5.30.6.m6.2.2.cmml"><mo id="S4.SS1.p5.30.6.m6.2.2.4.2.1" mathcolor="#000099" stretchy="false" xref="S4.SS1.p5.30.6.m6.2.2.cmml">(</mo><mi id="S4.SS1.p5.30.6.m6.1.1" mathcolor="#000099" xref="S4.SS1.p5.30.6.m6.1.1.cmml">s</mi><mo id="S4.SS1.p5.30.6.m6.2.2.4.2.2" mathcolor="#000099" stretchy="false" xref="S4.SS1.p5.30.6.m6.2.2.cmml">)</mo></mrow><mo id="S4.SS1.p5.30.6.m6.2.2.2a" xref="S4.SS1.p5.30.6.m6.2.2.2.cmml">⁢</mo><mrow id="S4.SS1.p5.30.6.m6.2.2.1.1" xref="S4.SS1.p5.30.6.m6.2.2.1.2.cmml"><mo id="S4.SS1.p5.30.6.m6.2.2.1.1.2" mathcolor="#000099" stretchy="false" xref="S4.SS1.p5.30.6.m6.2.2.1.2.1.cmml">[</mo><mrow id="S4.SS1.p5.30.6.m6.2.2.1.1.1" xref="S4.SS1.p5.30.6.m6.2.2.1.1.1.cmml"><mi id="S4.SS1.p5.30.6.m6.2.2.1.1.1.3" mathcolor="#000099" xref="S4.SS1.p5.30.6.m6.2.2.1.1.1.3.cmml">𝒒</mi><mo id="S4.SS1.p5.30.6.m6.2.2.1.1.1.2" xref="S4.SS1.p5.30.6.m6.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.p5.30.6.m6.2.2.1.1.1.1.1" xref="S4.SS1.p5.30.6.m6.2.2.1.1.1.1.1.1.cmml"><mo id="S4.SS1.p5.30.6.m6.2.2.1.1.1.1.1.2" mathcolor="#000099" stretchy="false" xref="S4.SS1.p5.30.6.m6.2.2.1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS1.p5.30.6.m6.2.2.1.1.1.1.1.1" xref="S4.SS1.p5.30.6.m6.2.2.1.1.1.1.1.1.cmml"><mi id="S4.SS1.p5.30.6.m6.2.2.1.1.1.1.1.1.2" mathcolor="#000099" xref="S4.SS1.p5.30.6.m6.2.2.1.1.1.1.1.1.2.cmml">t</mi><mi id="S4.SS1.p5.30.6.m6.2.2.1.1.1.1.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS1.p5.30.6.m6.2.2.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="S4.SS1.p5.30.6.m6.2.2.1.1.1.1.1.3" mathcolor="#000099" stretchy="false" xref="S4.SS1.p5.30.6.m6.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p5.30.6.m6.2.2.1.1.3" mathcolor="#000099" stretchy="false" xref="S4.SS1.p5.30.6.m6.2.2.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.30.6.m6.2b"><apply id="S4.SS1.p5.30.6.m6.2.2.cmml" xref="S4.SS1.p5.30.6.m6.2.2"><times id="S4.SS1.p5.30.6.m6.2.2.2.cmml" xref="S4.SS1.p5.30.6.m6.2.2.2"></times><ci id="S4.SS1.p5.30.6.m6.2.2.3.cmml" xref="S4.SS1.p5.30.6.m6.2.2.3">𝐻</ci><ci id="S4.SS1.p5.30.6.m6.1.1.cmml" xref="S4.SS1.p5.30.6.m6.1.1">𝑠</ci><apply id="S4.SS1.p5.30.6.m6.2.2.1.2.cmml" xref="S4.SS1.p5.30.6.m6.2.2.1.1"><csymbol cd="latexml" id="S4.SS1.p5.30.6.m6.2.2.1.2.1.cmml" xref="S4.SS1.p5.30.6.m6.2.2.1.1.2">delimited-[]</csymbol><apply id="S4.SS1.p5.30.6.m6.2.2.1.1.1.cmml" xref="S4.SS1.p5.30.6.m6.2.2.1.1.1"><times id="S4.SS1.p5.30.6.m6.2.2.1.1.1.2.cmml" xref="S4.SS1.p5.30.6.m6.2.2.1.1.1.2"></times><ci id="S4.SS1.p5.30.6.m6.2.2.1.1.1.3.cmml" xref="S4.SS1.p5.30.6.m6.2.2.1.1.1.3">𝒒</ci><apply id="S4.SS1.p5.30.6.m6.2.2.1.1.1.1.1.1.cmml" xref="S4.SS1.p5.30.6.m6.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p5.30.6.m6.2.2.1.1.1.1.1.1.1.cmml" xref="S4.SS1.p5.30.6.m6.2.2.1.1.1.1.1">subscript</csymbol><ci id="S4.SS1.p5.30.6.m6.2.2.1.1.1.1.1.1.2.cmml" xref="S4.SS1.p5.30.6.m6.2.2.1.1.1.1.1.1.2">𝑡</ci><ci id="S4.SS1.p5.30.6.m6.2.2.1.1.1.1.1.1.3.cmml" xref="S4.SS1.p5.30.6.m6.2.2.1.1.1.1.1.1.3">e</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.30.6.m6.2c">H(s)[{\bm{q}}(t_{\rm e})]</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.30.6.m6.2d">italic_H ( italic_s ) [ bold_italic_q ( italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) ]</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S4.SS1.p6"> <p class="ltx_p" id="S4.SS1.p6.19"><span class="ltx_text" id="S4.SS1.p6.11.11" style="color:#000099;">In this study, the system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S2.E4" title="In II Problem Formulation ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">4</span></a>) is continuous-time, and Algorithm 1 is discrete-time implemented at the intervals of the sampling time <math alttext="T_{\rm s}" class="ltx_Math" display="inline" id="S4.SS1.p6.1.1.m1.1"><semantics id="S4.SS1.p6.1.1.m1.1a"><msub id="S4.SS1.p6.1.1.m1.1.1" xref="S4.SS1.p6.1.1.m1.1.1.cmml"><mi id="S4.SS1.p6.1.1.m1.1.1.2" mathcolor="#000099" xref="S4.SS1.p6.1.1.m1.1.1.2.cmml">T</mi><mi id="S4.SS1.p6.1.1.m1.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS1.p6.1.1.m1.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.1.1.m1.1b"><apply id="S4.SS1.p6.1.1.m1.1.1.cmml" xref="S4.SS1.p6.1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p6.1.1.m1.1.1.1.cmml" xref="S4.SS1.p6.1.1.m1.1.1">subscript</csymbol><ci id="S4.SS1.p6.1.1.m1.1.1.2.cmml" xref="S4.SS1.p6.1.1.m1.1.1.2">𝑇</ci><ci id="S4.SS1.p6.1.1.m1.1.1.3.cmml" xref="S4.SS1.p6.1.1.m1.1.1.3">s</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.1.1.m1.1c">T_{\rm s}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.1.1.m1.1d">italic_T start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT</annotation></semantics></math>. Thus, <math alttext="\tau_{\rm d}" class="ltx_Math" display="inline" id="S4.SS1.p6.2.2.m2.1"><semantics id="S4.SS1.p6.2.2.m2.1a"><msub id="S4.SS1.p6.2.2.m2.1.1" xref="S4.SS1.p6.2.2.m2.1.1.cmml"><mi id="S4.SS1.p6.2.2.m2.1.1.2" mathcolor="#000099" xref="S4.SS1.p6.2.2.m2.1.1.2.cmml">τ</mi><mi id="S4.SS1.p6.2.2.m2.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS1.p6.2.2.m2.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.2.2.m2.1b"><apply id="S4.SS1.p6.2.2.m2.1.1.cmml" xref="S4.SS1.p6.2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p6.2.2.m2.1.1.1.cmml" xref="S4.SS1.p6.2.2.m2.1.1">subscript</csymbol><ci id="S4.SS1.p6.2.2.m2.1.1.2.cmml" xref="S4.SS1.p6.2.2.m2.1.1.2">𝜏</ci><ci id="S4.SS1.p6.2.2.m2.1.1.3.cmml" xref="S4.SS1.p6.2.2.m2.1.1.3">d</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.2.2.m2.1c">\tau_{\rm d}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.2.2.m2.1d">italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT</annotation></semantics></math> should be chosen to be greater than <math alttext="T_{\rm s}" class="ltx_Math" display="inline" id="S4.SS1.p6.3.3.m3.1"><semantics id="S4.SS1.p6.3.3.m3.1a"><msub id="S4.SS1.p6.3.3.m3.1.1" xref="S4.SS1.p6.3.3.m3.1.1.cmml"><mi id="S4.SS1.p6.3.3.m3.1.1.2" mathcolor="#000099" xref="S4.SS1.p6.3.3.m3.1.1.2.cmml">T</mi><mi id="S4.SS1.p6.3.3.m3.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS1.p6.3.3.m3.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.3.3.m3.1b"><apply id="S4.SS1.p6.3.3.m3.1.1.cmml" xref="S4.SS1.p6.3.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.p6.3.3.m3.1.1.1.cmml" xref="S4.SS1.p6.3.3.m3.1.1">subscript</csymbol><ci id="S4.SS1.p6.3.3.m3.1.1.2.cmml" xref="S4.SS1.p6.3.3.m3.1.1.2">𝑇</ci><ci id="S4.SS1.p6.3.3.m3.1.1.3.cmml" xref="S4.SS1.p6.3.3.m3.1.1.3">s</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.3.3.m3.1c">T_{\rm s}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.3.3.m3.1d">italic_T start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT</annotation></semantics></math> to ensure the correct functioning of Algorithm 1. In Algorithm 1, we first choose a sufficiently small threshold <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS1.p6.4.4.m4.1"><semantics id="S4.SS1.p6.4.4.m4.1a"><mi id="S4.SS1.p6.4.4.m4.1.1" mathcolor="#000099" xref="S4.SS1.p6.4.4.m4.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.4.4.m4.1b"><ci id="S4.SS1.p6.4.4.m4.1.1.cmml" xref="S4.SS1.p6.4.4.m4.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.4.4.m4.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.4.4.m4.1d">italic_σ</annotation></semantics></math> [see Line 1 in Algorithm 1]. Note that partial IE usually exists; otherwise, all channels will be deactivated. At the beginning of each partial IE stage, the maximal exciting strength <math alttext="\sigma_{\rm c}" class="ltx_Math" display="inline" id="S4.SS1.p6.5.5.m5.1"><semantics id="S4.SS1.p6.5.5.m5.1a"><msub id="S4.SS1.p6.5.5.m5.1.1" xref="S4.SS1.p6.5.5.m5.1.1.cmml"><mi id="S4.SS1.p6.5.5.m5.1.1.2" mathcolor="#000099" xref="S4.SS1.p6.5.5.m5.1.1.2.cmml">σ</mi><mi id="S4.SS1.p6.5.5.m5.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS1.p6.5.5.m5.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.5.5.m5.1b"><apply id="S4.SS1.p6.5.5.m5.1.1.cmml" xref="S4.SS1.p6.5.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS1.p6.5.5.m5.1.1.1.cmml" xref="S4.SS1.p6.5.5.m5.1.1">subscript</csymbol><ci id="S4.SS1.p6.5.5.m5.1.1.2.cmml" xref="S4.SS1.p6.5.5.m5.1.1.2">𝜎</ci><ci id="S4.SS1.p6.5.5.m5.1.1.3.cmml" xref="S4.SS1.p6.5.5.m5.1.1.3">c</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.5.5.m5.1c">\sigma_{\rm c}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.5.5.m5.1d">italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT</annotation></semantics></math> is reset to <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS1.p6.6.6.m6.1"><semantics id="S4.SS1.p6.6.6.m6.1a"><mi id="S4.SS1.p6.6.6.m6.1.1" mathcolor="#000099" xref="S4.SS1.p6.6.6.m6.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.6.6.m6.1b"><ci id="S4.SS1.p6.6.6.m6.1.1.cmml" xref="S4.SS1.p6.6.6.m6.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.6.6.m6.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.6.6.m6.1d">italic_σ</annotation></semantics></math> [see Line 7 in Algorithm 1]. If the current exciting strength <math alttext="\sigma_{\min}(\Psi_{\zeta}(t))" class="ltx_Math" display="inline" id="S4.SS1.p6.7.7.m7.2"><semantics id="S4.SS1.p6.7.7.m7.2a"><mrow id="S4.SS1.p6.7.7.m7.2.2" xref="S4.SS1.p6.7.7.m7.2.2.cmml"><msub id="S4.SS1.p6.7.7.m7.2.2.3" xref="S4.SS1.p6.7.7.m7.2.2.3.cmml"><mi id="S4.SS1.p6.7.7.m7.2.2.3.2" mathcolor="#000099" xref="S4.SS1.p6.7.7.m7.2.2.3.2.cmml">σ</mi><mi id="S4.SS1.p6.7.7.m7.2.2.3.3" mathcolor="#000099" xref="S4.SS1.p6.7.7.m7.2.2.3.3.cmml">min</mi></msub><mo id="S4.SS1.p6.7.7.m7.2.2.2" xref="S4.SS1.p6.7.7.m7.2.2.2.cmml">⁢</mo><mrow id="S4.SS1.p6.7.7.m7.2.2.1.1" xref="S4.SS1.p6.7.7.m7.2.2.1.1.1.cmml"><mo id="S4.SS1.p6.7.7.m7.2.2.1.1.2" mathcolor="#000099" stretchy="false" xref="S4.SS1.p6.7.7.m7.2.2.1.1.1.cmml">(</mo><mrow id="S4.SS1.p6.7.7.m7.2.2.1.1.1" xref="S4.SS1.p6.7.7.m7.2.2.1.1.1.cmml"><msub id="S4.SS1.p6.7.7.m7.2.2.1.1.1.2" xref="S4.SS1.p6.7.7.m7.2.2.1.1.1.2.cmml"><mi id="S4.SS1.p6.7.7.m7.2.2.1.1.1.2.2" mathcolor="#000099" mathvariant="normal" xref="S4.SS1.p6.7.7.m7.2.2.1.1.1.2.2.cmml">Ψ</mi><mi id="S4.SS1.p6.7.7.m7.2.2.1.1.1.2.3" mathcolor="#000099" xref="S4.SS1.p6.7.7.m7.2.2.1.1.1.2.3.cmml">ζ</mi></msub><mo id="S4.SS1.p6.7.7.m7.2.2.1.1.1.1" xref="S4.SS1.p6.7.7.m7.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS1.p6.7.7.m7.2.2.1.1.1.3.2" xref="S4.SS1.p6.7.7.m7.2.2.1.1.1.cmml"><mo id="S4.SS1.p6.7.7.m7.2.2.1.1.1.3.2.1" mathcolor="#000099" stretchy="false" xref="S4.SS1.p6.7.7.m7.2.2.1.1.1.cmml">(</mo><mi id="S4.SS1.p6.7.7.m7.1.1" mathcolor="#000099" xref="S4.SS1.p6.7.7.m7.1.1.cmml">t</mi><mo id="S4.SS1.p6.7.7.m7.2.2.1.1.1.3.2.2" mathcolor="#000099" stretchy="false" xref="S4.SS1.p6.7.7.m7.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p6.7.7.m7.2.2.1.1.3" mathcolor="#000099" stretchy="false" xref="S4.SS1.p6.7.7.m7.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.7.7.m7.2b"><apply id="S4.SS1.p6.7.7.m7.2.2.cmml" xref="S4.SS1.p6.7.7.m7.2.2"><times id="S4.SS1.p6.7.7.m7.2.2.2.cmml" xref="S4.SS1.p6.7.7.m7.2.2.2"></times><apply id="S4.SS1.p6.7.7.m7.2.2.3.cmml" xref="S4.SS1.p6.7.7.m7.2.2.3"><csymbol cd="ambiguous" id="S4.SS1.p6.7.7.m7.2.2.3.1.cmml" xref="S4.SS1.p6.7.7.m7.2.2.3">subscript</csymbol><ci id="S4.SS1.p6.7.7.m7.2.2.3.2.cmml" xref="S4.SS1.p6.7.7.m7.2.2.3.2">𝜎</ci><min id="S4.SS1.p6.7.7.m7.2.2.3.3.cmml" xref="S4.SS1.p6.7.7.m7.2.2.3.3"></min></apply><apply id="S4.SS1.p6.7.7.m7.2.2.1.1.1.cmml" xref="S4.SS1.p6.7.7.m7.2.2.1.1"><times id="S4.SS1.p6.7.7.m7.2.2.1.1.1.1.cmml" xref="S4.SS1.p6.7.7.m7.2.2.1.1.1.1"></times><apply id="S4.SS1.p6.7.7.m7.2.2.1.1.1.2.cmml" xref="S4.SS1.p6.7.7.m7.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p6.7.7.m7.2.2.1.1.1.2.1.cmml" xref="S4.SS1.p6.7.7.m7.2.2.1.1.1.2">subscript</csymbol><ci id="S4.SS1.p6.7.7.m7.2.2.1.1.1.2.2.cmml" xref="S4.SS1.p6.7.7.m7.2.2.1.1.1.2.2">Ψ</ci><ci id="S4.SS1.p6.7.7.m7.2.2.1.1.1.2.3.cmml" xref="S4.SS1.p6.7.7.m7.2.2.1.1.1.2.3">𝜁</ci></apply><ci id="S4.SS1.p6.7.7.m7.1.1.cmml" xref="S4.SS1.p6.7.7.m7.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.7.7.m7.2c">\sigma_{\min}(\Psi_{\zeta}(t))</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.7.7.m7.2d">italic_σ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Ψ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t ) )</annotation></semantics></math> is greater than <math alttext="\sigma_{\rm c}" class="ltx_Math" display="inline" id="S4.SS1.p6.8.8.m8.1"><semantics id="S4.SS1.p6.8.8.m8.1a"><msub id="S4.SS1.p6.8.8.m8.1.1" xref="S4.SS1.p6.8.8.m8.1.1.cmml"><mi id="S4.SS1.p6.8.8.m8.1.1.2" mathcolor="#000099" xref="S4.SS1.p6.8.8.m8.1.1.2.cmml">σ</mi><mi id="S4.SS1.p6.8.8.m8.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS1.p6.8.8.m8.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.8.8.m8.1b"><apply id="S4.SS1.p6.8.8.m8.1.1.cmml" xref="S4.SS1.p6.8.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS1.p6.8.8.m8.1.1.1.cmml" xref="S4.SS1.p6.8.8.m8.1.1">subscript</csymbol><ci id="S4.SS1.p6.8.8.m8.1.1.2.cmml" xref="S4.SS1.p6.8.8.m8.1.1.2">𝜎</ci><ci id="S4.SS1.p6.8.8.m8.1.1.3.cmml" xref="S4.SS1.p6.8.8.m8.1.1.3">c</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.8.8.m8.1c">\sigma_{\rm c}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.8.8.m8.1d">italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT</annotation></semantics></math>, the exciting time <math alttext="t_{\rm e}" class="ltx_Math" display="inline" id="S4.SS1.p6.9.9.m9.1"><semantics id="S4.SS1.p6.9.9.m9.1a"><msub id="S4.SS1.p6.9.9.m9.1.1" xref="S4.SS1.p6.9.9.m9.1.1.cmml"><mi id="S4.SS1.p6.9.9.m9.1.1.2" mathcolor="#000099" xref="S4.SS1.p6.9.9.m9.1.1.2.cmml">t</mi><mi id="S4.SS1.p6.9.9.m9.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS1.p6.9.9.m9.1.1.3.cmml">e</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.9.9.m9.1b"><apply id="S4.SS1.p6.9.9.m9.1.1.cmml" xref="S4.SS1.p6.9.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS1.p6.9.9.m9.1.1.1.cmml" xref="S4.SS1.p6.9.9.m9.1.1">subscript</csymbol><ci id="S4.SS1.p6.9.9.m9.1.1.2.cmml" xref="S4.SS1.p6.9.9.m9.1.1.2">𝑡</ci><ci id="S4.SS1.p6.9.9.m9.1.1.3.cmml" xref="S4.SS1.p6.9.9.m9.1.1.3">e</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.9.9.m9.1c">t_{\rm e}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.9.9.m9.1d">italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\sigma_{\rm c}" class="ltx_Math" display="inline" id="S4.SS1.p6.10.10.m10.1"><semantics id="S4.SS1.p6.10.10.m10.1a"><msub id="S4.SS1.p6.10.10.m10.1.1" xref="S4.SS1.p6.10.10.m10.1.1.cmml"><mi id="S4.SS1.p6.10.10.m10.1.1.2" mathcolor="#000099" xref="S4.SS1.p6.10.10.m10.1.1.2.cmml">σ</mi><mi id="S4.SS1.p6.10.10.m10.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS1.p6.10.10.m10.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.10.10.m10.1b"><apply id="S4.SS1.p6.10.10.m10.1.1.cmml" xref="S4.SS1.p6.10.10.m10.1.1"><csymbol cd="ambiguous" id="S4.SS1.p6.10.10.m10.1.1.1.cmml" xref="S4.SS1.p6.10.10.m10.1.1">subscript</csymbol><ci id="S4.SS1.p6.10.10.m10.1.1.2.cmml" xref="S4.SS1.p6.10.10.m10.1.1.2">𝜎</ci><ci id="S4.SS1.p6.10.10.m10.1.1.3.cmml" xref="S4.SS1.p6.10.10.m10.1.1.3">c</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.10.10.m10.1c">\sigma_{\rm c}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.10.10.m10.1d">italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT</annotation></semantics></math> are updated [see Lines 9–11 in Algorithm 1]; otherwise, they remain unchanged. Thus, Algorithm 1 ensures that the exciting strength <math alttext="\sigma_{\min}(\Psi_{\zeta}(t_{\rm e}))" class="ltx_Math" display="inline" id="S4.SS1.p6.11.11.m11.1"><semantics id="S4.SS1.p6.11.11.m11.1a"><mrow id="S4.SS1.p6.11.11.m11.1.1" xref="S4.SS1.p6.11.11.m11.1.1.cmml"><msub id="S4.SS1.p6.11.11.m11.1.1.3" xref="S4.SS1.p6.11.11.m11.1.1.3.cmml"><mi id="S4.SS1.p6.11.11.m11.1.1.3.2" mathcolor="#000099" xref="S4.SS1.p6.11.11.m11.1.1.3.2.cmml">σ</mi><mi id="S4.SS1.p6.11.11.m11.1.1.3.3" mathcolor="#000099" xref="S4.SS1.p6.11.11.m11.1.1.3.3.cmml">min</mi></msub><mo id="S4.SS1.p6.11.11.m11.1.1.2" xref="S4.SS1.p6.11.11.m11.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.p6.11.11.m11.1.1.1.1" xref="S4.SS1.p6.11.11.m11.1.1.1.1.1.cmml"><mo id="S4.SS1.p6.11.11.m11.1.1.1.1.2" mathcolor="#000099" stretchy="false" xref="S4.SS1.p6.11.11.m11.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS1.p6.11.11.m11.1.1.1.1.1" xref="S4.SS1.p6.11.11.m11.1.1.1.1.1.cmml"><msub id="S4.SS1.p6.11.11.m11.1.1.1.1.1.3" xref="S4.SS1.p6.11.11.m11.1.1.1.1.1.3.cmml"><mi id="S4.SS1.p6.11.11.m11.1.1.1.1.1.3.2" mathcolor="#000099" mathvariant="normal" xref="S4.SS1.p6.11.11.m11.1.1.1.1.1.3.2.cmml">Ψ</mi><mi id="S4.SS1.p6.11.11.m11.1.1.1.1.1.3.3" mathcolor="#000099" xref="S4.SS1.p6.11.11.m11.1.1.1.1.1.3.3.cmml">ζ</mi></msub><mo id="S4.SS1.p6.11.11.m11.1.1.1.1.1.2" xref="S4.SS1.p6.11.11.m11.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.p6.11.11.m11.1.1.1.1.1.1.1" xref="S4.SS1.p6.11.11.m11.1.1.1.1.1.1.1.1.cmml"><mo id="S4.SS1.p6.11.11.m11.1.1.1.1.1.1.1.2" mathcolor="#000099" stretchy="false" xref="S4.SS1.p6.11.11.m11.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS1.p6.11.11.m11.1.1.1.1.1.1.1.1" xref="S4.SS1.p6.11.11.m11.1.1.1.1.1.1.1.1.cmml"><mi id="S4.SS1.p6.11.11.m11.1.1.1.1.1.1.1.1.2" mathcolor="#000099" xref="S4.SS1.p6.11.11.m11.1.1.1.1.1.1.1.1.2.cmml">t</mi><mi id="S4.SS1.p6.11.11.m11.1.1.1.1.1.1.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS1.p6.11.11.m11.1.1.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="S4.SS1.p6.11.11.m11.1.1.1.1.1.1.1.3" mathcolor="#000099" stretchy="false" xref="S4.SS1.p6.11.11.m11.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p6.11.11.m11.1.1.1.1.3" mathcolor="#000099" stretchy="false" xref="S4.SS1.p6.11.11.m11.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.11.11.m11.1b"><apply id="S4.SS1.p6.11.11.m11.1.1.cmml" xref="S4.SS1.p6.11.11.m11.1.1"><times id="S4.SS1.p6.11.11.m11.1.1.2.cmml" xref="S4.SS1.p6.11.11.m11.1.1.2"></times><apply id="S4.SS1.p6.11.11.m11.1.1.3.cmml" xref="S4.SS1.p6.11.11.m11.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p6.11.11.m11.1.1.3.1.cmml" xref="S4.SS1.p6.11.11.m11.1.1.3">subscript</csymbol><ci id="S4.SS1.p6.11.11.m11.1.1.3.2.cmml" xref="S4.SS1.p6.11.11.m11.1.1.3.2">𝜎</ci><min id="S4.SS1.p6.11.11.m11.1.1.3.3.cmml" xref="S4.SS1.p6.11.11.m11.1.1.3.3"></min></apply><apply id="S4.SS1.p6.11.11.m11.1.1.1.1.1.cmml" xref="S4.SS1.p6.11.11.m11.1.1.1.1"><times id="S4.SS1.p6.11.11.m11.1.1.1.1.1.2.cmml" xref="S4.SS1.p6.11.11.m11.1.1.1.1.1.2"></times><apply id="S4.SS1.p6.11.11.m11.1.1.1.1.1.3.cmml" xref="S4.SS1.p6.11.11.m11.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p6.11.11.m11.1.1.1.1.1.3.1.cmml" xref="S4.SS1.p6.11.11.m11.1.1.1.1.1.3">subscript</csymbol><ci id="S4.SS1.p6.11.11.m11.1.1.1.1.1.3.2.cmml" xref="S4.SS1.p6.11.11.m11.1.1.1.1.1.3.2">Ψ</ci><ci id="S4.SS1.p6.11.11.m11.1.1.1.1.1.3.3.cmml" xref="S4.SS1.p6.11.11.m11.1.1.1.1.1.3.3">𝜁</ci></apply><apply id="S4.SS1.p6.11.11.m11.1.1.1.1.1.1.1.1.cmml" xref="S4.SS1.p6.11.11.m11.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p6.11.11.m11.1.1.1.1.1.1.1.1.1.cmml" xref="S4.SS1.p6.11.11.m11.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.SS1.p6.11.11.m11.1.1.1.1.1.1.1.1.2.cmml" xref="S4.SS1.p6.11.11.m11.1.1.1.1.1.1.1.1.2">𝑡</ci><ci id="S4.SS1.p6.11.11.m11.1.1.1.1.1.1.1.1.3.cmml" xref="S4.SS1.p6.11.11.m11.1.1.1.1.1.1.1.1.3">e</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.11.11.m11.1c">\sigma_{\min}(\Psi_{\zeta}(t_{\rm e}))</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.11.11.m11.1d">italic_σ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Ψ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) )</annotation></semantics></math> is monotonically non-decreasing at each partial IE stage.</span> Fig. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.F1" title="Figure 1 ‣ IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">1</span></a> illustrates the current maximal exciting strength <math alttext="\sigma_{\rm c}(t)" class="ltx_Math" display="inline" id="S4.SS1.p6.12.m1.1"><semantics id="S4.SS1.p6.12.m1.1a"><mrow id="S4.SS1.p6.12.m1.1.2" xref="S4.SS1.p6.12.m1.1.2.cmml"><msub id="S4.SS1.p6.12.m1.1.2.2" xref="S4.SS1.p6.12.m1.1.2.2.cmml"><mi id="S4.SS1.p6.12.m1.1.2.2.2" xref="S4.SS1.p6.12.m1.1.2.2.2.cmml">σ</mi><mi id="S4.SS1.p6.12.m1.1.2.2.3" mathvariant="normal" xref="S4.SS1.p6.12.m1.1.2.2.3.cmml">c</mi></msub><mo id="S4.SS1.p6.12.m1.1.2.1" xref="S4.SS1.p6.12.m1.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p6.12.m1.1.2.3.2" xref="S4.SS1.p6.12.m1.1.2.cmml"><mo id="S4.SS1.p6.12.m1.1.2.3.2.1" stretchy="false" xref="S4.SS1.p6.12.m1.1.2.cmml">(</mo><mi id="S4.SS1.p6.12.m1.1.1" xref="S4.SS1.p6.12.m1.1.1.cmml">t</mi><mo id="S4.SS1.p6.12.m1.1.2.3.2.2" stretchy="false" xref="S4.SS1.p6.12.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.12.m1.1b"><apply id="S4.SS1.p6.12.m1.1.2.cmml" xref="S4.SS1.p6.12.m1.1.2"><times id="S4.SS1.p6.12.m1.1.2.1.cmml" xref="S4.SS1.p6.12.m1.1.2.1"></times><apply id="S4.SS1.p6.12.m1.1.2.2.cmml" xref="S4.SS1.p6.12.m1.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.p6.12.m1.1.2.2.1.cmml" xref="S4.SS1.p6.12.m1.1.2.2">subscript</csymbol><ci id="S4.SS1.p6.12.m1.1.2.2.2.cmml" xref="S4.SS1.p6.12.m1.1.2.2.2">𝜎</ci><ci id="S4.SS1.p6.12.m1.1.2.2.3.cmml" xref="S4.SS1.p6.12.m1.1.2.2.3">c</ci></apply><ci id="S4.SS1.p6.12.m1.1.1.cmml" xref="S4.SS1.p6.12.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.12.m1.1c">\sigma_{\rm c}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.12.m1.1d">italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> in Algorithm 1 for a simple case with two partial IE stages. As new active channels <math alttext="\bm{\phi}_{{\rm s},k_{j}}(t)" class="ltx_Math" display="inline" id="S4.SS1.p6.13.m2.3"><semantics id="S4.SS1.p6.13.m2.3a"><mrow id="S4.SS1.p6.13.m2.3.4" xref="S4.SS1.p6.13.m2.3.4.cmml"><msub id="S4.SS1.p6.13.m2.3.4.2" xref="S4.SS1.p6.13.m2.3.4.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS1.p6.13.m2.3.4.2.2" mathvariant="bold-italic" xref="S4.SS1.p6.13.m2.3.4.2.2.cmml">ϕ</mi><mrow id="S4.SS1.p6.13.m2.2.2.2.2" xref="S4.SS1.p6.13.m2.2.2.2.3.cmml"><mi id="S4.SS1.p6.13.m2.1.1.1.1" mathvariant="normal" xref="S4.SS1.p6.13.m2.1.1.1.1.cmml">s</mi><mo id="S4.SS1.p6.13.m2.2.2.2.2.2" xref="S4.SS1.p6.13.m2.2.2.2.3.cmml">,</mo><msub id="S4.SS1.p6.13.m2.2.2.2.2.1" xref="S4.SS1.p6.13.m2.2.2.2.2.1.cmml"><mi id="S4.SS1.p6.13.m2.2.2.2.2.1.2" xref="S4.SS1.p6.13.m2.2.2.2.2.1.2.cmml">k</mi><mi id="S4.SS1.p6.13.m2.2.2.2.2.1.3" xref="S4.SS1.p6.13.m2.2.2.2.2.1.3.cmml">j</mi></msub></mrow></msub><mo id="S4.SS1.p6.13.m2.3.4.1" xref="S4.SS1.p6.13.m2.3.4.1.cmml">⁢</mo><mrow id="S4.SS1.p6.13.m2.3.4.3.2" xref="S4.SS1.p6.13.m2.3.4.cmml"><mo id="S4.SS1.p6.13.m2.3.4.3.2.1" stretchy="false" xref="S4.SS1.p6.13.m2.3.4.cmml">(</mo><mi id="S4.SS1.p6.13.m2.3.3" xref="S4.SS1.p6.13.m2.3.3.cmml">t</mi><mo id="S4.SS1.p6.13.m2.3.4.3.2.2" stretchy="false" xref="S4.SS1.p6.13.m2.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.13.m2.3b"><apply id="S4.SS1.p6.13.m2.3.4.cmml" xref="S4.SS1.p6.13.m2.3.4"><times id="S4.SS1.p6.13.m2.3.4.1.cmml" xref="S4.SS1.p6.13.m2.3.4.1"></times><apply id="S4.SS1.p6.13.m2.3.4.2.cmml" xref="S4.SS1.p6.13.m2.3.4.2"><csymbol cd="ambiguous" id="S4.SS1.p6.13.m2.3.4.2.1.cmml" xref="S4.SS1.p6.13.m2.3.4.2">subscript</csymbol><ci id="S4.SS1.p6.13.m2.3.4.2.2.cmml" xref="S4.SS1.p6.13.m2.3.4.2.2">bold-italic-ϕ</ci><list id="S4.SS1.p6.13.m2.2.2.2.3.cmml" xref="S4.SS1.p6.13.m2.2.2.2.2"><ci id="S4.SS1.p6.13.m2.1.1.1.1.cmml" xref="S4.SS1.p6.13.m2.1.1.1.1">s</ci><apply id="S4.SS1.p6.13.m2.2.2.2.2.1.cmml" xref="S4.SS1.p6.13.m2.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.SS1.p6.13.m2.2.2.2.2.1.1.cmml" xref="S4.SS1.p6.13.m2.2.2.2.2.1">subscript</csymbol><ci id="S4.SS1.p6.13.m2.2.2.2.2.1.2.cmml" xref="S4.SS1.p6.13.m2.2.2.2.2.1.2">𝑘</ci><ci id="S4.SS1.p6.13.m2.2.2.2.2.1.3.cmml" xref="S4.SS1.p6.13.m2.2.2.2.2.1.3">𝑗</ci></apply></list></apply><ci id="S4.SS1.p6.13.m2.3.3.cmml" xref="S4.SS1.p6.13.m2.3.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.13.m2.3c">\bm{\phi}_{{\rm s},k_{j}}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.13.m2.3d">bold_italic_ϕ start_POSTSUBSCRIPT roman_s , italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> exist at <math alttext="t=T_{\rm a}" class="ltx_Math" display="inline" id="S4.SS1.p6.14.m3.1"><semantics id="S4.SS1.p6.14.m3.1a"><mrow id="S4.SS1.p6.14.m3.1.1" xref="S4.SS1.p6.14.m3.1.1.cmml"><mi id="S4.SS1.p6.14.m3.1.1.2" xref="S4.SS1.p6.14.m3.1.1.2.cmml">t</mi><mo id="S4.SS1.p6.14.m3.1.1.1" xref="S4.SS1.p6.14.m3.1.1.1.cmml">=</mo><msub id="S4.SS1.p6.14.m3.1.1.3" xref="S4.SS1.p6.14.m3.1.1.3.cmml"><mi id="S4.SS1.p6.14.m3.1.1.3.2" xref="S4.SS1.p6.14.m3.1.1.3.2.cmml">T</mi><mi id="S4.SS1.p6.14.m3.1.1.3.3" mathvariant="normal" xref="S4.SS1.p6.14.m3.1.1.3.3.cmml">a</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.14.m3.1b"><apply id="S4.SS1.p6.14.m3.1.1.cmml" xref="S4.SS1.p6.14.m3.1.1"><eq id="S4.SS1.p6.14.m3.1.1.1.cmml" xref="S4.SS1.p6.14.m3.1.1.1"></eq><ci id="S4.SS1.p6.14.m3.1.1.2.cmml" xref="S4.SS1.p6.14.m3.1.1.2">𝑡</ci><apply id="S4.SS1.p6.14.m3.1.1.3.cmml" xref="S4.SS1.p6.14.m3.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p6.14.m3.1.1.3.1.cmml" xref="S4.SS1.p6.14.m3.1.1.3">subscript</csymbol><ci id="S4.SS1.p6.14.m3.1.1.3.2.cmml" xref="S4.SS1.p6.14.m3.1.1.3.2">𝑇</ci><ci id="S4.SS1.p6.14.m3.1.1.3.3.cmml" xref="S4.SS1.p6.14.m3.1.1.3.3">a</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.14.m3.1c">t=T_{\rm a}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.14.m3.1d">italic_t = italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT</annotation></semantics></math>, the sub-regressor <math alttext="\Phi_{{\rm s},\zeta}" class="ltx_Math" display="inline" id="S4.SS1.p6.15.m4.2"><semantics id="S4.SS1.p6.15.m4.2a"><msub id="S4.SS1.p6.15.m4.2.3" xref="S4.SS1.p6.15.m4.2.3.cmml"><mi id="S4.SS1.p6.15.m4.2.3.2" mathvariant="normal" xref="S4.SS1.p6.15.m4.2.3.2.cmml">Φ</mi><mrow id="S4.SS1.p6.15.m4.2.2.2.4" xref="S4.SS1.p6.15.m4.2.2.2.3.cmml"><mi id="S4.SS1.p6.15.m4.1.1.1.1" mathvariant="normal" xref="S4.SS1.p6.15.m4.1.1.1.1.cmml">s</mi><mo id="S4.SS1.p6.15.m4.2.2.2.4.1" xref="S4.SS1.p6.15.m4.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p6.15.m4.2.2.2.2" xref="S4.SS1.p6.15.m4.2.2.2.2.cmml">ζ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.15.m4.2b"><apply id="S4.SS1.p6.15.m4.2.3.cmml" xref="S4.SS1.p6.15.m4.2.3"><csymbol cd="ambiguous" id="S4.SS1.p6.15.m4.2.3.1.cmml" xref="S4.SS1.p6.15.m4.2.3">subscript</csymbol><ci id="S4.SS1.p6.15.m4.2.3.2.cmml" xref="S4.SS1.p6.15.m4.2.3.2">Φ</ci><list id="S4.SS1.p6.15.m4.2.2.2.3.cmml" xref="S4.SS1.p6.15.m4.2.2.2.4"><ci id="S4.SS1.p6.15.m4.1.1.1.1.cmml" xref="S4.SS1.p6.15.m4.1.1.1.1">s</ci><ci id="S4.SS1.p6.15.m4.2.2.2.2.cmml" xref="S4.SS1.p6.15.m4.2.2.2.2">𝜁</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.15.m4.2c">\Phi_{{\rm s},\zeta}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.15.m4.2d">roman_Φ start_POSTSUBSCRIPT roman_s , italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math> with the excitation matrix <math alttext="\Psi_{\zeta}(t)" class="ltx_Math" display="inline" id="S4.SS1.p6.16.m5.1"><semantics id="S4.SS1.p6.16.m5.1a"><mrow id="S4.SS1.p6.16.m5.1.2" xref="S4.SS1.p6.16.m5.1.2.cmml"><msub id="S4.SS1.p6.16.m5.1.2.2" xref="S4.SS1.p6.16.m5.1.2.2.cmml"><mi id="S4.SS1.p6.16.m5.1.2.2.2" mathvariant="normal" xref="S4.SS1.p6.16.m5.1.2.2.2.cmml">Ψ</mi><mi id="S4.SS1.p6.16.m5.1.2.2.3" xref="S4.SS1.p6.16.m5.1.2.2.3.cmml">ζ</mi></msub><mo id="S4.SS1.p6.16.m5.1.2.1" xref="S4.SS1.p6.16.m5.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p6.16.m5.1.2.3.2" xref="S4.SS1.p6.16.m5.1.2.cmml"><mo id="S4.SS1.p6.16.m5.1.2.3.2.1" stretchy="false" xref="S4.SS1.p6.16.m5.1.2.cmml">(</mo><mi id="S4.SS1.p6.16.m5.1.1" xref="S4.SS1.p6.16.m5.1.1.cmml">t</mi><mo id="S4.SS1.p6.16.m5.1.2.3.2.2" stretchy="false" xref="S4.SS1.p6.16.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.16.m5.1b"><apply id="S4.SS1.p6.16.m5.1.2.cmml" xref="S4.SS1.p6.16.m5.1.2"><times id="S4.SS1.p6.16.m5.1.2.1.cmml" xref="S4.SS1.p6.16.m5.1.2.1"></times><apply id="S4.SS1.p6.16.m5.1.2.2.cmml" xref="S4.SS1.p6.16.m5.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.p6.16.m5.1.2.2.1.cmml" xref="S4.SS1.p6.16.m5.1.2.2">subscript</csymbol><ci id="S4.SS1.p6.16.m5.1.2.2.2.cmml" xref="S4.SS1.p6.16.m5.1.2.2.2">Ψ</ci><ci id="S4.SS1.p6.16.m5.1.2.2.3.cmml" xref="S4.SS1.p6.16.m5.1.2.2.3">𝜁</ci></apply><ci id="S4.SS1.p6.16.m5.1.1.cmml" xref="S4.SS1.p6.16.m5.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.16.m5.1c">\Psi_{\zeta}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.16.m5.1d">roman_Ψ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E19" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">19</span></a>) is reconstructed by all new active channels [see Lines 4–8 in Algorithm 1]. In this partial IE stage, as the exciting strength <math alttext="\sigma_{\min}(\Psi_{\zeta}(t))" class="ltx_Math" display="inline" id="S4.SS1.p6.17.m6.2"><semantics id="S4.SS1.p6.17.m6.2a"><mrow id="S4.SS1.p6.17.m6.2.2" xref="S4.SS1.p6.17.m6.2.2.cmml"><msub id="S4.SS1.p6.17.m6.2.2.3" xref="S4.SS1.p6.17.m6.2.2.3.cmml"><mi id="S4.SS1.p6.17.m6.2.2.3.2" xref="S4.SS1.p6.17.m6.2.2.3.2.cmml">σ</mi><mi id="S4.SS1.p6.17.m6.2.2.3.3" xref="S4.SS1.p6.17.m6.2.2.3.3.cmml">min</mi></msub><mo id="S4.SS1.p6.17.m6.2.2.2" xref="S4.SS1.p6.17.m6.2.2.2.cmml">⁢</mo><mrow id="S4.SS1.p6.17.m6.2.2.1.1" xref="S4.SS1.p6.17.m6.2.2.1.1.1.cmml"><mo id="S4.SS1.p6.17.m6.2.2.1.1.2" stretchy="false" xref="S4.SS1.p6.17.m6.2.2.1.1.1.cmml">(</mo><mrow id="S4.SS1.p6.17.m6.2.2.1.1.1" xref="S4.SS1.p6.17.m6.2.2.1.1.1.cmml"><msub id="S4.SS1.p6.17.m6.2.2.1.1.1.2" xref="S4.SS1.p6.17.m6.2.2.1.1.1.2.cmml"><mi id="S4.SS1.p6.17.m6.2.2.1.1.1.2.2" mathvariant="normal" xref="S4.SS1.p6.17.m6.2.2.1.1.1.2.2.cmml">Ψ</mi><mi id="S4.SS1.p6.17.m6.2.2.1.1.1.2.3" xref="S4.SS1.p6.17.m6.2.2.1.1.1.2.3.cmml">ζ</mi></msub><mo id="S4.SS1.p6.17.m6.2.2.1.1.1.1" xref="S4.SS1.p6.17.m6.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS1.p6.17.m6.2.2.1.1.1.3.2" xref="S4.SS1.p6.17.m6.2.2.1.1.1.cmml"><mo id="S4.SS1.p6.17.m6.2.2.1.1.1.3.2.1" stretchy="false" xref="S4.SS1.p6.17.m6.2.2.1.1.1.cmml">(</mo><mi id="S4.SS1.p6.17.m6.1.1" xref="S4.SS1.p6.17.m6.1.1.cmml">t</mi><mo id="S4.SS1.p6.17.m6.2.2.1.1.1.3.2.2" stretchy="false" xref="S4.SS1.p6.17.m6.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p6.17.m6.2.2.1.1.3" stretchy="false" xref="S4.SS1.p6.17.m6.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.17.m6.2b"><apply id="S4.SS1.p6.17.m6.2.2.cmml" xref="S4.SS1.p6.17.m6.2.2"><times id="S4.SS1.p6.17.m6.2.2.2.cmml" xref="S4.SS1.p6.17.m6.2.2.2"></times><apply id="S4.SS1.p6.17.m6.2.2.3.cmml" xref="S4.SS1.p6.17.m6.2.2.3"><csymbol cd="ambiguous" id="S4.SS1.p6.17.m6.2.2.3.1.cmml" xref="S4.SS1.p6.17.m6.2.2.3">subscript</csymbol><ci id="S4.SS1.p6.17.m6.2.2.3.2.cmml" xref="S4.SS1.p6.17.m6.2.2.3.2">𝜎</ci><min id="S4.SS1.p6.17.m6.2.2.3.3.cmml" xref="S4.SS1.p6.17.m6.2.2.3.3"></min></apply><apply id="S4.SS1.p6.17.m6.2.2.1.1.1.cmml" xref="S4.SS1.p6.17.m6.2.2.1.1"><times id="S4.SS1.p6.17.m6.2.2.1.1.1.1.cmml" xref="S4.SS1.p6.17.m6.2.2.1.1.1.1"></times><apply id="S4.SS1.p6.17.m6.2.2.1.1.1.2.cmml" xref="S4.SS1.p6.17.m6.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p6.17.m6.2.2.1.1.1.2.1.cmml" xref="S4.SS1.p6.17.m6.2.2.1.1.1.2">subscript</csymbol><ci id="S4.SS1.p6.17.m6.2.2.1.1.1.2.2.cmml" xref="S4.SS1.p6.17.m6.2.2.1.1.1.2.2">Ψ</ci><ci id="S4.SS1.p6.17.m6.2.2.1.1.1.2.3.cmml" xref="S4.SS1.p6.17.m6.2.2.1.1.1.2.3">𝜁</ci></apply><ci id="S4.SS1.p6.17.m6.1.1.cmml" xref="S4.SS1.p6.17.m6.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.17.m6.2c">\sigma_{\min}(\Psi_{\zeta}(t))</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.17.m6.2d">italic_σ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Ψ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t ) )</annotation></semantics></math> can be time-varying [see the green dash line in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.F1" title="Figure 1 ‣ IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">1</span></a>], the update of the exciting time <math alttext="t_{\rm e}" class="ltx_Math" display="inline" id="S4.SS1.p6.18.m7.1"><semantics id="S4.SS1.p6.18.m7.1a"><msub id="S4.SS1.p6.18.m7.1.1" xref="S4.SS1.p6.18.m7.1.1.cmml"><mi id="S4.SS1.p6.18.m7.1.1.2" xref="S4.SS1.p6.18.m7.1.1.2.cmml">t</mi><mi id="S4.SS1.p6.18.m7.1.1.3" mathvariant="normal" xref="S4.SS1.p6.18.m7.1.1.3.cmml">e</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.18.m7.1b"><apply id="S4.SS1.p6.18.m7.1.1.cmml" xref="S4.SS1.p6.18.m7.1.1"><csymbol cd="ambiguous" id="S4.SS1.p6.18.m7.1.1.1.cmml" xref="S4.SS1.p6.18.m7.1.1">subscript</csymbol><ci id="S4.SS1.p6.18.m7.1.1.2.cmml" xref="S4.SS1.p6.18.m7.1.1.2">𝑡</ci><ci id="S4.SS1.p6.18.m7.1.1.3.cmml" xref="S4.SS1.p6.18.m7.1.1.3">e</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.18.m7.1c">t_{\rm e}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.18.m7.1d">italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT</annotation></semantics></math> is based on exciting strength maximization, i.e., <math alttext="\max_{\tau\in[T_{\rm a},t]}\sigma_{\min}(\Psi_{\zeta}(\tau))" class="ltx_Math" display="inline" id="S4.SS1.p6.19.m8.4"><semantics id="S4.SS1.p6.19.m8.4a"><mrow id="S4.SS1.p6.19.m8.4.4" xref="S4.SS1.p6.19.m8.4.4.cmml"><mrow id="S4.SS1.p6.19.m8.4.4.3" xref="S4.SS1.p6.19.m8.4.4.3.cmml"><msub id="S4.SS1.p6.19.m8.4.4.3.1" xref="S4.SS1.p6.19.m8.4.4.3.1.cmml"><mi id="S4.SS1.p6.19.m8.4.4.3.1.2" xref="S4.SS1.p6.19.m8.4.4.3.1.2.cmml">max</mi><mrow id="S4.SS1.p6.19.m8.2.2.2" xref="S4.SS1.p6.19.m8.2.2.2.cmml"><mi id="S4.SS1.p6.19.m8.2.2.2.4" xref="S4.SS1.p6.19.m8.2.2.2.4.cmml">τ</mi><mo id="S4.SS1.p6.19.m8.2.2.2.3" xref="S4.SS1.p6.19.m8.2.2.2.3.cmml">∈</mo><mrow id="S4.SS1.p6.19.m8.2.2.2.2.1" xref="S4.SS1.p6.19.m8.2.2.2.2.2.cmml"><mo id="S4.SS1.p6.19.m8.2.2.2.2.1.2" stretchy="false" xref="S4.SS1.p6.19.m8.2.2.2.2.2.cmml">[</mo><msub id="S4.SS1.p6.19.m8.2.2.2.2.1.1" xref="S4.SS1.p6.19.m8.2.2.2.2.1.1.cmml"><mi id="S4.SS1.p6.19.m8.2.2.2.2.1.1.2" xref="S4.SS1.p6.19.m8.2.2.2.2.1.1.2.cmml">T</mi><mi id="S4.SS1.p6.19.m8.2.2.2.2.1.1.3" mathvariant="normal" xref="S4.SS1.p6.19.m8.2.2.2.2.1.1.3.cmml">a</mi></msub><mo id="S4.SS1.p6.19.m8.2.2.2.2.1.3" xref="S4.SS1.p6.19.m8.2.2.2.2.2.cmml">,</mo><mi id="S4.SS1.p6.19.m8.1.1.1.1" xref="S4.SS1.p6.19.m8.1.1.1.1.cmml">t</mi><mo id="S4.SS1.p6.19.m8.2.2.2.2.1.4" stretchy="false" xref="S4.SS1.p6.19.m8.2.2.2.2.2.cmml">]</mo></mrow></mrow></msub><mo id="S4.SS1.p6.19.m8.4.4.3a" lspace="0.167em" xref="S4.SS1.p6.19.m8.4.4.3.cmml">⁡</mo><msub id="S4.SS1.p6.19.m8.4.4.3.2" xref="S4.SS1.p6.19.m8.4.4.3.2.cmml"><mi id="S4.SS1.p6.19.m8.4.4.3.2.2" xref="S4.SS1.p6.19.m8.4.4.3.2.2.cmml">σ</mi><mi id="S4.SS1.p6.19.m8.4.4.3.2.3" xref="S4.SS1.p6.19.m8.4.4.3.2.3.cmml">min</mi></msub></mrow><mo id="S4.SS1.p6.19.m8.4.4.2" xref="S4.SS1.p6.19.m8.4.4.2.cmml">⁢</mo><mrow id="S4.SS1.p6.19.m8.4.4.1.1" xref="S4.SS1.p6.19.m8.4.4.1.1.1.cmml"><mo id="S4.SS1.p6.19.m8.4.4.1.1.2" stretchy="false" xref="S4.SS1.p6.19.m8.4.4.1.1.1.cmml">(</mo><mrow id="S4.SS1.p6.19.m8.4.4.1.1.1" xref="S4.SS1.p6.19.m8.4.4.1.1.1.cmml"><msub id="S4.SS1.p6.19.m8.4.4.1.1.1.2" xref="S4.SS1.p6.19.m8.4.4.1.1.1.2.cmml"><mi id="S4.SS1.p6.19.m8.4.4.1.1.1.2.2" mathvariant="normal" xref="S4.SS1.p6.19.m8.4.4.1.1.1.2.2.cmml">Ψ</mi><mi id="S4.SS1.p6.19.m8.4.4.1.1.1.2.3" xref="S4.SS1.p6.19.m8.4.4.1.1.1.2.3.cmml">ζ</mi></msub><mo id="S4.SS1.p6.19.m8.4.4.1.1.1.1" xref="S4.SS1.p6.19.m8.4.4.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS1.p6.19.m8.4.4.1.1.1.3.2" xref="S4.SS1.p6.19.m8.4.4.1.1.1.cmml"><mo id="S4.SS1.p6.19.m8.4.4.1.1.1.3.2.1" stretchy="false" xref="S4.SS1.p6.19.m8.4.4.1.1.1.cmml">(</mo><mi id="S4.SS1.p6.19.m8.3.3" xref="S4.SS1.p6.19.m8.3.3.cmml">τ</mi><mo id="S4.SS1.p6.19.m8.4.4.1.1.1.3.2.2" stretchy="false" xref="S4.SS1.p6.19.m8.4.4.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p6.19.m8.4.4.1.1.3" stretchy="false" xref="S4.SS1.p6.19.m8.4.4.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.19.m8.4b"><apply id="S4.SS1.p6.19.m8.4.4.cmml" xref="S4.SS1.p6.19.m8.4.4"><times id="S4.SS1.p6.19.m8.4.4.2.cmml" xref="S4.SS1.p6.19.m8.4.4.2"></times><apply id="S4.SS1.p6.19.m8.4.4.3.cmml" xref="S4.SS1.p6.19.m8.4.4.3"><apply id="S4.SS1.p6.19.m8.4.4.3.1.cmml" xref="S4.SS1.p6.19.m8.4.4.3.1"><csymbol cd="ambiguous" id="S4.SS1.p6.19.m8.4.4.3.1.1.cmml" xref="S4.SS1.p6.19.m8.4.4.3.1">subscript</csymbol><max id="S4.SS1.p6.19.m8.4.4.3.1.2.cmml" xref="S4.SS1.p6.19.m8.4.4.3.1.2"></max><apply id="S4.SS1.p6.19.m8.2.2.2.cmml" xref="S4.SS1.p6.19.m8.2.2.2"><in id="S4.SS1.p6.19.m8.2.2.2.3.cmml" xref="S4.SS1.p6.19.m8.2.2.2.3"></in><ci id="S4.SS1.p6.19.m8.2.2.2.4.cmml" xref="S4.SS1.p6.19.m8.2.2.2.4">𝜏</ci><interval closure="closed" id="S4.SS1.p6.19.m8.2.2.2.2.2.cmml" xref="S4.SS1.p6.19.m8.2.2.2.2.1"><apply id="S4.SS1.p6.19.m8.2.2.2.2.1.1.cmml" xref="S4.SS1.p6.19.m8.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p6.19.m8.2.2.2.2.1.1.1.cmml" xref="S4.SS1.p6.19.m8.2.2.2.2.1.1">subscript</csymbol><ci id="S4.SS1.p6.19.m8.2.2.2.2.1.1.2.cmml" xref="S4.SS1.p6.19.m8.2.2.2.2.1.1.2">𝑇</ci><ci id="S4.SS1.p6.19.m8.2.2.2.2.1.1.3.cmml" xref="S4.SS1.p6.19.m8.2.2.2.2.1.1.3">a</ci></apply><ci id="S4.SS1.p6.19.m8.1.1.1.1.cmml" xref="S4.SS1.p6.19.m8.1.1.1.1">𝑡</ci></interval></apply></apply><apply id="S4.SS1.p6.19.m8.4.4.3.2.cmml" xref="S4.SS1.p6.19.m8.4.4.3.2"><csymbol cd="ambiguous" id="S4.SS1.p6.19.m8.4.4.3.2.1.cmml" xref="S4.SS1.p6.19.m8.4.4.3.2">subscript</csymbol><ci id="S4.SS1.p6.19.m8.4.4.3.2.2.cmml" xref="S4.SS1.p6.19.m8.4.4.3.2.2">𝜎</ci><min id="S4.SS1.p6.19.m8.4.4.3.2.3.cmml" xref="S4.SS1.p6.19.m8.4.4.3.2.3"></min></apply></apply><apply id="S4.SS1.p6.19.m8.4.4.1.1.1.cmml" xref="S4.SS1.p6.19.m8.4.4.1.1"><times id="S4.SS1.p6.19.m8.4.4.1.1.1.1.cmml" xref="S4.SS1.p6.19.m8.4.4.1.1.1.1"></times><apply id="S4.SS1.p6.19.m8.4.4.1.1.1.2.cmml" xref="S4.SS1.p6.19.m8.4.4.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p6.19.m8.4.4.1.1.1.2.1.cmml" xref="S4.SS1.p6.19.m8.4.4.1.1.1.2">subscript</csymbol><ci id="S4.SS1.p6.19.m8.4.4.1.1.1.2.2.cmml" xref="S4.SS1.p6.19.m8.4.4.1.1.1.2.2">Ψ</ci><ci id="S4.SS1.p6.19.m8.4.4.1.1.1.2.3.cmml" xref="S4.SS1.p6.19.m8.4.4.1.1.1.2.3">𝜁</ci></apply><ci id="S4.SS1.p6.19.m8.3.3.cmml" xref="S4.SS1.p6.19.m8.3.3">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.19.m8.4c">\max_{\tau\in[T_{\rm a},t]}\sigma_{\min}(\Psi_{\zeta}(\tau))</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.19.m8.4d">roman_max start_POSTSUBSCRIPT italic_τ ∈ [ italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT , italic_t ] end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Ψ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_τ ) )</annotation></semantics></math> [see Lines 9–11 in Algorithm 1 and the black solid line in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.F1" title="Figure 1 ‣ IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">1</span></a>]. This is the same for the IE stage [see Lines 13–15 in Algorithm 1 and the red area in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.F1" title="Figure 1 ‣ IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">1</span></a>].</p> </div> <figure class="ltx_float ltx_float_algorithm ltx_framed ltx_framed_top" id="alg1"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg1.2.1.1">Algorithm 1</span> </span> Staged exciting strength maximization</figcaption> <div class="ltx_listing ltx_listing" id="alg1.3"> <div class="ltx_listingline" id="alg1.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l1.1.1.1" style="font-size:80%;">1:</span></span><span class="ltx_text ltx_font_bold" id="alg1.l1.2">Initialize</span>: <math alttext="\mathcal{I}^{\prime}\leftarrow\emptyset" class="ltx_Math" display="inline" id="alg1.l1.m1.1"><semantics id="alg1.l1.m1.1a"><mrow id="alg1.l1.m1.1.1" xref="alg1.l1.m1.1.1.cmml"><msup id="alg1.l1.m1.1.1.2" xref="alg1.l1.m1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="alg1.l1.m1.1.1.2.2" xref="alg1.l1.m1.1.1.2.2.cmml">ℐ</mi><mo id="alg1.l1.m1.1.1.2.3" xref="alg1.l1.m1.1.1.2.3.cmml">′</mo></msup><mo id="alg1.l1.m1.1.1.1" stretchy="false" xref="alg1.l1.m1.1.1.1.cmml">←</mo><mi id="alg1.l1.m1.1.1.3" mathvariant="normal" xref="alg1.l1.m1.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l1.m1.1b"><apply id="alg1.l1.m1.1.1.cmml" xref="alg1.l1.m1.1.1"><ci id="alg1.l1.m1.1.1.1.cmml" xref="alg1.l1.m1.1.1.1">←</ci><apply id="alg1.l1.m1.1.1.2.cmml" xref="alg1.l1.m1.1.1.2"><csymbol cd="ambiguous" id="alg1.l1.m1.1.1.2.1.cmml" xref="alg1.l1.m1.1.1.2">superscript</csymbol><ci id="alg1.l1.m1.1.1.2.2.cmml" xref="alg1.l1.m1.1.1.2.2">ℐ</ci><ci id="alg1.l1.m1.1.1.2.3.cmml" xref="alg1.l1.m1.1.1.2.3">′</ci></apply><emptyset id="alg1.l1.m1.1.1.3.cmml" xref="alg1.l1.m1.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m1.1c">\mathcal{I}^{\prime}\leftarrow\emptyset</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m1.1d">caligraphic_I start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ← ∅</annotation></semantics></math>, <math alttext="T_{\rm a}=0" class="ltx_Math" display="inline" id="alg1.l1.m2.1"><semantics id="alg1.l1.m2.1a"><mrow id="alg1.l1.m2.1.1" xref="alg1.l1.m2.1.1.cmml"><msub id="alg1.l1.m2.1.1.2" xref="alg1.l1.m2.1.1.2.cmml"><mi id="alg1.l1.m2.1.1.2.2" xref="alg1.l1.m2.1.1.2.2.cmml">T</mi><mi id="alg1.l1.m2.1.1.2.3" mathvariant="normal" xref="alg1.l1.m2.1.1.2.3.cmml">a</mi></msub><mo id="alg1.l1.m2.1.1.1" xref="alg1.l1.m2.1.1.1.cmml">=</mo><mn id="alg1.l1.m2.1.1.3" xref="alg1.l1.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg1.l1.m2.1b"><apply id="alg1.l1.m2.1.1.cmml" xref="alg1.l1.m2.1.1"><eq id="alg1.l1.m2.1.1.1.cmml" xref="alg1.l1.m2.1.1.1"></eq><apply id="alg1.l1.m2.1.1.2.cmml" xref="alg1.l1.m2.1.1.2"><csymbol cd="ambiguous" id="alg1.l1.m2.1.1.2.1.cmml" xref="alg1.l1.m2.1.1.2">subscript</csymbol><ci id="alg1.l1.m2.1.1.2.2.cmml" xref="alg1.l1.m2.1.1.2.2">𝑇</ci><ci id="alg1.l1.m2.1.1.2.3.cmml" xref="alg1.l1.m2.1.1.2.3">a</ci></apply><cn id="alg1.l1.m2.1.1.3.cmml" type="integer" xref="alg1.l1.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m2.1c">T_{\rm a}=0</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m2.1d">italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT = 0</annotation></semantics></math>, <math alttext="\sigma_{\rm c}=\sigma" class="ltx_Math" display="inline" id="alg1.l1.m3.1"><semantics id="alg1.l1.m3.1a"><mrow id="alg1.l1.m3.1.1" xref="alg1.l1.m3.1.1.cmml"><msub id="alg1.l1.m3.1.1.2" xref="alg1.l1.m3.1.1.2.cmml"><mi id="alg1.l1.m3.1.1.2.2" xref="alg1.l1.m3.1.1.2.2.cmml">σ</mi><mi id="alg1.l1.m3.1.1.2.3" mathvariant="normal" xref="alg1.l1.m3.1.1.2.3.cmml">c</mi></msub><mo id="alg1.l1.m3.1.1.1" xref="alg1.l1.m3.1.1.1.cmml">=</mo><mi id="alg1.l1.m3.1.1.3" xref="alg1.l1.m3.1.1.3.cmml">σ</mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l1.m3.1b"><apply id="alg1.l1.m3.1.1.cmml" xref="alg1.l1.m3.1.1"><eq id="alg1.l1.m3.1.1.1.cmml" xref="alg1.l1.m3.1.1.1"></eq><apply id="alg1.l1.m3.1.1.2.cmml" xref="alg1.l1.m3.1.1.2"><csymbol cd="ambiguous" id="alg1.l1.m3.1.1.2.1.cmml" xref="alg1.l1.m3.1.1.2">subscript</csymbol><ci id="alg1.l1.m3.1.1.2.2.cmml" xref="alg1.l1.m3.1.1.2.2">𝜎</ci><ci id="alg1.l1.m3.1.1.2.3.cmml" xref="alg1.l1.m3.1.1.2.3">c</ci></apply><ci id="alg1.l1.m3.1.1.3.cmml" xref="alg1.l1.m3.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m3.1c">\sigma_{\rm c}=\sigma</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m3.1d">italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT = italic_σ</annotation></semantics></math>, <math alttext="t_{\rm e}=0" class="ltx_Math" display="inline" id="alg1.l1.m4.1"><semantics id="alg1.l1.m4.1a"><mrow id="alg1.l1.m4.1.1" xref="alg1.l1.m4.1.1.cmml"><msub id="alg1.l1.m4.1.1.2" xref="alg1.l1.m4.1.1.2.cmml"><mi id="alg1.l1.m4.1.1.2.2" xref="alg1.l1.m4.1.1.2.2.cmml">t</mi><mi id="alg1.l1.m4.1.1.2.3" mathvariant="normal" xref="alg1.l1.m4.1.1.2.3.cmml">e</mi></msub><mo id="alg1.l1.m4.1.1.1" xref="alg1.l1.m4.1.1.1.cmml">=</mo><mn id="alg1.l1.m4.1.1.3" xref="alg1.l1.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg1.l1.m4.1b"><apply id="alg1.l1.m4.1.1.cmml" xref="alg1.l1.m4.1.1"><eq id="alg1.l1.m4.1.1.1.cmml" xref="alg1.l1.m4.1.1.1"></eq><apply id="alg1.l1.m4.1.1.2.cmml" xref="alg1.l1.m4.1.1.2"><csymbol cd="ambiguous" id="alg1.l1.m4.1.1.2.1.cmml" xref="alg1.l1.m4.1.1.2">subscript</csymbol><ci id="alg1.l1.m4.1.1.2.2.cmml" xref="alg1.l1.m4.1.1.2.2">𝑡</ci><ci id="alg1.l1.m4.1.1.2.3.cmml" xref="alg1.l1.m4.1.1.2.3">e</ci></apply><cn id="alg1.l1.m4.1.1.3.cmml" type="integer" xref="alg1.l1.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m4.1c">t_{\rm e}=0</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m4.1d">italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT = 0</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l2"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l2.1.1.1" style="font-size:80%;">2:</span></span><span class="ltx_text ltx_font_bold" id="alg1.l2.2">for</span> <math alttext="t=0" class="ltx_Math" display="inline" id="alg1.l2.m1.1"><semantics id="alg1.l2.m1.1a"><mrow id="alg1.l2.m1.1.1" xref="alg1.l2.m1.1.1.cmml"><mi id="alg1.l2.m1.1.1.2" xref="alg1.l2.m1.1.1.2.cmml">t</mi><mo id="alg1.l2.m1.1.1.1" xref="alg1.l2.m1.1.1.1.cmml">=</mo><mn id="alg1.l2.m1.1.1.3" xref="alg1.l2.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg1.l2.m1.1b"><apply id="alg1.l2.m1.1.1.cmml" xref="alg1.l2.m1.1.1"><eq id="alg1.l2.m1.1.1.1.cmml" xref="alg1.l2.m1.1.1.1"></eq><ci id="alg1.l2.m1.1.1.2.cmml" xref="alg1.l2.m1.1.1.2">𝑡</ci><cn id="alg1.l2.m1.1.1.3.cmml" type="integer" xref="alg1.l2.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l2.m1.1c">t=0</annotation><annotation encoding="application/x-llamapun" id="alg1.l2.m1.1d">italic_t = 0</annotation></semantics></math> with a step size of <math alttext="T_{\rm s}" class="ltx_Math" display="inline" id="alg1.l2.m2.1"><semantics id="alg1.l2.m2.1a"><msub id="alg1.l2.m2.1.1" xref="alg1.l2.m2.1.1.cmml"><mi id="alg1.l2.m2.1.1.2" xref="alg1.l2.m2.1.1.2.cmml">T</mi><mi id="alg1.l2.m2.1.1.3" mathvariant="normal" xref="alg1.l2.m2.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="alg1.l2.m2.1b"><apply id="alg1.l2.m2.1.1.cmml" xref="alg1.l2.m2.1.1"><csymbol cd="ambiguous" id="alg1.l2.m2.1.1.1.cmml" xref="alg1.l2.m2.1.1">subscript</csymbol><ci id="alg1.l2.m2.1.1.2.cmml" xref="alg1.l2.m2.1.1.2">𝑇</ci><ci id="alg1.l2.m2.1.1.3.cmml" xref="alg1.l2.m2.1.1.3">s</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l2.m2.1c">T_{\rm s}</annotation><annotation encoding="application/x-llamapun" id="alg1.l2.m2.1d">italic_T start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg1.l2.3">do</span> </div> <div class="ltx_listingline" id="alg1.l3"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l3.1.1.1" style="font-size:80%;">3:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l3.2">if</span> length<math alttext="(\mathcal{I}^{\prime})<N" class="ltx_Math" display="inline" id="alg1.l3.m1.1"><semantics id="alg1.l3.m1.1a"><mrow id="alg1.l3.m1.1.1" xref="alg1.l3.m1.1.1.cmml"><mrow id="alg1.l3.m1.1.1.1.1" xref="alg1.l3.m1.1.1.1.1.1.cmml"><mo id="alg1.l3.m1.1.1.1.1.2" stretchy="false" xref="alg1.l3.m1.1.1.1.1.1.cmml">(</mo><msup id="alg1.l3.m1.1.1.1.1.1" xref="alg1.l3.m1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="alg1.l3.m1.1.1.1.1.1.2" xref="alg1.l3.m1.1.1.1.1.1.2.cmml">ℐ</mi><mo id="alg1.l3.m1.1.1.1.1.1.3" xref="alg1.l3.m1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="alg1.l3.m1.1.1.1.1.3" stretchy="false" xref="alg1.l3.m1.1.1.1.1.1.cmml">)</mo></mrow><mo id="alg1.l3.m1.1.1.2" xref="alg1.l3.m1.1.1.2.cmml"><</mo><mi id="alg1.l3.m1.1.1.3" xref="alg1.l3.m1.1.1.3.cmml">N</mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l3.m1.1b"><apply id="alg1.l3.m1.1.1.cmml" xref="alg1.l3.m1.1.1"><lt id="alg1.l3.m1.1.1.2.cmml" xref="alg1.l3.m1.1.1.2"></lt><apply id="alg1.l3.m1.1.1.1.1.1.cmml" xref="alg1.l3.m1.1.1.1.1"><csymbol cd="ambiguous" id="alg1.l3.m1.1.1.1.1.1.1.cmml" xref="alg1.l3.m1.1.1.1.1">superscript</csymbol><ci id="alg1.l3.m1.1.1.1.1.1.2.cmml" xref="alg1.l3.m1.1.1.1.1.1.2">ℐ</ci><ci id="alg1.l3.m1.1.1.1.1.1.3.cmml" xref="alg1.l3.m1.1.1.1.1.1.3">′</ci></apply><ci id="alg1.l3.m1.1.1.3.cmml" xref="alg1.l3.m1.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l3.m1.1c">(\mathcal{I}^{\prime})<N</annotation><annotation encoding="application/x-llamapun" id="alg1.l3.m1.1d">( caligraphic_I start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) < italic_N</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg1.l3.3">then</span> </div> <div class="ltx_listingline" id="alg1.l4"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l4.1.1.1" style="font-size:80%;">4:</span></span> Find the indexes <math alttext="k_{j}" class="ltx_Math" display="inline" id="alg1.l4.m1.1"><semantics id="alg1.l4.m1.1a"><msub id="alg1.l4.m1.1.1" xref="alg1.l4.m1.1.1.cmml"><mi id="alg1.l4.m1.1.1.2" xref="alg1.l4.m1.1.1.2.cmml">k</mi><mi id="alg1.l4.m1.1.1.3" xref="alg1.l4.m1.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="alg1.l4.m1.1b"><apply id="alg1.l4.m1.1.1.cmml" xref="alg1.l4.m1.1.1"><csymbol cd="ambiguous" id="alg1.l4.m1.1.1.1.cmml" xref="alg1.l4.m1.1.1">subscript</csymbol><ci id="alg1.l4.m1.1.1.2.cmml" xref="alg1.l4.m1.1.1.2">𝑘</ci><ci id="alg1.l4.m1.1.1.3.cmml" xref="alg1.l4.m1.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l4.m1.1c">k_{j}</annotation><annotation encoding="application/x-llamapun" id="alg1.l4.m1.1d">italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> satisfying <math alttext="\Psi_{k_{j},k_{j}}(t)>0" class="ltx_Math" display="inline" id="alg1.l4.m2.3"><semantics id="alg1.l4.m2.3a"><mrow id="alg1.l4.m2.3.4" xref="alg1.l4.m2.3.4.cmml"><mrow id="alg1.l4.m2.3.4.2" xref="alg1.l4.m2.3.4.2.cmml"><msub id="alg1.l4.m2.3.4.2.2" xref="alg1.l4.m2.3.4.2.2.cmml"><mi id="alg1.l4.m2.3.4.2.2.2" mathvariant="normal" xref="alg1.l4.m2.3.4.2.2.2.cmml">Ψ</mi><mrow id="alg1.l4.m2.2.2.2.2" xref="alg1.l4.m2.2.2.2.3.cmml"><msub id="alg1.l4.m2.1.1.1.1.1" xref="alg1.l4.m2.1.1.1.1.1.cmml"><mi id="alg1.l4.m2.1.1.1.1.1.2" xref="alg1.l4.m2.1.1.1.1.1.2.cmml">k</mi><mi id="alg1.l4.m2.1.1.1.1.1.3" xref="alg1.l4.m2.1.1.1.1.1.3.cmml">j</mi></msub><mo id="alg1.l4.m2.2.2.2.2.3" xref="alg1.l4.m2.2.2.2.3.cmml">,</mo><msub id="alg1.l4.m2.2.2.2.2.2" xref="alg1.l4.m2.2.2.2.2.2.cmml"><mi id="alg1.l4.m2.2.2.2.2.2.2" xref="alg1.l4.m2.2.2.2.2.2.2.cmml">k</mi><mi id="alg1.l4.m2.2.2.2.2.2.3" xref="alg1.l4.m2.2.2.2.2.2.3.cmml">j</mi></msub></mrow></msub><mo id="alg1.l4.m2.3.4.2.1" xref="alg1.l4.m2.3.4.2.1.cmml">⁢</mo><mrow id="alg1.l4.m2.3.4.2.3.2" xref="alg1.l4.m2.3.4.2.cmml"><mo id="alg1.l4.m2.3.4.2.3.2.1" stretchy="false" xref="alg1.l4.m2.3.4.2.cmml">(</mo><mi id="alg1.l4.m2.3.3" xref="alg1.l4.m2.3.3.cmml">t</mi><mo id="alg1.l4.m2.3.4.2.3.2.2" stretchy="false" xref="alg1.l4.m2.3.4.2.cmml">)</mo></mrow></mrow><mo id="alg1.l4.m2.3.4.1" xref="alg1.l4.m2.3.4.1.cmml">></mo><mn id="alg1.l4.m2.3.4.3" xref="alg1.l4.m2.3.4.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg1.l4.m2.3b"><apply id="alg1.l4.m2.3.4.cmml" xref="alg1.l4.m2.3.4"><gt id="alg1.l4.m2.3.4.1.cmml" xref="alg1.l4.m2.3.4.1"></gt><apply id="alg1.l4.m2.3.4.2.cmml" xref="alg1.l4.m2.3.4.2"><times id="alg1.l4.m2.3.4.2.1.cmml" xref="alg1.l4.m2.3.4.2.1"></times><apply id="alg1.l4.m2.3.4.2.2.cmml" xref="alg1.l4.m2.3.4.2.2"><csymbol cd="ambiguous" id="alg1.l4.m2.3.4.2.2.1.cmml" xref="alg1.l4.m2.3.4.2.2">subscript</csymbol><ci id="alg1.l4.m2.3.4.2.2.2.cmml" xref="alg1.l4.m2.3.4.2.2.2">Ψ</ci><list id="alg1.l4.m2.2.2.2.3.cmml" xref="alg1.l4.m2.2.2.2.2"><apply id="alg1.l4.m2.1.1.1.1.1.cmml" xref="alg1.l4.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="alg1.l4.m2.1.1.1.1.1.1.cmml" xref="alg1.l4.m2.1.1.1.1.1">subscript</csymbol><ci id="alg1.l4.m2.1.1.1.1.1.2.cmml" xref="alg1.l4.m2.1.1.1.1.1.2">𝑘</ci><ci id="alg1.l4.m2.1.1.1.1.1.3.cmml" xref="alg1.l4.m2.1.1.1.1.1.3">𝑗</ci></apply><apply id="alg1.l4.m2.2.2.2.2.2.cmml" xref="alg1.l4.m2.2.2.2.2.2"><csymbol cd="ambiguous" id="alg1.l4.m2.2.2.2.2.2.1.cmml" xref="alg1.l4.m2.2.2.2.2.2">subscript</csymbol><ci id="alg1.l4.m2.2.2.2.2.2.2.cmml" xref="alg1.l4.m2.2.2.2.2.2.2">𝑘</ci><ci id="alg1.l4.m2.2.2.2.2.2.3.cmml" xref="alg1.l4.m2.2.2.2.2.2.3">𝑗</ci></apply></list></apply><ci id="alg1.l4.m2.3.3.cmml" xref="alg1.l4.m2.3.3">𝑡</ci></apply><cn id="alg1.l4.m2.3.4.3.cmml" type="integer" xref="alg1.l4.m2.3.4.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l4.m2.3c">\Psi_{k_{j},k_{j}}(t)>0</annotation><annotation encoding="application/x-llamapun" id="alg1.l4.m2.3d">roman_Ψ start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_t ) > 0</annotation></semantics></math>, <math alttext="k_{j}\in\{1,2,\cdots,N\}" class="ltx_Math" display="inline" id="alg1.l4.m3.4"><semantics id="alg1.l4.m3.4a"><mrow id="alg1.l4.m3.4.5" xref="alg1.l4.m3.4.5.cmml"><msub id="alg1.l4.m3.4.5.2" xref="alg1.l4.m3.4.5.2.cmml"><mi id="alg1.l4.m3.4.5.2.2" xref="alg1.l4.m3.4.5.2.2.cmml">k</mi><mi id="alg1.l4.m3.4.5.2.3" xref="alg1.l4.m3.4.5.2.3.cmml">j</mi></msub><mo id="alg1.l4.m3.4.5.1" xref="alg1.l4.m3.4.5.1.cmml">∈</mo><mrow id="alg1.l4.m3.4.5.3.2" xref="alg1.l4.m3.4.5.3.1.cmml"><mo id="alg1.l4.m3.4.5.3.2.1" stretchy="false" xref="alg1.l4.m3.4.5.3.1.cmml">{</mo><mn id="alg1.l4.m3.1.1" xref="alg1.l4.m3.1.1.cmml">1</mn><mo id="alg1.l4.m3.4.5.3.2.2" xref="alg1.l4.m3.4.5.3.1.cmml">,</mo><mn id="alg1.l4.m3.2.2" xref="alg1.l4.m3.2.2.cmml">2</mn><mo id="alg1.l4.m3.4.5.3.2.3" xref="alg1.l4.m3.4.5.3.1.cmml">,</mo><mi id="alg1.l4.m3.3.3" mathvariant="normal" xref="alg1.l4.m3.3.3.cmml">⋯</mi><mo id="alg1.l4.m3.4.5.3.2.4" xref="alg1.l4.m3.4.5.3.1.cmml">,</mo><mi id="alg1.l4.m3.4.4" xref="alg1.l4.m3.4.4.cmml">N</mi><mo id="alg1.l4.m3.4.5.3.2.5" stretchy="false" xref="alg1.l4.m3.4.5.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l4.m3.4b"><apply id="alg1.l4.m3.4.5.cmml" xref="alg1.l4.m3.4.5"><in id="alg1.l4.m3.4.5.1.cmml" xref="alg1.l4.m3.4.5.1"></in><apply id="alg1.l4.m3.4.5.2.cmml" xref="alg1.l4.m3.4.5.2"><csymbol cd="ambiguous" id="alg1.l4.m3.4.5.2.1.cmml" xref="alg1.l4.m3.4.5.2">subscript</csymbol><ci id="alg1.l4.m3.4.5.2.2.cmml" xref="alg1.l4.m3.4.5.2.2">𝑘</ci><ci id="alg1.l4.m3.4.5.2.3.cmml" xref="alg1.l4.m3.4.5.2.3">𝑗</ci></apply><set id="alg1.l4.m3.4.5.3.1.cmml" xref="alg1.l4.m3.4.5.3.2"><cn id="alg1.l4.m3.1.1.cmml" type="integer" xref="alg1.l4.m3.1.1">1</cn><cn id="alg1.l4.m3.2.2.cmml" type="integer" xref="alg1.l4.m3.2.2">2</cn><ci id="alg1.l4.m3.3.3.cmml" xref="alg1.l4.m3.3.3">⋯</ci><ci id="alg1.l4.m3.4.4.cmml" xref="alg1.l4.m3.4.4">𝑁</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l4.m3.4c">k_{j}\in\{1,2,\cdots,N\}</annotation><annotation encoding="application/x-llamapun" id="alg1.l4.m3.4d">italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ { 1 , 2 , ⋯ , italic_N }</annotation></semantics></math> and set <math alttext="\mathcal{I}\leftarrow\{k_{1},k_{2},\cdots,k_{N_{\zeta}}\}" class="ltx_Math" display="inline" id="alg1.l4.m4.4"><semantics id="alg1.l4.m4.4a"><mrow id="alg1.l4.m4.4.4" xref="alg1.l4.m4.4.4.cmml"><mi class="ltx_font_mathcaligraphic" id="alg1.l4.m4.4.4.5" xref="alg1.l4.m4.4.4.5.cmml">ℐ</mi><mo id="alg1.l4.m4.4.4.4" stretchy="false" xref="alg1.l4.m4.4.4.4.cmml">←</mo><mrow id="alg1.l4.m4.4.4.3.3" xref="alg1.l4.m4.4.4.3.4.cmml"><mo id="alg1.l4.m4.4.4.3.3.4" stretchy="false" xref="alg1.l4.m4.4.4.3.4.cmml">{</mo><msub id="alg1.l4.m4.2.2.1.1.1" xref="alg1.l4.m4.2.2.1.1.1.cmml"><mi id="alg1.l4.m4.2.2.1.1.1.2" xref="alg1.l4.m4.2.2.1.1.1.2.cmml">k</mi><mn id="alg1.l4.m4.2.2.1.1.1.3" xref="alg1.l4.m4.2.2.1.1.1.3.cmml">1</mn></msub><mo id="alg1.l4.m4.4.4.3.3.5" xref="alg1.l4.m4.4.4.3.4.cmml">,</mo><msub id="alg1.l4.m4.3.3.2.2.2" xref="alg1.l4.m4.3.3.2.2.2.cmml"><mi id="alg1.l4.m4.3.3.2.2.2.2" xref="alg1.l4.m4.3.3.2.2.2.2.cmml">k</mi><mn id="alg1.l4.m4.3.3.2.2.2.3" xref="alg1.l4.m4.3.3.2.2.2.3.cmml">2</mn></msub><mo id="alg1.l4.m4.4.4.3.3.6" xref="alg1.l4.m4.4.4.3.4.cmml">,</mo><mi id="alg1.l4.m4.1.1" mathvariant="normal" xref="alg1.l4.m4.1.1.cmml">⋯</mi><mo id="alg1.l4.m4.4.4.3.3.7" xref="alg1.l4.m4.4.4.3.4.cmml">,</mo><msub id="alg1.l4.m4.4.4.3.3.3" xref="alg1.l4.m4.4.4.3.3.3.cmml"><mi id="alg1.l4.m4.4.4.3.3.3.2" xref="alg1.l4.m4.4.4.3.3.3.2.cmml">k</mi><msub id="alg1.l4.m4.4.4.3.3.3.3" xref="alg1.l4.m4.4.4.3.3.3.3.cmml"><mi id="alg1.l4.m4.4.4.3.3.3.3.2" xref="alg1.l4.m4.4.4.3.3.3.3.2.cmml">N</mi><mi id="alg1.l4.m4.4.4.3.3.3.3.3" xref="alg1.l4.m4.4.4.3.3.3.3.3.cmml">ζ</mi></msub></msub><mo id="alg1.l4.m4.4.4.3.3.8" stretchy="false" xref="alg1.l4.m4.4.4.3.4.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l4.m4.4b"><apply id="alg1.l4.m4.4.4.cmml" xref="alg1.l4.m4.4.4"><ci id="alg1.l4.m4.4.4.4.cmml" xref="alg1.l4.m4.4.4.4">←</ci><ci id="alg1.l4.m4.4.4.5.cmml" xref="alg1.l4.m4.4.4.5">ℐ</ci><set id="alg1.l4.m4.4.4.3.4.cmml" xref="alg1.l4.m4.4.4.3.3"><apply id="alg1.l4.m4.2.2.1.1.1.cmml" xref="alg1.l4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="alg1.l4.m4.2.2.1.1.1.1.cmml" xref="alg1.l4.m4.2.2.1.1.1">subscript</csymbol><ci id="alg1.l4.m4.2.2.1.1.1.2.cmml" xref="alg1.l4.m4.2.2.1.1.1.2">𝑘</ci><cn id="alg1.l4.m4.2.2.1.1.1.3.cmml" type="integer" xref="alg1.l4.m4.2.2.1.1.1.3">1</cn></apply><apply id="alg1.l4.m4.3.3.2.2.2.cmml" xref="alg1.l4.m4.3.3.2.2.2"><csymbol cd="ambiguous" id="alg1.l4.m4.3.3.2.2.2.1.cmml" xref="alg1.l4.m4.3.3.2.2.2">subscript</csymbol><ci id="alg1.l4.m4.3.3.2.2.2.2.cmml" xref="alg1.l4.m4.3.3.2.2.2.2">𝑘</ci><cn id="alg1.l4.m4.3.3.2.2.2.3.cmml" type="integer" xref="alg1.l4.m4.3.3.2.2.2.3">2</cn></apply><ci id="alg1.l4.m4.1.1.cmml" xref="alg1.l4.m4.1.1">⋯</ci><apply id="alg1.l4.m4.4.4.3.3.3.cmml" xref="alg1.l4.m4.4.4.3.3.3"><csymbol cd="ambiguous" id="alg1.l4.m4.4.4.3.3.3.1.cmml" xref="alg1.l4.m4.4.4.3.3.3">subscript</csymbol><ci id="alg1.l4.m4.4.4.3.3.3.2.cmml" xref="alg1.l4.m4.4.4.3.3.3.2">𝑘</ci><apply id="alg1.l4.m4.4.4.3.3.3.3.cmml" xref="alg1.l4.m4.4.4.3.3.3.3"><csymbol cd="ambiguous" id="alg1.l4.m4.4.4.3.3.3.3.1.cmml" xref="alg1.l4.m4.4.4.3.3.3.3">subscript</csymbol><ci id="alg1.l4.m4.4.4.3.3.3.3.2.cmml" xref="alg1.l4.m4.4.4.3.3.3.3.2">𝑁</ci><ci id="alg1.l4.m4.4.4.3.3.3.3.3.cmml" xref="alg1.l4.m4.4.4.3.3.3.3.3">𝜁</ci></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l4.m4.4c">\mathcal{I}\leftarrow\{k_{1},k_{2},\cdots,k_{N_{\zeta}}\}</annotation><annotation encoding="application/x-llamapun" id="alg1.l4.m4.4d">caligraphic_I ← { italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , ⋯ , italic_k start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT end_POSTSUBSCRIPT }</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l5"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l5.1.1.1" style="font-size:80%;">5:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l5.2">if</span> <math alttext="\exists k_{j}\in\mathcal{I}" class="ltx_Math" display="inline" id="alg1.l5.m1.1"><semantics id="alg1.l5.m1.1a"><mrow id="alg1.l5.m1.1.1" xref="alg1.l5.m1.1.1.cmml"><mrow id="alg1.l5.m1.1.1.2" xref="alg1.l5.m1.1.1.2.cmml"><mo id="alg1.l5.m1.1.1.2.1" rspace="0.167em" xref="alg1.l5.m1.1.1.2.1.cmml">∃</mo><msub id="alg1.l5.m1.1.1.2.2" xref="alg1.l5.m1.1.1.2.2.cmml"><mi id="alg1.l5.m1.1.1.2.2.2" xref="alg1.l5.m1.1.1.2.2.2.cmml">k</mi><mi id="alg1.l5.m1.1.1.2.2.3" xref="alg1.l5.m1.1.1.2.2.3.cmml">j</mi></msub></mrow><mo id="alg1.l5.m1.1.1.1" xref="alg1.l5.m1.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="alg1.l5.m1.1.1.3" xref="alg1.l5.m1.1.1.3.cmml">ℐ</mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l5.m1.1b"><apply id="alg1.l5.m1.1.1.cmml" xref="alg1.l5.m1.1.1"><in id="alg1.l5.m1.1.1.1.cmml" xref="alg1.l5.m1.1.1.1"></in><apply id="alg1.l5.m1.1.1.2.cmml" xref="alg1.l5.m1.1.1.2"><exists id="alg1.l5.m1.1.1.2.1.cmml" xref="alg1.l5.m1.1.1.2.1"></exists><apply id="alg1.l5.m1.1.1.2.2.cmml" xref="alg1.l5.m1.1.1.2.2"><csymbol cd="ambiguous" id="alg1.l5.m1.1.1.2.2.1.cmml" xref="alg1.l5.m1.1.1.2.2">subscript</csymbol><ci id="alg1.l5.m1.1.1.2.2.2.cmml" xref="alg1.l5.m1.1.1.2.2.2">𝑘</ci><ci id="alg1.l5.m1.1.1.2.2.3.cmml" xref="alg1.l5.m1.1.1.2.2.3">𝑗</ci></apply></apply><ci id="alg1.l5.m1.1.1.3.cmml" xref="alg1.l5.m1.1.1.3">ℐ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l5.m1.1c">\exists k_{j}\in\mathcal{I}</annotation><annotation encoding="application/x-llamapun" id="alg1.l5.m1.1d">∃ italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ caligraphic_I</annotation></semantics></math> such that <math alttext="k_{j}\notin\mathcal{I}^{\prime}" class="ltx_Math" display="inline" id="alg1.l5.m2.1"><semantics id="alg1.l5.m2.1a"><mrow id="alg1.l5.m2.1.1" xref="alg1.l5.m2.1.1.cmml"><msub id="alg1.l5.m2.1.1.2" xref="alg1.l5.m2.1.1.2.cmml"><mi id="alg1.l5.m2.1.1.2.2" xref="alg1.l5.m2.1.1.2.2.cmml">k</mi><mi id="alg1.l5.m2.1.1.2.3" xref="alg1.l5.m2.1.1.2.3.cmml">j</mi></msub><mo id="alg1.l5.m2.1.1.1" xref="alg1.l5.m2.1.1.1.cmml">∉</mo><msup id="alg1.l5.m2.1.1.3" xref="alg1.l5.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="alg1.l5.m2.1.1.3.2" xref="alg1.l5.m2.1.1.3.2.cmml">ℐ</mi><mo id="alg1.l5.m2.1.1.3.3" xref="alg1.l5.m2.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="alg1.l5.m2.1b"><apply id="alg1.l5.m2.1.1.cmml" xref="alg1.l5.m2.1.1"><notin id="alg1.l5.m2.1.1.1.cmml" xref="alg1.l5.m2.1.1.1"></notin><apply id="alg1.l5.m2.1.1.2.cmml" xref="alg1.l5.m2.1.1.2"><csymbol cd="ambiguous" id="alg1.l5.m2.1.1.2.1.cmml" xref="alg1.l5.m2.1.1.2">subscript</csymbol><ci id="alg1.l5.m2.1.1.2.2.cmml" xref="alg1.l5.m2.1.1.2.2">𝑘</ci><ci id="alg1.l5.m2.1.1.2.3.cmml" xref="alg1.l5.m2.1.1.2.3">𝑗</ci></apply><apply id="alg1.l5.m2.1.1.3.cmml" xref="alg1.l5.m2.1.1.3"><csymbol cd="ambiguous" id="alg1.l5.m2.1.1.3.1.cmml" xref="alg1.l5.m2.1.1.3">superscript</csymbol><ci id="alg1.l5.m2.1.1.3.2.cmml" xref="alg1.l5.m2.1.1.3.2">ℐ</ci><ci id="alg1.l5.m2.1.1.3.3.cmml" xref="alg1.l5.m2.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l5.m2.1c">k_{j}\notin\mathcal{I}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="alg1.l5.m2.1d">italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∉ caligraphic_I start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg1.l5.3">then</span> </div> <div class="ltx_listingline" id="alg1.l6"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l6.1.1.1" style="font-size:80%;">6:</span></span> Reconstruct <math alttext="\Phi_{\rm s,\zeta}(t)" class="ltx_Math" display="inline" id="alg1.l6.m1.3"><semantics id="alg1.l6.m1.3a"><mrow id="alg1.l6.m1.3.4" xref="alg1.l6.m1.3.4.cmml"><msub id="alg1.l6.m1.3.4.2" xref="alg1.l6.m1.3.4.2.cmml"><mi id="alg1.l6.m1.3.4.2.2" mathvariant="normal" xref="alg1.l6.m1.3.4.2.2.cmml">Φ</mi><mrow id="alg1.l6.m1.2.2.2.4" xref="alg1.l6.m1.2.2.2.3.cmml"><mi id="alg1.l6.m1.1.1.1.1" mathvariant="normal" xref="alg1.l6.m1.1.1.1.1.cmml">s</mi><mo id="alg1.l6.m1.2.2.2.4.1" xref="alg1.l6.m1.2.2.2.3.cmml">,</mo><mi id="alg1.l6.m1.2.2.2.2" xref="alg1.l6.m1.2.2.2.2.cmml">ζ</mi></mrow></msub><mo id="alg1.l6.m1.3.4.1" xref="alg1.l6.m1.3.4.1.cmml">⁢</mo><mrow id="alg1.l6.m1.3.4.3.2" xref="alg1.l6.m1.3.4.cmml"><mo id="alg1.l6.m1.3.4.3.2.1" stretchy="false" xref="alg1.l6.m1.3.4.cmml">(</mo><mi id="alg1.l6.m1.3.3" xref="alg1.l6.m1.3.3.cmml">t</mi><mo id="alg1.l6.m1.3.4.3.2.2" stretchy="false" xref="alg1.l6.m1.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l6.m1.3b"><apply id="alg1.l6.m1.3.4.cmml" xref="alg1.l6.m1.3.4"><times id="alg1.l6.m1.3.4.1.cmml" xref="alg1.l6.m1.3.4.1"></times><apply id="alg1.l6.m1.3.4.2.cmml" xref="alg1.l6.m1.3.4.2"><csymbol cd="ambiguous" id="alg1.l6.m1.3.4.2.1.cmml" xref="alg1.l6.m1.3.4.2">subscript</csymbol><ci id="alg1.l6.m1.3.4.2.2.cmml" xref="alg1.l6.m1.3.4.2.2">Φ</ci><list id="alg1.l6.m1.2.2.2.3.cmml" xref="alg1.l6.m1.2.2.2.4"><ci id="alg1.l6.m1.1.1.1.1.cmml" xref="alg1.l6.m1.1.1.1.1">s</ci><ci id="alg1.l6.m1.2.2.2.2.cmml" xref="alg1.l6.m1.2.2.2.2">𝜁</ci></list></apply><ci id="alg1.l6.m1.3.3.cmml" xref="alg1.l6.m1.3.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l6.m1.3c">\Phi_{\rm s,\zeta}(t)</annotation><annotation encoding="application/x-llamapun" id="alg1.l6.m1.3d">roman_Φ start_POSTSUBSCRIPT roman_s , italic_ζ end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> and <math alttext="\Psi_{\zeta}(t)" class="ltx_Math" display="inline" id="alg1.l6.m2.1"><semantics id="alg1.l6.m2.1a"><mrow id="alg1.l6.m2.1.2" xref="alg1.l6.m2.1.2.cmml"><msub id="alg1.l6.m2.1.2.2" xref="alg1.l6.m2.1.2.2.cmml"><mi id="alg1.l6.m2.1.2.2.2" mathvariant="normal" xref="alg1.l6.m2.1.2.2.2.cmml">Ψ</mi><mi id="alg1.l6.m2.1.2.2.3" xref="alg1.l6.m2.1.2.2.3.cmml">ζ</mi></msub><mo id="alg1.l6.m2.1.2.1" xref="alg1.l6.m2.1.2.1.cmml">⁢</mo><mrow id="alg1.l6.m2.1.2.3.2" xref="alg1.l6.m2.1.2.cmml"><mo id="alg1.l6.m2.1.2.3.2.1" stretchy="false" xref="alg1.l6.m2.1.2.cmml">(</mo><mi id="alg1.l6.m2.1.1" xref="alg1.l6.m2.1.1.cmml">t</mi><mo id="alg1.l6.m2.1.2.3.2.2" stretchy="false" xref="alg1.l6.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l6.m2.1b"><apply id="alg1.l6.m2.1.2.cmml" xref="alg1.l6.m2.1.2"><times id="alg1.l6.m2.1.2.1.cmml" xref="alg1.l6.m2.1.2.1"></times><apply id="alg1.l6.m2.1.2.2.cmml" xref="alg1.l6.m2.1.2.2"><csymbol cd="ambiguous" id="alg1.l6.m2.1.2.2.1.cmml" xref="alg1.l6.m2.1.2.2">subscript</csymbol><ci id="alg1.l6.m2.1.2.2.2.cmml" xref="alg1.l6.m2.1.2.2.2">Ψ</ci><ci id="alg1.l6.m2.1.2.2.3.cmml" xref="alg1.l6.m2.1.2.2.3">𝜁</ci></apply><ci id="alg1.l6.m2.1.1.cmml" xref="alg1.l6.m2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l6.m2.1c">\Psi_{\zeta}(t)</annotation><annotation encoding="application/x-llamapun" id="alg1.l6.m2.1d">roman_Ψ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> by <math alttext="\mathcal{I}" class="ltx_Math" display="inline" id="alg1.l6.m3.1"><semantics id="alg1.l6.m3.1a"><mi class="ltx_font_mathcaligraphic" id="alg1.l6.m3.1.1" xref="alg1.l6.m3.1.1.cmml">ℐ</mi><annotation-xml encoding="MathML-Content" id="alg1.l6.m3.1b"><ci id="alg1.l6.m3.1.1.cmml" xref="alg1.l6.m3.1.1">ℐ</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l6.m3.1c">\mathcal{I}</annotation><annotation encoding="application/x-llamapun" id="alg1.l6.m3.1d">caligraphic_I</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l7"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l7.1.1.1" style="font-size:80%;">7:</span></span> <math alttext="\sigma_{\rm c}\leftarrow\sigma" class="ltx_Math" display="inline" id="alg1.l7.m1.1"><semantics id="alg1.l7.m1.1a"><mrow id="alg1.l7.m1.1.1" xref="alg1.l7.m1.1.1.cmml"><msub id="alg1.l7.m1.1.1.2" xref="alg1.l7.m1.1.1.2.cmml"><mi id="alg1.l7.m1.1.1.2.2" xref="alg1.l7.m1.1.1.2.2.cmml">σ</mi><mi id="alg1.l7.m1.1.1.2.3" mathvariant="normal" xref="alg1.l7.m1.1.1.2.3.cmml">c</mi></msub><mo id="alg1.l7.m1.1.1.1" stretchy="false" xref="alg1.l7.m1.1.1.1.cmml">←</mo><mi id="alg1.l7.m1.1.1.3" xref="alg1.l7.m1.1.1.3.cmml">σ</mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l7.m1.1b"><apply id="alg1.l7.m1.1.1.cmml" xref="alg1.l7.m1.1.1"><ci id="alg1.l7.m1.1.1.1.cmml" xref="alg1.l7.m1.1.1.1">←</ci><apply id="alg1.l7.m1.1.1.2.cmml" xref="alg1.l7.m1.1.1.2"><csymbol cd="ambiguous" id="alg1.l7.m1.1.1.2.1.cmml" xref="alg1.l7.m1.1.1.2">subscript</csymbol><ci id="alg1.l7.m1.1.1.2.2.cmml" xref="alg1.l7.m1.1.1.2.2">𝜎</ci><ci id="alg1.l7.m1.1.1.2.3.cmml" xref="alg1.l7.m1.1.1.2.3">c</ci></apply><ci id="alg1.l7.m1.1.1.3.cmml" xref="alg1.l7.m1.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l7.m1.1c">\sigma_{\rm c}\leftarrow\sigma</annotation><annotation encoding="application/x-llamapun" id="alg1.l7.m1.1d">italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ← italic_σ</annotation></semantics></math>, <math alttext="T_{\rm a}\leftarrow t" class="ltx_Math" display="inline" id="alg1.l7.m2.1"><semantics id="alg1.l7.m2.1a"><mrow id="alg1.l7.m2.1.1" xref="alg1.l7.m2.1.1.cmml"><msub id="alg1.l7.m2.1.1.2" xref="alg1.l7.m2.1.1.2.cmml"><mi id="alg1.l7.m2.1.1.2.2" xref="alg1.l7.m2.1.1.2.2.cmml">T</mi><mi id="alg1.l7.m2.1.1.2.3" mathvariant="normal" xref="alg1.l7.m2.1.1.2.3.cmml">a</mi></msub><mo id="alg1.l7.m2.1.1.1" stretchy="false" xref="alg1.l7.m2.1.1.1.cmml">←</mo><mi id="alg1.l7.m2.1.1.3" xref="alg1.l7.m2.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l7.m2.1b"><apply id="alg1.l7.m2.1.1.cmml" xref="alg1.l7.m2.1.1"><ci id="alg1.l7.m2.1.1.1.cmml" xref="alg1.l7.m2.1.1.1">←</ci><apply id="alg1.l7.m2.1.1.2.cmml" xref="alg1.l7.m2.1.1.2"><csymbol cd="ambiguous" id="alg1.l7.m2.1.1.2.1.cmml" xref="alg1.l7.m2.1.1.2">subscript</csymbol><ci id="alg1.l7.m2.1.1.2.2.cmml" xref="alg1.l7.m2.1.1.2.2">𝑇</ci><ci id="alg1.l7.m2.1.1.2.3.cmml" xref="alg1.l7.m2.1.1.2.3">a</ci></apply><ci id="alg1.l7.m2.1.1.3.cmml" xref="alg1.l7.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l7.m2.1c">T_{\rm a}\leftarrow t</annotation><annotation encoding="application/x-llamapun" id="alg1.l7.m2.1d">italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT ← italic_t</annotation></semantics></math>, <math alttext="\mathcal{I}^{\prime}\leftarrow\mathcal{I}" class="ltx_Math" display="inline" id="alg1.l7.m3.1"><semantics id="alg1.l7.m3.1a"><mrow id="alg1.l7.m3.1.1" xref="alg1.l7.m3.1.1.cmml"><msup id="alg1.l7.m3.1.1.2" xref="alg1.l7.m3.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="alg1.l7.m3.1.1.2.2" xref="alg1.l7.m3.1.1.2.2.cmml">ℐ</mi><mo id="alg1.l7.m3.1.1.2.3" xref="alg1.l7.m3.1.1.2.3.cmml">′</mo></msup><mo id="alg1.l7.m3.1.1.1" stretchy="false" xref="alg1.l7.m3.1.1.1.cmml">←</mo><mi class="ltx_font_mathcaligraphic" id="alg1.l7.m3.1.1.3" xref="alg1.l7.m3.1.1.3.cmml">ℐ</mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l7.m3.1b"><apply id="alg1.l7.m3.1.1.cmml" xref="alg1.l7.m3.1.1"><ci id="alg1.l7.m3.1.1.1.cmml" xref="alg1.l7.m3.1.1.1">←</ci><apply id="alg1.l7.m3.1.1.2.cmml" xref="alg1.l7.m3.1.1.2"><csymbol cd="ambiguous" id="alg1.l7.m3.1.1.2.1.cmml" xref="alg1.l7.m3.1.1.2">superscript</csymbol><ci id="alg1.l7.m3.1.1.2.2.cmml" xref="alg1.l7.m3.1.1.2.2">ℐ</ci><ci id="alg1.l7.m3.1.1.2.3.cmml" xref="alg1.l7.m3.1.1.2.3">′</ci></apply><ci id="alg1.l7.m3.1.1.3.cmml" xref="alg1.l7.m3.1.1.3">ℐ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l7.m3.1c">\mathcal{I}^{\prime}\leftarrow\mathcal{I}</annotation><annotation encoding="application/x-llamapun" id="alg1.l7.m3.1d">caligraphic_I start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ← caligraphic_I</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l8"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l8.1.1.1" style="font-size:80%;">8:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l8.2">end</span> <span class="ltx_text ltx_font_bold" id="alg1.l8.3">if</span> </div> <div class="ltx_listingline" id="alg1.l9"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l9.1.1.1" style="font-size:80%;">9:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l9.2">if</span> <math alttext="\sigma_{\min}(\Psi_{\zeta}(t))\geq\sigma_{\rm c}" class="ltx_Math" display="inline" id="alg1.l9.m1.2"><semantics id="alg1.l9.m1.2a"><mrow id="alg1.l9.m1.2.2" xref="alg1.l9.m1.2.2.cmml"><mrow id="alg1.l9.m1.2.2.1" xref="alg1.l9.m1.2.2.1.cmml"><msub id="alg1.l9.m1.2.2.1.3" xref="alg1.l9.m1.2.2.1.3.cmml"><mi id="alg1.l9.m1.2.2.1.3.2" xref="alg1.l9.m1.2.2.1.3.2.cmml">σ</mi><mi id="alg1.l9.m1.2.2.1.3.3" xref="alg1.l9.m1.2.2.1.3.3.cmml">min</mi></msub><mo id="alg1.l9.m1.2.2.1.2" xref="alg1.l9.m1.2.2.1.2.cmml">⁢</mo><mrow id="alg1.l9.m1.2.2.1.1.1" xref="alg1.l9.m1.2.2.1.1.1.1.cmml"><mo id="alg1.l9.m1.2.2.1.1.1.2" stretchy="false" xref="alg1.l9.m1.2.2.1.1.1.1.cmml">(</mo><mrow id="alg1.l9.m1.2.2.1.1.1.1" xref="alg1.l9.m1.2.2.1.1.1.1.cmml"><msub id="alg1.l9.m1.2.2.1.1.1.1.2" xref="alg1.l9.m1.2.2.1.1.1.1.2.cmml"><mi id="alg1.l9.m1.2.2.1.1.1.1.2.2" mathvariant="normal" xref="alg1.l9.m1.2.2.1.1.1.1.2.2.cmml">Ψ</mi><mi id="alg1.l9.m1.2.2.1.1.1.1.2.3" xref="alg1.l9.m1.2.2.1.1.1.1.2.3.cmml">ζ</mi></msub><mo id="alg1.l9.m1.2.2.1.1.1.1.1" xref="alg1.l9.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="alg1.l9.m1.2.2.1.1.1.1.3.2" xref="alg1.l9.m1.2.2.1.1.1.1.cmml"><mo id="alg1.l9.m1.2.2.1.1.1.1.3.2.1" stretchy="false" xref="alg1.l9.m1.2.2.1.1.1.1.cmml">(</mo><mi id="alg1.l9.m1.1.1" xref="alg1.l9.m1.1.1.cmml">t</mi><mo id="alg1.l9.m1.2.2.1.1.1.1.3.2.2" stretchy="false" xref="alg1.l9.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="alg1.l9.m1.2.2.1.1.1.3" stretchy="false" xref="alg1.l9.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="alg1.l9.m1.2.2.2" xref="alg1.l9.m1.2.2.2.cmml">≥</mo><msub id="alg1.l9.m1.2.2.3" xref="alg1.l9.m1.2.2.3.cmml"><mi id="alg1.l9.m1.2.2.3.2" xref="alg1.l9.m1.2.2.3.2.cmml">σ</mi><mi id="alg1.l9.m1.2.2.3.3" mathvariant="normal" xref="alg1.l9.m1.2.2.3.3.cmml">c</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="alg1.l9.m1.2b"><apply id="alg1.l9.m1.2.2.cmml" xref="alg1.l9.m1.2.2"><geq id="alg1.l9.m1.2.2.2.cmml" xref="alg1.l9.m1.2.2.2"></geq><apply id="alg1.l9.m1.2.2.1.cmml" xref="alg1.l9.m1.2.2.1"><times id="alg1.l9.m1.2.2.1.2.cmml" xref="alg1.l9.m1.2.2.1.2"></times><apply id="alg1.l9.m1.2.2.1.3.cmml" xref="alg1.l9.m1.2.2.1.3"><csymbol cd="ambiguous" id="alg1.l9.m1.2.2.1.3.1.cmml" xref="alg1.l9.m1.2.2.1.3">subscript</csymbol><ci id="alg1.l9.m1.2.2.1.3.2.cmml" xref="alg1.l9.m1.2.2.1.3.2">𝜎</ci><min id="alg1.l9.m1.2.2.1.3.3.cmml" xref="alg1.l9.m1.2.2.1.3.3"></min></apply><apply id="alg1.l9.m1.2.2.1.1.1.1.cmml" xref="alg1.l9.m1.2.2.1.1.1"><times id="alg1.l9.m1.2.2.1.1.1.1.1.cmml" xref="alg1.l9.m1.2.2.1.1.1.1.1"></times><apply id="alg1.l9.m1.2.2.1.1.1.1.2.cmml" xref="alg1.l9.m1.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="alg1.l9.m1.2.2.1.1.1.1.2.1.cmml" xref="alg1.l9.m1.2.2.1.1.1.1.2">subscript</csymbol><ci id="alg1.l9.m1.2.2.1.1.1.1.2.2.cmml" xref="alg1.l9.m1.2.2.1.1.1.1.2.2">Ψ</ci><ci id="alg1.l9.m1.2.2.1.1.1.1.2.3.cmml" xref="alg1.l9.m1.2.2.1.1.1.1.2.3">𝜁</ci></apply><ci id="alg1.l9.m1.1.1.cmml" xref="alg1.l9.m1.1.1">𝑡</ci></apply></apply><apply id="alg1.l9.m1.2.2.3.cmml" xref="alg1.l9.m1.2.2.3"><csymbol cd="ambiguous" id="alg1.l9.m1.2.2.3.1.cmml" xref="alg1.l9.m1.2.2.3">subscript</csymbol><ci id="alg1.l9.m1.2.2.3.2.cmml" xref="alg1.l9.m1.2.2.3.2">𝜎</ci><ci id="alg1.l9.m1.2.2.3.3.cmml" xref="alg1.l9.m1.2.2.3.3">c</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l9.m1.2c">\sigma_{\min}(\Psi_{\zeta}(t))\geq\sigma_{\rm c}</annotation><annotation encoding="application/x-llamapun" id="alg1.l9.m1.2d">italic_σ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Ψ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t ) ) ≥ italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg1.l9.3">then</span> </div> <div class="ltx_listingline" id="alg1.l10"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l10.1.1.1" style="font-size:80%;">10:</span></span> <math alttext="\sigma_{\rm c}\leftarrow\sigma_{\min}(\Psi_{\zeta}(t))" class="ltx_Math" display="inline" id="alg1.l10.m1.2"><semantics id="alg1.l10.m1.2a"><mrow id="alg1.l10.m1.2.2" xref="alg1.l10.m1.2.2.cmml"><msub id="alg1.l10.m1.2.2.3" xref="alg1.l10.m1.2.2.3.cmml"><mi id="alg1.l10.m1.2.2.3.2" xref="alg1.l10.m1.2.2.3.2.cmml">σ</mi><mi id="alg1.l10.m1.2.2.3.3" mathvariant="normal" xref="alg1.l10.m1.2.2.3.3.cmml">c</mi></msub><mo id="alg1.l10.m1.2.2.2" stretchy="false" xref="alg1.l10.m1.2.2.2.cmml">←</mo><mrow id="alg1.l10.m1.2.2.1" xref="alg1.l10.m1.2.2.1.cmml"><msub id="alg1.l10.m1.2.2.1.3" xref="alg1.l10.m1.2.2.1.3.cmml"><mi id="alg1.l10.m1.2.2.1.3.2" xref="alg1.l10.m1.2.2.1.3.2.cmml">σ</mi><mi id="alg1.l10.m1.2.2.1.3.3" xref="alg1.l10.m1.2.2.1.3.3.cmml">min</mi></msub><mo id="alg1.l10.m1.2.2.1.2" xref="alg1.l10.m1.2.2.1.2.cmml">⁢</mo><mrow id="alg1.l10.m1.2.2.1.1.1" xref="alg1.l10.m1.2.2.1.1.1.1.cmml"><mo id="alg1.l10.m1.2.2.1.1.1.2" stretchy="false" xref="alg1.l10.m1.2.2.1.1.1.1.cmml">(</mo><mrow id="alg1.l10.m1.2.2.1.1.1.1" xref="alg1.l10.m1.2.2.1.1.1.1.cmml"><msub id="alg1.l10.m1.2.2.1.1.1.1.2" xref="alg1.l10.m1.2.2.1.1.1.1.2.cmml"><mi id="alg1.l10.m1.2.2.1.1.1.1.2.2" mathvariant="normal" xref="alg1.l10.m1.2.2.1.1.1.1.2.2.cmml">Ψ</mi><mi id="alg1.l10.m1.2.2.1.1.1.1.2.3" xref="alg1.l10.m1.2.2.1.1.1.1.2.3.cmml">ζ</mi></msub><mo id="alg1.l10.m1.2.2.1.1.1.1.1" xref="alg1.l10.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="alg1.l10.m1.2.2.1.1.1.1.3.2" xref="alg1.l10.m1.2.2.1.1.1.1.cmml"><mo id="alg1.l10.m1.2.2.1.1.1.1.3.2.1" stretchy="false" xref="alg1.l10.m1.2.2.1.1.1.1.cmml">(</mo><mi id="alg1.l10.m1.1.1" xref="alg1.l10.m1.1.1.cmml">t</mi><mo id="alg1.l10.m1.2.2.1.1.1.1.3.2.2" stretchy="false" xref="alg1.l10.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="alg1.l10.m1.2.2.1.1.1.3" stretchy="false" xref="alg1.l10.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l10.m1.2b"><apply id="alg1.l10.m1.2.2.cmml" xref="alg1.l10.m1.2.2"><ci id="alg1.l10.m1.2.2.2.cmml" xref="alg1.l10.m1.2.2.2">←</ci><apply id="alg1.l10.m1.2.2.3.cmml" xref="alg1.l10.m1.2.2.3"><csymbol cd="ambiguous" id="alg1.l10.m1.2.2.3.1.cmml" xref="alg1.l10.m1.2.2.3">subscript</csymbol><ci id="alg1.l10.m1.2.2.3.2.cmml" xref="alg1.l10.m1.2.2.3.2">𝜎</ci><ci id="alg1.l10.m1.2.2.3.3.cmml" xref="alg1.l10.m1.2.2.3.3">c</ci></apply><apply id="alg1.l10.m1.2.2.1.cmml" xref="alg1.l10.m1.2.2.1"><times id="alg1.l10.m1.2.2.1.2.cmml" xref="alg1.l10.m1.2.2.1.2"></times><apply id="alg1.l10.m1.2.2.1.3.cmml" xref="alg1.l10.m1.2.2.1.3"><csymbol cd="ambiguous" id="alg1.l10.m1.2.2.1.3.1.cmml" xref="alg1.l10.m1.2.2.1.3">subscript</csymbol><ci id="alg1.l10.m1.2.2.1.3.2.cmml" xref="alg1.l10.m1.2.2.1.3.2">𝜎</ci><min id="alg1.l10.m1.2.2.1.3.3.cmml" xref="alg1.l10.m1.2.2.1.3.3"></min></apply><apply id="alg1.l10.m1.2.2.1.1.1.1.cmml" xref="alg1.l10.m1.2.2.1.1.1"><times id="alg1.l10.m1.2.2.1.1.1.1.1.cmml" xref="alg1.l10.m1.2.2.1.1.1.1.1"></times><apply id="alg1.l10.m1.2.2.1.1.1.1.2.cmml" xref="alg1.l10.m1.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="alg1.l10.m1.2.2.1.1.1.1.2.1.cmml" xref="alg1.l10.m1.2.2.1.1.1.1.2">subscript</csymbol><ci id="alg1.l10.m1.2.2.1.1.1.1.2.2.cmml" xref="alg1.l10.m1.2.2.1.1.1.1.2.2">Ψ</ci><ci id="alg1.l10.m1.2.2.1.1.1.1.2.3.cmml" xref="alg1.l10.m1.2.2.1.1.1.1.2.3">𝜁</ci></apply><ci id="alg1.l10.m1.1.1.cmml" xref="alg1.l10.m1.1.1">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l10.m1.2c">\sigma_{\rm c}\leftarrow\sigma_{\min}(\Psi_{\zeta}(t))</annotation><annotation encoding="application/x-llamapun" id="alg1.l10.m1.2d">italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ← italic_σ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Ψ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t ) )</annotation></semantics></math>, <math alttext="t_{\rm e}\leftarrow t" class="ltx_Math" display="inline" id="alg1.l10.m2.1"><semantics id="alg1.l10.m2.1a"><mrow id="alg1.l10.m2.1.1" xref="alg1.l10.m2.1.1.cmml"><msub id="alg1.l10.m2.1.1.2" xref="alg1.l10.m2.1.1.2.cmml"><mi id="alg1.l10.m2.1.1.2.2" xref="alg1.l10.m2.1.1.2.2.cmml">t</mi><mi id="alg1.l10.m2.1.1.2.3" mathvariant="normal" xref="alg1.l10.m2.1.1.2.3.cmml">e</mi></msub><mo id="alg1.l10.m2.1.1.1" stretchy="false" xref="alg1.l10.m2.1.1.1.cmml">←</mo><mi id="alg1.l10.m2.1.1.3" xref="alg1.l10.m2.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l10.m2.1b"><apply id="alg1.l10.m2.1.1.cmml" xref="alg1.l10.m2.1.1"><ci id="alg1.l10.m2.1.1.1.cmml" xref="alg1.l10.m2.1.1.1">←</ci><apply id="alg1.l10.m2.1.1.2.cmml" xref="alg1.l10.m2.1.1.2"><csymbol cd="ambiguous" id="alg1.l10.m2.1.1.2.1.cmml" xref="alg1.l10.m2.1.1.2">subscript</csymbol><ci id="alg1.l10.m2.1.1.2.2.cmml" xref="alg1.l10.m2.1.1.2.2">𝑡</ci><ci id="alg1.l10.m2.1.1.2.3.cmml" xref="alg1.l10.m2.1.1.2.3">e</ci></apply><ci id="alg1.l10.m2.1.1.3.cmml" xref="alg1.l10.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l10.m2.1c">t_{\rm e}\leftarrow t</annotation><annotation encoding="application/x-llamapun" id="alg1.l10.m2.1d">italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ← italic_t</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l11"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l11.1.1.1" style="font-size:80%;">11:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l11.2">end</span> <span class="ltx_text ltx_font_bold" id="alg1.l11.3">if</span> </div> <div class="ltx_listingline" id="alg1.l12"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l12.1.1.1" style="font-size:80%;">12:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l12.2">else</span> </div> <div class="ltx_listingline" id="alg1.l13"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l13.1.1.1" style="font-size:80%;">13:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l13.2">if</span> <math alttext="\sigma_{\min}(\Psi(t))\geq\sigma_{\rm c}" class="ltx_Math" display="inline" id="alg1.l13.m1.2"><semantics id="alg1.l13.m1.2a"><mrow id="alg1.l13.m1.2.2" xref="alg1.l13.m1.2.2.cmml"><mrow id="alg1.l13.m1.2.2.1" xref="alg1.l13.m1.2.2.1.cmml"><msub id="alg1.l13.m1.2.2.1.3" xref="alg1.l13.m1.2.2.1.3.cmml"><mi id="alg1.l13.m1.2.2.1.3.2" xref="alg1.l13.m1.2.2.1.3.2.cmml">σ</mi><mi id="alg1.l13.m1.2.2.1.3.3" xref="alg1.l13.m1.2.2.1.3.3.cmml">min</mi></msub><mo id="alg1.l13.m1.2.2.1.2" xref="alg1.l13.m1.2.2.1.2.cmml">⁢</mo><mrow id="alg1.l13.m1.2.2.1.1.1" xref="alg1.l13.m1.2.2.1.1.1.1.cmml"><mo id="alg1.l13.m1.2.2.1.1.1.2" stretchy="false" xref="alg1.l13.m1.2.2.1.1.1.1.cmml">(</mo><mrow id="alg1.l13.m1.2.2.1.1.1.1" xref="alg1.l13.m1.2.2.1.1.1.1.cmml"><mi id="alg1.l13.m1.2.2.1.1.1.1.2" mathvariant="normal" xref="alg1.l13.m1.2.2.1.1.1.1.2.cmml">Ψ</mi><mo id="alg1.l13.m1.2.2.1.1.1.1.1" xref="alg1.l13.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="alg1.l13.m1.2.2.1.1.1.1.3.2" xref="alg1.l13.m1.2.2.1.1.1.1.cmml"><mo id="alg1.l13.m1.2.2.1.1.1.1.3.2.1" stretchy="false" xref="alg1.l13.m1.2.2.1.1.1.1.cmml">(</mo><mi id="alg1.l13.m1.1.1" xref="alg1.l13.m1.1.1.cmml">t</mi><mo id="alg1.l13.m1.2.2.1.1.1.1.3.2.2" stretchy="false" xref="alg1.l13.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="alg1.l13.m1.2.2.1.1.1.3" stretchy="false" xref="alg1.l13.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="alg1.l13.m1.2.2.2" xref="alg1.l13.m1.2.2.2.cmml">≥</mo><msub id="alg1.l13.m1.2.2.3" xref="alg1.l13.m1.2.2.3.cmml"><mi id="alg1.l13.m1.2.2.3.2" xref="alg1.l13.m1.2.2.3.2.cmml">σ</mi><mi id="alg1.l13.m1.2.2.3.3" mathvariant="normal" xref="alg1.l13.m1.2.2.3.3.cmml">c</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="alg1.l13.m1.2b"><apply id="alg1.l13.m1.2.2.cmml" xref="alg1.l13.m1.2.2"><geq id="alg1.l13.m1.2.2.2.cmml" xref="alg1.l13.m1.2.2.2"></geq><apply id="alg1.l13.m1.2.2.1.cmml" xref="alg1.l13.m1.2.2.1"><times id="alg1.l13.m1.2.2.1.2.cmml" xref="alg1.l13.m1.2.2.1.2"></times><apply id="alg1.l13.m1.2.2.1.3.cmml" xref="alg1.l13.m1.2.2.1.3"><csymbol cd="ambiguous" id="alg1.l13.m1.2.2.1.3.1.cmml" xref="alg1.l13.m1.2.2.1.3">subscript</csymbol><ci id="alg1.l13.m1.2.2.1.3.2.cmml" xref="alg1.l13.m1.2.2.1.3.2">𝜎</ci><min id="alg1.l13.m1.2.2.1.3.3.cmml" xref="alg1.l13.m1.2.2.1.3.3"></min></apply><apply id="alg1.l13.m1.2.2.1.1.1.1.cmml" xref="alg1.l13.m1.2.2.1.1.1"><times id="alg1.l13.m1.2.2.1.1.1.1.1.cmml" xref="alg1.l13.m1.2.2.1.1.1.1.1"></times><ci id="alg1.l13.m1.2.2.1.1.1.1.2.cmml" xref="alg1.l13.m1.2.2.1.1.1.1.2">Ψ</ci><ci id="alg1.l13.m1.1.1.cmml" xref="alg1.l13.m1.1.1">𝑡</ci></apply></apply><apply id="alg1.l13.m1.2.2.3.cmml" xref="alg1.l13.m1.2.2.3"><csymbol cd="ambiguous" id="alg1.l13.m1.2.2.3.1.cmml" xref="alg1.l13.m1.2.2.3">subscript</csymbol><ci id="alg1.l13.m1.2.2.3.2.cmml" xref="alg1.l13.m1.2.2.3.2">𝜎</ci><ci id="alg1.l13.m1.2.2.3.3.cmml" xref="alg1.l13.m1.2.2.3.3">c</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l13.m1.2c">\sigma_{\min}(\Psi(t))\geq\sigma_{\rm c}</annotation><annotation encoding="application/x-llamapun" id="alg1.l13.m1.2d">italic_σ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Ψ ( italic_t ) ) ≥ italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg1.l13.3">then</span> </div> <div class="ltx_listingline" id="alg1.l14"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l14.1.1.1" style="font-size:80%;">14:</span></span> <math alttext="\sigma_{\rm c}\leftarrow\sigma_{\min}(\Psi(t))" class="ltx_Math" display="inline" id="alg1.l14.m1.2"><semantics id="alg1.l14.m1.2a"><mrow id="alg1.l14.m1.2.2" xref="alg1.l14.m1.2.2.cmml"><msub id="alg1.l14.m1.2.2.3" xref="alg1.l14.m1.2.2.3.cmml"><mi id="alg1.l14.m1.2.2.3.2" xref="alg1.l14.m1.2.2.3.2.cmml">σ</mi><mi id="alg1.l14.m1.2.2.3.3" mathvariant="normal" xref="alg1.l14.m1.2.2.3.3.cmml">c</mi></msub><mo id="alg1.l14.m1.2.2.2" stretchy="false" xref="alg1.l14.m1.2.2.2.cmml">←</mo><mrow id="alg1.l14.m1.2.2.1" xref="alg1.l14.m1.2.2.1.cmml"><msub id="alg1.l14.m1.2.2.1.3" xref="alg1.l14.m1.2.2.1.3.cmml"><mi id="alg1.l14.m1.2.2.1.3.2" xref="alg1.l14.m1.2.2.1.3.2.cmml">σ</mi><mi id="alg1.l14.m1.2.2.1.3.3" xref="alg1.l14.m1.2.2.1.3.3.cmml">min</mi></msub><mo id="alg1.l14.m1.2.2.1.2" xref="alg1.l14.m1.2.2.1.2.cmml">⁢</mo><mrow id="alg1.l14.m1.2.2.1.1.1" xref="alg1.l14.m1.2.2.1.1.1.1.cmml"><mo id="alg1.l14.m1.2.2.1.1.1.2" stretchy="false" xref="alg1.l14.m1.2.2.1.1.1.1.cmml">(</mo><mrow id="alg1.l14.m1.2.2.1.1.1.1" xref="alg1.l14.m1.2.2.1.1.1.1.cmml"><mi id="alg1.l14.m1.2.2.1.1.1.1.2" mathvariant="normal" xref="alg1.l14.m1.2.2.1.1.1.1.2.cmml">Ψ</mi><mo id="alg1.l14.m1.2.2.1.1.1.1.1" xref="alg1.l14.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="alg1.l14.m1.2.2.1.1.1.1.3.2" xref="alg1.l14.m1.2.2.1.1.1.1.cmml"><mo id="alg1.l14.m1.2.2.1.1.1.1.3.2.1" stretchy="false" xref="alg1.l14.m1.2.2.1.1.1.1.cmml">(</mo><mi id="alg1.l14.m1.1.1" xref="alg1.l14.m1.1.1.cmml">t</mi><mo id="alg1.l14.m1.2.2.1.1.1.1.3.2.2" stretchy="false" xref="alg1.l14.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="alg1.l14.m1.2.2.1.1.1.3" stretchy="false" xref="alg1.l14.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l14.m1.2b"><apply id="alg1.l14.m1.2.2.cmml" xref="alg1.l14.m1.2.2"><ci id="alg1.l14.m1.2.2.2.cmml" xref="alg1.l14.m1.2.2.2">←</ci><apply id="alg1.l14.m1.2.2.3.cmml" xref="alg1.l14.m1.2.2.3"><csymbol cd="ambiguous" id="alg1.l14.m1.2.2.3.1.cmml" xref="alg1.l14.m1.2.2.3">subscript</csymbol><ci id="alg1.l14.m1.2.2.3.2.cmml" xref="alg1.l14.m1.2.2.3.2">𝜎</ci><ci id="alg1.l14.m1.2.2.3.3.cmml" xref="alg1.l14.m1.2.2.3.3">c</ci></apply><apply id="alg1.l14.m1.2.2.1.cmml" xref="alg1.l14.m1.2.2.1"><times id="alg1.l14.m1.2.2.1.2.cmml" xref="alg1.l14.m1.2.2.1.2"></times><apply id="alg1.l14.m1.2.2.1.3.cmml" xref="alg1.l14.m1.2.2.1.3"><csymbol cd="ambiguous" id="alg1.l14.m1.2.2.1.3.1.cmml" xref="alg1.l14.m1.2.2.1.3">subscript</csymbol><ci id="alg1.l14.m1.2.2.1.3.2.cmml" xref="alg1.l14.m1.2.2.1.3.2">𝜎</ci><min id="alg1.l14.m1.2.2.1.3.3.cmml" xref="alg1.l14.m1.2.2.1.3.3"></min></apply><apply id="alg1.l14.m1.2.2.1.1.1.1.cmml" xref="alg1.l14.m1.2.2.1.1.1"><times id="alg1.l14.m1.2.2.1.1.1.1.1.cmml" xref="alg1.l14.m1.2.2.1.1.1.1.1"></times><ci id="alg1.l14.m1.2.2.1.1.1.1.2.cmml" xref="alg1.l14.m1.2.2.1.1.1.1.2">Ψ</ci><ci id="alg1.l14.m1.1.1.cmml" xref="alg1.l14.m1.1.1">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l14.m1.2c">\sigma_{\rm c}\leftarrow\sigma_{\min}(\Psi(t))</annotation><annotation encoding="application/x-llamapun" id="alg1.l14.m1.2d">italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ← italic_σ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Ψ ( italic_t ) )</annotation></semantics></math>, <math alttext="t_{\rm e}\leftarrow t" class="ltx_Math" display="inline" id="alg1.l14.m2.1"><semantics id="alg1.l14.m2.1a"><mrow id="alg1.l14.m2.1.1" xref="alg1.l14.m2.1.1.cmml"><msub id="alg1.l14.m2.1.1.2" xref="alg1.l14.m2.1.1.2.cmml"><mi id="alg1.l14.m2.1.1.2.2" xref="alg1.l14.m2.1.1.2.2.cmml">t</mi><mi id="alg1.l14.m2.1.1.2.3" mathvariant="normal" xref="alg1.l14.m2.1.1.2.3.cmml">e</mi></msub><mo id="alg1.l14.m2.1.1.1" stretchy="false" xref="alg1.l14.m2.1.1.1.cmml">←</mo><mi id="alg1.l14.m2.1.1.3" xref="alg1.l14.m2.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l14.m2.1b"><apply id="alg1.l14.m2.1.1.cmml" xref="alg1.l14.m2.1.1"><ci id="alg1.l14.m2.1.1.1.cmml" xref="alg1.l14.m2.1.1.1">←</ci><apply id="alg1.l14.m2.1.1.2.cmml" xref="alg1.l14.m2.1.1.2"><csymbol cd="ambiguous" id="alg1.l14.m2.1.1.2.1.cmml" xref="alg1.l14.m2.1.1.2">subscript</csymbol><ci id="alg1.l14.m2.1.1.2.2.cmml" xref="alg1.l14.m2.1.1.2.2">𝑡</ci><ci id="alg1.l14.m2.1.1.2.3.cmml" xref="alg1.l14.m2.1.1.2.3">e</ci></apply><ci id="alg1.l14.m2.1.1.3.cmml" xref="alg1.l14.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l14.m2.1c">t_{\rm e}\leftarrow t</annotation><annotation encoding="application/x-llamapun" id="alg1.l14.m2.1d">italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ← italic_t</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l15"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l15.1.1.1" style="font-size:80%;">15:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l15.2">end</span> <span class="ltx_text ltx_font_bold" id="alg1.l15.3">if</span> </div> <div class="ltx_listingline" id="alg1.l16"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l16.1.1.1" style="font-size:80%;">16:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l16.2">end</span> <span class="ltx_text ltx_font_bold" id="alg1.l16.3">if</span> </div> <div class="ltx_listingline" id="alg1.l17"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l17.1.1.1" style="font-size:80%;">17:</span></span><span class="ltx_text ltx_font_bold" id="alg1.l17.2">end</span> <span class="ltx_text ltx_font_bold" id="alg1.l17.3">for</span> </div> </div> </figure> <figure class="ltx_figure" id="S4.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="211" id="S4.F1.g1" src="x1.png" width="484"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>An illustration of the current maximal exciting strength <math alttext="\sigma_{\rm c}" class="ltx_Math" display="inline" id="S4.F1.5.m1.1"><semantics id="S4.F1.5.m1.1b"><msub id="S4.F1.5.m1.1.1" xref="S4.F1.5.m1.1.1.cmml"><mi id="S4.F1.5.m1.1.1.2" xref="S4.F1.5.m1.1.1.2.cmml">σ</mi><mi id="S4.F1.5.m1.1.1.3" mathvariant="normal" xref="S4.F1.5.m1.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S4.F1.5.m1.1c"><apply id="S4.F1.5.m1.1.1.cmml" xref="S4.F1.5.m1.1.1"><csymbol cd="ambiguous" id="S4.F1.5.m1.1.1.1.cmml" xref="S4.F1.5.m1.1.1">subscript</csymbol><ci id="S4.F1.5.m1.1.1.2.cmml" xref="S4.F1.5.m1.1.1.2">𝜎</ci><ci id="S4.F1.5.m1.1.1.3.cmml" xref="S4.F1.5.m1.1.1.3">c</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F1.5.m1.1d">\sigma_{\rm c}</annotation><annotation encoding="application/x-llamapun" id="S4.F1.5.m1.1e">italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT</annotation></semantics></math> in Algorithm 1. Note that the black solid line denotes <math alttext="\sigma_{\rm c}" class="ltx_Math" display="inline" id="S4.F1.6.m2.1"><semantics id="S4.F1.6.m2.1b"><msub id="S4.F1.6.m2.1.1" xref="S4.F1.6.m2.1.1.cmml"><mi id="S4.F1.6.m2.1.1.2" xref="S4.F1.6.m2.1.1.2.cmml">σ</mi><mi id="S4.F1.6.m2.1.1.3" mathvariant="normal" xref="S4.F1.6.m2.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S4.F1.6.m2.1c"><apply id="S4.F1.6.m2.1.1.cmml" xref="S4.F1.6.m2.1.1"><csymbol cd="ambiguous" id="S4.F1.6.m2.1.1.1.cmml" xref="S4.F1.6.m2.1.1">subscript</csymbol><ci id="S4.F1.6.m2.1.1.2.cmml" xref="S4.F1.6.m2.1.1.2">𝜎</ci><ci id="S4.F1.6.m2.1.1.3.cmml" xref="S4.F1.6.m2.1.1.3">c</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F1.6.m2.1d">\sigma_{\rm c}</annotation><annotation encoding="application/x-llamapun" id="S4.F1.6.m2.1e">italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT</annotation></semantics></math>, the blue and green dash lines are <math alttext="\sigma_{\min}(\Psi_{\zeta})" class="ltx_Math" display="inline" id="S4.F1.7.m3.1"><semantics id="S4.F1.7.m3.1b"><mrow id="S4.F1.7.m3.1.1" xref="S4.F1.7.m3.1.1.cmml"><msub id="S4.F1.7.m3.1.1.3" xref="S4.F1.7.m3.1.1.3.cmml"><mi id="S4.F1.7.m3.1.1.3.2" xref="S4.F1.7.m3.1.1.3.2.cmml">σ</mi><mi id="S4.F1.7.m3.1.1.3.3" xref="S4.F1.7.m3.1.1.3.3.cmml">min</mi></msub><mo id="S4.F1.7.m3.1.1.2" xref="S4.F1.7.m3.1.1.2.cmml">⁢</mo><mrow id="S4.F1.7.m3.1.1.1.1" xref="S4.F1.7.m3.1.1.1.1.1.cmml"><mo id="S4.F1.7.m3.1.1.1.1.2" stretchy="false" xref="S4.F1.7.m3.1.1.1.1.1.cmml">(</mo><msub id="S4.F1.7.m3.1.1.1.1.1" xref="S4.F1.7.m3.1.1.1.1.1.cmml"><mi id="S4.F1.7.m3.1.1.1.1.1.2" mathvariant="normal" xref="S4.F1.7.m3.1.1.1.1.1.2.cmml">Ψ</mi><mi id="S4.F1.7.m3.1.1.1.1.1.3" xref="S4.F1.7.m3.1.1.1.1.1.3.cmml">ζ</mi></msub><mo id="S4.F1.7.m3.1.1.1.1.3" stretchy="false" xref="S4.F1.7.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F1.7.m3.1c"><apply id="S4.F1.7.m3.1.1.cmml" xref="S4.F1.7.m3.1.1"><times id="S4.F1.7.m3.1.1.2.cmml" xref="S4.F1.7.m3.1.1.2"></times><apply id="S4.F1.7.m3.1.1.3.cmml" xref="S4.F1.7.m3.1.1.3"><csymbol cd="ambiguous" id="S4.F1.7.m3.1.1.3.1.cmml" xref="S4.F1.7.m3.1.1.3">subscript</csymbol><ci id="S4.F1.7.m3.1.1.3.2.cmml" xref="S4.F1.7.m3.1.1.3.2">𝜎</ci><min id="S4.F1.7.m3.1.1.3.3.cmml" xref="S4.F1.7.m3.1.1.3.3"></min></apply><apply id="S4.F1.7.m3.1.1.1.1.1.cmml" xref="S4.F1.7.m3.1.1.1.1"><csymbol cd="ambiguous" id="S4.F1.7.m3.1.1.1.1.1.1.cmml" xref="S4.F1.7.m3.1.1.1.1">subscript</csymbol><ci id="S4.F1.7.m3.1.1.1.1.1.2.cmml" xref="S4.F1.7.m3.1.1.1.1.1.2">Ψ</ci><ci id="S4.F1.7.m3.1.1.1.1.1.3.cmml" xref="S4.F1.7.m3.1.1.1.1.1.3">𝜁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F1.7.m3.1d">\sigma_{\min}(\Psi_{\zeta})</annotation><annotation encoding="application/x-llamapun" id="S4.F1.7.m3.1e">italic_σ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Ψ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT )</annotation></semantics></math> in two partial IE stages, and the red dotted line is <math alttext="\sigma_{\min}(\Psi)" class="ltx_Math" display="inline" id="S4.F1.8.m4.1"><semantics id="S4.F1.8.m4.1b"><mrow id="S4.F1.8.m4.1.2" xref="S4.F1.8.m4.1.2.cmml"><msub id="S4.F1.8.m4.1.2.2" xref="S4.F1.8.m4.1.2.2.cmml"><mi id="S4.F1.8.m4.1.2.2.2" xref="S4.F1.8.m4.1.2.2.2.cmml">σ</mi><mi id="S4.F1.8.m4.1.2.2.3" xref="S4.F1.8.m4.1.2.2.3.cmml">min</mi></msub><mo id="S4.F1.8.m4.1.2.1" xref="S4.F1.8.m4.1.2.1.cmml">⁢</mo><mrow id="S4.F1.8.m4.1.2.3.2" xref="S4.F1.8.m4.1.2.cmml"><mo id="S4.F1.8.m4.1.2.3.2.1" stretchy="false" xref="S4.F1.8.m4.1.2.cmml">(</mo><mi id="S4.F1.8.m4.1.1" mathvariant="normal" xref="S4.F1.8.m4.1.1.cmml">Ψ</mi><mo id="S4.F1.8.m4.1.2.3.2.2" stretchy="false" xref="S4.F1.8.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F1.8.m4.1c"><apply id="S4.F1.8.m4.1.2.cmml" xref="S4.F1.8.m4.1.2"><times id="S4.F1.8.m4.1.2.1.cmml" xref="S4.F1.8.m4.1.2.1"></times><apply id="S4.F1.8.m4.1.2.2.cmml" xref="S4.F1.8.m4.1.2.2"><csymbol cd="ambiguous" id="S4.F1.8.m4.1.2.2.1.cmml" xref="S4.F1.8.m4.1.2.2">subscript</csymbol><ci id="S4.F1.8.m4.1.2.2.2.cmml" xref="S4.F1.8.m4.1.2.2.2">𝜎</ci><min id="S4.F1.8.m4.1.2.2.3.cmml" xref="S4.F1.8.m4.1.2.2.3"></min></apply><ci id="S4.F1.8.m4.1.1.cmml" xref="S4.F1.8.m4.1.1">Ψ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F1.8.m4.1d">\sigma_{\min}(\Psi)</annotation><annotation encoding="application/x-llamapun" id="S4.F1.8.m4.1e">italic_σ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Ψ )</annotation></semantics></math>.</figcaption> </figure> <div class="ltx_para" id="S4.SS1.p7"> <p class="ltx_p" id="S4.SS1.p7.5">To counteract the transient process of the modeling error term <math alttext="\Phi^{T}\tilde{\bm{\theta}}" class="ltx_Math" display="inline" id="S4.SS1.p7.1.m1.1"><semantics id="S4.SS1.p7.1.m1.1a"><mrow id="S4.SS1.p7.1.m1.1.1" xref="S4.SS1.p7.1.m1.1.1.cmml"><msup id="S4.SS1.p7.1.m1.1.1.2" xref="S4.SS1.p7.1.m1.1.1.2.cmml"><mi id="S4.SS1.p7.1.m1.1.1.2.2" mathvariant="normal" xref="S4.SS1.p7.1.m1.1.1.2.2.cmml">Φ</mi><mi id="S4.SS1.p7.1.m1.1.1.2.3" xref="S4.SS1.p7.1.m1.1.1.2.3.cmml">T</mi></msup><mo id="S4.SS1.p7.1.m1.1.1.1" xref="S4.SS1.p7.1.m1.1.1.1.cmml">⁢</mo><mover accent="true" id="S4.SS1.p7.1.m1.1.1.3" xref="S4.SS1.p7.1.m1.1.1.3.cmml"><mi id="S4.SS1.p7.1.m1.1.1.3.2" xref="S4.SS1.p7.1.m1.1.1.3.2.cmml">𝜽</mi><mo id="S4.SS1.p7.1.m1.1.1.3.1" xref="S4.SS1.p7.1.m1.1.1.3.1.cmml">~</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p7.1.m1.1b"><apply id="S4.SS1.p7.1.m1.1.1.cmml" xref="S4.SS1.p7.1.m1.1.1"><times id="S4.SS1.p7.1.m1.1.1.1.cmml" xref="S4.SS1.p7.1.m1.1.1.1"></times><apply id="S4.SS1.p7.1.m1.1.1.2.cmml" xref="S4.SS1.p7.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p7.1.m1.1.1.2.1.cmml" xref="S4.SS1.p7.1.m1.1.1.2">superscript</csymbol><ci id="S4.SS1.p7.1.m1.1.1.2.2.cmml" xref="S4.SS1.p7.1.m1.1.1.2.2">Φ</ci><ci id="S4.SS1.p7.1.m1.1.1.2.3.cmml" xref="S4.SS1.p7.1.m1.1.1.2.3">𝑇</ci></apply><apply id="S4.SS1.p7.1.m1.1.1.3.cmml" xref="S4.SS1.p7.1.m1.1.1.3"><ci id="S4.SS1.p7.1.m1.1.1.3.1.cmml" xref="S4.SS1.p7.1.m1.1.1.3.1">~</ci><ci id="S4.SS1.p7.1.m1.1.1.3.2.cmml" xref="S4.SS1.p7.1.m1.1.1.3.2">𝜽</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p7.1.m1.1c">\Phi^{T}\tilde{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p7.1.m1.1d">roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E7" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">7</span></a>) such that closed-loop stability can be ensured without resorting to the nonlinear damping terms <math alttext="k_{\rm{d}\it i}\|\bm{\psi}_{i}\|^{2}" class="ltx_Math" display="inline" id="S4.SS1.p7.2.m2.1"><semantics id="S4.SS1.p7.2.m2.1a"><mrow id="S4.SS1.p7.2.m2.1.1" xref="S4.SS1.p7.2.m2.1.1.cmml"><msub id="S4.SS1.p7.2.m2.1.1.3" xref="S4.SS1.p7.2.m2.1.1.3.cmml"><mi id="S4.SS1.p7.2.m2.1.1.3.2" xref="S4.SS1.p7.2.m2.1.1.3.2.cmml">k</mi><mrow id="S4.SS1.p7.2.m2.1.1.3.3" xref="S4.SS1.p7.2.m2.1.1.3.3.cmml"><mi id="S4.SS1.p7.2.m2.1.1.3.3.2" mathvariant="normal" xref="S4.SS1.p7.2.m2.1.1.3.3.2.cmml">d</mi><mo id="S4.SS1.p7.2.m2.1.1.3.3.1" xref="S4.SS1.p7.2.m2.1.1.3.3.1.cmml">⁢</mo><mi id="S4.SS1.p7.2.m2.1.1.3.3.3" xref="S4.SS1.p7.2.m2.1.1.3.3.3.cmml">i</mi></mrow></msub><mo id="S4.SS1.p7.2.m2.1.1.2" xref="S4.SS1.p7.2.m2.1.1.2.cmml">⁢</mo><msup id="S4.SS1.p7.2.m2.1.1.1" xref="S4.SS1.p7.2.m2.1.1.1.cmml"><mrow id="S4.SS1.p7.2.m2.1.1.1.1.1" xref="S4.SS1.p7.2.m2.1.1.1.1.2.cmml"><mo id="S4.SS1.p7.2.m2.1.1.1.1.1.2" stretchy="false" xref="S4.SS1.p7.2.m2.1.1.1.1.2.1.cmml">‖</mo><msub id="S4.SS1.p7.2.m2.1.1.1.1.1.1" xref="S4.SS1.p7.2.m2.1.1.1.1.1.1.cmml"><mi id="S4.SS1.p7.2.m2.1.1.1.1.1.1.2" xref="S4.SS1.p7.2.m2.1.1.1.1.1.1.2.cmml">𝝍</mi><mi id="S4.SS1.p7.2.m2.1.1.1.1.1.1.3" xref="S4.SS1.p7.2.m2.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S4.SS1.p7.2.m2.1.1.1.1.1.3" stretchy="false" xref="S4.SS1.p7.2.m2.1.1.1.1.2.1.cmml">‖</mo></mrow><mn id="S4.SS1.p7.2.m2.1.1.1.3" xref="S4.SS1.p7.2.m2.1.1.1.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p7.2.m2.1b"><apply id="S4.SS1.p7.2.m2.1.1.cmml" xref="S4.SS1.p7.2.m2.1.1"><times id="S4.SS1.p7.2.m2.1.1.2.cmml" xref="S4.SS1.p7.2.m2.1.1.2"></times><apply id="S4.SS1.p7.2.m2.1.1.3.cmml" xref="S4.SS1.p7.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p7.2.m2.1.1.3.1.cmml" xref="S4.SS1.p7.2.m2.1.1.3">subscript</csymbol><ci id="S4.SS1.p7.2.m2.1.1.3.2.cmml" xref="S4.SS1.p7.2.m2.1.1.3.2">𝑘</ci><apply id="S4.SS1.p7.2.m2.1.1.3.3.cmml" xref="S4.SS1.p7.2.m2.1.1.3.3"><times id="S4.SS1.p7.2.m2.1.1.3.3.1.cmml" xref="S4.SS1.p7.2.m2.1.1.3.3.1"></times><ci id="S4.SS1.p7.2.m2.1.1.3.3.2.cmml" xref="S4.SS1.p7.2.m2.1.1.3.3.2">d</ci><ci id="S4.SS1.p7.2.m2.1.1.3.3.3.cmml" xref="S4.SS1.p7.2.m2.1.1.3.3.3">𝑖</ci></apply></apply><apply id="S4.SS1.p7.2.m2.1.1.1.cmml" xref="S4.SS1.p7.2.m2.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p7.2.m2.1.1.1.2.cmml" xref="S4.SS1.p7.2.m2.1.1.1">superscript</csymbol><apply id="S4.SS1.p7.2.m2.1.1.1.1.2.cmml" xref="S4.SS1.p7.2.m2.1.1.1.1.1"><csymbol cd="latexml" id="S4.SS1.p7.2.m2.1.1.1.1.2.1.cmml" xref="S4.SS1.p7.2.m2.1.1.1.1.1.2">norm</csymbol><apply id="S4.SS1.p7.2.m2.1.1.1.1.1.1.cmml" xref="S4.SS1.p7.2.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p7.2.m2.1.1.1.1.1.1.1.cmml" xref="S4.SS1.p7.2.m2.1.1.1.1.1.1">subscript</csymbol><ci id="S4.SS1.p7.2.m2.1.1.1.1.1.1.2.cmml" xref="S4.SS1.p7.2.m2.1.1.1.1.1.1.2">𝝍</ci><ci id="S4.SS1.p7.2.m2.1.1.1.1.1.1.3.cmml" xref="S4.SS1.p7.2.m2.1.1.1.1.1.1.3">𝑖</ci></apply></apply><cn id="S4.SS1.p7.2.m2.1.1.1.3.cmml" type="integer" xref="S4.SS1.p7.2.m2.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p7.2.m2.1c">k_{\rm{d}\it i}\|\bm{\psi}_{i}\|^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p7.2.m2.1d">italic_k start_POSTSUBSCRIPT roman_d italic_i end_POSTSUBSCRIPT ∥ bold_italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> (<math alttext="i=" class="ltx_Math" display="inline" id="S4.SS1.p7.3.m3.1"><semantics id="S4.SS1.p7.3.m3.1a"><mrow id="S4.SS1.p7.3.m3.1.1" xref="S4.SS1.p7.3.m3.1.1.cmml"><mi id="S4.SS1.p7.3.m3.1.1.2" xref="S4.SS1.p7.3.m3.1.1.2.cmml">i</mi><mo id="S4.SS1.p7.3.m3.1.1.1" xref="S4.SS1.p7.3.m3.1.1.1.cmml">=</mo><mi id="S4.SS1.p7.3.m3.1.1.3" xref="S4.SS1.p7.3.m3.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p7.3.m3.1b"><apply id="S4.SS1.p7.3.m3.1.1.cmml" xref="S4.SS1.p7.3.m3.1.1"><eq id="S4.SS1.p7.3.m3.1.1.1.cmml" xref="S4.SS1.p7.3.m3.1.1.1"></eq><ci id="S4.SS1.p7.3.m3.1.1.2.cmml" xref="S4.SS1.p7.3.m3.1.1.2">𝑖</ci><csymbol cd="latexml" id="S4.SS1.p7.3.m3.1.1.3.cmml" xref="S4.SS1.p7.3.m3.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p7.3.m3.1c">i=</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p7.3.m3.1d">italic_i =</annotation></semantics></math> 1 to <math alttext="n" class="ltx_Math" display="inline" id="S4.SS1.p7.4.m4.1"><semantics id="S4.SS1.p7.4.m4.1a"><mi id="S4.SS1.p7.4.m4.1.1" xref="S4.SS1.p7.4.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p7.4.m4.1b"><ci id="S4.SS1.p7.4.m4.1.1.cmml" xref="S4.SS1.p7.4.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p7.4.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p7.4.m4.1d">italic_n</annotation></semantics></math>), we introduce extra prediction error feedback. Applying <math alttext="H(s)" class="ltx_Math" display="inline" id="S4.SS1.p7.5.m5.1"><semantics id="S4.SS1.p7.5.m5.1a"><mrow id="S4.SS1.p7.5.m5.1.2" xref="S4.SS1.p7.5.m5.1.2.cmml"><mi id="S4.SS1.p7.5.m5.1.2.2" xref="S4.SS1.p7.5.m5.1.2.2.cmml">H</mi><mo id="S4.SS1.p7.5.m5.1.2.1" xref="S4.SS1.p7.5.m5.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p7.5.m5.1.2.3.2" xref="S4.SS1.p7.5.m5.1.2.cmml"><mo id="S4.SS1.p7.5.m5.1.2.3.2.1" stretchy="false" xref="S4.SS1.p7.5.m5.1.2.cmml">(</mo><mi id="S4.SS1.p7.5.m5.1.1" xref="S4.SS1.p7.5.m5.1.1.cmml">s</mi><mo id="S4.SS1.p7.5.m5.1.2.3.2.2" stretchy="false" xref="S4.SS1.p7.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p7.5.m5.1b"><apply id="S4.SS1.p7.5.m5.1.2.cmml" xref="S4.SS1.p7.5.m5.1.2"><times id="S4.SS1.p7.5.m5.1.2.1.cmml" xref="S4.SS1.p7.5.m5.1.2.1"></times><ci id="S4.SS1.p7.5.m5.1.2.2.cmml" xref="S4.SS1.p7.5.m5.1.2.2">𝐻</ci><ci id="S4.SS1.p7.5.m5.1.1.cmml" xref="S4.SS1.p7.5.m5.1.1">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p7.5.m5.1c">H(s)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p7.5.m5.1d">italic_H ( italic_s )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E17" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">17</span></a>) to (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E7" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">7</span></a>) results in a filtered regression equation</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx15"> <tbody id="S4.E21"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\bm{z}(t)=\Phi_{\rm f}^{T}(t)\bm{\theta}" class="ltx_Math" display="inline" id="S4.E21.m1.2"><semantics id="S4.E21.m1.2a"><mrow id="S4.E21.m1.2.3" xref="S4.E21.m1.2.3.cmml"><mrow id="S4.E21.m1.2.3.2" xref="S4.E21.m1.2.3.2.cmml"><mi id="S4.E21.m1.2.3.2.2" xref="S4.E21.m1.2.3.2.2.cmml">𝒛</mi><mo id="S4.E21.m1.2.3.2.1" xref="S4.E21.m1.2.3.2.1.cmml">⁢</mo><mrow id="S4.E21.m1.2.3.2.3.2" xref="S4.E21.m1.2.3.2.cmml"><mo id="S4.E21.m1.2.3.2.3.2.1" stretchy="false" xref="S4.E21.m1.2.3.2.cmml">(</mo><mi id="S4.E21.m1.1.1" xref="S4.E21.m1.1.1.cmml">t</mi><mo id="S4.E21.m1.2.3.2.3.2.2" stretchy="false" xref="S4.E21.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.E21.m1.2.3.1" xref="S4.E21.m1.2.3.1.cmml">=</mo><mrow id="S4.E21.m1.2.3.3" xref="S4.E21.m1.2.3.3.cmml"><msubsup id="S4.E21.m1.2.3.3.2" xref="S4.E21.m1.2.3.3.2.cmml"><mi id="S4.E21.m1.2.3.3.2.2.2" mathvariant="normal" xref="S4.E21.m1.2.3.3.2.2.2.cmml">Φ</mi><mi id="S4.E21.m1.2.3.3.2.2.3" mathvariant="normal" xref="S4.E21.m1.2.3.3.2.2.3.cmml">f</mi><mi id="S4.E21.m1.2.3.3.2.3" xref="S4.E21.m1.2.3.3.2.3.cmml">T</mi></msubsup><mo id="S4.E21.m1.2.3.3.1" xref="S4.E21.m1.2.3.3.1.cmml">⁢</mo><mrow id="S4.E21.m1.2.3.3.3.2" xref="S4.E21.m1.2.3.3.cmml"><mo id="S4.E21.m1.2.3.3.3.2.1" stretchy="false" xref="S4.E21.m1.2.3.3.cmml">(</mo><mi id="S4.E21.m1.2.2" xref="S4.E21.m1.2.2.cmml">t</mi><mo id="S4.E21.m1.2.3.3.3.2.2" stretchy="false" xref="S4.E21.m1.2.3.3.cmml">)</mo></mrow><mo id="S4.E21.m1.2.3.3.1a" xref="S4.E21.m1.2.3.3.1.cmml">⁢</mo><mi id="S4.E21.m1.2.3.3.4" xref="S4.E21.m1.2.3.3.4.cmml">𝜽</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E21.m1.2b"><apply id="S4.E21.m1.2.3.cmml" xref="S4.E21.m1.2.3"><eq id="S4.E21.m1.2.3.1.cmml" xref="S4.E21.m1.2.3.1"></eq><apply id="S4.E21.m1.2.3.2.cmml" xref="S4.E21.m1.2.3.2"><times id="S4.E21.m1.2.3.2.1.cmml" xref="S4.E21.m1.2.3.2.1"></times><ci id="S4.E21.m1.2.3.2.2.cmml" xref="S4.E21.m1.2.3.2.2">𝒛</ci><ci id="S4.E21.m1.1.1.cmml" xref="S4.E21.m1.1.1">𝑡</ci></apply><apply id="S4.E21.m1.2.3.3.cmml" xref="S4.E21.m1.2.3.3"><times id="S4.E21.m1.2.3.3.1.cmml" xref="S4.E21.m1.2.3.3.1"></times><apply id="S4.E21.m1.2.3.3.2.cmml" xref="S4.E21.m1.2.3.3.2"><csymbol cd="ambiguous" id="S4.E21.m1.2.3.3.2.1.cmml" xref="S4.E21.m1.2.3.3.2">superscript</csymbol><apply id="S4.E21.m1.2.3.3.2.2.cmml" xref="S4.E21.m1.2.3.3.2"><csymbol cd="ambiguous" id="S4.E21.m1.2.3.3.2.2.1.cmml" xref="S4.E21.m1.2.3.3.2">subscript</csymbol><ci id="S4.E21.m1.2.3.3.2.2.2.cmml" xref="S4.E21.m1.2.3.3.2.2.2">Φ</ci><ci id="S4.E21.m1.2.3.3.2.2.3.cmml" xref="S4.E21.m1.2.3.3.2.2.3">f</ci></apply><ci id="S4.E21.m1.2.3.3.2.3.cmml" xref="S4.E21.m1.2.3.3.2.3">𝑇</ci></apply><ci id="S4.E21.m1.2.2.cmml" xref="S4.E21.m1.2.2">𝑡</ci><ci id="S4.E21.m1.2.3.3.4.cmml" xref="S4.E21.m1.2.3.3.4">𝜽</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E21.m1.2c">\displaystyle\bm{z}(t)=\Phi_{\rm f}^{T}(t)\bm{\theta}</annotation><annotation encoding="application/x-llamapun" id="S4.E21.m1.2d">bold_italic_z ( italic_t ) = roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_t ) bold_italic_θ</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(21)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p7.11">with <math alttext="\bm{z}(t)" class="ltx_Math" display="inline" id="S4.SS1.p7.6.m1.1"><semantics id="S4.SS1.p7.6.m1.1a"><mrow id="S4.SS1.p7.6.m1.1.2" xref="S4.SS1.p7.6.m1.1.2.cmml"><mi id="S4.SS1.p7.6.m1.1.2.2" xref="S4.SS1.p7.6.m1.1.2.2.cmml">𝒛</mi><mo id="S4.SS1.p7.6.m1.1.2.1" xref="S4.SS1.p7.6.m1.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p7.6.m1.1.2.3.2" xref="S4.SS1.p7.6.m1.1.2.cmml"><mo id="S4.SS1.p7.6.m1.1.2.3.2.1" stretchy="false" xref="S4.SS1.p7.6.m1.1.2.cmml">(</mo><mi id="S4.SS1.p7.6.m1.1.1" xref="S4.SS1.p7.6.m1.1.1.cmml">t</mi><mo id="S4.SS1.p7.6.m1.1.2.3.2.2" stretchy="false" xref="S4.SS1.p7.6.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p7.6.m1.1b"><apply id="S4.SS1.p7.6.m1.1.2.cmml" xref="S4.SS1.p7.6.m1.1.2"><times id="S4.SS1.p7.6.m1.1.2.1.cmml" xref="S4.SS1.p7.6.m1.1.2.1"></times><ci id="S4.SS1.p7.6.m1.1.2.2.cmml" xref="S4.SS1.p7.6.m1.1.2.2">𝒛</ci><ci id="S4.SS1.p7.6.m1.1.1.cmml" xref="S4.SS1.p7.6.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p7.6.m1.1c">\bm{z}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p7.6.m1.1d">bold_italic_z ( italic_t )</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S4.SS1.p7.7.m2.1"><semantics id="S4.SS1.p7.7.m2.1a"><mo id="S4.SS1.p7.7.m2.1.1" xref="S4.SS1.p7.7.m2.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.p7.7.m2.1b"><csymbol cd="latexml" id="S4.SS1.p7.7.m2.1.1.cmml" xref="S4.SS1.p7.7.m2.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p7.7.m2.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p7.7.m2.1d">:=</annotation></semantics></math> <math alttext="sH(s)[\bm{e}]+H(s)[\Phi^{T}\hat{\bm{\theta}}-\Lambda\bm{e}]" class="ltx_Math" display="inline" id="S4.SS1.p7.8.m3.4"><semantics id="S4.SS1.p7.8.m3.4a"><mrow id="S4.SS1.p7.8.m3.4.4" xref="S4.SS1.p7.8.m3.4.4.cmml"><mrow id="S4.SS1.p7.8.m3.4.4.3" xref="S4.SS1.p7.8.m3.4.4.3.cmml"><mi id="S4.SS1.p7.8.m3.4.4.3.2" xref="S4.SS1.p7.8.m3.4.4.3.2.cmml">s</mi><mo id="S4.SS1.p7.8.m3.4.4.3.1" xref="S4.SS1.p7.8.m3.4.4.3.1.cmml">⁢</mo><mi id="S4.SS1.p7.8.m3.4.4.3.3" xref="S4.SS1.p7.8.m3.4.4.3.3.cmml">H</mi><mo id="S4.SS1.p7.8.m3.4.4.3.1a" xref="S4.SS1.p7.8.m3.4.4.3.1.cmml">⁢</mo><mrow id="S4.SS1.p7.8.m3.4.4.3.4.2" xref="S4.SS1.p7.8.m3.4.4.3.cmml"><mo id="S4.SS1.p7.8.m3.4.4.3.4.2.1" stretchy="false" xref="S4.SS1.p7.8.m3.4.4.3.cmml">(</mo><mi id="S4.SS1.p7.8.m3.1.1" xref="S4.SS1.p7.8.m3.1.1.cmml">s</mi><mo id="S4.SS1.p7.8.m3.4.4.3.4.2.2" stretchy="false" xref="S4.SS1.p7.8.m3.4.4.3.cmml">)</mo></mrow><mo id="S4.SS1.p7.8.m3.4.4.3.1b" xref="S4.SS1.p7.8.m3.4.4.3.1.cmml">⁢</mo><mrow id="S4.SS1.p7.8.m3.4.4.3.5.2" xref="S4.SS1.p7.8.m3.4.4.3.5.1.cmml"><mo id="S4.SS1.p7.8.m3.4.4.3.5.2.1" stretchy="false" xref="S4.SS1.p7.8.m3.4.4.3.5.1.1.cmml">[</mo><mi id="S4.SS1.p7.8.m3.2.2" xref="S4.SS1.p7.8.m3.2.2.cmml">𝒆</mi><mo id="S4.SS1.p7.8.m3.4.4.3.5.2.2" stretchy="false" xref="S4.SS1.p7.8.m3.4.4.3.5.1.1.cmml">]</mo></mrow></mrow><mo id="S4.SS1.p7.8.m3.4.4.2" xref="S4.SS1.p7.8.m3.4.4.2.cmml">+</mo><mrow id="S4.SS1.p7.8.m3.4.4.1" xref="S4.SS1.p7.8.m3.4.4.1.cmml"><mi id="S4.SS1.p7.8.m3.4.4.1.3" xref="S4.SS1.p7.8.m3.4.4.1.3.cmml">H</mi><mo id="S4.SS1.p7.8.m3.4.4.1.2" xref="S4.SS1.p7.8.m3.4.4.1.2.cmml">⁢</mo><mrow id="S4.SS1.p7.8.m3.4.4.1.4.2" xref="S4.SS1.p7.8.m3.4.4.1.cmml"><mo id="S4.SS1.p7.8.m3.4.4.1.4.2.1" stretchy="false" xref="S4.SS1.p7.8.m3.4.4.1.cmml">(</mo><mi id="S4.SS1.p7.8.m3.3.3" xref="S4.SS1.p7.8.m3.3.3.cmml">s</mi><mo id="S4.SS1.p7.8.m3.4.4.1.4.2.2" stretchy="false" xref="S4.SS1.p7.8.m3.4.4.1.cmml">)</mo></mrow><mo id="S4.SS1.p7.8.m3.4.4.1.2a" xref="S4.SS1.p7.8.m3.4.4.1.2.cmml">⁢</mo><mrow id="S4.SS1.p7.8.m3.4.4.1.1.1" xref="S4.SS1.p7.8.m3.4.4.1.1.2.cmml"><mo id="S4.SS1.p7.8.m3.4.4.1.1.1.2" stretchy="false" xref="S4.SS1.p7.8.m3.4.4.1.1.2.1.cmml">[</mo><mrow id="S4.SS1.p7.8.m3.4.4.1.1.1.1" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.cmml"><mrow id="S4.SS1.p7.8.m3.4.4.1.1.1.1.2" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.cmml"><msup id="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.2" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.2.cmml"><mi id="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.2.2" mathvariant="normal" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.2.2.cmml">Φ</mi><mi id="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.2.3" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.2.3.cmml">T</mi></msup><mo id="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.1" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.1.cmml">⁢</mo><mover accent="true" id="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.3" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.3.cmml"><mi id="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.3.2" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.3.2.cmml">𝜽</mi><mo id="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.3.1" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.3.1.cmml">^</mo></mover></mrow><mo id="S4.SS1.p7.8.m3.4.4.1.1.1.1.1" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.1.cmml">−</mo><mrow id="S4.SS1.p7.8.m3.4.4.1.1.1.1.3" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.3.cmml"><mi id="S4.SS1.p7.8.m3.4.4.1.1.1.1.3.2" mathvariant="normal" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.3.2.cmml">Λ</mi><mo id="S4.SS1.p7.8.m3.4.4.1.1.1.1.3.1" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.3.1.cmml">⁢</mo><mi id="S4.SS1.p7.8.m3.4.4.1.1.1.1.3.3" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.3.3.cmml">𝒆</mi></mrow></mrow><mo id="S4.SS1.p7.8.m3.4.4.1.1.1.3" stretchy="false" xref="S4.SS1.p7.8.m3.4.4.1.1.2.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p7.8.m3.4b"><apply id="S4.SS1.p7.8.m3.4.4.cmml" xref="S4.SS1.p7.8.m3.4.4"><plus id="S4.SS1.p7.8.m3.4.4.2.cmml" xref="S4.SS1.p7.8.m3.4.4.2"></plus><apply id="S4.SS1.p7.8.m3.4.4.3.cmml" xref="S4.SS1.p7.8.m3.4.4.3"><times id="S4.SS1.p7.8.m3.4.4.3.1.cmml" xref="S4.SS1.p7.8.m3.4.4.3.1"></times><ci id="S4.SS1.p7.8.m3.4.4.3.2.cmml" xref="S4.SS1.p7.8.m3.4.4.3.2">𝑠</ci><ci id="S4.SS1.p7.8.m3.4.4.3.3.cmml" xref="S4.SS1.p7.8.m3.4.4.3.3">𝐻</ci><ci id="S4.SS1.p7.8.m3.1.1.cmml" xref="S4.SS1.p7.8.m3.1.1">𝑠</ci><apply id="S4.SS1.p7.8.m3.4.4.3.5.1.cmml" xref="S4.SS1.p7.8.m3.4.4.3.5.2"><csymbol cd="latexml" id="S4.SS1.p7.8.m3.4.4.3.5.1.1.cmml" xref="S4.SS1.p7.8.m3.4.4.3.5.2.1">delimited-[]</csymbol><ci id="S4.SS1.p7.8.m3.2.2.cmml" xref="S4.SS1.p7.8.m3.2.2">𝒆</ci></apply></apply><apply id="S4.SS1.p7.8.m3.4.4.1.cmml" xref="S4.SS1.p7.8.m3.4.4.1"><times id="S4.SS1.p7.8.m3.4.4.1.2.cmml" xref="S4.SS1.p7.8.m3.4.4.1.2"></times><ci id="S4.SS1.p7.8.m3.4.4.1.3.cmml" xref="S4.SS1.p7.8.m3.4.4.1.3">𝐻</ci><ci id="S4.SS1.p7.8.m3.3.3.cmml" xref="S4.SS1.p7.8.m3.3.3">𝑠</ci><apply id="S4.SS1.p7.8.m3.4.4.1.1.2.cmml" xref="S4.SS1.p7.8.m3.4.4.1.1.1"><csymbol cd="latexml" id="S4.SS1.p7.8.m3.4.4.1.1.2.1.cmml" xref="S4.SS1.p7.8.m3.4.4.1.1.1.2">delimited-[]</csymbol><apply id="S4.SS1.p7.8.m3.4.4.1.1.1.1.cmml" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1"><minus id="S4.SS1.p7.8.m3.4.4.1.1.1.1.1.cmml" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.1"></minus><apply id="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.cmml" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.2"><times id="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.1.cmml" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.1"></times><apply id="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.2.cmml" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.2.1.cmml" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.2">superscript</csymbol><ci id="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.2.2.cmml" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.2.2">Φ</ci><ci id="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.2.3.cmml" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.2.3">𝑇</ci></apply><apply id="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.3.cmml" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.3"><ci id="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.3.1.cmml" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.3.1">^</ci><ci id="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.3.2.cmml" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.2.3.2">𝜽</ci></apply></apply><apply id="S4.SS1.p7.8.m3.4.4.1.1.1.1.3.cmml" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.3"><times id="S4.SS1.p7.8.m3.4.4.1.1.1.1.3.1.cmml" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.3.1"></times><ci id="S4.SS1.p7.8.m3.4.4.1.1.1.1.3.2.cmml" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.3.2">Λ</ci><ci id="S4.SS1.p7.8.m3.4.4.1.1.1.1.3.3.cmml" xref="S4.SS1.p7.8.m3.4.4.1.1.1.1.3.3">𝒆</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p7.8.m3.4c">sH(s)[\bm{e}]+H(s)[\Phi^{T}\hat{\bm{\theta}}-\Lambda\bm{e}]</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p7.8.m3.4d">italic_s italic_H ( italic_s ) [ bold_italic_e ] + italic_H ( italic_s ) [ roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over^ start_ARG bold_italic_θ end_ARG - roman_Λ bold_italic_e ]</annotation></semantics></math> and <math alttext="\Phi_{\rm f}" class="ltx_Math" display="inline" id="S4.SS1.p7.9.m4.1"><semantics id="S4.SS1.p7.9.m4.1a"><msub id="S4.SS1.p7.9.m4.1.1" xref="S4.SS1.p7.9.m4.1.1.cmml"><mi id="S4.SS1.p7.9.m4.1.1.2" mathvariant="normal" xref="S4.SS1.p7.9.m4.1.1.2.cmml">Φ</mi><mi id="S4.SS1.p7.9.m4.1.1.3" mathvariant="normal" xref="S4.SS1.p7.9.m4.1.1.3.cmml">f</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p7.9.m4.1b"><apply id="S4.SS1.p7.9.m4.1.1.cmml" xref="S4.SS1.p7.9.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.p7.9.m4.1.1.1.cmml" xref="S4.SS1.p7.9.m4.1.1">subscript</csymbol><ci id="S4.SS1.p7.9.m4.1.1.2.cmml" xref="S4.SS1.p7.9.m4.1.1.2">Φ</ci><ci id="S4.SS1.p7.9.m4.1.1.3.cmml" xref="S4.SS1.p7.9.m4.1.1.3">f</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p7.9.m4.1c">\Phi_{\rm f}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p7.9.m4.1d">roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S4.SS1.p7.10.m5.1"><semantics id="S4.SS1.p7.10.m5.1a"><mo id="S4.SS1.p7.10.m5.1.1" xref="S4.SS1.p7.10.m5.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.p7.10.m5.1b"><csymbol cd="latexml" id="S4.SS1.p7.10.m5.1.1.cmml" xref="S4.SS1.p7.10.m5.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p7.10.m5.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p7.10.m5.1d">:=</annotation></semantics></math> <math alttext="H(s)[\Phi]" class="ltx_Math" display="inline" id="S4.SS1.p7.11.m6.2"><semantics id="S4.SS1.p7.11.m6.2a"><mrow id="S4.SS1.p7.11.m6.2.3" xref="S4.SS1.p7.11.m6.2.3.cmml"><mi id="S4.SS1.p7.11.m6.2.3.2" xref="S4.SS1.p7.11.m6.2.3.2.cmml">H</mi><mo id="S4.SS1.p7.11.m6.2.3.1" xref="S4.SS1.p7.11.m6.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.p7.11.m6.2.3.3.2" xref="S4.SS1.p7.11.m6.2.3.cmml"><mo id="S4.SS1.p7.11.m6.2.3.3.2.1" stretchy="false" xref="S4.SS1.p7.11.m6.2.3.cmml">(</mo><mi id="S4.SS1.p7.11.m6.1.1" xref="S4.SS1.p7.11.m6.1.1.cmml">s</mi><mo id="S4.SS1.p7.11.m6.2.3.3.2.2" stretchy="false" xref="S4.SS1.p7.11.m6.2.3.cmml">)</mo></mrow><mo id="S4.SS1.p7.11.m6.2.3.1a" xref="S4.SS1.p7.11.m6.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.p7.11.m6.2.3.4.2" xref="S4.SS1.p7.11.m6.2.3.4.1.cmml"><mo id="S4.SS1.p7.11.m6.2.3.4.2.1" stretchy="false" xref="S4.SS1.p7.11.m6.2.3.4.1.1.cmml">[</mo><mi id="S4.SS1.p7.11.m6.2.2" mathvariant="normal" xref="S4.SS1.p7.11.m6.2.2.cmml">Φ</mi><mo id="S4.SS1.p7.11.m6.2.3.4.2.2" stretchy="false" xref="S4.SS1.p7.11.m6.2.3.4.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p7.11.m6.2b"><apply id="S4.SS1.p7.11.m6.2.3.cmml" xref="S4.SS1.p7.11.m6.2.3"><times id="S4.SS1.p7.11.m6.2.3.1.cmml" xref="S4.SS1.p7.11.m6.2.3.1"></times><ci id="S4.SS1.p7.11.m6.2.3.2.cmml" xref="S4.SS1.p7.11.m6.2.3.2">𝐻</ci><ci id="S4.SS1.p7.11.m6.1.1.cmml" xref="S4.SS1.p7.11.m6.1.1">𝑠</ci><apply id="S4.SS1.p7.11.m6.2.3.4.1.cmml" xref="S4.SS1.p7.11.m6.2.3.4.2"><csymbol cd="latexml" id="S4.SS1.p7.11.m6.2.3.4.1.1.cmml" xref="S4.SS1.p7.11.m6.2.3.4.2.1">delimited-[]</csymbol><ci id="S4.SS1.p7.11.m6.2.2.cmml" xref="S4.SS1.p7.11.m6.2.2">Φ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p7.11.m6.2c">H(s)[\Phi]</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p7.11.m6.2d">italic_H ( italic_s ) [ roman_Φ ]</annotation></semantics></math>. Then, giving a filtered prediction model</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx16"> <tbody id="S4.E22"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\hat{\bm{z}}(t)=\Phi_{\rm f}^{T}(t)\hat{\bm{\theta}}" class="ltx_Math" display="inline" id="S4.E22.m1.2"><semantics id="S4.E22.m1.2a"><mrow id="S4.E22.m1.2.3" xref="S4.E22.m1.2.3.cmml"><mrow id="S4.E22.m1.2.3.2" xref="S4.E22.m1.2.3.2.cmml"><mover accent="true" id="S4.E22.m1.2.3.2.2" xref="S4.E22.m1.2.3.2.2.cmml"><mi id="S4.E22.m1.2.3.2.2.2" xref="S4.E22.m1.2.3.2.2.2.cmml">𝒛</mi><mo id="S4.E22.m1.2.3.2.2.1" xref="S4.E22.m1.2.3.2.2.1.cmml">^</mo></mover><mo id="S4.E22.m1.2.3.2.1" xref="S4.E22.m1.2.3.2.1.cmml">⁢</mo><mrow id="S4.E22.m1.2.3.2.3.2" xref="S4.E22.m1.2.3.2.cmml"><mo id="S4.E22.m1.2.3.2.3.2.1" stretchy="false" xref="S4.E22.m1.2.3.2.cmml">(</mo><mi id="S4.E22.m1.1.1" xref="S4.E22.m1.1.1.cmml">t</mi><mo id="S4.E22.m1.2.3.2.3.2.2" stretchy="false" xref="S4.E22.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.E22.m1.2.3.1" xref="S4.E22.m1.2.3.1.cmml">=</mo><mrow id="S4.E22.m1.2.3.3" xref="S4.E22.m1.2.3.3.cmml"><msubsup id="S4.E22.m1.2.3.3.2" xref="S4.E22.m1.2.3.3.2.cmml"><mi id="S4.E22.m1.2.3.3.2.2.2" mathvariant="normal" xref="S4.E22.m1.2.3.3.2.2.2.cmml">Φ</mi><mi id="S4.E22.m1.2.3.3.2.2.3" mathvariant="normal" xref="S4.E22.m1.2.3.3.2.2.3.cmml">f</mi><mi id="S4.E22.m1.2.3.3.2.3" xref="S4.E22.m1.2.3.3.2.3.cmml">T</mi></msubsup><mo id="S4.E22.m1.2.3.3.1" xref="S4.E22.m1.2.3.3.1.cmml">⁢</mo><mrow id="S4.E22.m1.2.3.3.3.2" xref="S4.E22.m1.2.3.3.cmml"><mo id="S4.E22.m1.2.3.3.3.2.1" stretchy="false" xref="S4.E22.m1.2.3.3.cmml">(</mo><mi id="S4.E22.m1.2.2" xref="S4.E22.m1.2.2.cmml">t</mi><mo id="S4.E22.m1.2.3.3.3.2.2" stretchy="false" xref="S4.E22.m1.2.3.3.cmml">)</mo></mrow><mo id="S4.E22.m1.2.3.3.1a" xref="S4.E22.m1.2.3.3.1.cmml">⁢</mo><mover accent="true" id="S4.E22.m1.2.3.3.4" xref="S4.E22.m1.2.3.3.4.cmml"><mi id="S4.E22.m1.2.3.3.4.2" xref="S4.E22.m1.2.3.3.4.2.cmml">𝜽</mi><mo id="S4.E22.m1.2.3.3.4.1" xref="S4.E22.m1.2.3.3.4.1.cmml">^</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E22.m1.2b"><apply id="S4.E22.m1.2.3.cmml" xref="S4.E22.m1.2.3"><eq id="S4.E22.m1.2.3.1.cmml" xref="S4.E22.m1.2.3.1"></eq><apply id="S4.E22.m1.2.3.2.cmml" xref="S4.E22.m1.2.3.2"><times id="S4.E22.m1.2.3.2.1.cmml" xref="S4.E22.m1.2.3.2.1"></times><apply id="S4.E22.m1.2.3.2.2.cmml" xref="S4.E22.m1.2.3.2.2"><ci id="S4.E22.m1.2.3.2.2.1.cmml" xref="S4.E22.m1.2.3.2.2.1">^</ci><ci id="S4.E22.m1.2.3.2.2.2.cmml" xref="S4.E22.m1.2.3.2.2.2">𝒛</ci></apply><ci id="S4.E22.m1.1.1.cmml" xref="S4.E22.m1.1.1">𝑡</ci></apply><apply id="S4.E22.m1.2.3.3.cmml" xref="S4.E22.m1.2.3.3"><times id="S4.E22.m1.2.3.3.1.cmml" xref="S4.E22.m1.2.3.3.1"></times><apply id="S4.E22.m1.2.3.3.2.cmml" xref="S4.E22.m1.2.3.3.2"><csymbol cd="ambiguous" id="S4.E22.m1.2.3.3.2.1.cmml" xref="S4.E22.m1.2.3.3.2">superscript</csymbol><apply id="S4.E22.m1.2.3.3.2.2.cmml" xref="S4.E22.m1.2.3.3.2"><csymbol cd="ambiguous" id="S4.E22.m1.2.3.3.2.2.1.cmml" xref="S4.E22.m1.2.3.3.2">subscript</csymbol><ci id="S4.E22.m1.2.3.3.2.2.2.cmml" xref="S4.E22.m1.2.3.3.2.2.2">Φ</ci><ci id="S4.E22.m1.2.3.3.2.2.3.cmml" xref="S4.E22.m1.2.3.3.2.2.3">f</ci></apply><ci id="S4.E22.m1.2.3.3.2.3.cmml" xref="S4.E22.m1.2.3.3.2.3">𝑇</ci></apply><ci id="S4.E22.m1.2.2.cmml" xref="S4.E22.m1.2.2">𝑡</ci><apply id="S4.E22.m1.2.3.3.4.cmml" xref="S4.E22.m1.2.3.3.4"><ci id="S4.E22.m1.2.3.3.4.1.cmml" xref="S4.E22.m1.2.3.3.4.1">^</ci><ci id="S4.E22.m1.2.3.3.4.2.cmml" xref="S4.E22.m1.2.3.3.4.2">𝜽</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E22.m1.2c">\displaystyle\hat{\bm{z}}(t)=\Phi_{\rm f}^{T}(t)\hat{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S4.E22.m1.2d">over^ start_ARG bold_italic_z end_ARG ( italic_t ) = roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_t ) over^ start_ARG bold_italic_θ end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(22)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p7.14">define a general filtered prediction error</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx17"> <tbody id="S4.E23"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\bm{\epsilon}(t):=\bm{z}(t)-\Phi_{\rm f}^{T}(t)\hat{\bm{\theta}}(t)" class="ltx_Math" display="inline" id="S4.E23.m1.4"><semantics id="S4.E23.m1.4a"><mrow id="S4.E23.m1.4.5" xref="S4.E23.m1.4.5.cmml"><mrow id="S4.E23.m1.4.5.2" xref="S4.E23.m1.4.5.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E23.m1.4.5.2.2" mathvariant="bold-italic" xref="S4.E23.m1.4.5.2.2.cmml">ϵ</mi><mo id="S4.E23.m1.4.5.2.1" xref="S4.E23.m1.4.5.2.1.cmml">⁢</mo><mrow id="S4.E23.m1.4.5.2.3.2" xref="S4.E23.m1.4.5.2.cmml"><mo id="S4.E23.m1.4.5.2.3.2.1" stretchy="false" xref="S4.E23.m1.4.5.2.cmml">(</mo><mi id="S4.E23.m1.1.1" xref="S4.E23.m1.1.1.cmml">t</mi><mo id="S4.E23.m1.4.5.2.3.2.2" rspace="0.278em" stretchy="false" xref="S4.E23.m1.4.5.2.cmml">)</mo></mrow></mrow><mo id="S4.E23.m1.4.5.1" rspace="0.278em" xref="S4.E23.m1.4.5.1.cmml">:=</mo><mrow id="S4.E23.m1.4.5.3" xref="S4.E23.m1.4.5.3.cmml"><mrow id="S4.E23.m1.4.5.3.2" xref="S4.E23.m1.4.5.3.2.cmml"><mi id="S4.E23.m1.4.5.3.2.2" xref="S4.E23.m1.4.5.3.2.2.cmml">𝒛</mi><mo id="S4.E23.m1.4.5.3.2.1" xref="S4.E23.m1.4.5.3.2.1.cmml">⁢</mo><mrow id="S4.E23.m1.4.5.3.2.3.2" xref="S4.E23.m1.4.5.3.2.cmml"><mo id="S4.E23.m1.4.5.3.2.3.2.1" stretchy="false" xref="S4.E23.m1.4.5.3.2.cmml">(</mo><mi id="S4.E23.m1.2.2" xref="S4.E23.m1.2.2.cmml">t</mi><mo id="S4.E23.m1.4.5.3.2.3.2.2" stretchy="false" xref="S4.E23.m1.4.5.3.2.cmml">)</mo></mrow></mrow><mo id="S4.E23.m1.4.5.3.1" xref="S4.E23.m1.4.5.3.1.cmml">−</mo><mrow id="S4.E23.m1.4.5.3.3" xref="S4.E23.m1.4.5.3.3.cmml"><msubsup id="S4.E23.m1.4.5.3.3.2" xref="S4.E23.m1.4.5.3.3.2.cmml"><mi id="S4.E23.m1.4.5.3.3.2.2.2" mathvariant="normal" xref="S4.E23.m1.4.5.3.3.2.2.2.cmml">Φ</mi><mi id="S4.E23.m1.4.5.3.3.2.2.3" mathvariant="normal" xref="S4.E23.m1.4.5.3.3.2.2.3.cmml">f</mi><mi id="S4.E23.m1.4.5.3.3.2.3" xref="S4.E23.m1.4.5.3.3.2.3.cmml">T</mi></msubsup><mo id="S4.E23.m1.4.5.3.3.1" xref="S4.E23.m1.4.5.3.3.1.cmml">⁢</mo><mrow id="S4.E23.m1.4.5.3.3.3.2" xref="S4.E23.m1.4.5.3.3.cmml"><mo id="S4.E23.m1.4.5.3.3.3.2.1" stretchy="false" xref="S4.E23.m1.4.5.3.3.cmml">(</mo><mi id="S4.E23.m1.3.3" xref="S4.E23.m1.3.3.cmml">t</mi><mo id="S4.E23.m1.4.5.3.3.3.2.2" stretchy="false" xref="S4.E23.m1.4.5.3.3.cmml">)</mo></mrow><mo id="S4.E23.m1.4.5.3.3.1a" xref="S4.E23.m1.4.5.3.3.1.cmml">⁢</mo><mover accent="true" id="S4.E23.m1.4.5.3.3.4" xref="S4.E23.m1.4.5.3.3.4.cmml"><mi id="S4.E23.m1.4.5.3.3.4.2" xref="S4.E23.m1.4.5.3.3.4.2.cmml">𝜽</mi><mo id="S4.E23.m1.4.5.3.3.4.1" xref="S4.E23.m1.4.5.3.3.4.1.cmml">^</mo></mover><mo id="S4.E23.m1.4.5.3.3.1b" xref="S4.E23.m1.4.5.3.3.1.cmml">⁢</mo><mrow id="S4.E23.m1.4.5.3.3.5.2" xref="S4.E23.m1.4.5.3.3.cmml"><mo id="S4.E23.m1.4.5.3.3.5.2.1" stretchy="false" xref="S4.E23.m1.4.5.3.3.cmml">(</mo><mi id="S4.E23.m1.4.4" xref="S4.E23.m1.4.4.cmml">t</mi><mo id="S4.E23.m1.4.5.3.3.5.2.2" stretchy="false" xref="S4.E23.m1.4.5.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E23.m1.4b"><apply id="S4.E23.m1.4.5.cmml" xref="S4.E23.m1.4.5"><csymbol cd="latexml" id="S4.E23.m1.4.5.1.cmml" xref="S4.E23.m1.4.5.1">assign</csymbol><apply id="S4.E23.m1.4.5.2.cmml" xref="S4.E23.m1.4.5.2"><times id="S4.E23.m1.4.5.2.1.cmml" xref="S4.E23.m1.4.5.2.1"></times><ci id="S4.E23.m1.4.5.2.2.cmml" xref="S4.E23.m1.4.5.2.2">bold-italic-ϵ</ci><ci id="S4.E23.m1.1.1.cmml" xref="S4.E23.m1.1.1">𝑡</ci></apply><apply id="S4.E23.m1.4.5.3.cmml" xref="S4.E23.m1.4.5.3"><minus id="S4.E23.m1.4.5.3.1.cmml" xref="S4.E23.m1.4.5.3.1"></minus><apply id="S4.E23.m1.4.5.3.2.cmml" xref="S4.E23.m1.4.5.3.2"><times id="S4.E23.m1.4.5.3.2.1.cmml" xref="S4.E23.m1.4.5.3.2.1"></times><ci id="S4.E23.m1.4.5.3.2.2.cmml" xref="S4.E23.m1.4.5.3.2.2">𝒛</ci><ci id="S4.E23.m1.2.2.cmml" xref="S4.E23.m1.2.2">𝑡</ci></apply><apply id="S4.E23.m1.4.5.3.3.cmml" xref="S4.E23.m1.4.5.3.3"><times id="S4.E23.m1.4.5.3.3.1.cmml" xref="S4.E23.m1.4.5.3.3.1"></times><apply id="S4.E23.m1.4.5.3.3.2.cmml" xref="S4.E23.m1.4.5.3.3.2"><csymbol cd="ambiguous" id="S4.E23.m1.4.5.3.3.2.1.cmml" xref="S4.E23.m1.4.5.3.3.2">superscript</csymbol><apply id="S4.E23.m1.4.5.3.3.2.2.cmml" xref="S4.E23.m1.4.5.3.3.2"><csymbol cd="ambiguous" id="S4.E23.m1.4.5.3.3.2.2.1.cmml" xref="S4.E23.m1.4.5.3.3.2">subscript</csymbol><ci id="S4.E23.m1.4.5.3.3.2.2.2.cmml" xref="S4.E23.m1.4.5.3.3.2.2.2">Φ</ci><ci id="S4.E23.m1.4.5.3.3.2.2.3.cmml" xref="S4.E23.m1.4.5.3.3.2.2.3">f</ci></apply><ci id="S4.E23.m1.4.5.3.3.2.3.cmml" xref="S4.E23.m1.4.5.3.3.2.3">𝑇</ci></apply><ci id="S4.E23.m1.3.3.cmml" xref="S4.E23.m1.3.3">𝑡</ci><apply id="S4.E23.m1.4.5.3.3.4.cmml" xref="S4.E23.m1.4.5.3.3.4"><ci id="S4.E23.m1.4.5.3.3.4.1.cmml" xref="S4.E23.m1.4.5.3.3.4.1">^</ci><ci id="S4.E23.m1.4.5.3.3.4.2.cmml" xref="S4.E23.m1.4.5.3.3.4.2">𝜽</ci></apply><ci id="S4.E23.m1.4.4.cmml" xref="S4.E23.m1.4.4">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E23.m1.4c">\displaystyle\bm{\epsilon}(t):=\bm{z}(t)-\Phi_{\rm f}^{T}(t)\hat{\bm{\theta}}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.E23.m1.4d">bold_italic_ϵ ( italic_t ) := bold_italic_z ( italic_t ) - roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_t ) over^ start_ARG bold_italic_θ end_ARG ( italic_t )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(23)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p7.13">where <math alttext="\hat{\bm{z}}(t)\in\mathbb{R}^{n}" class="ltx_Math" display="inline" id="S4.SS1.p7.12.m1.1"><semantics id="S4.SS1.p7.12.m1.1a"><mrow id="S4.SS1.p7.12.m1.1.2" xref="S4.SS1.p7.12.m1.1.2.cmml"><mrow id="S4.SS1.p7.12.m1.1.2.2" xref="S4.SS1.p7.12.m1.1.2.2.cmml"><mover accent="true" id="S4.SS1.p7.12.m1.1.2.2.2" xref="S4.SS1.p7.12.m1.1.2.2.2.cmml"><mi id="S4.SS1.p7.12.m1.1.2.2.2.2" xref="S4.SS1.p7.12.m1.1.2.2.2.2.cmml">𝒛</mi><mo id="S4.SS1.p7.12.m1.1.2.2.2.1" xref="S4.SS1.p7.12.m1.1.2.2.2.1.cmml">^</mo></mover><mo id="S4.SS1.p7.12.m1.1.2.2.1" xref="S4.SS1.p7.12.m1.1.2.2.1.cmml">⁢</mo><mrow id="S4.SS1.p7.12.m1.1.2.2.3.2" xref="S4.SS1.p7.12.m1.1.2.2.cmml"><mo id="S4.SS1.p7.12.m1.1.2.2.3.2.1" stretchy="false" xref="S4.SS1.p7.12.m1.1.2.2.cmml">(</mo><mi id="S4.SS1.p7.12.m1.1.1" xref="S4.SS1.p7.12.m1.1.1.cmml">t</mi><mo id="S4.SS1.p7.12.m1.1.2.2.3.2.2" stretchy="false" xref="S4.SS1.p7.12.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p7.12.m1.1.2.1" xref="S4.SS1.p7.12.m1.1.2.1.cmml">∈</mo><msup id="S4.SS1.p7.12.m1.1.2.3" xref="S4.SS1.p7.12.m1.1.2.3.cmml"><mi id="S4.SS1.p7.12.m1.1.2.3.2" xref="S4.SS1.p7.12.m1.1.2.3.2.cmml">ℝ</mi><mi id="S4.SS1.p7.12.m1.1.2.3.3" xref="S4.SS1.p7.12.m1.1.2.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p7.12.m1.1b"><apply id="S4.SS1.p7.12.m1.1.2.cmml" xref="S4.SS1.p7.12.m1.1.2"><in id="S4.SS1.p7.12.m1.1.2.1.cmml" xref="S4.SS1.p7.12.m1.1.2.1"></in><apply id="S4.SS1.p7.12.m1.1.2.2.cmml" xref="S4.SS1.p7.12.m1.1.2.2"><times id="S4.SS1.p7.12.m1.1.2.2.1.cmml" xref="S4.SS1.p7.12.m1.1.2.2.1"></times><apply id="S4.SS1.p7.12.m1.1.2.2.2.cmml" xref="S4.SS1.p7.12.m1.1.2.2.2"><ci id="S4.SS1.p7.12.m1.1.2.2.2.1.cmml" xref="S4.SS1.p7.12.m1.1.2.2.2.1">^</ci><ci id="S4.SS1.p7.12.m1.1.2.2.2.2.cmml" xref="S4.SS1.p7.12.m1.1.2.2.2.2">𝒛</ci></apply><ci id="S4.SS1.p7.12.m1.1.1.cmml" xref="S4.SS1.p7.12.m1.1.1">𝑡</ci></apply><apply id="S4.SS1.p7.12.m1.1.2.3.cmml" xref="S4.SS1.p7.12.m1.1.2.3"><csymbol cd="ambiguous" id="S4.SS1.p7.12.m1.1.2.3.1.cmml" xref="S4.SS1.p7.12.m1.1.2.3">superscript</csymbol><ci id="S4.SS1.p7.12.m1.1.2.3.2.cmml" xref="S4.SS1.p7.12.m1.1.2.3.2">ℝ</ci><ci id="S4.SS1.p7.12.m1.1.2.3.3.cmml" xref="S4.SS1.p7.12.m1.1.2.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p7.12.m1.1c">\hat{\bm{z}}(t)\in\mathbb{R}^{n}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p7.12.m1.1d">over^ start_ARG bold_italic_z end_ARG ( italic_t ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> is a predicted value of <math alttext="\bm{z}(t)" class="ltx_Math" display="inline" id="S4.SS1.p7.13.m2.1"><semantics id="S4.SS1.p7.13.m2.1a"><mrow id="S4.SS1.p7.13.m2.1.2" xref="S4.SS1.p7.13.m2.1.2.cmml"><mi id="S4.SS1.p7.13.m2.1.2.2" xref="S4.SS1.p7.13.m2.1.2.2.cmml">𝒛</mi><mo id="S4.SS1.p7.13.m2.1.2.1" xref="S4.SS1.p7.13.m2.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p7.13.m2.1.2.3.2" xref="S4.SS1.p7.13.m2.1.2.cmml"><mo id="S4.SS1.p7.13.m2.1.2.3.2.1" stretchy="false" xref="S4.SS1.p7.13.m2.1.2.cmml">(</mo><mi id="S4.SS1.p7.13.m2.1.1" xref="S4.SS1.p7.13.m2.1.1.cmml">t</mi><mo id="S4.SS1.p7.13.m2.1.2.3.2.2" stretchy="false" xref="S4.SS1.p7.13.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p7.13.m2.1b"><apply id="S4.SS1.p7.13.m2.1.2.cmml" xref="S4.SS1.p7.13.m2.1.2"><times id="S4.SS1.p7.13.m2.1.2.1.cmml" xref="S4.SS1.p7.13.m2.1.2.1"></times><ci id="S4.SS1.p7.13.m2.1.2.2.cmml" xref="S4.SS1.p7.13.m2.1.2.2">𝒛</ci><ci id="S4.SS1.p7.13.m2.1.1.cmml" xref="S4.SS1.p7.13.m2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p7.13.m2.1c">\bm{z}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p7.13.m2.1d">bold_italic_z ( italic_t )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS1.p8"> <p class="ltx_p" id="S4.SS1.p8.2">The generalized prediction error <math alttext="\bm{\xi}(t)" class="ltx_Math" display="inline" id="S4.SS1.p8.1.m1.1"><semantics id="S4.SS1.p8.1.m1.1a"><mrow id="S4.SS1.p8.1.m1.1.2" xref="S4.SS1.p8.1.m1.1.2.cmml"><mi id="S4.SS1.p8.1.m1.1.2.2" xref="S4.SS1.p8.1.m1.1.2.2.cmml">𝝃</mi><mo id="S4.SS1.p8.1.m1.1.2.1" xref="S4.SS1.p8.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p8.1.m1.1.2.3.2" xref="S4.SS1.p8.1.m1.1.2.cmml"><mo id="S4.SS1.p8.1.m1.1.2.3.2.1" stretchy="false" xref="S4.SS1.p8.1.m1.1.2.cmml">(</mo><mi id="S4.SS1.p8.1.m1.1.1" xref="S4.SS1.p8.1.m1.1.1.cmml">t</mi><mo id="S4.SS1.p8.1.m1.1.2.3.2.2" stretchy="false" xref="S4.SS1.p8.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p8.1.m1.1b"><apply id="S4.SS1.p8.1.m1.1.2.cmml" xref="S4.SS1.p8.1.m1.1.2"><times id="S4.SS1.p8.1.m1.1.2.1.cmml" xref="S4.SS1.p8.1.m1.1.2.1"></times><ci id="S4.SS1.p8.1.m1.1.2.2.cmml" xref="S4.SS1.p8.1.m1.1.2.2">𝝃</ci><ci id="S4.SS1.p8.1.m1.1.1.cmml" xref="S4.SS1.p8.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p8.1.m1.1c">\bm{\xi}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p8.1.m1.1d">bold_italic_ξ ( italic_t )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E20" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">20</span></a>) is combined with the prediction error <math alttext="\bm{\epsilon}(t)" class="ltx_Math" display="inline" id="S4.SS1.p8.2.m2.1"><semantics id="S4.SS1.p8.2.m2.1a"><mrow id="S4.SS1.p8.2.m2.1.2" xref="S4.SS1.p8.2.m2.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS1.p8.2.m2.1.2.2" mathvariant="bold-italic" xref="S4.SS1.p8.2.m2.1.2.2.cmml">ϵ</mi><mo id="S4.SS1.p8.2.m2.1.2.1" xref="S4.SS1.p8.2.m2.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p8.2.m2.1.2.3.2" xref="S4.SS1.p8.2.m2.1.2.cmml"><mo id="S4.SS1.p8.2.m2.1.2.3.2.1" stretchy="false" xref="S4.SS1.p8.2.m2.1.2.cmml">(</mo><mi id="S4.SS1.p8.2.m2.1.1" xref="S4.SS1.p8.2.m2.1.1.cmml">t</mi><mo id="S4.SS1.p8.2.m2.1.2.3.2.2" stretchy="false" xref="S4.SS1.p8.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p8.2.m2.1b"><apply id="S4.SS1.p8.2.m2.1.2.cmml" xref="S4.SS1.p8.2.m2.1.2"><times id="S4.SS1.p8.2.m2.1.2.1.cmml" xref="S4.SS1.p8.2.m2.1.2.1"></times><ci id="S4.SS1.p8.2.m2.1.2.2.cmml" xref="S4.SS1.p8.2.m2.1.2.2">bold-italic-ϵ</ci><ci id="S4.SS1.p8.2.m2.1.1.cmml" xref="S4.SS1.p8.2.m2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p8.2.m2.1c">\bm{\epsilon}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p8.2.m2.1d">bold_italic_ϵ ( italic_t )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E23" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">23</span></a>) to design a composite learning HOT</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx18"> <tbody id="S4.E24"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{\hat{\bm{\theta}}}=\Gamma\left(\Phi_{\rm f}\bm{\epsilon}(t)+% \kappa\bm{\xi}(t)\right)" class="ltx_Math" display="inline" id="S4.E24.m1.3"><semantics id="S4.E24.m1.3a"><mrow id="S4.E24.m1.3.3" xref="S4.E24.m1.3.3.cmml"><mover accent="true" id="S4.E24.m1.3.3.3" xref="S4.E24.m1.3.3.3.cmml"><mover accent="true" id="S4.E24.m1.3.3.3.2" xref="S4.E24.m1.3.3.3.2.cmml"><mi id="S4.E24.m1.3.3.3.2.2" xref="S4.E24.m1.3.3.3.2.2.cmml">𝜽</mi><mo id="S4.E24.m1.3.3.3.2.1" xref="S4.E24.m1.3.3.3.2.1.cmml">^</mo></mover><mo id="S4.E24.m1.3.3.3.1" xref="S4.E24.m1.3.3.3.1.cmml">˙</mo></mover><mo id="S4.E24.m1.3.3.2" xref="S4.E24.m1.3.3.2.cmml">=</mo><mrow id="S4.E24.m1.3.3.1" xref="S4.E24.m1.3.3.1.cmml"><mi id="S4.E24.m1.3.3.1.3" mathvariant="normal" xref="S4.E24.m1.3.3.1.3.cmml">Γ</mi><mo id="S4.E24.m1.3.3.1.2" xref="S4.E24.m1.3.3.1.2.cmml">⁢</mo><mrow id="S4.E24.m1.3.3.1.1.1" xref="S4.E24.m1.3.3.1.1.1.1.cmml"><mo id="S4.E24.m1.3.3.1.1.1.2" xref="S4.E24.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="S4.E24.m1.3.3.1.1.1.1" xref="S4.E24.m1.3.3.1.1.1.1.cmml"><mrow id="S4.E24.m1.3.3.1.1.1.1.2" xref="S4.E24.m1.3.3.1.1.1.1.2.cmml"><msub id="S4.E24.m1.3.3.1.1.1.1.2.2" xref="S4.E24.m1.3.3.1.1.1.1.2.2.cmml"><mi id="S4.E24.m1.3.3.1.1.1.1.2.2.2" mathvariant="normal" xref="S4.E24.m1.3.3.1.1.1.1.2.2.2.cmml">Φ</mi><mi id="S4.E24.m1.3.3.1.1.1.1.2.2.3" mathvariant="normal" xref="S4.E24.m1.3.3.1.1.1.1.2.2.3.cmml">f</mi></msub><mo id="S4.E24.m1.3.3.1.1.1.1.2.1" xref="S4.E24.m1.3.3.1.1.1.1.2.1.cmml">⁢</mo><mi class="ltx_mathvariant_bold-italic" id="S4.E24.m1.3.3.1.1.1.1.2.3" mathvariant="bold-italic" xref="S4.E24.m1.3.3.1.1.1.1.2.3.cmml">ϵ</mi><mo id="S4.E24.m1.3.3.1.1.1.1.2.1a" xref="S4.E24.m1.3.3.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S4.E24.m1.3.3.1.1.1.1.2.4.2" xref="S4.E24.m1.3.3.1.1.1.1.2.cmml"><mo id="S4.E24.m1.3.3.1.1.1.1.2.4.2.1" stretchy="false" xref="S4.E24.m1.3.3.1.1.1.1.2.cmml">(</mo><mi id="S4.E24.m1.1.1" xref="S4.E24.m1.1.1.cmml">t</mi><mo id="S4.E24.m1.3.3.1.1.1.1.2.4.2.2" stretchy="false" xref="S4.E24.m1.3.3.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.E24.m1.3.3.1.1.1.1.1" xref="S4.E24.m1.3.3.1.1.1.1.1.cmml">+</mo><mrow id="S4.E24.m1.3.3.1.1.1.1.3" xref="S4.E24.m1.3.3.1.1.1.1.3.cmml"><mi id="S4.E24.m1.3.3.1.1.1.1.3.2" xref="S4.E24.m1.3.3.1.1.1.1.3.2.cmml">κ</mi><mo id="S4.E24.m1.3.3.1.1.1.1.3.1" xref="S4.E24.m1.3.3.1.1.1.1.3.1.cmml">⁢</mo><mi id="S4.E24.m1.3.3.1.1.1.1.3.3" xref="S4.E24.m1.3.3.1.1.1.1.3.3.cmml">𝝃</mi><mo id="S4.E24.m1.3.3.1.1.1.1.3.1a" xref="S4.E24.m1.3.3.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S4.E24.m1.3.3.1.1.1.1.3.4.2" xref="S4.E24.m1.3.3.1.1.1.1.3.cmml"><mo id="S4.E24.m1.3.3.1.1.1.1.3.4.2.1" stretchy="false" xref="S4.E24.m1.3.3.1.1.1.1.3.cmml">(</mo><mi id="S4.E24.m1.2.2" xref="S4.E24.m1.2.2.cmml">t</mi><mo id="S4.E24.m1.3.3.1.1.1.1.3.4.2.2" stretchy="false" xref="S4.E24.m1.3.3.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E24.m1.3.3.1.1.1.3" xref="S4.E24.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E24.m1.3b"><apply id="S4.E24.m1.3.3.cmml" xref="S4.E24.m1.3.3"><eq id="S4.E24.m1.3.3.2.cmml" xref="S4.E24.m1.3.3.2"></eq><apply id="S4.E24.m1.3.3.3.cmml" xref="S4.E24.m1.3.3.3"><ci id="S4.E24.m1.3.3.3.1.cmml" xref="S4.E24.m1.3.3.3.1">˙</ci><apply id="S4.E24.m1.3.3.3.2.cmml" xref="S4.E24.m1.3.3.3.2"><ci id="S4.E24.m1.3.3.3.2.1.cmml" xref="S4.E24.m1.3.3.3.2.1">^</ci><ci id="S4.E24.m1.3.3.3.2.2.cmml" xref="S4.E24.m1.3.3.3.2.2">𝜽</ci></apply></apply><apply id="S4.E24.m1.3.3.1.cmml" xref="S4.E24.m1.3.3.1"><times id="S4.E24.m1.3.3.1.2.cmml" xref="S4.E24.m1.3.3.1.2"></times><ci id="S4.E24.m1.3.3.1.3.cmml" xref="S4.E24.m1.3.3.1.3">Γ</ci><apply id="S4.E24.m1.3.3.1.1.1.1.cmml" xref="S4.E24.m1.3.3.1.1.1"><plus id="S4.E24.m1.3.3.1.1.1.1.1.cmml" xref="S4.E24.m1.3.3.1.1.1.1.1"></plus><apply id="S4.E24.m1.3.3.1.1.1.1.2.cmml" xref="S4.E24.m1.3.3.1.1.1.1.2"><times id="S4.E24.m1.3.3.1.1.1.1.2.1.cmml" xref="S4.E24.m1.3.3.1.1.1.1.2.1"></times><apply id="S4.E24.m1.3.3.1.1.1.1.2.2.cmml" xref="S4.E24.m1.3.3.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S4.E24.m1.3.3.1.1.1.1.2.2.1.cmml" xref="S4.E24.m1.3.3.1.1.1.1.2.2">subscript</csymbol><ci id="S4.E24.m1.3.3.1.1.1.1.2.2.2.cmml" xref="S4.E24.m1.3.3.1.1.1.1.2.2.2">Φ</ci><ci id="S4.E24.m1.3.3.1.1.1.1.2.2.3.cmml" xref="S4.E24.m1.3.3.1.1.1.1.2.2.3">f</ci></apply><ci id="S4.E24.m1.3.3.1.1.1.1.2.3.cmml" xref="S4.E24.m1.3.3.1.1.1.1.2.3">bold-italic-ϵ</ci><ci id="S4.E24.m1.1.1.cmml" xref="S4.E24.m1.1.1">𝑡</ci></apply><apply id="S4.E24.m1.3.3.1.1.1.1.3.cmml" xref="S4.E24.m1.3.3.1.1.1.1.3"><times id="S4.E24.m1.3.3.1.1.1.1.3.1.cmml" xref="S4.E24.m1.3.3.1.1.1.1.3.1"></times><ci id="S4.E24.m1.3.3.1.1.1.1.3.2.cmml" xref="S4.E24.m1.3.3.1.1.1.1.3.2">𝜅</ci><ci id="S4.E24.m1.3.3.1.1.1.1.3.3.cmml" xref="S4.E24.m1.3.3.1.1.1.1.3.3">𝝃</ci><ci id="S4.E24.m1.2.2.cmml" xref="S4.E24.m1.2.2">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E24.m1.3c">\displaystyle\dot{\hat{\bm{\theta}}}=\Gamma\left(\Phi_{\rm f}\bm{\epsilon}(t)+% \kappa\bm{\xi}(t)\right)</annotation><annotation encoding="application/x-llamapun" id="S4.E24.m1.3d">over˙ start_ARG over^ start_ARG bold_italic_θ end_ARG end_ARG = roman_Γ ( roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT bold_italic_ϵ ( italic_t ) + italic_κ bold_italic_ξ ( italic_t ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(24)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p8.9">in which <math alttext="\Gamma\in\mathbb{R}^{N\times N}" class="ltx_Math" display="inline" id="S4.SS1.p8.3.m1.1"><semantics id="S4.SS1.p8.3.m1.1a"><mrow id="S4.SS1.p8.3.m1.1.1" xref="S4.SS1.p8.3.m1.1.1.cmml"><mi id="S4.SS1.p8.3.m1.1.1.2" mathvariant="normal" xref="S4.SS1.p8.3.m1.1.1.2.cmml">Γ</mi><mo id="S4.SS1.p8.3.m1.1.1.1" xref="S4.SS1.p8.3.m1.1.1.1.cmml">∈</mo><msup id="S4.SS1.p8.3.m1.1.1.3" xref="S4.SS1.p8.3.m1.1.1.3.cmml"><mi id="S4.SS1.p8.3.m1.1.1.3.2" xref="S4.SS1.p8.3.m1.1.1.3.2.cmml">ℝ</mi><mrow id="S4.SS1.p8.3.m1.1.1.3.3" xref="S4.SS1.p8.3.m1.1.1.3.3.cmml"><mi id="S4.SS1.p8.3.m1.1.1.3.3.2" xref="S4.SS1.p8.3.m1.1.1.3.3.2.cmml">N</mi><mo id="S4.SS1.p8.3.m1.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S4.SS1.p8.3.m1.1.1.3.3.1.cmml">×</mo><mi id="S4.SS1.p8.3.m1.1.1.3.3.3" xref="S4.SS1.p8.3.m1.1.1.3.3.3.cmml">N</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p8.3.m1.1b"><apply id="S4.SS1.p8.3.m1.1.1.cmml" xref="S4.SS1.p8.3.m1.1.1"><in id="S4.SS1.p8.3.m1.1.1.1.cmml" xref="S4.SS1.p8.3.m1.1.1.1"></in><ci id="S4.SS1.p8.3.m1.1.1.2.cmml" xref="S4.SS1.p8.3.m1.1.1.2">Γ</ci><apply id="S4.SS1.p8.3.m1.1.1.3.cmml" xref="S4.SS1.p8.3.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p8.3.m1.1.1.3.1.cmml" xref="S4.SS1.p8.3.m1.1.1.3">superscript</csymbol><ci id="S4.SS1.p8.3.m1.1.1.3.2.cmml" xref="S4.SS1.p8.3.m1.1.1.3.2">ℝ</ci><apply id="S4.SS1.p8.3.m1.1.1.3.3.cmml" xref="S4.SS1.p8.3.m1.1.1.3.3"><times id="S4.SS1.p8.3.m1.1.1.3.3.1.cmml" xref="S4.SS1.p8.3.m1.1.1.3.3.1"></times><ci id="S4.SS1.p8.3.m1.1.1.3.3.2.cmml" xref="S4.SS1.p8.3.m1.1.1.3.3.2">𝑁</ci><ci id="S4.SS1.p8.3.m1.1.1.3.3.3.cmml" xref="S4.SS1.p8.3.m1.1.1.3.3.3">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p8.3.m1.1c">\Gamma\in\mathbb{R}^{N\times N}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p8.3.m1.1d">roman_Γ ∈ blackboard_R start_POSTSUPERSCRIPT italic_N × italic_N end_POSTSUPERSCRIPT</annotation></semantics></math> is a positive-definite diagonal matrix of learning rates, and <math alttext="\kappa\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S4.SS1.p8.4.m2.1"><semantics id="S4.SS1.p8.4.m2.1a"><mrow id="S4.SS1.p8.4.m2.1.1" xref="S4.SS1.p8.4.m2.1.1.cmml"><mi id="S4.SS1.p8.4.m2.1.1.2" xref="S4.SS1.p8.4.m2.1.1.2.cmml">κ</mi><mo id="S4.SS1.p8.4.m2.1.1.1" xref="S4.SS1.p8.4.m2.1.1.1.cmml">∈</mo><msup id="S4.SS1.p8.4.m2.1.1.3" xref="S4.SS1.p8.4.m2.1.1.3.cmml"><mi id="S4.SS1.p8.4.m2.1.1.3.2" xref="S4.SS1.p8.4.m2.1.1.3.2.cmml">ℝ</mi><mo id="S4.SS1.p8.4.m2.1.1.3.3" xref="S4.SS1.p8.4.m2.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p8.4.m2.1b"><apply id="S4.SS1.p8.4.m2.1.1.cmml" xref="S4.SS1.p8.4.m2.1.1"><in id="S4.SS1.p8.4.m2.1.1.1.cmml" xref="S4.SS1.p8.4.m2.1.1.1"></in><ci id="S4.SS1.p8.4.m2.1.1.2.cmml" xref="S4.SS1.p8.4.m2.1.1.2">𝜅</ci><apply id="S4.SS1.p8.4.m2.1.1.3.cmml" xref="S4.SS1.p8.4.m2.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p8.4.m2.1.1.3.1.cmml" xref="S4.SS1.p8.4.m2.1.1.3">superscript</csymbol><ci id="S4.SS1.p8.4.m2.1.1.3.2.cmml" xref="S4.SS1.p8.4.m2.1.1.3.2">ℝ</ci><plus id="S4.SS1.p8.4.m2.1.1.3.3.cmml" xref="S4.SS1.p8.4.m2.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p8.4.m2.1c">\kappa\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p8.4.m2.1d">italic_κ ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> is a weighting factor. Since <math alttext="H(s)" class="ltx_Math" display="inline" id="S4.SS1.p8.5.m3.1"><semantics id="S4.SS1.p8.5.m3.1a"><mrow id="S4.SS1.p8.5.m3.1.2" xref="S4.SS1.p8.5.m3.1.2.cmml"><mi id="S4.SS1.p8.5.m3.1.2.2" xref="S4.SS1.p8.5.m3.1.2.2.cmml">H</mi><mo id="S4.SS1.p8.5.m3.1.2.1" xref="S4.SS1.p8.5.m3.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p8.5.m3.1.2.3.2" xref="S4.SS1.p8.5.m3.1.2.cmml"><mo id="S4.SS1.p8.5.m3.1.2.3.2.1" stretchy="false" xref="S4.SS1.p8.5.m3.1.2.cmml">(</mo><mi id="S4.SS1.p8.5.m3.1.1" xref="S4.SS1.p8.5.m3.1.1.cmml">s</mi><mo id="S4.SS1.p8.5.m3.1.2.3.2.2" stretchy="false" xref="S4.SS1.p8.5.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p8.5.m3.1b"><apply id="S4.SS1.p8.5.m3.1.2.cmml" xref="S4.SS1.p8.5.m3.1.2"><times id="S4.SS1.p8.5.m3.1.2.1.cmml" xref="S4.SS1.p8.5.m3.1.2.1"></times><ci id="S4.SS1.p8.5.m3.1.2.2.cmml" xref="S4.SS1.p8.5.m3.1.2.2">𝐻</ci><ci id="S4.SS1.p8.5.m3.1.1.cmml" xref="S4.SS1.p8.5.m3.1.1">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p8.5.m3.1c">H(s)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p8.5.m3.1d">italic_H ( italic_s )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E17" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">17</span></a>) owns <math alttext="n-1" class="ltx_Math" display="inline" id="S4.SS1.p8.6.m4.1"><semantics id="S4.SS1.p8.6.m4.1a"><mrow id="S4.SS1.p8.6.m4.1.1" xref="S4.SS1.p8.6.m4.1.1.cmml"><mi id="S4.SS1.p8.6.m4.1.1.2" xref="S4.SS1.p8.6.m4.1.1.2.cmml">n</mi><mo id="S4.SS1.p8.6.m4.1.1.1" xref="S4.SS1.p8.6.m4.1.1.1.cmml">−</mo><mn id="S4.SS1.p8.6.m4.1.1.3" xref="S4.SS1.p8.6.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p8.6.m4.1b"><apply id="S4.SS1.p8.6.m4.1.1.cmml" xref="S4.SS1.p8.6.m4.1.1"><minus id="S4.SS1.p8.6.m4.1.1.1.cmml" xref="S4.SS1.p8.6.m4.1.1.1"></minus><ci id="S4.SS1.p8.6.m4.1.1.2.cmml" xref="S4.SS1.p8.6.m4.1.1.2">𝑛</ci><cn id="S4.SS1.p8.6.m4.1.1.3.cmml" type="integer" xref="S4.SS1.p8.6.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p8.6.m4.1c">n-1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p8.6.m4.1d">italic_n - 1</annotation></semantics></math> relative degrees, the time derivatives of <math alttext="\hat{\bm{\theta}}" class="ltx_Math" display="inline" id="S4.SS1.p8.7.m5.1"><semantics id="S4.SS1.p8.7.m5.1a"><mover accent="true" id="S4.SS1.p8.7.m5.1.1" xref="S4.SS1.p8.7.m5.1.1.cmml"><mi id="S4.SS1.p8.7.m5.1.1.2" xref="S4.SS1.p8.7.m5.1.1.2.cmml">𝜽</mi><mo id="S4.SS1.p8.7.m5.1.1.1" xref="S4.SS1.p8.7.m5.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS1.p8.7.m5.1b"><apply id="S4.SS1.p8.7.m5.1.1.cmml" xref="S4.SS1.p8.7.m5.1.1"><ci id="S4.SS1.p8.7.m5.1.1.1.cmml" xref="S4.SS1.p8.7.m5.1.1.1">^</ci><ci id="S4.SS1.p8.7.m5.1.1.2.cmml" xref="S4.SS1.p8.7.m5.1.1.2">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p8.7.m5.1c">\hat{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p8.7.m5.1d">over^ start_ARG bold_italic_θ end_ARG</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>) up to the (<math alttext="n-1" class="ltx_Math" display="inline" id="S4.SS1.p8.8.m6.1"><semantics id="S4.SS1.p8.8.m6.1a"><mrow id="S4.SS1.p8.8.m6.1.1" xref="S4.SS1.p8.8.m6.1.1.cmml"><mi id="S4.SS1.p8.8.m6.1.1.2" xref="S4.SS1.p8.8.m6.1.1.2.cmml">n</mi><mo id="S4.SS1.p8.8.m6.1.1.1" xref="S4.SS1.p8.8.m6.1.1.1.cmml">−</mo><mn id="S4.SS1.p8.8.m6.1.1.3" xref="S4.SS1.p8.8.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p8.8.m6.1b"><apply id="S4.SS1.p8.8.m6.1.1.cmml" xref="S4.SS1.p8.8.m6.1.1"><minus id="S4.SS1.p8.8.m6.1.1.1.cmml" xref="S4.SS1.p8.8.m6.1.1.1"></minus><ci id="S4.SS1.p8.8.m6.1.1.2.cmml" xref="S4.SS1.p8.8.m6.1.1.2">𝑛</ci><cn id="S4.SS1.p8.8.m6.1.1.3.cmml" type="integer" xref="S4.SS1.p8.8.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p8.8.m6.1c">n-1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p8.8.m6.1d">italic_n - 1</annotation></semantics></math>)th order can be implemented physically by a direct differentiation scheme <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib28" title="">28</a>]</cite>. More specifically, the high-order time derivatives of <math alttext="\hat{\bm{\theta}}" class="ltx_Math" display="inline" id="S4.SS1.p8.9.m7.1"><semantics id="S4.SS1.p8.9.m7.1a"><mover accent="true" id="S4.SS1.p8.9.m7.1.1" xref="S4.SS1.p8.9.m7.1.1.cmml"><mi id="S4.SS1.p8.9.m7.1.1.2" xref="S4.SS1.p8.9.m7.1.1.2.cmml">𝜽</mi><mo id="S4.SS1.p8.9.m7.1.1.1" xref="S4.SS1.p8.9.m7.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS1.p8.9.m7.1b"><apply id="S4.SS1.p8.9.m7.1.1.cmml" xref="S4.SS1.p8.9.m7.1.1"><ci id="S4.SS1.p8.9.m7.1.1.1.cmml" xref="S4.SS1.p8.9.m7.1.1.1">^</ci><ci id="S4.SS1.p8.9.m7.1.1.2.cmml" xref="S4.SS1.p8.9.m7.1.1.2">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p8.9.m7.1c">\hat{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p8.9.m7.1d">over^ start_ARG bold_italic_θ end_ARG</annotation></semantics></math> are calculated by</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx19"> <tbody id="S4.E25"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\hat{\bm{\theta}}^{(k+1)}=\Gamma\left(\sum_{i=0}^{k}C_{k}^{i}s^{k% -i}H(s)[\Phi]\bm{\epsilon}^{(i)}+\kappa\bm{\xi}^{(k)}\right)" class="ltx_Math" display="inline" id="S4.E25.m1.6"><semantics id="S4.E25.m1.6a"><mrow id="S4.E25.m1.6.6" xref="S4.E25.m1.6.6.cmml"><msup id="S4.E25.m1.6.6.3" xref="S4.E25.m1.6.6.3.cmml"><mover accent="true" id="S4.E25.m1.6.6.3.2" xref="S4.E25.m1.6.6.3.2.cmml"><mi id="S4.E25.m1.6.6.3.2.2" xref="S4.E25.m1.6.6.3.2.2.cmml">𝜽</mi><mo id="S4.E25.m1.6.6.3.2.1" xref="S4.E25.m1.6.6.3.2.1.cmml">^</mo></mover><mrow id="S4.E25.m1.1.1.1.1" xref="S4.E25.m1.1.1.1.1.1.cmml"><mo id="S4.E25.m1.1.1.1.1.2" stretchy="false" xref="S4.E25.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E25.m1.1.1.1.1.1" xref="S4.E25.m1.1.1.1.1.1.cmml"><mi id="S4.E25.m1.1.1.1.1.1.2" xref="S4.E25.m1.1.1.1.1.1.2.cmml">k</mi><mo id="S4.E25.m1.1.1.1.1.1.1" xref="S4.E25.m1.1.1.1.1.1.1.cmml">+</mo><mn id="S4.E25.m1.1.1.1.1.1.3" xref="S4.E25.m1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S4.E25.m1.1.1.1.1.3" stretchy="false" xref="S4.E25.m1.1.1.1.1.1.cmml">)</mo></mrow></msup><mo id="S4.E25.m1.6.6.2" xref="S4.E25.m1.6.6.2.cmml">=</mo><mrow id="S4.E25.m1.6.6.1" xref="S4.E25.m1.6.6.1.cmml"><mi id="S4.E25.m1.6.6.1.3" mathvariant="normal" xref="S4.E25.m1.6.6.1.3.cmml">Γ</mi><mo id="S4.E25.m1.6.6.1.2" xref="S4.E25.m1.6.6.1.2.cmml">⁢</mo><mrow id="S4.E25.m1.6.6.1.1.1" xref="S4.E25.m1.6.6.1.1.1.1.cmml"><mo id="S4.E25.m1.6.6.1.1.1.2" xref="S4.E25.m1.6.6.1.1.1.1.cmml">(</mo><mrow id="S4.E25.m1.6.6.1.1.1.1" xref="S4.E25.m1.6.6.1.1.1.1.cmml"><mrow id="S4.E25.m1.6.6.1.1.1.1.2" xref="S4.E25.m1.6.6.1.1.1.1.2.cmml"><mstyle displaystyle="true" id="S4.E25.m1.6.6.1.1.1.1.2.1" xref="S4.E25.m1.6.6.1.1.1.1.2.1.cmml"><munderover id="S4.E25.m1.6.6.1.1.1.1.2.1a" xref="S4.E25.m1.6.6.1.1.1.1.2.1.cmml"><mo id="S4.E25.m1.6.6.1.1.1.1.2.1.2.2" movablelimits="false" xref="S4.E25.m1.6.6.1.1.1.1.2.1.2.2.cmml">∑</mo><mrow id="S4.E25.m1.6.6.1.1.1.1.2.1.2.3" xref="S4.E25.m1.6.6.1.1.1.1.2.1.2.3.cmml"><mi id="S4.E25.m1.6.6.1.1.1.1.2.1.2.3.2" xref="S4.E25.m1.6.6.1.1.1.1.2.1.2.3.2.cmml">i</mi><mo id="S4.E25.m1.6.6.1.1.1.1.2.1.2.3.1" xref="S4.E25.m1.6.6.1.1.1.1.2.1.2.3.1.cmml">=</mo><mn id="S4.E25.m1.6.6.1.1.1.1.2.1.2.3.3" xref="S4.E25.m1.6.6.1.1.1.1.2.1.2.3.3.cmml">0</mn></mrow><mi id="S4.E25.m1.6.6.1.1.1.1.2.1.3" xref="S4.E25.m1.6.6.1.1.1.1.2.1.3.cmml">k</mi></munderover></mstyle><mrow id="S4.E25.m1.6.6.1.1.1.1.2.2" xref="S4.E25.m1.6.6.1.1.1.1.2.2.cmml"><msubsup id="S4.E25.m1.6.6.1.1.1.1.2.2.2" xref="S4.E25.m1.6.6.1.1.1.1.2.2.2.cmml"><mi id="S4.E25.m1.6.6.1.1.1.1.2.2.2.2.2" xref="S4.E25.m1.6.6.1.1.1.1.2.2.2.2.2.cmml">C</mi><mi id="S4.E25.m1.6.6.1.1.1.1.2.2.2.2.3" xref="S4.E25.m1.6.6.1.1.1.1.2.2.2.2.3.cmml">k</mi><mi id="S4.E25.m1.6.6.1.1.1.1.2.2.2.3" xref="S4.E25.m1.6.6.1.1.1.1.2.2.2.3.cmml">i</mi></msubsup><mo id="S4.E25.m1.6.6.1.1.1.1.2.2.1" xref="S4.E25.m1.6.6.1.1.1.1.2.2.1.cmml">⁢</mo><msup id="S4.E25.m1.6.6.1.1.1.1.2.2.3" xref="S4.E25.m1.6.6.1.1.1.1.2.2.3.cmml"><mi id="S4.E25.m1.6.6.1.1.1.1.2.2.3.2" xref="S4.E25.m1.6.6.1.1.1.1.2.2.3.2.cmml">s</mi><mrow id="S4.E25.m1.6.6.1.1.1.1.2.2.3.3" xref="S4.E25.m1.6.6.1.1.1.1.2.2.3.3.cmml"><mi id="S4.E25.m1.6.6.1.1.1.1.2.2.3.3.2" xref="S4.E25.m1.6.6.1.1.1.1.2.2.3.3.2.cmml">k</mi><mo id="S4.E25.m1.6.6.1.1.1.1.2.2.3.3.1" xref="S4.E25.m1.6.6.1.1.1.1.2.2.3.3.1.cmml">−</mo><mi id="S4.E25.m1.6.6.1.1.1.1.2.2.3.3.3" xref="S4.E25.m1.6.6.1.1.1.1.2.2.3.3.3.cmml">i</mi></mrow></msup><mo id="S4.E25.m1.6.6.1.1.1.1.2.2.1a" xref="S4.E25.m1.6.6.1.1.1.1.2.2.1.cmml">⁢</mo><mi id="S4.E25.m1.6.6.1.1.1.1.2.2.4" xref="S4.E25.m1.6.6.1.1.1.1.2.2.4.cmml">H</mi><mo id="S4.E25.m1.6.6.1.1.1.1.2.2.1b" xref="S4.E25.m1.6.6.1.1.1.1.2.2.1.cmml">⁢</mo><mrow id="S4.E25.m1.6.6.1.1.1.1.2.2.5.2" xref="S4.E25.m1.6.6.1.1.1.1.2.2.cmml"><mo id="S4.E25.m1.6.6.1.1.1.1.2.2.5.2.1" stretchy="false" xref="S4.E25.m1.6.6.1.1.1.1.2.2.cmml">(</mo><mi id="S4.E25.m1.4.4" xref="S4.E25.m1.4.4.cmml">s</mi><mo id="S4.E25.m1.6.6.1.1.1.1.2.2.5.2.2" stretchy="false" xref="S4.E25.m1.6.6.1.1.1.1.2.2.cmml">)</mo></mrow><mo id="S4.E25.m1.6.6.1.1.1.1.2.2.1c" xref="S4.E25.m1.6.6.1.1.1.1.2.2.1.cmml">⁢</mo><mrow id="S4.E25.m1.6.6.1.1.1.1.2.2.6.2" xref="S4.E25.m1.6.6.1.1.1.1.2.2.6.1.cmml"><mo id="S4.E25.m1.6.6.1.1.1.1.2.2.6.2.1" stretchy="false" xref="S4.E25.m1.6.6.1.1.1.1.2.2.6.1.1.cmml">[</mo><mi id="S4.E25.m1.5.5" mathvariant="normal" xref="S4.E25.m1.5.5.cmml">Φ</mi><mo id="S4.E25.m1.6.6.1.1.1.1.2.2.6.2.2" stretchy="false" xref="S4.E25.m1.6.6.1.1.1.1.2.2.6.1.1.cmml">]</mo></mrow><mo id="S4.E25.m1.6.6.1.1.1.1.2.2.1d" xref="S4.E25.m1.6.6.1.1.1.1.2.2.1.cmml">⁢</mo><msup id="S4.E25.m1.6.6.1.1.1.1.2.2.7" xref="S4.E25.m1.6.6.1.1.1.1.2.2.7.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E25.m1.6.6.1.1.1.1.2.2.7.2" mathvariant="bold-italic" xref="S4.E25.m1.6.6.1.1.1.1.2.2.7.2.cmml">ϵ</mi><mrow id="S4.E25.m1.2.2.1.3" xref="S4.E25.m1.6.6.1.1.1.1.2.2.7.cmml"><mo id="S4.E25.m1.2.2.1.3.1" stretchy="false" xref="S4.E25.m1.6.6.1.1.1.1.2.2.7.cmml">(</mo><mi id="S4.E25.m1.2.2.1.1" xref="S4.E25.m1.2.2.1.1.cmml">i</mi><mo id="S4.E25.m1.2.2.1.3.2" stretchy="false" xref="S4.E25.m1.6.6.1.1.1.1.2.2.7.cmml">)</mo></mrow></msup></mrow></mrow><mo id="S4.E25.m1.6.6.1.1.1.1.1" xref="S4.E25.m1.6.6.1.1.1.1.1.cmml">+</mo><mrow id="S4.E25.m1.6.6.1.1.1.1.3" xref="S4.E25.m1.6.6.1.1.1.1.3.cmml"><mi id="S4.E25.m1.6.6.1.1.1.1.3.2" xref="S4.E25.m1.6.6.1.1.1.1.3.2.cmml">κ</mi><mo id="S4.E25.m1.6.6.1.1.1.1.3.1" xref="S4.E25.m1.6.6.1.1.1.1.3.1.cmml">⁢</mo><msup id="S4.E25.m1.6.6.1.1.1.1.3.3" xref="S4.E25.m1.6.6.1.1.1.1.3.3.cmml"><mi id="S4.E25.m1.6.6.1.1.1.1.3.3.2" xref="S4.E25.m1.6.6.1.1.1.1.3.3.2.cmml">𝝃</mi><mrow id="S4.E25.m1.3.3.1.3" xref="S4.E25.m1.6.6.1.1.1.1.3.3.cmml"><mo id="S4.E25.m1.3.3.1.3.1" stretchy="false" xref="S4.E25.m1.6.6.1.1.1.1.3.3.cmml">(</mo><mi id="S4.E25.m1.3.3.1.1" xref="S4.E25.m1.3.3.1.1.cmml">k</mi><mo id="S4.E25.m1.3.3.1.3.2" stretchy="false" xref="S4.E25.m1.6.6.1.1.1.1.3.3.cmml">)</mo></mrow></msup></mrow></mrow><mo id="S4.E25.m1.6.6.1.1.1.3" xref="S4.E25.m1.6.6.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E25.m1.6b"><apply id="S4.E25.m1.6.6.cmml" xref="S4.E25.m1.6.6"><eq id="S4.E25.m1.6.6.2.cmml" xref="S4.E25.m1.6.6.2"></eq><apply id="S4.E25.m1.6.6.3.cmml" xref="S4.E25.m1.6.6.3"><csymbol cd="ambiguous" id="S4.E25.m1.6.6.3.1.cmml" xref="S4.E25.m1.6.6.3">superscript</csymbol><apply id="S4.E25.m1.6.6.3.2.cmml" xref="S4.E25.m1.6.6.3.2"><ci id="S4.E25.m1.6.6.3.2.1.cmml" xref="S4.E25.m1.6.6.3.2.1">^</ci><ci id="S4.E25.m1.6.6.3.2.2.cmml" xref="S4.E25.m1.6.6.3.2.2">𝜽</ci></apply><apply id="S4.E25.m1.1.1.1.1.1.cmml" xref="S4.E25.m1.1.1.1.1"><plus id="S4.E25.m1.1.1.1.1.1.1.cmml" xref="S4.E25.m1.1.1.1.1.1.1"></plus><ci id="S4.E25.m1.1.1.1.1.1.2.cmml" xref="S4.E25.m1.1.1.1.1.1.2">𝑘</ci><cn id="S4.E25.m1.1.1.1.1.1.3.cmml" type="integer" xref="S4.E25.m1.1.1.1.1.1.3">1</cn></apply></apply><apply id="S4.E25.m1.6.6.1.cmml" xref="S4.E25.m1.6.6.1"><times id="S4.E25.m1.6.6.1.2.cmml" xref="S4.E25.m1.6.6.1.2"></times><ci id="S4.E25.m1.6.6.1.3.cmml" xref="S4.E25.m1.6.6.1.3">Γ</ci><apply id="S4.E25.m1.6.6.1.1.1.1.cmml" xref="S4.E25.m1.6.6.1.1.1"><plus id="S4.E25.m1.6.6.1.1.1.1.1.cmml" xref="S4.E25.m1.6.6.1.1.1.1.1"></plus><apply id="S4.E25.m1.6.6.1.1.1.1.2.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2"><apply id="S4.E25.m1.6.6.1.1.1.1.2.1.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.1"><csymbol cd="ambiguous" id="S4.E25.m1.6.6.1.1.1.1.2.1.1.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.1">superscript</csymbol><apply id="S4.E25.m1.6.6.1.1.1.1.2.1.2.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.1"><csymbol cd="ambiguous" id="S4.E25.m1.6.6.1.1.1.1.2.1.2.1.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.1">subscript</csymbol><sum id="S4.E25.m1.6.6.1.1.1.1.2.1.2.2.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.1.2.2"></sum><apply id="S4.E25.m1.6.6.1.1.1.1.2.1.2.3.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.1.2.3"><eq id="S4.E25.m1.6.6.1.1.1.1.2.1.2.3.1.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.1.2.3.1"></eq><ci id="S4.E25.m1.6.6.1.1.1.1.2.1.2.3.2.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.1.2.3.2">𝑖</ci><cn id="S4.E25.m1.6.6.1.1.1.1.2.1.2.3.3.cmml" type="integer" xref="S4.E25.m1.6.6.1.1.1.1.2.1.2.3.3">0</cn></apply></apply><ci id="S4.E25.m1.6.6.1.1.1.1.2.1.3.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.1.3">𝑘</ci></apply><apply id="S4.E25.m1.6.6.1.1.1.1.2.2.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.2"><times id="S4.E25.m1.6.6.1.1.1.1.2.2.1.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.2.1"></times><apply id="S4.E25.m1.6.6.1.1.1.1.2.2.2.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="S4.E25.m1.6.6.1.1.1.1.2.2.2.1.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.2.2">superscript</csymbol><apply id="S4.E25.m1.6.6.1.1.1.1.2.2.2.2.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="S4.E25.m1.6.6.1.1.1.1.2.2.2.2.1.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.2.2">subscript</csymbol><ci id="S4.E25.m1.6.6.1.1.1.1.2.2.2.2.2.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.2.2.2.2">𝐶</ci><ci id="S4.E25.m1.6.6.1.1.1.1.2.2.2.2.3.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.2.2.2.3">𝑘</ci></apply><ci id="S4.E25.m1.6.6.1.1.1.1.2.2.2.3.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.2.2.3">𝑖</ci></apply><apply id="S4.E25.m1.6.6.1.1.1.1.2.2.3.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.2.3"><csymbol cd="ambiguous" id="S4.E25.m1.6.6.1.1.1.1.2.2.3.1.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.2.3">superscript</csymbol><ci id="S4.E25.m1.6.6.1.1.1.1.2.2.3.2.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.2.3.2">𝑠</ci><apply id="S4.E25.m1.6.6.1.1.1.1.2.2.3.3.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.2.3.3"><minus id="S4.E25.m1.6.6.1.1.1.1.2.2.3.3.1.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.2.3.3.1"></minus><ci id="S4.E25.m1.6.6.1.1.1.1.2.2.3.3.2.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.2.3.3.2">𝑘</ci><ci id="S4.E25.m1.6.6.1.1.1.1.2.2.3.3.3.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.2.3.3.3">𝑖</ci></apply></apply><ci id="S4.E25.m1.6.6.1.1.1.1.2.2.4.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.2.4">𝐻</ci><ci id="S4.E25.m1.4.4.cmml" xref="S4.E25.m1.4.4">𝑠</ci><apply id="S4.E25.m1.6.6.1.1.1.1.2.2.6.1.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.2.6.2"><csymbol cd="latexml" id="S4.E25.m1.6.6.1.1.1.1.2.2.6.1.1.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.2.6.2.1">delimited-[]</csymbol><ci id="S4.E25.m1.5.5.cmml" xref="S4.E25.m1.5.5">Φ</ci></apply><apply id="S4.E25.m1.6.6.1.1.1.1.2.2.7.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.2.7"><csymbol cd="ambiguous" id="S4.E25.m1.6.6.1.1.1.1.2.2.7.1.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.2.7">superscript</csymbol><ci id="S4.E25.m1.6.6.1.1.1.1.2.2.7.2.cmml" xref="S4.E25.m1.6.6.1.1.1.1.2.2.7.2">bold-italic-ϵ</ci><ci id="S4.E25.m1.2.2.1.1.cmml" xref="S4.E25.m1.2.2.1.1">𝑖</ci></apply></apply></apply><apply id="S4.E25.m1.6.6.1.1.1.1.3.cmml" xref="S4.E25.m1.6.6.1.1.1.1.3"><times id="S4.E25.m1.6.6.1.1.1.1.3.1.cmml" xref="S4.E25.m1.6.6.1.1.1.1.3.1"></times><ci id="S4.E25.m1.6.6.1.1.1.1.3.2.cmml" xref="S4.E25.m1.6.6.1.1.1.1.3.2">𝜅</ci><apply id="S4.E25.m1.6.6.1.1.1.1.3.3.cmml" xref="S4.E25.m1.6.6.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S4.E25.m1.6.6.1.1.1.1.3.3.1.cmml" xref="S4.E25.m1.6.6.1.1.1.1.3.3">superscript</csymbol><ci id="S4.E25.m1.6.6.1.1.1.1.3.3.2.cmml" xref="S4.E25.m1.6.6.1.1.1.1.3.3.2">𝝃</ci><ci id="S4.E25.m1.3.3.1.1.cmml" xref="S4.E25.m1.3.3.1.1">𝑘</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E25.m1.6c">\displaystyle\hat{\bm{\theta}}^{(k+1)}=\Gamma\left(\sum_{i=0}^{k}C_{k}^{i}s^{k% -i}H(s)[\Phi]\bm{\epsilon}^{(i)}+\kappa\bm{\xi}^{(k)}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.E25.m1.6d">over^ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT ( italic_k + 1 ) end_POSTSUPERSCRIPT = roman_Γ ( ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_s start_POSTSUPERSCRIPT italic_k - italic_i end_POSTSUPERSCRIPT italic_H ( italic_s ) [ roman_Φ ] bold_italic_ϵ start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT + italic_κ bold_italic_ξ start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(25)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p8.12">with <math alttext="\hat{\bm{\theta}}^{(i)}(0)" class="ltx_Math" display="inline" id="S4.SS1.p8.10.m1.2"><semantics id="S4.SS1.p8.10.m1.2a"><mrow id="S4.SS1.p8.10.m1.2.3" xref="S4.SS1.p8.10.m1.2.3.cmml"><msup id="S4.SS1.p8.10.m1.2.3.2" xref="S4.SS1.p8.10.m1.2.3.2.cmml"><mover accent="true" id="S4.SS1.p8.10.m1.2.3.2.2" xref="S4.SS1.p8.10.m1.2.3.2.2.cmml"><mi id="S4.SS1.p8.10.m1.2.3.2.2.2" xref="S4.SS1.p8.10.m1.2.3.2.2.2.cmml">𝜽</mi><mo id="S4.SS1.p8.10.m1.2.3.2.2.1" xref="S4.SS1.p8.10.m1.2.3.2.2.1.cmml">^</mo></mover><mrow id="S4.SS1.p8.10.m1.1.1.1.3" xref="S4.SS1.p8.10.m1.2.3.2.cmml"><mo id="S4.SS1.p8.10.m1.1.1.1.3.1" stretchy="false" xref="S4.SS1.p8.10.m1.2.3.2.cmml">(</mo><mi id="S4.SS1.p8.10.m1.1.1.1.1" xref="S4.SS1.p8.10.m1.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p8.10.m1.1.1.1.3.2" stretchy="false" xref="S4.SS1.p8.10.m1.2.3.2.cmml">)</mo></mrow></msup><mo id="S4.SS1.p8.10.m1.2.3.1" xref="S4.SS1.p8.10.m1.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.p8.10.m1.2.3.3.2" xref="S4.SS1.p8.10.m1.2.3.cmml"><mo id="S4.SS1.p8.10.m1.2.3.3.2.1" stretchy="false" xref="S4.SS1.p8.10.m1.2.3.cmml">(</mo><mn id="S4.SS1.p8.10.m1.2.2" xref="S4.SS1.p8.10.m1.2.2.cmml">0</mn><mo id="S4.SS1.p8.10.m1.2.3.3.2.2" stretchy="false" xref="S4.SS1.p8.10.m1.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p8.10.m1.2b"><apply id="S4.SS1.p8.10.m1.2.3.cmml" xref="S4.SS1.p8.10.m1.2.3"><times id="S4.SS1.p8.10.m1.2.3.1.cmml" xref="S4.SS1.p8.10.m1.2.3.1"></times><apply id="S4.SS1.p8.10.m1.2.3.2.cmml" xref="S4.SS1.p8.10.m1.2.3.2"><csymbol cd="ambiguous" id="S4.SS1.p8.10.m1.2.3.2.1.cmml" xref="S4.SS1.p8.10.m1.2.3.2">superscript</csymbol><apply id="S4.SS1.p8.10.m1.2.3.2.2.cmml" xref="S4.SS1.p8.10.m1.2.3.2.2"><ci id="S4.SS1.p8.10.m1.2.3.2.2.1.cmml" xref="S4.SS1.p8.10.m1.2.3.2.2.1">^</ci><ci id="S4.SS1.p8.10.m1.2.3.2.2.2.cmml" xref="S4.SS1.p8.10.m1.2.3.2.2.2">𝜽</ci></apply><ci id="S4.SS1.p8.10.m1.1.1.1.1.cmml" xref="S4.SS1.p8.10.m1.1.1.1.1">𝑖</ci></apply><cn id="S4.SS1.p8.10.m1.2.2.cmml" type="integer" xref="S4.SS1.p8.10.m1.2.2">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p8.10.m1.2c">\hat{\bm{\theta}}^{(i)}(0)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p8.10.m1.2d">over^ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT ( 0 )</annotation></semantics></math> <math alttext="=" class="ltx_Math" display="inline" id="S4.SS1.p8.11.m2.1"><semantics id="S4.SS1.p8.11.m2.1a"><mo id="S4.SS1.p8.11.m2.1.1" xref="S4.SS1.p8.11.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.p8.11.m2.1b"><eq id="S4.SS1.p8.11.m2.1.1.cmml" xref="S4.SS1.p8.11.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p8.11.m2.1c">=</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p8.11.m2.1d">=</annotation></semantics></math> <math alttext="\bm{0}" class="ltx_Math" display="inline" id="S4.SS1.p8.12.m3.1"><semantics id="S4.SS1.p8.12.m3.1a"><mn id="S4.SS1.p8.12.m3.1.1" xref="S4.SS1.p8.12.m3.1.1.cmml">𝟎</mn><annotation-xml encoding="MathML-Content" id="S4.SS1.p8.12.m3.1b"><cn id="S4.SS1.p8.12.m3.1.1.cmml" type="integer" xref="S4.SS1.p8.12.m3.1.1">0</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p8.12.m3.1c">\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p8.12.m3.1d">bold_0</annotation></semantics></math> and intermediate time derivatives</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx20"> <tbody id="S4.Ex6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\bm{\epsilon}^{(i)}=\bm{z}^{(i)}-\sum_{j=0}^{i}C_{i}^{j}s^{i-j}H(% s)[\Phi]\hat{\bm{\theta}}^{(j)}," class="ltx_Math" display="inline" id="S4.Ex6.m1.6"><semantics id="S4.Ex6.m1.6a"><mrow id="S4.Ex6.m1.6.6.1" xref="S4.Ex6.m1.6.6.1.1.cmml"><mrow id="S4.Ex6.m1.6.6.1.1" xref="S4.Ex6.m1.6.6.1.1.cmml"><msup id="S4.Ex6.m1.6.6.1.1.2" xref="S4.Ex6.m1.6.6.1.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.Ex6.m1.6.6.1.1.2.2" mathvariant="bold-italic" xref="S4.Ex6.m1.6.6.1.1.2.2.cmml">ϵ</mi><mrow id="S4.Ex6.m1.1.1.1.3" xref="S4.Ex6.m1.6.6.1.1.2.cmml"><mo id="S4.Ex6.m1.1.1.1.3.1" stretchy="false" xref="S4.Ex6.m1.6.6.1.1.2.cmml">(</mo><mi id="S4.Ex6.m1.1.1.1.1" xref="S4.Ex6.m1.1.1.1.1.cmml">i</mi><mo id="S4.Ex6.m1.1.1.1.3.2" stretchy="false" xref="S4.Ex6.m1.6.6.1.1.2.cmml">)</mo></mrow></msup><mo id="S4.Ex6.m1.6.6.1.1.1" xref="S4.Ex6.m1.6.6.1.1.1.cmml">=</mo><mrow id="S4.Ex6.m1.6.6.1.1.3" xref="S4.Ex6.m1.6.6.1.1.3.cmml"><msup id="S4.Ex6.m1.6.6.1.1.3.2" xref="S4.Ex6.m1.6.6.1.1.3.2.cmml"><mi id="S4.Ex6.m1.6.6.1.1.3.2.2" xref="S4.Ex6.m1.6.6.1.1.3.2.2.cmml">𝒛</mi><mrow id="S4.Ex6.m1.2.2.1.3" xref="S4.Ex6.m1.6.6.1.1.3.2.cmml"><mo id="S4.Ex6.m1.2.2.1.3.1" stretchy="false" xref="S4.Ex6.m1.6.6.1.1.3.2.cmml">(</mo><mi id="S4.Ex6.m1.2.2.1.1" xref="S4.Ex6.m1.2.2.1.1.cmml">i</mi><mo id="S4.Ex6.m1.2.2.1.3.2" stretchy="false" xref="S4.Ex6.m1.6.6.1.1.3.2.cmml">)</mo></mrow></msup><mo id="S4.Ex6.m1.6.6.1.1.3.1" xref="S4.Ex6.m1.6.6.1.1.3.1.cmml">−</mo><mrow id="S4.Ex6.m1.6.6.1.1.3.3" xref="S4.Ex6.m1.6.6.1.1.3.3.cmml"><mstyle displaystyle="true" id="S4.Ex6.m1.6.6.1.1.3.3.1" xref="S4.Ex6.m1.6.6.1.1.3.3.1.cmml"><munderover id="S4.Ex6.m1.6.6.1.1.3.3.1a" xref="S4.Ex6.m1.6.6.1.1.3.3.1.cmml"><mo id="S4.Ex6.m1.6.6.1.1.3.3.1.2.2" movablelimits="false" xref="S4.Ex6.m1.6.6.1.1.3.3.1.2.2.cmml">∑</mo><mrow id="S4.Ex6.m1.6.6.1.1.3.3.1.2.3" xref="S4.Ex6.m1.6.6.1.1.3.3.1.2.3.cmml"><mi id="S4.Ex6.m1.6.6.1.1.3.3.1.2.3.2" xref="S4.Ex6.m1.6.6.1.1.3.3.1.2.3.2.cmml">j</mi><mo id="S4.Ex6.m1.6.6.1.1.3.3.1.2.3.1" xref="S4.Ex6.m1.6.6.1.1.3.3.1.2.3.1.cmml">=</mo><mn id="S4.Ex6.m1.6.6.1.1.3.3.1.2.3.3" xref="S4.Ex6.m1.6.6.1.1.3.3.1.2.3.3.cmml">0</mn></mrow><mi id="S4.Ex6.m1.6.6.1.1.3.3.1.3" xref="S4.Ex6.m1.6.6.1.1.3.3.1.3.cmml">i</mi></munderover></mstyle><mrow id="S4.Ex6.m1.6.6.1.1.3.3.2" xref="S4.Ex6.m1.6.6.1.1.3.3.2.cmml"><msubsup id="S4.Ex6.m1.6.6.1.1.3.3.2.2" xref="S4.Ex6.m1.6.6.1.1.3.3.2.2.cmml"><mi id="S4.Ex6.m1.6.6.1.1.3.3.2.2.2.2" xref="S4.Ex6.m1.6.6.1.1.3.3.2.2.2.2.cmml">C</mi><mi id="S4.Ex6.m1.6.6.1.1.3.3.2.2.2.3" xref="S4.Ex6.m1.6.6.1.1.3.3.2.2.2.3.cmml">i</mi><mi id="S4.Ex6.m1.6.6.1.1.3.3.2.2.3" xref="S4.Ex6.m1.6.6.1.1.3.3.2.2.3.cmml">j</mi></msubsup><mo id="S4.Ex6.m1.6.6.1.1.3.3.2.1" xref="S4.Ex6.m1.6.6.1.1.3.3.2.1.cmml">⁢</mo><msup id="S4.Ex6.m1.6.6.1.1.3.3.2.3" xref="S4.Ex6.m1.6.6.1.1.3.3.2.3.cmml"><mi id="S4.Ex6.m1.6.6.1.1.3.3.2.3.2" xref="S4.Ex6.m1.6.6.1.1.3.3.2.3.2.cmml">s</mi><mrow id="S4.Ex6.m1.6.6.1.1.3.3.2.3.3" xref="S4.Ex6.m1.6.6.1.1.3.3.2.3.3.cmml"><mi id="S4.Ex6.m1.6.6.1.1.3.3.2.3.3.2" xref="S4.Ex6.m1.6.6.1.1.3.3.2.3.3.2.cmml">i</mi><mo id="S4.Ex6.m1.6.6.1.1.3.3.2.3.3.1" xref="S4.Ex6.m1.6.6.1.1.3.3.2.3.3.1.cmml">−</mo><mi id="S4.Ex6.m1.6.6.1.1.3.3.2.3.3.3" xref="S4.Ex6.m1.6.6.1.1.3.3.2.3.3.3.cmml">j</mi></mrow></msup><mo id="S4.Ex6.m1.6.6.1.1.3.3.2.1a" xref="S4.Ex6.m1.6.6.1.1.3.3.2.1.cmml">⁢</mo><mi id="S4.Ex6.m1.6.6.1.1.3.3.2.4" xref="S4.Ex6.m1.6.6.1.1.3.3.2.4.cmml">H</mi><mo id="S4.Ex6.m1.6.6.1.1.3.3.2.1b" xref="S4.Ex6.m1.6.6.1.1.3.3.2.1.cmml">⁢</mo><mrow id="S4.Ex6.m1.6.6.1.1.3.3.2.5.2" xref="S4.Ex6.m1.6.6.1.1.3.3.2.cmml"><mo id="S4.Ex6.m1.6.6.1.1.3.3.2.5.2.1" stretchy="false" xref="S4.Ex6.m1.6.6.1.1.3.3.2.cmml">(</mo><mi id="S4.Ex6.m1.4.4" xref="S4.Ex6.m1.4.4.cmml">s</mi><mo id="S4.Ex6.m1.6.6.1.1.3.3.2.5.2.2" stretchy="false" xref="S4.Ex6.m1.6.6.1.1.3.3.2.cmml">)</mo></mrow><mo id="S4.Ex6.m1.6.6.1.1.3.3.2.1c" xref="S4.Ex6.m1.6.6.1.1.3.3.2.1.cmml">⁢</mo><mrow id="S4.Ex6.m1.6.6.1.1.3.3.2.6.2" xref="S4.Ex6.m1.6.6.1.1.3.3.2.6.1.cmml"><mo id="S4.Ex6.m1.6.6.1.1.3.3.2.6.2.1" stretchy="false" xref="S4.Ex6.m1.6.6.1.1.3.3.2.6.1.1.cmml">[</mo><mi id="S4.Ex6.m1.5.5" mathvariant="normal" xref="S4.Ex6.m1.5.5.cmml">Φ</mi><mo id="S4.Ex6.m1.6.6.1.1.3.3.2.6.2.2" stretchy="false" xref="S4.Ex6.m1.6.6.1.1.3.3.2.6.1.1.cmml">]</mo></mrow><mo id="S4.Ex6.m1.6.6.1.1.3.3.2.1d" xref="S4.Ex6.m1.6.6.1.1.3.3.2.1.cmml">⁢</mo><msup id="S4.Ex6.m1.6.6.1.1.3.3.2.7" xref="S4.Ex6.m1.6.6.1.1.3.3.2.7.cmml"><mover accent="true" id="S4.Ex6.m1.6.6.1.1.3.3.2.7.2" xref="S4.Ex6.m1.6.6.1.1.3.3.2.7.2.cmml"><mi id="S4.Ex6.m1.6.6.1.1.3.3.2.7.2.2" xref="S4.Ex6.m1.6.6.1.1.3.3.2.7.2.2.cmml">𝜽</mi><mo id="S4.Ex6.m1.6.6.1.1.3.3.2.7.2.1" xref="S4.Ex6.m1.6.6.1.1.3.3.2.7.2.1.cmml">^</mo></mover><mrow id="S4.Ex6.m1.3.3.1.3" xref="S4.Ex6.m1.6.6.1.1.3.3.2.7.cmml"><mo id="S4.Ex6.m1.3.3.1.3.1" stretchy="false" xref="S4.Ex6.m1.6.6.1.1.3.3.2.7.cmml">(</mo><mi id="S4.Ex6.m1.3.3.1.1" xref="S4.Ex6.m1.3.3.1.1.cmml">j</mi><mo id="S4.Ex6.m1.3.3.1.3.2" stretchy="false" xref="S4.Ex6.m1.6.6.1.1.3.3.2.7.cmml">)</mo></mrow></msup></mrow></mrow></mrow></mrow><mo id="S4.Ex6.m1.6.6.1.2" xref="S4.Ex6.m1.6.6.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex6.m1.6b"><apply id="S4.Ex6.m1.6.6.1.1.cmml" xref="S4.Ex6.m1.6.6.1"><eq id="S4.Ex6.m1.6.6.1.1.1.cmml" xref="S4.Ex6.m1.6.6.1.1.1"></eq><apply id="S4.Ex6.m1.6.6.1.1.2.cmml" xref="S4.Ex6.m1.6.6.1.1.2"><csymbol cd="ambiguous" id="S4.Ex6.m1.6.6.1.1.2.1.cmml" xref="S4.Ex6.m1.6.6.1.1.2">superscript</csymbol><ci id="S4.Ex6.m1.6.6.1.1.2.2.cmml" xref="S4.Ex6.m1.6.6.1.1.2.2">bold-italic-ϵ</ci><ci id="S4.Ex6.m1.1.1.1.1.cmml" xref="S4.Ex6.m1.1.1.1.1">𝑖</ci></apply><apply id="S4.Ex6.m1.6.6.1.1.3.cmml" xref="S4.Ex6.m1.6.6.1.1.3"><minus id="S4.Ex6.m1.6.6.1.1.3.1.cmml" xref="S4.Ex6.m1.6.6.1.1.3.1"></minus><apply id="S4.Ex6.m1.6.6.1.1.3.2.cmml" xref="S4.Ex6.m1.6.6.1.1.3.2"><csymbol cd="ambiguous" id="S4.Ex6.m1.6.6.1.1.3.2.1.cmml" xref="S4.Ex6.m1.6.6.1.1.3.2">superscript</csymbol><ci id="S4.Ex6.m1.6.6.1.1.3.2.2.cmml" xref="S4.Ex6.m1.6.6.1.1.3.2.2">𝒛</ci><ci id="S4.Ex6.m1.2.2.1.1.cmml" xref="S4.Ex6.m1.2.2.1.1">𝑖</ci></apply><apply id="S4.Ex6.m1.6.6.1.1.3.3.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3"><apply id="S4.Ex6.m1.6.6.1.1.3.3.1.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.1"><csymbol cd="ambiguous" id="S4.Ex6.m1.6.6.1.1.3.3.1.1.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.1">superscript</csymbol><apply id="S4.Ex6.m1.6.6.1.1.3.3.1.2.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.1"><csymbol cd="ambiguous" id="S4.Ex6.m1.6.6.1.1.3.3.1.2.1.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.1">subscript</csymbol><sum id="S4.Ex6.m1.6.6.1.1.3.3.1.2.2.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.1.2.2"></sum><apply id="S4.Ex6.m1.6.6.1.1.3.3.1.2.3.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.1.2.3"><eq id="S4.Ex6.m1.6.6.1.1.3.3.1.2.3.1.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.1.2.3.1"></eq><ci id="S4.Ex6.m1.6.6.1.1.3.3.1.2.3.2.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.1.2.3.2">𝑗</ci><cn id="S4.Ex6.m1.6.6.1.1.3.3.1.2.3.3.cmml" type="integer" xref="S4.Ex6.m1.6.6.1.1.3.3.1.2.3.3">0</cn></apply></apply><ci id="S4.Ex6.m1.6.6.1.1.3.3.1.3.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.1.3">𝑖</ci></apply><apply id="S4.Ex6.m1.6.6.1.1.3.3.2.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.2"><times id="S4.Ex6.m1.6.6.1.1.3.3.2.1.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.2.1"></times><apply id="S4.Ex6.m1.6.6.1.1.3.3.2.2.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.2.2"><csymbol cd="ambiguous" id="S4.Ex6.m1.6.6.1.1.3.3.2.2.1.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.2.2">superscript</csymbol><apply id="S4.Ex6.m1.6.6.1.1.3.3.2.2.2.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.2.2"><csymbol cd="ambiguous" id="S4.Ex6.m1.6.6.1.1.3.3.2.2.2.1.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.2.2">subscript</csymbol><ci id="S4.Ex6.m1.6.6.1.1.3.3.2.2.2.2.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.2.2.2.2">𝐶</ci><ci id="S4.Ex6.m1.6.6.1.1.3.3.2.2.2.3.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.2.2.2.3">𝑖</ci></apply><ci id="S4.Ex6.m1.6.6.1.1.3.3.2.2.3.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.2.2.3">𝑗</ci></apply><apply id="S4.Ex6.m1.6.6.1.1.3.3.2.3.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.2.3"><csymbol cd="ambiguous" id="S4.Ex6.m1.6.6.1.1.3.3.2.3.1.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.2.3">superscript</csymbol><ci id="S4.Ex6.m1.6.6.1.1.3.3.2.3.2.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.2.3.2">𝑠</ci><apply id="S4.Ex6.m1.6.6.1.1.3.3.2.3.3.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.2.3.3"><minus id="S4.Ex6.m1.6.6.1.1.3.3.2.3.3.1.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.2.3.3.1"></minus><ci id="S4.Ex6.m1.6.6.1.1.3.3.2.3.3.2.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.2.3.3.2">𝑖</ci><ci id="S4.Ex6.m1.6.6.1.1.3.3.2.3.3.3.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.2.3.3.3">𝑗</ci></apply></apply><ci id="S4.Ex6.m1.6.6.1.1.3.3.2.4.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.2.4">𝐻</ci><ci id="S4.Ex6.m1.4.4.cmml" xref="S4.Ex6.m1.4.4">𝑠</ci><apply id="S4.Ex6.m1.6.6.1.1.3.3.2.6.1.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.2.6.2"><csymbol cd="latexml" id="S4.Ex6.m1.6.6.1.1.3.3.2.6.1.1.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.2.6.2.1">delimited-[]</csymbol><ci id="S4.Ex6.m1.5.5.cmml" xref="S4.Ex6.m1.5.5">Φ</ci></apply><apply id="S4.Ex6.m1.6.6.1.1.3.3.2.7.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.2.7"><csymbol cd="ambiguous" id="S4.Ex6.m1.6.6.1.1.3.3.2.7.1.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.2.7">superscript</csymbol><apply id="S4.Ex6.m1.6.6.1.1.3.3.2.7.2.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.2.7.2"><ci id="S4.Ex6.m1.6.6.1.1.3.3.2.7.2.1.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.2.7.2.1">^</ci><ci id="S4.Ex6.m1.6.6.1.1.3.3.2.7.2.2.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.2.7.2.2">𝜽</ci></apply><ci id="S4.Ex6.m1.3.3.1.1.cmml" xref="S4.Ex6.m1.3.3.1.1">𝑗</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex6.m1.6c">\displaystyle\bm{\epsilon}^{(i)}=\bm{z}^{(i)}-\sum_{j=0}^{i}C_{i}^{j}s^{i-j}H(% s)[\Phi]\hat{\bm{\theta}}^{(j)},</annotation><annotation encoding="application/x-llamapun" id="S4.Ex6.m1.6d">bold_italic_ϵ start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT = bold_italic_z start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT - ∑ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_s start_POSTSUPERSCRIPT italic_i - italic_j end_POSTSUPERSCRIPT italic_H ( italic_s ) [ roman_Φ ] over^ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle{\color[rgb]{0.00,0.00,0.60}\bm{\xi}^{(k)}={\bm{q}}_{\rm f}^{(k)}% (t,t_{\rm e})-\sum_{i=0}^{k}C_{k}^{i}Q^{(k-i)}(t,t_{\rm e})\hat{\bm{\theta}}^{% (i)}}" class="ltx_Math" display="inline" id="S4.Ex7.m1.8"><semantics id="S4.Ex7.m1.8a"><mrow id="S4.Ex7.m1.8.8" xref="S4.Ex7.m1.8.8.cmml"><msup id="S4.Ex7.m1.8.8.4" xref="S4.Ex7.m1.8.8.4.cmml"><mi id="S4.Ex7.m1.8.8.4.2" mathcolor="#000099" xref="S4.Ex7.m1.8.8.4.2.cmml">𝝃</mi><mrow id="S4.Ex7.m1.1.1.1.3" xref="S4.Ex7.m1.8.8.4.cmml"><mo id="S4.Ex7.m1.1.1.1.3.1" mathcolor="#000099" stretchy="false" xref="S4.Ex7.m1.8.8.4.cmml">(</mo><mi id="S4.Ex7.m1.1.1.1.1" mathcolor="#000099" xref="S4.Ex7.m1.1.1.1.1.cmml">k</mi><mo id="S4.Ex7.m1.1.1.1.3.2" mathcolor="#000099" stretchy="false" xref="S4.Ex7.m1.8.8.4.cmml">)</mo></mrow></msup><mo id="S4.Ex7.m1.8.8.3" mathcolor="#000099" xref="S4.Ex7.m1.8.8.3.cmml">=</mo><mrow id="S4.Ex7.m1.8.8.2" xref="S4.Ex7.m1.8.8.2.cmml"><mrow id="S4.Ex7.m1.7.7.1.1" xref="S4.Ex7.m1.7.7.1.1.cmml"><msubsup id="S4.Ex7.m1.7.7.1.1.3" xref="S4.Ex7.m1.7.7.1.1.3.cmml"><mi id="S4.Ex7.m1.7.7.1.1.3.2.2" mathcolor="#000099" xref="S4.Ex7.m1.7.7.1.1.3.2.2.cmml">𝒒</mi><mi id="S4.Ex7.m1.7.7.1.1.3.2.3" mathcolor="#000099" mathvariant="normal" xref="S4.Ex7.m1.7.7.1.1.3.2.3.cmml">f</mi><mrow id="S4.Ex7.m1.2.2.1.3" xref="S4.Ex7.m1.7.7.1.1.3.cmml"><mo id="S4.Ex7.m1.2.2.1.3.1" mathcolor="#000099" stretchy="false" xref="S4.Ex7.m1.7.7.1.1.3.cmml">(</mo><mi id="S4.Ex7.m1.2.2.1.1" mathcolor="#000099" xref="S4.Ex7.m1.2.2.1.1.cmml">k</mi><mo id="S4.Ex7.m1.2.2.1.3.2" mathcolor="#000099" stretchy="false" xref="S4.Ex7.m1.7.7.1.1.3.cmml">)</mo></mrow></msubsup><mo id="S4.Ex7.m1.7.7.1.1.2" xref="S4.Ex7.m1.7.7.1.1.2.cmml">⁢</mo><mrow id="S4.Ex7.m1.7.7.1.1.1.1" xref="S4.Ex7.m1.7.7.1.1.1.2.cmml"><mo id="S4.Ex7.m1.7.7.1.1.1.1.2" mathcolor="#000099" stretchy="false" xref="S4.Ex7.m1.7.7.1.1.1.2.cmml">(</mo><mi id="S4.Ex7.m1.5.5" mathcolor="#000099" xref="S4.Ex7.m1.5.5.cmml">t</mi><mo id="S4.Ex7.m1.7.7.1.1.1.1.3" mathcolor="#000099" xref="S4.Ex7.m1.7.7.1.1.1.2.cmml">,</mo><msub id="S4.Ex7.m1.7.7.1.1.1.1.1" xref="S4.Ex7.m1.7.7.1.1.1.1.1.cmml"><mi id="S4.Ex7.m1.7.7.1.1.1.1.1.2" mathcolor="#000099" xref="S4.Ex7.m1.7.7.1.1.1.1.1.2.cmml">t</mi><mi id="S4.Ex7.m1.7.7.1.1.1.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S4.Ex7.m1.7.7.1.1.1.1.1.3.cmml">e</mi></msub><mo id="S4.Ex7.m1.7.7.1.1.1.1.4" mathcolor="#000099" stretchy="false" xref="S4.Ex7.m1.7.7.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex7.m1.8.8.2.3" mathcolor="#000099" xref="S4.Ex7.m1.8.8.2.3.cmml">−</mo><mrow id="S4.Ex7.m1.8.8.2.2" xref="S4.Ex7.m1.8.8.2.2.cmml"><mstyle displaystyle="true" id="S4.Ex7.m1.8.8.2.2.2" xref="S4.Ex7.m1.8.8.2.2.2.cmml"><munderover id="S4.Ex7.m1.8.8.2.2.2a" xref="S4.Ex7.m1.8.8.2.2.2.cmml"><mo id="S4.Ex7.m1.8.8.2.2.2.2.2" mathcolor="#000099" movablelimits="false" xref="S4.Ex7.m1.8.8.2.2.2.2.2.cmml">∑</mo><mrow id="S4.Ex7.m1.8.8.2.2.2.2.3" xref="S4.Ex7.m1.8.8.2.2.2.2.3.cmml"><mi id="S4.Ex7.m1.8.8.2.2.2.2.3.2" mathcolor="#000099" xref="S4.Ex7.m1.8.8.2.2.2.2.3.2.cmml">i</mi><mo id="S4.Ex7.m1.8.8.2.2.2.2.3.1" mathcolor="#000099" xref="S4.Ex7.m1.8.8.2.2.2.2.3.1.cmml">=</mo><mn id="S4.Ex7.m1.8.8.2.2.2.2.3.3" mathcolor="#000099" xref="S4.Ex7.m1.8.8.2.2.2.2.3.3.cmml">0</mn></mrow><mi id="S4.Ex7.m1.8.8.2.2.2.3" mathcolor="#000099" xref="S4.Ex7.m1.8.8.2.2.2.3.cmml">k</mi></munderover></mstyle><mrow id="S4.Ex7.m1.8.8.2.2.1" xref="S4.Ex7.m1.8.8.2.2.1.cmml"><msubsup id="S4.Ex7.m1.8.8.2.2.1.3" xref="S4.Ex7.m1.8.8.2.2.1.3.cmml"><mi id="S4.Ex7.m1.8.8.2.2.1.3.2.2" mathcolor="#000099" xref="S4.Ex7.m1.8.8.2.2.1.3.2.2.cmml">C</mi><mi id="S4.Ex7.m1.8.8.2.2.1.3.2.3" mathcolor="#000099" xref="S4.Ex7.m1.8.8.2.2.1.3.2.3.cmml">k</mi><mi id="S4.Ex7.m1.8.8.2.2.1.3.3" mathcolor="#000099" xref="S4.Ex7.m1.8.8.2.2.1.3.3.cmml">i</mi></msubsup><mo id="S4.Ex7.m1.8.8.2.2.1.2" xref="S4.Ex7.m1.8.8.2.2.1.2.cmml">⁢</mo><msup id="S4.Ex7.m1.8.8.2.2.1.4" xref="S4.Ex7.m1.8.8.2.2.1.4.cmml"><mi id="S4.Ex7.m1.8.8.2.2.1.4.2" mathcolor="#000099" xref="S4.Ex7.m1.8.8.2.2.1.4.2.cmml">Q</mi><mrow id="S4.Ex7.m1.3.3.1.1" xref="S4.Ex7.m1.3.3.1.1.1.cmml"><mo id="S4.Ex7.m1.3.3.1.1.2" mathcolor="#000099" stretchy="false" xref="S4.Ex7.m1.3.3.1.1.1.cmml">(</mo><mrow id="S4.Ex7.m1.3.3.1.1.1" xref="S4.Ex7.m1.3.3.1.1.1.cmml"><mi id="S4.Ex7.m1.3.3.1.1.1.2" mathcolor="#000099" xref="S4.Ex7.m1.3.3.1.1.1.2.cmml">k</mi><mo id="S4.Ex7.m1.3.3.1.1.1.1" mathcolor="#000099" xref="S4.Ex7.m1.3.3.1.1.1.1.cmml">−</mo><mi id="S4.Ex7.m1.3.3.1.1.1.3" mathcolor="#000099" xref="S4.Ex7.m1.3.3.1.1.1.3.cmml">i</mi></mrow><mo id="S4.Ex7.m1.3.3.1.1.3" mathcolor="#000099" stretchy="false" xref="S4.Ex7.m1.3.3.1.1.1.cmml">)</mo></mrow></msup><mo id="S4.Ex7.m1.8.8.2.2.1.2a" xref="S4.Ex7.m1.8.8.2.2.1.2.cmml">⁢</mo><mrow id="S4.Ex7.m1.8.8.2.2.1.1.1" xref="S4.Ex7.m1.8.8.2.2.1.1.2.cmml"><mo id="S4.Ex7.m1.8.8.2.2.1.1.1.2" mathcolor="#000099" stretchy="false" xref="S4.Ex7.m1.8.8.2.2.1.1.2.cmml">(</mo><mi id="S4.Ex7.m1.6.6" mathcolor="#000099" xref="S4.Ex7.m1.6.6.cmml">t</mi><mo id="S4.Ex7.m1.8.8.2.2.1.1.1.3" mathcolor="#000099" xref="S4.Ex7.m1.8.8.2.2.1.1.2.cmml">,</mo><msub id="S4.Ex7.m1.8.8.2.2.1.1.1.1" xref="S4.Ex7.m1.8.8.2.2.1.1.1.1.cmml"><mi id="S4.Ex7.m1.8.8.2.2.1.1.1.1.2" mathcolor="#000099" xref="S4.Ex7.m1.8.8.2.2.1.1.1.1.2.cmml">t</mi><mi id="S4.Ex7.m1.8.8.2.2.1.1.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S4.Ex7.m1.8.8.2.2.1.1.1.1.3.cmml">e</mi></msub><mo id="S4.Ex7.m1.8.8.2.2.1.1.1.4" mathcolor="#000099" stretchy="false" xref="S4.Ex7.m1.8.8.2.2.1.1.2.cmml">)</mo></mrow><mo id="S4.Ex7.m1.8.8.2.2.1.2b" xref="S4.Ex7.m1.8.8.2.2.1.2.cmml">⁢</mo><msup id="S4.Ex7.m1.8.8.2.2.1.5" xref="S4.Ex7.m1.8.8.2.2.1.5.cmml"><mover accent="true" id="S4.Ex7.m1.8.8.2.2.1.5.2" xref="S4.Ex7.m1.8.8.2.2.1.5.2.cmml"><mi id="S4.Ex7.m1.8.8.2.2.1.5.2.2" mathcolor="#000099" xref="S4.Ex7.m1.8.8.2.2.1.5.2.2.cmml">𝜽</mi><mo id="S4.Ex7.m1.8.8.2.2.1.5.2.1" mathcolor="#000099" xref="S4.Ex7.m1.8.8.2.2.1.5.2.1.cmml">^</mo></mover><mrow id="S4.Ex7.m1.4.4.1.3" xref="S4.Ex7.m1.8.8.2.2.1.5.cmml"><mo id="S4.Ex7.m1.4.4.1.3.1" mathcolor="#000099" stretchy="false" xref="S4.Ex7.m1.8.8.2.2.1.5.cmml">(</mo><mi id="S4.Ex7.m1.4.4.1.1" mathcolor="#000099" xref="S4.Ex7.m1.4.4.1.1.cmml">i</mi><mo id="S4.Ex7.m1.4.4.1.3.2" mathcolor="#000099" stretchy="false" xref="S4.Ex7.m1.8.8.2.2.1.5.cmml">)</mo></mrow></msup></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex7.m1.8b"><apply id="S4.Ex7.m1.8.8.cmml" xref="S4.Ex7.m1.8.8"><eq id="S4.Ex7.m1.8.8.3.cmml" xref="S4.Ex7.m1.8.8.3"></eq><apply id="S4.Ex7.m1.8.8.4.cmml" xref="S4.Ex7.m1.8.8.4"><csymbol cd="ambiguous" id="S4.Ex7.m1.8.8.4.1.cmml" xref="S4.Ex7.m1.8.8.4">superscript</csymbol><ci id="S4.Ex7.m1.8.8.4.2.cmml" xref="S4.Ex7.m1.8.8.4.2">𝝃</ci><ci id="S4.Ex7.m1.1.1.1.1.cmml" xref="S4.Ex7.m1.1.1.1.1">𝑘</ci></apply><apply id="S4.Ex7.m1.8.8.2.cmml" xref="S4.Ex7.m1.8.8.2"><minus id="S4.Ex7.m1.8.8.2.3.cmml" xref="S4.Ex7.m1.8.8.2.3"></minus><apply id="S4.Ex7.m1.7.7.1.1.cmml" xref="S4.Ex7.m1.7.7.1.1"><times id="S4.Ex7.m1.7.7.1.1.2.cmml" xref="S4.Ex7.m1.7.7.1.1.2"></times><apply id="S4.Ex7.m1.7.7.1.1.3.cmml" xref="S4.Ex7.m1.7.7.1.1.3"><csymbol cd="ambiguous" id="S4.Ex7.m1.7.7.1.1.3.1.cmml" xref="S4.Ex7.m1.7.7.1.1.3">superscript</csymbol><apply id="S4.Ex7.m1.7.7.1.1.3.2.cmml" xref="S4.Ex7.m1.7.7.1.1.3"><csymbol cd="ambiguous" id="S4.Ex7.m1.7.7.1.1.3.2.1.cmml" xref="S4.Ex7.m1.7.7.1.1.3">subscript</csymbol><ci id="S4.Ex7.m1.7.7.1.1.3.2.2.cmml" xref="S4.Ex7.m1.7.7.1.1.3.2.2">𝒒</ci><ci id="S4.Ex7.m1.7.7.1.1.3.2.3.cmml" xref="S4.Ex7.m1.7.7.1.1.3.2.3">f</ci></apply><ci id="S4.Ex7.m1.2.2.1.1.cmml" xref="S4.Ex7.m1.2.2.1.1">𝑘</ci></apply><interval closure="open" id="S4.Ex7.m1.7.7.1.1.1.2.cmml" xref="S4.Ex7.m1.7.7.1.1.1.1"><ci id="S4.Ex7.m1.5.5.cmml" xref="S4.Ex7.m1.5.5">𝑡</ci><apply id="S4.Ex7.m1.7.7.1.1.1.1.1.cmml" xref="S4.Ex7.m1.7.7.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex7.m1.7.7.1.1.1.1.1.1.cmml" xref="S4.Ex7.m1.7.7.1.1.1.1.1">subscript</csymbol><ci id="S4.Ex7.m1.7.7.1.1.1.1.1.2.cmml" xref="S4.Ex7.m1.7.7.1.1.1.1.1.2">𝑡</ci><ci id="S4.Ex7.m1.7.7.1.1.1.1.1.3.cmml" xref="S4.Ex7.m1.7.7.1.1.1.1.1.3">e</ci></apply></interval></apply><apply id="S4.Ex7.m1.8.8.2.2.cmml" xref="S4.Ex7.m1.8.8.2.2"><apply id="S4.Ex7.m1.8.8.2.2.2.cmml" xref="S4.Ex7.m1.8.8.2.2.2"><csymbol cd="ambiguous" id="S4.Ex7.m1.8.8.2.2.2.1.cmml" xref="S4.Ex7.m1.8.8.2.2.2">superscript</csymbol><apply id="S4.Ex7.m1.8.8.2.2.2.2.cmml" xref="S4.Ex7.m1.8.8.2.2.2"><csymbol cd="ambiguous" id="S4.Ex7.m1.8.8.2.2.2.2.1.cmml" xref="S4.Ex7.m1.8.8.2.2.2">subscript</csymbol><sum id="S4.Ex7.m1.8.8.2.2.2.2.2.cmml" xref="S4.Ex7.m1.8.8.2.2.2.2.2"></sum><apply id="S4.Ex7.m1.8.8.2.2.2.2.3.cmml" xref="S4.Ex7.m1.8.8.2.2.2.2.3"><eq id="S4.Ex7.m1.8.8.2.2.2.2.3.1.cmml" xref="S4.Ex7.m1.8.8.2.2.2.2.3.1"></eq><ci id="S4.Ex7.m1.8.8.2.2.2.2.3.2.cmml" xref="S4.Ex7.m1.8.8.2.2.2.2.3.2">𝑖</ci><cn id="S4.Ex7.m1.8.8.2.2.2.2.3.3.cmml" type="integer" xref="S4.Ex7.m1.8.8.2.2.2.2.3.3">0</cn></apply></apply><ci id="S4.Ex7.m1.8.8.2.2.2.3.cmml" xref="S4.Ex7.m1.8.8.2.2.2.3">𝑘</ci></apply><apply id="S4.Ex7.m1.8.8.2.2.1.cmml" xref="S4.Ex7.m1.8.8.2.2.1"><times id="S4.Ex7.m1.8.8.2.2.1.2.cmml" xref="S4.Ex7.m1.8.8.2.2.1.2"></times><apply id="S4.Ex7.m1.8.8.2.2.1.3.cmml" xref="S4.Ex7.m1.8.8.2.2.1.3"><csymbol cd="ambiguous" id="S4.Ex7.m1.8.8.2.2.1.3.1.cmml" xref="S4.Ex7.m1.8.8.2.2.1.3">superscript</csymbol><apply id="S4.Ex7.m1.8.8.2.2.1.3.2.cmml" xref="S4.Ex7.m1.8.8.2.2.1.3"><csymbol cd="ambiguous" id="S4.Ex7.m1.8.8.2.2.1.3.2.1.cmml" xref="S4.Ex7.m1.8.8.2.2.1.3">subscript</csymbol><ci id="S4.Ex7.m1.8.8.2.2.1.3.2.2.cmml" xref="S4.Ex7.m1.8.8.2.2.1.3.2.2">𝐶</ci><ci id="S4.Ex7.m1.8.8.2.2.1.3.2.3.cmml" xref="S4.Ex7.m1.8.8.2.2.1.3.2.3">𝑘</ci></apply><ci id="S4.Ex7.m1.8.8.2.2.1.3.3.cmml" xref="S4.Ex7.m1.8.8.2.2.1.3.3">𝑖</ci></apply><apply id="S4.Ex7.m1.8.8.2.2.1.4.cmml" xref="S4.Ex7.m1.8.8.2.2.1.4"><csymbol cd="ambiguous" id="S4.Ex7.m1.8.8.2.2.1.4.1.cmml" xref="S4.Ex7.m1.8.8.2.2.1.4">superscript</csymbol><ci id="S4.Ex7.m1.8.8.2.2.1.4.2.cmml" xref="S4.Ex7.m1.8.8.2.2.1.4.2">𝑄</ci><apply id="S4.Ex7.m1.3.3.1.1.1.cmml" xref="S4.Ex7.m1.3.3.1.1"><minus id="S4.Ex7.m1.3.3.1.1.1.1.cmml" xref="S4.Ex7.m1.3.3.1.1.1.1"></minus><ci id="S4.Ex7.m1.3.3.1.1.1.2.cmml" xref="S4.Ex7.m1.3.3.1.1.1.2">𝑘</ci><ci id="S4.Ex7.m1.3.3.1.1.1.3.cmml" xref="S4.Ex7.m1.3.3.1.1.1.3">𝑖</ci></apply></apply><interval closure="open" id="S4.Ex7.m1.8.8.2.2.1.1.2.cmml" xref="S4.Ex7.m1.8.8.2.2.1.1.1"><ci id="S4.Ex7.m1.6.6.cmml" xref="S4.Ex7.m1.6.6">𝑡</ci><apply id="S4.Ex7.m1.8.8.2.2.1.1.1.1.cmml" xref="S4.Ex7.m1.8.8.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex7.m1.8.8.2.2.1.1.1.1.1.cmml" xref="S4.Ex7.m1.8.8.2.2.1.1.1.1">subscript</csymbol><ci id="S4.Ex7.m1.8.8.2.2.1.1.1.1.2.cmml" xref="S4.Ex7.m1.8.8.2.2.1.1.1.1.2">𝑡</ci><ci id="S4.Ex7.m1.8.8.2.2.1.1.1.1.3.cmml" xref="S4.Ex7.m1.8.8.2.2.1.1.1.1.3">e</ci></apply></interval><apply id="S4.Ex7.m1.8.8.2.2.1.5.cmml" xref="S4.Ex7.m1.8.8.2.2.1.5"><csymbol cd="ambiguous" id="S4.Ex7.m1.8.8.2.2.1.5.1.cmml" xref="S4.Ex7.m1.8.8.2.2.1.5">superscript</csymbol><apply id="S4.Ex7.m1.8.8.2.2.1.5.2.cmml" xref="S4.Ex7.m1.8.8.2.2.1.5.2"><ci id="S4.Ex7.m1.8.8.2.2.1.5.2.1.cmml" xref="S4.Ex7.m1.8.8.2.2.1.5.2.1">^</ci><ci id="S4.Ex7.m1.8.8.2.2.1.5.2.2.cmml" xref="S4.Ex7.m1.8.8.2.2.1.5.2.2">𝜽</ci></apply><ci id="S4.Ex7.m1.4.4.1.1.cmml" xref="S4.Ex7.m1.4.4.1.1">𝑖</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex7.m1.8c">\displaystyle{\color[rgb]{0.00,0.00,0.60}\bm{\xi}^{(k)}={\bm{q}}_{\rm f}^{(k)}% (t,t_{\rm e})-\sum_{i=0}^{k}C_{k}^{i}Q^{(k-i)}(t,t_{\rm e})\hat{\bm{\theta}}^{% (i)}}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex7.m1.8d">bold_italic_ξ start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT = bold_italic_q start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) - ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_Q start_POSTSUPERSCRIPT ( italic_k - italic_i ) end_POSTSUPERSCRIPT ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) over^ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p8.15">in which <math alttext="C_{k}^{i}=k!/(i!(k-i)!)" class="ltx_Math" display="inline" id="S4.SS1.p8.13.m1.1"><semantics id="S4.SS1.p8.13.m1.1a"><mrow id="S4.SS1.p8.13.m1.1.1" xref="S4.SS1.p8.13.m1.1.1.cmml"><msubsup id="S4.SS1.p8.13.m1.1.1.3" xref="S4.SS1.p8.13.m1.1.1.3.cmml"><mi id="S4.SS1.p8.13.m1.1.1.3.2.2" xref="S4.SS1.p8.13.m1.1.1.3.2.2.cmml">C</mi><mi id="S4.SS1.p8.13.m1.1.1.3.2.3" xref="S4.SS1.p8.13.m1.1.1.3.2.3.cmml">k</mi><mi id="S4.SS1.p8.13.m1.1.1.3.3" xref="S4.SS1.p8.13.m1.1.1.3.3.cmml">i</mi></msubsup><mo id="S4.SS1.p8.13.m1.1.1.2" xref="S4.SS1.p8.13.m1.1.1.2.cmml">=</mo><mrow id="S4.SS1.p8.13.m1.1.1.1" xref="S4.SS1.p8.13.m1.1.1.1.cmml"><mrow id="S4.SS1.p8.13.m1.1.1.1.3" xref="S4.SS1.p8.13.m1.1.1.1.3.cmml"><mi id="S4.SS1.p8.13.m1.1.1.1.3.2" xref="S4.SS1.p8.13.m1.1.1.1.3.2.cmml">k</mi><mo id="S4.SS1.p8.13.m1.1.1.1.3.1" xref="S4.SS1.p8.13.m1.1.1.1.3.1.cmml">!</mo></mrow><mo id="S4.SS1.p8.13.m1.1.1.1.2" xref="S4.SS1.p8.13.m1.1.1.1.2.cmml">/</mo><mrow id="S4.SS1.p8.13.m1.1.1.1.1.1" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.cmml"><mo id="S4.SS1.p8.13.m1.1.1.1.1.1.2" stretchy="false" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS1.p8.13.m1.1.1.1.1.1.1" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.cmml"><mrow id="S4.SS1.p8.13.m1.1.1.1.1.1.1.3" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.3.cmml"><mi id="S4.SS1.p8.13.m1.1.1.1.1.1.1.3.2" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.3.2.cmml">i</mi><mo id="S4.SS1.p8.13.m1.1.1.1.1.1.1.3.1" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.3.1.cmml">!</mo></mrow><mo id="S4.SS1.p8.13.m1.1.1.1.1.1.1.2" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.p8.13.m1.1.1.1.1.1.1.1" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.cmml"><mrow id="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.1.1" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.1.1.1" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.1.1.1.2" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.1.1.1.2.cmml">k</mi><mo id="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.1.1.1.1" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.1.1.1.3" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.1.1.1.3.cmml">i</mi></mrow><mo id="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.2" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.2.cmml">!</mo></mrow></mrow><mo id="S4.SS1.p8.13.m1.1.1.1.1.1.3" stretchy="false" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p8.13.m1.1b"><apply id="S4.SS1.p8.13.m1.1.1.cmml" xref="S4.SS1.p8.13.m1.1.1"><eq id="S4.SS1.p8.13.m1.1.1.2.cmml" xref="S4.SS1.p8.13.m1.1.1.2"></eq><apply id="S4.SS1.p8.13.m1.1.1.3.cmml" xref="S4.SS1.p8.13.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p8.13.m1.1.1.3.1.cmml" xref="S4.SS1.p8.13.m1.1.1.3">superscript</csymbol><apply id="S4.SS1.p8.13.m1.1.1.3.2.cmml" xref="S4.SS1.p8.13.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p8.13.m1.1.1.3.2.1.cmml" xref="S4.SS1.p8.13.m1.1.1.3">subscript</csymbol><ci id="S4.SS1.p8.13.m1.1.1.3.2.2.cmml" xref="S4.SS1.p8.13.m1.1.1.3.2.2">𝐶</ci><ci id="S4.SS1.p8.13.m1.1.1.3.2.3.cmml" xref="S4.SS1.p8.13.m1.1.1.3.2.3">𝑘</ci></apply><ci id="S4.SS1.p8.13.m1.1.1.3.3.cmml" xref="S4.SS1.p8.13.m1.1.1.3.3">𝑖</ci></apply><apply id="S4.SS1.p8.13.m1.1.1.1.cmml" xref="S4.SS1.p8.13.m1.1.1.1"><divide id="S4.SS1.p8.13.m1.1.1.1.2.cmml" xref="S4.SS1.p8.13.m1.1.1.1.2"></divide><apply id="S4.SS1.p8.13.m1.1.1.1.3.cmml" xref="S4.SS1.p8.13.m1.1.1.1.3"><factorial id="S4.SS1.p8.13.m1.1.1.1.3.1.cmml" xref="S4.SS1.p8.13.m1.1.1.1.3.1"></factorial><ci id="S4.SS1.p8.13.m1.1.1.1.3.2.cmml" xref="S4.SS1.p8.13.m1.1.1.1.3.2">𝑘</ci></apply><apply id="S4.SS1.p8.13.m1.1.1.1.1.1.1.cmml" xref="S4.SS1.p8.13.m1.1.1.1.1.1"><times id="S4.SS1.p8.13.m1.1.1.1.1.1.1.2.cmml" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.2"></times><apply id="S4.SS1.p8.13.m1.1.1.1.1.1.1.3.cmml" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.3"><factorial id="S4.SS1.p8.13.m1.1.1.1.1.1.1.3.1.cmml" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.3.1"></factorial><ci id="S4.SS1.p8.13.m1.1.1.1.1.1.1.3.2.cmml" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.3.2">𝑖</ci></apply><apply id="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.cmml" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.1"><factorial id="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.2.cmml" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.2"></factorial><apply id="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.1.1"><minus id="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.1.1.1.1"></minus><ci id="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.1.1.1.2">𝑘</ci><ci id="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.SS1.p8.13.m1.1.1.1.1.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p8.13.m1.1c">C_{k}^{i}=k!/(i!(k-i)!)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p8.13.m1.1d">italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = italic_k ! / ( italic_i ! ( italic_k - italic_i ) ! )</annotation></semantics></math> are binomial coefficients with <math alttext="0\leq i\leq k" class="ltx_Math" display="inline" id="S4.SS1.p8.14.m2.1"><semantics id="S4.SS1.p8.14.m2.1a"><mrow id="S4.SS1.p8.14.m2.1.1" xref="S4.SS1.p8.14.m2.1.1.cmml"><mn id="S4.SS1.p8.14.m2.1.1.2" xref="S4.SS1.p8.14.m2.1.1.2.cmml">0</mn><mo id="S4.SS1.p8.14.m2.1.1.3" xref="S4.SS1.p8.14.m2.1.1.3.cmml">≤</mo><mi id="S4.SS1.p8.14.m2.1.1.4" xref="S4.SS1.p8.14.m2.1.1.4.cmml">i</mi><mo id="S4.SS1.p8.14.m2.1.1.5" xref="S4.SS1.p8.14.m2.1.1.5.cmml">≤</mo><mi id="S4.SS1.p8.14.m2.1.1.6" xref="S4.SS1.p8.14.m2.1.1.6.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p8.14.m2.1b"><apply id="S4.SS1.p8.14.m2.1.1.cmml" xref="S4.SS1.p8.14.m2.1.1"><and id="S4.SS1.p8.14.m2.1.1a.cmml" xref="S4.SS1.p8.14.m2.1.1"></and><apply id="S4.SS1.p8.14.m2.1.1b.cmml" xref="S4.SS1.p8.14.m2.1.1"><leq id="S4.SS1.p8.14.m2.1.1.3.cmml" xref="S4.SS1.p8.14.m2.1.1.3"></leq><cn id="S4.SS1.p8.14.m2.1.1.2.cmml" type="integer" xref="S4.SS1.p8.14.m2.1.1.2">0</cn><ci id="S4.SS1.p8.14.m2.1.1.4.cmml" xref="S4.SS1.p8.14.m2.1.1.4">𝑖</ci></apply><apply id="S4.SS1.p8.14.m2.1.1c.cmml" xref="S4.SS1.p8.14.m2.1.1"><leq id="S4.SS1.p8.14.m2.1.1.5.cmml" xref="S4.SS1.p8.14.m2.1.1.5"></leq><share href="https://arxiv.org/html/2401.10785v2#S4.SS1.p8.14.m2.1.1.4.cmml" id="S4.SS1.p8.14.m2.1.1d.cmml" xref="S4.SS1.p8.14.m2.1.1"></share><ci id="S4.SS1.p8.14.m2.1.1.6.cmml" xref="S4.SS1.p8.14.m2.1.1.6">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p8.14.m2.1c">0\leq i\leq k</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p8.14.m2.1d">0 ≤ italic_i ≤ italic_k</annotation></semantics></math> and <math alttext="1\leq k\leq n-2" class="ltx_Math" display="inline" id="S4.SS1.p8.15.m3.1"><semantics id="S4.SS1.p8.15.m3.1a"><mrow id="S4.SS1.p8.15.m3.1.1" xref="S4.SS1.p8.15.m3.1.1.cmml"><mn id="S4.SS1.p8.15.m3.1.1.2" xref="S4.SS1.p8.15.m3.1.1.2.cmml">1</mn><mo id="S4.SS1.p8.15.m3.1.1.3" xref="S4.SS1.p8.15.m3.1.1.3.cmml">≤</mo><mi id="S4.SS1.p8.15.m3.1.1.4" xref="S4.SS1.p8.15.m3.1.1.4.cmml">k</mi><mo id="S4.SS1.p8.15.m3.1.1.5" xref="S4.SS1.p8.15.m3.1.1.5.cmml">≤</mo><mrow id="S4.SS1.p8.15.m3.1.1.6" xref="S4.SS1.p8.15.m3.1.1.6.cmml"><mi id="S4.SS1.p8.15.m3.1.1.6.2" xref="S4.SS1.p8.15.m3.1.1.6.2.cmml">n</mi><mo id="S4.SS1.p8.15.m3.1.1.6.1" xref="S4.SS1.p8.15.m3.1.1.6.1.cmml">−</mo><mn id="S4.SS1.p8.15.m3.1.1.6.3" xref="S4.SS1.p8.15.m3.1.1.6.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p8.15.m3.1b"><apply id="S4.SS1.p8.15.m3.1.1.cmml" xref="S4.SS1.p8.15.m3.1.1"><and id="S4.SS1.p8.15.m3.1.1a.cmml" xref="S4.SS1.p8.15.m3.1.1"></and><apply id="S4.SS1.p8.15.m3.1.1b.cmml" xref="S4.SS1.p8.15.m3.1.1"><leq id="S4.SS1.p8.15.m3.1.1.3.cmml" xref="S4.SS1.p8.15.m3.1.1.3"></leq><cn id="S4.SS1.p8.15.m3.1.1.2.cmml" type="integer" xref="S4.SS1.p8.15.m3.1.1.2">1</cn><ci id="S4.SS1.p8.15.m3.1.1.4.cmml" xref="S4.SS1.p8.15.m3.1.1.4">𝑘</ci></apply><apply id="S4.SS1.p8.15.m3.1.1c.cmml" xref="S4.SS1.p8.15.m3.1.1"><leq id="S4.SS1.p8.15.m3.1.1.5.cmml" xref="S4.SS1.p8.15.m3.1.1.5"></leq><share href="https://arxiv.org/html/2401.10785v2#S4.SS1.p8.15.m3.1.1.4.cmml" id="S4.SS1.p8.15.m3.1.1d.cmml" xref="S4.SS1.p8.15.m3.1.1"></share><apply id="S4.SS1.p8.15.m3.1.1.6.cmml" xref="S4.SS1.p8.15.m3.1.1.6"><minus id="S4.SS1.p8.15.m3.1.1.6.1.cmml" xref="S4.SS1.p8.15.m3.1.1.6.1"></minus><ci id="S4.SS1.p8.15.m3.1.1.6.2.cmml" xref="S4.SS1.p8.15.m3.1.1.6.2">𝑛</ci><cn id="S4.SS1.p8.15.m3.1.1.6.3.cmml" type="integer" xref="S4.SS1.p8.15.m3.1.1.6.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p8.15.m3.1c">1\leq k\leq n-2</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p8.15.m3.1d">1 ≤ italic_k ≤ italic_n - 2</annotation></semantics></math>.</p> </div> <figure class="ltx_figure" id="S4.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="245" id="S4.F2.g1" src="x2.png" width="484"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 2: </span> An illustration of the relationship between the channels <math alttext="\bm{\phi}_{{\rm s},i}" class="ltx_Math" display="inline" id="S4.F2.5.m1.2"><semantics id="S4.F2.5.m1.2b"><msub id="S4.F2.5.m1.2.3" xref="S4.F2.5.m1.2.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.F2.5.m1.2.3.2" mathvariant="bold-italic" xref="S4.F2.5.m1.2.3.2.cmml">ϕ</mi><mrow id="S4.F2.5.m1.2.2.2.4" xref="S4.F2.5.m1.2.2.2.3.cmml"><mi id="S4.F2.5.m1.1.1.1.1" mathvariant="normal" xref="S4.F2.5.m1.1.1.1.1.cmml">s</mi><mo id="S4.F2.5.m1.2.2.2.4.1" xref="S4.F2.5.m1.2.2.2.3.cmml">,</mo><mi id="S4.F2.5.m1.2.2.2.2" xref="S4.F2.5.m1.2.2.2.2.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.F2.5.m1.2c"><apply id="S4.F2.5.m1.2.3.cmml" xref="S4.F2.5.m1.2.3"><csymbol cd="ambiguous" id="S4.F2.5.m1.2.3.1.cmml" xref="S4.F2.5.m1.2.3">subscript</csymbol><ci id="S4.F2.5.m1.2.3.2.cmml" xref="S4.F2.5.m1.2.3.2">bold-italic-ϕ</ci><list id="S4.F2.5.m1.2.2.2.3.cmml" xref="S4.F2.5.m1.2.2.2.4"><ci id="S4.F2.5.m1.1.1.1.1.cmml" xref="S4.F2.5.m1.1.1.1.1">s</ci><ci id="S4.F2.5.m1.2.2.2.2.cmml" xref="S4.F2.5.m1.2.2.2.2">𝑖</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F2.5.m1.2d">\bm{\phi}_{{\rm s},i}</annotation><annotation encoding="application/x-llamapun" id="S4.F2.5.m1.2e">bold_italic_ϕ start_POSTSUBSCRIPT roman_s , italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and the excitation matrix <math alttext="\Psi" class="ltx_Math" display="inline" id="S4.F2.6.m2.1"><semantics id="S4.F2.6.m2.1b"><mi id="S4.F2.6.m2.1.1" mathvariant="normal" xref="S4.F2.6.m2.1.1.cmml">Ψ</mi><annotation-xml encoding="MathML-Content" id="S4.F2.6.m2.1c"><ci id="S4.F2.6.m2.1.1.cmml" xref="S4.F2.6.m2.1.1">Ψ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F2.6.m2.1d">\Psi</annotation><annotation encoding="application/x-llamapun" id="S4.F2.6.m2.1e">roman_Ψ</annotation></semantics></math> with <math alttext="N=7" class="ltx_Math" display="inline" id="S4.F2.7.m3.1"><semantics id="S4.F2.7.m3.1b"><mrow id="S4.F2.7.m3.1.1" xref="S4.F2.7.m3.1.1.cmml"><mi id="S4.F2.7.m3.1.1.2" xref="S4.F2.7.m3.1.1.2.cmml">N</mi><mo id="S4.F2.7.m3.1.1.1" xref="S4.F2.7.m3.1.1.1.cmml">=</mo><mn id="S4.F2.7.m3.1.1.3" xref="S4.F2.7.m3.1.1.3.cmml">7</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F2.7.m3.1c"><apply id="S4.F2.7.m3.1.1.cmml" xref="S4.F2.7.m3.1.1"><eq id="S4.F2.7.m3.1.1.1.cmml" xref="S4.F2.7.m3.1.1.1"></eq><ci id="S4.F2.7.m3.1.1.2.cmml" xref="S4.F2.7.m3.1.1.2">𝑁</ci><cn id="S4.F2.7.m3.1.1.3.cmml" type="integer" xref="S4.F2.7.m3.1.1.3">7</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F2.7.m3.1d">N=7</annotation><annotation encoding="application/x-llamapun" id="S4.F2.7.m3.1e">italic_N = 7</annotation></semantics></math>. Note that the blue modules denote active channels, the white modules are inactive channels, and <math alttext="\bm{p}" class="ltx_Math" display="inline" id="S4.F2.8.m4.1"><semantics id="S4.F2.8.m4.1b"><mi id="S4.F2.8.m4.1.1" xref="S4.F2.8.m4.1.1.cmml">𝒑</mi><annotation-xml encoding="MathML-Content" id="S4.F2.8.m4.1c"><ci id="S4.F2.8.m4.1.1.cmml" xref="S4.F2.8.m4.1.1">𝒑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F2.8.m4.1d">\bm{p}</annotation><annotation encoding="application/x-llamapun" id="S4.F2.8.m4.1e">bold_italic_p</annotation></semantics></math> is defined in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E12" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">12</span></a>). </figcaption> </figure> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S4.SS2.4.1.1">IV-B</span> </span><span class="ltx_text ltx_font_italic" id="S4.SS2.5.2">Some Discussions</span> </h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.37">Regarding the regression equation (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E12" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">12</span></a>), changes in the state <math alttext="\bm{x}" class="ltx_Math" display="inline" id="S4.SS2.p1.1.m1.1"><semantics id="S4.SS2.p1.1.m1.1a"><mi id="S4.SS2.p1.1.m1.1.1" xref="S4.SS2.p1.1.m1.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.1.m1.1b"><ci id="S4.SS2.p1.1.m1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.1.m1.1c">\bm{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.1.m1.1d">bold_italic_x</annotation></semantics></math> within a time window [<math alttext="t-\tau_{\rm d}" class="ltx_Math" display="inline" id="S4.SS2.p1.2.m2.1"><semantics id="S4.SS2.p1.2.m2.1a"><mrow id="S4.SS2.p1.2.m2.1.1" xref="S4.SS2.p1.2.m2.1.1.cmml"><mi id="S4.SS2.p1.2.m2.1.1.2" xref="S4.SS2.p1.2.m2.1.1.2.cmml">t</mi><mo id="S4.SS2.p1.2.m2.1.1.1" xref="S4.SS2.p1.2.m2.1.1.1.cmml">−</mo><msub id="S4.SS2.p1.2.m2.1.1.3" xref="S4.SS2.p1.2.m2.1.1.3.cmml"><mi id="S4.SS2.p1.2.m2.1.1.3.2" xref="S4.SS2.p1.2.m2.1.1.3.2.cmml">τ</mi><mi id="S4.SS2.p1.2.m2.1.1.3.3" mathvariant="normal" xref="S4.SS2.p1.2.m2.1.1.3.3.cmml">d</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.2.m2.1b"><apply id="S4.SS2.p1.2.m2.1.1.cmml" xref="S4.SS2.p1.2.m2.1.1"><minus id="S4.SS2.p1.2.m2.1.1.1.cmml" xref="S4.SS2.p1.2.m2.1.1.1"></minus><ci id="S4.SS2.p1.2.m2.1.1.2.cmml" xref="S4.SS2.p1.2.m2.1.1.2">𝑡</ci><apply id="S4.SS2.p1.2.m2.1.1.3.cmml" xref="S4.SS2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p1.2.m2.1.1.3.1.cmml" xref="S4.SS2.p1.2.m2.1.1.3">subscript</csymbol><ci id="S4.SS2.p1.2.m2.1.1.3.2.cmml" xref="S4.SS2.p1.2.m2.1.1.3.2">𝜏</ci><ci id="S4.SS2.p1.2.m2.1.1.3.3.cmml" xref="S4.SS2.p1.2.m2.1.1.3.3">d</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.2.m2.1c">t-\tau_{\rm d}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.2.m2.1d">italic_t - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="t" class="ltx_Math" display="inline" id="S4.SS2.p1.3.m3.1"><semantics id="S4.SS2.p1.3.m3.1a"><mi id="S4.SS2.p1.3.m3.1.1" xref="S4.SS2.p1.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.3.m3.1b"><ci id="S4.SS2.p1.3.m3.1.1.cmml" xref="S4.SS2.p1.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.3.m3.1d">italic_t</annotation></semantics></math>] must cause a set of regressor channels <math alttext="{\bm{\phi}}_{{\rm s},k_{j}}" class="ltx_Math" display="inline" id="S4.SS2.p1.4.m4.2"><semantics id="S4.SS2.p1.4.m4.2a"><msub id="S4.SS2.p1.4.m4.2.3" xref="S4.SS2.p1.4.m4.2.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS2.p1.4.m4.2.3.2" mathvariant="bold-italic" xref="S4.SS2.p1.4.m4.2.3.2.cmml">ϕ</mi><mrow id="S4.SS2.p1.4.m4.2.2.2.2" xref="S4.SS2.p1.4.m4.2.2.2.3.cmml"><mi id="S4.SS2.p1.4.m4.1.1.1.1" mathvariant="normal" xref="S4.SS2.p1.4.m4.1.1.1.1.cmml">s</mi><mo id="S4.SS2.p1.4.m4.2.2.2.2.2" xref="S4.SS2.p1.4.m4.2.2.2.3.cmml">,</mo><msub id="S4.SS2.p1.4.m4.2.2.2.2.1" xref="S4.SS2.p1.4.m4.2.2.2.2.1.cmml"><mi id="S4.SS2.p1.4.m4.2.2.2.2.1.2" xref="S4.SS2.p1.4.m4.2.2.2.2.1.2.cmml">k</mi><mi id="S4.SS2.p1.4.m4.2.2.2.2.1.3" xref="S4.SS2.p1.4.m4.2.2.2.2.1.3.cmml">j</mi></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.4.m4.2b"><apply id="S4.SS2.p1.4.m4.2.3.cmml" xref="S4.SS2.p1.4.m4.2.3"><csymbol cd="ambiguous" id="S4.SS2.p1.4.m4.2.3.1.cmml" xref="S4.SS2.p1.4.m4.2.3">subscript</csymbol><ci id="S4.SS2.p1.4.m4.2.3.2.cmml" xref="S4.SS2.p1.4.m4.2.3.2">bold-italic-ϕ</ci><list id="S4.SS2.p1.4.m4.2.2.2.3.cmml" xref="S4.SS2.p1.4.m4.2.2.2.2"><ci id="S4.SS2.p1.4.m4.1.1.1.1.cmml" xref="S4.SS2.p1.4.m4.1.1.1.1">s</ci><apply id="S4.SS2.p1.4.m4.2.2.2.2.1.cmml" xref="S4.SS2.p1.4.m4.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.SS2.p1.4.m4.2.2.2.2.1.1.cmml" xref="S4.SS2.p1.4.m4.2.2.2.2.1">subscript</csymbol><ci id="S4.SS2.p1.4.m4.2.2.2.2.1.2.cmml" xref="S4.SS2.p1.4.m4.2.2.2.2.1.2">𝑘</ci><ci id="S4.SS2.p1.4.m4.2.2.2.2.1.3.cmml" xref="S4.SS2.p1.4.m4.2.2.2.2.1.3">𝑗</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.4.m4.2c">{\bm{\phi}}_{{\rm s},k_{j}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.4.m4.2d">bold_italic_ϕ start_POSTSUBSCRIPT roman_s , italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> to activate in an uncorrelated manner<span class="ltx_note ltx_role_footnote" id="footnote2"><sup class="ltx_note_mark">2</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">2</sup><span class="ltx_tag ltx_tag_note">2</span><span class="ltx_text" id="footnote2.15" style="color:#000099;">Each regressor channel <math alttext="{\bm{\phi}}_{{\rm s},k_{j}}" class="ltx_Math" display="inline" id="footnote2.1.m1.2"><semantics id="footnote2.1.m1.2b"><msub id="footnote2.1.m1.2.3" xref="footnote2.1.m1.2.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="footnote2.1.m1.2.3.2" mathcolor="#000099" mathvariant="bold-italic" xref="footnote2.1.m1.2.3.2.cmml">ϕ</mi><mrow id="footnote2.1.m1.2.2.2.2" xref="footnote2.1.m1.2.2.2.3.cmml"><mi id="footnote2.1.m1.1.1.1.1" mathcolor="#000099" mathvariant="normal" xref="footnote2.1.m1.1.1.1.1.cmml">s</mi><mo id="footnote2.1.m1.2.2.2.2.2" mathcolor="#000099" xref="footnote2.1.m1.2.2.2.3.cmml">,</mo><msub id="footnote2.1.m1.2.2.2.2.1" xref="footnote2.1.m1.2.2.2.2.1.cmml"><mi id="footnote2.1.m1.2.2.2.2.1.2" mathcolor="#000099" xref="footnote2.1.m1.2.2.2.2.1.2.cmml">k</mi><mi id="footnote2.1.m1.2.2.2.2.1.3" mathcolor="#000099" xref="footnote2.1.m1.2.2.2.2.1.3.cmml">j</mi></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="footnote2.1.m1.2c"><apply id="footnote2.1.m1.2.3.cmml" xref="footnote2.1.m1.2.3"><csymbol cd="ambiguous" id="footnote2.1.m1.2.3.1.cmml" xref="footnote2.1.m1.2.3">subscript</csymbol><ci id="footnote2.1.m1.2.3.2.cmml" xref="footnote2.1.m1.2.3.2">bold-italic-ϕ</ci><list id="footnote2.1.m1.2.2.2.3.cmml" xref="footnote2.1.m1.2.2.2.2"><ci id="footnote2.1.m1.1.1.1.1.cmml" xref="footnote2.1.m1.1.1.1.1">s</ci><apply id="footnote2.1.m1.2.2.2.2.1.cmml" xref="footnote2.1.m1.2.2.2.2.1"><csymbol cd="ambiguous" id="footnote2.1.m1.2.2.2.2.1.1.cmml" xref="footnote2.1.m1.2.2.2.2.1">subscript</csymbol><ci id="footnote2.1.m1.2.2.2.2.1.2.cmml" xref="footnote2.1.m1.2.2.2.2.1.2">𝑘</ci><ci id="footnote2.1.m1.2.2.2.2.1.3.cmml" xref="footnote2.1.m1.2.2.2.2.1.3">𝑗</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.1.m1.2d">{\bm{\phi}}_{{\rm s},k_{j}}</annotation><annotation encoding="application/x-llamapun" id="footnote2.1.m1.2e">bold_italic_ϕ start_POSTSUBSCRIPT roman_s , italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is a function of the state variable <math alttext="\bm{x}" class="ltx_Math" display="inline" id="footnote2.2.m2.1"><semantics id="footnote2.2.m2.1b"><mi id="footnote2.2.m2.1.1" mathcolor="#000099" xref="footnote2.2.m2.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="footnote2.2.m2.1c"><ci id="footnote2.2.m2.1.1.cmml" xref="footnote2.2.m2.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote2.2.m2.1d">\bm{x}</annotation><annotation encoding="application/x-llamapun" id="footnote2.2.m2.1e">bold_italic_x</annotation></semantics></math>, so changes in <math alttext="\bm{x}" class="ltx_Math" display="inline" id="footnote2.3.m3.1"><semantics id="footnote2.3.m3.1b"><mi id="footnote2.3.m3.1.1" mathcolor="#000099" xref="footnote2.3.m3.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="footnote2.3.m3.1c"><ci id="footnote2.3.m3.1.1.cmml" xref="footnote2.3.m3.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote2.3.m3.1d">\bm{x}</annotation><annotation encoding="application/x-llamapun" id="footnote2.3.m3.1e">bold_italic_x</annotation></semantics></math> must lead to the corresponding changes in <math alttext="{\bm{\phi}}_{{\rm s},k_{j}}" class="ltx_Math" display="inline" id="footnote2.4.m4.2"><semantics id="footnote2.4.m4.2b"><msub id="footnote2.4.m4.2.3" xref="footnote2.4.m4.2.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="footnote2.4.m4.2.3.2" mathcolor="#000099" mathvariant="bold-italic" xref="footnote2.4.m4.2.3.2.cmml">ϕ</mi><mrow id="footnote2.4.m4.2.2.2.2" xref="footnote2.4.m4.2.2.2.3.cmml"><mi id="footnote2.4.m4.1.1.1.1" mathcolor="#000099" mathvariant="normal" xref="footnote2.4.m4.1.1.1.1.cmml">s</mi><mo id="footnote2.4.m4.2.2.2.2.2" mathcolor="#000099" xref="footnote2.4.m4.2.2.2.3.cmml">,</mo><msub id="footnote2.4.m4.2.2.2.2.1" xref="footnote2.4.m4.2.2.2.2.1.cmml"><mi id="footnote2.4.m4.2.2.2.2.1.2" mathcolor="#000099" xref="footnote2.4.m4.2.2.2.2.1.2.cmml">k</mi><mi id="footnote2.4.m4.2.2.2.2.1.3" mathcolor="#000099" xref="footnote2.4.m4.2.2.2.2.1.3.cmml">j</mi></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="footnote2.4.m4.2c"><apply id="footnote2.4.m4.2.3.cmml" xref="footnote2.4.m4.2.3"><csymbol cd="ambiguous" id="footnote2.4.m4.2.3.1.cmml" xref="footnote2.4.m4.2.3">subscript</csymbol><ci id="footnote2.4.m4.2.3.2.cmml" xref="footnote2.4.m4.2.3.2">bold-italic-ϕ</ci><list id="footnote2.4.m4.2.2.2.3.cmml" xref="footnote2.4.m4.2.2.2.2"><ci id="footnote2.4.m4.1.1.1.1.cmml" xref="footnote2.4.m4.1.1.1.1">s</ci><apply id="footnote2.4.m4.2.2.2.2.1.cmml" xref="footnote2.4.m4.2.2.2.2.1"><csymbol cd="ambiguous" id="footnote2.4.m4.2.2.2.2.1.1.cmml" xref="footnote2.4.m4.2.2.2.2.1">subscript</csymbol><ci id="footnote2.4.m4.2.2.2.2.1.2.cmml" xref="footnote2.4.m4.2.2.2.2.1.2">𝑘</ci><ci id="footnote2.4.m4.2.2.2.2.1.3.cmml" xref="footnote2.4.m4.2.2.2.2.1.3">𝑗</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.4.m4.2d">{\bm{\phi}}_{{\rm s},k_{j}}</annotation><annotation encoding="application/x-llamapun" id="footnote2.4.m4.2e">bold_italic_ϕ start_POSTSUBSCRIPT roman_s , italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. If changes in <math alttext="\bm{x}" class="ltx_Math" display="inline" id="footnote2.5.m5.1"><semantics id="footnote2.5.m5.1b"><mi id="footnote2.5.m5.1.1" mathcolor="#000099" xref="footnote2.5.m5.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="footnote2.5.m5.1c"><ci id="footnote2.5.m5.1.1.cmml" xref="footnote2.5.m5.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote2.5.m5.1d">\bm{x}</annotation><annotation encoding="application/x-llamapun" id="footnote2.5.m5.1e">bold_italic_x</annotation></semantics></math> within a time window <math alttext="[t-\tau_{\rm d},t]" class="ltx_Math" display="inline" id="footnote2.6.m6.2"><semantics id="footnote2.6.m6.2b"><mrow id="footnote2.6.m6.2.2.1" xref="footnote2.6.m6.2.2.2.cmml"><mo id="footnote2.6.m6.2.2.1.2" mathcolor="#000099" stretchy="false" xref="footnote2.6.m6.2.2.2.cmml">[</mo><mrow id="footnote2.6.m6.2.2.1.1" xref="footnote2.6.m6.2.2.1.1.cmml"><mi id="footnote2.6.m6.2.2.1.1.2" mathcolor="#000099" xref="footnote2.6.m6.2.2.1.1.2.cmml">t</mi><mo id="footnote2.6.m6.2.2.1.1.1" mathcolor="#000099" xref="footnote2.6.m6.2.2.1.1.1.cmml">−</mo><msub id="footnote2.6.m6.2.2.1.1.3" xref="footnote2.6.m6.2.2.1.1.3.cmml"><mi id="footnote2.6.m6.2.2.1.1.3.2" mathcolor="#000099" xref="footnote2.6.m6.2.2.1.1.3.2.cmml">τ</mi><mi id="footnote2.6.m6.2.2.1.1.3.3" mathcolor="#000099" mathvariant="normal" xref="footnote2.6.m6.2.2.1.1.3.3.cmml">d</mi></msub></mrow><mo id="footnote2.6.m6.2.2.1.3" mathcolor="#000099" xref="footnote2.6.m6.2.2.2.cmml">,</mo><mi id="footnote2.6.m6.1.1" mathcolor="#000099" xref="footnote2.6.m6.1.1.cmml">t</mi><mo id="footnote2.6.m6.2.2.1.4" mathcolor="#000099" stretchy="false" xref="footnote2.6.m6.2.2.2.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="footnote2.6.m6.2c"><interval closure="closed" id="footnote2.6.m6.2.2.2.cmml" xref="footnote2.6.m6.2.2.1"><apply id="footnote2.6.m6.2.2.1.1.cmml" xref="footnote2.6.m6.2.2.1.1"><minus id="footnote2.6.m6.2.2.1.1.1.cmml" xref="footnote2.6.m6.2.2.1.1.1"></minus><ci id="footnote2.6.m6.2.2.1.1.2.cmml" xref="footnote2.6.m6.2.2.1.1.2">𝑡</ci><apply id="footnote2.6.m6.2.2.1.1.3.cmml" xref="footnote2.6.m6.2.2.1.1.3"><csymbol cd="ambiguous" id="footnote2.6.m6.2.2.1.1.3.1.cmml" xref="footnote2.6.m6.2.2.1.1.3">subscript</csymbol><ci id="footnote2.6.m6.2.2.1.1.3.2.cmml" xref="footnote2.6.m6.2.2.1.1.3.2">𝜏</ci><ci id="footnote2.6.m6.2.2.1.1.3.3.cmml" xref="footnote2.6.m6.2.2.1.1.3.3">d</ci></apply></apply><ci id="footnote2.6.m6.1.1.cmml" xref="footnote2.6.m6.1.1">𝑡</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="footnote2.6.m6.2d">[t-\tau_{\rm d},t]</annotation><annotation encoding="application/x-llamapun" id="footnote2.6.m6.2e">[ italic_t - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT , italic_t ]</annotation></semantics></math> cause two channels, <math alttext="{\bm{\phi}}_{{\rm s},1}" class="ltx_Math" display="inline" id="footnote2.7.m7.2"><semantics id="footnote2.7.m7.2b"><msub id="footnote2.7.m7.2.3" xref="footnote2.7.m7.2.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="footnote2.7.m7.2.3.2" mathcolor="#000099" mathvariant="bold-italic" xref="footnote2.7.m7.2.3.2.cmml">ϕ</mi><mrow id="footnote2.7.m7.2.2.2.4" xref="footnote2.7.m7.2.2.2.3.cmml"><mi id="footnote2.7.m7.1.1.1.1" mathcolor="#000099" mathvariant="normal" xref="footnote2.7.m7.1.1.1.1.cmml">s</mi><mo id="footnote2.7.m7.2.2.2.4.1" mathcolor="#000099" xref="footnote2.7.m7.2.2.2.3.cmml">,</mo><mn id="footnote2.7.m7.2.2.2.2" mathcolor="#000099" xref="footnote2.7.m7.2.2.2.2.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="footnote2.7.m7.2c"><apply id="footnote2.7.m7.2.3.cmml" xref="footnote2.7.m7.2.3"><csymbol cd="ambiguous" id="footnote2.7.m7.2.3.1.cmml" xref="footnote2.7.m7.2.3">subscript</csymbol><ci id="footnote2.7.m7.2.3.2.cmml" xref="footnote2.7.m7.2.3.2">bold-italic-ϕ</ci><list id="footnote2.7.m7.2.2.2.3.cmml" xref="footnote2.7.m7.2.2.2.4"><ci id="footnote2.7.m7.1.1.1.1.cmml" xref="footnote2.7.m7.1.1.1.1">s</ci><cn id="footnote2.7.m7.2.2.2.2.cmml" type="integer" xref="footnote2.7.m7.2.2.2.2">1</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.7.m7.2d">{\bm{\phi}}_{{\rm s},1}</annotation><annotation encoding="application/x-llamapun" id="footnote2.7.m7.2e">bold_italic_ϕ start_POSTSUBSCRIPT roman_s , 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="{\bm{\phi}}_{{\rm s},2}" class="ltx_Math" display="inline" id="footnote2.8.m8.2"><semantics id="footnote2.8.m8.2b"><msub id="footnote2.8.m8.2.3" xref="footnote2.8.m8.2.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="footnote2.8.m8.2.3.2" mathcolor="#000099" mathvariant="bold-italic" xref="footnote2.8.m8.2.3.2.cmml">ϕ</mi><mrow id="footnote2.8.m8.2.2.2.4" xref="footnote2.8.m8.2.2.2.3.cmml"><mi id="footnote2.8.m8.1.1.1.1" mathcolor="#000099" mathvariant="normal" xref="footnote2.8.m8.1.1.1.1.cmml">s</mi><mo id="footnote2.8.m8.2.2.2.4.1" mathcolor="#000099" xref="footnote2.8.m8.2.2.2.3.cmml">,</mo><mn id="footnote2.8.m8.2.2.2.2" mathcolor="#000099" xref="footnote2.8.m8.2.2.2.2.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="footnote2.8.m8.2c"><apply id="footnote2.8.m8.2.3.cmml" xref="footnote2.8.m8.2.3"><csymbol cd="ambiguous" id="footnote2.8.m8.2.3.1.cmml" xref="footnote2.8.m8.2.3">subscript</csymbol><ci id="footnote2.8.m8.2.3.2.cmml" xref="footnote2.8.m8.2.3.2">bold-italic-ϕ</ci><list id="footnote2.8.m8.2.2.2.3.cmml" xref="footnote2.8.m8.2.2.2.4"><ci id="footnote2.8.m8.1.1.1.1.cmml" xref="footnote2.8.m8.1.1.1.1">s</ci><cn id="footnote2.8.m8.2.2.2.2.cmml" type="integer" xref="footnote2.8.m8.2.2.2.2">2</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.8.m8.2d">{\bm{\phi}}_{{\rm s},2}</annotation><annotation encoding="application/x-llamapun" id="footnote2.8.m8.2e">bold_italic_ϕ start_POSTSUBSCRIPT roman_s , 2 end_POSTSUBSCRIPT</annotation></semantics></math>, to activate in a correlated manner, then <math alttext="{\bm{\phi}}_{{\rm s},1}" class="ltx_Math" display="inline" id="footnote2.9.m9.2"><semantics id="footnote2.9.m9.2b"><msub id="footnote2.9.m9.2.3" xref="footnote2.9.m9.2.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="footnote2.9.m9.2.3.2" mathcolor="#000099" mathvariant="bold-italic" xref="footnote2.9.m9.2.3.2.cmml">ϕ</mi><mrow id="footnote2.9.m9.2.2.2.4" xref="footnote2.9.m9.2.2.2.3.cmml"><mi id="footnote2.9.m9.1.1.1.1" mathcolor="#000099" mathvariant="normal" xref="footnote2.9.m9.1.1.1.1.cmml">s</mi><mo id="footnote2.9.m9.2.2.2.4.1" mathcolor="#000099" xref="footnote2.9.m9.2.2.2.3.cmml">,</mo><mn id="footnote2.9.m9.2.2.2.2" mathcolor="#000099" xref="footnote2.9.m9.2.2.2.2.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="footnote2.9.m9.2c"><apply id="footnote2.9.m9.2.3.cmml" xref="footnote2.9.m9.2.3"><csymbol cd="ambiguous" id="footnote2.9.m9.2.3.1.cmml" xref="footnote2.9.m9.2.3">subscript</csymbol><ci id="footnote2.9.m9.2.3.2.cmml" xref="footnote2.9.m9.2.3.2">bold-italic-ϕ</ci><list id="footnote2.9.m9.2.2.2.3.cmml" xref="footnote2.9.m9.2.2.2.4"><ci id="footnote2.9.m9.1.1.1.1.cmml" xref="footnote2.9.m9.1.1.1.1">s</ci><cn id="footnote2.9.m9.2.2.2.2.cmml" type="integer" xref="footnote2.9.m9.2.2.2.2">1</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.9.m9.2d">{\bm{\phi}}_{{\rm s},1}</annotation><annotation encoding="application/x-llamapun" id="footnote2.9.m9.2e">bold_italic_ϕ start_POSTSUBSCRIPT roman_s , 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="{\bm{\phi}}_{{\rm s},2}" class="ltx_Math" display="inline" id="footnote2.10.m10.2"><semantics id="footnote2.10.m10.2b"><msub id="footnote2.10.m10.2.3" xref="footnote2.10.m10.2.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="footnote2.10.m10.2.3.2" mathcolor="#000099" mathvariant="bold-italic" xref="footnote2.10.m10.2.3.2.cmml">ϕ</mi><mrow id="footnote2.10.m10.2.2.2.4" xref="footnote2.10.m10.2.2.2.3.cmml"><mi id="footnote2.10.m10.1.1.1.1" mathcolor="#000099" mathvariant="normal" xref="footnote2.10.m10.1.1.1.1.cmml">s</mi><mo id="footnote2.10.m10.2.2.2.4.1" mathcolor="#000099" xref="footnote2.10.m10.2.2.2.3.cmml">,</mo><mn id="footnote2.10.m10.2.2.2.2" mathcolor="#000099" xref="footnote2.10.m10.2.2.2.2.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="footnote2.10.m10.2c"><apply id="footnote2.10.m10.2.3.cmml" xref="footnote2.10.m10.2.3"><csymbol cd="ambiguous" id="footnote2.10.m10.2.3.1.cmml" xref="footnote2.10.m10.2.3">subscript</csymbol><ci id="footnote2.10.m10.2.3.2.cmml" xref="footnote2.10.m10.2.3.2">bold-italic-ϕ</ci><list id="footnote2.10.m10.2.2.2.3.cmml" xref="footnote2.10.m10.2.2.2.4"><ci id="footnote2.10.m10.1.1.1.1.cmml" xref="footnote2.10.m10.1.1.1.1">s</ci><cn id="footnote2.10.m10.2.2.2.2.cmml" type="integer" xref="footnote2.10.m10.2.2.2.2">2</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.10.m10.2d">{\bm{\phi}}_{{\rm s},2}</annotation><annotation encoding="application/x-llamapun" id="footnote2.10.m10.2e">bold_italic_ϕ start_POSTSUBSCRIPT roman_s , 2 end_POSTSUBSCRIPT</annotation></semantics></math> are linearly dependent. That is, there exists a constant <math alttext="c" class="ltx_Math" display="inline" id="footnote2.11.m11.1"><semantics id="footnote2.11.m11.1b"><mi id="footnote2.11.m11.1.1" mathcolor="#000099" xref="footnote2.11.m11.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="footnote2.11.m11.1c"><ci id="footnote2.11.m11.1.1.cmml" xref="footnote2.11.m11.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote2.11.m11.1d">c</annotation><annotation encoding="application/x-llamapun" id="footnote2.11.m11.1e">italic_c</annotation></semantics></math> such that <math alttext="{\bm{\phi}}_{{\rm s},1}(\tau)" class="ltx_Math" display="inline" id="footnote2.12.m12.3"><semantics id="footnote2.12.m12.3b"><mrow id="footnote2.12.m12.3.4" xref="footnote2.12.m12.3.4.cmml"><msub id="footnote2.12.m12.3.4.2" xref="footnote2.12.m12.3.4.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="footnote2.12.m12.3.4.2.2" mathcolor="#000099" mathvariant="bold-italic" xref="footnote2.12.m12.3.4.2.2.cmml">ϕ</mi><mrow id="footnote2.12.m12.2.2.2.4" xref="footnote2.12.m12.2.2.2.3.cmml"><mi id="footnote2.12.m12.1.1.1.1" mathcolor="#000099" mathvariant="normal" xref="footnote2.12.m12.1.1.1.1.cmml">s</mi><mo id="footnote2.12.m12.2.2.2.4.1" mathcolor="#000099" xref="footnote2.12.m12.2.2.2.3.cmml">,</mo><mn id="footnote2.12.m12.2.2.2.2" mathcolor="#000099" xref="footnote2.12.m12.2.2.2.2.cmml">1</mn></mrow></msub><mo id="footnote2.12.m12.3.4.1" xref="footnote2.12.m12.3.4.1.cmml">⁢</mo><mrow id="footnote2.12.m12.3.4.3.2" xref="footnote2.12.m12.3.4.cmml"><mo id="footnote2.12.m12.3.4.3.2.1" mathcolor="#000099" stretchy="false" xref="footnote2.12.m12.3.4.cmml">(</mo><mi id="footnote2.12.m12.3.3" mathcolor="#000099" xref="footnote2.12.m12.3.3.cmml">τ</mi><mo id="footnote2.12.m12.3.4.3.2.2" mathcolor="#000099" stretchy="false" xref="footnote2.12.m12.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote2.12.m12.3c"><apply id="footnote2.12.m12.3.4.cmml" xref="footnote2.12.m12.3.4"><times id="footnote2.12.m12.3.4.1.cmml" xref="footnote2.12.m12.3.4.1"></times><apply id="footnote2.12.m12.3.4.2.cmml" xref="footnote2.12.m12.3.4.2"><csymbol cd="ambiguous" id="footnote2.12.m12.3.4.2.1.cmml" xref="footnote2.12.m12.3.4.2">subscript</csymbol><ci id="footnote2.12.m12.3.4.2.2.cmml" xref="footnote2.12.m12.3.4.2.2">bold-italic-ϕ</ci><list id="footnote2.12.m12.2.2.2.3.cmml" xref="footnote2.12.m12.2.2.2.4"><ci id="footnote2.12.m12.1.1.1.1.cmml" xref="footnote2.12.m12.1.1.1.1">s</ci><cn id="footnote2.12.m12.2.2.2.2.cmml" type="integer" xref="footnote2.12.m12.2.2.2.2">1</cn></list></apply><ci id="footnote2.12.m12.3.3.cmml" xref="footnote2.12.m12.3.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.12.m12.3d">{\bm{\phi}}_{{\rm s},1}(\tau)</annotation><annotation encoding="application/x-llamapun" id="footnote2.12.m12.3e">bold_italic_ϕ start_POSTSUBSCRIPT roman_s , 1 end_POSTSUBSCRIPT ( italic_τ )</annotation></semantics></math> <math alttext="=" class="ltx_Math" display="inline" id="footnote2.13.m13.1"><semantics id="footnote2.13.m13.1b"><mo id="footnote2.13.m13.1.1" mathcolor="#000099" xref="footnote2.13.m13.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="footnote2.13.m13.1c"><eq id="footnote2.13.m13.1.1.cmml" xref="footnote2.13.m13.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="footnote2.13.m13.1d">=</annotation><annotation encoding="application/x-llamapun" id="footnote2.13.m13.1e">=</annotation></semantics></math> <math alttext="c{\bm{\phi}}_{{\rm s},2}(\tau)" class="ltx_Math" display="inline" id="footnote2.14.m14.3"><semantics id="footnote2.14.m14.3b"><mrow id="footnote2.14.m14.3.4" xref="footnote2.14.m14.3.4.cmml"><mi id="footnote2.14.m14.3.4.2" mathcolor="#000099" xref="footnote2.14.m14.3.4.2.cmml">c</mi><mo id="footnote2.14.m14.3.4.1" xref="footnote2.14.m14.3.4.1.cmml">⁢</mo><msub id="footnote2.14.m14.3.4.3" xref="footnote2.14.m14.3.4.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="footnote2.14.m14.3.4.3.2" mathcolor="#000099" mathvariant="bold-italic" xref="footnote2.14.m14.3.4.3.2.cmml">ϕ</mi><mrow id="footnote2.14.m14.2.2.2.4" xref="footnote2.14.m14.2.2.2.3.cmml"><mi id="footnote2.14.m14.1.1.1.1" mathcolor="#000099" mathvariant="normal" xref="footnote2.14.m14.1.1.1.1.cmml">s</mi><mo id="footnote2.14.m14.2.2.2.4.1" mathcolor="#000099" xref="footnote2.14.m14.2.2.2.3.cmml">,</mo><mn id="footnote2.14.m14.2.2.2.2" mathcolor="#000099" xref="footnote2.14.m14.2.2.2.2.cmml">2</mn></mrow></msub><mo id="footnote2.14.m14.3.4.1b" xref="footnote2.14.m14.3.4.1.cmml">⁢</mo><mrow id="footnote2.14.m14.3.4.4.2" xref="footnote2.14.m14.3.4.cmml"><mo id="footnote2.14.m14.3.4.4.2.1" mathcolor="#000099" stretchy="false" xref="footnote2.14.m14.3.4.cmml">(</mo><mi id="footnote2.14.m14.3.3" mathcolor="#000099" xref="footnote2.14.m14.3.3.cmml">τ</mi><mo id="footnote2.14.m14.3.4.4.2.2" mathcolor="#000099" stretchy="false" xref="footnote2.14.m14.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote2.14.m14.3c"><apply id="footnote2.14.m14.3.4.cmml" xref="footnote2.14.m14.3.4"><times id="footnote2.14.m14.3.4.1.cmml" xref="footnote2.14.m14.3.4.1"></times><ci id="footnote2.14.m14.3.4.2.cmml" xref="footnote2.14.m14.3.4.2">𝑐</ci><apply id="footnote2.14.m14.3.4.3.cmml" xref="footnote2.14.m14.3.4.3"><csymbol cd="ambiguous" id="footnote2.14.m14.3.4.3.1.cmml" xref="footnote2.14.m14.3.4.3">subscript</csymbol><ci id="footnote2.14.m14.3.4.3.2.cmml" xref="footnote2.14.m14.3.4.3.2">bold-italic-ϕ</ci><list id="footnote2.14.m14.2.2.2.3.cmml" xref="footnote2.14.m14.2.2.2.4"><ci id="footnote2.14.m14.1.1.1.1.cmml" xref="footnote2.14.m14.1.1.1.1">s</ci><cn id="footnote2.14.m14.2.2.2.2.cmml" type="integer" xref="footnote2.14.m14.2.2.2.2">2</cn></list></apply><ci id="footnote2.14.m14.3.3.cmml" xref="footnote2.14.m14.3.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.14.m14.3d">c{\bm{\phi}}_{{\rm s},2}(\tau)</annotation><annotation encoding="application/x-llamapun" id="footnote2.14.m14.3e">italic_c bold_italic_ϕ start_POSTSUBSCRIPT roman_s , 2 end_POSTSUBSCRIPT ( italic_τ )</annotation></semantics></math>, <math alttext="\forall\tau\in[t-\tau_{d},t]" class="ltx_Math" display="inline" id="footnote2.15.m15.2"><semantics id="footnote2.15.m15.2b"><mrow id="footnote2.15.m15.2.2" xref="footnote2.15.m15.2.2.cmml"><mrow id="footnote2.15.m15.2.2.3" xref="footnote2.15.m15.2.2.3.cmml"><mo id="footnote2.15.m15.2.2.3.1" mathcolor="#000099" rspace="0.167em" xref="footnote2.15.m15.2.2.3.1.cmml">∀</mo><mi id="footnote2.15.m15.2.2.3.2" mathcolor="#000099" xref="footnote2.15.m15.2.2.3.2.cmml">τ</mi></mrow><mo id="footnote2.15.m15.2.2.2" mathcolor="#000099" xref="footnote2.15.m15.2.2.2.cmml">∈</mo><mrow id="footnote2.15.m15.2.2.1.1" xref="footnote2.15.m15.2.2.1.2.cmml"><mo id="footnote2.15.m15.2.2.1.1.2" mathcolor="#000099" stretchy="false" xref="footnote2.15.m15.2.2.1.2.cmml">[</mo><mrow id="footnote2.15.m15.2.2.1.1.1" xref="footnote2.15.m15.2.2.1.1.1.cmml"><mi id="footnote2.15.m15.2.2.1.1.1.2" mathcolor="#000099" xref="footnote2.15.m15.2.2.1.1.1.2.cmml">t</mi><mo id="footnote2.15.m15.2.2.1.1.1.1" mathcolor="#000099" xref="footnote2.15.m15.2.2.1.1.1.1.cmml">−</mo><msub id="footnote2.15.m15.2.2.1.1.1.3" xref="footnote2.15.m15.2.2.1.1.1.3.cmml"><mi id="footnote2.15.m15.2.2.1.1.1.3.2" mathcolor="#000099" xref="footnote2.15.m15.2.2.1.1.1.3.2.cmml">τ</mi><mi id="footnote2.15.m15.2.2.1.1.1.3.3" mathcolor="#000099" xref="footnote2.15.m15.2.2.1.1.1.3.3.cmml">d</mi></msub></mrow><mo id="footnote2.15.m15.2.2.1.1.3" mathcolor="#000099" xref="footnote2.15.m15.2.2.1.2.cmml">,</mo><mi id="footnote2.15.m15.1.1" mathcolor="#000099" xref="footnote2.15.m15.1.1.cmml">t</mi><mo id="footnote2.15.m15.2.2.1.1.4" mathcolor="#000099" stretchy="false" xref="footnote2.15.m15.2.2.1.2.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote2.15.m15.2c"><apply id="footnote2.15.m15.2.2.cmml" xref="footnote2.15.m15.2.2"><in id="footnote2.15.m15.2.2.2.cmml" xref="footnote2.15.m15.2.2.2"></in><apply id="footnote2.15.m15.2.2.3.cmml" xref="footnote2.15.m15.2.2.3"><csymbol cd="latexml" id="footnote2.15.m15.2.2.3.1.cmml" xref="footnote2.15.m15.2.2.3.1">for-all</csymbol><ci id="footnote2.15.m15.2.2.3.2.cmml" xref="footnote2.15.m15.2.2.3.2">𝜏</ci></apply><interval closure="closed" id="footnote2.15.m15.2.2.1.2.cmml" xref="footnote2.15.m15.2.2.1.1"><apply id="footnote2.15.m15.2.2.1.1.1.cmml" xref="footnote2.15.m15.2.2.1.1.1"><minus id="footnote2.15.m15.2.2.1.1.1.1.cmml" xref="footnote2.15.m15.2.2.1.1.1.1"></minus><ci id="footnote2.15.m15.2.2.1.1.1.2.cmml" xref="footnote2.15.m15.2.2.1.1.1.2">𝑡</ci><apply id="footnote2.15.m15.2.2.1.1.1.3.cmml" xref="footnote2.15.m15.2.2.1.1.1.3"><csymbol cd="ambiguous" id="footnote2.15.m15.2.2.1.1.1.3.1.cmml" xref="footnote2.15.m15.2.2.1.1.1.3">subscript</csymbol><ci id="footnote2.15.m15.2.2.1.1.1.3.2.cmml" xref="footnote2.15.m15.2.2.1.1.1.3.2">𝜏</ci><ci id="footnote2.15.m15.2.2.1.1.1.3.3.cmml" xref="footnote2.15.m15.2.2.1.1.1.3.3">𝑑</ci></apply></apply><ci id="footnote2.15.m15.1.1.cmml" xref="footnote2.15.m15.1.1">𝑡</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.15.m15.2d">\forall\tau\in[t-\tau_{d},t]</annotation><annotation encoding="application/x-llamapun" id="footnote2.15.m15.2e">∀ italic_τ ∈ [ italic_t - italic_τ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT , italic_t ]</annotation></semantics></math>. This situation would violate Assumption 3.</span></span></span></span>, and the (sub-regressor-based) excitation matrix <math alttext="\Psi_{\zeta}" class="ltx_Math" display="inline" id="S4.SS2.p1.5.m5.1"><semantics id="S4.SS2.p1.5.m5.1a"><msub id="S4.SS2.p1.5.m5.1.1" xref="S4.SS2.p1.5.m5.1.1.cmml"><mi id="S4.SS2.p1.5.m5.1.1.2" mathvariant="normal" xref="S4.SS2.p1.5.m5.1.1.2.cmml">Ψ</mi><mi id="S4.SS2.p1.5.m5.1.1.3" xref="S4.SS2.p1.5.m5.1.1.3.cmml">ζ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.5.m5.1b"><apply id="S4.SS2.p1.5.m5.1.1.cmml" xref="S4.SS2.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.5.m5.1.1.1.cmml" xref="S4.SS2.p1.5.m5.1.1">subscript</csymbol><ci id="S4.SS2.p1.5.m5.1.1.2.cmml" xref="S4.SS2.p1.5.m5.1.1.2">Ψ</ci><ci id="S4.SS2.p1.5.m5.1.1.3.cmml" xref="S4.SS2.p1.5.m5.1.1.3">𝜁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.5.m5.1c">\Psi_{\zeta}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.5.m5.1d">roman_Ψ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E19" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">19</span></a>) composed of these active channels <math alttext="{\bm{\phi}}_{{\rm s},k_{j}}" class="ltx_Math" display="inline" id="S4.SS2.p1.6.m6.2"><semantics id="S4.SS2.p1.6.m6.2a"><msub id="S4.SS2.p1.6.m6.2.3" xref="S4.SS2.p1.6.m6.2.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS2.p1.6.m6.2.3.2" mathvariant="bold-italic" xref="S4.SS2.p1.6.m6.2.3.2.cmml">ϕ</mi><mrow id="S4.SS2.p1.6.m6.2.2.2.2" xref="S4.SS2.p1.6.m6.2.2.2.3.cmml"><mi id="S4.SS2.p1.6.m6.1.1.1.1" mathvariant="normal" xref="S4.SS2.p1.6.m6.1.1.1.1.cmml">s</mi><mo id="S4.SS2.p1.6.m6.2.2.2.2.2" xref="S4.SS2.p1.6.m6.2.2.2.3.cmml">,</mo><msub id="S4.SS2.p1.6.m6.2.2.2.2.1" xref="S4.SS2.p1.6.m6.2.2.2.2.1.cmml"><mi id="S4.SS2.p1.6.m6.2.2.2.2.1.2" xref="S4.SS2.p1.6.m6.2.2.2.2.1.2.cmml">k</mi><mi id="S4.SS2.p1.6.m6.2.2.2.2.1.3" xref="S4.SS2.p1.6.m6.2.2.2.2.1.3.cmml">j</mi></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.6.m6.2b"><apply id="S4.SS2.p1.6.m6.2.3.cmml" xref="S4.SS2.p1.6.m6.2.3"><csymbol cd="ambiguous" id="S4.SS2.p1.6.m6.2.3.1.cmml" xref="S4.SS2.p1.6.m6.2.3">subscript</csymbol><ci id="S4.SS2.p1.6.m6.2.3.2.cmml" xref="S4.SS2.p1.6.m6.2.3.2">bold-italic-ϕ</ci><list id="S4.SS2.p1.6.m6.2.2.2.3.cmml" xref="S4.SS2.p1.6.m6.2.2.2.2"><ci id="S4.SS2.p1.6.m6.1.1.1.1.cmml" xref="S4.SS2.p1.6.m6.1.1.1.1">s</ci><apply id="S4.SS2.p1.6.m6.2.2.2.2.1.cmml" xref="S4.SS2.p1.6.m6.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.SS2.p1.6.m6.2.2.2.2.1.1.cmml" xref="S4.SS2.p1.6.m6.2.2.2.2.1">subscript</csymbol><ci id="S4.SS2.p1.6.m6.2.2.2.2.1.2.cmml" xref="S4.SS2.p1.6.m6.2.2.2.2.1.2">𝑘</ci><ci id="S4.SS2.p1.6.m6.2.2.2.2.1.3.cmml" xref="S4.SS2.p1.6.m6.2.2.2.2.1.3">𝑗</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.6.m6.2c">{\bm{\phi}}_{{\rm s},k_{j}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.6.m6.2d">bold_italic_ϕ start_POSTSUBSCRIPT roman_s , italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is always positive-definite, which ensures the existence of the parameters <math alttext="\tau_{\rm d}" class="ltx_Math" display="inline" id="S4.SS2.p1.7.m7.1"><semantics id="S4.SS2.p1.7.m7.1a"><msub id="S4.SS2.p1.7.m7.1.1" xref="S4.SS2.p1.7.m7.1.1.cmml"><mi id="S4.SS2.p1.7.m7.1.1.2" xref="S4.SS2.p1.7.m7.1.1.2.cmml">τ</mi><mi id="S4.SS2.p1.7.m7.1.1.3" mathvariant="normal" xref="S4.SS2.p1.7.m7.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.7.m7.1b"><apply id="S4.SS2.p1.7.m7.1.1.cmml" xref="S4.SS2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.7.m7.1.1.1.cmml" xref="S4.SS2.p1.7.m7.1.1">subscript</csymbol><ci id="S4.SS2.p1.7.m7.1.1.2.cmml" xref="S4.SS2.p1.7.m7.1.1.2">𝜏</ci><ci id="S4.SS2.p1.7.m7.1.1.3.cmml" xref="S4.SS2.p1.7.m7.1.1.3">d</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.7.m7.1c">\tau_{\rm d}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.7.m7.1d">italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS2.p1.8.m8.1"><semantics id="S4.SS2.p1.8.m8.1a"><mi id="S4.SS2.p1.8.m8.1.1" xref="S4.SS2.p1.8.m8.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.8.m8.1b"><ci id="S4.SS2.p1.8.m8.1.1.cmml" xref="S4.SS2.p1.8.m8.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.8.m8.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.8.m8.1d">italic_σ</annotation></semantics></math> for the partial IE condition in Definition 3. This fact is also suitable for the IE case. <span class="ltx_text" id="S4.SS2.p1.10.2" style="color:#000099;">Assumption 3 about partial identifiability is weaker than the identifiability assumption, as it only requires that partial channels be activated in an uncorrelated manner within a time window <math alttext="[T_{\rm a}-\tau_{\rm d},T_{\rm a}]" class="ltx_Math" display="inline" id="S4.SS2.p1.9.1.m1.2"><semantics id="S4.SS2.p1.9.1.m1.2a"><mrow id="S4.SS2.p1.9.1.m1.2.2.2" xref="S4.SS2.p1.9.1.m1.2.2.3.cmml"><mo id="S4.SS2.p1.9.1.m1.2.2.2.3" mathcolor="#000099" stretchy="false" xref="S4.SS2.p1.9.1.m1.2.2.3.cmml">[</mo><mrow id="S4.SS2.p1.9.1.m1.1.1.1.1" xref="S4.SS2.p1.9.1.m1.1.1.1.1.cmml"><msub id="S4.SS2.p1.9.1.m1.1.1.1.1.2" xref="S4.SS2.p1.9.1.m1.1.1.1.1.2.cmml"><mi id="S4.SS2.p1.9.1.m1.1.1.1.1.2.2" mathcolor="#000099" xref="S4.SS2.p1.9.1.m1.1.1.1.1.2.2.cmml">T</mi><mi id="S4.SS2.p1.9.1.m1.1.1.1.1.2.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS2.p1.9.1.m1.1.1.1.1.2.3.cmml">a</mi></msub><mo id="S4.SS2.p1.9.1.m1.1.1.1.1.1" mathcolor="#000099" xref="S4.SS2.p1.9.1.m1.1.1.1.1.1.cmml">−</mo><msub id="S4.SS2.p1.9.1.m1.1.1.1.1.3" xref="S4.SS2.p1.9.1.m1.1.1.1.1.3.cmml"><mi id="S4.SS2.p1.9.1.m1.1.1.1.1.3.2" mathcolor="#000099" xref="S4.SS2.p1.9.1.m1.1.1.1.1.3.2.cmml">τ</mi><mi id="S4.SS2.p1.9.1.m1.1.1.1.1.3.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS2.p1.9.1.m1.1.1.1.1.3.3.cmml">d</mi></msub></mrow><mo id="S4.SS2.p1.9.1.m1.2.2.2.4" mathcolor="#000099" xref="S4.SS2.p1.9.1.m1.2.2.3.cmml">,</mo><msub id="S4.SS2.p1.9.1.m1.2.2.2.2" xref="S4.SS2.p1.9.1.m1.2.2.2.2.cmml"><mi id="S4.SS2.p1.9.1.m1.2.2.2.2.2" mathcolor="#000099" xref="S4.SS2.p1.9.1.m1.2.2.2.2.2.cmml">T</mi><mi id="S4.SS2.p1.9.1.m1.2.2.2.2.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS2.p1.9.1.m1.2.2.2.2.3.cmml">a</mi></msub><mo id="S4.SS2.p1.9.1.m1.2.2.2.5" mathcolor="#000099" stretchy="false" xref="S4.SS2.p1.9.1.m1.2.2.3.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.9.1.m1.2b"><interval closure="closed" id="S4.SS2.p1.9.1.m1.2.2.3.cmml" xref="S4.SS2.p1.9.1.m1.2.2.2"><apply id="S4.SS2.p1.9.1.m1.1.1.1.1.cmml" xref="S4.SS2.p1.9.1.m1.1.1.1.1"><minus id="S4.SS2.p1.9.1.m1.1.1.1.1.1.cmml" xref="S4.SS2.p1.9.1.m1.1.1.1.1.1"></minus><apply id="S4.SS2.p1.9.1.m1.1.1.1.1.2.cmml" xref="S4.SS2.p1.9.1.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.9.1.m1.1.1.1.1.2.1.cmml" xref="S4.SS2.p1.9.1.m1.1.1.1.1.2">subscript</csymbol><ci id="S4.SS2.p1.9.1.m1.1.1.1.1.2.2.cmml" xref="S4.SS2.p1.9.1.m1.1.1.1.1.2.2">𝑇</ci><ci id="S4.SS2.p1.9.1.m1.1.1.1.1.2.3.cmml" xref="S4.SS2.p1.9.1.m1.1.1.1.1.2.3">a</ci></apply><apply id="S4.SS2.p1.9.1.m1.1.1.1.1.3.cmml" xref="S4.SS2.p1.9.1.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p1.9.1.m1.1.1.1.1.3.1.cmml" xref="S4.SS2.p1.9.1.m1.1.1.1.1.3">subscript</csymbol><ci id="S4.SS2.p1.9.1.m1.1.1.1.1.3.2.cmml" xref="S4.SS2.p1.9.1.m1.1.1.1.1.3.2">𝜏</ci><ci id="S4.SS2.p1.9.1.m1.1.1.1.1.3.3.cmml" xref="S4.SS2.p1.9.1.m1.1.1.1.1.3.3">d</ci></apply></apply><apply id="S4.SS2.p1.9.1.m1.2.2.2.2.cmml" xref="S4.SS2.p1.9.1.m1.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.p1.9.1.m1.2.2.2.2.1.cmml" xref="S4.SS2.p1.9.1.m1.2.2.2.2">subscript</csymbol><ci id="S4.SS2.p1.9.1.m1.2.2.2.2.2.cmml" xref="S4.SS2.p1.9.1.m1.2.2.2.2.2">𝑇</ci><ci id="S4.SS2.p1.9.1.m1.2.2.2.2.3.cmml" xref="S4.SS2.p1.9.1.m1.2.2.2.2.3">a</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.9.1.m1.2c">[T_{\rm a}-\tau_{\rm d},T_{\rm a}]</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.9.1.m1.2d">[ italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT ]</annotation></semantics></math> (i.e., partial IE exists). The identifiability assumption, on the other hand, requires that all channels be activated in an uncorrelated manner within <math alttext="[T_{\rm e}-\tau_{\rm d},T_{\rm e}]" class="ltx_Math" display="inline" id="S4.SS2.p1.10.2.m2.2"><semantics id="S4.SS2.p1.10.2.m2.2a"><mrow id="S4.SS2.p1.10.2.m2.2.2.2" xref="S4.SS2.p1.10.2.m2.2.2.3.cmml"><mo id="S4.SS2.p1.10.2.m2.2.2.2.3" mathcolor="#000099" stretchy="false" xref="S4.SS2.p1.10.2.m2.2.2.3.cmml">[</mo><mrow id="S4.SS2.p1.10.2.m2.1.1.1.1" xref="S4.SS2.p1.10.2.m2.1.1.1.1.cmml"><msub id="S4.SS2.p1.10.2.m2.1.1.1.1.2" xref="S4.SS2.p1.10.2.m2.1.1.1.1.2.cmml"><mi id="S4.SS2.p1.10.2.m2.1.1.1.1.2.2" mathcolor="#000099" xref="S4.SS2.p1.10.2.m2.1.1.1.1.2.2.cmml">T</mi><mi id="S4.SS2.p1.10.2.m2.1.1.1.1.2.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS2.p1.10.2.m2.1.1.1.1.2.3.cmml">e</mi></msub><mo id="S4.SS2.p1.10.2.m2.1.1.1.1.1" mathcolor="#000099" xref="S4.SS2.p1.10.2.m2.1.1.1.1.1.cmml">−</mo><msub id="S4.SS2.p1.10.2.m2.1.1.1.1.3" xref="S4.SS2.p1.10.2.m2.1.1.1.1.3.cmml"><mi id="S4.SS2.p1.10.2.m2.1.1.1.1.3.2" mathcolor="#000099" xref="S4.SS2.p1.10.2.m2.1.1.1.1.3.2.cmml">τ</mi><mi id="S4.SS2.p1.10.2.m2.1.1.1.1.3.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS2.p1.10.2.m2.1.1.1.1.3.3.cmml">d</mi></msub></mrow><mo id="S4.SS2.p1.10.2.m2.2.2.2.4" mathcolor="#000099" xref="S4.SS2.p1.10.2.m2.2.2.3.cmml">,</mo><msub id="S4.SS2.p1.10.2.m2.2.2.2.2" xref="S4.SS2.p1.10.2.m2.2.2.2.2.cmml"><mi id="S4.SS2.p1.10.2.m2.2.2.2.2.2" mathcolor="#000099" xref="S4.SS2.p1.10.2.m2.2.2.2.2.2.cmml">T</mi><mi id="S4.SS2.p1.10.2.m2.2.2.2.2.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS2.p1.10.2.m2.2.2.2.2.3.cmml">e</mi></msub><mo id="S4.SS2.p1.10.2.m2.2.2.2.5" mathcolor="#000099" stretchy="false" xref="S4.SS2.p1.10.2.m2.2.2.3.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.10.2.m2.2b"><interval closure="closed" id="S4.SS2.p1.10.2.m2.2.2.3.cmml" xref="S4.SS2.p1.10.2.m2.2.2.2"><apply id="S4.SS2.p1.10.2.m2.1.1.1.1.cmml" xref="S4.SS2.p1.10.2.m2.1.1.1.1"><minus id="S4.SS2.p1.10.2.m2.1.1.1.1.1.cmml" xref="S4.SS2.p1.10.2.m2.1.1.1.1.1"></minus><apply id="S4.SS2.p1.10.2.m2.1.1.1.1.2.cmml" xref="S4.SS2.p1.10.2.m2.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.10.2.m2.1.1.1.1.2.1.cmml" xref="S4.SS2.p1.10.2.m2.1.1.1.1.2">subscript</csymbol><ci id="S4.SS2.p1.10.2.m2.1.1.1.1.2.2.cmml" xref="S4.SS2.p1.10.2.m2.1.1.1.1.2.2">𝑇</ci><ci id="S4.SS2.p1.10.2.m2.1.1.1.1.2.3.cmml" xref="S4.SS2.p1.10.2.m2.1.1.1.1.2.3">e</ci></apply><apply id="S4.SS2.p1.10.2.m2.1.1.1.1.3.cmml" xref="S4.SS2.p1.10.2.m2.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p1.10.2.m2.1.1.1.1.3.1.cmml" xref="S4.SS2.p1.10.2.m2.1.1.1.1.3">subscript</csymbol><ci id="S4.SS2.p1.10.2.m2.1.1.1.1.3.2.cmml" xref="S4.SS2.p1.10.2.m2.1.1.1.1.3.2">𝜏</ci><ci id="S4.SS2.p1.10.2.m2.1.1.1.1.3.3.cmml" xref="S4.SS2.p1.10.2.m2.1.1.1.1.3.3">d</ci></apply></apply><apply id="S4.SS2.p1.10.2.m2.2.2.2.2.cmml" xref="S4.SS2.p1.10.2.m2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.p1.10.2.m2.2.2.2.2.1.cmml" xref="S4.SS2.p1.10.2.m2.2.2.2.2">subscript</csymbol><ci id="S4.SS2.p1.10.2.m2.2.2.2.2.2.cmml" xref="S4.SS2.p1.10.2.m2.2.2.2.2.2">𝑇</ci><ci id="S4.SS2.p1.10.2.m2.2.2.2.2.3.cmml" xref="S4.SS2.p1.10.2.m2.2.2.2.2.3">e</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.10.2.m2.2c">[T_{\rm e}-\tau_{\rm d},T_{\rm e}]</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.10.2.m2.2d">[ italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ]</annotation></semantics></math> (i.e., IE exists).</span> <span class="ltx_text" id="S4.SS2.p1.14.6" style="color:#000099;">The regressors <math alttext="{\bm{\phi}}_{i}" class="ltx_Math" display="inline" id="S4.SS2.p1.11.3.m1.1"><semantics id="S4.SS2.p1.11.3.m1.1a"><msub id="S4.SS2.p1.11.3.m1.1.1" xref="S4.SS2.p1.11.3.m1.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS2.p1.11.3.m1.1.1.2" mathcolor="#000099" mathvariant="bold-italic" xref="S4.SS2.p1.11.3.m1.1.1.2.cmml">ϕ</mi><mi id="S4.SS2.p1.11.3.m1.1.1.3" mathcolor="#000099" xref="S4.SS2.p1.11.3.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.11.3.m1.1b"><apply id="S4.SS2.p1.11.3.m1.1.1.cmml" xref="S4.SS2.p1.11.3.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.11.3.m1.1.1.1.cmml" xref="S4.SS2.p1.11.3.m1.1.1">subscript</csymbol><ci id="S4.SS2.p1.11.3.m1.1.1.2.cmml" xref="S4.SS2.p1.11.3.m1.1.1.2">bold-italic-ϕ</ci><ci id="S4.SS2.p1.11.3.m1.1.1.3.cmml" xref="S4.SS2.p1.11.3.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.11.3.m1.1c">{\bm{\phi}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.11.3.m1.1d">bold_italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> can be checked a priori to determine whether the sub-regressor constructed by active channels satisfies Assumption 3. In fact, if the analytical expressions of <math alttext="\bm{\phi}_{i}" class="ltx_Math" display="inline" id="S4.SS2.p1.12.4.m2.1"><semantics id="S4.SS2.p1.12.4.m2.1a"><msub id="S4.SS2.p1.12.4.m2.1.1" xref="S4.SS2.p1.12.4.m2.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS2.p1.12.4.m2.1.1.2" mathcolor="#000099" mathvariant="bold-italic" xref="S4.SS2.p1.12.4.m2.1.1.2.cmml">ϕ</mi><mi id="S4.SS2.p1.12.4.m2.1.1.3" mathcolor="#000099" xref="S4.SS2.p1.12.4.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.12.4.m2.1b"><apply id="S4.SS2.p1.12.4.m2.1.1.cmml" xref="S4.SS2.p1.12.4.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.12.4.m2.1.1.1.cmml" xref="S4.SS2.p1.12.4.m2.1.1">subscript</csymbol><ci id="S4.SS2.p1.12.4.m2.1.1.2.cmml" xref="S4.SS2.p1.12.4.m2.1.1.2">bold-italic-ϕ</ci><ci id="S4.SS2.p1.12.4.m2.1.1.3.cmml" xref="S4.SS2.p1.12.4.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.12.4.m2.1c">\bm{\phi}_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.12.4.m2.1d">bold_italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> (<math alttext="i=1" class="ltx_Math" display="inline" id="S4.SS2.p1.13.5.m3.1"><semantics id="S4.SS2.p1.13.5.m3.1a"><mrow id="S4.SS2.p1.13.5.m3.1.1" xref="S4.SS2.p1.13.5.m3.1.1.cmml"><mi id="S4.SS2.p1.13.5.m3.1.1.2" mathcolor="#000099" xref="S4.SS2.p1.13.5.m3.1.1.2.cmml">i</mi><mo id="S4.SS2.p1.13.5.m3.1.1.1" mathcolor="#000099" xref="S4.SS2.p1.13.5.m3.1.1.1.cmml">=</mo><mn id="S4.SS2.p1.13.5.m3.1.1.3" mathcolor="#000099" xref="S4.SS2.p1.13.5.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.13.5.m3.1b"><apply id="S4.SS2.p1.13.5.m3.1.1.cmml" xref="S4.SS2.p1.13.5.m3.1.1"><eq id="S4.SS2.p1.13.5.m3.1.1.1.cmml" xref="S4.SS2.p1.13.5.m3.1.1.1"></eq><ci id="S4.SS2.p1.13.5.m3.1.1.2.cmml" xref="S4.SS2.p1.13.5.m3.1.1.2">𝑖</ci><cn id="S4.SS2.p1.13.5.m3.1.1.3.cmml" type="integer" xref="S4.SS2.p1.13.5.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.13.5.m3.1c">i=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.13.5.m3.1d">italic_i = 1</annotation></semantics></math> to <math alttext="n" class="ltx_Math" display="inline" id="S4.SS2.p1.14.6.m4.1"><semantics id="S4.SS2.p1.14.6.m4.1a"><mi id="S4.SS2.p1.14.6.m4.1.1" mathcolor="#000099" xref="S4.SS2.p1.14.6.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.14.6.m4.1b"><ci id="S4.SS2.p1.14.6.m4.1.1.cmml" xref="S4.SS2.p1.14.6.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.14.6.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.14.6.m4.1d">italic_n</annotation></semantics></math>) are linearly independent, Assumption 3 will necessarily hold.</span> Note that if Assumption 3 is not true, i.e., there are some linearly dependent active channels <math alttext="\bm{\phi}_{{\rm s},j}" class="ltx_Math" display="inline" id="S4.SS2.p1.15.m9.2"><semantics id="S4.SS2.p1.15.m9.2a"><msub id="S4.SS2.p1.15.m9.2.3" xref="S4.SS2.p1.15.m9.2.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS2.p1.15.m9.2.3.2" mathvariant="bold-italic" xref="S4.SS2.p1.15.m9.2.3.2.cmml">ϕ</mi><mrow id="S4.SS2.p1.15.m9.2.2.2.4" xref="S4.SS2.p1.15.m9.2.2.2.3.cmml"><mi id="S4.SS2.p1.15.m9.1.1.1.1" mathvariant="normal" xref="S4.SS2.p1.15.m9.1.1.1.1.cmml">s</mi><mo id="S4.SS2.p1.15.m9.2.2.2.4.1" xref="S4.SS2.p1.15.m9.2.2.2.3.cmml">,</mo><mi id="S4.SS2.p1.15.m9.2.2.2.2" xref="S4.SS2.p1.15.m9.2.2.2.2.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.15.m9.2b"><apply id="S4.SS2.p1.15.m9.2.3.cmml" xref="S4.SS2.p1.15.m9.2.3"><csymbol cd="ambiguous" id="S4.SS2.p1.15.m9.2.3.1.cmml" xref="S4.SS2.p1.15.m9.2.3">subscript</csymbol><ci id="S4.SS2.p1.15.m9.2.3.2.cmml" xref="S4.SS2.p1.15.m9.2.3.2">bold-italic-ϕ</ci><list id="S4.SS2.p1.15.m9.2.2.2.3.cmml" xref="S4.SS2.p1.15.m9.2.2.2.4"><ci id="S4.SS2.p1.15.m9.1.1.1.1.cmml" xref="S4.SS2.p1.15.m9.1.1.1.1">s</ci><ci id="S4.SS2.p1.15.m9.2.2.2.2.cmml" xref="S4.SS2.p1.15.m9.2.2.2.2">𝑗</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.15.m9.2c">\bm{\phi}_{{\rm s},j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.15.m9.2d">bold_italic_ϕ start_POSTSUBSCRIPT roman_s , italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E12" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">12</span></a>) must be reconstructed such that all channels are linearly independent <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib29" title="">29</a>]</cite>. For example, if there exist two linearly dependent channels <math alttext="{\bm{\phi}}_{1}(t)" class="ltx_Math" display="inline" id="S4.SS2.p1.16.m10.1"><semantics id="S4.SS2.p1.16.m10.1a"><mrow id="S4.SS2.p1.16.m10.1.2" xref="S4.SS2.p1.16.m10.1.2.cmml"><msub id="S4.SS2.p1.16.m10.1.2.2" xref="S4.SS2.p1.16.m10.1.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS2.p1.16.m10.1.2.2.2" mathvariant="bold-italic" xref="S4.SS2.p1.16.m10.1.2.2.2.cmml">ϕ</mi><mn id="S4.SS2.p1.16.m10.1.2.2.3" xref="S4.SS2.p1.16.m10.1.2.2.3.cmml">1</mn></msub><mo id="S4.SS2.p1.16.m10.1.2.1" xref="S4.SS2.p1.16.m10.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p1.16.m10.1.2.3.2" xref="S4.SS2.p1.16.m10.1.2.cmml"><mo id="S4.SS2.p1.16.m10.1.2.3.2.1" stretchy="false" xref="S4.SS2.p1.16.m10.1.2.cmml">(</mo><mi id="S4.SS2.p1.16.m10.1.1" xref="S4.SS2.p1.16.m10.1.1.cmml">t</mi><mo id="S4.SS2.p1.16.m10.1.2.3.2.2" stretchy="false" xref="S4.SS2.p1.16.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.16.m10.1b"><apply id="S4.SS2.p1.16.m10.1.2.cmml" xref="S4.SS2.p1.16.m10.1.2"><times id="S4.SS2.p1.16.m10.1.2.1.cmml" xref="S4.SS2.p1.16.m10.1.2.1"></times><apply id="S4.SS2.p1.16.m10.1.2.2.cmml" xref="S4.SS2.p1.16.m10.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.p1.16.m10.1.2.2.1.cmml" xref="S4.SS2.p1.16.m10.1.2.2">subscript</csymbol><ci id="S4.SS2.p1.16.m10.1.2.2.2.cmml" xref="S4.SS2.p1.16.m10.1.2.2.2">bold-italic-ϕ</ci><cn id="S4.SS2.p1.16.m10.1.2.2.3.cmml" type="integer" xref="S4.SS2.p1.16.m10.1.2.2.3">1</cn></apply><ci id="S4.SS2.p1.16.m10.1.1.cmml" xref="S4.SS2.p1.16.m10.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.16.m10.1c">{\bm{\phi}}_{1}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.16.m10.1d">bold_italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math>, <math alttext="{\bm{\phi}}_{2}(t)" class="ltx_Math" display="inline" id="S4.SS2.p1.17.m11.1"><semantics id="S4.SS2.p1.17.m11.1a"><mrow id="S4.SS2.p1.17.m11.1.2" xref="S4.SS2.p1.17.m11.1.2.cmml"><msub id="S4.SS2.p1.17.m11.1.2.2" xref="S4.SS2.p1.17.m11.1.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS2.p1.17.m11.1.2.2.2" mathvariant="bold-italic" xref="S4.SS2.p1.17.m11.1.2.2.2.cmml">ϕ</mi><mn id="S4.SS2.p1.17.m11.1.2.2.3" xref="S4.SS2.p1.17.m11.1.2.2.3.cmml">2</mn></msub><mo id="S4.SS2.p1.17.m11.1.2.1" xref="S4.SS2.p1.17.m11.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p1.17.m11.1.2.3.2" xref="S4.SS2.p1.17.m11.1.2.cmml"><mo id="S4.SS2.p1.17.m11.1.2.3.2.1" stretchy="false" xref="S4.SS2.p1.17.m11.1.2.cmml">(</mo><mi id="S4.SS2.p1.17.m11.1.1" xref="S4.SS2.p1.17.m11.1.1.cmml">t</mi><mo id="S4.SS2.p1.17.m11.1.2.3.2.2" stretchy="false" xref="S4.SS2.p1.17.m11.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.17.m11.1b"><apply id="S4.SS2.p1.17.m11.1.2.cmml" xref="S4.SS2.p1.17.m11.1.2"><times id="S4.SS2.p1.17.m11.1.2.1.cmml" xref="S4.SS2.p1.17.m11.1.2.1"></times><apply id="S4.SS2.p1.17.m11.1.2.2.cmml" xref="S4.SS2.p1.17.m11.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.p1.17.m11.1.2.2.1.cmml" xref="S4.SS2.p1.17.m11.1.2.2">subscript</csymbol><ci id="S4.SS2.p1.17.m11.1.2.2.2.cmml" xref="S4.SS2.p1.17.m11.1.2.2.2">bold-italic-ϕ</ci><cn id="S4.SS2.p1.17.m11.1.2.2.3.cmml" type="integer" xref="S4.SS2.p1.17.m11.1.2.2.3">2</cn></apply><ci id="S4.SS2.p1.17.m11.1.1.cmml" xref="S4.SS2.p1.17.m11.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.17.m11.1c">{\bm{\phi}}_{2}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.17.m11.1d">bold_italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math>, i.e., <math alttext="c{\bm{\phi}}_{1}(t)={\bm{\phi}}_{2}(t)" class="ltx_Math" display="inline" id="S4.SS2.p1.18.m12.2"><semantics id="S4.SS2.p1.18.m12.2a"><mrow id="S4.SS2.p1.18.m12.2.3" xref="S4.SS2.p1.18.m12.2.3.cmml"><mrow id="S4.SS2.p1.18.m12.2.3.2" xref="S4.SS2.p1.18.m12.2.3.2.cmml"><mi id="S4.SS2.p1.18.m12.2.3.2.2" xref="S4.SS2.p1.18.m12.2.3.2.2.cmml">c</mi><mo id="S4.SS2.p1.18.m12.2.3.2.1" xref="S4.SS2.p1.18.m12.2.3.2.1.cmml">⁢</mo><msub id="S4.SS2.p1.18.m12.2.3.2.3" xref="S4.SS2.p1.18.m12.2.3.2.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS2.p1.18.m12.2.3.2.3.2" mathvariant="bold-italic" xref="S4.SS2.p1.18.m12.2.3.2.3.2.cmml">ϕ</mi><mn id="S4.SS2.p1.18.m12.2.3.2.3.3" xref="S4.SS2.p1.18.m12.2.3.2.3.3.cmml">1</mn></msub><mo id="S4.SS2.p1.18.m12.2.3.2.1a" xref="S4.SS2.p1.18.m12.2.3.2.1.cmml">⁢</mo><mrow id="S4.SS2.p1.18.m12.2.3.2.4.2" xref="S4.SS2.p1.18.m12.2.3.2.cmml"><mo id="S4.SS2.p1.18.m12.2.3.2.4.2.1" stretchy="false" xref="S4.SS2.p1.18.m12.2.3.2.cmml">(</mo><mi id="S4.SS2.p1.18.m12.1.1" xref="S4.SS2.p1.18.m12.1.1.cmml">t</mi><mo id="S4.SS2.p1.18.m12.2.3.2.4.2.2" stretchy="false" xref="S4.SS2.p1.18.m12.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.SS2.p1.18.m12.2.3.1" xref="S4.SS2.p1.18.m12.2.3.1.cmml">=</mo><mrow id="S4.SS2.p1.18.m12.2.3.3" xref="S4.SS2.p1.18.m12.2.3.3.cmml"><msub id="S4.SS2.p1.18.m12.2.3.3.2" xref="S4.SS2.p1.18.m12.2.3.3.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS2.p1.18.m12.2.3.3.2.2" mathvariant="bold-italic" xref="S4.SS2.p1.18.m12.2.3.3.2.2.cmml">ϕ</mi><mn id="S4.SS2.p1.18.m12.2.3.3.2.3" xref="S4.SS2.p1.18.m12.2.3.3.2.3.cmml">2</mn></msub><mo id="S4.SS2.p1.18.m12.2.3.3.1" xref="S4.SS2.p1.18.m12.2.3.3.1.cmml">⁢</mo><mrow id="S4.SS2.p1.18.m12.2.3.3.3.2" xref="S4.SS2.p1.18.m12.2.3.3.cmml"><mo id="S4.SS2.p1.18.m12.2.3.3.3.2.1" stretchy="false" xref="S4.SS2.p1.18.m12.2.3.3.cmml">(</mo><mi id="S4.SS2.p1.18.m12.2.2" xref="S4.SS2.p1.18.m12.2.2.cmml">t</mi><mo id="S4.SS2.p1.18.m12.2.3.3.3.2.2" stretchy="false" xref="S4.SS2.p1.18.m12.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.18.m12.2b"><apply id="S4.SS2.p1.18.m12.2.3.cmml" xref="S4.SS2.p1.18.m12.2.3"><eq id="S4.SS2.p1.18.m12.2.3.1.cmml" xref="S4.SS2.p1.18.m12.2.3.1"></eq><apply id="S4.SS2.p1.18.m12.2.3.2.cmml" xref="S4.SS2.p1.18.m12.2.3.2"><times id="S4.SS2.p1.18.m12.2.3.2.1.cmml" xref="S4.SS2.p1.18.m12.2.3.2.1"></times><ci id="S4.SS2.p1.18.m12.2.3.2.2.cmml" xref="S4.SS2.p1.18.m12.2.3.2.2">𝑐</ci><apply id="S4.SS2.p1.18.m12.2.3.2.3.cmml" xref="S4.SS2.p1.18.m12.2.3.2.3"><csymbol cd="ambiguous" id="S4.SS2.p1.18.m12.2.3.2.3.1.cmml" xref="S4.SS2.p1.18.m12.2.3.2.3">subscript</csymbol><ci id="S4.SS2.p1.18.m12.2.3.2.3.2.cmml" xref="S4.SS2.p1.18.m12.2.3.2.3.2">bold-italic-ϕ</ci><cn id="S4.SS2.p1.18.m12.2.3.2.3.3.cmml" type="integer" xref="S4.SS2.p1.18.m12.2.3.2.3.3">1</cn></apply><ci id="S4.SS2.p1.18.m12.1.1.cmml" xref="S4.SS2.p1.18.m12.1.1">𝑡</ci></apply><apply id="S4.SS2.p1.18.m12.2.3.3.cmml" xref="S4.SS2.p1.18.m12.2.3.3"><times id="S4.SS2.p1.18.m12.2.3.3.1.cmml" xref="S4.SS2.p1.18.m12.2.3.3.1"></times><apply id="S4.SS2.p1.18.m12.2.3.3.2.cmml" xref="S4.SS2.p1.18.m12.2.3.3.2"><csymbol cd="ambiguous" id="S4.SS2.p1.18.m12.2.3.3.2.1.cmml" xref="S4.SS2.p1.18.m12.2.3.3.2">subscript</csymbol><ci id="S4.SS2.p1.18.m12.2.3.3.2.2.cmml" xref="S4.SS2.p1.18.m12.2.3.3.2.2">bold-italic-ϕ</ci><cn id="S4.SS2.p1.18.m12.2.3.3.2.3.cmml" type="integer" xref="S4.SS2.p1.18.m12.2.3.3.2.3">2</cn></apply><ci id="S4.SS2.p1.18.m12.2.2.cmml" xref="S4.SS2.p1.18.m12.2.2">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.18.m12.2c">c{\bm{\phi}}_{1}(t)={\bm{\phi}}_{2}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.18.m12.2d">italic_c bold_italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) = bold_italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> with <math alttext="c" class="ltx_Math" display="inline" id="S4.SS2.p1.19.m13.1"><semantics id="S4.SS2.p1.19.m13.1a"><mi id="S4.SS2.p1.19.m13.1.1" xref="S4.SS2.p1.19.m13.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.19.m13.1b"><ci id="S4.SS2.p1.19.m13.1.1.cmml" xref="S4.SS2.p1.19.m13.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.19.m13.1c">c</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.19.m13.1d">italic_c</annotation></semantics></math> being a non-zero constant, (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E12" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">12</span></a>) must be reconstructed by <math alttext="\bm{p}(t)" class="ltx_Math" display="inline" id="S4.SS2.p1.20.m14.1"><semantics id="S4.SS2.p1.20.m14.1a"><mrow id="S4.SS2.p1.20.m14.1.2" xref="S4.SS2.p1.20.m14.1.2.cmml"><mi id="S4.SS2.p1.20.m14.1.2.2" xref="S4.SS2.p1.20.m14.1.2.2.cmml">𝒑</mi><mo id="S4.SS2.p1.20.m14.1.2.1" xref="S4.SS2.p1.20.m14.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p1.20.m14.1.2.3.2" xref="S4.SS2.p1.20.m14.1.2.cmml"><mo id="S4.SS2.p1.20.m14.1.2.3.2.1" stretchy="false" xref="S4.SS2.p1.20.m14.1.2.cmml">(</mo><mi id="S4.SS2.p1.20.m14.1.1" xref="S4.SS2.p1.20.m14.1.1.cmml">t</mi><mo id="S4.SS2.p1.20.m14.1.2.3.2.2" stretchy="false" xref="S4.SS2.p1.20.m14.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.20.m14.1b"><apply id="S4.SS2.p1.20.m14.1.2.cmml" xref="S4.SS2.p1.20.m14.1.2"><times id="S4.SS2.p1.20.m14.1.2.1.cmml" xref="S4.SS2.p1.20.m14.1.2.1"></times><ci id="S4.SS2.p1.20.m14.1.2.2.cmml" xref="S4.SS2.p1.20.m14.1.2.2">𝒑</ci><ci id="S4.SS2.p1.20.m14.1.1.cmml" xref="S4.SS2.p1.20.m14.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.20.m14.1c">\bm{p}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.20.m14.1d">bold_italic_p ( italic_t )</annotation></semantics></math> <math alttext="=" class="ltx_Math" display="inline" id="S4.SS2.p1.21.m15.1"><semantics id="S4.SS2.p1.21.m15.1a"><mo id="S4.SS2.p1.21.m15.1.1" xref="S4.SS2.p1.21.m15.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.21.m15.1b"><eq id="S4.SS2.p1.21.m15.1.1.cmml" xref="S4.SS2.p1.21.m15.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.21.m15.1c">=</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.21.m15.1d">=</annotation></semantics></math> <math alttext="\Phi^{T}_{\rm sr}(t)\bm{\theta}_{\rm r}" class="ltx_Math" display="inline" id="S4.SS2.p1.22.m16.1"><semantics id="S4.SS2.p1.22.m16.1a"><mrow id="S4.SS2.p1.22.m16.1.2" xref="S4.SS2.p1.22.m16.1.2.cmml"><msubsup id="S4.SS2.p1.22.m16.1.2.2" xref="S4.SS2.p1.22.m16.1.2.2.cmml"><mi id="S4.SS2.p1.22.m16.1.2.2.2.2" mathvariant="normal" xref="S4.SS2.p1.22.m16.1.2.2.2.2.cmml">Φ</mi><mi id="S4.SS2.p1.22.m16.1.2.2.3" xref="S4.SS2.p1.22.m16.1.2.2.3.cmml">sr</mi><mi id="S4.SS2.p1.22.m16.1.2.2.2.3" xref="S4.SS2.p1.22.m16.1.2.2.2.3.cmml">T</mi></msubsup><mo id="S4.SS2.p1.22.m16.1.2.1" xref="S4.SS2.p1.22.m16.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p1.22.m16.1.2.3.2" xref="S4.SS2.p1.22.m16.1.2.cmml"><mo id="S4.SS2.p1.22.m16.1.2.3.2.1" stretchy="false" xref="S4.SS2.p1.22.m16.1.2.cmml">(</mo><mi id="S4.SS2.p1.22.m16.1.1" xref="S4.SS2.p1.22.m16.1.1.cmml">t</mi><mo id="S4.SS2.p1.22.m16.1.2.3.2.2" stretchy="false" xref="S4.SS2.p1.22.m16.1.2.cmml">)</mo></mrow><mo id="S4.SS2.p1.22.m16.1.2.1a" xref="S4.SS2.p1.22.m16.1.2.1.cmml">⁢</mo><msub id="S4.SS2.p1.22.m16.1.2.4" xref="S4.SS2.p1.22.m16.1.2.4.cmml"><mi id="S4.SS2.p1.22.m16.1.2.4.2" xref="S4.SS2.p1.22.m16.1.2.4.2.cmml">𝜽</mi><mi id="S4.SS2.p1.22.m16.1.2.4.3" mathvariant="normal" xref="S4.SS2.p1.22.m16.1.2.4.3.cmml">r</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.22.m16.1b"><apply id="S4.SS2.p1.22.m16.1.2.cmml" xref="S4.SS2.p1.22.m16.1.2"><times id="S4.SS2.p1.22.m16.1.2.1.cmml" xref="S4.SS2.p1.22.m16.1.2.1"></times><apply id="S4.SS2.p1.22.m16.1.2.2.cmml" xref="S4.SS2.p1.22.m16.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.p1.22.m16.1.2.2.1.cmml" xref="S4.SS2.p1.22.m16.1.2.2">subscript</csymbol><apply id="S4.SS2.p1.22.m16.1.2.2.2.cmml" xref="S4.SS2.p1.22.m16.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.p1.22.m16.1.2.2.2.1.cmml" xref="S4.SS2.p1.22.m16.1.2.2">superscript</csymbol><ci id="S4.SS2.p1.22.m16.1.2.2.2.2.cmml" xref="S4.SS2.p1.22.m16.1.2.2.2.2">Φ</ci><ci id="S4.SS2.p1.22.m16.1.2.2.2.3.cmml" xref="S4.SS2.p1.22.m16.1.2.2.2.3">𝑇</ci></apply><ci id="S4.SS2.p1.22.m16.1.2.2.3.cmml" xref="S4.SS2.p1.22.m16.1.2.2.3">sr</ci></apply><ci id="S4.SS2.p1.22.m16.1.1.cmml" xref="S4.SS2.p1.22.m16.1.1">𝑡</ci><apply id="S4.SS2.p1.22.m16.1.2.4.cmml" xref="S4.SS2.p1.22.m16.1.2.4"><csymbol cd="ambiguous" id="S4.SS2.p1.22.m16.1.2.4.1.cmml" xref="S4.SS2.p1.22.m16.1.2.4">subscript</csymbol><ci id="S4.SS2.p1.22.m16.1.2.4.2.cmml" xref="S4.SS2.p1.22.m16.1.2.4.2">𝜽</ci><ci id="S4.SS2.p1.22.m16.1.2.4.3.cmml" xref="S4.SS2.p1.22.m16.1.2.4.3">r</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.22.m16.1c">\Phi^{T}_{\rm sr}(t)\bm{\theta}_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.22.m16.1d">roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_sr end_POSTSUBSCRIPT ( italic_t ) bold_italic_θ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\Phi_{\rm sr}(t)" class="ltx_Math" display="inline" id="S4.SS2.p1.23.m17.1"><semantics id="S4.SS2.p1.23.m17.1a"><mrow id="S4.SS2.p1.23.m17.1.2" xref="S4.SS2.p1.23.m17.1.2.cmml"><msub id="S4.SS2.p1.23.m17.1.2.2" xref="S4.SS2.p1.23.m17.1.2.2.cmml"><mi id="S4.SS2.p1.23.m17.1.2.2.2" mathvariant="normal" xref="S4.SS2.p1.23.m17.1.2.2.2.cmml">Φ</mi><mi id="S4.SS2.p1.23.m17.1.2.2.3" xref="S4.SS2.p1.23.m17.1.2.2.3.cmml">sr</mi></msub><mo id="S4.SS2.p1.23.m17.1.2.1" xref="S4.SS2.p1.23.m17.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p1.23.m17.1.2.3.2" xref="S4.SS2.p1.23.m17.1.2.cmml"><mo id="S4.SS2.p1.23.m17.1.2.3.2.1" stretchy="false" xref="S4.SS2.p1.23.m17.1.2.cmml">(</mo><mi id="S4.SS2.p1.23.m17.1.1" xref="S4.SS2.p1.23.m17.1.1.cmml">t</mi><mo id="S4.SS2.p1.23.m17.1.2.3.2.2" stretchy="false" xref="S4.SS2.p1.23.m17.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.23.m17.1b"><apply id="S4.SS2.p1.23.m17.1.2.cmml" xref="S4.SS2.p1.23.m17.1.2"><times id="S4.SS2.p1.23.m17.1.2.1.cmml" xref="S4.SS2.p1.23.m17.1.2.1"></times><apply id="S4.SS2.p1.23.m17.1.2.2.cmml" xref="S4.SS2.p1.23.m17.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.p1.23.m17.1.2.2.1.cmml" xref="S4.SS2.p1.23.m17.1.2.2">subscript</csymbol><ci id="S4.SS2.p1.23.m17.1.2.2.2.cmml" xref="S4.SS2.p1.23.m17.1.2.2.2">Φ</ci><ci id="S4.SS2.p1.23.m17.1.2.2.3.cmml" xref="S4.SS2.p1.23.m17.1.2.2.3">sr</ci></apply><ci id="S4.SS2.p1.23.m17.1.1.cmml" xref="S4.SS2.p1.23.m17.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.23.m17.1c">\Phi_{\rm sr}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.23.m17.1d">roman_Φ start_POSTSUBSCRIPT roman_sr end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S4.SS2.p1.24.m18.1"><semantics id="S4.SS2.p1.24.m18.1a"><mo id="S4.SS2.p1.24.m18.1.1" xref="S4.SS2.p1.24.m18.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.24.m18.1b"><csymbol cd="latexml" id="S4.SS2.p1.24.m18.1.1.cmml" xref="S4.SS2.p1.24.m18.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.24.m18.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.24.m18.1d">:=</annotation></semantics></math> <math alttext="[{\bm{\phi}}_{1}(t),{\bm{\phi}}_{3}(t),\cdots,{\bm{\phi}}_{N}(t)]^{T}" class="ltx_Math" display="inline" id="S4.SS2.p1.25.m19.7"><semantics id="S4.SS2.p1.25.m19.7a"><msup id="S4.SS2.p1.25.m19.7.7" xref="S4.SS2.p1.25.m19.7.7.cmml"><mrow id="S4.SS2.p1.25.m19.7.7.3.3" xref="S4.SS2.p1.25.m19.7.7.3.4.cmml"><mo id="S4.SS2.p1.25.m19.7.7.3.3.4" stretchy="false" xref="S4.SS2.p1.25.m19.7.7.3.4.cmml">[</mo><mrow id="S4.SS2.p1.25.m19.5.5.1.1.1" xref="S4.SS2.p1.25.m19.5.5.1.1.1.cmml"><msub id="S4.SS2.p1.25.m19.5.5.1.1.1.2" xref="S4.SS2.p1.25.m19.5.5.1.1.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS2.p1.25.m19.5.5.1.1.1.2.2" mathvariant="bold-italic" xref="S4.SS2.p1.25.m19.5.5.1.1.1.2.2.cmml">ϕ</mi><mn id="S4.SS2.p1.25.m19.5.5.1.1.1.2.3" xref="S4.SS2.p1.25.m19.5.5.1.1.1.2.3.cmml">1</mn></msub><mo id="S4.SS2.p1.25.m19.5.5.1.1.1.1" xref="S4.SS2.p1.25.m19.5.5.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS2.p1.25.m19.5.5.1.1.1.3.2" xref="S4.SS2.p1.25.m19.5.5.1.1.1.cmml"><mo id="S4.SS2.p1.25.m19.5.5.1.1.1.3.2.1" stretchy="false" xref="S4.SS2.p1.25.m19.5.5.1.1.1.cmml">(</mo><mi id="S4.SS2.p1.25.m19.1.1" xref="S4.SS2.p1.25.m19.1.1.cmml">t</mi><mo id="S4.SS2.p1.25.m19.5.5.1.1.1.3.2.2" stretchy="false" xref="S4.SS2.p1.25.m19.5.5.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.p1.25.m19.7.7.3.3.5" xref="S4.SS2.p1.25.m19.7.7.3.4.cmml">,</mo><mrow id="S4.SS2.p1.25.m19.6.6.2.2.2" xref="S4.SS2.p1.25.m19.6.6.2.2.2.cmml"><msub id="S4.SS2.p1.25.m19.6.6.2.2.2.2" xref="S4.SS2.p1.25.m19.6.6.2.2.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS2.p1.25.m19.6.6.2.2.2.2.2" mathvariant="bold-italic" xref="S4.SS2.p1.25.m19.6.6.2.2.2.2.2.cmml">ϕ</mi><mn id="S4.SS2.p1.25.m19.6.6.2.2.2.2.3" xref="S4.SS2.p1.25.m19.6.6.2.2.2.2.3.cmml">3</mn></msub><mo id="S4.SS2.p1.25.m19.6.6.2.2.2.1" xref="S4.SS2.p1.25.m19.6.6.2.2.2.1.cmml">⁢</mo><mrow id="S4.SS2.p1.25.m19.6.6.2.2.2.3.2" xref="S4.SS2.p1.25.m19.6.6.2.2.2.cmml"><mo id="S4.SS2.p1.25.m19.6.6.2.2.2.3.2.1" stretchy="false" xref="S4.SS2.p1.25.m19.6.6.2.2.2.cmml">(</mo><mi id="S4.SS2.p1.25.m19.2.2" xref="S4.SS2.p1.25.m19.2.2.cmml">t</mi><mo id="S4.SS2.p1.25.m19.6.6.2.2.2.3.2.2" stretchy="false" xref="S4.SS2.p1.25.m19.6.6.2.2.2.cmml">)</mo></mrow></mrow><mo id="S4.SS2.p1.25.m19.7.7.3.3.6" xref="S4.SS2.p1.25.m19.7.7.3.4.cmml">,</mo><mi id="S4.SS2.p1.25.m19.4.4" mathvariant="normal" xref="S4.SS2.p1.25.m19.4.4.cmml">⋯</mi><mo id="S4.SS2.p1.25.m19.7.7.3.3.7" xref="S4.SS2.p1.25.m19.7.7.3.4.cmml">,</mo><mrow id="S4.SS2.p1.25.m19.7.7.3.3.3" xref="S4.SS2.p1.25.m19.7.7.3.3.3.cmml"><msub id="S4.SS2.p1.25.m19.7.7.3.3.3.2" xref="S4.SS2.p1.25.m19.7.7.3.3.3.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS2.p1.25.m19.7.7.3.3.3.2.2" mathvariant="bold-italic" xref="S4.SS2.p1.25.m19.7.7.3.3.3.2.2.cmml">ϕ</mi><mi id="S4.SS2.p1.25.m19.7.7.3.3.3.2.3" xref="S4.SS2.p1.25.m19.7.7.3.3.3.2.3.cmml">N</mi></msub><mo id="S4.SS2.p1.25.m19.7.7.3.3.3.1" xref="S4.SS2.p1.25.m19.7.7.3.3.3.1.cmml">⁢</mo><mrow id="S4.SS2.p1.25.m19.7.7.3.3.3.3.2" xref="S4.SS2.p1.25.m19.7.7.3.3.3.cmml"><mo id="S4.SS2.p1.25.m19.7.7.3.3.3.3.2.1" stretchy="false" xref="S4.SS2.p1.25.m19.7.7.3.3.3.cmml">(</mo><mi id="S4.SS2.p1.25.m19.3.3" xref="S4.SS2.p1.25.m19.3.3.cmml">t</mi><mo id="S4.SS2.p1.25.m19.7.7.3.3.3.3.2.2" stretchy="false" xref="S4.SS2.p1.25.m19.7.7.3.3.3.cmml">)</mo></mrow></mrow><mo id="S4.SS2.p1.25.m19.7.7.3.3.8" stretchy="false" xref="S4.SS2.p1.25.m19.7.7.3.4.cmml">]</mo></mrow><mi id="S4.SS2.p1.25.m19.7.7.5" xref="S4.SS2.p1.25.m19.7.7.5.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.25.m19.7b"><apply id="S4.SS2.p1.25.m19.7.7.cmml" xref="S4.SS2.p1.25.m19.7.7"><csymbol cd="ambiguous" id="S4.SS2.p1.25.m19.7.7.4.cmml" xref="S4.SS2.p1.25.m19.7.7">superscript</csymbol><list id="S4.SS2.p1.25.m19.7.7.3.4.cmml" xref="S4.SS2.p1.25.m19.7.7.3.3"><apply id="S4.SS2.p1.25.m19.5.5.1.1.1.cmml" xref="S4.SS2.p1.25.m19.5.5.1.1.1"><times id="S4.SS2.p1.25.m19.5.5.1.1.1.1.cmml" xref="S4.SS2.p1.25.m19.5.5.1.1.1.1"></times><apply id="S4.SS2.p1.25.m19.5.5.1.1.1.2.cmml" xref="S4.SS2.p1.25.m19.5.5.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.25.m19.5.5.1.1.1.2.1.cmml" xref="S4.SS2.p1.25.m19.5.5.1.1.1.2">subscript</csymbol><ci id="S4.SS2.p1.25.m19.5.5.1.1.1.2.2.cmml" xref="S4.SS2.p1.25.m19.5.5.1.1.1.2.2">bold-italic-ϕ</ci><cn id="S4.SS2.p1.25.m19.5.5.1.1.1.2.3.cmml" type="integer" xref="S4.SS2.p1.25.m19.5.5.1.1.1.2.3">1</cn></apply><ci id="S4.SS2.p1.25.m19.1.1.cmml" xref="S4.SS2.p1.25.m19.1.1">𝑡</ci></apply><apply id="S4.SS2.p1.25.m19.6.6.2.2.2.cmml" xref="S4.SS2.p1.25.m19.6.6.2.2.2"><times id="S4.SS2.p1.25.m19.6.6.2.2.2.1.cmml" xref="S4.SS2.p1.25.m19.6.6.2.2.2.1"></times><apply id="S4.SS2.p1.25.m19.6.6.2.2.2.2.cmml" xref="S4.SS2.p1.25.m19.6.6.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.p1.25.m19.6.6.2.2.2.2.1.cmml" xref="S4.SS2.p1.25.m19.6.6.2.2.2.2">subscript</csymbol><ci id="S4.SS2.p1.25.m19.6.6.2.2.2.2.2.cmml" xref="S4.SS2.p1.25.m19.6.6.2.2.2.2.2">bold-italic-ϕ</ci><cn id="S4.SS2.p1.25.m19.6.6.2.2.2.2.3.cmml" type="integer" xref="S4.SS2.p1.25.m19.6.6.2.2.2.2.3">3</cn></apply><ci id="S4.SS2.p1.25.m19.2.2.cmml" xref="S4.SS2.p1.25.m19.2.2">𝑡</ci></apply><ci id="S4.SS2.p1.25.m19.4.4.cmml" xref="S4.SS2.p1.25.m19.4.4">⋯</ci><apply id="S4.SS2.p1.25.m19.7.7.3.3.3.cmml" xref="S4.SS2.p1.25.m19.7.7.3.3.3"><times id="S4.SS2.p1.25.m19.7.7.3.3.3.1.cmml" xref="S4.SS2.p1.25.m19.7.7.3.3.3.1"></times><apply id="S4.SS2.p1.25.m19.7.7.3.3.3.2.cmml" xref="S4.SS2.p1.25.m19.7.7.3.3.3.2"><csymbol cd="ambiguous" id="S4.SS2.p1.25.m19.7.7.3.3.3.2.1.cmml" xref="S4.SS2.p1.25.m19.7.7.3.3.3.2">subscript</csymbol><ci id="S4.SS2.p1.25.m19.7.7.3.3.3.2.2.cmml" xref="S4.SS2.p1.25.m19.7.7.3.3.3.2.2">bold-italic-ϕ</ci><ci id="S4.SS2.p1.25.m19.7.7.3.3.3.2.3.cmml" xref="S4.SS2.p1.25.m19.7.7.3.3.3.2.3">𝑁</ci></apply><ci id="S4.SS2.p1.25.m19.3.3.cmml" xref="S4.SS2.p1.25.m19.3.3">𝑡</ci></apply></list><ci id="S4.SS2.p1.25.m19.7.7.5.cmml" xref="S4.SS2.p1.25.m19.7.7.5">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.25.m19.7c">[{\bm{\phi}}_{1}(t),{\bm{\phi}}_{3}(t),\cdots,{\bm{\phi}}_{N}(t)]^{T}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.25.m19.7d">[ bold_italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) , bold_italic_ϕ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) , ⋯ , bold_italic_ϕ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_t ) ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> <math alttext="\in\mathbb{R}^{(N-1)\times n}" class="ltx_Math" display="inline" id="S4.SS2.p1.26.m20.1"><semantics id="S4.SS2.p1.26.m20.1a"><mrow id="S4.SS2.p1.26.m20.1.2" xref="S4.SS2.p1.26.m20.1.2.cmml"><mi id="S4.SS2.p1.26.m20.1.2.2" xref="S4.SS2.p1.26.m20.1.2.2.cmml"></mi><mo id="S4.SS2.p1.26.m20.1.2.1" xref="S4.SS2.p1.26.m20.1.2.1.cmml">∈</mo><msup id="S4.SS2.p1.26.m20.1.2.3" xref="S4.SS2.p1.26.m20.1.2.3.cmml"><mi id="S4.SS2.p1.26.m20.1.2.3.2" xref="S4.SS2.p1.26.m20.1.2.3.2.cmml">ℝ</mi><mrow id="S4.SS2.p1.26.m20.1.1.1" xref="S4.SS2.p1.26.m20.1.1.1.cmml"><mrow id="S4.SS2.p1.26.m20.1.1.1.1.1" xref="S4.SS2.p1.26.m20.1.1.1.1.1.1.cmml"><mo id="S4.SS2.p1.26.m20.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.p1.26.m20.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.p1.26.m20.1.1.1.1.1.1" xref="S4.SS2.p1.26.m20.1.1.1.1.1.1.cmml"><mi id="S4.SS2.p1.26.m20.1.1.1.1.1.1.2" xref="S4.SS2.p1.26.m20.1.1.1.1.1.1.2.cmml">N</mi><mo id="S4.SS2.p1.26.m20.1.1.1.1.1.1.1" xref="S4.SS2.p1.26.m20.1.1.1.1.1.1.1.cmml">−</mo><mn id="S4.SS2.p1.26.m20.1.1.1.1.1.1.3" xref="S4.SS2.p1.26.m20.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S4.SS2.p1.26.m20.1.1.1.1.1.3" rspace="0.055em" stretchy="false" xref="S4.SS2.p1.26.m20.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.SS2.p1.26.m20.1.1.1.2" rspace="0.222em" xref="S4.SS2.p1.26.m20.1.1.1.2.cmml">×</mo><mi id="S4.SS2.p1.26.m20.1.1.1.3" xref="S4.SS2.p1.26.m20.1.1.1.3.cmml">n</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.26.m20.1b"><apply id="S4.SS2.p1.26.m20.1.2.cmml" xref="S4.SS2.p1.26.m20.1.2"><in id="S4.SS2.p1.26.m20.1.2.1.cmml" xref="S4.SS2.p1.26.m20.1.2.1"></in><csymbol cd="latexml" id="S4.SS2.p1.26.m20.1.2.2.cmml" xref="S4.SS2.p1.26.m20.1.2.2">absent</csymbol><apply id="S4.SS2.p1.26.m20.1.2.3.cmml" xref="S4.SS2.p1.26.m20.1.2.3"><csymbol cd="ambiguous" id="S4.SS2.p1.26.m20.1.2.3.1.cmml" xref="S4.SS2.p1.26.m20.1.2.3">superscript</csymbol><ci id="S4.SS2.p1.26.m20.1.2.3.2.cmml" xref="S4.SS2.p1.26.m20.1.2.3.2">ℝ</ci><apply id="S4.SS2.p1.26.m20.1.1.1.cmml" xref="S4.SS2.p1.26.m20.1.1.1"><times id="S4.SS2.p1.26.m20.1.1.1.2.cmml" xref="S4.SS2.p1.26.m20.1.1.1.2"></times><apply id="S4.SS2.p1.26.m20.1.1.1.1.1.1.cmml" xref="S4.SS2.p1.26.m20.1.1.1.1.1"><minus id="S4.SS2.p1.26.m20.1.1.1.1.1.1.1.cmml" xref="S4.SS2.p1.26.m20.1.1.1.1.1.1.1"></minus><ci id="S4.SS2.p1.26.m20.1.1.1.1.1.1.2.cmml" xref="S4.SS2.p1.26.m20.1.1.1.1.1.1.2">𝑁</ci><cn id="S4.SS2.p1.26.m20.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.p1.26.m20.1.1.1.1.1.1.3">1</cn></apply><ci id="S4.SS2.p1.26.m20.1.1.1.3.cmml" xref="S4.SS2.p1.26.m20.1.1.1.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.26.m20.1c">\in\mathbb{R}^{(N-1)\times n}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.26.m20.1d">∈ blackboard_R start_POSTSUPERSCRIPT ( italic_N - 1 ) × italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\bm{\theta}_{\rm r}" class="ltx_Math" display="inline" id="S4.SS2.p1.27.m21.1"><semantics id="S4.SS2.p1.27.m21.1a"><msub id="S4.SS2.p1.27.m21.1.1" xref="S4.SS2.p1.27.m21.1.1.cmml"><mi id="S4.SS2.p1.27.m21.1.1.2" xref="S4.SS2.p1.27.m21.1.1.2.cmml">𝜽</mi><mi id="S4.SS2.p1.27.m21.1.1.3" mathvariant="normal" xref="S4.SS2.p1.27.m21.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.27.m21.1b"><apply id="S4.SS2.p1.27.m21.1.1.cmml" xref="S4.SS2.p1.27.m21.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.27.m21.1.1.1.cmml" xref="S4.SS2.p1.27.m21.1.1">subscript</csymbol><ci id="S4.SS2.p1.27.m21.1.1.2.cmml" xref="S4.SS2.p1.27.m21.1.1.2">𝜽</ci><ci id="S4.SS2.p1.27.m21.1.1.3.cmml" xref="S4.SS2.p1.27.m21.1.1.3">r</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.27.m21.1c">\bm{\theta}_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.27.m21.1d">bold_italic_θ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S4.SS2.p1.28.m22.1"><semantics id="S4.SS2.p1.28.m22.1a"><mo id="S4.SS2.p1.28.m22.1.1" xref="S4.SS2.p1.28.m22.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.28.m22.1b"><csymbol cd="latexml" id="S4.SS2.p1.28.m22.1.1.cmml" xref="S4.SS2.p1.28.m22.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.28.m22.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.28.m22.1d">:=</annotation></semantics></math> <math alttext="[\theta_{1}+c\theta_{2},\theta_{3},\cdots,\theta_{N}]^{T}" class="ltx_Math" display="inline" id="S4.SS2.p1.29.m23.4"><semantics id="S4.SS2.p1.29.m23.4a"><msup id="S4.SS2.p1.29.m23.4.4" xref="S4.SS2.p1.29.m23.4.4.cmml"><mrow id="S4.SS2.p1.29.m23.4.4.3.3" xref="S4.SS2.p1.29.m23.4.4.3.4.cmml"><mo id="S4.SS2.p1.29.m23.4.4.3.3.4" stretchy="false" xref="S4.SS2.p1.29.m23.4.4.3.4.cmml">[</mo><mrow id="S4.SS2.p1.29.m23.2.2.1.1.1" xref="S4.SS2.p1.29.m23.2.2.1.1.1.cmml"><msub id="S4.SS2.p1.29.m23.2.2.1.1.1.2" xref="S4.SS2.p1.29.m23.2.2.1.1.1.2.cmml"><mi id="S4.SS2.p1.29.m23.2.2.1.1.1.2.2" xref="S4.SS2.p1.29.m23.2.2.1.1.1.2.2.cmml">θ</mi><mn id="S4.SS2.p1.29.m23.2.2.1.1.1.2.3" xref="S4.SS2.p1.29.m23.2.2.1.1.1.2.3.cmml">1</mn></msub><mo id="S4.SS2.p1.29.m23.2.2.1.1.1.1" xref="S4.SS2.p1.29.m23.2.2.1.1.1.1.cmml">+</mo><mrow id="S4.SS2.p1.29.m23.2.2.1.1.1.3" xref="S4.SS2.p1.29.m23.2.2.1.1.1.3.cmml"><mi id="S4.SS2.p1.29.m23.2.2.1.1.1.3.2" xref="S4.SS2.p1.29.m23.2.2.1.1.1.3.2.cmml">c</mi><mo id="S4.SS2.p1.29.m23.2.2.1.1.1.3.1" xref="S4.SS2.p1.29.m23.2.2.1.1.1.3.1.cmml">⁢</mo><msub id="S4.SS2.p1.29.m23.2.2.1.1.1.3.3" xref="S4.SS2.p1.29.m23.2.2.1.1.1.3.3.cmml"><mi id="S4.SS2.p1.29.m23.2.2.1.1.1.3.3.2" xref="S4.SS2.p1.29.m23.2.2.1.1.1.3.3.2.cmml">θ</mi><mn id="S4.SS2.p1.29.m23.2.2.1.1.1.3.3.3" xref="S4.SS2.p1.29.m23.2.2.1.1.1.3.3.3.cmml">2</mn></msub></mrow></mrow><mo id="S4.SS2.p1.29.m23.4.4.3.3.5" xref="S4.SS2.p1.29.m23.4.4.3.4.cmml">,</mo><msub id="S4.SS2.p1.29.m23.3.3.2.2.2" xref="S4.SS2.p1.29.m23.3.3.2.2.2.cmml"><mi id="S4.SS2.p1.29.m23.3.3.2.2.2.2" xref="S4.SS2.p1.29.m23.3.3.2.2.2.2.cmml">θ</mi><mn id="S4.SS2.p1.29.m23.3.3.2.2.2.3" xref="S4.SS2.p1.29.m23.3.3.2.2.2.3.cmml">3</mn></msub><mo id="S4.SS2.p1.29.m23.4.4.3.3.6" xref="S4.SS2.p1.29.m23.4.4.3.4.cmml">,</mo><mi id="S4.SS2.p1.29.m23.1.1" mathvariant="normal" xref="S4.SS2.p1.29.m23.1.1.cmml">⋯</mi><mo id="S4.SS2.p1.29.m23.4.4.3.3.7" xref="S4.SS2.p1.29.m23.4.4.3.4.cmml">,</mo><msub id="S4.SS2.p1.29.m23.4.4.3.3.3" xref="S4.SS2.p1.29.m23.4.4.3.3.3.cmml"><mi id="S4.SS2.p1.29.m23.4.4.3.3.3.2" xref="S4.SS2.p1.29.m23.4.4.3.3.3.2.cmml">θ</mi><mi id="S4.SS2.p1.29.m23.4.4.3.3.3.3" xref="S4.SS2.p1.29.m23.4.4.3.3.3.3.cmml">N</mi></msub><mo id="S4.SS2.p1.29.m23.4.4.3.3.8" stretchy="false" xref="S4.SS2.p1.29.m23.4.4.3.4.cmml">]</mo></mrow><mi id="S4.SS2.p1.29.m23.4.4.5" xref="S4.SS2.p1.29.m23.4.4.5.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.29.m23.4b"><apply id="S4.SS2.p1.29.m23.4.4.cmml" xref="S4.SS2.p1.29.m23.4.4"><csymbol cd="ambiguous" id="S4.SS2.p1.29.m23.4.4.4.cmml" xref="S4.SS2.p1.29.m23.4.4">superscript</csymbol><list id="S4.SS2.p1.29.m23.4.4.3.4.cmml" xref="S4.SS2.p1.29.m23.4.4.3.3"><apply id="S4.SS2.p1.29.m23.2.2.1.1.1.cmml" xref="S4.SS2.p1.29.m23.2.2.1.1.1"><plus id="S4.SS2.p1.29.m23.2.2.1.1.1.1.cmml" xref="S4.SS2.p1.29.m23.2.2.1.1.1.1"></plus><apply id="S4.SS2.p1.29.m23.2.2.1.1.1.2.cmml" xref="S4.SS2.p1.29.m23.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.29.m23.2.2.1.1.1.2.1.cmml" xref="S4.SS2.p1.29.m23.2.2.1.1.1.2">subscript</csymbol><ci id="S4.SS2.p1.29.m23.2.2.1.1.1.2.2.cmml" xref="S4.SS2.p1.29.m23.2.2.1.1.1.2.2">𝜃</ci><cn id="S4.SS2.p1.29.m23.2.2.1.1.1.2.3.cmml" type="integer" xref="S4.SS2.p1.29.m23.2.2.1.1.1.2.3">1</cn></apply><apply id="S4.SS2.p1.29.m23.2.2.1.1.1.3.cmml" xref="S4.SS2.p1.29.m23.2.2.1.1.1.3"><times id="S4.SS2.p1.29.m23.2.2.1.1.1.3.1.cmml" xref="S4.SS2.p1.29.m23.2.2.1.1.1.3.1"></times><ci id="S4.SS2.p1.29.m23.2.2.1.1.1.3.2.cmml" xref="S4.SS2.p1.29.m23.2.2.1.1.1.3.2">𝑐</ci><apply id="S4.SS2.p1.29.m23.2.2.1.1.1.3.3.cmml" xref="S4.SS2.p1.29.m23.2.2.1.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS2.p1.29.m23.2.2.1.1.1.3.3.1.cmml" xref="S4.SS2.p1.29.m23.2.2.1.1.1.3.3">subscript</csymbol><ci id="S4.SS2.p1.29.m23.2.2.1.1.1.3.3.2.cmml" xref="S4.SS2.p1.29.m23.2.2.1.1.1.3.3.2">𝜃</ci><cn id="S4.SS2.p1.29.m23.2.2.1.1.1.3.3.3.cmml" type="integer" xref="S4.SS2.p1.29.m23.2.2.1.1.1.3.3.3">2</cn></apply></apply></apply><apply id="S4.SS2.p1.29.m23.3.3.2.2.2.cmml" xref="S4.SS2.p1.29.m23.3.3.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.p1.29.m23.3.3.2.2.2.1.cmml" xref="S4.SS2.p1.29.m23.3.3.2.2.2">subscript</csymbol><ci id="S4.SS2.p1.29.m23.3.3.2.2.2.2.cmml" xref="S4.SS2.p1.29.m23.3.3.2.2.2.2">𝜃</ci><cn id="S4.SS2.p1.29.m23.3.3.2.2.2.3.cmml" type="integer" xref="S4.SS2.p1.29.m23.3.3.2.2.2.3">3</cn></apply><ci id="S4.SS2.p1.29.m23.1.1.cmml" xref="S4.SS2.p1.29.m23.1.1">⋯</ci><apply id="S4.SS2.p1.29.m23.4.4.3.3.3.cmml" xref="S4.SS2.p1.29.m23.4.4.3.3.3"><csymbol cd="ambiguous" id="S4.SS2.p1.29.m23.4.4.3.3.3.1.cmml" xref="S4.SS2.p1.29.m23.4.4.3.3.3">subscript</csymbol><ci id="S4.SS2.p1.29.m23.4.4.3.3.3.2.cmml" xref="S4.SS2.p1.29.m23.4.4.3.3.3.2">𝜃</ci><ci id="S4.SS2.p1.29.m23.4.4.3.3.3.3.cmml" xref="S4.SS2.p1.29.m23.4.4.3.3.3.3">𝑁</ci></apply></list><ci id="S4.SS2.p1.29.m23.4.4.5.cmml" xref="S4.SS2.p1.29.m23.4.4.5">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.29.m23.4c">[\theta_{1}+c\theta_{2},\theta_{3},\cdots,\theta_{N}]^{T}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.29.m23.4d">[ italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_c italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , ⋯ , italic_θ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> <math alttext="\in\mathbb{R}^{N-1}" class="ltx_Math" display="inline" id="S4.SS2.p1.30.m24.1"><semantics id="S4.SS2.p1.30.m24.1a"><mrow id="S4.SS2.p1.30.m24.1.1" xref="S4.SS2.p1.30.m24.1.1.cmml"><mi id="S4.SS2.p1.30.m24.1.1.2" xref="S4.SS2.p1.30.m24.1.1.2.cmml"></mi><mo id="S4.SS2.p1.30.m24.1.1.1" xref="S4.SS2.p1.30.m24.1.1.1.cmml">∈</mo><msup id="S4.SS2.p1.30.m24.1.1.3" xref="S4.SS2.p1.30.m24.1.1.3.cmml"><mi id="S4.SS2.p1.30.m24.1.1.3.2" xref="S4.SS2.p1.30.m24.1.1.3.2.cmml">ℝ</mi><mrow id="S4.SS2.p1.30.m24.1.1.3.3" xref="S4.SS2.p1.30.m24.1.1.3.3.cmml"><mi id="S4.SS2.p1.30.m24.1.1.3.3.2" xref="S4.SS2.p1.30.m24.1.1.3.3.2.cmml">N</mi><mo id="S4.SS2.p1.30.m24.1.1.3.3.1" xref="S4.SS2.p1.30.m24.1.1.3.3.1.cmml">−</mo><mn id="S4.SS2.p1.30.m24.1.1.3.3.3" xref="S4.SS2.p1.30.m24.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.30.m24.1b"><apply id="S4.SS2.p1.30.m24.1.1.cmml" xref="S4.SS2.p1.30.m24.1.1"><in id="S4.SS2.p1.30.m24.1.1.1.cmml" xref="S4.SS2.p1.30.m24.1.1.1"></in><csymbol cd="latexml" id="S4.SS2.p1.30.m24.1.1.2.cmml" xref="S4.SS2.p1.30.m24.1.1.2">absent</csymbol><apply id="S4.SS2.p1.30.m24.1.1.3.cmml" xref="S4.SS2.p1.30.m24.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p1.30.m24.1.1.3.1.cmml" xref="S4.SS2.p1.30.m24.1.1.3">superscript</csymbol><ci id="S4.SS2.p1.30.m24.1.1.3.2.cmml" xref="S4.SS2.p1.30.m24.1.1.3.2">ℝ</ci><apply id="S4.SS2.p1.30.m24.1.1.3.3.cmml" xref="S4.SS2.p1.30.m24.1.1.3.3"><minus id="S4.SS2.p1.30.m24.1.1.3.3.1.cmml" xref="S4.SS2.p1.30.m24.1.1.3.3.1"></minus><ci id="S4.SS2.p1.30.m24.1.1.3.3.2.cmml" xref="S4.SS2.p1.30.m24.1.1.3.3.2">𝑁</ci><cn id="S4.SS2.p1.30.m24.1.1.3.3.3.cmml" type="integer" xref="S4.SS2.p1.30.m24.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.30.m24.1c">\in\mathbb{R}^{N-1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.30.m24.1d">∈ blackboard_R start_POSTSUPERSCRIPT italic_N - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, such that one has the excitation matrix <math alttext="\int_{t-\tau_{\rm d}}^{t}\Phi_{\rm sr}(\tau)\Phi_{\rm sr}^{T}(\tau)d\tau\geq\sigma I" class="ltx_Math" display="inline" id="S4.SS2.p1.31.m25.2"><semantics id="S4.SS2.p1.31.m25.2a"><mrow id="S4.SS2.p1.31.m25.2.3" xref="S4.SS2.p1.31.m25.2.3.cmml"><mrow id="S4.SS2.p1.31.m25.2.3.2" xref="S4.SS2.p1.31.m25.2.3.2.cmml"><msubsup id="S4.SS2.p1.31.m25.2.3.2.1" xref="S4.SS2.p1.31.m25.2.3.2.1.cmml"><mo id="S4.SS2.p1.31.m25.2.3.2.1.2.2" xref="S4.SS2.p1.31.m25.2.3.2.1.2.2.cmml">∫</mo><mrow id="S4.SS2.p1.31.m25.2.3.2.1.2.3" xref="S4.SS2.p1.31.m25.2.3.2.1.2.3.cmml"><mi id="S4.SS2.p1.31.m25.2.3.2.1.2.3.2" xref="S4.SS2.p1.31.m25.2.3.2.1.2.3.2.cmml">t</mi><mo id="S4.SS2.p1.31.m25.2.3.2.1.2.3.1" xref="S4.SS2.p1.31.m25.2.3.2.1.2.3.1.cmml">−</mo><msub id="S4.SS2.p1.31.m25.2.3.2.1.2.3.3" xref="S4.SS2.p1.31.m25.2.3.2.1.2.3.3.cmml"><mi id="S4.SS2.p1.31.m25.2.3.2.1.2.3.3.2" xref="S4.SS2.p1.31.m25.2.3.2.1.2.3.3.2.cmml">τ</mi><mi id="S4.SS2.p1.31.m25.2.3.2.1.2.3.3.3" mathvariant="normal" xref="S4.SS2.p1.31.m25.2.3.2.1.2.3.3.3.cmml">d</mi></msub></mrow><mi id="S4.SS2.p1.31.m25.2.3.2.1.3" xref="S4.SS2.p1.31.m25.2.3.2.1.3.cmml">t</mi></msubsup><mrow id="S4.SS2.p1.31.m25.2.3.2.2" xref="S4.SS2.p1.31.m25.2.3.2.2.cmml"><msub id="S4.SS2.p1.31.m25.2.3.2.2.2" xref="S4.SS2.p1.31.m25.2.3.2.2.2.cmml"><mi id="S4.SS2.p1.31.m25.2.3.2.2.2.2" mathvariant="normal" xref="S4.SS2.p1.31.m25.2.3.2.2.2.2.cmml">Φ</mi><mi id="S4.SS2.p1.31.m25.2.3.2.2.2.3" xref="S4.SS2.p1.31.m25.2.3.2.2.2.3.cmml">sr</mi></msub><mo id="S4.SS2.p1.31.m25.2.3.2.2.1" xref="S4.SS2.p1.31.m25.2.3.2.2.1.cmml">⁢</mo><mrow id="S4.SS2.p1.31.m25.2.3.2.2.3.2" xref="S4.SS2.p1.31.m25.2.3.2.2.cmml"><mo id="S4.SS2.p1.31.m25.2.3.2.2.3.2.1" stretchy="false" xref="S4.SS2.p1.31.m25.2.3.2.2.cmml">(</mo><mi id="S4.SS2.p1.31.m25.1.1" xref="S4.SS2.p1.31.m25.1.1.cmml">τ</mi><mo id="S4.SS2.p1.31.m25.2.3.2.2.3.2.2" stretchy="false" xref="S4.SS2.p1.31.m25.2.3.2.2.cmml">)</mo></mrow><mo id="S4.SS2.p1.31.m25.2.3.2.2.1a" xref="S4.SS2.p1.31.m25.2.3.2.2.1.cmml">⁢</mo><msubsup id="S4.SS2.p1.31.m25.2.3.2.2.4" xref="S4.SS2.p1.31.m25.2.3.2.2.4.cmml"><mi id="S4.SS2.p1.31.m25.2.3.2.2.4.2.2" mathvariant="normal" xref="S4.SS2.p1.31.m25.2.3.2.2.4.2.2.cmml">Φ</mi><mi id="S4.SS2.p1.31.m25.2.3.2.2.4.2.3" xref="S4.SS2.p1.31.m25.2.3.2.2.4.2.3.cmml">sr</mi><mi id="S4.SS2.p1.31.m25.2.3.2.2.4.3" xref="S4.SS2.p1.31.m25.2.3.2.2.4.3.cmml">T</mi></msubsup><mo id="S4.SS2.p1.31.m25.2.3.2.2.1b" xref="S4.SS2.p1.31.m25.2.3.2.2.1.cmml">⁢</mo><mrow id="S4.SS2.p1.31.m25.2.3.2.2.5.2" xref="S4.SS2.p1.31.m25.2.3.2.2.cmml"><mo id="S4.SS2.p1.31.m25.2.3.2.2.5.2.1" stretchy="false" xref="S4.SS2.p1.31.m25.2.3.2.2.cmml">(</mo><mi id="S4.SS2.p1.31.m25.2.2" xref="S4.SS2.p1.31.m25.2.2.cmml">τ</mi><mo id="S4.SS2.p1.31.m25.2.3.2.2.5.2.2" stretchy="false" xref="S4.SS2.p1.31.m25.2.3.2.2.cmml">)</mo></mrow><mo id="S4.SS2.p1.31.m25.2.3.2.2.1c" lspace="0em" xref="S4.SS2.p1.31.m25.2.3.2.2.1.cmml">⁢</mo><mrow id="S4.SS2.p1.31.m25.2.3.2.2.6" xref="S4.SS2.p1.31.m25.2.3.2.2.6.cmml"><mo id="S4.SS2.p1.31.m25.2.3.2.2.6.1" rspace="0em" xref="S4.SS2.p1.31.m25.2.3.2.2.6.1.cmml">𝑑</mo><mi id="S4.SS2.p1.31.m25.2.3.2.2.6.2" xref="S4.SS2.p1.31.m25.2.3.2.2.6.2.cmml">τ</mi></mrow></mrow></mrow><mo id="S4.SS2.p1.31.m25.2.3.1" xref="S4.SS2.p1.31.m25.2.3.1.cmml">≥</mo><mrow id="S4.SS2.p1.31.m25.2.3.3" xref="S4.SS2.p1.31.m25.2.3.3.cmml"><mi id="S4.SS2.p1.31.m25.2.3.3.2" xref="S4.SS2.p1.31.m25.2.3.3.2.cmml">σ</mi><mo id="S4.SS2.p1.31.m25.2.3.3.1" xref="S4.SS2.p1.31.m25.2.3.3.1.cmml">⁢</mo><mi id="S4.SS2.p1.31.m25.2.3.3.3" xref="S4.SS2.p1.31.m25.2.3.3.3.cmml">I</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.31.m25.2b"><apply id="S4.SS2.p1.31.m25.2.3.cmml" xref="S4.SS2.p1.31.m25.2.3"><geq id="S4.SS2.p1.31.m25.2.3.1.cmml" xref="S4.SS2.p1.31.m25.2.3.1"></geq><apply id="S4.SS2.p1.31.m25.2.3.2.cmml" xref="S4.SS2.p1.31.m25.2.3.2"><apply id="S4.SS2.p1.31.m25.2.3.2.1.cmml" xref="S4.SS2.p1.31.m25.2.3.2.1"><csymbol cd="ambiguous" id="S4.SS2.p1.31.m25.2.3.2.1.1.cmml" xref="S4.SS2.p1.31.m25.2.3.2.1">superscript</csymbol><apply id="S4.SS2.p1.31.m25.2.3.2.1.2.cmml" xref="S4.SS2.p1.31.m25.2.3.2.1"><csymbol cd="ambiguous" id="S4.SS2.p1.31.m25.2.3.2.1.2.1.cmml" xref="S4.SS2.p1.31.m25.2.3.2.1">subscript</csymbol><int id="S4.SS2.p1.31.m25.2.3.2.1.2.2.cmml" xref="S4.SS2.p1.31.m25.2.3.2.1.2.2"></int><apply id="S4.SS2.p1.31.m25.2.3.2.1.2.3.cmml" xref="S4.SS2.p1.31.m25.2.3.2.1.2.3"><minus id="S4.SS2.p1.31.m25.2.3.2.1.2.3.1.cmml" xref="S4.SS2.p1.31.m25.2.3.2.1.2.3.1"></minus><ci id="S4.SS2.p1.31.m25.2.3.2.1.2.3.2.cmml" xref="S4.SS2.p1.31.m25.2.3.2.1.2.3.2">𝑡</ci><apply id="S4.SS2.p1.31.m25.2.3.2.1.2.3.3.cmml" xref="S4.SS2.p1.31.m25.2.3.2.1.2.3.3"><csymbol cd="ambiguous" id="S4.SS2.p1.31.m25.2.3.2.1.2.3.3.1.cmml" xref="S4.SS2.p1.31.m25.2.3.2.1.2.3.3">subscript</csymbol><ci id="S4.SS2.p1.31.m25.2.3.2.1.2.3.3.2.cmml" xref="S4.SS2.p1.31.m25.2.3.2.1.2.3.3.2">𝜏</ci><ci id="S4.SS2.p1.31.m25.2.3.2.1.2.3.3.3.cmml" xref="S4.SS2.p1.31.m25.2.3.2.1.2.3.3.3">d</ci></apply></apply></apply><ci id="S4.SS2.p1.31.m25.2.3.2.1.3.cmml" xref="S4.SS2.p1.31.m25.2.3.2.1.3">𝑡</ci></apply><apply id="S4.SS2.p1.31.m25.2.3.2.2.cmml" xref="S4.SS2.p1.31.m25.2.3.2.2"><times id="S4.SS2.p1.31.m25.2.3.2.2.1.cmml" xref="S4.SS2.p1.31.m25.2.3.2.2.1"></times><apply id="S4.SS2.p1.31.m25.2.3.2.2.2.cmml" xref="S4.SS2.p1.31.m25.2.3.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.p1.31.m25.2.3.2.2.2.1.cmml" xref="S4.SS2.p1.31.m25.2.3.2.2.2">subscript</csymbol><ci id="S4.SS2.p1.31.m25.2.3.2.2.2.2.cmml" xref="S4.SS2.p1.31.m25.2.3.2.2.2.2">Φ</ci><ci id="S4.SS2.p1.31.m25.2.3.2.2.2.3.cmml" xref="S4.SS2.p1.31.m25.2.3.2.2.2.3">sr</ci></apply><ci id="S4.SS2.p1.31.m25.1.1.cmml" xref="S4.SS2.p1.31.m25.1.1">𝜏</ci><apply id="S4.SS2.p1.31.m25.2.3.2.2.4.cmml" xref="S4.SS2.p1.31.m25.2.3.2.2.4"><csymbol cd="ambiguous" id="S4.SS2.p1.31.m25.2.3.2.2.4.1.cmml" xref="S4.SS2.p1.31.m25.2.3.2.2.4">superscript</csymbol><apply id="S4.SS2.p1.31.m25.2.3.2.2.4.2.cmml" xref="S4.SS2.p1.31.m25.2.3.2.2.4"><csymbol cd="ambiguous" id="S4.SS2.p1.31.m25.2.3.2.2.4.2.1.cmml" xref="S4.SS2.p1.31.m25.2.3.2.2.4">subscript</csymbol><ci id="S4.SS2.p1.31.m25.2.3.2.2.4.2.2.cmml" xref="S4.SS2.p1.31.m25.2.3.2.2.4.2.2">Φ</ci><ci id="S4.SS2.p1.31.m25.2.3.2.2.4.2.3.cmml" xref="S4.SS2.p1.31.m25.2.3.2.2.4.2.3">sr</ci></apply><ci id="S4.SS2.p1.31.m25.2.3.2.2.4.3.cmml" xref="S4.SS2.p1.31.m25.2.3.2.2.4.3">𝑇</ci></apply><ci id="S4.SS2.p1.31.m25.2.2.cmml" xref="S4.SS2.p1.31.m25.2.2">𝜏</ci><apply id="S4.SS2.p1.31.m25.2.3.2.2.6.cmml" xref="S4.SS2.p1.31.m25.2.3.2.2.6"><csymbol cd="latexml" id="S4.SS2.p1.31.m25.2.3.2.2.6.1.cmml" xref="S4.SS2.p1.31.m25.2.3.2.2.6.1">differential-d</csymbol><ci id="S4.SS2.p1.31.m25.2.3.2.2.6.2.cmml" xref="S4.SS2.p1.31.m25.2.3.2.2.6.2">𝜏</ci></apply></apply></apply><apply id="S4.SS2.p1.31.m25.2.3.3.cmml" xref="S4.SS2.p1.31.m25.2.3.3"><times id="S4.SS2.p1.31.m25.2.3.3.1.cmml" xref="S4.SS2.p1.31.m25.2.3.3.1"></times><ci id="S4.SS2.p1.31.m25.2.3.3.2.cmml" xref="S4.SS2.p1.31.m25.2.3.3.2">𝜎</ci><ci id="S4.SS2.p1.31.m25.2.3.3.3.cmml" xref="S4.SS2.p1.31.m25.2.3.3.3">𝐼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.31.m25.2c">\int_{t-\tau_{\rm d}}^{t}\Phi_{\rm sr}(\tau)\Phi_{\rm sr}^{T}(\tau)d\tau\geq\sigma I</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.31.m25.2d">∫ start_POSTSUBSCRIPT italic_t - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_sr end_POSTSUBSCRIPT ( italic_τ ) roman_Φ start_POSTSUBSCRIPT roman_sr end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_τ ) italic_d italic_τ ≥ italic_σ italic_I</annotation></semantics></math> if the regressor vectors <math alttext="{\bm{\phi}}_{1}(t)" class="ltx_Math" display="inline" id="S4.SS2.p1.32.m26.1"><semantics id="S4.SS2.p1.32.m26.1a"><mrow id="S4.SS2.p1.32.m26.1.2" xref="S4.SS2.p1.32.m26.1.2.cmml"><msub id="S4.SS2.p1.32.m26.1.2.2" xref="S4.SS2.p1.32.m26.1.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS2.p1.32.m26.1.2.2.2" mathvariant="bold-italic" xref="S4.SS2.p1.32.m26.1.2.2.2.cmml">ϕ</mi><mn id="S4.SS2.p1.32.m26.1.2.2.3" xref="S4.SS2.p1.32.m26.1.2.2.3.cmml">1</mn></msub><mo id="S4.SS2.p1.32.m26.1.2.1" xref="S4.SS2.p1.32.m26.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p1.32.m26.1.2.3.2" xref="S4.SS2.p1.32.m26.1.2.cmml"><mo id="S4.SS2.p1.32.m26.1.2.3.2.1" stretchy="false" xref="S4.SS2.p1.32.m26.1.2.cmml">(</mo><mi id="S4.SS2.p1.32.m26.1.1" xref="S4.SS2.p1.32.m26.1.1.cmml">t</mi><mo id="S4.SS2.p1.32.m26.1.2.3.2.2" stretchy="false" xref="S4.SS2.p1.32.m26.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.32.m26.1b"><apply id="S4.SS2.p1.32.m26.1.2.cmml" xref="S4.SS2.p1.32.m26.1.2"><times id="S4.SS2.p1.32.m26.1.2.1.cmml" xref="S4.SS2.p1.32.m26.1.2.1"></times><apply id="S4.SS2.p1.32.m26.1.2.2.cmml" xref="S4.SS2.p1.32.m26.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.p1.32.m26.1.2.2.1.cmml" xref="S4.SS2.p1.32.m26.1.2.2">subscript</csymbol><ci id="S4.SS2.p1.32.m26.1.2.2.2.cmml" xref="S4.SS2.p1.32.m26.1.2.2.2">bold-italic-ϕ</ci><cn id="S4.SS2.p1.32.m26.1.2.2.3.cmml" type="integer" xref="S4.SS2.p1.32.m26.1.2.2.3">1</cn></apply><ci id="S4.SS2.p1.32.m26.1.1.cmml" xref="S4.SS2.p1.32.m26.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.32.m26.1c">{\bm{\phi}}_{1}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.32.m26.1d">bold_italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math>, <math alttext="{\bm{\phi}}_{3}(t)" class="ltx_Math" display="inline" id="S4.SS2.p1.33.m27.1"><semantics id="S4.SS2.p1.33.m27.1a"><mrow id="S4.SS2.p1.33.m27.1.2" xref="S4.SS2.p1.33.m27.1.2.cmml"><msub id="S4.SS2.p1.33.m27.1.2.2" xref="S4.SS2.p1.33.m27.1.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS2.p1.33.m27.1.2.2.2" mathvariant="bold-italic" xref="S4.SS2.p1.33.m27.1.2.2.2.cmml">ϕ</mi><mn id="S4.SS2.p1.33.m27.1.2.2.3" xref="S4.SS2.p1.33.m27.1.2.2.3.cmml">3</mn></msub><mo id="S4.SS2.p1.33.m27.1.2.1" xref="S4.SS2.p1.33.m27.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p1.33.m27.1.2.3.2" xref="S4.SS2.p1.33.m27.1.2.cmml"><mo id="S4.SS2.p1.33.m27.1.2.3.2.1" stretchy="false" xref="S4.SS2.p1.33.m27.1.2.cmml">(</mo><mi id="S4.SS2.p1.33.m27.1.1" xref="S4.SS2.p1.33.m27.1.1.cmml">t</mi><mo id="S4.SS2.p1.33.m27.1.2.3.2.2" stretchy="false" xref="S4.SS2.p1.33.m27.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.33.m27.1b"><apply id="S4.SS2.p1.33.m27.1.2.cmml" xref="S4.SS2.p1.33.m27.1.2"><times id="S4.SS2.p1.33.m27.1.2.1.cmml" xref="S4.SS2.p1.33.m27.1.2.1"></times><apply id="S4.SS2.p1.33.m27.1.2.2.cmml" xref="S4.SS2.p1.33.m27.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.p1.33.m27.1.2.2.1.cmml" xref="S4.SS2.p1.33.m27.1.2.2">subscript</csymbol><ci id="S4.SS2.p1.33.m27.1.2.2.2.cmml" xref="S4.SS2.p1.33.m27.1.2.2.2">bold-italic-ϕ</ci><cn id="S4.SS2.p1.33.m27.1.2.2.3.cmml" type="integer" xref="S4.SS2.p1.33.m27.1.2.2.3">3</cn></apply><ci id="S4.SS2.p1.33.m27.1.1.cmml" xref="S4.SS2.p1.33.m27.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.33.m27.1c">{\bm{\phi}}_{3}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.33.m27.1d">bold_italic_ϕ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math>, <math alttext="\cdots" class="ltx_Math" display="inline" id="S4.SS2.p1.34.m28.1"><semantics id="S4.SS2.p1.34.m28.1a"><mi id="S4.SS2.p1.34.m28.1.1" mathvariant="normal" xref="S4.SS2.p1.34.m28.1.1.cmml">⋯</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.34.m28.1b"><ci id="S4.SS2.p1.34.m28.1.1.cmml" xref="S4.SS2.p1.34.m28.1.1">⋯</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.34.m28.1c">\cdots</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.34.m28.1d">⋯</annotation></semantics></math>, <math alttext="{\bm{\phi}}_{N}(t)" class="ltx_Math" display="inline" id="S4.SS2.p1.35.m29.1"><semantics id="S4.SS2.p1.35.m29.1a"><mrow id="S4.SS2.p1.35.m29.1.2" xref="S4.SS2.p1.35.m29.1.2.cmml"><msub id="S4.SS2.p1.35.m29.1.2.2" xref="S4.SS2.p1.35.m29.1.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS2.p1.35.m29.1.2.2.2" mathvariant="bold-italic" xref="S4.SS2.p1.35.m29.1.2.2.2.cmml">ϕ</mi><mi id="S4.SS2.p1.35.m29.1.2.2.3" xref="S4.SS2.p1.35.m29.1.2.2.3.cmml">N</mi></msub><mo id="S4.SS2.p1.35.m29.1.2.1" xref="S4.SS2.p1.35.m29.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p1.35.m29.1.2.3.2" xref="S4.SS2.p1.35.m29.1.2.cmml"><mo id="S4.SS2.p1.35.m29.1.2.3.2.1" stretchy="false" xref="S4.SS2.p1.35.m29.1.2.cmml">(</mo><mi id="S4.SS2.p1.35.m29.1.1" xref="S4.SS2.p1.35.m29.1.1.cmml">t</mi><mo id="S4.SS2.p1.35.m29.1.2.3.2.2" stretchy="false" xref="S4.SS2.p1.35.m29.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.35.m29.1b"><apply id="S4.SS2.p1.35.m29.1.2.cmml" xref="S4.SS2.p1.35.m29.1.2"><times id="S4.SS2.p1.35.m29.1.2.1.cmml" xref="S4.SS2.p1.35.m29.1.2.1"></times><apply id="S4.SS2.p1.35.m29.1.2.2.cmml" xref="S4.SS2.p1.35.m29.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.p1.35.m29.1.2.2.1.cmml" xref="S4.SS2.p1.35.m29.1.2.2">subscript</csymbol><ci id="S4.SS2.p1.35.m29.1.2.2.2.cmml" xref="S4.SS2.p1.35.m29.1.2.2.2">bold-italic-ϕ</ci><ci id="S4.SS2.p1.35.m29.1.2.2.3.cmml" xref="S4.SS2.p1.35.m29.1.2.2.3">𝑁</ci></apply><ci id="S4.SS2.p1.35.m29.1.1.cmml" xref="S4.SS2.p1.35.m29.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.35.m29.1c">{\bm{\phi}}_{N}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.35.m29.1d">bold_italic_ϕ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> are activated in [<math alttext="t-\tau_{\rm d}" class="ltx_Math" display="inline" id="S4.SS2.p1.36.m30.1"><semantics id="S4.SS2.p1.36.m30.1a"><mrow id="S4.SS2.p1.36.m30.1.1" xref="S4.SS2.p1.36.m30.1.1.cmml"><mi id="S4.SS2.p1.36.m30.1.1.2" xref="S4.SS2.p1.36.m30.1.1.2.cmml">t</mi><mo id="S4.SS2.p1.36.m30.1.1.1" xref="S4.SS2.p1.36.m30.1.1.1.cmml">−</mo><msub id="S4.SS2.p1.36.m30.1.1.3" xref="S4.SS2.p1.36.m30.1.1.3.cmml"><mi id="S4.SS2.p1.36.m30.1.1.3.2" xref="S4.SS2.p1.36.m30.1.1.3.2.cmml">τ</mi><mi id="S4.SS2.p1.36.m30.1.1.3.3" mathvariant="normal" xref="S4.SS2.p1.36.m30.1.1.3.3.cmml">d</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.36.m30.1b"><apply id="S4.SS2.p1.36.m30.1.1.cmml" xref="S4.SS2.p1.36.m30.1.1"><minus id="S4.SS2.p1.36.m30.1.1.1.cmml" xref="S4.SS2.p1.36.m30.1.1.1"></minus><ci id="S4.SS2.p1.36.m30.1.1.2.cmml" xref="S4.SS2.p1.36.m30.1.1.2">𝑡</ci><apply id="S4.SS2.p1.36.m30.1.1.3.cmml" xref="S4.SS2.p1.36.m30.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p1.36.m30.1.1.3.1.cmml" xref="S4.SS2.p1.36.m30.1.1.3">subscript</csymbol><ci id="S4.SS2.p1.36.m30.1.1.3.2.cmml" xref="S4.SS2.p1.36.m30.1.1.3.2">𝜏</ci><ci id="S4.SS2.p1.36.m30.1.1.3.3.cmml" xref="S4.SS2.p1.36.m30.1.1.3.3">d</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.36.m30.1c">t-\tau_{\rm d}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.36.m30.1d">italic_t - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="t" class="ltx_Math" display="inline" id="S4.SS2.p1.37.m31.1"><semantics id="S4.SS2.p1.37.m31.1a"><mi id="S4.SS2.p1.37.m31.1.1" xref="S4.SS2.p1.37.m31.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.37.m31.1b"><ci id="S4.SS2.p1.37.m31.1.1.cmml" xref="S4.SS2.p1.37.m31.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.37.m31.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.37.m31.1d">italic_t</annotation></semantics></math>].</p> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.20">In the proposed composite learning HOT (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>), the novel generalized prediction error <math alttext="\bm{\xi}(t)" class="ltx_Math" display="inline" id="S4.SS2.p2.1.m1.1"><semantics id="S4.SS2.p2.1.m1.1a"><mrow id="S4.SS2.p2.1.m1.1.2" xref="S4.SS2.p2.1.m1.1.2.cmml"><mi id="S4.SS2.p2.1.m1.1.2.2" xref="S4.SS2.p2.1.m1.1.2.2.cmml">𝝃</mi><mo id="S4.SS2.p2.1.m1.1.2.1" xref="S4.SS2.p2.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p2.1.m1.1.2.3.2" xref="S4.SS2.p2.1.m1.1.2.cmml"><mo id="S4.SS2.p2.1.m1.1.2.3.2.1" stretchy="false" xref="S4.SS2.p2.1.m1.1.2.cmml">(</mo><mi id="S4.SS2.p2.1.m1.1.1" xref="S4.SS2.p2.1.m1.1.1.cmml">t</mi><mo id="S4.SS2.p2.1.m1.1.2.3.2.2" stretchy="false" xref="S4.SS2.p2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.1.m1.1b"><apply id="S4.SS2.p2.1.m1.1.2.cmml" xref="S4.SS2.p2.1.m1.1.2"><times id="S4.SS2.p2.1.m1.1.2.1.cmml" xref="S4.SS2.p2.1.m1.1.2.1"></times><ci id="S4.SS2.p2.1.m1.1.2.2.cmml" xref="S4.SS2.p2.1.m1.1.2.2">𝝃</ci><ci id="S4.SS2.p2.1.m1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.1.m1.1c">\bm{\xi}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.1.m1.1d">bold_italic_ξ ( italic_t )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E20" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">20</span></a>) is updated by a newly designed algorithm termed staged exciting strength maximization, which enables online data memory to be achieved under different partial IE stages, whereas the online data memory of the classical composite learning can only be achieved under IE as its exciting strength is obtained directly by calculating the minimum singular value of the excitation matrix <math alttext="\Psi(t)" class="ltx_Math" display="inline" id="S4.SS2.p2.2.m2.1"><semantics id="S4.SS2.p2.2.m2.1a"><mrow id="S4.SS2.p2.2.m2.1.2" xref="S4.SS2.p2.2.m2.1.2.cmml"><mi id="S4.SS2.p2.2.m2.1.2.2" mathvariant="normal" xref="S4.SS2.p2.2.m2.1.2.2.cmml">Ψ</mi><mo id="S4.SS2.p2.2.m2.1.2.1" xref="S4.SS2.p2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p2.2.m2.1.2.3.2" xref="S4.SS2.p2.2.m2.1.2.cmml"><mo id="S4.SS2.p2.2.m2.1.2.3.2.1" stretchy="false" xref="S4.SS2.p2.2.m2.1.2.cmml">(</mo><mi id="S4.SS2.p2.2.m2.1.1" xref="S4.SS2.p2.2.m2.1.1.cmml">t</mi><mo id="S4.SS2.p2.2.m2.1.2.3.2.2" stretchy="false" xref="S4.SS2.p2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.2.m2.1b"><apply id="S4.SS2.p2.2.m2.1.2.cmml" xref="S4.SS2.p2.2.m2.1.2"><times id="S4.SS2.p2.2.m2.1.2.1.cmml" xref="S4.SS2.p2.2.m2.1.2.1"></times><ci id="S4.SS2.p2.2.m2.1.2.2.cmml" xref="S4.SS2.p2.2.m2.1.2.2">Ψ</ci><ci id="S4.SS2.p2.2.m2.1.1.cmml" xref="S4.SS2.p2.2.m2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.2.m2.1c">\Psi(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.2.m2.1d">roman_Ψ ( italic_t )</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib20" title="">20</a>]</cite>, where IE requires all regressor channels to be activated in an uncorrelated manner at a time window <math alttext="[T_{\rm e}-\tau_{\rm d},T_{\rm e}]" class="ltx_Math" display="inline" id="S4.SS2.p2.3.m3.2"><semantics id="S4.SS2.p2.3.m3.2a"><mrow id="S4.SS2.p2.3.m3.2.2.2" xref="S4.SS2.p2.3.m3.2.2.3.cmml"><mo id="S4.SS2.p2.3.m3.2.2.2.3" stretchy="false" xref="S4.SS2.p2.3.m3.2.2.3.cmml">[</mo><mrow id="S4.SS2.p2.3.m3.1.1.1.1" xref="S4.SS2.p2.3.m3.1.1.1.1.cmml"><msub id="S4.SS2.p2.3.m3.1.1.1.1.2" xref="S4.SS2.p2.3.m3.1.1.1.1.2.cmml"><mi id="S4.SS2.p2.3.m3.1.1.1.1.2.2" xref="S4.SS2.p2.3.m3.1.1.1.1.2.2.cmml">T</mi><mi id="S4.SS2.p2.3.m3.1.1.1.1.2.3" mathvariant="normal" xref="S4.SS2.p2.3.m3.1.1.1.1.2.3.cmml">e</mi></msub><mo id="S4.SS2.p2.3.m3.1.1.1.1.1" xref="S4.SS2.p2.3.m3.1.1.1.1.1.cmml">−</mo><msub id="S4.SS2.p2.3.m3.1.1.1.1.3" xref="S4.SS2.p2.3.m3.1.1.1.1.3.cmml"><mi id="S4.SS2.p2.3.m3.1.1.1.1.3.2" xref="S4.SS2.p2.3.m3.1.1.1.1.3.2.cmml">τ</mi><mi id="S4.SS2.p2.3.m3.1.1.1.1.3.3" mathvariant="normal" xref="S4.SS2.p2.3.m3.1.1.1.1.3.3.cmml">d</mi></msub></mrow><mo id="S4.SS2.p2.3.m3.2.2.2.4" xref="S4.SS2.p2.3.m3.2.2.3.cmml">,</mo><msub id="S4.SS2.p2.3.m3.2.2.2.2" xref="S4.SS2.p2.3.m3.2.2.2.2.cmml"><mi id="S4.SS2.p2.3.m3.2.2.2.2.2" xref="S4.SS2.p2.3.m3.2.2.2.2.2.cmml">T</mi><mi id="S4.SS2.p2.3.m3.2.2.2.2.3" mathvariant="normal" xref="S4.SS2.p2.3.m3.2.2.2.2.3.cmml">e</mi></msub><mo id="S4.SS2.p2.3.m3.2.2.2.5" stretchy="false" xref="S4.SS2.p2.3.m3.2.2.3.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.3.m3.2b"><interval closure="closed" id="S4.SS2.p2.3.m3.2.2.3.cmml" xref="S4.SS2.p2.3.m3.2.2.2"><apply id="S4.SS2.p2.3.m3.1.1.1.1.cmml" xref="S4.SS2.p2.3.m3.1.1.1.1"><minus id="S4.SS2.p2.3.m3.1.1.1.1.1.cmml" xref="S4.SS2.p2.3.m3.1.1.1.1.1"></minus><apply id="S4.SS2.p2.3.m3.1.1.1.1.2.cmml" xref="S4.SS2.p2.3.m3.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.3.m3.1.1.1.1.2.1.cmml" xref="S4.SS2.p2.3.m3.1.1.1.1.2">subscript</csymbol><ci id="S4.SS2.p2.3.m3.1.1.1.1.2.2.cmml" xref="S4.SS2.p2.3.m3.1.1.1.1.2.2">𝑇</ci><ci id="S4.SS2.p2.3.m3.1.1.1.1.2.3.cmml" xref="S4.SS2.p2.3.m3.1.1.1.1.2.3">e</ci></apply><apply id="S4.SS2.p2.3.m3.1.1.1.1.3.cmml" xref="S4.SS2.p2.3.m3.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p2.3.m3.1.1.1.1.3.1.cmml" xref="S4.SS2.p2.3.m3.1.1.1.1.3">subscript</csymbol><ci id="S4.SS2.p2.3.m3.1.1.1.1.3.2.cmml" xref="S4.SS2.p2.3.m3.1.1.1.1.3.2">𝜏</ci><ci id="S4.SS2.p2.3.m3.1.1.1.1.3.3.cmml" xref="S4.SS2.p2.3.m3.1.1.1.1.3.3">d</ci></apply></apply><apply id="S4.SS2.p2.3.m3.2.2.2.2.cmml" xref="S4.SS2.p2.3.m3.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.p2.3.m3.2.2.2.2.1.cmml" xref="S4.SS2.p2.3.m3.2.2.2.2">subscript</csymbol><ci id="S4.SS2.p2.3.m3.2.2.2.2.2.cmml" xref="S4.SS2.p2.3.m3.2.2.2.2.2">𝑇</ci><ci id="S4.SS2.p2.3.m3.2.2.2.2.3.cmml" xref="S4.SS2.p2.3.m3.2.2.2.2.3">e</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.3.m3.2c">[T_{\rm e}-\tau_{\rm d},T_{\rm e}]</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.3.m3.2d">[ italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ]</annotation></semantics></math>, but partial IE only requires partial regressor channels to be activated at a time window <math alttext="[T_{\rm a}-\tau_{\rm d},T_{\rm a}]" class="ltx_Math" display="inline" id="S4.SS2.p2.4.m4.2"><semantics id="S4.SS2.p2.4.m4.2a"><mrow id="S4.SS2.p2.4.m4.2.2.2" xref="S4.SS2.p2.4.m4.2.2.3.cmml"><mo id="S4.SS2.p2.4.m4.2.2.2.3" stretchy="false" xref="S4.SS2.p2.4.m4.2.2.3.cmml">[</mo><mrow id="S4.SS2.p2.4.m4.1.1.1.1" xref="S4.SS2.p2.4.m4.1.1.1.1.cmml"><msub id="S4.SS2.p2.4.m4.1.1.1.1.2" xref="S4.SS2.p2.4.m4.1.1.1.1.2.cmml"><mi id="S4.SS2.p2.4.m4.1.1.1.1.2.2" xref="S4.SS2.p2.4.m4.1.1.1.1.2.2.cmml">T</mi><mi id="S4.SS2.p2.4.m4.1.1.1.1.2.3" mathvariant="normal" xref="S4.SS2.p2.4.m4.1.1.1.1.2.3.cmml">a</mi></msub><mo id="S4.SS2.p2.4.m4.1.1.1.1.1" xref="S4.SS2.p2.4.m4.1.1.1.1.1.cmml">−</mo><msub id="S4.SS2.p2.4.m4.1.1.1.1.3" xref="S4.SS2.p2.4.m4.1.1.1.1.3.cmml"><mi id="S4.SS2.p2.4.m4.1.1.1.1.3.2" xref="S4.SS2.p2.4.m4.1.1.1.1.3.2.cmml">τ</mi><mi id="S4.SS2.p2.4.m4.1.1.1.1.3.3" mathvariant="normal" xref="S4.SS2.p2.4.m4.1.1.1.1.3.3.cmml">d</mi></msub></mrow><mo id="S4.SS2.p2.4.m4.2.2.2.4" xref="S4.SS2.p2.4.m4.2.2.3.cmml">,</mo><msub id="S4.SS2.p2.4.m4.2.2.2.2" xref="S4.SS2.p2.4.m4.2.2.2.2.cmml"><mi id="S4.SS2.p2.4.m4.2.2.2.2.2" xref="S4.SS2.p2.4.m4.2.2.2.2.2.cmml">T</mi><mi id="S4.SS2.p2.4.m4.2.2.2.2.3" mathvariant="normal" xref="S4.SS2.p2.4.m4.2.2.2.2.3.cmml">a</mi></msub><mo id="S4.SS2.p2.4.m4.2.2.2.5" stretchy="false" xref="S4.SS2.p2.4.m4.2.2.3.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.4.m4.2b"><interval closure="closed" id="S4.SS2.p2.4.m4.2.2.3.cmml" xref="S4.SS2.p2.4.m4.2.2.2"><apply id="S4.SS2.p2.4.m4.1.1.1.1.cmml" xref="S4.SS2.p2.4.m4.1.1.1.1"><minus id="S4.SS2.p2.4.m4.1.1.1.1.1.cmml" xref="S4.SS2.p2.4.m4.1.1.1.1.1"></minus><apply id="S4.SS2.p2.4.m4.1.1.1.1.2.cmml" xref="S4.SS2.p2.4.m4.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.4.m4.1.1.1.1.2.1.cmml" xref="S4.SS2.p2.4.m4.1.1.1.1.2">subscript</csymbol><ci id="S4.SS2.p2.4.m4.1.1.1.1.2.2.cmml" xref="S4.SS2.p2.4.m4.1.1.1.1.2.2">𝑇</ci><ci id="S4.SS2.p2.4.m4.1.1.1.1.2.3.cmml" xref="S4.SS2.p2.4.m4.1.1.1.1.2.3">a</ci></apply><apply id="S4.SS2.p2.4.m4.1.1.1.1.3.cmml" xref="S4.SS2.p2.4.m4.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p2.4.m4.1.1.1.1.3.1.cmml" xref="S4.SS2.p2.4.m4.1.1.1.1.3">subscript</csymbol><ci id="S4.SS2.p2.4.m4.1.1.1.1.3.2.cmml" xref="S4.SS2.p2.4.m4.1.1.1.1.3.2">𝜏</ci><ci id="S4.SS2.p2.4.m4.1.1.1.1.3.3.cmml" xref="S4.SS2.p2.4.m4.1.1.1.1.3.3">d</ci></apply></apply><apply id="S4.SS2.p2.4.m4.2.2.2.2.cmml" xref="S4.SS2.p2.4.m4.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.p2.4.m4.2.2.2.2.1.cmml" xref="S4.SS2.p2.4.m4.2.2.2.2">subscript</csymbol><ci id="S4.SS2.p2.4.m4.2.2.2.2.2.cmml" xref="S4.SS2.p2.4.m4.2.2.2.2.2">𝑇</ci><ci id="S4.SS2.p2.4.m4.2.2.2.2.3.cmml" xref="S4.SS2.p2.4.m4.2.2.2.2.3">a</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.4.m4.2c">[T_{\rm a}-\tau_{\rm d},T_{\rm a}]</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.4.m4.2d">[ italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT ]</annotation></semantics></math>. If only the partial IE condition holds, there exists at least one inactive channel <math alttext="\bm{\phi}_{{\rm s},j}" class="ltx_Math" display="inline" id="S4.SS2.p2.5.m5.2"><semantics id="S4.SS2.p2.5.m5.2a"><msub id="S4.SS2.p2.5.m5.2.3" xref="S4.SS2.p2.5.m5.2.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS2.p2.5.m5.2.3.2" mathvariant="bold-italic" xref="S4.SS2.p2.5.m5.2.3.2.cmml">ϕ</mi><mrow id="S4.SS2.p2.5.m5.2.2.2.4" xref="S4.SS2.p2.5.m5.2.2.2.3.cmml"><mi id="S4.SS2.p2.5.m5.1.1.1.1" mathvariant="normal" xref="S4.SS2.p2.5.m5.1.1.1.1.cmml">s</mi><mo id="S4.SS2.p2.5.m5.2.2.2.4.1" xref="S4.SS2.p2.5.m5.2.2.2.3.cmml">,</mo><mi id="S4.SS2.p2.5.m5.2.2.2.2" xref="S4.SS2.p2.5.m5.2.2.2.2.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.5.m5.2b"><apply id="S4.SS2.p2.5.m5.2.3.cmml" xref="S4.SS2.p2.5.m5.2.3"><csymbol cd="ambiguous" id="S4.SS2.p2.5.m5.2.3.1.cmml" xref="S4.SS2.p2.5.m5.2.3">subscript</csymbol><ci id="S4.SS2.p2.5.m5.2.3.2.cmml" xref="S4.SS2.p2.5.m5.2.3.2">bold-italic-ϕ</ci><list id="S4.SS2.p2.5.m5.2.2.2.3.cmml" xref="S4.SS2.p2.5.m5.2.2.2.4"><ci id="S4.SS2.p2.5.m5.1.1.1.1.cmml" xref="S4.SS2.p2.5.m5.1.1.1.1">s</ci><ci id="S4.SS2.p2.5.m5.2.2.2.2.cmml" xref="S4.SS2.p2.5.m5.2.2.2.2">𝑗</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.5.m5.2c">\bm{\phi}_{{\rm s},j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.5.m5.2d">bold_italic_ϕ start_POSTSUBSCRIPT roman_s , italic_j end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="j\in\{1,2,\cdots,N\}" class="ltx_Math" display="inline" id="S4.SS2.p2.6.m6.4"><semantics id="S4.SS2.p2.6.m6.4a"><mrow id="S4.SS2.p2.6.m6.4.5" xref="S4.SS2.p2.6.m6.4.5.cmml"><mi id="S4.SS2.p2.6.m6.4.5.2" xref="S4.SS2.p2.6.m6.4.5.2.cmml">j</mi><mo id="S4.SS2.p2.6.m6.4.5.1" xref="S4.SS2.p2.6.m6.4.5.1.cmml">∈</mo><mrow id="S4.SS2.p2.6.m6.4.5.3.2" xref="S4.SS2.p2.6.m6.4.5.3.1.cmml"><mo id="S4.SS2.p2.6.m6.4.5.3.2.1" stretchy="false" xref="S4.SS2.p2.6.m6.4.5.3.1.cmml">{</mo><mn id="S4.SS2.p2.6.m6.1.1" xref="S4.SS2.p2.6.m6.1.1.cmml">1</mn><mo id="S4.SS2.p2.6.m6.4.5.3.2.2" xref="S4.SS2.p2.6.m6.4.5.3.1.cmml">,</mo><mn id="S4.SS2.p2.6.m6.2.2" xref="S4.SS2.p2.6.m6.2.2.cmml">2</mn><mo id="S4.SS2.p2.6.m6.4.5.3.2.3" xref="S4.SS2.p2.6.m6.4.5.3.1.cmml">,</mo><mi id="S4.SS2.p2.6.m6.3.3" mathvariant="normal" xref="S4.SS2.p2.6.m6.3.3.cmml">⋯</mi><mo id="S4.SS2.p2.6.m6.4.5.3.2.4" xref="S4.SS2.p2.6.m6.4.5.3.1.cmml">,</mo><mi id="S4.SS2.p2.6.m6.4.4" xref="S4.SS2.p2.6.m6.4.4.cmml">N</mi><mo id="S4.SS2.p2.6.m6.4.5.3.2.5" stretchy="false" xref="S4.SS2.p2.6.m6.4.5.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.6.m6.4b"><apply id="S4.SS2.p2.6.m6.4.5.cmml" xref="S4.SS2.p2.6.m6.4.5"><in id="S4.SS2.p2.6.m6.4.5.1.cmml" xref="S4.SS2.p2.6.m6.4.5.1"></in><ci id="S4.SS2.p2.6.m6.4.5.2.cmml" xref="S4.SS2.p2.6.m6.4.5.2">𝑗</ci><set id="S4.SS2.p2.6.m6.4.5.3.1.cmml" xref="S4.SS2.p2.6.m6.4.5.3.2"><cn id="S4.SS2.p2.6.m6.1.1.cmml" type="integer" xref="S4.SS2.p2.6.m6.1.1">1</cn><cn id="S4.SS2.p2.6.m6.2.2.cmml" type="integer" xref="S4.SS2.p2.6.m6.2.2">2</cn><ci id="S4.SS2.p2.6.m6.3.3.cmml" xref="S4.SS2.p2.6.m6.3.3">⋯</ci><ci id="S4.SS2.p2.6.m6.4.4.cmml" xref="S4.SS2.p2.6.m6.4.4">𝑁</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.6.m6.4c">j\in\{1,2,\cdots,N\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.6.m6.4d">italic_j ∈ { 1 , 2 , ⋯ , italic_N }</annotation></semantics></math>, i.e., <math alttext="\|\bm{\phi}_{{\rm s},j}\|\equiv 0" class="ltx_Math" display="inline" id="S4.SS2.p2.7.m7.3"><semantics id="S4.SS2.p2.7.m7.3a"><mrow id="S4.SS2.p2.7.m7.3.3" xref="S4.SS2.p2.7.m7.3.3.cmml"><mrow id="S4.SS2.p2.7.m7.3.3.1.1" xref="S4.SS2.p2.7.m7.3.3.1.2.cmml"><mo id="S4.SS2.p2.7.m7.3.3.1.1.2" stretchy="false" xref="S4.SS2.p2.7.m7.3.3.1.2.1.cmml">‖</mo><msub id="S4.SS2.p2.7.m7.3.3.1.1.1" xref="S4.SS2.p2.7.m7.3.3.1.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS2.p2.7.m7.3.3.1.1.1.2" mathvariant="bold-italic" xref="S4.SS2.p2.7.m7.3.3.1.1.1.2.cmml">ϕ</mi><mrow id="S4.SS2.p2.7.m7.2.2.2.4" xref="S4.SS2.p2.7.m7.2.2.2.3.cmml"><mi id="S4.SS2.p2.7.m7.1.1.1.1" mathvariant="normal" xref="S4.SS2.p2.7.m7.1.1.1.1.cmml">s</mi><mo id="S4.SS2.p2.7.m7.2.2.2.4.1" xref="S4.SS2.p2.7.m7.2.2.2.3.cmml">,</mo><mi id="S4.SS2.p2.7.m7.2.2.2.2" xref="S4.SS2.p2.7.m7.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S4.SS2.p2.7.m7.3.3.1.1.3" stretchy="false" xref="S4.SS2.p2.7.m7.3.3.1.2.1.cmml">‖</mo></mrow><mo id="S4.SS2.p2.7.m7.3.3.2" xref="S4.SS2.p2.7.m7.3.3.2.cmml">≡</mo><mn id="S4.SS2.p2.7.m7.3.3.3" xref="S4.SS2.p2.7.m7.3.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.7.m7.3b"><apply id="S4.SS2.p2.7.m7.3.3.cmml" xref="S4.SS2.p2.7.m7.3.3"><equivalent id="S4.SS2.p2.7.m7.3.3.2.cmml" xref="S4.SS2.p2.7.m7.3.3.2"></equivalent><apply id="S4.SS2.p2.7.m7.3.3.1.2.cmml" xref="S4.SS2.p2.7.m7.3.3.1.1"><csymbol cd="latexml" id="S4.SS2.p2.7.m7.3.3.1.2.1.cmml" xref="S4.SS2.p2.7.m7.3.3.1.1.2">norm</csymbol><apply id="S4.SS2.p2.7.m7.3.3.1.1.1.cmml" xref="S4.SS2.p2.7.m7.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.7.m7.3.3.1.1.1.1.cmml" xref="S4.SS2.p2.7.m7.3.3.1.1.1">subscript</csymbol><ci id="S4.SS2.p2.7.m7.3.3.1.1.1.2.cmml" xref="S4.SS2.p2.7.m7.3.3.1.1.1.2">bold-italic-ϕ</ci><list id="S4.SS2.p2.7.m7.2.2.2.3.cmml" xref="S4.SS2.p2.7.m7.2.2.2.4"><ci id="S4.SS2.p2.7.m7.1.1.1.1.cmml" xref="S4.SS2.p2.7.m7.1.1.1.1">s</ci><ci id="S4.SS2.p2.7.m7.2.2.2.2.cmml" xref="S4.SS2.p2.7.m7.2.2.2.2">𝑗</ci></list></apply></apply><cn id="S4.SS2.p2.7.m7.3.3.3.cmml" type="integer" xref="S4.SS2.p2.7.m7.3.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.7.m7.3c">\|\bm{\phi}_{{\rm s},j}\|\equiv 0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.7.m7.3d">∥ bold_italic_ϕ start_POSTSUBSCRIPT roman_s , italic_j end_POSTSUBSCRIPT ∥ ≡ 0</annotation></semantics></math>, and all elements in the <math alttext="j" class="ltx_Math" display="inline" id="S4.SS2.p2.8.m8.1"><semantics id="S4.SS2.p2.8.m8.1a"><mi id="S4.SS2.p2.8.m8.1.1" xref="S4.SS2.p2.8.m8.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.8.m8.1b"><ci id="S4.SS2.p2.8.m8.1.1.cmml" xref="S4.SS2.p2.8.m8.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.8.m8.1c">j</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.8.m8.1d">italic_j</annotation></semantics></math>th row and the <math alttext="j" class="ltx_Math" display="inline" id="S4.SS2.p2.9.m9.1"><semantics id="S4.SS2.p2.9.m9.1a"><mi id="S4.SS2.p2.9.m9.1.1" xref="S4.SS2.p2.9.m9.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.9.m9.1b"><ci id="S4.SS2.p2.9.m9.1.1.cmml" xref="S4.SS2.p2.9.m9.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.9.m9.1c">j</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.9.m9.1d">italic_j</annotation></semantics></math>th column of <math alttext="\Psi(t)" class="ltx_Math" display="inline" id="S4.SS2.p2.10.m10.1"><semantics id="S4.SS2.p2.10.m10.1a"><mrow id="S4.SS2.p2.10.m10.1.2" xref="S4.SS2.p2.10.m10.1.2.cmml"><mi id="S4.SS2.p2.10.m10.1.2.2" mathvariant="normal" xref="S4.SS2.p2.10.m10.1.2.2.cmml">Ψ</mi><mo id="S4.SS2.p2.10.m10.1.2.1" xref="S4.SS2.p2.10.m10.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p2.10.m10.1.2.3.2" xref="S4.SS2.p2.10.m10.1.2.cmml"><mo id="S4.SS2.p2.10.m10.1.2.3.2.1" stretchy="false" xref="S4.SS2.p2.10.m10.1.2.cmml">(</mo><mi id="S4.SS2.p2.10.m10.1.1" xref="S4.SS2.p2.10.m10.1.1.cmml">t</mi><mo id="S4.SS2.p2.10.m10.1.2.3.2.2" stretchy="false" xref="S4.SS2.p2.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.10.m10.1b"><apply id="S4.SS2.p2.10.m10.1.2.cmml" xref="S4.SS2.p2.10.m10.1.2"><times id="S4.SS2.p2.10.m10.1.2.1.cmml" xref="S4.SS2.p2.10.m10.1.2.1"></times><ci id="S4.SS2.p2.10.m10.1.2.2.cmml" xref="S4.SS2.p2.10.m10.1.2.2">Ψ</ci><ci id="S4.SS2.p2.10.m10.1.1.cmml" xref="S4.SS2.p2.10.m10.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.10.m10.1c">\Psi(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.10.m10.1d">roman_Ψ ( italic_t )</annotation></semantics></math> are 0. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.F2" title="Figure 2 ‣ IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">2</span></a> illustrates the relationship between the channels <math alttext="\bm{\phi}_{{\rm s},j}(t)" class="ltx_Math" display="inline" id="S4.SS2.p2.11.m11.3"><semantics id="S4.SS2.p2.11.m11.3a"><mrow id="S4.SS2.p2.11.m11.3.4" xref="S4.SS2.p2.11.m11.3.4.cmml"><msub id="S4.SS2.p2.11.m11.3.4.2" xref="S4.SS2.p2.11.m11.3.4.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS2.p2.11.m11.3.4.2.2" mathvariant="bold-italic" xref="S4.SS2.p2.11.m11.3.4.2.2.cmml">ϕ</mi><mrow id="S4.SS2.p2.11.m11.2.2.2.4" xref="S4.SS2.p2.11.m11.2.2.2.3.cmml"><mi id="S4.SS2.p2.11.m11.1.1.1.1" mathvariant="normal" xref="S4.SS2.p2.11.m11.1.1.1.1.cmml">s</mi><mo id="S4.SS2.p2.11.m11.2.2.2.4.1" xref="S4.SS2.p2.11.m11.2.2.2.3.cmml">,</mo><mi id="S4.SS2.p2.11.m11.2.2.2.2" xref="S4.SS2.p2.11.m11.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S4.SS2.p2.11.m11.3.4.1" xref="S4.SS2.p2.11.m11.3.4.1.cmml">⁢</mo><mrow id="S4.SS2.p2.11.m11.3.4.3.2" xref="S4.SS2.p2.11.m11.3.4.cmml"><mo id="S4.SS2.p2.11.m11.3.4.3.2.1" stretchy="false" xref="S4.SS2.p2.11.m11.3.4.cmml">(</mo><mi id="S4.SS2.p2.11.m11.3.3" xref="S4.SS2.p2.11.m11.3.3.cmml">t</mi><mo id="S4.SS2.p2.11.m11.3.4.3.2.2" stretchy="false" xref="S4.SS2.p2.11.m11.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.11.m11.3b"><apply id="S4.SS2.p2.11.m11.3.4.cmml" xref="S4.SS2.p2.11.m11.3.4"><times id="S4.SS2.p2.11.m11.3.4.1.cmml" xref="S4.SS2.p2.11.m11.3.4.1"></times><apply id="S4.SS2.p2.11.m11.3.4.2.cmml" xref="S4.SS2.p2.11.m11.3.4.2"><csymbol cd="ambiguous" id="S4.SS2.p2.11.m11.3.4.2.1.cmml" xref="S4.SS2.p2.11.m11.3.4.2">subscript</csymbol><ci id="S4.SS2.p2.11.m11.3.4.2.2.cmml" xref="S4.SS2.p2.11.m11.3.4.2.2">bold-italic-ϕ</ci><list id="S4.SS2.p2.11.m11.2.2.2.3.cmml" xref="S4.SS2.p2.11.m11.2.2.2.4"><ci id="S4.SS2.p2.11.m11.1.1.1.1.cmml" xref="S4.SS2.p2.11.m11.1.1.1.1">s</ci><ci id="S4.SS2.p2.11.m11.2.2.2.2.cmml" xref="S4.SS2.p2.11.m11.2.2.2.2">𝑗</ci></list></apply><ci id="S4.SS2.p2.11.m11.3.3.cmml" xref="S4.SS2.p2.11.m11.3.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.11.m11.3c">\bm{\phi}_{{\rm s},j}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.11.m11.3d">bold_italic_ϕ start_POSTSUBSCRIPT roman_s , italic_j end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> and the excitation matrix <math alttext="\Psi(t)" class="ltx_Math" display="inline" id="S4.SS2.p2.12.m12.1"><semantics id="S4.SS2.p2.12.m12.1a"><mrow id="S4.SS2.p2.12.m12.1.2" xref="S4.SS2.p2.12.m12.1.2.cmml"><mi id="S4.SS2.p2.12.m12.1.2.2" mathvariant="normal" xref="S4.SS2.p2.12.m12.1.2.2.cmml">Ψ</mi><mo id="S4.SS2.p2.12.m12.1.2.1" xref="S4.SS2.p2.12.m12.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p2.12.m12.1.2.3.2" xref="S4.SS2.p2.12.m12.1.2.cmml"><mo id="S4.SS2.p2.12.m12.1.2.3.2.1" stretchy="false" xref="S4.SS2.p2.12.m12.1.2.cmml">(</mo><mi id="S4.SS2.p2.12.m12.1.1" xref="S4.SS2.p2.12.m12.1.1.cmml">t</mi><mo id="S4.SS2.p2.12.m12.1.2.3.2.2" stretchy="false" xref="S4.SS2.p2.12.m12.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.12.m12.1b"><apply id="S4.SS2.p2.12.m12.1.2.cmml" xref="S4.SS2.p2.12.m12.1.2"><times id="S4.SS2.p2.12.m12.1.2.1.cmml" xref="S4.SS2.p2.12.m12.1.2.1"></times><ci id="S4.SS2.p2.12.m12.1.2.2.cmml" xref="S4.SS2.p2.12.m12.1.2.2">Ψ</ci><ci id="S4.SS2.p2.12.m12.1.1.cmml" xref="S4.SS2.p2.12.m12.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.12.m12.1c">\Psi(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.12.m12.1d">roman_Ψ ( italic_t )</annotation></semantics></math> with <math alttext="N=7" class="ltx_Math" display="inline" id="S4.SS2.p2.13.m13.1"><semantics id="S4.SS2.p2.13.m13.1a"><mrow id="S4.SS2.p2.13.m13.1.1" xref="S4.SS2.p2.13.m13.1.1.cmml"><mi id="S4.SS2.p2.13.m13.1.1.2" xref="S4.SS2.p2.13.m13.1.1.2.cmml">N</mi><mo id="S4.SS2.p2.13.m13.1.1.1" xref="S4.SS2.p2.13.m13.1.1.1.cmml">=</mo><mn id="S4.SS2.p2.13.m13.1.1.3" xref="S4.SS2.p2.13.m13.1.1.3.cmml">7</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.13.m13.1b"><apply id="S4.SS2.p2.13.m13.1.1.cmml" xref="S4.SS2.p2.13.m13.1.1"><eq id="S4.SS2.p2.13.m13.1.1.1.cmml" xref="S4.SS2.p2.13.m13.1.1.1"></eq><ci id="S4.SS2.p2.13.m13.1.1.2.cmml" xref="S4.SS2.p2.13.m13.1.1.2">𝑁</ci><cn id="S4.SS2.p2.13.m13.1.1.3.cmml" type="integer" xref="S4.SS2.p2.13.m13.1.1.3">7</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.13.m13.1c">N=7</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.13.m13.1d">italic_N = 7</annotation></semantics></math>. In this case, the full exciting strength <math alttext="\sigma_{\min}(\Psi(t))\equiv 0" class="ltx_Math" display="inline" id="S4.SS2.p2.14.m14.2"><semantics id="S4.SS2.p2.14.m14.2a"><mrow id="S4.SS2.p2.14.m14.2.2" xref="S4.SS2.p2.14.m14.2.2.cmml"><mrow id="S4.SS2.p2.14.m14.2.2.1" xref="S4.SS2.p2.14.m14.2.2.1.cmml"><msub id="S4.SS2.p2.14.m14.2.2.1.3" xref="S4.SS2.p2.14.m14.2.2.1.3.cmml"><mi id="S4.SS2.p2.14.m14.2.2.1.3.2" xref="S4.SS2.p2.14.m14.2.2.1.3.2.cmml">σ</mi><mi id="S4.SS2.p2.14.m14.2.2.1.3.3" xref="S4.SS2.p2.14.m14.2.2.1.3.3.cmml">min</mi></msub><mo id="S4.SS2.p2.14.m14.2.2.1.2" xref="S4.SS2.p2.14.m14.2.2.1.2.cmml">⁢</mo><mrow id="S4.SS2.p2.14.m14.2.2.1.1.1" xref="S4.SS2.p2.14.m14.2.2.1.1.1.1.cmml"><mo id="S4.SS2.p2.14.m14.2.2.1.1.1.2" stretchy="false" xref="S4.SS2.p2.14.m14.2.2.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.p2.14.m14.2.2.1.1.1.1" xref="S4.SS2.p2.14.m14.2.2.1.1.1.1.cmml"><mi id="S4.SS2.p2.14.m14.2.2.1.1.1.1.2" mathvariant="normal" xref="S4.SS2.p2.14.m14.2.2.1.1.1.1.2.cmml">Ψ</mi><mo id="S4.SS2.p2.14.m14.2.2.1.1.1.1.1" xref="S4.SS2.p2.14.m14.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS2.p2.14.m14.2.2.1.1.1.1.3.2" xref="S4.SS2.p2.14.m14.2.2.1.1.1.1.cmml"><mo id="S4.SS2.p2.14.m14.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S4.SS2.p2.14.m14.2.2.1.1.1.1.cmml">(</mo><mi id="S4.SS2.p2.14.m14.1.1" xref="S4.SS2.p2.14.m14.1.1.cmml">t</mi><mo id="S4.SS2.p2.14.m14.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S4.SS2.p2.14.m14.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.p2.14.m14.2.2.1.1.1.3" stretchy="false" xref="S4.SS2.p2.14.m14.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.p2.14.m14.2.2.2" xref="S4.SS2.p2.14.m14.2.2.2.cmml">≡</mo><mn id="S4.SS2.p2.14.m14.2.2.3" xref="S4.SS2.p2.14.m14.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.14.m14.2b"><apply id="S4.SS2.p2.14.m14.2.2.cmml" xref="S4.SS2.p2.14.m14.2.2"><equivalent id="S4.SS2.p2.14.m14.2.2.2.cmml" xref="S4.SS2.p2.14.m14.2.2.2"></equivalent><apply id="S4.SS2.p2.14.m14.2.2.1.cmml" xref="S4.SS2.p2.14.m14.2.2.1"><times id="S4.SS2.p2.14.m14.2.2.1.2.cmml" xref="S4.SS2.p2.14.m14.2.2.1.2"></times><apply id="S4.SS2.p2.14.m14.2.2.1.3.cmml" xref="S4.SS2.p2.14.m14.2.2.1.3"><csymbol cd="ambiguous" id="S4.SS2.p2.14.m14.2.2.1.3.1.cmml" xref="S4.SS2.p2.14.m14.2.2.1.3">subscript</csymbol><ci id="S4.SS2.p2.14.m14.2.2.1.3.2.cmml" xref="S4.SS2.p2.14.m14.2.2.1.3.2">𝜎</ci><min id="S4.SS2.p2.14.m14.2.2.1.3.3.cmml" xref="S4.SS2.p2.14.m14.2.2.1.3.3"></min></apply><apply id="S4.SS2.p2.14.m14.2.2.1.1.1.1.cmml" xref="S4.SS2.p2.14.m14.2.2.1.1.1"><times id="S4.SS2.p2.14.m14.2.2.1.1.1.1.1.cmml" xref="S4.SS2.p2.14.m14.2.2.1.1.1.1.1"></times><ci id="S4.SS2.p2.14.m14.2.2.1.1.1.1.2.cmml" xref="S4.SS2.p2.14.m14.2.2.1.1.1.1.2">Ψ</ci><ci id="S4.SS2.p2.14.m14.1.1.cmml" xref="S4.SS2.p2.14.m14.1.1">𝑡</ci></apply></apply><cn id="S4.SS2.p2.14.m14.2.2.3.cmml" type="integer" xref="S4.SS2.p2.14.m14.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.14.m14.2c">\sigma_{\min}(\Psi(t))\equiv 0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.14.m14.2d">italic_σ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Ψ ( italic_t ) ) ≡ 0</annotation></semantics></math> [see the red dotted line in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.F1" title="Figure 1 ‣ IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">1</span></a>], such that online data need to be stored and updated by the excitation matrix <math alttext="\Psi_{\zeta}" class="ltx_Math" display="inline" id="S4.SS2.p2.15.m15.1"><semantics id="S4.SS2.p2.15.m15.1a"><msub id="S4.SS2.p2.15.m15.1.1" xref="S4.SS2.p2.15.m15.1.1.cmml"><mi id="S4.SS2.p2.15.m15.1.1.2" mathvariant="normal" xref="S4.SS2.p2.15.m15.1.1.2.cmml">Ψ</mi><mi id="S4.SS2.p2.15.m15.1.1.3" xref="S4.SS2.p2.15.m15.1.1.3.cmml">ζ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.15.m15.1b"><apply id="S4.SS2.p2.15.m15.1.1.cmml" xref="S4.SS2.p2.15.m15.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.15.m15.1.1.1.cmml" xref="S4.SS2.p2.15.m15.1.1">subscript</csymbol><ci id="S4.SS2.p2.15.m15.1.1.2.cmml" xref="S4.SS2.p2.15.m15.1.1.2">Ψ</ci><ci id="S4.SS2.p2.15.m15.1.1.3.cmml" xref="S4.SS2.p2.15.m15.1.1.3">𝜁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.15.m15.1c">\Psi_{\zeta}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.15.m15.1d">roman_Ψ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math>, which removes the effect that the current exciting strength <math alttext="\sigma_{\min}(\Psi(t))" class="ltx_Math" display="inline" id="S4.SS2.p2.16.m16.2"><semantics id="S4.SS2.p2.16.m16.2a"><mrow id="S4.SS2.p2.16.m16.2.2" xref="S4.SS2.p2.16.m16.2.2.cmml"><msub id="S4.SS2.p2.16.m16.2.2.3" xref="S4.SS2.p2.16.m16.2.2.3.cmml"><mi id="S4.SS2.p2.16.m16.2.2.3.2" xref="S4.SS2.p2.16.m16.2.2.3.2.cmml">σ</mi><mi id="S4.SS2.p2.16.m16.2.2.3.3" xref="S4.SS2.p2.16.m16.2.2.3.3.cmml">min</mi></msub><mo id="S4.SS2.p2.16.m16.2.2.2" xref="S4.SS2.p2.16.m16.2.2.2.cmml">⁢</mo><mrow id="S4.SS2.p2.16.m16.2.2.1.1" xref="S4.SS2.p2.16.m16.2.2.1.1.1.cmml"><mo id="S4.SS2.p2.16.m16.2.2.1.1.2" stretchy="false" xref="S4.SS2.p2.16.m16.2.2.1.1.1.cmml">(</mo><mrow id="S4.SS2.p2.16.m16.2.2.1.1.1" xref="S4.SS2.p2.16.m16.2.2.1.1.1.cmml"><mi id="S4.SS2.p2.16.m16.2.2.1.1.1.2" mathvariant="normal" xref="S4.SS2.p2.16.m16.2.2.1.1.1.2.cmml">Ψ</mi><mo id="S4.SS2.p2.16.m16.2.2.1.1.1.1" xref="S4.SS2.p2.16.m16.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS2.p2.16.m16.2.2.1.1.1.3.2" xref="S4.SS2.p2.16.m16.2.2.1.1.1.cmml"><mo id="S4.SS2.p2.16.m16.2.2.1.1.1.3.2.1" stretchy="false" xref="S4.SS2.p2.16.m16.2.2.1.1.1.cmml">(</mo><mi id="S4.SS2.p2.16.m16.1.1" xref="S4.SS2.p2.16.m16.1.1.cmml">t</mi><mo id="S4.SS2.p2.16.m16.2.2.1.1.1.3.2.2" stretchy="false" xref="S4.SS2.p2.16.m16.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.p2.16.m16.2.2.1.1.3" stretchy="false" xref="S4.SS2.p2.16.m16.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.16.m16.2b"><apply id="S4.SS2.p2.16.m16.2.2.cmml" xref="S4.SS2.p2.16.m16.2.2"><times id="S4.SS2.p2.16.m16.2.2.2.cmml" xref="S4.SS2.p2.16.m16.2.2.2"></times><apply id="S4.SS2.p2.16.m16.2.2.3.cmml" xref="S4.SS2.p2.16.m16.2.2.3"><csymbol cd="ambiguous" id="S4.SS2.p2.16.m16.2.2.3.1.cmml" xref="S4.SS2.p2.16.m16.2.2.3">subscript</csymbol><ci id="S4.SS2.p2.16.m16.2.2.3.2.cmml" xref="S4.SS2.p2.16.m16.2.2.3.2">𝜎</ci><min id="S4.SS2.p2.16.m16.2.2.3.3.cmml" xref="S4.SS2.p2.16.m16.2.2.3.3"></min></apply><apply id="S4.SS2.p2.16.m16.2.2.1.1.1.cmml" xref="S4.SS2.p2.16.m16.2.2.1.1"><times id="S4.SS2.p2.16.m16.2.2.1.1.1.1.cmml" xref="S4.SS2.p2.16.m16.2.2.1.1.1.1"></times><ci id="S4.SS2.p2.16.m16.2.2.1.1.1.2.cmml" xref="S4.SS2.p2.16.m16.2.2.1.1.1.2">Ψ</ci><ci id="S4.SS2.p2.16.m16.1.1.cmml" xref="S4.SS2.p2.16.m16.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.16.m16.2c">\sigma_{\min}(\Psi(t))</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.16.m16.2d">italic_σ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Ψ ( italic_t ) )</annotation></semantics></math> is decided by inactive channels. Note that inactive channels can be determined by <math alttext="\|\bm{\phi}_{{\rm s},j}\|<\mu" class="ltx_Math" display="inline" id="S4.SS2.p2.17.m17.3"><semantics id="S4.SS2.p2.17.m17.3a"><mrow id="S4.SS2.p2.17.m17.3.3" xref="S4.SS2.p2.17.m17.3.3.cmml"><mrow id="S4.SS2.p2.17.m17.3.3.1.1" xref="S4.SS2.p2.17.m17.3.3.1.2.cmml"><mo id="S4.SS2.p2.17.m17.3.3.1.1.2" stretchy="false" xref="S4.SS2.p2.17.m17.3.3.1.2.1.cmml">‖</mo><msub id="S4.SS2.p2.17.m17.3.3.1.1.1" xref="S4.SS2.p2.17.m17.3.3.1.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS2.p2.17.m17.3.3.1.1.1.2" mathvariant="bold-italic" xref="S4.SS2.p2.17.m17.3.3.1.1.1.2.cmml">ϕ</mi><mrow id="S4.SS2.p2.17.m17.2.2.2.4" xref="S4.SS2.p2.17.m17.2.2.2.3.cmml"><mi id="S4.SS2.p2.17.m17.1.1.1.1" mathvariant="normal" xref="S4.SS2.p2.17.m17.1.1.1.1.cmml">s</mi><mo id="S4.SS2.p2.17.m17.2.2.2.4.1" xref="S4.SS2.p2.17.m17.2.2.2.3.cmml">,</mo><mi id="S4.SS2.p2.17.m17.2.2.2.2" xref="S4.SS2.p2.17.m17.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S4.SS2.p2.17.m17.3.3.1.1.3" stretchy="false" xref="S4.SS2.p2.17.m17.3.3.1.2.1.cmml">‖</mo></mrow><mo id="S4.SS2.p2.17.m17.3.3.2" xref="S4.SS2.p2.17.m17.3.3.2.cmml"><</mo><mi id="S4.SS2.p2.17.m17.3.3.3" xref="S4.SS2.p2.17.m17.3.3.3.cmml">μ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.17.m17.3b"><apply id="S4.SS2.p2.17.m17.3.3.cmml" xref="S4.SS2.p2.17.m17.3.3"><lt id="S4.SS2.p2.17.m17.3.3.2.cmml" xref="S4.SS2.p2.17.m17.3.3.2"></lt><apply id="S4.SS2.p2.17.m17.3.3.1.2.cmml" xref="S4.SS2.p2.17.m17.3.3.1.1"><csymbol cd="latexml" id="S4.SS2.p2.17.m17.3.3.1.2.1.cmml" xref="S4.SS2.p2.17.m17.3.3.1.1.2">norm</csymbol><apply id="S4.SS2.p2.17.m17.3.3.1.1.1.cmml" xref="S4.SS2.p2.17.m17.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.17.m17.3.3.1.1.1.1.cmml" xref="S4.SS2.p2.17.m17.3.3.1.1.1">subscript</csymbol><ci id="S4.SS2.p2.17.m17.3.3.1.1.1.2.cmml" xref="S4.SS2.p2.17.m17.3.3.1.1.1.2">bold-italic-ϕ</ci><list id="S4.SS2.p2.17.m17.2.2.2.3.cmml" xref="S4.SS2.p2.17.m17.2.2.2.4"><ci id="S4.SS2.p2.17.m17.1.1.1.1.cmml" xref="S4.SS2.p2.17.m17.1.1.1.1">s</ci><ci id="S4.SS2.p2.17.m17.2.2.2.2.cmml" xref="S4.SS2.p2.17.m17.2.2.2.2">𝑗</ci></list></apply></apply><ci id="S4.SS2.p2.17.m17.3.3.3.cmml" xref="S4.SS2.p2.17.m17.3.3.3">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.17.m17.3c">\|\bm{\phi}_{{\rm s},j}\|<\mu</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.17.m17.3d">∥ bold_italic_ϕ start_POSTSUBSCRIPT roman_s , italic_j end_POSTSUBSCRIPT ∥ < italic_μ</annotation></semantics></math> with a small threshold <math alttext="\mu\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S4.SS2.p2.18.m18.1"><semantics id="S4.SS2.p2.18.m18.1a"><mrow id="S4.SS2.p2.18.m18.1.1" xref="S4.SS2.p2.18.m18.1.1.cmml"><mi id="S4.SS2.p2.18.m18.1.1.2" xref="S4.SS2.p2.18.m18.1.1.2.cmml">μ</mi><mo id="S4.SS2.p2.18.m18.1.1.1" xref="S4.SS2.p2.18.m18.1.1.1.cmml">∈</mo><msup id="S4.SS2.p2.18.m18.1.1.3" xref="S4.SS2.p2.18.m18.1.1.3.cmml"><mi id="S4.SS2.p2.18.m18.1.1.3.2" xref="S4.SS2.p2.18.m18.1.1.3.2.cmml">ℝ</mi><mo id="S4.SS2.p2.18.m18.1.1.3.3" xref="S4.SS2.p2.18.m18.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.18.m18.1b"><apply id="S4.SS2.p2.18.m18.1.1.cmml" xref="S4.SS2.p2.18.m18.1.1"><in id="S4.SS2.p2.18.m18.1.1.1.cmml" xref="S4.SS2.p2.18.m18.1.1.1"></in><ci id="S4.SS2.p2.18.m18.1.1.2.cmml" xref="S4.SS2.p2.18.m18.1.1.2">𝜇</ci><apply id="S4.SS2.p2.18.m18.1.1.3.cmml" xref="S4.SS2.p2.18.m18.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p2.18.m18.1.1.3.1.cmml" xref="S4.SS2.p2.18.m18.1.1.3">superscript</csymbol><ci id="S4.SS2.p2.18.m18.1.1.3.2.cmml" xref="S4.SS2.p2.18.m18.1.1.3.2">ℝ</ci><plus id="S4.SS2.p2.18.m18.1.1.3.3.cmml" xref="S4.SS2.p2.18.m18.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.18.m18.1c">\mu\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.18.m18.1d">italic_μ ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> instead of <math alttext="\|\bm{\phi}_{{\rm s},j}\|=0" class="ltx_Math" display="inline" id="S4.SS2.p2.19.m19.3"><semantics id="S4.SS2.p2.19.m19.3a"><mrow id="S4.SS2.p2.19.m19.3.3" xref="S4.SS2.p2.19.m19.3.3.cmml"><mrow id="S4.SS2.p2.19.m19.3.3.1.1" xref="S4.SS2.p2.19.m19.3.3.1.2.cmml"><mo id="S4.SS2.p2.19.m19.3.3.1.1.2" stretchy="false" xref="S4.SS2.p2.19.m19.3.3.1.2.1.cmml">‖</mo><msub id="S4.SS2.p2.19.m19.3.3.1.1.1" xref="S4.SS2.p2.19.m19.3.3.1.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS2.p2.19.m19.3.3.1.1.1.2" mathvariant="bold-italic" xref="S4.SS2.p2.19.m19.3.3.1.1.1.2.cmml">ϕ</mi><mrow id="S4.SS2.p2.19.m19.2.2.2.4" xref="S4.SS2.p2.19.m19.2.2.2.3.cmml"><mi id="S4.SS2.p2.19.m19.1.1.1.1" mathvariant="normal" xref="S4.SS2.p2.19.m19.1.1.1.1.cmml">s</mi><mo id="S4.SS2.p2.19.m19.2.2.2.4.1" xref="S4.SS2.p2.19.m19.2.2.2.3.cmml">,</mo><mi id="S4.SS2.p2.19.m19.2.2.2.2" xref="S4.SS2.p2.19.m19.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S4.SS2.p2.19.m19.3.3.1.1.3" stretchy="false" xref="S4.SS2.p2.19.m19.3.3.1.2.1.cmml">‖</mo></mrow><mo id="S4.SS2.p2.19.m19.3.3.2" xref="S4.SS2.p2.19.m19.3.3.2.cmml">=</mo><mn id="S4.SS2.p2.19.m19.3.3.3" xref="S4.SS2.p2.19.m19.3.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.19.m19.3b"><apply id="S4.SS2.p2.19.m19.3.3.cmml" xref="S4.SS2.p2.19.m19.3.3"><eq id="S4.SS2.p2.19.m19.3.3.2.cmml" xref="S4.SS2.p2.19.m19.3.3.2"></eq><apply id="S4.SS2.p2.19.m19.3.3.1.2.cmml" xref="S4.SS2.p2.19.m19.3.3.1.1"><csymbol cd="latexml" id="S4.SS2.p2.19.m19.3.3.1.2.1.cmml" xref="S4.SS2.p2.19.m19.3.3.1.1.2">norm</csymbol><apply id="S4.SS2.p2.19.m19.3.3.1.1.1.cmml" xref="S4.SS2.p2.19.m19.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.19.m19.3.3.1.1.1.1.cmml" xref="S4.SS2.p2.19.m19.3.3.1.1.1">subscript</csymbol><ci id="S4.SS2.p2.19.m19.3.3.1.1.1.2.cmml" xref="S4.SS2.p2.19.m19.3.3.1.1.1.2">bold-italic-ϕ</ci><list id="S4.SS2.p2.19.m19.2.2.2.3.cmml" xref="S4.SS2.p2.19.m19.2.2.2.4"><ci id="S4.SS2.p2.19.m19.1.1.1.1.cmml" xref="S4.SS2.p2.19.m19.1.1.1.1">s</ci><ci id="S4.SS2.p2.19.m19.2.2.2.2.cmml" xref="S4.SS2.p2.19.m19.2.2.2.2">𝑗</ci></list></apply></apply><cn id="S4.SS2.p2.19.m19.3.3.3.cmml" type="integer" xref="S4.SS2.p2.19.m19.3.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.19.m19.3c">\|\bm{\phi}_{{\rm s},j}\|=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.19.m19.3d">∥ bold_italic_ϕ start_POSTSUBSCRIPT roman_s , italic_j end_POSTSUBSCRIPT ∥ = 0</annotation></semantics></math> in practice, as measurement noise may exist such that it is impossible to have <math alttext="\|\bm{\phi}_{{\rm s},j}\|\neq 0" class="ltx_Math" display="inline" id="S4.SS2.p2.20.m20.3"><semantics id="S4.SS2.p2.20.m20.3a"><mrow id="S4.SS2.p2.20.m20.3.3" xref="S4.SS2.p2.20.m20.3.3.cmml"><mrow id="S4.SS2.p2.20.m20.3.3.1.1" xref="S4.SS2.p2.20.m20.3.3.1.2.cmml"><mo id="S4.SS2.p2.20.m20.3.3.1.1.2" stretchy="false" xref="S4.SS2.p2.20.m20.3.3.1.2.1.cmml">‖</mo><msub id="S4.SS2.p2.20.m20.3.3.1.1.1" xref="S4.SS2.p2.20.m20.3.3.1.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.SS2.p2.20.m20.3.3.1.1.1.2" mathvariant="bold-italic" xref="S4.SS2.p2.20.m20.3.3.1.1.1.2.cmml">ϕ</mi><mrow id="S4.SS2.p2.20.m20.2.2.2.4" xref="S4.SS2.p2.20.m20.2.2.2.3.cmml"><mi id="S4.SS2.p2.20.m20.1.1.1.1" mathvariant="normal" xref="S4.SS2.p2.20.m20.1.1.1.1.cmml">s</mi><mo id="S4.SS2.p2.20.m20.2.2.2.4.1" xref="S4.SS2.p2.20.m20.2.2.2.3.cmml">,</mo><mi id="S4.SS2.p2.20.m20.2.2.2.2" xref="S4.SS2.p2.20.m20.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S4.SS2.p2.20.m20.3.3.1.1.3" stretchy="false" xref="S4.SS2.p2.20.m20.3.3.1.2.1.cmml">‖</mo></mrow><mo id="S4.SS2.p2.20.m20.3.3.2" xref="S4.SS2.p2.20.m20.3.3.2.cmml">≠</mo><mn id="S4.SS2.p2.20.m20.3.3.3" xref="S4.SS2.p2.20.m20.3.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.20.m20.3b"><apply id="S4.SS2.p2.20.m20.3.3.cmml" xref="S4.SS2.p2.20.m20.3.3"><neq id="S4.SS2.p2.20.m20.3.3.2.cmml" xref="S4.SS2.p2.20.m20.3.3.2"></neq><apply id="S4.SS2.p2.20.m20.3.3.1.2.cmml" xref="S4.SS2.p2.20.m20.3.3.1.1"><csymbol cd="latexml" id="S4.SS2.p2.20.m20.3.3.1.2.1.cmml" xref="S4.SS2.p2.20.m20.3.3.1.1.2">norm</csymbol><apply id="S4.SS2.p2.20.m20.3.3.1.1.1.cmml" xref="S4.SS2.p2.20.m20.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.20.m20.3.3.1.1.1.1.cmml" xref="S4.SS2.p2.20.m20.3.3.1.1.1">subscript</csymbol><ci id="S4.SS2.p2.20.m20.3.3.1.1.1.2.cmml" xref="S4.SS2.p2.20.m20.3.3.1.1.1.2">bold-italic-ϕ</ci><list id="S4.SS2.p2.20.m20.2.2.2.3.cmml" xref="S4.SS2.p2.20.m20.2.2.2.4"><ci id="S4.SS2.p2.20.m20.1.1.1.1.cmml" xref="S4.SS2.p2.20.m20.1.1.1.1">s</ci><ci id="S4.SS2.p2.20.m20.2.2.2.2.cmml" xref="S4.SS2.p2.20.m20.2.2.2.2">𝑗</ci></list></apply></apply><cn id="S4.SS2.p2.20.m20.3.3.3.cmml" type="integer" xref="S4.SS2.p2.20.m20.3.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.20.m20.3c">\|\bm{\phi}_{{\rm s},j}\|\neq 0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.20.m20.3d">∥ bold_italic_ϕ start_POSTSUBSCRIPT roman_s , italic_j end_POSTSUBSCRIPT ∥ ≠ 0</annotation></semantics></math> even for inactive channels.</p> </div> <div class="ltx_para" id="S4.SS2.p3"> <p class="ltx_p" id="S4.SS2.p3.16"><span class="ltx_text ltx_font_italic" id="S4.SS2.p3.16.16" style="color:#000099;">Remark 5:<span class="ltx_text ltx_font_upright" id="S4.SS2.p3.16.16.16" style="color:#000099;"> The selection of <math alttext="\tau_{\rm d}" class="ltx_Math" display="inline" id="S4.SS2.p3.1.1.1.m1.1"><semantics id="S4.SS2.p3.1.1.1.m1.1a"><msub id="S4.SS2.p3.1.1.1.m1.1.1" xref="S4.SS2.p3.1.1.1.m1.1.1.cmml"><mi id="S4.SS2.p3.1.1.1.m1.1.1.2" mathcolor="#000099" xref="S4.SS2.p3.1.1.1.m1.1.1.2.cmml">τ</mi><mi id="S4.SS2.p3.1.1.1.m1.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS2.p3.1.1.1.m1.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.1.1.1.m1.1b"><apply id="S4.SS2.p3.1.1.1.m1.1.1.cmml" xref="S4.SS2.p3.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.1.1.1.m1.1.1.1.cmml" xref="S4.SS2.p3.1.1.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.p3.1.1.1.m1.1.1.2.cmml" xref="S4.SS2.p3.1.1.1.m1.1.1.2">𝜏</ci><ci id="S4.SS2.p3.1.1.1.m1.1.1.3.cmml" xref="S4.SS2.p3.1.1.1.m1.1.1.3">d</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.1.1.1.m1.1c">\tau_{\rm d}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.1.1.1.m1.1d">italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS2.p3.2.2.2.m2.1"><semantics id="S4.SS2.p3.2.2.2.m2.1a"><mi id="S4.SS2.p3.2.2.2.m2.1.1" mathcolor="#000099" xref="S4.SS2.p3.2.2.2.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.2.2.2.m2.1b"><ci id="S4.SS2.p3.2.2.2.m2.1.1.cmml" xref="S4.SS2.p3.2.2.2.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.2.2.2.m2.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.2.2.2.m2.1d">italic_σ</annotation></semantics></math> in Algorithm 1 is flexible, but they should not be chosen arbitrarily, as improper parameters directly affect the system performance. Choosing a proper <math alttext="\tau_{\rm d}" class="ltx_Math" display="inline" id="S4.SS2.p3.3.3.3.m3.1"><semantics id="S4.SS2.p3.3.3.3.m3.1a"><msub id="S4.SS2.p3.3.3.3.m3.1.1" xref="S4.SS2.p3.3.3.3.m3.1.1.cmml"><mi id="S4.SS2.p3.3.3.3.m3.1.1.2" mathcolor="#000099" xref="S4.SS2.p3.3.3.3.m3.1.1.2.cmml">τ</mi><mi id="S4.SS2.p3.3.3.3.m3.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS2.p3.3.3.3.m3.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.3.3.3.m3.1b"><apply id="S4.SS2.p3.3.3.3.m3.1.1.cmml" xref="S4.SS2.p3.3.3.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.3.3.3.m3.1.1.1.cmml" xref="S4.SS2.p3.3.3.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.p3.3.3.3.m3.1.1.2.cmml" xref="S4.SS2.p3.3.3.3.m3.1.1.2">𝜏</ci><ci id="S4.SS2.p3.3.3.3.m3.1.1.3.cmml" xref="S4.SS2.p3.3.3.3.m3.1.1.3">d</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.3.3.3.m3.1c">\tau_{\rm d}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.3.3.3.m3.1d">italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT</annotation></semantics></math> ensures that a sufficient amount of uncorrelated data <math alttext="\Phi_{\rm s}(t_{1})" class="ltx_Math" display="inline" id="S4.SS2.p3.4.4.4.m4.1"><semantics id="S4.SS2.p3.4.4.4.m4.1a"><mrow id="S4.SS2.p3.4.4.4.m4.1.1" xref="S4.SS2.p3.4.4.4.m4.1.1.cmml"><msub id="S4.SS2.p3.4.4.4.m4.1.1.3" xref="S4.SS2.p3.4.4.4.m4.1.1.3.cmml"><mi id="S4.SS2.p3.4.4.4.m4.1.1.3.2" mathcolor="#000099" mathvariant="normal" xref="S4.SS2.p3.4.4.4.m4.1.1.3.2.cmml">Φ</mi><mi id="S4.SS2.p3.4.4.4.m4.1.1.3.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS2.p3.4.4.4.m4.1.1.3.3.cmml">s</mi></msub><mo id="S4.SS2.p3.4.4.4.m4.1.1.2" xref="S4.SS2.p3.4.4.4.m4.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p3.4.4.4.m4.1.1.1.1" xref="S4.SS2.p3.4.4.4.m4.1.1.1.1.1.cmml"><mo id="S4.SS2.p3.4.4.4.m4.1.1.1.1.2" mathcolor="#000099" stretchy="false" xref="S4.SS2.p3.4.4.4.m4.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.p3.4.4.4.m4.1.1.1.1.1" xref="S4.SS2.p3.4.4.4.m4.1.1.1.1.1.cmml"><mi id="S4.SS2.p3.4.4.4.m4.1.1.1.1.1.2" mathcolor="#000099" xref="S4.SS2.p3.4.4.4.m4.1.1.1.1.1.2.cmml">t</mi><mn id="S4.SS2.p3.4.4.4.m4.1.1.1.1.1.3" mathcolor="#000099" xref="S4.SS2.p3.4.4.4.m4.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.SS2.p3.4.4.4.m4.1.1.1.1.3" mathcolor="#000099" stretchy="false" xref="S4.SS2.p3.4.4.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.4.4.4.m4.1b"><apply id="S4.SS2.p3.4.4.4.m4.1.1.cmml" xref="S4.SS2.p3.4.4.4.m4.1.1"><times id="S4.SS2.p3.4.4.4.m4.1.1.2.cmml" xref="S4.SS2.p3.4.4.4.m4.1.1.2"></times><apply id="S4.SS2.p3.4.4.4.m4.1.1.3.cmml" xref="S4.SS2.p3.4.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p3.4.4.4.m4.1.1.3.1.cmml" xref="S4.SS2.p3.4.4.4.m4.1.1.3">subscript</csymbol><ci id="S4.SS2.p3.4.4.4.m4.1.1.3.2.cmml" xref="S4.SS2.p3.4.4.4.m4.1.1.3.2">Φ</ci><ci id="S4.SS2.p3.4.4.4.m4.1.1.3.3.cmml" xref="S4.SS2.p3.4.4.4.m4.1.1.3.3">s</ci></apply><apply id="S4.SS2.p3.4.4.4.m4.1.1.1.1.1.cmml" xref="S4.SS2.p3.4.4.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.4.4.4.m4.1.1.1.1.1.1.cmml" xref="S4.SS2.p3.4.4.4.m4.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p3.4.4.4.m4.1.1.1.1.1.2.cmml" xref="S4.SS2.p3.4.4.4.m4.1.1.1.1.1.2">𝑡</ci><cn id="S4.SS2.p3.4.4.4.m4.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.p3.4.4.4.m4.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.4.4.4.m4.1c">\Phi_{\rm s}(t_{1})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.4.4.4.m4.1d">roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>, <math alttext="\Phi_{\rm s}(t_{2})" class="ltx_Math" display="inline" id="S4.SS2.p3.5.5.5.m5.1"><semantics id="S4.SS2.p3.5.5.5.m5.1a"><mrow id="S4.SS2.p3.5.5.5.m5.1.1" xref="S4.SS2.p3.5.5.5.m5.1.1.cmml"><msub id="S4.SS2.p3.5.5.5.m5.1.1.3" xref="S4.SS2.p3.5.5.5.m5.1.1.3.cmml"><mi id="S4.SS2.p3.5.5.5.m5.1.1.3.2" mathcolor="#000099" mathvariant="normal" xref="S4.SS2.p3.5.5.5.m5.1.1.3.2.cmml">Φ</mi><mi id="S4.SS2.p3.5.5.5.m5.1.1.3.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS2.p3.5.5.5.m5.1.1.3.3.cmml">s</mi></msub><mo id="S4.SS2.p3.5.5.5.m5.1.1.2" xref="S4.SS2.p3.5.5.5.m5.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p3.5.5.5.m5.1.1.1.1" xref="S4.SS2.p3.5.5.5.m5.1.1.1.1.1.cmml"><mo id="S4.SS2.p3.5.5.5.m5.1.1.1.1.2" mathcolor="#000099" stretchy="false" xref="S4.SS2.p3.5.5.5.m5.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.p3.5.5.5.m5.1.1.1.1.1" xref="S4.SS2.p3.5.5.5.m5.1.1.1.1.1.cmml"><mi id="S4.SS2.p3.5.5.5.m5.1.1.1.1.1.2" mathcolor="#000099" xref="S4.SS2.p3.5.5.5.m5.1.1.1.1.1.2.cmml">t</mi><mn id="S4.SS2.p3.5.5.5.m5.1.1.1.1.1.3" mathcolor="#000099" xref="S4.SS2.p3.5.5.5.m5.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S4.SS2.p3.5.5.5.m5.1.1.1.1.3" mathcolor="#000099" stretchy="false" xref="S4.SS2.p3.5.5.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.5.5.5.m5.1b"><apply id="S4.SS2.p3.5.5.5.m5.1.1.cmml" xref="S4.SS2.p3.5.5.5.m5.1.1"><times id="S4.SS2.p3.5.5.5.m5.1.1.2.cmml" xref="S4.SS2.p3.5.5.5.m5.1.1.2"></times><apply id="S4.SS2.p3.5.5.5.m5.1.1.3.cmml" xref="S4.SS2.p3.5.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p3.5.5.5.m5.1.1.3.1.cmml" xref="S4.SS2.p3.5.5.5.m5.1.1.3">subscript</csymbol><ci id="S4.SS2.p3.5.5.5.m5.1.1.3.2.cmml" xref="S4.SS2.p3.5.5.5.m5.1.1.3.2">Φ</ci><ci id="S4.SS2.p3.5.5.5.m5.1.1.3.3.cmml" xref="S4.SS2.p3.5.5.5.m5.1.1.3.3">s</ci></apply><apply id="S4.SS2.p3.5.5.5.m5.1.1.1.1.1.cmml" xref="S4.SS2.p3.5.5.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.5.5.5.m5.1.1.1.1.1.1.cmml" xref="S4.SS2.p3.5.5.5.m5.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p3.5.5.5.m5.1.1.1.1.1.2.cmml" xref="S4.SS2.p3.5.5.5.m5.1.1.1.1.1.2">𝑡</ci><cn id="S4.SS2.p3.5.5.5.m5.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.p3.5.5.5.m5.1.1.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.5.5.5.m5.1c">\Phi_{\rm s}(t_{2})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.5.5.5.m5.1d">roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math>, <math alttext="\cdots" class="ltx_Math" display="inline" id="S4.SS2.p3.6.6.6.m6.1"><semantics id="S4.SS2.p3.6.6.6.m6.1a"><mi id="S4.SS2.p3.6.6.6.m6.1.1" mathcolor="#000099" mathvariant="normal" xref="S4.SS2.p3.6.6.6.m6.1.1.cmml">⋯</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.6.6.6.m6.1b"><ci id="S4.SS2.p3.6.6.6.m6.1.1.cmml" xref="S4.SS2.p3.6.6.6.m6.1.1">⋯</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.6.6.6.m6.1c">\cdots</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.6.6.6.m6.1d">⋯</annotation></semantics></math>, <math alttext="\Phi_{\rm s}(t_{N})" class="ltx_Math" display="inline" id="S4.SS2.p3.7.7.7.m7.1"><semantics id="S4.SS2.p3.7.7.7.m7.1a"><mrow id="S4.SS2.p3.7.7.7.m7.1.1" xref="S4.SS2.p3.7.7.7.m7.1.1.cmml"><msub id="S4.SS2.p3.7.7.7.m7.1.1.3" xref="S4.SS2.p3.7.7.7.m7.1.1.3.cmml"><mi id="S4.SS2.p3.7.7.7.m7.1.1.3.2" mathcolor="#000099" mathvariant="normal" xref="S4.SS2.p3.7.7.7.m7.1.1.3.2.cmml">Φ</mi><mi id="S4.SS2.p3.7.7.7.m7.1.1.3.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS2.p3.7.7.7.m7.1.1.3.3.cmml">s</mi></msub><mo id="S4.SS2.p3.7.7.7.m7.1.1.2" xref="S4.SS2.p3.7.7.7.m7.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p3.7.7.7.m7.1.1.1.1" xref="S4.SS2.p3.7.7.7.m7.1.1.1.1.1.cmml"><mo id="S4.SS2.p3.7.7.7.m7.1.1.1.1.2" mathcolor="#000099" stretchy="false" xref="S4.SS2.p3.7.7.7.m7.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.p3.7.7.7.m7.1.1.1.1.1" xref="S4.SS2.p3.7.7.7.m7.1.1.1.1.1.cmml"><mi id="S4.SS2.p3.7.7.7.m7.1.1.1.1.1.2" mathcolor="#000099" xref="S4.SS2.p3.7.7.7.m7.1.1.1.1.1.2.cmml">t</mi><mi id="S4.SS2.p3.7.7.7.m7.1.1.1.1.1.3" mathcolor="#000099" xref="S4.SS2.p3.7.7.7.m7.1.1.1.1.1.3.cmml">N</mi></msub><mo id="S4.SS2.p3.7.7.7.m7.1.1.1.1.3" mathcolor="#000099" stretchy="false" xref="S4.SS2.p3.7.7.7.m7.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.7.7.7.m7.1b"><apply id="S4.SS2.p3.7.7.7.m7.1.1.cmml" xref="S4.SS2.p3.7.7.7.m7.1.1"><times id="S4.SS2.p3.7.7.7.m7.1.1.2.cmml" xref="S4.SS2.p3.7.7.7.m7.1.1.2"></times><apply id="S4.SS2.p3.7.7.7.m7.1.1.3.cmml" xref="S4.SS2.p3.7.7.7.m7.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p3.7.7.7.m7.1.1.3.1.cmml" xref="S4.SS2.p3.7.7.7.m7.1.1.3">subscript</csymbol><ci id="S4.SS2.p3.7.7.7.m7.1.1.3.2.cmml" xref="S4.SS2.p3.7.7.7.m7.1.1.3.2">Φ</ci><ci id="S4.SS2.p3.7.7.7.m7.1.1.3.3.cmml" xref="S4.SS2.p3.7.7.7.m7.1.1.3.3">s</ci></apply><apply id="S4.SS2.p3.7.7.7.m7.1.1.1.1.1.cmml" xref="S4.SS2.p3.7.7.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.7.7.7.m7.1.1.1.1.1.1.cmml" xref="S4.SS2.p3.7.7.7.m7.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p3.7.7.7.m7.1.1.1.1.1.2.cmml" xref="S4.SS2.p3.7.7.7.m7.1.1.1.1.1.2">𝑡</ci><ci id="S4.SS2.p3.7.7.7.m7.1.1.1.1.1.3.cmml" xref="S4.SS2.p3.7.7.7.m7.1.1.1.1.1.3">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.7.7.7.m7.1c">\Phi_{\rm s}(t_{N})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.7.7.7.m7.1d">roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT )</annotation></semantics></math> with <math alttext="t_{j}\in[T_{\rm d}-\tau_{\rm d},T_{\rm d}]" class="ltx_Math" display="inline" id="S4.SS2.p3.8.8.8.m8.2"><semantics id="S4.SS2.p3.8.8.8.m8.2a"><mrow id="S4.SS2.p3.8.8.8.m8.2.2" xref="S4.SS2.p3.8.8.8.m8.2.2.cmml"><msub id="S4.SS2.p3.8.8.8.m8.2.2.4" xref="S4.SS2.p3.8.8.8.m8.2.2.4.cmml"><mi id="S4.SS2.p3.8.8.8.m8.2.2.4.2" mathcolor="#000099" xref="S4.SS2.p3.8.8.8.m8.2.2.4.2.cmml">t</mi><mi id="S4.SS2.p3.8.8.8.m8.2.2.4.3" mathcolor="#000099" xref="S4.SS2.p3.8.8.8.m8.2.2.4.3.cmml">j</mi></msub><mo id="S4.SS2.p3.8.8.8.m8.2.2.3" mathcolor="#000099" xref="S4.SS2.p3.8.8.8.m8.2.2.3.cmml">∈</mo><mrow id="S4.SS2.p3.8.8.8.m8.2.2.2.2" xref="S4.SS2.p3.8.8.8.m8.2.2.2.3.cmml"><mo id="S4.SS2.p3.8.8.8.m8.2.2.2.2.3" mathcolor="#000099" stretchy="false" xref="S4.SS2.p3.8.8.8.m8.2.2.2.3.cmml">[</mo><mrow id="S4.SS2.p3.8.8.8.m8.1.1.1.1.1" xref="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.cmml"><msub id="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.2" xref="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.2.cmml"><mi id="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.2.2" mathcolor="#000099" xref="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.2.2.cmml">T</mi><mi id="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.2.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.2.3.cmml">d</mi></msub><mo id="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.1" mathcolor="#000099" xref="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.1.cmml">−</mo><msub id="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.3" xref="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.3.cmml"><mi id="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.3.2" mathcolor="#000099" xref="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.3.2.cmml">τ</mi><mi id="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.3.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.3.3.cmml">d</mi></msub></mrow><mo id="S4.SS2.p3.8.8.8.m8.2.2.2.2.4" mathcolor="#000099" xref="S4.SS2.p3.8.8.8.m8.2.2.2.3.cmml">,</mo><msub id="S4.SS2.p3.8.8.8.m8.2.2.2.2.2" xref="S4.SS2.p3.8.8.8.m8.2.2.2.2.2.cmml"><mi id="S4.SS2.p3.8.8.8.m8.2.2.2.2.2.2" mathcolor="#000099" xref="S4.SS2.p3.8.8.8.m8.2.2.2.2.2.2.cmml">T</mi><mi id="S4.SS2.p3.8.8.8.m8.2.2.2.2.2.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS2.p3.8.8.8.m8.2.2.2.2.2.3.cmml">d</mi></msub><mo id="S4.SS2.p3.8.8.8.m8.2.2.2.2.5" mathcolor="#000099" stretchy="false" xref="S4.SS2.p3.8.8.8.m8.2.2.2.3.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.8.8.8.m8.2b"><apply id="S4.SS2.p3.8.8.8.m8.2.2.cmml" xref="S4.SS2.p3.8.8.8.m8.2.2"><in id="S4.SS2.p3.8.8.8.m8.2.2.3.cmml" xref="S4.SS2.p3.8.8.8.m8.2.2.3"></in><apply id="S4.SS2.p3.8.8.8.m8.2.2.4.cmml" xref="S4.SS2.p3.8.8.8.m8.2.2.4"><csymbol cd="ambiguous" id="S4.SS2.p3.8.8.8.m8.2.2.4.1.cmml" xref="S4.SS2.p3.8.8.8.m8.2.2.4">subscript</csymbol><ci id="S4.SS2.p3.8.8.8.m8.2.2.4.2.cmml" xref="S4.SS2.p3.8.8.8.m8.2.2.4.2">𝑡</ci><ci id="S4.SS2.p3.8.8.8.m8.2.2.4.3.cmml" xref="S4.SS2.p3.8.8.8.m8.2.2.4.3">𝑗</ci></apply><interval closure="closed" id="S4.SS2.p3.8.8.8.m8.2.2.2.3.cmml" xref="S4.SS2.p3.8.8.8.m8.2.2.2.2"><apply id="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.cmml" xref="S4.SS2.p3.8.8.8.m8.1.1.1.1.1"><minus id="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.1.cmml" xref="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.1"></minus><apply id="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.2.cmml" xref="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.2.1.cmml" xref="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.2">subscript</csymbol><ci id="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.2.2.cmml" xref="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.2.2">𝑇</ci><ci id="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.2.3.cmml" xref="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.2.3">d</ci></apply><apply id="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.3.cmml" xref="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.3.1.cmml" xref="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.3">subscript</csymbol><ci id="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.3.2.cmml" xref="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.3.2">𝜏</ci><ci id="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.3.3.cmml" xref="S4.SS2.p3.8.8.8.m8.1.1.1.1.1.3.3">d</ci></apply></apply><apply id="S4.SS2.p3.8.8.8.m8.2.2.2.2.2.cmml" xref="S4.SS2.p3.8.8.8.m8.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.p3.8.8.8.m8.2.2.2.2.2.1.cmml" xref="S4.SS2.p3.8.8.8.m8.2.2.2.2.2">subscript</csymbol><ci id="S4.SS2.p3.8.8.8.m8.2.2.2.2.2.2.cmml" xref="S4.SS2.p3.8.8.8.m8.2.2.2.2.2.2">𝑇</ci><ci id="S4.SS2.p3.8.8.8.m8.2.2.2.2.2.3.cmml" xref="S4.SS2.p3.8.8.8.m8.2.2.2.2.2.3">d</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.8.8.8.m8.2c">t_{j}\in[T_{\rm d}-\tau_{\rm d},T_{\rm d}]</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.8.8.8.m8.2d">italic_t start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ [ italic_T start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT ]</annotation></semantics></math> and <math alttext="j=1" class="ltx_Math" display="inline" id="S4.SS2.p3.9.9.9.m9.1"><semantics id="S4.SS2.p3.9.9.9.m9.1a"><mrow id="S4.SS2.p3.9.9.9.m9.1.1" xref="S4.SS2.p3.9.9.9.m9.1.1.cmml"><mi id="S4.SS2.p3.9.9.9.m9.1.1.2" mathcolor="#000099" xref="S4.SS2.p3.9.9.9.m9.1.1.2.cmml">j</mi><mo id="S4.SS2.p3.9.9.9.m9.1.1.1" mathcolor="#000099" xref="S4.SS2.p3.9.9.9.m9.1.1.1.cmml">=</mo><mn id="S4.SS2.p3.9.9.9.m9.1.1.3" mathcolor="#000099" xref="S4.SS2.p3.9.9.9.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.9.9.9.m9.1b"><apply id="S4.SS2.p3.9.9.9.m9.1.1.cmml" xref="S4.SS2.p3.9.9.9.m9.1.1"><eq id="S4.SS2.p3.9.9.9.m9.1.1.1.cmml" xref="S4.SS2.p3.9.9.9.m9.1.1.1"></eq><ci id="S4.SS2.p3.9.9.9.m9.1.1.2.cmml" xref="S4.SS2.p3.9.9.9.m9.1.1.2">𝑗</ci><cn id="S4.SS2.p3.9.9.9.m9.1.1.3.cmml" type="integer" xref="S4.SS2.p3.9.9.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.9.9.9.m9.1c">j=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.9.9.9.m9.1d">italic_j = 1</annotation></semantics></math> to <math alttext="N" class="ltx_Math" display="inline" id="S4.SS2.p3.10.10.10.m10.1"><semantics id="S4.SS2.p3.10.10.10.m10.1a"><mi id="S4.SS2.p3.10.10.10.m10.1.1" mathcolor="#000099" xref="S4.SS2.p3.10.10.10.m10.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.10.10.10.m10.1b"><ci id="S4.SS2.p3.10.10.10.m10.1.1.cmml" xref="S4.SS2.p3.10.10.10.m10.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.10.10.10.m10.1c">N</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.10.10.10.m10.1d">italic_N</annotation></semantics></math> can be stored, ensuring that the excitation matrix <math alttext="\Psi(T_{\rm d})" class="ltx_Math" display="inline" id="S4.SS2.p3.11.11.11.m11.1"><semantics id="S4.SS2.p3.11.11.11.m11.1a"><mrow id="S4.SS2.p3.11.11.11.m11.1.1" xref="S4.SS2.p3.11.11.11.m11.1.1.cmml"><mi id="S4.SS2.p3.11.11.11.m11.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS2.p3.11.11.11.m11.1.1.3.cmml">Ψ</mi><mo id="S4.SS2.p3.11.11.11.m11.1.1.2" xref="S4.SS2.p3.11.11.11.m11.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p3.11.11.11.m11.1.1.1.1" xref="S4.SS2.p3.11.11.11.m11.1.1.1.1.1.cmml"><mo id="S4.SS2.p3.11.11.11.m11.1.1.1.1.2" mathcolor="#000099" stretchy="false" xref="S4.SS2.p3.11.11.11.m11.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.p3.11.11.11.m11.1.1.1.1.1" xref="S4.SS2.p3.11.11.11.m11.1.1.1.1.1.cmml"><mi id="S4.SS2.p3.11.11.11.m11.1.1.1.1.1.2" mathcolor="#000099" xref="S4.SS2.p3.11.11.11.m11.1.1.1.1.1.2.cmml">T</mi><mi id="S4.SS2.p3.11.11.11.m11.1.1.1.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS2.p3.11.11.11.m11.1.1.1.1.1.3.cmml">d</mi></msub><mo id="S4.SS2.p3.11.11.11.m11.1.1.1.1.3" mathcolor="#000099" stretchy="false" xref="S4.SS2.p3.11.11.11.m11.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.11.11.11.m11.1b"><apply id="S4.SS2.p3.11.11.11.m11.1.1.cmml" xref="S4.SS2.p3.11.11.11.m11.1.1"><times id="S4.SS2.p3.11.11.11.m11.1.1.2.cmml" xref="S4.SS2.p3.11.11.11.m11.1.1.2"></times><ci id="S4.SS2.p3.11.11.11.m11.1.1.3.cmml" xref="S4.SS2.p3.11.11.11.m11.1.1.3">Ψ</ci><apply id="S4.SS2.p3.11.11.11.m11.1.1.1.1.1.cmml" xref="S4.SS2.p3.11.11.11.m11.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.11.11.11.m11.1.1.1.1.1.1.cmml" xref="S4.SS2.p3.11.11.11.m11.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p3.11.11.11.m11.1.1.1.1.1.2.cmml" xref="S4.SS2.p3.11.11.11.m11.1.1.1.1.1.2">𝑇</ci><ci id="S4.SS2.p3.11.11.11.m11.1.1.1.1.1.3.cmml" xref="S4.SS2.p3.11.11.11.m11.1.1.1.1.1.3">d</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.11.11.11.m11.1c">\Psi(T_{\rm d})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.11.11.11.m11.1d">roman_Ψ ( italic_T start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT )</annotation></semantics></math> in (10) remains positive-definite, i.e., the minimal singular value <math alttext="\sigma_{\min}(\Psi(T_{\rm d}))>0" class="ltx_Math" display="inline" id="S4.SS2.p3.12.12.12.m12.1"><semantics id="S4.SS2.p3.12.12.12.m12.1a"><mrow id="S4.SS2.p3.12.12.12.m12.1.1" xref="S4.SS2.p3.12.12.12.m12.1.1.cmml"><mrow id="S4.SS2.p3.12.12.12.m12.1.1.1" xref="S4.SS2.p3.12.12.12.m12.1.1.1.cmml"><msub id="S4.SS2.p3.12.12.12.m12.1.1.1.3" xref="S4.SS2.p3.12.12.12.m12.1.1.1.3.cmml"><mi id="S4.SS2.p3.12.12.12.m12.1.1.1.3.2" mathcolor="#000099" xref="S4.SS2.p3.12.12.12.m12.1.1.1.3.2.cmml">σ</mi><mi id="S4.SS2.p3.12.12.12.m12.1.1.1.3.3" mathcolor="#000099" xref="S4.SS2.p3.12.12.12.m12.1.1.1.3.3.cmml">min</mi></msub><mo id="S4.SS2.p3.12.12.12.m12.1.1.1.2" xref="S4.SS2.p3.12.12.12.m12.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p3.12.12.12.m12.1.1.1.1.1" xref="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.cmml"><mo id="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.2" mathcolor="#000099" stretchy="false" xref="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1" xref="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.cmml"><mi id="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.3.cmml">Ψ</mi><mo id="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.2" xref="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.1.1" xref="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.1.1.2" mathcolor="#000099" stretchy="false" xref="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.1.1.1" xref="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.1.1.1.2" mathcolor="#000099" xref="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.1.1.1.2.cmml">T</mi><mi id="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.1.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.1.1.1.3.cmml">d</mi></msub><mo id="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.1.1.3" mathcolor="#000099" stretchy="false" xref="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.3" mathcolor="#000099" stretchy="false" xref="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.p3.12.12.12.m12.1.1.2" mathcolor="#000099" xref="S4.SS2.p3.12.12.12.m12.1.1.2.cmml">></mo><mn id="S4.SS2.p3.12.12.12.m12.1.1.3" mathcolor="#000099" xref="S4.SS2.p3.12.12.12.m12.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.12.12.12.m12.1b"><apply id="S4.SS2.p3.12.12.12.m12.1.1.cmml" xref="S4.SS2.p3.12.12.12.m12.1.1"><gt id="S4.SS2.p3.12.12.12.m12.1.1.2.cmml" xref="S4.SS2.p3.12.12.12.m12.1.1.2"></gt><apply id="S4.SS2.p3.12.12.12.m12.1.1.1.cmml" xref="S4.SS2.p3.12.12.12.m12.1.1.1"><times id="S4.SS2.p3.12.12.12.m12.1.1.1.2.cmml" xref="S4.SS2.p3.12.12.12.m12.1.1.1.2"></times><apply id="S4.SS2.p3.12.12.12.m12.1.1.1.3.cmml" xref="S4.SS2.p3.12.12.12.m12.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p3.12.12.12.m12.1.1.1.3.1.cmml" xref="S4.SS2.p3.12.12.12.m12.1.1.1.3">subscript</csymbol><ci id="S4.SS2.p3.12.12.12.m12.1.1.1.3.2.cmml" xref="S4.SS2.p3.12.12.12.m12.1.1.1.3.2">𝜎</ci><min id="S4.SS2.p3.12.12.12.m12.1.1.1.3.3.cmml" xref="S4.SS2.p3.12.12.12.m12.1.1.1.3.3"></min></apply><apply id="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.cmml" xref="S4.SS2.p3.12.12.12.m12.1.1.1.1.1"><times id="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.2.cmml" xref="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.2"></times><ci id="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.3.cmml" xref="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.3">Ψ</ci><apply id="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.1.1.1.2">𝑇</ci><ci id="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.p3.12.12.12.m12.1.1.1.1.1.1.1.1.1.3">d</ci></apply></apply></apply><cn id="S4.SS2.p3.12.12.12.m12.1.1.3.cmml" type="integer" xref="S4.SS2.p3.12.12.12.m12.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.12.12.12.m12.1c">\sigma_{\min}(\Psi(T_{\rm d}))>0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.12.12.12.m12.1d">italic_σ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Ψ ( italic_T start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT ) ) > 0</annotation></semantics></math>. The parameter <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS2.p3.13.13.13.m13.1"><semantics id="S4.SS2.p3.13.13.13.m13.1a"><mi id="S4.SS2.p3.13.13.13.m13.1.1" mathcolor="#000099" xref="S4.SS2.p3.13.13.13.m13.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.13.13.13.m13.1b"><ci id="S4.SS2.p3.13.13.13.m13.1.1.cmml" xref="S4.SS2.p3.13.13.13.m13.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.13.13.13.m13.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.13.13.13.m13.1d">italic_σ</annotation></semantics></math> in Algorithm 1 serves as a threshold of exciting strength, which ensures that the convergence rate of <math alttext="\tilde{\bm{\theta}}_{\zeta}" class="ltx_Math" display="inline" id="S4.SS2.p3.14.14.14.m14.1"><semantics id="S4.SS2.p3.14.14.14.m14.1a"><msub id="S4.SS2.p3.14.14.14.m14.1.1" xref="S4.SS2.p3.14.14.14.m14.1.1.cmml"><mover accent="true" id="S4.SS2.p3.14.14.14.m14.1.1.2" xref="S4.SS2.p3.14.14.14.m14.1.1.2.cmml"><mi id="S4.SS2.p3.14.14.14.m14.1.1.2.2" mathcolor="#000099" xref="S4.SS2.p3.14.14.14.m14.1.1.2.2.cmml">𝜽</mi><mo id="S4.SS2.p3.14.14.14.m14.1.1.2.1" mathcolor="#000099" xref="S4.SS2.p3.14.14.14.m14.1.1.2.1.cmml">~</mo></mover><mi id="S4.SS2.p3.14.14.14.m14.1.1.3" mathcolor="#000099" xref="S4.SS2.p3.14.14.14.m14.1.1.3.cmml">ζ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.14.14.14.m14.1b"><apply id="S4.SS2.p3.14.14.14.m14.1.1.cmml" xref="S4.SS2.p3.14.14.14.m14.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.14.14.14.m14.1.1.1.cmml" xref="S4.SS2.p3.14.14.14.m14.1.1">subscript</csymbol><apply id="S4.SS2.p3.14.14.14.m14.1.1.2.cmml" xref="S4.SS2.p3.14.14.14.m14.1.1.2"><ci id="S4.SS2.p3.14.14.14.m14.1.1.2.1.cmml" xref="S4.SS2.p3.14.14.14.m14.1.1.2.1">~</ci><ci id="S4.SS2.p3.14.14.14.m14.1.1.2.2.cmml" xref="S4.SS2.p3.14.14.14.m14.1.1.2.2">𝜽</ci></apply><ci id="S4.SS2.p3.14.14.14.m14.1.1.3.cmml" xref="S4.SS2.p3.14.14.14.m14.1.1.3">𝜁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.14.14.14.m14.1c">\tilde{\bm{\theta}}_{\zeta}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.14.14.14.m14.1d">over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math> (or <math alttext="\tilde{\bm{\theta}}" class="ltx_Math" display="inline" id="S4.SS2.p3.15.15.15.m15.1"><semantics id="S4.SS2.p3.15.15.15.m15.1a"><mover accent="true" id="S4.SS2.p3.15.15.15.m15.1.1" xref="S4.SS2.p3.15.15.15.m15.1.1.cmml"><mi id="S4.SS2.p3.15.15.15.m15.1.1.2" mathcolor="#000099" xref="S4.SS2.p3.15.15.15.m15.1.1.2.cmml">𝜽</mi><mo id="S4.SS2.p3.15.15.15.m15.1.1.1" mathcolor="#000099" xref="S4.SS2.p3.15.15.15.m15.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.15.15.15.m15.1b"><apply id="S4.SS2.p3.15.15.15.m15.1.1.cmml" xref="S4.SS2.p3.15.15.15.m15.1.1"><ci id="S4.SS2.p3.15.15.15.m15.1.1.1.cmml" xref="S4.SS2.p3.15.15.15.m15.1.1.1">~</ci><ci id="S4.SS2.p3.15.15.15.m15.1.1.2.cmml" xref="S4.SS2.p3.15.15.15.m15.1.1.2">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.15.15.15.m15.1c">\tilde{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.15.15.15.m15.1d">over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math> when the IE condition holds) is always greater than a positive constant. Note that the threshold <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS2.p3.16.16.16.m16.1"><semantics id="S4.SS2.p3.16.16.16.m16.1a"><mi id="S4.SS2.p3.16.16.16.m16.1.1" mathcolor="#000099" xref="S4.SS2.p3.16.16.16.m16.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.16.16.16.m16.1b"><ci id="S4.SS2.p3.16.16.16.m16.1.1.cmml" xref="S4.SS2.p3.16.16.16.m16.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.16.16.16.m16.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.16.16.16.m16.1d">italic_σ</annotation></semantics></math> is typically chosen to be a small constant.</span></span></p> </div> <div class="ltx_para" id="S4.SS2.p4"> <p class="ltx_p" id="S4.SS2.p4.11"><span class="ltx_text ltx_font_italic" id="S4.SS2.p4.11.1">Remark 6:</span> The previous composite learning methods only consider the IE case and utilize the excitation matrix <math alttext="\Psi(t_{\rm e})" class="ltx_Math" display="inline" id="S4.SS2.p4.1.m1.1"><semantics id="S4.SS2.p4.1.m1.1a"><mrow id="S4.SS2.p4.1.m1.1.1" xref="S4.SS2.p4.1.m1.1.1.cmml"><mi id="S4.SS2.p4.1.m1.1.1.3" mathvariant="normal" xref="S4.SS2.p4.1.m1.1.1.3.cmml">Ψ</mi><mo id="S4.SS2.p4.1.m1.1.1.2" xref="S4.SS2.p4.1.m1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p4.1.m1.1.1.1.1" xref="S4.SS2.p4.1.m1.1.1.1.1.1.cmml"><mo id="S4.SS2.p4.1.m1.1.1.1.1.2" stretchy="false" xref="S4.SS2.p4.1.m1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.p4.1.m1.1.1.1.1.1" xref="S4.SS2.p4.1.m1.1.1.1.1.1.cmml"><mi id="S4.SS2.p4.1.m1.1.1.1.1.1.2" xref="S4.SS2.p4.1.m1.1.1.1.1.1.2.cmml">t</mi><mi id="S4.SS2.p4.1.m1.1.1.1.1.1.3" mathvariant="normal" xref="S4.SS2.p4.1.m1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="S4.SS2.p4.1.m1.1.1.1.1.3" stretchy="false" xref="S4.SS2.p4.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.1.m1.1b"><apply id="S4.SS2.p4.1.m1.1.1.cmml" xref="S4.SS2.p4.1.m1.1.1"><times id="S4.SS2.p4.1.m1.1.1.2.cmml" xref="S4.SS2.p4.1.m1.1.1.2"></times><ci id="S4.SS2.p4.1.m1.1.1.3.cmml" xref="S4.SS2.p4.1.m1.1.1.3">Ψ</ci><apply id="S4.SS2.p4.1.m1.1.1.1.1.1.cmml" xref="S4.SS2.p4.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.1.m1.1.1.1.1.1.1.cmml" xref="S4.SS2.p4.1.m1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p4.1.m1.1.1.1.1.1.2.cmml" xref="S4.SS2.p4.1.m1.1.1.1.1.1.2">𝑡</ci><ci id="S4.SS2.p4.1.m1.1.1.1.1.1.3.cmml" xref="S4.SS2.p4.1.m1.1.1.1.1.1.3">e</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.1.m1.1c">\Psi(t_{\rm e})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.1.m1.1d">roman_Ψ ( italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E15" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">15</span></a>) with the auxiliary variable <math alttext="\bm{q}(t_{\rm e})" class="ltx_Math" display="inline" id="S4.SS2.p4.2.m2.1"><semantics id="S4.SS2.p4.2.m2.1a"><mrow id="S4.SS2.p4.2.m2.1.1" xref="S4.SS2.p4.2.m2.1.1.cmml"><mi id="S4.SS2.p4.2.m2.1.1.3" xref="S4.SS2.p4.2.m2.1.1.3.cmml">𝒒</mi><mo id="S4.SS2.p4.2.m2.1.1.2" xref="S4.SS2.p4.2.m2.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p4.2.m2.1.1.1.1" xref="S4.SS2.p4.2.m2.1.1.1.1.1.cmml"><mo id="S4.SS2.p4.2.m2.1.1.1.1.2" stretchy="false" xref="S4.SS2.p4.2.m2.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.p4.2.m2.1.1.1.1.1" xref="S4.SS2.p4.2.m2.1.1.1.1.1.cmml"><mi id="S4.SS2.p4.2.m2.1.1.1.1.1.2" xref="S4.SS2.p4.2.m2.1.1.1.1.1.2.cmml">t</mi><mi id="S4.SS2.p4.2.m2.1.1.1.1.1.3" mathvariant="normal" xref="S4.SS2.p4.2.m2.1.1.1.1.1.3.cmml">e</mi></msub><mo id="S4.SS2.p4.2.m2.1.1.1.1.3" stretchy="false" xref="S4.SS2.p4.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.2.m2.1b"><apply id="S4.SS2.p4.2.m2.1.1.cmml" xref="S4.SS2.p4.2.m2.1.1"><times id="S4.SS2.p4.2.m2.1.1.2.cmml" xref="S4.SS2.p4.2.m2.1.1.2"></times><ci id="S4.SS2.p4.2.m2.1.1.3.cmml" xref="S4.SS2.p4.2.m2.1.1.3">𝒒</ci><apply id="S4.SS2.p4.2.m2.1.1.1.1.1.cmml" xref="S4.SS2.p4.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.2.m2.1.1.1.1.1.1.cmml" xref="S4.SS2.p4.2.m2.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p4.2.m2.1.1.1.1.1.2.cmml" xref="S4.SS2.p4.2.m2.1.1.1.1.1.2">𝑡</ci><ci id="S4.SS2.p4.2.m2.1.1.1.1.1.3.cmml" xref="S4.SS2.p4.2.m2.1.1.1.1.1.3">e</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.2.m2.1c">\bm{q}(t_{\rm e})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.2.m2.1d">bold_italic_q ( italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E16" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">16</span></a>) to directly compute the generalized prediction error <math alttext="\bm{\xi}(t)" class="ltx_Math" display="inline" id="S4.SS2.p4.3.m3.1"><semantics id="S4.SS2.p4.3.m3.1a"><mrow id="S4.SS2.p4.3.m3.1.2" xref="S4.SS2.p4.3.m3.1.2.cmml"><mi id="S4.SS2.p4.3.m3.1.2.2" xref="S4.SS2.p4.3.m3.1.2.2.cmml">𝝃</mi><mo id="S4.SS2.p4.3.m3.1.2.1" xref="S4.SS2.p4.3.m3.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p4.3.m3.1.2.3.2" xref="S4.SS2.p4.3.m3.1.2.cmml"><mo id="S4.SS2.p4.3.m3.1.2.3.2.1" stretchy="false" xref="S4.SS2.p4.3.m3.1.2.cmml">(</mo><mi id="S4.SS2.p4.3.m3.1.1" xref="S4.SS2.p4.3.m3.1.1.cmml">t</mi><mo id="S4.SS2.p4.3.m3.1.2.3.2.2" stretchy="false" xref="S4.SS2.p4.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.3.m3.1b"><apply id="S4.SS2.p4.3.m3.1.2.cmml" xref="S4.SS2.p4.3.m3.1.2"><times id="S4.SS2.p4.3.m3.1.2.1.cmml" xref="S4.SS2.p4.3.m3.1.2.1"></times><ci id="S4.SS2.p4.3.m3.1.2.2.cmml" xref="S4.SS2.p4.3.m3.1.2.2">𝝃</ci><ci id="S4.SS2.p4.3.m3.1.1.cmml" xref="S4.SS2.p4.3.m3.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.3.m3.1c">\bm{\xi}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.3.m3.1d">bold_italic_ξ ( italic_t )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E20" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">20</span></a>) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib6" title="">6</a>]</cite>. However, the excitation matrix <math alttext="\Psi(t_{\rm e})" class="ltx_Math" display="inline" id="S4.SS2.p4.4.m4.1"><semantics id="S4.SS2.p4.4.m4.1a"><mrow id="S4.SS2.p4.4.m4.1.1" xref="S4.SS2.p4.4.m4.1.1.cmml"><mi id="S4.SS2.p4.4.m4.1.1.3" mathvariant="normal" xref="S4.SS2.p4.4.m4.1.1.3.cmml">Ψ</mi><mo id="S4.SS2.p4.4.m4.1.1.2" xref="S4.SS2.p4.4.m4.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p4.4.m4.1.1.1.1" xref="S4.SS2.p4.4.m4.1.1.1.1.1.cmml"><mo id="S4.SS2.p4.4.m4.1.1.1.1.2" stretchy="false" xref="S4.SS2.p4.4.m4.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.p4.4.m4.1.1.1.1.1" xref="S4.SS2.p4.4.m4.1.1.1.1.1.cmml"><mi id="S4.SS2.p4.4.m4.1.1.1.1.1.2" xref="S4.SS2.p4.4.m4.1.1.1.1.1.2.cmml">t</mi><mi id="S4.SS2.p4.4.m4.1.1.1.1.1.3" mathvariant="normal" xref="S4.SS2.p4.4.m4.1.1.1.1.1.3.cmml">e</mi></msub><mo id="S4.SS2.p4.4.m4.1.1.1.1.3" stretchy="false" xref="S4.SS2.p4.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.4.m4.1b"><apply id="S4.SS2.p4.4.m4.1.1.cmml" xref="S4.SS2.p4.4.m4.1.1"><times id="S4.SS2.p4.4.m4.1.1.2.cmml" xref="S4.SS2.p4.4.m4.1.1.2"></times><ci id="S4.SS2.p4.4.m4.1.1.3.cmml" xref="S4.SS2.p4.4.m4.1.1.3">Ψ</ci><apply id="S4.SS2.p4.4.m4.1.1.1.1.1.cmml" xref="S4.SS2.p4.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.4.m4.1.1.1.1.1.1.cmml" xref="S4.SS2.p4.4.m4.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p4.4.m4.1.1.1.1.1.2.cmml" xref="S4.SS2.p4.4.m4.1.1.1.1.1.2">𝑡</ci><ci id="S4.SS2.p4.4.m4.1.1.1.1.1.3.cmml" xref="S4.SS2.p4.4.m4.1.1.1.1.1.3">e</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.4.m4.1c">\Psi(t_{\rm e})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.4.m4.1d">roman_Ψ ( italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT )</annotation></semantics></math> can be discontinuous at some moments even the excitation strength <math alttext="\sigma_{\rm c}(t)" class="ltx_Math" display="inline" id="S4.SS2.p4.5.m5.1"><semantics id="S4.SS2.p4.5.m5.1a"><mrow id="S4.SS2.p4.5.m5.1.2" xref="S4.SS2.p4.5.m5.1.2.cmml"><msub id="S4.SS2.p4.5.m5.1.2.2" xref="S4.SS2.p4.5.m5.1.2.2.cmml"><mi id="S4.SS2.p4.5.m5.1.2.2.2" xref="S4.SS2.p4.5.m5.1.2.2.2.cmml">σ</mi><mi id="S4.SS2.p4.5.m5.1.2.2.3" mathvariant="normal" xref="S4.SS2.p4.5.m5.1.2.2.3.cmml">c</mi></msub><mo id="S4.SS2.p4.5.m5.1.2.1" xref="S4.SS2.p4.5.m5.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p4.5.m5.1.2.3.2" xref="S4.SS2.p4.5.m5.1.2.cmml"><mo id="S4.SS2.p4.5.m5.1.2.3.2.1" stretchy="false" xref="S4.SS2.p4.5.m5.1.2.cmml">(</mo><mi id="S4.SS2.p4.5.m5.1.1" xref="S4.SS2.p4.5.m5.1.1.cmml">t</mi><mo id="S4.SS2.p4.5.m5.1.2.3.2.2" stretchy="false" xref="S4.SS2.p4.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.5.m5.1b"><apply id="S4.SS2.p4.5.m5.1.2.cmml" xref="S4.SS2.p4.5.m5.1.2"><times id="S4.SS2.p4.5.m5.1.2.1.cmml" xref="S4.SS2.p4.5.m5.1.2.1"></times><apply id="S4.SS2.p4.5.m5.1.2.2.cmml" xref="S4.SS2.p4.5.m5.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.p4.5.m5.1.2.2.1.cmml" xref="S4.SS2.p4.5.m5.1.2.2">subscript</csymbol><ci id="S4.SS2.p4.5.m5.1.2.2.2.cmml" xref="S4.SS2.p4.5.m5.1.2.2.2">𝜎</ci><ci id="S4.SS2.p4.5.m5.1.2.2.3.cmml" xref="S4.SS2.p4.5.m5.1.2.2.3">c</ci></apply><ci id="S4.SS2.p4.5.m5.1.1.cmml" xref="S4.SS2.p4.5.m5.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.5.m5.1c">\sigma_{\rm c}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.5.m5.1d">italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> is continuous, which makes it impossible to obtain the high-order time derivatives of <math alttext="\hat{\bm{\theta}}" class="ltx_Math" display="inline" id="S4.SS2.p4.6.m6.1"><semantics id="S4.SS2.p4.6.m6.1a"><mover accent="true" id="S4.SS2.p4.6.m6.1.1" xref="S4.SS2.p4.6.m6.1.1.cmml"><mi id="S4.SS2.p4.6.m6.1.1.2" xref="S4.SS2.p4.6.m6.1.1.2.cmml">𝜽</mi><mo id="S4.SS2.p4.6.m6.1.1.1" xref="S4.SS2.p4.6.m6.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.6.m6.1b"><apply id="S4.SS2.p4.6.m6.1.1.cmml" xref="S4.SS2.p4.6.m6.1.1"><ci id="S4.SS2.p4.6.m6.1.1.1.cmml" xref="S4.SS2.p4.6.m6.1.1.1">^</ci><ci id="S4.SS2.p4.6.m6.1.1.2.cmml" xref="S4.SS2.p4.6.m6.1.1.2">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.6.m6.1c">\hat{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.6.m6.1d">over^ start_ARG bold_italic_θ end_ARG</annotation></semantics></math> directly, and the control law (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E6" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">6</span></a>) may generate a discontinuous signal. Differently in the proposed composite learning HOT (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>), the excitation matrix <math alttext="\Psi(t_{\rm e})" class="ltx_Math" display="inline" id="S4.SS2.p4.7.m7.1"><semantics id="S4.SS2.p4.7.m7.1a"><mrow id="S4.SS2.p4.7.m7.1.1" xref="S4.SS2.p4.7.m7.1.1.cmml"><mi id="S4.SS2.p4.7.m7.1.1.3" mathvariant="normal" xref="S4.SS2.p4.7.m7.1.1.3.cmml">Ψ</mi><mo id="S4.SS2.p4.7.m7.1.1.2" xref="S4.SS2.p4.7.m7.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p4.7.m7.1.1.1.1" xref="S4.SS2.p4.7.m7.1.1.1.1.1.cmml"><mo id="S4.SS2.p4.7.m7.1.1.1.1.2" stretchy="false" xref="S4.SS2.p4.7.m7.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.p4.7.m7.1.1.1.1.1" xref="S4.SS2.p4.7.m7.1.1.1.1.1.cmml"><mi id="S4.SS2.p4.7.m7.1.1.1.1.1.2" xref="S4.SS2.p4.7.m7.1.1.1.1.1.2.cmml">t</mi><mi id="S4.SS2.p4.7.m7.1.1.1.1.1.3" mathvariant="normal" xref="S4.SS2.p4.7.m7.1.1.1.1.1.3.cmml">e</mi></msub><mo id="S4.SS2.p4.7.m7.1.1.1.1.3" stretchy="false" xref="S4.SS2.p4.7.m7.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.7.m7.1b"><apply id="S4.SS2.p4.7.m7.1.1.cmml" xref="S4.SS2.p4.7.m7.1.1"><times id="S4.SS2.p4.7.m7.1.1.2.cmml" xref="S4.SS2.p4.7.m7.1.1.2"></times><ci id="S4.SS2.p4.7.m7.1.1.3.cmml" xref="S4.SS2.p4.7.m7.1.1.3">Ψ</ci><apply id="S4.SS2.p4.7.m7.1.1.1.1.1.cmml" xref="S4.SS2.p4.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.7.m7.1.1.1.1.1.1.cmml" xref="S4.SS2.p4.7.m7.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p4.7.m7.1.1.1.1.1.2.cmml" xref="S4.SS2.p4.7.m7.1.1.1.1.1.2">𝑡</ci><ci id="S4.SS2.p4.7.m7.1.1.1.1.1.3.cmml" xref="S4.SS2.p4.7.m7.1.1.1.1.1.3">e</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.7.m7.1c">\Psi(t_{\rm e})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.7.m7.1d">roman_Ψ ( italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E15" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">15</span></a>) is embedded in the stable filter <math alttext="H(s)" class="ltx_Math" display="inline" id="S4.SS2.p4.8.m8.1"><semantics id="S4.SS2.p4.8.m8.1a"><mrow id="S4.SS2.p4.8.m8.1.2" xref="S4.SS2.p4.8.m8.1.2.cmml"><mi id="S4.SS2.p4.8.m8.1.2.2" xref="S4.SS2.p4.8.m8.1.2.2.cmml">H</mi><mo id="S4.SS2.p4.8.m8.1.2.1" xref="S4.SS2.p4.8.m8.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p4.8.m8.1.2.3.2" xref="S4.SS2.p4.8.m8.1.2.cmml"><mo id="S4.SS2.p4.8.m8.1.2.3.2.1" stretchy="false" xref="S4.SS2.p4.8.m8.1.2.cmml">(</mo><mi id="S4.SS2.p4.8.m8.1.1" xref="S4.SS2.p4.8.m8.1.1.cmml">s</mi><mo id="S4.SS2.p4.8.m8.1.2.3.2.2" stretchy="false" xref="S4.SS2.p4.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.8.m8.1b"><apply id="S4.SS2.p4.8.m8.1.2.cmml" xref="S4.SS2.p4.8.m8.1.2"><times id="S4.SS2.p4.8.m8.1.2.1.cmml" xref="S4.SS2.p4.8.m8.1.2.1"></times><ci id="S4.SS2.p4.8.m8.1.2.2.cmml" xref="S4.SS2.p4.8.m8.1.2.2">𝐻</ci><ci id="S4.SS2.p4.8.m8.1.1.cmml" xref="S4.SS2.p4.8.m8.1.1">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.8.m8.1c">H(s)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.8.m8.1d">italic_H ( italic_s )</annotation></semantics></math> to generate the filtered excitation matrix <math alttext="Q(t,t_{\rm e})" class="ltx_Math" display="inline" id="S4.SS2.p4.9.m9.2"><semantics id="S4.SS2.p4.9.m9.2a"><mrow id="S4.SS2.p4.9.m9.2.2" xref="S4.SS2.p4.9.m9.2.2.cmml"><mi id="S4.SS2.p4.9.m9.2.2.3" xref="S4.SS2.p4.9.m9.2.2.3.cmml">Q</mi><mo id="S4.SS2.p4.9.m9.2.2.2" xref="S4.SS2.p4.9.m9.2.2.2.cmml">⁢</mo><mrow id="S4.SS2.p4.9.m9.2.2.1.1" xref="S4.SS2.p4.9.m9.2.2.1.2.cmml"><mo id="S4.SS2.p4.9.m9.2.2.1.1.2" stretchy="false" xref="S4.SS2.p4.9.m9.2.2.1.2.cmml">(</mo><mi id="S4.SS2.p4.9.m9.1.1" xref="S4.SS2.p4.9.m9.1.1.cmml">t</mi><mo id="S4.SS2.p4.9.m9.2.2.1.1.3" xref="S4.SS2.p4.9.m9.2.2.1.2.cmml">,</mo><msub id="S4.SS2.p4.9.m9.2.2.1.1.1" xref="S4.SS2.p4.9.m9.2.2.1.1.1.cmml"><mi id="S4.SS2.p4.9.m9.2.2.1.1.1.2" xref="S4.SS2.p4.9.m9.2.2.1.1.1.2.cmml">t</mi><mi id="S4.SS2.p4.9.m9.2.2.1.1.1.3" mathvariant="normal" xref="S4.SS2.p4.9.m9.2.2.1.1.1.3.cmml">e</mi></msub><mo id="S4.SS2.p4.9.m9.2.2.1.1.4" stretchy="false" xref="S4.SS2.p4.9.m9.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.9.m9.2b"><apply id="S4.SS2.p4.9.m9.2.2.cmml" xref="S4.SS2.p4.9.m9.2.2"><times id="S4.SS2.p4.9.m9.2.2.2.cmml" xref="S4.SS2.p4.9.m9.2.2.2"></times><ci id="S4.SS2.p4.9.m9.2.2.3.cmml" xref="S4.SS2.p4.9.m9.2.2.3">𝑄</ci><interval closure="open" id="S4.SS2.p4.9.m9.2.2.1.2.cmml" xref="S4.SS2.p4.9.m9.2.2.1.1"><ci id="S4.SS2.p4.9.m9.1.1.cmml" xref="S4.SS2.p4.9.m9.1.1">𝑡</ci><apply id="S4.SS2.p4.9.m9.2.2.1.1.1.cmml" xref="S4.SS2.p4.9.m9.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.9.m9.2.2.1.1.1.1.cmml" xref="S4.SS2.p4.9.m9.2.2.1.1.1">subscript</csymbol><ci id="S4.SS2.p4.9.m9.2.2.1.1.1.2.cmml" xref="S4.SS2.p4.9.m9.2.2.1.1.1.2">𝑡</ci><ci id="S4.SS2.p4.9.m9.2.2.1.1.1.3.cmml" xref="S4.SS2.p4.9.m9.2.2.1.1.1.3">e</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.9.m9.2c">Q(t,t_{\rm e})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.9.m9.2d">italic_Q ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E18" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">18</span></a>) and to construct the novel generalized prediction error <math alttext="\bm{\xi}(t)" class="ltx_Math" display="inline" id="S4.SS2.p4.10.m10.1"><semantics id="S4.SS2.p4.10.m10.1a"><mrow id="S4.SS2.p4.10.m10.1.2" xref="S4.SS2.p4.10.m10.1.2.cmml"><mi id="S4.SS2.p4.10.m10.1.2.2" xref="S4.SS2.p4.10.m10.1.2.2.cmml">𝝃</mi><mo id="S4.SS2.p4.10.m10.1.2.1" xref="S4.SS2.p4.10.m10.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p4.10.m10.1.2.3.2" xref="S4.SS2.p4.10.m10.1.2.cmml"><mo id="S4.SS2.p4.10.m10.1.2.3.2.1" stretchy="false" xref="S4.SS2.p4.10.m10.1.2.cmml">(</mo><mi id="S4.SS2.p4.10.m10.1.1" xref="S4.SS2.p4.10.m10.1.1.cmml">t</mi><mo id="S4.SS2.p4.10.m10.1.2.3.2.2" stretchy="false" xref="S4.SS2.p4.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.10.m10.1b"><apply id="S4.SS2.p4.10.m10.1.2.cmml" xref="S4.SS2.p4.10.m10.1.2"><times id="S4.SS2.p4.10.m10.1.2.1.cmml" xref="S4.SS2.p4.10.m10.1.2.1"></times><ci id="S4.SS2.p4.10.m10.1.2.2.cmml" xref="S4.SS2.p4.10.m10.1.2.2">𝝃</ci><ci id="S4.SS2.p4.10.m10.1.1.cmml" xref="S4.SS2.p4.10.m10.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.10.m10.1c">\bm{\xi}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.10.m10.1d">bold_italic_ξ ( italic_t )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E20" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">20</span></a>). In this manner, the high-order time derivatives <math alttext="\hat{\bm{\theta}}^{(k+1)}" class="ltx_Math" display="inline" id="S4.SS2.p4.11.m11.1"><semantics id="S4.SS2.p4.11.m11.1a"><msup id="S4.SS2.p4.11.m11.1.2" xref="S4.SS2.p4.11.m11.1.2.cmml"><mover accent="true" id="S4.SS2.p4.11.m11.1.2.2" xref="S4.SS2.p4.11.m11.1.2.2.cmml"><mi id="S4.SS2.p4.11.m11.1.2.2.2" xref="S4.SS2.p4.11.m11.1.2.2.2.cmml">𝜽</mi><mo id="S4.SS2.p4.11.m11.1.2.2.1" xref="S4.SS2.p4.11.m11.1.2.2.1.cmml">^</mo></mover><mrow id="S4.SS2.p4.11.m11.1.1.1.1" xref="S4.SS2.p4.11.m11.1.1.1.1.1.cmml"><mo id="S4.SS2.p4.11.m11.1.1.1.1.2" stretchy="false" xref="S4.SS2.p4.11.m11.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.p4.11.m11.1.1.1.1.1" xref="S4.SS2.p4.11.m11.1.1.1.1.1.cmml"><mi id="S4.SS2.p4.11.m11.1.1.1.1.1.2" xref="S4.SS2.p4.11.m11.1.1.1.1.1.2.cmml">k</mi><mo id="S4.SS2.p4.11.m11.1.1.1.1.1.1" xref="S4.SS2.p4.11.m11.1.1.1.1.1.1.cmml">+</mo><mn id="S4.SS2.p4.11.m11.1.1.1.1.1.3" xref="S4.SS2.p4.11.m11.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S4.SS2.p4.11.m11.1.1.1.1.3" stretchy="false" xref="S4.SS2.p4.11.m11.1.1.1.1.1.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.11.m11.1b"><apply id="S4.SS2.p4.11.m11.1.2.cmml" xref="S4.SS2.p4.11.m11.1.2"><csymbol cd="ambiguous" id="S4.SS2.p4.11.m11.1.2.1.cmml" xref="S4.SS2.p4.11.m11.1.2">superscript</csymbol><apply id="S4.SS2.p4.11.m11.1.2.2.cmml" xref="S4.SS2.p4.11.m11.1.2.2"><ci id="S4.SS2.p4.11.m11.1.2.2.1.cmml" xref="S4.SS2.p4.11.m11.1.2.2.1">^</ci><ci id="S4.SS2.p4.11.m11.1.2.2.2.cmml" xref="S4.SS2.p4.11.m11.1.2.2.2">𝜽</ci></apply><apply id="S4.SS2.p4.11.m11.1.1.1.1.1.cmml" xref="S4.SS2.p4.11.m11.1.1.1.1"><plus id="S4.SS2.p4.11.m11.1.1.1.1.1.1.cmml" xref="S4.SS2.p4.11.m11.1.1.1.1.1.1"></plus><ci id="S4.SS2.p4.11.m11.1.1.1.1.1.2.cmml" xref="S4.SS2.p4.11.m11.1.1.1.1.1.2">𝑘</ci><cn id="S4.SS2.p4.11.m11.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.p4.11.m11.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.11.m11.1c">\hat{\bm{\theta}}^{(k+1)}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.11.m11.1d">over^ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT ( italic_k + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E25" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">25</span></a>) and the control law (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E6" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">6</span></a>) can be implemented continuously.</p> </div> <div class="ltx_para" id="S4.SS2.p5"> <p class="ltx_p" id="S4.SS2.p5.13"><span class="ltx_text ltx_font_italic" id="S4.SS2.p5.13.1">Remark 7:</span> For the system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S2.E4" title="In II Problem Formulation ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">4</span></a>) with slowly time-varying parameter case, forgetting previous information and using new data are essential to ensure closed-loop stability with parameter convergence <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib30" title="">30</a>]</cite>. In the composite learning-based HOT (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>), the generalized prediction error <math alttext="\bm{\xi}(t)" class="ltx_Math" display="inline" id="S4.SS2.p5.1.m1.1"><semantics id="S4.SS2.p5.1.m1.1a"><mrow id="S4.SS2.p5.1.m1.1.2" xref="S4.SS2.p5.1.m1.1.2.cmml"><mi id="S4.SS2.p5.1.m1.1.2.2" xref="S4.SS2.p5.1.m1.1.2.2.cmml">𝝃</mi><mo id="S4.SS2.p5.1.m1.1.2.1" xref="S4.SS2.p5.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p5.1.m1.1.2.3.2" xref="S4.SS2.p5.1.m1.1.2.cmml"><mo id="S4.SS2.p5.1.m1.1.2.3.2.1" stretchy="false" xref="S4.SS2.p5.1.m1.1.2.cmml">(</mo><mi id="S4.SS2.p5.1.m1.1.1" xref="S4.SS2.p5.1.m1.1.1.cmml">t</mi><mo id="S4.SS2.p5.1.m1.1.2.3.2.2" stretchy="false" xref="S4.SS2.p5.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.1.m1.1b"><apply id="S4.SS2.p5.1.m1.1.2.cmml" xref="S4.SS2.p5.1.m1.1.2"><times id="S4.SS2.p5.1.m1.1.2.1.cmml" xref="S4.SS2.p5.1.m1.1.2.1"></times><ci id="S4.SS2.p5.1.m1.1.2.2.cmml" xref="S4.SS2.p5.1.m1.1.2.2">𝝃</ci><ci id="S4.SS2.p5.1.m1.1.1.cmml" xref="S4.SS2.p5.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.1.m1.1c">\bm{\xi}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.1.m1.1d">bold_italic_ξ ( italic_t )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E20" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">20</span></a>) should be calculated using the current data <math alttext="Q(t)" class="ltx_Math" display="inline" id="S4.SS2.p5.2.m2.1"><semantics id="S4.SS2.p5.2.m2.1a"><mrow id="S4.SS2.p5.2.m2.1.2" xref="S4.SS2.p5.2.m2.1.2.cmml"><mi id="S4.SS2.p5.2.m2.1.2.2" xref="S4.SS2.p5.2.m2.1.2.2.cmml">Q</mi><mo id="S4.SS2.p5.2.m2.1.2.1" xref="S4.SS2.p5.2.m2.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p5.2.m2.1.2.3.2" xref="S4.SS2.p5.2.m2.1.2.cmml"><mo id="S4.SS2.p5.2.m2.1.2.3.2.1" stretchy="false" xref="S4.SS2.p5.2.m2.1.2.cmml">(</mo><mi id="S4.SS2.p5.2.m2.1.1" xref="S4.SS2.p5.2.m2.1.1.cmml">t</mi><mo id="S4.SS2.p5.2.m2.1.2.3.2.2" stretchy="false" xref="S4.SS2.p5.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.2.m2.1b"><apply id="S4.SS2.p5.2.m2.1.2.cmml" xref="S4.SS2.p5.2.m2.1.2"><times id="S4.SS2.p5.2.m2.1.2.1.cmml" xref="S4.SS2.p5.2.m2.1.2.1"></times><ci id="S4.SS2.p5.2.m2.1.2.2.cmml" xref="S4.SS2.p5.2.m2.1.2.2">𝑄</ci><ci id="S4.SS2.p5.2.m2.1.1.cmml" xref="S4.SS2.p5.2.m2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.2.m2.1c">Q(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.2.m2.1d">italic_Q ( italic_t )</annotation></semantics></math> and <math alttext="{\bm{q}}_{\rm f}(t)" class="ltx_Math" display="inline" id="S4.SS2.p5.3.m3.1"><semantics id="S4.SS2.p5.3.m3.1a"><mrow id="S4.SS2.p5.3.m3.1.2" xref="S4.SS2.p5.3.m3.1.2.cmml"><msub id="S4.SS2.p5.3.m3.1.2.2" xref="S4.SS2.p5.3.m3.1.2.2.cmml"><mi id="S4.SS2.p5.3.m3.1.2.2.2" xref="S4.SS2.p5.3.m3.1.2.2.2.cmml">𝒒</mi><mi id="S4.SS2.p5.3.m3.1.2.2.3" mathvariant="normal" xref="S4.SS2.p5.3.m3.1.2.2.3.cmml">f</mi></msub><mo id="S4.SS2.p5.3.m3.1.2.1" xref="S4.SS2.p5.3.m3.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p5.3.m3.1.2.3.2" xref="S4.SS2.p5.3.m3.1.2.cmml"><mo id="S4.SS2.p5.3.m3.1.2.3.2.1" stretchy="false" xref="S4.SS2.p5.3.m3.1.2.cmml">(</mo><mi id="S4.SS2.p5.3.m3.1.1" xref="S4.SS2.p5.3.m3.1.1.cmml">t</mi><mo id="S4.SS2.p5.3.m3.1.2.3.2.2" stretchy="false" xref="S4.SS2.p5.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.3.m3.1b"><apply id="S4.SS2.p5.3.m3.1.2.cmml" xref="S4.SS2.p5.3.m3.1.2"><times id="S4.SS2.p5.3.m3.1.2.1.cmml" xref="S4.SS2.p5.3.m3.1.2.1"></times><apply id="S4.SS2.p5.3.m3.1.2.2.cmml" xref="S4.SS2.p5.3.m3.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.p5.3.m3.1.2.2.1.cmml" xref="S4.SS2.p5.3.m3.1.2.2">subscript</csymbol><ci id="S4.SS2.p5.3.m3.1.2.2.2.cmml" xref="S4.SS2.p5.3.m3.1.2.2.2">𝒒</ci><ci id="S4.SS2.p5.3.m3.1.2.2.3.cmml" xref="S4.SS2.p5.3.m3.1.2.2.3">f</ci></apply><ci id="S4.SS2.p5.3.m3.1.1.cmml" xref="S4.SS2.p5.3.m3.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.3.m3.1c">{\bm{q}}_{\rm f}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.3.m3.1d">bold_italic_q start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> with the current exciting strength <math alttext="\sigma_{\min}(\Psi(t))" class="ltx_Math" display="inline" id="S4.SS2.p5.4.m4.2"><semantics id="S4.SS2.p5.4.m4.2a"><mrow id="S4.SS2.p5.4.m4.2.2" xref="S4.SS2.p5.4.m4.2.2.cmml"><msub id="S4.SS2.p5.4.m4.2.2.3" xref="S4.SS2.p5.4.m4.2.2.3.cmml"><mi id="S4.SS2.p5.4.m4.2.2.3.2" xref="S4.SS2.p5.4.m4.2.2.3.2.cmml">σ</mi><mi id="S4.SS2.p5.4.m4.2.2.3.3" xref="S4.SS2.p5.4.m4.2.2.3.3.cmml">min</mi></msub><mo id="S4.SS2.p5.4.m4.2.2.2" xref="S4.SS2.p5.4.m4.2.2.2.cmml">⁢</mo><mrow id="S4.SS2.p5.4.m4.2.2.1.1" xref="S4.SS2.p5.4.m4.2.2.1.1.1.cmml"><mo id="S4.SS2.p5.4.m4.2.2.1.1.2" stretchy="false" xref="S4.SS2.p5.4.m4.2.2.1.1.1.cmml">(</mo><mrow id="S4.SS2.p5.4.m4.2.2.1.1.1" xref="S4.SS2.p5.4.m4.2.2.1.1.1.cmml"><mi id="S4.SS2.p5.4.m4.2.2.1.1.1.2" mathvariant="normal" xref="S4.SS2.p5.4.m4.2.2.1.1.1.2.cmml">Ψ</mi><mo id="S4.SS2.p5.4.m4.2.2.1.1.1.1" xref="S4.SS2.p5.4.m4.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS2.p5.4.m4.2.2.1.1.1.3.2" xref="S4.SS2.p5.4.m4.2.2.1.1.1.cmml"><mo id="S4.SS2.p5.4.m4.2.2.1.1.1.3.2.1" stretchy="false" xref="S4.SS2.p5.4.m4.2.2.1.1.1.cmml">(</mo><mi id="S4.SS2.p5.4.m4.1.1" xref="S4.SS2.p5.4.m4.1.1.cmml">t</mi><mo id="S4.SS2.p5.4.m4.2.2.1.1.1.3.2.2" stretchy="false" xref="S4.SS2.p5.4.m4.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.p5.4.m4.2.2.1.1.3" stretchy="false" xref="S4.SS2.p5.4.m4.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.4.m4.2b"><apply id="S4.SS2.p5.4.m4.2.2.cmml" xref="S4.SS2.p5.4.m4.2.2"><times id="S4.SS2.p5.4.m4.2.2.2.cmml" xref="S4.SS2.p5.4.m4.2.2.2"></times><apply id="S4.SS2.p5.4.m4.2.2.3.cmml" xref="S4.SS2.p5.4.m4.2.2.3"><csymbol cd="ambiguous" id="S4.SS2.p5.4.m4.2.2.3.1.cmml" xref="S4.SS2.p5.4.m4.2.2.3">subscript</csymbol><ci id="S4.SS2.p5.4.m4.2.2.3.2.cmml" xref="S4.SS2.p5.4.m4.2.2.3.2">𝜎</ci><min id="S4.SS2.p5.4.m4.2.2.3.3.cmml" xref="S4.SS2.p5.4.m4.2.2.3.3"></min></apply><apply id="S4.SS2.p5.4.m4.2.2.1.1.1.cmml" xref="S4.SS2.p5.4.m4.2.2.1.1"><times id="S4.SS2.p5.4.m4.2.2.1.1.1.1.cmml" xref="S4.SS2.p5.4.m4.2.2.1.1.1.1"></times><ci id="S4.SS2.p5.4.m4.2.2.1.1.1.2.cmml" xref="S4.SS2.p5.4.m4.2.2.1.1.1.2">Ψ</ci><ci id="S4.SS2.p5.4.m4.1.1.cmml" xref="S4.SS2.p5.4.m4.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.4.m4.2c">\sigma_{\min}(\Psi(t))</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.4.m4.2d">italic_σ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Ψ ( italic_t ) )</annotation></semantics></math>, rather than relying on history data <math alttext="Q(t,t_{\rm e})" class="ltx_Math" display="inline" id="S4.SS2.p5.5.m5.2"><semantics id="S4.SS2.p5.5.m5.2a"><mrow id="S4.SS2.p5.5.m5.2.2" xref="S4.SS2.p5.5.m5.2.2.cmml"><mi id="S4.SS2.p5.5.m5.2.2.3" xref="S4.SS2.p5.5.m5.2.2.3.cmml">Q</mi><mo id="S4.SS2.p5.5.m5.2.2.2" xref="S4.SS2.p5.5.m5.2.2.2.cmml">⁢</mo><mrow id="S4.SS2.p5.5.m5.2.2.1.1" xref="S4.SS2.p5.5.m5.2.2.1.2.cmml"><mo id="S4.SS2.p5.5.m5.2.2.1.1.2" stretchy="false" xref="S4.SS2.p5.5.m5.2.2.1.2.cmml">(</mo><mi id="S4.SS2.p5.5.m5.1.1" xref="S4.SS2.p5.5.m5.1.1.cmml">t</mi><mo id="S4.SS2.p5.5.m5.2.2.1.1.3" xref="S4.SS2.p5.5.m5.2.2.1.2.cmml">,</mo><msub id="S4.SS2.p5.5.m5.2.2.1.1.1" xref="S4.SS2.p5.5.m5.2.2.1.1.1.cmml"><mi id="S4.SS2.p5.5.m5.2.2.1.1.1.2" xref="S4.SS2.p5.5.m5.2.2.1.1.1.2.cmml">t</mi><mi id="S4.SS2.p5.5.m5.2.2.1.1.1.3" mathvariant="normal" xref="S4.SS2.p5.5.m5.2.2.1.1.1.3.cmml">e</mi></msub><mo id="S4.SS2.p5.5.m5.2.2.1.1.4" stretchy="false" xref="S4.SS2.p5.5.m5.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.5.m5.2b"><apply id="S4.SS2.p5.5.m5.2.2.cmml" xref="S4.SS2.p5.5.m5.2.2"><times id="S4.SS2.p5.5.m5.2.2.2.cmml" xref="S4.SS2.p5.5.m5.2.2.2"></times><ci id="S4.SS2.p5.5.m5.2.2.3.cmml" xref="S4.SS2.p5.5.m5.2.2.3">𝑄</ci><interval closure="open" id="S4.SS2.p5.5.m5.2.2.1.2.cmml" xref="S4.SS2.p5.5.m5.2.2.1.1"><ci id="S4.SS2.p5.5.m5.1.1.cmml" xref="S4.SS2.p5.5.m5.1.1">𝑡</ci><apply id="S4.SS2.p5.5.m5.2.2.1.1.1.cmml" xref="S4.SS2.p5.5.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.5.m5.2.2.1.1.1.1.cmml" xref="S4.SS2.p5.5.m5.2.2.1.1.1">subscript</csymbol><ci id="S4.SS2.p5.5.m5.2.2.1.1.1.2.cmml" xref="S4.SS2.p5.5.m5.2.2.1.1.1.2">𝑡</ci><ci id="S4.SS2.p5.5.m5.2.2.1.1.1.3.cmml" xref="S4.SS2.p5.5.m5.2.2.1.1.1.3">e</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.5.m5.2c">Q(t,t_{\rm e})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.5.m5.2d">italic_Q ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="{\bm{q}}_{\rm f}(t_{\rm e})" class="ltx_Math" display="inline" id="S4.SS2.p5.6.m6.1"><semantics id="S4.SS2.p5.6.m6.1a"><mrow id="S4.SS2.p5.6.m6.1.1" xref="S4.SS2.p5.6.m6.1.1.cmml"><msub id="S4.SS2.p5.6.m6.1.1.3" xref="S4.SS2.p5.6.m6.1.1.3.cmml"><mi id="S4.SS2.p5.6.m6.1.1.3.2" xref="S4.SS2.p5.6.m6.1.1.3.2.cmml">𝒒</mi><mi id="S4.SS2.p5.6.m6.1.1.3.3" mathvariant="normal" xref="S4.SS2.p5.6.m6.1.1.3.3.cmml">f</mi></msub><mo id="S4.SS2.p5.6.m6.1.1.2" xref="S4.SS2.p5.6.m6.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p5.6.m6.1.1.1.1" xref="S4.SS2.p5.6.m6.1.1.1.1.1.cmml"><mo id="S4.SS2.p5.6.m6.1.1.1.1.2" stretchy="false" xref="S4.SS2.p5.6.m6.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.p5.6.m6.1.1.1.1.1" xref="S4.SS2.p5.6.m6.1.1.1.1.1.cmml"><mi id="S4.SS2.p5.6.m6.1.1.1.1.1.2" xref="S4.SS2.p5.6.m6.1.1.1.1.1.2.cmml">t</mi><mi id="S4.SS2.p5.6.m6.1.1.1.1.1.3" mathvariant="normal" xref="S4.SS2.p5.6.m6.1.1.1.1.1.3.cmml">e</mi></msub><mo id="S4.SS2.p5.6.m6.1.1.1.1.3" stretchy="false" xref="S4.SS2.p5.6.m6.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.6.m6.1b"><apply id="S4.SS2.p5.6.m6.1.1.cmml" xref="S4.SS2.p5.6.m6.1.1"><times id="S4.SS2.p5.6.m6.1.1.2.cmml" xref="S4.SS2.p5.6.m6.1.1.2"></times><apply id="S4.SS2.p5.6.m6.1.1.3.cmml" xref="S4.SS2.p5.6.m6.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p5.6.m6.1.1.3.1.cmml" xref="S4.SS2.p5.6.m6.1.1.3">subscript</csymbol><ci id="S4.SS2.p5.6.m6.1.1.3.2.cmml" xref="S4.SS2.p5.6.m6.1.1.3.2">𝒒</ci><ci id="S4.SS2.p5.6.m6.1.1.3.3.cmml" xref="S4.SS2.p5.6.m6.1.1.3.3">f</ci></apply><apply id="S4.SS2.p5.6.m6.1.1.1.1.1.cmml" xref="S4.SS2.p5.6.m6.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.6.m6.1.1.1.1.1.1.cmml" xref="S4.SS2.p5.6.m6.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p5.6.m6.1.1.1.1.1.2.cmml" xref="S4.SS2.p5.6.m6.1.1.1.1.1.2">𝑡</ci><ci id="S4.SS2.p5.6.m6.1.1.1.1.1.3.cmml" xref="S4.SS2.p5.6.m6.1.1.1.1.1.3">e</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.6.m6.1c">{\bm{q}}_{\rm f}(t_{\rm e})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.6.m6.1d">bold_italic_q start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT )</annotation></semantics></math> to maximal exciting strength <math alttext="\sigma_{\rm c}" class="ltx_Math" display="inline" id="S4.SS2.p5.7.m7.1"><semantics id="S4.SS2.p5.7.m7.1a"><msub id="S4.SS2.p5.7.m7.1.1" xref="S4.SS2.p5.7.m7.1.1.cmml"><mi id="S4.SS2.p5.7.m7.1.1.2" xref="S4.SS2.p5.7.m7.1.1.2.cmml">σ</mi><mi id="S4.SS2.p5.7.m7.1.1.3" mathvariant="normal" xref="S4.SS2.p5.7.m7.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.7.m7.1b"><apply id="S4.SS2.p5.7.m7.1.1.cmml" xref="S4.SS2.p5.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.7.m7.1.1.1.cmml" xref="S4.SS2.p5.7.m7.1.1">subscript</csymbol><ci id="S4.SS2.p5.7.m7.1.1.2.cmml" xref="S4.SS2.p5.7.m7.1.1.2">𝜎</ci><ci id="S4.SS2.p5.7.m7.1.1.3.cmml" xref="S4.SS2.p5.7.m7.1.1.3">c</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.7.m7.1c">\sigma_{\rm c}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.7.m7.1d">italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT</annotation></semantics></math>. Further, the proposed CLBC with online data memory can be adapted automatically by observing the general filtered prediction error <math alttext="\bm{\epsilon}" class="ltx_Math" display="inline" id="S4.SS2.p5.8.m8.1"><semantics id="S4.SS2.p5.8.m8.1a"><mi class="ltx_mathvariant_bold-italic" id="S4.SS2.p5.8.m8.1.1" mathvariant="bold-italic" xref="S4.SS2.p5.8.m8.1.1.cmml">ϵ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.8.m8.1b"><ci id="S4.SS2.p5.8.m8.1.1.cmml" xref="S4.SS2.p5.8.m8.1.1">bold-italic-ϵ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.8.m8.1c">\bm{\epsilon}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.8.m8.1d">bold_italic_ϵ</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E23" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">23</span></a>) to distinguish between the constant <math alttext="\bm{\theta}" class="ltx_Math" display="inline" id="S4.SS2.p5.9.m9.1"><semantics id="S4.SS2.p5.9.m9.1a"><mi id="S4.SS2.p5.9.m9.1.1" xref="S4.SS2.p5.9.m9.1.1.cmml">𝜽</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.9.m9.1b"><ci id="S4.SS2.p5.9.m9.1.1.cmml" xref="S4.SS2.p5.9.m9.1.1">𝜽</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.9.m9.1c">\bm{\theta}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.9.m9.1d">bold_italic_θ</annotation></semantics></math> and the time-varying <math alttext="{\bm{\theta}}(t)" class="ltx_Math" display="inline" id="S4.SS2.p5.10.m10.1"><semantics id="S4.SS2.p5.10.m10.1a"><mrow id="S4.SS2.p5.10.m10.1.2" xref="S4.SS2.p5.10.m10.1.2.cmml"><mi id="S4.SS2.p5.10.m10.1.2.2" xref="S4.SS2.p5.10.m10.1.2.2.cmml">𝜽</mi><mo id="S4.SS2.p5.10.m10.1.2.1" xref="S4.SS2.p5.10.m10.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p5.10.m10.1.2.3.2" xref="S4.SS2.p5.10.m10.1.2.cmml"><mo id="S4.SS2.p5.10.m10.1.2.3.2.1" stretchy="false" xref="S4.SS2.p5.10.m10.1.2.cmml">(</mo><mi id="S4.SS2.p5.10.m10.1.1" xref="S4.SS2.p5.10.m10.1.1.cmml">t</mi><mo id="S4.SS2.p5.10.m10.1.2.3.2.2" stretchy="false" xref="S4.SS2.p5.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.10.m10.1b"><apply id="S4.SS2.p5.10.m10.1.2.cmml" xref="S4.SS2.p5.10.m10.1.2"><times id="S4.SS2.p5.10.m10.1.2.1.cmml" xref="S4.SS2.p5.10.m10.1.2.1"></times><ci id="S4.SS2.p5.10.m10.1.2.2.cmml" xref="S4.SS2.p5.10.m10.1.2.2">𝜽</ci><ci id="S4.SS2.p5.10.m10.1.1.cmml" xref="S4.SS2.p5.10.m10.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.10.m10.1c">{\bm{\theta}}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.10.m10.1d">bold_italic_θ ( italic_t )</annotation></semantics></math>. For the constant parameter case, the prediction error <math alttext="\bm{\epsilon}" class="ltx_Math" display="inline" id="S4.SS2.p5.11.m11.1"><semantics id="S4.SS2.p5.11.m11.1a"><mi class="ltx_mathvariant_bold-italic" id="S4.SS2.p5.11.m11.1.1" mathvariant="bold-italic" xref="S4.SS2.p5.11.m11.1.1.cmml">ϵ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.11.m11.1b"><ci id="S4.SS2.p5.11.m11.1.1.cmml" xref="S4.SS2.p5.11.m11.1.1">bold-italic-ϵ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.11.m11.1c">\bm{\epsilon}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.11.m11.1d">bold_italic_ϵ</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E23" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">23</span></a>) always converges to <math alttext="\bm{0}" class="ltx_Math" display="inline" id="S4.SS2.p5.12.m12.1"><semantics id="S4.SS2.p5.12.m12.1a"><mn id="S4.SS2.p5.12.m12.1.1" xref="S4.SS2.p5.12.m12.1.1.cmml">𝟎</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.12.m12.1b"><cn id="S4.SS2.p5.12.m12.1.1.cmml" type="integer" xref="S4.SS2.p5.12.m12.1.1">0</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.12.m12.1c">\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.12.m12.1d">bold_0</annotation></semantics></math>, whereas for the time-varying parameter case, the prediction error <math alttext="\bm{\epsilon}" class="ltx_Math" display="inline" id="S4.SS2.p5.13.m13.1"><semantics id="S4.SS2.p5.13.m13.1a"><mi class="ltx_mathvariant_bold-italic" id="S4.SS2.p5.13.m13.1.1" mathvariant="bold-italic" xref="S4.SS2.p5.13.m13.1.1.cmml">ϵ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.13.m13.1b"><ci id="S4.SS2.p5.13.m13.1.1.cmml" xref="S4.SS2.p5.13.m13.1.1">bold-italic-ϵ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.13.m13.1c">\bm{\epsilon}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.13.m13.1d">bold_italic_ϵ</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E23" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">23</span></a>) fluctuates around zero.</p> </div> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">V </span><span class="ltx_text ltx_font_smallcaps" id="S5.1.1">Theoretical Guarantees</span> </h2> <section class="ltx_subsection" id="S5.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S5.SS1.4.1.1">V-A</span> </span><span class="ltx_text ltx_font_italic" id="S5.SS1.5.2">Parameter Convergence Results</span> </h3> <div class="ltx_para" id="S5.SS1.p1"> <p class="ltx_p" id="S5.SS1.p1.11">Let <math alttext="\Phi_{{\rm f},\zeta}" class="ltx_Math" display="inline" id="S5.SS1.p1.1.m1.2"><semantics id="S5.SS1.p1.1.m1.2a"><msub id="S5.SS1.p1.1.m1.2.3" xref="S5.SS1.p1.1.m1.2.3.cmml"><mi id="S5.SS1.p1.1.m1.2.3.2" mathvariant="normal" xref="S5.SS1.p1.1.m1.2.3.2.cmml">Φ</mi><mrow id="S5.SS1.p1.1.m1.2.2.2.4" xref="S5.SS1.p1.1.m1.2.2.2.3.cmml"><mi id="S5.SS1.p1.1.m1.1.1.1.1" mathvariant="normal" xref="S5.SS1.p1.1.m1.1.1.1.1.cmml">f</mi><mo id="S5.SS1.p1.1.m1.2.2.2.4.1" xref="S5.SS1.p1.1.m1.2.2.2.3.cmml">,</mo><mi id="S5.SS1.p1.1.m1.2.2.2.2" xref="S5.SS1.p1.1.m1.2.2.2.2.cmml">ζ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.1.m1.2b"><apply id="S5.SS1.p1.1.m1.2.3.cmml" xref="S5.SS1.p1.1.m1.2.3"><csymbol cd="ambiguous" id="S5.SS1.p1.1.m1.2.3.1.cmml" xref="S5.SS1.p1.1.m1.2.3">subscript</csymbol><ci id="S5.SS1.p1.1.m1.2.3.2.cmml" xref="S5.SS1.p1.1.m1.2.3.2">Φ</ci><list id="S5.SS1.p1.1.m1.2.2.2.3.cmml" xref="S5.SS1.p1.1.m1.2.2.2.4"><ci id="S5.SS1.p1.1.m1.1.1.1.1.cmml" xref="S5.SS1.p1.1.m1.1.1.1.1">f</ci><ci id="S5.SS1.p1.1.m1.2.2.2.2.cmml" xref="S5.SS1.p1.1.m1.2.2.2.2">𝜁</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.1.m1.2c">\Phi_{{\rm f},\zeta}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.1.m1.2d">roman_Φ start_POSTSUBSCRIPT roman_f , italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S5.SS1.p1.2.m2.1"><semantics id="S5.SS1.p1.2.m2.1a"><mo id="S5.SS1.p1.2.m2.1.1" xref="S5.SS1.p1.2.m2.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.2.m2.1b"><csymbol cd="latexml" id="S5.SS1.p1.2.m2.1.1.cmml" xref="S5.SS1.p1.2.m2.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.2.m2.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.2.m2.1d">:=</annotation></semantics></math> <math alttext="[\bm{\phi}_{{\rm f},k_{1}},\bm{\phi}_{{\rm f},k_{2}},\cdots,\bm{\phi}_{{\rm f}% ,k_{N_{\zeta}}}]^{T}" class="ltx_Math" display="inline" id="S5.SS1.p1.3.m3.10"><semantics id="S5.SS1.p1.3.m3.10a"><msup id="S5.SS1.p1.3.m3.10.10" xref="S5.SS1.p1.3.m3.10.10.cmml"><mrow id="S5.SS1.p1.3.m3.10.10.3.3" xref="S5.SS1.p1.3.m3.10.10.3.4.cmml"><mo id="S5.SS1.p1.3.m3.10.10.3.3.4" stretchy="false" xref="S5.SS1.p1.3.m3.10.10.3.4.cmml">[</mo><msub id="S5.SS1.p1.3.m3.8.8.1.1.1" xref="S5.SS1.p1.3.m3.8.8.1.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S5.SS1.p1.3.m3.8.8.1.1.1.2" mathvariant="bold-italic" xref="S5.SS1.p1.3.m3.8.8.1.1.1.2.cmml">ϕ</mi><mrow id="S5.SS1.p1.3.m3.2.2.2.2" xref="S5.SS1.p1.3.m3.2.2.2.3.cmml"><mi id="S5.SS1.p1.3.m3.1.1.1.1" mathvariant="normal" xref="S5.SS1.p1.3.m3.1.1.1.1.cmml">f</mi><mo id="S5.SS1.p1.3.m3.2.2.2.2.2" xref="S5.SS1.p1.3.m3.2.2.2.3.cmml">,</mo><msub id="S5.SS1.p1.3.m3.2.2.2.2.1" xref="S5.SS1.p1.3.m3.2.2.2.2.1.cmml"><mi id="S5.SS1.p1.3.m3.2.2.2.2.1.2" xref="S5.SS1.p1.3.m3.2.2.2.2.1.2.cmml">k</mi><mn id="S5.SS1.p1.3.m3.2.2.2.2.1.3" xref="S5.SS1.p1.3.m3.2.2.2.2.1.3.cmml">1</mn></msub></mrow></msub><mo id="S5.SS1.p1.3.m3.10.10.3.3.5" xref="S5.SS1.p1.3.m3.10.10.3.4.cmml">,</mo><msub id="S5.SS1.p1.3.m3.9.9.2.2.2" xref="S5.SS1.p1.3.m3.9.9.2.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S5.SS1.p1.3.m3.9.9.2.2.2.2" mathvariant="bold-italic" xref="S5.SS1.p1.3.m3.9.9.2.2.2.2.cmml">ϕ</mi><mrow id="S5.SS1.p1.3.m3.4.4.2.2" xref="S5.SS1.p1.3.m3.4.4.2.3.cmml"><mi id="S5.SS1.p1.3.m3.3.3.1.1" mathvariant="normal" xref="S5.SS1.p1.3.m3.3.3.1.1.cmml">f</mi><mo id="S5.SS1.p1.3.m3.4.4.2.2.2" xref="S5.SS1.p1.3.m3.4.4.2.3.cmml">,</mo><msub id="S5.SS1.p1.3.m3.4.4.2.2.1" xref="S5.SS1.p1.3.m3.4.4.2.2.1.cmml"><mi id="S5.SS1.p1.3.m3.4.4.2.2.1.2" xref="S5.SS1.p1.3.m3.4.4.2.2.1.2.cmml">k</mi><mn id="S5.SS1.p1.3.m3.4.4.2.2.1.3" xref="S5.SS1.p1.3.m3.4.4.2.2.1.3.cmml">2</mn></msub></mrow></msub><mo id="S5.SS1.p1.3.m3.10.10.3.3.6" xref="S5.SS1.p1.3.m3.10.10.3.4.cmml">,</mo><mi id="S5.SS1.p1.3.m3.7.7" mathvariant="normal" xref="S5.SS1.p1.3.m3.7.7.cmml">⋯</mi><mo id="S5.SS1.p1.3.m3.10.10.3.3.7" xref="S5.SS1.p1.3.m3.10.10.3.4.cmml">,</mo><msub id="S5.SS1.p1.3.m3.10.10.3.3.3" xref="S5.SS1.p1.3.m3.10.10.3.3.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S5.SS1.p1.3.m3.10.10.3.3.3.2" mathvariant="bold-italic" xref="S5.SS1.p1.3.m3.10.10.3.3.3.2.cmml">ϕ</mi><mrow id="S5.SS1.p1.3.m3.6.6.2.2" xref="S5.SS1.p1.3.m3.6.6.2.3.cmml"><mi id="S5.SS1.p1.3.m3.5.5.1.1" mathvariant="normal" xref="S5.SS1.p1.3.m3.5.5.1.1.cmml">f</mi><mo id="S5.SS1.p1.3.m3.6.6.2.2.2" xref="S5.SS1.p1.3.m3.6.6.2.3.cmml">,</mo><msub id="S5.SS1.p1.3.m3.6.6.2.2.1" xref="S5.SS1.p1.3.m3.6.6.2.2.1.cmml"><mi id="S5.SS1.p1.3.m3.6.6.2.2.1.2" xref="S5.SS1.p1.3.m3.6.6.2.2.1.2.cmml">k</mi><msub id="S5.SS1.p1.3.m3.6.6.2.2.1.3" xref="S5.SS1.p1.3.m3.6.6.2.2.1.3.cmml"><mi id="S5.SS1.p1.3.m3.6.6.2.2.1.3.2" xref="S5.SS1.p1.3.m3.6.6.2.2.1.3.2.cmml">N</mi><mi id="S5.SS1.p1.3.m3.6.6.2.2.1.3.3" xref="S5.SS1.p1.3.m3.6.6.2.2.1.3.3.cmml">ζ</mi></msub></msub></mrow></msub><mo id="S5.SS1.p1.3.m3.10.10.3.3.8" stretchy="false" xref="S5.SS1.p1.3.m3.10.10.3.4.cmml">]</mo></mrow><mi id="S5.SS1.p1.3.m3.10.10.5" xref="S5.SS1.p1.3.m3.10.10.5.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.3.m3.10b"><apply id="S5.SS1.p1.3.m3.10.10.cmml" xref="S5.SS1.p1.3.m3.10.10"><csymbol cd="ambiguous" id="S5.SS1.p1.3.m3.10.10.4.cmml" xref="S5.SS1.p1.3.m3.10.10">superscript</csymbol><list id="S5.SS1.p1.3.m3.10.10.3.4.cmml" xref="S5.SS1.p1.3.m3.10.10.3.3"><apply id="S5.SS1.p1.3.m3.8.8.1.1.1.cmml" xref="S5.SS1.p1.3.m3.8.8.1.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.3.m3.8.8.1.1.1.1.cmml" xref="S5.SS1.p1.3.m3.8.8.1.1.1">subscript</csymbol><ci id="S5.SS1.p1.3.m3.8.8.1.1.1.2.cmml" xref="S5.SS1.p1.3.m3.8.8.1.1.1.2">bold-italic-ϕ</ci><list id="S5.SS1.p1.3.m3.2.2.2.3.cmml" xref="S5.SS1.p1.3.m3.2.2.2.2"><ci id="S5.SS1.p1.3.m3.1.1.1.1.cmml" xref="S5.SS1.p1.3.m3.1.1.1.1">f</ci><apply id="S5.SS1.p1.3.m3.2.2.2.2.1.cmml" xref="S5.SS1.p1.3.m3.2.2.2.2.1"><csymbol cd="ambiguous" id="S5.SS1.p1.3.m3.2.2.2.2.1.1.cmml" xref="S5.SS1.p1.3.m3.2.2.2.2.1">subscript</csymbol><ci id="S5.SS1.p1.3.m3.2.2.2.2.1.2.cmml" xref="S5.SS1.p1.3.m3.2.2.2.2.1.2">𝑘</ci><cn id="S5.SS1.p1.3.m3.2.2.2.2.1.3.cmml" type="integer" xref="S5.SS1.p1.3.m3.2.2.2.2.1.3">1</cn></apply></list></apply><apply id="S5.SS1.p1.3.m3.9.9.2.2.2.cmml" xref="S5.SS1.p1.3.m3.9.9.2.2.2"><csymbol cd="ambiguous" id="S5.SS1.p1.3.m3.9.9.2.2.2.1.cmml" xref="S5.SS1.p1.3.m3.9.9.2.2.2">subscript</csymbol><ci id="S5.SS1.p1.3.m3.9.9.2.2.2.2.cmml" xref="S5.SS1.p1.3.m3.9.9.2.2.2.2">bold-italic-ϕ</ci><list id="S5.SS1.p1.3.m3.4.4.2.3.cmml" xref="S5.SS1.p1.3.m3.4.4.2.2"><ci id="S5.SS1.p1.3.m3.3.3.1.1.cmml" xref="S5.SS1.p1.3.m3.3.3.1.1">f</ci><apply id="S5.SS1.p1.3.m3.4.4.2.2.1.cmml" xref="S5.SS1.p1.3.m3.4.4.2.2.1"><csymbol cd="ambiguous" id="S5.SS1.p1.3.m3.4.4.2.2.1.1.cmml" xref="S5.SS1.p1.3.m3.4.4.2.2.1">subscript</csymbol><ci id="S5.SS1.p1.3.m3.4.4.2.2.1.2.cmml" xref="S5.SS1.p1.3.m3.4.4.2.2.1.2">𝑘</ci><cn id="S5.SS1.p1.3.m3.4.4.2.2.1.3.cmml" type="integer" xref="S5.SS1.p1.3.m3.4.4.2.2.1.3">2</cn></apply></list></apply><ci id="S5.SS1.p1.3.m3.7.7.cmml" xref="S5.SS1.p1.3.m3.7.7">⋯</ci><apply id="S5.SS1.p1.3.m3.10.10.3.3.3.cmml" xref="S5.SS1.p1.3.m3.10.10.3.3.3"><csymbol cd="ambiguous" id="S5.SS1.p1.3.m3.10.10.3.3.3.1.cmml" xref="S5.SS1.p1.3.m3.10.10.3.3.3">subscript</csymbol><ci id="S5.SS1.p1.3.m3.10.10.3.3.3.2.cmml" xref="S5.SS1.p1.3.m3.10.10.3.3.3.2">bold-italic-ϕ</ci><list id="S5.SS1.p1.3.m3.6.6.2.3.cmml" xref="S5.SS1.p1.3.m3.6.6.2.2"><ci id="S5.SS1.p1.3.m3.5.5.1.1.cmml" xref="S5.SS1.p1.3.m3.5.5.1.1">f</ci><apply id="S5.SS1.p1.3.m3.6.6.2.2.1.cmml" xref="S5.SS1.p1.3.m3.6.6.2.2.1"><csymbol cd="ambiguous" id="S5.SS1.p1.3.m3.6.6.2.2.1.1.cmml" xref="S5.SS1.p1.3.m3.6.6.2.2.1">subscript</csymbol><ci id="S5.SS1.p1.3.m3.6.6.2.2.1.2.cmml" xref="S5.SS1.p1.3.m3.6.6.2.2.1.2">𝑘</ci><apply id="S5.SS1.p1.3.m3.6.6.2.2.1.3.cmml" xref="S5.SS1.p1.3.m3.6.6.2.2.1.3"><csymbol cd="ambiguous" id="S5.SS1.p1.3.m3.6.6.2.2.1.3.1.cmml" xref="S5.SS1.p1.3.m3.6.6.2.2.1.3">subscript</csymbol><ci id="S5.SS1.p1.3.m3.6.6.2.2.1.3.2.cmml" xref="S5.SS1.p1.3.m3.6.6.2.2.1.3.2">𝑁</ci><ci id="S5.SS1.p1.3.m3.6.6.2.2.1.3.3.cmml" xref="S5.SS1.p1.3.m3.6.6.2.2.1.3.3">𝜁</ci></apply></apply></list></apply></list><ci id="S5.SS1.p1.3.m3.10.10.5.cmml" xref="S5.SS1.p1.3.m3.10.10.5">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.3.m3.10c">[\bm{\phi}_{{\rm f},k_{1}},\bm{\phi}_{{\rm f},k_{2}},\cdots,\bm{\phi}_{{\rm f}% ,k_{N_{\zeta}}}]^{T}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.3.m3.10d">[ bold_italic_ϕ start_POSTSUBSCRIPT roman_f , italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , bold_italic_ϕ start_POSTSUBSCRIPT roman_f , italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , ⋯ , bold_italic_ϕ start_POSTSUBSCRIPT roman_f , italic_k start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> <math alttext="\in\mathbb{R}^{N_{\zeta}\times n}" class="ltx_Math" display="inline" id="S5.SS1.p1.4.m4.1"><semantics id="S5.SS1.p1.4.m4.1a"><mrow id="S5.SS1.p1.4.m4.1.1" xref="S5.SS1.p1.4.m4.1.1.cmml"><mi id="S5.SS1.p1.4.m4.1.1.2" xref="S5.SS1.p1.4.m4.1.1.2.cmml"></mi><mo id="S5.SS1.p1.4.m4.1.1.1" xref="S5.SS1.p1.4.m4.1.1.1.cmml">∈</mo><msup id="S5.SS1.p1.4.m4.1.1.3" xref="S5.SS1.p1.4.m4.1.1.3.cmml"><mi id="S5.SS1.p1.4.m4.1.1.3.2" xref="S5.SS1.p1.4.m4.1.1.3.2.cmml">ℝ</mi><mrow id="S5.SS1.p1.4.m4.1.1.3.3" xref="S5.SS1.p1.4.m4.1.1.3.3.cmml"><msub id="S5.SS1.p1.4.m4.1.1.3.3.2" xref="S5.SS1.p1.4.m4.1.1.3.3.2.cmml"><mi id="S5.SS1.p1.4.m4.1.1.3.3.2.2" xref="S5.SS1.p1.4.m4.1.1.3.3.2.2.cmml">N</mi><mi id="S5.SS1.p1.4.m4.1.1.3.3.2.3" xref="S5.SS1.p1.4.m4.1.1.3.3.2.3.cmml">ζ</mi></msub><mo id="S5.SS1.p1.4.m4.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S5.SS1.p1.4.m4.1.1.3.3.1.cmml">×</mo><mi id="S5.SS1.p1.4.m4.1.1.3.3.3" xref="S5.SS1.p1.4.m4.1.1.3.3.3.cmml">n</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.4.m4.1b"><apply id="S5.SS1.p1.4.m4.1.1.cmml" xref="S5.SS1.p1.4.m4.1.1"><in id="S5.SS1.p1.4.m4.1.1.1.cmml" xref="S5.SS1.p1.4.m4.1.1.1"></in><csymbol cd="latexml" id="S5.SS1.p1.4.m4.1.1.2.cmml" xref="S5.SS1.p1.4.m4.1.1.2">absent</csymbol><apply id="S5.SS1.p1.4.m4.1.1.3.cmml" xref="S5.SS1.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S5.SS1.p1.4.m4.1.1.3.1.cmml" xref="S5.SS1.p1.4.m4.1.1.3">superscript</csymbol><ci id="S5.SS1.p1.4.m4.1.1.3.2.cmml" xref="S5.SS1.p1.4.m4.1.1.3.2">ℝ</ci><apply id="S5.SS1.p1.4.m4.1.1.3.3.cmml" xref="S5.SS1.p1.4.m4.1.1.3.3"><times id="S5.SS1.p1.4.m4.1.1.3.3.1.cmml" xref="S5.SS1.p1.4.m4.1.1.3.3.1"></times><apply id="S5.SS1.p1.4.m4.1.1.3.3.2.cmml" xref="S5.SS1.p1.4.m4.1.1.3.3.2"><csymbol cd="ambiguous" id="S5.SS1.p1.4.m4.1.1.3.3.2.1.cmml" xref="S5.SS1.p1.4.m4.1.1.3.3.2">subscript</csymbol><ci id="S5.SS1.p1.4.m4.1.1.3.3.2.2.cmml" xref="S5.SS1.p1.4.m4.1.1.3.3.2.2">𝑁</ci><ci id="S5.SS1.p1.4.m4.1.1.3.3.2.3.cmml" xref="S5.SS1.p1.4.m4.1.1.3.3.2.3">𝜁</ci></apply><ci id="S5.SS1.p1.4.m4.1.1.3.3.3.cmml" xref="S5.SS1.p1.4.m4.1.1.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.4.m4.1c">\in\mathbb{R}^{N_{\zeta}\times n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.4.m4.1d">∈ blackboard_R start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT × italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> denote an active sub-regressor of <math alttext="\Phi_{\rm f}" class="ltx_Math" display="inline" id="S5.SS1.p1.5.m5.1"><semantics id="S5.SS1.p1.5.m5.1a"><msub id="S5.SS1.p1.5.m5.1.1" xref="S5.SS1.p1.5.m5.1.1.cmml"><mi id="S5.SS1.p1.5.m5.1.1.2" mathvariant="normal" xref="S5.SS1.p1.5.m5.1.1.2.cmml">Φ</mi><mi id="S5.SS1.p1.5.m5.1.1.3" mathvariant="normal" xref="S5.SS1.p1.5.m5.1.1.3.cmml">f</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.5.m5.1b"><apply id="S5.SS1.p1.5.m5.1.1.cmml" xref="S5.SS1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.5.m5.1.1.1.cmml" xref="S5.SS1.p1.5.m5.1.1">subscript</csymbol><ci id="S5.SS1.p1.5.m5.1.1.2.cmml" xref="S5.SS1.p1.5.m5.1.1.2">Φ</ci><ci id="S5.SS1.p1.5.m5.1.1.3.cmml" xref="S5.SS1.p1.5.m5.1.1.3">f</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.5.m5.1c">\Phi_{\rm f}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.5.m5.1d">roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E22" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">22</span></a>) with <math alttext="\bm{\phi}_{{\rm f},k_{j}}" class="ltx_Math" display="inline" id="S5.SS1.p1.6.m6.2"><semantics id="S5.SS1.p1.6.m6.2a"><msub id="S5.SS1.p1.6.m6.2.3" xref="S5.SS1.p1.6.m6.2.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S5.SS1.p1.6.m6.2.3.2" mathvariant="bold-italic" xref="S5.SS1.p1.6.m6.2.3.2.cmml">ϕ</mi><mrow id="S5.SS1.p1.6.m6.2.2.2.2" xref="S5.SS1.p1.6.m6.2.2.2.3.cmml"><mi id="S5.SS1.p1.6.m6.1.1.1.1" mathvariant="normal" xref="S5.SS1.p1.6.m6.1.1.1.1.cmml">f</mi><mo id="S5.SS1.p1.6.m6.2.2.2.2.2" xref="S5.SS1.p1.6.m6.2.2.2.3.cmml">,</mo><msub id="S5.SS1.p1.6.m6.2.2.2.2.1" xref="S5.SS1.p1.6.m6.2.2.2.2.1.cmml"><mi id="S5.SS1.p1.6.m6.2.2.2.2.1.2" xref="S5.SS1.p1.6.m6.2.2.2.2.1.2.cmml">k</mi><mi id="S5.SS1.p1.6.m6.2.2.2.2.1.3" xref="S5.SS1.p1.6.m6.2.2.2.2.1.3.cmml">j</mi></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.6.m6.2b"><apply id="S5.SS1.p1.6.m6.2.3.cmml" xref="S5.SS1.p1.6.m6.2.3"><csymbol cd="ambiguous" id="S5.SS1.p1.6.m6.2.3.1.cmml" xref="S5.SS1.p1.6.m6.2.3">subscript</csymbol><ci id="S5.SS1.p1.6.m6.2.3.2.cmml" xref="S5.SS1.p1.6.m6.2.3.2">bold-italic-ϕ</ci><list id="S5.SS1.p1.6.m6.2.2.2.3.cmml" xref="S5.SS1.p1.6.m6.2.2.2.2"><ci id="S5.SS1.p1.6.m6.1.1.1.1.cmml" xref="S5.SS1.p1.6.m6.1.1.1.1">f</ci><apply id="S5.SS1.p1.6.m6.2.2.2.2.1.cmml" xref="S5.SS1.p1.6.m6.2.2.2.2.1"><csymbol cd="ambiguous" id="S5.SS1.p1.6.m6.2.2.2.2.1.1.cmml" xref="S5.SS1.p1.6.m6.2.2.2.2.1">subscript</csymbol><ci id="S5.SS1.p1.6.m6.2.2.2.2.1.2.cmml" xref="S5.SS1.p1.6.m6.2.2.2.2.1.2">𝑘</ci><ci id="S5.SS1.p1.6.m6.2.2.2.2.1.3.cmml" xref="S5.SS1.p1.6.m6.2.2.2.2.1.3">𝑗</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.6.m6.2c">\bm{\phi}_{{\rm f},k_{j}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.6.m6.2d">bold_italic_ϕ start_POSTSUBSCRIPT roman_f , italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> being the <math alttext="k_{j}" class="ltx_Math" display="inline" id="S5.SS1.p1.7.m7.1"><semantics id="S5.SS1.p1.7.m7.1a"><msub id="S5.SS1.p1.7.m7.1.1" xref="S5.SS1.p1.7.m7.1.1.cmml"><mi id="S5.SS1.p1.7.m7.1.1.2" xref="S5.SS1.p1.7.m7.1.1.2.cmml">k</mi><mi id="S5.SS1.p1.7.m7.1.1.3" xref="S5.SS1.p1.7.m7.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.7.m7.1b"><apply id="S5.SS1.p1.7.m7.1.1.cmml" xref="S5.SS1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.7.m7.1.1.1.cmml" xref="S5.SS1.p1.7.m7.1.1">subscript</csymbol><ci id="S5.SS1.p1.7.m7.1.1.2.cmml" xref="S5.SS1.p1.7.m7.1.1.2">𝑘</ci><ci id="S5.SS1.p1.7.m7.1.1.3.cmml" xref="S5.SS1.p1.7.m7.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.7.m7.1c">k_{j}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.7.m7.1d">italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>th channel of <math alttext="\Phi_{\rm f}" class="ltx_Math" display="inline" id="S5.SS1.p1.8.m8.1"><semantics id="S5.SS1.p1.8.m8.1a"><msub id="S5.SS1.p1.8.m8.1.1" xref="S5.SS1.p1.8.m8.1.1.cmml"><mi id="S5.SS1.p1.8.m8.1.1.2" mathvariant="normal" xref="S5.SS1.p1.8.m8.1.1.2.cmml">Φ</mi><mi id="S5.SS1.p1.8.m8.1.1.3" mathvariant="normal" xref="S5.SS1.p1.8.m8.1.1.3.cmml">f</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.8.m8.1b"><apply id="S5.SS1.p1.8.m8.1.1.cmml" xref="S5.SS1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.8.m8.1.1.1.cmml" xref="S5.SS1.p1.8.m8.1.1">subscript</csymbol><ci id="S5.SS1.p1.8.m8.1.1.2.cmml" xref="S5.SS1.p1.8.m8.1.1.2">Φ</ci><ci id="S5.SS1.p1.8.m8.1.1.3.cmml" xref="S5.SS1.p1.8.m8.1.1.3">f</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.8.m8.1c">\Phi_{\rm f}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.8.m8.1d">roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\tilde{\bm{\theta}}_{\zeta}" class="ltx_Math" display="inline" id="S5.SS1.p1.9.m9.1"><semantics id="S5.SS1.p1.9.m9.1a"><msub id="S5.SS1.p1.9.m9.1.1" xref="S5.SS1.p1.9.m9.1.1.cmml"><mover accent="true" id="S5.SS1.p1.9.m9.1.1.2" xref="S5.SS1.p1.9.m9.1.1.2.cmml"><mi id="S5.SS1.p1.9.m9.1.1.2.2" xref="S5.SS1.p1.9.m9.1.1.2.2.cmml">𝜽</mi><mo id="S5.SS1.p1.9.m9.1.1.2.1" xref="S5.SS1.p1.9.m9.1.1.2.1.cmml">~</mo></mover><mi id="S5.SS1.p1.9.m9.1.1.3" xref="S5.SS1.p1.9.m9.1.1.3.cmml">ζ</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.9.m9.1b"><apply id="S5.SS1.p1.9.m9.1.1.cmml" xref="S5.SS1.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.9.m9.1.1.1.cmml" xref="S5.SS1.p1.9.m9.1.1">subscript</csymbol><apply id="S5.SS1.p1.9.m9.1.1.2.cmml" xref="S5.SS1.p1.9.m9.1.1.2"><ci id="S5.SS1.p1.9.m9.1.1.2.1.cmml" xref="S5.SS1.p1.9.m9.1.1.2.1">~</ci><ci id="S5.SS1.p1.9.m9.1.1.2.2.cmml" xref="S5.SS1.p1.9.m9.1.1.2.2">𝜽</ci></apply><ci id="S5.SS1.p1.9.m9.1.1.3.cmml" xref="S5.SS1.p1.9.m9.1.1.3">𝜁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.9.m9.1c">\tilde{\bm{\theta}}_{\zeta}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.9.m9.1d">over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S5.SS1.p1.10.m10.1"><semantics id="S5.SS1.p1.10.m10.1a"><mo id="S5.SS1.p1.10.m10.1.1" xref="S5.SS1.p1.10.m10.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.10.m10.1b"><csymbol cd="latexml" id="S5.SS1.p1.10.m10.1.1.cmml" xref="S5.SS1.p1.10.m10.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.10.m10.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.10.m10.1d">:=</annotation></semantics></math> <math alttext="[\tilde{\theta}_{k_{1}},\tilde{\theta}_{k_{2}},\cdots,\tilde{\theta}_{k_{N_{% \zeta}}}]^{T}\in\mathbb{R}^{N_{\zeta}}" class="ltx_Math" display="inline" id="S5.SS1.p1.11.m11.4"><semantics id="S5.SS1.p1.11.m11.4a"><mrow id="S5.SS1.p1.11.m11.4.4" xref="S5.SS1.p1.11.m11.4.4.cmml"><msup id="S5.SS1.p1.11.m11.4.4.3" xref="S5.SS1.p1.11.m11.4.4.3.cmml"><mrow id="S5.SS1.p1.11.m11.4.4.3.3.3" xref="S5.SS1.p1.11.m11.4.4.3.3.4.cmml"><mo id="S5.SS1.p1.11.m11.4.4.3.3.3.4" stretchy="false" xref="S5.SS1.p1.11.m11.4.4.3.3.4.cmml">[</mo><msub id="S5.SS1.p1.11.m11.2.2.1.1.1.1" xref="S5.SS1.p1.11.m11.2.2.1.1.1.1.cmml"><mover accent="true" id="S5.SS1.p1.11.m11.2.2.1.1.1.1.2" xref="S5.SS1.p1.11.m11.2.2.1.1.1.1.2.cmml"><mi id="S5.SS1.p1.11.m11.2.2.1.1.1.1.2.2" xref="S5.SS1.p1.11.m11.2.2.1.1.1.1.2.2.cmml">θ</mi><mo id="S5.SS1.p1.11.m11.2.2.1.1.1.1.2.1" xref="S5.SS1.p1.11.m11.2.2.1.1.1.1.2.1.cmml">~</mo></mover><msub id="S5.SS1.p1.11.m11.2.2.1.1.1.1.3" xref="S5.SS1.p1.11.m11.2.2.1.1.1.1.3.cmml"><mi id="S5.SS1.p1.11.m11.2.2.1.1.1.1.3.2" xref="S5.SS1.p1.11.m11.2.2.1.1.1.1.3.2.cmml">k</mi><mn id="S5.SS1.p1.11.m11.2.2.1.1.1.1.3.3" xref="S5.SS1.p1.11.m11.2.2.1.1.1.1.3.3.cmml">1</mn></msub></msub><mo id="S5.SS1.p1.11.m11.4.4.3.3.3.5" xref="S5.SS1.p1.11.m11.4.4.3.3.4.cmml">,</mo><msub id="S5.SS1.p1.11.m11.3.3.2.2.2.2" xref="S5.SS1.p1.11.m11.3.3.2.2.2.2.cmml"><mover accent="true" id="S5.SS1.p1.11.m11.3.3.2.2.2.2.2" xref="S5.SS1.p1.11.m11.3.3.2.2.2.2.2.cmml"><mi id="S5.SS1.p1.11.m11.3.3.2.2.2.2.2.2" xref="S5.SS1.p1.11.m11.3.3.2.2.2.2.2.2.cmml">θ</mi><mo id="S5.SS1.p1.11.m11.3.3.2.2.2.2.2.1" xref="S5.SS1.p1.11.m11.3.3.2.2.2.2.2.1.cmml">~</mo></mover><msub id="S5.SS1.p1.11.m11.3.3.2.2.2.2.3" xref="S5.SS1.p1.11.m11.3.3.2.2.2.2.3.cmml"><mi id="S5.SS1.p1.11.m11.3.3.2.2.2.2.3.2" xref="S5.SS1.p1.11.m11.3.3.2.2.2.2.3.2.cmml">k</mi><mn id="S5.SS1.p1.11.m11.3.3.2.2.2.2.3.3" xref="S5.SS1.p1.11.m11.3.3.2.2.2.2.3.3.cmml">2</mn></msub></msub><mo id="S5.SS1.p1.11.m11.4.4.3.3.3.6" xref="S5.SS1.p1.11.m11.4.4.3.3.4.cmml">,</mo><mi id="S5.SS1.p1.11.m11.1.1" mathvariant="normal" xref="S5.SS1.p1.11.m11.1.1.cmml">⋯</mi><mo id="S5.SS1.p1.11.m11.4.4.3.3.3.7" xref="S5.SS1.p1.11.m11.4.4.3.3.4.cmml">,</mo><msub id="S5.SS1.p1.11.m11.4.4.3.3.3.3" xref="S5.SS1.p1.11.m11.4.4.3.3.3.3.cmml"><mover accent="true" id="S5.SS1.p1.11.m11.4.4.3.3.3.3.2" xref="S5.SS1.p1.11.m11.4.4.3.3.3.3.2.cmml"><mi id="S5.SS1.p1.11.m11.4.4.3.3.3.3.2.2" xref="S5.SS1.p1.11.m11.4.4.3.3.3.3.2.2.cmml">θ</mi><mo id="S5.SS1.p1.11.m11.4.4.3.3.3.3.2.1" xref="S5.SS1.p1.11.m11.4.4.3.3.3.3.2.1.cmml">~</mo></mover><msub id="S5.SS1.p1.11.m11.4.4.3.3.3.3.3" xref="S5.SS1.p1.11.m11.4.4.3.3.3.3.3.cmml"><mi id="S5.SS1.p1.11.m11.4.4.3.3.3.3.3.2" xref="S5.SS1.p1.11.m11.4.4.3.3.3.3.3.2.cmml">k</mi><msub id="S5.SS1.p1.11.m11.4.4.3.3.3.3.3.3" xref="S5.SS1.p1.11.m11.4.4.3.3.3.3.3.3.cmml"><mi id="S5.SS1.p1.11.m11.4.4.3.3.3.3.3.3.2" xref="S5.SS1.p1.11.m11.4.4.3.3.3.3.3.3.2.cmml">N</mi><mi id="S5.SS1.p1.11.m11.4.4.3.3.3.3.3.3.3" xref="S5.SS1.p1.11.m11.4.4.3.3.3.3.3.3.3.cmml">ζ</mi></msub></msub></msub><mo id="S5.SS1.p1.11.m11.4.4.3.3.3.8" stretchy="false" xref="S5.SS1.p1.11.m11.4.4.3.3.4.cmml">]</mo></mrow><mi id="S5.SS1.p1.11.m11.4.4.3.5" xref="S5.SS1.p1.11.m11.4.4.3.5.cmml">T</mi></msup><mo id="S5.SS1.p1.11.m11.4.4.4" xref="S5.SS1.p1.11.m11.4.4.4.cmml">∈</mo><msup id="S5.SS1.p1.11.m11.4.4.5" xref="S5.SS1.p1.11.m11.4.4.5.cmml"><mi id="S5.SS1.p1.11.m11.4.4.5.2" xref="S5.SS1.p1.11.m11.4.4.5.2.cmml">ℝ</mi><msub id="S5.SS1.p1.11.m11.4.4.5.3" xref="S5.SS1.p1.11.m11.4.4.5.3.cmml"><mi id="S5.SS1.p1.11.m11.4.4.5.3.2" xref="S5.SS1.p1.11.m11.4.4.5.3.2.cmml">N</mi><mi id="S5.SS1.p1.11.m11.4.4.5.3.3" xref="S5.SS1.p1.11.m11.4.4.5.3.3.cmml">ζ</mi></msub></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.11.m11.4b"><apply id="S5.SS1.p1.11.m11.4.4.cmml" xref="S5.SS1.p1.11.m11.4.4"><in id="S5.SS1.p1.11.m11.4.4.4.cmml" xref="S5.SS1.p1.11.m11.4.4.4"></in><apply id="S5.SS1.p1.11.m11.4.4.3.cmml" xref="S5.SS1.p1.11.m11.4.4.3"><csymbol cd="ambiguous" id="S5.SS1.p1.11.m11.4.4.3.4.cmml" xref="S5.SS1.p1.11.m11.4.4.3">superscript</csymbol><list id="S5.SS1.p1.11.m11.4.4.3.3.4.cmml" xref="S5.SS1.p1.11.m11.4.4.3.3.3"><apply id="S5.SS1.p1.11.m11.2.2.1.1.1.1.cmml" xref="S5.SS1.p1.11.m11.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.11.m11.2.2.1.1.1.1.1.cmml" xref="S5.SS1.p1.11.m11.2.2.1.1.1.1">subscript</csymbol><apply id="S5.SS1.p1.11.m11.2.2.1.1.1.1.2.cmml" xref="S5.SS1.p1.11.m11.2.2.1.1.1.1.2"><ci id="S5.SS1.p1.11.m11.2.2.1.1.1.1.2.1.cmml" xref="S5.SS1.p1.11.m11.2.2.1.1.1.1.2.1">~</ci><ci id="S5.SS1.p1.11.m11.2.2.1.1.1.1.2.2.cmml" xref="S5.SS1.p1.11.m11.2.2.1.1.1.1.2.2">𝜃</ci></apply><apply id="S5.SS1.p1.11.m11.2.2.1.1.1.1.3.cmml" xref="S5.SS1.p1.11.m11.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.SS1.p1.11.m11.2.2.1.1.1.1.3.1.cmml" xref="S5.SS1.p1.11.m11.2.2.1.1.1.1.3">subscript</csymbol><ci id="S5.SS1.p1.11.m11.2.2.1.1.1.1.3.2.cmml" xref="S5.SS1.p1.11.m11.2.2.1.1.1.1.3.2">𝑘</ci><cn id="S5.SS1.p1.11.m11.2.2.1.1.1.1.3.3.cmml" type="integer" xref="S5.SS1.p1.11.m11.2.2.1.1.1.1.3.3">1</cn></apply></apply><apply id="S5.SS1.p1.11.m11.3.3.2.2.2.2.cmml" xref="S5.SS1.p1.11.m11.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S5.SS1.p1.11.m11.3.3.2.2.2.2.1.cmml" xref="S5.SS1.p1.11.m11.3.3.2.2.2.2">subscript</csymbol><apply id="S5.SS1.p1.11.m11.3.3.2.2.2.2.2.cmml" xref="S5.SS1.p1.11.m11.3.3.2.2.2.2.2"><ci id="S5.SS1.p1.11.m11.3.3.2.2.2.2.2.1.cmml" xref="S5.SS1.p1.11.m11.3.3.2.2.2.2.2.1">~</ci><ci id="S5.SS1.p1.11.m11.3.3.2.2.2.2.2.2.cmml" xref="S5.SS1.p1.11.m11.3.3.2.2.2.2.2.2">𝜃</ci></apply><apply id="S5.SS1.p1.11.m11.3.3.2.2.2.2.3.cmml" xref="S5.SS1.p1.11.m11.3.3.2.2.2.2.3"><csymbol cd="ambiguous" id="S5.SS1.p1.11.m11.3.3.2.2.2.2.3.1.cmml" xref="S5.SS1.p1.11.m11.3.3.2.2.2.2.3">subscript</csymbol><ci id="S5.SS1.p1.11.m11.3.3.2.2.2.2.3.2.cmml" xref="S5.SS1.p1.11.m11.3.3.2.2.2.2.3.2">𝑘</ci><cn id="S5.SS1.p1.11.m11.3.3.2.2.2.2.3.3.cmml" type="integer" xref="S5.SS1.p1.11.m11.3.3.2.2.2.2.3.3">2</cn></apply></apply><ci id="S5.SS1.p1.11.m11.1.1.cmml" xref="S5.SS1.p1.11.m11.1.1">⋯</ci><apply id="S5.SS1.p1.11.m11.4.4.3.3.3.3.cmml" xref="S5.SS1.p1.11.m11.4.4.3.3.3.3"><csymbol cd="ambiguous" id="S5.SS1.p1.11.m11.4.4.3.3.3.3.1.cmml" xref="S5.SS1.p1.11.m11.4.4.3.3.3.3">subscript</csymbol><apply id="S5.SS1.p1.11.m11.4.4.3.3.3.3.2.cmml" xref="S5.SS1.p1.11.m11.4.4.3.3.3.3.2"><ci id="S5.SS1.p1.11.m11.4.4.3.3.3.3.2.1.cmml" xref="S5.SS1.p1.11.m11.4.4.3.3.3.3.2.1">~</ci><ci id="S5.SS1.p1.11.m11.4.4.3.3.3.3.2.2.cmml" xref="S5.SS1.p1.11.m11.4.4.3.3.3.3.2.2">𝜃</ci></apply><apply id="S5.SS1.p1.11.m11.4.4.3.3.3.3.3.cmml" xref="S5.SS1.p1.11.m11.4.4.3.3.3.3.3"><csymbol cd="ambiguous" id="S5.SS1.p1.11.m11.4.4.3.3.3.3.3.1.cmml" xref="S5.SS1.p1.11.m11.4.4.3.3.3.3.3">subscript</csymbol><ci id="S5.SS1.p1.11.m11.4.4.3.3.3.3.3.2.cmml" xref="S5.SS1.p1.11.m11.4.4.3.3.3.3.3.2">𝑘</ci><apply id="S5.SS1.p1.11.m11.4.4.3.3.3.3.3.3.cmml" xref="S5.SS1.p1.11.m11.4.4.3.3.3.3.3.3"><csymbol cd="ambiguous" id="S5.SS1.p1.11.m11.4.4.3.3.3.3.3.3.1.cmml" xref="S5.SS1.p1.11.m11.4.4.3.3.3.3.3.3">subscript</csymbol><ci id="S5.SS1.p1.11.m11.4.4.3.3.3.3.3.3.2.cmml" xref="S5.SS1.p1.11.m11.4.4.3.3.3.3.3.3.2">𝑁</ci><ci id="S5.SS1.p1.11.m11.4.4.3.3.3.3.3.3.3.cmml" xref="S5.SS1.p1.11.m11.4.4.3.3.3.3.3.3.3">𝜁</ci></apply></apply></apply></list><ci id="S5.SS1.p1.11.m11.4.4.3.5.cmml" xref="S5.SS1.p1.11.m11.4.4.3.5">𝑇</ci></apply><apply id="S5.SS1.p1.11.m11.4.4.5.cmml" xref="S5.SS1.p1.11.m11.4.4.5"><csymbol cd="ambiguous" id="S5.SS1.p1.11.m11.4.4.5.1.cmml" xref="S5.SS1.p1.11.m11.4.4.5">superscript</csymbol><ci id="S5.SS1.p1.11.m11.4.4.5.2.cmml" xref="S5.SS1.p1.11.m11.4.4.5.2">ℝ</ci><apply id="S5.SS1.p1.11.m11.4.4.5.3.cmml" xref="S5.SS1.p1.11.m11.4.4.5.3"><csymbol cd="ambiguous" id="S5.SS1.p1.11.m11.4.4.5.3.1.cmml" xref="S5.SS1.p1.11.m11.4.4.5.3">subscript</csymbol><ci id="S5.SS1.p1.11.m11.4.4.5.3.2.cmml" xref="S5.SS1.p1.11.m11.4.4.5.3.2">𝑁</ci><ci id="S5.SS1.p1.11.m11.4.4.5.3.3.cmml" xref="S5.SS1.p1.11.m11.4.4.5.3.3">𝜁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.11.m11.4c">[\tilde{\theta}_{k_{1}},\tilde{\theta}_{k_{2}},\cdots,\tilde{\theta}_{k_{N_{% \zeta}}}]^{T}\in\mathbb{R}^{N_{\zeta}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.11.m11.4d">[ over~ start_ARG italic_θ end_ARG start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , over~ start_ARG italic_θ end_ARG start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , ⋯ , over~ start_ARG italic_θ end_ARG start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> be a parameter estimation error corresponding to active channels. The following theorem shows parameter convergence results for the proposed estimation scheme.</p> </div> <div class="ltx_para" id="S5.SS1.p2"> <p class="ltx_p" id="S5.SS1.p2.5"><span class="ltx_text ltx_font_italic" id="S5.SS1.p2.5.1">Theorem 1:</span> Let <math alttext="[0,t_{\rm f})" class="ltx_Math" display="inline" id="S5.SS1.p2.1.m1.2"><semantics id="S5.SS1.p2.1.m1.2a"><mrow id="S5.SS1.p2.1.m1.2.2.1" xref="S5.SS1.p2.1.m1.2.2.2.cmml"><mo id="S5.SS1.p2.1.m1.2.2.1.2" stretchy="false" xref="S5.SS1.p2.1.m1.2.2.2.cmml">[</mo><mn id="S5.SS1.p2.1.m1.1.1" xref="S5.SS1.p2.1.m1.1.1.cmml">0</mn><mo id="S5.SS1.p2.1.m1.2.2.1.3" xref="S5.SS1.p2.1.m1.2.2.2.cmml">,</mo><msub id="S5.SS1.p2.1.m1.2.2.1.1" xref="S5.SS1.p2.1.m1.2.2.1.1.cmml"><mi id="S5.SS1.p2.1.m1.2.2.1.1.2" xref="S5.SS1.p2.1.m1.2.2.1.1.2.cmml">t</mi><mi id="S5.SS1.p2.1.m1.2.2.1.1.3" mathvariant="normal" xref="S5.SS1.p2.1.m1.2.2.1.1.3.cmml">f</mi></msub><mo id="S5.SS1.p2.1.m1.2.2.1.4" stretchy="false" xref="S5.SS1.p2.1.m1.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.1.m1.2b"><interval closure="closed-open" id="S5.SS1.p2.1.m1.2.2.2.cmml" xref="S5.SS1.p2.1.m1.2.2.1"><cn id="S5.SS1.p2.1.m1.1.1.cmml" type="integer" xref="S5.SS1.p2.1.m1.1.1">0</cn><apply id="S5.SS1.p2.1.m1.2.2.1.1.cmml" xref="S5.SS1.p2.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S5.SS1.p2.1.m1.2.2.1.1.1.cmml" xref="S5.SS1.p2.1.m1.2.2.1.1">subscript</csymbol><ci id="S5.SS1.p2.1.m1.2.2.1.1.2.cmml" xref="S5.SS1.p2.1.m1.2.2.1.1.2">𝑡</ci><ci id="S5.SS1.p2.1.m1.2.2.1.1.3.cmml" xref="S5.SS1.p2.1.m1.2.2.1.1.3">f</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.1.m1.2c">[0,t_{\rm f})</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.1.m1.2d">[ 0 , italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT )</annotation></semantics></math> with <math alttext="t_{\rm f}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S5.SS1.p2.2.m2.1"><semantics id="S5.SS1.p2.2.m2.1a"><mrow id="S5.SS1.p2.2.m2.1.1" xref="S5.SS1.p2.2.m2.1.1.cmml"><msub id="S5.SS1.p2.2.m2.1.1.2" xref="S5.SS1.p2.2.m2.1.1.2.cmml"><mi id="S5.SS1.p2.2.m2.1.1.2.2" xref="S5.SS1.p2.2.m2.1.1.2.2.cmml">t</mi><mi id="S5.SS1.p2.2.m2.1.1.2.3" mathvariant="normal" xref="S5.SS1.p2.2.m2.1.1.2.3.cmml">f</mi></msub><mo id="S5.SS1.p2.2.m2.1.1.1" xref="S5.SS1.p2.2.m2.1.1.1.cmml">∈</mo><msup id="S5.SS1.p2.2.m2.1.1.3" xref="S5.SS1.p2.2.m2.1.1.3.cmml"><mi id="S5.SS1.p2.2.m2.1.1.3.2" xref="S5.SS1.p2.2.m2.1.1.3.2.cmml">ℝ</mi><mo id="S5.SS1.p2.2.m2.1.1.3.3" xref="S5.SS1.p2.2.m2.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.2.m2.1b"><apply id="S5.SS1.p2.2.m2.1.1.cmml" xref="S5.SS1.p2.2.m2.1.1"><in id="S5.SS1.p2.2.m2.1.1.1.cmml" xref="S5.SS1.p2.2.m2.1.1.1"></in><apply id="S5.SS1.p2.2.m2.1.1.2.cmml" xref="S5.SS1.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.SS1.p2.2.m2.1.1.2.1.cmml" xref="S5.SS1.p2.2.m2.1.1.2">subscript</csymbol><ci id="S5.SS1.p2.2.m2.1.1.2.2.cmml" xref="S5.SS1.p2.2.m2.1.1.2.2">𝑡</ci><ci id="S5.SS1.p2.2.m2.1.1.2.3.cmml" xref="S5.SS1.p2.2.m2.1.1.2.3">f</ci></apply><apply id="S5.SS1.p2.2.m2.1.1.3.cmml" xref="S5.SS1.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.SS1.p2.2.m2.1.1.3.1.cmml" xref="S5.SS1.p2.2.m2.1.1.3">superscript</csymbol><ci id="S5.SS1.p2.2.m2.1.1.3.2.cmml" xref="S5.SS1.p2.2.m2.1.1.3.2">ℝ</ci><plus id="S5.SS1.p2.2.m2.1.1.3.3.cmml" xref="S5.SS1.p2.2.m2.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.2.m2.1c">t_{\rm f}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.2.m2.1d">italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> be the maximal time interval for the existence of solutions of the system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S2.E4" title="In II Problem Formulation ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">4</span></a>). For any given <math alttext="\bm{\theta}\in\Omega_{{\rm c}_{\theta}}" class="ltx_Math" display="inline" id="S5.SS1.p2.3.m3.1"><semantics id="S5.SS1.p2.3.m3.1a"><mrow id="S5.SS1.p2.3.m3.1.1" xref="S5.SS1.p2.3.m3.1.1.cmml"><mi id="S5.SS1.p2.3.m3.1.1.2" xref="S5.SS1.p2.3.m3.1.1.2.cmml">𝜽</mi><mo id="S5.SS1.p2.3.m3.1.1.1" xref="S5.SS1.p2.3.m3.1.1.1.cmml">∈</mo><msub id="S5.SS1.p2.3.m3.1.1.3" xref="S5.SS1.p2.3.m3.1.1.3.cmml"><mi id="S5.SS1.p2.3.m3.1.1.3.2" mathvariant="normal" xref="S5.SS1.p2.3.m3.1.1.3.2.cmml">Ω</mi><msub id="S5.SS1.p2.3.m3.1.1.3.3" xref="S5.SS1.p2.3.m3.1.1.3.3.cmml"><mi id="S5.SS1.p2.3.m3.1.1.3.3.2" mathvariant="normal" xref="S5.SS1.p2.3.m3.1.1.3.3.2.cmml">c</mi><mi id="S5.SS1.p2.3.m3.1.1.3.3.3" xref="S5.SS1.p2.3.m3.1.1.3.3.3.cmml">θ</mi></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.3.m3.1b"><apply id="S5.SS1.p2.3.m3.1.1.cmml" xref="S5.SS1.p2.3.m3.1.1"><in id="S5.SS1.p2.3.m3.1.1.1.cmml" xref="S5.SS1.p2.3.m3.1.1.1"></in><ci id="S5.SS1.p2.3.m3.1.1.2.cmml" xref="S5.SS1.p2.3.m3.1.1.2">𝜽</ci><apply id="S5.SS1.p2.3.m3.1.1.3.cmml" xref="S5.SS1.p2.3.m3.1.1.3"><csymbol cd="ambiguous" id="S5.SS1.p2.3.m3.1.1.3.1.cmml" xref="S5.SS1.p2.3.m3.1.1.3">subscript</csymbol><ci id="S5.SS1.p2.3.m3.1.1.3.2.cmml" xref="S5.SS1.p2.3.m3.1.1.3.2">Ω</ci><apply id="S5.SS1.p2.3.m3.1.1.3.3.cmml" xref="S5.SS1.p2.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S5.SS1.p2.3.m3.1.1.3.3.1.cmml" xref="S5.SS1.p2.3.m3.1.1.3.3">subscript</csymbol><ci id="S5.SS1.p2.3.m3.1.1.3.3.2.cmml" xref="S5.SS1.p2.3.m3.1.1.3.3.2">c</ci><ci id="S5.SS1.p2.3.m3.1.1.3.3.3.cmml" xref="S5.SS1.p2.3.m3.1.1.3.3.3">𝜃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.3.m3.1c">\bm{\theta}\in\Omega_{{\rm c}_{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.3.m3.1d">bold_italic_θ ∈ roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\Gamma=\Gamma^{T}>0" class="ltx_Math" display="inline" id="S5.SS1.p2.4.m4.1"><semantics id="S5.SS1.p2.4.m4.1a"><mrow id="S5.SS1.p2.4.m4.1.1" xref="S5.SS1.p2.4.m4.1.1.cmml"><mi id="S5.SS1.p2.4.m4.1.1.2" mathvariant="normal" xref="S5.SS1.p2.4.m4.1.1.2.cmml">Γ</mi><mo id="S5.SS1.p2.4.m4.1.1.3" xref="S5.SS1.p2.4.m4.1.1.3.cmml">=</mo><msup id="S5.SS1.p2.4.m4.1.1.4" xref="S5.SS1.p2.4.m4.1.1.4.cmml"><mi id="S5.SS1.p2.4.m4.1.1.4.2" mathvariant="normal" xref="S5.SS1.p2.4.m4.1.1.4.2.cmml">Γ</mi><mi id="S5.SS1.p2.4.m4.1.1.4.3" xref="S5.SS1.p2.4.m4.1.1.4.3.cmml">T</mi></msup><mo id="S5.SS1.p2.4.m4.1.1.5" xref="S5.SS1.p2.4.m4.1.1.5.cmml">></mo><mn id="S5.SS1.p2.4.m4.1.1.6" xref="S5.SS1.p2.4.m4.1.1.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.4.m4.1b"><apply id="S5.SS1.p2.4.m4.1.1.cmml" xref="S5.SS1.p2.4.m4.1.1"><and id="S5.SS1.p2.4.m4.1.1a.cmml" xref="S5.SS1.p2.4.m4.1.1"></and><apply id="S5.SS1.p2.4.m4.1.1b.cmml" xref="S5.SS1.p2.4.m4.1.1"><eq id="S5.SS1.p2.4.m4.1.1.3.cmml" xref="S5.SS1.p2.4.m4.1.1.3"></eq><ci id="S5.SS1.p2.4.m4.1.1.2.cmml" xref="S5.SS1.p2.4.m4.1.1.2">Γ</ci><apply id="S5.SS1.p2.4.m4.1.1.4.cmml" xref="S5.SS1.p2.4.m4.1.1.4"><csymbol cd="ambiguous" id="S5.SS1.p2.4.m4.1.1.4.1.cmml" xref="S5.SS1.p2.4.m4.1.1.4">superscript</csymbol><ci id="S5.SS1.p2.4.m4.1.1.4.2.cmml" xref="S5.SS1.p2.4.m4.1.1.4.2">Γ</ci><ci id="S5.SS1.p2.4.m4.1.1.4.3.cmml" xref="S5.SS1.p2.4.m4.1.1.4.3">𝑇</ci></apply></apply><apply id="S5.SS1.p2.4.m4.1.1c.cmml" xref="S5.SS1.p2.4.m4.1.1"><gt id="S5.SS1.p2.4.m4.1.1.5.cmml" xref="S5.SS1.p2.4.m4.1.1.5"></gt><share href="https://arxiv.org/html/2401.10785v2#S5.SS1.p2.4.m4.1.1.4.cmml" id="S5.SS1.p2.4.m4.1.1d.cmml" xref="S5.SS1.p2.4.m4.1.1"></share><cn id="S5.SS1.p2.4.m4.1.1.6.cmml" type="integer" xref="S5.SS1.p2.4.m4.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.4.m4.1c">\Gamma=\Gamma^{T}>0</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.4.m4.1d">roman_Γ = roman_Γ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT > 0</annotation></semantics></math>, the composite learning law of <math alttext="\hat{\bm{\theta}}" class="ltx_Math" display="inline" id="S5.SS1.p2.5.m5.1"><semantics id="S5.SS1.p2.5.m5.1a"><mover accent="true" id="S5.SS1.p2.5.m5.1.1" xref="S5.SS1.p2.5.m5.1.1.cmml"><mi id="S5.SS1.p2.5.m5.1.1.2" xref="S5.SS1.p2.5.m5.1.1.2.cmml">𝜽</mi><mo id="S5.SS1.p2.5.m5.1.1.1" xref="S5.SS1.p2.5.m5.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.5.m5.1b"><apply id="S5.SS1.p2.5.m5.1.1.cmml" xref="S5.SS1.p2.5.m5.1.1"><ci id="S5.SS1.p2.5.m5.1.1.1.cmml" xref="S5.SS1.p2.5.m5.1.1.1">^</ci><ci id="S5.SS1.p2.5.m5.1.1.2.cmml" xref="S5.SS1.p2.5.m5.1.1.2">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.5.m5.1c">\hat{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.5.m5.1d">over^ start_ARG bold_italic_θ end_ARG</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>) ensures:</p> <ol class="ltx_enumerate" id="S5.I1"> <li class="ltx_item" id="S5.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S5.I1.i1.p1"> <p class="ltx_p" id="S5.I1.i1.p1.6">The parameter estimation error <math alttext="\tilde{\bm{\theta}}(t)" class="ltx_Math" display="inline" id="S5.I1.i1.p1.1.m1.1"><semantics id="S5.I1.i1.p1.1.m1.1a"><mrow id="S5.I1.i1.p1.1.m1.1.2" xref="S5.I1.i1.p1.1.m1.1.2.cmml"><mover accent="true" id="S5.I1.i1.p1.1.m1.1.2.2" xref="S5.I1.i1.p1.1.m1.1.2.2.cmml"><mi id="S5.I1.i1.p1.1.m1.1.2.2.2" xref="S5.I1.i1.p1.1.m1.1.2.2.2.cmml">𝜽</mi><mo id="S5.I1.i1.p1.1.m1.1.2.2.1" xref="S5.I1.i1.p1.1.m1.1.2.2.1.cmml">~</mo></mover><mo id="S5.I1.i1.p1.1.m1.1.2.1" xref="S5.I1.i1.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S5.I1.i1.p1.1.m1.1.2.3.2" xref="S5.I1.i1.p1.1.m1.1.2.cmml"><mo id="S5.I1.i1.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S5.I1.i1.p1.1.m1.1.2.cmml">(</mo><mi id="S5.I1.i1.p1.1.m1.1.1" xref="S5.I1.i1.p1.1.m1.1.1.cmml">t</mi><mo id="S5.I1.i1.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S5.I1.i1.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.1.m1.1b"><apply id="S5.I1.i1.p1.1.m1.1.2.cmml" xref="S5.I1.i1.p1.1.m1.1.2"><times id="S5.I1.i1.p1.1.m1.1.2.1.cmml" xref="S5.I1.i1.p1.1.m1.1.2.1"></times><apply id="S5.I1.i1.p1.1.m1.1.2.2.cmml" xref="S5.I1.i1.p1.1.m1.1.2.2"><ci id="S5.I1.i1.p1.1.m1.1.2.2.1.cmml" xref="S5.I1.i1.p1.1.m1.1.2.2.1">~</ci><ci id="S5.I1.i1.p1.1.m1.1.2.2.2.cmml" xref="S5.I1.i1.p1.1.m1.1.2.2.2">𝜽</ci></apply><ci id="S5.I1.i1.p1.1.m1.1.1.cmml" xref="S5.I1.i1.p1.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.1.m1.1c">\tilde{\bm{\theta}}(t)</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.1.m1.1d">over~ start_ARG bold_italic_θ end_ARG ( italic_t )</annotation></semantics></math> is of <math alttext="L_{\infty}" class="ltx_Math" display="inline" id="S5.I1.i1.p1.2.m2.1"><semantics id="S5.I1.i1.p1.2.m2.1a"><msub id="S5.I1.i1.p1.2.m2.1.1" xref="S5.I1.i1.p1.2.m2.1.1.cmml"><mi id="S5.I1.i1.p1.2.m2.1.1.2" xref="S5.I1.i1.p1.2.m2.1.1.2.cmml">L</mi><mi id="S5.I1.i1.p1.2.m2.1.1.3" mathvariant="normal" xref="S5.I1.i1.p1.2.m2.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.2.m2.1b"><apply id="S5.I1.i1.p1.2.m2.1.1.cmml" xref="S5.I1.i1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.I1.i1.p1.2.m2.1.1.1.cmml" xref="S5.I1.i1.p1.2.m2.1.1">subscript</csymbol><ci id="S5.I1.i1.p1.2.m2.1.1.2.cmml" xref="S5.I1.i1.p1.2.m2.1.1.2">𝐿</ci><infinity id="S5.I1.i1.p1.2.m2.1.1.3.cmml" xref="S5.I1.i1.p1.2.m2.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.2.m2.1c">L_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.2.m2.1d">italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\forall t\geq 0" class="ltx_Math" display="inline" id="S5.I1.i1.p1.3.m3.1"><semantics id="S5.I1.i1.p1.3.m3.1a"><mrow id="S5.I1.i1.p1.3.m3.1.1" xref="S5.I1.i1.p1.3.m3.1.1.cmml"><mrow id="S5.I1.i1.p1.3.m3.1.1.2" xref="S5.I1.i1.p1.3.m3.1.1.2.cmml"><mo id="S5.I1.i1.p1.3.m3.1.1.2.1" rspace="0.167em" xref="S5.I1.i1.p1.3.m3.1.1.2.1.cmml">∀</mo><mi id="S5.I1.i1.p1.3.m3.1.1.2.2" xref="S5.I1.i1.p1.3.m3.1.1.2.2.cmml">t</mi></mrow><mo id="S5.I1.i1.p1.3.m3.1.1.1" xref="S5.I1.i1.p1.3.m3.1.1.1.cmml">≥</mo><mn id="S5.I1.i1.p1.3.m3.1.1.3" xref="S5.I1.i1.p1.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.3.m3.1b"><apply id="S5.I1.i1.p1.3.m3.1.1.cmml" xref="S5.I1.i1.p1.3.m3.1.1"><geq id="S5.I1.i1.p1.3.m3.1.1.1.cmml" xref="S5.I1.i1.p1.3.m3.1.1.1"></geq><apply id="S5.I1.i1.p1.3.m3.1.1.2.cmml" xref="S5.I1.i1.p1.3.m3.1.1.2"><csymbol cd="latexml" id="S5.I1.i1.p1.3.m3.1.1.2.1.cmml" xref="S5.I1.i1.p1.3.m3.1.1.2.1">for-all</csymbol><ci id="S5.I1.i1.p1.3.m3.1.1.2.2.cmml" xref="S5.I1.i1.p1.3.m3.1.1.2.2">𝑡</ci></apply><cn id="S5.I1.i1.p1.3.m3.1.1.3.cmml" type="integer" xref="S5.I1.i1.p1.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.3.m3.1c">\forall t\geq 0</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.3.m3.1d">∀ italic_t ≥ 0</annotation></semantics></math>, and <math alttext="\bm{\epsilon}(t)" class="ltx_Math" display="inline" id="S5.I1.i1.p1.4.m4.1"><semantics id="S5.I1.i1.p1.4.m4.1a"><mrow id="S5.I1.i1.p1.4.m4.1.2" xref="S5.I1.i1.p1.4.m4.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S5.I1.i1.p1.4.m4.1.2.2" mathvariant="bold-italic" xref="S5.I1.i1.p1.4.m4.1.2.2.cmml">ϵ</mi><mo id="S5.I1.i1.p1.4.m4.1.2.1" xref="S5.I1.i1.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="S5.I1.i1.p1.4.m4.1.2.3.2" xref="S5.I1.i1.p1.4.m4.1.2.cmml"><mo id="S5.I1.i1.p1.4.m4.1.2.3.2.1" stretchy="false" xref="S5.I1.i1.p1.4.m4.1.2.cmml">(</mo><mi id="S5.I1.i1.p1.4.m4.1.1" xref="S5.I1.i1.p1.4.m4.1.1.cmml">t</mi><mo id="S5.I1.i1.p1.4.m4.1.2.3.2.2" stretchy="false" xref="S5.I1.i1.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.4.m4.1b"><apply id="S5.I1.i1.p1.4.m4.1.2.cmml" xref="S5.I1.i1.p1.4.m4.1.2"><times id="S5.I1.i1.p1.4.m4.1.2.1.cmml" xref="S5.I1.i1.p1.4.m4.1.2.1"></times><ci id="S5.I1.i1.p1.4.m4.1.2.2.cmml" xref="S5.I1.i1.p1.4.m4.1.2.2">bold-italic-ϵ</ci><ci id="S5.I1.i1.p1.4.m4.1.1.cmml" xref="S5.I1.i1.p1.4.m4.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.4.m4.1c">\bm{\epsilon}(t)</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.4.m4.1d">bold_italic_ϵ ( italic_t )</annotation></semantics></math> is of <math alttext="L_{2}\cap L_{\infty}" class="ltx_Math" display="inline" id="S5.I1.i1.p1.5.m5.1"><semantics id="S5.I1.i1.p1.5.m5.1a"><mrow id="S5.I1.i1.p1.5.m5.1.1" xref="S5.I1.i1.p1.5.m5.1.1.cmml"><msub id="S5.I1.i1.p1.5.m5.1.1.2" xref="S5.I1.i1.p1.5.m5.1.1.2.cmml"><mi id="S5.I1.i1.p1.5.m5.1.1.2.2" xref="S5.I1.i1.p1.5.m5.1.1.2.2.cmml">L</mi><mn id="S5.I1.i1.p1.5.m5.1.1.2.3" xref="S5.I1.i1.p1.5.m5.1.1.2.3.cmml">2</mn></msub><mo id="S5.I1.i1.p1.5.m5.1.1.1" xref="S5.I1.i1.p1.5.m5.1.1.1.cmml">∩</mo><msub id="S5.I1.i1.p1.5.m5.1.1.3" xref="S5.I1.i1.p1.5.m5.1.1.3.cmml"><mi id="S5.I1.i1.p1.5.m5.1.1.3.2" xref="S5.I1.i1.p1.5.m5.1.1.3.2.cmml">L</mi><mi id="S5.I1.i1.p1.5.m5.1.1.3.3" mathvariant="normal" xref="S5.I1.i1.p1.5.m5.1.1.3.3.cmml">∞</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.5.m5.1b"><apply id="S5.I1.i1.p1.5.m5.1.1.cmml" xref="S5.I1.i1.p1.5.m5.1.1"><intersect id="S5.I1.i1.p1.5.m5.1.1.1.cmml" xref="S5.I1.i1.p1.5.m5.1.1.1"></intersect><apply id="S5.I1.i1.p1.5.m5.1.1.2.cmml" xref="S5.I1.i1.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S5.I1.i1.p1.5.m5.1.1.2.1.cmml" xref="S5.I1.i1.p1.5.m5.1.1.2">subscript</csymbol><ci id="S5.I1.i1.p1.5.m5.1.1.2.2.cmml" xref="S5.I1.i1.p1.5.m5.1.1.2.2">𝐿</ci><cn id="S5.I1.i1.p1.5.m5.1.1.2.3.cmml" type="integer" xref="S5.I1.i1.p1.5.m5.1.1.2.3">2</cn></apply><apply id="S5.I1.i1.p1.5.m5.1.1.3.cmml" xref="S5.I1.i1.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S5.I1.i1.p1.5.m5.1.1.3.1.cmml" xref="S5.I1.i1.p1.5.m5.1.1.3">subscript</csymbol><ci id="S5.I1.i1.p1.5.m5.1.1.3.2.cmml" xref="S5.I1.i1.p1.5.m5.1.1.3.2">𝐿</ci><infinity id="S5.I1.i1.p1.5.m5.1.1.3.3.cmml" xref="S5.I1.i1.p1.5.m5.1.1.3.3"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.5.m5.1c">L_{2}\cap L_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.5.m5.1d">italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∩ italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\forall t\in[0,t_{\rm f})" class="ltx_Math" display="inline" id="S5.I1.i1.p1.6.m6.2"><semantics id="S5.I1.i1.p1.6.m6.2a"><mrow id="S5.I1.i1.p1.6.m6.2.2" xref="S5.I1.i1.p1.6.m6.2.2.cmml"><mrow id="S5.I1.i1.p1.6.m6.2.2.3" xref="S5.I1.i1.p1.6.m6.2.2.3.cmml"><mo id="S5.I1.i1.p1.6.m6.2.2.3.1" rspace="0.167em" xref="S5.I1.i1.p1.6.m6.2.2.3.1.cmml">∀</mo><mi id="S5.I1.i1.p1.6.m6.2.2.3.2" xref="S5.I1.i1.p1.6.m6.2.2.3.2.cmml">t</mi></mrow><mo id="S5.I1.i1.p1.6.m6.2.2.2" xref="S5.I1.i1.p1.6.m6.2.2.2.cmml">∈</mo><mrow id="S5.I1.i1.p1.6.m6.2.2.1.1" xref="S5.I1.i1.p1.6.m6.2.2.1.2.cmml"><mo id="S5.I1.i1.p1.6.m6.2.2.1.1.2" stretchy="false" xref="S5.I1.i1.p1.6.m6.2.2.1.2.cmml">[</mo><mn id="S5.I1.i1.p1.6.m6.1.1" xref="S5.I1.i1.p1.6.m6.1.1.cmml">0</mn><mo id="S5.I1.i1.p1.6.m6.2.2.1.1.3" xref="S5.I1.i1.p1.6.m6.2.2.1.2.cmml">,</mo><msub id="S5.I1.i1.p1.6.m6.2.2.1.1.1" xref="S5.I1.i1.p1.6.m6.2.2.1.1.1.cmml"><mi id="S5.I1.i1.p1.6.m6.2.2.1.1.1.2" xref="S5.I1.i1.p1.6.m6.2.2.1.1.1.2.cmml">t</mi><mi id="S5.I1.i1.p1.6.m6.2.2.1.1.1.3" mathvariant="normal" xref="S5.I1.i1.p1.6.m6.2.2.1.1.1.3.cmml">f</mi></msub><mo id="S5.I1.i1.p1.6.m6.2.2.1.1.4" stretchy="false" xref="S5.I1.i1.p1.6.m6.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.6.m6.2b"><apply id="S5.I1.i1.p1.6.m6.2.2.cmml" xref="S5.I1.i1.p1.6.m6.2.2"><in id="S5.I1.i1.p1.6.m6.2.2.2.cmml" xref="S5.I1.i1.p1.6.m6.2.2.2"></in><apply id="S5.I1.i1.p1.6.m6.2.2.3.cmml" xref="S5.I1.i1.p1.6.m6.2.2.3"><csymbol cd="latexml" id="S5.I1.i1.p1.6.m6.2.2.3.1.cmml" xref="S5.I1.i1.p1.6.m6.2.2.3.1">for-all</csymbol><ci id="S5.I1.i1.p1.6.m6.2.2.3.2.cmml" xref="S5.I1.i1.p1.6.m6.2.2.3.2">𝑡</ci></apply><interval closure="closed-open" id="S5.I1.i1.p1.6.m6.2.2.1.2.cmml" xref="S5.I1.i1.p1.6.m6.2.2.1.1"><cn id="S5.I1.i1.p1.6.m6.1.1.cmml" type="integer" xref="S5.I1.i1.p1.6.m6.1.1">0</cn><apply id="S5.I1.i1.p1.6.m6.2.2.1.1.1.cmml" xref="S5.I1.i1.p1.6.m6.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.I1.i1.p1.6.m6.2.2.1.1.1.1.cmml" xref="S5.I1.i1.p1.6.m6.2.2.1.1.1">subscript</csymbol><ci id="S5.I1.i1.p1.6.m6.2.2.1.1.1.2.cmml" xref="S5.I1.i1.p1.6.m6.2.2.1.1.1.2">𝑡</ci><ci id="S5.I1.i1.p1.6.m6.2.2.1.1.1.3.cmml" xref="S5.I1.i1.p1.6.m6.2.2.1.1.1.3">f</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.6.m6.2c">\forall t\in[0,t_{\rm f})</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.6.m6.2d">∀ italic_t ∈ [ 0 , italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT )</annotation></semantics></math>;</p> </div> </li> <li class="ltx_item" id="S5.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S5.I1.i2.p1"> <p class="ltx_p" id="S5.I1.i2.p1.7">The partial parameter estimation error <math alttext="\tilde{\bm{\theta}}_{\zeta}(t)" class="ltx_Math" display="inline" id="S5.I1.i2.p1.1.m1.1"><semantics id="S5.I1.i2.p1.1.m1.1a"><mrow id="S5.I1.i2.p1.1.m1.1.2" xref="S5.I1.i2.p1.1.m1.1.2.cmml"><msub id="S5.I1.i2.p1.1.m1.1.2.2" xref="S5.I1.i2.p1.1.m1.1.2.2.cmml"><mover accent="true" id="S5.I1.i2.p1.1.m1.1.2.2.2" xref="S5.I1.i2.p1.1.m1.1.2.2.2.cmml"><mi id="S5.I1.i2.p1.1.m1.1.2.2.2.2" xref="S5.I1.i2.p1.1.m1.1.2.2.2.2.cmml">𝜽</mi><mo id="S5.I1.i2.p1.1.m1.1.2.2.2.1" xref="S5.I1.i2.p1.1.m1.1.2.2.2.1.cmml">~</mo></mover><mi id="S5.I1.i2.p1.1.m1.1.2.2.3" xref="S5.I1.i2.p1.1.m1.1.2.2.3.cmml">ζ</mi></msub><mo id="S5.I1.i2.p1.1.m1.1.2.1" xref="S5.I1.i2.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S5.I1.i2.p1.1.m1.1.2.3.2" xref="S5.I1.i2.p1.1.m1.1.2.cmml"><mo id="S5.I1.i2.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S5.I1.i2.p1.1.m1.1.2.cmml">(</mo><mi id="S5.I1.i2.p1.1.m1.1.1" xref="S5.I1.i2.p1.1.m1.1.1.cmml">t</mi><mo id="S5.I1.i2.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S5.I1.i2.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.1.m1.1b"><apply id="S5.I1.i2.p1.1.m1.1.2.cmml" xref="S5.I1.i2.p1.1.m1.1.2"><times id="S5.I1.i2.p1.1.m1.1.2.1.cmml" xref="S5.I1.i2.p1.1.m1.1.2.1"></times><apply id="S5.I1.i2.p1.1.m1.1.2.2.cmml" xref="S5.I1.i2.p1.1.m1.1.2.2"><csymbol cd="ambiguous" id="S5.I1.i2.p1.1.m1.1.2.2.1.cmml" xref="S5.I1.i2.p1.1.m1.1.2.2">subscript</csymbol><apply id="S5.I1.i2.p1.1.m1.1.2.2.2.cmml" xref="S5.I1.i2.p1.1.m1.1.2.2.2"><ci id="S5.I1.i2.p1.1.m1.1.2.2.2.1.cmml" xref="S5.I1.i2.p1.1.m1.1.2.2.2.1">~</ci><ci id="S5.I1.i2.p1.1.m1.1.2.2.2.2.cmml" xref="S5.I1.i2.p1.1.m1.1.2.2.2.2">𝜽</ci></apply><ci id="S5.I1.i2.p1.1.m1.1.2.2.3.cmml" xref="S5.I1.i2.p1.1.m1.1.2.2.3">𝜁</ci></apply><ci id="S5.I1.i2.p1.1.m1.1.1.cmml" xref="S5.I1.i2.p1.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.1.m1.1c">\tilde{\bm{\theta}}_{\zeta}(t)</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.1.m1.1d">over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> exponentially converges to <math alttext="\bm{0}" class="ltx_Math" display="inline" id="S5.I1.i2.p1.2.m2.1"><semantics id="S5.I1.i2.p1.2.m2.1a"><mn id="S5.I1.i2.p1.2.m2.1.1" xref="S5.I1.i2.p1.2.m2.1.1.cmml">𝟎</mn><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.2.m2.1b"><cn id="S5.I1.i2.p1.2.m2.1.1.cmml" type="integer" xref="S5.I1.i2.p1.2.m2.1.1">0</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.2.m2.1c">\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.2.m2.1d">bold_0</annotation></semantics></math> if <math alttext="t_{\rm f}\rightarrow\infty" class="ltx_Math" display="inline" id="S5.I1.i2.p1.3.m3.1"><semantics id="S5.I1.i2.p1.3.m3.1a"><mrow id="S5.I1.i2.p1.3.m3.1.1" xref="S5.I1.i2.p1.3.m3.1.1.cmml"><msub id="S5.I1.i2.p1.3.m3.1.1.2" xref="S5.I1.i2.p1.3.m3.1.1.2.cmml"><mi id="S5.I1.i2.p1.3.m3.1.1.2.2" xref="S5.I1.i2.p1.3.m3.1.1.2.2.cmml">t</mi><mi id="S5.I1.i2.p1.3.m3.1.1.2.3" mathvariant="normal" xref="S5.I1.i2.p1.3.m3.1.1.2.3.cmml">f</mi></msub><mo id="S5.I1.i2.p1.3.m3.1.1.1" stretchy="false" xref="S5.I1.i2.p1.3.m3.1.1.1.cmml">→</mo><mi id="S5.I1.i2.p1.3.m3.1.1.3" mathvariant="normal" xref="S5.I1.i2.p1.3.m3.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.3.m3.1b"><apply id="S5.I1.i2.p1.3.m3.1.1.cmml" xref="S5.I1.i2.p1.3.m3.1.1"><ci id="S5.I1.i2.p1.3.m3.1.1.1.cmml" xref="S5.I1.i2.p1.3.m3.1.1.1">→</ci><apply id="S5.I1.i2.p1.3.m3.1.1.2.cmml" xref="S5.I1.i2.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S5.I1.i2.p1.3.m3.1.1.2.1.cmml" xref="S5.I1.i2.p1.3.m3.1.1.2">subscript</csymbol><ci id="S5.I1.i2.p1.3.m3.1.1.2.2.cmml" xref="S5.I1.i2.p1.3.m3.1.1.2.2">𝑡</ci><ci id="S5.I1.i2.p1.3.m3.1.1.2.3.cmml" xref="S5.I1.i2.p1.3.m3.1.1.2.3">f</ci></apply><infinity id="S5.I1.i2.p1.3.m3.1.1.3.cmml" xref="S5.I1.i2.p1.3.m3.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.3.m3.1c">t_{\rm f}\rightarrow\infty</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.3.m3.1d">italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT → ∞</annotation></semantics></math>, partial IEexists for some constants <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.I1.i2.p1.4.m4.1"><semantics id="S5.I1.i2.p1.4.m4.1a"><mi id="S5.I1.i2.p1.4.m4.1.1" xref="S5.I1.i2.p1.4.m4.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.4.m4.1b"><ci id="S5.I1.i2.p1.4.m4.1.1.cmml" xref="S5.I1.i2.p1.4.m4.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.4.m4.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.4.m4.1d">italic_σ</annotation></semantics></math>, <math alttext="T_{\rm a}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S5.I1.i2.p1.5.m5.1"><semantics id="S5.I1.i2.p1.5.m5.1a"><mrow id="S5.I1.i2.p1.5.m5.1.1" xref="S5.I1.i2.p1.5.m5.1.1.cmml"><msub id="S5.I1.i2.p1.5.m5.1.1.2" xref="S5.I1.i2.p1.5.m5.1.1.2.cmml"><mi id="S5.I1.i2.p1.5.m5.1.1.2.2" xref="S5.I1.i2.p1.5.m5.1.1.2.2.cmml">T</mi><mi id="S5.I1.i2.p1.5.m5.1.1.2.3" mathvariant="normal" xref="S5.I1.i2.p1.5.m5.1.1.2.3.cmml">a</mi></msub><mo id="S5.I1.i2.p1.5.m5.1.1.1" xref="S5.I1.i2.p1.5.m5.1.1.1.cmml">∈</mo><msup id="S5.I1.i2.p1.5.m5.1.1.3" xref="S5.I1.i2.p1.5.m5.1.1.3.cmml"><mi id="S5.I1.i2.p1.5.m5.1.1.3.2" xref="S5.I1.i2.p1.5.m5.1.1.3.2.cmml">ℝ</mi><mo id="S5.I1.i2.p1.5.m5.1.1.3.3" xref="S5.I1.i2.p1.5.m5.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.5.m5.1b"><apply id="S5.I1.i2.p1.5.m5.1.1.cmml" xref="S5.I1.i2.p1.5.m5.1.1"><in id="S5.I1.i2.p1.5.m5.1.1.1.cmml" xref="S5.I1.i2.p1.5.m5.1.1.1"></in><apply id="S5.I1.i2.p1.5.m5.1.1.2.cmml" xref="S5.I1.i2.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S5.I1.i2.p1.5.m5.1.1.2.1.cmml" xref="S5.I1.i2.p1.5.m5.1.1.2">subscript</csymbol><ci id="S5.I1.i2.p1.5.m5.1.1.2.2.cmml" xref="S5.I1.i2.p1.5.m5.1.1.2.2">𝑇</ci><ci id="S5.I1.i2.p1.5.m5.1.1.2.3.cmml" xref="S5.I1.i2.p1.5.m5.1.1.2.3">a</ci></apply><apply id="S5.I1.i2.p1.5.m5.1.1.3.cmml" xref="S5.I1.i2.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S5.I1.i2.p1.5.m5.1.1.3.1.cmml" xref="S5.I1.i2.p1.5.m5.1.1.3">superscript</csymbol><ci id="S5.I1.i2.p1.5.m5.1.1.3.2.cmml" xref="S5.I1.i2.p1.5.m5.1.1.3.2">ℝ</ci><plus id="S5.I1.i2.p1.5.m5.1.1.3.3.cmml" xref="S5.I1.i2.p1.5.m5.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.5.m5.1c">T_{\rm a}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.5.m5.1d">italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, and the index set <math alttext="\mathcal{I}" class="ltx_Math" display="inline" id="S5.I1.i2.p1.6.m6.1"><semantics id="S5.I1.i2.p1.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S5.I1.i2.p1.6.m6.1.1" xref="S5.I1.i2.p1.6.m6.1.1.cmml">ℐ</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.6.m6.1b"><ci id="S5.I1.i2.p1.6.m6.1.1.cmml" xref="S5.I1.i2.p1.6.m6.1.1">ℐ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.6.m6.1c">\mathcal{I}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.6.m6.1d">caligraphic_I</annotation></semantics></math> no longer changes on <math alttext="t\in(T_{\rm a},\infty)" class="ltx_Math" display="inline" id="S5.I1.i2.p1.7.m7.2"><semantics id="S5.I1.i2.p1.7.m7.2a"><mrow id="S5.I1.i2.p1.7.m7.2.2" xref="S5.I1.i2.p1.7.m7.2.2.cmml"><mi id="S5.I1.i2.p1.7.m7.2.2.3" xref="S5.I1.i2.p1.7.m7.2.2.3.cmml">t</mi><mo id="S5.I1.i2.p1.7.m7.2.2.2" xref="S5.I1.i2.p1.7.m7.2.2.2.cmml">∈</mo><mrow id="S5.I1.i2.p1.7.m7.2.2.1.1" xref="S5.I1.i2.p1.7.m7.2.2.1.2.cmml"><mo id="S5.I1.i2.p1.7.m7.2.2.1.1.2" stretchy="false" xref="S5.I1.i2.p1.7.m7.2.2.1.2.cmml">(</mo><msub id="S5.I1.i2.p1.7.m7.2.2.1.1.1" xref="S5.I1.i2.p1.7.m7.2.2.1.1.1.cmml"><mi id="S5.I1.i2.p1.7.m7.2.2.1.1.1.2" xref="S5.I1.i2.p1.7.m7.2.2.1.1.1.2.cmml">T</mi><mi id="S5.I1.i2.p1.7.m7.2.2.1.1.1.3" mathvariant="normal" xref="S5.I1.i2.p1.7.m7.2.2.1.1.1.3.cmml">a</mi></msub><mo id="S5.I1.i2.p1.7.m7.2.2.1.1.3" xref="S5.I1.i2.p1.7.m7.2.2.1.2.cmml">,</mo><mi id="S5.I1.i2.p1.7.m7.1.1" mathvariant="normal" xref="S5.I1.i2.p1.7.m7.1.1.cmml">∞</mi><mo id="S5.I1.i2.p1.7.m7.2.2.1.1.4" stretchy="false" xref="S5.I1.i2.p1.7.m7.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.7.m7.2b"><apply id="S5.I1.i2.p1.7.m7.2.2.cmml" xref="S5.I1.i2.p1.7.m7.2.2"><in id="S5.I1.i2.p1.7.m7.2.2.2.cmml" xref="S5.I1.i2.p1.7.m7.2.2.2"></in><ci id="S5.I1.i2.p1.7.m7.2.2.3.cmml" xref="S5.I1.i2.p1.7.m7.2.2.3">𝑡</ci><interval closure="open" id="S5.I1.i2.p1.7.m7.2.2.1.2.cmml" xref="S5.I1.i2.p1.7.m7.2.2.1.1"><apply id="S5.I1.i2.p1.7.m7.2.2.1.1.1.cmml" xref="S5.I1.i2.p1.7.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.I1.i2.p1.7.m7.2.2.1.1.1.1.cmml" xref="S5.I1.i2.p1.7.m7.2.2.1.1.1">subscript</csymbol><ci id="S5.I1.i2.p1.7.m7.2.2.1.1.1.2.cmml" xref="S5.I1.i2.p1.7.m7.2.2.1.1.1.2">𝑇</ci><ci id="S5.I1.i2.p1.7.m7.2.2.1.1.1.3.cmml" xref="S5.I1.i2.p1.7.m7.2.2.1.1.1.3">a</ci></apply><infinity id="S5.I1.i2.p1.7.m7.1.1.cmml" xref="S5.I1.i2.p1.7.m7.1.1"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.7.m7.2c">t\in(T_{\rm a},\infty)</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.7.m7.2d">italic_t ∈ ( italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT , ∞ )</annotation></semantics></math>;</p> </div> </li> <li class="ltx_item" id="S5.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">3.</span> <div class="ltx_para" id="S5.I1.i3.p1"> <p class="ltx_p" id="S5.I1.i3.p1.5">The parameter estimation error <math alttext="\tilde{\bm{\theta}}(t)" class="ltx_Math" display="inline" id="S5.I1.i3.p1.1.m1.1"><semantics id="S5.I1.i3.p1.1.m1.1a"><mrow id="S5.I1.i3.p1.1.m1.1.2" xref="S5.I1.i3.p1.1.m1.1.2.cmml"><mover accent="true" id="S5.I1.i3.p1.1.m1.1.2.2" xref="S5.I1.i3.p1.1.m1.1.2.2.cmml"><mi id="S5.I1.i3.p1.1.m1.1.2.2.2" xref="S5.I1.i3.p1.1.m1.1.2.2.2.cmml">𝜽</mi><mo id="S5.I1.i3.p1.1.m1.1.2.2.1" xref="S5.I1.i3.p1.1.m1.1.2.2.1.cmml">~</mo></mover><mo id="S5.I1.i3.p1.1.m1.1.2.1" xref="S5.I1.i3.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S5.I1.i3.p1.1.m1.1.2.3.2" xref="S5.I1.i3.p1.1.m1.1.2.cmml"><mo id="S5.I1.i3.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S5.I1.i3.p1.1.m1.1.2.cmml">(</mo><mi id="S5.I1.i3.p1.1.m1.1.1" xref="S5.I1.i3.p1.1.m1.1.1.cmml">t</mi><mo id="S5.I1.i3.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S5.I1.i3.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i3.p1.1.m1.1b"><apply id="S5.I1.i3.p1.1.m1.1.2.cmml" xref="S5.I1.i3.p1.1.m1.1.2"><times id="S5.I1.i3.p1.1.m1.1.2.1.cmml" xref="S5.I1.i3.p1.1.m1.1.2.1"></times><apply id="S5.I1.i3.p1.1.m1.1.2.2.cmml" xref="S5.I1.i3.p1.1.m1.1.2.2"><ci id="S5.I1.i3.p1.1.m1.1.2.2.1.cmml" xref="S5.I1.i3.p1.1.m1.1.2.2.1">~</ci><ci id="S5.I1.i3.p1.1.m1.1.2.2.2.cmml" xref="S5.I1.i3.p1.1.m1.1.2.2.2">𝜽</ci></apply><ci id="S5.I1.i3.p1.1.m1.1.1.cmml" xref="S5.I1.i3.p1.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i3.p1.1.m1.1c">\tilde{\bm{\theta}}(t)</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i3.p1.1.m1.1d">over~ start_ARG bold_italic_θ end_ARG ( italic_t )</annotation></semantics></math> exponentially converges to <math alttext="\bm{0}" class="ltx_Math" display="inline" id="S5.I1.i3.p1.2.m2.1"><semantics id="S5.I1.i3.p1.2.m2.1a"><mn id="S5.I1.i3.p1.2.m2.1.1" xref="S5.I1.i3.p1.2.m2.1.1.cmml">𝟎</mn><annotation-xml encoding="MathML-Content" id="S5.I1.i3.p1.2.m2.1b"><cn id="S5.I1.i3.p1.2.m2.1.1.cmml" type="integer" xref="S5.I1.i3.p1.2.m2.1.1">0</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i3.p1.2.m2.1c">\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i3.p1.2.m2.1d">bold_0</annotation></semantics></math> if <math alttext="t_{\rm f}\rightarrow\infty" class="ltx_Math" display="inline" id="S5.I1.i3.p1.3.m3.1"><semantics id="S5.I1.i3.p1.3.m3.1a"><mrow id="S5.I1.i3.p1.3.m3.1.1" xref="S5.I1.i3.p1.3.m3.1.1.cmml"><msub id="S5.I1.i3.p1.3.m3.1.1.2" xref="S5.I1.i3.p1.3.m3.1.1.2.cmml"><mi id="S5.I1.i3.p1.3.m3.1.1.2.2" xref="S5.I1.i3.p1.3.m3.1.1.2.2.cmml">t</mi><mi id="S5.I1.i3.p1.3.m3.1.1.2.3" mathvariant="normal" xref="S5.I1.i3.p1.3.m3.1.1.2.3.cmml">f</mi></msub><mo id="S5.I1.i3.p1.3.m3.1.1.1" stretchy="false" xref="S5.I1.i3.p1.3.m3.1.1.1.cmml">→</mo><mi id="S5.I1.i3.p1.3.m3.1.1.3" mathvariant="normal" xref="S5.I1.i3.p1.3.m3.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i3.p1.3.m3.1b"><apply id="S5.I1.i3.p1.3.m3.1.1.cmml" xref="S5.I1.i3.p1.3.m3.1.1"><ci id="S5.I1.i3.p1.3.m3.1.1.1.cmml" xref="S5.I1.i3.p1.3.m3.1.1.1">→</ci><apply id="S5.I1.i3.p1.3.m3.1.1.2.cmml" xref="S5.I1.i3.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S5.I1.i3.p1.3.m3.1.1.2.1.cmml" xref="S5.I1.i3.p1.3.m3.1.1.2">subscript</csymbol><ci id="S5.I1.i3.p1.3.m3.1.1.2.2.cmml" xref="S5.I1.i3.p1.3.m3.1.1.2.2">𝑡</ci><ci id="S5.I1.i3.p1.3.m3.1.1.2.3.cmml" xref="S5.I1.i3.p1.3.m3.1.1.2.3">f</ci></apply><infinity id="S5.I1.i3.p1.3.m3.1.1.3.cmml" xref="S5.I1.i3.p1.3.m3.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i3.p1.3.m3.1c">t_{\rm f}\rightarrow\infty</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i3.p1.3.m3.1d">italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT → ∞</annotation></semantics></math> and IE exists for some constants <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.I1.i3.p1.4.m4.1"><semantics id="S5.I1.i3.p1.4.m4.1a"><mi id="S5.I1.i3.p1.4.m4.1.1" xref="S5.I1.i3.p1.4.m4.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i3.p1.4.m4.1b"><ci id="S5.I1.i3.p1.4.m4.1.1.cmml" xref="S5.I1.i3.p1.4.m4.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i3.p1.4.m4.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i3.p1.4.m4.1d">italic_σ</annotation></semantics></math>, <math alttext="T_{\rm e}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S5.I1.i3.p1.5.m5.1"><semantics id="S5.I1.i3.p1.5.m5.1a"><mrow id="S5.I1.i3.p1.5.m5.1.1" xref="S5.I1.i3.p1.5.m5.1.1.cmml"><msub id="S5.I1.i3.p1.5.m5.1.1.2" xref="S5.I1.i3.p1.5.m5.1.1.2.cmml"><mi id="S5.I1.i3.p1.5.m5.1.1.2.2" xref="S5.I1.i3.p1.5.m5.1.1.2.2.cmml">T</mi><mi id="S5.I1.i3.p1.5.m5.1.1.2.3" mathvariant="normal" xref="S5.I1.i3.p1.5.m5.1.1.2.3.cmml">e</mi></msub><mo id="S5.I1.i3.p1.5.m5.1.1.1" xref="S5.I1.i3.p1.5.m5.1.1.1.cmml">∈</mo><msup id="S5.I1.i3.p1.5.m5.1.1.3" xref="S5.I1.i3.p1.5.m5.1.1.3.cmml"><mi id="S5.I1.i3.p1.5.m5.1.1.3.2" xref="S5.I1.i3.p1.5.m5.1.1.3.2.cmml">ℝ</mi><mo id="S5.I1.i3.p1.5.m5.1.1.3.3" xref="S5.I1.i3.p1.5.m5.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i3.p1.5.m5.1b"><apply id="S5.I1.i3.p1.5.m5.1.1.cmml" xref="S5.I1.i3.p1.5.m5.1.1"><in id="S5.I1.i3.p1.5.m5.1.1.1.cmml" xref="S5.I1.i3.p1.5.m5.1.1.1"></in><apply id="S5.I1.i3.p1.5.m5.1.1.2.cmml" xref="S5.I1.i3.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S5.I1.i3.p1.5.m5.1.1.2.1.cmml" xref="S5.I1.i3.p1.5.m5.1.1.2">subscript</csymbol><ci id="S5.I1.i3.p1.5.m5.1.1.2.2.cmml" xref="S5.I1.i3.p1.5.m5.1.1.2.2">𝑇</ci><ci id="S5.I1.i3.p1.5.m5.1.1.2.3.cmml" xref="S5.I1.i3.p1.5.m5.1.1.2.3">e</ci></apply><apply id="S5.I1.i3.p1.5.m5.1.1.3.cmml" xref="S5.I1.i3.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S5.I1.i3.p1.5.m5.1.1.3.1.cmml" xref="S5.I1.i3.p1.5.m5.1.1.3">superscript</csymbol><ci id="S5.I1.i3.p1.5.m5.1.1.3.2.cmml" xref="S5.I1.i3.p1.5.m5.1.1.3.2">ℝ</ci><plus id="S5.I1.i3.p1.5.m5.1.1.3.3.cmml" xref="S5.I1.i3.p1.5.m5.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i3.p1.5.m5.1c">T_{\rm e}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i3.p1.5.m5.1d">italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </li> </ol> </div> </section> <section class="ltx_subsection" id="S5.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S5.SS2.4.1.1">V-B</span> </span><span class="ltx_text ltx_font_italic" id="S5.SS2.5.2">Closed-Loop Stability Results</span> </h3> <div class="ltx_para" id="S5.SS2.p1"> <p class="ltx_p" id="S5.SS2.p1.1">We introduce the following Lemma 1 (local existence of solutions) and Lemma 2 (convergence of stable filters) to facilitate the analysis of closed-loop stability, where Lemma 2 can be obtained by a simple extension of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib31" title="">31</a>, Lemma 4]</cite>.</p> </div> <div class="ltx_para" id="S5.SS2.p2"> <p class="ltx_p" id="S5.SS2.p2.5"><span class="ltx_text ltx_font_italic" id="S5.SS2.p2.5.1">Lemma 1</span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib27" title="">27</a>]</cite>: Consider the system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S2.E4" title="In II Problem Formulation ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">4</span></a>) under Assumptions 1–2 with the control law (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E6" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">6</span></a>). For any given <math alttext="\bm{x}(0)\in\Omega_{{\rm c}_{0}}\subset\mathbb{R}^{n}" class="ltx_Math" display="inline" id="S5.SS2.p2.1.m1.1"><semantics id="S5.SS2.p2.1.m1.1a"><mrow id="S5.SS2.p2.1.m1.1.2" xref="S5.SS2.p2.1.m1.1.2.cmml"><mrow id="S5.SS2.p2.1.m1.1.2.2" xref="S5.SS2.p2.1.m1.1.2.2.cmml"><mi id="S5.SS2.p2.1.m1.1.2.2.2" xref="S5.SS2.p2.1.m1.1.2.2.2.cmml">𝒙</mi><mo id="S5.SS2.p2.1.m1.1.2.2.1" xref="S5.SS2.p2.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S5.SS2.p2.1.m1.1.2.2.3.2" xref="S5.SS2.p2.1.m1.1.2.2.cmml"><mo id="S5.SS2.p2.1.m1.1.2.2.3.2.1" stretchy="false" xref="S5.SS2.p2.1.m1.1.2.2.cmml">(</mo><mn id="S5.SS2.p2.1.m1.1.1" xref="S5.SS2.p2.1.m1.1.1.cmml">0</mn><mo id="S5.SS2.p2.1.m1.1.2.2.3.2.2" stretchy="false" xref="S5.SS2.p2.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.SS2.p2.1.m1.1.2.3" xref="S5.SS2.p2.1.m1.1.2.3.cmml">∈</mo><msub id="S5.SS2.p2.1.m1.1.2.4" xref="S5.SS2.p2.1.m1.1.2.4.cmml"><mi id="S5.SS2.p2.1.m1.1.2.4.2" mathvariant="normal" xref="S5.SS2.p2.1.m1.1.2.4.2.cmml">Ω</mi><msub id="S5.SS2.p2.1.m1.1.2.4.3" xref="S5.SS2.p2.1.m1.1.2.4.3.cmml"><mi id="S5.SS2.p2.1.m1.1.2.4.3.2" mathvariant="normal" xref="S5.SS2.p2.1.m1.1.2.4.3.2.cmml">c</mi><mn id="S5.SS2.p2.1.m1.1.2.4.3.3" xref="S5.SS2.p2.1.m1.1.2.4.3.3.cmml">0</mn></msub></msub><mo id="S5.SS2.p2.1.m1.1.2.5" xref="S5.SS2.p2.1.m1.1.2.5.cmml">⊂</mo><msup id="S5.SS2.p2.1.m1.1.2.6" xref="S5.SS2.p2.1.m1.1.2.6.cmml"><mi id="S5.SS2.p2.1.m1.1.2.6.2" xref="S5.SS2.p2.1.m1.1.2.6.2.cmml">ℝ</mi><mi id="S5.SS2.p2.1.m1.1.2.6.3" xref="S5.SS2.p2.1.m1.1.2.6.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.1.m1.1b"><apply id="S5.SS2.p2.1.m1.1.2.cmml" xref="S5.SS2.p2.1.m1.1.2"><and id="S5.SS2.p2.1.m1.1.2a.cmml" xref="S5.SS2.p2.1.m1.1.2"></and><apply id="S5.SS2.p2.1.m1.1.2b.cmml" xref="S5.SS2.p2.1.m1.1.2"><in id="S5.SS2.p2.1.m1.1.2.3.cmml" xref="S5.SS2.p2.1.m1.1.2.3"></in><apply id="S5.SS2.p2.1.m1.1.2.2.cmml" xref="S5.SS2.p2.1.m1.1.2.2"><times id="S5.SS2.p2.1.m1.1.2.2.1.cmml" xref="S5.SS2.p2.1.m1.1.2.2.1"></times><ci id="S5.SS2.p2.1.m1.1.2.2.2.cmml" xref="S5.SS2.p2.1.m1.1.2.2.2">𝒙</ci><cn id="S5.SS2.p2.1.m1.1.1.cmml" type="integer" xref="S5.SS2.p2.1.m1.1.1">0</cn></apply><apply id="S5.SS2.p2.1.m1.1.2.4.cmml" xref="S5.SS2.p2.1.m1.1.2.4"><csymbol cd="ambiguous" id="S5.SS2.p2.1.m1.1.2.4.1.cmml" xref="S5.SS2.p2.1.m1.1.2.4">subscript</csymbol><ci id="S5.SS2.p2.1.m1.1.2.4.2.cmml" xref="S5.SS2.p2.1.m1.1.2.4.2">Ω</ci><apply id="S5.SS2.p2.1.m1.1.2.4.3.cmml" xref="S5.SS2.p2.1.m1.1.2.4.3"><csymbol cd="ambiguous" id="S5.SS2.p2.1.m1.1.2.4.3.1.cmml" xref="S5.SS2.p2.1.m1.1.2.4.3">subscript</csymbol><ci id="S5.SS2.p2.1.m1.1.2.4.3.2.cmml" xref="S5.SS2.p2.1.m1.1.2.4.3.2">c</ci><cn id="S5.SS2.p2.1.m1.1.2.4.3.3.cmml" type="integer" xref="S5.SS2.p2.1.m1.1.2.4.3.3">0</cn></apply></apply></apply><apply id="S5.SS2.p2.1.m1.1.2c.cmml" xref="S5.SS2.p2.1.m1.1.2"><subset id="S5.SS2.p2.1.m1.1.2.5.cmml" xref="S5.SS2.p2.1.m1.1.2.5"></subset><share href="https://arxiv.org/html/2401.10785v2#S5.SS2.p2.1.m1.1.2.4.cmml" id="S5.SS2.p2.1.m1.1.2d.cmml" xref="S5.SS2.p2.1.m1.1.2"></share><apply id="S5.SS2.p2.1.m1.1.2.6.cmml" xref="S5.SS2.p2.1.m1.1.2.6"><csymbol cd="ambiguous" id="S5.SS2.p2.1.m1.1.2.6.1.cmml" xref="S5.SS2.p2.1.m1.1.2.6">superscript</csymbol><ci id="S5.SS2.p2.1.m1.1.2.6.2.cmml" xref="S5.SS2.p2.1.m1.1.2.6.2">ℝ</ci><ci id="S5.SS2.p2.1.m1.1.2.6.3.cmml" xref="S5.SS2.p2.1.m1.1.2.6.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.1.m1.1c">\bm{x}(0)\in\Omega_{{\rm c}_{0}}\subset\mathbb{R}^{n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.1.m1.1d">bold_italic_x ( 0 ) ∈ roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊂ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="c_{0}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S5.SS2.p2.2.m2.1"><semantics id="S5.SS2.p2.2.m2.1a"><mrow id="S5.SS2.p2.2.m2.1.1" xref="S5.SS2.p2.2.m2.1.1.cmml"><msub id="S5.SS2.p2.2.m2.1.1.2" xref="S5.SS2.p2.2.m2.1.1.2.cmml"><mi id="S5.SS2.p2.2.m2.1.1.2.2" xref="S5.SS2.p2.2.m2.1.1.2.2.cmml">c</mi><mn id="S5.SS2.p2.2.m2.1.1.2.3" xref="S5.SS2.p2.2.m2.1.1.2.3.cmml">0</mn></msub><mo id="S5.SS2.p2.2.m2.1.1.1" xref="S5.SS2.p2.2.m2.1.1.1.cmml">∈</mo><msup id="S5.SS2.p2.2.m2.1.1.3" xref="S5.SS2.p2.2.m2.1.1.3.cmml"><mi id="S5.SS2.p2.2.m2.1.1.3.2" xref="S5.SS2.p2.2.m2.1.1.3.2.cmml">ℝ</mi><mo id="S5.SS2.p2.2.m2.1.1.3.3" xref="S5.SS2.p2.2.m2.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.2.m2.1b"><apply id="S5.SS2.p2.2.m2.1.1.cmml" xref="S5.SS2.p2.2.m2.1.1"><in id="S5.SS2.p2.2.m2.1.1.1.cmml" xref="S5.SS2.p2.2.m2.1.1.1"></in><apply id="S5.SS2.p2.2.m2.1.1.2.cmml" xref="S5.SS2.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.p2.2.m2.1.1.2.1.cmml" xref="S5.SS2.p2.2.m2.1.1.2">subscript</csymbol><ci id="S5.SS2.p2.2.m2.1.1.2.2.cmml" xref="S5.SS2.p2.2.m2.1.1.2.2">𝑐</ci><cn id="S5.SS2.p2.2.m2.1.1.2.3.cmml" type="integer" xref="S5.SS2.p2.2.m2.1.1.2.3">0</cn></apply><apply id="S5.SS2.p2.2.m2.1.1.3.cmml" xref="S5.SS2.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.p2.2.m2.1.1.3.1.cmml" xref="S5.SS2.p2.2.m2.1.1.3">superscript</csymbol><ci id="S5.SS2.p2.2.m2.1.1.3.2.cmml" xref="S5.SS2.p2.2.m2.1.1.3.2">ℝ</ci><plus id="S5.SS2.p2.2.m2.1.1.3.3.cmml" xref="S5.SS2.p2.2.m2.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.2.m2.1c">c_{0}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.2.m2.1d">italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, there exists <math alttext="c_{x}>c_{0}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S5.SS2.p2.3.m3.1"><semantics id="S5.SS2.p2.3.m3.1a"><mrow id="S5.SS2.p2.3.m3.1.1" xref="S5.SS2.p2.3.m3.1.1.cmml"><msub id="S5.SS2.p2.3.m3.1.1.2" xref="S5.SS2.p2.3.m3.1.1.2.cmml"><mi id="S5.SS2.p2.3.m3.1.1.2.2" xref="S5.SS2.p2.3.m3.1.1.2.2.cmml">c</mi><mi id="S5.SS2.p2.3.m3.1.1.2.3" xref="S5.SS2.p2.3.m3.1.1.2.3.cmml">x</mi></msub><mo id="S5.SS2.p2.3.m3.1.1.3" xref="S5.SS2.p2.3.m3.1.1.3.cmml">></mo><msub id="S5.SS2.p2.3.m3.1.1.4" xref="S5.SS2.p2.3.m3.1.1.4.cmml"><mi id="S5.SS2.p2.3.m3.1.1.4.2" xref="S5.SS2.p2.3.m3.1.1.4.2.cmml">c</mi><mn id="S5.SS2.p2.3.m3.1.1.4.3" xref="S5.SS2.p2.3.m3.1.1.4.3.cmml">0</mn></msub><mo id="S5.SS2.p2.3.m3.1.1.5" xref="S5.SS2.p2.3.m3.1.1.5.cmml">∈</mo><msup id="S5.SS2.p2.3.m3.1.1.6" xref="S5.SS2.p2.3.m3.1.1.6.cmml"><mi id="S5.SS2.p2.3.m3.1.1.6.2" xref="S5.SS2.p2.3.m3.1.1.6.2.cmml">ℝ</mi><mo id="S5.SS2.p2.3.m3.1.1.6.3" xref="S5.SS2.p2.3.m3.1.1.6.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.3.m3.1b"><apply id="S5.SS2.p2.3.m3.1.1.cmml" xref="S5.SS2.p2.3.m3.1.1"><and id="S5.SS2.p2.3.m3.1.1a.cmml" xref="S5.SS2.p2.3.m3.1.1"></and><apply id="S5.SS2.p2.3.m3.1.1b.cmml" xref="S5.SS2.p2.3.m3.1.1"><gt id="S5.SS2.p2.3.m3.1.1.3.cmml" xref="S5.SS2.p2.3.m3.1.1.3"></gt><apply id="S5.SS2.p2.3.m3.1.1.2.cmml" xref="S5.SS2.p2.3.m3.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.p2.3.m3.1.1.2.1.cmml" xref="S5.SS2.p2.3.m3.1.1.2">subscript</csymbol><ci id="S5.SS2.p2.3.m3.1.1.2.2.cmml" xref="S5.SS2.p2.3.m3.1.1.2.2">𝑐</ci><ci id="S5.SS2.p2.3.m3.1.1.2.3.cmml" xref="S5.SS2.p2.3.m3.1.1.2.3">𝑥</ci></apply><apply id="S5.SS2.p2.3.m3.1.1.4.cmml" xref="S5.SS2.p2.3.m3.1.1.4"><csymbol cd="ambiguous" id="S5.SS2.p2.3.m3.1.1.4.1.cmml" xref="S5.SS2.p2.3.m3.1.1.4">subscript</csymbol><ci id="S5.SS2.p2.3.m3.1.1.4.2.cmml" xref="S5.SS2.p2.3.m3.1.1.4.2">𝑐</ci><cn id="S5.SS2.p2.3.m3.1.1.4.3.cmml" type="integer" xref="S5.SS2.p2.3.m3.1.1.4.3">0</cn></apply></apply><apply id="S5.SS2.p2.3.m3.1.1c.cmml" xref="S5.SS2.p2.3.m3.1.1"><in id="S5.SS2.p2.3.m3.1.1.5.cmml" xref="S5.SS2.p2.3.m3.1.1.5"></in><share href="https://arxiv.org/html/2401.10785v2#S5.SS2.p2.3.m3.1.1.4.cmml" id="S5.SS2.p2.3.m3.1.1d.cmml" xref="S5.SS2.p2.3.m3.1.1"></share><apply id="S5.SS2.p2.3.m3.1.1.6.cmml" xref="S5.SS2.p2.3.m3.1.1.6"><csymbol cd="ambiguous" id="S5.SS2.p2.3.m3.1.1.6.1.cmml" xref="S5.SS2.p2.3.m3.1.1.6">superscript</csymbol><ci id="S5.SS2.p2.3.m3.1.1.6.2.cmml" xref="S5.SS2.p2.3.m3.1.1.6.2">ℝ</ci><plus id="S5.SS2.p2.3.m3.1.1.6.3.cmml" xref="S5.SS2.p2.3.m3.1.1.6.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.3.m3.1c">c_{x}>c_{0}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.3.m3.1d">italic_c start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT > italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, such that <math alttext="\bm{x}(t)\in\Omega_{{\rm c}_{x}}\subset\mathbb{R}^{n}" class="ltx_Math" display="inline" id="S5.SS2.p2.4.m4.1"><semantics id="S5.SS2.p2.4.m4.1a"><mrow id="S5.SS2.p2.4.m4.1.2" xref="S5.SS2.p2.4.m4.1.2.cmml"><mrow id="S5.SS2.p2.4.m4.1.2.2" xref="S5.SS2.p2.4.m4.1.2.2.cmml"><mi id="S5.SS2.p2.4.m4.1.2.2.2" xref="S5.SS2.p2.4.m4.1.2.2.2.cmml">𝒙</mi><mo id="S5.SS2.p2.4.m4.1.2.2.1" xref="S5.SS2.p2.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S5.SS2.p2.4.m4.1.2.2.3.2" xref="S5.SS2.p2.4.m4.1.2.2.cmml"><mo id="S5.SS2.p2.4.m4.1.2.2.3.2.1" stretchy="false" xref="S5.SS2.p2.4.m4.1.2.2.cmml">(</mo><mi id="S5.SS2.p2.4.m4.1.1" xref="S5.SS2.p2.4.m4.1.1.cmml">t</mi><mo id="S5.SS2.p2.4.m4.1.2.2.3.2.2" stretchy="false" xref="S5.SS2.p2.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.SS2.p2.4.m4.1.2.3" xref="S5.SS2.p2.4.m4.1.2.3.cmml">∈</mo><msub id="S5.SS2.p2.4.m4.1.2.4" xref="S5.SS2.p2.4.m4.1.2.4.cmml"><mi id="S5.SS2.p2.4.m4.1.2.4.2" mathvariant="normal" xref="S5.SS2.p2.4.m4.1.2.4.2.cmml">Ω</mi><msub id="S5.SS2.p2.4.m4.1.2.4.3" xref="S5.SS2.p2.4.m4.1.2.4.3.cmml"><mi id="S5.SS2.p2.4.m4.1.2.4.3.2" mathvariant="normal" xref="S5.SS2.p2.4.m4.1.2.4.3.2.cmml">c</mi><mi id="S5.SS2.p2.4.m4.1.2.4.3.3" xref="S5.SS2.p2.4.m4.1.2.4.3.3.cmml">x</mi></msub></msub><mo id="S5.SS2.p2.4.m4.1.2.5" xref="S5.SS2.p2.4.m4.1.2.5.cmml">⊂</mo><msup id="S5.SS2.p2.4.m4.1.2.6" xref="S5.SS2.p2.4.m4.1.2.6.cmml"><mi id="S5.SS2.p2.4.m4.1.2.6.2" xref="S5.SS2.p2.4.m4.1.2.6.2.cmml">ℝ</mi><mi id="S5.SS2.p2.4.m4.1.2.6.3" xref="S5.SS2.p2.4.m4.1.2.6.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.4.m4.1b"><apply id="S5.SS2.p2.4.m4.1.2.cmml" xref="S5.SS2.p2.4.m4.1.2"><and id="S5.SS2.p2.4.m4.1.2a.cmml" xref="S5.SS2.p2.4.m4.1.2"></and><apply id="S5.SS2.p2.4.m4.1.2b.cmml" xref="S5.SS2.p2.4.m4.1.2"><in id="S5.SS2.p2.4.m4.1.2.3.cmml" xref="S5.SS2.p2.4.m4.1.2.3"></in><apply id="S5.SS2.p2.4.m4.1.2.2.cmml" xref="S5.SS2.p2.4.m4.1.2.2"><times id="S5.SS2.p2.4.m4.1.2.2.1.cmml" xref="S5.SS2.p2.4.m4.1.2.2.1"></times><ci id="S5.SS2.p2.4.m4.1.2.2.2.cmml" xref="S5.SS2.p2.4.m4.1.2.2.2">𝒙</ci><ci id="S5.SS2.p2.4.m4.1.1.cmml" xref="S5.SS2.p2.4.m4.1.1">𝑡</ci></apply><apply id="S5.SS2.p2.4.m4.1.2.4.cmml" xref="S5.SS2.p2.4.m4.1.2.4"><csymbol cd="ambiguous" id="S5.SS2.p2.4.m4.1.2.4.1.cmml" xref="S5.SS2.p2.4.m4.1.2.4">subscript</csymbol><ci id="S5.SS2.p2.4.m4.1.2.4.2.cmml" xref="S5.SS2.p2.4.m4.1.2.4.2">Ω</ci><apply id="S5.SS2.p2.4.m4.1.2.4.3.cmml" xref="S5.SS2.p2.4.m4.1.2.4.3"><csymbol cd="ambiguous" id="S5.SS2.p2.4.m4.1.2.4.3.1.cmml" xref="S5.SS2.p2.4.m4.1.2.4.3">subscript</csymbol><ci id="S5.SS2.p2.4.m4.1.2.4.3.2.cmml" xref="S5.SS2.p2.4.m4.1.2.4.3.2">c</ci><ci id="S5.SS2.p2.4.m4.1.2.4.3.3.cmml" xref="S5.SS2.p2.4.m4.1.2.4.3.3">𝑥</ci></apply></apply></apply><apply id="S5.SS2.p2.4.m4.1.2c.cmml" xref="S5.SS2.p2.4.m4.1.2"><subset id="S5.SS2.p2.4.m4.1.2.5.cmml" xref="S5.SS2.p2.4.m4.1.2.5"></subset><share href="https://arxiv.org/html/2401.10785v2#S5.SS2.p2.4.m4.1.2.4.cmml" id="S5.SS2.p2.4.m4.1.2d.cmml" xref="S5.SS2.p2.4.m4.1.2"></share><apply id="S5.SS2.p2.4.m4.1.2.6.cmml" xref="S5.SS2.p2.4.m4.1.2.6"><csymbol cd="ambiguous" id="S5.SS2.p2.4.m4.1.2.6.1.cmml" xref="S5.SS2.p2.4.m4.1.2.6">superscript</csymbol><ci id="S5.SS2.p2.4.m4.1.2.6.2.cmml" xref="S5.SS2.p2.4.m4.1.2.6.2">ℝ</ci><ci id="S5.SS2.p2.4.m4.1.2.6.3.cmml" xref="S5.SS2.p2.4.m4.1.2.6.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.4.m4.1c">\bm{x}(t)\in\Omega_{{\rm c}_{x}}\subset\mathbb{R}^{n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.4.m4.1d">bold_italic_x ( italic_t ) ∈ roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊂ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\forall t\in[0,t_{\rm f})" class="ltx_Math" display="inline" id="S5.SS2.p2.5.m5.2"><semantics id="S5.SS2.p2.5.m5.2a"><mrow id="S5.SS2.p2.5.m5.2.2" xref="S5.SS2.p2.5.m5.2.2.cmml"><mrow id="S5.SS2.p2.5.m5.2.2.3" xref="S5.SS2.p2.5.m5.2.2.3.cmml"><mo id="S5.SS2.p2.5.m5.2.2.3.1" rspace="0.167em" xref="S5.SS2.p2.5.m5.2.2.3.1.cmml">∀</mo><mi id="S5.SS2.p2.5.m5.2.2.3.2" xref="S5.SS2.p2.5.m5.2.2.3.2.cmml">t</mi></mrow><mo id="S5.SS2.p2.5.m5.2.2.2" xref="S5.SS2.p2.5.m5.2.2.2.cmml">∈</mo><mrow id="S5.SS2.p2.5.m5.2.2.1.1" xref="S5.SS2.p2.5.m5.2.2.1.2.cmml"><mo id="S5.SS2.p2.5.m5.2.2.1.1.2" stretchy="false" xref="S5.SS2.p2.5.m5.2.2.1.2.cmml">[</mo><mn id="S5.SS2.p2.5.m5.1.1" xref="S5.SS2.p2.5.m5.1.1.cmml">0</mn><mo id="S5.SS2.p2.5.m5.2.2.1.1.3" xref="S5.SS2.p2.5.m5.2.2.1.2.cmml">,</mo><msub id="S5.SS2.p2.5.m5.2.2.1.1.1" xref="S5.SS2.p2.5.m5.2.2.1.1.1.cmml"><mi id="S5.SS2.p2.5.m5.2.2.1.1.1.2" xref="S5.SS2.p2.5.m5.2.2.1.1.1.2.cmml">t</mi><mi id="S5.SS2.p2.5.m5.2.2.1.1.1.3" mathvariant="normal" xref="S5.SS2.p2.5.m5.2.2.1.1.1.3.cmml">f</mi></msub><mo id="S5.SS2.p2.5.m5.2.2.1.1.4" stretchy="false" xref="S5.SS2.p2.5.m5.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.5.m5.2b"><apply id="S5.SS2.p2.5.m5.2.2.cmml" xref="S5.SS2.p2.5.m5.2.2"><in id="S5.SS2.p2.5.m5.2.2.2.cmml" xref="S5.SS2.p2.5.m5.2.2.2"></in><apply id="S5.SS2.p2.5.m5.2.2.3.cmml" xref="S5.SS2.p2.5.m5.2.2.3"><csymbol cd="latexml" id="S5.SS2.p2.5.m5.2.2.3.1.cmml" xref="S5.SS2.p2.5.m5.2.2.3.1">for-all</csymbol><ci id="S5.SS2.p2.5.m5.2.2.3.2.cmml" xref="S5.SS2.p2.5.m5.2.2.3.2">𝑡</ci></apply><interval closure="closed-open" id="S5.SS2.p2.5.m5.2.2.1.2.cmml" xref="S5.SS2.p2.5.m5.2.2.1.1"><cn id="S5.SS2.p2.5.m5.1.1.cmml" type="integer" xref="S5.SS2.p2.5.m5.1.1">0</cn><apply id="S5.SS2.p2.5.m5.2.2.1.1.1.cmml" xref="S5.SS2.p2.5.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.p2.5.m5.2.2.1.1.1.1.cmml" xref="S5.SS2.p2.5.m5.2.2.1.1.1">subscript</csymbol><ci id="S5.SS2.p2.5.m5.2.2.1.1.1.2.cmml" xref="S5.SS2.p2.5.m5.2.2.1.1.1.2">𝑡</ci><ci id="S5.SS2.p2.5.m5.2.2.1.1.1.3.cmml" xref="S5.SS2.p2.5.m5.2.2.1.1.1.3">f</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.5.m5.2c">\forall t\in[0,t_{\rm f})</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.5.m5.2d">∀ italic_t ∈ [ 0 , italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.SS2.p3"> <p class="ltx_p" id="S5.SS2.p3.10"><span class="ltx_text ltx_font_italic" id="S5.SS2.p3.10.1">Lemma 2</span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib31" title="">31</a>]</cite>: Consider <math alttext="H(s)" class="ltx_Math" display="inline" id="S5.SS2.p3.1.m1.1"><semantics id="S5.SS2.p3.1.m1.1a"><mrow id="S5.SS2.p3.1.m1.1.2" xref="S5.SS2.p3.1.m1.1.2.cmml"><mi id="S5.SS2.p3.1.m1.1.2.2" xref="S5.SS2.p3.1.m1.1.2.2.cmml">H</mi><mo id="S5.SS2.p3.1.m1.1.2.1" xref="S5.SS2.p3.1.m1.1.2.1.cmml">⁢</mo><mrow id="S5.SS2.p3.1.m1.1.2.3.2" xref="S5.SS2.p3.1.m1.1.2.cmml"><mo id="S5.SS2.p3.1.m1.1.2.3.2.1" stretchy="false" xref="S5.SS2.p3.1.m1.1.2.cmml">(</mo><mi id="S5.SS2.p3.1.m1.1.1" xref="S5.SS2.p3.1.m1.1.1.cmml">s</mi><mo id="S5.SS2.p3.1.m1.1.2.3.2.2" stretchy="false" xref="S5.SS2.p3.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.1.m1.1b"><apply id="S5.SS2.p3.1.m1.1.2.cmml" xref="S5.SS2.p3.1.m1.1.2"><times id="S5.SS2.p3.1.m1.1.2.1.cmml" xref="S5.SS2.p3.1.m1.1.2.1"></times><ci id="S5.SS2.p3.1.m1.1.2.2.cmml" xref="S5.SS2.p3.1.m1.1.2.2">𝐻</ci><ci id="S5.SS2.p3.1.m1.1.1.cmml" xref="S5.SS2.p3.1.m1.1.1">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.1.m1.1c">H(s)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.1.m1.1d">italic_H ( italic_s )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E17" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">17</span></a>) on <math alttext="t\in[0,t_{\rm f})" class="ltx_Math" display="inline" id="S5.SS2.p3.2.m2.2"><semantics id="S5.SS2.p3.2.m2.2a"><mrow id="S5.SS2.p3.2.m2.2.2" xref="S5.SS2.p3.2.m2.2.2.cmml"><mi id="S5.SS2.p3.2.m2.2.2.3" xref="S5.SS2.p3.2.m2.2.2.3.cmml">t</mi><mo id="S5.SS2.p3.2.m2.2.2.2" xref="S5.SS2.p3.2.m2.2.2.2.cmml">∈</mo><mrow id="S5.SS2.p3.2.m2.2.2.1.1" xref="S5.SS2.p3.2.m2.2.2.1.2.cmml"><mo id="S5.SS2.p3.2.m2.2.2.1.1.2" stretchy="false" xref="S5.SS2.p3.2.m2.2.2.1.2.cmml">[</mo><mn id="S5.SS2.p3.2.m2.1.1" xref="S5.SS2.p3.2.m2.1.1.cmml">0</mn><mo id="S5.SS2.p3.2.m2.2.2.1.1.3" xref="S5.SS2.p3.2.m2.2.2.1.2.cmml">,</mo><msub id="S5.SS2.p3.2.m2.2.2.1.1.1" xref="S5.SS2.p3.2.m2.2.2.1.1.1.cmml"><mi id="S5.SS2.p3.2.m2.2.2.1.1.1.2" xref="S5.SS2.p3.2.m2.2.2.1.1.1.2.cmml">t</mi><mi id="S5.SS2.p3.2.m2.2.2.1.1.1.3" mathvariant="normal" xref="S5.SS2.p3.2.m2.2.2.1.1.1.3.cmml">f</mi></msub><mo id="S5.SS2.p3.2.m2.2.2.1.1.4" stretchy="false" xref="S5.SS2.p3.2.m2.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.2.m2.2b"><apply id="S5.SS2.p3.2.m2.2.2.cmml" xref="S5.SS2.p3.2.m2.2.2"><in id="S5.SS2.p3.2.m2.2.2.2.cmml" xref="S5.SS2.p3.2.m2.2.2.2"></in><ci id="S5.SS2.p3.2.m2.2.2.3.cmml" xref="S5.SS2.p3.2.m2.2.2.3">𝑡</ci><interval closure="closed-open" id="S5.SS2.p3.2.m2.2.2.1.2.cmml" xref="S5.SS2.p3.2.m2.2.2.1.1"><cn id="S5.SS2.p3.2.m2.1.1.cmml" type="integer" xref="S5.SS2.p3.2.m2.1.1">0</cn><apply id="S5.SS2.p3.2.m2.2.2.1.1.1.cmml" xref="S5.SS2.p3.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.p3.2.m2.2.2.1.1.1.1.cmml" xref="S5.SS2.p3.2.m2.2.2.1.1.1">subscript</csymbol><ci id="S5.SS2.p3.2.m2.2.2.1.1.1.2.cmml" xref="S5.SS2.p3.2.m2.2.2.1.1.1.2">𝑡</ci><ci id="S5.SS2.p3.2.m2.2.2.1.1.1.3.cmml" xref="S5.SS2.p3.2.m2.2.2.1.1.1.3">f</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.2.m2.2c">t\in[0,t_{\rm f})</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.2.m2.2d">italic_t ∈ [ 0 , italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="\Phi_{\rm f}" class="ltx_Math" display="inline" id="S5.SS2.p3.3.m3.1"><semantics id="S5.SS2.p3.3.m3.1a"><msub id="S5.SS2.p3.3.m3.1.1" xref="S5.SS2.p3.3.m3.1.1.cmml"><mi id="S5.SS2.p3.3.m3.1.1.2" mathvariant="normal" xref="S5.SS2.p3.3.m3.1.1.2.cmml">Φ</mi><mi id="S5.SS2.p3.3.m3.1.1.3" mathvariant="normal" xref="S5.SS2.p3.3.m3.1.1.3.cmml">f</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.3.m3.1b"><apply id="S5.SS2.p3.3.m3.1.1.cmml" xref="S5.SS2.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S5.SS2.p3.3.m3.1.1.1.cmml" xref="S5.SS2.p3.3.m3.1.1">subscript</csymbol><ci id="S5.SS2.p3.3.m3.1.1.2.cmml" xref="S5.SS2.p3.3.m3.1.1.2">Φ</ci><ci id="S5.SS2.p3.3.m3.1.1.3.cmml" xref="S5.SS2.p3.3.m3.1.1.3">f</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.3.m3.1c">\Phi_{\rm f}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.3.m3.1d">roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E21" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">21</span></a>). For any given small <math alttext="\delta\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S5.SS2.p3.4.m4.1"><semantics id="S5.SS2.p3.4.m4.1a"><mrow id="S5.SS2.p3.4.m4.1.1" xref="S5.SS2.p3.4.m4.1.1.cmml"><mi id="S5.SS2.p3.4.m4.1.1.2" xref="S5.SS2.p3.4.m4.1.1.2.cmml">δ</mi><mo id="S5.SS2.p3.4.m4.1.1.1" xref="S5.SS2.p3.4.m4.1.1.1.cmml">∈</mo><msup id="S5.SS2.p3.4.m4.1.1.3" xref="S5.SS2.p3.4.m4.1.1.3.cmml"><mi id="S5.SS2.p3.4.m4.1.1.3.2" xref="S5.SS2.p3.4.m4.1.1.3.2.cmml">ℝ</mi><mo id="S5.SS2.p3.4.m4.1.1.3.3" xref="S5.SS2.p3.4.m4.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.4.m4.1b"><apply id="S5.SS2.p3.4.m4.1.1.cmml" xref="S5.SS2.p3.4.m4.1.1"><in id="S5.SS2.p3.4.m4.1.1.1.cmml" xref="S5.SS2.p3.4.m4.1.1.1"></in><ci id="S5.SS2.p3.4.m4.1.1.2.cmml" xref="S5.SS2.p3.4.m4.1.1.2">𝛿</ci><apply id="S5.SS2.p3.4.m4.1.1.3.cmml" xref="S5.SS2.p3.4.m4.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.p3.4.m4.1.1.3.1.cmml" xref="S5.SS2.p3.4.m4.1.1.3">superscript</csymbol><ci id="S5.SS2.p3.4.m4.1.1.3.2.cmml" xref="S5.SS2.p3.4.m4.1.1.3.2">ℝ</ci><plus id="S5.SS2.p3.4.m4.1.1.3.3.cmml" xref="S5.SS2.p3.4.m4.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.4.m4.1c">\delta\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.4.m4.1d">italic_δ ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, there exists a sufficiently large <math alttext="\alpha\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S5.SS2.p3.5.m5.1"><semantics id="S5.SS2.p3.5.m5.1a"><mrow id="S5.SS2.p3.5.m5.1.1" xref="S5.SS2.p3.5.m5.1.1.cmml"><mi id="S5.SS2.p3.5.m5.1.1.2" xref="S5.SS2.p3.5.m5.1.1.2.cmml">α</mi><mo id="S5.SS2.p3.5.m5.1.1.1" xref="S5.SS2.p3.5.m5.1.1.1.cmml">∈</mo><msup id="S5.SS2.p3.5.m5.1.1.3" xref="S5.SS2.p3.5.m5.1.1.3.cmml"><mi id="S5.SS2.p3.5.m5.1.1.3.2" xref="S5.SS2.p3.5.m5.1.1.3.2.cmml">ℝ</mi><mo id="S5.SS2.p3.5.m5.1.1.3.3" xref="S5.SS2.p3.5.m5.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.5.m5.1b"><apply id="S5.SS2.p3.5.m5.1.1.cmml" xref="S5.SS2.p3.5.m5.1.1"><in id="S5.SS2.p3.5.m5.1.1.1.cmml" xref="S5.SS2.p3.5.m5.1.1.1"></in><ci id="S5.SS2.p3.5.m5.1.1.2.cmml" xref="S5.SS2.p3.5.m5.1.1.2">𝛼</ci><apply id="S5.SS2.p3.5.m5.1.1.3.cmml" xref="S5.SS2.p3.5.m5.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.p3.5.m5.1.1.3.1.cmml" xref="S5.SS2.p3.5.m5.1.1.3">superscript</csymbol><ci id="S5.SS2.p3.5.m5.1.1.3.2.cmml" xref="S5.SS2.p3.5.m5.1.1.3.2">ℝ</ci><plus id="S5.SS2.p3.5.m5.1.1.3.3.cmml" xref="S5.SS2.p3.5.m5.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.5.m5.1c">\alpha\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.5.m5.1d">italic_α ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\alpha_{i}>\alpha" class="ltx_Math" display="inline" id="S5.SS2.p3.6.m6.1"><semantics id="S5.SS2.p3.6.m6.1a"><mrow id="S5.SS2.p3.6.m6.1.1" xref="S5.SS2.p3.6.m6.1.1.cmml"><msub id="S5.SS2.p3.6.m6.1.1.2" xref="S5.SS2.p3.6.m6.1.1.2.cmml"><mi id="S5.SS2.p3.6.m6.1.1.2.2" xref="S5.SS2.p3.6.m6.1.1.2.2.cmml">α</mi><mi id="S5.SS2.p3.6.m6.1.1.2.3" xref="S5.SS2.p3.6.m6.1.1.2.3.cmml">i</mi></msub><mo id="S5.SS2.p3.6.m6.1.1.1" xref="S5.SS2.p3.6.m6.1.1.1.cmml">></mo><mi id="S5.SS2.p3.6.m6.1.1.3" xref="S5.SS2.p3.6.m6.1.1.3.cmml">α</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.6.m6.1b"><apply id="S5.SS2.p3.6.m6.1.1.cmml" xref="S5.SS2.p3.6.m6.1.1"><gt id="S5.SS2.p3.6.m6.1.1.1.cmml" xref="S5.SS2.p3.6.m6.1.1.1"></gt><apply id="S5.SS2.p3.6.m6.1.1.2.cmml" xref="S5.SS2.p3.6.m6.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.p3.6.m6.1.1.2.1.cmml" xref="S5.SS2.p3.6.m6.1.1.2">subscript</csymbol><ci id="S5.SS2.p3.6.m6.1.1.2.2.cmml" xref="S5.SS2.p3.6.m6.1.1.2.2">𝛼</ci><ci id="S5.SS2.p3.6.m6.1.1.2.3.cmml" xref="S5.SS2.p3.6.m6.1.1.2.3">𝑖</ci></apply><ci id="S5.SS2.p3.6.m6.1.1.3.cmml" xref="S5.SS2.p3.6.m6.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.6.m6.1c">\alpha_{i}>\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.6.m6.1d">italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT > italic_α</annotation></semantics></math> with <math alttext="i=1" class="ltx_Math" display="inline" id="S5.SS2.p3.7.m7.1"><semantics id="S5.SS2.p3.7.m7.1a"><mrow id="S5.SS2.p3.7.m7.1.1" xref="S5.SS2.p3.7.m7.1.1.cmml"><mi id="S5.SS2.p3.7.m7.1.1.2" xref="S5.SS2.p3.7.m7.1.1.2.cmml">i</mi><mo id="S5.SS2.p3.7.m7.1.1.1" xref="S5.SS2.p3.7.m7.1.1.1.cmml">=</mo><mn id="S5.SS2.p3.7.m7.1.1.3" xref="S5.SS2.p3.7.m7.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.7.m7.1b"><apply id="S5.SS2.p3.7.m7.1.1.cmml" xref="S5.SS2.p3.7.m7.1.1"><eq id="S5.SS2.p3.7.m7.1.1.1.cmml" xref="S5.SS2.p3.7.m7.1.1.1"></eq><ci id="S5.SS2.p3.7.m7.1.1.2.cmml" xref="S5.SS2.p3.7.m7.1.1.2">𝑖</ci><cn id="S5.SS2.p3.7.m7.1.1.3.cmml" type="integer" xref="S5.SS2.p3.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.7.m7.1c">i=1</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.7.m7.1d">italic_i = 1</annotation></semantics></math> to <math alttext="n-1" class="ltx_Math" display="inline" id="S5.SS2.p3.8.m8.1"><semantics id="S5.SS2.p3.8.m8.1a"><mrow id="S5.SS2.p3.8.m8.1.1" xref="S5.SS2.p3.8.m8.1.1.cmml"><mi id="S5.SS2.p3.8.m8.1.1.2" xref="S5.SS2.p3.8.m8.1.1.2.cmml">n</mi><mo id="S5.SS2.p3.8.m8.1.1.1" xref="S5.SS2.p3.8.m8.1.1.1.cmml">−</mo><mn id="S5.SS2.p3.8.m8.1.1.3" xref="S5.SS2.p3.8.m8.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.8.m8.1b"><apply id="S5.SS2.p3.8.m8.1.1.cmml" xref="S5.SS2.p3.8.m8.1.1"><minus id="S5.SS2.p3.8.m8.1.1.1.cmml" xref="S5.SS2.p3.8.m8.1.1.1"></minus><ci id="S5.SS2.p3.8.m8.1.1.2.cmml" xref="S5.SS2.p3.8.m8.1.1.2">𝑛</ci><cn id="S5.SS2.p3.8.m8.1.1.3.cmml" type="integer" xref="S5.SS2.p3.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.8.m8.1c">n-1</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.8.m8.1d">italic_n - 1</annotation></semantics></math>, so that <math alttext="\|\Phi-\Phi_{\rm f}\|\leq\delta" class="ltx_Math" display="inline" id="S5.SS2.p3.9.m9.1"><semantics id="S5.SS2.p3.9.m9.1a"><mrow id="S5.SS2.p3.9.m9.1.1" xref="S5.SS2.p3.9.m9.1.1.cmml"><mrow id="S5.SS2.p3.9.m9.1.1.1.1" xref="S5.SS2.p3.9.m9.1.1.1.2.cmml"><mo id="S5.SS2.p3.9.m9.1.1.1.1.2" stretchy="false" xref="S5.SS2.p3.9.m9.1.1.1.2.1.cmml">‖</mo><mrow id="S5.SS2.p3.9.m9.1.1.1.1.1" xref="S5.SS2.p3.9.m9.1.1.1.1.1.cmml"><mi id="S5.SS2.p3.9.m9.1.1.1.1.1.2" mathvariant="normal" xref="S5.SS2.p3.9.m9.1.1.1.1.1.2.cmml">Φ</mi><mo id="S5.SS2.p3.9.m9.1.1.1.1.1.1" xref="S5.SS2.p3.9.m9.1.1.1.1.1.1.cmml">−</mo><msub id="S5.SS2.p3.9.m9.1.1.1.1.1.3" xref="S5.SS2.p3.9.m9.1.1.1.1.1.3.cmml"><mi id="S5.SS2.p3.9.m9.1.1.1.1.1.3.2" mathvariant="normal" xref="S5.SS2.p3.9.m9.1.1.1.1.1.3.2.cmml">Φ</mi><mi id="S5.SS2.p3.9.m9.1.1.1.1.1.3.3" mathvariant="normal" xref="S5.SS2.p3.9.m9.1.1.1.1.1.3.3.cmml">f</mi></msub></mrow><mo id="S5.SS2.p3.9.m9.1.1.1.1.3" stretchy="false" xref="S5.SS2.p3.9.m9.1.1.1.2.1.cmml">‖</mo></mrow><mo id="S5.SS2.p3.9.m9.1.1.2" xref="S5.SS2.p3.9.m9.1.1.2.cmml">≤</mo><mi id="S5.SS2.p3.9.m9.1.1.3" xref="S5.SS2.p3.9.m9.1.1.3.cmml">δ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.9.m9.1b"><apply id="S5.SS2.p3.9.m9.1.1.cmml" xref="S5.SS2.p3.9.m9.1.1"><leq id="S5.SS2.p3.9.m9.1.1.2.cmml" xref="S5.SS2.p3.9.m9.1.1.2"></leq><apply id="S5.SS2.p3.9.m9.1.1.1.2.cmml" xref="S5.SS2.p3.9.m9.1.1.1.1"><csymbol cd="latexml" id="S5.SS2.p3.9.m9.1.1.1.2.1.cmml" xref="S5.SS2.p3.9.m9.1.1.1.1.2">norm</csymbol><apply id="S5.SS2.p3.9.m9.1.1.1.1.1.cmml" xref="S5.SS2.p3.9.m9.1.1.1.1.1"><minus id="S5.SS2.p3.9.m9.1.1.1.1.1.1.cmml" xref="S5.SS2.p3.9.m9.1.1.1.1.1.1"></minus><ci id="S5.SS2.p3.9.m9.1.1.1.1.1.2.cmml" xref="S5.SS2.p3.9.m9.1.1.1.1.1.2">Φ</ci><apply id="S5.SS2.p3.9.m9.1.1.1.1.1.3.cmml" xref="S5.SS2.p3.9.m9.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.p3.9.m9.1.1.1.1.1.3.1.cmml" xref="S5.SS2.p3.9.m9.1.1.1.1.1.3">subscript</csymbol><ci id="S5.SS2.p3.9.m9.1.1.1.1.1.3.2.cmml" xref="S5.SS2.p3.9.m9.1.1.1.1.1.3.2">Φ</ci><ci id="S5.SS2.p3.9.m9.1.1.1.1.1.3.3.cmml" xref="S5.SS2.p3.9.m9.1.1.1.1.1.3.3">f</ci></apply></apply></apply><ci id="S5.SS2.p3.9.m9.1.1.3.cmml" xref="S5.SS2.p3.9.m9.1.1.3">𝛿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.9.m9.1c">\|\Phi-\Phi_{\rm f}\|\leq\delta</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.9.m9.1d">∥ roman_Φ - roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ∥ ≤ italic_δ</annotation></semantics></math>, <math alttext="\forall t\in[0,t_{\rm f})" class="ltx_Math" display="inline" id="S5.SS2.p3.10.m10.2"><semantics id="S5.SS2.p3.10.m10.2a"><mrow id="S5.SS2.p3.10.m10.2.2" xref="S5.SS2.p3.10.m10.2.2.cmml"><mrow id="S5.SS2.p3.10.m10.2.2.3" xref="S5.SS2.p3.10.m10.2.2.3.cmml"><mo id="S5.SS2.p3.10.m10.2.2.3.1" rspace="0.167em" xref="S5.SS2.p3.10.m10.2.2.3.1.cmml">∀</mo><mi id="S5.SS2.p3.10.m10.2.2.3.2" xref="S5.SS2.p3.10.m10.2.2.3.2.cmml">t</mi></mrow><mo id="S5.SS2.p3.10.m10.2.2.2" xref="S5.SS2.p3.10.m10.2.2.2.cmml">∈</mo><mrow id="S5.SS2.p3.10.m10.2.2.1.1" xref="S5.SS2.p3.10.m10.2.2.1.2.cmml"><mo id="S5.SS2.p3.10.m10.2.2.1.1.2" stretchy="false" xref="S5.SS2.p3.10.m10.2.2.1.2.cmml">[</mo><mn id="S5.SS2.p3.10.m10.1.1" xref="S5.SS2.p3.10.m10.1.1.cmml">0</mn><mo id="S5.SS2.p3.10.m10.2.2.1.1.3" xref="S5.SS2.p3.10.m10.2.2.1.2.cmml">,</mo><msub id="S5.SS2.p3.10.m10.2.2.1.1.1" xref="S5.SS2.p3.10.m10.2.2.1.1.1.cmml"><mi id="S5.SS2.p3.10.m10.2.2.1.1.1.2" xref="S5.SS2.p3.10.m10.2.2.1.1.1.2.cmml">t</mi><mi id="S5.SS2.p3.10.m10.2.2.1.1.1.3" mathvariant="normal" xref="S5.SS2.p3.10.m10.2.2.1.1.1.3.cmml">f</mi></msub><mo id="S5.SS2.p3.10.m10.2.2.1.1.4" stretchy="false" xref="S5.SS2.p3.10.m10.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.10.m10.2b"><apply id="S5.SS2.p3.10.m10.2.2.cmml" xref="S5.SS2.p3.10.m10.2.2"><in id="S5.SS2.p3.10.m10.2.2.2.cmml" xref="S5.SS2.p3.10.m10.2.2.2"></in><apply id="S5.SS2.p3.10.m10.2.2.3.cmml" xref="S5.SS2.p3.10.m10.2.2.3"><csymbol cd="latexml" id="S5.SS2.p3.10.m10.2.2.3.1.cmml" xref="S5.SS2.p3.10.m10.2.2.3.1">for-all</csymbol><ci id="S5.SS2.p3.10.m10.2.2.3.2.cmml" xref="S5.SS2.p3.10.m10.2.2.3.2">𝑡</ci></apply><interval closure="closed-open" id="S5.SS2.p3.10.m10.2.2.1.2.cmml" xref="S5.SS2.p3.10.m10.2.2.1.1"><cn id="S5.SS2.p3.10.m10.1.1.cmml" type="integer" xref="S5.SS2.p3.10.m10.1.1">0</cn><apply id="S5.SS2.p3.10.m10.2.2.1.1.1.cmml" xref="S5.SS2.p3.10.m10.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.p3.10.m10.2.2.1.1.1.1.cmml" xref="S5.SS2.p3.10.m10.2.2.1.1.1">subscript</csymbol><ci id="S5.SS2.p3.10.m10.2.2.1.1.1.2.cmml" xref="S5.SS2.p3.10.m10.2.2.1.1.1.2">𝑡</ci><ci id="S5.SS2.p3.10.m10.2.2.1.1.1.3.cmml" xref="S5.SS2.p3.10.m10.2.2.1.1.1.3">f</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.10.m10.2c">\forall t\in[0,t_{\rm f})</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.10.m10.2d">∀ italic_t ∈ [ 0 , italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.SS2.p4"> <p class="ltx_p" id="S5.SS2.p4.1">The following theorem is established to show the stability of the closed-loop system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E7" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">7</span></a>) with (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>).</p> </div> <div class="ltx_para" id="S5.SS2.p5"> <p class="ltx_p" id="S5.SS2.p5.8"><span class="ltx_text ltx_font_italic" id="S5.SS2.p5.8.3">Theorem 2:</span> For the system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S2.E4" title="In II Problem Formulation ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">4</span></a>) under Assumptions 1–3 driven by the CLBC law (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E6" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">6</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>) with <math alttext="\bm{x}(0)\in\Omega_{{\rm c}_{0}}" class="ltx_Math" display="inline" id="S5.SS2.p5.1.m1.1"><semantics id="S5.SS2.p5.1.m1.1a"><mrow id="S5.SS2.p5.1.m1.1.2" xref="S5.SS2.p5.1.m1.1.2.cmml"><mrow id="S5.SS2.p5.1.m1.1.2.2" xref="S5.SS2.p5.1.m1.1.2.2.cmml"><mi id="S5.SS2.p5.1.m1.1.2.2.2" xref="S5.SS2.p5.1.m1.1.2.2.2.cmml">𝒙</mi><mo id="S5.SS2.p5.1.m1.1.2.2.1" xref="S5.SS2.p5.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S5.SS2.p5.1.m1.1.2.2.3.2" xref="S5.SS2.p5.1.m1.1.2.2.cmml"><mo id="S5.SS2.p5.1.m1.1.2.2.3.2.1" stretchy="false" xref="S5.SS2.p5.1.m1.1.2.2.cmml">(</mo><mn id="S5.SS2.p5.1.m1.1.1" xref="S5.SS2.p5.1.m1.1.1.cmml">0</mn><mo id="S5.SS2.p5.1.m1.1.2.2.3.2.2" stretchy="false" xref="S5.SS2.p5.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.SS2.p5.1.m1.1.2.1" xref="S5.SS2.p5.1.m1.1.2.1.cmml">∈</mo><msub id="S5.SS2.p5.1.m1.1.2.3" xref="S5.SS2.p5.1.m1.1.2.3.cmml"><mi id="S5.SS2.p5.1.m1.1.2.3.2" mathvariant="normal" xref="S5.SS2.p5.1.m1.1.2.3.2.cmml">Ω</mi><msub id="S5.SS2.p5.1.m1.1.2.3.3" xref="S5.SS2.p5.1.m1.1.2.3.3.cmml"><mi id="S5.SS2.p5.1.m1.1.2.3.3.2" mathvariant="normal" xref="S5.SS2.p5.1.m1.1.2.3.3.2.cmml">c</mi><mn id="S5.SS2.p5.1.m1.1.2.3.3.3" xref="S5.SS2.p5.1.m1.1.2.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p5.1.m1.1b"><apply id="S5.SS2.p5.1.m1.1.2.cmml" xref="S5.SS2.p5.1.m1.1.2"><in id="S5.SS2.p5.1.m1.1.2.1.cmml" xref="S5.SS2.p5.1.m1.1.2.1"></in><apply id="S5.SS2.p5.1.m1.1.2.2.cmml" xref="S5.SS2.p5.1.m1.1.2.2"><times id="S5.SS2.p5.1.m1.1.2.2.1.cmml" xref="S5.SS2.p5.1.m1.1.2.2.1"></times><ci id="S5.SS2.p5.1.m1.1.2.2.2.cmml" xref="S5.SS2.p5.1.m1.1.2.2.2">𝒙</ci><cn id="S5.SS2.p5.1.m1.1.1.cmml" type="integer" xref="S5.SS2.p5.1.m1.1.1">0</cn></apply><apply id="S5.SS2.p5.1.m1.1.2.3.cmml" xref="S5.SS2.p5.1.m1.1.2.3"><csymbol cd="ambiguous" id="S5.SS2.p5.1.m1.1.2.3.1.cmml" xref="S5.SS2.p5.1.m1.1.2.3">subscript</csymbol><ci id="S5.SS2.p5.1.m1.1.2.3.2.cmml" xref="S5.SS2.p5.1.m1.1.2.3.2">Ω</ci><apply id="S5.SS2.p5.1.m1.1.2.3.3.cmml" xref="S5.SS2.p5.1.m1.1.2.3.3"><csymbol cd="ambiguous" id="S5.SS2.p5.1.m1.1.2.3.3.1.cmml" xref="S5.SS2.p5.1.m1.1.2.3.3">subscript</csymbol><ci id="S5.SS2.p5.1.m1.1.2.3.3.2.cmml" xref="S5.SS2.p5.1.m1.1.2.3.3.2">c</ci><cn id="S5.SS2.p5.1.m1.1.2.3.3.3.cmml" type="integer" xref="S5.SS2.p5.1.m1.1.2.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p5.1.m1.1c">\bm{x}(0)\in\Omega_{{\rm c}_{0}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p5.1.m1.1d">bold_italic_x ( 0 ) ∈ roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\hat{\bm{\theta}}(0)\in\Omega_{{\rm c}_{\theta}}" class="ltx_Math" display="inline" id="S5.SS2.p5.2.m2.1"><semantics id="S5.SS2.p5.2.m2.1a"><mrow id="S5.SS2.p5.2.m2.1.2" xref="S5.SS2.p5.2.m2.1.2.cmml"><mrow id="S5.SS2.p5.2.m2.1.2.2" xref="S5.SS2.p5.2.m2.1.2.2.cmml"><mover accent="true" id="S5.SS2.p5.2.m2.1.2.2.2" xref="S5.SS2.p5.2.m2.1.2.2.2.cmml"><mi id="S5.SS2.p5.2.m2.1.2.2.2.2" xref="S5.SS2.p5.2.m2.1.2.2.2.2.cmml">𝜽</mi><mo id="S5.SS2.p5.2.m2.1.2.2.2.1" xref="S5.SS2.p5.2.m2.1.2.2.2.1.cmml">^</mo></mover><mo id="S5.SS2.p5.2.m2.1.2.2.1" xref="S5.SS2.p5.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S5.SS2.p5.2.m2.1.2.2.3.2" xref="S5.SS2.p5.2.m2.1.2.2.cmml"><mo id="S5.SS2.p5.2.m2.1.2.2.3.2.1" stretchy="false" xref="S5.SS2.p5.2.m2.1.2.2.cmml">(</mo><mn id="S5.SS2.p5.2.m2.1.1" xref="S5.SS2.p5.2.m2.1.1.cmml">0</mn><mo id="S5.SS2.p5.2.m2.1.2.2.3.2.2" stretchy="false" xref="S5.SS2.p5.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.SS2.p5.2.m2.1.2.1" xref="S5.SS2.p5.2.m2.1.2.1.cmml">∈</mo><msub id="S5.SS2.p5.2.m2.1.2.3" xref="S5.SS2.p5.2.m2.1.2.3.cmml"><mi id="S5.SS2.p5.2.m2.1.2.3.2" mathvariant="normal" xref="S5.SS2.p5.2.m2.1.2.3.2.cmml">Ω</mi><msub id="S5.SS2.p5.2.m2.1.2.3.3" xref="S5.SS2.p5.2.m2.1.2.3.3.cmml"><mi id="S5.SS2.p5.2.m2.1.2.3.3.2" mathvariant="normal" xref="S5.SS2.p5.2.m2.1.2.3.3.2.cmml">c</mi><mi id="S5.SS2.p5.2.m2.1.2.3.3.3" xref="S5.SS2.p5.2.m2.1.2.3.3.3.cmml">θ</mi></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p5.2.m2.1b"><apply id="S5.SS2.p5.2.m2.1.2.cmml" xref="S5.SS2.p5.2.m2.1.2"><in id="S5.SS2.p5.2.m2.1.2.1.cmml" xref="S5.SS2.p5.2.m2.1.2.1"></in><apply id="S5.SS2.p5.2.m2.1.2.2.cmml" xref="S5.SS2.p5.2.m2.1.2.2"><times id="S5.SS2.p5.2.m2.1.2.2.1.cmml" xref="S5.SS2.p5.2.m2.1.2.2.1"></times><apply id="S5.SS2.p5.2.m2.1.2.2.2.cmml" xref="S5.SS2.p5.2.m2.1.2.2.2"><ci id="S5.SS2.p5.2.m2.1.2.2.2.1.cmml" xref="S5.SS2.p5.2.m2.1.2.2.2.1">^</ci><ci id="S5.SS2.p5.2.m2.1.2.2.2.2.cmml" xref="S5.SS2.p5.2.m2.1.2.2.2.2">𝜽</ci></apply><cn id="S5.SS2.p5.2.m2.1.1.cmml" type="integer" xref="S5.SS2.p5.2.m2.1.1">0</cn></apply><apply id="S5.SS2.p5.2.m2.1.2.3.cmml" xref="S5.SS2.p5.2.m2.1.2.3"><csymbol cd="ambiguous" id="S5.SS2.p5.2.m2.1.2.3.1.cmml" xref="S5.SS2.p5.2.m2.1.2.3">subscript</csymbol><ci id="S5.SS2.p5.2.m2.1.2.3.2.cmml" xref="S5.SS2.p5.2.m2.1.2.3.2">Ω</ci><apply id="S5.SS2.p5.2.m2.1.2.3.3.cmml" xref="S5.SS2.p5.2.m2.1.2.3.3"><csymbol cd="ambiguous" id="S5.SS2.p5.2.m2.1.2.3.3.1.cmml" xref="S5.SS2.p5.2.m2.1.2.3.3">subscript</csymbol><ci id="S5.SS2.p5.2.m2.1.2.3.3.2.cmml" xref="S5.SS2.p5.2.m2.1.2.3.3.2">c</ci><ci id="S5.SS2.p5.2.m2.1.2.3.3.3.cmml" xref="S5.SS2.p5.2.m2.1.2.3.3.3">𝜃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p5.2.m2.1c">\hat{\bm{\theta}}(0)\in\Omega_{{\rm c}_{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p5.2.m2.1d">over^ start_ARG bold_italic_θ end_ARG ( 0 ) ∈ roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, there exist suitable control parameters <math alttext="k_{{\rm c}1}" class="ltx_Math" display="inline" id="S5.SS2.p5.3.m3.1"><semantics id="S5.SS2.p5.3.m3.1a"><msub id="S5.SS2.p5.3.m3.1.1" xref="S5.SS2.p5.3.m3.1.1.cmml"><mi id="S5.SS2.p5.3.m3.1.1.2" xref="S5.SS2.p5.3.m3.1.1.2.cmml">k</mi><mi id="S5.SS2.p5.3.m3.1.1.3" xref="S5.SS2.p5.3.m3.1.1.3.cmml">c1</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.p5.3.m3.1b"><apply id="S5.SS2.p5.3.m3.1.1.cmml" xref="S5.SS2.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S5.SS2.p5.3.m3.1.1.1.cmml" xref="S5.SS2.p5.3.m3.1.1">subscript</csymbol><ci id="S5.SS2.p5.3.m3.1.1.2.cmml" xref="S5.SS2.p5.3.m3.1.1.2">𝑘</ci><ci id="S5.SS2.p5.3.m3.1.1.3.cmml" xref="S5.SS2.p5.3.m3.1.1.3">c1</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p5.3.m3.1c">k_{{\rm c}1}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p5.3.m3.1d">italic_k start_POSTSUBSCRIPT c1 end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="k_{{\rm c}n}" class="ltx_Math" display="inline" id="S5.SS2.p5.4.m4.1"><semantics id="S5.SS2.p5.4.m4.1a"><msub id="S5.SS2.p5.4.m4.1.1" xref="S5.SS2.p5.4.m4.1.1.cmml"><mi id="S5.SS2.p5.4.m4.1.1.2" xref="S5.SS2.p5.4.m4.1.1.2.cmml">k</mi><mrow id="S5.SS2.p5.4.m4.1.1.3" xref="S5.SS2.p5.4.m4.1.1.3.cmml"><mi id="S5.SS2.p5.4.m4.1.1.3.2" mathvariant="normal" xref="S5.SS2.p5.4.m4.1.1.3.2.cmml">c</mi><mo id="S5.SS2.p5.4.m4.1.1.3.1" xref="S5.SS2.p5.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S5.SS2.p5.4.m4.1.1.3.3" xref="S5.SS2.p5.4.m4.1.1.3.3.cmml">n</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.p5.4.m4.1b"><apply id="S5.SS2.p5.4.m4.1.1.cmml" xref="S5.SS2.p5.4.m4.1.1"><csymbol cd="ambiguous" id="S5.SS2.p5.4.m4.1.1.1.cmml" xref="S5.SS2.p5.4.m4.1.1">subscript</csymbol><ci id="S5.SS2.p5.4.m4.1.1.2.cmml" xref="S5.SS2.p5.4.m4.1.1.2">𝑘</ci><apply id="S5.SS2.p5.4.m4.1.1.3.cmml" xref="S5.SS2.p5.4.m4.1.1.3"><times id="S5.SS2.p5.4.m4.1.1.3.1.cmml" xref="S5.SS2.p5.4.m4.1.1.3.1"></times><ci id="S5.SS2.p5.4.m4.1.1.3.2.cmml" xref="S5.SS2.p5.4.m4.1.1.3.2">c</ci><ci id="S5.SS2.p5.4.m4.1.1.3.3.cmml" xref="S5.SS2.p5.4.m4.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p5.4.m4.1c">k_{{\rm c}n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p5.4.m4.1d">italic_k start_POSTSUBSCRIPT roman_c italic_n end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E8" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">8</span></a>) and filtered parameters <math alttext="\alpha_{1}" class="ltx_Math" display="inline" id="S5.SS2.p5.5.m5.1"><semantics id="S5.SS2.p5.5.m5.1a"><msub id="S5.SS2.p5.5.m5.1.1" xref="S5.SS2.p5.5.m5.1.1.cmml"><mi id="S5.SS2.p5.5.m5.1.1.2" xref="S5.SS2.p5.5.m5.1.1.2.cmml">α</mi><mn id="S5.SS2.p5.5.m5.1.1.3" xref="S5.SS2.p5.5.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.p5.5.m5.1b"><apply id="S5.SS2.p5.5.m5.1.1.cmml" xref="S5.SS2.p5.5.m5.1.1"><csymbol cd="ambiguous" id="S5.SS2.p5.5.m5.1.1.1.cmml" xref="S5.SS2.p5.5.m5.1.1">subscript</csymbol><ci id="S5.SS2.p5.5.m5.1.1.2.cmml" xref="S5.SS2.p5.5.m5.1.1.2">𝛼</ci><cn id="S5.SS2.p5.5.m5.1.1.3.cmml" type="integer" xref="S5.SS2.p5.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p5.5.m5.1c">\alpha_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p5.5.m5.1d">italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="\alpha_{n-1}" class="ltx_Math" display="inline" id="S5.SS2.p5.6.m6.1"><semantics id="S5.SS2.p5.6.m6.1a"><msub id="S5.SS2.p5.6.m6.1.1" xref="S5.SS2.p5.6.m6.1.1.cmml"><mi id="S5.SS2.p5.6.m6.1.1.2" xref="S5.SS2.p5.6.m6.1.1.2.cmml">α</mi><mrow id="S5.SS2.p5.6.m6.1.1.3" xref="S5.SS2.p5.6.m6.1.1.3.cmml"><mi id="S5.SS2.p5.6.m6.1.1.3.2" xref="S5.SS2.p5.6.m6.1.1.3.2.cmml">n</mi><mo id="S5.SS2.p5.6.m6.1.1.3.1" xref="S5.SS2.p5.6.m6.1.1.3.1.cmml">−</mo><mn id="S5.SS2.p5.6.m6.1.1.3.3" xref="S5.SS2.p5.6.m6.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.p5.6.m6.1b"><apply id="S5.SS2.p5.6.m6.1.1.cmml" xref="S5.SS2.p5.6.m6.1.1"><csymbol cd="ambiguous" id="S5.SS2.p5.6.m6.1.1.1.cmml" xref="S5.SS2.p5.6.m6.1.1">subscript</csymbol><ci id="S5.SS2.p5.6.m6.1.1.2.cmml" xref="S5.SS2.p5.6.m6.1.1.2">𝛼</ci><apply id="S5.SS2.p5.6.m6.1.1.3.cmml" xref="S5.SS2.p5.6.m6.1.1.3"><minus id="S5.SS2.p5.6.m6.1.1.3.1.cmml" xref="S5.SS2.p5.6.m6.1.1.3.1"></minus><ci id="S5.SS2.p5.6.m6.1.1.3.2.cmml" xref="S5.SS2.p5.6.m6.1.1.3.2">𝑛</ci><cn id="S5.SS2.p5.6.m6.1.1.3.3.cmml" type="integer" xref="S5.SS2.p5.6.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p5.6.m6.1c">\alpha_{n-1}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p5.6.m6.1d">italic_α start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E17" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">17</span></a>), <span class="ltx_text" id="S5.SS2.p5.8.2" style="color:#000099;">such that the equilibrium point <math alttext="(\bm{e}" class="ltx_math_unparsed" display="inline" id="S5.SS2.p5.7.1.m1.1"><semantics id="S5.SS2.p5.7.1.m1.1a"><mrow id="S5.SS2.p5.7.1.m1.1b"><mo id="S5.SS2.p5.7.1.m1.1.1" mathcolor="#000099" stretchy="false">(</mo><mi id="S5.SS2.p5.7.1.m1.1.2" mathcolor="#000099">𝒆</mi></mrow><annotation encoding="application/x-tex" id="S5.SS2.p5.7.1.m1.1c">(\bm{e}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p5.7.1.m1.1d">( bold_italic_e</annotation></semantics></math>, <math alttext="\tilde{\bm{\theta}})=\bm{0}" class="ltx_math_unparsed" display="inline" id="S5.SS2.p5.8.2.m2.1"><semantics id="S5.SS2.p5.8.2.m2.1a"><mrow id="S5.SS2.p5.8.2.m2.1b"><mover accent="true" id="S5.SS2.p5.8.2.m2.1.1"><mi id="S5.SS2.p5.8.2.m2.1.1.2" mathcolor="#000099">𝜽</mi><mo id="S5.SS2.p5.8.2.m2.1.1.1" mathcolor="#000099">~</mo></mover><mo id="S5.SS2.p5.8.2.m2.1.2" mathcolor="#000099" stretchy="false">)</mo><mo id="S5.SS2.p5.8.2.m2.1.3" mathcolor="#000099">=</mo><mn id="S5.SS2.p5.8.2.m2.1.4" mathcolor="#000099">𝟎</mn></mrow><annotation encoding="application/x-tex" id="S5.SS2.p5.8.2.m2.1c">\tilde{\bm{\theta}})=\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p5.8.2.m2.1d">over~ start_ARG bold_italic_θ end_ARG ) = bold_0</annotation></semantics></math> of the closed-loop system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E7" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">7</span></a>) with (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>) has:</span></p> <ol class="ltx_enumerate" id="S5.I2"> <li class="ltx_item" id="S5.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S5.I2.i1.p1"> <p class="ltx_p" id="S5.I2.i1.p1.3"><span class="ltx_text" id="S5.I2.i1.p1.3.1" style="color:#000099;">Uniformly ultimately bounded (UUB) stability on </span><math alttext="t" class="ltx_Math" display="inline" id="S5.I2.i1.p1.1.m1.1"><semantics id="S5.I2.i1.p1.1.m1.1a"><mi id="S5.I2.i1.p1.1.m1.1.1" mathcolor="#000099" xref="S5.I2.i1.p1.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S5.I2.i1.p1.1.m1.1b"><ci id="S5.I2.i1.p1.1.m1.1.1.cmml" xref="S5.I2.i1.p1.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.1.m1.1d">italic_t</annotation></semantics></math><span class="ltx_text" id="S5.I2.i1.p1.3.2" style="color:#000099;"> </span><math alttext="\in" class="ltx_Math" display="inline" id="S5.I2.i1.p1.2.m2.1"><semantics id="S5.I2.i1.p1.2.m2.1a"><mo id="S5.I2.i1.p1.2.m2.1.1" mathcolor="#000099" xref="S5.I2.i1.p1.2.m2.1.1.cmml">∈</mo><annotation-xml encoding="MathML-Content" id="S5.I2.i1.p1.2.m2.1b"><in id="S5.I2.i1.p1.2.m2.1.1.cmml" xref="S5.I2.i1.p1.2.m2.1.1"></in></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.2.m2.1c">\in</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.2.m2.1d">∈</annotation></semantics></math><span class="ltx_text" id="S5.I2.i1.p1.3.3" style="color:#000099;"> </span><math alttext="[0,\infty)" class="ltx_Math" display="inline" id="S5.I2.i1.p1.3.m3.2"><semantics id="S5.I2.i1.p1.3.m3.2a"><mrow id="S5.I2.i1.p1.3.m3.2.3.2" xref="S5.I2.i1.p1.3.m3.2.3.1.cmml"><mo id="S5.I2.i1.p1.3.m3.2.3.2.1" mathcolor="#000099" stretchy="false" xref="S5.I2.i1.p1.3.m3.2.3.1.cmml">[</mo><mn id="S5.I2.i1.p1.3.m3.1.1" mathcolor="#000099" xref="S5.I2.i1.p1.3.m3.1.1.cmml">0</mn><mo id="S5.I2.i1.p1.3.m3.2.3.2.2" mathcolor="#000099" xref="S5.I2.i1.p1.3.m3.2.3.1.cmml">,</mo><mi id="S5.I2.i1.p1.3.m3.2.2" mathcolor="#000099" mathvariant="normal" xref="S5.I2.i1.p1.3.m3.2.2.cmml">∞</mi><mo id="S5.I2.i1.p1.3.m3.2.3.2.3" mathcolor="#000099" stretchy="false" xref="S5.I2.i1.p1.3.m3.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i1.p1.3.m3.2b"><interval closure="closed-open" id="S5.I2.i1.p1.3.m3.2.3.1.cmml" xref="S5.I2.i1.p1.3.m3.2.3.2"><cn id="S5.I2.i1.p1.3.m3.1.1.cmml" type="integer" xref="S5.I2.i1.p1.3.m3.1.1">0</cn><infinity id="S5.I2.i1.p1.3.m3.2.2.cmml" xref="S5.I2.i1.p1.3.m3.2.2"></infinity></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.3.m3.2c">[0,\infty)</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.3.m3.2d">[ 0 , ∞ )</annotation></semantics></math><span class="ltx_text" id="S5.I2.i1.p1.3.4" style="color:#000099;">;</span></p> </div> </li> <li class="ltx_item" id="S5.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S5.I2.i2.p1"> <p class="ltx_p" id="S5.I2.i2.p1.9"><span class="ltx_text" id="S5.I2.i2.p1.9.1" style="color:#000099;">Partial exponential stability on </span><math alttext="t\in[T_{\rm a},\infty)" class="ltx_Math" display="inline" id="S5.I2.i2.p1.1.m1.2"><semantics id="S5.I2.i2.p1.1.m1.2a"><mrow id="S5.I2.i2.p1.1.m1.2.2" xref="S5.I2.i2.p1.1.m1.2.2.cmml"><mi id="S5.I2.i2.p1.1.m1.2.2.3" mathcolor="#000099" xref="S5.I2.i2.p1.1.m1.2.2.3.cmml">t</mi><mo id="S5.I2.i2.p1.1.m1.2.2.2" mathcolor="#000099" xref="S5.I2.i2.p1.1.m1.2.2.2.cmml">∈</mo><mrow id="S5.I2.i2.p1.1.m1.2.2.1.1" xref="S5.I2.i2.p1.1.m1.2.2.1.2.cmml"><mo id="S5.I2.i2.p1.1.m1.2.2.1.1.2" mathcolor="#000099" stretchy="false" xref="S5.I2.i2.p1.1.m1.2.2.1.2.cmml">[</mo><msub id="S5.I2.i2.p1.1.m1.2.2.1.1.1" xref="S5.I2.i2.p1.1.m1.2.2.1.1.1.cmml"><mi id="S5.I2.i2.p1.1.m1.2.2.1.1.1.2" mathcolor="#000099" xref="S5.I2.i2.p1.1.m1.2.2.1.1.1.2.cmml">T</mi><mi id="S5.I2.i2.p1.1.m1.2.2.1.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S5.I2.i2.p1.1.m1.2.2.1.1.1.3.cmml">a</mi></msub><mo id="S5.I2.i2.p1.1.m1.2.2.1.1.3" mathcolor="#000099" xref="S5.I2.i2.p1.1.m1.2.2.1.2.cmml">,</mo><mi id="S5.I2.i2.p1.1.m1.1.1" mathcolor="#000099" mathvariant="normal" xref="S5.I2.i2.p1.1.m1.1.1.cmml">∞</mi><mo id="S5.I2.i2.p1.1.m1.2.2.1.1.4" mathcolor="#000099" stretchy="false" xref="S5.I2.i2.p1.1.m1.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.1.m1.2b"><apply id="S5.I2.i2.p1.1.m1.2.2.cmml" xref="S5.I2.i2.p1.1.m1.2.2"><in id="S5.I2.i2.p1.1.m1.2.2.2.cmml" xref="S5.I2.i2.p1.1.m1.2.2.2"></in><ci id="S5.I2.i2.p1.1.m1.2.2.3.cmml" xref="S5.I2.i2.p1.1.m1.2.2.3">𝑡</ci><interval closure="closed-open" id="S5.I2.i2.p1.1.m1.2.2.1.2.cmml" xref="S5.I2.i2.p1.1.m1.2.2.1.1"><apply id="S5.I2.i2.p1.1.m1.2.2.1.1.1.cmml" xref="S5.I2.i2.p1.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.I2.i2.p1.1.m1.2.2.1.1.1.1.cmml" xref="S5.I2.i2.p1.1.m1.2.2.1.1.1">subscript</csymbol><ci id="S5.I2.i2.p1.1.m1.2.2.1.1.1.2.cmml" xref="S5.I2.i2.p1.1.m1.2.2.1.1.1.2">𝑇</ci><ci id="S5.I2.i2.p1.1.m1.2.2.1.1.1.3.cmml" xref="S5.I2.i2.p1.1.m1.2.2.1.1.1.3">a</ci></apply><infinity id="S5.I2.i2.p1.1.m1.1.1.cmml" xref="S5.I2.i2.p1.1.m1.1.1"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.1.m1.2c">t\in[T_{\rm a},\infty)</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.1.m1.2d">italic_t ∈ [ italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT , ∞ )</annotation></semantics></math><span class="ltx_text" id="S5.I2.i2.p1.9.2" style="color:#000099;"> when partial IE in Definition 3 exists for some constants </span><math alttext="T_{\rm a}" class="ltx_Math" display="inline" id="S5.I2.i2.p1.2.m2.1"><semantics id="S5.I2.i2.p1.2.m2.1a"><msub id="S5.I2.i2.p1.2.m2.1.1" xref="S5.I2.i2.p1.2.m2.1.1.cmml"><mi id="S5.I2.i2.p1.2.m2.1.1.2" mathcolor="#000099" xref="S5.I2.i2.p1.2.m2.1.1.2.cmml">T</mi><mi id="S5.I2.i2.p1.2.m2.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S5.I2.i2.p1.2.m2.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.2.m2.1b"><apply id="S5.I2.i2.p1.2.m2.1.1.cmml" xref="S5.I2.i2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.I2.i2.p1.2.m2.1.1.1.cmml" xref="S5.I2.i2.p1.2.m2.1.1">subscript</csymbol><ci id="S5.I2.i2.p1.2.m2.1.1.2.cmml" xref="S5.I2.i2.p1.2.m2.1.1.2">𝑇</ci><ci id="S5.I2.i2.p1.2.m2.1.1.3.cmml" xref="S5.I2.i2.p1.2.m2.1.1.3">a</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.2.m2.1c">T_{\rm a}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.2.m2.1d">italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text" id="S5.I2.i2.p1.9.3" style="color:#000099;">, </span><math alttext="\sigma" class="ltx_Math" display="inline" id="S5.I2.i2.p1.3.m3.1"><semantics id="S5.I2.i2.p1.3.m3.1a"><mi id="S5.I2.i2.p1.3.m3.1.1" mathcolor="#000099" xref="S5.I2.i2.p1.3.m3.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.3.m3.1b"><ci id="S5.I2.i2.p1.3.m3.1.1.cmml" xref="S5.I2.i2.p1.3.m3.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.3.m3.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.3.m3.1d">italic_σ</annotation></semantics></math><span class="ltx_text" id="S5.I2.i2.p1.9.4" style="color:#000099;"> </span><math alttext="\in" class="ltx_Math" display="inline" id="S5.I2.i2.p1.4.m4.1"><semantics id="S5.I2.i2.p1.4.m4.1a"><mo id="S5.I2.i2.p1.4.m4.1.1" mathcolor="#000099" xref="S5.I2.i2.p1.4.m4.1.1.cmml">∈</mo><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.4.m4.1b"><in id="S5.I2.i2.p1.4.m4.1.1.cmml" xref="S5.I2.i2.p1.4.m4.1.1"></in></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.4.m4.1c">\in</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.4.m4.1d">∈</annotation></semantics></math><span class="ltx_text" id="S5.I2.i2.p1.9.5" style="color:#000099;"> </span><math alttext="\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S5.I2.i2.p1.5.m5.1"><semantics id="S5.I2.i2.p1.5.m5.1a"><msup id="S5.I2.i2.p1.5.m5.1.1" xref="S5.I2.i2.p1.5.m5.1.1.cmml"><mi id="S5.I2.i2.p1.5.m5.1.1.2" mathcolor="#000099" xref="S5.I2.i2.p1.5.m5.1.1.2.cmml">ℝ</mi><mo id="S5.I2.i2.p1.5.m5.1.1.3" mathcolor="#000099" xref="S5.I2.i2.p1.5.m5.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.5.m5.1b"><apply id="S5.I2.i2.p1.5.m5.1.1.cmml" xref="S5.I2.i2.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S5.I2.i2.p1.5.m5.1.1.1.cmml" xref="S5.I2.i2.p1.5.m5.1.1">superscript</csymbol><ci id="S5.I2.i2.p1.5.m5.1.1.2.cmml" xref="S5.I2.i2.p1.5.m5.1.1.2">ℝ</ci><plus id="S5.I2.i2.p1.5.m5.1.1.3.cmml" xref="S5.I2.i2.p1.5.m5.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.5.m5.1c">\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.5.m5.1d">blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text" id="S5.I2.i2.p1.9.6" style="color:#000099;"> and the set </span><math alttext="\mathcal{I}" class="ltx_Math" display="inline" id="S5.I2.i2.p1.6.m6.1"><semantics id="S5.I2.i2.p1.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S5.I2.i2.p1.6.m6.1.1" mathcolor="#000099" xref="S5.I2.i2.p1.6.m6.1.1.cmml">ℐ</mi><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.6.m6.1b"><ci id="S5.I2.i2.p1.6.m6.1.1.cmml" xref="S5.I2.i2.p1.6.m6.1.1">ℐ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.6.m6.1c">\mathcal{I}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.6.m6.1d">caligraphic_I</annotation></semantics></math><span class="ltx_text" id="S5.I2.i2.p1.9.7" style="color:#000099;"> no longer changes, where the tracking error </span><math alttext="\bm{e}(t)" class="ltx_Math" display="inline" id="S5.I2.i2.p1.7.m7.1"><semantics id="S5.I2.i2.p1.7.m7.1a"><mrow id="S5.I2.i2.p1.7.m7.1.2" xref="S5.I2.i2.p1.7.m7.1.2.cmml"><mi id="S5.I2.i2.p1.7.m7.1.2.2" mathcolor="#000099" xref="S5.I2.i2.p1.7.m7.1.2.2.cmml">𝒆</mi><mo id="S5.I2.i2.p1.7.m7.1.2.1" xref="S5.I2.i2.p1.7.m7.1.2.1.cmml">⁢</mo><mrow id="S5.I2.i2.p1.7.m7.1.2.3.2" xref="S5.I2.i2.p1.7.m7.1.2.cmml"><mo id="S5.I2.i2.p1.7.m7.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="S5.I2.i2.p1.7.m7.1.2.cmml">(</mo><mi id="S5.I2.i2.p1.7.m7.1.1" mathcolor="#000099" xref="S5.I2.i2.p1.7.m7.1.1.cmml">t</mi><mo id="S5.I2.i2.p1.7.m7.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="S5.I2.i2.p1.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.7.m7.1b"><apply id="S5.I2.i2.p1.7.m7.1.2.cmml" xref="S5.I2.i2.p1.7.m7.1.2"><times id="S5.I2.i2.p1.7.m7.1.2.1.cmml" xref="S5.I2.i2.p1.7.m7.1.2.1"></times><ci id="S5.I2.i2.p1.7.m7.1.2.2.cmml" xref="S5.I2.i2.p1.7.m7.1.2.2">𝒆</ci><ci id="S5.I2.i2.p1.7.m7.1.1.cmml" xref="S5.I2.i2.p1.7.m7.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.7.m7.1c">\bm{e}(t)</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.7.m7.1d">bold_italic_e ( italic_t )</annotation></semantics></math><span class="ltx_text" id="S5.I2.i2.p1.9.8" style="color:#000099;"> and the partial parameter estimation error </span><math alttext="\tilde{\bm{\theta}}_{\zeta}(t)" class="ltx_Math" display="inline" id="S5.I2.i2.p1.8.m8.1"><semantics id="S5.I2.i2.p1.8.m8.1a"><mrow id="S5.I2.i2.p1.8.m8.1.2" xref="S5.I2.i2.p1.8.m8.1.2.cmml"><msub id="S5.I2.i2.p1.8.m8.1.2.2" xref="S5.I2.i2.p1.8.m8.1.2.2.cmml"><mover accent="true" id="S5.I2.i2.p1.8.m8.1.2.2.2" xref="S5.I2.i2.p1.8.m8.1.2.2.2.cmml"><mi id="S5.I2.i2.p1.8.m8.1.2.2.2.2" mathcolor="#000099" xref="S5.I2.i2.p1.8.m8.1.2.2.2.2.cmml">𝜽</mi><mo id="S5.I2.i2.p1.8.m8.1.2.2.2.1" mathcolor="#000099" xref="S5.I2.i2.p1.8.m8.1.2.2.2.1.cmml">~</mo></mover><mi id="S5.I2.i2.p1.8.m8.1.2.2.3" mathcolor="#000099" xref="S5.I2.i2.p1.8.m8.1.2.2.3.cmml">ζ</mi></msub><mo id="S5.I2.i2.p1.8.m8.1.2.1" xref="S5.I2.i2.p1.8.m8.1.2.1.cmml">⁢</mo><mrow id="S5.I2.i2.p1.8.m8.1.2.3.2" xref="S5.I2.i2.p1.8.m8.1.2.cmml"><mo id="S5.I2.i2.p1.8.m8.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="S5.I2.i2.p1.8.m8.1.2.cmml">(</mo><mi id="S5.I2.i2.p1.8.m8.1.1" mathcolor="#000099" xref="S5.I2.i2.p1.8.m8.1.1.cmml">t</mi><mo id="S5.I2.i2.p1.8.m8.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="S5.I2.i2.p1.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.8.m8.1b"><apply id="S5.I2.i2.p1.8.m8.1.2.cmml" xref="S5.I2.i2.p1.8.m8.1.2"><times id="S5.I2.i2.p1.8.m8.1.2.1.cmml" xref="S5.I2.i2.p1.8.m8.1.2.1"></times><apply id="S5.I2.i2.p1.8.m8.1.2.2.cmml" xref="S5.I2.i2.p1.8.m8.1.2.2"><csymbol cd="ambiguous" id="S5.I2.i2.p1.8.m8.1.2.2.1.cmml" xref="S5.I2.i2.p1.8.m8.1.2.2">subscript</csymbol><apply id="S5.I2.i2.p1.8.m8.1.2.2.2.cmml" xref="S5.I2.i2.p1.8.m8.1.2.2.2"><ci id="S5.I2.i2.p1.8.m8.1.2.2.2.1.cmml" xref="S5.I2.i2.p1.8.m8.1.2.2.2.1">~</ci><ci id="S5.I2.i2.p1.8.m8.1.2.2.2.2.cmml" xref="S5.I2.i2.p1.8.m8.1.2.2.2.2">𝜽</ci></apply><ci id="S5.I2.i2.p1.8.m8.1.2.2.3.cmml" xref="S5.I2.i2.p1.8.m8.1.2.2.3">𝜁</ci></apply><ci id="S5.I2.i2.p1.8.m8.1.1.cmml" xref="S5.I2.i2.p1.8.m8.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.8.m8.1c">\tilde{\bm{\theta}}_{\zeta}(t)</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.8.m8.1d">over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math><span class="ltx_text" id="S5.I2.i2.p1.9.9" style="color:#000099;"> exponentially converge to </span><math alttext="\bm{0}" class="ltx_Math" display="inline" id="S5.I2.i2.p1.9.m9.1"><semantics id="S5.I2.i2.p1.9.m9.1a"><mn id="S5.I2.i2.p1.9.m9.1.1" mathcolor="#000099" xref="S5.I2.i2.p1.9.m9.1.1.cmml">𝟎</mn><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.9.m9.1b"><cn id="S5.I2.i2.p1.9.m9.1.1.cmml" type="integer" xref="S5.I2.i2.p1.9.m9.1.1">0</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.9.m9.1c">\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.9.m9.1d">bold_0</annotation></semantics></math><span class="ltx_text" id="S5.I2.i2.p1.9.10" style="color:#000099;">;</span></p> </div> </li> <li class="ltx_item" id="S5.I2.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">3.</span> <div class="ltx_para" id="S5.I2.i3.p1"> <p class="ltx_p" id="S5.I2.i3.p1.8"><span class="ltx_text" id="S5.I2.i3.p1.8.1" style="color:#000099;">Exponential stability on </span><math alttext="t" class="ltx_Math" display="inline" id="S5.I2.i3.p1.1.m1.1"><semantics id="S5.I2.i3.p1.1.m1.1a"><mi id="S5.I2.i3.p1.1.m1.1.1" mathcolor="#000099" xref="S5.I2.i3.p1.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S5.I2.i3.p1.1.m1.1b"><ci id="S5.I2.i3.p1.1.m1.1.1.cmml" xref="S5.I2.i3.p1.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i3.p1.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i3.p1.1.m1.1d">italic_t</annotation></semantics></math><span class="ltx_text" id="S5.I2.i3.p1.8.2" style="color:#000099;"> </span><math alttext="\in" class="ltx_Math" display="inline" id="S5.I2.i3.p1.2.m2.1"><semantics id="S5.I2.i3.p1.2.m2.1a"><mo id="S5.I2.i3.p1.2.m2.1.1" mathcolor="#000099" xref="S5.I2.i3.p1.2.m2.1.1.cmml">∈</mo><annotation-xml encoding="MathML-Content" id="S5.I2.i3.p1.2.m2.1b"><in id="S5.I2.i3.p1.2.m2.1.1.cmml" xref="S5.I2.i3.p1.2.m2.1.1"></in></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i3.p1.2.m2.1c">\in</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i3.p1.2.m2.1d">∈</annotation></semantics></math><span class="ltx_text" id="S5.I2.i3.p1.8.3" style="color:#000099;"> </span><math alttext="[T_{\rm e},\infty)" class="ltx_Math" display="inline" id="S5.I2.i3.p1.3.m3.2"><semantics id="S5.I2.i3.p1.3.m3.2a"><mrow id="S5.I2.i3.p1.3.m3.2.2.1" xref="S5.I2.i3.p1.3.m3.2.2.2.cmml"><mo id="S5.I2.i3.p1.3.m3.2.2.1.2" mathcolor="#000099" stretchy="false" xref="S5.I2.i3.p1.3.m3.2.2.2.cmml">[</mo><msub id="S5.I2.i3.p1.3.m3.2.2.1.1" xref="S5.I2.i3.p1.3.m3.2.2.1.1.cmml"><mi id="S5.I2.i3.p1.3.m3.2.2.1.1.2" mathcolor="#000099" xref="S5.I2.i3.p1.3.m3.2.2.1.1.2.cmml">T</mi><mi id="S5.I2.i3.p1.3.m3.2.2.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S5.I2.i3.p1.3.m3.2.2.1.1.3.cmml">e</mi></msub><mo id="S5.I2.i3.p1.3.m3.2.2.1.3" mathcolor="#000099" xref="S5.I2.i3.p1.3.m3.2.2.2.cmml">,</mo><mi id="S5.I2.i3.p1.3.m3.1.1" mathcolor="#000099" mathvariant="normal" xref="S5.I2.i3.p1.3.m3.1.1.cmml">∞</mi><mo id="S5.I2.i3.p1.3.m3.2.2.1.4" mathcolor="#000099" stretchy="false" xref="S5.I2.i3.p1.3.m3.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i3.p1.3.m3.2b"><interval closure="closed-open" id="S5.I2.i3.p1.3.m3.2.2.2.cmml" xref="S5.I2.i3.p1.3.m3.2.2.1"><apply id="S5.I2.i3.p1.3.m3.2.2.1.1.cmml" xref="S5.I2.i3.p1.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S5.I2.i3.p1.3.m3.2.2.1.1.1.cmml" xref="S5.I2.i3.p1.3.m3.2.2.1.1">subscript</csymbol><ci id="S5.I2.i3.p1.3.m3.2.2.1.1.2.cmml" xref="S5.I2.i3.p1.3.m3.2.2.1.1.2">𝑇</ci><ci id="S5.I2.i3.p1.3.m3.2.2.1.1.3.cmml" xref="S5.I2.i3.p1.3.m3.2.2.1.1.3">e</ci></apply><infinity id="S5.I2.i3.p1.3.m3.1.1.cmml" xref="S5.I2.i3.p1.3.m3.1.1"></infinity></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i3.p1.3.m3.2c">[T_{\rm e},\infty)</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i3.p1.3.m3.2d">[ italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT , ∞ )</annotation></semantics></math><span class="ltx_text" id="S5.I2.i3.p1.8.4" style="color:#000099;"> if IE in Definition 2 exists for some constants </span><math alttext="T_{\rm e}" class="ltx_Math" display="inline" id="S5.I2.i3.p1.4.m4.1"><semantics id="S5.I2.i3.p1.4.m4.1a"><msub id="S5.I2.i3.p1.4.m4.1.1" xref="S5.I2.i3.p1.4.m4.1.1.cmml"><mi id="S5.I2.i3.p1.4.m4.1.1.2" mathcolor="#000099" xref="S5.I2.i3.p1.4.m4.1.1.2.cmml">T</mi><mi id="S5.I2.i3.p1.4.m4.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S5.I2.i3.p1.4.m4.1.1.3.cmml">e</mi></msub><annotation-xml encoding="MathML-Content" id="S5.I2.i3.p1.4.m4.1b"><apply id="S5.I2.i3.p1.4.m4.1.1.cmml" xref="S5.I2.i3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S5.I2.i3.p1.4.m4.1.1.1.cmml" xref="S5.I2.i3.p1.4.m4.1.1">subscript</csymbol><ci id="S5.I2.i3.p1.4.m4.1.1.2.cmml" xref="S5.I2.i3.p1.4.m4.1.1.2">𝑇</ci><ci id="S5.I2.i3.p1.4.m4.1.1.3.cmml" xref="S5.I2.i3.p1.4.m4.1.1.3">e</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i3.p1.4.m4.1c">T_{\rm e}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i3.p1.4.m4.1d">italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text" id="S5.I2.i3.p1.8.5" style="color:#000099;">, </span><math alttext="\sigma\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S5.I2.i3.p1.5.m5.1"><semantics id="S5.I2.i3.p1.5.m5.1a"><mrow id="S5.I2.i3.p1.5.m5.1.1" xref="S5.I2.i3.p1.5.m5.1.1.cmml"><mi id="S5.I2.i3.p1.5.m5.1.1.2" mathcolor="#000099" xref="S5.I2.i3.p1.5.m5.1.1.2.cmml">σ</mi><mo id="S5.I2.i3.p1.5.m5.1.1.1" mathcolor="#000099" xref="S5.I2.i3.p1.5.m5.1.1.1.cmml">∈</mo><msup id="S5.I2.i3.p1.5.m5.1.1.3" xref="S5.I2.i3.p1.5.m5.1.1.3.cmml"><mi id="S5.I2.i3.p1.5.m5.1.1.3.2" mathcolor="#000099" xref="S5.I2.i3.p1.5.m5.1.1.3.2.cmml">ℝ</mi><mo id="S5.I2.i3.p1.5.m5.1.1.3.3" mathcolor="#000099" xref="S5.I2.i3.p1.5.m5.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i3.p1.5.m5.1b"><apply id="S5.I2.i3.p1.5.m5.1.1.cmml" xref="S5.I2.i3.p1.5.m5.1.1"><in id="S5.I2.i3.p1.5.m5.1.1.1.cmml" xref="S5.I2.i3.p1.5.m5.1.1.1"></in><ci id="S5.I2.i3.p1.5.m5.1.1.2.cmml" xref="S5.I2.i3.p1.5.m5.1.1.2">𝜎</ci><apply id="S5.I2.i3.p1.5.m5.1.1.3.cmml" xref="S5.I2.i3.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S5.I2.i3.p1.5.m5.1.1.3.1.cmml" xref="S5.I2.i3.p1.5.m5.1.1.3">superscript</csymbol><ci id="S5.I2.i3.p1.5.m5.1.1.3.2.cmml" xref="S5.I2.i3.p1.5.m5.1.1.3.2">ℝ</ci><plus id="S5.I2.i3.p1.5.m5.1.1.3.3.cmml" xref="S5.I2.i3.p1.5.m5.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i3.p1.5.m5.1c">\sigma\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i3.p1.5.m5.1d">italic_σ ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text" id="S5.I2.i3.p1.8.6" style="color:#000099;">, where the tracking error </span><math alttext="\bm{e}(t)" class="ltx_Math" display="inline" id="S5.I2.i3.p1.6.m6.1"><semantics id="S5.I2.i3.p1.6.m6.1a"><mrow id="S5.I2.i3.p1.6.m6.1.2" xref="S5.I2.i3.p1.6.m6.1.2.cmml"><mi id="S5.I2.i3.p1.6.m6.1.2.2" mathcolor="#000099" xref="S5.I2.i3.p1.6.m6.1.2.2.cmml">𝒆</mi><mo id="S5.I2.i3.p1.6.m6.1.2.1" xref="S5.I2.i3.p1.6.m6.1.2.1.cmml">⁢</mo><mrow id="S5.I2.i3.p1.6.m6.1.2.3.2" xref="S5.I2.i3.p1.6.m6.1.2.cmml"><mo id="S5.I2.i3.p1.6.m6.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="S5.I2.i3.p1.6.m6.1.2.cmml">(</mo><mi id="S5.I2.i3.p1.6.m6.1.1" mathcolor="#000099" xref="S5.I2.i3.p1.6.m6.1.1.cmml">t</mi><mo id="S5.I2.i3.p1.6.m6.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="S5.I2.i3.p1.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i3.p1.6.m6.1b"><apply id="S5.I2.i3.p1.6.m6.1.2.cmml" xref="S5.I2.i3.p1.6.m6.1.2"><times id="S5.I2.i3.p1.6.m6.1.2.1.cmml" xref="S5.I2.i3.p1.6.m6.1.2.1"></times><ci id="S5.I2.i3.p1.6.m6.1.2.2.cmml" xref="S5.I2.i3.p1.6.m6.1.2.2">𝒆</ci><ci id="S5.I2.i3.p1.6.m6.1.1.cmml" xref="S5.I2.i3.p1.6.m6.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i3.p1.6.m6.1c">\bm{e}(t)</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i3.p1.6.m6.1d">bold_italic_e ( italic_t )</annotation></semantics></math><span class="ltx_text" id="S5.I2.i3.p1.8.7" style="color:#000099;"> and the parameter estimation error </span><math alttext="\tilde{\bm{\theta}}(t)" class="ltx_Math" display="inline" id="S5.I2.i3.p1.7.m7.1"><semantics id="S5.I2.i3.p1.7.m7.1a"><mrow id="S5.I2.i3.p1.7.m7.1.2" xref="S5.I2.i3.p1.7.m7.1.2.cmml"><mover accent="true" id="S5.I2.i3.p1.7.m7.1.2.2" xref="S5.I2.i3.p1.7.m7.1.2.2.cmml"><mi id="S5.I2.i3.p1.7.m7.1.2.2.2" mathcolor="#000099" xref="S5.I2.i3.p1.7.m7.1.2.2.2.cmml">𝜽</mi><mo id="S5.I2.i3.p1.7.m7.1.2.2.1" mathcolor="#000099" xref="S5.I2.i3.p1.7.m7.1.2.2.1.cmml">~</mo></mover><mo id="S5.I2.i3.p1.7.m7.1.2.1" xref="S5.I2.i3.p1.7.m7.1.2.1.cmml">⁢</mo><mrow id="S5.I2.i3.p1.7.m7.1.2.3.2" xref="S5.I2.i3.p1.7.m7.1.2.cmml"><mo id="S5.I2.i3.p1.7.m7.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="S5.I2.i3.p1.7.m7.1.2.cmml">(</mo><mi id="S5.I2.i3.p1.7.m7.1.1" mathcolor="#000099" xref="S5.I2.i3.p1.7.m7.1.1.cmml">t</mi><mo id="S5.I2.i3.p1.7.m7.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="S5.I2.i3.p1.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i3.p1.7.m7.1b"><apply id="S5.I2.i3.p1.7.m7.1.2.cmml" xref="S5.I2.i3.p1.7.m7.1.2"><times id="S5.I2.i3.p1.7.m7.1.2.1.cmml" xref="S5.I2.i3.p1.7.m7.1.2.1"></times><apply id="S5.I2.i3.p1.7.m7.1.2.2.cmml" xref="S5.I2.i3.p1.7.m7.1.2.2"><ci id="S5.I2.i3.p1.7.m7.1.2.2.1.cmml" xref="S5.I2.i3.p1.7.m7.1.2.2.1">~</ci><ci id="S5.I2.i3.p1.7.m7.1.2.2.2.cmml" xref="S5.I2.i3.p1.7.m7.1.2.2.2">𝜽</ci></apply><ci id="S5.I2.i3.p1.7.m7.1.1.cmml" xref="S5.I2.i3.p1.7.m7.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i3.p1.7.m7.1c">\tilde{\bm{\theta}}(t)</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i3.p1.7.m7.1d">over~ start_ARG bold_italic_θ end_ARG ( italic_t )</annotation></semantics></math><span class="ltx_text" id="S5.I2.i3.p1.8.8" style="color:#000099;"> exponentially converge to </span><math alttext="\bm{0}" class="ltx_Math" display="inline" id="S5.I2.i3.p1.8.m8.1"><semantics id="S5.I2.i3.p1.8.m8.1a"><mn id="S5.I2.i3.p1.8.m8.1.1" mathcolor="#000099" xref="S5.I2.i3.p1.8.m8.1.1.cmml">𝟎</mn><annotation-xml encoding="MathML-Content" id="S5.I2.i3.p1.8.m8.1b"><cn id="S5.I2.i3.p1.8.m8.1.1.cmml" type="integer" xref="S5.I2.i3.p1.8.m8.1.1">0</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i3.p1.8.m8.1c">\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i3.p1.8.m8.1d">bold_0</annotation></semantics></math><span class="ltx_text" id="S5.I2.i3.p1.8.9" style="color:#000099;">.</span></p> </div> </li> </ol> </div> <div class="ltx_para" id="S5.SS2.p6"> <p class="ltx_p" id="S5.SS2.p6.8"><span class="ltx_text ltx_font_italic" id="S5.SS2.p6.8.1">Remark 8:</span> In existing modular backstepping control methods, the nonlinear damping terms <math alttext="k_{\rm{d}\it i}\|\bm{\psi}_{i}\|^{2}" class="ltx_Math" display="inline" id="S5.SS2.p6.1.m1.1"><semantics id="S5.SS2.p6.1.m1.1a"><mrow id="S5.SS2.p6.1.m1.1.1" xref="S5.SS2.p6.1.m1.1.1.cmml"><msub id="S5.SS2.p6.1.m1.1.1.3" xref="S5.SS2.p6.1.m1.1.1.3.cmml"><mi id="S5.SS2.p6.1.m1.1.1.3.2" xref="S5.SS2.p6.1.m1.1.1.3.2.cmml">k</mi><mrow id="S5.SS2.p6.1.m1.1.1.3.3" xref="S5.SS2.p6.1.m1.1.1.3.3.cmml"><mi id="S5.SS2.p6.1.m1.1.1.3.3.2" mathvariant="normal" xref="S5.SS2.p6.1.m1.1.1.3.3.2.cmml">d</mi><mo id="S5.SS2.p6.1.m1.1.1.3.3.1" xref="S5.SS2.p6.1.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S5.SS2.p6.1.m1.1.1.3.3.3" xref="S5.SS2.p6.1.m1.1.1.3.3.3.cmml">i</mi></mrow></msub><mo id="S5.SS2.p6.1.m1.1.1.2" xref="S5.SS2.p6.1.m1.1.1.2.cmml">⁢</mo><msup id="S5.SS2.p6.1.m1.1.1.1" xref="S5.SS2.p6.1.m1.1.1.1.cmml"><mrow id="S5.SS2.p6.1.m1.1.1.1.1.1" xref="S5.SS2.p6.1.m1.1.1.1.1.2.cmml"><mo id="S5.SS2.p6.1.m1.1.1.1.1.1.2" stretchy="false" xref="S5.SS2.p6.1.m1.1.1.1.1.2.1.cmml">‖</mo><msub id="S5.SS2.p6.1.m1.1.1.1.1.1.1" xref="S5.SS2.p6.1.m1.1.1.1.1.1.1.cmml"><mi id="S5.SS2.p6.1.m1.1.1.1.1.1.1.2" xref="S5.SS2.p6.1.m1.1.1.1.1.1.1.2.cmml">𝝍</mi><mi id="S5.SS2.p6.1.m1.1.1.1.1.1.1.3" xref="S5.SS2.p6.1.m1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.SS2.p6.1.m1.1.1.1.1.1.3" stretchy="false" xref="S5.SS2.p6.1.m1.1.1.1.1.2.1.cmml">‖</mo></mrow><mn id="S5.SS2.p6.1.m1.1.1.1.3" xref="S5.SS2.p6.1.m1.1.1.1.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p6.1.m1.1b"><apply id="S5.SS2.p6.1.m1.1.1.cmml" xref="S5.SS2.p6.1.m1.1.1"><times id="S5.SS2.p6.1.m1.1.1.2.cmml" xref="S5.SS2.p6.1.m1.1.1.2"></times><apply id="S5.SS2.p6.1.m1.1.1.3.cmml" xref="S5.SS2.p6.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.p6.1.m1.1.1.3.1.cmml" xref="S5.SS2.p6.1.m1.1.1.3">subscript</csymbol><ci id="S5.SS2.p6.1.m1.1.1.3.2.cmml" xref="S5.SS2.p6.1.m1.1.1.3.2">𝑘</ci><apply id="S5.SS2.p6.1.m1.1.1.3.3.cmml" xref="S5.SS2.p6.1.m1.1.1.3.3"><times id="S5.SS2.p6.1.m1.1.1.3.3.1.cmml" xref="S5.SS2.p6.1.m1.1.1.3.3.1"></times><ci id="S5.SS2.p6.1.m1.1.1.3.3.2.cmml" xref="S5.SS2.p6.1.m1.1.1.3.3.2">d</ci><ci id="S5.SS2.p6.1.m1.1.1.3.3.3.cmml" xref="S5.SS2.p6.1.m1.1.1.3.3.3">𝑖</ci></apply></apply><apply id="S5.SS2.p6.1.m1.1.1.1.cmml" xref="S5.SS2.p6.1.m1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.p6.1.m1.1.1.1.2.cmml" xref="S5.SS2.p6.1.m1.1.1.1">superscript</csymbol><apply id="S5.SS2.p6.1.m1.1.1.1.1.2.cmml" xref="S5.SS2.p6.1.m1.1.1.1.1.1"><csymbol cd="latexml" id="S5.SS2.p6.1.m1.1.1.1.1.2.1.cmml" xref="S5.SS2.p6.1.m1.1.1.1.1.1.2">norm</csymbol><apply id="S5.SS2.p6.1.m1.1.1.1.1.1.1.cmml" xref="S5.SS2.p6.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.p6.1.m1.1.1.1.1.1.1.1.cmml" xref="S5.SS2.p6.1.m1.1.1.1.1.1.1">subscript</csymbol><ci id="S5.SS2.p6.1.m1.1.1.1.1.1.1.2.cmml" xref="S5.SS2.p6.1.m1.1.1.1.1.1.1.2">𝝍</ci><ci id="S5.SS2.p6.1.m1.1.1.1.1.1.1.3.cmml" xref="S5.SS2.p6.1.m1.1.1.1.1.1.1.3">𝑖</ci></apply></apply><cn id="S5.SS2.p6.1.m1.1.1.1.3.cmml" type="integer" xref="S5.SS2.p6.1.m1.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p6.1.m1.1c">k_{\rm{d}\it i}\|\bm{\psi}_{i}\|^{2}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p6.1.m1.1d">italic_k start_POSTSUBSCRIPT roman_d italic_i end_POSTSUBSCRIPT ∥ bold_italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> (<math alttext="i=1" class="ltx_Math" display="inline" id="S5.SS2.p6.2.m2.1"><semantics id="S5.SS2.p6.2.m2.1a"><mrow id="S5.SS2.p6.2.m2.1.1" xref="S5.SS2.p6.2.m2.1.1.cmml"><mi id="S5.SS2.p6.2.m2.1.1.2" xref="S5.SS2.p6.2.m2.1.1.2.cmml">i</mi><mo id="S5.SS2.p6.2.m2.1.1.1" xref="S5.SS2.p6.2.m2.1.1.1.cmml">=</mo><mn id="S5.SS2.p6.2.m2.1.1.3" xref="S5.SS2.p6.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p6.2.m2.1b"><apply id="S5.SS2.p6.2.m2.1.1.cmml" xref="S5.SS2.p6.2.m2.1.1"><eq id="S5.SS2.p6.2.m2.1.1.1.cmml" xref="S5.SS2.p6.2.m2.1.1.1"></eq><ci id="S5.SS2.p6.2.m2.1.1.2.cmml" xref="S5.SS2.p6.2.m2.1.1.2">𝑖</ci><cn id="S5.SS2.p6.2.m2.1.1.3.cmml" type="integer" xref="S5.SS2.p6.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p6.2.m2.1c">i=1</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p6.2.m2.1d">italic_i = 1</annotation></semantics></math> to <math alttext="n" class="ltx_Math" display="inline" id="S5.SS2.p6.3.m3.1"><semantics id="S5.SS2.p6.3.m3.1a"><mi id="S5.SS2.p6.3.m3.1.1" xref="S5.SS2.p6.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p6.3.m3.1b"><ci id="S5.SS2.p6.3.m3.1.1.cmml" xref="S5.SS2.p6.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p6.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p6.3.m3.1d">italic_n</annotation></semantics></math>) can guarantee the boundedness of the system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S2.E4" title="In II Problem Formulation ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">4</span></a>) in the absence of adaptation. In the proposed CLBC in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E6" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">6</span></a>) with (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>), the no updating of some elements of the parameter estimate <math alttext="\hat{\bm{\theta}}" class="ltx_Math" display="inline" id="S5.SS2.p6.4.m4.1"><semantics id="S5.SS2.p6.4.m4.1a"><mover accent="true" id="S5.SS2.p6.4.m4.1.1" xref="S5.SS2.p6.4.m4.1.1.cmml"><mi id="S5.SS2.p6.4.m4.1.1.2" xref="S5.SS2.p6.4.m4.1.1.2.cmml">𝜽</mi><mo id="S5.SS2.p6.4.m4.1.1.1" xref="S5.SS2.p6.4.m4.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S5.SS2.p6.4.m4.1b"><apply id="S5.SS2.p6.4.m4.1.1.cmml" xref="S5.SS2.p6.4.m4.1.1"><ci id="S5.SS2.p6.4.m4.1.1.1.cmml" xref="S5.SS2.p6.4.m4.1.1.1">^</ci><ci id="S5.SS2.p6.4.m4.1.1.2.cmml" xref="S5.SS2.p6.4.m4.1.1.2">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p6.4.m4.1c">\hat{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p6.4.m4.1d">over^ start_ARG bold_italic_θ end_ARG</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>) implies that the corresponding channels are inactive. In this case, the partial IE condition in Definition 3 holds, and the partial exponential stability of the closed-loop system without resorting to the nonlinear damping terms <math alttext="k_{\rm{d}\it i}\|\bm{\psi}_{i}\|^{2}" class="ltx_Math" display="inline" id="S5.SS2.p6.5.m5.1"><semantics id="S5.SS2.p6.5.m5.1a"><mrow id="S5.SS2.p6.5.m5.1.1" xref="S5.SS2.p6.5.m5.1.1.cmml"><msub id="S5.SS2.p6.5.m5.1.1.3" xref="S5.SS2.p6.5.m5.1.1.3.cmml"><mi id="S5.SS2.p6.5.m5.1.1.3.2" xref="S5.SS2.p6.5.m5.1.1.3.2.cmml">k</mi><mrow id="S5.SS2.p6.5.m5.1.1.3.3" xref="S5.SS2.p6.5.m5.1.1.3.3.cmml"><mi id="S5.SS2.p6.5.m5.1.1.3.3.2" mathvariant="normal" xref="S5.SS2.p6.5.m5.1.1.3.3.2.cmml">d</mi><mo id="S5.SS2.p6.5.m5.1.1.3.3.1" xref="S5.SS2.p6.5.m5.1.1.3.3.1.cmml">⁢</mo><mi id="S5.SS2.p6.5.m5.1.1.3.3.3" xref="S5.SS2.p6.5.m5.1.1.3.3.3.cmml">i</mi></mrow></msub><mo id="S5.SS2.p6.5.m5.1.1.2" xref="S5.SS2.p6.5.m5.1.1.2.cmml">⁢</mo><msup id="S5.SS2.p6.5.m5.1.1.1" xref="S5.SS2.p6.5.m5.1.1.1.cmml"><mrow id="S5.SS2.p6.5.m5.1.1.1.1.1" xref="S5.SS2.p6.5.m5.1.1.1.1.2.cmml"><mo id="S5.SS2.p6.5.m5.1.1.1.1.1.2" stretchy="false" xref="S5.SS2.p6.5.m5.1.1.1.1.2.1.cmml">‖</mo><msub id="S5.SS2.p6.5.m5.1.1.1.1.1.1" xref="S5.SS2.p6.5.m5.1.1.1.1.1.1.cmml"><mi id="S5.SS2.p6.5.m5.1.1.1.1.1.1.2" xref="S5.SS2.p6.5.m5.1.1.1.1.1.1.2.cmml">𝝍</mi><mi id="S5.SS2.p6.5.m5.1.1.1.1.1.1.3" xref="S5.SS2.p6.5.m5.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.SS2.p6.5.m5.1.1.1.1.1.3" stretchy="false" xref="S5.SS2.p6.5.m5.1.1.1.1.2.1.cmml">‖</mo></mrow><mn id="S5.SS2.p6.5.m5.1.1.1.3" xref="S5.SS2.p6.5.m5.1.1.1.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p6.5.m5.1b"><apply id="S5.SS2.p6.5.m5.1.1.cmml" xref="S5.SS2.p6.5.m5.1.1"><times id="S5.SS2.p6.5.m5.1.1.2.cmml" xref="S5.SS2.p6.5.m5.1.1.2"></times><apply id="S5.SS2.p6.5.m5.1.1.3.cmml" xref="S5.SS2.p6.5.m5.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.p6.5.m5.1.1.3.1.cmml" xref="S5.SS2.p6.5.m5.1.1.3">subscript</csymbol><ci id="S5.SS2.p6.5.m5.1.1.3.2.cmml" xref="S5.SS2.p6.5.m5.1.1.3.2">𝑘</ci><apply id="S5.SS2.p6.5.m5.1.1.3.3.cmml" xref="S5.SS2.p6.5.m5.1.1.3.3"><times id="S5.SS2.p6.5.m5.1.1.3.3.1.cmml" xref="S5.SS2.p6.5.m5.1.1.3.3.1"></times><ci id="S5.SS2.p6.5.m5.1.1.3.3.2.cmml" xref="S5.SS2.p6.5.m5.1.1.3.3.2">d</ci><ci id="S5.SS2.p6.5.m5.1.1.3.3.3.cmml" xref="S5.SS2.p6.5.m5.1.1.3.3.3">𝑖</ci></apply></apply><apply id="S5.SS2.p6.5.m5.1.1.1.cmml" xref="S5.SS2.p6.5.m5.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.p6.5.m5.1.1.1.2.cmml" xref="S5.SS2.p6.5.m5.1.1.1">superscript</csymbol><apply id="S5.SS2.p6.5.m5.1.1.1.1.2.cmml" xref="S5.SS2.p6.5.m5.1.1.1.1.1"><csymbol cd="latexml" id="S5.SS2.p6.5.m5.1.1.1.1.2.1.cmml" xref="S5.SS2.p6.5.m5.1.1.1.1.1.2">norm</csymbol><apply id="S5.SS2.p6.5.m5.1.1.1.1.1.1.cmml" xref="S5.SS2.p6.5.m5.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.p6.5.m5.1.1.1.1.1.1.1.cmml" xref="S5.SS2.p6.5.m5.1.1.1.1.1.1">subscript</csymbol><ci id="S5.SS2.p6.5.m5.1.1.1.1.1.1.2.cmml" xref="S5.SS2.p6.5.m5.1.1.1.1.1.1.2">𝝍</ci><ci id="S5.SS2.p6.5.m5.1.1.1.1.1.1.3.cmml" xref="S5.SS2.p6.5.m5.1.1.1.1.1.1.3">𝑖</ci></apply></apply><cn id="S5.SS2.p6.5.m5.1.1.1.3.cmml" type="integer" xref="S5.SS2.p6.5.m5.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p6.5.m5.1c">k_{\rm{d}\it i}\|\bm{\psi}_{i}\|^{2}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p6.5.m5.1d">italic_k start_POSTSUBSCRIPT roman_d italic_i end_POSTSUBSCRIPT ∥ bold_italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> can be established under Algorithm 1. It is worth noting in Theorem 2 that if the condition “the index set <math alttext="\mathcal{I}" class="ltx_Math" display="inline" id="S5.SS2.p6.6.m6.1"><semantics id="S5.SS2.p6.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S5.SS2.p6.6.m6.1.1" xref="S5.SS2.p6.6.m6.1.1.cmml">ℐ</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p6.6.m6.1b"><ci id="S5.SS2.p6.6.m6.1.1.cmml" xref="S5.SS2.p6.6.m6.1.1">ℐ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p6.6.m6.1c">\mathcal{I}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p6.6.m6.1d">caligraphic_I</annotation></semantics></math> no longer changes on <math alttext="t\in(T_{\rm a},\infty)" class="ltx_Math" display="inline" id="S5.SS2.p6.7.m7.2"><semantics id="S5.SS2.p6.7.m7.2a"><mrow id="S5.SS2.p6.7.m7.2.2" xref="S5.SS2.p6.7.m7.2.2.cmml"><mi id="S5.SS2.p6.7.m7.2.2.3" xref="S5.SS2.p6.7.m7.2.2.3.cmml">t</mi><mo id="S5.SS2.p6.7.m7.2.2.2" xref="S5.SS2.p6.7.m7.2.2.2.cmml">∈</mo><mrow id="S5.SS2.p6.7.m7.2.2.1.1" xref="S5.SS2.p6.7.m7.2.2.1.2.cmml"><mo id="S5.SS2.p6.7.m7.2.2.1.1.2" stretchy="false" xref="S5.SS2.p6.7.m7.2.2.1.2.cmml">(</mo><msub id="S5.SS2.p6.7.m7.2.2.1.1.1" xref="S5.SS2.p6.7.m7.2.2.1.1.1.cmml"><mi id="S5.SS2.p6.7.m7.2.2.1.1.1.2" xref="S5.SS2.p6.7.m7.2.2.1.1.1.2.cmml">T</mi><mi id="S5.SS2.p6.7.m7.2.2.1.1.1.3" mathvariant="normal" xref="S5.SS2.p6.7.m7.2.2.1.1.1.3.cmml">a</mi></msub><mo id="S5.SS2.p6.7.m7.2.2.1.1.3" xref="S5.SS2.p6.7.m7.2.2.1.2.cmml">,</mo><mi id="S5.SS2.p6.7.m7.1.1" mathvariant="normal" xref="S5.SS2.p6.7.m7.1.1.cmml">∞</mi><mo id="S5.SS2.p6.7.m7.2.2.1.1.4" stretchy="false" xref="S5.SS2.p6.7.m7.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p6.7.m7.2b"><apply id="S5.SS2.p6.7.m7.2.2.cmml" xref="S5.SS2.p6.7.m7.2.2"><in id="S5.SS2.p6.7.m7.2.2.2.cmml" xref="S5.SS2.p6.7.m7.2.2.2"></in><ci id="S5.SS2.p6.7.m7.2.2.3.cmml" xref="S5.SS2.p6.7.m7.2.2.3">𝑡</ci><interval closure="open" id="S5.SS2.p6.7.m7.2.2.1.2.cmml" xref="S5.SS2.p6.7.m7.2.2.1.1"><apply id="S5.SS2.p6.7.m7.2.2.1.1.1.cmml" xref="S5.SS2.p6.7.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.p6.7.m7.2.2.1.1.1.1.cmml" xref="S5.SS2.p6.7.m7.2.2.1.1.1">subscript</csymbol><ci id="S5.SS2.p6.7.m7.2.2.1.1.1.2.cmml" xref="S5.SS2.p6.7.m7.2.2.1.1.1.2">𝑇</ci><ci id="S5.SS2.p6.7.m7.2.2.1.1.1.3.cmml" xref="S5.SS2.p6.7.m7.2.2.1.1.1.3">a</ci></apply><infinity id="S5.SS2.p6.7.m7.1.1.cmml" xref="S5.SS2.p6.7.m7.1.1"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p6.7.m7.2c">t\in(T_{\rm a},\infty)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p6.7.m7.2d">italic_t ∈ ( italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT , ∞ )</annotation></semantics></math>” is not satisfied, the partial exponential stability of the closed-loop system can still be established. Actually, the endless changes of some indexes <math alttext="k_{j}" class="ltx_Math" display="inline" id="S5.SS2.p6.8.m8.1"><semantics id="S5.SS2.p6.8.m8.1a"><msub id="S5.SS2.p6.8.m8.1.1" xref="S5.SS2.p6.8.m8.1.1.cmml"><mi id="S5.SS2.p6.8.m8.1.1.2" xref="S5.SS2.p6.8.m8.1.1.2.cmml">k</mi><mi id="S5.SS2.p6.8.m8.1.1.3" xref="S5.SS2.p6.8.m8.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.p6.8.m8.1b"><apply id="S5.SS2.p6.8.m8.1.1.cmml" xref="S5.SS2.p6.8.m8.1.1"><csymbol cd="ambiguous" id="S5.SS2.p6.8.m8.1.1.1.cmml" xref="S5.SS2.p6.8.m8.1.1">subscript</csymbol><ci id="S5.SS2.p6.8.m8.1.1.2.cmml" xref="S5.SS2.p6.8.m8.1.1.2">𝑘</ci><ci id="S5.SS2.p6.8.m8.1.1.3.cmml" xref="S5.SS2.p6.8.m8.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p6.8.m8.1c">k_{j}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p6.8.m8.1d">italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> imply the establishment of the partial PE condition.</p> </div> <div class="ltx_para" id="S5.SS2.p7"> <p class="ltx_p" id="S5.SS2.p7.2"><span class="ltx_text ltx_font_italic" id="S5.SS2.p7.2.1">Remark 9:</span> The proposed CLBC in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E6" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">6</span></a>) with (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>) has several connections and distinctions compared to the MRE-HOT in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib24" title="">24</a>]</cite>. Similarly to the MRE-HOT, the swapping and the stable filter <math alttext="H(s)" class="ltx_Math" display="inline" id="S5.SS2.p7.1.m1.1"><semantics id="S5.SS2.p7.1.m1.1a"><mrow id="S5.SS2.p7.1.m1.1.2" xref="S5.SS2.p7.1.m1.1.2.cmml"><mi id="S5.SS2.p7.1.m1.1.2.2" xref="S5.SS2.p7.1.m1.1.2.2.cmml">H</mi><mo id="S5.SS2.p7.1.m1.1.2.1" xref="S5.SS2.p7.1.m1.1.2.1.cmml">⁢</mo><mrow id="S5.SS2.p7.1.m1.1.2.3.2" xref="S5.SS2.p7.1.m1.1.2.cmml"><mo id="S5.SS2.p7.1.m1.1.2.3.2.1" stretchy="false" xref="S5.SS2.p7.1.m1.1.2.cmml">(</mo><mi id="S5.SS2.p7.1.m1.1.1" xref="S5.SS2.p7.1.m1.1.1.cmml">s</mi><mo id="S5.SS2.p7.1.m1.1.2.3.2.2" stretchy="false" xref="S5.SS2.p7.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p7.1.m1.1b"><apply id="S5.SS2.p7.1.m1.1.2.cmml" xref="S5.SS2.p7.1.m1.1.2"><times id="S5.SS2.p7.1.m1.1.2.1.cmml" xref="S5.SS2.p7.1.m1.1.2.1"></times><ci id="S5.SS2.p7.1.m1.1.2.2.cmml" xref="S5.SS2.p7.1.m1.1.2.2">𝐻</ci><ci id="S5.SS2.p7.1.m1.1.1.cmml" xref="S5.SS2.p7.1.m1.1.1">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p7.1.m1.1c">H(s)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p7.1.m1.1d">italic_H ( italic_s )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E17" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">17</span></a>) are applied to generate the high-order time derivatives <math alttext="\hat{\bm{\theta}}^{(k+1)}" class="ltx_Math" display="inline" id="S5.SS2.p7.2.m2.1"><semantics id="S5.SS2.p7.2.m2.1a"><msup id="S5.SS2.p7.2.m2.1.2" xref="S5.SS2.p7.2.m2.1.2.cmml"><mover accent="true" id="S5.SS2.p7.2.m2.1.2.2" xref="S5.SS2.p7.2.m2.1.2.2.cmml"><mi id="S5.SS2.p7.2.m2.1.2.2.2" xref="S5.SS2.p7.2.m2.1.2.2.2.cmml">𝜽</mi><mo id="S5.SS2.p7.2.m2.1.2.2.1" xref="S5.SS2.p7.2.m2.1.2.2.1.cmml">^</mo></mover><mrow id="S5.SS2.p7.2.m2.1.1.1.1" xref="S5.SS2.p7.2.m2.1.1.1.1.1.cmml"><mo id="S5.SS2.p7.2.m2.1.1.1.1.2" stretchy="false" xref="S5.SS2.p7.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S5.SS2.p7.2.m2.1.1.1.1.1" xref="S5.SS2.p7.2.m2.1.1.1.1.1.cmml"><mi id="S5.SS2.p7.2.m2.1.1.1.1.1.2" xref="S5.SS2.p7.2.m2.1.1.1.1.1.2.cmml">k</mi><mo id="S5.SS2.p7.2.m2.1.1.1.1.1.1" xref="S5.SS2.p7.2.m2.1.1.1.1.1.1.cmml">+</mo><mn id="S5.SS2.p7.2.m2.1.1.1.1.1.3" xref="S5.SS2.p7.2.m2.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S5.SS2.p7.2.m2.1.1.1.1.3" stretchy="false" xref="S5.SS2.p7.2.m2.1.1.1.1.1.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S5.SS2.p7.2.m2.1b"><apply id="S5.SS2.p7.2.m2.1.2.cmml" xref="S5.SS2.p7.2.m2.1.2"><csymbol cd="ambiguous" id="S5.SS2.p7.2.m2.1.2.1.cmml" xref="S5.SS2.p7.2.m2.1.2">superscript</csymbol><apply id="S5.SS2.p7.2.m2.1.2.2.cmml" xref="S5.SS2.p7.2.m2.1.2.2"><ci id="S5.SS2.p7.2.m2.1.2.2.1.cmml" xref="S5.SS2.p7.2.m2.1.2.2.1">^</ci><ci id="S5.SS2.p7.2.m2.1.2.2.2.cmml" xref="S5.SS2.p7.2.m2.1.2.2.2">𝜽</ci></apply><apply id="S5.SS2.p7.2.m2.1.1.1.1.1.cmml" xref="S5.SS2.p7.2.m2.1.1.1.1"><plus id="S5.SS2.p7.2.m2.1.1.1.1.1.1.cmml" xref="S5.SS2.p7.2.m2.1.1.1.1.1.1"></plus><ci id="S5.SS2.p7.2.m2.1.1.1.1.1.2.cmml" xref="S5.SS2.p7.2.m2.1.1.1.1.1.2">𝑘</ci><cn id="S5.SS2.p7.2.m2.1.1.1.1.1.3.cmml" type="integer" xref="S5.SS2.p7.2.m2.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p7.2.m2.1c">\hat{\bm{\theta}}^{(k+1)}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p7.2.m2.1d">over^ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT ( italic_k + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E25" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">25</span></a>) for counteracting the negative effect due to the transient of parameter estimation. The distinctions between the two controllers include:</p> <ol class="ltx_enumerate" id="S5.I3"> <li class="ltx_item" id="S5.I3.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S5.I3.i1.p1"> <p class="ltx_p" id="S5.I3.i1.p1.7">The proposed CLBC introduces the filtered prediction error <math alttext="\bm{\epsilon}" class="ltx_Math" display="inline" id="S5.I3.i1.p1.1.m1.1"><semantics id="S5.I3.i1.p1.1.m1.1a"><mi class="ltx_mathvariant_bold-italic" id="S5.I3.i1.p1.1.m1.1.1" mathvariant="bold-italic" xref="S5.I3.i1.p1.1.m1.1.1.cmml">ϵ</mi><annotation-xml encoding="MathML-Content" id="S5.I3.i1.p1.1.m1.1b"><ci id="S5.I3.i1.p1.1.m1.1.1.cmml" xref="S5.I3.i1.p1.1.m1.1.1">bold-italic-ϵ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i1.p1.1.m1.1c">\bm{\epsilon}</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i1.p1.1.m1.1d">bold_italic_ϵ</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E23" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">23</span></a>) to counteract the transient process caused by the modeling error <math alttext="\Phi^{T}\tilde{\bm{\theta}}" class="ltx_Math" display="inline" id="S5.I3.i1.p1.2.m2.1"><semantics id="S5.I3.i1.p1.2.m2.1a"><mrow id="S5.I3.i1.p1.2.m2.1.1" xref="S5.I3.i1.p1.2.m2.1.1.cmml"><msup id="S5.I3.i1.p1.2.m2.1.1.2" xref="S5.I3.i1.p1.2.m2.1.1.2.cmml"><mi id="S5.I3.i1.p1.2.m2.1.1.2.2" mathvariant="normal" xref="S5.I3.i1.p1.2.m2.1.1.2.2.cmml">Φ</mi><mi id="S5.I3.i1.p1.2.m2.1.1.2.3" xref="S5.I3.i1.p1.2.m2.1.1.2.3.cmml">T</mi></msup><mo id="S5.I3.i1.p1.2.m2.1.1.1" xref="S5.I3.i1.p1.2.m2.1.1.1.cmml">⁢</mo><mover accent="true" id="S5.I3.i1.p1.2.m2.1.1.3" xref="S5.I3.i1.p1.2.m2.1.1.3.cmml"><mi id="S5.I3.i1.p1.2.m2.1.1.3.2" xref="S5.I3.i1.p1.2.m2.1.1.3.2.cmml">𝜽</mi><mo id="S5.I3.i1.p1.2.m2.1.1.3.1" xref="S5.I3.i1.p1.2.m2.1.1.3.1.cmml">~</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S5.I3.i1.p1.2.m2.1b"><apply id="S5.I3.i1.p1.2.m2.1.1.cmml" xref="S5.I3.i1.p1.2.m2.1.1"><times id="S5.I3.i1.p1.2.m2.1.1.1.cmml" xref="S5.I3.i1.p1.2.m2.1.1.1"></times><apply id="S5.I3.i1.p1.2.m2.1.1.2.cmml" xref="S5.I3.i1.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.I3.i1.p1.2.m2.1.1.2.1.cmml" xref="S5.I3.i1.p1.2.m2.1.1.2">superscript</csymbol><ci id="S5.I3.i1.p1.2.m2.1.1.2.2.cmml" xref="S5.I3.i1.p1.2.m2.1.1.2.2">Φ</ci><ci id="S5.I3.i1.p1.2.m2.1.1.2.3.cmml" xref="S5.I3.i1.p1.2.m2.1.1.2.3">𝑇</ci></apply><apply id="S5.I3.i1.p1.2.m2.1.1.3.cmml" xref="S5.I3.i1.p1.2.m2.1.1.3"><ci id="S5.I3.i1.p1.2.m2.1.1.3.1.cmml" xref="S5.I3.i1.p1.2.m2.1.1.3.1">~</ci><ci id="S5.I3.i1.p1.2.m2.1.1.3.2.cmml" xref="S5.I3.i1.p1.2.m2.1.1.3.2">𝜽</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i1.p1.2.m2.1c">\Phi^{T}\tilde{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i1.p1.2.m2.1d">roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E7" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">7</span></a>) such that the stability of the closed-loop system is established without the nonlinear damping terms <math alttext="k_{\rm{d}\it i}\|\bm{\psi}_{i}\|^{2}" class="ltx_Math" display="inline" id="S5.I3.i1.p1.3.m3.1"><semantics id="S5.I3.i1.p1.3.m3.1a"><mrow id="S5.I3.i1.p1.3.m3.1.1" xref="S5.I3.i1.p1.3.m3.1.1.cmml"><msub id="S5.I3.i1.p1.3.m3.1.1.3" xref="S5.I3.i1.p1.3.m3.1.1.3.cmml"><mi id="S5.I3.i1.p1.3.m3.1.1.3.2" xref="S5.I3.i1.p1.3.m3.1.1.3.2.cmml">k</mi><mrow id="S5.I3.i1.p1.3.m3.1.1.3.3" xref="S5.I3.i1.p1.3.m3.1.1.3.3.cmml"><mi id="S5.I3.i1.p1.3.m3.1.1.3.3.2" mathvariant="normal" xref="S5.I3.i1.p1.3.m3.1.1.3.3.2.cmml">d</mi><mo id="S5.I3.i1.p1.3.m3.1.1.3.3.1" xref="S5.I3.i1.p1.3.m3.1.1.3.3.1.cmml">⁢</mo><mi id="S5.I3.i1.p1.3.m3.1.1.3.3.3" xref="S5.I3.i1.p1.3.m3.1.1.3.3.3.cmml">i</mi></mrow></msub><mo id="S5.I3.i1.p1.3.m3.1.1.2" xref="S5.I3.i1.p1.3.m3.1.1.2.cmml">⁢</mo><msup id="S5.I3.i1.p1.3.m3.1.1.1" xref="S5.I3.i1.p1.3.m3.1.1.1.cmml"><mrow id="S5.I3.i1.p1.3.m3.1.1.1.1.1" xref="S5.I3.i1.p1.3.m3.1.1.1.1.2.cmml"><mo id="S5.I3.i1.p1.3.m3.1.1.1.1.1.2" stretchy="false" xref="S5.I3.i1.p1.3.m3.1.1.1.1.2.1.cmml">‖</mo><msub id="S5.I3.i1.p1.3.m3.1.1.1.1.1.1" xref="S5.I3.i1.p1.3.m3.1.1.1.1.1.1.cmml"><mi id="S5.I3.i1.p1.3.m3.1.1.1.1.1.1.2" xref="S5.I3.i1.p1.3.m3.1.1.1.1.1.1.2.cmml">𝝍</mi><mi id="S5.I3.i1.p1.3.m3.1.1.1.1.1.1.3" xref="S5.I3.i1.p1.3.m3.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.I3.i1.p1.3.m3.1.1.1.1.1.3" stretchy="false" xref="S5.I3.i1.p1.3.m3.1.1.1.1.2.1.cmml">‖</mo></mrow><mn id="S5.I3.i1.p1.3.m3.1.1.1.3" xref="S5.I3.i1.p1.3.m3.1.1.1.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.I3.i1.p1.3.m3.1b"><apply id="S5.I3.i1.p1.3.m3.1.1.cmml" xref="S5.I3.i1.p1.3.m3.1.1"><times id="S5.I3.i1.p1.3.m3.1.1.2.cmml" xref="S5.I3.i1.p1.3.m3.1.1.2"></times><apply id="S5.I3.i1.p1.3.m3.1.1.3.cmml" xref="S5.I3.i1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S5.I3.i1.p1.3.m3.1.1.3.1.cmml" xref="S5.I3.i1.p1.3.m3.1.1.3">subscript</csymbol><ci id="S5.I3.i1.p1.3.m3.1.1.3.2.cmml" xref="S5.I3.i1.p1.3.m3.1.1.3.2">𝑘</ci><apply id="S5.I3.i1.p1.3.m3.1.1.3.3.cmml" xref="S5.I3.i1.p1.3.m3.1.1.3.3"><times id="S5.I3.i1.p1.3.m3.1.1.3.3.1.cmml" xref="S5.I3.i1.p1.3.m3.1.1.3.3.1"></times><ci id="S5.I3.i1.p1.3.m3.1.1.3.3.2.cmml" xref="S5.I3.i1.p1.3.m3.1.1.3.3.2">d</ci><ci id="S5.I3.i1.p1.3.m3.1.1.3.3.3.cmml" xref="S5.I3.i1.p1.3.m3.1.1.3.3.3">𝑖</ci></apply></apply><apply id="S5.I3.i1.p1.3.m3.1.1.1.cmml" xref="S5.I3.i1.p1.3.m3.1.1.1"><csymbol cd="ambiguous" id="S5.I3.i1.p1.3.m3.1.1.1.2.cmml" xref="S5.I3.i1.p1.3.m3.1.1.1">superscript</csymbol><apply id="S5.I3.i1.p1.3.m3.1.1.1.1.2.cmml" xref="S5.I3.i1.p1.3.m3.1.1.1.1.1"><csymbol cd="latexml" id="S5.I3.i1.p1.3.m3.1.1.1.1.2.1.cmml" xref="S5.I3.i1.p1.3.m3.1.1.1.1.1.2">norm</csymbol><apply id="S5.I3.i1.p1.3.m3.1.1.1.1.1.1.cmml" xref="S5.I3.i1.p1.3.m3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.I3.i1.p1.3.m3.1.1.1.1.1.1.1.cmml" xref="S5.I3.i1.p1.3.m3.1.1.1.1.1.1">subscript</csymbol><ci id="S5.I3.i1.p1.3.m3.1.1.1.1.1.1.2.cmml" xref="S5.I3.i1.p1.3.m3.1.1.1.1.1.1.2">𝝍</ci><ci id="S5.I3.i1.p1.3.m3.1.1.1.1.1.1.3.cmml" xref="S5.I3.i1.p1.3.m3.1.1.1.1.1.1.3">𝑖</ci></apply></apply><cn id="S5.I3.i1.p1.3.m3.1.1.1.3.cmml" type="integer" xref="S5.I3.i1.p1.3.m3.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i1.p1.3.m3.1c">k_{\rm{d}\it i}\|\bm{\psi}_{i}\|^{2}</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i1.p1.3.m3.1d">italic_k start_POSTSUBSCRIPT roman_d italic_i end_POSTSUBSCRIPT ∥ bold_italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> (<math alttext="i=1" class="ltx_Math" display="inline" id="S5.I3.i1.p1.4.m4.1"><semantics id="S5.I3.i1.p1.4.m4.1a"><mrow id="S5.I3.i1.p1.4.m4.1.1" xref="S5.I3.i1.p1.4.m4.1.1.cmml"><mi id="S5.I3.i1.p1.4.m4.1.1.2" xref="S5.I3.i1.p1.4.m4.1.1.2.cmml">i</mi><mo id="S5.I3.i1.p1.4.m4.1.1.1" xref="S5.I3.i1.p1.4.m4.1.1.1.cmml">=</mo><mn id="S5.I3.i1.p1.4.m4.1.1.3" xref="S5.I3.i1.p1.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.I3.i1.p1.4.m4.1b"><apply id="S5.I3.i1.p1.4.m4.1.1.cmml" xref="S5.I3.i1.p1.4.m4.1.1"><eq id="S5.I3.i1.p1.4.m4.1.1.1.cmml" xref="S5.I3.i1.p1.4.m4.1.1.1"></eq><ci id="S5.I3.i1.p1.4.m4.1.1.2.cmml" xref="S5.I3.i1.p1.4.m4.1.1.2">𝑖</ci><cn id="S5.I3.i1.p1.4.m4.1.1.3.cmml" type="integer" xref="S5.I3.i1.p1.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i1.p1.4.m4.1c">i=1</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i1.p1.4.m4.1d">italic_i = 1</annotation></semantics></math> to <math alttext="n" class="ltx_Math" display="inline" id="S5.I3.i1.p1.5.m5.1"><semantics id="S5.I3.i1.p1.5.m5.1a"><mi id="S5.I3.i1.p1.5.m5.1.1" xref="S5.I3.i1.p1.5.m5.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S5.I3.i1.p1.5.m5.1b"><ci id="S5.I3.i1.p1.5.m5.1.1.cmml" xref="S5.I3.i1.p1.5.m5.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i1.p1.5.m5.1c">n</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i1.p1.5.m5.1d">italic_n</annotation></semantics></math>)<span class="ltx_note ltx_role_footnote" id="footnote3"><sup class="ltx_note_mark">3</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">3</sup><span class="ltx_tag ltx_tag_note">3</span>The establishment of stability in the sense that all closed-loop signals are bounded also can be termed as guaranteeing transient performance <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib8" title="">8</a>]</cite>.</span></span></span>, whereas the MRE-HOT resorts to <math alttext="k_{\rm{d}\it i}\|\bm{\psi}_{i}\|^{2}" class="ltx_Math" display="inline" id="S5.I3.i1.p1.6.m6.1"><semantics id="S5.I3.i1.p1.6.m6.1a"><mrow id="S5.I3.i1.p1.6.m6.1.1" xref="S5.I3.i1.p1.6.m6.1.1.cmml"><msub id="S5.I3.i1.p1.6.m6.1.1.3" xref="S5.I3.i1.p1.6.m6.1.1.3.cmml"><mi id="S5.I3.i1.p1.6.m6.1.1.3.2" xref="S5.I3.i1.p1.6.m6.1.1.3.2.cmml">k</mi><mrow id="S5.I3.i1.p1.6.m6.1.1.3.3" xref="S5.I3.i1.p1.6.m6.1.1.3.3.cmml"><mi id="S5.I3.i1.p1.6.m6.1.1.3.3.2" mathvariant="normal" xref="S5.I3.i1.p1.6.m6.1.1.3.3.2.cmml">d</mi><mo id="S5.I3.i1.p1.6.m6.1.1.3.3.1" xref="S5.I3.i1.p1.6.m6.1.1.3.3.1.cmml">⁢</mo><mi id="S5.I3.i1.p1.6.m6.1.1.3.3.3" xref="S5.I3.i1.p1.6.m6.1.1.3.3.3.cmml">i</mi></mrow></msub><mo id="S5.I3.i1.p1.6.m6.1.1.2" xref="S5.I3.i1.p1.6.m6.1.1.2.cmml">⁢</mo><msup id="S5.I3.i1.p1.6.m6.1.1.1" xref="S5.I3.i1.p1.6.m6.1.1.1.cmml"><mrow id="S5.I3.i1.p1.6.m6.1.1.1.1.1" xref="S5.I3.i1.p1.6.m6.1.1.1.1.2.cmml"><mo id="S5.I3.i1.p1.6.m6.1.1.1.1.1.2" stretchy="false" xref="S5.I3.i1.p1.6.m6.1.1.1.1.2.1.cmml">‖</mo><msub id="S5.I3.i1.p1.6.m6.1.1.1.1.1.1" xref="S5.I3.i1.p1.6.m6.1.1.1.1.1.1.cmml"><mi id="S5.I3.i1.p1.6.m6.1.1.1.1.1.1.2" xref="S5.I3.i1.p1.6.m6.1.1.1.1.1.1.2.cmml">𝝍</mi><mi id="S5.I3.i1.p1.6.m6.1.1.1.1.1.1.3" xref="S5.I3.i1.p1.6.m6.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.I3.i1.p1.6.m6.1.1.1.1.1.3" stretchy="false" xref="S5.I3.i1.p1.6.m6.1.1.1.1.2.1.cmml">‖</mo></mrow><mn id="S5.I3.i1.p1.6.m6.1.1.1.3" xref="S5.I3.i1.p1.6.m6.1.1.1.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.I3.i1.p1.6.m6.1b"><apply id="S5.I3.i1.p1.6.m6.1.1.cmml" xref="S5.I3.i1.p1.6.m6.1.1"><times id="S5.I3.i1.p1.6.m6.1.1.2.cmml" xref="S5.I3.i1.p1.6.m6.1.1.2"></times><apply id="S5.I3.i1.p1.6.m6.1.1.3.cmml" xref="S5.I3.i1.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="S5.I3.i1.p1.6.m6.1.1.3.1.cmml" xref="S5.I3.i1.p1.6.m6.1.1.3">subscript</csymbol><ci id="S5.I3.i1.p1.6.m6.1.1.3.2.cmml" xref="S5.I3.i1.p1.6.m6.1.1.3.2">𝑘</ci><apply id="S5.I3.i1.p1.6.m6.1.1.3.3.cmml" xref="S5.I3.i1.p1.6.m6.1.1.3.3"><times id="S5.I3.i1.p1.6.m6.1.1.3.3.1.cmml" xref="S5.I3.i1.p1.6.m6.1.1.3.3.1"></times><ci id="S5.I3.i1.p1.6.m6.1.1.3.3.2.cmml" xref="S5.I3.i1.p1.6.m6.1.1.3.3.2">d</ci><ci id="S5.I3.i1.p1.6.m6.1.1.3.3.3.cmml" xref="S5.I3.i1.p1.6.m6.1.1.3.3.3">𝑖</ci></apply></apply><apply id="S5.I3.i1.p1.6.m6.1.1.1.cmml" xref="S5.I3.i1.p1.6.m6.1.1.1"><csymbol cd="ambiguous" id="S5.I3.i1.p1.6.m6.1.1.1.2.cmml" xref="S5.I3.i1.p1.6.m6.1.1.1">superscript</csymbol><apply id="S5.I3.i1.p1.6.m6.1.1.1.1.2.cmml" xref="S5.I3.i1.p1.6.m6.1.1.1.1.1"><csymbol cd="latexml" id="S5.I3.i1.p1.6.m6.1.1.1.1.2.1.cmml" xref="S5.I3.i1.p1.6.m6.1.1.1.1.1.2">norm</csymbol><apply id="S5.I3.i1.p1.6.m6.1.1.1.1.1.1.cmml" xref="S5.I3.i1.p1.6.m6.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.I3.i1.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="S5.I3.i1.p1.6.m6.1.1.1.1.1.1">subscript</csymbol><ci id="S5.I3.i1.p1.6.m6.1.1.1.1.1.1.2.cmml" xref="S5.I3.i1.p1.6.m6.1.1.1.1.1.1.2">𝝍</ci><ci id="S5.I3.i1.p1.6.m6.1.1.1.1.1.1.3.cmml" xref="S5.I3.i1.p1.6.m6.1.1.1.1.1.1.3">𝑖</ci></apply></apply><cn id="S5.I3.i1.p1.6.m6.1.1.1.3.cmml" type="integer" xref="S5.I3.i1.p1.6.m6.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i1.p1.6.m6.1c">k_{\rm{d}\it i}\|\bm{\psi}_{i}\|^{2}</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i1.p1.6.m6.1d">italic_k start_POSTSUBSCRIPT roman_d italic_i end_POSTSUBSCRIPT ∥ bold_italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> to counteract <math alttext="\Phi^{T}\tilde{\bm{\theta}}" class="ltx_Math" display="inline" id="S5.I3.i1.p1.7.m7.1"><semantics id="S5.I3.i1.p1.7.m7.1a"><mrow id="S5.I3.i1.p1.7.m7.1.1" xref="S5.I3.i1.p1.7.m7.1.1.cmml"><msup id="S5.I3.i1.p1.7.m7.1.1.2" xref="S5.I3.i1.p1.7.m7.1.1.2.cmml"><mi id="S5.I3.i1.p1.7.m7.1.1.2.2" mathvariant="normal" xref="S5.I3.i1.p1.7.m7.1.1.2.2.cmml">Φ</mi><mi id="S5.I3.i1.p1.7.m7.1.1.2.3" xref="S5.I3.i1.p1.7.m7.1.1.2.3.cmml">T</mi></msup><mo id="S5.I3.i1.p1.7.m7.1.1.1" xref="S5.I3.i1.p1.7.m7.1.1.1.cmml">⁢</mo><mover accent="true" id="S5.I3.i1.p1.7.m7.1.1.3" xref="S5.I3.i1.p1.7.m7.1.1.3.cmml"><mi id="S5.I3.i1.p1.7.m7.1.1.3.2" xref="S5.I3.i1.p1.7.m7.1.1.3.2.cmml">𝜽</mi><mo id="S5.I3.i1.p1.7.m7.1.1.3.1" xref="S5.I3.i1.p1.7.m7.1.1.3.1.cmml">~</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S5.I3.i1.p1.7.m7.1b"><apply id="S5.I3.i1.p1.7.m7.1.1.cmml" xref="S5.I3.i1.p1.7.m7.1.1"><times id="S5.I3.i1.p1.7.m7.1.1.1.cmml" xref="S5.I3.i1.p1.7.m7.1.1.1"></times><apply id="S5.I3.i1.p1.7.m7.1.1.2.cmml" xref="S5.I3.i1.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S5.I3.i1.p1.7.m7.1.1.2.1.cmml" xref="S5.I3.i1.p1.7.m7.1.1.2">superscript</csymbol><ci id="S5.I3.i1.p1.7.m7.1.1.2.2.cmml" xref="S5.I3.i1.p1.7.m7.1.1.2.2">Φ</ci><ci id="S5.I3.i1.p1.7.m7.1.1.2.3.cmml" xref="S5.I3.i1.p1.7.m7.1.1.2.3">𝑇</ci></apply><apply id="S5.I3.i1.p1.7.m7.1.1.3.cmml" xref="S5.I3.i1.p1.7.m7.1.1.3"><ci id="S5.I3.i1.p1.7.m7.1.1.3.1.cmml" xref="S5.I3.i1.p1.7.m7.1.1.3.1">~</ci><ci id="S5.I3.i1.p1.7.m7.1.1.3.2.cmml" xref="S5.I3.i1.p1.7.m7.1.1.3.2">𝜽</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i1.p1.7.m7.1c">\Phi^{T}\tilde{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i1.p1.7.m7.1d">roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math> so as to establish the boundedness stability;</p> </div> </li> <li class="ltx_item" id="S5.I3.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S5.I3.i2.p1"> <p class="ltx_p" id="S5.I3.i2.p1.5">The proposed CLBC integrates the extended regression equation (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E13" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">13</span></a>) to obtain the generalized regression equation (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E14" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">14</span></a>), applies <math alttext="H(s)" class="ltx_Math" display="inline" id="S5.I3.i2.p1.1.m1.1"><semantics id="S5.I3.i2.p1.1.m1.1a"><mrow id="S5.I3.i2.p1.1.m1.1.2" xref="S5.I3.i2.p1.1.m1.1.2.cmml"><mi id="S5.I3.i2.p1.1.m1.1.2.2" xref="S5.I3.i2.p1.1.m1.1.2.2.cmml">H</mi><mo id="S5.I3.i2.p1.1.m1.1.2.1" xref="S5.I3.i2.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S5.I3.i2.p1.1.m1.1.2.3.2" xref="S5.I3.i2.p1.1.m1.1.2.cmml"><mo id="S5.I3.i2.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S5.I3.i2.p1.1.m1.1.2.cmml">(</mo><mi id="S5.I3.i2.p1.1.m1.1.1" xref="S5.I3.i2.p1.1.m1.1.1.cmml">s</mi><mo id="S5.I3.i2.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S5.I3.i2.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I3.i2.p1.1.m1.1b"><apply id="S5.I3.i2.p1.1.m1.1.2.cmml" xref="S5.I3.i2.p1.1.m1.1.2"><times id="S5.I3.i2.p1.1.m1.1.2.1.cmml" xref="S5.I3.i2.p1.1.m1.1.2.1"></times><ci id="S5.I3.i2.p1.1.m1.1.2.2.cmml" xref="S5.I3.i2.p1.1.m1.1.2.2">𝐻</ci><ci id="S5.I3.i2.p1.1.m1.1.1.cmml" xref="S5.I3.i2.p1.1.m1.1.1">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i2.p1.1.m1.1c">H(s)</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i2.p1.1.m1.1d">italic_H ( italic_s )</annotation></semantics></math> to (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E14" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">14</span></a>) to construct the novel generalized prediction error <math alttext="\bm{\xi}" class="ltx_Math" display="inline" id="S5.I3.i2.p1.2.m2.1"><semantics id="S5.I3.i2.p1.2.m2.1a"><mi id="S5.I3.i2.p1.2.m2.1.1" xref="S5.I3.i2.p1.2.m2.1.1.cmml">𝝃</mi><annotation-xml encoding="MathML-Content" id="S5.I3.i2.p1.2.m2.1b"><ci id="S5.I3.i2.p1.2.m2.1.1.cmml" xref="S5.I3.i2.p1.2.m2.1.1">𝝃</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i2.p1.2.m2.1c">\bm{\xi}</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i2.p1.2.m2.1d">bold_italic_ξ</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E20" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">20</span></a>), and combines the two prediction errors <math alttext="\bm{\xi}" class="ltx_Math" display="inline" id="S5.I3.i2.p1.3.m3.1"><semantics id="S5.I3.i2.p1.3.m3.1a"><mi id="S5.I3.i2.p1.3.m3.1.1" xref="S5.I3.i2.p1.3.m3.1.1.cmml">𝝃</mi><annotation-xml encoding="MathML-Content" id="S5.I3.i2.p1.3.m3.1b"><ci id="S5.I3.i2.p1.3.m3.1.1.cmml" xref="S5.I3.i2.p1.3.m3.1.1">𝝃</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i2.p1.3.m3.1c">\bm{\xi}</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i2.p1.3.m3.1d">bold_italic_ξ</annotation></semantics></math> and <math alttext="\bm{\epsilon}" class="ltx_Math" display="inline" id="S5.I3.i2.p1.4.m4.1"><semantics id="S5.I3.i2.p1.4.m4.1a"><mi class="ltx_mathvariant_bold-italic" id="S5.I3.i2.p1.4.m4.1.1" mathvariant="bold-italic" xref="S5.I3.i2.p1.4.m4.1.1.cmml">ϵ</mi><annotation-xml encoding="MathML-Content" id="S5.I3.i2.p1.4.m4.1b"><ci id="S5.I3.i2.p1.4.m4.1.1.cmml" xref="S5.I3.i2.p1.4.m4.1.1">bold-italic-ϵ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i2.p1.4.m4.1c">\bm{\epsilon}</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i2.p1.4.m4.1d">bold_italic_ϵ</annotation></semantics></math> for the composite learning HOT (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>), whereas the MRE-HOT applies <math alttext="H(s)" class="ltx_Math" display="inline" id="S5.I3.i2.p1.5.m5.1"><semantics id="S5.I3.i2.p1.5.m5.1a"><mrow id="S5.I3.i2.p1.5.m5.1.2" xref="S5.I3.i2.p1.5.m5.1.2.cmml"><mi id="S5.I3.i2.p1.5.m5.1.2.2" xref="S5.I3.i2.p1.5.m5.1.2.2.cmml">H</mi><mo id="S5.I3.i2.p1.5.m5.1.2.1" xref="S5.I3.i2.p1.5.m5.1.2.1.cmml">⁢</mo><mrow id="S5.I3.i2.p1.5.m5.1.2.3.2" xref="S5.I3.i2.p1.5.m5.1.2.cmml"><mo id="S5.I3.i2.p1.5.m5.1.2.3.2.1" stretchy="false" xref="S5.I3.i2.p1.5.m5.1.2.cmml">(</mo><mi id="S5.I3.i2.p1.5.m5.1.1" xref="S5.I3.i2.p1.5.m5.1.1.cmml">s</mi><mo id="S5.I3.i2.p1.5.m5.1.2.3.2.2" stretchy="false" xref="S5.I3.i2.p1.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I3.i2.p1.5.m5.1b"><apply id="S5.I3.i2.p1.5.m5.1.2.cmml" xref="S5.I3.i2.p1.5.m5.1.2"><times id="S5.I3.i2.p1.5.m5.1.2.1.cmml" xref="S5.I3.i2.p1.5.m5.1.2.1"></times><ci id="S5.I3.i2.p1.5.m5.1.2.2.cmml" xref="S5.I3.i2.p1.5.m5.1.2.2">𝐻</ci><ci id="S5.I3.i2.p1.5.m5.1.1.cmml" xref="S5.I3.i2.p1.5.m5.1.1">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i2.p1.5.m5.1c">H(s)</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i2.p1.5.m5.1d">italic_H ( italic_s )</annotation></semantics></math> directly to (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E13" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">13</span></a>) to design an indirect adaptive law;</p> </div> </li> <li class="ltx_item" id="S5.I3.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">3.</span> <div class="ltx_para" id="S5.I3.i3.p1"> <p class="ltx_p" id="S5.I3.i3.p1.2">The proposed CLBC storages and forgets online data based on Algorithm 1 to ensure that the exciting strength <math alttext="\sigma_{\rm c}" class="ltx_Math" display="inline" id="S5.I3.i3.p1.1.m1.1"><semantics id="S5.I3.i3.p1.1.m1.1a"><msub id="S5.I3.i3.p1.1.m1.1.1" xref="S5.I3.i3.p1.1.m1.1.1.cmml"><mi id="S5.I3.i3.p1.1.m1.1.1.2" xref="S5.I3.i3.p1.1.m1.1.1.2.cmml">σ</mi><mi id="S5.I3.i3.p1.1.m1.1.1.3" mathvariant="normal" xref="S5.I3.i3.p1.1.m1.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S5.I3.i3.p1.1.m1.1b"><apply id="S5.I3.i3.p1.1.m1.1.1.cmml" xref="S5.I3.i3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.I3.i3.p1.1.m1.1.1.1.cmml" xref="S5.I3.i3.p1.1.m1.1.1">subscript</csymbol><ci id="S5.I3.i3.p1.1.m1.1.1.2.cmml" xref="S5.I3.i3.p1.1.m1.1.1.2">𝜎</ci><ci id="S5.I3.i3.p1.1.m1.1.1.3.cmml" xref="S5.I3.i3.p1.1.m1.1.1.3">c</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i3.p1.1.m1.1c">\sigma_{\rm c}</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i3.p1.1.m1.1d">italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT</annotation></semantics></math> is monotonously non-decreasing in each excitation stage, which enhances exponential stability and robustness under partial IE or IE, whereas the MRE-HOT is hard to ensure monotonously non-decreasing of <math alttext="\sigma_{\rm c}" class="ltx_Math" display="inline" id="S5.I3.i3.p1.2.m2.1"><semantics id="S5.I3.i3.p1.2.m2.1a"><msub id="S5.I3.i3.p1.2.m2.1.1" xref="S5.I3.i3.p1.2.m2.1.1.cmml"><mi id="S5.I3.i3.p1.2.m2.1.1.2" xref="S5.I3.i3.p1.2.m2.1.1.2.cmml">σ</mi><mi id="S5.I3.i3.p1.2.m2.1.1.3" mathvariant="normal" xref="S5.I3.i3.p1.2.m2.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S5.I3.i3.p1.2.m2.1b"><apply id="S5.I3.i3.p1.2.m2.1.1.cmml" xref="S5.I3.i3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.I3.i3.p1.2.m2.1.1.1.cmml" xref="S5.I3.i3.p1.2.m2.1.1">subscript</csymbol><ci id="S5.I3.i3.p1.2.m2.1.1.2.cmml" xref="S5.I3.i3.p1.2.m2.1.1.2">𝜎</ci><ci id="S5.I3.i3.p1.2.m2.1.1.3.cmml" xref="S5.I3.i3.p1.2.m2.1.1.3">c</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i3.p1.2.m2.1c">\sigma_{\rm c}</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i3.p1.2.m2.1d">italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT</annotation></semantics></math>, and its exponential stability depends on PE.</p> </div> </li> </ol> </div> </section> <section class="ltx_subsection" id="S5.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S5.SS3.4.1.1">V-C</span> </span><span class="ltx_text ltx_font_italic" id="S5.SS3.5.2">Robustness Results</span> </h3> <div class="ltx_para" id="S5.SS3.p1"> <p class="ltx_p" id="S5.SS3.p1.9">In this section, an external disturbance <math alttext="\bm{d}(t)" class="ltx_Math" display="inline" id="S5.SS3.p1.1.m1.1"><semantics id="S5.SS3.p1.1.m1.1a"><mrow id="S5.SS3.p1.1.m1.1.2" xref="S5.SS3.p1.1.m1.1.2.cmml"><mi id="S5.SS3.p1.1.m1.1.2.2" xref="S5.SS3.p1.1.m1.1.2.2.cmml">𝒅</mi><mo id="S5.SS3.p1.1.m1.1.2.1" xref="S5.SS3.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S5.SS3.p1.1.m1.1.2.3.2" xref="S5.SS3.p1.1.m1.1.2.cmml"><mo id="S5.SS3.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S5.SS3.p1.1.m1.1.2.cmml">(</mo><mi id="S5.SS3.p1.1.m1.1.1" xref="S5.SS3.p1.1.m1.1.1.cmml">t</mi><mo id="S5.SS3.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S5.SS3.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.1.m1.1b"><apply id="S5.SS3.p1.1.m1.1.2.cmml" xref="S5.SS3.p1.1.m1.1.2"><times id="S5.SS3.p1.1.m1.1.2.1.cmml" xref="S5.SS3.p1.1.m1.1.2.1"></times><ci id="S5.SS3.p1.1.m1.1.2.2.cmml" xref="S5.SS3.p1.1.m1.1.2.2">𝒅</ci><ci id="S5.SS3.p1.1.m1.1.1.cmml" xref="S5.SS3.p1.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.1.m1.1c">\bm{d}(t)</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.1.m1.1d">bold_italic_d ( italic_t )</annotation></semantics></math> <math alttext="\in" class="ltx_Math" display="inline" id="S5.SS3.p1.2.m2.1"><semantics id="S5.SS3.p1.2.m2.1a"><mo id="S5.SS3.p1.2.m2.1.1" xref="S5.SS3.p1.2.m2.1.1.cmml">∈</mo><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.2.m2.1b"><in id="S5.SS3.p1.2.m2.1.1.cmml" xref="S5.SS3.p1.2.m2.1.1"></in></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.2.m2.1c">\in</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.2.m2.1d">∈</annotation></semantics></math> <math alttext="\mathbb{R}^{n}" class="ltx_Math" display="inline" id="S5.SS3.p1.3.m3.1"><semantics id="S5.SS3.p1.3.m3.1a"><msup id="S5.SS3.p1.3.m3.1.1" xref="S5.SS3.p1.3.m3.1.1.cmml"><mi id="S5.SS3.p1.3.m3.1.1.2" xref="S5.SS3.p1.3.m3.1.1.2.cmml">ℝ</mi><mi id="S5.SS3.p1.3.m3.1.1.3" xref="S5.SS3.p1.3.m3.1.1.3.cmml">n</mi></msup><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.3.m3.1b"><apply id="S5.SS3.p1.3.m3.1.1.cmml" xref="S5.SS3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.SS3.p1.3.m3.1.1.1.cmml" xref="S5.SS3.p1.3.m3.1.1">superscript</csymbol><ci id="S5.SS3.p1.3.m3.1.1.2.cmml" xref="S5.SS3.p1.3.m3.1.1.2">ℝ</ci><ci id="S5.SS3.p1.3.m3.1.1.3.cmml" xref="S5.SS3.p1.3.m3.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.3.m3.1c">\mathbb{R}^{n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.3.m3.1d">blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> satisfying <math alttext="\|\bm{d}(t)\|" class="ltx_Math" display="inline" id="S5.SS3.p1.4.m4.2"><semantics id="S5.SS3.p1.4.m4.2a"><mrow id="S5.SS3.p1.4.m4.2.2.1" xref="S5.SS3.p1.4.m4.2.2.2.cmml"><mo id="S5.SS3.p1.4.m4.2.2.1.2" stretchy="false" xref="S5.SS3.p1.4.m4.2.2.2.1.cmml">‖</mo><mrow id="S5.SS3.p1.4.m4.2.2.1.1" xref="S5.SS3.p1.4.m4.2.2.1.1.cmml"><mi id="S5.SS3.p1.4.m4.2.2.1.1.2" xref="S5.SS3.p1.4.m4.2.2.1.1.2.cmml">𝒅</mi><mo id="S5.SS3.p1.4.m4.2.2.1.1.1" xref="S5.SS3.p1.4.m4.2.2.1.1.1.cmml">⁢</mo><mrow id="S5.SS3.p1.4.m4.2.2.1.1.3.2" xref="S5.SS3.p1.4.m4.2.2.1.1.cmml"><mo id="S5.SS3.p1.4.m4.2.2.1.1.3.2.1" stretchy="false" xref="S5.SS3.p1.4.m4.2.2.1.1.cmml">(</mo><mi id="S5.SS3.p1.4.m4.1.1" xref="S5.SS3.p1.4.m4.1.1.cmml">t</mi><mo id="S5.SS3.p1.4.m4.2.2.1.1.3.2.2" stretchy="false" xref="S5.SS3.p1.4.m4.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="S5.SS3.p1.4.m4.2.2.1.3" stretchy="false" xref="S5.SS3.p1.4.m4.2.2.2.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.4.m4.2b"><apply id="S5.SS3.p1.4.m4.2.2.2.cmml" xref="S5.SS3.p1.4.m4.2.2.1"><csymbol cd="latexml" id="S5.SS3.p1.4.m4.2.2.2.1.cmml" xref="S5.SS3.p1.4.m4.2.2.1.2">norm</csymbol><apply id="S5.SS3.p1.4.m4.2.2.1.1.cmml" xref="S5.SS3.p1.4.m4.2.2.1.1"><times id="S5.SS3.p1.4.m4.2.2.1.1.1.cmml" xref="S5.SS3.p1.4.m4.2.2.1.1.1"></times><ci id="S5.SS3.p1.4.m4.2.2.1.1.2.cmml" xref="S5.SS3.p1.4.m4.2.2.1.1.2">𝒅</ci><ci id="S5.SS3.p1.4.m4.1.1.cmml" xref="S5.SS3.p1.4.m4.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.4.m4.2c">\|\bm{d}(t)\|</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.4.m4.2d">∥ bold_italic_d ( italic_t ) ∥</annotation></semantics></math> <math alttext="\leq" class="ltx_Math" display="inline" id="S5.SS3.p1.5.m5.1"><semantics id="S5.SS3.p1.5.m5.1a"><mo id="S5.SS3.p1.5.m5.1.1" xref="S5.SS3.p1.5.m5.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.5.m5.1b"><leq id="S5.SS3.p1.5.m5.1.1.cmml" xref="S5.SS3.p1.5.m5.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.5.m5.1c">\leq</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.5.m5.1d">≤</annotation></semantics></math> <math alttext="\bar{d}" class="ltx_Math" display="inline" id="S5.SS3.p1.6.m6.1"><semantics id="S5.SS3.p1.6.m6.1a"><mover accent="true" id="S5.SS3.p1.6.m6.1.1" xref="S5.SS3.p1.6.m6.1.1.cmml"><mi id="S5.SS3.p1.6.m6.1.1.2" xref="S5.SS3.p1.6.m6.1.1.2.cmml">d</mi><mo id="S5.SS3.p1.6.m6.1.1.1" xref="S5.SS3.p1.6.m6.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.6.m6.1b"><apply id="S5.SS3.p1.6.m6.1.1.cmml" xref="S5.SS3.p1.6.m6.1.1"><ci id="S5.SS3.p1.6.m6.1.1.1.cmml" xref="S5.SS3.p1.6.m6.1.1.1">¯</ci><ci id="S5.SS3.p1.6.m6.1.1.2.cmml" xref="S5.SS3.p1.6.m6.1.1.2">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.6.m6.1c">\bar{d}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.6.m6.1d">over¯ start_ARG italic_d end_ARG</annotation></semantics></math> is introduced to the system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S2.E4" title="In II Problem Formulation ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">4</span></a>), where <math alttext="\bar{d}" class="ltx_Math" display="inline" id="S5.SS3.p1.7.m7.1"><semantics id="S5.SS3.p1.7.m7.1a"><mover accent="true" id="S5.SS3.p1.7.m7.1.1" xref="S5.SS3.p1.7.m7.1.1.cmml"><mi id="S5.SS3.p1.7.m7.1.1.2" xref="S5.SS3.p1.7.m7.1.1.2.cmml">d</mi><mo id="S5.SS3.p1.7.m7.1.1.1" xref="S5.SS3.p1.7.m7.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.7.m7.1b"><apply id="S5.SS3.p1.7.m7.1.1.cmml" xref="S5.SS3.p1.7.m7.1.1"><ci id="S5.SS3.p1.7.m7.1.1.1.cmml" xref="S5.SS3.p1.7.m7.1.1.1">¯</ci><ci id="S5.SS3.p1.7.m7.1.1.2.cmml" xref="S5.SS3.p1.7.m7.1.1.2">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.7.m7.1c">\bar{d}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.7.m7.1d">over¯ start_ARG italic_d end_ARG</annotation></semantics></math> <math alttext="\in" class="ltx_Math" display="inline" id="S5.SS3.p1.8.m8.1"><semantics id="S5.SS3.p1.8.m8.1a"><mo id="S5.SS3.p1.8.m8.1.1" xref="S5.SS3.p1.8.m8.1.1.cmml">∈</mo><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.8.m8.1b"><in id="S5.SS3.p1.8.m8.1.1.cmml" xref="S5.SS3.p1.8.m8.1.1"></in></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.8.m8.1c">\in</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.8.m8.1d">∈</annotation></semantics></math> <math alttext="\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S5.SS3.p1.9.m9.1"><semantics id="S5.SS3.p1.9.m9.1a"><msup id="S5.SS3.p1.9.m9.1.1" xref="S5.SS3.p1.9.m9.1.1.cmml"><mi id="S5.SS3.p1.9.m9.1.1.2" xref="S5.SS3.p1.9.m9.1.1.2.cmml">ℝ</mi><mo id="S5.SS3.p1.9.m9.1.1.3" xref="S5.SS3.p1.9.m9.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.9.m9.1b"><apply id="S5.SS3.p1.9.m9.1.1.cmml" xref="S5.SS3.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S5.SS3.p1.9.m9.1.1.1.cmml" xref="S5.SS3.p1.9.m9.1.1">superscript</csymbol><ci id="S5.SS3.p1.9.m9.1.1.2.cmml" xref="S5.SS3.p1.9.m9.1.1.2">ℝ</ci><plus id="S5.SS3.p1.9.m9.1.1.3.cmml" xref="S5.SS3.p1.9.m9.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.9.m9.1c">\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.9.m9.1d">blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> is a constant. The closed-loop tracking error system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E7" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">7</span></a>) can be rewritten into</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx21"> <tbody id="S5.E26"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{\bm{e}}=\Lambda\bm{e}+\Phi^{T}(\bm{x},\Theta_{n-1},{\bm{y}}_% {{\rm r}n})\tilde{\bm{\theta}}+\bm{d}(t)." class="ltx_Math" display="inline" id="S5.E26.m1.3"><semantics id="S5.E26.m1.3a"><mrow id="S5.E26.m1.3.3.1" xref="S5.E26.m1.3.3.1.1.cmml"><mrow id="S5.E26.m1.3.3.1.1" xref="S5.E26.m1.3.3.1.1.cmml"><mover accent="true" id="S5.E26.m1.3.3.1.1.4" xref="S5.E26.m1.3.3.1.1.4.cmml"><mi id="S5.E26.m1.3.3.1.1.4.2" xref="S5.E26.m1.3.3.1.1.4.2.cmml">𝒆</mi><mo id="S5.E26.m1.3.3.1.1.4.1" xref="S5.E26.m1.3.3.1.1.4.1.cmml">˙</mo></mover><mo id="S5.E26.m1.3.3.1.1.3" xref="S5.E26.m1.3.3.1.1.3.cmml">=</mo><mrow id="S5.E26.m1.3.3.1.1.2" xref="S5.E26.m1.3.3.1.1.2.cmml"><mrow id="S5.E26.m1.3.3.1.1.2.4" xref="S5.E26.m1.3.3.1.1.2.4.cmml"><mi id="S5.E26.m1.3.3.1.1.2.4.2" mathvariant="normal" xref="S5.E26.m1.3.3.1.1.2.4.2.cmml">Λ</mi><mo id="S5.E26.m1.3.3.1.1.2.4.1" xref="S5.E26.m1.3.3.1.1.2.4.1.cmml">⁢</mo><mi id="S5.E26.m1.3.3.1.1.2.4.3" xref="S5.E26.m1.3.3.1.1.2.4.3.cmml">𝒆</mi></mrow><mo id="S5.E26.m1.3.3.1.1.2.3" xref="S5.E26.m1.3.3.1.1.2.3.cmml">+</mo><mrow id="S5.E26.m1.3.3.1.1.2.2" xref="S5.E26.m1.3.3.1.1.2.2.cmml"><msup id="S5.E26.m1.3.3.1.1.2.2.4" xref="S5.E26.m1.3.3.1.1.2.2.4.cmml"><mi id="S5.E26.m1.3.3.1.1.2.2.4.2" mathvariant="normal" xref="S5.E26.m1.3.3.1.1.2.2.4.2.cmml">Φ</mi><mi id="S5.E26.m1.3.3.1.1.2.2.4.3" xref="S5.E26.m1.3.3.1.1.2.2.4.3.cmml">T</mi></msup><mo id="S5.E26.m1.3.3.1.1.2.2.3" xref="S5.E26.m1.3.3.1.1.2.2.3.cmml">⁢</mo><mrow id="S5.E26.m1.3.3.1.1.2.2.2.2" xref="S5.E26.m1.3.3.1.1.2.2.2.3.cmml"><mo id="S5.E26.m1.3.3.1.1.2.2.2.2.3" stretchy="false" xref="S5.E26.m1.3.3.1.1.2.2.2.3.cmml">(</mo><mi id="S5.E26.m1.1.1" xref="S5.E26.m1.1.1.cmml">𝒙</mi><mo id="S5.E26.m1.3.3.1.1.2.2.2.2.4" xref="S5.E26.m1.3.3.1.1.2.2.2.3.cmml">,</mo><msub id="S5.E26.m1.3.3.1.1.1.1.1.1.1" xref="S5.E26.m1.3.3.1.1.1.1.1.1.1.cmml"><mi id="S5.E26.m1.3.3.1.1.1.1.1.1.1.2" mathvariant="normal" xref="S5.E26.m1.3.3.1.1.1.1.1.1.1.2.cmml">Θ</mi><mrow id="S5.E26.m1.3.3.1.1.1.1.1.1.1.3" xref="S5.E26.m1.3.3.1.1.1.1.1.1.1.3.cmml"><mi id="S5.E26.m1.3.3.1.1.1.1.1.1.1.3.2" xref="S5.E26.m1.3.3.1.1.1.1.1.1.1.3.2.cmml">n</mi><mo id="S5.E26.m1.3.3.1.1.1.1.1.1.1.3.1" xref="S5.E26.m1.3.3.1.1.1.1.1.1.1.3.1.cmml">−</mo><mn id="S5.E26.m1.3.3.1.1.1.1.1.1.1.3.3" xref="S5.E26.m1.3.3.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S5.E26.m1.3.3.1.1.2.2.2.2.5" xref="S5.E26.m1.3.3.1.1.2.2.2.3.cmml">,</mo><msub id="S5.E26.m1.3.3.1.1.2.2.2.2.2" xref="S5.E26.m1.3.3.1.1.2.2.2.2.2.cmml"><mi id="S5.E26.m1.3.3.1.1.2.2.2.2.2.2" xref="S5.E26.m1.3.3.1.1.2.2.2.2.2.2.cmml">𝒚</mi><mrow id="S5.E26.m1.3.3.1.1.2.2.2.2.2.3" xref="S5.E26.m1.3.3.1.1.2.2.2.2.2.3.cmml"><mi id="S5.E26.m1.3.3.1.1.2.2.2.2.2.3.2" mathvariant="normal" xref="S5.E26.m1.3.3.1.1.2.2.2.2.2.3.2.cmml">r</mi><mo id="S5.E26.m1.3.3.1.1.2.2.2.2.2.3.1" xref="S5.E26.m1.3.3.1.1.2.2.2.2.2.3.1.cmml">⁢</mo><mi id="S5.E26.m1.3.3.1.1.2.2.2.2.2.3.3" xref="S5.E26.m1.3.3.1.1.2.2.2.2.2.3.3.cmml">n</mi></mrow></msub><mo id="S5.E26.m1.3.3.1.1.2.2.2.2.6" stretchy="false" xref="S5.E26.m1.3.3.1.1.2.2.2.3.cmml">)</mo></mrow><mo id="S5.E26.m1.3.3.1.1.2.2.3a" xref="S5.E26.m1.3.3.1.1.2.2.3.cmml">⁢</mo><mover accent="true" id="S5.E26.m1.3.3.1.1.2.2.5" xref="S5.E26.m1.3.3.1.1.2.2.5.cmml"><mi id="S5.E26.m1.3.3.1.1.2.2.5.2" xref="S5.E26.m1.3.3.1.1.2.2.5.2.cmml">𝜽</mi><mo id="S5.E26.m1.3.3.1.1.2.2.5.1" xref="S5.E26.m1.3.3.1.1.2.2.5.1.cmml">~</mo></mover></mrow><mo id="S5.E26.m1.3.3.1.1.2.3a" xref="S5.E26.m1.3.3.1.1.2.3.cmml">+</mo><mrow id="S5.E26.m1.3.3.1.1.2.5" xref="S5.E26.m1.3.3.1.1.2.5.cmml"><mi id="S5.E26.m1.3.3.1.1.2.5.2" xref="S5.E26.m1.3.3.1.1.2.5.2.cmml">𝒅</mi><mo id="S5.E26.m1.3.3.1.1.2.5.1" xref="S5.E26.m1.3.3.1.1.2.5.1.cmml">⁢</mo><mrow id="S5.E26.m1.3.3.1.1.2.5.3.2" xref="S5.E26.m1.3.3.1.1.2.5.cmml"><mo id="S5.E26.m1.3.3.1.1.2.5.3.2.1" stretchy="false" xref="S5.E26.m1.3.3.1.1.2.5.cmml">(</mo><mi id="S5.E26.m1.2.2" xref="S5.E26.m1.2.2.cmml">t</mi><mo id="S5.E26.m1.3.3.1.1.2.5.3.2.2" stretchy="false" xref="S5.E26.m1.3.3.1.1.2.5.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S5.E26.m1.3.3.1.2" lspace="0em" xref="S5.E26.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.E26.m1.3b"><apply id="S5.E26.m1.3.3.1.1.cmml" xref="S5.E26.m1.3.3.1"><eq id="S5.E26.m1.3.3.1.1.3.cmml" xref="S5.E26.m1.3.3.1.1.3"></eq><apply id="S5.E26.m1.3.3.1.1.4.cmml" xref="S5.E26.m1.3.3.1.1.4"><ci id="S5.E26.m1.3.3.1.1.4.1.cmml" xref="S5.E26.m1.3.3.1.1.4.1">˙</ci><ci id="S5.E26.m1.3.3.1.1.4.2.cmml" xref="S5.E26.m1.3.3.1.1.4.2">𝒆</ci></apply><apply id="S5.E26.m1.3.3.1.1.2.cmml" xref="S5.E26.m1.3.3.1.1.2"><plus id="S5.E26.m1.3.3.1.1.2.3.cmml" xref="S5.E26.m1.3.3.1.1.2.3"></plus><apply id="S5.E26.m1.3.3.1.1.2.4.cmml" xref="S5.E26.m1.3.3.1.1.2.4"><times id="S5.E26.m1.3.3.1.1.2.4.1.cmml" xref="S5.E26.m1.3.3.1.1.2.4.1"></times><ci id="S5.E26.m1.3.3.1.1.2.4.2.cmml" xref="S5.E26.m1.3.3.1.1.2.4.2">Λ</ci><ci id="S5.E26.m1.3.3.1.1.2.4.3.cmml" xref="S5.E26.m1.3.3.1.1.2.4.3">𝒆</ci></apply><apply id="S5.E26.m1.3.3.1.1.2.2.cmml" xref="S5.E26.m1.3.3.1.1.2.2"><times id="S5.E26.m1.3.3.1.1.2.2.3.cmml" xref="S5.E26.m1.3.3.1.1.2.2.3"></times><apply id="S5.E26.m1.3.3.1.1.2.2.4.cmml" xref="S5.E26.m1.3.3.1.1.2.2.4"><csymbol cd="ambiguous" id="S5.E26.m1.3.3.1.1.2.2.4.1.cmml" xref="S5.E26.m1.3.3.1.1.2.2.4">superscript</csymbol><ci id="S5.E26.m1.3.3.1.1.2.2.4.2.cmml" xref="S5.E26.m1.3.3.1.1.2.2.4.2">Φ</ci><ci id="S5.E26.m1.3.3.1.1.2.2.4.3.cmml" xref="S5.E26.m1.3.3.1.1.2.2.4.3">𝑇</ci></apply><vector id="S5.E26.m1.3.3.1.1.2.2.2.3.cmml" xref="S5.E26.m1.3.3.1.1.2.2.2.2"><ci id="S5.E26.m1.1.1.cmml" xref="S5.E26.m1.1.1">𝒙</ci><apply id="S5.E26.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S5.E26.m1.3.3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.E26.m1.3.3.1.1.1.1.1.1.1.1.cmml" xref="S5.E26.m1.3.3.1.1.1.1.1.1.1">subscript</csymbol><ci id="S5.E26.m1.3.3.1.1.1.1.1.1.1.2.cmml" xref="S5.E26.m1.3.3.1.1.1.1.1.1.1.2">Θ</ci><apply id="S5.E26.m1.3.3.1.1.1.1.1.1.1.3.cmml" xref="S5.E26.m1.3.3.1.1.1.1.1.1.1.3"><minus id="S5.E26.m1.3.3.1.1.1.1.1.1.1.3.1.cmml" xref="S5.E26.m1.3.3.1.1.1.1.1.1.1.3.1"></minus><ci id="S5.E26.m1.3.3.1.1.1.1.1.1.1.3.2.cmml" xref="S5.E26.m1.3.3.1.1.1.1.1.1.1.3.2">𝑛</ci><cn id="S5.E26.m1.3.3.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S5.E26.m1.3.3.1.1.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="S5.E26.m1.3.3.1.1.2.2.2.2.2.cmml" xref="S5.E26.m1.3.3.1.1.2.2.2.2.2"><csymbol cd="ambiguous" id="S5.E26.m1.3.3.1.1.2.2.2.2.2.1.cmml" xref="S5.E26.m1.3.3.1.1.2.2.2.2.2">subscript</csymbol><ci id="S5.E26.m1.3.3.1.1.2.2.2.2.2.2.cmml" xref="S5.E26.m1.3.3.1.1.2.2.2.2.2.2">𝒚</ci><apply id="S5.E26.m1.3.3.1.1.2.2.2.2.2.3.cmml" xref="S5.E26.m1.3.3.1.1.2.2.2.2.2.3"><times id="S5.E26.m1.3.3.1.1.2.2.2.2.2.3.1.cmml" xref="S5.E26.m1.3.3.1.1.2.2.2.2.2.3.1"></times><ci id="S5.E26.m1.3.3.1.1.2.2.2.2.2.3.2.cmml" xref="S5.E26.m1.3.3.1.1.2.2.2.2.2.3.2">r</ci><ci id="S5.E26.m1.3.3.1.1.2.2.2.2.2.3.3.cmml" xref="S5.E26.m1.3.3.1.1.2.2.2.2.2.3.3">𝑛</ci></apply></apply></vector><apply id="S5.E26.m1.3.3.1.1.2.2.5.cmml" xref="S5.E26.m1.3.3.1.1.2.2.5"><ci id="S5.E26.m1.3.3.1.1.2.2.5.1.cmml" xref="S5.E26.m1.3.3.1.1.2.2.5.1">~</ci><ci id="S5.E26.m1.3.3.1.1.2.2.5.2.cmml" xref="S5.E26.m1.3.3.1.1.2.2.5.2">𝜽</ci></apply></apply><apply id="S5.E26.m1.3.3.1.1.2.5.cmml" xref="S5.E26.m1.3.3.1.1.2.5"><times id="S5.E26.m1.3.3.1.1.2.5.1.cmml" xref="S5.E26.m1.3.3.1.1.2.5.1"></times><ci id="S5.E26.m1.3.3.1.1.2.5.2.cmml" xref="S5.E26.m1.3.3.1.1.2.5.2">𝒅</ci><ci id="S5.E26.m1.2.2.cmml" xref="S5.E26.m1.2.2">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E26.m1.3c">\displaystyle\dot{\bm{e}}=\Lambda\bm{e}+\Phi^{T}(\bm{x},\Theta_{n-1},{\bm{y}}_% {{\rm r}n})\tilde{\bm{\theta}}+\bm{d}(t).</annotation><annotation encoding="application/x-llamapun" id="S5.E26.m1.3d">over˙ start_ARG bold_italic_e end_ARG = roman_Λ bold_italic_e + roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_italic_x , roman_Θ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT roman_r italic_n end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG + bold_italic_d ( italic_t ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(26)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.SS3.p1.53">Noting (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E11" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">11</span></a>), consider the linear filtering operation</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx22"> <tbody id="S5.E27"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{\bm{d}}_{\rm s}(t)=\Lambda\bm{d}_{\rm s}(t)+\bm{d}(t)" class="ltx_Math" display="inline" id="S5.E27.m1.3"><semantics id="S5.E27.m1.3a"><mrow id="S5.E27.m1.3.4" xref="S5.E27.m1.3.4.cmml"><mrow id="S5.E27.m1.3.4.2" xref="S5.E27.m1.3.4.2.cmml"><msub id="S5.E27.m1.3.4.2.2" xref="S5.E27.m1.3.4.2.2.cmml"><mover accent="true" id="S5.E27.m1.3.4.2.2.2" xref="S5.E27.m1.3.4.2.2.2.cmml"><mi id="S5.E27.m1.3.4.2.2.2.2" xref="S5.E27.m1.3.4.2.2.2.2.cmml">𝒅</mi><mo id="S5.E27.m1.3.4.2.2.2.1" xref="S5.E27.m1.3.4.2.2.2.1.cmml">˙</mo></mover><mi id="S5.E27.m1.3.4.2.2.3" mathvariant="normal" xref="S5.E27.m1.3.4.2.2.3.cmml">s</mi></msub><mo id="S5.E27.m1.3.4.2.1" xref="S5.E27.m1.3.4.2.1.cmml">⁢</mo><mrow id="S5.E27.m1.3.4.2.3.2" xref="S5.E27.m1.3.4.2.cmml"><mo id="S5.E27.m1.3.4.2.3.2.1" stretchy="false" xref="S5.E27.m1.3.4.2.cmml">(</mo><mi id="S5.E27.m1.1.1" xref="S5.E27.m1.1.1.cmml">t</mi><mo id="S5.E27.m1.3.4.2.3.2.2" stretchy="false" xref="S5.E27.m1.3.4.2.cmml">)</mo></mrow></mrow><mo id="S5.E27.m1.3.4.1" xref="S5.E27.m1.3.4.1.cmml">=</mo><mrow id="S5.E27.m1.3.4.3" xref="S5.E27.m1.3.4.3.cmml"><mrow id="S5.E27.m1.3.4.3.2" xref="S5.E27.m1.3.4.3.2.cmml"><mi id="S5.E27.m1.3.4.3.2.2" mathvariant="normal" xref="S5.E27.m1.3.4.3.2.2.cmml">Λ</mi><mo id="S5.E27.m1.3.4.3.2.1" xref="S5.E27.m1.3.4.3.2.1.cmml">⁢</mo><msub id="S5.E27.m1.3.4.3.2.3" xref="S5.E27.m1.3.4.3.2.3.cmml"><mi id="S5.E27.m1.3.4.3.2.3.2" xref="S5.E27.m1.3.4.3.2.3.2.cmml">𝒅</mi><mi id="S5.E27.m1.3.4.3.2.3.3" mathvariant="normal" xref="S5.E27.m1.3.4.3.2.3.3.cmml">s</mi></msub><mo id="S5.E27.m1.3.4.3.2.1a" xref="S5.E27.m1.3.4.3.2.1.cmml">⁢</mo><mrow id="S5.E27.m1.3.4.3.2.4.2" xref="S5.E27.m1.3.4.3.2.cmml"><mo id="S5.E27.m1.3.4.3.2.4.2.1" stretchy="false" xref="S5.E27.m1.3.4.3.2.cmml">(</mo><mi id="S5.E27.m1.2.2" xref="S5.E27.m1.2.2.cmml">t</mi><mo id="S5.E27.m1.3.4.3.2.4.2.2" stretchy="false" xref="S5.E27.m1.3.4.3.2.cmml">)</mo></mrow></mrow><mo id="S5.E27.m1.3.4.3.1" xref="S5.E27.m1.3.4.3.1.cmml">+</mo><mrow id="S5.E27.m1.3.4.3.3" xref="S5.E27.m1.3.4.3.3.cmml"><mi id="S5.E27.m1.3.4.3.3.2" xref="S5.E27.m1.3.4.3.3.2.cmml">𝒅</mi><mo id="S5.E27.m1.3.4.3.3.1" xref="S5.E27.m1.3.4.3.3.1.cmml">⁢</mo><mrow id="S5.E27.m1.3.4.3.3.3.2" xref="S5.E27.m1.3.4.3.3.cmml"><mo id="S5.E27.m1.3.4.3.3.3.2.1" stretchy="false" xref="S5.E27.m1.3.4.3.3.cmml">(</mo><mi id="S5.E27.m1.3.3" xref="S5.E27.m1.3.3.cmml">t</mi><mo id="S5.E27.m1.3.4.3.3.3.2.2" stretchy="false" xref="S5.E27.m1.3.4.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.E27.m1.3b"><apply id="S5.E27.m1.3.4.cmml" xref="S5.E27.m1.3.4"><eq id="S5.E27.m1.3.4.1.cmml" xref="S5.E27.m1.3.4.1"></eq><apply id="S5.E27.m1.3.4.2.cmml" xref="S5.E27.m1.3.4.2"><times id="S5.E27.m1.3.4.2.1.cmml" xref="S5.E27.m1.3.4.2.1"></times><apply id="S5.E27.m1.3.4.2.2.cmml" xref="S5.E27.m1.3.4.2.2"><csymbol cd="ambiguous" id="S5.E27.m1.3.4.2.2.1.cmml" xref="S5.E27.m1.3.4.2.2">subscript</csymbol><apply id="S5.E27.m1.3.4.2.2.2.cmml" xref="S5.E27.m1.3.4.2.2.2"><ci id="S5.E27.m1.3.4.2.2.2.1.cmml" xref="S5.E27.m1.3.4.2.2.2.1">˙</ci><ci id="S5.E27.m1.3.4.2.2.2.2.cmml" xref="S5.E27.m1.3.4.2.2.2.2">𝒅</ci></apply><ci id="S5.E27.m1.3.4.2.2.3.cmml" xref="S5.E27.m1.3.4.2.2.3">s</ci></apply><ci id="S5.E27.m1.1.1.cmml" xref="S5.E27.m1.1.1">𝑡</ci></apply><apply id="S5.E27.m1.3.4.3.cmml" xref="S5.E27.m1.3.4.3"><plus id="S5.E27.m1.3.4.3.1.cmml" xref="S5.E27.m1.3.4.3.1"></plus><apply id="S5.E27.m1.3.4.3.2.cmml" xref="S5.E27.m1.3.4.3.2"><times id="S5.E27.m1.3.4.3.2.1.cmml" xref="S5.E27.m1.3.4.3.2.1"></times><ci id="S5.E27.m1.3.4.3.2.2.cmml" xref="S5.E27.m1.3.4.3.2.2">Λ</ci><apply id="S5.E27.m1.3.4.3.2.3.cmml" xref="S5.E27.m1.3.4.3.2.3"><csymbol cd="ambiguous" id="S5.E27.m1.3.4.3.2.3.1.cmml" xref="S5.E27.m1.3.4.3.2.3">subscript</csymbol><ci id="S5.E27.m1.3.4.3.2.3.2.cmml" xref="S5.E27.m1.3.4.3.2.3.2">𝒅</ci><ci id="S5.E27.m1.3.4.3.2.3.3.cmml" xref="S5.E27.m1.3.4.3.2.3.3">s</ci></apply><ci id="S5.E27.m1.2.2.cmml" xref="S5.E27.m1.2.2">𝑡</ci></apply><apply id="S5.E27.m1.3.4.3.3.cmml" xref="S5.E27.m1.3.4.3.3"><times id="S5.E27.m1.3.4.3.3.1.cmml" xref="S5.E27.m1.3.4.3.3.1"></times><ci id="S5.E27.m1.3.4.3.3.2.cmml" xref="S5.E27.m1.3.4.3.3.2">𝒅</ci><ci id="S5.E27.m1.3.3.cmml" xref="S5.E27.m1.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E27.m1.3c">\displaystyle\dot{\bm{d}}_{\rm s}(t)=\Lambda\bm{d}_{\rm s}(t)+\bm{d}(t)</annotation><annotation encoding="application/x-llamapun" id="S5.E27.m1.3d">over˙ start_ARG bold_italic_d end_ARG start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_t ) = roman_Λ bold_italic_d start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_t ) + bold_italic_d ( italic_t )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(27)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.SS3.p1.11">in which <math alttext="\bm{d}_{\rm s}\in\mathbb{R}^{n}" class="ltx_Math" display="inline" id="S5.SS3.p1.10.m1.1"><semantics id="S5.SS3.p1.10.m1.1a"><mrow id="S5.SS3.p1.10.m1.1.1" xref="S5.SS3.p1.10.m1.1.1.cmml"><msub id="S5.SS3.p1.10.m1.1.1.2" xref="S5.SS3.p1.10.m1.1.1.2.cmml"><mi id="S5.SS3.p1.10.m1.1.1.2.2" xref="S5.SS3.p1.10.m1.1.1.2.2.cmml">𝒅</mi><mi id="S5.SS3.p1.10.m1.1.1.2.3" mathvariant="normal" xref="S5.SS3.p1.10.m1.1.1.2.3.cmml">s</mi></msub><mo id="S5.SS3.p1.10.m1.1.1.1" xref="S5.SS3.p1.10.m1.1.1.1.cmml">∈</mo><msup id="S5.SS3.p1.10.m1.1.1.3" xref="S5.SS3.p1.10.m1.1.1.3.cmml"><mi id="S5.SS3.p1.10.m1.1.1.3.2" xref="S5.SS3.p1.10.m1.1.1.3.2.cmml">ℝ</mi><mi id="S5.SS3.p1.10.m1.1.1.3.3" xref="S5.SS3.p1.10.m1.1.1.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.10.m1.1b"><apply id="S5.SS3.p1.10.m1.1.1.cmml" xref="S5.SS3.p1.10.m1.1.1"><in id="S5.SS3.p1.10.m1.1.1.1.cmml" xref="S5.SS3.p1.10.m1.1.1.1"></in><apply id="S5.SS3.p1.10.m1.1.1.2.cmml" xref="S5.SS3.p1.10.m1.1.1.2"><csymbol cd="ambiguous" id="S5.SS3.p1.10.m1.1.1.2.1.cmml" xref="S5.SS3.p1.10.m1.1.1.2">subscript</csymbol><ci id="S5.SS3.p1.10.m1.1.1.2.2.cmml" xref="S5.SS3.p1.10.m1.1.1.2.2">𝒅</ci><ci id="S5.SS3.p1.10.m1.1.1.2.3.cmml" xref="S5.SS3.p1.10.m1.1.1.2.3">s</ci></apply><apply id="S5.SS3.p1.10.m1.1.1.3.cmml" xref="S5.SS3.p1.10.m1.1.1.3"><csymbol cd="ambiguous" id="S5.SS3.p1.10.m1.1.1.3.1.cmml" xref="S5.SS3.p1.10.m1.1.1.3">superscript</csymbol><ci id="S5.SS3.p1.10.m1.1.1.3.2.cmml" xref="S5.SS3.p1.10.m1.1.1.3.2">ℝ</ci><ci id="S5.SS3.p1.10.m1.1.1.3.3.cmml" xref="S5.SS3.p1.10.m1.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.10.m1.1c">\bm{d}_{\rm s}\in\mathbb{R}^{n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.10.m1.1d">bold_italic_d start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> is a filtered counterpart of <math alttext="\bm{d}" class="ltx_Math" display="inline" id="S5.SS3.p1.11.m2.1"><semantics id="S5.SS3.p1.11.m2.1a"><mi id="S5.SS3.p1.11.m2.1.1" xref="S5.SS3.p1.11.m2.1.1.cmml">𝒅</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.11.m2.1b"><ci id="S5.SS3.p1.11.m2.1.1.cmml" xref="S5.SS3.p1.11.m2.1.1">𝒅</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.11.m2.1c">\bm{d}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.11.m2.1d">bold_italic_d</annotation></semantics></math>. Then, (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E12" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">12</span></a>) becomes</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx23"> <tbody id="S5.E28"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\bm{p}(t)=\Phi^{T}_{\rm s}(t)\bm{\theta}+\bm{d}_{\rm s}(t)." class="ltx_Math" display="inline" id="S5.E28.m1.4"><semantics id="S5.E28.m1.4a"><mrow id="S5.E28.m1.4.4.1" xref="S5.E28.m1.4.4.1.1.cmml"><mrow id="S5.E28.m1.4.4.1.1" xref="S5.E28.m1.4.4.1.1.cmml"><mrow id="S5.E28.m1.4.4.1.1.2" xref="S5.E28.m1.4.4.1.1.2.cmml"><mi id="S5.E28.m1.4.4.1.1.2.2" xref="S5.E28.m1.4.4.1.1.2.2.cmml">𝒑</mi><mo id="S5.E28.m1.4.4.1.1.2.1" xref="S5.E28.m1.4.4.1.1.2.1.cmml">⁢</mo><mrow id="S5.E28.m1.4.4.1.1.2.3.2" xref="S5.E28.m1.4.4.1.1.2.cmml"><mo id="S5.E28.m1.4.4.1.1.2.3.2.1" stretchy="false" xref="S5.E28.m1.4.4.1.1.2.cmml">(</mo><mi id="S5.E28.m1.1.1" xref="S5.E28.m1.1.1.cmml">t</mi><mo id="S5.E28.m1.4.4.1.1.2.3.2.2" stretchy="false" xref="S5.E28.m1.4.4.1.1.2.cmml">)</mo></mrow></mrow><mo id="S5.E28.m1.4.4.1.1.1" xref="S5.E28.m1.4.4.1.1.1.cmml">=</mo><mrow id="S5.E28.m1.4.4.1.1.3" xref="S5.E28.m1.4.4.1.1.3.cmml"><mrow id="S5.E28.m1.4.4.1.1.3.2" xref="S5.E28.m1.4.4.1.1.3.2.cmml"><msubsup id="S5.E28.m1.4.4.1.1.3.2.2" xref="S5.E28.m1.4.4.1.1.3.2.2.cmml"><mi id="S5.E28.m1.4.4.1.1.3.2.2.2.2" mathvariant="normal" xref="S5.E28.m1.4.4.1.1.3.2.2.2.2.cmml">Φ</mi><mi id="S5.E28.m1.4.4.1.1.3.2.2.3" mathvariant="normal" xref="S5.E28.m1.4.4.1.1.3.2.2.3.cmml">s</mi><mi id="S5.E28.m1.4.4.1.1.3.2.2.2.3" xref="S5.E28.m1.4.4.1.1.3.2.2.2.3.cmml">T</mi></msubsup><mo id="S5.E28.m1.4.4.1.1.3.2.1" xref="S5.E28.m1.4.4.1.1.3.2.1.cmml">⁢</mo><mrow id="S5.E28.m1.4.4.1.1.3.2.3.2" xref="S5.E28.m1.4.4.1.1.3.2.cmml"><mo id="S5.E28.m1.4.4.1.1.3.2.3.2.1" stretchy="false" xref="S5.E28.m1.4.4.1.1.3.2.cmml">(</mo><mi id="S5.E28.m1.2.2" xref="S5.E28.m1.2.2.cmml">t</mi><mo id="S5.E28.m1.4.4.1.1.3.2.3.2.2" stretchy="false" xref="S5.E28.m1.4.4.1.1.3.2.cmml">)</mo></mrow><mo id="S5.E28.m1.4.4.1.1.3.2.1a" xref="S5.E28.m1.4.4.1.1.3.2.1.cmml">⁢</mo><mi id="S5.E28.m1.4.4.1.1.3.2.4" xref="S5.E28.m1.4.4.1.1.3.2.4.cmml">𝜽</mi></mrow><mo id="S5.E28.m1.4.4.1.1.3.1" xref="S5.E28.m1.4.4.1.1.3.1.cmml">+</mo><mrow id="S5.E28.m1.4.4.1.1.3.3" xref="S5.E28.m1.4.4.1.1.3.3.cmml"><msub id="S5.E28.m1.4.4.1.1.3.3.2" xref="S5.E28.m1.4.4.1.1.3.3.2.cmml"><mi id="S5.E28.m1.4.4.1.1.3.3.2.2" xref="S5.E28.m1.4.4.1.1.3.3.2.2.cmml">𝒅</mi><mi id="S5.E28.m1.4.4.1.1.3.3.2.3" mathvariant="normal" xref="S5.E28.m1.4.4.1.1.3.3.2.3.cmml">s</mi></msub><mo id="S5.E28.m1.4.4.1.1.3.3.1" xref="S5.E28.m1.4.4.1.1.3.3.1.cmml">⁢</mo><mrow id="S5.E28.m1.4.4.1.1.3.3.3.2" xref="S5.E28.m1.4.4.1.1.3.3.cmml"><mo id="S5.E28.m1.4.4.1.1.3.3.3.2.1" stretchy="false" xref="S5.E28.m1.4.4.1.1.3.3.cmml">(</mo><mi id="S5.E28.m1.3.3" xref="S5.E28.m1.3.3.cmml">t</mi><mo id="S5.E28.m1.4.4.1.1.3.3.3.2.2" stretchy="false" xref="S5.E28.m1.4.4.1.1.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S5.E28.m1.4.4.1.2" lspace="0em" xref="S5.E28.m1.4.4.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.E28.m1.4b"><apply id="S5.E28.m1.4.4.1.1.cmml" xref="S5.E28.m1.4.4.1"><eq id="S5.E28.m1.4.4.1.1.1.cmml" xref="S5.E28.m1.4.4.1.1.1"></eq><apply id="S5.E28.m1.4.4.1.1.2.cmml" xref="S5.E28.m1.4.4.1.1.2"><times id="S5.E28.m1.4.4.1.1.2.1.cmml" xref="S5.E28.m1.4.4.1.1.2.1"></times><ci id="S5.E28.m1.4.4.1.1.2.2.cmml" xref="S5.E28.m1.4.4.1.1.2.2">𝒑</ci><ci id="S5.E28.m1.1.1.cmml" xref="S5.E28.m1.1.1">𝑡</ci></apply><apply id="S5.E28.m1.4.4.1.1.3.cmml" xref="S5.E28.m1.4.4.1.1.3"><plus id="S5.E28.m1.4.4.1.1.3.1.cmml" xref="S5.E28.m1.4.4.1.1.3.1"></plus><apply id="S5.E28.m1.4.4.1.1.3.2.cmml" xref="S5.E28.m1.4.4.1.1.3.2"><times id="S5.E28.m1.4.4.1.1.3.2.1.cmml" xref="S5.E28.m1.4.4.1.1.3.2.1"></times><apply id="S5.E28.m1.4.4.1.1.3.2.2.cmml" xref="S5.E28.m1.4.4.1.1.3.2.2"><csymbol cd="ambiguous" id="S5.E28.m1.4.4.1.1.3.2.2.1.cmml" xref="S5.E28.m1.4.4.1.1.3.2.2">subscript</csymbol><apply id="S5.E28.m1.4.4.1.1.3.2.2.2.cmml" xref="S5.E28.m1.4.4.1.1.3.2.2"><csymbol cd="ambiguous" id="S5.E28.m1.4.4.1.1.3.2.2.2.1.cmml" xref="S5.E28.m1.4.4.1.1.3.2.2">superscript</csymbol><ci id="S5.E28.m1.4.4.1.1.3.2.2.2.2.cmml" xref="S5.E28.m1.4.4.1.1.3.2.2.2.2">Φ</ci><ci id="S5.E28.m1.4.4.1.1.3.2.2.2.3.cmml" xref="S5.E28.m1.4.4.1.1.3.2.2.2.3">𝑇</ci></apply><ci id="S5.E28.m1.4.4.1.1.3.2.2.3.cmml" xref="S5.E28.m1.4.4.1.1.3.2.2.3">s</ci></apply><ci id="S5.E28.m1.2.2.cmml" xref="S5.E28.m1.2.2">𝑡</ci><ci id="S5.E28.m1.4.4.1.1.3.2.4.cmml" xref="S5.E28.m1.4.4.1.1.3.2.4">𝜽</ci></apply><apply id="S5.E28.m1.4.4.1.1.3.3.cmml" xref="S5.E28.m1.4.4.1.1.3.3"><times id="S5.E28.m1.4.4.1.1.3.3.1.cmml" xref="S5.E28.m1.4.4.1.1.3.3.1"></times><apply id="S5.E28.m1.4.4.1.1.3.3.2.cmml" xref="S5.E28.m1.4.4.1.1.3.3.2"><csymbol cd="ambiguous" id="S5.E28.m1.4.4.1.1.3.3.2.1.cmml" xref="S5.E28.m1.4.4.1.1.3.3.2">subscript</csymbol><ci id="S5.E28.m1.4.4.1.1.3.3.2.2.cmml" xref="S5.E28.m1.4.4.1.1.3.3.2.2">𝒅</ci><ci id="S5.E28.m1.4.4.1.1.3.3.2.3.cmml" xref="S5.E28.m1.4.4.1.1.3.3.2.3">s</ci></apply><ci id="S5.E28.m1.3.3.cmml" xref="S5.E28.m1.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E28.m1.4c">\displaystyle\bm{p}(t)=\Phi^{T}_{\rm s}(t)\bm{\theta}+\bm{d}_{\rm s}(t).</annotation><annotation encoding="application/x-llamapun" id="S5.E28.m1.4d">bold_italic_p ( italic_t ) = roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_t ) bold_italic_θ + bold_italic_d start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_t ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(28)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.SS3.p1.24">Note that <math alttext="\bm{d}_{\rm s}" class="ltx_Math" display="inline" id="S5.SS3.p1.12.m1.1"><semantics id="S5.SS3.p1.12.m1.1a"><msub id="S5.SS3.p1.12.m1.1.1" xref="S5.SS3.p1.12.m1.1.1.cmml"><mi id="S5.SS3.p1.12.m1.1.1.2" xref="S5.SS3.p1.12.m1.1.1.2.cmml">𝒅</mi><mi id="S5.SS3.p1.12.m1.1.1.3" mathvariant="normal" xref="S5.SS3.p1.12.m1.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.12.m1.1b"><apply id="S5.SS3.p1.12.m1.1.1.cmml" xref="S5.SS3.p1.12.m1.1.1"><csymbol cd="ambiguous" id="S5.SS3.p1.12.m1.1.1.1.cmml" xref="S5.SS3.p1.12.m1.1.1">subscript</csymbol><ci id="S5.SS3.p1.12.m1.1.1.2.cmml" xref="S5.SS3.p1.12.m1.1.1.2">𝒅</ci><ci id="S5.SS3.p1.12.m1.1.1.3.cmml" xref="S5.SS3.p1.12.m1.1.1.3">s</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.12.m1.1c">\bm{d}_{\rm s}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.12.m1.1d">bold_italic_d start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT</annotation></semantics></math> satisfies <math alttext="\|\bm{d}_{\rm s}\|" class="ltx_Math" display="inline" id="S5.SS3.p1.13.m2.1"><semantics id="S5.SS3.p1.13.m2.1a"><mrow id="S5.SS3.p1.13.m2.1.1.1" xref="S5.SS3.p1.13.m2.1.1.2.cmml"><mo id="S5.SS3.p1.13.m2.1.1.1.2" stretchy="false" xref="S5.SS3.p1.13.m2.1.1.2.1.cmml">‖</mo><msub id="S5.SS3.p1.13.m2.1.1.1.1" xref="S5.SS3.p1.13.m2.1.1.1.1.cmml"><mi id="S5.SS3.p1.13.m2.1.1.1.1.2" xref="S5.SS3.p1.13.m2.1.1.1.1.2.cmml">𝒅</mi><mi id="S5.SS3.p1.13.m2.1.1.1.1.3" mathvariant="normal" xref="S5.SS3.p1.13.m2.1.1.1.1.3.cmml">s</mi></msub><mo id="S5.SS3.p1.13.m2.1.1.1.3" stretchy="false" xref="S5.SS3.p1.13.m2.1.1.2.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.13.m2.1b"><apply id="S5.SS3.p1.13.m2.1.1.2.cmml" xref="S5.SS3.p1.13.m2.1.1.1"><csymbol cd="latexml" id="S5.SS3.p1.13.m2.1.1.2.1.cmml" xref="S5.SS3.p1.13.m2.1.1.1.2">norm</csymbol><apply id="S5.SS3.p1.13.m2.1.1.1.1.cmml" xref="S5.SS3.p1.13.m2.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.p1.13.m2.1.1.1.1.1.cmml" xref="S5.SS3.p1.13.m2.1.1.1.1">subscript</csymbol><ci id="S5.SS3.p1.13.m2.1.1.1.1.2.cmml" xref="S5.SS3.p1.13.m2.1.1.1.1.2">𝒅</ci><ci id="S5.SS3.p1.13.m2.1.1.1.1.3.cmml" xref="S5.SS3.p1.13.m2.1.1.1.1.3">s</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.13.m2.1c">\|\bm{d}_{\rm s}\|</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.13.m2.1d">∥ bold_italic_d start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ∥</annotation></semantics></math> <math alttext="\leq" class="ltx_Math" display="inline" id="S5.SS3.p1.14.m3.1"><semantics id="S5.SS3.p1.14.m3.1a"><mo id="S5.SS3.p1.14.m3.1.1" xref="S5.SS3.p1.14.m3.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.14.m3.1b"><leq id="S5.SS3.p1.14.m3.1.1.cmml" xref="S5.SS3.p1.14.m3.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.14.m3.1c">\leq</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.14.m3.1d">≤</annotation></semantics></math> <math alttext="\bar{d}/\sqrt{k_{\min}}" class="ltx_Math" display="inline" id="S5.SS3.p1.15.m4.1"><semantics id="S5.SS3.p1.15.m4.1a"><mrow id="S5.SS3.p1.15.m4.1.1" xref="S5.SS3.p1.15.m4.1.1.cmml"><mover accent="true" id="S5.SS3.p1.15.m4.1.1.2" xref="S5.SS3.p1.15.m4.1.1.2.cmml"><mi id="S5.SS3.p1.15.m4.1.1.2.2" xref="S5.SS3.p1.15.m4.1.1.2.2.cmml">d</mi><mo id="S5.SS3.p1.15.m4.1.1.2.1" xref="S5.SS3.p1.15.m4.1.1.2.1.cmml">¯</mo></mover><mo id="S5.SS3.p1.15.m4.1.1.1" xref="S5.SS3.p1.15.m4.1.1.1.cmml">/</mo><msqrt id="S5.SS3.p1.15.m4.1.1.3" xref="S5.SS3.p1.15.m4.1.1.3.cmml"><msub id="S5.SS3.p1.15.m4.1.1.3.2" xref="S5.SS3.p1.15.m4.1.1.3.2.cmml"><mi id="S5.SS3.p1.15.m4.1.1.3.2.2" xref="S5.SS3.p1.15.m4.1.1.3.2.2.cmml">k</mi><mi id="S5.SS3.p1.15.m4.1.1.3.2.3" xref="S5.SS3.p1.15.m4.1.1.3.2.3.cmml">min</mi></msub></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.15.m4.1b"><apply id="S5.SS3.p1.15.m4.1.1.cmml" xref="S5.SS3.p1.15.m4.1.1"><divide id="S5.SS3.p1.15.m4.1.1.1.cmml" xref="S5.SS3.p1.15.m4.1.1.1"></divide><apply id="S5.SS3.p1.15.m4.1.1.2.cmml" xref="S5.SS3.p1.15.m4.1.1.2"><ci id="S5.SS3.p1.15.m4.1.1.2.1.cmml" xref="S5.SS3.p1.15.m4.1.1.2.1">¯</ci><ci id="S5.SS3.p1.15.m4.1.1.2.2.cmml" xref="S5.SS3.p1.15.m4.1.1.2.2">𝑑</ci></apply><apply id="S5.SS3.p1.15.m4.1.1.3.cmml" xref="S5.SS3.p1.15.m4.1.1.3"><root id="S5.SS3.p1.15.m4.1.1.3a.cmml" xref="S5.SS3.p1.15.m4.1.1.3"></root><apply id="S5.SS3.p1.15.m4.1.1.3.2.cmml" xref="S5.SS3.p1.15.m4.1.1.3.2"><csymbol cd="ambiguous" id="S5.SS3.p1.15.m4.1.1.3.2.1.cmml" xref="S5.SS3.p1.15.m4.1.1.3.2">subscript</csymbol><ci id="S5.SS3.p1.15.m4.1.1.3.2.2.cmml" xref="S5.SS3.p1.15.m4.1.1.3.2.2">𝑘</ci><min id="S5.SS3.p1.15.m4.1.1.3.2.3.cmml" xref="S5.SS3.p1.15.m4.1.1.3.2.3"></min></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.15.m4.1c">\bar{d}/\sqrt{k_{\min}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.15.m4.1d">over¯ start_ARG italic_d end_ARG / square-root start_ARG italic_k start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT end_ARG</annotation></semantics></math> with <math alttext="k_{\min}" class="ltx_Math" display="inline" id="S5.SS3.p1.16.m5.1"><semantics id="S5.SS3.p1.16.m5.1a"><msub id="S5.SS3.p1.16.m5.1.1" xref="S5.SS3.p1.16.m5.1.1.cmml"><mi id="S5.SS3.p1.16.m5.1.1.2" xref="S5.SS3.p1.16.m5.1.1.2.cmml">k</mi><mi id="S5.SS3.p1.16.m5.1.1.3" xref="S5.SS3.p1.16.m5.1.1.3.cmml">min</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.16.m5.1b"><apply id="S5.SS3.p1.16.m5.1.1.cmml" xref="S5.SS3.p1.16.m5.1.1"><csymbol cd="ambiguous" id="S5.SS3.p1.16.m5.1.1.1.cmml" xref="S5.SS3.p1.16.m5.1.1">subscript</csymbol><ci id="S5.SS3.p1.16.m5.1.1.2.cmml" xref="S5.SS3.p1.16.m5.1.1.2">𝑘</ci><min id="S5.SS3.p1.16.m5.1.1.3.cmml" xref="S5.SS3.p1.16.m5.1.1.3"></min></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.16.m5.1c">k_{\min}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.16.m5.1d">italic_k start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S5.SS3.p1.17.m6.1"><semantics id="S5.SS3.p1.17.m6.1a"><mo id="S5.SS3.p1.17.m6.1.1" xref="S5.SS3.p1.17.m6.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.17.m6.1b"><csymbol cd="latexml" id="S5.SS3.p1.17.m6.1.1.cmml" xref="S5.SS3.p1.17.m6.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.17.m6.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.17.m6.1d">:=</annotation></semantics></math> <math alttext="\min_{1\leq i\leq n}\{k_{{\rm c}i}\}" class="ltx_Math" display="inline" id="S5.SS3.p1.18.m7.2"><semantics id="S5.SS3.p1.18.m7.2a"><mrow id="S5.SS3.p1.18.m7.2.2.2" xref="S5.SS3.p1.18.m7.2.2.3.cmml"><msub id="S5.SS3.p1.18.m7.1.1.1.1" xref="S5.SS3.p1.18.m7.1.1.1.1.cmml"><mi id="S5.SS3.p1.18.m7.1.1.1.1.2" xref="S5.SS3.p1.18.m7.1.1.1.1.2.cmml">min</mi><mrow id="S5.SS3.p1.18.m7.1.1.1.1.3" xref="S5.SS3.p1.18.m7.1.1.1.1.3.cmml"><mn id="S5.SS3.p1.18.m7.1.1.1.1.3.2" xref="S5.SS3.p1.18.m7.1.1.1.1.3.2.cmml">1</mn><mo id="S5.SS3.p1.18.m7.1.1.1.1.3.3" xref="S5.SS3.p1.18.m7.1.1.1.1.3.3.cmml">≤</mo><mi id="S5.SS3.p1.18.m7.1.1.1.1.3.4" xref="S5.SS3.p1.18.m7.1.1.1.1.3.4.cmml">i</mi><mo id="S5.SS3.p1.18.m7.1.1.1.1.3.5" xref="S5.SS3.p1.18.m7.1.1.1.1.3.5.cmml">≤</mo><mi id="S5.SS3.p1.18.m7.1.1.1.1.3.6" xref="S5.SS3.p1.18.m7.1.1.1.1.3.6.cmml">n</mi></mrow></msub><mo id="S5.SS3.p1.18.m7.2.2.2a" xref="S5.SS3.p1.18.m7.2.2.3.cmml">⁡</mo><mrow id="S5.SS3.p1.18.m7.2.2.2.2" xref="S5.SS3.p1.18.m7.2.2.3.cmml"><mo id="S5.SS3.p1.18.m7.2.2.2.2.2" stretchy="false" xref="S5.SS3.p1.18.m7.2.2.3.cmml">{</mo><msub id="S5.SS3.p1.18.m7.2.2.2.2.1" xref="S5.SS3.p1.18.m7.2.2.2.2.1.cmml"><mi id="S5.SS3.p1.18.m7.2.2.2.2.1.2" xref="S5.SS3.p1.18.m7.2.2.2.2.1.2.cmml">k</mi><mrow id="S5.SS3.p1.18.m7.2.2.2.2.1.3" xref="S5.SS3.p1.18.m7.2.2.2.2.1.3.cmml"><mi id="S5.SS3.p1.18.m7.2.2.2.2.1.3.2" mathvariant="normal" xref="S5.SS3.p1.18.m7.2.2.2.2.1.3.2.cmml">c</mi><mo id="S5.SS3.p1.18.m7.2.2.2.2.1.3.1" xref="S5.SS3.p1.18.m7.2.2.2.2.1.3.1.cmml">⁢</mo><mi id="S5.SS3.p1.18.m7.2.2.2.2.1.3.3" xref="S5.SS3.p1.18.m7.2.2.2.2.1.3.3.cmml">i</mi></mrow></msub><mo id="S5.SS3.p1.18.m7.2.2.2.2.3" stretchy="false" xref="S5.SS3.p1.18.m7.2.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.18.m7.2b"><apply id="S5.SS3.p1.18.m7.2.2.3.cmml" xref="S5.SS3.p1.18.m7.2.2.2"><apply id="S5.SS3.p1.18.m7.1.1.1.1.cmml" xref="S5.SS3.p1.18.m7.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.p1.18.m7.1.1.1.1.1.cmml" xref="S5.SS3.p1.18.m7.1.1.1.1">subscript</csymbol><min id="S5.SS3.p1.18.m7.1.1.1.1.2.cmml" xref="S5.SS3.p1.18.m7.1.1.1.1.2"></min><apply id="S5.SS3.p1.18.m7.1.1.1.1.3.cmml" xref="S5.SS3.p1.18.m7.1.1.1.1.3"><and id="S5.SS3.p1.18.m7.1.1.1.1.3a.cmml" xref="S5.SS3.p1.18.m7.1.1.1.1.3"></and><apply id="S5.SS3.p1.18.m7.1.1.1.1.3b.cmml" xref="S5.SS3.p1.18.m7.1.1.1.1.3"><leq id="S5.SS3.p1.18.m7.1.1.1.1.3.3.cmml" xref="S5.SS3.p1.18.m7.1.1.1.1.3.3"></leq><cn id="S5.SS3.p1.18.m7.1.1.1.1.3.2.cmml" type="integer" xref="S5.SS3.p1.18.m7.1.1.1.1.3.2">1</cn><ci id="S5.SS3.p1.18.m7.1.1.1.1.3.4.cmml" xref="S5.SS3.p1.18.m7.1.1.1.1.3.4">𝑖</ci></apply><apply id="S5.SS3.p1.18.m7.1.1.1.1.3c.cmml" xref="S5.SS3.p1.18.m7.1.1.1.1.3"><leq id="S5.SS3.p1.18.m7.1.1.1.1.3.5.cmml" xref="S5.SS3.p1.18.m7.1.1.1.1.3.5"></leq><share href="https://arxiv.org/html/2401.10785v2#S5.SS3.p1.18.m7.1.1.1.1.3.4.cmml" id="S5.SS3.p1.18.m7.1.1.1.1.3d.cmml" xref="S5.SS3.p1.18.m7.1.1.1.1.3"></share><ci id="S5.SS3.p1.18.m7.1.1.1.1.3.6.cmml" xref="S5.SS3.p1.18.m7.1.1.1.1.3.6">𝑛</ci></apply></apply></apply><apply id="S5.SS3.p1.18.m7.2.2.2.2.1.cmml" xref="S5.SS3.p1.18.m7.2.2.2.2.1"><csymbol cd="ambiguous" id="S5.SS3.p1.18.m7.2.2.2.2.1.1.cmml" xref="S5.SS3.p1.18.m7.2.2.2.2.1">subscript</csymbol><ci id="S5.SS3.p1.18.m7.2.2.2.2.1.2.cmml" xref="S5.SS3.p1.18.m7.2.2.2.2.1.2">𝑘</ci><apply id="S5.SS3.p1.18.m7.2.2.2.2.1.3.cmml" xref="S5.SS3.p1.18.m7.2.2.2.2.1.3"><times id="S5.SS3.p1.18.m7.2.2.2.2.1.3.1.cmml" xref="S5.SS3.p1.18.m7.2.2.2.2.1.3.1"></times><ci id="S5.SS3.p1.18.m7.2.2.2.2.1.3.2.cmml" xref="S5.SS3.p1.18.m7.2.2.2.2.1.3.2">c</ci><ci id="S5.SS3.p1.18.m7.2.2.2.2.1.3.3.cmml" xref="S5.SS3.p1.18.m7.2.2.2.2.1.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.18.m7.2c">\min_{1\leq i\leq n}\{k_{{\rm c}i}\}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.18.m7.2d">roman_min start_POSTSUBSCRIPT 1 ≤ italic_i ≤ italic_n end_POSTSUBSCRIPT { italic_k start_POSTSUBSCRIPT roman_c italic_i end_POSTSUBSCRIPT }</annotation></semantics></math> <math alttext="\in" class="ltx_Math" display="inline" id="S5.SS3.p1.19.m8.1"><semantics id="S5.SS3.p1.19.m8.1a"><mo id="S5.SS3.p1.19.m8.1.1" xref="S5.SS3.p1.19.m8.1.1.cmml">∈</mo><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.19.m8.1b"><in id="S5.SS3.p1.19.m8.1.1.cmml" xref="S5.SS3.p1.19.m8.1.1"></in></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.19.m8.1c">\in</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.19.m8.1d">∈</annotation></semantics></math> <math alttext="\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S5.SS3.p1.20.m9.1"><semantics id="S5.SS3.p1.20.m9.1a"><msup id="S5.SS3.p1.20.m9.1.1" xref="S5.SS3.p1.20.m9.1.1.cmml"><mi id="S5.SS3.p1.20.m9.1.1.2" xref="S5.SS3.p1.20.m9.1.1.2.cmml">ℝ</mi><mo id="S5.SS3.p1.20.m9.1.1.3" xref="S5.SS3.p1.20.m9.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.20.m9.1b"><apply id="S5.SS3.p1.20.m9.1.1.cmml" xref="S5.SS3.p1.20.m9.1.1"><csymbol cd="ambiguous" id="S5.SS3.p1.20.m9.1.1.1.cmml" xref="S5.SS3.p1.20.m9.1.1">superscript</csymbol><ci id="S5.SS3.p1.20.m9.1.1.2.cmml" xref="S5.SS3.p1.20.m9.1.1.2">ℝ</ci><plus id="S5.SS3.p1.20.m9.1.1.3.cmml" xref="S5.SS3.p1.20.m9.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.20.m9.1c">\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.20.m9.1d">blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> as (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S5.E27" title="In V-C Robustness Results ‣ V Theoretical Guarantees ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">27</span></a>) is stable. Multiplying (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S5.E28" title="In V-C Robustness Results ‣ V Theoretical Guarantees ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">28</span></a>) by <math alttext="\Phi_{\rm s}" class="ltx_Math" display="inline" id="S5.SS3.p1.21.m10.1"><semantics id="S5.SS3.p1.21.m10.1a"><msub id="S5.SS3.p1.21.m10.1.1" xref="S5.SS3.p1.21.m10.1.1.cmml"><mi id="S5.SS3.p1.21.m10.1.1.2" mathvariant="normal" xref="S5.SS3.p1.21.m10.1.1.2.cmml">Φ</mi><mi id="S5.SS3.p1.21.m10.1.1.3" mathvariant="normal" xref="S5.SS3.p1.21.m10.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.21.m10.1b"><apply id="S5.SS3.p1.21.m10.1.1.cmml" xref="S5.SS3.p1.21.m10.1.1"><csymbol cd="ambiguous" id="S5.SS3.p1.21.m10.1.1.1.cmml" xref="S5.SS3.p1.21.m10.1.1">subscript</csymbol><ci id="S5.SS3.p1.21.m10.1.1.2.cmml" xref="S5.SS3.p1.21.m10.1.1.2">Φ</ci><ci id="S5.SS3.p1.21.m10.1.1.3.cmml" xref="S5.SS3.p1.21.m10.1.1.3">s</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.21.m10.1c">\Phi_{\rm s}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.21.m10.1d">roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT</annotation></semantics></math>, integrating the resultant equality at [<math alttext="t-\tau_{\rm d}" class="ltx_Math" display="inline" id="S5.SS3.p1.22.m11.1"><semantics id="S5.SS3.p1.22.m11.1a"><mrow id="S5.SS3.p1.22.m11.1.1" xref="S5.SS3.p1.22.m11.1.1.cmml"><mi id="S5.SS3.p1.22.m11.1.1.2" xref="S5.SS3.p1.22.m11.1.1.2.cmml">t</mi><mo id="S5.SS3.p1.22.m11.1.1.1" xref="S5.SS3.p1.22.m11.1.1.1.cmml">−</mo><msub id="S5.SS3.p1.22.m11.1.1.3" xref="S5.SS3.p1.22.m11.1.1.3.cmml"><mi id="S5.SS3.p1.22.m11.1.1.3.2" xref="S5.SS3.p1.22.m11.1.1.3.2.cmml">τ</mi><mi id="S5.SS3.p1.22.m11.1.1.3.3" mathvariant="normal" xref="S5.SS3.p1.22.m11.1.1.3.3.cmml">d</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.22.m11.1b"><apply id="S5.SS3.p1.22.m11.1.1.cmml" xref="S5.SS3.p1.22.m11.1.1"><minus id="S5.SS3.p1.22.m11.1.1.1.cmml" xref="S5.SS3.p1.22.m11.1.1.1"></minus><ci id="S5.SS3.p1.22.m11.1.1.2.cmml" xref="S5.SS3.p1.22.m11.1.1.2">𝑡</ci><apply id="S5.SS3.p1.22.m11.1.1.3.cmml" xref="S5.SS3.p1.22.m11.1.1.3"><csymbol cd="ambiguous" id="S5.SS3.p1.22.m11.1.1.3.1.cmml" xref="S5.SS3.p1.22.m11.1.1.3">subscript</csymbol><ci id="S5.SS3.p1.22.m11.1.1.3.2.cmml" xref="S5.SS3.p1.22.m11.1.1.3.2">𝜏</ci><ci id="S5.SS3.p1.22.m11.1.1.3.3.cmml" xref="S5.SS3.p1.22.m11.1.1.3.3">d</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.22.m11.1c">t-\tau_{\rm d}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.22.m11.1d">italic_t - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="t" class="ltx_Math" display="inline" id="S5.SS3.p1.23.m12.1"><semantics id="S5.SS3.p1.23.m12.1a"><mi id="S5.SS3.p1.23.m12.1.1" xref="S5.SS3.p1.23.m12.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.23.m12.1b"><ci id="S5.SS3.p1.23.m12.1.1.cmml" xref="S5.SS3.p1.23.m12.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.23.m12.1c">t</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.23.m12.1d">italic_t</annotation></semantics></math>], and using <math alttext="\Psi" class="ltx_Math" display="inline" id="S5.SS3.p1.24.m13.1"><semantics id="S5.SS3.p1.24.m13.1a"><mi id="S5.SS3.p1.24.m13.1.1" mathvariant="normal" xref="S5.SS3.p1.24.m13.1.1.cmml">Ψ</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.24.m13.1b"><ci id="S5.SS3.p1.24.m13.1.1.cmml" xref="S5.SS3.p1.24.m13.1.1">Ψ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.24.m13.1c">\Psi</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.24.m13.1d">roman_Ψ</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E15" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">15</span></a>), one obtains</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx24"> <tbody id="S5.E29"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\Psi(t)\bm{\theta}+\int_{t-\tau_{\rm d}}^{t}\Phi_{\rm s}(\tau)\bm% {d}_{\rm s}(\tau)d\tau=\int_{t-\tau_{\rm d}}^{t}\Phi_{\rm s}(\tau)\bm{p}(\tau)% d\tau." class="ltx_Math" display="inline" id="S5.E29.m1.6"><semantics id="S5.E29.m1.6a"><mrow id="S5.E29.m1.6.6.1" xref="S5.E29.m1.6.6.1.1.cmml"><mrow id="S5.E29.m1.6.6.1.1" xref="S5.E29.m1.6.6.1.1.cmml"><mrow id="S5.E29.m1.6.6.1.1.2" xref="S5.E29.m1.6.6.1.1.2.cmml"><mrow id="S5.E29.m1.6.6.1.1.2.2" xref="S5.E29.m1.6.6.1.1.2.2.cmml"><mi id="S5.E29.m1.6.6.1.1.2.2.2" mathvariant="normal" xref="S5.E29.m1.6.6.1.1.2.2.2.cmml">Ψ</mi><mo id="S5.E29.m1.6.6.1.1.2.2.1" xref="S5.E29.m1.6.6.1.1.2.2.1.cmml">⁢</mo><mrow id="S5.E29.m1.6.6.1.1.2.2.3.2" xref="S5.E29.m1.6.6.1.1.2.2.cmml"><mo id="S5.E29.m1.6.6.1.1.2.2.3.2.1" stretchy="false" xref="S5.E29.m1.6.6.1.1.2.2.cmml">(</mo><mi id="S5.E29.m1.1.1" xref="S5.E29.m1.1.1.cmml">t</mi><mo id="S5.E29.m1.6.6.1.1.2.2.3.2.2" stretchy="false" xref="S5.E29.m1.6.6.1.1.2.2.cmml">)</mo></mrow><mo id="S5.E29.m1.6.6.1.1.2.2.1a" xref="S5.E29.m1.6.6.1.1.2.2.1.cmml">⁢</mo><mi id="S5.E29.m1.6.6.1.1.2.2.4" xref="S5.E29.m1.6.6.1.1.2.2.4.cmml">𝜽</mi></mrow><mo id="S5.E29.m1.6.6.1.1.2.1" xref="S5.E29.m1.6.6.1.1.2.1.cmml">+</mo><mrow id="S5.E29.m1.6.6.1.1.2.3" xref="S5.E29.m1.6.6.1.1.2.3.cmml"><mstyle displaystyle="true" id="S5.E29.m1.6.6.1.1.2.3.1" xref="S5.E29.m1.6.6.1.1.2.3.1.cmml"><msubsup id="S5.E29.m1.6.6.1.1.2.3.1a" xref="S5.E29.m1.6.6.1.1.2.3.1.cmml"><mo id="S5.E29.m1.6.6.1.1.2.3.1.2.2" xref="S5.E29.m1.6.6.1.1.2.3.1.2.2.cmml">∫</mo><mrow id="S5.E29.m1.6.6.1.1.2.3.1.2.3" xref="S5.E29.m1.6.6.1.1.2.3.1.2.3.cmml"><mi id="S5.E29.m1.6.6.1.1.2.3.1.2.3.2" xref="S5.E29.m1.6.6.1.1.2.3.1.2.3.2.cmml">t</mi><mo id="S5.E29.m1.6.6.1.1.2.3.1.2.3.1" xref="S5.E29.m1.6.6.1.1.2.3.1.2.3.1.cmml">−</mo><msub id="S5.E29.m1.6.6.1.1.2.3.1.2.3.3" xref="S5.E29.m1.6.6.1.1.2.3.1.2.3.3.cmml"><mi id="S5.E29.m1.6.6.1.1.2.3.1.2.3.3.2" xref="S5.E29.m1.6.6.1.1.2.3.1.2.3.3.2.cmml">τ</mi><mi id="S5.E29.m1.6.6.1.1.2.3.1.2.3.3.3" mathvariant="normal" xref="S5.E29.m1.6.6.1.1.2.3.1.2.3.3.3.cmml">d</mi></msub></mrow><mi id="S5.E29.m1.6.6.1.1.2.3.1.3" xref="S5.E29.m1.6.6.1.1.2.3.1.3.cmml">t</mi></msubsup></mstyle><mrow id="S5.E29.m1.6.6.1.1.2.3.2" xref="S5.E29.m1.6.6.1.1.2.3.2.cmml"><msub id="S5.E29.m1.6.6.1.1.2.3.2.2" xref="S5.E29.m1.6.6.1.1.2.3.2.2.cmml"><mi id="S5.E29.m1.6.6.1.1.2.3.2.2.2" mathvariant="normal" xref="S5.E29.m1.6.6.1.1.2.3.2.2.2.cmml">Φ</mi><mi id="S5.E29.m1.6.6.1.1.2.3.2.2.3" mathvariant="normal" xref="S5.E29.m1.6.6.1.1.2.3.2.2.3.cmml">s</mi></msub><mo id="S5.E29.m1.6.6.1.1.2.3.2.1" xref="S5.E29.m1.6.6.1.1.2.3.2.1.cmml">⁢</mo><mrow id="S5.E29.m1.6.6.1.1.2.3.2.3.2" xref="S5.E29.m1.6.6.1.1.2.3.2.cmml"><mo id="S5.E29.m1.6.6.1.1.2.3.2.3.2.1" stretchy="false" xref="S5.E29.m1.6.6.1.1.2.3.2.cmml">(</mo><mi id="S5.E29.m1.2.2" xref="S5.E29.m1.2.2.cmml">τ</mi><mo id="S5.E29.m1.6.6.1.1.2.3.2.3.2.2" stretchy="false" xref="S5.E29.m1.6.6.1.1.2.3.2.cmml">)</mo></mrow><mo id="S5.E29.m1.6.6.1.1.2.3.2.1a" xref="S5.E29.m1.6.6.1.1.2.3.2.1.cmml">⁢</mo><msub id="S5.E29.m1.6.6.1.1.2.3.2.4" xref="S5.E29.m1.6.6.1.1.2.3.2.4.cmml"><mi id="S5.E29.m1.6.6.1.1.2.3.2.4.2" xref="S5.E29.m1.6.6.1.1.2.3.2.4.2.cmml">𝒅</mi><mi id="S5.E29.m1.6.6.1.1.2.3.2.4.3" mathvariant="normal" xref="S5.E29.m1.6.6.1.1.2.3.2.4.3.cmml">s</mi></msub><mo id="S5.E29.m1.6.6.1.1.2.3.2.1b" xref="S5.E29.m1.6.6.1.1.2.3.2.1.cmml">⁢</mo><mrow id="S5.E29.m1.6.6.1.1.2.3.2.5.2" xref="S5.E29.m1.6.6.1.1.2.3.2.cmml"><mo id="S5.E29.m1.6.6.1.1.2.3.2.5.2.1" stretchy="false" xref="S5.E29.m1.6.6.1.1.2.3.2.cmml">(</mo><mi id="S5.E29.m1.3.3" xref="S5.E29.m1.3.3.cmml">τ</mi><mo id="S5.E29.m1.6.6.1.1.2.3.2.5.2.2" stretchy="false" xref="S5.E29.m1.6.6.1.1.2.3.2.cmml">)</mo></mrow><mo id="S5.E29.m1.6.6.1.1.2.3.2.1c" lspace="0em" xref="S5.E29.m1.6.6.1.1.2.3.2.1.cmml">⁢</mo><mrow id="S5.E29.m1.6.6.1.1.2.3.2.6" xref="S5.E29.m1.6.6.1.1.2.3.2.6.cmml"><mo id="S5.E29.m1.6.6.1.1.2.3.2.6.1" rspace="0em" xref="S5.E29.m1.6.6.1.1.2.3.2.6.1.cmml">𝑑</mo><mi id="S5.E29.m1.6.6.1.1.2.3.2.6.2" xref="S5.E29.m1.6.6.1.1.2.3.2.6.2.cmml">τ</mi></mrow></mrow></mrow></mrow><mo id="S5.E29.m1.6.6.1.1.1" xref="S5.E29.m1.6.6.1.1.1.cmml">=</mo><mrow id="S5.E29.m1.6.6.1.1.3" xref="S5.E29.m1.6.6.1.1.3.cmml"><mstyle displaystyle="true" id="S5.E29.m1.6.6.1.1.3.1" xref="S5.E29.m1.6.6.1.1.3.1.cmml"><msubsup id="S5.E29.m1.6.6.1.1.3.1a" xref="S5.E29.m1.6.6.1.1.3.1.cmml"><mo id="S5.E29.m1.6.6.1.1.3.1.2.2" xref="S5.E29.m1.6.6.1.1.3.1.2.2.cmml">∫</mo><mrow id="S5.E29.m1.6.6.1.1.3.1.2.3" xref="S5.E29.m1.6.6.1.1.3.1.2.3.cmml"><mi id="S5.E29.m1.6.6.1.1.3.1.2.3.2" xref="S5.E29.m1.6.6.1.1.3.1.2.3.2.cmml">t</mi><mo id="S5.E29.m1.6.6.1.1.3.1.2.3.1" xref="S5.E29.m1.6.6.1.1.3.1.2.3.1.cmml">−</mo><msub id="S5.E29.m1.6.6.1.1.3.1.2.3.3" xref="S5.E29.m1.6.6.1.1.3.1.2.3.3.cmml"><mi id="S5.E29.m1.6.6.1.1.3.1.2.3.3.2" xref="S5.E29.m1.6.6.1.1.3.1.2.3.3.2.cmml">τ</mi><mi id="S5.E29.m1.6.6.1.1.3.1.2.3.3.3" mathvariant="normal" xref="S5.E29.m1.6.6.1.1.3.1.2.3.3.3.cmml">d</mi></msub></mrow><mi id="S5.E29.m1.6.6.1.1.3.1.3" xref="S5.E29.m1.6.6.1.1.3.1.3.cmml">t</mi></msubsup></mstyle><mrow id="S5.E29.m1.6.6.1.1.3.2" xref="S5.E29.m1.6.6.1.1.3.2.cmml"><msub id="S5.E29.m1.6.6.1.1.3.2.2" xref="S5.E29.m1.6.6.1.1.3.2.2.cmml"><mi id="S5.E29.m1.6.6.1.1.3.2.2.2" mathvariant="normal" xref="S5.E29.m1.6.6.1.1.3.2.2.2.cmml">Φ</mi><mi id="S5.E29.m1.6.6.1.1.3.2.2.3" mathvariant="normal" xref="S5.E29.m1.6.6.1.1.3.2.2.3.cmml">s</mi></msub><mo id="S5.E29.m1.6.6.1.1.3.2.1" xref="S5.E29.m1.6.6.1.1.3.2.1.cmml">⁢</mo><mrow id="S5.E29.m1.6.6.1.1.3.2.3.2" xref="S5.E29.m1.6.6.1.1.3.2.cmml"><mo id="S5.E29.m1.6.6.1.1.3.2.3.2.1" stretchy="false" xref="S5.E29.m1.6.6.1.1.3.2.cmml">(</mo><mi id="S5.E29.m1.4.4" xref="S5.E29.m1.4.4.cmml">τ</mi><mo id="S5.E29.m1.6.6.1.1.3.2.3.2.2" stretchy="false" xref="S5.E29.m1.6.6.1.1.3.2.cmml">)</mo></mrow><mo id="S5.E29.m1.6.6.1.1.3.2.1a" xref="S5.E29.m1.6.6.1.1.3.2.1.cmml">⁢</mo><mi id="S5.E29.m1.6.6.1.1.3.2.4" xref="S5.E29.m1.6.6.1.1.3.2.4.cmml">𝒑</mi><mo id="S5.E29.m1.6.6.1.1.3.2.1b" xref="S5.E29.m1.6.6.1.1.3.2.1.cmml">⁢</mo><mrow id="S5.E29.m1.6.6.1.1.3.2.5.2" xref="S5.E29.m1.6.6.1.1.3.2.cmml"><mo id="S5.E29.m1.6.6.1.1.3.2.5.2.1" stretchy="false" xref="S5.E29.m1.6.6.1.1.3.2.cmml">(</mo><mi id="S5.E29.m1.5.5" xref="S5.E29.m1.5.5.cmml">τ</mi><mo id="S5.E29.m1.6.6.1.1.3.2.5.2.2" stretchy="false" xref="S5.E29.m1.6.6.1.1.3.2.cmml">)</mo></mrow><mo id="S5.E29.m1.6.6.1.1.3.2.1c" lspace="0em" xref="S5.E29.m1.6.6.1.1.3.2.1.cmml">⁢</mo><mrow id="S5.E29.m1.6.6.1.1.3.2.6" xref="S5.E29.m1.6.6.1.1.3.2.6.cmml"><mo id="S5.E29.m1.6.6.1.1.3.2.6.1" rspace="0em" xref="S5.E29.m1.6.6.1.1.3.2.6.1.cmml">𝑑</mo><mi id="S5.E29.m1.6.6.1.1.3.2.6.2" xref="S5.E29.m1.6.6.1.1.3.2.6.2.cmml">τ</mi></mrow></mrow></mrow></mrow><mo id="S5.E29.m1.6.6.1.2" lspace="0em" xref="S5.E29.m1.6.6.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.E29.m1.6b"><apply id="S5.E29.m1.6.6.1.1.cmml" xref="S5.E29.m1.6.6.1"><eq id="S5.E29.m1.6.6.1.1.1.cmml" xref="S5.E29.m1.6.6.1.1.1"></eq><apply id="S5.E29.m1.6.6.1.1.2.cmml" xref="S5.E29.m1.6.6.1.1.2"><plus id="S5.E29.m1.6.6.1.1.2.1.cmml" xref="S5.E29.m1.6.6.1.1.2.1"></plus><apply id="S5.E29.m1.6.6.1.1.2.2.cmml" xref="S5.E29.m1.6.6.1.1.2.2"><times id="S5.E29.m1.6.6.1.1.2.2.1.cmml" xref="S5.E29.m1.6.6.1.1.2.2.1"></times><ci id="S5.E29.m1.6.6.1.1.2.2.2.cmml" xref="S5.E29.m1.6.6.1.1.2.2.2">Ψ</ci><ci id="S5.E29.m1.1.1.cmml" xref="S5.E29.m1.1.1">𝑡</ci><ci id="S5.E29.m1.6.6.1.1.2.2.4.cmml" xref="S5.E29.m1.6.6.1.1.2.2.4">𝜽</ci></apply><apply id="S5.E29.m1.6.6.1.1.2.3.cmml" xref="S5.E29.m1.6.6.1.1.2.3"><apply id="S5.E29.m1.6.6.1.1.2.3.1.cmml" xref="S5.E29.m1.6.6.1.1.2.3.1"><csymbol cd="ambiguous" id="S5.E29.m1.6.6.1.1.2.3.1.1.cmml" xref="S5.E29.m1.6.6.1.1.2.3.1">superscript</csymbol><apply id="S5.E29.m1.6.6.1.1.2.3.1.2.cmml" xref="S5.E29.m1.6.6.1.1.2.3.1"><csymbol cd="ambiguous" id="S5.E29.m1.6.6.1.1.2.3.1.2.1.cmml" xref="S5.E29.m1.6.6.1.1.2.3.1">subscript</csymbol><int id="S5.E29.m1.6.6.1.1.2.3.1.2.2.cmml" xref="S5.E29.m1.6.6.1.1.2.3.1.2.2"></int><apply id="S5.E29.m1.6.6.1.1.2.3.1.2.3.cmml" xref="S5.E29.m1.6.6.1.1.2.3.1.2.3"><minus id="S5.E29.m1.6.6.1.1.2.3.1.2.3.1.cmml" xref="S5.E29.m1.6.6.1.1.2.3.1.2.3.1"></minus><ci id="S5.E29.m1.6.6.1.1.2.3.1.2.3.2.cmml" xref="S5.E29.m1.6.6.1.1.2.3.1.2.3.2">𝑡</ci><apply id="S5.E29.m1.6.6.1.1.2.3.1.2.3.3.cmml" xref="S5.E29.m1.6.6.1.1.2.3.1.2.3.3"><csymbol cd="ambiguous" id="S5.E29.m1.6.6.1.1.2.3.1.2.3.3.1.cmml" xref="S5.E29.m1.6.6.1.1.2.3.1.2.3.3">subscript</csymbol><ci id="S5.E29.m1.6.6.1.1.2.3.1.2.3.3.2.cmml" xref="S5.E29.m1.6.6.1.1.2.3.1.2.3.3.2">𝜏</ci><ci id="S5.E29.m1.6.6.1.1.2.3.1.2.3.3.3.cmml" xref="S5.E29.m1.6.6.1.1.2.3.1.2.3.3.3">d</ci></apply></apply></apply><ci id="S5.E29.m1.6.6.1.1.2.3.1.3.cmml" xref="S5.E29.m1.6.6.1.1.2.3.1.3">𝑡</ci></apply><apply id="S5.E29.m1.6.6.1.1.2.3.2.cmml" xref="S5.E29.m1.6.6.1.1.2.3.2"><times id="S5.E29.m1.6.6.1.1.2.3.2.1.cmml" xref="S5.E29.m1.6.6.1.1.2.3.2.1"></times><apply id="S5.E29.m1.6.6.1.1.2.3.2.2.cmml" xref="S5.E29.m1.6.6.1.1.2.3.2.2"><csymbol cd="ambiguous" id="S5.E29.m1.6.6.1.1.2.3.2.2.1.cmml" xref="S5.E29.m1.6.6.1.1.2.3.2.2">subscript</csymbol><ci id="S5.E29.m1.6.6.1.1.2.3.2.2.2.cmml" xref="S5.E29.m1.6.6.1.1.2.3.2.2.2">Φ</ci><ci id="S5.E29.m1.6.6.1.1.2.3.2.2.3.cmml" xref="S5.E29.m1.6.6.1.1.2.3.2.2.3">s</ci></apply><ci id="S5.E29.m1.2.2.cmml" xref="S5.E29.m1.2.2">𝜏</ci><apply id="S5.E29.m1.6.6.1.1.2.3.2.4.cmml" xref="S5.E29.m1.6.6.1.1.2.3.2.4"><csymbol cd="ambiguous" id="S5.E29.m1.6.6.1.1.2.3.2.4.1.cmml" xref="S5.E29.m1.6.6.1.1.2.3.2.4">subscript</csymbol><ci id="S5.E29.m1.6.6.1.1.2.3.2.4.2.cmml" xref="S5.E29.m1.6.6.1.1.2.3.2.4.2">𝒅</ci><ci id="S5.E29.m1.6.6.1.1.2.3.2.4.3.cmml" xref="S5.E29.m1.6.6.1.1.2.3.2.4.3">s</ci></apply><ci id="S5.E29.m1.3.3.cmml" xref="S5.E29.m1.3.3">𝜏</ci><apply id="S5.E29.m1.6.6.1.1.2.3.2.6.cmml" xref="S5.E29.m1.6.6.1.1.2.3.2.6"><csymbol cd="latexml" id="S5.E29.m1.6.6.1.1.2.3.2.6.1.cmml" xref="S5.E29.m1.6.6.1.1.2.3.2.6.1">differential-d</csymbol><ci id="S5.E29.m1.6.6.1.1.2.3.2.6.2.cmml" xref="S5.E29.m1.6.6.1.1.2.3.2.6.2">𝜏</ci></apply></apply></apply></apply><apply id="S5.E29.m1.6.6.1.1.3.cmml" xref="S5.E29.m1.6.6.1.1.3"><apply id="S5.E29.m1.6.6.1.1.3.1.cmml" xref="S5.E29.m1.6.6.1.1.3.1"><csymbol cd="ambiguous" id="S5.E29.m1.6.6.1.1.3.1.1.cmml" xref="S5.E29.m1.6.6.1.1.3.1">superscript</csymbol><apply id="S5.E29.m1.6.6.1.1.3.1.2.cmml" xref="S5.E29.m1.6.6.1.1.3.1"><csymbol cd="ambiguous" id="S5.E29.m1.6.6.1.1.3.1.2.1.cmml" xref="S5.E29.m1.6.6.1.1.3.1">subscript</csymbol><int id="S5.E29.m1.6.6.1.1.3.1.2.2.cmml" xref="S5.E29.m1.6.6.1.1.3.1.2.2"></int><apply id="S5.E29.m1.6.6.1.1.3.1.2.3.cmml" xref="S5.E29.m1.6.6.1.1.3.1.2.3"><minus id="S5.E29.m1.6.6.1.1.3.1.2.3.1.cmml" xref="S5.E29.m1.6.6.1.1.3.1.2.3.1"></minus><ci id="S5.E29.m1.6.6.1.1.3.1.2.3.2.cmml" xref="S5.E29.m1.6.6.1.1.3.1.2.3.2">𝑡</ci><apply id="S5.E29.m1.6.6.1.1.3.1.2.3.3.cmml" xref="S5.E29.m1.6.6.1.1.3.1.2.3.3"><csymbol cd="ambiguous" id="S5.E29.m1.6.6.1.1.3.1.2.3.3.1.cmml" xref="S5.E29.m1.6.6.1.1.3.1.2.3.3">subscript</csymbol><ci id="S5.E29.m1.6.6.1.1.3.1.2.3.3.2.cmml" xref="S5.E29.m1.6.6.1.1.3.1.2.3.3.2">𝜏</ci><ci id="S5.E29.m1.6.6.1.1.3.1.2.3.3.3.cmml" xref="S5.E29.m1.6.6.1.1.3.1.2.3.3.3">d</ci></apply></apply></apply><ci id="S5.E29.m1.6.6.1.1.3.1.3.cmml" xref="S5.E29.m1.6.6.1.1.3.1.3">𝑡</ci></apply><apply id="S5.E29.m1.6.6.1.1.3.2.cmml" xref="S5.E29.m1.6.6.1.1.3.2"><times id="S5.E29.m1.6.6.1.1.3.2.1.cmml" xref="S5.E29.m1.6.6.1.1.3.2.1"></times><apply id="S5.E29.m1.6.6.1.1.3.2.2.cmml" xref="S5.E29.m1.6.6.1.1.3.2.2"><csymbol cd="ambiguous" id="S5.E29.m1.6.6.1.1.3.2.2.1.cmml" xref="S5.E29.m1.6.6.1.1.3.2.2">subscript</csymbol><ci id="S5.E29.m1.6.6.1.1.3.2.2.2.cmml" xref="S5.E29.m1.6.6.1.1.3.2.2.2">Φ</ci><ci id="S5.E29.m1.6.6.1.1.3.2.2.3.cmml" xref="S5.E29.m1.6.6.1.1.3.2.2.3">s</ci></apply><ci id="S5.E29.m1.4.4.cmml" xref="S5.E29.m1.4.4">𝜏</ci><ci id="S5.E29.m1.6.6.1.1.3.2.4.cmml" xref="S5.E29.m1.6.6.1.1.3.2.4">𝒑</ci><ci id="S5.E29.m1.5.5.cmml" xref="S5.E29.m1.5.5">𝜏</ci><apply id="S5.E29.m1.6.6.1.1.3.2.6.cmml" xref="S5.E29.m1.6.6.1.1.3.2.6"><csymbol cd="latexml" id="S5.E29.m1.6.6.1.1.3.2.6.1.cmml" xref="S5.E29.m1.6.6.1.1.3.2.6.1">differential-d</csymbol><ci id="S5.E29.m1.6.6.1.1.3.2.6.2.cmml" xref="S5.E29.m1.6.6.1.1.3.2.6.2">𝜏</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E29.m1.6c">\displaystyle\Psi(t)\bm{\theta}+\int_{t-\tau_{\rm d}}^{t}\Phi_{\rm s}(\tau)\bm% {d}_{\rm s}(\tau)d\tau=\int_{t-\tau_{\rm d}}^{t}\Phi_{\rm s}(\tau)\bm{p}(\tau)% d\tau.</annotation><annotation encoding="application/x-llamapun" id="S5.E29.m1.6d">roman_Ψ ( italic_t ) bold_italic_θ + ∫ start_POSTSUBSCRIPT italic_t - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_τ ) bold_italic_d start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_τ ) italic_d italic_τ = ∫ start_POSTSUBSCRIPT italic_t - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_τ ) bold_italic_p ( italic_τ ) italic_d italic_τ .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(29)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.SS3.p1.25">Applying the filter <math alttext="H(s)" class="ltx_Math" display="inline" id="S5.SS3.p1.25.m1.1"><semantics id="S5.SS3.p1.25.m1.1a"><mrow id="S5.SS3.p1.25.m1.1.2" xref="S5.SS3.p1.25.m1.1.2.cmml"><mi id="S5.SS3.p1.25.m1.1.2.2" xref="S5.SS3.p1.25.m1.1.2.2.cmml">H</mi><mo id="S5.SS3.p1.25.m1.1.2.1" xref="S5.SS3.p1.25.m1.1.2.1.cmml">⁢</mo><mrow id="S5.SS3.p1.25.m1.1.2.3.2" xref="S5.SS3.p1.25.m1.1.2.cmml"><mo id="S5.SS3.p1.25.m1.1.2.3.2.1" stretchy="false" xref="S5.SS3.p1.25.m1.1.2.cmml">(</mo><mi id="S5.SS3.p1.25.m1.1.1" xref="S5.SS3.p1.25.m1.1.1.cmml">s</mi><mo id="S5.SS3.p1.25.m1.1.2.3.2.2" stretchy="false" xref="S5.SS3.p1.25.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.25.m1.1b"><apply id="S5.SS3.p1.25.m1.1.2.cmml" xref="S5.SS3.p1.25.m1.1.2"><times id="S5.SS3.p1.25.m1.1.2.1.cmml" xref="S5.SS3.p1.25.m1.1.2.1"></times><ci id="S5.SS3.p1.25.m1.1.2.2.cmml" xref="S5.SS3.p1.25.m1.1.2.2">𝐻</ci><ci id="S5.SS3.p1.25.m1.1.1.cmml" xref="S5.SS3.p1.25.m1.1.1">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.25.m1.1c">H(s)</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.25.m1.1d">italic_H ( italic_s )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E17" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">17</span></a>) to (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S5.E29" title="In V-C Robustness Results ‣ V Theoretical Guarantees ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">29</span></a>) and noting the generalized linearly parameterized model (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E18" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">18</span></a>), one gets</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx25"> <tbody id="S5.E30"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle Q(t)\bm{\theta}+{\bm{d}}_{\rm g}(t)={\bm{q}}_{\rm f}(t)" class="ltx_Math" display="inline" id="S5.E30.m1.3"><semantics id="S5.E30.m1.3a"><mrow id="S5.E30.m1.3.4" xref="S5.E30.m1.3.4.cmml"><mrow id="S5.E30.m1.3.4.2" xref="S5.E30.m1.3.4.2.cmml"><mrow id="S5.E30.m1.3.4.2.2" xref="S5.E30.m1.3.4.2.2.cmml"><mi id="S5.E30.m1.3.4.2.2.2" xref="S5.E30.m1.3.4.2.2.2.cmml">Q</mi><mo id="S5.E30.m1.3.4.2.2.1" xref="S5.E30.m1.3.4.2.2.1.cmml">⁢</mo><mrow id="S5.E30.m1.3.4.2.2.3.2" xref="S5.E30.m1.3.4.2.2.cmml"><mo id="S5.E30.m1.3.4.2.2.3.2.1" stretchy="false" xref="S5.E30.m1.3.4.2.2.cmml">(</mo><mi id="S5.E30.m1.1.1" xref="S5.E30.m1.1.1.cmml">t</mi><mo id="S5.E30.m1.3.4.2.2.3.2.2" stretchy="false" xref="S5.E30.m1.3.4.2.2.cmml">)</mo></mrow><mo id="S5.E30.m1.3.4.2.2.1a" xref="S5.E30.m1.3.4.2.2.1.cmml">⁢</mo><mi id="S5.E30.m1.3.4.2.2.4" xref="S5.E30.m1.3.4.2.2.4.cmml">𝜽</mi></mrow><mo id="S5.E30.m1.3.4.2.1" xref="S5.E30.m1.3.4.2.1.cmml">+</mo><mrow id="S5.E30.m1.3.4.2.3" xref="S5.E30.m1.3.4.2.3.cmml"><msub id="S5.E30.m1.3.4.2.3.2" xref="S5.E30.m1.3.4.2.3.2.cmml"><mi id="S5.E30.m1.3.4.2.3.2.2" xref="S5.E30.m1.3.4.2.3.2.2.cmml">𝒅</mi><mi id="S5.E30.m1.3.4.2.3.2.3" mathvariant="normal" xref="S5.E30.m1.3.4.2.3.2.3.cmml">g</mi></msub><mo id="S5.E30.m1.3.4.2.3.1" xref="S5.E30.m1.3.4.2.3.1.cmml">⁢</mo><mrow id="S5.E30.m1.3.4.2.3.3.2" xref="S5.E30.m1.3.4.2.3.cmml"><mo id="S5.E30.m1.3.4.2.3.3.2.1" stretchy="false" xref="S5.E30.m1.3.4.2.3.cmml">(</mo><mi id="S5.E30.m1.2.2" xref="S5.E30.m1.2.2.cmml">t</mi><mo id="S5.E30.m1.3.4.2.3.3.2.2" stretchy="false" xref="S5.E30.m1.3.4.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S5.E30.m1.3.4.1" xref="S5.E30.m1.3.4.1.cmml">=</mo><mrow id="S5.E30.m1.3.4.3" xref="S5.E30.m1.3.4.3.cmml"><msub id="S5.E30.m1.3.4.3.2" xref="S5.E30.m1.3.4.3.2.cmml"><mi id="S5.E30.m1.3.4.3.2.2" xref="S5.E30.m1.3.4.3.2.2.cmml">𝒒</mi><mi id="S5.E30.m1.3.4.3.2.3" mathvariant="normal" xref="S5.E30.m1.3.4.3.2.3.cmml">f</mi></msub><mo id="S5.E30.m1.3.4.3.1" xref="S5.E30.m1.3.4.3.1.cmml">⁢</mo><mrow id="S5.E30.m1.3.4.3.3.2" xref="S5.E30.m1.3.4.3.cmml"><mo id="S5.E30.m1.3.4.3.3.2.1" stretchy="false" xref="S5.E30.m1.3.4.3.cmml">(</mo><mi id="S5.E30.m1.3.3" xref="S5.E30.m1.3.3.cmml">t</mi><mo id="S5.E30.m1.3.4.3.3.2.2" stretchy="false" xref="S5.E30.m1.3.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.E30.m1.3b"><apply id="S5.E30.m1.3.4.cmml" xref="S5.E30.m1.3.4"><eq id="S5.E30.m1.3.4.1.cmml" xref="S5.E30.m1.3.4.1"></eq><apply id="S5.E30.m1.3.4.2.cmml" xref="S5.E30.m1.3.4.2"><plus id="S5.E30.m1.3.4.2.1.cmml" xref="S5.E30.m1.3.4.2.1"></plus><apply id="S5.E30.m1.3.4.2.2.cmml" xref="S5.E30.m1.3.4.2.2"><times id="S5.E30.m1.3.4.2.2.1.cmml" xref="S5.E30.m1.3.4.2.2.1"></times><ci id="S5.E30.m1.3.4.2.2.2.cmml" xref="S5.E30.m1.3.4.2.2.2">𝑄</ci><ci id="S5.E30.m1.1.1.cmml" xref="S5.E30.m1.1.1">𝑡</ci><ci id="S5.E30.m1.3.4.2.2.4.cmml" xref="S5.E30.m1.3.4.2.2.4">𝜽</ci></apply><apply id="S5.E30.m1.3.4.2.3.cmml" xref="S5.E30.m1.3.4.2.3"><times id="S5.E30.m1.3.4.2.3.1.cmml" xref="S5.E30.m1.3.4.2.3.1"></times><apply id="S5.E30.m1.3.4.2.3.2.cmml" xref="S5.E30.m1.3.4.2.3.2"><csymbol cd="ambiguous" id="S5.E30.m1.3.4.2.3.2.1.cmml" xref="S5.E30.m1.3.4.2.3.2">subscript</csymbol><ci id="S5.E30.m1.3.4.2.3.2.2.cmml" xref="S5.E30.m1.3.4.2.3.2.2">𝒅</ci><ci id="S5.E30.m1.3.4.2.3.2.3.cmml" xref="S5.E30.m1.3.4.2.3.2.3">g</ci></apply><ci id="S5.E30.m1.2.2.cmml" xref="S5.E30.m1.2.2">𝑡</ci></apply></apply><apply id="S5.E30.m1.3.4.3.cmml" xref="S5.E30.m1.3.4.3"><times id="S5.E30.m1.3.4.3.1.cmml" xref="S5.E30.m1.3.4.3.1"></times><apply id="S5.E30.m1.3.4.3.2.cmml" xref="S5.E30.m1.3.4.3.2"><csymbol cd="ambiguous" id="S5.E30.m1.3.4.3.2.1.cmml" xref="S5.E30.m1.3.4.3.2">subscript</csymbol><ci id="S5.E30.m1.3.4.3.2.2.cmml" xref="S5.E30.m1.3.4.3.2.2">𝒒</ci><ci id="S5.E30.m1.3.4.3.2.3.cmml" xref="S5.E30.m1.3.4.3.2.3">f</ci></apply><ci id="S5.E30.m1.3.3.cmml" xref="S5.E30.m1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E30.m1.3c">\displaystyle Q(t)\bm{\theta}+{\bm{d}}_{\rm g}(t)={\bm{q}}_{\rm f}(t)</annotation><annotation encoding="application/x-llamapun" id="S5.E30.m1.3d">italic_Q ( italic_t ) bold_italic_θ + bold_italic_d start_POSTSUBSCRIPT roman_g end_POSTSUBSCRIPT ( italic_t ) = bold_italic_q start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(30)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.SS3.p1.46">with <math alttext="{\bm{d}}_{\rm g}(t)" class="ltx_Math" display="inline" id="S5.SS3.p1.26.m1.1"><semantics id="S5.SS3.p1.26.m1.1a"><mrow id="S5.SS3.p1.26.m1.1.2" xref="S5.SS3.p1.26.m1.1.2.cmml"><msub id="S5.SS3.p1.26.m1.1.2.2" xref="S5.SS3.p1.26.m1.1.2.2.cmml"><mi id="S5.SS3.p1.26.m1.1.2.2.2" xref="S5.SS3.p1.26.m1.1.2.2.2.cmml">𝒅</mi><mi id="S5.SS3.p1.26.m1.1.2.2.3" mathvariant="normal" xref="S5.SS3.p1.26.m1.1.2.2.3.cmml">g</mi></msub><mo id="S5.SS3.p1.26.m1.1.2.1" xref="S5.SS3.p1.26.m1.1.2.1.cmml">⁢</mo><mrow id="S5.SS3.p1.26.m1.1.2.3.2" xref="S5.SS3.p1.26.m1.1.2.cmml"><mo id="S5.SS3.p1.26.m1.1.2.3.2.1" stretchy="false" xref="S5.SS3.p1.26.m1.1.2.cmml">(</mo><mi id="S5.SS3.p1.26.m1.1.1" xref="S5.SS3.p1.26.m1.1.1.cmml">t</mi><mo id="S5.SS3.p1.26.m1.1.2.3.2.2" stretchy="false" xref="S5.SS3.p1.26.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.26.m1.1b"><apply id="S5.SS3.p1.26.m1.1.2.cmml" xref="S5.SS3.p1.26.m1.1.2"><times id="S5.SS3.p1.26.m1.1.2.1.cmml" xref="S5.SS3.p1.26.m1.1.2.1"></times><apply id="S5.SS3.p1.26.m1.1.2.2.cmml" xref="S5.SS3.p1.26.m1.1.2.2"><csymbol cd="ambiguous" id="S5.SS3.p1.26.m1.1.2.2.1.cmml" xref="S5.SS3.p1.26.m1.1.2.2">subscript</csymbol><ci id="S5.SS3.p1.26.m1.1.2.2.2.cmml" xref="S5.SS3.p1.26.m1.1.2.2.2">𝒅</ci><ci id="S5.SS3.p1.26.m1.1.2.2.3.cmml" xref="S5.SS3.p1.26.m1.1.2.2.3">g</ci></apply><ci id="S5.SS3.p1.26.m1.1.1.cmml" xref="S5.SS3.p1.26.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.26.m1.1c">{\bm{d}}_{\rm g}(t)</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.26.m1.1d">bold_italic_d start_POSTSUBSCRIPT roman_g end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S5.SS3.p1.27.m2.1"><semantics id="S5.SS3.p1.27.m2.1a"><mo id="S5.SS3.p1.27.m2.1.1" xref="S5.SS3.p1.27.m2.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.27.m2.1b"><csymbol cd="latexml" id="S5.SS3.p1.27.m2.1.1.cmml" xref="S5.SS3.p1.27.m2.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.27.m2.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.27.m2.1d">:=</annotation></semantics></math> <math alttext="H(s)[\int_{t-\tau_{\rm d}}^{t}\Phi_{\rm s}(\tau)\bm{d}_{\rm s}(\tau)d\tau]\in% \mathbb{R}^{N}" class="ltx_Math" display="inline" id="S5.SS3.p1.28.m3.4"><semantics id="S5.SS3.p1.28.m3.4a"><mrow id="S5.SS3.p1.28.m3.4.4" xref="S5.SS3.p1.28.m3.4.4.cmml"><mrow id="S5.SS3.p1.28.m3.4.4.1" xref="S5.SS3.p1.28.m3.4.4.1.cmml"><mi id="S5.SS3.p1.28.m3.4.4.1.3" xref="S5.SS3.p1.28.m3.4.4.1.3.cmml">H</mi><mo id="S5.SS3.p1.28.m3.4.4.1.2" xref="S5.SS3.p1.28.m3.4.4.1.2.cmml">⁢</mo><mrow id="S5.SS3.p1.28.m3.4.4.1.4.2" xref="S5.SS3.p1.28.m3.4.4.1.cmml"><mo id="S5.SS3.p1.28.m3.4.4.1.4.2.1" stretchy="false" xref="S5.SS3.p1.28.m3.4.4.1.cmml">(</mo><mi id="S5.SS3.p1.28.m3.1.1" xref="S5.SS3.p1.28.m3.1.1.cmml">s</mi><mo id="S5.SS3.p1.28.m3.4.4.1.4.2.2" stretchy="false" xref="S5.SS3.p1.28.m3.4.4.1.cmml">)</mo></mrow><mo id="S5.SS3.p1.28.m3.4.4.1.2a" xref="S5.SS3.p1.28.m3.4.4.1.2.cmml">⁢</mo><mrow id="S5.SS3.p1.28.m3.4.4.1.1.1" xref="S5.SS3.p1.28.m3.4.4.1.1.2.cmml"><mo id="S5.SS3.p1.28.m3.4.4.1.1.1.2" stretchy="false" xref="S5.SS3.p1.28.m3.4.4.1.1.2.1.cmml">[</mo><mrow id="S5.SS3.p1.28.m3.4.4.1.1.1.1" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.cmml"><msubsup id="S5.SS3.p1.28.m3.4.4.1.1.1.1.1" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.cmml"><mo id="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.2" lspace="0em" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.2.cmml">∫</mo><mrow id="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3.cmml"><mi id="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3.2" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3.2.cmml">t</mi><mo id="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3.1" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3.1.cmml">−</mo><msub id="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3.3" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3.3.cmml"><mi id="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3.3.2" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3.3.2.cmml">τ</mi><mi id="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3.3.3" mathvariant="normal" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3.3.3.cmml">d</mi></msub></mrow><mi id="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.3" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.3.cmml">t</mi></msubsup><mrow id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.cmml"><msub id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.2" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.2.cmml"><mi id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.2.2" mathvariant="normal" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.2.2.cmml">Φ</mi><mi id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.2.3" mathvariant="normal" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.2.3.cmml">s</mi></msub><mo id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.1" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.3.2" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.cmml"><mo id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.3.2.1" stretchy="false" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.cmml">(</mo><mi id="S5.SS3.p1.28.m3.2.2" xref="S5.SS3.p1.28.m3.2.2.cmml">τ</mi><mo id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.3.2.2" stretchy="false" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.cmml">)</mo></mrow><mo id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.1a" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.1.cmml">⁢</mo><msub id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.4" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.4.cmml"><mi id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.4.2" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.4.2.cmml">𝒅</mi><mi id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.4.3" mathvariant="normal" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.4.3.cmml">s</mi></msub><mo id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.1b" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.5.2" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.cmml"><mo id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.5.2.1" stretchy="false" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.cmml">(</mo><mi id="S5.SS3.p1.28.m3.3.3" xref="S5.SS3.p1.28.m3.3.3.cmml">τ</mi><mo id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.5.2.2" stretchy="false" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.cmml">)</mo></mrow><mo id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.1c" lspace="0em" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.6" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.6.cmml"><mo id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.6.1" rspace="0em" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.6.1.cmml">𝑑</mo><mi id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.6.2" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.6.2.cmml">τ</mi></mrow></mrow></mrow><mo id="S5.SS3.p1.28.m3.4.4.1.1.1.3" stretchy="false" xref="S5.SS3.p1.28.m3.4.4.1.1.2.1.cmml">]</mo></mrow></mrow><mo id="S5.SS3.p1.28.m3.4.4.2" xref="S5.SS3.p1.28.m3.4.4.2.cmml">∈</mo><msup id="S5.SS3.p1.28.m3.4.4.3" xref="S5.SS3.p1.28.m3.4.4.3.cmml"><mi id="S5.SS3.p1.28.m3.4.4.3.2" xref="S5.SS3.p1.28.m3.4.4.3.2.cmml">ℝ</mi><mi id="S5.SS3.p1.28.m3.4.4.3.3" xref="S5.SS3.p1.28.m3.4.4.3.3.cmml">N</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.28.m3.4b"><apply id="S5.SS3.p1.28.m3.4.4.cmml" xref="S5.SS3.p1.28.m3.4.4"><in id="S5.SS3.p1.28.m3.4.4.2.cmml" xref="S5.SS3.p1.28.m3.4.4.2"></in><apply id="S5.SS3.p1.28.m3.4.4.1.cmml" xref="S5.SS3.p1.28.m3.4.4.1"><times id="S5.SS3.p1.28.m3.4.4.1.2.cmml" xref="S5.SS3.p1.28.m3.4.4.1.2"></times><ci id="S5.SS3.p1.28.m3.4.4.1.3.cmml" xref="S5.SS3.p1.28.m3.4.4.1.3">𝐻</ci><ci id="S5.SS3.p1.28.m3.1.1.cmml" xref="S5.SS3.p1.28.m3.1.1">𝑠</ci><apply id="S5.SS3.p1.28.m3.4.4.1.1.2.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1"><csymbol cd="latexml" id="S5.SS3.p1.28.m3.4.4.1.1.2.1.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.2">delimited-[]</csymbol><apply id="S5.SS3.p1.28.m3.4.4.1.1.1.1.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1"><apply id="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.1.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.1">superscript</csymbol><apply id="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.1.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.1">subscript</csymbol><int id="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.2.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.2"></int><apply id="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3"><minus id="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3.1.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3.1"></minus><ci id="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3.2.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3.2">𝑡</ci><apply id="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3.3.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3.3"><csymbol cd="ambiguous" id="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3.3.1.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3.3">subscript</csymbol><ci id="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3.3.2.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3.3.2">𝜏</ci><ci id="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3.3.3.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.2.3.3.3">d</ci></apply></apply></apply><ci id="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.3.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.1.3">𝑡</ci></apply><apply id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2"><times id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.1.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.1"></times><apply id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.2.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.2.1.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.2">subscript</csymbol><ci id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.2.2.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.2.2">Φ</ci><ci id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.2.3.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.2.3">s</ci></apply><ci id="S5.SS3.p1.28.m3.2.2.cmml" xref="S5.SS3.p1.28.m3.2.2">𝜏</ci><apply id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.4.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.4"><csymbol cd="ambiguous" id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.4.1.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.4">subscript</csymbol><ci id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.4.2.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.4.2">𝒅</ci><ci id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.4.3.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.4.3">s</ci></apply><ci id="S5.SS3.p1.28.m3.3.3.cmml" xref="S5.SS3.p1.28.m3.3.3">𝜏</ci><apply id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.6.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.6"><csymbol cd="latexml" id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.6.1.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.6.1">differential-d</csymbol><ci id="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.6.2.cmml" xref="S5.SS3.p1.28.m3.4.4.1.1.1.1.2.6.2">𝜏</ci></apply></apply></apply></apply></apply><apply id="S5.SS3.p1.28.m3.4.4.3.cmml" xref="S5.SS3.p1.28.m3.4.4.3"><csymbol cd="ambiguous" id="S5.SS3.p1.28.m3.4.4.3.1.cmml" xref="S5.SS3.p1.28.m3.4.4.3">superscript</csymbol><ci id="S5.SS3.p1.28.m3.4.4.3.2.cmml" xref="S5.SS3.p1.28.m3.4.4.3.2">ℝ</ci><ci id="S5.SS3.p1.28.m3.4.4.3.3.cmml" xref="S5.SS3.p1.28.m3.4.4.3.3">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.28.m3.4c">H(s)[\int_{t-\tau_{\rm d}}^{t}\Phi_{\rm s}(\tau)\bm{d}_{\rm s}(\tau)d\tau]\in% \mathbb{R}^{N}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.28.m3.4d">italic_H ( italic_s ) [ ∫ start_POSTSUBSCRIPT italic_t - italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_τ ) bold_italic_d start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_τ ) italic_d italic_τ ] ∈ blackboard_R start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT</annotation></semantics></math>. Let <math alttext="\Omega_{{\rm c}_{x}}" class="ltx_Math" display="inline" id="S5.SS3.p1.29.m4.1"><semantics id="S5.SS3.p1.29.m4.1a"><msub id="S5.SS3.p1.29.m4.1.1" xref="S5.SS3.p1.29.m4.1.1.cmml"><mi id="S5.SS3.p1.29.m4.1.1.2" mathvariant="normal" xref="S5.SS3.p1.29.m4.1.1.2.cmml">Ω</mi><msub id="S5.SS3.p1.29.m4.1.1.3" xref="S5.SS3.p1.29.m4.1.1.3.cmml"><mi id="S5.SS3.p1.29.m4.1.1.3.2" mathvariant="normal" xref="S5.SS3.p1.29.m4.1.1.3.2.cmml">c</mi><mi id="S5.SS3.p1.29.m4.1.1.3.3" xref="S5.SS3.p1.29.m4.1.1.3.3.cmml">x</mi></msub></msub><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.29.m4.1b"><apply id="S5.SS3.p1.29.m4.1.1.cmml" xref="S5.SS3.p1.29.m4.1.1"><csymbol cd="ambiguous" id="S5.SS3.p1.29.m4.1.1.1.cmml" xref="S5.SS3.p1.29.m4.1.1">subscript</csymbol><ci id="S5.SS3.p1.29.m4.1.1.2.cmml" xref="S5.SS3.p1.29.m4.1.1.2">Ω</ci><apply id="S5.SS3.p1.29.m4.1.1.3.cmml" xref="S5.SS3.p1.29.m4.1.1.3"><csymbol cd="ambiguous" id="S5.SS3.p1.29.m4.1.1.3.1.cmml" xref="S5.SS3.p1.29.m4.1.1.3">subscript</csymbol><ci id="S5.SS3.p1.29.m4.1.1.3.2.cmml" xref="S5.SS3.p1.29.m4.1.1.3.2">c</ci><ci id="S5.SS3.p1.29.m4.1.1.3.3.cmml" xref="S5.SS3.p1.29.m4.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.29.m4.1c">\Omega_{{\rm c}_{x}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.29.m4.1d">roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="{\rm c}_{x}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S5.SS3.p1.30.m5.1"><semantics id="S5.SS3.p1.30.m5.1a"><mrow id="S5.SS3.p1.30.m5.1.1" xref="S5.SS3.p1.30.m5.1.1.cmml"><msub id="S5.SS3.p1.30.m5.1.1.2" xref="S5.SS3.p1.30.m5.1.1.2.cmml"><mi id="S5.SS3.p1.30.m5.1.1.2.2" mathvariant="normal" xref="S5.SS3.p1.30.m5.1.1.2.2.cmml">c</mi><mi id="S5.SS3.p1.30.m5.1.1.2.3" xref="S5.SS3.p1.30.m5.1.1.2.3.cmml">x</mi></msub><mo id="S5.SS3.p1.30.m5.1.1.1" xref="S5.SS3.p1.30.m5.1.1.1.cmml">∈</mo><msup id="S5.SS3.p1.30.m5.1.1.3" xref="S5.SS3.p1.30.m5.1.1.3.cmml"><mi id="S5.SS3.p1.30.m5.1.1.3.2" xref="S5.SS3.p1.30.m5.1.1.3.2.cmml">ℝ</mi><mo id="S5.SS3.p1.30.m5.1.1.3.3" xref="S5.SS3.p1.30.m5.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.30.m5.1b"><apply id="S5.SS3.p1.30.m5.1.1.cmml" xref="S5.SS3.p1.30.m5.1.1"><in id="S5.SS3.p1.30.m5.1.1.1.cmml" xref="S5.SS3.p1.30.m5.1.1.1"></in><apply id="S5.SS3.p1.30.m5.1.1.2.cmml" xref="S5.SS3.p1.30.m5.1.1.2"><csymbol cd="ambiguous" id="S5.SS3.p1.30.m5.1.1.2.1.cmml" xref="S5.SS3.p1.30.m5.1.1.2">subscript</csymbol><ci id="S5.SS3.p1.30.m5.1.1.2.2.cmml" xref="S5.SS3.p1.30.m5.1.1.2.2">c</ci><ci id="S5.SS3.p1.30.m5.1.1.2.3.cmml" xref="S5.SS3.p1.30.m5.1.1.2.3">𝑥</ci></apply><apply id="S5.SS3.p1.30.m5.1.1.3.cmml" xref="S5.SS3.p1.30.m5.1.1.3"><csymbol cd="ambiguous" id="S5.SS3.p1.30.m5.1.1.3.1.cmml" xref="S5.SS3.p1.30.m5.1.1.3">superscript</csymbol><ci id="S5.SS3.p1.30.m5.1.1.3.2.cmml" xref="S5.SS3.p1.30.m5.1.1.3.2">ℝ</ci><plus id="S5.SS3.p1.30.m5.1.1.3.3.cmml" xref="S5.SS3.p1.30.m5.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.30.m5.1c">{\rm c}_{x}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.30.m5.1d">roman_c start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> be a compact set of <math alttext="\bm{x}" class="ltx_Math" display="inline" id="S5.SS3.p1.31.m6.1"><semantics id="S5.SS3.p1.31.m6.1a"><mi id="S5.SS3.p1.31.m6.1.1" xref="S5.SS3.p1.31.m6.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.31.m6.1b"><ci id="S5.SS3.p1.31.m6.1.1.cmml" xref="S5.SS3.p1.31.m6.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.31.m6.1c">\bm{x}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.31.m6.1d">bold_italic_x</annotation></semantics></math>. Then, there exists a constant <math alttext="\phi\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S5.SS3.p1.32.m7.1"><semantics id="S5.SS3.p1.32.m7.1a"><mrow id="S5.SS3.p1.32.m7.1.1" xref="S5.SS3.p1.32.m7.1.1.cmml"><mi id="S5.SS3.p1.32.m7.1.1.2" xref="S5.SS3.p1.32.m7.1.1.2.cmml">ϕ</mi><mo id="S5.SS3.p1.32.m7.1.1.1" xref="S5.SS3.p1.32.m7.1.1.1.cmml">∈</mo><msup id="S5.SS3.p1.32.m7.1.1.3" xref="S5.SS3.p1.32.m7.1.1.3.cmml"><mi id="S5.SS3.p1.32.m7.1.1.3.2" xref="S5.SS3.p1.32.m7.1.1.3.2.cmml">ℝ</mi><mo id="S5.SS3.p1.32.m7.1.1.3.3" xref="S5.SS3.p1.32.m7.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.32.m7.1b"><apply id="S5.SS3.p1.32.m7.1.1.cmml" xref="S5.SS3.p1.32.m7.1.1"><in id="S5.SS3.p1.32.m7.1.1.1.cmml" xref="S5.SS3.p1.32.m7.1.1.1"></in><ci id="S5.SS3.p1.32.m7.1.1.2.cmml" xref="S5.SS3.p1.32.m7.1.1.2">italic-ϕ</ci><apply id="S5.SS3.p1.32.m7.1.1.3.cmml" xref="S5.SS3.p1.32.m7.1.1.3"><csymbol cd="ambiguous" id="S5.SS3.p1.32.m7.1.1.3.1.cmml" xref="S5.SS3.p1.32.m7.1.1.3">superscript</csymbol><ci id="S5.SS3.p1.32.m7.1.1.3.2.cmml" xref="S5.SS3.p1.32.m7.1.1.3.2">ℝ</ci><plus id="S5.SS3.p1.32.m7.1.1.3.3.cmml" xref="S5.SS3.p1.32.m7.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.32.m7.1c">\phi\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.32.m7.1d">italic_ϕ ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="\|\Phi_{\rm s}(t)\|\leq\phi" class="ltx_Math" display="inline" id="S5.SS3.p1.33.m8.2"><semantics id="S5.SS3.p1.33.m8.2a"><mrow id="S5.SS3.p1.33.m8.2.2" xref="S5.SS3.p1.33.m8.2.2.cmml"><mrow id="S5.SS3.p1.33.m8.2.2.1.1" xref="S5.SS3.p1.33.m8.2.2.1.2.cmml"><mo id="S5.SS3.p1.33.m8.2.2.1.1.2" stretchy="false" xref="S5.SS3.p1.33.m8.2.2.1.2.1.cmml">‖</mo><mrow id="S5.SS3.p1.33.m8.2.2.1.1.1" xref="S5.SS3.p1.33.m8.2.2.1.1.1.cmml"><msub id="S5.SS3.p1.33.m8.2.2.1.1.1.2" xref="S5.SS3.p1.33.m8.2.2.1.1.1.2.cmml"><mi id="S5.SS3.p1.33.m8.2.2.1.1.1.2.2" mathvariant="normal" xref="S5.SS3.p1.33.m8.2.2.1.1.1.2.2.cmml">Φ</mi><mi id="S5.SS3.p1.33.m8.2.2.1.1.1.2.3" mathvariant="normal" xref="S5.SS3.p1.33.m8.2.2.1.1.1.2.3.cmml">s</mi></msub><mo id="S5.SS3.p1.33.m8.2.2.1.1.1.1" xref="S5.SS3.p1.33.m8.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S5.SS3.p1.33.m8.2.2.1.1.1.3.2" xref="S5.SS3.p1.33.m8.2.2.1.1.1.cmml"><mo id="S5.SS3.p1.33.m8.2.2.1.1.1.3.2.1" stretchy="false" xref="S5.SS3.p1.33.m8.2.2.1.1.1.cmml">(</mo><mi id="S5.SS3.p1.33.m8.1.1" xref="S5.SS3.p1.33.m8.1.1.cmml">t</mi><mo id="S5.SS3.p1.33.m8.2.2.1.1.1.3.2.2" stretchy="false" xref="S5.SS3.p1.33.m8.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.SS3.p1.33.m8.2.2.1.1.3" stretchy="false" xref="S5.SS3.p1.33.m8.2.2.1.2.1.cmml">‖</mo></mrow><mo id="S5.SS3.p1.33.m8.2.2.2" xref="S5.SS3.p1.33.m8.2.2.2.cmml">≤</mo><mi id="S5.SS3.p1.33.m8.2.2.3" xref="S5.SS3.p1.33.m8.2.2.3.cmml">ϕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.33.m8.2b"><apply id="S5.SS3.p1.33.m8.2.2.cmml" xref="S5.SS3.p1.33.m8.2.2"><leq id="S5.SS3.p1.33.m8.2.2.2.cmml" xref="S5.SS3.p1.33.m8.2.2.2"></leq><apply id="S5.SS3.p1.33.m8.2.2.1.2.cmml" xref="S5.SS3.p1.33.m8.2.2.1.1"><csymbol cd="latexml" id="S5.SS3.p1.33.m8.2.2.1.2.1.cmml" xref="S5.SS3.p1.33.m8.2.2.1.1.2">norm</csymbol><apply id="S5.SS3.p1.33.m8.2.2.1.1.1.cmml" xref="S5.SS3.p1.33.m8.2.2.1.1.1"><times id="S5.SS3.p1.33.m8.2.2.1.1.1.1.cmml" xref="S5.SS3.p1.33.m8.2.2.1.1.1.1"></times><apply id="S5.SS3.p1.33.m8.2.2.1.1.1.2.cmml" xref="S5.SS3.p1.33.m8.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S5.SS3.p1.33.m8.2.2.1.1.1.2.1.cmml" xref="S5.SS3.p1.33.m8.2.2.1.1.1.2">subscript</csymbol><ci id="S5.SS3.p1.33.m8.2.2.1.1.1.2.2.cmml" xref="S5.SS3.p1.33.m8.2.2.1.1.1.2.2">Φ</ci><ci id="S5.SS3.p1.33.m8.2.2.1.1.1.2.3.cmml" xref="S5.SS3.p1.33.m8.2.2.1.1.1.2.3">s</ci></apply><ci id="S5.SS3.p1.33.m8.1.1.cmml" xref="S5.SS3.p1.33.m8.1.1">𝑡</ci></apply></apply><ci id="S5.SS3.p1.33.m8.2.2.3.cmml" xref="S5.SS3.p1.33.m8.2.2.3">italic-ϕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.33.m8.2c">\|\Phi_{\rm s}(t)\|\leq\phi</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.33.m8.2d">∥ roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_t ) ∥ ≤ italic_ϕ</annotation></semantics></math>, <math alttext="\forall\bm{x}\in\Omega_{{\rm c}_{x}}" class="ltx_Math" display="inline" id="S5.SS3.p1.34.m9.1"><semantics id="S5.SS3.p1.34.m9.1a"><mrow id="S5.SS3.p1.34.m9.1.1" xref="S5.SS3.p1.34.m9.1.1.cmml"><mrow id="S5.SS3.p1.34.m9.1.1.2" xref="S5.SS3.p1.34.m9.1.1.2.cmml"><mo id="S5.SS3.p1.34.m9.1.1.2.1" rspace="0.167em" xref="S5.SS3.p1.34.m9.1.1.2.1.cmml">∀</mo><mi id="S5.SS3.p1.34.m9.1.1.2.2" xref="S5.SS3.p1.34.m9.1.1.2.2.cmml">𝒙</mi></mrow><mo id="S5.SS3.p1.34.m9.1.1.1" xref="S5.SS3.p1.34.m9.1.1.1.cmml">∈</mo><msub id="S5.SS3.p1.34.m9.1.1.3" xref="S5.SS3.p1.34.m9.1.1.3.cmml"><mi id="S5.SS3.p1.34.m9.1.1.3.2" mathvariant="normal" xref="S5.SS3.p1.34.m9.1.1.3.2.cmml">Ω</mi><msub id="S5.SS3.p1.34.m9.1.1.3.3" xref="S5.SS3.p1.34.m9.1.1.3.3.cmml"><mi id="S5.SS3.p1.34.m9.1.1.3.3.2" mathvariant="normal" xref="S5.SS3.p1.34.m9.1.1.3.3.2.cmml">c</mi><mi id="S5.SS3.p1.34.m9.1.1.3.3.3" xref="S5.SS3.p1.34.m9.1.1.3.3.3.cmml">x</mi></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.34.m9.1b"><apply id="S5.SS3.p1.34.m9.1.1.cmml" xref="S5.SS3.p1.34.m9.1.1"><in id="S5.SS3.p1.34.m9.1.1.1.cmml" xref="S5.SS3.p1.34.m9.1.1.1"></in><apply id="S5.SS3.p1.34.m9.1.1.2.cmml" xref="S5.SS3.p1.34.m9.1.1.2"><csymbol cd="latexml" id="S5.SS3.p1.34.m9.1.1.2.1.cmml" xref="S5.SS3.p1.34.m9.1.1.2.1">for-all</csymbol><ci id="S5.SS3.p1.34.m9.1.1.2.2.cmml" xref="S5.SS3.p1.34.m9.1.1.2.2">𝒙</ci></apply><apply id="S5.SS3.p1.34.m9.1.1.3.cmml" xref="S5.SS3.p1.34.m9.1.1.3"><csymbol cd="ambiguous" id="S5.SS3.p1.34.m9.1.1.3.1.cmml" xref="S5.SS3.p1.34.m9.1.1.3">subscript</csymbol><ci id="S5.SS3.p1.34.m9.1.1.3.2.cmml" xref="S5.SS3.p1.34.m9.1.1.3.2">Ω</ci><apply id="S5.SS3.p1.34.m9.1.1.3.3.cmml" xref="S5.SS3.p1.34.m9.1.1.3.3"><csymbol cd="ambiguous" id="S5.SS3.p1.34.m9.1.1.3.3.1.cmml" xref="S5.SS3.p1.34.m9.1.1.3.3">subscript</csymbol><ci id="S5.SS3.p1.34.m9.1.1.3.3.2.cmml" xref="S5.SS3.p1.34.m9.1.1.3.3.2">c</ci><ci id="S5.SS3.p1.34.m9.1.1.3.3.3.cmml" xref="S5.SS3.p1.34.m9.1.1.3.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.34.m9.1c">\forall\bm{x}\in\Omega_{{\rm c}_{x}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.34.m9.1d">∀ bold_italic_x ∈ roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Noting that <math alttext="H(s)" class="ltx_Math" display="inline" id="S5.SS3.p1.35.m10.1"><semantics id="S5.SS3.p1.35.m10.1a"><mrow id="S5.SS3.p1.35.m10.1.2" xref="S5.SS3.p1.35.m10.1.2.cmml"><mi id="S5.SS3.p1.35.m10.1.2.2" xref="S5.SS3.p1.35.m10.1.2.2.cmml">H</mi><mo id="S5.SS3.p1.35.m10.1.2.1" xref="S5.SS3.p1.35.m10.1.2.1.cmml">⁢</mo><mrow id="S5.SS3.p1.35.m10.1.2.3.2" xref="S5.SS3.p1.35.m10.1.2.cmml"><mo id="S5.SS3.p1.35.m10.1.2.3.2.1" stretchy="false" xref="S5.SS3.p1.35.m10.1.2.cmml">(</mo><mi id="S5.SS3.p1.35.m10.1.1" xref="S5.SS3.p1.35.m10.1.1.cmml">s</mi><mo id="S5.SS3.p1.35.m10.1.2.3.2.2" stretchy="false" xref="S5.SS3.p1.35.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.35.m10.1b"><apply id="S5.SS3.p1.35.m10.1.2.cmml" xref="S5.SS3.p1.35.m10.1.2"><times id="S5.SS3.p1.35.m10.1.2.1.cmml" xref="S5.SS3.p1.35.m10.1.2.1"></times><ci id="S5.SS3.p1.35.m10.1.2.2.cmml" xref="S5.SS3.p1.35.m10.1.2.2">𝐻</ci><ci id="S5.SS3.p1.35.m10.1.1.cmml" xref="S5.SS3.p1.35.m10.1.1">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.35.m10.1c">H(s)</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.35.m10.1d">italic_H ( italic_s )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E17" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">17</span></a>) is a stable filter with unit DC gain and combining the above results with <math alttext="\|\bm{d}_{\rm s}\|" class="ltx_Math" display="inline" id="S5.SS3.p1.36.m11.1"><semantics id="S5.SS3.p1.36.m11.1a"><mrow id="S5.SS3.p1.36.m11.1.1.1" xref="S5.SS3.p1.36.m11.1.1.2.cmml"><mo id="S5.SS3.p1.36.m11.1.1.1.2" stretchy="false" xref="S5.SS3.p1.36.m11.1.1.2.1.cmml">‖</mo><msub id="S5.SS3.p1.36.m11.1.1.1.1" xref="S5.SS3.p1.36.m11.1.1.1.1.cmml"><mi id="S5.SS3.p1.36.m11.1.1.1.1.2" xref="S5.SS3.p1.36.m11.1.1.1.1.2.cmml">𝒅</mi><mi id="S5.SS3.p1.36.m11.1.1.1.1.3" mathvariant="normal" xref="S5.SS3.p1.36.m11.1.1.1.1.3.cmml">s</mi></msub><mo id="S5.SS3.p1.36.m11.1.1.1.3" stretchy="false" xref="S5.SS3.p1.36.m11.1.1.2.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.36.m11.1b"><apply id="S5.SS3.p1.36.m11.1.1.2.cmml" xref="S5.SS3.p1.36.m11.1.1.1"><csymbol cd="latexml" id="S5.SS3.p1.36.m11.1.1.2.1.cmml" xref="S5.SS3.p1.36.m11.1.1.1.2">norm</csymbol><apply id="S5.SS3.p1.36.m11.1.1.1.1.cmml" xref="S5.SS3.p1.36.m11.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.p1.36.m11.1.1.1.1.1.cmml" xref="S5.SS3.p1.36.m11.1.1.1.1">subscript</csymbol><ci id="S5.SS3.p1.36.m11.1.1.1.1.2.cmml" xref="S5.SS3.p1.36.m11.1.1.1.1.2">𝒅</ci><ci id="S5.SS3.p1.36.m11.1.1.1.1.3.cmml" xref="S5.SS3.p1.36.m11.1.1.1.1.3">s</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.36.m11.1c">\|\bm{d}_{\rm s}\|</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.36.m11.1d">∥ bold_italic_d start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ∥</annotation></semantics></math> <math alttext="\leq" class="ltx_Math" display="inline" id="S5.SS3.p1.37.m12.1"><semantics id="S5.SS3.p1.37.m12.1a"><mo id="S5.SS3.p1.37.m12.1.1" xref="S5.SS3.p1.37.m12.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.37.m12.1b"><leq id="S5.SS3.p1.37.m12.1.1.cmml" xref="S5.SS3.p1.37.m12.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.37.m12.1c">\leq</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.37.m12.1d">≤</annotation></semantics></math> <math alttext="\bar{d}/\sqrt{k_{\min}}" class="ltx_Math" display="inline" id="S5.SS3.p1.38.m13.1"><semantics id="S5.SS3.p1.38.m13.1a"><mrow id="S5.SS3.p1.38.m13.1.1" xref="S5.SS3.p1.38.m13.1.1.cmml"><mover accent="true" id="S5.SS3.p1.38.m13.1.1.2" xref="S5.SS3.p1.38.m13.1.1.2.cmml"><mi id="S5.SS3.p1.38.m13.1.1.2.2" xref="S5.SS3.p1.38.m13.1.1.2.2.cmml">d</mi><mo id="S5.SS3.p1.38.m13.1.1.2.1" xref="S5.SS3.p1.38.m13.1.1.2.1.cmml">¯</mo></mover><mo id="S5.SS3.p1.38.m13.1.1.1" xref="S5.SS3.p1.38.m13.1.1.1.cmml">/</mo><msqrt id="S5.SS3.p1.38.m13.1.1.3" xref="S5.SS3.p1.38.m13.1.1.3.cmml"><msub id="S5.SS3.p1.38.m13.1.1.3.2" xref="S5.SS3.p1.38.m13.1.1.3.2.cmml"><mi id="S5.SS3.p1.38.m13.1.1.3.2.2" xref="S5.SS3.p1.38.m13.1.1.3.2.2.cmml">k</mi><mi id="S5.SS3.p1.38.m13.1.1.3.2.3" xref="S5.SS3.p1.38.m13.1.1.3.2.3.cmml">min</mi></msub></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.38.m13.1b"><apply id="S5.SS3.p1.38.m13.1.1.cmml" xref="S5.SS3.p1.38.m13.1.1"><divide id="S5.SS3.p1.38.m13.1.1.1.cmml" xref="S5.SS3.p1.38.m13.1.1.1"></divide><apply id="S5.SS3.p1.38.m13.1.1.2.cmml" xref="S5.SS3.p1.38.m13.1.1.2"><ci id="S5.SS3.p1.38.m13.1.1.2.1.cmml" xref="S5.SS3.p1.38.m13.1.1.2.1">¯</ci><ci id="S5.SS3.p1.38.m13.1.1.2.2.cmml" xref="S5.SS3.p1.38.m13.1.1.2.2">𝑑</ci></apply><apply id="S5.SS3.p1.38.m13.1.1.3.cmml" xref="S5.SS3.p1.38.m13.1.1.3"><root id="S5.SS3.p1.38.m13.1.1.3a.cmml" xref="S5.SS3.p1.38.m13.1.1.3"></root><apply id="S5.SS3.p1.38.m13.1.1.3.2.cmml" xref="S5.SS3.p1.38.m13.1.1.3.2"><csymbol cd="ambiguous" id="S5.SS3.p1.38.m13.1.1.3.2.1.cmml" xref="S5.SS3.p1.38.m13.1.1.3.2">subscript</csymbol><ci id="S5.SS3.p1.38.m13.1.1.3.2.2.cmml" xref="S5.SS3.p1.38.m13.1.1.3.2.2">𝑘</ci><min id="S5.SS3.p1.38.m13.1.1.3.2.3.cmml" xref="S5.SS3.p1.38.m13.1.1.3.2.3"></min></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.38.m13.1c">\bar{d}/\sqrt{k_{\min}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.38.m13.1d">over¯ start_ARG italic_d end_ARG / square-root start_ARG italic_k start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT end_ARG</annotation></semantics></math>, one gets <math alttext="\|{\bm{d}}_{\rm g}(t)\|" class="ltx_Math" display="inline" id="S5.SS3.p1.39.m14.2"><semantics id="S5.SS3.p1.39.m14.2a"><mrow id="S5.SS3.p1.39.m14.2.2.1" xref="S5.SS3.p1.39.m14.2.2.2.cmml"><mo id="S5.SS3.p1.39.m14.2.2.1.2" stretchy="false" xref="S5.SS3.p1.39.m14.2.2.2.1.cmml">‖</mo><mrow id="S5.SS3.p1.39.m14.2.2.1.1" xref="S5.SS3.p1.39.m14.2.2.1.1.cmml"><msub id="S5.SS3.p1.39.m14.2.2.1.1.2" xref="S5.SS3.p1.39.m14.2.2.1.1.2.cmml"><mi id="S5.SS3.p1.39.m14.2.2.1.1.2.2" xref="S5.SS3.p1.39.m14.2.2.1.1.2.2.cmml">𝒅</mi><mi id="S5.SS3.p1.39.m14.2.2.1.1.2.3" mathvariant="normal" xref="S5.SS3.p1.39.m14.2.2.1.1.2.3.cmml">g</mi></msub><mo id="S5.SS3.p1.39.m14.2.2.1.1.1" xref="S5.SS3.p1.39.m14.2.2.1.1.1.cmml">⁢</mo><mrow id="S5.SS3.p1.39.m14.2.2.1.1.3.2" xref="S5.SS3.p1.39.m14.2.2.1.1.cmml"><mo id="S5.SS3.p1.39.m14.2.2.1.1.3.2.1" stretchy="false" xref="S5.SS3.p1.39.m14.2.2.1.1.cmml">(</mo><mi id="S5.SS3.p1.39.m14.1.1" xref="S5.SS3.p1.39.m14.1.1.cmml">t</mi><mo id="S5.SS3.p1.39.m14.2.2.1.1.3.2.2" stretchy="false" xref="S5.SS3.p1.39.m14.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="S5.SS3.p1.39.m14.2.2.1.3" stretchy="false" xref="S5.SS3.p1.39.m14.2.2.2.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.39.m14.2b"><apply id="S5.SS3.p1.39.m14.2.2.2.cmml" xref="S5.SS3.p1.39.m14.2.2.1"><csymbol cd="latexml" id="S5.SS3.p1.39.m14.2.2.2.1.cmml" xref="S5.SS3.p1.39.m14.2.2.1.2">norm</csymbol><apply id="S5.SS3.p1.39.m14.2.2.1.1.cmml" xref="S5.SS3.p1.39.m14.2.2.1.1"><times id="S5.SS3.p1.39.m14.2.2.1.1.1.cmml" xref="S5.SS3.p1.39.m14.2.2.1.1.1"></times><apply id="S5.SS3.p1.39.m14.2.2.1.1.2.cmml" xref="S5.SS3.p1.39.m14.2.2.1.1.2"><csymbol cd="ambiguous" id="S5.SS3.p1.39.m14.2.2.1.1.2.1.cmml" xref="S5.SS3.p1.39.m14.2.2.1.1.2">subscript</csymbol><ci id="S5.SS3.p1.39.m14.2.2.1.1.2.2.cmml" xref="S5.SS3.p1.39.m14.2.2.1.1.2.2">𝒅</ci><ci id="S5.SS3.p1.39.m14.2.2.1.1.2.3.cmml" xref="S5.SS3.p1.39.m14.2.2.1.1.2.3">g</ci></apply><ci id="S5.SS3.p1.39.m14.1.1.cmml" xref="S5.SS3.p1.39.m14.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.39.m14.2c">\|{\bm{d}}_{\rm g}(t)\|</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.39.m14.2d">∥ bold_italic_d start_POSTSUBSCRIPT roman_g end_POSTSUBSCRIPT ( italic_t ) ∥</annotation></semantics></math> <math alttext="\leq" class="ltx_Math" display="inline" id="S5.SS3.p1.40.m15.1"><semantics id="S5.SS3.p1.40.m15.1a"><mo id="S5.SS3.p1.40.m15.1.1" xref="S5.SS3.p1.40.m15.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.40.m15.1b"><leq id="S5.SS3.p1.40.m15.1.1.cmml" xref="S5.SS3.p1.40.m15.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.40.m15.1c">\leq</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.40.m15.1d">≤</annotation></semantics></math> <math alttext="{\bar{d}}_{\rm g}" class="ltx_Math" display="inline" id="S5.SS3.p1.41.m16.1"><semantics id="S5.SS3.p1.41.m16.1a"><msub id="S5.SS3.p1.41.m16.1.1" xref="S5.SS3.p1.41.m16.1.1.cmml"><mover accent="true" id="S5.SS3.p1.41.m16.1.1.2" xref="S5.SS3.p1.41.m16.1.1.2.cmml"><mi id="S5.SS3.p1.41.m16.1.1.2.2" xref="S5.SS3.p1.41.m16.1.1.2.2.cmml">d</mi><mo id="S5.SS3.p1.41.m16.1.1.2.1" xref="S5.SS3.p1.41.m16.1.1.2.1.cmml">¯</mo></mover><mi id="S5.SS3.p1.41.m16.1.1.3" mathvariant="normal" xref="S5.SS3.p1.41.m16.1.1.3.cmml">g</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.41.m16.1b"><apply id="S5.SS3.p1.41.m16.1.1.cmml" xref="S5.SS3.p1.41.m16.1.1"><csymbol cd="ambiguous" id="S5.SS3.p1.41.m16.1.1.1.cmml" xref="S5.SS3.p1.41.m16.1.1">subscript</csymbol><apply id="S5.SS3.p1.41.m16.1.1.2.cmml" xref="S5.SS3.p1.41.m16.1.1.2"><ci id="S5.SS3.p1.41.m16.1.1.2.1.cmml" xref="S5.SS3.p1.41.m16.1.1.2.1">¯</ci><ci id="S5.SS3.p1.41.m16.1.1.2.2.cmml" xref="S5.SS3.p1.41.m16.1.1.2.2">𝑑</ci></apply><ci id="S5.SS3.p1.41.m16.1.1.3.cmml" xref="S5.SS3.p1.41.m16.1.1.3">g</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.41.m16.1c">{\bar{d}}_{\rm g}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.41.m16.1d">over¯ start_ARG italic_d end_ARG start_POSTSUBSCRIPT roman_g end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\forall\bm{x}\in\Omega_{{\rm c}_{x}}" class="ltx_Math" display="inline" id="S5.SS3.p1.42.m17.1"><semantics id="S5.SS3.p1.42.m17.1a"><mrow id="S5.SS3.p1.42.m17.1.1" xref="S5.SS3.p1.42.m17.1.1.cmml"><mrow id="S5.SS3.p1.42.m17.1.1.2" xref="S5.SS3.p1.42.m17.1.1.2.cmml"><mo id="S5.SS3.p1.42.m17.1.1.2.1" rspace="0.167em" xref="S5.SS3.p1.42.m17.1.1.2.1.cmml">∀</mo><mi id="S5.SS3.p1.42.m17.1.1.2.2" xref="S5.SS3.p1.42.m17.1.1.2.2.cmml">𝒙</mi></mrow><mo id="S5.SS3.p1.42.m17.1.1.1" xref="S5.SS3.p1.42.m17.1.1.1.cmml">∈</mo><msub id="S5.SS3.p1.42.m17.1.1.3" xref="S5.SS3.p1.42.m17.1.1.3.cmml"><mi id="S5.SS3.p1.42.m17.1.1.3.2" mathvariant="normal" xref="S5.SS3.p1.42.m17.1.1.3.2.cmml">Ω</mi><msub id="S5.SS3.p1.42.m17.1.1.3.3" xref="S5.SS3.p1.42.m17.1.1.3.3.cmml"><mi id="S5.SS3.p1.42.m17.1.1.3.3.2" mathvariant="normal" xref="S5.SS3.p1.42.m17.1.1.3.3.2.cmml">c</mi><mi id="S5.SS3.p1.42.m17.1.1.3.3.3" xref="S5.SS3.p1.42.m17.1.1.3.3.3.cmml">x</mi></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.42.m17.1b"><apply id="S5.SS3.p1.42.m17.1.1.cmml" xref="S5.SS3.p1.42.m17.1.1"><in id="S5.SS3.p1.42.m17.1.1.1.cmml" xref="S5.SS3.p1.42.m17.1.1.1"></in><apply id="S5.SS3.p1.42.m17.1.1.2.cmml" xref="S5.SS3.p1.42.m17.1.1.2"><csymbol cd="latexml" id="S5.SS3.p1.42.m17.1.1.2.1.cmml" xref="S5.SS3.p1.42.m17.1.1.2.1">for-all</csymbol><ci id="S5.SS3.p1.42.m17.1.1.2.2.cmml" xref="S5.SS3.p1.42.m17.1.1.2.2">𝒙</ci></apply><apply id="S5.SS3.p1.42.m17.1.1.3.cmml" xref="S5.SS3.p1.42.m17.1.1.3"><csymbol cd="ambiguous" id="S5.SS3.p1.42.m17.1.1.3.1.cmml" xref="S5.SS3.p1.42.m17.1.1.3">subscript</csymbol><ci id="S5.SS3.p1.42.m17.1.1.3.2.cmml" xref="S5.SS3.p1.42.m17.1.1.3.2">Ω</ci><apply id="S5.SS3.p1.42.m17.1.1.3.3.cmml" xref="S5.SS3.p1.42.m17.1.1.3.3"><csymbol cd="ambiguous" id="S5.SS3.p1.42.m17.1.1.3.3.1.cmml" xref="S5.SS3.p1.42.m17.1.1.3.3">subscript</csymbol><ci id="S5.SS3.p1.42.m17.1.1.3.3.2.cmml" xref="S5.SS3.p1.42.m17.1.1.3.3.2">c</ci><ci id="S5.SS3.p1.42.m17.1.1.3.3.3.cmml" xref="S5.SS3.p1.42.m17.1.1.3.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.42.m17.1c">\forall\bm{x}\in\Omega_{{\rm c}_{x}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.42.m17.1d">∀ bold_italic_x ∈ roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="{\bar{d}}_{\rm g}" class="ltx_Math" display="inline" id="S5.SS3.p1.43.m18.1"><semantics id="S5.SS3.p1.43.m18.1a"><msub id="S5.SS3.p1.43.m18.1.1" xref="S5.SS3.p1.43.m18.1.1.cmml"><mover accent="true" id="S5.SS3.p1.43.m18.1.1.2" xref="S5.SS3.p1.43.m18.1.1.2.cmml"><mi id="S5.SS3.p1.43.m18.1.1.2.2" xref="S5.SS3.p1.43.m18.1.1.2.2.cmml">d</mi><mo id="S5.SS3.p1.43.m18.1.1.2.1" xref="S5.SS3.p1.43.m18.1.1.2.1.cmml">¯</mo></mover><mi id="S5.SS3.p1.43.m18.1.1.3" mathvariant="normal" xref="S5.SS3.p1.43.m18.1.1.3.cmml">g</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.43.m18.1b"><apply id="S5.SS3.p1.43.m18.1.1.cmml" xref="S5.SS3.p1.43.m18.1.1"><csymbol cd="ambiguous" id="S5.SS3.p1.43.m18.1.1.1.cmml" xref="S5.SS3.p1.43.m18.1.1">subscript</csymbol><apply id="S5.SS3.p1.43.m18.1.1.2.cmml" xref="S5.SS3.p1.43.m18.1.1.2"><ci id="S5.SS3.p1.43.m18.1.1.2.1.cmml" xref="S5.SS3.p1.43.m18.1.1.2.1">¯</ci><ci id="S5.SS3.p1.43.m18.1.1.2.2.cmml" xref="S5.SS3.p1.43.m18.1.1.2.2">𝑑</ci></apply><ci id="S5.SS3.p1.43.m18.1.1.3.cmml" xref="S5.SS3.p1.43.m18.1.1.3">g</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.43.m18.1c">{\bar{d}}_{\rm g}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.43.m18.1d">over¯ start_ARG italic_d end_ARG start_POSTSUBSCRIPT roman_g end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S5.SS3.p1.44.m19.1"><semantics id="S5.SS3.p1.44.m19.1a"><mo id="S5.SS3.p1.44.m19.1.1" xref="S5.SS3.p1.44.m19.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.44.m19.1b"><csymbol cd="latexml" id="S5.SS3.p1.44.m19.1.1.cmml" xref="S5.SS3.p1.44.m19.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.44.m19.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.44.m19.1d">:=</annotation></semantics></math> <math alttext="\tau_{\rm d}\phi\bar{d}/\sqrt{k_{\min}}" class="ltx_Math" display="inline" id="S5.SS3.p1.45.m20.1"><semantics id="S5.SS3.p1.45.m20.1a"><mrow id="S5.SS3.p1.45.m20.1.1" xref="S5.SS3.p1.45.m20.1.1.cmml"><mrow id="S5.SS3.p1.45.m20.1.1.2" xref="S5.SS3.p1.45.m20.1.1.2.cmml"><msub id="S5.SS3.p1.45.m20.1.1.2.2" xref="S5.SS3.p1.45.m20.1.1.2.2.cmml"><mi id="S5.SS3.p1.45.m20.1.1.2.2.2" xref="S5.SS3.p1.45.m20.1.1.2.2.2.cmml">τ</mi><mi id="S5.SS3.p1.45.m20.1.1.2.2.3" mathvariant="normal" xref="S5.SS3.p1.45.m20.1.1.2.2.3.cmml">d</mi></msub><mo id="S5.SS3.p1.45.m20.1.1.2.1" xref="S5.SS3.p1.45.m20.1.1.2.1.cmml">⁢</mo><mi id="S5.SS3.p1.45.m20.1.1.2.3" xref="S5.SS3.p1.45.m20.1.1.2.3.cmml">ϕ</mi><mo id="S5.SS3.p1.45.m20.1.1.2.1a" xref="S5.SS3.p1.45.m20.1.1.2.1.cmml">⁢</mo><mover accent="true" id="S5.SS3.p1.45.m20.1.1.2.4" xref="S5.SS3.p1.45.m20.1.1.2.4.cmml"><mi id="S5.SS3.p1.45.m20.1.1.2.4.2" xref="S5.SS3.p1.45.m20.1.1.2.4.2.cmml">d</mi><mo id="S5.SS3.p1.45.m20.1.1.2.4.1" xref="S5.SS3.p1.45.m20.1.1.2.4.1.cmml">¯</mo></mover></mrow><mo id="S5.SS3.p1.45.m20.1.1.1" xref="S5.SS3.p1.45.m20.1.1.1.cmml">/</mo><msqrt id="S5.SS3.p1.45.m20.1.1.3" xref="S5.SS3.p1.45.m20.1.1.3.cmml"><msub id="S5.SS3.p1.45.m20.1.1.3.2" xref="S5.SS3.p1.45.m20.1.1.3.2.cmml"><mi id="S5.SS3.p1.45.m20.1.1.3.2.2" xref="S5.SS3.p1.45.m20.1.1.3.2.2.cmml">k</mi><mi id="S5.SS3.p1.45.m20.1.1.3.2.3" xref="S5.SS3.p1.45.m20.1.1.3.2.3.cmml">min</mi></msub></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.45.m20.1b"><apply id="S5.SS3.p1.45.m20.1.1.cmml" xref="S5.SS3.p1.45.m20.1.1"><divide id="S5.SS3.p1.45.m20.1.1.1.cmml" xref="S5.SS3.p1.45.m20.1.1.1"></divide><apply id="S5.SS3.p1.45.m20.1.1.2.cmml" xref="S5.SS3.p1.45.m20.1.1.2"><times id="S5.SS3.p1.45.m20.1.1.2.1.cmml" xref="S5.SS3.p1.45.m20.1.1.2.1"></times><apply id="S5.SS3.p1.45.m20.1.1.2.2.cmml" xref="S5.SS3.p1.45.m20.1.1.2.2"><csymbol cd="ambiguous" id="S5.SS3.p1.45.m20.1.1.2.2.1.cmml" xref="S5.SS3.p1.45.m20.1.1.2.2">subscript</csymbol><ci id="S5.SS3.p1.45.m20.1.1.2.2.2.cmml" xref="S5.SS3.p1.45.m20.1.1.2.2.2">𝜏</ci><ci id="S5.SS3.p1.45.m20.1.1.2.2.3.cmml" xref="S5.SS3.p1.45.m20.1.1.2.2.3">d</ci></apply><ci id="S5.SS3.p1.45.m20.1.1.2.3.cmml" xref="S5.SS3.p1.45.m20.1.1.2.3">italic-ϕ</ci><apply id="S5.SS3.p1.45.m20.1.1.2.4.cmml" xref="S5.SS3.p1.45.m20.1.1.2.4"><ci id="S5.SS3.p1.45.m20.1.1.2.4.1.cmml" xref="S5.SS3.p1.45.m20.1.1.2.4.1">¯</ci><ci id="S5.SS3.p1.45.m20.1.1.2.4.2.cmml" xref="S5.SS3.p1.45.m20.1.1.2.4.2">𝑑</ci></apply></apply><apply id="S5.SS3.p1.45.m20.1.1.3.cmml" xref="S5.SS3.p1.45.m20.1.1.3"><root id="S5.SS3.p1.45.m20.1.1.3a.cmml" xref="S5.SS3.p1.45.m20.1.1.3"></root><apply id="S5.SS3.p1.45.m20.1.1.3.2.cmml" xref="S5.SS3.p1.45.m20.1.1.3.2"><csymbol cd="ambiguous" id="S5.SS3.p1.45.m20.1.1.3.2.1.cmml" xref="S5.SS3.p1.45.m20.1.1.3.2">subscript</csymbol><ci id="S5.SS3.p1.45.m20.1.1.3.2.2.cmml" xref="S5.SS3.p1.45.m20.1.1.3.2.2">𝑘</ci><min id="S5.SS3.p1.45.m20.1.1.3.2.3.cmml" xref="S5.SS3.p1.45.m20.1.1.3.2.3"></min></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.45.m20.1c">\tau_{\rm d}\phi\bar{d}/\sqrt{k_{\min}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.45.m20.1d">italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT italic_ϕ over¯ start_ARG italic_d end_ARG / square-root start_ARG italic_k start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT end_ARG</annotation></semantics></math>. In addition, applying <math alttext="H(s)" class="ltx_Math" display="inline" id="S5.SS3.p1.46.m21.1"><semantics id="S5.SS3.p1.46.m21.1a"><mrow id="S5.SS3.p1.46.m21.1.2" xref="S5.SS3.p1.46.m21.1.2.cmml"><mi id="S5.SS3.p1.46.m21.1.2.2" xref="S5.SS3.p1.46.m21.1.2.2.cmml">H</mi><mo id="S5.SS3.p1.46.m21.1.2.1" xref="S5.SS3.p1.46.m21.1.2.1.cmml">⁢</mo><mrow id="S5.SS3.p1.46.m21.1.2.3.2" xref="S5.SS3.p1.46.m21.1.2.cmml"><mo id="S5.SS3.p1.46.m21.1.2.3.2.1" stretchy="false" xref="S5.SS3.p1.46.m21.1.2.cmml">(</mo><mi id="S5.SS3.p1.46.m21.1.1" xref="S5.SS3.p1.46.m21.1.1.cmml">s</mi><mo id="S5.SS3.p1.46.m21.1.2.3.2.2" stretchy="false" xref="S5.SS3.p1.46.m21.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.46.m21.1b"><apply id="S5.SS3.p1.46.m21.1.2.cmml" xref="S5.SS3.p1.46.m21.1.2"><times id="S5.SS3.p1.46.m21.1.2.1.cmml" xref="S5.SS3.p1.46.m21.1.2.1"></times><ci id="S5.SS3.p1.46.m21.1.2.2.cmml" xref="S5.SS3.p1.46.m21.1.2.2">𝐻</ci><ci id="S5.SS3.p1.46.m21.1.1.cmml" xref="S5.SS3.p1.46.m21.1.1">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.46.m21.1c">H(s)</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.46.m21.1d">italic_H ( italic_s )</annotation></semantics></math> to the closed-loop tracking error system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E7" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">7</span></a>) and noting the filtered regression equation (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E21" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">21</span></a>), one gets</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx26"> <tbody id="S5.E31"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\bm{z}(t)=\Phi_{\rm f}^{T}(t)\bm{\theta}+{\bm{d}}_{\rm f}(t)" class="ltx_Math" display="inline" id="S5.E31.m1.3"><semantics id="S5.E31.m1.3a"><mrow id="S5.E31.m1.3.4" xref="S5.E31.m1.3.4.cmml"><mrow id="S5.E31.m1.3.4.2" xref="S5.E31.m1.3.4.2.cmml"><mi id="S5.E31.m1.3.4.2.2" xref="S5.E31.m1.3.4.2.2.cmml">𝒛</mi><mo id="S5.E31.m1.3.4.2.1" xref="S5.E31.m1.3.4.2.1.cmml">⁢</mo><mrow id="S5.E31.m1.3.4.2.3.2" xref="S5.E31.m1.3.4.2.cmml"><mo id="S5.E31.m1.3.4.2.3.2.1" stretchy="false" xref="S5.E31.m1.3.4.2.cmml">(</mo><mi id="S5.E31.m1.1.1" xref="S5.E31.m1.1.1.cmml">t</mi><mo id="S5.E31.m1.3.4.2.3.2.2" stretchy="false" xref="S5.E31.m1.3.4.2.cmml">)</mo></mrow></mrow><mo id="S5.E31.m1.3.4.1" xref="S5.E31.m1.3.4.1.cmml">=</mo><mrow id="S5.E31.m1.3.4.3" xref="S5.E31.m1.3.4.3.cmml"><mrow id="S5.E31.m1.3.4.3.2" xref="S5.E31.m1.3.4.3.2.cmml"><msubsup id="S5.E31.m1.3.4.3.2.2" xref="S5.E31.m1.3.4.3.2.2.cmml"><mi id="S5.E31.m1.3.4.3.2.2.2.2" mathvariant="normal" xref="S5.E31.m1.3.4.3.2.2.2.2.cmml">Φ</mi><mi id="S5.E31.m1.3.4.3.2.2.2.3" mathvariant="normal" xref="S5.E31.m1.3.4.3.2.2.2.3.cmml">f</mi><mi id="S5.E31.m1.3.4.3.2.2.3" xref="S5.E31.m1.3.4.3.2.2.3.cmml">T</mi></msubsup><mo id="S5.E31.m1.3.4.3.2.1" xref="S5.E31.m1.3.4.3.2.1.cmml">⁢</mo><mrow id="S5.E31.m1.3.4.3.2.3.2" xref="S5.E31.m1.3.4.3.2.cmml"><mo id="S5.E31.m1.3.4.3.2.3.2.1" stretchy="false" xref="S5.E31.m1.3.4.3.2.cmml">(</mo><mi id="S5.E31.m1.2.2" xref="S5.E31.m1.2.2.cmml">t</mi><mo id="S5.E31.m1.3.4.3.2.3.2.2" stretchy="false" xref="S5.E31.m1.3.4.3.2.cmml">)</mo></mrow><mo id="S5.E31.m1.3.4.3.2.1a" xref="S5.E31.m1.3.4.3.2.1.cmml">⁢</mo><mi id="S5.E31.m1.3.4.3.2.4" xref="S5.E31.m1.3.4.3.2.4.cmml">𝜽</mi></mrow><mo id="S5.E31.m1.3.4.3.1" xref="S5.E31.m1.3.4.3.1.cmml">+</mo><mrow id="S5.E31.m1.3.4.3.3" xref="S5.E31.m1.3.4.3.3.cmml"><msub id="S5.E31.m1.3.4.3.3.2" xref="S5.E31.m1.3.4.3.3.2.cmml"><mi id="S5.E31.m1.3.4.3.3.2.2" xref="S5.E31.m1.3.4.3.3.2.2.cmml">𝒅</mi><mi id="S5.E31.m1.3.4.3.3.2.3" mathvariant="normal" xref="S5.E31.m1.3.4.3.3.2.3.cmml">f</mi></msub><mo id="S5.E31.m1.3.4.3.3.1" xref="S5.E31.m1.3.4.3.3.1.cmml">⁢</mo><mrow id="S5.E31.m1.3.4.3.3.3.2" xref="S5.E31.m1.3.4.3.3.cmml"><mo id="S5.E31.m1.3.4.3.3.3.2.1" stretchy="false" xref="S5.E31.m1.3.4.3.3.cmml">(</mo><mi id="S5.E31.m1.3.3" xref="S5.E31.m1.3.3.cmml">t</mi><mo id="S5.E31.m1.3.4.3.3.3.2.2" stretchy="false" xref="S5.E31.m1.3.4.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.E31.m1.3b"><apply id="S5.E31.m1.3.4.cmml" xref="S5.E31.m1.3.4"><eq id="S5.E31.m1.3.4.1.cmml" xref="S5.E31.m1.3.4.1"></eq><apply id="S5.E31.m1.3.4.2.cmml" xref="S5.E31.m1.3.4.2"><times id="S5.E31.m1.3.4.2.1.cmml" xref="S5.E31.m1.3.4.2.1"></times><ci id="S5.E31.m1.3.4.2.2.cmml" xref="S5.E31.m1.3.4.2.2">𝒛</ci><ci id="S5.E31.m1.1.1.cmml" xref="S5.E31.m1.1.1">𝑡</ci></apply><apply id="S5.E31.m1.3.4.3.cmml" xref="S5.E31.m1.3.4.3"><plus id="S5.E31.m1.3.4.3.1.cmml" xref="S5.E31.m1.3.4.3.1"></plus><apply id="S5.E31.m1.3.4.3.2.cmml" xref="S5.E31.m1.3.4.3.2"><times id="S5.E31.m1.3.4.3.2.1.cmml" xref="S5.E31.m1.3.4.3.2.1"></times><apply id="S5.E31.m1.3.4.3.2.2.cmml" xref="S5.E31.m1.3.4.3.2.2"><csymbol cd="ambiguous" id="S5.E31.m1.3.4.3.2.2.1.cmml" xref="S5.E31.m1.3.4.3.2.2">superscript</csymbol><apply id="S5.E31.m1.3.4.3.2.2.2.cmml" xref="S5.E31.m1.3.4.3.2.2"><csymbol cd="ambiguous" id="S5.E31.m1.3.4.3.2.2.2.1.cmml" xref="S5.E31.m1.3.4.3.2.2">subscript</csymbol><ci id="S5.E31.m1.3.4.3.2.2.2.2.cmml" xref="S5.E31.m1.3.4.3.2.2.2.2">Φ</ci><ci id="S5.E31.m1.3.4.3.2.2.2.3.cmml" xref="S5.E31.m1.3.4.3.2.2.2.3">f</ci></apply><ci id="S5.E31.m1.3.4.3.2.2.3.cmml" xref="S5.E31.m1.3.4.3.2.2.3">𝑇</ci></apply><ci id="S5.E31.m1.2.2.cmml" xref="S5.E31.m1.2.2">𝑡</ci><ci id="S5.E31.m1.3.4.3.2.4.cmml" xref="S5.E31.m1.3.4.3.2.4">𝜽</ci></apply><apply id="S5.E31.m1.3.4.3.3.cmml" xref="S5.E31.m1.3.4.3.3"><times id="S5.E31.m1.3.4.3.3.1.cmml" xref="S5.E31.m1.3.4.3.3.1"></times><apply id="S5.E31.m1.3.4.3.3.2.cmml" xref="S5.E31.m1.3.4.3.3.2"><csymbol cd="ambiguous" id="S5.E31.m1.3.4.3.3.2.1.cmml" xref="S5.E31.m1.3.4.3.3.2">subscript</csymbol><ci id="S5.E31.m1.3.4.3.3.2.2.cmml" xref="S5.E31.m1.3.4.3.3.2.2">𝒅</ci><ci id="S5.E31.m1.3.4.3.3.2.3.cmml" xref="S5.E31.m1.3.4.3.3.2.3">f</ci></apply><ci id="S5.E31.m1.3.3.cmml" xref="S5.E31.m1.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E31.m1.3c">\displaystyle\bm{z}(t)=\Phi_{\rm f}^{T}(t)\bm{\theta}+{\bm{d}}_{\rm f}(t)</annotation><annotation encoding="application/x-llamapun" id="S5.E31.m1.3d">bold_italic_z ( italic_t ) = roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_t ) bold_italic_θ + bold_italic_d start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(31)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.SS3.p1.52">with <math alttext="{\bm{d}}_{\rm f}" class="ltx_Math" display="inline" id="S5.SS3.p1.47.m1.1"><semantics id="S5.SS3.p1.47.m1.1a"><msub id="S5.SS3.p1.47.m1.1.1" xref="S5.SS3.p1.47.m1.1.1.cmml"><mi id="S5.SS3.p1.47.m1.1.1.2" xref="S5.SS3.p1.47.m1.1.1.2.cmml">𝒅</mi><mi id="S5.SS3.p1.47.m1.1.1.3" mathvariant="normal" xref="S5.SS3.p1.47.m1.1.1.3.cmml">f</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.47.m1.1b"><apply id="S5.SS3.p1.47.m1.1.1.cmml" xref="S5.SS3.p1.47.m1.1.1"><csymbol cd="ambiguous" id="S5.SS3.p1.47.m1.1.1.1.cmml" xref="S5.SS3.p1.47.m1.1.1">subscript</csymbol><ci id="S5.SS3.p1.47.m1.1.1.2.cmml" xref="S5.SS3.p1.47.m1.1.1.2">𝒅</ci><ci id="S5.SS3.p1.47.m1.1.1.3.cmml" xref="S5.SS3.p1.47.m1.1.1.3">f</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.47.m1.1c">{\bm{d}}_{\rm f}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.47.m1.1d">bold_italic_d start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S5.SS3.p1.48.m2.1"><semantics id="S5.SS3.p1.48.m2.1a"><mo id="S5.SS3.p1.48.m2.1.1" xref="S5.SS3.p1.48.m2.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.48.m2.1b"><csymbol cd="latexml" id="S5.SS3.p1.48.m2.1.1.cmml" xref="S5.SS3.p1.48.m2.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.48.m2.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.48.m2.1d">:=</annotation></semantics></math> <math alttext="H(s)[\bm{d}]" class="ltx_Math" display="inline" id="S5.SS3.p1.49.m3.2"><semantics id="S5.SS3.p1.49.m3.2a"><mrow id="S5.SS3.p1.49.m3.2.3" xref="S5.SS3.p1.49.m3.2.3.cmml"><mi id="S5.SS3.p1.49.m3.2.3.2" xref="S5.SS3.p1.49.m3.2.3.2.cmml">H</mi><mo id="S5.SS3.p1.49.m3.2.3.1" xref="S5.SS3.p1.49.m3.2.3.1.cmml">⁢</mo><mrow id="S5.SS3.p1.49.m3.2.3.3.2" xref="S5.SS3.p1.49.m3.2.3.cmml"><mo id="S5.SS3.p1.49.m3.2.3.3.2.1" stretchy="false" xref="S5.SS3.p1.49.m3.2.3.cmml">(</mo><mi id="S5.SS3.p1.49.m3.1.1" xref="S5.SS3.p1.49.m3.1.1.cmml">s</mi><mo id="S5.SS3.p1.49.m3.2.3.3.2.2" stretchy="false" xref="S5.SS3.p1.49.m3.2.3.cmml">)</mo></mrow><mo id="S5.SS3.p1.49.m3.2.3.1a" xref="S5.SS3.p1.49.m3.2.3.1.cmml">⁢</mo><mrow id="S5.SS3.p1.49.m3.2.3.4.2" xref="S5.SS3.p1.49.m3.2.3.4.1.cmml"><mo id="S5.SS3.p1.49.m3.2.3.4.2.1" stretchy="false" xref="S5.SS3.p1.49.m3.2.3.4.1.1.cmml">[</mo><mi id="S5.SS3.p1.49.m3.2.2" xref="S5.SS3.p1.49.m3.2.2.cmml">𝒅</mi><mo id="S5.SS3.p1.49.m3.2.3.4.2.2" stretchy="false" xref="S5.SS3.p1.49.m3.2.3.4.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.49.m3.2b"><apply id="S5.SS3.p1.49.m3.2.3.cmml" xref="S5.SS3.p1.49.m3.2.3"><times id="S5.SS3.p1.49.m3.2.3.1.cmml" xref="S5.SS3.p1.49.m3.2.3.1"></times><ci id="S5.SS3.p1.49.m3.2.3.2.cmml" xref="S5.SS3.p1.49.m3.2.3.2">𝐻</ci><ci id="S5.SS3.p1.49.m3.1.1.cmml" xref="S5.SS3.p1.49.m3.1.1">𝑠</ci><apply id="S5.SS3.p1.49.m3.2.3.4.1.cmml" xref="S5.SS3.p1.49.m3.2.3.4.2"><csymbol cd="latexml" id="S5.SS3.p1.49.m3.2.3.4.1.1.cmml" xref="S5.SS3.p1.49.m3.2.3.4.2.1">delimited-[]</csymbol><ci id="S5.SS3.p1.49.m3.2.2.cmml" xref="S5.SS3.p1.49.m3.2.2">𝒅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.49.m3.2c">H(s)[\bm{d}]</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.49.m3.2d">italic_H ( italic_s ) [ bold_italic_d ]</annotation></semantics></math>. It is clear that <math alttext="\bm{d}_{\rm f}" class="ltx_Math" display="inline" id="S5.SS3.p1.50.m4.1"><semantics id="S5.SS3.p1.50.m4.1a"><msub id="S5.SS3.p1.50.m4.1.1" xref="S5.SS3.p1.50.m4.1.1.cmml"><mi id="S5.SS3.p1.50.m4.1.1.2" xref="S5.SS3.p1.50.m4.1.1.2.cmml">𝒅</mi><mi id="S5.SS3.p1.50.m4.1.1.3" mathvariant="normal" xref="S5.SS3.p1.50.m4.1.1.3.cmml">f</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.50.m4.1b"><apply id="S5.SS3.p1.50.m4.1.1.cmml" xref="S5.SS3.p1.50.m4.1.1"><csymbol cd="ambiguous" id="S5.SS3.p1.50.m4.1.1.1.cmml" xref="S5.SS3.p1.50.m4.1.1">subscript</csymbol><ci id="S5.SS3.p1.50.m4.1.1.2.cmml" xref="S5.SS3.p1.50.m4.1.1.2">𝒅</ci><ci id="S5.SS3.p1.50.m4.1.1.3.cmml" xref="S5.SS3.p1.50.m4.1.1.3">f</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.50.m4.1c">\bm{d}_{\rm f}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.50.m4.1d">bold_italic_d start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT</annotation></semantics></math> satisfies <math alttext="\|\bm{d}_{\rm f}\|\leq\bar{d}" class="ltx_Math" display="inline" id="S5.SS3.p1.51.m5.1"><semantics id="S5.SS3.p1.51.m5.1a"><mrow id="S5.SS3.p1.51.m5.1.1" xref="S5.SS3.p1.51.m5.1.1.cmml"><mrow id="S5.SS3.p1.51.m5.1.1.1.1" xref="S5.SS3.p1.51.m5.1.1.1.2.cmml"><mo id="S5.SS3.p1.51.m5.1.1.1.1.2" stretchy="false" xref="S5.SS3.p1.51.m5.1.1.1.2.1.cmml">‖</mo><msub id="S5.SS3.p1.51.m5.1.1.1.1.1" xref="S5.SS3.p1.51.m5.1.1.1.1.1.cmml"><mi id="S5.SS3.p1.51.m5.1.1.1.1.1.2" xref="S5.SS3.p1.51.m5.1.1.1.1.1.2.cmml">𝒅</mi><mi id="S5.SS3.p1.51.m5.1.1.1.1.1.3" mathvariant="normal" xref="S5.SS3.p1.51.m5.1.1.1.1.1.3.cmml">f</mi></msub><mo id="S5.SS3.p1.51.m5.1.1.1.1.3" stretchy="false" xref="S5.SS3.p1.51.m5.1.1.1.2.1.cmml">‖</mo></mrow><mo id="S5.SS3.p1.51.m5.1.1.2" xref="S5.SS3.p1.51.m5.1.1.2.cmml">≤</mo><mover accent="true" id="S5.SS3.p1.51.m5.1.1.3" xref="S5.SS3.p1.51.m5.1.1.3.cmml"><mi id="S5.SS3.p1.51.m5.1.1.3.2" xref="S5.SS3.p1.51.m5.1.1.3.2.cmml">d</mi><mo id="S5.SS3.p1.51.m5.1.1.3.1" xref="S5.SS3.p1.51.m5.1.1.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.51.m5.1b"><apply id="S5.SS3.p1.51.m5.1.1.cmml" xref="S5.SS3.p1.51.m5.1.1"><leq id="S5.SS3.p1.51.m5.1.1.2.cmml" xref="S5.SS3.p1.51.m5.1.1.2"></leq><apply id="S5.SS3.p1.51.m5.1.1.1.2.cmml" xref="S5.SS3.p1.51.m5.1.1.1.1"><csymbol cd="latexml" id="S5.SS3.p1.51.m5.1.1.1.2.1.cmml" xref="S5.SS3.p1.51.m5.1.1.1.1.2">norm</csymbol><apply id="S5.SS3.p1.51.m5.1.1.1.1.1.cmml" xref="S5.SS3.p1.51.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.p1.51.m5.1.1.1.1.1.1.cmml" xref="S5.SS3.p1.51.m5.1.1.1.1.1">subscript</csymbol><ci id="S5.SS3.p1.51.m5.1.1.1.1.1.2.cmml" xref="S5.SS3.p1.51.m5.1.1.1.1.1.2">𝒅</ci><ci id="S5.SS3.p1.51.m5.1.1.1.1.1.3.cmml" xref="S5.SS3.p1.51.m5.1.1.1.1.1.3">f</ci></apply></apply><apply id="S5.SS3.p1.51.m5.1.1.3.cmml" xref="S5.SS3.p1.51.m5.1.1.3"><ci id="S5.SS3.p1.51.m5.1.1.3.1.cmml" xref="S5.SS3.p1.51.m5.1.1.3.1">¯</ci><ci id="S5.SS3.p1.51.m5.1.1.3.2.cmml" xref="S5.SS3.p1.51.m5.1.1.3.2">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.51.m5.1c">\|\bm{d}_{\rm f}\|\leq\bar{d}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.51.m5.1d">∥ bold_italic_d start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ∥ ≤ over¯ start_ARG italic_d end_ARG</annotation></semantics></math> due to the properties of <math alttext="H(s)" class="ltx_Math" display="inline" id="S5.SS3.p1.52.m6.1"><semantics id="S5.SS3.p1.52.m6.1a"><mrow id="S5.SS3.p1.52.m6.1.2" xref="S5.SS3.p1.52.m6.1.2.cmml"><mi id="S5.SS3.p1.52.m6.1.2.2" xref="S5.SS3.p1.52.m6.1.2.2.cmml">H</mi><mo id="S5.SS3.p1.52.m6.1.2.1" xref="S5.SS3.p1.52.m6.1.2.1.cmml">⁢</mo><mrow id="S5.SS3.p1.52.m6.1.2.3.2" xref="S5.SS3.p1.52.m6.1.2.cmml"><mo id="S5.SS3.p1.52.m6.1.2.3.2.1" stretchy="false" xref="S5.SS3.p1.52.m6.1.2.cmml">(</mo><mi id="S5.SS3.p1.52.m6.1.1" xref="S5.SS3.p1.52.m6.1.1.cmml">s</mi><mo id="S5.SS3.p1.52.m6.1.2.3.2.2" stretchy="false" xref="S5.SS3.p1.52.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.52.m6.1b"><apply id="S5.SS3.p1.52.m6.1.2.cmml" xref="S5.SS3.p1.52.m6.1.2"><times id="S5.SS3.p1.52.m6.1.2.1.cmml" xref="S5.SS3.p1.52.m6.1.2.1"></times><ci id="S5.SS3.p1.52.m6.1.2.2.cmml" xref="S5.SS3.p1.52.m6.1.2.2">𝐻</ci><ci id="S5.SS3.p1.52.m6.1.1.cmml" xref="S5.SS3.p1.52.m6.1.1">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.52.m6.1c">H(s)</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.52.m6.1d">italic_H ( italic_s )</annotation></semantics></math>. The following theorem is established to show the robustness results of the proposed CLBC law (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E6" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">6</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>).</p> </div> <div class="ltx_para" id="S5.SS3.p2"> <p class="ltx_p" id="S5.SS3.p2.9"><span class="ltx_text ltx_font_italic" id="S5.SS3.p2.9.3">Theorem 3:</span> For the system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S2.E4" title="In II Problem Formulation ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">4</span></a>) with the disturbance <math alttext="\bm{d}" class="ltx_Math" display="inline" id="S5.SS3.p2.1.m1.1"><semantics id="S5.SS3.p2.1.m1.1a"><mi id="S5.SS3.p2.1.m1.1.1" xref="S5.SS3.p2.1.m1.1.1.cmml">𝒅</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.1.m1.1b"><ci id="S5.SS3.p2.1.m1.1.1.cmml" xref="S5.SS3.p2.1.m1.1.1">𝒅</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.1.m1.1c">\bm{d}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.1.m1.1d">bold_italic_d</annotation></semantics></math> and Assumptions 1–3 driven by the CLBC law (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E6" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">6</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>) with <math alttext="\bm{x}(0)\in\Omega_{{\rm c}_{0}}" class="ltx_Math" display="inline" id="S5.SS3.p2.2.m2.1"><semantics id="S5.SS3.p2.2.m2.1a"><mrow id="S5.SS3.p2.2.m2.1.2" xref="S5.SS3.p2.2.m2.1.2.cmml"><mrow id="S5.SS3.p2.2.m2.1.2.2" xref="S5.SS3.p2.2.m2.1.2.2.cmml"><mi id="S5.SS3.p2.2.m2.1.2.2.2" xref="S5.SS3.p2.2.m2.1.2.2.2.cmml">𝒙</mi><mo id="S5.SS3.p2.2.m2.1.2.2.1" xref="S5.SS3.p2.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S5.SS3.p2.2.m2.1.2.2.3.2" xref="S5.SS3.p2.2.m2.1.2.2.cmml"><mo id="S5.SS3.p2.2.m2.1.2.2.3.2.1" stretchy="false" xref="S5.SS3.p2.2.m2.1.2.2.cmml">(</mo><mn id="S5.SS3.p2.2.m2.1.1" xref="S5.SS3.p2.2.m2.1.1.cmml">0</mn><mo id="S5.SS3.p2.2.m2.1.2.2.3.2.2" stretchy="false" xref="S5.SS3.p2.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.SS3.p2.2.m2.1.2.1" xref="S5.SS3.p2.2.m2.1.2.1.cmml">∈</mo><msub id="S5.SS3.p2.2.m2.1.2.3" xref="S5.SS3.p2.2.m2.1.2.3.cmml"><mi id="S5.SS3.p2.2.m2.1.2.3.2" mathvariant="normal" xref="S5.SS3.p2.2.m2.1.2.3.2.cmml">Ω</mi><msub id="S5.SS3.p2.2.m2.1.2.3.3" xref="S5.SS3.p2.2.m2.1.2.3.3.cmml"><mi id="S5.SS3.p2.2.m2.1.2.3.3.2" mathvariant="normal" xref="S5.SS3.p2.2.m2.1.2.3.3.2.cmml">c</mi><mn id="S5.SS3.p2.2.m2.1.2.3.3.3" xref="S5.SS3.p2.2.m2.1.2.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.2.m2.1b"><apply id="S5.SS3.p2.2.m2.1.2.cmml" xref="S5.SS3.p2.2.m2.1.2"><in id="S5.SS3.p2.2.m2.1.2.1.cmml" xref="S5.SS3.p2.2.m2.1.2.1"></in><apply id="S5.SS3.p2.2.m2.1.2.2.cmml" xref="S5.SS3.p2.2.m2.1.2.2"><times id="S5.SS3.p2.2.m2.1.2.2.1.cmml" xref="S5.SS3.p2.2.m2.1.2.2.1"></times><ci id="S5.SS3.p2.2.m2.1.2.2.2.cmml" xref="S5.SS3.p2.2.m2.1.2.2.2">𝒙</ci><cn id="S5.SS3.p2.2.m2.1.1.cmml" type="integer" xref="S5.SS3.p2.2.m2.1.1">0</cn></apply><apply id="S5.SS3.p2.2.m2.1.2.3.cmml" xref="S5.SS3.p2.2.m2.1.2.3"><csymbol cd="ambiguous" id="S5.SS3.p2.2.m2.1.2.3.1.cmml" xref="S5.SS3.p2.2.m2.1.2.3">subscript</csymbol><ci id="S5.SS3.p2.2.m2.1.2.3.2.cmml" xref="S5.SS3.p2.2.m2.1.2.3.2">Ω</ci><apply id="S5.SS3.p2.2.m2.1.2.3.3.cmml" xref="S5.SS3.p2.2.m2.1.2.3.3"><csymbol cd="ambiguous" id="S5.SS3.p2.2.m2.1.2.3.3.1.cmml" xref="S5.SS3.p2.2.m2.1.2.3.3">subscript</csymbol><ci id="S5.SS3.p2.2.m2.1.2.3.3.2.cmml" xref="S5.SS3.p2.2.m2.1.2.3.3.2">c</ci><cn id="S5.SS3.p2.2.m2.1.2.3.3.3.cmml" type="integer" xref="S5.SS3.p2.2.m2.1.2.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.2.m2.1c">\bm{x}(0)\in\Omega_{{\rm c}_{0}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.2.m2.1d">bold_italic_x ( 0 ) ∈ roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\hat{\bm{\theta}}(0)\in\Omega_{{\rm c}_{\theta}}" class="ltx_Math" display="inline" id="S5.SS3.p2.3.m3.1"><semantics id="S5.SS3.p2.3.m3.1a"><mrow id="S5.SS3.p2.3.m3.1.2" xref="S5.SS3.p2.3.m3.1.2.cmml"><mrow id="S5.SS3.p2.3.m3.1.2.2" xref="S5.SS3.p2.3.m3.1.2.2.cmml"><mover accent="true" id="S5.SS3.p2.3.m3.1.2.2.2" xref="S5.SS3.p2.3.m3.1.2.2.2.cmml"><mi id="S5.SS3.p2.3.m3.1.2.2.2.2" xref="S5.SS3.p2.3.m3.1.2.2.2.2.cmml">𝜽</mi><mo id="S5.SS3.p2.3.m3.1.2.2.2.1" xref="S5.SS3.p2.3.m3.1.2.2.2.1.cmml">^</mo></mover><mo id="S5.SS3.p2.3.m3.1.2.2.1" xref="S5.SS3.p2.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S5.SS3.p2.3.m3.1.2.2.3.2" xref="S5.SS3.p2.3.m3.1.2.2.cmml"><mo id="S5.SS3.p2.3.m3.1.2.2.3.2.1" stretchy="false" xref="S5.SS3.p2.3.m3.1.2.2.cmml">(</mo><mn id="S5.SS3.p2.3.m3.1.1" xref="S5.SS3.p2.3.m3.1.1.cmml">0</mn><mo id="S5.SS3.p2.3.m3.1.2.2.3.2.2" stretchy="false" xref="S5.SS3.p2.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.SS3.p2.3.m3.1.2.1" xref="S5.SS3.p2.3.m3.1.2.1.cmml">∈</mo><msub id="S5.SS3.p2.3.m3.1.2.3" xref="S5.SS3.p2.3.m3.1.2.3.cmml"><mi id="S5.SS3.p2.3.m3.1.2.3.2" mathvariant="normal" xref="S5.SS3.p2.3.m3.1.2.3.2.cmml">Ω</mi><msub id="S5.SS3.p2.3.m3.1.2.3.3" xref="S5.SS3.p2.3.m3.1.2.3.3.cmml"><mi id="S5.SS3.p2.3.m3.1.2.3.3.2" mathvariant="normal" xref="S5.SS3.p2.3.m3.1.2.3.3.2.cmml">c</mi><mi id="S5.SS3.p2.3.m3.1.2.3.3.3" xref="S5.SS3.p2.3.m3.1.2.3.3.3.cmml">θ</mi></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.3.m3.1b"><apply id="S5.SS3.p2.3.m3.1.2.cmml" xref="S5.SS3.p2.3.m3.1.2"><in id="S5.SS3.p2.3.m3.1.2.1.cmml" xref="S5.SS3.p2.3.m3.1.2.1"></in><apply id="S5.SS3.p2.3.m3.1.2.2.cmml" xref="S5.SS3.p2.3.m3.1.2.2"><times id="S5.SS3.p2.3.m3.1.2.2.1.cmml" xref="S5.SS3.p2.3.m3.1.2.2.1"></times><apply id="S5.SS3.p2.3.m3.1.2.2.2.cmml" xref="S5.SS3.p2.3.m3.1.2.2.2"><ci id="S5.SS3.p2.3.m3.1.2.2.2.1.cmml" xref="S5.SS3.p2.3.m3.1.2.2.2.1">^</ci><ci id="S5.SS3.p2.3.m3.1.2.2.2.2.cmml" xref="S5.SS3.p2.3.m3.1.2.2.2.2">𝜽</ci></apply><cn id="S5.SS3.p2.3.m3.1.1.cmml" type="integer" xref="S5.SS3.p2.3.m3.1.1">0</cn></apply><apply id="S5.SS3.p2.3.m3.1.2.3.cmml" xref="S5.SS3.p2.3.m3.1.2.3"><csymbol cd="ambiguous" id="S5.SS3.p2.3.m3.1.2.3.1.cmml" xref="S5.SS3.p2.3.m3.1.2.3">subscript</csymbol><ci id="S5.SS3.p2.3.m3.1.2.3.2.cmml" xref="S5.SS3.p2.3.m3.1.2.3.2">Ω</ci><apply id="S5.SS3.p2.3.m3.1.2.3.3.cmml" xref="S5.SS3.p2.3.m3.1.2.3.3"><csymbol cd="ambiguous" id="S5.SS3.p2.3.m3.1.2.3.3.1.cmml" xref="S5.SS3.p2.3.m3.1.2.3.3">subscript</csymbol><ci id="S5.SS3.p2.3.m3.1.2.3.3.2.cmml" xref="S5.SS3.p2.3.m3.1.2.3.3.2">c</ci><ci id="S5.SS3.p2.3.m3.1.2.3.3.3.cmml" xref="S5.SS3.p2.3.m3.1.2.3.3.3">𝜃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.3.m3.1c">\hat{\bm{\theta}}(0)\in\Omega_{{\rm c}_{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.3.m3.1d">over^ start_ARG bold_italic_θ end_ARG ( 0 ) ∈ roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, there exist suitable control parameters <math alttext="k_{{\rm c}1}" class="ltx_Math" display="inline" id="S5.SS3.p2.4.m4.1"><semantics id="S5.SS3.p2.4.m4.1a"><msub id="S5.SS3.p2.4.m4.1.1" xref="S5.SS3.p2.4.m4.1.1.cmml"><mi id="S5.SS3.p2.4.m4.1.1.2" xref="S5.SS3.p2.4.m4.1.1.2.cmml">k</mi><mi id="S5.SS3.p2.4.m4.1.1.3" xref="S5.SS3.p2.4.m4.1.1.3.cmml">c1</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.4.m4.1b"><apply id="S5.SS3.p2.4.m4.1.1.cmml" xref="S5.SS3.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S5.SS3.p2.4.m4.1.1.1.cmml" xref="S5.SS3.p2.4.m4.1.1">subscript</csymbol><ci id="S5.SS3.p2.4.m4.1.1.2.cmml" xref="S5.SS3.p2.4.m4.1.1.2">𝑘</ci><ci id="S5.SS3.p2.4.m4.1.1.3.cmml" xref="S5.SS3.p2.4.m4.1.1.3">c1</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.4.m4.1c">k_{{\rm c}1}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.4.m4.1d">italic_k start_POSTSUBSCRIPT c1 end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="k_{{\rm c}n}" class="ltx_Math" display="inline" id="S5.SS3.p2.5.m5.1"><semantics id="S5.SS3.p2.5.m5.1a"><msub id="S5.SS3.p2.5.m5.1.1" xref="S5.SS3.p2.5.m5.1.1.cmml"><mi id="S5.SS3.p2.5.m5.1.1.2" xref="S5.SS3.p2.5.m5.1.1.2.cmml">k</mi><mrow id="S5.SS3.p2.5.m5.1.1.3" xref="S5.SS3.p2.5.m5.1.1.3.cmml"><mi id="S5.SS3.p2.5.m5.1.1.3.2" mathvariant="normal" xref="S5.SS3.p2.5.m5.1.1.3.2.cmml">c</mi><mo id="S5.SS3.p2.5.m5.1.1.3.1" xref="S5.SS3.p2.5.m5.1.1.3.1.cmml">⁢</mo><mi id="S5.SS3.p2.5.m5.1.1.3.3" xref="S5.SS3.p2.5.m5.1.1.3.3.cmml">n</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.5.m5.1b"><apply id="S5.SS3.p2.5.m5.1.1.cmml" xref="S5.SS3.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S5.SS3.p2.5.m5.1.1.1.cmml" xref="S5.SS3.p2.5.m5.1.1">subscript</csymbol><ci id="S5.SS3.p2.5.m5.1.1.2.cmml" xref="S5.SS3.p2.5.m5.1.1.2">𝑘</ci><apply id="S5.SS3.p2.5.m5.1.1.3.cmml" xref="S5.SS3.p2.5.m5.1.1.3"><times id="S5.SS3.p2.5.m5.1.1.3.1.cmml" xref="S5.SS3.p2.5.m5.1.1.3.1"></times><ci id="S5.SS3.p2.5.m5.1.1.3.2.cmml" xref="S5.SS3.p2.5.m5.1.1.3.2">c</ci><ci id="S5.SS3.p2.5.m5.1.1.3.3.cmml" xref="S5.SS3.p2.5.m5.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.5.m5.1c">k_{{\rm c}n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.5.m5.1d">italic_k start_POSTSUBSCRIPT roman_c italic_n end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E8" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">8</span></a>) and filtered parameters <math alttext="\alpha_{1}" class="ltx_Math" display="inline" id="S5.SS3.p2.6.m6.1"><semantics id="S5.SS3.p2.6.m6.1a"><msub id="S5.SS3.p2.6.m6.1.1" xref="S5.SS3.p2.6.m6.1.1.cmml"><mi id="S5.SS3.p2.6.m6.1.1.2" xref="S5.SS3.p2.6.m6.1.1.2.cmml">α</mi><mn id="S5.SS3.p2.6.m6.1.1.3" xref="S5.SS3.p2.6.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.6.m6.1b"><apply id="S5.SS3.p2.6.m6.1.1.cmml" xref="S5.SS3.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S5.SS3.p2.6.m6.1.1.1.cmml" xref="S5.SS3.p2.6.m6.1.1">subscript</csymbol><ci id="S5.SS3.p2.6.m6.1.1.2.cmml" xref="S5.SS3.p2.6.m6.1.1.2">𝛼</ci><cn id="S5.SS3.p2.6.m6.1.1.3.cmml" type="integer" xref="S5.SS3.p2.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.6.m6.1c">\alpha_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.6.m6.1d">italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="\alpha_{n-1}" class="ltx_Math" display="inline" id="S5.SS3.p2.7.m7.1"><semantics id="S5.SS3.p2.7.m7.1a"><msub id="S5.SS3.p2.7.m7.1.1" xref="S5.SS3.p2.7.m7.1.1.cmml"><mi id="S5.SS3.p2.7.m7.1.1.2" xref="S5.SS3.p2.7.m7.1.1.2.cmml">α</mi><mrow id="S5.SS3.p2.7.m7.1.1.3" xref="S5.SS3.p2.7.m7.1.1.3.cmml"><mi id="S5.SS3.p2.7.m7.1.1.3.2" xref="S5.SS3.p2.7.m7.1.1.3.2.cmml">n</mi><mo id="S5.SS3.p2.7.m7.1.1.3.1" xref="S5.SS3.p2.7.m7.1.1.3.1.cmml">−</mo><mn id="S5.SS3.p2.7.m7.1.1.3.3" xref="S5.SS3.p2.7.m7.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.7.m7.1b"><apply id="S5.SS3.p2.7.m7.1.1.cmml" xref="S5.SS3.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S5.SS3.p2.7.m7.1.1.1.cmml" xref="S5.SS3.p2.7.m7.1.1">subscript</csymbol><ci id="S5.SS3.p2.7.m7.1.1.2.cmml" xref="S5.SS3.p2.7.m7.1.1.2">𝛼</ci><apply id="S5.SS3.p2.7.m7.1.1.3.cmml" xref="S5.SS3.p2.7.m7.1.1.3"><minus id="S5.SS3.p2.7.m7.1.1.3.1.cmml" xref="S5.SS3.p2.7.m7.1.1.3.1"></minus><ci id="S5.SS3.p2.7.m7.1.1.3.2.cmml" xref="S5.SS3.p2.7.m7.1.1.3.2">𝑛</ci><cn id="S5.SS3.p2.7.m7.1.1.3.3.cmml" type="integer" xref="S5.SS3.p2.7.m7.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.7.m7.1c">\alpha_{n-1}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.7.m7.1d">italic_α start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E17" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">17</span></a>), <span class="ltx_text" id="S5.SS3.p2.9.2" style="color:#000099;">such that the equilibrium point <math alttext="(\bm{e}" class="ltx_math_unparsed" display="inline" id="S5.SS3.p2.8.1.m1.1"><semantics id="S5.SS3.p2.8.1.m1.1a"><mrow id="S5.SS3.p2.8.1.m1.1b"><mo id="S5.SS3.p2.8.1.m1.1.1" mathcolor="#000099" stretchy="false">(</mo><mi id="S5.SS3.p2.8.1.m1.1.2" mathcolor="#000099">𝒆</mi></mrow><annotation encoding="application/x-tex" id="S5.SS3.p2.8.1.m1.1c">(\bm{e}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.8.1.m1.1d">( bold_italic_e</annotation></semantics></math>, <math alttext="\tilde{\bm{\theta}})=\bm{0}" class="ltx_math_unparsed" display="inline" id="S5.SS3.p2.9.2.m2.1"><semantics id="S5.SS3.p2.9.2.m2.1a"><mrow id="S5.SS3.p2.9.2.m2.1b"><mover accent="true" id="S5.SS3.p2.9.2.m2.1.1"><mi id="S5.SS3.p2.9.2.m2.1.1.2" mathcolor="#000099">𝜽</mi><mo id="S5.SS3.p2.9.2.m2.1.1.1" mathcolor="#000099">~</mo></mover><mo id="S5.SS3.p2.9.2.m2.1.2" mathcolor="#000099" stretchy="false">)</mo><mo id="S5.SS3.p2.9.2.m2.1.3" mathcolor="#000099">=</mo><mn id="S5.SS3.p2.9.2.m2.1.4" mathcolor="#000099">𝟎</mn></mrow><annotation encoding="application/x-tex" id="S5.SS3.p2.9.2.m2.1c">\tilde{\bm{\theta}})=\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.9.2.m2.1d">over~ start_ARG bold_italic_θ end_ARG ) = bold_0</annotation></semantics></math> of the closed-loop system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S5.E26" title="In V-C Robustness Results ‣ V Theoretical Guarantees ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">26</span></a>) with (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>) has:</span></p> <ol class="ltx_enumerate" id="S5.I4"> <li class="ltx_item" id="S5.I4.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S5.I4.i1.p1"> <p class="ltx_p" id="S5.I4.i1.p1.3"><span class="ltx_text" id="S5.I4.i1.p1.3.1" style="color:#000099;">UUB stability on </span><math alttext="t" class="ltx_Math" display="inline" id="S5.I4.i1.p1.1.m1.1"><semantics id="S5.I4.i1.p1.1.m1.1a"><mi id="S5.I4.i1.p1.1.m1.1.1" mathcolor="#000099" xref="S5.I4.i1.p1.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S5.I4.i1.p1.1.m1.1b"><ci id="S5.I4.i1.p1.1.m1.1.1.cmml" xref="S5.I4.i1.p1.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i1.p1.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i1.p1.1.m1.1d">italic_t</annotation></semantics></math><span class="ltx_text" id="S5.I4.i1.p1.3.2" style="color:#000099;"> </span><math alttext="\in" class="ltx_Math" display="inline" id="S5.I4.i1.p1.2.m2.1"><semantics id="S5.I4.i1.p1.2.m2.1a"><mo id="S5.I4.i1.p1.2.m2.1.1" mathcolor="#000099" xref="S5.I4.i1.p1.2.m2.1.1.cmml">∈</mo><annotation-xml encoding="MathML-Content" id="S5.I4.i1.p1.2.m2.1b"><in id="S5.I4.i1.p1.2.m2.1.1.cmml" xref="S5.I4.i1.p1.2.m2.1.1"></in></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i1.p1.2.m2.1c">\in</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i1.p1.2.m2.1d">∈</annotation></semantics></math><span class="ltx_text" id="S5.I4.i1.p1.3.3" style="color:#000099;"> </span><math alttext="[0,\infty)" class="ltx_Math" display="inline" id="S5.I4.i1.p1.3.m3.2"><semantics id="S5.I4.i1.p1.3.m3.2a"><mrow id="S5.I4.i1.p1.3.m3.2.3.2" xref="S5.I4.i1.p1.3.m3.2.3.1.cmml"><mo id="S5.I4.i1.p1.3.m3.2.3.2.1" mathcolor="#000099" stretchy="false" xref="S5.I4.i1.p1.3.m3.2.3.1.cmml">[</mo><mn id="S5.I4.i1.p1.3.m3.1.1" mathcolor="#000099" xref="S5.I4.i1.p1.3.m3.1.1.cmml">0</mn><mo id="S5.I4.i1.p1.3.m3.2.3.2.2" mathcolor="#000099" xref="S5.I4.i1.p1.3.m3.2.3.1.cmml">,</mo><mi id="S5.I4.i1.p1.3.m3.2.2" mathcolor="#000099" mathvariant="normal" xref="S5.I4.i1.p1.3.m3.2.2.cmml">∞</mi><mo id="S5.I4.i1.p1.3.m3.2.3.2.3" mathcolor="#000099" stretchy="false" xref="S5.I4.i1.p1.3.m3.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.I4.i1.p1.3.m3.2b"><interval closure="closed-open" id="S5.I4.i1.p1.3.m3.2.3.1.cmml" xref="S5.I4.i1.p1.3.m3.2.3.2"><cn id="S5.I4.i1.p1.3.m3.1.1.cmml" type="integer" xref="S5.I4.i1.p1.3.m3.1.1">0</cn><infinity id="S5.I4.i1.p1.3.m3.2.2.cmml" xref="S5.I4.i1.p1.3.m3.2.2"></infinity></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i1.p1.3.m3.2c">[0,\infty)</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i1.p1.3.m3.2d">[ 0 , ∞ )</annotation></semantics></math><span class="ltx_text" id="S5.I4.i1.p1.3.4" style="color:#000099;">;</span></p> </div> </li> <li class="ltx_item" id="S5.I4.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S5.I4.i2.p1"> <p class="ltx_p" id="S5.I4.i2.p1.12"><span class="ltx_text" id="S5.I4.i2.p1.12.1" style="color:#000099;">Partial practical exponential stability on </span><math alttext="t" class="ltx_Math" display="inline" id="S5.I4.i2.p1.1.m1.1"><semantics id="S5.I4.i2.p1.1.m1.1a"><mi id="S5.I4.i2.p1.1.m1.1.1" mathcolor="#000099" xref="S5.I4.i2.p1.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S5.I4.i2.p1.1.m1.1b"><ci id="S5.I4.i2.p1.1.m1.1.1.cmml" xref="S5.I4.i2.p1.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i2.p1.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i2.p1.1.m1.1d">italic_t</annotation></semantics></math><span class="ltx_text" id="S5.I4.i2.p1.12.2" style="color:#000099;"> </span><math alttext="\in" class="ltx_Math" display="inline" id="S5.I4.i2.p1.2.m2.1"><semantics id="S5.I4.i2.p1.2.m2.1a"><mo id="S5.I4.i2.p1.2.m2.1.1" mathcolor="#000099" xref="S5.I4.i2.p1.2.m2.1.1.cmml">∈</mo><annotation-xml encoding="MathML-Content" id="S5.I4.i2.p1.2.m2.1b"><in id="S5.I4.i2.p1.2.m2.1.1.cmml" xref="S5.I4.i2.p1.2.m2.1.1"></in></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i2.p1.2.m2.1c">\in</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i2.p1.2.m2.1d">∈</annotation></semantics></math><span class="ltx_text" id="S5.I4.i2.p1.12.3" style="color:#000099;"> </span><math alttext="[T_{\rm a},\infty)" class="ltx_Math" display="inline" id="S5.I4.i2.p1.3.m3.2"><semantics id="S5.I4.i2.p1.3.m3.2a"><mrow id="S5.I4.i2.p1.3.m3.2.2.1" xref="S5.I4.i2.p1.3.m3.2.2.2.cmml"><mo id="S5.I4.i2.p1.3.m3.2.2.1.2" mathcolor="#000099" stretchy="false" xref="S5.I4.i2.p1.3.m3.2.2.2.cmml">[</mo><msub id="S5.I4.i2.p1.3.m3.2.2.1.1" xref="S5.I4.i2.p1.3.m3.2.2.1.1.cmml"><mi id="S5.I4.i2.p1.3.m3.2.2.1.1.2" mathcolor="#000099" xref="S5.I4.i2.p1.3.m3.2.2.1.1.2.cmml">T</mi><mi id="S5.I4.i2.p1.3.m3.2.2.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S5.I4.i2.p1.3.m3.2.2.1.1.3.cmml">a</mi></msub><mo id="S5.I4.i2.p1.3.m3.2.2.1.3" mathcolor="#000099" xref="S5.I4.i2.p1.3.m3.2.2.2.cmml">,</mo><mi id="S5.I4.i2.p1.3.m3.1.1" mathcolor="#000099" mathvariant="normal" xref="S5.I4.i2.p1.3.m3.1.1.cmml">∞</mi><mo id="S5.I4.i2.p1.3.m3.2.2.1.4" mathcolor="#000099" stretchy="false" xref="S5.I4.i2.p1.3.m3.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.I4.i2.p1.3.m3.2b"><interval closure="closed-open" id="S5.I4.i2.p1.3.m3.2.2.2.cmml" xref="S5.I4.i2.p1.3.m3.2.2.1"><apply id="S5.I4.i2.p1.3.m3.2.2.1.1.cmml" xref="S5.I4.i2.p1.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S5.I4.i2.p1.3.m3.2.2.1.1.1.cmml" xref="S5.I4.i2.p1.3.m3.2.2.1.1">subscript</csymbol><ci id="S5.I4.i2.p1.3.m3.2.2.1.1.2.cmml" xref="S5.I4.i2.p1.3.m3.2.2.1.1.2">𝑇</ci><ci id="S5.I4.i2.p1.3.m3.2.2.1.1.3.cmml" xref="S5.I4.i2.p1.3.m3.2.2.1.1.3">a</ci></apply><infinity id="S5.I4.i2.p1.3.m3.1.1.cmml" xref="S5.I4.i2.p1.3.m3.1.1"></infinity></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i2.p1.3.m3.2c">[T_{\rm a},\infty)</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i2.p1.3.m3.2d">[ italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT , ∞ )</annotation></semantics></math><span class="ltx_text" id="S5.I4.i2.p1.12.4" style="color:#000099;"> if partial IE in Definition 3 exists for some constants </span><math alttext="T_{\rm a}" class="ltx_Math" display="inline" id="S5.I4.i2.p1.4.m4.1"><semantics id="S5.I4.i2.p1.4.m4.1a"><msub id="S5.I4.i2.p1.4.m4.1.1" xref="S5.I4.i2.p1.4.m4.1.1.cmml"><mi id="S5.I4.i2.p1.4.m4.1.1.2" mathcolor="#000099" xref="S5.I4.i2.p1.4.m4.1.1.2.cmml">T</mi><mi id="S5.I4.i2.p1.4.m4.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S5.I4.i2.p1.4.m4.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S5.I4.i2.p1.4.m4.1b"><apply id="S5.I4.i2.p1.4.m4.1.1.cmml" xref="S5.I4.i2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S5.I4.i2.p1.4.m4.1.1.1.cmml" xref="S5.I4.i2.p1.4.m4.1.1">subscript</csymbol><ci id="S5.I4.i2.p1.4.m4.1.1.2.cmml" xref="S5.I4.i2.p1.4.m4.1.1.2">𝑇</ci><ci id="S5.I4.i2.p1.4.m4.1.1.3.cmml" xref="S5.I4.i2.p1.4.m4.1.1.3">a</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i2.p1.4.m4.1c">T_{\rm a}</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i2.p1.4.m4.1d">italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text" id="S5.I4.i2.p1.12.5" style="color:#000099;">, </span><math alttext="\sigma\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S5.I4.i2.p1.5.m5.1"><semantics id="S5.I4.i2.p1.5.m5.1a"><mrow id="S5.I4.i2.p1.5.m5.1.1" xref="S5.I4.i2.p1.5.m5.1.1.cmml"><mi id="S5.I4.i2.p1.5.m5.1.1.2" mathcolor="#000099" xref="S5.I4.i2.p1.5.m5.1.1.2.cmml">σ</mi><mo id="S5.I4.i2.p1.5.m5.1.1.1" mathcolor="#000099" xref="S5.I4.i2.p1.5.m5.1.1.1.cmml">∈</mo><msup id="S5.I4.i2.p1.5.m5.1.1.3" xref="S5.I4.i2.p1.5.m5.1.1.3.cmml"><mi id="S5.I4.i2.p1.5.m5.1.1.3.2" mathcolor="#000099" xref="S5.I4.i2.p1.5.m5.1.1.3.2.cmml">ℝ</mi><mo id="S5.I4.i2.p1.5.m5.1.1.3.3" mathcolor="#000099" xref="S5.I4.i2.p1.5.m5.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.I4.i2.p1.5.m5.1b"><apply id="S5.I4.i2.p1.5.m5.1.1.cmml" xref="S5.I4.i2.p1.5.m5.1.1"><in id="S5.I4.i2.p1.5.m5.1.1.1.cmml" xref="S5.I4.i2.p1.5.m5.1.1.1"></in><ci id="S5.I4.i2.p1.5.m5.1.1.2.cmml" xref="S5.I4.i2.p1.5.m5.1.1.2">𝜎</ci><apply id="S5.I4.i2.p1.5.m5.1.1.3.cmml" xref="S5.I4.i2.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S5.I4.i2.p1.5.m5.1.1.3.1.cmml" xref="S5.I4.i2.p1.5.m5.1.1.3">superscript</csymbol><ci id="S5.I4.i2.p1.5.m5.1.1.3.2.cmml" xref="S5.I4.i2.p1.5.m5.1.1.3.2">ℝ</ci><plus id="S5.I4.i2.p1.5.m5.1.1.3.3.cmml" xref="S5.I4.i2.p1.5.m5.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i2.p1.5.m5.1c">\sigma\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i2.p1.5.m5.1d">italic_σ ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text" id="S5.I4.i2.p1.12.6" style="color:#000099;"> and the set </span><math alttext="\mathcal{I}" class="ltx_Math" display="inline" id="S5.I4.i2.p1.6.m6.1"><semantics id="S5.I4.i2.p1.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S5.I4.i2.p1.6.m6.1.1" mathcolor="#000099" xref="S5.I4.i2.p1.6.m6.1.1.cmml">ℐ</mi><annotation-xml encoding="MathML-Content" id="S5.I4.i2.p1.6.m6.1b"><ci id="S5.I4.i2.p1.6.m6.1.1.cmml" xref="S5.I4.i2.p1.6.m6.1.1">ℐ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i2.p1.6.m6.1c">\mathcal{I}</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i2.p1.6.m6.1d">caligraphic_I</annotation></semantics></math><span class="ltx_text" id="S5.I4.i2.p1.12.7" style="color:#000099;"> no longer changes, where the tracking error </span><math alttext="\bm{e}(t)" class="ltx_Math" display="inline" id="S5.I4.i2.p1.7.m7.1"><semantics id="S5.I4.i2.p1.7.m7.1a"><mrow id="S5.I4.i2.p1.7.m7.1.2" xref="S5.I4.i2.p1.7.m7.1.2.cmml"><mi id="S5.I4.i2.p1.7.m7.1.2.2" mathcolor="#000099" xref="S5.I4.i2.p1.7.m7.1.2.2.cmml">𝒆</mi><mo id="S5.I4.i2.p1.7.m7.1.2.1" xref="S5.I4.i2.p1.7.m7.1.2.1.cmml">⁢</mo><mrow id="S5.I4.i2.p1.7.m7.1.2.3.2" xref="S5.I4.i2.p1.7.m7.1.2.cmml"><mo id="S5.I4.i2.p1.7.m7.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="S5.I4.i2.p1.7.m7.1.2.cmml">(</mo><mi id="S5.I4.i2.p1.7.m7.1.1" mathcolor="#000099" xref="S5.I4.i2.p1.7.m7.1.1.cmml">t</mi><mo id="S5.I4.i2.p1.7.m7.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="S5.I4.i2.p1.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I4.i2.p1.7.m7.1b"><apply id="S5.I4.i2.p1.7.m7.1.2.cmml" xref="S5.I4.i2.p1.7.m7.1.2"><times id="S5.I4.i2.p1.7.m7.1.2.1.cmml" xref="S5.I4.i2.p1.7.m7.1.2.1"></times><ci id="S5.I4.i2.p1.7.m7.1.2.2.cmml" xref="S5.I4.i2.p1.7.m7.1.2.2">𝒆</ci><ci id="S5.I4.i2.p1.7.m7.1.1.cmml" xref="S5.I4.i2.p1.7.m7.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i2.p1.7.m7.1c">\bm{e}(t)</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i2.p1.7.m7.1d">bold_italic_e ( italic_t )</annotation></semantics></math><span class="ltx_text" id="S5.I4.i2.p1.12.8" style="color:#000099;"> and the partial parameter estimation error </span><math alttext="\tilde{\bm{\theta}}_{\zeta}(t)" class="ltx_Math" display="inline" id="S5.I4.i2.p1.8.m8.1"><semantics id="S5.I4.i2.p1.8.m8.1a"><mrow id="S5.I4.i2.p1.8.m8.1.2" xref="S5.I4.i2.p1.8.m8.1.2.cmml"><msub id="S5.I4.i2.p1.8.m8.1.2.2" xref="S5.I4.i2.p1.8.m8.1.2.2.cmml"><mover accent="true" id="S5.I4.i2.p1.8.m8.1.2.2.2" xref="S5.I4.i2.p1.8.m8.1.2.2.2.cmml"><mi id="S5.I4.i2.p1.8.m8.1.2.2.2.2" mathcolor="#000099" xref="S5.I4.i2.p1.8.m8.1.2.2.2.2.cmml">𝜽</mi><mo id="S5.I4.i2.p1.8.m8.1.2.2.2.1" mathcolor="#000099" xref="S5.I4.i2.p1.8.m8.1.2.2.2.1.cmml">~</mo></mover><mi id="S5.I4.i2.p1.8.m8.1.2.2.3" mathcolor="#000099" xref="S5.I4.i2.p1.8.m8.1.2.2.3.cmml">ζ</mi></msub><mo id="S5.I4.i2.p1.8.m8.1.2.1" xref="S5.I4.i2.p1.8.m8.1.2.1.cmml">⁢</mo><mrow id="S5.I4.i2.p1.8.m8.1.2.3.2" xref="S5.I4.i2.p1.8.m8.1.2.cmml"><mo id="S5.I4.i2.p1.8.m8.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="S5.I4.i2.p1.8.m8.1.2.cmml">(</mo><mi id="S5.I4.i2.p1.8.m8.1.1" mathcolor="#000099" xref="S5.I4.i2.p1.8.m8.1.1.cmml">t</mi><mo id="S5.I4.i2.p1.8.m8.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="S5.I4.i2.p1.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I4.i2.p1.8.m8.1b"><apply id="S5.I4.i2.p1.8.m8.1.2.cmml" xref="S5.I4.i2.p1.8.m8.1.2"><times id="S5.I4.i2.p1.8.m8.1.2.1.cmml" xref="S5.I4.i2.p1.8.m8.1.2.1"></times><apply id="S5.I4.i2.p1.8.m8.1.2.2.cmml" xref="S5.I4.i2.p1.8.m8.1.2.2"><csymbol cd="ambiguous" id="S5.I4.i2.p1.8.m8.1.2.2.1.cmml" xref="S5.I4.i2.p1.8.m8.1.2.2">subscript</csymbol><apply id="S5.I4.i2.p1.8.m8.1.2.2.2.cmml" xref="S5.I4.i2.p1.8.m8.1.2.2.2"><ci id="S5.I4.i2.p1.8.m8.1.2.2.2.1.cmml" xref="S5.I4.i2.p1.8.m8.1.2.2.2.1">~</ci><ci id="S5.I4.i2.p1.8.m8.1.2.2.2.2.cmml" xref="S5.I4.i2.p1.8.m8.1.2.2.2.2">𝜽</ci></apply><ci id="S5.I4.i2.p1.8.m8.1.2.2.3.cmml" xref="S5.I4.i2.p1.8.m8.1.2.2.3">𝜁</ci></apply><ci id="S5.I4.i2.p1.8.m8.1.1.cmml" xref="S5.I4.i2.p1.8.m8.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i2.p1.8.m8.1c">\tilde{\bm{\theta}}_{\zeta}(t)</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i2.p1.8.m8.1d">over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math><span class="ltx_text" id="S5.I4.i2.p1.12.9" style="color:#000099;"> exponentially converge to a small neighborhood of </span><math alttext="\bm{0}" class="ltx_Math" display="inline" id="S5.I4.i2.p1.9.m9.1"><semantics id="S5.I4.i2.p1.9.m9.1a"><mn id="S5.I4.i2.p1.9.m9.1.1" mathcolor="#000099" xref="S5.I4.i2.p1.9.m9.1.1.cmml">𝟎</mn><annotation-xml encoding="MathML-Content" id="S5.I4.i2.p1.9.m9.1b"><cn id="S5.I4.i2.p1.9.m9.1.1.cmml" type="integer" xref="S5.I4.i2.p1.9.m9.1.1">0</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i2.p1.9.m9.1c">\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i2.p1.9.m9.1d">bold_0</annotation></semantics></math><span class="ltx_text" id="S5.I4.i2.p1.12.10" style="color:#000099;"> dominated by </span><math alttext="k_{{\rm c}i}" class="ltx_Math" display="inline" id="S5.I4.i2.p1.10.m10.1"><semantics id="S5.I4.i2.p1.10.m10.1a"><msub id="S5.I4.i2.p1.10.m10.1.1" xref="S5.I4.i2.p1.10.m10.1.1.cmml"><mi id="S5.I4.i2.p1.10.m10.1.1.2" mathcolor="#000099" xref="S5.I4.i2.p1.10.m10.1.1.2.cmml">k</mi><mrow id="S5.I4.i2.p1.10.m10.1.1.3" xref="S5.I4.i2.p1.10.m10.1.1.3.cmml"><mi id="S5.I4.i2.p1.10.m10.1.1.3.2" mathcolor="#000099" mathvariant="normal" xref="S5.I4.i2.p1.10.m10.1.1.3.2.cmml">c</mi><mo id="S5.I4.i2.p1.10.m10.1.1.3.1" xref="S5.I4.i2.p1.10.m10.1.1.3.1.cmml">⁢</mo><mi id="S5.I4.i2.p1.10.m10.1.1.3.3" mathcolor="#000099" xref="S5.I4.i2.p1.10.m10.1.1.3.3.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.I4.i2.p1.10.m10.1b"><apply id="S5.I4.i2.p1.10.m10.1.1.cmml" xref="S5.I4.i2.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S5.I4.i2.p1.10.m10.1.1.1.cmml" xref="S5.I4.i2.p1.10.m10.1.1">subscript</csymbol><ci id="S5.I4.i2.p1.10.m10.1.1.2.cmml" xref="S5.I4.i2.p1.10.m10.1.1.2">𝑘</ci><apply id="S5.I4.i2.p1.10.m10.1.1.3.cmml" xref="S5.I4.i2.p1.10.m10.1.1.3"><times id="S5.I4.i2.p1.10.m10.1.1.3.1.cmml" xref="S5.I4.i2.p1.10.m10.1.1.3.1"></times><ci id="S5.I4.i2.p1.10.m10.1.1.3.2.cmml" xref="S5.I4.i2.p1.10.m10.1.1.3.2">c</ci><ci id="S5.I4.i2.p1.10.m10.1.1.3.3.cmml" xref="S5.I4.i2.p1.10.m10.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i2.p1.10.m10.1c">k_{{\rm c}i}</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i2.p1.10.m10.1d">italic_k start_POSTSUBSCRIPT roman_c italic_i end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text" id="S5.I4.i2.p1.12.11" style="color:#000099;">, </span><math alttext="\kappa" class="ltx_Math" display="inline" id="S5.I4.i2.p1.11.m11.1"><semantics id="S5.I4.i2.p1.11.m11.1a"><mi id="S5.I4.i2.p1.11.m11.1.1" mathcolor="#000099" xref="S5.I4.i2.p1.11.m11.1.1.cmml">κ</mi><annotation-xml encoding="MathML-Content" id="S5.I4.i2.p1.11.m11.1b"><ci id="S5.I4.i2.p1.11.m11.1.1.cmml" xref="S5.I4.i2.p1.11.m11.1.1">𝜅</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i2.p1.11.m11.1c">\kappa</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i2.p1.11.m11.1d">italic_κ</annotation></semantics></math><span class="ltx_text" id="S5.I4.i2.p1.12.12" style="color:#000099;">, and </span><math alttext="\bar{d}" class="ltx_Math" display="inline" id="S5.I4.i2.p1.12.m12.1"><semantics id="S5.I4.i2.p1.12.m12.1a"><mover accent="true" id="S5.I4.i2.p1.12.m12.1.1" xref="S5.I4.i2.p1.12.m12.1.1.cmml"><mi id="S5.I4.i2.p1.12.m12.1.1.2" mathcolor="#000099" xref="S5.I4.i2.p1.12.m12.1.1.2.cmml">d</mi><mo id="S5.I4.i2.p1.12.m12.1.1.1" mathcolor="#000099" xref="S5.I4.i2.p1.12.m12.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S5.I4.i2.p1.12.m12.1b"><apply id="S5.I4.i2.p1.12.m12.1.1.cmml" xref="S5.I4.i2.p1.12.m12.1.1"><ci id="S5.I4.i2.p1.12.m12.1.1.1.cmml" xref="S5.I4.i2.p1.12.m12.1.1.1">¯</ci><ci id="S5.I4.i2.p1.12.m12.1.1.2.cmml" xref="S5.I4.i2.p1.12.m12.1.1.2">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i2.p1.12.m12.1c">\bar{d}</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i2.p1.12.m12.1d">over¯ start_ARG italic_d end_ARG</annotation></semantics></math><span class="ltx_text" id="S5.I4.i2.p1.12.13" style="color:#000099;">;</span></p> </div> </li> <li class="ltx_item" id="S5.I4.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">3.</span> <div class="ltx_para" id="S5.I4.i3.p1"> <p class="ltx_p" id="S5.I4.i3.p1.13"><span class="ltx_text" id="S5.I4.i3.p1.13.1" style="color:#000099;">Practical exponential stability on </span><math alttext="t" class="ltx_Math" display="inline" id="S5.I4.i3.p1.1.m1.1"><semantics id="S5.I4.i3.p1.1.m1.1a"><mi id="S5.I4.i3.p1.1.m1.1.1" mathcolor="#000099" xref="S5.I4.i3.p1.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S5.I4.i3.p1.1.m1.1b"><ci id="S5.I4.i3.p1.1.m1.1.1.cmml" xref="S5.I4.i3.p1.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i3.p1.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i3.p1.1.m1.1d">italic_t</annotation></semantics></math><span class="ltx_text" id="S5.I4.i3.p1.13.2" style="color:#000099;"> </span><math alttext="\in" class="ltx_Math" display="inline" id="S5.I4.i3.p1.2.m2.1"><semantics id="S5.I4.i3.p1.2.m2.1a"><mo id="S5.I4.i3.p1.2.m2.1.1" mathcolor="#000099" xref="S5.I4.i3.p1.2.m2.1.1.cmml">∈</mo><annotation-xml encoding="MathML-Content" id="S5.I4.i3.p1.2.m2.1b"><in id="S5.I4.i3.p1.2.m2.1.1.cmml" xref="S5.I4.i3.p1.2.m2.1.1"></in></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i3.p1.2.m2.1c">\in</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i3.p1.2.m2.1d">∈</annotation></semantics></math><span class="ltx_text" id="S5.I4.i3.p1.13.3" style="color:#000099;"> </span><math alttext="[T_{\rm e},\infty)" class="ltx_Math" display="inline" id="S5.I4.i3.p1.3.m3.2"><semantics id="S5.I4.i3.p1.3.m3.2a"><mrow id="S5.I4.i3.p1.3.m3.2.2.1" xref="S5.I4.i3.p1.3.m3.2.2.2.cmml"><mo id="S5.I4.i3.p1.3.m3.2.2.1.2" mathcolor="#000099" stretchy="false" xref="S5.I4.i3.p1.3.m3.2.2.2.cmml">[</mo><msub id="S5.I4.i3.p1.3.m3.2.2.1.1" xref="S5.I4.i3.p1.3.m3.2.2.1.1.cmml"><mi id="S5.I4.i3.p1.3.m3.2.2.1.1.2" mathcolor="#000099" xref="S5.I4.i3.p1.3.m3.2.2.1.1.2.cmml">T</mi><mi id="S5.I4.i3.p1.3.m3.2.2.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S5.I4.i3.p1.3.m3.2.2.1.1.3.cmml">e</mi></msub><mo id="S5.I4.i3.p1.3.m3.2.2.1.3" mathcolor="#000099" xref="S5.I4.i3.p1.3.m3.2.2.2.cmml">,</mo><mi id="S5.I4.i3.p1.3.m3.1.1" mathcolor="#000099" mathvariant="normal" xref="S5.I4.i3.p1.3.m3.1.1.cmml">∞</mi><mo id="S5.I4.i3.p1.3.m3.2.2.1.4" mathcolor="#000099" stretchy="false" xref="S5.I4.i3.p1.3.m3.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.I4.i3.p1.3.m3.2b"><interval closure="closed-open" id="S5.I4.i3.p1.3.m3.2.2.2.cmml" xref="S5.I4.i3.p1.3.m3.2.2.1"><apply id="S5.I4.i3.p1.3.m3.2.2.1.1.cmml" xref="S5.I4.i3.p1.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S5.I4.i3.p1.3.m3.2.2.1.1.1.cmml" xref="S5.I4.i3.p1.3.m3.2.2.1.1">subscript</csymbol><ci id="S5.I4.i3.p1.3.m3.2.2.1.1.2.cmml" xref="S5.I4.i3.p1.3.m3.2.2.1.1.2">𝑇</ci><ci id="S5.I4.i3.p1.3.m3.2.2.1.1.3.cmml" xref="S5.I4.i3.p1.3.m3.2.2.1.1.3">e</ci></apply><infinity id="S5.I4.i3.p1.3.m3.1.1.cmml" xref="S5.I4.i3.p1.3.m3.1.1"></infinity></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i3.p1.3.m3.2c">[T_{\rm e},\infty)</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i3.p1.3.m3.2d">[ italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT , ∞ )</annotation></semantics></math><span class="ltx_text" id="S5.I4.i3.p1.13.4" style="color:#000099;"> if IE in Definition 2 exists for some constants </span><math alttext="T_{\rm e}" class="ltx_Math" display="inline" id="S5.I4.i3.p1.4.m4.1"><semantics id="S5.I4.i3.p1.4.m4.1a"><msub id="S5.I4.i3.p1.4.m4.1.1" xref="S5.I4.i3.p1.4.m4.1.1.cmml"><mi id="S5.I4.i3.p1.4.m4.1.1.2" mathcolor="#000099" xref="S5.I4.i3.p1.4.m4.1.1.2.cmml">T</mi><mi id="S5.I4.i3.p1.4.m4.1.1.3" mathcolor="#000099" mathvariant="normal" xref="S5.I4.i3.p1.4.m4.1.1.3.cmml">e</mi></msub><annotation-xml encoding="MathML-Content" id="S5.I4.i3.p1.4.m4.1b"><apply id="S5.I4.i3.p1.4.m4.1.1.cmml" xref="S5.I4.i3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S5.I4.i3.p1.4.m4.1.1.1.cmml" xref="S5.I4.i3.p1.4.m4.1.1">subscript</csymbol><ci id="S5.I4.i3.p1.4.m4.1.1.2.cmml" xref="S5.I4.i3.p1.4.m4.1.1.2">𝑇</ci><ci id="S5.I4.i3.p1.4.m4.1.1.3.cmml" xref="S5.I4.i3.p1.4.m4.1.1.3">e</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i3.p1.4.m4.1c">T_{\rm e}</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i3.p1.4.m4.1d">italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text" id="S5.I4.i3.p1.13.5" style="color:#000099;">, </span><math alttext="\sigma" class="ltx_Math" display="inline" id="S5.I4.i3.p1.5.m5.1"><semantics id="S5.I4.i3.p1.5.m5.1a"><mi id="S5.I4.i3.p1.5.m5.1.1" mathcolor="#000099" xref="S5.I4.i3.p1.5.m5.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.I4.i3.p1.5.m5.1b"><ci id="S5.I4.i3.p1.5.m5.1.1.cmml" xref="S5.I4.i3.p1.5.m5.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i3.p1.5.m5.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i3.p1.5.m5.1d">italic_σ</annotation></semantics></math><span class="ltx_text" id="S5.I4.i3.p1.13.6" style="color:#000099;"> </span><math alttext="\in" class="ltx_Math" display="inline" id="S5.I4.i3.p1.6.m6.1"><semantics id="S5.I4.i3.p1.6.m6.1a"><mo id="S5.I4.i3.p1.6.m6.1.1" mathcolor="#000099" xref="S5.I4.i3.p1.6.m6.1.1.cmml">∈</mo><annotation-xml encoding="MathML-Content" id="S5.I4.i3.p1.6.m6.1b"><in id="S5.I4.i3.p1.6.m6.1.1.cmml" xref="S5.I4.i3.p1.6.m6.1.1"></in></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i3.p1.6.m6.1c">\in</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i3.p1.6.m6.1d">∈</annotation></semantics></math><span class="ltx_text" id="S5.I4.i3.p1.13.7" style="color:#000099;"> </span><math alttext="\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S5.I4.i3.p1.7.m7.1"><semantics id="S5.I4.i3.p1.7.m7.1a"><msup id="S5.I4.i3.p1.7.m7.1.1" xref="S5.I4.i3.p1.7.m7.1.1.cmml"><mi id="S5.I4.i3.p1.7.m7.1.1.2" mathcolor="#000099" xref="S5.I4.i3.p1.7.m7.1.1.2.cmml">ℝ</mi><mo id="S5.I4.i3.p1.7.m7.1.1.3" mathcolor="#000099" xref="S5.I4.i3.p1.7.m7.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S5.I4.i3.p1.7.m7.1b"><apply id="S5.I4.i3.p1.7.m7.1.1.cmml" xref="S5.I4.i3.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S5.I4.i3.p1.7.m7.1.1.1.cmml" xref="S5.I4.i3.p1.7.m7.1.1">superscript</csymbol><ci id="S5.I4.i3.p1.7.m7.1.1.2.cmml" xref="S5.I4.i3.p1.7.m7.1.1.2">ℝ</ci><plus id="S5.I4.i3.p1.7.m7.1.1.3.cmml" xref="S5.I4.i3.p1.7.m7.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i3.p1.7.m7.1c">\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i3.p1.7.m7.1d">blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text" id="S5.I4.i3.p1.13.8" style="color:#000099;">, where the tracking error </span><math alttext="\bm{e}(t)" class="ltx_Math" display="inline" id="S5.I4.i3.p1.8.m8.1"><semantics id="S5.I4.i3.p1.8.m8.1a"><mrow id="S5.I4.i3.p1.8.m8.1.2" xref="S5.I4.i3.p1.8.m8.1.2.cmml"><mi id="S5.I4.i3.p1.8.m8.1.2.2" mathcolor="#000099" xref="S5.I4.i3.p1.8.m8.1.2.2.cmml">𝒆</mi><mo id="S5.I4.i3.p1.8.m8.1.2.1" xref="S5.I4.i3.p1.8.m8.1.2.1.cmml">⁢</mo><mrow id="S5.I4.i3.p1.8.m8.1.2.3.2" xref="S5.I4.i3.p1.8.m8.1.2.cmml"><mo id="S5.I4.i3.p1.8.m8.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="S5.I4.i3.p1.8.m8.1.2.cmml">(</mo><mi id="S5.I4.i3.p1.8.m8.1.1" mathcolor="#000099" xref="S5.I4.i3.p1.8.m8.1.1.cmml">t</mi><mo id="S5.I4.i3.p1.8.m8.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="S5.I4.i3.p1.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I4.i3.p1.8.m8.1b"><apply id="S5.I4.i3.p1.8.m8.1.2.cmml" xref="S5.I4.i3.p1.8.m8.1.2"><times id="S5.I4.i3.p1.8.m8.1.2.1.cmml" xref="S5.I4.i3.p1.8.m8.1.2.1"></times><ci id="S5.I4.i3.p1.8.m8.1.2.2.cmml" xref="S5.I4.i3.p1.8.m8.1.2.2">𝒆</ci><ci id="S5.I4.i3.p1.8.m8.1.1.cmml" xref="S5.I4.i3.p1.8.m8.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i3.p1.8.m8.1c">\bm{e}(t)</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i3.p1.8.m8.1d">bold_italic_e ( italic_t )</annotation></semantics></math><span class="ltx_text" id="S5.I4.i3.p1.13.9" style="color:#000099;"> and the parameter estimation error </span><math alttext="\tilde{\bm{\theta}}(t)" class="ltx_Math" display="inline" id="S5.I4.i3.p1.9.m9.1"><semantics id="S5.I4.i3.p1.9.m9.1a"><mrow id="S5.I4.i3.p1.9.m9.1.2" xref="S5.I4.i3.p1.9.m9.1.2.cmml"><mover accent="true" id="S5.I4.i3.p1.9.m9.1.2.2" xref="S5.I4.i3.p1.9.m9.1.2.2.cmml"><mi id="S5.I4.i3.p1.9.m9.1.2.2.2" mathcolor="#000099" xref="S5.I4.i3.p1.9.m9.1.2.2.2.cmml">𝜽</mi><mo id="S5.I4.i3.p1.9.m9.1.2.2.1" mathcolor="#000099" xref="S5.I4.i3.p1.9.m9.1.2.2.1.cmml">~</mo></mover><mo id="S5.I4.i3.p1.9.m9.1.2.1" xref="S5.I4.i3.p1.9.m9.1.2.1.cmml">⁢</mo><mrow id="S5.I4.i3.p1.9.m9.1.2.3.2" xref="S5.I4.i3.p1.9.m9.1.2.cmml"><mo id="S5.I4.i3.p1.9.m9.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="S5.I4.i3.p1.9.m9.1.2.cmml">(</mo><mi id="S5.I4.i3.p1.9.m9.1.1" mathcolor="#000099" xref="S5.I4.i3.p1.9.m9.1.1.cmml">t</mi><mo id="S5.I4.i3.p1.9.m9.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="S5.I4.i3.p1.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I4.i3.p1.9.m9.1b"><apply id="S5.I4.i3.p1.9.m9.1.2.cmml" xref="S5.I4.i3.p1.9.m9.1.2"><times id="S5.I4.i3.p1.9.m9.1.2.1.cmml" xref="S5.I4.i3.p1.9.m9.1.2.1"></times><apply id="S5.I4.i3.p1.9.m9.1.2.2.cmml" xref="S5.I4.i3.p1.9.m9.1.2.2"><ci id="S5.I4.i3.p1.9.m9.1.2.2.1.cmml" xref="S5.I4.i3.p1.9.m9.1.2.2.1">~</ci><ci id="S5.I4.i3.p1.9.m9.1.2.2.2.cmml" xref="S5.I4.i3.p1.9.m9.1.2.2.2">𝜽</ci></apply><ci id="S5.I4.i3.p1.9.m9.1.1.cmml" xref="S5.I4.i3.p1.9.m9.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i3.p1.9.m9.1c">\tilde{\bm{\theta}}(t)</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i3.p1.9.m9.1d">over~ start_ARG bold_italic_θ end_ARG ( italic_t )</annotation></semantics></math><span class="ltx_text" id="S5.I4.i3.p1.13.10" style="color:#000099;"> exponentially converge to a small neighborhood of </span><math alttext="\bm{0}" class="ltx_Math" display="inline" id="S5.I4.i3.p1.10.m10.1"><semantics id="S5.I4.i3.p1.10.m10.1a"><mn id="S5.I4.i3.p1.10.m10.1.1" mathcolor="#000099" xref="S5.I4.i3.p1.10.m10.1.1.cmml">𝟎</mn><annotation-xml encoding="MathML-Content" id="S5.I4.i3.p1.10.m10.1b"><cn id="S5.I4.i3.p1.10.m10.1.1.cmml" type="integer" xref="S5.I4.i3.p1.10.m10.1.1">0</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i3.p1.10.m10.1c">\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i3.p1.10.m10.1d">bold_0</annotation></semantics></math><span class="ltx_text" id="S5.I4.i3.p1.13.11" style="color:#000099;"> dominated by </span><math alttext="k_{{\rm c}i}" class="ltx_Math" display="inline" id="S5.I4.i3.p1.11.m11.1"><semantics id="S5.I4.i3.p1.11.m11.1a"><msub id="S5.I4.i3.p1.11.m11.1.1" xref="S5.I4.i3.p1.11.m11.1.1.cmml"><mi id="S5.I4.i3.p1.11.m11.1.1.2" mathcolor="#000099" xref="S5.I4.i3.p1.11.m11.1.1.2.cmml">k</mi><mrow id="S5.I4.i3.p1.11.m11.1.1.3" xref="S5.I4.i3.p1.11.m11.1.1.3.cmml"><mi id="S5.I4.i3.p1.11.m11.1.1.3.2" mathcolor="#000099" mathvariant="normal" xref="S5.I4.i3.p1.11.m11.1.1.3.2.cmml">c</mi><mo id="S5.I4.i3.p1.11.m11.1.1.3.1" xref="S5.I4.i3.p1.11.m11.1.1.3.1.cmml">⁢</mo><mi id="S5.I4.i3.p1.11.m11.1.1.3.3" mathcolor="#000099" xref="S5.I4.i3.p1.11.m11.1.1.3.3.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.I4.i3.p1.11.m11.1b"><apply id="S5.I4.i3.p1.11.m11.1.1.cmml" xref="S5.I4.i3.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S5.I4.i3.p1.11.m11.1.1.1.cmml" xref="S5.I4.i3.p1.11.m11.1.1">subscript</csymbol><ci id="S5.I4.i3.p1.11.m11.1.1.2.cmml" xref="S5.I4.i3.p1.11.m11.1.1.2">𝑘</ci><apply id="S5.I4.i3.p1.11.m11.1.1.3.cmml" xref="S5.I4.i3.p1.11.m11.1.1.3"><times id="S5.I4.i3.p1.11.m11.1.1.3.1.cmml" xref="S5.I4.i3.p1.11.m11.1.1.3.1"></times><ci id="S5.I4.i3.p1.11.m11.1.1.3.2.cmml" xref="S5.I4.i3.p1.11.m11.1.1.3.2">c</ci><ci id="S5.I4.i3.p1.11.m11.1.1.3.3.cmml" xref="S5.I4.i3.p1.11.m11.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i3.p1.11.m11.1c">k_{{\rm c}i}</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i3.p1.11.m11.1d">italic_k start_POSTSUBSCRIPT roman_c italic_i end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text" id="S5.I4.i3.p1.13.12" style="color:#000099;">, </span><math alttext="\kappa" class="ltx_Math" display="inline" id="S5.I4.i3.p1.12.m12.1"><semantics id="S5.I4.i3.p1.12.m12.1a"><mi id="S5.I4.i3.p1.12.m12.1.1" mathcolor="#000099" xref="S5.I4.i3.p1.12.m12.1.1.cmml">κ</mi><annotation-xml encoding="MathML-Content" id="S5.I4.i3.p1.12.m12.1b"><ci id="S5.I4.i3.p1.12.m12.1.1.cmml" xref="S5.I4.i3.p1.12.m12.1.1">𝜅</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i3.p1.12.m12.1c">\kappa</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i3.p1.12.m12.1d">italic_κ</annotation></semantics></math><span class="ltx_text" id="S5.I4.i3.p1.13.13" style="color:#000099;">, and </span><math alttext="\bar{d}" class="ltx_Math" display="inline" id="S5.I4.i3.p1.13.m13.1"><semantics id="S5.I4.i3.p1.13.m13.1a"><mover accent="true" id="S5.I4.i3.p1.13.m13.1.1" xref="S5.I4.i3.p1.13.m13.1.1.cmml"><mi id="S5.I4.i3.p1.13.m13.1.1.2" mathcolor="#000099" xref="S5.I4.i3.p1.13.m13.1.1.2.cmml">d</mi><mo id="S5.I4.i3.p1.13.m13.1.1.1" mathcolor="#000099" xref="S5.I4.i3.p1.13.m13.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S5.I4.i3.p1.13.m13.1b"><apply id="S5.I4.i3.p1.13.m13.1.1.cmml" xref="S5.I4.i3.p1.13.m13.1.1"><ci id="S5.I4.i3.p1.13.m13.1.1.1.cmml" xref="S5.I4.i3.p1.13.m13.1.1.1">¯</ci><ci id="S5.I4.i3.p1.13.m13.1.1.2.cmml" xref="S5.I4.i3.p1.13.m13.1.1.2">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i3.p1.13.m13.1c">\bar{d}</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i3.p1.13.m13.1d">over¯ start_ARG italic_d end_ARG</annotation></semantics></math><span class="ltx_text" id="S5.I4.i3.p1.13.14" style="color:#000099;">.</span></p> </div> </li> </ol> </div> <div class="ltx_para" id="S5.SS3.p3"> <p class="ltx_p" id="S5.SS3.p3.2"><span class="ltx_text ltx_font_italic" id="S5.SS3.p3.2.1">Remark 10:</span> Composite learning is an innovative methodology that exploits memory regressor extension to achieve the exponential stability of adaptive systems with guaranteed parameter convergence without the PE condition and has been widely applied to uncertain nonlinear systems<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib6" title="">6</a>, <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib32" title="">32</a>, <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib33" title="">33</a>, <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib34" title="">34</a>, <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib35" title="">35</a>, <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib36" title="">36</a>]</cite>. It follows from Theorem 2 that the (partial) exponential stability of the closed-loop system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E7" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">7</span></a>) with (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>) can be achieved under the (partial) IE condition. Actually, the exponential stability obtained not only implies the rapid convergence of both the tracking error <math alttext="\bm{e}" class="ltx_Math" display="inline" id="S5.SS3.p3.1.m1.1"><semantics id="S5.SS3.p3.1.m1.1a"><mi id="S5.SS3.p3.1.m1.1.1" xref="S5.SS3.p3.1.m1.1.1.cmml">𝒆</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.p3.1.m1.1b"><ci id="S5.SS3.p3.1.m1.1.1.cmml" xref="S5.SS3.p3.1.m1.1.1">𝒆</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p3.1.m1.1c">\bm{e}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p3.1.m1.1d">bold_italic_e</annotation></semantics></math> and the estimation error <math alttext="\tilde{\bm{\theta}}" class="ltx_Math" display="inline" id="S5.SS3.p3.2.m2.1"><semantics id="S5.SS3.p3.2.m2.1a"><mover accent="true" id="S5.SS3.p3.2.m2.1.1" xref="S5.SS3.p3.2.m2.1.1.cmml"><mi id="S5.SS3.p3.2.m2.1.1.2" xref="S5.SS3.p3.2.m2.1.1.2.cmml">𝜽</mi><mo id="S5.SS3.p3.2.m2.1.1.1" xref="S5.SS3.p3.2.m2.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S5.SS3.p3.2.m2.1b"><apply id="S5.SS3.p3.2.m2.1.1.cmml" xref="S5.SS3.p3.2.m2.1.1"><ci id="S5.SS3.p3.2.m2.1.1.1.cmml" xref="S5.SS3.p3.2.m2.1.1.1">~</ci><ci id="S5.SS3.p3.2.m2.1.1.2.cmml" xref="S5.SS3.p3.2.m2.1.1.2">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p3.2.m2.1c">\tilde{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p3.2.m2.1d">over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math> but endows robustness against system perturbations resulting from, e.g., external disturbances, unmodeled dynamics, and measurement noise <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib37" title="">37</a>]</cite>.</p> </div> <div class="ltx_para" id="S5.SS3.p4"> <p class="ltx_p" id="S5.SS3.p4.16"><span class="ltx_text ltx_font_italic" id="S5.SS3.p4.16.5">Remark 11:</span> The parameter selection of the proposed CLBC in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E6" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">6</span></a>) with (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>) follows the following rules: 1) The control gains <math alttext="k_{{\rm c}i}" class="ltx_Math" display="inline" id="S5.SS3.p4.1.m1.1"><semantics id="S5.SS3.p4.1.m1.1a"><msub id="S5.SS3.p4.1.m1.1.1" xref="S5.SS3.p4.1.m1.1.1.cmml"><mi id="S5.SS3.p4.1.m1.1.1.2" xref="S5.SS3.p4.1.m1.1.1.2.cmml">k</mi><mrow id="S5.SS3.p4.1.m1.1.1.3" xref="S5.SS3.p4.1.m1.1.1.3.cmml"><mi id="S5.SS3.p4.1.m1.1.1.3.2" mathvariant="normal" xref="S5.SS3.p4.1.m1.1.1.3.2.cmml">c</mi><mo id="S5.SS3.p4.1.m1.1.1.3.1" xref="S5.SS3.p4.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S5.SS3.p4.1.m1.1.1.3.3" xref="S5.SS3.p4.1.m1.1.1.3.3.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.SS3.p4.1.m1.1b"><apply id="S5.SS3.p4.1.m1.1.1.cmml" xref="S5.SS3.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS3.p4.1.m1.1.1.1.cmml" xref="S5.SS3.p4.1.m1.1.1">subscript</csymbol><ci id="S5.SS3.p4.1.m1.1.1.2.cmml" xref="S5.SS3.p4.1.m1.1.1.2">𝑘</ci><apply id="S5.SS3.p4.1.m1.1.1.3.cmml" xref="S5.SS3.p4.1.m1.1.1.3"><times id="S5.SS3.p4.1.m1.1.1.3.1.cmml" xref="S5.SS3.p4.1.m1.1.1.3.1"></times><ci id="S5.SS3.p4.1.m1.1.1.3.2.cmml" xref="S5.SS3.p4.1.m1.1.1.3.2">c</ci><ci id="S5.SS3.p4.1.m1.1.1.3.3.cmml" xref="S5.SS3.p4.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p4.1.m1.1c">k_{{\rm c}i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p4.1.m1.1d">italic_k start_POSTSUBSCRIPT roman_c italic_i end_POSTSUBSCRIPT</annotation></semantics></math> (<math alttext="i=1" class="ltx_Math" display="inline" id="S5.SS3.p4.2.m2.1"><semantics id="S5.SS3.p4.2.m2.1a"><mrow id="S5.SS3.p4.2.m2.1.1" xref="S5.SS3.p4.2.m2.1.1.cmml"><mi id="S5.SS3.p4.2.m2.1.1.2" xref="S5.SS3.p4.2.m2.1.1.2.cmml">i</mi><mo id="S5.SS3.p4.2.m2.1.1.1" xref="S5.SS3.p4.2.m2.1.1.1.cmml">=</mo><mn id="S5.SS3.p4.2.m2.1.1.3" xref="S5.SS3.p4.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p4.2.m2.1b"><apply id="S5.SS3.p4.2.m2.1.1.cmml" xref="S5.SS3.p4.2.m2.1.1"><eq id="S5.SS3.p4.2.m2.1.1.1.cmml" xref="S5.SS3.p4.2.m2.1.1.1"></eq><ci id="S5.SS3.p4.2.m2.1.1.2.cmml" xref="S5.SS3.p4.2.m2.1.1.2">𝑖</ci><cn id="S5.SS3.p4.2.m2.1.1.3.cmml" type="integer" xref="S5.SS3.p4.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p4.2.m2.1c">i=1</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p4.2.m2.1d">italic_i = 1</annotation></semantics></math> to <math alttext="n" class="ltx_Math" display="inline" id="S5.SS3.p4.3.m3.1"><semantics id="S5.SS3.p4.3.m3.1a"><mi id="S5.SS3.p4.3.m3.1.1" xref="S5.SS3.p4.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.p4.3.m3.1b"><ci id="S5.SS3.p4.3.m3.1.1.cmml" xref="S5.SS3.p4.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p4.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p4.3.m3.1d">italic_n</annotation></semantics></math>) in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E8" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">8</span></a>) should be chosen to be greater than <math alttext="1/4" class="ltx_Math" display="inline" id="S5.SS3.p4.4.m4.1"><semantics id="S5.SS3.p4.4.m4.1a"><mrow id="S5.SS3.p4.4.m4.1.1" xref="S5.SS3.p4.4.m4.1.1.cmml"><mn id="S5.SS3.p4.4.m4.1.1.2" xref="S5.SS3.p4.4.m4.1.1.2.cmml">1</mn><mo id="S5.SS3.p4.4.m4.1.1.1" xref="S5.SS3.p4.4.m4.1.1.1.cmml">/</mo><mn id="S5.SS3.p4.4.m4.1.1.3" xref="S5.SS3.p4.4.m4.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p4.4.m4.1b"><apply id="S5.SS3.p4.4.m4.1.1.cmml" xref="S5.SS3.p4.4.m4.1.1"><divide id="S5.SS3.p4.4.m4.1.1.1.cmml" xref="S5.SS3.p4.4.m4.1.1.1"></divide><cn id="S5.SS3.p4.4.m4.1.1.2.cmml" type="integer" xref="S5.SS3.p4.4.m4.1.1.2">1</cn><cn id="S5.SS3.p4.4.m4.1.1.3.cmml" type="integer" xref="S5.SS3.p4.4.m4.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p4.4.m4.1c">1/4</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p4.4.m4.1d">1 / 4</annotation></semantics></math>, as shown in the proof of Item 1 of Theorem 2, and are subject to control saturation and measurement noise; 2) the filtering constants <math alttext="\alpha_{i}" class="ltx_Math" display="inline" id="S5.SS3.p4.5.m5.1"><semantics id="S5.SS3.p4.5.m5.1a"><msub id="S5.SS3.p4.5.m5.1.1" xref="S5.SS3.p4.5.m5.1.1.cmml"><mi id="S5.SS3.p4.5.m5.1.1.2" xref="S5.SS3.p4.5.m5.1.1.2.cmml">α</mi><mi id="S5.SS3.p4.5.m5.1.1.3" xref="S5.SS3.p4.5.m5.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS3.p4.5.m5.1b"><apply id="S5.SS3.p4.5.m5.1.1.cmml" xref="S5.SS3.p4.5.m5.1.1"><csymbol cd="ambiguous" id="S5.SS3.p4.5.m5.1.1.1.cmml" xref="S5.SS3.p4.5.m5.1.1">subscript</csymbol><ci id="S5.SS3.p4.5.m5.1.1.2.cmml" xref="S5.SS3.p4.5.m5.1.1.2">𝛼</ci><ci id="S5.SS3.p4.5.m5.1.1.3.cmml" xref="S5.SS3.p4.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p4.5.m5.1c">\alpha_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p4.5.m5.1d">italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> (<math alttext="i=1" class="ltx_Math" display="inline" id="S5.SS3.p4.6.m6.1"><semantics id="S5.SS3.p4.6.m6.1a"><mrow id="S5.SS3.p4.6.m6.1.1" xref="S5.SS3.p4.6.m6.1.1.cmml"><mi id="S5.SS3.p4.6.m6.1.1.2" xref="S5.SS3.p4.6.m6.1.1.2.cmml">i</mi><mo id="S5.SS3.p4.6.m6.1.1.1" xref="S5.SS3.p4.6.m6.1.1.1.cmml">=</mo><mn id="S5.SS3.p4.6.m6.1.1.3" xref="S5.SS3.p4.6.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p4.6.m6.1b"><apply id="S5.SS3.p4.6.m6.1.1.cmml" xref="S5.SS3.p4.6.m6.1.1"><eq id="S5.SS3.p4.6.m6.1.1.1.cmml" xref="S5.SS3.p4.6.m6.1.1.1"></eq><ci id="S5.SS3.p4.6.m6.1.1.2.cmml" xref="S5.SS3.p4.6.m6.1.1.2">𝑖</ci><cn id="S5.SS3.p4.6.m6.1.1.3.cmml" type="integer" xref="S5.SS3.p4.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p4.6.m6.1c">i=1</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p4.6.m6.1d">italic_i = 1</annotation></semantics></math> to <math alttext="n-1" class="ltx_Math" display="inline" id="S5.SS3.p4.7.m7.1"><semantics id="S5.SS3.p4.7.m7.1a"><mrow id="S5.SS3.p4.7.m7.1.1" xref="S5.SS3.p4.7.m7.1.1.cmml"><mi id="S5.SS3.p4.7.m7.1.1.2" xref="S5.SS3.p4.7.m7.1.1.2.cmml">n</mi><mo id="S5.SS3.p4.7.m7.1.1.1" xref="S5.SS3.p4.7.m7.1.1.1.cmml">−</mo><mn id="S5.SS3.p4.7.m7.1.1.3" xref="S5.SS3.p4.7.m7.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p4.7.m7.1b"><apply id="S5.SS3.p4.7.m7.1.1.cmml" xref="S5.SS3.p4.7.m7.1.1"><minus id="S5.SS3.p4.7.m7.1.1.1.cmml" xref="S5.SS3.p4.7.m7.1.1.1"></minus><ci id="S5.SS3.p4.7.m7.1.1.2.cmml" xref="S5.SS3.p4.7.m7.1.1.2">𝑛</ci><cn id="S5.SS3.p4.7.m7.1.1.3.cmml" type="integer" xref="S5.SS3.p4.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p4.7.m7.1c">n-1</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p4.7.m7.1d">italic_n - 1</annotation></semantics></math>) in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E17" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">17</span></a>) are chosen to be suitably large such that the filtered regressor <math alttext="\Phi_{\rm f}" class="ltx_Math" display="inline" id="S5.SS3.p4.8.m8.1"><semantics id="S5.SS3.p4.8.m8.1a"><msub id="S5.SS3.p4.8.m8.1.1" xref="S5.SS3.p4.8.m8.1.1.cmml"><mi id="S5.SS3.p4.8.m8.1.1.2" mathvariant="normal" xref="S5.SS3.p4.8.m8.1.1.2.cmml">Φ</mi><mi id="S5.SS3.p4.8.m8.1.1.3" mathvariant="normal" xref="S5.SS3.p4.8.m8.1.1.3.cmml">f</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS3.p4.8.m8.1b"><apply id="S5.SS3.p4.8.m8.1.1.cmml" xref="S5.SS3.p4.8.m8.1.1"><csymbol cd="ambiguous" id="S5.SS3.p4.8.m8.1.1.1.cmml" xref="S5.SS3.p4.8.m8.1.1">subscript</csymbol><ci id="S5.SS3.p4.8.m8.1.1.2.cmml" xref="S5.SS3.p4.8.m8.1.1.2">Φ</ci><ci id="S5.SS3.p4.8.m8.1.1.3.cmml" xref="S5.SS3.p4.8.m8.1.1.3">f</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p4.8.m8.1c">\Phi_{\rm f}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p4.8.m8.1d">roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT</annotation></semantics></math> can approach the regressor <math alttext="\Phi" class="ltx_Math" display="inline" id="S5.SS3.p4.9.m9.1"><semantics id="S5.SS3.p4.9.m9.1a"><mi id="S5.SS3.p4.9.m9.1.1" mathvariant="normal" xref="S5.SS3.p4.9.m9.1.1.cmml">Φ</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.p4.9.m9.1b"><ci id="S5.SS3.p4.9.m9.1.1.cmml" xref="S5.SS3.p4.9.m9.1.1">Φ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p4.9.m9.1c">\Phi</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p4.9.m9.1d">roman_Φ</annotation></semantics></math> while filtering out high-frequency unmodelled dynamics; 3) increasing the learning parameters <math alttext="\kappa" class="ltx_Math" display="inline" id="S5.SS3.p4.10.m10.1"><semantics id="S5.SS3.p4.10.m10.1a"><mi id="S5.SS3.p4.10.m10.1.1" xref="S5.SS3.p4.10.m10.1.1.cmml">κ</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.p4.10.m10.1b"><ci id="S5.SS3.p4.10.m10.1.1.cmml" xref="S5.SS3.p4.10.m10.1.1">𝜅</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p4.10.m10.1c">\kappa</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p4.10.m10.1d">italic_κ</annotation></semantics></math> and <math alttext="\Gamma" class="ltx_Math" display="inline" id="S5.SS3.p4.11.m11.1"><semantics id="S5.SS3.p4.11.m11.1a"><mi id="S5.SS3.p4.11.m11.1.1" mathvariant="normal" xref="S5.SS3.p4.11.m11.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.p4.11.m11.1b"><ci id="S5.SS3.p4.11.m11.1.1.cmml" xref="S5.SS3.p4.11.m11.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p4.11.m11.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p4.11.m11.1d">roman_Γ</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>) can speed up learning while increasing the required sampling frequency and noise sensibility; 4) increasing the integral duration <math alttext="\tau_{\rm d}" class="ltx_Math" display="inline" id="S5.SS3.p4.12.m12.1"><semantics id="S5.SS3.p4.12.m12.1a"><msub id="S5.SS3.p4.12.m12.1.1" xref="S5.SS3.p4.12.m12.1.1.cmml"><mi id="S5.SS3.p4.12.m12.1.1.2" xref="S5.SS3.p4.12.m12.1.1.2.cmml">τ</mi><mi id="S5.SS3.p4.12.m12.1.1.3" mathvariant="normal" xref="S5.SS3.p4.12.m12.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS3.p4.12.m12.1b"><apply id="S5.SS3.p4.12.m12.1.1.cmml" xref="S5.SS3.p4.12.m12.1.1"><csymbol cd="ambiguous" id="S5.SS3.p4.12.m12.1.1.1.cmml" xref="S5.SS3.p4.12.m12.1.1">subscript</csymbol><ci id="S5.SS3.p4.12.m12.1.1.2.cmml" xref="S5.SS3.p4.12.m12.1.1.2">𝜏</ci><ci id="S5.SS3.p4.12.m12.1.1.3.cmml" xref="S5.SS3.p4.12.m12.1.1.3">d</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p4.12.m12.1c">\tau_{\rm d}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p4.12.m12.1d">italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E15" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">15</span></a>) can improve parameter convergence while increasing memory usage and enlarging low-frequency unmodelled effects. <span class="ltx_text" id="S5.SS3.p4.16.4" style="color:#000099;">Based on <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib38" title="">38</a>, <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib33" title="">33</a>]</cite>, we usually set <math alttext="\tau_{\rm d}\in" class="ltx_Math" display="inline" id="S5.SS3.p4.13.1.m1.1"><semantics id="S5.SS3.p4.13.1.m1.1a"><mrow id="S5.SS3.p4.13.1.m1.1.1" xref="S5.SS3.p4.13.1.m1.1.1.cmml"><msub id="S5.SS3.p4.13.1.m1.1.1.2" xref="S5.SS3.p4.13.1.m1.1.1.2.cmml"><mi id="S5.SS3.p4.13.1.m1.1.1.2.2" mathcolor="#000099" xref="S5.SS3.p4.13.1.m1.1.1.2.2.cmml">τ</mi><mi id="S5.SS3.p4.13.1.m1.1.1.2.3" mathcolor="#000099" mathvariant="normal" xref="S5.SS3.p4.13.1.m1.1.1.2.3.cmml">d</mi></msub><mo id="S5.SS3.p4.13.1.m1.1.1.1" mathcolor="#000099" xref="S5.SS3.p4.13.1.m1.1.1.1.cmml">∈</mo><mi id="S5.SS3.p4.13.1.m1.1.1.3" xref="S5.SS3.p4.13.1.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p4.13.1.m1.1b"><apply id="S5.SS3.p4.13.1.m1.1.1.cmml" xref="S5.SS3.p4.13.1.m1.1.1"><in id="S5.SS3.p4.13.1.m1.1.1.1.cmml" xref="S5.SS3.p4.13.1.m1.1.1.1"></in><apply id="S5.SS3.p4.13.1.m1.1.1.2.cmml" xref="S5.SS3.p4.13.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.SS3.p4.13.1.m1.1.1.2.1.cmml" xref="S5.SS3.p4.13.1.m1.1.1.2">subscript</csymbol><ci id="S5.SS3.p4.13.1.m1.1.1.2.2.cmml" xref="S5.SS3.p4.13.1.m1.1.1.2.2">𝜏</ci><ci id="S5.SS3.p4.13.1.m1.1.1.2.3.cmml" xref="S5.SS3.p4.13.1.m1.1.1.2.3">d</ci></apply><csymbol cd="latexml" id="S5.SS3.p4.13.1.m1.1.1.3.cmml" xref="S5.SS3.p4.13.1.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p4.13.1.m1.1c">\tau_{\rm d}\in</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p4.13.1.m1.1d">italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT ∈</annotation></semantics></math> [1, 25] s and <math alttext="\sigma\in" class="ltx_Math" display="inline" id="S5.SS3.p4.14.2.m2.1"><semantics id="S5.SS3.p4.14.2.m2.1a"><mrow id="S5.SS3.p4.14.2.m2.1.1" xref="S5.SS3.p4.14.2.m2.1.1.cmml"><mi id="S5.SS3.p4.14.2.m2.1.1.2" mathcolor="#000099" xref="S5.SS3.p4.14.2.m2.1.1.2.cmml">σ</mi><mo id="S5.SS3.p4.14.2.m2.1.1.1" mathcolor="#000099" xref="S5.SS3.p4.14.2.m2.1.1.1.cmml">∈</mo><mi id="S5.SS3.p4.14.2.m2.1.1.3" xref="S5.SS3.p4.14.2.m2.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p4.14.2.m2.1b"><apply id="S5.SS3.p4.14.2.m2.1.1.cmml" xref="S5.SS3.p4.14.2.m2.1.1"><in id="S5.SS3.p4.14.2.m2.1.1.1.cmml" xref="S5.SS3.p4.14.2.m2.1.1.1"></in><ci id="S5.SS3.p4.14.2.m2.1.1.2.cmml" xref="S5.SS3.p4.14.2.m2.1.1.2">𝜎</ci><csymbol cd="latexml" id="S5.SS3.p4.14.2.m2.1.1.3.cmml" xref="S5.SS3.p4.14.2.m2.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p4.14.2.m2.1c">\sigma\in</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p4.14.2.m2.1d">italic_σ ∈</annotation></semantics></math> [<math alttext="10^{-4}" class="ltx_Math" display="inline" id="S5.SS3.p4.15.3.m3.1"><semantics id="S5.SS3.p4.15.3.m3.1a"><msup id="S5.SS3.p4.15.3.m3.1.1" xref="S5.SS3.p4.15.3.m3.1.1.cmml"><mn id="S5.SS3.p4.15.3.m3.1.1.2" mathcolor="#000099" xref="S5.SS3.p4.15.3.m3.1.1.2.cmml">10</mn><mrow id="S5.SS3.p4.15.3.m3.1.1.3" xref="S5.SS3.p4.15.3.m3.1.1.3.cmml"><mo id="S5.SS3.p4.15.3.m3.1.1.3a" mathcolor="#000099" xref="S5.SS3.p4.15.3.m3.1.1.3.cmml">−</mo><mn id="S5.SS3.p4.15.3.m3.1.1.3.2" mathcolor="#000099" xref="S5.SS3.p4.15.3.m3.1.1.3.2.cmml">4</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S5.SS3.p4.15.3.m3.1b"><apply id="S5.SS3.p4.15.3.m3.1.1.cmml" xref="S5.SS3.p4.15.3.m3.1.1"><csymbol cd="ambiguous" id="S5.SS3.p4.15.3.m3.1.1.1.cmml" xref="S5.SS3.p4.15.3.m3.1.1">superscript</csymbol><cn id="S5.SS3.p4.15.3.m3.1.1.2.cmml" type="integer" xref="S5.SS3.p4.15.3.m3.1.1.2">10</cn><apply id="S5.SS3.p4.15.3.m3.1.1.3.cmml" xref="S5.SS3.p4.15.3.m3.1.1.3"><minus id="S5.SS3.p4.15.3.m3.1.1.3.1.cmml" xref="S5.SS3.p4.15.3.m3.1.1.3"></minus><cn id="S5.SS3.p4.15.3.m3.1.1.3.2.cmml" type="integer" xref="S5.SS3.p4.15.3.m3.1.1.3.2">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p4.15.3.m3.1c">10^{-4}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p4.15.3.m3.1d">10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="10^{-2}" class="ltx_Math" display="inline" id="S5.SS3.p4.16.4.m4.1"><semantics id="S5.SS3.p4.16.4.m4.1a"><msup id="S5.SS3.p4.16.4.m4.1.1" xref="S5.SS3.p4.16.4.m4.1.1.cmml"><mn id="S5.SS3.p4.16.4.m4.1.1.2" mathcolor="#000099" xref="S5.SS3.p4.16.4.m4.1.1.2.cmml">10</mn><mrow id="S5.SS3.p4.16.4.m4.1.1.3" xref="S5.SS3.p4.16.4.m4.1.1.3.cmml"><mo id="S5.SS3.p4.16.4.m4.1.1.3a" mathcolor="#000099" xref="S5.SS3.p4.16.4.m4.1.1.3.cmml">−</mo><mn id="S5.SS3.p4.16.4.m4.1.1.3.2" mathcolor="#000099" xref="S5.SS3.p4.16.4.m4.1.1.3.2.cmml">2</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S5.SS3.p4.16.4.m4.1b"><apply id="S5.SS3.p4.16.4.m4.1.1.cmml" xref="S5.SS3.p4.16.4.m4.1.1"><csymbol cd="ambiguous" id="S5.SS3.p4.16.4.m4.1.1.1.cmml" xref="S5.SS3.p4.16.4.m4.1.1">superscript</csymbol><cn id="S5.SS3.p4.16.4.m4.1.1.2.cmml" type="integer" xref="S5.SS3.p4.16.4.m4.1.1.2">10</cn><apply id="S5.SS3.p4.16.4.m4.1.1.3.cmml" xref="S5.SS3.p4.16.4.m4.1.1.3"><minus id="S5.SS3.p4.16.4.m4.1.1.3.1.cmml" xref="S5.SS3.p4.16.4.m4.1.1.3"></minus><cn id="S5.SS3.p4.16.4.m4.1.1.3.2.cmml" type="integer" xref="S5.SS3.p4.16.4.m4.1.1.3.2">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p4.16.4.m4.1c">10^{-2}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p4.16.4.m4.1d">10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT</annotation></semantics></math>].</span></p> </div> </section> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">VI </span><span class="ltx_text ltx_font_smallcaps" id="S6.1.1">Simulation Studies</span> </h2> <section class="ltx_subsection" id="S6.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S6.SS1.4.1.1">VI-A</span> </span><span class="ltx_text ltx_font_italic" id="S6.SS1.5.2">Stability and Convergence Comparison</span> </h3> <div class="ltx_para" id="S6.SS1.p1"> <p class="ltx_p" id="S6.SS1.p1.6">This section is devoted to verifying the exponential stability and parameter convergence of the proposed CLBC in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E6" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">6</span></a>) with (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>) under various excitation conditions. Consider a mass-spring-damping model as follows <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib39" title="">39</a>]</cite>:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx27"> <tbody id="S6.Ex12"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\left\{\begin{array}[]{l}\dot{x}_{1}=x_{2},\\ \dot{x}_{2}=x_{3}+\bm{\varphi}_{2}^{T}(\bm{x}_{2})\bm{\theta},\\ \dot{x}_{3}=u,\\ y=x_{1}\end{array}\right." class="ltx_Math" display="inline" id="S6.Ex12.m1.1"><semantics id="S6.Ex12.m1.1a"><mrow id="S6.Ex12.m1.1.2.2" xref="S6.Ex12.m1.1.2.1.cmml"><mo id="S6.Ex12.m1.1.2.2.1" xref="S6.Ex12.m1.1.2.1.1.cmml">{</mo><mtable id="S6.Ex12.m1.1.1" rowspacing="0pt" xref="S6.Ex12.m1.1.1.cmml"><mtr id="S6.Ex12.m1.1.1a" xref="S6.Ex12.m1.1.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S6.Ex12.m1.1.1b" xref="S6.Ex12.m1.1.1.cmml"><mrow id="S6.Ex8.1.1.1" xref="S6.Ex8.1.1.1.1.cmml"><mrow id="S6.Ex8.1.1.1.1" xref="S6.Ex8.1.1.1.1.cmml"><msub id="S6.Ex8.1.1.1.1.2" xref="S6.Ex8.1.1.1.1.2.cmml"><mover accent="true" id="S6.Ex8.1.1.1.1.2.2" xref="S6.Ex8.1.1.1.1.2.2.cmml"><mi id="S6.Ex8.1.1.1.1.2.2.2" xref="S6.Ex8.1.1.1.1.2.2.2.cmml">x</mi><mo id="S6.Ex8.1.1.1.1.2.2.1" xref="S6.Ex8.1.1.1.1.2.2.1.cmml">˙</mo></mover><mn id="S6.Ex8.1.1.1.1.2.3" xref="S6.Ex8.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S6.Ex8.1.1.1.1.1" xref="S6.Ex8.1.1.1.1.1.cmml">=</mo><msub id="S6.Ex8.1.1.1.1.3" xref="S6.Ex8.1.1.1.1.3.cmml"><mi id="S6.Ex8.1.1.1.1.3.2" xref="S6.Ex8.1.1.1.1.3.2.cmml">x</mi><mn id="S6.Ex8.1.1.1.1.3.3" xref="S6.Ex8.1.1.1.1.3.3.cmml">2</mn></msub></mrow><mo id="S6.Ex8.1.1.1.2" xref="S6.Ex8.1.1.1.1.cmml">,</mo></mrow></mtd></mtr><mtr id="S6.Ex12.m1.1.1c" xref="S6.Ex12.m1.1.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S6.Ex12.m1.1.1d" xref="S6.Ex12.m1.1.1.cmml"><mrow id="S6.Ex9.1.1.1" xref="S6.Ex9.1.1.1.1.cmml"><mrow id="S6.Ex9.1.1.1.1" xref="S6.Ex9.1.1.1.1.cmml"><msub id="S6.Ex9.1.1.1.1.3" xref="S6.Ex9.1.1.1.1.3.cmml"><mover accent="true" id="S6.Ex9.1.1.1.1.3.2" xref="S6.Ex9.1.1.1.1.3.2.cmml"><mi id="S6.Ex9.1.1.1.1.3.2.2" xref="S6.Ex9.1.1.1.1.3.2.2.cmml">x</mi><mo id="S6.Ex9.1.1.1.1.3.2.1" xref="S6.Ex9.1.1.1.1.3.2.1.cmml">˙</mo></mover><mn id="S6.Ex9.1.1.1.1.3.3" xref="S6.Ex9.1.1.1.1.3.3.cmml">2</mn></msub><mo id="S6.Ex9.1.1.1.1.2" xref="S6.Ex9.1.1.1.1.2.cmml">=</mo><mrow id="S6.Ex9.1.1.1.1.1" xref="S6.Ex9.1.1.1.1.1.cmml"><msub id="S6.Ex9.1.1.1.1.1.3" xref="S6.Ex9.1.1.1.1.1.3.cmml"><mi id="S6.Ex9.1.1.1.1.1.3.2" xref="S6.Ex9.1.1.1.1.1.3.2.cmml">x</mi><mn id="S6.Ex9.1.1.1.1.1.3.3" xref="S6.Ex9.1.1.1.1.1.3.3.cmml">3</mn></msub><mo id="S6.Ex9.1.1.1.1.1.2" xref="S6.Ex9.1.1.1.1.1.2.cmml">+</mo><mrow id="S6.Ex9.1.1.1.1.1.1" xref="S6.Ex9.1.1.1.1.1.1.cmml"><msubsup id="S6.Ex9.1.1.1.1.1.1.3" xref="S6.Ex9.1.1.1.1.1.1.3.cmml"><mi id="S6.Ex9.1.1.1.1.1.1.3.2.2" xref="S6.Ex9.1.1.1.1.1.1.3.2.2.cmml">𝝋</mi><mn id="S6.Ex9.1.1.1.1.1.1.3.2.3" xref="S6.Ex9.1.1.1.1.1.1.3.2.3.cmml">2</mn><mi id="S6.Ex9.1.1.1.1.1.1.3.3" xref="S6.Ex9.1.1.1.1.1.1.3.3.cmml">T</mi></msubsup><mo id="S6.Ex9.1.1.1.1.1.1.2" xref="S6.Ex9.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S6.Ex9.1.1.1.1.1.1.1.1" xref="S6.Ex9.1.1.1.1.1.1.1.1.1.cmml"><mo id="S6.Ex9.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S6.Ex9.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S6.Ex9.1.1.1.1.1.1.1.1.1" xref="S6.Ex9.1.1.1.1.1.1.1.1.1.cmml"><mi id="S6.Ex9.1.1.1.1.1.1.1.1.1.2" xref="S6.Ex9.1.1.1.1.1.1.1.1.1.2.cmml">𝒙</mi><mn id="S6.Ex9.1.1.1.1.1.1.1.1.1.3" xref="S6.Ex9.1.1.1.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S6.Ex9.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S6.Ex9.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S6.Ex9.1.1.1.1.1.1.2a" xref="S6.Ex9.1.1.1.1.1.1.2.cmml">⁢</mo><mi id="S6.Ex9.1.1.1.1.1.1.4" xref="S6.Ex9.1.1.1.1.1.1.4.cmml">𝜽</mi></mrow></mrow></mrow><mo id="S6.Ex9.1.1.1.2" xref="S6.Ex9.1.1.1.1.cmml">,</mo></mrow></mtd></mtr><mtr id="S6.Ex12.m1.1.1e" xref="S6.Ex12.m1.1.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S6.Ex12.m1.1.1f" xref="S6.Ex12.m1.1.1.cmml"><mrow id="S6.Ex10.1.1.1" xref="S6.Ex10.1.1.1.1.cmml"><mrow id="S6.Ex10.1.1.1.1" xref="S6.Ex10.1.1.1.1.cmml"><msub id="S6.Ex10.1.1.1.1.2" xref="S6.Ex10.1.1.1.1.2.cmml"><mover accent="true" id="S6.Ex10.1.1.1.1.2.2" xref="S6.Ex10.1.1.1.1.2.2.cmml"><mi id="S6.Ex10.1.1.1.1.2.2.2" xref="S6.Ex10.1.1.1.1.2.2.2.cmml">x</mi><mo id="S6.Ex10.1.1.1.1.2.2.1" xref="S6.Ex10.1.1.1.1.2.2.1.cmml">˙</mo></mover><mn id="S6.Ex10.1.1.1.1.2.3" xref="S6.Ex10.1.1.1.1.2.3.cmml">3</mn></msub><mo id="S6.Ex10.1.1.1.1.1" xref="S6.Ex10.1.1.1.1.1.cmml">=</mo><mi id="S6.Ex10.1.1.1.1.3" xref="S6.Ex10.1.1.1.1.3.cmml">u</mi></mrow><mo id="S6.Ex10.1.1.1.2" xref="S6.Ex10.1.1.1.1.cmml">,</mo></mrow></mtd></mtr><mtr id="S6.Ex12.m1.1.1g" xref="S6.Ex12.m1.1.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S6.Ex12.m1.1.1h" xref="S6.Ex12.m1.1.1.cmml"><mrow id="S6.Ex11.1.1" xref="S6.Ex11.1.1.cmml"><mi id="S6.Ex11.1.1.2" xref="S6.Ex11.1.1.2.cmml">y</mi><mo id="S6.Ex11.1.1.1" xref="S6.Ex11.1.1.1.cmml">=</mo><msub id="S6.Ex11.1.1.3" xref="S6.Ex11.1.1.3.cmml"><mi id="S6.Ex11.1.1.3.2" xref="S6.Ex11.1.1.3.2.cmml">x</mi><mn id="S6.Ex11.1.1.3.3" xref="S6.Ex11.1.1.3.3.cmml">1</mn></msub></mrow></mtd></mtr></mtable><mi id="S6.Ex12.m1.1.2.2.2" xref="S6.Ex12.m1.1.2.1.1.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex12.m1.1b"><apply id="S6.Ex12.m1.1.2.1.cmml" xref="S6.Ex12.m1.1.2.2"><csymbol cd="latexml" id="S6.Ex12.m1.1.2.1.1.cmml" xref="S6.Ex12.m1.1.2.2.1">cases</csymbol><matrix id="S6.Ex12.m1.1.1.cmml" xref="S6.Ex12.m1.1.1"><matrixrow id="S6.Ex12.m1.1.1a.cmml" xref="S6.Ex12.m1.1.1"><apply id="S6.Ex8.1.1.1.1.cmml" xref="S6.Ex8.1.1.1"><eq id="S6.Ex8.1.1.1.1.1.cmml" xref="S6.Ex8.1.1.1.1.1"></eq><apply id="S6.Ex8.1.1.1.1.2.cmml" xref="S6.Ex8.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.Ex8.1.1.1.1.2.1.cmml" xref="S6.Ex8.1.1.1.1.2">subscript</csymbol><apply id="S6.Ex8.1.1.1.1.2.2.cmml" xref="S6.Ex8.1.1.1.1.2.2"><ci id="S6.Ex8.1.1.1.1.2.2.1.cmml" xref="S6.Ex8.1.1.1.1.2.2.1">˙</ci><ci id="S6.Ex8.1.1.1.1.2.2.2.cmml" xref="S6.Ex8.1.1.1.1.2.2.2">𝑥</ci></apply><cn id="S6.Ex8.1.1.1.1.2.3.cmml" type="integer" xref="S6.Ex8.1.1.1.1.2.3">1</cn></apply><apply id="S6.Ex8.1.1.1.1.3.cmml" xref="S6.Ex8.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.Ex8.1.1.1.1.3.1.cmml" xref="S6.Ex8.1.1.1.1.3">subscript</csymbol><ci id="S6.Ex8.1.1.1.1.3.2.cmml" xref="S6.Ex8.1.1.1.1.3.2">𝑥</ci><cn id="S6.Ex8.1.1.1.1.3.3.cmml" type="integer" xref="S6.Ex8.1.1.1.1.3.3">2</cn></apply></apply></matrixrow><matrixrow id="S6.Ex12.m1.1.1b.cmml" xref="S6.Ex12.m1.1.1"><apply id="S6.Ex9.1.1.1.1.cmml" xref="S6.Ex9.1.1.1"><eq id="S6.Ex9.1.1.1.1.2.cmml" xref="S6.Ex9.1.1.1.1.2"></eq><apply id="S6.Ex9.1.1.1.1.3.cmml" xref="S6.Ex9.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.Ex9.1.1.1.1.3.1.cmml" xref="S6.Ex9.1.1.1.1.3">subscript</csymbol><apply id="S6.Ex9.1.1.1.1.3.2.cmml" xref="S6.Ex9.1.1.1.1.3.2"><ci id="S6.Ex9.1.1.1.1.3.2.1.cmml" xref="S6.Ex9.1.1.1.1.3.2.1">˙</ci><ci id="S6.Ex9.1.1.1.1.3.2.2.cmml" xref="S6.Ex9.1.1.1.1.3.2.2">𝑥</ci></apply><cn id="S6.Ex9.1.1.1.1.3.3.cmml" type="integer" xref="S6.Ex9.1.1.1.1.3.3">2</cn></apply><apply id="S6.Ex9.1.1.1.1.1.cmml" xref="S6.Ex9.1.1.1.1.1"><plus id="S6.Ex9.1.1.1.1.1.2.cmml" xref="S6.Ex9.1.1.1.1.1.2"></plus><apply id="S6.Ex9.1.1.1.1.1.3.cmml" xref="S6.Ex9.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.Ex9.1.1.1.1.1.3.1.cmml" xref="S6.Ex9.1.1.1.1.1.3">subscript</csymbol><ci id="S6.Ex9.1.1.1.1.1.3.2.cmml" xref="S6.Ex9.1.1.1.1.1.3.2">𝑥</ci><cn id="S6.Ex9.1.1.1.1.1.3.3.cmml" type="integer" xref="S6.Ex9.1.1.1.1.1.3.3">3</cn></apply><apply id="S6.Ex9.1.1.1.1.1.1.cmml" xref="S6.Ex9.1.1.1.1.1.1"><times id="S6.Ex9.1.1.1.1.1.1.2.cmml" xref="S6.Ex9.1.1.1.1.1.1.2"></times><apply id="S6.Ex9.1.1.1.1.1.1.3.cmml" xref="S6.Ex9.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.Ex9.1.1.1.1.1.1.3.1.cmml" xref="S6.Ex9.1.1.1.1.1.1.3">superscript</csymbol><apply id="S6.Ex9.1.1.1.1.1.1.3.2.cmml" xref="S6.Ex9.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.Ex9.1.1.1.1.1.1.3.2.1.cmml" xref="S6.Ex9.1.1.1.1.1.1.3">subscript</csymbol><ci id="S6.Ex9.1.1.1.1.1.1.3.2.2.cmml" xref="S6.Ex9.1.1.1.1.1.1.3.2.2">𝝋</ci><cn id="S6.Ex9.1.1.1.1.1.1.3.2.3.cmml" type="integer" xref="S6.Ex9.1.1.1.1.1.1.3.2.3">2</cn></apply><ci id="S6.Ex9.1.1.1.1.1.1.3.3.cmml" xref="S6.Ex9.1.1.1.1.1.1.3.3">𝑇</ci></apply><apply id="S6.Ex9.1.1.1.1.1.1.1.1.1.cmml" xref="S6.Ex9.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.Ex9.1.1.1.1.1.1.1.1.1.1.cmml" xref="S6.Ex9.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S6.Ex9.1.1.1.1.1.1.1.1.1.2.cmml" xref="S6.Ex9.1.1.1.1.1.1.1.1.1.2">𝒙</ci><cn id="S6.Ex9.1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.Ex9.1.1.1.1.1.1.1.1.1.3">2</cn></apply><ci id="S6.Ex9.1.1.1.1.1.1.4.cmml" xref="S6.Ex9.1.1.1.1.1.1.4">𝜽</ci></apply></apply></apply></matrixrow><matrixrow id="S6.Ex12.m1.1.1c.cmml" xref="S6.Ex12.m1.1.1"><apply id="S6.Ex10.1.1.1.1.cmml" xref="S6.Ex10.1.1.1"><eq id="S6.Ex10.1.1.1.1.1.cmml" xref="S6.Ex10.1.1.1.1.1"></eq><apply id="S6.Ex10.1.1.1.1.2.cmml" xref="S6.Ex10.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.Ex10.1.1.1.1.2.1.cmml" xref="S6.Ex10.1.1.1.1.2">subscript</csymbol><apply id="S6.Ex10.1.1.1.1.2.2.cmml" xref="S6.Ex10.1.1.1.1.2.2"><ci id="S6.Ex10.1.1.1.1.2.2.1.cmml" xref="S6.Ex10.1.1.1.1.2.2.1">˙</ci><ci id="S6.Ex10.1.1.1.1.2.2.2.cmml" xref="S6.Ex10.1.1.1.1.2.2.2">𝑥</ci></apply><cn id="S6.Ex10.1.1.1.1.2.3.cmml" type="integer" xref="S6.Ex10.1.1.1.1.2.3">3</cn></apply><ci id="S6.Ex10.1.1.1.1.3.cmml" xref="S6.Ex10.1.1.1.1.3">𝑢</ci></apply></matrixrow><matrixrow id="S6.Ex12.m1.1.1d.cmml" xref="S6.Ex12.m1.1.1"><apply id="S6.Ex11.1.1.cmml" xref="S6.Ex11.1.1"><eq id="S6.Ex11.1.1.1.cmml" xref="S6.Ex11.1.1.1"></eq><ci id="S6.Ex11.1.1.2.cmml" xref="S6.Ex11.1.1.2">𝑦</ci><apply id="S6.Ex11.1.1.3.cmml" xref="S6.Ex11.1.1.3"><csymbol cd="ambiguous" id="S6.Ex11.1.1.3.1.cmml" xref="S6.Ex11.1.1.3">subscript</csymbol><ci id="S6.Ex11.1.1.3.2.cmml" xref="S6.Ex11.1.1.3.2">𝑥</ci><cn id="S6.Ex11.1.1.3.3.cmml" type="integer" xref="S6.Ex11.1.1.3.3">1</cn></apply></apply></matrixrow></matrix></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex12.m1.1c">\displaystyle\left\{\begin{array}[]{l}\dot{x}_{1}=x_{2},\\ \dot{x}_{2}=x_{3}+\bm{\varphi}_{2}^{T}(\bm{x}_{2})\bm{\theta},\\ \dot{x}_{3}=u,\\ y=x_{1}\end{array}\right.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex12.m1.1d">{ start_ARRAY start_ROW start_CELL over˙ start_ARG italic_x end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL over˙ start_ARG italic_x end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_x start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + bold_italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) bold_italic_θ , end_CELL end_ROW start_ROW start_CELL over˙ start_ARG italic_x end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_u , end_CELL end_ROW start_ROW start_CELL italic_y = italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS1.p1.5">where <math alttext="x_{1}\in\mathbb{R}" class="ltx_Math" display="inline" id="S6.SS1.p1.1.m1.1"><semantics id="S6.SS1.p1.1.m1.1a"><mrow id="S6.SS1.p1.1.m1.1.1" xref="S6.SS1.p1.1.m1.1.1.cmml"><msub id="S6.SS1.p1.1.m1.1.1.2" xref="S6.SS1.p1.1.m1.1.1.2.cmml"><mi id="S6.SS1.p1.1.m1.1.1.2.2" xref="S6.SS1.p1.1.m1.1.1.2.2.cmml">x</mi><mn id="S6.SS1.p1.1.m1.1.1.2.3" xref="S6.SS1.p1.1.m1.1.1.2.3.cmml">1</mn></msub><mo id="S6.SS1.p1.1.m1.1.1.1" xref="S6.SS1.p1.1.m1.1.1.1.cmml">∈</mo><mi id="S6.SS1.p1.1.m1.1.1.3" xref="S6.SS1.p1.1.m1.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p1.1.m1.1b"><apply id="S6.SS1.p1.1.m1.1.1.cmml" xref="S6.SS1.p1.1.m1.1.1"><in id="S6.SS1.p1.1.m1.1.1.1.cmml" xref="S6.SS1.p1.1.m1.1.1.1"></in><apply id="S6.SS1.p1.1.m1.1.1.2.cmml" xref="S6.SS1.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.p1.1.m1.1.1.2.1.cmml" xref="S6.SS1.p1.1.m1.1.1.2">subscript</csymbol><ci id="S6.SS1.p1.1.m1.1.1.2.2.cmml" xref="S6.SS1.p1.1.m1.1.1.2.2">𝑥</ci><cn id="S6.SS1.p1.1.m1.1.1.2.3.cmml" type="integer" xref="S6.SS1.p1.1.m1.1.1.2.3">1</cn></apply><ci id="S6.SS1.p1.1.m1.1.1.3.cmml" xref="S6.SS1.p1.1.m1.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p1.1.m1.1c">x_{1}\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p1.1.m1.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ blackboard_R</annotation></semantics></math> denotes a mass position, <math alttext="u\in\mathbb{R}" class="ltx_Math" display="inline" id="S6.SS1.p1.2.m2.1"><semantics id="S6.SS1.p1.2.m2.1a"><mrow id="S6.SS1.p1.2.m2.1.1" xref="S6.SS1.p1.2.m2.1.1.cmml"><mi id="S6.SS1.p1.2.m2.1.1.2" xref="S6.SS1.p1.2.m2.1.1.2.cmml">u</mi><mo id="S6.SS1.p1.2.m2.1.1.1" xref="S6.SS1.p1.2.m2.1.1.1.cmml">∈</mo><mi id="S6.SS1.p1.2.m2.1.1.3" xref="S6.SS1.p1.2.m2.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p1.2.m2.1b"><apply id="S6.SS1.p1.2.m2.1.1.cmml" xref="S6.SS1.p1.2.m2.1.1"><in id="S6.SS1.p1.2.m2.1.1.1.cmml" xref="S6.SS1.p1.2.m2.1.1.1"></in><ci id="S6.SS1.p1.2.m2.1.1.2.cmml" xref="S6.SS1.p1.2.m2.1.1.2">𝑢</ci><ci id="S6.SS1.p1.2.m2.1.1.3.cmml" xref="S6.SS1.p1.2.m2.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p1.2.m2.1c">u\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p1.2.m2.1d">italic_u ∈ blackboard_R</annotation></semantics></math> is a control input, and <math alttext="\bm{\theta}\in\mathbb{R}^{3}" class="ltx_Math" display="inline" id="S6.SS1.p1.3.m3.1"><semantics id="S6.SS1.p1.3.m3.1a"><mrow id="S6.SS1.p1.3.m3.1.1" xref="S6.SS1.p1.3.m3.1.1.cmml"><mi id="S6.SS1.p1.3.m3.1.1.2" xref="S6.SS1.p1.3.m3.1.1.2.cmml">𝜽</mi><mo id="S6.SS1.p1.3.m3.1.1.1" xref="S6.SS1.p1.3.m3.1.1.1.cmml">∈</mo><msup id="S6.SS1.p1.3.m3.1.1.3" xref="S6.SS1.p1.3.m3.1.1.3.cmml"><mi id="S6.SS1.p1.3.m3.1.1.3.2" xref="S6.SS1.p1.3.m3.1.1.3.2.cmml">ℝ</mi><mn id="S6.SS1.p1.3.m3.1.1.3.3" xref="S6.SS1.p1.3.m3.1.1.3.3.cmml">3</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p1.3.m3.1b"><apply id="S6.SS1.p1.3.m3.1.1.cmml" xref="S6.SS1.p1.3.m3.1.1"><in id="S6.SS1.p1.3.m3.1.1.1.cmml" xref="S6.SS1.p1.3.m3.1.1.1"></in><ci id="S6.SS1.p1.3.m3.1.1.2.cmml" xref="S6.SS1.p1.3.m3.1.1.2">𝜽</ci><apply id="S6.SS1.p1.3.m3.1.1.3.cmml" xref="S6.SS1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.p1.3.m3.1.1.3.1.cmml" xref="S6.SS1.p1.3.m3.1.1.3">superscript</csymbol><ci id="S6.SS1.p1.3.m3.1.1.3.2.cmml" xref="S6.SS1.p1.3.m3.1.1.3.2">ℝ</ci><cn id="S6.SS1.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S6.SS1.p1.3.m3.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p1.3.m3.1c">\bm{\theta}\in\mathbb{R}^{3}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p1.3.m3.1d">bold_italic_θ ∈ blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math> is a unknown parameter vector. Noting (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S2.E4" title="In II Problem Formulation ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">4</span></a>), one has <math alttext="\bm{\varphi}_{2}({\bm{x}}_{2})=[-x_{2},-x_{1},-x_{2}^{3}]^{T}" class="ltx_Math" display="inline" id="S6.SS1.p1.4.m4.4"><semantics id="S6.SS1.p1.4.m4.4a"><mrow id="S6.SS1.p1.4.m4.4.4" xref="S6.SS1.p1.4.m4.4.4.cmml"><mrow id="S6.SS1.p1.4.m4.1.1.1" xref="S6.SS1.p1.4.m4.1.1.1.cmml"><msub id="S6.SS1.p1.4.m4.1.1.1.3" xref="S6.SS1.p1.4.m4.1.1.1.3.cmml"><mi id="S6.SS1.p1.4.m4.1.1.1.3.2" xref="S6.SS1.p1.4.m4.1.1.1.3.2.cmml">𝝋</mi><mn id="S6.SS1.p1.4.m4.1.1.1.3.3" xref="S6.SS1.p1.4.m4.1.1.1.3.3.cmml">2</mn></msub><mo id="S6.SS1.p1.4.m4.1.1.1.2" xref="S6.SS1.p1.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S6.SS1.p1.4.m4.1.1.1.1.1" xref="S6.SS1.p1.4.m4.1.1.1.1.1.1.cmml"><mo id="S6.SS1.p1.4.m4.1.1.1.1.1.2" stretchy="false" xref="S6.SS1.p1.4.m4.1.1.1.1.1.1.cmml">(</mo><msub id="S6.SS1.p1.4.m4.1.1.1.1.1.1" xref="S6.SS1.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="S6.SS1.p1.4.m4.1.1.1.1.1.1.2" xref="S6.SS1.p1.4.m4.1.1.1.1.1.1.2.cmml">𝒙</mi><mn id="S6.SS1.p1.4.m4.1.1.1.1.1.1.3" xref="S6.SS1.p1.4.m4.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S6.SS1.p1.4.m4.1.1.1.1.1.3" stretchy="false" xref="S6.SS1.p1.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS1.p1.4.m4.4.4.5" xref="S6.SS1.p1.4.m4.4.4.5.cmml">=</mo><msup id="S6.SS1.p1.4.m4.4.4.4" xref="S6.SS1.p1.4.m4.4.4.4.cmml"><mrow id="S6.SS1.p1.4.m4.4.4.4.3.3" xref="S6.SS1.p1.4.m4.4.4.4.3.4.cmml"><mo id="S6.SS1.p1.4.m4.4.4.4.3.3.4" stretchy="false" xref="S6.SS1.p1.4.m4.4.4.4.3.4.cmml">[</mo><mrow id="S6.SS1.p1.4.m4.2.2.2.1.1.1" xref="S6.SS1.p1.4.m4.2.2.2.1.1.1.cmml"><mo id="S6.SS1.p1.4.m4.2.2.2.1.1.1a" xref="S6.SS1.p1.4.m4.2.2.2.1.1.1.cmml">−</mo><msub id="S6.SS1.p1.4.m4.2.2.2.1.1.1.2" xref="S6.SS1.p1.4.m4.2.2.2.1.1.1.2.cmml"><mi id="S6.SS1.p1.4.m4.2.2.2.1.1.1.2.2" xref="S6.SS1.p1.4.m4.2.2.2.1.1.1.2.2.cmml">x</mi><mn id="S6.SS1.p1.4.m4.2.2.2.1.1.1.2.3" xref="S6.SS1.p1.4.m4.2.2.2.1.1.1.2.3.cmml">2</mn></msub></mrow><mo id="S6.SS1.p1.4.m4.4.4.4.3.3.5" xref="S6.SS1.p1.4.m4.4.4.4.3.4.cmml">,</mo><mrow id="S6.SS1.p1.4.m4.3.3.3.2.2.2" xref="S6.SS1.p1.4.m4.3.3.3.2.2.2.cmml"><mo id="S6.SS1.p1.4.m4.3.3.3.2.2.2a" xref="S6.SS1.p1.4.m4.3.3.3.2.2.2.cmml">−</mo><msub id="S6.SS1.p1.4.m4.3.3.3.2.2.2.2" xref="S6.SS1.p1.4.m4.3.3.3.2.2.2.2.cmml"><mi id="S6.SS1.p1.4.m4.3.3.3.2.2.2.2.2" xref="S6.SS1.p1.4.m4.3.3.3.2.2.2.2.2.cmml">x</mi><mn id="S6.SS1.p1.4.m4.3.3.3.2.2.2.2.3" xref="S6.SS1.p1.4.m4.3.3.3.2.2.2.2.3.cmml">1</mn></msub></mrow><mo id="S6.SS1.p1.4.m4.4.4.4.3.3.6" xref="S6.SS1.p1.4.m4.4.4.4.3.4.cmml">,</mo><mrow id="S6.SS1.p1.4.m4.4.4.4.3.3.3" xref="S6.SS1.p1.4.m4.4.4.4.3.3.3.cmml"><mo id="S6.SS1.p1.4.m4.4.4.4.3.3.3a" xref="S6.SS1.p1.4.m4.4.4.4.3.3.3.cmml">−</mo><msubsup id="S6.SS1.p1.4.m4.4.4.4.3.3.3.2" xref="S6.SS1.p1.4.m4.4.4.4.3.3.3.2.cmml"><mi id="S6.SS1.p1.4.m4.4.4.4.3.3.3.2.2.2" xref="S6.SS1.p1.4.m4.4.4.4.3.3.3.2.2.2.cmml">x</mi><mn id="S6.SS1.p1.4.m4.4.4.4.3.3.3.2.2.3" xref="S6.SS1.p1.4.m4.4.4.4.3.3.3.2.2.3.cmml">2</mn><mn id="S6.SS1.p1.4.m4.4.4.4.3.3.3.2.3" xref="S6.SS1.p1.4.m4.4.4.4.3.3.3.2.3.cmml">3</mn></msubsup></mrow><mo id="S6.SS1.p1.4.m4.4.4.4.3.3.7" stretchy="false" xref="S6.SS1.p1.4.m4.4.4.4.3.4.cmml">]</mo></mrow><mi id="S6.SS1.p1.4.m4.4.4.4.5" xref="S6.SS1.p1.4.m4.4.4.4.5.cmml">T</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p1.4.m4.4b"><apply id="S6.SS1.p1.4.m4.4.4.cmml" xref="S6.SS1.p1.4.m4.4.4"><eq id="S6.SS1.p1.4.m4.4.4.5.cmml" xref="S6.SS1.p1.4.m4.4.4.5"></eq><apply id="S6.SS1.p1.4.m4.1.1.1.cmml" xref="S6.SS1.p1.4.m4.1.1.1"><times id="S6.SS1.p1.4.m4.1.1.1.2.cmml" xref="S6.SS1.p1.4.m4.1.1.1.2"></times><apply id="S6.SS1.p1.4.m4.1.1.1.3.cmml" xref="S6.SS1.p1.4.m4.1.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.p1.4.m4.1.1.1.3.1.cmml" xref="S6.SS1.p1.4.m4.1.1.1.3">subscript</csymbol><ci id="S6.SS1.p1.4.m4.1.1.1.3.2.cmml" xref="S6.SS1.p1.4.m4.1.1.1.3.2">𝝋</ci><cn id="S6.SS1.p1.4.m4.1.1.1.3.3.cmml" type="integer" xref="S6.SS1.p1.4.m4.1.1.1.3.3">2</cn></apply><apply id="S6.SS1.p1.4.m4.1.1.1.1.1.1.cmml" xref="S6.SS1.p1.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS1.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S6.SS1.p1.4.m4.1.1.1.1.1">subscript</csymbol><ci id="S6.SS1.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="S6.SS1.p1.4.m4.1.1.1.1.1.1.2">𝒙</ci><cn id="S6.SS1.p1.4.m4.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.SS1.p1.4.m4.1.1.1.1.1.1.3">2</cn></apply></apply><apply id="S6.SS1.p1.4.m4.4.4.4.cmml" xref="S6.SS1.p1.4.m4.4.4.4"><csymbol cd="ambiguous" id="S6.SS1.p1.4.m4.4.4.4.4.cmml" xref="S6.SS1.p1.4.m4.4.4.4">superscript</csymbol><list id="S6.SS1.p1.4.m4.4.4.4.3.4.cmml" xref="S6.SS1.p1.4.m4.4.4.4.3.3"><apply id="S6.SS1.p1.4.m4.2.2.2.1.1.1.cmml" xref="S6.SS1.p1.4.m4.2.2.2.1.1.1"><minus id="S6.SS1.p1.4.m4.2.2.2.1.1.1.1.cmml" xref="S6.SS1.p1.4.m4.2.2.2.1.1.1"></minus><apply id="S6.SS1.p1.4.m4.2.2.2.1.1.1.2.cmml" xref="S6.SS1.p1.4.m4.2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.p1.4.m4.2.2.2.1.1.1.2.1.cmml" xref="S6.SS1.p1.4.m4.2.2.2.1.1.1.2">subscript</csymbol><ci id="S6.SS1.p1.4.m4.2.2.2.1.1.1.2.2.cmml" xref="S6.SS1.p1.4.m4.2.2.2.1.1.1.2.2">𝑥</ci><cn id="S6.SS1.p1.4.m4.2.2.2.1.1.1.2.3.cmml" type="integer" xref="S6.SS1.p1.4.m4.2.2.2.1.1.1.2.3">2</cn></apply></apply><apply id="S6.SS1.p1.4.m4.3.3.3.2.2.2.cmml" xref="S6.SS1.p1.4.m4.3.3.3.2.2.2"><minus id="S6.SS1.p1.4.m4.3.3.3.2.2.2.1.cmml" xref="S6.SS1.p1.4.m4.3.3.3.2.2.2"></minus><apply id="S6.SS1.p1.4.m4.3.3.3.2.2.2.2.cmml" xref="S6.SS1.p1.4.m4.3.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS1.p1.4.m4.3.3.3.2.2.2.2.1.cmml" xref="S6.SS1.p1.4.m4.3.3.3.2.2.2.2">subscript</csymbol><ci id="S6.SS1.p1.4.m4.3.3.3.2.2.2.2.2.cmml" xref="S6.SS1.p1.4.m4.3.3.3.2.2.2.2.2">𝑥</ci><cn id="S6.SS1.p1.4.m4.3.3.3.2.2.2.2.3.cmml" type="integer" xref="S6.SS1.p1.4.m4.3.3.3.2.2.2.2.3">1</cn></apply></apply><apply id="S6.SS1.p1.4.m4.4.4.4.3.3.3.cmml" xref="S6.SS1.p1.4.m4.4.4.4.3.3.3"><minus id="S6.SS1.p1.4.m4.4.4.4.3.3.3.1.cmml" xref="S6.SS1.p1.4.m4.4.4.4.3.3.3"></minus><apply id="S6.SS1.p1.4.m4.4.4.4.3.3.3.2.cmml" xref="S6.SS1.p1.4.m4.4.4.4.3.3.3.2"><csymbol cd="ambiguous" id="S6.SS1.p1.4.m4.4.4.4.3.3.3.2.1.cmml" xref="S6.SS1.p1.4.m4.4.4.4.3.3.3.2">superscript</csymbol><apply id="S6.SS1.p1.4.m4.4.4.4.3.3.3.2.2.cmml" xref="S6.SS1.p1.4.m4.4.4.4.3.3.3.2"><csymbol cd="ambiguous" id="S6.SS1.p1.4.m4.4.4.4.3.3.3.2.2.1.cmml" xref="S6.SS1.p1.4.m4.4.4.4.3.3.3.2">subscript</csymbol><ci id="S6.SS1.p1.4.m4.4.4.4.3.3.3.2.2.2.cmml" xref="S6.SS1.p1.4.m4.4.4.4.3.3.3.2.2.2">𝑥</ci><cn id="S6.SS1.p1.4.m4.4.4.4.3.3.3.2.2.3.cmml" type="integer" xref="S6.SS1.p1.4.m4.4.4.4.3.3.3.2.2.3">2</cn></apply><cn id="S6.SS1.p1.4.m4.4.4.4.3.3.3.2.3.cmml" type="integer" xref="S6.SS1.p1.4.m4.4.4.4.3.3.3.2.3">3</cn></apply></apply></list><ci id="S6.SS1.p1.4.m4.4.4.4.5.cmml" xref="S6.SS1.p1.4.m4.4.4.4.5">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p1.4.m4.4c">\bm{\varphi}_{2}({\bm{x}}_{2})=[-x_{2},-x_{1},-x_{2}^{3}]^{T}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p1.4.m4.4d">bold_italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( bold_italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = [ - italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , - italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , - italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\bm{\varphi}_{1}(x_{1})=\bm{\varphi}_{3}({\bm{x}})=\bm{0}" class="ltx_Math" display="inline" id="S6.SS1.p1.5.m5.2"><semantics id="S6.SS1.p1.5.m5.2a"><mrow id="S6.SS1.p1.5.m5.2.2" xref="S6.SS1.p1.5.m5.2.2.cmml"><mrow id="S6.SS1.p1.5.m5.2.2.1" xref="S6.SS1.p1.5.m5.2.2.1.cmml"><msub id="S6.SS1.p1.5.m5.2.2.1.3" xref="S6.SS1.p1.5.m5.2.2.1.3.cmml"><mi id="S6.SS1.p1.5.m5.2.2.1.3.2" xref="S6.SS1.p1.5.m5.2.2.1.3.2.cmml">𝝋</mi><mn id="S6.SS1.p1.5.m5.2.2.1.3.3" xref="S6.SS1.p1.5.m5.2.2.1.3.3.cmml">1</mn></msub><mo id="S6.SS1.p1.5.m5.2.2.1.2" xref="S6.SS1.p1.5.m5.2.2.1.2.cmml">⁢</mo><mrow id="S6.SS1.p1.5.m5.2.2.1.1.1" xref="S6.SS1.p1.5.m5.2.2.1.1.1.1.cmml"><mo id="S6.SS1.p1.5.m5.2.2.1.1.1.2" stretchy="false" xref="S6.SS1.p1.5.m5.2.2.1.1.1.1.cmml">(</mo><msub id="S6.SS1.p1.5.m5.2.2.1.1.1.1" xref="S6.SS1.p1.5.m5.2.2.1.1.1.1.cmml"><mi id="S6.SS1.p1.5.m5.2.2.1.1.1.1.2" xref="S6.SS1.p1.5.m5.2.2.1.1.1.1.2.cmml">x</mi><mn id="S6.SS1.p1.5.m5.2.2.1.1.1.1.3" xref="S6.SS1.p1.5.m5.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="S6.SS1.p1.5.m5.2.2.1.1.1.3" stretchy="false" xref="S6.SS1.p1.5.m5.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS1.p1.5.m5.2.2.3" xref="S6.SS1.p1.5.m5.2.2.3.cmml">=</mo><mrow id="S6.SS1.p1.5.m5.2.2.4" xref="S6.SS1.p1.5.m5.2.2.4.cmml"><msub id="S6.SS1.p1.5.m5.2.2.4.2" xref="S6.SS1.p1.5.m5.2.2.4.2.cmml"><mi id="S6.SS1.p1.5.m5.2.2.4.2.2" xref="S6.SS1.p1.5.m5.2.2.4.2.2.cmml">𝝋</mi><mn id="S6.SS1.p1.5.m5.2.2.4.2.3" xref="S6.SS1.p1.5.m5.2.2.4.2.3.cmml">3</mn></msub><mo id="S6.SS1.p1.5.m5.2.2.4.1" xref="S6.SS1.p1.5.m5.2.2.4.1.cmml">⁢</mo><mrow id="S6.SS1.p1.5.m5.2.2.4.3.2" xref="S6.SS1.p1.5.m5.2.2.4.cmml"><mo id="S6.SS1.p1.5.m5.2.2.4.3.2.1" stretchy="false" xref="S6.SS1.p1.5.m5.2.2.4.cmml">(</mo><mi id="S6.SS1.p1.5.m5.1.1" xref="S6.SS1.p1.5.m5.1.1.cmml">𝒙</mi><mo id="S6.SS1.p1.5.m5.2.2.4.3.2.2" stretchy="false" xref="S6.SS1.p1.5.m5.2.2.4.cmml">)</mo></mrow></mrow><mo id="S6.SS1.p1.5.m5.2.2.5" xref="S6.SS1.p1.5.m5.2.2.5.cmml">=</mo><mn id="S6.SS1.p1.5.m5.2.2.6" xref="S6.SS1.p1.5.m5.2.2.6.cmml">𝟎</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p1.5.m5.2b"><apply id="S6.SS1.p1.5.m5.2.2.cmml" xref="S6.SS1.p1.5.m5.2.2"><and id="S6.SS1.p1.5.m5.2.2a.cmml" xref="S6.SS1.p1.5.m5.2.2"></and><apply id="S6.SS1.p1.5.m5.2.2b.cmml" xref="S6.SS1.p1.5.m5.2.2"><eq id="S6.SS1.p1.5.m5.2.2.3.cmml" xref="S6.SS1.p1.5.m5.2.2.3"></eq><apply id="S6.SS1.p1.5.m5.2.2.1.cmml" xref="S6.SS1.p1.5.m5.2.2.1"><times id="S6.SS1.p1.5.m5.2.2.1.2.cmml" xref="S6.SS1.p1.5.m5.2.2.1.2"></times><apply id="S6.SS1.p1.5.m5.2.2.1.3.cmml" xref="S6.SS1.p1.5.m5.2.2.1.3"><csymbol cd="ambiguous" id="S6.SS1.p1.5.m5.2.2.1.3.1.cmml" xref="S6.SS1.p1.5.m5.2.2.1.3">subscript</csymbol><ci id="S6.SS1.p1.5.m5.2.2.1.3.2.cmml" xref="S6.SS1.p1.5.m5.2.2.1.3.2">𝝋</ci><cn id="S6.SS1.p1.5.m5.2.2.1.3.3.cmml" type="integer" xref="S6.SS1.p1.5.m5.2.2.1.3.3">1</cn></apply><apply id="S6.SS1.p1.5.m5.2.2.1.1.1.1.cmml" xref="S6.SS1.p1.5.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.SS1.p1.5.m5.2.2.1.1.1.1.1.cmml" xref="S6.SS1.p1.5.m5.2.2.1.1.1">subscript</csymbol><ci id="S6.SS1.p1.5.m5.2.2.1.1.1.1.2.cmml" xref="S6.SS1.p1.5.m5.2.2.1.1.1.1.2">𝑥</ci><cn id="S6.SS1.p1.5.m5.2.2.1.1.1.1.3.cmml" type="integer" xref="S6.SS1.p1.5.m5.2.2.1.1.1.1.3">1</cn></apply></apply><apply id="S6.SS1.p1.5.m5.2.2.4.cmml" xref="S6.SS1.p1.5.m5.2.2.4"><times id="S6.SS1.p1.5.m5.2.2.4.1.cmml" xref="S6.SS1.p1.5.m5.2.2.4.1"></times><apply id="S6.SS1.p1.5.m5.2.2.4.2.cmml" xref="S6.SS1.p1.5.m5.2.2.4.2"><csymbol cd="ambiguous" id="S6.SS1.p1.5.m5.2.2.4.2.1.cmml" xref="S6.SS1.p1.5.m5.2.2.4.2">subscript</csymbol><ci id="S6.SS1.p1.5.m5.2.2.4.2.2.cmml" xref="S6.SS1.p1.5.m5.2.2.4.2.2">𝝋</ci><cn id="S6.SS1.p1.5.m5.2.2.4.2.3.cmml" type="integer" xref="S6.SS1.p1.5.m5.2.2.4.2.3">3</cn></apply><ci id="S6.SS1.p1.5.m5.1.1.cmml" xref="S6.SS1.p1.5.m5.1.1">𝒙</ci></apply></apply><apply id="S6.SS1.p1.5.m5.2.2c.cmml" xref="S6.SS1.p1.5.m5.2.2"><eq id="S6.SS1.p1.5.m5.2.2.5.cmml" xref="S6.SS1.p1.5.m5.2.2.5"></eq><share href="https://arxiv.org/html/2401.10785v2#S6.SS1.p1.5.m5.2.2.4.cmml" id="S6.SS1.p1.5.m5.2.2d.cmml" xref="S6.SS1.p1.5.m5.2.2"></share><cn id="S6.SS1.p1.5.m5.2.2.6.cmml" type="integer" xref="S6.SS1.p1.5.m5.2.2.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p1.5.m5.2c">\bm{\varphi}_{1}(x_{1})=\bm{\varphi}_{3}({\bm{x}})=\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p1.5.m5.2d">bold_italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = bold_italic_φ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( bold_italic_x ) = bold_0</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.SS1.p2"> <p class="ltx_p" id="S6.SS1.p2.15">Choose the control parameters <math alttext="k_{{\rm c}i}=1" class="ltx_Math" display="inline" id="S6.SS1.p2.1.m1.1"><semantics id="S6.SS1.p2.1.m1.1a"><mrow id="S6.SS1.p2.1.m1.1.1" xref="S6.SS1.p2.1.m1.1.1.cmml"><msub id="S6.SS1.p2.1.m1.1.1.2" xref="S6.SS1.p2.1.m1.1.1.2.cmml"><mi id="S6.SS1.p2.1.m1.1.1.2.2" xref="S6.SS1.p2.1.m1.1.1.2.2.cmml">k</mi><mrow id="S6.SS1.p2.1.m1.1.1.2.3" xref="S6.SS1.p2.1.m1.1.1.2.3.cmml"><mi id="S6.SS1.p2.1.m1.1.1.2.3.2" mathvariant="normal" xref="S6.SS1.p2.1.m1.1.1.2.3.2.cmml">c</mi><mo id="S6.SS1.p2.1.m1.1.1.2.3.1" xref="S6.SS1.p2.1.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S6.SS1.p2.1.m1.1.1.2.3.3" xref="S6.SS1.p2.1.m1.1.1.2.3.3.cmml">i</mi></mrow></msub><mo id="S6.SS1.p2.1.m1.1.1.1" xref="S6.SS1.p2.1.m1.1.1.1.cmml">=</mo><mn id="S6.SS1.p2.1.m1.1.1.3" xref="S6.SS1.p2.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.1.m1.1b"><apply id="S6.SS1.p2.1.m1.1.1.cmml" xref="S6.SS1.p2.1.m1.1.1"><eq id="S6.SS1.p2.1.m1.1.1.1.cmml" xref="S6.SS1.p2.1.m1.1.1.1"></eq><apply id="S6.SS1.p2.1.m1.1.1.2.cmml" xref="S6.SS1.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.p2.1.m1.1.1.2.1.cmml" xref="S6.SS1.p2.1.m1.1.1.2">subscript</csymbol><ci id="S6.SS1.p2.1.m1.1.1.2.2.cmml" xref="S6.SS1.p2.1.m1.1.1.2.2">𝑘</ci><apply id="S6.SS1.p2.1.m1.1.1.2.3.cmml" xref="S6.SS1.p2.1.m1.1.1.2.3"><times id="S6.SS1.p2.1.m1.1.1.2.3.1.cmml" xref="S6.SS1.p2.1.m1.1.1.2.3.1"></times><ci id="S6.SS1.p2.1.m1.1.1.2.3.2.cmml" xref="S6.SS1.p2.1.m1.1.1.2.3.2">c</ci><ci id="S6.SS1.p2.1.m1.1.1.2.3.3.cmml" xref="S6.SS1.p2.1.m1.1.1.2.3.3">𝑖</ci></apply></apply><cn id="S6.SS1.p2.1.m1.1.1.3.cmml" type="integer" xref="S6.SS1.p2.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.1.m1.1c">k_{{\rm c}i}=1</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.1.m1.1d">italic_k start_POSTSUBSCRIPT roman_c italic_i end_POSTSUBSCRIPT = 1</annotation></semantics></math> with <math alttext="i=1,2,3" class="ltx_Math" display="inline" id="S6.SS1.p2.2.m2.3"><semantics id="S6.SS1.p2.2.m2.3a"><mrow id="S6.SS1.p2.2.m2.3.4" xref="S6.SS1.p2.2.m2.3.4.cmml"><mi id="S6.SS1.p2.2.m2.3.4.2" xref="S6.SS1.p2.2.m2.3.4.2.cmml">i</mi><mo id="S6.SS1.p2.2.m2.3.4.1" xref="S6.SS1.p2.2.m2.3.4.1.cmml">=</mo><mrow id="S6.SS1.p2.2.m2.3.4.3.2" xref="S6.SS1.p2.2.m2.3.4.3.1.cmml"><mn id="S6.SS1.p2.2.m2.1.1" xref="S6.SS1.p2.2.m2.1.1.cmml">1</mn><mo id="S6.SS1.p2.2.m2.3.4.3.2.1" xref="S6.SS1.p2.2.m2.3.4.3.1.cmml">,</mo><mn id="S6.SS1.p2.2.m2.2.2" xref="S6.SS1.p2.2.m2.2.2.cmml">2</mn><mo id="S6.SS1.p2.2.m2.3.4.3.2.2" xref="S6.SS1.p2.2.m2.3.4.3.1.cmml">,</mo><mn id="S6.SS1.p2.2.m2.3.3" xref="S6.SS1.p2.2.m2.3.3.cmml">3</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.2.m2.3b"><apply id="S6.SS1.p2.2.m2.3.4.cmml" xref="S6.SS1.p2.2.m2.3.4"><eq id="S6.SS1.p2.2.m2.3.4.1.cmml" xref="S6.SS1.p2.2.m2.3.4.1"></eq><ci id="S6.SS1.p2.2.m2.3.4.2.cmml" xref="S6.SS1.p2.2.m2.3.4.2">𝑖</ci><list id="S6.SS1.p2.2.m2.3.4.3.1.cmml" xref="S6.SS1.p2.2.m2.3.4.3.2"><cn id="S6.SS1.p2.2.m2.1.1.cmml" type="integer" xref="S6.SS1.p2.2.m2.1.1">1</cn><cn id="S6.SS1.p2.2.m2.2.2.cmml" type="integer" xref="S6.SS1.p2.2.m2.2.2">2</cn><cn id="S6.SS1.p2.2.m2.3.3.cmml" type="integer" xref="S6.SS1.p2.2.m2.3.3">3</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.2.m2.3c">i=1,2,3</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.2.m2.3d">italic_i = 1 , 2 , 3</annotation></semantics></math>, <math alttext="\Gamma" class="ltx_Math" display="inline" id="S6.SS1.p2.3.m3.1"><semantics id="S6.SS1.p2.3.m3.1a"><mi id="S6.SS1.p2.3.m3.1.1" mathvariant="normal" xref="S6.SS1.p2.3.m3.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.3.m3.1b"><ci id="S6.SS1.p2.3.m3.1.1.cmml" xref="S6.SS1.p2.3.m3.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.3.m3.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.3.m3.1d">roman_Γ</annotation></semantics></math> <math alttext="=" class="ltx_Math" display="inline" id="S6.SS1.p2.4.m4.1"><semantics id="S6.SS1.p2.4.m4.1a"><mo id="S6.SS1.p2.4.m4.1.1" xref="S6.SS1.p2.4.m4.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.4.m4.1b"><eq id="S6.SS1.p2.4.m4.1.1.cmml" xref="S6.SS1.p2.4.m4.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.4.m4.1c">=</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.4.m4.1d">=</annotation></semantics></math> <math alttext="3I" class="ltx_Math" display="inline" id="S6.SS1.p2.5.m5.1"><semantics id="S6.SS1.p2.5.m5.1a"><mrow id="S6.SS1.p2.5.m5.1.1" xref="S6.SS1.p2.5.m5.1.1.cmml"><mn id="S6.SS1.p2.5.m5.1.1.2" xref="S6.SS1.p2.5.m5.1.1.2.cmml">3</mn><mo id="S6.SS1.p2.5.m5.1.1.1" xref="S6.SS1.p2.5.m5.1.1.1.cmml">⁢</mo><mi id="S6.SS1.p2.5.m5.1.1.3" xref="S6.SS1.p2.5.m5.1.1.3.cmml">I</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.5.m5.1b"><apply id="S6.SS1.p2.5.m5.1.1.cmml" xref="S6.SS1.p2.5.m5.1.1"><times id="S6.SS1.p2.5.m5.1.1.1.cmml" xref="S6.SS1.p2.5.m5.1.1.1"></times><cn id="S6.SS1.p2.5.m5.1.1.2.cmml" type="integer" xref="S6.SS1.p2.5.m5.1.1.2">3</cn><ci id="S6.SS1.p2.5.m5.1.1.3.cmml" xref="S6.SS1.p2.5.m5.1.1.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.5.m5.1c">3I</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.5.m5.1d">3 italic_I</annotation></semantics></math>, <math alttext="\kappa=1" class="ltx_Math" display="inline" id="S6.SS1.p2.6.m6.1"><semantics id="S6.SS1.p2.6.m6.1a"><mrow id="S6.SS1.p2.6.m6.1.1" xref="S6.SS1.p2.6.m6.1.1.cmml"><mi id="S6.SS1.p2.6.m6.1.1.2" xref="S6.SS1.p2.6.m6.1.1.2.cmml">κ</mi><mo id="S6.SS1.p2.6.m6.1.1.1" xref="S6.SS1.p2.6.m6.1.1.1.cmml">=</mo><mn id="S6.SS1.p2.6.m6.1.1.3" xref="S6.SS1.p2.6.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.6.m6.1b"><apply id="S6.SS1.p2.6.m6.1.1.cmml" xref="S6.SS1.p2.6.m6.1.1"><eq id="S6.SS1.p2.6.m6.1.1.1.cmml" xref="S6.SS1.p2.6.m6.1.1.1"></eq><ci id="S6.SS1.p2.6.m6.1.1.2.cmml" xref="S6.SS1.p2.6.m6.1.1.2">𝜅</ci><cn id="S6.SS1.p2.6.m6.1.1.3.cmml" type="integer" xref="S6.SS1.p2.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.6.m6.1c">\kappa=1</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.6.m6.1d">italic_κ = 1</annotation></semantics></math>, <math alttext="\tau_{\rm d}=3" class="ltx_Math" display="inline" id="S6.SS1.p2.7.m7.1"><semantics id="S6.SS1.p2.7.m7.1a"><mrow id="S6.SS1.p2.7.m7.1.1" xref="S6.SS1.p2.7.m7.1.1.cmml"><msub id="S6.SS1.p2.7.m7.1.1.2" xref="S6.SS1.p2.7.m7.1.1.2.cmml"><mi id="S6.SS1.p2.7.m7.1.1.2.2" xref="S6.SS1.p2.7.m7.1.1.2.2.cmml">τ</mi><mi id="S6.SS1.p2.7.m7.1.1.2.3" mathvariant="normal" xref="S6.SS1.p2.7.m7.1.1.2.3.cmml">d</mi></msub><mo id="S6.SS1.p2.7.m7.1.1.1" xref="S6.SS1.p2.7.m7.1.1.1.cmml">=</mo><mn id="S6.SS1.p2.7.m7.1.1.3" xref="S6.SS1.p2.7.m7.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.7.m7.1b"><apply id="S6.SS1.p2.7.m7.1.1.cmml" xref="S6.SS1.p2.7.m7.1.1"><eq id="S6.SS1.p2.7.m7.1.1.1.cmml" xref="S6.SS1.p2.7.m7.1.1.1"></eq><apply id="S6.SS1.p2.7.m7.1.1.2.cmml" xref="S6.SS1.p2.7.m7.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.p2.7.m7.1.1.2.1.cmml" xref="S6.SS1.p2.7.m7.1.1.2">subscript</csymbol><ci id="S6.SS1.p2.7.m7.1.1.2.2.cmml" xref="S6.SS1.p2.7.m7.1.1.2.2">𝜏</ci><ci id="S6.SS1.p2.7.m7.1.1.2.3.cmml" xref="S6.SS1.p2.7.m7.1.1.2.3">d</ci></apply><cn id="S6.SS1.p2.7.m7.1.1.3.cmml" type="integer" xref="S6.SS1.p2.7.m7.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.7.m7.1c">\tau_{\rm d}=3</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.7.m7.1d">italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT = 3</annotation></semantics></math>, <math alttext="\hat{\bm{\theta}}(0)=\bm{0}" class="ltx_Math" display="inline" id="S6.SS1.p2.8.m8.1"><semantics id="S6.SS1.p2.8.m8.1a"><mrow id="S6.SS1.p2.8.m8.1.2" xref="S6.SS1.p2.8.m8.1.2.cmml"><mrow id="S6.SS1.p2.8.m8.1.2.2" xref="S6.SS1.p2.8.m8.1.2.2.cmml"><mover accent="true" id="S6.SS1.p2.8.m8.1.2.2.2" xref="S6.SS1.p2.8.m8.1.2.2.2.cmml"><mi id="S6.SS1.p2.8.m8.1.2.2.2.2" xref="S6.SS1.p2.8.m8.1.2.2.2.2.cmml">𝜽</mi><mo id="S6.SS1.p2.8.m8.1.2.2.2.1" xref="S6.SS1.p2.8.m8.1.2.2.2.1.cmml">^</mo></mover><mo id="S6.SS1.p2.8.m8.1.2.2.1" xref="S6.SS1.p2.8.m8.1.2.2.1.cmml">⁢</mo><mrow id="S6.SS1.p2.8.m8.1.2.2.3.2" xref="S6.SS1.p2.8.m8.1.2.2.cmml"><mo id="S6.SS1.p2.8.m8.1.2.2.3.2.1" stretchy="false" xref="S6.SS1.p2.8.m8.1.2.2.cmml">(</mo><mn id="S6.SS1.p2.8.m8.1.1" xref="S6.SS1.p2.8.m8.1.1.cmml">0</mn><mo id="S6.SS1.p2.8.m8.1.2.2.3.2.2" stretchy="false" xref="S6.SS1.p2.8.m8.1.2.2.cmml">)</mo></mrow></mrow><mo id="S6.SS1.p2.8.m8.1.2.1" xref="S6.SS1.p2.8.m8.1.2.1.cmml">=</mo><mn id="S6.SS1.p2.8.m8.1.2.3" xref="S6.SS1.p2.8.m8.1.2.3.cmml">𝟎</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.8.m8.1b"><apply id="S6.SS1.p2.8.m8.1.2.cmml" xref="S6.SS1.p2.8.m8.1.2"><eq id="S6.SS1.p2.8.m8.1.2.1.cmml" xref="S6.SS1.p2.8.m8.1.2.1"></eq><apply id="S6.SS1.p2.8.m8.1.2.2.cmml" xref="S6.SS1.p2.8.m8.1.2.2"><times id="S6.SS1.p2.8.m8.1.2.2.1.cmml" xref="S6.SS1.p2.8.m8.1.2.2.1"></times><apply id="S6.SS1.p2.8.m8.1.2.2.2.cmml" xref="S6.SS1.p2.8.m8.1.2.2.2"><ci id="S6.SS1.p2.8.m8.1.2.2.2.1.cmml" xref="S6.SS1.p2.8.m8.1.2.2.2.1">^</ci><ci id="S6.SS1.p2.8.m8.1.2.2.2.2.cmml" xref="S6.SS1.p2.8.m8.1.2.2.2.2">𝜽</ci></apply><cn id="S6.SS1.p2.8.m8.1.1.cmml" type="integer" xref="S6.SS1.p2.8.m8.1.1">0</cn></apply><cn id="S6.SS1.p2.8.m8.1.2.3.cmml" type="integer" xref="S6.SS1.p2.8.m8.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.8.m8.1c">\hat{\bm{\theta}}(0)=\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.8.m8.1d">over^ start_ARG bold_italic_θ end_ARG ( 0 ) = bold_0</annotation></semantics></math>, and <math alttext="\sigma=10^{-4}" class="ltx_Math" display="inline" id="S6.SS1.p2.9.m9.1"><semantics id="S6.SS1.p2.9.m9.1a"><mrow id="S6.SS1.p2.9.m9.1.1" xref="S6.SS1.p2.9.m9.1.1.cmml"><mi id="S6.SS1.p2.9.m9.1.1.2" xref="S6.SS1.p2.9.m9.1.1.2.cmml">σ</mi><mo id="S6.SS1.p2.9.m9.1.1.1" xref="S6.SS1.p2.9.m9.1.1.1.cmml">=</mo><msup id="S6.SS1.p2.9.m9.1.1.3" xref="S6.SS1.p2.9.m9.1.1.3.cmml"><mn id="S6.SS1.p2.9.m9.1.1.3.2" xref="S6.SS1.p2.9.m9.1.1.3.2.cmml">10</mn><mrow id="S6.SS1.p2.9.m9.1.1.3.3" xref="S6.SS1.p2.9.m9.1.1.3.3.cmml"><mo id="S6.SS1.p2.9.m9.1.1.3.3a" xref="S6.SS1.p2.9.m9.1.1.3.3.cmml">−</mo><mn id="S6.SS1.p2.9.m9.1.1.3.3.2" xref="S6.SS1.p2.9.m9.1.1.3.3.2.cmml">4</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.9.m9.1b"><apply id="S6.SS1.p2.9.m9.1.1.cmml" xref="S6.SS1.p2.9.m9.1.1"><eq id="S6.SS1.p2.9.m9.1.1.1.cmml" xref="S6.SS1.p2.9.m9.1.1.1"></eq><ci id="S6.SS1.p2.9.m9.1.1.2.cmml" xref="S6.SS1.p2.9.m9.1.1.2">𝜎</ci><apply id="S6.SS1.p2.9.m9.1.1.3.cmml" xref="S6.SS1.p2.9.m9.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.p2.9.m9.1.1.3.1.cmml" xref="S6.SS1.p2.9.m9.1.1.3">superscript</csymbol><cn id="S6.SS1.p2.9.m9.1.1.3.2.cmml" type="integer" xref="S6.SS1.p2.9.m9.1.1.3.2">10</cn><apply id="S6.SS1.p2.9.m9.1.1.3.3.cmml" xref="S6.SS1.p2.9.m9.1.1.3.3"><minus id="S6.SS1.p2.9.m9.1.1.3.3.1.cmml" xref="S6.SS1.p2.9.m9.1.1.3.3"></minus><cn id="S6.SS1.p2.9.m9.1.1.3.3.2.cmml" type="integer" xref="S6.SS1.p2.9.m9.1.1.3.3.2">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.9.m9.1c">\sigma=10^{-4}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.9.m9.1d">italic_σ = 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math>, and the stable filter <math alttext="H(s)=25/(s^{2}+10s+25)" class="ltx_Math" display="inline" id="S6.SS1.p2.10.m10.2"><semantics id="S6.SS1.p2.10.m10.2a"><mrow id="S6.SS1.p2.10.m10.2.2" xref="S6.SS1.p2.10.m10.2.2.cmml"><mrow id="S6.SS1.p2.10.m10.2.2.3" xref="S6.SS1.p2.10.m10.2.2.3.cmml"><mi id="S6.SS1.p2.10.m10.2.2.3.2" xref="S6.SS1.p2.10.m10.2.2.3.2.cmml">H</mi><mo id="S6.SS1.p2.10.m10.2.2.3.1" xref="S6.SS1.p2.10.m10.2.2.3.1.cmml">⁢</mo><mrow id="S6.SS1.p2.10.m10.2.2.3.3.2" xref="S6.SS1.p2.10.m10.2.2.3.cmml"><mo id="S6.SS1.p2.10.m10.2.2.3.3.2.1" stretchy="false" xref="S6.SS1.p2.10.m10.2.2.3.cmml">(</mo><mi id="S6.SS1.p2.10.m10.1.1" xref="S6.SS1.p2.10.m10.1.1.cmml">s</mi><mo id="S6.SS1.p2.10.m10.2.2.3.3.2.2" stretchy="false" xref="S6.SS1.p2.10.m10.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS1.p2.10.m10.2.2.2" xref="S6.SS1.p2.10.m10.2.2.2.cmml">=</mo><mrow id="S6.SS1.p2.10.m10.2.2.1" xref="S6.SS1.p2.10.m10.2.2.1.cmml"><mn id="S6.SS1.p2.10.m10.2.2.1.3" xref="S6.SS1.p2.10.m10.2.2.1.3.cmml">25</mn><mo id="S6.SS1.p2.10.m10.2.2.1.2" xref="S6.SS1.p2.10.m10.2.2.1.2.cmml">/</mo><mrow id="S6.SS1.p2.10.m10.2.2.1.1.1" xref="S6.SS1.p2.10.m10.2.2.1.1.1.1.cmml"><mo id="S6.SS1.p2.10.m10.2.2.1.1.1.2" stretchy="false" xref="S6.SS1.p2.10.m10.2.2.1.1.1.1.cmml">(</mo><mrow id="S6.SS1.p2.10.m10.2.2.1.1.1.1" xref="S6.SS1.p2.10.m10.2.2.1.1.1.1.cmml"><msup id="S6.SS1.p2.10.m10.2.2.1.1.1.1.2" xref="S6.SS1.p2.10.m10.2.2.1.1.1.1.2.cmml"><mi id="S6.SS1.p2.10.m10.2.2.1.1.1.1.2.2" xref="S6.SS1.p2.10.m10.2.2.1.1.1.1.2.2.cmml">s</mi><mn id="S6.SS1.p2.10.m10.2.2.1.1.1.1.2.3" xref="S6.SS1.p2.10.m10.2.2.1.1.1.1.2.3.cmml">2</mn></msup><mo id="S6.SS1.p2.10.m10.2.2.1.1.1.1.1" xref="S6.SS1.p2.10.m10.2.2.1.1.1.1.1.cmml">+</mo><mrow id="S6.SS1.p2.10.m10.2.2.1.1.1.1.3" xref="S6.SS1.p2.10.m10.2.2.1.1.1.1.3.cmml"><mn id="S6.SS1.p2.10.m10.2.2.1.1.1.1.3.2" xref="S6.SS1.p2.10.m10.2.2.1.1.1.1.3.2.cmml">10</mn><mo id="S6.SS1.p2.10.m10.2.2.1.1.1.1.3.1" xref="S6.SS1.p2.10.m10.2.2.1.1.1.1.3.1.cmml">⁢</mo><mi id="S6.SS1.p2.10.m10.2.2.1.1.1.1.3.3" xref="S6.SS1.p2.10.m10.2.2.1.1.1.1.3.3.cmml">s</mi></mrow><mo id="S6.SS1.p2.10.m10.2.2.1.1.1.1.1a" xref="S6.SS1.p2.10.m10.2.2.1.1.1.1.1.cmml">+</mo><mn id="S6.SS1.p2.10.m10.2.2.1.1.1.1.4" xref="S6.SS1.p2.10.m10.2.2.1.1.1.1.4.cmml">25</mn></mrow><mo id="S6.SS1.p2.10.m10.2.2.1.1.1.3" stretchy="false" xref="S6.SS1.p2.10.m10.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.10.m10.2b"><apply id="S6.SS1.p2.10.m10.2.2.cmml" xref="S6.SS1.p2.10.m10.2.2"><eq id="S6.SS1.p2.10.m10.2.2.2.cmml" xref="S6.SS1.p2.10.m10.2.2.2"></eq><apply id="S6.SS1.p2.10.m10.2.2.3.cmml" xref="S6.SS1.p2.10.m10.2.2.3"><times id="S6.SS1.p2.10.m10.2.2.3.1.cmml" xref="S6.SS1.p2.10.m10.2.2.3.1"></times><ci id="S6.SS1.p2.10.m10.2.2.3.2.cmml" xref="S6.SS1.p2.10.m10.2.2.3.2">𝐻</ci><ci id="S6.SS1.p2.10.m10.1.1.cmml" xref="S6.SS1.p2.10.m10.1.1">𝑠</ci></apply><apply id="S6.SS1.p2.10.m10.2.2.1.cmml" xref="S6.SS1.p2.10.m10.2.2.1"><divide id="S6.SS1.p2.10.m10.2.2.1.2.cmml" xref="S6.SS1.p2.10.m10.2.2.1.2"></divide><cn id="S6.SS1.p2.10.m10.2.2.1.3.cmml" type="integer" xref="S6.SS1.p2.10.m10.2.2.1.3">25</cn><apply id="S6.SS1.p2.10.m10.2.2.1.1.1.1.cmml" xref="S6.SS1.p2.10.m10.2.2.1.1.1"><plus id="S6.SS1.p2.10.m10.2.2.1.1.1.1.1.cmml" xref="S6.SS1.p2.10.m10.2.2.1.1.1.1.1"></plus><apply id="S6.SS1.p2.10.m10.2.2.1.1.1.1.2.cmml" xref="S6.SS1.p2.10.m10.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.p2.10.m10.2.2.1.1.1.1.2.1.cmml" xref="S6.SS1.p2.10.m10.2.2.1.1.1.1.2">superscript</csymbol><ci id="S6.SS1.p2.10.m10.2.2.1.1.1.1.2.2.cmml" xref="S6.SS1.p2.10.m10.2.2.1.1.1.1.2.2">𝑠</ci><cn id="S6.SS1.p2.10.m10.2.2.1.1.1.1.2.3.cmml" type="integer" xref="S6.SS1.p2.10.m10.2.2.1.1.1.1.2.3">2</cn></apply><apply id="S6.SS1.p2.10.m10.2.2.1.1.1.1.3.cmml" xref="S6.SS1.p2.10.m10.2.2.1.1.1.1.3"><times id="S6.SS1.p2.10.m10.2.2.1.1.1.1.3.1.cmml" xref="S6.SS1.p2.10.m10.2.2.1.1.1.1.3.1"></times><cn id="S6.SS1.p2.10.m10.2.2.1.1.1.1.3.2.cmml" type="integer" xref="S6.SS1.p2.10.m10.2.2.1.1.1.1.3.2">10</cn><ci id="S6.SS1.p2.10.m10.2.2.1.1.1.1.3.3.cmml" xref="S6.SS1.p2.10.m10.2.2.1.1.1.1.3.3">𝑠</ci></apply><cn id="S6.SS1.p2.10.m10.2.2.1.1.1.1.4.cmml" type="integer" xref="S6.SS1.p2.10.m10.2.2.1.1.1.1.4">25</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.10.m10.2c">H(s)=25/(s^{2}+10s+25)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.10.m10.2d">italic_H ( italic_s ) = 25 / ( italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 10 italic_s + 25 )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E17" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">17</span></a>). Gaussian white noise with 0 mean and 0.001 standard deviation is added to the measurement of the states <math alttext="x_{i}" class="ltx_Math" display="inline" id="S6.SS1.p2.11.m11.1"><semantics id="S6.SS1.p2.11.m11.1a"><msub id="S6.SS1.p2.11.m11.1.1" xref="S6.SS1.p2.11.m11.1.1.cmml"><mi id="S6.SS1.p2.11.m11.1.1.2" xref="S6.SS1.p2.11.m11.1.1.2.cmml">x</mi><mi id="S6.SS1.p2.11.m11.1.1.3" xref="S6.SS1.p2.11.m11.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.11.m11.1b"><apply id="S6.SS1.p2.11.m11.1.1.cmml" xref="S6.SS1.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S6.SS1.p2.11.m11.1.1.1.cmml" xref="S6.SS1.p2.11.m11.1.1">subscript</csymbol><ci id="S6.SS1.p2.11.m11.1.1.2.cmml" xref="S6.SS1.p2.11.m11.1.1.2">𝑥</ci><ci id="S6.SS1.p2.11.m11.1.1.3.cmml" xref="S6.SS1.p2.11.m11.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.11.m11.1c">x_{i}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.11.m11.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. The MRE-HOT in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib24" title="">24</a>]</cite> and the composite learning dynamic surface control (CL-DSC) in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib10" title="">10</a>]</cite> are selected as baseline controllers, where we set the damping parameters <math alttext="k_{\rm{d}\it i}=" class="ltx_Math" display="inline" id="S6.SS1.p2.12.m12.1"><semantics id="S6.SS1.p2.12.m12.1a"><mrow id="S6.SS1.p2.12.m12.1.1" xref="S6.SS1.p2.12.m12.1.1.cmml"><msub id="S6.SS1.p2.12.m12.1.1.2" xref="S6.SS1.p2.12.m12.1.1.2.cmml"><mi id="S6.SS1.p2.12.m12.1.1.2.2" xref="S6.SS1.p2.12.m12.1.1.2.2.cmml">k</mi><mrow id="S6.SS1.p2.12.m12.1.1.2.3" xref="S6.SS1.p2.12.m12.1.1.2.3.cmml"><mi id="S6.SS1.p2.12.m12.1.1.2.3.2" mathvariant="normal" xref="S6.SS1.p2.12.m12.1.1.2.3.2.cmml">d</mi><mo id="S6.SS1.p2.12.m12.1.1.2.3.1" xref="S6.SS1.p2.12.m12.1.1.2.3.1.cmml">⁢</mo><mi id="S6.SS1.p2.12.m12.1.1.2.3.3" xref="S6.SS1.p2.12.m12.1.1.2.3.3.cmml">i</mi></mrow></msub><mo id="S6.SS1.p2.12.m12.1.1.1" xref="S6.SS1.p2.12.m12.1.1.1.cmml">=</mo><mi id="S6.SS1.p2.12.m12.1.1.3" xref="S6.SS1.p2.12.m12.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.12.m12.1b"><apply id="S6.SS1.p2.12.m12.1.1.cmml" xref="S6.SS1.p2.12.m12.1.1"><eq id="S6.SS1.p2.12.m12.1.1.1.cmml" xref="S6.SS1.p2.12.m12.1.1.1"></eq><apply id="S6.SS1.p2.12.m12.1.1.2.cmml" xref="S6.SS1.p2.12.m12.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.p2.12.m12.1.1.2.1.cmml" xref="S6.SS1.p2.12.m12.1.1.2">subscript</csymbol><ci id="S6.SS1.p2.12.m12.1.1.2.2.cmml" xref="S6.SS1.p2.12.m12.1.1.2.2">𝑘</ci><apply id="S6.SS1.p2.12.m12.1.1.2.3.cmml" xref="S6.SS1.p2.12.m12.1.1.2.3"><times id="S6.SS1.p2.12.m12.1.1.2.3.1.cmml" xref="S6.SS1.p2.12.m12.1.1.2.3.1"></times><ci id="S6.SS1.p2.12.m12.1.1.2.3.2.cmml" xref="S6.SS1.p2.12.m12.1.1.2.3.2">d</ci><ci id="S6.SS1.p2.12.m12.1.1.2.3.3.cmml" xref="S6.SS1.p2.12.m12.1.1.2.3.3">𝑖</ci></apply></apply><csymbol cd="latexml" id="S6.SS1.p2.12.m12.1.1.3.cmml" xref="S6.SS1.p2.12.m12.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.12.m12.1c">k_{\rm{d}\it i}=</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.12.m12.1d">italic_k start_POSTSUBSCRIPT roman_d italic_i end_POSTSUBSCRIPT =</annotation></semantics></math> 0.1 for the MRE-HOT, the stable filter <math alttext="L(s)" class="ltx_Math" display="inline" id="S6.SS1.p2.13.m13.1"><semantics id="S6.SS1.p2.13.m13.1a"><mrow id="S6.SS1.p2.13.m13.1.2" xref="S6.SS1.p2.13.m13.1.2.cmml"><mi id="S6.SS1.p2.13.m13.1.2.2" xref="S6.SS1.p2.13.m13.1.2.2.cmml">L</mi><mo id="S6.SS1.p2.13.m13.1.2.1" xref="S6.SS1.p2.13.m13.1.2.1.cmml">⁢</mo><mrow id="S6.SS1.p2.13.m13.1.2.3.2" xref="S6.SS1.p2.13.m13.1.2.cmml"><mo id="S6.SS1.p2.13.m13.1.2.3.2.1" stretchy="false" xref="S6.SS1.p2.13.m13.1.2.cmml">(</mo><mi id="S6.SS1.p2.13.m13.1.1" xref="S6.SS1.p2.13.m13.1.1.cmml">s</mi><mo id="S6.SS1.p2.13.m13.1.2.3.2.2" stretchy="false" xref="S6.SS1.p2.13.m13.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.13.m13.1b"><apply id="S6.SS1.p2.13.m13.1.2.cmml" xref="S6.SS1.p2.13.m13.1.2"><times id="S6.SS1.p2.13.m13.1.2.1.cmml" xref="S6.SS1.p2.13.m13.1.2.1"></times><ci id="S6.SS1.p2.13.m13.1.2.2.cmml" xref="S6.SS1.p2.13.m13.1.2.2">𝐿</ci><ci id="S6.SS1.p2.13.m13.1.1.cmml" xref="S6.SS1.p2.13.m13.1.1">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.13.m13.1c">L(s)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.13.m13.1d">italic_L ( italic_s )</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S6.SS1.p2.14.m14.1"><semantics id="S6.SS1.p2.14.m14.1a"><mo id="S6.SS1.p2.14.m14.1.1" xref="S6.SS1.p2.14.m14.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.14.m14.1b"><csymbol cd="latexml" id="S6.SS1.p2.14.m14.1.1.cmml" xref="S6.SS1.p2.14.m14.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.14.m14.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.14.m14.1d">:=</annotation></semantics></math> <math alttext="20/(s+20)" class="ltx_Math" display="inline" id="S6.SS1.p2.15.m15.1"><semantics id="S6.SS1.p2.15.m15.1a"><mrow id="S6.SS1.p2.15.m15.1.1" xref="S6.SS1.p2.15.m15.1.1.cmml"><mn id="S6.SS1.p2.15.m15.1.1.3" xref="S6.SS1.p2.15.m15.1.1.3.cmml">20</mn><mo id="S6.SS1.p2.15.m15.1.1.2" xref="S6.SS1.p2.15.m15.1.1.2.cmml">/</mo><mrow id="S6.SS1.p2.15.m15.1.1.1.1" xref="S6.SS1.p2.15.m15.1.1.1.1.1.cmml"><mo id="S6.SS1.p2.15.m15.1.1.1.1.2" stretchy="false" xref="S6.SS1.p2.15.m15.1.1.1.1.1.cmml">(</mo><mrow id="S6.SS1.p2.15.m15.1.1.1.1.1" xref="S6.SS1.p2.15.m15.1.1.1.1.1.cmml"><mi id="S6.SS1.p2.15.m15.1.1.1.1.1.2" xref="S6.SS1.p2.15.m15.1.1.1.1.1.2.cmml">s</mi><mo id="S6.SS1.p2.15.m15.1.1.1.1.1.1" xref="S6.SS1.p2.15.m15.1.1.1.1.1.1.cmml">+</mo><mn id="S6.SS1.p2.15.m15.1.1.1.1.1.3" xref="S6.SS1.p2.15.m15.1.1.1.1.1.3.cmml">20</mn></mrow><mo id="S6.SS1.p2.15.m15.1.1.1.1.3" stretchy="false" xref="S6.SS1.p2.15.m15.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.15.m15.1b"><apply id="S6.SS1.p2.15.m15.1.1.cmml" xref="S6.SS1.p2.15.m15.1.1"><divide id="S6.SS1.p2.15.m15.1.1.2.cmml" xref="S6.SS1.p2.15.m15.1.1.2"></divide><cn id="S6.SS1.p2.15.m15.1.1.3.cmml" type="integer" xref="S6.SS1.p2.15.m15.1.1.3">20</cn><apply id="S6.SS1.p2.15.m15.1.1.1.1.1.cmml" xref="S6.SS1.p2.15.m15.1.1.1.1"><plus id="S6.SS1.p2.15.m15.1.1.1.1.1.1.cmml" xref="S6.SS1.p2.15.m15.1.1.1.1.1.1"></plus><ci id="S6.SS1.p2.15.m15.1.1.1.1.1.2.cmml" xref="S6.SS1.p2.15.m15.1.1.1.1.1.2">𝑠</ci><cn id="S6.SS1.p2.15.m15.1.1.1.1.1.3.cmml" type="integer" xref="S6.SS1.p2.15.m15.1.1.1.1.1.3">20</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.15.m15.1c">20/(s+20)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.15.m15.1d">20 / ( italic_s + 20 )</annotation></semantics></math> for the CL-DSC, and the other shared parameters to be the same values for fair comparisons.</p> </div> <figure class="ltx_figure" id="S6.F3"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="447" id="S6.F3.g1" src="x3.png" width="470"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 3: </span>Various performance comparisons of three controllers for the regulation problem under partial IE or IE condition. (a) The partial estimation errors <math alttext="\|\tilde{\bm{\theta}}_{\zeta}\|" class="ltx_Math" display="inline" id="S6.F3.6.m1.1"><semantics id="S6.F3.6.m1.1b"><mrow id="S6.F3.6.m1.1.1.1" xref="S6.F3.6.m1.1.1.2.cmml"><mo id="S6.F3.6.m1.1.1.1.2" stretchy="false" xref="S6.F3.6.m1.1.1.2.1.cmml">‖</mo><msub id="S6.F3.6.m1.1.1.1.1" xref="S6.F3.6.m1.1.1.1.1.cmml"><mover accent="true" id="S6.F3.6.m1.1.1.1.1.2" xref="S6.F3.6.m1.1.1.1.1.2.cmml"><mi id="S6.F3.6.m1.1.1.1.1.2.2" xref="S6.F3.6.m1.1.1.1.1.2.2.cmml">𝜽</mi><mo id="S6.F3.6.m1.1.1.1.1.2.1" xref="S6.F3.6.m1.1.1.1.1.2.1.cmml">~</mo></mover><mi id="S6.F3.6.m1.1.1.1.1.3" xref="S6.F3.6.m1.1.1.1.1.3.cmml">ζ</mi></msub><mo id="S6.F3.6.m1.1.1.1.3" stretchy="false" xref="S6.F3.6.m1.1.1.2.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.F3.6.m1.1c"><apply id="S6.F3.6.m1.1.1.2.cmml" xref="S6.F3.6.m1.1.1.1"><csymbol cd="latexml" id="S6.F3.6.m1.1.1.2.1.cmml" xref="S6.F3.6.m1.1.1.1.2">norm</csymbol><apply id="S6.F3.6.m1.1.1.1.1.cmml" xref="S6.F3.6.m1.1.1.1.1"><csymbol cd="ambiguous" id="S6.F3.6.m1.1.1.1.1.1.cmml" xref="S6.F3.6.m1.1.1.1.1">subscript</csymbol><apply id="S6.F3.6.m1.1.1.1.1.2.cmml" xref="S6.F3.6.m1.1.1.1.1.2"><ci id="S6.F3.6.m1.1.1.1.1.2.1.cmml" xref="S6.F3.6.m1.1.1.1.1.2.1">~</ci><ci id="S6.F3.6.m1.1.1.1.1.2.2.cmml" xref="S6.F3.6.m1.1.1.1.1.2.2">𝜽</ci></apply><ci id="S6.F3.6.m1.1.1.1.1.3.cmml" xref="S6.F3.6.m1.1.1.1.1.3">𝜁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F3.6.m1.1d">\|\tilde{\bm{\theta}}_{\zeta}\|</annotation><annotation encoding="application/x-llamapun" id="S6.F3.6.m1.1e">∥ over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ∥</annotation></semantics></math>. (b) The estimation errors <math alttext="\|\tilde{\bm{\theta}}\|" class="ltx_Math" display="inline" id="S6.F3.7.m2.1"><semantics id="S6.F3.7.m2.1b"><mrow id="S6.F3.7.m2.1.2.2" xref="S6.F3.7.m2.1.2.1.cmml"><mo id="S6.F3.7.m2.1.2.2.1" stretchy="false" xref="S6.F3.7.m2.1.2.1.1.cmml">‖</mo><mover accent="true" id="S6.F3.7.m2.1.1" xref="S6.F3.7.m2.1.1.cmml"><mi id="S6.F3.7.m2.1.1.2" xref="S6.F3.7.m2.1.1.2.cmml">𝜽</mi><mo id="S6.F3.7.m2.1.1.1" xref="S6.F3.7.m2.1.1.1.cmml">~</mo></mover><mo id="S6.F3.7.m2.1.2.2.2" stretchy="false" xref="S6.F3.7.m2.1.2.1.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.F3.7.m2.1c"><apply id="S6.F3.7.m2.1.2.1.cmml" xref="S6.F3.7.m2.1.2.2"><csymbol cd="latexml" id="S6.F3.7.m2.1.2.1.1.cmml" xref="S6.F3.7.m2.1.2.2.1">norm</csymbol><apply id="S6.F3.7.m2.1.1.cmml" xref="S6.F3.7.m2.1.1"><ci id="S6.F3.7.m2.1.1.1.cmml" xref="S6.F3.7.m2.1.1.1">~</ci><ci id="S6.F3.7.m2.1.1.2.cmml" xref="S6.F3.7.m2.1.1.2">𝜽</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F3.7.m2.1d">\|\tilde{\bm{\theta}}\|</annotation><annotation encoding="application/x-llamapun" id="S6.F3.7.m2.1e">∥ over~ start_ARG bold_italic_θ end_ARG ∥</annotation></semantics></math>. (c) The absolute tracking error norms <math alttext="|e_{1}|" class="ltx_Math" display="inline" id="S6.F3.8.m3.1"><semantics id="S6.F3.8.m3.1b"><mrow id="S6.F3.8.m3.1.1.1" xref="S6.F3.8.m3.1.1.2.cmml"><mo id="S6.F3.8.m3.1.1.1.2" stretchy="false" xref="S6.F3.8.m3.1.1.2.1.cmml">|</mo><msub id="S6.F3.8.m3.1.1.1.1" xref="S6.F3.8.m3.1.1.1.1.cmml"><mi id="S6.F3.8.m3.1.1.1.1.2" xref="S6.F3.8.m3.1.1.1.1.2.cmml">e</mi><mn id="S6.F3.8.m3.1.1.1.1.3" xref="S6.F3.8.m3.1.1.1.1.3.cmml">1</mn></msub><mo id="S6.F3.8.m3.1.1.1.3" stretchy="false" xref="S6.F3.8.m3.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.F3.8.m3.1c"><apply id="S6.F3.8.m3.1.1.2.cmml" xref="S6.F3.8.m3.1.1.1"><abs id="S6.F3.8.m3.1.1.2.1.cmml" xref="S6.F3.8.m3.1.1.1.2"></abs><apply id="S6.F3.8.m3.1.1.1.1.cmml" xref="S6.F3.8.m3.1.1.1.1"><csymbol cd="ambiguous" id="S6.F3.8.m3.1.1.1.1.1.cmml" xref="S6.F3.8.m3.1.1.1.1">subscript</csymbol><ci id="S6.F3.8.m3.1.1.1.1.2.cmml" xref="S6.F3.8.m3.1.1.1.1.2">𝑒</ci><cn id="S6.F3.8.m3.1.1.1.1.3.cmml" type="integer" xref="S6.F3.8.m3.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F3.8.m3.1d">|e_{1}|</annotation><annotation encoding="application/x-llamapun" id="S6.F3.8.m3.1e">| italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT |</annotation></semantics></math>. (d) The control inputs <math alttext="u" class="ltx_Math" display="inline" id="S6.F3.9.m4.1"><semantics id="S6.F3.9.m4.1b"><mi id="S6.F3.9.m4.1.1" xref="S6.F3.9.m4.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S6.F3.9.m4.1c"><ci id="S6.F3.9.m4.1.1.cmml" xref="S6.F3.9.m4.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.F3.9.m4.1d">u</annotation><annotation encoding="application/x-llamapun" id="S6.F3.9.m4.1e">italic_u</annotation></semantics></math>. (e) The exciting strengths <math alttext="\sigma_{\rm c}" class="ltx_Math" display="inline" id="S6.F3.10.m5.1"><semantics id="S6.F3.10.m5.1b"><msub id="S6.F3.10.m5.1.1" xref="S6.F3.10.m5.1.1.cmml"><mi id="S6.F3.10.m5.1.1.2" xref="S6.F3.10.m5.1.1.2.cmml">σ</mi><mi id="S6.F3.10.m5.1.1.3" mathvariant="normal" xref="S6.F3.10.m5.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S6.F3.10.m5.1c"><apply id="S6.F3.10.m5.1.1.cmml" xref="S6.F3.10.m5.1.1"><csymbol cd="ambiguous" id="S6.F3.10.m5.1.1.1.cmml" xref="S6.F3.10.m5.1.1">subscript</csymbol><ci id="S6.F3.10.m5.1.1.2.cmml" xref="S6.F3.10.m5.1.1.2">𝜎</ci><ci id="S6.F3.10.m5.1.1.3.cmml" xref="S6.F3.10.m5.1.1.3">c</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F3.10.m5.1d">\sigma_{\rm c}</annotation><annotation encoding="application/x-llamapun" id="S6.F3.10.m5.1e">italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT</annotation></semantics></math>.</figcaption> </figure> <figure class="ltx_figure" id="S6.F4"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="363" id="S6.F4.g1" src="x4.png" width="470"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 4: </span>Various performance comparisons of three controllers for the slowly time-varying parameter learning under the PE condition. (a) The estimation error norms <math alttext="\|\tilde{\bm{\theta}}\|" class="ltx_Math" display="inline" id="S6.F4.5.m1.1"><semantics id="S6.F4.5.m1.1b"><mrow id="S6.F4.5.m1.1.2.2" xref="S6.F4.5.m1.1.2.1.cmml"><mo id="S6.F4.5.m1.1.2.2.1" stretchy="false" xref="S6.F4.5.m1.1.2.1.1.cmml">‖</mo><mover accent="true" id="S6.F4.5.m1.1.1" xref="S6.F4.5.m1.1.1.cmml"><mi id="S6.F4.5.m1.1.1.2" xref="S6.F4.5.m1.1.1.2.cmml">𝜽</mi><mo id="S6.F4.5.m1.1.1.1" xref="S6.F4.5.m1.1.1.1.cmml">~</mo></mover><mo id="S6.F4.5.m1.1.2.2.2" stretchy="false" xref="S6.F4.5.m1.1.2.1.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.F4.5.m1.1c"><apply id="S6.F4.5.m1.1.2.1.cmml" xref="S6.F4.5.m1.1.2.2"><csymbol cd="latexml" id="S6.F4.5.m1.1.2.1.1.cmml" xref="S6.F4.5.m1.1.2.2.1">norm</csymbol><apply id="S6.F4.5.m1.1.1.cmml" xref="S6.F4.5.m1.1.1"><ci id="S6.F4.5.m1.1.1.1.cmml" xref="S6.F4.5.m1.1.1.1">~</ci><ci id="S6.F4.5.m1.1.1.2.cmml" xref="S6.F4.5.m1.1.1.2">𝜽</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F4.5.m1.1d">\|\tilde{\bm{\theta}}\|</annotation><annotation encoding="application/x-llamapun" id="S6.F4.5.m1.1e">∥ over~ start_ARG bold_italic_θ end_ARG ∥</annotation></semantics></math>. (b) The absolute tracking errors <math alttext="|e_{1}|" class="ltx_Math" display="inline" id="S6.F4.6.m2.1"><semantics id="S6.F4.6.m2.1b"><mrow id="S6.F4.6.m2.1.1.1" xref="S6.F4.6.m2.1.1.2.cmml"><mo id="S6.F4.6.m2.1.1.1.2" stretchy="false" xref="S6.F4.6.m2.1.1.2.1.cmml">|</mo><msub id="S6.F4.6.m2.1.1.1.1" xref="S6.F4.6.m2.1.1.1.1.cmml"><mi id="S6.F4.6.m2.1.1.1.1.2" xref="S6.F4.6.m2.1.1.1.1.2.cmml">e</mi><mn id="S6.F4.6.m2.1.1.1.1.3" xref="S6.F4.6.m2.1.1.1.1.3.cmml">1</mn></msub><mo id="S6.F4.6.m2.1.1.1.3" stretchy="false" xref="S6.F4.6.m2.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.F4.6.m2.1c"><apply id="S6.F4.6.m2.1.1.2.cmml" xref="S6.F4.6.m2.1.1.1"><abs id="S6.F4.6.m2.1.1.2.1.cmml" xref="S6.F4.6.m2.1.1.1.2"></abs><apply id="S6.F4.6.m2.1.1.1.1.cmml" xref="S6.F4.6.m2.1.1.1.1"><csymbol cd="ambiguous" id="S6.F4.6.m2.1.1.1.1.1.cmml" xref="S6.F4.6.m2.1.1.1.1">subscript</csymbol><ci id="S6.F4.6.m2.1.1.1.1.2.cmml" xref="S6.F4.6.m2.1.1.1.1.2">𝑒</ci><cn id="S6.F4.6.m2.1.1.1.1.3.cmml" type="integer" xref="S6.F4.6.m2.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F4.6.m2.1d">|e_{1}|</annotation><annotation encoding="application/x-llamapun" id="S6.F4.6.m2.1e">| italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT |</annotation></semantics></math>. (c) The control inputs <math alttext="u" class="ltx_Math" display="inline" id="S6.F4.7.m3.1"><semantics id="S6.F4.7.m3.1b"><mi id="S6.F4.7.m3.1.1" xref="S6.F4.7.m3.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S6.F4.7.m3.1c"><ci id="S6.F4.7.m3.1.1.cmml" xref="S6.F4.7.m3.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.F4.7.m3.1d">u</annotation><annotation encoding="application/x-llamapun" id="S6.F4.7.m3.1e">italic_u</annotation></semantics></math>. (d) The exciting strengths <math alttext="\sigma_{\rm c}" class="ltx_Math" display="inline" id="S6.F4.8.m4.1"><semantics id="S6.F4.8.m4.1b"><msub id="S6.F4.8.m4.1.1" xref="S6.F4.8.m4.1.1.cmml"><mi id="S6.F4.8.m4.1.1.2" xref="S6.F4.8.m4.1.1.2.cmml">σ</mi><mi id="S6.F4.8.m4.1.1.3" mathvariant="normal" xref="S6.F4.8.m4.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S6.F4.8.m4.1c"><apply id="S6.F4.8.m4.1.1.cmml" xref="S6.F4.8.m4.1.1"><csymbol cd="ambiguous" id="S6.F4.8.m4.1.1.1.cmml" xref="S6.F4.8.m4.1.1">subscript</csymbol><ci id="S6.F4.8.m4.1.1.2.cmml" xref="S6.F4.8.m4.1.1.2">𝜎</ci><ci id="S6.F4.8.m4.1.1.3.cmml" xref="S6.F4.8.m4.1.1.3">c</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F4.8.m4.1d">\sigma_{\rm c}</annotation><annotation encoding="application/x-llamapun" id="S6.F4.8.m4.1e">italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT</annotation></semantics></math>.</figcaption> </figure> <div class="ltx_para" id="S6.SS1.p3"> <p class="ltx_p" id="S6.SS1.p3.11"><span class="ltx_text ltx_font_italic" id="S6.SS1.p3.11.1">Case 1: Regulation with partial IE or IE.</span> Consider a regulation problem under the partial IE or IE condition with the desired parameter <math alttext="\bm{\theta}=[0.4,0.5,0.1]^{T}" class="ltx_Math" display="inline" id="S6.SS1.p3.1.m1.3"><semantics id="S6.SS1.p3.1.m1.3a"><mrow id="S6.SS1.p3.1.m1.3.4" xref="S6.SS1.p3.1.m1.3.4.cmml"><mi id="S6.SS1.p3.1.m1.3.4.2" xref="S6.SS1.p3.1.m1.3.4.2.cmml">𝜽</mi><mo id="S6.SS1.p3.1.m1.3.4.1" xref="S6.SS1.p3.1.m1.3.4.1.cmml">=</mo><msup id="S6.SS1.p3.1.m1.3.4.3" xref="S6.SS1.p3.1.m1.3.4.3.cmml"><mrow id="S6.SS1.p3.1.m1.3.4.3.2.2" xref="S6.SS1.p3.1.m1.3.4.3.2.1.cmml"><mo id="S6.SS1.p3.1.m1.3.4.3.2.2.1" stretchy="false" xref="S6.SS1.p3.1.m1.3.4.3.2.1.cmml">[</mo><mn id="S6.SS1.p3.1.m1.1.1" xref="S6.SS1.p3.1.m1.1.1.cmml">0.4</mn><mo id="S6.SS1.p3.1.m1.3.4.3.2.2.2" xref="S6.SS1.p3.1.m1.3.4.3.2.1.cmml">,</mo><mn id="S6.SS1.p3.1.m1.2.2" xref="S6.SS1.p3.1.m1.2.2.cmml">0.5</mn><mo id="S6.SS1.p3.1.m1.3.4.3.2.2.3" xref="S6.SS1.p3.1.m1.3.4.3.2.1.cmml">,</mo><mn id="S6.SS1.p3.1.m1.3.3" xref="S6.SS1.p3.1.m1.3.3.cmml">0.1</mn><mo id="S6.SS1.p3.1.m1.3.4.3.2.2.4" stretchy="false" xref="S6.SS1.p3.1.m1.3.4.3.2.1.cmml">]</mo></mrow><mi id="S6.SS1.p3.1.m1.3.4.3.3" xref="S6.SS1.p3.1.m1.3.4.3.3.cmml">T</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.1.m1.3b"><apply id="S6.SS1.p3.1.m1.3.4.cmml" xref="S6.SS1.p3.1.m1.3.4"><eq id="S6.SS1.p3.1.m1.3.4.1.cmml" xref="S6.SS1.p3.1.m1.3.4.1"></eq><ci id="S6.SS1.p3.1.m1.3.4.2.cmml" xref="S6.SS1.p3.1.m1.3.4.2">𝜽</ci><apply id="S6.SS1.p3.1.m1.3.4.3.cmml" xref="S6.SS1.p3.1.m1.3.4.3"><csymbol cd="ambiguous" id="S6.SS1.p3.1.m1.3.4.3.1.cmml" xref="S6.SS1.p3.1.m1.3.4.3">superscript</csymbol><list id="S6.SS1.p3.1.m1.3.4.3.2.1.cmml" xref="S6.SS1.p3.1.m1.3.4.3.2.2"><cn id="S6.SS1.p3.1.m1.1.1.cmml" type="float" xref="S6.SS1.p3.1.m1.1.1">0.4</cn><cn id="S6.SS1.p3.1.m1.2.2.cmml" type="float" xref="S6.SS1.p3.1.m1.2.2">0.5</cn><cn id="S6.SS1.p3.1.m1.3.3.cmml" type="float" xref="S6.SS1.p3.1.m1.3.3">0.1</cn></list><ci id="S6.SS1.p3.1.m1.3.4.3.3.cmml" xref="S6.SS1.p3.1.m1.3.4.3.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.1.m1.3c">\bm{\theta}=[0.4,0.5,0.1]^{T}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.1.m1.3d">bold_italic_θ = [ 0.4 , 0.5 , 0.1 ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> and the initial state <math alttext="\bm{x}(0)=\bm{0}" class="ltx_Math" display="inline" id="S6.SS1.p3.2.m2.1"><semantics id="S6.SS1.p3.2.m2.1a"><mrow id="S6.SS1.p3.2.m2.1.2" xref="S6.SS1.p3.2.m2.1.2.cmml"><mrow id="S6.SS1.p3.2.m2.1.2.2" xref="S6.SS1.p3.2.m2.1.2.2.cmml"><mi id="S6.SS1.p3.2.m2.1.2.2.2" xref="S6.SS1.p3.2.m2.1.2.2.2.cmml">𝒙</mi><mo id="S6.SS1.p3.2.m2.1.2.2.1" xref="S6.SS1.p3.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S6.SS1.p3.2.m2.1.2.2.3.2" xref="S6.SS1.p3.2.m2.1.2.2.cmml"><mo id="S6.SS1.p3.2.m2.1.2.2.3.2.1" stretchy="false" xref="S6.SS1.p3.2.m2.1.2.2.cmml">(</mo><mn id="S6.SS1.p3.2.m2.1.1" xref="S6.SS1.p3.2.m2.1.1.cmml">0</mn><mo id="S6.SS1.p3.2.m2.1.2.2.3.2.2" stretchy="false" xref="S6.SS1.p3.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S6.SS1.p3.2.m2.1.2.1" xref="S6.SS1.p3.2.m2.1.2.1.cmml">=</mo><mn id="S6.SS1.p3.2.m2.1.2.3" xref="S6.SS1.p3.2.m2.1.2.3.cmml">𝟎</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.2.m2.1b"><apply id="S6.SS1.p3.2.m2.1.2.cmml" xref="S6.SS1.p3.2.m2.1.2"><eq id="S6.SS1.p3.2.m2.1.2.1.cmml" xref="S6.SS1.p3.2.m2.1.2.1"></eq><apply id="S6.SS1.p3.2.m2.1.2.2.cmml" xref="S6.SS1.p3.2.m2.1.2.2"><times id="S6.SS1.p3.2.m2.1.2.2.1.cmml" xref="S6.SS1.p3.2.m2.1.2.2.1"></times><ci id="S6.SS1.p3.2.m2.1.2.2.2.cmml" xref="S6.SS1.p3.2.m2.1.2.2.2">𝒙</ci><cn id="S6.SS1.p3.2.m2.1.1.cmml" type="integer" xref="S6.SS1.p3.2.m2.1.1">0</cn></apply><cn id="S6.SS1.p3.2.m2.1.2.3.cmml" type="integer" xref="S6.SS1.p3.2.m2.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.2.m2.1c">\bm{x}(0)=\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.2.m2.1d">bold_italic_x ( 0 ) = bold_0</annotation></semantics></math>, where the reference trajectories <math alttext="y_{\rm r}" class="ltx_Math" display="inline" id="S6.SS1.p3.3.m3.1"><semantics id="S6.SS1.p3.3.m3.1a"><msub id="S6.SS1.p3.3.m3.1.1" xref="S6.SS1.p3.3.m3.1.1.cmml"><mi id="S6.SS1.p3.3.m3.1.1.2" xref="S6.SS1.p3.3.m3.1.1.2.cmml">y</mi><mi id="S6.SS1.p3.3.m3.1.1.3" mathvariant="normal" xref="S6.SS1.p3.3.m3.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.3.m3.1b"><apply id="S6.SS1.p3.3.m3.1.1.cmml" xref="S6.SS1.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S6.SS1.p3.3.m3.1.1.1.cmml" xref="S6.SS1.p3.3.m3.1.1">subscript</csymbol><ci id="S6.SS1.p3.3.m3.1.1.2.cmml" xref="S6.SS1.p3.3.m3.1.1.2">𝑦</ci><ci id="S6.SS1.p3.3.m3.1.1.3.cmml" xref="S6.SS1.p3.3.m3.1.1.3">r</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.3.m3.1c">y_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.3.m3.1d">italic_y start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\dot{y}_{\rm r}" class="ltx_Math" display="inline" id="S6.SS1.p3.4.m4.1"><semantics id="S6.SS1.p3.4.m4.1a"><msub id="S6.SS1.p3.4.m4.1.1" xref="S6.SS1.p3.4.m4.1.1.cmml"><mover accent="true" id="S6.SS1.p3.4.m4.1.1.2" xref="S6.SS1.p3.4.m4.1.1.2.cmml"><mi id="S6.SS1.p3.4.m4.1.1.2.2" xref="S6.SS1.p3.4.m4.1.1.2.2.cmml">y</mi><mo id="S6.SS1.p3.4.m4.1.1.2.1" xref="S6.SS1.p3.4.m4.1.1.2.1.cmml">˙</mo></mover><mi id="S6.SS1.p3.4.m4.1.1.3" mathvariant="normal" xref="S6.SS1.p3.4.m4.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.4.m4.1b"><apply id="S6.SS1.p3.4.m4.1.1.cmml" xref="S6.SS1.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S6.SS1.p3.4.m4.1.1.1.cmml" xref="S6.SS1.p3.4.m4.1.1">subscript</csymbol><apply id="S6.SS1.p3.4.m4.1.1.2.cmml" xref="S6.SS1.p3.4.m4.1.1.2"><ci id="S6.SS1.p3.4.m4.1.1.2.1.cmml" xref="S6.SS1.p3.4.m4.1.1.2.1">˙</ci><ci id="S6.SS1.p3.4.m4.1.1.2.2.cmml" xref="S6.SS1.p3.4.m4.1.1.2.2">𝑦</ci></apply><ci id="S6.SS1.p3.4.m4.1.1.3.cmml" xref="S6.SS1.p3.4.m4.1.1.3">r</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.4.m4.1c">\dot{y}_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.4.m4.1d">over˙ start_ARG italic_y end_ARG start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\ddot{y}_{\rm r}" class="ltx_Math" display="inline" id="S6.SS1.p3.5.m5.1"><semantics id="S6.SS1.p3.5.m5.1a"><msub id="S6.SS1.p3.5.m5.1.1" xref="S6.SS1.p3.5.m5.1.1.cmml"><mover accent="true" id="S6.SS1.p3.5.m5.1.1.2" xref="S6.SS1.p3.5.m5.1.1.2.cmml"><mi id="S6.SS1.p3.5.m5.1.1.2.2" xref="S6.SS1.p3.5.m5.1.1.2.2.cmml">y</mi><mo id="S6.SS1.p3.5.m5.1.1.2.1" xref="S6.SS1.p3.5.m5.1.1.2.1.cmml">¨</mo></mover><mi id="S6.SS1.p3.5.m5.1.1.3" mathvariant="normal" xref="S6.SS1.p3.5.m5.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.5.m5.1b"><apply id="S6.SS1.p3.5.m5.1.1.cmml" xref="S6.SS1.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S6.SS1.p3.5.m5.1.1.1.cmml" xref="S6.SS1.p3.5.m5.1.1">subscript</csymbol><apply id="S6.SS1.p3.5.m5.1.1.2.cmml" xref="S6.SS1.p3.5.m5.1.1.2"><ci id="S6.SS1.p3.5.m5.1.1.2.1.cmml" xref="S6.SS1.p3.5.m5.1.1.2.1">¨</ci><ci id="S6.SS1.p3.5.m5.1.1.2.2.cmml" xref="S6.SS1.p3.5.m5.1.1.2.2">𝑦</ci></apply><ci id="S6.SS1.p3.5.m5.1.1.3.cmml" xref="S6.SS1.p3.5.m5.1.1.3">r</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.5.m5.1c">\ddot{y}_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.5.m5.1d">over¨ start_ARG italic_y end_ARG start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\dddot{y}_{\rm r}" class="ltx_Math" display="inline" id="S6.SS1.p3.6.m6.1"><semantics id="S6.SS1.p3.6.m6.1a"><msub id="S6.SS1.p3.6.m6.1.1" xref="S6.SS1.p3.6.m6.1.1.cmml"><mover accent="true" id="S6.SS1.p3.6.m6.1.1.2" xref="S6.SS1.p3.6.m6.1.1.2.cmml"><mi id="S6.SS1.p3.6.m6.1.1.2.2" xref="S6.SS1.p3.6.m6.1.1.2.2.cmml">y</mi><mo id="S6.SS1.p3.6.m6.1.1.2.1" xref="S6.SS1.p3.6.m6.1.1.2.1.cmml">˙˙˙</mo></mover><mi id="S6.SS1.p3.6.m6.1.1.3" mathvariant="normal" xref="S6.SS1.p3.6.m6.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.6.m6.1b"><apply id="S6.SS1.p3.6.m6.1.1.cmml" xref="S6.SS1.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S6.SS1.p3.6.m6.1.1.1.cmml" xref="S6.SS1.p3.6.m6.1.1">subscript</csymbol><apply id="S6.SS1.p3.6.m6.1.1.2.cmml" xref="S6.SS1.p3.6.m6.1.1.2"><ci id="S6.SS1.p3.6.m6.1.1.2.1.cmml" xref="S6.SS1.p3.6.m6.1.1.2.1">˙˙˙</ci><ci id="S6.SS1.p3.6.m6.1.1.2.2.cmml" xref="S6.SS1.p3.6.m6.1.1.2.2">𝑦</ci></apply><ci id="S6.SS1.p3.6.m6.1.1.3.cmml" xref="S6.SS1.p3.6.m6.1.1.3">r</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.6.m6.1c">\dddot{y}_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.6.m6.1d">over˙˙˙ start_ARG italic_y end_ARG start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math> are generated by a reference model <math alttext="y_{\rm r}(t)" class="ltx_Math" display="inline" id="S6.SS1.p3.7.m7.1"><semantics id="S6.SS1.p3.7.m7.1a"><mrow id="S6.SS1.p3.7.m7.1.2" xref="S6.SS1.p3.7.m7.1.2.cmml"><msub id="S6.SS1.p3.7.m7.1.2.2" xref="S6.SS1.p3.7.m7.1.2.2.cmml"><mi id="S6.SS1.p3.7.m7.1.2.2.2" xref="S6.SS1.p3.7.m7.1.2.2.2.cmml">y</mi><mi id="S6.SS1.p3.7.m7.1.2.2.3" mathvariant="normal" xref="S6.SS1.p3.7.m7.1.2.2.3.cmml">r</mi></msub><mo id="S6.SS1.p3.7.m7.1.2.1" xref="S6.SS1.p3.7.m7.1.2.1.cmml">⁢</mo><mrow id="S6.SS1.p3.7.m7.1.2.3.2" xref="S6.SS1.p3.7.m7.1.2.cmml"><mo id="S6.SS1.p3.7.m7.1.2.3.2.1" stretchy="false" xref="S6.SS1.p3.7.m7.1.2.cmml">(</mo><mi id="S6.SS1.p3.7.m7.1.1" xref="S6.SS1.p3.7.m7.1.1.cmml">t</mi><mo id="S6.SS1.p3.7.m7.1.2.3.2.2" stretchy="false" xref="S6.SS1.p3.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.7.m7.1b"><apply id="S6.SS1.p3.7.m7.1.2.cmml" xref="S6.SS1.p3.7.m7.1.2"><times id="S6.SS1.p3.7.m7.1.2.1.cmml" xref="S6.SS1.p3.7.m7.1.2.1"></times><apply id="S6.SS1.p3.7.m7.1.2.2.cmml" xref="S6.SS1.p3.7.m7.1.2.2"><csymbol cd="ambiguous" id="S6.SS1.p3.7.m7.1.2.2.1.cmml" xref="S6.SS1.p3.7.m7.1.2.2">subscript</csymbol><ci id="S6.SS1.p3.7.m7.1.2.2.2.cmml" xref="S6.SS1.p3.7.m7.1.2.2.2">𝑦</ci><ci id="S6.SS1.p3.7.m7.1.2.2.3.cmml" xref="S6.SS1.p3.7.m7.1.2.2.3">r</ci></apply><ci id="S6.SS1.p3.7.m7.1.1.cmml" xref="S6.SS1.p3.7.m7.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.7.m7.1c">y_{\rm r}(t)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.7.m7.1d">italic_y start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> <math alttext="=" class="ltx_Math" display="inline" id="S6.SS1.p3.8.m8.1"><semantics id="S6.SS1.p3.8.m8.1a"><mo id="S6.SS1.p3.8.m8.1.1" xref="S6.SS1.p3.8.m8.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.8.m8.1b"><eq id="S6.SS1.p3.8.m8.1.1.cmml" xref="S6.SS1.p3.8.m8.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.8.m8.1c">=</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.8.m8.1d">=</annotation></semantics></math> <math alttext="\frac{a_{0}}{a(s)}[r(t)]" class="ltx_Math" display="inline" id="S6.SS1.p3.9.m9.3"><semantics id="S6.SS1.p3.9.m9.3a"><mrow id="S6.SS1.p3.9.m9.3.3" xref="S6.SS1.p3.9.m9.3.3.cmml"><mfrac id="S6.SS1.p3.9.m9.1.1" xref="S6.SS1.p3.9.m9.1.1.cmml"><msub id="S6.SS1.p3.9.m9.1.1.3" xref="S6.SS1.p3.9.m9.1.1.3.cmml"><mi id="S6.SS1.p3.9.m9.1.1.3.2" xref="S6.SS1.p3.9.m9.1.1.3.2.cmml">a</mi><mn id="S6.SS1.p3.9.m9.1.1.3.3" xref="S6.SS1.p3.9.m9.1.1.3.3.cmml">0</mn></msub><mrow id="S6.SS1.p3.9.m9.1.1.1" xref="S6.SS1.p3.9.m9.1.1.1.cmml"><mi id="S6.SS1.p3.9.m9.1.1.1.3" xref="S6.SS1.p3.9.m9.1.1.1.3.cmml">a</mi><mo id="S6.SS1.p3.9.m9.1.1.1.2" xref="S6.SS1.p3.9.m9.1.1.1.2.cmml">⁢</mo><mrow id="S6.SS1.p3.9.m9.1.1.1.4.2" xref="S6.SS1.p3.9.m9.1.1.1.cmml"><mo id="S6.SS1.p3.9.m9.1.1.1.4.2.1" stretchy="false" xref="S6.SS1.p3.9.m9.1.1.1.cmml">(</mo><mi id="S6.SS1.p3.9.m9.1.1.1.1" xref="S6.SS1.p3.9.m9.1.1.1.1.cmml">s</mi><mo id="S6.SS1.p3.9.m9.1.1.1.4.2.2" stretchy="false" xref="S6.SS1.p3.9.m9.1.1.1.cmml">)</mo></mrow></mrow></mfrac><mo id="S6.SS1.p3.9.m9.3.3.2" xref="S6.SS1.p3.9.m9.3.3.2.cmml">⁢</mo><mrow id="S6.SS1.p3.9.m9.3.3.1.1" xref="S6.SS1.p3.9.m9.3.3.1.2.cmml"><mo id="S6.SS1.p3.9.m9.3.3.1.1.2" stretchy="false" xref="S6.SS1.p3.9.m9.3.3.1.2.1.cmml">[</mo><mrow id="S6.SS1.p3.9.m9.3.3.1.1.1" xref="S6.SS1.p3.9.m9.3.3.1.1.1.cmml"><mi id="S6.SS1.p3.9.m9.3.3.1.1.1.2" xref="S6.SS1.p3.9.m9.3.3.1.1.1.2.cmml">r</mi><mo id="S6.SS1.p3.9.m9.3.3.1.1.1.1" xref="S6.SS1.p3.9.m9.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S6.SS1.p3.9.m9.3.3.1.1.1.3.2" xref="S6.SS1.p3.9.m9.3.3.1.1.1.cmml"><mo id="S6.SS1.p3.9.m9.3.3.1.1.1.3.2.1" stretchy="false" xref="S6.SS1.p3.9.m9.3.3.1.1.1.cmml">(</mo><mi id="S6.SS1.p3.9.m9.2.2" xref="S6.SS1.p3.9.m9.2.2.cmml">t</mi><mo id="S6.SS1.p3.9.m9.3.3.1.1.1.3.2.2" stretchy="false" xref="S6.SS1.p3.9.m9.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS1.p3.9.m9.3.3.1.1.3" stretchy="false" xref="S6.SS1.p3.9.m9.3.3.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.9.m9.3b"><apply id="S6.SS1.p3.9.m9.3.3.cmml" xref="S6.SS1.p3.9.m9.3.3"><times id="S6.SS1.p3.9.m9.3.3.2.cmml" xref="S6.SS1.p3.9.m9.3.3.2"></times><apply id="S6.SS1.p3.9.m9.1.1.cmml" xref="S6.SS1.p3.9.m9.1.1"><divide id="S6.SS1.p3.9.m9.1.1.2.cmml" xref="S6.SS1.p3.9.m9.1.1"></divide><apply id="S6.SS1.p3.9.m9.1.1.3.cmml" xref="S6.SS1.p3.9.m9.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.p3.9.m9.1.1.3.1.cmml" xref="S6.SS1.p3.9.m9.1.1.3">subscript</csymbol><ci id="S6.SS1.p3.9.m9.1.1.3.2.cmml" xref="S6.SS1.p3.9.m9.1.1.3.2">𝑎</ci><cn id="S6.SS1.p3.9.m9.1.1.3.3.cmml" type="integer" xref="S6.SS1.p3.9.m9.1.1.3.3">0</cn></apply><apply id="S6.SS1.p3.9.m9.1.1.1.cmml" xref="S6.SS1.p3.9.m9.1.1.1"><times id="S6.SS1.p3.9.m9.1.1.1.2.cmml" xref="S6.SS1.p3.9.m9.1.1.1.2"></times><ci id="S6.SS1.p3.9.m9.1.1.1.3.cmml" xref="S6.SS1.p3.9.m9.1.1.1.3">𝑎</ci><ci id="S6.SS1.p3.9.m9.1.1.1.1.cmml" xref="S6.SS1.p3.9.m9.1.1.1.1">𝑠</ci></apply></apply><apply id="S6.SS1.p3.9.m9.3.3.1.2.cmml" xref="S6.SS1.p3.9.m9.3.3.1.1"><csymbol cd="latexml" id="S6.SS1.p3.9.m9.3.3.1.2.1.cmml" xref="S6.SS1.p3.9.m9.3.3.1.1.2">delimited-[]</csymbol><apply id="S6.SS1.p3.9.m9.3.3.1.1.1.cmml" xref="S6.SS1.p3.9.m9.3.3.1.1.1"><times id="S6.SS1.p3.9.m9.3.3.1.1.1.1.cmml" xref="S6.SS1.p3.9.m9.3.3.1.1.1.1"></times><ci id="S6.SS1.p3.9.m9.3.3.1.1.1.2.cmml" xref="S6.SS1.p3.9.m9.3.3.1.1.1.2">𝑟</ci><ci id="S6.SS1.p3.9.m9.2.2.cmml" xref="S6.SS1.p3.9.m9.2.2">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.9.m9.3c">\frac{a_{0}}{a(s)}[r(t)]</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.9.m9.3d">divide start_ARG italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG italic_a ( italic_s ) end_ARG [ italic_r ( italic_t ) ]</annotation></semantics></math> with <math alttext="a_{0}=16" class="ltx_Math" display="inline" id="S6.SS1.p3.10.m10.1"><semantics id="S6.SS1.p3.10.m10.1a"><mrow id="S6.SS1.p3.10.m10.1.1" xref="S6.SS1.p3.10.m10.1.1.cmml"><msub id="S6.SS1.p3.10.m10.1.1.2" xref="S6.SS1.p3.10.m10.1.1.2.cmml"><mi id="S6.SS1.p3.10.m10.1.1.2.2" xref="S6.SS1.p3.10.m10.1.1.2.2.cmml">a</mi><mn id="S6.SS1.p3.10.m10.1.1.2.3" xref="S6.SS1.p3.10.m10.1.1.2.3.cmml">0</mn></msub><mo id="S6.SS1.p3.10.m10.1.1.1" xref="S6.SS1.p3.10.m10.1.1.1.cmml">=</mo><mn id="S6.SS1.p3.10.m10.1.1.3" xref="S6.SS1.p3.10.m10.1.1.3.cmml">16</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.10.m10.1b"><apply id="S6.SS1.p3.10.m10.1.1.cmml" xref="S6.SS1.p3.10.m10.1.1"><eq id="S6.SS1.p3.10.m10.1.1.1.cmml" xref="S6.SS1.p3.10.m10.1.1.1"></eq><apply id="S6.SS1.p3.10.m10.1.1.2.cmml" xref="S6.SS1.p3.10.m10.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.p3.10.m10.1.1.2.1.cmml" xref="S6.SS1.p3.10.m10.1.1.2">subscript</csymbol><ci id="S6.SS1.p3.10.m10.1.1.2.2.cmml" xref="S6.SS1.p3.10.m10.1.1.2.2">𝑎</ci><cn id="S6.SS1.p3.10.m10.1.1.2.3.cmml" type="integer" xref="S6.SS1.p3.10.m10.1.1.2.3">0</cn></apply><cn id="S6.SS1.p3.10.m10.1.1.3.cmml" type="integer" xref="S6.SS1.p3.10.m10.1.1.3">16</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.10.m10.1c">a_{0}=16</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.10.m10.1d">italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 16</annotation></semantics></math>, <math alttext="a(s)=s^{4}+8s^{3}+24s^{2}+32s+16" class="ltx_Math" display="inline" id="S6.SS1.p3.11.m11.1"><semantics id="S6.SS1.p3.11.m11.1a"><mrow id="S6.SS1.p3.11.m11.1.2" xref="S6.SS1.p3.11.m11.1.2.cmml"><mrow id="S6.SS1.p3.11.m11.1.2.2" xref="S6.SS1.p3.11.m11.1.2.2.cmml"><mi id="S6.SS1.p3.11.m11.1.2.2.2" xref="S6.SS1.p3.11.m11.1.2.2.2.cmml">a</mi><mo id="S6.SS1.p3.11.m11.1.2.2.1" xref="S6.SS1.p3.11.m11.1.2.2.1.cmml">⁢</mo><mrow id="S6.SS1.p3.11.m11.1.2.2.3.2" xref="S6.SS1.p3.11.m11.1.2.2.cmml"><mo id="S6.SS1.p3.11.m11.1.2.2.3.2.1" stretchy="false" xref="S6.SS1.p3.11.m11.1.2.2.cmml">(</mo><mi id="S6.SS1.p3.11.m11.1.1" xref="S6.SS1.p3.11.m11.1.1.cmml">s</mi><mo id="S6.SS1.p3.11.m11.1.2.2.3.2.2" stretchy="false" xref="S6.SS1.p3.11.m11.1.2.2.cmml">)</mo></mrow></mrow><mo id="S6.SS1.p3.11.m11.1.2.1" xref="S6.SS1.p3.11.m11.1.2.1.cmml">=</mo><mrow id="S6.SS1.p3.11.m11.1.2.3" xref="S6.SS1.p3.11.m11.1.2.3.cmml"><msup id="S6.SS1.p3.11.m11.1.2.3.2" xref="S6.SS1.p3.11.m11.1.2.3.2.cmml"><mi id="S6.SS1.p3.11.m11.1.2.3.2.2" xref="S6.SS1.p3.11.m11.1.2.3.2.2.cmml">s</mi><mn id="S6.SS1.p3.11.m11.1.2.3.2.3" xref="S6.SS1.p3.11.m11.1.2.3.2.3.cmml">4</mn></msup><mo id="S6.SS1.p3.11.m11.1.2.3.1" xref="S6.SS1.p3.11.m11.1.2.3.1.cmml">+</mo><mrow id="S6.SS1.p3.11.m11.1.2.3.3" xref="S6.SS1.p3.11.m11.1.2.3.3.cmml"><mn id="S6.SS1.p3.11.m11.1.2.3.3.2" xref="S6.SS1.p3.11.m11.1.2.3.3.2.cmml">8</mn><mo id="S6.SS1.p3.11.m11.1.2.3.3.1" xref="S6.SS1.p3.11.m11.1.2.3.3.1.cmml">⁢</mo><msup id="S6.SS1.p3.11.m11.1.2.3.3.3" xref="S6.SS1.p3.11.m11.1.2.3.3.3.cmml"><mi id="S6.SS1.p3.11.m11.1.2.3.3.3.2" xref="S6.SS1.p3.11.m11.1.2.3.3.3.2.cmml">s</mi><mn id="S6.SS1.p3.11.m11.1.2.3.3.3.3" xref="S6.SS1.p3.11.m11.1.2.3.3.3.3.cmml">3</mn></msup></mrow><mo id="S6.SS1.p3.11.m11.1.2.3.1a" xref="S6.SS1.p3.11.m11.1.2.3.1.cmml">+</mo><mrow id="S6.SS1.p3.11.m11.1.2.3.4" xref="S6.SS1.p3.11.m11.1.2.3.4.cmml"><mn id="S6.SS1.p3.11.m11.1.2.3.4.2" xref="S6.SS1.p3.11.m11.1.2.3.4.2.cmml">24</mn><mo id="S6.SS1.p3.11.m11.1.2.3.4.1" xref="S6.SS1.p3.11.m11.1.2.3.4.1.cmml">⁢</mo><msup id="S6.SS1.p3.11.m11.1.2.3.4.3" xref="S6.SS1.p3.11.m11.1.2.3.4.3.cmml"><mi id="S6.SS1.p3.11.m11.1.2.3.4.3.2" xref="S6.SS1.p3.11.m11.1.2.3.4.3.2.cmml">s</mi><mn id="S6.SS1.p3.11.m11.1.2.3.4.3.3" xref="S6.SS1.p3.11.m11.1.2.3.4.3.3.cmml">2</mn></msup></mrow><mo id="S6.SS1.p3.11.m11.1.2.3.1b" xref="S6.SS1.p3.11.m11.1.2.3.1.cmml">+</mo><mrow id="S6.SS1.p3.11.m11.1.2.3.5" xref="S6.SS1.p3.11.m11.1.2.3.5.cmml"><mn id="S6.SS1.p3.11.m11.1.2.3.5.2" xref="S6.SS1.p3.11.m11.1.2.3.5.2.cmml">32</mn><mo id="S6.SS1.p3.11.m11.1.2.3.5.1" xref="S6.SS1.p3.11.m11.1.2.3.5.1.cmml">⁢</mo><mi id="S6.SS1.p3.11.m11.1.2.3.5.3" xref="S6.SS1.p3.11.m11.1.2.3.5.3.cmml">s</mi></mrow><mo id="S6.SS1.p3.11.m11.1.2.3.1c" xref="S6.SS1.p3.11.m11.1.2.3.1.cmml">+</mo><mn id="S6.SS1.p3.11.m11.1.2.3.6" xref="S6.SS1.p3.11.m11.1.2.3.6.cmml">16</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.11.m11.1b"><apply id="S6.SS1.p3.11.m11.1.2.cmml" xref="S6.SS1.p3.11.m11.1.2"><eq id="S6.SS1.p3.11.m11.1.2.1.cmml" xref="S6.SS1.p3.11.m11.1.2.1"></eq><apply id="S6.SS1.p3.11.m11.1.2.2.cmml" xref="S6.SS1.p3.11.m11.1.2.2"><times id="S6.SS1.p3.11.m11.1.2.2.1.cmml" xref="S6.SS1.p3.11.m11.1.2.2.1"></times><ci id="S6.SS1.p3.11.m11.1.2.2.2.cmml" xref="S6.SS1.p3.11.m11.1.2.2.2">𝑎</ci><ci id="S6.SS1.p3.11.m11.1.1.cmml" xref="S6.SS1.p3.11.m11.1.1">𝑠</ci></apply><apply id="S6.SS1.p3.11.m11.1.2.3.cmml" xref="S6.SS1.p3.11.m11.1.2.3"><plus id="S6.SS1.p3.11.m11.1.2.3.1.cmml" xref="S6.SS1.p3.11.m11.1.2.3.1"></plus><apply id="S6.SS1.p3.11.m11.1.2.3.2.cmml" xref="S6.SS1.p3.11.m11.1.2.3.2"><csymbol cd="ambiguous" id="S6.SS1.p3.11.m11.1.2.3.2.1.cmml" xref="S6.SS1.p3.11.m11.1.2.3.2">superscript</csymbol><ci id="S6.SS1.p3.11.m11.1.2.3.2.2.cmml" xref="S6.SS1.p3.11.m11.1.2.3.2.2">𝑠</ci><cn id="S6.SS1.p3.11.m11.1.2.3.2.3.cmml" type="integer" xref="S6.SS1.p3.11.m11.1.2.3.2.3">4</cn></apply><apply id="S6.SS1.p3.11.m11.1.2.3.3.cmml" xref="S6.SS1.p3.11.m11.1.2.3.3"><times id="S6.SS1.p3.11.m11.1.2.3.3.1.cmml" xref="S6.SS1.p3.11.m11.1.2.3.3.1"></times><cn id="S6.SS1.p3.11.m11.1.2.3.3.2.cmml" type="integer" xref="S6.SS1.p3.11.m11.1.2.3.3.2">8</cn><apply id="S6.SS1.p3.11.m11.1.2.3.3.3.cmml" xref="S6.SS1.p3.11.m11.1.2.3.3.3"><csymbol cd="ambiguous" id="S6.SS1.p3.11.m11.1.2.3.3.3.1.cmml" xref="S6.SS1.p3.11.m11.1.2.3.3.3">superscript</csymbol><ci id="S6.SS1.p3.11.m11.1.2.3.3.3.2.cmml" xref="S6.SS1.p3.11.m11.1.2.3.3.3.2">𝑠</ci><cn id="S6.SS1.p3.11.m11.1.2.3.3.3.3.cmml" type="integer" xref="S6.SS1.p3.11.m11.1.2.3.3.3.3">3</cn></apply></apply><apply id="S6.SS1.p3.11.m11.1.2.3.4.cmml" xref="S6.SS1.p3.11.m11.1.2.3.4"><times id="S6.SS1.p3.11.m11.1.2.3.4.1.cmml" xref="S6.SS1.p3.11.m11.1.2.3.4.1"></times><cn id="S6.SS1.p3.11.m11.1.2.3.4.2.cmml" type="integer" xref="S6.SS1.p3.11.m11.1.2.3.4.2">24</cn><apply id="S6.SS1.p3.11.m11.1.2.3.4.3.cmml" xref="S6.SS1.p3.11.m11.1.2.3.4.3"><csymbol cd="ambiguous" id="S6.SS1.p3.11.m11.1.2.3.4.3.1.cmml" xref="S6.SS1.p3.11.m11.1.2.3.4.3">superscript</csymbol><ci id="S6.SS1.p3.11.m11.1.2.3.4.3.2.cmml" xref="S6.SS1.p3.11.m11.1.2.3.4.3.2">𝑠</ci><cn id="S6.SS1.p3.11.m11.1.2.3.4.3.3.cmml" type="integer" xref="S6.SS1.p3.11.m11.1.2.3.4.3.3">2</cn></apply></apply><apply id="S6.SS1.p3.11.m11.1.2.3.5.cmml" xref="S6.SS1.p3.11.m11.1.2.3.5"><times id="S6.SS1.p3.11.m11.1.2.3.5.1.cmml" xref="S6.SS1.p3.11.m11.1.2.3.5.1"></times><cn id="S6.SS1.p3.11.m11.1.2.3.5.2.cmml" type="integer" xref="S6.SS1.p3.11.m11.1.2.3.5.2">32</cn><ci id="S6.SS1.p3.11.m11.1.2.3.5.3.cmml" xref="S6.SS1.p3.11.m11.1.2.3.5.3">𝑠</ci></apply><cn id="S6.SS1.p3.11.m11.1.2.3.6.cmml" type="integer" xref="S6.SS1.p3.11.m11.1.2.3.6">16</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.11.m11.1c">a(s)=s^{4}+8s^{3}+24s^{2}+32s+16</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.11.m11.1d">italic_a ( italic_s ) = italic_s start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT + 8 italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + 24 italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 32 italic_s + 16</annotation></semantics></math>, and</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx28"> <tbody id="S6.Ex16"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle r(t)=\left\{\begin{array}[]{ll}-0.3,&t<60\\ -1.5,&60\leq t<100\\ 0,&t\geq 100\\ \end{array}\right." class="ltx_Math" display="inline" id="S6.Ex16.m1.2"><semantics id="S6.Ex16.m1.2a"><mrow id="S6.Ex16.m1.2.3" xref="S6.Ex16.m1.2.3.cmml"><mrow id="S6.Ex16.m1.2.3.2" xref="S6.Ex16.m1.2.3.2.cmml"><mi id="S6.Ex16.m1.2.3.2.2" xref="S6.Ex16.m1.2.3.2.2.cmml">r</mi><mo id="S6.Ex16.m1.2.3.2.1" xref="S6.Ex16.m1.2.3.2.1.cmml">⁢</mo><mrow id="S6.Ex16.m1.2.3.2.3.2" xref="S6.Ex16.m1.2.3.2.cmml"><mo id="S6.Ex16.m1.2.3.2.3.2.1" stretchy="false" xref="S6.Ex16.m1.2.3.2.cmml">(</mo><mi id="S6.Ex16.m1.1.1" xref="S6.Ex16.m1.1.1.cmml">t</mi><mo id="S6.Ex16.m1.2.3.2.3.2.2" stretchy="false" xref="S6.Ex16.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S6.Ex16.m1.2.3.1" xref="S6.Ex16.m1.2.3.1.cmml">=</mo><mrow id="S6.Ex16.m1.2.3.3.2" xref="S6.Ex16.m1.2.3.3.1.cmml"><mo id="S6.Ex16.m1.2.3.3.2.1" xref="S6.Ex16.m1.2.3.3.1.1.cmml">{</mo><mtable columnspacing="5pt" id="S6.Ex16.m1.2.2" rowspacing="0pt" xref="S6.Ex16.m1.2.2.cmml"><mtr id="S6.Ex16.m1.2.2a" xref="S6.Ex16.m1.2.2.cmml"><mtd class="ltx_align_left" columnalign="left" id="S6.Ex16.m1.2.2b" xref="S6.Ex16.m1.2.2.cmml"><mrow id="S6.Ex13.1.1.1" xref="S6.Ex13.1.1.1.1.cmml"><mrow id="S6.Ex13.1.1.1.1" xref="S6.Ex13.1.1.1.1.cmml"><mo id="S6.Ex13.1.1.1.1a" xref="S6.Ex13.1.1.1.1.cmml">−</mo><mn id="S6.Ex13.1.1.1.1.2" xref="S6.Ex13.1.1.1.1.2.cmml">0.3</mn></mrow><mo id="S6.Ex13.1.1.1.2" xref="S6.Ex13.1.1.1.1.cmml">,</mo></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="S6.Ex16.m1.2.2c" xref="S6.Ex16.m1.2.2.cmml"><mrow id="S6.Ex13.2.1" xref="S6.Ex13.2.1.cmml"><mi id="S6.Ex13.2.1.2" xref="S6.Ex13.2.1.2.cmml">t</mi><mo id="S6.Ex13.2.1.1" xref="S6.Ex13.2.1.1.cmml"><</mo><mn id="S6.Ex13.2.1.3" xref="S6.Ex13.2.1.3.cmml">60</mn></mrow></mtd></mtr><mtr id="S6.Ex16.m1.2.2d" xref="S6.Ex16.m1.2.2.cmml"><mtd class="ltx_align_left" columnalign="left" id="S6.Ex16.m1.2.2e" xref="S6.Ex16.m1.2.2.cmml"><mrow id="S6.Ex14.1.1.1" xref="S6.Ex14.1.1.1.1.cmml"><mrow id="S6.Ex14.1.1.1.1" xref="S6.Ex14.1.1.1.1.cmml"><mo id="S6.Ex14.1.1.1.1a" xref="S6.Ex14.1.1.1.1.cmml">−</mo><mn id="S6.Ex14.1.1.1.1.2" xref="S6.Ex14.1.1.1.1.2.cmml">1.5</mn></mrow><mo id="S6.Ex14.1.1.1.2" xref="S6.Ex14.1.1.1.1.cmml">,</mo></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="S6.Ex16.m1.2.2f" xref="S6.Ex16.m1.2.2.cmml"><mrow id="S6.Ex14.2.1" xref="S6.Ex14.2.1.cmml"><mn id="S6.Ex14.2.1.2" xref="S6.Ex14.2.1.2.cmml">60</mn><mo id="S6.Ex14.2.1.3" xref="S6.Ex14.2.1.3.cmml">≤</mo><mi id="S6.Ex14.2.1.4" xref="S6.Ex14.2.1.4.cmml">t</mi><mo id="S6.Ex14.2.1.5" xref="S6.Ex14.2.1.5.cmml"><</mo><mn id="S6.Ex14.2.1.6" xref="S6.Ex14.2.1.6.cmml">100</mn></mrow></mtd></mtr><mtr id="S6.Ex16.m1.2.2g" xref="S6.Ex16.m1.2.2.cmml"><mtd class="ltx_align_left" columnalign="left" id="S6.Ex16.m1.2.2h" xref="S6.Ex16.m1.2.2.cmml"><mrow id="S6.Ex15.1.1.3" xref="S6.Ex16.m1.2.2.cmml"><mn id="S6.Ex15.1.1.1" xref="S6.Ex15.1.1.1.cmml">0</mn><mo id="S6.Ex15.1.1.3.1" xref="S6.Ex16.m1.2.2.cmml">,</mo></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="S6.Ex16.m1.2.2i" xref="S6.Ex16.m1.2.2.cmml"><mrow id="S6.Ex15.2.1" xref="S6.Ex15.2.1.cmml"><mi id="S6.Ex15.2.1.2" xref="S6.Ex15.2.1.2.cmml">t</mi><mo id="S6.Ex15.2.1.1" xref="S6.Ex15.2.1.1.cmml">≥</mo><mn id="S6.Ex15.2.1.3" xref="S6.Ex15.2.1.3.cmml">100</mn></mrow></mtd></mtr></mtable><mi id="S6.Ex16.m1.2.3.3.2.2" xref="S6.Ex16.m1.2.3.3.1.1.cmml"></mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex16.m1.2b"><apply id="S6.Ex16.m1.2.3.cmml" xref="S6.Ex16.m1.2.3"><eq id="S6.Ex16.m1.2.3.1.cmml" xref="S6.Ex16.m1.2.3.1"></eq><apply id="S6.Ex16.m1.2.3.2.cmml" xref="S6.Ex16.m1.2.3.2"><times id="S6.Ex16.m1.2.3.2.1.cmml" xref="S6.Ex16.m1.2.3.2.1"></times><ci id="S6.Ex16.m1.2.3.2.2.cmml" xref="S6.Ex16.m1.2.3.2.2">𝑟</ci><ci id="S6.Ex16.m1.1.1.cmml" xref="S6.Ex16.m1.1.1">𝑡</ci></apply><apply id="S6.Ex16.m1.2.3.3.1.cmml" xref="S6.Ex16.m1.2.3.3.2"><csymbol cd="latexml" id="S6.Ex16.m1.2.3.3.1.1.cmml" xref="S6.Ex16.m1.2.3.3.2.1">cases</csymbol><matrix id="S6.Ex16.m1.2.2.cmml" xref="S6.Ex16.m1.2.2"><matrixrow id="S6.Ex16.m1.2.2a.cmml" xref="S6.Ex16.m1.2.2"><apply id="S6.Ex13.1.1.1.1.cmml" xref="S6.Ex13.1.1.1"><minus id="S6.Ex13.1.1.1.1.1.cmml" xref="S6.Ex13.1.1.1"></minus><cn id="S6.Ex13.1.1.1.1.2.cmml" type="float" xref="S6.Ex13.1.1.1.1.2">0.3</cn></apply><apply id="S6.Ex13.2.1.cmml" xref="S6.Ex13.2.1"><lt id="S6.Ex13.2.1.1.cmml" xref="S6.Ex13.2.1.1"></lt><ci id="S6.Ex13.2.1.2.cmml" xref="S6.Ex13.2.1.2">𝑡</ci><cn id="S6.Ex13.2.1.3.cmml" type="integer" xref="S6.Ex13.2.1.3">60</cn></apply></matrixrow><matrixrow id="S6.Ex16.m1.2.2b.cmml" xref="S6.Ex16.m1.2.2"><apply id="S6.Ex14.1.1.1.1.cmml" xref="S6.Ex14.1.1.1"><minus id="S6.Ex14.1.1.1.1.1.cmml" xref="S6.Ex14.1.1.1"></minus><cn id="S6.Ex14.1.1.1.1.2.cmml" type="float" xref="S6.Ex14.1.1.1.1.2">1.5</cn></apply><apply id="S6.Ex14.2.1.cmml" xref="S6.Ex14.2.1"><and id="S6.Ex14.2.1a.cmml" xref="S6.Ex14.2.1"></and><apply id="S6.Ex14.2.1b.cmml" xref="S6.Ex14.2.1"><leq id="S6.Ex14.2.1.3.cmml" xref="S6.Ex14.2.1.3"></leq><cn id="S6.Ex14.2.1.2.cmml" type="integer" xref="S6.Ex14.2.1.2">60</cn><ci id="S6.Ex14.2.1.4.cmml" xref="S6.Ex14.2.1.4">𝑡</ci></apply><apply id="S6.Ex14.2.1c.cmml" xref="S6.Ex14.2.1"><lt id="S6.Ex14.2.1.5.cmml" xref="S6.Ex14.2.1.5"></lt><share href="https://arxiv.org/html/2401.10785v2#S6.Ex14.2.1.4.cmml" id="S6.Ex14.2.1d.cmml" xref="S6.Ex14.2.1"></share><cn id="S6.Ex14.2.1.6.cmml" type="integer" xref="S6.Ex14.2.1.6">100</cn></apply></apply></matrixrow><matrixrow id="S6.Ex16.m1.2.2c.cmml" xref="S6.Ex16.m1.2.2"><cn id="S6.Ex15.1.1.1.cmml" type="integer" xref="S6.Ex15.1.1.1">0</cn><apply id="S6.Ex15.2.1.cmml" xref="S6.Ex15.2.1"><geq id="S6.Ex15.2.1.1.cmml" xref="S6.Ex15.2.1.1"></geq><ci id="S6.Ex15.2.1.2.cmml" xref="S6.Ex15.2.1.2">𝑡</ci><cn id="S6.Ex15.2.1.3.cmml" type="integer" xref="S6.Ex15.2.1.3">100</cn></apply></matrixrow></matrix></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex16.m1.2c">\displaystyle r(t)=\left\{\begin{array}[]{ll}-0.3,&t<60\\ -1.5,&60\leq t<100\\ 0,&t\geq 100\\ \end{array}\right.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex16.m1.2d">italic_r ( italic_t ) = { start_ARRAY start_ROW start_CELL - 0.3 , end_CELL start_CELL italic_t < 60 end_CELL end_ROW start_ROW start_CELL - 1.5 , end_CELL start_CELL 60 ≤ italic_t < 100 end_CELL end_ROW start_ROW start_CELL 0 , end_CELL start_CELL italic_t ≥ 100 end_CELL end_ROW end_ARRAY</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS1.p3.35">such that partial IE exists in <math alttext="t\in[0,60)" class="ltx_Math" display="inline" id="S6.SS1.p3.12.m1.2"><semantics id="S6.SS1.p3.12.m1.2a"><mrow id="S6.SS1.p3.12.m1.2.3" xref="S6.SS1.p3.12.m1.2.3.cmml"><mi id="S6.SS1.p3.12.m1.2.3.2" xref="S6.SS1.p3.12.m1.2.3.2.cmml">t</mi><mo id="S6.SS1.p3.12.m1.2.3.1" xref="S6.SS1.p3.12.m1.2.3.1.cmml">∈</mo><mrow id="S6.SS1.p3.12.m1.2.3.3.2" xref="S6.SS1.p3.12.m1.2.3.3.1.cmml"><mo id="S6.SS1.p3.12.m1.2.3.3.2.1" stretchy="false" xref="S6.SS1.p3.12.m1.2.3.3.1.cmml">[</mo><mn id="S6.SS1.p3.12.m1.1.1" xref="S6.SS1.p3.12.m1.1.1.cmml">0</mn><mo id="S6.SS1.p3.12.m1.2.3.3.2.2" xref="S6.SS1.p3.12.m1.2.3.3.1.cmml">,</mo><mn id="S6.SS1.p3.12.m1.2.2" xref="S6.SS1.p3.12.m1.2.2.cmml">60</mn><mo id="S6.SS1.p3.12.m1.2.3.3.2.3" stretchy="false" xref="S6.SS1.p3.12.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.12.m1.2b"><apply id="S6.SS1.p3.12.m1.2.3.cmml" xref="S6.SS1.p3.12.m1.2.3"><in id="S6.SS1.p3.12.m1.2.3.1.cmml" xref="S6.SS1.p3.12.m1.2.3.1"></in><ci id="S6.SS1.p3.12.m1.2.3.2.cmml" xref="S6.SS1.p3.12.m1.2.3.2">𝑡</ci><interval closure="closed-open" id="S6.SS1.p3.12.m1.2.3.3.1.cmml" xref="S6.SS1.p3.12.m1.2.3.3.2"><cn id="S6.SS1.p3.12.m1.1.1.cmml" type="integer" xref="S6.SS1.p3.12.m1.1.1">0</cn><cn id="S6.SS1.p3.12.m1.2.2.cmml" type="integer" xref="S6.SS1.p3.12.m1.2.2">60</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.12.m1.2c">t\in[0,60)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.12.m1.2d">italic_t ∈ [ 0 , 60 )</annotation></semantics></math> s, and IE exists in <math alttext="t\in[60,120]" class="ltx_Math" display="inline" id="S6.SS1.p3.13.m2.2"><semantics id="S6.SS1.p3.13.m2.2a"><mrow id="S6.SS1.p3.13.m2.2.3" xref="S6.SS1.p3.13.m2.2.3.cmml"><mi id="S6.SS1.p3.13.m2.2.3.2" xref="S6.SS1.p3.13.m2.2.3.2.cmml">t</mi><mo id="S6.SS1.p3.13.m2.2.3.1" xref="S6.SS1.p3.13.m2.2.3.1.cmml">∈</mo><mrow id="S6.SS1.p3.13.m2.2.3.3.2" xref="S6.SS1.p3.13.m2.2.3.3.1.cmml"><mo id="S6.SS1.p3.13.m2.2.3.3.2.1" stretchy="false" xref="S6.SS1.p3.13.m2.2.3.3.1.cmml">[</mo><mn id="S6.SS1.p3.13.m2.1.1" xref="S6.SS1.p3.13.m2.1.1.cmml">60</mn><mo id="S6.SS1.p3.13.m2.2.3.3.2.2" xref="S6.SS1.p3.13.m2.2.3.3.1.cmml">,</mo><mn id="S6.SS1.p3.13.m2.2.2" xref="S6.SS1.p3.13.m2.2.2.cmml">120</mn><mo id="S6.SS1.p3.13.m2.2.3.3.2.3" stretchy="false" xref="S6.SS1.p3.13.m2.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.13.m2.2b"><apply id="S6.SS1.p3.13.m2.2.3.cmml" xref="S6.SS1.p3.13.m2.2.3"><in id="S6.SS1.p3.13.m2.2.3.1.cmml" xref="S6.SS1.p3.13.m2.2.3.1"></in><ci id="S6.SS1.p3.13.m2.2.3.2.cmml" xref="S6.SS1.p3.13.m2.2.3.2">𝑡</ci><interval closure="closed" id="S6.SS1.p3.13.m2.2.3.3.1.cmml" xref="S6.SS1.p3.13.m2.2.3.3.2"><cn id="S6.SS1.p3.13.m2.1.1.cmml" type="integer" xref="S6.SS1.p3.13.m2.1.1">60</cn><cn id="S6.SS1.p3.13.m2.2.2.cmml" type="integer" xref="S6.SS1.p3.13.m2.2.2">120</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.13.m2.2c">t\in[60,120]</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.13.m2.2d">italic_t ∈ [ 60 , 120 ]</annotation></semantics></math> s. Performance comparisons of the three controllers are depicted in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S6.F3" title="Figure 3 ‣ VI-A Stability and Convergence Comparison ‣ VI Simulation Studies ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">3</span></a> with <math alttext="\tilde{\bm{\theta}}_{\zeta}" class="ltx_Math" display="inline" id="S6.SS1.p3.14.m3.1"><semantics id="S6.SS1.p3.14.m3.1a"><msub id="S6.SS1.p3.14.m3.1.1" xref="S6.SS1.p3.14.m3.1.1.cmml"><mover accent="true" id="S6.SS1.p3.14.m3.1.1.2" xref="S6.SS1.p3.14.m3.1.1.2.cmml"><mi id="S6.SS1.p3.14.m3.1.1.2.2" xref="S6.SS1.p3.14.m3.1.1.2.2.cmml">𝜽</mi><mo id="S6.SS1.p3.14.m3.1.1.2.1" xref="S6.SS1.p3.14.m3.1.1.2.1.cmml">~</mo></mover><mi id="S6.SS1.p3.14.m3.1.1.3" xref="S6.SS1.p3.14.m3.1.1.3.cmml">ζ</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.14.m3.1b"><apply id="S6.SS1.p3.14.m3.1.1.cmml" xref="S6.SS1.p3.14.m3.1.1"><csymbol cd="ambiguous" id="S6.SS1.p3.14.m3.1.1.1.cmml" xref="S6.SS1.p3.14.m3.1.1">subscript</csymbol><apply id="S6.SS1.p3.14.m3.1.1.2.cmml" xref="S6.SS1.p3.14.m3.1.1.2"><ci id="S6.SS1.p3.14.m3.1.1.2.1.cmml" xref="S6.SS1.p3.14.m3.1.1.2.1">~</ci><ci id="S6.SS1.p3.14.m3.1.1.2.2.cmml" xref="S6.SS1.p3.14.m3.1.1.2.2">𝜽</ci></apply><ci id="S6.SS1.p3.14.m3.1.1.3.cmml" xref="S6.SS1.p3.14.m3.1.1.3">𝜁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.14.m3.1c">\tilde{\bm{\theta}}_{\zeta}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.14.m3.1d">over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="S6.SS1.p3.15.m4.1"><semantics id="S6.SS1.p3.15.m4.1a"><mo id="S6.SS1.p3.15.m4.1.1" xref="S6.SS1.p3.15.m4.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.15.m4.1b"><csymbol cd="latexml" id="S6.SS1.p3.15.m4.1.1.cmml" xref="S6.SS1.p3.15.m4.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.15.m4.1c">:=</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.15.m4.1d">:=</annotation></semantics></math> <math alttext="[\tilde{\theta}_{1},\tilde{\theta}_{2}]^{T}" class="ltx_Math" display="inline" id="S6.SS1.p3.16.m5.2"><semantics id="S6.SS1.p3.16.m5.2a"><msup id="S6.SS1.p3.16.m5.2.2" xref="S6.SS1.p3.16.m5.2.2.cmml"><mrow id="S6.SS1.p3.16.m5.2.2.2.2" xref="S6.SS1.p3.16.m5.2.2.2.3.cmml"><mo id="S6.SS1.p3.16.m5.2.2.2.2.3" stretchy="false" xref="S6.SS1.p3.16.m5.2.2.2.3.cmml">[</mo><msub id="S6.SS1.p3.16.m5.1.1.1.1.1" xref="S6.SS1.p3.16.m5.1.1.1.1.1.cmml"><mover accent="true" id="S6.SS1.p3.16.m5.1.1.1.1.1.2" xref="S6.SS1.p3.16.m5.1.1.1.1.1.2.cmml"><mi id="S6.SS1.p3.16.m5.1.1.1.1.1.2.2" xref="S6.SS1.p3.16.m5.1.1.1.1.1.2.2.cmml">θ</mi><mo id="S6.SS1.p3.16.m5.1.1.1.1.1.2.1" xref="S6.SS1.p3.16.m5.1.1.1.1.1.2.1.cmml">~</mo></mover><mn id="S6.SS1.p3.16.m5.1.1.1.1.1.3" xref="S6.SS1.p3.16.m5.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S6.SS1.p3.16.m5.2.2.2.2.4" xref="S6.SS1.p3.16.m5.2.2.2.3.cmml">,</mo><msub id="S6.SS1.p3.16.m5.2.2.2.2.2" xref="S6.SS1.p3.16.m5.2.2.2.2.2.cmml"><mover accent="true" id="S6.SS1.p3.16.m5.2.2.2.2.2.2" xref="S6.SS1.p3.16.m5.2.2.2.2.2.2.cmml"><mi id="S6.SS1.p3.16.m5.2.2.2.2.2.2.2" xref="S6.SS1.p3.16.m5.2.2.2.2.2.2.2.cmml">θ</mi><mo id="S6.SS1.p3.16.m5.2.2.2.2.2.2.1" xref="S6.SS1.p3.16.m5.2.2.2.2.2.2.1.cmml">~</mo></mover><mn id="S6.SS1.p3.16.m5.2.2.2.2.2.3" xref="S6.SS1.p3.16.m5.2.2.2.2.2.3.cmml">2</mn></msub><mo id="S6.SS1.p3.16.m5.2.2.2.2.5" stretchy="false" xref="S6.SS1.p3.16.m5.2.2.2.3.cmml">]</mo></mrow><mi id="S6.SS1.p3.16.m5.2.2.4" xref="S6.SS1.p3.16.m5.2.2.4.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.16.m5.2b"><apply id="S6.SS1.p3.16.m5.2.2.cmml" xref="S6.SS1.p3.16.m5.2.2"><csymbol cd="ambiguous" id="S6.SS1.p3.16.m5.2.2.3.cmml" xref="S6.SS1.p3.16.m5.2.2">superscript</csymbol><interval closure="closed" id="S6.SS1.p3.16.m5.2.2.2.3.cmml" xref="S6.SS1.p3.16.m5.2.2.2.2"><apply id="S6.SS1.p3.16.m5.1.1.1.1.1.cmml" xref="S6.SS1.p3.16.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS1.p3.16.m5.1.1.1.1.1.1.cmml" xref="S6.SS1.p3.16.m5.1.1.1.1.1">subscript</csymbol><apply id="S6.SS1.p3.16.m5.1.1.1.1.1.2.cmml" xref="S6.SS1.p3.16.m5.1.1.1.1.1.2"><ci id="S6.SS1.p3.16.m5.1.1.1.1.1.2.1.cmml" xref="S6.SS1.p3.16.m5.1.1.1.1.1.2.1">~</ci><ci id="S6.SS1.p3.16.m5.1.1.1.1.1.2.2.cmml" xref="S6.SS1.p3.16.m5.1.1.1.1.1.2.2">𝜃</ci></apply><cn id="S6.SS1.p3.16.m5.1.1.1.1.1.3.cmml" type="integer" xref="S6.SS1.p3.16.m5.1.1.1.1.1.3">1</cn></apply><apply id="S6.SS1.p3.16.m5.2.2.2.2.2.cmml" xref="S6.SS1.p3.16.m5.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS1.p3.16.m5.2.2.2.2.2.1.cmml" xref="S6.SS1.p3.16.m5.2.2.2.2.2">subscript</csymbol><apply id="S6.SS1.p3.16.m5.2.2.2.2.2.2.cmml" xref="S6.SS1.p3.16.m5.2.2.2.2.2.2"><ci id="S6.SS1.p3.16.m5.2.2.2.2.2.2.1.cmml" xref="S6.SS1.p3.16.m5.2.2.2.2.2.2.1">~</ci><ci id="S6.SS1.p3.16.m5.2.2.2.2.2.2.2.cmml" xref="S6.SS1.p3.16.m5.2.2.2.2.2.2.2">𝜃</ci></apply><cn id="S6.SS1.p3.16.m5.2.2.2.2.2.3.cmml" type="integer" xref="S6.SS1.p3.16.m5.2.2.2.2.2.3">2</cn></apply></interval><ci id="S6.SS1.p3.16.m5.2.2.4.cmml" xref="S6.SS1.p3.16.m5.2.2.4">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.16.m5.2c">[\tilde{\theta}_{1},\tilde{\theta}_{2}]^{T}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.16.m5.2d">[ over~ start_ARG italic_θ end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , over~ start_ARG italic_θ end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math>. It is observed that the CL-DSC exhibits the rapid convergence of the estimation error <math alttext="\tilde{\bm{\theta}}" class="ltx_Math" display="inline" id="S6.SS1.p3.17.m6.1"><semantics id="S6.SS1.p3.17.m6.1a"><mover accent="true" id="S6.SS1.p3.17.m6.1.1" xref="S6.SS1.p3.17.m6.1.1.cmml"><mi id="S6.SS1.p3.17.m6.1.1.2" xref="S6.SS1.p3.17.m6.1.1.2.cmml">𝜽</mi><mo id="S6.SS1.p3.17.m6.1.1.1" xref="S6.SS1.p3.17.m6.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.17.m6.1b"><apply id="S6.SS1.p3.17.m6.1.1.cmml" xref="S6.SS1.p3.17.m6.1.1"><ci id="S6.SS1.p3.17.m6.1.1.1.cmml" xref="S6.SS1.p3.17.m6.1.1.1">~</ci><ci id="S6.SS1.p3.17.m6.1.1.2.cmml" xref="S6.SS1.p3.17.m6.1.1.2">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.17.m6.1c">\tilde{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.17.m6.1d">over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math> at <math alttext="t\in[60,120]" class="ltx_Math" display="inline" id="S6.SS1.p3.18.m7.2"><semantics id="S6.SS1.p3.18.m7.2a"><mrow id="S6.SS1.p3.18.m7.2.3" xref="S6.SS1.p3.18.m7.2.3.cmml"><mi id="S6.SS1.p3.18.m7.2.3.2" xref="S6.SS1.p3.18.m7.2.3.2.cmml">t</mi><mo id="S6.SS1.p3.18.m7.2.3.1" xref="S6.SS1.p3.18.m7.2.3.1.cmml">∈</mo><mrow id="S6.SS1.p3.18.m7.2.3.3.2" xref="S6.SS1.p3.18.m7.2.3.3.1.cmml"><mo id="S6.SS1.p3.18.m7.2.3.3.2.1" stretchy="false" xref="S6.SS1.p3.18.m7.2.3.3.1.cmml">[</mo><mn id="S6.SS1.p3.18.m7.1.1" xref="S6.SS1.p3.18.m7.1.1.cmml">60</mn><mo id="S6.SS1.p3.18.m7.2.3.3.2.2" xref="S6.SS1.p3.18.m7.2.3.3.1.cmml">,</mo><mn id="S6.SS1.p3.18.m7.2.2" xref="S6.SS1.p3.18.m7.2.2.cmml">120</mn><mo id="S6.SS1.p3.18.m7.2.3.3.2.3" stretchy="false" xref="S6.SS1.p3.18.m7.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.18.m7.2b"><apply id="S6.SS1.p3.18.m7.2.3.cmml" xref="S6.SS1.p3.18.m7.2.3"><in id="S6.SS1.p3.18.m7.2.3.1.cmml" xref="S6.SS1.p3.18.m7.2.3.1"></in><ci id="S6.SS1.p3.18.m7.2.3.2.cmml" xref="S6.SS1.p3.18.m7.2.3.2">𝑡</ci><interval closure="closed" id="S6.SS1.p3.18.m7.2.3.3.1.cmml" xref="S6.SS1.p3.18.m7.2.3.3.2"><cn id="S6.SS1.p3.18.m7.1.1.cmml" type="integer" xref="S6.SS1.p3.18.m7.1.1">60</cn><cn id="S6.SS1.p3.18.m7.2.2.cmml" type="integer" xref="S6.SS1.p3.18.m7.2.2">120</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.18.m7.2c">t\in[60,120]</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.18.m7.2d">italic_t ∈ [ 60 , 120 ]</annotation></semantics></math> s due to the establishment of IE but still has a steady-state error [see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S6.F3" title="Figure 3 ‣ VI-A Stability and Convergence Comparison ‣ VI Simulation Studies ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">3</span></a>(b)], the MRE-HOT does not show the convergence of <math alttext="\tilde{\bm{\theta}}" class="ltx_Math" display="inline" id="S6.SS1.p3.19.m8.1"><semantics id="S6.SS1.p3.19.m8.1a"><mover accent="true" id="S6.SS1.p3.19.m8.1.1" xref="S6.SS1.p3.19.m8.1.1.cmml"><mi id="S6.SS1.p3.19.m8.1.1.2" xref="S6.SS1.p3.19.m8.1.1.2.cmml">𝜽</mi><mo id="S6.SS1.p3.19.m8.1.1.1" xref="S6.SS1.p3.19.m8.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.19.m8.1b"><apply id="S6.SS1.p3.19.m8.1.1.cmml" xref="S6.SS1.p3.19.m8.1.1"><ci id="S6.SS1.p3.19.m8.1.1.1.cmml" xref="S6.SS1.p3.19.m8.1.1.1">~</ci><ci id="S6.SS1.p3.19.m8.1.1.2.cmml" xref="S6.SS1.p3.19.m8.1.1.2">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.19.m8.1c">\tilde{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.19.m8.1d">over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math> to <math alttext="\bm{0}" class="ltx_Math" display="inline" id="S6.SS1.p3.20.m9.1"><semantics id="S6.SS1.p3.20.m9.1a"><mn id="S6.SS1.p3.20.m9.1.1" xref="S6.SS1.p3.20.m9.1.1.cmml">𝟎</mn><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.20.m9.1b"><cn id="S6.SS1.p3.20.m9.1.1.cmml" type="integer" xref="S6.SS1.p3.20.m9.1.1">0</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.20.m9.1c">\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.20.m9.1d">bold_0</annotation></semantics></math> after 60 s [see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S6.F3" title="Figure 3 ‣ VI-A Stability and Convergence Comparison ‣ VI Simulation Studies ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">3</span></a>(b)] due to the lack of PE, and the proposed CLBC shows the convergence of partial elements <math alttext="\tilde{\theta}_{1}" class="ltx_Math" display="inline" id="S6.SS1.p3.21.m10.1"><semantics id="S6.SS1.p3.21.m10.1a"><msub id="S6.SS1.p3.21.m10.1.1" xref="S6.SS1.p3.21.m10.1.1.cmml"><mover accent="true" id="S6.SS1.p3.21.m10.1.1.2" xref="S6.SS1.p3.21.m10.1.1.2.cmml"><mi id="S6.SS1.p3.21.m10.1.1.2.2" xref="S6.SS1.p3.21.m10.1.1.2.2.cmml">θ</mi><mo id="S6.SS1.p3.21.m10.1.1.2.1" xref="S6.SS1.p3.21.m10.1.1.2.1.cmml">~</mo></mover><mn id="S6.SS1.p3.21.m10.1.1.3" xref="S6.SS1.p3.21.m10.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.21.m10.1b"><apply id="S6.SS1.p3.21.m10.1.1.cmml" xref="S6.SS1.p3.21.m10.1.1"><csymbol cd="ambiguous" id="S6.SS1.p3.21.m10.1.1.1.cmml" xref="S6.SS1.p3.21.m10.1.1">subscript</csymbol><apply id="S6.SS1.p3.21.m10.1.1.2.cmml" xref="S6.SS1.p3.21.m10.1.1.2"><ci id="S6.SS1.p3.21.m10.1.1.2.1.cmml" xref="S6.SS1.p3.21.m10.1.1.2.1">~</ci><ci id="S6.SS1.p3.21.m10.1.1.2.2.cmml" xref="S6.SS1.p3.21.m10.1.1.2.2">𝜃</ci></apply><cn id="S6.SS1.p3.21.m10.1.1.3.cmml" type="integer" xref="S6.SS1.p3.21.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.21.m10.1c">\tilde{\theta}_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.21.m10.1d">over~ start_ARG italic_θ end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\tilde{\theta}_{2}" class="ltx_Math" display="inline" id="S6.SS1.p3.22.m11.1"><semantics id="S6.SS1.p3.22.m11.1a"><msub id="S6.SS1.p3.22.m11.1.1" xref="S6.SS1.p3.22.m11.1.1.cmml"><mover accent="true" id="S6.SS1.p3.22.m11.1.1.2" xref="S6.SS1.p3.22.m11.1.1.2.cmml"><mi id="S6.SS1.p3.22.m11.1.1.2.2" xref="S6.SS1.p3.22.m11.1.1.2.2.cmml">θ</mi><mo id="S6.SS1.p3.22.m11.1.1.2.1" xref="S6.SS1.p3.22.m11.1.1.2.1.cmml">~</mo></mover><mn id="S6.SS1.p3.22.m11.1.1.3" xref="S6.SS1.p3.22.m11.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.22.m11.1b"><apply id="S6.SS1.p3.22.m11.1.1.cmml" xref="S6.SS1.p3.22.m11.1.1"><csymbol cd="ambiguous" id="S6.SS1.p3.22.m11.1.1.1.cmml" xref="S6.SS1.p3.22.m11.1.1">subscript</csymbol><apply id="S6.SS1.p3.22.m11.1.1.2.cmml" xref="S6.SS1.p3.22.m11.1.1.2"><ci id="S6.SS1.p3.22.m11.1.1.2.1.cmml" xref="S6.SS1.p3.22.m11.1.1.2.1">~</ci><ci id="S6.SS1.p3.22.m11.1.1.2.2.cmml" xref="S6.SS1.p3.22.m11.1.1.2.2">𝜃</ci></apply><cn id="S6.SS1.p3.22.m11.1.1.3.cmml" type="integer" xref="S6.SS1.p3.22.m11.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.22.m11.1c">\tilde{\theta}_{2}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.22.m11.1d">over~ start_ARG italic_θ end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> to 0 at <math alttext="t\in[0,60)" class="ltx_Math" display="inline" id="S6.SS1.p3.23.m12.2"><semantics id="S6.SS1.p3.23.m12.2a"><mrow id="S6.SS1.p3.23.m12.2.3" xref="S6.SS1.p3.23.m12.2.3.cmml"><mi id="S6.SS1.p3.23.m12.2.3.2" xref="S6.SS1.p3.23.m12.2.3.2.cmml">t</mi><mo id="S6.SS1.p3.23.m12.2.3.1" xref="S6.SS1.p3.23.m12.2.3.1.cmml">∈</mo><mrow id="S6.SS1.p3.23.m12.2.3.3.2" xref="S6.SS1.p3.23.m12.2.3.3.1.cmml"><mo id="S6.SS1.p3.23.m12.2.3.3.2.1" stretchy="false" xref="S6.SS1.p3.23.m12.2.3.3.1.cmml">[</mo><mn id="S6.SS1.p3.23.m12.1.1" xref="S6.SS1.p3.23.m12.1.1.cmml">0</mn><mo id="S6.SS1.p3.23.m12.2.3.3.2.2" xref="S6.SS1.p3.23.m12.2.3.3.1.cmml">,</mo><mn id="S6.SS1.p3.23.m12.2.2" xref="S6.SS1.p3.23.m12.2.2.cmml">60</mn><mo id="S6.SS1.p3.23.m12.2.3.3.2.3" stretchy="false" xref="S6.SS1.p3.23.m12.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.23.m12.2b"><apply id="S6.SS1.p3.23.m12.2.3.cmml" xref="S6.SS1.p3.23.m12.2.3"><in id="S6.SS1.p3.23.m12.2.3.1.cmml" xref="S6.SS1.p3.23.m12.2.3.1"></in><ci id="S6.SS1.p3.23.m12.2.3.2.cmml" xref="S6.SS1.p3.23.m12.2.3.2">𝑡</ci><interval closure="closed-open" id="S6.SS1.p3.23.m12.2.3.3.1.cmml" xref="S6.SS1.p3.23.m12.2.3.3.2"><cn id="S6.SS1.p3.23.m12.1.1.cmml" type="integer" xref="S6.SS1.p3.23.m12.1.1">0</cn><cn id="S6.SS1.p3.23.m12.2.2.cmml" type="integer" xref="S6.SS1.p3.23.m12.2.2">60</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.23.m12.2c">t\in[0,60)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.23.m12.2d">italic_t ∈ [ 0 , 60 )</annotation></semantics></math> s and then all elements <math alttext="\tilde{\theta}_{i}" class="ltx_Math" display="inline" id="S6.SS1.p3.24.m13.1"><semantics id="S6.SS1.p3.24.m13.1a"><msub id="S6.SS1.p3.24.m13.1.1" xref="S6.SS1.p3.24.m13.1.1.cmml"><mover accent="true" id="S6.SS1.p3.24.m13.1.1.2" xref="S6.SS1.p3.24.m13.1.1.2.cmml"><mi id="S6.SS1.p3.24.m13.1.1.2.2" xref="S6.SS1.p3.24.m13.1.1.2.2.cmml">θ</mi><mo id="S6.SS1.p3.24.m13.1.1.2.1" xref="S6.SS1.p3.24.m13.1.1.2.1.cmml">~</mo></mover><mi id="S6.SS1.p3.24.m13.1.1.3" xref="S6.SS1.p3.24.m13.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.24.m13.1b"><apply id="S6.SS1.p3.24.m13.1.1.cmml" xref="S6.SS1.p3.24.m13.1.1"><csymbol cd="ambiguous" id="S6.SS1.p3.24.m13.1.1.1.cmml" xref="S6.SS1.p3.24.m13.1.1">subscript</csymbol><apply id="S6.SS1.p3.24.m13.1.1.2.cmml" xref="S6.SS1.p3.24.m13.1.1.2"><ci id="S6.SS1.p3.24.m13.1.1.2.1.cmml" xref="S6.SS1.p3.24.m13.1.1.2.1">~</ci><ci id="S6.SS1.p3.24.m13.1.1.2.2.cmml" xref="S6.SS1.p3.24.m13.1.1.2.2">𝜃</ci></apply><ci id="S6.SS1.p3.24.m13.1.1.3.cmml" xref="S6.SS1.p3.24.m13.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.24.m13.1c">\tilde{\theta}_{i}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.24.m13.1d">over~ start_ARG italic_θ end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> (<math alttext="i=1" class="ltx_Math" display="inline" id="S6.SS1.p3.25.m14.1"><semantics id="S6.SS1.p3.25.m14.1a"><mrow id="S6.SS1.p3.25.m14.1.1" xref="S6.SS1.p3.25.m14.1.1.cmml"><mi id="S6.SS1.p3.25.m14.1.1.2" xref="S6.SS1.p3.25.m14.1.1.2.cmml">i</mi><mo id="S6.SS1.p3.25.m14.1.1.1" xref="S6.SS1.p3.25.m14.1.1.1.cmml">=</mo><mn id="S6.SS1.p3.25.m14.1.1.3" xref="S6.SS1.p3.25.m14.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.25.m14.1b"><apply id="S6.SS1.p3.25.m14.1.1.cmml" xref="S6.SS1.p3.25.m14.1.1"><eq id="S6.SS1.p3.25.m14.1.1.1.cmml" xref="S6.SS1.p3.25.m14.1.1.1"></eq><ci id="S6.SS1.p3.25.m14.1.1.2.cmml" xref="S6.SS1.p3.25.m14.1.1.2">𝑖</ci><cn id="S6.SS1.p3.25.m14.1.1.3.cmml" type="integer" xref="S6.SS1.p3.25.m14.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.25.m14.1c">i=1</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.25.m14.1d">italic_i = 1</annotation></semantics></math> to <math alttext="3" class="ltx_Math" display="inline" id="S6.SS1.p3.26.m15.1"><semantics id="S6.SS1.p3.26.m15.1a"><mn id="S6.SS1.p3.26.m15.1.1" xref="S6.SS1.p3.26.m15.1.1.cmml">3</mn><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.26.m15.1b"><cn id="S6.SS1.p3.26.m15.1.1.cmml" type="integer" xref="S6.SS1.p3.26.m15.1.1">3</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.26.m15.1c">3</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.26.m15.1d">3</annotation></semantics></math>) to 0 after <math alttext="t=60" class="ltx_Math" display="inline" id="S6.SS1.p3.27.m16.1"><semantics id="S6.SS1.p3.27.m16.1a"><mrow id="S6.SS1.p3.27.m16.1.1" xref="S6.SS1.p3.27.m16.1.1.cmml"><mi id="S6.SS1.p3.27.m16.1.1.2" xref="S6.SS1.p3.27.m16.1.1.2.cmml">t</mi><mo id="S6.SS1.p3.27.m16.1.1.1" xref="S6.SS1.p3.27.m16.1.1.1.cmml">=</mo><mn id="S6.SS1.p3.27.m16.1.1.3" xref="S6.SS1.p3.27.m16.1.1.3.cmml">60</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.27.m16.1b"><apply id="S6.SS1.p3.27.m16.1.1.cmml" xref="S6.SS1.p3.27.m16.1.1"><eq id="S6.SS1.p3.27.m16.1.1.1.cmml" xref="S6.SS1.p3.27.m16.1.1.1"></eq><ci id="S6.SS1.p3.27.m16.1.1.2.cmml" xref="S6.SS1.p3.27.m16.1.1.2">𝑡</ci><cn id="S6.SS1.p3.27.m16.1.1.3.cmml" type="integer" xref="S6.SS1.p3.27.m16.1.1.3">60</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.27.m16.1c">t=60</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.27.m16.1d">italic_t = 60</annotation></semantics></math> s [see Figs. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S6.F3" title="Figure 3 ‣ VI-A Stability and Convergence Comparison ‣ VI Simulation Studies ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">3</span></a>(a) and (b)], which is consistent with the theoretical result in Theorem 1. Regarding the tracking performance, the proposed CLBC owns the highest tracking accuracy after <math alttext="\tilde{\bm{\theta}}" class="ltx_Math" display="inline" id="S6.SS1.p3.28.m17.1"><semantics id="S6.SS1.p3.28.m17.1a"><mover accent="true" id="S6.SS1.p3.28.m17.1.1" xref="S6.SS1.p3.28.m17.1.1.cmml"><mi id="S6.SS1.p3.28.m17.1.1.2" xref="S6.SS1.p3.28.m17.1.1.2.cmml">𝜽</mi><mo id="S6.SS1.p3.28.m17.1.1.1" xref="S6.SS1.p3.28.m17.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.28.m17.1b"><apply id="S6.SS1.p3.28.m17.1.1.cmml" xref="S6.SS1.p3.28.m17.1.1"><ci id="S6.SS1.p3.28.m17.1.1.1.cmml" xref="S6.SS1.p3.28.m17.1.1.1">~</ci><ci id="S6.SS1.p3.28.m17.1.1.2.cmml" xref="S6.SS1.p3.28.m17.1.1.2">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.28.m17.1c">\tilde{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.28.m17.1d">over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math> converges to <math alttext="\bm{0}" class="ltx_Math" display="inline" id="S6.SS1.p3.29.m18.1"><semantics id="S6.SS1.p3.29.m18.1a"><mn id="S6.SS1.p3.29.m18.1.1" xref="S6.SS1.p3.29.m18.1.1.cmml">𝟎</mn><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.29.m18.1b"><cn id="S6.SS1.p3.29.m18.1.1.cmml" type="integer" xref="S6.SS1.p3.29.m18.1.1">0</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.29.m18.1c">\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.29.m18.1d">bold_0</annotation></semantics></math> [see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S6.F3" title="Figure 3 ‣ VI-A Stability and Convergence Comparison ‣ VI Simulation Studies ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">3</span></a>(c)]. Moreover, the control inputs <math alttext="u" class="ltx_Math" display="inline" id="S6.SS1.p3.30.m19.1"><semantics id="S6.SS1.p3.30.m19.1a"><mi id="S6.SS1.p3.30.m19.1.1" xref="S6.SS1.p3.30.m19.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.30.m19.1b"><ci id="S6.SS1.p3.30.m19.1.1.cmml" xref="S6.SS1.p3.30.m19.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.30.m19.1c">u</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.30.m19.1d">italic_u</annotation></semantics></math> by the proposed CLBC and the MRE-HOT are comparable in this case, and the control input <math alttext="u" class="ltx_Math" display="inline" id="S6.SS1.p3.31.m20.1"><semantics id="S6.SS1.p3.31.m20.1a"><mi id="S6.SS1.p3.31.m20.1.1" xref="S6.SS1.p3.31.m20.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.31.m20.1b"><ci id="S6.SS1.p3.31.m20.1.1.cmml" xref="S6.SS1.p3.31.m20.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.31.m20.1c">u</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.31.m20.1d">italic_u</annotation></semantics></math> by the CL-DSC is sensitive to the measurement noise [see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S6.F3" title="Figure 3 ‣ VI-A Stability and Convergence Comparison ‣ VI Simulation Studies ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">3</span></a>(d)]. It is worth noting that the exciting strengths <math alttext="\sigma_{\rm c}" class="ltx_Math" display="inline" id="S6.SS1.p3.32.m21.1"><semantics id="S6.SS1.p3.32.m21.1a"><msub id="S6.SS1.p3.32.m21.1.1" xref="S6.SS1.p3.32.m21.1.1.cmml"><mi id="S6.SS1.p3.32.m21.1.1.2" xref="S6.SS1.p3.32.m21.1.1.2.cmml">σ</mi><mi id="S6.SS1.p3.32.m21.1.1.3" mathvariant="normal" xref="S6.SS1.p3.32.m21.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.32.m21.1b"><apply id="S6.SS1.p3.32.m21.1.1.cmml" xref="S6.SS1.p3.32.m21.1.1"><csymbol cd="ambiguous" id="S6.SS1.p3.32.m21.1.1.1.cmml" xref="S6.SS1.p3.32.m21.1.1">subscript</csymbol><ci id="S6.SS1.p3.32.m21.1.1.2.cmml" xref="S6.SS1.p3.32.m21.1.1.2">𝜎</ci><ci id="S6.SS1.p3.32.m21.1.1.3.cmml" xref="S6.SS1.p3.32.m21.1.1.3">c</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.32.m21.1c">\sigma_{\rm c}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.32.m21.1d">italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT</annotation></semantics></math> by the CL-DSC and MRE-HOT are <math alttext="0" class="ltx_Math" display="inline" id="S6.SS1.p3.33.m22.1"><semantics id="S6.SS1.p3.33.m22.1a"><mn id="S6.SS1.p3.33.m22.1.1" xref="S6.SS1.p3.33.m22.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.33.m22.1b"><cn id="S6.SS1.p3.33.m22.1.1.cmml" type="integer" xref="S6.SS1.p3.33.m22.1.1">0</cn></annotation-xml></semantics></math> at <math alttext="t\in[0,60)" class="ltx_Math" display="inline" id="S6.SS1.p3.34.m23.2"><semantics id="S6.SS1.p3.34.m23.2a"><mrow id="S6.SS1.p3.34.m23.2.3" xref="S6.SS1.p3.34.m23.2.3.cmml"><mi id="S6.SS1.p3.34.m23.2.3.2" xref="S6.SS1.p3.34.m23.2.3.2.cmml">t</mi><mo id="S6.SS1.p3.34.m23.2.3.1" xref="S6.SS1.p3.34.m23.2.3.1.cmml">∈</mo><mrow id="S6.SS1.p3.34.m23.2.3.3.2" xref="S6.SS1.p3.34.m23.2.3.3.1.cmml"><mo id="S6.SS1.p3.34.m23.2.3.3.2.1" stretchy="false" xref="S6.SS1.p3.34.m23.2.3.3.1.cmml">[</mo><mn id="S6.SS1.p3.34.m23.1.1" xref="S6.SS1.p3.34.m23.1.1.cmml">0</mn><mo id="S6.SS1.p3.34.m23.2.3.3.2.2" xref="S6.SS1.p3.34.m23.2.3.3.1.cmml">,</mo><mn id="S6.SS1.p3.34.m23.2.2" xref="S6.SS1.p3.34.m23.2.2.cmml">60</mn><mo id="S6.SS1.p3.34.m23.2.3.3.2.3" stretchy="false" xref="S6.SS1.p3.34.m23.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.34.m23.2b"><apply id="S6.SS1.p3.34.m23.2.3.cmml" xref="S6.SS1.p3.34.m23.2.3"><in id="S6.SS1.p3.34.m23.2.3.1.cmml" xref="S6.SS1.p3.34.m23.2.3.1"></in><ci id="S6.SS1.p3.34.m23.2.3.2.cmml" xref="S6.SS1.p3.34.m23.2.3.2">𝑡</ci><interval closure="closed-open" id="S6.SS1.p3.34.m23.2.3.3.1.cmml" xref="S6.SS1.p3.34.m23.2.3.3.2"><cn id="S6.SS1.p3.34.m23.1.1.cmml" type="integer" xref="S6.SS1.p3.34.m23.1.1">0</cn><cn id="S6.SS1.p3.34.m23.2.2.cmml" type="integer" xref="S6.SS1.p3.34.m23.2.2">60</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.34.m23.2c">t\in[0,60)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.34.m23.2d">italic_t ∈ [ 0 , 60 )</annotation></semantics></math> s due to the inactive channel <math alttext="\bm{\phi}_{3}" class="ltx_Math" display="inline" id="S6.SS1.p3.35.m24.1"><semantics id="S6.SS1.p3.35.m24.1a"><msub id="S6.SS1.p3.35.m24.1.1" xref="S6.SS1.p3.35.m24.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S6.SS1.p3.35.m24.1.1.2" mathvariant="bold-italic" xref="S6.SS1.p3.35.m24.1.1.2.cmml">ϕ</mi><mn id="S6.SS1.p3.35.m24.1.1.3" xref="S6.SS1.p3.35.m24.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.35.m24.1b"><apply id="S6.SS1.p3.35.m24.1.1.cmml" xref="S6.SS1.p3.35.m24.1.1"><csymbol cd="ambiguous" id="S6.SS1.p3.35.m24.1.1.1.cmml" xref="S6.SS1.p3.35.m24.1.1">subscript</csymbol><ci id="S6.SS1.p3.35.m24.1.1.2.cmml" xref="S6.SS1.p3.35.m24.1.1.2">bold-italic-ϕ</ci><cn id="S6.SS1.p3.35.m24.1.1.3.cmml" type="integer" xref="S6.SS1.p3.35.m24.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.35.m24.1c">\bm{\phi}_{3}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.35.m24.1d">bold_italic_ϕ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> [see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S6.F3" title="Figure 3 ‣ VI-A Stability and Convergence Comparison ‣ VI Simulation Studies ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">3</span></a>(e)], but their different control designs lead in obviously different tracking results as shown above.</p> </div> <div class="ltx_para" id="S6.SS1.p4"> <p class="ltx_p" id="S6.SS1.p4.7"><span class="ltx_text ltx_font_italic" id="S6.SS1.p4.7.1">Case 2: Slowly time-varying parameter learning.</span> Let the reference trajectory <math alttext="y_{\rm r}=\sin(0.5t)" class="ltx_Math" display="inline" id="S6.SS1.p4.1.m1.2"><semantics id="S6.SS1.p4.1.m1.2a"><mrow id="S6.SS1.p4.1.m1.2.2" xref="S6.SS1.p4.1.m1.2.2.cmml"><msub id="S6.SS1.p4.1.m1.2.2.3" xref="S6.SS1.p4.1.m1.2.2.3.cmml"><mi id="S6.SS1.p4.1.m1.2.2.3.2" xref="S6.SS1.p4.1.m1.2.2.3.2.cmml">y</mi><mi id="S6.SS1.p4.1.m1.2.2.3.3" mathvariant="normal" xref="S6.SS1.p4.1.m1.2.2.3.3.cmml">r</mi></msub><mo id="S6.SS1.p4.1.m1.2.2.2" xref="S6.SS1.p4.1.m1.2.2.2.cmml">=</mo><mrow id="S6.SS1.p4.1.m1.2.2.1.1" xref="S6.SS1.p4.1.m1.2.2.1.2.cmml"><mi id="S6.SS1.p4.1.m1.1.1" xref="S6.SS1.p4.1.m1.1.1.cmml">sin</mi><mo id="S6.SS1.p4.1.m1.2.2.1.1a" xref="S6.SS1.p4.1.m1.2.2.1.2.cmml">⁡</mo><mrow id="S6.SS1.p4.1.m1.2.2.1.1.1" xref="S6.SS1.p4.1.m1.2.2.1.2.cmml"><mo id="S6.SS1.p4.1.m1.2.2.1.1.1.2" stretchy="false" xref="S6.SS1.p4.1.m1.2.2.1.2.cmml">(</mo><mrow id="S6.SS1.p4.1.m1.2.2.1.1.1.1" xref="S6.SS1.p4.1.m1.2.2.1.1.1.1.cmml"><mn id="S6.SS1.p4.1.m1.2.2.1.1.1.1.2" xref="S6.SS1.p4.1.m1.2.2.1.1.1.1.2.cmml">0.5</mn><mo id="S6.SS1.p4.1.m1.2.2.1.1.1.1.1" xref="S6.SS1.p4.1.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S6.SS1.p4.1.m1.2.2.1.1.1.1.3" xref="S6.SS1.p4.1.m1.2.2.1.1.1.1.3.cmml">t</mi></mrow><mo id="S6.SS1.p4.1.m1.2.2.1.1.1.3" stretchy="false" xref="S6.SS1.p4.1.m1.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p4.1.m1.2b"><apply id="S6.SS1.p4.1.m1.2.2.cmml" xref="S6.SS1.p4.1.m1.2.2"><eq id="S6.SS1.p4.1.m1.2.2.2.cmml" xref="S6.SS1.p4.1.m1.2.2.2"></eq><apply id="S6.SS1.p4.1.m1.2.2.3.cmml" xref="S6.SS1.p4.1.m1.2.2.3"><csymbol cd="ambiguous" id="S6.SS1.p4.1.m1.2.2.3.1.cmml" xref="S6.SS1.p4.1.m1.2.2.3">subscript</csymbol><ci id="S6.SS1.p4.1.m1.2.2.3.2.cmml" xref="S6.SS1.p4.1.m1.2.2.3.2">𝑦</ci><ci id="S6.SS1.p4.1.m1.2.2.3.3.cmml" xref="S6.SS1.p4.1.m1.2.2.3.3">r</ci></apply><apply id="S6.SS1.p4.1.m1.2.2.1.2.cmml" xref="S6.SS1.p4.1.m1.2.2.1.1"><sin id="S6.SS1.p4.1.m1.1.1.cmml" xref="S6.SS1.p4.1.m1.1.1"></sin><apply id="S6.SS1.p4.1.m1.2.2.1.1.1.1.cmml" xref="S6.SS1.p4.1.m1.2.2.1.1.1.1"><times id="S6.SS1.p4.1.m1.2.2.1.1.1.1.1.cmml" xref="S6.SS1.p4.1.m1.2.2.1.1.1.1.1"></times><cn id="S6.SS1.p4.1.m1.2.2.1.1.1.1.2.cmml" type="float" xref="S6.SS1.p4.1.m1.2.2.1.1.1.1.2">0.5</cn><ci id="S6.SS1.p4.1.m1.2.2.1.1.1.1.3.cmml" xref="S6.SS1.p4.1.m1.2.2.1.1.1.1.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p4.1.m1.2c">y_{\rm r}=\sin(0.5t)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p4.1.m1.2d">italic_y start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT = roman_sin ( 0.5 italic_t )</annotation></semantics></math> such that PE exists and the desired parameter <math alttext="\bm{\theta}(t)=[0.1,0.5,1.5]^{T}(1+0.2\sin(\pi/100t))" class="ltx_Math" display="inline" id="S6.SS1.p4.2.m2.6"><semantics id="S6.SS1.p4.2.m2.6a"><mrow id="S6.SS1.p4.2.m2.6.6" xref="S6.SS1.p4.2.m2.6.6.cmml"><mrow id="S6.SS1.p4.2.m2.6.6.3" xref="S6.SS1.p4.2.m2.6.6.3.cmml"><mi id="S6.SS1.p4.2.m2.6.6.3.2" xref="S6.SS1.p4.2.m2.6.6.3.2.cmml">𝜽</mi><mo id="S6.SS1.p4.2.m2.6.6.3.1" xref="S6.SS1.p4.2.m2.6.6.3.1.cmml">⁢</mo><mrow id="S6.SS1.p4.2.m2.6.6.3.3.2" xref="S6.SS1.p4.2.m2.6.6.3.cmml"><mo id="S6.SS1.p4.2.m2.6.6.3.3.2.1" stretchy="false" xref="S6.SS1.p4.2.m2.6.6.3.cmml">(</mo><mi id="S6.SS1.p4.2.m2.1.1" xref="S6.SS1.p4.2.m2.1.1.cmml">t</mi><mo id="S6.SS1.p4.2.m2.6.6.3.3.2.2" stretchy="false" xref="S6.SS1.p4.2.m2.6.6.3.cmml">)</mo></mrow></mrow><mo id="S6.SS1.p4.2.m2.6.6.2" xref="S6.SS1.p4.2.m2.6.6.2.cmml">=</mo><mrow id="S6.SS1.p4.2.m2.6.6.1" xref="S6.SS1.p4.2.m2.6.6.1.cmml"><msup id="S6.SS1.p4.2.m2.6.6.1.3" xref="S6.SS1.p4.2.m2.6.6.1.3.cmml"><mrow id="S6.SS1.p4.2.m2.6.6.1.3.2.2" xref="S6.SS1.p4.2.m2.6.6.1.3.2.1.cmml"><mo id="S6.SS1.p4.2.m2.6.6.1.3.2.2.1" stretchy="false" xref="S6.SS1.p4.2.m2.6.6.1.3.2.1.cmml">[</mo><mn id="S6.SS1.p4.2.m2.2.2" xref="S6.SS1.p4.2.m2.2.2.cmml">0.1</mn><mo id="S6.SS1.p4.2.m2.6.6.1.3.2.2.2" xref="S6.SS1.p4.2.m2.6.6.1.3.2.1.cmml">,</mo><mn id="S6.SS1.p4.2.m2.3.3" xref="S6.SS1.p4.2.m2.3.3.cmml">0.5</mn><mo id="S6.SS1.p4.2.m2.6.6.1.3.2.2.3" xref="S6.SS1.p4.2.m2.6.6.1.3.2.1.cmml">,</mo><mn id="S6.SS1.p4.2.m2.4.4" xref="S6.SS1.p4.2.m2.4.4.cmml">1.5</mn><mo id="S6.SS1.p4.2.m2.6.6.1.3.2.2.4" stretchy="false" xref="S6.SS1.p4.2.m2.6.6.1.3.2.1.cmml">]</mo></mrow><mi id="S6.SS1.p4.2.m2.6.6.1.3.3" xref="S6.SS1.p4.2.m2.6.6.1.3.3.cmml">T</mi></msup><mo id="S6.SS1.p4.2.m2.6.6.1.2" xref="S6.SS1.p4.2.m2.6.6.1.2.cmml">⁢</mo><mrow id="S6.SS1.p4.2.m2.6.6.1.1.1" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.cmml"><mo id="S6.SS1.p4.2.m2.6.6.1.1.1.2" stretchy="false" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.cmml">(</mo><mrow id="S6.SS1.p4.2.m2.6.6.1.1.1.1" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.cmml"><mn id="S6.SS1.p4.2.m2.6.6.1.1.1.1.3" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.3.cmml">1</mn><mo id="S6.SS1.p4.2.m2.6.6.1.1.1.1.2" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.2.cmml">+</mo><mrow id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.cmml"><mn id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.3" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.3.cmml">0.2</mn><mo id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.2" lspace="0.167em" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.2.cmml"><mi id="S6.SS1.p4.2.m2.5.5" xref="S6.SS1.p4.2.m2.5.5.cmml">sin</mi><mo id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1a" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.2.cmml">⁡</mo><mrow id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.2.cmml"><mo id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.2.cmml">(</mo><mrow id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.cmml"><mrow id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.2" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.2.2" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.2.2.cmml">π</mi><mo id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.2.1" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.2.1.cmml">/</mo><mn id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.2.3" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.2.3.cmml">100</mn></mrow><mo id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.1" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.3" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.3.cmml">t</mi></mrow><mo id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S6.SS1.p4.2.m2.6.6.1.1.1.3" stretchy="false" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p4.2.m2.6b"><apply id="S6.SS1.p4.2.m2.6.6.cmml" xref="S6.SS1.p4.2.m2.6.6"><eq id="S6.SS1.p4.2.m2.6.6.2.cmml" xref="S6.SS1.p4.2.m2.6.6.2"></eq><apply id="S6.SS1.p4.2.m2.6.6.3.cmml" xref="S6.SS1.p4.2.m2.6.6.3"><times id="S6.SS1.p4.2.m2.6.6.3.1.cmml" xref="S6.SS1.p4.2.m2.6.6.3.1"></times><ci id="S6.SS1.p4.2.m2.6.6.3.2.cmml" xref="S6.SS1.p4.2.m2.6.6.3.2">𝜽</ci><ci id="S6.SS1.p4.2.m2.1.1.cmml" xref="S6.SS1.p4.2.m2.1.1">𝑡</ci></apply><apply id="S6.SS1.p4.2.m2.6.6.1.cmml" xref="S6.SS1.p4.2.m2.6.6.1"><times id="S6.SS1.p4.2.m2.6.6.1.2.cmml" xref="S6.SS1.p4.2.m2.6.6.1.2"></times><apply id="S6.SS1.p4.2.m2.6.6.1.3.cmml" xref="S6.SS1.p4.2.m2.6.6.1.3"><csymbol cd="ambiguous" id="S6.SS1.p4.2.m2.6.6.1.3.1.cmml" xref="S6.SS1.p4.2.m2.6.6.1.3">superscript</csymbol><list id="S6.SS1.p4.2.m2.6.6.1.3.2.1.cmml" xref="S6.SS1.p4.2.m2.6.6.1.3.2.2"><cn id="S6.SS1.p4.2.m2.2.2.cmml" type="float" xref="S6.SS1.p4.2.m2.2.2">0.1</cn><cn id="S6.SS1.p4.2.m2.3.3.cmml" type="float" xref="S6.SS1.p4.2.m2.3.3">0.5</cn><cn id="S6.SS1.p4.2.m2.4.4.cmml" type="float" xref="S6.SS1.p4.2.m2.4.4">1.5</cn></list><ci id="S6.SS1.p4.2.m2.6.6.1.3.3.cmml" xref="S6.SS1.p4.2.m2.6.6.1.3.3">𝑇</ci></apply><apply id="S6.SS1.p4.2.m2.6.6.1.1.1.1.cmml" xref="S6.SS1.p4.2.m2.6.6.1.1.1"><plus id="S6.SS1.p4.2.m2.6.6.1.1.1.1.2.cmml" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.2"></plus><cn id="S6.SS1.p4.2.m2.6.6.1.1.1.1.3.cmml" type="integer" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.3">1</cn><apply id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.cmml" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1"><times id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.2.cmml" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.2"></times><cn id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.3.cmml" type="float" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.3">0.2</cn><apply id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.2.cmml" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1"><sin id="S6.SS1.p4.2.m2.5.5.cmml" xref="S6.SS1.p4.2.m2.5.5"></sin><apply id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.cmml" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1"><times id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.1.cmml" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.1"></times><apply id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.2.cmml" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.2"><divide id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.2.1"></divide><ci id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.2.2">𝜋</ci><cn id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.2.3">100</cn></apply><ci id="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.3.cmml" xref="S6.SS1.p4.2.m2.6.6.1.1.1.1.1.1.1.1.1.3">𝑡</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p4.2.m2.6c">\bm{\theta}(t)=[0.1,0.5,1.5]^{T}(1+0.2\sin(\pi/100t))</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p4.2.m2.6d">bold_italic_θ ( italic_t ) = [ 0.1 , 0.5 , 1.5 ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( 1 + 0.2 roman_sin ( italic_π / 100 italic_t ) )</annotation></semantics></math> so that it is slowly time-varying. We use the exciting information corresponding to the current exciting strength <math alttext="\sigma_{\min}(\Psi(t))" class="ltx_Math" display="inline" id="S6.SS1.p4.3.m3.2"><semantics id="S6.SS1.p4.3.m3.2a"><mrow id="S6.SS1.p4.3.m3.2.2" xref="S6.SS1.p4.3.m3.2.2.cmml"><msub id="S6.SS1.p4.3.m3.2.2.3" xref="S6.SS1.p4.3.m3.2.2.3.cmml"><mi id="S6.SS1.p4.3.m3.2.2.3.2" xref="S6.SS1.p4.3.m3.2.2.3.2.cmml">σ</mi><mi id="S6.SS1.p4.3.m3.2.2.3.3" xref="S6.SS1.p4.3.m3.2.2.3.3.cmml">min</mi></msub><mo id="S6.SS1.p4.3.m3.2.2.2" xref="S6.SS1.p4.3.m3.2.2.2.cmml">⁢</mo><mrow id="S6.SS1.p4.3.m3.2.2.1.1" xref="S6.SS1.p4.3.m3.2.2.1.1.1.cmml"><mo id="S6.SS1.p4.3.m3.2.2.1.1.2" stretchy="false" xref="S6.SS1.p4.3.m3.2.2.1.1.1.cmml">(</mo><mrow id="S6.SS1.p4.3.m3.2.2.1.1.1" xref="S6.SS1.p4.3.m3.2.2.1.1.1.cmml"><mi id="S6.SS1.p4.3.m3.2.2.1.1.1.2" mathvariant="normal" xref="S6.SS1.p4.3.m3.2.2.1.1.1.2.cmml">Ψ</mi><mo id="S6.SS1.p4.3.m3.2.2.1.1.1.1" xref="S6.SS1.p4.3.m3.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S6.SS1.p4.3.m3.2.2.1.1.1.3.2" xref="S6.SS1.p4.3.m3.2.2.1.1.1.cmml"><mo id="S6.SS1.p4.3.m3.2.2.1.1.1.3.2.1" stretchy="false" xref="S6.SS1.p4.3.m3.2.2.1.1.1.cmml">(</mo><mi id="S6.SS1.p4.3.m3.1.1" xref="S6.SS1.p4.3.m3.1.1.cmml">t</mi><mo id="S6.SS1.p4.3.m3.2.2.1.1.1.3.2.2" stretchy="false" xref="S6.SS1.p4.3.m3.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS1.p4.3.m3.2.2.1.1.3" stretchy="false" xref="S6.SS1.p4.3.m3.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p4.3.m3.2b"><apply id="S6.SS1.p4.3.m3.2.2.cmml" xref="S6.SS1.p4.3.m3.2.2"><times id="S6.SS1.p4.3.m3.2.2.2.cmml" xref="S6.SS1.p4.3.m3.2.2.2"></times><apply id="S6.SS1.p4.3.m3.2.2.3.cmml" xref="S6.SS1.p4.3.m3.2.2.3"><csymbol cd="ambiguous" id="S6.SS1.p4.3.m3.2.2.3.1.cmml" xref="S6.SS1.p4.3.m3.2.2.3">subscript</csymbol><ci id="S6.SS1.p4.3.m3.2.2.3.2.cmml" xref="S6.SS1.p4.3.m3.2.2.3.2">𝜎</ci><min id="S6.SS1.p4.3.m3.2.2.3.3.cmml" xref="S6.SS1.p4.3.m3.2.2.3.3"></min></apply><apply id="S6.SS1.p4.3.m3.2.2.1.1.1.cmml" xref="S6.SS1.p4.3.m3.2.2.1.1"><times id="S6.SS1.p4.3.m3.2.2.1.1.1.1.cmml" xref="S6.SS1.p4.3.m3.2.2.1.1.1.1"></times><ci id="S6.SS1.p4.3.m3.2.2.1.1.1.2.cmml" xref="S6.SS1.p4.3.m3.2.2.1.1.1.2">Ψ</ci><ci id="S6.SS1.p4.3.m3.1.1.cmml" xref="S6.SS1.p4.3.m3.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p4.3.m3.2c">\sigma_{\min}(\Psi(t))</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p4.3.m3.2d">italic_σ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Ψ ( italic_t ) )</annotation></semantics></math> instead of the maximal exciting strength <math alttext="\sigma_{\rm c}" class="ltx_Math" display="inline" id="S6.SS1.p4.4.m4.1"><semantics id="S6.SS1.p4.4.m4.1a"><msub id="S6.SS1.p4.4.m4.1.1" xref="S6.SS1.p4.4.m4.1.1.cmml"><mi id="S6.SS1.p4.4.m4.1.1.2" xref="S6.SS1.p4.4.m4.1.1.2.cmml">σ</mi><mi id="S6.SS1.p4.4.m4.1.1.3" mathvariant="normal" xref="S6.SS1.p4.4.m4.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.p4.4.m4.1b"><apply id="S6.SS1.p4.4.m4.1.1.cmml" xref="S6.SS1.p4.4.m4.1.1"><csymbol cd="ambiguous" id="S6.SS1.p4.4.m4.1.1.1.cmml" xref="S6.SS1.p4.4.m4.1.1">subscript</csymbol><ci id="S6.SS1.p4.4.m4.1.1.2.cmml" xref="S6.SS1.p4.4.m4.1.1.2">𝜎</ci><ci id="S6.SS1.p4.4.m4.1.1.3.cmml" xref="S6.SS1.p4.4.m4.1.1.3">c</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p4.4.m4.1c">\sigma_{\rm c}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p4.4.m4.1d">italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT</annotation></semantics></math> defined in Algorithm 1 in this case. <span class="ltx_text" id="S6.SS1.p4.7.2" style="color:#000099;">Consequently, the proposed CLBC requires PE to achieve learning for slowly time-varying parameters and can no longer function solely under the IE (or partial IE) condition due to the forgetting that occurs.</span> Performance comparisons of the three controllers are exhibited in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S6.F4" title="Figure 4 ‣ VI-A Stability and Convergence Comparison ‣ VI Simulation Studies ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">4</span></a>. In this case, the parameter estimates <math alttext="\hat{\bm{\theta}}" class="ltx_Math" display="inline" id="S6.SS1.p4.5.m5.1"><semantics id="S6.SS1.p4.5.m5.1a"><mover accent="true" id="S6.SS1.p4.5.m5.1.1" xref="S6.SS1.p4.5.m5.1.1.cmml"><mi id="S6.SS1.p4.5.m5.1.1.2" xref="S6.SS1.p4.5.m5.1.1.2.cmml">𝜽</mi><mo id="S6.SS1.p4.5.m5.1.1.1" xref="S6.SS1.p4.5.m5.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S6.SS1.p4.5.m5.1b"><apply id="S6.SS1.p4.5.m5.1.1.cmml" xref="S6.SS1.p4.5.m5.1.1"><ci id="S6.SS1.p4.5.m5.1.1.1.cmml" xref="S6.SS1.p4.5.m5.1.1.1">^</ci><ci id="S6.SS1.p4.5.m5.1.1.2.cmml" xref="S6.SS1.p4.5.m5.1.1.2">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p4.5.m5.1c">\hat{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p4.5.m5.1d">over^ start_ARG bold_italic_θ end_ARG</annotation></semantics></math> by all controllers can follow the desired time-varying parameter <math alttext="\bm{\theta}(t)" class="ltx_Math" display="inline" id="S6.SS1.p4.6.m6.1"><semantics id="S6.SS1.p4.6.m6.1a"><mrow id="S6.SS1.p4.6.m6.1.2" xref="S6.SS1.p4.6.m6.1.2.cmml"><mi id="S6.SS1.p4.6.m6.1.2.2" xref="S6.SS1.p4.6.m6.1.2.2.cmml">𝜽</mi><mo id="S6.SS1.p4.6.m6.1.2.1" xref="S6.SS1.p4.6.m6.1.2.1.cmml">⁢</mo><mrow id="S6.SS1.p4.6.m6.1.2.3.2" xref="S6.SS1.p4.6.m6.1.2.cmml"><mo id="S6.SS1.p4.6.m6.1.2.3.2.1" stretchy="false" xref="S6.SS1.p4.6.m6.1.2.cmml">(</mo><mi id="S6.SS1.p4.6.m6.1.1" xref="S6.SS1.p4.6.m6.1.1.cmml">t</mi><mo id="S6.SS1.p4.6.m6.1.2.3.2.2" stretchy="false" xref="S6.SS1.p4.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p4.6.m6.1b"><apply id="S6.SS1.p4.6.m6.1.2.cmml" xref="S6.SS1.p4.6.m6.1.2"><times id="S6.SS1.p4.6.m6.1.2.1.cmml" xref="S6.SS1.p4.6.m6.1.2.1"></times><ci id="S6.SS1.p4.6.m6.1.2.2.cmml" xref="S6.SS1.p4.6.m6.1.2.2">𝜽</ci><ci id="S6.SS1.p4.6.m6.1.1.cmml" xref="S6.SS1.p4.6.m6.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p4.6.m6.1c">\bm{\theta}(t)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p4.6.m6.1d">bold_italic_θ ( italic_t )</annotation></semantics></math> [see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S6.F4" title="Figure 4 ‣ VI-A Stability and Convergence Comparison ‣ VI Simulation Studies ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">4</span></a>(a)] due to the establishment of PE, but the proposed CLBC achieves the best performance of parameter estimation [see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S6.F4" title="Figure 4 ‣ VI-A Stability and Convergence Comparison ‣ VI Simulation Studies ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">4</span></a>(a)] as its exciting strength <math alttext="\sigma_{\rm c}" class="ltx_Math" display="inline" id="S6.SS1.p4.7.m7.1"><semantics id="S6.SS1.p4.7.m7.1a"><msub id="S6.SS1.p4.7.m7.1.1" xref="S6.SS1.p4.7.m7.1.1.cmml"><mi id="S6.SS1.p4.7.m7.1.1.2" xref="S6.SS1.p4.7.m7.1.1.2.cmml">σ</mi><mi id="S6.SS1.p4.7.m7.1.1.3" mathvariant="normal" xref="S6.SS1.p4.7.m7.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.p4.7.m7.1b"><apply id="S6.SS1.p4.7.m7.1.1.cmml" xref="S6.SS1.p4.7.m7.1.1"><csymbol cd="ambiguous" id="S6.SS1.p4.7.m7.1.1.1.cmml" xref="S6.SS1.p4.7.m7.1.1">subscript</csymbol><ci id="S6.SS1.p4.7.m7.1.1.2.cmml" xref="S6.SS1.p4.7.m7.1.1.2">𝜎</ci><ci id="S6.SS1.p4.7.m7.1.1.3.cmml" xref="S6.SS1.p4.7.m7.1.1.3">c</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p4.7.m7.1c">\sigma_{\rm c}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p4.7.m7.1d">italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT</annotation></semantics></math> is greatest [see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S6.F4" title="Figure 4 ‣ VI-A Stability and Convergence Comparison ‣ VI Simulation Studies ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">4</span></a>(d)]. Besides, the proposed CLBC also shows the best tracking performance [see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S6.F4" title="Figure 4 ‣ VI-A Stability and Convergence Comparison ‣ VI Simulation Studies ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">4</span></a>(b)], which validates that: 1) it is also suitable for the case of slowly time-varying parameter learning; 2) it can enhance both the tracking performance and parameter convergence by combining two different prediction errors; 3) it guarantees strong robustness against perturbations resulting from measurement noise.</p> </div> </section> <section class="ltx_subsection" id="S6.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S6.SS2.4.1.1">VI-B</span> </span><span class="ltx_text ltx_font_italic" id="S6.SS2.5.2">Transient Performance Comparisons</span> </h3> <div class="ltx_para" id="S6.SS2.p1"> <p class="ltx_p" id="S6.SS2.p1.3">This section is devoted to demonstrating that the transient performance of the proposed CLBC in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E6" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">6</span></a>) with (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>) can still be guaranteed without the nonlinear damping terms <math alttext="k_{\rm{d}\it i}\|\bm{\psi}_{i}\|^{2}" class="ltx_Math" display="inline" id="S6.SS2.p1.1.m1.1"><semantics id="S6.SS2.p1.1.m1.1a"><mrow id="S6.SS2.p1.1.m1.1.1" xref="S6.SS2.p1.1.m1.1.1.cmml"><msub id="S6.SS2.p1.1.m1.1.1.3" xref="S6.SS2.p1.1.m1.1.1.3.cmml"><mi id="S6.SS2.p1.1.m1.1.1.3.2" xref="S6.SS2.p1.1.m1.1.1.3.2.cmml">k</mi><mrow id="S6.SS2.p1.1.m1.1.1.3.3" xref="S6.SS2.p1.1.m1.1.1.3.3.cmml"><mi id="S6.SS2.p1.1.m1.1.1.3.3.2" mathvariant="normal" xref="S6.SS2.p1.1.m1.1.1.3.3.2.cmml">d</mi><mo id="S6.SS2.p1.1.m1.1.1.3.3.1" xref="S6.SS2.p1.1.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S6.SS2.p1.1.m1.1.1.3.3.3" xref="S6.SS2.p1.1.m1.1.1.3.3.3.cmml">i</mi></mrow></msub><mo id="S6.SS2.p1.1.m1.1.1.2" xref="S6.SS2.p1.1.m1.1.1.2.cmml">⁢</mo><msup id="S6.SS2.p1.1.m1.1.1.1" xref="S6.SS2.p1.1.m1.1.1.1.cmml"><mrow id="S6.SS2.p1.1.m1.1.1.1.1.1" xref="S6.SS2.p1.1.m1.1.1.1.1.2.cmml"><mo id="S6.SS2.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.p1.1.m1.1.1.1.1.2.1.cmml">‖</mo><msub id="S6.SS2.p1.1.m1.1.1.1.1.1.1" xref="S6.SS2.p1.1.m1.1.1.1.1.1.1.cmml"><mi id="S6.SS2.p1.1.m1.1.1.1.1.1.1.2" xref="S6.SS2.p1.1.m1.1.1.1.1.1.1.2.cmml">𝝍</mi><mi id="S6.SS2.p1.1.m1.1.1.1.1.1.1.3" xref="S6.SS2.p1.1.m1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S6.SS2.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S6.SS2.p1.1.m1.1.1.1.1.2.1.cmml">‖</mo></mrow><mn id="S6.SS2.p1.1.m1.1.1.1.3" xref="S6.SS2.p1.1.m1.1.1.1.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.1.m1.1b"><apply id="S6.SS2.p1.1.m1.1.1.cmml" xref="S6.SS2.p1.1.m1.1.1"><times id="S6.SS2.p1.1.m1.1.1.2.cmml" xref="S6.SS2.p1.1.m1.1.1.2"></times><apply id="S6.SS2.p1.1.m1.1.1.3.cmml" xref="S6.SS2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.p1.1.m1.1.1.3.1.cmml" xref="S6.SS2.p1.1.m1.1.1.3">subscript</csymbol><ci id="S6.SS2.p1.1.m1.1.1.3.2.cmml" xref="S6.SS2.p1.1.m1.1.1.3.2">𝑘</ci><apply id="S6.SS2.p1.1.m1.1.1.3.3.cmml" xref="S6.SS2.p1.1.m1.1.1.3.3"><times id="S6.SS2.p1.1.m1.1.1.3.3.1.cmml" xref="S6.SS2.p1.1.m1.1.1.3.3.1"></times><ci id="S6.SS2.p1.1.m1.1.1.3.3.2.cmml" xref="S6.SS2.p1.1.m1.1.1.3.3.2">d</ci><ci id="S6.SS2.p1.1.m1.1.1.3.3.3.cmml" xref="S6.SS2.p1.1.m1.1.1.3.3.3">𝑖</ci></apply></apply><apply id="S6.SS2.p1.1.m1.1.1.1.cmml" xref="S6.SS2.p1.1.m1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.p1.1.m1.1.1.1.2.cmml" xref="S6.SS2.p1.1.m1.1.1.1">superscript</csymbol><apply id="S6.SS2.p1.1.m1.1.1.1.1.2.cmml" xref="S6.SS2.p1.1.m1.1.1.1.1.1"><csymbol cd="latexml" id="S6.SS2.p1.1.m1.1.1.1.1.2.1.cmml" xref="S6.SS2.p1.1.m1.1.1.1.1.1.2">norm</csymbol><apply id="S6.SS2.p1.1.m1.1.1.1.1.1.1.cmml" xref="S6.SS2.p1.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.p1.1.m1.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="S6.SS2.p1.1.m1.1.1.1.1.1.1.2">𝝍</ci><ci id="S6.SS2.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="S6.SS2.p1.1.m1.1.1.1.1.1.1.3">𝑖</ci></apply></apply><cn id="S6.SS2.p1.1.m1.1.1.1.3.cmml" type="integer" xref="S6.SS2.p1.1.m1.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.1.m1.1c">k_{\rm{d}\it i}\|\bm{\psi}_{i}\|^{2}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.1.m1.1d">italic_k start_POSTSUBSCRIPT roman_d italic_i end_POSTSUBSCRIPT ∥ bold_italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> (<math alttext="i=1" class="ltx_Math" display="inline" id="S6.SS2.p1.2.m2.1"><semantics id="S6.SS2.p1.2.m2.1a"><mrow id="S6.SS2.p1.2.m2.1.1" xref="S6.SS2.p1.2.m2.1.1.cmml"><mi id="S6.SS2.p1.2.m2.1.1.2" xref="S6.SS2.p1.2.m2.1.1.2.cmml">i</mi><mo id="S6.SS2.p1.2.m2.1.1.1" xref="S6.SS2.p1.2.m2.1.1.1.cmml">=</mo><mn id="S6.SS2.p1.2.m2.1.1.3" xref="S6.SS2.p1.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.2.m2.1b"><apply id="S6.SS2.p1.2.m2.1.1.cmml" xref="S6.SS2.p1.2.m2.1.1"><eq id="S6.SS2.p1.2.m2.1.1.1.cmml" xref="S6.SS2.p1.2.m2.1.1.1"></eq><ci id="S6.SS2.p1.2.m2.1.1.2.cmml" xref="S6.SS2.p1.2.m2.1.1.2">𝑖</ci><cn id="S6.SS2.p1.2.m2.1.1.3.cmml" type="integer" xref="S6.SS2.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.2.m2.1c">i=1</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.2.m2.1d">italic_i = 1</annotation></semantics></math> to <math alttext="n" class="ltx_Math" display="inline" id="S6.SS2.p1.3.m3.1"><semantics id="S6.SS2.p1.3.m3.1a"><mi id="S6.SS2.p1.3.m3.1.1" xref="S6.SS2.p1.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.3.m3.1b"><ci id="S6.SS2.p1.3.m3.1.1.cmml" xref="S6.SS2.p1.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.3.m3.1d">italic_n</annotation></semantics></math>). Consider a second-order system <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib8" title="">8</a>]</cite></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx29"> <tbody id="S6.Ex20"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\left\{\begin{array}[]{l}\dot{x}_{1}=x_{2}+\varphi_{1}(x_{1})% \theta,\\ \dot{x}_{2}=u,\\ y=x_{1}\end{array}\right." class="ltx_Math" display="inline" id="S6.Ex20.m1.1"><semantics id="S6.Ex20.m1.1a"><mrow id="S6.Ex20.m1.1.2.2" xref="S6.Ex20.m1.1.2.1.cmml"><mo id="S6.Ex20.m1.1.2.2.1" xref="S6.Ex20.m1.1.2.1.1.cmml">{</mo><mtable id="S6.Ex20.m1.1.1" rowspacing="0pt" xref="S6.Ex20.m1.1.1.cmml"><mtr id="S6.Ex20.m1.1.1a" xref="S6.Ex20.m1.1.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S6.Ex20.m1.1.1b" xref="S6.Ex20.m1.1.1.cmml"><mrow id="S6.Ex17.1.1.1" xref="S6.Ex17.1.1.1.1.cmml"><mrow id="S6.Ex17.1.1.1.1" xref="S6.Ex17.1.1.1.1.cmml"><msub id="S6.Ex17.1.1.1.1.3" xref="S6.Ex17.1.1.1.1.3.cmml"><mover accent="true" id="S6.Ex17.1.1.1.1.3.2" xref="S6.Ex17.1.1.1.1.3.2.cmml"><mi id="S6.Ex17.1.1.1.1.3.2.2" xref="S6.Ex17.1.1.1.1.3.2.2.cmml">x</mi><mo id="S6.Ex17.1.1.1.1.3.2.1" xref="S6.Ex17.1.1.1.1.3.2.1.cmml">˙</mo></mover><mn id="S6.Ex17.1.1.1.1.3.3" xref="S6.Ex17.1.1.1.1.3.3.cmml">1</mn></msub><mo id="S6.Ex17.1.1.1.1.2" xref="S6.Ex17.1.1.1.1.2.cmml">=</mo><mrow id="S6.Ex17.1.1.1.1.1" xref="S6.Ex17.1.1.1.1.1.cmml"><msub id="S6.Ex17.1.1.1.1.1.3" xref="S6.Ex17.1.1.1.1.1.3.cmml"><mi id="S6.Ex17.1.1.1.1.1.3.2" xref="S6.Ex17.1.1.1.1.1.3.2.cmml">x</mi><mn id="S6.Ex17.1.1.1.1.1.3.3" xref="S6.Ex17.1.1.1.1.1.3.3.cmml">2</mn></msub><mo id="S6.Ex17.1.1.1.1.1.2" xref="S6.Ex17.1.1.1.1.1.2.cmml">+</mo><mrow id="S6.Ex17.1.1.1.1.1.1" xref="S6.Ex17.1.1.1.1.1.1.cmml"><msub id="S6.Ex17.1.1.1.1.1.1.3" xref="S6.Ex17.1.1.1.1.1.1.3.cmml"><mi id="S6.Ex17.1.1.1.1.1.1.3.2" xref="S6.Ex17.1.1.1.1.1.1.3.2.cmml">φ</mi><mn id="S6.Ex17.1.1.1.1.1.1.3.3" xref="S6.Ex17.1.1.1.1.1.1.3.3.cmml">1</mn></msub><mo id="S6.Ex17.1.1.1.1.1.1.2" xref="S6.Ex17.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S6.Ex17.1.1.1.1.1.1.1.1" xref="S6.Ex17.1.1.1.1.1.1.1.1.1.cmml"><mo id="S6.Ex17.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S6.Ex17.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S6.Ex17.1.1.1.1.1.1.1.1.1" xref="S6.Ex17.1.1.1.1.1.1.1.1.1.cmml"><mi id="S6.Ex17.1.1.1.1.1.1.1.1.1.2" xref="S6.Ex17.1.1.1.1.1.1.1.1.1.2.cmml">x</mi><mn id="S6.Ex17.1.1.1.1.1.1.1.1.1.3" xref="S6.Ex17.1.1.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S6.Ex17.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S6.Ex17.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S6.Ex17.1.1.1.1.1.1.2a" xref="S6.Ex17.1.1.1.1.1.1.2.cmml">⁢</mo><mi id="S6.Ex17.1.1.1.1.1.1.4" xref="S6.Ex17.1.1.1.1.1.1.4.cmml">θ</mi></mrow></mrow></mrow><mo id="S6.Ex17.1.1.1.2" xref="S6.Ex17.1.1.1.1.cmml">,</mo></mrow></mtd></mtr><mtr id="S6.Ex20.m1.1.1c" xref="S6.Ex20.m1.1.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S6.Ex20.m1.1.1d" xref="S6.Ex20.m1.1.1.cmml"><mrow id="S6.Ex18.1.1.1" xref="S6.Ex18.1.1.1.1.cmml"><mrow id="S6.Ex18.1.1.1.1" xref="S6.Ex18.1.1.1.1.cmml"><msub id="S6.Ex18.1.1.1.1.2" xref="S6.Ex18.1.1.1.1.2.cmml"><mover accent="true" id="S6.Ex18.1.1.1.1.2.2" xref="S6.Ex18.1.1.1.1.2.2.cmml"><mi id="S6.Ex18.1.1.1.1.2.2.2" xref="S6.Ex18.1.1.1.1.2.2.2.cmml">x</mi><mo id="S6.Ex18.1.1.1.1.2.2.1" xref="S6.Ex18.1.1.1.1.2.2.1.cmml">˙</mo></mover><mn id="S6.Ex18.1.1.1.1.2.3" xref="S6.Ex18.1.1.1.1.2.3.cmml">2</mn></msub><mo id="S6.Ex18.1.1.1.1.1" xref="S6.Ex18.1.1.1.1.1.cmml">=</mo><mi id="S6.Ex18.1.1.1.1.3" xref="S6.Ex18.1.1.1.1.3.cmml">u</mi></mrow><mo id="S6.Ex18.1.1.1.2" xref="S6.Ex18.1.1.1.1.cmml">,</mo></mrow></mtd></mtr><mtr id="S6.Ex20.m1.1.1e" xref="S6.Ex20.m1.1.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S6.Ex20.m1.1.1f" xref="S6.Ex20.m1.1.1.cmml"><mrow id="S6.Ex19.1.1" xref="S6.Ex19.1.1.cmml"><mi id="S6.Ex19.1.1.2" xref="S6.Ex19.1.1.2.cmml">y</mi><mo id="S6.Ex19.1.1.1" xref="S6.Ex19.1.1.1.cmml">=</mo><msub id="S6.Ex19.1.1.3" xref="S6.Ex19.1.1.3.cmml"><mi id="S6.Ex19.1.1.3.2" xref="S6.Ex19.1.1.3.2.cmml">x</mi><mn id="S6.Ex19.1.1.3.3" xref="S6.Ex19.1.1.3.3.cmml">1</mn></msub></mrow></mtd></mtr></mtable><mi id="S6.Ex20.m1.1.2.2.2" xref="S6.Ex20.m1.1.2.1.1.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex20.m1.1b"><apply id="S6.Ex20.m1.1.2.1.cmml" xref="S6.Ex20.m1.1.2.2"><csymbol cd="latexml" id="S6.Ex20.m1.1.2.1.1.cmml" xref="S6.Ex20.m1.1.2.2.1">cases</csymbol><matrix id="S6.Ex20.m1.1.1.cmml" xref="S6.Ex20.m1.1.1"><matrixrow id="S6.Ex20.m1.1.1a.cmml" xref="S6.Ex20.m1.1.1"><apply id="S6.Ex17.1.1.1.1.cmml" xref="S6.Ex17.1.1.1"><eq id="S6.Ex17.1.1.1.1.2.cmml" xref="S6.Ex17.1.1.1.1.2"></eq><apply id="S6.Ex17.1.1.1.1.3.cmml" xref="S6.Ex17.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.Ex17.1.1.1.1.3.1.cmml" xref="S6.Ex17.1.1.1.1.3">subscript</csymbol><apply id="S6.Ex17.1.1.1.1.3.2.cmml" xref="S6.Ex17.1.1.1.1.3.2"><ci id="S6.Ex17.1.1.1.1.3.2.1.cmml" xref="S6.Ex17.1.1.1.1.3.2.1">˙</ci><ci id="S6.Ex17.1.1.1.1.3.2.2.cmml" xref="S6.Ex17.1.1.1.1.3.2.2">𝑥</ci></apply><cn id="S6.Ex17.1.1.1.1.3.3.cmml" type="integer" xref="S6.Ex17.1.1.1.1.3.3">1</cn></apply><apply id="S6.Ex17.1.1.1.1.1.cmml" xref="S6.Ex17.1.1.1.1.1"><plus id="S6.Ex17.1.1.1.1.1.2.cmml" xref="S6.Ex17.1.1.1.1.1.2"></plus><apply id="S6.Ex17.1.1.1.1.1.3.cmml" xref="S6.Ex17.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.Ex17.1.1.1.1.1.3.1.cmml" xref="S6.Ex17.1.1.1.1.1.3">subscript</csymbol><ci id="S6.Ex17.1.1.1.1.1.3.2.cmml" xref="S6.Ex17.1.1.1.1.1.3.2">𝑥</ci><cn id="S6.Ex17.1.1.1.1.1.3.3.cmml" type="integer" xref="S6.Ex17.1.1.1.1.1.3.3">2</cn></apply><apply id="S6.Ex17.1.1.1.1.1.1.cmml" xref="S6.Ex17.1.1.1.1.1.1"><times id="S6.Ex17.1.1.1.1.1.1.2.cmml" xref="S6.Ex17.1.1.1.1.1.1.2"></times><apply id="S6.Ex17.1.1.1.1.1.1.3.cmml" xref="S6.Ex17.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.Ex17.1.1.1.1.1.1.3.1.cmml" xref="S6.Ex17.1.1.1.1.1.1.3">subscript</csymbol><ci id="S6.Ex17.1.1.1.1.1.1.3.2.cmml" xref="S6.Ex17.1.1.1.1.1.1.3.2">𝜑</ci><cn id="S6.Ex17.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S6.Ex17.1.1.1.1.1.1.3.3">1</cn></apply><apply id="S6.Ex17.1.1.1.1.1.1.1.1.1.cmml" xref="S6.Ex17.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.Ex17.1.1.1.1.1.1.1.1.1.1.cmml" xref="S6.Ex17.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S6.Ex17.1.1.1.1.1.1.1.1.1.2.cmml" xref="S6.Ex17.1.1.1.1.1.1.1.1.1.2">𝑥</ci><cn id="S6.Ex17.1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.Ex17.1.1.1.1.1.1.1.1.1.3">1</cn></apply><ci id="S6.Ex17.1.1.1.1.1.1.4.cmml" xref="S6.Ex17.1.1.1.1.1.1.4">𝜃</ci></apply></apply></apply></matrixrow><matrixrow id="S6.Ex20.m1.1.1b.cmml" xref="S6.Ex20.m1.1.1"><apply id="S6.Ex18.1.1.1.1.cmml" xref="S6.Ex18.1.1.1"><eq id="S6.Ex18.1.1.1.1.1.cmml" xref="S6.Ex18.1.1.1.1.1"></eq><apply id="S6.Ex18.1.1.1.1.2.cmml" xref="S6.Ex18.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.Ex18.1.1.1.1.2.1.cmml" xref="S6.Ex18.1.1.1.1.2">subscript</csymbol><apply id="S6.Ex18.1.1.1.1.2.2.cmml" xref="S6.Ex18.1.1.1.1.2.2"><ci id="S6.Ex18.1.1.1.1.2.2.1.cmml" xref="S6.Ex18.1.1.1.1.2.2.1">˙</ci><ci id="S6.Ex18.1.1.1.1.2.2.2.cmml" xref="S6.Ex18.1.1.1.1.2.2.2">𝑥</ci></apply><cn id="S6.Ex18.1.1.1.1.2.3.cmml" type="integer" xref="S6.Ex18.1.1.1.1.2.3">2</cn></apply><ci id="S6.Ex18.1.1.1.1.3.cmml" xref="S6.Ex18.1.1.1.1.3">𝑢</ci></apply></matrixrow><matrixrow id="S6.Ex20.m1.1.1c.cmml" xref="S6.Ex20.m1.1.1"><apply id="S6.Ex19.1.1.cmml" xref="S6.Ex19.1.1"><eq id="S6.Ex19.1.1.1.cmml" xref="S6.Ex19.1.1.1"></eq><ci id="S6.Ex19.1.1.2.cmml" xref="S6.Ex19.1.1.2">𝑦</ci><apply id="S6.Ex19.1.1.3.cmml" xref="S6.Ex19.1.1.3"><csymbol cd="ambiguous" id="S6.Ex19.1.1.3.1.cmml" xref="S6.Ex19.1.1.3">subscript</csymbol><ci id="S6.Ex19.1.1.3.2.cmml" xref="S6.Ex19.1.1.3.2">𝑥</ci><cn id="S6.Ex19.1.1.3.3.cmml" type="integer" xref="S6.Ex19.1.1.3.3">1</cn></apply></apply></matrixrow></matrix></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex20.m1.1c">\displaystyle\left\{\begin{array}[]{l}\dot{x}_{1}=x_{2}+\varphi_{1}(x_{1})% \theta,\\ \dot{x}_{2}=u,\\ y=x_{1}\end{array}\right.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex20.m1.1d">{ start_ARRAY start_ROW start_CELL over˙ start_ARG italic_x end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) italic_θ , end_CELL end_ROW start_ROW start_CELL over˙ start_ARG italic_x end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_u , end_CELL end_ROW start_ROW start_CELL italic_y = italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS2.p1.26">with <math alttext="u\in\mathbb{R}" class="ltx_Math" display="inline" id="S6.SS2.p1.4.m1.1"><semantics id="S6.SS2.p1.4.m1.1a"><mrow id="S6.SS2.p1.4.m1.1.1" xref="S6.SS2.p1.4.m1.1.1.cmml"><mi id="S6.SS2.p1.4.m1.1.1.2" xref="S6.SS2.p1.4.m1.1.1.2.cmml">u</mi><mo id="S6.SS2.p1.4.m1.1.1.1" xref="S6.SS2.p1.4.m1.1.1.1.cmml">∈</mo><mi id="S6.SS2.p1.4.m1.1.1.3" xref="S6.SS2.p1.4.m1.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.4.m1.1b"><apply id="S6.SS2.p1.4.m1.1.1.cmml" xref="S6.SS2.p1.4.m1.1.1"><in id="S6.SS2.p1.4.m1.1.1.1.cmml" xref="S6.SS2.p1.4.m1.1.1.1"></in><ci id="S6.SS2.p1.4.m1.1.1.2.cmml" xref="S6.SS2.p1.4.m1.1.1.2">𝑢</ci><ci id="S6.SS2.p1.4.m1.1.1.3.cmml" xref="S6.SS2.p1.4.m1.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.4.m1.1c">u\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.4.m1.1d">italic_u ∈ blackboard_R</annotation></semantics></math>, <math alttext="\varphi_{1}(x_{1})=x_{1}^{2}" class="ltx_Math" display="inline" id="S6.SS2.p1.5.m2.1"><semantics id="S6.SS2.p1.5.m2.1a"><mrow id="S6.SS2.p1.5.m2.1.1" xref="S6.SS2.p1.5.m2.1.1.cmml"><mrow id="S6.SS2.p1.5.m2.1.1.1" xref="S6.SS2.p1.5.m2.1.1.1.cmml"><msub id="S6.SS2.p1.5.m2.1.1.1.3" xref="S6.SS2.p1.5.m2.1.1.1.3.cmml"><mi id="S6.SS2.p1.5.m2.1.1.1.3.2" xref="S6.SS2.p1.5.m2.1.1.1.3.2.cmml">φ</mi><mn id="S6.SS2.p1.5.m2.1.1.1.3.3" xref="S6.SS2.p1.5.m2.1.1.1.3.3.cmml">1</mn></msub><mo id="S6.SS2.p1.5.m2.1.1.1.2" xref="S6.SS2.p1.5.m2.1.1.1.2.cmml">⁢</mo><mrow id="S6.SS2.p1.5.m2.1.1.1.1.1" xref="S6.SS2.p1.5.m2.1.1.1.1.1.1.cmml"><mo id="S6.SS2.p1.5.m2.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.p1.5.m2.1.1.1.1.1.1.cmml">(</mo><msub id="S6.SS2.p1.5.m2.1.1.1.1.1.1" xref="S6.SS2.p1.5.m2.1.1.1.1.1.1.cmml"><mi id="S6.SS2.p1.5.m2.1.1.1.1.1.1.2" xref="S6.SS2.p1.5.m2.1.1.1.1.1.1.2.cmml">x</mi><mn id="S6.SS2.p1.5.m2.1.1.1.1.1.1.3" xref="S6.SS2.p1.5.m2.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S6.SS2.p1.5.m2.1.1.1.1.1.3" stretchy="false" xref="S6.SS2.p1.5.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.p1.5.m2.1.1.2" xref="S6.SS2.p1.5.m2.1.1.2.cmml">=</mo><msubsup id="S6.SS2.p1.5.m2.1.1.3" xref="S6.SS2.p1.5.m2.1.1.3.cmml"><mi id="S6.SS2.p1.5.m2.1.1.3.2.2" xref="S6.SS2.p1.5.m2.1.1.3.2.2.cmml">x</mi><mn id="S6.SS2.p1.5.m2.1.1.3.2.3" xref="S6.SS2.p1.5.m2.1.1.3.2.3.cmml">1</mn><mn id="S6.SS2.p1.5.m2.1.1.3.3" xref="S6.SS2.p1.5.m2.1.1.3.3.cmml">2</mn></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.5.m2.1b"><apply id="S6.SS2.p1.5.m2.1.1.cmml" xref="S6.SS2.p1.5.m2.1.1"><eq id="S6.SS2.p1.5.m2.1.1.2.cmml" xref="S6.SS2.p1.5.m2.1.1.2"></eq><apply id="S6.SS2.p1.5.m2.1.1.1.cmml" xref="S6.SS2.p1.5.m2.1.1.1"><times id="S6.SS2.p1.5.m2.1.1.1.2.cmml" xref="S6.SS2.p1.5.m2.1.1.1.2"></times><apply id="S6.SS2.p1.5.m2.1.1.1.3.cmml" xref="S6.SS2.p1.5.m2.1.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.p1.5.m2.1.1.1.3.1.cmml" xref="S6.SS2.p1.5.m2.1.1.1.3">subscript</csymbol><ci id="S6.SS2.p1.5.m2.1.1.1.3.2.cmml" xref="S6.SS2.p1.5.m2.1.1.1.3.2">𝜑</ci><cn id="S6.SS2.p1.5.m2.1.1.1.3.3.cmml" type="integer" xref="S6.SS2.p1.5.m2.1.1.1.3.3">1</cn></apply><apply id="S6.SS2.p1.5.m2.1.1.1.1.1.1.cmml" xref="S6.SS2.p1.5.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.p1.5.m2.1.1.1.1.1.1.1.cmml" xref="S6.SS2.p1.5.m2.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.p1.5.m2.1.1.1.1.1.1.2.cmml" xref="S6.SS2.p1.5.m2.1.1.1.1.1.1.2">𝑥</ci><cn id="S6.SS2.p1.5.m2.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.SS2.p1.5.m2.1.1.1.1.1.1.3">1</cn></apply></apply><apply id="S6.SS2.p1.5.m2.1.1.3.cmml" xref="S6.SS2.p1.5.m2.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.p1.5.m2.1.1.3.1.cmml" xref="S6.SS2.p1.5.m2.1.1.3">superscript</csymbol><apply id="S6.SS2.p1.5.m2.1.1.3.2.cmml" xref="S6.SS2.p1.5.m2.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.p1.5.m2.1.1.3.2.1.cmml" xref="S6.SS2.p1.5.m2.1.1.3">subscript</csymbol><ci id="S6.SS2.p1.5.m2.1.1.3.2.2.cmml" xref="S6.SS2.p1.5.m2.1.1.3.2.2">𝑥</ci><cn id="S6.SS2.p1.5.m2.1.1.3.2.3.cmml" type="integer" xref="S6.SS2.p1.5.m2.1.1.3.2.3">1</cn></apply><cn id="S6.SS2.p1.5.m2.1.1.3.3.cmml" type="integer" xref="S6.SS2.p1.5.m2.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.5.m2.1c">\varphi_{1}(x_{1})=x_{1}^{2}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.5.m2.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\theta=2" class="ltx_Math" display="inline" id="S6.SS2.p1.6.m3.1"><semantics id="S6.SS2.p1.6.m3.1a"><mrow id="S6.SS2.p1.6.m3.1.1" xref="S6.SS2.p1.6.m3.1.1.cmml"><mi id="S6.SS2.p1.6.m3.1.1.2" xref="S6.SS2.p1.6.m3.1.1.2.cmml">θ</mi><mo id="S6.SS2.p1.6.m3.1.1.1" xref="S6.SS2.p1.6.m3.1.1.1.cmml">=</mo><mn id="S6.SS2.p1.6.m3.1.1.3" xref="S6.SS2.p1.6.m3.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.6.m3.1b"><apply id="S6.SS2.p1.6.m3.1.1.cmml" xref="S6.SS2.p1.6.m3.1.1"><eq id="S6.SS2.p1.6.m3.1.1.1.cmml" xref="S6.SS2.p1.6.m3.1.1.1"></eq><ci id="S6.SS2.p1.6.m3.1.1.2.cmml" xref="S6.SS2.p1.6.m3.1.1.2">𝜃</ci><cn id="S6.SS2.p1.6.m3.1.1.3.cmml" type="integer" xref="S6.SS2.p1.6.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.6.m3.1c">\theta=2</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.6.m3.1d">italic_θ = 2</annotation></semantics></math>, and <math alttext="\bm{x}(0)=[0.6,0]^{T}" class="ltx_Math" display="inline" id="S6.SS2.p1.7.m4.3"><semantics id="S6.SS2.p1.7.m4.3a"><mrow id="S6.SS2.p1.7.m4.3.4" xref="S6.SS2.p1.7.m4.3.4.cmml"><mrow id="S6.SS2.p1.7.m4.3.4.2" xref="S6.SS2.p1.7.m4.3.4.2.cmml"><mi id="S6.SS2.p1.7.m4.3.4.2.2" xref="S6.SS2.p1.7.m4.3.4.2.2.cmml">𝒙</mi><mo id="S6.SS2.p1.7.m4.3.4.2.1" xref="S6.SS2.p1.7.m4.3.4.2.1.cmml">⁢</mo><mrow id="S6.SS2.p1.7.m4.3.4.2.3.2" xref="S6.SS2.p1.7.m4.3.4.2.cmml"><mo id="S6.SS2.p1.7.m4.3.4.2.3.2.1" stretchy="false" xref="S6.SS2.p1.7.m4.3.4.2.cmml">(</mo><mn id="S6.SS2.p1.7.m4.1.1" xref="S6.SS2.p1.7.m4.1.1.cmml">0</mn><mo id="S6.SS2.p1.7.m4.3.4.2.3.2.2" stretchy="false" xref="S6.SS2.p1.7.m4.3.4.2.cmml">)</mo></mrow></mrow><mo id="S6.SS2.p1.7.m4.3.4.1" xref="S6.SS2.p1.7.m4.3.4.1.cmml">=</mo><msup id="S6.SS2.p1.7.m4.3.4.3" xref="S6.SS2.p1.7.m4.3.4.3.cmml"><mrow id="S6.SS2.p1.7.m4.3.4.3.2.2" xref="S6.SS2.p1.7.m4.3.4.3.2.1.cmml"><mo id="S6.SS2.p1.7.m4.3.4.3.2.2.1" stretchy="false" xref="S6.SS2.p1.7.m4.3.4.3.2.1.cmml">[</mo><mn id="S6.SS2.p1.7.m4.2.2" xref="S6.SS2.p1.7.m4.2.2.cmml">0.6</mn><mo id="S6.SS2.p1.7.m4.3.4.3.2.2.2" xref="S6.SS2.p1.7.m4.3.4.3.2.1.cmml">,</mo><mn id="S6.SS2.p1.7.m4.3.3" xref="S6.SS2.p1.7.m4.3.3.cmml">0</mn><mo id="S6.SS2.p1.7.m4.3.4.3.2.2.3" stretchy="false" xref="S6.SS2.p1.7.m4.3.4.3.2.1.cmml">]</mo></mrow><mi id="S6.SS2.p1.7.m4.3.4.3.3" xref="S6.SS2.p1.7.m4.3.4.3.3.cmml">T</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.7.m4.3b"><apply id="S6.SS2.p1.7.m4.3.4.cmml" xref="S6.SS2.p1.7.m4.3.4"><eq id="S6.SS2.p1.7.m4.3.4.1.cmml" xref="S6.SS2.p1.7.m4.3.4.1"></eq><apply id="S6.SS2.p1.7.m4.3.4.2.cmml" xref="S6.SS2.p1.7.m4.3.4.2"><times id="S6.SS2.p1.7.m4.3.4.2.1.cmml" xref="S6.SS2.p1.7.m4.3.4.2.1"></times><ci id="S6.SS2.p1.7.m4.3.4.2.2.cmml" xref="S6.SS2.p1.7.m4.3.4.2.2">𝒙</ci><cn id="S6.SS2.p1.7.m4.1.1.cmml" type="integer" xref="S6.SS2.p1.7.m4.1.1">0</cn></apply><apply id="S6.SS2.p1.7.m4.3.4.3.cmml" xref="S6.SS2.p1.7.m4.3.4.3"><csymbol cd="ambiguous" id="S6.SS2.p1.7.m4.3.4.3.1.cmml" xref="S6.SS2.p1.7.m4.3.4.3">superscript</csymbol><interval closure="closed" id="S6.SS2.p1.7.m4.3.4.3.2.1.cmml" xref="S6.SS2.p1.7.m4.3.4.3.2.2"><cn id="S6.SS2.p1.7.m4.2.2.cmml" type="float" xref="S6.SS2.p1.7.m4.2.2">0.6</cn><cn id="S6.SS2.p1.7.m4.3.3.cmml" type="integer" xref="S6.SS2.p1.7.m4.3.3">0</cn></interval><ci id="S6.SS2.p1.7.m4.3.4.3.3.cmml" xref="S6.SS2.p1.7.m4.3.4.3.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.7.m4.3c">\bm{x}(0)=[0.6,0]^{T}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.7.m4.3d">bold_italic_x ( 0 ) = [ 0.6 , 0 ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math>. Note that for existing modular backstepping control methods, if the damping coefficients <math alttext="k_{\rm{d}\it i}" class="ltx_Math" display="inline" id="S6.SS2.p1.8.m5.1"><semantics id="S6.SS2.p1.8.m5.1a"><msub id="S6.SS2.p1.8.m5.1.1" xref="S6.SS2.p1.8.m5.1.1.cmml"><mi id="S6.SS2.p1.8.m5.1.1.2" xref="S6.SS2.p1.8.m5.1.1.2.cmml">k</mi><mrow id="S6.SS2.p1.8.m5.1.1.3" xref="S6.SS2.p1.8.m5.1.1.3.cmml"><mi id="S6.SS2.p1.8.m5.1.1.3.2" mathvariant="normal" xref="S6.SS2.p1.8.m5.1.1.3.2.cmml">d</mi><mo id="S6.SS2.p1.8.m5.1.1.3.1" xref="S6.SS2.p1.8.m5.1.1.3.1.cmml">⁢</mo><mi id="S6.SS2.p1.8.m5.1.1.3.3" xref="S6.SS2.p1.8.m5.1.1.3.3.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.8.m5.1b"><apply id="S6.SS2.p1.8.m5.1.1.cmml" xref="S6.SS2.p1.8.m5.1.1"><csymbol cd="ambiguous" id="S6.SS2.p1.8.m5.1.1.1.cmml" xref="S6.SS2.p1.8.m5.1.1">subscript</csymbol><ci id="S6.SS2.p1.8.m5.1.1.2.cmml" xref="S6.SS2.p1.8.m5.1.1.2">𝑘</ci><apply id="S6.SS2.p1.8.m5.1.1.3.cmml" xref="S6.SS2.p1.8.m5.1.1.3"><times id="S6.SS2.p1.8.m5.1.1.3.1.cmml" xref="S6.SS2.p1.8.m5.1.1.3.1"></times><ci id="S6.SS2.p1.8.m5.1.1.3.2.cmml" xref="S6.SS2.p1.8.m5.1.1.3.2">d</ci><ci id="S6.SS2.p1.8.m5.1.1.3.3.cmml" xref="S6.SS2.p1.8.m5.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.8.m5.1c">k_{\rm{d}\it i}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.8.m5.1d">italic_k start_POSTSUBSCRIPT roman_d italic_i end_POSTSUBSCRIPT</annotation></semantics></math> are close to 0, the transient performance of the above system can be deteriorated by the modeling error term <math alttext="\Phi^{T}\tilde{\theta}" class="ltx_Math" display="inline" id="S6.SS2.p1.9.m6.1"><semantics id="S6.SS2.p1.9.m6.1a"><mrow id="S6.SS2.p1.9.m6.1.1" xref="S6.SS2.p1.9.m6.1.1.cmml"><msup id="S6.SS2.p1.9.m6.1.1.2" xref="S6.SS2.p1.9.m6.1.1.2.cmml"><mi id="S6.SS2.p1.9.m6.1.1.2.2" mathvariant="normal" xref="S6.SS2.p1.9.m6.1.1.2.2.cmml">Φ</mi><mi id="S6.SS2.p1.9.m6.1.1.2.3" xref="S6.SS2.p1.9.m6.1.1.2.3.cmml">T</mi></msup><mo id="S6.SS2.p1.9.m6.1.1.1" xref="S6.SS2.p1.9.m6.1.1.1.cmml">⁢</mo><mover accent="true" id="S6.SS2.p1.9.m6.1.1.3" xref="S6.SS2.p1.9.m6.1.1.3.cmml"><mi id="S6.SS2.p1.9.m6.1.1.3.2" xref="S6.SS2.p1.9.m6.1.1.3.2.cmml">θ</mi><mo id="S6.SS2.p1.9.m6.1.1.3.1" xref="S6.SS2.p1.9.m6.1.1.3.1.cmml">~</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.9.m6.1b"><apply id="S6.SS2.p1.9.m6.1.1.cmml" xref="S6.SS2.p1.9.m6.1.1"><times id="S6.SS2.p1.9.m6.1.1.1.cmml" xref="S6.SS2.p1.9.m6.1.1.1"></times><apply id="S6.SS2.p1.9.m6.1.1.2.cmml" xref="S6.SS2.p1.9.m6.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.p1.9.m6.1.1.2.1.cmml" xref="S6.SS2.p1.9.m6.1.1.2">superscript</csymbol><ci id="S6.SS2.p1.9.m6.1.1.2.2.cmml" xref="S6.SS2.p1.9.m6.1.1.2.2">Φ</ci><ci id="S6.SS2.p1.9.m6.1.1.2.3.cmml" xref="S6.SS2.p1.9.m6.1.1.2.3">𝑇</ci></apply><apply id="S6.SS2.p1.9.m6.1.1.3.cmml" xref="S6.SS2.p1.9.m6.1.1.3"><ci id="S6.SS2.p1.9.m6.1.1.3.1.cmml" xref="S6.SS2.p1.9.m6.1.1.3.1">~</ci><ci id="S6.SS2.p1.9.m6.1.1.3.2.cmml" xref="S6.SS2.p1.9.m6.1.1.3.2">𝜃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.9.m6.1c">\Phi^{T}\tilde{\theta}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.9.m6.1d">roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG italic_θ end_ARG</annotation></semantics></math> regardless of the convergence of the estimation error <math alttext="\tilde{\theta}" class="ltx_Math" display="inline" id="S6.SS2.p1.10.m7.1"><semantics id="S6.SS2.p1.10.m7.1a"><mover accent="true" id="S6.SS2.p1.10.m7.1.1" xref="S6.SS2.p1.10.m7.1.1.cmml"><mi id="S6.SS2.p1.10.m7.1.1.2" xref="S6.SS2.p1.10.m7.1.1.2.cmml">θ</mi><mo id="S6.SS2.p1.10.m7.1.1.1" xref="S6.SS2.p1.10.m7.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.10.m7.1b"><apply id="S6.SS2.p1.10.m7.1.1.cmml" xref="S6.SS2.p1.10.m7.1.1"><ci id="S6.SS2.p1.10.m7.1.1.1.cmml" xref="S6.SS2.p1.10.m7.1.1.1">~</ci><ci id="S6.SS2.p1.10.m7.1.1.2.cmml" xref="S6.SS2.p1.10.m7.1.1.2">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.10.m7.1c">\tilde{\theta}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.10.m7.1d">over~ start_ARG italic_θ end_ARG</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib8" title="">8</a>]</cite>. The reference trajectories <math alttext="y_{\rm r}" class="ltx_Math" display="inline" id="S6.SS2.p1.11.m8.1"><semantics id="S6.SS2.p1.11.m8.1a"><msub id="S6.SS2.p1.11.m8.1.1" xref="S6.SS2.p1.11.m8.1.1.cmml"><mi id="S6.SS2.p1.11.m8.1.1.2" xref="S6.SS2.p1.11.m8.1.1.2.cmml">y</mi><mi id="S6.SS2.p1.11.m8.1.1.3" mathvariant="normal" xref="S6.SS2.p1.11.m8.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.11.m8.1b"><apply id="S6.SS2.p1.11.m8.1.1.cmml" xref="S6.SS2.p1.11.m8.1.1"><csymbol cd="ambiguous" id="S6.SS2.p1.11.m8.1.1.1.cmml" xref="S6.SS2.p1.11.m8.1.1">subscript</csymbol><ci id="S6.SS2.p1.11.m8.1.1.2.cmml" xref="S6.SS2.p1.11.m8.1.1.2">𝑦</ci><ci id="S6.SS2.p1.11.m8.1.1.3.cmml" xref="S6.SS2.p1.11.m8.1.1.3">r</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.11.m8.1c">y_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.11.m8.1d">italic_y start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\dot{y}_{\rm r}" class="ltx_Math" display="inline" id="S6.SS2.p1.12.m9.1"><semantics id="S6.SS2.p1.12.m9.1a"><msub id="S6.SS2.p1.12.m9.1.1" xref="S6.SS2.p1.12.m9.1.1.cmml"><mover accent="true" id="S6.SS2.p1.12.m9.1.1.2" xref="S6.SS2.p1.12.m9.1.1.2.cmml"><mi id="S6.SS2.p1.12.m9.1.1.2.2" xref="S6.SS2.p1.12.m9.1.1.2.2.cmml">y</mi><mo id="S6.SS2.p1.12.m9.1.1.2.1" xref="S6.SS2.p1.12.m9.1.1.2.1.cmml">˙</mo></mover><mi id="S6.SS2.p1.12.m9.1.1.3" mathvariant="normal" xref="S6.SS2.p1.12.m9.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.12.m9.1b"><apply id="S6.SS2.p1.12.m9.1.1.cmml" xref="S6.SS2.p1.12.m9.1.1"><csymbol cd="ambiguous" id="S6.SS2.p1.12.m9.1.1.1.cmml" xref="S6.SS2.p1.12.m9.1.1">subscript</csymbol><apply id="S6.SS2.p1.12.m9.1.1.2.cmml" xref="S6.SS2.p1.12.m9.1.1.2"><ci id="S6.SS2.p1.12.m9.1.1.2.1.cmml" xref="S6.SS2.p1.12.m9.1.1.2.1">˙</ci><ci id="S6.SS2.p1.12.m9.1.1.2.2.cmml" xref="S6.SS2.p1.12.m9.1.1.2.2">𝑦</ci></apply><ci id="S6.SS2.p1.12.m9.1.1.3.cmml" xref="S6.SS2.p1.12.m9.1.1.3">r</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.12.m9.1c">\dot{y}_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.12.m9.1d">over˙ start_ARG italic_y end_ARG start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\ddot{y}_{\rm r}" class="ltx_Math" display="inline" id="S6.SS2.p1.13.m10.1"><semantics id="S6.SS2.p1.13.m10.1a"><msub id="S6.SS2.p1.13.m10.1.1" xref="S6.SS2.p1.13.m10.1.1.cmml"><mover accent="true" id="S6.SS2.p1.13.m10.1.1.2" xref="S6.SS2.p1.13.m10.1.1.2.cmml"><mi id="S6.SS2.p1.13.m10.1.1.2.2" xref="S6.SS2.p1.13.m10.1.1.2.2.cmml">y</mi><mo id="S6.SS2.p1.13.m10.1.1.2.1" xref="S6.SS2.p1.13.m10.1.1.2.1.cmml">¨</mo></mover><mi id="S6.SS2.p1.13.m10.1.1.3" mathvariant="normal" xref="S6.SS2.p1.13.m10.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.13.m10.1b"><apply id="S6.SS2.p1.13.m10.1.1.cmml" xref="S6.SS2.p1.13.m10.1.1"><csymbol cd="ambiguous" id="S6.SS2.p1.13.m10.1.1.1.cmml" xref="S6.SS2.p1.13.m10.1.1">subscript</csymbol><apply id="S6.SS2.p1.13.m10.1.1.2.cmml" xref="S6.SS2.p1.13.m10.1.1.2"><ci id="S6.SS2.p1.13.m10.1.1.2.1.cmml" xref="S6.SS2.p1.13.m10.1.1.2.1">¨</ci><ci id="S6.SS2.p1.13.m10.1.1.2.2.cmml" xref="S6.SS2.p1.13.m10.1.1.2.2">𝑦</ci></apply><ci id="S6.SS2.p1.13.m10.1.1.3.cmml" xref="S6.SS2.p1.13.m10.1.1.3">r</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.13.m10.1c">\ddot{y}_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.13.m10.1d">over¨ start_ARG italic_y end_ARG start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math> are generated by <math alttext="y_{\rm r}(t)" class="ltx_Math" display="inline" id="S6.SS2.p1.14.m11.1"><semantics id="S6.SS2.p1.14.m11.1a"><mrow id="S6.SS2.p1.14.m11.1.2" xref="S6.SS2.p1.14.m11.1.2.cmml"><msub id="S6.SS2.p1.14.m11.1.2.2" xref="S6.SS2.p1.14.m11.1.2.2.cmml"><mi id="S6.SS2.p1.14.m11.1.2.2.2" xref="S6.SS2.p1.14.m11.1.2.2.2.cmml">y</mi><mi id="S6.SS2.p1.14.m11.1.2.2.3" mathvariant="normal" xref="S6.SS2.p1.14.m11.1.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.p1.14.m11.1.2.1" xref="S6.SS2.p1.14.m11.1.2.1.cmml">⁢</mo><mrow id="S6.SS2.p1.14.m11.1.2.3.2" xref="S6.SS2.p1.14.m11.1.2.cmml"><mo id="S6.SS2.p1.14.m11.1.2.3.2.1" stretchy="false" xref="S6.SS2.p1.14.m11.1.2.cmml">(</mo><mi id="S6.SS2.p1.14.m11.1.1" xref="S6.SS2.p1.14.m11.1.1.cmml">t</mi><mo id="S6.SS2.p1.14.m11.1.2.3.2.2" stretchy="false" xref="S6.SS2.p1.14.m11.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.14.m11.1b"><apply id="S6.SS2.p1.14.m11.1.2.cmml" xref="S6.SS2.p1.14.m11.1.2"><times id="S6.SS2.p1.14.m11.1.2.1.cmml" xref="S6.SS2.p1.14.m11.1.2.1"></times><apply id="S6.SS2.p1.14.m11.1.2.2.cmml" xref="S6.SS2.p1.14.m11.1.2.2"><csymbol cd="ambiguous" id="S6.SS2.p1.14.m11.1.2.2.1.cmml" xref="S6.SS2.p1.14.m11.1.2.2">subscript</csymbol><ci id="S6.SS2.p1.14.m11.1.2.2.2.cmml" xref="S6.SS2.p1.14.m11.1.2.2.2">𝑦</ci><ci id="S6.SS2.p1.14.m11.1.2.2.3.cmml" xref="S6.SS2.p1.14.m11.1.2.2.3">r</ci></apply><ci id="S6.SS2.p1.14.m11.1.1.cmml" xref="S6.SS2.p1.14.m11.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.14.m11.1c">y_{\rm r}(t)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.14.m11.1d">italic_y start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> <math alttext="=" class="ltx_Math" display="inline" id="S6.SS2.p1.15.m12.1"><semantics id="S6.SS2.p1.15.m12.1a"><mo id="S6.SS2.p1.15.m12.1.1" xref="S6.SS2.p1.15.m12.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.15.m12.1b"><eq id="S6.SS2.p1.15.m12.1.1.cmml" xref="S6.SS2.p1.15.m12.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.15.m12.1c">=</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.15.m12.1d">=</annotation></semantics></math> <math alttext="\frac{a_{0}}{a(s)}[r(t)]" class="ltx_Math" display="inline" id="S6.SS2.p1.16.m13.3"><semantics id="S6.SS2.p1.16.m13.3a"><mrow id="S6.SS2.p1.16.m13.3.3" xref="S6.SS2.p1.16.m13.3.3.cmml"><mfrac id="S6.SS2.p1.16.m13.1.1" xref="S6.SS2.p1.16.m13.1.1.cmml"><msub id="S6.SS2.p1.16.m13.1.1.3" xref="S6.SS2.p1.16.m13.1.1.3.cmml"><mi id="S6.SS2.p1.16.m13.1.1.3.2" xref="S6.SS2.p1.16.m13.1.1.3.2.cmml">a</mi><mn id="S6.SS2.p1.16.m13.1.1.3.3" xref="S6.SS2.p1.16.m13.1.1.3.3.cmml">0</mn></msub><mrow id="S6.SS2.p1.16.m13.1.1.1" xref="S6.SS2.p1.16.m13.1.1.1.cmml"><mi id="S6.SS2.p1.16.m13.1.1.1.3" xref="S6.SS2.p1.16.m13.1.1.1.3.cmml">a</mi><mo id="S6.SS2.p1.16.m13.1.1.1.2" xref="S6.SS2.p1.16.m13.1.1.1.2.cmml">⁢</mo><mrow id="S6.SS2.p1.16.m13.1.1.1.4.2" xref="S6.SS2.p1.16.m13.1.1.1.cmml"><mo id="S6.SS2.p1.16.m13.1.1.1.4.2.1" stretchy="false" xref="S6.SS2.p1.16.m13.1.1.1.cmml">(</mo><mi id="S6.SS2.p1.16.m13.1.1.1.1" xref="S6.SS2.p1.16.m13.1.1.1.1.cmml">s</mi><mo id="S6.SS2.p1.16.m13.1.1.1.4.2.2" stretchy="false" xref="S6.SS2.p1.16.m13.1.1.1.cmml">)</mo></mrow></mrow></mfrac><mo id="S6.SS2.p1.16.m13.3.3.2" xref="S6.SS2.p1.16.m13.3.3.2.cmml">⁢</mo><mrow id="S6.SS2.p1.16.m13.3.3.1.1" xref="S6.SS2.p1.16.m13.3.3.1.2.cmml"><mo id="S6.SS2.p1.16.m13.3.3.1.1.2" stretchy="false" xref="S6.SS2.p1.16.m13.3.3.1.2.1.cmml">[</mo><mrow id="S6.SS2.p1.16.m13.3.3.1.1.1" xref="S6.SS2.p1.16.m13.3.3.1.1.1.cmml"><mi id="S6.SS2.p1.16.m13.3.3.1.1.1.2" xref="S6.SS2.p1.16.m13.3.3.1.1.1.2.cmml">r</mi><mo id="S6.SS2.p1.16.m13.3.3.1.1.1.1" xref="S6.SS2.p1.16.m13.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S6.SS2.p1.16.m13.3.3.1.1.1.3.2" xref="S6.SS2.p1.16.m13.3.3.1.1.1.cmml"><mo id="S6.SS2.p1.16.m13.3.3.1.1.1.3.2.1" stretchy="false" xref="S6.SS2.p1.16.m13.3.3.1.1.1.cmml">(</mo><mi id="S6.SS2.p1.16.m13.2.2" xref="S6.SS2.p1.16.m13.2.2.cmml">t</mi><mo id="S6.SS2.p1.16.m13.3.3.1.1.1.3.2.2" stretchy="false" xref="S6.SS2.p1.16.m13.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.p1.16.m13.3.3.1.1.3" stretchy="false" xref="S6.SS2.p1.16.m13.3.3.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.16.m13.3b"><apply id="S6.SS2.p1.16.m13.3.3.cmml" xref="S6.SS2.p1.16.m13.3.3"><times id="S6.SS2.p1.16.m13.3.3.2.cmml" xref="S6.SS2.p1.16.m13.3.3.2"></times><apply id="S6.SS2.p1.16.m13.1.1.cmml" xref="S6.SS2.p1.16.m13.1.1"><divide id="S6.SS2.p1.16.m13.1.1.2.cmml" xref="S6.SS2.p1.16.m13.1.1"></divide><apply id="S6.SS2.p1.16.m13.1.1.3.cmml" xref="S6.SS2.p1.16.m13.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.p1.16.m13.1.1.3.1.cmml" xref="S6.SS2.p1.16.m13.1.1.3">subscript</csymbol><ci id="S6.SS2.p1.16.m13.1.1.3.2.cmml" xref="S6.SS2.p1.16.m13.1.1.3.2">𝑎</ci><cn id="S6.SS2.p1.16.m13.1.1.3.3.cmml" type="integer" xref="S6.SS2.p1.16.m13.1.1.3.3">0</cn></apply><apply id="S6.SS2.p1.16.m13.1.1.1.cmml" xref="S6.SS2.p1.16.m13.1.1.1"><times id="S6.SS2.p1.16.m13.1.1.1.2.cmml" xref="S6.SS2.p1.16.m13.1.1.1.2"></times><ci id="S6.SS2.p1.16.m13.1.1.1.3.cmml" xref="S6.SS2.p1.16.m13.1.1.1.3">𝑎</ci><ci id="S6.SS2.p1.16.m13.1.1.1.1.cmml" xref="S6.SS2.p1.16.m13.1.1.1.1">𝑠</ci></apply></apply><apply id="S6.SS2.p1.16.m13.3.3.1.2.cmml" xref="S6.SS2.p1.16.m13.3.3.1.1"><csymbol cd="latexml" id="S6.SS2.p1.16.m13.3.3.1.2.1.cmml" xref="S6.SS2.p1.16.m13.3.3.1.1.2">delimited-[]</csymbol><apply id="S6.SS2.p1.16.m13.3.3.1.1.1.cmml" xref="S6.SS2.p1.16.m13.3.3.1.1.1"><times id="S6.SS2.p1.16.m13.3.3.1.1.1.1.cmml" xref="S6.SS2.p1.16.m13.3.3.1.1.1.1"></times><ci id="S6.SS2.p1.16.m13.3.3.1.1.1.2.cmml" xref="S6.SS2.p1.16.m13.3.3.1.1.1.2">𝑟</ci><ci id="S6.SS2.p1.16.m13.2.2.cmml" xref="S6.SS2.p1.16.m13.2.2">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.16.m13.3c">\frac{a_{0}}{a(s)}[r(t)]</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.16.m13.3d">divide start_ARG italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG italic_a ( italic_s ) end_ARG [ italic_r ( italic_t ) ]</annotation></semantics></math> with <math alttext="a_{0}=1" class="ltx_Math" display="inline" id="S6.SS2.p1.17.m14.1"><semantics id="S6.SS2.p1.17.m14.1a"><mrow id="S6.SS2.p1.17.m14.1.1" xref="S6.SS2.p1.17.m14.1.1.cmml"><msub id="S6.SS2.p1.17.m14.1.1.2" xref="S6.SS2.p1.17.m14.1.1.2.cmml"><mi id="S6.SS2.p1.17.m14.1.1.2.2" xref="S6.SS2.p1.17.m14.1.1.2.2.cmml">a</mi><mn id="S6.SS2.p1.17.m14.1.1.2.3" xref="S6.SS2.p1.17.m14.1.1.2.3.cmml">0</mn></msub><mo id="S6.SS2.p1.17.m14.1.1.1" xref="S6.SS2.p1.17.m14.1.1.1.cmml">=</mo><mn id="S6.SS2.p1.17.m14.1.1.3" xref="S6.SS2.p1.17.m14.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.17.m14.1b"><apply id="S6.SS2.p1.17.m14.1.1.cmml" xref="S6.SS2.p1.17.m14.1.1"><eq id="S6.SS2.p1.17.m14.1.1.1.cmml" xref="S6.SS2.p1.17.m14.1.1.1"></eq><apply id="S6.SS2.p1.17.m14.1.1.2.cmml" xref="S6.SS2.p1.17.m14.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.p1.17.m14.1.1.2.1.cmml" xref="S6.SS2.p1.17.m14.1.1.2">subscript</csymbol><ci id="S6.SS2.p1.17.m14.1.1.2.2.cmml" xref="S6.SS2.p1.17.m14.1.1.2.2">𝑎</ci><cn id="S6.SS2.p1.17.m14.1.1.2.3.cmml" type="integer" xref="S6.SS2.p1.17.m14.1.1.2.3">0</cn></apply><cn id="S6.SS2.p1.17.m14.1.1.3.cmml" type="integer" xref="S6.SS2.p1.17.m14.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.17.m14.1c">a_{0}=1</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.17.m14.1d">italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 1</annotation></semantics></math>, <math alttext="a(s)=s^{3}+3s^{2}+3s+1" class="ltx_Math" display="inline" id="S6.SS2.p1.18.m15.1"><semantics id="S6.SS2.p1.18.m15.1a"><mrow id="S6.SS2.p1.18.m15.1.2" xref="S6.SS2.p1.18.m15.1.2.cmml"><mrow id="S6.SS2.p1.18.m15.1.2.2" xref="S6.SS2.p1.18.m15.1.2.2.cmml"><mi id="S6.SS2.p1.18.m15.1.2.2.2" xref="S6.SS2.p1.18.m15.1.2.2.2.cmml">a</mi><mo id="S6.SS2.p1.18.m15.1.2.2.1" xref="S6.SS2.p1.18.m15.1.2.2.1.cmml">⁢</mo><mrow id="S6.SS2.p1.18.m15.1.2.2.3.2" xref="S6.SS2.p1.18.m15.1.2.2.cmml"><mo id="S6.SS2.p1.18.m15.1.2.2.3.2.1" stretchy="false" xref="S6.SS2.p1.18.m15.1.2.2.cmml">(</mo><mi id="S6.SS2.p1.18.m15.1.1" xref="S6.SS2.p1.18.m15.1.1.cmml">s</mi><mo id="S6.SS2.p1.18.m15.1.2.2.3.2.2" stretchy="false" xref="S6.SS2.p1.18.m15.1.2.2.cmml">)</mo></mrow></mrow><mo id="S6.SS2.p1.18.m15.1.2.1" xref="S6.SS2.p1.18.m15.1.2.1.cmml">=</mo><mrow id="S6.SS2.p1.18.m15.1.2.3" xref="S6.SS2.p1.18.m15.1.2.3.cmml"><msup id="S6.SS2.p1.18.m15.1.2.3.2" xref="S6.SS2.p1.18.m15.1.2.3.2.cmml"><mi id="S6.SS2.p1.18.m15.1.2.3.2.2" xref="S6.SS2.p1.18.m15.1.2.3.2.2.cmml">s</mi><mn id="S6.SS2.p1.18.m15.1.2.3.2.3" xref="S6.SS2.p1.18.m15.1.2.3.2.3.cmml">3</mn></msup><mo id="S6.SS2.p1.18.m15.1.2.3.1" xref="S6.SS2.p1.18.m15.1.2.3.1.cmml">+</mo><mrow id="S6.SS2.p1.18.m15.1.2.3.3" xref="S6.SS2.p1.18.m15.1.2.3.3.cmml"><mn id="S6.SS2.p1.18.m15.1.2.3.3.2" xref="S6.SS2.p1.18.m15.1.2.3.3.2.cmml">3</mn><mo id="S6.SS2.p1.18.m15.1.2.3.3.1" xref="S6.SS2.p1.18.m15.1.2.3.3.1.cmml">⁢</mo><msup id="S6.SS2.p1.18.m15.1.2.3.3.3" xref="S6.SS2.p1.18.m15.1.2.3.3.3.cmml"><mi id="S6.SS2.p1.18.m15.1.2.3.3.3.2" xref="S6.SS2.p1.18.m15.1.2.3.3.3.2.cmml">s</mi><mn id="S6.SS2.p1.18.m15.1.2.3.3.3.3" xref="S6.SS2.p1.18.m15.1.2.3.3.3.3.cmml">2</mn></msup></mrow><mo id="S6.SS2.p1.18.m15.1.2.3.1a" xref="S6.SS2.p1.18.m15.1.2.3.1.cmml">+</mo><mrow id="S6.SS2.p1.18.m15.1.2.3.4" xref="S6.SS2.p1.18.m15.1.2.3.4.cmml"><mn id="S6.SS2.p1.18.m15.1.2.3.4.2" xref="S6.SS2.p1.18.m15.1.2.3.4.2.cmml">3</mn><mo id="S6.SS2.p1.18.m15.1.2.3.4.1" xref="S6.SS2.p1.18.m15.1.2.3.4.1.cmml">⁢</mo><mi id="S6.SS2.p1.18.m15.1.2.3.4.3" xref="S6.SS2.p1.18.m15.1.2.3.4.3.cmml">s</mi></mrow><mo id="S6.SS2.p1.18.m15.1.2.3.1b" xref="S6.SS2.p1.18.m15.1.2.3.1.cmml">+</mo><mn id="S6.SS2.p1.18.m15.1.2.3.5" xref="S6.SS2.p1.18.m15.1.2.3.5.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.18.m15.1b"><apply id="S6.SS2.p1.18.m15.1.2.cmml" xref="S6.SS2.p1.18.m15.1.2"><eq id="S6.SS2.p1.18.m15.1.2.1.cmml" xref="S6.SS2.p1.18.m15.1.2.1"></eq><apply id="S6.SS2.p1.18.m15.1.2.2.cmml" xref="S6.SS2.p1.18.m15.1.2.2"><times id="S6.SS2.p1.18.m15.1.2.2.1.cmml" xref="S6.SS2.p1.18.m15.1.2.2.1"></times><ci id="S6.SS2.p1.18.m15.1.2.2.2.cmml" xref="S6.SS2.p1.18.m15.1.2.2.2">𝑎</ci><ci id="S6.SS2.p1.18.m15.1.1.cmml" xref="S6.SS2.p1.18.m15.1.1">𝑠</ci></apply><apply id="S6.SS2.p1.18.m15.1.2.3.cmml" xref="S6.SS2.p1.18.m15.1.2.3"><plus id="S6.SS2.p1.18.m15.1.2.3.1.cmml" xref="S6.SS2.p1.18.m15.1.2.3.1"></plus><apply id="S6.SS2.p1.18.m15.1.2.3.2.cmml" xref="S6.SS2.p1.18.m15.1.2.3.2"><csymbol cd="ambiguous" id="S6.SS2.p1.18.m15.1.2.3.2.1.cmml" xref="S6.SS2.p1.18.m15.1.2.3.2">superscript</csymbol><ci id="S6.SS2.p1.18.m15.1.2.3.2.2.cmml" xref="S6.SS2.p1.18.m15.1.2.3.2.2">𝑠</ci><cn id="S6.SS2.p1.18.m15.1.2.3.2.3.cmml" type="integer" xref="S6.SS2.p1.18.m15.1.2.3.2.3">3</cn></apply><apply id="S6.SS2.p1.18.m15.1.2.3.3.cmml" xref="S6.SS2.p1.18.m15.1.2.3.3"><times id="S6.SS2.p1.18.m15.1.2.3.3.1.cmml" xref="S6.SS2.p1.18.m15.1.2.3.3.1"></times><cn id="S6.SS2.p1.18.m15.1.2.3.3.2.cmml" type="integer" xref="S6.SS2.p1.18.m15.1.2.3.3.2">3</cn><apply id="S6.SS2.p1.18.m15.1.2.3.3.3.cmml" xref="S6.SS2.p1.18.m15.1.2.3.3.3"><csymbol cd="ambiguous" id="S6.SS2.p1.18.m15.1.2.3.3.3.1.cmml" xref="S6.SS2.p1.18.m15.1.2.3.3.3">superscript</csymbol><ci id="S6.SS2.p1.18.m15.1.2.3.3.3.2.cmml" xref="S6.SS2.p1.18.m15.1.2.3.3.3.2">𝑠</ci><cn id="S6.SS2.p1.18.m15.1.2.3.3.3.3.cmml" type="integer" xref="S6.SS2.p1.18.m15.1.2.3.3.3.3">2</cn></apply></apply><apply id="S6.SS2.p1.18.m15.1.2.3.4.cmml" xref="S6.SS2.p1.18.m15.1.2.3.4"><times id="S6.SS2.p1.18.m15.1.2.3.4.1.cmml" xref="S6.SS2.p1.18.m15.1.2.3.4.1"></times><cn id="S6.SS2.p1.18.m15.1.2.3.4.2.cmml" type="integer" xref="S6.SS2.p1.18.m15.1.2.3.4.2">3</cn><ci id="S6.SS2.p1.18.m15.1.2.3.4.3.cmml" xref="S6.SS2.p1.18.m15.1.2.3.4.3">𝑠</ci></apply><cn id="S6.SS2.p1.18.m15.1.2.3.5.cmml" type="integer" xref="S6.SS2.p1.18.m15.1.2.3.5">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.18.m15.1c">a(s)=s^{3}+3s^{2}+3s+1</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.18.m15.1d">italic_a ( italic_s ) = italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + 3 italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 3 italic_s + 1</annotation></semantics></math>, and <math alttext="r(t)=\sin(2t)" class="ltx_Math" display="inline" id="S6.SS2.p1.19.m16.3"><semantics id="S6.SS2.p1.19.m16.3a"><mrow id="S6.SS2.p1.19.m16.3.3" xref="S6.SS2.p1.19.m16.3.3.cmml"><mrow id="S6.SS2.p1.19.m16.3.3.3" xref="S6.SS2.p1.19.m16.3.3.3.cmml"><mi id="S6.SS2.p1.19.m16.3.3.3.2" xref="S6.SS2.p1.19.m16.3.3.3.2.cmml">r</mi><mo id="S6.SS2.p1.19.m16.3.3.3.1" xref="S6.SS2.p1.19.m16.3.3.3.1.cmml">⁢</mo><mrow id="S6.SS2.p1.19.m16.3.3.3.3.2" xref="S6.SS2.p1.19.m16.3.3.3.cmml"><mo id="S6.SS2.p1.19.m16.3.3.3.3.2.1" stretchy="false" xref="S6.SS2.p1.19.m16.3.3.3.cmml">(</mo><mi id="S6.SS2.p1.19.m16.1.1" xref="S6.SS2.p1.19.m16.1.1.cmml">t</mi><mo id="S6.SS2.p1.19.m16.3.3.3.3.2.2" stretchy="false" xref="S6.SS2.p1.19.m16.3.3.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.p1.19.m16.3.3.2" xref="S6.SS2.p1.19.m16.3.3.2.cmml">=</mo><mrow id="S6.SS2.p1.19.m16.3.3.1.1" xref="S6.SS2.p1.19.m16.3.3.1.2.cmml"><mi id="S6.SS2.p1.19.m16.2.2" xref="S6.SS2.p1.19.m16.2.2.cmml">sin</mi><mo id="S6.SS2.p1.19.m16.3.3.1.1a" xref="S6.SS2.p1.19.m16.3.3.1.2.cmml">⁡</mo><mrow id="S6.SS2.p1.19.m16.3.3.1.1.1" xref="S6.SS2.p1.19.m16.3.3.1.2.cmml"><mo id="S6.SS2.p1.19.m16.3.3.1.1.1.2" stretchy="false" xref="S6.SS2.p1.19.m16.3.3.1.2.cmml">(</mo><mrow id="S6.SS2.p1.19.m16.3.3.1.1.1.1" xref="S6.SS2.p1.19.m16.3.3.1.1.1.1.cmml"><mn id="S6.SS2.p1.19.m16.3.3.1.1.1.1.2" xref="S6.SS2.p1.19.m16.3.3.1.1.1.1.2.cmml">2</mn><mo id="S6.SS2.p1.19.m16.3.3.1.1.1.1.1" xref="S6.SS2.p1.19.m16.3.3.1.1.1.1.1.cmml">⁢</mo><mi id="S6.SS2.p1.19.m16.3.3.1.1.1.1.3" xref="S6.SS2.p1.19.m16.3.3.1.1.1.1.3.cmml">t</mi></mrow><mo id="S6.SS2.p1.19.m16.3.3.1.1.1.3" stretchy="false" xref="S6.SS2.p1.19.m16.3.3.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.19.m16.3b"><apply id="S6.SS2.p1.19.m16.3.3.cmml" xref="S6.SS2.p1.19.m16.3.3"><eq id="S6.SS2.p1.19.m16.3.3.2.cmml" xref="S6.SS2.p1.19.m16.3.3.2"></eq><apply id="S6.SS2.p1.19.m16.3.3.3.cmml" xref="S6.SS2.p1.19.m16.3.3.3"><times id="S6.SS2.p1.19.m16.3.3.3.1.cmml" xref="S6.SS2.p1.19.m16.3.3.3.1"></times><ci id="S6.SS2.p1.19.m16.3.3.3.2.cmml" xref="S6.SS2.p1.19.m16.3.3.3.2">𝑟</ci><ci id="S6.SS2.p1.19.m16.1.1.cmml" xref="S6.SS2.p1.19.m16.1.1">𝑡</ci></apply><apply id="S6.SS2.p1.19.m16.3.3.1.2.cmml" xref="S6.SS2.p1.19.m16.3.3.1.1"><sin id="S6.SS2.p1.19.m16.2.2.cmml" xref="S6.SS2.p1.19.m16.2.2"></sin><apply id="S6.SS2.p1.19.m16.3.3.1.1.1.1.cmml" xref="S6.SS2.p1.19.m16.3.3.1.1.1.1"><times id="S6.SS2.p1.19.m16.3.3.1.1.1.1.1.cmml" xref="S6.SS2.p1.19.m16.3.3.1.1.1.1.1"></times><cn id="S6.SS2.p1.19.m16.3.3.1.1.1.1.2.cmml" type="integer" xref="S6.SS2.p1.19.m16.3.3.1.1.1.1.2">2</cn><ci id="S6.SS2.p1.19.m16.3.3.1.1.1.1.3.cmml" xref="S6.SS2.p1.19.m16.3.3.1.1.1.1.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.19.m16.3c">r(t)=\sin(2t)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.19.m16.3d">italic_r ( italic_t ) = roman_sin ( 2 italic_t )</annotation></semantics></math>. Choose the control parameters <math alttext="k_{{\rm c}1}=k_{{\rm c}2}=1" class="ltx_Math" display="inline" id="S6.SS2.p1.20.m17.1"><semantics id="S6.SS2.p1.20.m17.1a"><mrow id="S6.SS2.p1.20.m17.1.1" xref="S6.SS2.p1.20.m17.1.1.cmml"><msub id="S6.SS2.p1.20.m17.1.1.2" xref="S6.SS2.p1.20.m17.1.1.2.cmml"><mi id="S6.SS2.p1.20.m17.1.1.2.2" xref="S6.SS2.p1.20.m17.1.1.2.2.cmml">k</mi><mi id="S6.SS2.p1.20.m17.1.1.2.3" xref="S6.SS2.p1.20.m17.1.1.2.3.cmml">c1</mi></msub><mo id="S6.SS2.p1.20.m17.1.1.3" xref="S6.SS2.p1.20.m17.1.1.3.cmml">=</mo><msub id="S6.SS2.p1.20.m17.1.1.4" xref="S6.SS2.p1.20.m17.1.1.4.cmml"><mi id="S6.SS2.p1.20.m17.1.1.4.2" xref="S6.SS2.p1.20.m17.1.1.4.2.cmml">k</mi><mi id="S6.SS2.p1.20.m17.1.1.4.3" xref="S6.SS2.p1.20.m17.1.1.4.3.cmml">c2</mi></msub><mo id="S6.SS2.p1.20.m17.1.1.5" xref="S6.SS2.p1.20.m17.1.1.5.cmml">=</mo><mn id="S6.SS2.p1.20.m17.1.1.6" xref="S6.SS2.p1.20.m17.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.20.m17.1b"><apply id="S6.SS2.p1.20.m17.1.1.cmml" xref="S6.SS2.p1.20.m17.1.1"><and id="S6.SS2.p1.20.m17.1.1a.cmml" xref="S6.SS2.p1.20.m17.1.1"></and><apply id="S6.SS2.p1.20.m17.1.1b.cmml" xref="S6.SS2.p1.20.m17.1.1"><eq id="S6.SS2.p1.20.m17.1.1.3.cmml" xref="S6.SS2.p1.20.m17.1.1.3"></eq><apply id="S6.SS2.p1.20.m17.1.1.2.cmml" xref="S6.SS2.p1.20.m17.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.p1.20.m17.1.1.2.1.cmml" xref="S6.SS2.p1.20.m17.1.1.2">subscript</csymbol><ci id="S6.SS2.p1.20.m17.1.1.2.2.cmml" xref="S6.SS2.p1.20.m17.1.1.2.2">𝑘</ci><ci id="S6.SS2.p1.20.m17.1.1.2.3.cmml" xref="S6.SS2.p1.20.m17.1.1.2.3">c1</ci></apply><apply id="S6.SS2.p1.20.m17.1.1.4.cmml" xref="S6.SS2.p1.20.m17.1.1.4"><csymbol cd="ambiguous" id="S6.SS2.p1.20.m17.1.1.4.1.cmml" xref="S6.SS2.p1.20.m17.1.1.4">subscript</csymbol><ci id="S6.SS2.p1.20.m17.1.1.4.2.cmml" xref="S6.SS2.p1.20.m17.1.1.4.2">𝑘</ci><ci id="S6.SS2.p1.20.m17.1.1.4.3.cmml" xref="S6.SS2.p1.20.m17.1.1.4.3">c2</ci></apply></apply><apply id="S6.SS2.p1.20.m17.1.1c.cmml" xref="S6.SS2.p1.20.m17.1.1"><eq id="S6.SS2.p1.20.m17.1.1.5.cmml" xref="S6.SS2.p1.20.m17.1.1.5"></eq><share href="https://arxiv.org/html/2401.10785v2#S6.SS2.p1.20.m17.1.1.4.cmml" id="S6.SS2.p1.20.m17.1.1d.cmml" xref="S6.SS2.p1.20.m17.1.1"></share><cn id="S6.SS2.p1.20.m17.1.1.6.cmml" type="integer" xref="S6.SS2.p1.20.m17.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.20.m17.1c">k_{{\rm c}1}=k_{{\rm c}2}=1</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.20.m17.1d">italic_k start_POSTSUBSCRIPT c1 end_POSTSUBSCRIPT = italic_k start_POSTSUBSCRIPT c2 end_POSTSUBSCRIPT = 1</annotation></semantics></math>, <math alttext="\Gamma=1" class="ltx_Math" display="inline" id="S6.SS2.p1.21.m18.1"><semantics id="S6.SS2.p1.21.m18.1a"><mrow id="S6.SS2.p1.21.m18.1.1" xref="S6.SS2.p1.21.m18.1.1.cmml"><mi id="S6.SS2.p1.21.m18.1.1.2" mathvariant="normal" xref="S6.SS2.p1.21.m18.1.1.2.cmml">Γ</mi><mo id="S6.SS2.p1.21.m18.1.1.1" xref="S6.SS2.p1.21.m18.1.1.1.cmml">=</mo><mn id="S6.SS2.p1.21.m18.1.1.3" xref="S6.SS2.p1.21.m18.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.21.m18.1b"><apply id="S6.SS2.p1.21.m18.1.1.cmml" xref="S6.SS2.p1.21.m18.1.1"><eq id="S6.SS2.p1.21.m18.1.1.1.cmml" xref="S6.SS2.p1.21.m18.1.1.1"></eq><ci id="S6.SS2.p1.21.m18.1.1.2.cmml" xref="S6.SS2.p1.21.m18.1.1.2">Γ</ci><cn id="S6.SS2.p1.21.m18.1.1.3.cmml" type="integer" xref="S6.SS2.p1.21.m18.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.21.m18.1c">\Gamma=1</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.21.m18.1d">roman_Γ = 1</annotation></semantics></math>, <math alttext="\kappa=1" class="ltx_Math" display="inline" id="S6.SS2.p1.22.m19.1"><semantics id="S6.SS2.p1.22.m19.1a"><mrow id="S6.SS2.p1.22.m19.1.1" xref="S6.SS2.p1.22.m19.1.1.cmml"><mi id="S6.SS2.p1.22.m19.1.1.2" xref="S6.SS2.p1.22.m19.1.1.2.cmml">κ</mi><mo id="S6.SS2.p1.22.m19.1.1.1" xref="S6.SS2.p1.22.m19.1.1.1.cmml">=</mo><mn id="S6.SS2.p1.22.m19.1.1.3" xref="S6.SS2.p1.22.m19.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.22.m19.1b"><apply id="S6.SS2.p1.22.m19.1.1.cmml" xref="S6.SS2.p1.22.m19.1.1"><eq id="S6.SS2.p1.22.m19.1.1.1.cmml" xref="S6.SS2.p1.22.m19.1.1.1"></eq><ci id="S6.SS2.p1.22.m19.1.1.2.cmml" xref="S6.SS2.p1.22.m19.1.1.2">𝜅</ci><cn id="S6.SS2.p1.22.m19.1.1.3.cmml" type="integer" xref="S6.SS2.p1.22.m19.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.22.m19.1c">\kappa=1</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.22.m19.1d">italic_κ = 1</annotation></semantics></math>, <math alttext="\tau_{\rm d}=1" class="ltx_Math" display="inline" id="S6.SS2.p1.23.m20.1"><semantics id="S6.SS2.p1.23.m20.1a"><mrow id="S6.SS2.p1.23.m20.1.1" xref="S6.SS2.p1.23.m20.1.1.cmml"><msub id="S6.SS2.p1.23.m20.1.1.2" xref="S6.SS2.p1.23.m20.1.1.2.cmml"><mi id="S6.SS2.p1.23.m20.1.1.2.2" xref="S6.SS2.p1.23.m20.1.1.2.2.cmml">τ</mi><mi id="S6.SS2.p1.23.m20.1.1.2.3" mathvariant="normal" xref="S6.SS2.p1.23.m20.1.1.2.3.cmml">d</mi></msub><mo id="S6.SS2.p1.23.m20.1.1.1" xref="S6.SS2.p1.23.m20.1.1.1.cmml">=</mo><mn id="S6.SS2.p1.23.m20.1.1.3" xref="S6.SS2.p1.23.m20.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.23.m20.1b"><apply id="S6.SS2.p1.23.m20.1.1.cmml" xref="S6.SS2.p1.23.m20.1.1"><eq id="S6.SS2.p1.23.m20.1.1.1.cmml" xref="S6.SS2.p1.23.m20.1.1.1"></eq><apply id="S6.SS2.p1.23.m20.1.1.2.cmml" xref="S6.SS2.p1.23.m20.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.p1.23.m20.1.1.2.1.cmml" xref="S6.SS2.p1.23.m20.1.1.2">subscript</csymbol><ci id="S6.SS2.p1.23.m20.1.1.2.2.cmml" xref="S6.SS2.p1.23.m20.1.1.2.2">𝜏</ci><ci id="S6.SS2.p1.23.m20.1.1.2.3.cmml" xref="S6.SS2.p1.23.m20.1.1.2.3">d</ci></apply><cn id="S6.SS2.p1.23.m20.1.1.3.cmml" type="integer" xref="S6.SS2.p1.23.m20.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.23.m20.1c">\tau_{\rm d}=1</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.23.m20.1d">italic_τ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT = 1</annotation></semantics></math>, <math alttext="\hat{\theta}(0)=0" class="ltx_Math" display="inline" id="S6.SS2.p1.24.m21.1"><semantics id="S6.SS2.p1.24.m21.1a"><mrow id="S6.SS2.p1.24.m21.1.2" xref="S6.SS2.p1.24.m21.1.2.cmml"><mrow id="S6.SS2.p1.24.m21.1.2.2" xref="S6.SS2.p1.24.m21.1.2.2.cmml"><mover accent="true" id="S6.SS2.p1.24.m21.1.2.2.2" xref="S6.SS2.p1.24.m21.1.2.2.2.cmml"><mi id="S6.SS2.p1.24.m21.1.2.2.2.2" xref="S6.SS2.p1.24.m21.1.2.2.2.2.cmml">θ</mi><mo id="S6.SS2.p1.24.m21.1.2.2.2.1" xref="S6.SS2.p1.24.m21.1.2.2.2.1.cmml">^</mo></mover><mo id="S6.SS2.p1.24.m21.1.2.2.1" xref="S6.SS2.p1.24.m21.1.2.2.1.cmml">⁢</mo><mrow id="S6.SS2.p1.24.m21.1.2.2.3.2" xref="S6.SS2.p1.24.m21.1.2.2.cmml"><mo id="S6.SS2.p1.24.m21.1.2.2.3.2.1" stretchy="false" xref="S6.SS2.p1.24.m21.1.2.2.cmml">(</mo><mn id="S6.SS2.p1.24.m21.1.1" xref="S6.SS2.p1.24.m21.1.1.cmml">0</mn><mo id="S6.SS2.p1.24.m21.1.2.2.3.2.2" stretchy="false" xref="S6.SS2.p1.24.m21.1.2.2.cmml">)</mo></mrow></mrow><mo id="S6.SS2.p1.24.m21.1.2.1" xref="S6.SS2.p1.24.m21.1.2.1.cmml">=</mo><mn id="S6.SS2.p1.24.m21.1.2.3" xref="S6.SS2.p1.24.m21.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.24.m21.1b"><apply id="S6.SS2.p1.24.m21.1.2.cmml" xref="S6.SS2.p1.24.m21.1.2"><eq id="S6.SS2.p1.24.m21.1.2.1.cmml" xref="S6.SS2.p1.24.m21.1.2.1"></eq><apply id="S6.SS2.p1.24.m21.1.2.2.cmml" xref="S6.SS2.p1.24.m21.1.2.2"><times id="S6.SS2.p1.24.m21.1.2.2.1.cmml" xref="S6.SS2.p1.24.m21.1.2.2.1"></times><apply id="S6.SS2.p1.24.m21.1.2.2.2.cmml" xref="S6.SS2.p1.24.m21.1.2.2.2"><ci id="S6.SS2.p1.24.m21.1.2.2.2.1.cmml" xref="S6.SS2.p1.24.m21.1.2.2.2.1">^</ci><ci id="S6.SS2.p1.24.m21.1.2.2.2.2.cmml" xref="S6.SS2.p1.24.m21.1.2.2.2.2">𝜃</ci></apply><cn id="S6.SS2.p1.24.m21.1.1.cmml" type="integer" xref="S6.SS2.p1.24.m21.1.1">0</cn></apply><cn id="S6.SS2.p1.24.m21.1.2.3.cmml" type="integer" xref="S6.SS2.p1.24.m21.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.24.m21.1c">\hat{\theta}(0)=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.24.m21.1d">over^ start_ARG italic_θ end_ARG ( 0 ) = 0</annotation></semantics></math>, and <math alttext="\sigma=10^{-4}" class="ltx_Math" display="inline" id="S6.SS2.p1.25.m22.1"><semantics id="S6.SS2.p1.25.m22.1a"><mrow id="S6.SS2.p1.25.m22.1.1" xref="S6.SS2.p1.25.m22.1.1.cmml"><mi id="S6.SS2.p1.25.m22.1.1.2" xref="S6.SS2.p1.25.m22.1.1.2.cmml">σ</mi><mo id="S6.SS2.p1.25.m22.1.1.1" xref="S6.SS2.p1.25.m22.1.1.1.cmml">=</mo><msup id="S6.SS2.p1.25.m22.1.1.3" xref="S6.SS2.p1.25.m22.1.1.3.cmml"><mn id="S6.SS2.p1.25.m22.1.1.3.2" xref="S6.SS2.p1.25.m22.1.1.3.2.cmml">10</mn><mrow id="S6.SS2.p1.25.m22.1.1.3.3" xref="S6.SS2.p1.25.m22.1.1.3.3.cmml"><mo id="S6.SS2.p1.25.m22.1.1.3.3a" xref="S6.SS2.p1.25.m22.1.1.3.3.cmml">−</mo><mn id="S6.SS2.p1.25.m22.1.1.3.3.2" xref="S6.SS2.p1.25.m22.1.1.3.3.2.cmml">4</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.25.m22.1b"><apply id="S6.SS2.p1.25.m22.1.1.cmml" xref="S6.SS2.p1.25.m22.1.1"><eq id="S6.SS2.p1.25.m22.1.1.1.cmml" xref="S6.SS2.p1.25.m22.1.1.1"></eq><ci id="S6.SS2.p1.25.m22.1.1.2.cmml" xref="S6.SS2.p1.25.m22.1.1.2">𝜎</ci><apply id="S6.SS2.p1.25.m22.1.1.3.cmml" xref="S6.SS2.p1.25.m22.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.p1.25.m22.1.1.3.1.cmml" xref="S6.SS2.p1.25.m22.1.1.3">superscript</csymbol><cn id="S6.SS2.p1.25.m22.1.1.3.2.cmml" type="integer" xref="S6.SS2.p1.25.m22.1.1.3.2">10</cn><apply id="S6.SS2.p1.25.m22.1.1.3.3.cmml" xref="S6.SS2.p1.25.m22.1.1.3.3"><minus id="S6.SS2.p1.25.m22.1.1.3.3.1.cmml" xref="S6.SS2.p1.25.m22.1.1.3.3"></minus><cn id="S6.SS2.p1.25.m22.1.1.3.3.2.cmml" type="integer" xref="S6.SS2.p1.25.m22.1.1.3.3.2">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.25.m22.1c">\sigma=10^{-4}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.25.m22.1d">italic_σ = 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math>, and the stable filter <math alttext="H(s)=25/(s+25)" class="ltx_Math" display="inline" id="S6.SS2.p1.26.m23.2"><semantics id="S6.SS2.p1.26.m23.2a"><mrow id="S6.SS2.p1.26.m23.2.2" xref="S6.SS2.p1.26.m23.2.2.cmml"><mrow id="S6.SS2.p1.26.m23.2.2.3" xref="S6.SS2.p1.26.m23.2.2.3.cmml"><mi id="S6.SS2.p1.26.m23.2.2.3.2" xref="S6.SS2.p1.26.m23.2.2.3.2.cmml">H</mi><mo id="S6.SS2.p1.26.m23.2.2.3.1" xref="S6.SS2.p1.26.m23.2.2.3.1.cmml">⁢</mo><mrow id="S6.SS2.p1.26.m23.2.2.3.3.2" xref="S6.SS2.p1.26.m23.2.2.3.cmml"><mo id="S6.SS2.p1.26.m23.2.2.3.3.2.1" stretchy="false" xref="S6.SS2.p1.26.m23.2.2.3.cmml">(</mo><mi id="S6.SS2.p1.26.m23.1.1" xref="S6.SS2.p1.26.m23.1.1.cmml">s</mi><mo id="S6.SS2.p1.26.m23.2.2.3.3.2.2" stretchy="false" xref="S6.SS2.p1.26.m23.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.p1.26.m23.2.2.2" xref="S6.SS2.p1.26.m23.2.2.2.cmml">=</mo><mrow id="S6.SS2.p1.26.m23.2.2.1" xref="S6.SS2.p1.26.m23.2.2.1.cmml"><mn id="S6.SS2.p1.26.m23.2.2.1.3" xref="S6.SS2.p1.26.m23.2.2.1.3.cmml">25</mn><mo id="S6.SS2.p1.26.m23.2.2.1.2" xref="S6.SS2.p1.26.m23.2.2.1.2.cmml">/</mo><mrow id="S6.SS2.p1.26.m23.2.2.1.1.1" xref="S6.SS2.p1.26.m23.2.2.1.1.1.1.cmml"><mo id="S6.SS2.p1.26.m23.2.2.1.1.1.2" stretchy="false" xref="S6.SS2.p1.26.m23.2.2.1.1.1.1.cmml">(</mo><mrow id="S6.SS2.p1.26.m23.2.2.1.1.1.1" xref="S6.SS2.p1.26.m23.2.2.1.1.1.1.cmml"><mi id="S6.SS2.p1.26.m23.2.2.1.1.1.1.2" xref="S6.SS2.p1.26.m23.2.2.1.1.1.1.2.cmml">s</mi><mo id="S6.SS2.p1.26.m23.2.2.1.1.1.1.1" xref="S6.SS2.p1.26.m23.2.2.1.1.1.1.1.cmml">+</mo><mn id="S6.SS2.p1.26.m23.2.2.1.1.1.1.3" xref="S6.SS2.p1.26.m23.2.2.1.1.1.1.3.cmml">25</mn></mrow><mo id="S6.SS2.p1.26.m23.2.2.1.1.1.3" stretchy="false" xref="S6.SS2.p1.26.m23.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.26.m23.2b"><apply id="S6.SS2.p1.26.m23.2.2.cmml" xref="S6.SS2.p1.26.m23.2.2"><eq id="S6.SS2.p1.26.m23.2.2.2.cmml" xref="S6.SS2.p1.26.m23.2.2.2"></eq><apply id="S6.SS2.p1.26.m23.2.2.3.cmml" xref="S6.SS2.p1.26.m23.2.2.3"><times id="S6.SS2.p1.26.m23.2.2.3.1.cmml" xref="S6.SS2.p1.26.m23.2.2.3.1"></times><ci id="S6.SS2.p1.26.m23.2.2.3.2.cmml" xref="S6.SS2.p1.26.m23.2.2.3.2">𝐻</ci><ci id="S6.SS2.p1.26.m23.1.1.cmml" xref="S6.SS2.p1.26.m23.1.1">𝑠</ci></apply><apply id="S6.SS2.p1.26.m23.2.2.1.cmml" xref="S6.SS2.p1.26.m23.2.2.1"><divide id="S6.SS2.p1.26.m23.2.2.1.2.cmml" xref="S6.SS2.p1.26.m23.2.2.1.2"></divide><cn id="S6.SS2.p1.26.m23.2.2.1.3.cmml" type="integer" xref="S6.SS2.p1.26.m23.2.2.1.3">25</cn><apply id="S6.SS2.p1.26.m23.2.2.1.1.1.1.cmml" xref="S6.SS2.p1.26.m23.2.2.1.1.1"><plus id="S6.SS2.p1.26.m23.2.2.1.1.1.1.1.cmml" xref="S6.SS2.p1.26.m23.2.2.1.1.1.1.1"></plus><ci id="S6.SS2.p1.26.m23.2.2.1.1.1.1.2.cmml" xref="S6.SS2.p1.26.m23.2.2.1.1.1.1.2">𝑠</ci><cn id="S6.SS2.p1.26.m23.2.2.1.1.1.1.3.cmml" type="integer" xref="S6.SS2.p1.26.m23.2.2.1.1.1.1.3">25</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.26.m23.2c">H(s)=25/(s+25)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.26.m23.2d">italic_H ( italic_s ) = 25 / ( italic_s + 25 )</annotation></semantics></math>. The HOT adaptive backstepping control (HOT-ABC) in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib23" title="">23</a>]</cite> and the MRE-HOT in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib24" title="">24</a>]</cite> are selected as baseline controllers, where their shared parameters are set to be the same values for fair comparisons.</p> </div> <div class="ltx_para" id="S6.SS2.p2"> <p class="ltx_p" id="S6.SS2.p2.16">Performance comparisons of transient tracking and parameter convergence of the three controllers are exhibited in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S6.F5" title="Figure 5 ‣ VI-B Transient Performance Comparisons ‣ VI Simulation Studies ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">5</span></a>, where both the damping parameters <math alttext="k_{\rm{d}\it 1}" class="ltx_Math" display="inline" id="S6.SS2.p2.1.m1.1"><semantics id="S6.SS2.p2.1.m1.1a"><msub id="S6.SS2.p2.1.m1.1.1" xref="S6.SS2.p2.1.m1.1.1.cmml"><mi id="S6.SS2.p2.1.m1.1.1.2" xref="S6.SS2.p2.1.m1.1.1.2.cmml">k</mi><mrow id="S6.SS2.p2.1.m1.1.1.3" xref="S6.SS2.p2.1.m1.1.1.3.cmml"><mi id="S6.SS2.p2.1.m1.1.1.3.2" mathvariant="normal" xref="S6.SS2.p2.1.m1.1.1.3.2.cmml">d</mi><mo id="S6.SS2.p2.1.m1.1.1.3.1" xref="S6.SS2.p2.1.m1.1.1.3.1.cmml">⁢</mo><mn class="ltx_mathvariant_italic" id="S6.SS2.p2.1.m1.1.1.3.3" mathvariant="italic" xref="S6.SS2.p2.1.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.1.m1.1b"><apply id="S6.SS2.p2.1.m1.1.1.cmml" xref="S6.SS2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S6.SS2.p2.1.m1.1.1.1.cmml" xref="S6.SS2.p2.1.m1.1.1">subscript</csymbol><ci id="S6.SS2.p2.1.m1.1.1.2.cmml" xref="S6.SS2.p2.1.m1.1.1.2">𝑘</ci><apply id="S6.SS2.p2.1.m1.1.1.3.cmml" xref="S6.SS2.p2.1.m1.1.1.3"><times id="S6.SS2.p2.1.m1.1.1.3.1.cmml" xref="S6.SS2.p2.1.m1.1.1.3.1"></times><ci id="S6.SS2.p2.1.m1.1.1.3.2.cmml" xref="S6.SS2.p2.1.m1.1.1.3.2">d</ci><cn id="S6.SS2.p2.1.m1.1.1.3.3.cmml" type="integer" xref="S6.SS2.p2.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.1.m1.1c">k_{\rm{d}\it 1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.1.m1.1d">italic_k start_POSTSUBSCRIPT roman_d italic_1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="k_{\rm{d}\it 2}" class="ltx_Math" display="inline" id="S6.SS2.p2.2.m2.1"><semantics id="S6.SS2.p2.2.m2.1a"><msub id="S6.SS2.p2.2.m2.1.1" xref="S6.SS2.p2.2.m2.1.1.cmml"><mi id="S6.SS2.p2.2.m2.1.1.2" xref="S6.SS2.p2.2.m2.1.1.2.cmml">k</mi><mrow id="S6.SS2.p2.2.m2.1.1.3" xref="S6.SS2.p2.2.m2.1.1.3.cmml"><mi id="S6.SS2.p2.2.m2.1.1.3.2" mathvariant="normal" xref="S6.SS2.p2.2.m2.1.1.3.2.cmml">d</mi><mo id="S6.SS2.p2.2.m2.1.1.3.1" xref="S6.SS2.p2.2.m2.1.1.3.1.cmml">⁢</mo><mn class="ltx_mathvariant_italic" id="S6.SS2.p2.2.m2.1.1.3.3" mathvariant="italic" xref="S6.SS2.p2.2.m2.1.1.3.3.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.2.m2.1b"><apply id="S6.SS2.p2.2.m2.1.1.cmml" xref="S6.SS2.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S6.SS2.p2.2.m2.1.1.1.cmml" xref="S6.SS2.p2.2.m2.1.1">subscript</csymbol><ci id="S6.SS2.p2.2.m2.1.1.2.cmml" xref="S6.SS2.p2.2.m2.1.1.2">𝑘</ci><apply id="S6.SS2.p2.2.m2.1.1.3.cmml" xref="S6.SS2.p2.2.m2.1.1.3"><times id="S6.SS2.p2.2.m2.1.1.3.1.cmml" xref="S6.SS2.p2.2.m2.1.1.3.1"></times><ci id="S6.SS2.p2.2.m2.1.1.3.2.cmml" xref="S6.SS2.p2.2.m2.1.1.3.2">d</ci><cn id="S6.SS2.p2.2.m2.1.1.3.3.cmml" type="integer" xref="S6.SS2.p2.2.m2.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.2.m2.1c">k_{\rm{d}\it 2}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.2.m2.1d">italic_k start_POSTSUBSCRIPT roman_d italic_2 end_POSTSUBSCRIPT</annotation></semantics></math> increase from <math alttext="0.01" class="ltx_Math" display="inline" id="S6.SS2.p2.3.m3.1"><semantics id="S6.SS2.p2.3.m3.1a"><mn id="S6.SS2.p2.3.m3.1.1" xref="S6.SS2.p2.3.m3.1.1.cmml">0.01</mn><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.3.m3.1b"><cn id="S6.SS2.p2.3.m3.1.1.cmml" type="float" xref="S6.SS2.p2.3.m3.1.1">0.01</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.3.m3.1c">0.01</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.3.m3.1d">0.01</annotation></semantics></math> to <math alttext="0.19" class="ltx_Math" display="inline" id="S6.SS2.p2.4.m4.1"><semantics id="S6.SS2.p2.4.m4.1a"><mn id="S6.SS2.p2.4.m4.1.1" xref="S6.SS2.p2.4.m4.1.1.cmml">0.19</mn><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.4.m4.1b"><cn id="S6.SS2.p2.4.m4.1.1.cmml" type="float" xref="S6.SS2.p2.4.m4.1.1">0.19</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.4.m4.1c">0.19</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.4.m4.1d">0.19</annotation></semantics></math> with a step size of <math alttext="0.03" class="ltx_Math" display="inline" id="S6.SS2.p2.5.m5.1"><semantics id="S6.SS2.p2.5.m5.1a"><mn id="S6.SS2.p2.5.m5.1.1" xref="S6.SS2.p2.5.m5.1.1.cmml">0.03</mn><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.5.m5.1b"><cn id="S6.SS2.p2.5.m5.1.1.cmml" type="float" xref="S6.SS2.p2.5.m5.1.1">0.03</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.5.m5.1c">0.03</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.5.m5.1d">0.03</annotation></semantics></math>, and the arrows indicate the increasing direction <math alttext="k_{\rm{d}\it 1}" class="ltx_Math" display="inline" id="S6.SS2.p2.6.m6.1"><semantics id="S6.SS2.p2.6.m6.1a"><msub id="S6.SS2.p2.6.m6.1.1" xref="S6.SS2.p2.6.m6.1.1.cmml"><mi id="S6.SS2.p2.6.m6.1.1.2" xref="S6.SS2.p2.6.m6.1.1.2.cmml">k</mi><mrow id="S6.SS2.p2.6.m6.1.1.3" xref="S6.SS2.p2.6.m6.1.1.3.cmml"><mi id="S6.SS2.p2.6.m6.1.1.3.2" mathvariant="normal" xref="S6.SS2.p2.6.m6.1.1.3.2.cmml">d</mi><mo id="S6.SS2.p2.6.m6.1.1.3.1" xref="S6.SS2.p2.6.m6.1.1.3.1.cmml">⁢</mo><mn class="ltx_mathvariant_italic" id="S6.SS2.p2.6.m6.1.1.3.3" mathvariant="italic" xref="S6.SS2.p2.6.m6.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.6.m6.1b"><apply id="S6.SS2.p2.6.m6.1.1.cmml" xref="S6.SS2.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S6.SS2.p2.6.m6.1.1.1.cmml" xref="S6.SS2.p2.6.m6.1.1">subscript</csymbol><ci id="S6.SS2.p2.6.m6.1.1.2.cmml" xref="S6.SS2.p2.6.m6.1.1.2">𝑘</ci><apply id="S6.SS2.p2.6.m6.1.1.3.cmml" xref="S6.SS2.p2.6.m6.1.1.3"><times id="S6.SS2.p2.6.m6.1.1.3.1.cmml" xref="S6.SS2.p2.6.m6.1.1.3.1"></times><ci id="S6.SS2.p2.6.m6.1.1.3.2.cmml" xref="S6.SS2.p2.6.m6.1.1.3.2">d</ci><cn id="S6.SS2.p2.6.m6.1.1.3.3.cmml" type="integer" xref="S6.SS2.p2.6.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.6.m6.1c">k_{\rm{d}\it 1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.6.m6.1d">italic_k start_POSTSUBSCRIPT roman_d italic_1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="k_{\rm{d}\it 2}" class="ltx_Math" display="inline" id="S6.SS2.p2.7.m7.1"><semantics id="S6.SS2.p2.7.m7.1a"><msub id="S6.SS2.p2.7.m7.1.1" xref="S6.SS2.p2.7.m7.1.1.cmml"><mi id="S6.SS2.p2.7.m7.1.1.2" xref="S6.SS2.p2.7.m7.1.1.2.cmml">k</mi><mrow id="S6.SS2.p2.7.m7.1.1.3" xref="S6.SS2.p2.7.m7.1.1.3.cmml"><mi id="S6.SS2.p2.7.m7.1.1.3.2" mathvariant="normal" xref="S6.SS2.p2.7.m7.1.1.3.2.cmml">d</mi><mo id="S6.SS2.p2.7.m7.1.1.3.1" xref="S6.SS2.p2.7.m7.1.1.3.1.cmml">⁢</mo><mn class="ltx_mathvariant_italic" id="S6.SS2.p2.7.m7.1.1.3.3" mathvariant="italic" xref="S6.SS2.p2.7.m7.1.1.3.3.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.7.m7.1b"><apply id="S6.SS2.p2.7.m7.1.1.cmml" xref="S6.SS2.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S6.SS2.p2.7.m7.1.1.1.cmml" xref="S6.SS2.p2.7.m7.1.1">subscript</csymbol><ci id="S6.SS2.p2.7.m7.1.1.2.cmml" xref="S6.SS2.p2.7.m7.1.1.2">𝑘</ci><apply id="S6.SS2.p2.7.m7.1.1.3.cmml" xref="S6.SS2.p2.7.m7.1.1.3"><times id="S6.SS2.p2.7.m7.1.1.3.1.cmml" xref="S6.SS2.p2.7.m7.1.1.3.1"></times><ci id="S6.SS2.p2.7.m7.1.1.3.2.cmml" xref="S6.SS2.p2.7.m7.1.1.3.2">d</ci><cn id="S6.SS2.p2.7.m7.1.1.3.3.cmml" type="integer" xref="S6.SS2.p2.7.m7.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.7.m7.1c">k_{\rm{d}\it 2}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.7.m7.1d">italic_k start_POSTSUBSCRIPT roman_d italic_2 end_POSTSUBSCRIPT</annotation></semantics></math>. It is observed that the transient results of the HOT-ABC and MRE-HOT are deteriorated by decreasing <math alttext="k_{\rm{d}\it 1}" class="ltx_Math" display="inline" id="S6.SS2.p2.8.m8.1"><semantics id="S6.SS2.p2.8.m8.1a"><msub id="S6.SS2.p2.8.m8.1.1" xref="S6.SS2.p2.8.m8.1.1.cmml"><mi id="S6.SS2.p2.8.m8.1.1.2" xref="S6.SS2.p2.8.m8.1.1.2.cmml">k</mi><mrow id="S6.SS2.p2.8.m8.1.1.3" xref="S6.SS2.p2.8.m8.1.1.3.cmml"><mi id="S6.SS2.p2.8.m8.1.1.3.2" mathvariant="normal" xref="S6.SS2.p2.8.m8.1.1.3.2.cmml">d</mi><mo id="S6.SS2.p2.8.m8.1.1.3.1" xref="S6.SS2.p2.8.m8.1.1.3.1.cmml">⁢</mo><mn class="ltx_mathvariant_italic" id="S6.SS2.p2.8.m8.1.1.3.3" mathvariant="italic" xref="S6.SS2.p2.8.m8.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.8.m8.1b"><apply id="S6.SS2.p2.8.m8.1.1.cmml" xref="S6.SS2.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S6.SS2.p2.8.m8.1.1.1.cmml" xref="S6.SS2.p2.8.m8.1.1">subscript</csymbol><ci id="S6.SS2.p2.8.m8.1.1.2.cmml" xref="S6.SS2.p2.8.m8.1.1.2">𝑘</ci><apply id="S6.SS2.p2.8.m8.1.1.3.cmml" xref="S6.SS2.p2.8.m8.1.1.3"><times id="S6.SS2.p2.8.m8.1.1.3.1.cmml" xref="S6.SS2.p2.8.m8.1.1.3.1"></times><ci id="S6.SS2.p2.8.m8.1.1.3.2.cmml" xref="S6.SS2.p2.8.m8.1.1.3.2">d</ci><cn id="S6.SS2.p2.8.m8.1.1.3.3.cmml" type="integer" xref="S6.SS2.p2.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.8.m8.1c">k_{\rm{d}\it 1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.8.m8.1d">italic_k start_POSTSUBSCRIPT roman_d italic_1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="k_{\rm{d}\it 2}" class="ltx_Math" display="inline" id="S6.SS2.p2.9.m9.1"><semantics id="S6.SS2.p2.9.m9.1a"><msub id="S6.SS2.p2.9.m9.1.1" xref="S6.SS2.p2.9.m9.1.1.cmml"><mi id="S6.SS2.p2.9.m9.1.1.2" xref="S6.SS2.p2.9.m9.1.1.2.cmml">k</mi><mrow id="S6.SS2.p2.9.m9.1.1.3" xref="S6.SS2.p2.9.m9.1.1.3.cmml"><mi id="S6.SS2.p2.9.m9.1.1.3.2" mathvariant="normal" xref="S6.SS2.p2.9.m9.1.1.3.2.cmml">d</mi><mo id="S6.SS2.p2.9.m9.1.1.3.1" xref="S6.SS2.p2.9.m9.1.1.3.1.cmml">⁢</mo><mn class="ltx_mathvariant_italic" id="S6.SS2.p2.9.m9.1.1.3.3" mathvariant="italic" xref="S6.SS2.p2.9.m9.1.1.3.3.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.9.m9.1b"><apply id="S6.SS2.p2.9.m9.1.1.cmml" xref="S6.SS2.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S6.SS2.p2.9.m9.1.1.1.cmml" xref="S6.SS2.p2.9.m9.1.1">subscript</csymbol><ci id="S6.SS2.p2.9.m9.1.1.2.cmml" xref="S6.SS2.p2.9.m9.1.1.2">𝑘</ci><apply id="S6.SS2.p2.9.m9.1.1.3.cmml" xref="S6.SS2.p2.9.m9.1.1.3"><times id="S6.SS2.p2.9.m9.1.1.3.1.cmml" xref="S6.SS2.p2.9.m9.1.1.3.1"></times><ci id="S6.SS2.p2.9.m9.1.1.3.2.cmml" xref="S6.SS2.p2.9.m9.1.1.3.2">d</ci><cn id="S6.SS2.p2.9.m9.1.1.3.3.cmml" type="integer" xref="S6.SS2.p2.9.m9.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.9.m9.1c">k_{\rm{d}\it 2}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.9.m9.1d">italic_k start_POSTSUBSCRIPT roman_d italic_2 end_POSTSUBSCRIPT</annotation></semantics></math> [see green and red dash lines in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S6.F5" title="Figure 5 ‣ VI-B Transient Performance Comparisons ‣ VI Simulation Studies ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">5</span></a>(a)]. For the MRE-HOT, increasing <math alttext="k_{\rm{d}\it 1}" class="ltx_Math" display="inline" id="S6.SS2.p2.10.m10.1"><semantics id="S6.SS2.p2.10.m10.1a"><msub id="S6.SS2.p2.10.m10.1.1" xref="S6.SS2.p2.10.m10.1.1.cmml"><mi id="S6.SS2.p2.10.m10.1.1.2" xref="S6.SS2.p2.10.m10.1.1.2.cmml">k</mi><mrow id="S6.SS2.p2.10.m10.1.1.3" xref="S6.SS2.p2.10.m10.1.1.3.cmml"><mi id="S6.SS2.p2.10.m10.1.1.3.2" mathvariant="normal" xref="S6.SS2.p2.10.m10.1.1.3.2.cmml">d</mi><mo id="S6.SS2.p2.10.m10.1.1.3.1" xref="S6.SS2.p2.10.m10.1.1.3.1.cmml">⁢</mo><mn class="ltx_mathvariant_italic" id="S6.SS2.p2.10.m10.1.1.3.3" mathvariant="italic" xref="S6.SS2.p2.10.m10.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.10.m10.1b"><apply id="S6.SS2.p2.10.m10.1.1.cmml" xref="S6.SS2.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S6.SS2.p2.10.m10.1.1.1.cmml" xref="S6.SS2.p2.10.m10.1.1">subscript</csymbol><ci id="S6.SS2.p2.10.m10.1.1.2.cmml" xref="S6.SS2.p2.10.m10.1.1.2">𝑘</ci><apply id="S6.SS2.p2.10.m10.1.1.3.cmml" xref="S6.SS2.p2.10.m10.1.1.3"><times id="S6.SS2.p2.10.m10.1.1.3.1.cmml" xref="S6.SS2.p2.10.m10.1.1.3.1"></times><ci id="S6.SS2.p2.10.m10.1.1.3.2.cmml" xref="S6.SS2.p2.10.m10.1.1.3.2">d</ci><cn id="S6.SS2.p2.10.m10.1.1.3.3.cmml" type="integer" xref="S6.SS2.p2.10.m10.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.10.m10.1c">k_{\rm{d}\it 1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.10.m10.1d">italic_k start_POSTSUBSCRIPT roman_d italic_1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="k_{\rm{d}\it 2}" class="ltx_Math" display="inline" id="S6.SS2.p2.11.m11.1"><semantics id="S6.SS2.p2.11.m11.1a"><msub id="S6.SS2.p2.11.m11.1.1" xref="S6.SS2.p2.11.m11.1.1.cmml"><mi id="S6.SS2.p2.11.m11.1.1.2" xref="S6.SS2.p2.11.m11.1.1.2.cmml">k</mi><mrow id="S6.SS2.p2.11.m11.1.1.3" xref="S6.SS2.p2.11.m11.1.1.3.cmml"><mi id="S6.SS2.p2.11.m11.1.1.3.2" mathvariant="normal" xref="S6.SS2.p2.11.m11.1.1.3.2.cmml">d</mi><mo id="S6.SS2.p2.11.m11.1.1.3.1" xref="S6.SS2.p2.11.m11.1.1.3.1.cmml">⁢</mo><mn class="ltx_mathvariant_italic" id="S6.SS2.p2.11.m11.1.1.3.3" mathvariant="italic" xref="S6.SS2.p2.11.m11.1.1.3.3.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.11.m11.1b"><apply id="S6.SS2.p2.11.m11.1.1.cmml" xref="S6.SS2.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S6.SS2.p2.11.m11.1.1.1.cmml" xref="S6.SS2.p2.11.m11.1.1">subscript</csymbol><ci id="S6.SS2.p2.11.m11.1.1.2.cmml" xref="S6.SS2.p2.11.m11.1.1.2">𝑘</ci><apply id="S6.SS2.p2.11.m11.1.1.3.cmml" xref="S6.SS2.p2.11.m11.1.1.3"><times id="S6.SS2.p2.11.m11.1.1.3.1.cmml" xref="S6.SS2.p2.11.m11.1.1.3.1"></times><ci id="S6.SS2.p2.11.m11.1.1.3.2.cmml" xref="S6.SS2.p2.11.m11.1.1.3.2">d</ci><cn id="S6.SS2.p2.11.m11.1.1.3.3.cmml" type="integer" xref="S6.SS2.p2.11.m11.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.11.m11.1c">k_{\rm{d}\it 2}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.11.m11.1d">italic_k start_POSTSUBSCRIPT roman_d italic_2 end_POSTSUBSCRIPT</annotation></semantics></math> reduces the initial oscillations of the tracking error <math alttext="|e_{1}|" class="ltx_Math" display="inline" id="S6.SS2.p2.12.m12.1"><semantics id="S6.SS2.p2.12.m12.1a"><mrow id="S6.SS2.p2.12.m12.1.1.1" xref="S6.SS2.p2.12.m12.1.1.2.cmml"><mo id="S6.SS2.p2.12.m12.1.1.1.2" stretchy="false" xref="S6.SS2.p2.12.m12.1.1.2.1.cmml">|</mo><msub id="S6.SS2.p2.12.m12.1.1.1.1" xref="S6.SS2.p2.12.m12.1.1.1.1.cmml"><mi id="S6.SS2.p2.12.m12.1.1.1.1.2" xref="S6.SS2.p2.12.m12.1.1.1.1.2.cmml">e</mi><mn id="S6.SS2.p2.12.m12.1.1.1.1.3" xref="S6.SS2.p2.12.m12.1.1.1.1.3.cmml">1</mn></msub><mo id="S6.SS2.p2.12.m12.1.1.1.3" stretchy="false" xref="S6.SS2.p2.12.m12.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.12.m12.1b"><apply id="S6.SS2.p2.12.m12.1.1.2.cmml" xref="S6.SS2.p2.12.m12.1.1.1"><abs id="S6.SS2.p2.12.m12.1.1.2.1.cmml" xref="S6.SS2.p2.12.m12.1.1.1.2"></abs><apply id="S6.SS2.p2.12.m12.1.1.1.1.cmml" xref="S6.SS2.p2.12.m12.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.p2.12.m12.1.1.1.1.1.cmml" xref="S6.SS2.p2.12.m12.1.1.1.1">subscript</csymbol><ci id="S6.SS2.p2.12.m12.1.1.1.1.2.cmml" xref="S6.SS2.p2.12.m12.1.1.1.1.2">𝑒</ci><cn id="S6.SS2.p2.12.m12.1.1.1.1.3.cmml" type="integer" xref="S6.SS2.p2.12.m12.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.12.m12.1c">|e_{1}|</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.12.m12.1d">| italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT |</annotation></semantics></math> [see green dash lines in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S6.F5" title="Figure 5 ‣ VI-B Transient Performance Comparisons ‣ VI Simulation Studies ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">5</span></a>(a)] but also slows down parameter convergence [see green dash lines in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S6.F5" title="Figure 5 ‣ VI-B Transient Performance Comparisons ‣ VI Simulation Studies ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">5</span></a>(b)]. The proposed CLBC provides the convergence of <math alttext="|e_{1}|" class="ltx_Math" display="inline" id="S6.SS2.p2.13.m13.1"><semantics id="S6.SS2.p2.13.m13.1a"><mrow id="S6.SS2.p2.13.m13.1.1.1" xref="S6.SS2.p2.13.m13.1.1.2.cmml"><mo id="S6.SS2.p2.13.m13.1.1.1.2" stretchy="false" xref="S6.SS2.p2.13.m13.1.1.2.1.cmml">|</mo><msub id="S6.SS2.p2.13.m13.1.1.1.1" xref="S6.SS2.p2.13.m13.1.1.1.1.cmml"><mi id="S6.SS2.p2.13.m13.1.1.1.1.2" xref="S6.SS2.p2.13.m13.1.1.1.1.2.cmml">e</mi><mn id="S6.SS2.p2.13.m13.1.1.1.1.3" xref="S6.SS2.p2.13.m13.1.1.1.1.3.cmml">1</mn></msub><mo id="S6.SS2.p2.13.m13.1.1.1.3" stretchy="false" xref="S6.SS2.p2.13.m13.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.13.m13.1b"><apply id="S6.SS2.p2.13.m13.1.1.2.cmml" xref="S6.SS2.p2.13.m13.1.1.1"><abs id="S6.SS2.p2.13.m13.1.1.2.1.cmml" xref="S6.SS2.p2.13.m13.1.1.1.2"></abs><apply id="S6.SS2.p2.13.m13.1.1.1.1.cmml" xref="S6.SS2.p2.13.m13.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.p2.13.m13.1.1.1.1.1.cmml" xref="S6.SS2.p2.13.m13.1.1.1.1">subscript</csymbol><ci id="S6.SS2.p2.13.m13.1.1.1.1.2.cmml" xref="S6.SS2.p2.13.m13.1.1.1.1.2">𝑒</ci><cn id="S6.SS2.p2.13.m13.1.1.1.1.3.cmml" type="integer" xref="S6.SS2.p2.13.m13.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.13.m13.1c">|e_{1}|</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.13.m13.1d">| italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT |</annotation></semantics></math> to 0 with the smallest initial oscillations [see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S6.F5" title="Figure 5 ‣ VI-B Transient Performance Comparisons ‣ VI Simulation Studies ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">5</span></a>(a)] and achieves the fastest parameter convergence [see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S6.F5" title="Figure 5 ‣ VI-B Transient Performance Comparisons ‣ VI Simulation Studies ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">5</span></a>(b)], which implies that: 1) the transient tracking and parameter convergence of the HOT-ABC and MRE-HOT are sensitive to damping parameters; 2) the transient performance of the proposed CLBC is enhanced by introducing the extra prediction error <math alttext="\bm{\epsilon}" class="ltx_Math" display="inline" id="S6.SS2.p2.14.m14.1"><semantics id="S6.SS2.p2.14.m14.1a"><mi class="ltx_mathvariant_bold-italic" id="S6.SS2.p2.14.m14.1.1" mathvariant="bold-italic" xref="S6.SS2.p2.14.m14.1.1.cmml">ϵ</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.14.m14.1b"><ci id="S6.SS2.p2.14.m14.1.1.cmml" xref="S6.SS2.p2.14.m14.1.1">bold-italic-ϵ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.14.m14.1c">\bm{\epsilon}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.14.m14.1d">bold_italic_ϵ</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E23" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">23</span></a>) without resorting to nonlinear damping terms; 3) The combination of two prediction errors <math alttext="\bm{\xi}" class="ltx_Math" display="inline" id="S6.SS2.p2.15.m15.1"><semantics id="S6.SS2.p2.15.m15.1a"><mi id="S6.SS2.p2.15.m15.1.1" xref="S6.SS2.p2.15.m15.1.1.cmml">𝝃</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.15.m15.1b"><ci id="S6.SS2.p2.15.m15.1.1.cmml" xref="S6.SS2.p2.15.m15.1.1">𝝃</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.15.m15.1c">\bm{\xi}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.15.m15.1d">bold_italic_ξ</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E20" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">20</span></a>) and <math alttext="\bm{\epsilon}" class="ltx_Math" display="inline" id="S6.SS2.p2.16.m16.1"><semantics id="S6.SS2.p2.16.m16.1a"><mi class="ltx_mathvariant_bold-italic" id="S6.SS2.p2.16.m16.1.1" mathvariant="bold-italic" xref="S6.SS2.p2.16.m16.1.1.cmml">ϵ</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.16.m16.1b"><ci id="S6.SS2.p2.16.m16.1.1.cmml" xref="S6.SS2.p2.16.m16.1.1">bold-italic-ϵ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.16.m16.1c">\bm{\epsilon}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.16.m16.1d">bold_italic_ϵ</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E23" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">23</span></a>) accelerates parameter convergence.</p> </div> <figure class="ltx_figure" id="S6.F5"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="205" id="S6.F5.g1" src="x5.png" width="470"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 5: </span>Performance comparisons of three controllers for the tracking problem under different values of the damping parameters <math alttext="k_{\rm{d}\it 1}" class="ltx_Math" display="inline" id="S6.F5.7.m1.1"><semantics id="S6.F5.7.m1.1b"><msub id="S6.F5.7.m1.1.1" xref="S6.F5.7.m1.1.1.cmml"><mi id="S6.F5.7.m1.1.1.2" xref="S6.F5.7.m1.1.1.2.cmml">k</mi><mrow id="S6.F5.7.m1.1.1.3" xref="S6.F5.7.m1.1.1.3.cmml"><mi id="S6.F5.7.m1.1.1.3.2" mathvariant="normal" xref="S6.F5.7.m1.1.1.3.2.cmml">d</mi><mo id="S6.F5.7.m1.1.1.3.1" xref="S6.F5.7.m1.1.1.3.1.cmml">⁢</mo><mn class="ltx_mathvariant_italic" id="S6.F5.7.m1.1.1.3.3" mathvariant="italic" xref="S6.F5.7.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.F5.7.m1.1c"><apply id="S6.F5.7.m1.1.1.cmml" xref="S6.F5.7.m1.1.1"><csymbol cd="ambiguous" id="S6.F5.7.m1.1.1.1.cmml" xref="S6.F5.7.m1.1.1">subscript</csymbol><ci id="S6.F5.7.m1.1.1.2.cmml" xref="S6.F5.7.m1.1.1.2">𝑘</ci><apply id="S6.F5.7.m1.1.1.3.cmml" xref="S6.F5.7.m1.1.1.3"><times id="S6.F5.7.m1.1.1.3.1.cmml" xref="S6.F5.7.m1.1.1.3.1"></times><ci id="S6.F5.7.m1.1.1.3.2.cmml" xref="S6.F5.7.m1.1.1.3.2">d</ci><cn id="S6.F5.7.m1.1.1.3.3.cmml" type="integer" xref="S6.F5.7.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F5.7.m1.1d">k_{\rm{d}\it 1}</annotation><annotation encoding="application/x-llamapun" id="S6.F5.7.m1.1e">italic_k start_POSTSUBSCRIPT roman_d italic_1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="k_{\rm{d}\it 2}" class="ltx_Math" display="inline" id="S6.F5.8.m2.1"><semantics id="S6.F5.8.m2.1b"><msub id="S6.F5.8.m2.1.1" xref="S6.F5.8.m2.1.1.cmml"><mi id="S6.F5.8.m2.1.1.2" xref="S6.F5.8.m2.1.1.2.cmml">k</mi><mrow id="S6.F5.8.m2.1.1.3" xref="S6.F5.8.m2.1.1.3.cmml"><mi id="S6.F5.8.m2.1.1.3.2" mathvariant="normal" xref="S6.F5.8.m2.1.1.3.2.cmml">d</mi><mo id="S6.F5.8.m2.1.1.3.1" xref="S6.F5.8.m2.1.1.3.1.cmml">⁢</mo><mn class="ltx_mathvariant_italic" id="S6.F5.8.m2.1.1.3.3" mathvariant="italic" xref="S6.F5.8.m2.1.1.3.3.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.F5.8.m2.1c"><apply id="S6.F5.8.m2.1.1.cmml" xref="S6.F5.8.m2.1.1"><csymbol cd="ambiguous" id="S6.F5.8.m2.1.1.1.cmml" xref="S6.F5.8.m2.1.1">subscript</csymbol><ci id="S6.F5.8.m2.1.1.2.cmml" xref="S6.F5.8.m2.1.1.2">𝑘</ci><apply id="S6.F5.8.m2.1.1.3.cmml" xref="S6.F5.8.m2.1.1.3"><times id="S6.F5.8.m2.1.1.3.1.cmml" xref="S6.F5.8.m2.1.1.3.1"></times><ci id="S6.F5.8.m2.1.1.3.2.cmml" xref="S6.F5.8.m2.1.1.3.2">d</ci><cn id="S6.F5.8.m2.1.1.3.3.cmml" type="integer" xref="S6.F5.8.m2.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F5.8.m2.1d">k_{\rm{d}\it 2}</annotation><annotation encoding="application/x-llamapun" id="S6.F5.8.m2.1e">italic_k start_POSTSUBSCRIPT roman_d italic_2 end_POSTSUBSCRIPT</annotation></semantics></math>. (a) The absolute tracking errors <math alttext="|e_{1}|" class="ltx_Math" display="inline" id="S6.F5.9.m3.1"><semantics id="S6.F5.9.m3.1b"><mrow id="S6.F5.9.m3.1.1.1" xref="S6.F5.9.m3.1.1.2.cmml"><mo id="S6.F5.9.m3.1.1.1.2" stretchy="false" xref="S6.F5.9.m3.1.1.2.1.cmml">|</mo><msub id="S6.F5.9.m3.1.1.1.1" xref="S6.F5.9.m3.1.1.1.1.cmml"><mi id="S6.F5.9.m3.1.1.1.1.2" xref="S6.F5.9.m3.1.1.1.1.2.cmml">e</mi><mn id="S6.F5.9.m3.1.1.1.1.3" xref="S6.F5.9.m3.1.1.1.1.3.cmml">1</mn></msub><mo id="S6.F5.9.m3.1.1.1.3" stretchy="false" xref="S6.F5.9.m3.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.F5.9.m3.1c"><apply id="S6.F5.9.m3.1.1.2.cmml" xref="S6.F5.9.m3.1.1.1"><abs id="S6.F5.9.m3.1.1.2.1.cmml" xref="S6.F5.9.m3.1.1.1.2"></abs><apply id="S6.F5.9.m3.1.1.1.1.cmml" xref="S6.F5.9.m3.1.1.1.1"><csymbol cd="ambiguous" id="S6.F5.9.m3.1.1.1.1.1.cmml" xref="S6.F5.9.m3.1.1.1.1">subscript</csymbol><ci id="S6.F5.9.m3.1.1.1.1.2.cmml" xref="S6.F5.9.m3.1.1.1.1.2">𝑒</ci><cn id="S6.F5.9.m3.1.1.1.1.3.cmml" type="integer" xref="S6.F5.9.m3.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F5.9.m3.1d">|e_{1}|</annotation><annotation encoding="application/x-llamapun" id="S6.F5.9.m3.1e">| italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT |</annotation></semantics></math>. (b) The absolute estimation errors <math alttext="|\tilde{\theta}|" class="ltx_Math" display="inline" id="S6.F5.10.m4.1"><semantics id="S6.F5.10.m4.1b"><mrow id="S6.F5.10.m4.1.2.2" xref="S6.F5.10.m4.1.2.1.cmml"><mo id="S6.F5.10.m4.1.2.2.1" stretchy="false" xref="S6.F5.10.m4.1.2.1.1.cmml">|</mo><mover accent="true" id="S6.F5.10.m4.1.1" xref="S6.F5.10.m4.1.1.cmml"><mi id="S6.F5.10.m4.1.1.2" xref="S6.F5.10.m4.1.1.2.cmml">θ</mi><mo id="S6.F5.10.m4.1.1.1" xref="S6.F5.10.m4.1.1.1.cmml">~</mo></mover><mo id="S6.F5.10.m4.1.2.2.2" stretchy="false" xref="S6.F5.10.m4.1.2.1.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.F5.10.m4.1c"><apply id="S6.F5.10.m4.1.2.1.cmml" xref="S6.F5.10.m4.1.2.2"><abs id="S6.F5.10.m4.1.2.1.1.cmml" xref="S6.F5.10.m4.1.2.2.1"></abs><apply id="S6.F5.10.m4.1.1.cmml" xref="S6.F5.10.m4.1.1"><ci id="S6.F5.10.m4.1.1.1.cmml" xref="S6.F5.10.m4.1.1.1">~</ci><ci id="S6.F5.10.m4.1.1.2.cmml" xref="S6.F5.10.m4.1.1.2">𝜃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F5.10.m4.1d">|\tilde{\theta}|</annotation><annotation encoding="application/x-llamapun" id="S6.F5.10.m4.1e">| over~ start_ARG italic_θ end_ARG |</annotation></semantics></math>. Note that the arrows indicate the increasing direction <math alttext="k_{\rm{d}\it 1}" class="ltx_Math" display="inline" id="S6.F5.11.m5.1"><semantics id="S6.F5.11.m5.1b"><msub id="S6.F5.11.m5.1.1" xref="S6.F5.11.m5.1.1.cmml"><mi id="S6.F5.11.m5.1.1.2" xref="S6.F5.11.m5.1.1.2.cmml">k</mi><mrow id="S6.F5.11.m5.1.1.3" xref="S6.F5.11.m5.1.1.3.cmml"><mi id="S6.F5.11.m5.1.1.3.2" mathvariant="normal" xref="S6.F5.11.m5.1.1.3.2.cmml">d</mi><mo id="S6.F5.11.m5.1.1.3.1" xref="S6.F5.11.m5.1.1.3.1.cmml">⁢</mo><mn class="ltx_mathvariant_italic" id="S6.F5.11.m5.1.1.3.3" mathvariant="italic" xref="S6.F5.11.m5.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.F5.11.m5.1c"><apply id="S6.F5.11.m5.1.1.cmml" xref="S6.F5.11.m5.1.1"><csymbol cd="ambiguous" id="S6.F5.11.m5.1.1.1.cmml" xref="S6.F5.11.m5.1.1">subscript</csymbol><ci id="S6.F5.11.m5.1.1.2.cmml" xref="S6.F5.11.m5.1.1.2">𝑘</ci><apply id="S6.F5.11.m5.1.1.3.cmml" xref="S6.F5.11.m5.1.1.3"><times id="S6.F5.11.m5.1.1.3.1.cmml" xref="S6.F5.11.m5.1.1.3.1"></times><ci id="S6.F5.11.m5.1.1.3.2.cmml" xref="S6.F5.11.m5.1.1.3.2">d</ci><cn id="S6.F5.11.m5.1.1.3.3.cmml" type="integer" xref="S6.F5.11.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F5.11.m5.1d">k_{\rm{d}\it 1}</annotation><annotation encoding="application/x-llamapun" id="S6.F5.11.m5.1e">italic_k start_POSTSUBSCRIPT roman_d italic_1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="k_{\rm{d}\it 2}" class="ltx_Math" display="inline" id="S6.F5.12.m6.1"><semantics id="S6.F5.12.m6.1b"><msub id="S6.F5.12.m6.1.1" xref="S6.F5.12.m6.1.1.cmml"><mi id="S6.F5.12.m6.1.1.2" xref="S6.F5.12.m6.1.1.2.cmml">k</mi><mrow id="S6.F5.12.m6.1.1.3" xref="S6.F5.12.m6.1.1.3.cmml"><mi id="S6.F5.12.m6.1.1.3.2" mathvariant="normal" xref="S6.F5.12.m6.1.1.3.2.cmml">d</mi><mo id="S6.F5.12.m6.1.1.3.1" xref="S6.F5.12.m6.1.1.3.1.cmml">⁢</mo><mn class="ltx_mathvariant_italic" id="S6.F5.12.m6.1.1.3.3" mathvariant="italic" xref="S6.F5.12.m6.1.1.3.3.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.F5.12.m6.1c"><apply id="S6.F5.12.m6.1.1.cmml" xref="S6.F5.12.m6.1.1"><csymbol cd="ambiguous" id="S6.F5.12.m6.1.1.1.cmml" xref="S6.F5.12.m6.1.1">subscript</csymbol><ci id="S6.F5.12.m6.1.1.2.cmml" xref="S6.F5.12.m6.1.1.2">𝑘</ci><apply id="S6.F5.12.m6.1.1.3.cmml" xref="S6.F5.12.m6.1.1.3"><times id="S6.F5.12.m6.1.1.3.1.cmml" xref="S6.F5.12.m6.1.1.3.1"></times><ci id="S6.F5.12.m6.1.1.3.2.cmml" xref="S6.F5.12.m6.1.1.3.2">d</ci><cn id="S6.F5.12.m6.1.1.3.3.cmml" type="integer" xref="S6.F5.12.m6.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F5.12.m6.1d">k_{\rm{d}\it 2}</annotation><annotation encoding="application/x-llamapun" id="S6.F5.12.m6.1e">italic_k start_POSTSUBSCRIPT roman_d italic_2 end_POSTSUBSCRIPT</annotation></semantics></math>.</figcaption> </figure> </section> </section> <section class="ltx_section" id="S7"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">VII </span><span class="ltx_text ltx_font_smallcaps" id="S7.1.1">Conclusions</span> </h2> <div class="ltx_para" id="S7.p1"> <p class="ltx_p" id="S7.p1.1">This paper has presented a feasible modular backstepping control strategy named CLBC for strict-feedback uncertain nonlinear systems. The proposed composite learning HOT allows the exact implementation of the high-order time derivatives of parameter estimates and the offset of modeling errors, such that the transient performance can be guaranteed without resorting to nonlinear damping terms or high control gains. The proposed algorithm of staged exciting strength maximization ensures that the exciting strength is monotonously non-decreasing in each excitation stage, such that the exponential stability of the closed-loop system with parameter convergence is obtainable under the much weaker condition of IE or partial IE. Simulation studies have validated that the proposed CLBC greatly outperforms two state-of-the-art modular backstepping controllers, namely HOT-ABC and MRE-HOT, in both parameter estimation and control. <span class="ltx_text" id="S7.p1.1.1" style="color:#000099;">Further work would focus on a formal analysis of the time-varying parameter case and robot control based on the proposed method.</span></p> </div> </section> <section class="ltx_appendix" id="A1"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix A </span><span class="ltx_text" id="A1.1.1" style="color:#000099;">The derivation of (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E12" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">12</span></a>)</span> </h2> <div class="ltx_para" id="A1.p1"> <p class="ltx_p" id="A1.p1.2"><span class="ltx_text" id="A1.p1.2.2" style="color:#000099;">Differentiating <math alttext="\bm{p}" class="ltx_Math" display="inline" id="A1.p1.1.1.m1.1"><semantics id="A1.p1.1.1.m1.1a"><mi id="A1.p1.1.1.m1.1.1" mathcolor="#000099" xref="A1.p1.1.1.m1.1.1.cmml">𝒑</mi><annotation-xml encoding="MathML-Content" id="A1.p1.1.1.m1.1b"><ci id="A1.p1.1.1.m1.1.1.cmml" xref="A1.p1.1.1.m1.1.1">𝒑</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.1.1.m1.1c">\bm{p}</annotation><annotation encoding="application/x-llamapun" id="A1.p1.1.1.m1.1d">bold_italic_p</annotation></semantics></math> with respect to <math alttext="t" class="ltx_Math" display="inline" id="A1.p1.2.2.m2.1"><semantics id="A1.p1.2.2.m2.1a"><mi id="A1.p1.2.2.m2.1.1" mathcolor="#000099" xref="A1.p1.2.2.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A1.p1.2.2.m2.1b"><ci id="A1.p1.2.2.m2.1.1.cmml" xref="A1.p1.2.2.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.2.2.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="A1.p1.2.2.m2.1d">italic_t</annotation></semantics></math>, and applying (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E7" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">7</span></a>) and the first equation of (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E11" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">11</span></a>) to the resulting expression, one obtains</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx30"> <tbody id="A1.Ex21"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{\bm{p}}(t)" class="ltx_Math" display="inline" id="A1.Ex21.m1.1"><semantics id="A1.Ex21.m1.1a"><mrow id="A1.Ex21.m1.1.2" xref="A1.Ex21.m1.1.2.cmml"><mover accent="true" id="A1.Ex21.m1.1.2.2" xref="A1.Ex21.m1.1.2.2.cmml"><mi id="A1.Ex21.m1.1.2.2.2" mathcolor="#000099" xref="A1.Ex21.m1.1.2.2.2.cmml">𝒑</mi><mo id="A1.Ex21.m1.1.2.2.1" mathcolor="#000099" xref="A1.Ex21.m1.1.2.2.1.cmml">˙</mo></mover><mo id="A1.Ex21.m1.1.2.1" xref="A1.Ex21.m1.1.2.1.cmml">⁢</mo><mrow id="A1.Ex21.m1.1.2.3.2" xref="A1.Ex21.m1.1.2.cmml"><mo id="A1.Ex21.m1.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="A1.Ex21.m1.1.2.cmml">(</mo><mi id="A1.Ex21.m1.1.1" mathcolor="#000099" xref="A1.Ex21.m1.1.1.cmml">t</mi><mo id="A1.Ex21.m1.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="A1.Ex21.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex21.m1.1b"><apply id="A1.Ex21.m1.1.2.cmml" xref="A1.Ex21.m1.1.2"><times id="A1.Ex21.m1.1.2.1.cmml" xref="A1.Ex21.m1.1.2.1"></times><apply id="A1.Ex21.m1.1.2.2.cmml" xref="A1.Ex21.m1.1.2.2"><ci id="A1.Ex21.m1.1.2.2.1.cmml" xref="A1.Ex21.m1.1.2.2.1">˙</ci><ci id="A1.Ex21.m1.1.2.2.2.cmml" xref="A1.Ex21.m1.1.2.2.2">𝒑</ci></apply><ci id="A1.Ex21.m1.1.1.cmml" xref="A1.Ex21.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex21.m1.1c">\displaystyle\dot{\bm{p}}(t)</annotation><annotation encoding="application/x-llamapun" id="A1.Ex21.m1.1d">over˙ start_ARG bold_italic_p end_ARG ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\dot{\bm{e}}(t)+\dot{\bm{\zeta}}(t)" class="ltx_Math" display="inline" id="A1.Ex21.m2.2"><semantics id="A1.Ex21.m2.2a"><mrow id="A1.Ex21.m2.2.3" xref="A1.Ex21.m2.2.3.cmml"><mi id="A1.Ex21.m2.2.3.2" xref="A1.Ex21.m2.2.3.2.cmml"></mi><mo id="A1.Ex21.m2.2.3.1" mathcolor="#000099" xref="A1.Ex21.m2.2.3.1.cmml">=</mo><mrow id="A1.Ex21.m2.2.3.3" xref="A1.Ex21.m2.2.3.3.cmml"><mrow id="A1.Ex21.m2.2.3.3.2" xref="A1.Ex21.m2.2.3.3.2.cmml"><mover accent="true" id="A1.Ex21.m2.2.3.3.2.2" xref="A1.Ex21.m2.2.3.3.2.2.cmml"><mi id="A1.Ex21.m2.2.3.3.2.2.2" mathcolor="#000099" xref="A1.Ex21.m2.2.3.3.2.2.2.cmml">𝒆</mi><mo id="A1.Ex21.m2.2.3.3.2.2.1" mathcolor="#000099" xref="A1.Ex21.m2.2.3.3.2.2.1.cmml">˙</mo></mover><mo id="A1.Ex21.m2.2.3.3.2.1" xref="A1.Ex21.m2.2.3.3.2.1.cmml">⁢</mo><mrow id="A1.Ex21.m2.2.3.3.2.3.2" xref="A1.Ex21.m2.2.3.3.2.cmml"><mo id="A1.Ex21.m2.2.3.3.2.3.2.1" mathcolor="#000099" stretchy="false" xref="A1.Ex21.m2.2.3.3.2.cmml">(</mo><mi id="A1.Ex21.m2.1.1" mathcolor="#000099" xref="A1.Ex21.m2.1.1.cmml">t</mi><mo id="A1.Ex21.m2.2.3.3.2.3.2.2" mathcolor="#000099" stretchy="false" xref="A1.Ex21.m2.2.3.3.2.cmml">)</mo></mrow></mrow><mo id="A1.Ex21.m2.2.3.3.1" mathcolor="#000099" xref="A1.Ex21.m2.2.3.3.1.cmml">+</mo><mrow id="A1.Ex21.m2.2.3.3.3" xref="A1.Ex21.m2.2.3.3.3.cmml"><mover accent="true" id="A1.Ex21.m2.2.3.3.3.2" xref="A1.Ex21.m2.2.3.3.3.2.cmml"><mi id="A1.Ex21.m2.2.3.3.3.2.2" mathcolor="#000099" xref="A1.Ex21.m2.2.3.3.3.2.2.cmml">𝜻</mi><mo id="A1.Ex21.m2.2.3.3.3.2.1" mathcolor="#000099" xref="A1.Ex21.m2.2.3.3.3.2.1.cmml">˙</mo></mover><mo id="A1.Ex21.m2.2.3.3.3.1" xref="A1.Ex21.m2.2.3.3.3.1.cmml">⁢</mo><mrow id="A1.Ex21.m2.2.3.3.3.3.2" xref="A1.Ex21.m2.2.3.3.3.cmml"><mo id="A1.Ex21.m2.2.3.3.3.3.2.1" mathcolor="#000099" stretchy="false" xref="A1.Ex21.m2.2.3.3.3.cmml">(</mo><mi id="A1.Ex21.m2.2.2" mathcolor="#000099" xref="A1.Ex21.m2.2.2.cmml">t</mi><mo id="A1.Ex21.m2.2.3.3.3.3.2.2" mathcolor="#000099" stretchy="false" xref="A1.Ex21.m2.2.3.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex21.m2.2b"><apply id="A1.Ex21.m2.2.3.cmml" xref="A1.Ex21.m2.2.3"><eq id="A1.Ex21.m2.2.3.1.cmml" xref="A1.Ex21.m2.2.3.1"></eq><csymbol cd="latexml" id="A1.Ex21.m2.2.3.2.cmml" xref="A1.Ex21.m2.2.3.2">absent</csymbol><apply id="A1.Ex21.m2.2.3.3.cmml" xref="A1.Ex21.m2.2.3.3"><plus id="A1.Ex21.m2.2.3.3.1.cmml" xref="A1.Ex21.m2.2.3.3.1"></plus><apply id="A1.Ex21.m2.2.3.3.2.cmml" xref="A1.Ex21.m2.2.3.3.2"><times id="A1.Ex21.m2.2.3.3.2.1.cmml" xref="A1.Ex21.m2.2.3.3.2.1"></times><apply id="A1.Ex21.m2.2.3.3.2.2.cmml" xref="A1.Ex21.m2.2.3.3.2.2"><ci id="A1.Ex21.m2.2.3.3.2.2.1.cmml" xref="A1.Ex21.m2.2.3.3.2.2.1">˙</ci><ci id="A1.Ex21.m2.2.3.3.2.2.2.cmml" xref="A1.Ex21.m2.2.3.3.2.2.2">𝒆</ci></apply><ci id="A1.Ex21.m2.1.1.cmml" xref="A1.Ex21.m2.1.1">𝑡</ci></apply><apply id="A1.Ex21.m2.2.3.3.3.cmml" xref="A1.Ex21.m2.2.3.3.3"><times id="A1.Ex21.m2.2.3.3.3.1.cmml" xref="A1.Ex21.m2.2.3.3.3.1"></times><apply id="A1.Ex21.m2.2.3.3.3.2.cmml" xref="A1.Ex21.m2.2.3.3.3.2"><ci id="A1.Ex21.m2.2.3.3.3.2.1.cmml" xref="A1.Ex21.m2.2.3.3.3.2.1">˙</ci><ci id="A1.Ex21.m2.2.3.3.3.2.2.cmml" xref="A1.Ex21.m2.2.3.3.3.2.2">𝜻</ci></apply><ci id="A1.Ex21.m2.2.2.cmml" xref="A1.Ex21.m2.2.2">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex21.m2.2c">\displaystyle=\dot{\bm{e}}(t)+\dot{\bm{\zeta}}(t)</annotation><annotation encoding="application/x-llamapun" id="A1.Ex21.m2.2d">= over˙ start_ARG bold_italic_e end_ARG ( italic_t ) + over˙ start_ARG bold_italic_ζ end_ARG ( italic_t )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A1.Ex22"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\Lambda\bm{e}+\Phi^{T}(t)\tilde{\bm{\theta}}(t)+\Lambda\bm{\zeta% }(t)+\Phi^{T}(t)\hat{\bm{\theta}}(t)" class="ltx_Math" display="inline" id="A1.Ex22.m1.5"><semantics id="A1.Ex22.m1.5a"><mrow id="A1.Ex22.m1.5.6" xref="A1.Ex22.m1.5.6.cmml"><mi id="A1.Ex22.m1.5.6.2" xref="A1.Ex22.m1.5.6.2.cmml"></mi><mo id="A1.Ex22.m1.5.6.1" mathcolor="#000099" xref="A1.Ex22.m1.5.6.1.cmml">=</mo><mrow id="A1.Ex22.m1.5.6.3" xref="A1.Ex22.m1.5.6.3.cmml"><mrow id="A1.Ex22.m1.5.6.3.2" xref="A1.Ex22.m1.5.6.3.2.cmml"><mi id="A1.Ex22.m1.5.6.3.2.2" mathcolor="#000099" mathvariant="normal" xref="A1.Ex22.m1.5.6.3.2.2.cmml">Λ</mi><mo id="A1.Ex22.m1.5.6.3.2.1" xref="A1.Ex22.m1.5.6.3.2.1.cmml">⁢</mo><mi id="A1.Ex22.m1.5.6.3.2.3" mathcolor="#000099" xref="A1.Ex22.m1.5.6.3.2.3.cmml">𝒆</mi></mrow><mo id="A1.Ex22.m1.5.6.3.1" mathcolor="#000099" xref="A1.Ex22.m1.5.6.3.1.cmml">+</mo><mrow id="A1.Ex22.m1.5.6.3.3" xref="A1.Ex22.m1.5.6.3.3.cmml"><msup id="A1.Ex22.m1.5.6.3.3.2" xref="A1.Ex22.m1.5.6.3.3.2.cmml"><mi id="A1.Ex22.m1.5.6.3.3.2.2" mathcolor="#000099" mathvariant="normal" xref="A1.Ex22.m1.5.6.3.3.2.2.cmml">Φ</mi><mi id="A1.Ex22.m1.5.6.3.3.2.3" mathcolor="#000099" xref="A1.Ex22.m1.5.6.3.3.2.3.cmml">T</mi></msup><mo id="A1.Ex22.m1.5.6.3.3.1" xref="A1.Ex22.m1.5.6.3.3.1.cmml">⁢</mo><mrow id="A1.Ex22.m1.5.6.3.3.3.2" xref="A1.Ex22.m1.5.6.3.3.cmml"><mo id="A1.Ex22.m1.5.6.3.3.3.2.1" mathcolor="#000099" stretchy="false" xref="A1.Ex22.m1.5.6.3.3.cmml">(</mo><mi id="A1.Ex22.m1.1.1" mathcolor="#000099" xref="A1.Ex22.m1.1.1.cmml">t</mi><mo id="A1.Ex22.m1.5.6.3.3.3.2.2" mathcolor="#000099" stretchy="false" xref="A1.Ex22.m1.5.6.3.3.cmml">)</mo></mrow><mo id="A1.Ex22.m1.5.6.3.3.1a" xref="A1.Ex22.m1.5.6.3.3.1.cmml">⁢</mo><mover accent="true" id="A1.Ex22.m1.5.6.3.3.4" xref="A1.Ex22.m1.5.6.3.3.4.cmml"><mi id="A1.Ex22.m1.5.6.3.3.4.2" mathcolor="#000099" xref="A1.Ex22.m1.5.6.3.3.4.2.cmml">𝜽</mi><mo id="A1.Ex22.m1.5.6.3.3.4.1" mathcolor="#000099" xref="A1.Ex22.m1.5.6.3.3.4.1.cmml">~</mo></mover><mo id="A1.Ex22.m1.5.6.3.3.1b" xref="A1.Ex22.m1.5.6.3.3.1.cmml">⁢</mo><mrow id="A1.Ex22.m1.5.6.3.3.5.2" xref="A1.Ex22.m1.5.6.3.3.cmml"><mo id="A1.Ex22.m1.5.6.3.3.5.2.1" mathcolor="#000099" stretchy="false" xref="A1.Ex22.m1.5.6.3.3.cmml">(</mo><mi id="A1.Ex22.m1.2.2" mathcolor="#000099" xref="A1.Ex22.m1.2.2.cmml">t</mi><mo id="A1.Ex22.m1.5.6.3.3.5.2.2" mathcolor="#000099" stretchy="false" xref="A1.Ex22.m1.5.6.3.3.cmml">)</mo></mrow></mrow><mo id="A1.Ex22.m1.5.6.3.1a" mathcolor="#000099" xref="A1.Ex22.m1.5.6.3.1.cmml">+</mo><mrow id="A1.Ex22.m1.5.6.3.4" xref="A1.Ex22.m1.5.6.3.4.cmml"><mi id="A1.Ex22.m1.5.6.3.4.2" mathcolor="#000099" mathvariant="normal" xref="A1.Ex22.m1.5.6.3.4.2.cmml">Λ</mi><mo id="A1.Ex22.m1.5.6.3.4.1" xref="A1.Ex22.m1.5.6.3.4.1.cmml">⁢</mo><mi id="A1.Ex22.m1.5.6.3.4.3" mathcolor="#000099" xref="A1.Ex22.m1.5.6.3.4.3.cmml">𝜻</mi><mo id="A1.Ex22.m1.5.6.3.4.1a" xref="A1.Ex22.m1.5.6.3.4.1.cmml">⁢</mo><mrow id="A1.Ex22.m1.5.6.3.4.4.2" xref="A1.Ex22.m1.5.6.3.4.cmml"><mo id="A1.Ex22.m1.5.6.3.4.4.2.1" mathcolor="#000099" stretchy="false" xref="A1.Ex22.m1.5.6.3.4.cmml">(</mo><mi id="A1.Ex22.m1.3.3" mathcolor="#000099" xref="A1.Ex22.m1.3.3.cmml">t</mi><mo id="A1.Ex22.m1.5.6.3.4.4.2.2" mathcolor="#000099" stretchy="false" xref="A1.Ex22.m1.5.6.3.4.cmml">)</mo></mrow></mrow><mo id="A1.Ex22.m1.5.6.3.1b" mathcolor="#000099" xref="A1.Ex22.m1.5.6.3.1.cmml">+</mo><mrow id="A1.Ex22.m1.5.6.3.5" xref="A1.Ex22.m1.5.6.3.5.cmml"><msup id="A1.Ex22.m1.5.6.3.5.2" xref="A1.Ex22.m1.5.6.3.5.2.cmml"><mi id="A1.Ex22.m1.5.6.3.5.2.2" mathcolor="#000099" mathvariant="normal" xref="A1.Ex22.m1.5.6.3.5.2.2.cmml">Φ</mi><mi id="A1.Ex22.m1.5.6.3.5.2.3" mathcolor="#000099" xref="A1.Ex22.m1.5.6.3.5.2.3.cmml">T</mi></msup><mo id="A1.Ex22.m1.5.6.3.5.1" xref="A1.Ex22.m1.5.6.3.5.1.cmml">⁢</mo><mrow id="A1.Ex22.m1.5.6.3.5.3.2" xref="A1.Ex22.m1.5.6.3.5.cmml"><mo id="A1.Ex22.m1.5.6.3.5.3.2.1" mathcolor="#000099" stretchy="false" xref="A1.Ex22.m1.5.6.3.5.cmml">(</mo><mi id="A1.Ex22.m1.4.4" mathcolor="#000099" xref="A1.Ex22.m1.4.4.cmml">t</mi><mo id="A1.Ex22.m1.5.6.3.5.3.2.2" mathcolor="#000099" stretchy="false" xref="A1.Ex22.m1.5.6.3.5.cmml">)</mo></mrow><mo id="A1.Ex22.m1.5.6.3.5.1a" xref="A1.Ex22.m1.5.6.3.5.1.cmml">⁢</mo><mover accent="true" id="A1.Ex22.m1.5.6.3.5.4" xref="A1.Ex22.m1.5.6.3.5.4.cmml"><mi id="A1.Ex22.m1.5.6.3.5.4.2" mathcolor="#000099" xref="A1.Ex22.m1.5.6.3.5.4.2.cmml">𝜽</mi><mo id="A1.Ex22.m1.5.6.3.5.4.1" mathcolor="#000099" xref="A1.Ex22.m1.5.6.3.5.4.1.cmml">^</mo></mover><mo id="A1.Ex22.m1.5.6.3.5.1b" xref="A1.Ex22.m1.5.6.3.5.1.cmml">⁢</mo><mrow id="A1.Ex22.m1.5.6.3.5.5.2" xref="A1.Ex22.m1.5.6.3.5.cmml"><mo id="A1.Ex22.m1.5.6.3.5.5.2.1" mathcolor="#000099" stretchy="false" xref="A1.Ex22.m1.5.6.3.5.cmml">(</mo><mi id="A1.Ex22.m1.5.5" mathcolor="#000099" xref="A1.Ex22.m1.5.5.cmml">t</mi><mo id="A1.Ex22.m1.5.6.3.5.5.2.2" mathcolor="#000099" stretchy="false" xref="A1.Ex22.m1.5.6.3.5.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex22.m1.5b"><apply id="A1.Ex22.m1.5.6.cmml" xref="A1.Ex22.m1.5.6"><eq id="A1.Ex22.m1.5.6.1.cmml" xref="A1.Ex22.m1.5.6.1"></eq><csymbol cd="latexml" id="A1.Ex22.m1.5.6.2.cmml" xref="A1.Ex22.m1.5.6.2">absent</csymbol><apply id="A1.Ex22.m1.5.6.3.cmml" xref="A1.Ex22.m1.5.6.3"><plus id="A1.Ex22.m1.5.6.3.1.cmml" xref="A1.Ex22.m1.5.6.3.1"></plus><apply id="A1.Ex22.m1.5.6.3.2.cmml" xref="A1.Ex22.m1.5.6.3.2"><times id="A1.Ex22.m1.5.6.3.2.1.cmml" xref="A1.Ex22.m1.5.6.3.2.1"></times><ci id="A1.Ex22.m1.5.6.3.2.2.cmml" xref="A1.Ex22.m1.5.6.3.2.2">Λ</ci><ci id="A1.Ex22.m1.5.6.3.2.3.cmml" xref="A1.Ex22.m1.5.6.3.2.3">𝒆</ci></apply><apply id="A1.Ex22.m1.5.6.3.3.cmml" xref="A1.Ex22.m1.5.6.3.3"><times id="A1.Ex22.m1.5.6.3.3.1.cmml" xref="A1.Ex22.m1.5.6.3.3.1"></times><apply id="A1.Ex22.m1.5.6.3.3.2.cmml" xref="A1.Ex22.m1.5.6.3.3.2"><csymbol cd="ambiguous" id="A1.Ex22.m1.5.6.3.3.2.1.cmml" xref="A1.Ex22.m1.5.6.3.3.2">superscript</csymbol><ci id="A1.Ex22.m1.5.6.3.3.2.2.cmml" xref="A1.Ex22.m1.5.6.3.3.2.2">Φ</ci><ci id="A1.Ex22.m1.5.6.3.3.2.3.cmml" xref="A1.Ex22.m1.5.6.3.3.2.3">𝑇</ci></apply><ci id="A1.Ex22.m1.1.1.cmml" xref="A1.Ex22.m1.1.1">𝑡</ci><apply id="A1.Ex22.m1.5.6.3.3.4.cmml" xref="A1.Ex22.m1.5.6.3.3.4"><ci id="A1.Ex22.m1.5.6.3.3.4.1.cmml" xref="A1.Ex22.m1.5.6.3.3.4.1">~</ci><ci id="A1.Ex22.m1.5.6.3.3.4.2.cmml" xref="A1.Ex22.m1.5.6.3.3.4.2">𝜽</ci></apply><ci id="A1.Ex22.m1.2.2.cmml" xref="A1.Ex22.m1.2.2">𝑡</ci></apply><apply id="A1.Ex22.m1.5.6.3.4.cmml" xref="A1.Ex22.m1.5.6.3.4"><times id="A1.Ex22.m1.5.6.3.4.1.cmml" xref="A1.Ex22.m1.5.6.3.4.1"></times><ci id="A1.Ex22.m1.5.6.3.4.2.cmml" xref="A1.Ex22.m1.5.6.3.4.2">Λ</ci><ci id="A1.Ex22.m1.5.6.3.4.3.cmml" xref="A1.Ex22.m1.5.6.3.4.3">𝜻</ci><ci id="A1.Ex22.m1.3.3.cmml" xref="A1.Ex22.m1.3.3">𝑡</ci></apply><apply id="A1.Ex22.m1.5.6.3.5.cmml" xref="A1.Ex22.m1.5.6.3.5"><times id="A1.Ex22.m1.5.6.3.5.1.cmml" xref="A1.Ex22.m1.5.6.3.5.1"></times><apply id="A1.Ex22.m1.5.6.3.5.2.cmml" xref="A1.Ex22.m1.5.6.3.5.2"><csymbol cd="ambiguous" id="A1.Ex22.m1.5.6.3.5.2.1.cmml" xref="A1.Ex22.m1.5.6.3.5.2">superscript</csymbol><ci id="A1.Ex22.m1.5.6.3.5.2.2.cmml" xref="A1.Ex22.m1.5.6.3.5.2.2">Φ</ci><ci id="A1.Ex22.m1.5.6.3.5.2.3.cmml" xref="A1.Ex22.m1.5.6.3.5.2.3">𝑇</ci></apply><ci id="A1.Ex22.m1.4.4.cmml" xref="A1.Ex22.m1.4.4">𝑡</ci><apply id="A1.Ex22.m1.5.6.3.5.4.cmml" xref="A1.Ex22.m1.5.6.3.5.4"><ci id="A1.Ex22.m1.5.6.3.5.4.1.cmml" xref="A1.Ex22.m1.5.6.3.5.4.1">^</ci><ci id="A1.Ex22.m1.5.6.3.5.4.2.cmml" xref="A1.Ex22.m1.5.6.3.5.4.2">𝜽</ci></apply><ci id="A1.Ex22.m1.5.5.cmml" xref="A1.Ex22.m1.5.5">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex22.m1.5c">\displaystyle=\Lambda\bm{e}+\Phi^{T}(t)\tilde{\bm{\theta}}(t)+\Lambda\bm{\zeta% }(t)+\Phi^{T}(t)\hat{\bm{\theta}}(t)</annotation><annotation encoding="application/x-llamapun" id="A1.Ex22.m1.5d">= roman_Λ bold_italic_e + roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_t ) over~ start_ARG bold_italic_θ end_ARG ( italic_t ) + roman_Λ bold_italic_ζ ( italic_t ) + roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_t ) over^ start_ARG bold_italic_θ end_ARG ( italic_t )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A1.Ex23"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\Lambda(\bm{e}+\bm{\zeta})+\Phi^{T}(t)\bm{\theta}(t)" class="ltx_Math" display="inline" id="A1.Ex23.m1.3"><semantics id="A1.Ex23.m1.3a"><mrow id="A1.Ex23.m1.3.3" xref="A1.Ex23.m1.3.3.cmml"><mi id="A1.Ex23.m1.3.3.3" xref="A1.Ex23.m1.3.3.3.cmml"></mi><mo id="A1.Ex23.m1.3.3.2" mathcolor="#000099" xref="A1.Ex23.m1.3.3.2.cmml">=</mo><mrow id="A1.Ex23.m1.3.3.1" xref="A1.Ex23.m1.3.3.1.cmml"><mrow id="A1.Ex23.m1.3.3.1.1" xref="A1.Ex23.m1.3.3.1.1.cmml"><mi id="A1.Ex23.m1.3.3.1.1.3" mathcolor="#000099" mathvariant="normal" xref="A1.Ex23.m1.3.3.1.1.3.cmml">Λ</mi><mo id="A1.Ex23.m1.3.3.1.1.2" xref="A1.Ex23.m1.3.3.1.1.2.cmml">⁢</mo><mrow id="A1.Ex23.m1.3.3.1.1.1.1" xref="A1.Ex23.m1.3.3.1.1.1.1.1.cmml"><mo id="A1.Ex23.m1.3.3.1.1.1.1.2" mathcolor="#000099" stretchy="false" xref="A1.Ex23.m1.3.3.1.1.1.1.1.cmml">(</mo><mrow id="A1.Ex23.m1.3.3.1.1.1.1.1" xref="A1.Ex23.m1.3.3.1.1.1.1.1.cmml"><mi id="A1.Ex23.m1.3.3.1.1.1.1.1.2" mathcolor="#000099" xref="A1.Ex23.m1.3.3.1.1.1.1.1.2.cmml">𝒆</mi><mo id="A1.Ex23.m1.3.3.1.1.1.1.1.1" mathcolor="#000099" xref="A1.Ex23.m1.3.3.1.1.1.1.1.1.cmml">+</mo><mi id="A1.Ex23.m1.3.3.1.1.1.1.1.3" mathcolor="#000099" xref="A1.Ex23.m1.3.3.1.1.1.1.1.3.cmml">𝜻</mi></mrow><mo id="A1.Ex23.m1.3.3.1.1.1.1.3" mathcolor="#000099" stretchy="false" xref="A1.Ex23.m1.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.Ex23.m1.3.3.1.2" mathcolor="#000099" xref="A1.Ex23.m1.3.3.1.2.cmml">+</mo><mrow id="A1.Ex23.m1.3.3.1.3" xref="A1.Ex23.m1.3.3.1.3.cmml"><msup id="A1.Ex23.m1.3.3.1.3.2" xref="A1.Ex23.m1.3.3.1.3.2.cmml"><mi id="A1.Ex23.m1.3.3.1.3.2.2" mathcolor="#000099" mathvariant="normal" xref="A1.Ex23.m1.3.3.1.3.2.2.cmml">Φ</mi><mi id="A1.Ex23.m1.3.3.1.3.2.3" mathcolor="#000099" xref="A1.Ex23.m1.3.3.1.3.2.3.cmml">T</mi></msup><mo id="A1.Ex23.m1.3.3.1.3.1" xref="A1.Ex23.m1.3.3.1.3.1.cmml">⁢</mo><mrow id="A1.Ex23.m1.3.3.1.3.3.2" xref="A1.Ex23.m1.3.3.1.3.cmml"><mo id="A1.Ex23.m1.3.3.1.3.3.2.1" mathcolor="#000099" stretchy="false" xref="A1.Ex23.m1.3.3.1.3.cmml">(</mo><mi id="A1.Ex23.m1.1.1" mathcolor="#000099" xref="A1.Ex23.m1.1.1.cmml">t</mi><mo id="A1.Ex23.m1.3.3.1.3.3.2.2" mathcolor="#000099" stretchy="false" xref="A1.Ex23.m1.3.3.1.3.cmml">)</mo></mrow><mo id="A1.Ex23.m1.3.3.1.3.1a" xref="A1.Ex23.m1.3.3.1.3.1.cmml">⁢</mo><mi id="A1.Ex23.m1.3.3.1.3.4" mathcolor="#000099" xref="A1.Ex23.m1.3.3.1.3.4.cmml">𝜽</mi><mo id="A1.Ex23.m1.3.3.1.3.1b" xref="A1.Ex23.m1.3.3.1.3.1.cmml">⁢</mo><mrow id="A1.Ex23.m1.3.3.1.3.5.2" xref="A1.Ex23.m1.3.3.1.3.cmml"><mo id="A1.Ex23.m1.3.3.1.3.5.2.1" mathcolor="#000099" stretchy="false" xref="A1.Ex23.m1.3.3.1.3.cmml">(</mo><mi id="A1.Ex23.m1.2.2" mathcolor="#000099" xref="A1.Ex23.m1.2.2.cmml">t</mi><mo id="A1.Ex23.m1.3.3.1.3.5.2.2" mathcolor="#000099" stretchy="false" xref="A1.Ex23.m1.3.3.1.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex23.m1.3b"><apply id="A1.Ex23.m1.3.3.cmml" xref="A1.Ex23.m1.3.3"><eq id="A1.Ex23.m1.3.3.2.cmml" xref="A1.Ex23.m1.3.3.2"></eq><csymbol cd="latexml" id="A1.Ex23.m1.3.3.3.cmml" xref="A1.Ex23.m1.3.3.3">absent</csymbol><apply id="A1.Ex23.m1.3.3.1.cmml" xref="A1.Ex23.m1.3.3.1"><plus id="A1.Ex23.m1.3.3.1.2.cmml" xref="A1.Ex23.m1.3.3.1.2"></plus><apply id="A1.Ex23.m1.3.3.1.1.cmml" xref="A1.Ex23.m1.3.3.1.1"><times id="A1.Ex23.m1.3.3.1.1.2.cmml" xref="A1.Ex23.m1.3.3.1.1.2"></times><ci id="A1.Ex23.m1.3.3.1.1.3.cmml" xref="A1.Ex23.m1.3.3.1.1.3">Λ</ci><apply id="A1.Ex23.m1.3.3.1.1.1.1.1.cmml" xref="A1.Ex23.m1.3.3.1.1.1.1"><plus id="A1.Ex23.m1.3.3.1.1.1.1.1.1.cmml" xref="A1.Ex23.m1.3.3.1.1.1.1.1.1"></plus><ci id="A1.Ex23.m1.3.3.1.1.1.1.1.2.cmml" xref="A1.Ex23.m1.3.3.1.1.1.1.1.2">𝒆</ci><ci id="A1.Ex23.m1.3.3.1.1.1.1.1.3.cmml" xref="A1.Ex23.m1.3.3.1.1.1.1.1.3">𝜻</ci></apply></apply><apply id="A1.Ex23.m1.3.3.1.3.cmml" xref="A1.Ex23.m1.3.3.1.3"><times id="A1.Ex23.m1.3.3.1.3.1.cmml" xref="A1.Ex23.m1.3.3.1.3.1"></times><apply id="A1.Ex23.m1.3.3.1.3.2.cmml" xref="A1.Ex23.m1.3.3.1.3.2"><csymbol cd="ambiguous" id="A1.Ex23.m1.3.3.1.3.2.1.cmml" xref="A1.Ex23.m1.3.3.1.3.2">superscript</csymbol><ci id="A1.Ex23.m1.3.3.1.3.2.2.cmml" xref="A1.Ex23.m1.3.3.1.3.2.2">Φ</ci><ci id="A1.Ex23.m1.3.3.1.3.2.3.cmml" xref="A1.Ex23.m1.3.3.1.3.2.3">𝑇</ci></apply><ci id="A1.Ex23.m1.1.1.cmml" xref="A1.Ex23.m1.1.1">𝑡</ci><ci id="A1.Ex23.m1.3.3.1.3.4.cmml" xref="A1.Ex23.m1.3.3.1.3.4">𝜽</ci><ci id="A1.Ex23.m1.2.2.cmml" xref="A1.Ex23.m1.2.2">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex23.m1.3c">\displaystyle=\Lambda(\bm{e}+\bm{\zeta})+\Phi^{T}(t)\bm{\theta}(t)</annotation><annotation encoding="application/x-llamapun" id="A1.Ex23.m1.3d">= roman_Λ ( bold_italic_e + bold_italic_ζ ) + roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_t ) bold_italic_θ ( italic_t )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.p1.6"><span class="ltx_text" id="A1.p1.6.1" style="color:#000099;">which can be rewritten into</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx31"> <tbody id="A1.E32"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{\bm{p}}=\Lambda\bm{p}+\Phi^{T}(t)\bm{\theta}(t)" class="ltx_Math" display="inline" id="A1.E32.m1.2"><semantics id="A1.E32.m1.2a"><mrow id="A1.E32.m1.2.3" xref="A1.E32.m1.2.3.cmml"><mover accent="true" id="A1.E32.m1.2.3.2" xref="A1.E32.m1.2.3.2.cmml"><mi id="A1.E32.m1.2.3.2.2" mathcolor="#000099" xref="A1.E32.m1.2.3.2.2.cmml">𝒑</mi><mo id="A1.E32.m1.2.3.2.1" mathcolor="#000099" xref="A1.E32.m1.2.3.2.1.cmml">˙</mo></mover><mo id="A1.E32.m1.2.3.1" mathcolor="#000099" xref="A1.E32.m1.2.3.1.cmml">=</mo><mrow id="A1.E32.m1.2.3.3" xref="A1.E32.m1.2.3.3.cmml"><mrow id="A1.E32.m1.2.3.3.2" xref="A1.E32.m1.2.3.3.2.cmml"><mi id="A1.E32.m1.2.3.3.2.2" mathcolor="#000099" mathvariant="normal" xref="A1.E32.m1.2.3.3.2.2.cmml">Λ</mi><mo id="A1.E32.m1.2.3.3.2.1" xref="A1.E32.m1.2.3.3.2.1.cmml">⁢</mo><mi id="A1.E32.m1.2.3.3.2.3" mathcolor="#000099" xref="A1.E32.m1.2.3.3.2.3.cmml">𝒑</mi></mrow><mo id="A1.E32.m1.2.3.3.1" mathcolor="#000099" xref="A1.E32.m1.2.3.3.1.cmml">+</mo><mrow id="A1.E32.m1.2.3.3.3" xref="A1.E32.m1.2.3.3.3.cmml"><msup id="A1.E32.m1.2.3.3.3.2" xref="A1.E32.m1.2.3.3.3.2.cmml"><mi id="A1.E32.m1.2.3.3.3.2.2" mathcolor="#000099" mathvariant="normal" xref="A1.E32.m1.2.3.3.3.2.2.cmml">Φ</mi><mi id="A1.E32.m1.2.3.3.3.2.3" mathcolor="#000099" xref="A1.E32.m1.2.3.3.3.2.3.cmml">T</mi></msup><mo id="A1.E32.m1.2.3.3.3.1" xref="A1.E32.m1.2.3.3.3.1.cmml">⁢</mo><mrow id="A1.E32.m1.2.3.3.3.3.2" xref="A1.E32.m1.2.3.3.3.cmml"><mo id="A1.E32.m1.2.3.3.3.3.2.1" mathcolor="#000099" stretchy="false" xref="A1.E32.m1.2.3.3.3.cmml">(</mo><mi id="A1.E32.m1.1.1" mathcolor="#000099" xref="A1.E32.m1.1.1.cmml">t</mi><mo id="A1.E32.m1.2.3.3.3.3.2.2" mathcolor="#000099" stretchy="false" xref="A1.E32.m1.2.3.3.3.cmml">)</mo></mrow><mo id="A1.E32.m1.2.3.3.3.1a" xref="A1.E32.m1.2.3.3.3.1.cmml">⁢</mo><mi id="A1.E32.m1.2.3.3.3.4" mathcolor="#000099" xref="A1.E32.m1.2.3.3.3.4.cmml">𝜽</mi><mo id="A1.E32.m1.2.3.3.3.1b" xref="A1.E32.m1.2.3.3.3.1.cmml">⁢</mo><mrow id="A1.E32.m1.2.3.3.3.5.2" xref="A1.E32.m1.2.3.3.3.cmml"><mo id="A1.E32.m1.2.3.3.3.5.2.1" mathcolor="#000099" stretchy="false" xref="A1.E32.m1.2.3.3.3.cmml">(</mo><mi id="A1.E32.m1.2.2" mathcolor="#000099" xref="A1.E32.m1.2.2.cmml">t</mi><mo id="A1.E32.m1.2.3.3.3.5.2.2" mathcolor="#000099" stretchy="false" xref="A1.E32.m1.2.3.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.E32.m1.2b"><apply id="A1.E32.m1.2.3.cmml" xref="A1.E32.m1.2.3"><eq id="A1.E32.m1.2.3.1.cmml" xref="A1.E32.m1.2.3.1"></eq><apply id="A1.E32.m1.2.3.2.cmml" xref="A1.E32.m1.2.3.2"><ci id="A1.E32.m1.2.3.2.1.cmml" xref="A1.E32.m1.2.3.2.1">˙</ci><ci id="A1.E32.m1.2.3.2.2.cmml" xref="A1.E32.m1.2.3.2.2">𝒑</ci></apply><apply id="A1.E32.m1.2.3.3.cmml" xref="A1.E32.m1.2.3.3"><plus id="A1.E32.m1.2.3.3.1.cmml" xref="A1.E32.m1.2.3.3.1"></plus><apply id="A1.E32.m1.2.3.3.2.cmml" xref="A1.E32.m1.2.3.3.2"><times id="A1.E32.m1.2.3.3.2.1.cmml" xref="A1.E32.m1.2.3.3.2.1"></times><ci id="A1.E32.m1.2.3.3.2.2.cmml" xref="A1.E32.m1.2.3.3.2.2">Λ</ci><ci id="A1.E32.m1.2.3.3.2.3.cmml" xref="A1.E32.m1.2.3.3.2.3">𝒑</ci></apply><apply id="A1.E32.m1.2.3.3.3.cmml" xref="A1.E32.m1.2.3.3.3"><times id="A1.E32.m1.2.3.3.3.1.cmml" xref="A1.E32.m1.2.3.3.3.1"></times><apply id="A1.E32.m1.2.3.3.3.2.cmml" xref="A1.E32.m1.2.3.3.3.2"><csymbol cd="ambiguous" id="A1.E32.m1.2.3.3.3.2.1.cmml" xref="A1.E32.m1.2.3.3.3.2">superscript</csymbol><ci id="A1.E32.m1.2.3.3.3.2.2.cmml" xref="A1.E32.m1.2.3.3.3.2.2">Φ</ci><ci id="A1.E32.m1.2.3.3.3.2.3.cmml" xref="A1.E32.m1.2.3.3.3.2.3">𝑇</ci></apply><ci id="A1.E32.m1.1.1.cmml" xref="A1.E32.m1.1.1">𝑡</ci><ci id="A1.E32.m1.2.3.3.3.4.cmml" xref="A1.E32.m1.2.3.3.3.4">𝜽</ci><ci id="A1.E32.m1.2.2.cmml" xref="A1.E32.m1.2.2">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E32.m1.2c">\displaystyle\dot{\bm{p}}=\Lambda\bm{p}+\Phi^{T}(t)\bm{\theta}(t)</annotation><annotation encoding="application/x-llamapun" id="A1.E32.m1.2d">over˙ start_ARG bold_italic_p end_ARG = roman_Λ bold_italic_p + roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_t ) bold_italic_θ ( italic_t )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(32)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A1.p1.3"><span class="ltx_text" id="A1.p1.3.1" style="color:#000099;">Multiplying the second equation of (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E11" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">11</span></a>) by <math alttext="\bm{\theta}" class="ltx_Math" display="inline" id="A1.p1.3.1.m1.1"><semantics id="A1.p1.3.1.m1.1a"><mi id="A1.p1.3.1.m1.1.1" mathcolor="#000099" xref="A1.p1.3.1.m1.1.1.cmml">𝜽</mi><annotation-xml encoding="MathML-Content" id="A1.p1.3.1.m1.1b"><ci id="A1.p1.3.1.m1.1.1.cmml" xref="A1.p1.3.1.m1.1.1">𝜽</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.3.1.m1.1c">\bm{\theta}</annotation><annotation encoding="application/x-llamapun" id="A1.p1.3.1.m1.1d">bold_italic_θ</annotation></semantics></math> and subtracting the result from (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A1.E32" title="In Appendix A The derivation of (12) ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">32</span></a>), one obtains</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx32"> <tbody id="A1.Ex24"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{\bm{p}}(t)-\dot{\Phi}^{T}_{\rm s}(t)\bm{\theta}=\Lambda(\bm{% p}(t)-{\Phi}_{\rm s}^{T}(t)\bm{\theta})." class="ltx_Math" display="inline" id="A1.Ex24.m1.5"><semantics id="A1.Ex24.m1.5a"><mrow id="A1.Ex24.m1.5.5.1" xref="A1.Ex24.m1.5.5.1.1.cmml"><mrow id="A1.Ex24.m1.5.5.1.1" xref="A1.Ex24.m1.5.5.1.1.cmml"><mrow id="A1.Ex24.m1.5.5.1.1.3" xref="A1.Ex24.m1.5.5.1.1.3.cmml"><mrow id="A1.Ex24.m1.5.5.1.1.3.2" xref="A1.Ex24.m1.5.5.1.1.3.2.cmml"><mover accent="true" id="A1.Ex24.m1.5.5.1.1.3.2.2" xref="A1.Ex24.m1.5.5.1.1.3.2.2.cmml"><mi id="A1.Ex24.m1.5.5.1.1.3.2.2.2" mathcolor="#000099" xref="A1.Ex24.m1.5.5.1.1.3.2.2.2.cmml">𝒑</mi><mo id="A1.Ex24.m1.5.5.1.1.3.2.2.1" mathcolor="#000099" xref="A1.Ex24.m1.5.5.1.1.3.2.2.1.cmml">˙</mo></mover><mo id="A1.Ex24.m1.5.5.1.1.3.2.1" xref="A1.Ex24.m1.5.5.1.1.3.2.1.cmml">⁢</mo><mrow id="A1.Ex24.m1.5.5.1.1.3.2.3.2" xref="A1.Ex24.m1.5.5.1.1.3.2.cmml"><mo id="A1.Ex24.m1.5.5.1.1.3.2.3.2.1" mathcolor="#000099" stretchy="false" xref="A1.Ex24.m1.5.5.1.1.3.2.cmml">(</mo><mi id="A1.Ex24.m1.1.1" mathcolor="#000099" xref="A1.Ex24.m1.1.1.cmml">t</mi><mo id="A1.Ex24.m1.5.5.1.1.3.2.3.2.2" mathcolor="#000099" stretchy="false" xref="A1.Ex24.m1.5.5.1.1.3.2.cmml">)</mo></mrow></mrow><mo id="A1.Ex24.m1.5.5.1.1.3.1" mathcolor="#000099" xref="A1.Ex24.m1.5.5.1.1.3.1.cmml">−</mo><mrow id="A1.Ex24.m1.5.5.1.1.3.3" xref="A1.Ex24.m1.5.5.1.1.3.3.cmml"><msubsup id="A1.Ex24.m1.5.5.1.1.3.3.2" xref="A1.Ex24.m1.5.5.1.1.3.3.2.cmml"><mover accent="true" id="A1.Ex24.m1.5.5.1.1.3.3.2.2.2" xref="A1.Ex24.m1.5.5.1.1.3.3.2.2.2.cmml"><mi id="A1.Ex24.m1.5.5.1.1.3.3.2.2.2.2" mathcolor="#000099" mathvariant="normal" xref="A1.Ex24.m1.5.5.1.1.3.3.2.2.2.2.cmml">Φ</mi><mo id="A1.Ex24.m1.5.5.1.1.3.3.2.2.2.1" mathcolor="#000099" xref="A1.Ex24.m1.5.5.1.1.3.3.2.2.2.1.cmml">˙</mo></mover><mi id="A1.Ex24.m1.5.5.1.1.3.3.2.3" mathcolor="#000099" mathvariant="normal" xref="A1.Ex24.m1.5.5.1.1.3.3.2.3.cmml">s</mi><mi id="A1.Ex24.m1.5.5.1.1.3.3.2.2.3" mathcolor="#000099" xref="A1.Ex24.m1.5.5.1.1.3.3.2.2.3.cmml">T</mi></msubsup><mo id="A1.Ex24.m1.5.5.1.1.3.3.1" xref="A1.Ex24.m1.5.5.1.1.3.3.1.cmml">⁢</mo><mrow id="A1.Ex24.m1.5.5.1.1.3.3.3.2" xref="A1.Ex24.m1.5.5.1.1.3.3.cmml"><mo id="A1.Ex24.m1.5.5.1.1.3.3.3.2.1" mathcolor="#000099" stretchy="false" xref="A1.Ex24.m1.5.5.1.1.3.3.cmml">(</mo><mi id="A1.Ex24.m1.2.2" mathcolor="#000099" xref="A1.Ex24.m1.2.2.cmml">t</mi><mo id="A1.Ex24.m1.5.5.1.1.3.3.3.2.2" mathcolor="#000099" stretchy="false" xref="A1.Ex24.m1.5.5.1.1.3.3.cmml">)</mo></mrow><mo id="A1.Ex24.m1.5.5.1.1.3.3.1a" xref="A1.Ex24.m1.5.5.1.1.3.3.1.cmml">⁢</mo><mi id="A1.Ex24.m1.5.5.1.1.3.3.4" mathcolor="#000099" xref="A1.Ex24.m1.5.5.1.1.3.3.4.cmml">𝜽</mi></mrow></mrow><mo id="A1.Ex24.m1.5.5.1.1.2" mathcolor="#000099" xref="A1.Ex24.m1.5.5.1.1.2.cmml">=</mo><mrow id="A1.Ex24.m1.5.5.1.1.1" xref="A1.Ex24.m1.5.5.1.1.1.cmml"><mi id="A1.Ex24.m1.5.5.1.1.1.3" mathcolor="#000099" mathvariant="normal" xref="A1.Ex24.m1.5.5.1.1.1.3.cmml">Λ</mi><mo id="A1.Ex24.m1.5.5.1.1.1.2" xref="A1.Ex24.m1.5.5.1.1.1.2.cmml">⁢</mo><mrow id="A1.Ex24.m1.5.5.1.1.1.1.1" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.cmml"><mo id="A1.Ex24.m1.5.5.1.1.1.1.1.2" mathcolor="#000099" stretchy="false" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.Ex24.m1.5.5.1.1.1.1.1.1" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.cmml"><mrow id="A1.Ex24.m1.5.5.1.1.1.1.1.1.2" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.2.cmml"><mi id="A1.Ex24.m1.5.5.1.1.1.1.1.1.2.2" mathcolor="#000099" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.2.2.cmml">𝒑</mi><mo id="A1.Ex24.m1.5.5.1.1.1.1.1.1.2.1" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.2.1.cmml">⁢</mo><mrow id="A1.Ex24.m1.5.5.1.1.1.1.1.1.2.3.2" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.2.cmml"><mo id="A1.Ex24.m1.5.5.1.1.1.1.1.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.2.cmml">(</mo><mi id="A1.Ex24.m1.3.3" mathcolor="#000099" xref="A1.Ex24.m1.3.3.cmml">t</mi><mo id="A1.Ex24.m1.5.5.1.1.1.1.1.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="A1.Ex24.m1.5.5.1.1.1.1.1.1.1" mathcolor="#000099" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.1.cmml">−</mo><mrow id="A1.Ex24.m1.5.5.1.1.1.1.1.1.3" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.cmml"><msubsup id="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.2" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.2.cmml"><mi id="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.2.2.2" mathcolor="#000099" mathvariant="normal" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.2.2.2.cmml">Φ</mi><mi id="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.2.2.3" mathcolor="#000099" mathvariant="normal" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.2.2.3.cmml">s</mi><mi id="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.2.3" mathcolor="#000099" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.2.3.cmml">T</mi></msubsup><mo id="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.1" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.3.2" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.cmml"><mo id="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.3.2.1" mathcolor="#000099" stretchy="false" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.cmml">(</mo><mi id="A1.Ex24.m1.4.4" mathcolor="#000099" xref="A1.Ex24.m1.4.4.cmml">t</mi><mo id="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.3.2.2" mathcolor="#000099" stretchy="false" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.cmml">)</mo></mrow><mo id="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.1a" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.4" mathcolor="#000099" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.4.cmml">𝜽</mi></mrow></mrow><mo id="A1.Ex24.m1.5.5.1.1.1.1.1.3" mathcolor="#000099" stretchy="false" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A1.Ex24.m1.5.5.1.2" lspace="0em" mathcolor="#000099" xref="A1.Ex24.m1.5.5.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex24.m1.5b"><apply id="A1.Ex24.m1.5.5.1.1.cmml" xref="A1.Ex24.m1.5.5.1"><eq id="A1.Ex24.m1.5.5.1.1.2.cmml" xref="A1.Ex24.m1.5.5.1.1.2"></eq><apply id="A1.Ex24.m1.5.5.1.1.3.cmml" xref="A1.Ex24.m1.5.5.1.1.3"><minus id="A1.Ex24.m1.5.5.1.1.3.1.cmml" xref="A1.Ex24.m1.5.5.1.1.3.1"></minus><apply id="A1.Ex24.m1.5.5.1.1.3.2.cmml" xref="A1.Ex24.m1.5.5.1.1.3.2"><times id="A1.Ex24.m1.5.5.1.1.3.2.1.cmml" xref="A1.Ex24.m1.5.5.1.1.3.2.1"></times><apply id="A1.Ex24.m1.5.5.1.1.3.2.2.cmml" xref="A1.Ex24.m1.5.5.1.1.3.2.2"><ci id="A1.Ex24.m1.5.5.1.1.3.2.2.1.cmml" xref="A1.Ex24.m1.5.5.1.1.3.2.2.1">˙</ci><ci id="A1.Ex24.m1.5.5.1.1.3.2.2.2.cmml" xref="A1.Ex24.m1.5.5.1.1.3.2.2.2">𝒑</ci></apply><ci id="A1.Ex24.m1.1.1.cmml" xref="A1.Ex24.m1.1.1">𝑡</ci></apply><apply id="A1.Ex24.m1.5.5.1.1.3.3.cmml" xref="A1.Ex24.m1.5.5.1.1.3.3"><times id="A1.Ex24.m1.5.5.1.1.3.3.1.cmml" xref="A1.Ex24.m1.5.5.1.1.3.3.1"></times><apply id="A1.Ex24.m1.5.5.1.1.3.3.2.cmml" xref="A1.Ex24.m1.5.5.1.1.3.3.2"><csymbol cd="ambiguous" id="A1.Ex24.m1.5.5.1.1.3.3.2.1.cmml" xref="A1.Ex24.m1.5.5.1.1.3.3.2">subscript</csymbol><apply id="A1.Ex24.m1.5.5.1.1.3.3.2.2.cmml" xref="A1.Ex24.m1.5.5.1.1.3.3.2"><csymbol cd="ambiguous" id="A1.Ex24.m1.5.5.1.1.3.3.2.2.1.cmml" xref="A1.Ex24.m1.5.5.1.1.3.3.2">superscript</csymbol><apply id="A1.Ex24.m1.5.5.1.1.3.3.2.2.2.cmml" xref="A1.Ex24.m1.5.5.1.1.3.3.2.2.2"><ci id="A1.Ex24.m1.5.5.1.1.3.3.2.2.2.1.cmml" xref="A1.Ex24.m1.5.5.1.1.3.3.2.2.2.1">˙</ci><ci id="A1.Ex24.m1.5.5.1.1.3.3.2.2.2.2.cmml" xref="A1.Ex24.m1.5.5.1.1.3.3.2.2.2.2">Φ</ci></apply><ci id="A1.Ex24.m1.5.5.1.1.3.3.2.2.3.cmml" xref="A1.Ex24.m1.5.5.1.1.3.3.2.2.3">𝑇</ci></apply><ci id="A1.Ex24.m1.5.5.1.1.3.3.2.3.cmml" xref="A1.Ex24.m1.5.5.1.1.3.3.2.3">s</ci></apply><ci id="A1.Ex24.m1.2.2.cmml" xref="A1.Ex24.m1.2.2">𝑡</ci><ci id="A1.Ex24.m1.5.5.1.1.3.3.4.cmml" xref="A1.Ex24.m1.5.5.1.1.3.3.4">𝜽</ci></apply></apply><apply id="A1.Ex24.m1.5.5.1.1.1.cmml" xref="A1.Ex24.m1.5.5.1.1.1"><times id="A1.Ex24.m1.5.5.1.1.1.2.cmml" xref="A1.Ex24.m1.5.5.1.1.1.2"></times><ci id="A1.Ex24.m1.5.5.1.1.1.3.cmml" xref="A1.Ex24.m1.5.5.1.1.1.3">Λ</ci><apply id="A1.Ex24.m1.5.5.1.1.1.1.1.1.cmml" xref="A1.Ex24.m1.5.5.1.1.1.1.1"><minus id="A1.Ex24.m1.5.5.1.1.1.1.1.1.1.cmml" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.1"></minus><apply id="A1.Ex24.m1.5.5.1.1.1.1.1.1.2.cmml" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.2"><times id="A1.Ex24.m1.5.5.1.1.1.1.1.1.2.1.cmml" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.2.1"></times><ci id="A1.Ex24.m1.5.5.1.1.1.1.1.1.2.2.cmml" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.2.2">𝒑</ci><ci id="A1.Ex24.m1.3.3.cmml" xref="A1.Ex24.m1.3.3">𝑡</ci></apply><apply id="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.cmml" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.3"><times id="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.1.cmml" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.1"></times><apply id="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.2.cmml" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.2.1.cmml" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.2">superscript</csymbol><apply id="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.2.2.cmml" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.2.2.1.cmml" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.2">subscript</csymbol><ci id="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.2.2.2.cmml" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.2.2.2">Φ</ci><ci id="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.2.2.3.cmml" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.2.2.3">s</ci></apply><ci id="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.2.3.cmml" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.2.3">𝑇</ci></apply><ci id="A1.Ex24.m1.4.4.cmml" xref="A1.Ex24.m1.4.4">𝑡</ci><ci id="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.4.cmml" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.3.4">𝜽</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex24.m1.5c">\displaystyle\dot{\bm{p}}(t)-\dot{\Phi}^{T}_{\rm s}(t)\bm{\theta}=\Lambda(\bm{% p}(t)-{\Phi}_{\rm s}^{T}(t)\bm{\theta}).</annotation><annotation encoding="application/x-llamapun" id="A1.Ex24.m1.5d">over˙ start_ARG bold_italic_p end_ARG ( italic_t ) - over˙ start_ARG roman_Φ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_t ) bold_italic_θ = roman_Λ ( bold_italic_p ( italic_t ) - roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_t ) bold_italic_θ ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.p1.5"><span class="ltx_text" id="A1.p1.5.2" style="color:#000099;">Integrating the above resultant equality at [<math alttext="0" class="ltx_Math" display="inline" id="A1.p1.4.1.m1.1"><semantics id="A1.p1.4.1.m1.1a"><mn id="A1.p1.4.1.m1.1.1" mathcolor="#000099" xref="A1.p1.4.1.m1.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="A1.p1.4.1.m1.1b"><cn id="A1.p1.4.1.m1.1.1.cmml" type="integer" xref="A1.p1.4.1.m1.1.1">0</cn></annotation-xml></semantics></math>, <math alttext="t" class="ltx_Math" display="inline" id="A1.p1.5.2.m2.1"><semantics id="A1.p1.5.2.m2.1a"><mi id="A1.p1.5.2.m2.1.1" mathcolor="#000099" xref="A1.p1.5.2.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A1.p1.5.2.m2.1b"><ci id="A1.p1.5.2.m2.1.1.cmml" xref="A1.p1.5.2.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.5.2.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="A1.p1.5.2.m2.1d">italic_t</annotation></semantics></math>], one has</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx33"> <tbody id="A1.Ex25"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle{\color[rgb]{0.00,0.00,0.60}\bm{p}(t)-{\Phi}_{\rm s}^{T}(t)\bm{% \theta}=(\bm{p}(0)-{\Phi}_{\rm s}^{T}(0)\bm{\theta})e^{-\Lambda t}=\bm{% \epsilon}(t).}" class="ltx_Math" display="inline" id="A1.Ex25.m1.6"><semantics id="A1.Ex25.m1.6a"><mrow id="A1.Ex25.m1.6.6.1" xref="A1.Ex25.m1.6.6.1.1.cmml"><mrow id="A1.Ex25.m1.6.6.1.1" xref="A1.Ex25.m1.6.6.1.1.cmml"><mrow id="A1.Ex25.m1.6.6.1.1.3" xref="A1.Ex25.m1.6.6.1.1.3.cmml"><mrow id="A1.Ex25.m1.6.6.1.1.3.2" xref="A1.Ex25.m1.6.6.1.1.3.2.cmml"><mi id="A1.Ex25.m1.6.6.1.1.3.2.2" mathcolor="#000099" xref="A1.Ex25.m1.6.6.1.1.3.2.2.cmml">𝒑</mi><mo id="A1.Ex25.m1.6.6.1.1.3.2.1" xref="A1.Ex25.m1.6.6.1.1.3.2.1.cmml">⁢</mo><mrow id="A1.Ex25.m1.6.6.1.1.3.2.3.2" xref="A1.Ex25.m1.6.6.1.1.3.2.cmml"><mo id="A1.Ex25.m1.6.6.1.1.3.2.3.2.1" mathcolor="#000099" stretchy="false" xref="A1.Ex25.m1.6.6.1.1.3.2.cmml">(</mo><mi id="A1.Ex25.m1.1.1" mathcolor="#000099" xref="A1.Ex25.m1.1.1.cmml">t</mi><mo id="A1.Ex25.m1.6.6.1.1.3.2.3.2.2" mathcolor="#000099" stretchy="false" xref="A1.Ex25.m1.6.6.1.1.3.2.cmml">)</mo></mrow></mrow><mo id="A1.Ex25.m1.6.6.1.1.3.1" mathcolor="#000099" xref="A1.Ex25.m1.6.6.1.1.3.1.cmml">−</mo><mrow id="A1.Ex25.m1.6.6.1.1.3.3" xref="A1.Ex25.m1.6.6.1.1.3.3.cmml"><msubsup id="A1.Ex25.m1.6.6.1.1.3.3.2" xref="A1.Ex25.m1.6.6.1.1.3.3.2.cmml"><mi id="A1.Ex25.m1.6.6.1.1.3.3.2.2.2" mathcolor="#000099" mathvariant="normal" xref="A1.Ex25.m1.6.6.1.1.3.3.2.2.2.cmml">Φ</mi><mi id="A1.Ex25.m1.6.6.1.1.3.3.2.2.3" mathcolor="#000099" mathvariant="normal" xref="A1.Ex25.m1.6.6.1.1.3.3.2.2.3.cmml">s</mi><mi id="A1.Ex25.m1.6.6.1.1.3.3.2.3" mathcolor="#000099" xref="A1.Ex25.m1.6.6.1.1.3.3.2.3.cmml">T</mi></msubsup><mo id="A1.Ex25.m1.6.6.1.1.3.3.1" xref="A1.Ex25.m1.6.6.1.1.3.3.1.cmml">⁢</mo><mrow id="A1.Ex25.m1.6.6.1.1.3.3.3.2" xref="A1.Ex25.m1.6.6.1.1.3.3.cmml"><mo id="A1.Ex25.m1.6.6.1.1.3.3.3.2.1" mathcolor="#000099" stretchy="false" xref="A1.Ex25.m1.6.6.1.1.3.3.cmml">(</mo><mi id="A1.Ex25.m1.2.2" mathcolor="#000099" xref="A1.Ex25.m1.2.2.cmml">t</mi><mo id="A1.Ex25.m1.6.6.1.1.3.3.3.2.2" mathcolor="#000099" stretchy="false" xref="A1.Ex25.m1.6.6.1.1.3.3.cmml">)</mo></mrow><mo id="A1.Ex25.m1.6.6.1.1.3.3.1a" xref="A1.Ex25.m1.6.6.1.1.3.3.1.cmml">⁢</mo><mi id="A1.Ex25.m1.6.6.1.1.3.3.4" mathcolor="#000099" xref="A1.Ex25.m1.6.6.1.1.3.3.4.cmml">𝜽</mi></mrow></mrow><mo id="A1.Ex25.m1.6.6.1.1.4" mathcolor="#000099" xref="A1.Ex25.m1.6.6.1.1.4.cmml">=</mo><mrow id="A1.Ex25.m1.6.6.1.1.1" xref="A1.Ex25.m1.6.6.1.1.1.cmml"><mrow id="A1.Ex25.m1.6.6.1.1.1.1.1" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.cmml"><mo id="A1.Ex25.m1.6.6.1.1.1.1.1.2" mathcolor="#000099" stretchy="false" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.Ex25.m1.6.6.1.1.1.1.1.1" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.cmml"><mrow id="A1.Ex25.m1.6.6.1.1.1.1.1.1.2" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.2.cmml"><mi id="A1.Ex25.m1.6.6.1.1.1.1.1.1.2.2" mathcolor="#000099" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.2.2.cmml">𝒑</mi><mo id="A1.Ex25.m1.6.6.1.1.1.1.1.1.2.1" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.2.1.cmml">⁢</mo><mrow id="A1.Ex25.m1.6.6.1.1.1.1.1.1.2.3.2" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.2.cmml"><mo id="A1.Ex25.m1.6.6.1.1.1.1.1.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.2.cmml">(</mo><mn id="A1.Ex25.m1.3.3" mathcolor="#000099" xref="A1.Ex25.m1.3.3.cmml">0</mn><mo id="A1.Ex25.m1.6.6.1.1.1.1.1.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="A1.Ex25.m1.6.6.1.1.1.1.1.1.1" mathcolor="#000099" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.1.cmml">−</mo><mrow id="A1.Ex25.m1.6.6.1.1.1.1.1.1.3" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.cmml"><msubsup id="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.2" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.2.cmml"><mi id="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.2.2.2" mathcolor="#000099" mathvariant="normal" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.2.2.2.cmml">Φ</mi><mi id="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.2.2.3" mathcolor="#000099" mathvariant="normal" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.2.2.3.cmml">s</mi><mi id="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.2.3" mathcolor="#000099" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.2.3.cmml">T</mi></msubsup><mo id="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.1" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.3.2" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.cmml"><mo id="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.3.2.1" mathcolor="#000099" stretchy="false" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.cmml">(</mo><mn id="A1.Ex25.m1.4.4" mathcolor="#000099" xref="A1.Ex25.m1.4.4.cmml">0</mn><mo id="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.3.2.2" mathcolor="#000099" stretchy="false" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.cmml">)</mo></mrow><mo id="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.1a" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.4" mathcolor="#000099" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.4.cmml">𝜽</mi></mrow></mrow><mo id="A1.Ex25.m1.6.6.1.1.1.1.1.3" mathcolor="#000099" stretchy="false" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A1.Ex25.m1.6.6.1.1.1.2" xref="A1.Ex25.m1.6.6.1.1.1.2.cmml">⁢</mo><msup id="A1.Ex25.m1.6.6.1.1.1.3" xref="A1.Ex25.m1.6.6.1.1.1.3.cmml"><mi id="A1.Ex25.m1.6.6.1.1.1.3.2" mathcolor="#000099" xref="A1.Ex25.m1.6.6.1.1.1.3.2.cmml">e</mi><mrow id="A1.Ex25.m1.6.6.1.1.1.3.3" xref="A1.Ex25.m1.6.6.1.1.1.3.3.cmml"><mo id="A1.Ex25.m1.6.6.1.1.1.3.3a" mathcolor="#000099" xref="A1.Ex25.m1.6.6.1.1.1.3.3.cmml">−</mo><mrow id="A1.Ex25.m1.6.6.1.1.1.3.3.2" xref="A1.Ex25.m1.6.6.1.1.1.3.3.2.cmml"><mi id="A1.Ex25.m1.6.6.1.1.1.3.3.2.2" mathcolor="#000099" mathvariant="normal" xref="A1.Ex25.m1.6.6.1.1.1.3.3.2.2.cmml">Λ</mi><mo id="A1.Ex25.m1.6.6.1.1.1.3.3.2.1" xref="A1.Ex25.m1.6.6.1.1.1.3.3.2.1.cmml">⁢</mo><mi id="A1.Ex25.m1.6.6.1.1.1.3.3.2.3" mathcolor="#000099" xref="A1.Ex25.m1.6.6.1.1.1.3.3.2.3.cmml">t</mi></mrow></mrow></msup></mrow><mo id="A1.Ex25.m1.6.6.1.1.5" mathcolor="#000099" xref="A1.Ex25.m1.6.6.1.1.5.cmml">=</mo><mrow id="A1.Ex25.m1.6.6.1.1.6" xref="A1.Ex25.m1.6.6.1.1.6.cmml"><mi class="ltx_mathvariant_bold-italic" id="A1.Ex25.m1.6.6.1.1.6.2" mathcolor="#000099" mathvariant="bold-italic" xref="A1.Ex25.m1.6.6.1.1.6.2.cmml">ϵ</mi><mo id="A1.Ex25.m1.6.6.1.1.6.1" xref="A1.Ex25.m1.6.6.1.1.6.1.cmml">⁢</mo><mrow id="A1.Ex25.m1.6.6.1.1.6.3.2" xref="A1.Ex25.m1.6.6.1.1.6.cmml"><mo id="A1.Ex25.m1.6.6.1.1.6.3.2.1" mathcolor="#000099" stretchy="false" xref="A1.Ex25.m1.6.6.1.1.6.cmml">(</mo><mi id="A1.Ex25.m1.5.5" mathcolor="#000099" xref="A1.Ex25.m1.5.5.cmml">t</mi><mo id="A1.Ex25.m1.6.6.1.1.6.3.2.2" mathcolor="#000099" stretchy="false" xref="A1.Ex25.m1.6.6.1.1.6.cmml">)</mo></mrow></mrow></mrow><mo id="A1.Ex25.m1.6.6.1.2" lspace="0em" mathcolor="#000099" xref="A1.Ex25.m1.6.6.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex25.m1.6b"><apply id="A1.Ex25.m1.6.6.1.1.cmml" xref="A1.Ex25.m1.6.6.1"><and id="A1.Ex25.m1.6.6.1.1a.cmml" xref="A1.Ex25.m1.6.6.1"></and><apply id="A1.Ex25.m1.6.6.1.1b.cmml" xref="A1.Ex25.m1.6.6.1"><eq id="A1.Ex25.m1.6.6.1.1.4.cmml" xref="A1.Ex25.m1.6.6.1.1.4"></eq><apply id="A1.Ex25.m1.6.6.1.1.3.cmml" xref="A1.Ex25.m1.6.6.1.1.3"><minus id="A1.Ex25.m1.6.6.1.1.3.1.cmml" xref="A1.Ex25.m1.6.6.1.1.3.1"></minus><apply id="A1.Ex25.m1.6.6.1.1.3.2.cmml" xref="A1.Ex25.m1.6.6.1.1.3.2"><times id="A1.Ex25.m1.6.6.1.1.3.2.1.cmml" xref="A1.Ex25.m1.6.6.1.1.3.2.1"></times><ci id="A1.Ex25.m1.6.6.1.1.3.2.2.cmml" xref="A1.Ex25.m1.6.6.1.1.3.2.2">𝒑</ci><ci id="A1.Ex25.m1.1.1.cmml" xref="A1.Ex25.m1.1.1">𝑡</ci></apply><apply id="A1.Ex25.m1.6.6.1.1.3.3.cmml" xref="A1.Ex25.m1.6.6.1.1.3.3"><times id="A1.Ex25.m1.6.6.1.1.3.3.1.cmml" xref="A1.Ex25.m1.6.6.1.1.3.3.1"></times><apply id="A1.Ex25.m1.6.6.1.1.3.3.2.cmml" xref="A1.Ex25.m1.6.6.1.1.3.3.2"><csymbol cd="ambiguous" id="A1.Ex25.m1.6.6.1.1.3.3.2.1.cmml" xref="A1.Ex25.m1.6.6.1.1.3.3.2">superscript</csymbol><apply id="A1.Ex25.m1.6.6.1.1.3.3.2.2.cmml" xref="A1.Ex25.m1.6.6.1.1.3.3.2"><csymbol cd="ambiguous" id="A1.Ex25.m1.6.6.1.1.3.3.2.2.1.cmml" xref="A1.Ex25.m1.6.6.1.1.3.3.2">subscript</csymbol><ci id="A1.Ex25.m1.6.6.1.1.3.3.2.2.2.cmml" xref="A1.Ex25.m1.6.6.1.1.3.3.2.2.2">Φ</ci><ci id="A1.Ex25.m1.6.6.1.1.3.3.2.2.3.cmml" xref="A1.Ex25.m1.6.6.1.1.3.3.2.2.3">s</ci></apply><ci id="A1.Ex25.m1.6.6.1.1.3.3.2.3.cmml" xref="A1.Ex25.m1.6.6.1.1.3.3.2.3">𝑇</ci></apply><ci id="A1.Ex25.m1.2.2.cmml" xref="A1.Ex25.m1.2.2">𝑡</ci><ci id="A1.Ex25.m1.6.6.1.1.3.3.4.cmml" xref="A1.Ex25.m1.6.6.1.1.3.3.4">𝜽</ci></apply></apply><apply id="A1.Ex25.m1.6.6.1.1.1.cmml" xref="A1.Ex25.m1.6.6.1.1.1"><times id="A1.Ex25.m1.6.6.1.1.1.2.cmml" xref="A1.Ex25.m1.6.6.1.1.1.2"></times><apply id="A1.Ex25.m1.6.6.1.1.1.1.1.1.cmml" xref="A1.Ex25.m1.6.6.1.1.1.1.1"><minus id="A1.Ex25.m1.6.6.1.1.1.1.1.1.1.cmml" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.1"></minus><apply id="A1.Ex25.m1.6.6.1.1.1.1.1.1.2.cmml" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.2"><times id="A1.Ex25.m1.6.6.1.1.1.1.1.1.2.1.cmml" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.2.1"></times><ci id="A1.Ex25.m1.6.6.1.1.1.1.1.1.2.2.cmml" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.2.2">𝒑</ci><cn id="A1.Ex25.m1.3.3.cmml" type="integer" xref="A1.Ex25.m1.3.3">0</cn></apply><apply id="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.cmml" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.3"><times id="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.1.cmml" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.1"></times><apply id="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.2.cmml" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.2.1.cmml" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.2">superscript</csymbol><apply id="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.2.2.cmml" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.2.2.1.cmml" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.2">subscript</csymbol><ci id="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.2.2.2.cmml" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.2.2.2">Φ</ci><ci id="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.2.2.3.cmml" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.2.2.3">s</ci></apply><ci id="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.2.3.cmml" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.2.3">𝑇</ci></apply><cn id="A1.Ex25.m1.4.4.cmml" type="integer" xref="A1.Ex25.m1.4.4">0</cn><ci id="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.4.cmml" xref="A1.Ex25.m1.6.6.1.1.1.1.1.1.3.4">𝜽</ci></apply></apply><apply id="A1.Ex25.m1.6.6.1.1.1.3.cmml" xref="A1.Ex25.m1.6.6.1.1.1.3"><csymbol cd="ambiguous" id="A1.Ex25.m1.6.6.1.1.1.3.1.cmml" xref="A1.Ex25.m1.6.6.1.1.1.3">superscript</csymbol><ci id="A1.Ex25.m1.6.6.1.1.1.3.2.cmml" xref="A1.Ex25.m1.6.6.1.1.1.3.2">𝑒</ci><apply id="A1.Ex25.m1.6.6.1.1.1.3.3.cmml" xref="A1.Ex25.m1.6.6.1.1.1.3.3"><minus id="A1.Ex25.m1.6.6.1.1.1.3.3.1.cmml" xref="A1.Ex25.m1.6.6.1.1.1.3.3"></minus><apply id="A1.Ex25.m1.6.6.1.1.1.3.3.2.cmml" xref="A1.Ex25.m1.6.6.1.1.1.3.3.2"><times id="A1.Ex25.m1.6.6.1.1.1.3.3.2.1.cmml" xref="A1.Ex25.m1.6.6.1.1.1.3.3.2.1"></times><ci id="A1.Ex25.m1.6.6.1.1.1.3.3.2.2.cmml" xref="A1.Ex25.m1.6.6.1.1.1.3.3.2.2">Λ</ci><ci id="A1.Ex25.m1.6.6.1.1.1.3.3.2.3.cmml" xref="A1.Ex25.m1.6.6.1.1.1.3.3.2.3">𝑡</ci></apply></apply></apply></apply></apply><apply id="A1.Ex25.m1.6.6.1.1c.cmml" xref="A1.Ex25.m1.6.6.1"><eq id="A1.Ex25.m1.6.6.1.1.5.cmml" xref="A1.Ex25.m1.6.6.1.1.5"></eq><share href="https://arxiv.org/html/2401.10785v2#A1.Ex25.m1.6.6.1.1.1.cmml" id="A1.Ex25.m1.6.6.1.1d.cmml" xref="A1.Ex25.m1.6.6.1"></share><apply id="A1.Ex25.m1.6.6.1.1.6.cmml" xref="A1.Ex25.m1.6.6.1.1.6"><times id="A1.Ex25.m1.6.6.1.1.6.1.cmml" xref="A1.Ex25.m1.6.6.1.1.6.1"></times><ci id="A1.Ex25.m1.6.6.1.1.6.2.cmml" xref="A1.Ex25.m1.6.6.1.1.6.2">bold-italic-ϵ</ci><ci id="A1.Ex25.m1.5.5.cmml" xref="A1.Ex25.m1.5.5">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex25.m1.6c">\displaystyle{\color[rgb]{0.00,0.00,0.60}\bm{p}(t)-{\Phi}_{\rm s}^{T}(t)\bm{% \theta}=(\bm{p}(0)-{\Phi}_{\rm s}^{T}(0)\bm{\theta})e^{-\Lambda t}=\bm{% \epsilon}(t).}</annotation><annotation encoding="application/x-llamapun" id="A1.Ex25.m1.6d">bold_italic_p ( italic_t ) - roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_t ) bold_italic_θ = ( bold_italic_p ( 0 ) - roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( 0 ) bold_italic_θ ) italic_e start_POSTSUPERSCRIPT - roman_Λ italic_t end_POSTSUPERSCRIPT = bold_italic_ϵ ( italic_t ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </section> <section class="ltx_appendix" id="A2"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix B </span>The proof of Theorem 1</h2> <div class="ltx_proof" id="A2.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A2.1.p1"> <p class="ltx_p" id="A2.1.p1.3">1) Since one has <math alttext="\bm{\xi}=Q(t,t_{\rm e})\tilde{\bm{\theta}}" class="ltx_Math" display="inline" id="A2.1.p1.1.m1.2"><semantics id="A2.1.p1.1.m1.2a"><mrow id="A2.1.p1.1.m1.2.2" xref="A2.1.p1.1.m1.2.2.cmml"><mi id="A2.1.p1.1.m1.2.2.3" xref="A2.1.p1.1.m1.2.2.3.cmml">𝝃</mi><mo id="A2.1.p1.1.m1.2.2.2" xref="A2.1.p1.1.m1.2.2.2.cmml">=</mo><mrow id="A2.1.p1.1.m1.2.2.1" xref="A2.1.p1.1.m1.2.2.1.cmml"><mi id="A2.1.p1.1.m1.2.2.1.3" xref="A2.1.p1.1.m1.2.2.1.3.cmml">Q</mi><mo id="A2.1.p1.1.m1.2.2.1.2" xref="A2.1.p1.1.m1.2.2.1.2.cmml">⁢</mo><mrow id="A2.1.p1.1.m1.2.2.1.1.1" xref="A2.1.p1.1.m1.2.2.1.1.2.cmml"><mo id="A2.1.p1.1.m1.2.2.1.1.1.2" stretchy="false" xref="A2.1.p1.1.m1.2.2.1.1.2.cmml">(</mo><mi id="A2.1.p1.1.m1.1.1" xref="A2.1.p1.1.m1.1.1.cmml">t</mi><mo id="A2.1.p1.1.m1.2.2.1.1.1.3" xref="A2.1.p1.1.m1.2.2.1.1.2.cmml">,</mo><msub id="A2.1.p1.1.m1.2.2.1.1.1.1" xref="A2.1.p1.1.m1.2.2.1.1.1.1.cmml"><mi id="A2.1.p1.1.m1.2.2.1.1.1.1.2" xref="A2.1.p1.1.m1.2.2.1.1.1.1.2.cmml">t</mi><mi id="A2.1.p1.1.m1.2.2.1.1.1.1.3" mathvariant="normal" xref="A2.1.p1.1.m1.2.2.1.1.1.1.3.cmml">e</mi></msub><mo id="A2.1.p1.1.m1.2.2.1.1.1.4" stretchy="false" xref="A2.1.p1.1.m1.2.2.1.1.2.cmml">)</mo></mrow><mo id="A2.1.p1.1.m1.2.2.1.2a" xref="A2.1.p1.1.m1.2.2.1.2.cmml">⁢</mo><mover accent="true" id="A2.1.p1.1.m1.2.2.1.4" xref="A2.1.p1.1.m1.2.2.1.4.cmml"><mi id="A2.1.p1.1.m1.2.2.1.4.2" xref="A2.1.p1.1.m1.2.2.1.4.2.cmml">𝜽</mi><mo id="A2.1.p1.1.m1.2.2.1.4.1" xref="A2.1.p1.1.m1.2.2.1.4.1.cmml">~</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.1.m1.2b"><apply id="A2.1.p1.1.m1.2.2.cmml" xref="A2.1.p1.1.m1.2.2"><eq id="A2.1.p1.1.m1.2.2.2.cmml" xref="A2.1.p1.1.m1.2.2.2"></eq><ci id="A2.1.p1.1.m1.2.2.3.cmml" xref="A2.1.p1.1.m1.2.2.3">𝝃</ci><apply id="A2.1.p1.1.m1.2.2.1.cmml" xref="A2.1.p1.1.m1.2.2.1"><times id="A2.1.p1.1.m1.2.2.1.2.cmml" xref="A2.1.p1.1.m1.2.2.1.2"></times><ci id="A2.1.p1.1.m1.2.2.1.3.cmml" xref="A2.1.p1.1.m1.2.2.1.3">𝑄</ci><interval closure="open" id="A2.1.p1.1.m1.2.2.1.1.2.cmml" xref="A2.1.p1.1.m1.2.2.1.1.1"><ci id="A2.1.p1.1.m1.1.1.cmml" xref="A2.1.p1.1.m1.1.1">𝑡</ci><apply id="A2.1.p1.1.m1.2.2.1.1.1.1.cmml" xref="A2.1.p1.1.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="A2.1.p1.1.m1.2.2.1.1.1.1.1.cmml" xref="A2.1.p1.1.m1.2.2.1.1.1.1">subscript</csymbol><ci id="A2.1.p1.1.m1.2.2.1.1.1.1.2.cmml" xref="A2.1.p1.1.m1.2.2.1.1.1.1.2">𝑡</ci><ci id="A2.1.p1.1.m1.2.2.1.1.1.1.3.cmml" xref="A2.1.p1.1.m1.2.2.1.1.1.1.3">e</ci></apply></interval><apply id="A2.1.p1.1.m1.2.2.1.4.cmml" xref="A2.1.p1.1.m1.2.2.1.4"><ci id="A2.1.p1.1.m1.2.2.1.4.1.cmml" xref="A2.1.p1.1.m1.2.2.1.4.1">~</ci><ci id="A2.1.p1.1.m1.2.2.1.4.2.cmml" xref="A2.1.p1.1.m1.2.2.1.4.2">𝜽</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.1.m1.2c">\bm{\xi}=Q(t,t_{\rm e})\tilde{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.1.m1.2d">bold_italic_ξ = italic_Q ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math>, <math alttext="\bm{\epsilon}=\Phi_{\rm f}^{T}\tilde{\bm{\theta}}" class="ltx_Math" display="inline" id="A2.1.p1.2.m2.1"><semantics id="A2.1.p1.2.m2.1a"><mrow id="A2.1.p1.2.m2.1.1" xref="A2.1.p1.2.m2.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="A2.1.p1.2.m2.1.1.2" mathvariant="bold-italic" xref="A2.1.p1.2.m2.1.1.2.cmml">ϵ</mi><mo id="A2.1.p1.2.m2.1.1.1" xref="A2.1.p1.2.m2.1.1.1.cmml">=</mo><mrow id="A2.1.p1.2.m2.1.1.3" xref="A2.1.p1.2.m2.1.1.3.cmml"><msubsup id="A2.1.p1.2.m2.1.1.3.2" xref="A2.1.p1.2.m2.1.1.3.2.cmml"><mi id="A2.1.p1.2.m2.1.1.3.2.2.2" mathvariant="normal" xref="A2.1.p1.2.m2.1.1.3.2.2.2.cmml">Φ</mi><mi id="A2.1.p1.2.m2.1.1.3.2.2.3" mathvariant="normal" xref="A2.1.p1.2.m2.1.1.3.2.2.3.cmml">f</mi><mi id="A2.1.p1.2.m2.1.1.3.2.3" xref="A2.1.p1.2.m2.1.1.3.2.3.cmml">T</mi></msubsup><mo id="A2.1.p1.2.m2.1.1.3.1" xref="A2.1.p1.2.m2.1.1.3.1.cmml">⁢</mo><mover accent="true" id="A2.1.p1.2.m2.1.1.3.3" xref="A2.1.p1.2.m2.1.1.3.3.cmml"><mi id="A2.1.p1.2.m2.1.1.3.3.2" xref="A2.1.p1.2.m2.1.1.3.3.2.cmml">𝜽</mi><mo id="A2.1.p1.2.m2.1.1.3.3.1" xref="A2.1.p1.2.m2.1.1.3.3.1.cmml">~</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.2.m2.1b"><apply id="A2.1.p1.2.m2.1.1.cmml" xref="A2.1.p1.2.m2.1.1"><eq id="A2.1.p1.2.m2.1.1.1.cmml" xref="A2.1.p1.2.m2.1.1.1"></eq><ci id="A2.1.p1.2.m2.1.1.2.cmml" xref="A2.1.p1.2.m2.1.1.2">bold-italic-ϵ</ci><apply id="A2.1.p1.2.m2.1.1.3.cmml" xref="A2.1.p1.2.m2.1.1.3"><times id="A2.1.p1.2.m2.1.1.3.1.cmml" xref="A2.1.p1.2.m2.1.1.3.1"></times><apply id="A2.1.p1.2.m2.1.1.3.2.cmml" xref="A2.1.p1.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="A2.1.p1.2.m2.1.1.3.2.1.cmml" xref="A2.1.p1.2.m2.1.1.3.2">superscript</csymbol><apply id="A2.1.p1.2.m2.1.1.3.2.2.cmml" xref="A2.1.p1.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="A2.1.p1.2.m2.1.1.3.2.2.1.cmml" xref="A2.1.p1.2.m2.1.1.3.2">subscript</csymbol><ci id="A2.1.p1.2.m2.1.1.3.2.2.2.cmml" xref="A2.1.p1.2.m2.1.1.3.2.2.2">Φ</ci><ci id="A2.1.p1.2.m2.1.1.3.2.2.3.cmml" xref="A2.1.p1.2.m2.1.1.3.2.2.3">f</ci></apply><ci id="A2.1.p1.2.m2.1.1.3.2.3.cmml" xref="A2.1.p1.2.m2.1.1.3.2.3">𝑇</ci></apply><apply id="A2.1.p1.2.m2.1.1.3.3.cmml" xref="A2.1.p1.2.m2.1.1.3.3"><ci id="A2.1.p1.2.m2.1.1.3.3.1.cmml" xref="A2.1.p1.2.m2.1.1.3.3.1">~</ci><ci id="A2.1.p1.2.m2.1.1.3.3.2.cmml" xref="A2.1.p1.2.m2.1.1.3.3.2">𝜽</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.2.m2.1c">\bm{\epsilon}=\Phi_{\rm f}^{T}\tilde{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.2.m2.1d">bold_italic_ϵ = roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math>, and <math alttext="\dot{\tilde{\bm{\theta}}}=-\dot{\hat{\bm{\theta}}}" class="ltx_Math" display="inline" id="A2.1.p1.3.m3.1"><semantics id="A2.1.p1.3.m3.1a"><mrow id="A2.1.p1.3.m3.1.1" xref="A2.1.p1.3.m3.1.1.cmml"><mover accent="true" id="A2.1.p1.3.m3.1.1.2" xref="A2.1.p1.3.m3.1.1.2.cmml"><mover accent="true" id="A2.1.p1.3.m3.1.1.2.2" xref="A2.1.p1.3.m3.1.1.2.2.cmml"><mi id="A2.1.p1.3.m3.1.1.2.2.2" xref="A2.1.p1.3.m3.1.1.2.2.2.cmml">𝜽</mi><mo id="A2.1.p1.3.m3.1.1.2.2.1" xref="A2.1.p1.3.m3.1.1.2.2.1.cmml">~</mo></mover><mo id="A2.1.p1.3.m3.1.1.2.1" xref="A2.1.p1.3.m3.1.1.2.1.cmml">˙</mo></mover><mo id="A2.1.p1.3.m3.1.1.1" xref="A2.1.p1.3.m3.1.1.1.cmml">=</mo><mrow id="A2.1.p1.3.m3.1.1.3" xref="A2.1.p1.3.m3.1.1.3.cmml"><mo id="A2.1.p1.3.m3.1.1.3a" xref="A2.1.p1.3.m3.1.1.3.cmml">−</mo><mover accent="true" id="A2.1.p1.3.m3.1.1.3.2" xref="A2.1.p1.3.m3.1.1.3.2.cmml"><mover accent="true" id="A2.1.p1.3.m3.1.1.3.2.2" xref="A2.1.p1.3.m3.1.1.3.2.2.cmml"><mi id="A2.1.p1.3.m3.1.1.3.2.2.2" xref="A2.1.p1.3.m3.1.1.3.2.2.2.cmml">𝜽</mi><mo id="A2.1.p1.3.m3.1.1.3.2.2.1" xref="A2.1.p1.3.m3.1.1.3.2.2.1.cmml">^</mo></mover><mo id="A2.1.p1.3.m3.1.1.3.2.1" xref="A2.1.p1.3.m3.1.1.3.2.1.cmml">˙</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.3.m3.1b"><apply id="A2.1.p1.3.m3.1.1.cmml" xref="A2.1.p1.3.m3.1.1"><eq id="A2.1.p1.3.m3.1.1.1.cmml" xref="A2.1.p1.3.m3.1.1.1"></eq><apply id="A2.1.p1.3.m3.1.1.2.cmml" xref="A2.1.p1.3.m3.1.1.2"><ci id="A2.1.p1.3.m3.1.1.2.1.cmml" xref="A2.1.p1.3.m3.1.1.2.1">˙</ci><apply id="A2.1.p1.3.m3.1.1.2.2.cmml" xref="A2.1.p1.3.m3.1.1.2.2"><ci id="A2.1.p1.3.m3.1.1.2.2.1.cmml" xref="A2.1.p1.3.m3.1.1.2.2.1">~</ci><ci id="A2.1.p1.3.m3.1.1.2.2.2.cmml" xref="A2.1.p1.3.m3.1.1.2.2.2">𝜽</ci></apply></apply><apply id="A2.1.p1.3.m3.1.1.3.cmml" xref="A2.1.p1.3.m3.1.1.3"><minus id="A2.1.p1.3.m3.1.1.3.1.cmml" xref="A2.1.p1.3.m3.1.1.3"></minus><apply id="A2.1.p1.3.m3.1.1.3.2.cmml" xref="A2.1.p1.3.m3.1.1.3.2"><ci id="A2.1.p1.3.m3.1.1.3.2.1.cmml" xref="A2.1.p1.3.m3.1.1.3.2.1">˙</ci><apply id="A2.1.p1.3.m3.1.1.3.2.2.cmml" xref="A2.1.p1.3.m3.1.1.3.2.2"><ci id="A2.1.p1.3.m3.1.1.3.2.2.1.cmml" xref="A2.1.p1.3.m3.1.1.3.2.2.1">^</ci><ci id="A2.1.p1.3.m3.1.1.3.2.2.2.cmml" xref="A2.1.p1.3.m3.1.1.3.2.2.2">𝜽</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.3.m3.1c">\dot{\tilde{\bm{\theta}}}=-\dot{\hat{\bm{\theta}}}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.3.m3.1d">over˙ start_ARG over~ start_ARG bold_italic_θ end_ARG end_ARG = - over˙ start_ARG over^ start_ARG bold_italic_θ end_ARG end_ARG</annotation></semantics></math>, the composite learning law (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>) becomes</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx34"> <tbody id="A2.E33"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{\tilde{\bm{\theta}}}=-\Gamma(\Phi_{\rm f}\Phi_{\rm f}^{T}% \tilde{\bm{\theta}}+\kappa Q(t,t_{\rm e})\tilde{\bm{\theta}})." class="ltx_Math" display="inline" id="A2.E33.m1.2"><semantics id="A2.E33.m1.2a"><mrow id="A2.E33.m1.2.2.1" xref="A2.E33.m1.2.2.1.1.cmml"><mrow id="A2.E33.m1.2.2.1.1" xref="A2.E33.m1.2.2.1.1.cmml"><mover accent="true" id="A2.E33.m1.2.2.1.1.3" xref="A2.E33.m1.2.2.1.1.3.cmml"><mover accent="true" id="A2.E33.m1.2.2.1.1.3.2" xref="A2.E33.m1.2.2.1.1.3.2.cmml"><mi id="A2.E33.m1.2.2.1.1.3.2.2" xref="A2.E33.m1.2.2.1.1.3.2.2.cmml">𝜽</mi><mo id="A2.E33.m1.2.2.1.1.3.2.1" xref="A2.E33.m1.2.2.1.1.3.2.1.cmml">~</mo></mover><mo id="A2.E33.m1.2.2.1.1.3.1" xref="A2.E33.m1.2.2.1.1.3.1.cmml">˙</mo></mover><mo id="A2.E33.m1.2.2.1.1.2" xref="A2.E33.m1.2.2.1.1.2.cmml">=</mo><mrow id="A2.E33.m1.2.2.1.1.1" xref="A2.E33.m1.2.2.1.1.1.cmml"><mo id="A2.E33.m1.2.2.1.1.1a" xref="A2.E33.m1.2.2.1.1.1.cmml">−</mo><mrow id="A2.E33.m1.2.2.1.1.1.1" xref="A2.E33.m1.2.2.1.1.1.1.cmml"><mi id="A2.E33.m1.2.2.1.1.1.1.3" mathvariant="normal" xref="A2.E33.m1.2.2.1.1.1.1.3.cmml">Γ</mi><mo id="A2.E33.m1.2.2.1.1.1.1.2" xref="A2.E33.m1.2.2.1.1.1.1.2.cmml">⁢</mo><mrow id="A2.E33.m1.2.2.1.1.1.1.1.1" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.cmml"><mo id="A2.E33.m1.2.2.1.1.1.1.1.1.2" stretchy="false" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A2.E33.m1.2.2.1.1.1.1.1.1.1" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.cmml"><mrow id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.cmml"><msub id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.2" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.2.cmml"><mi id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.2.2" mathvariant="normal" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.2.2.cmml">Φ</mi><mi id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.2.3" mathvariant="normal" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.2.3.cmml">f</mi></msub><mo id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.1" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><msubsup id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.3" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.3.cmml"><mi id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.3.2.2" mathvariant="normal" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.3.2.2.cmml">Φ</mi><mi id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.3.2.3" mathvariant="normal" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.3.2.3.cmml">f</mi><mi id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.3.3" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.3.3.cmml">T</mi></msubsup><mo id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.1a" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mover accent="true" id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.4" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.4.cmml"><mi id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.4.2" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.4.2.cmml">𝜽</mi><mo id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.4.1" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.4.1.cmml">~</mo></mover></mrow><mo id="A2.E33.m1.2.2.1.1.1.1.1.1.1.2" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.2.cmml">+</mo><mrow id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.cmml"><mi id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.3" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.3.cmml">κ</mi><mo id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.2" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mi id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.4" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.4.cmml">Q</mi><mo id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.2a" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.1.1" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.1.2.cmml"><mo id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.1.2.cmml">(</mo><mi id="A2.E33.m1.1.1" xref="A2.E33.m1.1.1.cmml">t</mi><mo id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.1.1.3" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.1.2.cmml">,</mo><msub id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.1.1.1" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.1.1.1.2" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.1.1.1.2.cmml">t</mi><mi id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.1.1.1.3" mathvariant="normal" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.1.1.4" stretchy="false" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.1.2.cmml">)</mo></mrow><mo id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.2b" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mover accent="true" id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.5" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.5.cmml"><mi id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.5.2" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.5.2.cmml">𝜽</mi><mo id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.5.1" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.5.1.cmml">~</mo></mover></mrow></mrow><mo id="A2.E33.m1.2.2.1.1.1.1.1.1.3" stretchy="false" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="A2.E33.m1.2.2.1.2" lspace="0em" xref="A2.E33.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.E33.m1.2b"><apply id="A2.E33.m1.2.2.1.1.cmml" xref="A2.E33.m1.2.2.1"><eq id="A2.E33.m1.2.2.1.1.2.cmml" xref="A2.E33.m1.2.2.1.1.2"></eq><apply id="A2.E33.m1.2.2.1.1.3.cmml" xref="A2.E33.m1.2.2.1.1.3"><ci id="A2.E33.m1.2.2.1.1.3.1.cmml" xref="A2.E33.m1.2.2.1.1.3.1">˙</ci><apply id="A2.E33.m1.2.2.1.1.3.2.cmml" xref="A2.E33.m1.2.2.1.1.3.2"><ci id="A2.E33.m1.2.2.1.1.3.2.1.cmml" xref="A2.E33.m1.2.2.1.1.3.2.1">~</ci><ci id="A2.E33.m1.2.2.1.1.3.2.2.cmml" xref="A2.E33.m1.2.2.1.1.3.2.2">𝜽</ci></apply></apply><apply id="A2.E33.m1.2.2.1.1.1.cmml" xref="A2.E33.m1.2.2.1.1.1"><minus id="A2.E33.m1.2.2.1.1.1.2.cmml" xref="A2.E33.m1.2.2.1.1.1"></minus><apply id="A2.E33.m1.2.2.1.1.1.1.cmml" xref="A2.E33.m1.2.2.1.1.1.1"><times id="A2.E33.m1.2.2.1.1.1.1.2.cmml" xref="A2.E33.m1.2.2.1.1.1.1.2"></times><ci id="A2.E33.m1.2.2.1.1.1.1.3.cmml" xref="A2.E33.m1.2.2.1.1.1.1.3">Γ</ci><apply id="A2.E33.m1.2.2.1.1.1.1.1.1.1.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1"><plus id="A2.E33.m1.2.2.1.1.1.1.1.1.1.2.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.2"></plus><apply id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3"><times id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.1.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.1"></times><apply id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.2.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.2.1.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.2">subscript</csymbol><ci id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.2.2.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.2.2">Φ</ci><ci id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.2.3.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.2.3">f</ci></apply><apply id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.3.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.3.1.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.3">superscript</csymbol><apply id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.3.2.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.3.2.1.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.3">subscript</csymbol><ci id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.3.2.2.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.3.2.2">Φ</ci><ci id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.3.2.3.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.3.2.3">f</ci></apply><ci id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.3.3.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.3.3">𝑇</ci></apply><apply id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.4.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.4"><ci id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.4.1.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.4.1">~</ci><ci id="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.4.2.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.3.4.2">𝜽</ci></apply></apply><apply id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1"><times id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.2.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.2"></times><ci id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.3.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.3">𝜅</ci><ci id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.4.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.4">𝑄</ci><interval closure="open" id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.1.2.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.1.1"><ci id="A2.E33.m1.1.1.cmml" xref="A2.E33.m1.1.1">𝑡</ci><apply id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.1.1.1.2">𝑡</ci><ci id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.1.1.1.3">e</ci></apply></interval><apply id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.5.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.5"><ci id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.5.1.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.5.1">~</ci><ci id="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.5.2.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1.1.1.1.5.2">𝜽</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E33.m1.2c">\displaystyle\dot{\tilde{\bm{\theta}}}=-\Gamma(\Phi_{\rm f}\Phi_{\rm f}^{T}% \tilde{\bm{\theta}}+\kappa Q(t,t_{\rm e})\tilde{\bm{\theta}}).</annotation><annotation encoding="application/x-llamapun" id="A2.E33.m1.2d">over˙ start_ARG over~ start_ARG bold_italic_θ end_ARG end_ARG = - roman_Γ ( roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG + italic_κ italic_Q ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(33)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.1.p1.36">Choose a Lyapunov function candidate</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx35"> <tbody id="A2.E34"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle V_{\theta}=\tilde{\bm{\theta}}^{T}\Gamma^{-1}\tilde{\bm{\theta}}." class="ltx_Math" display="inline" id="A2.E34.m1.1"><semantics id="A2.E34.m1.1a"><mrow id="A2.E34.m1.1.1.1" xref="A2.E34.m1.1.1.1.1.cmml"><mrow id="A2.E34.m1.1.1.1.1" xref="A2.E34.m1.1.1.1.1.cmml"><msub id="A2.E34.m1.1.1.1.1.2" xref="A2.E34.m1.1.1.1.1.2.cmml"><mi id="A2.E34.m1.1.1.1.1.2.2" xref="A2.E34.m1.1.1.1.1.2.2.cmml">V</mi><mi id="A2.E34.m1.1.1.1.1.2.3" xref="A2.E34.m1.1.1.1.1.2.3.cmml">θ</mi></msub><mo id="A2.E34.m1.1.1.1.1.1" xref="A2.E34.m1.1.1.1.1.1.cmml">=</mo><mrow id="A2.E34.m1.1.1.1.1.3" xref="A2.E34.m1.1.1.1.1.3.cmml"><msup id="A2.E34.m1.1.1.1.1.3.2" xref="A2.E34.m1.1.1.1.1.3.2.cmml"><mover accent="true" id="A2.E34.m1.1.1.1.1.3.2.2" xref="A2.E34.m1.1.1.1.1.3.2.2.cmml"><mi id="A2.E34.m1.1.1.1.1.3.2.2.2" xref="A2.E34.m1.1.1.1.1.3.2.2.2.cmml">𝜽</mi><mo id="A2.E34.m1.1.1.1.1.3.2.2.1" xref="A2.E34.m1.1.1.1.1.3.2.2.1.cmml">~</mo></mover><mi id="A2.E34.m1.1.1.1.1.3.2.3" xref="A2.E34.m1.1.1.1.1.3.2.3.cmml">T</mi></msup><mo id="A2.E34.m1.1.1.1.1.3.1" xref="A2.E34.m1.1.1.1.1.3.1.cmml">⁢</mo><msup id="A2.E34.m1.1.1.1.1.3.3" xref="A2.E34.m1.1.1.1.1.3.3.cmml"><mi id="A2.E34.m1.1.1.1.1.3.3.2" mathvariant="normal" xref="A2.E34.m1.1.1.1.1.3.3.2.cmml">Γ</mi><mrow id="A2.E34.m1.1.1.1.1.3.3.3" xref="A2.E34.m1.1.1.1.1.3.3.3.cmml"><mo id="A2.E34.m1.1.1.1.1.3.3.3a" xref="A2.E34.m1.1.1.1.1.3.3.3.cmml">−</mo><mn id="A2.E34.m1.1.1.1.1.3.3.3.2" xref="A2.E34.m1.1.1.1.1.3.3.3.2.cmml">1</mn></mrow></msup><mo id="A2.E34.m1.1.1.1.1.3.1a" xref="A2.E34.m1.1.1.1.1.3.1.cmml">⁢</mo><mover accent="true" id="A2.E34.m1.1.1.1.1.3.4" xref="A2.E34.m1.1.1.1.1.3.4.cmml"><mi id="A2.E34.m1.1.1.1.1.3.4.2" xref="A2.E34.m1.1.1.1.1.3.4.2.cmml">𝜽</mi><mo id="A2.E34.m1.1.1.1.1.3.4.1" xref="A2.E34.m1.1.1.1.1.3.4.1.cmml">~</mo></mover></mrow></mrow><mo id="A2.E34.m1.1.1.1.2" lspace="0em" xref="A2.E34.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.E34.m1.1b"><apply id="A2.E34.m1.1.1.1.1.cmml" xref="A2.E34.m1.1.1.1"><eq id="A2.E34.m1.1.1.1.1.1.cmml" xref="A2.E34.m1.1.1.1.1.1"></eq><apply id="A2.E34.m1.1.1.1.1.2.cmml" xref="A2.E34.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="A2.E34.m1.1.1.1.1.2.1.cmml" xref="A2.E34.m1.1.1.1.1.2">subscript</csymbol><ci id="A2.E34.m1.1.1.1.1.2.2.cmml" xref="A2.E34.m1.1.1.1.1.2.2">𝑉</ci><ci id="A2.E34.m1.1.1.1.1.2.3.cmml" xref="A2.E34.m1.1.1.1.1.2.3">𝜃</ci></apply><apply id="A2.E34.m1.1.1.1.1.3.cmml" xref="A2.E34.m1.1.1.1.1.3"><times id="A2.E34.m1.1.1.1.1.3.1.cmml" xref="A2.E34.m1.1.1.1.1.3.1"></times><apply id="A2.E34.m1.1.1.1.1.3.2.cmml" xref="A2.E34.m1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A2.E34.m1.1.1.1.1.3.2.1.cmml" xref="A2.E34.m1.1.1.1.1.3.2">superscript</csymbol><apply id="A2.E34.m1.1.1.1.1.3.2.2.cmml" xref="A2.E34.m1.1.1.1.1.3.2.2"><ci id="A2.E34.m1.1.1.1.1.3.2.2.1.cmml" xref="A2.E34.m1.1.1.1.1.3.2.2.1">~</ci><ci id="A2.E34.m1.1.1.1.1.3.2.2.2.cmml" xref="A2.E34.m1.1.1.1.1.3.2.2.2">𝜽</ci></apply><ci id="A2.E34.m1.1.1.1.1.3.2.3.cmml" xref="A2.E34.m1.1.1.1.1.3.2.3">𝑇</ci></apply><apply id="A2.E34.m1.1.1.1.1.3.3.cmml" xref="A2.E34.m1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="A2.E34.m1.1.1.1.1.3.3.1.cmml" xref="A2.E34.m1.1.1.1.1.3.3">superscript</csymbol><ci id="A2.E34.m1.1.1.1.1.3.3.2.cmml" xref="A2.E34.m1.1.1.1.1.3.3.2">Γ</ci><apply id="A2.E34.m1.1.1.1.1.3.3.3.cmml" xref="A2.E34.m1.1.1.1.1.3.3.3"><minus id="A2.E34.m1.1.1.1.1.3.3.3.1.cmml" xref="A2.E34.m1.1.1.1.1.3.3.3"></minus><cn id="A2.E34.m1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="A2.E34.m1.1.1.1.1.3.3.3.2">1</cn></apply></apply><apply id="A2.E34.m1.1.1.1.1.3.4.cmml" xref="A2.E34.m1.1.1.1.1.3.4"><ci id="A2.E34.m1.1.1.1.1.3.4.1.cmml" xref="A2.E34.m1.1.1.1.1.3.4.1">~</ci><ci id="A2.E34.m1.1.1.1.1.3.4.2.cmml" xref="A2.E34.m1.1.1.1.1.3.4.2">𝜽</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E34.m1.1c">\displaystyle V_{\theta}=\tilde{\bm{\theta}}^{T}\Gamma^{-1}\tilde{\bm{\theta}}.</annotation><annotation encoding="application/x-llamapun" id="A2.E34.m1.1d">italic_V start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT = over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Γ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(34)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.1.p1.5">Differentiating <math alttext="V_{\theta}" class="ltx_Math" display="inline" id="A2.1.p1.4.m1.1"><semantics id="A2.1.p1.4.m1.1a"><msub id="A2.1.p1.4.m1.1.1" xref="A2.1.p1.4.m1.1.1.cmml"><mi id="A2.1.p1.4.m1.1.1.2" xref="A2.1.p1.4.m1.1.1.2.cmml">V</mi><mi id="A2.1.p1.4.m1.1.1.3" xref="A2.1.p1.4.m1.1.1.3.cmml">θ</mi></msub><annotation-xml encoding="MathML-Content" id="A2.1.p1.4.m1.1b"><apply id="A2.1.p1.4.m1.1.1.cmml" xref="A2.1.p1.4.m1.1.1"><csymbol cd="ambiguous" id="A2.1.p1.4.m1.1.1.1.cmml" xref="A2.1.p1.4.m1.1.1">subscript</csymbol><ci id="A2.1.p1.4.m1.1.1.2.cmml" xref="A2.1.p1.4.m1.1.1.2">𝑉</ci><ci id="A2.1.p1.4.m1.1.1.3.cmml" xref="A2.1.p1.4.m1.1.1.3">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.4.m1.1c">V_{\theta}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.4.m1.1d">italic_V start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT</annotation></semantics></math> regarding <math alttext="t" class="ltx_Math" display="inline" id="A2.1.p1.5.m2.1"><semantics id="A2.1.p1.5.m2.1a"><mi id="A2.1.p1.5.m2.1.1" xref="A2.1.p1.5.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A2.1.p1.5.m2.1b"><ci id="A2.1.p1.5.m2.1.1.cmml" xref="A2.1.p1.5.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.5.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.5.m2.1d">italic_t</annotation></semantics></math> and using (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A2.E33" title="In Proof. ‣ Appendix B The proof of Theorem 1 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">33</span></a>) yields</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx36"> <tbody id="A2.Ex26"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}_{\theta}\leq-2\kappa\tilde{\bm{\theta}}^{T}Q(t,t_{\rm e})% \tilde{\bm{\theta}}-2\tilde{\bm{\theta}}^{T}\Phi_{\rm f}\Phi_{\rm f}^{T}\tilde% {\bm{\theta}}." class="ltx_Math" display="inline" id="A2.Ex26.m1.2"><semantics id="A2.Ex26.m1.2a"><mrow id="A2.Ex26.m1.2.2.1" xref="A2.Ex26.m1.2.2.1.1.cmml"><mrow id="A2.Ex26.m1.2.2.1.1" xref="A2.Ex26.m1.2.2.1.1.cmml"><msub id="A2.Ex26.m1.2.2.1.1.3" xref="A2.Ex26.m1.2.2.1.1.3.cmml"><mover accent="true" id="A2.Ex26.m1.2.2.1.1.3.2" xref="A2.Ex26.m1.2.2.1.1.3.2.cmml"><mi id="A2.Ex26.m1.2.2.1.1.3.2.2" xref="A2.Ex26.m1.2.2.1.1.3.2.2.cmml">V</mi><mo id="A2.Ex26.m1.2.2.1.1.3.2.1" xref="A2.Ex26.m1.2.2.1.1.3.2.1.cmml">˙</mo></mover><mi id="A2.Ex26.m1.2.2.1.1.3.3" xref="A2.Ex26.m1.2.2.1.1.3.3.cmml">θ</mi></msub><mo id="A2.Ex26.m1.2.2.1.1.2" xref="A2.Ex26.m1.2.2.1.1.2.cmml">≤</mo><mrow id="A2.Ex26.m1.2.2.1.1.1" xref="A2.Ex26.m1.2.2.1.1.1.cmml"><mrow id="A2.Ex26.m1.2.2.1.1.1.1" xref="A2.Ex26.m1.2.2.1.1.1.1.cmml"><mo id="A2.Ex26.m1.2.2.1.1.1.1a" xref="A2.Ex26.m1.2.2.1.1.1.1.cmml">−</mo><mrow id="A2.Ex26.m1.2.2.1.1.1.1.1" xref="A2.Ex26.m1.2.2.1.1.1.1.1.cmml"><mn id="A2.Ex26.m1.2.2.1.1.1.1.1.3" xref="A2.Ex26.m1.2.2.1.1.1.1.1.3.cmml">2</mn><mo id="A2.Ex26.m1.2.2.1.1.1.1.1.2" xref="A2.Ex26.m1.2.2.1.1.1.1.1.2.cmml">⁢</mo><mi id="A2.Ex26.m1.2.2.1.1.1.1.1.4" xref="A2.Ex26.m1.2.2.1.1.1.1.1.4.cmml">κ</mi><mo id="A2.Ex26.m1.2.2.1.1.1.1.1.2a" xref="A2.Ex26.m1.2.2.1.1.1.1.1.2.cmml">⁢</mo><msup id="A2.Ex26.m1.2.2.1.1.1.1.1.5" xref="A2.Ex26.m1.2.2.1.1.1.1.1.5.cmml"><mover accent="true" id="A2.Ex26.m1.2.2.1.1.1.1.1.5.2" xref="A2.Ex26.m1.2.2.1.1.1.1.1.5.2.cmml"><mi id="A2.Ex26.m1.2.2.1.1.1.1.1.5.2.2" xref="A2.Ex26.m1.2.2.1.1.1.1.1.5.2.2.cmml">𝜽</mi><mo id="A2.Ex26.m1.2.2.1.1.1.1.1.5.2.1" xref="A2.Ex26.m1.2.2.1.1.1.1.1.5.2.1.cmml">~</mo></mover><mi id="A2.Ex26.m1.2.2.1.1.1.1.1.5.3" xref="A2.Ex26.m1.2.2.1.1.1.1.1.5.3.cmml">T</mi></msup><mo id="A2.Ex26.m1.2.2.1.1.1.1.1.2b" xref="A2.Ex26.m1.2.2.1.1.1.1.1.2.cmml">⁢</mo><mi id="A2.Ex26.m1.2.2.1.1.1.1.1.6" xref="A2.Ex26.m1.2.2.1.1.1.1.1.6.cmml">Q</mi><mo id="A2.Ex26.m1.2.2.1.1.1.1.1.2c" xref="A2.Ex26.m1.2.2.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A2.Ex26.m1.2.2.1.1.1.1.1.1.1" xref="A2.Ex26.m1.2.2.1.1.1.1.1.1.2.cmml"><mo id="A2.Ex26.m1.2.2.1.1.1.1.1.1.1.2" stretchy="false" xref="A2.Ex26.m1.2.2.1.1.1.1.1.1.2.cmml">(</mo><mi id="A2.Ex26.m1.1.1" xref="A2.Ex26.m1.1.1.cmml">t</mi><mo id="A2.Ex26.m1.2.2.1.1.1.1.1.1.1.3" xref="A2.Ex26.m1.2.2.1.1.1.1.1.1.2.cmml">,</mo><msub id="A2.Ex26.m1.2.2.1.1.1.1.1.1.1.1" xref="A2.Ex26.m1.2.2.1.1.1.1.1.1.1.1.cmml"><mi id="A2.Ex26.m1.2.2.1.1.1.1.1.1.1.1.2" xref="A2.Ex26.m1.2.2.1.1.1.1.1.1.1.1.2.cmml">t</mi><mi id="A2.Ex26.m1.2.2.1.1.1.1.1.1.1.1.3" mathvariant="normal" xref="A2.Ex26.m1.2.2.1.1.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="A2.Ex26.m1.2.2.1.1.1.1.1.1.1.4" stretchy="false" xref="A2.Ex26.m1.2.2.1.1.1.1.1.1.2.cmml">)</mo></mrow><mo id="A2.Ex26.m1.2.2.1.1.1.1.1.2d" xref="A2.Ex26.m1.2.2.1.1.1.1.1.2.cmml">⁢</mo><mover accent="true" id="A2.Ex26.m1.2.2.1.1.1.1.1.7" xref="A2.Ex26.m1.2.2.1.1.1.1.1.7.cmml"><mi id="A2.Ex26.m1.2.2.1.1.1.1.1.7.2" xref="A2.Ex26.m1.2.2.1.1.1.1.1.7.2.cmml">𝜽</mi><mo id="A2.Ex26.m1.2.2.1.1.1.1.1.7.1" xref="A2.Ex26.m1.2.2.1.1.1.1.1.7.1.cmml">~</mo></mover></mrow></mrow><mo id="A2.Ex26.m1.2.2.1.1.1.2" xref="A2.Ex26.m1.2.2.1.1.1.2.cmml">−</mo><mrow id="A2.Ex26.m1.2.2.1.1.1.3" xref="A2.Ex26.m1.2.2.1.1.1.3.cmml"><mn id="A2.Ex26.m1.2.2.1.1.1.3.2" xref="A2.Ex26.m1.2.2.1.1.1.3.2.cmml">2</mn><mo id="A2.Ex26.m1.2.2.1.1.1.3.1" xref="A2.Ex26.m1.2.2.1.1.1.3.1.cmml">⁢</mo><msup id="A2.Ex26.m1.2.2.1.1.1.3.3" xref="A2.Ex26.m1.2.2.1.1.1.3.3.cmml"><mover accent="true" id="A2.Ex26.m1.2.2.1.1.1.3.3.2" xref="A2.Ex26.m1.2.2.1.1.1.3.3.2.cmml"><mi id="A2.Ex26.m1.2.2.1.1.1.3.3.2.2" xref="A2.Ex26.m1.2.2.1.1.1.3.3.2.2.cmml">𝜽</mi><mo id="A2.Ex26.m1.2.2.1.1.1.3.3.2.1" xref="A2.Ex26.m1.2.2.1.1.1.3.3.2.1.cmml">~</mo></mover><mi id="A2.Ex26.m1.2.2.1.1.1.3.3.3" xref="A2.Ex26.m1.2.2.1.1.1.3.3.3.cmml">T</mi></msup><mo id="A2.Ex26.m1.2.2.1.1.1.3.1a" xref="A2.Ex26.m1.2.2.1.1.1.3.1.cmml">⁢</mo><msub id="A2.Ex26.m1.2.2.1.1.1.3.4" xref="A2.Ex26.m1.2.2.1.1.1.3.4.cmml"><mi id="A2.Ex26.m1.2.2.1.1.1.3.4.2" mathvariant="normal" xref="A2.Ex26.m1.2.2.1.1.1.3.4.2.cmml">Φ</mi><mi id="A2.Ex26.m1.2.2.1.1.1.3.4.3" mathvariant="normal" xref="A2.Ex26.m1.2.2.1.1.1.3.4.3.cmml">f</mi></msub><mo id="A2.Ex26.m1.2.2.1.1.1.3.1b" xref="A2.Ex26.m1.2.2.1.1.1.3.1.cmml">⁢</mo><msubsup id="A2.Ex26.m1.2.2.1.1.1.3.5" xref="A2.Ex26.m1.2.2.1.1.1.3.5.cmml"><mi id="A2.Ex26.m1.2.2.1.1.1.3.5.2.2" mathvariant="normal" xref="A2.Ex26.m1.2.2.1.1.1.3.5.2.2.cmml">Φ</mi><mi id="A2.Ex26.m1.2.2.1.1.1.3.5.2.3" mathvariant="normal" xref="A2.Ex26.m1.2.2.1.1.1.3.5.2.3.cmml">f</mi><mi id="A2.Ex26.m1.2.2.1.1.1.3.5.3" xref="A2.Ex26.m1.2.2.1.1.1.3.5.3.cmml">T</mi></msubsup><mo id="A2.Ex26.m1.2.2.1.1.1.3.1c" xref="A2.Ex26.m1.2.2.1.1.1.3.1.cmml">⁢</mo><mover accent="true" id="A2.Ex26.m1.2.2.1.1.1.3.6" xref="A2.Ex26.m1.2.2.1.1.1.3.6.cmml"><mi id="A2.Ex26.m1.2.2.1.1.1.3.6.2" xref="A2.Ex26.m1.2.2.1.1.1.3.6.2.cmml">𝜽</mi><mo id="A2.Ex26.m1.2.2.1.1.1.3.6.1" xref="A2.Ex26.m1.2.2.1.1.1.3.6.1.cmml">~</mo></mover></mrow></mrow></mrow><mo id="A2.Ex26.m1.2.2.1.2" lspace="0em" xref="A2.Ex26.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex26.m1.2b"><apply id="A2.Ex26.m1.2.2.1.1.cmml" xref="A2.Ex26.m1.2.2.1"><leq id="A2.Ex26.m1.2.2.1.1.2.cmml" xref="A2.Ex26.m1.2.2.1.1.2"></leq><apply id="A2.Ex26.m1.2.2.1.1.3.cmml" xref="A2.Ex26.m1.2.2.1.1.3"><csymbol cd="ambiguous" id="A2.Ex26.m1.2.2.1.1.3.1.cmml" xref="A2.Ex26.m1.2.2.1.1.3">subscript</csymbol><apply id="A2.Ex26.m1.2.2.1.1.3.2.cmml" xref="A2.Ex26.m1.2.2.1.1.3.2"><ci id="A2.Ex26.m1.2.2.1.1.3.2.1.cmml" xref="A2.Ex26.m1.2.2.1.1.3.2.1">˙</ci><ci id="A2.Ex26.m1.2.2.1.1.3.2.2.cmml" xref="A2.Ex26.m1.2.2.1.1.3.2.2">𝑉</ci></apply><ci id="A2.Ex26.m1.2.2.1.1.3.3.cmml" xref="A2.Ex26.m1.2.2.1.1.3.3">𝜃</ci></apply><apply id="A2.Ex26.m1.2.2.1.1.1.cmml" xref="A2.Ex26.m1.2.2.1.1.1"><minus id="A2.Ex26.m1.2.2.1.1.1.2.cmml" xref="A2.Ex26.m1.2.2.1.1.1.2"></minus><apply id="A2.Ex26.m1.2.2.1.1.1.1.cmml" xref="A2.Ex26.m1.2.2.1.1.1.1"><minus id="A2.Ex26.m1.2.2.1.1.1.1.2.cmml" xref="A2.Ex26.m1.2.2.1.1.1.1"></minus><apply id="A2.Ex26.m1.2.2.1.1.1.1.1.cmml" xref="A2.Ex26.m1.2.2.1.1.1.1.1"><times id="A2.Ex26.m1.2.2.1.1.1.1.1.2.cmml" xref="A2.Ex26.m1.2.2.1.1.1.1.1.2"></times><cn id="A2.Ex26.m1.2.2.1.1.1.1.1.3.cmml" type="integer" xref="A2.Ex26.m1.2.2.1.1.1.1.1.3">2</cn><ci id="A2.Ex26.m1.2.2.1.1.1.1.1.4.cmml" xref="A2.Ex26.m1.2.2.1.1.1.1.1.4">𝜅</ci><apply id="A2.Ex26.m1.2.2.1.1.1.1.1.5.cmml" xref="A2.Ex26.m1.2.2.1.1.1.1.1.5"><csymbol cd="ambiguous" id="A2.Ex26.m1.2.2.1.1.1.1.1.5.1.cmml" xref="A2.Ex26.m1.2.2.1.1.1.1.1.5">superscript</csymbol><apply id="A2.Ex26.m1.2.2.1.1.1.1.1.5.2.cmml" xref="A2.Ex26.m1.2.2.1.1.1.1.1.5.2"><ci id="A2.Ex26.m1.2.2.1.1.1.1.1.5.2.1.cmml" xref="A2.Ex26.m1.2.2.1.1.1.1.1.5.2.1">~</ci><ci id="A2.Ex26.m1.2.2.1.1.1.1.1.5.2.2.cmml" xref="A2.Ex26.m1.2.2.1.1.1.1.1.5.2.2">𝜽</ci></apply><ci id="A2.Ex26.m1.2.2.1.1.1.1.1.5.3.cmml" xref="A2.Ex26.m1.2.2.1.1.1.1.1.5.3">𝑇</ci></apply><ci id="A2.Ex26.m1.2.2.1.1.1.1.1.6.cmml" xref="A2.Ex26.m1.2.2.1.1.1.1.1.6">𝑄</ci><interval closure="open" id="A2.Ex26.m1.2.2.1.1.1.1.1.1.2.cmml" xref="A2.Ex26.m1.2.2.1.1.1.1.1.1.1"><ci id="A2.Ex26.m1.1.1.cmml" xref="A2.Ex26.m1.1.1">𝑡</ci><apply id="A2.Ex26.m1.2.2.1.1.1.1.1.1.1.1.cmml" xref="A2.Ex26.m1.2.2.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex26.m1.2.2.1.1.1.1.1.1.1.1.1.cmml" xref="A2.Ex26.m1.2.2.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex26.m1.2.2.1.1.1.1.1.1.1.1.2.cmml" xref="A2.Ex26.m1.2.2.1.1.1.1.1.1.1.1.2">𝑡</ci><ci id="A2.Ex26.m1.2.2.1.1.1.1.1.1.1.1.3.cmml" xref="A2.Ex26.m1.2.2.1.1.1.1.1.1.1.1.3">e</ci></apply></interval><apply id="A2.Ex26.m1.2.2.1.1.1.1.1.7.cmml" xref="A2.Ex26.m1.2.2.1.1.1.1.1.7"><ci id="A2.Ex26.m1.2.2.1.1.1.1.1.7.1.cmml" xref="A2.Ex26.m1.2.2.1.1.1.1.1.7.1">~</ci><ci id="A2.Ex26.m1.2.2.1.1.1.1.1.7.2.cmml" xref="A2.Ex26.m1.2.2.1.1.1.1.1.7.2">𝜽</ci></apply></apply></apply><apply id="A2.Ex26.m1.2.2.1.1.1.3.cmml" xref="A2.Ex26.m1.2.2.1.1.1.3"><times id="A2.Ex26.m1.2.2.1.1.1.3.1.cmml" xref="A2.Ex26.m1.2.2.1.1.1.3.1"></times><cn id="A2.Ex26.m1.2.2.1.1.1.3.2.cmml" type="integer" xref="A2.Ex26.m1.2.2.1.1.1.3.2">2</cn><apply id="A2.Ex26.m1.2.2.1.1.1.3.3.cmml" xref="A2.Ex26.m1.2.2.1.1.1.3.3"><csymbol cd="ambiguous" id="A2.Ex26.m1.2.2.1.1.1.3.3.1.cmml" xref="A2.Ex26.m1.2.2.1.1.1.3.3">superscript</csymbol><apply id="A2.Ex26.m1.2.2.1.1.1.3.3.2.cmml" xref="A2.Ex26.m1.2.2.1.1.1.3.3.2"><ci id="A2.Ex26.m1.2.2.1.1.1.3.3.2.1.cmml" xref="A2.Ex26.m1.2.2.1.1.1.3.3.2.1">~</ci><ci id="A2.Ex26.m1.2.2.1.1.1.3.3.2.2.cmml" xref="A2.Ex26.m1.2.2.1.1.1.3.3.2.2">𝜽</ci></apply><ci id="A2.Ex26.m1.2.2.1.1.1.3.3.3.cmml" xref="A2.Ex26.m1.2.2.1.1.1.3.3.3">𝑇</ci></apply><apply id="A2.Ex26.m1.2.2.1.1.1.3.4.cmml" xref="A2.Ex26.m1.2.2.1.1.1.3.4"><csymbol cd="ambiguous" id="A2.Ex26.m1.2.2.1.1.1.3.4.1.cmml" xref="A2.Ex26.m1.2.2.1.1.1.3.4">subscript</csymbol><ci id="A2.Ex26.m1.2.2.1.1.1.3.4.2.cmml" xref="A2.Ex26.m1.2.2.1.1.1.3.4.2">Φ</ci><ci id="A2.Ex26.m1.2.2.1.1.1.3.4.3.cmml" xref="A2.Ex26.m1.2.2.1.1.1.3.4.3">f</ci></apply><apply id="A2.Ex26.m1.2.2.1.1.1.3.5.cmml" xref="A2.Ex26.m1.2.2.1.1.1.3.5"><csymbol cd="ambiguous" id="A2.Ex26.m1.2.2.1.1.1.3.5.1.cmml" xref="A2.Ex26.m1.2.2.1.1.1.3.5">superscript</csymbol><apply id="A2.Ex26.m1.2.2.1.1.1.3.5.2.cmml" xref="A2.Ex26.m1.2.2.1.1.1.3.5"><csymbol cd="ambiguous" id="A2.Ex26.m1.2.2.1.1.1.3.5.2.1.cmml" xref="A2.Ex26.m1.2.2.1.1.1.3.5">subscript</csymbol><ci id="A2.Ex26.m1.2.2.1.1.1.3.5.2.2.cmml" xref="A2.Ex26.m1.2.2.1.1.1.3.5.2.2">Φ</ci><ci id="A2.Ex26.m1.2.2.1.1.1.3.5.2.3.cmml" xref="A2.Ex26.m1.2.2.1.1.1.3.5.2.3">f</ci></apply><ci id="A2.Ex26.m1.2.2.1.1.1.3.5.3.cmml" xref="A2.Ex26.m1.2.2.1.1.1.3.5.3">𝑇</ci></apply><apply id="A2.Ex26.m1.2.2.1.1.1.3.6.cmml" xref="A2.Ex26.m1.2.2.1.1.1.3.6"><ci id="A2.Ex26.m1.2.2.1.1.1.3.6.1.cmml" xref="A2.Ex26.m1.2.2.1.1.1.3.6.1">~</ci><ci id="A2.Ex26.m1.2.2.1.1.1.3.6.2.cmml" xref="A2.Ex26.m1.2.2.1.1.1.3.6.2">𝜽</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex26.m1.2c">\displaystyle\dot{V}_{\theta}\leq-2\kappa\tilde{\bm{\theta}}^{T}Q(t,t_{\rm e})% \tilde{\bm{\theta}}-2\tilde{\bm{\theta}}^{T}\Phi_{\rm f}\Phi_{\rm f}^{T}\tilde% {\bm{\theta}}.</annotation><annotation encoding="application/x-llamapun" id="A2.Ex26.m1.2d">over˙ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ≤ - 2 italic_κ over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT italic_Q ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG - 2 over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A2.1.p1.8">Because <math alttext="Q(t,t_{\rm e})" class="ltx_Math" display="inline" id="A2.1.p1.6.m1.2"><semantics id="A2.1.p1.6.m1.2a"><mrow id="A2.1.p1.6.m1.2.2" xref="A2.1.p1.6.m1.2.2.cmml"><mi id="A2.1.p1.6.m1.2.2.3" xref="A2.1.p1.6.m1.2.2.3.cmml">Q</mi><mo id="A2.1.p1.6.m1.2.2.2" xref="A2.1.p1.6.m1.2.2.2.cmml">⁢</mo><mrow id="A2.1.p1.6.m1.2.2.1.1" xref="A2.1.p1.6.m1.2.2.1.2.cmml"><mo id="A2.1.p1.6.m1.2.2.1.1.2" stretchy="false" xref="A2.1.p1.6.m1.2.2.1.2.cmml">(</mo><mi id="A2.1.p1.6.m1.1.1" xref="A2.1.p1.6.m1.1.1.cmml">t</mi><mo id="A2.1.p1.6.m1.2.2.1.1.3" xref="A2.1.p1.6.m1.2.2.1.2.cmml">,</mo><msub id="A2.1.p1.6.m1.2.2.1.1.1" xref="A2.1.p1.6.m1.2.2.1.1.1.cmml"><mi id="A2.1.p1.6.m1.2.2.1.1.1.2" xref="A2.1.p1.6.m1.2.2.1.1.1.2.cmml">t</mi><mi id="A2.1.p1.6.m1.2.2.1.1.1.3" mathvariant="normal" xref="A2.1.p1.6.m1.2.2.1.1.1.3.cmml">e</mi></msub><mo id="A2.1.p1.6.m1.2.2.1.1.4" stretchy="false" xref="A2.1.p1.6.m1.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.6.m1.2b"><apply id="A2.1.p1.6.m1.2.2.cmml" xref="A2.1.p1.6.m1.2.2"><times id="A2.1.p1.6.m1.2.2.2.cmml" xref="A2.1.p1.6.m1.2.2.2"></times><ci id="A2.1.p1.6.m1.2.2.3.cmml" xref="A2.1.p1.6.m1.2.2.3">𝑄</ci><interval closure="open" id="A2.1.p1.6.m1.2.2.1.2.cmml" xref="A2.1.p1.6.m1.2.2.1.1"><ci id="A2.1.p1.6.m1.1.1.cmml" xref="A2.1.p1.6.m1.1.1">𝑡</ci><apply id="A2.1.p1.6.m1.2.2.1.1.1.cmml" xref="A2.1.p1.6.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="A2.1.p1.6.m1.2.2.1.1.1.1.cmml" xref="A2.1.p1.6.m1.2.2.1.1.1">subscript</csymbol><ci id="A2.1.p1.6.m1.2.2.1.1.1.2.cmml" xref="A2.1.p1.6.m1.2.2.1.1.1.2">𝑡</ci><ci id="A2.1.p1.6.m1.2.2.1.1.1.3.cmml" xref="A2.1.p1.6.m1.2.2.1.1.1.3">e</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.6.m1.2c">Q(t,t_{\rm e})</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.6.m1.2d">italic_Q ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT )</annotation></semantics></math> is positive semidefinite, i.e., <math alttext="Q(t,t_{\rm e})\geq 0" class="ltx_Math" display="inline" id="A2.1.p1.7.m2.2"><semantics id="A2.1.p1.7.m2.2a"><mrow id="A2.1.p1.7.m2.2.2" xref="A2.1.p1.7.m2.2.2.cmml"><mrow id="A2.1.p1.7.m2.2.2.1" xref="A2.1.p1.7.m2.2.2.1.cmml"><mi id="A2.1.p1.7.m2.2.2.1.3" xref="A2.1.p1.7.m2.2.2.1.3.cmml">Q</mi><mo id="A2.1.p1.7.m2.2.2.1.2" xref="A2.1.p1.7.m2.2.2.1.2.cmml">⁢</mo><mrow id="A2.1.p1.7.m2.2.2.1.1.1" xref="A2.1.p1.7.m2.2.2.1.1.2.cmml"><mo id="A2.1.p1.7.m2.2.2.1.1.1.2" stretchy="false" xref="A2.1.p1.7.m2.2.2.1.1.2.cmml">(</mo><mi id="A2.1.p1.7.m2.1.1" xref="A2.1.p1.7.m2.1.1.cmml">t</mi><mo id="A2.1.p1.7.m2.2.2.1.1.1.3" xref="A2.1.p1.7.m2.2.2.1.1.2.cmml">,</mo><msub id="A2.1.p1.7.m2.2.2.1.1.1.1" xref="A2.1.p1.7.m2.2.2.1.1.1.1.cmml"><mi id="A2.1.p1.7.m2.2.2.1.1.1.1.2" xref="A2.1.p1.7.m2.2.2.1.1.1.1.2.cmml">t</mi><mi id="A2.1.p1.7.m2.2.2.1.1.1.1.3" mathvariant="normal" xref="A2.1.p1.7.m2.2.2.1.1.1.1.3.cmml">e</mi></msub><mo id="A2.1.p1.7.m2.2.2.1.1.1.4" stretchy="false" xref="A2.1.p1.7.m2.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="A2.1.p1.7.m2.2.2.2" xref="A2.1.p1.7.m2.2.2.2.cmml">≥</mo><mn id="A2.1.p1.7.m2.2.2.3" xref="A2.1.p1.7.m2.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.7.m2.2b"><apply id="A2.1.p1.7.m2.2.2.cmml" xref="A2.1.p1.7.m2.2.2"><geq id="A2.1.p1.7.m2.2.2.2.cmml" xref="A2.1.p1.7.m2.2.2.2"></geq><apply id="A2.1.p1.7.m2.2.2.1.cmml" xref="A2.1.p1.7.m2.2.2.1"><times id="A2.1.p1.7.m2.2.2.1.2.cmml" xref="A2.1.p1.7.m2.2.2.1.2"></times><ci id="A2.1.p1.7.m2.2.2.1.3.cmml" xref="A2.1.p1.7.m2.2.2.1.3">𝑄</ci><interval closure="open" id="A2.1.p1.7.m2.2.2.1.1.2.cmml" xref="A2.1.p1.7.m2.2.2.1.1.1"><ci id="A2.1.p1.7.m2.1.1.cmml" xref="A2.1.p1.7.m2.1.1">𝑡</ci><apply id="A2.1.p1.7.m2.2.2.1.1.1.1.cmml" xref="A2.1.p1.7.m2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="A2.1.p1.7.m2.2.2.1.1.1.1.1.cmml" xref="A2.1.p1.7.m2.2.2.1.1.1.1">subscript</csymbol><ci id="A2.1.p1.7.m2.2.2.1.1.1.1.2.cmml" xref="A2.1.p1.7.m2.2.2.1.1.1.1.2">𝑡</ci><ci id="A2.1.p1.7.m2.2.2.1.1.1.1.3.cmml" xref="A2.1.p1.7.m2.2.2.1.1.1.1.3">e</ci></apply></interval></apply><cn id="A2.1.p1.7.m2.2.2.3.cmml" type="integer" xref="A2.1.p1.7.m2.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.7.m2.2c">Q(t,t_{\rm e})\geq 0</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.7.m2.2d">italic_Q ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) ≥ 0</annotation></semantics></math>, one has <math alttext="\tilde{\bm{\theta}}^{T}Q(t,t_{\rm e})\tilde{\bm{\theta}}\geq 0" class="ltx_Math" display="inline" id="A2.1.p1.8.m3.2"><semantics id="A2.1.p1.8.m3.2a"><mrow id="A2.1.p1.8.m3.2.2" xref="A2.1.p1.8.m3.2.2.cmml"><mrow id="A2.1.p1.8.m3.2.2.1" xref="A2.1.p1.8.m3.2.2.1.cmml"><msup id="A2.1.p1.8.m3.2.2.1.3" xref="A2.1.p1.8.m3.2.2.1.3.cmml"><mover accent="true" id="A2.1.p1.8.m3.2.2.1.3.2" xref="A2.1.p1.8.m3.2.2.1.3.2.cmml"><mi id="A2.1.p1.8.m3.2.2.1.3.2.2" xref="A2.1.p1.8.m3.2.2.1.3.2.2.cmml">𝜽</mi><mo id="A2.1.p1.8.m3.2.2.1.3.2.1" xref="A2.1.p1.8.m3.2.2.1.3.2.1.cmml">~</mo></mover><mi id="A2.1.p1.8.m3.2.2.1.3.3" xref="A2.1.p1.8.m3.2.2.1.3.3.cmml">T</mi></msup><mo id="A2.1.p1.8.m3.2.2.1.2" xref="A2.1.p1.8.m3.2.2.1.2.cmml">⁢</mo><mi id="A2.1.p1.8.m3.2.2.1.4" xref="A2.1.p1.8.m3.2.2.1.4.cmml">Q</mi><mo id="A2.1.p1.8.m3.2.2.1.2a" xref="A2.1.p1.8.m3.2.2.1.2.cmml">⁢</mo><mrow id="A2.1.p1.8.m3.2.2.1.1.1" xref="A2.1.p1.8.m3.2.2.1.1.2.cmml"><mo id="A2.1.p1.8.m3.2.2.1.1.1.2" stretchy="false" xref="A2.1.p1.8.m3.2.2.1.1.2.cmml">(</mo><mi id="A2.1.p1.8.m3.1.1" xref="A2.1.p1.8.m3.1.1.cmml">t</mi><mo id="A2.1.p1.8.m3.2.2.1.1.1.3" xref="A2.1.p1.8.m3.2.2.1.1.2.cmml">,</mo><msub id="A2.1.p1.8.m3.2.2.1.1.1.1" xref="A2.1.p1.8.m3.2.2.1.1.1.1.cmml"><mi id="A2.1.p1.8.m3.2.2.1.1.1.1.2" xref="A2.1.p1.8.m3.2.2.1.1.1.1.2.cmml">t</mi><mi id="A2.1.p1.8.m3.2.2.1.1.1.1.3" mathvariant="normal" xref="A2.1.p1.8.m3.2.2.1.1.1.1.3.cmml">e</mi></msub><mo id="A2.1.p1.8.m3.2.2.1.1.1.4" stretchy="false" xref="A2.1.p1.8.m3.2.2.1.1.2.cmml">)</mo></mrow><mo id="A2.1.p1.8.m3.2.2.1.2b" xref="A2.1.p1.8.m3.2.2.1.2.cmml">⁢</mo><mover accent="true" id="A2.1.p1.8.m3.2.2.1.5" xref="A2.1.p1.8.m3.2.2.1.5.cmml"><mi id="A2.1.p1.8.m3.2.2.1.5.2" xref="A2.1.p1.8.m3.2.2.1.5.2.cmml">𝜽</mi><mo id="A2.1.p1.8.m3.2.2.1.5.1" xref="A2.1.p1.8.m3.2.2.1.5.1.cmml">~</mo></mover></mrow><mo id="A2.1.p1.8.m3.2.2.2" xref="A2.1.p1.8.m3.2.2.2.cmml">≥</mo><mn id="A2.1.p1.8.m3.2.2.3" xref="A2.1.p1.8.m3.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.8.m3.2b"><apply id="A2.1.p1.8.m3.2.2.cmml" xref="A2.1.p1.8.m3.2.2"><geq id="A2.1.p1.8.m3.2.2.2.cmml" xref="A2.1.p1.8.m3.2.2.2"></geq><apply id="A2.1.p1.8.m3.2.2.1.cmml" xref="A2.1.p1.8.m3.2.2.1"><times id="A2.1.p1.8.m3.2.2.1.2.cmml" xref="A2.1.p1.8.m3.2.2.1.2"></times><apply id="A2.1.p1.8.m3.2.2.1.3.cmml" xref="A2.1.p1.8.m3.2.2.1.3"><csymbol cd="ambiguous" id="A2.1.p1.8.m3.2.2.1.3.1.cmml" xref="A2.1.p1.8.m3.2.2.1.3">superscript</csymbol><apply id="A2.1.p1.8.m3.2.2.1.3.2.cmml" xref="A2.1.p1.8.m3.2.2.1.3.2"><ci id="A2.1.p1.8.m3.2.2.1.3.2.1.cmml" xref="A2.1.p1.8.m3.2.2.1.3.2.1">~</ci><ci id="A2.1.p1.8.m3.2.2.1.3.2.2.cmml" xref="A2.1.p1.8.m3.2.2.1.3.2.2">𝜽</ci></apply><ci id="A2.1.p1.8.m3.2.2.1.3.3.cmml" xref="A2.1.p1.8.m3.2.2.1.3.3">𝑇</ci></apply><ci id="A2.1.p1.8.m3.2.2.1.4.cmml" xref="A2.1.p1.8.m3.2.2.1.4">𝑄</ci><interval closure="open" id="A2.1.p1.8.m3.2.2.1.1.2.cmml" xref="A2.1.p1.8.m3.2.2.1.1.1"><ci id="A2.1.p1.8.m3.1.1.cmml" xref="A2.1.p1.8.m3.1.1">𝑡</ci><apply id="A2.1.p1.8.m3.2.2.1.1.1.1.cmml" xref="A2.1.p1.8.m3.2.2.1.1.1.1"><csymbol cd="ambiguous" id="A2.1.p1.8.m3.2.2.1.1.1.1.1.cmml" xref="A2.1.p1.8.m3.2.2.1.1.1.1">subscript</csymbol><ci id="A2.1.p1.8.m3.2.2.1.1.1.1.2.cmml" xref="A2.1.p1.8.m3.2.2.1.1.1.1.2">𝑡</ci><ci id="A2.1.p1.8.m3.2.2.1.1.1.1.3.cmml" xref="A2.1.p1.8.m3.2.2.1.1.1.1.3">e</ci></apply></interval><apply id="A2.1.p1.8.m3.2.2.1.5.cmml" xref="A2.1.p1.8.m3.2.2.1.5"><ci id="A2.1.p1.8.m3.2.2.1.5.1.cmml" xref="A2.1.p1.8.m3.2.2.1.5.1">~</ci><ci id="A2.1.p1.8.m3.2.2.1.5.2.cmml" xref="A2.1.p1.8.m3.2.2.1.5.2">𝜽</ci></apply></apply><cn id="A2.1.p1.8.m3.2.2.3.cmml" type="integer" xref="A2.1.p1.8.m3.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.8.m3.2c">\tilde{\bm{\theta}}^{T}Q(t,t_{\rm e})\tilde{\bm{\theta}}\geq 0</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.8.m3.2d">over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT italic_Q ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG ≥ 0</annotation></semantics></math> such that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx37"> <tbody id="A2.E35"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}_{\theta}\leq-2\tilde{\bm{\theta}}^{T}\Phi_{\rm f}\Phi_{% \rm f}^{T}\tilde{\bm{\theta}},\forall t\in[0,t_{\rm f})." class="ltx_Math" display="inline" id="A2.E35.m1.2"><semantics id="A2.E35.m1.2a"><mrow id="A2.E35.m1.2.2.1"><mrow id="A2.E35.m1.2.2.1.1.2" xref="A2.E35.m1.2.2.1.1.3.cmml"><mrow id="A2.E35.m1.2.2.1.1.1.1" xref="A2.E35.m1.2.2.1.1.1.1.cmml"><msub id="A2.E35.m1.2.2.1.1.1.1.2" xref="A2.E35.m1.2.2.1.1.1.1.2.cmml"><mover accent="true" id="A2.E35.m1.2.2.1.1.1.1.2.2" xref="A2.E35.m1.2.2.1.1.1.1.2.2.cmml"><mi id="A2.E35.m1.2.2.1.1.1.1.2.2.2" xref="A2.E35.m1.2.2.1.1.1.1.2.2.2.cmml">V</mi><mo id="A2.E35.m1.2.2.1.1.1.1.2.2.1" xref="A2.E35.m1.2.2.1.1.1.1.2.2.1.cmml">˙</mo></mover><mi id="A2.E35.m1.2.2.1.1.1.1.2.3" xref="A2.E35.m1.2.2.1.1.1.1.2.3.cmml">θ</mi></msub><mo id="A2.E35.m1.2.2.1.1.1.1.1" xref="A2.E35.m1.2.2.1.1.1.1.1.cmml">≤</mo><mrow id="A2.E35.m1.2.2.1.1.1.1.3" xref="A2.E35.m1.2.2.1.1.1.1.3.cmml"><mo id="A2.E35.m1.2.2.1.1.1.1.3a" xref="A2.E35.m1.2.2.1.1.1.1.3.cmml">−</mo><mrow id="A2.E35.m1.2.2.1.1.1.1.3.2" xref="A2.E35.m1.2.2.1.1.1.1.3.2.cmml"><mn id="A2.E35.m1.2.2.1.1.1.1.3.2.2" xref="A2.E35.m1.2.2.1.1.1.1.3.2.2.cmml">2</mn><mo id="A2.E35.m1.2.2.1.1.1.1.3.2.1" xref="A2.E35.m1.2.2.1.1.1.1.3.2.1.cmml">⁢</mo><msup id="A2.E35.m1.2.2.1.1.1.1.3.2.3" xref="A2.E35.m1.2.2.1.1.1.1.3.2.3.cmml"><mover accent="true" id="A2.E35.m1.2.2.1.1.1.1.3.2.3.2" xref="A2.E35.m1.2.2.1.1.1.1.3.2.3.2.cmml"><mi id="A2.E35.m1.2.2.1.1.1.1.3.2.3.2.2" xref="A2.E35.m1.2.2.1.1.1.1.3.2.3.2.2.cmml">𝜽</mi><mo id="A2.E35.m1.2.2.1.1.1.1.3.2.3.2.1" xref="A2.E35.m1.2.2.1.1.1.1.3.2.3.2.1.cmml">~</mo></mover><mi id="A2.E35.m1.2.2.1.1.1.1.3.2.3.3" xref="A2.E35.m1.2.2.1.1.1.1.3.2.3.3.cmml">T</mi></msup><mo id="A2.E35.m1.2.2.1.1.1.1.3.2.1a" xref="A2.E35.m1.2.2.1.1.1.1.3.2.1.cmml">⁢</mo><msub id="A2.E35.m1.2.2.1.1.1.1.3.2.4" xref="A2.E35.m1.2.2.1.1.1.1.3.2.4.cmml"><mi id="A2.E35.m1.2.2.1.1.1.1.3.2.4.2" mathvariant="normal" xref="A2.E35.m1.2.2.1.1.1.1.3.2.4.2.cmml">Φ</mi><mi id="A2.E35.m1.2.2.1.1.1.1.3.2.4.3" mathvariant="normal" xref="A2.E35.m1.2.2.1.1.1.1.3.2.4.3.cmml">f</mi></msub><mo id="A2.E35.m1.2.2.1.1.1.1.3.2.1b" xref="A2.E35.m1.2.2.1.1.1.1.3.2.1.cmml">⁢</mo><msubsup id="A2.E35.m1.2.2.1.1.1.1.3.2.5" xref="A2.E35.m1.2.2.1.1.1.1.3.2.5.cmml"><mi id="A2.E35.m1.2.2.1.1.1.1.3.2.5.2.2" mathvariant="normal" xref="A2.E35.m1.2.2.1.1.1.1.3.2.5.2.2.cmml">Φ</mi><mi id="A2.E35.m1.2.2.1.1.1.1.3.2.5.2.3" mathvariant="normal" xref="A2.E35.m1.2.2.1.1.1.1.3.2.5.2.3.cmml">f</mi><mi id="A2.E35.m1.2.2.1.1.1.1.3.2.5.3" xref="A2.E35.m1.2.2.1.1.1.1.3.2.5.3.cmml">T</mi></msubsup><mo id="A2.E35.m1.2.2.1.1.1.1.3.2.1c" xref="A2.E35.m1.2.2.1.1.1.1.3.2.1.cmml">⁢</mo><mover accent="true" id="A2.E35.m1.2.2.1.1.1.1.3.2.6" xref="A2.E35.m1.2.2.1.1.1.1.3.2.6.cmml"><mi id="A2.E35.m1.2.2.1.1.1.1.3.2.6.2" xref="A2.E35.m1.2.2.1.1.1.1.3.2.6.2.cmml">𝜽</mi><mo id="A2.E35.m1.2.2.1.1.1.1.3.2.6.1" xref="A2.E35.m1.2.2.1.1.1.1.3.2.6.1.cmml">~</mo></mover></mrow></mrow></mrow><mo id="A2.E35.m1.2.2.1.1.2.3" xref="A2.E35.m1.2.2.1.1.3a.cmml">,</mo><mrow id="A2.E35.m1.2.2.1.1.2.2" xref="A2.E35.m1.2.2.1.1.2.2.cmml"><mrow id="A2.E35.m1.2.2.1.1.2.2.3" xref="A2.E35.m1.2.2.1.1.2.2.3.cmml"><mo id="A2.E35.m1.2.2.1.1.2.2.3.1" rspace="0.167em" xref="A2.E35.m1.2.2.1.1.2.2.3.1.cmml">∀</mo><mi id="A2.E35.m1.2.2.1.1.2.2.3.2" xref="A2.E35.m1.2.2.1.1.2.2.3.2.cmml">t</mi></mrow><mo id="A2.E35.m1.2.2.1.1.2.2.2" xref="A2.E35.m1.2.2.1.1.2.2.2.cmml">∈</mo><mrow id="A2.E35.m1.2.2.1.1.2.2.1.1" xref="A2.E35.m1.2.2.1.1.2.2.1.2.cmml"><mo id="A2.E35.m1.2.2.1.1.2.2.1.1.2" stretchy="false" xref="A2.E35.m1.2.2.1.1.2.2.1.2.cmml">[</mo><mn id="A2.E35.m1.1.1" xref="A2.E35.m1.1.1.cmml">0</mn><mo id="A2.E35.m1.2.2.1.1.2.2.1.1.3" xref="A2.E35.m1.2.2.1.1.2.2.1.2.cmml">,</mo><msub id="A2.E35.m1.2.2.1.1.2.2.1.1.1" xref="A2.E35.m1.2.2.1.1.2.2.1.1.1.cmml"><mi id="A2.E35.m1.2.2.1.1.2.2.1.1.1.2" xref="A2.E35.m1.2.2.1.1.2.2.1.1.1.2.cmml">t</mi><mi id="A2.E35.m1.2.2.1.1.2.2.1.1.1.3" mathvariant="normal" xref="A2.E35.m1.2.2.1.1.2.2.1.1.1.3.cmml">f</mi></msub><mo id="A2.E35.m1.2.2.1.1.2.2.1.1.4" stretchy="false" xref="A2.E35.m1.2.2.1.1.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="A2.E35.m1.2.2.1.2" lspace="0em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.E35.m1.2b"><apply id="A2.E35.m1.2.2.1.1.3.cmml" xref="A2.E35.m1.2.2.1.1.2"><csymbol cd="ambiguous" id="A2.E35.m1.2.2.1.1.3a.cmml" xref="A2.E35.m1.2.2.1.1.2.3">formulae-sequence</csymbol><apply id="A2.E35.m1.2.2.1.1.1.1.cmml" xref="A2.E35.m1.2.2.1.1.1.1"><leq id="A2.E35.m1.2.2.1.1.1.1.1.cmml" xref="A2.E35.m1.2.2.1.1.1.1.1"></leq><apply id="A2.E35.m1.2.2.1.1.1.1.2.cmml" xref="A2.E35.m1.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="A2.E35.m1.2.2.1.1.1.1.2.1.cmml" xref="A2.E35.m1.2.2.1.1.1.1.2">subscript</csymbol><apply id="A2.E35.m1.2.2.1.1.1.1.2.2.cmml" xref="A2.E35.m1.2.2.1.1.1.1.2.2"><ci id="A2.E35.m1.2.2.1.1.1.1.2.2.1.cmml" xref="A2.E35.m1.2.2.1.1.1.1.2.2.1">˙</ci><ci id="A2.E35.m1.2.2.1.1.1.1.2.2.2.cmml" xref="A2.E35.m1.2.2.1.1.1.1.2.2.2">𝑉</ci></apply><ci id="A2.E35.m1.2.2.1.1.1.1.2.3.cmml" xref="A2.E35.m1.2.2.1.1.1.1.2.3">𝜃</ci></apply><apply id="A2.E35.m1.2.2.1.1.1.1.3.cmml" xref="A2.E35.m1.2.2.1.1.1.1.3"><minus id="A2.E35.m1.2.2.1.1.1.1.3.1.cmml" xref="A2.E35.m1.2.2.1.1.1.1.3"></minus><apply id="A2.E35.m1.2.2.1.1.1.1.3.2.cmml" xref="A2.E35.m1.2.2.1.1.1.1.3.2"><times id="A2.E35.m1.2.2.1.1.1.1.3.2.1.cmml" xref="A2.E35.m1.2.2.1.1.1.1.3.2.1"></times><cn id="A2.E35.m1.2.2.1.1.1.1.3.2.2.cmml" type="integer" xref="A2.E35.m1.2.2.1.1.1.1.3.2.2">2</cn><apply id="A2.E35.m1.2.2.1.1.1.1.3.2.3.cmml" xref="A2.E35.m1.2.2.1.1.1.1.3.2.3"><csymbol cd="ambiguous" id="A2.E35.m1.2.2.1.1.1.1.3.2.3.1.cmml" xref="A2.E35.m1.2.2.1.1.1.1.3.2.3">superscript</csymbol><apply id="A2.E35.m1.2.2.1.1.1.1.3.2.3.2.cmml" xref="A2.E35.m1.2.2.1.1.1.1.3.2.3.2"><ci id="A2.E35.m1.2.2.1.1.1.1.3.2.3.2.1.cmml" xref="A2.E35.m1.2.2.1.1.1.1.3.2.3.2.1">~</ci><ci id="A2.E35.m1.2.2.1.1.1.1.3.2.3.2.2.cmml" xref="A2.E35.m1.2.2.1.1.1.1.3.2.3.2.2">𝜽</ci></apply><ci id="A2.E35.m1.2.2.1.1.1.1.3.2.3.3.cmml" xref="A2.E35.m1.2.2.1.1.1.1.3.2.3.3">𝑇</ci></apply><apply id="A2.E35.m1.2.2.1.1.1.1.3.2.4.cmml" xref="A2.E35.m1.2.2.1.1.1.1.3.2.4"><csymbol cd="ambiguous" id="A2.E35.m1.2.2.1.1.1.1.3.2.4.1.cmml" xref="A2.E35.m1.2.2.1.1.1.1.3.2.4">subscript</csymbol><ci id="A2.E35.m1.2.2.1.1.1.1.3.2.4.2.cmml" xref="A2.E35.m1.2.2.1.1.1.1.3.2.4.2">Φ</ci><ci id="A2.E35.m1.2.2.1.1.1.1.3.2.4.3.cmml" xref="A2.E35.m1.2.2.1.1.1.1.3.2.4.3">f</ci></apply><apply id="A2.E35.m1.2.2.1.1.1.1.3.2.5.cmml" xref="A2.E35.m1.2.2.1.1.1.1.3.2.5"><csymbol cd="ambiguous" id="A2.E35.m1.2.2.1.1.1.1.3.2.5.1.cmml" xref="A2.E35.m1.2.2.1.1.1.1.3.2.5">superscript</csymbol><apply id="A2.E35.m1.2.2.1.1.1.1.3.2.5.2.cmml" xref="A2.E35.m1.2.2.1.1.1.1.3.2.5"><csymbol cd="ambiguous" id="A2.E35.m1.2.2.1.1.1.1.3.2.5.2.1.cmml" xref="A2.E35.m1.2.2.1.1.1.1.3.2.5">subscript</csymbol><ci id="A2.E35.m1.2.2.1.1.1.1.3.2.5.2.2.cmml" xref="A2.E35.m1.2.2.1.1.1.1.3.2.5.2.2">Φ</ci><ci id="A2.E35.m1.2.2.1.1.1.1.3.2.5.2.3.cmml" xref="A2.E35.m1.2.2.1.1.1.1.3.2.5.2.3">f</ci></apply><ci id="A2.E35.m1.2.2.1.1.1.1.3.2.5.3.cmml" xref="A2.E35.m1.2.2.1.1.1.1.3.2.5.3">𝑇</ci></apply><apply id="A2.E35.m1.2.2.1.1.1.1.3.2.6.cmml" xref="A2.E35.m1.2.2.1.1.1.1.3.2.6"><ci id="A2.E35.m1.2.2.1.1.1.1.3.2.6.1.cmml" xref="A2.E35.m1.2.2.1.1.1.1.3.2.6.1">~</ci><ci id="A2.E35.m1.2.2.1.1.1.1.3.2.6.2.cmml" xref="A2.E35.m1.2.2.1.1.1.1.3.2.6.2">𝜽</ci></apply></apply></apply></apply><apply id="A2.E35.m1.2.2.1.1.2.2.cmml" xref="A2.E35.m1.2.2.1.1.2.2"><in id="A2.E35.m1.2.2.1.1.2.2.2.cmml" xref="A2.E35.m1.2.2.1.1.2.2.2"></in><apply id="A2.E35.m1.2.2.1.1.2.2.3.cmml" xref="A2.E35.m1.2.2.1.1.2.2.3"><csymbol cd="latexml" id="A2.E35.m1.2.2.1.1.2.2.3.1.cmml" xref="A2.E35.m1.2.2.1.1.2.2.3.1">for-all</csymbol><ci id="A2.E35.m1.2.2.1.1.2.2.3.2.cmml" xref="A2.E35.m1.2.2.1.1.2.2.3.2">𝑡</ci></apply><interval closure="closed-open" id="A2.E35.m1.2.2.1.1.2.2.1.2.cmml" xref="A2.E35.m1.2.2.1.1.2.2.1.1"><cn id="A2.E35.m1.1.1.cmml" type="integer" xref="A2.E35.m1.1.1">0</cn><apply id="A2.E35.m1.2.2.1.1.2.2.1.1.1.cmml" xref="A2.E35.m1.2.2.1.1.2.2.1.1.1"><csymbol cd="ambiguous" id="A2.E35.m1.2.2.1.1.2.2.1.1.1.1.cmml" xref="A2.E35.m1.2.2.1.1.2.2.1.1.1">subscript</csymbol><ci id="A2.E35.m1.2.2.1.1.2.2.1.1.1.2.cmml" xref="A2.E35.m1.2.2.1.1.2.2.1.1.1.2">𝑡</ci><ci id="A2.E35.m1.2.2.1.1.2.2.1.1.1.3.cmml" xref="A2.E35.m1.2.2.1.1.2.2.1.1.1.3">f</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E35.m1.2c">\displaystyle\dot{V}_{\theta}\leq-2\tilde{\bm{\theta}}^{T}\Phi_{\rm f}\Phi_{% \rm f}^{T}\tilde{\bm{\theta}},\forall t\in[0,t_{\rm f}).</annotation><annotation encoding="application/x-llamapun" id="A2.E35.m1.2d">over˙ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ≤ - 2 over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG , ∀ italic_t ∈ [ 0 , italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(35)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.1.p1.14">Noting <math alttext="V_{\theta}(t)\geq 0" class="ltx_Math" display="inline" id="A2.1.p1.9.m1.1"><semantics id="A2.1.p1.9.m1.1a"><mrow id="A2.1.p1.9.m1.1.2" xref="A2.1.p1.9.m1.1.2.cmml"><mrow id="A2.1.p1.9.m1.1.2.2" xref="A2.1.p1.9.m1.1.2.2.cmml"><msub id="A2.1.p1.9.m1.1.2.2.2" xref="A2.1.p1.9.m1.1.2.2.2.cmml"><mi id="A2.1.p1.9.m1.1.2.2.2.2" xref="A2.1.p1.9.m1.1.2.2.2.2.cmml">V</mi><mi id="A2.1.p1.9.m1.1.2.2.2.3" xref="A2.1.p1.9.m1.1.2.2.2.3.cmml">θ</mi></msub><mo id="A2.1.p1.9.m1.1.2.2.1" xref="A2.1.p1.9.m1.1.2.2.1.cmml">⁢</mo><mrow id="A2.1.p1.9.m1.1.2.2.3.2" xref="A2.1.p1.9.m1.1.2.2.cmml"><mo id="A2.1.p1.9.m1.1.2.2.3.2.1" stretchy="false" xref="A2.1.p1.9.m1.1.2.2.cmml">(</mo><mi id="A2.1.p1.9.m1.1.1" xref="A2.1.p1.9.m1.1.1.cmml">t</mi><mo id="A2.1.p1.9.m1.1.2.2.3.2.2" stretchy="false" xref="A2.1.p1.9.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="A2.1.p1.9.m1.1.2.1" xref="A2.1.p1.9.m1.1.2.1.cmml">≥</mo><mn id="A2.1.p1.9.m1.1.2.3" xref="A2.1.p1.9.m1.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.9.m1.1b"><apply id="A2.1.p1.9.m1.1.2.cmml" xref="A2.1.p1.9.m1.1.2"><geq id="A2.1.p1.9.m1.1.2.1.cmml" xref="A2.1.p1.9.m1.1.2.1"></geq><apply id="A2.1.p1.9.m1.1.2.2.cmml" xref="A2.1.p1.9.m1.1.2.2"><times id="A2.1.p1.9.m1.1.2.2.1.cmml" xref="A2.1.p1.9.m1.1.2.2.1"></times><apply id="A2.1.p1.9.m1.1.2.2.2.cmml" xref="A2.1.p1.9.m1.1.2.2.2"><csymbol cd="ambiguous" id="A2.1.p1.9.m1.1.2.2.2.1.cmml" xref="A2.1.p1.9.m1.1.2.2.2">subscript</csymbol><ci id="A2.1.p1.9.m1.1.2.2.2.2.cmml" xref="A2.1.p1.9.m1.1.2.2.2.2">𝑉</ci><ci id="A2.1.p1.9.m1.1.2.2.2.3.cmml" xref="A2.1.p1.9.m1.1.2.2.2.3">𝜃</ci></apply><ci id="A2.1.p1.9.m1.1.1.cmml" xref="A2.1.p1.9.m1.1.1">𝑡</ci></apply><cn id="A2.1.p1.9.m1.1.2.3.cmml" type="integer" xref="A2.1.p1.9.m1.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.9.m1.1c">V_{\theta}(t)\geq 0</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.9.m1.1d">italic_V start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( italic_t ) ≥ 0</annotation></semantics></math> and <math alttext="\dot{V}_{\theta}(t)\leq 0" class="ltx_Math" display="inline" id="A2.1.p1.10.m2.1"><semantics id="A2.1.p1.10.m2.1a"><mrow id="A2.1.p1.10.m2.1.2" xref="A2.1.p1.10.m2.1.2.cmml"><mrow id="A2.1.p1.10.m2.1.2.2" xref="A2.1.p1.10.m2.1.2.2.cmml"><msub id="A2.1.p1.10.m2.1.2.2.2" xref="A2.1.p1.10.m2.1.2.2.2.cmml"><mover accent="true" id="A2.1.p1.10.m2.1.2.2.2.2" xref="A2.1.p1.10.m2.1.2.2.2.2.cmml"><mi id="A2.1.p1.10.m2.1.2.2.2.2.2" xref="A2.1.p1.10.m2.1.2.2.2.2.2.cmml">V</mi><mo id="A2.1.p1.10.m2.1.2.2.2.2.1" xref="A2.1.p1.10.m2.1.2.2.2.2.1.cmml">˙</mo></mover><mi id="A2.1.p1.10.m2.1.2.2.2.3" xref="A2.1.p1.10.m2.1.2.2.2.3.cmml">θ</mi></msub><mo id="A2.1.p1.10.m2.1.2.2.1" xref="A2.1.p1.10.m2.1.2.2.1.cmml">⁢</mo><mrow id="A2.1.p1.10.m2.1.2.2.3.2" xref="A2.1.p1.10.m2.1.2.2.cmml"><mo id="A2.1.p1.10.m2.1.2.2.3.2.1" stretchy="false" xref="A2.1.p1.10.m2.1.2.2.cmml">(</mo><mi id="A2.1.p1.10.m2.1.1" xref="A2.1.p1.10.m2.1.1.cmml">t</mi><mo id="A2.1.p1.10.m2.1.2.2.3.2.2" stretchy="false" xref="A2.1.p1.10.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="A2.1.p1.10.m2.1.2.1" xref="A2.1.p1.10.m2.1.2.1.cmml">≤</mo><mn id="A2.1.p1.10.m2.1.2.3" xref="A2.1.p1.10.m2.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.10.m2.1b"><apply id="A2.1.p1.10.m2.1.2.cmml" xref="A2.1.p1.10.m2.1.2"><leq id="A2.1.p1.10.m2.1.2.1.cmml" xref="A2.1.p1.10.m2.1.2.1"></leq><apply id="A2.1.p1.10.m2.1.2.2.cmml" xref="A2.1.p1.10.m2.1.2.2"><times id="A2.1.p1.10.m2.1.2.2.1.cmml" xref="A2.1.p1.10.m2.1.2.2.1"></times><apply id="A2.1.p1.10.m2.1.2.2.2.cmml" xref="A2.1.p1.10.m2.1.2.2.2"><csymbol cd="ambiguous" id="A2.1.p1.10.m2.1.2.2.2.1.cmml" xref="A2.1.p1.10.m2.1.2.2.2">subscript</csymbol><apply id="A2.1.p1.10.m2.1.2.2.2.2.cmml" xref="A2.1.p1.10.m2.1.2.2.2.2"><ci id="A2.1.p1.10.m2.1.2.2.2.2.1.cmml" xref="A2.1.p1.10.m2.1.2.2.2.2.1">˙</ci><ci id="A2.1.p1.10.m2.1.2.2.2.2.2.cmml" xref="A2.1.p1.10.m2.1.2.2.2.2.2">𝑉</ci></apply><ci id="A2.1.p1.10.m2.1.2.2.2.3.cmml" xref="A2.1.p1.10.m2.1.2.2.2.3">𝜃</ci></apply><ci id="A2.1.p1.10.m2.1.1.cmml" xref="A2.1.p1.10.m2.1.1">𝑡</ci></apply><cn id="A2.1.p1.10.m2.1.2.3.cmml" type="integer" xref="A2.1.p1.10.m2.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.10.m2.1c">\dot{V}_{\theta}(t)\leq 0</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.10.m2.1d">over˙ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( italic_t ) ≤ 0</annotation></semantics></math> from (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A2.E35" title="In Proof. ‣ Appendix B The proof of Theorem 1 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">35</span></a>), one obtains <math alttext="0\leq V_{\theta}(t)\leq V_{\theta}(0)" class="ltx_Math" display="inline" id="A2.1.p1.11.m3.2"><semantics id="A2.1.p1.11.m3.2a"><mrow id="A2.1.p1.11.m3.2.3" xref="A2.1.p1.11.m3.2.3.cmml"><mn id="A2.1.p1.11.m3.2.3.2" xref="A2.1.p1.11.m3.2.3.2.cmml">0</mn><mo id="A2.1.p1.11.m3.2.3.3" xref="A2.1.p1.11.m3.2.3.3.cmml">≤</mo><mrow id="A2.1.p1.11.m3.2.3.4" xref="A2.1.p1.11.m3.2.3.4.cmml"><msub id="A2.1.p1.11.m3.2.3.4.2" xref="A2.1.p1.11.m3.2.3.4.2.cmml"><mi id="A2.1.p1.11.m3.2.3.4.2.2" xref="A2.1.p1.11.m3.2.3.4.2.2.cmml">V</mi><mi id="A2.1.p1.11.m3.2.3.4.2.3" xref="A2.1.p1.11.m3.2.3.4.2.3.cmml">θ</mi></msub><mo id="A2.1.p1.11.m3.2.3.4.1" xref="A2.1.p1.11.m3.2.3.4.1.cmml">⁢</mo><mrow id="A2.1.p1.11.m3.2.3.4.3.2" xref="A2.1.p1.11.m3.2.3.4.cmml"><mo id="A2.1.p1.11.m3.2.3.4.3.2.1" stretchy="false" xref="A2.1.p1.11.m3.2.3.4.cmml">(</mo><mi id="A2.1.p1.11.m3.1.1" xref="A2.1.p1.11.m3.1.1.cmml">t</mi><mo id="A2.1.p1.11.m3.2.3.4.3.2.2" stretchy="false" xref="A2.1.p1.11.m3.2.3.4.cmml">)</mo></mrow></mrow><mo id="A2.1.p1.11.m3.2.3.5" xref="A2.1.p1.11.m3.2.3.5.cmml">≤</mo><mrow id="A2.1.p1.11.m3.2.3.6" xref="A2.1.p1.11.m3.2.3.6.cmml"><msub id="A2.1.p1.11.m3.2.3.6.2" xref="A2.1.p1.11.m3.2.3.6.2.cmml"><mi id="A2.1.p1.11.m3.2.3.6.2.2" xref="A2.1.p1.11.m3.2.3.6.2.2.cmml">V</mi><mi id="A2.1.p1.11.m3.2.3.6.2.3" xref="A2.1.p1.11.m3.2.3.6.2.3.cmml">θ</mi></msub><mo id="A2.1.p1.11.m3.2.3.6.1" xref="A2.1.p1.11.m3.2.3.6.1.cmml">⁢</mo><mrow id="A2.1.p1.11.m3.2.3.6.3.2" xref="A2.1.p1.11.m3.2.3.6.cmml"><mo id="A2.1.p1.11.m3.2.3.6.3.2.1" stretchy="false" xref="A2.1.p1.11.m3.2.3.6.cmml">(</mo><mn id="A2.1.p1.11.m3.2.2" xref="A2.1.p1.11.m3.2.2.cmml">0</mn><mo id="A2.1.p1.11.m3.2.3.6.3.2.2" stretchy="false" xref="A2.1.p1.11.m3.2.3.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.11.m3.2b"><apply id="A2.1.p1.11.m3.2.3.cmml" xref="A2.1.p1.11.m3.2.3"><and id="A2.1.p1.11.m3.2.3a.cmml" xref="A2.1.p1.11.m3.2.3"></and><apply id="A2.1.p1.11.m3.2.3b.cmml" xref="A2.1.p1.11.m3.2.3"><leq id="A2.1.p1.11.m3.2.3.3.cmml" xref="A2.1.p1.11.m3.2.3.3"></leq><cn id="A2.1.p1.11.m3.2.3.2.cmml" type="integer" xref="A2.1.p1.11.m3.2.3.2">0</cn><apply id="A2.1.p1.11.m3.2.3.4.cmml" xref="A2.1.p1.11.m3.2.3.4"><times id="A2.1.p1.11.m3.2.3.4.1.cmml" xref="A2.1.p1.11.m3.2.3.4.1"></times><apply id="A2.1.p1.11.m3.2.3.4.2.cmml" xref="A2.1.p1.11.m3.2.3.4.2"><csymbol cd="ambiguous" id="A2.1.p1.11.m3.2.3.4.2.1.cmml" xref="A2.1.p1.11.m3.2.3.4.2">subscript</csymbol><ci id="A2.1.p1.11.m3.2.3.4.2.2.cmml" xref="A2.1.p1.11.m3.2.3.4.2.2">𝑉</ci><ci id="A2.1.p1.11.m3.2.3.4.2.3.cmml" xref="A2.1.p1.11.m3.2.3.4.2.3">𝜃</ci></apply><ci id="A2.1.p1.11.m3.1.1.cmml" xref="A2.1.p1.11.m3.1.1">𝑡</ci></apply></apply><apply id="A2.1.p1.11.m3.2.3c.cmml" xref="A2.1.p1.11.m3.2.3"><leq id="A2.1.p1.11.m3.2.3.5.cmml" xref="A2.1.p1.11.m3.2.3.5"></leq><share href="https://arxiv.org/html/2401.10785v2#A2.1.p1.11.m3.2.3.4.cmml" id="A2.1.p1.11.m3.2.3d.cmml" xref="A2.1.p1.11.m3.2.3"></share><apply id="A2.1.p1.11.m3.2.3.6.cmml" xref="A2.1.p1.11.m3.2.3.6"><times id="A2.1.p1.11.m3.2.3.6.1.cmml" xref="A2.1.p1.11.m3.2.3.6.1"></times><apply id="A2.1.p1.11.m3.2.3.6.2.cmml" xref="A2.1.p1.11.m3.2.3.6.2"><csymbol cd="ambiguous" id="A2.1.p1.11.m3.2.3.6.2.1.cmml" xref="A2.1.p1.11.m3.2.3.6.2">subscript</csymbol><ci id="A2.1.p1.11.m3.2.3.6.2.2.cmml" xref="A2.1.p1.11.m3.2.3.6.2.2">𝑉</ci><ci id="A2.1.p1.11.m3.2.3.6.2.3.cmml" xref="A2.1.p1.11.m3.2.3.6.2.3">𝜃</ci></apply><cn id="A2.1.p1.11.m3.2.2.cmml" type="integer" xref="A2.1.p1.11.m3.2.2">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.11.m3.2c">0\leq V_{\theta}(t)\leq V_{\theta}(0)</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.11.m3.2d">0 ≤ italic_V start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( italic_t ) ≤ italic_V start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( 0 )</annotation></semantics></math>, <math alttext="\forall t\geq 0" class="ltx_Math" display="inline" id="A2.1.p1.12.m4.1"><semantics id="A2.1.p1.12.m4.1a"><mrow id="A2.1.p1.12.m4.1.1" xref="A2.1.p1.12.m4.1.1.cmml"><mrow id="A2.1.p1.12.m4.1.1.2" xref="A2.1.p1.12.m4.1.1.2.cmml"><mo id="A2.1.p1.12.m4.1.1.2.1" rspace="0.167em" xref="A2.1.p1.12.m4.1.1.2.1.cmml">∀</mo><mi id="A2.1.p1.12.m4.1.1.2.2" xref="A2.1.p1.12.m4.1.1.2.2.cmml">t</mi></mrow><mo id="A2.1.p1.12.m4.1.1.1" xref="A2.1.p1.12.m4.1.1.1.cmml">≥</mo><mn id="A2.1.p1.12.m4.1.1.3" xref="A2.1.p1.12.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.12.m4.1b"><apply id="A2.1.p1.12.m4.1.1.cmml" xref="A2.1.p1.12.m4.1.1"><geq id="A2.1.p1.12.m4.1.1.1.cmml" xref="A2.1.p1.12.m4.1.1.1"></geq><apply id="A2.1.p1.12.m4.1.1.2.cmml" xref="A2.1.p1.12.m4.1.1.2"><csymbol cd="latexml" id="A2.1.p1.12.m4.1.1.2.1.cmml" xref="A2.1.p1.12.m4.1.1.2.1">for-all</csymbol><ci id="A2.1.p1.12.m4.1.1.2.2.cmml" xref="A2.1.p1.12.m4.1.1.2.2">𝑡</ci></apply><cn id="A2.1.p1.12.m4.1.1.3.cmml" type="integer" xref="A2.1.p1.12.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.12.m4.1c">\forall t\geq 0</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.12.m4.1d">∀ italic_t ≥ 0</annotation></semantics></math>. Integrating each side of (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A2.E35" title="In Proof. ‣ Appendix B The proof of Theorem 1 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">35</span></a>) over <math alttext="[0,t]" class="ltx_Math" display="inline" id="A2.1.p1.13.m5.2"><semantics id="A2.1.p1.13.m5.2a"><mrow id="A2.1.p1.13.m5.2.3.2" xref="A2.1.p1.13.m5.2.3.1.cmml"><mo id="A2.1.p1.13.m5.2.3.2.1" stretchy="false" xref="A2.1.p1.13.m5.2.3.1.cmml">[</mo><mn id="A2.1.p1.13.m5.1.1" xref="A2.1.p1.13.m5.1.1.cmml">0</mn><mo id="A2.1.p1.13.m5.2.3.2.2" xref="A2.1.p1.13.m5.2.3.1.cmml">,</mo><mi id="A2.1.p1.13.m5.2.2" xref="A2.1.p1.13.m5.2.2.cmml">t</mi><mo id="A2.1.p1.13.m5.2.3.2.3" stretchy="false" xref="A2.1.p1.13.m5.2.3.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.13.m5.2b"><interval closure="closed" id="A2.1.p1.13.m5.2.3.1.cmml" xref="A2.1.p1.13.m5.2.3.2"><cn id="A2.1.p1.13.m5.1.1.cmml" type="integer" xref="A2.1.p1.13.m5.1.1">0</cn><ci id="A2.1.p1.13.m5.2.2.cmml" xref="A2.1.p1.13.m5.2.2">𝑡</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.13.m5.2c">[0,t]</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.13.m5.2d">[ 0 , italic_t ]</annotation></semantics></math> and using <math alttext="\bm{\epsilon}=\Phi_{\rm f}^{T}\tilde{\bm{\theta}}" class="ltx_Math" display="inline" id="A2.1.p1.14.m6.1"><semantics id="A2.1.p1.14.m6.1a"><mrow id="A2.1.p1.14.m6.1.1" xref="A2.1.p1.14.m6.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="A2.1.p1.14.m6.1.1.2" mathvariant="bold-italic" xref="A2.1.p1.14.m6.1.1.2.cmml">ϵ</mi><mo id="A2.1.p1.14.m6.1.1.1" xref="A2.1.p1.14.m6.1.1.1.cmml">=</mo><mrow id="A2.1.p1.14.m6.1.1.3" xref="A2.1.p1.14.m6.1.1.3.cmml"><msubsup id="A2.1.p1.14.m6.1.1.3.2" xref="A2.1.p1.14.m6.1.1.3.2.cmml"><mi id="A2.1.p1.14.m6.1.1.3.2.2.2" mathvariant="normal" xref="A2.1.p1.14.m6.1.1.3.2.2.2.cmml">Φ</mi><mi id="A2.1.p1.14.m6.1.1.3.2.2.3" mathvariant="normal" xref="A2.1.p1.14.m6.1.1.3.2.2.3.cmml">f</mi><mi id="A2.1.p1.14.m6.1.1.3.2.3" xref="A2.1.p1.14.m6.1.1.3.2.3.cmml">T</mi></msubsup><mo id="A2.1.p1.14.m6.1.1.3.1" xref="A2.1.p1.14.m6.1.1.3.1.cmml">⁢</mo><mover accent="true" id="A2.1.p1.14.m6.1.1.3.3" xref="A2.1.p1.14.m6.1.1.3.3.cmml"><mi id="A2.1.p1.14.m6.1.1.3.3.2" xref="A2.1.p1.14.m6.1.1.3.3.2.cmml">𝜽</mi><mo id="A2.1.p1.14.m6.1.1.3.3.1" xref="A2.1.p1.14.m6.1.1.3.3.1.cmml">~</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.14.m6.1b"><apply id="A2.1.p1.14.m6.1.1.cmml" xref="A2.1.p1.14.m6.1.1"><eq id="A2.1.p1.14.m6.1.1.1.cmml" xref="A2.1.p1.14.m6.1.1.1"></eq><ci id="A2.1.p1.14.m6.1.1.2.cmml" xref="A2.1.p1.14.m6.1.1.2">bold-italic-ϵ</ci><apply id="A2.1.p1.14.m6.1.1.3.cmml" xref="A2.1.p1.14.m6.1.1.3"><times id="A2.1.p1.14.m6.1.1.3.1.cmml" xref="A2.1.p1.14.m6.1.1.3.1"></times><apply id="A2.1.p1.14.m6.1.1.3.2.cmml" xref="A2.1.p1.14.m6.1.1.3.2"><csymbol cd="ambiguous" id="A2.1.p1.14.m6.1.1.3.2.1.cmml" xref="A2.1.p1.14.m6.1.1.3.2">superscript</csymbol><apply id="A2.1.p1.14.m6.1.1.3.2.2.cmml" xref="A2.1.p1.14.m6.1.1.3.2"><csymbol cd="ambiguous" id="A2.1.p1.14.m6.1.1.3.2.2.1.cmml" xref="A2.1.p1.14.m6.1.1.3.2">subscript</csymbol><ci id="A2.1.p1.14.m6.1.1.3.2.2.2.cmml" xref="A2.1.p1.14.m6.1.1.3.2.2.2">Φ</ci><ci id="A2.1.p1.14.m6.1.1.3.2.2.3.cmml" xref="A2.1.p1.14.m6.1.1.3.2.2.3">f</ci></apply><ci id="A2.1.p1.14.m6.1.1.3.2.3.cmml" xref="A2.1.p1.14.m6.1.1.3.2.3">𝑇</ci></apply><apply id="A2.1.p1.14.m6.1.1.3.3.cmml" xref="A2.1.p1.14.m6.1.1.3.3"><ci id="A2.1.p1.14.m6.1.1.3.3.1.cmml" xref="A2.1.p1.14.m6.1.1.3.3.1">~</ci><ci id="A2.1.p1.14.m6.1.1.3.3.2.cmml" xref="A2.1.p1.14.m6.1.1.3.3.2">𝜽</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.14.m6.1c">\bm{\epsilon}=\Phi_{\rm f}^{T}\tilde{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.14.m6.1d">bold_italic_ϵ = roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math>, one obtains</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx38"> <tbody id="A2.Ex27"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\int_{0}^{t}\bm{\epsilon}^{2}(\tau)d\tau\leq(V_{\theta}(0)-V_{% \theta}(t))/2." class="ltx_Math" display="inline" id="A2.Ex27.m1.4"><semantics id="A2.Ex27.m1.4a"><mrow id="A2.Ex27.m1.4.4.1" xref="A2.Ex27.m1.4.4.1.1.cmml"><mrow id="A2.Ex27.m1.4.4.1.1" xref="A2.Ex27.m1.4.4.1.1.cmml"><mrow id="A2.Ex27.m1.4.4.1.1.3" xref="A2.Ex27.m1.4.4.1.1.3.cmml"><mstyle displaystyle="true" id="A2.Ex27.m1.4.4.1.1.3.1" xref="A2.Ex27.m1.4.4.1.1.3.1.cmml"><msubsup id="A2.Ex27.m1.4.4.1.1.3.1a" xref="A2.Ex27.m1.4.4.1.1.3.1.cmml"><mo id="A2.Ex27.m1.4.4.1.1.3.1.2.2" xref="A2.Ex27.m1.4.4.1.1.3.1.2.2.cmml">∫</mo><mn id="A2.Ex27.m1.4.4.1.1.3.1.2.3" xref="A2.Ex27.m1.4.4.1.1.3.1.2.3.cmml">0</mn><mi id="A2.Ex27.m1.4.4.1.1.3.1.3" xref="A2.Ex27.m1.4.4.1.1.3.1.3.cmml">t</mi></msubsup></mstyle><mrow id="A2.Ex27.m1.4.4.1.1.3.2" xref="A2.Ex27.m1.4.4.1.1.3.2.cmml"><msup id="A2.Ex27.m1.4.4.1.1.3.2.2" xref="A2.Ex27.m1.4.4.1.1.3.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="A2.Ex27.m1.4.4.1.1.3.2.2.2" mathvariant="bold-italic" xref="A2.Ex27.m1.4.4.1.1.3.2.2.2.cmml">ϵ</mi><mn id="A2.Ex27.m1.4.4.1.1.3.2.2.3" xref="A2.Ex27.m1.4.4.1.1.3.2.2.3.cmml">2</mn></msup><mo id="A2.Ex27.m1.4.4.1.1.3.2.1" xref="A2.Ex27.m1.4.4.1.1.3.2.1.cmml">⁢</mo><mrow id="A2.Ex27.m1.4.4.1.1.3.2.3.2" xref="A2.Ex27.m1.4.4.1.1.3.2.cmml"><mo id="A2.Ex27.m1.4.4.1.1.3.2.3.2.1" stretchy="false" xref="A2.Ex27.m1.4.4.1.1.3.2.cmml">(</mo><mi id="A2.Ex27.m1.1.1" xref="A2.Ex27.m1.1.1.cmml">τ</mi><mo id="A2.Ex27.m1.4.4.1.1.3.2.3.2.2" stretchy="false" xref="A2.Ex27.m1.4.4.1.1.3.2.cmml">)</mo></mrow><mo id="A2.Ex27.m1.4.4.1.1.3.2.1a" lspace="0em" xref="A2.Ex27.m1.4.4.1.1.3.2.1.cmml">⁢</mo><mrow id="A2.Ex27.m1.4.4.1.1.3.2.4" xref="A2.Ex27.m1.4.4.1.1.3.2.4.cmml"><mo id="A2.Ex27.m1.4.4.1.1.3.2.4.1" rspace="0em" xref="A2.Ex27.m1.4.4.1.1.3.2.4.1.cmml">𝑑</mo><mi id="A2.Ex27.m1.4.4.1.1.3.2.4.2" xref="A2.Ex27.m1.4.4.1.1.3.2.4.2.cmml">τ</mi></mrow></mrow></mrow><mo id="A2.Ex27.m1.4.4.1.1.2" xref="A2.Ex27.m1.4.4.1.1.2.cmml">≤</mo><mrow id="A2.Ex27.m1.4.4.1.1.1" xref="A2.Ex27.m1.4.4.1.1.1.cmml"><mrow id="A2.Ex27.m1.4.4.1.1.1.1.1" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.cmml"><mo id="A2.Ex27.m1.4.4.1.1.1.1.1.2" stretchy="false" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.cmml">(</mo><mrow id="A2.Ex27.m1.4.4.1.1.1.1.1.1" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.cmml"><mrow id="A2.Ex27.m1.4.4.1.1.1.1.1.1.2" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.cmml"><msub id="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.2" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.2.cmml"><mi id="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.2.2" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.2.2.cmml">V</mi><mi id="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.2.3" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.2.3.cmml">θ</mi></msub><mo id="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.1" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.1.cmml">⁢</mo><mrow id="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.3.2" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.cmml"><mo id="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.3.2.1" stretchy="false" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.cmml">(</mo><mn id="A2.Ex27.m1.2.2" xref="A2.Ex27.m1.2.2.cmml">0</mn><mo id="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.3.2.2" stretchy="false" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="A2.Ex27.m1.4.4.1.1.1.1.1.1.1" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.1.cmml">−</mo><mrow id="A2.Ex27.m1.4.4.1.1.1.1.1.1.3" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.cmml"><msub id="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.2" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.2.cmml"><mi id="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.2.2" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.2.2.cmml">V</mi><mi id="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.2.3" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.2.3.cmml">θ</mi></msub><mo id="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.1" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.3.2" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.cmml"><mo id="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.3.2.1" stretchy="false" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.cmml">(</mo><mi id="A2.Ex27.m1.3.3" xref="A2.Ex27.m1.3.3.cmml">t</mi><mo id="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.3.2.2" stretchy="false" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="A2.Ex27.m1.4.4.1.1.1.1.1.3" stretchy="false" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A2.Ex27.m1.4.4.1.1.1.2" xref="A2.Ex27.m1.4.4.1.1.1.2.cmml">/</mo><mn id="A2.Ex27.m1.4.4.1.1.1.3" xref="A2.Ex27.m1.4.4.1.1.1.3.cmml">2</mn></mrow></mrow><mo id="A2.Ex27.m1.4.4.1.2" lspace="0em" xref="A2.Ex27.m1.4.4.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex27.m1.4b"><apply id="A2.Ex27.m1.4.4.1.1.cmml" xref="A2.Ex27.m1.4.4.1"><leq id="A2.Ex27.m1.4.4.1.1.2.cmml" xref="A2.Ex27.m1.4.4.1.1.2"></leq><apply id="A2.Ex27.m1.4.4.1.1.3.cmml" xref="A2.Ex27.m1.4.4.1.1.3"><apply id="A2.Ex27.m1.4.4.1.1.3.1.cmml" xref="A2.Ex27.m1.4.4.1.1.3.1"><csymbol cd="ambiguous" id="A2.Ex27.m1.4.4.1.1.3.1.1.cmml" xref="A2.Ex27.m1.4.4.1.1.3.1">superscript</csymbol><apply id="A2.Ex27.m1.4.4.1.1.3.1.2.cmml" xref="A2.Ex27.m1.4.4.1.1.3.1"><csymbol cd="ambiguous" id="A2.Ex27.m1.4.4.1.1.3.1.2.1.cmml" xref="A2.Ex27.m1.4.4.1.1.3.1">subscript</csymbol><int id="A2.Ex27.m1.4.4.1.1.3.1.2.2.cmml" xref="A2.Ex27.m1.4.4.1.1.3.1.2.2"></int><cn id="A2.Ex27.m1.4.4.1.1.3.1.2.3.cmml" type="integer" xref="A2.Ex27.m1.4.4.1.1.3.1.2.3">0</cn></apply><ci id="A2.Ex27.m1.4.4.1.1.3.1.3.cmml" xref="A2.Ex27.m1.4.4.1.1.3.1.3">𝑡</ci></apply><apply id="A2.Ex27.m1.4.4.1.1.3.2.cmml" xref="A2.Ex27.m1.4.4.1.1.3.2"><times id="A2.Ex27.m1.4.4.1.1.3.2.1.cmml" xref="A2.Ex27.m1.4.4.1.1.3.2.1"></times><apply id="A2.Ex27.m1.4.4.1.1.3.2.2.cmml" xref="A2.Ex27.m1.4.4.1.1.3.2.2"><csymbol cd="ambiguous" id="A2.Ex27.m1.4.4.1.1.3.2.2.1.cmml" xref="A2.Ex27.m1.4.4.1.1.3.2.2">superscript</csymbol><ci id="A2.Ex27.m1.4.4.1.1.3.2.2.2.cmml" xref="A2.Ex27.m1.4.4.1.1.3.2.2.2">bold-italic-ϵ</ci><cn id="A2.Ex27.m1.4.4.1.1.3.2.2.3.cmml" type="integer" xref="A2.Ex27.m1.4.4.1.1.3.2.2.3">2</cn></apply><ci id="A2.Ex27.m1.1.1.cmml" xref="A2.Ex27.m1.1.1">𝜏</ci><apply id="A2.Ex27.m1.4.4.1.1.3.2.4.cmml" xref="A2.Ex27.m1.4.4.1.1.3.2.4"><csymbol cd="latexml" id="A2.Ex27.m1.4.4.1.1.3.2.4.1.cmml" xref="A2.Ex27.m1.4.4.1.1.3.2.4.1">differential-d</csymbol><ci id="A2.Ex27.m1.4.4.1.1.3.2.4.2.cmml" xref="A2.Ex27.m1.4.4.1.1.3.2.4.2">𝜏</ci></apply></apply></apply><apply id="A2.Ex27.m1.4.4.1.1.1.cmml" xref="A2.Ex27.m1.4.4.1.1.1"><divide id="A2.Ex27.m1.4.4.1.1.1.2.cmml" xref="A2.Ex27.m1.4.4.1.1.1.2"></divide><apply id="A2.Ex27.m1.4.4.1.1.1.1.1.1.cmml" xref="A2.Ex27.m1.4.4.1.1.1.1.1"><minus id="A2.Ex27.m1.4.4.1.1.1.1.1.1.1.cmml" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.1"></minus><apply id="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.cmml" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.2"><times id="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.1.cmml" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.1"></times><apply id="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.2.cmml" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.2.1.cmml" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.2">subscript</csymbol><ci id="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.2.2.cmml" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.2.2">𝑉</ci><ci id="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.2.3.cmml" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.2.2.3">𝜃</ci></apply><cn id="A2.Ex27.m1.2.2.cmml" type="integer" xref="A2.Ex27.m1.2.2">0</cn></apply><apply id="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.cmml" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.3"><times id="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.1.cmml" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.1"></times><apply id="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.2.cmml" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.2.1.cmml" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.2">subscript</csymbol><ci id="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.2.2.cmml" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.2.2">𝑉</ci><ci id="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.2.3.cmml" xref="A2.Ex27.m1.4.4.1.1.1.1.1.1.3.2.3">𝜃</ci></apply><ci id="A2.Ex27.m1.3.3.cmml" xref="A2.Ex27.m1.3.3">𝑡</ci></apply></apply><cn id="A2.Ex27.m1.4.4.1.1.1.3.cmml" type="integer" xref="A2.Ex27.m1.4.4.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex27.m1.4c">\displaystyle\int_{0}^{t}\bm{\epsilon}^{2}(\tau)d\tau\leq(V_{\theta}(0)-V_{% \theta}(t))/2.</annotation><annotation encoding="application/x-llamapun" id="A2.Ex27.m1.4d">∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT bold_italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_τ ) italic_d italic_τ ≤ ( italic_V start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( 0 ) - italic_V start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( italic_t ) ) / 2 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A2.1.p1.19">Noting <math alttext="V_{\theta}(t)\in L_{\infty}" class="ltx_Math" display="inline" id="A2.1.p1.15.m1.1"><semantics id="A2.1.p1.15.m1.1a"><mrow id="A2.1.p1.15.m1.1.2" xref="A2.1.p1.15.m1.1.2.cmml"><mrow id="A2.1.p1.15.m1.1.2.2" xref="A2.1.p1.15.m1.1.2.2.cmml"><msub id="A2.1.p1.15.m1.1.2.2.2" xref="A2.1.p1.15.m1.1.2.2.2.cmml"><mi id="A2.1.p1.15.m1.1.2.2.2.2" xref="A2.1.p1.15.m1.1.2.2.2.2.cmml">V</mi><mi id="A2.1.p1.15.m1.1.2.2.2.3" xref="A2.1.p1.15.m1.1.2.2.2.3.cmml">θ</mi></msub><mo id="A2.1.p1.15.m1.1.2.2.1" xref="A2.1.p1.15.m1.1.2.2.1.cmml">⁢</mo><mrow id="A2.1.p1.15.m1.1.2.2.3.2" xref="A2.1.p1.15.m1.1.2.2.cmml"><mo id="A2.1.p1.15.m1.1.2.2.3.2.1" stretchy="false" xref="A2.1.p1.15.m1.1.2.2.cmml">(</mo><mi id="A2.1.p1.15.m1.1.1" xref="A2.1.p1.15.m1.1.1.cmml">t</mi><mo id="A2.1.p1.15.m1.1.2.2.3.2.2" stretchy="false" xref="A2.1.p1.15.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="A2.1.p1.15.m1.1.2.1" xref="A2.1.p1.15.m1.1.2.1.cmml">∈</mo><msub id="A2.1.p1.15.m1.1.2.3" xref="A2.1.p1.15.m1.1.2.3.cmml"><mi id="A2.1.p1.15.m1.1.2.3.2" xref="A2.1.p1.15.m1.1.2.3.2.cmml">L</mi><mi id="A2.1.p1.15.m1.1.2.3.3" mathvariant="normal" xref="A2.1.p1.15.m1.1.2.3.3.cmml">∞</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.15.m1.1b"><apply id="A2.1.p1.15.m1.1.2.cmml" xref="A2.1.p1.15.m1.1.2"><in id="A2.1.p1.15.m1.1.2.1.cmml" xref="A2.1.p1.15.m1.1.2.1"></in><apply id="A2.1.p1.15.m1.1.2.2.cmml" xref="A2.1.p1.15.m1.1.2.2"><times id="A2.1.p1.15.m1.1.2.2.1.cmml" xref="A2.1.p1.15.m1.1.2.2.1"></times><apply id="A2.1.p1.15.m1.1.2.2.2.cmml" xref="A2.1.p1.15.m1.1.2.2.2"><csymbol cd="ambiguous" id="A2.1.p1.15.m1.1.2.2.2.1.cmml" xref="A2.1.p1.15.m1.1.2.2.2">subscript</csymbol><ci id="A2.1.p1.15.m1.1.2.2.2.2.cmml" xref="A2.1.p1.15.m1.1.2.2.2.2">𝑉</ci><ci id="A2.1.p1.15.m1.1.2.2.2.3.cmml" xref="A2.1.p1.15.m1.1.2.2.2.3">𝜃</ci></apply><ci id="A2.1.p1.15.m1.1.1.cmml" xref="A2.1.p1.15.m1.1.1">𝑡</ci></apply><apply id="A2.1.p1.15.m1.1.2.3.cmml" xref="A2.1.p1.15.m1.1.2.3"><csymbol cd="ambiguous" id="A2.1.p1.15.m1.1.2.3.1.cmml" xref="A2.1.p1.15.m1.1.2.3">subscript</csymbol><ci id="A2.1.p1.15.m1.1.2.3.2.cmml" xref="A2.1.p1.15.m1.1.2.3.2">𝐿</ci><infinity id="A2.1.p1.15.m1.1.2.3.3.cmml" xref="A2.1.p1.15.m1.1.2.3.3"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.15.m1.1c">V_{\theta}(t)\in L_{\infty}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.15.m1.1d">italic_V start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( italic_t ) ∈ italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="0\leq V_{\theta}(t)\leq V_{\theta}(0)" class="ltx_Math" display="inline" id="A2.1.p1.16.m2.2"><semantics id="A2.1.p1.16.m2.2a"><mrow id="A2.1.p1.16.m2.2.3" xref="A2.1.p1.16.m2.2.3.cmml"><mn id="A2.1.p1.16.m2.2.3.2" xref="A2.1.p1.16.m2.2.3.2.cmml">0</mn><mo id="A2.1.p1.16.m2.2.3.3" xref="A2.1.p1.16.m2.2.3.3.cmml">≤</mo><mrow id="A2.1.p1.16.m2.2.3.4" xref="A2.1.p1.16.m2.2.3.4.cmml"><msub id="A2.1.p1.16.m2.2.3.4.2" xref="A2.1.p1.16.m2.2.3.4.2.cmml"><mi id="A2.1.p1.16.m2.2.3.4.2.2" xref="A2.1.p1.16.m2.2.3.4.2.2.cmml">V</mi><mi id="A2.1.p1.16.m2.2.3.4.2.3" xref="A2.1.p1.16.m2.2.3.4.2.3.cmml">θ</mi></msub><mo id="A2.1.p1.16.m2.2.3.4.1" xref="A2.1.p1.16.m2.2.3.4.1.cmml">⁢</mo><mrow id="A2.1.p1.16.m2.2.3.4.3.2" xref="A2.1.p1.16.m2.2.3.4.cmml"><mo id="A2.1.p1.16.m2.2.3.4.3.2.1" stretchy="false" xref="A2.1.p1.16.m2.2.3.4.cmml">(</mo><mi id="A2.1.p1.16.m2.1.1" xref="A2.1.p1.16.m2.1.1.cmml">t</mi><mo id="A2.1.p1.16.m2.2.3.4.3.2.2" stretchy="false" xref="A2.1.p1.16.m2.2.3.4.cmml">)</mo></mrow></mrow><mo id="A2.1.p1.16.m2.2.3.5" xref="A2.1.p1.16.m2.2.3.5.cmml">≤</mo><mrow id="A2.1.p1.16.m2.2.3.6" xref="A2.1.p1.16.m2.2.3.6.cmml"><msub id="A2.1.p1.16.m2.2.3.6.2" xref="A2.1.p1.16.m2.2.3.6.2.cmml"><mi id="A2.1.p1.16.m2.2.3.6.2.2" xref="A2.1.p1.16.m2.2.3.6.2.2.cmml">V</mi><mi id="A2.1.p1.16.m2.2.3.6.2.3" xref="A2.1.p1.16.m2.2.3.6.2.3.cmml">θ</mi></msub><mo id="A2.1.p1.16.m2.2.3.6.1" xref="A2.1.p1.16.m2.2.3.6.1.cmml">⁢</mo><mrow id="A2.1.p1.16.m2.2.3.6.3.2" xref="A2.1.p1.16.m2.2.3.6.cmml"><mo id="A2.1.p1.16.m2.2.3.6.3.2.1" stretchy="false" xref="A2.1.p1.16.m2.2.3.6.cmml">(</mo><mn id="A2.1.p1.16.m2.2.2" xref="A2.1.p1.16.m2.2.2.cmml">0</mn><mo id="A2.1.p1.16.m2.2.3.6.3.2.2" stretchy="false" xref="A2.1.p1.16.m2.2.3.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.16.m2.2b"><apply id="A2.1.p1.16.m2.2.3.cmml" xref="A2.1.p1.16.m2.2.3"><and id="A2.1.p1.16.m2.2.3a.cmml" xref="A2.1.p1.16.m2.2.3"></and><apply id="A2.1.p1.16.m2.2.3b.cmml" xref="A2.1.p1.16.m2.2.3"><leq id="A2.1.p1.16.m2.2.3.3.cmml" xref="A2.1.p1.16.m2.2.3.3"></leq><cn id="A2.1.p1.16.m2.2.3.2.cmml" type="integer" xref="A2.1.p1.16.m2.2.3.2">0</cn><apply id="A2.1.p1.16.m2.2.3.4.cmml" xref="A2.1.p1.16.m2.2.3.4"><times id="A2.1.p1.16.m2.2.3.4.1.cmml" xref="A2.1.p1.16.m2.2.3.4.1"></times><apply id="A2.1.p1.16.m2.2.3.4.2.cmml" xref="A2.1.p1.16.m2.2.3.4.2"><csymbol cd="ambiguous" id="A2.1.p1.16.m2.2.3.4.2.1.cmml" xref="A2.1.p1.16.m2.2.3.4.2">subscript</csymbol><ci id="A2.1.p1.16.m2.2.3.4.2.2.cmml" xref="A2.1.p1.16.m2.2.3.4.2.2">𝑉</ci><ci id="A2.1.p1.16.m2.2.3.4.2.3.cmml" xref="A2.1.p1.16.m2.2.3.4.2.3">𝜃</ci></apply><ci id="A2.1.p1.16.m2.1.1.cmml" xref="A2.1.p1.16.m2.1.1">𝑡</ci></apply></apply><apply id="A2.1.p1.16.m2.2.3c.cmml" xref="A2.1.p1.16.m2.2.3"><leq id="A2.1.p1.16.m2.2.3.5.cmml" xref="A2.1.p1.16.m2.2.3.5"></leq><share href="https://arxiv.org/html/2401.10785v2#A2.1.p1.16.m2.2.3.4.cmml" id="A2.1.p1.16.m2.2.3d.cmml" xref="A2.1.p1.16.m2.2.3"></share><apply id="A2.1.p1.16.m2.2.3.6.cmml" xref="A2.1.p1.16.m2.2.3.6"><times id="A2.1.p1.16.m2.2.3.6.1.cmml" xref="A2.1.p1.16.m2.2.3.6.1"></times><apply id="A2.1.p1.16.m2.2.3.6.2.cmml" xref="A2.1.p1.16.m2.2.3.6.2"><csymbol cd="ambiguous" id="A2.1.p1.16.m2.2.3.6.2.1.cmml" xref="A2.1.p1.16.m2.2.3.6.2">subscript</csymbol><ci id="A2.1.p1.16.m2.2.3.6.2.2.cmml" xref="A2.1.p1.16.m2.2.3.6.2.2">𝑉</ci><ci id="A2.1.p1.16.m2.2.3.6.2.3.cmml" xref="A2.1.p1.16.m2.2.3.6.2.3">𝜃</ci></apply><cn id="A2.1.p1.16.m2.2.2.cmml" type="integer" xref="A2.1.p1.16.m2.2.2">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.16.m2.2c">0\leq V_{\theta}(t)\leq V_{\theta}(0)</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.16.m2.2d">0 ≤ italic_V start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( italic_t ) ≤ italic_V start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( 0 )</annotation></semantics></math>, the above result implies <math alttext="\bm{\epsilon}\in L_{2}" class="ltx_Math" display="inline" id="A2.1.p1.17.m3.1"><semantics id="A2.1.p1.17.m3.1a"><mrow id="A2.1.p1.17.m3.1.1" xref="A2.1.p1.17.m3.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="A2.1.p1.17.m3.1.1.2" mathvariant="bold-italic" xref="A2.1.p1.17.m3.1.1.2.cmml">ϵ</mi><mo id="A2.1.p1.17.m3.1.1.1" xref="A2.1.p1.17.m3.1.1.1.cmml">∈</mo><msub id="A2.1.p1.17.m3.1.1.3" xref="A2.1.p1.17.m3.1.1.3.cmml"><mi id="A2.1.p1.17.m3.1.1.3.2" xref="A2.1.p1.17.m3.1.1.3.2.cmml">L</mi><mn id="A2.1.p1.17.m3.1.1.3.3" xref="A2.1.p1.17.m3.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.17.m3.1b"><apply id="A2.1.p1.17.m3.1.1.cmml" xref="A2.1.p1.17.m3.1.1"><in id="A2.1.p1.17.m3.1.1.1.cmml" xref="A2.1.p1.17.m3.1.1.1"></in><ci id="A2.1.p1.17.m3.1.1.2.cmml" xref="A2.1.p1.17.m3.1.1.2">bold-italic-ϵ</ci><apply id="A2.1.p1.17.m3.1.1.3.cmml" xref="A2.1.p1.17.m3.1.1.3"><csymbol cd="ambiguous" id="A2.1.p1.17.m3.1.1.3.1.cmml" xref="A2.1.p1.17.m3.1.1.3">subscript</csymbol><ci id="A2.1.p1.17.m3.1.1.3.2.cmml" xref="A2.1.p1.17.m3.1.1.3.2">𝐿</ci><cn id="A2.1.p1.17.m3.1.1.3.3.cmml" type="integer" xref="A2.1.p1.17.m3.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.17.m3.1c">\bm{\epsilon}\in L_{2}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.17.m3.1d">bold_italic_ϵ ∈ italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\forall t\in[0,t_{\rm f})" class="ltx_Math" display="inline" id="A2.1.p1.18.m4.2"><semantics id="A2.1.p1.18.m4.2a"><mrow id="A2.1.p1.18.m4.2.2" xref="A2.1.p1.18.m4.2.2.cmml"><mrow id="A2.1.p1.18.m4.2.2.3" xref="A2.1.p1.18.m4.2.2.3.cmml"><mo id="A2.1.p1.18.m4.2.2.3.1" rspace="0.167em" xref="A2.1.p1.18.m4.2.2.3.1.cmml">∀</mo><mi id="A2.1.p1.18.m4.2.2.3.2" xref="A2.1.p1.18.m4.2.2.3.2.cmml">t</mi></mrow><mo id="A2.1.p1.18.m4.2.2.2" xref="A2.1.p1.18.m4.2.2.2.cmml">∈</mo><mrow id="A2.1.p1.18.m4.2.2.1.1" xref="A2.1.p1.18.m4.2.2.1.2.cmml"><mo id="A2.1.p1.18.m4.2.2.1.1.2" stretchy="false" xref="A2.1.p1.18.m4.2.2.1.2.cmml">[</mo><mn id="A2.1.p1.18.m4.1.1" xref="A2.1.p1.18.m4.1.1.cmml">0</mn><mo id="A2.1.p1.18.m4.2.2.1.1.3" xref="A2.1.p1.18.m4.2.2.1.2.cmml">,</mo><msub id="A2.1.p1.18.m4.2.2.1.1.1" xref="A2.1.p1.18.m4.2.2.1.1.1.cmml"><mi id="A2.1.p1.18.m4.2.2.1.1.1.2" xref="A2.1.p1.18.m4.2.2.1.1.1.2.cmml">t</mi><mi id="A2.1.p1.18.m4.2.2.1.1.1.3" mathvariant="normal" xref="A2.1.p1.18.m4.2.2.1.1.1.3.cmml">f</mi></msub><mo id="A2.1.p1.18.m4.2.2.1.1.4" stretchy="false" xref="A2.1.p1.18.m4.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.18.m4.2b"><apply id="A2.1.p1.18.m4.2.2.cmml" xref="A2.1.p1.18.m4.2.2"><in id="A2.1.p1.18.m4.2.2.2.cmml" xref="A2.1.p1.18.m4.2.2.2"></in><apply id="A2.1.p1.18.m4.2.2.3.cmml" xref="A2.1.p1.18.m4.2.2.3"><csymbol cd="latexml" id="A2.1.p1.18.m4.2.2.3.1.cmml" xref="A2.1.p1.18.m4.2.2.3.1">for-all</csymbol><ci id="A2.1.p1.18.m4.2.2.3.2.cmml" xref="A2.1.p1.18.m4.2.2.3.2">𝑡</ci></apply><interval closure="closed-open" id="A2.1.p1.18.m4.2.2.1.2.cmml" xref="A2.1.p1.18.m4.2.2.1.1"><cn id="A2.1.p1.18.m4.1.1.cmml" type="integer" xref="A2.1.p1.18.m4.1.1">0</cn><apply id="A2.1.p1.18.m4.2.2.1.1.1.cmml" xref="A2.1.p1.18.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="A2.1.p1.18.m4.2.2.1.1.1.1.cmml" xref="A2.1.p1.18.m4.2.2.1.1.1">subscript</csymbol><ci id="A2.1.p1.18.m4.2.2.1.1.1.2.cmml" xref="A2.1.p1.18.m4.2.2.1.1.1.2">𝑡</ci><ci id="A2.1.p1.18.m4.2.2.1.1.1.3.cmml" xref="A2.1.p1.18.m4.2.2.1.1.1.3">f</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.18.m4.2c">\forall t\in[0,t_{\rm f})</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.18.m4.2d">∀ italic_t ∈ [ 0 , italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT )</annotation></semantics></math>. Using <math alttext="V_{\theta}(t)\leq V_{\theta}(0)" class="ltx_Math" display="inline" id="A2.1.p1.19.m5.2"><semantics id="A2.1.p1.19.m5.2a"><mrow id="A2.1.p1.19.m5.2.3" xref="A2.1.p1.19.m5.2.3.cmml"><mrow id="A2.1.p1.19.m5.2.3.2" xref="A2.1.p1.19.m5.2.3.2.cmml"><msub id="A2.1.p1.19.m5.2.3.2.2" xref="A2.1.p1.19.m5.2.3.2.2.cmml"><mi id="A2.1.p1.19.m5.2.3.2.2.2" xref="A2.1.p1.19.m5.2.3.2.2.2.cmml">V</mi><mi id="A2.1.p1.19.m5.2.3.2.2.3" xref="A2.1.p1.19.m5.2.3.2.2.3.cmml">θ</mi></msub><mo id="A2.1.p1.19.m5.2.3.2.1" xref="A2.1.p1.19.m5.2.3.2.1.cmml">⁢</mo><mrow id="A2.1.p1.19.m5.2.3.2.3.2" xref="A2.1.p1.19.m5.2.3.2.cmml"><mo id="A2.1.p1.19.m5.2.3.2.3.2.1" stretchy="false" xref="A2.1.p1.19.m5.2.3.2.cmml">(</mo><mi id="A2.1.p1.19.m5.1.1" xref="A2.1.p1.19.m5.1.1.cmml">t</mi><mo id="A2.1.p1.19.m5.2.3.2.3.2.2" stretchy="false" xref="A2.1.p1.19.m5.2.3.2.cmml">)</mo></mrow></mrow><mo id="A2.1.p1.19.m5.2.3.1" xref="A2.1.p1.19.m5.2.3.1.cmml">≤</mo><mrow id="A2.1.p1.19.m5.2.3.3" xref="A2.1.p1.19.m5.2.3.3.cmml"><msub id="A2.1.p1.19.m5.2.3.3.2" xref="A2.1.p1.19.m5.2.3.3.2.cmml"><mi id="A2.1.p1.19.m5.2.3.3.2.2" xref="A2.1.p1.19.m5.2.3.3.2.2.cmml">V</mi><mi id="A2.1.p1.19.m5.2.3.3.2.3" xref="A2.1.p1.19.m5.2.3.3.2.3.cmml">θ</mi></msub><mo id="A2.1.p1.19.m5.2.3.3.1" xref="A2.1.p1.19.m5.2.3.3.1.cmml">⁢</mo><mrow id="A2.1.p1.19.m5.2.3.3.3.2" xref="A2.1.p1.19.m5.2.3.3.cmml"><mo id="A2.1.p1.19.m5.2.3.3.3.2.1" stretchy="false" xref="A2.1.p1.19.m5.2.3.3.cmml">(</mo><mn id="A2.1.p1.19.m5.2.2" xref="A2.1.p1.19.m5.2.2.cmml">0</mn><mo id="A2.1.p1.19.m5.2.3.3.3.2.2" stretchy="false" xref="A2.1.p1.19.m5.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.19.m5.2b"><apply id="A2.1.p1.19.m5.2.3.cmml" xref="A2.1.p1.19.m5.2.3"><leq id="A2.1.p1.19.m5.2.3.1.cmml" xref="A2.1.p1.19.m5.2.3.1"></leq><apply id="A2.1.p1.19.m5.2.3.2.cmml" xref="A2.1.p1.19.m5.2.3.2"><times id="A2.1.p1.19.m5.2.3.2.1.cmml" xref="A2.1.p1.19.m5.2.3.2.1"></times><apply id="A2.1.p1.19.m5.2.3.2.2.cmml" xref="A2.1.p1.19.m5.2.3.2.2"><csymbol cd="ambiguous" id="A2.1.p1.19.m5.2.3.2.2.1.cmml" xref="A2.1.p1.19.m5.2.3.2.2">subscript</csymbol><ci id="A2.1.p1.19.m5.2.3.2.2.2.cmml" xref="A2.1.p1.19.m5.2.3.2.2.2">𝑉</ci><ci id="A2.1.p1.19.m5.2.3.2.2.3.cmml" xref="A2.1.p1.19.m5.2.3.2.2.3">𝜃</ci></apply><ci id="A2.1.p1.19.m5.1.1.cmml" xref="A2.1.p1.19.m5.1.1">𝑡</ci></apply><apply id="A2.1.p1.19.m5.2.3.3.cmml" xref="A2.1.p1.19.m5.2.3.3"><times id="A2.1.p1.19.m5.2.3.3.1.cmml" xref="A2.1.p1.19.m5.2.3.3.1"></times><apply id="A2.1.p1.19.m5.2.3.3.2.cmml" xref="A2.1.p1.19.m5.2.3.3.2"><csymbol cd="ambiguous" id="A2.1.p1.19.m5.2.3.3.2.1.cmml" xref="A2.1.p1.19.m5.2.3.3.2">subscript</csymbol><ci id="A2.1.p1.19.m5.2.3.3.2.2.cmml" xref="A2.1.p1.19.m5.2.3.3.2.2">𝑉</ci><ci id="A2.1.p1.19.m5.2.3.3.2.3.cmml" xref="A2.1.p1.19.m5.2.3.3.2.3">𝜃</ci></apply><cn id="A2.1.p1.19.m5.2.2.cmml" type="integer" xref="A2.1.p1.19.m5.2.2">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.19.m5.2c">V_{\theta}(t)\leq V_{\theta}(0)</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.19.m5.2d">italic_V start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( italic_t ) ≤ italic_V start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( 0 )</annotation></semantics></math> yields</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx39"> <tbody id="A2.Ex28"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\tilde{\bm{\theta}}^{T}(t)\Gamma^{-1}\tilde{\bm{\theta}}(t)\leq% \tilde{\bm{\theta}}^{T}(0)\Gamma^{-1}\tilde{\bm{\theta}}(0)." class="ltx_Math" display="inline" id="A2.Ex28.m1.5"><semantics id="A2.Ex28.m1.5a"><mrow id="A2.Ex28.m1.5.5.1" xref="A2.Ex28.m1.5.5.1.1.cmml"><mrow id="A2.Ex28.m1.5.5.1.1" xref="A2.Ex28.m1.5.5.1.1.cmml"><mrow id="A2.Ex28.m1.5.5.1.1.2" xref="A2.Ex28.m1.5.5.1.1.2.cmml"><msup id="A2.Ex28.m1.5.5.1.1.2.2" xref="A2.Ex28.m1.5.5.1.1.2.2.cmml"><mover accent="true" id="A2.Ex28.m1.5.5.1.1.2.2.2" xref="A2.Ex28.m1.5.5.1.1.2.2.2.cmml"><mi id="A2.Ex28.m1.5.5.1.1.2.2.2.2" xref="A2.Ex28.m1.5.5.1.1.2.2.2.2.cmml">𝜽</mi><mo id="A2.Ex28.m1.5.5.1.1.2.2.2.1" xref="A2.Ex28.m1.5.5.1.1.2.2.2.1.cmml">~</mo></mover><mi id="A2.Ex28.m1.5.5.1.1.2.2.3" xref="A2.Ex28.m1.5.5.1.1.2.2.3.cmml">T</mi></msup><mo id="A2.Ex28.m1.5.5.1.1.2.1" xref="A2.Ex28.m1.5.5.1.1.2.1.cmml">⁢</mo><mrow id="A2.Ex28.m1.5.5.1.1.2.3.2" xref="A2.Ex28.m1.5.5.1.1.2.cmml"><mo id="A2.Ex28.m1.5.5.1.1.2.3.2.1" stretchy="false" xref="A2.Ex28.m1.5.5.1.1.2.cmml">(</mo><mi id="A2.Ex28.m1.1.1" xref="A2.Ex28.m1.1.1.cmml">t</mi><mo id="A2.Ex28.m1.5.5.1.1.2.3.2.2" stretchy="false" xref="A2.Ex28.m1.5.5.1.1.2.cmml">)</mo></mrow><mo id="A2.Ex28.m1.5.5.1.1.2.1a" xref="A2.Ex28.m1.5.5.1.1.2.1.cmml">⁢</mo><msup id="A2.Ex28.m1.5.5.1.1.2.4" xref="A2.Ex28.m1.5.5.1.1.2.4.cmml"><mi id="A2.Ex28.m1.5.5.1.1.2.4.2" mathvariant="normal" xref="A2.Ex28.m1.5.5.1.1.2.4.2.cmml">Γ</mi><mrow id="A2.Ex28.m1.5.5.1.1.2.4.3" xref="A2.Ex28.m1.5.5.1.1.2.4.3.cmml"><mo id="A2.Ex28.m1.5.5.1.1.2.4.3a" xref="A2.Ex28.m1.5.5.1.1.2.4.3.cmml">−</mo><mn id="A2.Ex28.m1.5.5.1.1.2.4.3.2" xref="A2.Ex28.m1.5.5.1.1.2.4.3.2.cmml">1</mn></mrow></msup><mo id="A2.Ex28.m1.5.5.1.1.2.1b" xref="A2.Ex28.m1.5.5.1.1.2.1.cmml">⁢</mo><mover accent="true" id="A2.Ex28.m1.5.5.1.1.2.5" xref="A2.Ex28.m1.5.5.1.1.2.5.cmml"><mi id="A2.Ex28.m1.5.5.1.1.2.5.2" xref="A2.Ex28.m1.5.5.1.1.2.5.2.cmml">𝜽</mi><mo id="A2.Ex28.m1.5.5.1.1.2.5.1" xref="A2.Ex28.m1.5.5.1.1.2.5.1.cmml">~</mo></mover><mo id="A2.Ex28.m1.5.5.1.1.2.1c" xref="A2.Ex28.m1.5.5.1.1.2.1.cmml">⁢</mo><mrow id="A2.Ex28.m1.5.5.1.1.2.6.2" xref="A2.Ex28.m1.5.5.1.1.2.cmml"><mo id="A2.Ex28.m1.5.5.1.1.2.6.2.1" stretchy="false" xref="A2.Ex28.m1.5.5.1.1.2.cmml">(</mo><mi id="A2.Ex28.m1.2.2" xref="A2.Ex28.m1.2.2.cmml">t</mi><mo id="A2.Ex28.m1.5.5.1.1.2.6.2.2" stretchy="false" xref="A2.Ex28.m1.5.5.1.1.2.cmml">)</mo></mrow></mrow><mo id="A2.Ex28.m1.5.5.1.1.1" xref="A2.Ex28.m1.5.5.1.1.1.cmml">≤</mo><mrow id="A2.Ex28.m1.5.5.1.1.3" xref="A2.Ex28.m1.5.5.1.1.3.cmml"><msup id="A2.Ex28.m1.5.5.1.1.3.2" xref="A2.Ex28.m1.5.5.1.1.3.2.cmml"><mover accent="true" id="A2.Ex28.m1.5.5.1.1.3.2.2" xref="A2.Ex28.m1.5.5.1.1.3.2.2.cmml"><mi id="A2.Ex28.m1.5.5.1.1.3.2.2.2" xref="A2.Ex28.m1.5.5.1.1.3.2.2.2.cmml">𝜽</mi><mo id="A2.Ex28.m1.5.5.1.1.3.2.2.1" xref="A2.Ex28.m1.5.5.1.1.3.2.2.1.cmml">~</mo></mover><mi id="A2.Ex28.m1.5.5.1.1.3.2.3" xref="A2.Ex28.m1.5.5.1.1.3.2.3.cmml">T</mi></msup><mo id="A2.Ex28.m1.5.5.1.1.3.1" xref="A2.Ex28.m1.5.5.1.1.3.1.cmml">⁢</mo><mrow id="A2.Ex28.m1.5.5.1.1.3.3.2" xref="A2.Ex28.m1.5.5.1.1.3.cmml"><mo id="A2.Ex28.m1.5.5.1.1.3.3.2.1" stretchy="false" xref="A2.Ex28.m1.5.5.1.1.3.cmml">(</mo><mn id="A2.Ex28.m1.3.3" xref="A2.Ex28.m1.3.3.cmml">0</mn><mo id="A2.Ex28.m1.5.5.1.1.3.3.2.2" stretchy="false" xref="A2.Ex28.m1.5.5.1.1.3.cmml">)</mo></mrow><mo id="A2.Ex28.m1.5.5.1.1.3.1a" xref="A2.Ex28.m1.5.5.1.1.3.1.cmml">⁢</mo><msup id="A2.Ex28.m1.5.5.1.1.3.4" xref="A2.Ex28.m1.5.5.1.1.3.4.cmml"><mi id="A2.Ex28.m1.5.5.1.1.3.4.2" mathvariant="normal" xref="A2.Ex28.m1.5.5.1.1.3.4.2.cmml">Γ</mi><mrow id="A2.Ex28.m1.5.5.1.1.3.4.3" xref="A2.Ex28.m1.5.5.1.1.3.4.3.cmml"><mo id="A2.Ex28.m1.5.5.1.1.3.4.3a" xref="A2.Ex28.m1.5.5.1.1.3.4.3.cmml">−</mo><mn id="A2.Ex28.m1.5.5.1.1.3.4.3.2" xref="A2.Ex28.m1.5.5.1.1.3.4.3.2.cmml">1</mn></mrow></msup><mo id="A2.Ex28.m1.5.5.1.1.3.1b" xref="A2.Ex28.m1.5.5.1.1.3.1.cmml">⁢</mo><mover accent="true" id="A2.Ex28.m1.5.5.1.1.3.5" xref="A2.Ex28.m1.5.5.1.1.3.5.cmml"><mi id="A2.Ex28.m1.5.5.1.1.3.5.2" xref="A2.Ex28.m1.5.5.1.1.3.5.2.cmml">𝜽</mi><mo id="A2.Ex28.m1.5.5.1.1.3.5.1" xref="A2.Ex28.m1.5.5.1.1.3.5.1.cmml">~</mo></mover><mo id="A2.Ex28.m1.5.5.1.1.3.1c" xref="A2.Ex28.m1.5.5.1.1.3.1.cmml">⁢</mo><mrow id="A2.Ex28.m1.5.5.1.1.3.6.2" xref="A2.Ex28.m1.5.5.1.1.3.cmml"><mo id="A2.Ex28.m1.5.5.1.1.3.6.2.1" stretchy="false" xref="A2.Ex28.m1.5.5.1.1.3.cmml">(</mo><mn id="A2.Ex28.m1.4.4" xref="A2.Ex28.m1.4.4.cmml">0</mn><mo id="A2.Ex28.m1.5.5.1.1.3.6.2.2" stretchy="false" xref="A2.Ex28.m1.5.5.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="A2.Ex28.m1.5.5.1.2" lspace="0em" xref="A2.Ex28.m1.5.5.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex28.m1.5b"><apply id="A2.Ex28.m1.5.5.1.1.cmml" xref="A2.Ex28.m1.5.5.1"><leq id="A2.Ex28.m1.5.5.1.1.1.cmml" xref="A2.Ex28.m1.5.5.1.1.1"></leq><apply id="A2.Ex28.m1.5.5.1.1.2.cmml" xref="A2.Ex28.m1.5.5.1.1.2"><times id="A2.Ex28.m1.5.5.1.1.2.1.cmml" xref="A2.Ex28.m1.5.5.1.1.2.1"></times><apply id="A2.Ex28.m1.5.5.1.1.2.2.cmml" xref="A2.Ex28.m1.5.5.1.1.2.2"><csymbol cd="ambiguous" id="A2.Ex28.m1.5.5.1.1.2.2.1.cmml" xref="A2.Ex28.m1.5.5.1.1.2.2">superscript</csymbol><apply id="A2.Ex28.m1.5.5.1.1.2.2.2.cmml" xref="A2.Ex28.m1.5.5.1.1.2.2.2"><ci id="A2.Ex28.m1.5.5.1.1.2.2.2.1.cmml" xref="A2.Ex28.m1.5.5.1.1.2.2.2.1">~</ci><ci id="A2.Ex28.m1.5.5.1.1.2.2.2.2.cmml" xref="A2.Ex28.m1.5.5.1.1.2.2.2.2">𝜽</ci></apply><ci id="A2.Ex28.m1.5.5.1.1.2.2.3.cmml" xref="A2.Ex28.m1.5.5.1.1.2.2.3">𝑇</ci></apply><ci id="A2.Ex28.m1.1.1.cmml" xref="A2.Ex28.m1.1.1">𝑡</ci><apply id="A2.Ex28.m1.5.5.1.1.2.4.cmml" xref="A2.Ex28.m1.5.5.1.1.2.4"><csymbol cd="ambiguous" id="A2.Ex28.m1.5.5.1.1.2.4.1.cmml" xref="A2.Ex28.m1.5.5.1.1.2.4">superscript</csymbol><ci id="A2.Ex28.m1.5.5.1.1.2.4.2.cmml" xref="A2.Ex28.m1.5.5.1.1.2.4.2">Γ</ci><apply id="A2.Ex28.m1.5.5.1.1.2.4.3.cmml" xref="A2.Ex28.m1.5.5.1.1.2.4.3"><minus id="A2.Ex28.m1.5.5.1.1.2.4.3.1.cmml" xref="A2.Ex28.m1.5.5.1.1.2.4.3"></minus><cn id="A2.Ex28.m1.5.5.1.1.2.4.3.2.cmml" type="integer" xref="A2.Ex28.m1.5.5.1.1.2.4.3.2">1</cn></apply></apply><apply id="A2.Ex28.m1.5.5.1.1.2.5.cmml" xref="A2.Ex28.m1.5.5.1.1.2.5"><ci id="A2.Ex28.m1.5.5.1.1.2.5.1.cmml" xref="A2.Ex28.m1.5.5.1.1.2.5.1">~</ci><ci id="A2.Ex28.m1.5.5.1.1.2.5.2.cmml" xref="A2.Ex28.m1.5.5.1.1.2.5.2">𝜽</ci></apply><ci id="A2.Ex28.m1.2.2.cmml" xref="A2.Ex28.m1.2.2">𝑡</ci></apply><apply id="A2.Ex28.m1.5.5.1.1.3.cmml" xref="A2.Ex28.m1.5.5.1.1.3"><times id="A2.Ex28.m1.5.5.1.1.3.1.cmml" xref="A2.Ex28.m1.5.5.1.1.3.1"></times><apply id="A2.Ex28.m1.5.5.1.1.3.2.cmml" xref="A2.Ex28.m1.5.5.1.1.3.2"><csymbol cd="ambiguous" id="A2.Ex28.m1.5.5.1.1.3.2.1.cmml" xref="A2.Ex28.m1.5.5.1.1.3.2">superscript</csymbol><apply id="A2.Ex28.m1.5.5.1.1.3.2.2.cmml" xref="A2.Ex28.m1.5.5.1.1.3.2.2"><ci id="A2.Ex28.m1.5.5.1.1.3.2.2.1.cmml" xref="A2.Ex28.m1.5.5.1.1.3.2.2.1">~</ci><ci id="A2.Ex28.m1.5.5.1.1.3.2.2.2.cmml" xref="A2.Ex28.m1.5.5.1.1.3.2.2.2">𝜽</ci></apply><ci id="A2.Ex28.m1.5.5.1.1.3.2.3.cmml" xref="A2.Ex28.m1.5.5.1.1.3.2.3">𝑇</ci></apply><cn id="A2.Ex28.m1.3.3.cmml" type="integer" xref="A2.Ex28.m1.3.3">0</cn><apply id="A2.Ex28.m1.5.5.1.1.3.4.cmml" xref="A2.Ex28.m1.5.5.1.1.3.4"><csymbol cd="ambiguous" id="A2.Ex28.m1.5.5.1.1.3.4.1.cmml" xref="A2.Ex28.m1.5.5.1.1.3.4">superscript</csymbol><ci id="A2.Ex28.m1.5.5.1.1.3.4.2.cmml" xref="A2.Ex28.m1.5.5.1.1.3.4.2">Γ</ci><apply id="A2.Ex28.m1.5.5.1.1.3.4.3.cmml" xref="A2.Ex28.m1.5.5.1.1.3.4.3"><minus id="A2.Ex28.m1.5.5.1.1.3.4.3.1.cmml" xref="A2.Ex28.m1.5.5.1.1.3.4.3"></minus><cn id="A2.Ex28.m1.5.5.1.1.3.4.3.2.cmml" type="integer" xref="A2.Ex28.m1.5.5.1.1.3.4.3.2">1</cn></apply></apply><apply id="A2.Ex28.m1.5.5.1.1.3.5.cmml" xref="A2.Ex28.m1.5.5.1.1.3.5"><ci id="A2.Ex28.m1.5.5.1.1.3.5.1.cmml" xref="A2.Ex28.m1.5.5.1.1.3.5.1">~</ci><ci id="A2.Ex28.m1.5.5.1.1.3.5.2.cmml" xref="A2.Ex28.m1.5.5.1.1.3.5.2">𝜽</ci></apply><cn id="A2.Ex28.m1.4.4.cmml" type="integer" xref="A2.Ex28.m1.4.4">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex28.m1.5c">\displaystyle\tilde{\bm{\theta}}^{T}(t)\Gamma^{-1}\tilde{\bm{\theta}}(t)\leq% \tilde{\bm{\theta}}^{T}(0)\Gamma^{-1}\tilde{\bm{\theta}}(0).</annotation><annotation encoding="application/x-llamapun" id="A2.Ex28.m1.5d">over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_t ) roman_Γ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG ( italic_t ) ≤ over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( 0 ) roman_Γ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG ( 0 ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A2.1.p1.37">After some algebraic operations, the above result leads to</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx40"> <tbody id="A2.Ex29"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\tilde{\bm{\theta}}(t)\|\leq\sqrt{\lambda_{\max}(\Gamma)/% \lambda_{\min}(\Gamma)}\|\tilde{\bm{\theta}}(0)\|" class="ltx_Math" display="inline" id="A2.Ex29.m1.6"><semantics id="A2.Ex29.m1.6a"><mrow id="A2.Ex29.m1.6.6" xref="A2.Ex29.m1.6.6.cmml"><mrow id="A2.Ex29.m1.5.5.1.1" xref="A2.Ex29.m1.5.5.1.2.cmml"><mo id="A2.Ex29.m1.5.5.1.1.2" stretchy="false" xref="A2.Ex29.m1.5.5.1.2.1.cmml">‖</mo><mrow id="A2.Ex29.m1.5.5.1.1.1" xref="A2.Ex29.m1.5.5.1.1.1.cmml"><mover accent="true" id="A2.Ex29.m1.5.5.1.1.1.2" xref="A2.Ex29.m1.5.5.1.1.1.2.cmml"><mi id="A2.Ex29.m1.5.5.1.1.1.2.2" xref="A2.Ex29.m1.5.5.1.1.1.2.2.cmml">𝜽</mi><mo id="A2.Ex29.m1.5.5.1.1.1.2.1" xref="A2.Ex29.m1.5.5.1.1.1.2.1.cmml">~</mo></mover><mo id="A2.Ex29.m1.5.5.1.1.1.1" xref="A2.Ex29.m1.5.5.1.1.1.1.cmml">⁢</mo><mrow id="A2.Ex29.m1.5.5.1.1.1.3.2" xref="A2.Ex29.m1.5.5.1.1.1.cmml"><mo id="A2.Ex29.m1.5.5.1.1.1.3.2.1" stretchy="false" xref="A2.Ex29.m1.5.5.1.1.1.cmml">(</mo><mi id="A2.Ex29.m1.3.3" xref="A2.Ex29.m1.3.3.cmml">t</mi><mo id="A2.Ex29.m1.5.5.1.1.1.3.2.2" stretchy="false" xref="A2.Ex29.m1.5.5.1.1.1.cmml">)</mo></mrow></mrow><mo id="A2.Ex29.m1.5.5.1.1.3" stretchy="false" xref="A2.Ex29.m1.5.5.1.2.1.cmml">‖</mo></mrow><mo id="A2.Ex29.m1.6.6.3" xref="A2.Ex29.m1.6.6.3.cmml">≤</mo><mrow id="A2.Ex29.m1.6.6.2" xref="A2.Ex29.m1.6.6.2.cmml"><msqrt id="A2.Ex29.m1.2.2" xref="A2.Ex29.m1.2.2.cmml"><mrow id="A2.Ex29.m1.2.2.2" xref="A2.Ex29.m1.2.2.2.cmml"><mrow id="A2.Ex29.m1.2.2.2.4" xref="A2.Ex29.m1.2.2.2.4.cmml"><mrow id="A2.Ex29.m1.2.2.2.4.2" xref="A2.Ex29.m1.2.2.2.4.2.cmml"><msub id="A2.Ex29.m1.2.2.2.4.2.2" xref="A2.Ex29.m1.2.2.2.4.2.2.cmml"><mi id="A2.Ex29.m1.2.2.2.4.2.2.2" xref="A2.Ex29.m1.2.2.2.4.2.2.2.cmml">λ</mi><mi id="A2.Ex29.m1.2.2.2.4.2.2.3" xref="A2.Ex29.m1.2.2.2.4.2.2.3.cmml">max</mi></msub><mo id="A2.Ex29.m1.2.2.2.4.2.1" xref="A2.Ex29.m1.2.2.2.4.2.1.cmml">⁢</mo><mrow id="A2.Ex29.m1.2.2.2.4.2.3.2" xref="A2.Ex29.m1.2.2.2.4.2.cmml"><mo id="A2.Ex29.m1.2.2.2.4.2.3.2.1" stretchy="false" xref="A2.Ex29.m1.2.2.2.4.2.cmml">(</mo><mi id="A2.Ex29.m1.1.1.1.1" mathvariant="normal" xref="A2.Ex29.m1.1.1.1.1.cmml">Γ</mi><mo id="A2.Ex29.m1.2.2.2.4.2.3.2.2" stretchy="false" xref="A2.Ex29.m1.2.2.2.4.2.cmml">)</mo></mrow></mrow><mo id="A2.Ex29.m1.2.2.2.4.1" xref="A2.Ex29.m1.2.2.2.4.1.cmml">/</mo><msub id="A2.Ex29.m1.2.2.2.4.3" xref="A2.Ex29.m1.2.2.2.4.3.cmml"><mi id="A2.Ex29.m1.2.2.2.4.3.2" xref="A2.Ex29.m1.2.2.2.4.3.2.cmml">λ</mi><mi id="A2.Ex29.m1.2.2.2.4.3.3" xref="A2.Ex29.m1.2.2.2.4.3.3.cmml">min</mi></msub></mrow><mo id="A2.Ex29.m1.2.2.2.3" xref="A2.Ex29.m1.2.2.2.3.cmml">⁢</mo><mrow id="A2.Ex29.m1.2.2.2.5.2" xref="A2.Ex29.m1.2.2.2.cmml"><mo id="A2.Ex29.m1.2.2.2.5.2.1" stretchy="false" xref="A2.Ex29.m1.2.2.2.cmml">(</mo><mi id="A2.Ex29.m1.2.2.2.2" mathvariant="normal" xref="A2.Ex29.m1.2.2.2.2.cmml">Γ</mi><mo id="A2.Ex29.m1.2.2.2.5.2.2" stretchy="false" xref="A2.Ex29.m1.2.2.2.cmml">)</mo></mrow></mrow></msqrt><mo id="A2.Ex29.m1.6.6.2.2" xref="A2.Ex29.m1.6.6.2.2.cmml">⁢</mo><mrow id="A2.Ex29.m1.6.6.2.1.1" xref="A2.Ex29.m1.6.6.2.1.2.cmml"><mo id="A2.Ex29.m1.6.6.2.1.1.2" stretchy="false" xref="A2.Ex29.m1.6.6.2.1.2.1.cmml">‖</mo><mrow id="A2.Ex29.m1.6.6.2.1.1.1" xref="A2.Ex29.m1.6.6.2.1.1.1.cmml"><mover accent="true" id="A2.Ex29.m1.6.6.2.1.1.1.2" xref="A2.Ex29.m1.6.6.2.1.1.1.2.cmml"><mi id="A2.Ex29.m1.6.6.2.1.1.1.2.2" xref="A2.Ex29.m1.6.6.2.1.1.1.2.2.cmml">𝜽</mi><mo id="A2.Ex29.m1.6.6.2.1.1.1.2.1" xref="A2.Ex29.m1.6.6.2.1.1.1.2.1.cmml">~</mo></mover><mo id="A2.Ex29.m1.6.6.2.1.1.1.1" xref="A2.Ex29.m1.6.6.2.1.1.1.1.cmml">⁢</mo><mrow id="A2.Ex29.m1.6.6.2.1.1.1.3.2" xref="A2.Ex29.m1.6.6.2.1.1.1.cmml"><mo id="A2.Ex29.m1.6.6.2.1.1.1.3.2.1" stretchy="false" xref="A2.Ex29.m1.6.6.2.1.1.1.cmml">(</mo><mn id="A2.Ex29.m1.4.4" xref="A2.Ex29.m1.4.4.cmml">0</mn><mo id="A2.Ex29.m1.6.6.2.1.1.1.3.2.2" stretchy="false" xref="A2.Ex29.m1.6.6.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="A2.Ex29.m1.6.6.2.1.1.3" stretchy="false" xref="A2.Ex29.m1.6.6.2.1.2.1.cmml">‖</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex29.m1.6b"><apply id="A2.Ex29.m1.6.6.cmml" xref="A2.Ex29.m1.6.6"><leq id="A2.Ex29.m1.6.6.3.cmml" xref="A2.Ex29.m1.6.6.3"></leq><apply id="A2.Ex29.m1.5.5.1.2.cmml" xref="A2.Ex29.m1.5.5.1.1"><csymbol cd="latexml" id="A2.Ex29.m1.5.5.1.2.1.cmml" xref="A2.Ex29.m1.5.5.1.1.2">norm</csymbol><apply id="A2.Ex29.m1.5.5.1.1.1.cmml" xref="A2.Ex29.m1.5.5.1.1.1"><times id="A2.Ex29.m1.5.5.1.1.1.1.cmml" xref="A2.Ex29.m1.5.5.1.1.1.1"></times><apply id="A2.Ex29.m1.5.5.1.1.1.2.cmml" xref="A2.Ex29.m1.5.5.1.1.1.2"><ci id="A2.Ex29.m1.5.5.1.1.1.2.1.cmml" xref="A2.Ex29.m1.5.5.1.1.1.2.1">~</ci><ci id="A2.Ex29.m1.5.5.1.1.1.2.2.cmml" xref="A2.Ex29.m1.5.5.1.1.1.2.2">𝜽</ci></apply><ci id="A2.Ex29.m1.3.3.cmml" xref="A2.Ex29.m1.3.3">𝑡</ci></apply></apply><apply id="A2.Ex29.m1.6.6.2.cmml" xref="A2.Ex29.m1.6.6.2"><times id="A2.Ex29.m1.6.6.2.2.cmml" xref="A2.Ex29.m1.6.6.2.2"></times><apply id="A2.Ex29.m1.2.2.cmml" xref="A2.Ex29.m1.2.2"><root id="A2.Ex29.m1.2.2a.cmml" xref="A2.Ex29.m1.2.2"></root><apply id="A2.Ex29.m1.2.2.2.cmml" xref="A2.Ex29.m1.2.2.2"><times id="A2.Ex29.m1.2.2.2.3.cmml" xref="A2.Ex29.m1.2.2.2.3"></times><apply id="A2.Ex29.m1.2.2.2.4.cmml" xref="A2.Ex29.m1.2.2.2.4"><divide id="A2.Ex29.m1.2.2.2.4.1.cmml" xref="A2.Ex29.m1.2.2.2.4.1"></divide><apply id="A2.Ex29.m1.2.2.2.4.2.cmml" xref="A2.Ex29.m1.2.2.2.4.2"><times id="A2.Ex29.m1.2.2.2.4.2.1.cmml" xref="A2.Ex29.m1.2.2.2.4.2.1"></times><apply id="A2.Ex29.m1.2.2.2.4.2.2.cmml" xref="A2.Ex29.m1.2.2.2.4.2.2"><csymbol cd="ambiguous" id="A2.Ex29.m1.2.2.2.4.2.2.1.cmml" xref="A2.Ex29.m1.2.2.2.4.2.2">subscript</csymbol><ci id="A2.Ex29.m1.2.2.2.4.2.2.2.cmml" xref="A2.Ex29.m1.2.2.2.4.2.2.2">𝜆</ci><max id="A2.Ex29.m1.2.2.2.4.2.2.3.cmml" xref="A2.Ex29.m1.2.2.2.4.2.2.3"></max></apply><ci id="A2.Ex29.m1.1.1.1.1.cmml" xref="A2.Ex29.m1.1.1.1.1">Γ</ci></apply><apply id="A2.Ex29.m1.2.2.2.4.3.cmml" xref="A2.Ex29.m1.2.2.2.4.3"><csymbol cd="ambiguous" id="A2.Ex29.m1.2.2.2.4.3.1.cmml" xref="A2.Ex29.m1.2.2.2.4.3">subscript</csymbol><ci id="A2.Ex29.m1.2.2.2.4.3.2.cmml" xref="A2.Ex29.m1.2.2.2.4.3.2">𝜆</ci><min id="A2.Ex29.m1.2.2.2.4.3.3.cmml" xref="A2.Ex29.m1.2.2.2.4.3.3"></min></apply></apply><ci id="A2.Ex29.m1.2.2.2.2.cmml" xref="A2.Ex29.m1.2.2.2.2">Γ</ci></apply></apply><apply id="A2.Ex29.m1.6.6.2.1.2.cmml" xref="A2.Ex29.m1.6.6.2.1.1"><csymbol cd="latexml" id="A2.Ex29.m1.6.6.2.1.2.1.cmml" xref="A2.Ex29.m1.6.6.2.1.1.2">norm</csymbol><apply id="A2.Ex29.m1.6.6.2.1.1.1.cmml" xref="A2.Ex29.m1.6.6.2.1.1.1"><times id="A2.Ex29.m1.6.6.2.1.1.1.1.cmml" xref="A2.Ex29.m1.6.6.2.1.1.1.1"></times><apply id="A2.Ex29.m1.6.6.2.1.1.1.2.cmml" xref="A2.Ex29.m1.6.6.2.1.1.1.2"><ci id="A2.Ex29.m1.6.6.2.1.1.1.2.1.cmml" xref="A2.Ex29.m1.6.6.2.1.1.1.2.1">~</ci><ci id="A2.Ex29.m1.6.6.2.1.1.1.2.2.cmml" xref="A2.Ex29.m1.6.6.2.1.1.1.2.2">𝜽</ci></apply><cn id="A2.Ex29.m1.4.4.cmml" type="integer" xref="A2.Ex29.m1.4.4">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex29.m1.6c">\displaystyle\|\tilde{\bm{\theta}}(t)\|\leq\sqrt{\lambda_{\max}(\Gamma)/% \lambda_{\min}(\Gamma)}\|\tilde{\bm{\theta}}(0)\|</annotation><annotation encoding="application/x-llamapun" id="A2.Ex29.m1.6d">∥ over~ start_ARG bold_italic_θ end_ARG ( italic_t ) ∥ ≤ square-root start_ARG italic_λ start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ( roman_Γ ) / italic_λ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Γ ) end_ARG ∥ over~ start_ARG bold_italic_θ end_ARG ( 0 ) ∥</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A2.1.p1.35">implying <math alttext="\tilde{\bm{\theta}}(t)\in L_{\infty}" class="ltx_Math" display="inline" id="A2.1.p1.20.m1.1"><semantics id="A2.1.p1.20.m1.1a"><mrow id="A2.1.p1.20.m1.1.2" xref="A2.1.p1.20.m1.1.2.cmml"><mrow id="A2.1.p1.20.m1.1.2.2" xref="A2.1.p1.20.m1.1.2.2.cmml"><mover accent="true" id="A2.1.p1.20.m1.1.2.2.2" xref="A2.1.p1.20.m1.1.2.2.2.cmml"><mi id="A2.1.p1.20.m1.1.2.2.2.2" xref="A2.1.p1.20.m1.1.2.2.2.2.cmml">𝜽</mi><mo id="A2.1.p1.20.m1.1.2.2.2.1" xref="A2.1.p1.20.m1.1.2.2.2.1.cmml">~</mo></mover><mo id="A2.1.p1.20.m1.1.2.2.1" xref="A2.1.p1.20.m1.1.2.2.1.cmml">⁢</mo><mrow id="A2.1.p1.20.m1.1.2.2.3.2" xref="A2.1.p1.20.m1.1.2.2.cmml"><mo id="A2.1.p1.20.m1.1.2.2.3.2.1" stretchy="false" xref="A2.1.p1.20.m1.1.2.2.cmml">(</mo><mi id="A2.1.p1.20.m1.1.1" xref="A2.1.p1.20.m1.1.1.cmml">t</mi><mo id="A2.1.p1.20.m1.1.2.2.3.2.2" stretchy="false" xref="A2.1.p1.20.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="A2.1.p1.20.m1.1.2.1" xref="A2.1.p1.20.m1.1.2.1.cmml">∈</mo><msub id="A2.1.p1.20.m1.1.2.3" xref="A2.1.p1.20.m1.1.2.3.cmml"><mi id="A2.1.p1.20.m1.1.2.3.2" xref="A2.1.p1.20.m1.1.2.3.2.cmml">L</mi><mi id="A2.1.p1.20.m1.1.2.3.3" mathvariant="normal" xref="A2.1.p1.20.m1.1.2.3.3.cmml">∞</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.20.m1.1b"><apply id="A2.1.p1.20.m1.1.2.cmml" xref="A2.1.p1.20.m1.1.2"><in id="A2.1.p1.20.m1.1.2.1.cmml" xref="A2.1.p1.20.m1.1.2.1"></in><apply id="A2.1.p1.20.m1.1.2.2.cmml" xref="A2.1.p1.20.m1.1.2.2"><times id="A2.1.p1.20.m1.1.2.2.1.cmml" xref="A2.1.p1.20.m1.1.2.2.1"></times><apply id="A2.1.p1.20.m1.1.2.2.2.cmml" xref="A2.1.p1.20.m1.1.2.2.2"><ci id="A2.1.p1.20.m1.1.2.2.2.1.cmml" xref="A2.1.p1.20.m1.1.2.2.2.1">~</ci><ci id="A2.1.p1.20.m1.1.2.2.2.2.cmml" xref="A2.1.p1.20.m1.1.2.2.2.2">𝜽</ci></apply><ci id="A2.1.p1.20.m1.1.1.cmml" xref="A2.1.p1.20.m1.1.1">𝑡</ci></apply><apply id="A2.1.p1.20.m1.1.2.3.cmml" xref="A2.1.p1.20.m1.1.2.3"><csymbol cd="ambiguous" id="A2.1.p1.20.m1.1.2.3.1.cmml" xref="A2.1.p1.20.m1.1.2.3">subscript</csymbol><ci id="A2.1.p1.20.m1.1.2.3.2.cmml" xref="A2.1.p1.20.m1.1.2.3.2">𝐿</ci><infinity id="A2.1.p1.20.m1.1.2.3.3.cmml" xref="A2.1.p1.20.m1.1.2.3.3"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.20.m1.1c">\tilde{\bm{\theta}}(t)\in L_{\infty}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.20.m1.1d">over~ start_ARG bold_italic_θ end_ARG ( italic_t ) ∈ italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>. Since the solutions of (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S2.E4" title="In II Problem Formulation ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">4</span></a>) only exist on <math alttext="t\in[0,t_{\rm f})" class="ltx_Math" display="inline" id="A2.1.p1.21.m2.2"><semantics id="A2.1.p1.21.m2.2a"><mrow id="A2.1.p1.21.m2.2.2" xref="A2.1.p1.21.m2.2.2.cmml"><mi id="A2.1.p1.21.m2.2.2.3" xref="A2.1.p1.21.m2.2.2.3.cmml">t</mi><mo id="A2.1.p1.21.m2.2.2.2" xref="A2.1.p1.21.m2.2.2.2.cmml">∈</mo><mrow id="A2.1.p1.21.m2.2.2.1.1" xref="A2.1.p1.21.m2.2.2.1.2.cmml"><mo id="A2.1.p1.21.m2.2.2.1.1.2" stretchy="false" xref="A2.1.p1.21.m2.2.2.1.2.cmml">[</mo><mn id="A2.1.p1.21.m2.1.1" xref="A2.1.p1.21.m2.1.1.cmml">0</mn><mo id="A2.1.p1.21.m2.2.2.1.1.3" xref="A2.1.p1.21.m2.2.2.1.2.cmml">,</mo><msub id="A2.1.p1.21.m2.2.2.1.1.1" xref="A2.1.p1.21.m2.2.2.1.1.1.cmml"><mi id="A2.1.p1.21.m2.2.2.1.1.1.2" xref="A2.1.p1.21.m2.2.2.1.1.1.2.cmml">t</mi><mi id="A2.1.p1.21.m2.2.2.1.1.1.3" mathvariant="normal" xref="A2.1.p1.21.m2.2.2.1.1.1.3.cmml">f</mi></msub><mo id="A2.1.p1.21.m2.2.2.1.1.4" stretchy="false" xref="A2.1.p1.21.m2.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.21.m2.2b"><apply id="A2.1.p1.21.m2.2.2.cmml" xref="A2.1.p1.21.m2.2.2"><in id="A2.1.p1.21.m2.2.2.2.cmml" xref="A2.1.p1.21.m2.2.2.2"></in><ci id="A2.1.p1.21.m2.2.2.3.cmml" xref="A2.1.p1.21.m2.2.2.3">𝑡</ci><interval closure="closed-open" id="A2.1.p1.21.m2.2.2.1.2.cmml" xref="A2.1.p1.21.m2.2.2.1.1"><cn id="A2.1.p1.21.m2.1.1.cmml" type="integer" xref="A2.1.p1.21.m2.1.1">0</cn><apply id="A2.1.p1.21.m2.2.2.1.1.1.cmml" xref="A2.1.p1.21.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="A2.1.p1.21.m2.2.2.1.1.1.1.cmml" xref="A2.1.p1.21.m2.2.2.1.1.1">subscript</csymbol><ci id="A2.1.p1.21.m2.2.2.1.1.1.2.cmml" xref="A2.1.p1.21.m2.2.2.1.1.1.2">𝑡</ci><ci id="A2.1.p1.21.m2.2.2.1.1.1.3.cmml" xref="A2.1.p1.21.m2.2.2.1.1.1.3">f</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.21.m2.2c">t\in[0,t_{\rm f})</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.21.m2.2d">italic_t ∈ [ 0 , italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT )</annotation></semantics></math>, one obtains <math alttext="\bm{x}(t)" class="ltx_Math" display="inline" id="A2.1.p1.22.m3.1"><semantics id="A2.1.p1.22.m3.1a"><mrow id="A2.1.p1.22.m3.1.2" xref="A2.1.p1.22.m3.1.2.cmml"><mi id="A2.1.p1.22.m3.1.2.2" xref="A2.1.p1.22.m3.1.2.2.cmml">𝒙</mi><mo id="A2.1.p1.22.m3.1.2.1" xref="A2.1.p1.22.m3.1.2.1.cmml">⁢</mo><mrow id="A2.1.p1.22.m3.1.2.3.2" xref="A2.1.p1.22.m3.1.2.cmml"><mo id="A2.1.p1.22.m3.1.2.3.2.1" stretchy="false" xref="A2.1.p1.22.m3.1.2.cmml">(</mo><mi id="A2.1.p1.22.m3.1.1" xref="A2.1.p1.22.m3.1.1.cmml">t</mi><mo id="A2.1.p1.22.m3.1.2.3.2.2" stretchy="false" xref="A2.1.p1.22.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.22.m3.1b"><apply id="A2.1.p1.22.m3.1.2.cmml" xref="A2.1.p1.22.m3.1.2"><times id="A2.1.p1.22.m3.1.2.1.cmml" xref="A2.1.p1.22.m3.1.2.1"></times><ci id="A2.1.p1.22.m3.1.2.2.cmml" xref="A2.1.p1.22.m3.1.2.2">𝒙</ci><ci id="A2.1.p1.22.m3.1.1.cmml" xref="A2.1.p1.22.m3.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.22.m3.1c">\bm{x}(t)</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.22.m3.1d">bold_italic_x ( italic_t )</annotation></semantics></math> and <math alttext="\hat{\bm{\theta}}^{(k)}(t)\in L_{\infty}" class="ltx_Math" display="inline" id="A2.1.p1.23.m4.2"><semantics id="A2.1.p1.23.m4.2a"><mrow id="A2.1.p1.23.m4.2.3" xref="A2.1.p1.23.m4.2.3.cmml"><mrow id="A2.1.p1.23.m4.2.3.2" xref="A2.1.p1.23.m4.2.3.2.cmml"><msup id="A2.1.p1.23.m4.2.3.2.2" xref="A2.1.p1.23.m4.2.3.2.2.cmml"><mover accent="true" id="A2.1.p1.23.m4.2.3.2.2.2" xref="A2.1.p1.23.m4.2.3.2.2.2.cmml"><mi id="A2.1.p1.23.m4.2.3.2.2.2.2" xref="A2.1.p1.23.m4.2.3.2.2.2.2.cmml">𝜽</mi><mo id="A2.1.p1.23.m4.2.3.2.2.2.1" xref="A2.1.p1.23.m4.2.3.2.2.2.1.cmml">^</mo></mover><mrow id="A2.1.p1.23.m4.1.1.1.3" xref="A2.1.p1.23.m4.2.3.2.2.cmml"><mo id="A2.1.p1.23.m4.1.1.1.3.1" stretchy="false" xref="A2.1.p1.23.m4.2.3.2.2.cmml">(</mo><mi id="A2.1.p1.23.m4.1.1.1.1" xref="A2.1.p1.23.m4.1.1.1.1.cmml">k</mi><mo id="A2.1.p1.23.m4.1.1.1.3.2" stretchy="false" xref="A2.1.p1.23.m4.2.3.2.2.cmml">)</mo></mrow></msup><mo id="A2.1.p1.23.m4.2.3.2.1" xref="A2.1.p1.23.m4.2.3.2.1.cmml">⁢</mo><mrow id="A2.1.p1.23.m4.2.3.2.3.2" xref="A2.1.p1.23.m4.2.3.2.cmml"><mo id="A2.1.p1.23.m4.2.3.2.3.2.1" stretchy="false" xref="A2.1.p1.23.m4.2.3.2.cmml">(</mo><mi id="A2.1.p1.23.m4.2.2" xref="A2.1.p1.23.m4.2.2.cmml">t</mi><mo id="A2.1.p1.23.m4.2.3.2.3.2.2" stretchy="false" xref="A2.1.p1.23.m4.2.3.2.cmml">)</mo></mrow></mrow><mo id="A2.1.p1.23.m4.2.3.1" xref="A2.1.p1.23.m4.2.3.1.cmml">∈</mo><msub id="A2.1.p1.23.m4.2.3.3" xref="A2.1.p1.23.m4.2.3.3.cmml"><mi id="A2.1.p1.23.m4.2.3.3.2" xref="A2.1.p1.23.m4.2.3.3.2.cmml">L</mi><mi id="A2.1.p1.23.m4.2.3.3.3" mathvariant="normal" xref="A2.1.p1.23.m4.2.3.3.3.cmml">∞</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.23.m4.2b"><apply id="A2.1.p1.23.m4.2.3.cmml" xref="A2.1.p1.23.m4.2.3"><in id="A2.1.p1.23.m4.2.3.1.cmml" xref="A2.1.p1.23.m4.2.3.1"></in><apply id="A2.1.p1.23.m4.2.3.2.cmml" xref="A2.1.p1.23.m4.2.3.2"><times id="A2.1.p1.23.m4.2.3.2.1.cmml" xref="A2.1.p1.23.m4.2.3.2.1"></times><apply id="A2.1.p1.23.m4.2.3.2.2.cmml" xref="A2.1.p1.23.m4.2.3.2.2"><csymbol cd="ambiguous" id="A2.1.p1.23.m4.2.3.2.2.1.cmml" xref="A2.1.p1.23.m4.2.3.2.2">superscript</csymbol><apply id="A2.1.p1.23.m4.2.3.2.2.2.cmml" xref="A2.1.p1.23.m4.2.3.2.2.2"><ci id="A2.1.p1.23.m4.2.3.2.2.2.1.cmml" xref="A2.1.p1.23.m4.2.3.2.2.2.1">^</ci><ci id="A2.1.p1.23.m4.2.3.2.2.2.2.cmml" xref="A2.1.p1.23.m4.2.3.2.2.2.2">𝜽</ci></apply><ci id="A2.1.p1.23.m4.1.1.1.1.cmml" xref="A2.1.p1.23.m4.1.1.1.1">𝑘</ci></apply><ci id="A2.1.p1.23.m4.2.2.cmml" xref="A2.1.p1.23.m4.2.2">𝑡</ci></apply><apply id="A2.1.p1.23.m4.2.3.3.cmml" xref="A2.1.p1.23.m4.2.3.3"><csymbol cd="ambiguous" id="A2.1.p1.23.m4.2.3.3.1.cmml" xref="A2.1.p1.23.m4.2.3.3">subscript</csymbol><ci id="A2.1.p1.23.m4.2.3.3.2.cmml" xref="A2.1.p1.23.m4.2.3.3.2">𝐿</ci><infinity id="A2.1.p1.23.m4.2.3.3.3.cmml" xref="A2.1.p1.23.m4.2.3.3.3"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.23.m4.2c">\hat{\bm{\theta}}^{(k)}(t)\in L_{\infty}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.23.m4.2d">over^ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( italic_t ) ∈ italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="k=1" class="ltx_Math" display="inline" id="A2.1.p1.24.m5.1"><semantics id="A2.1.p1.24.m5.1a"><mrow id="A2.1.p1.24.m5.1.1" xref="A2.1.p1.24.m5.1.1.cmml"><mi id="A2.1.p1.24.m5.1.1.2" xref="A2.1.p1.24.m5.1.1.2.cmml">k</mi><mo id="A2.1.p1.24.m5.1.1.1" xref="A2.1.p1.24.m5.1.1.1.cmml">=</mo><mn id="A2.1.p1.24.m5.1.1.3" xref="A2.1.p1.24.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.24.m5.1b"><apply id="A2.1.p1.24.m5.1.1.cmml" xref="A2.1.p1.24.m5.1.1"><eq id="A2.1.p1.24.m5.1.1.1.cmml" xref="A2.1.p1.24.m5.1.1.1"></eq><ci id="A2.1.p1.24.m5.1.1.2.cmml" xref="A2.1.p1.24.m5.1.1.2">𝑘</ci><cn id="A2.1.p1.24.m5.1.1.3.cmml" type="integer" xref="A2.1.p1.24.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.24.m5.1c">k=1</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.24.m5.1d">italic_k = 1</annotation></semantics></math> to <math alttext="n-1" class="ltx_Math" display="inline" id="A2.1.p1.25.m6.1"><semantics id="A2.1.p1.25.m6.1a"><mrow id="A2.1.p1.25.m6.1.1" xref="A2.1.p1.25.m6.1.1.cmml"><mi id="A2.1.p1.25.m6.1.1.2" xref="A2.1.p1.25.m6.1.1.2.cmml">n</mi><mo id="A2.1.p1.25.m6.1.1.1" xref="A2.1.p1.25.m6.1.1.1.cmml">−</mo><mn id="A2.1.p1.25.m6.1.1.3" xref="A2.1.p1.25.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.25.m6.1b"><apply id="A2.1.p1.25.m6.1.1.cmml" xref="A2.1.p1.25.m6.1.1"><minus id="A2.1.p1.25.m6.1.1.1.cmml" xref="A2.1.p1.25.m6.1.1.1"></minus><ci id="A2.1.p1.25.m6.1.1.2.cmml" xref="A2.1.p1.25.m6.1.1.2">𝑛</ci><cn id="A2.1.p1.25.m6.1.1.3.cmml" type="integer" xref="A2.1.p1.25.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.25.m6.1c">n-1</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.25.m6.1d">italic_n - 1</annotation></semantics></math>, <math alttext="\forall t\in[0,t_{\rm f})" class="ltx_Math" display="inline" id="A2.1.p1.26.m7.2"><semantics id="A2.1.p1.26.m7.2a"><mrow id="A2.1.p1.26.m7.2.2" xref="A2.1.p1.26.m7.2.2.cmml"><mrow id="A2.1.p1.26.m7.2.2.3" xref="A2.1.p1.26.m7.2.2.3.cmml"><mo id="A2.1.p1.26.m7.2.2.3.1" rspace="0.167em" xref="A2.1.p1.26.m7.2.2.3.1.cmml">∀</mo><mi id="A2.1.p1.26.m7.2.2.3.2" xref="A2.1.p1.26.m7.2.2.3.2.cmml">t</mi></mrow><mo id="A2.1.p1.26.m7.2.2.2" xref="A2.1.p1.26.m7.2.2.2.cmml">∈</mo><mrow id="A2.1.p1.26.m7.2.2.1.1" xref="A2.1.p1.26.m7.2.2.1.2.cmml"><mo id="A2.1.p1.26.m7.2.2.1.1.2" stretchy="false" xref="A2.1.p1.26.m7.2.2.1.2.cmml">[</mo><mn id="A2.1.p1.26.m7.1.1" xref="A2.1.p1.26.m7.1.1.cmml">0</mn><mo id="A2.1.p1.26.m7.2.2.1.1.3" xref="A2.1.p1.26.m7.2.2.1.2.cmml">,</mo><msub id="A2.1.p1.26.m7.2.2.1.1.1" xref="A2.1.p1.26.m7.2.2.1.1.1.cmml"><mi id="A2.1.p1.26.m7.2.2.1.1.1.2" xref="A2.1.p1.26.m7.2.2.1.1.1.2.cmml">t</mi><mi id="A2.1.p1.26.m7.2.2.1.1.1.3" mathvariant="normal" xref="A2.1.p1.26.m7.2.2.1.1.1.3.cmml">f</mi></msub><mo id="A2.1.p1.26.m7.2.2.1.1.4" stretchy="false" xref="A2.1.p1.26.m7.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.26.m7.2b"><apply id="A2.1.p1.26.m7.2.2.cmml" xref="A2.1.p1.26.m7.2.2"><in id="A2.1.p1.26.m7.2.2.2.cmml" xref="A2.1.p1.26.m7.2.2.2"></in><apply id="A2.1.p1.26.m7.2.2.3.cmml" xref="A2.1.p1.26.m7.2.2.3"><csymbol cd="latexml" id="A2.1.p1.26.m7.2.2.3.1.cmml" xref="A2.1.p1.26.m7.2.2.3.1">for-all</csymbol><ci id="A2.1.p1.26.m7.2.2.3.2.cmml" xref="A2.1.p1.26.m7.2.2.3.2">𝑡</ci></apply><interval closure="closed-open" id="A2.1.p1.26.m7.2.2.1.2.cmml" xref="A2.1.p1.26.m7.2.2.1.1"><cn id="A2.1.p1.26.m7.1.1.cmml" type="integer" xref="A2.1.p1.26.m7.1.1">0</cn><apply id="A2.1.p1.26.m7.2.2.1.1.1.cmml" xref="A2.1.p1.26.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="A2.1.p1.26.m7.2.2.1.1.1.1.cmml" xref="A2.1.p1.26.m7.2.2.1.1.1">subscript</csymbol><ci id="A2.1.p1.26.m7.2.2.1.1.1.2.cmml" xref="A2.1.p1.26.m7.2.2.1.1.1.2">𝑡</ci><ci id="A2.1.p1.26.m7.2.2.1.1.1.3.cmml" xref="A2.1.p1.26.m7.2.2.1.1.1.3">f</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.26.m7.2c">\forall t\in[0,t_{\rm f})</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.26.m7.2d">∀ italic_t ∈ [ 0 , italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT )</annotation></semantics></math>, implying <math alttext="\Phi_{\rm f}\in L_{\infty}" class="ltx_Math" display="inline" id="A2.1.p1.27.m8.1"><semantics id="A2.1.p1.27.m8.1a"><mrow id="A2.1.p1.27.m8.1.1" xref="A2.1.p1.27.m8.1.1.cmml"><msub id="A2.1.p1.27.m8.1.1.2" xref="A2.1.p1.27.m8.1.1.2.cmml"><mi id="A2.1.p1.27.m8.1.1.2.2" mathvariant="normal" xref="A2.1.p1.27.m8.1.1.2.2.cmml">Φ</mi><mi id="A2.1.p1.27.m8.1.1.2.3" mathvariant="normal" xref="A2.1.p1.27.m8.1.1.2.3.cmml">f</mi></msub><mo id="A2.1.p1.27.m8.1.1.1" xref="A2.1.p1.27.m8.1.1.1.cmml">∈</mo><msub id="A2.1.p1.27.m8.1.1.3" xref="A2.1.p1.27.m8.1.1.3.cmml"><mi id="A2.1.p1.27.m8.1.1.3.2" xref="A2.1.p1.27.m8.1.1.3.2.cmml">L</mi><mi id="A2.1.p1.27.m8.1.1.3.3" mathvariant="normal" xref="A2.1.p1.27.m8.1.1.3.3.cmml">∞</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.27.m8.1b"><apply id="A2.1.p1.27.m8.1.1.cmml" xref="A2.1.p1.27.m8.1.1"><in id="A2.1.p1.27.m8.1.1.1.cmml" xref="A2.1.p1.27.m8.1.1.1"></in><apply id="A2.1.p1.27.m8.1.1.2.cmml" xref="A2.1.p1.27.m8.1.1.2"><csymbol cd="ambiguous" id="A2.1.p1.27.m8.1.1.2.1.cmml" xref="A2.1.p1.27.m8.1.1.2">subscript</csymbol><ci id="A2.1.p1.27.m8.1.1.2.2.cmml" xref="A2.1.p1.27.m8.1.1.2.2">Φ</ci><ci id="A2.1.p1.27.m8.1.1.2.3.cmml" xref="A2.1.p1.27.m8.1.1.2.3">f</ci></apply><apply id="A2.1.p1.27.m8.1.1.3.cmml" xref="A2.1.p1.27.m8.1.1.3"><csymbol cd="ambiguous" id="A2.1.p1.27.m8.1.1.3.1.cmml" xref="A2.1.p1.27.m8.1.1.3">subscript</csymbol><ci id="A2.1.p1.27.m8.1.1.3.2.cmml" xref="A2.1.p1.27.m8.1.1.3.2">𝐿</ci><infinity id="A2.1.p1.27.m8.1.1.3.3.cmml" xref="A2.1.p1.27.m8.1.1.3.3"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.27.m8.1c">\Phi_{\rm f}\in L_{\infty}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.27.m8.1d">roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ∈ italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\forall t\in[0,t_{\rm f})" class="ltx_Math" display="inline" id="A2.1.p1.28.m9.2"><semantics id="A2.1.p1.28.m9.2a"><mrow id="A2.1.p1.28.m9.2.2" xref="A2.1.p1.28.m9.2.2.cmml"><mrow id="A2.1.p1.28.m9.2.2.3" xref="A2.1.p1.28.m9.2.2.3.cmml"><mo id="A2.1.p1.28.m9.2.2.3.1" rspace="0.167em" xref="A2.1.p1.28.m9.2.2.3.1.cmml">∀</mo><mi id="A2.1.p1.28.m9.2.2.3.2" xref="A2.1.p1.28.m9.2.2.3.2.cmml">t</mi></mrow><mo id="A2.1.p1.28.m9.2.2.2" xref="A2.1.p1.28.m9.2.2.2.cmml">∈</mo><mrow id="A2.1.p1.28.m9.2.2.1.1" xref="A2.1.p1.28.m9.2.2.1.2.cmml"><mo id="A2.1.p1.28.m9.2.2.1.1.2" stretchy="false" xref="A2.1.p1.28.m9.2.2.1.2.cmml">[</mo><mn id="A2.1.p1.28.m9.1.1" xref="A2.1.p1.28.m9.1.1.cmml">0</mn><mo id="A2.1.p1.28.m9.2.2.1.1.3" xref="A2.1.p1.28.m9.2.2.1.2.cmml">,</mo><msub id="A2.1.p1.28.m9.2.2.1.1.1" xref="A2.1.p1.28.m9.2.2.1.1.1.cmml"><mi id="A2.1.p1.28.m9.2.2.1.1.1.2" xref="A2.1.p1.28.m9.2.2.1.1.1.2.cmml">t</mi><mi id="A2.1.p1.28.m9.2.2.1.1.1.3" mathvariant="normal" xref="A2.1.p1.28.m9.2.2.1.1.1.3.cmml">f</mi></msub><mo id="A2.1.p1.28.m9.2.2.1.1.4" stretchy="false" xref="A2.1.p1.28.m9.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.28.m9.2b"><apply id="A2.1.p1.28.m9.2.2.cmml" xref="A2.1.p1.28.m9.2.2"><in id="A2.1.p1.28.m9.2.2.2.cmml" xref="A2.1.p1.28.m9.2.2.2"></in><apply id="A2.1.p1.28.m9.2.2.3.cmml" xref="A2.1.p1.28.m9.2.2.3"><csymbol cd="latexml" id="A2.1.p1.28.m9.2.2.3.1.cmml" xref="A2.1.p1.28.m9.2.2.3.1">for-all</csymbol><ci id="A2.1.p1.28.m9.2.2.3.2.cmml" xref="A2.1.p1.28.m9.2.2.3.2">𝑡</ci></apply><interval closure="closed-open" id="A2.1.p1.28.m9.2.2.1.2.cmml" xref="A2.1.p1.28.m9.2.2.1.1"><cn id="A2.1.p1.28.m9.1.1.cmml" type="integer" xref="A2.1.p1.28.m9.1.1">0</cn><apply id="A2.1.p1.28.m9.2.2.1.1.1.cmml" xref="A2.1.p1.28.m9.2.2.1.1.1"><csymbol cd="ambiguous" id="A2.1.p1.28.m9.2.2.1.1.1.1.cmml" xref="A2.1.p1.28.m9.2.2.1.1.1">subscript</csymbol><ci id="A2.1.p1.28.m9.2.2.1.1.1.2.cmml" xref="A2.1.p1.28.m9.2.2.1.1.1.2">𝑡</ci><ci id="A2.1.p1.28.m9.2.2.1.1.1.3.cmml" xref="A2.1.p1.28.m9.2.2.1.1.1.3">f</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.28.m9.2c">\forall t\in[0,t_{\rm f})</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.28.m9.2d">∀ italic_t ∈ [ 0 , italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT )</annotation></semantics></math>. It follows from <math alttext="\bm{\epsilon}=\Phi_{\rm f}^{T}\tilde{\bm{\theta}}" class="ltx_Math" display="inline" id="A2.1.p1.29.m10.1"><semantics id="A2.1.p1.29.m10.1a"><mrow id="A2.1.p1.29.m10.1.1" xref="A2.1.p1.29.m10.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="A2.1.p1.29.m10.1.1.2" mathvariant="bold-italic" xref="A2.1.p1.29.m10.1.1.2.cmml">ϵ</mi><mo id="A2.1.p1.29.m10.1.1.1" xref="A2.1.p1.29.m10.1.1.1.cmml">=</mo><mrow id="A2.1.p1.29.m10.1.1.3" xref="A2.1.p1.29.m10.1.1.3.cmml"><msubsup id="A2.1.p1.29.m10.1.1.3.2" xref="A2.1.p1.29.m10.1.1.3.2.cmml"><mi id="A2.1.p1.29.m10.1.1.3.2.2.2" mathvariant="normal" xref="A2.1.p1.29.m10.1.1.3.2.2.2.cmml">Φ</mi><mi id="A2.1.p1.29.m10.1.1.3.2.2.3" mathvariant="normal" xref="A2.1.p1.29.m10.1.1.3.2.2.3.cmml">f</mi><mi id="A2.1.p1.29.m10.1.1.3.2.3" xref="A2.1.p1.29.m10.1.1.3.2.3.cmml">T</mi></msubsup><mo id="A2.1.p1.29.m10.1.1.3.1" xref="A2.1.p1.29.m10.1.1.3.1.cmml">⁢</mo><mover accent="true" id="A2.1.p1.29.m10.1.1.3.3" xref="A2.1.p1.29.m10.1.1.3.3.cmml"><mi id="A2.1.p1.29.m10.1.1.3.3.2" xref="A2.1.p1.29.m10.1.1.3.3.2.cmml">𝜽</mi><mo id="A2.1.p1.29.m10.1.1.3.3.1" xref="A2.1.p1.29.m10.1.1.3.3.1.cmml">~</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.29.m10.1b"><apply id="A2.1.p1.29.m10.1.1.cmml" xref="A2.1.p1.29.m10.1.1"><eq id="A2.1.p1.29.m10.1.1.1.cmml" xref="A2.1.p1.29.m10.1.1.1"></eq><ci id="A2.1.p1.29.m10.1.1.2.cmml" xref="A2.1.p1.29.m10.1.1.2">bold-italic-ϵ</ci><apply id="A2.1.p1.29.m10.1.1.3.cmml" xref="A2.1.p1.29.m10.1.1.3"><times id="A2.1.p1.29.m10.1.1.3.1.cmml" xref="A2.1.p1.29.m10.1.1.3.1"></times><apply id="A2.1.p1.29.m10.1.1.3.2.cmml" xref="A2.1.p1.29.m10.1.1.3.2"><csymbol cd="ambiguous" id="A2.1.p1.29.m10.1.1.3.2.1.cmml" xref="A2.1.p1.29.m10.1.1.3.2">superscript</csymbol><apply id="A2.1.p1.29.m10.1.1.3.2.2.cmml" xref="A2.1.p1.29.m10.1.1.3.2"><csymbol cd="ambiguous" id="A2.1.p1.29.m10.1.1.3.2.2.1.cmml" xref="A2.1.p1.29.m10.1.1.3.2">subscript</csymbol><ci id="A2.1.p1.29.m10.1.1.3.2.2.2.cmml" xref="A2.1.p1.29.m10.1.1.3.2.2.2">Φ</ci><ci id="A2.1.p1.29.m10.1.1.3.2.2.3.cmml" xref="A2.1.p1.29.m10.1.1.3.2.2.3">f</ci></apply><ci id="A2.1.p1.29.m10.1.1.3.2.3.cmml" xref="A2.1.p1.29.m10.1.1.3.2.3">𝑇</ci></apply><apply id="A2.1.p1.29.m10.1.1.3.3.cmml" xref="A2.1.p1.29.m10.1.1.3.3"><ci id="A2.1.p1.29.m10.1.1.3.3.1.cmml" xref="A2.1.p1.29.m10.1.1.3.3.1">~</ci><ci id="A2.1.p1.29.m10.1.1.3.3.2.cmml" xref="A2.1.p1.29.m10.1.1.3.3.2">𝜽</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.29.m10.1c">\bm{\epsilon}=\Phi_{\rm f}^{T}\tilde{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.29.m10.1d">bold_italic_ϵ = roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math> that <math alttext="\bm{\epsilon}\in L_{\infty}" class="ltx_Math" display="inline" id="A2.1.p1.30.m11.1"><semantics id="A2.1.p1.30.m11.1a"><mrow id="A2.1.p1.30.m11.1.1" xref="A2.1.p1.30.m11.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="A2.1.p1.30.m11.1.1.2" mathvariant="bold-italic" xref="A2.1.p1.30.m11.1.1.2.cmml">ϵ</mi><mo id="A2.1.p1.30.m11.1.1.1" xref="A2.1.p1.30.m11.1.1.1.cmml">∈</mo><msub id="A2.1.p1.30.m11.1.1.3" xref="A2.1.p1.30.m11.1.1.3.cmml"><mi id="A2.1.p1.30.m11.1.1.3.2" xref="A2.1.p1.30.m11.1.1.3.2.cmml">L</mi><mi id="A2.1.p1.30.m11.1.1.3.3" mathvariant="normal" xref="A2.1.p1.30.m11.1.1.3.3.cmml">∞</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.30.m11.1b"><apply id="A2.1.p1.30.m11.1.1.cmml" xref="A2.1.p1.30.m11.1.1"><in id="A2.1.p1.30.m11.1.1.1.cmml" xref="A2.1.p1.30.m11.1.1.1"></in><ci id="A2.1.p1.30.m11.1.1.2.cmml" xref="A2.1.p1.30.m11.1.1.2">bold-italic-ϵ</ci><apply id="A2.1.p1.30.m11.1.1.3.cmml" xref="A2.1.p1.30.m11.1.1.3"><csymbol cd="ambiguous" id="A2.1.p1.30.m11.1.1.3.1.cmml" xref="A2.1.p1.30.m11.1.1.3">subscript</csymbol><ci id="A2.1.p1.30.m11.1.1.3.2.cmml" xref="A2.1.p1.30.m11.1.1.3.2">𝐿</ci><infinity id="A2.1.p1.30.m11.1.1.3.3.cmml" xref="A2.1.p1.30.m11.1.1.3.3"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.30.m11.1c">\bm{\epsilon}\in L_{\infty}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.30.m11.1d">bold_italic_ϵ ∈ italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\forall t\in[0,t_{\rm f})" class="ltx_Math" display="inline" id="A2.1.p1.31.m12.2"><semantics id="A2.1.p1.31.m12.2a"><mrow id="A2.1.p1.31.m12.2.2" xref="A2.1.p1.31.m12.2.2.cmml"><mrow id="A2.1.p1.31.m12.2.2.3" xref="A2.1.p1.31.m12.2.2.3.cmml"><mo id="A2.1.p1.31.m12.2.2.3.1" rspace="0.167em" xref="A2.1.p1.31.m12.2.2.3.1.cmml">∀</mo><mi id="A2.1.p1.31.m12.2.2.3.2" xref="A2.1.p1.31.m12.2.2.3.2.cmml">t</mi></mrow><mo id="A2.1.p1.31.m12.2.2.2" xref="A2.1.p1.31.m12.2.2.2.cmml">∈</mo><mrow id="A2.1.p1.31.m12.2.2.1.1" xref="A2.1.p1.31.m12.2.2.1.2.cmml"><mo id="A2.1.p1.31.m12.2.2.1.1.2" stretchy="false" xref="A2.1.p1.31.m12.2.2.1.2.cmml">[</mo><mn id="A2.1.p1.31.m12.1.1" xref="A2.1.p1.31.m12.1.1.cmml">0</mn><mo id="A2.1.p1.31.m12.2.2.1.1.3" xref="A2.1.p1.31.m12.2.2.1.2.cmml">,</mo><msub id="A2.1.p1.31.m12.2.2.1.1.1" xref="A2.1.p1.31.m12.2.2.1.1.1.cmml"><mi id="A2.1.p1.31.m12.2.2.1.1.1.2" xref="A2.1.p1.31.m12.2.2.1.1.1.2.cmml">t</mi><mi id="A2.1.p1.31.m12.2.2.1.1.1.3" mathvariant="normal" xref="A2.1.p1.31.m12.2.2.1.1.1.3.cmml">f</mi></msub><mo id="A2.1.p1.31.m12.2.2.1.1.4" stretchy="false" xref="A2.1.p1.31.m12.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.31.m12.2b"><apply id="A2.1.p1.31.m12.2.2.cmml" xref="A2.1.p1.31.m12.2.2"><in id="A2.1.p1.31.m12.2.2.2.cmml" xref="A2.1.p1.31.m12.2.2.2"></in><apply id="A2.1.p1.31.m12.2.2.3.cmml" xref="A2.1.p1.31.m12.2.2.3"><csymbol cd="latexml" id="A2.1.p1.31.m12.2.2.3.1.cmml" xref="A2.1.p1.31.m12.2.2.3.1">for-all</csymbol><ci id="A2.1.p1.31.m12.2.2.3.2.cmml" xref="A2.1.p1.31.m12.2.2.3.2">𝑡</ci></apply><interval closure="closed-open" id="A2.1.p1.31.m12.2.2.1.2.cmml" xref="A2.1.p1.31.m12.2.2.1.1"><cn id="A2.1.p1.31.m12.1.1.cmml" type="integer" xref="A2.1.p1.31.m12.1.1">0</cn><apply id="A2.1.p1.31.m12.2.2.1.1.1.cmml" xref="A2.1.p1.31.m12.2.2.1.1.1"><csymbol cd="ambiguous" id="A2.1.p1.31.m12.2.2.1.1.1.1.cmml" xref="A2.1.p1.31.m12.2.2.1.1.1">subscript</csymbol><ci id="A2.1.p1.31.m12.2.2.1.1.1.2.cmml" xref="A2.1.p1.31.m12.2.2.1.1.1.2">𝑡</ci><ci id="A2.1.p1.31.m12.2.2.1.1.1.3.cmml" xref="A2.1.p1.31.m12.2.2.1.1.1.3">f</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.31.m12.2c">\forall t\in[0,t_{\rm f})</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.31.m12.2d">∀ italic_t ∈ [ 0 , italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT )</annotation></semantics></math>. In summary, one has <math alttext="\tilde{\bm{\theta}}(t)\in L_{\infty}" class="ltx_Math" display="inline" id="A2.1.p1.32.m13.1"><semantics id="A2.1.p1.32.m13.1a"><mrow id="A2.1.p1.32.m13.1.2" xref="A2.1.p1.32.m13.1.2.cmml"><mrow id="A2.1.p1.32.m13.1.2.2" xref="A2.1.p1.32.m13.1.2.2.cmml"><mover accent="true" id="A2.1.p1.32.m13.1.2.2.2" xref="A2.1.p1.32.m13.1.2.2.2.cmml"><mi id="A2.1.p1.32.m13.1.2.2.2.2" xref="A2.1.p1.32.m13.1.2.2.2.2.cmml">𝜽</mi><mo id="A2.1.p1.32.m13.1.2.2.2.1" xref="A2.1.p1.32.m13.1.2.2.2.1.cmml">~</mo></mover><mo id="A2.1.p1.32.m13.1.2.2.1" xref="A2.1.p1.32.m13.1.2.2.1.cmml">⁢</mo><mrow id="A2.1.p1.32.m13.1.2.2.3.2" xref="A2.1.p1.32.m13.1.2.2.cmml"><mo id="A2.1.p1.32.m13.1.2.2.3.2.1" stretchy="false" xref="A2.1.p1.32.m13.1.2.2.cmml">(</mo><mi id="A2.1.p1.32.m13.1.1" xref="A2.1.p1.32.m13.1.1.cmml">t</mi><mo id="A2.1.p1.32.m13.1.2.2.3.2.2" stretchy="false" xref="A2.1.p1.32.m13.1.2.2.cmml">)</mo></mrow></mrow><mo id="A2.1.p1.32.m13.1.2.1" xref="A2.1.p1.32.m13.1.2.1.cmml">∈</mo><msub id="A2.1.p1.32.m13.1.2.3" xref="A2.1.p1.32.m13.1.2.3.cmml"><mi id="A2.1.p1.32.m13.1.2.3.2" xref="A2.1.p1.32.m13.1.2.3.2.cmml">L</mi><mi id="A2.1.p1.32.m13.1.2.3.3" mathvariant="normal" xref="A2.1.p1.32.m13.1.2.3.3.cmml">∞</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.32.m13.1b"><apply id="A2.1.p1.32.m13.1.2.cmml" xref="A2.1.p1.32.m13.1.2"><in id="A2.1.p1.32.m13.1.2.1.cmml" xref="A2.1.p1.32.m13.1.2.1"></in><apply id="A2.1.p1.32.m13.1.2.2.cmml" xref="A2.1.p1.32.m13.1.2.2"><times id="A2.1.p1.32.m13.1.2.2.1.cmml" xref="A2.1.p1.32.m13.1.2.2.1"></times><apply id="A2.1.p1.32.m13.1.2.2.2.cmml" xref="A2.1.p1.32.m13.1.2.2.2"><ci id="A2.1.p1.32.m13.1.2.2.2.1.cmml" xref="A2.1.p1.32.m13.1.2.2.2.1">~</ci><ci id="A2.1.p1.32.m13.1.2.2.2.2.cmml" xref="A2.1.p1.32.m13.1.2.2.2.2">𝜽</ci></apply><ci id="A2.1.p1.32.m13.1.1.cmml" xref="A2.1.p1.32.m13.1.1">𝑡</ci></apply><apply id="A2.1.p1.32.m13.1.2.3.cmml" xref="A2.1.p1.32.m13.1.2.3"><csymbol cd="ambiguous" id="A2.1.p1.32.m13.1.2.3.1.cmml" xref="A2.1.p1.32.m13.1.2.3">subscript</csymbol><ci id="A2.1.p1.32.m13.1.2.3.2.cmml" xref="A2.1.p1.32.m13.1.2.3.2">𝐿</ci><infinity id="A2.1.p1.32.m13.1.2.3.3.cmml" xref="A2.1.p1.32.m13.1.2.3.3"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.32.m13.1c">\tilde{\bm{\theta}}(t)\in L_{\infty}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.32.m13.1d">over~ start_ARG bold_italic_θ end_ARG ( italic_t ) ∈ italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\forall t\geq 0" class="ltx_Math" display="inline" id="A2.1.p1.33.m14.1"><semantics id="A2.1.p1.33.m14.1a"><mrow id="A2.1.p1.33.m14.1.1" xref="A2.1.p1.33.m14.1.1.cmml"><mrow id="A2.1.p1.33.m14.1.1.2" xref="A2.1.p1.33.m14.1.1.2.cmml"><mo id="A2.1.p1.33.m14.1.1.2.1" rspace="0.167em" xref="A2.1.p1.33.m14.1.1.2.1.cmml">∀</mo><mi id="A2.1.p1.33.m14.1.1.2.2" xref="A2.1.p1.33.m14.1.1.2.2.cmml">t</mi></mrow><mo id="A2.1.p1.33.m14.1.1.1" xref="A2.1.p1.33.m14.1.1.1.cmml">≥</mo><mn id="A2.1.p1.33.m14.1.1.3" xref="A2.1.p1.33.m14.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.33.m14.1b"><apply id="A2.1.p1.33.m14.1.1.cmml" xref="A2.1.p1.33.m14.1.1"><geq id="A2.1.p1.33.m14.1.1.1.cmml" xref="A2.1.p1.33.m14.1.1.1"></geq><apply id="A2.1.p1.33.m14.1.1.2.cmml" xref="A2.1.p1.33.m14.1.1.2"><csymbol cd="latexml" id="A2.1.p1.33.m14.1.1.2.1.cmml" xref="A2.1.p1.33.m14.1.1.2.1">for-all</csymbol><ci id="A2.1.p1.33.m14.1.1.2.2.cmml" xref="A2.1.p1.33.m14.1.1.2.2">𝑡</ci></apply><cn id="A2.1.p1.33.m14.1.1.3.cmml" type="integer" xref="A2.1.p1.33.m14.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.33.m14.1c">\forall t\geq 0</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.33.m14.1d">∀ italic_t ≥ 0</annotation></semantics></math> and <math alttext="\bm{\epsilon}(t)\in L_{2}\cap L_{\infty}" class="ltx_Math" display="inline" id="A2.1.p1.34.m15.1"><semantics id="A2.1.p1.34.m15.1a"><mrow id="A2.1.p1.34.m15.1.2" xref="A2.1.p1.34.m15.1.2.cmml"><mrow id="A2.1.p1.34.m15.1.2.2" xref="A2.1.p1.34.m15.1.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="A2.1.p1.34.m15.1.2.2.2" mathvariant="bold-italic" xref="A2.1.p1.34.m15.1.2.2.2.cmml">ϵ</mi><mo id="A2.1.p1.34.m15.1.2.2.1" xref="A2.1.p1.34.m15.1.2.2.1.cmml">⁢</mo><mrow id="A2.1.p1.34.m15.1.2.2.3.2" xref="A2.1.p1.34.m15.1.2.2.cmml"><mo id="A2.1.p1.34.m15.1.2.2.3.2.1" stretchy="false" xref="A2.1.p1.34.m15.1.2.2.cmml">(</mo><mi id="A2.1.p1.34.m15.1.1" xref="A2.1.p1.34.m15.1.1.cmml">t</mi><mo id="A2.1.p1.34.m15.1.2.2.3.2.2" stretchy="false" xref="A2.1.p1.34.m15.1.2.2.cmml">)</mo></mrow></mrow><mo id="A2.1.p1.34.m15.1.2.1" xref="A2.1.p1.34.m15.1.2.1.cmml">∈</mo><mrow id="A2.1.p1.34.m15.1.2.3" xref="A2.1.p1.34.m15.1.2.3.cmml"><msub id="A2.1.p1.34.m15.1.2.3.2" xref="A2.1.p1.34.m15.1.2.3.2.cmml"><mi id="A2.1.p1.34.m15.1.2.3.2.2" xref="A2.1.p1.34.m15.1.2.3.2.2.cmml">L</mi><mn id="A2.1.p1.34.m15.1.2.3.2.3" xref="A2.1.p1.34.m15.1.2.3.2.3.cmml">2</mn></msub><mo id="A2.1.p1.34.m15.1.2.3.1" xref="A2.1.p1.34.m15.1.2.3.1.cmml">∩</mo><msub id="A2.1.p1.34.m15.1.2.3.3" xref="A2.1.p1.34.m15.1.2.3.3.cmml"><mi id="A2.1.p1.34.m15.1.2.3.3.2" xref="A2.1.p1.34.m15.1.2.3.3.2.cmml">L</mi><mi id="A2.1.p1.34.m15.1.2.3.3.3" mathvariant="normal" xref="A2.1.p1.34.m15.1.2.3.3.3.cmml">∞</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.34.m15.1b"><apply id="A2.1.p1.34.m15.1.2.cmml" xref="A2.1.p1.34.m15.1.2"><in id="A2.1.p1.34.m15.1.2.1.cmml" xref="A2.1.p1.34.m15.1.2.1"></in><apply id="A2.1.p1.34.m15.1.2.2.cmml" xref="A2.1.p1.34.m15.1.2.2"><times id="A2.1.p1.34.m15.1.2.2.1.cmml" xref="A2.1.p1.34.m15.1.2.2.1"></times><ci id="A2.1.p1.34.m15.1.2.2.2.cmml" xref="A2.1.p1.34.m15.1.2.2.2">bold-italic-ϵ</ci><ci id="A2.1.p1.34.m15.1.1.cmml" xref="A2.1.p1.34.m15.1.1">𝑡</ci></apply><apply id="A2.1.p1.34.m15.1.2.3.cmml" xref="A2.1.p1.34.m15.1.2.3"><intersect id="A2.1.p1.34.m15.1.2.3.1.cmml" xref="A2.1.p1.34.m15.1.2.3.1"></intersect><apply id="A2.1.p1.34.m15.1.2.3.2.cmml" xref="A2.1.p1.34.m15.1.2.3.2"><csymbol cd="ambiguous" id="A2.1.p1.34.m15.1.2.3.2.1.cmml" xref="A2.1.p1.34.m15.1.2.3.2">subscript</csymbol><ci id="A2.1.p1.34.m15.1.2.3.2.2.cmml" xref="A2.1.p1.34.m15.1.2.3.2.2">𝐿</ci><cn id="A2.1.p1.34.m15.1.2.3.2.3.cmml" type="integer" xref="A2.1.p1.34.m15.1.2.3.2.3">2</cn></apply><apply id="A2.1.p1.34.m15.1.2.3.3.cmml" xref="A2.1.p1.34.m15.1.2.3.3"><csymbol cd="ambiguous" id="A2.1.p1.34.m15.1.2.3.3.1.cmml" xref="A2.1.p1.34.m15.1.2.3.3">subscript</csymbol><ci id="A2.1.p1.34.m15.1.2.3.3.2.cmml" xref="A2.1.p1.34.m15.1.2.3.3.2">𝐿</ci><infinity id="A2.1.p1.34.m15.1.2.3.3.3.cmml" xref="A2.1.p1.34.m15.1.2.3.3.3"></infinity></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.34.m15.1c">\bm{\epsilon}(t)\in L_{2}\cap L_{\infty}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.34.m15.1d">bold_italic_ϵ ( italic_t ) ∈ italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∩ italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\forall t\in[0,t_{\rm f})" class="ltx_Math" display="inline" id="A2.1.p1.35.m16.2"><semantics id="A2.1.p1.35.m16.2a"><mrow id="A2.1.p1.35.m16.2.2" xref="A2.1.p1.35.m16.2.2.cmml"><mrow id="A2.1.p1.35.m16.2.2.3" xref="A2.1.p1.35.m16.2.2.3.cmml"><mo id="A2.1.p1.35.m16.2.2.3.1" rspace="0.167em" xref="A2.1.p1.35.m16.2.2.3.1.cmml">∀</mo><mi id="A2.1.p1.35.m16.2.2.3.2" xref="A2.1.p1.35.m16.2.2.3.2.cmml">t</mi></mrow><mo id="A2.1.p1.35.m16.2.2.2" xref="A2.1.p1.35.m16.2.2.2.cmml">∈</mo><mrow id="A2.1.p1.35.m16.2.2.1.1" xref="A2.1.p1.35.m16.2.2.1.2.cmml"><mo id="A2.1.p1.35.m16.2.2.1.1.2" stretchy="false" xref="A2.1.p1.35.m16.2.2.1.2.cmml">[</mo><mn id="A2.1.p1.35.m16.1.1" xref="A2.1.p1.35.m16.1.1.cmml">0</mn><mo id="A2.1.p1.35.m16.2.2.1.1.3" xref="A2.1.p1.35.m16.2.2.1.2.cmml">,</mo><msub id="A2.1.p1.35.m16.2.2.1.1.1" xref="A2.1.p1.35.m16.2.2.1.1.1.cmml"><mi id="A2.1.p1.35.m16.2.2.1.1.1.2" xref="A2.1.p1.35.m16.2.2.1.1.1.2.cmml">t</mi><mi id="A2.1.p1.35.m16.2.2.1.1.1.3" mathvariant="normal" xref="A2.1.p1.35.m16.2.2.1.1.1.3.cmml">f</mi></msub><mo id="A2.1.p1.35.m16.2.2.1.1.4" stretchy="false" xref="A2.1.p1.35.m16.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.35.m16.2b"><apply id="A2.1.p1.35.m16.2.2.cmml" xref="A2.1.p1.35.m16.2.2"><in id="A2.1.p1.35.m16.2.2.2.cmml" xref="A2.1.p1.35.m16.2.2.2"></in><apply id="A2.1.p1.35.m16.2.2.3.cmml" xref="A2.1.p1.35.m16.2.2.3"><csymbol cd="latexml" id="A2.1.p1.35.m16.2.2.3.1.cmml" xref="A2.1.p1.35.m16.2.2.3.1">for-all</csymbol><ci id="A2.1.p1.35.m16.2.2.3.2.cmml" xref="A2.1.p1.35.m16.2.2.3.2">𝑡</ci></apply><interval closure="closed-open" id="A2.1.p1.35.m16.2.2.1.2.cmml" xref="A2.1.p1.35.m16.2.2.1.1"><cn id="A2.1.p1.35.m16.1.1.cmml" type="integer" xref="A2.1.p1.35.m16.1.1">0</cn><apply id="A2.1.p1.35.m16.2.2.1.1.1.cmml" xref="A2.1.p1.35.m16.2.2.1.1.1"><csymbol cd="ambiguous" id="A2.1.p1.35.m16.2.2.1.1.1.1.cmml" xref="A2.1.p1.35.m16.2.2.1.1.1">subscript</csymbol><ci id="A2.1.p1.35.m16.2.2.1.1.1.2.cmml" xref="A2.1.p1.35.m16.2.2.1.1.1.2">𝑡</ci><ci id="A2.1.p1.35.m16.2.2.1.1.1.3.cmml" xref="A2.1.p1.35.m16.2.2.1.1.1.3">f</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.35.m16.2c">\forall t\in[0,t_{\rm f})</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.35.m16.2d">∀ italic_t ∈ [ 0 , italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A2.2.p2"> <p class="ltx_p" id="A2.2.p2.1">2) Consider the parameter convergence problem under partial IE and <math alttext="t_{\rm f}\rightarrow\infty" class="ltx_Math" display="inline" id="A2.2.p2.1.m1.1"><semantics id="A2.2.p2.1.m1.1a"><mrow id="A2.2.p2.1.m1.1.1" xref="A2.2.p2.1.m1.1.1.cmml"><msub id="A2.2.p2.1.m1.1.1.2" xref="A2.2.p2.1.m1.1.1.2.cmml"><mi id="A2.2.p2.1.m1.1.1.2.2" xref="A2.2.p2.1.m1.1.1.2.2.cmml">t</mi><mi id="A2.2.p2.1.m1.1.1.2.3" mathvariant="normal" xref="A2.2.p2.1.m1.1.1.2.3.cmml">f</mi></msub><mo id="A2.2.p2.1.m1.1.1.1" stretchy="false" xref="A2.2.p2.1.m1.1.1.1.cmml">→</mo><mi id="A2.2.p2.1.m1.1.1.3" mathvariant="normal" xref="A2.2.p2.1.m1.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.1.m1.1b"><apply id="A2.2.p2.1.m1.1.1.cmml" xref="A2.2.p2.1.m1.1.1"><ci id="A2.2.p2.1.m1.1.1.1.cmml" xref="A2.2.p2.1.m1.1.1.1">→</ci><apply id="A2.2.p2.1.m1.1.1.2.cmml" xref="A2.2.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="A2.2.p2.1.m1.1.1.2.1.cmml" xref="A2.2.p2.1.m1.1.1.2">subscript</csymbol><ci id="A2.2.p2.1.m1.1.1.2.2.cmml" xref="A2.2.p2.1.m1.1.1.2.2">𝑡</ci><ci id="A2.2.p2.1.m1.1.1.2.3.cmml" xref="A2.2.p2.1.m1.1.1.2.3">f</ci></apply><infinity id="A2.2.p2.1.m1.1.1.3.cmml" xref="A2.2.p2.1.m1.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.1.m1.1c">t_{\rm f}\rightarrow\infty</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.1.m1.1d">italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT → ∞</annotation></semantics></math>. The composite learning law (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>) with respect to active channels is given by</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx41"> <tbody id="A2.E36"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{\tilde{\bm{\theta}}}_{\zeta}=-\Gamma_{\zeta}\left(\Phi_{{\rm f% },\zeta}\Phi_{{\rm f},\zeta}^{T}\tilde{\bm{\theta}}_{\zeta}+\kappa Q_{\zeta}(t% ,t_{\rm e}){\tilde{\bm{\theta}}}_{\zeta}\right),t\geq T_{\rm a}" class="ltx_Math" display="inline" id="A2.E36.m1.7"><semantics id="A2.E36.m1.7a"><mrow id="A2.E36.m1.7.7.2" xref="A2.E36.m1.7.7.3.cmml"><mrow id="A2.E36.m1.6.6.1.1" xref="A2.E36.m1.6.6.1.1.cmml"><msub id="A2.E36.m1.6.6.1.1.3" xref="A2.E36.m1.6.6.1.1.3.cmml"><mover accent="true" id="A2.E36.m1.6.6.1.1.3.2" xref="A2.E36.m1.6.6.1.1.3.2.cmml"><mover accent="true" id="A2.E36.m1.6.6.1.1.3.2.2" xref="A2.E36.m1.6.6.1.1.3.2.2.cmml"><mi id="A2.E36.m1.6.6.1.1.3.2.2.2" xref="A2.E36.m1.6.6.1.1.3.2.2.2.cmml">𝜽</mi><mo id="A2.E36.m1.6.6.1.1.3.2.2.1" xref="A2.E36.m1.6.6.1.1.3.2.2.1.cmml">~</mo></mover><mo id="A2.E36.m1.6.6.1.1.3.2.1" xref="A2.E36.m1.6.6.1.1.3.2.1.cmml">˙</mo></mover><mi id="A2.E36.m1.6.6.1.1.3.3" xref="A2.E36.m1.6.6.1.1.3.3.cmml">ζ</mi></msub><mo id="A2.E36.m1.6.6.1.1.2" xref="A2.E36.m1.6.6.1.1.2.cmml">=</mo><mrow id="A2.E36.m1.6.6.1.1.1" xref="A2.E36.m1.6.6.1.1.1.cmml"><mo id="A2.E36.m1.6.6.1.1.1a" xref="A2.E36.m1.6.6.1.1.1.cmml">−</mo><mrow id="A2.E36.m1.6.6.1.1.1.1" xref="A2.E36.m1.6.6.1.1.1.1.cmml"><msub id="A2.E36.m1.6.6.1.1.1.1.3" xref="A2.E36.m1.6.6.1.1.1.1.3.cmml"><mi id="A2.E36.m1.6.6.1.1.1.1.3.2" mathvariant="normal" xref="A2.E36.m1.6.6.1.1.1.1.3.2.cmml">Γ</mi><mi id="A2.E36.m1.6.6.1.1.1.1.3.3" xref="A2.E36.m1.6.6.1.1.1.1.3.3.cmml">ζ</mi></msub><mo id="A2.E36.m1.6.6.1.1.1.1.2" xref="A2.E36.m1.6.6.1.1.1.1.2.cmml">⁢</mo><mrow id="A2.E36.m1.6.6.1.1.1.1.1.1" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.cmml"><mo id="A2.E36.m1.6.6.1.1.1.1.1.1.2" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A2.E36.m1.6.6.1.1.1.1.1.1.1" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.cmml"><mrow id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.cmml"><msub id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.2" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.2.cmml"><mi id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.2.2" mathvariant="normal" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.2.2.cmml">Φ</mi><mrow id="A2.E36.m1.2.2.2.4" xref="A2.E36.m1.2.2.2.3.cmml"><mi id="A2.E36.m1.1.1.1.1" mathvariant="normal" xref="A2.E36.m1.1.1.1.1.cmml">f</mi><mo id="A2.E36.m1.2.2.2.4.1" xref="A2.E36.m1.2.2.2.3.cmml">,</mo><mi id="A2.E36.m1.2.2.2.2" xref="A2.E36.m1.2.2.2.2.cmml">ζ</mi></mrow></msub><mo id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.1" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><msubsup id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.3" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.3.cmml"><mi id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.3.2.2" mathvariant="normal" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.3.2.2.cmml">Φ</mi><mrow id="A2.E36.m1.4.4.2.4" xref="A2.E36.m1.4.4.2.3.cmml"><mi id="A2.E36.m1.3.3.1.1" mathvariant="normal" xref="A2.E36.m1.3.3.1.1.cmml">f</mi><mo id="A2.E36.m1.4.4.2.4.1" xref="A2.E36.m1.4.4.2.3.cmml">,</mo><mi id="A2.E36.m1.4.4.2.2" xref="A2.E36.m1.4.4.2.2.cmml">ζ</mi></mrow><mi id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.3.3" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.3.3.cmml">T</mi></msubsup><mo id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.1a" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><msub id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.4" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.4.cmml"><mover accent="true" id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.4.2" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.4.2.cmml"><mi id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.4.2.2" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.4.2.2.cmml">𝜽</mi><mo id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.4.2.1" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.4.2.1.cmml">~</mo></mover><mi id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.4.3" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.4.3.cmml">ζ</mi></msub></mrow><mo id="A2.E36.m1.6.6.1.1.1.1.1.1.1.2" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.2.cmml">+</mo><mrow id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.cmml"><mi id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.3" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.3.cmml">κ</mi><mo id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.2" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><msub id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.4" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.4.cmml"><mi id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.4.2" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.4.2.cmml">Q</mi><mi id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.4.3" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.4.3.cmml">ζ</mi></msub><mo id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.2a" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.1.1" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.1.2.cmml"><mo id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.1.2.cmml">(</mo><mi id="A2.E36.m1.5.5" xref="A2.E36.m1.5.5.cmml">t</mi><mo id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.1.1.3" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.1.2.cmml">,</mo><msub id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.1.1.1" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.2" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.2.cmml">t</mi><mi id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.3" mathvariant="normal" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.1.1.4" stretchy="false" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.1.2.cmml">)</mo></mrow><mo id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.2b" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><msub id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.5" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.5.cmml"><mover accent="true" id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.5.2" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.5.2.cmml"><mi id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.5.2.2" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.5.2.2.cmml">𝜽</mi><mo id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.5.2.1" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.5.2.1.cmml">~</mo></mover><mi id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.5.3" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.5.3.cmml">ζ</mi></msub></mrow></mrow><mo id="A2.E36.m1.6.6.1.1.1.1.1.1.3" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="A2.E36.m1.7.7.2.3" xref="A2.E36.m1.7.7.3a.cmml">,</mo><mrow id="A2.E36.m1.7.7.2.2" xref="A2.E36.m1.7.7.2.2.cmml"><mi id="A2.E36.m1.7.7.2.2.2" xref="A2.E36.m1.7.7.2.2.2.cmml">t</mi><mo id="A2.E36.m1.7.7.2.2.1" xref="A2.E36.m1.7.7.2.2.1.cmml">≥</mo><msub id="A2.E36.m1.7.7.2.2.3" xref="A2.E36.m1.7.7.2.2.3.cmml"><mi id="A2.E36.m1.7.7.2.2.3.2" xref="A2.E36.m1.7.7.2.2.3.2.cmml">T</mi><mi id="A2.E36.m1.7.7.2.2.3.3" mathvariant="normal" xref="A2.E36.m1.7.7.2.2.3.3.cmml">a</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.E36.m1.7b"><apply id="A2.E36.m1.7.7.3.cmml" xref="A2.E36.m1.7.7.2"><csymbol cd="ambiguous" id="A2.E36.m1.7.7.3a.cmml" xref="A2.E36.m1.7.7.2.3">formulae-sequence</csymbol><apply id="A2.E36.m1.6.6.1.1.cmml" xref="A2.E36.m1.6.6.1.1"><eq id="A2.E36.m1.6.6.1.1.2.cmml" xref="A2.E36.m1.6.6.1.1.2"></eq><apply id="A2.E36.m1.6.6.1.1.3.cmml" xref="A2.E36.m1.6.6.1.1.3"><csymbol cd="ambiguous" id="A2.E36.m1.6.6.1.1.3.1.cmml" xref="A2.E36.m1.6.6.1.1.3">subscript</csymbol><apply id="A2.E36.m1.6.6.1.1.3.2.cmml" xref="A2.E36.m1.6.6.1.1.3.2"><ci id="A2.E36.m1.6.6.1.1.3.2.1.cmml" xref="A2.E36.m1.6.6.1.1.3.2.1">˙</ci><apply id="A2.E36.m1.6.6.1.1.3.2.2.cmml" xref="A2.E36.m1.6.6.1.1.3.2.2"><ci id="A2.E36.m1.6.6.1.1.3.2.2.1.cmml" xref="A2.E36.m1.6.6.1.1.3.2.2.1">~</ci><ci id="A2.E36.m1.6.6.1.1.3.2.2.2.cmml" xref="A2.E36.m1.6.6.1.1.3.2.2.2">𝜽</ci></apply></apply><ci id="A2.E36.m1.6.6.1.1.3.3.cmml" xref="A2.E36.m1.6.6.1.1.3.3">𝜁</ci></apply><apply id="A2.E36.m1.6.6.1.1.1.cmml" xref="A2.E36.m1.6.6.1.1.1"><minus id="A2.E36.m1.6.6.1.1.1.2.cmml" xref="A2.E36.m1.6.6.1.1.1"></minus><apply id="A2.E36.m1.6.6.1.1.1.1.cmml" xref="A2.E36.m1.6.6.1.1.1.1"><times id="A2.E36.m1.6.6.1.1.1.1.2.cmml" xref="A2.E36.m1.6.6.1.1.1.1.2"></times><apply id="A2.E36.m1.6.6.1.1.1.1.3.cmml" xref="A2.E36.m1.6.6.1.1.1.1.3"><csymbol cd="ambiguous" id="A2.E36.m1.6.6.1.1.1.1.3.1.cmml" xref="A2.E36.m1.6.6.1.1.1.1.3">subscript</csymbol><ci id="A2.E36.m1.6.6.1.1.1.1.3.2.cmml" xref="A2.E36.m1.6.6.1.1.1.1.3.2">Γ</ci><ci id="A2.E36.m1.6.6.1.1.1.1.3.3.cmml" xref="A2.E36.m1.6.6.1.1.1.1.3.3">𝜁</ci></apply><apply id="A2.E36.m1.6.6.1.1.1.1.1.1.1.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1"><plus id="A2.E36.m1.6.6.1.1.1.1.1.1.1.2.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.2"></plus><apply id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3"><times id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.1.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.1"></times><apply id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.2.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.2.1.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.2">subscript</csymbol><ci id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.2.2.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.2.2">Φ</ci><list id="A2.E36.m1.2.2.2.3.cmml" xref="A2.E36.m1.2.2.2.4"><ci id="A2.E36.m1.1.1.1.1.cmml" xref="A2.E36.m1.1.1.1.1">f</ci><ci id="A2.E36.m1.2.2.2.2.cmml" xref="A2.E36.m1.2.2.2.2">𝜁</ci></list></apply><apply id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.3.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.3.1.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.3">superscript</csymbol><apply id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.3.2.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.3.2.1.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.3">subscript</csymbol><ci id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.3.2.2.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.3.2.2">Φ</ci><list id="A2.E36.m1.4.4.2.3.cmml" xref="A2.E36.m1.4.4.2.4"><ci id="A2.E36.m1.3.3.1.1.cmml" xref="A2.E36.m1.3.3.1.1">f</ci><ci id="A2.E36.m1.4.4.2.2.cmml" xref="A2.E36.m1.4.4.2.2">𝜁</ci></list></apply><ci id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.3.3.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.3.3">𝑇</ci></apply><apply id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.4.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.4"><csymbol cd="ambiguous" id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.4.1.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.4">subscript</csymbol><apply id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.4.2.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.4.2"><ci id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.4.2.1.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.4.2.1">~</ci><ci id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.4.2.2.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.4.2.2">𝜽</ci></apply><ci id="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.4.3.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.3.4.3">𝜁</ci></apply></apply><apply id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1"><times id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.2.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.2"></times><ci id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.3.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.3">𝜅</ci><apply id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.4.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.4"><csymbol cd="ambiguous" id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.4.1.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.4">subscript</csymbol><ci id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.4.2.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.4.2">𝑄</ci><ci id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.4.3.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.4.3">𝜁</ci></apply><interval closure="open" id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.1.2.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.1.1"><ci id="A2.E36.m1.5.5.cmml" xref="A2.E36.m1.5.5">𝑡</ci><apply id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.2">𝑡</ci><ci id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.3">e</ci></apply></interval><apply id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.5.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.5"><csymbol cd="ambiguous" id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.5.1.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.5">subscript</csymbol><apply id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.5.2.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.5.2"><ci id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.5.2.1.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.5.2.1">~</ci><ci id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.5.2.2.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.5.2.2">𝜽</ci></apply><ci id="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.5.3.cmml" xref="A2.E36.m1.6.6.1.1.1.1.1.1.1.1.5.3">𝜁</ci></apply></apply></apply></apply></apply></apply><apply id="A2.E36.m1.7.7.2.2.cmml" xref="A2.E36.m1.7.7.2.2"><geq id="A2.E36.m1.7.7.2.2.1.cmml" xref="A2.E36.m1.7.7.2.2.1"></geq><ci id="A2.E36.m1.7.7.2.2.2.cmml" xref="A2.E36.m1.7.7.2.2.2">𝑡</ci><apply id="A2.E36.m1.7.7.2.2.3.cmml" xref="A2.E36.m1.7.7.2.2.3"><csymbol cd="ambiguous" id="A2.E36.m1.7.7.2.2.3.1.cmml" xref="A2.E36.m1.7.7.2.2.3">subscript</csymbol><ci id="A2.E36.m1.7.7.2.2.3.2.cmml" xref="A2.E36.m1.7.7.2.2.3.2">𝑇</ci><ci id="A2.E36.m1.7.7.2.2.3.3.cmml" xref="A2.E36.m1.7.7.2.2.3.3">a</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E36.m1.7c">\displaystyle\dot{\tilde{\bm{\theta}}}_{\zeta}=-\Gamma_{\zeta}\left(\Phi_{{\rm f% },\zeta}\Phi_{{\rm f},\zeta}^{T}\tilde{\bm{\theta}}_{\zeta}+\kappa Q_{\zeta}(t% ,t_{\rm e}){\tilde{\bm{\theta}}}_{\zeta}\right),t\geq T_{\rm a}</annotation><annotation encoding="application/x-llamapun" id="A2.E36.m1.7d">over˙ start_ARG over~ start_ARG bold_italic_θ end_ARG end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT = - roman_Γ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( roman_Φ start_POSTSUBSCRIPT roman_f , italic_ζ end_POSTSUBSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f , italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT + italic_κ italic_Q start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ) , italic_t ≥ italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(36)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.2.p2.5">with <math alttext="Q_{\zeta}(t,t_{\rm e})" class="ltx_Math" display="inline" id="A2.2.p2.2.m1.2"><semantics id="A2.2.p2.2.m1.2a"><mrow id="A2.2.p2.2.m1.2.2" xref="A2.2.p2.2.m1.2.2.cmml"><msub id="A2.2.p2.2.m1.2.2.3" xref="A2.2.p2.2.m1.2.2.3.cmml"><mi id="A2.2.p2.2.m1.2.2.3.2" xref="A2.2.p2.2.m1.2.2.3.2.cmml">Q</mi><mi id="A2.2.p2.2.m1.2.2.3.3" xref="A2.2.p2.2.m1.2.2.3.3.cmml">ζ</mi></msub><mo id="A2.2.p2.2.m1.2.2.2" xref="A2.2.p2.2.m1.2.2.2.cmml">⁢</mo><mrow id="A2.2.p2.2.m1.2.2.1.1" xref="A2.2.p2.2.m1.2.2.1.2.cmml"><mo id="A2.2.p2.2.m1.2.2.1.1.2" stretchy="false" xref="A2.2.p2.2.m1.2.2.1.2.cmml">(</mo><mi id="A2.2.p2.2.m1.1.1" xref="A2.2.p2.2.m1.1.1.cmml">t</mi><mo id="A2.2.p2.2.m1.2.2.1.1.3" xref="A2.2.p2.2.m1.2.2.1.2.cmml">,</mo><msub id="A2.2.p2.2.m1.2.2.1.1.1" xref="A2.2.p2.2.m1.2.2.1.1.1.cmml"><mi id="A2.2.p2.2.m1.2.2.1.1.1.2" xref="A2.2.p2.2.m1.2.2.1.1.1.2.cmml">t</mi><mi id="A2.2.p2.2.m1.2.2.1.1.1.3" mathvariant="normal" xref="A2.2.p2.2.m1.2.2.1.1.1.3.cmml">e</mi></msub><mo id="A2.2.p2.2.m1.2.2.1.1.4" stretchy="false" xref="A2.2.p2.2.m1.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.2.m1.2b"><apply id="A2.2.p2.2.m1.2.2.cmml" xref="A2.2.p2.2.m1.2.2"><times id="A2.2.p2.2.m1.2.2.2.cmml" xref="A2.2.p2.2.m1.2.2.2"></times><apply id="A2.2.p2.2.m1.2.2.3.cmml" xref="A2.2.p2.2.m1.2.2.3"><csymbol cd="ambiguous" id="A2.2.p2.2.m1.2.2.3.1.cmml" xref="A2.2.p2.2.m1.2.2.3">subscript</csymbol><ci id="A2.2.p2.2.m1.2.2.3.2.cmml" xref="A2.2.p2.2.m1.2.2.3.2">𝑄</ci><ci id="A2.2.p2.2.m1.2.2.3.3.cmml" xref="A2.2.p2.2.m1.2.2.3.3">𝜁</ci></apply><interval closure="open" id="A2.2.p2.2.m1.2.2.1.2.cmml" xref="A2.2.p2.2.m1.2.2.1.1"><ci id="A2.2.p2.2.m1.1.1.cmml" xref="A2.2.p2.2.m1.1.1">𝑡</ci><apply id="A2.2.p2.2.m1.2.2.1.1.1.cmml" xref="A2.2.p2.2.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="A2.2.p2.2.m1.2.2.1.1.1.1.cmml" xref="A2.2.p2.2.m1.2.2.1.1.1">subscript</csymbol><ci id="A2.2.p2.2.m1.2.2.1.1.1.2.cmml" xref="A2.2.p2.2.m1.2.2.1.1.1.2">𝑡</ci><ci id="A2.2.p2.2.m1.2.2.1.1.1.3.cmml" xref="A2.2.p2.2.m1.2.2.1.1.1.3">e</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.2.m1.2c">Q_{\zeta}(t,t_{\rm e})</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.2.m1.2d">italic_Q start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT )</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="A2.2.p2.3.m2.1"><semantics id="A2.2.p2.3.m2.1a"><mo id="A2.2.p2.3.m2.1.1" xref="A2.2.p2.3.m2.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="A2.2.p2.3.m2.1b"><csymbol cd="latexml" id="A2.2.p2.3.m2.1.1.cmml" xref="A2.2.p2.3.m2.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.3.m2.1c">:=</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.3.m2.1d">:=</annotation></semantics></math> <math alttext="H(s)[\Psi_{\zeta}(t_{\rm e})]" class="ltx_Math" display="inline" id="A2.2.p2.4.m3.2"><semantics id="A2.2.p2.4.m3.2a"><mrow id="A2.2.p2.4.m3.2.2" xref="A2.2.p2.4.m3.2.2.cmml"><mi id="A2.2.p2.4.m3.2.2.3" xref="A2.2.p2.4.m3.2.2.3.cmml">H</mi><mo id="A2.2.p2.4.m3.2.2.2" xref="A2.2.p2.4.m3.2.2.2.cmml">⁢</mo><mrow id="A2.2.p2.4.m3.2.2.4.2" xref="A2.2.p2.4.m3.2.2.cmml"><mo id="A2.2.p2.4.m3.2.2.4.2.1" stretchy="false" xref="A2.2.p2.4.m3.2.2.cmml">(</mo><mi id="A2.2.p2.4.m3.1.1" xref="A2.2.p2.4.m3.1.1.cmml">s</mi><mo id="A2.2.p2.4.m3.2.2.4.2.2" stretchy="false" xref="A2.2.p2.4.m3.2.2.cmml">)</mo></mrow><mo id="A2.2.p2.4.m3.2.2.2a" xref="A2.2.p2.4.m3.2.2.2.cmml">⁢</mo><mrow id="A2.2.p2.4.m3.2.2.1.1" xref="A2.2.p2.4.m3.2.2.1.2.cmml"><mo id="A2.2.p2.4.m3.2.2.1.1.2" stretchy="false" xref="A2.2.p2.4.m3.2.2.1.2.1.cmml">[</mo><mrow id="A2.2.p2.4.m3.2.2.1.1.1" xref="A2.2.p2.4.m3.2.2.1.1.1.cmml"><msub id="A2.2.p2.4.m3.2.2.1.1.1.3" xref="A2.2.p2.4.m3.2.2.1.1.1.3.cmml"><mi id="A2.2.p2.4.m3.2.2.1.1.1.3.2" mathvariant="normal" xref="A2.2.p2.4.m3.2.2.1.1.1.3.2.cmml">Ψ</mi><mi id="A2.2.p2.4.m3.2.2.1.1.1.3.3" xref="A2.2.p2.4.m3.2.2.1.1.1.3.3.cmml">ζ</mi></msub><mo id="A2.2.p2.4.m3.2.2.1.1.1.2" xref="A2.2.p2.4.m3.2.2.1.1.1.2.cmml">⁢</mo><mrow id="A2.2.p2.4.m3.2.2.1.1.1.1.1" xref="A2.2.p2.4.m3.2.2.1.1.1.1.1.1.cmml"><mo id="A2.2.p2.4.m3.2.2.1.1.1.1.1.2" stretchy="false" xref="A2.2.p2.4.m3.2.2.1.1.1.1.1.1.cmml">(</mo><msub id="A2.2.p2.4.m3.2.2.1.1.1.1.1.1" xref="A2.2.p2.4.m3.2.2.1.1.1.1.1.1.cmml"><mi id="A2.2.p2.4.m3.2.2.1.1.1.1.1.1.2" xref="A2.2.p2.4.m3.2.2.1.1.1.1.1.1.2.cmml">t</mi><mi id="A2.2.p2.4.m3.2.2.1.1.1.1.1.1.3" mathvariant="normal" xref="A2.2.p2.4.m3.2.2.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="A2.2.p2.4.m3.2.2.1.1.1.1.1.3" stretchy="false" xref="A2.2.p2.4.m3.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A2.2.p2.4.m3.2.2.1.1.3" stretchy="false" xref="A2.2.p2.4.m3.2.2.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.4.m3.2b"><apply id="A2.2.p2.4.m3.2.2.cmml" xref="A2.2.p2.4.m3.2.2"><times id="A2.2.p2.4.m3.2.2.2.cmml" xref="A2.2.p2.4.m3.2.2.2"></times><ci id="A2.2.p2.4.m3.2.2.3.cmml" xref="A2.2.p2.4.m3.2.2.3">𝐻</ci><ci id="A2.2.p2.4.m3.1.1.cmml" xref="A2.2.p2.4.m3.1.1">𝑠</ci><apply id="A2.2.p2.4.m3.2.2.1.2.cmml" xref="A2.2.p2.4.m3.2.2.1.1"><csymbol cd="latexml" id="A2.2.p2.4.m3.2.2.1.2.1.cmml" xref="A2.2.p2.4.m3.2.2.1.1.2">delimited-[]</csymbol><apply id="A2.2.p2.4.m3.2.2.1.1.1.cmml" xref="A2.2.p2.4.m3.2.2.1.1.1"><times id="A2.2.p2.4.m3.2.2.1.1.1.2.cmml" xref="A2.2.p2.4.m3.2.2.1.1.1.2"></times><apply id="A2.2.p2.4.m3.2.2.1.1.1.3.cmml" xref="A2.2.p2.4.m3.2.2.1.1.1.3"><csymbol cd="ambiguous" id="A2.2.p2.4.m3.2.2.1.1.1.3.1.cmml" xref="A2.2.p2.4.m3.2.2.1.1.1.3">subscript</csymbol><ci id="A2.2.p2.4.m3.2.2.1.1.1.3.2.cmml" xref="A2.2.p2.4.m3.2.2.1.1.1.3.2">Ψ</ci><ci id="A2.2.p2.4.m3.2.2.1.1.1.3.3.cmml" xref="A2.2.p2.4.m3.2.2.1.1.1.3.3">𝜁</ci></apply><apply id="A2.2.p2.4.m3.2.2.1.1.1.1.1.1.cmml" xref="A2.2.p2.4.m3.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.2.p2.4.m3.2.2.1.1.1.1.1.1.1.cmml" xref="A2.2.p2.4.m3.2.2.1.1.1.1.1">subscript</csymbol><ci id="A2.2.p2.4.m3.2.2.1.1.1.1.1.1.2.cmml" xref="A2.2.p2.4.m3.2.2.1.1.1.1.1.1.2">𝑡</ci><ci id="A2.2.p2.4.m3.2.2.1.1.1.1.1.1.3.cmml" xref="A2.2.p2.4.m3.2.2.1.1.1.1.1.1.3">e</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.4.m3.2c">H(s)[\Psi_{\zeta}(t_{\rm e})]</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.4.m3.2d">italic_H ( italic_s ) [ roman_Ψ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) ]</annotation></semantics></math>, in which <math alttext="\Gamma_{\zeta}\in\mathbb{R}^{N_{\zeta}\times N_{\zeta}}" class="ltx_Math" display="inline" id="A2.2.p2.5.m4.1"><semantics id="A2.2.p2.5.m4.1a"><mrow id="A2.2.p2.5.m4.1.1" xref="A2.2.p2.5.m4.1.1.cmml"><msub id="A2.2.p2.5.m4.1.1.2" xref="A2.2.p2.5.m4.1.1.2.cmml"><mi id="A2.2.p2.5.m4.1.1.2.2" mathvariant="normal" xref="A2.2.p2.5.m4.1.1.2.2.cmml">Γ</mi><mi id="A2.2.p2.5.m4.1.1.2.3" xref="A2.2.p2.5.m4.1.1.2.3.cmml">ζ</mi></msub><mo id="A2.2.p2.5.m4.1.1.1" xref="A2.2.p2.5.m4.1.1.1.cmml">∈</mo><msup id="A2.2.p2.5.m4.1.1.3" xref="A2.2.p2.5.m4.1.1.3.cmml"><mi id="A2.2.p2.5.m4.1.1.3.2" xref="A2.2.p2.5.m4.1.1.3.2.cmml">ℝ</mi><mrow id="A2.2.p2.5.m4.1.1.3.3" xref="A2.2.p2.5.m4.1.1.3.3.cmml"><msub id="A2.2.p2.5.m4.1.1.3.3.2" xref="A2.2.p2.5.m4.1.1.3.3.2.cmml"><mi id="A2.2.p2.5.m4.1.1.3.3.2.2" xref="A2.2.p2.5.m4.1.1.3.3.2.2.cmml">N</mi><mi id="A2.2.p2.5.m4.1.1.3.3.2.3" xref="A2.2.p2.5.m4.1.1.3.3.2.3.cmml">ζ</mi></msub><mo id="A2.2.p2.5.m4.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="A2.2.p2.5.m4.1.1.3.3.1.cmml">×</mo><msub id="A2.2.p2.5.m4.1.1.3.3.3" xref="A2.2.p2.5.m4.1.1.3.3.3.cmml"><mi id="A2.2.p2.5.m4.1.1.3.3.3.2" xref="A2.2.p2.5.m4.1.1.3.3.3.2.cmml">N</mi><mi id="A2.2.p2.5.m4.1.1.3.3.3.3" xref="A2.2.p2.5.m4.1.1.3.3.3.3.cmml">ζ</mi></msub></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.5.m4.1b"><apply id="A2.2.p2.5.m4.1.1.cmml" xref="A2.2.p2.5.m4.1.1"><in id="A2.2.p2.5.m4.1.1.1.cmml" xref="A2.2.p2.5.m4.1.1.1"></in><apply id="A2.2.p2.5.m4.1.1.2.cmml" xref="A2.2.p2.5.m4.1.1.2"><csymbol cd="ambiguous" id="A2.2.p2.5.m4.1.1.2.1.cmml" xref="A2.2.p2.5.m4.1.1.2">subscript</csymbol><ci id="A2.2.p2.5.m4.1.1.2.2.cmml" xref="A2.2.p2.5.m4.1.1.2.2">Γ</ci><ci id="A2.2.p2.5.m4.1.1.2.3.cmml" xref="A2.2.p2.5.m4.1.1.2.3">𝜁</ci></apply><apply id="A2.2.p2.5.m4.1.1.3.cmml" xref="A2.2.p2.5.m4.1.1.3"><csymbol cd="ambiguous" id="A2.2.p2.5.m4.1.1.3.1.cmml" xref="A2.2.p2.5.m4.1.1.3">superscript</csymbol><ci id="A2.2.p2.5.m4.1.1.3.2.cmml" xref="A2.2.p2.5.m4.1.1.3.2">ℝ</ci><apply id="A2.2.p2.5.m4.1.1.3.3.cmml" xref="A2.2.p2.5.m4.1.1.3.3"><times id="A2.2.p2.5.m4.1.1.3.3.1.cmml" xref="A2.2.p2.5.m4.1.1.3.3.1"></times><apply id="A2.2.p2.5.m4.1.1.3.3.2.cmml" xref="A2.2.p2.5.m4.1.1.3.3.2"><csymbol cd="ambiguous" id="A2.2.p2.5.m4.1.1.3.3.2.1.cmml" xref="A2.2.p2.5.m4.1.1.3.3.2">subscript</csymbol><ci id="A2.2.p2.5.m4.1.1.3.3.2.2.cmml" xref="A2.2.p2.5.m4.1.1.3.3.2.2">𝑁</ci><ci id="A2.2.p2.5.m4.1.1.3.3.2.3.cmml" xref="A2.2.p2.5.m4.1.1.3.3.2.3">𝜁</ci></apply><apply id="A2.2.p2.5.m4.1.1.3.3.3.cmml" xref="A2.2.p2.5.m4.1.1.3.3.3"><csymbol cd="ambiguous" id="A2.2.p2.5.m4.1.1.3.3.3.1.cmml" xref="A2.2.p2.5.m4.1.1.3.3.3">subscript</csymbol><ci id="A2.2.p2.5.m4.1.1.3.3.3.2.cmml" xref="A2.2.p2.5.m4.1.1.3.3.3.2">𝑁</ci><ci id="A2.2.p2.5.m4.1.1.3.3.3.3.cmml" xref="A2.2.p2.5.m4.1.1.3.3.3.3">𝜁</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.5.m4.1c">\Gamma_{\zeta}\in\mathbb{R}^{N_{\zeta}\times N_{\zeta}}</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.5.m4.1d">roman_Γ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT × italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> is a positive-definite diagonal matrix of learning rates with respect to active channels. Let a Lyapunov function candidate be</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx42"> <tbody id="A2.E37"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle V_{\theta,\zeta}=\tilde{\bm{\theta}}_{\zeta}^{T}\Gamma^{-1}_{% \zeta}\tilde{\bm{\theta}}_{\zeta}." class="ltx_Math" display="inline" id="A2.E37.m1.3"><semantics id="A2.E37.m1.3a"><mrow id="A2.E37.m1.3.3.1" xref="A2.E37.m1.3.3.1.1.cmml"><mrow id="A2.E37.m1.3.3.1.1" xref="A2.E37.m1.3.3.1.1.cmml"><msub id="A2.E37.m1.3.3.1.1.2" xref="A2.E37.m1.3.3.1.1.2.cmml"><mi id="A2.E37.m1.3.3.1.1.2.2" xref="A2.E37.m1.3.3.1.1.2.2.cmml">V</mi><mrow id="A2.E37.m1.2.2.2.4" xref="A2.E37.m1.2.2.2.3.cmml"><mi id="A2.E37.m1.1.1.1.1" xref="A2.E37.m1.1.1.1.1.cmml">θ</mi><mo id="A2.E37.m1.2.2.2.4.1" xref="A2.E37.m1.2.2.2.3.cmml">,</mo><mi id="A2.E37.m1.2.2.2.2" xref="A2.E37.m1.2.2.2.2.cmml">ζ</mi></mrow></msub><mo id="A2.E37.m1.3.3.1.1.1" xref="A2.E37.m1.3.3.1.1.1.cmml">=</mo><mrow id="A2.E37.m1.3.3.1.1.3" xref="A2.E37.m1.3.3.1.1.3.cmml"><msubsup id="A2.E37.m1.3.3.1.1.3.2" xref="A2.E37.m1.3.3.1.1.3.2.cmml"><mover accent="true" id="A2.E37.m1.3.3.1.1.3.2.2.2" xref="A2.E37.m1.3.3.1.1.3.2.2.2.cmml"><mi id="A2.E37.m1.3.3.1.1.3.2.2.2.2" xref="A2.E37.m1.3.3.1.1.3.2.2.2.2.cmml">𝜽</mi><mo id="A2.E37.m1.3.3.1.1.3.2.2.2.1" xref="A2.E37.m1.3.3.1.1.3.2.2.2.1.cmml">~</mo></mover><mi id="A2.E37.m1.3.3.1.1.3.2.2.3" xref="A2.E37.m1.3.3.1.1.3.2.2.3.cmml">ζ</mi><mi id="A2.E37.m1.3.3.1.1.3.2.3" xref="A2.E37.m1.3.3.1.1.3.2.3.cmml">T</mi></msubsup><mo id="A2.E37.m1.3.3.1.1.3.1" xref="A2.E37.m1.3.3.1.1.3.1.cmml">⁢</mo><msubsup id="A2.E37.m1.3.3.1.1.3.3" xref="A2.E37.m1.3.3.1.1.3.3.cmml"><mi id="A2.E37.m1.3.3.1.1.3.3.2.2" mathvariant="normal" xref="A2.E37.m1.3.3.1.1.3.3.2.2.cmml">Γ</mi><mi id="A2.E37.m1.3.3.1.1.3.3.3" xref="A2.E37.m1.3.3.1.1.3.3.3.cmml">ζ</mi><mrow id="A2.E37.m1.3.3.1.1.3.3.2.3" xref="A2.E37.m1.3.3.1.1.3.3.2.3.cmml"><mo id="A2.E37.m1.3.3.1.1.3.3.2.3a" xref="A2.E37.m1.3.3.1.1.3.3.2.3.cmml">−</mo><mn id="A2.E37.m1.3.3.1.1.3.3.2.3.2" xref="A2.E37.m1.3.3.1.1.3.3.2.3.2.cmml">1</mn></mrow></msubsup><mo id="A2.E37.m1.3.3.1.1.3.1a" xref="A2.E37.m1.3.3.1.1.3.1.cmml">⁢</mo><msub id="A2.E37.m1.3.3.1.1.3.4" xref="A2.E37.m1.3.3.1.1.3.4.cmml"><mover accent="true" id="A2.E37.m1.3.3.1.1.3.4.2" xref="A2.E37.m1.3.3.1.1.3.4.2.cmml"><mi id="A2.E37.m1.3.3.1.1.3.4.2.2" xref="A2.E37.m1.3.3.1.1.3.4.2.2.cmml">𝜽</mi><mo id="A2.E37.m1.3.3.1.1.3.4.2.1" xref="A2.E37.m1.3.3.1.1.3.4.2.1.cmml">~</mo></mover><mi id="A2.E37.m1.3.3.1.1.3.4.3" xref="A2.E37.m1.3.3.1.1.3.4.3.cmml">ζ</mi></msub></mrow></mrow><mo id="A2.E37.m1.3.3.1.2" lspace="0em" xref="A2.E37.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.E37.m1.3b"><apply id="A2.E37.m1.3.3.1.1.cmml" xref="A2.E37.m1.3.3.1"><eq id="A2.E37.m1.3.3.1.1.1.cmml" xref="A2.E37.m1.3.3.1.1.1"></eq><apply id="A2.E37.m1.3.3.1.1.2.cmml" xref="A2.E37.m1.3.3.1.1.2"><csymbol cd="ambiguous" id="A2.E37.m1.3.3.1.1.2.1.cmml" xref="A2.E37.m1.3.3.1.1.2">subscript</csymbol><ci id="A2.E37.m1.3.3.1.1.2.2.cmml" xref="A2.E37.m1.3.3.1.1.2.2">𝑉</ci><list id="A2.E37.m1.2.2.2.3.cmml" xref="A2.E37.m1.2.2.2.4"><ci id="A2.E37.m1.1.1.1.1.cmml" xref="A2.E37.m1.1.1.1.1">𝜃</ci><ci id="A2.E37.m1.2.2.2.2.cmml" xref="A2.E37.m1.2.2.2.2">𝜁</ci></list></apply><apply id="A2.E37.m1.3.3.1.1.3.cmml" xref="A2.E37.m1.3.3.1.1.3"><times id="A2.E37.m1.3.3.1.1.3.1.cmml" xref="A2.E37.m1.3.3.1.1.3.1"></times><apply id="A2.E37.m1.3.3.1.1.3.2.cmml" xref="A2.E37.m1.3.3.1.1.3.2"><csymbol cd="ambiguous" id="A2.E37.m1.3.3.1.1.3.2.1.cmml" xref="A2.E37.m1.3.3.1.1.3.2">superscript</csymbol><apply id="A2.E37.m1.3.3.1.1.3.2.2.cmml" xref="A2.E37.m1.3.3.1.1.3.2"><csymbol cd="ambiguous" id="A2.E37.m1.3.3.1.1.3.2.2.1.cmml" xref="A2.E37.m1.3.3.1.1.3.2">subscript</csymbol><apply id="A2.E37.m1.3.3.1.1.3.2.2.2.cmml" xref="A2.E37.m1.3.3.1.1.3.2.2.2"><ci id="A2.E37.m1.3.3.1.1.3.2.2.2.1.cmml" xref="A2.E37.m1.3.3.1.1.3.2.2.2.1">~</ci><ci id="A2.E37.m1.3.3.1.1.3.2.2.2.2.cmml" xref="A2.E37.m1.3.3.1.1.3.2.2.2.2">𝜽</ci></apply><ci id="A2.E37.m1.3.3.1.1.3.2.2.3.cmml" xref="A2.E37.m1.3.3.1.1.3.2.2.3">𝜁</ci></apply><ci id="A2.E37.m1.3.3.1.1.3.2.3.cmml" xref="A2.E37.m1.3.3.1.1.3.2.3">𝑇</ci></apply><apply id="A2.E37.m1.3.3.1.1.3.3.cmml" xref="A2.E37.m1.3.3.1.1.3.3"><csymbol cd="ambiguous" id="A2.E37.m1.3.3.1.1.3.3.1.cmml" xref="A2.E37.m1.3.3.1.1.3.3">subscript</csymbol><apply id="A2.E37.m1.3.3.1.1.3.3.2.cmml" xref="A2.E37.m1.3.3.1.1.3.3"><csymbol cd="ambiguous" id="A2.E37.m1.3.3.1.1.3.3.2.1.cmml" xref="A2.E37.m1.3.3.1.1.3.3">superscript</csymbol><ci id="A2.E37.m1.3.3.1.1.3.3.2.2.cmml" xref="A2.E37.m1.3.3.1.1.3.3.2.2">Γ</ci><apply id="A2.E37.m1.3.3.1.1.3.3.2.3.cmml" xref="A2.E37.m1.3.3.1.1.3.3.2.3"><minus id="A2.E37.m1.3.3.1.1.3.3.2.3.1.cmml" xref="A2.E37.m1.3.3.1.1.3.3.2.3"></minus><cn id="A2.E37.m1.3.3.1.1.3.3.2.3.2.cmml" type="integer" xref="A2.E37.m1.3.3.1.1.3.3.2.3.2">1</cn></apply></apply><ci id="A2.E37.m1.3.3.1.1.3.3.3.cmml" xref="A2.E37.m1.3.3.1.1.3.3.3">𝜁</ci></apply><apply id="A2.E37.m1.3.3.1.1.3.4.cmml" xref="A2.E37.m1.3.3.1.1.3.4"><csymbol cd="ambiguous" id="A2.E37.m1.3.3.1.1.3.4.1.cmml" xref="A2.E37.m1.3.3.1.1.3.4">subscript</csymbol><apply id="A2.E37.m1.3.3.1.1.3.4.2.cmml" xref="A2.E37.m1.3.3.1.1.3.4.2"><ci id="A2.E37.m1.3.3.1.1.3.4.2.1.cmml" xref="A2.E37.m1.3.3.1.1.3.4.2.1">~</ci><ci id="A2.E37.m1.3.3.1.1.3.4.2.2.cmml" xref="A2.E37.m1.3.3.1.1.3.4.2.2">𝜽</ci></apply><ci id="A2.E37.m1.3.3.1.1.3.4.3.cmml" xref="A2.E37.m1.3.3.1.1.3.4.3">𝜁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E37.m1.3c">\displaystyle V_{\theta,\zeta}=\tilde{\bm{\theta}}_{\zeta}^{T}\Gamma^{-1}_{% \zeta}\tilde{\bm{\theta}}_{\zeta}.</annotation><annotation encoding="application/x-llamapun" id="A2.E37.m1.3d">italic_V start_POSTSUBSCRIPT italic_θ , italic_ζ end_POSTSUBSCRIPT = over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Γ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(37)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.2.p2.7">Differentiating <math alttext="V_{\theta,\zeta}" class="ltx_Math" display="inline" id="A2.2.p2.6.m1.2"><semantics id="A2.2.p2.6.m1.2a"><msub id="A2.2.p2.6.m1.2.3" xref="A2.2.p2.6.m1.2.3.cmml"><mi id="A2.2.p2.6.m1.2.3.2" xref="A2.2.p2.6.m1.2.3.2.cmml">V</mi><mrow id="A2.2.p2.6.m1.2.2.2.4" xref="A2.2.p2.6.m1.2.2.2.3.cmml"><mi id="A2.2.p2.6.m1.1.1.1.1" xref="A2.2.p2.6.m1.1.1.1.1.cmml">θ</mi><mo id="A2.2.p2.6.m1.2.2.2.4.1" xref="A2.2.p2.6.m1.2.2.2.3.cmml">,</mo><mi id="A2.2.p2.6.m1.2.2.2.2" xref="A2.2.p2.6.m1.2.2.2.2.cmml">ζ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A2.2.p2.6.m1.2b"><apply id="A2.2.p2.6.m1.2.3.cmml" xref="A2.2.p2.6.m1.2.3"><csymbol cd="ambiguous" id="A2.2.p2.6.m1.2.3.1.cmml" xref="A2.2.p2.6.m1.2.3">subscript</csymbol><ci id="A2.2.p2.6.m1.2.3.2.cmml" xref="A2.2.p2.6.m1.2.3.2">𝑉</ci><list id="A2.2.p2.6.m1.2.2.2.3.cmml" xref="A2.2.p2.6.m1.2.2.2.4"><ci id="A2.2.p2.6.m1.1.1.1.1.cmml" xref="A2.2.p2.6.m1.1.1.1.1">𝜃</ci><ci id="A2.2.p2.6.m1.2.2.2.2.cmml" xref="A2.2.p2.6.m1.2.2.2.2">𝜁</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.6.m1.2c">V_{\theta,\zeta}</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.6.m1.2d">italic_V start_POSTSUBSCRIPT italic_θ , italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math> with respect to <math alttext="t" class="ltx_Math" display="inline" id="A2.2.p2.7.m2.1"><semantics id="A2.2.p2.7.m2.1a"><mi id="A2.2.p2.7.m2.1.1" xref="A2.2.p2.7.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A2.2.p2.7.m2.1b"><ci id="A2.2.p2.7.m2.1.1.cmml" xref="A2.2.p2.7.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.7.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.7.m2.1d">italic_t</annotation></semantics></math> and applying (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A2.E36" title="In Proof. ‣ Appendix B The proof of Theorem 1 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">36</span></a>) to the resulting expression, one obtains</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx43"> <tbody id="A2.Ex30"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle{\dot{V}}_{\theta,\zeta}\leq" class="ltx_Math" display="inline" id="A2.Ex30.m1.2"><semantics id="A2.Ex30.m1.2a"><mrow id="A2.Ex30.m1.2.3" xref="A2.Ex30.m1.2.3.cmml"><msub id="A2.Ex30.m1.2.3.2" xref="A2.Ex30.m1.2.3.2.cmml"><mover accent="true" id="A2.Ex30.m1.2.3.2.2" xref="A2.Ex30.m1.2.3.2.2.cmml"><mi id="A2.Ex30.m1.2.3.2.2.2" xref="A2.Ex30.m1.2.3.2.2.2.cmml">V</mi><mo id="A2.Ex30.m1.2.3.2.2.1" xref="A2.Ex30.m1.2.3.2.2.1.cmml">˙</mo></mover><mrow id="A2.Ex30.m1.2.2.2.4" xref="A2.Ex30.m1.2.2.2.3.cmml"><mi id="A2.Ex30.m1.1.1.1.1" xref="A2.Ex30.m1.1.1.1.1.cmml">θ</mi><mo id="A2.Ex30.m1.2.2.2.4.1" xref="A2.Ex30.m1.2.2.2.3.cmml">,</mo><mi id="A2.Ex30.m1.2.2.2.2" xref="A2.Ex30.m1.2.2.2.2.cmml">ζ</mi></mrow></msub><mo id="A2.Ex30.m1.2.3.1" xref="A2.Ex30.m1.2.3.1.cmml">≤</mo><mi id="A2.Ex30.m1.2.3.3" xref="A2.Ex30.m1.2.3.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex30.m1.2b"><apply id="A2.Ex30.m1.2.3.cmml" xref="A2.Ex30.m1.2.3"><leq id="A2.Ex30.m1.2.3.1.cmml" xref="A2.Ex30.m1.2.3.1"></leq><apply id="A2.Ex30.m1.2.3.2.cmml" xref="A2.Ex30.m1.2.3.2"><csymbol cd="ambiguous" id="A2.Ex30.m1.2.3.2.1.cmml" xref="A2.Ex30.m1.2.3.2">subscript</csymbol><apply id="A2.Ex30.m1.2.3.2.2.cmml" xref="A2.Ex30.m1.2.3.2.2"><ci id="A2.Ex30.m1.2.3.2.2.1.cmml" xref="A2.Ex30.m1.2.3.2.2.1">˙</ci><ci id="A2.Ex30.m1.2.3.2.2.2.cmml" xref="A2.Ex30.m1.2.3.2.2.2">𝑉</ci></apply><list id="A2.Ex30.m1.2.2.2.3.cmml" xref="A2.Ex30.m1.2.2.2.4"><ci id="A2.Ex30.m1.1.1.1.1.cmml" xref="A2.Ex30.m1.1.1.1.1">𝜃</ci><ci id="A2.Ex30.m1.2.2.2.2.cmml" xref="A2.Ex30.m1.2.2.2.2">𝜁</ci></list></apply><csymbol cd="latexml" id="A2.Ex30.m1.2.3.3.cmml" xref="A2.Ex30.m1.2.3.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex30.m1.2c">\displaystyle{\dot{V}}_{\theta,\zeta}\leq</annotation><annotation encoding="application/x-llamapun" id="A2.Ex30.m1.2d">over˙ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_θ , italic_ζ end_POSTSUBSCRIPT ≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-2\tilde{\bm{\theta}}_{\zeta}^{T}\Phi_{{\rm f},\zeta}\Phi_{{\rm f% },\zeta}^{T}\tilde{\bm{\theta}}_{\zeta}-2\kappa\tilde{\bm{\theta}}_{\zeta}^{T}% Q_{\zeta}(t,t_{\rm e}){\tilde{\bm{\theta}}}_{\zeta}" class="ltx_Math" display="inline" id="A2.Ex30.m2.6"><semantics id="A2.Ex30.m2.6a"><mrow id="A2.Ex30.m2.6.6" xref="A2.Ex30.m2.6.6.cmml"><mrow id="A2.Ex30.m2.6.6.3" xref="A2.Ex30.m2.6.6.3.cmml"><mo id="A2.Ex30.m2.6.6.3a" xref="A2.Ex30.m2.6.6.3.cmml">−</mo><mrow id="A2.Ex30.m2.6.6.3.2" xref="A2.Ex30.m2.6.6.3.2.cmml"><mn id="A2.Ex30.m2.6.6.3.2.2" xref="A2.Ex30.m2.6.6.3.2.2.cmml">2</mn><mo id="A2.Ex30.m2.6.6.3.2.1" xref="A2.Ex30.m2.6.6.3.2.1.cmml">⁢</mo><msubsup id="A2.Ex30.m2.6.6.3.2.3" xref="A2.Ex30.m2.6.6.3.2.3.cmml"><mover accent="true" id="A2.Ex30.m2.6.6.3.2.3.2.2" xref="A2.Ex30.m2.6.6.3.2.3.2.2.cmml"><mi id="A2.Ex30.m2.6.6.3.2.3.2.2.2" xref="A2.Ex30.m2.6.6.3.2.3.2.2.2.cmml">𝜽</mi><mo id="A2.Ex30.m2.6.6.3.2.3.2.2.1" xref="A2.Ex30.m2.6.6.3.2.3.2.2.1.cmml">~</mo></mover><mi id="A2.Ex30.m2.6.6.3.2.3.2.3" xref="A2.Ex30.m2.6.6.3.2.3.2.3.cmml">ζ</mi><mi id="A2.Ex30.m2.6.6.3.2.3.3" xref="A2.Ex30.m2.6.6.3.2.3.3.cmml">T</mi></msubsup><mo id="A2.Ex30.m2.6.6.3.2.1a" xref="A2.Ex30.m2.6.6.3.2.1.cmml">⁢</mo><msub id="A2.Ex30.m2.6.6.3.2.4" xref="A2.Ex30.m2.6.6.3.2.4.cmml"><mi id="A2.Ex30.m2.6.6.3.2.4.2" mathvariant="normal" xref="A2.Ex30.m2.6.6.3.2.4.2.cmml">Φ</mi><mrow id="A2.Ex30.m2.2.2.2.4" xref="A2.Ex30.m2.2.2.2.3.cmml"><mi id="A2.Ex30.m2.1.1.1.1" mathvariant="normal" xref="A2.Ex30.m2.1.1.1.1.cmml">f</mi><mo id="A2.Ex30.m2.2.2.2.4.1" xref="A2.Ex30.m2.2.2.2.3.cmml">,</mo><mi id="A2.Ex30.m2.2.2.2.2" xref="A2.Ex30.m2.2.2.2.2.cmml">ζ</mi></mrow></msub><mo id="A2.Ex30.m2.6.6.3.2.1b" xref="A2.Ex30.m2.6.6.3.2.1.cmml">⁢</mo><msubsup id="A2.Ex30.m2.6.6.3.2.5" xref="A2.Ex30.m2.6.6.3.2.5.cmml"><mi id="A2.Ex30.m2.6.6.3.2.5.2.2" mathvariant="normal" xref="A2.Ex30.m2.6.6.3.2.5.2.2.cmml">Φ</mi><mrow id="A2.Ex30.m2.4.4.2.4" xref="A2.Ex30.m2.4.4.2.3.cmml"><mi id="A2.Ex30.m2.3.3.1.1" mathvariant="normal" xref="A2.Ex30.m2.3.3.1.1.cmml">f</mi><mo id="A2.Ex30.m2.4.4.2.4.1" xref="A2.Ex30.m2.4.4.2.3.cmml">,</mo><mi id="A2.Ex30.m2.4.4.2.2" xref="A2.Ex30.m2.4.4.2.2.cmml">ζ</mi></mrow><mi id="A2.Ex30.m2.6.6.3.2.5.3" xref="A2.Ex30.m2.6.6.3.2.5.3.cmml">T</mi></msubsup><mo id="A2.Ex30.m2.6.6.3.2.1c" xref="A2.Ex30.m2.6.6.3.2.1.cmml">⁢</mo><msub id="A2.Ex30.m2.6.6.3.2.6" xref="A2.Ex30.m2.6.6.3.2.6.cmml"><mover accent="true" id="A2.Ex30.m2.6.6.3.2.6.2" xref="A2.Ex30.m2.6.6.3.2.6.2.cmml"><mi id="A2.Ex30.m2.6.6.3.2.6.2.2" xref="A2.Ex30.m2.6.6.3.2.6.2.2.cmml">𝜽</mi><mo id="A2.Ex30.m2.6.6.3.2.6.2.1" xref="A2.Ex30.m2.6.6.3.2.6.2.1.cmml">~</mo></mover><mi id="A2.Ex30.m2.6.6.3.2.6.3" xref="A2.Ex30.m2.6.6.3.2.6.3.cmml">ζ</mi></msub></mrow></mrow><mo id="A2.Ex30.m2.6.6.2" xref="A2.Ex30.m2.6.6.2.cmml">−</mo><mrow id="A2.Ex30.m2.6.6.1" xref="A2.Ex30.m2.6.6.1.cmml"><mn id="A2.Ex30.m2.6.6.1.3" xref="A2.Ex30.m2.6.6.1.3.cmml">2</mn><mo id="A2.Ex30.m2.6.6.1.2" xref="A2.Ex30.m2.6.6.1.2.cmml">⁢</mo><mi id="A2.Ex30.m2.6.6.1.4" xref="A2.Ex30.m2.6.6.1.4.cmml">κ</mi><mo id="A2.Ex30.m2.6.6.1.2a" xref="A2.Ex30.m2.6.6.1.2.cmml">⁢</mo><msubsup id="A2.Ex30.m2.6.6.1.5" xref="A2.Ex30.m2.6.6.1.5.cmml"><mover accent="true" id="A2.Ex30.m2.6.6.1.5.2.2" xref="A2.Ex30.m2.6.6.1.5.2.2.cmml"><mi id="A2.Ex30.m2.6.6.1.5.2.2.2" xref="A2.Ex30.m2.6.6.1.5.2.2.2.cmml">𝜽</mi><mo id="A2.Ex30.m2.6.6.1.5.2.2.1" xref="A2.Ex30.m2.6.6.1.5.2.2.1.cmml">~</mo></mover><mi id="A2.Ex30.m2.6.6.1.5.2.3" xref="A2.Ex30.m2.6.6.1.5.2.3.cmml">ζ</mi><mi id="A2.Ex30.m2.6.6.1.5.3" xref="A2.Ex30.m2.6.6.1.5.3.cmml">T</mi></msubsup><mo id="A2.Ex30.m2.6.6.1.2b" xref="A2.Ex30.m2.6.6.1.2.cmml">⁢</mo><msub id="A2.Ex30.m2.6.6.1.6" xref="A2.Ex30.m2.6.6.1.6.cmml"><mi id="A2.Ex30.m2.6.6.1.6.2" xref="A2.Ex30.m2.6.6.1.6.2.cmml">Q</mi><mi id="A2.Ex30.m2.6.6.1.6.3" xref="A2.Ex30.m2.6.6.1.6.3.cmml">ζ</mi></msub><mo id="A2.Ex30.m2.6.6.1.2c" xref="A2.Ex30.m2.6.6.1.2.cmml">⁢</mo><mrow id="A2.Ex30.m2.6.6.1.1.1" xref="A2.Ex30.m2.6.6.1.1.2.cmml"><mo id="A2.Ex30.m2.6.6.1.1.1.2" stretchy="false" xref="A2.Ex30.m2.6.6.1.1.2.cmml">(</mo><mi id="A2.Ex30.m2.5.5" xref="A2.Ex30.m2.5.5.cmml">t</mi><mo id="A2.Ex30.m2.6.6.1.1.1.3" xref="A2.Ex30.m2.6.6.1.1.2.cmml">,</mo><msub id="A2.Ex30.m2.6.6.1.1.1.1" xref="A2.Ex30.m2.6.6.1.1.1.1.cmml"><mi id="A2.Ex30.m2.6.6.1.1.1.1.2" xref="A2.Ex30.m2.6.6.1.1.1.1.2.cmml">t</mi><mi id="A2.Ex30.m2.6.6.1.1.1.1.3" mathvariant="normal" xref="A2.Ex30.m2.6.6.1.1.1.1.3.cmml">e</mi></msub><mo id="A2.Ex30.m2.6.6.1.1.1.4" stretchy="false" xref="A2.Ex30.m2.6.6.1.1.2.cmml">)</mo></mrow><mo id="A2.Ex30.m2.6.6.1.2d" xref="A2.Ex30.m2.6.6.1.2.cmml">⁢</mo><msub id="A2.Ex30.m2.6.6.1.7" xref="A2.Ex30.m2.6.6.1.7.cmml"><mover accent="true" id="A2.Ex30.m2.6.6.1.7.2" xref="A2.Ex30.m2.6.6.1.7.2.cmml"><mi id="A2.Ex30.m2.6.6.1.7.2.2" xref="A2.Ex30.m2.6.6.1.7.2.2.cmml">𝜽</mi><mo id="A2.Ex30.m2.6.6.1.7.2.1" xref="A2.Ex30.m2.6.6.1.7.2.1.cmml">~</mo></mover><mi id="A2.Ex30.m2.6.6.1.7.3" xref="A2.Ex30.m2.6.6.1.7.3.cmml">ζ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex30.m2.6b"><apply id="A2.Ex30.m2.6.6.cmml" xref="A2.Ex30.m2.6.6"><minus id="A2.Ex30.m2.6.6.2.cmml" xref="A2.Ex30.m2.6.6.2"></minus><apply id="A2.Ex30.m2.6.6.3.cmml" xref="A2.Ex30.m2.6.6.3"><minus id="A2.Ex30.m2.6.6.3.1.cmml" xref="A2.Ex30.m2.6.6.3"></minus><apply id="A2.Ex30.m2.6.6.3.2.cmml" xref="A2.Ex30.m2.6.6.3.2"><times id="A2.Ex30.m2.6.6.3.2.1.cmml" xref="A2.Ex30.m2.6.6.3.2.1"></times><cn id="A2.Ex30.m2.6.6.3.2.2.cmml" type="integer" xref="A2.Ex30.m2.6.6.3.2.2">2</cn><apply id="A2.Ex30.m2.6.6.3.2.3.cmml" xref="A2.Ex30.m2.6.6.3.2.3"><csymbol cd="ambiguous" id="A2.Ex30.m2.6.6.3.2.3.1.cmml" xref="A2.Ex30.m2.6.6.3.2.3">superscript</csymbol><apply id="A2.Ex30.m2.6.6.3.2.3.2.cmml" xref="A2.Ex30.m2.6.6.3.2.3"><csymbol cd="ambiguous" id="A2.Ex30.m2.6.6.3.2.3.2.1.cmml" xref="A2.Ex30.m2.6.6.3.2.3">subscript</csymbol><apply id="A2.Ex30.m2.6.6.3.2.3.2.2.cmml" xref="A2.Ex30.m2.6.6.3.2.3.2.2"><ci id="A2.Ex30.m2.6.6.3.2.3.2.2.1.cmml" xref="A2.Ex30.m2.6.6.3.2.3.2.2.1">~</ci><ci id="A2.Ex30.m2.6.6.3.2.3.2.2.2.cmml" xref="A2.Ex30.m2.6.6.3.2.3.2.2.2">𝜽</ci></apply><ci id="A2.Ex30.m2.6.6.3.2.3.2.3.cmml" xref="A2.Ex30.m2.6.6.3.2.3.2.3">𝜁</ci></apply><ci id="A2.Ex30.m2.6.6.3.2.3.3.cmml" xref="A2.Ex30.m2.6.6.3.2.3.3">𝑇</ci></apply><apply id="A2.Ex30.m2.6.6.3.2.4.cmml" xref="A2.Ex30.m2.6.6.3.2.4"><csymbol cd="ambiguous" id="A2.Ex30.m2.6.6.3.2.4.1.cmml" xref="A2.Ex30.m2.6.6.3.2.4">subscript</csymbol><ci id="A2.Ex30.m2.6.6.3.2.4.2.cmml" xref="A2.Ex30.m2.6.6.3.2.4.2">Φ</ci><list id="A2.Ex30.m2.2.2.2.3.cmml" xref="A2.Ex30.m2.2.2.2.4"><ci id="A2.Ex30.m2.1.1.1.1.cmml" xref="A2.Ex30.m2.1.1.1.1">f</ci><ci id="A2.Ex30.m2.2.2.2.2.cmml" xref="A2.Ex30.m2.2.2.2.2">𝜁</ci></list></apply><apply id="A2.Ex30.m2.6.6.3.2.5.cmml" xref="A2.Ex30.m2.6.6.3.2.5"><csymbol cd="ambiguous" id="A2.Ex30.m2.6.6.3.2.5.1.cmml" xref="A2.Ex30.m2.6.6.3.2.5">superscript</csymbol><apply id="A2.Ex30.m2.6.6.3.2.5.2.cmml" xref="A2.Ex30.m2.6.6.3.2.5"><csymbol cd="ambiguous" id="A2.Ex30.m2.6.6.3.2.5.2.1.cmml" xref="A2.Ex30.m2.6.6.3.2.5">subscript</csymbol><ci id="A2.Ex30.m2.6.6.3.2.5.2.2.cmml" xref="A2.Ex30.m2.6.6.3.2.5.2.2">Φ</ci><list id="A2.Ex30.m2.4.4.2.3.cmml" xref="A2.Ex30.m2.4.4.2.4"><ci id="A2.Ex30.m2.3.3.1.1.cmml" xref="A2.Ex30.m2.3.3.1.1">f</ci><ci id="A2.Ex30.m2.4.4.2.2.cmml" xref="A2.Ex30.m2.4.4.2.2">𝜁</ci></list></apply><ci id="A2.Ex30.m2.6.6.3.2.5.3.cmml" xref="A2.Ex30.m2.6.6.3.2.5.3">𝑇</ci></apply><apply id="A2.Ex30.m2.6.6.3.2.6.cmml" xref="A2.Ex30.m2.6.6.3.2.6"><csymbol cd="ambiguous" id="A2.Ex30.m2.6.6.3.2.6.1.cmml" xref="A2.Ex30.m2.6.6.3.2.6">subscript</csymbol><apply id="A2.Ex30.m2.6.6.3.2.6.2.cmml" xref="A2.Ex30.m2.6.6.3.2.6.2"><ci id="A2.Ex30.m2.6.6.3.2.6.2.1.cmml" xref="A2.Ex30.m2.6.6.3.2.6.2.1">~</ci><ci id="A2.Ex30.m2.6.6.3.2.6.2.2.cmml" xref="A2.Ex30.m2.6.6.3.2.6.2.2">𝜽</ci></apply><ci id="A2.Ex30.m2.6.6.3.2.6.3.cmml" xref="A2.Ex30.m2.6.6.3.2.6.3">𝜁</ci></apply></apply></apply><apply id="A2.Ex30.m2.6.6.1.cmml" xref="A2.Ex30.m2.6.6.1"><times id="A2.Ex30.m2.6.6.1.2.cmml" xref="A2.Ex30.m2.6.6.1.2"></times><cn id="A2.Ex30.m2.6.6.1.3.cmml" type="integer" xref="A2.Ex30.m2.6.6.1.3">2</cn><ci id="A2.Ex30.m2.6.6.1.4.cmml" xref="A2.Ex30.m2.6.6.1.4">𝜅</ci><apply id="A2.Ex30.m2.6.6.1.5.cmml" xref="A2.Ex30.m2.6.6.1.5"><csymbol cd="ambiguous" id="A2.Ex30.m2.6.6.1.5.1.cmml" xref="A2.Ex30.m2.6.6.1.5">superscript</csymbol><apply id="A2.Ex30.m2.6.6.1.5.2.cmml" xref="A2.Ex30.m2.6.6.1.5"><csymbol cd="ambiguous" id="A2.Ex30.m2.6.6.1.5.2.1.cmml" xref="A2.Ex30.m2.6.6.1.5">subscript</csymbol><apply id="A2.Ex30.m2.6.6.1.5.2.2.cmml" xref="A2.Ex30.m2.6.6.1.5.2.2"><ci id="A2.Ex30.m2.6.6.1.5.2.2.1.cmml" xref="A2.Ex30.m2.6.6.1.5.2.2.1">~</ci><ci id="A2.Ex30.m2.6.6.1.5.2.2.2.cmml" xref="A2.Ex30.m2.6.6.1.5.2.2.2">𝜽</ci></apply><ci id="A2.Ex30.m2.6.6.1.5.2.3.cmml" xref="A2.Ex30.m2.6.6.1.5.2.3">𝜁</ci></apply><ci id="A2.Ex30.m2.6.6.1.5.3.cmml" xref="A2.Ex30.m2.6.6.1.5.3">𝑇</ci></apply><apply id="A2.Ex30.m2.6.6.1.6.cmml" xref="A2.Ex30.m2.6.6.1.6"><csymbol cd="ambiguous" id="A2.Ex30.m2.6.6.1.6.1.cmml" xref="A2.Ex30.m2.6.6.1.6">subscript</csymbol><ci id="A2.Ex30.m2.6.6.1.6.2.cmml" xref="A2.Ex30.m2.6.6.1.6.2">𝑄</ci><ci id="A2.Ex30.m2.6.6.1.6.3.cmml" xref="A2.Ex30.m2.6.6.1.6.3">𝜁</ci></apply><interval closure="open" id="A2.Ex30.m2.6.6.1.1.2.cmml" xref="A2.Ex30.m2.6.6.1.1.1"><ci id="A2.Ex30.m2.5.5.cmml" xref="A2.Ex30.m2.5.5">𝑡</ci><apply id="A2.Ex30.m2.6.6.1.1.1.1.cmml" xref="A2.Ex30.m2.6.6.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex30.m2.6.6.1.1.1.1.1.cmml" xref="A2.Ex30.m2.6.6.1.1.1.1">subscript</csymbol><ci id="A2.Ex30.m2.6.6.1.1.1.1.2.cmml" xref="A2.Ex30.m2.6.6.1.1.1.1.2">𝑡</ci><ci id="A2.Ex30.m2.6.6.1.1.1.1.3.cmml" xref="A2.Ex30.m2.6.6.1.1.1.1.3">e</ci></apply></interval><apply id="A2.Ex30.m2.6.6.1.7.cmml" xref="A2.Ex30.m2.6.6.1.7"><csymbol cd="ambiguous" id="A2.Ex30.m2.6.6.1.7.1.cmml" xref="A2.Ex30.m2.6.6.1.7">subscript</csymbol><apply id="A2.Ex30.m2.6.6.1.7.2.cmml" xref="A2.Ex30.m2.6.6.1.7.2"><ci id="A2.Ex30.m2.6.6.1.7.2.1.cmml" xref="A2.Ex30.m2.6.6.1.7.2.1">~</ci><ci id="A2.Ex30.m2.6.6.1.7.2.2.cmml" xref="A2.Ex30.m2.6.6.1.7.2.2">𝜽</ci></apply><ci id="A2.Ex30.m2.6.6.1.7.3.cmml" xref="A2.Ex30.m2.6.6.1.7.3">𝜁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex30.m2.6c">\displaystyle-2\tilde{\bm{\theta}}_{\zeta}^{T}\Phi_{{\rm f},\zeta}\Phi_{{\rm f% },\zeta}^{T}\tilde{\bm{\theta}}_{\zeta}-2\kappa\tilde{\bm{\theta}}_{\zeta}^{T}% Q_{\zeta}(t,t_{\rm e}){\tilde{\bm{\theta}}}_{\zeta}</annotation><annotation encoding="application/x-llamapun" id="A2.Ex30.m2.6d">- 2 over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f , italic_ζ end_POSTSUBSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f , italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT - 2 italic_κ over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT italic_Q start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A2.Ex31"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\leq" class="ltx_Math" display="inline" id="A2.Ex31.m1.1"><semantics id="A2.Ex31.m1.1a"><mo id="A2.Ex31.m1.1.1" xref="A2.Ex31.m1.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="A2.Ex31.m1.1b"><leq id="A2.Ex31.m1.1.1.cmml" xref="A2.Ex31.m1.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex31.m1.1c">\displaystyle\leq</annotation><annotation encoding="application/x-llamapun" id="A2.Ex31.m1.1d">≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-2\kappa\tilde{\bm{\theta}}_{\zeta}^{T}Q_{\zeta}(t,t_{\rm e}){% \tilde{\bm{\theta}}}_{\zeta}." class="ltx_Math" display="inline" id="A2.Ex31.m2.2"><semantics id="A2.Ex31.m2.2a"><mrow id="A2.Ex31.m2.2.2.1" xref="A2.Ex31.m2.2.2.1.1.cmml"><mrow id="A2.Ex31.m2.2.2.1.1" xref="A2.Ex31.m2.2.2.1.1.cmml"><mo id="A2.Ex31.m2.2.2.1.1a" xref="A2.Ex31.m2.2.2.1.1.cmml">−</mo><mrow id="A2.Ex31.m2.2.2.1.1.1" xref="A2.Ex31.m2.2.2.1.1.1.cmml"><mn id="A2.Ex31.m2.2.2.1.1.1.3" xref="A2.Ex31.m2.2.2.1.1.1.3.cmml">2</mn><mo id="A2.Ex31.m2.2.2.1.1.1.2" xref="A2.Ex31.m2.2.2.1.1.1.2.cmml">⁢</mo><mi id="A2.Ex31.m2.2.2.1.1.1.4" xref="A2.Ex31.m2.2.2.1.1.1.4.cmml">κ</mi><mo id="A2.Ex31.m2.2.2.1.1.1.2a" xref="A2.Ex31.m2.2.2.1.1.1.2.cmml">⁢</mo><msubsup id="A2.Ex31.m2.2.2.1.1.1.5" xref="A2.Ex31.m2.2.2.1.1.1.5.cmml"><mover accent="true" id="A2.Ex31.m2.2.2.1.1.1.5.2.2" xref="A2.Ex31.m2.2.2.1.1.1.5.2.2.cmml"><mi id="A2.Ex31.m2.2.2.1.1.1.5.2.2.2" xref="A2.Ex31.m2.2.2.1.1.1.5.2.2.2.cmml">𝜽</mi><mo id="A2.Ex31.m2.2.2.1.1.1.5.2.2.1" xref="A2.Ex31.m2.2.2.1.1.1.5.2.2.1.cmml">~</mo></mover><mi id="A2.Ex31.m2.2.2.1.1.1.5.2.3" xref="A2.Ex31.m2.2.2.1.1.1.5.2.3.cmml">ζ</mi><mi id="A2.Ex31.m2.2.2.1.1.1.5.3" xref="A2.Ex31.m2.2.2.1.1.1.5.3.cmml">T</mi></msubsup><mo id="A2.Ex31.m2.2.2.1.1.1.2b" xref="A2.Ex31.m2.2.2.1.1.1.2.cmml">⁢</mo><msub id="A2.Ex31.m2.2.2.1.1.1.6" xref="A2.Ex31.m2.2.2.1.1.1.6.cmml"><mi id="A2.Ex31.m2.2.2.1.1.1.6.2" xref="A2.Ex31.m2.2.2.1.1.1.6.2.cmml">Q</mi><mi id="A2.Ex31.m2.2.2.1.1.1.6.3" xref="A2.Ex31.m2.2.2.1.1.1.6.3.cmml">ζ</mi></msub><mo id="A2.Ex31.m2.2.2.1.1.1.2c" xref="A2.Ex31.m2.2.2.1.1.1.2.cmml">⁢</mo><mrow id="A2.Ex31.m2.2.2.1.1.1.1.1" xref="A2.Ex31.m2.2.2.1.1.1.1.2.cmml"><mo id="A2.Ex31.m2.2.2.1.1.1.1.1.2" stretchy="false" xref="A2.Ex31.m2.2.2.1.1.1.1.2.cmml">(</mo><mi id="A2.Ex31.m2.1.1" xref="A2.Ex31.m2.1.1.cmml">t</mi><mo id="A2.Ex31.m2.2.2.1.1.1.1.1.3" xref="A2.Ex31.m2.2.2.1.1.1.1.2.cmml">,</mo><msub id="A2.Ex31.m2.2.2.1.1.1.1.1.1" xref="A2.Ex31.m2.2.2.1.1.1.1.1.1.cmml"><mi id="A2.Ex31.m2.2.2.1.1.1.1.1.1.2" xref="A2.Ex31.m2.2.2.1.1.1.1.1.1.2.cmml">t</mi><mi id="A2.Ex31.m2.2.2.1.1.1.1.1.1.3" mathvariant="normal" xref="A2.Ex31.m2.2.2.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="A2.Ex31.m2.2.2.1.1.1.1.1.4" stretchy="false" xref="A2.Ex31.m2.2.2.1.1.1.1.2.cmml">)</mo></mrow><mo id="A2.Ex31.m2.2.2.1.1.1.2d" xref="A2.Ex31.m2.2.2.1.1.1.2.cmml">⁢</mo><msub id="A2.Ex31.m2.2.2.1.1.1.7" xref="A2.Ex31.m2.2.2.1.1.1.7.cmml"><mover accent="true" id="A2.Ex31.m2.2.2.1.1.1.7.2" xref="A2.Ex31.m2.2.2.1.1.1.7.2.cmml"><mi id="A2.Ex31.m2.2.2.1.1.1.7.2.2" xref="A2.Ex31.m2.2.2.1.1.1.7.2.2.cmml">𝜽</mi><mo id="A2.Ex31.m2.2.2.1.1.1.7.2.1" xref="A2.Ex31.m2.2.2.1.1.1.7.2.1.cmml">~</mo></mover><mi id="A2.Ex31.m2.2.2.1.1.1.7.3" xref="A2.Ex31.m2.2.2.1.1.1.7.3.cmml">ζ</mi></msub></mrow></mrow><mo id="A2.Ex31.m2.2.2.1.2" lspace="0em" xref="A2.Ex31.m2.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex31.m2.2b"><apply id="A2.Ex31.m2.2.2.1.1.cmml" xref="A2.Ex31.m2.2.2.1"><minus id="A2.Ex31.m2.2.2.1.1.2.cmml" xref="A2.Ex31.m2.2.2.1"></minus><apply id="A2.Ex31.m2.2.2.1.1.1.cmml" xref="A2.Ex31.m2.2.2.1.1.1"><times id="A2.Ex31.m2.2.2.1.1.1.2.cmml" xref="A2.Ex31.m2.2.2.1.1.1.2"></times><cn id="A2.Ex31.m2.2.2.1.1.1.3.cmml" type="integer" xref="A2.Ex31.m2.2.2.1.1.1.3">2</cn><ci id="A2.Ex31.m2.2.2.1.1.1.4.cmml" xref="A2.Ex31.m2.2.2.1.1.1.4">𝜅</ci><apply id="A2.Ex31.m2.2.2.1.1.1.5.cmml" xref="A2.Ex31.m2.2.2.1.1.1.5"><csymbol cd="ambiguous" id="A2.Ex31.m2.2.2.1.1.1.5.1.cmml" xref="A2.Ex31.m2.2.2.1.1.1.5">superscript</csymbol><apply id="A2.Ex31.m2.2.2.1.1.1.5.2.cmml" xref="A2.Ex31.m2.2.2.1.1.1.5"><csymbol cd="ambiguous" id="A2.Ex31.m2.2.2.1.1.1.5.2.1.cmml" xref="A2.Ex31.m2.2.2.1.1.1.5">subscript</csymbol><apply id="A2.Ex31.m2.2.2.1.1.1.5.2.2.cmml" xref="A2.Ex31.m2.2.2.1.1.1.5.2.2"><ci id="A2.Ex31.m2.2.2.1.1.1.5.2.2.1.cmml" xref="A2.Ex31.m2.2.2.1.1.1.5.2.2.1">~</ci><ci id="A2.Ex31.m2.2.2.1.1.1.5.2.2.2.cmml" xref="A2.Ex31.m2.2.2.1.1.1.5.2.2.2">𝜽</ci></apply><ci id="A2.Ex31.m2.2.2.1.1.1.5.2.3.cmml" xref="A2.Ex31.m2.2.2.1.1.1.5.2.3">𝜁</ci></apply><ci id="A2.Ex31.m2.2.2.1.1.1.5.3.cmml" xref="A2.Ex31.m2.2.2.1.1.1.5.3">𝑇</ci></apply><apply id="A2.Ex31.m2.2.2.1.1.1.6.cmml" xref="A2.Ex31.m2.2.2.1.1.1.6"><csymbol cd="ambiguous" id="A2.Ex31.m2.2.2.1.1.1.6.1.cmml" xref="A2.Ex31.m2.2.2.1.1.1.6">subscript</csymbol><ci id="A2.Ex31.m2.2.2.1.1.1.6.2.cmml" xref="A2.Ex31.m2.2.2.1.1.1.6.2">𝑄</ci><ci id="A2.Ex31.m2.2.2.1.1.1.6.3.cmml" xref="A2.Ex31.m2.2.2.1.1.1.6.3">𝜁</ci></apply><interval closure="open" id="A2.Ex31.m2.2.2.1.1.1.1.2.cmml" xref="A2.Ex31.m2.2.2.1.1.1.1.1"><ci id="A2.Ex31.m2.1.1.cmml" xref="A2.Ex31.m2.1.1">𝑡</ci><apply id="A2.Ex31.m2.2.2.1.1.1.1.1.1.cmml" xref="A2.Ex31.m2.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex31.m2.2.2.1.1.1.1.1.1.1.cmml" xref="A2.Ex31.m2.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex31.m2.2.2.1.1.1.1.1.1.2.cmml" xref="A2.Ex31.m2.2.2.1.1.1.1.1.1.2">𝑡</ci><ci id="A2.Ex31.m2.2.2.1.1.1.1.1.1.3.cmml" xref="A2.Ex31.m2.2.2.1.1.1.1.1.1.3">e</ci></apply></interval><apply id="A2.Ex31.m2.2.2.1.1.1.7.cmml" xref="A2.Ex31.m2.2.2.1.1.1.7"><csymbol cd="ambiguous" id="A2.Ex31.m2.2.2.1.1.1.7.1.cmml" xref="A2.Ex31.m2.2.2.1.1.1.7">subscript</csymbol><apply id="A2.Ex31.m2.2.2.1.1.1.7.2.cmml" xref="A2.Ex31.m2.2.2.1.1.1.7.2"><ci id="A2.Ex31.m2.2.2.1.1.1.7.2.1.cmml" xref="A2.Ex31.m2.2.2.1.1.1.7.2.1">~</ci><ci id="A2.Ex31.m2.2.2.1.1.1.7.2.2.cmml" xref="A2.Ex31.m2.2.2.1.1.1.7.2.2">𝜽</ci></apply><ci id="A2.Ex31.m2.2.2.1.1.1.7.3.cmml" xref="A2.Ex31.m2.2.2.1.1.1.7.3">𝜁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex31.m2.2c">\displaystyle-2\kappa\tilde{\bm{\theta}}_{\zeta}^{T}Q_{\zeta}(t,t_{\rm e}){% \tilde{\bm{\theta}}}_{\zeta}.</annotation><annotation encoding="application/x-llamapun" id="A2.Ex31.m2.2d">- 2 italic_κ over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT italic_Q start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A2.2.p2.14">As partial IE exists for the constant <math alttext="\sigma" class="ltx_Math" display="inline" id="A2.2.p2.8.m1.1"><semantics id="A2.2.p2.8.m1.1a"><mi id="A2.2.p2.8.m1.1.1" xref="A2.2.p2.8.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="A2.2.p2.8.m1.1b"><ci id="A2.2.p2.8.m1.1.1.cmml" xref="A2.2.p2.8.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.8.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.8.m1.1d">italic_σ</annotation></semantics></math>, one gets <math alttext="\Psi_{\zeta}(t_{\rm e})\geq\sigma_{\rm c}(T_{\rm a})I\geq\sigma I" class="ltx_Math" display="inline" id="A2.2.p2.9.m2.2"><semantics id="A2.2.p2.9.m2.2a"><mrow id="A2.2.p2.9.m2.2.2" xref="A2.2.p2.9.m2.2.2.cmml"><mrow id="A2.2.p2.9.m2.1.1.1" xref="A2.2.p2.9.m2.1.1.1.cmml"><msub id="A2.2.p2.9.m2.1.1.1.3" xref="A2.2.p2.9.m2.1.1.1.3.cmml"><mi id="A2.2.p2.9.m2.1.1.1.3.2" mathvariant="normal" xref="A2.2.p2.9.m2.1.1.1.3.2.cmml">Ψ</mi><mi id="A2.2.p2.9.m2.1.1.1.3.3" xref="A2.2.p2.9.m2.1.1.1.3.3.cmml">ζ</mi></msub><mo id="A2.2.p2.9.m2.1.1.1.2" xref="A2.2.p2.9.m2.1.1.1.2.cmml">⁢</mo><mrow id="A2.2.p2.9.m2.1.1.1.1.1" xref="A2.2.p2.9.m2.1.1.1.1.1.1.cmml"><mo id="A2.2.p2.9.m2.1.1.1.1.1.2" stretchy="false" xref="A2.2.p2.9.m2.1.1.1.1.1.1.cmml">(</mo><msub id="A2.2.p2.9.m2.1.1.1.1.1.1" xref="A2.2.p2.9.m2.1.1.1.1.1.1.cmml"><mi id="A2.2.p2.9.m2.1.1.1.1.1.1.2" xref="A2.2.p2.9.m2.1.1.1.1.1.1.2.cmml">t</mi><mi id="A2.2.p2.9.m2.1.1.1.1.1.1.3" mathvariant="normal" xref="A2.2.p2.9.m2.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="A2.2.p2.9.m2.1.1.1.1.1.3" stretchy="false" xref="A2.2.p2.9.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A2.2.p2.9.m2.2.2.4" xref="A2.2.p2.9.m2.2.2.4.cmml">≥</mo><mrow id="A2.2.p2.9.m2.2.2.2" xref="A2.2.p2.9.m2.2.2.2.cmml"><msub id="A2.2.p2.9.m2.2.2.2.3" xref="A2.2.p2.9.m2.2.2.2.3.cmml"><mi id="A2.2.p2.9.m2.2.2.2.3.2" xref="A2.2.p2.9.m2.2.2.2.3.2.cmml">σ</mi><mi id="A2.2.p2.9.m2.2.2.2.3.3" mathvariant="normal" xref="A2.2.p2.9.m2.2.2.2.3.3.cmml">c</mi></msub><mo id="A2.2.p2.9.m2.2.2.2.2" xref="A2.2.p2.9.m2.2.2.2.2.cmml">⁢</mo><mrow id="A2.2.p2.9.m2.2.2.2.1.1" xref="A2.2.p2.9.m2.2.2.2.1.1.1.cmml"><mo id="A2.2.p2.9.m2.2.2.2.1.1.2" stretchy="false" xref="A2.2.p2.9.m2.2.2.2.1.1.1.cmml">(</mo><msub id="A2.2.p2.9.m2.2.2.2.1.1.1" xref="A2.2.p2.9.m2.2.2.2.1.1.1.cmml"><mi id="A2.2.p2.9.m2.2.2.2.1.1.1.2" xref="A2.2.p2.9.m2.2.2.2.1.1.1.2.cmml">T</mi><mi id="A2.2.p2.9.m2.2.2.2.1.1.1.3" mathvariant="normal" xref="A2.2.p2.9.m2.2.2.2.1.1.1.3.cmml">a</mi></msub><mo id="A2.2.p2.9.m2.2.2.2.1.1.3" stretchy="false" xref="A2.2.p2.9.m2.2.2.2.1.1.1.cmml">)</mo></mrow><mo id="A2.2.p2.9.m2.2.2.2.2a" xref="A2.2.p2.9.m2.2.2.2.2.cmml">⁢</mo><mi id="A2.2.p2.9.m2.2.2.2.4" xref="A2.2.p2.9.m2.2.2.2.4.cmml">I</mi></mrow><mo id="A2.2.p2.9.m2.2.2.5" xref="A2.2.p2.9.m2.2.2.5.cmml">≥</mo><mrow id="A2.2.p2.9.m2.2.2.6" xref="A2.2.p2.9.m2.2.2.6.cmml"><mi id="A2.2.p2.9.m2.2.2.6.2" xref="A2.2.p2.9.m2.2.2.6.2.cmml">σ</mi><mo id="A2.2.p2.9.m2.2.2.6.1" xref="A2.2.p2.9.m2.2.2.6.1.cmml">⁢</mo><mi id="A2.2.p2.9.m2.2.2.6.3" xref="A2.2.p2.9.m2.2.2.6.3.cmml">I</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.9.m2.2b"><apply id="A2.2.p2.9.m2.2.2.cmml" xref="A2.2.p2.9.m2.2.2"><and id="A2.2.p2.9.m2.2.2a.cmml" xref="A2.2.p2.9.m2.2.2"></and><apply id="A2.2.p2.9.m2.2.2b.cmml" xref="A2.2.p2.9.m2.2.2"><geq id="A2.2.p2.9.m2.2.2.4.cmml" xref="A2.2.p2.9.m2.2.2.4"></geq><apply id="A2.2.p2.9.m2.1.1.1.cmml" xref="A2.2.p2.9.m2.1.1.1"><times id="A2.2.p2.9.m2.1.1.1.2.cmml" xref="A2.2.p2.9.m2.1.1.1.2"></times><apply id="A2.2.p2.9.m2.1.1.1.3.cmml" xref="A2.2.p2.9.m2.1.1.1.3"><csymbol cd="ambiguous" id="A2.2.p2.9.m2.1.1.1.3.1.cmml" xref="A2.2.p2.9.m2.1.1.1.3">subscript</csymbol><ci id="A2.2.p2.9.m2.1.1.1.3.2.cmml" xref="A2.2.p2.9.m2.1.1.1.3.2">Ψ</ci><ci id="A2.2.p2.9.m2.1.1.1.3.3.cmml" xref="A2.2.p2.9.m2.1.1.1.3.3">𝜁</ci></apply><apply id="A2.2.p2.9.m2.1.1.1.1.1.1.cmml" xref="A2.2.p2.9.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.2.p2.9.m2.1.1.1.1.1.1.1.cmml" xref="A2.2.p2.9.m2.1.1.1.1.1">subscript</csymbol><ci id="A2.2.p2.9.m2.1.1.1.1.1.1.2.cmml" xref="A2.2.p2.9.m2.1.1.1.1.1.1.2">𝑡</ci><ci id="A2.2.p2.9.m2.1.1.1.1.1.1.3.cmml" xref="A2.2.p2.9.m2.1.1.1.1.1.1.3">e</ci></apply></apply><apply id="A2.2.p2.9.m2.2.2.2.cmml" xref="A2.2.p2.9.m2.2.2.2"><times id="A2.2.p2.9.m2.2.2.2.2.cmml" xref="A2.2.p2.9.m2.2.2.2.2"></times><apply id="A2.2.p2.9.m2.2.2.2.3.cmml" xref="A2.2.p2.9.m2.2.2.2.3"><csymbol cd="ambiguous" id="A2.2.p2.9.m2.2.2.2.3.1.cmml" xref="A2.2.p2.9.m2.2.2.2.3">subscript</csymbol><ci id="A2.2.p2.9.m2.2.2.2.3.2.cmml" xref="A2.2.p2.9.m2.2.2.2.3.2">𝜎</ci><ci id="A2.2.p2.9.m2.2.2.2.3.3.cmml" xref="A2.2.p2.9.m2.2.2.2.3.3">c</ci></apply><apply id="A2.2.p2.9.m2.2.2.2.1.1.1.cmml" xref="A2.2.p2.9.m2.2.2.2.1.1"><csymbol cd="ambiguous" id="A2.2.p2.9.m2.2.2.2.1.1.1.1.cmml" xref="A2.2.p2.9.m2.2.2.2.1.1">subscript</csymbol><ci id="A2.2.p2.9.m2.2.2.2.1.1.1.2.cmml" xref="A2.2.p2.9.m2.2.2.2.1.1.1.2">𝑇</ci><ci id="A2.2.p2.9.m2.2.2.2.1.1.1.3.cmml" xref="A2.2.p2.9.m2.2.2.2.1.1.1.3">a</ci></apply><ci id="A2.2.p2.9.m2.2.2.2.4.cmml" xref="A2.2.p2.9.m2.2.2.2.4">𝐼</ci></apply></apply><apply id="A2.2.p2.9.m2.2.2c.cmml" xref="A2.2.p2.9.m2.2.2"><geq id="A2.2.p2.9.m2.2.2.5.cmml" xref="A2.2.p2.9.m2.2.2.5"></geq><share href="https://arxiv.org/html/2401.10785v2#A2.2.p2.9.m2.2.2.2.cmml" id="A2.2.p2.9.m2.2.2d.cmml" xref="A2.2.p2.9.m2.2.2"></share><apply id="A2.2.p2.9.m2.2.2.6.cmml" xref="A2.2.p2.9.m2.2.2.6"><times id="A2.2.p2.9.m2.2.2.6.1.cmml" xref="A2.2.p2.9.m2.2.2.6.1"></times><ci id="A2.2.p2.9.m2.2.2.6.2.cmml" xref="A2.2.p2.9.m2.2.2.6.2">𝜎</ci><ci id="A2.2.p2.9.m2.2.2.6.3.cmml" xref="A2.2.p2.9.m2.2.2.6.3">𝐼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.9.m2.2c">\Psi_{\zeta}(t_{\rm e})\geq\sigma_{\rm c}(T_{\rm a})I\geq\sigma I</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.9.m2.2d">roman_Ψ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) ≥ italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ( italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT ) italic_I ≥ italic_σ italic_I</annotation></semantics></math> from (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E19" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">19</span></a>). Since <math alttext="H(s)" class="ltx_Math" display="inline" id="A2.2.p2.10.m3.1"><semantics id="A2.2.p2.10.m3.1a"><mrow id="A2.2.p2.10.m3.1.2" xref="A2.2.p2.10.m3.1.2.cmml"><mi id="A2.2.p2.10.m3.1.2.2" xref="A2.2.p2.10.m3.1.2.2.cmml">H</mi><mo id="A2.2.p2.10.m3.1.2.1" xref="A2.2.p2.10.m3.1.2.1.cmml">⁢</mo><mrow id="A2.2.p2.10.m3.1.2.3.2" xref="A2.2.p2.10.m3.1.2.cmml"><mo id="A2.2.p2.10.m3.1.2.3.2.1" stretchy="false" xref="A2.2.p2.10.m3.1.2.cmml">(</mo><mi id="A2.2.p2.10.m3.1.1" xref="A2.2.p2.10.m3.1.1.cmml">s</mi><mo id="A2.2.p2.10.m3.1.2.3.2.2" stretchy="false" xref="A2.2.p2.10.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.10.m3.1b"><apply id="A2.2.p2.10.m3.1.2.cmml" xref="A2.2.p2.10.m3.1.2"><times id="A2.2.p2.10.m3.1.2.1.cmml" xref="A2.2.p2.10.m3.1.2.1"></times><ci id="A2.2.p2.10.m3.1.2.2.cmml" xref="A2.2.p2.10.m3.1.2.2">𝐻</ci><ci id="A2.2.p2.10.m3.1.1.cmml" xref="A2.2.p2.10.m3.1.1">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.10.m3.1c">H(s)</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.10.m3.1d">italic_H ( italic_s )</annotation></semantics></math> is stable with unit DC gain, there exists a constant <math alttext="\sigma^{*}" class="ltx_Math" display="inline" id="A2.2.p2.11.m4.1"><semantics id="A2.2.p2.11.m4.1a"><msup id="A2.2.p2.11.m4.1.1" xref="A2.2.p2.11.m4.1.1.cmml"><mi id="A2.2.p2.11.m4.1.1.2" xref="A2.2.p2.11.m4.1.1.2.cmml">σ</mi><mo id="A2.2.p2.11.m4.1.1.3" xref="A2.2.p2.11.m4.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="A2.2.p2.11.m4.1b"><apply id="A2.2.p2.11.m4.1.1.cmml" xref="A2.2.p2.11.m4.1.1"><csymbol cd="ambiguous" id="A2.2.p2.11.m4.1.1.1.cmml" xref="A2.2.p2.11.m4.1.1">superscript</csymbol><ci id="A2.2.p2.11.m4.1.1.2.cmml" xref="A2.2.p2.11.m4.1.1.2">𝜎</ci><times id="A2.2.p2.11.m4.1.1.3.cmml" xref="A2.2.p2.11.m4.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.11.m4.1c">\sigma^{*}</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.11.m4.1d">italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> <math alttext="\in" class="ltx_Math" display="inline" id="A2.2.p2.12.m5.1"><semantics id="A2.2.p2.12.m5.1a"><mo id="A2.2.p2.12.m5.1.1" xref="A2.2.p2.12.m5.1.1.cmml">∈</mo><annotation-xml encoding="MathML-Content" id="A2.2.p2.12.m5.1b"><in id="A2.2.p2.12.m5.1.1.cmml" xref="A2.2.p2.12.m5.1.1"></in></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.12.m5.1c">\in</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.12.m5.1d">∈</annotation></semantics></math> <math alttext="\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A2.2.p2.13.m6.1"><semantics id="A2.2.p2.13.m6.1a"><msup id="A2.2.p2.13.m6.1.1" xref="A2.2.p2.13.m6.1.1.cmml"><mi id="A2.2.p2.13.m6.1.1.2" xref="A2.2.p2.13.m6.1.1.2.cmml">ℝ</mi><mo id="A2.2.p2.13.m6.1.1.3" xref="A2.2.p2.13.m6.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="A2.2.p2.13.m6.1b"><apply id="A2.2.p2.13.m6.1.1.cmml" xref="A2.2.p2.13.m6.1.1"><csymbol cd="ambiguous" id="A2.2.p2.13.m6.1.1.1.cmml" xref="A2.2.p2.13.m6.1.1">superscript</csymbol><ci id="A2.2.p2.13.m6.1.1.2.cmml" xref="A2.2.p2.13.m6.1.1.2">ℝ</ci><plus id="A2.2.p2.13.m6.1.1.3.cmml" xref="A2.2.p2.13.m6.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.13.m6.1c">\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.13.m6.1d">blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="\sigma^{*}\leq\sigma" class="ltx_Math" display="inline" id="A2.2.p2.14.m7.1"><semantics id="A2.2.p2.14.m7.1a"><mrow id="A2.2.p2.14.m7.1.1" xref="A2.2.p2.14.m7.1.1.cmml"><msup id="A2.2.p2.14.m7.1.1.2" xref="A2.2.p2.14.m7.1.1.2.cmml"><mi id="A2.2.p2.14.m7.1.1.2.2" xref="A2.2.p2.14.m7.1.1.2.2.cmml">σ</mi><mo id="A2.2.p2.14.m7.1.1.2.3" xref="A2.2.p2.14.m7.1.1.2.3.cmml">∗</mo></msup><mo id="A2.2.p2.14.m7.1.1.1" xref="A2.2.p2.14.m7.1.1.1.cmml">≤</mo><mi id="A2.2.p2.14.m7.1.1.3" xref="A2.2.p2.14.m7.1.1.3.cmml">σ</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.14.m7.1b"><apply id="A2.2.p2.14.m7.1.1.cmml" xref="A2.2.p2.14.m7.1.1"><leq id="A2.2.p2.14.m7.1.1.1.cmml" xref="A2.2.p2.14.m7.1.1.1"></leq><apply id="A2.2.p2.14.m7.1.1.2.cmml" xref="A2.2.p2.14.m7.1.1.2"><csymbol cd="ambiguous" id="A2.2.p2.14.m7.1.1.2.1.cmml" xref="A2.2.p2.14.m7.1.1.2">superscript</csymbol><ci id="A2.2.p2.14.m7.1.1.2.2.cmml" xref="A2.2.p2.14.m7.1.1.2.2">𝜎</ci><times id="A2.2.p2.14.m7.1.1.2.3.cmml" xref="A2.2.p2.14.m7.1.1.2.3"></times></apply><ci id="A2.2.p2.14.m7.1.1.3.cmml" xref="A2.2.p2.14.m7.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.14.m7.1c">\sigma^{*}\leq\sigma</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.14.m7.1d">italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ≤ italic_σ</annotation></semantics></math> such that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx44"> <tbody id="A2.Ex32"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle Q_{\zeta}(t,t_{\rm e})=H(s)[\Psi_{\zeta}(t_{\rm e})]\geq\sigma^{% *}I." class="ltx_Math" display="inline" id="A2.Ex32.m1.3"><semantics id="A2.Ex32.m1.3a"><mrow id="A2.Ex32.m1.3.3.1" xref="A2.Ex32.m1.3.3.1.1.cmml"><mrow id="A2.Ex32.m1.3.3.1.1" xref="A2.Ex32.m1.3.3.1.1.cmml"><mrow id="A2.Ex32.m1.3.3.1.1.1" xref="A2.Ex32.m1.3.3.1.1.1.cmml"><msub id="A2.Ex32.m1.3.3.1.1.1.3" xref="A2.Ex32.m1.3.3.1.1.1.3.cmml"><mi id="A2.Ex32.m1.3.3.1.1.1.3.2" xref="A2.Ex32.m1.3.3.1.1.1.3.2.cmml">Q</mi><mi id="A2.Ex32.m1.3.3.1.1.1.3.3" xref="A2.Ex32.m1.3.3.1.1.1.3.3.cmml">ζ</mi></msub><mo id="A2.Ex32.m1.3.3.1.1.1.2" xref="A2.Ex32.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="A2.Ex32.m1.3.3.1.1.1.1.1" xref="A2.Ex32.m1.3.3.1.1.1.1.2.cmml"><mo id="A2.Ex32.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="A2.Ex32.m1.3.3.1.1.1.1.2.cmml">(</mo><mi id="A2.Ex32.m1.1.1" xref="A2.Ex32.m1.1.1.cmml">t</mi><mo id="A2.Ex32.m1.3.3.1.1.1.1.1.3" xref="A2.Ex32.m1.3.3.1.1.1.1.2.cmml">,</mo><msub id="A2.Ex32.m1.3.3.1.1.1.1.1.1" xref="A2.Ex32.m1.3.3.1.1.1.1.1.1.cmml"><mi id="A2.Ex32.m1.3.3.1.1.1.1.1.1.2" xref="A2.Ex32.m1.3.3.1.1.1.1.1.1.2.cmml">t</mi><mi id="A2.Ex32.m1.3.3.1.1.1.1.1.1.3" mathvariant="normal" xref="A2.Ex32.m1.3.3.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="A2.Ex32.m1.3.3.1.1.1.1.1.4" stretchy="false" xref="A2.Ex32.m1.3.3.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="A2.Ex32.m1.3.3.1.1.4" xref="A2.Ex32.m1.3.3.1.1.4.cmml">=</mo><mrow id="A2.Ex32.m1.3.3.1.1.2" xref="A2.Ex32.m1.3.3.1.1.2.cmml"><mi id="A2.Ex32.m1.3.3.1.1.2.3" xref="A2.Ex32.m1.3.3.1.1.2.3.cmml">H</mi><mo id="A2.Ex32.m1.3.3.1.1.2.2" xref="A2.Ex32.m1.3.3.1.1.2.2.cmml">⁢</mo><mrow id="A2.Ex32.m1.3.3.1.1.2.4.2" xref="A2.Ex32.m1.3.3.1.1.2.cmml"><mo id="A2.Ex32.m1.3.3.1.1.2.4.2.1" stretchy="false" xref="A2.Ex32.m1.3.3.1.1.2.cmml">(</mo><mi id="A2.Ex32.m1.2.2" xref="A2.Ex32.m1.2.2.cmml">s</mi><mo id="A2.Ex32.m1.3.3.1.1.2.4.2.2" stretchy="false" xref="A2.Ex32.m1.3.3.1.1.2.cmml">)</mo></mrow><mo id="A2.Ex32.m1.3.3.1.1.2.2a" xref="A2.Ex32.m1.3.3.1.1.2.2.cmml">⁢</mo><mrow id="A2.Ex32.m1.3.3.1.1.2.1.1" xref="A2.Ex32.m1.3.3.1.1.2.1.2.cmml"><mo id="A2.Ex32.m1.3.3.1.1.2.1.1.2" stretchy="false" xref="A2.Ex32.m1.3.3.1.1.2.1.2.1.cmml">[</mo><mrow id="A2.Ex32.m1.3.3.1.1.2.1.1.1" xref="A2.Ex32.m1.3.3.1.1.2.1.1.1.cmml"><msub id="A2.Ex32.m1.3.3.1.1.2.1.1.1.3" xref="A2.Ex32.m1.3.3.1.1.2.1.1.1.3.cmml"><mi id="A2.Ex32.m1.3.3.1.1.2.1.1.1.3.2" mathvariant="normal" xref="A2.Ex32.m1.3.3.1.1.2.1.1.1.3.2.cmml">Ψ</mi><mi id="A2.Ex32.m1.3.3.1.1.2.1.1.1.3.3" xref="A2.Ex32.m1.3.3.1.1.2.1.1.1.3.3.cmml">ζ</mi></msub><mo id="A2.Ex32.m1.3.3.1.1.2.1.1.1.2" xref="A2.Ex32.m1.3.3.1.1.2.1.1.1.2.cmml">⁢</mo><mrow id="A2.Ex32.m1.3.3.1.1.2.1.1.1.1.1" xref="A2.Ex32.m1.3.3.1.1.2.1.1.1.1.1.1.cmml"><mo id="A2.Ex32.m1.3.3.1.1.2.1.1.1.1.1.2" stretchy="false" xref="A2.Ex32.m1.3.3.1.1.2.1.1.1.1.1.1.cmml">(</mo><msub id="A2.Ex32.m1.3.3.1.1.2.1.1.1.1.1.1" xref="A2.Ex32.m1.3.3.1.1.2.1.1.1.1.1.1.cmml"><mi id="A2.Ex32.m1.3.3.1.1.2.1.1.1.1.1.1.2" xref="A2.Ex32.m1.3.3.1.1.2.1.1.1.1.1.1.2.cmml">t</mi><mi id="A2.Ex32.m1.3.3.1.1.2.1.1.1.1.1.1.3" mathvariant="normal" xref="A2.Ex32.m1.3.3.1.1.2.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="A2.Ex32.m1.3.3.1.1.2.1.1.1.1.1.3" stretchy="false" xref="A2.Ex32.m1.3.3.1.1.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A2.Ex32.m1.3.3.1.1.2.1.1.3" stretchy="false" xref="A2.Ex32.m1.3.3.1.1.2.1.2.1.cmml">]</mo></mrow></mrow><mo id="A2.Ex32.m1.3.3.1.1.5" xref="A2.Ex32.m1.3.3.1.1.5.cmml">≥</mo><mrow id="A2.Ex32.m1.3.3.1.1.6" xref="A2.Ex32.m1.3.3.1.1.6.cmml"><msup id="A2.Ex32.m1.3.3.1.1.6.2" xref="A2.Ex32.m1.3.3.1.1.6.2.cmml"><mi id="A2.Ex32.m1.3.3.1.1.6.2.2" xref="A2.Ex32.m1.3.3.1.1.6.2.2.cmml">σ</mi><mo id="A2.Ex32.m1.3.3.1.1.6.2.3" xref="A2.Ex32.m1.3.3.1.1.6.2.3.cmml">∗</mo></msup><mo id="A2.Ex32.m1.3.3.1.1.6.1" xref="A2.Ex32.m1.3.3.1.1.6.1.cmml">⁢</mo><mi id="A2.Ex32.m1.3.3.1.1.6.3" xref="A2.Ex32.m1.3.3.1.1.6.3.cmml">I</mi></mrow></mrow><mo id="A2.Ex32.m1.3.3.1.2" lspace="0em" xref="A2.Ex32.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex32.m1.3b"><apply id="A2.Ex32.m1.3.3.1.1.cmml" xref="A2.Ex32.m1.3.3.1"><and id="A2.Ex32.m1.3.3.1.1a.cmml" xref="A2.Ex32.m1.3.3.1"></and><apply id="A2.Ex32.m1.3.3.1.1b.cmml" xref="A2.Ex32.m1.3.3.1"><eq id="A2.Ex32.m1.3.3.1.1.4.cmml" xref="A2.Ex32.m1.3.3.1.1.4"></eq><apply id="A2.Ex32.m1.3.3.1.1.1.cmml" xref="A2.Ex32.m1.3.3.1.1.1"><times id="A2.Ex32.m1.3.3.1.1.1.2.cmml" xref="A2.Ex32.m1.3.3.1.1.1.2"></times><apply id="A2.Ex32.m1.3.3.1.1.1.3.cmml" xref="A2.Ex32.m1.3.3.1.1.1.3"><csymbol cd="ambiguous" id="A2.Ex32.m1.3.3.1.1.1.3.1.cmml" xref="A2.Ex32.m1.3.3.1.1.1.3">subscript</csymbol><ci id="A2.Ex32.m1.3.3.1.1.1.3.2.cmml" xref="A2.Ex32.m1.3.3.1.1.1.3.2">𝑄</ci><ci id="A2.Ex32.m1.3.3.1.1.1.3.3.cmml" xref="A2.Ex32.m1.3.3.1.1.1.3.3">𝜁</ci></apply><interval closure="open" id="A2.Ex32.m1.3.3.1.1.1.1.2.cmml" xref="A2.Ex32.m1.3.3.1.1.1.1.1"><ci id="A2.Ex32.m1.1.1.cmml" xref="A2.Ex32.m1.1.1">𝑡</ci><apply id="A2.Ex32.m1.3.3.1.1.1.1.1.1.cmml" xref="A2.Ex32.m1.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex32.m1.3.3.1.1.1.1.1.1.1.cmml" xref="A2.Ex32.m1.3.3.1.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex32.m1.3.3.1.1.1.1.1.1.2.cmml" xref="A2.Ex32.m1.3.3.1.1.1.1.1.1.2">𝑡</ci><ci id="A2.Ex32.m1.3.3.1.1.1.1.1.1.3.cmml" xref="A2.Ex32.m1.3.3.1.1.1.1.1.1.3">e</ci></apply></interval></apply><apply id="A2.Ex32.m1.3.3.1.1.2.cmml" xref="A2.Ex32.m1.3.3.1.1.2"><times id="A2.Ex32.m1.3.3.1.1.2.2.cmml" xref="A2.Ex32.m1.3.3.1.1.2.2"></times><ci id="A2.Ex32.m1.3.3.1.1.2.3.cmml" xref="A2.Ex32.m1.3.3.1.1.2.3">𝐻</ci><ci id="A2.Ex32.m1.2.2.cmml" xref="A2.Ex32.m1.2.2">𝑠</ci><apply id="A2.Ex32.m1.3.3.1.1.2.1.2.cmml" xref="A2.Ex32.m1.3.3.1.1.2.1.1"><csymbol cd="latexml" id="A2.Ex32.m1.3.3.1.1.2.1.2.1.cmml" xref="A2.Ex32.m1.3.3.1.1.2.1.1.2">delimited-[]</csymbol><apply id="A2.Ex32.m1.3.3.1.1.2.1.1.1.cmml" xref="A2.Ex32.m1.3.3.1.1.2.1.1.1"><times id="A2.Ex32.m1.3.3.1.1.2.1.1.1.2.cmml" xref="A2.Ex32.m1.3.3.1.1.2.1.1.1.2"></times><apply id="A2.Ex32.m1.3.3.1.1.2.1.1.1.3.cmml" xref="A2.Ex32.m1.3.3.1.1.2.1.1.1.3"><csymbol cd="ambiguous" id="A2.Ex32.m1.3.3.1.1.2.1.1.1.3.1.cmml" xref="A2.Ex32.m1.3.3.1.1.2.1.1.1.3">subscript</csymbol><ci id="A2.Ex32.m1.3.3.1.1.2.1.1.1.3.2.cmml" xref="A2.Ex32.m1.3.3.1.1.2.1.1.1.3.2">Ψ</ci><ci id="A2.Ex32.m1.3.3.1.1.2.1.1.1.3.3.cmml" xref="A2.Ex32.m1.3.3.1.1.2.1.1.1.3.3">𝜁</ci></apply><apply id="A2.Ex32.m1.3.3.1.1.2.1.1.1.1.1.1.cmml" xref="A2.Ex32.m1.3.3.1.1.2.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex32.m1.3.3.1.1.2.1.1.1.1.1.1.1.cmml" xref="A2.Ex32.m1.3.3.1.1.2.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex32.m1.3.3.1.1.2.1.1.1.1.1.1.2.cmml" xref="A2.Ex32.m1.3.3.1.1.2.1.1.1.1.1.1.2">𝑡</ci><ci id="A2.Ex32.m1.3.3.1.1.2.1.1.1.1.1.1.3.cmml" xref="A2.Ex32.m1.3.3.1.1.2.1.1.1.1.1.1.3">e</ci></apply></apply></apply></apply></apply><apply id="A2.Ex32.m1.3.3.1.1c.cmml" xref="A2.Ex32.m1.3.3.1"><geq id="A2.Ex32.m1.3.3.1.1.5.cmml" xref="A2.Ex32.m1.3.3.1.1.5"></geq><share href="https://arxiv.org/html/2401.10785v2#A2.Ex32.m1.3.3.1.1.2.cmml" id="A2.Ex32.m1.3.3.1.1d.cmml" xref="A2.Ex32.m1.3.3.1"></share><apply id="A2.Ex32.m1.3.3.1.1.6.cmml" xref="A2.Ex32.m1.3.3.1.1.6"><times id="A2.Ex32.m1.3.3.1.1.6.1.cmml" xref="A2.Ex32.m1.3.3.1.1.6.1"></times><apply id="A2.Ex32.m1.3.3.1.1.6.2.cmml" xref="A2.Ex32.m1.3.3.1.1.6.2"><csymbol cd="ambiguous" id="A2.Ex32.m1.3.3.1.1.6.2.1.cmml" xref="A2.Ex32.m1.3.3.1.1.6.2">superscript</csymbol><ci id="A2.Ex32.m1.3.3.1.1.6.2.2.cmml" xref="A2.Ex32.m1.3.3.1.1.6.2.2">𝜎</ci><times id="A2.Ex32.m1.3.3.1.1.6.2.3.cmml" xref="A2.Ex32.m1.3.3.1.1.6.2.3"></times></apply><ci id="A2.Ex32.m1.3.3.1.1.6.3.cmml" xref="A2.Ex32.m1.3.3.1.1.6.3">𝐼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex32.m1.3c">\displaystyle Q_{\zeta}(t,t_{\rm e})=H(s)[\Psi_{\zeta}(t_{\rm e})]\geq\sigma^{% *}I.</annotation><annotation encoding="application/x-llamapun" id="A2.Ex32.m1.3d">italic_Q start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) = italic_H ( italic_s ) [ roman_Ψ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) ] ≥ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_I .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A2.2.p2.21">It follows from the above two results that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx45"> <tbody id="A2.Ex33"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}_{\theta,\zeta}\leq-2\kappa\sigma^{*}\tilde{\bm{\theta}}_{% \zeta}^{T}{\tilde{\bm{\theta}}}_{\zeta}\leq-2\kappa\sigma^{*}\lambda_{\min}(% \Gamma_{\zeta})\tilde{\bm{\theta}}_{\zeta}^{T}\Gamma^{-1}_{\zeta}{\tilde{\bm{% \theta}}}_{\zeta}." class="ltx_Math" display="inline" id="A2.Ex33.m1.3"><semantics id="A2.Ex33.m1.3a"><mrow id="A2.Ex33.m1.3.3.1" xref="A2.Ex33.m1.3.3.1.1.cmml"><mrow id="A2.Ex33.m1.3.3.1.1" xref="A2.Ex33.m1.3.3.1.1.cmml"><msub id="A2.Ex33.m1.3.3.1.1.3" xref="A2.Ex33.m1.3.3.1.1.3.cmml"><mover accent="true" id="A2.Ex33.m1.3.3.1.1.3.2" xref="A2.Ex33.m1.3.3.1.1.3.2.cmml"><mi id="A2.Ex33.m1.3.3.1.1.3.2.2" xref="A2.Ex33.m1.3.3.1.1.3.2.2.cmml">V</mi><mo id="A2.Ex33.m1.3.3.1.1.3.2.1" xref="A2.Ex33.m1.3.3.1.1.3.2.1.cmml">˙</mo></mover><mrow id="A2.Ex33.m1.2.2.2.4" xref="A2.Ex33.m1.2.2.2.3.cmml"><mi id="A2.Ex33.m1.1.1.1.1" xref="A2.Ex33.m1.1.1.1.1.cmml">θ</mi><mo id="A2.Ex33.m1.2.2.2.4.1" xref="A2.Ex33.m1.2.2.2.3.cmml">,</mo><mi id="A2.Ex33.m1.2.2.2.2" xref="A2.Ex33.m1.2.2.2.2.cmml">ζ</mi></mrow></msub><mo id="A2.Ex33.m1.3.3.1.1.4" xref="A2.Ex33.m1.3.3.1.1.4.cmml">≤</mo><mrow id="A2.Ex33.m1.3.3.1.1.5" xref="A2.Ex33.m1.3.3.1.1.5.cmml"><mo id="A2.Ex33.m1.3.3.1.1.5a" xref="A2.Ex33.m1.3.3.1.1.5.cmml">−</mo><mrow id="A2.Ex33.m1.3.3.1.1.5.2" xref="A2.Ex33.m1.3.3.1.1.5.2.cmml"><mn id="A2.Ex33.m1.3.3.1.1.5.2.2" xref="A2.Ex33.m1.3.3.1.1.5.2.2.cmml">2</mn><mo id="A2.Ex33.m1.3.3.1.1.5.2.1" xref="A2.Ex33.m1.3.3.1.1.5.2.1.cmml">⁢</mo><mi id="A2.Ex33.m1.3.3.1.1.5.2.3" xref="A2.Ex33.m1.3.3.1.1.5.2.3.cmml">κ</mi><mo id="A2.Ex33.m1.3.3.1.1.5.2.1a" xref="A2.Ex33.m1.3.3.1.1.5.2.1.cmml">⁢</mo><msup id="A2.Ex33.m1.3.3.1.1.5.2.4" xref="A2.Ex33.m1.3.3.1.1.5.2.4.cmml"><mi id="A2.Ex33.m1.3.3.1.1.5.2.4.2" xref="A2.Ex33.m1.3.3.1.1.5.2.4.2.cmml">σ</mi><mo id="A2.Ex33.m1.3.3.1.1.5.2.4.3" xref="A2.Ex33.m1.3.3.1.1.5.2.4.3.cmml">∗</mo></msup><mo id="A2.Ex33.m1.3.3.1.1.5.2.1b" xref="A2.Ex33.m1.3.3.1.1.5.2.1.cmml">⁢</mo><msubsup id="A2.Ex33.m1.3.3.1.1.5.2.5" xref="A2.Ex33.m1.3.3.1.1.5.2.5.cmml"><mover accent="true" id="A2.Ex33.m1.3.3.1.1.5.2.5.2.2" xref="A2.Ex33.m1.3.3.1.1.5.2.5.2.2.cmml"><mi id="A2.Ex33.m1.3.3.1.1.5.2.5.2.2.2" xref="A2.Ex33.m1.3.3.1.1.5.2.5.2.2.2.cmml">𝜽</mi><mo id="A2.Ex33.m1.3.3.1.1.5.2.5.2.2.1" xref="A2.Ex33.m1.3.3.1.1.5.2.5.2.2.1.cmml">~</mo></mover><mi id="A2.Ex33.m1.3.3.1.1.5.2.5.2.3" xref="A2.Ex33.m1.3.3.1.1.5.2.5.2.3.cmml">ζ</mi><mi id="A2.Ex33.m1.3.3.1.1.5.2.5.3" xref="A2.Ex33.m1.3.3.1.1.5.2.5.3.cmml">T</mi></msubsup><mo id="A2.Ex33.m1.3.3.1.1.5.2.1c" xref="A2.Ex33.m1.3.3.1.1.5.2.1.cmml">⁢</mo><msub id="A2.Ex33.m1.3.3.1.1.5.2.6" xref="A2.Ex33.m1.3.3.1.1.5.2.6.cmml"><mover accent="true" id="A2.Ex33.m1.3.3.1.1.5.2.6.2" xref="A2.Ex33.m1.3.3.1.1.5.2.6.2.cmml"><mi id="A2.Ex33.m1.3.3.1.1.5.2.6.2.2" xref="A2.Ex33.m1.3.3.1.1.5.2.6.2.2.cmml">𝜽</mi><mo id="A2.Ex33.m1.3.3.1.1.5.2.6.2.1" xref="A2.Ex33.m1.3.3.1.1.5.2.6.2.1.cmml">~</mo></mover><mi id="A2.Ex33.m1.3.3.1.1.5.2.6.3" xref="A2.Ex33.m1.3.3.1.1.5.2.6.3.cmml">ζ</mi></msub></mrow></mrow><mo id="A2.Ex33.m1.3.3.1.1.6" xref="A2.Ex33.m1.3.3.1.1.6.cmml">≤</mo><mrow id="A2.Ex33.m1.3.3.1.1.1" xref="A2.Ex33.m1.3.3.1.1.1.cmml"><mo id="A2.Ex33.m1.3.3.1.1.1a" xref="A2.Ex33.m1.3.3.1.1.1.cmml">−</mo><mrow id="A2.Ex33.m1.3.3.1.1.1.1" xref="A2.Ex33.m1.3.3.1.1.1.1.cmml"><mn id="A2.Ex33.m1.3.3.1.1.1.1.3" xref="A2.Ex33.m1.3.3.1.1.1.1.3.cmml">2</mn><mo id="A2.Ex33.m1.3.3.1.1.1.1.2" xref="A2.Ex33.m1.3.3.1.1.1.1.2.cmml">⁢</mo><mi id="A2.Ex33.m1.3.3.1.1.1.1.4" xref="A2.Ex33.m1.3.3.1.1.1.1.4.cmml">κ</mi><mo id="A2.Ex33.m1.3.3.1.1.1.1.2a" xref="A2.Ex33.m1.3.3.1.1.1.1.2.cmml">⁢</mo><msup id="A2.Ex33.m1.3.3.1.1.1.1.5" xref="A2.Ex33.m1.3.3.1.1.1.1.5.cmml"><mi id="A2.Ex33.m1.3.3.1.1.1.1.5.2" xref="A2.Ex33.m1.3.3.1.1.1.1.5.2.cmml">σ</mi><mo id="A2.Ex33.m1.3.3.1.1.1.1.5.3" xref="A2.Ex33.m1.3.3.1.1.1.1.5.3.cmml">∗</mo></msup><mo id="A2.Ex33.m1.3.3.1.1.1.1.2b" xref="A2.Ex33.m1.3.3.1.1.1.1.2.cmml">⁢</mo><msub id="A2.Ex33.m1.3.3.1.1.1.1.6" xref="A2.Ex33.m1.3.3.1.1.1.1.6.cmml"><mi id="A2.Ex33.m1.3.3.1.1.1.1.6.2" xref="A2.Ex33.m1.3.3.1.1.1.1.6.2.cmml">λ</mi><mi id="A2.Ex33.m1.3.3.1.1.1.1.6.3" xref="A2.Ex33.m1.3.3.1.1.1.1.6.3.cmml">min</mi></msub><mo id="A2.Ex33.m1.3.3.1.1.1.1.2c" xref="A2.Ex33.m1.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="A2.Ex33.m1.3.3.1.1.1.1.1.1" xref="A2.Ex33.m1.3.3.1.1.1.1.1.1.1.cmml"><mo id="A2.Ex33.m1.3.3.1.1.1.1.1.1.2" stretchy="false" xref="A2.Ex33.m1.3.3.1.1.1.1.1.1.1.cmml">(</mo><msub id="A2.Ex33.m1.3.3.1.1.1.1.1.1.1" xref="A2.Ex33.m1.3.3.1.1.1.1.1.1.1.cmml"><mi id="A2.Ex33.m1.3.3.1.1.1.1.1.1.1.2" mathvariant="normal" xref="A2.Ex33.m1.3.3.1.1.1.1.1.1.1.2.cmml">Γ</mi><mi id="A2.Ex33.m1.3.3.1.1.1.1.1.1.1.3" xref="A2.Ex33.m1.3.3.1.1.1.1.1.1.1.3.cmml">ζ</mi></msub><mo id="A2.Ex33.m1.3.3.1.1.1.1.1.1.3" stretchy="false" xref="A2.Ex33.m1.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A2.Ex33.m1.3.3.1.1.1.1.2d" xref="A2.Ex33.m1.3.3.1.1.1.1.2.cmml">⁢</mo><msubsup id="A2.Ex33.m1.3.3.1.1.1.1.7" xref="A2.Ex33.m1.3.3.1.1.1.1.7.cmml"><mover accent="true" id="A2.Ex33.m1.3.3.1.1.1.1.7.2.2" xref="A2.Ex33.m1.3.3.1.1.1.1.7.2.2.cmml"><mi id="A2.Ex33.m1.3.3.1.1.1.1.7.2.2.2" xref="A2.Ex33.m1.3.3.1.1.1.1.7.2.2.2.cmml">𝜽</mi><mo id="A2.Ex33.m1.3.3.1.1.1.1.7.2.2.1" xref="A2.Ex33.m1.3.3.1.1.1.1.7.2.2.1.cmml">~</mo></mover><mi id="A2.Ex33.m1.3.3.1.1.1.1.7.2.3" xref="A2.Ex33.m1.3.3.1.1.1.1.7.2.3.cmml">ζ</mi><mi id="A2.Ex33.m1.3.3.1.1.1.1.7.3" xref="A2.Ex33.m1.3.3.1.1.1.1.7.3.cmml">T</mi></msubsup><mo id="A2.Ex33.m1.3.3.1.1.1.1.2e" xref="A2.Ex33.m1.3.3.1.1.1.1.2.cmml">⁢</mo><msubsup id="A2.Ex33.m1.3.3.1.1.1.1.8" xref="A2.Ex33.m1.3.3.1.1.1.1.8.cmml"><mi id="A2.Ex33.m1.3.3.1.1.1.1.8.2.2" mathvariant="normal" xref="A2.Ex33.m1.3.3.1.1.1.1.8.2.2.cmml">Γ</mi><mi id="A2.Ex33.m1.3.3.1.1.1.1.8.3" xref="A2.Ex33.m1.3.3.1.1.1.1.8.3.cmml">ζ</mi><mrow id="A2.Ex33.m1.3.3.1.1.1.1.8.2.3" xref="A2.Ex33.m1.3.3.1.1.1.1.8.2.3.cmml"><mo id="A2.Ex33.m1.3.3.1.1.1.1.8.2.3a" xref="A2.Ex33.m1.3.3.1.1.1.1.8.2.3.cmml">−</mo><mn id="A2.Ex33.m1.3.3.1.1.1.1.8.2.3.2" xref="A2.Ex33.m1.3.3.1.1.1.1.8.2.3.2.cmml">1</mn></mrow></msubsup><mo id="A2.Ex33.m1.3.3.1.1.1.1.2f" xref="A2.Ex33.m1.3.3.1.1.1.1.2.cmml">⁢</mo><msub id="A2.Ex33.m1.3.3.1.1.1.1.9" xref="A2.Ex33.m1.3.3.1.1.1.1.9.cmml"><mover accent="true" id="A2.Ex33.m1.3.3.1.1.1.1.9.2" xref="A2.Ex33.m1.3.3.1.1.1.1.9.2.cmml"><mi id="A2.Ex33.m1.3.3.1.1.1.1.9.2.2" xref="A2.Ex33.m1.3.3.1.1.1.1.9.2.2.cmml">𝜽</mi><mo id="A2.Ex33.m1.3.3.1.1.1.1.9.2.1" xref="A2.Ex33.m1.3.3.1.1.1.1.9.2.1.cmml">~</mo></mover><mi id="A2.Ex33.m1.3.3.1.1.1.1.9.3" xref="A2.Ex33.m1.3.3.1.1.1.1.9.3.cmml">ζ</mi></msub></mrow></mrow></mrow><mo id="A2.Ex33.m1.3.3.1.2" lspace="0em" xref="A2.Ex33.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex33.m1.3b"><apply id="A2.Ex33.m1.3.3.1.1.cmml" xref="A2.Ex33.m1.3.3.1"><and id="A2.Ex33.m1.3.3.1.1a.cmml" xref="A2.Ex33.m1.3.3.1"></and><apply id="A2.Ex33.m1.3.3.1.1b.cmml" xref="A2.Ex33.m1.3.3.1"><leq id="A2.Ex33.m1.3.3.1.1.4.cmml" xref="A2.Ex33.m1.3.3.1.1.4"></leq><apply id="A2.Ex33.m1.3.3.1.1.3.cmml" xref="A2.Ex33.m1.3.3.1.1.3"><csymbol cd="ambiguous" id="A2.Ex33.m1.3.3.1.1.3.1.cmml" xref="A2.Ex33.m1.3.3.1.1.3">subscript</csymbol><apply id="A2.Ex33.m1.3.3.1.1.3.2.cmml" xref="A2.Ex33.m1.3.3.1.1.3.2"><ci id="A2.Ex33.m1.3.3.1.1.3.2.1.cmml" xref="A2.Ex33.m1.3.3.1.1.3.2.1">˙</ci><ci id="A2.Ex33.m1.3.3.1.1.3.2.2.cmml" xref="A2.Ex33.m1.3.3.1.1.3.2.2">𝑉</ci></apply><list id="A2.Ex33.m1.2.2.2.3.cmml" xref="A2.Ex33.m1.2.2.2.4"><ci id="A2.Ex33.m1.1.1.1.1.cmml" xref="A2.Ex33.m1.1.1.1.1">𝜃</ci><ci id="A2.Ex33.m1.2.2.2.2.cmml" xref="A2.Ex33.m1.2.2.2.2">𝜁</ci></list></apply><apply id="A2.Ex33.m1.3.3.1.1.5.cmml" xref="A2.Ex33.m1.3.3.1.1.5"><minus id="A2.Ex33.m1.3.3.1.1.5.1.cmml" xref="A2.Ex33.m1.3.3.1.1.5"></minus><apply id="A2.Ex33.m1.3.3.1.1.5.2.cmml" xref="A2.Ex33.m1.3.3.1.1.5.2"><times id="A2.Ex33.m1.3.3.1.1.5.2.1.cmml" xref="A2.Ex33.m1.3.3.1.1.5.2.1"></times><cn id="A2.Ex33.m1.3.3.1.1.5.2.2.cmml" type="integer" xref="A2.Ex33.m1.3.3.1.1.5.2.2">2</cn><ci id="A2.Ex33.m1.3.3.1.1.5.2.3.cmml" xref="A2.Ex33.m1.3.3.1.1.5.2.3">𝜅</ci><apply id="A2.Ex33.m1.3.3.1.1.5.2.4.cmml" xref="A2.Ex33.m1.3.3.1.1.5.2.4"><csymbol cd="ambiguous" id="A2.Ex33.m1.3.3.1.1.5.2.4.1.cmml" xref="A2.Ex33.m1.3.3.1.1.5.2.4">superscript</csymbol><ci id="A2.Ex33.m1.3.3.1.1.5.2.4.2.cmml" xref="A2.Ex33.m1.3.3.1.1.5.2.4.2">𝜎</ci><times id="A2.Ex33.m1.3.3.1.1.5.2.4.3.cmml" xref="A2.Ex33.m1.3.3.1.1.5.2.4.3"></times></apply><apply id="A2.Ex33.m1.3.3.1.1.5.2.5.cmml" xref="A2.Ex33.m1.3.3.1.1.5.2.5"><csymbol cd="ambiguous" id="A2.Ex33.m1.3.3.1.1.5.2.5.1.cmml" xref="A2.Ex33.m1.3.3.1.1.5.2.5">superscript</csymbol><apply id="A2.Ex33.m1.3.3.1.1.5.2.5.2.cmml" xref="A2.Ex33.m1.3.3.1.1.5.2.5"><csymbol cd="ambiguous" id="A2.Ex33.m1.3.3.1.1.5.2.5.2.1.cmml" xref="A2.Ex33.m1.3.3.1.1.5.2.5">subscript</csymbol><apply id="A2.Ex33.m1.3.3.1.1.5.2.5.2.2.cmml" xref="A2.Ex33.m1.3.3.1.1.5.2.5.2.2"><ci id="A2.Ex33.m1.3.3.1.1.5.2.5.2.2.1.cmml" xref="A2.Ex33.m1.3.3.1.1.5.2.5.2.2.1">~</ci><ci id="A2.Ex33.m1.3.3.1.1.5.2.5.2.2.2.cmml" xref="A2.Ex33.m1.3.3.1.1.5.2.5.2.2.2">𝜽</ci></apply><ci id="A2.Ex33.m1.3.3.1.1.5.2.5.2.3.cmml" xref="A2.Ex33.m1.3.3.1.1.5.2.5.2.3">𝜁</ci></apply><ci id="A2.Ex33.m1.3.3.1.1.5.2.5.3.cmml" xref="A2.Ex33.m1.3.3.1.1.5.2.5.3">𝑇</ci></apply><apply id="A2.Ex33.m1.3.3.1.1.5.2.6.cmml" xref="A2.Ex33.m1.3.3.1.1.5.2.6"><csymbol cd="ambiguous" id="A2.Ex33.m1.3.3.1.1.5.2.6.1.cmml" xref="A2.Ex33.m1.3.3.1.1.5.2.6">subscript</csymbol><apply id="A2.Ex33.m1.3.3.1.1.5.2.6.2.cmml" xref="A2.Ex33.m1.3.3.1.1.5.2.6.2"><ci id="A2.Ex33.m1.3.3.1.1.5.2.6.2.1.cmml" xref="A2.Ex33.m1.3.3.1.1.5.2.6.2.1">~</ci><ci id="A2.Ex33.m1.3.3.1.1.5.2.6.2.2.cmml" xref="A2.Ex33.m1.3.3.1.1.5.2.6.2.2">𝜽</ci></apply><ci id="A2.Ex33.m1.3.3.1.1.5.2.6.3.cmml" xref="A2.Ex33.m1.3.3.1.1.5.2.6.3">𝜁</ci></apply></apply></apply></apply><apply id="A2.Ex33.m1.3.3.1.1c.cmml" xref="A2.Ex33.m1.3.3.1"><leq id="A2.Ex33.m1.3.3.1.1.6.cmml" xref="A2.Ex33.m1.3.3.1.1.6"></leq><share href="https://arxiv.org/html/2401.10785v2#A2.Ex33.m1.3.3.1.1.5.cmml" id="A2.Ex33.m1.3.3.1.1d.cmml" xref="A2.Ex33.m1.3.3.1"></share><apply id="A2.Ex33.m1.3.3.1.1.1.cmml" xref="A2.Ex33.m1.3.3.1.1.1"><minus id="A2.Ex33.m1.3.3.1.1.1.2.cmml" xref="A2.Ex33.m1.3.3.1.1.1"></minus><apply id="A2.Ex33.m1.3.3.1.1.1.1.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1"><times id="A2.Ex33.m1.3.3.1.1.1.1.2.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.2"></times><cn id="A2.Ex33.m1.3.3.1.1.1.1.3.cmml" type="integer" xref="A2.Ex33.m1.3.3.1.1.1.1.3">2</cn><ci id="A2.Ex33.m1.3.3.1.1.1.1.4.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.4">𝜅</ci><apply id="A2.Ex33.m1.3.3.1.1.1.1.5.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.5"><csymbol cd="ambiguous" id="A2.Ex33.m1.3.3.1.1.1.1.5.1.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.5">superscript</csymbol><ci id="A2.Ex33.m1.3.3.1.1.1.1.5.2.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.5.2">𝜎</ci><times id="A2.Ex33.m1.3.3.1.1.1.1.5.3.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.5.3"></times></apply><apply id="A2.Ex33.m1.3.3.1.1.1.1.6.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.6"><csymbol cd="ambiguous" id="A2.Ex33.m1.3.3.1.1.1.1.6.1.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.6">subscript</csymbol><ci id="A2.Ex33.m1.3.3.1.1.1.1.6.2.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.6.2">𝜆</ci><min id="A2.Ex33.m1.3.3.1.1.1.1.6.3.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.6.3"></min></apply><apply id="A2.Ex33.m1.3.3.1.1.1.1.1.1.1.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex33.m1.3.3.1.1.1.1.1.1.1.1.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex33.m1.3.3.1.1.1.1.1.1.1.2.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.1.1.1.2">Γ</ci><ci id="A2.Ex33.m1.3.3.1.1.1.1.1.1.1.3.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.1.1.1.3">𝜁</ci></apply><apply id="A2.Ex33.m1.3.3.1.1.1.1.7.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.7"><csymbol cd="ambiguous" id="A2.Ex33.m1.3.3.1.1.1.1.7.1.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.7">superscript</csymbol><apply id="A2.Ex33.m1.3.3.1.1.1.1.7.2.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.7"><csymbol cd="ambiguous" id="A2.Ex33.m1.3.3.1.1.1.1.7.2.1.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.7">subscript</csymbol><apply id="A2.Ex33.m1.3.3.1.1.1.1.7.2.2.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.7.2.2"><ci id="A2.Ex33.m1.3.3.1.1.1.1.7.2.2.1.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.7.2.2.1">~</ci><ci id="A2.Ex33.m1.3.3.1.1.1.1.7.2.2.2.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.7.2.2.2">𝜽</ci></apply><ci id="A2.Ex33.m1.3.3.1.1.1.1.7.2.3.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.7.2.3">𝜁</ci></apply><ci id="A2.Ex33.m1.3.3.1.1.1.1.7.3.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.7.3">𝑇</ci></apply><apply id="A2.Ex33.m1.3.3.1.1.1.1.8.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.8"><csymbol cd="ambiguous" id="A2.Ex33.m1.3.3.1.1.1.1.8.1.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.8">subscript</csymbol><apply id="A2.Ex33.m1.3.3.1.1.1.1.8.2.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.8"><csymbol cd="ambiguous" id="A2.Ex33.m1.3.3.1.1.1.1.8.2.1.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.8">superscript</csymbol><ci id="A2.Ex33.m1.3.3.1.1.1.1.8.2.2.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.8.2.2">Γ</ci><apply id="A2.Ex33.m1.3.3.1.1.1.1.8.2.3.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.8.2.3"><minus id="A2.Ex33.m1.3.3.1.1.1.1.8.2.3.1.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.8.2.3"></minus><cn id="A2.Ex33.m1.3.3.1.1.1.1.8.2.3.2.cmml" type="integer" xref="A2.Ex33.m1.3.3.1.1.1.1.8.2.3.2">1</cn></apply></apply><ci id="A2.Ex33.m1.3.3.1.1.1.1.8.3.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.8.3">𝜁</ci></apply><apply id="A2.Ex33.m1.3.3.1.1.1.1.9.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.9"><csymbol cd="ambiguous" id="A2.Ex33.m1.3.3.1.1.1.1.9.1.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.9">subscript</csymbol><apply id="A2.Ex33.m1.3.3.1.1.1.1.9.2.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.9.2"><ci id="A2.Ex33.m1.3.3.1.1.1.1.9.2.1.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.9.2.1">~</ci><ci id="A2.Ex33.m1.3.3.1.1.1.1.9.2.2.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.9.2.2">𝜽</ci></apply><ci id="A2.Ex33.m1.3.3.1.1.1.1.9.3.cmml" xref="A2.Ex33.m1.3.3.1.1.1.1.9.3">𝜁</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex33.m1.3c">\displaystyle\dot{V}_{\theta,\zeta}\leq-2\kappa\sigma^{*}\tilde{\bm{\theta}}_{% \zeta}^{T}{\tilde{\bm{\theta}}}_{\zeta}\leq-2\kappa\sigma^{*}\lambda_{\min}(% \Gamma_{\zeta})\tilde{\bm{\theta}}_{\zeta}^{T}\Gamma^{-1}_{\zeta}{\tilde{\bm{% \theta}}}_{\zeta}.</annotation><annotation encoding="application/x-llamapun" id="A2.Ex33.m1.3d">over˙ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_θ , italic_ζ end_POSTSUBSCRIPT ≤ - 2 italic_κ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ≤ - 2 italic_κ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Γ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Γ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A2.2.p2.22">Thus, one immediately obtains</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx46"> <tbody id="A2.Ex34"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}_{\theta,\zeta}(t)\leq-k_{\zeta}V_{\theta,\zeta}(t),t\geq T% _{\rm a}" class="ltx_Math" display="inline" id="A2.Ex34.m1.8"><semantics id="A2.Ex34.m1.8a"><mrow id="A2.Ex34.m1.8.8.2" xref="A2.Ex34.m1.8.8.3.cmml"><mrow id="A2.Ex34.m1.7.7.1.1" xref="A2.Ex34.m1.7.7.1.1.cmml"><mrow id="A2.Ex34.m1.7.7.1.1.2" xref="A2.Ex34.m1.7.7.1.1.2.cmml"><msub id="A2.Ex34.m1.7.7.1.1.2.2" xref="A2.Ex34.m1.7.7.1.1.2.2.cmml"><mover accent="true" id="A2.Ex34.m1.7.7.1.1.2.2.2" xref="A2.Ex34.m1.7.7.1.1.2.2.2.cmml"><mi id="A2.Ex34.m1.7.7.1.1.2.2.2.2" xref="A2.Ex34.m1.7.7.1.1.2.2.2.2.cmml">V</mi><mo id="A2.Ex34.m1.7.7.1.1.2.2.2.1" xref="A2.Ex34.m1.7.7.1.1.2.2.2.1.cmml">˙</mo></mover><mrow id="A2.Ex34.m1.2.2.2.4" xref="A2.Ex34.m1.2.2.2.3.cmml"><mi id="A2.Ex34.m1.1.1.1.1" xref="A2.Ex34.m1.1.1.1.1.cmml">θ</mi><mo id="A2.Ex34.m1.2.2.2.4.1" xref="A2.Ex34.m1.2.2.2.3.cmml">,</mo><mi id="A2.Ex34.m1.2.2.2.2" xref="A2.Ex34.m1.2.2.2.2.cmml">ζ</mi></mrow></msub><mo id="A2.Ex34.m1.7.7.1.1.2.1" xref="A2.Ex34.m1.7.7.1.1.2.1.cmml">⁢</mo><mrow id="A2.Ex34.m1.7.7.1.1.2.3.2" xref="A2.Ex34.m1.7.7.1.1.2.cmml"><mo id="A2.Ex34.m1.7.7.1.1.2.3.2.1" stretchy="false" xref="A2.Ex34.m1.7.7.1.1.2.cmml">(</mo><mi id="A2.Ex34.m1.5.5" xref="A2.Ex34.m1.5.5.cmml">t</mi><mo id="A2.Ex34.m1.7.7.1.1.2.3.2.2" stretchy="false" xref="A2.Ex34.m1.7.7.1.1.2.cmml">)</mo></mrow></mrow><mo id="A2.Ex34.m1.7.7.1.1.1" xref="A2.Ex34.m1.7.7.1.1.1.cmml">≤</mo><mrow id="A2.Ex34.m1.7.7.1.1.3" xref="A2.Ex34.m1.7.7.1.1.3.cmml"><mo id="A2.Ex34.m1.7.7.1.1.3a" xref="A2.Ex34.m1.7.7.1.1.3.cmml">−</mo><mrow id="A2.Ex34.m1.7.7.1.1.3.2" xref="A2.Ex34.m1.7.7.1.1.3.2.cmml"><msub id="A2.Ex34.m1.7.7.1.1.3.2.2" xref="A2.Ex34.m1.7.7.1.1.3.2.2.cmml"><mi id="A2.Ex34.m1.7.7.1.1.3.2.2.2" xref="A2.Ex34.m1.7.7.1.1.3.2.2.2.cmml">k</mi><mi id="A2.Ex34.m1.7.7.1.1.3.2.2.3" xref="A2.Ex34.m1.7.7.1.1.3.2.2.3.cmml">ζ</mi></msub><mo id="A2.Ex34.m1.7.7.1.1.3.2.1" xref="A2.Ex34.m1.7.7.1.1.3.2.1.cmml">⁢</mo><msub id="A2.Ex34.m1.7.7.1.1.3.2.3" xref="A2.Ex34.m1.7.7.1.1.3.2.3.cmml"><mi id="A2.Ex34.m1.7.7.1.1.3.2.3.2" xref="A2.Ex34.m1.7.7.1.1.3.2.3.2.cmml">V</mi><mrow id="A2.Ex34.m1.4.4.2.4" xref="A2.Ex34.m1.4.4.2.3.cmml"><mi id="A2.Ex34.m1.3.3.1.1" xref="A2.Ex34.m1.3.3.1.1.cmml">θ</mi><mo id="A2.Ex34.m1.4.4.2.4.1" xref="A2.Ex34.m1.4.4.2.3.cmml">,</mo><mi id="A2.Ex34.m1.4.4.2.2" xref="A2.Ex34.m1.4.4.2.2.cmml">ζ</mi></mrow></msub><mo id="A2.Ex34.m1.7.7.1.1.3.2.1a" xref="A2.Ex34.m1.7.7.1.1.3.2.1.cmml">⁢</mo><mrow id="A2.Ex34.m1.7.7.1.1.3.2.4.2" xref="A2.Ex34.m1.7.7.1.1.3.2.cmml"><mo id="A2.Ex34.m1.7.7.1.1.3.2.4.2.1" stretchy="false" xref="A2.Ex34.m1.7.7.1.1.3.2.cmml">(</mo><mi id="A2.Ex34.m1.6.6" xref="A2.Ex34.m1.6.6.cmml">t</mi><mo id="A2.Ex34.m1.7.7.1.1.3.2.4.2.2" stretchy="false" xref="A2.Ex34.m1.7.7.1.1.3.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="A2.Ex34.m1.8.8.2.3" xref="A2.Ex34.m1.8.8.3a.cmml">,</mo><mrow id="A2.Ex34.m1.8.8.2.2" xref="A2.Ex34.m1.8.8.2.2.cmml"><mi id="A2.Ex34.m1.8.8.2.2.2" xref="A2.Ex34.m1.8.8.2.2.2.cmml">t</mi><mo id="A2.Ex34.m1.8.8.2.2.1" xref="A2.Ex34.m1.8.8.2.2.1.cmml">≥</mo><msub id="A2.Ex34.m1.8.8.2.2.3" xref="A2.Ex34.m1.8.8.2.2.3.cmml"><mi id="A2.Ex34.m1.8.8.2.2.3.2" xref="A2.Ex34.m1.8.8.2.2.3.2.cmml">T</mi><mi id="A2.Ex34.m1.8.8.2.2.3.3" mathvariant="normal" xref="A2.Ex34.m1.8.8.2.2.3.3.cmml">a</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex34.m1.8b"><apply id="A2.Ex34.m1.8.8.3.cmml" xref="A2.Ex34.m1.8.8.2"><csymbol cd="ambiguous" id="A2.Ex34.m1.8.8.3a.cmml" xref="A2.Ex34.m1.8.8.2.3">formulae-sequence</csymbol><apply id="A2.Ex34.m1.7.7.1.1.cmml" xref="A2.Ex34.m1.7.7.1.1"><leq id="A2.Ex34.m1.7.7.1.1.1.cmml" xref="A2.Ex34.m1.7.7.1.1.1"></leq><apply id="A2.Ex34.m1.7.7.1.1.2.cmml" xref="A2.Ex34.m1.7.7.1.1.2"><times id="A2.Ex34.m1.7.7.1.1.2.1.cmml" xref="A2.Ex34.m1.7.7.1.1.2.1"></times><apply id="A2.Ex34.m1.7.7.1.1.2.2.cmml" xref="A2.Ex34.m1.7.7.1.1.2.2"><csymbol cd="ambiguous" id="A2.Ex34.m1.7.7.1.1.2.2.1.cmml" xref="A2.Ex34.m1.7.7.1.1.2.2">subscript</csymbol><apply id="A2.Ex34.m1.7.7.1.1.2.2.2.cmml" xref="A2.Ex34.m1.7.7.1.1.2.2.2"><ci id="A2.Ex34.m1.7.7.1.1.2.2.2.1.cmml" xref="A2.Ex34.m1.7.7.1.1.2.2.2.1">˙</ci><ci id="A2.Ex34.m1.7.7.1.1.2.2.2.2.cmml" xref="A2.Ex34.m1.7.7.1.1.2.2.2.2">𝑉</ci></apply><list id="A2.Ex34.m1.2.2.2.3.cmml" xref="A2.Ex34.m1.2.2.2.4"><ci id="A2.Ex34.m1.1.1.1.1.cmml" xref="A2.Ex34.m1.1.1.1.1">𝜃</ci><ci id="A2.Ex34.m1.2.2.2.2.cmml" xref="A2.Ex34.m1.2.2.2.2">𝜁</ci></list></apply><ci id="A2.Ex34.m1.5.5.cmml" xref="A2.Ex34.m1.5.5">𝑡</ci></apply><apply id="A2.Ex34.m1.7.7.1.1.3.cmml" xref="A2.Ex34.m1.7.7.1.1.3"><minus id="A2.Ex34.m1.7.7.1.1.3.1.cmml" xref="A2.Ex34.m1.7.7.1.1.3"></minus><apply id="A2.Ex34.m1.7.7.1.1.3.2.cmml" xref="A2.Ex34.m1.7.7.1.1.3.2"><times id="A2.Ex34.m1.7.7.1.1.3.2.1.cmml" xref="A2.Ex34.m1.7.7.1.1.3.2.1"></times><apply id="A2.Ex34.m1.7.7.1.1.3.2.2.cmml" xref="A2.Ex34.m1.7.7.1.1.3.2.2"><csymbol cd="ambiguous" id="A2.Ex34.m1.7.7.1.1.3.2.2.1.cmml" xref="A2.Ex34.m1.7.7.1.1.3.2.2">subscript</csymbol><ci id="A2.Ex34.m1.7.7.1.1.3.2.2.2.cmml" xref="A2.Ex34.m1.7.7.1.1.3.2.2.2">𝑘</ci><ci id="A2.Ex34.m1.7.7.1.1.3.2.2.3.cmml" xref="A2.Ex34.m1.7.7.1.1.3.2.2.3">𝜁</ci></apply><apply id="A2.Ex34.m1.7.7.1.1.3.2.3.cmml" xref="A2.Ex34.m1.7.7.1.1.3.2.3"><csymbol cd="ambiguous" id="A2.Ex34.m1.7.7.1.1.3.2.3.1.cmml" xref="A2.Ex34.m1.7.7.1.1.3.2.3">subscript</csymbol><ci id="A2.Ex34.m1.7.7.1.1.3.2.3.2.cmml" xref="A2.Ex34.m1.7.7.1.1.3.2.3.2">𝑉</ci><list id="A2.Ex34.m1.4.4.2.3.cmml" xref="A2.Ex34.m1.4.4.2.4"><ci id="A2.Ex34.m1.3.3.1.1.cmml" xref="A2.Ex34.m1.3.3.1.1">𝜃</ci><ci id="A2.Ex34.m1.4.4.2.2.cmml" xref="A2.Ex34.m1.4.4.2.2">𝜁</ci></list></apply><ci id="A2.Ex34.m1.6.6.cmml" xref="A2.Ex34.m1.6.6">𝑡</ci></apply></apply></apply><apply id="A2.Ex34.m1.8.8.2.2.cmml" xref="A2.Ex34.m1.8.8.2.2"><geq id="A2.Ex34.m1.8.8.2.2.1.cmml" xref="A2.Ex34.m1.8.8.2.2.1"></geq><ci id="A2.Ex34.m1.8.8.2.2.2.cmml" xref="A2.Ex34.m1.8.8.2.2.2">𝑡</ci><apply id="A2.Ex34.m1.8.8.2.2.3.cmml" xref="A2.Ex34.m1.8.8.2.2.3"><csymbol cd="ambiguous" id="A2.Ex34.m1.8.8.2.2.3.1.cmml" xref="A2.Ex34.m1.8.8.2.2.3">subscript</csymbol><ci id="A2.Ex34.m1.8.8.2.2.3.2.cmml" xref="A2.Ex34.m1.8.8.2.2.3.2">𝑇</ci><ci id="A2.Ex34.m1.8.8.2.2.3.3.cmml" xref="A2.Ex34.m1.8.8.2.2.3.3">a</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex34.m1.8c">\displaystyle\dot{V}_{\theta,\zeta}(t)\leq-k_{\zeta}V_{\theta,\zeta}(t),t\geq T% _{\rm a}</annotation><annotation encoding="application/x-llamapun" id="A2.Ex34.m1.8d">over˙ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_θ , italic_ζ end_POSTSUBSCRIPT ( italic_t ) ≤ - italic_k start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT italic_V start_POSTSUBSCRIPT italic_θ , italic_ζ end_POSTSUBSCRIPT ( italic_t ) , italic_t ≥ italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A2.2.p2.17">with <math alttext="k_{\zeta}" class="ltx_Math" display="inline" id="A2.2.p2.15.m1.1"><semantics id="A2.2.p2.15.m1.1a"><msub id="A2.2.p2.15.m1.1.1" xref="A2.2.p2.15.m1.1.1.cmml"><mi id="A2.2.p2.15.m1.1.1.2" xref="A2.2.p2.15.m1.1.1.2.cmml">k</mi><mi id="A2.2.p2.15.m1.1.1.3" xref="A2.2.p2.15.m1.1.1.3.cmml">ζ</mi></msub><annotation-xml encoding="MathML-Content" id="A2.2.p2.15.m1.1b"><apply id="A2.2.p2.15.m1.1.1.cmml" xref="A2.2.p2.15.m1.1.1"><csymbol cd="ambiguous" id="A2.2.p2.15.m1.1.1.1.cmml" xref="A2.2.p2.15.m1.1.1">subscript</csymbol><ci id="A2.2.p2.15.m1.1.1.2.cmml" xref="A2.2.p2.15.m1.1.1.2">𝑘</ci><ci id="A2.2.p2.15.m1.1.1.3.cmml" xref="A2.2.p2.15.m1.1.1.3">𝜁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.15.m1.1c">k_{\zeta}</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.15.m1.1d">italic_k start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="A2.2.p2.16.m2.1"><semantics id="A2.2.p2.16.m2.1a"><mo id="A2.2.p2.16.m2.1.1" xref="A2.2.p2.16.m2.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="A2.2.p2.16.m2.1b"><csymbol cd="latexml" id="A2.2.p2.16.m2.1.1.cmml" xref="A2.2.p2.16.m2.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.16.m2.1c">:=</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.16.m2.1d">:=</annotation></semantics></math> <math alttext="2\kappa\sigma^{*}\lambda_{\min}(\Gamma_{\zeta})\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A2.2.p2.17.m3.1"><semantics id="A2.2.p2.17.m3.1a"><mrow id="A2.2.p2.17.m3.1.1" xref="A2.2.p2.17.m3.1.1.cmml"><mrow id="A2.2.p2.17.m3.1.1.1" xref="A2.2.p2.17.m3.1.1.1.cmml"><mn id="A2.2.p2.17.m3.1.1.1.3" xref="A2.2.p2.17.m3.1.1.1.3.cmml">2</mn><mo id="A2.2.p2.17.m3.1.1.1.2" xref="A2.2.p2.17.m3.1.1.1.2.cmml">⁢</mo><mi id="A2.2.p2.17.m3.1.1.1.4" xref="A2.2.p2.17.m3.1.1.1.4.cmml">κ</mi><mo id="A2.2.p2.17.m3.1.1.1.2a" xref="A2.2.p2.17.m3.1.1.1.2.cmml">⁢</mo><msup id="A2.2.p2.17.m3.1.1.1.5" xref="A2.2.p2.17.m3.1.1.1.5.cmml"><mi id="A2.2.p2.17.m3.1.1.1.5.2" xref="A2.2.p2.17.m3.1.1.1.5.2.cmml">σ</mi><mo id="A2.2.p2.17.m3.1.1.1.5.3" xref="A2.2.p2.17.m3.1.1.1.5.3.cmml">∗</mo></msup><mo id="A2.2.p2.17.m3.1.1.1.2b" xref="A2.2.p2.17.m3.1.1.1.2.cmml">⁢</mo><msub id="A2.2.p2.17.m3.1.1.1.6" xref="A2.2.p2.17.m3.1.1.1.6.cmml"><mi id="A2.2.p2.17.m3.1.1.1.6.2" xref="A2.2.p2.17.m3.1.1.1.6.2.cmml">λ</mi><mi id="A2.2.p2.17.m3.1.1.1.6.3" xref="A2.2.p2.17.m3.1.1.1.6.3.cmml">min</mi></msub><mo id="A2.2.p2.17.m3.1.1.1.2c" xref="A2.2.p2.17.m3.1.1.1.2.cmml">⁢</mo><mrow id="A2.2.p2.17.m3.1.1.1.1.1" xref="A2.2.p2.17.m3.1.1.1.1.1.1.cmml"><mo id="A2.2.p2.17.m3.1.1.1.1.1.2" stretchy="false" xref="A2.2.p2.17.m3.1.1.1.1.1.1.cmml">(</mo><msub id="A2.2.p2.17.m3.1.1.1.1.1.1" xref="A2.2.p2.17.m3.1.1.1.1.1.1.cmml"><mi id="A2.2.p2.17.m3.1.1.1.1.1.1.2" mathvariant="normal" xref="A2.2.p2.17.m3.1.1.1.1.1.1.2.cmml">Γ</mi><mi id="A2.2.p2.17.m3.1.1.1.1.1.1.3" xref="A2.2.p2.17.m3.1.1.1.1.1.1.3.cmml">ζ</mi></msub><mo id="A2.2.p2.17.m3.1.1.1.1.1.3" stretchy="false" xref="A2.2.p2.17.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A2.2.p2.17.m3.1.1.2" xref="A2.2.p2.17.m3.1.1.2.cmml">∈</mo><msup id="A2.2.p2.17.m3.1.1.3" xref="A2.2.p2.17.m3.1.1.3.cmml"><mi id="A2.2.p2.17.m3.1.1.3.2" xref="A2.2.p2.17.m3.1.1.3.2.cmml">ℝ</mi><mo id="A2.2.p2.17.m3.1.1.3.3" xref="A2.2.p2.17.m3.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.17.m3.1b"><apply id="A2.2.p2.17.m3.1.1.cmml" xref="A2.2.p2.17.m3.1.1"><in id="A2.2.p2.17.m3.1.1.2.cmml" xref="A2.2.p2.17.m3.1.1.2"></in><apply id="A2.2.p2.17.m3.1.1.1.cmml" xref="A2.2.p2.17.m3.1.1.1"><times id="A2.2.p2.17.m3.1.1.1.2.cmml" xref="A2.2.p2.17.m3.1.1.1.2"></times><cn id="A2.2.p2.17.m3.1.1.1.3.cmml" type="integer" xref="A2.2.p2.17.m3.1.1.1.3">2</cn><ci id="A2.2.p2.17.m3.1.1.1.4.cmml" xref="A2.2.p2.17.m3.1.1.1.4">𝜅</ci><apply id="A2.2.p2.17.m3.1.1.1.5.cmml" xref="A2.2.p2.17.m3.1.1.1.5"><csymbol cd="ambiguous" id="A2.2.p2.17.m3.1.1.1.5.1.cmml" xref="A2.2.p2.17.m3.1.1.1.5">superscript</csymbol><ci id="A2.2.p2.17.m3.1.1.1.5.2.cmml" xref="A2.2.p2.17.m3.1.1.1.5.2">𝜎</ci><times id="A2.2.p2.17.m3.1.1.1.5.3.cmml" xref="A2.2.p2.17.m3.1.1.1.5.3"></times></apply><apply id="A2.2.p2.17.m3.1.1.1.6.cmml" xref="A2.2.p2.17.m3.1.1.1.6"><csymbol cd="ambiguous" id="A2.2.p2.17.m3.1.1.1.6.1.cmml" xref="A2.2.p2.17.m3.1.1.1.6">subscript</csymbol><ci id="A2.2.p2.17.m3.1.1.1.6.2.cmml" xref="A2.2.p2.17.m3.1.1.1.6.2">𝜆</ci><min id="A2.2.p2.17.m3.1.1.1.6.3.cmml" xref="A2.2.p2.17.m3.1.1.1.6.3"></min></apply><apply id="A2.2.p2.17.m3.1.1.1.1.1.1.cmml" xref="A2.2.p2.17.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.2.p2.17.m3.1.1.1.1.1.1.1.cmml" xref="A2.2.p2.17.m3.1.1.1.1.1">subscript</csymbol><ci id="A2.2.p2.17.m3.1.1.1.1.1.1.2.cmml" xref="A2.2.p2.17.m3.1.1.1.1.1.1.2">Γ</ci><ci id="A2.2.p2.17.m3.1.1.1.1.1.1.3.cmml" xref="A2.2.p2.17.m3.1.1.1.1.1.1.3">𝜁</ci></apply></apply><apply id="A2.2.p2.17.m3.1.1.3.cmml" xref="A2.2.p2.17.m3.1.1.3"><csymbol cd="ambiguous" id="A2.2.p2.17.m3.1.1.3.1.cmml" xref="A2.2.p2.17.m3.1.1.3">superscript</csymbol><ci id="A2.2.p2.17.m3.1.1.3.2.cmml" xref="A2.2.p2.17.m3.1.1.3.2">ℝ</ci><plus id="A2.2.p2.17.m3.1.1.3.3.cmml" xref="A2.2.p2.17.m3.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.17.m3.1c">2\kappa\sigma^{*}\lambda_{\min}(\Gamma_{\zeta})\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.17.m3.1d">2 italic_κ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Γ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ) ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>. Applying <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib40" title="">40</a>, Lemma A.3.2]</cite> to solve the above inequality, one obtains</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx47"> <tbody id="A2.Ex35"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle V_{\theta,\zeta}(t)\leq V_{\theta,\zeta}(T_{\rm a})e^{-k_{\zeta}% (t-T_{\rm a})}." class="ltx_Math" display="inline" id="A2.Ex35.m1.7"><semantics id="A2.Ex35.m1.7a"><mrow id="A2.Ex35.m1.7.7.1" xref="A2.Ex35.m1.7.7.1.1.cmml"><mrow id="A2.Ex35.m1.7.7.1.1" xref="A2.Ex35.m1.7.7.1.1.cmml"><mrow id="A2.Ex35.m1.7.7.1.1.3" xref="A2.Ex35.m1.7.7.1.1.3.cmml"><msub id="A2.Ex35.m1.7.7.1.1.3.2" xref="A2.Ex35.m1.7.7.1.1.3.2.cmml"><mi id="A2.Ex35.m1.7.7.1.1.3.2.2" xref="A2.Ex35.m1.7.7.1.1.3.2.2.cmml">V</mi><mrow id="A2.Ex35.m1.2.2.2.4" xref="A2.Ex35.m1.2.2.2.3.cmml"><mi id="A2.Ex35.m1.1.1.1.1" xref="A2.Ex35.m1.1.1.1.1.cmml">θ</mi><mo id="A2.Ex35.m1.2.2.2.4.1" xref="A2.Ex35.m1.2.2.2.3.cmml">,</mo><mi id="A2.Ex35.m1.2.2.2.2" xref="A2.Ex35.m1.2.2.2.2.cmml">ζ</mi></mrow></msub><mo id="A2.Ex35.m1.7.7.1.1.3.1" xref="A2.Ex35.m1.7.7.1.1.3.1.cmml">⁢</mo><mrow id="A2.Ex35.m1.7.7.1.1.3.3.2" xref="A2.Ex35.m1.7.7.1.1.3.cmml"><mo id="A2.Ex35.m1.7.7.1.1.3.3.2.1" stretchy="false" xref="A2.Ex35.m1.7.7.1.1.3.cmml">(</mo><mi id="A2.Ex35.m1.6.6" xref="A2.Ex35.m1.6.6.cmml">t</mi><mo id="A2.Ex35.m1.7.7.1.1.3.3.2.2" stretchy="false" xref="A2.Ex35.m1.7.7.1.1.3.cmml">)</mo></mrow></mrow><mo id="A2.Ex35.m1.7.7.1.1.2" xref="A2.Ex35.m1.7.7.1.1.2.cmml">≤</mo><mrow id="A2.Ex35.m1.7.7.1.1.1" xref="A2.Ex35.m1.7.7.1.1.1.cmml"><msub id="A2.Ex35.m1.7.7.1.1.1.3" xref="A2.Ex35.m1.7.7.1.1.1.3.cmml"><mi id="A2.Ex35.m1.7.7.1.1.1.3.2" xref="A2.Ex35.m1.7.7.1.1.1.3.2.cmml">V</mi><mrow id="A2.Ex35.m1.4.4.2.4" xref="A2.Ex35.m1.4.4.2.3.cmml"><mi id="A2.Ex35.m1.3.3.1.1" xref="A2.Ex35.m1.3.3.1.1.cmml">θ</mi><mo id="A2.Ex35.m1.4.4.2.4.1" xref="A2.Ex35.m1.4.4.2.3.cmml">,</mo><mi id="A2.Ex35.m1.4.4.2.2" xref="A2.Ex35.m1.4.4.2.2.cmml">ζ</mi></mrow></msub><mo id="A2.Ex35.m1.7.7.1.1.1.2" xref="A2.Ex35.m1.7.7.1.1.1.2.cmml">⁢</mo><mrow id="A2.Ex35.m1.7.7.1.1.1.1.1" xref="A2.Ex35.m1.7.7.1.1.1.1.1.1.cmml"><mo id="A2.Ex35.m1.7.7.1.1.1.1.1.2" stretchy="false" xref="A2.Ex35.m1.7.7.1.1.1.1.1.1.cmml">(</mo><msub id="A2.Ex35.m1.7.7.1.1.1.1.1.1" xref="A2.Ex35.m1.7.7.1.1.1.1.1.1.cmml"><mi id="A2.Ex35.m1.7.7.1.1.1.1.1.1.2" xref="A2.Ex35.m1.7.7.1.1.1.1.1.1.2.cmml">T</mi><mi id="A2.Ex35.m1.7.7.1.1.1.1.1.1.3" mathvariant="normal" xref="A2.Ex35.m1.7.7.1.1.1.1.1.1.3.cmml">a</mi></msub><mo id="A2.Ex35.m1.7.7.1.1.1.1.1.3" stretchy="false" xref="A2.Ex35.m1.7.7.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A2.Ex35.m1.7.7.1.1.1.2a" xref="A2.Ex35.m1.7.7.1.1.1.2.cmml">⁢</mo><msup id="A2.Ex35.m1.7.7.1.1.1.4" xref="A2.Ex35.m1.7.7.1.1.1.4.cmml"><mi id="A2.Ex35.m1.7.7.1.1.1.4.2" xref="A2.Ex35.m1.7.7.1.1.1.4.2.cmml">e</mi><mrow id="A2.Ex35.m1.5.5.1" xref="A2.Ex35.m1.5.5.1.cmml"><mo id="A2.Ex35.m1.5.5.1a" xref="A2.Ex35.m1.5.5.1.cmml">−</mo><mrow id="A2.Ex35.m1.5.5.1.1" xref="A2.Ex35.m1.5.5.1.1.cmml"><msub id="A2.Ex35.m1.5.5.1.1.3" xref="A2.Ex35.m1.5.5.1.1.3.cmml"><mi id="A2.Ex35.m1.5.5.1.1.3.2" xref="A2.Ex35.m1.5.5.1.1.3.2.cmml">k</mi><mi id="A2.Ex35.m1.5.5.1.1.3.3" xref="A2.Ex35.m1.5.5.1.1.3.3.cmml">ζ</mi></msub><mo id="A2.Ex35.m1.5.5.1.1.2" xref="A2.Ex35.m1.5.5.1.1.2.cmml">⁢</mo><mrow id="A2.Ex35.m1.5.5.1.1.1.1" xref="A2.Ex35.m1.5.5.1.1.1.1.1.cmml"><mo id="A2.Ex35.m1.5.5.1.1.1.1.2" stretchy="false" xref="A2.Ex35.m1.5.5.1.1.1.1.1.cmml">(</mo><mrow id="A2.Ex35.m1.5.5.1.1.1.1.1" xref="A2.Ex35.m1.5.5.1.1.1.1.1.cmml"><mi id="A2.Ex35.m1.5.5.1.1.1.1.1.2" xref="A2.Ex35.m1.5.5.1.1.1.1.1.2.cmml">t</mi><mo id="A2.Ex35.m1.5.5.1.1.1.1.1.1" xref="A2.Ex35.m1.5.5.1.1.1.1.1.1.cmml">−</mo><msub id="A2.Ex35.m1.5.5.1.1.1.1.1.3" xref="A2.Ex35.m1.5.5.1.1.1.1.1.3.cmml"><mi id="A2.Ex35.m1.5.5.1.1.1.1.1.3.2" xref="A2.Ex35.m1.5.5.1.1.1.1.1.3.2.cmml">T</mi><mi id="A2.Ex35.m1.5.5.1.1.1.1.1.3.3" mathvariant="normal" xref="A2.Ex35.m1.5.5.1.1.1.1.1.3.3.cmml">a</mi></msub></mrow><mo id="A2.Ex35.m1.5.5.1.1.1.1.3" stretchy="false" xref="A2.Ex35.m1.5.5.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow></mrow><mo id="A2.Ex35.m1.7.7.1.2" lspace="0em" xref="A2.Ex35.m1.7.7.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex35.m1.7b"><apply id="A2.Ex35.m1.7.7.1.1.cmml" xref="A2.Ex35.m1.7.7.1"><leq id="A2.Ex35.m1.7.7.1.1.2.cmml" xref="A2.Ex35.m1.7.7.1.1.2"></leq><apply id="A2.Ex35.m1.7.7.1.1.3.cmml" xref="A2.Ex35.m1.7.7.1.1.3"><times id="A2.Ex35.m1.7.7.1.1.3.1.cmml" xref="A2.Ex35.m1.7.7.1.1.3.1"></times><apply id="A2.Ex35.m1.7.7.1.1.3.2.cmml" xref="A2.Ex35.m1.7.7.1.1.3.2"><csymbol cd="ambiguous" id="A2.Ex35.m1.7.7.1.1.3.2.1.cmml" xref="A2.Ex35.m1.7.7.1.1.3.2">subscript</csymbol><ci id="A2.Ex35.m1.7.7.1.1.3.2.2.cmml" xref="A2.Ex35.m1.7.7.1.1.3.2.2">𝑉</ci><list id="A2.Ex35.m1.2.2.2.3.cmml" xref="A2.Ex35.m1.2.2.2.4"><ci id="A2.Ex35.m1.1.1.1.1.cmml" xref="A2.Ex35.m1.1.1.1.1">𝜃</ci><ci id="A2.Ex35.m1.2.2.2.2.cmml" xref="A2.Ex35.m1.2.2.2.2">𝜁</ci></list></apply><ci id="A2.Ex35.m1.6.6.cmml" xref="A2.Ex35.m1.6.6">𝑡</ci></apply><apply id="A2.Ex35.m1.7.7.1.1.1.cmml" xref="A2.Ex35.m1.7.7.1.1.1"><times id="A2.Ex35.m1.7.7.1.1.1.2.cmml" xref="A2.Ex35.m1.7.7.1.1.1.2"></times><apply id="A2.Ex35.m1.7.7.1.1.1.3.cmml" xref="A2.Ex35.m1.7.7.1.1.1.3"><csymbol cd="ambiguous" id="A2.Ex35.m1.7.7.1.1.1.3.1.cmml" xref="A2.Ex35.m1.7.7.1.1.1.3">subscript</csymbol><ci id="A2.Ex35.m1.7.7.1.1.1.3.2.cmml" xref="A2.Ex35.m1.7.7.1.1.1.3.2">𝑉</ci><list id="A2.Ex35.m1.4.4.2.3.cmml" xref="A2.Ex35.m1.4.4.2.4"><ci id="A2.Ex35.m1.3.3.1.1.cmml" xref="A2.Ex35.m1.3.3.1.1">𝜃</ci><ci id="A2.Ex35.m1.4.4.2.2.cmml" xref="A2.Ex35.m1.4.4.2.2">𝜁</ci></list></apply><apply id="A2.Ex35.m1.7.7.1.1.1.1.1.1.cmml" xref="A2.Ex35.m1.7.7.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex35.m1.7.7.1.1.1.1.1.1.1.cmml" xref="A2.Ex35.m1.7.7.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex35.m1.7.7.1.1.1.1.1.1.2.cmml" xref="A2.Ex35.m1.7.7.1.1.1.1.1.1.2">𝑇</ci><ci id="A2.Ex35.m1.7.7.1.1.1.1.1.1.3.cmml" xref="A2.Ex35.m1.7.7.1.1.1.1.1.1.3">a</ci></apply><apply id="A2.Ex35.m1.7.7.1.1.1.4.cmml" xref="A2.Ex35.m1.7.7.1.1.1.4"><csymbol cd="ambiguous" id="A2.Ex35.m1.7.7.1.1.1.4.1.cmml" xref="A2.Ex35.m1.7.7.1.1.1.4">superscript</csymbol><ci id="A2.Ex35.m1.7.7.1.1.1.4.2.cmml" xref="A2.Ex35.m1.7.7.1.1.1.4.2">𝑒</ci><apply id="A2.Ex35.m1.5.5.1.cmml" xref="A2.Ex35.m1.5.5.1"><minus id="A2.Ex35.m1.5.5.1.2.cmml" xref="A2.Ex35.m1.5.5.1"></minus><apply id="A2.Ex35.m1.5.5.1.1.cmml" xref="A2.Ex35.m1.5.5.1.1"><times id="A2.Ex35.m1.5.5.1.1.2.cmml" xref="A2.Ex35.m1.5.5.1.1.2"></times><apply id="A2.Ex35.m1.5.5.1.1.3.cmml" xref="A2.Ex35.m1.5.5.1.1.3"><csymbol cd="ambiguous" id="A2.Ex35.m1.5.5.1.1.3.1.cmml" xref="A2.Ex35.m1.5.5.1.1.3">subscript</csymbol><ci id="A2.Ex35.m1.5.5.1.1.3.2.cmml" xref="A2.Ex35.m1.5.5.1.1.3.2">𝑘</ci><ci id="A2.Ex35.m1.5.5.1.1.3.3.cmml" xref="A2.Ex35.m1.5.5.1.1.3.3">𝜁</ci></apply><apply id="A2.Ex35.m1.5.5.1.1.1.1.1.cmml" xref="A2.Ex35.m1.5.5.1.1.1.1"><minus id="A2.Ex35.m1.5.5.1.1.1.1.1.1.cmml" xref="A2.Ex35.m1.5.5.1.1.1.1.1.1"></minus><ci id="A2.Ex35.m1.5.5.1.1.1.1.1.2.cmml" xref="A2.Ex35.m1.5.5.1.1.1.1.1.2">𝑡</ci><apply id="A2.Ex35.m1.5.5.1.1.1.1.1.3.cmml" xref="A2.Ex35.m1.5.5.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A2.Ex35.m1.5.5.1.1.1.1.1.3.1.cmml" xref="A2.Ex35.m1.5.5.1.1.1.1.1.3">subscript</csymbol><ci id="A2.Ex35.m1.5.5.1.1.1.1.1.3.2.cmml" xref="A2.Ex35.m1.5.5.1.1.1.1.1.3.2">𝑇</ci><ci id="A2.Ex35.m1.5.5.1.1.1.1.1.3.3.cmml" xref="A2.Ex35.m1.5.5.1.1.1.1.1.3.3">a</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex35.m1.7c">\displaystyle V_{\theta,\zeta}(t)\leq V_{\theta,\zeta}(T_{\rm a})e^{-k_{\zeta}% (t-T_{\rm a})}.</annotation><annotation encoding="application/x-llamapun" id="A2.Ex35.m1.7d">italic_V start_POSTSUBSCRIPT italic_θ , italic_ζ end_POSTSUBSCRIPT ( italic_t ) ≤ italic_V start_POSTSUBSCRIPT italic_θ , italic_ζ end_POSTSUBSCRIPT ( italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT ) italic_e start_POSTSUPERSCRIPT - italic_k start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t - italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A2.2.p2.23">Combining (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A2.E37" title="In Proof. ‣ Appendix B The proof of Theorem 1 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">37</span></a>) with the above result yields</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx48"> <tbody id="A2.Ex36"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\tilde{\bm{\theta}}_{\zeta}(t)\|^{2}\leq" class="ltx_Math" display="inline" id="A2.Ex36.m1.2"><semantics id="A2.Ex36.m1.2a"><mrow id="A2.Ex36.m1.2.2" xref="A2.Ex36.m1.2.2.cmml"><msup id="A2.Ex36.m1.2.2.1" xref="A2.Ex36.m1.2.2.1.cmml"><mrow id="A2.Ex36.m1.2.2.1.1.1" xref="A2.Ex36.m1.2.2.1.1.2.cmml"><mo id="A2.Ex36.m1.2.2.1.1.1.2" stretchy="false" xref="A2.Ex36.m1.2.2.1.1.2.1.cmml">‖</mo><mrow id="A2.Ex36.m1.2.2.1.1.1.1" xref="A2.Ex36.m1.2.2.1.1.1.1.cmml"><msub id="A2.Ex36.m1.2.2.1.1.1.1.2" xref="A2.Ex36.m1.2.2.1.1.1.1.2.cmml"><mover accent="true" id="A2.Ex36.m1.2.2.1.1.1.1.2.2" xref="A2.Ex36.m1.2.2.1.1.1.1.2.2.cmml"><mi id="A2.Ex36.m1.2.2.1.1.1.1.2.2.2" xref="A2.Ex36.m1.2.2.1.1.1.1.2.2.2.cmml">𝜽</mi><mo id="A2.Ex36.m1.2.2.1.1.1.1.2.2.1" xref="A2.Ex36.m1.2.2.1.1.1.1.2.2.1.cmml">~</mo></mover><mi id="A2.Ex36.m1.2.2.1.1.1.1.2.3" xref="A2.Ex36.m1.2.2.1.1.1.1.2.3.cmml">ζ</mi></msub><mo id="A2.Ex36.m1.2.2.1.1.1.1.1" xref="A2.Ex36.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="A2.Ex36.m1.2.2.1.1.1.1.3.2" xref="A2.Ex36.m1.2.2.1.1.1.1.cmml"><mo id="A2.Ex36.m1.2.2.1.1.1.1.3.2.1" stretchy="false" xref="A2.Ex36.m1.2.2.1.1.1.1.cmml">(</mo><mi id="A2.Ex36.m1.1.1" xref="A2.Ex36.m1.1.1.cmml">t</mi><mo id="A2.Ex36.m1.2.2.1.1.1.1.3.2.2" stretchy="false" xref="A2.Ex36.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A2.Ex36.m1.2.2.1.1.1.3" stretchy="false" xref="A2.Ex36.m1.2.2.1.1.2.1.cmml">‖</mo></mrow><mn id="A2.Ex36.m1.2.2.1.3" xref="A2.Ex36.m1.2.2.1.3.cmml">2</mn></msup><mo id="A2.Ex36.m1.2.2.2" xref="A2.Ex36.m1.2.2.2.cmml">≤</mo><mi id="A2.Ex36.m1.2.2.3" xref="A2.Ex36.m1.2.2.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex36.m1.2b"><apply id="A2.Ex36.m1.2.2.cmml" xref="A2.Ex36.m1.2.2"><leq id="A2.Ex36.m1.2.2.2.cmml" xref="A2.Ex36.m1.2.2.2"></leq><apply id="A2.Ex36.m1.2.2.1.cmml" xref="A2.Ex36.m1.2.2.1"><csymbol cd="ambiguous" id="A2.Ex36.m1.2.2.1.2.cmml" xref="A2.Ex36.m1.2.2.1">superscript</csymbol><apply id="A2.Ex36.m1.2.2.1.1.2.cmml" xref="A2.Ex36.m1.2.2.1.1.1"><csymbol cd="latexml" id="A2.Ex36.m1.2.2.1.1.2.1.cmml" xref="A2.Ex36.m1.2.2.1.1.1.2">norm</csymbol><apply id="A2.Ex36.m1.2.2.1.1.1.1.cmml" xref="A2.Ex36.m1.2.2.1.1.1.1"><times id="A2.Ex36.m1.2.2.1.1.1.1.1.cmml" xref="A2.Ex36.m1.2.2.1.1.1.1.1"></times><apply id="A2.Ex36.m1.2.2.1.1.1.1.2.cmml" xref="A2.Ex36.m1.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="A2.Ex36.m1.2.2.1.1.1.1.2.1.cmml" xref="A2.Ex36.m1.2.2.1.1.1.1.2">subscript</csymbol><apply id="A2.Ex36.m1.2.2.1.1.1.1.2.2.cmml" xref="A2.Ex36.m1.2.2.1.1.1.1.2.2"><ci id="A2.Ex36.m1.2.2.1.1.1.1.2.2.1.cmml" xref="A2.Ex36.m1.2.2.1.1.1.1.2.2.1">~</ci><ci id="A2.Ex36.m1.2.2.1.1.1.1.2.2.2.cmml" xref="A2.Ex36.m1.2.2.1.1.1.1.2.2.2">𝜽</ci></apply><ci id="A2.Ex36.m1.2.2.1.1.1.1.2.3.cmml" xref="A2.Ex36.m1.2.2.1.1.1.1.2.3">𝜁</ci></apply><ci id="A2.Ex36.m1.1.1.cmml" xref="A2.Ex36.m1.1.1">𝑡</ci></apply></apply><cn id="A2.Ex36.m1.2.2.1.3.cmml" type="integer" xref="A2.Ex36.m1.2.2.1.3">2</cn></apply><csymbol cd="latexml" id="A2.Ex36.m1.2.2.3.cmml" xref="A2.Ex36.m1.2.2.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex36.m1.2c">\displaystyle\|\tilde{\bm{\theta}}_{\zeta}(t)\|^{2}\leq</annotation><annotation encoding="application/x-llamapun" id="A2.Ex36.m1.2d">∥ over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t ) ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\lambda_{\max}(\Gamma_{\zeta})\tilde{\bm{\theta}}_{\zeta}^{T}(t)% \Gamma_{\zeta}^{-1}\tilde{\bm{\theta}}_{\zeta}(t)=\lambda_{\max}(\Gamma_{\zeta% })V_{\theta,\zeta}(t)" class="ltx_Math" display="inline" id="A2.Ex36.m2.7"><semantics id="A2.Ex36.m2.7a"><mrow id="A2.Ex36.m2.7.7" xref="A2.Ex36.m2.7.7.cmml"><mrow id="A2.Ex36.m2.6.6.1" xref="A2.Ex36.m2.6.6.1.cmml"><msub id="A2.Ex36.m2.6.6.1.3" xref="A2.Ex36.m2.6.6.1.3.cmml"><mi id="A2.Ex36.m2.6.6.1.3.2" xref="A2.Ex36.m2.6.6.1.3.2.cmml">λ</mi><mi id="A2.Ex36.m2.6.6.1.3.3" xref="A2.Ex36.m2.6.6.1.3.3.cmml">max</mi></msub><mo id="A2.Ex36.m2.6.6.1.2" xref="A2.Ex36.m2.6.6.1.2.cmml">⁢</mo><mrow id="A2.Ex36.m2.6.6.1.1.1" xref="A2.Ex36.m2.6.6.1.1.1.1.cmml"><mo id="A2.Ex36.m2.6.6.1.1.1.2" stretchy="false" xref="A2.Ex36.m2.6.6.1.1.1.1.cmml">(</mo><msub id="A2.Ex36.m2.6.6.1.1.1.1" xref="A2.Ex36.m2.6.6.1.1.1.1.cmml"><mi id="A2.Ex36.m2.6.6.1.1.1.1.2" mathvariant="normal" xref="A2.Ex36.m2.6.6.1.1.1.1.2.cmml">Γ</mi><mi id="A2.Ex36.m2.6.6.1.1.1.1.3" xref="A2.Ex36.m2.6.6.1.1.1.1.3.cmml">ζ</mi></msub><mo id="A2.Ex36.m2.6.6.1.1.1.3" stretchy="false" xref="A2.Ex36.m2.6.6.1.1.1.1.cmml">)</mo></mrow><mo id="A2.Ex36.m2.6.6.1.2a" xref="A2.Ex36.m2.6.6.1.2.cmml">⁢</mo><msubsup id="A2.Ex36.m2.6.6.1.4" xref="A2.Ex36.m2.6.6.1.4.cmml"><mover accent="true" id="A2.Ex36.m2.6.6.1.4.2.2" xref="A2.Ex36.m2.6.6.1.4.2.2.cmml"><mi id="A2.Ex36.m2.6.6.1.4.2.2.2" xref="A2.Ex36.m2.6.6.1.4.2.2.2.cmml">𝜽</mi><mo id="A2.Ex36.m2.6.6.1.4.2.2.1" xref="A2.Ex36.m2.6.6.1.4.2.2.1.cmml">~</mo></mover><mi id="A2.Ex36.m2.6.6.1.4.2.3" xref="A2.Ex36.m2.6.6.1.4.2.3.cmml">ζ</mi><mi id="A2.Ex36.m2.6.6.1.4.3" xref="A2.Ex36.m2.6.6.1.4.3.cmml">T</mi></msubsup><mo id="A2.Ex36.m2.6.6.1.2b" xref="A2.Ex36.m2.6.6.1.2.cmml">⁢</mo><mrow id="A2.Ex36.m2.6.6.1.5.2" xref="A2.Ex36.m2.6.6.1.cmml"><mo id="A2.Ex36.m2.6.6.1.5.2.1" stretchy="false" xref="A2.Ex36.m2.6.6.1.cmml">(</mo><mi id="A2.Ex36.m2.3.3" xref="A2.Ex36.m2.3.3.cmml">t</mi><mo id="A2.Ex36.m2.6.6.1.5.2.2" stretchy="false" xref="A2.Ex36.m2.6.6.1.cmml">)</mo></mrow><mo id="A2.Ex36.m2.6.6.1.2c" xref="A2.Ex36.m2.6.6.1.2.cmml">⁢</mo><msubsup id="A2.Ex36.m2.6.6.1.6" xref="A2.Ex36.m2.6.6.1.6.cmml"><mi id="A2.Ex36.m2.6.6.1.6.2.2" mathvariant="normal" xref="A2.Ex36.m2.6.6.1.6.2.2.cmml">Γ</mi><mi id="A2.Ex36.m2.6.6.1.6.2.3" xref="A2.Ex36.m2.6.6.1.6.2.3.cmml">ζ</mi><mrow id="A2.Ex36.m2.6.6.1.6.3" xref="A2.Ex36.m2.6.6.1.6.3.cmml"><mo id="A2.Ex36.m2.6.6.1.6.3a" xref="A2.Ex36.m2.6.6.1.6.3.cmml">−</mo><mn id="A2.Ex36.m2.6.6.1.6.3.2" xref="A2.Ex36.m2.6.6.1.6.3.2.cmml">1</mn></mrow></msubsup><mo id="A2.Ex36.m2.6.6.1.2d" xref="A2.Ex36.m2.6.6.1.2.cmml">⁢</mo><msub id="A2.Ex36.m2.6.6.1.7" xref="A2.Ex36.m2.6.6.1.7.cmml"><mover accent="true" id="A2.Ex36.m2.6.6.1.7.2" xref="A2.Ex36.m2.6.6.1.7.2.cmml"><mi id="A2.Ex36.m2.6.6.1.7.2.2" xref="A2.Ex36.m2.6.6.1.7.2.2.cmml">𝜽</mi><mo id="A2.Ex36.m2.6.6.1.7.2.1" xref="A2.Ex36.m2.6.6.1.7.2.1.cmml">~</mo></mover><mi id="A2.Ex36.m2.6.6.1.7.3" xref="A2.Ex36.m2.6.6.1.7.3.cmml">ζ</mi></msub><mo id="A2.Ex36.m2.6.6.1.2e" xref="A2.Ex36.m2.6.6.1.2.cmml">⁢</mo><mrow id="A2.Ex36.m2.6.6.1.8.2" xref="A2.Ex36.m2.6.6.1.cmml"><mo id="A2.Ex36.m2.6.6.1.8.2.1" stretchy="false" xref="A2.Ex36.m2.6.6.1.cmml">(</mo><mi id="A2.Ex36.m2.4.4" xref="A2.Ex36.m2.4.4.cmml">t</mi><mo id="A2.Ex36.m2.6.6.1.8.2.2" stretchy="false" xref="A2.Ex36.m2.6.6.1.cmml">)</mo></mrow></mrow><mo id="A2.Ex36.m2.7.7.3" xref="A2.Ex36.m2.7.7.3.cmml">=</mo><mrow id="A2.Ex36.m2.7.7.2" xref="A2.Ex36.m2.7.7.2.cmml"><msub id="A2.Ex36.m2.7.7.2.3" xref="A2.Ex36.m2.7.7.2.3.cmml"><mi id="A2.Ex36.m2.7.7.2.3.2" xref="A2.Ex36.m2.7.7.2.3.2.cmml">λ</mi><mi id="A2.Ex36.m2.7.7.2.3.3" xref="A2.Ex36.m2.7.7.2.3.3.cmml">max</mi></msub><mo id="A2.Ex36.m2.7.7.2.2" xref="A2.Ex36.m2.7.7.2.2.cmml">⁢</mo><mrow id="A2.Ex36.m2.7.7.2.1.1" xref="A2.Ex36.m2.7.7.2.1.1.1.cmml"><mo id="A2.Ex36.m2.7.7.2.1.1.2" stretchy="false" xref="A2.Ex36.m2.7.7.2.1.1.1.cmml">(</mo><msub id="A2.Ex36.m2.7.7.2.1.1.1" xref="A2.Ex36.m2.7.7.2.1.1.1.cmml"><mi id="A2.Ex36.m2.7.7.2.1.1.1.2" mathvariant="normal" xref="A2.Ex36.m2.7.7.2.1.1.1.2.cmml">Γ</mi><mi id="A2.Ex36.m2.7.7.2.1.1.1.3" xref="A2.Ex36.m2.7.7.2.1.1.1.3.cmml">ζ</mi></msub><mo id="A2.Ex36.m2.7.7.2.1.1.3" stretchy="false" xref="A2.Ex36.m2.7.7.2.1.1.1.cmml">)</mo></mrow><mo id="A2.Ex36.m2.7.7.2.2a" xref="A2.Ex36.m2.7.7.2.2.cmml">⁢</mo><msub id="A2.Ex36.m2.7.7.2.4" xref="A2.Ex36.m2.7.7.2.4.cmml"><mi id="A2.Ex36.m2.7.7.2.4.2" xref="A2.Ex36.m2.7.7.2.4.2.cmml">V</mi><mrow id="A2.Ex36.m2.2.2.2.4" xref="A2.Ex36.m2.2.2.2.3.cmml"><mi id="A2.Ex36.m2.1.1.1.1" xref="A2.Ex36.m2.1.1.1.1.cmml">θ</mi><mo id="A2.Ex36.m2.2.2.2.4.1" xref="A2.Ex36.m2.2.2.2.3.cmml">,</mo><mi id="A2.Ex36.m2.2.2.2.2" xref="A2.Ex36.m2.2.2.2.2.cmml">ζ</mi></mrow></msub><mo id="A2.Ex36.m2.7.7.2.2b" xref="A2.Ex36.m2.7.7.2.2.cmml">⁢</mo><mrow id="A2.Ex36.m2.7.7.2.5.2" xref="A2.Ex36.m2.7.7.2.cmml"><mo id="A2.Ex36.m2.7.7.2.5.2.1" stretchy="false" xref="A2.Ex36.m2.7.7.2.cmml">(</mo><mi id="A2.Ex36.m2.5.5" xref="A2.Ex36.m2.5.5.cmml">t</mi><mo id="A2.Ex36.m2.7.7.2.5.2.2" stretchy="false" xref="A2.Ex36.m2.7.7.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex36.m2.7b"><apply id="A2.Ex36.m2.7.7.cmml" xref="A2.Ex36.m2.7.7"><eq id="A2.Ex36.m2.7.7.3.cmml" xref="A2.Ex36.m2.7.7.3"></eq><apply id="A2.Ex36.m2.6.6.1.cmml" xref="A2.Ex36.m2.6.6.1"><times id="A2.Ex36.m2.6.6.1.2.cmml" xref="A2.Ex36.m2.6.6.1.2"></times><apply id="A2.Ex36.m2.6.6.1.3.cmml" xref="A2.Ex36.m2.6.6.1.3"><csymbol cd="ambiguous" id="A2.Ex36.m2.6.6.1.3.1.cmml" xref="A2.Ex36.m2.6.6.1.3">subscript</csymbol><ci id="A2.Ex36.m2.6.6.1.3.2.cmml" xref="A2.Ex36.m2.6.6.1.3.2">𝜆</ci><max id="A2.Ex36.m2.6.6.1.3.3.cmml" xref="A2.Ex36.m2.6.6.1.3.3"></max></apply><apply id="A2.Ex36.m2.6.6.1.1.1.1.cmml" xref="A2.Ex36.m2.6.6.1.1.1"><csymbol cd="ambiguous" id="A2.Ex36.m2.6.6.1.1.1.1.1.cmml" xref="A2.Ex36.m2.6.6.1.1.1">subscript</csymbol><ci id="A2.Ex36.m2.6.6.1.1.1.1.2.cmml" xref="A2.Ex36.m2.6.6.1.1.1.1.2">Γ</ci><ci id="A2.Ex36.m2.6.6.1.1.1.1.3.cmml" xref="A2.Ex36.m2.6.6.1.1.1.1.3">𝜁</ci></apply><apply id="A2.Ex36.m2.6.6.1.4.cmml" xref="A2.Ex36.m2.6.6.1.4"><csymbol cd="ambiguous" id="A2.Ex36.m2.6.6.1.4.1.cmml" xref="A2.Ex36.m2.6.6.1.4">superscript</csymbol><apply id="A2.Ex36.m2.6.6.1.4.2.cmml" xref="A2.Ex36.m2.6.6.1.4"><csymbol cd="ambiguous" id="A2.Ex36.m2.6.6.1.4.2.1.cmml" xref="A2.Ex36.m2.6.6.1.4">subscript</csymbol><apply id="A2.Ex36.m2.6.6.1.4.2.2.cmml" xref="A2.Ex36.m2.6.6.1.4.2.2"><ci id="A2.Ex36.m2.6.6.1.4.2.2.1.cmml" xref="A2.Ex36.m2.6.6.1.4.2.2.1">~</ci><ci id="A2.Ex36.m2.6.6.1.4.2.2.2.cmml" xref="A2.Ex36.m2.6.6.1.4.2.2.2">𝜽</ci></apply><ci id="A2.Ex36.m2.6.6.1.4.2.3.cmml" xref="A2.Ex36.m2.6.6.1.4.2.3">𝜁</ci></apply><ci id="A2.Ex36.m2.6.6.1.4.3.cmml" xref="A2.Ex36.m2.6.6.1.4.3">𝑇</ci></apply><ci id="A2.Ex36.m2.3.3.cmml" xref="A2.Ex36.m2.3.3">𝑡</ci><apply id="A2.Ex36.m2.6.6.1.6.cmml" xref="A2.Ex36.m2.6.6.1.6"><csymbol cd="ambiguous" id="A2.Ex36.m2.6.6.1.6.1.cmml" xref="A2.Ex36.m2.6.6.1.6">superscript</csymbol><apply id="A2.Ex36.m2.6.6.1.6.2.cmml" xref="A2.Ex36.m2.6.6.1.6"><csymbol cd="ambiguous" id="A2.Ex36.m2.6.6.1.6.2.1.cmml" xref="A2.Ex36.m2.6.6.1.6">subscript</csymbol><ci id="A2.Ex36.m2.6.6.1.6.2.2.cmml" xref="A2.Ex36.m2.6.6.1.6.2.2">Γ</ci><ci id="A2.Ex36.m2.6.6.1.6.2.3.cmml" xref="A2.Ex36.m2.6.6.1.6.2.3">𝜁</ci></apply><apply id="A2.Ex36.m2.6.6.1.6.3.cmml" xref="A2.Ex36.m2.6.6.1.6.3"><minus id="A2.Ex36.m2.6.6.1.6.3.1.cmml" xref="A2.Ex36.m2.6.6.1.6.3"></minus><cn id="A2.Ex36.m2.6.6.1.6.3.2.cmml" type="integer" xref="A2.Ex36.m2.6.6.1.6.3.2">1</cn></apply></apply><apply id="A2.Ex36.m2.6.6.1.7.cmml" xref="A2.Ex36.m2.6.6.1.7"><csymbol cd="ambiguous" id="A2.Ex36.m2.6.6.1.7.1.cmml" xref="A2.Ex36.m2.6.6.1.7">subscript</csymbol><apply id="A2.Ex36.m2.6.6.1.7.2.cmml" xref="A2.Ex36.m2.6.6.1.7.2"><ci id="A2.Ex36.m2.6.6.1.7.2.1.cmml" xref="A2.Ex36.m2.6.6.1.7.2.1">~</ci><ci id="A2.Ex36.m2.6.6.1.7.2.2.cmml" xref="A2.Ex36.m2.6.6.1.7.2.2">𝜽</ci></apply><ci id="A2.Ex36.m2.6.6.1.7.3.cmml" xref="A2.Ex36.m2.6.6.1.7.3">𝜁</ci></apply><ci id="A2.Ex36.m2.4.4.cmml" xref="A2.Ex36.m2.4.4">𝑡</ci></apply><apply id="A2.Ex36.m2.7.7.2.cmml" xref="A2.Ex36.m2.7.7.2"><times id="A2.Ex36.m2.7.7.2.2.cmml" xref="A2.Ex36.m2.7.7.2.2"></times><apply id="A2.Ex36.m2.7.7.2.3.cmml" xref="A2.Ex36.m2.7.7.2.3"><csymbol cd="ambiguous" id="A2.Ex36.m2.7.7.2.3.1.cmml" xref="A2.Ex36.m2.7.7.2.3">subscript</csymbol><ci id="A2.Ex36.m2.7.7.2.3.2.cmml" xref="A2.Ex36.m2.7.7.2.3.2">𝜆</ci><max id="A2.Ex36.m2.7.7.2.3.3.cmml" xref="A2.Ex36.m2.7.7.2.3.3"></max></apply><apply id="A2.Ex36.m2.7.7.2.1.1.1.cmml" xref="A2.Ex36.m2.7.7.2.1.1"><csymbol cd="ambiguous" id="A2.Ex36.m2.7.7.2.1.1.1.1.cmml" xref="A2.Ex36.m2.7.7.2.1.1">subscript</csymbol><ci id="A2.Ex36.m2.7.7.2.1.1.1.2.cmml" xref="A2.Ex36.m2.7.7.2.1.1.1.2">Γ</ci><ci id="A2.Ex36.m2.7.7.2.1.1.1.3.cmml" xref="A2.Ex36.m2.7.7.2.1.1.1.3">𝜁</ci></apply><apply id="A2.Ex36.m2.7.7.2.4.cmml" xref="A2.Ex36.m2.7.7.2.4"><csymbol cd="ambiguous" id="A2.Ex36.m2.7.7.2.4.1.cmml" xref="A2.Ex36.m2.7.7.2.4">subscript</csymbol><ci id="A2.Ex36.m2.7.7.2.4.2.cmml" xref="A2.Ex36.m2.7.7.2.4.2">𝑉</ci><list id="A2.Ex36.m2.2.2.2.3.cmml" xref="A2.Ex36.m2.2.2.2.4"><ci id="A2.Ex36.m2.1.1.1.1.cmml" xref="A2.Ex36.m2.1.1.1.1">𝜃</ci><ci id="A2.Ex36.m2.2.2.2.2.cmml" xref="A2.Ex36.m2.2.2.2.2">𝜁</ci></list></apply><ci id="A2.Ex36.m2.5.5.cmml" xref="A2.Ex36.m2.5.5">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex36.m2.7c">\displaystyle\lambda_{\max}(\Gamma_{\zeta})\tilde{\bm{\theta}}_{\zeta}^{T}(t)% \Gamma_{\zeta}^{-1}\tilde{\bm{\theta}}_{\zeta}(t)=\lambda_{\max}(\Gamma_{\zeta% })V_{\theta,\zeta}(t)</annotation><annotation encoding="application/x-llamapun" id="A2.Ex36.m2.7d">italic_λ start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ( roman_Γ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_t ) roman_Γ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t ) = italic_λ start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ( roman_Γ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ) italic_V start_POSTSUBSCRIPT italic_θ , italic_ζ end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A2.Ex37"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\leq" class="ltx_Math" display="inline" id="A2.Ex37.m1.1"><semantics id="A2.Ex37.m1.1a"><mo id="A2.Ex37.m1.1.1" xref="A2.Ex37.m1.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="A2.Ex37.m1.1b"><leq id="A2.Ex37.m1.1.1.cmml" xref="A2.Ex37.m1.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex37.m1.1c">\displaystyle\leq</annotation><annotation encoding="application/x-llamapun" id="A2.Ex37.m1.1d">≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\lambda_{\max}(\Gamma_{\zeta})V_{\theta,\zeta}(T_{\rm a})e^{-k_{% \zeta}(t-T_{\rm a})},\forall t\geq T_{\rm a}" class="ltx_Math" display="inline" id="A2.Ex37.m2.5"><semantics id="A2.Ex37.m2.5a"><mrow id="A2.Ex37.m2.5.5" xref="A2.Ex37.m2.5.5.cmml"><mrow id="A2.Ex37.m2.5.5.2.2" xref="A2.Ex37.m2.5.5.2.3.cmml"><mrow id="A2.Ex37.m2.4.4.1.1.1" xref="A2.Ex37.m2.4.4.1.1.1.cmml"><msub id="A2.Ex37.m2.4.4.1.1.1.4" xref="A2.Ex37.m2.4.4.1.1.1.4.cmml"><mi id="A2.Ex37.m2.4.4.1.1.1.4.2" xref="A2.Ex37.m2.4.4.1.1.1.4.2.cmml">λ</mi><mi id="A2.Ex37.m2.4.4.1.1.1.4.3" xref="A2.Ex37.m2.4.4.1.1.1.4.3.cmml">max</mi></msub><mo id="A2.Ex37.m2.4.4.1.1.1.3" xref="A2.Ex37.m2.4.4.1.1.1.3.cmml">⁢</mo><mrow id="A2.Ex37.m2.4.4.1.1.1.1.1" xref="A2.Ex37.m2.4.4.1.1.1.1.1.1.cmml"><mo id="A2.Ex37.m2.4.4.1.1.1.1.1.2" stretchy="false" xref="A2.Ex37.m2.4.4.1.1.1.1.1.1.cmml">(</mo><msub id="A2.Ex37.m2.4.4.1.1.1.1.1.1" xref="A2.Ex37.m2.4.4.1.1.1.1.1.1.cmml"><mi id="A2.Ex37.m2.4.4.1.1.1.1.1.1.2" mathvariant="normal" xref="A2.Ex37.m2.4.4.1.1.1.1.1.1.2.cmml">Γ</mi><mi id="A2.Ex37.m2.4.4.1.1.1.1.1.1.3" xref="A2.Ex37.m2.4.4.1.1.1.1.1.1.3.cmml">ζ</mi></msub><mo id="A2.Ex37.m2.4.4.1.1.1.1.1.3" stretchy="false" xref="A2.Ex37.m2.4.4.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A2.Ex37.m2.4.4.1.1.1.3a" xref="A2.Ex37.m2.4.4.1.1.1.3.cmml">⁢</mo><msub id="A2.Ex37.m2.4.4.1.1.1.5" xref="A2.Ex37.m2.4.4.1.1.1.5.cmml"><mi id="A2.Ex37.m2.4.4.1.1.1.5.2" xref="A2.Ex37.m2.4.4.1.1.1.5.2.cmml">V</mi><mrow id="A2.Ex37.m2.2.2.2.4" xref="A2.Ex37.m2.2.2.2.3.cmml"><mi id="A2.Ex37.m2.1.1.1.1" xref="A2.Ex37.m2.1.1.1.1.cmml">θ</mi><mo id="A2.Ex37.m2.2.2.2.4.1" xref="A2.Ex37.m2.2.2.2.3.cmml">,</mo><mi id="A2.Ex37.m2.2.2.2.2" xref="A2.Ex37.m2.2.2.2.2.cmml">ζ</mi></mrow></msub><mo id="A2.Ex37.m2.4.4.1.1.1.3b" xref="A2.Ex37.m2.4.4.1.1.1.3.cmml">⁢</mo><mrow id="A2.Ex37.m2.4.4.1.1.1.2.1" xref="A2.Ex37.m2.4.4.1.1.1.2.1.1.cmml"><mo id="A2.Ex37.m2.4.4.1.1.1.2.1.2" stretchy="false" xref="A2.Ex37.m2.4.4.1.1.1.2.1.1.cmml">(</mo><msub id="A2.Ex37.m2.4.4.1.1.1.2.1.1" xref="A2.Ex37.m2.4.4.1.1.1.2.1.1.cmml"><mi id="A2.Ex37.m2.4.4.1.1.1.2.1.1.2" xref="A2.Ex37.m2.4.4.1.1.1.2.1.1.2.cmml">T</mi><mi id="A2.Ex37.m2.4.4.1.1.1.2.1.1.3" mathvariant="normal" xref="A2.Ex37.m2.4.4.1.1.1.2.1.1.3.cmml">a</mi></msub><mo id="A2.Ex37.m2.4.4.1.1.1.2.1.3" stretchy="false" xref="A2.Ex37.m2.4.4.1.1.1.2.1.1.cmml">)</mo></mrow><mo id="A2.Ex37.m2.4.4.1.1.1.3c" xref="A2.Ex37.m2.4.4.1.1.1.3.cmml">⁢</mo><msup id="A2.Ex37.m2.4.4.1.1.1.6" xref="A2.Ex37.m2.4.4.1.1.1.6.cmml"><mi id="A2.Ex37.m2.4.4.1.1.1.6.2" xref="A2.Ex37.m2.4.4.1.1.1.6.2.cmml">e</mi><mrow id="A2.Ex37.m2.3.3.1" xref="A2.Ex37.m2.3.3.1.cmml"><mo id="A2.Ex37.m2.3.3.1a" xref="A2.Ex37.m2.3.3.1.cmml">−</mo><mrow id="A2.Ex37.m2.3.3.1.1" xref="A2.Ex37.m2.3.3.1.1.cmml"><msub id="A2.Ex37.m2.3.3.1.1.3" xref="A2.Ex37.m2.3.3.1.1.3.cmml"><mi id="A2.Ex37.m2.3.3.1.1.3.2" xref="A2.Ex37.m2.3.3.1.1.3.2.cmml">k</mi><mi id="A2.Ex37.m2.3.3.1.1.3.3" xref="A2.Ex37.m2.3.3.1.1.3.3.cmml">ζ</mi></msub><mo id="A2.Ex37.m2.3.3.1.1.2" xref="A2.Ex37.m2.3.3.1.1.2.cmml">⁢</mo><mrow id="A2.Ex37.m2.3.3.1.1.1.1" xref="A2.Ex37.m2.3.3.1.1.1.1.1.cmml"><mo id="A2.Ex37.m2.3.3.1.1.1.1.2" stretchy="false" xref="A2.Ex37.m2.3.3.1.1.1.1.1.cmml">(</mo><mrow id="A2.Ex37.m2.3.3.1.1.1.1.1" xref="A2.Ex37.m2.3.3.1.1.1.1.1.cmml"><mi id="A2.Ex37.m2.3.3.1.1.1.1.1.2" xref="A2.Ex37.m2.3.3.1.1.1.1.1.2.cmml">t</mi><mo id="A2.Ex37.m2.3.3.1.1.1.1.1.1" xref="A2.Ex37.m2.3.3.1.1.1.1.1.1.cmml">−</mo><msub id="A2.Ex37.m2.3.3.1.1.1.1.1.3" xref="A2.Ex37.m2.3.3.1.1.1.1.1.3.cmml"><mi id="A2.Ex37.m2.3.3.1.1.1.1.1.3.2" xref="A2.Ex37.m2.3.3.1.1.1.1.1.3.2.cmml">T</mi><mi id="A2.Ex37.m2.3.3.1.1.1.1.1.3.3" mathvariant="normal" xref="A2.Ex37.m2.3.3.1.1.1.1.1.3.3.cmml">a</mi></msub></mrow><mo id="A2.Ex37.m2.3.3.1.1.1.1.3" stretchy="false" xref="A2.Ex37.m2.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow><mo id="A2.Ex37.m2.5.5.2.2.3" xref="A2.Ex37.m2.5.5.2.3.cmml">,</mo><mrow id="A2.Ex37.m2.5.5.2.2.2" xref="A2.Ex37.m2.5.5.2.2.2.cmml"><mo id="A2.Ex37.m2.5.5.2.2.2.1" rspace="0.167em" xref="A2.Ex37.m2.5.5.2.2.2.1.cmml">∀</mo><mi id="A2.Ex37.m2.5.5.2.2.2.2" xref="A2.Ex37.m2.5.5.2.2.2.2.cmml">t</mi></mrow></mrow><mo id="A2.Ex37.m2.5.5.3" xref="A2.Ex37.m2.5.5.3.cmml">≥</mo><msub id="A2.Ex37.m2.5.5.4" xref="A2.Ex37.m2.5.5.4.cmml"><mi id="A2.Ex37.m2.5.5.4.2" xref="A2.Ex37.m2.5.5.4.2.cmml">T</mi><mi id="A2.Ex37.m2.5.5.4.3" mathvariant="normal" xref="A2.Ex37.m2.5.5.4.3.cmml">a</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex37.m2.5b"><apply id="A2.Ex37.m2.5.5.cmml" xref="A2.Ex37.m2.5.5"><geq id="A2.Ex37.m2.5.5.3.cmml" xref="A2.Ex37.m2.5.5.3"></geq><list id="A2.Ex37.m2.5.5.2.3.cmml" xref="A2.Ex37.m2.5.5.2.2"><apply id="A2.Ex37.m2.4.4.1.1.1.cmml" xref="A2.Ex37.m2.4.4.1.1.1"><times id="A2.Ex37.m2.4.4.1.1.1.3.cmml" xref="A2.Ex37.m2.4.4.1.1.1.3"></times><apply id="A2.Ex37.m2.4.4.1.1.1.4.cmml" xref="A2.Ex37.m2.4.4.1.1.1.4"><csymbol cd="ambiguous" id="A2.Ex37.m2.4.4.1.1.1.4.1.cmml" xref="A2.Ex37.m2.4.4.1.1.1.4">subscript</csymbol><ci id="A2.Ex37.m2.4.4.1.1.1.4.2.cmml" xref="A2.Ex37.m2.4.4.1.1.1.4.2">𝜆</ci><max id="A2.Ex37.m2.4.4.1.1.1.4.3.cmml" xref="A2.Ex37.m2.4.4.1.1.1.4.3"></max></apply><apply id="A2.Ex37.m2.4.4.1.1.1.1.1.1.cmml" xref="A2.Ex37.m2.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex37.m2.4.4.1.1.1.1.1.1.1.cmml" xref="A2.Ex37.m2.4.4.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex37.m2.4.4.1.1.1.1.1.1.2.cmml" xref="A2.Ex37.m2.4.4.1.1.1.1.1.1.2">Γ</ci><ci id="A2.Ex37.m2.4.4.1.1.1.1.1.1.3.cmml" xref="A2.Ex37.m2.4.4.1.1.1.1.1.1.3">𝜁</ci></apply><apply id="A2.Ex37.m2.4.4.1.1.1.5.cmml" xref="A2.Ex37.m2.4.4.1.1.1.5"><csymbol cd="ambiguous" id="A2.Ex37.m2.4.4.1.1.1.5.1.cmml" xref="A2.Ex37.m2.4.4.1.1.1.5">subscript</csymbol><ci id="A2.Ex37.m2.4.4.1.1.1.5.2.cmml" xref="A2.Ex37.m2.4.4.1.1.1.5.2">𝑉</ci><list id="A2.Ex37.m2.2.2.2.3.cmml" xref="A2.Ex37.m2.2.2.2.4"><ci id="A2.Ex37.m2.1.1.1.1.cmml" xref="A2.Ex37.m2.1.1.1.1">𝜃</ci><ci id="A2.Ex37.m2.2.2.2.2.cmml" xref="A2.Ex37.m2.2.2.2.2">𝜁</ci></list></apply><apply id="A2.Ex37.m2.4.4.1.1.1.2.1.1.cmml" xref="A2.Ex37.m2.4.4.1.1.1.2.1"><csymbol cd="ambiguous" id="A2.Ex37.m2.4.4.1.1.1.2.1.1.1.cmml" xref="A2.Ex37.m2.4.4.1.1.1.2.1">subscript</csymbol><ci id="A2.Ex37.m2.4.4.1.1.1.2.1.1.2.cmml" xref="A2.Ex37.m2.4.4.1.1.1.2.1.1.2">𝑇</ci><ci id="A2.Ex37.m2.4.4.1.1.1.2.1.1.3.cmml" xref="A2.Ex37.m2.4.4.1.1.1.2.1.1.3">a</ci></apply><apply id="A2.Ex37.m2.4.4.1.1.1.6.cmml" xref="A2.Ex37.m2.4.4.1.1.1.6"><csymbol cd="ambiguous" id="A2.Ex37.m2.4.4.1.1.1.6.1.cmml" xref="A2.Ex37.m2.4.4.1.1.1.6">superscript</csymbol><ci id="A2.Ex37.m2.4.4.1.1.1.6.2.cmml" xref="A2.Ex37.m2.4.4.1.1.1.6.2">𝑒</ci><apply id="A2.Ex37.m2.3.3.1.cmml" xref="A2.Ex37.m2.3.3.1"><minus id="A2.Ex37.m2.3.3.1.2.cmml" xref="A2.Ex37.m2.3.3.1"></minus><apply id="A2.Ex37.m2.3.3.1.1.cmml" xref="A2.Ex37.m2.3.3.1.1"><times id="A2.Ex37.m2.3.3.1.1.2.cmml" xref="A2.Ex37.m2.3.3.1.1.2"></times><apply id="A2.Ex37.m2.3.3.1.1.3.cmml" xref="A2.Ex37.m2.3.3.1.1.3"><csymbol cd="ambiguous" id="A2.Ex37.m2.3.3.1.1.3.1.cmml" xref="A2.Ex37.m2.3.3.1.1.3">subscript</csymbol><ci id="A2.Ex37.m2.3.3.1.1.3.2.cmml" xref="A2.Ex37.m2.3.3.1.1.3.2">𝑘</ci><ci id="A2.Ex37.m2.3.3.1.1.3.3.cmml" xref="A2.Ex37.m2.3.3.1.1.3.3">𝜁</ci></apply><apply id="A2.Ex37.m2.3.3.1.1.1.1.1.cmml" xref="A2.Ex37.m2.3.3.1.1.1.1"><minus id="A2.Ex37.m2.3.3.1.1.1.1.1.1.cmml" xref="A2.Ex37.m2.3.3.1.1.1.1.1.1"></minus><ci id="A2.Ex37.m2.3.3.1.1.1.1.1.2.cmml" xref="A2.Ex37.m2.3.3.1.1.1.1.1.2">𝑡</ci><apply id="A2.Ex37.m2.3.3.1.1.1.1.1.3.cmml" xref="A2.Ex37.m2.3.3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A2.Ex37.m2.3.3.1.1.1.1.1.3.1.cmml" xref="A2.Ex37.m2.3.3.1.1.1.1.1.3">subscript</csymbol><ci id="A2.Ex37.m2.3.3.1.1.1.1.1.3.2.cmml" xref="A2.Ex37.m2.3.3.1.1.1.1.1.3.2">𝑇</ci><ci id="A2.Ex37.m2.3.3.1.1.1.1.1.3.3.cmml" xref="A2.Ex37.m2.3.3.1.1.1.1.1.3.3">a</ci></apply></apply></apply></apply></apply></apply><apply id="A2.Ex37.m2.5.5.2.2.2.cmml" xref="A2.Ex37.m2.5.5.2.2.2"><csymbol cd="latexml" id="A2.Ex37.m2.5.5.2.2.2.1.cmml" xref="A2.Ex37.m2.5.5.2.2.2.1">for-all</csymbol><ci id="A2.Ex37.m2.5.5.2.2.2.2.cmml" xref="A2.Ex37.m2.5.5.2.2.2.2">𝑡</ci></apply></list><apply id="A2.Ex37.m2.5.5.4.cmml" xref="A2.Ex37.m2.5.5.4"><csymbol cd="ambiguous" id="A2.Ex37.m2.5.5.4.1.cmml" xref="A2.Ex37.m2.5.5.4">subscript</csymbol><ci id="A2.Ex37.m2.5.5.4.2.cmml" xref="A2.Ex37.m2.5.5.4.2">𝑇</ci><ci id="A2.Ex37.m2.5.5.4.3.cmml" xref="A2.Ex37.m2.5.5.4.3">a</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex37.m2.5c">\displaystyle\lambda_{\max}(\Gamma_{\zeta})V_{\theta,\zeta}(T_{\rm a})e^{-k_{% \zeta}(t-T_{\rm a})},\forall t\geq T_{\rm a}</annotation><annotation encoding="application/x-llamapun" id="A2.Ex37.m2.5d">italic_λ start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ( roman_Γ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ) italic_V start_POSTSUBSCRIPT italic_θ , italic_ζ end_POSTSUBSCRIPT ( italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT ) italic_e start_POSTSUPERSCRIPT - italic_k start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t - italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT , ∀ italic_t ≥ italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A2.2.p2.20">which implies that the partial estimation error <math alttext="\tilde{\bm{\theta}}_{\zeta}(t)" class="ltx_Math" display="inline" id="A2.2.p2.18.m1.1"><semantics id="A2.2.p2.18.m1.1a"><mrow id="A2.2.p2.18.m1.1.2" xref="A2.2.p2.18.m1.1.2.cmml"><msub id="A2.2.p2.18.m1.1.2.2" xref="A2.2.p2.18.m1.1.2.2.cmml"><mover accent="true" id="A2.2.p2.18.m1.1.2.2.2" xref="A2.2.p2.18.m1.1.2.2.2.cmml"><mi id="A2.2.p2.18.m1.1.2.2.2.2" xref="A2.2.p2.18.m1.1.2.2.2.2.cmml">𝜽</mi><mo id="A2.2.p2.18.m1.1.2.2.2.1" xref="A2.2.p2.18.m1.1.2.2.2.1.cmml">~</mo></mover><mi id="A2.2.p2.18.m1.1.2.2.3" xref="A2.2.p2.18.m1.1.2.2.3.cmml">ζ</mi></msub><mo id="A2.2.p2.18.m1.1.2.1" xref="A2.2.p2.18.m1.1.2.1.cmml">⁢</mo><mrow id="A2.2.p2.18.m1.1.2.3.2" xref="A2.2.p2.18.m1.1.2.cmml"><mo id="A2.2.p2.18.m1.1.2.3.2.1" stretchy="false" xref="A2.2.p2.18.m1.1.2.cmml">(</mo><mi id="A2.2.p2.18.m1.1.1" xref="A2.2.p2.18.m1.1.1.cmml">t</mi><mo id="A2.2.p2.18.m1.1.2.3.2.2" stretchy="false" xref="A2.2.p2.18.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.18.m1.1b"><apply id="A2.2.p2.18.m1.1.2.cmml" xref="A2.2.p2.18.m1.1.2"><times id="A2.2.p2.18.m1.1.2.1.cmml" xref="A2.2.p2.18.m1.1.2.1"></times><apply id="A2.2.p2.18.m1.1.2.2.cmml" xref="A2.2.p2.18.m1.1.2.2"><csymbol cd="ambiguous" id="A2.2.p2.18.m1.1.2.2.1.cmml" xref="A2.2.p2.18.m1.1.2.2">subscript</csymbol><apply id="A2.2.p2.18.m1.1.2.2.2.cmml" xref="A2.2.p2.18.m1.1.2.2.2"><ci id="A2.2.p2.18.m1.1.2.2.2.1.cmml" xref="A2.2.p2.18.m1.1.2.2.2.1">~</ci><ci id="A2.2.p2.18.m1.1.2.2.2.2.cmml" xref="A2.2.p2.18.m1.1.2.2.2.2">𝜽</ci></apply><ci id="A2.2.p2.18.m1.1.2.2.3.cmml" xref="A2.2.p2.18.m1.1.2.2.3">𝜁</ci></apply><ci id="A2.2.p2.18.m1.1.1.cmml" xref="A2.2.p2.18.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.18.m1.1c">\tilde{\bm{\theta}}_{\zeta}(t)</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.18.m1.1d">over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> exponentially converges to <math alttext="\bm{0}" class="ltx_Math" display="inline" id="A2.2.p2.19.m2.1"><semantics id="A2.2.p2.19.m2.1a"><mn id="A2.2.p2.19.m2.1.1" xref="A2.2.p2.19.m2.1.1.cmml">𝟎</mn><annotation-xml encoding="MathML-Content" id="A2.2.p2.19.m2.1b"><cn id="A2.2.p2.19.m2.1.1.cmml" type="integer" xref="A2.2.p2.19.m2.1.1">0</cn></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.19.m2.1c">\bm{0}</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.19.m2.1d">bold_0</annotation></semantics></math> on <math alttext="t\in[T_{\rm a},\infty)" class="ltx_Math" display="inline" id="A2.2.p2.20.m3.2"><semantics id="A2.2.p2.20.m3.2a"><mrow id="A2.2.p2.20.m3.2.2" xref="A2.2.p2.20.m3.2.2.cmml"><mi id="A2.2.p2.20.m3.2.2.3" xref="A2.2.p2.20.m3.2.2.3.cmml">t</mi><mo id="A2.2.p2.20.m3.2.2.2" xref="A2.2.p2.20.m3.2.2.2.cmml">∈</mo><mrow id="A2.2.p2.20.m3.2.2.1.1" xref="A2.2.p2.20.m3.2.2.1.2.cmml"><mo id="A2.2.p2.20.m3.2.2.1.1.2" stretchy="false" xref="A2.2.p2.20.m3.2.2.1.2.cmml">[</mo><msub id="A2.2.p2.20.m3.2.2.1.1.1" xref="A2.2.p2.20.m3.2.2.1.1.1.cmml"><mi id="A2.2.p2.20.m3.2.2.1.1.1.2" xref="A2.2.p2.20.m3.2.2.1.1.1.2.cmml">T</mi><mi id="A2.2.p2.20.m3.2.2.1.1.1.3" mathvariant="normal" xref="A2.2.p2.20.m3.2.2.1.1.1.3.cmml">a</mi></msub><mo id="A2.2.p2.20.m3.2.2.1.1.3" xref="A2.2.p2.20.m3.2.2.1.2.cmml">,</mo><mi id="A2.2.p2.20.m3.1.1" mathvariant="normal" xref="A2.2.p2.20.m3.1.1.cmml">∞</mi><mo id="A2.2.p2.20.m3.2.2.1.1.4" stretchy="false" xref="A2.2.p2.20.m3.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.20.m3.2b"><apply id="A2.2.p2.20.m3.2.2.cmml" xref="A2.2.p2.20.m3.2.2"><in id="A2.2.p2.20.m3.2.2.2.cmml" xref="A2.2.p2.20.m3.2.2.2"></in><ci id="A2.2.p2.20.m3.2.2.3.cmml" xref="A2.2.p2.20.m3.2.2.3">𝑡</ci><interval closure="closed-open" id="A2.2.p2.20.m3.2.2.1.2.cmml" xref="A2.2.p2.20.m3.2.2.1.1"><apply id="A2.2.p2.20.m3.2.2.1.1.1.cmml" xref="A2.2.p2.20.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="A2.2.p2.20.m3.2.2.1.1.1.1.cmml" xref="A2.2.p2.20.m3.2.2.1.1.1">subscript</csymbol><ci id="A2.2.p2.20.m3.2.2.1.1.1.2.cmml" xref="A2.2.p2.20.m3.2.2.1.1.1.2">𝑇</ci><ci id="A2.2.p2.20.m3.2.2.1.1.1.3.cmml" xref="A2.2.p2.20.m3.2.2.1.1.1.3">a</ci></apply><infinity id="A2.2.p2.20.m3.1.1.cmml" xref="A2.2.p2.20.m3.1.1"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.20.m3.2c">t\in[T_{\rm a},\infty)</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.20.m3.2d">italic_t ∈ [ italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT , ∞ )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A2.3.p3"> <p class="ltx_p" id="A2.3.p3.2">3) The proof of parameter convergence under IE is similar to those under partial IE, so we omit some similar steps. Applying the Lyapunov function <math alttext="V_{\theta}" class="ltx_Math" display="inline" id="A2.3.p3.1.m1.1"><semantics id="A2.3.p3.1.m1.1a"><msub id="A2.3.p3.1.m1.1.1" xref="A2.3.p3.1.m1.1.1.cmml"><mi id="A2.3.p3.1.m1.1.1.2" xref="A2.3.p3.1.m1.1.1.2.cmml">V</mi><mi id="A2.3.p3.1.m1.1.1.3" xref="A2.3.p3.1.m1.1.1.3.cmml">θ</mi></msub><annotation-xml encoding="MathML-Content" id="A2.3.p3.1.m1.1b"><apply id="A2.3.p3.1.m1.1.1.cmml" xref="A2.3.p3.1.m1.1.1"><csymbol cd="ambiguous" id="A2.3.p3.1.m1.1.1.1.cmml" xref="A2.3.p3.1.m1.1.1">subscript</csymbol><ci id="A2.3.p3.1.m1.1.1.2.cmml" xref="A2.3.p3.1.m1.1.1.2">𝑉</ci><ci id="A2.3.p3.1.m1.1.1.3.cmml" xref="A2.3.p3.1.m1.1.1.3">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.3.p3.1.m1.1c">V_{\theta}</annotation><annotation encoding="application/x-llamapun" id="A2.3.p3.1.m1.1d">italic_V start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A2.E34" title="In Proof. ‣ Appendix B The proof of Theorem 1 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">34</span></a>) and the IE condition <math alttext="\Psi(T_{\rm e})\geq\sigma I" class="ltx_Math" display="inline" id="A2.3.p3.2.m2.1"><semantics id="A2.3.p3.2.m2.1a"><mrow id="A2.3.p3.2.m2.1.1" xref="A2.3.p3.2.m2.1.1.cmml"><mrow id="A2.3.p3.2.m2.1.1.1" xref="A2.3.p3.2.m2.1.1.1.cmml"><mi id="A2.3.p3.2.m2.1.1.1.3" mathvariant="normal" xref="A2.3.p3.2.m2.1.1.1.3.cmml">Ψ</mi><mo id="A2.3.p3.2.m2.1.1.1.2" xref="A2.3.p3.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="A2.3.p3.2.m2.1.1.1.1.1" xref="A2.3.p3.2.m2.1.1.1.1.1.1.cmml"><mo id="A2.3.p3.2.m2.1.1.1.1.1.2" stretchy="false" xref="A2.3.p3.2.m2.1.1.1.1.1.1.cmml">(</mo><msub id="A2.3.p3.2.m2.1.1.1.1.1.1" xref="A2.3.p3.2.m2.1.1.1.1.1.1.cmml"><mi id="A2.3.p3.2.m2.1.1.1.1.1.1.2" xref="A2.3.p3.2.m2.1.1.1.1.1.1.2.cmml">T</mi><mi id="A2.3.p3.2.m2.1.1.1.1.1.1.3" mathvariant="normal" xref="A2.3.p3.2.m2.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="A2.3.p3.2.m2.1.1.1.1.1.3" stretchy="false" xref="A2.3.p3.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A2.3.p3.2.m2.1.1.2" xref="A2.3.p3.2.m2.1.1.2.cmml">≥</mo><mrow id="A2.3.p3.2.m2.1.1.3" xref="A2.3.p3.2.m2.1.1.3.cmml"><mi id="A2.3.p3.2.m2.1.1.3.2" xref="A2.3.p3.2.m2.1.1.3.2.cmml">σ</mi><mo id="A2.3.p3.2.m2.1.1.3.1" xref="A2.3.p3.2.m2.1.1.3.1.cmml">⁢</mo><mi id="A2.3.p3.2.m2.1.1.3.3" xref="A2.3.p3.2.m2.1.1.3.3.cmml">I</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.3.p3.2.m2.1b"><apply id="A2.3.p3.2.m2.1.1.cmml" xref="A2.3.p3.2.m2.1.1"><geq id="A2.3.p3.2.m2.1.1.2.cmml" xref="A2.3.p3.2.m2.1.1.2"></geq><apply id="A2.3.p3.2.m2.1.1.1.cmml" xref="A2.3.p3.2.m2.1.1.1"><times id="A2.3.p3.2.m2.1.1.1.2.cmml" xref="A2.3.p3.2.m2.1.1.1.2"></times><ci id="A2.3.p3.2.m2.1.1.1.3.cmml" xref="A2.3.p3.2.m2.1.1.1.3">Ψ</ci><apply id="A2.3.p3.2.m2.1.1.1.1.1.1.cmml" xref="A2.3.p3.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.3.p3.2.m2.1.1.1.1.1.1.1.cmml" xref="A2.3.p3.2.m2.1.1.1.1.1">subscript</csymbol><ci id="A2.3.p3.2.m2.1.1.1.1.1.1.2.cmml" xref="A2.3.p3.2.m2.1.1.1.1.1.1.2">𝑇</ci><ci id="A2.3.p3.2.m2.1.1.1.1.1.1.3.cmml" xref="A2.3.p3.2.m2.1.1.1.1.1.1.3">e</ci></apply></apply><apply id="A2.3.p3.2.m2.1.1.3.cmml" xref="A2.3.p3.2.m2.1.1.3"><times id="A2.3.p3.2.m2.1.1.3.1.cmml" xref="A2.3.p3.2.m2.1.1.3.1"></times><ci id="A2.3.p3.2.m2.1.1.3.2.cmml" xref="A2.3.p3.2.m2.1.1.3.2">𝜎</ci><ci id="A2.3.p3.2.m2.1.1.3.3.cmml" xref="A2.3.p3.2.m2.1.1.3.3">𝐼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.3.p3.2.m2.1c">\Psi(T_{\rm e})\geq\sigma I</annotation><annotation encoding="application/x-llamapun" id="A2.3.p3.2.m2.1d">roman_Ψ ( italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) ≥ italic_σ italic_I</annotation></semantics></math>, one obtains</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx49"> <tbody id="A2.Ex38"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\tilde{\bm{\theta}}(t)\|^{2}\leq" class="ltx_Math" display="inline" id="A2.Ex38.m1.2"><semantics id="A2.Ex38.m1.2a"><mrow id="A2.Ex38.m1.2.2" xref="A2.Ex38.m1.2.2.cmml"><msup id="A2.Ex38.m1.2.2.1" xref="A2.Ex38.m1.2.2.1.cmml"><mrow id="A2.Ex38.m1.2.2.1.1.1" xref="A2.Ex38.m1.2.2.1.1.2.cmml"><mo id="A2.Ex38.m1.2.2.1.1.1.2" stretchy="false" xref="A2.Ex38.m1.2.2.1.1.2.1.cmml">‖</mo><mrow id="A2.Ex38.m1.2.2.1.1.1.1" xref="A2.Ex38.m1.2.2.1.1.1.1.cmml"><mover accent="true" id="A2.Ex38.m1.2.2.1.1.1.1.2" xref="A2.Ex38.m1.2.2.1.1.1.1.2.cmml"><mi id="A2.Ex38.m1.2.2.1.1.1.1.2.2" xref="A2.Ex38.m1.2.2.1.1.1.1.2.2.cmml">𝜽</mi><mo id="A2.Ex38.m1.2.2.1.1.1.1.2.1" xref="A2.Ex38.m1.2.2.1.1.1.1.2.1.cmml">~</mo></mover><mo id="A2.Ex38.m1.2.2.1.1.1.1.1" xref="A2.Ex38.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="A2.Ex38.m1.2.2.1.1.1.1.3.2" xref="A2.Ex38.m1.2.2.1.1.1.1.cmml"><mo id="A2.Ex38.m1.2.2.1.1.1.1.3.2.1" stretchy="false" xref="A2.Ex38.m1.2.2.1.1.1.1.cmml">(</mo><mi id="A2.Ex38.m1.1.1" xref="A2.Ex38.m1.1.1.cmml">t</mi><mo id="A2.Ex38.m1.2.2.1.1.1.1.3.2.2" stretchy="false" xref="A2.Ex38.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A2.Ex38.m1.2.2.1.1.1.3" stretchy="false" xref="A2.Ex38.m1.2.2.1.1.2.1.cmml">‖</mo></mrow><mn id="A2.Ex38.m1.2.2.1.3" xref="A2.Ex38.m1.2.2.1.3.cmml">2</mn></msup><mo id="A2.Ex38.m1.2.2.2" xref="A2.Ex38.m1.2.2.2.cmml">≤</mo><mi id="A2.Ex38.m1.2.2.3" xref="A2.Ex38.m1.2.2.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex38.m1.2b"><apply id="A2.Ex38.m1.2.2.cmml" xref="A2.Ex38.m1.2.2"><leq id="A2.Ex38.m1.2.2.2.cmml" xref="A2.Ex38.m1.2.2.2"></leq><apply id="A2.Ex38.m1.2.2.1.cmml" xref="A2.Ex38.m1.2.2.1"><csymbol cd="ambiguous" id="A2.Ex38.m1.2.2.1.2.cmml" xref="A2.Ex38.m1.2.2.1">superscript</csymbol><apply id="A2.Ex38.m1.2.2.1.1.2.cmml" xref="A2.Ex38.m1.2.2.1.1.1"><csymbol cd="latexml" id="A2.Ex38.m1.2.2.1.1.2.1.cmml" xref="A2.Ex38.m1.2.2.1.1.1.2">norm</csymbol><apply id="A2.Ex38.m1.2.2.1.1.1.1.cmml" xref="A2.Ex38.m1.2.2.1.1.1.1"><times id="A2.Ex38.m1.2.2.1.1.1.1.1.cmml" xref="A2.Ex38.m1.2.2.1.1.1.1.1"></times><apply id="A2.Ex38.m1.2.2.1.1.1.1.2.cmml" xref="A2.Ex38.m1.2.2.1.1.1.1.2"><ci id="A2.Ex38.m1.2.2.1.1.1.1.2.1.cmml" xref="A2.Ex38.m1.2.2.1.1.1.1.2.1">~</ci><ci id="A2.Ex38.m1.2.2.1.1.1.1.2.2.cmml" xref="A2.Ex38.m1.2.2.1.1.1.1.2.2">𝜽</ci></apply><ci id="A2.Ex38.m1.1.1.cmml" xref="A2.Ex38.m1.1.1">𝑡</ci></apply></apply><cn id="A2.Ex38.m1.2.2.1.3.cmml" type="integer" xref="A2.Ex38.m1.2.2.1.3">2</cn></apply><csymbol cd="latexml" id="A2.Ex38.m1.2.2.3.cmml" xref="A2.Ex38.m1.2.2.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex38.m1.2c">\displaystyle\|\tilde{\bm{\theta}}(t)\|^{2}\leq</annotation><annotation encoding="application/x-llamapun" id="A2.Ex38.m1.2d">∥ over~ start_ARG bold_italic_θ end_ARG ( italic_t ) ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\lambda_{\max}(\Gamma)\tilde{\bm{\theta}}^{T}\Gamma^{-1}\tilde{% \bm{\theta}}=\lambda_{\max}(\Gamma)V_{\theta}(t)" class="ltx_Math" display="inline" id="A2.Ex38.m2.3"><semantics id="A2.Ex38.m2.3a"><mrow id="A2.Ex38.m2.3.4" xref="A2.Ex38.m2.3.4.cmml"><mrow id="A2.Ex38.m2.3.4.2" xref="A2.Ex38.m2.3.4.2.cmml"><msub id="A2.Ex38.m2.3.4.2.2" xref="A2.Ex38.m2.3.4.2.2.cmml"><mi id="A2.Ex38.m2.3.4.2.2.2" xref="A2.Ex38.m2.3.4.2.2.2.cmml">λ</mi><mi id="A2.Ex38.m2.3.4.2.2.3" xref="A2.Ex38.m2.3.4.2.2.3.cmml">max</mi></msub><mo id="A2.Ex38.m2.3.4.2.1" xref="A2.Ex38.m2.3.4.2.1.cmml">⁢</mo><mrow id="A2.Ex38.m2.3.4.2.3.2" xref="A2.Ex38.m2.3.4.2.cmml"><mo id="A2.Ex38.m2.3.4.2.3.2.1" stretchy="false" xref="A2.Ex38.m2.3.4.2.cmml">(</mo><mi id="A2.Ex38.m2.1.1" mathvariant="normal" xref="A2.Ex38.m2.1.1.cmml">Γ</mi><mo id="A2.Ex38.m2.3.4.2.3.2.2" stretchy="false" xref="A2.Ex38.m2.3.4.2.cmml">)</mo></mrow><mo id="A2.Ex38.m2.3.4.2.1a" xref="A2.Ex38.m2.3.4.2.1.cmml">⁢</mo><msup id="A2.Ex38.m2.3.4.2.4" xref="A2.Ex38.m2.3.4.2.4.cmml"><mover accent="true" id="A2.Ex38.m2.3.4.2.4.2" xref="A2.Ex38.m2.3.4.2.4.2.cmml"><mi id="A2.Ex38.m2.3.4.2.4.2.2" xref="A2.Ex38.m2.3.4.2.4.2.2.cmml">𝜽</mi><mo id="A2.Ex38.m2.3.4.2.4.2.1" xref="A2.Ex38.m2.3.4.2.4.2.1.cmml">~</mo></mover><mi id="A2.Ex38.m2.3.4.2.4.3" xref="A2.Ex38.m2.3.4.2.4.3.cmml">T</mi></msup><mo id="A2.Ex38.m2.3.4.2.1b" xref="A2.Ex38.m2.3.4.2.1.cmml">⁢</mo><msup id="A2.Ex38.m2.3.4.2.5" xref="A2.Ex38.m2.3.4.2.5.cmml"><mi id="A2.Ex38.m2.3.4.2.5.2" mathvariant="normal" xref="A2.Ex38.m2.3.4.2.5.2.cmml">Γ</mi><mrow id="A2.Ex38.m2.3.4.2.5.3" xref="A2.Ex38.m2.3.4.2.5.3.cmml"><mo id="A2.Ex38.m2.3.4.2.5.3a" xref="A2.Ex38.m2.3.4.2.5.3.cmml">−</mo><mn id="A2.Ex38.m2.3.4.2.5.3.2" xref="A2.Ex38.m2.3.4.2.5.3.2.cmml">1</mn></mrow></msup><mo id="A2.Ex38.m2.3.4.2.1c" xref="A2.Ex38.m2.3.4.2.1.cmml">⁢</mo><mover accent="true" id="A2.Ex38.m2.3.4.2.6" xref="A2.Ex38.m2.3.4.2.6.cmml"><mi id="A2.Ex38.m2.3.4.2.6.2" xref="A2.Ex38.m2.3.4.2.6.2.cmml">𝜽</mi><mo id="A2.Ex38.m2.3.4.2.6.1" xref="A2.Ex38.m2.3.4.2.6.1.cmml">~</mo></mover></mrow><mo id="A2.Ex38.m2.3.4.1" xref="A2.Ex38.m2.3.4.1.cmml">=</mo><mrow id="A2.Ex38.m2.3.4.3" xref="A2.Ex38.m2.3.4.3.cmml"><msub id="A2.Ex38.m2.3.4.3.2" xref="A2.Ex38.m2.3.4.3.2.cmml"><mi id="A2.Ex38.m2.3.4.3.2.2" xref="A2.Ex38.m2.3.4.3.2.2.cmml">λ</mi><mi id="A2.Ex38.m2.3.4.3.2.3" xref="A2.Ex38.m2.3.4.3.2.3.cmml">max</mi></msub><mo id="A2.Ex38.m2.3.4.3.1" xref="A2.Ex38.m2.3.4.3.1.cmml">⁢</mo><mrow id="A2.Ex38.m2.3.4.3.3.2" xref="A2.Ex38.m2.3.4.3.cmml"><mo id="A2.Ex38.m2.3.4.3.3.2.1" stretchy="false" xref="A2.Ex38.m2.3.4.3.cmml">(</mo><mi id="A2.Ex38.m2.2.2" mathvariant="normal" xref="A2.Ex38.m2.2.2.cmml">Γ</mi><mo id="A2.Ex38.m2.3.4.3.3.2.2" stretchy="false" xref="A2.Ex38.m2.3.4.3.cmml">)</mo></mrow><mo id="A2.Ex38.m2.3.4.3.1a" xref="A2.Ex38.m2.3.4.3.1.cmml">⁢</mo><msub id="A2.Ex38.m2.3.4.3.4" xref="A2.Ex38.m2.3.4.3.4.cmml"><mi id="A2.Ex38.m2.3.4.3.4.2" xref="A2.Ex38.m2.3.4.3.4.2.cmml">V</mi><mi id="A2.Ex38.m2.3.4.3.4.3" xref="A2.Ex38.m2.3.4.3.4.3.cmml">θ</mi></msub><mo id="A2.Ex38.m2.3.4.3.1b" xref="A2.Ex38.m2.3.4.3.1.cmml">⁢</mo><mrow id="A2.Ex38.m2.3.4.3.5.2" xref="A2.Ex38.m2.3.4.3.cmml"><mo id="A2.Ex38.m2.3.4.3.5.2.1" stretchy="false" xref="A2.Ex38.m2.3.4.3.cmml">(</mo><mi id="A2.Ex38.m2.3.3" xref="A2.Ex38.m2.3.3.cmml">t</mi><mo id="A2.Ex38.m2.3.4.3.5.2.2" stretchy="false" xref="A2.Ex38.m2.3.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex38.m2.3b"><apply id="A2.Ex38.m2.3.4.cmml" xref="A2.Ex38.m2.3.4"><eq id="A2.Ex38.m2.3.4.1.cmml" xref="A2.Ex38.m2.3.4.1"></eq><apply id="A2.Ex38.m2.3.4.2.cmml" xref="A2.Ex38.m2.3.4.2"><times id="A2.Ex38.m2.3.4.2.1.cmml" xref="A2.Ex38.m2.3.4.2.1"></times><apply id="A2.Ex38.m2.3.4.2.2.cmml" xref="A2.Ex38.m2.3.4.2.2"><csymbol cd="ambiguous" id="A2.Ex38.m2.3.4.2.2.1.cmml" xref="A2.Ex38.m2.3.4.2.2">subscript</csymbol><ci id="A2.Ex38.m2.3.4.2.2.2.cmml" xref="A2.Ex38.m2.3.4.2.2.2">𝜆</ci><max id="A2.Ex38.m2.3.4.2.2.3.cmml" xref="A2.Ex38.m2.3.4.2.2.3"></max></apply><ci id="A2.Ex38.m2.1.1.cmml" xref="A2.Ex38.m2.1.1">Γ</ci><apply id="A2.Ex38.m2.3.4.2.4.cmml" xref="A2.Ex38.m2.3.4.2.4"><csymbol cd="ambiguous" id="A2.Ex38.m2.3.4.2.4.1.cmml" xref="A2.Ex38.m2.3.4.2.4">superscript</csymbol><apply id="A2.Ex38.m2.3.4.2.4.2.cmml" xref="A2.Ex38.m2.3.4.2.4.2"><ci id="A2.Ex38.m2.3.4.2.4.2.1.cmml" xref="A2.Ex38.m2.3.4.2.4.2.1">~</ci><ci id="A2.Ex38.m2.3.4.2.4.2.2.cmml" xref="A2.Ex38.m2.3.4.2.4.2.2">𝜽</ci></apply><ci id="A2.Ex38.m2.3.4.2.4.3.cmml" xref="A2.Ex38.m2.3.4.2.4.3">𝑇</ci></apply><apply id="A2.Ex38.m2.3.4.2.5.cmml" xref="A2.Ex38.m2.3.4.2.5"><csymbol cd="ambiguous" id="A2.Ex38.m2.3.4.2.5.1.cmml" xref="A2.Ex38.m2.3.4.2.5">superscript</csymbol><ci id="A2.Ex38.m2.3.4.2.5.2.cmml" xref="A2.Ex38.m2.3.4.2.5.2">Γ</ci><apply id="A2.Ex38.m2.3.4.2.5.3.cmml" xref="A2.Ex38.m2.3.4.2.5.3"><minus id="A2.Ex38.m2.3.4.2.5.3.1.cmml" xref="A2.Ex38.m2.3.4.2.5.3"></minus><cn id="A2.Ex38.m2.3.4.2.5.3.2.cmml" type="integer" xref="A2.Ex38.m2.3.4.2.5.3.2">1</cn></apply></apply><apply id="A2.Ex38.m2.3.4.2.6.cmml" xref="A2.Ex38.m2.3.4.2.6"><ci id="A2.Ex38.m2.3.4.2.6.1.cmml" xref="A2.Ex38.m2.3.4.2.6.1">~</ci><ci id="A2.Ex38.m2.3.4.2.6.2.cmml" xref="A2.Ex38.m2.3.4.2.6.2">𝜽</ci></apply></apply><apply id="A2.Ex38.m2.3.4.3.cmml" xref="A2.Ex38.m2.3.4.3"><times id="A2.Ex38.m2.3.4.3.1.cmml" xref="A2.Ex38.m2.3.4.3.1"></times><apply id="A2.Ex38.m2.3.4.3.2.cmml" xref="A2.Ex38.m2.3.4.3.2"><csymbol cd="ambiguous" id="A2.Ex38.m2.3.4.3.2.1.cmml" xref="A2.Ex38.m2.3.4.3.2">subscript</csymbol><ci id="A2.Ex38.m2.3.4.3.2.2.cmml" xref="A2.Ex38.m2.3.4.3.2.2">𝜆</ci><max id="A2.Ex38.m2.3.4.3.2.3.cmml" xref="A2.Ex38.m2.3.4.3.2.3"></max></apply><ci id="A2.Ex38.m2.2.2.cmml" xref="A2.Ex38.m2.2.2">Γ</ci><apply id="A2.Ex38.m2.3.4.3.4.cmml" xref="A2.Ex38.m2.3.4.3.4"><csymbol cd="ambiguous" id="A2.Ex38.m2.3.4.3.4.1.cmml" xref="A2.Ex38.m2.3.4.3.4">subscript</csymbol><ci id="A2.Ex38.m2.3.4.3.4.2.cmml" xref="A2.Ex38.m2.3.4.3.4.2">𝑉</ci><ci id="A2.Ex38.m2.3.4.3.4.3.cmml" xref="A2.Ex38.m2.3.4.3.4.3">𝜃</ci></apply><ci id="A2.Ex38.m2.3.3.cmml" xref="A2.Ex38.m2.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex38.m2.3c">\displaystyle\lambda_{\max}(\Gamma)\tilde{\bm{\theta}}^{T}\Gamma^{-1}\tilde{% \bm{\theta}}=\lambda_{\max}(\Gamma)V_{\theta}(t)</annotation><annotation encoding="application/x-llamapun" id="A2.Ex38.m2.3d">italic_λ start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ( roman_Γ ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Γ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG = italic_λ start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ( roman_Γ ) italic_V start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A2.Ex39"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\leq" class="ltx_Math" display="inline" id="A2.Ex39.m1.1"><semantics id="A2.Ex39.m1.1a"><mo id="A2.Ex39.m1.1.1" xref="A2.Ex39.m1.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="A2.Ex39.m1.1b"><leq id="A2.Ex39.m1.1.1.cmml" xref="A2.Ex39.m1.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex39.m1.1c">\displaystyle\leq</annotation><annotation encoding="application/x-llamapun" id="A2.Ex39.m1.1d">≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\lambda_{\max}(\Gamma)V_{\theta}(T_{\rm e})e^{-k_{\theta}(t-T_{% \rm e})},\forall t\geq T_{\rm e}" class="ltx_Math" display="inline" id="A2.Ex39.m2.4"><semantics id="A2.Ex39.m2.4a"><mrow id="A2.Ex39.m2.4.4" xref="A2.Ex39.m2.4.4.cmml"><mrow id="A2.Ex39.m2.4.4.2.2" xref="A2.Ex39.m2.4.4.2.3.cmml"><mrow id="A2.Ex39.m2.3.3.1.1.1" xref="A2.Ex39.m2.3.3.1.1.1.cmml"><msub id="A2.Ex39.m2.3.3.1.1.1.3" xref="A2.Ex39.m2.3.3.1.1.1.3.cmml"><mi id="A2.Ex39.m2.3.3.1.1.1.3.2" xref="A2.Ex39.m2.3.3.1.1.1.3.2.cmml">λ</mi><mi id="A2.Ex39.m2.3.3.1.1.1.3.3" xref="A2.Ex39.m2.3.3.1.1.1.3.3.cmml">max</mi></msub><mo id="A2.Ex39.m2.3.3.1.1.1.2" xref="A2.Ex39.m2.3.3.1.1.1.2.cmml">⁢</mo><mrow id="A2.Ex39.m2.3.3.1.1.1.4.2" xref="A2.Ex39.m2.3.3.1.1.1.cmml"><mo id="A2.Ex39.m2.3.3.1.1.1.4.2.1" stretchy="false" xref="A2.Ex39.m2.3.3.1.1.1.cmml">(</mo><mi id="A2.Ex39.m2.2.2" mathvariant="normal" xref="A2.Ex39.m2.2.2.cmml">Γ</mi><mo id="A2.Ex39.m2.3.3.1.1.1.4.2.2" stretchy="false" xref="A2.Ex39.m2.3.3.1.1.1.cmml">)</mo></mrow><mo id="A2.Ex39.m2.3.3.1.1.1.2a" xref="A2.Ex39.m2.3.3.1.1.1.2.cmml">⁢</mo><msub id="A2.Ex39.m2.3.3.1.1.1.5" xref="A2.Ex39.m2.3.3.1.1.1.5.cmml"><mi id="A2.Ex39.m2.3.3.1.1.1.5.2" xref="A2.Ex39.m2.3.3.1.1.1.5.2.cmml">V</mi><mi id="A2.Ex39.m2.3.3.1.1.1.5.3" xref="A2.Ex39.m2.3.3.1.1.1.5.3.cmml">θ</mi></msub><mo id="A2.Ex39.m2.3.3.1.1.1.2b" xref="A2.Ex39.m2.3.3.1.1.1.2.cmml">⁢</mo><mrow id="A2.Ex39.m2.3.3.1.1.1.1.1" xref="A2.Ex39.m2.3.3.1.1.1.1.1.1.cmml"><mo id="A2.Ex39.m2.3.3.1.1.1.1.1.2" stretchy="false" xref="A2.Ex39.m2.3.3.1.1.1.1.1.1.cmml">(</mo><msub id="A2.Ex39.m2.3.3.1.1.1.1.1.1" xref="A2.Ex39.m2.3.3.1.1.1.1.1.1.cmml"><mi id="A2.Ex39.m2.3.3.1.1.1.1.1.1.2" xref="A2.Ex39.m2.3.3.1.1.1.1.1.1.2.cmml">T</mi><mi id="A2.Ex39.m2.3.3.1.1.1.1.1.1.3" mathvariant="normal" xref="A2.Ex39.m2.3.3.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="A2.Ex39.m2.3.3.1.1.1.1.1.3" stretchy="false" xref="A2.Ex39.m2.3.3.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A2.Ex39.m2.3.3.1.1.1.2c" xref="A2.Ex39.m2.3.3.1.1.1.2.cmml">⁢</mo><msup id="A2.Ex39.m2.3.3.1.1.1.6" xref="A2.Ex39.m2.3.3.1.1.1.6.cmml"><mi id="A2.Ex39.m2.3.3.1.1.1.6.2" xref="A2.Ex39.m2.3.3.1.1.1.6.2.cmml">e</mi><mrow id="A2.Ex39.m2.1.1.1" xref="A2.Ex39.m2.1.1.1.cmml"><mo id="A2.Ex39.m2.1.1.1a" xref="A2.Ex39.m2.1.1.1.cmml">−</mo><mrow id="A2.Ex39.m2.1.1.1.1" xref="A2.Ex39.m2.1.1.1.1.cmml"><msub id="A2.Ex39.m2.1.1.1.1.3" xref="A2.Ex39.m2.1.1.1.1.3.cmml"><mi id="A2.Ex39.m2.1.1.1.1.3.2" xref="A2.Ex39.m2.1.1.1.1.3.2.cmml">k</mi><mi id="A2.Ex39.m2.1.1.1.1.3.3" xref="A2.Ex39.m2.1.1.1.1.3.3.cmml">θ</mi></msub><mo id="A2.Ex39.m2.1.1.1.1.2" xref="A2.Ex39.m2.1.1.1.1.2.cmml">⁢</mo><mrow id="A2.Ex39.m2.1.1.1.1.1.1" xref="A2.Ex39.m2.1.1.1.1.1.1.1.cmml"><mo id="A2.Ex39.m2.1.1.1.1.1.1.2" stretchy="false" xref="A2.Ex39.m2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A2.Ex39.m2.1.1.1.1.1.1.1" xref="A2.Ex39.m2.1.1.1.1.1.1.1.cmml"><mi id="A2.Ex39.m2.1.1.1.1.1.1.1.2" xref="A2.Ex39.m2.1.1.1.1.1.1.1.2.cmml">t</mi><mo id="A2.Ex39.m2.1.1.1.1.1.1.1.1" xref="A2.Ex39.m2.1.1.1.1.1.1.1.1.cmml">−</mo><msub id="A2.Ex39.m2.1.1.1.1.1.1.1.3" xref="A2.Ex39.m2.1.1.1.1.1.1.1.3.cmml"><mi id="A2.Ex39.m2.1.1.1.1.1.1.1.3.2" xref="A2.Ex39.m2.1.1.1.1.1.1.1.3.2.cmml">T</mi><mi id="A2.Ex39.m2.1.1.1.1.1.1.1.3.3" mathvariant="normal" xref="A2.Ex39.m2.1.1.1.1.1.1.1.3.3.cmml">e</mi></msub></mrow><mo id="A2.Ex39.m2.1.1.1.1.1.1.3" stretchy="false" xref="A2.Ex39.m2.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow><mo id="A2.Ex39.m2.4.4.2.2.3" xref="A2.Ex39.m2.4.4.2.3.cmml">,</mo><mrow id="A2.Ex39.m2.4.4.2.2.2" xref="A2.Ex39.m2.4.4.2.2.2.cmml"><mo id="A2.Ex39.m2.4.4.2.2.2.1" rspace="0.167em" xref="A2.Ex39.m2.4.4.2.2.2.1.cmml">∀</mo><mi id="A2.Ex39.m2.4.4.2.2.2.2" xref="A2.Ex39.m2.4.4.2.2.2.2.cmml">t</mi></mrow></mrow><mo id="A2.Ex39.m2.4.4.3" xref="A2.Ex39.m2.4.4.3.cmml">≥</mo><msub id="A2.Ex39.m2.4.4.4" xref="A2.Ex39.m2.4.4.4.cmml"><mi id="A2.Ex39.m2.4.4.4.2" xref="A2.Ex39.m2.4.4.4.2.cmml">T</mi><mi id="A2.Ex39.m2.4.4.4.3" mathvariant="normal" xref="A2.Ex39.m2.4.4.4.3.cmml">e</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex39.m2.4b"><apply id="A2.Ex39.m2.4.4.cmml" xref="A2.Ex39.m2.4.4"><geq id="A2.Ex39.m2.4.4.3.cmml" xref="A2.Ex39.m2.4.4.3"></geq><list id="A2.Ex39.m2.4.4.2.3.cmml" xref="A2.Ex39.m2.4.4.2.2"><apply id="A2.Ex39.m2.3.3.1.1.1.cmml" xref="A2.Ex39.m2.3.3.1.1.1"><times id="A2.Ex39.m2.3.3.1.1.1.2.cmml" xref="A2.Ex39.m2.3.3.1.1.1.2"></times><apply id="A2.Ex39.m2.3.3.1.1.1.3.cmml" xref="A2.Ex39.m2.3.3.1.1.1.3"><csymbol cd="ambiguous" id="A2.Ex39.m2.3.3.1.1.1.3.1.cmml" xref="A2.Ex39.m2.3.3.1.1.1.3">subscript</csymbol><ci id="A2.Ex39.m2.3.3.1.1.1.3.2.cmml" xref="A2.Ex39.m2.3.3.1.1.1.3.2">𝜆</ci><max id="A2.Ex39.m2.3.3.1.1.1.3.3.cmml" xref="A2.Ex39.m2.3.3.1.1.1.3.3"></max></apply><ci id="A2.Ex39.m2.2.2.cmml" xref="A2.Ex39.m2.2.2">Γ</ci><apply id="A2.Ex39.m2.3.3.1.1.1.5.cmml" xref="A2.Ex39.m2.3.3.1.1.1.5"><csymbol cd="ambiguous" id="A2.Ex39.m2.3.3.1.1.1.5.1.cmml" xref="A2.Ex39.m2.3.3.1.1.1.5">subscript</csymbol><ci id="A2.Ex39.m2.3.3.1.1.1.5.2.cmml" xref="A2.Ex39.m2.3.3.1.1.1.5.2">𝑉</ci><ci id="A2.Ex39.m2.3.3.1.1.1.5.3.cmml" xref="A2.Ex39.m2.3.3.1.1.1.5.3">𝜃</ci></apply><apply id="A2.Ex39.m2.3.3.1.1.1.1.1.1.cmml" xref="A2.Ex39.m2.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex39.m2.3.3.1.1.1.1.1.1.1.cmml" xref="A2.Ex39.m2.3.3.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex39.m2.3.3.1.1.1.1.1.1.2.cmml" xref="A2.Ex39.m2.3.3.1.1.1.1.1.1.2">𝑇</ci><ci id="A2.Ex39.m2.3.3.1.1.1.1.1.1.3.cmml" xref="A2.Ex39.m2.3.3.1.1.1.1.1.1.3">e</ci></apply><apply id="A2.Ex39.m2.3.3.1.1.1.6.cmml" xref="A2.Ex39.m2.3.3.1.1.1.6"><csymbol cd="ambiguous" id="A2.Ex39.m2.3.3.1.1.1.6.1.cmml" xref="A2.Ex39.m2.3.3.1.1.1.6">superscript</csymbol><ci id="A2.Ex39.m2.3.3.1.1.1.6.2.cmml" xref="A2.Ex39.m2.3.3.1.1.1.6.2">𝑒</ci><apply id="A2.Ex39.m2.1.1.1.cmml" xref="A2.Ex39.m2.1.1.1"><minus id="A2.Ex39.m2.1.1.1.2.cmml" xref="A2.Ex39.m2.1.1.1"></minus><apply id="A2.Ex39.m2.1.1.1.1.cmml" xref="A2.Ex39.m2.1.1.1.1"><times id="A2.Ex39.m2.1.1.1.1.2.cmml" xref="A2.Ex39.m2.1.1.1.1.2"></times><apply id="A2.Ex39.m2.1.1.1.1.3.cmml" xref="A2.Ex39.m2.1.1.1.1.3"><csymbol cd="ambiguous" id="A2.Ex39.m2.1.1.1.1.3.1.cmml" xref="A2.Ex39.m2.1.1.1.1.3">subscript</csymbol><ci id="A2.Ex39.m2.1.1.1.1.3.2.cmml" xref="A2.Ex39.m2.1.1.1.1.3.2">𝑘</ci><ci id="A2.Ex39.m2.1.1.1.1.3.3.cmml" xref="A2.Ex39.m2.1.1.1.1.3.3">𝜃</ci></apply><apply id="A2.Ex39.m2.1.1.1.1.1.1.1.cmml" xref="A2.Ex39.m2.1.1.1.1.1.1"><minus id="A2.Ex39.m2.1.1.1.1.1.1.1.1.cmml" xref="A2.Ex39.m2.1.1.1.1.1.1.1.1"></minus><ci id="A2.Ex39.m2.1.1.1.1.1.1.1.2.cmml" xref="A2.Ex39.m2.1.1.1.1.1.1.1.2">𝑡</ci><apply id="A2.Ex39.m2.1.1.1.1.1.1.1.3.cmml" xref="A2.Ex39.m2.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A2.Ex39.m2.1.1.1.1.1.1.1.3.1.cmml" xref="A2.Ex39.m2.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="A2.Ex39.m2.1.1.1.1.1.1.1.3.2.cmml" xref="A2.Ex39.m2.1.1.1.1.1.1.1.3.2">𝑇</ci><ci id="A2.Ex39.m2.1.1.1.1.1.1.1.3.3.cmml" xref="A2.Ex39.m2.1.1.1.1.1.1.1.3.3">e</ci></apply></apply></apply></apply></apply></apply><apply id="A2.Ex39.m2.4.4.2.2.2.cmml" xref="A2.Ex39.m2.4.4.2.2.2"><csymbol cd="latexml" id="A2.Ex39.m2.4.4.2.2.2.1.cmml" xref="A2.Ex39.m2.4.4.2.2.2.1">for-all</csymbol><ci id="A2.Ex39.m2.4.4.2.2.2.2.cmml" xref="A2.Ex39.m2.4.4.2.2.2.2">𝑡</ci></apply></list><apply id="A2.Ex39.m2.4.4.4.cmml" xref="A2.Ex39.m2.4.4.4"><csymbol cd="ambiguous" id="A2.Ex39.m2.4.4.4.1.cmml" xref="A2.Ex39.m2.4.4.4">subscript</csymbol><ci id="A2.Ex39.m2.4.4.4.2.cmml" xref="A2.Ex39.m2.4.4.4.2">𝑇</ci><ci id="A2.Ex39.m2.4.4.4.3.cmml" xref="A2.Ex39.m2.4.4.4.3">e</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex39.m2.4c">\displaystyle\lambda_{\max}(\Gamma)V_{\theta}(T_{\rm e})e^{-k_{\theta}(t-T_{% \rm e})},\forall t\geq T_{\rm e}</annotation><annotation encoding="application/x-llamapun" id="A2.Ex39.m2.4d">italic_λ start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ( roman_Γ ) italic_V start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) italic_e start_POSTSUPERSCRIPT - italic_k start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( italic_t - italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT , ∀ italic_t ≥ italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A2.3.p3.7">where <math alttext="k_{\theta}" class="ltx_Math" display="inline" id="A2.3.p3.3.m1.1"><semantics id="A2.3.p3.3.m1.1a"><msub id="A2.3.p3.3.m1.1.1" xref="A2.3.p3.3.m1.1.1.cmml"><mi id="A2.3.p3.3.m1.1.1.2" xref="A2.3.p3.3.m1.1.1.2.cmml">k</mi><mi id="A2.3.p3.3.m1.1.1.3" xref="A2.3.p3.3.m1.1.1.3.cmml">θ</mi></msub><annotation-xml encoding="MathML-Content" id="A2.3.p3.3.m1.1b"><apply id="A2.3.p3.3.m1.1.1.cmml" xref="A2.3.p3.3.m1.1.1"><csymbol cd="ambiguous" id="A2.3.p3.3.m1.1.1.1.cmml" xref="A2.3.p3.3.m1.1.1">subscript</csymbol><ci id="A2.3.p3.3.m1.1.1.2.cmml" xref="A2.3.p3.3.m1.1.1.2">𝑘</ci><ci id="A2.3.p3.3.m1.1.1.3.cmml" xref="A2.3.p3.3.m1.1.1.3">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.3.p3.3.m1.1c">k_{\theta}</annotation><annotation encoding="application/x-llamapun" id="A2.3.p3.3.m1.1d">italic_k start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="A2.3.p3.4.m2.1"><semantics id="A2.3.p3.4.m2.1a"><mo id="A2.3.p3.4.m2.1.1" xref="A2.3.p3.4.m2.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="A2.3.p3.4.m2.1b"><csymbol cd="latexml" id="A2.3.p3.4.m2.1.1.cmml" xref="A2.3.p3.4.m2.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A2.3.p3.4.m2.1c">:=</annotation><annotation encoding="application/x-llamapun" id="A2.3.p3.4.m2.1d">:=</annotation></semantics></math> <math alttext="2\kappa\sigma^{*}\lambda_{\min}(\Gamma)" class="ltx_Math" display="inline" id="A2.3.p3.5.m3.1"><semantics id="A2.3.p3.5.m3.1a"><mrow id="A2.3.p3.5.m3.1.2" xref="A2.3.p3.5.m3.1.2.cmml"><mn id="A2.3.p3.5.m3.1.2.2" xref="A2.3.p3.5.m3.1.2.2.cmml">2</mn><mo id="A2.3.p3.5.m3.1.2.1" xref="A2.3.p3.5.m3.1.2.1.cmml">⁢</mo><mi id="A2.3.p3.5.m3.1.2.3" xref="A2.3.p3.5.m3.1.2.3.cmml">κ</mi><mo id="A2.3.p3.5.m3.1.2.1a" xref="A2.3.p3.5.m3.1.2.1.cmml">⁢</mo><msup id="A2.3.p3.5.m3.1.2.4" xref="A2.3.p3.5.m3.1.2.4.cmml"><mi id="A2.3.p3.5.m3.1.2.4.2" xref="A2.3.p3.5.m3.1.2.4.2.cmml">σ</mi><mo id="A2.3.p3.5.m3.1.2.4.3" xref="A2.3.p3.5.m3.1.2.4.3.cmml">∗</mo></msup><mo id="A2.3.p3.5.m3.1.2.1b" xref="A2.3.p3.5.m3.1.2.1.cmml">⁢</mo><msub id="A2.3.p3.5.m3.1.2.5" xref="A2.3.p3.5.m3.1.2.5.cmml"><mi id="A2.3.p3.5.m3.1.2.5.2" xref="A2.3.p3.5.m3.1.2.5.2.cmml">λ</mi><mi id="A2.3.p3.5.m3.1.2.5.3" xref="A2.3.p3.5.m3.1.2.5.3.cmml">min</mi></msub><mo id="A2.3.p3.5.m3.1.2.1c" xref="A2.3.p3.5.m3.1.2.1.cmml">⁢</mo><mrow id="A2.3.p3.5.m3.1.2.6.2" xref="A2.3.p3.5.m3.1.2.cmml"><mo id="A2.3.p3.5.m3.1.2.6.2.1" stretchy="false" xref="A2.3.p3.5.m3.1.2.cmml">(</mo><mi id="A2.3.p3.5.m3.1.1" mathvariant="normal" xref="A2.3.p3.5.m3.1.1.cmml">Γ</mi><mo id="A2.3.p3.5.m3.1.2.6.2.2" stretchy="false" xref="A2.3.p3.5.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.3.p3.5.m3.1b"><apply id="A2.3.p3.5.m3.1.2.cmml" xref="A2.3.p3.5.m3.1.2"><times id="A2.3.p3.5.m3.1.2.1.cmml" xref="A2.3.p3.5.m3.1.2.1"></times><cn id="A2.3.p3.5.m3.1.2.2.cmml" type="integer" xref="A2.3.p3.5.m3.1.2.2">2</cn><ci id="A2.3.p3.5.m3.1.2.3.cmml" xref="A2.3.p3.5.m3.1.2.3">𝜅</ci><apply id="A2.3.p3.5.m3.1.2.4.cmml" xref="A2.3.p3.5.m3.1.2.4"><csymbol cd="ambiguous" id="A2.3.p3.5.m3.1.2.4.1.cmml" xref="A2.3.p3.5.m3.1.2.4">superscript</csymbol><ci id="A2.3.p3.5.m3.1.2.4.2.cmml" xref="A2.3.p3.5.m3.1.2.4.2">𝜎</ci><times id="A2.3.p3.5.m3.1.2.4.3.cmml" xref="A2.3.p3.5.m3.1.2.4.3"></times></apply><apply id="A2.3.p3.5.m3.1.2.5.cmml" xref="A2.3.p3.5.m3.1.2.5"><csymbol cd="ambiguous" id="A2.3.p3.5.m3.1.2.5.1.cmml" xref="A2.3.p3.5.m3.1.2.5">subscript</csymbol><ci id="A2.3.p3.5.m3.1.2.5.2.cmml" xref="A2.3.p3.5.m3.1.2.5.2">𝜆</ci><min id="A2.3.p3.5.m3.1.2.5.3.cmml" xref="A2.3.p3.5.m3.1.2.5.3"></min></apply><ci id="A2.3.p3.5.m3.1.1.cmml" xref="A2.3.p3.5.m3.1.1">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.3.p3.5.m3.1c">2\kappa\sigma^{*}\lambda_{\min}(\Gamma)</annotation><annotation encoding="application/x-llamapun" id="A2.3.p3.5.m3.1d">2 italic_κ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Γ )</annotation></semantics></math>, and <math alttext="\sigma^{*}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A2.3.p3.6.m4.1"><semantics id="A2.3.p3.6.m4.1a"><mrow id="A2.3.p3.6.m4.1.1" xref="A2.3.p3.6.m4.1.1.cmml"><msup id="A2.3.p3.6.m4.1.1.2" xref="A2.3.p3.6.m4.1.1.2.cmml"><mi id="A2.3.p3.6.m4.1.1.2.2" xref="A2.3.p3.6.m4.1.1.2.2.cmml">σ</mi><mo id="A2.3.p3.6.m4.1.1.2.3" xref="A2.3.p3.6.m4.1.1.2.3.cmml">∗</mo></msup><mo id="A2.3.p3.6.m4.1.1.1" xref="A2.3.p3.6.m4.1.1.1.cmml">∈</mo><msup id="A2.3.p3.6.m4.1.1.3" xref="A2.3.p3.6.m4.1.1.3.cmml"><mi id="A2.3.p3.6.m4.1.1.3.2" xref="A2.3.p3.6.m4.1.1.3.2.cmml">ℝ</mi><mo id="A2.3.p3.6.m4.1.1.3.3" xref="A2.3.p3.6.m4.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A2.3.p3.6.m4.1b"><apply id="A2.3.p3.6.m4.1.1.cmml" xref="A2.3.p3.6.m4.1.1"><in id="A2.3.p3.6.m4.1.1.1.cmml" xref="A2.3.p3.6.m4.1.1.1"></in><apply id="A2.3.p3.6.m4.1.1.2.cmml" xref="A2.3.p3.6.m4.1.1.2"><csymbol cd="ambiguous" id="A2.3.p3.6.m4.1.1.2.1.cmml" xref="A2.3.p3.6.m4.1.1.2">superscript</csymbol><ci id="A2.3.p3.6.m4.1.1.2.2.cmml" xref="A2.3.p3.6.m4.1.1.2.2">𝜎</ci><times id="A2.3.p3.6.m4.1.1.2.3.cmml" xref="A2.3.p3.6.m4.1.1.2.3"></times></apply><apply id="A2.3.p3.6.m4.1.1.3.cmml" xref="A2.3.p3.6.m4.1.1.3"><csymbol cd="ambiguous" id="A2.3.p3.6.m4.1.1.3.1.cmml" xref="A2.3.p3.6.m4.1.1.3">superscript</csymbol><ci id="A2.3.p3.6.m4.1.1.3.2.cmml" xref="A2.3.p3.6.m4.1.1.3.2">ℝ</ci><plus id="A2.3.p3.6.m4.1.1.3.3.cmml" xref="A2.3.p3.6.m4.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.3.p3.6.m4.1c">\sigma^{*}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A2.3.p3.6.m4.1d">italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="\sigma^{*}\leq\sigma" class="ltx_Math" display="inline" id="A2.3.p3.7.m5.1"><semantics id="A2.3.p3.7.m5.1a"><mrow id="A2.3.p3.7.m5.1.1" xref="A2.3.p3.7.m5.1.1.cmml"><msup id="A2.3.p3.7.m5.1.1.2" xref="A2.3.p3.7.m5.1.1.2.cmml"><mi id="A2.3.p3.7.m5.1.1.2.2" xref="A2.3.p3.7.m5.1.1.2.2.cmml">σ</mi><mo id="A2.3.p3.7.m5.1.1.2.3" xref="A2.3.p3.7.m5.1.1.2.3.cmml">∗</mo></msup><mo id="A2.3.p3.7.m5.1.1.1" xref="A2.3.p3.7.m5.1.1.1.cmml">≤</mo><mi id="A2.3.p3.7.m5.1.1.3" xref="A2.3.p3.7.m5.1.1.3.cmml">σ</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.3.p3.7.m5.1b"><apply id="A2.3.p3.7.m5.1.1.cmml" xref="A2.3.p3.7.m5.1.1"><leq id="A2.3.p3.7.m5.1.1.1.cmml" xref="A2.3.p3.7.m5.1.1.1"></leq><apply id="A2.3.p3.7.m5.1.1.2.cmml" xref="A2.3.p3.7.m5.1.1.2"><csymbol cd="ambiguous" id="A2.3.p3.7.m5.1.1.2.1.cmml" xref="A2.3.p3.7.m5.1.1.2">superscript</csymbol><ci id="A2.3.p3.7.m5.1.1.2.2.cmml" xref="A2.3.p3.7.m5.1.1.2.2">𝜎</ci><times id="A2.3.p3.7.m5.1.1.2.3.cmml" xref="A2.3.p3.7.m5.1.1.2.3"></times></apply><ci id="A2.3.p3.7.m5.1.1.3.cmml" xref="A2.3.p3.7.m5.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.3.p3.7.m5.1c">\sigma^{*}\leq\sigma</annotation><annotation encoding="application/x-llamapun" id="A2.3.p3.7.m5.1d">italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ≤ italic_σ</annotation></semantics></math> satisfies</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx50"> <tbody id="A2.E38"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle Q(t,t_{\rm e})=H(s)[\Psi(t_{\rm e})]\geq H(s)[\Psi(T_{\rm e})]% \geq\sigma^{*}I" class="ltx_Math" display="inline" id="A2.E38.m1.6"><semantics id="A2.E38.m1.6a"><mrow id="A2.E38.m1.6.6" xref="A2.E38.m1.6.6.cmml"><mrow id="A2.E38.m1.4.4.1" xref="A2.E38.m1.4.4.1.cmml"><mi id="A2.E38.m1.4.4.1.3" xref="A2.E38.m1.4.4.1.3.cmml">Q</mi><mo id="A2.E38.m1.4.4.1.2" xref="A2.E38.m1.4.4.1.2.cmml">⁢</mo><mrow id="A2.E38.m1.4.4.1.1.1" xref="A2.E38.m1.4.4.1.1.2.cmml"><mo id="A2.E38.m1.4.4.1.1.1.2" stretchy="false" xref="A2.E38.m1.4.4.1.1.2.cmml">(</mo><mi id="A2.E38.m1.1.1" xref="A2.E38.m1.1.1.cmml">t</mi><mo id="A2.E38.m1.4.4.1.1.1.3" xref="A2.E38.m1.4.4.1.1.2.cmml">,</mo><msub id="A2.E38.m1.4.4.1.1.1.1" xref="A2.E38.m1.4.4.1.1.1.1.cmml"><mi id="A2.E38.m1.4.4.1.1.1.1.2" xref="A2.E38.m1.4.4.1.1.1.1.2.cmml">t</mi><mi id="A2.E38.m1.4.4.1.1.1.1.3" mathvariant="normal" xref="A2.E38.m1.4.4.1.1.1.1.3.cmml">e</mi></msub><mo id="A2.E38.m1.4.4.1.1.1.4" stretchy="false" xref="A2.E38.m1.4.4.1.1.2.cmml">)</mo></mrow></mrow><mo id="A2.E38.m1.6.6.5" xref="A2.E38.m1.6.6.5.cmml">=</mo><mrow id="A2.E38.m1.5.5.2" xref="A2.E38.m1.5.5.2.cmml"><mi id="A2.E38.m1.5.5.2.3" xref="A2.E38.m1.5.5.2.3.cmml">H</mi><mo id="A2.E38.m1.5.5.2.2" xref="A2.E38.m1.5.5.2.2.cmml">⁢</mo><mrow id="A2.E38.m1.5.5.2.4.2" xref="A2.E38.m1.5.5.2.cmml"><mo id="A2.E38.m1.5.5.2.4.2.1" stretchy="false" xref="A2.E38.m1.5.5.2.cmml">(</mo><mi id="A2.E38.m1.2.2" xref="A2.E38.m1.2.2.cmml">s</mi><mo id="A2.E38.m1.5.5.2.4.2.2" stretchy="false" xref="A2.E38.m1.5.5.2.cmml">)</mo></mrow><mo id="A2.E38.m1.5.5.2.2a" xref="A2.E38.m1.5.5.2.2.cmml">⁢</mo><mrow id="A2.E38.m1.5.5.2.1.1" xref="A2.E38.m1.5.5.2.1.2.cmml"><mo id="A2.E38.m1.5.5.2.1.1.2" stretchy="false" xref="A2.E38.m1.5.5.2.1.2.1.cmml">[</mo><mrow id="A2.E38.m1.5.5.2.1.1.1" xref="A2.E38.m1.5.5.2.1.1.1.cmml"><mi id="A2.E38.m1.5.5.2.1.1.1.3" mathvariant="normal" xref="A2.E38.m1.5.5.2.1.1.1.3.cmml">Ψ</mi><mo id="A2.E38.m1.5.5.2.1.1.1.2" xref="A2.E38.m1.5.5.2.1.1.1.2.cmml">⁢</mo><mrow id="A2.E38.m1.5.5.2.1.1.1.1.1" xref="A2.E38.m1.5.5.2.1.1.1.1.1.1.cmml"><mo id="A2.E38.m1.5.5.2.1.1.1.1.1.2" stretchy="false" xref="A2.E38.m1.5.5.2.1.1.1.1.1.1.cmml">(</mo><msub id="A2.E38.m1.5.5.2.1.1.1.1.1.1" xref="A2.E38.m1.5.5.2.1.1.1.1.1.1.cmml"><mi id="A2.E38.m1.5.5.2.1.1.1.1.1.1.2" xref="A2.E38.m1.5.5.2.1.1.1.1.1.1.2.cmml">t</mi><mi id="A2.E38.m1.5.5.2.1.1.1.1.1.1.3" mathvariant="normal" xref="A2.E38.m1.5.5.2.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="A2.E38.m1.5.5.2.1.1.1.1.1.3" stretchy="false" xref="A2.E38.m1.5.5.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A2.E38.m1.5.5.2.1.1.3" stretchy="false" xref="A2.E38.m1.5.5.2.1.2.1.cmml">]</mo></mrow></mrow><mo id="A2.E38.m1.6.6.6" xref="A2.E38.m1.6.6.6.cmml">≥</mo><mrow id="A2.E38.m1.6.6.3" xref="A2.E38.m1.6.6.3.cmml"><mi id="A2.E38.m1.6.6.3.3" xref="A2.E38.m1.6.6.3.3.cmml">H</mi><mo id="A2.E38.m1.6.6.3.2" xref="A2.E38.m1.6.6.3.2.cmml">⁢</mo><mrow id="A2.E38.m1.6.6.3.4.2" xref="A2.E38.m1.6.6.3.cmml"><mo id="A2.E38.m1.6.6.3.4.2.1" stretchy="false" xref="A2.E38.m1.6.6.3.cmml">(</mo><mi id="A2.E38.m1.3.3" xref="A2.E38.m1.3.3.cmml">s</mi><mo id="A2.E38.m1.6.6.3.4.2.2" stretchy="false" xref="A2.E38.m1.6.6.3.cmml">)</mo></mrow><mo id="A2.E38.m1.6.6.3.2a" xref="A2.E38.m1.6.6.3.2.cmml">⁢</mo><mrow id="A2.E38.m1.6.6.3.1.1" xref="A2.E38.m1.6.6.3.1.2.cmml"><mo id="A2.E38.m1.6.6.3.1.1.2" stretchy="false" xref="A2.E38.m1.6.6.3.1.2.1.cmml">[</mo><mrow id="A2.E38.m1.6.6.3.1.1.1" xref="A2.E38.m1.6.6.3.1.1.1.cmml"><mi id="A2.E38.m1.6.6.3.1.1.1.3" mathvariant="normal" xref="A2.E38.m1.6.6.3.1.1.1.3.cmml">Ψ</mi><mo id="A2.E38.m1.6.6.3.1.1.1.2" xref="A2.E38.m1.6.6.3.1.1.1.2.cmml">⁢</mo><mrow id="A2.E38.m1.6.6.3.1.1.1.1.1" xref="A2.E38.m1.6.6.3.1.1.1.1.1.1.cmml"><mo id="A2.E38.m1.6.6.3.1.1.1.1.1.2" stretchy="false" xref="A2.E38.m1.6.6.3.1.1.1.1.1.1.cmml">(</mo><msub id="A2.E38.m1.6.6.3.1.1.1.1.1.1" xref="A2.E38.m1.6.6.3.1.1.1.1.1.1.cmml"><mi id="A2.E38.m1.6.6.3.1.1.1.1.1.1.2" xref="A2.E38.m1.6.6.3.1.1.1.1.1.1.2.cmml">T</mi><mi id="A2.E38.m1.6.6.3.1.1.1.1.1.1.3" mathvariant="normal" xref="A2.E38.m1.6.6.3.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="A2.E38.m1.6.6.3.1.1.1.1.1.3" stretchy="false" xref="A2.E38.m1.6.6.3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A2.E38.m1.6.6.3.1.1.3" stretchy="false" xref="A2.E38.m1.6.6.3.1.2.1.cmml">]</mo></mrow></mrow><mo id="A2.E38.m1.6.6.7" xref="A2.E38.m1.6.6.7.cmml">≥</mo><mrow id="A2.E38.m1.6.6.8" xref="A2.E38.m1.6.6.8.cmml"><msup id="A2.E38.m1.6.6.8.2" xref="A2.E38.m1.6.6.8.2.cmml"><mi id="A2.E38.m1.6.6.8.2.2" xref="A2.E38.m1.6.6.8.2.2.cmml">σ</mi><mo id="A2.E38.m1.6.6.8.2.3" xref="A2.E38.m1.6.6.8.2.3.cmml">∗</mo></msup><mo id="A2.E38.m1.6.6.8.1" xref="A2.E38.m1.6.6.8.1.cmml">⁢</mo><mi id="A2.E38.m1.6.6.8.3" xref="A2.E38.m1.6.6.8.3.cmml">I</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.E38.m1.6b"><apply id="A2.E38.m1.6.6.cmml" xref="A2.E38.m1.6.6"><and id="A2.E38.m1.6.6a.cmml" xref="A2.E38.m1.6.6"></and><apply id="A2.E38.m1.6.6b.cmml" xref="A2.E38.m1.6.6"><eq id="A2.E38.m1.6.6.5.cmml" xref="A2.E38.m1.6.6.5"></eq><apply id="A2.E38.m1.4.4.1.cmml" xref="A2.E38.m1.4.4.1"><times id="A2.E38.m1.4.4.1.2.cmml" xref="A2.E38.m1.4.4.1.2"></times><ci id="A2.E38.m1.4.4.1.3.cmml" xref="A2.E38.m1.4.4.1.3">𝑄</ci><interval closure="open" id="A2.E38.m1.4.4.1.1.2.cmml" xref="A2.E38.m1.4.4.1.1.1"><ci id="A2.E38.m1.1.1.cmml" xref="A2.E38.m1.1.1">𝑡</ci><apply id="A2.E38.m1.4.4.1.1.1.1.cmml" xref="A2.E38.m1.4.4.1.1.1.1"><csymbol cd="ambiguous" id="A2.E38.m1.4.4.1.1.1.1.1.cmml" xref="A2.E38.m1.4.4.1.1.1.1">subscript</csymbol><ci id="A2.E38.m1.4.4.1.1.1.1.2.cmml" xref="A2.E38.m1.4.4.1.1.1.1.2">𝑡</ci><ci id="A2.E38.m1.4.4.1.1.1.1.3.cmml" xref="A2.E38.m1.4.4.1.1.1.1.3">e</ci></apply></interval></apply><apply id="A2.E38.m1.5.5.2.cmml" xref="A2.E38.m1.5.5.2"><times id="A2.E38.m1.5.5.2.2.cmml" xref="A2.E38.m1.5.5.2.2"></times><ci id="A2.E38.m1.5.5.2.3.cmml" xref="A2.E38.m1.5.5.2.3">𝐻</ci><ci id="A2.E38.m1.2.2.cmml" xref="A2.E38.m1.2.2">𝑠</ci><apply id="A2.E38.m1.5.5.2.1.2.cmml" xref="A2.E38.m1.5.5.2.1.1"><csymbol cd="latexml" id="A2.E38.m1.5.5.2.1.2.1.cmml" xref="A2.E38.m1.5.5.2.1.1.2">delimited-[]</csymbol><apply id="A2.E38.m1.5.5.2.1.1.1.cmml" xref="A2.E38.m1.5.5.2.1.1.1"><times id="A2.E38.m1.5.5.2.1.1.1.2.cmml" xref="A2.E38.m1.5.5.2.1.1.1.2"></times><ci id="A2.E38.m1.5.5.2.1.1.1.3.cmml" xref="A2.E38.m1.5.5.2.1.1.1.3">Ψ</ci><apply id="A2.E38.m1.5.5.2.1.1.1.1.1.1.cmml" xref="A2.E38.m1.5.5.2.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.E38.m1.5.5.2.1.1.1.1.1.1.1.cmml" xref="A2.E38.m1.5.5.2.1.1.1.1.1">subscript</csymbol><ci id="A2.E38.m1.5.5.2.1.1.1.1.1.1.2.cmml" xref="A2.E38.m1.5.5.2.1.1.1.1.1.1.2">𝑡</ci><ci id="A2.E38.m1.5.5.2.1.1.1.1.1.1.3.cmml" xref="A2.E38.m1.5.5.2.1.1.1.1.1.1.3">e</ci></apply></apply></apply></apply></apply><apply id="A2.E38.m1.6.6c.cmml" xref="A2.E38.m1.6.6"><geq id="A2.E38.m1.6.6.6.cmml" xref="A2.E38.m1.6.6.6"></geq><share href="https://arxiv.org/html/2401.10785v2#A2.E38.m1.5.5.2.cmml" id="A2.E38.m1.6.6d.cmml" xref="A2.E38.m1.6.6"></share><apply id="A2.E38.m1.6.6.3.cmml" xref="A2.E38.m1.6.6.3"><times id="A2.E38.m1.6.6.3.2.cmml" xref="A2.E38.m1.6.6.3.2"></times><ci id="A2.E38.m1.6.6.3.3.cmml" xref="A2.E38.m1.6.6.3.3">𝐻</ci><ci id="A2.E38.m1.3.3.cmml" xref="A2.E38.m1.3.3">𝑠</ci><apply id="A2.E38.m1.6.6.3.1.2.cmml" xref="A2.E38.m1.6.6.3.1.1"><csymbol cd="latexml" id="A2.E38.m1.6.6.3.1.2.1.cmml" xref="A2.E38.m1.6.6.3.1.1.2">delimited-[]</csymbol><apply id="A2.E38.m1.6.6.3.1.1.1.cmml" xref="A2.E38.m1.6.6.3.1.1.1"><times id="A2.E38.m1.6.6.3.1.1.1.2.cmml" xref="A2.E38.m1.6.6.3.1.1.1.2"></times><ci id="A2.E38.m1.6.6.3.1.1.1.3.cmml" xref="A2.E38.m1.6.6.3.1.1.1.3">Ψ</ci><apply id="A2.E38.m1.6.6.3.1.1.1.1.1.1.cmml" xref="A2.E38.m1.6.6.3.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.E38.m1.6.6.3.1.1.1.1.1.1.1.cmml" xref="A2.E38.m1.6.6.3.1.1.1.1.1">subscript</csymbol><ci id="A2.E38.m1.6.6.3.1.1.1.1.1.1.2.cmml" xref="A2.E38.m1.6.6.3.1.1.1.1.1.1.2">𝑇</ci><ci id="A2.E38.m1.6.6.3.1.1.1.1.1.1.3.cmml" xref="A2.E38.m1.6.6.3.1.1.1.1.1.1.3">e</ci></apply></apply></apply></apply></apply><apply id="A2.E38.m1.6.6e.cmml" xref="A2.E38.m1.6.6"><geq id="A2.E38.m1.6.6.7.cmml" xref="A2.E38.m1.6.6.7"></geq><share href="https://arxiv.org/html/2401.10785v2#A2.E38.m1.6.6.3.cmml" id="A2.E38.m1.6.6f.cmml" xref="A2.E38.m1.6.6"></share><apply id="A2.E38.m1.6.6.8.cmml" xref="A2.E38.m1.6.6.8"><times id="A2.E38.m1.6.6.8.1.cmml" xref="A2.E38.m1.6.6.8.1"></times><apply id="A2.E38.m1.6.6.8.2.cmml" xref="A2.E38.m1.6.6.8.2"><csymbol cd="ambiguous" id="A2.E38.m1.6.6.8.2.1.cmml" xref="A2.E38.m1.6.6.8.2">superscript</csymbol><ci id="A2.E38.m1.6.6.8.2.2.cmml" xref="A2.E38.m1.6.6.8.2.2">𝜎</ci><times id="A2.E38.m1.6.6.8.2.3.cmml" xref="A2.E38.m1.6.6.8.2.3"></times></apply><ci id="A2.E38.m1.6.6.8.3.cmml" xref="A2.E38.m1.6.6.8.3">𝐼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E38.m1.6c">\displaystyle Q(t,t_{\rm e})=H(s)[\Psi(t_{\rm e})]\geq H(s)[\Psi(T_{\rm e})]% \geq\sigma^{*}I</annotation><annotation encoding="application/x-llamapun" id="A2.E38.m1.6d">italic_Q ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) = italic_H ( italic_s ) [ roman_Ψ ( italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) ] ≥ italic_H ( italic_s ) [ roman_Ψ ( italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) ] ≥ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_I</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(38)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.3.p3.10">which implies that the parameter estimation error <math alttext="\tilde{\bm{\theta}}(t)" class="ltx_Math" display="inline" id="A2.3.p3.8.m1.1"><semantics id="A2.3.p3.8.m1.1a"><mrow id="A2.3.p3.8.m1.1.2" xref="A2.3.p3.8.m1.1.2.cmml"><mover accent="true" id="A2.3.p3.8.m1.1.2.2" xref="A2.3.p3.8.m1.1.2.2.cmml"><mi id="A2.3.p3.8.m1.1.2.2.2" xref="A2.3.p3.8.m1.1.2.2.2.cmml">𝜽</mi><mo id="A2.3.p3.8.m1.1.2.2.1" xref="A2.3.p3.8.m1.1.2.2.1.cmml">~</mo></mover><mo id="A2.3.p3.8.m1.1.2.1" xref="A2.3.p3.8.m1.1.2.1.cmml">⁢</mo><mrow id="A2.3.p3.8.m1.1.2.3.2" xref="A2.3.p3.8.m1.1.2.cmml"><mo id="A2.3.p3.8.m1.1.2.3.2.1" stretchy="false" xref="A2.3.p3.8.m1.1.2.cmml">(</mo><mi id="A2.3.p3.8.m1.1.1" xref="A2.3.p3.8.m1.1.1.cmml">t</mi><mo id="A2.3.p3.8.m1.1.2.3.2.2" stretchy="false" xref="A2.3.p3.8.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.3.p3.8.m1.1b"><apply id="A2.3.p3.8.m1.1.2.cmml" xref="A2.3.p3.8.m1.1.2"><times id="A2.3.p3.8.m1.1.2.1.cmml" xref="A2.3.p3.8.m1.1.2.1"></times><apply id="A2.3.p3.8.m1.1.2.2.cmml" xref="A2.3.p3.8.m1.1.2.2"><ci id="A2.3.p3.8.m1.1.2.2.1.cmml" xref="A2.3.p3.8.m1.1.2.2.1">~</ci><ci id="A2.3.p3.8.m1.1.2.2.2.cmml" xref="A2.3.p3.8.m1.1.2.2.2">𝜽</ci></apply><ci id="A2.3.p3.8.m1.1.1.cmml" xref="A2.3.p3.8.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.3.p3.8.m1.1c">\tilde{\bm{\theta}}(t)</annotation><annotation encoding="application/x-llamapun" id="A2.3.p3.8.m1.1d">over~ start_ARG bold_italic_θ end_ARG ( italic_t )</annotation></semantics></math> exponentially converge to <math alttext="\bm{0}" class="ltx_Math" display="inline" id="A2.3.p3.9.m2.1"><semantics id="A2.3.p3.9.m2.1a"><mn id="A2.3.p3.9.m2.1.1" xref="A2.3.p3.9.m2.1.1.cmml">𝟎</mn><annotation-xml encoding="MathML-Content" id="A2.3.p3.9.m2.1b"><cn id="A2.3.p3.9.m2.1.1.cmml" type="integer" xref="A2.3.p3.9.m2.1.1">0</cn></annotation-xml><annotation encoding="application/x-tex" id="A2.3.p3.9.m2.1c">\bm{0}</annotation><annotation encoding="application/x-llamapun" id="A2.3.p3.9.m2.1d">bold_0</annotation></semantics></math> on <math alttext="t\in[T_{\rm e},\infty)" class="ltx_Math" display="inline" id="A2.3.p3.10.m3.2"><semantics id="A2.3.p3.10.m3.2a"><mrow id="A2.3.p3.10.m3.2.2" xref="A2.3.p3.10.m3.2.2.cmml"><mi id="A2.3.p3.10.m3.2.2.3" xref="A2.3.p3.10.m3.2.2.3.cmml">t</mi><mo id="A2.3.p3.10.m3.2.2.2" xref="A2.3.p3.10.m3.2.2.2.cmml">∈</mo><mrow id="A2.3.p3.10.m3.2.2.1.1" xref="A2.3.p3.10.m3.2.2.1.2.cmml"><mo id="A2.3.p3.10.m3.2.2.1.1.2" stretchy="false" xref="A2.3.p3.10.m3.2.2.1.2.cmml">[</mo><msub id="A2.3.p3.10.m3.2.2.1.1.1" xref="A2.3.p3.10.m3.2.2.1.1.1.cmml"><mi id="A2.3.p3.10.m3.2.2.1.1.1.2" xref="A2.3.p3.10.m3.2.2.1.1.1.2.cmml">T</mi><mi id="A2.3.p3.10.m3.2.2.1.1.1.3" mathvariant="normal" xref="A2.3.p3.10.m3.2.2.1.1.1.3.cmml">e</mi></msub><mo id="A2.3.p3.10.m3.2.2.1.1.3" xref="A2.3.p3.10.m3.2.2.1.2.cmml">,</mo><mi id="A2.3.p3.10.m3.1.1" mathvariant="normal" xref="A2.3.p3.10.m3.1.1.cmml">∞</mi><mo id="A2.3.p3.10.m3.2.2.1.1.4" stretchy="false" xref="A2.3.p3.10.m3.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.3.p3.10.m3.2b"><apply id="A2.3.p3.10.m3.2.2.cmml" xref="A2.3.p3.10.m3.2.2"><in id="A2.3.p3.10.m3.2.2.2.cmml" xref="A2.3.p3.10.m3.2.2.2"></in><ci id="A2.3.p3.10.m3.2.2.3.cmml" xref="A2.3.p3.10.m3.2.2.3">𝑡</ci><interval closure="closed-open" id="A2.3.p3.10.m3.2.2.1.2.cmml" xref="A2.3.p3.10.m3.2.2.1.1"><apply id="A2.3.p3.10.m3.2.2.1.1.1.cmml" xref="A2.3.p3.10.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="A2.3.p3.10.m3.2.2.1.1.1.1.cmml" xref="A2.3.p3.10.m3.2.2.1.1.1">subscript</csymbol><ci id="A2.3.p3.10.m3.2.2.1.1.1.2.cmml" xref="A2.3.p3.10.m3.2.2.1.1.1.2">𝑇</ci><ci id="A2.3.p3.10.m3.2.2.1.1.1.3.cmml" xref="A2.3.p3.10.m3.2.2.1.1.1.3">e</ci></apply><infinity id="A2.3.p3.10.m3.1.1.cmml" xref="A2.3.p3.10.m3.1.1"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.3.p3.10.m3.2c">t\in[T_{\rm e},\infty)</annotation><annotation encoding="application/x-llamapun" id="A2.3.p3.10.m3.2d">italic_t ∈ [ italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT , ∞ )</annotation></semantics></math>. ∎</p> </div> </div> </section> <section class="ltx_appendix" id="A3"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix C </span>The proof of Theorem 2</h2> <div class="ltx_proof" id="A3.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A3.1.p1"> <p class="ltx_p" id="A3.1.p1.94">1) Choose a Lyapunov function candidate</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx51"> <tbody id="A3.E39"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle V(\bm{w})=\bm{e}^{T}\bm{e}/2+(1+p)\tilde{\bm{\theta}}^{T}\Gamma^% {-1}\tilde{\bm{\theta}}/2" class="ltx_Math" display="inline" id="A3.E39.m1.2"><semantics id="A3.E39.m1.2a"><mrow id="A3.E39.m1.2.2" xref="A3.E39.m1.2.2.cmml"><mrow id="A3.E39.m1.2.2.3" xref="A3.E39.m1.2.2.3.cmml"><mi id="A3.E39.m1.2.2.3.2" xref="A3.E39.m1.2.2.3.2.cmml">V</mi><mo id="A3.E39.m1.2.2.3.1" xref="A3.E39.m1.2.2.3.1.cmml">⁢</mo><mrow id="A3.E39.m1.2.2.3.3.2" xref="A3.E39.m1.2.2.3.cmml"><mo id="A3.E39.m1.2.2.3.3.2.1" stretchy="false" xref="A3.E39.m1.2.2.3.cmml">(</mo><mi id="A3.E39.m1.1.1" xref="A3.E39.m1.1.1.cmml">𝒘</mi><mo id="A3.E39.m1.2.2.3.3.2.2" stretchy="false" xref="A3.E39.m1.2.2.3.cmml">)</mo></mrow></mrow><mo id="A3.E39.m1.2.2.2" xref="A3.E39.m1.2.2.2.cmml">=</mo><mrow id="A3.E39.m1.2.2.1" xref="A3.E39.m1.2.2.1.cmml"><mrow id="A3.E39.m1.2.2.1.3" xref="A3.E39.m1.2.2.1.3.cmml"><mrow id="A3.E39.m1.2.2.1.3.2" xref="A3.E39.m1.2.2.1.3.2.cmml"><msup id="A3.E39.m1.2.2.1.3.2.2" xref="A3.E39.m1.2.2.1.3.2.2.cmml"><mi id="A3.E39.m1.2.2.1.3.2.2.2" xref="A3.E39.m1.2.2.1.3.2.2.2.cmml">𝒆</mi><mi id="A3.E39.m1.2.2.1.3.2.2.3" xref="A3.E39.m1.2.2.1.3.2.2.3.cmml">T</mi></msup><mo id="A3.E39.m1.2.2.1.3.2.1" xref="A3.E39.m1.2.2.1.3.2.1.cmml">⁢</mo><mi id="A3.E39.m1.2.2.1.3.2.3" xref="A3.E39.m1.2.2.1.3.2.3.cmml">𝒆</mi></mrow><mo id="A3.E39.m1.2.2.1.3.1" xref="A3.E39.m1.2.2.1.3.1.cmml">/</mo><mn id="A3.E39.m1.2.2.1.3.3" xref="A3.E39.m1.2.2.1.3.3.cmml">2</mn></mrow><mo id="A3.E39.m1.2.2.1.2" xref="A3.E39.m1.2.2.1.2.cmml">+</mo><mrow id="A3.E39.m1.2.2.1.1" xref="A3.E39.m1.2.2.1.1.cmml"><mrow id="A3.E39.m1.2.2.1.1.1" xref="A3.E39.m1.2.2.1.1.1.cmml"><mrow id="A3.E39.m1.2.2.1.1.1.1.1" xref="A3.E39.m1.2.2.1.1.1.1.1.1.cmml"><mo id="A3.E39.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="A3.E39.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.E39.m1.2.2.1.1.1.1.1.1" xref="A3.E39.m1.2.2.1.1.1.1.1.1.cmml"><mn id="A3.E39.m1.2.2.1.1.1.1.1.1.2" xref="A3.E39.m1.2.2.1.1.1.1.1.1.2.cmml">1</mn><mo id="A3.E39.m1.2.2.1.1.1.1.1.1.1" xref="A3.E39.m1.2.2.1.1.1.1.1.1.1.cmml">+</mo><mi id="A3.E39.m1.2.2.1.1.1.1.1.1.3" xref="A3.E39.m1.2.2.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A3.E39.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="A3.E39.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A3.E39.m1.2.2.1.1.1.2" xref="A3.E39.m1.2.2.1.1.1.2.cmml">⁢</mo><msup id="A3.E39.m1.2.2.1.1.1.3" xref="A3.E39.m1.2.2.1.1.1.3.cmml"><mover accent="true" id="A3.E39.m1.2.2.1.1.1.3.2" xref="A3.E39.m1.2.2.1.1.1.3.2.cmml"><mi id="A3.E39.m1.2.2.1.1.1.3.2.2" xref="A3.E39.m1.2.2.1.1.1.3.2.2.cmml">𝜽</mi><mo id="A3.E39.m1.2.2.1.1.1.3.2.1" xref="A3.E39.m1.2.2.1.1.1.3.2.1.cmml">~</mo></mover><mi id="A3.E39.m1.2.2.1.1.1.3.3" xref="A3.E39.m1.2.2.1.1.1.3.3.cmml">T</mi></msup><mo id="A3.E39.m1.2.2.1.1.1.2a" xref="A3.E39.m1.2.2.1.1.1.2.cmml">⁢</mo><msup id="A3.E39.m1.2.2.1.1.1.4" xref="A3.E39.m1.2.2.1.1.1.4.cmml"><mi id="A3.E39.m1.2.2.1.1.1.4.2" mathvariant="normal" xref="A3.E39.m1.2.2.1.1.1.4.2.cmml">Γ</mi><mrow id="A3.E39.m1.2.2.1.1.1.4.3" xref="A3.E39.m1.2.2.1.1.1.4.3.cmml"><mo id="A3.E39.m1.2.2.1.1.1.4.3a" xref="A3.E39.m1.2.2.1.1.1.4.3.cmml">−</mo><mn id="A3.E39.m1.2.2.1.1.1.4.3.2" xref="A3.E39.m1.2.2.1.1.1.4.3.2.cmml">1</mn></mrow></msup><mo id="A3.E39.m1.2.2.1.1.1.2b" xref="A3.E39.m1.2.2.1.1.1.2.cmml">⁢</mo><mover accent="true" id="A3.E39.m1.2.2.1.1.1.5" xref="A3.E39.m1.2.2.1.1.1.5.cmml"><mi id="A3.E39.m1.2.2.1.1.1.5.2" xref="A3.E39.m1.2.2.1.1.1.5.2.cmml">𝜽</mi><mo id="A3.E39.m1.2.2.1.1.1.5.1" xref="A3.E39.m1.2.2.1.1.1.5.1.cmml">~</mo></mover></mrow><mo id="A3.E39.m1.2.2.1.1.2" xref="A3.E39.m1.2.2.1.1.2.cmml">/</mo><mn id="A3.E39.m1.2.2.1.1.3" xref="A3.E39.m1.2.2.1.1.3.cmml">2</mn></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.E39.m1.2b"><apply id="A3.E39.m1.2.2.cmml" xref="A3.E39.m1.2.2"><eq id="A3.E39.m1.2.2.2.cmml" xref="A3.E39.m1.2.2.2"></eq><apply id="A3.E39.m1.2.2.3.cmml" xref="A3.E39.m1.2.2.3"><times id="A3.E39.m1.2.2.3.1.cmml" xref="A3.E39.m1.2.2.3.1"></times><ci id="A3.E39.m1.2.2.3.2.cmml" xref="A3.E39.m1.2.2.3.2">𝑉</ci><ci id="A3.E39.m1.1.1.cmml" xref="A3.E39.m1.1.1">𝒘</ci></apply><apply id="A3.E39.m1.2.2.1.cmml" xref="A3.E39.m1.2.2.1"><plus id="A3.E39.m1.2.2.1.2.cmml" xref="A3.E39.m1.2.2.1.2"></plus><apply id="A3.E39.m1.2.2.1.3.cmml" xref="A3.E39.m1.2.2.1.3"><divide id="A3.E39.m1.2.2.1.3.1.cmml" xref="A3.E39.m1.2.2.1.3.1"></divide><apply id="A3.E39.m1.2.2.1.3.2.cmml" xref="A3.E39.m1.2.2.1.3.2"><times id="A3.E39.m1.2.2.1.3.2.1.cmml" xref="A3.E39.m1.2.2.1.3.2.1"></times><apply id="A3.E39.m1.2.2.1.3.2.2.cmml" xref="A3.E39.m1.2.2.1.3.2.2"><csymbol cd="ambiguous" id="A3.E39.m1.2.2.1.3.2.2.1.cmml" xref="A3.E39.m1.2.2.1.3.2.2">superscript</csymbol><ci id="A3.E39.m1.2.2.1.3.2.2.2.cmml" xref="A3.E39.m1.2.2.1.3.2.2.2">𝒆</ci><ci id="A3.E39.m1.2.2.1.3.2.2.3.cmml" xref="A3.E39.m1.2.2.1.3.2.2.3">𝑇</ci></apply><ci id="A3.E39.m1.2.2.1.3.2.3.cmml" xref="A3.E39.m1.2.2.1.3.2.3">𝒆</ci></apply><cn id="A3.E39.m1.2.2.1.3.3.cmml" type="integer" xref="A3.E39.m1.2.2.1.3.3">2</cn></apply><apply id="A3.E39.m1.2.2.1.1.cmml" xref="A3.E39.m1.2.2.1.1"><divide id="A3.E39.m1.2.2.1.1.2.cmml" xref="A3.E39.m1.2.2.1.1.2"></divide><apply id="A3.E39.m1.2.2.1.1.1.cmml" xref="A3.E39.m1.2.2.1.1.1"><times id="A3.E39.m1.2.2.1.1.1.2.cmml" xref="A3.E39.m1.2.2.1.1.1.2"></times><apply id="A3.E39.m1.2.2.1.1.1.1.1.1.cmml" xref="A3.E39.m1.2.2.1.1.1.1.1"><plus id="A3.E39.m1.2.2.1.1.1.1.1.1.1.cmml" xref="A3.E39.m1.2.2.1.1.1.1.1.1.1"></plus><cn id="A3.E39.m1.2.2.1.1.1.1.1.1.2.cmml" type="integer" xref="A3.E39.m1.2.2.1.1.1.1.1.1.2">1</cn><ci id="A3.E39.m1.2.2.1.1.1.1.1.1.3.cmml" xref="A3.E39.m1.2.2.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="A3.E39.m1.2.2.1.1.1.3.cmml" xref="A3.E39.m1.2.2.1.1.1.3"><csymbol cd="ambiguous" id="A3.E39.m1.2.2.1.1.1.3.1.cmml" xref="A3.E39.m1.2.2.1.1.1.3">superscript</csymbol><apply id="A3.E39.m1.2.2.1.1.1.3.2.cmml" xref="A3.E39.m1.2.2.1.1.1.3.2"><ci id="A3.E39.m1.2.2.1.1.1.3.2.1.cmml" xref="A3.E39.m1.2.2.1.1.1.3.2.1">~</ci><ci id="A3.E39.m1.2.2.1.1.1.3.2.2.cmml" xref="A3.E39.m1.2.2.1.1.1.3.2.2">𝜽</ci></apply><ci id="A3.E39.m1.2.2.1.1.1.3.3.cmml" xref="A3.E39.m1.2.2.1.1.1.3.3">𝑇</ci></apply><apply id="A3.E39.m1.2.2.1.1.1.4.cmml" xref="A3.E39.m1.2.2.1.1.1.4"><csymbol cd="ambiguous" id="A3.E39.m1.2.2.1.1.1.4.1.cmml" xref="A3.E39.m1.2.2.1.1.1.4">superscript</csymbol><ci id="A3.E39.m1.2.2.1.1.1.4.2.cmml" xref="A3.E39.m1.2.2.1.1.1.4.2">Γ</ci><apply id="A3.E39.m1.2.2.1.1.1.4.3.cmml" xref="A3.E39.m1.2.2.1.1.1.4.3"><minus id="A3.E39.m1.2.2.1.1.1.4.3.1.cmml" xref="A3.E39.m1.2.2.1.1.1.4.3"></minus><cn id="A3.E39.m1.2.2.1.1.1.4.3.2.cmml" type="integer" xref="A3.E39.m1.2.2.1.1.1.4.3.2">1</cn></apply></apply><apply id="A3.E39.m1.2.2.1.1.1.5.cmml" xref="A3.E39.m1.2.2.1.1.1.5"><ci id="A3.E39.m1.2.2.1.1.1.5.1.cmml" xref="A3.E39.m1.2.2.1.1.1.5.1">~</ci><ci id="A3.E39.m1.2.2.1.1.1.5.2.cmml" xref="A3.E39.m1.2.2.1.1.1.5.2">𝜽</ci></apply></apply><cn id="A3.E39.m1.2.2.1.1.3.cmml" type="integer" xref="A3.E39.m1.2.2.1.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E39.m1.2c">\displaystyle V(\bm{w})=\bm{e}^{T}\bm{e}/2+(1+p)\tilde{\bm{\theta}}^{T}\Gamma^% {-1}\tilde{\bm{\theta}}/2</annotation><annotation encoding="application/x-llamapun" id="A3.E39.m1.2d">italic_V ( bold_italic_w ) = bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_e / 2 + ( 1 + italic_p ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Γ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG / 2</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(39)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.1.p1.10">with <math alttext="p\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A3.1.p1.1.m1.1"><semantics id="A3.1.p1.1.m1.1a"><mrow id="A3.1.p1.1.m1.1.1" xref="A3.1.p1.1.m1.1.1.cmml"><mi id="A3.1.p1.1.m1.1.1.2" xref="A3.1.p1.1.m1.1.1.2.cmml">p</mi><mo id="A3.1.p1.1.m1.1.1.1" xref="A3.1.p1.1.m1.1.1.1.cmml">∈</mo><msup id="A3.1.p1.1.m1.1.1.3" xref="A3.1.p1.1.m1.1.1.3.cmml"><mi id="A3.1.p1.1.m1.1.1.3.2" xref="A3.1.p1.1.m1.1.1.3.2.cmml">ℝ</mi><mo id="A3.1.p1.1.m1.1.1.3.3" xref="A3.1.p1.1.m1.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.1.m1.1b"><apply id="A3.1.p1.1.m1.1.1.cmml" xref="A3.1.p1.1.m1.1.1"><in id="A3.1.p1.1.m1.1.1.1.cmml" xref="A3.1.p1.1.m1.1.1.1"></in><ci id="A3.1.p1.1.m1.1.1.2.cmml" xref="A3.1.p1.1.m1.1.1.2">𝑝</ci><apply id="A3.1.p1.1.m1.1.1.3.cmml" xref="A3.1.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="A3.1.p1.1.m1.1.1.3.1.cmml" xref="A3.1.p1.1.m1.1.1.3">superscript</csymbol><ci id="A3.1.p1.1.m1.1.1.3.2.cmml" xref="A3.1.p1.1.m1.1.1.3.2">ℝ</ci><plus id="A3.1.p1.1.m1.1.1.3.3.cmml" xref="A3.1.p1.1.m1.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.1.m1.1c">p\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.1.m1.1d">italic_p ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> being a constant and <math alttext="\bm{w}" class="ltx_Math" display="inline" id="A3.1.p1.2.m2.1"><semantics id="A3.1.p1.2.m2.1a"><mi id="A3.1.p1.2.m2.1.1" xref="A3.1.p1.2.m2.1.1.cmml">𝒘</mi><annotation-xml encoding="MathML-Content" id="A3.1.p1.2.m2.1b"><ci id="A3.1.p1.2.m2.1.1.cmml" xref="A3.1.p1.2.m2.1.1">𝒘</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.2.m2.1c">\bm{w}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.2.m2.1d">bold_italic_w</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="A3.1.p1.3.m3.1"><semantics id="A3.1.p1.3.m3.1a"><mo id="A3.1.p1.3.m3.1.1" xref="A3.1.p1.3.m3.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="A3.1.p1.3.m3.1b"><csymbol cd="latexml" id="A3.1.p1.3.m3.1.1.cmml" xref="A3.1.p1.3.m3.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.3.m3.1c">:=</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.3.m3.1d">:=</annotation></semantics></math> <math alttext="[\bm{e}^{T},\tilde{\bm{\theta}}^{T}]^{T}\in\mathbb{R}^{n+N}" class="ltx_Math" display="inline" id="A3.1.p1.4.m4.2"><semantics id="A3.1.p1.4.m4.2a"><mrow id="A3.1.p1.4.m4.2.2" xref="A3.1.p1.4.m4.2.2.cmml"><msup id="A3.1.p1.4.m4.2.2.2" xref="A3.1.p1.4.m4.2.2.2.cmml"><mrow id="A3.1.p1.4.m4.2.2.2.2.2" xref="A3.1.p1.4.m4.2.2.2.2.3.cmml"><mo id="A3.1.p1.4.m4.2.2.2.2.2.3" stretchy="false" xref="A3.1.p1.4.m4.2.2.2.2.3.cmml">[</mo><msup id="A3.1.p1.4.m4.1.1.1.1.1.1" xref="A3.1.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="A3.1.p1.4.m4.1.1.1.1.1.1.2" xref="A3.1.p1.4.m4.1.1.1.1.1.1.2.cmml">𝒆</mi><mi id="A3.1.p1.4.m4.1.1.1.1.1.1.3" xref="A3.1.p1.4.m4.1.1.1.1.1.1.3.cmml">T</mi></msup><mo id="A3.1.p1.4.m4.2.2.2.2.2.4" xref="A3.1.p1.4.m4.2.2.2.2.3.cmml">,</mo><msup id="A3.1.p1.4.m4.2.2.2.2.2.2" xref="A3.1.p1.4.m4.2.2.2.2.2.2.cmml"><mover accent="true" id="A3.1.p1.4.m4.2.2.2.2.2.2.2" xref="A3.1.p1.4.m4.2.2.2.2.2.2.2.cmml"><mi id="A3.1.p1.4.m4.2.2.2.2.2.2.2.2" xref="A3.1.p1.4.m4.2.2.2.2.2.2.2.2.cmml">𝜽</mi><mo id="A3.1.p1.4.m4.2.2.2.2.2.2.2.1" xref="A3.1.p1.4.m4.2.2.2.2.2.2.2.1.cmml">~</mo></mover><mi id="A3.1.p1.4.m4.2.2.2.2.2.2.3" xref="A3.1.p1.4.m4.2.2.2.2.2.2.3.cmml">T</mi></msup><mo id="A3.1.p1.4.m4.2.2.2.2.2.5" stretchy="false" xref="A3.1.p1.4.m4.2.2.2.2.3.cmml">]</mo></mrow><mi id="A3.1.p1.4.m4.2.2.2.4" xref="A3.1.p1.4.m4.2.2.2.4.cmml">T</mi></msup><mo id="A3.1.p1.4.m4.2.2.3" xref="A3.1.p1.4.m4.2.2.3.cmml">∈</mo><msup id="A3.1.p1.4.m4.2.2.4" xref="A3.1.p1.4.m4.2.2.4.cmml"><mi id="A3.1.p1.4.m4.2.2.4.2" xref="A3.1.p1.4.m4.2.2.4.2.cmml">ℝ</mi><mrow id="A3.1.p1.4.m4.2.2.4.3" xref="A3.1.p1.4.m4.2.2.4.3.cmml"><mi id="A3.1.p1.4.m4.2.2.4.3.2" xref="A3.1.p1.4.m4.2.2.4.3.2.cmml">n</mi><mo id="A3.1.p1.4.m4.2.2.4.3.1" xref="A3.1.p1.4.m4.2.2.4.3.1.cmml">+</mo><mi id="A3.1.p1.4.m4.2.2.4.3.3" xref="A3.1.p1.4.m4.2.2.4.3.3.cmml">N</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.4.m4.2b"><apply id="A3.1.p1.4.m4.2.2.cmml" xref="A3.1.p1.4.m4.2.2"><in id="A3.1.p1.4.m4.2.2.3.cmml" xref="A3.1.p1.4.m4.2.2.3"></in><apply id="A3.1.p1.4.m4.2.2.2.cmml" xref="A3.1.p1.4.m4.2.2.2"><csymbol cd="ambiguous" id="A3.1.p1.4.m4.2.2.2.3.cmml" xref="A3.1.p1.4.m4.2.2.2">superscript</csymbol><interval closure="closed" id="A3.1.p1.4.m4.2.2.2.2.3.cmml" xref="A3.1.p1.4.m4.2.2.2.2.2"><apply id="A3.1.p1.4.m4.1.1.1.1.1.1.cmml" xref="A3.1.p1.4.m4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.1.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="A3.1.p1.4.m4.1.1.1.1.1.1">superscript</csymbol><ci id="A3.1.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="A3.1.p1.4.m4.1.1.1.1.1.1.2">𝒆</ci><ci id="A3.1.p1.4.m4.1.1.1.1.1.1.3.cmml" xref="A3.1.p1.4.m4.1.1.1.1.1.1.3">𝑇</ci></apply><apply id="A3.1.p1.4.m4.2.2.2.2.2.2.cmml" xref="A3.1.p1.4.m4.2.2.2.2.2.2"><csymbol cd="ambiguous" id="A3.1.p1.4.m4.2.2.2.2.2.2.1.cmml" xref="A3.1.p1.4.m4.2.2.2.2.2.2">superscript</csymbol><apply id="A3.1.p1.4.m4.2.2.2.2.2.2.2.cmml" xref="A3.1.p1.4.m4.2.2.2.2.2.2.2"><ci id="A3.1.p1.4.m4.2.2.2.2.2.2.2.1.cmml" xref="A3.1.p1.4.m4.2.2.2.2.2.2.2.1">~</ci><ci id="A3.1.p1.4.m4.2.2.2.2.2.2.2.2.cmml" xref="A3.1.p1.4.m4.2.2.2.2.2.2.2.2">𝜽</ci></apply><ci id="A3.1.p1.4.m4.2.2.2.2.2.2.3.cmml" xref="A3.1.p1.4.m4.2.2.2.2.2.2.3">𝑇</ci></apply></interval><ci id="A3.1.p1.4.m4.2.2.2.4.cmml" xref="A3.1.p1.4.m4.2.2.2.4">𝑇</ci></apply><apply id="A3.1.p1.4.m4.2.2.4.cmml" xref="A3.1.p1.4.m4.2.2.4"><csymbol cd="ambiguous" id="A3.1.p1.4.m4.2.2.4.1.cmml" xref="A3.1.p1.4.m4.2.2.4">superscript</csymbol><ci id="A3.1.p1.4.m4.2.2.4.2.cmml" xref="A3.1.p1.4.m4.2.2.4.2">ℝ</ci><apply id="A3.1.p1.4.m4.2.2.4.3.cmml" xref="A3.1.p1.4.m4.2.2.4.3"><plus id="A3.1.p1.4.m4.2.2.4.3.1.cmml" xref="A3.1.p1.4.m4.2.2.4.3.1"></plus><ci id="A3.1.p1.4.m4.2.2.4.3.2.cmml" xref="A3.1.p1.4.m4.2.2.4.3.2">𝑛</ci><ci id="A3.1.p1.4.m4.2.2.4.3.3.cmml" xref="A3.1.p1.4.m4.2.2.4.3.3">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.4.m4.2c">[\bm{e}^{T},\tilde{\bm{\theta}}^{T}]^{T}\in\mathbb{R}^{n+N}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.4.m4.2d">[ bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT , over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_n + italic_N end_POSTSUPERSCRIPT</annotation></semantics></math>. Thus, there exist constants <math alttext="\lambda_{\rm a}" class="ltx_Math" display="inline" id="A3.1.p1.5.m5.1"><semantics id="A3.1.p1.5.m5.1a"><msub id="A3.1.p1.5.m5.1.1" xref="A3.1.p1.5.m5.1.1.cmml"><mi id="A3.1.p1.5.m5.1.1.2" xref="A3.1.p1.5.m5.1.1.2.cmml">λ</mi><mi id="A3.1.p1.5.m5.1.1.3" mathvariant="normal" xref="A3.1.p1.5.m5.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="A3.1.p1.5.m5.1b"><apply id="A3.1.p1.5.m5.1.1.cmml" xref="A3.1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="A3.1.p1.5.m5.1.1.1.cmml" xref="A3.1.p1.5.m5.1.1">subscript</csymbol><ci id="A3.1.p1.5.m5.1.1.2.cmml" xref="A3.1.p1.5.m5.1.1.2">𝜆</ci><ci id="A3.1.p1.5.m5.1.1.3.cmml" xref="A3.1.p1.5.m5.1.1.3">a</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.5.m5.1c">\lambda_{\rm a}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.5.m5.1d">italic_λ start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="A3.1.p1.6.m6.1"><semantics id="A3.1.p1.6.m6.1a"><mo id="A3.1.p1.6.m6.1.1" xref="A3.1.p1.6.m6.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="A3.1.p1.6.m6.1b"><csymbol cd="latexml" id="A3.1.p1.6.m6.1.1.cmml" xref="A3.1.p1.6.m6.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.6.m6.1c">:=</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.6.m6.1d">:=</annotation></semantics></math> <math alttext="\min\{1/2,(1+p)/(2\lambda_{\max}(\Gamma))\}" class="ltx_Math" display="inline" id="A3.1.p1.7.m7.4"><semantics id="A3.1.p1.7.m7.4a"><mrow id="A3.1.p1.7.m7.4.4.2" xref="A3.1.p1.7.m7.4.4.3.cmml"><mi id="A3.1.p1.7.m7.2.2" xref="A3.1.p1.7.m7.2.2.cmml">min</mi><mo id="A3.1.p1.7.m7.4.4.2a" xref="A3.1.p1.7.m7.4.4.3.cmml">⁡</mo><mrow id="A3.1.p1.7.m7.4.4.2.2" xref="A3.1.p1.7.m7.4.4.3.cmml"><mo id="A3.1.p1.7.m7.4.4.2.2.3" stretchy="false" xref="A3.1.p1.7.m7.4.4.3.cmml">{</mo><mrow id="A3.1.p1.7.m7.3.3.1.1.1" xref="A3.1.p1.7.m7.3.3.1.1.1.cmml"><mn id="A3.1.p1.7.m7.3.3.1.1.1.2" xref="A3.1.p1.7.m7.3.3.1.1.1.2.cmml">1</mn><mo id="A3.1.p1.7.m7.3.3.1.1.1.1" xref="A3.1.p1.7.m7.3.3.1.1.1.1.cmml">/</mo><mn id="A3.1.p1.7.m7.3.3.1.1.1.3" xref="A3.1.p1.7.m7.3.3.1.1.1.3.cmml">2</mn></mrow><mo id="A3.1.p1.7.m7.4.4.2.2.4" xref="A3.1.p1.7.m7.4.4.3.cmml">,</mo><mrow id="A3.1.p1.7.m7.4.4.2.2.2" xref="A3.1.p1.7.m7.4.4.2.2.2.cmml"><mrow id="A3.1.p1.7.m7.4.4.2.2.2.1.1" xref="A3.1.p1.7.m7.4.4.2.2.2.1.1.1.cmml"><mo id="A3.1.p1.7.m7.4.4.2.2.2.1.1.2" stretchy="false" xref="A3.1.p1.7.m7.4.4.2.2.2.1.1.1.cmml">(</mo><mrow id="A3.1.p1.7.m7.4.4.2.2.2.1.1.1" xref="A3.1.p1.7.m7.4.4.2.2.2.1.1.1.cmml"><mn id="A3.1.p1.7.m7.4.4.2.2.2.1.1.1.2" xref="A3.1.p1.7.m7.4.4.2.2.2.1.1.1.2.cmml">1</mn><mo id="A3.1.p1.7.m7.4.4.2.2.2.1.1.1.1" xref="A3.1.p1.7.m7.4.4.2.2.2.1.1.1.1.cmml">+</mo><mi id="A3.1.p1.7.m7.4.4.2.2.2.1.1.1.3" xref="A3.1.p1.7.m7.4.4.2.2.2.1.1.1.3.cmml">p</mi></mrow><mo id="A3.1.p1.7.m7.4.4.2.2.2.1.1.3" stretchy="false" xref="A3.1.p1.7.m7.4.4.2.2.2.1.1.1.cmml">)</mo></mrow><mo id="A3.1.p1.7.m7.4.4.2.2.2.3" xref="A3.1.p1.7.m7.4.4.2.2.2.3.cmml">/</mo><mrow id="A3.1.p1.7.m7.4.4.2.2.2.2.1" xref="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.cmml"><mo id="A3.1.p1.7.m7.4.4.2.2.2.2.1.2" stretchy="false" xref="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.cmml">(</mo><mrow id="A3.1.p1.7.m7.4.4.2.2.2.2.1.1" xref="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.cmml"><mn id="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.2" xref="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.2.cmml">2</mn><mo id="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.1" xref="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.1.cmml">⁢</mo><msub id="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.3" xref="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.3.cmml"><mi id="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.3.2" xref="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.3.2.cmml">λ</mi><mi id="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.3.3" xref="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.3.3.cmml">max</mi></msub><mo id="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.1a" xref="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.1.cmml">⁢</mo><mrow id="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.4.2" xref="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.cmml"><mo id="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.4.2.1" stretchy="false" xref="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.cmml">(</mo><mi id="A3.1.p1.7.m7.1.1" mathvariant="normal" xref="A3.1.p1.7.m7.1.1.cmml">Γ</mi><mo id="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.4.2.2" stretchy="false" xref="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="A3.1.p1.7.m7.4.4.2.2.2.2.1.3" stretchy="false" xref="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="A3.1.p1.7.m7.4.4.2.2.5" stretchy="false" xref="A3.1.p1.7.m7.4.4.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.7.m7.4b"><apply id="A3.1.p1.7.m7.4.4.3.cmml" xref="A3.1.p1.7.m7.4.4.2"><min id="A3.1.p1.7.m7.2.2.cmml" xref="A3.1.p1.7.m7.2.2"></min><apply id="A3.1.p1.7.m7.3.3.1.1.1.cmml" xref="A3.1.p1.7.m7.3.3.1.1.1"><divide id="A3.1.p1.7.m7.3.3.1.1.1.1.cmml" xref="A3.1.p1.7.m7.3.3.1.1.1.1"></divide><cn id="A3.1.p1.7.m7.3.3.1.1.1.2.cmml" type="integer" xref="A3.1.p1.7.m7.3.3.1.1.1.2">1</cn><cn id="A3.1.p1.7.m7.3.3.1.1.1.3.cmml" type="integer" xref="A3.1.p1.7.m7.3.3.1.1.1.3">2</cn></apply><apply id="A3.1.p1.7.m7.4.4.2.2.2.cmml" xref="A3.1.p1.7.m7.4.4.2.2.2"><divide id="A3.1.p1.7.m7.4.4.2.2.2.3.cmml" xref="A3.1.p1.7.m7.4.4.2.2.2.3"></divide><apply id="A3.1.p1.7.m7.4.4.2.2.2.1.1.1.cmml" xref="A3.1.p1.7.m7.4.4.2.2.2.1.1"><plus id="A3.1.p1.7.m7.4.4.2.2.2.1.1.1.1.cmml" xref="A3.1.p1.7.m7.4.4.2.2.2.1.1.1.1"></plus><cn id="A3.1.p1.7.m7.4.4.2.2.2.1.1.1.2.cmml" type="integer" xref="A3.1.p1.7.m7.4.4.2.2.2.1.1.1.2">1</cn><ci id="A3.1.p1.7.m7.4.4.2.2.2.1.1.1.3.cmml" xref="A3.1.p1.7.m7.4.4.2.2.2.1.1.1.3">𝑝</ci></apply><apply id="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.cmml" xref="A3.1.p1.7.m7.4.4.2.2.2.2.1"><times id="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.1.cmml" xref="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.1"></times><cn id="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.2.cmml" type="integer" xref="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.2">2</cn><apply id="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.3.cmml" xref="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.3"><csymbol cd="ambiguous" id="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.3.1.cmml" xref="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.3">subscript</csymbol><ci id="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.3.2.cmml" xref="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.3.2">𝜆</ci><max id="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.3.3.cmml" xref="A3.1.p1.7.m7.4.4.2.2.2.2.1.1.3.3"></max></apply><ci id="A3.1.p1.7.m7.1.1.cmml" xref="A3.1.p1.7.m7.1.1">Γ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.7.m7.4c">\min\{1/2,(1+p)/(2\lambda_{\max}(\Gamma))\}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.7.m7.4d">roman_min { 1 / 2 , ( 1 + italic_p ) / ( 2 italic_λ start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ( roman_Γ ) ) }</annotation></semantics></math> and <math alttext="\lambda_{\rm b}" class="ltx_Math" display="inline" id="A3.1.p1.8.m8.1"><semantics id="A3.1.p1.8.m8.1a"><msub id="A3.1.p1.8.m8.1.1" xref="A3.1.p1.8.m8.1.1.cmml"><mi id="A3.1.p1.8.m8.1.1.2" xref="A3.1.p1.8.m8.1.1.2.cmml">λ</mi><mi id="A3.1.p1.8.m8.1.1.3" mathvariant="normal" xref="A3.1.p1.8.m8.1.1.3.cmml">b</mi></msub><annotation-xml encoding="MathML-Content" id="A3.1.p1.8.m8.1b"><apply id="A3.1.p1.8.m8.1.1.cmml" xref="A3.1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="A3.1.p1.8.m8.1.1.1.cmml" xref="A3.1.p1.8.m8.1.1">subscript</csymbol><ci id="A3.1.p1.8.m8.1.1.2.cmml" xref="A3.1.p1.8.m8.1.1.2">𝜆</ci><ci id="A3.1.p1.8.m8.1.1.3.cmml" xref="A3.1.p1.8.m8.1.1.3">b</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.8.m8.1c">\lambda_{\rm b}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.8.m8.1d">italic_λ start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="A3.1.p1.9.m9.1"><semantics id="A3.1.p1.9.m9.1a"><mo id="A3.1.p1.9.m9.1.1" xref="A3.1.p1.9.m9.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="A3.1.p1.9.m9.1b"><csymbol cd="latexml" id="A3.1.p1.9.m9.1.1.cmml" xref="A3.1.p1.9.m9.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.9.m9.1c">:=</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.9.m9.1d">:=</annotation></semantics></math> <math alttext="\min\{1/2,(1+p)/(2\lambda_{\min}(\Gamma))\}" class="ltx_Math" display="inline" id="A3.1.p1.10.m10.4"><semantics id="A3.1.p1.10.m10.4a"><mrow id="A3.1.p1.10.m10.4.4.2" xref="A3.1.p1.10.m10.4.4.3.cmml"><mi id="A3.1.p1.10.m10.2.2" xref="A3.1.p1.10.m10.2.2.cmml">min</mi><mo id="A3.1.p1.10.m10.4.4.2a" xref="A3.1.p1.10.m10.4.4.3.cmml">⁡</mo><mrow id="A3.1.p1.10.m10.4.4.2.2" xref="A3.1.p1.10.m10.4.4.3.cmml"><mo id="A3.1.p1.10.m10.4.4.2.2.3" stretchy="false" xref="A3.1.p1.10.m10.4.4.3.cmml">{</mo><mrow id="A3.1.p1.10.m10.3.3.1.1.1" xref="A3.1.p1.10.m10.3.3.1.1.1.cmml"><mn id="A3.1.p1.10.m10.3.3.1.1.1.2" xref="A3.1.p1.10.m10.3.3.1.1.1.2.cmml">1</mn><mo id="A3.1.p1.10.m10.3.3.1.1.1.1" xref="A3.1.p1.10.m10.3.3.1.1.1.1.cmml">/</mo><mn id="A3.1.p1.10.m10.3.3.1.1.1.3" xref="A3.1.p1.10.m10.3.3.1.1.1.3.cmml">2</mn></mrow><mo id="A3.1.p1.10.m10.4.4.2.2.4" xref="A3.1.p1.10.m10.4.4.3.cmml">,</mo><mrow id="A3.1.p1.10.m10.4.4.2.2.2" xref="A3.1.p1.10.m10.4.4.2.2.2.cmml"><mrow id="A3.1.p1.10.m10.4.4.2.2.2.1.1" xref="A3.1.p1.10.m10.4.4.2.2.2.1.1.1.cmml"><mo id="A3.1.p1.10.m10.4.4.2.2.2.1.1.2" stretchy="false" xref="A3.1.p1.10.m10.4.4.2.2.2.1.1.1.cmml">(</mo><mrow id="A3.1.p1.10.m10.4.4.2.2.2.1.1.1" xref="A3.1.p1.10.m10.4.4.2.2.2.1.1.1.cmml"><mn id="A3.1.p1.10.m10.4.4.2.2.2.1.1.1.2" xref="A3.1.p1.10.m10.4.4.2.2.2.1.1.1.2.cmml">1</mn><mo id="A3.1.p1.10.m10.4.4.2.2.2.1.1.1.1" xref="A3.1.p1.10.m10.4.4.2.2.2.1.1.1.1.cmml">+</mo><mi id="A3.1.p1.10.m10.4.4.2.2.2.1.1.1.3" xref="A3.1.p1.10.m10.4.4.2.2.2.1.1.1.3.cmml">p</mi></mrow><mo id="A3.1.p1.10.m10.4.4.2.2.2.1.1.3" stretchy="false" xref="A3.1.p1.10.m10.4.4.2.2.2.1.1.1.cmml">)</mo></mrow><mo id="A3.1.p1.10.m10.4.4.2.2.2.3" xref="A3.1.p1.10.m10.4.4.2.2.2.3.cmml">/</mo><mrow id="A3.1.p1.10.m10.4.4.2.2.2.2.1" xref="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.cmml"><mo id="A3.1.p1.10.m10.4.4.2.2.2.2.1.2" stretchy="false" xref="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.cmml">(</mo><mrow id="A3.1.p1.10.m10.4.4.2.2.2.2.1.1" xref="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.cmml"><mn id="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.2" xref="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.2.cmml">2</mn><mo id="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.1" xref="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.1.cmml">⁢</mo><msub id="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.3" xref="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.3.cmml"><mi id="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.3.2" xref="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.3.2.cmml">λ</mi><mi id="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.3.3" xref="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.3.3.cmml">min</mi></msub><mo id="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.1a" xref="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.1.cmml">⁢</mo><mrow id="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.4.2" xref="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.cmml"><mo id="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.4.2.1" stretchy="false" xref="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.cmml">(</mo><mi id="A3.1.p1.10.m10.1.1" mathvariant="normal" xref="A3.1.p1.10.m10.1.1.cmml">Γ</mi><mo id="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.4.2.2" stretchy="false" xref="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="A3.1.p1.10.m10.4.4.2.2.2.2.1.3" stretchy="false" xref="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="A3.1.p1.10.m10.4.4.2.2.5" stretchy="false" xref="A3.1.p1.10.m10.4.4.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.10.m10.4b"><apply id="A3.1.p1.10.m10.4.4.3.cmml" xref="A3.1.p1.10.m10.4.4.2"><min id="A3.1.p1.10.m10.2.2.cmml" xref="A3.1.p1.10.m10.2.2"></min><apply id="A3.1.p1.10.m10.3.3.1.1.1.cmml" xref="A3.1.p1.10.m10.3.3.1.1.1"><divide id="A3.1.p1.10.m10.3.3.1.1.1.1.cmml" xref="A3.1.p1.10.m10.3.3.1.1.1.1"></divide><cn id="A3.1.p1.10.m10.3.3.1.1.1.2.cmml" type="integer" xref="A3.1.p1.10.m10.3.3.1.1.1.2">1</cn><cn id="A3.1.p1.10.m10.3.3.1.1.1.3.cmml" type="integer" xref="A3.1.p1.10.m10.3.3.1.1.1.3">2</cn></apply><apply id="A3.1.p1.10.m10.4.4.2.2.2.cmml" xref="A3.1.p1.10.m10.4.4.2.2.2"><divide id="A3.1.p1.10.m10.4.4.2.2.2.3.cmml" xref="A3.1.p1.10.m10.4.4.2.2.2.3"></divide><apply id="A3.1.p1.10.m10.4.4.2.2.2.1.1.1.cmml" xref="A3.1.p1.10.m10.4.4.2.2.2.1.1"><plus id="A3.1.p1.10.m10.4.4.2.2.2.1.1.1.1.cmml" xref="A3.1.p1.10.m10.4.4.2.2.2.1.1.1.1"></plus><cn id="A3.1.p1.10.m10.4.4.2.2.2.1.1.1.2.cmml" type="integer" xref="A3.1.p1.10.m10.4.4.2.2.2.1.1.1.2">1</cn><ci id="A3.1.p1.10.m10.4.4.2.2.2.1.1.1.3.cmml" xref="A3.1.p1.10.m10.4.4.2.2.2.1.1.1.3">𝑝</ci></apply><apply id="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.cmml" xref="A3.1.p1.10.m10.4.4.2.2.2.2.1"><times id="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.1.cmml" xref="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.1"></times><cn id="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.2.cmml" type="integer" xref="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.2">2</cn><apply id="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.3.cmml" xref="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.3"><csymbol cd="ambiguous" id="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.3.1.cmml" xref="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.3">subscript</csymbol><ci id="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.3.2.cmml" xref="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.3.2">𝜆</ci><min id="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.3.3.cmml" xref="A3.1.p1.10.m10.4.4.2.2.2.2.1.1.3.3"></min></apply><ci id="A3.1.p1.10.m10.1.1.cmml" xref="A3.1.p1.10.m10.1.1">Γ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.10.m10.4c">\min\{1/2,(1+p)/(2\lambda_{\min}(\Gamma))\}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.10.m10.4d">roman_min { 1 / 2 , ( 1 + italic_p ) / ( 2 italic_λ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Γ ) ) }</annotation></semantics></math> such that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx52"> <tbody id="A3.E40"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\lambda_{\rm a}\|\bm{w}\|^{2}\leq V(\bm{w})\leq\lambda_{\rm b}\|% \bm{w}\|^{2}." class="ltx_Math" display="inline" id="A3.E40.m1.4"><semantics id="A3.E40.m1.4a"><mrow id="A3.E40.m1.4.4.1" xref="A3.E40.m1.4.4.1.1.cmml"><mrow id="A3.E40.m1.4.4.1.1" xref="A3.E40.m1.4.4.1.1.cmml"><mrow id="A3.E40.m1.4.4.1.1.2" xref="A3.E40.m1.4.4.1.1.2.cmml"><msub id="A3.E40.m1.4.4.1.1.2.2" xref="A3.E40.m1.4.4.1.1.2.2.cmml"><mi id="A3.E40.m1.4.4.1.1.2.2.2" xref="A3.E40.m1.4.4.1.1.2.2.2.cmml">λ</mi><mi id="A3.E40.m1.4.4.1.1.2.2.3" mathvariant="normal" xref="A3.E40.m1.4.4.1.1.2.2.3.cmml">a</mi></msub><mo id="A3.E40.m1.4.4.1.1.2.1" xref="A3.E40.m1.4.4.1.1.2.1.cmml">⁢</mo><msup id="A3.E40.m1.4.4.1.1.2.3" xref="A3.E40.m1.4.4.1.1.2.3.cmml"><mrow id="A3.E40.m1.4.4.1.1.2.3.2.2" xref="A3.E40.m1.4.4.1.1.2.3.2.1.cmml"><mo id="A3.E40.m1.4.4.1.1.2.3.2.2.1" stretchy="false" xref="A3.E40.m1.4.4.1.1.2.3.2.1.1.cmml">‖</mo><mi id="A3.E40.m1.1.1" xref="A3.E40.m1.1.1.cmml">𝒘</mi><mo id="A3.E40.m1.4.4.1.1.2.3.2.2.2" stretchy="false" xref="A3.E40.m1.4.4.1.1.2.3.2.1.1.cmml">‖</mo></mrow><mn id="A3.E40.m1.4.4.1.1.2.3.3" xref="A3.E40.m1.4.4.1.1.2.3.3.cmml">2</mn></msup></mrow><mo id="A3.E40.m1.4.4.1.1.3" xref="A3.E40.m1.4.4.1.1.3.cmml">≤</mo><mrow id="A3.E40.m1.4.4.1.1.4" xref="A3.E40.m1.4.4.1.1.4.cmml"><mi id="A3.E40.m1.4.4.1.1.4.2" xref="A3.E40.m1.4.4.1.1.4.2.cmml">V</mi><mo id="A3.E40.m1.4.4.1.1.4.1" xref="A3.E40.m1.4.4.1.1.4.1.cmml">⁢</mo><mrow id="A3.E40.m1.4.4.1.1.4.3.2" xref="A3.E40.m1.4.4.1.1.4.cmml"><mo id="A3.E40.m1.4.4.1.1.4.3.2.1" stretchy="false" xref="A3.E40.m1.4.4.1.1.4.cmml">(</mo><mi id="A3.E40.m1.2.2" xref="A3.E40.m1.2.2.cmml">𝒘</mi><mo id="A3.E40.m1.4.4.1.1.4.3.2.2" stretchy="false" xref="A3.E40.m1.4.4.1.1.4.cmml">)</mo></mrow></mrow><mo id="A3.E40.m1.4.4.1.1.5" xref="A3.E40.m1.4.4.1.1.5.cmml">≤</mo><mrow id="A3.E40.m1.4.4.1.1.6" xref="A3.E40.m1.4.4.1.1.6.cmml"><msub id="A3.E40.m1.4.4.1.1.6.2" xref="A3.E40.m1.4.4.1.1.6.2.cmml"><mi id="A3.E40.m1.4.4.1.1.6.2.2" xref="A3.E40.m1.4.4.1.1.6.2.2.cmml">λ</mi><mi id="A3.E40.m1.4.4.1.1.6.2.3" mathvariant="normal" xref="A3.E40.m1.4.4.1.1.6.2.3.cmml">b</mi></msub><mo id="A3.E40.m1.4.4.1.1.6.1" xref="A3.E40.m1.4.4.1.1.6.1.cmml">⁢</mo><msup id="A3.E40.m1.4.4.1.1.6.3" xref="A3.E40.m1.4.4.1.1.6.3.cmml"><mrow id="A3.E40.m1.4.4.1.1.6.3.2.2" xref="A3.E40.m1.4.4.1.1.6.3.2.1.cmml"><mo id="A3.E40.m1.4.4.1.1.6.3.2.2.1" stretchy="false" xref="A3.E40.m1.4.4.1.1.6.3.2.1.1.cmml">‖</mo><mi id="A3.E40.m1.3.3" xref="A3.E40.m1.3.3.cmml">𝒘</mi><mo id="A3.E40.m1.4.4.1.1.6.3.2.2.2" stretchy="false" xref="A3.E40.m1.4.4.1.1.6.3.2.1.1.cmml">‖</mo></mrow><mn id="A3.E40.m1.4.4.1.1.6.3.3" xref="A3.E40.m1.4.4.1.1.6.3.3.cmml">2</mn></msup></mrow></mrow><mo id="A3.E40.m1.4.4.1.2" lspace="0em" xref="A3.E40.m1.4.4.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.E40.m1.4b"><apply id="A3.E40.m1.4.4.1.1.cmml" xref="A3.E40.m1.4.4.1"><and id="A3.E40.m1.4.4.1.1a.cmml" xref="A3.E40.m1.4.4.1"></and><apply id="A3.E40.m1.4.4.1.1b.cmml" xref="A3.E40.m1.4.4.1"><leq id="A3.E40.m1.4.4.1.1.3.cmml" xref="A3.E40.m1.4.4.1.1.3"></leq><apply id="A3.E40.m1.4.4.1.1.2.cmml" xref="A3.E40.m1.4.4.1.1.2"><times id="A3.E40.m1.4.4.1.1.2.1.cmml" xref="A3.E40.m1.4.4.1.1.2.1"></times><apply id="A3.E40.m1.4.4.1.1.2.2.cmml" xref="A3.E40.m1.4.4.1.1.2.2"><csymbol cd="ambiguous" id="A3.E40.m1.4.4.1.1.2.2.1.cmml" xref="A3.E40.m1.4.4.1.1.2.2">subscript</csymbol><ci id="A3.E40.m1.4.4.1.1.2.2.2.cmml" xref="A3.E40.m1.4.4.1.1.2.2.2">𝜆</ci><ci id="A3.E40.m1.4.4.1.1.2.2.3.cmml" xref="A3.E40.m1.4.4.1.1.2.2.3">a</ci></apply><apply id="A3.E40.m1.4.4.1.1.2.3.cmml" xref="A3.E40.m1.4.4.1.1.2.3"><csymbol cd="ambiguous" id="A3.E40.m1.4.4.1.1.2.3.1.cmml" xref="A3.E40.m1.4.4.1.1.2.3">superscript</csymbol><apply id="A3.E40.m1.4.4.1.1.2.3.2.1.cmml" xref="A3.E40.m1.4.4.1.1.2.3.2.2"><csymbol cd="latexml" id="A3.E40.m1.4.4.1.1.2.3.2.1.1.cmml" xref="A3.E40.m1.4.4.1.1.2.3.2.2.1">norm</csymbol><ci id="A3.E40.m1.1.1.cmml" xref="A3.E40.m1.1.1">𝒘</ci></apply><cn id="A3.E40.m1.4.4.1.1.2.3.3.cmml" type="integer" xref="A3.E40.m1.4.4.1.1.2.3.3">2</cn></apply></apply><apply id="A3.E40.m1.4.4.1.1.4.cmml" xref="A3.E40.m1.4.4.1.1.4"><times id="A3.E40.m1.4.4.1.1.4.1.cmml" xref="A3.E40.m1.4.4.1.1.4.1"></times><ci id="A3.E40.m1.4.4.1.1.4.2.cmml" xref="A3.E40.m1.4.4.1.1.4.2">𝑉</ci><ci id="A3.E40.m1.2.2.cmml" xref="A3.E40.m1.2.2">𝒘</ci></apply></apply><apply id="A3.E40.m1.4.4.1.1c.cmml" xref="A3.E40.m1.4.4.1"><leq id="A3.E40.m1.4.4.1.1.5.cmml" xref="A3.E40.m1.4.4.1.1.5"></leq><share href="https://arxiv.org/html/2401.10785v2#A3.E40.m1.4.4.1.1.4.cmml" id="A3.E40.m1.4.4.1.1d.cmml" xref="A3.E40.m1.4.4.1"></share><apply id="A3.E40.m1.4.4.1.1.6.cmml" xref="A3.E40.m1.4.4.1.1.6"><times id="A3.E40.m1.4.4.1.1.6.1.cmml" xref="A3.E40.m1.4.4.1.1.6.1"></times><apply id="A3.E40.m1.4.4.1.1.6.2.cmml" xref="A3.E40.m1.4.4.1.1.6.2"><csymbol cd="ambiguous" id="A3.E40.m1.4.4.1.1.6.2.1.cmml" xref="A3.E40.m1.4.4.1.1.6.2">subscript</csymbol><ci id="A3.E40.m1.4.4.1.1.6.2.2.cmml" xref="A3.E40.m1.4.4.1.1.6.2.2">𝜆</ci><ci id="A3.E40.m1.4.4.1.1.6.2.3.cmml" xref="A3.E40.m1.4.4.1.1.6.2.3">b</ci></apply><apply id="A3.E40.m1.4.4.1.1.6.3.cmml" xref="A3.E40.m1.4.4.1.1.6.3"><csymbol cd="ambiguous" id="A3.E40.m1.4.4.1.1.6.3.1.cmml" xref="A3.E40.m1.4.4.1.1.6.3">superscript</csymbol><apply id="A3.E40.m1.4.4.1.1.6.3.2.1.cmml" xref="A3.E40.m1.4.4.1.1.6.3.2.2"><csymbol cd="latexml" id="A3.E40.m1.4.4.1.1.6.3.2.1.1.cmml" xref="A3.E40.m1.4.4.1.1.6.3.2.2.1">norm</csymbol><ci id="A3.E40.m1.3.3.cmml" xref="A3.E40.m1.3.3">𝒘</ci></apply><cn id="A3.E40.m1.4.4.1.1.6.3.3.cmml" type="integer" xref="A3.E40.m1.4.4.1.1.6.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E40.m1.4c">\displaystyle\lambda_{\rm a}\|\bm{w}\|^{2}\leq V(\bm{w})\leq\lambda_{\rm b}\|% \bm{w}\|^{2}.</annotation><annotation encoding="application/x-llamapun" id="A3.E40.m1.4d">italic_λ start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT ∥ bold_italic_w ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ italic_V ( bold_italic_w ) ≤ italic_λ start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT ∥ bold_italic_w ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(40)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.1.p1.95">Noting (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E7" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">7</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S5.E26" title="In V-C Robustness Results ‣ V Theoretical Guarantees ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">26</span></a>), one obtains the closed-loop system</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx53"> <tbody id="A3.E43"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\left\{\begin{array}[]{c}\dot{\bm{e}}=\Lambda\bm{e}+\Phi^{T}(\bm{% x},\Theta_{n-1},{\bm{y}}_{{\rm r}n})\tilde{\bm{\theta}}\\ \dot{\tilde{\bm{\theta}}}=-\Gamma\left(\Phi_{\rm f}\Phi_{\rm f}^{T}\tilde{\bm{% \theta}}+\kappa Q(t,t_{\rm e})\tilde{\bm{\theta}}\right)\end{array}.\right." class="ltx_Math" display="inline" id="A3.E43.m1.2"><semantics id="A3.E43.m1.2a"><mrow id="A3.E43.m1.2.2.1"><mrow id="A3.E43.m1.2.2.1.1.2" xref="A3.E43.m1.2.2.1.1.1.cmml"><mo id="A3.E43.m1.2.2.1.1.2.1" xref="A3.E43.m1.2.2.1.1.1.1.cmml">{</mo><mtable id="A3.E43.m1.1.1" rowspacing="0pt" xref="A3.E43.m1.1.1.cmml"><mtr id="A3.E43.m1.1.1a" xref="A3.E43.m1.1.1.cmml"><mtd id="A3.E43.m1.1.1b" xref="A3.E43.m1.1.1.cmml"><mrow id="A3.E41.3.3" xref="A3.E41.3.3.cmml"><mover accent="true" id="A3.E41.3.3.5" xref="A3.E41.3.3.5.cmml"><mi id="A3.E41.3.3.5.2" xref="A3.E41.3.3.5.2.cmml">𝒆</mi><mo id="A3.E41.3.3.5.1" xref="A3.E41.3.3.5.1.cmml">˙</mo></mover><mo id="A3.E41.3.3.4" xref="A3.E41.3.3.4.cmml">=</mo><mrow id="A3.E41.3.3.3" xref="A3.E41.3.3.3.cmml"><mrow id="A3.E41.3.3.3.4" xref="A3.E41.3.3.3.4.cmml"><mi id="A3.E41.3.3.3.4.2" mathvariant="normal" xref="A3.E41.3.3.3.4.2.cmml">Λ</mi><mo id="A3.E41.3.3.3.4.1" xref="A3.E41.3.3.3.4.1.cmml">⁢</mo><mi id="A3.E41.3.3.3.4.3" xref="A3.E41.3.3.3.4.3.cmml">𝒆</mi></mrow><mo id="A3.E41.3.3.3.3" xref="A3.E41.3.3.3.3.cmml">+</mo><mrow id="A3.E41.3.3.3.2" xref="A3.E41.3.3.3.2.cmml"><msup id="A3.E41.3.3.3.2.4" xref="A3.E41.3.3.3.2.4.cmml"><mi id="A3.E41.3.3.3.2.4.2" mathvariant="normal" xref="A3.E41.3.3.3.2.4.2.cmml">Φ</mi><mi id="A3.E41.3.3.3.2.4.3" xref="A3.E41.3.3.3.2.4.3.cmml">T</mi></msup><mo id="A3.E41.3.3.3.2.3" xref="A3.E41.3.3.3.2.3.cmml">⁢</mo><mrow id="A3.E41.3.3.3.2.2.2" xref="A3.E41.3.3.3.2.2.3.cmml"><mo id="A3.E41.3.3.3.2.2.2.3" stretchy="false" xref="A3.E41.3.3.3.2.2.3.cmml">(</mo><mi id="A3.E41.1.1.1" xref="A3.E41.1.1.1.cmml">𝒙</mi><mo id="A3.E41.3.3.3.2.2.2.4" xref="A3.E41.3.3.3.2.2.3.cmml">,</mo><msub id="A3.E41.2.2.2.1.1.1.1" xref="A3.E41.2.2.2.1.1.1.1.cmml"><mi id="A3.E41.2.2.2.1.1.1.1.2" mathvariant="normal" xref="A3.E41.2.2.2.1.1.1.1.2.cmml">Θ</mi><mrow id="A3.E41.2.2.2.1.1.1.1.3" xref="A3.E41.2.2.2.1.1.1.1.3.cmml"><mi id="A3.E41.2.2.2.1.1.1.1.3.2" xref="A3.E41.2.2.2.1.1.1.1.3.2.cmml">n</mi><mo id="A3.E41.2.2.2.1.1.1.1.3.1" xref="A3.E41.2.2.2.1.1.1.1.3.1.cmml">−</mo><mn id="A3.E41.2.2.2.1.1.1.1.3.3" xref="A3.E41.2.2.2.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="A3.E41.3.3.3.2.2.2.5" xref="A3.E41.3.3.3.2.2.3.cmml">,</mo><msub id="A3.E41.3.3.3.2.2.2.2" xref="A3.E41.3.3.3.2.2.2.2.cmml"><mi id="A3.E41.3.3.3.2.2.2.2.2" xref="A3.E41.3.3.3.2.2.2.2.2.cmml">𝒚</mi><mrow id="A3.E41.3.3.3.2.2.2.2.3" xref="A3.E41.3.3.3.2.2.2.2.3.cmml"><mi id="A3.E41.3.3.3.2.2.2.2.3.2" mathvariant="normal" xref="A3.E41.3.3.3.2.2.2.2.3.2.cmml">r</mi><mo id="A3.E41.3.3.3.2.2.2.2.3.1" xref="A3.E41.3.3.3.2.2.2.2.3.1.cmml">⁢</mo><mi id="A3.E41.3.3.3.2.2.2.2.3.3" xref="A3.E41.3.3.3.2.2.2.2.3.3.cmml">n</mi></mrow></msub><mo id="A3.E41.3.3.3.2.2.2.6" stretchy="false" xref="A3.E41.3.3.3.2.2.3.cmml">)</mo></mrow><mo id="A3.E41.3.3.3.2.3a" xref="A3.E41.3.3.3.2.3.cmml">⁢</mo><mover accent="true" id="A3.E41.3.3.3.2.5" xref="A3.E41.3.3.3.2.5.cmml"><mi id="A3.E41.3.3.3.2.5.2" xref="A3.E41.3.3.3.2.5.2.cmml">𝜽</mi><mo id="A3.E41.3.3.3.2.5.1" xref="A3.E41.3.3.3.2.5.1.cmml">~</mo></mover></mrow></mrow></mrow></mtd></mtr><mtr id="A3.E43.m1.1.1c" xref="A3.E43.m1.1.1.cmml"><mtd id="A3.E43.m1.1.1d" xref="A3.E43.m1.1.1.cmml"><mrow id="A3.E42.2.2" xref="A3.E42.2.2.cmml"><mover accent="true" id="A3.E42.2.2.4" xref="A3.E42.2.2.4.cmml"><mover accent="true" id="A3.E42.2.2.4.2" xref="A3.E42.2.2.4.2.cmml"><mi id="A3.E42.2.2.4.2.2" xref="A3.E42.2.2.4.2.2.cmml">𝜽</mi><mo id="A3.E42.2.2.4.2.1" xref="A3.E42.2.2.4.2.1.cmml">~</mo></mover><mo id="A3.E42.2.2.4.1" xref="A3.E42.2.2.4.1.cmml">˙</mo></mover><mo id="A3.E42.2.2.3" xref="A3.E42.2.2.3.cmml">=</mo><mrow id="A3.E42.2.2.2" xref="A3.E42.2.2.2.cmml"><mo id="A3.E42.2.2.2a" xref="A3.E42.2.2.2.cmml">−</mo><mrow id="A3.E42.2.2.2.1" xref="A3.E42.2.2.2.1.cmml"><mi id="A3.E42.2.2.2.1.3" mathvariant="normal" xref="A3.E42.2.2.2.1.3.cmml">Γ</mi><mo id="A3.E42.2.2.2.1.2" xref="A3.E42.2.2.2.1.2.cmml">⁢</mo><mrow id="A3.E42.2.2.2.1.1.1" xref="A3.E42.2.2.2.1.1.1.1.cmml"><mo id="A3.E42.2.2.2.1.1.1.2" xref="A3.E42.2.2.2.1.1.1.1.cmml">(</mo><mrow id="A3.E42.2.2.2.1.1.1.1" xref="A3.E42.2.2.2.1.1.1.1.cmml"><mrow id="A3.E42.2.2.2.1.1.1.1.3" xref="A3.E42.2.2.2.1.1.1.1.3.cmml"><msub id="A3.E42.2.2.2.1.1.1.1.3.2" xref="A3.E42.2.2.2.1.1.1.1.3.2.cmml"><mi id="A3.E42.2.2.2.1.1.1.1.3.2.2" mathvariant="normal" xref="A3.E42.2.2.2.1.1.1.1.3.2.2.cmml">Φ</mi><mi id="A3.E42.2.2.2.1.1.1.1.3.2.3" mathvariant="normal" xref="A3.E42.2.2.2.1.1.1.1.3.2.3.cmml">f</mi></msub><mo id="A3.E42.2.2.2.1.1.1.1.3.1" xref="A3.E42.2.2.2.1.1.1.1.3.1.cmml">⁢</mo><msubsup id="A3.E42.2.2.2.1.1.1.1.3.3" xref="A3.E42.2.2.2.1.1.1.1.3.3.cmml"><mi id="A3.E42.2.2.2.1.1.1.1.3.3.2.2" mathvariant="normal" xref="A3.E42.2.2.2.1.1.1.1.3.3.2.2.cmml">Φ</mi><mi id="A3.E42.2.2.2.1.1.1.1.3.3.2.3" mathvariant="normal" xref="A3.E42.2.2.2.1.1.1.1.3.3.2.3.cmml">f</mi><mi id="A3.E42.2.2.2.1.1.1.1.3.3.3" xref="A3.E42.2.2.2.1.1.1.1.3.3.3.cmml">T</mi></msubsup><mo id="A3.E42.2.2.2.1.1.1.1.3.1a" xref="A3.E42.2.2.2.1.1.1.1.3.1.cmml">⁢</mo><mover accent="true" id="A3.E42.2.2.2.1.1.1.1.3.4" xref="A3.E42.2.2.2.1.1.1.1.3.4.cmml"><mi id="A3.E42.2.2.2.1.1.1.1.3.4.2" xref="A3.E42.2.2.2.1.1.1.1.3.4.2.cmml">𝜽</mi><mo id="A3.E42.2.2.2.1.1.1.1.3.4.1" xref="A3.E42.2.2.2.1.1.1.1.3.4.1.cmml">~</mo></mover></mrow><mo id="A3.E42.2.2.2.1.1.1.1.2" xref="A3.E42.2.2.2.1.1.1.1.2.cmml">+</mo><mrow id="A3.E42.2.2.2.1.1.1.1.1" xref="A3.E42.2.2.2.1.1.1.1.1.cmml"><mi id="A3.E42.2.2.2.1.1.1.1.1.3" xref="A3.E42.2.2.2.1.1.1.1.1.3.cmml">κ</mi><mo id="A3.E42.2.2.2.1.1.1.1.1.2" xref="A3.E42.2.2.2.1.1.1.1.1.2.cmml">⁢</mo><mi id="A3.E42.2.2.2.1.1.1.1.1.4" xref="A3.E42.2.2.2.1.1.1.1.1.4.cmml">Q</mi><mo id="A3.E42.2.2.2.1.1.1.1.1.2a" xref="A3.E42.2.2.2.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A3.E42.2.2.2.1.1.1.1.1.1.1" xref="A3.E42.2.2.2.1.1.1.1.1.1.2.cmml"><mo id="A3.E42.2.2.2.1.1.1.1.1.1.1.2" stretchy="false" xref="A3.E42.2.2.2.1.1.1.1.1.1.2.cmml">(</mo><mi id="A3.E42.1.1.1" xref="A3.E42.1.1.1.cmml">t</mi><mo id="A3.E42.2.2.2.1.1.1.1.1.1.1.3" xref="A3.E42.2.2.2.1.1.1.1.1.1.2.cmml">,</mo><msub id="A3.E42.2.2.2.1.1.1.1.1.1.1.1" xref="A3.E42.2.2.2.1.1.1.1.1.1.1.1.cmml"><mi id="A3.E42.2.2.2.1.1.1.1.1.1.1.1.2" xref="A3.E42.2.2.2.1.1.1.1.1.1.1.1.2.cmml">t</mi><mi id="A3.E42.2.2.2.1.1.1.1.1.1.1.1.3" mathvariant="normal" xref="A3.E42.2.2.2.1.1.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="A3.E42.2.2.2.1.1.1.1.1.1.1.4" stretchy="false" xref="A3.E42.2.2.2.1.1.1.1.1.1.2.cmml">)</mo></mrow><mo id="A3.E42.2.2.2.1.1.1.1.1.2b" xref="A3.E42.2.2.2.1.1.1.1.1.2.cmml">⁢</mo><mover accent="true" id="A3.E42.2.2.2.1.1.1.1.1.5" xref="A3.E42.2.2.2.1.1.1.1.1.5.cmml"><mi id="A3.E42.2.2.2.1.1.1.1.1.5.2" xref="A3.E42.2.2.2.1.1.1.1.1.5.2.cmml">𝜽</mi><mo id="A3.E42.2.2.2.1.1.1.1.1.5.1" xref="A3.E42.2.2.2.1.1.1.1.1.5.1.cmml">~</mo></mover></mrow></mrow><mo id="A3.E42.2.2.2.1.1.1.3" xref="A3.E42.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow></mtd></mtr></mtable><mi id="A3.E43.m1.2.2.1.1.2.2" xref="A3.E43.m1.2.2.1.1.1.1.cmml"></mi></mrow><mo id="A3.E43.m1.2.2.1.2" lspace="0em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.E43.m1.2b"><apply id="A3.E43.m1.2.2.1.1.1.cmml" xref="A3.E43.m1.2.2.1.1.2"><csymbol cd="latexml" id="A3.E43.m1.2.2.1.1.1.1.cmml" xref="A3.E43.m1.2.2.1.1.2.1">cases</csymbol><matrix id="A3.E43.m1.1.1.cmml" xref="A3.E43.m1.1.1"><matrixrow id="A3.E43.m1.1.1a.cmml" xref="A3.E43.m1.1.1"><apply id="A3.E41.3.3.cmml" xref="A3.E41.3.3"><eq id="A3.E41.3.3.4.cmml" xref="A3.E41.3.3.4"></eq><apply id="A3.E41.3.3.5.cmml" xref="A3.E41.3.3.5"><ci id="A3.E41.3.3.5.1.cmml" xref="A3.E41.3.3.5.1">˙</ci><ci id="A3.E41.3.3.5.2.cmml" xref="A3.E41.3.3.5.2">𝒆</ci></apply><apply id="A3.E41.3.3.3.cmml" xref="A3.E41.3.3.3"><plus id="A3.E41.3.3.3.3.cmml" xref="A3.E41.3.3.3.3"></plus><apply id="A3.E41.3.3.3.4.cmml" xref="A3.E41.3.3.3.4"><times id="A3.E41.3.3.3.4.1.cmml" xref="A3.E41.3.3.3.4.1"></times><ci id="A3.E41.3.3.3.4.2.cmml" xref="A3.E41.3.3.3.4.2">Λ</ci><ci id="A3.E41.3.3.3.4.3.cmml" xref="A3.E41.3.3.3.4.3">𝒆</ci></apply><apply id="A3.E41.3.3.3.2.cmml" xref="A3.E41.3.3.3.2"><times id="A3.E41.3.3.3.2.3.cmml" xref="A3.E41.3.3.3.2.3"></times><apply id="A3.E41.3.3.3.2.4.cmml" xref="A3.E41.3.3.3.2.4"><csymbol cd="ambiguous" id="A3.E41.3.3.3.2.4.1.cmml" xref="A3.E41.3.3.3.2.4">superscript</csymbol><ci id="A3.E41.3.3.3.2.4.2.cmml" xref="A3.E41.3.3.3.2.4.2">Φ</ci><ci id="A3.E41.3.3.3.2.4.3.cmml" xref="A3.E41.3.3.3.2.4.3">𝑇</ci></apply><vector id="A3.E41.3.3.3.2.2.3.cmml" xref="A3.E41.3.3.3.2.2.2"><ci id="A3.E41.1.1.1.cmml" xref="A3.E41.1.1.1">𝒙</ci><apply id="A3.E41.2.2.2.1.1.1.1.cmml" xref="A3.E41.2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="A3.E41.2.2.2.1.1.1.1.1.cmml" xref="A3.E41.2.2.2.1.1.1.1">subscript</csymbol><ci id="A3.E41.2.2.2.1.1.1.1.2.cmml" xref="A3.E41.2.2.2.1.1.1.1.2">Θ</ci><apply id="A3.E41.2.2.2.1.1.1.1.3.cmml" xref="A3.E41.2.2.2.1.1.1.1.3"><minus id="A3.E41.2.2.2.1.1.1.1.3.1.cmml" xref="A3.E41.2.2.2.1.1.1.1.3.1"></minus><ci id="A3.E41.2.2.2.1.1.1.1.3.2.cmml" xref="A3.E41.2.2.2.1.1.1.1.3.2">𝑛</ci><cn id="A3.E41.2.2.2.1.1.1.1.3.3.cmml" type="integer" xref="A3.E41.2.2.2.1.1.1.1.3.3">1</cn></apply></apply><apply id="A3.E41.3.3.3.2.2.2.2.cmml" xref="A3.E41.3.3.3.2.2.2.2"><csymbol cd="ambiguous" id="A3.E41.3.3.3.2.2.2.2.1.cmml" xref="A3.E41.3.3.3.2.2.2.2">subscript</csymbol><ci id="A3.E41.3.3.3.2.2.2.2.2.cmml" xref="A3.E41.3.3.3.2.2.2.2.2">𝒚</ci><apply id="A3.E41.3.3.3.2.2.2.2.3.cmml" xref="A3.E41.3.3.3.2.2.2.2.3"><times id="A3.E41.3.3.3.2.2.2.2.3.1.cmml" xref="A3.E41.3.3.3.2.2.2.2.3.1"></times><ci id="A3.E41.3.3.3.2.2.2.2.3.2.cmml" xref="A3.E41.3.3.3.2.2.2.2.3.2">r</ci><ci id="A3.E41.3.3.3.2.2.2.2.3.3.cmml" xref="A3.E41.3.3.3.2.2.2.2.3.3">𝑛</ci></apply></apply></vector><apply id="A3.E41.3.3.3.2.5.cmml" xref="A3.E41.3.3.3.2.5"><ci id="A3.E41.3.3.3.2.5.1.cmml" xref="A3.E41.3.3.3.2.5.1">~</ci><ci id="A3.E41.3.3.3.2.5.2.cmml" xref="A3.E41.3.3.3.2.5.2">𝜽</ci></apply></apply></apply></apply></matrixrow><matrixrow id="A3.E43.m1.1.1b.cmml" xref="A3.E43.m1.1.1"><apply id="A3.E42.2.2.cmml" xref="A3.E42.2.2"><eq id="A3.E42.2.2.3.cmml" xref="A3.E42.2.2.3"></eq><apply id="A3.E42.2.2.4.cmml" xref="A3.E42.2.2.4"><ci id="A3.E42.2.2.4.1.cmml" xref="A3.E42.2.2.4.1">˙</ci><apply id="A3.E42.2.2.4.2.cmml" xref="A3.E42.2.2.4.2"><ci id="A3.E42.2.2.4.2.1.cmml" xref="A3.E42.2.2.4.2.1">~</ci><ci id="A3.E42.2.2.4.2.2.cmml" xref="A3.E42.2.2.4.2.2">𝜽</ci></apply></apply><apply id="A3.E42.2.2.2.cmml" xref="A3.E42.2.2.2"><minus id="A3.E42.2.2.2.2.cmml" xref="A3.E42.2.2.2"></minus><apply id="A3.E42.2.2.2.1.cmml" xref="A3.E42.2.2.2.1"><times id="A3.E42.2.2.2.1.2.cmml" xref="A3.E42.2.2.2.1.2"></times><ci id="A3.E42.2.2.2.1.3.cmml" xref="A3.E42.2.2.2.1.3">Γ</ci><apply id="A3.E42.2.2.2.1.1.1.1.cmml" xref="A3.E42.2.2.2.1.1.1"><plus id="A3.E42.2.2.2.1.1.1.1.2.cmml" xref="A3.E42.2.2.2.1.1.1.1.2"></plus><apply id="A3.E42.2.2.2.1.1.1.1.3.cmml" xref="A3.E42.2.2.2.1.1.1.1.3"><times id="A3.E42.2.2.2.1.1.1.1.3.1.cmml" xref="A3.E42.2.2.2.1.1.1.1.3.1"></times><apply id="A3.E42.2.2.2.1.1.1.1.3.2.cmml" xref="A3.E42.2.2.2.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A3.E42.2.2.2.1.1.1.1.3.2.1.cmml" xref="A3.E42.2.2.2.1.1.1.1.3.2">subscript</csymbol><ci id="A3.E42.2.2.2.1.1.1.1.3.2.2.cmml" xref="A3.E42.2.2.2.1.1.1.1.3.2.2">Φ</ci><ci id="A3.E42.2.2.2.1.1.1.1.3.2.3.cmml" xref="A3.E42.2.2.2.1.1.1.1.3.2.3">f</ci></apply><apply id="A3.E42.2.2.2.1.1.1.1.3.3.cmml" xref="A3.E42.2.2.2.1.1.1.1.3.3"><csymbol cd="ambiguous" id="A3.E42.2.2.2.1.1.1.1.3.3.1.cmml" xref="A3.E42.2.2.2.1.1.1.1.3.3">superscript</csymbol><apply id="A3.E42.2.2.2.1.1.1.1.3.3.2.cmml" xref="A3.E42.2.2.2.1.1.1.1.3.3"><csymbol cd="ambiguous" id="A3.E42.2.2.2.1.1.1.1.3.3.2.1.cmml" xref="A3.E42.2.2.2.1.1.1.1.3.3">subscript</csymbol><ci id="A3.E42.2.2.2.1.1.1.1.3.3.2.2.cmml" xref="A3.E42.2.2.2.1.1.1.1.3.3.2.2">Φ</ci><ci id="A3.E42.2.2.2.1.1.1.1.3.3.2.3.cmml" xref="A3.E42.2.2.2.1.1.1.1.3.3.2.3">f</ci></apply><ci id="A3.E42.2.2.2.1.1.1.1.3.3.3.cmml" xref="A3.E42.2.2.2.1.1.1.1.3.3.3">𝑇</ci></apply><apply id="A3.E42.2.2.2.1.1.1.1.3.4.cmml" xref="A3.E42.2.2.2.1.1.1.1.3.4"><ci id="A3.E42.2.2.2.1.1.1.1.3.4.1.cmml" xref="A3.E42.2.2.2.1.1.1.1.3.4.1">~</ci><ci id="A3.E42.2.2.2.1.1.1.1.3.4.2.cmml" xref="A3.E42.2.2.2.1.1.1.1.3.4.2">𝜽</ci></apply></apply><apply id="A3.E42.2.2.2.1.1.1.1.1.cmml" xref="A3.E42.2.2.2.1.1.1.1.1"><times id="A3.E42.2.2.2.1.1.1.1.1.2.cmml" xref="A3.E42.2.2.2.1.1.1.1.1.2"></times><ci id="A3.E42.2.2.2.1.1.1.1.1.3.cmml" xref="A3.E42.2.2.2.1.1.1.1.1.3">𝜅</ci><ci id="A3.E42.2.2.2.1.1.1.1.1.4.cmml" xref="A3.E42.2.2.2.1.1.1.1.1.4">𝑄</ci><interval closure="open" id="A3.E42.2.2.2.1.1.1.1.1.1.2.cmml" xref="A3.E42.2.2.2.1.1.1.1.1.1.1"><ci id="A3.E42.1.1.1.cmml" xref="A3.E42.1.1.1">𝑡</ci><apply id="A3.E42.2.2.2.1.1.1.1.1.1.1.1.cmml" xref="A3.E42.2.2.2.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.E42.2.2.2.1.1.1.1.1.1.1.1.1.cmml" xref="A3.E42.2.2.2.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A3.E42.2.2.2.1.1.1.1.1.1.1.1.2.cmml" xref="A3.E42.2.2.2.1.1.1.1.1.1.1.1.2">𝑡</ci><ci id="A3.E42.2.2.2.1.1.1.1.1.1.1.1.3.cmml" xref="A3.E42.2.2.2.1.1.1.1.1.1.1.1.3">e</ci></apply></interval><apply id="A3.E42.2.2.2.1.1.1.1.1.5.cmml" xref="A3.E42.2.2.2.1.1.1.1.1.5"><ci id="A3.E42.2.2.2.1.1.1.1.1.5.1.cmml" xref="A3.E42.2.2.2.1.1.1.1.1.5.1">~</ci><ci id="A3.E42.2.2.2.1.1.1.1.1.5.2.cmml" xref="A3.E42.2.2.2.1.1.1.1.1.5.2">𝜽</ci></apply></apply></apply></apply></apply></apply></matrixrow></matrix></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E43.m1.2c">\displaystyle\left\{\begin{array}[]{c}\dot{\bm{e}}=\Lambda\bm{e}+\Phi^{T}(\bm{% x},\Theta_{n-1},{\bm{y}}_{{\rm r}n})\tilde{\bm{\theta}}\\ \dot{\tilde{\bm{\theta}}}=-\Gamma\left(\Phi_{\rm f}\Phi_{\rm f}^{T}\tilde{\bm{% \theta}}+\kappa Q(t,t_{\rm e})\tilde{\bm{\theta}}\right)\end{array}.\right.</annotation><annotation encoding="application/x-llamapun" id="A3.E43.m1.2d">{ start_ARRAY start_ROW start_CELL over˙ start_ARG bold_italic_e end_ARG = roman_Λ bold_italic_e + roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_italic_x , roman_Θ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT roman_r italic_n end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG end_CELL end_ROW start_ROW start_CELL over˙ start_ARG over~ start_ARG bold_italic_θ end_ARG end_ARG = - roman_Γ ( roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG + italic_κ italic_Q ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG ) end_CELL end_ROW end_ARRAY .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(43)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.1.p1.12">Differentiating <math alttext="V" class="ltx_Math" display="inline" id="A3.1.p1.11.m1.1"><semantics id="A3.1.p1.11.m1.1a"><mi id="A3.1.p1.11.m1.1.1" xref="A3.1.p1.11.m1.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="A3.1.p1.11.m1.1b"><ci id="A3.1.p1.11.m1.1.1.cmml" xref="A3.1.p1.11.m1.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.11.m1.1c">V</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.11.m1.1d">italic_V</annotation></semantics></math> with respect to <math alttext="t" class="ltx_Math" display="inline" id="A3.1.p1.12.m2.1"><semantics id="A3.1.p1.12.m2.1a"><mi id="A3.1.p1.12.m2.1.1" xref="A3.1.p1.12.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A3.1.p1.12.m2.1b"><ci id="A3.1.p1.12.m2.1.1.cmml" xref="A3.1.p1.12.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.12.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.12.m2.1d">italic_t</annotation></semantics></math> yields</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx54"> <tbody id="A3.Ex40"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}=(\dot{\bm{e}}^{T}\bm{e}+\bm{e}^{T}\dot{\bm{e}})/2+(1+p)% \tilde{\bm{\theta}}^{T}\Gamma^{-1}\dot{\tilde{\bm{\theta}}}." class="ltx_Math" display="inline" id="A3.Ex40.m1.1"><semantics id="A3.Ex40.m1.1a"><mrow id="A3.Ex40.m1.1.1.1" xref="A3.Ex40.m1.1.1.1.1.cmml"><mrow id="A3.Ex40.m1.1.1.1.1" xref="A3.Ex40.m1.1.1.1.1.cmml"><mover accent="true" id="A3.Ex40.m1.1.1.1.1.4" xref="A3.Ex40.m1.1.1.1.1.4.cmml"><mi id="A3.Ex40.m1.1.1.1.1.4.2" xref="A3.Ex40.m1.1.1.1.1.4.2.cmml">V</mi><mo id="A3.Ex40.m1.1.1.1.1.4.1" xref="A3.Ex40.m1.1.1.1.1.4.1.cmml">˙</mo></mover><mo id="A3.Ex40.m1.1.1.1.1.3" xref="A3.Ex40.m1.1.1.1.1.3.cmml">=</mo><mrow id="A3.Ex40.m1.1.1.1.1.2" xref="A3.Ex40.m1.1.1.1.1.2.cmml"><mrow id="A3.Ex40.m1.1.1.1.1.1.1" xref="A3.Ex40.m1.1.1.1.1.1.1.cmml"><mrow id="A3.Ex40.m1.1.1.1.1.1.1.1.1" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="A3.Ex40.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.cmml"><mrow id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.cmml"><msup id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.2" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.2.cmml"><mover accent="true" id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.2.2" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.2.2.cmml"><mi id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.2.2.2" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.2.2.2.cmml">𝒆</mi><mo id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.2.2.1" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.2.2.1.cmml">˙</mo></mover><mi id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.2.3" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.2.3.cmml">T</mi></msup><mo id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.1" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.3" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.3.cmml">𝒆</mi></mrow><mo id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.1" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.1.cmml">+</mo><mrow id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.cmml"><msup id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.2" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.2.cmml"><mi id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.2.2" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.2.2.cmml">𝒆</mi><mi id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.2.3" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.2.3.cmml">T</mi></msup><mo id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.1" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mover accent="true" id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.3" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.3.cmml"><mi id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.3.2" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.3.2.cmml">𝒆</mi><mo id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.3.1" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.3.1.cmml">˙</mo></mover></mrow></mrow><mo id="A3.Ex40.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A3.Ex40.m1.1.1.1.1.1.1.2" xref="A3.Ex40.m1.1.1.1.1.1.1.2.cmml">/</mo><mn id="A3.Ex40.m1.1.1.1.1.1.1.3" xref="A3.Ex40.m1.1.1.1.1.1.1.3.cmml">2</mn></mrow><mo id="A3.Ex40.m1.1.1.1.1.2.3" xref="A3.Ex40.m1.1.1.1.1.2.3.cmml">+</mo><mrow id="A3.Ex40.m1.1.1.1.1.2.2" xref="A3.Ex40.m1.1.1.1.1.2.2.cmml"><mrow id="A3.Ex40.m1.1.1.1.1.2.2.1.1" xref="A3.Ex40.m1.1.1.1.1.2.2.1.1.1.cmml"><mo id="A3.Ex40.m1.1.1.1.1.2.2.1.1.2" stretchy="false" xref="A3.Ex40.m1.1.1.1.1.2.2.1.1.1.cmml">(</mo><mrow id="A3.Ex40.m1.1.1.1.1.2.2.1.1.1" xref="A3.Ex40.m1.1.1.1.1.2.2.1.1.1.cmml"><mn id="A3.Ex40.m1.1.1.1.1.2.2.1.1.1.2" xref="A3.Ex40.m1.1.1.1.1.2.2.1.1.1.2.cmml">1</mn><mo id="A3.Ex40.m1.1.1.1.1.2.2.1.1.1.1" xref="A3.Ex40.m1.1.1.1.1.2.2.1.1.1.1.cmml">+</mo><mi id="A3.Ex40.m1.1.1.1.1.2.2.1.1.1.3" xref="A3.Ex40.m1.1.1.1.1.2.2.1.1.1.3.cmml">p</mi></mrow><mo id="A3.Ex40.m1.1.1.1.1.2.2.1.1.3" stretchy="false" xref="A3.Ex40.m1.1.1.1.1.2.2.1.1.1.cmml">)</mo></mrow><mo id="A3.Ex40.m1.1.1.1.1.2.2.2" xref="A3.Ex40.m1.1.1.1.1.2.2.2.cmml">⁢</mo><msup id="A3.Ex40.m1.1.1.1.1.2.2.3" xref="A3.Ex40.m1.1.1.1.1.2.2.3.cmml"><mover accent="true" id="A3.Ex40.m1.1.1.1.1.2.2.3.2" xref="A3.Ex40.m1.1.1.1.1.2.2.3.2.cmml"><mi id="A3.Ex40.m1.1.1.1.1.2.2.3.2.2" xref="A3.Ex40.m1.1.1.1.1.2.2.3.2.2.cmml">𝜽</mi><mo id="A3.Ex40.m1.1.1.1.1.2.2.3.2.1" xref="A3.Ex40.m1.1.1.1.1.2.2.3.2.1.cmml">~</mo></mover><mi id="A3.Ex40.m1.1.1.1.1.2.2.3.3" xref="A3.Ex40.m1.1.1.1.1.2.2.3.3.cmml">T</mi></msup><mo id="A3.Ex40.m1.1.1.1.1.2.2.2a" xref="A3.Ex40.m1.1.1.1.1.2.2.2.cmml">⁢</mo><msup id="A3.Ex40.m1.1.1.1.1.2.2.4" xref="A3.Ex40.m1.1.1.1.1.2.2.4.cmml"><mi id="A3.Ex40.m1.1.1.1.1.2.2.4.2" mathvariant="normal" xref="A3.Ex40.m1.1.1.1.1.2.2.4.2.cmml">Γ</mi><mrow id="A3.Ex40.m1.1.1.1.1.2.2.4.3" xref="A3.Ex40.m1.1.1.1.1.2.2.4.3.cmml"><mo id="A3.Ex40.m1.1.1.1.1.2.2.4.3a" xref="A3.Ex40.m1.1.1.1.1.2.2.4.3.cmml">−</mo><mn id="A3.Ex40.m1.1.1.1.1.2.2.4.3.2" xref="A3.Ex40.m1.1.1.1.1.2.2.4.3.2.cmml">1</mn></mrow></msup><mo id="A3.Ex40.m1.1.1.1.1.2.2.2b" xref="A3.Ex40.m1.1.1.1.1.2.2.2.cmml">⁢</mo><mover accent="true" id="A3.Ex40.m1.1.1.1.1.2.2.5" xref="A3.Ex40.m1.1.1.1.1.2.2.5.cmml"><mover accent="true" id="A3.Ex40.m1.1.1.1.1.2.2.5.2" xref="A3.Ex40.m1.1.1.1.1.2.2.5.2.cmml"><mi id="A3.Ex40.m1.1.1.1.1.2.2.5.2.2" xref="A3.Ex40.m1.1.1.1.1.2.2.5.2.2.cmml">𝜽</mi><mo id="A3.Ex40.m1.1.1.1.1.2.2.5.2.1" xref="A3.Ex40.m1.1.1.1.1.2.2.5.2.1.cmml">~</mo></mover><mo id="A3.Ex40.m1.1.1.1.1.2.2.5.1" xref="A3.Ex40.m1.1.1.1.1.2.2.5.1.cmml">˙</mo></mover></mrow></mrow></mrow><mo id="A3.Ex40.m1.1.1.1.2" lspace="0em" xref="A3.Ex40.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex40.m1.1b"><apply id="A3.Ex40.m1.1.1.1.1.cmml" xref="A3.Ex40.m1.1.1.1"><eq id="A3.Ex40.m1.1.1.1.1.3.cmml" xref="A3.Ex40.m1.1.1.1.1.3"></eq><apply id="A3.Ex40.m1.1.1.1.1.4.cmml" xref="A3.Ex40.m1.1.1.1.1.4"><ci id="A3.Ex40.m1.1.1.1.1.4.1.cmml" xref="A3.Ex40.m1.1.1.1.1.4.1">˙</ci><ci id="A3.Ex40.m1.1.1.1.1.4.2.cmml" xref="A3.Ex40.m1.1.1.1.1.4.2">𝑉</ci></apply><apply id="A3.Ex40.m1.1.1.1.1.2.cmml" xref="A3.Ex40.m1.1.1.1.1.2"><plus id="A3.Ex40.m1.1.1.1.1.2.3.cmml" xref="A3.Ex40.m1.1.1.1.1.2.3"></plus><apply id="A3.Ex40.m1.1.1.1.1.1.1.cmml" xref="A3.Ex40.m1.1.1.1.1.1.1"><divide id="A3.Ex40.m1.1.1.1.1.1.1.2.cmml" xref="A3.Ex40.m1.1.1.1.1.1.1.2"></divide><apply id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.cmml" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1"><plus id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.1"></plus><apply id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2"><times id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.1"></times><apply id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.2.1.cmml" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.2">superscript</csymbol><apply id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.2.2.cmml" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.2.2"><ci id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.2.2.1.cmml" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.2.2.1">˙</ci><ci id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.2.2.2.cmml" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.2.2.2">𝒆</ci></apply><ci id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.2.3.cmml" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.2.3">𝑇</ci></apply><ci id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.3.cmml" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.2.3">𝒆</ci></apply><apply id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3"><times id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.1"></times><apply id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.2.1.cmml" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.2">superscript</csymbol><ci id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.2.2.cmml" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.2.2">𝒆</ci><ci id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.2.3.cmml" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.2.3">𝑇</ci></apply><apply id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.3"><ci id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.3.1.cmml" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.3.1">˙</ci><ci id="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.3.2.cmml" xref="A3.Ex40.m1.1.1.1.1.1.1.1.1.1.3.3.2">𝒆</ci></apply></apply></apply><cn id="A3.Ex40.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="A3.Ex40.m1.1.1.1.1.1.1.3">2</cn></apply><apply id="A3.Ex40.m1.1.1.1.1.2.2.cmml" xref="A3.Ex40.m1.1.1.1.1.2.2"><times id="A3.Ex40.m1.1.1.1.1.2.2.2.cmml" xref="A3.Ex40.m1.1.1.1.1.2.2.2"></times><apply id="A3.Ex40.m1.1.1.1.1.2.2.1.1.1.cmml" xref="A3.Ex40.m1.1.1.1.1.2.2.1.1"><plus id="A3.Ex40.m1.1.1.1.1.2.2.1.1.1.1.cmml" xref="A3.Ex40.m1.1.1.1.1.2.2.1.1.1.1"></plus><cn id="A3.Ex40.m1.1.1.1.1.2.2.1.1.1.2.cmml" type="integer" xref="A3.Ex40.m1.1.1.1.1.2.2.1.1.1.2">1</cn><ci id="A3.Ex40.m1.1.1.1.1.2.2.1.1.1.3.cmml" xref="A3.Ex40.m1.1.1.1.1.2.2.1.1.1.3">𝑝</ci></apply><apply id="A3.Ex40.m1.1.1.1.1.2.2.3.cmml" xref="A3.Ex40.m1.1.1.1.1.2.2.3"><csymbol cd="ambiguous" id="A3.Ex40.m1.1.1.1.1.2.2.3.1.cmml" xref="A3.Ex40.m1.1.1.1.1.2.2.3">superscript</csymbol><apply id="A3.Ex40.m1.1.1.1.1.2.2.3.2.cmml" xref="A3.Ex40.m1.1.1.1.1.2.2.3.2"><ci id="A3.Ex40.m1.1.1.1.1.2.2.3.2.1.cmml" xref="A3.Ex40.m1.1.1.1.1.2.2.3.2.1">~</ci><ci id="A3.Ex40.m1.1.1.1.1.2.2.3.2.2.cmml" xref="A3.Ex40.m1.1.1.1.1.2.2.3.2.2">𝜽</ci></apply><ci id="A3.Ex40.m1.1.1.1.1.2.2.3.3.cmml" xref="A3.Ex40.m1.1.1.1.1.2.2.3.3">𝑇</ci></apply><apply id="A3.Ex40.m1.1.1.1.1.2.2.4.cmml" xref="A3.Ex40.m1.1.1.1.1.2.2.4"><csymbol cd="ambiguous" id="A3.Ex40.m1.1.1.1.1.2.2.4.1.cmml" xref="A3.Ex40.m1.1.1.1.1.2.2.4">superscript</csymbol><ci id="A3.Ex40.m1.1.1.1.1.2.2.4.2.cmml" xref="A3.Ex40.m1.1.1.1.1.2.2.4.2">Γ</ci><apply id="A3.Ex40.m1.1.1.1.1.2.2.4.3.cmml" xref="A3.Ex40.m1.1.1.1.1.2.2.4.3"><minus id="A3.Ex40.m1.1.1.1.1.2.2.4.3.1.cmml" xref="A3.Ex40.m1.1.1.1.1.2.2.4.3"></minus><cn id="A3.Ex40.m1.1.1.1.1.2.2.4.3.2.cmml" type="integer" xref="A3.Ex40.m1.1.1.1.1.2.2.4.3.2">1</cn></apply></apply><apply id="A3.Ex40.m1.1.1.1.1.2.2.5.cmml" xref="A3.Ex40.m1.1.1.1.1.2.2.5"><ci id="A3.Ex40.m1.1.1.1.1.2.2.5.1.cmml" xref="A3.Ex40.m1.1.1.1.1.2.2.5.1">˙</ci><apply id="A3.Ex40.m1.1.1.1.1.2.2.5.2.cmml" xref="A3.Ex40.m1.1.1.1.1.2.2.5.2"><ci id="A3.Ex40.m1.1.1.1.1.2.2.5.2.1.cmml" xref="A3.Ex40.m1.1.1.1.1.2.2.5.2.1">~</ci><ci id="A3.Ex40.m1.1.1.1.1.2.2.5.2.2.cmml" xref="A3.Ex40.m1.1.1.1.1.2.2.5.2.2">𝜽</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex40.m1.1c">\displaystyle\dot{V}=(\dot{\bm{e}}^{T}\bm{e}+\bm{e}^{T}\dot{\bm{e}})/2+(1+p)% \tilde{\bm{\theta}}^{T}\Gamma^{-1}\dot{\tilde{\bm{\theta}}}.</annotation><annotation encoding="application/x-llamapun" id="A3.Ex40.m1.1d">over˙ start_ARG italic_V end_ARG = ( over˙ start_ARG bold_italic_e end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_e + bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over˙ start_ARG bold_italic_e end_ARG ) / 2 + ( 1 + italic_p ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Γ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT over˙ start_ARG over~ start_ARG bold_italic_θ end_ARG end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.1.p1.96">Applying (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A3.E43" title="In Proof. ‣ Appendix C The proof of Theorem 2 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">43</span></a>) to the above formula, one obtains</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx55"> <tbody id="A3.E44"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}=" class="ltx_Math" display="inline" id="A3.E44.m1.1"><semantics id="A3.E44.m1.1a"><mrow id="A3.E44.m1.1.1" xref="A3.E44.m1.1.1.cmml"><mover accent="true" id="A3.E44.m1.1.1.2" xref="A3.E44.m1.1.1.2.cmml"><mi id="A3.E44.m1.1.1.2.2" xref="A3.E44.m1.1.1.2.2.cmml">V</mi><mo id="A3.E44.m1.1.1.2.1" xref="A3.E44.m1.1.1.2.1.cmml">˙</mo></mover><mo id="A3.E44.m1.1.1.1" xref="A3.E44.m1.1.1.1.cmml">=</mo><mi id="A3.E44.m1.1.1.3" xref="A3.E44.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A3.E44.m1.1b"><apply id="A3.E44.m1.1.1.cmml" xref="A3.E44.m1.1.1"><eq id="A3.E44.m1.1.1.1.cmml" xref="A3.E44.m1.1.1.1"></eq><apply id="A3.E44.m1.1.1.2.cmml" xref="A3.E44.m1.1.1.2"><ci id="A3.E44.m1.1.1.2.1.cmml" xref="A3.E44.m1.1.1.2.1">˙</ci><ci id="A3.E44.m1.1.1.2.2.cmml" xref="A3.E44.m1.1.1.2.2">𝑉</ci></apply><csymbol cd="latexml" id="A3.E44.m1.1.1.3.cmml" xref="A3.E44.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E44.m1.1c">\displaystyle\dot{V}=</annotation><annotation encoding="application/x-llamapun" id="A3.E44.m1.1d">over˙ start_ARG italic_V end_ARG =</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\bm{e}^{T}(-K\bm{e}+\Phi^{T}\tilde{\bm{\theta}})-(1+p)\tilde{\bm{% \theta}}^{T}(\Phi_{{\rm f}}\Phi_{{\rm f}}^{T}+\kappa Q(t,t_{\rm e}))\tilde{\bm% {\theta}}" class="ltx_Math" display="inline" id="A3.E44.m2.4"><semantics id="A3.E44.m2.4a"><mrow id="A3.E44.m2.4.4" xref="A3.E44.m2.4.4.cmml"><mrow id="A3.E44.m2.2.2.1" xref="A3.E44.m2.2.2.1.cmml"><msup id="A3.E44.m2.2.2.1.3" xref="A3.E44.m2.2.2.1.3.cmml"><mi id="A3.E44.m2.2.2.1.3.2" xref="A3.E44.m2.2.2.1.3.2.cmml">𝒆</mi><mi id="A3.E44.m2.2.2.1.3.3" xref="A3.E44.m2.2.2.1.3.3.cmml">T</mi></msup><mo id="A3.E44.m2.2.2.1.2" xref="A3.E44.m2.2.2.1.2.cmml">⁢</mo><mrow id="A3.E44.m2.2.2.1.1.1" xref="A3.E44.m2.2.2.1.1.1.1.cmml"><mo id="A3.E44.m2.2.2.1.1.1.2" stretchy="false" xref="A3.E44.m2.2.2.1.1.1.1.cmml">(</mo><mrow id="A3.E44.m2.2.2.1.1.1.1" xref="A3.E44.m2.2.2.1.1.1.1.cmml"><mrow id="A3.E44.m2.2.2.1.1.1.1.2" xref="A3.E44.m2.2.2.1.1.1.1.2.cmml"><mo id="A3.E44.m2.2.2.1.1.1.1.2a" xref="A3.E44.m2.2.2.1.1.1.1.2.cmml">−</mo><mrow id="A3.E44.m2.2.2.1.1.1.1.2.2" xref="A3.E44.m2.2.2.1.1.1.1.2.2.cmml"><mi id="A3.E44.m2.2.2.1.1.1.1.2.2.2" xref="A3.E44.m2.2.2.1.1.1.1.2.2.2.cmml">K</mi><mo id="A3.E44.m2.2.2.1.1.1.1.2.2.1" xref="A3.E44.m2.2.2.1.1.1.1.2.2.1.cmml">⁢</mo><mi id="A3.E44.m2.2.2.1.1.1.1.2.2.3" xref="A3.E44.m2.2.2.1.1.1.1.2.2.3.cmml">𝒆</mi></mrow></mrow><mo id="A3.E44.m2.2.2.1.1.1.1.1" xref="A3.E44.m2.2.2.1.1.1.1.1.cmml">+</mo><mrow id="A3.E44.m2.2.2.1.1.1.1.3" xref="A3.E44.m2.2.2.1.1.1.1.3.cmml"><msup id="A3.E44.m2.2.2.1.1.1.1.3.2" xref="A3.E44.m2.2.2.1.1.1.1.3.2.cmml"><mi id="A3.E44.m2.2.2.1.1.1.1.3.2.2" mathvariant="normal" xref="A3.E44.m2.2.2.1.1.1.1.3.2.2.cmml">Φ</mi><mi id="A3.E44.m2.2.2.1.1.1.1.3.2.3" xref="A3.E44.m2.2.2.1.1.1.1.3.2.3.cmml">T</mi></msup><mo id="A3.E44.m2.2.2.1.1.1.1.3.1" xref="A3.E44.m2.2.2.1.1.1.1.3.1.cmml">⁢</mo><mover accent="true" id="A3.E44.m2.2.2.1.1.1.1.3.3" xref="A3.E44.m2.2.2.1.1.1.1.3.3.cmml"><mi id="A3.E44.m2.2.2.1.1.1.1.3.3.2" xref="A3.E44.m2.2.2.1.1.1.1.3.3.2.cmml">𝜽</mi><mo id="A3.E44.m2.2.2.1.1.1.1.3.3.1" xref="A3.E44.m2.2.2.1.1.1.1.3.3.1.cmml">~</mo></mover></mrow></mrow><mo id="A3.E44.m2.2.2.1.1.1.3" stretchy="false" xref="A3.E44.m2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A3.E44.m2.4.4.4" xref="A3.E44.m2.4.4.4.cmml">−</mo><mrow id="A3.E44.m2.4.4.3" xref="A3.E44.m2.4.4.3.cmml"><mrow id="A3.E44.m2.3.3.2.1.1" xref="A3.E44.m2.3.3.2.1.1.1.cmml"><mo id="A3.E44.m2.3.3.2.1.1.2" stretchy="false" xref="A3.E44.m2.3.3.2.1.1.1.cmml">(</mo><mrow id="A3.E44.m2.3.3.2.1.1.1" xref="A3.E44.m2.3.3.2.1.1.1.cmml"><mn id="A3.E44.m2.3.3.2.1.1.1.2" xref="A3.E44.m2.3.3.2.1.1.1.2.cmml">1</mn><mo id="A3.E44.m2.3.3.2.1.1.1.1" xref="A3.E44.m2.3.3.2.1.1.1.1.cmml">+</mo><mi id="A3.E44.m2.3.3.2.1.1.1.3" xref="A3.E44.m2.3.3.2.1.1.1.3.cmml">p</mi></mrow><mo id="A3.E44.m2.3.3.2.1.1.3" stretchy="false" xref="A3.E44.m2.3.3.2.1.1.1.cmml">)</mo></mrow><mo id="A3.E44.m2.4.4.3.3" xref="A3.E44.m2.4.4.3.3.cmml">⁢</mo><msup id="A3.E44.m2.4.4.3.4" xref="A3.E44.m2.4.4.3.4.cmml"><mover accent="true" id="A3.E44.m2.4.4.3.4.2" xref="A3.E44.m2.4.4.3.4.2.cmml"><mi id="A3.E44.m2.4.4.3.4.2.2" xref="A3.E44.m2.4.4.3.4.2.2.cmml">𝜽</mi><mo id="A3.E44.m2.4.4.3.4.2.1" xref="A3.E44.m2.4.4.3.4.2.1.cmml">~</mo></mover><mi id="A3.E44.m2.4.4.3.4.3" xref="A3.E44.m2.4.4.3.4.3.cmml">T</mi></msup><mo id="A3.E44.m2.4.4.3.3a" xref="A3.E44.m2.4.4.3.3.cmml">⁢</mo><mrow id="A3.E44.m2.4.4.3.2.1" xref="A3.E44.m2.4.4.3.2.1.1.cmml"><mo id="A3.E44.m2.4.4.3.2.1.2" stretchy="false" xref="A3.E44.m2.4.4.3.2.1.1.cmml">(</mo><mrow id="A3.E44.m2.4.4.3.2.1.1" xref="A3.E44.m2.4.4.3.2.1.1.cmml"><mrow id="A3.E44.m2.4.4.3.2.1.1.3" xref="A3.E44.m2.4.4.3.2.1.1.3.cmml"><msub id="A3.E44.m2.4.4.3.2.1.1.3.2" xref="A3.E44.m2.4.4.3.2.1.1.3.2.cmml"><mi id="A3.E44.m2.4.4.3.2.1.1.3.2.2" mathvariant="normal" xref="A3.E44.m2.4.4.3.2.1.1.3.2.2.cmml">Φ</mi><mi id="A3.E44.m2.4.4.3.2.1.1.3.2.3" mathvariant="normal" xref="A3.E44.m2.4.4.3.2.1.1.3.2.3.cmml">f</mi></msub><mo id="A3.E44.m2.4.4.3.2.1.1.3.1" xref="A3.E44.m2.4.4.3.2.1.1.3.1.cmml">⁢</mo><msubsup id="A3.E44.m2.4.4.3.2.1.1.3.3" xref="A3.E44.m2.4.4.3.2.1.1.3.3.cmml"><mi id="A3.E44.m2.4.4.3.2.1.1.3.3.2.2" mathvariant="normal" xref="A3.E44.m2.4.4.3.2.1.1.3.3.2.2.cmml">Φ</mi><mi id="A3.E44.m2.4.4.3.2.1.1.3.3.2.3" mathvariant="normal" xref="A3.E44.m2.4.4.3.2.1.1.3.3.2.3.cmml">f</mi><mi id="A3.E44.m2.4.4.3.2.1.1.3.3.3" xref="A3.E44.m2.4.4.3.2.1.1.3.3.3.cmml">T</mi></msubsup></mrow><mo id="A3.E44.m2.4.4.3.2.1.1.2" xref="A3.E44.m2.4.4.3.2.1.1.2.cmml">+</mo><mrow id="A3.E44.m2.4.4.3.2.1.1.1" xref="A3.E44.m2.4.4.3.2.1.1.1.cmml"><mi id="A3.E44.m2.4.4.3.2.1.1.1.3" xref="A3.E44.m2.4.4.3.2.1.1.1.3.cmml">κ</mi><mo id="A3.E44.m2.4.4.3.2.1.1.1.2" xref="A3.E44.m2.4.4.3.2.1.1.1.2.cmml">⁢</mo><mi id="A3.E44.m2.4.4.3.2.1.1.1.4" xref="A3.E44.m2.4.4.3.2.1.1.1.4.cmml">Q</mi><mo id="A3.E44.m2.4.4.3.2.1.1.1.2a" xref="A3.E44.m2.4.4.3.2.1.1.1.2.cmml">⁢</mo><mrow id="A3.E44.m2.4.4.3.2.1.1.1.1.1" xref="A3.E44.m2.4.4.3.2.1.1.1.1.2.cmml"><mo id="A3.E44.m2.4.4.3.2.1.1.1.1.1.2" stretchy="false" xref="A3.E44.m2.4.4.3.2.1.1.1.1.2.cmml">(</mo><mi id="A3.E44.m2.1.1" xref="A3.E44.m2.1.1.cmml">t</mi><mo id="A3.E44.m2.4.4.3.2.1.1.1.1.1.3" xref="A3.E44.m2.4.4.3.2.1.1.1.1.2.cmml">,</mo><msub id="A3.E44.m2.4.4.3.2.1.1.1.1.1.1" xref="A3.E44.m2.4.4.3.2.1.1.1.1.1.1.cmml"><mi id="A3.E44.m2.4.4.3.2.1.1.1.1.1.1.2" xref="A3.E44.m2.4.4.3.2.1.1.1.1.1.1.2.cmml">t</mi><mi id="A3.E44.m2.4.4.3.2.1.1.1.1.1.1.3" mathvariant="normal" xref="A3.E44.m2.4.4.3.2.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="A3.E44.m2.4.4.3.2.1.1.1.1.1.4" stretchy="false" xref="A3.E44.m2.4.4.3.2.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="A3.E44.m2.4.4.3.2.1.3" stretchy="false" xref="A3.E44.m2.4.4.3.2.1.1.cmml">)</mo></mrow><mo id="A3.E44.m2.4.4.3.3b" xref="A3.E44.m2.4.4.3.3.cmml">⁢</mo><mover accent="true" id="A3.E44.m2.4.4.3.5" xref="A3.E44.m2.4.4.3.5.cmml"><mi id="A3.E44.m2.4.4.3.5.2" xref="A3.E44.m2.4.4.3.5.2.cmml">𝜽</mi><mo id="A3.E44.m2.4.4.3.5.1" xref="A3.E44.m2.4.4.3.5.1.cmml">~</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.E44.m2.4b"><apply id="A3.E44.m2.4.4.cmml" xref="A3.E44.m2.4.4"><minus id="A3.E44.m2.4.4.4.cmml" xref="A3.E44.m2.4.4.4"></minus><apply id="A3.E44.m2.2.2.1.cmml" xref="A3.E44.m2.2.2.1"><times id="A3.E44.m2.2.2.1.2.cmml" xref="A3.E44.m2.2.2.1.2"></times><apply id="A3.E44.m2.2.2.1.3.cmml" xref="A3.E44.m2.2.2.1.3"><csymbol cd="ambiguous" id="A3.E44.m2.2.2.1.3.1.cmml" xref="A3.E44.m2.2.2.1.3">superscript</csymbol><ci id="A3.E44.m2.2.2.1.3.2.cmml" xref="A3.E44.m2.2.2.1.3.2">𝒆</ci><ci id="A3.E44.m2.2.2.1.3.3.cmml" xref="A3.E44.m2.2.2.1.3.3">𝑇</ci></apply><apply id="A3.E44.m2.2.2.1.1.1.1.cmml" xref="A3.E44.m2.2.2.1.1.1"><plus id="A3.E44.m2.2.2.1.1.1.1.1.cmml" xref="A3.E44.m2.2.2.1.1.1.1.1"></plus><apply id="A3.E44.m2.2.2.1.1.1.1.2.cmml" xref="A3.E44.m2.2.2.1.1.1.1.2"><minus id="A3.E44.m2.2.2.1.1.1.1.2.1.cmml" xref="A3.E44.m2.2.2.1.1.1.1.2"></minus><apply id="A3.E44.m2.2.2.1.1.1.1.2.2.cmml" xref="A3.E44.m2.2.2.1.1.1.1.2.2"><times id="A3.E44.m2.2.2.1.1.1.1.2.2.1.cmml" xref="A3.E44.m2.2.2.1.1.1.1.2.2.1"></times><ci id="A3.E44.m2.2.2.1.1.1.1.2.2.2.cmml" xref="A3.E44.m2.2.2.1.1.1.1.2.2.2">𝐾</ci><ci id="A3.E44.m2.2.2.1.1.1.1.2.2.3.cmml" xref="A3.E44.m2.2.2.1.1.1.1.2.2.3">𝒆</ci></apply></apply><apply id="A3.E44.m2.2.2.1.1.1.1.3.cmml" xref="A3.E44.m2.2.2.1.1.1.1.3"><times id="A3.E44.m2.2.2.1.1.1.1.3.1.cmml" xref="A3.E44.m2.2.2.1.1.1.1.3.1"></times><apply id="A3.E44.m2.2.2.1.1.1.1.3.2.cmml" xref="A3.E44.m2.2.2.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A3.E44.m2.2.2.1.1.1.1.3.2.1.cmml" xref="A3.E44.m2.2.2.1.1.1.1.3.2">superscript</csymbol><ci id="A3.E44.m2.2.2.1.1.1.1.3.2.2.cmml" xref="A3.E44.m2.2.2.1.1.1.1.3.2.2">Φ</ci><ci id="A3.E44.m2.2.2.1.1.1.1.3.2.3.cmml" xref="A3.E44.m2.2.2.1.1.1.1.3.2.3">𝑇</ci></apply><apply id="A3.E44.m2.2.2.1.1.1.1.3.3.cmml" xref="A3.E44.m2.2.2.1.1.1.1.3.3"><ci id="A3.E44.m2.2.2.1.1.1.1.3.3.1.cmml" xref="A3.E44.m2.2.2.1.1.1.1.3.3.1">~</ci><ci id="A3.E44.m2.2.2.1.1.1.1.3.3.2.cmml" xref="A3.E44.m2.2.2.1.1.1.1.3.3.2">𝜽</ci></apply></apply></apply></apply><apply id="A3.E44.m2.4.4.3.cmml" xref="A3.E44.m2.4.4.3"><times id="A3.E44.m2.4.4.3.3.cmml" xref="A3.E44.m2.4.4.3.3"></times><apply id="A3.E44.m2.3.3.2.1.1.1.cmml" xref="A3.E44.m2.3.3.2.1.1"><plus id="A3.E44.m2.3.3.2.1.1.1.1.cmml" xref="A3.E44.m2.3.3.2.1.1.1.1"></plus><cn id="A3.E44.m2.3.3.2.1.1.1.2.cmml" type="integer" xref="A3.E44.m2.3.3.2.1.1.1.2">1</cn><ci id="A3.E44.m2.3.3.2.1.1.1.3.cmml" xref="A3.E44.m2.3.3.2.1.1.1.3">𝑝</ci></apply><apply id="A3.E44.m2.4.4.3.4.cmml" xref="A3.E44.m2.4.4.3.4"><csymbol cd="ambiguous" id="A3.E44.m2.4.4.3.4.1.cmml" xref="A3.E44.m2.4.4.3.4">superscript</csymbol><apply id="A3.E44.m2.4.4.3.4.2.cmml" xref="A3.E44.m2.4.4.3.4.2"><ci id="A3.E44.m2.4.4.3.4.2.1.cmml" xref="A3.E44.m2.4.4.3.4.2.1">~</ci><ci id="A3.E44.m2.4.4.3.4.2.2.cmml" xref="A3.E44.m2.4.4.3.4.2.2">𝜽</ci></apply><ci id="A3.E44.m2.4.4.3.4.3.cmml" xref="A3.E44.m2.4.4.3.4.3">𝑇</ci></apply><apply id="A3.E44.m2.4.4.3.2.1.1.cmml" xref="A3.E44.m2.4.4.3.2.1"><plus id="A3.E44.m2.4.4.3.2.1.1.2.cmml" xref="A3.E44.m2.4.4.3.2.1.1.2"></plus><apply id="A3.E44.m2.4.4.3.2.1.1.3.cmml" xref="A3.E44.m2.4.4.3.2.1.1.3"><times id="A3.E44.m2.4.4.3.2.1.1.3.1.cmml" xref="A3.E44.m2.4.4.3.2.1.1.3.1"></times><apply id="A3.E44.m2.4.4.3.2.1.1.3.2.cmml" xref="A3.E44.m2.4.4.3.2.1.1.3.2"><csymbol cd="ambiguous" id="A3.E44.m2.4.4.3.2.1.1.3.2.1.cmml" xref="A3.E44.m2.4.4.3.2.1.1.3.2">subscript</csymbol><ci id="A3.E44.m2.4.4.3.2.1.1.3.2.2.cmml" xref="A3.E44.m2.4.4.3.2.1.1.3.2.2">Φ</ci><ci id="A3.E44.m2.4.4.3.2.1.1.3.2.3.cmml" xref="A3.E44.m2.4.4.3.2.1.1.3.2.3">f</ci></apply><apply id="A3.E44.m2.4.4.3.2.1.1.3.3.cmml" xref="A3.E44.m2.4.4.3.2.1.1.3.3"><csymbol cd="ambiguous" id="A3.E44.m2.4.4.3.2.1.1.3.3.1.cmml" xref="A3.E44.m2.4.4.3.2.1.1.3.3">superscript</csymbol><apply id="A3.E44.m2.4.4.3.2.1.1.3.3.2.cmml" xref="A3.E44.m2.4.4.3.2.1.1.3.3"><csymbol cd="ambiguous" id="A3.E44.m2.4.4.3.2.1.1.3.3.2.1.cmml" xref="A3.E44.m2.4.4.3.2.1.1.3.3">subscript</csymbol><ci id="A3.E44.m2.4.4.3.2.1.1.3.3.2.2.cmml" xref="A3.E44.m2.4.4.3.2.1.1.3.3.2.2">Φ</ci><ci id="A3.E44.m2.4.4.3.2.1.1.3.3.2.3.cmml" xref="A3.E44.m2.4.4.3.2.1.1.3.3.2.3">f</ci></apply><ci id="A3.E44.m2.4.4.3.2.1.1.3.3.3.cmml" xref="A3.E44.m2.4.4.3.2.1.1.3.3.3">𝑇</ci></apply></apply><apply id="A3.E44.m2.4.4.3.2.1.1.1.cmml" xref="A3.E44.m2.4.4.3.2.1.1.1"><times id="A3.E44.m2.4.4.3.2.1.1.1.2.cmml" xref="A3.E44.m2.4.4.3.2.1.1.1.2"></times><ci id="A3.E44.m2.4.4.3.2.1.1.1.3.cmml" xref="A3.E44.m2.4.4.3.2.1.1.1.3">𝜅</ci><ci id="A3.E44.m2.4.4.3.2.1.1.1.4.cmml" xref="A3.E44.m2.4.4.3.2.1.1.1.4">𝑄</ci><interval closure="open" id="A3.E44.m2.4.4.3.2.1.1.1.1.2.cmml" xref="A3.E44.m2.4.4.3.2.1.1.1.1.1"><ci id="A3.E44.m2.1.1.cmml" xref="A3.E44.m2.1.1">𝑡</ci><apply id="A3.E44.m2.4.4.3.2.1.1.1.1.1.1.cmml" xref="A3.E44.m2.4.4.3.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.E44.m2.4.4.3.2.1.1.1.1.1.1.1.cmml" xref="A3.E44.m2.4.4.3.2.1.1.1.1.1.1">subscript</csymbol><ci id="A3.E44.m2.4.4.3.2.1.1.1.1.1.1.2.cmml" xref="A3.E44.m2.4.4.3.2.1.1.1.1.1.1.2">𝑡</ci><ci id="A3.E44.m2.4.4.3.2.1.1.1.1.1.1.3.cmml" xref="A3.E44.m2.4.4.3.2.1.1.1.1.1.1.3">e</ci></apply></interval></apply></apply><apply id="A3.E44.m2.4.4.3.5.cmml" xref="A3.E44.m2.4.4.3.5"><ci id="A3.E44.m2.4.4.3.5.1.cmml" xref="A3.E44.m2.4.4.3.5.1">~</ci><ci id="A3.E44.m2.4.4.3.5.2.cmml" xref="A3.E44.m2.4.4.3.5.2">𝜽</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E44.m2.4c">\displaystyle\bm{e}^{T}(-K\bm{e}+\Phi^{T}\tilde{\bm{\theta}})-(1+p)\tilde{\bm{% \theta}}^{T}(\Phi_{{\rm f}}\Phi_{{\rm f}}^{T}+\kappa Q(t,t_{\rm e}))\tilde{\bm% {\theta}}</annotation><annotation encoding="application/x-llamapun" id="A3.E44.m2.4d">bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( - italic_K bold_italic_e + roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG ) - ( 1 + italic_p ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT + italic_κ italic_Q ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) ) over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(44)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.1.p1.14">with <math alttext="K:=" class="ltx_Math" display="inline" id="A3.1.p1.13.m1.1"><semantics id="A3.1.p1.13.m1.1a"><mrow id="A3.1.p1.13.m1.1.1" xref="A3.1.p1.13.m1.1.1.cmml"><mi id="A3.1.p1.13.m1.1.1.2" xref="A3.1.p1.13.m1.1.1.2.cmml">K</mi><mo id="A3.1.p1.13.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="A3.1.p1.13.m1.1.1.1.cmml">:=</mo><mi id="A3.1.p1.13.m1.1.1.3" xref="A3.1.p1.13.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.13.m1.1b"><apply id="A3.1.p1.13.m1.1.1.cmml" xref="A3.1.p1.13.m1.1.1"><csymbol cd="latexml" id="A3.1.p1.13.m1.1.1.1.cmml" xref="A3.1.p1.13.m1.1.1.1">assign</csymbol><ci id="A3.1.p1.13.m1.1.1.2.cmml" xref="A3.1.p1.13.m1.1.1.2">𝐾</ci><csymbol cd="latexml" id="A3.1.p1.13.m1.1.1.3.cmml" xref="A3.1.p1.13.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.13.m1.1c">K:=</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.13.m1.1d">italic_K :=</annotation></semantics></math> diag(<math alttext="k_{{\rm c}1},k_{{\rm c}2},\cdots,k_{{\rm c}n}" class="ltx_Math" display="inline" id="A3.1.p1.14.m2.4"><semantics id="A3.1.p1.14.m2.4a"><mrow id="A3.1.p1.14.m2.4.4.3" xref="A3.1.p1.14.m2.4.4.4.cmml"><msub id="A3.1.p1.14.m2.2.2.1.1" xref="A3.1.p1.14.m2.2.2.1.1.cmml"><mi id="A3.1.p1.14.m2.2.2.1.1.2" xref="A3.1.p1.14.m2.2.2.1.1.2.cmml">k</mi><mi id="A3.1.p1.14.m2.2.2.1.1.3" xref="A3.1.p1.14.m2.2.2.1.1.3.cmml">c1</mi></msub><mo id="A3.1.p1.14.m2.4.4.3.4" xref="A3.1.p1.14.m2.4.4.4.cmml">,</mo><msub id="A3.1.p1.14.m2.3.3.2.2" xref="A3.1.p1.14.m2.3.3.2.2.cmml"><mi id="A3.1.p1.14.m2.3.3.2.2.2" xref="A3.1.p1.14.m2.3.3.2.2.2.cmml">k</mi><mi id="A3.1.p1.14.m2.3.3.2.2.3" xref="A3.1.p1.14.m2.3.3.2.2.3.cmml">c2</mi></msub><mo id="A3.1.p1.14.m2.4.4.3.5" xref="A3.1.p1.14.m2.4.4.4.cmml">,</mo><mi id="A3.1.p1.14.m2.1.1" mathvariant="normal" xref="A3.1.p1.14.m2.1.1.cmml">⋯</mi><mo id="A3.1.p1.14.m2.4.4.3.6" xref="A3.1.p1.14.m2.4.4.4.cmml">,</mo><msub id="A3.1.p1.14.m2.4.4.3.3" xref="A3.1.p1.14.m2.4.4.3.3.cmml"><mi id="A3.1.p1.14.m2.4.4.3.3.2" xref="A3.1.p1.14.m2.4.4.3.3.2.cmml">k</mi><mrow id="A3.1.p1.14.m2.4.4.3.3.3" xref="A3.1.p1.14.m2.4.4.3.3.3.cmml"><mi id="A3.1.p1.14.m2.4.4.3.3.3.2" mathvariant="normal" xref="A3.1.p1.14.m2.4.4.3.3.3.2.cmml">c</mi><mo id="A3.1.p1.14.m2.4.4.3.3.3.1" xref="A3.1.p1.14.m2.4.4.3.3.3.1.cmml">⁢</mo><mi id="A3.1.p1.14.m2.4.4.3.3.3.3" xref="A3.1.p1.14.m2.4.4.3.3.3.3.cmml">n</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.14.m2.4b"><list id="A3.1.p1.14.m2.4.4.4.cmml" xref="A3.1.p1.14.m2.4.4.3"><apply id="A3.1.p1.14.m2.2.2.1.1.cmml" xref="A3.1.p1.14.m2.2.2.1.1"><csymbol cd="ambiguous" id="A3.1.p1.14.m2.2.2.1.1.1.cmml" xref="A3.1.p1.14.m2.2.2.1.1">subscript</csymbol><ci id="A3.1.p1.14.m2.2.2.1.1.2.cmml" xref="A3.1.p1.14.m2.2.2.1.1.2">𝑘</ci><ci id="A3.1.p1.14.m2.2.2.1.1.3.cmml" xref="A3.1.p1.14.m2.2.2.1.1.3">c1</ci></apply><apply id="A3.1.p1.14.m2.3.3.2.2.cmml" xref="A3.1.p1.14.m2.3.3.2.2"><csymbol cd="ambiguous" id="A3.1.p1.14.m2.3.3.2.2.1.cmml" xref="A3.1.p1.14.m2.3.3.2.2">subscript</csymbol><ci id="A3.1.p1.14.m2.3.3.2.2.2.cmml" xref="A3.1.p1.14.m2.3.3.2.2.2">𝑘</ci><ci id="A3.1.p1.14.m2.3.3.2.2.3.cmml" xref="A3.1.p1.14.m2.3.3.2.2.3">c2</ci></apply><ci id="A3.1.p1.14.m2.1.1.cmml" xref="A3.1.p1.14.m2.1.1">⋯</ci><apply id="A3.1.p1.14.m2.4.4.3.3.cmml" xref="A3.1.p1.14.m2.4.4.3.3"><csymbol cd="ambiguous" id="A3.1.p1.14.m2.4.4.3.3.1.cmml" xref="A3.1.p1.14.m2.4.4.3.3">subscript</csymbol><ci id="A3.1.p1.14.m2.4.4.3.3.2.cmml" xref="A3.1.p1.14.m2.4.4.3.3.2">𝑘</ci><apply id="A3.1.p1.14.m2.4.4.3.3.3.cmml" xref="A3.1.p1.14.m2.4.4.3.3.3"><times id="A3.1.p1.14.m2.4.4.3.3.3.1.cmml" xref="A3.1.p1.14.m2.4.4.3.3.3.1"></times><ci id="A3.1.p1.14.m2.4.4.3.3.3.2.cmml" xref="A3.1.p1.14.m2.4.4.3.3.3.2">c</ci><ci id="A3.1.p1.14.m2.4.4.3.3.3.3.cmml" xref="A3.1.p1.14.m2.4.4.3.3.3.3">𝑛</ci></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.14.m2.4c">k_{{\rm c}1},k_{{\rm c}2},\cdots,k_{{\rm c}n}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.14.m2.4d">italic_k start_POSTSUBSCRIPT c1 end_POSTSUBSCRIPT , italic_k start_POSTSUBSCRIPT c2 end_POSTSUBSCRIPT , ⋯ , italic_k start_POSTSUBSCRIPT roman_c italic_n end_POSTSUBSCRIPT</annotation></semantics></math>), which can be rewritten into</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx56"> <tbody id="A3.Ex41"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}=" class="ltx_Math" display="inline" id="A3.Ex41.m1.1"><semantics id="A3.Ex41.m1.1a"><mrow id="A3.Ex41.m1.1.1" xref="A3.Ex41.m1.1.1.cmml"><mover accent="true" id="A3.Ex41.m1.1.1.2" xref="A3.Ex41.m1.1.1.2.cmml"><mi id="A3.Ex41.m1.1.1.2.2" xref="A3.Ex41.m1.1.1.2.2.cmml">V</mi><mo id="A3.Ex41.m1.1.1.2.1" xref="A3.Ex41.m1.1.1.2.1.cmml">˙</mo></mover><mo id="A3.Ex41.m1.1.1.1" xref="A3.Ex41.m1.1.1.1.cmml">=</mo><mi id="A3.Ex41.m1.1.1.3" xref="A3.Ex41.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex41.m1.1b"><apply id="A3.Ex41.m1.1.1.cmml" xref="A3.Ex41.m1.1.1"><eq id="A3.Ex41.m1.1.1.1.cmml" xref="A3.Ex41.m1.1.1.1"></eq><apply id="A3.Ex41.m1.1.1.2.cmml" xref="A3.Ex41.m1.1.1.2"><ci id="A3.Ex41.m1.1.1.2.1.cmml" xref="A3.Ex41.m1.1.1.2.1">˙</ci><ci id="A3.Ex41.m1.1.1.2.2.cmml" xref="A3.Ex41.m1.1.1.2.2">𝑉</ci></apply><csymbol cd="latexml" id="A3.Ex41.m1.1.1.3.cmml" xref="A3.Ex41.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex41.m1.1c">\displaystyle\dot{V}=</annotation><annotation encoding="application/x-llamapun" id="A3.Ex41.m1.1d">over˙ start_ARG italic_V end_ARG =</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-\bm{e}^{T}(K-I/4)\bm{e}-\kappa(1+p)\tilde{\bm{\theta}}^{T}Q(t,t_% {\rm e}){\tilde{\bm{\theta}}}+\bm{e}^{T}(\Phi-\Phi_{\rm f})\tilde{\bm{\theta}}" class="ltx_Math" display="inline" id="A3.Ex41.m2.5"><semantics id="A3.Ex41.m2.5a"><mrow id="A3.Ex41.m2.5.5" xref="A3.Ex41.m2.5.5.cmml"><mrow id="A3.Ex41.m2.4.4.3" xref="A3.Ex41.m2.4.4.3.cmml"><mrow id="A3.Ex41.m2.2.2.1.1" xref="A3.Ex41.m2.2.2.1.1.cmml"><mo id="A3.Ex41.m2.2.2.1.1a" xref="A3.Ex41.m2.2.2.1.1.cmml">−</mo><mrow id="A3.Ex41.m2.2.2.1.1.1" xref="A3.Ex41.m2.2.2.1.1.1.cmml"><msup id="A3.Ex41.m2.2.2.1.1.1.3" xref="A3.Ex41.m2.2.2.1.1.1.3.cmml"><mi id="A3.Ex41.m2.2.2.1.1.1.3.2" xref="A3.Ex41.m2.2.2.1.1.1.3.2.cmml">𝒆</mi><mi id="A3.Ex41.m2.2.2.1.1.1.3.3" xref="A3.Ex41.m2.2.2.1.1.1.3.3.cmml">T</mi></msup><mo id="A3.Ex41.m2.2.2.1.1.1.2" xref="A3.Ex41.m2.2.2.1.1.1.2.cmml">⁢</mo><mrow id="A3.Ex41.m2.2.2.1.1.1.1.1" xref="A3.Ex41.m2.2.2.1.1.1.1.1.1.cmml"><mo id="A3.Ex41.m2.2.2.1.1.1.1.1.2" stretchy="false" xref="A3.Ex41.m2.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.Ex41.m2.2.2.1.1.1.1.1.1" xref="A3.Ex41.m2.2.2.1.1.1.1.1.1.cmml"><mi id="A3.Ex41.m2.2.2.1.1.1.1.1.1.2" xref="A3.Ex41.m2.2.2.1.1.1.1.1.1.2.cmml">K</mi><mo id="A3.Ex41.m2.2.2.1.1.1.1.1.1.1" xref="A3.Ex41.m2.2.2.1.1.1.1.1.1.1.cmml">−</mo><mrow id="A3.Ex41.m2.2.2.1.1.1.1.1.1.3" xref="A3.Ex41.m2.2.2.1.1.1.1.1.1.3.cmml"><mi id="A3.Ex41.m2.2.2.1.1.1.1.1.1.3.2" xref="A3.Ex41.m2.2.2.1.1.1.1.1.1.3.2.cmml">I</mi><mo id="A3.Ex41.m2.2.2.1.1.1.1.1.1.3.1" xref="A3.Ex41.m2.2.2.1.1.1.1.1.1.3.1.cmml">/</mo><mn id="A3.Ex41.m2.2.2.1.1.1.1.1.1.3.3" xref="A3.Ex41.m2.2.2.1.1.1.1.1.1.3.3.cmml">4</mn></mrow></mrow><mo id="A3.Ex41.m2.2.2.1.1.1.1.1.3" stretchy="false" xref="A3.Ex41.m2.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A3.Ex41.m2.2.2.1.1.1.2a" xref="A3.Ex41.m2.2.2.1.1.1.2.cmml">⁢</mo><mi id="A3.Ex41.m2.2.2.1.1.1.4" xref="A3.Ex41.m2.2.2.1.1.1.4.cmml">𝒆</mi></mrow></mrow><mo id="A3.Ex41.m2.4.4.3.4" xref="A3.Ex41.m2.4.4.3.4.cmml">−</mo><mrow id="A3.Ex41.m2.4.4.3.3" xref="A3.Ex41.m2.4.4.3.3.cmml"><mi id="A3.Ex41.m2.4.4.3.3.4" xref="A3.Ex41.m2.4.4.3.3.4.cmml">κ</mi><mo id="A3.Ex41.m2.4.4.3.3.3" xref="A3.Ex41.m2.4.4.3.3.3.cmml">⁢</mo><mrow id="A3.Ex41.m2.3.3.2.2.1.1" xref="A3.Ex41.m2.3.3.2.2.1.1.1.cmml"><mo id="A3.Ex41.m2.3.3.2.2.1.1.2" stretchy="false" xref="A3.Ex41.m2.3.3.2.2.1.1.1.cmml">(</mo><mrow id="A3.Ex41.m2.3.3.2.2.1.1.1" xref="A3.Ex41.m2.3.3.2.2.1.1.1.cmml"><mn id="A3.Ex41.m2.3.3.2.2.1.1.1.2" xref="A3.Ex41.m2.3.3.2.2.1.1.1.2.cmml">1</mn><mo id="A3.Ex41.m2.3.3.2.2.1.1.1.1" xref="A3.Ex41.m2.3.3.2.2.1.1.1.1.cmml">+</mo><mi id="A3.Ex41.m2.3.3.2.2.1.1.1.3" xref="A3.Ex41.m2.3.3.2.2.1.1.1.3.cmml">p</mi></mrow><mo id="A3.Ex41.m2.3.3.2.2.1.1.3" stretchy="false" xref="A3.Ex41.m2.3.3.2.2.1.1.1.cmml">)</mo></mrow><mo id="A3.Ex41.m2.4.4.3.3.3a" xref="A3.Ex41.m2.4.4.3.3.3.cmml">⁢</mo><msup id="A3.Ex41.m2.4.4.3.3.5" xref="A3.Ex41.m2.4.4.3.3.5.cmml"><mover accent="true" id="A3.Ex41.m2.4.4.3.3.5.2" xref="A3.Ex41.m2.4.4.3.3.5.2.cmml"><mi id="A3.Ex41.m2.4.4.3.3.5.2.2" xref="A3.Ex41.m2.4.4.3.3.5.2.2.cmml">𝜽</mi><mo id="A3.Ex41.m2.4.4.3.3.5.2.1" xref="A3.Ex41.m2.4.4.3.3.5.2.1.cmml">~</mo></mover><mi id="A3.Ex41.m2.4.4.3.3.5.3" xref="A3.Ex41.m2.4.4.3.3.5.3.cmml">T</mi></msup><mo id="A3.Ex41.m2.4.4.3.3.3b" xref="A3.Ex41.m2.4.4.3.3.3.cmml">⁢</mo><mi id="A3.Ex41.m2.4.4.3.3.6" xref="A3.Ex41.m2.4.4.3.3.6.cmml">Q</mi><mo id="A3.Ex41.m2.4.4.3.3.3c" xref="A3.Ex41.m2.4.4.3.3.3.cmml">⁢</mo><mrow id="A3.Ex41.m2.4.4.3.3.2.1" xref="A3.Ex41.m2.4.4.3.3.2.2.cmml"><mo id="A3.Ex41.m2.4.4.3.3.2.1.2" stretchy="false" xref="A3.Ex41.m2.4.4.3.3.2.2.cmml">(</mo><mi id="A3.Ex41.m2.1.1" xref="A3.Ex41.m2.1.1.cmml">t</mi><mo id="A3.Ex41.m2.4.4.3.3.2.1.3" xref="A3.Ex41.m2.4.4.3.3.2.2.cmml">,</mo><msub id="A3.Ex41.m2.4.4.3.3.2.1.1" xref="A3.Ex41.m2.4.4.3.3.2.1.1.cmml"><mi id="A3.Ex41.m2.4.4.3.3.2.1.1.2" xref="A3.Ex41.m2.4.4.3.3.2.1.1.2.cmml">t</mi><mi id="A3.Ex41.m2.4.4.3.3.2.1.1.3" mathvariant="normal" xref="A3.Ex41.m2.4.4.3.3.2.1.1.3.cmml">e</mi></msub><mo id="A3.Ex41.m2.4.4.3.3.2.1.4" stretchy="false" xref="A3.Ex41.m2.4.4.3.3.2.2.cmml">)</mo></mrow><mo id="A3.Ex41.m2.4.4.3.3.3d" xref="A3.Ex41.m2.4.4.3.3.3.cmml">⁢</mo><mover accent="true" id="A3.Ex41.m2.4.4.3.3.7" xref="A3.Ex41.m2.4.4.3.3.7.cmml"><mi id="A3.Ex41.m2.4.4.3.3.7.2" xref="A3.Ex41.m2.4.4.3.3.7.2.cmml">𝜽</mi><mo id="A3.Ex41.m2.4.4.3.3.7.1" xref="A3.Ex41.m2.4.4.3.3.7.1.cmml">~</mo></mover></mrow></mrow><mo id="A3.Ex41.m2.5.5.5" xref="A3.Ex41.m2.5.5.5.cmml">+</mo><mrow id="A3.Ex41.m2.5.5.4" xref="A3.Ex41.m2.5.5.4.cmml"><msup id="A3.Ex41.m2.5.5.4.3" xref="A3.Ex41.m2.5.5.4.3.cmml"><mi id="A3.Ex41.m2.5.5.4.3.2" xref="A3.Ex41.m2.5.5.4.3.2.cmml">𝒆</mi><mi id="A3.Ex41.m2.5.5.4.3.3" xref="A3.Ex41.m2.5.5.4.3.3.cmml">T</mi></msup><mo id="A3.Ex41.m2.5.5.4.2" xref="A3.Ex41.m2.5.5.4.2.cmml">⁢</mo><mrow id="A3.Ex41.m2.5.5.4.1.1" xref="A3.Ex41.m2.5.5.4.1.1.1.cmml"><mo id="A3.Ex41.m2.5.5.4.1.1.2" stretchy="false" xref="A3.Ex41.m2.5.5.4.1.1.1.cmml">(</mo><mrow id="A3.Ex41.m2.5.5.4.1.1.1" xref="A3.Ex41.m2.5.5.4.1.1.1.cmml"><mi id="A3.Ex41.m2.5.5.4.1.1.1.2" mathvariant="normal" xref="A3.Ex41.m2.5.5.4.1.1.1.2.cmml">Φ</mi><mo id="A3.Ex41.m2.5.5.4.1.1.1.1" xref="A3.Ex41.m2.5.5.4.1.1.1.1.cmml">−</mo><msub id="A3.Ex41.m2.5.5.4.1.1.1.3" xref="A3.Ex41.m2.5.5.4.1.1.1.3.cmml"><mi id="A3.Ex41.m2.5.5.4.1.1.1.3.2" mathvariant="normal" xref="A3.Ex41.m2.5.5.4.1.1.1.3.2.cmml">Φ</mi><mi id="A3.Ex41.m2.5.5.4.1.1.1.3.3" mathvariant="normal" xref="A3.Ex41.m2.5.5.4.1.1.1.3.3.cmml">f</mi></msub></mrow><mo id="A3.Ex41.m2.5.5.4.1.1.3" stretchy="false" xref="A3.Ex41.m2.5.5.4.1.1.1.cmml">)</mo></mrow><mo id="A3.Ex41.m2.5.5.4.2a" xref="A3.Ex41.m2.5.5.4.2.cmml">⁢</mo><mover accent="true" id="A3.Ex41.m2.5.5.4.4" xref="A3.Ex41.m2.5.5.4.4.cmml"><mi id="A3.Ex41.m2.5.5.4.4.2" xref="A3.Ex41.m2.5.5.4.4.2.cmml">𝜽</mi><mo id="A3.Ex41.m2.5.5.4.4.1" xref="A3.Ex41.m2.5.5.4.4.1.cmml">~</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex41.m2.5b"><apply id="A3.Ex41.m2.5.5.cmml" xref="A3.Ex41.m2.5.5"><plus id="A3.Ex41.m2.5.5.5.cmml" xref="A3.Ex41.m2.5.5.5"></plus><apply id="A3.Ex41.m2.4.4.3.cmml" xref="A3.Ex41.m2.4.4.3"><minus id="A3.Ex41.m2.4.4.3.4.cmml" xref="A3.Ex41.m2.4.4.3.4"></minus><apply id="A3.Ex41.m2.2.2.1.1.cmml" xref="A3.Ex41.m2.2.2.1.1"><minus id="A3.Ex41.m2.2.2.1.1.2.cmml" xref="A3.Ex41.m2.2.2.1.1"></minus><apply id="A3.Ex41.m2.2.2.1.1.1.cmml" xref="A3.Ex41.m2.2.2.1.1.1"><times id="A3.Ex41.m2.2.2.1.1.1.2.cmml" xref="A3.Ex41.m2.2.2.1.1.1.2"></times><apply id="A3.Ex41.m2.2.2.1.1.1.3.cmml" xref="A3.Ex41.m2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="A3.Ex41.m2.2.2.1.1.1.3.1.cmml" xref="A3.Ex41.m2.2.2.1.1.1.3">superscript</csymbol><ci id="A3.Ex41.m2.2.2.1.1.1.3.2.cmml" xref="A3.Ex41.m2.2.2.1.1.1.3.2">𝒆</ci><ci id="A3.Ex41.m2.2.2.1.1.1.3.3.cmml" xref="A3.Ex41.m2.2.2.1.1.1.3.3">𝑇</ci></apply><apply id="A3.Ex41.m2.2.2.1.1.1.1.1.1.cmml" xref="A3.Ex41.m2.2.2.1.1.1.1.1"><minus id="A3.Ex41.m2.2.2.1.1.1.1.1.1.1.cmml" xref="A3.Ex41.m2.2.2.1.1.1.1.1.1.1"></minus><ci id="A3.Ex41.m2.2.2.1.1.1.1.1.1.2.cmml" xref="A3.Ex41.m2.2.2.1.1.1.1.1.1.2">𝐾</ci><apply id="A3.Ex41.m2.2.2.1.1.1.1.1.1.3.cmml" xref="A3.Ex41.m2.2.2.1.1.1.1.1.1.3"><divide id="A3.Ex41.m2.2.2.1.1.1.1.1.1.3.1.cmml" xref="A3.Ex41.m2.2.2.1.1.1.1.1.1.3.1"></divide><ci id="A3.Ex41.m2.2.2.1.1.1.1.1.1.3.2.cmml" xref="A3.Ex41.m2.2.2.1.1.1.1.1.1.3.2">𝐼</ci><cn id="A3.Ex41.m2.2.2.1.1.1.1.1.1.3.3.cmml" type="integer" xref="A3.Ex41.m2.2.2.1.1.1.1.1.1.3.3">4</cn></apply></apply><ci id="A3.Ex41.m2.2.2.1.1.1.4.cmml" xref="A3.Ex41.m2.2.2.1.1.1.4">𝒆</ci></apply></apply><apply id="A3.Ex41.m2.4.4.3.3.cmml" xref="A3.Ex41.m2.4.4.3.3"><times id="A3.Ex41.m2.4.4.3.3.3.cmml" xref="A3.Ex41.m2.4.4.3.3.3"></times><ci id="A3.Ex41.m2.4.4.3.3.4.cmml" xref="A3.Ex41.m2.4.4.3.3.4">𝜅</ci><apply id="A3.Ex41.m2.3.3.2.2.1.1.1.cmml" xref="A3.Ex41.m2.3.3.2.2.1.1"><plus id="A3.Ex41.m2.3.3.2.2.1.1.1.1.cmml" xref="A3.Ex41.m2.3.3.2.2.1.1.1.1"></plus><cn id="A3.Ex41.m2.3.3.2.2.1.1.1.2.cmml" type="integer" xref="A3.Ex41.m2.3.3.2.2.1.1.1.2">1</cn><ci id="A3.Ex41.m2.3.3.2.2.1.1.1.3.cmml" xref="A3.Ex41.m2.3.3.2.2.1.1.1.3">𝑝</ci></apply><apply id="A3.Ex41.m2.4.4.3.3.5.cmml" xref="A3.Ex41.m2.4.4.3.3.5"><csymbol cd="ambiguous" id="A3.Ex41.m2.4.4.3.3.5.1.cmml" xref="A3.Ex41.m2.4.4.3.3.5">superscript</csymbol><apply id="A3.Ex41.m2.4.4.3.3.5.2.cmml" xref="A3.Ex41.m2.4.4.3.3.5.2"><ci id="A3.Ex41.m2.4.4.3.3.5.2.1.cmml" xref="A3.Ex41.m2.4.4.3.3.5.2.1">~</ci><ci id="A3.Ex41.m2.4.4.3.3.5.2.2.cmml" xref="A3.Ex41.m2.4.4.3.3.5.2.2">𝜽</ci></apply><ci id="A3.Ex41.m2.4.4.3.3.5.3.cmml" xref="A3.Ex41.m2.4.4.3.3.5.3">𝑇</ci></apply><ci id="A3.Ex41.m2.4.4.3.3.6.cmml" xref="A3.Ex41.m2.4.4.3.3.6">𝑄</ci><interval closure="open" id="A3.Ex41.m2.4.4.3.3.2.2.cmml" xref="A3.Ex41.m2.4.4.3.3.2.1"><ci id="A3.Ex41.m2.1.1.cmml" xref="A3.Ex41.m2.1.1">𝑡</ci><apply id="A3.Ex41.m2.4.4.3.3.2.1.1.cmml" xref="A3.Ex41.m2.4.4.3.3.2.1.1"><csymbol cd="ambiguous" id="A3.Ex41.m2.4.4.3.3.2.1.1.1.cmml" xref="A3.Ex41.m2.4.4.3.3.2.1.1">subscript</csymbol><ci id="A3.Ex41.m2.4.4.3.3.2.1.1.2.cmml" xref="A3.Ex41.m2.4.4.3.3.2.1.1.2">𝑡</ci><ci id="A3.Ex41.m2.4.4.3.3.2.1.1.3.cmml" xref="A3.Ex41.m2.4.4.3.3.2.1.1.3">e</ci></apply></interval><apply id="A3.Ex41.m2.4.4.3.3.7.cmml" xref="A3.Ex41.m2.4.4.3.3.7"><ci id="A3.Ex41.m2.4.4.3.3.7.1.cmml" xref="A3.Ex41.m2.4.4.3.3.7.1">~</ci><ci id="A3.Ex41.m2.4.4.3.3.7.2.cmml" xref="A3.Ex41.m2.4.4.3.3.7.2">𝜽</ci></apply></apply></apply><apply id="A3.Ex41.m2.5.5.4.cmml" xref="A3.Ex41.m2.5.5.4"><times id="A3.Ex41.m2.5.5.4.2.cmml" xref="A3.Ex41.m2.5.5.4.2"></times><apply id="A3.Ex41.m2.5.5.4.3.cmml" xref="A3.Ex41.m2.5.5.4.3"><csymbol cd="ambiguous" id="A3.Ex41.m2.5.5.4.3.1.cmml" xref="A3.Ex41.m2.5.5.4.3">superscript</csymbol><ci id="A3.Ex41.m2.5.5.4.3.2.cmml" xref="A3.Ex41.m2.5.5.4.3.2">𝒆</ci><ci id="A3.Ex41.m2.5.5.4.3.3.cmml" xref="A3.Ex41.m2.5.5.4.3.3">𝑇</ci></apply><apply id="A3.Ex41.m2.5.5.4.1.1.1.cmml" xref="A3.Ex41.m2.5.5.4.1.1"><minus id="A3.Ex41.m2.5.5.4.1.1.1.1.cmml" xref="A3.Ex41.m2.5.5.4.1.1.1.1"></minus><ci id="A3.Ex41.m2.5.5.4.1.1.1.2.cmml" xref="A3.Ex41.m2.5.5.4.1.1.1.2">Φ</ci><apply id="A3.Ex41.m2.5.5.4.1.1.1.3.cmml" xref="A3.Ex41.m2.5.5.4.1.1.1.3"><csymbol cd="ambiguous" id="A3.Ex41.m2.5.5.4.1.1.1.3.1.cmml" xref="A3.Ex41.m2.5.5.4.1.1.1.3">subscript</csymbol><ci id="A3.Ex41.m2.5.5.4.1.1.1.3.2.cmml" xref="A3.Ex41.m2.5.5.4.1.1.1.3.2">Φ</ci><ci id="A3.Ex41.m2.5.5.4.1.1.1.3.3.cmml" xref="A3.Ex41.m2.5.5.4.1.1.1.3.3">f</ci></apply></apply><apply id="A3.Ex41.m2.5.5.4.4.cmml" xref="A3.Ex41.m2.5.5.4.4"><ci id="A3.Ex41.m2.5.5.4.4.1.cmml" xref="A3.Ex41.m2.5.5.4.4.1">~</ci><ci id="A3.Ex41.m2.5.5.4.4.2.cmml" xref="A3.Ex41.m2.5.5.4.4.2">𝜽</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex41.m2.5c">\displaystyle-\bm{e}^{T}(K-I/4)\bm{e}-\kappa(1+p)\tilde{\bm{\theta}}^{T}Q(t,t_% {\rm e}){\tilde{\bm{\theta}}}+\bm{e}^{T}(\Phi-\Phi_{\rm f})\tilde{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex41.m2.5d">- bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_K - italic_I / 4 ) bold_italic_e - italic_κ ( 1 + italic_p ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT italic_Q ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG + bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( roman_Φ - roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex42"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-\bm{e}^{T}\bm{e}/4-\tilde{\bm{\theta}}^{T}\Phi_{\rm f}\Phi_{\rm f% }^{T}\tilde{\bm{\theta}}+\bm{e}^{T}\Phi_{\rm f}^{T}\tilde{\bm{\theta}}-p\tilde% {\bm{\theta}}^{T}\Phi_{\rm f}\Phi_{\rm f}^{T}\tilde{\bm{\theta}}" class="ltx_Math" display="inline" id="A3.Ex42.m1.1"><semantics id="A3.Ex42.m1.1a"><mrow id="A3.Ex42.m1.1.1" xref="A3.Ex42.m1.1.1.cmml"><mrow id="A3.Ex42.m1.1.1.2" xref="A3.Ex42.m1.1.1.2.cmml"><mrow id="A3.Ex42.m1.1.1.2.2" xref="A3.Ex42.m1.1.1.2.2.cmml"><mrow id="A3.Ex42.m1.1.1.2.2.2" xref="A3.Ex42.m1.1.1.2.2.2.cmml"><mo id="A3.Ex42.m1.1.1.2.2.2a" xref="A3.Ex42.m1.1.1.2.2.2.cmml">−</mo><mrow id="A3.Ex42.m1.1.1.2.2.2.2" xref="A3.Ex42.m1.1.1.2.2.2.2.cmml"><mrow id="A3.Ex42.m1.1.1.2.2.2.2.2" xref="A3.Ex42.m1.1.1.2.2.2.2.2.cmml"><msup id="A3.Ex42.m1.1.1.2.2.2.2.2.2" xref="A3.Ex42.m1.1.1.2.2.2.2.2.2.cmml"><mi id="A3.Ex42.m1.1.1.2.2.2.2.2.2.2" xref="A3.Ex42.m1.1.1.2.2.2.2.2.2.2.cmml">𝒆</mi><mi id="A3.Ex42.m1.1.1.2.2.2.2.2.2.3" xref="A3.Ex42.m1.1.1.2.2.2.2.2.2.3.cmml">T</mi></msup><mo id="A3.Ex42.m1.1.1.2.2.2.2.2.1" xref="A3.Ex42.m1.1.1.2.2.2.2.2.1.cmml">⁢</mo><mi id="A3.Ex42.m1.1.1.2.2.2.2.2.3" xref="A3.Ex42.m1.1.1.2.2.2.2.2.3.cmml">𝒆</mi></mrow><mo id="A3.Ex42.m1.1.1.2.2.2.2.1" xref="A3.Ex42.m1.1.1.2.2.2.2.1.cmml">/</mo><mn id="A3.Ex42.m1.1.1.2.2.2.2.3" xref="A3.Ex42.m1.1.1.2.2.2.2.3.cmml">4</mn></mrow></mrow><mo id="A3.Ex42.m1.1.1.2.2.1" xref="A3.Ex42.m1.1.1.2.2.1.cmml">−</mo><mrow id="A3.Ex42.m1.1.1.2.2.3" xref="A3.Ex42.m1.1.1.2.2.3.cmml"><msup id="A3.Ex42.m1.1.1.2.2.3.2" xref="A3.Ex42.m1.1.1.2.2.3.2.cmml"><mover accent="true" id="A3.Ex42.m1.1.1.2.2.3.2.2" xref="A3.Ex42.m1.1.1.2.2.3.2.2.cmml"><mi id="A3.Ex42.m1.1.1.2.2.3.2.2.2" xref="A3.Ex42.m1.1.1.2.2.3.2.2.2.cmml">𝜽</mi><mo id="A3.Ex42.m1.1.1.2.2.3.2.2.1" xref="A3.Ex42.m1.1.1.2.2.3.2.2.1.cmml">~</mo></mover><mi id="A3.Ex42.m1.1.1.2.2.3.2.3" xref="A3.Ex42.m1.1.1.2.2.3.2.3.cmml">T</mi></msup><mo id="A3.Ex42.m1.1.1.2.2.3.1" xref="A3.Ex42.m1.1.1.2.2.3.1.cmml">⁢</mo><msub id="A3.Ex42.m1.1.1.2.2.3.3" xref="A3.Ex42.m1.1.1.2.2.3.3.cmml"><mi id="A3.Ex42.m1.1.1.2.2.3.3.2" mathvariant="normal" xref="A3.Ex42.m1.1.1.2.2.3.3.2.cmml">Φ</mi><mi id="A3.Ex42.m1.1.1.2.2.3.3.3" mathvariant="normal" xref="A3.Ex42.m1.1.1.2.2.3.3.3.cmml">f</mi></msub><mo id="A3.Ex42.m1.1.1.2.2.3.1a" xref="A3.Ex42.m1.1.1.2.2.3.1.cmml">⁢</mo><msubsup id="A3.Ex42.m1.1.1.2.2.3.4" xref="A3.Ex42.m1.1.1.2.2.3.4.cmml"><mi id="A3.Ex42.m1.1.1.2.2.3.4.2.2" mathvariant="normal" xref="A3.Ex42.m1.1.1.2.2.3.4.2.2.cmml">Φ</mi><mi id="A3.Ex42.m1.1.1.2.2.3.4.2.3" mathvariant="normal" xref="A3.Ex42.m1.1.1.2.2.3.4.2.3.cmml">f</mi><mi id="A3.Ex42.m1.1.1.2.2.3.4.3" xref="A3.Ex42.m1.1.1.2.2.3.4.3.cmml">T</mi></msubsup><mo id="A3.Ex42.m1.1.1.2.2.3.1b" xref="A3.Ex42.m1.1.1.2.2.3.1.cmml">⁢</mo><mover accent="true" id="A3.Ex42.m1.1.1.2.2.3.5" xref="A3.Ex42.m1.1.1.2.2.3.5.cmml"><mi id="A3.Ex42.m1.1.1.2.2.3.5.2" xref="A3.Ex42.m1.1.1.2.2.3.5.2.cmml">𝜽</mi><mo id="A3.Ex42.m1.1.1.2.2.3.5.1" xref="A3.Ex42.m1.1.1.2.2.3.5.1.cmml">~</mo></mover></mrow></mrow><mo id="A3.Ex42.m1.1.1.2.1" xref="A3.Ex42.m1.1.1.2.1.cmml">+</mo><mrow id="A3.Ex42.m1.1.1.2.3" xref="A3.Ex42.m1.1.1.2.3.cmml"><msup id="A3.Ex42.m1.1.1.2.3.2" xref="A3.Ex42.m1.1.1.2.3.2.cmml"><mi id="A3.Ex42.m1.1.1.2.3.2.2" xref="A3.Ex42.m1.1.1.2.3.2.2.cmml">𝒆</mi><mi id="A3.Ex42.m1.1.1.2.3.2.3" xref="A3.Ex42.m1.1.1.2.3.2.3.cmml">T</mi></msup><mo id="A3.Ex42.m1.1.1.2.3.1" xref="A3.Ex42.m1.1.1.2.3.1.cmml">⁢</mo><msubsup id="A3.Ex42.m1.1.1.2.3.3" xref="A3.Ex42.m1.1.1.2.3.3.cmml"><mi id="A3.Ex42.m1.1.1.2.3.3.2.2" mathvariant="normal" xref="A3.Ex42.m1.1.1.2.3.3.2.2.cmml">Φ</mi><mi id="A3.Ex42.m1.1.1.2.3.3.2.3" mathvariant="normal" xref="A3.Ex42.m1.1.1.2.3.3.2.3.cmml">f</mi><mi id="A3.Ex42.m1.1.1.2.3.3.3" xref="A3.Ex42.m1.1.1.2.3.3.3.cmml">T</mi></msubsup><mo id="A3.Ex42.m1.1.1.2.3.1a" xref="A3.Ex42.m1.1.1.2.3.1.cmml">⁢</mo><mover accent="true" id="A3.Ex42.m1.1.1.2.3.4" xref="A3.Ex42.m1.1.1.2.3.4.cmml"><mi id="A3.Ex42.m1.1.1.2.3.4.2" xref="A3.Ex42.m1.1.1.2.3.4.2.cmml">𝜽</mi><mo id="A3.Ex42.m1.1.1.2.3.4.1" xref="A3.Ex42.m1.1.1.2.3.4.1.cmml">~</mo></mover></mrow></mrow><mo id="A3.Ex42.m1.1.1.1" xref="A3.Ex42.m1.1.1.1.cmml">−</mo><mrow id="A3.Ex42.m1.1.1.3" xref="A3.Ex42.m1.1.1.3.cmml"><mi id="A3.Ex42.m1.1.1.3.2" xref="A3.Ex42.m1.1.1.3.2.cmml">p</mi><mo id="A3.Ex42.m1.1.1.3.1" xref="A3.Ex42.m1.1.1.3.1.cmml">⁢</mo><msup id="A3.Ex42.m1.1.1.3.3" xref="A3.Ex42.m1.1.1.3.3.cmml"><mover accent="true" id="A3.Ex42.m1.1.1.3.3.2" xref="A3.Ex42.m1.1.1.3.3.2.cmml"><mi id="A3.Ex42.m1.1.1.3.3.2.2" xref="A3.Ex42.m1.1.1.3.3.2.2.cmml">𝜽</mi><mo id="A3.Ex42.m1.1.1.3.3.2.1" xref="A3.Ex42.m1.1.1.3.3.2.1.cmml">~</mo></mover><mi id="A3.Ex42.m1.1.1.3.3.3" xref="A3.Ex42.m1.1.1.3.3.3.cmml">T</mi></msup><mo id="A3.Ex42.m1.1.1.3.1a" xref="A3.Ex42.m1.1.1.3.1.cmml">⁢</mo><msub id="A3.Ex42.m1.1.1.3.4" xref="A3.Ex42.m1.1.1.3.4.cmml"><mi id="A3.Ex42.m1.1.1.3.4.2" mathvariant="normal" xref="A3.Ex42.m1.1.1.3.4.2.cmml">Φ</mi><mi id="A3.Ex42.m1.1.1.3.4.3" mathvariant="normal" xref="A3.Ex42.m1.1.1.3.4.3.cmml">f</mi></msub><mo id="A3.Ex42.m1.1.1.3.1b" xref="A3.Ex42.m1.1.1.3.1.cmml">⁢</mo><msubsup id="A3.Ex42.m1.1.1.3.5" xref="A3.Ex42.m1.1.1.3.5.cmml"><mi id="A3.Ex42.m1.1.1.3.5.2.2" mathvariant="normal" xref="A3.Ex42.m1.1.1.3.5.2.2.cmml">Φ</mi><mi id="A3.Ex42.m1.1.1.3.5.2.3" mathvariant="normal" xref="A3.Ex42.m1.1.1.3.5.2.3.cmml">f</mi><mi id="A3.Ex42.m1.1.1.3.5.3" xref="A3.Ex42.m1.1.1.3.5.3.cmml">T</mi></msubsup><mo id="A3.Ex42.m1.1.1.3.1c" xref="A3.Ex42.m1.1.1.3.1.cmml">⁢</mo><mover accent="true" id="A3.Ex42.m1.1.1.3.6" xref="A3.Ex42.m1.1.1.3.6.cmml"><mi id="A3.Ex42.m1.1.1.3.6.2" xref="A3.Ex42.m1.1.1.3.6.2.cmml">𝜽</mi><mo id="A3.Ex42.m1.1.1.3.6.1" xref="A3.Ex42.m1.1.1.3.6.1.cmml">~</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex42.m1.1b"><apply id="A3.Ex42.m1.1.1.cmml" xref="A3.Ex42.m1.1.1"><minus id="A3.Ex42.m1.1.1.1.cmml" xref="A3.Ex42.m1.1.1.1"></minus><apply id="A3.Ex42.m1.1.1.2.cmml" xref="A3.Ex42.m1.1.1.2"><plus id="A3.Ex42.m1.1.1.2.1.cmml" xref="A3.Ex42.m1.1.1.2.1"></plus><apply id="A3.Ex42.m1.1.1.2.2.cmml" xref="A3.Ex42.m1.1.1.2.2"><minus id="A3.Ex42.m1.1.1.2.2.1.cmml" xref="A3.Ex42.m1.1.1.2.2.1"></minus><apply id="A3.Ex42.m1.1.1.2.2.2.cmml" xref="A3.Ex42.m1.1.1.2.2.2"><minus id="A3.Ex42.m1.1.1.2.2.2.1.cmml" xref="A3.Ex42.m1.1.1.2.2.2"></minus><apply id="A3.Ex42.m1.1.1.2.2.2.2.cmml" xref="A3.Ex42.m1.1.1.2.2.2.2"><divide id="A3.Ex42.m1.1.1.2.2.2.2.1.cmml" xref="A3.Ex42.m1.1.1.2.2.2.2.1"></divide><apply id="A3.Ex42.m1.1.1.2.2.2.2.2.cmml" xref="A3.Ex42.m1.1.1.2.2.2.2.2"><times id="A3.Ex42.m1.1.1.2.2.2.2.2.1.cmml" xref="A3.Ex42.m1.1.1.2.2.2.2.2.1"></times><apply id="A3.Ex42.m1.1.1.2.2.2.2.2.2.cmml" xref="A3.Ex42.m1.1.1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="A3.Ex42.m1.1.1.2.2.2.2.2.2.1.cmml" xref="A3.Ex42.m1.1.1.2.2.2.2.2.2">superscript</csymbol><ci id="A3.Ex42.m1.1.1.2.2.2.2.2.2.2.cmml" xref="A3.Ex42.m1.1.1.2.2.2.2.2.2.2">𝒆</ci><ci id="A3.Ex42.m1.1.1.2.2.2.2.2.2.3.cmml" xref="A3.Ex42.m1.1.1.2.2.2.2.2.2.3">𝑇</ci></apply><ci id="A3.Ex42.m1.1.1.2.2.2.2.2.3.cmml" xref="A3.Ex42.m1.1.1.2.2.2.2.2.3">𝒆</ci></apply><cn id="A3.Ex42.m1.1.1.2.2.2.2.3.cmml" type="integer" xref="A3.Ex42.m1.1.1.2.2.2.2.3">4</cn></apply></apply><apply id="A3.Ex42.m1.1.1.2.2.3.cmml" xref="A3.Ex42.m1.1.1.2.2.3"><times id="A3.Ex42.m1.1.1.2.2.3.1.cmml" xref="A3.Ex42.m1.1.1.2.2.3.1"></times><apply id="A3.Ex42.m1.1.1.2.2.3.2.cmml" xref="A3.Ex42.m1.1.1.2.2.3.2"><csymbol cd="ambiguous" id="A3.Ex42.m1.1.1.2.2.3.2.1.cmml" xref="A3.Ex42.m1.1.1.2.2.3.2">superscript</csymbol><apply id="A3.Ex42.m1.1.1.2.2.3.2.2.cmml" xref="A3.Ex42.m1.1.1.2.2.3.2.2"><ci id="A3.Ex42.m1.1.1.2.2.3.2.2.1.cmml" xref="A3.Ex42.m1.1.1.2.2.3.2.2.1">~</ci><ci id="A3.Ex42.m1.1.1.2.2.3.2.2.2.cmml" xref="A3.Ex42.m1.1.1.2.2.3.2.2.2">𝜽</ci></apply><ci id="A3.Ex42.m1.1.1.2.2.3.2.3.cmml" xref="A3.Ex42.m1.1.1.2.2.3.2.3">𝑇</ci></apply><apply id="A3.Ex42.m1.1.1.2.2.3.3.cmml" xref="A3.Ex42.m1.1.1.2.2.3.3"><csymbol cd="ambiguous" id="A3.Ex42.m1.1.1.2.2.3.3.1.cmml" xref="A3.Ex42.m1.1.1.2.2.3.3">subscript</csymbol><ci id="A3.Ex42.m1.1.1.2.2.3.3.2.cmml" xref="A3.Ex42.m1.1.1.2.2.3.3.2">Φ</ci><ci id="A3.Ex42.m1.1.1.2.2.3.3.3.cmml" xref="A3.Ex42.m1.1.1.2.2.3.3.3">f</ci></apply><apply id="A3.Ex42.m1.1.1.2.2.3.4.cmml" xref="A3.Ex42.m1.1.1.2.2.3.4"><csymbol cd="ambiguous" id="A3.Ex42.m1.1.1.2.2.3.4.1.cmml" xref="A3.Ex42.m1.1.1.2.2.3.4">superscript</csymbol><apply id="A3.Ex42.m1.1.1.2.2.3.4.2.cmml" xref="A3.Ex42.m1.1.1.2.2.3.4"><csymbol cd="ambiguous" id="A3.Ex42.m1.1.1.2.2.3.4.2.1.cmml" xref="A3.Ex42.m1.1.1.2.2.3.4">subscript</csymbol><ci id="A3.Ex42.m1.1.1.2.2.3.4.2.2.cmml" xref="A3.Ex42.m1.1.1.2.2.3.4.2.2">Φ</ci><ci id="A3.Ex42.m1.1.1.2.2.3.4.2.3.cmml" xref="A3.Ex42.m1.1.1.2.2.3.4.2.3">f</ci></apply><ci id="A3.Ex42.m1.1.1.2.2.3.4.3.cmml" xref="A3.Ex42.m1.1.1.2.2.3.4.3">𝑇</ci></apply><apply id="A3.Ex42.m1.1.1.2.2.3.5.cmml" xref="A3.Ex42.m1.1.1.2.2.3.5"><ci id="A3.Ex42.m1.1.1.2.2.3.5.1.cmml" xref="A3.Ex42.m1.1.1.2.2.3.5.1">~</ci><ci id="A3.Ex42.m1.1.1.2.2.3.5.2.cmml" xref="A3.Ex42.m1.1.1.2.2.3.5.2">𝜽</ci></apply></apply></apply><apply id="A3.Ex42.m1.1.1.2.3.cmml" xref="A3.Ex42.m1.1.1.2.3"><times id="A3.Ex42.m1.1.1.2.3.1.cmml" xref="A3.Ex42.m1.1.1.2.3.1"></times><apply id="A3.Ex42.m1.1.1.2.3.2.cmml" xref="A3.Ex42.m1.1.1.2.3.2"><csymbol cd="ambiguous" id="A3.Ex42.m1.1.1.2.3.2.1.cmml" xref="A3.Ex42.m1.1.1.2.3.2">superscript</csymbol><ci id="A3.Ex42.m1.1.1.2.3.2.2.cmml" xref="A3.Ex42.m1.1.1.2.3.2.2">𝒆</ci><ci id="A3.Ex42.m1.1.1.2.3.2.3.cmml" xref="A3.Ex42.m1.1.1.2.3.2.3">𝑇</ci></apply><apply id="A3.Ex42.m1.1.1.2.3.3.cmml" xref="A3.Ex42.m1.1.1.2.3.3"><csymbol cd="ambiguous" id="A3.Ex42.m1.1.1.2.3.3.1.cmml" xref="A3.Ex42.m1.1.1.2.3.3">superscript</csymbol><apply id="A3.Ex42.m1.1.1.2.3.3.2.cmml" xref="A3.Ex42.m1.1.1.2.3.3"><csymbol cd="ambiguous" id="A3.Ex42.m1.1.1.2.3.3.2.1.cmml" xref="A3.Ex42.m1.1.1.2.3.3">subscript</csymbol><ci id="A3.Ex42.m1.1.1.2.3.3.2.2.cmml" xref="A3.Ex42.m1.1.1.2.3.3.2.2">Φ</ci><ci id="A3.Ex42.m1.1.1.2.3.3.2.3.cmml" xref="A3.Ex42.m1.1.1.2.3.3.2.3">f</ci></apply><ci id="A3.Ex42.m1.1.1.2.3.3.3.cmml" xref="A3.Ex42.m1.1.1.2.3.3.3">𝑇</ci></apply><apply id="A3.Ex42.m1.1.1.2.3.4.cmml" xref="A3.Ex42.m1.1.1.2.3.4"><ci id="A3.Ex42.m1.1.1.2.3.4.1.cmml" xref="A3.Ex42.m1.1.1.2.3.4.1">~</ci><ci id="A3.Ex42.m1.1.1.2.3.4.2.cmml" xref="A3.Ex42.m1.1.1.2.3.4.2">𝜽</ci></apply></apply></apply><apply id="A3.Ex42.m1.1.1.3.cmml" xref="A3.Ex42.m1.1.1.3"><times id="A3.Ex42.m1.1.1.3.1.cmml" xref="A3.Ex42.m1.1.1.3.1"></times><ci id="A3.Ex42.m1.1.1.3.2.cmml" xref="A3.Ex42.m1.1.1.3.2">𝑝</ci><apply id="A3.Ex42.m1.1.1.3.3.cmml" xref="A3.Ex42.m1.1.1.3.3"><csymbol cd="ambiguous" id="A3.Ex42.m1.1.1.3.3.1.cmml" xref="A3.Ex42.m1.1.1.3.3">superscript</csymbol><apply id="A3.Ex42.m1.1.1.3.3.2.cmml" xref="A3.Ex42.m1.1.1.3.3.2"><ci id="A3.Ex42.m1.1.1.3.3.2.1.cmml" xref="A3.Ex42.m1.1.1.3.3.2.1">~</ci><ci id="A3.Ex42.m1.1.1.3.3.2.2.cmml" xref="A3.Ex42.m1.1.1.3.3.2.2">𝜽</ci></apply><ci id="A3.Ex42.m1.1.1.3.3.3.cmml" xref="A3.Ex42.m1.1.1.3.3.3">𝑇</ci></apply><apply id="A3.Ex42.m1.1.1.3.4.cmml" xref="A3.Ex42.m1.1.1.3.4"><csymbol cd="ambiguous" id="A3.Ex42.m1.1.1.3.4.1.cmml" xref="A3.Ex42.m1.1.1.3.4">subscript</csymbol><ci id="A3.Ex42.m1.1.1.3.4.2.cmml" xref="A3.Ex42.m1.1.1.3.4.2">Φ</ci><ci id="A3.Ex42.m1.1.1.3.4.3.cmml" xref="A3.Ex42.m1.1.1.3.4.3">f</ci></apply><apply id="A3.Ex42.m1.1.1.3.5.cmml" xref="A3.Ex42.m1.1.1.3.5"><csymbol cd="ambiguous" id="A3.Ex42.m1.1.1.3.5.1.cmml" xref="A3.Ex42.m1.1.1.3.5">superscript</csymbol><apply id="A3.Ex42.m1.1.1.3.5.2.cmml" xref="A3.Ex42.m1.1.1.3.5"><csymbol cd="ambiguous" id="A3.Ex42.m1.1.1.3.5.2.1.cmml" xref="A3.Ex42.m1.1.1.3.5">subscript</csymbol><ci id="A3.Ex42.m1.1.1.3.5.2.2.cmml" xref="A3.Ex42.m1.1.1.3.5.2.2">Φ</ci><ci id="A3.Ex42.m1.1.1.3.5.2.3.cmml" xref="A3.Ex42.m1.1.1.3.5.2.3">f</ci></apply><ci id="A3.Ex42.m1.1.1.3.5.3.cmml" xref="A3.Ex42.m1.1.1.3.5.3">𝑇</ci></apply><apply id="A3.Ex42.m1.1.1.3.6.cmml" xref="A3.Ex42.m1.1.1.3.6"><ci id="A3.Ex42.m1.1.1.3.6.1.cmml" xref="A3.Ex42.m1.1.1.3.6.1">~</ci><ci id="A3.Ex42.m1.1.1.3.6.2.cmml" xref="A3.Ex42.m1.1.1.3.6.2">𝜽</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex42.m1.1c">\displaystyle-\bm{e}^{T}\bm{e}/4-\tilde{\bm{\theta}}^{T}\Phi_{\rm f}\Phi_{\rm f% }^{T}\tilde{\bm{\theta}}+\bm{e}^{T}\Phi_{\rm f}^{T}\tilde{\bm{\theta}}-p\tilde% {\bm{\theta}}^{T}\Phi_{\rm f}\Phi_{\rm f}^{T}\tilde{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex42.m1.1d">- bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_e / 4 - over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG + bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG - italic_p over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.1.p1.97">where the second line of the above formula satisfies</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx57"> <tbody id="A3.Ex43"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle-\bm{e}^{T}\bm{e}/4-\tilde{\bm{\theta}}^{T}\Phi_{\rm f}\Phi_{\rm f% }^{T}\tilde{\bm{\theta}}+\bm{e}^{T}\Phi_{\rm f}^{T}\tilde{\bm{\theta}}\leq-\|% \bm{e}/2-\Phi_{\rm f}^{T}\tilde{\bm{\theta}}\|^{2}\leq 0." class="ltx_Math" display="inline" id="A3.Ex43.m1.1"><semantics id="A3.Ex43.m1.1a"><mrow id="A3.Ex43.m1.1.1.1" xref="A3.Ex43.m1.1.1.1.1.cmml"><mrow id="A3.Ex43.m1.1.1.1.1" xref="A3.Ex43.m1.1.1.1.1.cmml"><mrow id="A3.Ex43.m1.1.1.1.1.3" xref="A3.Ex43.m1.1.1.1.1.3.cmml"><mrow id="A3.Ex43.m1.1.1.1.1.3.2" xref="A3.Ex43.m1.1.1.1.1.3.2.cmml"><mrow id="A3.Ex43.m1.1.1.1.1.3.2.2" xref="A3.Ex43.m1.1.1.1.1.3.2.2.cmml"><mo id="A3.Ex43.m1.1.1.1.1.3.2.2a" xref="A3.Ex43.m1.1.1.1.1.3.2.2.cmml">−</mo><mrow id="A3.Ex43.m1.1.1.1.1.3.2.2.2" xref="A3.Ex43.m1.1.1.1.1.3.2.2.2.cmml"><mrow id="A3.Ex43.m1.1.1.1.1.3.2.2.2.2" xref="A3.Ex43.m1.1.1.1.1.3.2.2.2.2.cmml"><msup id="A3.Ex43.m1.1.1.1.1.3.2.2.2.2.2" xref="A3.Ex43.m1.1.1.1.1.3.2.2.2.2.2.cmml"><mi id="A3.Ex43.m1.1.1.1.1.3.2.2.2.2.2.2" xref="A3.Ex43.m1.1.1.1.1.3.2.2.2.2.2.2.cmml">𝒆</mi><mi id="A3.Ex43.m1.1.1.1.1.3.2.2.2.2.2.3" xref="A3.Ex43.m1.1.1.1.1.3.2.2.2.2.2.3.cmml">T</mi></msup><mo id="A3.Ex43.m1.1.1.1.1.3.2.2.2.2.1" xref="A3.Ex43.m1.1.1.1.1.3.2.2.2.2.1.cmml">⁢</mo><mi id="A3.Ex43.m1.1.1.1.1.3.2.2.2.2.3" xref="A3.Ex43.m1.1.1.1.1.3.2.2.2.2.3.cmml">𝒆</mi></mrow><mo id="A3.Ex43.m1.1.1.1.1.3.2.2.2.1" xref="A3.Ex43.m1.1.1.1.1.3.2.2.2.1.cmml">/</mo><mn id="A3.Ex43.m1.1.1.1.1.3.2.2.2.3" xref="A3.Ex43.m1.1.1.1.1.3.2.2.2.3.cmml">4</mn></mrow></mrow><mo id="A3.Ex43.m1.1.1.1.1.3.2.1" xref="A3.Ex43.m1.1.1.1.1.3.2.1.cmml">−</mo><mrow id="A3.Ex43.m1.1.1.1.1.3.2.3" xref="A3.Ex43.m1.1.1.1.1.3.2.3.cmml"><msup id="A3.Ex43.m1.1.1.1.1.3.2.3.2" xref="A3.Ex43.m1.1.1.1.1.3.2.3.2.cmml"><mover accent="true" id="A3.Ex43.m1.1.1.1.1.3.2.3.2.2" xref="A3.Ex43.m1.1.1.1.1.3.2.3.2.2.cmml"><mi id="A3.Ex43.m1.1.1.1.1.3.2.3.2.2.2" xref="A3.Ex43.m1.1.1.1.1.3.2.3.2.2.2.cmml">𝜽</mi><mo id="A3.Ex43.m1.1.1.1.1.3.2.3.2.2.1" xref="A3.Ex43.m1.1.1.1.1.3.2.3.2.2.1.cmml">~</mo></mover><mi id="A3.Ex43.m1.1.1.1.1.3.2.3.2.3" xref="A3.Ex43.m1.1.1.1.1.3.2.3.2.3.cmml">T</mi></msup><mo id="A3.Ex43.m1.1.1.1.1.3.2.3.1" xref="A3.Ex43.m1.1.1.1.1.3.2.3.1.cmml">⁢</mo><msub id="A3.Ex43.m1.1.1.1.1.3.2.3.3" xref="A3.Ex43.m1.1.1.1.1.3.2.3.3.cmml"><mi id="A3.Ex43.m1.1.1.1.1.3.2.3.3.2" mathvariant="normal" xref="A3.Ex43.m1.1.1.1.1.3.2.3.3.2.cmml">Φ</mi><mi id="A3.Ex43.m1.1.1.1.1.3.2.3.3.3" mathvariant="normal" xref="A3.Ex43.m1.1.1.1.1.3.2.3.3.3.cmml">f</mi></msub><mo id="A3.Ex43.m1.1.1.1.1.3.2.3.1a" xref="A3.Ex43.m1.1.1.1.1.3.2.3.1.cmml">⁢</mo><msubsup id="A3.Ex43.m1.1.1.1.1.3.2.3.4" xref="A3.Ex43.m1.1.1.1.1.3.2.3.4.cmml"><mi id="A3.Ex43.m1.1.1.1.1.3.2.3.4.2.2" mathvariant="normal" xref="A3.Ex43.m1.1.1.1.1.3.2.3.4.2.2.cmml">Φ</mi><mi id="A3.Ex43.m1.1.1.1.1.3.2.3.4.2.3" mathvariant="normal" xref="A3.Ex43.m1.1.1.1.1.3.2.3.4.2.3.cmml">f</mi><mi id="A3.Ex43.m1.1.1.1.1.3.2.3.4.3" xref="A3.Ex43.m1.1.1.1.1.3.2.3.4.3.cmml">T</mi></msubsup><mo id="A3.Ex43.m1.1.1.1.1.3.2.3.1b" xref="A3.Ex43.m1.1.1.1.1.3.2.3.1.cmml">⁢</mo><mover accent="true" id="A3.Ex43.m1.1.1.1.1.3.2.3.5" xref="A3.Ex43.m1.1.1.1.1.3.2.3.5.cmml"><mi id="A3.Ex43.m1.1.1.1.1.3.2.3.5.2" xref="A3.Ex43.m1.1.1.1.1.3.2.3.5.2.cmml">𝜽</mi><mo id="A3.Ex43.m1.1.1.1.1.3.2.3.5.1" xref="A3.Ex43.m1.1.1.1.1.3.2.3.5.1.cmml">~</mo></mover></mrow></mrow><mo id="A3.Ex43.m1.1.1.1.1.3.1" xref="A3.Ex43.m1.1.1.1.1.3.1.cmml">+</mo><mrow id="A3.Ex43.m1.1.1.1.1.3.3" xref="A3.Ex43.m1.1.1.1.1.3.3.cmml"><msup id="A3.Ex43.m1.1.1.1.1.3.3.2" xref="A3.Ex43.m1.1.1.1.1.3.3.2.cmml"><mi id="A3.Ex43.m1.1.1.1.1.3.3.2.2" xref="A3.Ex43.m1.1.1.1.1.3.3.2.2.cmml">𝒆</mi><mi id="A3.Ex43.m1.1.1.1.1.3.3.2.3" xref="A3.Ex43.m1.1.1.1.1.3.3.2.3.cmml">T</mi></msup><mo id="A3.Ex43.m1.1.1.1.1.3.3.1" xref="A3.Ex43.m1.1.1.1.1.3.3.1.cmml">⁢</mo><msubsup id="A3.Ex43.m1.1.1.1.1.3.3.3" xref="A3.Ex43.m1.1.1.1.1.3.3.3.cmml"><mi id="A3.Ex43.m1.1.1.1.1.3.3.3.2.2" mathvariant="normal" xref="A3.Ex43.m1.1.1.1.1.3.3.3.2.2.cmml">Φ</mi><mi id="A3.Ex43.m1.1.1.1.1.3.3.3.2.3" mathvariant="normal" xref="A3.Ex43.m1.1.1.1.1.3.3.3.2.3.cmml">f</mi><mi id="A3.Ex43.m1.1.1.1.1.3.3.3.3" xref="A3.Ex43.m1.1.1.1.1.3.3.3.3.cmml">T</mi></msubsup><mo id="A3.Ex43.m1.1.1.1.1.3.3.1a" xref="A3.Ex43.m1.1.1.1.1.3.3.1.cmml">⁢</mo><mover accent="true" id="A3.Ex43.m1.1.1.1.1.3.3.4" xref="A3.Ex43.m1.1.1.1.1.3.3.4.cmml"><mi id="A3.Ex43.m1.1.1.1.1.3.3.4.2" xref="A3.Ex43.m1.1.1.1.1.3.3.4.2.cmml">𝜽</mi><mo id="A3.Ex43.m1.1.1.1.1.3.3.4.1" xref="A3.Ex43.m1.1.1.1.1.3.3.4.1.cmml">~</mo></mover></mrow></mrow><mo id="A3.Ex43.m1.1.1.1.1.4" xref="A3.Ex43.m1.1.1.1.1.4.cmml">≤</mo><mrow id="A3.Ex43.m1.1.1.1.1.1" xref="A3.Ex43.m1.1.1.1.1.1.cmml"><mo id="A3.Ex43.m1.1.1.1.1.1a" xref="A3.Ex43.m1.1.1.1.1.1.cmml">−</mo><msup id="A3.Ex43.m1.1.1.1.1.1.1" xref="A3.Ex43.m1.1.1.1.1.1.1.cmml"><mrow id="A3.Ex43.m1.1.1.1.1.1.1.1.1" xref="A3.Ex43.m1.1.1.1.1.1.1.1.2.cmml"><mo id="A3.Ex43.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A3.Ex43.m1.1.1.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.cmml"><mrow id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.2" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.2.2" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.2.2.cmml">𝒆</mi><mo id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.2.1" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.2.1.cmml">/</mo><mn id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.2.3" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.2.3.cmml">2</mn></mrow><mo id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.1" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mrow id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.cmml"><msubsup id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.2" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.2.cmml"><mi id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.2.2.2" mathvariant="normal" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.2.2.2.cmml">Φ</mi><mi id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.2.2.3" mathvariant="normal" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.2.2.3.cmml">f</mi><mi id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.2.3" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.2.3.cmml">T</mi></msubsup><mo id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.1" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mover accent="true" id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.3" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.3.cmml"><mi id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.3.2" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.3.2.cmml">𝜽</mi><mo id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.3.1" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.3.1.cmml">~</mo></mover></mrow></mrow><mo id="A3.Ex43.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="A3.Ex43.m1.1.1.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mn id="A3.Ex43.m1.1.1.1.1.1.1.3" xref="A3.Ex43.m1.1.1.1.1.1.1.3.cmml">2</mn></msup></mrow><mo id="A3.Ex43.m1.1.1.1.1.5" xref="A3.Ex43.m1.1.1.1.1.5.cmml">≤</mo><mn id="A3.Ex43.m1.1.1.1.1.6" xref="A3.Ex43.m1.1.1.1.1.6.cmml">0</mn></mrow><mo id="A3.Ex43.m1.1.1.1.2" lspace="0em" xref="A3.Ex43.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex43.m1.1b"><apply id="A3.Ex43.m1.1.1.1.1.cmml" xref="A3.Ex43.m1.1.1.1"><and id="A3.Ex43.m1.1.1.1.1a.cmml" xref="A3.Ex43.m1.1.1.1"></and><apply id="A3.Ex43.m1.1.1.1.1b.cmml" xref="A3.Ex43.m1.1.1.1"><leq id="A3.Ex43.m1.1.1.1.1.4.cmml" xref="A3.Ex43.m1.1.1.1.1.4"></leq><apply id="A3.Ex43.m1.1.1.1.1.3.cmml" xref="A3.Ex43.m1.1.1.1.1.3"><plus id="A3.Ex43.m1.1.1.1.1.3.1.cmml" xref="A3.Ex43.m1.1.1.1.1.3.1"></plus><apply id="A3.Ex43.m1.1.1.1.1.3.2.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2"><minus id="A3.Ex43.m1.1.1.1.1.3.2.1.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.1"></minus><apply id="A3.Ex43.m1.1.1.1.1.3.2.2.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.2"><minus id="A3.Ex43.m1.1.1.1.1.3.2.2.1.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.2"></minus><apply id="A3.Ex43.m1.1.1.1.1.3.2.2.2.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.2.2"><divide id="A3.Ex43.m1.1.1.1.1.3.2.2.2.1.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.2.2.1"></divide><apply id="A3.Ex43.m1.1.1.1.1.3.2.2.2.2.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.2.2.2"><times id="A3.Ex43.m1.1.1.1.1.3.2.2.2.2.1.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.2.2.2.1"></times><apply id="A3.Ex43.m1.1.1.1.1.3.2.2.2.2.2.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.2.2.2.2"><csymbol cd="ambiguous" id="A3.Ex43.m1.1.1.1.1.3.2.2.2.2.2.1.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.2.2.2.2">superscript</csymbol><ci id="A3.Ex43.m1.1.1.1.1.3.2.2.2.2.2.2.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.2.2.2.2.2">𝒆</ci><ci id="A3.Ex43.m1.1.1.1.1.3.2.2.2.2.2.3.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.2.2.2.2.3">𝑇</ci></apply><ci id="A3.Ex43.m1.1.1.1.1.3.2.2.2.2.3.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.2.2.2.3">𝒆</ci></apply><cn id="A3.Ex43.m1.1.1.1.1.3.2.2.2.3.cmml" type="integer" xref="A3.Ex43.m1.1.1.1.1.3.2.2.2.3">4</cn></apply></apply><apply id="A3.Ex43.m1.1.1.1.1.3.2.3.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.3"><times id="A3.Ex43.m1.1.1.1.1.3.2.3.1.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.3.1"></times><apply id="A3.Ex43.m1.1.1.1.1.3.2.3.2.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.3.2"><csymbol cd="ambiguous" id="A3.Ex43.m1.1.1.1.1.3.2.3.2.1.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.3.2">superscript</csymbol><apply id="A3.Ex43.m1.1.1.1.1.3.2.3.2.2.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.3.2.2"><ci id="A3.Ex43.m1.1.1.1.1.3.2.3.2.2.1.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.3.2.2.1">~</ci><ci id="A3.Ex43.m1.1.1.1.1.3.2.3.2.2.2.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.3.2.2.2">𝜽</ci></apply><ci id="A3.Ex43.m1.1.1.1.1.3.2.3.2.3.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.3.2.3">𝑇</ci></apply><apply id="A3.Ex43.m1.1.1.1.1.3.2.3.3.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.3.3"><csymbol cd="ambiguous" id="A3.Ex43.m1.1.1.1.1.3.2.3.3.1.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.3.3">subscript</csymbol><ci id="A3.Ex43.m1.1.1.1.1.3.2.3.3.2.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.3.3.2">Φ</ci><ci id="A3.Ex43.m1.1.1.1.1.3.2.3.3.3.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.3.3.3">f</ci></apply><apply id="A3.Ex43.m1.1.1.1.1.3.2.3.4.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.3.4"><csymbol cd="ambiguous" id="A3.Ex43.m1.1.1.1.1.3.2.3.4.1.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.3.4">superscript</csymbol><apply id="A3.Ex43.m1.1.1.1.1.3.2.3.4.2.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.3.4"><csymbol cd="ambiguous" id="A3.Ex43.m1.1.1.1.1.3.2.3.4.2.1.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.3.4">subscript</csymbol><ci id="A3.Ex43.m1.1.1.1.1.3.2.3.4.2.2.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.3.4.2.2">Φ</ci><ci id="A3.Ex43.m1.1.1.1.1.3.2.3.4.2.3.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.3.4.2.3">f</ci></apply><ci id="A3.Ex43.m1.1.1.1.1.3.2.3.4.3.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.3.4.3">𝑇</ci></apply><apply id="A3.Ex43.m1.1.1.1.1.3.2.3.5.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.3.5"><ci id="A3.Ex43.m1.1.1.1.1.3.2.3.5.1.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.3.5.1">~</ci><ci id="A3.Ex43.m1.1.1.1.1.3.2.3.5.2.cmml" xref="A3.Ex43.m1.1.1.1.1.3.2.3.5.2">𝜽</ci></apply></apply></apply><apply id="A3.Ex43.m1.1.1.1.1.3.3.cmml" xref="A3.Ex43.m1.1.1.1.1.3.3"><times id="A3.Ex43.m1.1.1.1.1.3.3.1.cmml" xref="A3.Ex43.m1.1.1.1.1.3.3.1"></times><apply id="A3.Ex43.m1.1.1.1.1.3.3.2.cmml" xref="A3.Ex43.m1.1.1.1.1.3.3.2"><csymbol cd="ambiguous" id="A3.Ex43.m1.1.1.1.1.3.3.2.1.cmml" xref="A3.Ex43.m1.1.1.1.1.3.3.2">superscript</csymbol><ci id="A3.Ex43.m1.1.1.1.1.3.3.2.2.cmml" xref="A3.Ex43.m1.1.1.1.1.3.3.2.2">𝒆</ci><ci id="A3.Ex43.m1.1.1.1.1.3.3.2.3.cmml" xref="A3.Ex43.m1.1.1.1.1.3.3.2.3">𝑇</ci></apply><apply id="A3.Ex43.m1.1.1.1.1.3.3.3.cmml" xref="A3.Ex43.m1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="A3.Ex43.m1.1.1.1.1.3.3.3.1.cmml" xref="A3.Ex43.m1.1.1.1.1.3.3.3">superscript</csymbol><apply id="A3.Ex43.m1.1.1.1.1.3.3.3.2.cmml" xref="A3.Ex43.m1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="A3.Ex43.m1.1.1.1.1.3.3.3.2.1.cmml" xref="A3.Ex43.m1.1.1.1.1.3.3.3">subscript</csymbol><ci id="A3.Ex43.m1.1.1.1.1.3.3.3.2.2.cmml" xref="A3.Ex43.m1.1.1.1.1.3.3.3.2.2">Φ</ci><ci id="A3.Ex43.m1.1.1.1.1.3.3.3.2.3.cmml" xref="A3.Ex43.m1.1.1.1.1.3.3.3.2.3">f</ci></apply><ci id="A3.Ex43.m1.1.1.1.1.3.3.3.3.cmml" xref="A3.Ex43.m1.1.1.1.1.3.3.3.3">𝑇</ci></apply><apply id="A3.Ex43.m1.1.1.1.1.3.3.4.cmml" xref="A3.Ex43.m1.1.1.1.1.3.3.4"><ci id="A3.Ex43.m1.1.1.1.1.3.3.4.1.cmml" xref="A3.Ex43.m1.1.1.1.1.3.3.4.1">~</ci><ci id="A3.Ex43.m1.1.1.1.1.3.3.4.2.cmml" xref="A3.Ex43.m1.1.1.1.1.3.3.4.2">𝜽</ci></apply></apply></apply><apply id="A3.Ex43.m1.1.1.1.1.1.cmml" xref="A3.Ex43.m1.1.1.1.1.1"><minus id="A3.Ex43.m1.1.1.1.1.1.2.cmml" xref="A3.Ex43.m1.1.1.1.1.1"></minus><apply id="A3.Ex43.m1.1.1.1.1.1.1.cmml" xref="A3.Ex43.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.Ex43.m1.1.1.1.1.1.1.2.cmml" xref="A3.Ex43.m1.1.1.1.1.1.1">superscript</csymbol><apply id="A3.Ex43.m1.1.1.1.1.1.1.1.2.cmml" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1"><csymbol cd="latexml" id="A3.Ex43.m1.1.1.1.1.1.1.1.2.1.cmml" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.2">norm</csymbol><apply id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.cmml" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1"><minus id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.1"></minus><apply id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.2"><divide id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.2.1"></divide><ci id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.2.2">𝒆</ci><cn id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.2.3">2</cn></apply><apply id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3"><times id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.1"></times><apply id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.2.1.cmml" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.2">superscript</csymbol><apply id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.2.2.cmml" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.2.2.1.cmml" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.2">subscript</csymbol><ci id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.2.2.2.cmml" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.2.2.2">Φ</ci><ci id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.2.2.3.cmml" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.2.2.3">f</ci></apply><ci id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.2.3.cmml" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.2.3">𝑇</ci></apply><apply id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.3"><ci id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.3.1.cmml" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.3.1">~</ci><ci id="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.3.2.cmml" xref="A3.Ex43.m1.1.1.1.1.1.1.1.1.1.3.3.2">𝜽</ci></apply></apply></apply></apply><cn id="A3.Ex43.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="A3.Ex43.m1.1.1.1.1.1.1.3">2</cn></apply></apply></apply><apply id="A3.Ex43.m1.1.1.1.1c.cmml" xref="A3.Ex43.m1.1.1.1"><leq id="A3.Ex43.m1.1.1.1.1.5.cmml" xref="A3.Ex43.m1.1.1.1.1.5"></leq><share href="https://arxiv.org/html/2401.10785v2#A3.Ex43.m1.1.1.1.1.1.cmml" id="A3.Ex43.m1.1.1.1.1d.cmml" xref="A3.Ex43.m1.1.1.1"></share><cn id="A3.Ex43.m1.1.1.1.1.6.cmml" type="integer" xref="A3.Ex43.m1.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex43.m1.1c">\displaystyle-\bm{e}^{T}\bm{e}/4-\tilde{\bm{\theta}}^{T}\Phi_{\rm f}\Phi_{\rm f% }^{T}\tilde{\bm{\theta}}+\bm{e}^{T}\Phi_{\rm f}^{T}\tilde{\bm{\theta}}\leq-\|% \bm{e}/2-\Phi_{\rm f}^{T}\tilde{\bm{\theta}}\|^{2}\leq 0.</annotation><annotation encoding="application/x-llamapun" id="A3.Ex43.m1.1d">- bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_e / 4 - over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG + bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG ≤ - ∥ bold_italic_e / 2 - roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ 0 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.1.p1.98">Then, it is straightforward to get</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx58"> <tbody id="A3.E45"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}\leq" class="ltx_Math" display="inline" id="A3.E45.m1.1"><semantics id="A3.E45.m1.1a"><mrow id="A3.E45.m1.1.1" xref="A3.E45.m1.1.1.cmml"><mover accent="true" id="A3.E45.m1.1.1.2" xref="A3.E45.m1.1.1.2.cmml"><mi id="A3.E45.m1.1.1.2.2" xref="A3.E45.m1.1.1.2.2.cmml">V</mi><mo id="A3.E45.m1.1.1.2.1" xref="A3.E45.m1.1.1.2.1.cmml">˙</mo></mover><mo id="A3.E45.m1.1.1.1" xref="A3.E45.m1.1.1.1.cmml">≤</mo><mi id="A3.E45.m1.1.1.3" xref="A3.E45.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A3.E45.m1.1b"><apply id="A3.E45.m1.1.1.cmml" xref="A3.E45.m1.1.1"><leq id="A3.E45.m1.1.1.1.cmml" xref="A3.E45.m1.1.1.1"></leq><apply id="A3.E45.m1.1.1.2.cmml" xref="A3.E45.m1.1.1.2"><ci id="A3.E45.m1.1.1.2.1.cmml" xref="A3.E45.m1.1.1.2.1">˙</ci><ci id="A3.E45.m1.1.1.2.2.cmml" xref="A3.E45.m1.1.1.2.2">𝑉</ci></apply><csymbol cd="latexml" id="A3.E45.m1.1.1.3.cmml" xref="A3.E45.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E45.m1.1c">\displaystyle\dot{V}\leq</annotation><annotation encoding="application/x-llamapun" id="A3.E45.m1.1d">over˙ start_ARG italic_V end_ARG ≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-\bm{e}^{T}(K-I/4)\bm{e}-\kappa(1+p)\tilde{\bm{\theta}}^{T}Q(t,t_% {\rm e}){\tilde{\bm{\theta}}}+\bm{e}^{T}(\Phi-\Phi_{\rm f})\tilde{\bm{\theta}}." class="ltx_Math" display="inline" id="A3.E45.m2.2"><semantics id="A3.E45.m2.2a"><mrow id="A3.E45.m2.2.2.1" xref="A3.E45.m2.2.2.1.1.cmml"><mrow id="A3.E45.m2.2.2.1.1" xref="A3.E45.m2.2.2.1.1.cmml"><mrow id="A3.E45.m2.2.2.1.1.3" xref="A3.E45.m2.2.2.1.1.3.cmml"><mrow id="A3.E45.m2.2.2.1.1.1.1" xref="A3.E45.m2.2.2.1.1.1.1.cmml"><mo id="A3.E45.m2.2.2.1.1.1.1a" xref="A3.E45.m2.2.2.1.1.1.1.cmml">−</mo><mrow id="A3.E45.m2.2.2.1.1.1.1.1" xref="A3.E45.m2.2.2.1.1.1.1.1.cmml"><msup id="A3.E45.m2.2.2.1.1.1.1.1.3" xref="A3.E45.m2.2.2.1.1.1.1.1.3.cmml"><mi id="A3.E45.m2.2.2.1.1.1.1.1.3.2" xref="A3.E45.m2.2.2.1.1.1.1.1.3.2.cmml">𝒆</mi><mi id="A3.E45.m2.2.2.1.1.1.1.1.3.3" xref="A3.E45.m2.2.2.1.1.1.1.1.3.3.cmml">T</mi></msup><mo id="A3.E45.m2.2.2.1.1.1.1.1.2" xref="A3.E45.m2.2.2.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A3.E45.m2.2.2.1.1.1.1.1.1.1" xref="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.cmml"><mo id="A3.E45.m2.2.2.1.1.1.1.1.1.1.2" stretchy="false" xref="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.E45.m2.2.2.1.1.1.1.1.1.1.1" xref="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.cmml"><mi id="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.2" xref="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.2.cmml">K</mi><mo id="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.1" xref="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.1.cmml">−</mo><mrow id="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.3" xref="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.3.cmml"><mi id="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.3.2" xref="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.3.2.cmml">I</mi><mo id="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.3.1" xref="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.3.1.cmml">/</mo><mn id="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.3.3" xref="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.3.3.cmml">4</mn></mrow></mrow><mo id="A3.E45.m2.2.2.1.1.1.1.1.1.1.3" stretchy="false" xref="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A3.E45.m2.2.2.1.1.1.1.1.2a" xref="A3.E45.m2.2.2.1.1.1.1.1.2.cmml">⁢</mo><mi id="A3.E45.m2.2.2.1.1.1.1.1.4" xref="A3.E45.m2.2.2.1.1.1.1.1.4.cmml">𝒆</mi></mrow></mrow><mo id="A3.E45.m2.2.2.1.1.3.4" xref="A3.E45.m2.2.2.1.1.3.4.cmml">−</mo><mrow id="A3.E45.m2.2.2.1.1.3.3" xref="A3.E45.m2.2.2.1.1.3.3.cmml"><mi id="A3.E45.m2.2.2.1.1.3.3.4" xref="A3.E45.m2.2.2.1.1.3.3.4.cmml">κ</mi><mo id="A3.E45.m2.2.2.1.1.3.3.3" xref="A3.E45.m2.2.2.1.1.3.3.3.cmml">⁢</mo><mrow id="A3.E45.m2.2.2.1.1.2.2.1.1" xref="A3.E45.m2.2.2.1.1.2.2.1.1.1.cmml"><mo id="A3.E45.m2.2.2.1.1.2.2.1.1.2" stretchy="false" xref="A3.E45.m2.2.2.1.1.2.2.1.1.1.cmml">(</mo><mrow id="A3.E45.m2.2.2.1.1.2.2.1.1.1" xref="A3.E45.m2.2.2.1.1.2.2.1.1.1.cmml"><mn id="A3.E45.m2.2.2.1.1.2.2.1.1.1.2" xref="A3.E45.m2.2.2.1.1.2.2.1.1.1.2.cmml">1</mn><mo id="A3.E45.m2.2.2.1.1.2.2.1.1.1.1" xref="A3.E45.m2.2.2.1.1.2.2.1.1.1.1.cmml">+</mo><mi id="A3.E45.m2.2.2.1.1.2.2.1.1.1.3" xref="A3.E45.m2.2.2.1.1.2.2.1.1.1.3.cmml">p</mi></mrow><mo id="A3.E45.m2.2.2.1.1.2.2.1.1.3" stretchy="false" xref="A3.E45.m2.2.2.1.1.2.2.1.1.1.cmml">)</mo></mrow><mo id="A3.E45.m2.2.2.1.1.3.3.3a" xref="A3.E45.m2.2.2.1.1.3.3.3.cmml">⁢</mo><msup id="A3.E45.m2.2.2.1.1.3.3.5" xref="A3.E45.m2.2.2.1.1.3.3.5.cmml"><mover accent="true" id="A3.E45.m2.2.2.1.1.3.3.5.2" xref="A3.E45.m2.2.2.1.1.3.3.5.2.cmml"><mi id="A3.E45.m2.2.2.1.1.3.3.5.2.2" xref="A3.E45.m2.2.2.1.1.3.3.5.2.2.cmml">𝜽</mi><mo id="A3.E45.m2.2.2.1.1.3.3.5.2.1" xref="A3.E45.m2.2.2.1.1.3.3.5.2.1.cmml">~</mo></mover><mi id="A3.E45.m2.2.2.1.1.3.3.5.3" xref="A3.E45.m2.2.2.1.1.3.3.5.3.cmml">T</mi></msup><mo id="A3.E45.m2.2.2.1.1.3.3.3b" xref="A3.E45.m2.2.2.1.1.3.3.3.cmml">⁢</mo><mi id="A3.E45.m2.2.2.1.1.3.3.6" xref="A3.E45.m2.2.2.1.1.3.3.6.cmml">Q</mi><mo id="A3.E45.m2.2.2.1.1.3.3.3c" xref="A3.E45.m2.2.2.1.1.3.3.3.cmml">⁢</mo><mrow id="A3.E45.m2.2.2.1.1.3.3.2.1" xref="A3.E45.m2.2.2.1.1.3.3.2.2.cmml"><mo id="A3.E45.m2.2.2.1.1.3.3.2.1.2" stretchy="false" xref="A3.E45.m2.2.2.1.1.3.3.2.2.cmml">(</mo><mi id="A3.E45.m2.1.1" xref="A3.E45.m2.1.1.cmml">t</mi><mo id="A3.E45.m2.2.2.1.1.3.3.2.1.3" xref="A3.E45.m2.2.2.1.1.3.3.2.2.cmml">,</mo><msub id="A3.E45.m2.2.2.1.1.3.3.2.1.1" xref="A3.E45.m2.2.2.1.1.3.3.2.1.1.cmml"><mi id="A3.E45.m2.2.2.1.1.3.3.2.1.1.2" xref="A3.E45.m2.2.2.1.1.3.3.2.1.1.2.cmml">t</mi><mi id="A3.E45.m2.2.2.1.1.3.3.2.1.1.3" mathvariant="normal" xref="A3.E45.m2.2.2.1.1.3.3.2.1.1.3.cmml">e</mi></msub><mo id="A3.E45.m2.2.2.1.1.3.3.2.1.4" stretchy="false" xref="A3.E45.m2.2.2.1.1.3.3.2.2.cmml">)</mo></mrow><mo id="A3.E45.m2.2.2.1.1.3.3.3d" xref="A3.E45.m2.2.2.1.1.3.3.3.cmml">⁢</mo><mover accent="true" id="A3.E45.m2.2.2.1.1.3.3.7" xref="A3.E45.m2.2.2.1.1.3.3.7.cmml"><mi id="A3.E45.m2.2.2.1.1.3.3.7.2" xref="A3.E45.m2.2.2.1.1.3.3.7.2.cmml">𝜽</mi><mo id="A3.E45.m2.2.2.1.1.3.3.7.1" xref="A3.E45.m2.2.2.1.1.3.3.7.1.cmml">~</mo></mover></mrow></mrow><mo id="A3.E45.m2.2.2.1.1.5" xref="A3.E45.m2.2.2.1.1.5.cmml">+</mo><mrow id="A3.E45.m2.2.2.1.1.4" xref="A3.E45.m2.2.2.1.1.4.cmml"><msup id="A3.E45.m2.2.2.1.1.4.3" xref="A3.E45.m2.2.2.1.1.4.3.cmml"><mi id="A3.E45.m2.2.2.1.1.4.3.2" xref="A3.E45.m2.2.2.1.1.4.3.2.cmml">𝒆</mi><mi id="A3.E45.m2.2.2.1.1.4.3.3" xref="A3.E45.m2.2.2.1.1.4.3.3.cmml">T</mi></msup><mo id="A3.E45.m2.2.2.1.1.4.2" xref="A3.E45.m2.2.2.1.1.4.2.cmml">⁢</mo><mrow id="A3.E45.m2.2.2.1.1.4.1.1" xref="A3.E45.m2.2.2.1.1.4.1.1.1.cmml"><mo id="A3.E45.m2.2.2.1.1.4.1.1.2" stretchy="false" xref="A3.E45.m2.2.2.1.1.4.1.1.1.cmml">(</mo><mrow id="A3.E45.m2.2.2.1.1.4.1.1.1" xref="A3.E45.m2.2.2.1.1.4.1.1.1.cmml"><mi id="A3.E45.m2.2.2.1.1.4.1.1.1.2" mathvariant="normal" xref="A3.E45.m2.2.2.1.1.4.1.1.1.2.cmml">Φ</mi><mo id="A3.E45.m2.2.2.1.1.4.1.1.1.1" xref="A3.E45.m2.2.2.1.1.4.1.1.1.1.cmml">−</mo><msub id="A3.E45.m2.2.2.1.1.4.1.1.1.3" xref="A3.E45.m2.2.2.1.1.4.1.1.1.3.cmml"><mi id="A3.E45.m2.2.2.1.1.4.1.1.1.3.2" mathvariant="normal" xref="A3.E45.m2.2.2.1.1.4.1.1.1.3.2.cmml">Φ</mi><mi id="A3.E45.m2.2.2.1.1.4.1.1.1.3.3" mathvariant="normal" xref="A3.E45.m2.2.2.1.1.4.1.1.1.3.3.cmml">f</mi></msub></mrow><mo id="A3.E45.m2.2.2.1.1.4.1.1.3" stretchy="false" xref="A3.E45.m2.2.2.1.1.4.1.1.1.cmml">)</mo></mrow><mo id="A3.E45.m2.2.2.1.1.4.2a" xref="A3.E45.m2.2.2.1.1.4.2.cmml">⁢</mo><mover accent="true" id="A3.E45.m2.2.2.1.1.4.4" xref="A3.E45.m2.2.2.1.1.4.4.cmml"><mi id="A3.E45.m2.2.2.1.1.4.4.2" xref="A3.E45.m2.2.2.1.1.4.4.2.cmml">𝜽</mi><mo id="A3.E45.m2.2.2.1.1.4.4.1" xref="A3.E45.m2.2.2.1.1.4.4.1.cmml">~</mo></mover></mrow></mrow><mo id="A3.E45.m2.2.2.1.2" lspace="0em" xref="A3.E45.m2.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.E45.m2.2b"><apply id="A3.E45.m2.2.2.1.1.cmml" xref="A3.E45.m2.2.2.1"><plus id="A3.E45.m2.2.2.1.1.5.cmml" xref="A3.E45.m2.2.2.1.1.5"></plus><apply id="A3.E45.m2.2.2.1.1.3.cmml" xref="A3.E45.m2.2.2.1.1.3"><minus id="A3.E45.m2.2.2.1.1.3.4.cmml" xref="A3.E45.m2.2.2.1.1.3.4"></minus><apply id="A3.E45.m2.2.2.1.1.1.1.cmml" xref="A3.E45.m2.2.2.1.1.1.1"><minus id="A3.E45.m2.2.2.1.1.1.1.2.cmml" xref="A3.E45.m2.2.2.1.1.1.1"></minus><apply id="A3.E45.m2.2.2.1.1.1.1.1.cmml" xref="A3.E45.m2.2.2.1.1.1.1.1"><times id="A3.E45.m2.2.2.1.1.1.1.1.2.cmml" xref="A3.E45.m2.2.2.1.1.1.1.1.2"></times><apply id="A3.E45.m2.2.2.1.1.1.1.1.3.cmml" xref="A3.E45.m2.2.2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A3.E45.m2.2.2.1.1.1.1.1.3.1.cmml" xref="A3.E45.m2.2.2.1.1.1.1.1.3">superscript</csymbol><ci id="A3.E45.m2.2.2.1.1.1.1.1.3.2.cmml" xref="A3.E45.m2.2.2.1.1.1.1.1.3.2">𝒆</ci><ci id="A3.E45.m2.2.2.1.1.1.1.1.3.3.cmml" xref="A3.E45.m2.2.2.1.1.1.1.1.3.3">𝑇</ci></apply><apply id="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.cmml" xref="A3.E45.m2.2.2.1.1.1.1.1.1.1"><minus id="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.1.cmml" xref="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.1"></minus><ci id="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.2.cmml" xref="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.2">𝐾</ci><apply id="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.3.cmml" xref="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.3"><divide id="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.3.1.cmml" xref="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.3.1"></divide><ci id="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.3.2.cmml" xref="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.3.2">𝐼</ci><cn id="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="A3.E45.m2.2.2.1.1.1.1.1.1.1.1.3.3">4</cn></apply></apply><ci id="A3.E45.m2.2.2.1.1.1.1.1.4.cmml" xref="A3.E45.m2.2.2.1.1.1.1.1.4">𝒆</ci></apply></apply><apply id="A3.E45.m2.2.2.1.1.3.3.cmml" xref="A3.E45.m2.2.2.1.1.3.3"><times id="A3.E45.m2.2.2.1.1.3.3.3.cmml" xref="A3.E45.m2.2.2.1.1.3.3.3"></times><ci id="A3.E45.m2.2.2.1.1.3.3.4.cmml" xref="A3.E45.m2.2.2.1.1.3.3.4">𝜅</ci><apply id="A3.E45.m2.2.2.1.1.2.2.1.1.1.cmml" xref="A3.E45.m2.2.2.1.1.2.2.1.1"><plus id="A3.E45.m2.2.2.1.1.2.2.1.1.1.1.cmml" xref="A3.E45.m2.2.2.1.1.2.2.1.1.1.1"></plus><cn id="A3.E45.m2.2.2.1.1.2.2.1.1.1.2.cmml" type="integer" xref="A3.E45.m2.2.2.1.1.2.2.1.1.1.2">1</cn><ci id="A3.E45.m2.2.2.1.1.2.2.1.1.1.3.cmml" xref="A3.E45.m2.2.2.1.1.2.2.1.1.1.3">𝑝</ci></apply><apply id="A3.E45.m2.2.2.1.1.3.3.5.cmml" xref="A3.E45.m2.2.2.1.1.3.3.5"><csymbol cd="ambiguous" id="A3.E45.m2.2.2.1.1.3.3.5.1.cmml" xref="A3.E45.m2.2.2.1.1.3.3.5">superscript</csymbol><apply id="A3.E45.m2.2.2.1.1.3.3.5.2.cmml" xref="A3.E45.m2.2.2.1.1.3.3.5.2"><ci id="A3.E45.m2.2.2.1.1.3.3.5.2.1.cmml" xref="A3.E45.m2.2.2.1.1.3.3.5.2.1">~</ci><ci id="A3.E45.m2.2.2.1.1.3.3.5.2.2.cmml" xref="A3.E45.m2.2.2.1.1.3.3.5.2.2">𝜽</ci></apply><ci id="A3.E45.m2.2.2.1.1.3.3.5.3.cmml" xref="A3.E45.m2.2.2.1.1.3.3.5.3">𝑇</ci></apply><ci id="A3.E45.m2.2.2.1.1.3.3.6.cmml" xref="A3.E45.m2.2.2.1.1.3.3.6">𝑄</ci><interval closure="open" id="A3.E45.m2.2.2.1.1.3.3.2.2.cmml" xref="A3.E45.m2.2.2.1.1.3.3.2.1"><ci id="A3.E45.m2.1.1.cmml" xref="A3.E45.m2.1.1">𝑡</ci><apply id="A3.E45.m2.2.2.1.1.3.3.2.1.1.cmml" xref="A3.E45.m2.2.2.1.1.3.3.2.1.1"><csymbol cd="ambiguous" id="A3.E45.m2.2.2.1.1.3.3.2.1.1.1.cmml" xref="A3.E45.m2.2.2.1.1.3.3.2.1.1">subscript</csymbol><ci id="A3.E45.m2.2.2.1.1.3.3.2.1.1.2.cmml" xref="A3.E45.m2.2.2.1.1.3.3.2.1.1.2">𝑡</ci><ci id="A3.E45.m2.2.2.1.1.3.3.2.1.1.3.cmml" xref="A3.E45.m2.2.2.1.1.3.3.2.1.1.3">e</ci></apply></interval><apply id="A3.E45.m2.2.2.1.1.3.3.7.cmml" xref="A3.E45.m2.2.2.1.1.3.3.7"><ci id="A3.E45.m2.2.2.1.1.3.3.7.1.cmml" xref="A3.E45.m2.2.2.1.1.3.3.7.1">~</ci><ci id="A3.E45.m2.2.2.1.1.3.3.7.2.cmml" xref="A3.E45.m2.2.2.1.1.3.3.7.2">𝜽</ci></apply></apply></apply><apply id="A3.E45.m2.2.2.1.1.4.cmml" xref="A3.E45.m2.2.2.1.1.4"><times id="A3.E45.m2.2.2.1.1.4.2.cmml" xref="A3.E45.m2.2.2.1.1.4.2"></times><apply id="A3.E45.m2.2.2.1.1.4.3.cmml" xref="A3.E45.m2.2.2.1.1.4.3"><csymbol cd="ambiguous" id="A3.E45.m2.2.2.1.1.4.3.1.cmml" xref="A3.E45.m2.2.2.1.1.4.3">superscript</csymbol><ci id="A3.E45.m2.2.2.1.1.4.3.2.cmml" xref="A3.E45.m2.2.2.1.1.4.3.2">𝒆</ci><ci id="A3.E45.m2.2.2.1.1.4.3.3.cmml" xref="A3.E45.m2.2.2.1.1.4.3.3">𝑇</ci></apply><apply id="A3.E45.m2.2.2.1.1.4.1.1.1.cmml" xref="A3.E45.m2.2.2.1.1.4.1.1"><minus id="A3.E45.m2.2.2.1.1.4.1.1.1.1.cmml" xref="A3.E45.m2.2.2.1.1.4.1.1.1.1"></minus><ci id="A3.E45.m2.2.2.1.1.4.1.1.1.2.cmml" xref="A3.E45.m2.2.2.1.1.4.1.1.1.2">Φ</ci><apply id="A3.E45.m2.2.2.1.1.4.1.1.1.3.cmml" xref="A3.E45.m2.2.2.1.1.4.1.1.1.3"><csymbol cd="ambiguous" id="A3.E45.m2.2.2.1.1.4.1.1.1.3.1.cmml" xref="A3.E45.m2.2.2.1.1.4.1.1.1.3">subscript</csymbol><ci id="A3.E45.m2.2.2.1.1.4.1.1.1.3.2.cmml" xref="A3.E45.m2.2.2.1.1.4.1.1.1.3.2">Φ</ci><ci id="A3.E45.m2.2.2.1.1.4.1.1.1.3.3.cmml" xref="A3.E45.m2.2.2.1.1.4.1.1.1.3.3">f</ci></apply></apply><apply id="A3.E45.m2.2.2.1.1.4.4.cmml" xref="A3.E45.m2.2.2.1.1.4.4"><ci id="A3.E45.m2.2.2.1.1.4.4.1.cmml" xref="A3.E45.m2.2.2.1.1.4.4.1">~</ci><ci id="A3.E45.m2.2.2.1.1.4.4.2.cmml" xref="A3.E45.m2.2.2.1.1.4.4.2">𝜽</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E45.m2.2c">\displaystyle-\bm{e}^{T}(K-I/4)\bm{e}-\kappa(1+p)\tilde{\bm{\theta}}^{T}Q(t,t_% {\rm e}){\tilde{\bm{\theta}}}+\bm{e}^{T}(\Phi-\Phi_{\rm f})\tilde{\bm{\theta}}.</annotation><annotation encoding="application/x-llamapun" id="A3.E45.m2.2d">- bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_K - italic_I / 4 ) bold_italic_e - italic_κ ( 1 + italic_p ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT italic_Q ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG + bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( roman_Φ - roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(45)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.1.p1.25">From Lemma 2, one gets <math alttext="\|\Phi-\Phi_{\rm f}\|\leq\delta" class="ltx_Math" display="inline" id="A3.1.p1.15.m1.1"><semantics id="A3.1.p1.15.m1.1a"><mrow id="A3.1.p1.15.m1.1.1" xref="A3.1.p1.15.m1.1.1.cmml"><mrow id="A3.1.p1.15.m1.1.1.1.1" xref="A3.1.p1.15.m1.1.1.1.2.cmml"><mo id="A3.1.p1.15.m1.1.1.1.1.2" stretchy="false" xref="A3.1.p1.15.m1.1.1.1.2.1.cmml">‖</mo><mrow id="A3.1.p1.15.m1.1.1.1.1.1" xref="A3.1.p1.15.m1.1.1.1.1.1.cmml"><mi id="A3.1.p1.15.m1.1.1.1.1.1.2" mathvariant="normal" xref="A3.1.p1.15.m1.1.1.1.1.1.2.cmml">Φ</mi><mo id="A3.1.p1.15.m1.1.1.1.1.1.1" xref="A3.1.p1.15.m1.1.1.1.1.1.1.cmml">−</mo><msub id="A3.1.p1.15.m1.1.1.1.1.1.3" xref="A3.1.p1.15.m1.1.1.1.1.1.3.cmml"><mi id="A3.1.p1.15.m1.1.1.1.1.1.3.2" mathvariant="normal" xref="A3.1.p1.15.m1.1.1.1.1.1.3.2.cmml">Φ</mi><mi id="A3.1.p1.15.m1.1.1.1.1.1.3.3" mathvariant="normal" xref="A3.1.p1.15.m1.1.1.1.1.1.3.3.cmml">f</mi></msub></mrow><mo id="A3.1.p1.15.m1.1.1.1.1.3" stretchy="false" xref="A3.1.p1.15.m1.1.1.1.2.1.cmml">‖</mo></mrow><mo id="A3.1.p1.15.m1.1.1.2" xref="A3.1.p1.15.m1.1.1.2.cmml">≤</mo><mi id="A3.1.p1.15.m1.1.1.3" xref="A3.1.p1.15.m1.1.1.3.cmml">δ</mi></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.15.m1.1b"><apply id="A3.1.p1.15.m1.1.1.cmml" xref="A3.1.p1.15.m1.1.1"><leq id="A3.1.p1.15.m1.1.1.2.cmml" xref="A3.1.p1.15.m1.1.1.2"></leq><apply id="A3.1.p1.15.m1.1.1.1.2.cmml" xref="A3.1.p1.15.m1.1.1.1.1"><csymbol cd="latexml" id="A3.1.p1.15.m1.1.1.1.2.1.cmml" xref="A3.1.p1.15.m1.1.1.1.1.2">norm</csymbol><apply id="A3.1.p1.15.m1.1.1.1.1.1.cmml" xref="A3.1.p1.15.m1.1.1.1.1.1"><minus id="A3.1.p1.15.m1.1.1.1.1.1.1.cmml" xref="A3.1.p1.15.m1.1.1.1.1.1.1"></minus><ci id="A3.1.p1.15.m1.1.1.1.1.1.2.cmml" xref="A3.1.p1.15.m1.1.1.1.1.1.2">Φ</ci><apply id="A3.1.p1.15.m1.1.1.1.1.1.3.cmml" xref="A3.1.p1.15.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A3.1.p1.15.m1.1.1.1.1.1.3.1.cmml" xref="A3.1.p1.15.m1.1.1.1.1.1.3">subscript</csymbol><ci id="A3.1.p1.15.m1.1.1.1.1.1.3.2.cmml" xref="A3.1.p1.15.m1.1.1.1.1.1.3.2">Φ</ci><ci id="A3.1.p1.15.m1.1.1.1.1.1.3.3.cmml" xref="A3.1.p1.15.m1.1.1.1.1.1.3.3">f</ci></apply></apply></apply><ci id="A3.1.p1.15.m1.1.1.3.cmml" xref="A3.1.p1.15.m1.1.1.3">𝛿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.15.m1.1c">\|\Phi-\Phi_{\rm f}\|\leq\delta</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.15.m1.1d">∥ roman_Φ - roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ∥ ≤ italic_δ</annotation></semantics></math>, <math alttext="\forall t\in[0,t_{\rm f})" class="ltx_Math" display="inline" id="A3.1.p1.16.m2.2"><semantics id="A3.1.p1.16.m2.2a"><mrow id="A3.1.p1.16.m2.2.2" xref="A3.1.p1.16.m2.2.2.cmml"><mrow id="A3.1.p1.16.m2.2.2.3" xref="A3.1.p1.16.m2.2.2.3.cmml"><mo id="A3.1.p1.16.m2.2.2.3.1" rspace="0.167em" xref="A3.1.p1.16.m2.2.2.3.1.cmml">∀</mo><mi id="A3.1.p1.16.m2.2.2.3.2" xref="A3.1.p1.16.m2.2.2.3.2.cmml">t</mi></mrow><mo id="A3.1.p1.16.m2.2.2.2" xref="A3.1.p1.16.m2.2.2.2.cmml">∈</mo><mrow id="A3.1.p1.16.m2.2.2.1.1" xref="A3.1.p1.16.m2.2.2.1.2.cmml"><mo id="A3.1.p1.16.m2.2.2.1.1.2" stretchy="false" xref="A3.1.p1.16.m2.2.2.1.2.cmml">[</mo><mn id="A3.1.p1.16.m2.1.1" xref="A3.1.p1.16.m2.1.1.cmml">0</mn><mo id="A3.1.p1.16.m2.2.2.1.1.3" xref="A3.1.p1.16.m2.2.2.1.2.cmml">,</mo><msub id="A3.1.p1.16.m2.2.2.1.1.1" xref="A3.1.p1.16.m2.2.2.1.1.1.cmml"><mi id="A3.1.p1.16.m2.2.2.1.1.1.2" xref="A3.1.p1.16.m2.2.2.1.1.1.2.cmml">t</mi><mi id="A3.1.p1.16.m2.2.2.1.1.1.3" mathvariant="normal" xref="A3.1.p1.16.m2.2.2.1.1.1.3.cmml">f</mi></msub><mo id="A3.1.p1.16.m2.2.2.1.1.4" stretchy="false" xref="A3.1.p1.16.m2.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.16.m2.2b"><apply id="A3.1.p1.16.m2.2.2.cmml" xref="A3.1.p1.16.m2.2.2"><in id="A3.1.p1.16.m2.2.2.2.cmml" xref="A3.1.p1.16.m2.2.2.2"></in><apply id="A3.1.p1.16.m2.2.2.3.cmml" xref="A3.1.p1.16.m2.2.2.3"><csymbol cd="latexml" id="A3.1.p1.16.m2.2.2.3.1.cmml" xref="A3.1.p1.16.m2.2.2.3.1">for-all</csymbol><ci id="A3.1.p1.16.m2.2.2.3.2.cmml" xref="A3.1.p1.16.m2.2.2.3.2">𝑡</ci></apply><interval closure="closed-open" id="A3.1.p1.16.m2.2.2.1.2.cmml" xref="A3.1.p1.16.m2.2.2.1.1"><cn id="A3.1.p1.16.m2.1.1.cmml" type="integer" xref="A3.1.p1.16.m2.1.1">0</cn><apply id="A3.1.p1.16.m2.2.2.1.1.1.cmml" xref="A3.1.p1.16.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="A3.1.p1.16.m2.2.2.1.1.1.1.cmml" xref="A3.1.p1.16.m2.2.2.1.1.1">subscript</csymbol><ci id="A3.1.p1.16.m2.2.2.1.1.1.2.cmml" xref="A3.1.p1.16.m2.2.2.1.1.1.2">𝑡</ci><ci id="A3.1.p1.16.m2.2.2.1.1.1.3.cmml" xref="A3.1.p1.16.m2.2.2.1.1.1.3">f</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.16.m2.2c">\forall t\in[0,t_{\rm f})</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.16.m2.2d">∀ italic_t ∈ [ 0 , italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT )</annotation></semantics></math>, in which <math alttext="t_{\rm f}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A3.1.p1.17.m3.1"><semantics id="A3.1.p1.17.m3.1a"><mrow id="A3.1.p1.17.m3.1.1" xref="A3.1.p1.17.m3.1.1.cmml"><msub id="A3.1.p1.17.m3.1.1.2" xref="A3.1.p1.17.m3.1.1.2.cmml"><mi id="A3.1.p1.17.m3.1.1.2.2" xref="A3.1.p1.17.m3.1.1.2.2.cmml">t</mi><mi id="A3.1.p1.17.m3.1.1.2.3" mathvariant="normal" xref="A3.1.p1.17.m3.1.1.2.3.cmml">f</mi></msub><mo id="A3.1.p1.17.m3.1.1.1" xref="A3.1.p1.17.m3.1.1.1.cmml">∈</mo><msup id="A3.1.p1.17.m3.1.1.3" xref="A3.1.p1.17.m3.1.1.3.cmml"><mi id="A3.1.p1.17.m3.1.1.3.2" xref="A3.1.p1.17.m3.1.1.3.2.cmml">ℝ</mi><mo id="A3.1.p1.17.m3.1.1.3.3" xref="A3.1.p1.17.m3.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.17.m3.1b"><apply id="A3.1.p1.17.m3.1.1.cmml" xref="A3.1.p1.17.m3.1.1"><in id="A3.1.p1.17.m3.1.1.1.cmml" xref="A3.1.p1.17.m3.1.1.1"></in><apply id="A3.1.p1.17.m3.1.1.2.cmml" xref="A3.1.p1.17.m3.1.1.2"><csymbol cd="ambiguous" id="A3.1.p1.17.m3.1.1.2.1.cmml" xref="A3.1.p1.17.m3.1.1.2">subscript</csymbol><ci id="A3.1.p1.17.m3.1.1.2.2.cmml" xref="A3.1.p1.17.m3.1.1.2.2">𝑡</ci><ci id="A3.1.p1.17.m3.1.1.2.3.cmml" xref="A3.1.p1.17.m3.1.1.2.3">f</ci></apply><apply id="A3.1.p1.17.m3.1.1.3.cmml" xref="A3.1.p1.17.m3.1.1.3"><csymbol cd="ambiguous" id="A3.1.p1.17.m3.1.1.3.1.cmml" xref="A3.1.p1.17.m3.1.1.3">superscript</csymbol><ci id="A3.1.p1.17.m3.1.1.3.2.cmml" xref="A3.1.p1.17.m3.1.1.3.2">ℝ</ci><plus id="A3.1.p1.17.m3.1.1.3.3.cmml" xref="A3.1.p1.17.m3.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.17.m3.1c">t_{\rm f}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.17.m3.1d">italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> is the moment that <math alttext="\bm{x}(t)" class="ltx_Math" display="inline" id="A3.1.p1.18.m4.1"><semantics id="A3.1.p1.18.m4.1a"><mrow id="A3.1.p1.18.m4.1.2" xref="A3.1.p1.18.m4.1.2.cmml"><mi id="A3.1.p1.18.m4.1.2.2" xref="A3.1.p1.18.m4.1.2.2.cmml">𝒙</mi><mo id="A3.1.p1.18.m4.1.2.1" xref="A3.1.p1.18.m4.1.2.1.cmml">⁢</mo><mrow id="A3.1.p1.18.m4.1.2.3.2" xref="A3.1.p1.18.m4.1.2.cmml"><mo id="A3.1.p1.18.m4.1.2.3.2.1" stretchy="false" xref="A3.1.p1.18.m4.1.2.cmml">(</mo><mi id="A3.1.p1.18.m4.1.1" xref="A3.1.p1.18.m4.1.1.cmml">t</mi><mo id="A3.1.p1.18.m4.1.2.3.2.2" stretchy="false" xref="A3.1.p1.18.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.18.m4.1b"><apply id="A3.1.p1.18.m4.1.2.cmml" xref="A3.1.p1.18.m4.1.2"><times id="A3.1.p1.18.m4.1.2.1.cmml" xref="A3.1.p1.18.m4.1.2.1"></times><ci id="A3.1.p1.18.m4.1.2.2.cmml" xref="A3.1.p1.18.m4.1.2.2">𝒙</ci><ci id="A3.1.p1.18.m4.1.1.cmml" xref="A3.1.p1.18.m4.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.18.m4.1c">\bm{x}(t)</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.18.m4.1d">bold_italic_x ( italic_t )</annotation></semantics></math> leaves <math alttext="\Omega_{{\rm c}_{x}}" class="ltx_Math" display="inline" id="A3.1.p1.19.m5.1"><semantics id="A3.1.p1.19.m5.1a"><msub id="A3.1.p1.19.m5.1.1" xref="A3.1.p1.19.m5.1.1.cmml"><mi id="A3.1.p1.19.m5.1.1.2" mathvariant="normal" xref="A3.1.p1.19.m5.1.1.2.cmml">Ω</mi><msub id="A3.1.p1.19.m5.1.1.3" xref="A3.1.p1.19.m5.1.1.3.cmml"><mi id="A3.1.p1.19.m5.1.1.3.2" mathvariant="normal" xref="A3.1.p1.19.m5.1.1.3.2.cmml">c</mi><mi id="A3.1.p1.19.m5.1.1.3.3" xref="A3.1.p1.19.m5.1.1.3.3.cmml">x</mi></msub></msub><annotation-xml encoding="MathML-Content" id="A3.1.p1.19.m5.1b"><apply id="A3.1.p1.19.m5.1.1.cmml" xref="A3.1.p1.19.m5.1.1"><csymbol cd="ambiguous" id="A3.1.p1.19.m5.1.1.1.cmml" xref="A3.1.p1.19.m5.1.1">subscript</csymbol><ci id="A3.1.p1.19.m5.1.1.2.cmml" xref="A3.1.p1.19.m5.1.1.2">Ω</ci><apply id="A3.1.p1.19.m5.1.1.3.cmml" xref="A3.1.p1.19.m5.1.1.3"><csymbol cd="ambiguous" id="A3.1.p1.19.m5.1.1.3.1.cmml" xref="A3.1.p1.19.m5.1.1.3">subscript</csymbol><ci id="A3.1.p1.19.m5.1.1.3.2.cmml" xref="A3.1.p1.19.m5.1.1.3.2">c</ci><ci id="A3.1.p1.19.m5.1.1.3.3.cmml" xref="A3.1.p1.19.m5.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.19.m5.1c">\Omega_{{\rm c}_{x}}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.19.m5.1d">roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> for the first time. According to Theorem 1, one gets <math alttext="\tilde{\bm{\theta}}(t)\in L_{\infty}" class="ltx_Math" display="inline" id="A3.1.p1.20.m6.1"><semantics id="A3.1.p1.20.m6.1a"><mrow id="A3.1.p1.20.m6.1.2" xref="A3.1.p1.20.m6.1.2.cmml"><mrow id="A3.1.p1.20.m6.1.2.2" xref="A3.1.p1.20.m6.1.2.2.cmml"><mover accent="true" id="A3.1.p1.20.m6.1.2.2.2" xref="A3.1.p1.20.m6.1.2.2.2.cmml"><mi id="A3.1.p1.20.m6.1.2.2.2.2" xref="A3.1.p1.20.m6.1.2.2.2.2.cmml">𝜽</mi><mo id="A3.1.p1.20.m6.1.2.2.2.1" xref="A3.1.p1.20.m6.1.2.2.2.1.cmml">~</mo></mover><mo id="A3.1.p1.20.m6.1.2.2.1" xref="A3.1.p1.20.m6.1.2.2.1.cmml">⁢</mo><mrow id="A3.1.p1.20.m6.1.2.2.3.2" xref="A3.1.p1.20.m6.1.2.2.cmml"><mo id="A3.1.p1.20.m6.1.2.2.3.2.1" stretchy="false" xref="A3.1.p1.20.m6.1.2.2.cmml">(</mo><mi id="A3.1.p1.20.m6.1.1" xref="A3.1.p1.20.m6.1.1.cmml">t</mi><mo id="A3.1.p1.20.m6.1.2.2.3.2.2" stretchy="false" xref="A3.1.p1.20.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="A3.1.p1.20.m6.1.2.1" xref="A3.1.p1.20.m6.1.2.1.cmml">∈</mo><msub id="A3.1.p1.20.m6.1.2.3" xref="A3.1.p1.20.m6.1.2.3.cmml"><mi id="A3.1.p1.20.m6.1.2.3.2" xref="A3.1.p1.20.m6.1.2.3.2.cmml">L</mi><mi id="A3.1.p1.20.m6.1.2.3.3" mathvariant="normal" xref="A3.1.p1.20.m6.1.2.3.3.cmml">∞</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.20.m6.1b"><apply id="A3.1.p1.20.m6.1.2.cmml" xref="A3.1.p1.20.m6.1.2"><in id="A3.1.p1.20.m6.1.2.1.cmml" xref="A3.1.p1.20.m6.1.2.1"></in><apply id="A3.1.p1.20.m6.1.2.2.cmml" xref="A3.1.p1.20.m6.1.2.2"><times id="A3.1.p1.20.m6.1.2.2.1.cmml" xref="A3.1.p1.20.m6.1.2.2.1"></times><apply id="A3.1.p1.20.m6.1.2.2.2.cmml" xref="A3.1.p1.20.m6.1.2.2.2"><ci id="A3.1.p1.20.m6.1.2.2.2.1.cmml" xref="A3.1.p1.20.m6.1.2.2.2.1">~</ci><ci id="A3.1.p1.20.m6.1.2.2.2.2.cmml" xref="A3.1.p1.20.m6.1.2.2.2.2">𝜽</ci></apply><ci id="A3.1.p1.20.m6.1.1.cmml" xref="A3.1.p1.20.m6.1.1">𝑡</ci></apply><apply id="A3.1.p1.20.m6.1.2.3.cmml" xref="A3.1.p1.20.m6.1.2.3"><csymbol cd="ambiguous" id="A3.1.p1.20.m6.1.2.3.1.cmml" xref="A3.1.p1.20.m6.1.2.3">subscript</csymbol><ci id="A3.1.p1.20.m6.1.2.3.2.cmml" xref="A3.1.p1.20.m6.1.2.3.2">𝐿</ci><infinity id="A3.1.p1.20.m6.1.2.3.3.cmml" xref="A3.1.p1.20.m6.1.2.3.3"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.20.m6.1c">\tilde{\bm{\theta}}(t)\in L_{\infty}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.20.m6.1d">over~ start_ARG bold_italic_θ end_ARG ( italic_t ) ∈ italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\forall t\in[0,\infty)" class="ltx_Math" display="inline" id="A3.1.p1.21.m7.2"><semantics id="A3.1.p1.21.m7.2a"><mrow id="A3.1.p1.21.m7.2.3" xref="A3.1.p1.21.m7.2.3.cmml"><mrow id="A3.1.p1.21.m7.2.3.2" xref="A3.1.p1.21.m7.2.3.2.cmml"><mo id="A3.1.p1.21.m7.2.3.2.1" rspace="0.167em" xref="A3.1.p1.21.m7.2.3.2.1.cmml">∀</mo><mi id="A3.1.p1.21.m7.2.3.2.2" xref="A3.1.p1.21.m7.2.3.2.2.cmml">t</mi></mrow><mo id="A3.1.p1.21.m7.2.3.1" xref="A3.1.p1.21.m7.2.3.1.cmml">∈</mo><mrow id="A3.1.p1.21.m7.2.3.3.2" xref="A3.1.p1.21.m7.2.3.3.1.cmml"><mo id="A3.1.p1.21.m7.2.3.3.2.1" stretchy="false" xref="A3.1.p1.21.m7.2.3.3.1.cmml">[</mo><mn id="A3.1.p1.21.m7.1.1" xref="A3.1.p1.21.m7.1.1.cmml">0</mn><mo id="A3.1.p1.21.m7.2.3.3.2.2" xref="A3.1.p1.21.m7.2.3.3.1.cmml">,</mo><mi id="A3.1.p1.21.m7.2.2" mathvariant="normal" xref="A3.1.p1.21.m7.2.2.cmml">∞</mi><mo id="A3.1.p1.21.m7.2.3.3.2.3" stretchy="false" xref="A3.1.p1.21.m7.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.21.m7.2b"><apply id="A3.1.p1.21.m7.2.3.cmml" xref="A3.1.p1.21.m7.2.3"><in id="A3.1.p1.21.m7.2.3.1.cmml" xref="A3.1.p1.21.m7.2.3.1"></in><apply id="A3.1.p1.21.m7.2.3.2.cmml" xref="A3.1.p1.21.m7.2.3.2"><csymbol cd="latexml" id="A3.1.p1.21.m7.2.3.2.1.cmml" xref="A3.1.p1.21.m7.2.3.2.1">for-all</csymbol><ci id="A3.1.p1.21.m7.2.3.2.2.cmml" xref="A3.1.p1.21.m7.2.3.2.2">𝑡</ci></apply><interval closure="closed-open" id="A3.1.p1.21.m7.2.3.3.1.cmml" xref="A3.1.p1.21.m7.2.3.3.2"><cn id="A3.1.p1.21.m7.1.1.cmml" type="integer" xref="A3.1.p1.21.m7.1.1">0</cn><infinity id="A3.1.p1.21.m7.2.2.cmml" xref="A3.1.p1.21.m7.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.21.m7.2c">\forall t\in[0,\infty)</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.21.m7.2d">∀ italic_t ∈ [ 0 , ∞ )</annotation></semantics></math>, such that there exists a constant <math alttext="c_{\theta}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A3.1.p1.22.m8.1"><semantics id="A3.1.p1.22.m8.1a"><mrow id="A3.1.p1.22.m8.1.1" xref="A3.1.p1.22.m8.1.1.cmml"><msub id="A3.1.p1.22.m8.1.1.2" xref="A3.1.p1.22.m8.1.1.2.cmml"><mi id="A3.1.p1.22.m8.1.1.2.2" xref="A3.1.p1.22.m8.1.1.2.2.cmml">c</mi><mi id="A3.1.p1.22.m8.1.1.2.3" xref="A3.1.p1.22.m8.1.1.2.3.cmml">θ</mi></msub><mo id="A3.1.p1.22.m8.1.1.1" xref="A3.1.p1.22.m8.1.1.1.cmml">∈</mo><msup id="A3.1.p1.22.m8.1.1.3" xref="A3.1.p1.22.m8.1.1.3.cmml"><mi id="A3.1.p1.22.m8.1.1.3.2" xref="A3.1.p1.22.m8.1.1.3.2.cmml">ℝ</mi><mo id="A3.1.p1.22.m8.1.1.3.3" xref="A3.1.p1.22.m8.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.22.m8.1b"><apply id="A3.1.p1.22.m8.1.1.cmml" xref="A3.1.p1.22.m8.1.1"><in id="A3.1.p1.22.m8.1.1.1.cmml" xref="A3.1.p1.22.m8.1.1.1"></in><apply id="A3.1.p1.22.m8.1.1.2.cmml" xref="A3.1.p1.22.m8.1.1.2"><csymbol cd="ambiguous" id="A3.1.p1.22.m8.1.1.2.1.cmml" xref="A3.1.p1.22.m8.1.1.2">subscript</csymbol><ci id="A3.1.p1.22.m8.1.1.2.2.cmml" xref="A3.1.p1.22.m8.1.1.2.2">𝑐</ci><ci id="A3.1.p1.22.m8.1.1.2.3.cmml" xref="A3.1.p1.22.m8.1.1.2.3">𝜃</ci></apply><apply id="A3.1.p1.22.m8.1.1.3.cmml" xref="A3.1.p1.22.m8.1.1.3"><csymbol cd="ambiguous" id="A3.1.p1.22.m8.1.1.3.1.cmml" xref="A3.1.p1.22.m8.1.1.3">superscript</csymbol><ci id="A3.1.p1.22.m8.1.1.3.2.cmml" xref="A3.1.p1.22.m8.1.1.3.2">ℝ</ci><plus id="A3.1.p1.22.m8.1.1.3.3.cmml" xref="A3.1.p1.22.m8.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.22.m8.1c">c_{\theta}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.22.m8.1d">italic_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> that satisfies <math alttext="\|\tilde{\bm{\theta}}(t)\|\leq c_{\theta}" class="ltx_Math" display="inline" id="A3.1.p1.23.m9.2"><semantics id="A3.1.p1.23.m9.2a"><mrow id="A3.1.p1.23.m9.2.2" xref="A3.1.p1.23.m9.2.2.cmml"><mrow id="A3.1.p1.23.m9.2.2.1.1" xref="A3.1.p1.23.m9.2.2.1.2.cmml"><mo id="A3.1.p1.23.m9.2.2.1.1.2" stretchy="false" xref="A3.1.p1.23.m9.2.2.1.2.1.cmml">‖</mo><mrow id="A3.1.p1.23.m9.2.2.1.1.1" xref="A3.1.p1.23.m9.2.2.1.1.1.cmml"><mover accent="true" id="A3.1.p1.23.m9.2.2.1.1.1.2" xref="A3.1.p1.23.m9.2.2.1.1.1.2.cmml"><mi id="A3.1.p1.23.m9.2.2.1.1.1.2.2" xref="A3.1.p1.23.m9.2.2.1.1.1.2.2.cmml">𝜽</mi><mo id="A3.1.p1.23.m9.2.2.1.1.1.2.1" xref="A3.1.p1.23.m9.2.2.1.1.1.2.1.cmml">~</mo></mover><mo id="A3.1.p1.23.m9.2.2.1.1.1.1" xref="A3.1.p1.23.m9.2.2.1.1.1.1.cmml">⁢</mo><mrow id="A3.1.p1.23.m9.2.2.1.1.1.3.2" xref="A3.1.p1.23.m9.2.2.1.1.1.cmml"><mo id="A3.1.p1.23.m9.2.2.1.1.1.3.2.1" stretchy="false" xref="A3.1.p1.23.m9.2.2.1.1.1.cmml">(</mo><mi id="A3.1.p1.23.m9.1.1" xref="A3.1.p1.23.m9.1.1.cmml">t</mi><mo id="A3.1.p1.23.m9.2.2.1.1.1.3.2.2" stretchy="false" xref="A3.1.p1.23.m9.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="A3.1.p1.23.m9.2.2.1.1.3" stretchy="false" xref="A3.1.p1.23.m9.2.2.1.2.1.cmml">‖</mo></mrow><mo id="A3.1.p1.23.m9.2.2.2" xref="A3.1.p1.23.m9.2.2.2.cmml">≤</mo><msub id="A3.1.p1.23.m9.2.2.3" xref="A3.1.p1.23.m9.2.2.3.cmml"><mi id="A3.1.p1.23.m9.2.2.3.2" xref="A3.1.p1.23.m9.2.2.3.2.cmml">c</mi><mi id="A3.1.p1.23.m9.2.2.3.3" xref="A3.1.p1.23.m9.2.2.3.3.cmml">θ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.23.m9.2b"><apply id="A3.1.p1.23.m9.2.2.cmml" xref="A3.1.p1.23.m9.2.2"><leq id="A3.1.p1.23.m9.2.2.2.cmml" xref="A3.1.p1.23.m9.2.2.2"></leq><apply id="A3.1.p1.23.m9.2.2.1.2.cmml" xref="A3.1.p1.23.m9.2.2.1.1"><csymbol cd="latexml" id="A3.1.p1.23.m9.2.2.1.2.1.cmml" xref="A3.1.p1.23.m9.2.2.1.1.2">norm</csymbol><apply id="A3.1.p1.23.m9.2.2.1.1.1.cmml" xref="A3.1.p1.23.m9.2.2.1.1.1"><times id="A3.1.p1.23.m9.2.2.1.1.1.1.cmml" xref="A3.1.p1.23.m9.2.2.1.1.1.1"></times><apply id="A3.1.p1.23.m9.2.2.1.1.1.2.cmml" xref="A3.1.p1.23.m9.2.2.1.1.1.2"><ci id="A3.1.p1.23.m9.2.2.1.1.1.2.1.cmml" xref="A3.1.p1.23.m9.2.2.1.1.1.2.1">~</ci><ci id="A3.1.p1.23.m9.2.2.1.1.1.2.2.cmml" xref="A3.1.p1.23.m9.2.2.1.1.1.2.2">𝜽</ci></apply><ci id="A3.1.p1.23.m9.1.1.cmml" xref="A3.1.p1.23.m9.1.1">𝑡</ci></apply></apply><apply id="A3.1.p1.23.m9.2.2.3.cmml" xref="A3.1.p1.23.m9.2.2.3"><csymbol cd="ambiguous" id="A3.1.p1.23.m9.2.2.3.1.cmml" xref="A3.1.p1.23.m9.2.2.3">subscript</csymbol><ci id="A3.1.p1.23.m9.2.2.3.2.cmml" xref="A3.1.p1.23.m9.2.2.3.2">𝑐</ci><ci id="A3.1.p1.23.m9.2.2.3.3.cmml" xref="A3.1.p1.23.m9.2.2.3.3">𝜃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.23.m9.2c">\|\tilde{\bm{\theta}}(t)\|\leq c_{\theta}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.23.m9.2d">∥ over~ start_ARG bold_italic_θ end_ARG ( italic_t ) ∥ ≤ italic_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\forall t\in[0,\infty)" class="ltx_Math" display="inline" id="A3.1.p1.24.m10.2"><semantics id="A3.1.p1.24.m10.2a"><mrow id="A3.1.p1.24.m10.2.3" xref="A3.1.p1.24.m10.2.3.cmml"><mrow id="A3.1.p1.24.m10.2.3.2" xref="A3.1.p1.24.m10.2.3.2.cmml"><mo id="A3.1.p1.24.m10.2.3.2.1" rspace="0.167em" xref="A3.1.p1.24.m10.2.3.2.1.cmml">∀</mo><mi id="A3.1.p1.24.m10.2.3.2.2" xref="A3.1.p1.24.m10.2.3.2.2.cmml">t</mi></mrow><mo id="A3.1.p1.24.m10.2.3.1" xref="A3.1.p1.24.m10.2.3.1.cmml">∈</mo><mrow id="A3.1.p1.24.m10.2.3.3.2" xref="A3.1.p1.24.m10.2.3.3.1.cmml"><mo id="A3.1.p1.24.m10.2.3.3.2.1" stretchy="false" xref="A3.1.p1.24.m10.2.3.3.1.cmml">[</mo><mn id="A3.1.p1.24.m10.1.1" xref="A3.1.p1.24.m10.1.1.cmml">0</mn><mo id="A3.1.p1.24.m10.2.3.3.2.2" xref="A3.1.p1.24.m10.2.3.3.1.cmml">,</mo><mi id="A3.1.p1.24.m10.2.2" mathvariant="normal" xref="A3.1.p1.24.m10.2.2.cmml">∞</mi><mo id="A3.1.p1.24.m10.2.3.3.2.3" stretchy="false" xref="A3.1.p1.24.m10.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.24.m10.2b"><apply id="A3.1.p1.24.m10.2.3.cmml" xref="A3.1.p1.24.m10.2.3"><in id="A3.1.p1.24.m10.2.3.1.cmml" xref="A3.1.p1.24.m10.2.3.1"></in><apply id="A3.1.p1.24.m10.2.3.2.cmml" xref="A3.1.p1.24.m10.2.3.2"><csymbol cd="latexml" id="A3.1.p1.24.m10.2.3.2.1.cmml" xref="A3.1.p1.24.m10.2.3.2.1">for-all</csymbol><ci id="A3.1.p1.24.m10.2.3.2.2.cmml" xref="A3.1.p1.24.m10.2.3.2.2">𝑡</ci></apply><interval closure="closed-open" id="A3.1.p1.24.m10.2.3.3.1.cmml" xref="A3.1.p1.24.m10.2.3.3.2"><cn id="A3.1.p1.24.m10.1.1.cmml" type="integer" xref="A3.1.p1.24.m10.1.1">0</cn><infinity id="A3.1.p1.24.m10.2.2.cmml" xref="A3.1.p1.24.m10.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.24.m10.2c">\forall t\in[0,\infty)</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.24.m10.2d">∀ italic_t ∈ [ 0 , ∞ )</annotation></semantics></math>. Applying these results to the above inequality and letting <math alttext="k_{{\rm c}i}>1/4" class="ltx_Math" display="inline" id="A3.1.p1.25.m11.1"><semantics id="A3.1.p1.25.m11.1a"><mrow id="A3.1.p1.25.m11.1.1" xref="A3.1.p1.25.m11.1.1.cmml"><msub id="A3.1.p1.25.m11.1.1.2" xref="A3.1.p1.25.m11.1.1.2.cmml"><mi id="A3.1.p1.25.m11.1.1.2.2" xref="A3.1.p1.25.m11.1.1.2.2.cmml">k</mi><mrow id="A3.1.p1.25.m11.1.1.2.3" xref="A3.1.p1.25.m11.1.1.2.3.cmml"><mi id="A3.1.p1.25.m11.1.1.2.3.2" mathvariant="normal" xref="A3.1.p1.25.m11.1.1.2.3.2.cmml">c</mi><mo id="A3.1.p1.25.m11.1.1.2.3.1" xref="A3.1.p1.25.m11.1.1.2.3.1.cmml">⁢</mo><mi id="A3.1.p1.25.m11.1.1.2.3.3" xref="A3.1.p1.25.m11.1.1.2.3.3.cmml">i</mi></mrow></msub><mo id="A3.1.p1.25.m11.1.1.1" xref="A3.1.p1.25.m11.1.1.1.cmml">></mo><mrow id="A3.1.p1.25.m11.1.1.3" xref="A3.1.p1.25.m11.1.1.3.cmml"><mn id="A3.1.p1.25.m11.1.1.3.2" xref="A3.1.p1.25.m11.1.1.3.2.cmml">1</mn><mo id="A3.1.p1.25.m11.1.1.3.1" xref="A3.1.p1.25.m11.1.1.3.1.cmml">/</mo><mn id="A3.1.p1.25.m11.1.1.3.3" xref="A3.1.p1.25.m11.1.1.3.3.cmml">4</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.25.m11.1b"><apply id="A3.1.p1.25.m11.1.1.cmml" xref="A3.1.p1.25.m11.1.1"><gt id="A3.1.p1.25.m11.1.1.1.cmml" xref="A3.1.p1.25.m11.1.1.1"></gt><apply id="A3.1.p1.25.m11.1.1.2.cmml" xref="A3.1.p1.25.m11.1.1.2"><csymbol cd="ambiguous" id="A3.1.p1.25.m11.1.1.2.1.cmml" xref="A3.1.p1.25.m11.1.1.2">subscript</csymbol><ci id="A3.1.p1.25.m11.1.1.2.2.cmml" xref="A3.1.p1.25.m11.1.1.2.2">𝑘</ci><apply id="A3.1.p1.25.m11.1.1.2.3.cmml" xref="A3.1.p1.25.m11.1.1.2.3"><times id="A3.1.p1.25.m11.1.1.2.3.1.cmml" xref="A3.1.p1.25.m11.1.1.2.3.1"></times><ci id="A3.1.p1.25.m11.1.1.2.3.2.cmml" xref="A3.1.p1.25.m11.1.1.2.3.2">c</ci><ci id="A3.1.p1.25.m11.1.1.2.3.3.cmml" xref="A3.1.p1.25.m11.1.1.2.3.3">𝑖</ci></apply></apply><apply id="A3.1.p1.25.m11.1.1.3.cmml" xref="A3.1.p1.25.m11.1.1.3"><divide id="A3.1.p1.25.m11.1.1.3.1.cmml" xref="A3.1.p1.25.m11.1.1.3.1"></divide><cn id="A3.1.p1.25.m11.1.1.3.2.cmml" type="integer" xref="A3.1.p1.25.m11.1.1.3.2">1</cn><cn id="A3.1.p1.25.m11.1.1.3.3.cmml" type="integer" xref="A3.1.p1.25.m11.1.1.3.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.25.m11.1c">k_{{\rm c}i}>1/4</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.25.m11.1d">italic_k start_POSTSUBSCRIPT roman_c italic_i end_POSTSUBSCRIPT > 1 / 4</annotation></semantics></math>, one has</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx59"> <tbody id="A3.Ex44"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}\leq" class="ltx_Math" display="inline" id="A3.Ex44.m1.1"><semantics id="A3.Ex44.m1.1a"><mrow id="A3.Ex44.m1.1.1" xref="A3.Ex44.m1.1.1.cmml"><mover accent="true" id="A3.Ex44.m1.1.1.2" xref="A3.Ex44.m1.1.1.2.cmml"><mi id="A3.Ex44.m1.1.1.2.2" xref="A3.Ex44.m1.1.1.2.2.cmml">V</mi><mo id="A3.Ex44.m1.1.1.2.1" xref="A3.Ex44.m1.1.1.2.1.cmml">˙</mo></mover><mo id="A3.Ex44.m1.1.1.1" xref="A3.Ex44.m1.1.1.1.cmml">≤</mo><mi id="A3.Ex44.m1.1.1.3" xref="A3.Ex44.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex44.m1.1b"><apply id="A3.Ex44.m1.1.1.cmml" xref="A3.Ex44.m1.1.1"><leq id="A3.Ex44.m1.1.1.1.cmml" xref="A3.Ex44.m1.1.1.1"></leq><apply id="A3.Ex44.m1.1.1.2.cmml" xref="A3.Ex44.m1.1.1.2"><ci id="A3.Ex44.m1.1.1.2.1.cmml" xref="A3.Ex44.m1.1.1.2.1">˙</ci><ci id="A3.Ex44.m1.1.1.2.2.cmml" xref="A3.Ex44.m1.1.1.2.2">𝑉</ci></apply><csymbol cd="latexml" id="A3.Ex44.m1.1.1.3.cmml" xref="A3.Ex44.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex44.m1.1c">\displaystyle\dot{V}\leq</annotation><annotation encoding="application/x-llamapun" id="A3.Ex44.m1.1d">over˙ start_ARG italic_V end_ARG ≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-k_{\rm c}\|\bm{e}\|^{2}+\delta c_{\theta}\|\bm{e}\|" class="ltx_Math" display="inline" id="A3.Ex44.m2.2"><semantics id="A3.Ex44.m2.2a"><mrow id="A3.Ex44.m2.2.3" xref="A3.Ex44.m2.2.3.cmml"><mrow id="A3.Ex44.m2.2.3.2" xref="A3.Ex44.m2.2.3.2.cmml"><mo id="A3.Ex44.m2.2.3.2a" xref="A3.Ex44.m2.2.3.2.cmml">−</mo><mrow id="A3.Ex44.m2.2.3.2.2" xref="A3.Ex44.m2.2.3.2.2.cmml"><msub id="A3.Ex44.m2.2.3.2.2.2" xref="A3.Ex44.m2.2.3.2.2.2.cmml"><mi id="A3.Ex44.m2.2.3.2.2.2.2" xref="A3.Ex44.m2.2.3.2.2.2.2.cmml">k</mi><mi id="A3.Ex44.m2.2.3.2.2.2.3" mathvariant="normal" xref="A3.Ex44.m2.2.3.2.2.2.3.cmml">c</mi></msub><mo id="A3.Ex44.m2.2.3.2.2.1" xref="A3.Ex44.m2.2.3.2.2.1.cmml">⁢</mo><msup id="A3.Ex44.m2.2.3.2.2.3" xref="A3.Ex44.m2.2.3.2.2.3.cmml"><mrow id="A3.Ex44.m2.2.3.2.2.3.2.2" xref="A3.Ex44.m2.2.3.2.2.3.2.1.cmml"><mo id="A3.Ex44.m2.2.3.2.2.3.2.2.1" stretchy="false" xref="A3.Ex44.m2.2.3.2.2.3.2.1.1.cmml">‖</mo><mi id="A3.Ex44.m2.1.1" xref="A3.Ex44.m2.1.1.cmml">𝒆</mi><mo id="A3.Ex44.m2.2.3.2.2.3.2.2.2" stretchy="false" xref="A3.Ex44.m2.2.3.2.2.3.2.1.1.cmml">‖</mo></mrow><mn id="A3.Ex44.m2.2.3.2.2.3.3" xref="A3.Ex44.m2.2.3.2.2.3.3.cmml">2</mn></msup></mrow></mrow><mo id="A3.Ex44.m2.2.3.1" xref="A3.Ex44.m2.2.3.1.cmml">+</mo><mrow id="A3.Ex44.m2.2.3.3" xref="A3.Ex44.m2.2.3.3.cmml"><mi id="A3.Ex44.m2.2.3.3.2" xref="A3.Ex44.m2.2.3.3.2.cmml">δ</mi><mo id="A3.Ex44.m2.2.3.3.1" xref="A3.Ex44.m2.2.3.3.1.cmml">⁢</mo><msub id="A3.Ex44.m2.2.3.3.3" xref="A3.Ex44.m2.2.3.3.3.cmml"><mi id="A3.Ex44.m2.2.3.3.3.2" xref="A3.Ex44.m2.2.3.3.3.2.cmml">c</mi><mi id="A3.Ex44.m2.2.3.3.3.3" xref="A3.Ex44.m2.2.3.3.3.3.cmml">θ</mi></msub><mo id="A3.Ex44.m2.2.3.3.1a" xref="A3.Ex44.m2.2.3.3.1.cmml">⁢</mo><mrow id="A3.Ex44.m2.2.3.3.4.2" xref="A3.Ex44.m2.2.3.3.4.1.cmml"><mo id="A3.Ex44.m2.2.3.3.4.2.1" stretchy="false" xref="A3.Ex44.m2.2.3.3.4.1.1.cmml">‖</mo><mi id="A3.Ex44.m2.2.2" xref="A3.Ex44.m2.2.2.cmml">𝒆</mi><mo id="A3.Ex44.m2.2.3.3.4.2.2" stretchy="false" xref="A3.Ex44.m2.2.3.3.4.1.1.cmml">‖</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex44.m2.2b"><apply id="A3.Ex44.m2.2.3.cmml" xref="A3.Ex44.m2.2.3"><plus id="A3.Ex44.m2.2.3.1.cmml" xref="A3.Ex44.m2.2.3.1"></plus><apply id="A3.Ex44.m2.2.3.2.cmml" xref="A3.Ex44.m2.2.3.2"><minus id="A3.Ex44.m2.2.3.2.1.cmml" xref="A3.Ex44.m2.2.3.2"></minus><apply id="A3.Ex44.m2.2.3.2.2.cmml" xref="A3.Ex44.m2.2.3.2.2"><times id="A3.Ex44.m2.2.3.2.2.1.cmml" xref="A3.Ex44.m2.2.3.2.2.1"></times><apply id="A3.Ex44.m2.2.3.2.2.2.cmml" xref="A3.Ex44.m2.2.3.2.2.2"><csymbol cd="ambiguous" id="A3.Ex44.m2.2.3.2.2.2.1.cmml" xref="A3.Ex44.m2.2.3.2.2.2">subscript</csymbol><ci id="A3.Ex44.m2.2.3.2.2.2.2.cmml" xref="A3.Ex44.m2.2.3.2.2.2.2">𝑘</ci><ci id="A3.Ex44.m2.2.3.2.2.2.3.cmml" xref="A3.Ex44.m2.2.3.2.2.2.3">c</ci></apply><apply id="A3.Ex44.m2.2.3.2.2.3.cmml" xref="A3.Ex44.m2.2.3.2.2.3"><csymbol cd="ambiguous" id="A3.Ex44.m2.2.3.2.2.3.1.cmml" xref="A3.Ex44.m2.2.3.2.2.3">superscript</csymbol><apply id="A3.Ex44.m2.2.3.2.2.3.2.1.cmml" xref="A3.Ex44.m2.2.3.2.2.3.2.2"><csymbol cd="latexml" id="A3.Ex44.m2.2.3.2.2.3.2.1.1.cmml" xref="A3.Ex44.m2.2.3.2.2.3.2.2.1">norm</csymbol><ci id="A3.Ex44.m2.1.1.cmml" xref="A3.Ex44.m2.1.1">𝒆</ci></apply><cn id="A3.Ex44.m2.2.3.2.2.3.3.cmml" type="integer" xref="A3.Ex44.m2.2.3.2.2.3.3">2</cn></apply></apply></apply><apply id="A3.Ex44.m2.2.3.3.cmml" xref="A3.Ex44.m2.2.3.3"><times id="A3.Ex44.m2.2.3.3.1.cmml" xref="A3.Ex44.m2.2.3.3.1"></times><ci id="A3.Ex44.m2.2.3.3.2.cmml" xref="A3.Ex44.m2.2.3.3.2">𝛿</ci><apply id="A3.Ex44.m2.2.3.3.3.cmml" xref="A3.Ex44.m2.2.3.3.3"><csymbol cd="ambiguous" id="A3.Ex44.m2.2.3.3.3.1.cmml" xref="A3.Ex44.m2.2.3.3.3">subscript</csymbol><ci id="A3.Ex44.m2.2.3.3.3.2.cmml" xref="A3.Ex44.m2.2.3.3.3.2">𝑐</ci><ci id="A3.Ex44.m2.2.3.3.3.3.cmml" xref="A3.Ex44.m2.2.3.3.3.3">𝜃</ci></apply><apply id="A3.Ex44.m2.2.3.3.4.1.cmml" xref="A3.Ex44.m2.2.3.3.4.2"><csymbol cd="latexml" id="A3.Ex44.m2.2.3.3.4.1.1.cmml" xref="A3.Ex44.m2.2.3.3.4.2.1">norm</csymbol><ci id="A3.Ex44.m2.2.2.cmml" xref="A3.Ex44.m2.2.2">𝒆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex44.m2.2c">\displaystyle-k_{\rm c}\|\bm{e}\|^{2}+\delta c_{\theta}\|\bm{e}\|</annotation><annotation encoding="application/x-llamapun" id="A3.Ex44.m2.2d">- italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ∥ bold_italic_e ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_δ italic_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ∥ bold_italic_e ∥</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.1.p1.31">with <math alttext="k_{\rm c}" class="ltx_Math" display="inline" id="A3.1.p1.26.m1.1"><semantics id="A3.1.p1.26.m1.1a"><msub id="A3.1.p1.26.m1.1.1" xref="A3.1.p1.26.m1.1.1.cmml"><mi id="A3.1.p1.26.m1.1.1.2" xref="A3.1.p1.26.m1.1.1.2.cmml">k</mi><mi id="A3.1.p1.26.m1.1.1.3" mathvariant="normal" xref="A3.1.p1.26.m1.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="A3.1.p1.26.m1.1b"><apply id="A3.1.p1.26.m1.1.1.cmml" xref="A3.1.p1.26.m1.1.1"><csymbol cd="ambiguous" id="A3.1.p1.26.m1.1.1.1.cmml" xref="A3.1.p1.26.m1.1.1">subscript</csymbol><ci id="A3.1.p1.26.m1.1.1.2.cmml" xref="A3.1.p1.26.m1.1.1.2">𝑘</ci><ci id="A3.1.p1.26.m1.1.1.3.cmml" xref="A3.1.p1.26.m1.1.1.3">c</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.26.m1.1c">k_{\rm c}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.26.m1.1d">italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="A3.1.p1.27.m2.1"><semantics id="A3.1.p1.27.m2.1a"><mo id="A3.1.p1.27.m2.1.1" xref="A3.1.p1.27.m2.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="A3.1.p1.27.m2.1b"><csymbol cd="latexml" id="A3.1.p1.27.m2.1.1.cmml" xref="A3.1.p1.27.m2.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.27.m2.1c">:=</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.27.m2.1d">:=</annotation></semantics></math> <math alttext="\min_{1\leq i\leq n}\{k_{{\rm c}i}\}-1/4\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A3.1.p1.28.m3.2"><semantics id="A3.1.p1.28.m3.2a"><mrow id="A3.1.p1.28.m3.2.2" xref="A3.1.p1.28.m3.2.2.cmml"><mrow id="A3.1.p1.28.m3.2.2.2" xref="A3.1.p1.28.m3.2.2.2.cmml"><mrow id="A3.1.p1.28.m3.2.2.2.2.2" xref="A3.1.p1.28.m3.2.2.2.2.3.cmml"><msub id="A3.1.p1.28.m3.1.1.1.1.1.1" xref="A3.1.p1.28.m3.1.1.1.1.1.1.cmml"><mi id="A3.1.p1.28.m3.1.1.1.1.1.1.2" xref="A3.1.p1.28.m3.1.1.1.1.1.1.2.cmml">min</mi><mrow id="A3.1.p1.28.m3.1.1.1.1.1.1.3" xref="A3.1.p1.28.m3.1.1.1.1.1.1.3.cmml"><mn id="A3.1.p1.28.m3.1.1.1.1.1.1.3.2" xref="A3.1.p1.28.m3.1.1.1.1.1.1.3.2.cmml">1</mn><mo id="A3.1.p1.28.m3.1.1.1.1.1.1.3.3" xref="A3.1.p1.28.m3.1.1.1.1.1.1.3.3.cmml">≤</mo><mi id="A3.1.p1.28.m3.1.1.1.1.1.1.3.4" xref="A3.1.p1.28.m3.1.1.1.1.1.1.3.4.cmml">i</mi><mo id="A3.1.p1.28.m3.1.1.1.1.1.1.3.5" xref="A3.1.p1.28.m3.1.1.1.1.1.1.3.5.cmml">≤</mo><mi id="A3.1.p1.28.m3.1.1.1.1.1.1.3.6" xref="A3.1.p1.28.m3.1.1.1.1.1.1.3.6.cmml">n</mi></mrow></msub><mo id="A3.1.p1.28.m3.2.2.2.2.2a" xref="A3.1.p1.28.m3.2.2.2.2.3.cmml">⁡</mo><mrow id="A3.1.p1.28.m3.2.2.2.2.2.2" xref="A3.1.p1.28.m3.2.2.2.2.3.cmml"><mo id="A3.1.p1.28.m3.2.2.2.2.2.2.2" stretchy="false" xref="A3.1.p1.28.m3.2.2.2.2.3.cmml">{</mo><msub id="A3.1.p1.28.m3.2.2.2.2.2.2.1" xref="A3.1.p1.28.m3.2.2.2.2.2.2.1.cmml"><mi id="A3.1.p1.28.m3.2.2.2.2.2.2.1.2" xref="A3.1.p1.28.m3.2.2.2.2.2.2.1.2.cmml">k</mi><mrow id="A3.1.p1.28.m3.2.2.2.2.2.2.1.3" xref="A3.1.p1.28.m3.2.2.2.2.2.2.1.3.cmml"><mi id="A3.1.p1.28.m3.2.2.2.2.2.2.1.3.2" mathvariant="normal" xref="A3.1.p1.28.m3.2.2.2.2.2.2.1.3.2.cmml">c</mi><mo id="A3.1.p1.28.m3.2.2.2.2.2.2.1.3.1" xref="A3.1.p1.28.m3.2.2.2.2.2.2.1.3.1.cmml">⁢</mo><mi id="A3.1.p1.28.m3.2.2.2.2.2.2.1.3.3" xref="A3.1.p1.28.m3.2.2.2.2.2.2.1.3.3.cmml">i</mi></mrow></msub><mo id="A3.1.p1.28.m3.2.2.2.2.2.2.3" stretchy="false" xref="A3.1.p1.28.m3.2.2.2.2.3.cmml">}</mo></mrow></mrow><mo id="A3.1.p1.28.m3.2.2.2.3" xref="A3.1.p1.28.m3.2.2.2.3.cmml">−</mo><mrow id="A3.1.p1.28.m3.2.2.2.4" xref="A3.1.p1.28.m3.2.2.2.4.cmml"><mn id="A3.1.p1.28.m3.2.2.2.4.2" xref="A3.1.p1.28.m3.2.2.2.4.2.cmml">1</mn><mo id="A3.1.p1.28.m3.2.2.2.4.1" xref="A3.1.p1.28.m3.2.2.2.4.1.cmml">/</mo><mn id="A3.1.p1.28.m3.2.2.2.4.3" xref="A3.1.p1.28.m3.2.2.2.4.3.cmml">4</mn></mrow></mrow><mo id="A3.1.p1.28.m3.2.2.3" xref="A3.1.p1.28.m3.2.2.3.cmml">∈</mo><msup id="A3.1.p1.28.m3.2.2.4" xref="A3.1.p1.28.m3.2.2.4.cmml"><mi id="A3.1.p1.28.m3.2.2.4.2" xref="A3.1.p1.28.m3.2.2.4.2.cmml">ℝ</mi><mo id="A3.1.p1.28.m3.2.2.4.3" xref="A3.1.p1.28.m3.2.2.4.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.28.m3.2b"><apply id="A3.1.p1.28.m3.2.2.cmml" xref="A3.1.p1.28.m3.2.2"><in id="A3.1.p1.28.m3.2.2.3.cmml" xref="A3.1.p1.28.m3.2.2.3"></in><apply id="A3.1.p1.28.m3.2.2.2.cmml" xref="A3.1.p1.28.m3.2.2.2"><minus id="A3.1.p1.28.m3.2.2.2.3.cmml" xref="A3.1.p1.28.m3.2.2.2.3"></minus><apply id="A3.1.p1.28.m3.2.2.2.2.3.cmml" xref="A3.1.p1.28.m3.2.2.2.2.2"><apply id="A3.1.p1.28.m3.1.1.1.1.1.1.cmml" xref="A3.1.p1.28.m3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.1.p1.28.m3.1.1.1.1.1.1.1.cmml" xref="A3.1.p1.28.m3.1.1.1.1.1.1">subscript</csymbol><min id="A3.1.p1.28.m3.1.1.1.1.1.1.2.cmml" xref="A3.1.p1.28.m3.1.1.1.1.1.1.2"></min><apply id="A3.1.p1.28.m3.1.1.1.1.1.1.3.cmml" xref="A3.1.p1.28.m3.1.1.1.1.1.1.3"><and id="A3.1.p1.28.m3.1.1.1.1.1.1.3a.cmml" xref="A3.1.p1.28.m3.1.1.1.1.1.1.3"></and><apply id="A3.1.p1.28.m3.1.1.1.1.1.1.3b.cmml" xref="A3.1.p1.28.m3.1.1.1.1.1.1.3"><leq id="A3.1.p1.28.m3.1.1.1.1.1.1.3.3.cmml" xref="A3.1.p1.28.m3.1.1.1.1.1.1.3.3"></leq><cn id="A3.1.p1.28.m3.1.1.1.1.1.1.3.2.cmml" type="integer" xref="A3.1.p1.28.m3.1.1.1.1.1.1.3.2">1</cn><ci id="A3.1.p1.28.m3.1.1.1.1.1.1.3.4.cmml" xref="A3.1.p1.28.m3.1.1.1.1.1.1.3.4">𝑖</ci></apply><apply id="A3.1.p1.28.m3.1.1.1.1.1.1.3c.cmml" xref="A3.1.p1.28.m3.1.1.1.1.1.1.3"><leq id="A3.1.p1.28.m3.1.1.1.1.1.1.3.5.cmml" xref="A3.1.p1.28.m3.1.1.1.1.1.1.3.5"></leq><share href="https://arxiv.org/html/2401.10785v2#A3.1.p1.28.m3.1.1.1.1.1.1.3.4.cmml" id="A3.1.p1.28.m3.1.1.1.1.1.1.3d.cmml" xref="A3.1.p1.28.m3.1.1.1.1.1.1.3"></share><ci id="A3.1.p1.28.m3.1.1.1.1.1.1.3.6.cmml" xref="A3.1.p1.28.m3.1.1.1.1.1.1.3.6">𝑛</ci></apply></apply></apply><apply id="A3.1.p1.28.m3.2.2.2.2.2.2.1.cmml" xref="A3.1.p1.28.m3.2.2.2.2.2.2.1"><csymbol cd="ambiguous" id="A3.1.p1.28.m3.2.2.2.2.2.2.1.1.cmml" xref="A3.1.p1.28.m3.2.2.2.2.2.2.1">subscript</csymbol><ci id="A3.1.p1.28.m3.2.2.2.2.2.2.1.2.cmml" xref="A3.1.p1.28.m3.2.2.2.2.2.2.1.2">𝑘</ci><apply id="A3.1.p1.28.m3.2.2.2.2.2.2.1.3.cmml" xref="A3.1.p1.28.m3.2.2.2.2.2.2.1.3"><times id="A3.1.p1.28.m3.2.2.2.2.2.2.1.3.1.cmml" xref="A3.1.p1.28.m3.2.2.2.2.2.2.1.3.1"></times><ci id="A3.1.p1.28.m3.2.2.2.2.2.2.1.3.2.cmml" xref="A3.1.p1.28.m3.2.2.2.2.2.2.1.3.2">c</ci><ci id="A3.1.p1.28.m3.2.2.2.2.2.2.1.3.3.cmml" xref="A3.1.p1.28.m3.2.2.2.2.2.2.1.3.3">𝑖</ci></apply></apply></apply><apply id="A3.1.p1.28.m3.2.2.2.4.cmml" xref="A3.1.p1.28.m3.2.2.2.4"><divide id="A3.1.p1.28.m3.2.2.2.4.1.cmml" xref="A3.1.p1.28.m3.2.2.2.4.1"></divide><cn id="A3.1.p1.28.m3.2.2.2.4.2.cmml" type="integer" xref="A3.1.p1.28.m3.2.2.2.4.2">1</cn><cn id="A3.1.p1.28.m3.2.2.2.4.3.cmml" type="integer" xref="A3.1.p1.28.m3.2.2.2.4.3">4</cn></apply></apply><apply id="A3.1.p1.28.m3.2.2.4.cmml" xref="A3.1.p1.28.m3.2.2.4"><csymbol cd="ambiguous" id="A3.1.p1.28.m3.2.2.4.1.cmml" xref="A3.1.p1.28.m3.2.2.4">superscript</csymbol><ci id="A3.1.p1.28.m3.2.2.4.2.cmml" xref="A3.1.p1.28.m3.2.2.4.2">ℝ</ci><plus id="A3.1.p1.28.m3.2.2.4.3.cmml" xref="A3.1.p1.28.m3.2.2.4.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.28.m3.2c">\min_{1\leq i\leq n}\{k_{{\rm c}i}\}-1/4\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.28.m3.2d">roman_min start_POSTSUBSCRIPT 1 ≤ italic_i ≤ italic_n end_POSTSUBSCRIPT { italic_k start_POSTSUBSCRIPT roman_c italic_i end_POSTSUBSCRIPT } - 1 / 4 ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>. Applying Young’s inequality <math alttext="ab\leq a^{2}/2+b^{2}/2" class="ltx_Math" display="inline" id="A3.1.p1.29.m4.1"><semantics id="A3.1.p1.29.m4.1a"><mrow id="A3.1.p1.29.m4.1.1" xref="A3.1.p1.29.m4.1.1.cmml"><mrow id="A3.1.p1.29.m4.1.1.2" xref="A3.1.p1.29.m4.1.1.2.cmml"><mi id="A3.1.p1.29.m4.1.1.2.2" xref="A3.1.p1.29.m4.1.1.2.2.cmml">a</mi><mo id="A3.1.p1.29.m4.1.1.2.1" xref="A3.1.p1.29.m4.1.1.2.1.cmml">⁢</mo><mi id="A3.1.p1.29.m4.1.1.2.3" xref="A3.1.p1.29.m4.1.1.2.3.cmml">b</mi></mrow><mo id="A3.1.p1.29.m4.1.1.1" xref="A3.1.p1.29.m4.1.1.1.cmml">≤</mo><mrow id="A3.1.p1.29.m4.1.1.3" xref="A3.1.p1.29.m4.1.1.3.cmml"><mrow id="A3.1.p1.29.m4.1.1.3.2" xref="A3.1.p1.29.m4.1.1.3.2.cmml"><msup id="A3.1.p1.29.m4.1.1.3.2.2" xref="A3.1.p1.29.m4.1.1.3.2.2.cmml"><mi id="A3.1.p1.29.m4.1.1.3.2.2.2" xref="A3.1.p1.29.m4.1.1.3.2.2.2.cmml">a</mi><mn id="A3.1.p1.29.m4.1.1.3.2.2.3" xref="A3.1.p1.29.m4.1.1.3.2.2.3.cmml">2</mn></msup><mo id="A3.1.p1.29.m4.1.1.3.2.1" xref="A3.1.p1.29.m4.1.1.3.2.1.cmml">/</mo><mn id="A3.1.p1.29.m4.1.1.3.2.3" xref="A3.1.p1.29.m4.1.1.3.2.3.cmml">2</mn></mrow><mo id="A3.1.p1.29.m4.1.1.3.1" xref="A3.1.p1.29.m4.1.1.3.1.cmml">+</mo><mrow id="A3.1.p1.29.m4.1.1.3.3" xref="A3.1.p1.29.m4.1.1.3.3.cmml"><msup id="A3.1.p1.29.m4.1.1.3.3.2" xref="A3.1.p1.29.m4.1.1.3.3.2.cmml"><mi id="A3.1.p1.29.m4.1.1.3.3.2.2" xref="A3.1.p1.29.m4.1.1.3.3.2.2.cmml">b</mi><mn id="A3.1.p1.29.m4.1.1.3.3.2.3" xref="A3.1.p1.29.m4.1.1.3.3.2.3.cmml">2</mn></msup><mo id="A3.1.p1.29.m4.1.1.3.3.1" xref="A3.1.p1.29.m4.1.1.3.3.1.cmml">/</mo><mn id="A3.1.p1.29.m4.1.1.3.3.3" xref="A3.1.p1.29.m4.1.1.3.3.3.cmml">2</mn></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.29.m4.1b"><apply id="A3.1.p1.29.m4.1.1.cmml" xref="A3.1.p1.29.m4.1.1"><leq id="A3.1.p1.29.m4.1.1.1.cmml" xref="A3.1.p1.29.m4.1.1.1"></leq><apply id="A3.1.p1.29.m4.1.1.2.cmml" xref="A3.1.p1.29.m4.1.1.2"><times id="A3.1.p1.29.m4.1.1.2.1.cmml" xref="A3.1.p1.29.m4.1.1.2.1"></times><ci id="A3.1.p1.29.m4.1.1.2.2.cmml" xref="A3.1.p1.29.m4.1.1.2.2">𝑎</ci><ci id="A3.1.p1.29.m4.1.1.2.3.cmml" xref="A3.1.p1.29.m4.1.1.2.3">𝑏</ci></apply><apply id="A3.1.p1.29.m4.1.1.3.cmml" xref="A3.1.p1.29.m4.1.1.3"><plus id="A3.1.p1.29.m4.1.1.3.1.cmml" xref="A3.1.p1.29.m4.1.1.3.1"></plus><apply id="A3.1.p1.29.m4.1.1.3.2.cmml" xref="A3.1.p1.29.m4.1.1.3.2"><divide id="A3.1.p1.29.m4.1.1.3.2.1.cmml" xref="A3.1.p1.29.m4.1.1.3.2.1"></divide><apply id="A3.1.p1.29.m4.1.1.3.2.2.cmml" xref="A3.1.p1.29.m4.1.1.3.2.2"><csymbol cd="ambiguous" id="A3.1.p1.29.m4.1.1.3.2.2.1.cmml" xref="A3.1.p1.29.m4.1.1.3.2.2">superscript</csymbol><ci id="A3.1.p1.29.m4.1.1.3.2.2.2.cmml" xref="A3.1.p1.29.m4.1.1.3.2.2.2">𝑎</ci><cn id="A3.1.p1.29.m4.1.1.3.2.2.3.cmml" type="integer" xref="A3.1.p1.29.m4.1.1.3.2.2.3">2</cn></apply><cn id="A3.1.p1.29.m4.1.1.3.2.3.cmml" type="integer" xref="A3.1.p1.29.m4.1.1.3.2.3">2</cn></apply><apply id="A3.1.p1.29.m4.1.1.3.3.cmml" xref="A3.1.p1.29.m4.1.1.3.3"><divide id="A3.1.p1.29.m4.1.1.3.3.1.cmml" xref="A3.1.p1.29.m4.1.1.3.3.1"></divide><apply id="A3.1.p1.29.m4.1.1.3.3.2.cmml" xref="A3.1.p1.29.m4.1.1.3.3.2"><csymbol cd="ambiguous" id="A3.1.p1.29.m4.1.1.3.3.2.1.cmml" xref="A3.1.p1.29.m4.1.1.3.3.2">superscript</csymbol><ci id="A3.1.p1.29.m4.1.1.3.3.2.2.cmml" xref="A3.1.p1.29.m4.1.1.3.3.2.2">𝑏</ci><cn id="A3.1.p1.29.m4.1.1.3.3.2.3.cmml" type="integer" xref="A3.1.p1.29.m4.1.1.3.3.2.3">2</cn></apply><cn id="A3.1.p1.29.m4.1.1.3.3.3.cmml" type="integer" xref="A3.1.p1.29.m4.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.29.m4.1c">ab\leq a^{2}/2+b^{2}/2</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.29.m4.1d">italic_a italic_b ≤ italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 + italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2</annotation></semantics></math> with <math alttext="a=\sqrt{k_{\rm c}}\|\bm{e}\|" class="ltx_Math" display="inline" id="A3.1.p1.30.m5.1"><semantics id="A3.1.p1.30.m5.1a"><mrow id="A3.1.p1.30.m5.1.2" xref="A3.1.p1.30.m5.1.2.cmml"><mi id="A3.1.p1.30.m5.1.2.2" xref="A3.1.p1.30.m5.1.2.2.cmml">a</mi><mo id="A3.1.p1.30.m5.1.2.1" xref="A3.1.p1.30.m5.1.2.1.cmml">=</mo><mrow id="A3.1.p1.30.m5.1.2.3" xref="A3.1.p1.30.m5.1.2.3.cmml"><msqrt id="A3.1.p1.30.m5.1.2.3.2" xref="A3.1.p1.30.m5.1.2.3.2.cmml"><msub id="A3.1.p1.30.m5.1.2.3.2.2" xref="A3.1.p1.30.m5.1.2.3.2.2.cmml"><mi id="A3.1.p1.30.m5.1.2.3.2.2.2" xref="A3.1.p1.30.m5.1.2.3.2.2.2.cmml">k</mi><mi id="A3.1.p1.30.m5.1.2.3.2.2.3" mathvariant="normal" xref="A3.1.p1.30.m5.1.2.3.2.2.3.cmml">c</mi></msub></msqrt><mo id="A3.1.p1.30.m5.1.2.3.1" xref="A3.1.p1.30.m5.1.2.3.1.cmml">⁢</mo><mrow id="A3.1.p1.30.m5.1.2.3.3.2" xref="A3.1.p1.30.m5.1.2.3.3.1.cmml"><mo id="A3.1.p1.30.m5.1.2.3.3.2.1" stretchy="false" xref="A3.1.p1.30.m5.1.2.3.3.1.1.cmml">‖</mo><mi id="A3.1.p1.30.m5.1.1" xref="A3.1.p1.30.m5.1.1.cmml">𝒆</mi><mo id="A3.1.p1.30.m5.1.2.3.3.2.2" stretchy="false" xref="A3.1.p1.30.m5.1.2.3.3.1.1.cmml">‖</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.30.m5.1b"><apply id="A3.1.p1.30.m5.1.2.cmml" xref="A3.1.p1.30.m5.1.2"><eq id="A3.1.p1.30.m5.1.2.1.cmml" xref="A3.1.p1.30.m5.1.2.1"></eq><ci id="A3.1.p1.30.m5.1.2.2.cmml" xref="A3.1.p1.30.m5.1.2.2">𝑎</ci><apply id="A3.1.p1.30.m5.1.2.3.cmml" xref="A3.1.p1.30.m5.1.2.3"><times id="A3.1.p1.30.m5.1.2.3.1.cmml" xref="A3.1.p1.30.m5.1.2.3.1"></times><apply id="A3.1.p1.30.m5.1.2.3.2.cmml" xref="A3.1.p1.30.m5.1.2.3.2"><root id="A3.1.p1.30.m5.1.2.3.2a.cmml" xref="A3.1.p1.30.m5.1.2.3.2"></root><apply id="A3.1.p1.30.m5.1.2.3.2.2.cmml" xref="A3.1.p1.30.m5.1.2.3.2.2"><csymbol cd="ambiguous" id="A3.1.p1.30.m5.1.2.3.2.2.1.cmml" xref="A3.1.p1.30.m5.1.2.3.2.2">subscript</csymbol><ci id="A3.1.p1.30.m5.1.2.3.2.2.2.cmml" xref="A3.1.p1.30.m5.1.2.3.2.2.2">𝑘</ci><ci id="A3.1.p1.30.m5.1.2.3.2.2.3.cmml" xref="A3.1.p1.30.m5.1.2.3.2.2.3">c</ci></apply></apply><apply id="A3.1.p1.30.m5.1.2.3.3.1.cmml" xref="A3.1.p1.30.m5.1.2.3.3.2"><csymbol cd="latexml" id="A3.1.p1.30.m5.1.2.3.3.1.1.cmml" xref="A3.1.p1.30.m5.1.2.3.3.2.1">norm</csymbol><ci id="A3.1.p1.30.m5.1.1.cmml" xref="A3.1.p1.30.m5.1.1">𝒆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.30.m5.1c">a=\sqrt{k_{\rm c}}\|\bm{e}\|</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.30.m5.1d">italic_a = square-root start_ARG italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT end_ARG ∥ bold_italic_e ∥</annotation></semantics></math> and <math alttext="b=\delta c_{\theta}/\sqrt{k_{\rm c}}" class="ltx_Math" display="inline" id="A3.1.p1.31.m6.1"><semantics id="A3.1.p1.31.m6.1a"><mrow id="A3.1.p1.31.m6.1.1" xref="A3.1.p1.31.m6.1.1.cmml"><mi id="A3.1.p1.31.m6.1.1.2" xref="A3.1.p1.31.m6.1.1.2.cmml">b</mi><mo id="A3.1.p1.31.m6.1.1.1" xref="A3.1.p1.31.m6.1.1.1.cmml">=</mo><mrow id="A3.1.p1.31.m6.1.1.3" xref="A3.1.p1.31.m6.1.1.3.cmml"><mrow id="A3.1.p1.31.m6.1.1.3.2" xref="A3.1.p1.31.m6.1.1.3.2.cmml"><mi id="A3.1.p1.31.m6.1.1.3.2.2" xref="A3.1.p1.31.m6.1.1.3.2.2.cmml">δ</mi><mo id="A3.1.p1.31.m6.1.1.3.2.1" xref="A3.1.p1.31.m6.1.1.3.2.1.cmml">⁢</mo><msub id="A3.1.p1.31.m6.1.1.3.2.3" xref="A3.1.p1.31.m6.1.1.3.2.3.cmml"><mi id="A3.1.p1.31.m6.1.1.3.2.3.2" xref="A3.1.p1.31.m6.1.1.3.2.3.2.cmml">c</mi><mi id="A3.1.p1.31.m6.1.1.3.2.3.3" xref="A3.1.p1.31.m6.1.1.3.2.3.3.cmml">θ</mi></msub></mrow><mo id="A3.1.p1.31.m6.1.1.3.1" xref="A3.1.p1.31.m6.1.1.3.1.cmml">/</mo><msqrt id="A3.1.p1.31.m6.1.1.3.3" xref="A3.1.p1.31.m6.1.1.3.3.cmml"><msub id="A3.1.p1.31.m6.1.1.3.3.2" xref="A3.1.p1.31.m6.1.1.3.3.2.cmml"><mi id="A3.1.p1.31.m6.1.1.3.3.2.2" xref="A3.1.p1.31.m6.1.1.3.3.2.2.cmml">k</mi><mi id="A3.1.p1.31.m6.1.1.3.3.2.3" mathvariant="normal" xref="A3.1.p1.31.m6.1.1.3.3.2.3.cmml">c</mi></msub></msqrt></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.31.m6.1b"><apply id="A3.1.p1.31.m6.1.1.cmml" xref="A3.1.p1.31.m6.1.1"><eq id="A3.1.p1.31.m6.1.1.1.cmml" xref="A3.1.p1.31.m6.1.1.1"></eq><ci id="A3.1.p1.31.m6.1.1.2.cmml" xref="A3.1.p1.31.m6.1.1.2">𝑏</ci><apply id="A3.1.p1.31.m6.1.1.3.cmml" xref="A3.1.p1.31.m6.1.1.3"><divide id="A3.1.p1.31.m6.1.1.3.1.cmml" xref="A3.1.p1.31.m6.1.1.3.1"></divide><apply id="A3.1.p1.31.m6.1.1.3.2.cmml" xref="A3.1.p1.31.m6.1.1.3.2"><times id="A3.1.p1.31.m6.1.1.3.2.1.cmml" xref="A3.1.p1.31.m6.1.1.3.2.1"></times><ci id="A3.1.p1.31.m6.1.1.3.2.2.cmml" xref="A3.1.p1.31.m6.1.1.3.2.2">𝛿</ci><apply id="A3.1.p1.31.m6.1.1.3.2.3.cmml" xref="A3.1.p1.31.m6.1.1.3.2.3"><csymbol cd="ambiguous" id="A3.1.p1.31.m6.1.1.3.2.3.1.cmml" xref="A3.1.p1.31.m6.1.1.3.2.3">subscript</csymbol><ci id="A3.1.p1.31.m6.1.1.3.2.3.2.cmml" xref="A3.1.p1.31.m6.1.1.3.2.3.2">𝑐</ci><ci id="A3.1.p1.31.m6.1.1.3.2.3.3.cmml" xref="A3.1.p1.31.m6.1.1.3.2.3.3">𝜃</ci></apply></apply><apply id="A3.1.p1.31.m6.1.1.3.3.cmml" xref="A3.1.p1.31.m6.1.1.3.3"><root id="A3.1.p1.31.m6.1.1.3.3a.cmml" xref="A3.1.p1.31.m6.1.1.3.3"></root><apply id="A3.1.p1.31.m6.1.1.3.3.2.cmml" xref="A3.1.p1.31.m6.1.1.3.3.2"><csymbol cd="ambiguous" id="A3.1.p1.31.m6.1.1.3.3.2.1.cmml" xref="A3.1.p1.31.m6.1.1.3.3.2">subscript</csymbol><ci id="A3.1.p1.31.m6.1.1.3.3.2.2.cmml" xref="A3.1.p1.31.m6.1.1.3.3.2.2">𝑘</ci><ci id="A3.1.p1.31.m6.1.1.3.3.2.3.cmml" xref="A3.1.p1.31.m6.1.1.3.3.2.3">c</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.31.m6.1c">b=\delta c_{\theta}/\sqrt{k_{\rm c}}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.31.m6.1d">italic_b = italic_δ italic_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT / square-root start_ARG italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT end_ARG</annotation></semantics></math> to the above inequality, one obtains</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx60"> <tbody id="A3.E46"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}\leq-k_{\rm c}\|\bm{e}\|^{2}/2+(\delta c_{\theta})^{2}/(2k% _{\rm c})" class="ltx_Math" display="inline" id="A3.E46.m1.3"><semantics id="A3.E46.m1.3a"><mrow id="A3.E46.m1.3.3" xref="A3.E46.m1.3.3.cmml"><mover accent="true" id="A3.E46.m1.3.3.4" xref="A3.E46.m1.3.3.4.cmml"><mi id="A3.E46.m1.3.3.4.2" xref="A3.E46.m1.3.3.4.2.cmml">V</mi><mo id="A3.E46.m1.3.3.4.1" xref="A3.E46.m1.3.3.4.1.cmml">˙</mo></mover><mo id="A3.E46.m1.3.3.3" xref="A3.E46.m1.3.3.3.cmml">≤</mo><mrow id="A3.E46.m1.3.3.2" xref="A3.E46.m1.3.3.2.cmml"><mrow id="A3.E46.m1.3.3.2.4" xref="A3.E46.m1.3.3.2.4.cmml"><mo id="A3.E46.m1.3.3.2.4a" xref="A3.E46.m1.3.3.2.4.cmml">−</mo><mrow id="A3.E46.m1.3.3.2.4.2" xref="A3.E46.m1.3.3.2.4.2.cmml"><mrow id="A3.E46.m1.3.3.2.4.2.2" xref="A3.E46.m1.3.3.2.4.2.2.cmml"><msub id="A3.E46.m1.3.3.2.4.2.2.2" xref="A3.E46.m1.3.3.2.4.2.2.2.cmml"><mi id="A3.E46.m1.3.3.2.4.2.2.2.2" xref="A3.E46.m1.3.3.2.4.2.2.2.2.cmml">k</mi><mi id="A3.E46.m1.3.3.2.4.2.2.2.3" mathvariant="normal" xref="A3.E46.m1.3.3.2.4.2.2.2.3.cmml">c</mi></msub><mo id="A3.E46.m1.3.3.2.4.2.2.1" xref="A3.E46.m1.3.3.2.4.2.2.1.cmml">⁢</mo><msup id="A3.E46.m1.3.3.2.4.2.2.3" xref="A3.E46.m1.3.3.2.4.2.2.3.cmml"><mrow id="A3.E46.m1.3.3.2.4.2.2.3.2.2" xref="A3.E46.m1.3.3.2.4.2.2.3.2.1.cmml"><mo id="A3.E46.m1.3.3.2.4.2.2.3.2.2.1" stretchy="false" xref="A3.E46.m1.3.3.2.4.2.2.3.2.1.1.cmml">‖</mo><mi id="A3.E46.m1.1.1" xref="A3.E46.m1.1.1.cmml">𝒆</mi><mo id="A3.E46.m1.3.3.2.4.2.2.3.2.2.2" stretchy="false" xref="A3.E46.m1.3.3.2.4.2.2.3.2.1.1.cmml">‖</mo></mrow><mn id="A3.E46.m1.3.3.2.4.2.2.3.3" xref="A3.E46.m1.3.3.2.4.2.2.3.3.cmml">2</mn></msup></mrow><mo id="A3.E46.m1.3.3.2.4.2.1" xref="A3.E46.m1.3.3.2.4.2.1.cmml">/</mo><mn id="A3.E46.m1.3.3.2.4.2.3" xref="A3.E46.m1.3.3.2.4.2.3.cmml">2</mn></mrow></mrow><mo id="A3.E46.m1.3.3.2.3" xref="A3.E46.m1.3.3.2.3.cmml">+</mo><mrow id="A3.E46.m1.3.3.2.2" xref="A3.E46.m1.3.3.2.2.cmml"><msup id="A3.E46.m1.2.2.1.1.1" xref="A3.E46.m1.2.2.1.1.1.cmml"><mrow id="A3.E46.m1.2.2.1.1.1.1.1" xref="A3.E46.m1.2.2.1.1.1.1.1.1.cmml"><mo id="A3.E46.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="A3.E46.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.E46.m1.2.2.1.1.1.1.1.1" xref="A3.E46.m1.2.2.1.1.1.1.1.1.cmml"><mi id="A3.E46.m1.2.2.1.1.1.1.1.1.2" xref="A3.E46.m1.2.2.1.1.1.1.1.1.2.cmml">δ</mi><mo id="A3.E46.m1.2.2.1.1.1.1.1.1.1" xref="A3.E46.m1.2.2.1.1.1.1.1.1.1.cmml">⁢</mo><msub id="A3.E46.m1.2.2.1.1.1.1.1.1.3" xref="A3.E46.m1.2.2.1.1.1.1.1.1.3.cmml"><mi id="A3.E46.m1.2.2.1.1.1.1.1.1.3.2" xref="A3.E46.m1.2.2.1.1.1.1.1.1.3.2.cmml">c</mi><mi id="A3.E46.m1.2.2.1.1.1.1.1.1.3.3" xref="A3.E46.m1.2.2.1.1.1.1.1.1.3.3.cmml">θ</mi></msub></mrow><mo id="A3.E46.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="A3.E46.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="A3.E46.m1.2.2.1.1.1.3" xref="A3.E46.m1.2.2.1.1.1.3.cmml">2</mn></msup><mo id="A3.E46.m1.3.3.2.2.3" xref="A3.E46.m1.3.3.2.2.3.cmml">/</mo><mrow id="A3.E46.m1.3.3.2.2.2.1" xref="A3.E46.m1.3.3.2.2.2.1.1.cmml"><mo id="A3.E46.m1.3.3.2.2.2.1.2" stretchy="false" xref="A3.E46.m1.3.3.2.2.2.1.1.cmml">(</mo><mrow id="A3.E46.m1.3.3.2.2.2.1.1" xref="A3.E46.m1.3.3.2.2.2.1.1.cmml"><mn id="A3.E46.m1.3.3.2.2.2.1.1.2" xref="A3.E46.m1.3.3.2.2.2.1.1.2.cmml">2</mn><mo id="A3.E46.m1.3.3.2.2.2.1.1.1" xref="A3.E46.m1.3.3.2.2.2.1.1.1.cmml">⁢</mo><msub id="A3.E46.m1.3.3.2.2.2.1.1.3" xref="A3.E46.m1.3.3.2.2.2.1.1.3.cmml"><mi id="A3.E46.m1.3.3.2.2.2.1.1.3.2" xref="A3.E46.m1.3.3.2.2.2.1.1.3.2.cmml">k</mi><mi id="A3.E46.m1.3.3.2.2.2.1.1.3.3" mathvariant="normal" xref="A3.E46.m1.3.3.2.2.2.1.1.3.3.cmml">c</mi></msub></mrow><mo id="A3.E46.m1.3.3.2.2.2.1.3" stretchy="false" xref="A3.E46.m1.3.3.2.2.2.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.E46.m1.3b"><apply id="A3.E46.m1.3.3.cmml" xref="A3.E46.m1.3.3"><leq id="A3.E46.m1.3.3.3.cmml" xref="A3.E46.m1.3.3.3"></leq><apply id="A3.E46.m1.3.3.4.cmml" xref="A3.E46.m1.3.3.4"><ci id="A3.E46.m1.3.3.4.1.cmml" xref="A3.E46.m1.3.3.4.1">˙</ci><ci id="A3.E46.m1.3.3.4.2.cmml" xref="A3.E46.m1.3.3.4.2">𝑉</ci></apply><apply id="A3.E46.m1.3.3.2.cmml" xref="A3.E46.m1.3.3.2"><plus id="A3.E46.m1.3.3.2.3.cmml" xref="A3.E46.m1.3.3.2.3"></plus><apply id="A3.E46.m1.3.3.2.4.cmml" xref="A3.E46.m1.3.3.2.4"><minus id="A3.E46.m1.3.3.2.4.1.cmml" xref="A3.E46.m1.3.3.2.4"></minus><apply id="A3.E46.m1.3.3.2.4.2.cmml" xref="A3.E46.m1.3.3.2.4.2"><divide id="A3.E46.m1.3.3.2.4.2.1.cmml" xref="A3.E46.m1.3.3.2.4.2.1"></divide><apply id="A3.E46.m1.3.3.2.4.2.2.cmml" xref="A3.E46.m1.3.3.2.4.2.2"><times id="A3.E46.m1.3.3.2.4.2.2.1.cmml" xref="A3.E46.m1.3.3.2.4.2.2.1"></times><apply id="A3.E46.m1.3.3.2.4.2.2.2.cmml" xref="A3.E46.m1.3.3.2.4.2.2.2"><csymbol cd="ambiguous" id="A3.E46.m1.3.3.2.4.2.2.2.1.cmml" xref="A3.E46.m1.3.3.2.4.2.2.2">subscript</csymbol><ci id="A3.E46.m1.3.3.2.4.2.2.2.2.cmml" xref="A3.E46.m1.3.3.2.4.2.2.2.2">𝑘</ci><ci id="A3.E46.m1.3.3.2.4.2.2.2.3.cmml" xref="A3.E46.m1.3.3.2.4.2.2.2.3">c</ci></apply><apply id="A3.E46.m1.3.3.2.4.2.2.3.cmml" xref="A3.E46.m1.3.3.2.4.2.2.3"><csymbol cd="ambiguous" id="A3.E46.m1.3.3.2.4.2.2.3.1.cmml" xref="A3.E46.m1.3.3.2.4.2.2.3">superscript</csymbol><apply id="A3.E46.m1.3.3.2.4.2.2.3.2.1.cmml" xref="A3.E46.m1.3.3.2.4.2.2.3.2.2"><csymbol cd="latexml" id="A3.E46.m1.3.3.2.4.2.2.3.2.1.1.cmml" xref="A3.E46.m1.3.3.2.4.2.2.3.2.2.1">norm</csymbol><ci id="A3.E46.m1.1.1.cmml" xref="A3.E46.m1.1.1">𝒆</ci></apply><cn id="A3.E46.m1.3.3.2.4.2.2.3.3.cmml" type="integer" xref="A3.E46.m1.3.3.2.4.2.2.3.3">2</cn></apply></apply><cn id="A3.E46.m1.3.3.2.4.2.3.cmml" type="integer" xref="A3.E46.m1.3.3.2.4.2.3">2</cn></apply></apply><apply id="A3.E46.m1.3.3.2.2.cmml" xref="A3.E46.m1.3.3.2.2"><divide id="A3.E46.m1.3.3.2.2.3.cmml" xref="A3.E46.m1.3.3.2.2.3"></divide><apply id="A3.E46.m1.2.2.1.1.1.cmml" xref="A3.E46.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="A3.E46.m1.2.2.1.1.1.2.cmml" xref="A3.E46.m1.2.2.1.1.1">superscript</csymbol><apply id="A3.E46.m1.2.2.1.1.1.1.1.1.cmml" xref="A3.E46.m1.2.2.1.1.1.1.1"><times id="A3.E46.m1.2.2.1.1.1.1.1.1.1.cmml" xref="A3.E46.m1.2.2.1.1.1.1.1.1.1"></times><ci id="A3.E46.m1.2.2.1.1.1.1.1.1.2.cmml" xref="A3.E46.m1.2.2.1.1.1.1.1.1.2">𝛿</ci><apply id="A3.E46.m1.2.2.1.1.1.1.1.1.3.cmml" xref="A3.E46.m1.2.2.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A3.E46.m1.2.2.1.1.1.1.1.1.3.1.cmml" xref="A3.E46.m1.2.2.1.1.1.1.1.1.3">subscript</csymbol><ci id="A3.E46.m1.2.2.1.1.1.1.1.1.3.2.cmml" xref="A3.E46.m1.2.2.1.1.1.1.1.1.3.2">𝑐</ci><ci id="A3.E46.m1.2.2.1.1.1.1.1.1.3.3.cmml" xref="A3.E46.m1.2.2.1.1.1.1.1.1.3.3">𝜃</ci></apply></apply><cn id="A3.E46.m1.2.2.1.1.1.3.cmml" type="integer" xref="A3.E46.m1.2.2.1.1.1.3">2</cn></apply><apply id="A3.E46.m1.3.3.2.2.2.1.1.cmml" xref="A3.E46.m1.3.3.2.2.2.1"><times id="A3.E46.m1.3.3.2.2.2.1.1.1.cmml" xref="A3.E46.m1.3.3.2.2.2.1.1.1"></times><cn id="A3.E46.m1.3.3.2.2.2.1.1.2.cmml" type="integer" xref="A3.E46.m1.3.3.2.2.2.1.1.2">2</cn><apply id="A3.E46.m1.3.3.2.2.2.1.1.3.cmml" xref="A3.E46.m1.3.3.2.2.2.1.1.3"><csymbol cd="ambiguous" id="A3.E46.m1.3.3.2.2.2.1.1.3.1.cmml" xref="A3.E46.m1.3.3.2.2.2.1.1.3">subscript</csymbol><ci id="A3.E46.m1.3.3.2.2.2.1.1.3.2.cmml" xref="A3.E46.m1.3.3.2.2.2.1.1.3.2">𝑘</ci><ci id="A3.E46.m1.3.3.2.2.2.1.1.3.3.cmml" xref="A3.E46.m1.3.3.2.2.2.1.1.3.3">c</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E46.m1.3c">\displaystyle\dot{V}\leq-k_{\rm c}\|\bm{e}\|^{2}/2+(\delta c_{\theta})^{2}/(2k% _{\rm c})</annotation><annotation encoding="application/x-llamapun" id="A3.E46.m1.3d">over˙ start_ARG italic_V end_ARG ≤ - italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ∥ bold_italic_e ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 + ( italic_δ italic_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( 2 italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(46)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.1.p1.40">which is valid on <math alttext="\bm{x}\in\Omega_{{\rm c}_{x}}" class="ltx_Math" display="inline" id="A3.1.p1.32.m1.1"><semantics id="A3.1.p1.32.m1.1a"><mrow id="A3.1.p1.32.m1.1.1" xref="A3.1.p1.32.m1.1.1.cmml"><mi id="A3.1.p1.32.m1.1.1.2" xref="A3.1.p1.32.m1.1.1.2.cmml">𝒙</mi><mo id="A3.1.p1.32.m1.1.1.1" xref="A3.1.p1.32.m1.1.1.1.cmml">∈</mo><msub id="A3.1.p1.32.m1.1.1.3" xref="A3.1.p1.32.m1.1.1.3.cmml"><mi id="A3.1.p1.32.m1.1.1.3.2" mathvariant="normal" xref="A3.1.p1.32.m1.1.1.3.2.cmml">Ω</mi><msub id="A3.1.p1.32.m1.1.1.3.3" xref="A3.1.p1.32.m1.1.1.3.3.cmml"><mi id="A3.1.p1.32.m1.1.1.3.3.2" mathvariant="normal" xref="A3.1.p1.32.m1.1.1.3.3.2.cmml">c</mi><mi id="A3.1.p1.32.m1.1.1.3.3.3" xref="A3.1.p1.32.m1.1.1.3.3.3.cmml">x</mi></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.32.m1.1b"><apply id="A3.1.p1.32.m1.1.1.cmml" xref="A3.1.p1.32.m1.1.1"><in id="A3.1.p1.32.m1.1.1.1.cmml" xref="A3.1.p1.32.m1.1.1.1"></in><ci id="A3.1.p1.32.m1.1.1.2.cmml" xref="A3.1.p1.32.m1.1.1.2">𝒙</ci><apply id="A3.1.p1.32.m1.1.1.3.cmml" xref="A3.1.p1.32.m1.1.1.3"><csymbol cd="ambiguous" id="A3.1.p1.32.m1.1.1.3.1.cmml" xref="A3.1.p1.32.m1.1.1.3">subscript</csymbol><ci id="A3.1.p1.32.m1.1.1.3.2.cmml" xref="A3.1.p1.32.m1.1.1.3.2">Ω</ci><apply id="A3.1.p1.32.m1.1.1.3.3.cmml" xref="A3.1.p1.32.m1.1.1.3.3"><csymbol cd="ambiguous" id="A3.1.p1.32.m1.1.1.3.3.1.cmml" xref="A3.1.p1.32.m1.1.1.3.3">subscript</csymbol><ci id="A3.1.p1.32.m1.1.1.3.3.2.cmml" xref="A3.1.p1.32.m1.1.1.3.3.2">c</ci><ci id="A3.1.p1.32.m1.1.1.3.3.3.cmml" xref="A3.1.p1.32.m1.1.1.3.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.32.m1.1c">\bm{x}\in\Omega_{{\rm c}_{x}}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.32.m1.1d">bold_italic_x ∈ roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="t\in[0,t_{\rm f})" class="ltx_Math" display="inline" id="A3.1.p1.33.m2.2"><semantics id="A3.1.p1.33.m2.2a"><mrow id="A3.1.p1.33.m2.2.2" xref="A3.1.p1.33.m2.2.2.cmml"><mi id="A3.1.p1.33.m2.2.2.3" xref="A3.1.p1.33.m2.2.2.3.cmml">t</mi><mo id="A3.1.p1.33.m2.2.2.2" xref="A3.1.p1.33.m2.2.2.2.cmml">∈</mo><mrow id="A3.1.p1.33.m2.2.2.1.1" xref="A3.1.p1.33.m2.2.2.1.2.cmml"><mo id="A3.1.p1.33.m2.2.2.1.1.2" stretchy="false" xref="A3.1.p1.33.m2.2.2.1.2.cmml">[</mo><mn id="A3.1.p1.33.m2.1.1" xref="A3.1.p1.33.m2.1.1.cmml">0</mn><mo id="A3.1.p1.33.m2.2.2.1.1.3" xref="A3.1.p1.33.m2.2.2.1.2.cmml">,</mo><msub id="A3.1.p1.33.m2.2.2.1.1.1" xref="A3.1.p1.33.m2.2.2.1.1.1.cmml"><mi id="A3.1.p1.33.m2.2.2.1.1.1.2" xref="A3.1.p1.33.m2.2.2.1.1.1.2.cmml">t</mi><mi id="A3.1.p1.33.m2.2.2.1.1.1.3" mathvariant="normal" xref="A3.1.p1.33.m2.2.2.1.1.1.3.cmml">f</mi></msub><mo id="A3.1.p1.33.m2.2.2.1.1.4" stretchy="false" xref="A3.1.p1.33.m2.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.33.m2.2b"><apply id="A3.1.p1.33.m2.2.2.cmml" xref="A3.1.p1.33.m2.2.2"><in id="A3.1.p1.33.m2.2.2.2.cmml" xref="A3.1.p1.33.m2.2.2.2"></in><ci id="A3.1.p1.33.m2.2.2.3.cmml" xref="A3.1.p1.33.m2.2.2.3">𝑡</ci><interval closure="closed-open" id="A3.1.p1.33.m2.2.2.1.2.cmml" xref="A3.1.p1.33.m2.2.2.1.1"><cn id="A3.1.p1.33.m2.1.1.cmml" type="integer" xref="A3.1.p1.33.m2.1.1">0</cn><apply id="A3.1.p1.33.m2.2.2.1.1.1.cmml" xref="A3.1.p1.33.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="A3.1.p1.33.m2.2.2.1.1.1.1.cmml" xref="A3.1.p1.33.m2.2.2.1.1.1">subscript</csymbol><ci id="A3.1.p1.33.m2.2.2.1.1.1.2.cmml" xref="A3.1.p1.33.m2.2.2.1.1.1.2">𝑡</ci><ci id="A3.1.p1.33.m2.2.2.1.1.1.3.cmml" xref="A3.1.p1.33.m2.2.2.1.1.1.3">f</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.33.m2.2c">t\in[0,t_{\rm f})</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.33.m2.2d">italic_t ∈ [ 0 , italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT )</annotation></semantics></math>. It is observed from (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A3.E45" title="In Proof. ‣ Appendix C The proof of Theorem 2 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">45</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A3.E46" title="In Proof. ‣ Appendix C The proof of Theorem 2 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">46</span></a>) that the general prediction error <math alttext="\bm{\epsilon}" class="ltx_Math" display="inline" id="A3.1.p1.34.m3.1"><semantics id="A3.1.p1.34.m3.1a"><mi class="ltx_mathvariant_bold-italic" id="A3.1.p1.34.m3.1.1" mathvariant="bold-italic" xref="A3.1.p1.34.m3.1.1.cmml">ϵ</mi><annotation-xml encoding="MathML-Content" id="A3.1.p1.34.m3.1b"><ci id="A3.1.p1.34.m3.1.1.cmml" xref="A3.1.p1.34.m3.1.1">bold-italic-ϵ</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.34.m3.1c">\bm{\epsilon}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.34.m3.1d">bold_italic_ϵ</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E23" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">23</span></a>) counteracts the modeling error term <math alttext="\Phi^{T}\tilde{\bm{\theta}}" class="ltx_Math" display="inline" id="A3.1.p1.35.m4.1"><semantics id="A3.1.p1.35.m4.1a"><mrow id="A3.1.p1.35.m4.1.1" xref="A3.1.p1.35.m4.1.1.cmml"><msup id="A3.1.p1.35.m4.1.1.2" xref="A3.1.p1.35.m4.1.1.2.cmml"><mi id="A3.1.p1.35.m4.1.1.2.2" mathvariant="normal" xref="A3.1.p1.35.m4.1.1.2.2.cmml">Φ</mi><mi id="A3.1.p1.35.m4.1.1.2.3" xref="A3.1.p1.35.m4.1.1.2.3.cmml">T</mi></msup><mo id="A3.1.p1.35.m4.1.1.1" xref="A3.1.p1.35.m4.1.1.1.cmml">⁢</mo><mover accent="true" id="A3.1.p1.35.m4.1.1.3" xref="A3.1.p1.35.m4.1.1.3.cmml"><mi id="A3.1.p1.35.m4.1.1.3.2" xref="A3.1.p1.35.m4.1.1.3.2.cmml">𝜽</mi><mo id="A3.1.p1.35.m4.1.1.3.1" xref="A3.1.p1.35.m4.1.1.3.1.cmml">~</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.35.m4.1b"><apply id="A3.1.p1.35.m4.1.1.cmml" xref="A3.1.p1.35.m4.1.1"><times id="A3.1.p1.35.m4.1.1.1.cmml" xref="A3.1.p1.35.m4.1.1.1"></times><apply id="A3.1.p1.35.m4.1.1.2.cmml" xref="A3.1.p1.35.m4.1.1.2"><csymbol cd="ambiguous" id="A3.1.p1.35.m4.1.1.2.1.cmml" xref="A3.1.p1.35.m4.1.1.2">superscript</csymbol><ci id="A3.1.p1.35.m4.1.1.2.2.cmml" xref="A3.1.p1.35.m4.1.1.2.2">Φ</ci><ci id="A3.1.p1.35.m4.1.1.2.3.cmml" xref="A3.1.p1.35.m4.1.1.2.3">𝑇</ci></apply><apply id="A3.1.p1.35.m4.1.1.3.cmml" xref="A3.1.p1.35.m4.1.1.3"><ci id="A3.1.p1.35.m4.1.1.3.1.cmml" xref="A3.1.p1.35.m4.1.1.3.1">~</ci><ci id="A3.1.p1.35.m4.1.1.3.2.cmml" xref="A3.1.p1.35.m4.1.1.3.2">𝜽</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.35.m4.1c">\Phi^{T}\tilde{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.35.m4.1d">roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math>, which leads to (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A3.E46" title="In Proof. ‣ Appendix C The proof of Theorem 2 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">46</span></a>). Thus, <math alttext="\bm{e}(t)" class="ltx_Math" display="inline" id="A3.1.p1.36.m5.1"><semantics id="A3.1.p1.36.m5.1a"><mrow id="A3.1.p1.36.m5.1.2" xref="A3.1.p1.36.m5.1.2.cmml"><mi id="A3.1.p1.36.m5.1.2.2" xref="A3.1.p1.36.m5.1.2.2.cmml">𝒆</mi><mo id="A3.1.p1.36.m5.1.2.1" xref="A3.1.p1.36.m5.1.2.1.cmml">⁢</mo><mrow id="A3.1.p1.36.m5.1.2.3.2" xref="A3.1.p1.36.m5.1.2.cmml"><mo id="A3.1.p1.36.m5.1.2.3.2.1" stretchy="false" xref="A3.1.p1.36.m5.1.2.cmml">(</mo><mi id="A3.1.p1.36.m5.1.1" xref="A3.1.p1.36.m5.1.1.cmml">t</mi><mo id="A3.1.p1.36.m5.1.2.3.2.2" stretchy="false" xref="A3.1.p1.36.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.36.m5.1b"><apply id="A3.1.p1.36.m5.1.2.cmml" xref="A3.1.p1.36.m5.1.2"><times id="A3.1.p1.36.m5.1.2.1.cmml" xref="A3.1.p1.36.m5.1.2.1"></times><ci id="A3.1.p1.36.m5.1.2.2.cmml" xref="A3.1.p1.36.m5.1.2.2">𝒆</ci><ci id="A3.1.p1.36.m5.1.1.cmml" xref="A3.1.p1.36.m5.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.36.m5.1c">\bm{e}(t)</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.36.m5.1d">bold_italic_e ( italic_t )</annotation></semantics></math> converges to a steady-state bound subject to <math alttext="\delta" class="ltx_Math" display="inline" id="A3.1.p1.37.m6.1"><semantics id="A3.1.p1.37.m6.1a"><mi id="A3.1.p1.37.m6.1.1" xref="A3.1.p1.37.m6.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="A3.1.p1.37.m6.1b"><ci id="A3.1.p1.37.m6.1.1.cmml" xref="A3.1.p1.37.m6.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.37.m6.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.37.m6.1d">italic_δ</annotation></semantics></math> on <math alttext="t\in[0,t_{\rm f})" class="ltx_Math" display="inline" id="A3.1.p1.38.m7.2"><semantics id="A3.1.p1.38.m7.2a"><mrow id="A3.1.p1.38.m7.2.2" xref="A3.1.p1.38.m7.2.2.cmml"><mi id="A3.1.p1.38.m7.2.2.3" xref="A3.1.p1.38.m7.2.2.3.cmml">t</mi><mo id="A3.1.p1.38.m7.2.2.2" xref="A3.1.p1.38.m7.2.2.2.cmml">∈</mo><mrow id="A3.1.p1.38.m7.2.2.1.1" xref="A3.1.p1.38.m7.2.2.1.2.cmml"><mo id="A3.1.p1.38.m7.2.2.1.1.2" stretchy="false" xref="A3.1.p1.38.m7.2.2.1.2.cmml">[</mo><mn id="A3.1.p1.38.m7.1.1" xref="A3.1.p1.38.m7.1.1.cmml">0</mn><mo id="A3.1.p1.38.m7.2.2.1.1.3" xref="A3.1.p1.38.m7.2.2.1.2.cmml">,</mo><msub id="A3.1.p1.38.m7.2.2.1.1.1" xref="A3.1.p1.38.m7.2.2.1.1.1.cmml"><mi id="A3.1.p1.38.m7.2.2.1.1.1.2" xref="A3.1.p1.38.m7.2.2.1.1.1.2.cmml">t</mi><mi id="A3.1.p1.38.m7.2.2.1.1.1.3" mathvariant="normal" xref="A3.1.p1.38.m7.2.2.1.1.1.3.cmml">f</mi></msub><mo id="A3.1.p1.38.m7.2.2.1.1.4" stretchy="false" xref="A3.1.p1.38.m7.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.38.m7.2b"><apply id="A3.1.p1.38.m7.2.2.cmml" xref="A3.1.p1.38.m7.2.2"><in id="A3.1.p1.38.m7.2.2.2.cmml" xref="A3.1.p1.38.m7.2.2.2"></in><ci id="A3.1.p1.38.m7.2.2.3.cmml" xref="A3.1.p1.38.m7.2.2.3">𝑡</ci><interval closure="closed-open" id="A3.1.p1.38.m7.2.2.1.2.cmml" xref="A3.1.p1.38.m7.2.2.1.1"><cn id="A3.1.p1.38.m7.1.1.cmml" type="integer" xref="A3.1.p1.38.m7.1.1">0</cn><apply id="A3.1.p1.38.m7.2.2.1.1.1.cmml" xref="A3.1.p1.38.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="A3.1.p1.38.m7.2.2.1.1.1.1.cmml" xref="A3.1.p1.38.m7.2.2.1.1.1">subscript</csymbol><ci id="A3.1.p1.38.m7.2.2.1.1.1.2.cmml" xref="A3.1.p1.38.m7.2.2.1.1.1.2">𝑡</ci><ci id="A3.1.p1.38.m7.2.2.1.1.1.3.cmml" xref="A3.1.p1.38.m7.2.2.1.1.1.3">f</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.38.m7.2c">t\in[0,t_{\rm f})</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.38.m7.2d">italic_t ∈ [ 0 , italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT )</annotation></semantics></math>. Based on (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A3.E39" title="In Proof. ‣ Appendix C The proof of Theorem 2 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">39</span></a>) and <math alttext="\|\tilde{\bm{\theta}}(t)\|\leq c_{\theta}" class="ltx_Math" display="inline" id="A3.1.p1.39.m8.2"><semantics id="A3.1.p1.39.m8.2a"><mrow id="A3.1.p1.39.m8.2.2" xref="A3.1.p1.39.m8.2.2.cmml"><mrow id="A3.1.p1.39.m8.2.2.1.1" xref="A3.1.p1.39.m8.2.2.1.2.cmml"><mo id="A3.1.p1.39.m8.2.2.1.1.2" stretchy="false" xref="A3.1.p1.39.m8.2.2.1.2.1.cmml">‖</mo><mrow id="A3.1.p1.39.m8.2.2.1.1.1" xref="A3.1.p1.39.m8.2.2.1.1.1.cmml"><mover accent="true" id="A3.1.p1.39.m8.2.2.1.1.1.2" xref="A3.1.p1.39.m8.2.2.1.1.1.2.cmml"><mi id="A3.1.p1.39.m8.2.2.1.1.1.2.2" xref="A3.1.p1.39.m8.2.2.1.1.1.2.2.cmml">𝜽</mi><mo id="A3.1.p1.39.m8.2.2.1.1.1.2.1" xref="A3.1.p1.39.m8.2.2.1.1.1.2.1.cmml">~</mo></mover><mo id="A3.1.p1.39.m8.2.2.1.1.1.1" xref="A3.1.p1.39.m8.2.2.1.1.1.1.cmml">⁢</mo><mrow id="A3.1.p1.39.m8.2.2.1.1.1.3.2" xref="A3.1.p1.39.m8.2.2.1.1.1.cmml"><mo id="A3.1.p1.39.m8.2.2.1.1.1.3.2.1" stretchy="false" xref="A3.1.p1.39.m8.2.2.1.1.1.cmml">(</mo><mi id="A3.1.p1.39.m8.1.1" xref="A3.1.p1.39.m8.1.1.cmml">t</mi><mo id="A3.1.p1.39.m8.2.2.1.1.1.3.2.2" stretchy="false" xref="A3.1.p1.39.m8.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="A3.1.p1.39.m8.2.2.1.1.3" stretchy="false" xref="A3.1.p1.39.m8.2.2.1.2.1.cmml">‖</mo></mrow><mo id="A3.1.p1.39.m8.2.2.2" xref="A3.1.p1.39.m8.2.2.2.cmml">≤</mo><msub id="A3.1.p1.39.m8.2.2.3" xref="A3.1.p1.39.m8.2.2.3.cmml"><mi id="A3.1.p1.39.m8.2.2.3.2" xref="A3.1.p1.39.m8.2.2.3.2.cmml">c</mi><mi id="A3.1.p1.39.m8.2.2.3.3" xref="A3.1.p1.39.m8.2.2.3.3.cmml">θ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.39.m8.2b"><apply id="A3.1.p1.39.m8.2.2.cmml" xref="A3.1.p1.39.m8.2.2"><leq id="A3.1.p1.39.m8.2.2.2.cmml" xref="A3.1.p1.39.m8.2.2.2"></leq><apply id="A3.1.p1.39.m8.2.2.1.2.cmml" xref="A3.1.p1.39.m8.2.2.1.1"><csymbol cd="latexml" id="A3.1.p1.39.m8.2.2.1.2.1.cmml" xref="A3.1.p1.39.m8.2.2.1.1.2">norm</csymbol><apply id="A3.1.p1.39.m8.2.2.1.1.1.cmml" xref="A3.1.p1.39.m8.2.2.1.1.1"><times id="A3.1.p1.39.m8.2.2.1.1.1.1.cmml" xref="A3.1.p1.39.m8.2.2.1.1.1.1"></times><apply id="A3.1.p1.39.m8.2.2.1.1.1.2.cmml" xref="A3.1.p1.39.m8.2.2.1.1.1.2"><ci id="A3.1.p1.39.m8.2.2.1.1.1.2.1.cmml" xref="A3.1.p1.39.m8.2.2.1.1.1.2.1">~</ci><ci id="A3.1.p1.39.m8.2.2.1.1.1.2.2.cmml" xref="A3.1.p1.39.m8.2.2.1.1.1.2.2">𝜽</ci></apply><ci id="A3.1.p1.39.m8.1.1.cmml" xref="A3.1.p1.39.m8.1.1">𝑡</ci></apply></apply><apply id="A3.1.p1.39.m8.2.2.3.cmml" xref="A3.1.p1.39.m8.2.2.3"><csymbol cd="ambiguous" id="A3.1.p1.39.m8.2.2.3.1.cmml" xref="A3.1.p1.39.m8.2.2.3">subscript</csymbol><ci id="A3.1.p1.39.m8.2.2.3.2.cmml" xref="A3.1.p1.39.m8.2.2.3.2">𝑐</ci><ci id="A3.1.p1.39.m8.2.2.3.3.cmml" xref="A3.1.p1.39.m8.2.2.3.3">𝜃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.39.m8.2c">\|\tilde{\bm{\theta}}(t)\|\leq c_{\theta}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.39.m8.2d">∥ over~ start_ARG bold_italic_θ end_ARG ( italic_t ) ∥ ≤ italic_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\forall t\in[0,\infty)" class="ltx_Math" display="inline" id="A3.1.p1.40.m9.2"><semantics id="A3.1.p1.40.m9.2a"><mrow id="A3.1.p1.40.m9.2.3" xref="A3.1.p1.40.m9.2.3.cmml"><mrow id="A3.1.p1.40.m9.2.3.2" xref="A3.1.p1.40.m9.2.3.2.cmml"><mo id="A3.1.p1.40.m9.2.3.2.1" rspace="0.167em" xref="A3.1.p1.40.m9.2.3.2.1.cmml">∀</mo><mi id="A3.1.p1.40.m9.2.3.2.2" xref="A3.1.p1.40.m9.2.3.2.2.cmml">t</mi></mrow><mo id="A3.1.p1.40.m9.2.3.1" xref="A3.1.p1.40.m9.2.3.1.cmml">∈</mo><mrow id="A3.1.p1.40.m9.2.3.3.2" xref="A3.1.p1.40.m9.2.3.3.1.cmml"><mo id="A3.1.p1.40.m9.2.3.3.2.1" stretchy="false" xref="A3.1.p1.40.m9.2.3.3.1.cmml">[</mo><mn id="A3.1.p1.40.m9.1.1" xref="A3.1.p1.40.m9.1.1.cmml">0</mn><mo id="A3.1.p1.40.m9.2.3.3.2.2" xref="A3.1.p1.40.m9.2.3.3.1.cmml">,</mo><mi id="A3.1.p1.40.m9.2.2" mathvariant="normal" xref="A3.1.p1.40.m9.2.2.cmml">∞</mi><mo id="A3.1.p1.40.m9.2.3.3.2.3" stretchy="false" xref="A3.1.p1.40.m9.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.40.m9.2b"><apply id="A3.1.p1.40.m9.2.3.cmml" xref="A3.1.p1.40.m9.2.3"><in id="A3.1.p1.40.m9.2.3.1.cmml" xref="A3.1.p1.40.m9.2.3.1"></in><apply id="A3.1.p1.40.m9.2.3.2.cmml" xref="A3.1.p1.40.m9.2.3.2"><csymbol cd="latexml" id="A3.1.p1.40.m9.2.3.2.1.cmml" xref="A3.1.p1.40.m9.2.3.2.1">for-all</csymbol><ci id="A3.1.p1.40.m9.2.3.2.2.cmml" xref="A3.1.p1.40.m9.2.3.2.2">𝑡</ci></apply><interval closure="closed-open" id="A3.1.p1.40.m9.2.3.3.1.cmml" xref="A3.1.p1.40.m9.2.3.3.2"><cn id="A3.1.p1.40.m9.1.1.cmml" type="integer" xref="A3.1.p1.40.m9.1.1">0</cn><infinity id="A3.1.p1.40.m9.2.2.cmml" xref="A3.1.p1.40.m9.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.40.m9.2c">\forall t\in[0,\infty)</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.40.m9.2d">∀ italic_t ∈ [ 0 , ∞ )</annotation></semantics></math>, the inequality (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A3.E46" title="In Proof. ‣ Appendix C The proof of Theorem 2 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">46</span></a>) can be rewritten into</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx61"> <tbody id="A3.Ex45"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}(t)\leq" class="ltx_Math" display="inline" id="A3.Ex45.m1.1"><semantics id="A3.Ex45.m1.1a"><mrow id="A3.Ex45.m1.1.2" xref="A3.Ex45.m1.1.2.cmml"><mrow id="A3.Ex45.m1.1.2.2" xref="A3.Ex45.m1.1.2.2.cmml"><mover accent="true" id="A3.Ex45.m1.1.2.2.2" xref="A3.Ex45.m1.1.2.2.2.cmml"><mi id="A3.Ex45.m1.1.2.2.2.2" xref="A3.Ex45.m1.1.2.2.2.2.cmml">V</mi><mo id="A3.Ex45.m1.1.2.2.2.1" xref="A3.Ex45.m1.1.2.2.2.1.cmml">˙</mo></mover><mo id="A3.Ex45.m1.1.2.2.1" xref="A3.Ex45.m1.1.2.2.1.cmml">⁢</mo><mrow id="A3.Ex45.m1.1.2.2.3.2" xref="A3.Ex45.m1.1.2.2.cmml"><mo id="A3.Ex45.m1.1.2.2.3.2.1" stretchy="false" xref="A3.Ex45.m1.1.2.2.cmml">(</mo><mi id="A3.Ex45.m1.1.1" xref="A3.Ex45.m1.1.1.cmml">t</mi><mo id="A3.Ex45.m1.1.2.2.3.2.2" stretchy="false" xref="A3.Ex45.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="A3.Ex45.m1.1.2.1" xref="A3.Ex45.m1.1.2.1.cmml">≤</mo><mi id="A3.Ex45.m1.1.2.3" xref="A3.Ex45.m1.1.2.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex45.m1.1b"><apply id="A3.Ex45.m1.1.2.cmml" xref="A3.Ex45.m1.1.2"><leq id="A3.Ex45.m1.1.2.1.cmml" xref="A3.Ex45.m1.1.2.1"></leq><apply id="A3.Ex45.m1.1.2.2.cmml" xref="A3.Ex45.m1.1.2.2"><times id="A3.Ex45.m1.1.2.2.1.cmml" xref="A3.Ex45.m1.1.2.2.1"></times><apply id="A3.Ex45.m1.1.2.2.2.cmml" xref="A3.Ex45.m1.1.2.2.2"><ci id="A3.Ex45.m1.1.2.2.2.1.cmml" xref="A3.Ex45.m1.1.2.2.2.1">˙</ci><ci id="A3.Ex45.m1.1.2.2.2.2.cmml" xref="A3.Ex45.m1.1.2.2.2.2">𝑉</ci></apply><ci id="A3.Ex45.m1.1.1.cmml" xref="A3.Ex45.m1.1.1">𝑡</ci></apply><csymbol cd="latexml" id="A3.Ex45.m1.1.2.3.cmml" xref="A3.Ex45.m1.1.2.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex45.m1.1c">\displaystyle\dot{V}(t)\leq</annotation><annotation encoding="application/x-llamapun" id="A3.Ex45.m1.1d">over˙ start_ARG italic_V end_ARG ( italic_t ) ≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-k_{\rm c}\|\bm{e}\|^{2}/2-k_{\rm c}(1+p)\tilde{\bm{\theta}}^{T}% \Gamma^{-1}\tilde{\bm{\theta}}/2" class="ltx_Math" display="inline" id="A3.Ex45.m2.2"><semantics id="A3.Ex45.m2.2a"><mrow id="A3.Ex45.m2.2.2" xref="A3.Ex45.m2.2.2.cmml"><mrow id="A3.Ex45.m2.2.2.3" xref="A3.Ex45.m2.2.2.3.cmml"><mo id="A3.Ex45.m2.2.2.3a" xref="A3.Ex45.m2.2.2.3.cmml">−</mo><mrow id="A3.Ex45.m2.2.2.3.2" xref="A3.Ex45.m2.2.2.3.2.cmml"><mrow id="A3.Ex45.m2.2.2.3.2.2" xref="A3.Ex45.m2.2.2.3.2.2.cmml"><msub id="A3.Ex45.m2.2.2.3.2.2.2" xref="A3.Ex45.m2.2.2.3.2.2.2.cmml"><mi id="A3.Ex45.m2.2.2.3.2.2.2.2" xref="A3.Ex45.m2.2.2.3.2.2.2.2.cmml">k</mi><mi id="A3.Ex45.m2.2.2.3.2.2.2.3" mathvariant="normal" xref="A3.Ex45.m2.2.2.3.2.2.2.3.cmml">c</mi></msub><mo id="A3.Ex45.m2.2.2.3.2.2.1" xref="A3.Ex45.m2.2.2.3.2.2.1.cmml">⁢</mo><msup id="A3.Ex45.m2.2.2.3.2.2.3" xref="A3.Ex45.m2.2.2.3.2.2.3.cmml"><mrow id="A3.Ex45.m2.2.2.3.2.2.3.2.2" xref="A3.Ex45.m2.2.2.3.2.2.3.2.1.cmml"><mo id="A3.Ex45.m2.2.2.3.2.2.3.2.2.1" stretchy="false" xref="A3.Ex45.m2.2.2.3.2.2.3.2.1.1.cmml">‖</mo><mi id="A3.Ex45.m2.1.1" xref="A3.Ex45.m2.1.1.cmml">𝒆</mi><mo id="A3.Ex45.m2.2.2.3.2.2.3.2.2.2" stretchy="false" xref="A3.Ex45.m2.2.2.3.2.2.3.2.1.1.cmml">‖</mo></mrow><mn id="A3.Ex45.m2.2.2.3.2.2.3.3" xref="A3.Ex45.m2.2.2.3.2.2.3.3.cmml">2</mn></msup></mrow><mo id="A3.Ex45.m2.2.2.3.2.1" xref="A3.Ex45.m2.2.2.3.2.1.cmml">/</mo><mn id="A3.Ex45.m2.2.2.3.2.3" xref="A3.Ex45.m2.2.2.3.2.3.cmml">2</mn></mrow></mrow><mo id="A3.Ex45.m2.2.2.2" xref="A3.Ex45.m2.2.2.2.cmml">−</mo><mrow id="A3.Ex45.m2.2.2.1" xref="A3.Ex45.m2.2.2.1.cmml"><mrow id="A3.Ex45.m2.2.2.1.1" xref="A3.Ex45.m2.2.2.1.1.cmml"><msub id="A3.Ex45.m2.2.2.1.1.3" xref="A3.Ex45.m2.2.2.1.1.3.cmml"><mi id="A3.Ex45.m2.2.2.1.1.3.2" xref="A3.Ex45.m2.2.2.1.1.3.2.cmml">k</mi><mi id="A3.Ex45.m2.2.2.1.1.3.3" mathvariant="normal" xref="A3.Ex45.m2.2.2.1.1.3.3.cmml">c</mi></msub><mo id="A3.Ex45.m2.2.2.1.1.2" xref="A3.Ex45.m2.2.2.1.1.2.cmml">⁢</mo><mrow id="A3.Ex45.m2.2.2.1.1.1.1" xref="A3.Ex45.m2.2.2.1.1.1.1.1.cmml"><mo id="A3.Ex45.m2.2.2.1.1.1.1.2" stretchy="false" xref="A3.Ex45.m2.2.2.1.1.1.1.1.cmml">(</mo><mrow id="A3.Ex45.m2.2.2.1.1.1.1.1" xref="A3.Ex45.m2.2.2.1.1.1.1.1.cmml"><mn id="A3.Ex45.m2.2.2.1.1.1.1.1.2" xref="A3.Ex45.m2.2.2.1.1.1.1.1.2.cmml">1</mn><mo id="A3.Ex45.m2.2.2.1.1.1.1.1.1" xref="A3.Ex45.m2.2.2.1.1.1.1.1.1.cmml">+</mo><mi id="A3.Ex45.m2.2.2.1.1.1.1.1.3" xref="A3.Ex45.m2.2.2.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A3.Ex45.m2.2.2.1.1.1.1.3" stretchy="false" xref="A3.Ex45.m2.2.2.1.1.1.1.1.cmml">)</mo></mrow><mo id="A3.Ex45.m2.2.2.1.1.2a" xref="A3.Ex45.m2.2.2.1.1.2.cmml">⁢</mo><msup id="A3.Ex45.m2.2.2.1.1.4" xref="A3.Ex45.m2.2.2.1.1.4.cmml"><mover accent="true" id="A3.Ex45.m2.2.2.1.1.4.2" xref="A3.Ex45.m2.2.2.1.1.4.2.cmml"><mi id="A3.Ex45.m2.2.2.1.1.4.2.2" xref="A3.Ex45.m2.2.2.1.1.4.2.2.cmml">𝜽</mi><mo id="A3.Ex45.m2.2.2.1.1.4.2.1" xref="A3.Ex45.m2.2.2.1.1.4.2.1.cmml">~</mo></mover><mi id="A3.Ex45.m2.2.2.1.1.4.3" xref="A3.Ex45.m2.2.2.1.1.4.3.cmml">T</mi></msup><mo id="A3.Ex45.m2.2.2.1.1.2b" xref="A3.Ex45.m2.2.2.1.1.2.cmml">⁢</mo><msup id="A3.Ex45.m2.2.2.1.1.5" xref="A3.Ex45.m2.2.2.1.1.5.cmml"><mi id="A3.Ex45.m2.2.2.1.1.5.2" mathvariant="normal" xref="A3.Ex45.m2.2.2.1.1.5.2.cmml">Γ</mi><mrow id="A3.Ex45.m2.2.2.1.1.5.3" xref="A3.Ex45.m2.2.2.1.1.5.3.cmml"><mo id="A3.Ex45.m2.2.2.1.1.5.3a" xref="A3.Ex45.m2.2.2.1.1.5.3.cmml">−</mo><mn id="A3.Ex45.m2.2.2.1.1.5.3.2" xref="A3.Ex45.m2.2.2.1.1.5.3.2.cmml">1</mn></mrow></msup><mo id="A3.Ex45.m2.2.2.1.1.2c" xref="A3.Ex45.m2.2.2.1.1.2.cmml">⁢</mo><mover accent="true" id="A3.Ex45.m2.2.2.1.1.6" xref="A3.Ex45.m2.2.2.1.1.6.cmml"><mi id="A3.Ex45.m2.2.2.1.1.6.2" xref="A3.Ex45.m2.2.2.1.1.6.2.cmml">𝜽</mi><mo id="A3.Ex45.m2.2.2.1.1.6.1" xref="A3.Ex45.m2.2.2.1.1.6.1.cmml">~</mo></mover></mrow><mo id="A3.Ex45.m2.2.2.1.2" xref="A3.Ex45.m2.2.2.1.2.cmml">/</mo><mn id="A3.Ex45.m2.2.2.1.3" xref="A3.Ex45.m2.2.2.1.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex45.m2.2b"><apply id="A3.Ex45.m2.2.2.cmml" xref="A3.Ex45.m2.2.2"><minus id="A3.Ex45.m2.2.2.2.cmml" xref="A3.Ex45.m2.2.2.2"></minus><apply id="A3.Ex45.m2.2.2.3.cmml" xref="A3.Ex45.m2.2.2.3"><minus id="A3.Ex45.m2.2.2.3.1.cmml" xref="A3.Ex45.m2.2.2.3"></minus><apply id="A3.Ex45.m2.2.2.3.2.cmml" xref="A3.Ex45.m2.2.2.3.2"><divide id="A3.Ex45.m2.2.2.3.2.1.cmml" xref="A3.Ex45.m2.2.2.3.2.1"></divide><apply id="A3.Ex45.m2.2.2.3.2.2.cmml" xref="A3.Ex45.m2.2.2.3.2.2"><times id="A3.Ex45.m2.2.2.3.2.2.1.cmml" xref="A3.Ex45.m2.2.2.3.2.2.1"></times><apply id="A3.Ex45.m2.2.2.3.2.2.2.cmml" xref="A3.Ex45.m2.2.2.3.2.2.2"><csymbol cd="ambiguous" id="A3.Ex45.m2.2.2.3.2.2.2.1.cmml" xref="A3.Ex45.m2.2.2.3.2.2.2">subscript</csymbol><ci id="A3.Ex45.m2.2.2.3.2.2.2.2.cmml" xref="A3.Ex45.m2.2.2.3.2.2.2.2">𝑘</ci><ci id="A3.Ex45.m2.2.2.3.2.2.2.3.cmml" xref="A3.Ex45.m2.2.2.3.2.2.2.3">c</ci></apply><apply id="A3.Ex45.m2.2.2.3.2.2.3.cmml" xref="A3.Ex45.m2.2.2.3.2.2.3"><csymbol cd="ambiguous" id="A3.Ex45.m2.2.2.3.2.2.3.1.cmml" xref="A3.Ex45.m2.2.2.3.2.2.3">superscript</csymbol><apply id="A3.Ex45.m2.2.2.3.2.2.3.2.1.cmml" xref="A3.Ex45.m2.2.2.3.2.2.3.2.2"><csymbol cd="latexml" id="A3.Ex45.m2.2.2.3.2.2.3.2.1.1.cmml" xref="A3.Ex45.m2.2.2.3.2.2.3.2.2.1">norm</csymbol><ci id="A3.Ex45.m2.1.1.cmml" xref="A3.Ex45.m2.1.1">𝒆</ci></apply><cn id="A3.Ex45.m2.2.2.3.2.2.3.3.cmml" type="integer" xref="A3.Ex45.m2.2.2.3.2.2.3.3">2</cn></apply></apply><cn id="A3.Ex45.m2.2.2.3.2.3.cmml" type="integer" xref="A3.Ex45.m2.2.2.3.2.3">2</cn></apply></apply><apply id="A3.Ex45.m2.2.2.1.cmml" xref="A3.Ex45.m2.2.2.1"><divide id="A3.Ex45.m2.2.2.1.2.cmml" xref="A3.Ex45.m2.2.2.1.2"></divide><apply id="A3.Ex45.m2.2.2.1.1.cmml" xref="A3.Ex45.m2.2.2.1.1"><times id="A3.Ex45.m2.2.2.1.1.2.cmml" xref="A3.Ex45.m2.2.2.1.1.2"></times><apply id="A3.Ex45.m2.2.2.1.1.3.cmml" xref="A3.Ex45.m2.2.2.1.1.3"><csymbol cd="ambiguous" id="A3.Ex45.m2.2.2.1.1.3.1.cmml" xref="A3.Ex45.m2.2.2.1.1.3">subscript</csymbol><ci id="A3.Ex45.m2.2.2.1.1.3.2.cmml" xref="A3.Ex45.m2.2.2.1.1.3.2">𝑘</ci><ci id="A3.Ex45.m2.2.2.1.1.3.3.cmml" xref="A3.Ex45.m2.2.2.1.1.3.3">c</ci></apply><apply id="A3.Ex45.m2.2.2.1.1.1.1.1.cmml" xref="A3.Ex45.m2.2.2.1.1.1.1"><plus id="A3.Ex45.m2.2.2.1.1.1.1.1.1.cmml" xref="A3.Ex45.m2.2.2.1.1.1.1.1.1"></plus><cn id="A3.Ex45.m2.2.2.1.1.1.1.1.2.cmml" type="integer" xref="A3.Ex45.m2.2.2.1.1.1.1.1.2">1</cn><ci id="A3.Ex45.m2.2.2.1.1.1.1.1.3.cmml" xref="A3.Ex45.m2.2.2.1.1.1.1.1.3">𝑝</ci></apply><apply id="A3.Ex45.m2.2.2.1.1.4.cmml" xref="A3.Ex45.m2.2.2.1.1.4"><csymbol cd="ambiguous" id="A3.Ex45.m2.2.2.1.1.4.1.cmml" xref="A3.Ex45.m2.2.2.1.1.4">superscript</csymbol><apply id="A3.Ex45.m2.2.2.1.1.4.2.cmml" xref="A3.Ex45.m2.2.2.1.1.4.2"><ci id="A3.Ex45.m2.2.2.1.1.4.2.1.cmml" xref="A3.Ex45.m2.2.2.1.1.4.2.1">~</ci><ci id="A3.Ex45.m2.2.2.1.1.4.2.2.cmml" xref="A3.Ex45.m2.2.2.1.1.4.2.2">𝜽</ci></apply><ci id="A3.Ex45.m2.2.2.1.1.4.3.cmml" xref="A3.Ex45.m2.2.2.1.1.4.3">𝑇</ci></apply><apply id="A3.Ex45.m2.2.2.1.1.5.cmml" xref="A3.Ex45.m2.2.2.1.1.5"><csymbol cd="ambiguous" id="A3.Ex45.m2.2.2.1.1.5.1.cmml" xref="A3.Ex45.m2.2.2.1.1.5">superscript</csymbol><ci id="A3.Ex45.m2.2.2.1.1.5.2.cmml" xref="A3.Ex45.m2.2.2.1.1.5.2">Γ</ci><apply id="A3.Ex45.m2.2.2.1.1.5.3.cmml" xref="A3.Ex45.m2.2.2.1.1.5.3"><minus id="A3.Ex45.m2.2.2.1.1.5.3.1.cmml" xref="A3.Ex45.m2.2.2.1.1.5.3"></minus><cn id="A3.Ex45.m2.2.2.1.1.5.3.2.cmml" type="integer" xref="A3.Ex45.m2.2.2.1.1.5.3.2">1</cn></apply></apply><apply id="A3.Ex45.m2.2.2.1.1.6.cmml" xref="A3.Ex45.m2.2.2.1.1.6"><ci id="A3.Ex45.m2.2.2.1.1.6.1.cmml" xref="A3.Ex45.m2.2.2.1.1.6.1">~</ci><ci id="A3.Ex45.m2.2.2.1.1.6.2.cmml" xref="A3.Ex45.m2.2.2.1.1.6.2">𝜽</ci></apply></apply><cn id="A3.Ex45.m2.2.2.1.3.cmml" type="integer" xref="A3.Ex45.m2.2.2.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex45.m2.2c">\displaystyle-k_{\rm c}\|\bm{e}\|^{2}/2-k_{\rm c}(1+p)\tilde{\bm{\theta}}^{T}% \Gamma^{-1}\tilde{\bm{\theta}}/2</annotation><annotation encoding="application/x-llamapun" id="A3.Ex45.m2.2d">- italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ∥ bold_italic_e ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 - italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ( 1 + italic_p ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Γ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG / 2</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex46"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle+k_{\rm c}(1+p)c_{\theta}^{2}/(2\lambda_{\min}(\Gamma))+(\delta c% _{\theta})^{2}/(2k_{\rm c})" class="ltx_Math" display="inline" id="A3.Ex46.m1.5"><semantics id="A3.Ex46.m1.5a"><mrow id="A3.Ex46.m1.5.5" xref="A3.Ex46.m1.5.5.cmml"><mrow id="A3.Ex46.m1.3.3.2" xref="A3.Ex46.m1.3.3.2.cmml"><mo id="A3.Ex46.m1.3.3.2a" xref="A3.Ex46.m1.3.3.2.cmml">+</mo><mrow id="A3.Ex46.m1.3.3.2.2" xref="A3.Ex46.m1.3.3.2.2.cmml"><mrow id="A3.Ex46.m1.2.2.1.1.1" xref="A3.Ex46.m1.2.2.1.1.1.cmml"><msub id="A3.Ex46.m1.2.2.1.1.1.3" xref="A3.Ex46.m1.2.2.1.1.1.3.cmml"><mi id="A3.Ex46.m1.2.2.1.1.1.3.2" xref="A3.Ex46.m1.2.2.1.1.1.3.2.cmml">k</mi><mi id="A3.Ex46.m1.2.2.1.1.1.3.3" mathvariant="normal" xref="A3.Ex46.m1.2.2.1.1.1.3.3.cmml">c</mi></msub><mo id="A3.Ex46.m1.2.2.1.1.1.2" xref="A3.Ex46.m1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="A3.Ex46.m1.2.2.1.1.1.1.1" xref="A3.Ex46.m1.2.2.1.1.1.1.1.1.cmml"><mo id="A3.Ex46.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="A3.Ex46.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.Ex46.m1.2.2.1.1.1.1.1.1" xref="A3.Ex46.m1.2.2.1.1.1.1.1.1.cmml"><mn id="A3.Ex46.m1.2.2.1.1.1.1.1.1.2" xref="A3.Ex46.m1.2.2.1.1.1.1.1.1.2.cmml">1</mn><mo id="A3.Ex46.m1.2.2.1.1.1.1.1.1.1" xref="A3.Ex46.m1.2.2.1.1.1.1.1.1.1.cmml">+</mo><mi id="A3.Ex46.m1.2.2.1.1.1.1.1.1.3" xref="A3.Ex46.m1.2.2.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A3.Ex46.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="A3.Ex46.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A3.Ex46.m1.2.2.1.1.1.2a" xref="A3.Ex46.m1.2.2.1.1.1.2.cmml">⁢</mo><msubsup id="A3.Ex46.m1.2.2.1.1.1.4" xref="A3.Ex46.m1.2.2.1.1.1.4.cmml"><mi id="A3.Ex46.m1.2.2.1.1.1.4.2.2" xref="A3.Ex46.m1.2.2.1.1.1.4.2.2.cmml">c</mi><mi id="A3.Ex46.m1.2.2.1.1.1.4.2.3" xref="A3.Ex46.m1.2.2.1.1.1.4.2.3.cmml">θ</mi><mn id="A3.Ex46.m1.2.2.1.1.1.4.3" xref="A3.Ex46.m1.2.2.1.1.1.4.3.cmml">2</mn></msubsup></mrow><mo id="A3.Ex46.m1.3.3.2.2.3" xref="A3.Ex46.m1.3.3.2.2.3.cmml">/</mo><mrow id="A3.Ex46.m1.3.3.2.2.2.1" xref="A3.Ex46.m1.3.3.2.2.2.1.1.cmml"><mo id="A3.Ex46.m1.3.3.2.2.2.1.2" stretchy="false" xref="A3.Ex46.m1.3.3.2.2.2.1.1.cmml">(</mo><mrow id="A3.Ex46.m1.3.3.2.2.2.1.1" xref="A3.Ex46.m1.3.3.2.2.2.1.1.cmml"><mn id="A3.Ex46.m1.3.3.2.2.2.1.1.2" xref="A3.Ex46.m1.3.3.2.2.2.1.1.2.cmml">2</mn><mo id="A3.Ex46.m1.3.3.2.2.2.1.1.1" xref="A3.Ex46.m1.3.3.2.2.2.1.1.1.cmml">⁢</mo><msub id="A3.Ex46.m1.3.3.2.2.2.1.1.3" xref="A3.Ex46.m1.3.3.2.2.2.1.1.3.cmml"><mi id="A3.Ex46.m1.3.3.2.2.2.1.1.3.2" xref="A3.Ex46.m1.3.3.2.2.2.1.1.3.2.cmml">λ</mi><mi id="A3.Ex46.m1.3.3.2.2.2.1.1.3.3" xref="A3.Ex46.m1.3.3.2.2.2.1.1.3.3.cmml">min</mi></msub><mo id="A3.Ex46.m1.3.3.2.2.2.1.1.1a" xref="A3.Ex46.m1.3.3.2.2.2.1.1.1.cmml">⁢</mo><mrow id="A3.Ex46.m1.3.3.2.2.2.1.1.4.2" xref="A3.Ex46.m1.3.3.2.2.2.1.1.cmml"><mo id="A3.Ex46.m1.3.3.2.2.2.1.1.4.2.1" stretchy="false" xref="A3.Ex46.m1.3.3.2.2.2.1.1.cmml">(</mo><mi id="A3.Ex46.m1.1.1" mathvariant="normal" xref="A3.Ex46.m1.1.1.cmml">Γ</mi><mo id="A3.Ex46.m1.3.3.2.2.2.1.1.4.2.2" stretchy="false" xref="A3.Ex46.m1.3.3.2.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="A3.Ex46.m1.3.3.2.2.2.1.3" stretchy="false" xref="A3.Ex46.m1.3.3.2.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A3.Ex46.m1.5.5.5" xref="A3.Ex46.m1.5.5.5.cmml">+</mo><mrow id="A3.Ex46.m1.5.5.4" xref="A3.Ex46.m1.5.5.4.cmml"><msup id="A3.Ex46.m1.4.4.3.1" xref="A3.Ex46.m1.4.4.3.1.cmml"><mrow id="A3.Ex46.m1.4.4.3.1.1.1" xref="A3.Ex46.m1.4.4.3.1.1.1.1.cmml"><mo id="A3.Ex46.m1.4.4.3.1.1.1.2" stretchy="false" xref="A3.Ex46.m1.4.4.3.1.1.1.1.cmml">(</mo><mrow id="A3.Ex46.m1.4.4.3.1.1.1.1" xref="A3.Ex46.m1.4.4.3.1.1.1.1.cmml"><mi id="A3.Ex46.m1.4.4.3.1.1.1.1.2" xref="A3.Ex46.m1.4.4.3.1.1.1.1.2.cmml">δ</mi><mo id="A3.Ex46.m1.4.4.3.1.1.1.1.1" xref="A3.Ex46.m1.4.4.3.1.1.1.1.1.cmml">⁢</mo><msub id="A3.Ex46.m1.4.4.3.1.1.1.1.3" xref="A3.Ex46.m1.4.4.3.1.1.1.1.3.cmml"><mi id="A3.Ex46.m1.4.4.3.1.1.1.1.3.2" xref="A3.Ex46.m1.4.4.3.1.1.1.1.3.2.cmml">c</mi><mi id="A3.Ex46.m1.4.4.3.1.1.1.1.3.3" xref="A3.Ex46.m1.4.4.3.1.1.1.1.3.3.cmml">θ</mi></msub></mrow><mo id="A3.Ex46.m1.4.4.3.1.1.1.3" stretchy="false" xref="A3.Ex46.m1.4.4.3.1.1.1.1.cmml">)</mo></mrow><mn id="A3.Ex46.m1.4.4.3.1.3" xref="A3.Ex46.m1.4.4.3.1.3.cmml">2</mn></msup><mo id="A3.Ex46.m1.5.5.4.3" xref="A3.Ex46.m1.5.5.4.3.cmml">/</mo><mrow id="A3.Ex46.m1.5.5.4.2.1" xref="A3.Ex46.m1.5.5.4.2.1.1.cmml"><mo id="A3.Ex46.m1.5.5.4.2.1.2" stretchy="false" xref="A3.Ex46.m1.5.5.4.2.1.1.cmml">(</mo><mrow id="A3.Ex46.m1.5.5.4.2.1.1" xref="A3.Ex46.m1.5.5.4.2.1.1.cmml"><mn id="A3.Ex46.m1.5.5.4.2.1.1.2" xref="A3.Ex46.m1.5.5.4.2.1.1.2.cmml">2</mn><mo id="A3.Ex46.m1.5.5.4.2.1.1.1" xref="A3.Ex46.m1.5.5.4.2.1.1.1.cmml">⁢</mo><msub id="A3.Ex46.m1.5.5.4.2.1.1.3" xref="A3.Ex46.m1.5.5.4.2.1.1.3.cmml"><mi id="A3.Ex46.m1.5.5.4.2.1.1.3.2" xref="A3.Ex46.m1.5.5.4.2.1.1.3.2.cmml">k</mi><mi id="A3.Ex46.m1.5.5.4.2.1.1.3.3" mathvariant="normal" xref="A3.Ex46.m1.5.5.4.2.1.1.3.3.cmml">c</mi></msub></mrow><mo id="A3.Ex46.m1.5.5.4.2.1.3" stretchy="false" xref="A3.Ex46.m1.5.5.4.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex46.m1.5b"><apply id="A3.Ex46.m1.5.5.cmml" xref="A3.Ex46.m1.5.5"><plus id="A3.Ex46.m1.5.5.5.cmml" xref="A3.Ex46.m1.5.5.5"></plus><apply id="A3.Ex46.m1.3.3.2.cmml" xref="A3.Ex46.m1.3.3.2"><plus id="A3.Ex46.m1.3.3.2.3.cmml" xref="A3.Ex46.m1.3.3.2"></plus><apply id="A3.Ex46.m1.3.3.2.2.cmml" xref="A3.Ex46.m1.3.3.2.2"><divide id="A3.Ex46.m1.3.3.2.2.3.cmml" xref="A3.Ex46.m1.3.3.2.2.3"></divide><apply id="A3.Ex46.m1.2.2.1.1.1.cmml" xref="A3.Ex46.m1.2.2.1.1.1"><times id="A3.Ex46.m1.2.2.1.1.1.2.cmml" xref="A3.Ex46.m1.2.2.1.1.1.2"></times><apply id="A3.Ex46.m1.2.2.1.1.1.3.cmml" xref="A3.Ex46.m1.2.2.1.1.1.3"><csymbol cd="ambiguous" id="A3.Ex46.m1.2.2.1.1.1.3.1.cmml" xref="A3.Ex46.m1.2.2.1.1.1.3">subscript</csymbol><ci id="A3.Ex46.m1.2.2.1.1.1.3.2.cmml" xref="A3.Ex46.m1.2.2.1.1.1.3.2">𝑘</ci><ci id="A3.Ex46.m1.2.2.1.1.1.3.3.cmml" xref="A3.Ex46.m1.2.2.1.1.1.3.3">c</ci></apply><apply id="A3.Ex46.m1.2.2.1.1.1.1.1.1.cmml" xref="A3.Ex46.m1.2.2.1.1.1.1.1"><plus id="A3.Ex46.m1.2.2.1.1.1.1.1.1.1.cmml" xref="A3.Ex46.m1.2.2.1.1.1.1.1.1.1"></plus><cn id="A3.Ex46.m1.2.2.1.1.1.1.1.1.2.cmml" type="integer" xref="A3.Ex46.m1.2.2.1.1.1.1.1.1.2">1</cn><ci id="A3.Ex46.m1.2.2.1.1.1.1.1.1.3.cmml" xref="A3.Ex46.m1.2.2.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="A3.Ex46.m1.2.2.1.1.1.4.cmml" xref="A3.Ex46.m1.2.2.1.1.1.4"><csymbol cd="ambiguous" id="A3.Ex46.m1.2.2.1.1.1.4.1.cmml" xref="A3.Ex46.m1.2.2.1.1.1.4">superscript</csymbol><apply id="A3.Ex46.m1.2.2.1.1.1.4.2.cmml" xref="A3.Ex46.m1.2.2.1.1.1.4"><csymbol cd="ambiguous" id="A3.Ex46.m1.2.2.1.1.1.4.2.1.cmml" xref="A3.Ex46.m1.2.2.1.1.1.4">subscript</csymbol><ci id="A3.Ex46.m1.2.2.1.1.1.4.2.2.cmml" xref="A3.Ex46.m1.2.2.1.1.1.4.2.2">𝑐</ci><ci id="A3.Ex46.m1.2.2.1.1.1.4.2.3.cmml" xref="A3.Ex46.m1.2.2.1.1.1.4.2.3">𝜃</ci></apply><cn id="A3.Ex46.m1.2.2.1.1.1.4.3.cmml" type="integer" xref="A3.Ex46.m1.2.2.1.1.1.4.3">2</cn></apply></apply><apply id="A3.Ex46.m1.3.3.2.2.2.1.1.cmml" xref="A3.Ex46.m1.3.3.2.2.2.1"><times id="A3.Ex46.m1.3.3.2.2.2.1.1.1.cmml" xref="A3.Ex46.m1.3.3.2.2.2.1.1.1"></times><cn id="A3.Ex46.m1.3.3.2.2.2.1.1.2.cmml" type="integer" xref="A3.Ex46.m1.3.3.2.2.2.1.1.2">2</cn><apply id="A3.Ex46.m1.3.3.2.2.2.1.1.3.cmml" xref="A3.Ex46.m1.3.3.2.2.2.1.1.3"><csymbol cd="ambiguous" id="A3.Ex46.m1.3.3.2.2.2.1.1.3.1.cmml" xref="A3.Ex46.m1.3.3.2.2.2.1.1.3">subscript</csymbol><ci id="A3.Ex46.m1.3.3.2.2.2.1.1.3.2.cmml" xref="A3.Ex46.m1.3.3.2.2.2.1.1.3.2">𝜆</ci><min id="A3.Ex46.m1.3.3.2.2.2.1.1.3.3.cmml" xref="A3.Ex46.m1.3.3.2.2.2.1.1.3.3"></min></apply><ci id="A3.Ex46.m1.1.1.cmml" xref="A3.Ex46.m1.1.1">Γ</ci></apply></apply></apply><apply id="A3.Ex46.m1.5.5.4.cmml" xref="A3.Ex46.m1.5.5.4"><divide id="A3.Ex46.m1.5.5.4.3.cmml" xref="A3.Ex46.m1.5.5.4.3"></divide><apply id="A3.Ex46.m1.4.4.3.1.cmml" xref="A3.Ex46.m1.4.4.3.1"><csymbol cd="ambiguous" id="A3.Ex46.m1.4.4.3.1.2.cmml" xref="A3.Ex46.m1.4.4.3.1">superscript</csymbol><apply id="A3.Ex46.m1.4.4.3.1.1.1.1.cmml" xref="A3.Ex46.m1.4.4.3.1.1.1"><times id="A3.Ex46.m1.4.4.3.1.1.1.1.1.cmml" xref="A3.Ex46.m1.4.4.3.1.1.1.1.1"></times><ci id="A3.Ex46.m1.4.4.3.1.1.1.1.2.cmml" xref="A3.Ex46.m1.4.4.3.1.1.1.1.2">𝛿</ci><apply id="A3.Ex46.m1.4.4.3.1.1.1.1.3.cmml" xref="A3.Ex46.m1.4.4.3.1.1.1.1.3"><csymbol cd="ambiguous" id="A3.Ex46.m1.4.4.3.1.1.1.1.3.1.cmml" xref="A3.Ex46.m1.4.4.3.1.1.1.1.3">subscript</csymbol><ci id="A3.Ex46.m1.4.4.3.1.1.1.1.3.2.cmml" xref="A3.Ex46.m1.4.4.3.1.1.1.1.3.2">𝑐</ci><ci id="A3.Ex46.m1.4.4.3.1.1.1.1.3.3.cmml" xref="A3.Ex46.m1.4.4.3.1.1.1.1.3.3">𝜃</ci></apply></apply><cn id="A3.Ex46.m1.4.4.3.1.3.cmml" type="integer" xref="A3.Ex46.m1.4.4.3.1.3">2</cn></apply><apply id="A3.Ex46.m1.5.5.4.2.1.1.cmml" xref="A3.Ex46.m1.5.5.4.2.1"><times id="A3.Ex46.m1.5.5.4.2.1.1.1.cmml" xref="A3.Ex46.m1.5.5.4.2.1.1.1"></times><cn id="A3.Ex46.m1.5.5.4.2.1.1.2.cmml" type="integer" xref="A3.Ex46.m1.5.5.4.2.1.1.2">2</cn><apply id="A3.Ex46.m1.5.5.4.2.1.1.3.cmml" xref="A3.Ex46.m1.5.5.4.2.1.1.3"><csymbol cd="ambiguous" id="A3.Ex46.m1.5.5.4.2.1.1.3.1.cmml" xref="A3.Ex46.m1.5.5.4.2.1.1.3">subscript</csymbol><ci id="A3.Ex46.m1.5.5.4.2.1.1.3.2.cmml" xref="A3.Ex46.m1.5.5.4.2.1.1.3.2">𝑘</ci><ci id="A3.Ex46.m1.5.5.4.2.1.1.3.3.cmml" xref="A3.Ex46.m1.5.5.4.2.1.1.3.3">c</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex46.m1.5c">\displaystyle+k_{\rm c}(1+p)c_{\theta}^{2}/(2\lambda_{\min}(\Gamma))+(\delta c% _{\theta})^{2}/(2k_{\rm c})</annotation><annotation encoding="application/x-llamapun" id="A3.Ex46.m1.5d">+ italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ( 1 + italic_p ) italic_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( 2 italic_λ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Γ ) ) + ( italic_δ italic_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( 2 italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex47"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="A3.Ex47.m1.1"><semantics id="A3.Ex47.m1.1a"><mo id="A3.Ex47.m1.1.1" xref="A3.Ex47.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="A3.Ex47.m1.1b"><eq id="A3.Ex47.m1.1.1.cmml" xref="A3.Ex47.m1.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex47.m1.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="A3.Ex47.m1.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-k_{\rm c}V(t)+k_{\rm c}(1+p)c_{\theta}^{2}/(2\lambda_{\min}(% \Gamma))+(\delta c_{\theta})^{2}/(2k_{\rm c})" class="ltx_Math" display="inline" id="A3.Ex47.m2.6"><semantics id="A3.Ex47.m2.6a"><mrow id="A3.Ex47.m2.6.6" xref="A3.Ex47.m2.6.6.cmml"><mrow id="A3.Ex47.m2.6.6.6" xref="A3.Ex47.m2.6.6.6.cmml"><mo id="A3.Ex47.m2.6.6.6a" xref="A3.Ex47.m2.6.6.6.cmml">−</mo><mrow id="A3.Ex47.m2.6.6.6.2" xref="A3.Ex47.m2.6.6.6.2.cmml"><msub id="A3.Ex47.m2.6.6.6.2.2" xref="A3.Ex47.m2.6.6.6.2.2.cmml"><mi id="A3.Ex47.m2.6.6.6.2.2.2" xref="A3.Ex47.m2.6.6.6.2.2.2.cmml">k</mi><mi id="A3.Ex47.m2.6.6.6.2.2.3" mathvariant="normal" xref="A3.Ex47.m2.6.6.6.2.2.3.cmml">c</mi></msub><mo id="A3.Ex47.m2.6.6.6.2.1" xref="A3.Ex47.m2.6.6.6.2.1.cmml">⁢</mo><mi id="A3.Ex47.m2.6.6.6.2.3" xref="A3.Ex47.m2.6.6.6.2.3.cmml">V</mi><mo id="A3.Ex47.m2.6.6.6.2.1a" xref="A3.Ex47.m2.6.6.6.2.1.cmml">⁢</mo><mrow id="A3.Ex47.m2.6.6.6.2.4.2" xref="A3.Ex47.m2.6.6.6.2.cmml"><mo id="A3.Ex47.m2.6.6.6.2.4.2.1" stretchy="false" xref="A3.Ex47.m2.6.6.6.2.cmml">(</mo><mi id="A3.Ex47.m2.1.1" xref="A3.Ex47.m2.1.1.cmml">t</mi><mo id="A3.Ex47.m2.6.6.6.2.4.2.2" stretchy="false" xref="A3.Ex47.m2.6.6.6.2.cmml">)</mo></mrow></mrow></mrow><mo id="A3.Ex47.m2.6.6.5" xref="A3.Ex47.m2.6.6.5.cmml">+</mo><mrow id="A3.Ex47.m2.4.4.2" xref="A3.Ex47.m2.4.4.2.cmml"><mrow id="A3.Ex47.m2.3.3.1.1" xref="A3.Ex47.m2.3.3.1.1.cmml"><msub id="A3.Ex47.m2.3.3.1.1.3" xref="A3.Ex47.m2.3.3.1.1.3.cmml"><mi id="A3.Ex47.m2.3.3.1.1.3.2" xref="A3.Ex47.m2.3.3.1.1.3.2.cmml">k</mi><mi id="A3.Ex47.m2.3.3.1.1.3.3" mathvariant="normal" xref="A3.Ex47.m2.3.3.1.1.3.3.cmml">c</mi></msub><mo id="A3.Ex47.m2.3.3.1.1.2" xref="A3.Ex47.m2.3.3.1.1.2.cmml">⁢</mo><mrow id="A3.Ex47.m2.3.3.1.1.1.1" xref="A3.Ex47.m2.3.3.1.1.1.1.1.cmml"><mo id="A3.Ex47.m2.3.3.1.1.1.1.2" stretchy="false" xref="A3.Ex47.m2.3.3.1.1.1.1.1.cmml">(</mo><mrow id="A3.Ex47.m2.3.3.1.1.1.1.1" xref="A3.Ex47.m2.3.3.1.1.1.1.1.cmml"><mn id="A3.Ex47.m2.3.3.1.1.1.1.1.2" xref="A3.Ex47.m2.3.3.1.1.1.1.1.2.cmml">1</mn><mo id="A3.Ex47.m2.3.3.1.1.1.1.1.1" xref="A3.Ex47.m2.3.3.1.1.1.1.1.1.cmml">+</mo><mi id="A3.Ex47.m2.3.3.1.1.1.1.1.3" xref="A3.Ex47.m2.3.3.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A3.Ex47.m2.3.3.1.1.1.1.3" stretchy="false" xref="A3.Ex47.m2.3.3.1.1.1.1.1.cmml">)</mo></mrow><mo id="A3.Ex47.m2.3.3.1.1.2a" xref="A3.Ex47.m2.3.3.1.1.2.cmml">⁢</mo><msubsup id="A3.Ex47.m2.3.3.1.1.4" xref="A3.Ex47.m2.3.3.1.1.4.cmml"><mi id="A3.Ex47.m2.3.3.1.1.4.2.2" xref="A3.Ex47.m2.3.3.1.1.4.2.2.cmml">c</mi><mi id="A3.Ex47.m2.3.3.1.1.4.2.3" xref="A3.Ex47.m2.3.3.1.1.4.2.3.cmml">θ</mi><mn id="A3.Ex47.m2.3.3.1.1.4.3" xref="A3.Ex47.m2.3.3.1.1.4.3.cmml">2</mn></msubsup></mrow><mo id="A3.Ex47.m2.4.4.2.3" xref="A3.Ex47.m2.4.4.2.3.cmml">/</mo><mrow id="A3.Ex47.m2.4.4.2.2.1" xref="A3.Ex47.m2.4.4.2.2.1.1.cmml"><mo id="A3.Ex47.m2.4.4.2.2.1.2" stretchy="false" xref="A3.Ex47.m2.4.4.2.2.1.1.cmml">(</mo><mrow id="A3.Ex47.m2.4.4.2.2.1.1" xref="A3.Ex47.m2.4.4.2.2.1.1.cmml"><mn id="A3.Ex47.m2.4.4.2.2.1.1.2" xref="A3.Ex47.m2.4.4.2.2.1.1.2.cmml">2</mn><mo id="A3.Ex47.m2.4.4.2.2.1.1.1" xref="A3.Ex47.m2.4.4.2.2.1.1.1.cmml">⁢</mo><msub id="A3.Ex47.m2.4.4.2.2.1.1.3" xref="A3.Ex47.m2.4.4.2.2.1.1.3.cmml"><mi id="A3.Ex47.m2.4.4.2.2.1.1.3.2" xref="A3.Ex47.m2.4.4.2.2.1.1.3.2.cmml">λ</mi><mi id="A3.Ex47.m2.4.4.2.2.1.1.3.3" xref="A3.Ex47.m2.4.4.2.2.1.1.3.3.cmml">min</mi></msub><mo id="A3.Ex47.m2.4.4.2.2.1.1.1a" xref="A3.Ex47.m2.4.4.2.2.1.1.1.cmml">⁢</mo><mrow id="A3.Ex47.m2.4.4.2.2.1.1.4.2" xref="A3.Ex47.m2.4.4.2.2.1.1.cmml"><mo id="A3.Ex47.m2.4.4.2.2.1.1.4.2.1" stretchy="false" xref="A3.Ex47.m2.4.4.2.2.1.1.cmml">(</mo><mi id="A3.Ex47.m2.2.2" mathvariant="normal" xref="A3.Ex47.m2.2.2.cmml">Γ</mi><mo id="A3.Ex47.m2.4.4.2.2.1.1.4.2.2" stretchy="false" xref="A3.Ex47.m2.4.4.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="A3.Ex47.m2.4.4.2.2.1.3" stretchy="false" xref="A3.Ex47.m2.4.4.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="A3.Ex47.m2.6.6.5a" xref="A3.Ex47.m2.6.6.5.cmml">+</mo><mrow id="A3.Ex47.m2.6.6.4" xref="A3.Ex47.m2.6.6.4.cmml"><msup id="A3.Ex47.m2.5.5.3.1" xref="A3.Ex47.m2.5.5.3.1.cmml"><mrow id="A3.Ex47.m2.5.5.3.1.1.1" xref="A3.Ex47.m2.5.5.3.1.1.1.1.cmml"><mo id="A3.Ex47.m2.5.5.3.1.1.1.2" stretchy="false" xref="A3.Ex47.m2.5.5.3.1.1.1.1.cmml">(</mo><mrow id="A3.Ex47.m2.5.5.3.1.1.1.1" xref="A3.Ex47.m2.5.5.3.1.1.1.1.cmml"><mi id="A3.Ex47.m2.5.5.3.1.1.1.1.2" xref="A3.Ex47.m2.5.5.3.1.1.1.1.2.cmml">δ</mi><mo id="A3.Ex47.m2.5.5.3.1.1.1.1.1" xref="A3.Ex47.m2.5.5.3.1.1.1.1.1.cmml">⁢</mo><msub id="A3.Ex47.m2.5.5.3.1.1.1.1.3" xref="A3.Ex47.m2.5.5.3.1.1.1.1.3.cmml"><mi id="A3.Ex47.m2.5.5.3.1.1.1.1.3.2" xref="A3.Ex47.m2.5.5.3.1.1.1.1.3.2.cmml">c</mi><mi id="A3.Ex47.m2.5.5.3.1.1.1.1.3.3" xref="A3.Ex47.m2.5.5.3.1.1.1.1.3.3.cmml">θ</mi></msub></mrow><mo id="A3.Ex47.m2.5.5.3.1.1.1.3" stretchy="false" xref="A3.Ex47.m2.5.5.3.1.1.1.1.cmml">)</mo></mrow><mn id="A3.Ex47.m2.5.5.3.1.3" xref="A3.Ex47.m2.5.5.3.1.3.cmml">2</mn></msup><mo id="A3.Ex47.m2.6.6.4.3" xref="A3.Ex47.m2.6.6.4.3.cmml">/</mo><mrow id="A3.Ex47.m2.6.6.4.2.1" xref="A3.Ex47.m2.6.6.4.2.1.1.cmml"><mo id="A3.Ex47.m2.6.6.4.2.1.2" stretchy="false" xref="A3.Ex47.m2.6.6.4.2.1.1.cmml">(</mo><mrow id="A3.Ex47.m2.6.6.4.2.1.1" xref="A3.Ex47.m2.6.6.4.2.1.1.cmml"><mn id="A3.Ex47.m2.6.6.4.2.1.1.2" xref="A3.Ex47.m2.6.6.4.2.1.1.2.cmml">2</mn><mo id="A3.Ex47.m2.6.6.4.2.1.1.1" xref="A3.Ex47.m2.6.6.4.2.1.1.1.cmml">⁢</mo><msub id="A3.Ex47.m2.6.6.4.2.1.1.3" xref="A3.Ex47.m2.6.6.4.2.1.1.3.cmml"><mi id="A3.Ex47.m2.6.6.4.2.1.1.3.2" xref="A3.Ex47.m2.6.6.4.2.1.1.3.2.cmml">k</mi><mi id="A3.Ex47.m2.6.6.4.2.1.1.3.3" mathvariant="normal" xref="A3.Ex47.m2.6.6.4.2.1.1.3.3.cmml">c</mi></msub></mrow><mo id="A3.Ex47.m2.6.6.4.2.1.3" stretchy="false" xref="A3.Ex47.m2.6.6.4.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex47.m2.6b"><apply id="A3.Ex47.m2.6.6.cmml" xref="A3.Ex47.m2.6.6"><plus id="A3.Ex47.m2.6.6.5.cmml" xref="A3.Ex47.m2.6.6.5"></plus><apply id="A3.Ex47.m2.6.6.6.cmml" xref="A3.Ex47.m2.6.6.6"><minus id="A3.Ex47.m2.6.6.6.1.cmml" xref="A3.Ex47.m2.6.6.6"></minus><apply id="A3.Ex47.m2.6.6.6.2.cmml" xref="A3.Ex47.m2.6.6.6.2"><times id="A3.Ex47.m2.6.6.6.2.1.cmml" xref="A3.Ex47.m2.6.6.6.2.1"></times><apply id="A3.Ex47.m2.6.6.6.2.2.cmml" xref="A3.Ex47.m2.6.6.6.2.2"><csymbol cd="ambiguous" id="A3.Ex47.m2.6.6.6.2.2.1.cmml" xref="A3.Ex47.m2.6.6.6.2.2">subscript</csymbol><ci id="A3.Ex47.m2.6.6.6.2.2.2.cmml" xref="A3.Ex47.m2.6.6.6.2.2.2">𝑘</ci><ci id="A3.Ex47.m2.6.6.6.2.2.3.cmml" xref="A3.Ex47.m2.6.6.6.2.2.3">c</ci></apply><ci id="A3.Ex47.m2.6.6.6.2.3.cmml" xref="A3.Ex47.m2.6.6.6.2.3">𝑉</ci><ci id="A3.Ex47.m2.1.1.cmml" xref="A3.Ex47.m2.1.1">𝑡</ci></apply></apply><apply id="A3.Ex47.m2.4.4.2.cmml" xref="A3.Ex47.m2.4.4.2"><divide id="A3.Ex47.m2.4.4.2.3.cmml" xref="A3.Ex47.m2.4.4.2.3"></divide><apply id="A3.Ex47.m2.3.3.1.1.cmml" xref="A3.Ex47.m2.3.3.1.1"><times id="A3.Ex47.m2.3.3.1.1.2.cmml" xref="A3.Ex47.m2.3.3.1.1.2"></times><apply id="A3.Ex47.m2.3.3.1.1.3.cmml" xref="A3.Ex47.m2.3.3.1.1.3"><csymbol cd="ambiguous" id="A3.Ex47.m2.3.3.1.1.3.1.cmml" xref="A3.Ex47.m2.3.3.1.1.3">subscript</csymbol><ci id="A3.Ex47.m2.3.3.1.1.3.2.cmml" xref="A3.Ex47.m2.3.3.1.1.3.2">𝑘</ci><ci id="A3.Ex47.m2.3.3.1.1.3.3.cmml" xref="A3.Ex47.m2.3.3.1.1.3.3">c</ci></apply><apply id="A3.Ex47.m2.3.3.1.1.1.1.1.cmml" xref="A3.Ex47.m2.3.3.1.1.1.1"><plus id="A3.Ex47.m2.3.3.1.1.1.1.1.1.cmml" xref="A3.Ex47.m2.3.3.1.1.1.1.1.1"></plus><cn id="A3.Ex47.m2.3.3.1.1.1.1.1.2.cmml" type="integer" xref="A3.Ex47.m2.3.3.1.1.1.1.1.2">1</cn><ci id="A3.Ex47.m2.3.3.1.1.1.1.1.3.cmml" xref="A3.Ex47.m2.3.3.1.1.1.1.1.3">𝑝</ci></apply><apply id="A3.Ex47.m2.3.3.1.1.4.cmml" xref="A3.Ex47.m2.3.3.1.1.4"><csymbol cd="ambiguous" id="A3.Ex47.m2.3.3.1.1.4.1.cmml" xref="A3.Ex47.m2.3.3.1.1.4">superscript</csymbol><apply id="A3.Ex47.m2.3.3.1.1.4.2.cmml" xref="A3.Ex47.m2.3.3.1.1.4"><csymbol cd="ambiguous" id="A3.Ex47.m2.3.3.1.1.4.2.1.cmml" xref="A3.Ex47.m2.3.3.1.1.4">subscript</csymbol><ci id="A3.Ex47.m2.3.3.1.1.4.2.2.cmml" xref="A3.Ex47.m2.3.3.1.1.4.2.2">𝑐</ci><ci id="A3.Ex47.m2.3.3.1.1.4.2.3.cmml" xref="A3.Ex47.m2.3.3.1.1.4.2.3">𝜃</ci></apply><cn id="A3.Ex47.m2.3.3.1.1.4.3.cmml" type="integer" xref="A3.Ex47.m2.3.3.1.1.4.3">2</cn></apply></apply><apply id="A3.Ex47.m2.4.4.2.2.1.1.cmml" xref="A3.Ex47.m2.4.4.2.2.1"><times id="A3.Ex47.m2.4.4.2.2.1.1.1.cmml" xref="A3.Ex47.m2.4.4.2.2.1.1.1"></times><cn id="A3.Ex47.m2.4.4.2.2.1.1.2.cmml" type="integer" xref="A3.Ex47.m2.4.4.2.2.1.1.2">2</cn><apply id="A3.Ex47.m2.4.4.2.2.1.1.3.cmml" xref="A3.Ex47.m2.4.4.2.2.1.1.3"><csymbol cd="ambiguous" id="A3.Ex47.m2.4.4.2.2.1.1.3.1.cmml" xref="A3.Ex47.m2.4.4.2.2.1.1.3">subscript</csymbol><ci id="A3.Ex47.m2.4.4.2.2.1.1.3.2.cmml" xref="A3.Ex47.m2.4.4.2.2.1.1.3.2">𝜆</ci><min id="A3.Ex47.m2.4.4.2.2.1.1.3.3.cmml" xref="A3.Ex47.m2.4.4.2.2.1.1.3.3"></min></apply><ci id="A3.Ex47.m2.2.2.cmml" xref="A3.Ex47.m2.2.2">Γ</ci></apply></apply><apply id="A3.Ex47.m2.6.6.4.cmml" xref="A3.Ex47.m2.6.6.4"><divide id="A3.Ex47.m2.6.6.4.3.cmml" xref="A3.Ex47.m2.6.6.4.3"></divide><apply id="A3.Ex47.m2.5.5.3.1.cmml" xref="A3.Ex47.m2.5.5.3.1"><csymbol cd="ambiguous" id="A3.Ex47.m2.5.5.3.1.2.cmml" xref="A3.Ex47.m2.5.5.3.1">superscript</csymbol><apply id="A3.Ex47.m2.5.5.3.1.1.1.1.cmml" xref="A3.Ex47.m2.5.5.3.1.1.1"><times id="A3.Ex47.m2.5.5.3.1.1.1.1.1.cmml" xref="A3.Ex47.m2.5.5.3.1.1.1.1.1"></times><ci id="A3.Ex47.m2.5.5.3.1.1.1.1.2.cmml" xref="A3.Ex47.m2.5.5.3.1.1.1.1.2">𝛿</ci><apply id="A3.Ex47.m2.5.5.3.1.1.1.1.3.cmml" xref="A3.Ex47.m2.5.5.3.1.1.1.1.3"><csymbol cd="ambiguous" id="A3.Ex47.m2.5.5.3.1.1.1.1.3.1.cmml" xref="A3.Ex47.m2.5.5.3.1.1.1.1.3">subscript</csymbol><ci id="A3.Ex47.m2.5.5.3.1.1.1.1.3.2.cmml" xref="A3.Ex47.m2.5.5.3.1.1.1.1.3.2">𝑐</ci><ci id="A3.Ex47.m2.5.5.3.1.1.1.1.3.3.cmml" xref="A3.Ex47.m2.5.5.3.1.1.1.1.3.3">𝜃</ci></apply></apply><cn id="A3.Ex47.m2.5.5.3.1.3.cmml" type="integer" xref="A3.Ex47.m2.5.5.3.1.3">2</cn></apply><apply id="A3.Ex47.m2.6.6.4.2.1.1.cmml" xref="A3.Ex47.m2.6.6.4.2.1"><times id="A3.Ex47.m2.6.6.4.2.1.1.1.cmml" xref="A3.Ex47.m2.6.6.4.2.1.1.1"></times><cn id="A3.Ex47.m2.6.6.4.2.1.1.2.cmml" type="integer" xref="A3.Ex47.m2.6.6.4.2.1.1.2">2</cn><apply id="A3.Ex47.m2.6.6.4.2.1.1.3.cmml" xref="A3.Ex47.m2.6.6.4.2.1.1.3"><csymbol cd="ambiguous" id="A3.Ex47.m2.6.6.4.2.1.1.3.1.cmml" xref="A3.Ex47.m2.6.6.4.2.1.1.3">subscript</csymbol><ci id="A3.Ex47.m2.6.6.4.2.1.1.3.2.cmml" xref="A3.Ex47.m2.6.6.4.2.1.1.3.2">𝑘</ci><ci id="A3.Ex47.m2.6.6.4.2.1.1.3.3.cmml" xref="A3.Ex47.m2.6.6.4.2.1.1.3.3">c</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex47.m2.6c">\displaystyle-k_{\rm c}V(t)+k_{\rm c}(1+p)c_{\theta}^{2}/(2\lambda_{\min}(% \Gamma))+(\delta c_{\theta})^{2}/(2k_{\rm c})</annotation><annotation encoding="application/x-llamapun" id="A3.Ex47.m2.6d">- italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT italic_V ( italic_t ) + italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ( 1 + italic_p ) italic_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( 2 italic_λ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Γ ) ) + ( italic_δ italic_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( 2 italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex48"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="A3.Ex48.m1.1"><semantics id="A3.Ex48.m1.1a"><mo id="A3.Ex48.m1.1.1" xref="A3.Ex48.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="A3.Ex48.m1.1b"><eq id="A3.Ex48.m1.1.1.cmml" xref="A3.Ex48.m1.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex48.m1.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="A3.Ex48.m1.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-k_{\rm c}V(t)/2-k_{\rm c}(V(t)-\eta(k_{\rm c},\alpha,\Gamma))/2" class="ltx_Math" display="inline" id="A3.Ex48.m2.5"><semantics id="A3.Ex48.m2.5a"><mrow id="A3.Ex48.m2.5.5" xref="A3.Ex48.m2.5.5.cmml"><mrow id="A3.Ex48.m2.5.5.3" xref="A3.Ex48.m2.5.5.3.cmml"><mo id="A3.Ex48.m2.5.5.3a" xref="A3.Ex48.m2.5.5.3.cmml">−</mo><mrow id="A3.Ex48.m2.5.5.3.2" xref="A3.Ex48.m2.5.5.3.2.cmml"><mrow id="A3.Ex48.m2.5.5.3.2.2" xref="A3.Ex48.m2.5.5.3.2.2.cmml"><msub id="A3.Ex48.m2.5.5.3.2.2.2" xref="A3.Ex48.m2.5.5.3.2.2.2.cmml"><mi id="A3.Ex48.m2.5.5.3.2.2.2.2" xref="A3.Ex48.m2.5.5.3.2.2.2.2.cmml">k</mi><mi id="A3.Ex48.m2.5.5.3.2.2.2.3" mathvariant="normal" xref="A3.Ex48.m2.5.5.3.2.2.2.3.cmml">c</mi></msub><mo id="A3.Ex48.m2.5.5.3.2.2.1" xref="A3.Ex48.m2.5.5.3.2.2.1.cmml">⁢</mo><mi id="A3.Ex48.m2.5.5.3.2.2.3" xref="A3.Ex48.m2.5.5.3.2.2.3.cmml">V</mi><mo id="A3.Ex48.m2.5.5.3.2.2.1a" xref="A3.Ex48.m2.5.5.3.2.2.1.cmml">⁢</mo><mrow id="A3.Ex48.m2.5.5.3.2.2.4.2" xref="A3.Ex48.m2.5.5.3.2.2.cmml"><mo id="A3.Ex48.m2.5.5.3.2.2.4.2.1" stretchy="false" xref="A3.Ex48.m2.5.5.3.2.2.cmml">(</mo><mi id="A3.Ex48.m2.1.1" xref="A3.Ex48.m2.1.1.cmml">t</mi><mo id="A3.Ex48.m2.5.5.3.2.2.4.2.2" stretchy="false" xref="A3.Ex48.m2.5.5.3.2.2.cmml">)</mo></mrow></mrow><mo id="A3.Ex48.m2.5.5.3.2.1" xref="A3.Ex48.m2.5.5.3.2.1.cmml">/</mo><mn id="A3.Ex48.m2.5.5.3.2.3" xref="A3.Ex48.m2.5.5.3.2.3.cmml">2</mn></mrow></mrow><mo id="A3.Ex48.m2.5.5.2" xref="A3.Ex48.m2.5.5.2.cmml">−</mo><mrow id="A3.Ex48.m2.5.5.1" xref="A3.Ex48.m2.5.5.1.cmml"><mrow id="A3.Ex48.m2.5.5.1.1" xref="A3.Ex48.m2.5.5.1.1.cmml"><msub id="A3.Ex48.m2.5.5.1.1.3" xref="A3.Ex48.m2.5.5.1.1.3.cmml"><mi id="A3.Ex48.m2.5.5.1.1.3.2" xref="A3.Ex48.m2.5.5.1.1.3.2.cmml">k</mi><mi id="A3.Ex48.m2.5.5.1.1.3.3" mathvariant="normal" xref="A3.Ex48.m2.5.5.1.1.3.3.cmml">c</mi></msub><mo id="A3.Ex48.m2.5.5.1.1.2" xref="A3.Ex48.m2.5.5.1.1.2.cmml">⁢</mo><mrow id="A3.Ex48.m2.5.5.1.1.1.1" xref="A3.Ex48.m2.5.5.1.1.1.1.1.cmml"><mo id="A3.Ex48.m2.5.5.1.1.1.1.2" stretchy="false" xref="A3.Ex48.m2.5.5.1.1.1.1.1.cmml">(</mo><mrow id="A3.Ex48.m2.5.5.1.1.1.1.1" xref="A3.Ex48.m2.5.5.1.1.1.1.1.cmml"><mrow id="A3.Ex48.m2.5.5.1.1.1.1.1.3" xref="A3.Ex48.m2.5.5.1.1.1.1.1.3.cmml"><mi id="A3.Ex48.m2.5.5.1.1.1.1.1.3.2" xref="A3.Ex48.m2.5.5.1.1.1.1.1.3.2.cmml">V</mi><mo id="A3.Ex48.m2.5.5.1.1.1.1.1.3.1" xref="A3.Ex48.m2.5.5.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="A3.Ex48.m2.5.5.1.1.1.1.1.3.3.2" xref="A3.Ex48.m2.5.5.1.1.1.1.1.3.cmml"><mo id="A3.Ex48.m2.5.5.1.1.1.1.1.3.3.2.1" stretchy="false" xref="A3.Ex48.m2.5.5.1.1.1.1.1.3.cmml">(</mo><mi id="A3.Ex48.m2.2.2" xref="A3.Ex48.m2.2.2.cmml">t</mi><mo id="A3.Ex48.m2.5.5.1.1.1.1.1.3.3.2.2" stretchy="false" xref="A3.Ex48.m2.5.5.1.1.1.1.1.3.cmml">)</mo></mrow></mrow><mo id="A3.Ex48.m2.5.5.1.1.1.1.1.2" xref="A3.Ex48.m2.5.5.1.1.1.1.1.2.cmml">−</mo><mrow id="A3.Ex48.m2.5.5.1.1.1.1.1.1" xref="A3.Ex48.m2.5.5.1.1.1.1.1.1.cmml"><mi id="A3.Ex48.m2.5.5.1.1.1.1.1.1.3" xref="A3.Ex48.m2.5.5.1.1.1.1.1.1.3.cmml">η</mi><mo id="A3.Ex48.m2.5.5.1.1.1.1.1.1.2" xref="A3.Ex48.m2.5.5.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.1" xref="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.2.cmml"><mo id="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.2.cmml">(</mo><msub id="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.1.1" xref="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.1.1.cmml"><mi id="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.1.1.2" xref="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.1.1.2.cmml">k</mi><mi id="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.1.1.3" mathvariant="normal" xref="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.1.1.3.cmml">c</mi></msub><mo id="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.1.3" xref="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.2.cmml">,</mo><mi id="A3.Ex48.m2.3.3" xref="A3.Ex48.m2.3.3.cmml">α</mi><mo id="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.1.4" xref="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.2.cmml">,</mo><mi id="A3.Ex48.m2.4.4" mathvariant="normal" xref="A3.Ex48.m2.4.4.cmml">Γ</mi><mo id="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.1.5" stretchy="false" xref="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="A3.Ex48.m2.5.5.1.1.1.1.3" stretchy="false" xref="A3.Ex48.m2.5.5.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A3.Ex48.m2.5.5.1.2" xref="A3.Ex48.m2.5.5.1.2.cmml">/</mo><mn id="A3.Ex48.m2.5.5.1.3" xref="A3.Ex48.m2.5.5.1.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex48.m2.5b"><apply id="A3.Ex48.m2.5.5.cmml" xref="A3.Ex48.m2.5.5"><minus id="A3.Ex48.m2.5.5.2.cmml" xref="A3.Ex48.m2.5.5.2"></minus><apply id="A3.Ex48.m2.5.5.3.cmml" xref="A3.Ex48.m2.5.5.3"><minus id="A3.Ex48.m2.5.5.3.1.cmml" xref="A3.Ex48.m2.5.5.3"></minus><apply id="A3.Ex48.m2.5.5.3.2.cmml" xref="A3.Ex48.m2.5.5.3.2"><divide id="A3.Ex48.m2.5.5.3.2.1.cmml" xref="A3.Ex48.m2.5.5.3.2.1"></divide><apply id="A3.Ex48.m2.5.5.3.2.2.cmml" xref="A3.Ex48.m2.5.5.3.2.2"><times id="A3.Ex48.m2.5.5.3.2.2.1.cmml" xref="A3.Ex48.m2.5.5.3.2.2.1"></times><apply id="A3.Ex48.m2.5.5.3.2.2.2.cmml" xref="A3.Ex48.m2.5.5.3.2.2.2"><csymbol cd="ambiguous" id="A3.Ex48.m2.5.5.3.2.2.2.1.cmml" xref="A3.Ex48.m2.5.5.3.2.2.2">subscript</csymbol><ci id="A3.Ex48.m2.5.5.3.2.2.2.2.cmml" xref="A3.Ex48.m2.5.5.3.2.2.2.2">𝑘</ci><ci id="A3.Ex48.m2.5.5.3.2.2.2.3.cmml" xref="A3.Ex48.m2.5.5.3.2.2.2.3">c</ci></apply><ci id="A3.Ex48.m2.5.5.3.2.2.3.cmml" xref="A3.Ex48.m2.5.5.3.2.2.3">𝑉</ci><ci id="A3.Ex48.m2.1.1.cmml" xref="A3.Ex48.m2.1.1">𝑡</ci></apply><cn id="A3.Ex48.m2.5.5.3.2.3.cmml" type="integer" xref="A3.Ex48.m2.5.5.3.2.3">2</cn></apply></apply><apply id="A3.Ex48.m2.5.5.1.cmml" xref="A3.Ex48.m2.5.5.1"><divide id="A3.Ex48.m2.5.5.1.2.cmml" xref="A3.Ex48.m2.5.5.1.2"></divide><apply id="A3.Ex48.m2.5.5.1.1.cmml" xref="A3.Ex48.m2.5.5.1.1"><times id="A3.Ex48.m2.5.5.1.1.2.cmml" xref="A3.Ex48.m2.5.5.1.1.2"></times><apply id="A3.Ex48.m2.5.5.1.1.3.cmml" xref="A3.Ex48.m2.5.5.1.1.3"><csymbol cd="ambiguous" id="A3.Ex48.m2.5.5.1.1.3.1.cmml" xref="A3.Ex48.m2.5.5.1.1.3">subscript</csymbol><ci id="A3.Ex48.m2.5.5.1.1.3.2.cmml" xref="A3.Ex48.m2.5.5.1.1.3.2">𝑘</ci><ci id="A3.Ex48.m2.5.5.1.1.3.3.cmml" xref="A3.Ex48.m2.5.5.1.1.3.3">c</ci></apply><apply id="A3.Ex48.m2.5.5.1.1.1.1.1.cmml" xref="A3.Ex48.m2.5.5.1.1.1.1"><minus id="A3.Ex48.m2.5.5.1.1.1.1.1.2.cmml" xref="A3.Ex48.m2.5.5.1.1.1.1.1.2"></minus><apply id="A3.Ex48.m2.5.5.1.1.1.1.1.3.cmml" xref="A3.Ex48.m2.5.5.1.1.1.1.1.3"><times id="A3.Ex48.m2.5.5.1.1.1.1.1.3.1.cmml" xref="A3.Ex48.m2.5.5.1.1.1.1.1.3.1"></times><ci id="A3.Ex48.m2.5.5.1.1.1.1.1.3.2.cmml" xref="A3.Ex48.m2.5.5.1.1.1.1.1.3.2">𝑉</ci><ci id="A3.Ex48.m2.2.2.cmml" xref="A3.Ex48.m2.2.2">𝑡</ci></apply><apply id="A3.Ex48.m2.5.5.1.1.1.1.1.1.cmml" xref="A3.Ex48.m2.5.5.1.1.1.1.1.1"><times id="A3.Ex48.m2.5.5.1.1.1.1.1.1.2.cmml" xref="A3.Ex48.m2.5.5.1.1.1.1.1.1.2"></times><ci id="A3.Ex48.m2.5.5.1.1.1.1.1.1.3.cmml" xref="A3.Ex48.m2.5.5.1.1.1.1.1.1.3">𝜂</ci><vector id="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.2.cmml" xref="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.1"><apply id="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.1.1.cmml" xref="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.1.1.1.cmml" xref="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.1.1.2.cmml" xref="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.1.1.2">𝑘</ci><ci id="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.1.1.3.cmml" xref="A3.Ex48.m2.5.5.1.1.1.1.1.1.1.1.1.3">c</ci></apply><ci id="A3.Ex48.m2.3.3.cmml" xref="A3.Ex48.m2.3.3">𝛼</ci><ci id="A3.Ex48.m2.4.4.cmml" xref="A3.Ex48.m2.4.4">Γ</ci></vector></apply></apply></apply><cn id="A3.Ex48.m2.5.5.1.3.cmml" type="integer" xref="A3.Ex48.m2.5.5.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex48.m2.5c">\displaystyle-k_{\rm c}V(t)/2-k_{\rm c}(V(t)-\eta(k_{\rm c},\alpha,\Gamma))/2</annotation><annotation encoding="application/x-llamapun" id="A3.Ex48.m2.5d">- italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT italic_V ( italic_t ) / 2 - italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ( italic_V ( italic_t ) - italic_η ( italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT , italic_α , roman_Γ ) ) / 2</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.1.p1.55">with <math alttext="\eta(k_{\rm c},\alpha,\Gamma)" class="ltx_Math" display="inline" id="A3.1.p1.41.m1.3"><semantics id="A3.1.p1.41.m1.3a"><mrow id="A3.1.p1.41.m1.3.3" xref="A3.1.p1.41.m1.3.3.cmml"><mi id="A3.1.p1.41.m1.3.3.3" xref="A3.1.p1.41.m1.3.3.3.cmml">η</mi><mo id="A3.1.p1.41.m1.3.3.2" xref="A3.1.p1.41.m1.3.3.2.cmml">⁢</mo><mrow id="A3.1.p1.41.m1.3.3.1.1" xref="A3.1.p1.41.m1.3.3.1.2.cmml"><mo id="A3.1.p1.41.m1.3.3.1.1.2" stretchy="false" xref="A3.1.p1.41.m1.3.3.1.2.cmml">(</mo><msub id="A3.1.p1.41.m1.3.3.1.1.1" xref="A3.1.p1.41.m1.3.3.1.1.1.cmml"><mi id="A3.1.p1.41.m1.3.3.1.1.1.2" xref="A3.1.p1.41.m1.3.3.1.1.1.2.cmml">k</mi><mi id="A3.1.p1.41.m1.3.3.1.1.1.3" mathvariant="normal" xref="A3.1.p1.41.m1.3.3.1.1.1.3.cmml">c</mi></msub><mo id="A3.1.p1.41.m1.3.3.1.1.3" xref="A3.1.p1.41.m1.3.3.1.2.cmml">,</mo><mi id="A3.1.p1.41.m1.1.1" xref="A3.1.p1.41.m1.1.1.cmml">α</mi><mo id="A3.1.p1.41.m1.3.3.1.1.4" xref="A3.1.p1.41.m1.3.3.1.2.cmml">,</mo><mi id="A3.1.p1.41.m1.2.2" mathvariant="normal" xref="A3.1.p1.41.m1.2.2.cmml">Γ</mi><mo id="A3.1.p1.41.m1.3.3.1.1.5" stretchy="false" xref="A3.1.p1.41.m1.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.41.m1.3b"><apply id="A3.1.p1.41.m1.3.3.cmml" xref="A3.1.p1.41.m1.3.3"><times id="A3.1.p1.41.m1.3.3.2.cmml" xref="A3.1.p1.41.m1.3.3.2"></times><ci id="A3.1.p1.41.m1.3.3.3.cmml" xref="A3.1.p1.41.m1.3.3.3">𝜂</ci><vector id="A3.1.p1.41.m1.3.3.1.2.cmml" xref="A3.1.p1.41.m1.3.3.1.1"><apply id="A3.1.p1.41.m1.3.3.1.1.1.cmml" xref="A3.1.p1.41.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="A3.1.p1.41.m1.3.3.1.1.1.1.cmml" xref="A3.1.p1.41.m1.3.3.1.1.1">subscript</csymbol><ci id="A3.1.p1.41.m1.3.3.1.1.1.2.cmml" xref="A3.1.p1.41.m1.3.3.1.1.1.2">𝑘</ci><ci id="A3.1.p1.41.m1.3.3.1.1.1.3.cmml" xref="A3.1.p1.41.m1.3.3.1.1.1.3">c</ci></apply><ci id="A3.1.p1.41.m1.1.1.cmml" xref="A3.1.p1.41.m1.1.1">𝛼</ci><ci id="A3.1.p1.41.m1.2.2.cmml" xref="A3.1.p1.41.m1.2.2">Γ</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.41.m1.3c">\eta(k_{\rm c},\alpha,\Gamma)</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.41.m1.3d">italic_η ( italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT , italic_α , roman_Γ )</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="A3.1.p1.42.m2.1"><semantics id="A3.1.p1.42.m2.1a"><mo id="A3.1.p1.42.m2.1.1" xref="A3.1.p1.42.m2.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="A3.1.p1.42.m2.1b"><csymbol cd="latexml" id="A3.1.p1.42.m2.1.1.cmml" xref="A3.1.p1.42.m2.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.42.m2.1c">:=</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.42.m2.1d">:=</annotation></semantics></math> <math alttext="(1+p)c_{\theta}^{2}/\lambda_{\min}(\Gamma)+(\delta c_{\theta}/k_{\rm c})^{2}% \in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A3.1.p1.43.m3.3"><semantics id="A3.1.p1.43.m3.3a"><mrow id="A3.1.p1.43.m3.3.3" xref="A3.1.p1.43.m3.3.3.cmml"><mrow id="A3.1.p1.43.m3.3.3.2" xref="A3.1.p1.43.m3.3.3.2.cmml"><mrow id="A3.1.p1.43.m3.2.2.1.1" xref="A3.1.p1.43.m3.2.2.1.1.cmml"><mrow id="A3.1.p1.43.m3.2.2.1.1.1" xref="A3.1.p1.43.m3.2.2.1.1.1.cmml"><mrow id="A3.1.p1.43.m3.2.2.1.1.1.1" xref="A3.1.p1.43.m3.2.2.1.1.1.1.cmml"><mrow id="A3.1.p1.43.m3.2.2.1.1.1.1.1.1" xref="A3.1.p1.43.m3.2.2.1.1.1.1.1.1.1.cmml"><mo id="A3.1.p1.43.m3.2.2.1.1.1.1.1.1.2" stretchy="false" xref="A3.1.p1.43.m3.2.2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.1.p1.43.m3.2.2.1.1.1.1.1.1.1" xref="A3.1.p1.43.m3.2.2.1.1.1.1.1.1.1.cmml"><mn id="A3.1.p1.43.m3.2.2.1.1.1.1.1.1.1.2" xref="A3.1.p1.43.m3.2.2.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A3.1.p1.43.m3.2.2.1.1.1.1.1.1.1.1" xref="A3.1.p1.43.m3.2.2.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="A3.1.p1.43.m3.2.2.1.1.1.1.1.1.1.3" xref="A3.1.p1.43.m3.2.2.1.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A3.1.p1.43.m3.2.2.1.1.1.1.1.1.3" stretchy="false" xref="A3.1.p1.43.m3.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A3.1.p1.43.m3.2.2.1.1.1.1.2" xref="A3.1.p1.43.m3.2.2.1.1.1.1.2.cmml">⁢</mo><msubsup id="A3.1.p1.43.m3.2.2.1.1.1.1.3" xref="A3.1.p1.43.m3.2.2.1.1.1.1.3.cmml"><mi id="A3.1.p1.43.m3.2.2.1.1.1.1.3.2.2" xref="A3.1.p1.43.m3.2.2.1.1.1.1.3.2.2.cmml">c</mi><mi id="A3.1.p1.43.m3.2.2.1.1.1.1.3.2.3" xref="A3.1.p1.43.m3.2.2.1.1.1.1.3.2.3.cmml">θ</mi><mn id="A3.1.p1.43.m3.2.2.1.1.1.1.3.3" xref="A3.1.p1.43.m3.2.2.1.1.1.1.3.3.cmml">2</mn></msubsup></mrow><mo id="A3.1.p1.43.m3.2.2.1.1.1.2" xref="A3.1.p1.43.m3.2.2.1.1.1.2.cmml">/</mo><msub id="A3.1.p1.43.m3.2.2.1.1.1.3" xref="A3.1.p1.43.m3.2.2.1.1.1.3.cmml"><mi id="A3.1.p1.43.m3.2.2.1.1.1.3.2" xref="A3.1.p1.43.m3.2.2.1.1.1.3.2.cmml">λ</mi><mi id="A3.1.p1.43.m3.2.2.1.1.1.3.3" xref="A3.1.p1.43.m3.2.2.1.1.1.3.3.cmml">min</mi></msub></mrow><mo id="A3.1.p1.43.m3.2.2.1.1.2" xref="A3.1.p1.43.m3.2.2.1.1.2.cmml">⁢</mo><mrow id="A3.1.p1.43.m3.2.2.1.1.3.2" xref="A3.1.p1.43.m3.2.2.1.1.cmml"><mo id="A3.1.p1.43.m3.2.2.1.1.3.2.1" stretchy="false" xref="A3.1.p1.43.m3.2.2.1.1.cmml">(</mo><mi id="A3.1.p1.43.m3.1.1" mathvariant="normal" xref="A3.1.p1.43.m3.1.1.cmml">Γ</mi><mo id="A3.1.p1.43.m3.2.2.1.1.3.2.2" stretchy="false" xref="A3.1.p1.43.m3.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="A3.1.p1.43.m3.3.3.2.3" xref="A3.1.p1.43.m3.3.3.2.3.cmml">+</mo><msup id="A3.1.p1.43.m3.3.3.2.2" xref="A3.1.p1.43.m3.3.3.2.2.cmml"><mrow id="A3.1.p1.43.m3.3.3.2.2.1.1" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.cmml"><mo id="A3.1.p1.43.m3.3.3.2.2.1.1.2" stretchy="false" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.cmml">(</mo><mrow id="A3.1.p1.43.m3.3.3.2.2.1.1.1" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.cmml"><mrow id="A3.1.p1.43.m3.3.3.2.2.1.1.1.2" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.2.cmml"><mi id="A3.1.p1.43.m3.3.3.2.2.1.1.1.2.2" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.2.2.cmml">δ</mi><mo id="A3.1.p1.43.m3.3.3.2.2.1.1.1.2.1" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.2.1.cmml">⁢</mo><msub id="A3.1.p1.43.m3.3.3.2.2.1.1.1.2.3" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.2.3.cmml"><mi id="A3.1.p1.43.m3.3.3.2.2.1.1.1.2.3.2" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.2.3.2.cmml">c</mi><mi id="A3.1.p1.43.m3.3.3.2.2.1.1.1.2.3.3" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.2.3.3.cmml">θ</mi></msub></mrow><mo id="A3.1.p1.43.m3.3.3.2.2.1.1.1.1" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.1.cmml">/</mo><msub id="A3.1.p1.43.m3.3.3.2.2.1.1.1.3" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.3.cmml"><mi id="A3.1.p1.43.m3.3.3.2.2.1.1.1.3.2" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.3.2.cmml">k</mi><mi id="A3.1.p1.43.m3.3.3.2.2.1.1.1.3.3" mathvariant="normal" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.3.3.cmml">c</mi></msub></mrow><mo id="A3.1.p1.43.m3.3.3.2.2.1.1.3" stretchy="false" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.cmml">)</mo></mrow><mn id="A3.1.p1.43.m3.3.3.2.2.3" xref="A3.1.p1.43.m3.3.3.2.2.3.cmml">2</mn></msup></mrow><mo id="A3.1.p1.43.m3.3.3.3" xref="A3.1.p1.43.m3.3.3.3.cmml">∈</mo><msup id="A3.1.p1.43.m3.3.3.4" xref="A3.1.p1.43.m3.3.3.4.cmml"><mi id="A3.1.p1.43.m3.3.3.4.2" xref="A3.1.p1.43.m3.3.3.4.2.cmml">ℝ</mi><mo id="A3.1.p1.43.m3.3.3.4.3" xref="A3.1.p1.43.m3.3.3.4.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.43.m3.3b"><apply id="A3.1.p1.43.m3.3.3.cmml" xref="A3.1.p1.43.m3.3.3"><in id="A3.1.p1.43.m3.3.3.3.cmml" xref="A3.1.p1.43.m3.3.3.3"></in><apply id="A3.1.p1.43.m3.3.3.2.cmml" xref="A3.1.p1.43.m3.3.3.2"><plus id="A3.1.p1.43.m3.3.3.2.3.cmml" xref="A3.1.p1.43.m3.3.3.2.3"></plus><apply id="A3.1.p1.43.m3.2.2.1.1.cmml" xref="A3.1.p1.43.m3.2.2.1.1"><times id="A3.1.p1.43.m3.2.2.1.1.2.cmml" xref="A3.1.p1.43.m3.2.2.1.1.2"></times><apply id="A3.1.p1.43.m3.2.2.1.1.1.cmml" xref="A3.1.p1.43.m3.2.2.1.1.1"><divide id="A3.1.p1.43.m3.2.2.1.1.1.2.cmml" xref="A3.1.p1.43.m3.2.2.1.1.1.2"></divide><apply id="A3.1.p1.43.m3.2.2.1.1.1.1.cmml" xref="A3.1.p1.43.m3.2.2.1.1.1.1"><times id="A3.1.p1.43.m3.2.2.1.1.1.1.2.cmml" xref="A3.1.p1.43.m3.2.2.1.1.1.1.2"></times><apply id="A3.1.p1.43.m3.2.2.1.1.1.1.1.1.1.cmml" xref="A3.1.p1.43.m3.2.2.1.1.1.1.1.1"><plus id="A3.1.p1.43.m3.2.2.1.1.1.1.1.1.1.1.cmml" xref="A3.1.p1.43.m3.2.2.1.1.1.1.1.1.1.1"></plus><cn id="A3.1.p1.43.m3.2.2.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A3.1.p1.43.m3.2.2.1.1.1.1.1.1.1.2">1</cn><ci id="A3.1.p1.43.m3.2.2.1.1.1.1.1.1.1.3.cmml" xref="A3.1.p1.43.m3.2.2.1.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="A3.1.p1.43.m3.2.2.1.1.1.1.3.cmml" xref="A3.1.p1.43.m3.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="A3.1.p1.43.m3.2.2.1.1.1.1.3.1.cmml" xref="A3.1.p1.43.m3.2.2.1.1.1.1.3">superscript</csymbol><apply id="A3.1.p1.43.m3.2.2.1.1.1.1.3.2.cmml" xref="A3.1.p1.43.m3.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="A3.1.p1.43.m3.2.2.1.1.1.1.3.2.1.cmml" xref="A3.1.p1.43.m3.2.2.1.1.1.1.3">subscript</csymbol><ci id="A3.1.p1.43.m3.2.2.1.1.1.1.3.2.2.cmml" xref="A3.1.p1.43.m3.2.2.1.1.1.1.3.2.2">𝑐</ci><ci id="A3.1.p1.43.m3.2.2.1.1.1.1.3.2.3.cmml" xref="A3.1.p1.43.m3.2.2.1.1.1.1.3.2.3">𝜃</ci></apply><cn id="A3.1.p1.43.m3.2.2.1.1.1.1.3.3.cmml" type="integer" xref="A3.1.p1.43.m3.2.2.1.1.1.1.3.3">2</cn></apply></apply><apply id="A3.1.p1.43.m3.2.2.1.1.1.3.cmml" xref="A3.1.p1.43.m3.2.2.1.1.1.3"><csymbol cd="ambiguous" id="A3.1.p1.43.m3.2.2.1.1.1.3.1.cmml" xref="A3.1.p1.43.m3.2.2.1.1.1.3">subscript</csymbol><ci id="A3.1.p1.43.m3.2.2.1.1.1.3.2.cmml" xref="A3.1.p1.43.m3.2.2.1.1.1.3.2">𝜆</ci><min id="A3.1.p1.43.m3.2.2.1.1.1.3.3.cmml" xref="A3.1.p1.43.m3.2.2.1.1.1.3.3"></min></apply></apply><ci id="A3.1.p1.43.m3.1.1.cmml" xref="A3.1.p1.43.m3.1.1">Γ</ci></apply><apply id="A3.1.p1.43.m3.3.3.2.2.cmml" xref="A3.1.p1.43.m3.3.3.2.2"><csymbol cd="ambiguous" id="A3.1.p1.43.m3.3.3.2.2.2.cmml" xref="A3.1.p1.43.m3.3.3.2.2">superscript</csymbol><apply id="A3.1.p1.43.m3.3.3.2.2.1.1.1.cmml" xref="A3.1.p1.43.m3.3.3.2.2.1.1"><divide id="A3.1.p1.43.m3.3.3.2.2.1.1.1.1.cmml" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.1"></divide><apply id="A3.1.p1.43.m3.3.3.2.2.1.1.1.2.cmml" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.2"><times id="A3.1.p1.43.m3.3.3.2.2.1.1.1.2.1.cmml" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.2.1"></times><ci id="A3.1.p1.43.m3.3.3.2.2.1.1.1.2.2.cmml" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.2.2">𝛿</ci><apply id="A3.1.p1.43.m3.3.3.2.2.1.1.1.2.3.cmml" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.2.3"><csymbol cd="ambiguous" id="A3.1.p1.43.m3.3.3.2.2.1.1.1.2.3.1.cmml" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.2.3">subscript</csymbol><ci id="A3.1.p1.43.m3.3.3.2.2.1.1.1.2.3.2.cmml" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.2.3.2">𝑐</ci><ci id="A3.1.p1.43.m3.3.3.2.2.1.1.1.2.3.3.cmml" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.2.3.3">𝜃</ci></apply></apply><apply id="A3.1.p1.43.m3.3.3.2.2.1.1.1.3.cmml" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.3"><csymbol cd="ambiguous" id="A3.1.p1.43.m3.3.3.2.2.1.1.1.3.1.cmml" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.3">subscript</csymbol><ci id="A3.1.p1.43.m3.3.3.2.2.1.1.1.3.2.cmml" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.3.2">𝑘</ci><ci id="A3.1.p1.43.m3.3.3.2.2.1.1.1.3.3.cmml" xref="A3.1.p1.43.m3.3.3.2.2.1.1.1.3.3">c</ci></apply></apply><cn id="A3.1.p1.43.m3.3.3.2.2.3.cmml" type="integer" xref="A3.1.p1.43.m3.3.3.2.2.3">2</cn></apply></apply><apply id="A3.1.p1.43.m3.3.3.4.cmml" xref="A3.1.p1.43.m3.3.3.4"><csymbol cd="ambiguous" id="A3.1.p1.43.m3.3.3.4.1.cmml" xref="A3.1.p1.43.m3.3.3.4">superscript</csymbol><ci id="A3.1.p1.43.m3.3.3.4.2.cmml" xref="A3.1.p1.43.m3.3.3.4.2">ℝ</ci><plus id="A3.1.p1.43.m3.3.3.4.3.cmml" xref="A3.1.p1.43.m3.3.3.4.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.43.m3.3c">(1+p)c_{\theta}^{2}/\lambda_{\min}(\Gamma)+(\delta c_{\theta}/k_{\rm c})^{2}% \in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.43.m3.3d">( 1 + italic_p ) italic_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / italic_λ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Γ ) + ( italic_δ italic_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT / italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>. Let <math alttext="\Omega_{r}" class="ltx_Math" display="inline" id="A3.1.p1.44.m4.1"><semantics id="A3.1.p1.44.m4.1a"><msub id="A3.1.p1.44.m4.1.1" xref="A3.1.p1.44.m4.1.1.cmml"><mi id="A3.1.p1.44.m4.1.1.2" mathvariant="normal" xref="A3.1.p1.44.m4.1.1.2.cmml">Ω</mi><mi id="A3.1.p1.44.m4.1.1.3" xref="A3.1.p1.44.m4.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="A3.1.p1.44.m4.1b"><apply id="A3.1.p1.44.m4.1.1.cmml" xref="A3.1.p1.44.m4.1.1"><csymbol cd="ambiguous" id="A3.1.p1.44.m4.1.1.1.cmml" xref="A3.1.p1.44.m4.1.1">subscript</csymbol><ci id="A3.1.p1.44.m4.1.1.2.cmml" xref="A3.1.p1.44.m4.1.1.2">Ω</ci><ci id="A3.1.p1.44.m4.1.1.3.cmml" xref="A3.1.p1.44.m4.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.44.m4.1c">\Omega_{r}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.44.m4.1d">roman_Ω start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="A3.1.p1.45.m5.1"><semantics id="A3.1.p1.45.m5.1a"><mo id="A3.1.p1.45.m5.1.1" xref="A3.1.p1.45.m5.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="A3.1.p1.45.m5.1b"><csymbol cd="latexml" id="A3.1.p1.45.m5.1.1.cmml" xref="A3.1.p1.45.m5.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.45.m5.1c">:=</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.45.m5.1d">:=</annotation></semantics></math> <math alttext="\{{\bm{y}}_{{\rm r}n}|\dot{y}_{\rm r}" class="ltx_math_unparsed" display="inline" id="A3.1.p1.46.m6.1"><semantics id="A3.1.p1.46.m6.1a"><mrow id="A3.1.p1.46.m6.1b"><mo id="A3.1.p1.46.m6.1.1" stretchy="false">{</mo><msub id="A3.1.p1.46.m6.1.2"><mi id="A3.1.p1.46.m6.1.2.2">𝒚</mi><mrow id="A3.1.p1.46.m6.1.2.3"><mi id="A3.1.p1.46.m6.1.2.3.2" mathvariant="normal">r</mi><mo id="A3.1.p1.46.m6.1.2.3.1">⁢</mo><mi id="A3.1.p1.46.m6.1.2.3.3">n</mi></mrow></msub><mo fence="false" id="A3.1.p1.46.m6.1.3" rspace="0.167em" stretchy="false">|</mo><msub id="A3.1.p1.46.m6.1.4"><mover accent="true" id="A3.1.p1.46.m6.1.4.2"><mi id="A3.1.p1.46.m6.1.4.2.2">y</mi><mo id="A3.1.p1.46.m6.1.4.2.1">˙</mo></mover><mi id="A3.1.p1.46.m6.1.4.3" mathvariant="normal">r</mi></msub></mrow><annotation encoding="application/x-tex" id="A3.1.p1.46.m6.1c">\{{\bm{y}}_{{\rm r}n}|\dot{y}_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.46.m6.1d">{ bold_italic_y start_POSTSUBSCRIPT roman_r italic_n end_POSTSUBSCRIPT | over˙ start_ARG italic_y end_ARG start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\cdots" class="ltx_Math" display="inline" id="A3.1.p1.47.m7.1"><semantics id="A3.1.p1.47.m7.1a"><mi id="A3.1.p1.47.m7.1.1" mathvariant="normal" xref="A3.1.p1.47.m7.1.1.cmml">⋯</mi><annotation-xml encoding="MathML-Content" id="A3.1.p1.47.m7.1b"><ci id="A3.1.p1.47.m7.1.1.cmml" xref="A3.1.p1.47.m7.1.1">⋯</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.47.m7.1c">\cdots</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.47.m7.1d">⋯</annotation></semantics></math>, <math alttext="y^{(n-1)}_{\rm r}\in\Omega_{{\rm c}_{\rm r}}\}\subset\mathbb{R}^{n}" class="ltx_math_unparsed" display="inline" id="A3.1.p1.48.m8.1"><semantics id="A3.1.p1.48.m8.1a"><mrow id="A3.1.p1.48.m8.1b"><msubsup id="A3.1.p1.48.m8.1.2"><mi id="A3.1.p1.48.m8.1.2.2.2">y</mi><mi id="A3.1.p1.48.m8.1.2.3" mathvariant="normal">r</mi><mrow id="A3.1.p1.48.m8.1.1.1.1"><mo id="A3.1.p1.48.m8.1.1.1.1.2" stretchy="false">(</mo><mrow id="A3.1.p1.48.m8.1.1.1.1.1"><mi id="A3.1.p1.48.m8.1.1.1.1.1.2">n</mi><mo id="A3.1.p1.48.m8.1.1.1.1.1.1">−</mo><mn id="A3.1.p1.48.m8.1.1.1.1.1.3">1</mn></mrow><mo id="A3.1.p1.48.m8.1.1.1.1.3" stretchy="false">)</mo></mrow></msubsup><mo id="A3.1.p1.48.m8.1.3">∈</mo><msub id="A3.1.p1.48.m8.1.4"><mi id="A3.1.p1.48.m8.1.4.2" mathvariant="normal">Ω</mi><msub id="A3.1.p1.48.m8.1.4.3"><mi id="A3.1.p1.48.m8.1.4.3.2" mathvariant="normal">c</mi><mi id="A3.1.p1.48.m8.1.4.3.3" mathvariant="normal">r</mi></msub></msub><mo id="A3.1.p1.48.m8.1.5" stretchy="false">}</mo><mo id="A3.1.p1.48.m8.1.6">⊂</mo><mi id="A3.1.p1.48.m8.1.7">ℝ</mi><msup id="A3.1.p1.48.m8.1.8"><mi id="A3.1.p1.48.m8.1.8a"></mi><mi id="A3.1.p1.48.m8.1.8.1">n</mi></msup></mrow><annotation encoding="application/x-tex" id="A3.1.p1.48.m8.1c">y^{(n-1)}_{\rm r}\in\Omega_{{\rm c}_{\rm r}}\}\subset\mathbb{R}^{n}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.48.m8.1d">italic_y start_POSTSUPERSCRIPT ( italic_n - 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT ∈ roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT end_POSTSUBSCRIPT } ⊂ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\Omega_{{\rm c}_{w}}" class="ltx_Math" display="inline" id="A3.1.p1.49.m9.1"><semantics id="A3.1.p1.49.m9.1a"><msub id="A3.1.p1.49.m9.1.1" xref="A3.1.p1.49.m9.1.1.cmml"><mi id="A3.1.p1.49.m9.1.1.2" mathvariant="normal" xref="A3.1.p1.49.m9.1.1.2.cmml">Ω</mi><msub id="A3.1.p1.49.m9.1.1.3" xref="A3.1.p1.49.m9.1.1.3.cmml"><mi id="A3.1.p1.49.m9.1.1.3.2" mathvariant="normal" xref="A3.1.p1.49.m9.1.1.3.2.cmml">c</mi><mi id="A3.1.p1.49.m9.1.1.3.3" xref="A3.1.p1.49.m9.1.1.3.3.cmml">w</mi></msub></msub><annotation-xml encoding="MathML-Content" id="A3.1.p1.49.m9.1b"><apply id="A3.1.p1.49.m9.1.1.cmml" xref="A3.1.p1.49.m9.1.1"><csymbol cd="ambiguous" id="A3.1.p1.49.m9.1.1.1.cmml" xref="A3.1.p1.49.m9.1.1">subscript</csymbol><ci id="A3.1.p1.49.m9.1.1.2.cmml" xref="A3.1.p1.49.m9.1.1.2">Ω</ci><apply id="A3.1.p1.49.m9.1.1.3.cmml" xref="A3.1.p1.49.m9.1.1.3"><csymbol cd="ambiguous" id="A3.1.p1.49.m9.1.1.3.1.cmml" xref="A3.1.p1.49.m9.1.1.3">subscript</csymbol><ci id="A3.1.p1.49.m9.1.1.3.2.cmml" xref="A3.1.p1.49.m9.1.1.3.2">c</ci><ci id="A3.1.p1.49.m9.1.1.3.3.cmml" xref="A3.1.p1.49.m9.1.1.3.3">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.49.m9.1c">\Omega_{{\rm c}_{w}}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.49.m9.1d">roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="A3.1.p1.50.m10.1"><semantics id="A3.1.p1.50.m10.1a"><mo id="A3.1.p1.50.m10.1.1" xref="A3.1.p1.50.m10.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="A3.1.p1.50.m10.1b"><csymbol cd="latexml" id="A3.1.p1.50.m10.1.1.cmml" xref="A3.1.p1.50.m10.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.50.m10.1c">:=</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.50.m10.1d">:=</annotation></semantics></math> <math alttext="\Omega_{{\rm c}_{x}}\cap\Omega_{r}\times\Omega_{{\rm c}_{\theta}}" class="ltx_Math" display="inline" id="A3.1.p1.51.m11.1"><semantics id="A3.1.p1.51.m11.1a"><mrow id="A3.1.p1.51.m11.1.1" xref="A3.1.p1.51.m11.1.1.cmml"><msub id="A3.1.p1.51.m11.1.1.2" xref="A3.1.p1.51.m11.1.1.2.cmml"><mi id="A3.1.p1.51.m11.1.1.2.2" mathvariant="normal" xref="A3.1.p1.51.m11.1.1.2.2.cmml">Ω</mi><msub id="A3.1.p1.51.m11.1.1.2.3" xref="A3.1.p1.51.m11.1.1.2.3.cmml"><mi id="A3.1.p1.51.m11.1.1.2.3.2" mathvariant="normal" xref="A3.1.p1.51.m11.1.1.2.3.2.cmml">c</mi><mi id="A3.1.p1.51.m11.1.1.2.3.3" xref="A3.1.p1.51.m11.1.1.2.3.3.cmml">x</mi></msub></msub><mo id="A3.1.p1.51.m11.1.1.1" xref="A3.1.p1.51.m11.1.1.1.cmml">∩</mo><mrow id="A3.1.p1.51.m11.1.1.3" xref="A3.1.p1.51.m11.1.1.3.cmml"><msub id="A3.1.p1.51.m11.1.1.3.2" xref="A3.1.p1.51.m11.1.1.3.2.cmml"><mi id="A3.1.p1.51.m11.1.1.3.2.2" mathvariant="normal" xref="A3.1.p1.51.m11.1.1.3.2.2.cmml">Ω</mi><mi id="A3.1.p1.51.m11.1.1.3.2.3" xref="A3.1.p1.51.m11.1.1.3.2.3.cmml">r</mi></msub><mo id="A3.1.p1.51.m11.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="A3.1.p1.51.m11.1.1.3.1.cmml">×</mo><msub id="A3.1.p1.51.m11.1.1.3.3" xref="A3.1.p1.51.m11.1.1.3.3.cmml"><mi id="A3.1.p1.51.m11.1.1.3.3.2" mathvariant="normal" xref="A3.1.p1.51.m11.1.1.3.3.2.cmml">Ω</mi><msub id="A3.1.p1.51.m11.1.1.3.3.3" xref="A3.1.p1.51.m11.1.1.3.3.3.cmml"><mi id="A3.1.p1.51.m11.1.1.3.3.3.2" mathvariant="normal" xref="A3.1.p1.51.m11.1.1.3.3.3.2.cmml">c</mi><mi id="A3.1.p1.51.m11.1.1.3.3.3.3" xref="A3.1.p1.51.m11.1.1.3.3.3.3.cmml">θ</mi></msub></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.51.m11.1b"><apply id="A3.1.p1.51.m11.1.1.cmml" xref="A3.1.p1.51.m11.1.1"><intersect id="A3.1.p1.51.m11.1.1.1.cmml" xref="A3.1.p1.51.m11.1.1.1"></intersect><apply id="A3.1.p1.51.m11.1.1.2.cmml" xref="A3.1.p1.51.m11.1.1.2"><csymbol cd="ambiguous" id="A3.1.p1.51.m11.1.1.2.1.cmml" xref="A3.1.p1.51.m11.1.1.2">subscript</csymbol><ci id="A3.1.p1.51.m11.1.1.2.2.cmml" xref="A3.1.p1.51.m11.1.1.2.2">Ω</ci><apply id="A3.1.p1.51.m11.1.1.2.3.cmml" xref="A3.1.p1.51.m11.1.1.2.3"><csymbol cd="ambiguous" id="A3.1.p1.51.m11.1.1.2.3.1.cmml" xref="A3.1.p1.51.m11.1.1.2.3">subscript</csymbol><ci id="A3.1.p1.51.m11.1.1.2.3.2.cmml" xref="A3.1.p1.51.m11.1.1.2.3.2">c</ci><ci id="A3.1.p1.51.m11.1.1.2.3.3.cmml" xref="A3.1.p1.51.m11.1.1.2.3.3">𝑥</ci></apply></apply><apply id="A3.1.p1.51.m11.1.1.3.cmml" xref="A3.1.p1.51.m11.1.1.3"><times id="A3.1.p1.51.m11.1.1.3.1.cmml" xref="A3.1.p1.51.m11.1.1.3.1"></times><apply id="A3.1.p1.51.m11.1.1.3.2.cmml" xref="A3.1.p1.51.m11.1.1.3.2"><csymbol cd="ambiguous" id="A3.1.p1.51.m11.1.1.3.2.1.cmml" xref="A3.1.p1.51.m11.1.1.3.2">subscript</csymbol><ci id="A3.1.p1.51.m11.1.1.3.2.2.cmml" xref="A3.1.p1.51.m11.1.1.3.2.2">Ω</ci><ci id="A3.1.p1.51.m11.1.1.3.2.3.cmml" xref="A3.1.p1.51.m11.1.1.3.2.3">𝑟</ci></apply><apply id="A3.1.p1.51.m11.1.1.3.3.cmml" xref="A3.1.p1.51.m11.1.1.3.3"><csymbol cd="ambiguous" id="A3.1.p1.51.m11.1.1.3.3.1.cmml" xref="A3.1.p1.51.m11.1.1.3.3">subscript</csymbol><ci id="A3.1.p1.51.m11.1.1.3.3.2.cmml" xref="A3.1.p1.51.m11.1.1.3.3.2">Ω</ci><apply id="A3.1.p1.51.m11.1.1.3.3.3.cmml" xref="A3.1.p1.51.m11.1.1.3.3.3"><csymbol cd="ambiguous" id="A3.1.p1.51.m11.1.1.3.3.3.1.cmml" xref="A3.1.p1.51.m11.1.1.3.3.3">subscript</csymbol><ci id="A3.1.p1.51.m11.1.1.3.3.3.2.cmml" xref="A3.1.p1.51.m11.1.1.3.3.3.2">c</ci><ci id="A3.1.p1.51.m11.1.1.3.3.3.3.cmml" xref="A3.1.p1.51.m11.1.1.3.3.3.3">𝜃</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.51.m11.1c">\Omega_{{\rm c}_{x}}\cap\Omega_{r}\times\Omega_{{\rm c}_{\theta}}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.51.m11.1d">roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∩ roman_Ω start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT × roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\Omega_{{\rm c}_{w0}}" class="ltx_Math" display="inline" id="A3.1.p1.52.m12.1"><semantics id="A3.1.p1.52.m12.1a"><msub id="A3.1.p1.52.m12.1.1" xref="A3.1.p1.52.m12.1.1.cmml"><mi id="A3.1.p1.52.m12.1.1.2" mathvariant="normal" xref="A3.1.p1.52.m12.1.1.2.cmml">Ω</mi><msub id="A3.1.p1.52.m12.1.1.3" xref="A3.1.p1.52.m12.1.1.3.cmml"><mi id="A3.1.p1.52.m12.1.1.3.2" mathvariant="normal" xref="A3.1.p1.52.m12.1.1.3.2.cmml">c</mi><mrow id="A3.1.p1.52.m12.1.1.3.3" xref="A3.1.p1.52.m12.1.1.3.3.cmml"><mi id="A3.1.p1.52.m12.1.1.3.3.2" xref="A3.1.p1.52.m12.1.1.3.3.2.cmml">w</mi><mo id="A3.1.p1.52.m12.1.1.3.3.1" xref="A3.1.p1.52.m12.1.1.3.3.1.cmml">⁢</mo><mn id="A3.1.p1.52.m12.1.1.3.3.3" xref="A3.1.p1.52.m12.1.1.3.3.3.cmml">0</mn></mrow></msub></msub><annotation-xml encoding="MathML-Content" id="A3.1.p1.52.m12.1b"><apply id="A3.1.p1.52.m12.1.1.cmml" xref="A3.1.p1.52.m12.1.1"><csymbol cd="ambiguous" id="A3.1.p1.52.m12.1.1.1.cmml" xref="A3.1.p1.52.m12.1.1">subscript</csymbol><ci id="A3.1.p1.52.m12.1.1.2.cmml" xref="A3.1.p1.52.m12.1.1.2">Ω</ci><apply id="A3.1.p1.52.m12.1.1.3.cmml" xref="A3.1.p1.52.m12.1.1.3"><csymbol cd="ambiguous" id="A3.1.p1.52.m12.1.1.3.1.cmml" xref="A3.1.p1.52.m12.1.1.3">subscript</csymbol><ci id="A3.1.p1.52.m12.1.1.3.2.cmml" xref="A3.1.p1.52.m12.1.1.3.2">c</ci><apply id="A3.1.p1.52.m12.1.1.3.3.cmml" xref="A3.1.p1.52.m12.1.1.3.3"><times id="A3.1.p1.52.m12.1.1.3.3.1.cmml" xref="A3.1.p1.52.m12.1.1.3.3.1"></times><ci id="A3.1.p1.52.m12.1.1.3.3.2.cmml" xref="A3.1.p1.52.m12.1.1.3.3.2">𝑤</ci><cn id="A3.1.p1.52.m12.1.1.3.3.3.cmml" type="integer" xref="A3.1.p1.52.m12.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.52.m12.1c">\Omega_{{\rm c}_{w0}}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.52.m12.1d">roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT italic_w 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="A3.1.p1.53.m13.1"><semantics id="A3.1.p1.53.m13.1a"><mo id="A3.1.p1.53.m13.1.1" xref="A3.1.p1.53.m13.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="A3.1.p1.53.m13.1b"><csymbol cd="latexml" id="A3.1.p1.53.m13.1.1.cmml" xref="A3.1.p1.53.m13.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.53.m13.1c">:=</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.53.m13.1d">:=</annotation></semantics></math> <math alttext="\Omega_{{\rm c}_{0}}\cap\Omega_{r}\times\Omega_{{\rm c}_{\theta}}" class="ltx_Math" display="inline" id="A3.1.p1.54.m14.1"><semantics id="A3.1.p1.54.m14.1a"><mrow id="A3.1.p1.54.m14.1.1" xref="A3.1.p1.54.m14.1.1.cmml"><msub id="A3.1.p1.54.m14.1.1.2" xref="A3.1.p1.54.m14.1.1.2.cmml"><mi id="A3.1.p1.54.m14.1.1.2.2" mathvariant="normal" xref="A3.1.p1.54.m14.1.1.2.2.cmml">Ω</mi><msub id="A3.1.p1.54.m14.1.1.2.3" xref="A3.1.p1.54.m14.1.1.2.3.cmml"><mi id="A3.1.p1.54.m14.1.1.2.3.2" mathvariant="normal" xref="A3.1.p1.54.m14.1.1.2.3.2.cmml">c</mi><mn id="A3.1.p1.54.m14.1.1.2.3.3" xref="A3.1.p1.54.m14.1.1.2.3.3.cmml">0</mn></msub></msub><mo id="A3.1.p1.54.m14.1.1.1" xref="A3.1.p1.54.m14.1.1.1.cmml">∩</mo><mrow id="A3.1.p1.54.m14.1.1.3" xref="A3.1.p1.54.m14.1.1.3.cmml"><msub id="A3.1.p1.54.m14.1.1.3.2" xref="A3.1.p1.54.m14.1.1.3.2.cmml"><mi id="A3.1.p1.54.m14.1.1.3.2.2" mathvariant="normal" xref="A3.1.p1.54.m14.1.1.3.2.2.cmml">Ω</mi><mi id="A3.1.p1.54.m14.1.1.3.2.3" xref="A3.1.p1.54.m14.1.1.3.2.3.cmml">r</mi></msub><mo id="A3.1.p1.54.m14.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="A3.1.p1.54.m14.1.1.3.1.cmml">×</mo><msub id="A3.1.p1.54.m14.1.1.3.3" xref="A3.1.p1.54.m14.1.1.3.3.cmml"><mi id="A3.1.p1.54.m14.1.1.3.3.2" mathvariant="normal" xref="A3.1.p1.54.m14.1.1.3.3.2.cmml">Ω</mi><msub id="A3.1.p1.54.m14.1.1.3.3.3" xref="A3.1.p1.54.m14.1.1.3.3.3.cmml"><mi id="A3.1.p1.54.m14.1.1.3.3.3.2" mathvariant="normal" xref="A3.1.p1.54.m14.1.1.3.3.3.2.cmml">c</mi><mi id="A3.1.p1.54.m14.1.1.3.3.3.3" xref="A3.1.p1.54.m14.1.1.3.3.3.3.cmml">θ</mi></msub></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.54.m14.1b"><apply id="A3.1.p1.54.m14.1.1.cmml" xref="A3.1.p1.54.m14.1.1"><intersect id="A3.1.p1.54.m14.1.1.1.cmml" xref="A3.1.p1.54.m14.1.1.1"></intersect><apply id="A3.1.p1.54.m14.1.1.2.cmml" xref="A3.1.p1.54.m14.1.1.2"><csymbol cd="ambiguous" id="A3.1.p1.54.m14.1.1.2.1.cmml" xref="A3.1.p1.54.m14.1.1.2">subscript</csymbol><ci id="A3.1.p1.54.m14.1.1.2.2.cmml" xref="A3.1.p1.54.m14.1.1.2.2">Ω</ci><apply id="A3.1.p1.54.m14.1.1.2.3.cmml" xref="A3.1.p1.54.m14.1.1.2.3"><csymbol cd="ambiguous" id="A3.1.p1.54.m14.1.1.2.3.1.cmml" xref="A3.1.p1.54.m14.1.1.2.3">subscript</csymbol><ci id="A3.1.p1.54.m14.1.1.2.3.2.cmml" xref="A3.1.p1.54.m14.1.1.2.3.2">c</ci><cn id="A3.1.p1.54.m14.1.1.2.3.3.cmml" type="integer" xref="A3.1.p1.54.m14.1.1.2.3.3">0</cn></apply></apply><apply id="A3.1.p1.54.m14.1.1.3.cmml" xref="A3.1.p1.54.m14.1.1.3"><times id="A3.1.p1.54.m14.1.1.3.1.cmml" xref="A3.1.p1.54.m14.1.1.3.1"></times><apply id="A3.1.p1.54.m14.1.1.3.2.cmml" xref="A3.1.p1.54.m14.1.1.3.2"><csymbol cd="ambiguous" id="A3.1.p1.54.m14.1.1.3.2.1.cmml" xref="A3.1.p1.54.m14.1.1.3.2">subscript</csymbol><ci id="A3.1.p1.54.m14.1.1.3.2.2.cmml" xref="A3.1.p1.54.m14.1.1.3.2.2">Ω</ci><ci id="A3.1.p1.54.m14.1.1.3.2.3.cmml" xref="A3.1.p1.54.m14.1.1.3.2.3">𝑟</ci></apply><apply id="A3.1.p1.54.m14.1.1.3.3.cmml" xref="A3.1.p1.54.m14.1.1.3.3"><csymbol cd="ambiguous" id="A3.1.p1.54.m14.1.1.3.3.1.cmml" xref="A3.1.p1.54.m14.1.1.3.3">subscript</csymbol><ci id="A3.1.p1.54.m14.1.1.3.3.2.cmml" xref="A3.1.p1.54.m14.1.1.3.3.2">Ω</ci><apply id="A3.1.p1.54.m14.1.1.3.3.3.cmml" xref="A3.1.p1.54.m14.1.1.3.3.3"><csymbol cd="ambiguous" id="A3.1.p1.54.m14.1.1.3.3.3.1.cmml" xref="A3.1.p1.54.m14.1.1.3.3.3">subscript</csymbol><ci id="A3.1.p1.54.m14.1.1.3.3.3.2.cmml" xref="A3.1.p1.54.m14.1.1.3.3.3.2">c</ci><ci id="A3.1.p1.54.m14.1.1.3.3.3.3.cmml" xref="A3.1.p1.54.m14.1.1.3.3.3.3">𝜃</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.54.m14.1c">\Omega_{{\rm c}_{0}}\cap\Omega_{r}\times\Omega_{{\rm c}_{\theta}}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.54.m14.1d">roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∩ roman_Ω start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT × roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="\Omega_{{\rm c}_{w0}}\subset\Omega_{{\rm c}_{w}}" class="ltx_Math" display="inline" id="A3.1.p1.55.m15.1"><semantics id="A3.1.p1.55.m15.1a"><mrow id="A3.1.p1.55.m15.1.1" xref="A3.1.p1.55.m15.1.1.cmml"><msub id="A3.1.p1.55.m15.1.1.2" xref="A3.1.p1.55.m15.1.1.2.cmml"><mi id="A3.1.p1.55.m15.1.1.2.2" mathvariant="normal" xref="A3.1.p1.55.m15.1.1.2.2.cmml">Ω</mi><msub id="A3.1.p1.55.m15.1.1.2.3" xref="A3.1.p1.55.m15.1.1.2.3.cmml"><mi id="A3.1.p1.55.m15.1.1.2.3.2" mathvariant="normal" xref="A3.1.p1.55.m15.1.1.2.3.2.cmml">c</mi><mrow id="A3.1.p1.55.m15.1.1.2.3.3" xref="A3.1.p1.55.m15.1.1.2.3.3.cmml"><mi id="A3.1.p1.55.m15.1.1.2.3.3.2" xref="A3.1.p1.55.m15.1.1.2.3.3.2.cmml">w</mi><mo id="A3.1.p1.55.m15.1.1.2.3.3.1" xref="A3.1.p1.55.m15.1.1.2.3.3.1.cmml">⁢</mo><mn id="A3.1.p1.55.m15.1.1.2.3.3.3" xref="A3.1.p1.55.m15.1.1.2.3.3.3.cmml">0</mn></mrow></msub></msub><mo id="A3.1.p1.55.m15.1.1.1" xref="A3.1.p1.55.m15.1.1.1.cmml">⊂</mo><msub id="A3.1.p1.55.m15.1.1.3" xref="A3.1.p1.55.m15.1.1.3.cmml"><mi id="A3.1.p1.55.m15.1.1.3.2" mathvariant="normal" xref="A3.1.p1.55.m15.1.1.3.2.cmml">Ω</mi><msub id="A3.1.p1.55.m15.1.1.3.3" xref="A3.1.p1.55.m15.1.1.3.3.cmml"><mi id="A3.1.p1.55.m15.1.1.3.3.2" mathvariant="normal" xref="A3.1.p1.55.m15.1.1.3.3.2.cmml">c</mi><mi id="A3.1.p1.55.m15.1.1.3.3.3" xref="A3.1.p1.55.m15.1.1.3.3.3.cmml">w</mi></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.55.m15.1b"><apply id="A3.1.p1.55.m15.1.1.cmml" xref="A3.1.p1.55.m15.1.1"><subset id="A3.1.p1.55.m15.1.1.1.cmml" xref="A3.1.p1.55.m15.1.1.1"></subset><apply id="A3.1.p1.55.m15.1.1.2.cmml" xref="A3.1.p1.55.m15.1.1.2"><csymbol cd="ambiguous" id="A3.1.p1.55.m15.1.1.2.1.cmml" xref="A3.1.p1.55.m15.1.1.2">subscript</csymbol><ci id="A3.1.p1.55.m15.1.1.2.2.cmml" xref="A3.1.p1.55.m15.1.1.2.2">Ω</ci><apply id="A3.1.p1.55.m15.1.1.2.3.cmml" xref="A3.1.p1.55.m15.1.1.2.3"><csymbol cd="ambiguous" id="A3.1.p1.55.m15.1.1.2.3.1.cmml" xref="A3.1.p1.55.m15.1.1.2.3">subscript</csymbol><ci id="A3.1.p1.55.m15.1.1.2.3.2.cmml" xref="A3.1.p1.55.m15.1.1.2.3.2">c</ci><apply id="A3.1.p1.55.m15.1.1.2.3.3.cmml" xref="A3.1.p1.55.m15.1.1.2.3.3"><times id="A3.1.p1.55.m15.1.1.2.3.3.1.cmml" xref="A3.1.p1.55.m15.1.1.2.3.3.1"></times><ci id="A3.1.p1.55.m15.1.1.2.3.3.2.cmml" xref="A3.1.p1.55.m15.1.1.2.3.3.2">𝑤</ci><cn id="A3.1.p1.55.m15.1.1.2.3.3.3.cmml" type="integer" xref="A3.1.p1.55.m15.1.1.2.3.3.3">0</cn></apply></apply></apply><apply id="A3.1.p1.55.m15.1.1.3.cmml" xref="A3.1.p1.55.m15.1.1.3"><csymbol cd="ambiguous" id="A3.1.p1.55.m15.1.1.3.1.cmml" xref="A3.1.p1.55.m15.1.1.3">subscript</csymbol><ci id="A3.1.p1.55.m15.1.1.3.2.cmml" xref="A3.1.p1.55.m15.1.1.3.2">Ω</ci><apply id="A3.1.p1.55.m15.1.1.3.3.cmml" xref="A3.1.p1.55.m15.1.1.3.3"><csymbol cd="ambiguous" id="A3.1.p1.55.m15.1.1.3.3.1.cmml" xref="A3.1.p1.55.m15.1.1.3.3">subscript</csymbol><ci id="A3.1.p1.55.m15.1.1.3.3.2.cmml" xref="A3.1.p1.55.m15.1.1.3.3.2">c</ci><ci id="A3.1.p1.55.m15.1.1.3.3.3.cmml" xref="A3.1.p1.55.m15.1.1.3.3.3">𝑤</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.55.m15.1c">\Omega_{{\rm c}_{w0}}\subset\Omega_{{\rm c}_{w}}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.55.m15.1d">roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT italic_w 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊂ roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. It is implied from the above inequality that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx62"> <tbody id="A3.E47"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}(t)\leq-k_{\rm c}V(t)/2,\forall V(t)\geq\eta" class="ltx_Math" display="inline" id="A3.E47.m1.5"><semantics id="A3.E47.m1.5a"><mrow id="A3.E47.m1.5.5.2" xref="A3.E47.m1.5.5.3.cmml"><mrow id="A3.E47.m1.4.4.1.1" xref="A3.E47.m1.4.4.1.1.cmml"><mrow id="A3.E47.m1.4.4.1.1.2" xref="A3.E47.m1.4.4.1.1.2.cmml"><mover accent="true" id="A3.E47.m1.4.4.1.1.2.2" xref="A3.E47.m1.4.4.1.1.2.2.cmml"><mi id="A3.E47.m1.4.4.1.1.2.2.2" xref="A3.E47.m1.4.4.1.1.2.2.2.cmml">V</mi><mo id="A3.E47.m1.4.4.1.1.2.2.1" xref="A3.E47.m1.4.4.1.1.2.2.1.cmml">˙</mo></mover><mo id="A3.E47.m1.4.4.1.1.2.1" xref="A3.E47.m1.4.4.1.1.2.1.cmml">⁢</mo><mrow id="A3.E47.m1.4.4.1.1.2.3.2" xref="A3.E47.m1.4.4.1.1.2.cmml"><mo id="A3.E47.m1.4.4.1.1.2.3.2.1" stretchy="false" xref="A3.E47.m1.4.4.1.1.2.cmml">(</mo><mi id="A3.E47.m1.1.1" xref="A3.E47.m1.1.1.cmml">t</mi><mo id="A3.E47.m1.4.4.1.1.2.3.2.2" stretchy="false" xref="A3.E47.m1.4.4.1.1.2.cmml">)</mo></mrow></mrow><mo id="A3.E47.m1.4.4.1.1.1" xref="A3.E47.m1.4.4.1.1.1.cmml">≤</mo><mrow id="A3.E47.m1.4.4.1.1.3" xref="A3.E47.m1.4.4.1.1.3.cmml"><mo id="A3.E47.m1.4.4.1.1.3a" xref="A3.E47.m1.4.4.1.1.3.cmml">−</mo><mrow id="A3.E47.m1.4.4.1.1.3.2" xref="A3.E47.m1.4.4.1.1.3.2.cmml"><mrow id="A3.E47.m1.4.4.1.1.3.2.2" xref="A3.E47.m1.4.4.1.1.3.2.2.cmml"><msub id="A3.E47.m1.4.4.1.1.3.2.2.2" xref="A3.E47.m1.4.4.1.1.3.2.2.2.cmml"><mi id="A3.E47.m1.4.4.1.1.3.2.2.2.2" xref="A3.E47.m1.4.4.1.1.3.2.2.2.2.cmml">k</mi><mi id="A3.E47.m1.4.4.1.1.3.2.2.2.3" mathvariant="normal" xref="A3.E47.m1.4.4.1.1.3.2.2.2.3.cmml">c</mi></msub><mo id="A3.E47.m1.4.4.1.1.3.2.2.1" xref="A3.E47.m1.4.4.1.1.3.2.2.1.cmml">⁢</mo><mi id="A3.E47.m1.4.4.1.1.3.2.2.3" xref="A3.E47.m1.4.4.1.1.3.2.2.3.cmml">V</mi><mo id="A3.E47.m1.4.4.1.1.3.2.2.1a" xref="A3.E47.m1.4.4.1.1.3.2.2.1.cmml">⁢</mo><mrow id="A3.E47.m1.4.4.1.1.3.2.2.4.2" xref="A3.E47.m1.4.4.1.1.3.2.2.cmml"><mo id="A3.E47.m1.4.4.1.1.3.2.2.4.2.1" stretchy="false" xref="A3.E47.m1.4.4.1.1.3.2.2.cmml">(</mo><mi id="A3.E47.m1.2.2" xref="A3.E47.m1.2.2.cmml">t</mi><mo id="A3.E47.m1.4.4.1.1.3.2.2.4.2.2" stretchy="false" xref="A3.E47.m1.4.4.1.1.3.2.2.cmml">)</mo></mrow></mrow><mo id="A3.E47.m1.4.4.1.1.3.2.1" xref="A3.E47.m1.4.4.1.1.3.2.1.cmml">/</mo><mn id="A3.E47.m1.4.4.1.1.3.2.3" xref="A3.E47.m1.4.4.1.1.3.2.3.cmml">2</mn></mrow></mrow></mrow><mo id="A3.E47.m1.5.5.2.3" xref="A3.E47.m1.5.5.3a.cmml">,</mo><mrow id="A3.E47.m1.5.5.2.2" xref="A3.E47.m1.5.5.2.2.cmml"><mrow id="A3.E47.m1.5.5.2.2.2" xref="A3.E47.m1.5.5.2.2.2.cmml"><mo id="A3.E47.m1.5.5.2.2.2.1" rspace="0.167em" xref="A3.E47.m1.5.5.2.2.2.1.cmml">∀</mo><mrow id="A3.E47.m1.5.5.2.2.2.2" xref="A3.E47.m1.5.5.2.2.2.2.cmml"><mi id="A3.E47.m1.5.5.2.2.2.2.2" xref="A3.E47.m1.5.5.2.2.2.2.2.cmml">V</mi><mo id="A3.E47.m1.5.5.2.2.2.2.1" xref="A3.E47.m1.5.5.2.2.2.2.1.cmml">⁢</mo><mrow id="A3.E47.m1.5.5.2.2.2.2.3.2" xref="A3.E47.m1.5.5.2.2.2.2.cmml"><mo id="A3.E47.m1.5.5.2.2.2.2.3.2.1" stretchy="false" xref="A3.E47.m1.5.5.2.2.2.2.cmml">(</mo><mi id="A3.E47.m1.3.3" xref="A3.E47.m1.3.3.cmml">t</mi><mo id="A3.E47.m1.5.5.2.2.2.2.3.2.2" stretchy="false" xref="A3.E47.m1.5.5.2.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="A3.E47.m1.5.5.2.2.1" xref="A3.E47.m1.5.5.2.2.1.cmml">≥</mo><mi id="A3.E47.m1.5.5.2.2.3" xref="A3.E47.m1.5.5.2.2.3.cmml">η</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.E47.m1.5b"><apply id="A3.E47.m1.5.5.3.cmml" xref="A3.E47.m1.5.5.2"><csymbol cd="ambiguous" id="A3.E47.m1.5.5.3a.cmml" xref="A3.E47.m1.5.5.2.3">formulae-sequence</csymbol><apply id="A3.E47.m1.4.4.1.1.cmml" xref="A3.E47.m1.4.4.1.1"><leq id="A3.E47.m1.4.4.1.1.1.cmml" xref="A3.E47.m1.4.4.1.1.1"></leq><apply id="A3.E47.m1.4.4.1.1.2.cmml" xref="A3.E47.m1.4.4.1.1.2"><times id="A3.E47.m1.4.4.1.1.2.1.cmml" xref="A3.E47.m1.4.4.1.1.2.1"></times><apply id="A3.E47.m1.4.4.1.1.2.2.cmml" xref="A3.E47.m1.4.4.1.1.2.2"><ci id="A3.E47.m1.4.4.1.1.2.2.1.cmml" xref="A3.E47.m1.4.4.1.1.2.2.1">˙</ci><ci id="A3.E47.m1.4.4.1.1.2.2.2.cmml" xref="A3.E47.m1.4.4.1.1.2.2.2">𝑉</ci></apply><ci id="A3.E47.m1.1.1.cmml" xref="A3.E47.m1.1.1">𝑡</ci></apply><apply id="A3.E47.m1.4.4.1.1.3.cmml" xref="A3.E47.m1.4.4.1.1.3"><minus id="A3.E47.m1.4.4.1.1.3.1.cmml" xref="A3.E47.m1.4.4.1.1.3"></minus><apply id="A3.E47.m1.4.4.1.1.3.2.cmml" xref="A3.E47.m1.4.4.1.1.3.2"><divide id="A3.E47.m1.4.4.1.1.3.2.1.cmml" xref="A3.E47.m1.4.4.1.1.3.2.1"></divide><apply id="A3.E47.m1.4.4.1.1.3.2.2.cmml" xref="A3.E47.m1.4.4.1.1.3.2.2"><times id="A3.E47.m1.4.4.1.1.3.2.2.1.cmml" xref="A3.E47.m1.4.4.1.1.3.2.2.1"></times><apply id="A3.E47.m1.4.4.1.1.3.2.2.2.cmml" xref="A3.E47.m1.4.4.1.1.3.2.2.2"><csymbol cd="ambiguous" id="A3.E47.m1.4.4.1.1.3.2.2.2.1.cmml" xref="A3.E47.m1.4.4.1.1.3.2.2.2">subscript</csymbol><ci id="A3.E47.m1.4.4.1.1.3.2.2.2.2.cmml" xref="A3.E47.m1.4.4.1.1.3.2.2.2.2">𝑘</ci><ci id="A3.E47.m1.4.4.1.1.3.2.2.2.3.cmml" xref="A3.E47.m1.4.4.1.1.3.2.2.2.3">c</ci></apply><ci id="A3.E47.m1.4.4.1.1.3.2.2.3.cmml" xref="A3.E47.m1.4.4.1.1.3.2.2.3">𝑉</ci><ci id="A3.E47.m1.2.2.cmml" xref="A3.E47.m1.2.2">𝑡</ci></apply><cn id="A3.E47.m1.4.4.1.1.3.2.3.cmml" type="integer" xref="A3.E47.m1.4.4.1.1.3.2.3">2</cn></apply></apply></apply><apply id="A3.E47.m1.5.5.2.2.cmml" xref="A3.E47.m1.5.5.2.2"><geq id="A3.E47.m1.5.5.2.2.1.cmml" xref="A3.E47.m1.5.5.2.2.1"></geq><apply id="A3.E47.m1.5.5.2.2.2.cmml" xref="A3.E47.m1.5.5.2.2.2"><csymbol cd="latexml" id="A3.E47.m1.5.5.2.2.2.1.cmml" xref="A3.E47.m1.5.5.2.2.2.1">for-all</csymbol><apply id="A3.E47.m1.5.5.2.2.2.2.cmml" xref="A3.E47.m1.5.5.2.2.2.2"><times id="A3.E47.m1.5.5.2.2.2.2.1.cmml" xref="A3.E47.m1.5.5.2.2.2.2.1"></times><ci id="A3.E47.m1.5.5.2.2.2.2.2.cmml" xref="A3.E47.m1.5.5.2.2.2.2.2">𝑉</ci><ci id="A3.E47.m1.3.3.cmml" xref="A3.E47.m1.3.3">𝑡</ci></apply></apply><ci id="A3.E47.m1.5.5.2.2.3.cmml" xref="A3.E47.m1.5.5.2.2.3">𝜂</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E47.m1.5c">\displaystyle\dot{V}(t)\leq-k_{\rm c}V(t)/2,\forall V(t)\geq\eta</annotation><annotation encoding="application/x-llamapun" id="A3.E47.m1.5d">over˙ start_ARG italic_V end_ARG ( italic_t ) ≤ - italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT italic_V ( italic_t ) / 2 , ∀ italic_V ( italic_t ) ≥ italic_η</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(47)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.1.p1.72">on <math alttext="\bm{w}(t)\in\Omega_{{\rm c}_{w}}" class="ltx_Math" display="inline" id="A3.1.p1.56.m1.1"><semantics id="A3.1.p1.56.m1.1a"><mrow id="A3.1.p1.56.m1.1.2" xref="A3.1.p1.56.m1.1.2.cmml"><mrow id="A3.1.p1.56.m1.1.2.2" xref="A3.1.p1.56.m1.1.2.2.cmml"><mi id="A3.1.p1.56.m1.1.2.2.2" xref="A3.1.p1.56.m1.1.2.2.2.cmml">𝒘</mi><mo id="A3.1.p1.56.m1.1.2.2.1" xref="A3.1.p1.56.m1.1.2.2.1.cmml">⁢</mo><mrow id="A3.1.p1.56.m1.1.2.2.3.2" xref="A3.1.p1.56.m1.1.2.2.cmml"><mo id="A3.1.p1.56.m1.1.2.2.3.2.1" stretchy="false" xref="A3.1.p1.56.m1.1.2.2.cmml">(</mo><mi id="A3.1.p1.56.m1.1.1" xref="A3.1.p1.56.m1.1.1.cmml">t</mi><mo id="A3.1.p1.56.m1.1.2.2.3.2.2" stretchy="false" xref="A3.1.p1.56.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="A3.1.p1.56.m1.1.2.1" xref="A3.1.p1.56.m1.1.2.1.cmml">∈</mo><msub id="A3.1.p1.56.m1.1.2.3" xref="A3.1.p1.56.m1.1.2.3.cmml"><mi id="A3.1.p1.56.m1.1.2.3.2" mathvariant="normal" xref="A3.1.p1.56.m1.1.2.3.2.cmml">Ω</mi><msub id="A3.1.p1.56.m1.1.2.3.3" xref="A3.1.p1.56.m1.1.2.3.3.cmml"><mi id="A3.1.p1.56.m1.1.2.3.3.2" mathvariant="normal" xref="A3.1.p1.56.m1.1.2.3.3.2.cmml">c</mi><mi id="A3.1.p1.56.m1.1.2.3.3.3" xref="A3.1.p1.56.m1.1.2.3.3.3.cmml">w</mi></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.56.m1.1b"><apply id="A3.1.p1.56.m1.1.2.cmml" xref="A3.1.p1.56.m1.1.2"><in id="A3.1.p1.56.m1.1.2.1.cmml" xref="A3.1.p1.56.m1.1.2.1"></in><apply id="A3.1.p1.56.m1.1.2.2.cmml" xref="A3.1.p1.56.m1.1.2.2"><times id="A3.1.p1.56.m1.1.2.2.1.cmml" xref="A3.1.p1.56.m1.1.2.2.1"></times><ci id="A3.1.p1.56.m1.1.2.2.2.cmml" xref="A3.1.p1.56.m1.1.2.2.2">𝒘</ci><ci id="A3.1.p1.56.m1.1.1.cmml" xref="A3.1.p1.56.m1.1.1">𝑡</ci></apply><apply id="A3.1.p1.56.m1.1.2.3.cmml" xref="A3.1.p1.56.m1.1.2.3"><csymbol cd="ambiguous" id="A3.1.p1.56.m1.1.2.3.1.cmml" xref="A3.1.p1.56.m1.1.2.3">subscript</csymbol><ci id="A3.1.p1.56.m1.1.2.3.2.cmml" xref="A3.1.p1.56.m1.1.2.3.2">Ω</ci><apply id="A3.1.p1.56.m1.1.2.3.3.cmml" xref="A3.1.p1.56.m1.1.2.3.3"><csymbol cd="ambiguous" id="A3.1.p1.56.m1.1.2.3.3.1.cmml" xref="A3.1.p1.56.m1.1.2.3.3">subscript</csymbol><ci id="A3.1.p1.56.m1.1.2.3.3.2.cmml" xref="A3.1.p1.56.m1.1.2.3.3.2">c</ci><ci id="A3.1.p1.56.m1.1.2.3.3.3.cmml" xref="A3.1.p1.56.m1.1.2.3.3.3">𝑤</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.56.m1.1c">\bm{w}(t)\in\Omega_{{\rm c}_{w}}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.56.m1.1d">bold_italic_w ( italic_t ) ∈ roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="t\in[0,t_{\rm f})" class="ltx_Math" display="inline" id="A3.1.p1.57.m2.2"><semantics id="A3.1.p1.57.m2.2a"><mrow id="A3.1.p1.57.m2.2.2" xref="A3.1.p1.57.m2.2.2.cmml"><mi id="A3.1.p1.57.m2.2.2.3" xref="A3.1.p1.57.m2.2.2.3.cmml">t</mi><mo id="A3.1.p1.57.m2.2.2.2" xref="A3.1.p1.57.m2.2.2.2.cmml">∈</mo><mrow id="A3.1.p1.57.m2.2.2.1.1" xref="A3.1.p1.57.m2.2.2.1.2.cmml"><mo id="A3.1.p1.57.m2.2.2.1.1.2" stretchy="false" xref="A3.1.p1.57.m2.2.2.1.2.cmml">[</mo><mn id="A3.1.p1.57.m2.1.1" xref="A3.1.p1.57.m2.1.1.cmml">0</mn><mo id="A3.1.p1.57.m2.2.2.1.1.3" xref="A3.1.p1.57.m2.2.2.1.2.cmml">,</mo><msub id="A3.1.p1.57.m2.2.2.1.1.1" xref="A3.1.p1.57.m2.2.2.1.1.1.cmml"><mi id="A3.1.p1.57.m2.2.2.1.1.1.2" xref="A3.1.p1.57.m2.2.2.1.1.1.2.cmml">t</mi><mi id="A3.1.p1.57.m2.2.2.1.1.1.3" mathvariant="normal" xref="A3.1.p1.57.m2.2.2.1.1.1.3.cmml">f</mi></msub><mo id="A3.1.p1.57.m2.2.2.1.1.4" stretchy="false" xref="A3.1.p1.57.m2.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.57.m2.2b"><apply id="A3.1.p1.57.m2.2.2.cmml" xref="A3.1.p1.57.m2.2.2"><in id="A3.1.p1.57.m2.2.2.2.cmml" xref="A3.1.p1.57.m2.2.2.2"></in><ci id="A3.1.p1.57.m2.2.2.3.cmml" xref="A3.1.p1.57.m2.2.2.3">𝑡</ci><interval closure="closed-open" id="A3.1.p1.57.m2.2.2.1.2.cmml" xref="A3.1.p1.57.m2.2.2.1.1"><cn id="A3.1.p1.57.m2.1.1.cmml" type="integer" xref="A3.1.p1.57.m2.1.1">0</cn><apply id="A3.1.p1.57.m2.2.2.1.1.1.cmml" xref="A3.1.p1.57.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="A3.1.p1.57.m2.2.2.1.1.1.1.cmml" xref="A3.1.p1.57.m2.2.2.1.1.1">subscript</csymbol><ci id="A3.1.p1.57.m2.2.2.1.1.1.2.cmml" xref="A3.1.p1.57.m2.2.2.1.1.1.2">𝑡</ci><ci id="A3.1.p1.57.m2.2.2.1.1.1.3.cmml" xref="A3.1.p1.57.m2.2.2.1.1.1.3">f</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.57.m2.2c">t\in[0,t_{\rm f})</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.57.m2.2d">italic_t ∈ [ 0 , italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT )</annotation></semantics></math>. Based on the results presented in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A3.E40" title="In Proof. ‣ Appendix C The proof of Theorem 2 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">40</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A3.E47" title="In Proof. ‣ Appendix C The proof of Theorem 2 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">47</span></a>), the UUB Theorem <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib41" title="">41</a>, Th. 4.5]</cite> is applied to conclude that if <math alttext="\eta<\sqrt{c_{w0}/\lambda_{\rm b}}" class="ltx_Math" display="inline" id="A3.1.p1.58.m3.1"><semantics id="A3.1.p1.58.m3.1a"><mrow id="A3.1.p1.58.m3.1.1" xref="A3.1.p1.58.m3.1.1.cmml"><mi id="A3.1.p1.58.m3.1.1.2" xref="A3.1.p1.58.m3.1.1.2.cmml">η</mi><mo id="A3.1.p1.58.m3.1.1.1" xref="A3.1.p1.58.m3.1.1.1.cmml"><</mo><msqrt id="A3.1.p1.58.m3.1.1.3" xref="A3.1.p1.58.m3.1.1.3.cmml"><mrow id="A3.1.p1.58.m3.1.1.3.2" xref="A3.1.p1.58.m3.1.1.3.2.cmml"><msub id="A3.1.p1.58.m3.1.1.3.2.2" xref="A3.1.p1.58.m3.1.1.3.2.2.cmml"><mi id="A3.1.p1.58.m3.1.1.3.2.2.2" xref="A3.1.p1.58.m3.1.1.3.2.2.2.cmml">c</mi><mrow id="A3.1.p1.58.m3.1.1.3.2.2.3" xref="A3.1.p1.58.m3.1.1.3.2.2.3.cmml"><mi id="A3.1.p1.58.m3.1.1.3.2.2.3.2" xref="A3.1.p1.58.m3.1.1.3.2.2.3.2.cmml">w</mi><mo id="A3.1.p1.58.m3.1.1.3.2.2.3.1" xref="A3.1.p1.58.m3.1.1.3.2.2.3.1.cmml">⁢</mo><mn id="A3.1.p1.58.m3.1.1.3.2.2.3.3" xref="A3.1.p1.58.m3.1.1.3.2.2.3.3.cmml">0</mn></mrow></msub><mo id="A3.1.p1.58.m3.1.1.3.2.1" xref="A3.1.p1.58.m3.1.1.3.2.1.cmml">/</mo><msub id="A3.1.p1.58.m3.1.1.3.2.3" xref="A3.1.p1.58.m3.1.1.3.2.3.cmml"><mi id="A3.1.p1.58.m3.1.1.3.2.3.2" xref="A3.1.p1.58.m3.1.1.3.2.3.2.cmml">λ</mi><mi id="A3.1.p1.58.m3.1.1.3.2.3.3" mathvariant="normal" xref="A3.1.p1.58.m3.1.1.3.2.3.3.cmml">b</mi></msub></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.58.m3.1b"><apply id="A3.1.p1.58.m3.1.1.cmml" xref="A3.1.p1.58.m3.1.1"><lt id="A3.1.p1.58.m3.1.1.1.cmml" xref="A3.1.p1.58.m3.1.1.1"></lt><ci id="A3.1.p1.58.m3.1.1.2.cmml" xref="A3.1.p1.58.m3.1.1.2">𝜂</ci><apply id="A3.1.p1.58.m3.1.1.3.cmml" xref="A3.1.p1.58.m3.1.1.3"><root id="A3.1.p1.58.m3.1.1.3a.cmml" xref="A3.1.p1.58.m3.1.1.3"></root><apply id="A3.1.p1.58.m3.1.1.3.2.cmml" xref="A3.1.p1.58.m3.1.1.3.2"><divide id="A3.1.p1.58.m3.1.1.3.2.1.cmml" xref="A3.1.p1.58.m3.1.1.3.2.1"></divide><apply id="A3.1.p1.58.m3.1.1.3.2.2.cmml" xref="A3.1.p1.58.m3.1.1.3.2.2"><csymbol cd="ambiguous" id="A3.1.p1.58.m3.1.1.3.2.2.1.cmml" xref="A3.1.p1.58.m3.1.1.3.2.2">subscript</csymbol><ci id="A3.1.p1.58.m3.1.1.3.2.2.2.cmml" xref="A3.1.p1.58.m3.1.1.3.2.2.2">𝑐</ci><apply id="A3.1.p1.58.m3.1.1.3.2.2.3.cmml" xref="A3.1.p1.58.m3.1.1.3.2.2.3"><times id="A3.1.p1.58.m3.1.1.3.2.2.3.1.cmml" xref="A3.1.p1.58.m3.1.1.3.2.2.3.1"></times><ci id="A3.1.p1.58.m3.1.1.3.2.2.3.2.cmml" xref="A3.1.p1.58.m3.1.1.3.2.2.3.2">𝑤</ci><cn id="A3.1.p1.58.m3.1.1.3.2.2.3.3.cmml" type="integer" xref="A3.1.p1.58.m3.1.1.3.2.2.3.3">0</cn></apply></apply><apply id="A3.1.p1.58.m3.1.1.3.2.3.cmml" xref="A3.1.p1.58.m3.1.1.3.2.3"><csymbol cd="ambiguous" id="A3.1.p1.58.m3.1.1.3.2.3.1.cmml" xref="A3.1.p1.58.m3.1.1.3.2.3">subscript</csymbol><ci id="A3.1.p1.58.m3.1.1.3.2.3.2.cmml" xref="A3.1.p1.58.m3.1.1.3.2.3.2">𝜆</ci><ci id="A3.1.p1.58.m3.1.1.3.2.3.3.cmml" xref="A3.1.p1.58.m3.1.1.3.2.3.3">b</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.58.m3.1c">\eta<\sqrt{c_{w0}/\lambda_{\rm b}}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.58.m3.1d">italic_η < square-root start_ARG italic_c start_POSTSUBSCRIPT italic_w 0 end_POSTSUBSCRIPT / italic_λ start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT end_ARG</annotation></semantics></math>, then <span class="ltx_text" id="A3.1.p1.65.7" style="color:#000099;">the closed-loop system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A3.E43" title="In Proof. ‣ Appendix C The proof of Theorem 2 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">43</span></a>) achieves UUB stability, where <math alttext="\bm{e}(t)" class="ltx_Math" display="inline" id="A3.1.p1.59.1.m1.1"><semantics id="A3.1.p1.59.1.m1.1a"><mrow id="A3.1.p1.59.1.m1.1.2" xref="A3.1.p1.59.1.m1.1.2.cmml"><mi id="A3.1.p1.59.1.m1.1.2.2" mathcolor="#000099" xref="A3.1.p1.59.1.m1.1.2.2.cmml">𝒆</mi><mo id="A3.1.p1.59.1.m1.1.2.1" xref="A3.1.p1.59.1.m1.1.2.1.cmml">⁢</mo><mrow id="A3.1.p1.59.1.m1.1.2.3.2" xref="A3.1.p1.59.1.m1.1.2.cmml"><mo id="A3.1.p1.59.1.m1.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="A3.1.p1.59.1.m1.1.2.cmml">(</mo><mi id="A3.1.p1.59.1.m1.1.1" mathcolor="#000099" xref="A3.1.p1.59.1.m1.1.1.cmml">t</mi><mo id="A3.1.p1.59.1.m1.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="A3.1.p1.59.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.59.1.m1.1b"><apply id="A3.1.p1.59.1.m1.1.2.cmml" xref="A3.1.p1.59.1.m1.1.2"><times id="A3.1.p1.59.1.m1.1.2.1.cmml" xref="A3.1.p1.59.1.m1.1.2.1"></times><ci id="A3.1.p1.59.1.m1.1.2.2.cmml" xref="A3.1.p1.59.1.m1.1.2.2">𝒆</ci><ci id="A3.1.p1.59.1.m1.1.1.cmml" xref="A3.1.p1.59.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.59.1.m1.1c">\bm{e}(t)</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.59.1.m1.1d">bold_italic_e ( italic_t )</annotation></semantics></math>, <math alttext="\tilde{\bm{\theta}}(t)\in" class="ltx_Math" display="inline" id="A3.1.p1.60.2.m2.1"><semantics id="A3.1.p1.60.2.m2.1a"><mrow id="A3.1.p1.60.2.m2.1.2" xref="A3.1.p1.60.2.m2.1.2.cmml"><mrow id="A3.1.p1.60.2.m2.1.2.2" xref="A3.1.p1.60.2.m2.1.2.2.cmml"><mover accent="true" id="A3.1.p1.60.2.m2.1.2.2.2" xref="A3.1.p1.60.2.m2.1.2.2.2.cmml"><mi id="A3.1.p1.60.2.m2.1.2.2.2.2" mathcolor="#000099" xref="A3.1.p1.60.2.m2.1.2.2.2.2.cmml">𝜽</mi><mo id="A3.1.p1.60.2.m2.1.2.2.2.1" mathcolor="#000099" xref="A3.1.p1.60.2.m2.1.2.2.2.1.cmml">~</mo></mover><mo id="A3.1.p1.60.2.m2.1.2.2.1" xref="A3.1.p1.60.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="A3.1.p1.60.2.m2.1.2.2.3.2" xref="A3.1.p1.60.2.m2.1.2.2.cmml"><mo id="A3.1.p1.60.2.m2.1.2.2.3.2.1" mathcolor="#000099" stretchy="false" xref="A3.1.p1.60.2.m2.1.2.2.cmml">(</mo><mi id="A3.1.p1.60.2.m2.1.1" mathcolor="#000099" xref="A3.1.p1.60.2.m2.1.1.cmml">t</mi><mo id="A3.1.p1.60.2.m2.1.2.2.3.2.2" mathcolor="#000099" stretchy="false" xref="A3.1.p1.60.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="A3.1.p1.60.2.m2.1.2.1" mathcolor="#000099" xref="A3.1.p1.60.2.m2.1.2.1.cmml">∈</mo><mi id="A3.1.p1.60.2.m2.1.2.3" xref="A3.1.p1.60.2.m2.1.2.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.60.2.m2.1b"><apply id="A3.1.p1.60.2.m2.1.2.cmml" xref="A3.1.p1.60.2.m2.1.2"><in id="A3.1.p1.60.2.m2.1.2.1.cmml" xref="A3.1.p1.60.2.m2.1.2.1"></in><apply id="A3.1.p1.60.2.m2.1.2.2.cmml" xref="A3.1.p1.60.2.m2.1.2.2"><times id="A3.1.p1.60.2.m2.1.2.2.1.cmml" xref="A3.1.p1.60.2.m2.1.2.2.1"></times><apply id="A3.1.p1.60.2.m2.1.2.2.2.cmml" xref="A3.1.p1.60.2.m2.1.2.2.2"><ci id="A3.1.p1.60.2.m2.1.2.2.2.1.cmml" xref="A3.1.p1.60.2.m2.1.2.2.2.1">~</ci><ci id="A3.1.p1.60.2.m2.1.2.2.2.2.cmml" xref="A3.1.p1.60.2.m2.1.2.2.2.2">𝜽</ci></apply><ci id="A3.1.p1.60.2.m2.1.1.cmml" xref="A3.1.p1.60.2.m2.1.1">𝑡</ci></apply><csymbol cd="latexml" id="A3.1.p1.60.2.m2.1.2.3.cmml" xref="A3.1.p1.60.2.m2.1.2.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.60.2.m2.1c">\tilde{\bm{\theta}}(t)\in</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.60.2.m2.1d">over~ start_ARG bold_italic_θ end_ARG ( italic_t ) ∈</annotation></semantics></math> <math alttext="L_{\infty}" class="ltx_Math" display="inline" id="A3.1.p1.61.3.m3.1"><semantics id="A3.1.p1.61.3.m3.1a"><msub id="A3.1.p1.61.3.m3.1.1" xref="A3.1.p1.61.3.m3.1.1.cmml"><mi id="A3.1.p1.61.3.m3.1.1.2" mathcolor="#000099" xref="A3.1.p1.61.3.m3.1.1.2.cmml">L</mi><mi id="A3.1.p1.61.3.m3.1.1.3" mathcolor="#000099" mathvariant="normal" xref="A3.1.p1.61.3.m3.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="A3.1.p1.61.3.m3.1b"><apply id="A3.1.p1.61.3.m3.1.1.cmml" xref="A3.1.p1.61.3.m3.1.1"><csymbol cd="ambiguous" id="A3.1.p1.61.3.m3.1.1.1.cmml" xref="A3.1.p1.61.3.m3.1.1">subscript</csymbol><ci id="A3.1.p1.61.3.m3.1.1.2.cmml" xref="A3.1.p1.61.3.m3.1.1.2">𝐿</ci><infinity id="A3.1.p1.61.3.m3.1.1.3.cmml" xref="A3.1.p1.61.3.m3.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.61.3.m3.1c">L_{\infty}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.61.3.m3.1d">italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\forall t\geq 0" class="ltx_Math" display="inline" id="A3.1.p1.62.4.m4.1"><semantics id="A3.1.p1.62.4.m4.1a"><mrow id="A3.1.p1.62.4.m4.1.1" xref="A3.1.p1.62.4.m4.1.1.cmml"><mrow id="A3.1.p1.62.4.m4.1.1.2" xref="A3.1.p1.62.4.m4.1.1.2.cmml"><mo id="A3.1.p1.62.4.m4.1.1.2.1" mathcolor="#000099" rspace="0.167em" xref="A3.1.p1.62.4.m4.1.1.2.1.cmml">∀</mo><mi id="A3.1.p1.62.4.m4.1.1.2.2" mathcolor="#000099" xref="A3.1.p1.62.4.m4.1.1.2.2.cmml">t</mi></mrow><mo id="A3.1.p1.62.4.m4.1.1.1" mathcolor="#000099" xref="A3.1.p1.62.4.m4.1.1.1.cmml">≥</mo><mn id="A3.1.p1.62.4.m4.1.1.3" mathcolor="#000099" xref="A3.1.p1.62.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.62.4.m4.1b"><apply id="A3.1.p1.62.4.m4.1.1.cmml" xref="A3.1.p1.62.4.m4.1.1"><geq id="A3.1.p1.62.4.m4.1.1.1.cmml" xref="A3.1.p1.62.4.m4.1.1.1"></geq><apply id="A3.1.p1.62.4.m4.1.1.2.cmml" xref="A3.1.p1.62.4.m4.1.1.2"><csymbol cd="latexml" id="A3.1.p1.62.4.m4.1.1.2.1.cmml" xref="A3.1.p1.62.4.m4.1.1.2.1">for-all</csymbol><ci id="A3.1.p1.62.4.m4.1.1.2.2.cmml" xref="A3.1.p1.62.4.m4.1.1.2.2">𝑡</ci></apply><cn id="A3.1.p1.62.4.m4.1.1.3.cmml" type="integer" xref="A3.1.p1.62.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.62.4.m4.1c">\forall t\geq 0</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.62.4.m4.1d">∀ italic_t ≥ 0</annotation></semantics></math> implying <math alttext="t_{\rm f}=\infty" class="ltx_Math" display="inline" id="A3.1.p1.63.5.m5.1"><semantics id="A3.1.p1.63.5.m5.1a"><mrow id="A3.1.p1.63.5.m5.1.1" xref="A3.1.p1.63.5.m5.1.1.cmml"><msub id="A3.1.p1.63.5.m5.1.1.2" xref="A3.1.p1.63.5.m5.1.1.2.cmml"><mi id="A3.1.p1.63.5.m5.1.1.2.2" mathcolor="#000099" xref="A3.1.p1.63.5.m5.1.1.2.2.cmml">t</mi><mi id="A3.1.p1.63.5.m5.1.1.2.3" mathcolor="#000099" mathvariant="normal" xref="A3.1.p1.63.5.m5.1.1.2.3.cmml">f</mi></msub><mo id="A3.1.p1.63.5.m5.1.1.1" mathcolor="#000099" xref="A3.1.p1.63.5.m5.1.1.1.cmml">=</mo><mi id="A3.1.p1.63.5.m5.1.1.3" mathcolor="#000099" mathvariant="normal" xref="A3.1.p1.63.5.m5.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.63.5.m5.1b"><apply id="A3.1.p1.63.5.m5.1.1.cmml" xref="A3.1.p1.63.5.m5.1.1"><eq id="A3.1.p1.63.5.m5.1.1.1.cmml" xref="A3.1.p1.63.5.m5.1.1.1"></eq><apply id="A3.1.p1.63.5.m5.1.1.2.cmml" xref="A3.1.p1.63.5.m5.1.1.2"><csymbol cd="ambiguous" id="A3.1.p1.63.5.m5.1.1.2.1.cmml" xref="A3.1.p1.63.5.m5.1.1.2">subscript</csymbol><ci id="A3.1.p1.63.5.m5.1.1.2.2.cmml" xref="A3.1.p1.63.5.m5.1.1.2.2">𝑡</ci><ci id="A3.1.p1.63.5.m5.1.1.2.3.cmml" xref="A3.1.p1.63.5.m5.1.1.2.3">f</ci></apply><infinity id="A3.1.p1.63.5.m5.1.1.3.cmml" xref="A3.1.p1.63.5.m5.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.63.5.m5.1c">t_{\rm f}=\infty</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.63.5.m5.1d">italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT = ∞</annotation></semantics></math> so that the solution <math alttext="\bm{w}(t)" class="ltx_Math" display="inline" id="A3.1.p1.64.6.m6.1"><semantics id="A3.1.p1.64.6.m6.1a"><mrow id="A3.1.p1.64.6.m6.1.2" xref="A3.1.p1.64.6.m6.1.2.cmml"><mi id="A3.1.p1.64.6.m6.1.2.2" mathcolor="#000099" xref="A3.1.p1.64.6.m6.1.2.2.cmml">𝒘</mi><mo id="A3.1.p1.64.6.m6.1.2.1" xref="A3.1.p1.64.6.m6.1.2.1.cmml">⁢</mo><mrow id="A3.1.p1.64.6.m6.1.2.3.2" xref="A3.1.p1.64.6.m6.1.2.cmml"><mo id="A3.1.p1.64.6.m6.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="A3.1.p1.64.6.m6.1.2.cmml">(</mo><mi id="A3.1.p1.64.6.m6.1.1" mathcolor="#000099" xref="A3.1.p1.64.6.m6.1.1.cmml">t</mi><mo id="A3.1.p1.64.6.m6.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="A3.1.p1.64.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.64.6.m6.1b"><apply id="A3.1.p1.64.6.m6.1.2.cmml" xref="A3.1.p1.64.6.m6.1.2"><times id="A3.1.p1.64.6.m6.1.2.1.cmml" xref="A3.1.p1.64.6.m6.1.2.1"></times><ci id="A3.1.p1.64.6.m6.1.2.2.cmml" xref="A3.1.p1.64.6.m6.1.2.2">𝒘</ci><ci id="A3.1.p1.64.6.m6.1.1.cmml" xref="A3.1.p1.64.6.m6.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.64.6.m6.1c">\bm{w}(t)</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.64.6.m6.1d">bold_italic_w ( italic_t )</annotation></semantics></math> is unique, <math alttext="\forall t\geq 0" class="ltx_Math" display="inline" id="A3.1.p1.65.7.m7.1"><semantics id="A3.1.p1.65.7.m7.1a"><mrow id="A3.1.p1.65.7.m7.1.1" xref="A3.1.p1.65.7.m7.1.1.cmml"><mrow id="A3.1.p1.65.7.m7.1.1.2" xref="A3.1.p1.65.7.m7.1.1.2.cmml"><mo id="A3.1.p1.65.7.m7.1.1.2.1" mathcolor="#000099" rspace="0.167em" xref="A3.1.p1.65.7.m7.1.1.2.1.cmml">∀</mo><mi id="A3.1.p1.65.7.m7.1.1.2.2" mathcolor="#000099" xref="A3.1.p1.65.7.m7.1.1.2.2.cmml">t</mi></mrow><mo id="A3.1.p1.65.7.m7.1.1.1" mathcolor="#000099" xref="A3.1.p1.65.7.m7.1.1.1.cmml">≥</mo><mn id="A3.1.p1.65.7.m7.1.1.3" mathcolor="#000099" xref="A3.1.p1.65.7.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.65.7.m7.1b"><apply id="A3.1.p1.65.7.m7.1.1.cmml" xref="A3.1.p1.65.7.m7.1.1"><geq id="A3.1.p1.65.7.m7.1.1.1.cmml" xref="A3.1.p1.65.7.m7.1.1.1"></geq><apply id="A3.1.p1.65.7.m7.1.1.2.cmml" xref="A3.1.p1.65.7.m7.1.1.2"><csymbol cd="latexml" id="A3.1.p1.65.7.m7.1.1.2.1.cmml" xref="A3.1.p1.65.7.m7.1.1.2.1">for-all</csymbol><ci id="A3.1.p1.65.7.m7.1.1.2.2.cmml" xref="A3.1.p1.65.7.m7.1.1.2.2">𝑡</ci></apply><cn id="A3.1.p1.65.7.m7.1.1.3.cmml" type="integer" xref="A3.1.p1.65.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.65.7.m7.1c">\forall t\geq 0</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.65.7.m7.1d">∀ italic_t ≥ 0</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib41" title="">41</a>, Lemma 3.1]</cite>.</span> It is clear from the definition of <math alttext="\eta" class="ltx_Math" display="inline" id="A3.1.p1.66.m4.1"><semantics id="A3.1.p1.66.m4.1a"><mi id="A3.1.p1.66.m4.1.1" xref="A3.1.p1.66.m4.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="A3.1.p1.66.m4.1b"><ci id="A3.1.p1.66.m4.1.1.cmml" xref="A3.1.p1.66.m4.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.66.m4.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.66.m4.1d">italic_η</annotation></semantics></math> that given any <math alttext="c_{0}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A3.1.p1.67.m5.1"><semantics id="A3.1.p1.67.m5.1a"><mrow id="A3.1.p1.67.m5.1.1" xref="A3.1.p1.67.m5.1.1.cmml"><msub id="A3.1.p1.67.m5.1.1.2" xref="A3.1.p1.67.m5.1.1.2.cmml"><mi id="A3.1.p1.67.m5.1.1.2.2" xref="A3.1.p1.67.m5.1.1.2.2.cmml">c</mi><mn id="A3.1.p1.67.m5.1.1.2.3" xref="A3.1.p1.67.m5.1.1.2.3.cmml">0</mn></msub><mo id="A3.1.p1.67.m5.1.1.1" xref="A3.1.p1.67.m5.1.1.1.cmml">∈</mo><msup id="A3.1.p1.67.m5.1.1.3" xref="A3.1.p1.67.m5.1.1.3.cmml"><mi id="A3.1.p1.67.m5.1.1.3.2" xref="A3.1.p1.67.m5.1.1.3.2.cmml">ℝ</mi><mo id="A3.1.p1.67.m5.1.1.3.3" xref="A3.1.p1.67.m5.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.67.m5.1b"><apply id="A3.1.p1.67.m5.1.1.cmml" xref="A3.1.p1.67.m5.1.1"><in id="A3.1.p1.67.m5.1.1.1.cmml" xref="A3.1.p1.67.m5.1.1.1"></in><apply id="A3.1.p1.67.m5.1.1.2.cmml" xref="A3.1.p1.67.m5.1.1.2"><csymbol cd="ambiguous" id="A3.1.p1.67.m5.1.1.2.1.cmml" xref="A3.1.p1.67.m5.1.1.2">subscript</csymbol><ci id="A3.1.p1.67.m5.1.1.2.2.cmml" xref="A3.1.p1.67.m5.1.1.2.2">𝑐</ci><cn id="A3.1.p1.67.m5.1.1.2.3.cmml" type="integer" xref="A3.1.p1.67.m5.1.1.2.3">0</cn></apply><apply id="A3.1.p1.67.m5.1.1.3.cmml" xref="A3.1.p1.67.m5.1.1.3"><csymbol cd="ambiguous" id="A3.1.p1.67.m5.1.1.3.1.cmml" xref="A3.1.p1.67.m5.1.1.3">superscript</csymbol><ci id="A3.1.p1.67.m5.1.1.3.2.cmml" xref="A3.1.p1.67.m5.1.1.3.2">ℝ</ci><plus id="A3.1.p1.67.m5.1.1.3.3.cmml" xref="A3.1.p1.67.m5.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.67.m5.1c">c_{0}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.67.m5.1d">italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, there certainly exist suitable <math alttext="k_{\rm c}" class="ltx_Math" display="inline" id="A3.1.p1.68.m6.1"><semantics id="A3.1.p1.68.m6.1a"><msub id="A3.1.p1.68.m6.1.1" xref="A3.1.p1.68.m6.1.1.cmml"><mi id="A3.1.p1.68.m6.1.1.2" xref="A3.1.p1.68.m6.1.1.2.cmml">k</mi><mi id="A3.1.p1.68.m6.1.1.3" mathvariant="normal" xref="A3.1.p1.68.m6.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="A3.1.p1.68.m6.1b"><apply id="A3.1.p1.68.m6.1.1.cmml" xref="A3.1.p1.68.m6.1.1"><csymbol cd="ambiguous" id="A3.1.p1.68.m6.1.1.1.cmml" xref="A3.1.p1.68.m6.1.1">subscript</csymbol><ci id="A3.1.p1.68.m6.1.1.2.cmml" xref="A3.1.p1.68.m6.1.1.2">𝑘</ci><ci id="A3.1.p1.68.m6.1.1.3.cmml" xref="A3.1.p1.68.m6.1.1.3">c</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.68.m6.1c">k_{\rm c}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.68.m6.1d">italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\alpha" class="ltx_Math" display="inline" id="A3.1.p1.69.m7.1"><semantics id="A3.1.p1.69.m7.1a"><mi id="A3.1.p1.69.m7.1.1" xref="A3.1.p1.69.m7.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A3.1.p1.69.m7.1b"><ci id="A3.1.p1.69.m7.1.1.cmml" xref="A3.1.p1.69.m7.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.69.m7.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.69.m7.1d">italic_α</annotation></semantics></math>, and <math alttext="\Gamma" class="ltx_Math" display="inline" id="A3.1.p1.70.m8.1"><semantics id="A3.1.p1.70.m8.1a"><mi id="A3.1.p1.70.m8.1.1" mathvariant="normal" xref="A3.1.p1.70.m8.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="A3.1.p1.70.m8.1b"><ci id="A3.1.p1.70.m8.1.1.cmml" xref="A3.1.p1.70.m8.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.70.m8.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.70.m8.1d">roman_Γ</annotation></semantics></math> such that <math alttext="\eta<\sqrt{c_{w0}/\lambda_{\rm b}}" class="ltx_Math" display="inline" id="A3.1.p1.71.m9.1"><semantics id="A3.1.p1.71.m9.1a"><mrow id="A3.1.p1.71.m9.1.1" xref="A3.1.p1.71.m9.1.1.cmml"><mi id="A3.1.p1.71.m9.1.1.2" xref="A3.1.p1.71.m9.1.1.2.cmml">η</mi><mo id="A3.1.p1.71.m9.1.1.1" xref="A3.1.p1.71.m9.1.1.1.cmml"><</mo><msqrt id="A3.1.p1.71.m9.1.1.3" xref="A3.1.p1.71.m9.1.1.3.cmml"><mrow id="A3.1.p1.71.m9.1.1.3.2" xref="A3.1.p1.71.m9.1.1.3.2.cmml"><msub id="A3.1.p1.71.m9.1.1.3.2.2" xref="A3.1.p1.71.m9.1.1.3.2.2.cmml"><mi id="A3.1.p1.71.m9.1.1.3.2.2.2" xref="A3.1.p1.71.m9.1.1.3.2.2.2.cmml">c</mi><mrow id="A3.1.p1.71.m9.1.1.3.2.2.3" xref="A3.1.p1.71.m9.1.1.3.2.2.3.cmml"><mi id="A3.1.p1.71.m9.1.1.3.2.2.3.2" xref="A3.1.p1.71.m9.1.1.3.2.2.3.2.cmml">w</mi><mo id="A3.1.p1.71.m9.1.1.3.2.2.3.1" xref="A3.1.p1.71.m9.1.1.3.2.2.3.1.cmml">⁢</mo><mn id="A3.1.p1.71.m9.1.1.3.2.2.3.3" xref="A3.1.p1.71.m9.1.1.3.2.2.3.3.cmml">0</mn></mrow></msub><mo id="A3.1.p1.71.m9.1.1.3.2.1" xref="A3.1.p1.71.m9.1.1.3.2.1.cmml">/</mo><msub id="A3.1.p1.71.m9.1.1.3.2.3" xref="A3.1.p1.71.m9.1.1.3.2.3.cmml"><mi id="A3.1.p1.71.m9.1.1.3.2.3.2" xref="A3.1.p1.71.m9.1.1.3.2.3.2.cmml">λ</mi><mi id="A3.1.p1.71.m9.1.1.3.2.3.3" mathvariant="normal" xref="A3.1.p1.71.m9.1.1.3.2.3.3.cmml">b</mi></msub></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.71.m9.1b"><apply id="A3.1.p1.71.m9.1.1.cmml" xref="A3.1.p1.71.m9.1.1"><lt id="A3.1.p1.71.m9.1.1.1.cmml" xref="A3.1.p1.71.m9.1.1.1"></lt><ci id="A3.1.p1.71.m9.1.1.2.cmml" xref="A3.1.p1.71.m9.1.1.2">𝜂</ci><apply id="A3.1.p1.71.m9.1.1.3.cmml" xref="A3.1.p1.71.m9.1.1.3"><root id="A3.1.p1.71.m9.1.1.3a.cmml" xref="A3.1.p1.71.m9.1.1.3"></root><apply id="A3.1.p1.71.m9.1.1.3.2.cmml" xref="A3.1.p1.71.m9.1.1.3.2"><divide id="A3.1.p1.71.m9.1.1.3.2.1.cmml" xref="A3.1.p1.71.m9.1.1.3.2.1"></divide><apply id="A3.1.p1.71.m9.1.1.3.2.2.cmml" xref="A3.1.p1.71.m9.1.1.3.2.2"><csymbol cd="ambiguous" id="A3.1.p1.71.m9.1.1.3.2.2.1.cmml" xref="A3.1.p1.71.m9.1.1.3.2.2">subscript</csymbol><ci id="A3.1.p1.71.m9.1.1.3.2.2.2.cmml" xref="A3.1.p1.71.m9.1.1.3.2.2.2">𝑐</ci><apply id="A3.1.p1.71.m9.1.1.3.2.2.3.cmml" xref="A3.1.p1.71.m9.1.1.3.2.2.3"><times id="A3.1.p1.71.m9.1.1.3.2.2.3.1.cmml" xref="A3.1.p1.71.m9.1.1.3.2.2.3.1"></times><ci id="A3.1.p1.71.m9.1.1.3.2.2.3.2.cmml" xref="A3.1.p1.71.m9.1.1.3.2.2.3.2">𝑤</ci><cn id="A3.1.p1.71.m9.1.1.3.2.2.3.3.cmml" type="integer" xref="A3.1.p1.71.m9.1.1.3.2.2.3.3">0</cn></apply></apply><apply id="A3.1.p1.71.m9.1.1.3.2.3.cmml" xref="A3.1.p1.71.m9.1.1.3.2.3"><csymbol cd="ambiguous" id="A3.1.p1.71.m9.1.1.3.2.3.1.cmml" xref="A3.1.p1.71.m9.1.1.3.2.3">subscript</csymbol><ci id="A3.1.p1.71.m9.1.1.3.2.3.2.cmml" xref="A3.1.p1.71.m9.1.1.3.2.3.2">𝜆</ci><ci id="A3.1.p1.71.m9.1.1.3.2.3.3.cmml" xref="A3.1.p1.71.m9.1.1.3.2.3.3">b</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.71.m9.1c">\eta<\sqrt{c_{w0}/\lambda_{\rm b}}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.71.m9.1d">italic_η < square-root start_ARG italic_c start_POSTSUBSCRIPT italic_w 0 end_POSTSUBSCRIPT / italic_λ start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT end_ARG</annotation></semantics></math> holds. Besides, the transient bound for <math alttext="\bm{e}" class="ltx_Math" display="inline" id="A3.1.p1.72.m10.1"><semantics id="A3.1.p1.72.m10.1a"><mi id="A3.1.p1.72.m10.1.1" xref="A3.1.p1.72.m10.1.1.cmml">𝒆</mi><annotation-xml encoding="MathML-Content" id="A3.1.p1.72.m10.1b"><ci id="A3.1.p1.72.m10.1.1.cmml" xref="A3.1.p1.72.m10.1.1">𝒆</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.72.m10.1c">\bm{e}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.72.m10.1d">bold_italic_e</annotation></semantics></math> derived by <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib41" title="">41</a>, Th. 4.5]</cite> is given as follows:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx63"> <tbody id="A3.Ex49"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\bm{e}(t)\|\leq" class="ltx_Math" display="inline" id="A3.Ex49.m1.2"><semantics id="A3.Ex49.m1.2a"><mrow id="A3.Ex49.m1.2.2" xref="A3.Ex49.m1.2.2.cmml"><mrow id="A3.Ex49.m1.2.2.1.1" xref="A3.Ex49.m1.2.2.1.2.cmml"><mo id="A3.Ex49.m1.2.2.1.1.2" stretchy="false" xref="A3.Ex49.m1.2.2.1.2.1.cmml">‖</mo><mrow id="A3.Ex49.m1.2.2.1.1.1" xref="A3.Ex49.m1.2.2.1.1.1.cmml"><mi id="A3.Ex49.m1.2.2.1.1.1.2" xref="A3.Ex49.m1.2.2.1.1.1.2.cmml">𝒆</mi><mo id="A3.Ex49.m1.2.2.1.1.1.1" xref="A3.Ex49.m1.2.2.1.1.1.1.cmml">⁢</mo><mrow id="A3.Ex49.m1.2.2.1.1.1.3.2" xref="A3.Ex49.m1.2.2.1.1.1.cmml"><mo id="A3.Ex49.m1.2.2.1.1.1.3.2.1" stretchy="false" xref="A3.Ex49.m1.2.2.1.1.1.cmml">(</mo><mi id="A3.Ex49.m1.1.1" xref="A3.Ex49.m1.1.1.cmml">t</mi><mo id="A3.Ex49.m1.2.2.1.1.1.3.2.2" stretchy="false" xref="A3.Ex49.m1.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="A3.Ex49.m1.2.2.1.1.3" stretchy="false" xref="A3.Ex49.m1.2.2.1.2.1.cmml">‖</mo></mrow><mo id="A3.Ex49.m1.2.2.2" xref="A3.Ex49.m1.2.2.2.cmml">≤</mo><mi id="A3.Ex49.m1.2.2.3" xref="A3.Ex49.m1.2.2.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex49.m1.2b"><apply id="A3.Ex49.m1.2.2.cmml" xref="A3.Ex49.m1.2.2"><leq id="A3.Ex49.m1.2.2.2.cmml" xref="A3.Ex49.m1.2.2.2"></leq><apply id="A3.Ex49.m1.2.2.1.2.cmml" xref="A3.Ex49.m1.2.2.1.1"><csymbol cd="latexml" id="A3.Ex49.m1.2.2.1.2.1.cmml" xref="A3.Ex49.m1.2.2.1.1.2">norm</csymbol><apply id="A3.Ex49.m1.2.2.1.1.1.cmml" xref="A3.Ex49.m1.2.2.1.1.1"><times id="A3.Ex49.m1.2.2.1.1.1.1.cmml" xref="A3.Ex49.m1.2.2.1.1.1.1"></times><ci id="A3.Ex49.m1.2.2.1.1.1.2.cmml" xref="A3.Ex49.m1.2.2.1.1.1.2">𝒆</ci><ci id="A3.Ex49.m1.1.1.cmml" xref="A3.Ex49.m1.1.1">𝑡</ci></apply></apply><csymbol cd="latexml" id="A3.Ex49.m1.2.2.3.cmml" xref="A3.Ex49.m1.2.2.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex49.m1.2c">\displaystyle\|\bm{e}(t)\|\leq</annotation><annotation encoding="application/x-llamapun" id="A3.Ex49.m1.2d">∥ bold_italic_e ( italic_t ) ∥ ≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\sqrt{\lambda_{\rm b}/\lambda_{\rm a}}\max\{(\|\bm{e}(0)\|+c_{% \theta})e^{-(k_{\rm c}\lambda_{\rm a})/(4\lambda_{\rm b})t}," class="ltx_math_unparsed" display="inline" id="A3.Ex49.m2.3"><semantics id="A3.Ex49.m2.3a"><mrow id="A3.Ex49.m2.3b"><msqrt id="A3.Ex49.m2.3.4"><mrow id="A3.Ex49.m2.3.4.2"><msub id="A3.Ex49.m2.3.4.2.2"><mi id="A3.Ex49.m2.3.4.2.2.2">λ</mi><mi id="A3.Ex49.m2.3.4.2.2.3" mathvariant="normal">b</mi></msub><mo id="A3.Ex49.m2.3.4.2.1">/</mo><msub id="A3.Ex49.m2.3.4.2.3"><mi id="A3.Ex49.m2.3.4.2.3.2">λ</mi><mi id="A3.Ex49.m2.3.4.2.3.3" mathvariant="normal">a</mi></msub></mrow></msqrt><mi id="A3.Ex49.m2.3.5">max</mi><mrow id="A3.Ex49.m2.3.6"><mo id="A3.Ex49.m2.3.6.1" stretchy="false">{</mo><mrow id="A3.Ex49.m2.3.6.2"><mo id="A3.Ex49.m2.3.6.2.1" stretchy="false">(</mo><mo id="A3.Ex49.m2.3.6.2.2" lspace="0em" rspace="0.167em">∥</mo><mi id="A3.Ex49.m2.3.6.2.3">𝒆</mi><mrow id="A3.Ex49.m2.3.6.2.4"><mo id="A3.Ex49.m2.3.6.2.4.1" stretchy="false">(</mo><mn id="A3.Ex49.m2.3.3">0</mn><mo id="A3.Ex49.m2.3.6.2.4.2" stretchy="false">)</mo></mrow><mo id="A3.Ex49.m2.3.6.2.5" lspace="0em" rspace="0em">∥</mo><mo id="A3.Ex49.m2.3.6.2.6" lspace="0em">+</mo><msub id="A3.Ex49.m2.3.6.2.7"><mi id="A3.Ex49.m2.3.6.2.7.2">c</mi><mi id="A3.Ex49.m2.3.6.2.7.3">θ</mi></msub><mo id="A3.Ex49.m2.3.6.2.8" stretchy="false">)</mo></mrow><msup id="A3.Ex49.m2.3.6.3"><mi id="A3.Ex49.m2.3.6.3.2">e</mi><mrow id="A3.Ex49.m2.2.2.2"><mo id="A3.Ex49.m2.2.2.2a">−</mo><mrow id="A3.Ex49.m2.2.2.2.2"><mrow id="A3.Ex49.m2.2.2.2.2.2"><mrow id="A3.Ex49.m2.1.1.1.1.1.1.1"><mo id="A3.Ex49.m2.1.1.1.1.1.1.1.2" stretchy="false">(</mo><mrow id="A3.Ex49.m2.1.1.1.1.1.1.1.1"><msub id="A3.Ex49.m2.1.1.1.1.1.1.1.1.2"><mi id="A3.Ex49.m2.1.1.1.1.1.1.1.1.2.2">k</mi><mi id="A3.Ex49.m2.1.1.1.1.1.1.1.1.2.3" mathvariant="normal">c</mi></msub><mo id="A3.Ex49.m2.1.1.1.1.1.1.1.1.1">⁢</mo><msub id="A3.Ex49.m2.1.1.1.1.1.1.1.1.3"><mi id="A3.Ex49.m2.1.1.1.1.1.1.1.1.3.2">λ</mi><mi id="A3.Ex49.m2.1.1.1.1.1.1.1.1.3.3" mathvariant="normal">a</mi></msub></mrow><mo id="A3.Ex49.m2.1.1.1.1.1.1.1.3" stretchy="false">)</mo></mrow><mo id="A3.Ex49.m2.2.2.2.2.2.3">/</mo><mrow id="A3.Ex49.m2.2.2.2.2.2.2.1"><mo id="A3.Ex49.m2.2.2.2.2.2.2.1.2" stretchy="false">(</mo><mrow id="A3.Ex49.m2.2.2.2.2.2.2.1.1"><mn id="A3.Ex49.m2.2.2.2.2.2.2.1.1.2">4</mn><mo id="A3.Ex49.m2.2.2.2.2.2.2.1.1.1">⁢</mo><msub id="A3.Ex49.m2.2.2.2.2.2.2.1.1.3"><mi id="A3.Ex49.m2.2.2.2.2.2.2.1.1.3.2">λ</mi><mi id="A3.Ex49.m2.2.2.2.2.2.2.1.1.3.3" mathvariant="normal">b</mi></msub></mrow><mo id="A3.Ex49.m2.2.2.2.2.2.2.1.3" stretchy="false">)</mo></mrow></mrow><mo id="A3.Ex49.m2.2.2.2.2.3">⁢</mo><mi id="A3.Ex49.m2.2.2.2.2.4">t</mi></mrow></mrow></msup><mo id="A3.Ex49.m2.3.6.4">,</mo></mrow></mrow><annotation encoding="application/x-tex" id="A3.Ex49.m2.3c">\displaystyle\sqrt{\lambda_{\rm b}/\lambda_{\rm a}}\max\{(\|\bm{e}(0)\|+c_{% \theta})e^{-(k_{\rm c}\lambda_{\rm a})/(4\lambda_{\rm b})t},</annotation><annotation encoding="application/x-llamapun" id="A3.Ex49.m2.3d">square-root start_ARG italic_λ start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT / italic_λ start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT end_ARG roman_max { ( ∥ bold_italic_e ( 0 ) ∥ + italic_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ) italic_e start_POSTSUPERSCRIPT - ( italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT ) / ( 4 italic_λ start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT ) italic_t end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.E48"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\sqrt{\eta/\lambda_{\rm b}}\},\forall t\geq 0." class="ltx_math_unparsed" display="inline" id="A3.E48.m1.1"><semantics id="A3.E48.m1.1a"><mrow id="A3.E48.m1.1b"><msqrt id="A3.E48.m1.1.1"><mrow id="A3.E48.m1.1.1.2"><mi id="A3.E48.m1.1.1.2.2">η</mi><mo id="A3.E48.m1.1.1.2.1">/</mo><msub id="A3.E48.m1.1.1.2.3"><mi id="A3.E48.m1.1.1.2.3.2">λ</mi><mi id="A3.E48.m1.1.1.2.3.3" mathvariant="normal">b</mi></msub></mrow></msqrt><mo id="A3.E48.m1.1.2" stretchy="false">}</mo><mo id="A3.E48.m1.1.3">,</mo><mo id="A3.E48.m1.1.4" rspace="0.167em">∀</mo><mi id="A3.E48.m1.1.5">t</mi><mo id="A3.E48.m1.1.6">≥</mo><mn id="A3.E48.m1.1.7">0</mn><mo id="A3.E48.m1.1.8" lspace="0em">.</mo></mrow><annotation encoding="application/x-tex" id="A3.E48.m1.1c">\displaystyle\sqrt{\eta/\lambda_{\rm b}}\},\forall t\geq 0.</annotation><annotation encoding="application/x-llamapun" id="A3.E48.m1.1d">square-root start_ARG italic_η / italic_λ start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT end_ARG } , ∀ italic_t ≥ 0 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(48)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.1.p1.93">Using Assumptions 1–2 and <math alttext="\bm{e}(t)" class="ltx_Math" display="inline" id="A3.1.p1.73.m1.1"><semantics id="A3.1.p1.73.m1.1a"><mrow id="A3.1.p1.73.m1.1.2" xref="A3.1.p1.73.m1.1.2.cmml"><mi id="A3.1.p1.73.m1.1.2.2" xref="A3.1.p1.73.m1.1.2.2.cmml">𝒆</mi><mo id="A3.1.p1.73.m1.1.2.1" xref="A3.1.p1.73.m1.1.2.1.cmml">⁢</mo><mrow id="A3.1.p1.73.m1.1.2.3.2" xref="A3.1.p1.73.m1.1.2.cmml"><mo id="A3.1.p1.73.m1.1.2.3.2.1" stretchy="false" xref="A3.1.p1.73.m1.1.2.cmml">(</mo><mi id="A3.1.p1.73.m1.1.1" xref="A3.1.p1.73.m1.1.1.cmml">t</mi><mo id="A3.1.p1.73.m1.1.2.3.2.2" stretchy="false" xref="A3.1.p1.73.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.73.m1.1b"><apply id="A3.1.p1.73.m1.1.2.cmml" xref="A3.1.p1.73.m1.1.2"><times id="A3.1.p1.73.m1.1.2.1.cmml" xref="A3.1.p1.73.m1.1.2.1"></times><ci id="A3.1.p1.73.m1.1.2.2.cmml" xref="A3.1.p1.73.m1.1.2.2">𝒆</ci><ci id="A3.1.p1.73.m1.1.1.cmml" xref="A3.1.p1.73.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.73.m1.1c">\bm{e}(t)</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.73.m1.1d">bold_italic_e ( italic_t )</annotation></semantics></math>, <math alttext="\tilde{\bm{\theta}}(t)\in" class="ltx_Math" display="inline" id="A3.1.p1.74.m2.1"><semantics id="A3.1.p1.74.m2.1a"><mrow id="A3.1.p1.74.m2.1.2" xref="A3.1.p1.74.m2.1.2.cmml"><mrow id="A3.1.p1.74.m2.1.2.2" xref="A3.1.p1.74.m2.1.2.2.cmml"><mover accent="true" id="A3.1.p1.74.m2.1.2.2.2" xref="A3.1.p1.74.m2.1.2.2.2.cmml"><mi id="A3.1.p1.74.m2.1.2.2.2.2" xref="A3.1.p1.74.m2.1.2.2.2.2.cmml">𝜽</mi><mo id="A3.1.p1.74.m2.1.2.2.2.1" xref="A3.1.p1.74.m2.1.2.2.2.1.cmml">~</mo></mover><mo id="A3.1.p1.74.m2.1.2.2.1" xref="A3.1.p1.74.m2.1.2.2.1.cmml">⁢</mo><mrow id="A3.1.p1.74.m2.1.2.2.3.2" xref="A3.1.p1.74.m2.1.2.2.cmml"><mo id="A3.1.p1.74.m2.1.2.2.3.2.1" stretchy="false" xref="A3.1.p1.74.m2.1.2.2.cmml">(</mo><mi id="A3.1.p1.74.m2.1.1" xref="A3.1.p1.74.m2.1.1.cmml">t</mi><mo id="A3.1.p1.74.m2.1.2.2.3.2.2" stretchy="false" xref="A3.1.p1.74.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="A3.1.p1.74.m2.1.2.1" xref="A3.1.p1.74.m2.1.2.1.cmml">∈</mo><mi id="A3.1.p1.74.m2.1.2.3" xref="A3.1.p1.74.m2.1.2.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.74.m2.1b"><apply id="A3.1.p1.74.m2.1.2.cmml" xref="A3.1.p1.74.m2.1.2"><in id="A3.1.p1.74.m2.1.2.1.cmml" xref="A3.1.p1.74.m2.1.2.1"></in><apply id="A3.1.p1.74.m2.1.2.2.cmml" xref="A3.1.p1.74.m2.1.2.2"><times id="A3.1.p1.74.m2.1.2.2.1.cmml" xref="A3.1.p1.74.m2.1.2.2.1"></times><apply id="A3.1.p1.74.m2.1.2.2.2.cmml" xref="A3.1.p1.74.m2.1.2.2.2"><ci id="A3.1.p1.74.m2.1.2.2.2.1.cmml" xref="A3.1.p1.74.m2.1.2.2.2.1">~</ci><ci id="A3.1.p1.74.m2.1.2.2.2.2.cmml" xref="A3.1.p1.74.m2.1.2.2.2.2">𝜽</ci></apply><ci id="A3.1.p1.74.m2.1.1.cmml" xref="A3.1.p1.74.m2.1.1">𝑡</ci></apply><csymbol cd="latexml" id="A3.1.p1.74.m2.1.2.3.cmml" xref="A3.1.p1.74.m2.1.2.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.74.m2.1c">\tilde{\bm{\theta}}(t)\in</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.74.m2.1d">over~ start_ARG bold_italic_θ end_ARG ( italic_t ) ∈</annotation></semantics></math> <math alttext="L_{\infty}" class="ltx_Math" display="inline" id="A3.1.p1.75.m3.1"><semantics id="A3.1.p1.75.m3.1a"><msub id="A3.1.p1.75.m3.1.1" xref="A3.1.p1.75.m3.1.1.cmml"><mi id="A3.1.p1.75.m3.1.1.2" xref="A3.1.p1.75.m3.1.1.2.cmml">L</mi><mi id="A3.1.p1.75.m3.1.1.3" mathvariant="normal" xref="A3.1.p1.75.m3.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="A3.1.p1.75.m3.1b"><apply id="A3.1.p1.75.m3.1.1.cmml" xref="A3.1.p1.75.m3.1.1"><csymbol cd="ambiguous" id="A3.1.p1.75.m3.1.1.1.cmml" xref="A3.1.p1.75.m3.1.1">subscript</csymbol><ci id="A3.1.p1.75.m3.1.1.2.cmml" xref="A3.1.p1.75.m3.1.1.2">𝐿</ci><infinity id="A3.1.p1.75.m3.1.1.3.cmml" xref="A3.1.p1.75.m3.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.75.m3.1c">L_{\infty}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.75.m3.1d">italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\forall t\geq 0" class="ltx_Math" display="inline" id="A3.1.p1.76.m4.1"><semantics id="A3.1.p1.76.m4.1a"><mrow id="A3.1.p1.76.m4.1.1" xref="A3.1.p1.76.m4.1.1.cmml"><mrow id="A3.1.p1.76.m4.1.1.2" xref="A3.1.p1.76.m4.1.1.2.cmml"><mo id="A3.1.p1.76.m4.1.1.2.1" rspace="0.167em" xref="A3.1.p1.76.m4.1.1.2.1.cmml">∀</mo><mi id="A3.1.p1.76.m4.1.1.2.2" xref="A3.1.p1.76.m4.1.1.2.2.cmml">t</mi></mrow><mo id="A3.1.p1.76.m4.1.1.1" xref="A3.1.p1.76.m4.1.1.1.cmml">≥</mo><mn id="A3.1.p1.76.m4.1.1.3" xref="A3.1.p1.76.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.76.m4.1b"><apply id="A3.1.p1.76.m4.1.1.cmml" xref="A3.1.p1.76.m4.1.1"><geq id="A3.1.p1.76.m4.1.1.1.cmml" xref="A3.1.p1.76.m4.1.1.1"></geq><apply id="A3.1.p1.76.m4.1.1.2.cmml" xref="A3.1.p1.76.m4.1.1.2"><csymbol cd="latexml" id="A3.1.p1.76.m4.1.1.2.1.cmml" xref="A3.1.p1.76.m4.1.1.2.1">for-all</csymbol><ci id="A3.1.p1.76.m4.1.1.2.2.cmml" xref="A3.1.p1.76.m4.1.1.2.2">𝑡</ci></apply><cn id="A3.1.p1.76.m4.1.1.3.cmml" type="integer" xref="A3.1.p1.76.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.76.m4.1c">\forall t\geq 0</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.76.m4.1d">∀ italic_t ≥ 0</annotation></semantics></math>, one obtains <math alttext="\bm{x}(t)" class="ltx_Math" display="inline" id="A3.1.p1.77.m5.1"><semantics id="A3.1.p1.77.m5.1a"><mrow id="A3.1.p1.77.m5.1.2" xref="A3.1.p1.77.m5.1.2.cmml"><mi id="A3.1.p1.77.m5.1.2.2" xref="A3.1.p1.77.m5.1.2.2.cmml">𝒙</mi><mo id="A3.1.p1.77.m5.1.2.1" xref="A3.1.p1.77.m5.1.2.1.cmml">⁢</mo><mrow id="A3.1.p1.77.m5.1.2.3.2" xref="A3.1.p1.77.m5.1.2.cmml"><mo id="A3.1.p1.77.m5.1.2.3.2.1" stretchy="false" xref="A3.1.p1.77.m5.1.2.cmml">(</mo><mi id="A3.1.p1.77.m5.1.1" xref="A3.1.p1.77.m5.1.1.cmml">t</mi><mo id="A3.1.p1.77.m5.1.2.3.2.2" stretchy="false" xref="A3.1.p1.77.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.77.m5.1b"><apply id="A3.1.p1.77.m5.1.2.cmml" xref="A3.1.p1.77.m5.1.2"><times id="A3.1.p1.77.m5.1.2.1.cmml" xref="A3.1.p1.77.m5.1.2.1"></times><ci id="A3.1.p1.77.m5.1.2.2.cmml" xref="A3.1.p1.77.m5.1.2.2">𝒙</ci><ci id="A3.1.p1.77.m5.1.1.cmml" xref="A3.1.p1.77.m5.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.77.m5.1c">\bm{x}(t)</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.77.m5.1d">bold_italic_x ( italic_t )</annotation></semantics></math> <math alttext="\in" class="ltx_Math" display="inline" id="A3.1.p1.78.m6.1"><semantics id="A3.1.p1.78.m6.1a"><mo id="A3.1.p1.78.m6.1.1" xref="A3.1.p1.78.m6.1.1.cmml">∈</mo><annotation-xml encoding="MathML-Content" id="A3.1.p1.78.m6.1b"><in id="A3.1.p1.78.m6.1.1.cmml" xref="A3.1.p1.78.m6.1.1"></in></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.78.m6.1c">\in</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.78.m6.1d">∈</annotation></semantics></math> <math alttext="L_{\infty}" class="ltx_Math" display="inline" id="A3.1.p1.79.m7.1"><semantics id="A3.1.p1.79.m7.1a"><msub id="A3.1.p1.79.m7.1.1" xref="A3.1.p1.79.m7.1.1.cmml"><mi id="A3.1.p1.79.m7.1.1.2" xref="A3.1.p1.79.m7.1.1.2.cmml">L</mi><mi id="A3.1.p1.79.m7.1.1.3" mathvariant="normal" xref="A3.1.p1.79.m7.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="A3.1.p1.79.m7.1b"><apply id="A3.1.p1.79.m7.1.1.cmml" xref="A3.1.p1.79.m7.1.1"><csymbol cd="ambiguous" id="A3.1.p1.79.m7.1.1.1.cmml" xref="A3.1.p1.79.m7.1.1">subscript</csymbol><ci id="A3.1.p1.79.m7.1.1.2.cmml" xref="A3.1.p1.79.m7.1.1.2">𝐿</ci><infinity id="A3.1.p1.79.m7.1.1.3.cmml" xref="A3.1.p1.79.m7.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.79.m7.1c">L_{\infty}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.79.m7.1d">italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\forall t\geq 0" class="ltx_Math" display="inline" id="A3.1.p1.80.m8.1"><semantics id="A3.1.p1.80.m8.1a"><mrow id="A3.1.p1.80.m8.1.1" xref="A3.1.p1.80.m8.1.1.cmml"><mrow id="A3.1.p1.80.m8.1.1.2" xref="A3.1.p1.80.m8.1.1.2.cmml"><mo id="A3.1.p1.80.m8.1.1.2.1" rspace="0.167em" xref="A3.1.p1.80.m8.1.1.2.1.cmml">∀</mo><mi id="A3.1.p1.80.m8.1.1.2.2" xref="A3.1.p1.80.m8.1.1.2.2.cmml">t</mi></mrow><mo id="A3.1.p1.80.m8.1.1.1" xref="A3.1.p1.80.m8.1.1.1.cmml">≥</mo><mn id="A3.1.p1.80.m8.1.1.3" xref="A3.1.p1.80.m8.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.80.m8.1b"><apply id="A3.1.p1.80.m8.1.1.cmml" xref="A3.1.p1.80.m8.1.1"><geq id="A3.1.p1.80.m8.1.1.1.cmml" xref="A3.1.p1.80.m8.1.1.1"></geq><apply id="A3.1.p1.80.m8.1.1.2.cmml" xref="A3.1.p1.80.m8.1.1.2"><csymbol cd="latexml" id="A3.1.p1.80.m8.1.1.2.1.cmml" xref="A3.1.p1.80.m8.1.1.2.1">for-all</csymbol><ci id="A3.1.p1.80.m8.1.1.2.2.cmml" xref="A3.1.p1.80.m8.1.1.2.2">𝑡</ci></apply><cn id="A3.1.p1.80.m8.1.1.3.cmml" type="integer" xref="A3.1.p1.80.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.80.m8.1c">\forall t\geq 0</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.80.m8.1d">∀ italic_t ≥ 0</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E7" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">7</span></a>). <span class="ltx_text" id="A3.1.p1.90.10" style="color:#000099;">As the signals <math alttext="\Phi_{\rm f}(t)" class="ltx_Math" display="inline" id="A3.1.p1.81.1.m1.1"><semantics id="A3.1.p1.81.1.m1.1a"><mrow id="A3.1.p1.81.1.m1.1.2" xref="A3.1.p1.81.1.m1.1.2.cmml"><msub id="A3.1.p1.81.1.m1.1.2.2" xref="A3.1.p1.81.1.m1.1.2.2.cmml"><mi id="A3.1.p1.81.1.m1.1.2.2.2" mathcolor="#000099" mathvariant="normal" xref="A3.1.p1.81.1.m1.1.2.2.2.cmml">Φ</mi><mi id="A3.1.p1.81.1.m1.1.2.2.3" mathcolor="#000099" mathvariant="normal" xref="A3.1.p1.81.1.m1.1.2.2.3.cmml">f</mi></msub><mo id="A3.1.p1.81.1.m1.1.2.1" xref="A3.1.p1.81.1.m1.1.2.1.cmml">⁢</mo><mrow id="A3.1.p1.81.1.m1.1.2.3.2" xref="A3.1.p1.81.1.m1.1.2.cmml"><mo id="A3.1.p1.81.1.m1.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="A3.1.p1.81.1.m1.1.2.cmml">(</mo><mi id="A3.1.p1.81.1.m1.1.1" mathcolor="#000099" xref="A3.1.p1.81.1.m1.1.1.cmml">t</mi><mo id="A3.1.p1.81.1.m1.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="A3.1.p1.81.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.81.1.m1.1b"><apply id="A3.1.p1.81.1.m1.1.2.cmml" xref="A3.1.p1.81.1.m1.1.2"><times id="A3.1.p1.81.1.m1.1.2.1.cmml" xref="A3.1.p1.81.1.m1.1.2.1"></times><apply id="A3.1.p1.81.1.m1.1.2.2.cmml" xref="A3.1.p1.81.1.m1.1.2.2"><csymbol cd="ambiguous" id="A3.1.p1.81.1.m1.1.2.2.1.cmml" xref="A3.1.p1.81.1.m1.1.2.2">subscript</csymbol><ci id="A3.1.p1.81.1.m1.1.2.2.2.cmml" xref="A3.1.p1.81.1.m1.1.2.2.2">Φ</ci><ci id="A3.1.p1.81.1.m1.1.2.2.3.cmml" xref="A3.1.p1.81.1.m1.1.2.2.3">f</ci></apply><ci id="A3.1.p1.81.1.m1.1.1.cmml" xref="A3.1.p1.81.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.81.1.m1.1c">\Phi_{\rm f}(t)</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.81.1.m1.1d">roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math>, <math alttext="\bm{z}(t)" class="ltx_Math" display="inline" id="A3.1.p1.82.2.m2.1"><semantics id="A3.1.p1.82.2.m2.1a"><mrow id="A3.1.p1.82.2.m2.1.2" xref="A3.1.p1.82.2.m2.1.2.cmml"><mi id="A3.1.p1.82.2.m2.1.2.2" mathcolor="#000099" xref="A3.1.p1.82.2.m2.1.2.2.cmml">𝒛</mi><mo id="A3.1.p1.82.2.m2.1.2.1" xref="A3.1.p1.82.2.m2.1.2.1.cmml">⁢</mo><mrow id="A3.1.p1.82.2.m2.1.2.3.2" xref="A3.1.p1.82.2.m2.1.2.cmml"><mo id="A3.1.p1.82.2.m2.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="A3.1.p1.82.2.m2.1.2.cmml">(</mo><mi id="A3.1.p1.82.2.m2.1.1" mathcolor="#000099" xref="A3.1.p1.82.2.m2.1.1.cmml">t</mi><mo id="A3.1.p1.82.2.m2.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="A3.1.p1.82.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.82.2.m2.1b"><apply id="A3.1.p1.82.2.m2.1.2.cmml" xref="A3.1.p1.82.2.m2.1.2"><times id="A3.1.p1.82.2.m2.1.2.1.cmml" xref="A3.1.p1.82.2.m2.1.2.1"></times><ci id="A3.1.p1.82.2.m2.1.2.2.cmml" xref="A3.1.p1.82.2.m2.1.2.2">𝒛</ci><ci id="A3.1.p1.82.2.m2.1.1.cmml" xref="A3.1.p1.82.2.m2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.82.2.m2.1c">\bm{z}(t)</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.82.2.m2.1d">bold_italic_z ( italic_t )</annotation></semantics></math>, <math alttext="Q(t)" class="ltx_Math" display="inline" id="A3.1.p1.83.3.m3.1"><semantics id="A3.1.p1.83.3.m3.1a"><mrow id="A3.1.p1.83.3.m3.1.2" xref="A3.1.p1.83.3.m3.1.2.cmml"><mi id="A3.1.p1.83.3.m3.1.2.2" mathcolor="#000099" xref="A3.1.p1.83.3.m3.1.2.2.cmml">Q</mi><mo id="A3.1.p1.83.3.m3.1.2.1" xref="A3.1.p1.83.3.m3.1.2.1.cmml">⁢</mo><mrow id="A3.1.p1.83.3.m3.1.2.3.2" xref="A3.1.p1.83.3.m3.1.2.cmml"><mo id="A3.1.p1.83.3.m3.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="A3.1.p1.83.3.m3.1.2.cmml">(</mo><mi id="A3.1.p1.83.3.m3.1.1" mathcolor="#000099" xref="A3.1.p1.83.3.m3.1.1.cmml">t</mi><mo id="A3.1.p1.83.3.m3.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="A3.1.p1.83.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.83.3.m3.1b"><apply id="A3.1.p1.83.3.m3.1.2.cmml" xref="A3.1.p1.83.3.m3.1.2"><times id="A3.1.p1.83.3.m3.1.2.1.cmml" xref="A3.1.p1.83.3.m3.1.2.1"></times><ci id="A3.1.p1.83.3.m3.1.2.2.cmml" xref="A3.1.p1.83.3.m3.1.2.2">𝑄</ci><ci id="A3.1.p1.83.3.m3.1.1.cmml" xref="A3.1.p1.83.3.m3.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.83.3.m3.1c">Q(t)</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.83.3.m3.1d">italic_Q ( italic_t )</annotation></semantics></math>, and <math alttext="\bm{q}_{\rm f}(t)" class="ltx_Math" display="inline" id="A3.1.p1.84.4.m4.1"><semantics id="A3.1.p1.84.4.m4.1a"><mrow id="A3.1.p1.84.4.m4.1.2" xref="A3.1.p1.84.4.m4.1.2.cmml"><msub id="A3.1.p1.84.4.m4.1.2.2" xref="A3.1.p1.84.4.m4.1.2.2.cmml"><mi id="A3.1.p1.84.4.m4.1.2.2.2" mathcolor="#000099" xref="A3.1.p1.84.4.m4.1.2.2.2.cmml">𝒒</mi><mi id="A3.1.p1.84.4.m4.1.2.2.3" mathcolor="#000099" mathvariant="normal" xref="A3.1.p1.84.4.m4.1.2.2.3.cmml">f</mi></msub><mo id="A3.1.p1.84.4.m4.1.2.1" xref="A3.1.p1.84.4.m4.1.2.1.cmml">⁢</mo><mrow id="A3.1.p1.84.4.m4.1.2.3.2" xref="A3.1.p1.84.4.m4.1.2.cmml"><mo id="A3.1.p1.84.4.m4.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="A3.1.p1.84.4.m4.1.2.cmml">(</mo><mi id="A3.1.p1.84.4.m4.1.1" mathcolor="#000099" xref="A3.1.p1.84.4.m4.1.1.cmml">t</mi><mo id="A3.1.p1.84.4.m4.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="A3.1.p1.84.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.84.4.m4.1b"><apply id="A3.1.p1.84.4.m4.1.2.cmml" xref="A3.1.p1.84.4.m4.1.2"><times id="A3.1.p1.84.4.m4.1.2.1.cmml" xref="A3.1.p1.84.4.m4.1.2.1"></times><apply id="A3.1.p1.84.4.m4.1.2.2.cmml" xref="A3.1.p1.84.4.m4.1.2.2"><csymbol cd="ambiguous" id="A3.1.p1.84.4.m4.1.2.2.1.cmml" xref="A3.1.p1.84.4.m4.1.2.2">subscript</csymbol><ci id="A3.1.p1.84.4.m4.1.2.2.2.cmml" xref="A3.1.p1.84.4.m4.1.2.2.2">𝒒</ci><ci id="A3.1.p1.84.4.m4.1.2.2.3.cmml" xref="A3.1.p1.84.4.m4.1.2.2.3">f</ci></apply><ci id="A3.1.p1.84.4.m4.1.1.cmml" xref="A3.1.p1.84.4.m4.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.84.4.m4.1c">\bm{q}_{\rm f}(t)</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.84.4.m4.1d">bold_italic_q start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>) are bounded, and are filtered by<math alttext="H(s)" class="ltx_Math" display="inline" id="A3.1.p1.85.5.m5.1"><semantics id="A3.1.p1.85.5.m5.1a"><mrow id="A3.1.p1.85.5.m5.1.2" xref="A3.1.p1.85.5.m5.1.2.cmml"><mi id="A3.1.p1.85.5.m5.1.2.2" mathcolor="#000099" xref="A3.1.p1.85.5.m5.1.2.2.cmml">H</mi><mo id="A3.1.p1.85.5.m5.1.2.1" xref="A3.1.p1.85.5.m5.1.2.1.cmml">⁢</mo><mrow id="A3.1.p1.85.5.m5.1.2.3.2" xref="A3.1.p1.85.5.m5.1.2.cmml"><mo id="A3.1.p1.85.5.m5.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="A3.1.p1.85.5.m5.1.2.cmml">(</mo><mi id="A3.1.p1.85.5.m5.1.1" mathcolor="#000099" xref="A3.1.p1.85.5.m5.1.1.cmml">s</mi><mo id="A3.1.p1.85.5.m5.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="A3.1.p1.85.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.85.5.m5.1b"><apply id="A3.1.p1.85.5.m5.1.2.cmml" xref="A3.1.p1.85.5.m5.1.2"><times id="A3.1.p1.85.5.m5.1.2.1.cmml" xref="A3.1.p1.85.5.m5.1.2.1"></times><ci id="A3.1.p1.85.5.m5.1.2.2.cmml" xref="A3.1.p1.85.5.m5.1.2.2">𝐻</ci><ci id="A3.1.p1.85.5.m5.1.1.cmml" xref="A3.1.p1.85.5.m5.1.1">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.85.5.m5.1c">H(s)</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.85.5.m5.1d">italic_H ( italic_s )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E17" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">17</span></a>), their time derivatives up to the (<math alttext="n-2" class="ltx_Math" display="inline" id="A3.1.p1.86.6.m6.1"><semantics id="A3.1.p1.86.6.m6.1a"><mrow id="A3.1.p1.86.6.m6.1.1" xref="A3.1.p1.86.6.m6.1.1.cmml"><mi id="A3.1.p1.86.6.m6.1.1.2" mathcolor="#000099" xref="A3.1.p1.86.6.m6.1.1.2.cmml">n</mi><mo id="A3.1.p1.86.6.m6.1.1.1" mathcolor="#000099" xref="A3.1.p1.86.6.m6.1.1.1.cmml">−</mo><mn id="A3.1.p1.86.6.m6.1.1.3" mathcolor="#000099" xref="A3.1.p1.86.6.m6.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.86.6.m6.1b"><apply id="A3.1.p1.86.6.m6.1.1.cmml" xref="A3.1.p1.86.6.m6.1.1"><minus id="A3.1.p1.86.6.m6.1.1.1.cmml" xref="A3.1.p1.86.6.m6.1.1.1"></minus><ci id="A3.1.p1.86.6.m6.1.1.2.cmml" xref="A3.1.p1.86.6.m6.1.1.2">𝑛</ci><cn id="A3.1.p1.86.6.m6.1.1.3.cmml" type="integer" xref="A3.1.p1.86.6.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.86.6.m6.1c">n-2</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.86.6.m6.1d">italic_n - 2</annotation></semantics></math>)th order are also bounded according to the input-to-state stability property of the stable filter <math alttext="H(s)" class="ltx_Math" display="inline" id="A3.1.p1.87.7.m7.1"><semantics id="A3.1.p1.87.7.m7.1a"><mrow id="A3.1.p1.87.7.m7.1.2" xref="A3.1.p1.87.7.m7.1.2.cmml"><mi id="A3.1.p1.87.7.m7.1.2.2" mathcolor="#000099" xref="A3.1.p1.87.7.m7.1.2.2.cmml">H</mi><mo id="A3.1.p1.87.7.m7.1.2.1" xref="A3.1.p1.87.7.m7.1.2.1.cmml">⁢</mo><mrow id="A3.1.p1.87.7.m7.1.2.3.2" xref="A3.1.p1.87.7.m7.1.2.cmml"><mo id="A3.1.p1.87.7.m7.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="A3.1.p1.87.7.m7.1.2.cmml">(</mo><mi id="A3.1.p1.87.7.m7.1.1" mathcolor="#000099" xref="A3.1.p1.87.7.m7.1.1.cmml">s</mi><mo id="A3.1.p1.87.7.m7.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="A3.1.p1.87.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.87.7.m7.1b"><apply id="A3.1.p1.87.7.m7.1.2.cmml" xref="A3.1.p1.87.7.m7.1.2"><times id="A3.1.p1.87.7.m7.1.2.1.cmml" xref="A3.1.p1.87.7.m7.1.2.1"></times><ci id="A3.1.p1.87.7.m7.1.2.2.cmml" xref="A3.1.p1.87.7.m7.1.2.2">𝐻</ci><ci id="A3.1.p1.87.7.m7.1.1.cmml" xref="A3.1.p1.87.7.m7.1.1">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.87.7.m7.1c">H(s)</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.87.7.m7.1d">italic_H ( italic_s )</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib27" title="">27</a>, <a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib24" title="">24</a>]</cite>. As the high-order time derivatives of <math alttext="\hat{\bm{\theta}}" class="ltx_Math" display="inline" id="A3.1.p1.88.8.m8.1"><semantics id="A3.1.p1.88.8.m8.1a"><mover accent="true" id="A3.1.p1.88.8.m8.1.1" xref="A3.1.p1.88.8.m8.1.1.cmml"><mi id="A3.1.p1.88.8.m8.1.1.2" mathcolor="#000099" xref="A3.1.p1.88.8.m8.1.1.2.cmml">𝜽</mi><mo id="A3.1.p1.88.8.m8.1.1.1" mathcolor="#000099" xref="A3.1.p1.88.8.m8.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="A3.1.p1.88.8.m8.1b"><apply id="A3.1.p1.88.8.m8.1.1.cmml" xref="A3.1.p1.88.8.m8.1.1"><ci id="A3.1.p1.88.8.m8.1.1.1.cmml" xref="A3.1.p1.88.8.m8.1.1.1">^</ci><ci id="A3.1.p1.88.8.m8.1.1.2.cmml" xref="A3.1.p1.88.8.m8.1.1.2">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.88.8.m8.1c">\hat{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.88.8.m8.1d">over^ start_ARG bold_italic_θ end_ARG</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E25" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">25</span></a>) are computed as a weighted sum of these signals and their high-order time derivatives, one concludes <math alttext="{\hat{\bm{\theta}}}^{(k)}\in L_{\infty}" class="ltx_Math" display="inline" id="A3.1.p1.89.9.m9.1"><semantics id="A3.1.p1.89.9.m9.1a"><mrow id="A3.1.p1.89.9.m9.1.2" xref="A3.1.p1.89.9.m9.1.2.cmml"><msup id="A3.1.p1.89.9.m9.1.2.2" xref="A3.1.p1.89.9.m9.1.2.2.cmml"><mover accent="true" id="A3.1.p1.89.9.m9.1.2.2.2" xref="A3.1.p1.89.9.m9.1.2.2.2.cmml"><mi id="A3.1.p1.89.9.m9.1.2.2.2.2" mathcolor="#000099" xref="A3.1.p1.89.9.m9.1.2.2.2.2.cmml">𝜽</mi><mo id="A3.1.p1.89.9.m9.1.2.2.2.1" mathcolor="#000099" xref="A3.1.p1.89.9.m9.1.2.2.2.1.cmml">^</mo></mover><mrow id="A3.1.p1.89.9.m9.1.1.1.3" xref="A3.1.p1.89.9.m9.1.2.2.cmml"><mo id="A3.1.p1.89.9.m9.1.1.1.3.1" mathcolor="#000099" stretchy="false" xref="A3.1.p1.89.9.m9.1.2.2.cmml">(</mo><mi id="A3.1.p1.89.9.m9.1.1.1.1" mathcolor="#000099" xref="A3.1.p1.89.9.m9.1.1.1.1.cmml">k</mi><mo id="A3.1.p1.89.9.m9.1.1.1.3.2" mathcolor="#000099" stretchy="false" xref="A3.1.p1.89.9.m9.1.2.2.cmml">)</mo></mrow></msup><mo id="A3.1.p1.89.9.m9.1.2.1" mathcolor="#000099" xref="A3.1.p1.89.9.m9.1.2.1.cmml">∈</mo><msub id="A3.1.p1.89.9.m9.1.2.3" xref="A3.1.p1.89.9.m9.1.2.3.cmml"><mi id="A3.1.p1.89.9.m9.1.2.3.2" mathcolor="#000099" xref="A3.1.p1.89.9.m9.1.2.3.2.cmml">L</mi><mi id="A3.1.p1.89.9.m9.1.2.3.3" mathcolor="#000099" mathvariant="normal" xref="A3.1.p1.89.9.m9.1.2.3.3.cmml">∞</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.89.9.m9.1b"><apply id="A3.1.p1.89.9.m9.1.2.cmml" xref="A3.1.p1.89.9.m9.1.2"><in id="A3.1.p1.89.9.m9.1.2.1.cmml" xref="A3.1.p1.89.9.m9.1.2.1"></in><apply id="A3.1.p1.89.9.m9.1.2.2.cmml" xref="A3.1.p1.89.9.m9.1.2.2"><csymbol cd="ambiguous" id="A3.1.p1.89.9.m9.1.2.2.1.cmml" xref="A3.1.p1.89.9.m9.1.2.2">superscript</csymbol><apply id="A3.1.p1.89.9.m9.1.2.2.2.cmml" xref="A3.1.p1.89.9.m9.1.2.2.2"><ci id="A3.1.p1.89.9.m9.1.2.2.2.1.cmml" xref="A3.1.p1.89.9.m9.1.2.2.2.1">^</ci><ci id="A3.1.p1.89.9.m9.1.2.2.2.2.cmml" xref="A3.1.p1.89.9.m9.1.2.2.2.2">𝜽</ci></apply><ci id="A3.1.p1.89.9.m9.1.1.1.1.cmml" xref="A3.1.p1.89.9.m9.1.1.1.1">𝑘</ci></apply><apply id="A3.1.p1.89.9.m9.1.2.3.cmml" xref="A3.1.p1.89.9.m9.1.2.3"><csymbol cd="ambiguous" id="A3.1.p1.89.9.m9.1.2.3.1.cmml" xref="A3.1.p1.89.9.m9.1.2.3">subscript</csymbol><ci id="A3.1.p1.89.9.m9.1.2.3.2.cmml" xref="A3.1.p1.89.9.m9.1.2.3.2">𝐿</ci><infinity id="A3.1.p1.89.9.m9.1.2.3.3.cmml" xref="A3.1.p1.89.9.m9.1.2.3.3"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.89.9.m9.1c">{\hat{\bm{\theta}}}^{(k)}\in L_{\infty}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.89.9.m9.1d">over^ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ∈ italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math> for all relevant <math alttext="k" class="ltx_Math" display="inline" id="A3.1.p1.90.10.m10.1"><semantics id="A3.1.p1.90.10.m10.1a"><mi id="A3.1.p1.90.10.m10.1.1" mathcolor="#000099" xref="A3.1.p1.90.10.m10.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A3.1.p1.90.10.m10.1b"><ci id="A3.1.p1.90.10.m10.1.1.cmml" xref="A3.1.p1.90.10.m10.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.90.10.m10.1c">k</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.90.10.m10.1d">italic_k</annotation></semantics></math>.</span> <span class="ltx_text" id="A3.1.p1.93.13" style="color:#000099;">Thus, the equilibrium point <math alttext="(\bm{e}" class="ltx_math_unparsed" display="inline" id="A3.1.p1.91.11.m1.1"><semantics id="A3.1.p1.91.11.m1.1a"><mrow id="A3.1.p1.91.11.m1.1b"><mo id="A3.1.p1.91.11.m1.1.1" mathcolor="#000099" stretchy="false">(</mo><mi id="A3.1.p1.91.11.m1.1.2" mathcolor="#000099">𝒆</mi></mrow><annotation encoding="application/x-tex" id="A3.1.p1.91.11.m1.1c">(\bm{e}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.91.11.m1.1d">( bold_italic_e</annotation></semantics></math>, <math alttext="\tilde{\bm{\theta}})=\bm{0}" class="ltx_math_unparsed" display="inline" id="A3.1.p1.92.12.m2.1"><semantics id="A3.1.p1.92.12.m2.1a"><mrow id="A3.1.p1.92.12.m2.1b"><mover accent="true" id="A3.1.p1.92.12.m2.1.1"><mi id="A3.1.p1.92.12.m2.1.1.2" mathcolor="#000099">𝜽</mi><mo id="A3.1.p1.92.12.m2.1.1.1" mathcolor="#000099">~</mo></mover><mo id="A3.1.p1.92.12.m2.1.2" mathcolor="#000099" stretchy="false">)</mo><mo id="A3.1.p1.92.12.m2.1.3" mathcolor="#000099">=</mo><mn id="A3.1.p1.92.12.m2.1.4" mathcolor="#000099">𝟎</mn></mrow><annotation encoding="application/x-tex" id="A3.1.p1.92.12.m2.1c">\tilde{\bm{\theta}})=\bm{0}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.92.12.m2.1d">over~ start_ARG bold_italic_θ end_ARG ) = bold_0</annotation></semantics></math> of the closed-loop system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E7" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">7</span></a>) with (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>) is UUB stable on <math alttext="t\in[0,\infty)" class="ltx_Math" display="inline" id="A3.1.p1.93.13.m3.2"><semantics id="A3.1.p1.93.13.m3.2a"><mrow id="A3.1.p1.93.13.m3.2.3" xref="A3.1.p1.93.13.m3.2.3.cmml"><mi id="A3.1.p1.93.13.m3.2.3.2" mathcolor="#000099" xref="A3.1.p1.93.13.m3.2.3.2.cmml">t</mi><mo id="A3.1.p1.93.13.m3.2.3.1" mathcolor="#000099" xref="A3.1.p1.93.13.m3.2.3.1.cmml">∈</mo><mrow id="A3.1.p1.93.13.m3.2.3.3.2" xref="A3.1.p1.93.13.m3.2.3.3.1.cmml"><mo id="A3.1.p1.93.13.m3.2.3.3.2.1" mathcolor="#000099" stretchy="false" xref="A3.1.p1.93.13.m3.2.3.3.1.cmml">[</mo><mn id="A3.1.p1.93.13.m3.1.1" mathcolor="#000099" xref="A3.1.p1.93.13.m3.1.1.cmml">0</mn><mo id="A3.1.p1.93.13.m3.2.3.3.2.2" mathcolor="#000099" xref="A3.1.p1.93.13.m3.2.3.3.1.cmml">,</mo><mi id="A3.1.p1.93.13.m3.2.2" mathcolor="#000099" mathvariant="normal" xref="A3.1.p1.93.13.m3.2.2.cmml">∞</mi><mo id="A3.1.p1.93.13.m3.2.3.3.2.3" mathcolor="#000099" stretchy="false" xref="A3.1.p1.93.13.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.93.13.m3.2b"><apply id="A3.1.p1.93.13.m3.2.3.cmml" xref="A3.1.p1.93.13.m3.2.3"><in id="A3.1.p1.93.13.m3.2.3.1.cmml" xref="A3.1.p1.93.13.m3.2.3.1"></in><ci id="A3.1.p1.93.13.m3.2.3.2.cmml" xref="A3.1.p1.93.13.m3.2.3.2">𝑡</ci><interval closure="closed-open" id="A3.1.p1.93.13.m3.2.3.3.1.cmml" xref="A3.1.p1.93.13.m3.2.3.3.2"><cn id="A3.1.p1.93.13.m3.1.1.cmml" type="integer" xref="A3.1.p1.93.13.m3.1.1">0</cn><infinity id="A3.1.p1.93.13.m3.2.2.cmml" xref="A3.1.p1.93.13.m3.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.93.13.m3.2c">t\in[0,\infty)</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.93.13.m3.2d">italic_t ∈ [ 0 , ∞ )</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="A3.2.p2"> <p class="ltx_p" id="A3.2.p2.34">2) Consider the control problem under partial IE on <math alttext="t" class="ltx_Math" display="inline" id="A3.2.p2.1.m1.1"><semantics id="A3.2.p2.1.m1.1a"><mi id="A3.2.p2.1.m1.1.1" xref="A3.2.p2.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A3.2.p2.1.m1.1b"><ci id="A3.2.p2.1.m1.1.1.cmml" xref="A3.2.p2.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.1.m1.1d">italic_t</annotation></semantics></math> <math alttext="\in" class="ltx_Math" display="inline" id="A3.2.p2.2.m2.1"><semantics id="A3.2.p2.2.m2.1a"><mo id="A3.2.p2.2.m2.1.1" xref="A3.2.p2.2.m2.1.1.cmml">∈</mo><annotation-xml encoding="MathML-Content" id="A3.2.p2.2.m2.1b"><in id="A3.2.p2.2.m2.1.1.cmml" xref="A3.2.p2.2.m2.1.1"></in></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.2.m2.1c">\in</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.2.m2.1d">∈</annotation></semantics></math> <math alttext="[T_{\rm a}" class="ltx_math_unparsed" display="inline" id="A3.2.p2.3.m3.1"><semantics id="A3.2.p2.3.m3.1a"><mrow id="A3.2.p2.3.m3.1b"><mo id="A3.2.p2.3.m3.1.1" stretchy="false">[</mo><msub id="A3.2.p2.3.m3.1.2"><mi id="A3.2.p2.3.m3.1.2.2">T</mi><mi id="A3.2.p2.3.m3.1.2.3" mathvariant="normal">a</mi></msub></mrow><annotation encoding="application/x-tex" id="A3.2.p2.3.m3.1c">[T_{\rm a}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.3.m3.1d">[ italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\infty)" class="ltx_math_unparsed" display="inline" id="A3.2.p2.4.m4.1"><semantics id="A3.2.p2.4.m4.1a"><mrow id="A3.2.p2.4.m4.1b"><mi id="A3.2.p2.4.m4.1.1" mathvariant="normal">∞</mi><mo id="A3.2.p2.4.m4.1.2" stretchy="false">)</mo></mrow><annotation encoding="application/x-tex" id="A3.2.p2.4.m4.1c">\infty)</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.4.m4.1d">∞ )</annotation></semantics></math>. If only partial IE in Definition 3 exists for some constants <math alttext="\sigma" class="ltx_Math" display="inline" id="A3.2.p2.5.m5.1"><semantics id="A3.2.p2.5.m5.1a"><mi id="A3.2.p2.5.m5.1.1" xref="A3.2.p2.5.m5.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="A3.2.p2.5.m5.1b"><ci id="A3.2.p2.5.m5.1.1.cmml" xref="A3.2.p2.5.m5.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.5.m5.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.5.m5.1d">italic_σ</annotation></semantics></math>, <math alttext="T_{\rm a}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A3.2.p2.6.m6.1"><semantics id="A3.2.p2.6.m6.1a"><mrow id="A3.2.p2.6.m6.1.1" xref="A3.2.p2.6.m6.1.1.cmml"><msub id="A3.2.p2.6.m6.1.1.2" xref="A3.2.p2.6.m6.1.1.2.cmml"><mi id="A3.2.p2.6.m6.1.1.2.2" xref="A3.2.p2.6.m6.1.1.2.2.cmml">T</mi><mi id="A3.2.p2.6.m6.1.1.2.3" mathvariant="normal" xref="A3.2.p2.6.m6.1.1.2.3.cmml">a</mi></msub><mo id="A3.2.p2.6.m6.1.1.1" xref="A3.2.p2.6.m6.1.1.1.cmml">∈</mo><msup id="A3.2.p2.6.m6.1.1.3" xref="A3.2.p2.6.m6.1.1.3.cmml"><mi id="A3.2.p2.6.m6.1.1.3.2" xref="A3.2.p2.6.m6.1.1.3.2.cmml">ℝ</mi><mo id="A3.2.p2.6.m6.1.1.3.3" xref="A3.2.p2.6.m6.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.6.m6.1b"><apply id="A3.2.p2.6.m6.1.1.cmml" xref="A3.2.p2.6.m6.1.1"><in id="A3.2.p2.6.m6.1.1.1.cmml" xref="A3.2.p2.6.m6.1.1.1"></in><apply id="A3.2.p2.6.m6.1.1.2.cmml" xref="A3.2.p2.6.m6.1.1.2"><csymbol cd="ambiguous" id="A3.2.p2.6.m6.1.1.2.1.cmml" xref="A3.2.p2.6.m6.1.1.2">subscript</csymbol><ci id="A3.2.p2.6.m6.1.1.2.2.cmml" xref="A3.2.p2.6.m6.1.1.2.2">𝑇</ci><ci id="A3.2.p2.6.m6.1.1.2.3.cmml" xref="A3.2.p2.6.m6.1.1.2.3">a</ci></apply><apply id="A3.2.p2.6.m6.1.1.3.cmml" xref="A3.2.p2.6.m6.1.1.3"><csymbol cd="ambiguous" id="A3.2.p2.6.m6.1.1.3.1.cmml" xref="A3.2.p2.6.m6.1.1.3">superscript</csymbol><ci id="A3.2.p2.6.m6.1.1.3.2.cmml" xref="A3.2.p2.6.m6.1.1.3.2">ℝ</ci><plus id="A3.2.p2.6.m6.1.1.3.3.cmml" xref="A3.2.p2.6.m6.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.6.m6.1c">T_{\rm a}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.6.m6.1d">italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, and the index set <math alttext="\mathcal{I}" class="ltx_Math" display="inline" id="A3.2.p2.7.m7.1"><semantics id="A3.2.p2.7.m7.1a"><mi class="ltx_font_mathcaligraphic" id="A3.2.p2.7.m7.1.1" xref="A3.2.p2.7.m7.1.1.cmml">ℐ</mi><annotation-xml encoding="MathML-Content" id="A3.2.p2.7.m7.1b"><ci id="A3.2.p2.7.m7.1.1.cmml" xref="A3.2.p2.7.m7.1.1">ℐ</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.7.m7.1c">\mathcal{I}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.7.m7.1d">caligraphic_I</annotation></semantics></math> no longer changes on <math alttext="t\in[T_{\rm a},\infty)" class="ltx_Math" display="inline" id="A3.2.p2.8.m8.2"><semantics id="A3.2.p2.8.m8.2a"><mrow id="A3.2.p2.8.m8.2.2" xref="A3.2.p2.8.m8.2.2.cmml"><mi id="A3.2.p2.8.m8.2.2.3" xref="A3.2.p2.8.m8.2.2.3.cmml">t</mi><mo id="A3.2.p2.8.m8.2.2.2" xref="A3.2.p2.8.m8.2.2.2.cmml">∈</mo><mrow id="A3.2.p2.8.m8.2.2.1.1" xref="A3.2.p2.8.m8.2.2.1.2.cmml"><mo id="A3.2.p2.8.m8.2.2.1.1.2" stretchy="false" xref="A3.2.p2.8.m8.2.2.1.2.cmml">[</mo><msub id="A3.2.p2.8.m8.2.2.1.1.1" xref="A3.2.p2.8.m8.2.2.1.1.1.cmml"><mi id="A3.2.p2.8.m8.2.2.1.1.1.2" xref="A3.2.p2.8.m8.2.2.1.1.1.2.cmml">T</mi><mi id="A3.2.p2.8.m8.2.2.1.1.1.3" mathvariant="normal" xref="A3.2.p2.8.m8.2.2.1.1.1.3.cmml">a</mi></msub><mo id="A3.2.p2.8.m8.2.2.1.1.3" xref="A3.2.p2.8.m8.2.2.1.2.cmml">,</mo><mi id="A3.2.p2.8.m8.1.1" mathvariant="normal" xref="A3.2.p2.8.m8.1.1.cmml">∞</mi><mo id="A3.2.p2.8.m8.2.2.1.1.4" stretchy="false" xref="A3.2.p2.8.m8.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.8.m8.2b"><apply id="A3.2.p2.8.m8.2.2.cmml" xref="A3.2.p2.8.m8.2.2"><in id="A3.2.p2.8.m8.2.2.2.cmml" xref="A3.2.p2.8.m8.2.2.2"></in><ci id="A3.2.p2.8.m8.2.2.3.cmml" xref="A3.2.p2.8.m8.2.2.3">𝑡</ci><interval closure="closed-open" id="A3.2.p2.8.m8.2.2.1.2.cmml" xref="A3.2.p2.8.m8.2.2.1.1"><apply id="A3.2.p2.8.m8.2.2.1.1.1.cmml" xref="A3.2.p2.8.m8.2.2.1.1.1"><csymbol cd="ambiguous" id="A3.2.p2.8.m8.2.2.1.1.1.1.cmml" xref="A3.2.p2.8.m8.2.2.1.1.1">subscript</csymbol><ci id="A3.2.p2.8.m8.2.2.1.1.1.2.cmml" xref="A3.2.p2.8.m8.2.2.1.1.1.2">𝑇</ci><ci id="A3.2.p2.8.m8.2.2.1.1.1.3.cmml" xref="A3.2.p2.8.m8.2.2.1.1.1.3">a</ci></apply><infinity id="A3.2.p2.8.m8.1.1.cmml" xref="A3.2.p2.8.m8.1.1"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.8.m8.2c">t\in[T_{\rm a},\infty)</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.8.m8.2d">italic_t ∈ [ italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT , ∞ )</annotation></semantics></math>, there exist some inactive channels <math alttext="\bm{\phi}_{j}(t)" class="ltx_Math" display="inline" id="A3.2.p2.9.m9.1"><semantics id="A3.2.p2.9.m9.1a"><mrow id="A3.2.p2.9.m9.1.2" xref="A3.2.p2.9.m9.1.2.cmml"><msub id="A3.2.p2.9.m9.1.2.2" xref="A3.2.p2.9.m9.1.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="A3.2.p2.9.m9.1.2.2.2" mathvariant="bold-italic" xref="A3.2.p2.9.m9.1.2.2.2.cmml">ϕ</mi><mi id="A3.2.p2.9.m9.1.2.2.3" xref="A3.2.p2.9.m9.1.2.2.3.cmml">j</mi></msub><mo id="A3.2.p2.9.m9.1.2.1" xref="A3.2.p2.9.m9.1.2.1.cmml">⁢</mo><mrow id="A3.2.p2.9.m9.1.2.3.2" xref="A3.2.p2.9.m9.1.2.cmml"><mo id="A3.2.p2.9.m9.1.2.3.2.1" stretchy="false" xref="A3.2.p2.9.m9.1.2.cmml">(</mo><mi id="A3.2.p2.9.m9.1.1" xref="A3.2.p2.9.m9.1.1.cmml">t</mi><mo id="A3.2.p2.9.m9.1.2.3.2.2" stretchy="false" xref="A3.2.p2.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.9.m9.1b"><apply id="A3.2.p2.9.m9.1.2.cmml" xref="A3.2.p2.9.m9.1.2"><times id="A3.2.p2.9.m9.1.2.1.cmml" xref="A3.2.p2.9.m9.1.2.1"></times><apply id="A3.2.p2.9.m9.1.2.2.cmml" xref="A3.2.p2.9.m9.1.2.2"><csymbol cd="ambiguous" id="A3.2.p2.9.m9.1.2.2.1.cmml" xref="A3.2.p2.9.m9.1.2.2">subscript</csymbol><ci id="A3.2.p2.9.m9.1.2.2.2.cmml" xref="A3.2.p2.9.m9.1.2.2.2">bold-italic-ϕ</ci><ci id="A3.2.p2.9.m9.1.2.2.3.cmml" xref="A3.2.p2.9.m9.1.2.2.3">𝑗</ci></apply><ci id="A3.2.p2.9.m9.1.1.cmml" xref="A3.2.p2.9.m9.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.9.m9.1c">\bm{\phi}_{j}(t)</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.9.m9.1d">bold_italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> such that <math alttext="\|\bm{\phi}_{j}(t)\|\equiv 0" class="ltx_Math" display="inline" id="A3.2.p2.10.m10.2"><semantics id="A3.2.p2.10.m10.2a"><mrow id="A3.2.p2.10.m10.2.2" xref="A3.2.p2.10.m10.2.2.cmml"><mrow id="A3.2.p2.10.m10.2.2.1.1" xref="A3.2.p2.10.m10.2.2.1.2.cmml"><mo id="A3.2.p2.10.m10.2.2.1.1.2" stretchy="false" xref="A3.2.p2.10.m10.2.2.1.2.1.cmml">‖</mo><mrow id="A3.2.p2.10.m10.2.2.1.1.1" xref="A3.2.p2.10.m10.2.2.1.1.1.cmml"><msub id="A3.2.p2.10.m10.2.2.1.1.1.2" xref="A3.2.p2.10.m10.2.2.1.1.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="A3.2.p2.10.m10.2.2.1.1.1.2.2" mathvariant="bold-italic" xref="A3.2.p2.10.m10.2.2.1.1.1.2.2.cmml">ϕ</mi><mi id="A3.2.p2.10.m10.2.2.1.1.1.2.3" xref="A3.2.p2.10.m10.2.2.1.1.1.2.3.cmml">j</mi></msub><mo id="A3.2.p2.10.m10.2.2.1.1.1.1" xref="A3.2.p2.10.m10.2.2.1.1.1.1.cmml">⁢</mo><mrow id="A3.2.p2.10.m10.2.2.1.1.1.3.2" xref="A3.2.p2.10.m10.2.2.1.1.1.cmml"><mo id="A3.2.p2.10.m10.2.2.1.1.1.3.2.1" stretchy="false" xref="A3.2.p2.10.m10.2.2.1.1.1.cmml">(</mo><mi id="A3.2.p2.10.m10.1.1" xref="A3.2.p2.10.m10.1.1.cmml">t</mi><mo id="A3.2.p2.10.m10.2.2.1.1.1.3.2.2" stretchy="false" xref="A3.2.p2.10.m10.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="A3.2.p2.10.m10.2.2.1.1.3" stretchy="false" xref="A3.2.p2.10.m10.2.2.1.2.1.cmml">‖</mo></mrow><mo id="A3.2.p2.10.m10.2.2.2" xref="A3.2.p2.10.m10.2.2.2.cmml">≡</mo><mn id="A3.2.p2.10.m10.2.2.3" xref="A3.2.p2.10.m10.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.10.m10.2b"><apply id="A3.2.p2.10.m10.2.2.cmml" xref="A3.2.p2.10.m10.2.2"><equivalent id="A3.2.p2.10.m10.2.2.2.cmml" xref="A3.2.p2.10.m10.2.2.2"></equivalent><apply id="A3.2.p2.10.m10.2.2.1.2.cmml" xref="A3.2.p2.10.m10.2.2.1.1"><csymbol cd="latexml" id="A3.2.p2.10.m10.2.2.1.2.1.cmml" xref="A3.2.p2.10.m10.2.2.1.1.2">norm</csymbol><apply id="A3.2.p2.10.m10.2.2.1.1.1.cmml" xref="A3.2.p2.10.m10.2.2.1.1.1"><times id="A3.2.p2.10.m10.2.2.1.1.1.1.cmml" xref="A3.2.p2.10.m10.2.2.1.1.1.1"></times><apply id="A3.2.p2.10.m10.2.2.1.1.1.2.cmml" xref="A3.2.p2.10.m10.2.2.1.1.1.2"><csymbol cd="ambiguous" id="A3.2.p2.10.m10.2.2.1.1.1.2.1.cmml" xref="A3.2.p2.10.m10.2.2.1.1.1.2">subscript</csymbol><ci id="A3.2.p2.10.m10.2.2.1.1.1.2.2.cmml" xref="A3.2.p2.10.m10.2.2.1.1.1.2.2">bold-italic-ϕ</ci><ci id="A3.2.p2.10.m10.2.2.1.1.1.2.3.cmml" xref="A3.2.p2.10.m10.2.2.1.1.1.2.3">𝑗</ci></apply><ci id="A3.2.p2.10.m10.1.1.cmml" xref="A3.2.p2.10.m10.1.1">𝑡</ci></apply></apply><cn id="A3.2.p2.10.m10.2.2.3.cmml" type="integer" xref="A3.2.p2.10.m10.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.10.m10.2c">\|\bm{\phi}_{j}(t)\|\equiv 0</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.10.m10.2d">∥ bold_italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) ∥ ≡ 0</annotation></semantics></math>, <math alttext="\forall t\in[T_{\rm a},\infty)" class="ltx_Math" display="inline" id="A3.2.p2.11.m11.2"><semantics id="A3.2.p2.11.m11.2a"><mrow id="A3.2.p2.11.m11.2.2" xref="A3.2.p2.11.m11.2.2.cmml"><mrow id="A3.2.p2.11.m11.2.2.3" xref="A3.2.p2.11.m11.2.2.3.cmml"><mo id="A3.2.p2.11.m11.2.2.3.1" rspace="0.167em" xref="A3.2.p2.11.m11.2.2.3.1.cmml">∀</mo><mi id="A3.2.p2.11.m11.2.2.3.2" xref="A3.2.p2.11.m11.2.2.3.2.cmml">t</mi></mrow><mo id="A3.2.p2.11.m11.2.2.2" xref="A3.2.p2.11.m11.2.2.2.cmml">∈</mo><mrow id="A3.2.p2.11.m11.2.2.1.1" xref="A3.2.p2.11.m11.2.2.1.2.cmml"><mo id="A3.2.p2.11.m11.2.2.1.1.2" stretchy="false" xref="A3.2.p2.11.m11.2.2.1.2.cmml">[</mo><msub id="A3.2.p2.11.m11.2.2.1.1.1" xref="A3.2.p2.11.m11.2.2.1.1.1.cmml"><mi id="A3.2.p2.11.m11.2.2.1.1.1.2" xref="A3.2.p2.11.m11.2.2.1.1.1.2.cmml">T</mi><mi id="A3.2.p2.11.m11.2.2.1.1.1.3" mathvariant="normal" xref="A3.2.p2.11.m11.2.2.1.1.1.3.cmml">a</mi></msub><mo id="A3.2.p2.11.m11.2.2.1.1.3" xref="A3.2.p2.11.m11.2.2.1.2.cmml">,</mo><mi id="A3.2.p2.11.m11.1.1" mathvariant="normal" xref="A3.2.p2.11.m11.1.1.cmml">∞</mi><mo id="A3.2.p2.11.m11.2.2.1.1.4" stretchy="false" xref="A3.2.p2.11.m11.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.11.m11.2b"><apply id="A3.2.p2.11.m11.2.2.cmml" xref="A3.2.p2.11.m11.2.2"><in id="A3.2.p2.11.m11.2.2.2.cmml" xref="A3.2.p2.11.m11.2.2.2"></in><apply id="A3.2.p2.11.m11.2.2.3.cmml" xref="A3.2.p2.11.m11.2.2.3"><csymbol cd="latexml" id="A3.2.p2.11.m11.2.2.3.1.cmml" xref="A3.2.p2.11.m11.2.2.3.1">for-all</csymbol><ci id="A3.2.p2.11.m11.2.2.3.2.cmml" xref="A3.2.p2.11.m11.2.2.3.2">𝑡</ci></apply><interval closure="closed-open" id="A3.2.p2.11.m11.2.2.1.2.cmml" xref="A3.2.p2.11.m11.2.2.1.1"><apply id="A3.2.p2.11.m11.2.2.1.1.1.cmml" xref="A3.2.p2.11.m11.2.2.1.1.1"><csymbol cd="ambiguous" id="A3.2.p2.11.m11.2.2.1.1.1.1.cmml" xref="A3.2.p2.11.m11.2.2.1.1.1">subscript</csymbol><ci id="A3.2.p2.11.m11.2.2.1.1.1.2.cmml" xref="A3.2.p2.11.m11.2.2.1.1.1.2">𝑇</ci><ci id="A3.2.p2.11.m11.2.2.1.1.1.3.cmml" xref="A3.2.p2.11.m11.2.2.1.1.1.3">a</ci></apply><infinity id="A3.2.p2.11.m11.1.1.cmml" xref="A3.2.p2.11.m11.1.1"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.11.m11.2c">\forall t\in[T_{\rm a},\infty)</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.11.m11.2d">∀ italic_t ∈ [ italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT , ∞ )</annotation></semantics></math>. Let <math alttext="\Phi_{\zeta}" class="ltx_Math" display="inline" id="A3.2.p2.12.m12.1"><semantics id="A3.2.p2.12.m12.1a"><msub id="A3.2.p2.12.m12.1.1" xref="A3.2.p2.12.m12.1.1.cmml"><mi id="A3.2.p2.12.m12.1.1.2" mathvariant="normal" xref="A3.2.p2.12.m12.1.1.2.cmml">Φ</mi><mi id="A3.2.p2.12.m12.1.1.3" xref="A3.2.p2.12.m12.1.1.3.cmml">ζ</mi></msub><annotation-xml encoding="MathML-Content" id="A3.2.p2.12.m12.1b"><apply id="A3.2.p2.12.m12.1.1.cmml" xref="A3.2.p2.12.m12.1.1"><csymbol cd="ambiguous" id="A3.2.p2.12.m12.1.1.1.cmml" xref="A3.2.p2.12.m12.1.1">subscript</csymbol><ci id="A3.2.p2.12.m12.1.1.2.cmml" xref="A3.2.p2.12.m12.1.1.2">Φ</ci><ci id="A3.2.p2.12.m12.1.1.3.cmml" xref="A3.2.p2.12.m12.1.1.3">𝜁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.12.m12.1c">\Phi_{\zeta}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.12.m12.1d">roman_Φ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="A3.2.p2.13.m13.1"><semantics id="A3.2.p2.13.m13.1a"><mo id="A3.2.p2.13.m13.1.1" xref="A3.2.p2.13.m13.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="A3.2.p2.13.m13.1b"><csymbol cd="latexml" id="A3.2.p2.13.m13.1.1.cmml" xref="A3.2.p2.13.m13.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.13.m13.1c">:=</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.13.m13.1d">:=</annotation></semantics></math> <math alttext="[\bm{\phi}_{1},\bm{\phi}_{2},\cdots,\bm{\phi}_{{N_{\zeta}}}]^{T}" class="ltx_Math" display="inline" id="A3.2.p2.14.m14.4"><semantics id="A3.2.p2.14.m14.4a"><msup id="A3.2.p2.14.m14.4.4" xref="A3.2.p2.14.m14.4.4.cmml"><mrow id="A3.2.p2.14.m14.4.4.3.3" xref="A3.2.p2.14.m14.4.4.3.4.cmml"><mo id="A3.2.p2.14.m14.4.4.3.3.4" stretchy="false" xref="A3.2.p2.14.m14.4.4.3.4.cmml">[</mo><msub id="A3.2.p2.14.m14.2.2.1.1.1" xref="A3.2.p2.14.m14.2.2.1.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="A3.2.p2.14.m14.2.2.1.1.1.2" mathvariant="bold-italic" xref="A3.2.p2.14.m14.2.2.1.1.1.2.cmml">ϕ</mi><mn id="A3.2.p2.14.m14.2.2.1.1.1.3" xref="A3.2.p2.14.m14.2.2.1.1.1.3.cmml">1</mn></msub><mo id="A3.2.p2.14.m14.4.4.3.3.5" xref="A3.2.p2.14.m14.4.4.3.4.cmml">,</mo><msub id="A3.2.p2.14.m14.3.3.2.2.2" xref="A3.2.p2.14.m14.3.3.2.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="A3.2.p2.14.m14.3.3.2.2.2.2" mathvariant="bold-italic" xref="A3.2.p2.14.m14.3.3.2.2.2.2.cmml">ϕ</mi><mn id="A3.2.p2.14.m14.3.3.2.2.2.3" xref="A3.2.p2.14.m14.3.3.2.2.2.3.cmml">2</mn></msub><mo id="A3.2.p2.14.m14.4.4.3.3.6" xref="A3.2.p2.14.m14.4.4.3.4.cmml">,</mo><mi id="A3.2.p2.14.m14.1.1" mathvariant="normal" xref="A3.2.p2.14.m14.1.1.cmml">⋯</mi><mo id="A3.2.p2.14.m14.4.4.3.3.7" xref="A3.2.p2.14.m14.4.4.3.4.cmml">,</mo><msub id="A3.2.p2.14.m14.4.4.3.3.3" xref="A3.2.p2.14.m14.4.4.3.3.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="A3.2.p2.14.m14.4.4.3.3.3.2" mathvariant="bold-italic" xref="A3.2.p2.14.m14.4.4.3.3.3.2.cmml">ϕ</mi><msub id="A3.2.p2.14.m14.4.4.3.3.3.3" xref="A3.2.p2.14.m14.4.4.3.3.3.3.cmml"><mi id="A3.2.p2.14.m14.4.4.3.3.3.3.2" xref="A3.2.p2.14.m14.4.4.3.3.3.3.2.cmml">N</mi><mi id="A3.2.p2.14.m14.4.4.3.3.3.3.3" xref="A3.2.p2.14.m14.4.4.3.3.3.3.3.cmml">ζ</mi></msub></msub><mo id="A3.2.p2.14.m14.4.4.3.3.8" stretchy="false" xref="A3.2.p2.14.m14.4.4.3.4.cmml">]</mo></mrow><mi id="A3.2.p2.14.m14.4.4.5" xref="A3.2.p2.14.m14.4.4.5.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="A3.2.p2.14.m14.4b"><apply id="A3.2.p2.14.m14.4.4.cmml" xref="A3.2.p2.14.m14.4.4"><csymbol cd="ambiguous" id="A3.2.p2.14.m14.4.4.4.cmml" xref="A3.2.p2.14.m14.4.4">superscript</csymbol><list id="A3.2.p2.14.m14.4.4.3.4.cmml" xref="A3.2.p2.14.m14.4.4.3.3"><apply id="A3.2.p2.14.m14.2.2.1.1.1.cmml" xref="A3.2.p2.14.m14.2.2.1.1.1"><csymbol cd="ambiguous" id="A3.2.p2.14.m14.2.2.1.1.1.1.cmml" xref="A3.2.p2.14.m14.2.2.1.1.1">subscript</csymbol><ci id="A3.2.p2.14.m14.2.2.1.1.1.2.cmml" xref="A3.2.p2.14.m14.2.2.1.1.1.2">bold-italic-ϕ</ci><cn id="A3.2.p2.14.m14.2.2.1.1.1.3.cmml" type="integer" xref="A3.2.p2.14.m14.2.2.1.1.1.3">1</cn></apply><apply id="A3.2.p2.14.m14.3.3.2.2.2.cmml" xref="A3.2.p2.14.m14.3.3.2.2.2"><csymbol cd="ambiguous" id="A3.2.p2.14.m14.3.3.2.2.2.1.cmml" xref="A3.2.p2.14.m14.3.3.2.2.2">subscript</csymbol><ci id="A3.2.p2.14.m14.3.3.2.2.2.2.cmml" xref="A3.2.p2.14.m14.3.3.2.2.2.2">bold-italic-ϕ</ci><cn id="A3.2.p2.14.m14.3.3.2.2.2.3.cmml" type="integer" xref="A3.2.p2.14.m14.3.3.2.2.2.3">2</cn></apply><ci id="A3.2.p2.14.m14.1.1.cmml" xref="A3.2.p2.14.m14.1.1">⋯</ci><apply id="A3.2.p2.14.m14.4.4.3.3.3.cmml" xref="A3.2.p2.14.m14.4.4.3.3.3"><csymbol cd="ambiguous" id="A3.2.p2.14.m14.4.4.3.3.3.1.cmml" xref="A3.2.p2.14.m14.4.4.3.3.3">subscript</csymbol><ci id="A3.2.p2.14.m14.4.4.3.3.3.2.cmml" xref="A3.2.p2.14.m14.4.4.3.3.3.2">bold-italic-ϕ</ci><apply id="A3.2.p2.14.m14.4.4.3.3.3.3.cmml" xref="A3.2.p2.14.m14.4.4.3.3.3.3"><csymbol cd="ambiguous" id="A3.2.p2.14.m14.4.4.3.3.3.3.1.cmml" xref="A3.2.p2.14.m14.4.4.3.3.3.3">subscript</csymbol><ci id="A3.2.p2.14.m14.4.4.3.3.3.3.2.cmml" xref="A3.2.p2.14.m14.4.4.3.3.3.3.2">𝑁</ci><ci id="A3.2.p2.14.m14.4.4.3.3.3.3.3.cmml" xref="A3.2.p2.14.m14.4.4.3.3.3.3.3">𝜁</ci></apply></apply></list><ci id="A3.2.p2.14.m14.4.4.5.cmml" xref="A3.2.p2.14.m14.4.4.5">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.14.m14.4c">[\bm{\phi}_{1},\bm{\phi}_{2},\cdots,\bm{\phi}_{{N_{\zeta}}}]^{T}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.14.m14.4d">[ bold_italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , ⋯ , bold_italic_ϕ start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> <math alttext="\in\mathbb{R}^{N_{\zeta}\times n}" class="ltx_Math" display="inline" id="A3.2.p2.15.m15.1"><semantics id="A3.2.p2.15.m15.1a"><mrow id="A3.2.p2.15.m15.1.1" xref="A3.2.p2.15.m15.1.1.cmml"><mi id="A3.2.p2.15.m15.1.1.2" xref="A3.2.p2.15.m15.1.1.2.cmml"></mi><mo id="A3.2.p2.15.m15.1.1.1" xref="A3.2.p2.15.m15.1.1.1.cmml">∈</mo><msup id="A3.2.p2.15.m15.1.1.3" xref="A3.2.p2.15.m15.1.1.3.cmml"><mi id="A3.2.p2.15.m15.1.1.3.2" xref="A3.2.p2.15.m15.1.1.3.2.cmml">ℝ</mi><mrow id="A3.2.p2.15.m15.1.1.3.3" xref="A3.2.p2.15.m15.1.1.3.3.cmml"><msub id="A3.2.p2.15.m15.1.1.3.3.2" xref="A3.2.p2.15.m15.1.1.3.3.2.cmml"><mi id="A3.2.p2.15.m15.1.1.3.3.2.2" xref="A3.2.p2.15.m15.1.1.3.3.2.2.cmml">N</mi><mi id="A3.2.p2.15.m15.1.1.3.3.2.3" xref="A3.2.p2.15.m15.1.1.3.3.2.3.cmml">ζ</mi></msub><mo id="A3.2.p2.15.m15.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="A3.2.p2.15.m15.1.1.3.3.1.cmml">×</mo><mi id="A3.2.p2.15.m15.1.1.3.3.3" xref="A3.2.p2.15.m15.1.1.3.3.3.cmml">n</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.15.m15.1b"><apply id="A3.2.p2.15.m15.1.1.cmml" xref="A3.2.p2.15.m15.1.1"><in id="A3.2.p2.15.m15.1.1.1.cmml" xref="A3.2.p2.15.m15.1.1.1"></in><csymbol cd="latexml" id="A3.2.p2.15.m15.1.1.2.cmml" xref="A3.2.p2.15.m15.1.1.2">absent</csymbol><apply id="A3.2.p2.15.m15.1.1.3.cmml" xref="A3.2.p2.15.m15.1.1.3"><csymbol cd="ambiguous" id="A3.2.p2.15.m15.1.1.3.1.cmml" xref="A3.2.p2.15.m15.1.1.3">superscript</csymbol><ci id="A3.2.p2.15.m15.1.1.3.2.cmml" xref="A3.2.p2.15.m15.1.1.3.2">ℝ</ci><apply id="A3.2.p2.15.m15.1.1.3.3.cmml" xref="A3.2.p2.15.m15.1.1.3.3"><times id="A3.2.p2.15.m15.1.1.3.3.1.cmml" xref="A3.2.p2.15.m15.1.1.3.3.1"></times><apply id="A3.2.p2.15.m15.1.1.3.3.2.cmml" xref="A3.2.p2.15.m15.1.1.3.3.2"><csymbol cd="ambiguous" id="A3.2.p2.15.m15.1.1.3.3.2.1.cmml" xref="A3.2.p2.15.m15.1.1.3.3.2">subscript</csymbol><ci id="A3.2.p2.15.m15.1.1.3.3.2.2.cmml" xref="A3.2.p2.15.m15.1.1.3.3.2.2">𝑁</ci><ci id="A3.2.p2.15.m15.1.1.3.3.2.3.cmml" xref="A3.2.p2.15.m15.1.1.3.3.2.3">𝜁</ci></apply><ci id="A3.2.p2.15.m15.1.1.3.3.3.cmml" xref="A3.2.p2.15.m15.1.1.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.15.m15.1c">\in\mathbb{R}^{N_{\zeta}\times n}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.15.m15.1d">∈ blackboard_R start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT × italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> be an active sub-regressor of <math alttext="\Phi" class="ltx_Math" display="inline" id="A3.2.p2.16.m16.1"><semantics id="A3.2.p2.16.m16.1a"><mi id="A3.2.p2.16.m16.1.1" mathvariant="normal" xref="A3.2.p2.16.m16.1.1.cmml">Φ</mi><annotation-xml encoding="MathML-Content" id="A3.2.p2.16.m16.1b"><ci id="A3.2.p2.16.m16.1.1.cmml" xref="A3.2.p2.16.m16.1.1">Φ</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.16.m16.1c">\Phi</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.16.m16.1d">roman_Φ</annotation></semantics></math> with <math alttext="\bm{\phi}_{j}" class="ltx_Math" display="inline" id="A3.2.p2.17.m17.1"><semantics id="A3.2.p2.17.m17.1a"><msub id="A3.2.p2.17.m17.1.1" xref="A3.2.p2.17.m17.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="A3.2.p2.17.m17.1.1.2" mathvariant="bold-italic" xref="A3.2.p2.17.m17.1.1.2.cmml">ϕ</mi><mi id="A3.2.p2.17.m17.1.1.3" xref="A3.2.p2.17.m17.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="A3.2.p2.17.m17.1b"><apply id="A3.2.p2.17.m17.1.1.cmml" xref="A3.2.p2.17.m17.1.1"><csymbol cd="ambiguous" id="A3.2.p2.17.m17.1.1.1.cmml" xref="A3.2.p2.17.m17.1.1">subscript</csymbol><ci id="A3.2.p2.17.m17.1.1.2.cmml" xref="A3.2.p2.17.m17.1.1.2">bold-italic-ϕ</ci><ci id="A3.2.p2.17.m17.1.1.3.cmml" xref="A3.2.p2.17.m17.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.17.m17.1c">\bm{\phi}_{j}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.17.m17.1d">bold_italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> being the <math alttext="j" class="ltx_Math" display="inline" id="A3.2.p2.18.m18.1"><semantics id="A3.2.p2.18.m18.1a"><mi id="A3.2.p2.18.m18.1.1" xref="A3.2.p2.18.m18.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="A3.2.p2.18.m18.1b"><ci id="A3.2.p2.18.m18.1.1.cmml" xref="A3.2.p2.18.m18.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.18.m18.1c">j</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.18.m18.1d">italic_j</annotation></semantics></math>th column of <math alttext="\Phi^{T}" class="ltx_Math" display="inline" id="A3.2.p2.19.m19.1"><semantics id="A3.2.p2.19.m19.1a"><msup id="A3.2.p2.19.m19.1.1" xref="A3.2.p2.19.m19.1.1.cmml"><mi id="A3.2.p2.19.m19.1.1.2" mathvariant="normal" xref="A3.2.p2.19.m19.1.1.2.cmml">Φ</mi><mi id="A3.2.p2.19.m19.1.1.3" xref="A3.2.p2.19.m19.1.1.3.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="A3.2.p2.19.m19.1b"><apply id="A3.2.p2.19.m19.1.1.cmml" xref="A3.2.p2.19.m19.1.1"><csymbol cd="ambiguous" id="A3.2.p2.19.m19.1.1.1.cmml" xref="A3.2.p2.19.m19.1.1">superscript</csymbol><ci id="A3.2.p2.19.m19.1.1.2.cmml" xref="A3.2.p2.19.m19.1.1.2">Φ</ci><ci id="A3.2.p2.19.m19.1.1.3.cmml" xref="A3.2.p2.19.m19.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.19.m19.1c">\Phi^{T}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.19.m19.1d">roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\Phi_{{\rm f},\zeta}" class="ltx_Math" display="inline" id="A3.2.p2.20.m20.2"><semantics id="A3.2.p2.20.m20.2a"><msub id="A3.2.p2.20.m20.2.3" xref="A3.2.p2.20.m20.2.3.cmml"><mi id="A3.2.p2.20.m20.2.3.2" mathvariant="normal" xref="A3.2.p2.20.m20.2.3.2.cmml">Φ</mi><mrow id="A3.2.p2.20.m20.2.2.2.4" xref="A3.2.p2.20.m20.2.2.2.3.cmml"><mi id="A3.2.p2.20.m20.1.1.1.1" mathvariant="normal" xref="A3.2.p2.20.m20.1.1.1.1.cmml">f</mi><mo id="A3.2.p2.20.m20.2.2.2.4.1" xref="A3.2.p2.20.m20.2.2.2.3.cmml">,</mo><mi id="A3.2.p2.20.m20.2.2.2.2" xref="A3.2.p2.20.m20.2.2.2.2.cmml">ζ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A3.2.p2.20.m20.2b"><apply id="A3.2.p2.20.m20.2.3.cmml" xref="A3.2.p2.20.m20.2.3"><csymbol cd="ambiguous" id="A3.2.p2.20.m20.2.3.1.cmml" xref="A3.2.p2.20.m20.2.3">subscript</csymbol><ci id="A3.2.p2.20.m20.2.3.2.cmml" xref="A3.2.p2.20.m20.2.3.2">Φ</ci><list id="A3.2.p2.20.m20.2.2.2.3.cmml" xref="A3.2.p2.20.m20.2.2.2.4"><ci id="A3.2.p2.20.m20.1.1.1.1.cmml" xref="A3.2.p2.20.m20.1.1.1.1">f</ci><ci id="A3.2.p2.20.m20.2.2.2.2.cmml" xref="A3.2.p2.20.m20.2.2.2.2">𝜁</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.20.m20.2c">\Phi_{{\rm f},\zeta}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.20.m20.2d">roman_Φ start_POSTSUBSCRIPT roman_f , italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="A3.2.p2.21.m21.1"><semantics id="A3.2.p2.21.m21.1a"><mo id="A3.2.p2.21.m21.1.1" xref="A3.2.p2.21.m21.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="A3.2.p2.21.m21.1b"><csymbol cd="latexml" id="A3.2.p2.21.m21.1.1.cmml" xref="A3.2.p2.21.m21.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.21.m21.1c">:=</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.21.m21.1d">:=</annotation></semantics></math> <math alttext="[\bm{\phi}_{{\rm f},1},\bm{\phi}_{{\rm f},2},\cdots,\bm{\phi}_{{\rm f},{N_{% \zeta}}}]^{T}" class="ltx_Math" display="inline" id="A3.2.p2.22.m22.10"><semantics id="A3.2.p2.22.m22.10a"><msup id="A3.2.p2.22.m22.10.10" xref="A3.2.p2.22.m22.10.10.cmml"><mrow id="A3.2.p2.22.m22.10.10.3.3" xref="A3.2.p2.22.m22.10.10.3.4.cmml"><mo id="A3.2.p2.22.m22.10.10.3.3.4" stretchy="false" xref="A3.2.p2.22.m22.10.10.3.4.cmml">[</mo><msub id="A3.2.p2.22.m22.8.8.1.1.1" xref="A3.2.p2.22.m22.8.8.1.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="A3.2.p2.22.m22.8.8.1.1.1.2" mathvariant="bold-italic" xref="A3.2.p2.22.m22.8.8.1.1.1.2.cmml">ϕ</mi><mrow id="A3.2.p2.22.m22.2.2.2.4" xref="A3.2.p2.22.m22.2.2.2.3.cmml"><mi id="A3.2.p2.22.m22.1.1.1.1" mathvariant="normal" xref="A3.2.p2.22.m22.1.1.1.1.cmml">f</mi><mo id="A3.2.p2.22.m22.2.2.2.4.1" xref="A3.2.p2.22.m22.2.2.2.3.cmml">,</mo><mn id="A3.2.p2.22.m22.2.2.2.2" xref="A3.2.p2.22.m22.2.2.2.2.cmml">1</mn></mrow></msub><mo id="A3.2.p2.22.m22.10.10.3.3.5" xref="A3.2.p2.22.m22.10.10.3.4.cmml">,</mo><msub id="A3.2.p2.22.m22.9.9.2.2.2" xref="A3.2.p2.22.m22.9.9.2.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="A3.2.p2.22.m22.9.9.2.2.2.2" mathvariant="bold-italic" xref="A3.2.p2.22.m22.9.9.2.2.2.2.cmml">ϕ</mi><mrow id="A3.2.p2.22.m22.4.4.2.4" xref="A3.2.p2.22.m22.4.4.2.3.cmml"><mi id="A3.2.p2.22.m22.3.3.1.1" mathvariant="normal" xref="A3.2.p2.22.m22.3.3.1.1.cmml">f</mi><mo id="A3.2.p2.22.m22.4.4.2.4.1" xref="A3.2.p2.22.m22.4.4.2.3.cmml">,</mo><mn id="A3.2.p2.22.m22.4.4.2.2" xref="A3.2.p2.22.m22.4.4.2.2.cmml">2</mn></mrow></msub><mo id="A3.2.p2.22.m22.10.10.3.3.6" xref="A3.2.p2.22.m22.10.10.3.4.cmml">,</mo><mi id="A3.2.p2.22.m22.7.7" mathvariant="normal" xref="A3.2.p2.22.m22.7.7.cmml">⋯</mi><mo id="A3.2.p2.22.m22.10.10.3.3.7" xref="A3.2.p2.22.m22.10.10.3.4.cmml">,</mo><msub id="A3.2.p2.22.m22.10.10.3.3.3" xref="A3.2.p2.22.m22.10.10.3.3.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="A3.2.p2.22.m22.10.10.3.3.3.2" mathvariant="bold-italic" xref="A3.2.p2.22.m22.10.10.3.3.3.2.cmml">ϕ</mi><mrow id="A3.2.p2.22.m22.6.6.2.2" xref="A3.2.p2.22.m22.6.6.2.3.cmml"><mi id="A3.2.p2.22.m22.5.5.1.1" mathvariant="normal" xref="A3.2.p2.22.m22.5.5.1.1.cmml">f</mi><mo id="A3.2.p2.22.m22.6.6.2.2.2" xref="A3.2.p2.22.m22.6.6.2.3.cmml">,</mo><msub id="A3.2.p2.22.m22.6.6.2.2.1" xref="A3.2.p2.22.m22.6.6.2.2.1.cmml"><mi id="A3.2.p2.22.m22.6.6.2.2.1.2" xref="A3.2.p2.22.m22.6.6.2.2.1.2.cmml">N</mi><mi id="A3.2.p2.22.m22.6.6.2.2.1.3" xref="A3.2.p2.22.m22.6.6.2.2.1.3.cmml">ζ</mi></msub></mrow></msub><mo id="A3.2.p2.22.m22.10.10.3.3.8" stretchy="false" xref="A3.2.p2.22.m22.10.10.3.4.cmml">]</mo></mrow><mi id="A3.2.p2.22.m22.10.10.5" xref="A3.2.p2.22.m22.10.10.5.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="A3.2.p2.22.m22.10b"><apply id="A3.2.p2.22.m22.10.10.cmml" xref="A3.2.p2.22.m22.10.10"><csymbol cd="ambiguous" id="A3.2.p2.22.m22.10.10.4.cmml" xref="A3.2.p2.22.m22.10.10">superscript</csymbol><list id="A3.2.p2.22.m22.10.10.3.4.cmml" xref="A3.2.p2.22.m22.10.10.3.3"><apply id="A3.2.p2.22.m22.8.8.1.1.1.cmml" xref="A3.2.p2.22.m22.8.8.1.1.1"><csymbol cd="ambiguous" id="A3.2.p2.22.m22.8.8.1.1.1.1.cmml" xref="A3.2.p2.22.m22.8.8.1.1.1">subscript</csymbol><ci id="A3.2.p2.22.m22.8.8.1.1.1.2.cmml" xref="A3.2.p2.22.m22.8.8.1.1.1.2">bold-italic-ϕ</ci><list id="A3.2.p2.22.m22.2.2.2.3.cmml" xref="A3.2.p2.22.m22.2.2.2.4"><ci id="A3.2.p2.22.m22.1.1.1.1.cmml" xref="A3.2.p2.22.m22.1.1.1.1">f</ci><cn id="A3.2.p2.22.m22.2.2.2.2.cmml" type="integer" xref="A3.2.p2.22.m22.2.2.2.2">1</cn></list></apply><apply id="A3.2.p2.22.m22.9.9.2.2.2.cmml" xref="A3.2.p2.22.m22.9.9.2.2.2"><csymbol cd="ambiguous" id="A3.2.p2.22.m22.9.9.2.2.2.1.cmml" xref="A3.2.p2.22.m22.9.9.2.2.2">subscript</csymbol><ci id="A3.2.p2.22.m22.9.9.2.2.2.2.cmml" xref="A3.2.p2.22.m22.9.9.2.2.2.2">bold-italic-ϕ</ci><list id="A3.2.p2.22.m22.4.4.2.3.cmml" xref="A3.2.p2.22.m22.4.4.2.4"><ci id="A3.2.p2.22.m22.3.3.1.1.cmml" xref="A3.2.p2.22.m22.3.3.1.1">f</ci><cn id="A3.2.p2.22.m22.4.4.2.2.cmml" type="integer" xref="A3.2.p2.22.m22.4.4.2.2">2</cn></list></apply><ci id="A3.2.p2.22.m22.7.7.cmml" xref="A3.2.p2.22.m22.7.7">⋯</ci><apply id="A3.2.p2.22.m22.10.10.3.3.3.cmml" xref="A3.2.p2.22.m22.10.10.3.3.3"><csymbol cd="ambiguous" id="A3.2.p2.22.m22.10.10.3.3.3.1.cmml" xref="A3.2.p2.22.m22.10.10.3.3.3">subscript</csymbol><ci id="A3.2.p2.22.m22.10.10.3.3.3.2.cmml" xref="A3.2.p2.22.m22.10.10.3.3.3.2">bold-italic-ϕ</ci><list id="A3.2.p2.22.m22.6.6.2.3.cmml" xref="A3.2.p2.22.m22.6.6.2.2"><ci id="A3.2.p2.22.m22.5.5.1.1.cmml" xref="A3.2.p2.22.m22.5.5.1.1">f</ci><apply id="A3.2.p2.22.m22.6.6.2.2.1.cmml" xref="A3.2.p2.22.m22.6.6.2.2.1"><csymbol cd="ambiguous" id="A3.2.p2.22.m22.6.6.2.2.1.1.cmml" xref="A3.2.p2.22.m22.6.6.2.2.1">subscript</csymbol><ci id="A3.2.p2.22.m22.6.6.2.2.1.2.cmml" xref="A3.2.p2.22.m22.6.6.2.2.1.2">𝑁</ci><ci id="A3.2.p2.22.m22.6.6.2.2.1.3.cmml" xref="A3.2.p2.22.m22.6.6.2.2.1.3">𝜁</ci></apply></list></apply></list><ci id="A3.2.p2.22.m22.10.10.5.cmml" xref="A3.2.p2.22.m22.10.10.5">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.22.m22.10c">[\bm{\phi}_{{\rm f},1},\bm{\phi}_{{\rm f},2},\cdots,\bm{\phi}_{{\rm f},{N_{% \zeta}}}]^{T}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.22.m22.10d">[ bold_italic_ϕ start_POSTSUBSCRIPT roman_f , 1 end_POSTSUBSCRIPT , bold_italic_ϕ start_POSTSUBSCRIPT roman_f , 2 end_POSTSUBSCRIPT , ⋯ , bold_italic_ϕ start_POSTSUBSCRIPT roman_f , italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> <math alttext="\in\mathbb{R}^{N_{\zeta}\times n}" class="ltx_Math" display="inline" id="A3.2.p2.23.m23.1"><semantics id="A3.2.p2.23.m23.1a"><mrow id="A3.2.p2.23.m23.1.1" xref="A3.2.p2.23.m23.1.1.cmml"><mi id="A3.2.p2.23.m23.1.1.2" xref="A3.2.p2.23.m23.1.1.2.cmml"></mi><mo id="A3.2.p2.23.m23.1.1.1" xref="A3.2.p2.23.m23.1.1.1.cmml">∈</mo><msup id="A3.2.p2.23.m23.1.1.3" xref="A3.2.p2.23.m23.1.1.3.cmml"><mi id="A3.2.p2.23.m23.1.1.3.2" xref="A3.2.p2.23.m23.1.1.3.2.cmml">ℝ</mi><mrow id="A3.2.p2.23.m23.1.1.3.3" xref="A3.2.p2.23.m23.1.1.3.3.cmml"><msub id="A3.2.p2.23.m23.1.1.3.3.2" xref="A3.2.p2.23.m23.1.1.3.3.2.cmml"><mi id="A3.2.p2.23.m23.1.1.3.3.2.2" xref="A3.2.p2.23.m23.1.1.3.3.2.2.cmml">N</mi><mi id="A3.2.p2.23.m23.1.1.3.3.2.3" xref="A3.2.p2.23.m23.1.1.3.3.2.3.cmml">ζ</mi></msub><mo id="A3.2.p2.23.m23.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="A3.2.p2.23.m23.1.1.3.3.1.cmml">×</mo><mi id="A3.2.p2.23.m23.1.1.3.3.3" xref="A3.2.p2.23.m23.1.1.3.3.3.cmml">n</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.23.m23.1b"><apply id="A3.2.p2.23.m23.1.1.cmml" xref="A3.2.p2.23.m23.1.1"><in id="A3.2.p2.23.m23.1.1.1.cmml" xref="A3.2.p2.23.m23.1.1.1"></in><csymbol cd="latexml" id="A3.2.p2.23.m23.1.1.2.cmml" xref="A3.2.p2.23.m23.1.1.2">absent</csymbol><apply id="A3.2.p2.23.m23.1.1.3.cmml" xref="A3.2.p2.23.m23.1.1.3"><csymbol cd="ambiguous" id="A3.2.p2.23.m23.1.1.3.1.cmml" xref="A3.2.p2.23.m23.1.1.3">superscript</csymbol><ci id="A3.2.p2.23.m23.1.1.3.2.cmml" xref="A3.2.p2.23.m23.1.1.3.2">ℝ</ci><apply id="A3.2.p2.23.m23.1.1.3.3.cmml" xref="A3.2.p2.23.m23.1.1.3.3"><times id="A3.2.p2.23.m23.1.1.3.3.1.cmml" xref="A3.2.p2.23.m23.1.1.3.3.1"></times><apply id="A3.2.p2.23.m23.1.1.3.3.2.cmml" xref="A3.2.p2.23.m23.1.1.3.3.2"><csymbol cd="ambiguous" id="A3.2.p2.23.m23.1.1.3.3.2.1.cmml" xref="A3.2.p2.23.m23.1.1.3.3.2">subscript</csymbol><ci id="A3.2.p2.23.m23.1.1.3.3.2.2.cmml" xref="A3.2.p2.23.m23.1.1.3.3.2.2">𝑁</ci><ci id="A3.2.p2.23.m23.1.1.3.3.2.3.cmml" xref="A3.2.p2.23.m23.1.1.3.3.2.3">𝜁</ci></apply><ci id="A3.2.p2.23.m23.1.1.3.3.3.cmml" xref="A3.2.p2.23.m23.1.1.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.23.m23.1c">\in\mathbb{R}^{N_{\zeta}\times n}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.23.m23.1d">∈ blackboard_R start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT × italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> be an active sub-regressor of <math alttext="\Phi_{\rm f}" class="ltx_Math" display="inline" id="A3.2.p2.24.m24.1"><semantics id="A3.2.p2.24.m24.1a"><msub id="A3.2.p2.24.m24.1.1" xref="A3.2.p2.24.m24.1.1.cmml"><mi id="A3.2.p2.24.m24.1.1.2" mathvariant="normal" xref="A3.2.p2.24.m24.1.1.2.cmml">Φ</mi><mi id="A3.2.p2.24.m24.1.1.3" mathvariant="normal" xref="A3.2.p2.24.m24.1.1.3.cmml">f</mi></msub><annotation-xml encoding="MathML-Content" id="A3.2.p2.24.m24.1b"><apply id="A3.2.p2.24.m24.1.1.cmml" xref="A3.2.p2.24.m24.1.1"><csymbol cd="ambiguous" id="A3.2.p2.24.m24.1.1.1.cmml" xref="A3.2.p2.24.m24.1.1">subscript</csymbol><ci id="A3.2.p2.24.m24.1.1.2.cmml" xref="A3.2.p2.24.m24.1.1.2">Φ</ci><ci id="A3.2.p2.24.m24.1.1.3.cmml" xref="A3.2.p2.24.m24.1.1.3">f</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.24.m24.1c">\Phi_{\rm f}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.24.m24.1d">roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\bm{\phi}_{{\rm f},j}" class="ltx_Math" display="inline" id="A3.2.p2.25.m25.2"><semantics id="A3.2.p2.25.m25.2a"><msub id="A3.2.p2.25.m25.2.3" xref="A3.2.p2.25.m25.2.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="A3.2.p2.25.m25.2.3.2" mathvariant="bold-italic" xref="A3.2.p2.25.m25.2.3.2.cmml">ϕ</mi><mrow id="A3.2.p2.25.m25.2.2.2.4" xref="A3.2.p2.25.m25.2.2.2.3.cmml"><mi id="A3.2.p2.25.m25.1.1.1.1" mathvariant="normal" xref="A3.2.p2.25.m25.1.1.1.1.cmml">f</mi><mo id="A3.2.p2.25.m25.2.2.2.4.1" xref="A3.2.p2.25.m25.2.2.2.3.cmml">,</mo><mi id="A3.2.p2.25.m25.2.2.2.2" xref="A3.2.p2.25.m25.2.2.2.2.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A3.2.p2.25.m25.2b"><apply id="A3.2.p2.25.m25.2.3.cmml" xref="A3.2.p2.25.m25.2.3"><csymbol cd="ambiguous" id="A3.2.p2.25.m25.2.3.1.cmml" xref="A3.2.p2.25.m25.2.3">subscript</csymbol><ci id="A3.2.p2.25.m25.2.3.2.cmml" xref="A3.2.p2.25.m25.2.3.2">bold-italic-ϕ</ci><list id="A3.2.p2.25.m25.2.2.2.3.cmml" xref="A3.2.p2.25.m25.2.2.2.4"><ci id="A3.2.p2.25.m25.1.1.1.1.cmml" xref="A3.2.p2.25.m25.1.1.1.1">f</ci><ci id="A3.2.p2.25.m25.2.2.2.2.cmml" xref="A3.2.p2.25.m25.2.2.2.2">𝑗</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.25.m25.2c">\bm{\phi}_{{\rm f},j}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.25.m25.2d">bold_italic_ϕ start_POSTSUBSCRIPT roman_f , italic_j end_POSTSUBSCRIPT</annotation></semantics></math> being the <math alttext="j" class="ltx_Math" display="inline" id="A3.2.p2.26.m26.1"><semantics id="A3.2.p2.26.m26.1a"><mi id="A3.2.p2.26.m26.1.1" xref="A3.2.p2.26.m26.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="A3.2.p2.26.m26.1b"><ci id="A3.2.p2.26.m26.1.1.cmml" xref="A3.2.p2.26.m26.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.26.m26.1c">j</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.26.m26.1d">italic_j</annotation></semantics></math>th column of <math alttext="\Phi_{\rm f}^{T}" class="ltx_Math" display="inline" id="A3.2.p2.27.m27.1"><semantics id="A3.2.p2.27.m27.1a"><msubsup id="A3.2.p2.27.m27.1.1" xref="A3.2.p2.27.m27.1.1.cmml"><mi id="A3.2.p2.27.m27.1.1.2.2" mathvariant="normal" xref="A3.2.p2.27.m27.1.1.2.2.cmml">Φ</mi><mi id="A3.2.p2.27.m27.1.1.2.3" mathvariant="normal" xref="A3.2.p2.27.m27.1.1.2.3.cmml">f</mi><mi id="A3.2.p2.27.m27.1.1.3" xref="A3.2.p2.27.m27.1.1.3.cmml">T</mi></msubsup><annotation-xml encoding="MathML-Content" id="A3.2.p2.27.m27.1b"><apply id="A3.2.p2.27.m27.1.1.cmml" xref="A3.2.p2.27.m27.1.1"><csymbol cd="ambiguous" id="A3.2.p2.27.m27.1.1.1.cmml" xref="A3.2.p2.27.m27.1.1">superscript</csymbol><apply id="A3.2.p2.27.m27.1.1.2.cmml" xref="A3.2.p2.27.m27.1.1"><csymbol cd="ambiguous" id="A3.2.p2.27.m27.1.1.2.1.cmml" xref="A3.2.p2.27.m27.1.1">subscript</csymbol><ci id="A3.2.p2.27.m27.1.1.2.2.cmml" xref="A3.2.p2.27.m27.1.1.2.2">Φ</ci><ci id="A3.2.p2.27.m27.1.1.2.3.cmml" xref="A3.2.p2.27.m27.1.1.2.3">f</ci></apply><ci id="A3.2.p2.27.m27.1.1.3.cmml" xref="A3.2.p2.27.m27.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.27.m27.1c">\Phi_{\rm f}^{T}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.27.m27.1d">roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="\tilde{\bm{\theta}}_{\zeta}" class="ltx_Math" display="inline" id="A3.2.p2.28.m28.1"><semantics id="A3.2.p2.28.m28.1a"><msub id="A3.2.p2.28.m28.1.1" xref="A3.2.p2.28.m28.1.1.cmml"><mover accent="true" id="A3.2.p2.28.m28.1.1.2" xref="A3.2.p2.28.m28.1.1.2.cmml"><mi id="A3.2.p2.28.m28.1.1.2.2" xref="A3.2.p2.28.m28.1.1.2.2.cmml">𝜽</mi><mo id="A3.2.p2.28.m28.1.1.2.1" xref="A3.2.p2.28.m28.1.1.2.1.cmml">~</mo></mover><mi id="A3.2.p2.28.m28.1.1.3" xref="A3.2.p2.28.m28.1.1.3.cmml">ζ</mi></msub><annotation-xml encoding="MathML-Content" id="A3.2.p2.28.m28.1b"><apply id="A3.2.p2.28.m28.1.1.cmml" xref="A3.2.p2.28.m28.1.1"><csymbol cd="ambiguous" id="A3.2.p2.28.m28.1.1.1.cmml" xref="A3.2.p2.28.m28.1.1">subscript</csymbol><apply id="A3.2.p2.28.m28.1.1.2.cmml" xref="A3.2.p2.28.m28.1.1.2"><ci id="A3.2.p2.28.m28.1.1.2.1.cmml" xref="A3.2.p2.28.m28.1.1.2.1">~</ci><ci id="A3.2.p2.28.m28.1.1.2.2.cmml" xref="A3.2.p2.28.m28.1.1.2.2">𝜽</ci></apply><ci id="A3.2.p2.28.m28.1.1.3.cmml" xref="A3.2.p2.28.m28.1.1.3">𝜁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.28.m28.1c">\tilde{\bm{\theta}}_{\zeta}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.28.m28.1d">over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="A3.2.p2.29.m29.1"><semantics id="A3.2.p2.29.m29.1a"><mo id="A3.2.p2.29.m29.1.1" xref="A3.2.p2.29.m29.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="A3.2.p2.29.m29.1b"><csymbol cd="latexml" id="A3.2.p2.29.m29.1.1.cmml" xref="A3.2.p2.29.m29.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.29.m29.1c">:=</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.29.m29.1d">:=</annotation></semantics></math> <math alttext="[\tilde{\theta}_{1},\tilde{\theta}_{2},\cdots,\tilde{\theta}_{N_{\zeta}}]^{T}% \in\mathbb{R}^{N_{\zeta}}" class="ltx_Math" display="inline" id="A3.2.p2.30.m30.4"><semantics id="A3.2.p2.30.m30.4a"><mrow id="A3.2.p2.30.m30.4.4" xref="A3.2.p2.30.m30.4.4.cmml"><msup id="A3.2.p2.30.m30.4.4.3" xref="A3.2.p2.30.m30.4.4.3.cmml"><mrow id="A3.2.p2.30.m30.4.4.3.3.3" xref="A3.2.p2.30.m30.4.4.3.3.4.cmml"><mo id="A3.2.p2.30.m30.4.4.3.3.3.4" stretchy="false" xref="A3.2.p2.30.m30.4.4.3.3.4.cmml">[</mo><msub id="A3.2.p2.30.m30.2.2.1.1.1.1" xref="A3.2.p2.30.m30.2.2.1.1.1.1.cmml"><mover accent="true" id="A3.2.p2.30.m30.2.2.1.1.1.1.2" xref="A3.2.p2.30.m30.2.2.1.1.1.1.2.cmml"><mi id="A3.2.p2.30.m30.2.2.1.1.1.1.2.2" xref="A3.2.p2.30.m30.2.2.1.1.1.1.2.2.cmml">θ</mi><mo id="A3.2.p2.30.m30.2.2.1.1.1.1.2.1" xref="A3.2.p2.30.m30.2.2.1.1.1.1.2.1.cmml">~</mo></mover><mn id="A3.2.p2.30.m30.2.2.1.1.1.1.3" xref="A3.2.p2.30.m30.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="A3.2.p2.30.m30.4.4.3.3.3.5" xref="A3.2.p2.30.m30.4.4.3.3.4.cmml">,</mo><msub id="A3.2.p2.30.m30.3.3.2.2.2.2" xref="A3.2.p2.30.m30.3.3.2.2.2.2.cmml"><mover accent="true" id="A3.2.p2.30.m30.3.3.2.2.2.2.2" xref="A3.2.p2.30.m30.3.3.2.2.2.2.2.cmml"><mi id="A3.2.p2.30.m30.3.3.2.2.2.2.2.2" xref="A3.2.p2.30.m30.3.3.2.2.2.2.2.2.cmml">θ</mi><mo id="A3.2.p2.30.m30.3.3.2.2.2.2.2.1" xref="A3.2.p2.30.m30.3.3.2.2.2.2.2.1.cmml">~</mo></mover><mn id="A3.2.p2.30.m30.3.3.2.2.2.2.3" xref="A3.2.p2.30.m30.3.3.2.2.2.2.3.cmml">2</mn></msub><mo id="A3.2.p2.30.m30.4.4.3.3.3.6" xref="A3.2.p2.30.m30.4.4.3.3.4.cmml">,</mo><mi id="A3.2.p2.30.m30.1.1" mathvariant="normal" xref="A3.2.p2.30.m30.1.1.cmml">⋯</mi><mo id="A3.2.p2.30.m30.4.4.3.3.3.7" xref="A3.2.p2.30.m30.4.4.3.3.4.cmml">,</mo><msub id="A3.2.p2.30.m30.4.4.3.3.3.3" xref="A3.2.p2.30.m30.4.4.3.3.3.3.cmml"><mover accent="true" id="A3.2.p2.30.m30.4.4.3.3.3.3.2" xref="A3.2.p2.30.m30.4.4.3.3.3.3.2.cmml"><mi id="A3.2.p2.30.m30.4.4.3.3.3.3.2.2" xref="A3.2.p2.30.m30.4.4.3.3.3.3.2.2.cmml">θ</mi><mo id="A3.2.p2.30.m30.4.4.3.3.3.3.2.1" xref="A3.2.p2.30.m30.4.4.3.3.3.3.2.1.cmml">~</mo></mover><msub id="A3.2.p2.30.m30.4.4.3.3.3.3.3" xref="A3.2.p2.30.m30.4.4.3.3.3.3.3.cmml"><mi id="A3.2.p2.30.m30.4.4.3.3.3.3.3.2" xref="A3.2.p2.30.m30.4.4.3.3.3.3.3.2.cmml">N</mi><mi id="A3.2.p2.30.m30.4.4.3.3.3.3.3.3" xref="A3.2.p2.30.m30.4.4.3.3.3.3.3.3.cmml">ζ</mi></msub></msub><mo id="A3.2.p2.30.m30.4.4.3.3.3.8" stretchy="false" xref="A3.2.p2.30.m30.4.4.3.3.4.cmml">]</mo></mrow><mi id="A3.2.p2.30.m30.4.4.3.5" xref="A3.2.p2.30.m30.4.4.3.5.cmml">T</mi></msup><mo id="A3.2.p2.30.m30.4.4.4" xref="A3.2.p2.30.m30.4.4.4.cmml">∈</mo><msup id="A3.2.p2.30.m30.4.4.5" xref="A3.2.p2.30.m30.4.4.5.cmml"><mi id="A3.2.p2.30.m30.4.4.5.2" xref="A3.2.p2.30.m30.4.4.5.2.cmml">ℝ</mi><msub id="A3.2.p2.30.m30.4.4.5.3" xref="A3.2.p2.30.m30.4.4.5.3.cmml"><mi id="A3.2.p2.30.m30.4.4.5.3.2" xref="A3.2.p2.30.m30.4.4.5.3.2.cmml">N</mi><mi id="A3.2.p2.30.m30.4.4.5.3.3" xref="A3.2.p2.30.m30.4.4.5.3.3.cmml">ζ</mi></msub></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.30.m30.4b"><apply id="A3.2.p2.30.m30.4.4.cmml" xref="A3.2.p2.30.m30.4.4"><in id="A3.2.p2.30.m30.4.4.4.cmml" xref="A3.2.p2.30.m30.4.4.4"></in><apply id="A3.2.p2.30.m30.4.4.3.cmml" xref="A3.2.p2.30.m30.4.4.3"><csymbol cd="ambiguous" id="A3.2.p2.30.m30.4.4.3.4.cmml" xref="A3.2.p2.30.m30.4.4.3">superscript</csymbol><list id="A3.2.p2.30.m30.4.4.3.3.4.cmml" xref="A3.2.p2.30.m30.4.4.3.3.3"><apply id="A3.2.p2.30.m30.2.2.1.1.1.1.cmml" xref="A3.2.p2.30.m30.2.2.1.1.1.1"><csymbol cd="ambiguous" id="A3.2.p2.30.m30.2.2.1.1.1.1.1.cmml" xref="A3.2.p2.30.m30.2.2.1.1.1.1">subscript</csymbol><apply id="A3.2.p2.30.m30.2.2.1.1.1.1.2.cmml" xref="A3.2.p2.30.m30.2.2.1.1.1.1.2"><ci id="A3.2.p2.30.m30.2.2.1.1.1.1.2.1.cmml" xref="A3.2.p2.30.m30.2.2.1.1.1.1.2.1">~</ci><ci id="A3.2.p2.30.m30.2.2.1.1.1.1.2.2.cmml" xref="A3.2.p2.30.m30.2.2.1.1.1.1.2.2">𝜃</ci></apply><cn id="A3.2.p2.30.m30.2.2.1.1.1.1.3.cmml" type="integer" xref="A3.2.p2.30.m30.2.2.1.1.1.1.3">1</cn></apply><apply id="A3.2.p2.30.m30.3.3.2.2.2.2.cmml" xref="A3.2.p2.30.m30.3.3.2.2.2.2"><csymbol cd="ambiguous" id="A3.2.p2.30.m30.3.3.2.2.2.2.1.cmml" xref="A3.2.p2.30.m30.3.3.2.2.2.2">subscript</csymbol><apply id="A3.2.p2.30.m30.3.3.2.2.2.2.2.cmml" xref="A3.2.p2.30.m30.3.3.2.2.2.2.2"><ci id="A3.2.p2.30.m30.3.3.2.2.2.2.2.1.cmml" xref="A3.2.p2.30.m30.3.3.2.2.2.2.2.1">~</ci><ci id="A3.2.p2.30.m30.3.3.2.2.2.2.2.2.cmml" xref="A3.2.p2.30.m30.3.3.2.2.2.2.2.2">𝜃</ci></apply><cn id="A3.2.p2.30.m30.3.3.2.2.2.2.3.cmml" type="integer" xref="A3.2.p2.30.m30.3.3.2.2.2.2.3">2</cn></apply><ci id="A3.2.p2.30.m30.1.1.cmml" xref="A3.2.p2.30.m30.1.1">⋯</ci><apply id="A3.2.p2.30.m30.4.4.3.3.3.3.cmml" xref="A3.2.p2.30.m30.4.4.3.3.3.3"><csymbol cd="ambiguous" id="A3.2.p2.30.m30.4.4.3.3.3.3.1.cmml" xref="A3.2.p2.30.m30.4.4.3.3.3.3">subscript</csymbol><apply id="A3.2.p2.30.m30.4.4.3.3.3.3.2.cmml" xref="A3.2.p2.30.m30.4.4.3.3.3.3.2"><ci id="A3.2.p2.30.m30.4.4.3.3.3.3.2.1.cmml" xref="A3.2.p2.30.m30.4.4.3.3.3.3.2.1">~</ci><ci id="A3.2.p2.30.m30.4.4.3.3.3.3.2.2.cmml" xref="A3.2.p2.30.m30.4.4.3.3.3.3.2.2">𝜃</ci></apply><apply id="A3.2.p2.30.m30.4.4.3.3.3.3.3.cmml" xref="A3.2.p2.30.m30.4.4.3.3.3.3.3"><csymbol cd="ambiguous" id="A3.2.p2.30.m30.4.4.3.3.3.3.3.1.cmml" xref="A3.2.p2.30.m30.4.4.3.3.3.3.3">subscript</csymbol><ci id="A3.2.p2.30.m30.4.4.3.3.3.3.3.2.cmml" xref="A3.2.p2.30.m30.4.4.3.3.3.3.3.2">𝑁</ci><ci id="A3.2.p2.30.m30.4.4.3.3.3.3.3.3.cmml" xref="A3.2.p2.30.m30.4.4.3.3.3.3.3.3">𝜁</ci></apply></apply></list><ci id="A3.2.p2.30.m30.4.4.3.5.cmml" xref="A3.2.p2.30.m30.4.4.3.5">𝑇</ci></apply><apply id="A3.2.p2.30.m30.4.4.5.cmml" xref="A3.2.p2.30.m30.4.4.5"><csymbol cd="ambiguous" id="A3.2.p2.30.m30.4.4.5.1.cmml" xref="A3.2.p2.30.m30.4.4.5">superscript</csymbol><ci id="A3.2.p2.30.m30.4.4.5.2.cmml" xref="A3.2.p2.30.m30.4.4.5.2">ℝ</ci><apply id="A3.2.p2.30.m30.4.4.5.3.cmml" xref="A3.2.p2.30.m30.4.4.5.3"><csymbol cd="ambiguous" id="A3.2.p2.30.m30.4.4.5.3.1.cmml" xref="A3.2.p2.30.m30.4.4.5.3">subscript</csymbol><ci id="A3.2.p2.30.m30.4.4.5.3.2.cmml" xref="A3.2.p2.30.m30.4.4.5.3.2">𝑁</ci><ci id="A3.2.p2.30.m30.4.4.5.3.3.cmml" xref="A3.2.p2.30.m30.4.4.5.3.3">𝜁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.30.m30.4c">[\tilde{\theta}_{1},\tilde{\theta}_{2},\cdots,\tilde{\theta}_{N_{\zeta}}]^{T}% \in\mathbb{R}^{N_{\zeta}}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.30.m30.4d">[ over~ start_ARG italic_θ end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , over~ start_ARG italic_θ end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , ⋯ , over~ start_ARG italic_θ end_ARG start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> be an estimation error corresponding to active channels. Thus, the regressors <math alttext="\Phi" class="ltx_Math" display="inline" id="A3.2.p2.31.m31.1"><semantics id="A3.2.p2.31.m31.1a"><mi id="A3.2.p2.31.m31.1.1" mathvariant="normal" xref="A3.2.p2.31.m31.1.1.cmml">Φ</mi><annotation-xml encoding="MathML-Content" id="A3.2.p2.31.m31.1b"><ci id="A3.2.p2.31.m31.1.1.cmml" xref="A3.2.p2.31.m31.1.1">Φ</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.31.m31.1c">\Phi</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.31.m31.1d">roman_Φ</annotation></semantics></math> and <math alttext="\Phi_{\rm f}" class="ltx_Math" display="inline" id="A3.2.p2.32.m32.1"><semantics id="A3.2.p2.32.m32.1a"><msub id="A3.2.p2.32.m32.1.1" xref="A3.2.p2.32.m32.1.1.cmml"><mi id="A3.2.p2.32.m32.1.1.2" mathvariant="normal" xref="A3.2.p2.32.m32.1.1.2.cmml">Φ</mi><mi id="A3.2.p2.32.m32.1.1.3" mathvariant="normal" xref="A3.2.p2.32.m32.1.1.3.cmml">f</mi></msub><annotation-xml encoding="MathML-Content" id="A3.2.p2.32.m32.1b"><apply id="A3.2.p2.32.m32.1.1.cmml" xref="A3.2.p2.32.m32.1.1"><csymbol cd="ambiguous" id="A3.2.p2.32.m32.1.1.1.cmml" xref="A3.2.p2.32.m32.1.1">subscript</csymbol><ci id="A3.2.p2.32.m32.1.1.2.cmml" xref="A3.2.p2.32.m32.1.1.2">Φ</ci><ci id="A3.2.p2.32.m32.1.1.3.cmml" xref="A3.2.p2.32.m32.1.1.3">f</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.32.m32.1c">\Phi_{\rm f}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.32.m32.1d">roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT</annotation></semantics></math> can be represented by sub-regressors <math alttext="\Phi_{\zeta}" class="ltx_Math" display="inline" id="A3.2.p2.33.m33.1"><semantics id="A3.2.p2.33.m33.1a"><msub id="A3.2.p2.33.m33.1.1" xref="A3.2.p2.33.m33.1.1.cmml"><mi id="A3.2.p2.33.m33.1.1.2" mathvariant="normal" xref="A3.2.p2.33.m33.1.1.2.cmml">Φ</mi><mi id="A3.2.p2.33.m33.1.1.3" xref="A3.2.p2.33.m33.1.1.3.cmml">ζ</mi></msub><annotation-xml encoding="MathML-Content" id="A3.2.p2.33.m33.1b"><apply id="A3.2.p2.33.m33.1.1.cmml" xref="A3.2.p2.33.m33.1.1"><csymbol cd="ambiguous" id="A3.2.p2.33.m33.1.1.1.cmml" xref="A3.2.p2.33.m33.1.1">subscript</csymbol><ci id="A3.2.p2.33.m33.1.1.2.cmml" xref="A3.2.p2.33.m33.1.1.2">Φ</ci><ci id="A3.2.p2.33.m33.1.1.3.cmml" xref="A3.2.p2.33.m33.1.1.3">𝜁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.33.m33.1c">\Phi_{\zeta}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.33.m33.1d">roman_Φ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\Phi_{{\rm f},\zeta}" class="ltx_Math" display="inline" id="A3.2.p2.34.m34.2"><semantics id="A3.2.p2.34.m34.2a"><msub id="A3.2.p2.34.m34.2.3" xref="A3.2.p2.34.m34.2.3.cmml"><mi id="A3.2.p2.34.m34.2.3.2" mathvariant="normal" xref="A3.2.p2.34.m34.2.3.2.cmml">Φ</mi><mrow id="A3.2.p2.34.m34.2.2.2.4" xref="A3.2.p2.34.m34.2.2.2.3.cmml"><mi id="A3.2.p2.34.m34.1.1.1.1" mathvariant="normal" xref="A3.2.p2.34.m34.1.1.1.1.cmml">f</mi><mo id="A3.2.p2.34.m34.2.2.2.4.1" xref="A3.2.p2.34.m34.2.2.2.3.cmml">,</mo><mi id="A3.2.p2.34.m34.2.2.2.2" xref="A3.2.p2.34.m34.2.2.2.2.cmml">ζ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A3.2.p2.34.m34.2b"><apply id="A3.2.p2.34.m34.2.3.cmml" xref="A3.2.p2.34.m34.2.3"><csymbol cd="ambiguous" id="A3.2.p2.34.m34.2.3.1.cmml" xref="A3.2.p2.34.m34.2.3">subscript</csymbol><ci id="A3.2.p2.34.m34.2.3.2.cmml" xref="A3.2.p2.34.m34.2.3.2">Φ</ci><list id="A3.2.p2.34.m34.2.2.2.3.cmml" xref="A3.2.p2.34.m34.2.2.2.4"><ci id="A3.2.p2.34.m34.1.1.1.1.cmml" xref="A3.2.p2.34.m34.1.1.1.1">f</ci><ci id="A3.2.p2.34.m34.2.2.2.2.cmml" xref="A3.2.p2.34.m34.2.2.2.2">𝜁</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.34.m34.2c">\Phi_{{\rm f},\zeta}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.34.m34.2d">roman_Φ start_POSTSUBSCRIPT roman_f , italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math>, respectively, i.e.,</p> <table class="ltx_equation ltx_eqn_table" id="A3.Ex50"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\Phi_{\rm f}=[\Phi_{{\rm f},\zeta}^{T},\underbrace{\bm{0},\bm{0},\cdots,\bm{0}% }_{N-N_{\zeta}}]^{T},\Phi=[\Phi_{\zeta}^{T},\underbrace{\bm{0},\bm{0},\cdots,% \bm{0}}_{N-N_{\zeta}}]^{T}," class="ltx_Math" display="block" id="A3.Ex50.m1.11"><semantics id="A3.Ex50.m1.11a"><mrow id="A3.Ex50.m1.11.11.1"><mrow id="A3.Ex50.m1.11.11.1.1.2" xref="A3.Ex50.m1.11.11.1.1.3.cmml"><mrow id="A3.Ex50.m1.11.11.1.1.1.1" xref="A3.Ex50.m1.11.11.1.1.1.1.cmml"><msub id="A3.Ex50.m1.11.11.1.1.1.1.4" xref="A3.Ex50.m1.11.11.1.1.1.1.4.cmml"><mi id="A3.Ex50.m1.11.11.1.1.1.1.4.2" mathvariant="normal" xref="A3.Ex50.m1.11.11.1.1.1.1.4.2.cmml">Φ</mi><mi id="A3.Ex50.m1.11.11.1.1.1.1.4.3" mathvariant="normal" xref="A3.Ex50.m1.11.11.1.1.1.1.4.3.cmml">f</mi></msub><mo id="A3.Ex50.m1.11.11.1.1.1.1.3" xref="A3.Ex50.m1.11.11.1.1.1.1.3.cmml">=</mo><msup id="A3.Ex50.m1.11.11.1.1.1.1.2" xref="A3.Ex50.m1.11.11.1.1.1.1.2.cmml"><mrow id="A3.Ex50.m1.11.11.1.1.1.1.2.2.2" xref="A3.Ex50.m1.11.11.1.1.1.1.2.2.3.cmml"><mo id="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.3" stretchy="false" xref="A3.Ex50.m1.11.11.1.1.1.1.2.2.3.cmml">[</mo><msubsup id="A3.Ex50.m1.11.11.1.1.1.1.1.1.1.1" xref="A3.Ex50.m1.11.11.1.1.1.1.1.1.1.1.cmml"><mi id="A3.Ex50.m1.11.11.1.1.1.1.1.1.1.1.2.2" mathvariant="normal" xref="A3.Ex50.m1.11.11.1.1.1.1.1.1.1.1.2.2.cmml">Φ</mi><mrow id="A3.Ex50.m1.2.2.2.4" xref="A3.Ex50.m1.2.2.2.3.cmml"><mi id="A3.Ex50.m1.1.1.1.1" mathvariant="normal" xref="A3.Ex50.m1.1.1.1.1.cmml">f</mi><mo id="A3.Ex50.m1.2.2.2.4.1" xref="A3.Ex50.m1.2.2.2.3.cmml">,</mo><mi id="A3.Ex50.m1.2.2.2.2" xref="A3.Ex50.m1.2.2.2.2.cmml">ζ</mi></mrow><mi id="A3.Ex50.m1.11.11.1.1.1.1.1.1.1.1.3" xref="A3.Ex50.m1.11.11.1.1.1.1.1.1.1.1.3.cmml">T</mi></msubsup><mo id="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.4" xref="A3.Ex50.m1.11.11.1.1.1.1.2.2.3.cmml">,</mo><munder id="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2" xref="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.cmml"><munder accentunder="true" id="A3.Ex50.m1.6.6" xref="A3.Ex50.m1.6.6.cmml"><mrow id="A3.Ex50.m1.6.6.4.6" xref="A3.Ex50.m1.6.6.4.5.cmml"><mn id="A3.Ex50.m1.3.3.1.1" xref="A3.Ex50.m1.3.3.1.1.cmml">𝟎</mn><mo id="A3.Ex50.m1.6.6.4.6.1" xref="A3.Ex50.m1.6.6.4.5.cmml">,</mo><mn id="A3.Ex50.m1.4.4.2.2" xref="A3.Ex50.m1.4.4.2.2.cmml">𝟎</mn><mo id="A3.Ex50.m1.6.6.4.6.2" xref="A3.Ex50.m1.6.6.4.5.cmml">,</mo><mi id="A3.Ex50.m1.5.5.3.3" mathvariant="normal" xref="A3.Ex50.m1.5.5.3.3.cmml">⋯</mi><mo id="A3.Ex50.m1.6.6.4.6.3" xref="A3.Ex50.m1.6.6.4.5.cmml">,</mo><mn id="A3.Ex50.m1.6.6.4.4" xref="A3.Ex50.m1.6.6.4.4.cmml">𝟎</mn></mrow><mo id="A3.Ex50.m1.6.6.5" xref="A3.Ex50.m1.6.6.5.cmml">⏟</mo></munder><mrow id="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2" xref="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2.cmml"><mi id="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2.2" xref="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2.2.cmml">N</mi><mo id="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2.1" xref="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2.1.cmml">−</mo><msub id="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2.3" xref="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2.3.cmml"><mi id="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2.3.2" xref="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2.3.2.cmml">N</mi><mi id="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2.3.3" xref="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2.3.3.cmml">ζ</mi></msub></mrow></munder><mo id="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.5" stretchy="false" xref="A3.Ex50.m1.11.11.1.1.1.1.2.2.3.cmml">]</mo></mrow><mi id="A3.Ex50.m1.11.11.1.1.1.1.2.4" xref="A3.Ex50.m1.11.11.1.1.1.1.2.4.cmml">T</mi></msup></mrow><mo id="A3.Ex50.m1.11.11.1.1.2.3" xref="A3.Ex50.m1.11.11.1.1.3a.cmml">,</mo><mrow id="A3.Ex50.m1.11.11.1.1.2.2" xref="A3.Ex50.m1.11.11.1.1.2.2.cmml"><mi id="A3.Ex50.m1.11.11.1.1.2.2.4" mathvariant="normal" xref="A3.Ex50.m1.11.11.1.1.2.2.4.cmml">Φ</mi><mo id="A3.Ex50.m1.11.11.1.1.2.2.3" xref="A3.Ex50.m1.11.11.1.1.2.2.3.cmml">=</mo><msup id="A3.Ex50.m1.11.11.1.1.2.2.2" xref="A3.Ex50.m1.11.11.1.1.2.2.2.cmml"><mrow id="A3.Ex50.m1.11.11.1.1.2.2.2.2.2" xref="A3.Ex50.m1.11.11.1.1.2.2.2.2.3.cmml"><mo id="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.3" stretchy="false" xref="A3.Ex50.m1.11.11.1.1.2.2.2.2.3.cmml">[</mo><msubsup id="A3.Ex50.m1.11.11.1.1.2.2.1.1.1.1" xref="A3.Ex50.m1.11.11.1.1.2.2.1.1.1.1.cmml"><mi id="A3.Ex50.m1.11.11.1.1.2.2.1.1.1.1.2.2" mathvariant="normal" xref="A3.Ex50.m1.11.11.1.1.2.2.1.1.1.1.2.2.cmml">Φ</mi><mi id="A3.Ex50.m1.11.11.1.1.2.2.1.1.1.1.2.3" xref="A3.Ex50.m1.11.11.1.1.2.2.1.1.1.1.2.3.cmml">ζ</mi><mi id="A3.Ex50.m1.11.11.1.1.2.2.1.1.1.1.3" xref="A3.Ex50.m1.11.11.1.1.2.2.1.1.1.1.3.cmml">T</mi></msubsup><mo id="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.4" xref="A3.Ex50.m1.11.11.1.1.2.2.2.2.3.cmml">,</mo><munder id="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2" xref="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.cmml"><munder accentunder="true" id="A3.Ex50.m1.10.10" xref="A3.Ex50.m1.10.10.cmml"><mrow id="A3.Ex50.m1.10.10.4.6" xref="A3.Ex50.m1.10.10.4.5.cmml"><mn id="A3.Ex50.m1.7.7.1.1" xref="A3.Ex50.m1.7.7.1.1.cmml">𝟎</mn><mo id="A3.Ex50.m1.10.10.4.6.1" xref="A3.Ex50.m1.10.10.4.5.cmml">,</mo><mn id="A3.Ex50.m1.8.8.2.2" xref="A3.Ex50.m1.8.8.2.2.cmml">𝟎</mn><mo id="A3.Ex50.m1.10.10.4.6.2" xref="A3.Ex50.m1.10.10.4.5.cmml">,</mo><mi id="A3.Ex50.m1.9.9.3.3" mathvariant="normal" xref="A3.Ex50.m1.9.9.3.3.cmml">⋯</mi><mo id="A3.Ex50.m1.10.10.4.6.3" xref="A3.Ex50.m1.10.10.4.5.cmml">,</mo><mn id="A3.Ex50.m1.10.10.4.4" xref="A3.Ex50.m1.10.10.4.4.cmml">𝟎</mn></mrow><mo id="A3.Ex50.m1.10.10.5" xref="A3.Ex50.m1.10.10.5.cmml">⏟</mo></munder><mrow id="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2" xref="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2.cmml"><mi id="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2.2" xref="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2.2.cmml">N</mi><mo id="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2.1" xref="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2.1.cmml">−</mo><msub id="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2.3" xref="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2.3.cmml"><mi id="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2.3.2" xref="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2.3.2.cmml">N</mi><mi id="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2.3.3" xref="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2.3.3.cmml">ζ</mi></msub></mrow></munder><mo id="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.5" stretchy="false" xref="A3.Ex50.m1.11.11.1.1.2.2.2.2.3.cmml">]</mo></mrow><mi id="A3.Ex50.m1.11.11.1.1.2.2.2.4" xref="A3.Ex50.m1.11.11.1.1.2.2.2.4.cmml">T</mi></msup></mrow></mrow><mo id="A3.Ex50.m1.11.11.1.2">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex50.m1.11b"><apply id="A3.Ex50.m1.11.11.1.1.3.cmml" xref="A3.Ex50.m1.11.11.1.1.2"><csymbol cd="ambiguous" id="A3.Ex50.m1.11.11.1.1.3a.cmml" xref="A3.Ex50.m1.11.11.1.1.2.3">formulae-sequence</csymbol><apply id="A3.Ex50.m1.11.11.1.1.1.1.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1"><eq id="A3.Ex50.m1.11.11.1.1.1.1.3.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1.3"></eq><apply id="A3.Ex50.m1.11.11.1.1.1.1.4.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1.4"><csymbol cd="ambiguous" id="A3.Ex50.m1.11.11.1.1.1.1.4.1.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1.4">subscript</csymbol><ci id="A3.Ex50.m1.11.11.1.1.1.1.4.2.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1.4.2">Φ</ci><ci id="A3.Ex50.m1.11.11.1.1.1.1.4.3.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1.4.3">f</ci></apply><apply id="A3.Ex50.m1.11.11.1.1.1.1.2.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1.2"><csymbol cd="ambiguous" id="A3.Ex50.m1.11.11.1.1.1.1.2.3.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1.2">superscript</csymbol><interval closure="closed" id="A3.Ex50.m1.11.11.1.1.1.1.2.2.3.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1.2.2.2"><apply id="A3.Ex50.m1.11.11.1.1.1.1.1.1.1.1.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.Ex50.m1.11.11.1.1.1.1.1.1.1.1.1.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1.1.1.1.1">superscript</csymbol><apply id="A3.Ex50.m1.11.11.1.1.1.1.1.1.1.1.2.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.Ex50.m1.11.11.1.1.1.1.1.1.1.1.2.1.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A3.Ex50.m1.11.11.1.1.1.1.1.1.1.1.2.2.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1.1.1.1.1.2.2">Φ</ci><list id="A3.Ex50.m1.2.2.2.3.cmml" xref="A3.Ex50.m1.2.2.2.4"><ci id="A3.Ex50.m1.1.1.1.1.cmml" xref="A3.Ex50.m1.1.1.1.1">f</ci><ci id="A3.Ex50.m1.2.2.2.2.cmml" xref="A3.Ex50.m1.2.2.2.2">𝜁</ci></list></apply><ci id="A3.Ex50.m1.11.11.1.1.1.1.1.1.1.1.3.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1.1.1.1.1.3">𝑇</ci></apply><apply id="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2"><csymbol cd="ambiguous" id="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.1.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2">subscript</csymbol><apply id="A3.Ex50.m1.6.6.cmml" xref="A3.Ex50.m1.6.6"><ci id="A3.Ex50.m1.6.6.5.cmml" xref="A3.Ex50.m1.6.6.5">⏟</ci><list id="A3.Ex50.m1.6.6.4.5.cmml" xref="A3.Ex50.m1.6.6.4.6"><cn id="A3.Ex50.m1.3.3.1.1.cmml" type="integer" xref="A3.Ex50.m1.3.3.1.1">0</cn><cn id="A3.Ex50.m1.4.4.2.2.cmml" type="integer" xref="A3.Ex50.m1.4.4.2.2">0</cn><ci id="A3.Ex50.m1.5.5.3.3.cmml" xref="A3.Ex50.m1.5.5.3.3">⋯</ci><cn id="A3.Ex50.m1.6.6.4.4.cmml" type="integer" xref="A3.Ex50.m1.6.6.4.4">0</cn></list></apply><apply id="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2"><minus id="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2.1.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2.1"></minus><ci id="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2.2.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2.2">𝑁</ci><apply id="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2.3.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2.3"><csymbol cd="ambiguous" id="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2.3.1.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2.3">subscript</csymbol><ci id="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2.3.2.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2.3.2">𝑁</ci><ci id="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2.3.3.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1.2.2.2.2.2.3.3">𝜁</ci></apply></apply></apply></interval><ci id="A3.Ex50.m1.11.11.1.1.1.1.2.4.cmml" xref="A3.Ex50.m1.11.11.1.1.1.1.2.4">𝑇</ci></apply></apply><apply id="A3.Ex50.m1.11.11.1.1.2.2.cmml" xref="A3.Ex50.m1.11.11.1.1.2.2"><eq id="A3.Ex50.m1.11.11.1.1.2.2.3.cmml" xref="A3.Ex50.m1.11.11.1.1.2.2.3"></eq><ci id="A3.Ex50.m1.11.11.1.1.2.2.4.cmml" xref="A3.Ex50.m1.11.11.1.1.2.2.4">Φ</ci><apply id="A3.Ex50.m1.11.11.1.1.2.2.2.cmml" xref="A3.Ex50.m1.11.11.1.1.2.2.2"><csymbol cd="ambiguous" id="A3.Ex50.m1.11.11.1.1.2.2.2.3.cmml" xref="A3.Ex50.m1.11.11.1.1.2.2.2">superscript</csymbol><interval closure="closed" id="A3.Ex50.m1.11.11.1.1.2.2.2.2.3.cmml" xref="A3.Ex50.m1.11.11.1.1.2.2.2.2.2"><apply id="A3.Ex50.m1.11.11.1.1.2.2.1.1.1.1.cmml" xref="A3.Ex50.m1.11.11.1.1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="A3.Ex50.m1.11.11.1.1.2.2.1.1.1.1.1.cmml" xref="A3.Ex50.m1.11.11.1.1.2.2.1.1.1.1">superscript</csymbol><apply id="A3.Ex50.m1.11.11.1.1.2.2.1.1.1.1.2.cmml" xref="A3.Ex50.m1.11.11.1.1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="A3.Ex50.m1.11.11.1.1.2.2.1.1.1.1.2.1.cmml" xref="A3.Ex50.m1.11.11.1.1.2.2.1.1.1.1">subscript</csymbol><ci id="A3.Ex50.m1.11.11.1.1.2.2.1.1.1.1.2.2.cmml" xref="A3.Ex50.m1.11.11.1.1.2.2.1.1.1.1.2.2">Φ</ci><ci id="A3.Ex50.m1.11.11.1.1.2.2.1.1.1.1.2.3.cmml" xref="A3.Ex50.m1.11.11.1.1.2.2.1.1.1.1.2.3">𝜁</ci></apply><ci id="A3.Ex50.m1.11.11.1.1.2.2.1.1.1.1.3.cmml" xref="A3.Ex50.m1.11.11.1.1.2.2.1.1.1.1.3">𝑇</ci></apply><apply id="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.cmml" xref="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.1.cmml" xref="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2">subscript</csymbol><apply id="A3.Ex50.m1.10.10.cmml" xref="A3.Ex50.m1.10.10"><ci id="A3.Ex50.m1.10.10.5.cmml" xref="A3.Ex50.m1.10.10.5">⏟</ci><list id="A3.Ex50.m1.10.10.4.5.cmml" xref="A3.Ex50.m1.10.10.4.6"><cn id="A3.Ex50.m1.7.7.1.1.cmml" type="integer" xref="A3.Ex50.m1.7.7.1.1">0</cn><cn id="A3.Ex50.m1.8.8.2.2.cmml" type="integer" xref="A3.Ex50.m1.8.8.2.2">0</cn><ci id="A3.Ex50.m1.9.9.3.3.cmml" xref="A3.Ex50.m1.9.9.3.3">⋯</ci><cn id="A3.Ex50.m1.10.10.4.4.cmml" type="integer" xref="A3.Ex50.m1.10.10.4.4">0</cn></list></apply><apply id="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2.cmml" xref="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2"><minus id="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2.1.cmml" xref="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2.1"></minus><ci id="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2.2.cmml" xref="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2.2">𝑁</ci><apply id="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2.3.cmml" xref="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2.3.1.cmml" xref="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2.3">subscript</csymbol><ci id="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2.3.2.cmml" xref="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2.3.2">𝑁</ci><ci id="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2.3.3.cmml" xref="A3.Ex50.m1.11.11.1.1.2.2.2.2.2.2.2.3.3">𝜁</ci></apply></apply></apply></interval><ci id="A3.Ex50.m1.11.11.1.1.2.2.2.4.cmml" xref="A3.Ex50.m1.11.11.1.1.2.2.2.4">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex50.m1.11c">\Phi_{\rm f}=[\Phi_{{\rm f},\zeta}^{T},\underbrace{\bm{0},\bm{0},\cdots,\bm{0}% }_{N-N_{\zeta}}]^{T},\Phi=[\Phi_{\zeta}^{T},\underbrace{\bm{0},\bm{0},\cdots,% \bm{0}}_{N-N_{\zeta}}]^{T},</annotation><annotation encoding="application/x-llamapun" id="A3.Ex50.m1.11d">roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT = [ roman_Φ start_POSTSUBSCRIPT roman_f , italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT , under⏟ start_ARG bold_0 , bold_0 , ⋯ , bold_0 end_ARG start_POSTSUBSCRIPT italic_N - italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT , roman_Φ = [ roman_Φ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT , under⏟ start_ARG bold_0 , bold_0 , ⋯ , bold_0 end_ARG start_POSTSUBSCRIPT italic_N - italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.2.p2.80">and the closed-loop system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A3.E43" title="In Proof. ‣ Appendix C The proof of Theorem 2 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">43</span></a>) can be rewritten into</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx64"> <tbody id="A3.E52"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\left\{\begin{array}[]{l}\dot{\bm{e}}=\Lambda\bm{e}+\Phi_{\zeta}^% {T}\tilde{\bm{\theta}}_{\zeta}\\ \dot{\tilde{\bm{\theta}}}_{\zeta}=-\Gamma_{\zeta}\left(\Phi_{{\rm f},\zeta}% \Phi_{{\rm f},\zeta}^{T}\tilde{\bm{\theta}}_{\zeta}+\kappa Q_{\zeta}(t,t_{\rm e% }){\tilde{\bm{\theta}}}_{\zeta}\right)\\ \dot{\tilde{\bm{\theta}}}_{0}=\bm{0}\in\mathbb{R}^{N-N_{\zeta}}\end{array}\right." class="ltx_Math" display="inline" id="A3.E52.m1.1"><semantics id="A3.E52.m1.1a"><mrow id="A3.E52.m1.1.2.2" xref="A3.E52.m1.1.2.1.cmml"><mo id="A3.E52.m1.1.2.2.1" xref="A3.E52.m1.1.2.1.1.cmml">{</mo><mtable id="A3.E52.m1.1.1" rowspacing="0pt" xref="A3.E52.m1.1.1.cmml"><mtr id="A3.E52.m1.1.1a" xref="A3.E52.m1.1.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="A3.E52.m1.1.1b" xref="A3.E52.m1.1.1.cmml"><mrow id="A3.E49.1.1" xref="A3.E49.1.1.cmml"><mover accent="true" id="A3.E49.1.1.2" xref="A3.E49.1.1.2.cmml"><mi id="A3.E49.1.1.2.2" xref="A3.E49.1.1.2.2.cmml">𝒆</mi><mo id="A3.E49.1.1.2.1" xref="A3.E49.1.1.2.1.cmml">˙</mo></mover><mo id="A3.E49.1.1.1" xref="A3.E49.1.1.1.cmml">=</mo><mrow id="A3.E49.1.1.3" xref="A3.E49.1.1.3.cmml"><mrow id="A3.E49.1.1.3.2" xref="A3.E49.1.1.3.2.cmml"><mi id="A3.E49.1.1.3.2.2" mathvariant="normal" xref="A3.E49.1.1.3.2.2.cmml">Λ</mi><mo id="A3.E49.1.1.3.2.1" xref="A3.E49.1.1.3.2.1.cmml">⁢</mo><mi id="A3.E49.1.1.3.2.3" xref="A3.E49.1.1.3.2.3.cmml">𝒆</mi></mrow><mo id="A3.E49.1.1.3.1" xref="A3.E49.1.1.3.1.cmml">+</mo><mrow id="A3.E49.1.1.3.3" xref="A3.E49.1.1.3.3.cmml"><msubsup id="A3.E49.1.1.3.3.2" xref="A3.E49.1.1.3.3.2.cmml"><mi id="A3.E49.1.1.3.3.2.2.2" mathvariant="normal" xref="A3.E49.1.1.3.3.2.2.2.cmml">Φ</mi><mi id="A3.E49.1.1.3.3.2.2.3" xref="A3.E49.1.1.3.3.2.2.3.cmml">ζ</mi><mi id="A3.E49.1.1.3.3.2.3" xref="A3.E49.1.1.3.3.2.3.cmml">T</mi></msubsup><mo id="A3.E49.1.1.3.3.1" xref="A3.E49.1.1.3.3.1.cmml">⁢</mo><msub id="A3.E49.1.1.3.3.3" xref="A3.E49.1.1.3.3.3.cmml"><mover accent="true" id="A3.E49.1.1.3.3.3.2" xref="A3.E49.1.1.3.3.3.2.cmml"><mi id="A3.E49.1.1.3.3.3.2.2" xref="A3.E49.1.1.3.3.3.2.2.cmml">𝜽</mi><mo id="A3.E49.1.1.3.3.3.2.1" xref="A3.E49.1.1.3.3.3.2.1.cmml">~</mo></mover><mi id="A3.E49.1.1.3.3.3.3" xref="A3.E49.1.1.3.3.3.3.cmml">ζ</mi></msub></mrow></mrow></mrow></mtd></mtr><mtr id="A3.E52.m1.1.1c" xref="A3.E52.m1.1.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="A3.E52.m1.1.1d" xref="A3.E52.m1.1.1.cmml"><mrow id="A3.E50.6.6" xref="A3.E50.6.6.cmml"><msub id="A3.E50.6.6.8" xref="A3.E50.6.6.8.cmml"><mover accent="true" id="A3.E50.6.6.8.2" xref="A3.E50.6.6.8.2.cmml"><mover accent="true" id="A3.E50.6.6.8.2.2" xref="A3.E50.6.6.8.2.2.cmml"><mi id="A3.E50.6.6.8.2.2.2" xref="A3.E50.6.6.8.2.2.2.cmml">𝜽</mi><mo id="A3.E50.6.6.8.2.2.1" xref="A3.E50.6.6.8.2.2.1.cmml">~</mo></mover><mo id="A3.E50.6.6.8.2.1" xref="A3.E50.6.6.8.2.1.cmml">˙</mo></mover><mi id="A3.E50.6.6.8.3" xref="A3.E50.6.6.8.3.cmml">ζ</mi></msub><mo id="A3.E50.6.6.7" xref="A3.E50.6.6.7.cmml">=</mo><mrow id="A3.E50.6.6.6" xref="A3.E50.6.6.6.cmml"><mo id="A3.E50.6.6.6a" xref="A3.E50.6.6.6.cmml">−</mo><mrow id="A3.E50.6.6.6.1" xref="A3.E50.6.6.6.1.cmml"><msub id="A3.E50.6.6.6.1.3" xref="A3.E50.6.6.6.1.3.cmml"><mi id="A3.E50.6.6.6.1.3.2" mathvariant="normal" xref="A3.E50.6.6.6.1.3.2.cmml">Γ</mi><mi id="A3.E50.6.6.6.1.3.3" xref="A3.E50.6.6.6.1.3.3.cmml">ζ</mi></msub><mo id="A3.E50.6.6.6.1.2" xref="A3.E50.6.6.6.1.2.cmml">⁢</mo><mrow id="A3.E50.6.6.6.1.1.1" xref="A3.E50.6.6.6.1.1.1.1.cmml"><mo id="A3.E50.6.6.6.1.1.1.2" xref="A3.E50.6.6.6.1.1.1.1.cmml">(</mo><mrow id="A3.E50.6.6.6.1.1.1.1" xref="A3.E50.6.6.6.1.1.1.1.cmml"><mrow id="A3.E50.6.6.6.1.1.1.1.3" xref="A3.E50.6.6.6.1.1.1.1.3.cmml"><msub id="A3.E50.6.6.6.1.1.1.1.3.2" xref="A3.E50.6.6.6.1.1.1.1.3.2.cmml"><mi id="A3.E50.6.6.6.1.1.1.1.3.2.2" mathvariant="normal" xref="A3.E50.6.6.6.1.1.1.1.3.2.2.cmml">Φ</mi><mrow id="A3.E50.2.2.2.2.4" xref="A3.E50.2.2.2.2.3.cmml"><mi id="A3.E50.1.1.1.1.1" mathvariant="normal" xref="A3.E50.1.1.1.1.1.cmml">f</mi><mo id="A3.E50.2.2.2.2.4.1" xref="A3.E50.2.2.2.2.3.cmml">,</mo><mi id="A3.E50.2.2.2.2.2" xref="A3.E50.2.2.2.2.2.cmml">ζ</mi></mrow></msub><mo id="A3.E50.6.6.6.1.1.1.1.3.1" xref="A3.E50.6.6.6.1.1.1.1.3.1.cmml">⁢</mo><msubsup id="A3.E50.6.6.6.1.1.1.1.3.3" xref="A3.E50.6.6.6.1.1.1.1.3.3.cmml"><mi id="A3.E50.6.6.6.1.1.1.1.3.3.2.2" mathvariant="normal" xref="A3.E50.6.6.6.1.1.1.1.3.3.2.2.cmml">Φ</mi><mrow id="A3.E50.4.4.4.2.4" xref="A3.E50.4.4.4.2.3.cmml"><mi id="A3.E50.3.3.3.1.1" mathvariant="normal" xref="A3.E50.3.3.3.1.1.cmml">f</mi><mo id="A3.E50.4.4.4.2.4.1" xref="A3.E50.4.4.4.2.3.cmml">,</mo><mi id="A3.E50.4.4.4.2.2" xref="A3.E50.4.4.4.2.2.cmml">ζ</mi></mrow><mi id="A3.E50.6.6.6.1.1.1.1.3.3.3" xref="A3.E50.6.6.6.1.1.1.1.3.3.3.cmml">T</mi></msubsup><mo id="A3.E50.6.6.6.1.1.1.1.3.1a" xref="A3.E50.6.6.6.1.1.1.1.3.1.cmml">⁢</mo><msub id="A3.E50.6.6.6.1.1.1.1.3.4" xref="A3.E50.6.6.6.1.1.1.1.3.4.cmml"><mover accent="true" id="A3.E50.6.6.6.1.1.1.1.3.4.2" xref="A3.E50.6.6.6.1.1.1.1.3.4.2.cmml"><mi id="A3.E50.6.6.6.1.1.1.1.3.4.2.2" xref="A3.E50.6.6.6.1.1.1.1.3.4.2.2.cmml">𝜽</mi><mo id="A3.E50.6.6.6.1.1.1.1.3.4.2.1" xref="A3.E50.6.6.6.1.1.1.1.3.4.2.1.cmml">~</mo></mover><mi id="A3.E50.6.6.6.1.1.1.1.3.4.3" xref="A3.E50.6.6.6.1.1.1.1.3.4.3.cmml">ζ</mi></msub></mrow><mo id="A3.E50.6.6.6.1.1.1.1.2" xref="A3.E50.6.6.6.1.1.1.1.2.cmml">+</mo><mrow id="A3.E50.6.6.6.1.1.1.1.1" xref="A3.E50.6.6.6.1.1.1.1.1.cmml"><mi id="A3.E50.6.6.6.1.1.1.1.1.3" xref="A3.E50.6.6.6.1.1.1.1.1.3.cmml">κ</mi><mo id="A3.E50.6.6.6.1.1.1.1.1.2" xref="A3.E50.6.6.6.1.1.1.1.1.2.cmml">⁢</mo><msub id="A3.E50.6.6.6.1.1.1.1.1.4" xref="A3.E50.6.6.6.1.1.1.1.1.4.cmml"><mi id="A3.E50.6.6.6.1.1.1.1.1.4.2" xref="A3.E50.6.6.6.1.1.1.1.1.4.2.cmml">Q</mi><mi id="A3.E50.6.6.6.1.1.1.1.1.4.3" xref="A3.E50.6.6.6.1.1.1.1.1.4.3.cmml">ζ</mi></msub><mo id="A3.E50.6.6.6.1.1.1.1.1.2a" xref="A3.E50.6.6.6.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A3.E50.6.6.6.1.1.1.1.1.1.1" xref="A3.E50.6.6.6.1.1.1.1.1.1.2.cmml"><mo id="A3.E50.6.6.6.1.1.1.1.1.1.1.2" stretchy="false" xref="A3.E50.6.6.6.1.1.1.1.1.1.2.cmml">(</mo><mi id="A3.E50.5.5.5" xref="A3.E50.5.5.5.cmml">t</mi><mo id="A3.E50.6.6.6.1.1.1.1.1.1.1.3" xref="A3.E50.6.6.6.1.1.1.1.1.1.2.cmml">,</mo><msub id="A3.E50.6.6.6.1.1.1.1.1.1.1.1" xref="A3.E50.6.6.6.1.1.1.1.1.1.1.1.cmml"><mi id="A3.E50.6.6.6.1.1.1.1.1.1.1.1.2" xref="A3.E50.6.6.6.1.1.1.1.1.1.1.1.2.cmml">t</mi><mi id="A3.E50.6.6.6.1.1.1.1.1.1.1.1.3" mathvariant="normal" xref="A3.E50.6.6.6.1.1.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="A3.E50.6.6.6.1.1.1.1.1.1.1.4" stretchy="false" xref="A3.E50.6.6.6.1.1.1.1.1.1.2.cmml">)</mo></mrow><mo id="A3.E50.6.6.6.1.1.1.1.1.2b" xref="A3.E50.6.6.6.1.1.1.1.1.2.cmml">⁢</mo><msub id="A3.E50.6.6.6.1.1.1.1.1.5" xref="A3.E50.6.6.6.1.1.1.1.1.5.cmml"><mover accent="true" id="A3.E50.6.6.6.1.1.1.1.1.5.2" xref="A3.E50.6.6.6.1.1.1.1.1.5.2.cmml"><mi id="A3.E50.6.6.6.1.1.1.1.1.5.2.2" xref="A3.E50.6.6.6.1.1.1.1.1.5.2.2.cmml">𝜽</mi><mo id="A3.E50.6.6.6.1.1.1.1.1.5.2.1" xref="A3.E50.6.6.6.1.1.1.1.1.5.2.1.cmml">~</mo></mover><mi id="A3.E50.6.6.6.1.1.1.1.1.5.3" xref="A3.E50.6.6.6.1.1.1.1.1.5.3.cmml">ζ</mi></msub></mrow></mrow><mo id="A3.E50.6.6.6.1.1.1.3" xref="A3.E50.6.6.6.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr id="A3.E52.m1.1.1e" xref="A3.E52.m1.1.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="A3.E52.m1.1.1f" xref="A3.E52.m1.1.1.cmml"><mrow id="A3.E51.1.1" xref="A3.E51.1.1.cmml"><msub id="A3.E51.1.1.2" xref="A3.E51.1.1.2.cmml"><mover accent="true" id="A3.E51.1.1.2.2" xref="A3.E51.1.1.2.2.cmml"><mover accent="true" id="A3.E51.1.1.2.2.2" xref="A3.E51.1.1.2.2.2.cmml"><mi id="A3.E51.1.1.2.2.2.2" xref="A3.E51.1.1.2.2.2.2.cmml">𝜽</mi><mo id="A3.E51.1.1.2.2.2.1" xref="A3.E51.1.1.2.2.2.1.cmml">~</mo></mover><mo id="A3.E51.1.1.2.2.1" xref="A3.E51.1.1.2.2.1.cmml">˙</mo></mover><mn id="A3.E51.1.1.2.3" xref="A3.E51.1.1.2.3.cmml">0</mn></msub><mo id="A3.E51.1.1.3" xref="A3.E51.1.1.3.cmml">=</mo><mn id="A3.E51.1.1.4" xref="A3.E51.1.1.4.cmml">𝟎</mn><mo id="A3.E51.1.1.5" xref="A3.E51.1.1.5.cmml">∈</mo><msup id="A3.E51.1.1.6" xref="A3.E51.1.1.6.cmml"><mi id="A3.E51.1.1.6.2" xref="A3.E51.1.1.6.2.cmml">ℝ</mi><mrow id="A3.E51.1.1.6.3" xref="A3.E51.1.1.6.3.cmml"><mi id="A3.E51.1.1.6.3.2" xref="A3.E51.1.1.6.3.2.cmml">N</mi><mo id="A3.E51.1.1.6.3.1" xref="A3.E51.1.1.6.3.1.cmml">−</mo><msub id="A3.E51.1.1.6.3.3" xref="A3.E51.1.1.6.3.3.cmml"><mi id="A3.E51.1.1.6.3.3.2" xref="A3.E51.1.1.6.3.3.2.cmml">N</mi><mi id="A3.E51.1.1.6.3.3.3" xref="A3.E51.1.1.6.3.3.3.cmml">ζ</mi></msub></mrow></msup></mrow></mtd></mtr></mtable><mi id="A3.E52.m1.1.2.2.2" xref="A3.E52.m1.1.2.1.1.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A3.E52.m1.1b"><apply id="A3.E52.m1.1.2.1.cmml" xref="A3.E52.m1.1.2.2"><csymbol cd="latexml" id="A3.E52.m1.1.2.1.1.cmml" xref="A3.E52.m1.1.2.2.1">cases</csymbol><matrix id="A3.E52.m1.1.1.cmml" xref="A3.E52.m1.1.1"><matrixrow id="A3.E52.m1.1.1a.cmml" xref="A3.E52.m1.1.1"><apply id="A3.E49.1.1.cmml" xref="A3.E49.1.1"><eq id="A3.E49.1.1.1.cmml" xref="A3.E49.1.1.1"></eq><apply id="A3.E49.1.1.2.cmml" xref="A3.E49.1.1.2"><ci id="A3.E49.1.1.2.1.cmml" xref="A3.E49.1.1.2.1">˙</ci><ci id="A3.E49.1.1.2.2.cmml" xref="A3.E49.1.1.2.2">𝒆</ci></apply><apply id="A3.E49.1.1.3.cmml" xref="A3.E49.1.1.3"><plus id="A3.E49.1.1.3.1.cmml" xref="A3.E49.1.1.3.1"></plus><apply id="A3.E49.1.1.3.2.cmml" xref="A3.E49.1.1.3.2"><times id="A3.E49.1.1.3.2.1.cmml" xref="A3.E49.1.1.3.2.1"></times><ci id="A3.E49.1.1.3.2.2.cmml" xref="A3.E49.1.1.3.2.2">Λ</ci><ci id="A3.E49.1.1.3.2.3.cmml" xref="A3.E49.1.1.3.2.3">𝒆</ci></apply><apply id="A3.E49.1.1.3.3.cmml" xref="A3.E49.1.1.3.3"><times id="A3.E49.1.1.3.3.1.cmml" xref="A3.E49.1.1.3.3.1"></times><apply id="A3.E49.1.1.3.3.2.cmml" xref="A3.E49.1.1.3.3.2"><csymbol cd="ambiguous" id="A3.E49.1.1.3.3.2.1.cmml" xref="A3.E49.1.1.3.3.2">superscript</csymbol><apply id="A3.E49.1.1.3.3.2.2.cmml" xref="A3.E49.1.1.3.3.2"><csymbol cd="ambiguous" id="A3.E49.1.1.3.3.2.2.1.cmml" xref="A3.E49.1.1.3.3.2">subscript</csymbol><ci id="A3.E49.1.1.3.3.2.2.2.cmml" xref="A3.E49.1.1.3.3.2.2.2">Φ</ci><ci id="A3.E49.1.1.3.3.2.2.3.cmml" xref="A3.E49.1.1.3.3.2.2.3">𝜁</ci></apply><ci id="A3.E49.1.1.3.3.2.3.cmml" xref="A3.E49.1.1.3.3.2.3">𝑇</ci></apply><apply id="A3.E49.1.1.3.3.3.cmml" xref="A3.E49.1.1.3.3.3"><csymbol cd="ambiguous" id="A3.E49.1.1.3.3.3.1.cmml" xref="A3.E49.1.1.3.3.3">subscript</csymbol><apply id="A3.E49.1.1.3.3.3.2.cmml" xref="A3.E49.1.1.3.3.3.2"><ci id="A3.E49.1.1.3.3.3.2.1.cmml" xref="A3.E49.1.1.3.3.3.2.1">~</ci><ci id="A3.E49.1.1.3.3.3.2.2.cmml" xref="A3.E49.1.1.3.3.3.2.2">𝜽</ci></apply><ci id="A3.E49.1.1.3.3.3.3.cmml" xref="A3.E49.1.1.3.3.3.3">𝜁</ci></apply></apply></apply></apply></matrixrow><matrixrow id="A3.E52.m1.1.1b.cmml" xref="A3.E52.m1.1.1"><apply id="A3.E50.6.6.cmml" xref="A3.E50.6.6"><eq id="A3.E50.6.6.7.cmml" xref="A3.E50.6.6.7"></eq><apply id="A3.E50.6.6.8.cmml" xref="A3.E50.6.6.8"><csymbol cd="ambiguous" id="A3.E50.6.6.8.1.cmml" xref="A3.E50.6.6.8">subscript</csymbol><apply id="A3.E50.6.6.8.2.cmml" xref="A3.E50.6.6.8.2"><ci id="A3.E50.6.6.8.2.1.cmml" xref="A3.E50.6.6.8.2.1">˙</ci><apply id="A3.E50.6.6.8.2.2.cmml" xref="A3.E50.6.6.8.2.2"><ci id="A3.E50.6.6.8.2.2.1.cmml" xref="A3.E50.6.6.8.2.2.1">~</ci><ci id="A3.E50.6.6.8.2.2.2.cmml" xref="A3.E50.6.6.8.2.2.2">𝜽</ci></apply></apply><ci id="A3.E50.6.6.8.3.cmml" xref="A3.E50.6.6.8.3">𝜁</ci></apply><apply id="A3.E50.6.6.6.cmml" xref="A3.E50.6.6.6"><minus id="A3.E50.6.6.6.2.cmml" xref="A3.E50.6.6.6"></minus><apply id="A3.E50.6.6.6.1.cmml" xref="A3.E50.6.6.6.1"><times id="A3.E50.6.6.6.1.2.cmml" xref="A3.E50.6.6.6.1.2"></times><apply id="A3.E50.6.6.6.1.3.cmml" xref="A3.E50.6.6.6.1.3"><csymbol cd="ambiguous" id="A3.E50.6.6.6.1.3.1.cmml" xref="A3.E50.6.6.6.1.3">subscript</csymbol><ci id="A3.E50.6.6.6.1.3.2.cmml" xref="A3.E50.6.6.6.1.3.2">Γ</ci><ci id="A3.E50.6.6.6.1.3.3.cmml" xref="A3.E50.6.6.6.1.3.3">𝜁</ci></apply><apply id="A3.E50.6.6.6.1.1.1.1.cmml" xref="A3.E50.6.6.6.1.1.1"><plus id="A3.E50.6.6.6.1.1.1.1.2.cmml" xref="A3.E50.6.6.6.1.1.1.1.2"></plus><apply id="A3.E50.6.6.6.1.1.1.1.3.cmml" xref="A3.E50.6.6.6.1.1.1.1.3"><times id="A3.E50.6.6.6.1.1.1.1.3.1.cmml" xref="A3.E50.6.6.6.1.1.1.1.3.1"></times><apply id="A3.E50.6.6.6.1.1.1.1.3.2.cmml" xref="A3.E50.6.6.6.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A3.E50.6.6.6.1.1.1.1.3.2.1.cmml" xref="A3.E50.6.6.6.1.1.1.1.3.2">subscript</csymbol><ci id="A3.E50.6.6.6.1.1.1.1.3.2.2.cmml" xref="A3.E50.6.6.6.1.1.1.1.3.2.2">Φ</ci><list id="A3.E50.2.2.2.2.3.cmml" xref="A3.E50.2.2.2.2.4"><ci id="A3.E50.1.1.1.1.1.cmml" xref="A3.E50.1.1.1.1.1">f</ci><ci id="A3.E50.2.2.2.2.2.cmml" xref="A3.E50.2.2.2.2.2">𝜁</ci></list></apply><apply id="A3.E50.6.6.6.1.1.1.1.3.3.cmml" xref="A3.E50.6.6.6.1.1.1.1.3.3"><csymbol cd="ambiguous" id="A3.E50.6.6.6.1.1.1.1.3.3.1.cmml" xref="A3.E50.6.6.6.1.1.1.1.3.3">superscript</csymbol><apply id="A3.E50.6.6.6.1.1.1.1.3.3.2.cmml" xref="A3.E50.6.6.6.1.1.1.1.3.3"><csymbol cd="ambiguous" id="A3.E50.6.6.6.1.1.1.1.3.3.2.1.cmml" xref="A3.E50.6.6.6.1.1.1.1.3.3">subscript</csymbol><ci id="A3.E50.6.6.6.1.1.1.1.3.3.2.2.cmml" xref="A3.E50.6.6.6.1.1.1.1.3.3.2.2">Φ</ci><list id="A3.E50.4.4.4.2.3.cmml" xref="A3.E50.4.4.4.2.4"><ci id="A3.E50.3.3.3.1.1.cmml" xref="A3.E50.3.3.3.1.1">f</ci><ci id="A3.E50.4.4.4.2.2.cmml" xref="A3.E50.4.4.4.2.2">𝜁</ci></list></apply><ci id="A3.E50.6.6.6.1.1.1.1.3.3.3.cmml" xref="A3.E50.6.6.6.1.1.1.1.3.3.3">𝑇</ci></apply><apply id="A3.E50.6.6.6.1.1.1.1.3.4.cmml" xref="A3.E50.6.6.6.1.1.1.1.3.4"><csymbol cd="ambiguous" id="A3.E50.6.6.6.1.1.1.1.3.4.1.cmml" xref="A3.E50.6.6.6.1.1.1.1.3.4">subscript</csymbol><apply id="A3.E50.6.6.6.1.1.1.1.3.4.2.cmml" xref="A3.E50.6.6.6.1.1.1.1.3.4.2"><ci id="A3.E50.6.6.6.1.1.1.1.3.4.2.1.cmml" xref="A3.E50.6.6.6.1.1.1.1.3.4.2.1">~</ci><ci id="A3.E50.6.6.6.1.1.1.1.3.4.2.2.cmml" xref="A3.E50.6.6.6.1.1.1.1.3.4.2.2">𝜽</ci></apply><ci id="A3.E50.6.6.6.1.1.1.1.3.4.3.cmml" xref="A3.E50.6.6.6.1.1.1.1.3.4.3">𝜁</ci></apply></apply><apply id="A3.E50.6.6.6.1.1.1.1.1.cmml" xref="A3.E50.6.6.6.1.1.1.1.1"><times id="A3.E50.6.6.6.1.1.1.1.1.2.cmml" xref="A3.E50.6.6.6.1.1.1.1.1.2"></times><ci id="A3.E50.6.6.6.1.1.1.1.1.3.cmml" xref="A3.E50.6.6.6.1.1.1.1.1.3">𝜅</ci><apply id="A3.E50.6.6.6.1.1.1.1.1.4.cmml" xref="A3.E50.6.6.6.1.1.1.1.1.4"><csymbol cd="ambiguous" id="A3.E50.6.6.6.1.1.1.1.1.4.1.cmml" xref="A3.E50.6.6.6.1.1.1.1.1.4">subscript</csymbol><ci id="A3.E50.6.6.6.1.1.1.1.1.4.2.cmml" xref="A3.E50.6.6.6.1.1.1.1.1.4.2">𝑄</ci><ci id="A3.E50.6.6.6.1.1.1.1.1.4.3.cmml" xref="A3.E50.6.6.6.1.1.1.1.1.4.3">𝜁</ci></apply><interval closure="open" id="A3.E50.6.6.6.1.1.1.1.1.1.2.cmml" xref="A3.E50.6.6.6.1.1.1.1.1.1.1"><ci id="A3.E50.5.5.5.cmml" xref="A3.E50.5.5.5">𝑡</ci><apply id="A3.E50.6.6.6.1.1.1.1.1.1.1.1.cmml" xref="A3.E50.6.6.6.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.E50.6.6.6.1.1.1.1.1.1.1.1.1.cmml" xref="A3.E50.6.6.6.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A3.E50.6.6.6.1.1.1.1.1.1.1.1.2.cmml" xref="A3.E50.6.6.6.1.1.1.1.1.1.1.1.2">𝑡</ci><ci id="A3.E50.6.6.6.1.1.1.1.1.1.1.1.3.cmml" xref="A3.E50.6.6.6.1.1.1.1.1.1.1.1.3">e</ci></apply></interval><apply id="A3.E50.6.6.6.1.1.1.1.1.5.cmml" xref="A3.E50.6.6.6.1.1.1.1.1.5"><csymbol cd="ambiguous" id="A3.E50.6.6.6.1.1.1.1.1.5.1.cmml" xref="A3.E50.6.6.6.1.1.1.1.1.5">subscript</csymbol><apply id="A3.E50.6.6.6.1.1.1.1.1.5.2.cmml" xref="A3.E50.6.6.6.1.1.1.1.1.5.2"><ci id="A3.E50.6.6.6.1.1.1.1.1.5.2.1.cmml" xref="A3.E50.6.6.6.1.1.1.1.1.5.2.1">~</ci><ci id="A3.E50.6.6.6.1.1.1.1.1.5.2.2.cmml" xref="A3.E50.6.6.6.1.1.1.1.1.5.2.2">𝜽</ci></apply><ci id="A3.E50.6.6.6.1.1.1.1.1.5.3.cmml" xref="A3.E50.6.6.6.1.1.1.1.1.5.3">𝜁</ci></apply></apply></apply></apply></apply></apply></matrixrow><matrixrow id="A3.E52.m1.1.1c.cmml" xref="A3.E52.m1.1.1"><apply id="A3.E51.1.1.cmml" xref="A3.E51.1.1"><and id="A3.E51.1.1a.cmml" xref="A3.E51.1.1"></and><apply id="A3.E51.1.1b.cmml" xref="A3.E51.1.1"><eq id="A3.E51.1.1.3.cmml" xref="A3.E51.1.1.3"></eq><apply id="A3.E51.1.1.2.cmml" xref="A3.E51.1.1.2"><csymbol cd="ambiguous" id="A3.E51.1.1.2.1.cmml" xref="A3.E51.1.1.2">subscript</csymbol><apply id="A3.E51.1.1.2.2.cmml" xref="A3.E51.1.1.2.2"><ci id="A3.E51.1.1.2.2.1.cmml" xref="A3.E51.1.1.2.2.1">˙</ci><apply id="A3.E51.1.1.2.2.2.cmml" xref="A3.E51.1.1.2.2.2"><ci id="A3.E51.1.1.2.2.2.1.cmml" xref="A3.E51.1.1.2.2.2.1">~</ci><ci id="A3.E51.1.1.2.2.2.2.cmml" xref="A3.E51.1.1.2.2.2.2">𝜽</ci></apply></apply><cn id="A3.E51.1.1.2.3.cmml" type="integer" xref="A3.E51.1.1.2.3">0</cn></apply><cn id="A3.E51.1.1.4.cmml" type="integer" xref="A3.E51.1.1.4">0</cn></apply><apply id="A3.E51.1.1c.cmml" xref="A3.E51.1.1"><in id="A3.E51.1.1.5.cmml" xref="A3.E51.1.1.5"></in><share href="https://arxiv.org/html/2401.10785v2#A3.E51.1.1.4.cmml" id="A3.E51.1.1d.cmml" xref="A3.E51.1.1"></share><apply id="A3.E51.1.1.6.cmml" xref="A3.E51.1.1.6"><csymbol cd="ambiguous" id="A3.E51.1.1.6.1.cmml" xref="A3.E51.1.1.6">superscript</csymbol><ci id="A3.E51.1.1.6.2.cmml" xref="A3.E51.1.1.6.2">ℝ</ci><apply id="A3.E51.1.1.6.3.cmml" xref="A3.E51.1.1.6.3"><minus id="A3.E51.1.1.6.3.1.cmml" xref="A3.E51.1.1.6.3.1"></minus><ci id="A3.E51.1.1.6.3.2.cmml" xref="A3.E51.1.1.6.3.2">𝑁</ci><apply id="A3.E51.1.1.6.3.3.cmml" xref="A3.E51.1.1.6.3.3"><csymbol cd="ambiguous" id="A3.E51.1.1.6.3.3.1.cmml" xref="A3.E51.1.1.6.3.3">subscript</csymbol><ci id="A3.E51.1.1.6.3.3.2.cmml" xref="A3.E51.1.1.6.3.3.2">𝑁</ci><ci id="A3.E51.1.1.6.3.3.3.cmml" xref="A3.E51.1.1.6.3.3.3">𝜁</ci></apply></apply></apply></apply></apply></matrixrow></matrix></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E52.m1.1c">\displaystyle\left\{\begin{array}[]{l}\dot{\bm{e}}=\Lambda\bm{e}+\Phi_{\zeta}^% {T}\tilde{\bm{\theta}}_{\zeta}\\ \dot{\tilde{\bm{\theta}}}_{\zeta}=-\Gamma_{\zeta}\left(\Phi_{{\rm f},\zeta}% \Phi_{{\rm f},\zeta}^{T}\tilde{\bm{\theta}}_{\zeta}+\kappa Q_{\zeta}(t,t_{\rm e% }){\tilde{\bm{\theta}}}_{\zeta}\right)\\ \dot{\tilde{\bm{\theta}}}_{0}=\bm{0}\in\mathbb{R}^{N-N_{\zeta}}\end{array}\right.</annotation><annotation encoding="application/x-llamapun" id="A3.E52.m1.1d">{ start_ARRAY start_ROW start_CELL over˙ start_ARG bold_italic_e end_ARG = roman_Λ bold_italic_e + roman_Φ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL over˙ start_ARG over~ start_ARG bold_italic_θ end_ARG end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT = - roman_Γ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( roman_Φ start_POSTSUBSCRIPT roman_f , italic_ζ end_POSTSUBSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f , italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT + italic_κ italic_Q start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL over˙ start_ARG over~ start_ARG bold_italic_θ end_ARG end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = bold_0 ∈ blackboard_R start_POSTSUPERSCRIPT italic_N - italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_CELL end_ROW end_ARRAY</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(52)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.2.p2.43">where <math alttext="\tilde{\bm{\theta}}_{0}" class="ltx_Math" display="inline" id="A3.2.p2.35.m1.1"><semantics id="A3.2.p2.35.m1.1a"><msub id="A3.2.p2.35.m1.1.1" xref="A3.2.p2.35.m1.1.1.cmml"><mover accent="true" id="A3.2.p2.35.m1.1.1.2" xref="A3.2.p2.35.m1.1.1.2.cmml"><mi id="A3.2.p2.35.m1.1.1.2.2" xref="A3.2.p2.35.m1.1.1.2.2.cmml">𝜽</mi><mo id="A3.2.p2.35.m1.1.1.2.1" xref="A3.2.p2.35.m1.1.1.2.1.cmml">~</mo></mover><mn id="A3.2.p2.35.m1.1.1.3" xref="A3.2.p2.35.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="A3.2.p2.35.m1.1b"><apply id="A3.2.p2.35.m1.1.1.cmml" xref="A3.2.p2.35.m1.1.1"><csymbol cd="ambiguous" id="A3.2.p2.35.m1.1.1.1.cmml" xref="A3.2.p2.35.m1.1.1">subscript</csymbol><apply id="A3.2.p2.35.m1.1.1.2.cmml" xref="A3.2.p2.35.m1.1.1.2"><ci id="A3.2.p2.35.m1.1.1.2.1.cmml" xref="A3.2.p2.35.m1.1.1.2.1">~</ci><ci id="A3.2.p2.35.m1.1.1.2.2.cmml" xref="A3.2.p2.35.m1.1.1.2.2">𝜽</ci></apply><cn id="A3.2.p2.35.m1.1.1.3.cmml" type="integer" xref="A3.2.p2.35.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.35.m1.1c">\tilde{\bm{\theta}}_{0}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.35.m1.1d">over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="A3.2.p2.36.m2.1"><semantics id="A3.2.p2.36.m2.1a"><mo id="A3.2.p2.36.m2.1.1" xref="A3.2.p2.36.m2.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="A3.2.p2.36.m2.1b"><csymbol cd="latexml" id="A3.2.p2.36.m2.1.1.cmml" xref="A3.2.p2.36.m2.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.36.m2.1c">:=</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.36.m2.1d">:=</annotation></semantics></math> <math alttext="[\tilde{\theta}_{N_{\zeta}+1},\tilde{\theta}_{N_{\zeta}+2},\cdots,\tilde{% \theta}_{N}]^{T}\in\mathbb{R}^{N-N_{\zeta}}" class="ltx_Math" display="inline" id="A3.2.p2.37.m3.4"><semantics id="A3.2.p2.37.m3.4a"><mrow id="A3.2.p2.37.m3.4.4" xref="A3.2.p2.37.m3.4.4.cmml"><msup id="A3.2.p2.37.m3.4.4.3" xref="A3.2.p2.37.m3.4.4.3.cmml"><mrow id="A3.2.p2.37.m3.4.4.3.3.3" xref="A3.2.p2.37.m3.4.4.3.3.4.cmml"><mo id="A3.2.p2.37.m3.4.4.3.3.3.4" stretchy="false" xref="A3.2.p2.37.m3.4.4.3.3.4.cmml">[</mo><msub id="A3.2.p2.37.m3.2.2.1.1.1.1" xref="A3.2.p2.37.m3.2.2.1.1.1.1.cmml"><mover accent="true" id="A3.2.p2.37.m3.2.2.1.1.1.1.2" xref="A3.2.p2.37.m3.2.2.1.1.1.1.2.cmml"><mi id="A3.2.p2.37.m3.2.2.1.1.1.1.2.2" xref="A3.2.p2.37.m3.2.2.1.1.1.1.2.2.cmml">θ</mi><mo id="A3.2.p2.37.m3.2.2.1.1.1.1.2.1" xref="A3.2.p2.37.m3.2.2.1.1.1.1.2.1.cmml">~</mo></mover><mrow id="A3.2.p2.37.m3.2.2.1.1.1.1.3" xref="A3.2.p2.37.m3.2.2.1.1.1.1.3.cmml"><msub id="A3.2.p2.37.m3.2.2.1.1.1.1.3.2" xref="A3.2.p2.37.m3.2.2.1.1.1.1.3.2.cmml"><mi id="A3.2.p2.37.m3.2.2.1.1.1.1.3.2.2" xref="A3.2.p2.37.m3.2.2.1.1.1.1.3.2.2.cmml">N</mi><mi id="A3.2.p2.37.m3.2.2.1.1.1.1.3.2.3" xref="A3.2.p2.37.m3.2.2.1.1.1.1.3.2.3.cmml">ζ</mi></msub><mo id="A3.2.p2.37.m3.2.2.1.1.1.1.3.1" xref="A3.2.p2.37.m3.2.2.1.1.1.1.3.1.cmml">+</mo><mn id="A3.2.p2.37.m3.2.2.1.1.1.1.3.3" xref="A3.2.p2.37.m3.2.2.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="A3.2.p2.37.m3.4.4.3.3.3.5" xref="A3.2.p2.37.m3.4.4.3.3.4.cmml">,</mo><msub id="A3.2.p2.37.m3.3.3.2.2.2.2" xref="A3.2.p2.37.m3.3.3.2.2.2.2.cmml"><mover accent="true" id="A3.2.p2.37.m3.3.3.2.2.2.2.2" xref="A3.2.p2.37.m3.3.3.2.2.2.2.2.cmml"><mi id="A3.2.p2.37.m3.3.3.2.2.2.2.2.2" xref="A3.2.p2.37.m3.3.3.2.2.2.2.2.2.cmml">θ</mi><mo id="A3.2.p2.37.m3.3.3.2.2.2.2.2.1" xref="A3.2.p2.37.m3.3.3.2.2.2.2.2.1.cmml">~</mo></mover><mrow id="A3.2.p2.37.m3.3.3.2.2.2.2.3" xref="A3.2.p2.37.m3.3.3.2.2.2.2.3.cmml"><msub id="A3.2.p2.37.m3.3.3.2.2.2.2.3.2" xref="A3.2.p2.37.m3.3.3.2.2.2.2.3.2.cmml"><mi id="A3.2.p2.37.m3.3.3.2.2.2.2.3.2.2" xref="A3.2.p2.37.m3.3.3.2.2.2.2.3.2.2.cmml">N</mi><mi id="A3.2.p2.37.m3.3.3.2.2.2.2.3.2.3" xref="A3.2.p2.37.m3.3.3.2.2.2.2.3.2.3.cmml">ζ</mi></msub><mo id="A3.2.p2.37.m3.3.3.2.2.2.2.3.1" xref="A3.2.p2.37.m3.3.3.2.2.2.2.3.1.cmml">+</mo><mn id="A3.2.p2.37.m3.3.3.2.2.2.2.3.3" xref="A3.2.p2.37.m3.3.3.2.2.2.2.3.3.cmml">2</mn></mrow></msub><mo id="A3.2.p2.37.m3.4.4.3.3.3.6" xref="A3.2.p2.37.m3.4.4.3.3.4.cmml">,</mo><mi id="A3.2.p2.37.m3.1.1" mathvariant="normal" xref="A3.2.p2.37.m3.1.1.cmml">⋯</mi><mo id="A3.2.p2.37.m3.4.4.3.3.3.7" xref="A3.2.p2.37.m3.4.4.3.3.4.cmml">,</mo><msub id="A3.2.p2.37.m3.4.4.3.3.3.3" xref="A3.2.p2.37.m3.4.4.3.3.3.3.cmml"><mover accent="true" id="A3.2.p2.37.m3.4.4.3.3.3.3.2" xref="A3.2.p2.37.m3.4.4.3.3.3.3.2.cmml"><mi id="A3.2.p2.37.m3.4.4.3.3.3.3.2.2" xref="A3.2.p2.37.m3.4.4.3.3.3.3.2.2.cmml">θ</mi><mo id="A3.2.p2.37.m3.4.4.3.3.3.3.2.1" xref="A3.2.p2.37.m3.4.4.3.3.3.3.2.1.cmml">~</mo></mover><mi id="A3.2.p2.37.m3.4.4.3.3.3.3.3" xref="A3.2.p2.37.m3.4.4.3.3.3.3.3.cmml">N</mi></msub><mo id="A3.2.p2.37.m3.4.4.3.3.3.8" stretchy="false" xref="A3.2.p2.37.m3.4.4.3.3.4.cmml">]</mo></mrow><mi id="A3.2.p2.37.m3.4.4.3.5" xref="A3.2.p2.37.m3.4.4.3.5.cmml">T</mi></msup><mo id="A3.2.p2.37.m3.4.4.4" xref="A3.2.p2.37.m3.4.4.4.cmml">∈</mo><msup id="A3.2.p2.37.m3.4.4.5" xref="A3.2.p2.37.m3.4.4.5.cmml"><mi id="A3.2.p2.37.m3.4.4.5.2" xref="A3.2.p2.37.m3.4.4.5.2.cmml">ℝ</mi><mrow id="A3.2.p2.37.m3.4.4.5.3" xref="A3.2.p2.37.m3.4.4.5.3.cmml"><mi id="A3.2.p2.37.m3.4.4.5.3.2" xref="A3.2.p2.37.m3.4.4.5.3.2.cmml">N</mi><mo id="A3.2.p2.37.m3.4.4.5.3.1" xref="A3.2.p2.37.m3.4.4.5.3.1.cmml">−</mo><msub id="A3.2.p2.37.m3.4.4.5.3.3" xref="A3.2.p2.37.m3.4.4.5.3.3.cmml"><mi id="A3.2.p2.37.m3.4.4.5.3.3.2" xref="A3.2.p2.37.m3.4.4.5.3.3.2.cmml">N</mi><mi id="A3.2.p2.37.m3.4.4.5.3.3.3" xref="A3.2.p2.37.m3.4.4.5.3.3.3.cmml">ζ</mi></msub></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.37.m3.4b"><apply id="A3.2.p2.37.m3.4.4.cmml" xref="A3.2.p2.37.m3.4.4"><in id="A3.2.p2.37.m3.4.4.4.cmml" xref="A3.2.p2.37.m3.4.4.4"></in><apply id="A3.2.p2.37.m3.4.4.3.cmml" xref="A3.2.p2.37.m3.4.4.3"><csymbol cd="ambiguous" id="A3.2.p2.37.m3.4.4.3.4.cmml" xref="A3.2.p2.37.m3.4.4.3">superscript</csymbol><list id="A3.2.p2.37.m3.4.4.3.3.4.cmml" xref="A3.2.p2.37.m3.4.4.3.3.3"><apply id="A3.2.p2.37.m3.2.2.1.1.1.1.cmml" xref="A3.2.p2.37.m3.2.2.1.1.1.1"><csymbol cd="ambiguous" id="A3.2.p2.37.m3.2.2.1.1.1.1.1.cmml" xref="A3.2.p2.37.m3.2.2.1.1.1.1">subscript</csymbol><apply id="A3.2.p2.37.m3.2.2.1.1.1.1.2.cmml" xref="A3.2.p2.37.m3.2.2.1.1.1.1.2"><ci id="A3.2.p2.37.m3.2.2.1.1.1.1.2.1.cmml" xref="A3.2.p2.37.m3.2.2.1.1.1.1.2.1">~</ci><ci id="A3.2.p2.37.m3.2.2.1.1.1.1.2.2.cmml" xref="A3.2.p2.37.m3.2.2.1.1.1.1.2.2">𝜃</ci></apply><apply id="A3.2.p2.37.m3.2.2.1.1.1.1.3.cmml" xref="A3.2.p2.37.m3.2.2.1.1.1.1.3"><plus id="A3.2.p2.37.m3.2.2.1.1.1.1.3.1.cmml" xref="A3.2.p2.37.m3.2.2.1.1.1.1.3.1"></plus><apply id="A3.2.p2.37.m3.2.2.1.1.1.1.3.2.cmml" xref="A3.2.p2.37.m3.2.2.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A3.2.p2.37.m3.2.2.1.1.1.1.3.2.1.cmml" xref="A3.2.p2.37.m3.2.2.1.1.1.1.3.2">subscript</csymbol><ci id="A3.2.p2.37.m3.2.2.1.1.1.1.3.2.2.cmml" xref="A3.2.p2.37.m3.2.2.1.1.1.1.3.2.2">𝑁</ci><ci id="A3.2.p2.37.m3.2.2.1.1.1.1.3.2.3.cmml" xref="A3.2.p2.37.m3.2.2.1.1.1.1.3.2.3">𝜁</ci></apply><cn id="A3.2.p2.37.m3.2.2.1.1.1.1.3.3.cmml" type="integer" xref="A3.2.p2.37.m3.2.2.1.1.1.1.3.3">1</cn></apply></apply><apply id="A3.2.p2.37.m3.3.3.2.2.2.2.cmml" xref="A3.2.p2.37.m3.3.3.2.2.2.2"><csymbol cd="ambiguous" id="A3.2.p2.37.m3.3.3.2.2.2.2.1.cmml" xref="A3.2.p2.37.m3.3.3.2.2.2.2">subscript</csymbol><apply id="A3.2.p2.37.m3.3.3.2.2.2.2.2.cmml" xref="A3.2.p2.37.m3.3.3.2.2.2.2.2"><ci id="A3.2.p2.37.m3.3.3.2.2.2.2.2.1.cmml" xref="A3.2.p2.37.m3.3.3.2.2.2.2.2.1">~</ci><ci id="A3.2.p2.37.m3.3.3.2.2.2.2.2.2.cmml" xref="A3.2.p2.37.m3.3.3.2.2.2.2.2.2">𝜃</ci></apply><apply id="A3.2.p2.37.m3.3.3.2.2.2.2.3.cmml" xref="A3.2.p2.37.m3.3.3.2.2.2.2.3"><plus id="A3.2.p2.37.m3.3.3.2.2.2.2.3.1.cmml" xref="A3.2.p2.37.m3.3.3.2.2.2.2.3.1"></plus><apply id="A3.2.p2.37.m3.3.3.2.2.2.2.3.2.cmml" xref="A3.2.p2.37.m3.3.3.2.2.2.2.3.2"><csymbol cd="ambiguous" id="A3.2.p2.37.m3.3.3.2.2.2.2.3.2.1.cmml" xref="A3.2.p2.37.m3.3.3.2.2.2.2.3.2">subscript</csymbol><ci id="A3.2.p2.37.m3.3.3.2.2.2.2.3.2.2.cmml" xref="A3.2.p2.37.m3.3.3.2.2.2.2.3.2.2">𝑁</ci><ci id="A3.2.p2.37.m3.3.3.2.2.2.2.3.2.3.cmml" xref="A3.2.p2.37.m3.3.3.2.2.2.2.3.2.3">𝜁</ci></apply><cn id="A3.2.p2.37.m3.3.3.2.2.2.2.3.3.cmml" type="integer" xref="A3.2.p2.37.m3.3.3.2.2.2.2.3.3">2</cn></apply></apply><ci id="A3.2.p2.37.m3.1.1.cmml" xref="A3.2.p2.37.m3.1.1">⋯</ci><apply id="A3.2.p2.37.m3.4.4.3.3.3.3.cmml" xref="A3.2.p2.37.m3.4.4.3.3.3.3"><csymbol cd="ambiguous" id="A3.2.p2.37.m3.4.4.3.3.3.3.1.cmml" xref="A3.2.p2.37.m3.4.4.3.3.3.3">subscript</csymbol><apply id="A3.2.p2.37.m3.4.4.3.3.3.3.2.cmml" xref="A3.2.p2.37.m3.4.4.3.3.3.3.2"><ci id="A3.2.p2.37.m3.4.4.3.3.3.3.2.1.cmml" xref="A3.2.p2.37.m3.4.4.3.3.3.3.2.1">~</ci><ci id="A3.2.p2.37.m3.4.4.3.3.3.3.2.2.cmml" xref="A3.2.p2.37.m3.4.4.3.3.3.3.2.2">𝜃</ci></apply><ci id="A3.2.p2.37.m3.4.4.3.3.3.3.3.cmml" xref="A3.2.p2.37.m3.4.4.3.3.3.3.3">𝑁</ci></apply></list><ci id="A3.2.p2.37.m3.4.4.3.5.cmml" xref="A3.2.p2.37.m3.4.4.3.5">𝑇</ci></apply><apply id="A3.2.p2.37.m3.4.4.5.cmml" xref="A3.2.p2.37.m3.4.4.5"><csymbol cd="ambiguous" id="A3.2.p2.37.m3.4.4.5.1.cmml" xref="A3.2.p2.37.m3.4.4.5">superscript</csymbol><ci id="A3.2.p2.37.m3.4.4.5.2.cmml" xref="A3.2.p2.37.m3.4.4.5.2">ℝ</ci><apply id="A3.2.p2.37.m3.4.4.5.3.cmml" xref="A3.2.p2.37.m3.4.4.5.3"><minus id="A3.2.p2.37.m3.4.4.5.3.1.cmml" xref="A3.2.p2.37.m3.4.4.5.3.1"></minus><ci id="A3.2.p2.37.m3.4.4.5.3.2.cmml" xref="A3.2.p2.37.m3.4.4.5.3.2">𝑁</ci><apply id="A3.2.p2.37.m3.4.4.5.3.3.cmml" xref="A3.2.p2.37.m3.4.4.5.3.3"><csymbol cd="ambiguous" id="A3.2.p2.37.m3.4.4.5.3.3.1.cmml" xref="A3.2.p2.37.m3.4.4.5.3.3">subscript</csymbol><ci id="A3.2.p2.37.m3.4.4.5.3.3.2.cmml" xref="A3.2.p2.37.m3.4.4.5.3.3.2">𝑁</ci><ci id="A3.2.p2.37.m3.4.4.5.3.3.3.cmml" xref="A3.2.p2.37.m3.4.4.5.3.3.3">𝜁</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.37.m3.4c">[\tilde{\theta}_{N_{\zeta}+1},\tilde{\theta}_{N_{\zeta}+2},\cdots,\tilde{% \theta}_{N}]^{T}\in\mathbb{R}^{N-N_{\zeta}}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.37.m3.4d">[ over~ start_ARG italic_θ end_ARG start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT + 1 end_POSTSUBSCRIPT , over~ start_ARG italic_θ end_ARG start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT + 2 end_POSTSUBSCRIPT , ⋯ , over~ start_ARG italic_θ end_ARG start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_N - italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> denotes an estimation error corresponding to inactive channels, and <math alttext="Q_{\zeta}(t,t_{\rm e})" class="ltx_Math" display="inline" id="A3.2.p2.38.m4.2"><semantics id="A3.2.p2.38.m4.2a"><mrow id="A3.2.p2.38.m4.2.2" xref="A3.2.p2.38.m4.2.2.cmml"><msub id="A3.2.p2.38.m4.2.2.3" xref="A3.2.p2.38.m4.2.2.3.cmml"><mi id="A3.2.p2.38.m4.2.2.3.2" xref="A3.2.p2.38.m4.2.2.3.2.cmml">Q</mi><mi id="A3.2.p2.38.m4.2.2.3.3" xref="A3.2.p2.38.m4.2.2.3.3.cmml">ζ</mi></msub><mo id="A3.2.p2.38.m4.2.2.2" xref="A3.2.p2.38.m4.2.2.2.cmml">⁢</mo><mrow id="A3.2.p2.38.m4.2.2.1.1" xref="A3.2.p2.38.m4.2.2.1.2.cmml"><mo id="A3.2.p2.38.m4.2.2.1.1.2" stretchy="false" xref="A3.2.p2.38.m4.2.2.1.2.cmml">(</mo><mi id="A3.2.p2.38.m4.1.1" xref="A3.2.p2.38.m4.1.1.cmml">t</mi><mo id="A3.2.p2.38.m4.2.2.1.1.3" xref="A3.2.p2.38.m4.2.2.1.2.cmml">,</mo><msub id="A3.2.p2.38.m4.2.2.1.1.1" xref="A3.2.p2.38.m4.2.2.1.1.1.cmml"><mi id="A3.2.p2.38.m4.2.2.1.1.1.2" xref="A3.2.p2.38.m4.2.2.1.1.1.2.cmml">t</mi><mi id="A3.2.p2.38.m4.2.2.1.1.1.3" mathvariant="normal" xref="A3.2.p2.38.m4.2.2.1.1.1.3.cmml">e</mi></msub><mo id="A3.2.p2.38.m4.2.2.1.1.4" stretchy="false" xref="A3.2.p2.38.m4.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.38.m4.2b"><apply id="A3.2.p2.38.m4.2.2.cmml" xref="A3.2.p2.38.m4.2.2"><times id="A3.2.p2.38.m4.2.2.2.cmml" xref="A3.2.p2.38.m4.2.2.2"></times><apply id="A3.2.p2.38.m4.2.2.3.cmml" xref="A3.2.p2.38.m4.2.2.3"><csymbol cd="ambiguous" id="A3.2.p2.38.m4.2.2.3.1.cmml" xref="A3.2.p2.38.m4.2.2.3">subscript</csymbol><ci id="A3.2.p2.38.m4.2.2.3.2.cmml" xref="A3.2.p2.38.m4.2.2.3.2">𝑄</ci><ci id="A3.2.p2.38.m4.2.2.3.3.cmml" xref="A3.2.p2.38.m4.2.2.3.3">𝜁</ci></apply><interval closure="open" id="A3.2.p2.38.m4.2.2.1.2.cmml" xref="A3.2.p2.38.m4.2.2.1.1"><ci id="A3.2.p2.38.m4.1.1.cmml" xref="A3.2.p2.38.m4.1.1">𝑡</ci><apply id="A3.2.p2.38.m4.2.2.1.1.1.cmml" xref="A3.2.p2.38.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="A3.2.p2.38.m4.2.2.1.1.1.1.cmml" xref="A3.2.p2.38.m4.2.2.1.1.1">subscript</csymbol><ci id="A3.2.p2.38.m4.2.2.1.1.1.2.cmml" xref="A3.2.p2.38.m4.2.2.1.1.1.2">𝑡</ci><ci id="A3.2.p2.38.m4.2.2.1.1.1.3.cmml" xref="A3.2.p2.38.m4.2.2.1.1.1.3">e</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.38.m4.2c">Q_{\zeta}(t,t_{\rm e})</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.38.m4.2d">italic_Q start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT )</annotation></semantics></math> is defined in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S5.E29" title="In V-C Robustness Results ‣ V Theoretical Guarantees ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">29</span></a>). Note that one has <math alttext="\tilde{\bm{\theta}}_{0}(t)\equiv\tilde{\bm{\theta}}_{0}(T_{\rm a})" class="ltx_Math" display="inline" id="A3.2.p2.39.m5.2"><semantics id="A3.2.p2.39.m5.2a"><mrow id="A3.2.p2.39.m5.2.2" xref="A3.2.p2.39.m5.2.2.cmml"><mrow id="A3.2.p2.39.m5.2.2.3" xref="A3.2.p2.39.m5.2.2.3.cmml"><msub id="A3.2.p2.39.m5.2.2.3.2" xref="A3.2.p2.39.m5.2.2.3.2.cmml"><mover accent="true" id="A3.2.p2.39.m5.2.2.3.2.2" xref="A3.2.p2.39.m5.2.2.3.2.2.cmml"><mi id="A3.2.p2.39.m5.2.2.3.2.2.2" xref="A3.2.p2.39.m5.2.2.3.2.2.2.cmml">𝜽</mi><mo id="A3.2.p2.39.m5.2.2.3.2.2.1" xref="A3.2.p2.39.m5.2.2.3.2.2.1.cmml">~</mo></mover><mn id="A3.2.p2.39.m5.2.2.3.2.3" xref="A3.2.p2.39.m5.2.2.3.2.3.cmml">0</mn></msub><mo id="A3.2.p2.39.m5.2.2.3.1" xref="A3.2.p2.39.m5.2.2.3.1.cmml">⁢</mo><mrow id="A3.2.p2.39.m5.2.2.3.3.2" xref="A3.2.p2.39.m5.2.2.3.cmml"><mo id="A3.2.p2.39.m5.2.2.3.3.2.1" stretchy="false" xref="A3.2.p2.39.m5.2.2.3.cmml">(</mo><mi id="A3.2.p2.39.m5.1.1" xref="A3.2.p2.39.m5.1.1.cmml">t</mi><mo id="A3.2.p2.39.m5.2.2.3.3.2.2" stretchy="false" xref="A3.2.p2.39.m5.2.2.3.cmml">)</mo></mrow></mrow><mo id="A3.2.p2.39.m5.2.2.2" xref="A3.2.p2.39.m5.2.2.2.cmml">≡</mo><mrow id="A3.2.p2.39.m5.2.2.1" xref="A3.2.p2.39.m5.2.2.1.cmml"><msub id="A3.2.p2.39.m5.2.2.1.3" xref="A3.2.p2.39.m5.2.2.1.3.cmml"><mover accent="true" id="A3.2.p2.39.m5.2.2.1.3.2" xref="A3.2.p2.39.m5.2.2.1.3.2.cmml"><mi id="A3.2.p2.39.m5.2.2.1.3.2.2" xref="A3.2.p2.39.m5.2.2.1.3.2.2.cmml">𝜽</mi><mo id="A3.2.p2.39.m5.2.2.1.3.2.1" xref="A3.2.p2.39.m5.2.2.1.3.2.1.cmml">~</mo></mover><mn id="A3.2.p2.39.m5.2.2.1.3.3" xref="A3.2.p2.39.m5.2.2.1.3.3.cmml">0</mn></msub><mo id="A3.2.p2.39.m5.2.2.1.2" xref="A3.2.p2.39.m5.2.2.1.2.cmml">⁢</mo><mrow id="A3.2.p2.39.m5.2.2.1.1.1" xref="A3.2.p2.39.m5.2.2.1.1.1.1.cmml"><mo id="A3.2.p2.39.m5.2.2.1.1.1.2" stretchy="false" xref="A3.2.p2.39.m5.2.2.1.1.1.1.cmml">(</mo><msub id="A3.2.p2.39.m5.2.2.1.1.1.1" xref="A3.2.p2.39.m5.2.2.1.1.1.1.cmml"><mi id="A3.2.p2.39.m5.2.2.1.1.1.1.2" xref="A3.2.p2.39.m5.2.2.1.1.1.1.2.cmml">T</mi><mi id="A3.2.p2.39.m5.2.2.1.1.1.1.3" mathvariant="normal" xref="A3.2.p2.39.m5.2.2.1.1.1.1.3.cmml">a</mi></msub><mo id="A3.2.p2.39.m5.2.2.1.1.1.3" stretchy="false" xref="A3.2.p2.39.m5.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.39.m5.2b"><apply id="A3.2.p2.39.m5.2.2.cmml" xref="A3.2.p2.39.m5.2.2"><equivalent id="A3.2.p2.39.m5.2.2.2.cmml" xref="A3.2.p2.39.m5.2.2.2"></equivalent><apply id="A3.2.p2.39.m5.2.2.3.cmml" xref="A3.2.p2.39.m5.2.2.3"><times id="A3.2.p2.39.m5.2.2.3.1.cmml" xref="A3.2.p2.39.m5.2.2.3.1"></times><apply id="A3.2.p2.39.m5.2.2.3.2.cmml" xref="A3.2.p2.39.m5.2.2.3.2"><csymbol cd="ambiguous" id="A3.2.p2.39.m5.2.2.3.2.1.cmml" xref="A3.2.p2.39.m5.2.2.3.2">subscript</csymbol><apply id="A3.2.p2.39.m5.2.2.3.2.2.cmml" xref="A3.2.p2.39.m5.2.2.3.2.2"><ci id="A3.2.p2.39.m5.2.2.3.2.2.1.cmml" xref="A3.2.p2.39.m5.2.2.3.2.2.1">~</ci><ci id="A3.2.p2.39.m5.2.2.3.2.2.2.cmml" xref="A3.2.p2.39.m5.2.2.3.2.2.2">𝜽</ci></apply><cn id="A3.2.p2.39.m5.2.2.3.2.3.cmml" type="integer" xref="A3.2.p2.39.m5.2.2.3.2.3">0</cn></apply><ci id="A3.2.p2.39.m5.1.1.cmml" xref="A3.2.p2.39.m5.1.1">𝑡</ci></apply><apply id="A3.2.p2.39.m5.2.2.1.cmml" xref="A3.2.p2.39.m5.2.2.1"><times id="A3.2.p2.39.m5.2.2.1.2.cmml" xref="A3.2.p2.39.m5.2.2.1.2"></times><apply id="A3.2.p2.39.m5.2.2.1.3.cmml" xref="A3.2.p2.39.m5.2.2.1.3"><csymbol cd="ambiguous" id="A3.2.p2.39.m5.2.2.1.3.1.cmml" xref="A3.2.p2.39.m5.2.2.1.3">subscript</csymbol><apply id="A3.2.p2.39.m5.2.2.1.3.2.cmml" xref="A3.2.p2.39.m5.2.2.1.3.2"><ci id="A3.2.p2.39.m5.2.2.1.3.2.1.cmml" xref="A3.2.p2.39.m5.2.2.1.3.2.1">~</ci><ci id="A3.2.p2.39.m5.2.2.1.3.2.2.cmml" xref="A3.2.p2.39.m5.2.2.1.3.2.2">𝜽</ci></apply><cn id="A3.2.p2.39.m5.2.2.1.3.3.cmml" type="integer" xref="A3.2.p2.39.m5.2.2.1.3.3">0</cn></apply><apply id="A3.2.p2.39.m5.2.2.1.1.1.1.cmml" xref="A3.2.p2.39.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="A3.2.p2.39.m5.2.2.1.1.1.1.1.cmml" xref="A3.2.p2.39.m5.2.2.1.1.1">subscript</csymbol><ci id="A3.2.p2.39.m5.2.2.1.1.1.1.2.cmml" xref="A3.2.p2.39.m5.2.2.1.1.1.1.2">𝑇</ci><ci id="A3.2.p2.39.m5.2.2.1.1.1.1.3.cmml" xref="A3.2.p2.39.m5.2.2.1.1.1.1.3">a</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.39.m5.2c">\tilde{\bm{\theta}}_{0}(t)\equiv\tilde{\bm{\theta}}_{0}(T_{\rm a})</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.39.m5.2d">over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t ) ≡ over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT )</annotation></semantics></math>, <math alttext="\forall t\geq T_{\rm a}" class="ltx_Math" display="inline" id="A3.2.p2.40.m6.1"><semantics id="A3.2.p2.40.m6.1a"><mrow id="A3.2.p2.40.m6.1.1" xref="A3.2.p2.40.m6.1.1.cmml"><mrow id="A3.2.p2.40.m6.1.1.2" xref="A3.2.p2.40.m6.1.1.2.cmml"><mo id="A3.2.p2.40.m6.1.1.2.1" rspace="0.167em" xref="A3.2.p2.40.m6.1.1.2.1.cmml">∀</mo><mi id="A3.2.p2.40.m6.1.1.2.2" xref="A3.2.p2.40.m6.1.1.2.2.cmml">t</mi></mrow><mo id="A3.2.p2.40.m6.1.1.1" xref="A3.2.p2.40.m6.1.1.1.cmml">≥</mo><msub id="A3.2.p2.40.m6.1.1.3" xref="A3.2.p2.40.m6.1.1.3.cmml"><mi id="A3.2.p2.40.m6.1.1.3.2" xref="A3.2.p2.40.m6.1.1.3.2.cmml">T</mi><mi id="A3.2.p2.40.m6.1.1.3.3" mathvariant="normal" xref="A3.2.p2.40.m6.1.1.3.3.cmml">a</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.40.m6.1b"><apply id="A3.2.p2.40.m6.1.1.cmml" xref="A3.2.p2.40.m6.1.1"><geq id="A3.2.p2.40.m6.1.1.1.cmml" xref="A3.2.p2.40.m6.1.1.1"></geq><apply id="A3.2.p2.40.m6.1.1.2.cmml" xref="A3.2.p2.40.m6.1.1.2"><csymbol cd="latexml" id="A3.2.p2.40.m6.1.1.2.1.cmml" xref="A3.2.p2.40.m6.1.1.2.1">for-all</csymbol><ci id="A3.2.p2.40.m6.1.1.2.2.cmml" xref="A3.2.p2.40.m6.1.1.2.2">𝑡</ci></apply><apply id="A3.2.p2.40.m6.1.1.3.cmml" xref="A3.2.p2.40.m6.1.1.3"><csymbol cd="ambiguous" id="A3.2.p2.40.m6.1.1.3.1.cmml" xref="A3.2.p2.40.m6.1.1.3">subscript</csymbol><ci id="A3.2.p2.40.m6.1.1.3.2.cmml" xref="A3.2.p2.40.m6.1.1.3.2">𝑇</ci><ci id="A3.2.p2.40.m6.1.1.3.3.cmml" xref="A3.2.p2.40.m6.1.1.3.3">a</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.40.m6.1c">\forall t\geq T_{\rm a}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.40.m6.1d">∀ italic_t ≥ italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT</annotation></semantics></math>, and the tracking error <math alttext="\bm{e}" class="ltx_Math" display="inline" id="A3.2.p2.41.m7.1"><semantics id="A3.2.p2.41.m7.1a"><mi id="A3.2.p2.41.m7.1.1" xref="A3.2.p2.41.m7.1.1.cmml">𝒆</mi><annotation-xml encoding="MathML-Content" id="A3.2.p2.41.m7.1b"><ci id="A3.2.p2.41.m7.1.1.cmml" xref="A3.2.p2.41.m7.1.1">𝒆</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.41.m7.1c">\bm{e}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.41.m7.1d">bold_italic_e</annotation></semantics></math> and the estimation error <math alttext="\tilde{\bm{\theta}}_{\zeta}" class="ltx_Math" display="inline" id="A3.2.p2.42.m8.1"><semantics id="A3.2.p2.42.m8.1a"><msub id="A3.2.p2.42.m8.1.1" xref="A3.2.p2.42.m8.1.1.cmml"><mover accent="true" id="A3.2.p2.42.m8.1.1.2" xref="A3.2.p2.42.m8.1.1.2.cmml"><mi id="A3.2.p2.42.m8.1.1.2.2" xref="A3.2.p2.42.m8.1.1.2.2.cmml">𝜽</mi><mo id="A3.2.p2.42.m8.1.1.2.1" xref="A3.2.p2.42.m8.1.1.2.1.cmml">~</mo></mover><mi id="A3.2.p2.42.m8.1.1.3" xref="A3.2.p2.42.m8.1.1.3.cmml">ζ</mi></msub><annotation-xml encoding="MathML-Content" id="A3.2.p2.42.m8.1b"><apply id="A3.2.p2.42.m8.1.1.cmml" xref="A3.2.p2.42.m8.1.1"><csymbol cd="ambiguous" id="A3.2.p2.42.m8.1.1.1.cmml" xref="A3.2.p2.42.m8.1.1">subscript</csymbol><apply id="A3.2.p2.42.m8.1.1.2.cmml" xref="A3.2.p2.42.m8.1.1.2"><ci id="A3.2.p2.42.m8.1.1.2.1.cmml" xref="A3.2.p2.42.m8.1.1.2.1">~</ci><ci id="A3.2.p2.42.m8.1.1.2.2.cmml" xref="A3.2.p2.42.m8.1.1.2.2">𝜽</ci></apply><ci id="A3.2.p2.42.m8.1.1.3.cmml" xref="A3.2.p2.42.m8.1.1.3">𝜁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.42.m8.1c">\tilde{\bm{\theta}}_{\zeta}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.42.m8.1d">over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math> are not affected by the estimation error <math alttext="\tilde{\bm{\theta}}_{0}" class="ltx_Math" display="inline" id="A3.2.p2.43.m9.1"><semantics id="A3.2.p2.43.m9.1a"><msub id="A3.2.p2.43.m9.1.1" xref="A3.2.p2.43.m9.1.1.cmml"><mover accent="true" id="A3.2.p2.43.m9.1.1.2" xref="A3.2.p2.43.m9.1.1.2.cmml"><mi id="A3.2.p2.43.m9.1.1.2.2" xref="A3.2.p2.43.m9.1.1.2.2.cmml">𝜽</mi><mo id="A3.2.p2.43.m9.1.1.2.1" xref="A3.2.p2.43.m9.1.1.2.1.cmml">~</mo></mover><mn id="A3.2.p2.43.m9.1.1.3" xref="A3.2.p2.43.m9.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="A3.2.p2.43.m9.1b"><apply id="A3.2.p2.43.m9.1.1.cmml" xref="A3.2.p2.43.m9.1.1"><csymbol cd="ambiguous" id="A3.2.p2.43.m9.1.1.1.cmml" xref="A3.2.p2.43.m9.1.1">subscript</csymbol><apply id="A3.2.p2.43.m9.1.1.2.cmml" xref="A3.2.p2.43.m9.1.1.2"><ci id="A3.2.p2.43.m9.1.1.2.1.cmml" xref="A3.2.p2.43.m9.1.1.2.1">~</ci><ci id="A3.2.p2.43.m9.1.1.2.2.cmml" xref="A3.2.p2.43.m9.1.1.2.2">𝜽</ci></apply><cn id="A3.2.p2.43.m9.1.1.3.cmml" type="integer" xref="A3.2.p2.43.m9.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.43.m9.1c">\tilde{\bm{\theta}}_{0}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.43.m9.1d">over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. Thus, choose a new Lyapunov function candidate</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx65"> <tbody id="A3.Ex51"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle V_{\zeta}=\frac{1}{2}\bm{e}^{T}\bm{e}+\frac{1}{2}(1+p)\tilde{\bm% {\theta}}_{\zeta}^{T}\Gamma^{-1}_{\zeta}\tilde{\bm{\theta}}_{\zeta}" class="ltx_Math" display="inline" id="A3.Ex51.m1.1"><semantics id="A3.Ex51.m1.1a"><mrow id="A3.Ex51.m1.1.1" xref="A3.Ex51.m1.1.1.cmml"><msub id="A3.Ex51.m1.1.1.3" xref="A3.Ex51.m1.1.1.3.cmml"><mi id="A3.Ex51.m1.1.1.3.2" xref="A3.Ex51.m1.1.1.3.2.cmml">V</mi><mi id="A3.Ex51.m1.1.1.3.3" xref="A3.Ex51.m1.1.1.3.3.cmml">ζ</mi></msub><mo id="A3.Ex51.m1.1.1.2" xref="A3.Ex51.m1.1.1.2.cmml">=</mo><mrow id="A3.Ex51.m1.1.1.1" xref="A3.Ex51.m1.1.1.1.cmml"><mrow id="A3.Ex51.m1.1.1.1.3" xref="A3.Ex51.m1.1.1.1.3.cmml"><mstyle displaystyle="true" id="A3.Ex51.m1.1.1.1.3.2" xref="A3.Ex51.m1.1.1.1.3.2.cmml"><mfrac id="A3.Ex51.m1.1.1.1.3.2a" xref="A3.Ex51.m1.1.1.1.3.2.cmml"><mn id="A3.Ex51.m1.1.1.1.3.2.2" xref="A3.Ex51.m1.1.1.1.3.2.2.cmml">1</mn><mn id="A3.Ex51.m1.1.1.1.3.2.3" xref="A3.Ex51.m1.1.1.1.3.2.3.cmml">2</mn></mfrac></mstyle><mo id="A3.Ex51.m1.1.1.1.3.1" xref="A3.Ex51.m1.1.1.1.3.1.cmml">⁢</mo><msup id="A3.Ex51.m1.1.1.1.3.3" xref="A3.Ex51.m1.1.1.1.3.3.cmml"><mi id="A3.Ex51.m1.1.1.1.3.3.2" xref="A3.Ex51.m1.1.1.1.3.3.2.cmml">𝒆</mi><mi id="A3.Ex51.m1.1.1.1.3.3.3" xref="A3.Ex51.m1.1.1.1.3.3.3.cmml">T</mi></msup><mo id="A3.Ex51.m1.1.1.1.3.1a" xref="A3.Ex51.m1.1.1.1.3.1.cmml">⁢</mo><mi id="A3.Ex51.m1.1.1.1.3.4" xref="A3.Ex51.m1.1.1.1.3.4.cmml">𝒆</mi></mrow><mo id="A3.Ex51.m1.1.1.1.2" xref="A3.Ex51.m1.1.1.1.2.cmml">+</mo><mrow id="A3.Ex51.m1.1.1.1.1" xref="A3.Ex51.m1.1.1.1.1.cmml"><mstyle displaystyle="true" id="A3.Ex51.m1.1.1.1.1.3" xref="A3.Ex51.m1.1.1.1.1.3.cmml"><mfrac id="A3.Ex51.m1.1.1.1.1.3a" xref="A3.Ex51.m1.1.1.1.1.3.cmml"><mn id="A3.Ex51.m1.1.1.1.1.3.2" xref="A3.Ex51.m1.1.1.1.1.3.2.cmml">1</mn><mn id="A3.Ex51.m1.1.1.1.1.3.3" xref="A3.Ex51.m1.1.1.1.1.3.3.cmml">2</mn></mfrac></mstyle><mo id="A3.Ex51.m1.1.1.1.1.2" xref="A3.Ex51.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="A3.Ex51.m1.1.1.1.1.1.1" xref="A3.Ex51.m1.1.1.1.1.1.1.1.cmml"><mo id="A3.Ex51.m1.1.1.1.1.1.1.2" stretchy="false" xref="A3.Ex51.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.Ex51.m1.1.1.1.1.1.1.1" xref="A3.Ex51.m1.1.1.1.1.1.1.1.cmml"><mn id="A3.Ex51.m1.1.1.1.1.1.1.1.2" xref="A3.Ex51.m1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A3.Ex51.m1.1.1.1.1.1.1.1.1" xref="A3.Ex51.m1.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="A3.Ex51.m1.1.1.1.1.1.1.1.3" xref="A3.Ex51.m1.1.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A3.Ex51.m1.1.1.1.1.1.1.3" stretchy="false" xref="A3.Ex51.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A3.Ex51.m1.1.1.1.1.2a" xref="A3.Ex51.m1.1.1.1.1.2.cmml">⁢</mo><msubsup id="A3.Ex51.m1.1.1.1.1.4" xref="A3.Ex51.m1.1.1.1.1.4.cmml"><mover accent="true" id="A3.Ex51.m1.1.1.1.1.4.2.2" xref="A3.Ex51.m1.1.1.1.1.4.2.2.cmml"><mi id="A3.Ex51.m1.1.1.1.1.4.2.2.2" xref="A3.Ex51.m1.1.1.1.1.4.2.2.2.cmml">𝜽</mi><mo id="A3.Ex51.m1.1.1.1.1.4.2.2.1" xref="A3.Ex51.m1.1.1.1.1.4.2.2.1.cmml">~</mo></mover><mi id="A3.Ex51.m1.1.1.1.1.4.2.3" xref="A3.Ex51.m1.1.1.1.1.4.2.3.cmml">ζ</mi><mi id="A3.Ex51.m1.1.1.1.1.4.3" xref="A3.Ex51.m1.1.1.1.1.4.3.cmml">T</mi></msubsup><mo id="A3.Ex51.m1.1.1.1.1.2b" xref="A3.Ex51.m1.1.1.1.1.2.cmml">⁢</mo><msubsup id="A3.Ex51.m1.1.1.1.1.5" xref="A3.Ex51.m1.1.1.1.1.5.cmml"><mi id="A3.Ex51.m1.1.1.1.1.5.2.2" mathvariant="normal" xref="A3.Ex51.m1.1.1.1.1.5.2.2.cmml">Γ</mi><mi id="A3.Ex51.m1.1.1.1.1.5.3" xref="A3.Ex51.m1.1.1.1.1.5.3.cmml">ζ</mi><mrow id="A3.Ex51.m1.1.1.1.1.5.2.3" xref="A3.Ex51.m1.1.1.1.1.5.2.3.cmml"><mo id="A3.Ex51.m1.1.1.1.1.5.2.3a" xref="A3.Ex51.m1.1.1.1.1.5.2.3.cmml">−</mo><mn id="A3.Ex51.m1.1.1.1.1.5.2.3.2" xref="A3.Ex51.m1.1.1.1.1.5.2.3.2.cmml">1</mn></mrow></msubsup><mo id="A3.Ex51.m1.1.1.1.1.2c" xref="A3.Ex51.m1.1.1.1.1.2.cmml">⁢</mo><msub id="A3.Ex51.m1.1.1.1.1.6" xref="A3.Ex51.m1.1.1.1.1.6.cmml"><mover accent="true" id="A3.Ex51.m1.1.1.1.1.6.2" xref="A3.Ex51.m1.1.1.1.1.6.2.cmml"><mi id="A3.Ex51.m1.1.1.1.1.6.2.2" xref="A3.Ex51.m1.1.1.1.1.6.2.2.cmml">𝜽</mi><mo id="A3.Ex51.m1.1.1.1.1.6.2.1" xref="A3.Ex51.m1.1.1.1.1.6.2.1.cmml">~</mo></mover><mi id="A3.Ex51.m1.1.1.1.1.6.3" xref="A3.Ex51.m1.1.1.1.1.6.3.cmml">ζ</mi></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex51.m1.1b"><apply id="A3.Ex51.m1.1.1.cmml" xref="A3.Ex51.m1.1.1"><eq id="A3.Ex51.m1.1.1.2.cmml" xref="A3.Ex51.m1.1.1.2"></eq><apply id="A3.Ex51.m1.1.1.3.cmml" xref="A3.Ex51.m1.1.1.3"><csymbol cd="ambiguous" id="A3.Ex51.m1.1.1.3.1.cmml" xref="A3.Ex51.m1.1.1.3">subscript</csymbol><ci id="A3.Ex51.m1.1.1.3.2.cmml" xref="A3.Ex51.m1.1.1.3.2">𝑉</ci><ci id="A3.Ex51.m1.1.1.3.3.cmml" xref="A3.Ex51.m1.1.1.3.3">𝜁</ci></apply><apply id="A3.Ex51.m1.1.1.1.cmml" xref="A3.Ex51.m1.1.1.1"><plus id="A3.Ex51.m1.1.1.1.2.cmml" xref="A3.Ex51.m1.1.1.1.2"></plus><apply id="A3.Ex51.m1.1.1.1.3.cmml" xref="A3.Ex51.m1.1.1.1.3"><times id="A3.Ex51.m1.1.1.1.3.1.cmml" xref="A3.Ex51.m1.1.1.1.3.1"></times><apply id="A3.Ex51.m1.1.1.1.3.2.cmml" xref="A3.Ex51.m1.1.1.1.3.2"><divide id="A3.Ex51.m1.1.1.1.3.2.1.cmml" xref="A3.Ex51.m1.1.1.1.3.2"></divide><cn id="A3.Ex51.m1.1.1.1.3.2.2.cmml" type="integer" xref="A3.Ex51.m1.1.1.1.3.2.2">1</cn><cn id="A3.Ex51.m1.1.1.1.3.2.3.cmml" type="integer" xref="A3.Ex51.m1.1.1.1.3.2.3">2</cn></apply><apply id="A3.Ex51.m1.1.1.1.3.3.cmml" xref="A3.Ex51.m1.1.1.1.3.3"><csymbol cd="ambiguous" id="A3.Ex51.m1.1.1.1.3.3.1.cmml" xref="A3.Ex51.m1.1.1.1.3.3">superscript</csymbol><ci id="A3.Ex51.m1.1.1.1.3.3.2.cmml" xref="A3.Ex51.m1.1.1.1.3.3.2">𝒆</ci><ci id="A3.Ex51.m1.1.1.1.3.3.3.cmml" xref="A3.Ex51.m1.1.1.1.3.3.3">𝑇</ci></apply><ci id="A3.Ex51.m1.1.1.1.3.4.cmml" xref="A3.Ex51.m1.1.1.1.3.4">𝒆</ci></apply><apply id="A3.Ex51.m1.1.1.1.1.cmml" xref="A3.Ex51.m1.1.1.1.1"><times id="A3.Ex51.m1.1.1.1.1.2.cmml" xref="A3.Ex51.m1.1.1.1.1.2"></times><apply id="A3.Ex51.m1.1.1.1.1.3.cmml" xref="A3.Ex51.m1.1.1.1.1.3"><divide id="A3.Ex51.m1.1.1.1.1.3.1.cmml" xref="A3.Ex51.m1.1.1.1.1.3"></divide><cn id="A3.Ex51.m1.1.1.1.1.3.2.cmml" type="integer" xref="A3.Ex51.m1.1.1.1.1.3.2">1</cn><cn id="A3.Ex51.m1.1.1.1.1.3.3.cmml" type="integer" xref="A3.Ex51.m1.1.1.1.1.3.3">2</cn></apply><apply id="A3.Ex51.m1.1.1.1.1.1.1.1.cmml" xref="A3.Ex51.m1.1.1.1.1.1.1"><plus id="A3.Ex51.m1.1.1.1.1.1.1.1.1.cmml" xref="A3.Ex51.m1.1.1.1.1.1.1.1.1"></plus><cn id="A3.Ex51.m1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A3.Ex51.m1.1.1.1.1.1.1.1.2">1</cn><ci id="A3.Ex51.m1.1.1.1.1.1.1.1.3.cmml" xref="A3.Ex51.m1.1.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="A3.Ex51.m1.1.1.1.1.4.cmml" xref="A3.Ex51.m1.1.1.1.1.4"><csymbol cd="ambiguous" id="A3.Ex51.m1.1.1.1.1.4.1.cmml" xref="A3.Ex51.m1.1.1.1.1.4">superscript</csymbol><apply id="A3.Ex51.m1.1.1.1.1.4.2.cmml" xref="A3.Ex51.m1.1.1.1.1.4"><csymbol cd="ambiguous" id="A3.Ex51.m1.1.1.1.1.4.2.1.cmml" xref="A3.Ex51.m1.1.1.1.1.4">subscript</csymbol><apply id="A3.Ex51.m1.1.1.1.1.4.2.2.cmml" xref="A3.Ex51.m1.1.1.1.1.4.2.2"><ci id="A3.Ex51.m1.1.1.1.1.4.2.2.1.cmml" xref="A3.Ex51.m1.1.1.1.1.4.2.2.1">~</ci><ci id="A3.Ex51.m1.1.1.1.1.4.2.2.2.cmml" xref="A3.Ex51.m1.1.1.1.1.4.2.2.2">𝜽</ci></apply><ci id="A3.Ex51.m1.1.1.1.1.4.2.3.cmml" xref="A3.Ex51.m1.1.1.1.1.4.2.3">𝜁</ci></apply><ci id="A3.Ex51.m1.1.1.1.1.4.3.cmml" xref="A3.Ex51.m1.1.1.1.1.4.3">𝑇</ci></apply><apply id="A3.Ex51.m1.1.1.1.1.5.cmml" xref="A3.Ex51.m1.1.1.1.1.5"><csymbol cd="ambiguous" id="A3.Ex51.m1.1.1.1.1.5.1.cmml" xref="A3.Ex51.m1.1.1.1.1.5">subscript</csymbol><apply id="A3.Ex51.m1.1.1.1.1.5.2.cmml" xref="A3.Ex51.m1.1.1.1.1.5"><csymbol cd="ambiguous" id="A3.Ex51.m1.1.1.1.1.5.2.1.cmml" xref="A3.Ex51.m1.1.1.1.1.5">superscript</csymbol><ci id="A3.Ex51.m1.1.1.1.1.5.2.2.cmml" xref="A3.Ex51.m1.1.1.1.1.5.2.2">Γ</ci><apply id="A3.Ex51.m1.1.1.1.1.5.2.3.cmml" xref="A3.Ex51.m1.1.1.1.1.5.2.3"><minus id="A3.Ex51.m1.1.1.1.1.5.2.3.1.cmml" xref="A3.Ex51.m1.1.1.1.1.5.2.3"></minus><cn id="A3.Ex51.m1.1.1.1.1.5.2.3.2.cmml" type="integer" xref="A3.Ex51.m1.1.1.1.1.5.2.3.2">1</cn></apply></apply><ci id="A3.Ex51.m1.1.1.1.1.5.3.cmml" xref="A3.Ex51.m1.1.1.1.1.5.3">𝜁</ci></apply><apply id="A3.Ex51.m1.1.1.1.1.6.cmml" xref="A3.Ex51.m1.1.1.1.1.6"><csymbol cd="ambiguous" id="A3.Ex51.m1.1.1.1.1.6.1.cmml" xref="A3.Ex51.m1.1.1.1.1.6">subscript</csymbol><apply id="A3.Ex51.m1.1.1.1.1.6.2.cmml" xref="A3.Ex51.m1.1.1.1.1.6.2"><ci id="A3.Ex51.m1.1.1.1.1.6.2.1.cmml" xref="A3.Ex51.m1.1.1.1.1.6.2.1">~</ci><ci id="A3.Ex51.m1.1.1.1.1.6.2.2.cmml" xref="A3.Ex51.m1.1.1.1.1.6.2.2">𝜽</ci></apply><ci id="A3.Ex51.m1.1.1.1.1.6.3.cmml" xref="A3.Ex51.m1.1.1.1.1.6.3">𝜁</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex51.m1.1c">\displaystyle V_{\zeta}=\frac{1}{2}\bm{e}^{T}\bm{e}+\frac{1}{2}(1+p)\tilde{\bm% {\theta}}_{\zeta}^{T}\Gamma^{-1}_{\zeta}\tilde{\bm{\theta}}_{\zeta}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex51.m1.1d">italic_V start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_e + divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( 1 + italic_p ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Γ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.2.p2.49">with <math alttext="p\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A3.2.p2.44.m1.1"><semantics id="A3.2.p2.44.m1.1a"><mrow id="A3.2.p2.44.m1.1.1" xref="A3.2.p2.44.m1.1.1.cmml"><mi id="A3.2.p2.44.m1.1.1.2" xref="A3.2.p2.44.m1.1.1.2.cmml">p</mi><mo id="A3.2.p2.44.m1.1.1.1" xref="A3.2.p2.44.m1.1.1.1.cmml">∈</mo><msup id="A3.2.p2.44.m1.1.1.3" xref="A3.2.p2.44.m1.1.1.3.cmml"><mi id="A3.2.p2.44.m1.1.1.3.2" xref="A3.2.p2.44.m1.1.1.3.2.cmml">ℝ</mi><mo id="A3.2.p2.44.m1.1.1.3.3" xref="A3.2.p2.44.m1.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.44.m1.1b"><apply id="A3.2.p2.44.m1.1.1.cmml" xref="A3.2.p2.44.m1.1.1"><in id="A3.2.p2.44.m1.1.1.1.cmml" xref="A3.2.p2.44.m1.1.1.1"></in><ci id="A3.2.p2.44.m1.1.1.2.cmml" xref="A3.2.p2.44.m1.1.1.2">𝑝</ci><apply id="A3.2.p2.44.m1.1.1.3.cmml" xref="A3.2.p2.44.m1.1.1.3"><csymbol cd="ambiguous" id="A3.2.p2.44.m1.1.1.3.1.cmml" xref="A3.2.p2.44.m1.1.1.3">superscript</csymbol><ci id="A3.2.p2.44.m1.1.1.3.2.cmml" xref="A3.2.p2.44.m1.1.1.3.2">ℝ</ci><plus id="A3.2.p2.44.m1.1.1.3.3.cmml" xref="A3.2.p2.44.m1.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.44.m1.1c">p\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.44.m1.1d">italic_p ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> being a constant, which contains only <math alttext="\bm{e}" class="ltx_Math" display="inline" id="A3.2.p2.45.m2.1"><semantics id="A3.2.p2.45.m2.1a"><mi id="A3.2.p2.45.m2.1.1" xref="A3.2.p2.45.m2.1.1.cmml">𝒆</mi><annotation-xml encoding="MathML-Content" id="A3.2.p2.45.m2.1b"><ci id="A3.2.p2.45.m2.1.1.cmml" xref="A3.2.p2.45.m2.1.1">𝒆</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.45.m2.1c">\bm{e}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.45.m2.1d">bold_italic_e</annotation></semantics></math> and <math alttext="\tilde{\bm{\theta}}_{\zeta}" class="ltx_Math" display="inline" id="A3.2.p2.46.m3.1"><semantics id="A3.2.p2.46.m3.1a"><msub id="A3.2.p2.46.m3.1.1" xref="A3.2.p2.46.m3.1.1.cmml"><mover accent="true" id="A3.2.p2.46.m3.1.1.2" xref="A3.2.p2.46.m3.1.1.2.cmml"><mi id="A3.2.p2.46.m3.1.1.2.2" xref="A3.2.p2.46.m3.1.1.2.2.cmml">𝜽</mi><mo id="A3.2.p2.46.m3.1.1.2.1" xref="A3.2.p2.46.m3.1.1.2.1.cmml">~</mo></mover><mi id="A3.2.p2.46.m3.1.1.3" xref="A3.2.p2.46.m3.1.1.3.cmml">ζ</mi></msub><annotation-xml encoding="MathML-Content" id="A3.2.p2.46.m3.1b"><apply id="A3.2.p2.46.m3.1.1.cmml" xref="A3.2.p2.46.m3.1.1"><csymbol cd="ambiguous" id="A3.2.p2.46.m3.1.1.1.cmml" xref="A3.2.p2.46.m3.1.1">subscript</csymbol><apply id="A3.2.p2.46.m3.1.1.2.cmml" xref="A3.2.p2.46.m3.1.1.2"><ci id="A3.2.p2.46.m3.1.1.2.1.cmml" xref="A3.2.p2.46.m3.1.1.2.1">~</ci><ci id="A3.2.p2.46.m3.1.1.2.2.cmml" xref="A3.2.p2.46.m3.1.1.2.2">𝜽</ci></apply><ci id="A3.2.p2.46.m3.1.1.3.cmml" xref="A3.2.p2.46.m3.1.1.3">𝜁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.46.m3.1c">\tilde{\bm{\theta}}_{\zeta}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.46.m3.1d">over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math>, rather than the whole estimation error <math alttext="\tilde{\bm{\theta}}" class="ltx_Math" display="inline" id="A3.2.p2.47.m4.1"><semantics id="A3.2.p2.47.m4.1a"><mover accent="true" id="A3.2.p2.47.m4.1.1" xref="A3.2.p2.47.m4.1.1.cmml"><mi id="A3.2.p2.47.m4.1.1.2" xref="A3.2.p2.47.m4.1.1.2.cmml">𝜽</mi><mo id="A3.2.p2.47.m4.1.1.1" xref="A3.2.p2.47.m4.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="A3.2.p2.47.m4.1b"><apply id="A3.2.p2.47.m4.1.1.cmml" xref="A3.2.p2.47.m4.1.1"><ci id="A3.2.p2.47.m4.1.1.1.cmml" xref="A3.2.p2.47.m4.1.1.1">~</ci><ci id="A3.2.p2.47.m4.1.1.2.cmml" xref="A3.2.p2.47.m4.1.1.2">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.47.m4.1c">\tilde{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.47.m4.1d">over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math>. Differentiating <math alttext="V_{\zeta}" class="ltx_Math" display="inline" id="A3.2.p2.48.m5.1"><semantics id="A3.2.p2.48.m5.1a"><msub id="A3.2.p2.48.m5.1.1" xref="A3.2.p2.48.m5.1.1.cmml"><mi id="A3.2.p2.48.m5.1.1.2" xref="A3.2.p2.48.m5.1.1.2.cmml">V</mi><mi id="A3.2.p2.48.m5.1.1.3" xref="A3.2.p2.48.m5.1.1.3.cmml">ζ</mi></msub><annotation-xml encoding="MathML-Content" id="A3.2.p2.48.m5.1b"><apply id="A3.2.p2.48.m5.1.1.cmml" xref="A3.2.p2.48.m5.1.1"><csymbol cd="ambiguous" id="A3.2.p2.48.m5.1.1.1.cmml" xref="A3.2.p2.48.m5.1.1">subscript</csymbol><ci id="A3.2.p2.48.m5.1.1.2.cmml" xref="A3.2.p2.48.m5.1.1.2">𝑉</ci><ci id="A3.2.p2.48.m5.1.1.3.cmml" xref="A3.2.p2.48.m5.1.1.3">𝜁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.48.m5.1c">V_{\zeta}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.48.m5.1d">italic_V start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math> on <math alttext="t" class="ltx_Math" display="inline" id="A3.2.p2.49.m6.1"><semantics id="A3.2.p2.49.m6.1a"><mi id="A3.2.p2.49.m6.1.1" xref="A3.2.p2.49.m6.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A3.2.p2.49.m6.1b"><ci id="A3.2.p2.49.m6.1.1.cmml" xref="A3.2.p2.49.m6.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.49.m6.1c">t</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.49.m6.1d">italic_t</annotation></semantics></math> yields</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx66"> <tbody id="A3.Ex52"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}_{\zeta}=(\dot{\bm{e}}^{T}\bm{e}+\bm{e}^{T}\dot{\bm{e}})/2% +(1+p){\tilde{\bm{\theta}}}_{\zeta}^{T}\Gamma^{-1}_{\zeta}\dot{\tilde{\bm{% \theta}}}_{\zeta}." class="ltx_Math" display="inline" id="A3.Ex52.m1.1"><semantics id="A3.Ex52.m1.1a"><mrow id="A3.Ex52.m1.1.1.1" xref="A3.Ex52.m1.1.1.1.1.cmml"><mrow id="A3.Ex52.m1.1.1.1.1" xref="A3.Ex52.m1.1.1.1.1.cmml"><msub id="A3.Ex52.m1.1.1.1.1.4" xref="A3.Ex52.m1.1.1.1.1.4.cmml"><mover accent="true" id="A3.Ex52.m1.1.1.1.1.4.2" xref="A3.Ex52.m1.1.1.1.1.4.2.cmml"><mi id="A3.Ex52.m1.1.1.1.1.4.2.2" xref="A3.Ex52.m1.1.1.1.1.4.2.2.cmml">V</mi><mo id="A3.Ex52.m1.1.1.1.1.4.2.1" xref="A3.Ex52.m1.1.1.1.1.4.2.1.cmml">˙</mo></mover><mi id="A3.Ex52.m1.1.1.1.1.4.3" xref="A3.Ex52.m1.1.1.1.1.4.3.cmml">ζ</mi></msub><mo id="A3.Ex52.m1.1.1.1.1.3" xref="A3.Ex52.m1.1.1.1.1.3.cmml">=</mo><mrow id="A3.Ex52.m1.1.1.1.1.2" xref="A3.Ex52.m1.1.1.1.1.2.cmml"><mrow id="A3.Ex52.m1.1.1.1.1.1.1" xref="A3.Ex52.m1.1.1.1.1.1.1.cmml"><mrow id="A3.Ex52.m1.1.1.1.1.1.1.1.1" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="A3.Ex52.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.cmml"><mrow id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.cmml"><msup id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.2" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.2.cmml"><mover accent="true" id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.2.2" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.2.2.cmml"><mi id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.2.2.2" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.2.2.2.cmml">𝒆</mi><mo id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.2.2.1" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.2.2.1.cmml">˙</mo></mover><mi id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.2.3" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.2.3.cmml">T</mi></msup><mo id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.1" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.3" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.3.cmml">𝒆</mi></mrow><mo id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.1" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.1.cmml">+</mo><mrow id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.cmml"><msup id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.2" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.2.cmml"><mi id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.2.2" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.2.2.cmml">𝒆</mi><mi id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.2.3" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.2.3.cmml">T</mi></msup><mo id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.1" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mover accent="true" id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.3" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.3.cmml"><mi id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.3.2" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.3.2.cmml">𝒆</mi><mo id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.3.1" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.3.1.cmml">˙</mo></mover></mrow></mrow><mo id="A3.Ex52.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A3.Ex52.m1.1.1.1.1.1.1.2" xref="A3.Ex52.m1.1.1.1.1.1.1.2.cmml">/</mo><mn id="A3.Ex52.m1.1.1.1.1.1.1.3" xref="A3.Ex52.m1.1.1.1.1.1.1.3.cmml">2</mn></mrow><mo id="A3.Ex52.m1.1.1.1.1.2.3" xref="A3.Ex52.m1.1.1.1.1.2.3.cmml">+</mo><mrow id="A3.Ex52.m1.1.1.1.1.2.2" xref="A3.Ex52.m1.1.1.1.1.2.2.cmml"><mrow id="A3.Ex52.m1.1.1.1.1.2.2.1.1" xref="A3.Ex52.m1.1.1.1.1.2.2.1.1.1.cmml"><mo id="A3.Ex52.m1.1.1.1.1.2.2.1.1.2" stretchy="false" xref="A3.Ex52.m1.1.1.1.1.2.2.1.1.1.cmml">(</mo><mrow id="A3.Ex52.m1.1.1.1.1.2.2.1.1.1" xref="A3.Ex52.m1.1.1.1.1.2.2.1.1.1.cmml"><mn id="A3.Ex52.m1.1.1.1.1.2.2.1.1.1.2" xref="A3.Ex52.m1.1.1.1.1.2.2.1.1.1.2.cmml">1</mn><mo id="A3.Ex52.m1.1.1.1.1.2.2.1.1.1.1" xref="A3.Ex52.m1.1.1.1.1.2.2.1.1.1.1.cmml">+</mo><mi id="A3.Ex52.m1.1.1.1.1.2.2.1.1.1.3" xref="A3.Ex52.m1.1.1.1.1.2.2.1.1.1.3.cmml">p</mi></mrow><mo id="A3.Ex52.m1.1.1.1.1.2.2.1.1.3" stretchy="false" xref="A3.Ex52.m1.1.1.1.1.2.2.1.1.1.cmml">)</mo></mrow><mo id="A3.Ex52.m1.1.1.1.1.2.2.2" xref="A3.Ex52.m1.1.1.1.1.2.2.2.cmml">⁢</mo><msubsup id="A3.Ex52.m1.1.1.1.1.2.2.3" xref="A3.Ex52.m1.1.1.1.1.2.2.3.cmml"><mover accent="true" id="A3.Ex52.m1.1.1.1.1.2.2.3.2.2" xref="A3.Ex52.m1.1.1.1.1.2.2.3.2.2.cmml"><mi id="A3.Ex52.m1.1.1.1.1.2.2.3.2.2.2" xref="A3.Ex52.m1.1.1.1.1.2.2.3.2.2.2.cmml">𝜽</mi><mo id="A3.Ex52.m1.1.1.1.1.2.2.3.2.2.1" xref="A3.Ex52.m1.1.1.1.1.2.2.3.2.2.1.cmml">~</mo></mover><mi id="A3.Ex52.m1.1.1.1.1.2.2.3.2.3" xref="A3.Ex52.m1.1.1.1.1.2.2.3.2.3.cmml">ζ</mi><mi id="A3.Ex52.m1.1.1.1.1.2.2.3.3" xref="A3.Ex52.m1.1.1.1.1.2.2.3.3.cmml">T</mi></msubsup><mo id="A3.Ex52.m1.1.1.1.1.2.2.2a" xref="A3.Ex52.m1.1.1.1.1.2.2.2.cmml">⁢</mo><msubsup id="A3.Ex52.m1.1.1.1.1.2.2.4" xref="A3.Ex52.m1.1.1.1.1.2.2.4.cmml"><mi id="A3.Ex52.m1.1.1.1.1.2.2.4.2.2" mathvariant="normal" xref="A3.Ex52.m1.1.1.1.1.2.2.4.2.2.cmml">Γ</mi><mi id="A3.Ex52.m1.1.1.1.1.2.2.4.3" xref="A3.Ex52.m1.1.1.1.1.2.2.4.3.cmml">ζ</mi><mrow id="A3.Ex52.m1.1.1.1.1.2.2.4.2.3" xref="A3.Ex52.m1.1.1.1.1.2.2.4.2.3.cmml"><mo id="A3.Ex52.m1.1.1.1.1.2.2.4.2.3a" xref="A3.Ex52.m1.1.1.1.1.2.2.4.2.3.cmml">−</mo><mn id="A3.Ex52.m1.1.1.1.1.2.2.4.2.3.2" xref="A3.Ex52.m1.1.1.1.1.2.2.4.2.3.2.cmml">1</mn></mrow></msubsup><mo id="A3.Ex52.m1.1.1.1.1.2.2.2b" xref="A3.Ex52.m1.1.1.1.1.2.2.2.cmml">⁢</mo><msub id="A3.Ex52.m1.1.1.1.1.2.2.5" xref="A3.Ex52.m1.1.1.1.1.2.2.5.cmml"><mover accent="true" id="A3.Ex52.m1.1.1.1.1.2.2.5.2" xref="A3.Ex52.m1.1.1.1.1.2.2.5.2.cmml"><mover accent="true" id="A3.Ex52.m1.1.1.1.1.2.2.5.2.2" xref="A3.Ex52.m1.1.1.1.1.2.2.5.2.2.cmml"><mi id="A3.Ex52.m1.1.1.1.1.2.2.5.2.2.2" xref="A3.Ex52.m1.1.1.1.1.2.2.5.2.2.2.cmml">𝜽</mi><mo id="A3.Ex52.m1.1.1.1.1.2.2.5.2.2.1" xref="A3.Ex52.m1.1.1.1.1.2.2.5.2.2.1.cmml">~</mo></mover><mo id="A3.Ex52.m1.1.1.1.1.2.2.5.2.1" xref="A3.Ex52.m1.1.1.1.1.2.2.5.2.1.cmml">˙</mo></mover><mi id="A3.Ex52.m1.1.1.1.1.2.2.5.3" xref="A3.Ex52.m1.1.1.1.1.2.2.5.3.cmml">ζ</mi></msub></mrow></mrow></mrow><mo id="A3.Ex52.m1.1.1.1.2" lspace="0em" xref="A3.Ex52.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex52.m1.1b"><apply id="A3.Ex52.m1.1.1.1.1.cmml" xref="A3.Ex52.m1.1.1.1"><eq id="A3.Ex52.m1.1.1.1.1.3.cmml" xref="A3.Ex52.m1.1.1.1.1.3"></eq><apply id="A3.Ex52.m1.1.1.1.1.4.cmml" xref="A3.Ex52.m1.1.1.1.1.4"><csymbol cd="ambiguous" id="A3.Ex52.m1.1.1.1.1.4.1.cmml" xref="A3.Ex52.m1.1.1.1.1.4">subscript</csymbol><apply id="A3.Ex52.m1.1.1.1.1.4.2.cmml" xref="A3.Ex52.m1.1.1.1.1.4.2"><ci id="A3.Ex52.m1.1.1.1.1.4.2.1.cmml" xref="A3.Ex52.m1.1.1.1.1.4.2.1">˙</ci><ci id="A3.Ex52.m1.1.1.1.1.4.2.2.cmml" xref="A3.Ex52.m1.1.1.1.1.4.2.2">𝑉</ci></apply><ci id="A3.Ex52.m1.1.1.1.1.4.3.cmml" xref="A3.Ex52.m1.1.1.1.1.4.3">𝜁</ci></apply><apply id="A3.Ex52.m1.1.1.1.1.2.cmml" xref="A3.Ex52.m1.1.1.1.1.2"><plus id="A3.Ex52.m1.1.1.1.1.2.3.cmml" xref="A3.Ex52.m1.1.1.1.1.2.3"></plus><apply id="A3.Ex52.m1.1.1.1.1.1.1.cmml" xref="A3.Ex52.m1.1.1.1.1.1.1"><divide id="A3.Ex52.m1.1.1.1.1.1.1.2.cmml" xref="A3.Ex52.m1.1.1.1.1.1.1.2"></divide><apply id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.cmml" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1"><plus id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.1"></plus><apply id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2"><times id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.1"></times><apply id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.2.1.cmml" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.2">superscript</csymbol><apply id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.2.2.cmml" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.2.2"><ci id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.2.2.1.cmml" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.2.2.1">˙</ci><ci id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.2.2.2.cmml" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.2.2.2">𝒆</ci></apply><ci id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.2.3.cmml" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.2.3">𝑇</ci></apply><ci id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.3.cmml" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.2.3">𝒆</ci></apply><apply id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3"><times id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.1"></times><apply id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.2.1.cmml" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.2">superscript</csymbol><ci id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.2.2.cmml" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.2.2">𝒆</ci><ci id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.2.3.cmml" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.2.3">𝑇</ci></apply><apply id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.3"><ci id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.3.1.cmml" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.3.1">˙</ci><ci id="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.3.2.cmml" xref="A3.Ex52.m1.1.1.1.1.1.1.1.1.1.3.3.2">𝒆</ci></apply></apply></apply><cn id="A3.Ex52.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="A3.Ex52.m1.1.1.1.1.1.1.3">2</cn></apply><apply id="A3.Ex52.m1.1.1.1.1.2.2.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2"><times id="A3.Ex52.m1.1.1.1.1.2.2.2.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.2"></times><apply id="A3.Ex52.m1.1.1.1.1.2.2.1.1.1.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.1.1"><plus id="A3.Ex52.m1.1.1.1.1.2.2.1.1.1.1.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.1.1.1.1"></plus><cn id="A3.Ex52.m1.1.1.1.1.2.2.1.1.1.2.cmml" type="integer" xref="A3.Ex52.m1.1.1.1.1.2.2.1.1.1.2">1</cn><ci id="A3.Ex52.m1.1.1.1.1.2.2.1.1.1.3.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.1.1.1.3">𝑝</ci></apply><apply id="A3.Ex52.m1.1.1.1.1.2.2.3.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.3"><csymbol cd="ambiguous" id="A3.Ex52.m1.1.1.1.1.2.2.3.1.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.3">superscript</csymbol><apply id="A3.Ex52.m1.1.1.1.1.2.2.3.2.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.3"><csymbol cd="ambiguous" id="A3.Ex52.m1.1.1.1.1.2.2.3.2.1.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.3">subscript</csymbol><apply id="A3.Ex52.m1.1.1.1.1.2.2.3.2.2.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.3.2.2"><ci id="A3.Ex52.m1.1.1.1.1.2.2.3.2.2.1.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.3.2.2.1">~</ci><ci id="A3.Ex52.m1.1.1.1.1.2.2.3.2.2.2.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.3.2.2.2">𝜽</ci></apply><ci id="A3.Ex52.m1.1.1.1.1.2.2.3.2.3.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.3.2.3">𝜁</ci></apply><ci id="A3.Ex52.m1.1.1.1.1.2.2.3.3.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.3.3">𝑇</ci></apply><apply id="A3.Ex52.m1.1.1.1.1.2.2.4.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.4"><csymbol cd="ambiguous" id="A3.Ex52.m1.1.1.1.1.2.2.4.1.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.4">subscript</csymbol><apply id="A3.Ex52.m1.1.1.1.1.2.2.4.2.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.4"><csymbol cd="ambiguous" id="A3.Ex52.m1.1.1.1.1.2.2.4.2.1.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.4">superscript</csymbol><ci id="A3.Ex52.m1.1.1.1.1.2.2.4.2.2.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.4.2.2">Γ</ci><apply id="A3.Ex52.m1.1.1.1.1.2.2.4.2.3.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.4.2.3"><minus id="A3.Ex52.m1.1.1.1.1.2.2.4.2.3.1.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.4.2.3"></minus><cn id="A3.Ex52.m1.1.1.1.1.2.2.4.2.3.2.cmml" type="integer" xref="A3.Ex52.m1.1.1.1.1.2.2.4.2.3.2">1</cn></apply></apply><ci id="A3.Ex52.m1.1.1.1.1.2.2.4.3.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.4.3">𝜁</ci></apply><apply id="A3.Ex52.m1.1.1.1.1.2.2.5.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.5"><csymbol cd="ambiguous" id="A3.Ex52.m1.1.1.1.1.2.2.5.1.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.5">subscript</csymbol><apply id="A3.Ex52.m1.1.1.1.1.2.2.5.2.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.5.2"><ci id="A3.Ex52.m1.1.1.1.1.2.2.5.2.1.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.5.2.1">˙</ci><apply id="A3.Ex52.m1.1.1.1.1.2.2.5.2.2.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.5.2.2"><ci id="A3.Ex52.m1.1.1.1.1.2.2.5.2.2.1.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.5.2.2.1">~</ci><ci id="A3.Ex52.m1.1.1.1.1.2.2.5.2.2.2.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.5.2.2.2">𝜽</ci></apply></apply><ci id="A3.Ex52.m1.1.1.1.1.2.2.5.3.cmml" xref="A3.Ex52.m1.1.1.1.1.2.2.5.3">𝜁</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex52.m1.1c">\displaystyle\dot{V}_{\zeta}=(\dot{\bm{e}}^{T}\bm{e}+\bm{e}^{T}\dot{\bm{e}})/2% +(1+p){\tilde{\bm{\theta}}}_{\zeta}^{T}\Gamma^{-1}_{\zeta}\dot{\tilde{\bm{% \theta}}}_{\zeta}.</annotation><annotation encoding="application/x-llamapun" id="A3.Ex52.m1.1d">over˙ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT = ( over˙ start_ARG bold_italic_e end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_e + bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over˙ start_ARG bold_italic_e end_ARG ) / 2 + ( 1 + italic_p ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Γ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT over˙ start_ARG over~ start_ARG bold_italic_θ end_ARG end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.2.p2.81">Applying (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A3.E52" title="In Proof. ‣ Appendix C The proof of Theorem 2 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">52</span></a>) to the above formula, one obtains</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx67"> <tbody id="A3.Ex53"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle{\dot{V}}_{\zeta}=" class="ltx_Math" display="inline" id="A3.Ex53.m1.1"><semantics id="A3.Ex53.m1.1a"><mrow id="A3.Ex53.m1.1.1" xref="A3.Ex53.m1.1.1.cmml"><msub id="A3.Ex53.m1.1.1.2" xref="A3.Ex53.m1.1.1.2.cmml"><mover accent="true" id="A3.Ex53.m1.1.1.2.2" xref="A3.Ex53.m1.1.1.2.2.cmml"><mi id="A3.Ex53.m1.1.1.2.2.2" xref="A3.Ex53.m1.1.1.2.2.2.cmml">V</mi><mo id="A3.Ex53.m1.1.1.2.2.1" xref="A3.Ex53.m1.1.1.2.2.1.cmml">˙</mo></mover><mi id="A3.Ex53.m1.1.1.2.3" xref="A3.Ex53.m1.1.1.2.3.cmml">ζ</mi></msub><mo id="A3.Ex53.m1.1.1.1" xref="A3.Ex53.m1.1.1.1.cmml">=</mo><mi id="A3.Ex53.m1.1.1.3" xref="A3.Ex53.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex53.m1.1b"><apply id="A3.Ex53.m1.1.1.cmml" xref="A3.Ex53.m1.1.1"><eq id="A3.Ex53.m1.1.1.1.cmml" xref="A3.Ex53.m1.1.1.1"></eq><apply id="A3.Ex53.m1.1.1.2.cmml" xref="A3.Ex53.m1.1.1.2"><csymbol cd="ambiguous" id="A3.Ex53.m1.1.1.2.1.cmml" xref="A3.Ex53.m1.1.1.2">subscript</csymbol><apply id="A3.Ex53.m1.1.1.2.2.cmml" xref="A3.Ex53.m1.1.1.2.2"><ci id="A3.Ex53.m1.1.1.2.2.1.cmml" xref="A3.Ex53.m1.1.1.2.2.1">˙</ci><ci id="A3.Ex53.m1.1.1.2.2.2.cmml" xref="A3.Ex53.m1.1.1.2.2.2">𝑉</ci></apply><ci id="A3.Ex53.m1.1.1.2.3.cmml" xref="A3.Ex53.m1.1.1.2.3">𝜁</ci></apply><csymbol cd="latexml" id="A3.Ex53.m1.1.1.3.cmml" xref="A3.Ex53.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex53.m1.1c">\displaystyle{\dot{V}}_{\zeta}=</annotation><annotation encoding="application/x-llamapun" id="A3.Ex53.m1.1d">over˙ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT =</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\bm{e}^{T}(-K\bm{e}+\Phi^{T}_{\zeta}{\tilde{\bm{\theta}}}_{\zeta})" class="ltx_Math" display="inline" id="A3.Ex53.m2.1"><semantics id="A3.Ex53.m2.1a"><mrow id="A3.Ex53.m2.1.1" xref="A3.Ex53.m2.1.1.cmml"><msup id="A3.Ex53.m2.1.1.3" xref="A3.Ex53.m2.1.1.3.cmml"><mi id="A3.Ex53.m2.1.1.3.2" xref="A3.Ex53.m2.1.1.3.2.cmml">𝒆</mi><mi id="A3.Ex53.m2.1.1.3.3" xref="A3.Ex53.m2.1.1.3.3.cmml">T</mi></msup><mo id="A3.Ex53.m2.1.1.2" xref="A3.Ex53.m2.1.1.2.cmml">⁢</mo><mrow id="A3.Ex53.m2.1.1.1.1" xref="A3.Ex53.m2.1.1.1.1.1.cmml"><mo id="A3.Ex53.m2.1.1.1.1.2" stretchy="false" xref="A3.Ex53.m2.1.1.1.1.1.cmml">(</mo><mrow id="A3.Ex53.m2.1.1.1.1.1" xref="A3.Ex53.m2.1.1.1.1.1.cmml"><mrow id="A3.Ex53.m2.1.1.1.1.1.2" xref="A3.Ex53.m2.1.1.1.1.1.2.cmml"><mo id="A3.Ex53.m2.1.1.1.1.1.2a" xref="A3.Ex53.m2.1.1.1.1.1.2.cmml">−</mo><mrow id="A3.Ex53.m2.1.1.1.1.1.2.2" xref="A3.Ex53.m2.1.1.1.1.1.2.2.cmml"><mi id="A3.Ex53.m2.1.1.1.1.1.2.2.2" xref="A3.Ex53.m2.1.1.1.1.1.2.2.2.cmml">K</mi><mo id="A3.Ex53.m2.1.1.1.1.1.2.2.1" xref="A3.Ex53.m2.1.1.1.1.1.2.2.1.cmml">⁢</mo><mi id="A3.Ex53.m2.1.1.1.1.1.2.2.3" xref="A3.Ex53.m2.1.1.1.1.1.2.2.3.cmml">𝒆</mi></mrow></mrow><mo id="A3.Ex53.m2.1.1.1.1.1.1" xref="A3.Ex53.m2.1.1.1.1.1.1.cmml">+</mo><mrow id="A3.Ex53.m2.1.1.1.1.1.3" xref="A3.Ex53.m2.1.1.1.1.1.3.cmml"><msubsup id="A3.Ex53.m2.1.1.1.1.1.3.2" xref="A3.Ex53.m2.1.1.1.1.1.3.2.cmml"><mi id="A3.Ex53.m2.1.1.1.1.1.3.2.2.2" mathvariant="normal" xref="A3.Ex53.m2.1.1.1.1.1.3.2.2.2.cmml">Φ</mi><mi id="A3.Ex53.m2.1.1.1.1.1.3.2.3" xref="A3.Ex53.m2.1.1.1.1.1.3.2.3.cmml">ζ</mi><mi id="A3.Ex53.m2.1.1.1.1.1.3.2.2.3" xref="A3.Ex53.m2.1.1.1.1.1.3.2.2.3.cmml">T</mi></msubsup><mo id="A3.Ex53.m2.1.1.1.1.1.3.1" xref="A3.Ex53.m2.1.1.1.1.1.3.1.cmml">⁢</mo><msub id="A3.Ex53.m2.1.1.1.1.1.3.3" xref="A3.Ex53.m2.1.1.1.1.1.3.3.cmml"><mover accent="true" id="A3.Ex53.m2.1.1.1.1.1.3.3.2" xref="A3.Ex53.m2.1.1.1.1.1.3.3.2.cmml"><mi id="A3.Ex53.m2.1.1.1.1.1.3.3.2.2" xref="A3.Ex53.m2.1.1.1.1.1.3.3.2.2.cmml">𝜽</mi><mo id="A3.Ex53.m2.1.1.1.1.1.3.3.2.1" xref="A3.Ex53.m2.1.1.1.1.1.3.3.2.1.cmml">~</mo></mover><mi id="A3.Ex53.m2.1.1.1.1.1.3.3.3" xref="A3.Ex53.m2.1.1.1.1.1.3.3.3.cmml">ζ</mi></msub></mrow></mrow><mo id="A3.Ex53.m2.1.1.1.1.3" stretchy="false" xref="A3.Ex53.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex53.m2.1b"><apply id="A3.Ex53.m2.1.1.cmml" xref="A3.Ex53.m2.1.1"><times id="A3.Ex53.m2.1.1.2.cmml" xref="A3.Ex53.m2.1.1.2"></times><apply id="A3.Ex53.m2.1.1.3.cmml" xref="A3.Ex53.m2.1.1.3"><csymbol cd="ambiguous" id="A3.Ex53.m2.1.1.3.1.cmml" xref="A3.Ex53.m2.1.1.3">superscript</csymbol><ci id="A3.Ex53.m2.1.1.3.2.cmml" xref="A3.Ex53.m2.1.1.3.2">𝒆</ci><ci id="A3.Ex53.m2.1.1.3.3.cmml" xref="A3.Ex53.m2.1.1.3.3">𝑇</ci></apply><apply id="A3.Ex53.m2.1.1.1.1.1.cmml" xref="A3.Ex53.m2.1.1.1.1"><plus id="A3.Ex53.m2.1.1.1.1.1.1.cmml" xref="A3.Ex53.m2.1.1.1.1.1.1"></plus><apply id="A3.Ex53.m2.1.1.1.1.1.2.cmml" xref="A3.Ex53.m2.1.1.1.1.1.2"><minus id="A3.Ex53.m2.1.1.1.1.1.2.1.cmml" xref="A3.Ex53.m2.1.1.1.1.1.2"></minus><apply id="A3.Ex53.m2.1.1.1.1.1.2.2.cmml" xref="A3.Ex53.m2.1.1.1.1.1.2.2"><times id="A3.Ex53.m2.1.1.1.1.1.2.2.1.cmml" xref="A3.Ex53.m2.1.1.1.1.1.2.2.1"></times><ci id="A3.Ex53.m2.1.1.1.1.1.2.2.2.cmml" xref="A3.Ex53.m2.1.1.1.1.1.2.2.2">𝐾</ci><ci id="A3.Ex53.m2.1.1.1.1.1.2.2.3.cmml" xref="A3.Ex53.m2.1.1.1.1.1.2.2.3">𝒆</ci></apply></apply><apply id="A3.Ex53.m2.1.1.1.1.1.3.cmml" xref="A3.Ex53.m2.1.1.1.1.1.3"><times id="A3.Ex53.m2.1.1.1.1.1.3.1.cmml" xref="A3.Ex53.m2.1.1.1.1.1.3.1"></times><apply id="A3.Ex53.m2.1.1.1.1.1.3.2.cmml" xref="A3.Ex53.m2.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A3.Ex53.m2.1.1.1.1.1.3.2.1.cmml" xref="A3.Ex53.m2.1.1.1.1.1.3.2">subscript</csymbol><apply id="A3.Ex53.m2.1.1.1.1.1.3.2.2.cmml" xref="A3.Ex53.m2.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A3.Ex53.m2.1.1.1.1.1.3.2.2.1.cmml" xref="A3.Ex53.m2.1.1.1.1.1.3.2">superscript</csymbol><ci id="A3.Ex53.m2.1.1.1.1.1.3.2.2.2.cmml" xref="A3.Ex53.m2.1.1.1.1.1.3.2.2.2">Φ</ci><ci id="A3.Ex53.m2.1.1.1.1.1.3.2.2.3.cmml" xref="A3.Ex53.m2.1.1.1.1.1.3.2.2.3">𝑇</ci></apply><ci id="A3.Ex53.m2.1.1.1.1.1.3.2.3.cmml" xref="A3.Ex53.m2.1.1.1.1.1.3.2.3">𝜁</ci></apply><apply id="A3.Ex53.m2.1.1.1.1.1.3.3.cmml" xref="A3.Ex53.m2.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="A3.Ex53.m2.1.1.1.1.1.3.3.1.cmml" xref="A3.Ex53.m2.1.1.1.1.1.3.3">subscript</csymbol><apply id="A3.Ex53.m2.1.1.1.1.1.3.3.2.cmml" xref="A3.Ex53.m2.1.1.1.1.1.3.3.2"><ci id="A3.Ex53.m2.1.1.1.1.1.3.3.2.1.cmml" xref="A3.Ex53.m2.1.1.1.1.1.3.3.2.1">~</ci><ci id="A3.Ex53.m2.1.1.1.1.1.3.3.2.2.cmml" xref="A3.Ex53.m2.1.1.1.1.1.3.3.2.2">𝜽</ci></apply><ci id="A3.Ex53.m2.1.1.1.1.1.3.3.3.cmml" xref="A3.Ex53.m2.1.1.1.1.1.3.3.3">𝜁</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex53.m2.1c">\displaystyle\bm{e}^{T}(-K\bm{e}+\Phi^{T}_{\zeta}{\tilde{\bm{\theta}}}_{\zeta})</annotation><annotation encoding="application/x-llamapun" id="A3.Ex53.m2.1d">bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( - italic_K bold_italic_e + roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.E53"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-(1+p)({\tilde{\bm{\theta}}}_{\zeta}^{T}\Phi_{{\rm f},\zeta}\Phi_% {{\rm f},\zeta}^{T}\tilde{\bm{\theta}}_{\zeta}+\kappa\tilde{\bm{\theta}}^{T}_{% \zeta}Q_{\zeta}(t,t_{\rm e}){\tilde{\bm{\theta}}}_{\zeta})." class="ltx_Math" display="inline" id="A3.E53.m1.6"><semantics id="A3.E53.m1.6a"><mrow id="A3.E53.m1.6.6.1" xref="A3.E53.m1.6.6.1.1.cmml"><mrow id="A3.E53.m1.6.6.1.1" xref="A3.E53.m1.6.6.1.1.cmml"><mo id="A3.E53.m1.6.6.1.1a" xref="A3.E53.m1.6.6.1.1.cmml">−</mo><mrow id="A3.E53.m1.6.6.1.1.2" xref="A3.E53.m1.6.6.1.1.2.cmml"><mrow id="A3.E53.m1.6.6.1.1.1.1.1" xref="A3.E53.m1.6.6.1.1.1.1.1.1.cmml"><mo id="A3.E53.m1.6.6.1.1.1.1.1.2" stretchy="false" xref="A3.E53.m1.6.6.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.E53.m1.6.6.1.1.1.1.1.1" xref="A3.E53.m1.6.6.1.1.1.1.1.1.cmml"><mn id="A3.E53.m1.6.6.1.1.1.1.1.1.2" xref="A3.E53.m1.6.6.1.1.1.1.1.1.2.cmml">1</mn><mo id="A3.E53.m1.6.6.1.1.1.1.1.1.1" xref="A3.E53.m1.6.6.1.1.1.1.1.1.1.cmml">+</mo><mi id="A3.E53.m1.6.6.1.1.1.1.1.1.3" xref="A3.E53.m1.6.6.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A3.E53.m1.6.6.1.1.1.1.1.3" stretchy="false" xref="A3.E53.m1.6.6.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A3.E53.m1.6.6.1.1.2.3" xref="A3.E53.m1.6.6.1.1.2.3.cmml">⁢</mo><mrow id="A3.E53.m1.6.6.1.1.2.2.1" xref="A3.E53.m1.6.6.1.1.2.2.1.1.cmml"><mo id="A3.E53.m1.6.6.1.1.2.2.1.2" stretchy="false" xref="A3.E53.m1.6.6.1.1.2.2.1.1.cmml">(</mo><mrow id="A3.E53.m1.6.6.1.1.2.2.1.1" xref="A3.E53.m1.6.6.1.1.2.2.1.1.cmml"><mrow id="A3.E53.m1.6.6.1.1.2.2.1.1.3" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.cmml"><msubsup id="A3.E53.m1.6.6.1.1.2.2.1.1.3.2" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.cmml"><mover accent="true" id="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.2.2" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.2.2.cmml"><mi id="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.2.2.2" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.2.2.2.cmml">𝜽</mi><mo id="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.2.2.1" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.2.2.1.cmml">~</mo></mover><mi id="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.2.3" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.2.3.cmml">ζ</mi><mi id="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.3" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.3.cmml">T</mi></msubsup><mo id="A3.E53.m1.6.6.1.1.2.2.1.1.3.1" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.1.cmml">⁢</mo><msub id="A3.E53.m1.6.6.1.1.2.2.1.1.3.3" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.3.cmml"><mi id="A3.E53.m1.6.6.1.1.2.2.1.1.3.3.2" mathvariant="normal" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.3.2.cmml">Φ</mi><mrow id="A3.E53.m1.2.2.2.4" xref="A3.E53.m1.2.2.2.3.cmml"><mi id="A3.E53.m1.1.1.1.1" mathvariant="normal" xref="A3.E53.m1.1.1.1.1.cmml">f</mi><mo id="A3.E53.m1.2.2.2.4.1" xref="A3.E53.m1.2.2.2.3.cmml">,</mo><mi id="A3.E53.m1.2.2.2.2" xref="A3.E53.m1.2.2.2.2.cmml">ζ</mi></mrow></msub><mo id="A3.E53.m1.6.6.1.1.2.2.1.1.3.1a" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.1.cmml">⁢</mo><msubsup id="A3.E53.m1.6.6.1.1.2.2.1.1.3.4" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.4.cmml"><mi id="A3.E53.m1.6.6.1.1.2.2.1.1.3.4.2.2" mathvariant="normal" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.4.2.2.cmml">Φ</mi><mrow id="A3.E53.m1.4.4.2.4" xref="A3.E53.m1.4.4.2.3.cmml"><mi id="A3.E53.m1.3.3.1.1" mathvariant="normal" xref="A3.E53.m1.3.3.1.1.cmml">f</mi><mo id="A3.E53.m1.4.4.2.4.1" xref="A3.E53.m1.4.4.2.3.cmml">,</mo><mi id="A3.E53.m1.4.4.2.2" xref="A3.E53.m1.4.4.2.2.cmml">ζ</mi></mrow><mi id="A3.E53.m1.6.6.1.1.2.2.1.1.3.4.3" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.4.3.cmml">T</mi></msubsup><mo id="A3.E53.m1.6.6.1.1.2.2.1.1.3.1b" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.1.cmml">⁢</mo><msub id="A3.E53.m1.6.6.1.1.2.2.1.1.3.5" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.5.cmml"><mover accent="true" id="A3.E53.m1.6.6.1.1.2.2.1.1.3.5.2" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.5.2.cmml"><mi id="A3.E53.m1.6.6.1.1.2.2.1.1.3.5.2.2" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.5.2.2.cmml">𝜽</mi><mo id="A3.E53.m1.6.6.1.1.2.2.1.1.3.5.2.1" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.5.2.1.cmml">~</mo></mover><mi id="A3.E53.m1.6.6.1.1.2.2.1.1.3.5.3" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.5.3.cmml">ζ</mi></msub></mrow><mo id="A3.E53.m1.6.6.1.1.2.2.1.1.2" xref="A3.E53.m1.6.6.1.1.2.2.1.1.2.cmml">+</mo><mrow id="A3.E53.m1.6.6.1.1.2.2.1.1.1" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.cmml"><mi id="A3.E53.m1.6.6.1.1.2.2.1.1.1.3" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.3.cmml">κ</mi><mo id="A3.E53.m1.6.6.1.1.2.2.1.1.1.2" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.2.cmml">⁢</mo><msubsup id="A3.E53.m1.6.6.1.1.2.2.1.1.1.4" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.cmml"><mover accent="true" id="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.2.2" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.2.2.cmml"><mi id="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.2.2.2" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.2.2.2.cmml">𝜽</mi><mo id="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.2.2.1" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.2.2.1.cmml">~</mo></mover><mi id="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.3" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.3.cmml">ζ</mi><mi id="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.2.3" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.2.3.cmml">T</mi></msubsup><mo id="A3.E53.m1.6.6.1.1.2.2.1.1.1.2a" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.2.cmml">⁢</mo><msub id="A3.E53.m1.6.6.1.1.2.2.1.1.1.5" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.5.cmml"><mi id="A3.E53.m1.6.6.1.1.2.2.1.1.1.5.2" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.5.2.cmml">Q</mi><mi id="A3.E53.m1.6.6.1.1.2.2.1.1.1.5.3" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.5.3.cmml">ζ</mi></msub><mo id="A3.E53.m1.6.6.1.1.2.2.1.1.1.2b" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="A3.E53.m1.6.6.1.1.2.2.1.1.1.1.1" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.1.2.cmml"><mo id="A3.E53.m1.6.6.1.1.2.2.1.1.1.1.1.2" stretchy="false" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.1.2.cmml">(</mo><mi id="A3.E53.m1.5.5" xref="A3.E53.m1.5.5.cmml">t</mi><mo id="A3.E53.m1.6.6.1.1.2.2.1.1.1.1.1.3" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.1.2.cmml">,</mo><msub id="A3.E53.m1.6.6.1.1.2.2.1.1.1.1.1.1" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.1.1.1.cmml"><mi id="A3.E53.m1.6.6.1.1.2.2.1.1.1.1.1.1.2" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.1.1.1.2.cmml">t</mi><mi id="A3.E53.m1.6.6.1.1.2.2.1.1.1.1.1.1.3" mathvariant="normal" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="A3.E53.m1.6.6.1.1.2.2.1.1.1.1.1.4" stretchy="false" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.1.2.cmml">)</mo></mrow><mo id="A3.E53.m1.6.6.1.1.2.2.1.1.1.2c" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.2.cmml">⁢</mo><msub id="A3.E53.m1.6.6.1.1.2.2.1.1.1.6" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.6.cmml"><mover accent="true" id="A3.E53.m1.6.6.1.1.2.2.1.1.1.6.2" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.6.2.cmml"><mi id="A3.E53.m1.6.6.1.1.2.2.1.1.1.6.2.2" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.6.2.2.cmml">𝜽</mi><mo id="A3.E53.m1.6.6.1.1.2.2.1.1.1.6.2.1" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.6.2.1.cmml">~</mo></mover><mi id="A3.E53.m1.6.6.1.1.2.2.1.1.1.6.3" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.6.3.cmml">ζ</mi></msub></mrow></mrow><mo id="A3.E53.m1.6.6.1.1.2.2.1.3" stretchy="false" xref="A3.E53.m1.6.6.1.1.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A3.E53.m1.6.6.1.2" lspace="0em" xref="A3.E53.m1.6.6.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.E53.m1.6b"><apply id="A3.E53.m1.6.6.1.1.cmml" xref="A3.E53.m1.6.6.1"><minus id="A3.E53.m1.6.6.1.1.3.cmml" xref="A3.E53.m1.6.6.1"></minus><apply id="A3.E53.m1.6.6.1.1.2.cmml" xref="A3.E53.m1.6.6.1.1.2"><times id="A3.E53.m1.6.6.1.1.2.3.cmml" xref="A3.E53.m1.6.6.1.1.2.3"></times><apply id="A3.E53.m1.6.6.1.1.1.1.1.1.cmml" xref="A3.E53.m1.6.6.1.1.1.1.1"><plus id="A3.E53.m1.6.6.1.1.1.1.1.1.1.cmml" xref="A3.E53.m1.6.6.1.1.1.1.1.1.1"></plus><cn id="A3.E53.m1.6.6.1.1.1.1.1.1.2.cmml" type="integer" xref="A3.E53.m1.6.6.1.1.1.1.1.1.2">1</cn><ci id="A3.E53.m1.6.6.1.1.1.1.1.1.3.cmml" xref="A3.E53.m1.6.6.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="A3.E53.m1.6.6.1.1.2.2.1.1.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1"><plus id="A3.E53.m1.6.6.1.1.2.2.1.1.2.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.2"></plus><apply id="A3.E53.m1.6.6.1.1.2.2.1.1.3.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3"><times id="A3.E53.m1.6.6.1.1.2.2.1.1.3.1.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.1"></times><apply id="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.2"><csymbol cd="ambiguous" id="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.1.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.2">superscript</csymbol><apply id="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.2.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.2"><csymbol cd="ambiguous" id="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.2.1.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.2">subscript</csymbol><apply id="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.2.2.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.2.2"><ci id="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.2.2.1.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.2.2.1">~</ci><ci id="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.2.2.2.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.2.2.2">𝜽</ci></apply><ci id="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.2.3.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.2.3">𝜁</ci></apply><ci id="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.3.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.2.3">𝑇</ci></apply><apply id="A3.E53.m1.6.6.1.1.2.2.1.1.3.3.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.3"><csymbol cd="ambiguous" id="A3.E53.m1.6.6.1.1.2.2.1.1.3.3.1.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.3">subscript</csymbol><ci id="A3.E53.m1.6.6.1.1.2.2.1.1.3.3.2.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.3.2">Φ</ci><list id="A3.E53.m1.2.2.2.3.cmml" xref="A3.E53.m1.2.2.2.4"><ci id="A3.E53.m1.1.1.1.1.cmml" xref="A3.E53.m1.1.1.1.1">f</ci><ci id="A3.E53.m1.2.2.2.2.cmml" xref="A3.E53.m1.2.2.2.2">𝜁</ci></list></apply><apply id="A3.E53.m1.6.6.1.1.2.2.1.1.3.4.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.4"><csymbol cd="ambiguous" id="A3.E53.m1.6.6.1.1.2.2.1.1.3.4.1.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.4">superscript</csymbol><apply id="A3.E53.m1.6.6.1.1.2.2.1.1.3.4.2.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.4"><csymbol cd="ambiguous" id="A3.E53.m1.6.6.1.1.2.2.1.1.3.4.2.1.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.4">subscript</csymbol><ci id="A3.E53.m1.6.6.1.1.2.2.1.1.3.4.2.2.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.4.2.2">Φ</ci><list id="A3.E53.m1.4.4.2.3.cmml" xref="A3.E53.m1.4.4.2.4"><ci id="A3.E53.m1.3.3.1.1.cmml" xref="A3.E53.m1.3.3.1.1">f</ci><ci id="A3.E53.m1.4.4.2.2.cmml" xref="A3.E53.m1.4.4.2.2">𝜁</ci></list></apply><ci id="A3.E53.m1.6.6.1.1.2.2.1.1.3.4.3.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.4.3">𝑇</ci></apply><apply id="A3.E53.m1.6.6.1.1.2.2.1.1.3.5.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.5"><csymbol cd="ambiguous" id="A3.E53.m1.6.6.1.1.2.2.1.1.3.5.1.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.5">subscript</csymbol><apply id="A3.E53.m1.6.6.1.1.2.2.1.1.3.5.2.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.5.2"><ci id="A3.E53.m1.6.6.1.1.2.2.1.1.3.5.2.1.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.5.2.1">~</ci><ci id="A3.E53.m1.6.6.1.1.2.2.1.1.3.5.2.2.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.5.2.2">𝜽</ci></apply><ci id="A3.E53.m1.6.6.1.1.2.2.1.1.3.5.3.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.3.5.3">𝜁</ci></apply></apply><apply id="A3.E53.m1.6.6.1.1.2.2.1.1.1.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1"><times id="A3.E53.m1.6.6.1.1.2.2.1.1.1.2.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.2"></times><ci id="A3.E53.m1.6.6.1.1.2.2.1.1.1.3.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.3">𝜅</ci><apply id="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.4"><csymbol cd="ambiguous" id="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.1.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.4">subscript</csymbol><apply id="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.2.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.4"><csymbol cd="ambiguous" id="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.2.1.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.4">superscript</csymbol><apply id="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.2.2.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.2.2"><ci id="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.2.2.1.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.2.2.1">~</ci><ci id="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.2.2.2.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.2.2.2">𝜽</ci></apply><ci id="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.2.3.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.2.3">𝑇</ci></apply><ci id="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.3.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.4.3">𝜁</ci></apply><apply id="A3.E53.m1.6.6.1.1.2.2.1.1.1.5.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.5"><csymbol cd="ambiguous" id="A3.E53.m1.6.6.1.1.2.2.1.1.1.5.1.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.5">subscript</csymbol><ci id="A3.E53.m1.6.6.1.1.2.2.1.1.1.5.2.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.5.2">𝑄</ci><ci id="A3.E53.m1.6.6.1.1.2.2.1.1.1.5.3.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.5.3">𝜁</ci></apply><interval closure="open" id="A3.E53.m1.6.6.1.1.2.2.1.1.1.1.2.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.1.1"><ci id="A3.E53.m1.5.5.cmml" xref="A3.E53.m1.5.5">𝑡</ci><apply id="A3.E53.m1.6.6.1.1.2.2.1.1.1.1.1.1.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.E53.m1.6.6.1.1.2.2.1.1.1.1.1.1.1.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="A3.E53.m1.6.6.1.1.2.2.1.1.1.1.1.1.2.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.1.1.1.2">𝑡</ci><ci id="A3.E53.m1.6.6.1.1.2.2.1.1.1.1.1.1.3.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.1.1.1.3">e</ci></apply></interval><apply id="A3.E53.m1.6.6.1.1.2.2.1.1.1.6.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.6"><csymbol cd="ambiguous" id="A3.E53.m1.6.6.1.1.2.2.1.1.1.6.1.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.6">subscript</csymbol><apply id="A3.E53.m1.6.6.1.1.2.2.1.1.1.6.2.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.6.2"><ci id="A3.E53.m1.6.6.1.1.2.2.1.1.1.6.2.1.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.6.2.1">~</ci><ci id="A3.E53.m1.6.6.1.1.2.2.1.1.1.6.2.2.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.6.2.2">𝜽</ci></apply><ci id="A3.E53.m1.6.6.1.1.2.2.1.1.1.6.3.cmml" xref="A3.E53.m1.6.6.1.1.2.2.1.1.1.6.3">𝜁</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E53.m1.6c">\displaystyle-(1+p)({\tilde{\bm{\theta}}}_{\zeta}^{T}\Phi_{{\rm f},\zeta}\Phi_% {{\rm f},\zeta}^{T}\tilde{\bm{\theta}}_{\zeta}+\kappa\tilde{\bm{\theta}}^{T}_{% \zeta}Q_{\zeta}(t,t_{\rm e}){\tilde{\bm{\theta}}}_{\zeta}).</annotation><annotation encoding="application/x-llamapun" id="A3.E53.m1.6d">- ( 1 + italic_p ) ( over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f , italic_ζ end_POSTSUBSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f , italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT + italic_κ over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(53)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.2.p2.82">Noting the derivation from (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A3.E44" title="In Proof. ‣ Appendix C The proof of Theorem 2 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">44</span></a>) to (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A3.E45" title="In Proof. ‣ Appendix C The proof of Theorem 2 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">45</span></a>), one omits the similar steps to directly give the following result:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx68"> <tbody id="A3.E54"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}\!\leq\!-k_{\rm c}\bm{e}^{T}\bm{e}\!-\!\kappa(1\!+\!p)% \tilde{\bm{\theta}}_{\zeta}^{T}Q_{\zeta}(t,t_{\rm e}){\tilde{\bm{\theta}}}_{% \zeta}\!+\!\bm{e}^{T}(\Phi_{\zeta}\!-\!\Phi_{{\rm f},\zeta})\tilde{\bm{\theta}% }_{\zeta}." class="ltx_Math" display="inline" id="A3.E54.m1.4"><semantics id="A3.E54.m1.4a"><mrow id="A3.E54.m1.4.4.1" xref="A3.E54.m1.4.4.1.1.cmml"><mrow id="A3.E54.m1.4.4.1.1" xref="A3.E54.m1.4.4.1.1.cmml"><mover accent="true" id="A3.E54.m1.4.4.1.1.5" xref="A3.E54.m1.4.4.1.1.5.cmml"><mi id="A3.E54.m1.4.4.1.1.5.2" xref="A3.E54.m1.4.4.1.1.5.2.cmml">V</mi><mo id="A3.E54.m1.4.4.1.1.5.1" xref="A3.E54.m1.4.4.1.1.5.1.cmml">˙</mo></mover><mo id="A3.E54.m1.4.4.1.1.4" lspace="0.108em" rspace="0.108em" xref="A3.E54.m1.4.4.1.1.4.cmml">≤</mo><mrow id="A3.E54.m1.4.4.1.1.3" xref="A3.E54.m1.4.4.1.1.3.cmml"><mrow id="A3.E54.m1.4.4.1.1.2.2" xref="A3.E54.m1.4.4.1.1.2.2.cmml"><mrow id="A3.E54.m1.4.4.1.1.2.2.4" xref="A3.E54.m1.4.4.1.1.2.2.4.cmml"><mo id="A3.E54.m1.4.4.1.1.2.2.4a" xref="A3.E54.m1.4.4.1.1.2.2.4.cmml">−</mo><mrow id="A3.E54.m1.4.4.1.1.2.2.4.2" xref="A3.E54.m1.4.4.1.1.2.2.4.2.cmml"><msub id="A3.E54.m1.4.4.1.1.2.2.4.2.2" xref="A3.E54.m1.4.4.1.1.2.2.4.2.2.cmml"><mi id="A3.E54.m1.4.4.1.1.2.2.4.2.2.2" xref="A3.E54.m1.4.4.1.1.2.2.4.2.2.2.cmml">k</mi><mi id="A3.E54.m1.4.4.1.1.2.2.4.2.2.3" mathvariant="normal" xref="A3.E54.m1.4.4.1.1.2.2.4.2.2.3.cmml">c</mi></msub><mo id="A3.E54.m1.4.4.1.1.2.2.4.2.1" xref="A3.E54.m1.4.4.1.1.2.2.4.2.1.cmml">⁢</mo><msup id="A3.E54.m1.4.4.1.1.2.2.4.2.3" xref="A3.E54.m1.4.4.1.1.2.2.4.2.3.cmml"><mi id="A3.E54.m1.4.4.1.1.2.2.4.2.3.2" xref="A3.E54.m1.4.4.1.1.2.2.4.2.3.2.cmml">𝒆</mi><mi id="A3.E54.m1.4.4.1.1.2.2.4.2.3.3" xref="A3.E54.m1.4.4.1.1.2.2.4.2.3.3.cmml">T</mi></msup><mo id="A3.E54.m1.4.4.1.1.2.2.4.2.1a" xref="A3.E54.m1.4.4.1.1.2.2.4.2.1.cmml">⁢</mo><mi id="A3.E54.m1.4.4.1.1.2.2.4.2.4" xref="A3.E54.m1.4.4.1.1.2.2.4.2.4.cmml">𝒆</mi></mrow></mrow><mo id="A3.E54.m1.4.4.1.1.2.2.3" lspace="0.052em" rspace="0.052em" xref="A3.E54.m1.4.4.1.1.2.2.3.cmml">−</mo><mrow id="A3.E54.m1.4.4.1.1.2.2.2" xref="A3.E54.m1.4.4.1.1.2.2.2.cmml"><mi id="A3.E54.m1.4.4.1.1.2.2.2.4" xref="A3.E54.m1.4.4.1.1.2.2.2.4.cmml">κ</mi><mo id="A3.E54.m1.4.4.1.1.2.2.2.3" xref="A3.E54.m1.4.4.1.1.2.2.2.3.cmml">⁢</mo><mrow id="A3.E54.m1.4.4.1.1.1.1.1.1.1" xref="A3.E54.m1.4.4.1.1.1.1.1.1.1.1.cmml"><mo id="A3.E54.m1.4.4.1.1.1.1.1.1.1.2" stretchy="false" xref="A3.E54.m1.4.4.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.E54.m1.4.4.1.1.1.1.1.1.1.1" xref="A3.E54.m1.4.4.1.1.1.1.1.1.1.1.cmml"><mn id="A3.E54.m1.4.4.1.1.1.1.1.1.1.1.2" xref="A3.E54.m1.4.4.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A3.E54.m1.4.4.1.1.1.1.1.1.1.1.1" lspace="0.052em" rspace="0.052em" xref="A3.E54.m1.4.4.1.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="A3.E54.m1.4.4.1.1.1.1.1.1.1.1.3" xref="A3.E54.m1.4.4.1.1.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A3.E54.m1.4.4.1.1.1.1.1.1.1.3" stretchy="false" xref="A3.E54.m1.4.4.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A3.E54.m1.4.4.1.1.2.2.2.3a" xref="A3.E54.m1.4.4.1.1.2.2.2.3.cmml">⁢</mo><msubsup id="A3.E54.m1.4.4.1.1.2.2.2.5" xref="A3.E54.m1.4.4.1.1.2.2.2.5.cmml"><mover accent="true" id="A3.E54.m1.4.4.1.1.2.2.2.5.2.2" xref="A3.E54.m1.4.4.1.1.2.2.2.5.2.2.cmml"><mi id="A3.E54.m1.4.4.1.1.2.2.2.5.2.2.2" xref="A3.E54.m1.4.4.1.1.2.2.2.5.2.2.2.cmml">𝜽</mi><mo id="A3.E54.m1.4.4.1.1.2.2.2.5.2.2.1" xref="A3.E54.m1.4.4.1.1.2.2.2.5.2.2.1.cmml">~</mo></mover><mi id="A3.E54.m1.4.4.1.1.2.2.2.5.2.3" xref="A3.E54.m1.4.4.1.1.2.2.2.5.2.3.cmml">ζ</mi><mi id="A3.E54.m1.4.4.1.1.2.2.2.5.3" xref="A3.E54.m1.4.4.1.1.2.2.2.5.3.cmml">T</mi></msubsup><mo id="A3.E54.m1.4.4.1.1.2.2.2.3b" xref="A3.E54.m1.4.4.1.1.2.2.2.3.cmml">⁢</mo><msub id="A3.E54.m1.4.4.1.1.2.2.2.6" xref="A3.E54.m1.4.4.1.1.2.2.2.6.cmml"><mi id="A3.E54.m1.4.4.1.1.2.2.2.6.2" xref="A3.E54.m1.4.4.1.1.2.2.2.6.2.cmml">Q</mi><mi id="A3.E54.m1.4.4.1.1.2.2.2.6.3" xref="A3.E54.m1.4.4.1.1.2.2.2.6.3.cmml">ζ</mi></msub><mo id="A3.E54.m1.4.4.1.1.2.2.2.3c" xref="A3.E54.m1.4.4.1.1.2.2.2.3.cmml">⁢</mo><mrow id="A3.E54.m1.4.4.1.1.2.2.2.2.1" xref="A3.E54.m1.4.4.1.1.2.2.2.2.2.cmml"><mo id="A3.E54.m1.4.4.1.1.2.2.2.2.1.2" stretchy="false" xref="A3.E54.m1.4.4.1.1.2.2.2.2.2.cmml">(</mo><mi id="A3.E54.m1.3.3" xref="A3.E54.m1.3.3.cmml">t</mi><mo id="A3.E54.m1.4.4.1.1.2.2.2.2.1.3" xref="A3.E54.m1.4.4.1.1.2.2.2.2.2.cmml">,</mo><msub id="A3.E54.m1.4.4.1.1.2.2.2.2.1.1" xref="A3.E54.m1.4.4.1.1.2.2.2.2.1.1.cmml"><mi id="A3.E54.m1.4.4.1.1.2.2.2.2.1.1.2" xref="A3.E54.m1.4.4.1.1.2.2.2.2.1.1.2.cmml">t</mi><mi id="A3.E54.m1.4.4.1.1.2.2.2.2.1.1.3" mathvariant="normal" xref="A3.E54.m1.4.4.1.1.2.2.2.2.1.1.3.cmml">e</mi></msub><mo id="A3.E54.m1.4.4.1.1.2.2.2.2.1.4" stretchy="false" xref="A3.E54.m1.4.4.1.1.2.2.2.2.2.cmml">)</mo></mrow><mo id="A3.E54.m1.4.4.1.1.2.2.2.3d" xref="A3.E54.m1.4.4.1.1.2.2.2.3.cmml">⁢</mo><msub id="A3.E54.m1.4.4.1.1.2.2.2.7" xref="A3.E54.m1.4.4.1.1.2.2.2.7.cmml"><mover accent="true" id="A3.E54.m1.4.4.1.1.2.2.2.7.2" xref="A3.E54.m1.4.4.1.1.2.2.2.7.2.cmml"><mi id="A3.E54.m1.4.4.1.1.2.2.2.7.2.2" xref="A3.E54.m1.4.4.1.1.2.2.2.7.2.2.cmml">𝜽</mi><mo id="A3.E54.m1.4.4.1.1.2.2.2.7.2.1" xref="A3.E54.m1.4.4.1.1.2.2.2.7.2.1.cmml">~</mo></mover><mi id="A3.E54.m1.4.4.1.1.2.2.2.7.3" xref="A3.E54.m1.4.4.1.1.2.2.2.7.3.cmml">ζ</mi></msub></mrow></mrow><mo id="A3.E54.m1.4.4.1.1.3.4" rspace="0.052em" xref="A3.E54.m1.4.4.1.1.3.4.cmml">+</mo><mrow id="A3.E54.m1.4.4.1.1.3.3" xref="A3.E54.m1.4.4.1.1.3.3.cmml"><msup id="A3.E54.m1.4.4.1.1.3.3.3" xref="A3.E54.m1.4.4.1.1.3.3.3.cmml"><mi id="A3.E54.m1.4.4.1.1.3.3.3.2" xref="A3.E54.m1.4.4.1.1.3.3.3.2.cmml">𝒆</mi><mi id="A3.E54.m1.4.4.1.1.3.3.3.3" xref="A3.E54.m1.4.4.1.1.3.3.3.3.cmml">T</mi></msup><mo id="A3.E54.m1.4.4.1.1.3.3.2" xref="A3.E54.m1.4.4.1.1.3.3.2.cmml">⁢</mo><mrow id="A3.E54.m1.4.4.1.1.3.3.1.1" xref="A3.E54.m1.4.4.1.1.3.3.1.1.1.cmml"><mo id="A3.E54.m1.4.4.1.1.3.3.1.1.2" stretchy="false" xref="A3.E54.m1.4.4.1.1.3.3.1.1.1.cmml">(</mo><mrow id="A3.E54.m1.4.4.1.1.3.3.1.1.1" xref="A3.E54.m1.4.4.1.1.3.3.1.1.1.cmml"><msub id="A3.E54.m1.4.4.1.1.3.3.1.1.1.2" xref="A3.E54.m1.4.4.1.1.3.3.1.1.1.2.cmml"><mi id="A3.E54.m1.4.4.1.1.3.3.1.1.1.2.2" mathvariant="normal" xref="A3.E54.m1.4.4.1.1.3.3.1.1.1.2.2.cmml">Φ</mi><mi id="A3.E54.m1.4.4.1.1.3.3.1.1.1.2.3" xref="A3.E54.m1.4.4.1.1.3.3.1.1.1.2.3.cmml">ζ</mi></msub><mo id="A3.E54.m1.4.4.1.1.3.3.1.1.1.1" rspace="0.052em" xref="A3.E54.m1.4.4.1.1.3.3.1.1.1.1.cmml">−</mo><msub id="A3.E54.m1.4.4.1.1.3.3.1.1.1.3" xref="A3.E54.m1.4.4.1.1.3.3.1.1.1.3.cmml"><mi id="A3.E54.m1.4.4.1.1.3.3.1.1.1.3.2" mathvariant="normal" xref="A3.E54.m1.4.4.1.1.3.3.1.1.1.3.2.cmml">Φ</mi><mrow id="A3.E54.m1.2.2.2.4" xref="A3.E54.m1.2.2.2.3.cmml"><mi id="A3.E54.m1.1.1.1.1" mathvariant="normal" xref="A3.E54.m1.1.1.1.1.cmml">f</mi><mo id="A3.E54.m1.2.2.2.4.1" xref="A3.E54.m1.2.2.2.3.cmml">,</mo><mi id="A3.E54.m1.2.2.2.2" xref="A3.E54.m1.2.2.2.2.cmml">ζ</mi></mrow></msub></mrow><mo id="A3.E54.m1.4.4.1.1.3.3.1.1.3" stretchy="false" xref="A3.E54.m1.4.4.1.1.3.3.1.1.1.cmml">)</mo></mrow><mo id="A3.E54.m1.4.4.1.1.3.3.2a" xref="A3.E54.m1.4.4.1.1.3.3.2.cmml">⁢</mo><msub id="A3.E54.m1.4.4.1.1.3.3.4" xref="A3.E54.m1.4.4.1.1.3.3.4.cmml"><mover accent="true" id="A3.E54.m1.4.4.1.1.3.3.4.2" xref="A3.E54.m1.4.4.1.1.3.3.4.2.cmml"><mi id="A3.E54.m1.4.4.1.1.3.3.4.2.2" xref="A3.E54.m1.4.4.1.1.3.3.4.2.2.cmml">𝜽</mi><mo id="A3.E54.m1.4.4.1.1.3.3.4.2.1" xref="A3.E54.m1.4.4.1.1.3.3.4.2.1.cmml">~</mo></mover><mi id="A3.E54.m1.4.4.1.1.3.3.4.3" xref="A3.E54.m1.4.4.1.1.3.3.4.3.cmml">ζ</mi></msub></mrow></mrow></mrow><mo id="A3.E54.m1.4.4.1.2" lspace="0em" xref="A3.E54.m1.4.4.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.E54.m1.4b"><apply id="A3.E54.m1.4.4.1.1.cmml" xref="A3.E54.m1.4.4.1"><leq id="A3.E54.m1.4.4.1.1.4.cmml" xref="A3.E54.m1.4.4.1.1.4"></leq><apply id="A3.E54.m1.4.4.1.1.5.cmml" xref="A3.E54.m1.4.4.1.1.5"><ci id="A3.E54.m1.4.4.1.1.5.1.cmml" xref="A3.E54.m1.4.4.1.1.5.1">˙</ci><ci id="A3.E54.m1.4.4.1.1.5.2.cmml" xref="A3.E54.m1.4.4.1.1.5.2">𝑉</ci></apply><apply id="A3.E54.m1.4.4.1.1.3.cmml" xref="A3.E54.m1.4.4.1.1.3"><plus id="A3.E54.m1.4.4.1.1.3.4.cmml" xref="A3.E54.m1.4.4.1.1.3.4"></plus><apply id="A3.E54.m1.4.4.1.1.2.2.cmml" xref="A3.E54.m1.4.4.1.1.2.2"><minus id="A3.E54.m1.4.4.1.1.2.2.3.cmml" xref="A3.E54.m1.4.4.1.1.2.2.3"></minus><apply id="A3.E54.m1.4.4.1.1.2.2.4.cmml" xref="A3.E54.m1.4.4.1.1.2.2.4"><minus id="A3.E54.m1.4.4.1.1.2.2.4.1.cmml" xref="A3.E54.m1.4.4.1.1.2.2.4"></minus><apply id="A3.E54.m1.4.4.1.1.2.2.4.2.cmml" xref="A3.E54.m1.4.4.1.1.2.2.4.2"><times id="A3.E54.m1.4.4.1.1.2.2.4.2.1.cmml" xref="A3.E54.m1.4.4.1.1.2.2.4.2.1"></times><apply id="A3.E54.m1.4.4.1.1.2.2.4.2.2.cmml" xref="A3.E54.m1.4.4.1.1.2.2.4.2.2"><csymbol cd="ambiguous" id="A3.E54.m1.4.4.1.1.2.2.4.2.2.1.cmml" xref="A3.E54.m1.4.4.1.1.2.2.4.2.2">subscript</csymbol><ci id="A3.E54.m1.4.4.1.1.2.2.4.2.2.2.cmml" xref="A3.E54.m1.4.4.1.1.2.2.4.2.2.2">𝑘</ci><ci id="A3.E54.m1.4.4.1.1.2.2.4.2.2.3.cmml" xref="A3.E54.m1.4.4.1.1.2.2.4.2.2.3">c</ci></apply><apply id="A3.E54.m1.4.4.1.1.2.2.4.2.3.cmml" xref="A3.E54.m1.4.4.1.1.2.2.4.2.3"><csymbol cd="ambiguous" id="A3.E54.m1.4.4.1.1.2.2.4.2.3.1.cmml" xref="A3.E54.m1.4.4.1.1.2.2.4.2.3">superscript</csymbol><ci id="A3.E54.m1.4.4.1.1.2.2.4.2.3.2.cmml" xref="A3.E54.m1.4.4.1.1.2.2.4.2.3.2">𝒆</ci><ci id="A3.E54.m1.4.4.1.1.2.2.4.2.3.3.cmml" xref="A3.E54.m1.4.4.1.1.2.2.4.2.3.3">𝑇</ci></apply><ci id="A3.E54.m1.4.4.1.1.2.2.4.2.4.cmml" xref="A3.E54.m1.4.4.1.1.2.2.4.2.4">𝒆</ci></apply></apply><apply id="A3.E54.m1.4.4.1.1.2.2.2.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2"><times id="A3.E54.m1.4.4.1.1.2.2.2.3.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.3"></times><ci id="A3.E54.m1.4.4.1.1.2.2.2.4.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.4">𝜅</ci><apply id="A3.E54.m1.4.4.1.1.1.1.1.1.1.1.cmml" xref="A3.E54.m1.4.4.1.1.1.1.1.1.1"><plus id="A3.E54.m1.4.4.1.1.1.1.1.1.1.1.1.cmml" xref="A3.E54.m1.4.4.1.1.1.1.1.1.1.1.1"></plus><cn id="A3.E54.m1.4.4.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A3.E54.m1.4.4.1.1.1.1.1.1.1.1.2">1</cn><ci id="A3.E54.m1.4.4.1.1.1.1.1.1.1.1.3.cmml" xref="A3.E54.m1.4.4.1.1.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="A3.E54.m1.4.4.1.1.2.2.2.5.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.5"><csymbol cd="ambiguous" id="A3.E54.m1.4.4.1.1.2.2.2.5.1.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.5">superscript</csymbol><apply id="A3.E54.m1.4.4.1.1.2.2.2.5.2.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.5"><csymbol cd="ambiguous" id="A3.E54.m1.4.4.1.1.2.2.2.5.2.1.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.5">subscript</csymbol><apply id="A3.E54.m1.4.4.1.1.2.2.2.5.2.2.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.5.2.2"><ci id="A3.E54.m1.4.4.1.1.2.2.2.5.2.2.1.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.5.2.2.1">~</ci><ci id="A3.E54.m1.4.4.1.1.2.2.2.5.2.2.2.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.5.2.2.2">𝜽</ci></apply><ci id="A3.E54.m1.4.4.1.1.2.2.2.5.2.3.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.5.2.3">𝜁</ci></apply><ci id="A3.E54.m1.4.4.1.1.2.2.2.5.3.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.5.3">𝑇</ci></apply><apply id="A3.E54.m1.4.4.1.1.2.2.2.6.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.6"><csymbol cd="ambiguous" id="A3.E54.m1.4.4.1.1.2.2.2.6.1.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.6">subscript</csymbol><ci id="A3.E54.m1.4.4.1.1.2.2.2.6.2.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.6.2">𝑄</ci><ci id="A3.E54.m1.4.4.1.1.2.2.2.6.3.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.6.3">𝜁</ci></apply><interval closure="open" id="A3.E54.m1.4.4.1.1.2.2.2.2.2.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.2.1"><ci id="A3.E54.m1.3.3.cmml" xref="A3.E54.m1.3.3">𝑡</ci><apply id="A3.E54.m1.4.4.1.1.2.2.2.2.1.1.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.2.1.1"><csymbol cd="ambiguous" id="A3.E54.m1.4.4.1.1.2.2.2.2.1.1.1.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.2.1.1">subscript</csymbol><ci id="A3.E54.m1.4.4.1.1.2.2.2.2.1.1.2.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.2.1.1.2">𝑡</ci><ci id="A3.E54.m1.4.4.1.1.2.2.2.2.1.1.3.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.2.1.1.3">e</ci></apply></interval><apply id="A3.E54.m1.4.4.1.1.2.2.2.7.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.7"><csymbol cd="ambiguous" id="A3.E54.m1.4.4.1.1.2.2.2.7.1.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.7">subscript</csymbol><apply id="A3.E54.m1.4.4.1.1.2.2.2.7.2.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.7.2"><ci id="A3.E54.m1.4.4.1.1.2.2.2.7.2.1.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.7.2.1">~</ci><ci id="A3.E54.m1.4.4.1.1.2.2.2.7.2.2.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.7.2.2">𝜽</ci></apply><ci id="A3.E54.m1.4.4.1.1.2.2.2.7.3.cmml" xref="A3.E54.m1.4.4.1.1.2.2.2.7.3">𝜁</ci></apply></apply></apply><apply id="A3.E54.m1.4.4.1.1.3.3.cmml" xref="A3.E54.m1.4.4.1.1.3.3"><times id="A3.E54.m1.4.4.1.1.3.3.2.cmml" xref="A3.E54.m1.4.4.1.1.3.3.2"></times><apply id="A3.E54.m1.4.4.1.1.3.3.3.cmml" xref="A3.E54.m1.4.4.1.1.3.3.3"><csymbol cd="ambiguous" id="A3.E54.m1.4.4.1.1.3.3.3.1.cmml" xref="A3.E54.m1.4.4.1.1.3.3.3">superscript</csymbol><ci id="A3.E54.m1.4.4.1.1.3.3.3.2.cmml" xref="A3.E54.m1.4.4.1.1.3.3.3.2">𝒆</ci><ci id="A3.E54.m1.4.4.1.1.3.3.3.3.cmml" xref="A3.E54.m1.4.4.1.1.3.3.3.3">𝑇</ci></apply><apply id="A3.E54.m1.4.4.1.1.3.3.1.1.1.cmml" xref="A3.E54.m1.4.4.1.1.3.3.1.1"><minus id="A3.E54.m1.4.4.1.1.3.3.1.1.1.1.cmml" xref="A3.E54.m1.4.4.1.1.3.3.1.1.1.1"></minus><apply id="A3.E54.m1.4.4.1.1.3.3.1.1.1.2.cmml" xref="A3.E54.m1.4.4.1.1.3.3.1.1.1.2"><csymbol cd="ambiguous" id="A3.E54.m1.4.4.1.1.3.3.1.1.1.2.1.cmml" xref="A3.E54.m1.4.4.1.1.3.3.1.1.1.2">subscript</csymbol><ci id="A3.E54.m1.4.4.1.1.3.3.1.1.1.2.2.cmml" xref="A3.E54.m1.4.4.1.1.3.3.1.1.1.2.2">Φ</ci><ci id="A3.E54.m1.4.4.1.1.3.3.1.1.1.2.3.cmml" xref="A3.E54.m1.4.4.1.1.3.3.1.1.1.2.3">𝜁</ci></apply><apply id="A3.E54.m1.4.4.1.1.3.3.1.1.1.3.cmml" xref="A3.E54.m1.4.4.1.1.3.3.1.1.1.3"><csymbol cd="ambiguous" id="A3.E54.m1.4.4.1.1.3.3.1.1.1.3.1.cmml" xref="A3.E54.m1.4.4.1.1.3.3.1.1.1.3">subscript</csymbol><ci id="A3.E54.m1.4.4.1.1.3.3.1.1.1.3.2.cmml" xref="A3.E54.m1.4.4.1.1.3.3.1.1.1.3.2">Φ</ci><list id="A3.E54.m1.2.2.2.3.cmml" xref="A3.E54.m1.2.2.2.4"><ci id="A3.E54.m1.1.1.1.1.cmml" xref="A3.E54.m1.1.1.1.1">f</ci><ci id="A3.E54.m1.2.2.2.2.cmml" xref="A3.E54.m1.2.2.2.2">𝜁</ci></list></apply></apply><apply id="A3.E54.m1.4.4.1.1.3.3.4.cmml" xref="A3.E54.m1.4.4.1.1.3.3.4"><csymbol cd="ambiguous" id="A3.E54.m1.4.4.1.1.3.3.4.1.cmml" xref="A3.E54.m1.4.4.1.1.3.3.4">subscript</csymbol><apply id="A3.E54.m1.4.4.1.1.3.3.4.2.cmml" xref="A3.E54.m1.4.4.1.1.3.3.4.2"><ci id="A3.E54.m1.4.4.1.1.3.3.4.2.1.cmml" xref="A3.E54.m1.4.4.1.1.3.3.4.2.1">~</ci><ci id="A3.E54.m1.4.4.1.1.3.3.4.2.2.cmml" xref="A3.E54.m1.4.4.1.1.3.3.4.2.2">𝜽</ci></apply><ci id="A3.E54.m1.4.4.1.1.3.3.4.3.cmml" xref="A3.E54.m1.4.4.1.1.3.3.4.3">𝜁</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E54.m1.4c">\displaystyle\dot{V}\!\leq\!-k_{\rm c}\bm{e}^{T}\bm{e}\!-\!\kappa(1\!+\!p)% \tilde{\bm{\theta}}_{\zeta}^{T}Q_{\zeta}(t,t_{\rm e}){\tilde{\bm{\theta}}}_{% \zeta}\!+\!\bm{e}^{T}(\Phi_{\zeta}\!-\!\Phi_{{\rm f},\zeta})\tilde{\bm{\theta}% }_{\zeta}.</annotation><annotation encoding="application/x-llamapun" id="A3.E54.m1.4d">over˙ start_ARG italic_V end_ARG ≤ - italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_e - italic_κ ( 1 + italic_p ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT italic_Q start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT + bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( roman_Φ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT - roman_Φ start_POSTSUBSCRIPT roman_f , italic_ζ end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(54)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.2.p2.57">As there exist <math alttext="T_{\rm a}" class="ltx_Math" display="inline" id="A3.2.p2.50.m1.1"><semantics id="A3.2.p2.50.m1.1a"><msub id="A3.2.p2.50.m1.1.1" xref="A3.2.p2.50.m1.1.1.cmml"><mi id="A3.2.p2.50.m1.1.1.2" xref="A3.2.p2.50.m1.1.1.2.cmml">T</mi><mi id="A3.2.p2.50.m1.1.1.3" mathvariant="normal" xref="A3.2.p2.50.m1.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="A3.2.p2.50.m1.1b"><apply id="A3.2.p2.50.m1.1.1.cmml" xref="A3.2.p2.50.m1.1.1"><csymbol cd="ambiguous" id="A3.2.p2.50.m1.1.1.1.cmml" xref="A3.2.p2.50.m1.1.1">subscript</csymbol><ci id="A3.2.p2.50.m1.1.1.2.cmml" xref="A3.2.p2.50.m1.1.1.2">𝑇</ci><ci id="A3.2.p2.50.m1.1.1.3.cmml" xref="A3.2.p2.50.m1.1.1.3">a</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.50.m1.1c">T_{\rm a}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.50.m1.1d">italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\sigma\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A3.2.p2.51.m2.1"><semantics id="A3.2.p2.51.m2.1a"><mrow id="A3.2.p2.51.m2.1.1" xref="A3.2.p2.51.m2.1.1.cmml"><mi id="A3.2.p2.51.m2.1.1.2" xref="A3.2.p2.51.m2.1.1.2.cmml">σ</mi><mo id="A3.2.p2.51.m2.1.1.1" xref="A3.2.p2.51.m2.1.1.1.cmml">∈</mo><msup id="A3.2.p2.51.m2.1.1.3" xref="A3.2.p2.51.m2.1.1.3.cmml"><mi id="A3.2.p2.51.m2.1.1.3.2" xref="A3.2.p2.51.m2.1.1.3.2.cmml">ℝ</mi><mo id="A3.2.p2.51.m2.1.1.3.3" xref="A3.2.p2.51.m2.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.51.m2.1b"><apply id="A3.2.p2.51.m2.1.1.cmml" xref="A3.2.p2.51.m2.1.1"><in id="A3.2.p2.51.m2.1.1.1.cmml" xref="A3.2.p2.51.m2.1.1.1"></in><ci id="A3.2.p2.51.m2.1.1.2.cmml" xref="A3.2.p2.51.m2.1.1.2">𝜎</ci><apply id="A3.2.p2.51.m2.1.1.3.cmml" xref="A3.2.p2.51.m2.1.1.3"><csymbol cd="ambiguous" id="A3.2.p2.51.m2.1.1.3.1.cmml" xref="A3.2.p2.51.m2.1.1.3">superscript</csymbol><ci id="A3.2.p2.51.m2.1.1.3.2.cmml" xref="A3.2.p2.51.m2.1.1.3.2">ℝ</ci><plus id="A3.2.p2.51.m2.1.1.3.3.cmml" xref="A3.2.p2.51.m2.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.51.m2.1c">\sigma\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.51.m2.1d">italic_σ ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> to satisfy the partial IE condition, one gets <math alttext="\Psi_{\zeta}(t_{\rm e})\geq\sigma_{\rm c}(T_{\rm a})I\geq\sigma I" class="ltx_Math" display="inline" id="A3.2.p2.52.m3.2"><semantics id="A3.2.p2.52.m3.2a"><mrow id="A3.2.p2.52.m3.2.2" xref="A3.2.p2.52.m3.2.2.cmml"><mrow id="A3.2.p2.52.m3.1.1.1" xref="A3.2.p2.52.m3.1.1.1.cmml"><msub id="A3.2.p2.52.m3.1.1.1.3" xref="A3.2.p2.52.m3.1.1.1.3.cmml"><mi id="A3.2.p2.52.m3.1.1.1.3.2" mathvariant="normal" xref="A3.2.p2.52.m3.1.1.1.3.2.cmml">Ψ</mi><mi id="A3.2.p2.52.m3.1.1.1.3.3" xref="A3.2.p2.52.m3.1.1.1.3.3.cmml">ζ</mi></msub><mo id="A3.2.p2.52.m3.1.1.1.2" xref="A3.2.p2.52.m3.1.1.1.2.cmml">⁢</mo><mrow id="A3.2.p2.52.m3.1.1.1.1.1" xref="A3.2.p2.52.m3.1.1.1.1.1.1.cmml"><mo id="A3.2.p2.52.m3.1.1.1.1.1.2" stretchy="false" xref="A3.2.p2.52.m3.1.1.1.1.1.1.cmml">(</mo><msub id="A3.2.p2.52.m3.1.1.1.1.1.1" xref="A3.2.p2.52.m3.1.1.1.1.1.1.cmml"><mi id="A3.2.p2.52.m3.1.1.1.1.1.1.2" xref="A3.2.p2.52.m3.1.1.1.1.1.1.2.cmml">t</mi><mi id="A3.2.p2.52.m3.1.1.1.1.1.1.3" mathvariant="normal" xref="A3.2.p2.52.m3.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="A3.2.p2.52.m3.1.1.1.1.1.3" stretchy="false" xref="A3.2.p2.52.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A3.2.p2.52.m3.2.2.4" xref="A3.2.p2.52.m3.2.2.4.cmml">≥</mo><mrow id="A3.2.p2.52.m3.2.2.2" xref="A3.2.p2.52.m3.2.2.2.cmml"><msub id="A3.2.p2.52.m3.2.2.2.3" xref="A3.2.p2.52.m3.2.2.2.3.cmml"><mi id="A3.2.p2.52.m3.2.2.2.3.2" xref="A3.2.p2.52.m3.2.2.2.3.2.cmml">σ</mi><mi id="A3.2.p2.52.m3.2.2.2.3.3" mathvariant="normal" xref="A3.2.p2.52.m3.2.2.2.3.3.cmml">c</mi></msub><mo id="A3.2.p2.52.m3.2.2.2.2" xref="A3.2.p2.52.m3.2.2.2.2.cmml">⁢</mo><mrow id="A3.2.p2.52.m3.2.2.2.1.1" xref="A3.2.p2.52.m3.2.2.2.1.1.1.cmml"><mo id="A3.2.p2.52.m3.2.2.2.1.1.2" stretchy="false" xref="A3.2.p2.52.m3.2.2.2.1.1.1.cmml">(</mo><msub id="A3.2.p2.52.m3.2.2.2.1.1.1" xref="A3.2.p2.52.m3.2.2.2.1.1.1.cmml"><mi id="A3.2.p2.52.m3.2.2.2.1.1.1.2" xref="A3.2.p2.52.m3.2.2.2.1.1.1.2.cmml">T</mi><mi id="A3.2.p2.52.m3.2.2.2.1.1.1.3" mathvariant="normal" xref="A3.2.p2.52.m3.2.2.2.1.1.1.3.cmml">a</mi></msub><mo id="A3.2.p2.52.m3.2.2.2.1.1.3" stretchy="false" xref="A3.2.p2.52.m3.2.2.2.1.1.1.cmml">)</mo></mrow><mo id="A3.2.p2.52.m3.2.2.2.2a" xref="A3.2.p2.52.m3.2.2.2.2.cmml">⁢</mo><mi id="A3.2.p2.52.m3.2.2.2.4" xref="A3.2.p2.52.m3.2.2.2.4.cmml">I</mi></mrow><mo id="A3.2.p2.52.m3.2.2.5" xref="A3.2.p2.52.m3.2.2.5.cmml">≥</mo><mrow id="A3.2.p2.52.m3.2.2.6" xref="A3.2.p2.52.m3.2.2.6.cmml"><mi id="A3.2.p2.52.m3.2.2.6.2" xref="A3.2.p2.52.m3.2.2.6.2.cmml">σ</mi><mo id="A3.2.p2.52.m3.2.2.6.1" xref="A3.2.p2.52.m3.2.2.6.1.cmml">⁢</mo><mi id="A3.2.p2.52.m3.2.2.6.3" xref="A3.2.p2.52.m3.2.2.6.3.cmml">I</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.52.m3.2b"><apply id="A3.2.p2.52.m3.2.2.cmml" xref="A3.2.p2.52.m3.2.2"><and id="A3.2.p2.52.m3.2.2a.cmml" xref="A3.2.p2.52.m3.2.2"></and><apply id="A3.2.p2.52.m3.2.2b.cmml" xref="A3.2.p2.52.m3.2.2"><geq id="A3.2.p2.52.m3.2.2.4.cmml" xref="A3.2.p2.52.m3.2.2.4"></geq><apply id="A3.2.p2.52.m3.1.1.1.cmml" xref="A3.2.p2.52.m3.1.1.1"><times id="A3.2.p2.52.m3.1.1.1.2.cmml" xref="A3.2.p2.52.m3.1.1.1.2"></times><apply id="A3.2.p2.52.m3.1.1.1.3.cmml" xref="A3.2.p2.52.m3.1.1.1.3"><csymbol cd="ambiguous" id="A3.2.p2.52.m3.1.1.1.3.1.cmml" xref="A3.2.p2.52.m3.1.1.1.3">subscript</csymbol><ci id="A3.2.p2.52.m3.1.1.1.3.2.cmml" xref="A3.2.p2.52.m3.1.1.1.3.2">Ψ</ci><ci id="A3.2.p2.52.m3.1.1.1.3.3.cmml" xref="A3.2.p2.52.m3.1.1.1.3.3">𝜁</ci></apply><apply id="A3.2.p2.52.m3.1.1.1.1.1.1.cmml" xref="A3.2.p2.52.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.2.p2.52.m3.1.1.1.1.1.1.1.cmml" xref="A3.2.p2.52.m3.1.1.1.1.1">subscript</csymbol><ci id="A3.2.p2.52.m3.1.1.1.1.1.1.2.cmml" xref="A3.2.p2.52.m3.1.1.1.1.1.1.2">𝑡</ci><ci id="A3.2.p2.52.m3.1.1.1.1.1.1.3.cmml" xref="A3.2.p2.52.m3.1.1.1.1.1.1.3">e</ci></apply></apply><apply id="A3.2.p2.52.m3.2.2.2.cmml" xref="A3.2.p2.52.m3.2.2.2"><times id="A3.2.p2.52.m3.2.2.2.2.cmml" xref="A3.2.p2.52.m3.2.2.2.2"></times><apply id="A3.2.p2.52.m3.2.2.2.3.cmml" xref="A3.2.p2.52.m3.2.2.2.3"><csymbol cd="ambiguous" id="A3.2.p2.52.m3.2.2.2.3.1.cmml" xref="A3.2.p2.52.m3.2.2.2.3">subscript</csymbol><ci id="A3.2.p2.52.m3.2.2.2.3.2.cmml" xref="A3.2.p2.52.m3.2.2.2.3.2">𝜎</ci><ci id="A3.2.p2.52.m3.2.2.2.3.3.cmml" xref="A3.2.p2.52.m3.2.2.2.3.3">c</ci></apply><apply id="A3.2.p2.52.m3.2.2.2.1.1.1.cmml" xref="A3.2.p2.52.m3.2.2.2.1.1"><csymbol cd="ambiguous" id="A3.2.p2.52.m3.2.2.2.1.1.1.1.cmml" xref="A3.2.p2.52.m3.2.2.2.1.1">subscript</csymbol><ci id="A3.2.p2.52.m3.2.2.2.1.1.1.2.cmml" xref="A3.2.p2.52.m3.2.2.2.1.1.1.2">𝑇</ci><ci id="A3.2.p2.52.m3.2.2.2.1.1.1.3.cmml" xref="A3.2.p2.52.m3.2.2.2.1.1.1.3">a</ci></apply><ci id="A3.2.p2.52.m3.2.2.2.4.cmml" xref="A3.2.p2.52.m3.2.2.2.4">𝐼</ci></apply></apply><apply id="A3.2.p2.52.m3.2.2c.cmml" xref="A3.2.p2.52.m3.2.2"><geq id="A3.2.p2.52.m3.2.2.5.cmml" xref="A3.2.p2.52.m3.2.2.5"></geq><share href="https://arxiv.org/html/2401.10785v2#A3.2.p2.52.m3.2.2.2.cmml" id="A3.2.p2.52.m3.2.2d.cmml" xref="A3.2.p2.52.m3.2.2"></share><apply id="A3.2.p2.52.m3.2.2.6.cmml" xref="A3.2.p2.52.m3.2.2.6"><times id="A3.2.p2.52.m3.2.2.6.1.cmml" xref="A3.2.p2.52.m3.2.2.6.1"></times><ci id="A3.2.p2.52.m3.2.2.6.2.cmml" xref="A3.2.p2.52.m3.2.2.6.2">𝜎</ci><ci id="A3.2.p2.52.m3.2.2.6.3.cmml" xref="A3.2.p2.52.m3.2.2.6.3">𝐼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.52.m3.2c">\Psi_{\zeta}(t_{\rm e})\geq\sigma_{\rm c}(T_{\rm a})I\geq\sigma I</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.52.m3.2d">roman_Ψ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) ≥ italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ( italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT ) italic_I ≥ italic_σ italic_I</annotation></semantics></math> from (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E19" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">19</span></a>). As <math alttext="H(s)" class="ltx_Math" display="inline" id="A3.2.p2.53.m4.1"><semantics id="A3.2.p2.53.m4.1a"><mrow id="A3.2.p2.53.m4.1.2" xref="A3.2.p2.53.m4.1.2.cmml"><mi id="A3.2.p2.53.m4.1.2.2" xref="A3.2.p2.53.m4.1.2.2.cmml">H</mi><mo id="A3.2.p2.53.m4.1.2.1" xref="A3.2.p2.53.m4.1.2.1.cmml">⁢</mo><mrow id="A3.2.p2.53.m4.1.2.3.2" xref="A3.2.p2.53.m4.1.2.cmml"><mo id="A3.2.p2.53.m4.1.2.3.2.1" stretchy="false" xref="A3.2.p2.53.m4.1.2.cmml">(</mo><mi id="A3.2.p2.53.m4.1.1" xref="A3.2.p2.53.m4.1.1.cmml">s</mi><mo id="A3.2.p2.53.m4.1.2.3.2.2" stretchy="false" xref="A3.2.p2.53.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.53.m4.1b"><apply id="A3.2.p2.53.m4.1.2.cmml" xref="A3.2.p2.53.m4.1.2"><times id="A3.2.p2.53.m4.1.2.1.cmml" xref="A3.2.p2.53.m4.1.2.1"></times><ci id="A3.2.p2.53.m4.1.2.2.cmml" xref="A3.2.p2.53.m4.1.2.2">𝐻</ci><ci id="A3.2.p2.53.m4.1.1.cmml" xref="A3.2.p2.53.m4.1.1">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.53.m4.1c">H(s)</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.53.m4.1d">italic_H ( italic_s )</annotation></semantics></math> is a stable filter with unit DC gain, there exists a constant <math alttext="\sigma^{*}" class="ltx_Math" display="inline" id="A3.2.p2.54.m5.1"><semantics id="A3.2.p2.54.m5.1a"><msup id="A3.2.p2.54.m5.1.1" xref="A3.2.p2.54.m5.1.1.cmml"><mi id="A3.2.p2.54.m5.1.1.2" xref="A3.2.p2.54.m5.1.1.2.cmml">σ</mi><mo id="A3.2.p2.54.m5.1.1.3" xref="A3.2.p2.54.m5.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="A3.2.p2.54.m5.1b"><apply id="A3.2.p2.54.m5.1.1.cmml" xref="A3.2.p2.54.m5.1.1"><csymbol cd="ambiguous" id="A3.2.p2.54.m5.1.1.1.cmml" xref="A3.2.p2.54.m5.1.1">superscript</csymbol><ci id="A3.2.p2.54.m5.1.1.2.cmml" xref="A3.2.p2.54.m5.1.1.2">𝜎</ci><times id="A3.2.p2.54.m5.1.1.3.cmml" xref="A3.2.p2.54.m5.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.54.m5.1c">\sigma^{*}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.54.m5.1d">italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> <math alttext="\in" class="ltx_Math" display="inline" id="A3.2.p2.55.m6.1"><semantics id="A3.2.p2.55.m6.1a"><mo id="A3.2.p2.55.m6.1.1" xref="A3.2.p2.55.m6.1.1.cmml">∈</mo><annotation-xml encoding="MathML-Content" id="A3.2.p2.55.m6.1b"><in id="A3.2.p2.55.m6.1.1.cmml" xref="A3.2.p2.55.m6.1.1"></in></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.55.m6.1c">\in</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.55.m6.1d">∈</annotation></semantics></math> <math alttext="\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A3.2.p2.56.m7.1"><semantics id="A3.2.p2.56.m7.1a"><msup id="A3.2.p2.56.m7.1.1" xref="A3.2.p2.56.m7.1.1.cmml"><mi id="A3.2.p2.56.m7.1.1.2" xref="A3.2.p2.56.m7.1.1.2.cmml">ℝ</mi><mo id="A3.2.p2.56.m7.1.1.3" xref="A3.2.p2.56.m7.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="A3.2.p2.56.m7.1b"><apply id="A3.2.p2.56.m7.1.1.cmml" xref="A3.2.p2.56.m7.1.1"><csymbol cd="ambiguous" id="A3.2.p2.56.m7.1.1.1.cmml" xref="A3.2.p2.56.m7.1.1">superscript</csymbol><ci id="A3.2.p2.56.m7.1.1.2.cmml" xref="A3.2.p2.56.m7.1.1.2">ℝ</ci><plus id="A3.2.p2.56.m7.1.1.3.cmml" xref="A3.2.p2.56.m7.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.56.m7.1c">\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.56.m7.1d">blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="\sigma^{*}\leq\sigma" class="ltx_Math" display="inline" id="A3.2.p2.57.m8.1"><semantics id="A3.2.p2.57.m8.1a"><mrow id="A3.2.p2.57.m8.1.1" xref="A3.2.p2.57.m8.1.1.cmml"><msup id="A3.2.p2.57.m8.1.1.2" xref="A3.2.p2.57.m8.1.1.2.cmml"><mi id="A3.2.p2.57.m8.1.1.2.2" xref="A3.2.p2.57.m8.1.1.2.2.cmml">σ</mi><mo id="A3.2.p2.57.m8.1.1.2.3" xref="A3.2.p2.57.m8.1.1.2.3.cmml">∗</mo></msup><mo id="A3.2.p2.57.m8.1.1.1" xref="A3.2.p2.57.m8.1.1.1.cmml">≤</mo><mi id="A3.2.p2.57.m8.1.1.3" xref="A3.2.p2.57.m8.1.1.3.cmml">σ</mi></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.57.m8.1b"><apply id="A3.2.p2.57.m8.1.1.cmml" xref="A3.2.p2.57.m8.1.1"><leq id="A3.2.p2.57.m8.1.1.1.cmml" xref="A3.2.p2.57.m8.1.1.1"></leq><apply id="A3.2.p2.57.m8.1.1.2.cmml" xref="A3.2.p2.57.m8.1.1.2"><csymbol cd="ambiguous" id="A3.2.p2.57.m8.1.1.2.1.cmml" xref="A3.2.p2.57.m8.1.1.2">superscript</csymbol><ci id="A3.2.p2.57.m8.1.1.2.2.cmml" xref="A3.2.p2.57.m8.1.1.2.2">𝜎</ci><times id="A3.2.p2.57.m8.1.1.2.3.cmml" xref="A3.2.p2.57.m8.1.1.2.3"></times></apply><ci id="A3.2.p2.57.m8.1.1.3.cmml" xref="A3.2.p2.57.m8.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.57.m8.1c">\sigma^{*}\leq\sigma</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.57.m8.1d">italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ≤ italic_σ</annotation></semantics></math> such that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx69"> <tbody id="A3.Ex54"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle Q_{\zeta}(t,t_{\rm e})=H(s)[\Psi_{\zeta}(t_{\rm e})]\geq\sigma^{% *}I." class="ltx_Math" display="inline" id="A3.Ex54.m1.3"><semantics id="A3.Ex54.m1.3a"><mrow id="A3.Ex54.m1.3.3.1" xref="A3.Ex54.m1.3.3.1.1.cmml"><mrow id="A3.Ex54.m1.3.3.1.1" xref="A3.Ex54.m1.3.3.1.1.cmml"><mrow id="A3.Ex54.m1.3.3.1.1.1" xref="A3.Ex54.m1.3.3.1.1.1.cmml"><msub id="A3.Ex54.m1.3.3.1.1.1.3" xref="A3.Ex54.m1.3.3.1.1.1.3.cmml"><mi id="A3.Ex54.m1.3.3.1.1.1.3.2" xref="A3.Ex54.m1.3.3.1.1.1.3.2.cmml">Q</mi><mi id="A3.Ex54.m1.3.3.1.1.1.3.3" xref="A3.Ex54.m1.3.3.1.1.1.3.3.cmml">ζ</mi></msub><mo id="A3.Ex54.m1.3.3.1.1.1.2" xref="A3.Ex54.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="A3.Ex54.m1.3.3.1.1.1.1.1" xref="A3.Ex54.m1.3.3.1.1.1.1.2.cmml"><mo id="A3.Ex54.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="A3.Ex54.m1.3.3.1.1.1.1.2.cmml">(</mo><mi id="A3.Ex54.m1.1.1" xref="A3.Ex54.m1.1.1.cmml">t</mi><mo id="A3.Ex54.m1.3.3.1.1.1.1.1.3" xref="A3.Ex54.m1.3.3.1.1.1.1.2.cmml">,</mo><msub id="A3.Ex54.m1.3.3.1.1.1.1.1.1" xref="A3.Ex54.m1.3.3.1.1.1.1.1.1.cmml"><mi id="A3.Ex54.m1.3.3.1.1.1.1.1.1.2" xref="A3.Ex54.m1.3.3.1.1.1.1.1.1.2.cmml">t</mi><mi id="A3.Ex54.m1.3.3.1.1.1.1.1.1.3" mathvariant="normal" xref="A3.Ex54.m1.3.3.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="A3.Ex54.m1.3.3.1.1.1.1.1.4" stretchy="false" xref="A3.Ex54.m1.3.3.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="A3.Ex54.m1.3.3.1.1.4" xref="A3.Ex54.m1.3.3.1.1.4.cmml">=</mo><mrow id="A3.Ex54.m1.3.3.1.1.2" xref="A3.Ex54.m1.3.3.1.1.2.cmml"><mi id="A3.Ex54.m1.3.3.1.1.2.3" xref="A3.Ex54.m1.3.3.1.1.2.3.cmml">H</mi><mo id="A3.Ex54.m1.3.3.1.1.2.2" xref="A3.Ex54.m1.3.3.1.1.2.2.cmml">⁢</mo><mrow id="A3.Ex54.m1.3.3.1.1.2.4.2" xref="A3.Ex54.m1.3.3.1.1.2.cmml"><mo id="A3.Ex54.m1.3.3.1.1.2.4.2.1" stretchy="false" xref="A3.Ex54.m1.3.3.1.1.2.cmml">(</mo><mi id="A3.Ex54.m1.2.2" xref="A3.Ex54.m1.2.2.cmml">s</mi><mo id="A3.Ex54.m1.3.3.1.1.2.4.2.2" stretchy="false" xref="A3.Ex54.m1.3.3.1.1.2.cmml">)</mo></mrow><mo id="A3.Ex54.m1.3.3.1.1.2.2a" xref="A3.Ex54.m1.3.3.1.1.2.2.cmml">⁢</mo><mrow id="A3.Ex54.m1.3.3.1.1.2.1.1" xref="A3.Ex54.m1.3.3.1.1.2.1.2.cmml"><mo id="A3.Ex54.m1.3.3.1.1.2.1.1.2" stretchy="false" xref="A3.Ex54.m1.3.3.1.1.2.1.2.1.cmml">[</mo><mrow id="A3.Ex54.m1.3.3.1.1.2.1.1.1" xref="A3.Ex54.m1.3.3.1.1.2.1.1.1.cmml"><msub id="A3.Ex54.m1.3.3.1.1.2.1.1.1.3" xref="A3.Ex54.m1.3.3.1.1.2.1.1.1.3.cmml"><mi id="A3.Ex54.m1.3.3.1.1.2.1.1.1.3.2" mathvariant="normal" xref="A3.Ex54.m1.3.3.1.1.2.1.1.1.3.2.cmml">Ψ</mi><mi id="A3.Ex54.m1.3.3.1.1.2.1.1.1.3.3" xref="A3.Ex54.m1.3.3.1.1.2.1.1.1.3.3.cmml">ζ</mi></msub><mo id="A3.Ex54.m1.3.3.1.1.2.1.1.1.2" xref="A3.Ex54.m1.3.3.1.1.2.1.1.1.2.cmml">⁢</mo><mrow id="A3.Ex54.m1.3.3.1.1.2.1.1.1.1.1" xref="A3.Ex54.m1.3.3.1.1.2.1.1.1.1.1.1.cmml"><mo id="A3.Ex54.m1.3.3.1.1.2.1.1.1.1.1.2" stretchy="false" xref="A3.Ex54.m1.3.3.1.1.2.1.1.1.1.1.1.cmml">(</mo><msub id="A3.Ex54.m1.3.3.1.1.2.1.1.1.1.1.1" xref="A3.Ex54.m1.3.3.1.1.2.1.1.1.1.1.1.cmml"><mi id="A3.Ex54.m1.3.3.1.1.2.1.1.1.1.1.1.2" xref="A3.Ex54.m1.3.3.1.1.2.1.1.1.1.1.1.2.cmml">t</mi><mi id="A3.Ex54.m1.3.3.1.1.2.1.1.1.1.1.1.3" mathvariant="normal" xref="A3.Ex54.m1.3.3.1.1.2.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="A3.Ex54.m1.3.3.1.1.2.1.1.1.1.1.3" stretchy="false" xref="A3.Ex54.m1.3.3.1.1.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A3.Ex54.m1.3.3.1.1.2.1.1.3" stretchy="false" xref="A3.Ex54.m1.3.3.1.1.2.1.2.1.cmml">]</mo></mrow></mrow><mo id="A3.Ex54.m1.3.3.1.1.5" xref="A3.Ex54.m1.3.3.1.1.5.cmml">≥</mo><mrow id="A3.Ex54.m1.3.3.1.1.6" xref="A3.Ex54.m1.3.3.1.1.6.cmml"><msup id="A3.Ex54.m1.3.3.1.1.6.2" xref="A3.Ex54.m1.3.3.1.1.6.2.cmml"><mi id="A3.Ex54.m1.3.3.1.1.6.2.2" xref="A3.Ex54.m1.3.3.1.1.6.2.2.cmml">σ</mi><mo id="A3.Ex54.m1.3.3.1.1.6.2.3" xref="A3.Ex54.m1.3.3.1.1.6.2.3.cmml">∗</mo></msup><mo id="A3.Ex54.m1.3.3.1.1.6.1" xref="A3.Ex54.m1.3.3.1.1.6.1.cmml">⁢</mo><mi id="A3.Ex54.m1.3.3.1.1.6.3" xref="A3.Ex54.m1.3.3.1.1.6.3.cmml">I</mi></mrow></mrow><mo id="A3.Ex54.m1.3.3.1.2" lspace="0em" xref="A3.Ex54.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex54.m1.3b"><apply id="A3.Ex54.m1.3.3.1.1.cmml" xref="A3.Ex54.m1.3.3.1"><and id="A3.Ex54.m1.3.3.1.1a.cmml" xref="A3.Ex54.m1.3.3.1"></and><apply id="A3.Ex54.m1.3.3.1.1b.cmml" xref="A3.Ex54.m1.3.3.1"><eq id="A3.Ex54.m1.3.3.1.1.4.cmml" xref="A3.Ex54.m1.3.3.1.1.4"></eq><apply id="A3.Ex54.m1.3.3.1.1.1.cmml" xref="A3.Ex54.m1.3.3.1.1.1"><times id="A3.Ex54.m1.3.3.1.1.1.2.cmml" xref="A3.Ex54.m1.3.3.1.1.1.2"></times><apply id="A3.Ex54.m1.3.3.1.1.1.3.cmml" xref="A3.Ex54.m1.3.3.1.1.1.3"><csymbol cd="ambiguous" id="A3.Ex54.m1.3.3.1.1.1.3.1.cmml" xref="A3.Ex54.m1.3.3.1.1.1.3">subscript</csymbol><ci id="A3.Ex54.m1.3.3.1.1.1.3.2.cmml" xref="A3.Ex54.m1.3.3.1.1.1.3.2">𝑄</ci><ci id="A3.Ex54.m1.3.3.1.1.1.3.3.cmml" xref="A3.Ex54.m1.3.3.1.1.1.3.3">𝜁</ci></apply><interval closure="open" id="A3.Ex54.m1.3.3.1.1.1.1.2.cmml" xref="A3.Ex54.m1.3.3.1.1.1.1.1"><ci id="A3.Ex54.m1.1.1.cmml" xref="A3.Ex54.m1.1.1">𝑡</ci><apply id="A3.Ex54.m1.3.3.1.1.1.1.1.1.cmml" xref="A3.Ex54.m1.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.Ex54.m1.3.3.1.1.1.1.1.1.1.cmml" xref="A3.Ex54.m1.3.3.1.1.1.1.1.1">subscript</csymbol><ci id="A3.Ex54.m1.3.3.1.1.1.1.1.1.2.cmml" xref="A3.Ex54.m1.3.3.1.1.1.1.1.1.2">𝑡</ci><ci id="A3.Ex54.m1.3.3.1.1.1.1.1.1.3.cmml" xref="A3.Ex54.m1.3.3.1.1.1.1.1.1.3">e</ci></apply></interval></apply><apply id="A3.Ex54.m1.3.3.1.1.2.cmml" xref="A3.Ex54.m1.3.3.1.1.2"><times id="A3.Ex54.m1.3.3.1.1.2.2.cmml" xref="A3.Ex54.m1.3.3.1.1.2.2"></times><ci id="A3.Ex54.m1.3.3.1.1.2.3.cmml" xref="A3.Ex54.m1.3.3.1.1.2.3">𝐻</ci><ci id="A3.Ex54.m1.2.2.cmml" xref="A3.Ex54.m1.2.2">𝑠</ci><apply id="A3.Ex54.m1.3.3.1.1.2.1.2.cmml" xref="A3.Ex54.m1.3.3.1.1.2.1.1"><csymbol cd="latexml" id="A3.Ex54.m1.3.3.1.1.2.1.2.1.cmml" xref="A3.Ex54.m1.3.3.1.1.2.1.1.2">delimited-[]</csymbol><apply id="A3.Ex54.m1.3.3.1.1.2.1.1.1.cmml" xref="A3.Ex54.m1.3.3.1.1.2.1.1.1"><times id="A3.Ex54.m1.3.3.1.1.2.1.1.1.2.cmml" xref="A3.Ex54.m1.3.3.1.1.2.1.1.1.2"></times><apply id="A3.Ex54.m1.3.3.1.1.2.1.1.1.3.cmml" xref="A3.Ex54.m1.3.3.1.1.2.1.1.1.3"><csymbol cd="ambiguous" id="A3.Ex54.m1.3.3.1.1.2.1.1.1.3.1.cmml" xref="A3.Ex54.m1.3.3.1.1.2.1.1.1.3">subscript</csymbol><ci id="A3.Ex54.m1.3.3.1.1.2.1.1.1.3.2.cmml" xref="A3.Ex54.m1.3.3.1.1.2.1.1.1.3.2">Ψ</ci><ci id="A3.Ex54.m1.3.3.1.1.2.1.1.1.3.3.cmml" xref="A3.Ex54.m1.3.3.1.1.2.1.1.1.3.3">𝜁</ci></apply><apply id="A3.Ex54.m1.3.3.1.1.2.1.1.1.1.1.1.cmml" xref="A3.Ex54.m1.3.3.1.1.2.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.Ex54.m1.3.3.1.1.2.1.1.1.1.1.1.1.cmml" xref="A3.Ex54.m1.3.3.1.1.2.1.1.1.1.1">subscript</csymbol><ci id="A3.Ex54.m1.3.3.1.1.2.1.1.1.1.1.1.2.cmml" xref="A3.Ex54.m1.3.3.1.1.2.1.1.1.1.1.1.2">𝑡</ci><ci id="A3.Ex54.m1.3.3.1.1.2.1.1.1.1.1.1.3.cmml" xref="A3.Ex54.m1.3.3.1.1.2.1.1.1.1.1.1.3">e</ci></apply></apply></apply></apply></apply><apply id="A3.Ex54.m1.3.3.1.1c.cmml" xref="A3.Ex54.m1.3.3.1"><geq id="A3.Ex54.m1.3.3.1.1.5.cmml" xref="A3.Ex54.m1.3.3.1.1.5"></geq><share href="https://arxiv.org/html/2401.10785v2#A3.Ex54.m1.3.3.1.1.2.cmml" id="A3.Ex54.m1.3.3.1.1d.cmml" xref="A3.Ex54.m1.3.3.1"></share><apply id="A3.Ex54.m1.3.3.1.1.6.cmml" xref="A3.Ex54.m1.3.3.1.1.6"><times id="A3.Ex54.m1.3.3.1.1.6.1.cmml" xref="A3.Ex54.m1.3.3.1.1.6.1"></times><apply id="A3.Ex54.m1.3.3.1.1.6.2.cmml" xref="A3.Ex54.m1.3.3.1.1.6.2"><csymbol cd="ambiguous" id="A3.Ex54.m1.3.3.1.1.6.2.1.cmml" xref="A3.Ex54.m1.3.3.1.1.6.2">superscript</csymbol><ci id="A3.Ex54.m1.3.3.1.1.6.2.2.cmml" xref="A3.Ex54.m1.3.3.1.1.6.2.2">𝜎</ci><times id="A3.Ex54.m1.3.3.1.1.6.2.3.cmml" xref="A3.Ex54.m1.3.3.1.1.6.2.3"></times></apply><ci id="A3.Ex54.m1.3.3.1.1.6.3.cmml" xref="A3.Ex54.m1.3.3.1.1.6.3">𝐼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex54.m1.3c">\displaystyle Q_{\zeta}(t,t_{\rm e})=H(s)[\Psi_{\zeta}(t_{\rm e})]\geq\sigma^{% *}I.</annotation><annotation encoding="application/x-llamapun" id="A3.Ex54.m1.3d">italic_Q start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) = italic_H ( italic_s ) [ roman_Ψ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) ] ≥ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_I .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.2.p2.83">It follows from the above result and (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A3.E54" title="In Proof. ‣ Appendix C The proof of Theorem 2 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">54</span></a>) that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx70"> <tbody id="A3.Ex55"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}_{\zeta}\leq-k_{\rm c}\bm{e}^{T}\bm{e}-\kappa\sigma^{*}(1+% p)\tilde{\bm{\theta}}^{T}_{\zeta}\tilde{\bm{\theta}}_{\zeta}+\bm{e}^{T}(\Phi_{% \zeta}-\Phi_{\rm f,\zeta})^{T}\tilde{\bm{\theta}}_{\zeta}." class="ltx_Math" display="inline" id="A3.Ex55.m1.3"><semantics id="A3.Ex55.m1.3a"><mrow id="A3.Ex55.m1.3.3.1" xref="A3.Ex55.m1.3.3.1.1.cmml"><mrow id="A3.Ex55.m1.3.3.1.1" xref="A3.Ex55.m1.3.3.1.1.cmml"><msub id="A3.Ex55.m1.3.3.1.1.4" xref="A3.Ex55.m1.3.3.1.1.4.cmml"><mover accent="true" id="A3.Ex55.m1.3.3.1.1.4.2" xref="A3.Ex55.m1.3.3.1.1.4.2.cmml"><mi id="A3.Ex55.m1.3.3.1.1.4.2.2" xref="A3.Ex55.m1.3.3.1.1.4.2.2.cmml">V</mi><mo id="A3.Ex55.m1.3.3.1.1.4.2.1" xref="A3.Ex55.m1.3.3.1.1.4.2.1.cmml">˙</mo></mover><mi id="A3.Ex55.m1.3.3.1.1.4.3" xref="A3.Ex55.m1.3.3.1.1.4.3.cmml">ζ</mi></msub><mo id="A3.Ex55.m1.3.3.1.1.3" xref="A3.Ex55.m1.3.3.1.1.3.cmml">≤</mo><mrow id="A3.Ex55.m1.3.3.1.1.2" xref="A3.Ex55.m1.3.3.1.1.2.cmml"><mrow id="A3.Ex55.m1.3.3.1.1.1.1" xref="A3.Ex55.m1.3.3.1.1.1.1.cmml"><mrow id="A3.Ex55.m1.3.3.1.1.1.1.3" xref="A3.Ex55.m1.3.3.1.1.1.1.3.cmml"><mo id="A3.Ex55.m1.3.3.1.1.1.1.3a" xref="A3.Ex55.m1.3.3.1.1.1.1.3.cmml">−</mo><mrow id="A3.Ex55.m1.3.3.1.1.1.1.3.2" xref="A3.Ex55.m1.3.3.1.1.1.1.3.2.cmml"><msub id="A3.Ex55.m1.3.3.1.1.1.1.3.2.2" xref="A3.Ex55.m1.3.3.1.1.1.1.3.2.2.cmml"><mi id="A3.Ex55.m1.3.3.1.1.1.1.3.2.2.2" xref="A3.Ex55.m1.3.3.1.1.1.1.3.2.2.2.cmml">k</mi><mi id="A3.Ex55.m1.3.3.1.1.1.1.3.2.2.3" mathvariant="normal" xref="A3.Ex55.m1.3.3.1.1.1.1.3.2.2.3.cmml">c</mi></msub><mo id="A3.Ex55.m1.3.3.1.1.1.1.3.2.1" xref="A3.Ex55.m1.3.3.1.1.1.1.3.2.1.cmml">⁢</mo><msup id="A3.Ex55.m1.3.3.1.1.1.1.3.2.3" xref="A3.Ex55.m1.3.3.1.1.1.1.3.2.3.cmml"><mi id="A3.Ex55.m1.3.3.1.1.1.1.3.2.3.2" xref="A3.Ex55.m1.3.3.1.1.1.1.3.2.3.2.cmml">𝒆</mi><mi id="A3.Ex55.m1.3.3.1.1.1.1.3.2.3.3" xref="A3.Ex55.m1.3.3.1.1.1.1.3.2.3.3.cmml">T</mi></msup><mo id="A3.Ex55.m1.3.3.1.1.1.1.3.2.1a" xref="A3.Ex55.m1.3.3.1.1.1.1.3.2.1.cmml">⁢</mo><mi id="A3.Ex55.m1.3.3.1.1.1.1.3.2.4" xref="A3.Ex55.m1.3.3.1.1.1.1.3.2.4.cmml">𝒆</mi></mrow></mrow><mo id="A3.Ex55.m1.3.3.1.1.1.1.2" xref="A3.Ex55.m1.3.3.1.1.1.1.2.cmml">−</mo><mrow id="A3.Ex55.m1.3.3.1.1.1.1.1" xref="A3.Ex55.m1.3.3.1.1.1.1.1.cmml"><mi id="A3.Ex55.m1.3.3.1.1.1.1.1.3" xref="A3.Ex55.m1.3.3.1.1.1.1.1.3.cmml">κ</mi><mo id="A3.Ex55.m1.3.3.1.1.1.1.1.2" xref="A3.Ex55.m1.3.3.1.1.1.1.1.2.cmml">⁢</mo><msup id="A3.Ex55.m1.3.3.1.1.1.1.1.4" xref="A3.Ex55.m1.3.3.1.1.1.1.1.4.cmml"><mi id="A3.Ex55.m1.3.3.1.1.1.1.1.4.2" xref="A3.Ex55.m1.3.3.1.1.1.1.1.4.2.cmml">σ</mi><mo id="A3.Ex55.m1.3.3.1.1.1.1.1.4.3" xref="A3.Ex55.m1.3.3.1.1.1.1.1.4.3.cmml">∗</mo></msup><mo id="A3.Ex55.m1.3.3.1.1.1.1.1.2a" xref="A3.Ex55.m1.3.3.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A3.Ex55.m1.3.3.1.1.1.1.1.1.1" xref="A3.Ex55.m1.3.3.1.1.1.1.1.1.1.1.cmml"><mo id="A3.Ex55.m1.3.3.1.1.1.1.1.1.1.2" stretchy="false" xref="A3.Ex55.m1.3.3.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.Ex55.m1.3.3.1.1.1.1.1.1.1.1" xref="A3.Ex55.m1.3.3.1.1.1.1.1.1.1.1.cmml"><mn id="A3.Ex55.m1.3.3.1.1.1.1.1.1.1.1.2" xref="A3.Ex55.m1.3.3.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A3.Ex55.m1.3.3.1.1.1.1.1.1.1.1.1" xref="A3.Ex55.m1.3.3.1.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="A3.Ex55.m1.3.3.1.1.1.1.1.1.1.1.3" xref="A3.Ex55.m1.3.3.1.1.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A3.Ex55.m1.3.3.1.1.1.1.1.1.1.3" stretchy="false" xref="A3.Ex55.m1.3.3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A3.Ex55.m1.3.3.1.1.1.1.1.2b" xref="A3.Ex55.m1.3.3.1.1.1.1.1.2.cmml">⁢</mo><msubsup id="A3.Ex55.m1.3.3.1.1.1.1.1.5" xref="A3.Ex55.m1.3.3.1.1.1.1.1.5.cmml"><mover accent="true" id="A3.Ex55.m1.3.3.1.1.1.1.1.5.2.2" xref="A3.Ex55.m1.3.3.1.1.1.1.1.5.2.2.cmml"><mi id="A3.Ex55.m1.3.3.1.1.1.1.1.5.2.2.2" xref="A3.Ex55.m1.3.3.1.1.1.1.1.5.2.2.2.cmml">𝜽</mi><mo id="A3.Ex55.m1.3.3.1.1.1.1.1.5.2.2.1" xref="A3.Ex55.m1.3.3.1.1.1.1.1.5.2.2.1.cmml">~</mo></mover><mi id="A3.Ex55.m1.3.3.1.1.1.1.1.5.3" xref="A3.Ex55.m1.3.3.1.1.1.1.1.5.3.cmml">ζ</mi><mi id="A3.Ex55.m1.3.3.1.1.1.1.1.5.2.3" xref="A3.Ex55.m1.3.3.1.1.1.1.1.5.2.3.cmml">T</mi></msubsup><mo id="A3.Ex55.m1.3.3.1.1.1.1.1.2c" xref="A3.Ex55.m1.3.3.1.1.1.1.1.2.cmml">⁢</mo><msub id="A3.Ex55.m1.3.3.1.1.1.1.1.6" xref="A3.Ex55.m1.3.3.1.1.1.1.1.6.cmml"><mover accent="true" id="A3.Ex55.m1.3.3.1.1.1.1.1.6.2" xref="A3.Ex55.m1.3.3.1.1.1.1.1.6.2.cmml"><mi id="A3.Ex55.m1.3.3.1.1.1.1.1.6.2.2" xref="A3.Ex55.m1.3.3.1.1.1.1.1.6.2.2.cmml">𝜽</mi><mo id="A3.Ex55.m1.3.3.1.1.1.1.1.6.2.1" xref="A3.Ex55.m1.3.3.1.1.1.1.1.6.2.1.cmml">~</mo></mover><mi id="A3.Ex55.m1.3.3.1.1.1.1.1.6.3" xref="A3.Ex55.m1.3.3.1.1.1.1.1.6.3.cmml">ζ</mi></msub></mrow></mrow><mo id="A3.Ex55.m1.3.3.1.1.2.3" xref="A3.Ex55.m1.3.3.1.1.2.3.cmml">+</mo><mrow id="A3.Ex55.m1.3.3.1.1.2.2" xref="A3.Ex55.m1.3.3.1.1.2.2.cmml"><msup id="A3.Ex55.m1.3.3.1.1.2.2.3" xref="A3.Ex55.m1.3.3.1.1.2.2.3.cmml"><mi id="A3.Ex55.m1.3.3.1.1.2.2.3.2" xref="A3.Ex55.m1.3.3.1.1.2.2.3.2.cmml">𝒆</mi><mi id="A3.Ex55.m1.3.3.1.1.2.2.3.3" xref="A3.Ex55.m1.3.3.1.1.2.2.3.3.cmml">T</mi></msup><mo id="A3.Ex55.m1.3.3.1.1.2.2.2" xref="A3.Ex55.m1.3.3.1.1.2.2.2.cmml">⁢</mo><msup id="A3.Ex55.m1.3.3.1.1.2.2.1" xref="A3.Ex55.m1.3.3.1.1.2.2.1.cmml"><mrow id="A3.Ex55.m1.3.3.1.1.2.2.1.1.1" xref="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.cmml"><mo id="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.2" stretchy="false" xref="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.cmml">(</mo><mrow id="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1" xref="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.cmml"><msub id="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.2" xref="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.2.cmml"><mi id="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.2.2" mathvariant="normal" xref="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.2.2.cmml">Φ</mi><mi id="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.2.3" xref="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.2.3.cmml">ζ</mi></msub><mo id="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.1" xref="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.1.cmml">−</mo><msub id="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.3" xref="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.3.cmml"><mi id="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.3.2" mathvariant="normal" xref="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.3.2.cmml">Φ</mi><mrow id="A3.Ex55.m1.2.2.2.4" xref="A3.Ex55.m1.2.2.2.3.cmml"><mi id="A3.Ex55.m1.1.1.1.1" mathvariant="normal" xref="A3.Ex55.m1.1.1.1.1.cmml">f</mi><mo id="A3.Ex55.m1.2.2.2.4.1" xref="A3.Ex55.m1.2.2.2.3.cmml">,</mo><mi id="A3.Ex55.m1.2.2.2.2" xref="A3.Ex55.m1.2.2.2.2.cmml">ζ</mi></mrow></msub></mrow><mo id="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.3" stretchy="false" xref="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.cmml">)</mo></mrow><mi id="A3.Ex55.m1.3.3.1.1.2.2.1.3" xref="A3.Ex55.m1.3.3.1.1.2.2.1.3.cmml">T</mi></msup><mo id="A3.Ex55.m1.3.3.1.1.2.2.2a" xref="A3.Ex55.m1.3.3.1.1.2.2.2.cmml">⁢</mo><msub id="A3.Ex55.m1.3.3.1.1.2.2.4" xref="A3.Ex55.m1.3.3.1.1.2.2.4.cmml"><mover accent="true" id="A3.Ex55.m1.3.3.1.1.2.2.4.2" xref="A3.Ex55.m1.3.3.1.1.2.2.4.2.cmml"><mi id="A3.Ex55.m1.3.3.1.1.2.2.4.2.2" xref="A3.Ex55.m1.3.3.1.1.2.2.4.2.2.cmml">𝜽</mi><mo id="A3.Ex55.m1.3.3.1.1.2.2.4.2.1" xref="A3.Ex55.m1.3.3.1.1.2.2.4.2.1.cmml">~</mo></mover><mi id="A3.Ex55.m1.3.3.1.1.2.2.4.3" xref="A3.Ex55.m1.3.3.1.1.2.2.4.3.cmml">ζ</mi></msub></mrow></mrow></mrow><mo id="A3.Ex55.m1.3.3.1.2" lspace="0em" xref="A3.Ex55.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex55.m1.3b"><apply id="A3.Ex55.m1.3.3.1.1.cmml" xref="A3.Ex55.m1.3.3.1"><leq id="A3.Ex55.m1.3.3.1.1.3.cmml" xref="A3.Ex55.m1.3.3.1.1.3"></leq><apply id="A3.Ex55.m1.3.3.1.1.4.cmml" xref="A3.Ex55.m1.3.3.1.1.4"><csymbol cd="ambiguous" id="A3.Ex55.m1.3.3.1.1.4.1.cmml" xref="A3.Ex55.m1.3.3.1.1.4">subscript</csymbol><apply id="A3.Ex55.m1.3.3.1.1.4.2.cmml" xref="A3.Ex55.m1.3.3.1.1.4.2"><ci id="A3.Ex55.m1.3.3.1.1.4.2.1.cmml" xref="A3.Ex55.m1.3.3.1.1.4.2.1">˙</ci><ci id="A3.Ex55.m1.3.3.1.1.4.2.2.cmml" xref="A3.Ex55.m1.3.3.1.1.4.2.2">𝑉</ci></apply><ci id="A3.Ex55.m1.3.3.1.1.4.3.cmml" xref="A3.Ex55.m1.3.3.1.1.4.3">𝜁</ci></apply><apply id="A3.Ex55.m1.3.3.1.1.2.cmml" xref="A3.Ex55.m1.3.3.1.1.2"><plus id="A3.Ex55.m1.3.3.1.1.2.3.cmml" xref="A3.Ex55.m1.3.3.1.1.2.3"></plus><apply id="A3.Ex55.m1.3.3.1.1.1.1.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1"><minus id="A3.Ex55.m1.3.3.1.1.1.1.2.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.2"></minus><apply id="A3.Ex55.m1.3.3.1.1.1.1.3.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.3"><minus id="A3.Ex55.m1.3.3.1.1.1.1.3.1.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.3"></minus><apply id="A3.Ex55.m1.3.3.1.1.1.1.3.2.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.3.2"><times id="A3.Ex55.m1.3.3.1.1.1.1.3.2.1.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.3.2.1"></times><apply id="A3.Ex55.m1.3.3.1.1.1.1.3.2.2.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.3.2.2"><csymbol cd="ambiguous" id="A3.Ex55.m1.3.3.1.1.1.1.3.2.2.1.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.3.2.2">subscript</csymbol><ci id="A3.Ex55.m1.3.3.1.1.1.1.3.2.2.2.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.3.2.2.2">𝑘</ci><ci id="A3.Ex55.m1.3.3.1.1.1.1.3.2.2.3.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.3.2.2.3">c</ci></apply><apply id="A3.Ex55.m1.3.3.1.1.1.1.3.2.3.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.3.2.3"><csymbol cd="ambiguous" id="A3.Ex55.m1.3.3.1.1.1.1.3.2.3.1.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.3.2.3">superscript</csymbol><ci id="A3.Ex55.m1.3.3.1.1.1.1.3.2.3.2.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.3.2.3.2">𝒆</ci><ci id="A3.Ex55.m1.3.3.1.1.1.1.3.2.3.3.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.3.2.3.3">𝑇</ci></apply><ci id="A3.Ex55.m1.3.3.1.1.1.1.3.2.4.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.3.2.4">𝒆</ci></apply></apply><apply id="A3.Ex55.m1.3.3.1.1.1.1.1.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1"><times id="A3.Ex55.m1.3.3.1.1.1.1.1.2.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1.2"></times><ci id="A3.Ex55.m1.3.3.1.1.1.1.1.3.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1.3">𝜅</ci><apply id="A3.Ex55.m1.3.3.1.1.1.1.1.4.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1.4"><csymbol cd="ambiguous" id="A3.Ex55.m1.3.3.1.1.1.1.1.4.1.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1.4">superscript</csymbol><ci id="A3.Ex55.m1.3.3.1.1.1.1.1.4.2.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1.4.2">𝜎</ci><times id="A3.Ex55.m1.3.3.1.1.1.1.1.4.3.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1.4.3"></times></apply><apply id="A3.Ex55.m1.3.3.1.1.1.1.1.1.1.1.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1.1.1"><plus id="A3.Ex55.m1.3.3.1.1.1.1.1.1.1.1.1.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1.1.1.1.1"></plus><cn id="A3.Ex55.m1.3.3.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A3.Ex55.m1.3.3.1.1.1.1.1.1.1.1.2">1</cn><ci id="A3.Ex55.m1.3.3.1.1.1.1.1.1.1.1.3.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="A3.Ex55.m1.3.3.1.1.1.1.1.5.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1.5"><csymbol cd="ambiguous" id="A3.Ex55.m1.3.3.1.1.1.1.1.5.1.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1.5">subscript</csymbol><apply id="A3.Ex55.m1.3.3.1.1.1.1.1.5.2.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1.5"><csymbol cd="ambiguous" id="A3.Ex55.m1.3.3.1.1.1.1.1.5.2.1.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1.5">superscript</csymbol><apply id="A3.Ex55.m1.3.3.1.1.1.1.1.5.2.2.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1.5.2.2"><ci id="A3.Ex55.m1.3.3.1.1.1.1.1.5.2.2.1.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1.5.2.2.1">~</ci><ci id="A3.Ex55.m1.3.3.1.1.1.1.1.5.2.2.2.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1.5.2.2.2">𝜽</ci></apply><ci id="A3.Ex55.m1.3.3.1.1.1.1.1.5.2.3.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1.5.2.3">𝑇</ci></apply><ci id="A3.Ex55.m1.3.3.1.1.1.1.1.5.3.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1.5.3">𝜁</ci></apply><apply id="A3.Ex55.m1.3.3.1.1.1.1.1.6.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1.6"><csymbol cd="ambiguous" id="A3.Ex55.m1.3.3.1.1.1.1.1.6.1.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1.6">subscript</csymbol><apply id="A3.Ex55.m1.3.3.1.1.1.1.1.6.2.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1.6.2"><ci id="A3.Ex55.m1.3.3.1.1.1.1.1.6.2.1.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1.6.2.1">~</ci><ci id="A3.Ex55.m1.3.3.1.1.1.1.1.6.2.2.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1.6.2.2">𝜽</ci></apply><ci id="A3.Ex55.m1.3.3.1.1.1.1.1.6.3.cmml" xref="A3.Ex55.m1.3.3.1.1.1.1.1.6.3">𝜁</ci></apply></apply></apply><apply id="A3.Ex55.m1.3.3.1.1.2.2.cmml" xref="A3.Ex55.m1.3.3.1.1.2.2"><times id="A3.Ex55.m1.3.3.1.1.2.2.2.cmml" xref="A3.Ex55.m1.3.3.1.1.2.2.2"></times><apply id="A3.Ex55.m1.3.3.1.1.2.2.3.cmml" xref="A3.Ex55.m1.3.3.1.1.2.2.3"><csymbol cd="ambiguous" id="A3.Ex55.m1.3.3.1.1.2.2.3.1.cmml" xref="A3.Ex55.m1.3.3.1.1.2.2.3">superscript</csymbol><ci id="A3.Ex55.m1.3.3.1.1.2.2.3.2.cmml" xref="A3.Ex55.m1.3.3.1.1.2.2.3.2">𝒆</ci><ci id="A3.Ex55.m1.3.3.1.1.2.2.3.3.cmml" xref="A3.Ex55.m1.3.3.1.1.2.2.3.3">𝑇</ci></apply><apply id="A3.Ex55.m1.3.3.1.1.2.2.1.cmml" xref="A3.Ex55.m1.3.3.1.1.2.2.1"><csymbol cd="ambiguous" id="A3.Ex55.m1.3.3.1.1.2.2.1.2.cmml" xref="A3.Ex55.m1.3.3.1.1.2.2.1">superscript</csymbol><apply id="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.cmml" xref="A3.Ex55.m1.3.3.1.1.2.2.1.1.1"><minus id="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.1.cmml" xref="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.1"></minus><apply id="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.2.cmml" xref="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.2.1.cmml" xref="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.2">subscript</csymbol><ci id="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.2.2.cmml" xref="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.2.2">Φ</ci><ci id="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.2.3.cmml" xref="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.2.3">𝜁</ci></apply><apply id="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.3.cmml" xref="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.3.1.cmml" xref="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.3">subscript</csymbol><ci id="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.3.2.cmml" xref="A3.Ex55.m1.3.3.1.1.2.2.1.1.1.1.3.2">Φ</ci><list id="A3.Ex55.m1.2.2.2.3.cmml" xref="A3.Ex55.m1.2.2.2.4"><ci id="A3.Ex55.m1.1.1.1.1.cmml" xref="A3.Ex55.m1.1.1.1.1">f</ci><ci id="A3.Ex55.m1.2.2.2.2.cmml" xref="A3.Ex55.m1.2.2.2.2">𝜁</ci></list></apply></apply><ci id="A3.Ex55.m1.3.3.1.1.2.2.1.3.cmml" xref="A3.Ex55.m1.3.3.1.1.2.2.1.3">𝑇</ci></apply><apply id="A3.Ex55.m1.3.3.1.1.2.2.4.cmml" xref="A3.Ex55.m1.3.3.1.1.2.2.4"><csymbol cd="ambiguous" id="A3.Ex55.m1.3.3.1.1.2.2.4.1.cmml" xref="A3.Ex55.m1.3.3.1.1.2.2.4">subscript</csymbol><apply id="A3.Ex55.m1.3.3.1.1.2.2.4.2.cmml" xref="A3.Ex55.m1.3.3.1.1.2.2.4.2"><ci id="A3.Ex55.m1.3.3.1.1.2.2.4.2.1.cmml" xref="A3.Ex55.m1.3.3.1.1.2.2.4.2.1">~</ci><ci id="A3.Ex55.m1.3.3.1.1.2.2.4.2.2.cmml" xref="A3.Ex55.m1.3.3.1.1.2.2.4.2.2">𝜽</ci></apply><ci id="A3.Ex55.m1.3.3.1.1.2.2.4.3.cmml" xref="A3.Ex55.m1.3.3.1.1.2.2.4.3">𝜁</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex55.m1.3c">\displaystyle\dot{V}_{\zeta}\leq-k_{\rm c}\bm{e}^{T}\bm{e}-\kappa\sigma^{*}(1+% p)\tilde{\bm{\theta}}^{T}_{\zeta}\tilde{\bm{\theta}}_{\zeta}+\bm{e}^{T}(\Phi_{% \zeta}-\Phi_{\rm f,\zeta})^{T}\tilde{\bm{\theta}}_{\zeta}.</annotation><annotation encoding="application/x-llamapun" id="A3.Ex55.m1.3d">over˙ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ≤ - italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_e - italic_κ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( 1 + italic_p ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT + bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( roman_Φ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT - roman_Φ start_POSTSUBSCRIPT roman_f , italic_ζ end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.2.p2.62">Noting Lemma 2 and <math alttext="\Phi" class="ltx_Math" display="inline" id="A3.2.p2.58.m1.1"><semantics id="A3.2.p2.58.m1.1a"><mi id="A3.2.p2.58.m1.1.1" mathvariant="normal" xref="A3.2.p2.58.m1.1.1.cmml">Φ</mi><annotation-xml encoding="MathML-Content" id="A3.2.p2.58.m1.1b"><ci id="A3.2.p2.58.m1.1.1.cmml" xref="A3.2.p2.58.m1.1.1">Φ</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.58.m1.1c">\Phi</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.58.m1.1d">roman_Φ</annotation></semantics></math>, <math alttext="\Phi_{\rm f}\in L_{\infty}" class="ltx_Math" display="inline" id="A3.2.p2.59.m2.1"><semantics id="A3.2.p2.59.m2.1a"><mrow id="A3.2.p2.59.m2.1.1" xref="A3.2.p2.59.m2.1.1.cmml"><msub id="A3.2.p2.59.m2.1.1.2" xref="A3.2.p2.59.m2.1.1.2.cmml"><mi id="A3.2.p2.59.m2.1.1.2.2" mathvariant="normal" xref="A3.2.p2.59.m2.1.1.2.2.cmml">Φ</mi><mi id="A3.2.p2.59.m2.1.1.2.3" mathvariant="normal" xref="A3.2.p2.59.m2.1.1.2.3.cmml">f</mi></msub><mo id="A3.2.p2.59.m2.1.1.1" xref="A3.2.p2.59.m2.1.1.1.cmml">∈</mo><msub id="A3.2.p2.59.m2.1.1.3" xref="A3.2.p2.59.m2.1.1.3.cmml"><mi id="A3.2.p2.59.m2.1.1.3.2" xref="A3.2.p2.59.m2.1.1.3.2.cmml">L</mi><mi id="A3.2.p2.59.m2.1.1.3.3" mathvariant="normal" xref="A3.2.p2.59.m2.1.1.3.3.cmml">∞</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.59.m2.1b"><apply id="A3.2.p2.59.m2.1.1.cmml" xref="A3.2.p2.59.m2.1.1"><in id="A3.2.p2.59.m2.1.1.1.cmml" xref="A3.2.p2.59.m2.1.1.1"></in><apply id="A3.2.p2.59.m2.1.1.2.cmml" xref="A3.2.p2.59.m2.1.1.2"><csymbol cd="ambiguous" id="A3.2.p2.59.m2.1.1.2.1.cmml" xref="A3.2.p2.59.m2.1.1.2">subscript</csymbol><ci id="A3.2.p2.59.m2.1.1.2.2.cmml" xref="A3.2.p2.59.m2.1.1.2.2">Φ</ci><ci id="A3.2.p2.59.m2.1.1.2.3.cmml" xref="A3.2.p2.59.m2.1.1.2.3">f</ci></apply><apply id="A3.2.p2.59.m2.1.1.3.cmml" xref="A3.2.p2.59.m2.1.1.3"><csymbol cd="ambiguous" id="A3.2.p2.59.m2.1.1.3.1.cmml" xref="A3.2.p2.59.m2.1.1.3">subscript</csymbol><ci id="A3.2.p2.59.m2.1.1.3.2.cmml" xref="A3.2.p2.59.m2.1.1.3.2">𝐿</ci><infinity id="A3.2.p2.59.m2.1.1.3.3.cmml" xref="A3.2.p2.59.m2.1.1.3.3"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.59.m2.1c">\Phi_{\rm f}\in L_{\infty}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.59.m2.1d">roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ∈ italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math> yields <math alttext="\|\Phi-\Phi_{\rm f}\|\leq\delta" class="ltx_Math" display="inline" id="A3.2.p2.60.m3.1"><semantics id="A3.2.p2.60.m3.1a"><mrow id="A3.2.p2.60.m3.1.1" xref="A3.2.p2.60.m3.1.1.cmml"><mrow id="A3.2.p2.60.m3.1.1.1.1" xref="A3.2.p2.60.m3.1.1.1.2.cmml"><mo id="A3.2.p2.60.m3.1.1.1.1.2" stretchy="false" xref="A3.2.p2.60.m3.1.1.1.2.1.cmml">‖</mo><mrow id="A3.2.p2.60.m3.1.1.1.1.1" xref="A3.2.p2.60.m3.1.1.1.1.1.cmml"><mi id="A3.2.p2.60.m3.1.1.1.1.1.2" mathvariant="normal" xref="A3.2.p2.60.m3.1.1.1.1.1.2.cmml">Φ</mi><mo id="A3.2.p2.60.m3.1.1.1.1.1.1" xref="A3.2.p2.60.m3.1.1.1.1.1.1.cmml">−</mo><msub id="A3.2.p2.60.m3.1.1.1.1.1.3" xref="A3.2.p2.60.m3.1.1.1.1.1.3.cmml"><mi id="A3.2.p2.60.m3.1.1.1.1.1.3.2" mathvariant="normal" xref="A3.2.p2.60.m3.1.1.1.1.1.3.2.cmml">Φ</mi><mi id="A3.2.p2.60.m3.1.1.1.1.1.3.3" mathvariant="normal" xref="A3.2.p2.60.m3.1.1.1.1.1.3.3.cmml">f</mi></msub></mrow><mo id="A3.2.p2.60.m3.1.1.1.1.3" stretchy="false" xref="A3.2.p2.60.m3.1.1.1.2.1.cmml">‖</mo></mrow><mo id="A3.2.p2.60.m3.1.1.2" xref="A3.2.p2.60.m3.1.1.2.cmml">≤</mo><mi id="A3.2.p2.60.m3.1.1.3" xref="A3.2.p2.60.m3.1.1.3.cmml">δ</mi></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.60.m3.1b"><apply id="A3.2.p2.60.m3.1.1.cmml" xref="A3.2.p2.60.m3.1.1"><leq id="A3.2.p2.60.m3.1.1.2.cmml" xref="A3.2.p2.60.m3.1.1.2"></leq><apply id="A3.2.p2.60.m3.1.1.1.2.cmml" xref="A3.2.p2.60.m3.1.1.1.1"><csymbol cd="latexml" id="A3.2.p2.60.m3.1.1.1.2.1.cmml" xref="A3.2.p2.60.m3.1.1.1.1.2">norm</csymbol><apply id="A3.2.p2.60.m3.1.1.1.1.1.cmml" xref="A3.2.p2.60.m3.1.1.1.1.1"><minus id="A3.2.p2.60.m3.1.1.1.1.1.1.cmml" xref="A3.2.p2.60.m3.1.1.1.1.1.1"></minus><ci id="A3.2.p2.60.m3.1.1.1.1.1.2.cmml" xref="A3.2.p2.60.m3.1.1.1.1.1.2">Φ</ci><apply id="A3.2.p2.60.m3.1.1.1.1.1.3.cmml" xref="A3.2.p2.60.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A3.2.p2.60.m3.1.1.1.1.1.3.1.cmml" xref="A3.2.p2.60.m3.1.1.1.1.1.3">subscript</csymbol><ci id="A3.2.p2.60.m3.1.1.1.1.1.3.2.cmml" xref="A3.2.p2.60.m3.1.1.1.1.1.3.2">Φ</ci><ci id="A3.2.p2.60.m3.1.1.1.1.1.3.3.cmml" xref="A3.2.p2.60.m3.1.1.1.1.1.3.3">f</ci></apply></apply></apply><ci id="A3.2.p2.60.m3.1.1.3.cmml" xref="A3.2.p2.60.m3.1.1.3">𝛿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.60.m3.1c">\|\Phi-\Phi_{\rm f}\|\leq\delta</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.60.m3.1d">∥ roman_Φ - roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ∥ ≤ italic_δ</annotation></semantics></math>, <math alttext="\forall t\geq 0" class="ltx_Math" display="inline" id="A3.2.p2.61.m4.1"><semantics id="A3.2.p2.61.m4.1a"><mrow id="A3.2.p2.61.m4.1.1" xref="A3.2.p2.61.m4.1.1.cmml"><mrow id="A3.2.p2.61.m4.1.1.2" xref="A3.2.p2.61.m4.1.1.2.cmml"><mo id="A3.2.p2.61.m4.1.1.2.1" rspace="0.167em" xref="A3.2.p2.61.m4.1.1.2.1.cmml">∀</mo><mi id="A3.2.p2.61.m4.1.1.2.2" xref="A3.2.p2.61.m4.1.1.2.2.cmml">t</mi></mrow><mo id="A3.2.p2.61.m4.1.1.1" xref="A3.2.p2.61.m4.1.1.1.cmml">≥</mo><mn id="A3.2.p2.61.m4.1.1.3" xref="A3.2.p2.61.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.61.m4.1b"><apply id="A3.2.p2.61.m4.1.1.cmml" xref="A3.2.p2.61.m4.1.1"><geq id="A3.2.p2.61.m4.1.1.1.cmml" xref="A3.2.p2.61.m4.1.1.1"></geq><apply id="A3.2.p2.61.m4.1.1.2.cmml" xref="A3.2.p2.61.m4.1.1.2"><csymbol cd="latexml" id="A3.2.p2.61.m4.1.1.2.1.cmml" xref="A3.2.p2.61.m4.1.1.2.1">for-all</csymbol><ci id="A3.2.p2.61.m4.1.1.2.2.cmml" xref="A3.2.p2.61.m4.1.1.2.2">𝑡</ci></apply><cn id="A3.2.p2.61.m4.1.1.3.cmml" type="integer" xref="A3.2.p2.61.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.61.m4.1c">\forall t\geq 0</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.61.m4.1d">∀ italic_t ≥ 0</annotation></semantics></math>. As <math alttext="\Phi-\Phi_{\rm f}=\Phi_{\zeta}-\Phi_{{\rm f},\zeta}" class="ltx_Math" display="inline" id="A3.2.p2.62.m5.2"><semantics id="A3.2.p2.62.m5.2a"><mrow id="A3.2.p2.62.m5.2.3" xref="A3.2.p2.62.m5.2.3.cmml"><mrow id="A3.2.p2.62.m5.2.3.2" xref="A3.2.p2.62.m5.2.3.2.cmml"><mi id="A3.2.p2.62.m5.2.3.2.2" mathvariant="normal" xref="A3.2.p2.62.m5.2.3.2.2.cmml">Φ</mi><mo id="A3.2.p2.62.m5.2.3.2.1" xref="A3.2.p2.62.m5.2.3.2.1.cmml">−</mo><msub id="A3.2.p2.62.m5.2.3.2.3" xref="A3.2.p2.62.m5.2.3.2.3.cmml"><mi id="A3.2.p2.62.m5.2.3.2.3.2" mathvariant="normal" xref="A3.2.p2.62.m5.2.3.2.3.2.cmml">Φ</mi><mi id="A3.2.p2.62.m5.2.3.2.3.3" mathvariant="normal" xref="A3.2.p2.62.m5.2.3.2.3.3.cmml">f</mi></msub></mrow><mo id="A3.2.p2.62.m5.2.3.1" xref="A3.2.p2.62.m5.2.3.1.cmml">=</mo><mrow id="A3.2.p2.62.m5.2.3.3" xref="A3.2.p2.62.m5.2.3.3.cmml"><msub id="A3.2.p2.62.m5.2.3.3.2" xref="A3.2.p2.62.m5.2.3.3.2.cmml"><mi id="A3.2.p2.62.m5.2.3.3.2.2" mathvariant="normal" xref="A3.2.p2.62.m5.2.3.3.2.2.cmml">Φ</mi><mi id="A3.2.p2.62.m5.2.3.3.2.3" xref="A3.2.p2.62.m5.2.3.3.2.3.cmml">ζ</mi></msub><mo id="A3.2.p2.62.m5.2.3.3.1" xref="A3.2.p2.62.m5.2.3.3.1.cmml">−</mo><msub id="A3.2.p2.62.m5.2.3.3.3" xref="A3.2.p2.62.m5.2.3.3.3.cmml"><mi id="A3.2.p2.62.m5.2.3.3.3.2" mathvariant="normal" xref="A3.2.p2.62.m5.2.3.3.3.2.cmml">Φ</mi><mrow id="A3.2.p2.62.m5.2.2.2.4" xref="A3.2.p2.62.m5.2.2.2.3.cmml"><mi id="A3.2.p2.62.m5.1.1.1.1" mathvariant="normal" xref="A3.2.p2.62.m5.1.1.1.1.cmml">f</mi><mo id="A3.2.p2.62.m5.2.2.2.4.1" xref="A3.2.p2.62.m5.2.2.2.3.cmml">,</mo><mi id="A3.2.p2.62.m5.2.2.2.2" xref="A3.2.p2.62.m5.2.2.2.2.cmml">ζ</mi></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.62.m5.2b"><apply id="A3.2.p2.62.m5.2.3.cmml" xref="A3.2.p2.62.m5.2.3"><eq id="A3.2.p2.62.m5.2.3.1.cmml" xref="A3.2.p2.62.m5.2.3.1"></eq><apply id="A3.2.p2.62.m5.2.3.2.cmml" xref="A3.2.p2.62.m5.2.3.2"><minus id="A3.2.p2.62.m5.2.3.2.1.cmml" xref="A3.2.p2.62.m5.2.3.2.1"></minus><ci id="A3.2.p2.62.m5.2.3.2.2.cmml" xref="A3.2.p2.62.m5.2.3.2.2">Φ</ci><apply id="A3.2.p2.62.m5.2.3.2.3.cmml" xref="A3.2.p2.62.m5.2.3.2.3"><csymbol cd="ambiguous" id="A3.2.p2.62.m5.2.3.2.3.1.cmml" xref="A3.2.p2.62.m5.2.3.2.3">subscript</csymbol><ci id="A3.2.p2.62.m5.2.3.2.3.2.cmml" xref="A3.2.p2.62.m5.2.3.2.3.2">Φ</ci><ci id="A3.2.p2.62.m5.2.3.2.3.3.cmml" xref="A3.2.p2.62.m5.2.3.2.3.3">f</ci></apply></apply><apply id="A3.2.p2.62.m5.2.3.3.cmml" xref="A3.2.p2.62.m5.2.3.3"><minus id="A3.2.p2.62.m5.2.3.3.1.cmml" xref="A3.2.p2.62.m5.2.3.3.1"></minus><apply id="A3.2.p2.62.m5.2.3.3.2.cmml" xref="A3.2.p2.62.m5.2.3.3.2"><csymbol cd="ambiguous" id="A3.2.p2.62.m5.2.3.3.2.1.cmml" xref="A3.2.p2.62.m5.2.3.3.2">subscript</csymbol><ci id="A3.2.p2.62.m5.2.3.3.2.2.cmml" xref="A3.2.p2.62.m5.2.3.3.2.2">Φ</ci><ci id="A3.2.p2.62.m5.2.3.3.2.3.cmml" xref="A3.2.p2.62.m5.2.3.3.2.3">𝜁</ci></apply><apply id="A3.2.p2.62.m5.2.3.3.3.cmml" xref="A3.2.p2.62.m5.2.3.3.3"><csymbol cd="ambiguous" id="A3.2.p2.62.m5.2.3.3.3.1.cmml" xref="A3.2.p2.62.m5.2.3.3.3">subscript</csymbol><ci id="A3.2.p2.62.m5.2.3.3.3.2.cmml" xref="A3.2.p2.62.m5.2.3.3.3.2">Φ</ci><list id="A3.2.p2.62.m5.2.2.2.3.cmml" xref="A3.2.p2.62.m5.2.2.2.4"><ci id="A3.2.p2.62.m5.1.1.1.1.cmml" xref="A3.2.p2.62.m5.1.1.1.1">f</ci><ci id="A3.2.p2.62.m5.2.2.2.2.cmml" xref="A3.2.p2.62.m5.2.2.2.2">𝜁</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.62.m5.2c">\Phi-\Phi_{\rm f}=\Phi_{\zeta}-\Phi_{{\rm f},\zeta}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.62.m5.2d">roman_Φ - roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT = roman_Φ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT - roman_Φ start_POSTSUBSCRIPT roman_f , italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math> due to the 0 norms of inactive channels, one gets</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx71"> <tbody id="A3.Ex56"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}_{\zeta}\leq" class="ltx_Math" display="inline" id="A3.Ex56.m1.1"><semantics id="A3.Ex56.m1.1a"><mrow id="A3.Ex56.m1.1.1" xref="A3.Ex56.m1.1.1.cmml"><msub id="A3.Ex56.m1.1.1.2" xref="A3.Ex56.m1.1.1.2.cmml"><mover accent="true" id="A3.Ex56.m1.1.1.2.2" xref="A3.Ex56.m1.1.1.2.2.cmml"><mi id="A3.Ex56.m1.1.1.2.2.2" xref="A3.Ex56.m1.1.1.2.2.2.cmml">V</mi><mo id="A3.Ex56.m1.1.1.2.2.1" xref="A3.Ex56.m1.1.1.2.2.1.cmml">˙</mo></mover><mi id="A3.Ex56.m1.1.1.2.3" xref="A3.Ex56.m1.1.1.2.3.cmml">ζ</mi></msub><mo id="A3.Ex56.m1.1.1.1" xref="A3.Ex56.m1.1.1.1.cmml">≤</mo><mi id="A3.Ex56.m1.1.1.3" xref="A3.Ex56.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex56.m1.1b"><apply id="A3.Ex56.m1.1.1.cmml" xref="A3.Ex56.m1.1.1"><leq id="A3.Ex56.m1.1.1.1.cmml" xref="A3.Ex56.m1.1.1.1"></leq><apply id="A3.Ex56.m1.1.1.2.cmml" xref="A3.Ex56.m1.1.1.2"><csymbol cd="ambiguous" id="A3.Ex56.m1.1.1.2.1.cmml" xref="A3.Ex56.m1.1.1.2">subscript</csymbol><apply id="A3.Ex56.m1.1.1.2.2.cmml" xref="A3.Ex56.m1.1.1.2.2"><ci id="A3.Ex56.m1.1.1.2.2.1.cmml" xref="A3.Ex56.m1.1.1.2.2.1">˙</ci><ci id="A3.Ex56.m1.1.1.2.2.2.cmml" xref="A3.Ex56.m1.1.1.2.2.2">𝑉</ci></apply><ci id="A3.Ex56.m1.1.1.2.3.cmml" xref="A3.Ex56.m1.1.1.2.3">𝜁</ci></apply><csymbol cd="latexml" id="A3.Ex56.m1.1.1.3.cmml" xref="A3.Ex56.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex56.m1.1c">\displaystyle\dot{V}_{\zeta}\leq</annotation><annotation encoding="application/x-llamapun" id="A3.Ex56.m1.1d">over˙ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-k_{\rm c}\bm{e}^{T}\bm{e}-\kappa\sigma^{*}(1+p)\tilde{\bm{\theta% }}^{T}_{\zeta}\tilde{\bm{\theta}}_{\zeta}+\delta\|\bm{e}\|\|\tilde{\bm{\theta}% }_{\zeta}\|." class="ltx_Math" display="inline" id="A3.Ex56.m2.2"><semantics id="A3.Ex56.m2.2a"><mrow id="A3.Ex56.m2.2.2.1" xref="A3.Ex56.m2.2.2.1.1.cmml"><mrow id="A3.Ex56.m2.2.2.1.1" xref="A3.Ex56.m2.2.2.1.1.cmml"><mrow id="A3.Ex56.m2.2.2.1.1.1" xref="A3.Ex56.m2.2.2.1.1.1.cmml"><mrow id="A3.Ex56.m2.2.2.1.1.1.3" xref="A3.Ex56.m2.2.2.1.1.1.3.cmml"><mo id="A3.Ex56.m2.2.2.1.1.1.3a" xref="A3.Ex56.m2.2.2.1.1.1.3.cmml">−</mo><mrow id="A3.Ex56.m2.2.2.1.1.1.3.2" xref="A3.Ex56.m2.2.2.1.1.1.3.2.cmml"><msub id="A3.Ex56.m2.2.2.1.1.1.3.2.2" xref="A3.Ex56.m2.2.2.1.1.1.3.2.2.cmml"><mi id="A3.Ex56.m2.2.2.1.1.1.3.2.2.2" xref="A3.Ex56.m2.2.2.1.1.1.3.2.2.2.cmml">k</mi><mi id="A3.Ex56.m2.2.2.1.1.1.3.2.2.3" mathvariant="normal" xref="A3.Ex56.m2.2.2.1.1.1.3.2.2.3.cmml">c</mi></msub><mo id="A3.Ex56.m2.2.2.1.1.1.3.2.1" xref="A3.Ex56.m2.2.2.1.1.1.3.2.1.cmml">⁢</mo><msup id="A3.Ex56.m2.2.2.1.1.1.3.2.3" xref="A3.Ex56.m2.2.2.1.1.1.3.2.3.cmml"><mi id="A3.Ex56.m2.2.2.1.1.1.3.2.3.2" xref="A3.Ex56.m2.2.2.1.1.1.3.2.3.2.cmml">𝒆</mi><mi id="A3.Ex56.m2.2.2.1.1.1.3.2.3.3" xref="A3.Ex56.m2.2.2.1.1.1.3.2.3.3.cmml">T</mi></msup><mo id="A3.Ex56.m2.2.2.1.1.1.3.2.1a" xref="A3.Ex56.m2.2.2.1.1.1.3.2.1.cmml">⁢</mo><mi id="A3.Ex56.m2.2.2.1.1.1.3.2.4" xref="A3.Ex56.m2.2.2.1.1.1.3.2.4.cmml">𝒆</mi></mrow></mrow><mo id="A3.Ex56.m2.2.2.1.1.1.2" xref="A3.Ex56.m2.2.2.1.1.1.2.cmml">−</mo><mrow id="A3.Ex56.m2.2.2.1.1.1.1" xref="A3.Ex56.m2.2.2.1.1.1.1.cmml"><mi id="A3.Ex56.m2.2.2.1.1.1.1.3" xref="A3.Ex56.m2.2.2.1.1.1.1.3.cmml">κ</mi><mo id="A3.Ex56.m2.2.2.1.1.1.1.2" xref="A3.Ex56.m2.2.2.1.1.1.1.2.cmml">⁢</mo><msup id="A3.Ex56.m2.2.2.1.1.1.1.4" xref="A3.Ex56.m2.2.2.1.1.1.1.4.cmml"><mi id="A3.Ex56.m2.2.2.1.1.1.1.4.2" xref="A3.Ex56.m2.2.2.1.1.1.1.4.2.cmml">σ</mi><mo id="A3.Ex56.m2.2.2.1.1.1.1.4.3" xref="A3.Ex56.m2.2.2.1.1.1.1.4.3.cmml">∗</mo></msup><mo id="A3.Ex56.m2.2.2.1.1.1.1.2a" xref="A3.Ex56.m2.2.2.1.1.1.1.2.cmml">⁢</mo><mrow id="A3.Ex56.m2.2.2.1.1.1.1.1.1" xref="A3.Ex56.m2.2.2.1.1.1.1.1.1.1.cmml"><mo id="A3.Ex56.m2.2.2.1.1.1.1.1.1.2" stretchy="false" xref="A3.Ex56.m2.2.2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.Ex56.m2.2.2.1.1.1.1.1.1.1" xref="A3.Ex56.m2.2.2.1.1.1.1.1.1.1.cmml"><mn id="A3.Ex56.m2.2.2.1.1.1.1.1.1.1.2" xref="A3.Ex56.m2.2.2.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A3.Ex56.m2.2.2.1.1.1.1.1.1.1.1" xref="A3.Ex56.m2.2.2.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="A3.Ex56.m2.2.2.1.1.1.1.1.1.1.3" xref="A3.Ex56.m2.2.2.1.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A3.Ex56.m2.2.2.1.1.1.1.1.1.3" stretchy="false" xref="A3.Ex56.m2.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A3.Ex56.m2.2.2.1.1.1.1.2b" xref="A3.Ex56.m2.2.2.1.1.1.1.2.cmml">⁢</mo><msubsup id="A3.Ex56.m2.2.2.1.1.1.1.5" xref="A3.Ex56.m2.2.2.1.1.1.1.5.cmml"><mover accent="true" id="A3.Ex56.m2.2.2.1.1.1.1.5.2.2" xref="A3.Ex56.m2.2.2.1.1.1.1.5.2.2.cmml"><mi id="A3.Ex56.m2.2.2.1.1.1.1.5.2.2.2" xref="A3.Ex56.m2.2.2.1.1.1.1.5.2.2.2.cmml">𝜽</mi><mo id="A3.Ex56.m2.2.2.1.1.1.1.5.2.2.1" xref="A3.Ex56.m2.2.2.1.1.1.1.5.2.2.1.cmml">~</mo></mover><mi id="A3.Ex56.m2.2.2.1.1.1.1.5.3" xref="A3.Ex56.m2.2.2.1.1.1.1.5.3.cmml">ζ</mi><mi id="A3.Ex56.m2.2.2.1.1.1.1.5.2.3" xref="A3.Ex56.m2.2.2.1.1.1.1.5.2.3.cmml">T</mi></msubsup><mo id="A3.Ex56.m2.2.2.1.1.1.1.2c" xref="A3.Ex56.m2.2.2.1.1.1.1.2.cmml">⁢</mo><msub id="A3.Ex56.m2.2.2.1.1.1.1.6" xref="A3.Ex56.m2.2.2.1.1.1.1.6.cmml"><mover accent="true" id="A3.Ex56.m2.2.2.1.1.1.1.6.2" xref="A3.Ex56.m2.2.2.1.1.1.1.6.2.cmml"><mi id="A3.Ex56.m2.2.2.1.1.1.1.6.2.2" xref="A3.Ex56.m2.2.2.1.1.1.1.6.2.2.cmml">𝜽</mi><mo id="A3.Ex56.m2.2.2.1.1.1.1.6.2.1" xref="A3.Ex56.m2.2.2.1.1.1.1.6.2.1.cmml">~</mo></mover><mi id="A3.Ex56.m2.2.2.1.1.1.1.6.3" xref="A3.Ex56.m2.2.2.1.1.1.1.6.3.cmml">ζ</mi></msub></mrow></mrow><mo id="A3.Ex56.m2.2.2.1.1.3" xref="A3.Ex56.m2.2.2.1.1.3.cmml">+</mo><mrow id="A3.Ex56.m2.2.2.1.1.2" xref="A3.Ex56.m2.2.2.1.1.2.cmml"><mi id="A3.Ex56.m2.2.2.1.1.2.3" xref="A3.Ex56.m2.2.2.1.1.2.3.cmml">δ</mi><mo id="A3.Ex56.m2.2.2.1.1.2.2" xref="A3.Ex56.m2.2.2.1.1.2.2.cmml">⁢</mo><mrow id="A3.Ex56.m2.2.2.1.1.2.4.2" xref="A3.Ex56.m2.2.2.1.1.2.4.1.cmml"><mo id="A3.Ex56.m2.2.2.1.1.2.4.2.1" stretchy="false" xref="A3.Ex56.m2.2.2.1.1.2.4.1.1.cmml">‖</mo><mi id="A3.Ex56.m2.1.1" xref="A3.Ex56.m2.1.1.cmml">𝒆</mi><mo id="A3.Ex56.m2.2.2.1.1.2.4.2.2" stretchy="false" xref="A3.Ex56.m2.2.2.1.1.2.4.1.1.cmml">‖</mo></mrow><mo id="A3.Ex56.m2.2.2.1.1.2.2a" xref="A3.Ex56.m2.2.2.1.1.2.2.cmml">⁢</mo><mrow id="A3.Ex56.m2.2.2.1.1.2.1.1" xref="A3.Ex56.m2.2.2.1.1.2.1.2.cmml"><mo id="A3.Ex56.m2.2.2.1.1.2.1.1.2" stretchy="false" xref="A3.Ex56.m2.2.2.1.1.2.1.2.1.cmml">‖</mo><msub id="A3.Ex56.m2.2.2.1.1.2.1.1.1" xref="A3.Ex56.m2.2.2.1.1.2.1.1.1.cmml"><mover accent="true" id="A3.Ex56.m2.2.2.1.1.2.1.1.1.2" xref="A3.Ex56.m2.2.2.1.1.2.1.1.1.2.cmml"><mi id="A3.Ex56.m2.2.2.1.1.2.1.1.1.2.2" xref="A3.Ex56.m2.2.2.1.1.2.1.1.1.2.2.cmml">𝜽</mi><mo id="A3.Ex56.m2.2.2.1.1.2.1.1.1.2.1" xref="A3.Ex56.m2.2.2.1.1.2.1.1.1.2.1.cmml">~</mo></mover><mi id="A3.Ex56.m2.2.2.1.1.2.1.1.1.3" xref="A3.Ex56.m2.2.2.1.1.2.1.1.1.3.cmml">ζ</mi></msub><mo id="A3.Ex56.m2.2.2.1.1.2.1.1.3" stretchy="false" xref="A3.Ex56.m2.2.2.1.1.2.1.2.1.cmml">‖</mo></mrow></mrow></mrow><mo id="A3.Ex56.m2.2.2.1.2" lspace="0em" xref="A3.Ex56.m2.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex56.m2.2b"><apply id="A3.Ex56.m2.2.2.1.1.cmml" xref="A3.Ex56.m2.2.2.1"><plus id="A3.Ex56.m2.2.2.1.1.3.cmml" xref="A3.Ex56.m2.2.2.1.1.3"></plus><apply id="A3.Ex56.m2.2.2.1.1.1.cmml" xref="A3.Ex56.m2.2.2.1.1.1"><minus id="A3.Ex56.m2.2.2.1.1.1.2.cmml" xref="A3.Ex56.m2.2.2.1.1.1.2"></minus><apply id="A3.Ex56.m2.2.2.1.1.1.3.cmml" xref="A3.Ex56.m2.2.2.1.1.1.3"><minus id="A3.Ex56.m2.2.2.1.1.1.3.1.cmml" xref="A3.Ex56.m2.2.2.1.1.1.3"></minus><apply id="A3.Ex56.m2.2.2.1.1.1.3.2.cmml" xref="A3.Ex56.m2.2.2.1.1.1.3.2"><times id="A3.Ex56.m2.2.2.1.1.1.3.2.1.cmml" xref="A3.Ex56.m2.2.2.1.1.1.3.2.1"></times><apply id="A3.Ex56.m2.2.2.1.1.1.3.2.2.cmml" xref="A3.Ex56.m2.2.2.1.1.1.3.2.2"><csymbol cd="ambiguous" id="A3.Ex56.m2.2.2.1.1.1.3.2.2.1.cmml" xref="A3.Ex56.m2.2.2.1.1.1.3.2.2">subscript</csymbol><ci id="A3.Ex56.m2.2.2.1.1.1.3.2.2.2.cmml" xref="A3.Ex56.m2.2.2.1.1.1.3.2.2.2">𝑘</ci><ci id="A3.Ex56.m2.2.2.1.1.1.3.2.2.3.cmml" xref="A3.Ex56.m2.2.2.1.1.1.3.2.2.3">c</ci></apply><apply id="A3.Ex56.m2.2.2.1.1.1.3.2.3.cmml" xref="A3.Ex56.m2.2.2.1.1.1.3.2.3"><csymbol cd="ambiguous" id="A3.Ex56.m2.2.2.1.1.1.3.2.3.1.cmml" xref="A3.Ex56.m2.2.2.1.1.1.3.2.3">superscript</csymbol><ci id="A3.Ex56.m2.2.2.1.1.1.3.2.3.2.cmml" xref="A3.Ex56.m2.2.2.1.1.1.3.2.3.2">𝒆</ci><ci id="A3.Ex56.m2.2.2.1.1.1.3.2.3.3.cmml" xref="A3.Ex56.m2.2.2.1.1.1.3.2.3.3">𝑇</ci></apply><ci id="A3.Ex56.m2.2.2.1.1.1.3.2.4.cmml" xref="A3.Ex56.m2.2.2.1.1.1.3.2.4">𝒆</ci></apply></apply><apply id="A3.Ex56.m2.2.2.1.1.1.1.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1"><times id="A3.Ex56.m2.2.2.1.1.1.1.2.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1.2"></times><ci id="A3.Ex56.m2.2.2.1.1.1.1.3.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1.3">𝜅</ci><apply id="A3.Ex56.m2.2.2.1.1.1.1.4.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1.4"><csymbol cd="ambiguous" id="A3.Ex56.m2.2.2.1.1.1.1.4.1.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1.4">superscript</csymbol><ci id="A3.Ex56.m2.2.2.1.1.1.1.4.2.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1.4.2">𝜎</ci><times id="A3.Ex56.m2.2.2.1.1.1.1.4.3.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1.4.3"></times></apply><apply id="A3.Ex56.m2.2.2.1.1.1.1.1.1.1.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1.1.1"><plus id="A3.Ex56.m2.2.2.1.1.1.1.1.1.1.1.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1.1.1.1.1"></plus><cn id="A3.Ex56.m2.2.2.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A3.Ex56.m2.2.2.1.1.1.1.1.1.1.2">1</cn><ci id="A3.Ex56.m2.2.2.1.1.1.1.1.1.1.3.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="A3.Ex56.m2.2.2.1.1.1.1.5.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1.5"><csymbol cd="ambiguous" id="A3.Ex56.m2.2.2.1.1.1.1.5.1.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1.5">subscript</csymbol><apply id="A3.Ex56.m2.2.2.1.1.1.1.5.2.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1.5"><csymbol cd="ambiguous" id="A3.Ex56.m2.2.2.1.1.1.1.5.2.1.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1.5">superscript</csymbol><apply id="A3.Ex56.m2.2.2.1.1.1.1.5.2.2.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1.5.2.2"><ci id="A3.Ex56.m2.2.2.1.1.1.1.5.2.2.1.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1.5.2.2.1">~</ci><ci id="A3.Ex56.m2.2.2.1.1.1.1.5.2.2.2.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1.5.2.2.2">𝜽</ci></apply><ci id="A3.Ex56.m2.2.2.1.1.1.1.5.2.3.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1.5.2.3">𝑇</ci></apply><ci id="A3.Ex56.m2.2.2.1.1.1.1.5.3.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1.5.3">𝜁</ci></apply><apply id="A3.Ex56.m2.2.2.1.1.1.1.6.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1.6"><csymbol cd="ambiguous" id="A3.Ex56.m2.2.2.1.1.1.1.6.1.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1.6">subscript</csymbol><apply id="A3.Ex56.m2.2.2.1.1.1.1.6.2.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1.6.2"><ci id="A3.Ex56.m2.2.2.1.1.1.1.6.2.1.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1.6.2.1">~</ci><ci id="A3.Ex56.m2.2.2.1.1.1.1.6.2.2.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1.6.2.2">𝜽</ci></apply><ci id="A3.Ex56.m2.2.2.1.1.1.1.6.3.cmml" xref="A3.Ex56.m2.2.2.1.1.1.1.6.3">𝜁</ci></apply></apply></apply><apply id="A3.Ex56.m2.2.2.1.1.2.cmml" xref="A3.Ex56.m2.2.2.1.1.2"><times id="A3.Ex56.m2.2.2.1.1.2.2.cmml" xref="A3.Ex56.m2.2.2.1.1.2.2"></times><ci id="A3.Ex56.m2.2.2.1.1.2.3.cmml" xref="A3.Ex56.m2.2.2.1.1.2.3">𝛿</ci><apply id="A3.Ex56.m2.2.2.1.1.2.4.1.cmml" xref="A3.Ex56.m2.2.2.1.1.2.4.2"><csymbol cd="latexml" id="A3.Ex56.m2.2.2.1.1.2.4.1.1.cmml" xref="A3.Ex56.m2.2.2.1.1.2.4.2.1">norm</csymbol><ci id="A3.Ex56.m2.1.1.cmml" xref="A3.Ex56.m2.1.1">𝒆</ci></apply><apply id="A3.Ex56.m2.2.2.1.1.2.1.2.cmml" xref="A3.Ex56.m2.2.2.1.1.2.1.1"><csymbol cd="latexml" id="A3.Ex56.m2.2.2.1.1.2.1.2.1.cmml" xref="A3.Ex56.m2.2.2.1.1.2.1.1.2">norm</csymbol><apply id="A3.Ex56.m2.2.2.1.1.2.1.1.1.cmml" xref="A3.Ex56.m2.2.2.1.1.2.1.1.1"><csymbol cd="ambiguous" id="A3.Ex56.m2.2.2.1.1.2.1.1.1.1.cmml" xref="A3.Ex56.m2.2.2.1.1.2.1.1.1">subscript</csymbol><apply id="A3.Ex56.m2.2.2.1.1.2.1.1.1.2.cmml" xref="A3.Ex56.m2.2.2.1.1.2.1.1.1.2"><ci id="A3.Ex56.m2.2.2.1.1.2.1.1.1.2.1.cmml" xref="A3.Ex56.m2.2.2.1.1.2.1.1.1.2.1">~</ci><ci id="A3.Ex56.m2.2.2.1.1.2.1.1.1.2.2.cmml" xref="A3.Ex56.m2.2.2.1.1.2.1.1.1.2.2">𝜽</ci></apply><ci id="A3.Ex56.m2.2.2.1.1.2.1.1.1.3.cmml" xref="A3.Ex56.m2.2.2.1.1.2.1.1.1.3">𝜁</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex56.m2.2c">\displaystyle-k_{\rm c}\bm{e}^{T}\bm{e}-\kappa\sigma^{*}(1+p)\tilde{\bm{\theta% }}^{T}_{\zeta}\tilde{\bm{\theta}}_{\zeta}+\delta\|\bm{e}\|\|\tilde{\bm{\theta}% }_{\zeta}\|.</annotation><annotation encoding="application/x-llamapun" id="A3.Ex56.m2.2d">- italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_e - italic_κ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( 1 + italic_p ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT + italic_δ ∥ bold_italic_e ∥ ∥ over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ∥ .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.2.p2.65">Applying Young’s inequality <math alttext="ab\leq a^{2}/2+b^{2}/2" class="ltx_Math" display="inline" id="A3.2.p2.63.m1.1"><semantics id="A3.2.p2.63.m1.1a"><mrow id="A3.2.p2.63.m1.1.1" xref="A3.2.p2.63.m1.1.1.cmml"><mrow id="A3.2.p2.63.m1.1.1.2" xref="A3.2.p2.63.m1.1.1.2.cmml"><mi id="A3.2.p2.63.m1.1.1.2.2" xref="A3.2.p2.63.m1.1.1.2.2.cmml">a</mi><mo id="A3.2.p2.63.m1.1.1.2.1" xref="A3.2.p2.63.m1.1.1.2.1.cmml">⁢</mo><mi id="A3.2.p2.63.m1.1.1.2.3" xref="A3.2.p2.63.m1.1.1.2.3.cmml">b</mi></mrow><mo id="A3.2.p2.63.m1.1.1.1" xref="A3.2.p2.63.m1.1.1.1.cmml">≤</mo><mrow id="A3.2.p2.63.m1.1.1.3" xref="A3.2.p2.63.m1.1.1.3.cmml"><mrow id="A3.2.p2.63.m1.1.1.3.2" xref="A3.2.p2.63.m1.1.1.3.2.cmml"><msup id="A3.2.p2.63.m1.1.1.3.2.2" xref="A3.2.p2.63.m1.1.1.3.2.2.cmml"><mi id="A3.2.p2.63.m1.1.1.3.2.2.2" xref="A3.2.p2.63.m1.1.1.3.2.2.2.cmml">a</mi><mn id="A3.2.p2.63.m1.1.1.3.2.2.3" xref="A3.2.p2.63.m1.1.1.3.2.2.3.cmml">2</mn></msup><mo id="A3.2.p2.63.m1.1.1.3.2.1" xref="A3.2.p2.63.m1.1.1.3.2.1.cmml">/</mo><mn id="A3.2.p2.63.m1.1.1.3.2.3" xref="A3.2.p2.63.m1.1.1.3.2.3.cmml">2</mn></mrow><mo id="A3.2.p2.63.m1.1.1.3.1" xref="A3.2.p2.63.m1.1.1.3.1.cmml">+</mo><mrow id="A3.2.p2.63.m1.1.1.3.3" xref="A3.2.p2.63.m1.1.1.3.3.cmml"><msup id="A3.2.p2.63.m1.1.1.3.3.2" xref="A3.2.p2.63.m1.1.1.3.3.2.cmml"><mi id="A3.2.p2.63.m1.1.1.3.3.2.2" xref="A3.2.p2.63.m1.1.1.3.3.2.2.cmml">b</mi><mn id="A3.2.p2.63.m1.1.1.3.3.2.3" xref="A3.2.p2.63.m1.1.1.3.3.2.3.cmml">2</mn></msup><mo id="A3.2.p2.63.m1.1.1.3.3.1" xref="A3.2.p2.63.m1.1.1.3.3.1.cmml">/</mo><mn id="A3.2.p2.63.m1.1.1.3.3.3" xref="A3.2.p2.63.m1.1.1.3.3.3.cmml">2</mn></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.63.m1.1b"><apply id="A3.2.p2.63.m1.1.1.cmml" xref="A3.2.p2.63.m1.1.1"><leq id="A3.2.p2.63.m1.1.1.1.cmml" xref="A3.2.p2.63.m1.1.1.1"></leq><apply id="A3.2.p2.63.m1.1.1.2.cmml" xref="A3.2.p2.63.m1.1.1.2"><times id="A3.2.p2.63.m1.1.1.2.1.cmml" xref="A3.2.p2.63.m1.1.1.2.1"></times><ci id="A3.2.p2.63.m1.1.1.2.2.cmml" xref="A3.2.p2.63.m1.1.1.2.2">𝑎</ci><ci id="A3.2.p2.63.m1.1.1.2.3.cmml" xref="A3.2.p2.63.m1.1.1.2.3">𝑏</ci></apply><apply id="A3.2.p2.63.m1.1.1.3.cmml" xref="A3.2.p2.63.m1.1.1.3"><plus id="A3.2.p2.63.m1.1.1.3.1.cmml" xref="A3.2.p2.63.m1.1.1.3.1"></plus><apply id="A3.2.p2.63.m1.1.1.3.2.cmml" xref="A3.2.p2.63.m1.1.1.3.2"><divide id="A3.2.p2.63.m1.1.1.3.2.1.cmml" xref="A3.2.p2.63.m1.1.1.3.2.1"></divide><apply id="A3.2.p2.63.m1.1.1.3.2.2.cmml" xref="A3.2.p2.63.m1.1.1.3.2.2"><csymbol cd="ambiguous" id="A3.2.p2.63.m1.1.1.3.2.2.1.cmml" xref="A3.2.p2.63.m1.1.1.3.2.2">superscript</csymbol><ci id="A3.2.p2.63.m1.1.1.3.2.2.2.cmml" xref="A3.2.p2.63.m1.1.1.3.2.2.2">𝑎</ci><cn id="A3.2.p2.63.m1.1.1.3.2.2.3.cmml" type="integer" xref="A3.2.p2.63.m1.1.1.3.2.2.3">2</cn></apply><cn id="A3.2.p2.63.m1.1.1.3.2.3.cmml" type="integer" xref="A3.2.p2.63.m1.1.1.3.2.3">2</cn></apply><apply id="A3.2.p2.63.m1.1.1.3.3.cmml" xref="A3.2.p2.63.m1.1.1.3.3"><divide id="A3.2.p2.63.m1.1.1.3.3.1.cmml" xref="A3.2.p2.63.m1.1.1.3.3.1"></divide><apply id="A3.2.p2.63.m1.1.1.3.3.2.cmml" xref="A3.2.p2.63.m1.1.1.3.3.2"><csymbol cd="ambiguous" id="A3.2.p2.63.m1.1.1.3.3.2.1.cmml" xref="A3.2.p2.63.m1.1.1.3.3.2">superscript</csymbol><ci id="A3.2.p2.63.m1.1.1.3.3.2.2.cmml" xref="A3.2.p2.63.m1.1.1.3.3.2.2">𝑏</ci><cn id="A3.2.p2.63.m1.1.1.3.3.2.3.cmml" type="integer" xref="A3.2.p2.63.m1.1.1.3.3.2.3">2</cn></apply><cn id="A3.2.p2.63.m1.1.1.3.3.3.cmml" type="integer" xref="A3.2.p2.63.m1.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.63.m1.1c">ab\leq a^{2}/2+b^{2}/2</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.63.m1.1d">italic_a italic_b ≤ italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 + italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2</annotation></semantics></math> with <math alttext="a=\sqrt{k_{\rm c}}\|\bm{e}\|" class="ltx_Math" display="inline" id="A3.2.p2.64.m2.1"><semantics id="A3.2.p2.64.m2.1a"><mrow id="A3.2.p2.64.m2.1.2" xref="A3.2.p2.64.m2.1.2.cmml"><mi id="A3.2.p2.64.m2.1.2.2" xref="A3.2.p2.64.m2.1.2.2.cmml">a</mi><mo id="A3.2.p2.64.m2.1.2.1" xref="A3.2.p2.64.m2.1.2.1.cmml">=</mo><mrow id="A3.2.p2.64.m2.1.2.3" xref="A3.2.p2.64.m2.1.2.3.cmml"><msqrt id="A3.2.p2.64.m2.1.2.3.2" xref="A3.2.p2.64.m2.1.2.3.2.cmml"><msub id="A3.2.p2.64.m2.1.2.3.2.2" xref="A3.2.p2.64.m2.1.2.3.2.2.cmml"><mi id="A3.2.p2.64.m2.1.2.3.2.2.2" xref="A3.2.p2.64.m2.1.2.3.2.2.2.cmml">k</mi><mi id="A3.2.p2.64.m2.1.2.3.2.2.3" mathvariant="normal" xref="A3.2.p2.64.m2.1.2.3.2.2.3.cmml">c</mi></msub></msqrt><mo id="A3.2.p2.64.m2.1.2.3.1" xref="A3.2.p2.64.m2.1.2.3.1.cmml">⁢</mo><mrow id="A3.2.p2.64.m2.1.2.3.3.2" xref="A3.2.p2.64.m2.1.2.3.3.1.cmml"><mo id="A3.2.p2.64.m2.1.2.3.3.2.1" stretchy="false" xref="A3.2.p2.64.m2.1.2.3.3.1.1.cmml">‖</mo><mi id="A3.2.p2.64.m2.1.1" xref="A3.2.p2.64.m2.1.1.cmml">𝒆</mi><mo id="A3.2.p2.64.m2.1.2.3.3.2.2" stretchy="false" xref="A3.2.p2.64.m2.1.2.3.3.1.1.cmml">‖</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.64.m2.1b"><apply id="A3.2.p2.64.m2.1.2.cmml" xref="A3.2.p2.64.m2.1.2"><eq id="A3.2.p2.64.m2.1.2.1.cmml" xref="A3.2.p2.64.m2.1.2.1"></eq><ci id="A3.2.p2.64.m2.1.2.2.cmml" xref="A3.2.p2.64.m2.1.2.2">𝑎</ci><apply id="A3.2.p2.64.m2.1.2.3.cmml" xref="A3.2.p2.64.m2.1.2.3"><times id="A3.2.p2.64.m2.1.2.3.1.cmml" xref="A3.2.p2.64.m2.1.2.3.1"></times><apply id="A3.2.p2.64.m2.1.2.3.2.cmml" xref="A3.2.p2.64.m2.1.2.3.2"><root id="A3.2.p2.64.m2.1.2.3.2a.cmml" xref="A3.2.p2.64.m2.1.2.3.2"></root><apply id="A3.2.p2.64.m2.1.2.3.2.2.cmml" xref="A3.2.p2.64.m2.1.2.3.2.2"><csymbol cd="ambiguous" id="A3.2.p2.64.m2.1.2.3.2.2.1.cmml" xref="A3.2.p2.64.m2.1.2.3.2.2">subscript</csymbol><ci id="A3.2.p2.64.m2.1.2.3.2.2.2.cmml" xref="A3.2.p2.64.m2.1.2.3.2.2.2">𝑘</ci><ci id="A3.2.p2.64.m2.1.2.3.2.2.3.cmml" xref="A3.2.p2.64.m2.1.2.3.2.2.3">c</ci></apply></apply><apply id="A3.2.p2.64.m2.1.2.3.3.1.cmml" xref="A3.2.p2.64.m2.1.2.3.3.2"><csymbol cd="latexml" id="A3.2.p2.64.m2.1.2.3.3.1.1.cmml" xref="A3.2.p2.64.m2.1.2.3.3.2.1">norm</csymbol><ci id="A3.2.p2.64.m2.1.1.cmml" xref="A3.2.p2.64.m2.1.1">𝒆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.64.m2.1c">a=\sqrt{k_{\rm c}}\|\bm{e}\|</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.64.m2.1d">italic_a = square-root start_ARG italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT end_ARG ∥ bold_italic_e ∥</annotation></semantics></math> and <math alttext="b=\delta\|\tilde{\bm{\theta}}_{\zeta}\|/\sqrt{k_{\rm c}}" class="ltx_Math" display="inline" id="A3.2.p2.65.m3.1"><semantics id="A3.2.p2.65.m3.1a"><mrow id="A3.2.p2.65.m3.1.1" xref="A3.2.p2.65.m3.1.1.cmml"><mi id="A3.2.p2.65.m3.1.1.3" xref="A3.2.p2.65.m3.1.1.3.cmml">b</mi><mo id="A3.2.p2.65.m3.1.1.2" xref="A3.2.p2.65.m3.1.1.2.cmml">=</mo><mrow id="A3.2.p2.65.m3.1.1.1" xref="A3.2.p2.65.m3.1.1.1.cmml"><mrow id="A3.2.p2.65.m3.1.1.1.1" xref="A3.2.p2.65.m3.1.1.1.1.cmml"><mi id="A3.2.p2.65.m3.1.1.1.1.3" xref="A3.2.p2.65.m3.1.1.1.1.3.cmml">δ</mi><mo id="A3.2.p2.65.m3.1.1.1.1.2" xref="A3.2.p2.65.m3.1.1.1.1.2.cmml">⁢</mo><mrow id="A3.2.p2.65.m3.1.1.1.1.1.1" xref="A3.2.p2.65.m3.1.1.1.1.1.2.cmml"><mo id="A3.2.p2.65.m3.1.1.1.1.1.1.2" stretchy="false" xref="A3.2.p2.65.m3.1.1.1.1.1.2.1.cmml">‖</mo><msub id="A3.2.p2.65.m3.1.1.1.1.1.1.1" xref="A3.2.p2.65.m3.1.1.1.1.1.1.1.cmml"><mover accent="true" id="A3.2.p2.65.m3.1.1.1.1.1.1.1.2" xref="A3.2.p2.65.m3.1.1.1.1.1.1.1.2.cmml"><mi id="A3.2.p2.65.m3.1.1.1.1.1.1.1.2.2" xref="A3.2.p2.65.m3.1.1.1.1.1.1.1.2.2.cmml">𝜽</mi><mo id="A3.2.p2.65.m3.1.1.1.1.1.1.1.2.1" xref="A3.2.p2.65.m3.1.1.1.1.1.1.1.2.1.cmml">~</mo></mover><mi id="A3.2.p2.65.m3.1.1.1.1.1.1.1.3" xref="A3.2.p2.65.m3.1.1.1.1.1.1.1.3.cmml">ζ</mi></msub><mo id="A3.2.p2.65.m3.1.1.1.1.1.1.3" stretchy="false" xref="A3.2.p2.65.m3.1.1.1.1.1.2.1.cmml">‖</mo></mrow></mrow><mo id="A3.2.p2.65.m3.1.1.1.2" xref="A3.2.p2.65.m3.1.1.1.2.cmml">/</mo><msqrt id="A3.2.p2.65.m3.1.1.1.3" xref="A3.2.p2.65.m3.1.1.1.3.cmml"><msub id="A3.2.p2.65.m3.1.1.1.3.2" xref="A3.2.p2.65.m3.1.1.1.3.2.cmml"><mi id="A3.2.p2.65.m3.1.1.1.3.2.2" xref="A3.2.p2.65.m3.1.1.1.3.2.2.cmml">k</mi><mi id="A3.2.p2.65.m3.1.1.1.3.2.3" mathvariant="normal" xref="A3.2.p2.65.m3.1.1.1.3.2.3.cmml">c</mi></msub></msqrt></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.65.m3.1b"><apply id="A3.2.p2.65.m3.1.1.cmml" xref="A3.2.p2.65.m3.1.1"><eq id="A3.2.p2.65.m3.1.1.2.cmml" xref="A3.2.p2.65.m3.1.1.2"></eq><ci id="A3.2.p2.65.m3.1.1.3.cmml" xref="A3.2.p2.65.m3.1.1.3">𝑏</ci><apply id="A3.2.p2.65.m3.1.1.1.cmml" xref="A3.2.p2.65.m3.1.1.1"><divide id="A3.2.p2.65.m3.1.1.1.2.cmml" xref="A3.2.p2.65.m3.1.1.1.2"></divide><apply id="A3.2.p2.65.m3.1.1.1.1.cmml" xref="A3.2.p2.65.m3.1.1.1.1"><times id="A3.2.p2.65.m3.1.1.1.1.2.cmml" xref="A3.2.p2.65.m3.1.1.1.1.2"></times><ci id="A3.2.p2.65.m3.1.1.1.1.3.cmml" xref="A3.2.p2.65.m3.1.1.1.1.3">𝛿</ci><apply id="A3.2.p2.65.m3.1.1.1.1.1.2.cmml" xref="A3.2.p2.65.m3.1.1.1.1.1.1"><csymbol cd="latexml" id="A3.2.p2.65.m3.1.1.1.1.1.2.1.cmml" xref="A3.2.p2.65.m3.1.1.1.1.1.1.2">norm</csymbol><apply id="A3.2.p2.65.m3.1.1.1.1.1.1.1.cmml" xref="A3.2.p2.65.m3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.2.p2.65.m3.1.1.1.1.1.1.1.1.cmml" xref="A3.2.p2.65.m3.1.1.1.1.1.1.1">subscript</csymbol><apply id="A3.2.p2.65.m3.1.1.1.1.1.1.1.2.cmml" xref="A3.2.p2.65.m3.1.1.1.1.1.1.1.2"><ci id="A3.2.p2.65.m3.1.1.1.1.1.1.1.2.1.cmml" xref="A3.2.p2.65.m3.1.1.1.1.1.1.1.2.1">~</ci><ci id="A3.2.p2.65.m3.1.1.1.1.1.1.1.2.2.cmml" xref="A3.2.p2.65.m3.1.1.1.1.1.1.1.2.2">𝜽</ci></apply><ci id="A3.2.p2.65.m3.1.1.1.1.1.1.1.3.cmml" xref="A3.2.p2.65.m3.1.1.1.1.1.1.1.3">𝜁</ci></apply></apply></apply><apply id="A3.2.p2.65.m3.1.1.1.3.cmml" xref="A3.2.p2.65.m3.1.1.1.3"><root id="A3.2.p2.65.m3.1.1.1.3a.cmml" xref="A3.2.p2.65.m3.1.1.1.3"></root><apply id="A3.2.p2.65.m3.1.1.1.3.2.cmml" xref="A3.2.p2.65.m3.1.1.1.3.2"><csymbol cd="ambiguous" id="A3.2.p2.65.m3.1.1.1.3.2.1.cmml" xref="A3.2.p2.65.m3.1.1.1.3.2">subscript</csymbol><ci id="A3.2.p2.65.m3.1.1.1.3.2.2.cmml" xref="A3.2.p2.65.m3.1.1.1.3.2.2">𝑘</ci><ci id="A3.2.p2.65.m3.1.1.1.3.2.3.cmml" xref="A3.2.p2.65.m3.1.1.1.3.2.3">c</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.65.m3.1c">b=\delta\|\tilde{\bm{\theta}}_{\zeta}\|/\sqrt{k_{\rm c}}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.65.m3.1d">italic_b = italic_δ ∥ over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ∥ / square-root start_ARG italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT end_ARG</annotation></semantics></math> to the above inequality yields</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx72"> <tbody id="A3.Ex57"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}_{\zeta}\leq" class="ltx_Math" display="inline" id="A3.Ex57.m1.1"><semantics id="A3.Ex57.m1.1a"><mrow id="A3.Ex57.m1.1.1" xref="A3.Ex57.m1.1.1.cmml"><msub id="A3.Ex57.m1.1.1.2" xref="A3.Ex57.m1.1.1.2.cmml"><mover accent="true" id="A3.Ex57.m1.1.1.2.2" xref="A3.Ex57.m1.1.1.2.2.cmml"><mi id="A3.Ex57.m1.1.1.2.2.2" xref="A3.Ex57.m1.1.1.2.2.2.cmml">V</mi><mo id="A3.Ex57.m1.1.1.2.2.1" xref="A3.Ex57.m1.1.1.2.2.1.cmml">˙</mo></mover><mi id="A3.Ex57.m1.1.1.2.3" xref="A3.Ex57.m1.1.1.2.3.cmml">ζ</mi></msub><mo id="A3.Ex57.m1.1.1.1" xref="A3.Ex57.m1.1.1.1.cmml">≤</mo><mi id="A3.Ex57.m1.1.1.3" xref="A3.Ex57.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex57.m1.1b"><apply id="A3.Ex57.m1.1.1.cmml" xref="A3.Ex57.m1.1.1"><leq id="A3.Ex57.m1.1.1.1.cmml" xref="A3.Ex57.m1.1.1.1"></leq><apply id="A3.Ex57.m1.1.1.2.cmml" xref="A3.Ex57.m1.1.1.2"><csymbol cd="ambiguous" id="A3.Ex57.m1.1.1.2.1.cmml" xref="A3.Ex57.m1.1.1.2">subscript</csymbol><apply id="A3.Ex57.m1.1.1.2.2.cmml" xref="A3.Ex57.m1.1.1.2.2"><ci id="A3.Ex57.m1.1.1.2.2.1.cmml" xref="A3.Ex57.m1.1.1.2.2.1">˙</ci><ci id="A3.Ex57.m1.1.1.2.2.2.cmml" xref="A3.Ex57.m1.1.1.2.2.2">𝑉</ci></apply><ci id="A3.Ex57.m1.1.1.2.3.cmml" xref="A3.Ex57.m1.1.1.2.3">𝜁</ci></apply><csymbol cd="latexml" id="A3.Ex57.m1.1.1.3.cmml" xref="A3.Ex57.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex57.m1.1c">\displaystyle\dot{V}_{\zeta}\leq</annotation><annotation encoding="application/x-llamapun" id="A3.Ex57.m1.1d">over˙ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-k_{\rm c}\bm{e}^{T}\bm{e}/2-\kappa\sigma^{*}(1+p)\tilde{\bm{% \theta}}^{T}_{\zeta}\tilde{\bm{\theta}}_{\zeta}+\delta^{2}\|\tilde{\bm{\theta}% }_{\zeta}\|^{2}/(2k_{\rm c})." class="ltx_Math" display="inline" id="A3.Ex57.m2.1"><semantics id="A3.Ex57.m2.1a"><mrow id="A3.Ex57.m2.1.1.1" xref="A3.Ex57.m2.1.1.1.1.cmml"><mrow id="A3.Ex57.m2.1.1.1.1" xref="A3.Ex57.m2.1.1.1.1.cmml"><mrow id="A3.Ex57.m2.1.1.1.1.1" xref="A3.Ex57.m2.1.1.1.1.1.cmml"><mrow id="A3.Ex57.m2.1.1.1.1.1.3" xref="A3.Ex57.m2.1.1.1.1.1.3.cmml"><mo id="A3.Ex57.m2.1.1.1.1.1.3a" xref="A3.Ex57.m2.1.1.1.1.1.3.cmml">−</mo><mrow id="A3.Ex57.m2.1.1.1.1.1.3.2" xref="A3.Ex57.m2.1.1.1.1.1.3.2.cmml"><mrow id="A3.Ex57.m2.1.1.1.1.1.3.2.2" xref="A3.Ex57.m2.1.1.1.1.1.3.2.2.cmml"><msub id="A3.Ex57.m2.1.1.1.1.1.3.2.2.2" xref="A3.Ex57.m2.1.1.1.1.1.3.2.2.2.cmml"><mi id="A3.Ex57.m2.1.1.1.1.1.3.2.2.2.2" xref="A3.Ex57.m2.1.1.1.1.1.3.2.2.2.2.cmml">k</mi><mi id="A3.Ex57.m2.1.1.1.1.1.3.2.2.2.3" mathvariant="normal" xref="A3.Ex57.m2.1.1.1.1.1.3.2.2.2.3.cmml">c</mi></msub><mo id="A3.Ex57.m2.1.1.1.1.1.3.2.2.1" xref="A3.Ex57.m2.1.1.1.1.1.3.2.2.1.cmml">⁢</mo><msup id="A3.Ex57.m2.1.1.1.1.1.3.2.2.3" xref="A3.Ex57.m2.1.1.1.1.1.3.2.2.3.cmml"><mi id="A3.Ex57.m2.1.1.1.1.1.3.2.2.3.2" xref="A3.Ex57.m2.1.1.1.1.1.3.2.2.3.2.cmml">𝒆</mi><mi id="A3.Ex57.m2.1.1.1.1.1.3.2.2.3.3" xref="A3.Ex57.m2.1.1.1.1.1.3.2.2.3.3.cmml">T</mi></msup><mo id="A3.Ex57.m2.1.1.1.1.1.3.2.2.1a" xref="A3.Ex57.m2.1.1.1.1.1.3.2.2.1.cmml">⁢</mo><mi id="A3.Ex57.m2.1.1.1.1.1.3.2.2.4" xref="A3.Ex57.m2.1.1.1.1.1.3.2.2.4.cmml">𝒆</mi></mrow><mo id="A3.Ex57.m2.1.1.1.1.1.3.2.1" xref="A3.Ex57.m2.1.1.1.1.1.3.2.1.cmml">/</mo><mn id="A3.Ex57.m2.1.1.1.1.1.3.2.3" xref="A3.Ex57.m2.1.1.1.1.1.3.2.3.cmml">2</mn></mrow></mrow><mo id="A3.Ex57.m2.1.1.1.1.1.2" xref="A3.Ex57.m2.1.1.1.1.1.2.cmml">−</mo><mrow id="A3.Ex57.m2.1.1.1.1.1.1" xref="A3.Ex57.m2.1.1.1.1.1.1.cmml"><mi id="A3.Ex57.m2.1.1.1.1.1.1.3" xref="A3.Ex57.m2.1.1.1.1.1.1.3.cmml">κ</mi><mo id="A3.Ex57.m2.1.1.1.1.1.1.2" xref="A3.Ex57.m2.1.1.1.1.1.1.2.cmml">⁢</mo><msup id="A3.Ex57.m2.1.1.1.1.1.1.4" xref="A3.Ex57.m2.1.1.1.1.1.1.4.cmml"><mi id="A3.Ex57.m2.1.1.1.1.1.1.4.2" xref="A3.Ex57.m2.1.1.1.1.1.1.4.2.cmml">σ</mi><mo id="A3.Ex57.m2.1.1.1.1.1.1.4.3" xref="A3.Ex57.m2.1.1.1.1.1.1.4.3.cmml">∗</mo></msup><mo id="A3.Ex57.m2.1.1.1.1.1.1.2a" xref="A3.Ex57.m2.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A3.Ex57.m2.1.1.1.1.1.1.1.1" xref="A3.Ex57.m2.1.1.1.1.1.1.1.1.1.cmml"><mo id="A3.Ex57.m2.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A3.Ex57.m2.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.Ex57.m2.1.1.1.1.1.1.1.1.1" xref="A3.Ex57.m2.1.1.1.1.1.1.1.1.1.cmml"><mn id="A3.Ex57.m2.1.1.1.1.1.1.1.1.1.2" xref="A3.Ex57.m2.1.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A3.Ex57.m2.1.1.1.1.1.1.1.1.1.1" xref="A3.Ex57.m2.1.1.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="A3.Ex57.m2.1.1.1.1.1.1.1.1.1.3" xref="A3.Ex57.m2.1.1.1.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A3.Ex57.m2.1.1.1.1.1.1.1.1.3" stretchy="false" xref="A3.Ex57.m2.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A3.Ex57.m2.1.1.1.1.1.1.2b" xref="A3.Ex57.m2.1.1.1.1.1.1.2.cmml">⁢</mo><msubsup id="A3.Ex57.m2.1.1.1.1.1.1.5" xref="A3.Ex57.m2.1.1.1.1.1.1.5.cmml"><mover accent="true" id="A3.Ex57.m2.1.1.1.1.1.1.5.2.2" xref="A3.Ex57.m2.1.1.1.1.1.1.5.2.2.cmml"><mi id="A3.Ex57.m2.1.1.1.1.1.1.5.2.2.2" xref="A3.Ex57.m2.1.1.1.1.1.1.5.2.2.2.cmml">𝜽</mi><mo id="A3.Ex57.m2.1.1.1.1.1.1.5.2.2.1" xref="A3.Ex57.m2.1.1.1.1.1.1.5.2.2.1.cmml">~</mo></mover><mi id="A3.Ex57.m2.1.1.1.1.1.1.5.3" xref="A3.Ex57.m2.1.1.1.1.1.1.5.3.cmml">ζ</mi><mi id="A3.Ex57.m2.1.1.1.1.1.1.5.2.3" xref="A3.Ex57.m2.1.1.1.1.1.1.5.2.3.cmml">T</mi></msubsup><mo id="A3.Ex57.m2.1.1.1.1.1.1.2c" xref="A3.Ex57.m2.1.1.1.1.1.1.2.cmml">⁢</mo><msub id="A3.Ex57.m2.1.1.1.1.1.1.6" xref="A3.Ex57.m2.1.1.1.1.1.1.6.cmml"><mover accent="true" id="A3.Ex57.m2.1.1.1.1.1.1.6.2" xref="A3.Ex57.m2.1.1.1.1.1.1.6.2.cmml"><mi id="A3.Ex57.m2.1.1.1.1.1.1.6.2.2" xref="A3.Ex57.m2.1.1.1.1.1.1.6.2.2.cmml">𝜽</mi><mo id="A3.Ex57.m2.1.1.1.1.1.1.6.2.1" xref="A3.Ex57.m2.1.1.1.1.1.1.6.2.1.cmml">~</mo></mover><mi id="A3.Ex57.m2.1.1.1.1.1.1.6.3" xref="A3.Ex57.m2.1.1.1.1.1.1.6.3.cmml">ζ</mi></msub></mrow></mrow><mo id="A3.Ex57.m2.1.1.1.1.4" xref="A3.Ex57.m2.1.1.1.1.4.cmml">+</mo><mrow id="A3.Ex57.m2.1.1.1.1.3" xref="A3.Ex57.m2.1.1.1.1.3.cmml"><mrow id="A3.Ex57.m2.1.1.1.1.2.1" xref="A3.Ex57.m2.1.1.1.1.2.1.cmml"><msup id="A3.Ex57.m2.1.1.1.1.2.1.3" xref="A3.Ex57.m2.1.1.1.1.2.1.3.cmml"><mi id="A3.Ex57.m2.1.1.1.1.2.1.3.2" xref="A3.Ex57.m2.1.1.1.1.2.1.3.2.cmml">δ</mi><mn id="A3.Ex57.m2.1.1.1.1.2.1.3.3" xref="A3.Ex57.m2.1.1.1.1.2.1.3.3.cmml">2</mn></msup><mo id="A3.Ex57.m2.1.1.1.1.2.1.2" xref="A3.Ex57.m2.1.1.1.1.2.1.2.cmml">⁢</mo><msup id="A3.Ex57.m2.1.1.1.1.2.1.1" xref="A3.Ex57.m2.1.1.1.1.2.1.1.cmml"><mrow id="A3.Ex57.m2.1.1.1.1.2.1.1.1.1" xref="A3.Ex57.m2.1.1.1.1.2.1.1.1.2.cmml"><mo id="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.2" stretchy="false" xref="A3.Ex57.m2.1.1.1.1.2.1.1.1.2.1.cmml">‖</mo><msub id="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.1" xref="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.1.cmml"><mover accent="true" id="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.1.2" xref="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.1.2.cmml"><mi id="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.1.2.2" xref="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.1.2.2.cmml">𝜽</mi><mo id="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.1.2.1" xref="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.1.2.1.cmml">~</mo></mover><mi id="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.1.3" xref="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.1.3.cmml">ζ</mi></msub><mo id="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.3" stretchy="false" xref="A3.Ex57.m2.1.1.1.1.2.1.1.1.2.1.cmml">‖</mo></mrow><mn id="A3.Ex57.m2.1.1.1.1.2.1.1.3" xref="A3.Ex57.m2.1.1.1.1.2.1.1.3.cmml">2</mn></msup></mrow><mo id="A3.Ex57.m2.1.1.1.1.3.3" xref="A3.Ex57.m2.1.1.1.1.3.3.cmml">/</mo><mrow id="A3.Ex57.m2.1.1.1.1.3.2.1" xref="A3.Ex57.m2.1.1.1.1.3.2.1.1.cmml"><mo id="A3.Ex57.m2.1.1.1.1.3.2.1.2" stretchy="false" xref="A3.Ex57.m2.1.1.1.1.3.2.1.1.cmml">(</mo><mrow id="A3.Ex57.m2.1.1.1.1.3.2.1.1" xref="A3.Ex57.m2.1.1.1.1.3.2.1.1.cmml"><mn id="A3.Ex57.m2.1.1.1.1.3.2.1.1.2" xref="A3.Ex57.m2.1.1.1.1.3.2.1.1.2.cmml">2</mn><mo id="A3.Ex57.m2.1.1.1.1.3.2.1.1.1" xref="A3.Ex57.m2.1.1.1.1.3.2.1.1.1.cmml">⁢</mo><msub id="A3.Ex57.m2.1.1.1.1.3.2.1.1.3" xref="A3.Ex57.m2.1.1.1.1.3.2.1.1.3.cmml"><mi id="A3.Ex57.m2.1.1.1.1.3.2.1.1.3.2" xref="A3.Ex57.m2.1.1.1.1.3.2.1.1.3.2.cmml">k</mi><mi id="A3.Ex57.m2.1.1.1.1.3.2.1.1.3.3" mathvariant="normal" xref="A3.Ex57.m2.1.1.1.1.3.2.1.1.3.3.cmml">c</mi></msub></mrow><mo id="A3.Ex57.m2.1.1.1.1.3.2.1.3" stretchy="false" xref="A3.Ex57.m2.1.1.1.1.3.2.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A3.Ex57.m2.1.1.1.2" lspace="0em" xref="A3.Ex57.m2.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex57.m2.1b"><apply id="A3.Ex57.m2.1.1.1.1.cmml" xref="A3.Ex57.m2.1.1.1"><plus id="A3.Ex57.m2.1.1.1.1.4.cmml" xref="A3.Ex57.m2.1.1.1.1.4"></plus><apply id="A3.Ex57.m2.1.1.1.1.1.cmml" xref="A3.Ex57.m2.1.1.1.1.1"><minus id="A3.Ex57.m2.1.1.1.1.1.2.cmml" xref="A3.Ex57.m2.1.1.1.1.1.2"></minus><apply id="A3.Ex57.m2.1.1.1.1.1.3.cmml" xref="A3.Ex57.m2.1.1.1.1.1.3"><minus id="A3.Ex57.m2.1.1.1.1.1.3.1.cmml" xref="A3.Ex57.m2.1.1.1.1.1.3"></minus><apply id="A3.Ex57.m2.1.1.1.1.1.3.2.cmml" xref="A3.Ex57.m2.1.1.1.1.1.3.2"><divide id="A3.Ex57.m2.1.1.1.1.1.3.2.1.cmml" xref="A3.Ex57.m2.1.1.1.1.1.3.2.1"></divide><apply id="A3.Ex57.m2.1.1.1.1.1.3.2.2.cmml" xref="A3.Ex57.m2.1.1.1.1.1.3.2.2"><times id="A3.Ex57.m2.1.1.1.1.1.3.2.2.1.cmml" xref="A3.Ex57.m2.1.1.1.1.1.3.2.2.1"></times><apply id="A3.Ex57.m2.1.1.1.1.1.3.2.2.2.cmml" xref="A3.Ex57.m2.1.1.1.1.1.3.2.2.2"><csymbol cd="ambiguous" id="A3.Ex57.m2.1.1.1.1.1.3.2.2.2.1.cmml" xref="A3.Ex57.m2.1.1.1.1.1.3.2.2.2">subscript</csymbol><ci id="A3.Ex57.m2.1.1.1.1.1.3.2.2.2.2.cmml" xref="A3.Ex57.m2.1.1.1.1.1.3.2.2.2.2">𝑘</ci><ci id="A3.Ex57.m2.1.1.1.1.1.3.2.2.2.3.cmml" xref="A3.Ex57.m2.1.1.1.1.1.3.2.2.2.3">c</ci></apply><apply id="A3.Ex57.m2.1.1.1.1.1.3.2.2.3.cmml" xref="A3.Ex57.m2.1.1.1.1.1.3.2.2.3"><csymbol cd="ambiguous" id="A3.Ex57.m2.1.1.1.1.1.3.2.2.3.1.cmml" xref="A3.Ex57.m2.1.1.1.1.1.3.2.2.3">superscript</csymbol><ci id="A3.Ex57.m2.1.1.1.1.1.3.2.2.3.2.cmml" xref="A3.Ex57.m2.1.1.1.1.1.3.2.2.3.2">𝒆</ci><ci id="A3.Ex57.m2.1.1.1.1.1.3.2.2.3.3.cmml" xref="A3.Ex57.m2.1.1.1.1.1.3.2.2.3.3">𝑇</ci></apply><ci id="A3.Ex57.m2.1.1.1.1.1.3.2.2.4.cmml" xref="A3.Ex57.m2.1.1.1.1.1.3.2.2.4">𝒆</ci></apply><cn id="A3.Ex57.m2.1.1.1.1.1.3.2.3.cmml" type="integer" xref="A3.Ex57.m2.1.1.1.1.1.3.2.3">2</cn></apply></apply><apply id="A3.Ex57.m2.1.1.1.1.1.1.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1"><times id="A3.Ex57.m2.1.1.1.1.1.1.2.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1.2"></times><ci id="A3.Ex57.m2.1.1.1.1.1.1.3.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1.3">𝜅</ci><apply id="A3.Ex57.m2.1.1.1.1.1.1.4.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1.4"><csymbol cd="ambiguous" id="A3.Ex57.m2.1.1.1.1.1.1.4.1.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1.4">superscript</csymbol><ci id="A3.Ex57.m2.1.1.1.1.1.1.4.2.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1.4.2">𝜎</ci><times id="A3.Ex57.m2.1.1.1.1.1.1.4.3.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1.4.3"></times></apply><apply id="A3.Ex57.m2.1.1.1.1.1.1.1.1.1.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1.1.1"><plus id="A3.Ex57.m2.1.1.1.1.1.1.1.1.1.1.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1.1.1.1.1"></plus><cn id="A3.Ex57.m2.1.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A3.Ex57.m2.1.1.1.1.1.1.1.1.1.2">1</cn><ci id="A3.Ex57.m2.1.1.1.1.1.1.1.1.1.3.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="A3.Ex57.m2.1.1.1.1.1.1.5.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1.5"><csymbol cd="ambiguous" id="A3.Ex57.m2.1.1.1.1.1.1.5.1.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1.5">subscript</csymbol><apply id="A3.Ex57.m2.1.1.1.1.1.1.5.2.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1.5"><csymbol cd="ambiguous" id="A3.Ex57.m2.1.1.1.1.1.1.5.2.1.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1.5">superscript</csymbol><apply id="A3.Ex57.m2.1.1.1.1.1.1.5.2.2.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1.5.2.2"><ci id="A3.Ex57.m2.1.1.1.1.1.1.5.2.2.1.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1.5.2.2.1">~</ci><ci id="A3.Ex57.m2.1.1.1.1.1.1.5.2.2.2.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1.5.2.2.2">𝜽</ci></apply><ci id="A3.Ex57.m2.1.1.1.1.1.1.5.2.3.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1.5.2.3">𝑇</ci></apply><ci id="A3.Ex57.m2.1.1.1.1.1.1.5.3.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1.5.3">𝜁</ci></apply><apply id="A3.Ex57.m2.1.1.1.1.1.1.6.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1.6"><csymbol cd="ambiguous" id="A3.Ex57.m2.1.1.1.1.1.1.6.1.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1.6">subscript</csymbol><apply id="A3.Ex57.m2.1.1.1.1.1.1.6.2.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1.6.2"><ci id="A3.Ex57.m2.1.1.1.1.1.1.6.2.1.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1.6.2.1">~</ci><ci id="A3.Ex57.m2.1.1.1.1.1.1.6.2.2.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1.6.2.2">𝜽</ci></apply><ci id="A3.Ex57.m2.1.1.1.1.1.1.6.3.cmml" xref="A3.Ex57.m2.1.1.1.1.1.1.6.3">𝜁</ci></apply></apply></apply><apply id="A3.Ex57.m2.1.1.1.1.3.cmml" xref="A3.Ex57.m2.1.1.1.1.3"><divide id="A3.Ex57.m2.1.1.1.1.3.3.cmml" xref="A3.Ex57.m2.1.1.1.1.3.3"></divide><apply id="A3.Ex57.m2.1.1.1.1.2.1.cmml" xref="A3.Ex57.m2.1.1.1.1.2.1"><times id="A3.Ex57.m2.1.1.1.1.2.1.2.cmml" xref="A3.Ex57.m2.1.1.1.1.2.1.2"></times><apply id="A3.Ex57.m2.1.1.1.1.2.1.3.cmml" xref="A3.Ex57.m2.1.1.1.1.2.1.3"><csymbol cd="ambiguous" id="A3.Ex57.m2.1.1.1.1.2.1.3.1.cmml" xref="A3.Ex57.m2.1.1.1.1.2.1.3">superscript</csymbol><ci id="A3.Ex57.m2.1.1.1.1.2.1.3.2.cmml" xref="A3.Ex57.m2.1.1.1.1.2.1.3.2">𝛿</ci><cn id="A3.Ex57.m2.1.1.1.1.2.1.3.3.cmml" type="integer" xref="A3.Ex57.m2.1.1.1.1.2.1.3.3">2</cn></apply><apply id="A3.Ex57.m2.1.1.1.1.2.1.1.cmml" xref="A3.Ex57.m2.1.1.1.1.2.1.1"><csymbol cd="ambiguous" id="A3.Ex57.m2.1.1.1.1.2.1.1.2.cmml" xref="A3.Ex57.m2.1.1.1.1.2.1.1">superscript</csymbol><apply id="A3.Ex57.m2.1.1.1.1.2.1.1.1.2.cmml" xref="A3.Ex57.m2.1.1.1.1.2.1.1.1.1"><csymbol cd="latexml" id="A3.Ex57.m2.1.1.1.1.2.1.1.1.2.1.cmml" xref="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.2">norm</csymbol><apply id="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.1.cmml" xref="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.1.1.cmml" xref="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.1">subscript</csymbol><apply id="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.1.2.cmml" xref="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.1.2"><ci id="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.1.2.1.cmml" xref="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.1.2.1">~</ci><ci id="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.1.2.2.cmml" xref="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.1.2.2">𝜽</ci></apply><ci id="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.1.3.cmml" xref="A3.Ex57.m2.1.1.1.1.2.1.1.1.1.1.3">𝜁</ci></apply></apply><cn id="A3.Ex57.m2.1.1.1.1.2.1.1.3.cmml" type="integer" xref="A3.Ex57.m2.1.1.1.1.2.1.1.3">2</cn></apply></apply><apply id="A3.Ex57.m2.1.1.1.1.3.2.1.1.cmml" xref="A3.Ex57.m2.1.1.1.1.3.2.1"><times id="A3.Ex57.m2.1.1.1.1.3.2.1.1.1.cmml" xref="A3.Ex57.m2.1.1.1.1.3.2.1.1.1"></times><cn id="A3.Ex57.m2.1.1.1.1.3.2.1.1.2.cmml" type="integer" xref="A3.Ex57.m2.1.1.1.1.3.2.1.1.2">2</cn><apply id="A3.Ex57.m2.1.1.1.1.3.2.1.1.3.cmml" xref="A3.Ex57.m2.1.1.1.1.3.2.1.1.3"><csymbol cd="ambiguous" id="A3.Ex57.m2.1.1.1.1.3.2.1.1.3.1.cmml" xref="A3.Ex57.m2.1.1.1.1.3.2.1.1.3">subscript</csymbol><ci id="A3.Ex57.m2.1.1.1.1.3.2.1.1.3.2.cmml" xref="A3.Ex57.m2.1.1.1.1.3.2.1.1.3.2">𝑘</ci><ci id="A3.Ex57.m2.1.1.1.1.3.2.1.1.3.3.cmml" xref="A3.Ex57.m2.1.1.1.1.3.2.1.1.3.3">c</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex57.m2.1c">\displaystyle-k_{\rm c}\bm{e}^{T}\bm{e}/2-\kappa\sigma^{*}(1+p)\tilde{\bm{% \theta}}^{T}_{\zeta}\tilde{\bm{\theta}}_{\zeta}+\delta^{2}\|\tilde{\bm{\theta}% }_{\zeta}\|^{2}/(2k_{\rm c}).</annotation><annotation encoding="application/x-llamapun" id="A3.Ex57.m2.1d">- italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_e / 2 - italic_κ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( 1 + italic_p ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT + italic_δ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( 2 italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.2.p2.68">Choosing <math alttext="p" class="ltx_Math" display="inline" id="A3.2.p2.66.m1.1"><semantics id="A3.2.p2.66.m1.1a"><mi id="A3.2.p2.66.m1.1.1" xref="A3.2.p2.66.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="A3.2.p2.66.m1.1b"><ci id="A3.2.p2.66.m1.1.1.cmml" xref="A3.2.p2.66.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.66.m1.1c">p</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.66.m1.1d">italic_p</annotation></semantics></math> <math alttext="=" class="ltx_Math" display="inline" id="A3.2.p2.67.m2.1"><semantics id="A3.2.p2.67.m2.1a"><mo id="A3.2.p2.67.m2.1.1" xref="A3.2.p2.67.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="A3.2.p2.67.m2.1b"><eq id="A3.2.p2.67.m2.1.1.cmml" xref="A3.2.p2.67.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.67.m2.1c">=</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.67.m2.1d">=</annotation></semantics></math> <math alttext="\delta^{2}/(2k_{\rm c}\kappa\sigma^{*})\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A3.2.p2.68.m3.1"><semantics id="A3.2.p2.68.m3.1a"><mrow id="A3.2.p2.68.m3.1.1" xref="A3.2.p2.68.m3.1.1.cmml"><mrow id="A3.2.p2.68.m3.1.1.1" xref="A3.2.p2.68.m3.1.1.1.cmml"><msup id="A3.2.p2.68.m3.1.1.1.3" xref="A3.2.p2.68.m3.1.1.1.3.cmml"><mi id="A3.2.p2.68.m3.1.1.1.3.2" xref="A3.2.p2.68.m3.1.1.1.3.2.cmml">δ</mi><mn id="A3.2.p2.68.m3.1.1.1.3.3" xref="A3.2.p2.68.m3.1.1.1.3.3.cmml">2</mn></msup><mo id="A3.2.p2.68.m3.1.1.1.2" xref="A3.2.p2.68.m3.1.1.1.2.cmml">/</mo><mrow id="A3.2.p2.68.m3.1.1.1.1.1" xref="A3.2.p2.68.m3.1.1.1.1.1.1.cmml"><mo id="A3.2.p2.68.m3.1.1.1.1.1.2" stretchy="false" xref="A3.2.p2.68.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.2.p2.68.m3.1.1.1.1.1.1" xref="A3.2.p2.68.m3.1.1.1.1.1.1.cmml"><mn id="A3.2.p2.68.m3.1.1.1.1.1.1.2" xref="A3.2.p2.68.m3.1.1.1.1.1.1.2.cmml">2</mn><mo id="A3.2.p2.68.m3.1.1.1.1.1.1.1" xref="A3.2.p2.68.m3.1.1.1.1.1.1.1.cmml">⁢</mo><msub id="A3.2.p2.68.m3.1.1.1.1.1.1.3" xref="A3.2.p2.68.m3.1.1.1.1.1.1.3.cmml"><mi id="A3.2.p2.68.m3.1.1.1.1.1.1.3.2" xref="A3.2.p2.68.m3.1.1.1.1.1.1.3.2.cmml">k</mi><mi id="A3.2.p2.68.m3.1.1.1.1.1.1.3.3" mathvariant="normal" xref="A3.2.p2.68.m3.1.1.1.1.1.1.3.3.cmml">c</mi></msub><mo id="A3.2.p2.68.m3.1.1.1.1.1.1.1a" xref="A3.2.p2.68.m3.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="A3.2.p2.68.m3.1.1.1.1.1.1.4" xref="A3.2.p2.68.m3.1.1.1.1.1.1.4.cmml">κ</mi><mo id="A3.2.p2.68.m3.1.1.1.1.1.1.1b" xref="A3.2.p2.68.m3.1.1.1.1.1.1.1.cmml">⁢</mo><msup id="A3.2.p2.68.m3.1.1.1.1.1.1.5" xref="A3.2.p2.68.m3.1.1.1.1.1.1.5.cmml"><mi id="A3.2.p2.68.m3.1.1.1.1.1.1.5.2" xref="A3.2.p2.68.m3.1.1.1.1.1.1.5.2.cmml">σ</mi><mo id="A3.2.p2.68.m3.1.1.1.1.1.1.5.3" xref="A3.2.p2.68.m3.1.1.1.1.1.1.5.3.cmml">∗</mo></msup></mrow><mo id="A3.2.p2.68.m3.1.1.1.1.1.3" stretchy="false" xref="A3.2.p2.68.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A3.2.p2.68.m3.1.1.2" xref="A3.2.p2.68.m3.1.1.2.cmml">∈</mo><msup id="A3.2.p2.68.m3.1.1.3" xref="A3.2.p2.68.m3.1.1.3.cmml"><mi id="A3.2.p2.68.m3.1.1.3.2" xref="A3.2.p2.68.m3.1.1.3.2.cmml">ℝ</mi><mo id="A3.2.p2.68.m3.1.1.3.3" xref="A3.2.p2.68.m3.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.68.m3.1b"><apply id="A3.2.p2.68.m3.1.1.cmml" xref="A3.2.p2.68.m3.1.1"><in id="A3.2.p2.68.m3.1.1.2.cmml" xref="A3.2.p2.68.m3.1.1.2"></in><apply id="A3.2.p2.68.m3.1.1.1.cmml" xref="A3.2.p2.68.m3.1.1.1"><divide id="A3.2.p2.68.m3.1.1.1.2.cmml" xref="A3.2.p2.68.m3.1.1.1.2"></divide><apply id="A3.2.p2.68.m3.1.1.1.3.cmml" xref="A3.2.p2.68.m3.1.1.1.3"><csymbol cd="ambiguous" id="A3.2.p2.68.m3.1.1.1.3.1.cmml" xref="A3.2.p2.68.m3.1.1.1.3">superscript</csymbol><ci id="A3.2.p2.68.m3.1.1.1.3.2.cmml" xref="A3.2.p2.68.m3.1.1.1.3.2">𝛿</ci><cn id="A3.2.p2.68.m3.1.1.1.3.3.cmml" type="integer" xref="A3.2.p2.68.m3.1.1.1.3.3">2</cn></apply><apply id="A3.2.p2.68.m3.1.1.1.1.1.1.cmml" xref="A3.2.p2.68.m3.1.1.1.1.1"><times id="A3.2.p2.68.m3.1.1.1.1.1.1.1.cmml" xref="A3.2.p2.68.m3.1.1.1.1.1.1.1"></times><cn id="A3.2.p2.68.m3.1.1.1.1.1.1.2.cmml" type="integer" xref="A3.2.p2.68.m3.1.1.1.1.1.1.2">2</cn><apply id="A3.2.p2.68.m3.1.1.1.1.1.1.3.cmml" xref="A3.2.p2.68.m3.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A3.2.p2.68.m3.1.1.1.1.1.1.3.1.cmml" xref="A3.2.p2.68.m3.1.1.1.1.1.1.3">subscript</csymbol><ci id="A3.2.p2.68.m3.1.1.1.1.1.1.3.2.cmml" xref="A3.2.p2.68.m3.1.1.1.1.1.1.3.2">𝑘</ci><ci id="A3.2.p2.68.m3.1.1.1.1.1.1.3.3.cmml" xref="A3.2.p2.68.m3.1.1.1.1.1.1.3.3">c</ci></apply><ci id="A3.2.p2.68.m3.1.1.1.1.1.1.4.cmml" xref="A3.2.p2.68.m3.1.1.1.1.1.1.4">𝜅</ci><apply id="A3.2.p2.68.m3.1.1.1.1.1.1.5.cmml" xref="A3.2.p2.68.m3.1.1.1.1.1.1.5"><csymbol cd="ambiguous" id="A3.2.p2.68.m3.1.1.1.1.1.1.5.1.cmml" xref="A3.2.p2.68.m3.1.1.1.1.1.1.5">superscript</csymbol><ci id="A3.2.p2.68.m3.1.1.1.1.1.1.5.2.cmml" xref="A3.2.p2.68.m3.1.1.1.1.1.1.5.2">𝜎</ci><times id="A3.2.p2.68.m3.1.1.1.1.1.1.5.3.cmml" xref="A3.2.p2.68.m3.1.1.1.1.1.1.5.3"></times></apply></apply></apply><apply id="A3.2.p2.68.m3.1.1.3.cmml" xref="A3.2.p2.68.m3.1.1.3"><csymbol cd="ambiguous" id="A3.2.p2.68.m3.1.1.3.1.cmml" xref="A3.2.p2.68.m3.1.1.3">superscript</csymbol><ci id="A3.2.p2.68.m3.1.1.3.2.cmml" xref="A3.2.p2.68.m3.1.1.3.2">ℝ</ci><plus id="A3.2.p2.68.m3.1.1.3.3.cmml" xref="A3.2.p2.68.m3.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.68.m3.1c">\delta^{2}/(2k_{\rm c}\kappa\sigma^{*})\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.68.m3.1d">italic_δ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( 2 italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT italic_κ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, one obtains</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx73"> <tbody id="A3.Ex58"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}_{\zeta}\leq" class="ltx_Math" display="inline" id="A3.Ex58.m1.1"><semantics id="A3.Ex58.m1.1a"><mrow id="A3.Ex58.m1.1.1" xref="A3.Ex58.m1.1.1.cmml"><msub id="A3.Ex58.m1.1.1.2" xref="A3.Ex58.m1.1.1.2.cmml"><mover accent="true" id="A3.Ex58.m1.1.1.2.2" xref="A3.Ex58.m1.1.1.2.2.cmml"><mi id="A3.Ex58.m1.1.1.2.2.2" xref="A3.Ex58.m1.1.1.2.2.2.cmml">V</mi><mo id="A3.Ex58.m1.1.1.2.2.1" xref="A3.Ex58.m1.1.1.2.2.1.cmml">˙</mo></mover><mi id="A3.Ex58.m1.1.1.2.3" xref="A3.Ex58.m1.1.1.2.3.cmml">ζ</mi></msub><mo id="A3.Ex58.m1.1.1.1" xref="A3.Ex58.m1.1.1.1.cmml">≤</mo><mi id="A3.Ex58.m1.1.1.3" xref="A3.Ex58.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex58.m1.1b"><apply id="A3.Ex58.m1.1.1.cmml" xref="A3.Ex58.m1.1.1"><leq id="A3.Ex58.m1.1.1.1.cmml" xref="A3.Ex58.m1.1.1.1"></leq><apply id="A3.Ex58.m1.1.1.2.cmml" xref="A3.Ex58.m1.1.1.2"><csymbol cd="ambiguous" id="A3.Ex58.m1.1.1.2.1.cmml" xref="A3.Ex58.m1.1.1.2">subscript</csymbol><apply id="A3.Ex58.m1.1.1.2.2.cmml" xref="A3.Ex58.m1.1.1.2.2"><ci id="A3.Ex58.m1.1.1.2.2.1.cmml" xref="A3.Ex58.m1.1.1.2.2.1">˙</ci><ci id="A3.Ex58.m1.1.1.2.2.2.cmml" xref="A3.Ex58.m1.1.1.2.2.2">𝑉</ci></apply><ci id="A3.Ex58.m1.1.1.2.3.cmml" xref="A3.Ex58.m1.1.1.2.3">𝜁</ci></apply><csymbol cd="latexml" id="A3.Ex58.m1.1.1.3.cmml" xref="A3.Ex58.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex58.m1.1c">\displaystyle\dot{V}_{\zeta}\leq</annotation><annotation encoding="application/x-llamapun" id="A3.Ex58.m1.1d">over˙ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-k_{\rm c}\bm{e}^{T}\bm{e}/2-\kappa\sigma^{*}\tilde{\bm{\theta}}^% {T}_{\zeta}\tilde{\bm{\theta}}_{\zeta}" class="ltx_Math" display="inline" id="A3.Ex58.m2.1"><semantics id="A3.Ex58.m2.1a"><mrow id="A3.Ex58.m2.1.1" xref="A3.Ex58.m2.1.1.cmml"><mrow id="A3.Ex58.m2.1.1.2" xref="A3.Ex58.m2.1.1.2.cmml"><mo id="A3.Ex58.m2.1.1.2a" xref="A3.Ex58.m2.1.1.2.cmml">−</mo><mrow id="A3.Ex58.m2.1.1.2.2" xref="A3.Ex58.m2.1.1.2.2.cmml"><mrow id="A3.Ex58.m2.1.1.2.2.2" xref="A3.Ex58.m2.1.1.2.2.2.cmml"><msub id="A3.Ex58.m2.1.1.2.2.2.2" xref="A3.Ex58.m2.1.1.2.2.2.2.cmml"><mi id="A3.Ex58.m2.1.1.2.2.2.2.2" xref="A3.Ex58.m2.1.1.2.2.2.2.2.cmml">k</mi><mi id="A3.Ex58.m2.1.1.2.2.2.2.3" mathvariant="normal" xref="A3.Ex58.m2.1.1.2.2.2.2.3.cmml">c</mi></msub><mo id="A3.Ex58.m2.1.1.2.2.2.1" xref="A3.Ex58.m2.1.1.2.2.2.1.cmml">⁢</mo><msup id="A3.Ex58.m2.1.1.2.2.2.3" xref="A3.Ex58.m2.1.1.2.2.2.3.cmml"><mi id="A3.Ex58.m2.1.1.2.2.2.3.2" xref="A3.Ex58.m2.1.1.2.2.2.3.2.cmml">𝒆</mi><mi id="A3.Ex58.m2.1.1.2.2.2.3.3" xref="A3.Ex58.m2.1.1.2.2.2.3.3.cmml">T</mi></msup><mo id="A3.Ex58.m2.1.1.2.2.2.1a" xref="A3.Ex58.m2.1.1.2.2.2.1.cmml">⁢</mo><mi id="A3.Ex58.m2.1.1.2.2.2.4" xref="A3.Ex58.m2.1.1.2.2.2.4.cmml">𝒆</mi></mrow><mo id="A3.Ex58.m2.1.1.2.2.1" xref="A3.Ex58.m2.1.1.2.2.1.cmml">/</mo><mn id="A3.Ex58.m2.1.1.2.2.3" xref="A3.Ex58.m2.1.1.2.2.3.cmml">2</mn></mrow></mrow><mo id="A3.Ex58.m2.1.1.1" xref="A3.Ex58.m2.1.1.1.cmml">−</mo><mrow id="A3.Ex58.m2.1.1.3" xref="A3.Ex58.m2.1.1.3.cmml"><mi id="A3.Ex58.m2.1.1.3.2" xref="A3.Ex58.m2.1.1.3.2.cmml">κ</mi><mo id="A3.Ex58.m2.1.1.3.1" xref="A3.Ex58.m2.1.1.3.1.cmml">⁢</mo><msup id="A3.Ex58.m2.1.1.3.3" xref="A3.Ex58.m2.1.1.3.3.cmml"><mi id="A3.Ex58.m2.1.1.3.3.2" xref="A3.Ex58.m2.1.1.3.3.2.cmml">σ</mi><mo id="A3.Ex58.m2.1.1.3.3.3" xref="A3.Ex58.m2.1.1.3.3.3.cmml">∗</mo></msup><mo id="A3.Ex58.m2.1.1.3.1a" xref="A3.Ex58.m2.1.1.3.1.cmml">⁢</mo><msubsup id="A3.Ex58.m2.1.1.3.4" xref="A3.Ex58.m2.1.1.3.4.cmml"><mover accent="true" id="A3.Ex58.m2.1.1.3.4.2.2" xref="A3.Ex58.m2.1.1.3.4.2.2.cmml"><mi id="A3.Ex58.m2.1.1.3.4.2.2.2" xref="A3.Ex58.m2.1.1.3.4.2.2.2.cmml">𝜽</mi><mo id="A3.Ex58.m2.1.1.3.4.2.2.1" xref="A3.Ex58.m2.1.1.3.4.2.2.1.cmml">~</mo></mover><mi id="A3.Ex58.m2.1.1.3.4.3" xref="A3.Ex58.m2.1.1.3.4.3.cmml">ζ</mi><mi id="A3.Ex58.m2.1.1.3.4.2.3" xref="A3.Ex58.m2.1.1.3.4.2.3.cmml">T</mi></msubsup><mo id="A3.Ex58.m2.1.1.3.1b" xref="A3.Ex58.m2.1.1.3.1.cmml">⁢</mo><msub id="A3.Ex58.m2.1.1.3.5" xref="A3.Ex58.m2.1.1.3.5.cmml"><mover accent="true" id="A3.Ex58.m2.1.1.3.5.2" xref="A3.Ex58.m2.1.1.3.5.2.cmml"><mi id="A3.Ex58.m2.1.1.3.5.2.2" xref="A3.Ex58.m2.1.1.3.5.2.2.cmml">𝜽</mi><mo id="A3.Ex58.m2.1.1.3.5.2.1" xref="A3.Ex58.m2.1.1.3.5.2.1.cmml">~</mo></mover><mi id="A3.Ex58.m2.1.1.3.5.3" xref="A3.Ex58.m2.1.1.3.5.3.cmml">ζ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex58.m2.1b"><apply id="A3.Ex58.m2.1.1.cmml" xref="A3.Ex58.m2.1.1"><minus id="A3.Ex58.m2.1.1.1.cmml" xref="A3.Ex58.m2.1.1.1"></minus><apply id="A3.Ex58.m2.1.1.2.cmml" xref="A3.Ex58.m2.1.1.2"><minus id="A3.Ex58.m2.1.1.2.1.cmml" xref="A3.Ex58.m2.1.1.2"></minus><apply id="A3.Ex58.m2.1.1.2.2.cmml" xref="A3.Ex58.m2.1.1.2.2"><divide id="A3.Ex58.m2.1.1.2.2.1.cmml" xref="A3.Ex58.m2.1.1.2.2.1"></divide><apply id="A3.Ex58.m2.1.1.2.2.2.cmml" xref="A3.Ex58.m2.1.1.2.2.2"><times id="A3.Ex58.m2.1.1.2.2.2.1.cmml" xref="A3.Ex58.m2.1.1.2.2.2.1"></times><apply id="A3.Ex58.m2.1.1.2.2.2.2.cmml" xref="A3.Ex58.m2.1.1.2.2.2.2"><csymbol cd="ambiguous" id="A3.Ex58.m2.1.1.2.2.2.2.1.cmml" xref="A3.Ex58.m2.1.1.2.2.2.2">subscript</csymbol><ci id="A3.Ex58.m2.1.1.2.2.2.2.2.cmml" xref="A3.Ex58.m2.1.1.2.2.2.2.2">𝑘</ci><ci id="A3.Ex58.m2.1.1.2.2.2.2.3.cmml" xref="A3.Ex58.m2.1.1.2.2.2.2.3">c</ci></apply><apply id="A3.Ex58.m2.1.1.2.2.2.3.cmml" xref="A3.Ex58.m2.1.1.2.2.2.3"><csymbol cd="ambiguous" id="A3.Ex58.m2.1.1.2.2.2.3.1.cmml" xref="A3.Ex58.m2.1.1.2.2.2.3">superscript</csymbol><ci id="A3.Ex58.m2.1.1.2.2.2.3.2.cmml" xref="A3.Ex58.m2.1.1.2.2.2.3.2">𝒆</ci><ci id="A3.Ex58.m2.1.1.2.2.2.3.3.cmml" xref="A3.Ex58.m2.1.1.2.2.2.3.3">𝑇</ci></apply><ci id="A3.Ex58.m2.1.1.2.2.2.4.cmml" xref="A3.Ex58.m2.1.1.2.2.2.4">𝒆</ci></apply><cn id="A3.Ex58.m2.1.1.2.2.3.cmml" type="integer" xref="A3.Ex58.m2.1.1.2.2.3">2</cn></apply></apply><apply id="A3.Ex58.m2.1.1.3.cmml" xref="A3.Ex58.m2.1.1.3"><times id="A3.Ex58.m2.1.1.3.1.cmml" xref="A3.Ex58.m2.1.1.3.1"></times><ci id="A3.Ex58.m2.1.1.3.2.cmml" xref="A3.Ex58.m2.1.1.3.2">𝜅</ci><apply id="A3.Ex58.m2.1.1.3.3.cmml" xref="A3.Ex58.m2.1.1.3.3"><csymbol cd="ambiguous" id="A3.Ex58.m2.1.1.3.3.1.cmml" xref="A3.Ex58.m2.1.1.3.3">superscript</csymbol><ci id="A3.Ex58.m2.1.1.3.3.2.cmml" xref="A3.Ex58.m2.1.1.3.3.2">𝜎</ci><times id="A3.Ex58.m2.1.1.3.3.3.cmml" xref="A3.Ex58.m2.1.1.3.3.3"></times></apply><apply id="A3.Ex58.m2.1.1.3.4.cmml" xref="A3.Ex58.m2.1.1.3.4"><csymbol cd="ambiguous" id="A3.Ex58.m2.1.1.3.4.1.cmml" xref="A3.Ex58.m2.1.1.3.4">subscript</csymbol><apply id="A3.Ex58.m2.1.1.3.4.2.cmml" xref="A3.Ex58.m2.1.1.3.4"><csymbol cd="ambiguous" id="A3.Ex58.m2.1.1.3.4.2.1.cmml" xref="A3.Ex58.m2.1.1.3.4">superscript</csymbol><apply id="A3.Ex58.m2.1.1.3.4.2.2.cmml" xref="A3.Ex58.m2.1.1.3.4.2.2"><ci id="A3.Ex58.m2.1.1.3.4.2.2.1.cmml" xref="A3.Ex58.m2.1.1.3.4.2.2.1">~</ci><ci id="A3.Ex58.m2.1.1.3.4.2.2.2.cmml" xref="A3.Ex58.m2.1.1.3.4.2.2.2">𝜽</ci></apply><ci id="A3.Ex58.m2.1.1.3.4.2.3.cmml" xref="A3.Ex58.m2.1.1.3.4.2.3">𝑇</ci></apply><ci id="A3.Ex58.m2.1.1.3.4.3.cmml" xref="A3.Ex58.m2.1.1.3.4.3">𝜁</ci></apply><apply id="A3.Ex58.m2.1.1.3.5.cmml" xref="A3.Ex58.m2.1.1.3.5"><csymbol cd="ambiguous" id="A3.Ex58.m2.1.1.3.5.1.cmml" xref="A3.Ex58.m2.1.1.3.5">subscript</csymbol><apply id="A3.Ex58.m2.1.1.3.5.2.cmml" xref="A3.Ex58.m2.1.1.3.5.2"><ci id="A3.Ex58.m2.1.1.3.5.2.1.cmml" xref="A3.Ex58.m2.1.1.3.5.2.1">~</ci><ci id="A3.Ex58.m2.1.1.3.5.2.2.cmml" xref="A3.Ex58.m2.1.1.3.5.2.2">𝜽</ci></apply><ci id="A3.Ex58.m2.1.1.3.5.3.cmml" xref="A3.Ex58.m2.1.1.3.5.3">𝜁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex58.m2.1c">\displaystyle-k_{\rm c}\bm{e}^{T}\bm{e}/2-\kappa\sigma^{*}\tilde{\bm{\theta}}^% {T}_{\zeta}\tilde{\bm{\theta}}_{\zeta}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex58.m2.1d">- italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_e / 2 - italic_κ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex59"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\leq" class="ltx_Math" display="inline" id="A3.Ex59.m1.1"><semantics id="A3.Ex59.m1.1a"><mo id="A3.Ex59.m1.1.1" xref="A3.Ex59.m1.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="A3.Ex59.m1.1b"><leq id="A3.Ex59.m1.1.1.cmml" xref="A3.Ex59.m1.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex59.m1.1c">\displaystyle\leq</annotation><annotation encoding="application/x-llamapun" id="A3.Ex59.m1.1d">≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-k_{\rm c}\bm{e}^{T}\bm{e}/2-\kappa\sigma^{*}\lambda_{\min}(% \Gamma_{\zeta})\tilde{\bm{\theta}}^{T}_{\zeta}\Gamma_{\zeta}^{-1}\tilde{\bm{% \theta}}_{\zeta}" class="ltx_Math" display="inline" id="A3.Ex59.m2.1"><semantics id="A3.Ex59.m2.1a"><mrow id="A3.Ex59.m2.1.1" xref="A3.Ex59.m2.1.1.cmml"><mrow id="A3.Ex59.m2.1.1.3" xref="A3.Ex59.m2.1.1.3.cmml"><mo id="A3.Ex59.m2.1.1.3a" xref="A3.Ex59.m2.1.1.3.cmml">−</mo><mrow id="A3.Ex59.m2.1.1.3.2" xref="A3.Ex59.m2.1.1.3.2.cmml"><mrow id="A3.Ex59.m2.1.1.3.2.2" xref="A3.Ex59.m2.1.1.3.2.2.cmml"><msub id="A3.Ex59.m2.1.1.3.2.2.2" xref="A3.Ex59.m2.1.1.3.2.2.2.cmml"><mi id="A3.Ex59.m2.1.1.3.2.2.2.2" xref="A3.Ex59.m2.1.1.3.2.2.2.2.cmml">k</mi><mi id="A3.Ex59.m2.1.1.3.2.2.2.3" mathvariant="normal" xref="A3.Ex59.m2.1.1.3.2.2.2.3.cmml">c</mi></msub><mo id="A3.Ex59.m2.1.1.3.2.2.1" xref="A3.Ex59.m2.1.1.3.2.2.1.cmml">⁢</mo><msup id="A3.Ex59.m2.1.1.3.2.2.3" xref="A3.Ex59.m2.1.1.3.2.2.3.cmml"><mi id="A3.Ex59.m2.1.1.3.2.2.3.2" xref="A3.Ex59.m2.1.1.3.2.2.3.2.cmml">𝒆</mi><mi id="A3.Ex59.m2.1.1.3.2.2.3.3" xref="A3.Ex59.m2.1.1.3.2.2.3.3.cmml">T</mi></msup><mo id="A3.Ex59.m2.1.1.3.2.2.1a" xref="A3.Ex59.m2.1.1.3.2.2.1.cmml">⁢</mo><mi id="A3.Ex59.m2.1.1.3.2.2.4" xref="A3.Ex59.m2.1.1.3.2.2.4.cmml">𝒆</mi></mrow><mo id="A3.Ex59.m2.1.1.3.2.1" xref="A3.Ex59.m2.1.1.3.2.1.cmml">/</mo><mn id="A3.Ex59.m2.1.1.3.2.3" xref="A3.Ex59.m2.1.1.3.2.3.cmml">2</mn></mrow></mrow><mo id="A3.Ex59.m2.1.1.2" xref="A3.Ex59.m2.1.1.2.cmml">−</mo><mrow id="A3.Ex59.m2.1.1.1" xref="A3.Ex59.m2.1.1.1.cmml"><mi id="A3.Ex59.m2.1.1.1.3" xref="A3.Ex59.m2.1.1.1.3.cmml">κ</mi><mo id="A3.Ex59.m2.1.1.1.2" xref="A3.Ex59.m2.1.1.1.2.cmml">⁢</mo><msup id="A3.Ex59.m2.1.1.1.4" xref="A3.Ex59.m2.1.1.1.4.cmml"><mi id="A3.Ex59.m2.1.1.1.4.2" xref="A3.Ex59.m2.1.1.1.4.2.cmml">σ</mi><mo id="A3.Ex59.m2.1.1.1.4.3" xref="A3.Ex59.m2.1.1.1.4.3.cmml">∗</mo></msup><mo id="A3.Ex59.m2.1.1.1.2a" xref="A3.Ex59.m2.1.1.1.2.cmml">⁢</mo><msub id="A3.Ex59.m2.1.1.1.5" xref="A3.Ex59.m2.1.1.1.5.cmml"><mi id="A3.Ex59.m2.1.1.1.5.2" xref="A3.Ex59.m2.1.1.1.5.2.cmml">λ</mi><mi id="A3.Ex59.m2.1.1.1.5.3" xref="A3.Ex59.m2.1.1.1.5.3.cmml">min</mi></msub><mo id="A3.Ex59.m2.1.1.1.2b" xref="A3.Ex59.m2.1.1.1.2.cmml">⁢</mo><mrow id="A3.Ex59.m2.1.1.1.1.1" xref="A3.Ex59.m2.1.1.1.1.1.1.cmml"><mo id="A3.Ex59.m2.1.1.1.1.1.2" stretchy="false" xref="A3.Ex59.m2.1.1.1.1.1.1.cmml">(</mo><msub id="A3.Ex59.m2.1.1.1.1.1.1" xref="A3.Ex59.m2.1.1.1.1.1.1.cmml"><mi id="A3.Ex59.m2.1.1.1.1.1.1.2" mathvariant="normal" xref="A3.Ex59.m2.1.1.1.1.1.1.2.cmml">Γ</mi><mi id="A3.Ex59.m2.1.1.1.1.1.1.3" xref="A3.Ex59.m2.1.1.1.1.1.1.3.cmml">ζ</mi></msub><mo id="A3.Ex59.m2.1.1.1.1.1.3" stretchy="false" xref="A3.Ex59.m2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A3.Ex59.m2.1.1.1.2c" xref="A3.Ex59.m2.1.1.1.2.cmml">⁢</mo><msubsup id="A3.Ex59.m2.1.1.1.6" xref="A3.Ex59.m2.1.1.1.6.cmml"><mover accent="true" id="A3.Ex59.m2.1.1.1.6.2.2" xref="A3.Ex59.m2.1.1.1.6.2.2.cmml"><mi id="A3.Ex59.m2.1.1.1.6.2.2.2" xref="A3.Ex59.m2.1.1.1.6.2.2.2.cmml">𝜽</mi><mo id="A3.Ex59.m2.1.1.1.6.2.2.1" xref="A3.Ex59.m2.1.1.1.6.2.2.1.cmml">~</mo></mover><mi id="A3.Ex59.m2.1.1.1.6.3" xref="A3.Ex59.m2.1.1.1.6.3.cmml">ζ</mi><mi id="A3.Ex59.m2.1.1.1.6.2.3" xref="A3.Ex59.m2.1.1.1.6.2.3.cmml">T</mi></msubsup><mo id="A3.Ex59.m2.1.1.1.2d" xref="A3.Ex59.m2.1.1.1.2.cmml">⁢</mo><msubsup id="A3.Ex59.m2.1.1.1.7" xref="A3.Ex59.m2.1.1.1.7.cmml"><mi id="A3.Ex59.m2.1.1.1.7.2.2" mathvariant="normal" xref="A3.Ex59.m2.1.1.1.7.2.2.cmml">Γ</mi><mi id="A3.Ex59.m2.1.1.1.7.2.3" xref="A3.Ex59.m2.1.1.1.7.2.3.cmml">ζ</mi><mrow id="A3.Ex59.m2.1.1.1.7.3" xref="A3.Ex59.m2.1.1.1.7.3.cmml"><mo id="A3.Ex59.m2.1.1.1.7.3a" xref="A3.Ex59.m2.1.1.1.7.3.cmml">−</mo><mn id="A3.Ex59.m2.1.1.1.7.3.2" xref="A3.Ex59.m2.1.1.1.7.3.2.cmml">1</mn></mrow></msubsup><mo id="A3.Ex59.m2.1.1.1.2e" xref="A3.Ex59.m2.1.1.1.2.cmml">⁢</mo><msub id="A3.Ex59.m2.1.1.1.8" xref="A3.Ex59.m2.1.1.1.8.cmml"><mover accent="true" id="A3.Ex59.m2.1.1.1.8.2" xref="A3.Ex59.m2.1.1.1.8.2.cmml"><mi id="A3.Ex59.m2.1.1.1.8.2.2" xref="A3.Ex59.m2.1.1.1.8.2.2.cmml">𝜽</mi><mo id="A3.Ex59.m2.1.1.1.8.2.1" xref="A3.Ex59.m2.1.1.1.8.2.1.cmml">~</mo></mover><mi id="A3.Ex59.m2.1.1.1.8.3" xref="A3.Ex59.m2.1.1.1.8.3.cmml">ζ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex59.m2.1b"><apply id="A3.Ex59.m2.1.1.cmml" xref="A3.Ex59.m2.1.1"><minus id="A3.Ex59.m2.1.1.2.cmml" xref="A3.Ex59.m2.1.1.2"></minus><apply id="A3.Ex59.m2.1.1.3.cmml" xref="A3.Ex59.m2.1.1.3"><minus id="A3.Ex59.m2.1.1.3.1.cmml" xref="A3.Ex59.m2.1.1.3"></minus><apply id="A3.Ex59.m2.1.1.3.2.cmml" xref="A3.Ex59.m2.1.1.3.2"><divide id="A3.Ex59.m2.1.1.3.2.1.cmml" xref="A3.Ex59.m2.1.1.3.2.1"></divide><apply id="A3.Ex59.m2.1.1.3.2.2.cmml" xref="A3.Ex59.m2.1.1.3.2.2"><times id="A3.Ex59.m2.1.1.3.2.2.1.cmml" xref="A3.Ex59.m2.1.1.3.2.2.1"></times><apply id="A3.Ex59.m2.1.1.3.2.2.2.cmml" xref="A3.Ex59.m2.1.1.3.2.2.2"><csymbol cd="ambiguous" id="A3.Ex59.m2.1.1.3.2.2.2.1.cmml" xref="A3.Ex59.m2.1.1.3.2.2.2">subscript</csymbol><ci id="A3.Ex59.m2.1.1.3.2.2.2.2.cmml" xref="A3.Ex59.m2.1.1.3.2.2.2.2">𝑘</ci><ci id="A3.Ex59.m2.1.1.3.2.2.2.3.cmml" xref="A3.Ex59.m2.1.1.3.2.2.2.3">c</ci></apply><apply id="A3.Ex59.m2.1.1.3.2.2.3.cmml" xref="A3.Ex59.m2.1.1.3.2.2.3"><csymbol cd="ambiguous" id="A3.Ex59.m2.1.1.3.2.2.3.1.cmml" xref="A3.Ex59.m2.1.1.3.2.2.3">superscript</csymbol><ci id="A3.Ex59.m2.1.1.3.2.2.3.2.cmml" xref="A3.Ex59.m2.1.1.3.2.2.3.2">𝒆</ci><ci id="A3.Ex59.m2.1.1.3.2.2.3.3.cmml" xref="A3.Ex59.m2.1.1.3.2.2.3.3">𝑇</ci></apply><ci id="A3.Ex59.m2.1.1.3.2.2.4.cmml" xref="A3.Ex59.m2.1.1.3.2.2.4">𝒆</ci></apply><cn id="A3.Ex59.m2.1.1.3.2.3.cmml" type="integer" xref="A3.Ex59.m2.1.1.3.2.3">2</cn></apply></apply><apply id="A3.Ex59.m2.1.1.1.cmml" xref="A3.Ex59.m2.1.1.1"><times id="A3.Ex59.m2.1.1.1.2.cmml" xref="A3.Ex59.m2.1.1.1.2"></times><ci id="A3.Ex59.m2.1.1.1.3.cmml" xref="A3.Ex59.m2.1.1.1.3">𝜅</ci><apply id="A3.Ex59.m2.1.1.1.4.cmml" xref="A3.Ex59.m2.1.1.1.4"><csymbol cd="ambiguous" id="A3.Ex59.m2.1.1.1.4.1.cmml" xref="A3.Ex59.m2.1.1.1.4">superscript</csymbol><ci id="A3.Ex59.m2.1.1.1.4.2.cmml" xref="A3.Ex59.m2.1.1.1.4.2">𝜎</ci><times id="A3.Ex59.m2.1.1.1.4.3.cmml" xref="A3.Ex59.m2.1.1.1.4.3"></times></apply><apply id="A3.Ex59.m2.1.1.1.5.cmml" xref="A3.Ex59.m2.1.1.1.5"><csymbol cd="ambiguous" id="A3.Ex59.m2.1.1.1.5.1.cmml" xref="A3.Ex59.m2.1.1.1.5">subscript</csymbol><ci id="A3.Ex59.m2.1.1.1.5.2.cmml" xref="A3.Ex59.m2.1.1.1.5.2">𝜆</ci><min id="A3.Ex59.m2.1.1.1.5.3.cmml" xref="A3.Ex59.m2.1.1.1.5.3"></min></apply><apply id="A3.Ex59.m2.1.1.1.1.1.1.cmml" xref="A3.Ex59.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.Ex59.m2.1.1.1.1.1.1.1.cmml" xref="A3.Ex59.m2.1.1.1.1.1">subscript</csymbol><ci id="A3.Ex59.m2.1.1.1.1.1.1.2.cmml" xref="A3.Ex59.m2.1.1.1.1.1.1.2">Γ</ci><ci id="A3.Ex59.m2.1.1.1.1.1.1.3.cmml" xref="A3.Ex59.m2.1.1.1.1.1.1.3">𝜁</ci></apply><apply id="A3.Ex59.m2.1.1.1.6.cmml" xref="A3.Ex59.m2.1.1.1.6"><csymbol cd="ambiguous" id="A3.Ex59.m2.1.1.1.6.1.cmml" xref="A3.Ex59.m2.1.1.1.6">subscript</csymbol><apply id="A3.Ex59.m2.1.1.1.6.2.cmml" xref="A3.Ex59.m2.1.1.1.6"><csymbol cd="ambiguous" id="A3.Ex59.m2.1.1.1.6.2.1.cmml" xref="A3.Ex59.m2.1.1.1.6">superscript</csymbol><apply id="A3.Ex59.m2.1.1.1.6.2.2.cmml" xref="A3.Ex59.m2.1.1.1.6.2.2"><ci id="A3.Ex59.m2.1.1.1.6.2.2.1.cmml" xref="A3.Ex59.m2.1.1.1.6.2.2.1">~</ci><ci id="A3.Ex59.m2.1.1.1.6.2.2.2.cmml" xref="A3.Ex59.m2.1.1.1.6.2.2.2">𝜽</ci></apply><ci id="A3.Ex59.m2.1.1.1.6.2.3.cmml" xref="A3.Ex59.m2.1.1.1.6.2.3">𝑇</ci></apply><ci id="A3.Ex59.m2.1.1.1.6.3.cmml" xref="A3.Ex59.m2.1.1.1.6.3">𝜁</ci></apply><apply id="A3.Ex59.m2.1.1.1.7.cmml" xref="A3.Ex59.m2.1.1.1.7"><csymbol cd="ambiguous" id="A3.Ex59.m2.1.1.1.7.1.cmml" xref="A3.Ex59.m2.1.1.1.7">superscript</csymbol><apply id="A3.Ex59.m2.1.1.1.7.2.cmml" xref="A3.Ex59.m2.1.1.1.7"><csymbol cd="ambiguous" id="A3.Ex59.m2.1.1.1.7.2.1.cmml" xref="A3.Ex59.m2.1.1.1.7">subscript</csymbol><ci id="A3.Ex59.m2.1.1.1.7.2.2.cmml" xref="A3.Ex59.m2.1.1.1.7.2.2">Γ</ci><ci id="A3.Ex59.m2.1.1.1.7.2.3.cmml" xref="A3.Ex59.m2.1.1.1.7.2.3">𝜁</ci></apply><apply id="A3.Ex59.m2.1.1.1.7.3.cmml" xref="A3.Ex59.m2.1.1.1.7.3"><minus id="A3.Ex59.m2.1.1.1.7.3.1.cmml" xref="A3.Ex59.m2.1.1.1.7.3"></minus><cn id="A3.Ex59.m2.1.1.1.7.3.2.cmml" type="integer" xref="A3.Ex59.m2.1.1.1.7.3.2">1</cn></apply></apply><apply id="A3.Ex59.m2.1.1.1.8.cmml" xref="A3.Ex59.m2.1.1.1.8"><csymbol cd="ambiguous" id="A3.Ex59.m2.1.1.1.8.1.cmml" xref="A3.Ex59.m2.1.1.1.8">subscript</csymbol><apply id="A3.Ex59.m2.1.1.1.8.2.cmml" xref="A3.Ex59.m2.1.1.1.8.2"><ci id="A3.Ex59.m2.1.1.1.8.2.1.cmml" xref="A3.Ex59.m2.1.1.1.8.2.1">~</ci><ci id="A3.Ex59.m2.1.1.1.8.2.2.cmml" xref="A3.Ex59.m2.1.1.1.8.2.2">𝜽</ci></apply><ci id="A3.Ex59.m2.1.1.1.8.3.cmml" xref="A3.Ex59.m2.1.1.1.8.3">𝜁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex59.m2.1c">\displaystyle-k_{\rm c}\bm{e}^{T}\bm{e}/2-\kappa\sigma^{*}\lambda_{\min}(% \Gamma_{\zeta})\tilde{\bm{\theta}}^{T}_{\zeta}\Gamma_{\zeta}^{-1}\tilde{\bm{% \theta}}_{\zeta}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex59.m2.1d">- italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_e / 2 - italic_κ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Γ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT roman_Γ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex60"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\leq" class="ltx_Math" display="inline" id="A3.Ex60.m1.1"><semantics id="A3.Ex60.m1.1a"><mo id="A3.Ex60.m1.1.1" xref="A3.Ex60.m1.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="A3.Ex60.m1.1b"><leq id="A3.Ex60.m1.1.1.cmml" xref="A3.Ex60.m1.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex60.m1.1c">\displaystyle\leq</annotation><annotation encoding="application/x-llamapun" id="A3.Ex60.m1.1d">≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-k_{\rm p}V_{\zeta},t\geq T_{\rm a}" class="ltx_Math" display="inline" id="A3.Ex60.m2.2"><semantics id="A3.Ex60.m2.2a"><mrow id="A3.Ex60.m2.2.2" xref="A3.Ex60.m2.2.2.cmml"><mrow id="A3.Ex60.m2.2.2.1.1" xref="A3.Ex60.m2.2.2.1.2.cmml"><mrow id="A3.Ex60.m2.2.2.1.1.1" xref="A3.Ex60.m2.2.2.1.1.1.cmml"><mo id="A3.Ex60.m2.2.2.1.1.1a" xref="A3.Ex60.m2.2.2.1.1.1.cmml">−</mo><mrow id="A3.Ex60.m2.2.2.1.1.1.2" xref="A3.Ex60.m2.2.2.1.1.1.2.cmml"><msub id="A3.Ex60.m2.2.2.1.1.1.2.2" xref="A3.Ex60.m2.2.2.1.1.1.2.2.cmml"><mi id="A3.Ex60.m2.2.2.1.1.1.2.2.2" xref="A3.Ex60.m2.2.2.1.1.1.2.2.2.cmml">k</mi><mi id="A3.Ex60.m2.2.2.1.1.1.2.2.3" mathvariant="normal" xref="A3.Ex60.m2.2.2.1.1.1.2.2.3.cmml">p</mi></msub><mo id="A3.Ex60.m2.2.2.1.1.1.2.1" xref="A3.Ex60.m2.2.2.1.1.1.2.1.cmml">⁢</mo><msub id="A3.Ex60.m2.2.2.1.1.1.2.3" xref="A3.Ex60.m2.2.2.1.1.1.2.3.cmml"><mi id="A3.Ex60.m2.2.2.1.1.1.2.3.2" xref="A3.Ex60.m2.2.2.1.1.1.2.3.2.cmml">V</mi><mi id="A3.Ex60.m2.2.2.1.1.1.2.3.3" xref="A3.Ex60.m2.2.2.1.1.1.2.3.3.cmml">ζ</mi></msub></mrow></mrow><mo id="A3.Ex60.m2.2.2.1.1.2" xref="A3.Ex60.m2.2.2.1.2.cmml">,</mo><mi id="A3.Ex60.m2.1.1" xref="A3.Ex60.m2.1.1.cmml">t</mi></mrow><mo id="A3.Ex60.m2.2.2.2" xref="A3.Ex60.m2.2.2.2.cmml">≥</mo><msub id="A3.Ex60.m2.2.2.3" xref="A3.Ex60.m2.2.2.3.cmml"><mi id="A3.Ex60.m2.2.2.3.2" xref="A3.Ex60.m2.2.2.3.2.cmml">T</mi><mi id="A3.Ex60.m2.2.2.3.3" mathvariant="normal" xref="A3.Ex60.m2.2.2.3.3.cmml">a</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex60.m2.2b"><apply id="A3.Ex60.m2.2.2.cmml" xref="A3.Ex60.m2.2.2"><geq id="A3.Ex60.m2.2.2.2.cmml" xref="A3.Ex60.m2.2.2.2"></geq><list id="A3.Ex60.m2.2.2.1.2.cmml" xref="A3.Ex60.m2.2.2.1.1"><apply id="A3.Ex60.m2.2.2.1.1.1.cmml" xref="A3.Ex60.m2.2.2.1.1.1"><minus id="A3.Ex60.m2.2.2.1.1.1.1.cmml" xref="A3.Ex60.m2.2.2.1.1.1"></minus><apply id="A3.Ex60.m2.2.2.1.1.1.2.cmml" xref="A3.Ex60.m2.2.2.1.1.1.2"><times id="A3.Ex60.m2.2.2.1.1.1.2.1.cmml" xref="A3.Ex60.m2.2.2.1.1.1.2.1"></times><apply id="A3.Ex60.m2.2.2.1.1.1.2.2.cmml" xref="A3.Ex60.m2.2.2.1.1.1.2.2"><csymbol cd="ambiguous" id="A3.Ex60.m2.2.2.1.1.1.2.2.1.cmml" xref="A3.Ex60.m2.2.2.1.1.1.2.2">subscript</csymbol><ci id="A3.Ex60.m2.2.2.1.1.1.2.2.2.cmml" xref="A3.Ex60.m2.2.2.1.1.1.2.2.2">𝑘</ci><ci id="A3.Ex60.m2.2.2.1.1.1.2.2.3.cmml" xref="A3.Ex60.m2.2.2.1.1.1.2.2.3">p</ci></apply><apply id="A3.Ex60.m2.2.2.1.1.1.2.3.cmml" xref="A3.Ex60.m2.2.2.1.1.1.2.3"><csymbol cd="ambiguous" id="A3.Ex60.m2.2.2.1.1.1.2.3.1.cmml" xref="A3.Ex60.m2.2.2.1.1.1.2.3">subscript</csymbol><ci id="A3.Ex60.m2.2.2.1.1.1.2.3.2.cmml" xref="A3.Ex60.m2.2.2.1.1.1.2.3.2">𝑉</ci><ci id="A3.Ex60.m2.2.2.1.1.1.2.3.3.cmml" xref="A3.Ex60.m2.2.2.1.1.1.2.3.3">𝜁</ci></apply></apply></apply><ci id="A3.Ex60.m2.1.1.cmml" xref="A3.Ex60.m2.1.1">𝑡</ci></list><apply id="A3.Ex60.m2.2.2.3.cmml" xref="A3.Ex60.m2.2.2.3"><csymbol cd="ambiguous" id="A3.Ex60.m2.2.2.3.1.cmml" xref="A3.Ex60.m2.2.2.3">subscript</csymbol><ci id="A3.Ex60.m2.2.2.3.2.cmml" xref="A3.Ex60.m2.2.2.3.2">𝑇</ci><ci id="A3.Ex60.m2.2.2.3.3.cmml" xref="A3.Ex60.m2.2.2.3.3">a</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex60.m2.2c">\displaystyle-k_{\rm p}V_{\zeta},t\geq T_{\rm a}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex60.m2.2d">- italic_k start_POSTSUBSCRIPT roman_p end_POSTSUBSCRIPT italic_V start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT , italic_t ≥ italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.2.p2.79">with <math alttext="k_{\rm p}" class="ltx_Math" display="inline" id="A3.2.p2.69.m1.1"><semantics id="A3.2.p2.69.m1.1a"><msub id="A3.2.p2.69.m1.1.1" xref="A3.2.p2.69.m1.1.1.cmml"><mi id="A3.2.p2.69.m1.1.1.2" xref="A3.2.p2.69.m1.1.1.2.cmml">k</mi><mi id="A3.2.p2.69.m1.1.1.3" mathvariant="normal" xref="A3.2.p2.69.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A3.2.p2.69.m1.1b"><apply id="A3.2.p2.69.m1.1.1.cmml" xref="A3.2.p2.69.m1.1.1"><csymbol cd="ambiguous" id="A3.2.p2.69.m1.1.1.1.cmml" xref="A3.2.p2.69.m1.1.1">subscript</csymbol><ci id="A3.2.p2.69.m1.1.1.2.cmml" xref="A3.2.p2.69.m1.1.1.2">𝑘</ci><ci id="A3.2.p2.69.m1.1.1.3.cmml" xref="A3.2.p2.69.m1.1.1.3">p</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.69.m1.1c">k_{\rm p}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.69.m1.1d">italic_k start_POSTSUBSCRIPT roman_p end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="A3.2.p2.70.m2.1"><semantics id="A3.2.p2.70.m2.1a"><mo id="A3.2.p2.70.m2.1.1" xref="A3.2.p2.70.m2.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="A3.2.p2.70.m2.1b"><csymbol cd="latexml" id="A3.2.p2.70.m2.1.1.cmml" xref="A3.2.p2.70.m2.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.70.m2.1c">:=</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.70.m2.1d">:=</annotation></semantics></math> <math alttext="\min\{k_{\rm c},2\kappa\sigma^{*}\lambda_{\min}(\Gamma_{\zeta})/(1+p)\}\in% \mathbb{R}^{+}" class="ltx_Math" display="inline" id="A3.2.p2.71.m3.3"><semantics id="A3.2.p2.71.m3.3a"><mrow id="A3.2.p2.71.m3.3.3" xref="A3.2.p2.71.m3.3.3.cmml"><mrow id="A3.2.p2.71.m3.3.3.2.2" xref="A3.2.p2.71.m3.3.3.2.3.cmml"><mi id="A3.2.p2.71.m3.1.1" xref="A3.2.p2.71.m3.1.1.cmml">min</mi><mo id="A3.2.p2.71.m3.3.3.2.2a" xref="A3.2.p2.71.m3.3.3.2.3.cmml">⁡</mo><mrow id="A3.2.p2.71.m3.3.3.2.2.2" xref="A3.2.p2.71.m3.3.3.2.3.cmml"><mo id="A3.2.p2.71.m3.3.3.2.2.2.3" stretchy="false" xref="A3.2.p2.71.m3.3.3.2.3.cmml">{</mo><msub id="A3.2.p2.71.m3.2.2.1.1.1.1" xref="A3.2.p2.71.m3.2.2.1.1.1.1.cmml"><mi id="A3.2.p2.71.m3.2.2.1.1.1.1.2" xref="A3.2.p2.71.m3.2.2.1.1.1.1.2.cmml">k</mi><mi id="A3.2.p2.71.m3.2.2.1.1.1.1.3" mathvariant="normal" xref="A3.2.p2.71.m3.2.2.1.1.1.1.3.cmml">c</mi></msub><mo id="A3.2.p2.71.m3.3.3.2.2.2.4" xref="A3.2.p2.71.m3.3.3.2.3.cmml">,</mo><mrow id="A3.2.p2.71.m3.3.3.2.2.2.2" xref="A3.2.p2.71.m3.3.3.2.2.2.2.cmml"><mrow id="A3.2.p2.71.m3.3.3.2.2.2.2.1" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.cmml"><mn id="A3.2.p2.71.m3.3.3.2.2.2.2.1.3" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.3.cmml">2</mn><mo id="A3.2.p2.71.m3.3.3.2.2.2.2.1.2" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.2.cmml">⁢</mo><mi id="A3.2.p2.71.m3.3.3.2.2.2.2.1.4" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.4.cmml">κ</mi><mo id="A3.2.p2.71.m3.3.3.2.2.2.2.1.2a" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.2.cmml">⁢</mo><msup id="A3.2.p2.71.m3.3.3.2.2.2.2.1.5" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.5.cmml"><mi id="A3.2.p2.71.m3.3.3.2.2.2.2.1.5.2" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.5.2.cmml">σ</mi><mo id="A3.2.p2.71.m3.3.3.2.2.2.2.1.5.3" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.5.3.cmml">∗</mo></msup><mo id="A3.2.p2.71.m3.3.3.2.2.2.2.1.2b" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.2.cmml">⁢</mo><msub id="A3.2.p2.71.m3.3.3.2.2.2.2.1.6" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.6.cmml"><mi id="A3.2.p2.71.m3.3.3.2.2.2.2.1.6.2" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.6.2.cmml">λ</mi><mi id="A3.2.p2.71.m3.3.3.2.2.2.2.1.6.3" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.6.3.cmml">min</mi></msub><mo id="A3.2.p2.71.m3.3.3.2.2.2.2.1.2c" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.2.cmml">⁢</mo><mrow id="A3.2.p2.71.m3.3.3.2.2.2.2.1.1.1" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.1.1.1.cmml"><mo id="A3.2.p2.71.m3.3.3.2.2.2.2.1.1.1.2" stretchy="false" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.1.1.1.cmml">(</mo><msub id="A3.2.p2.71.m3.3.3.2.2.2.2.1.1.1.1" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.1.1.1.cmml"><mi id="A3.2.p2.71.m3.3.3.2.2.2.2.1.1.1.1.2" mathvariant="normal" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.1.1.1.2.cmml">Γ</mi><mi id="A3.2.p2.71.m3.3.3.2.2.2.2.1.1.1.1.3" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.1.1.1.3.cmml">ζ</mi></msub><mo id="A3.2.p2.71.m3.3.3.2.2.2.2.1.1.1.3" stretchy="false" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A3.2.p2.71.m3.3.3.2.2.2.2.3" xref="A3.2.p2.71.m3.3.3.2.2.2.2.3.cmml">/</mo><mrow id="A3.2.p2.71.m3.3.3.2.2.2.2.2.1" xref="A3.2.p2.71.m3.3.3.2.2.2.2.2.1.1.cmml"><mo id="A3.2.p2.71.m3.3.3.2.2.2.2.2.1.2" stretchy="false" xref="A3.2.p2.71.m3.3.3.2.2.2.2.2.1.1.cmml">(</mo><mrow id="A3.2.p2.71.m3.3.3.2.2.2.2.2.1.1" xref="A3.2.p2.71.m3.3.3.2.2.2.2.2.1.1.cmml"><mn id="A3.2.p2.71.m3.3.3.2.2.2.2.2.1.1.2" xref="A3.2.p2.71.m3.3.3.2.2.2.2.2.1.1.2.cmml">1</mn><mo id="A3.2.p2.71.m3.3.3.2.2.2.2.2.1.1.1" xref="A3.2.p2.71.m3.3.3.2.2.2.2.2.1.1.1.cmml">+</mo><mi id="A3.2.p2.71.m3.3.3.2.2.2.2.2.1.1.3" xref="A3.2.p2.71.m3.3.3.2.2.2.2.2.1.1.3.cmml">p</mi></mrow><mo id="A3.2.p2.71.m3.3.3.2.2.2.2.2.1.3" stretchy="false" xref="A3.2.p2.71.m3.3.3.2.2.2.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="A3.2.p2.71.m3.3.3.2.2.2.5" stretchy="false" xref="A3.2.p2.71.m3.3.3.2.3.cmml">}</mo></mrow></mrow><mo id="A3.2.p2.71.m3.3.3.3" xref="A3.2.p2.71.m3.3.3.3.cmml">∈</mo><msup id="A3.2.p2.71.m3.3.3.4" xref="A3.2.p2.71.m3.3.3.4.cmml"><mi id="A3.2.p2.71.m3.3.3.4.2" xref="A3.2.p2.71.m3.3.3.4.2.cmml">ℝ</mi><mo id="A3.2.p2.71.m3.3.3.4.3" xref="A3.2.p2.71.m3.3.3.4.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.71.m3.3b"><apply id="A3.2.p2.71.m3.3.3.cmml" xref="A3.2.p2.71.m3.3.3"><in id="A3.2.p2.71.m3.3.3.3.cmml" xref="A3.2.p2.71.m3.3.3.3"></in><apply id="A3.2.p2.71.m3.3.3.2.3.cmml" xref="A3.2.p2.71.m3.3.3.2.2"><min id="A3.2.p2.71.m3.1.1.cmml" xref="A3.2.p2.71.m3.1.1"></min><apply id="A3.2.p2.71.m3.2.2.1.1.1.1.cmml" xref="A3.2.p2.71.m3.2.2.1.1.1.1"><csymbol cd="ambiguous" id="A3.2.p2.71.m3.2.2.1.1.1.1.1.cmml" xref="A3.2.p2.71.m3.2.2.1.1.1.1">subscript</csymbol><ci id="A3.2.p2.71.m3.2.2.1.1.1.1.2.cmml" xref="A3.2.p2.71.m3.2.2.1.1.1.1.2">𝑘</ci><ci id="A3.2.p2.71.m3.2.2.1.1.1.1.3.cmml" xref="A3.2.p2.71.m3.2.2.1.1.1.1.3">c</ci></apply><apply id="A3.2.p2.71.m3.3.3.2.2.2.2.cmml" xref="A3.2.p2.71.m3.3.3.2.2.2.2"><divide id="A3.2.p2.71.m3.3.3.2.2.2.2.3.cmml" xref="A3.2.p2.71.m3.3.3.2.2.2.2.3"></divide><apply id="A3.2.p2.71.m3.3.3.2.2.2.2.1.cmml" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1"><times id="A3.2.p2.71.m3.3.3.2.2.2.2.1.2.cmml" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.2"></times><cn id="A3.2.p2.71.m3.3.3.2.2.2.2.1.3.cmml" type="integer" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.3">2</cn><ci id="A3.2.p2.71.m3.3.3.2.2.2.2.1.4.cmml" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.4">𝜅</ci><apply id="A3.2.p2.71.m3.3.3.2.2.2.2.1.5.cmml" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.5"><csymbol cd="ambiguous" id="A3.2.p2.71.m3.3.3.2.2.2.2.1.5.1.cmml" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.5">superscript</csymbol><ci id="A3.2.p2.71.m3.3.3.2.2.2.2.1.5.2.cmml" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.5.2">𝜎</ci><times id="A3.2.p2.71.m3.3.3.2.2.2.2.1.5.3.cmml" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.5.3"></times></apply><apply id="A3.2.p2.71.m3.3.3.2.2.2.2.1.6.cmml" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.6"><csymbol cd="ambiguous" id="A3.2.p2.71.m3.3.3.2.2.2.2.1.6.1.cmml" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.6">subscript</csymbol><ci id="A3.2.p2.71.m3.3.3.2.2.2.2.1.6.2.cmml" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.6.2">𝜆</ci><min id="A3.2.p2.71.m3.3.3.2.2.2.2.1.6.3.cmml" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.6.3"></min></apply><apply id="A3.2.p2.71.m3.3.3.2.2.2.2.1.1.1.1.cmml" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="A3.2.p2.71.m3.3.3.2.2.2.2.1.1.1.1.1.cmml" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.1.1">subscript</csymbol><ci id="A3.2.p2.71.m3.3.3.2.2.2.2.1.1.1.1.2.cmml" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.1.1.1.2">Γ</ci><ci id="A3.2.p2.71.m3.3.3.2.2.2.2.1.1.1.1.3.cmml" xref="A3.2.p2.71.m3.3.3.2.2.2.2.1.1.1.1.3">𝜁</ci></apply></apply><apply id="A3.2.p2.71.m3.3.3.2.2.2.2.2.1.1.cmml" xref="A3.2.p2.71.m3.3.3.2.2.2.2.2.1"><plus id="A3.2.p2.71.m3.3.3.2.2.2.2.2.1.1.1.cmml" xref="A3.2.p2.71.m3.3.3.2.2.2.2.2.1.1.1"></plus><cn id="A3.2.p2.71.m3.3.3.2.2.2.2.2.1.1.2.cmml" type="integer" xref="A3.2.p2.71.m3.3.3.2.2.2.2.2.1.1.2">1</cn><ci id="A3.2.p2.71.m3.3.3.2.2.2.2.2.1.1.3.cmml" xref="A3.2.p2.71.m3.3.3.2.2.2.2.2.1.1.3">𝑝</ci></apply></apply></apply><apply id="A3.2.p2.71.m3.3.3.4.cmml" xref="A3.2.p2.71.m3.3.3.4"><csymbol cd="ambiguous" id="A3.2.p2.71.m3.3.3.4.1.cmml" xref="A3.2.p2.71.m3.3.3.4">superscript</csymbol><ci id="A3.2.p2.71.m3.3.3.4.2.cmml" xref="A3.2.p2.71.m3.3.3.4.2">ℝ</ci><plus id="A3.2.p2.71.m3.3.3.4.3.cmml" xref="A3.2.p2.71.m3.3.3.4.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.71.m3.3c">\min\{k_{\rm c},2\kappa\sigma^{*}\lambda_{\min}(\Gamma_{\zeta})/(1+p)\}\in% \mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.71.m3.3d">roman_min { italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT , 2 italic_κ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Γ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ) / ( 1 + italic_p ) } ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>. <span class="ltx_text" id="A3.2.p2.79.8" style="color:#000099;">So, the equilibrium point <math alttext="(\bm{e}" class="ltx_math_unparsed" display="inline" id="A3.2.p2.72.1.m1.1"><semantics id="A3.2.p2.72.1.m1.1a"><mrow id="A3.2.p2.72.1.m1.1b"><mo id="A3.2.p2.72.1.m1.1.1" mathcolor="#000099" stretchy="false">(</mo><mi id="A3.2.p2.72.1.m1.1.2" mathcolor="#000099">𝒆</mi></mrow><annotation encoding="application/x-tex" id="A3.2.p2.72.1.m1.1c">(\bm{e}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.72.1.m1.1d">( bold_italic_e</annotation></semantics></math>, <math alttext="\tilde{\bm{\theta}})=\bm{0}" class="ltx_math_unparsed" display="inline" id="A3.2.p2.73.2.m2.1"><semantics id="A3.2.p2.73.2.m2.1a"><mrow id="A3.2.p2.73.2.m2.1b"><mover accent="true" id="A3.2.p2.73.2.m2.1.1"><mi id="A3.2.p2.73.2.m2.1.1.2" mathcolor="#000099">𝜽</mi><mo id="A3.2.p2.73.2.m2.1.1.1" mathcolor="#000099">~</mo></mover><mo id="A3.2.p2.73.2.m2.1.2" mathcolor="#000099" stretchy="false">)</mo><mo id="A3.2.p2.73.2.m2.1.3" mathcolor="#000099">=</mo><mn id="A3.2.p2.73.2.m2.1.4" mathcolor="#000099">𝟎</mn></mrow><annotation encoding="application/x-tex" id="A3.2.p2.73.2.m2.1c">\tilde{\bm{\theta}})=\bm{0}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.73.2.m2.1d">over~ start_ARG bold_italic_θ end_ARG ) = bold_0</annotation></semantics></math> of the closed-loop system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E7" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">7</span></a>) with (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>) is partial exponential stable on <math alttext="t" class="ltx_Math" display="inline" id="A3.2.p2.74.3.m3.1"><semantics id="A3.2.p2.74.3.m3.1a"><mi id="A3.2.p2.74.3.m3.1.1" mathcolor="#000099" xref="A3.2.p2.74.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A3.2.p2.74.3.m3.1b"><ci id="A3.2.p2.74.3.m3.1.1.cmml" xref="A3.2.p2.74.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.74.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.74.3.m3.1d">italic_t</annotation></semantics></math> <math alttext="\in" class="ltx_Math" display="inline" id="A3.2.p2.75.4.m4.1"><semantics id="A3.2.p2.75.4.m4.1a"><mo id="A3.2.p2.75.4.m4.1.1" mathcolor="#000099" xref="A3.2.p2.75.4.m4.1.1.cmml">∈</mo><annotation-xml encoding="MathML-Content" id="A3.2.p2.75.4.m4.1b"><in id="A3.2.p2.75.4.m4.1.1.cmml" xref="A3.2.p2.75.4.m4.1.1"></in></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.75.4.m4.1c">\in</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.75.4.m4.1d">∈</annotation></semantics></math> <math alttext="[T_{\rm a},\infty)" class="ltx_Math" display="inline" id="A3.2.p2.76.5.m5.2"><semantics id="A3.2.p2.76.5.m5.2a"><mrow id="A3.2.p2.76.5.m5.2.2.1" xref="A3.2.p2.76.5.m5.2.2.2.cmml"><mo id="A3.2.p2.76.5.m5.2.2.1.2" mathcolor="#000099" stretchy="false" xref="A3.2.p2.76.5.m5.2.2.2.cmml">[</mo><msub id="A3.2.p2.76.5.m5.2.2.1.1" xref="A3.2.p2.76.5.m5.2.2.1.1.cmml"><mi id="A3.2.p2.76.5.m5.2.2.1.1.2" mathcolor="#000099" xref="A3.2.p2.76.5.m5.2.2.1.1.2.cmml">T</mi><mi id="A3.2.p2.76.5.m5.2.2.1.1.3" mathcolor="#000099" mathvariant="normal" xref="A3.2.p2.76.5.m5.2.2.1.1.3.cmml">a</mi></msub><mo id="A3.2.p2.76.5.m5.2.2.1.3" mathcolor="#000099" xref="A3.2.p2.76.5.m5.2.2.2.cmml">,</mo><mi id="A3.2.p2.76.5.m5.1.1" mathcolor="#000099" mathvariant="normal" xref="A3.2.p2.76.5.m5.1.1.cmml">∞</mi><mo id="A3.2.p2.76.5.m5.2.2.1.4" mathcolor="#000099" stretchy="false" xref="A3.2.p2.76.5.m5.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.76.5.m5.2b"><interval closure="closed-open" id="A3.2.p2.76.5.m5.2.2.2.cmml" xref="A3.2.p2.76.5.m5.2.2.1"><apply id="A3.2.p2.76.5.m5.2.2.1.1.cmml" xref="A3.2.p2.76.5.m5.2.2.1.1"><csymbol cd="ambiguous" id="A3.2.p2.76.5.m5.2.2.1.1.1.cmml" xref="A3.2.p2.76.5.m5.2.2.1.1">subscript</csymbol><ci id="A3.2.p2.76.5.m5.2.2.1.1.2.cmml" xref="A3.2.p2.76.5.m5.2.2.1.1.2">𝑇</ci><ci id="A3.2.p2.76.5.m5.2.2.1.1.3.cmml" xref="A3.2.p2.76.5.m5.2.2.1.1.3">a</ci></apply><infinity id="A3.2.p2.76.5.m5.1.1.cmml" xref="A3.2.p2.76.5.m5.1.1"></infinity></interval></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.76.5.m5.2c">[T_{\rm a},\infty)</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.76.5.m5.2d">[ italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT , ∞ )</annotation></semantics></math>, where the tracking error <math alttext="\bm{e}(t)" class="ltx_Math" display="inline" id="A3.2.p2.77.6.m6.1"><semantics id="A3.2.p2.77.6.m6.1a"><mrow id="A3.2.p2.77.6.m6.1.2" xref="A3.2.p2.77.6.m6.1.2.cmml"><mi id="A3.2.p2.77.6.m6.1.2.2" mathcolor="#000099" xref="A3.2.p2.77.6.m6.1.2.2.cmml">𝒆</mi><mo id="A3.2.p2.77.6.m6.1.2.1" xref="A3.2.p2.77.6.m6.1.2.1.cmml">⁢</mo><mrow id="A3.2.p2.77.6.m6.1.2.3.2" xref="A3.2.p2.77.6.m6.1.2.cmml"><mo id="A3.2.p2.77.6.m6.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="A3.2.p2.77.6.m6.1.2.cmml">(</mo><mi id="A3.2.p2.77.6.m6.1.1" mathcolor="#000099" xref="A3.2.p2.77.6.m6.1.1.cmml">t</mi><mo id="A3.2.p2.77.6.m6.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="A3.2.p2.77.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.77.6.m6.1b"><apply id="A3.2.p2.77.6.m6.1.2.cmml" xref="A3.2.p2.77.6.m6.1.2"><times id="A3.2.p2.77.6.m6.1.2.1.cmml" xref="A3.2.p2.77.6.m6.1.2.1"></times><ci id="A3.2.p2.77.6.m6.1.2.2.cmml" xref="A3.2.p2.77.6.m6.1.2.2">𝒆</ci><ci id="A3.2.p2.77.6.m6.1.1.cmml" xref="A3.2.p2.77.6.m6.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.77.6.m6.1c">\bm{e}(t)</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.77.6.m6.1d">bold_italic_e ( italic_t )</annotation></semantics></math> and the partial estimation error <math alttext="\tilde{\bm{\theta}}_{\zeta}(t)" class="ltx_Math" display="inline" id="A3.2.p2.78.7.m7.1"><semantics id="A3.2.p2.78.7.m7.1a"><mrow id="A3.2.p2.78.7.m7.1.2" xref="A3.2.p2.78.7.m7.1.2.cmml"><msub id="A3.2.p2.78.7.m7.1.2.2" xref="A3.2.p2.78.7.m7.1.2.2.cmml"><mover accent="true" id="A3.2.p2.78.7.m7.1.2.2.2" xref="A3.2.p2.78.7.m7.1.2.2.2.cmml"><mi id="A3.2.p2.78.7.m7.1.2.2.2.2" mathcolor="#000099" xref="A3.2.p2.78.7.m7.1.2.2.2.2.cmml">𝜽</mi><mo id="A3.2.p2.78.7.m7.1.2.2.2.1" mathcolor="#000099" xref="A3.2.p2.78.7.m7.1.2.2.2.1.cmml">~</mo></mover><mi id="A3.2.p2.78.7.m7.1.2.2.3" mathcolor="#000099" xref="A3.2.p2.78.7.m7.1.2.2.3.cmml">ζ</mi></msub><mo id="A3.2.p2.78.7.m7.1.2.1" xref="A3.2.p2.78.7.m7.1.2.1.cmml">⁢</mo><mrow id="A3.2.p2.78.7.m7.1.2.3.2" xref="A3.2.p2.78.7.m7.1.2.cmml"><mo id="A3.2.p2.78.7.m7.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="A3.2.p2.78.7.m7.1.2.cmml">(</mo><mi id="A3.2.p2.78.7.m7.1.1" mathcolor="#000099" xref="A3.2.p2.78.7.m7.1.1.cmml">t</mi><mo id="A3.2.p2.78.7.m7.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="A3.2.p2.78.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.2.p2.78.7.m7.1b"><apply id="A3.2.p2.78.7.m7.1.2.cmml" xref="A3.2.p2.78.7.m7.1.2"><times id="A3.2.p2.78.7.m7.1.2.1.cmml" xref="A3.2.p2.78.7.m7.1.2.1"></times><apply id="A3.2.p2.78.7.m7.1.2.2.cmml" xref="A3.2.p2.78.7.m7.1.2.2"><csymbol cd="ambiguous" id="A3.2.p2.78.7.m7.1.2.2.1.cmml" xref="A3.2.p2.78.7.m7.1.2.2">subscript</csymbol><apply id="A3.2.p2.78.7.m7.1.2.2.2.cmml" xref="A3.2.p2.78.7.m7.1.2.2.2"><ci id="A3.2.p2.78.7.m7.1.2.2.2.1.cmml" xref="A3.2.p2.78.7.m7.1.2.2.2.1">~</ci><ci id="A3.2.p2.78.7.m7.1.2.2.2.2.cmml" xref="A3.2.p2.78.7.m7.1.2.2.2.2">𝜽</ci></apply><ci id="A3.2.p2.78.7.m7.1.2.2.3.cmml" xref="A3.2.p2.78.7.m7.1.2.2.3">𝜁</ci></apply><ci id="A3.2.p2.78.7.m7.1.1.cmml" xref="A3.2.p2.78.7.m7.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.78.7.m7.1c">\tilde{\bm{\theta}}_{\zeta}(t)</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.78.7.m7.1d">over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> exponentially converge to <math alttext="\bm{0}" class="ltx_Math" display="inline" id="A3.2.p2.79.8.m8.1"><semantics id="A3.2.p2.79.8.m8.1a"><mn id="A3.2.p2.79.8.m8.1.1" mathcolor="#000099" xref="A3.2.p2.79.8.m8.1.1.cmml">𝟎</mn><annotation-xml encoding="MathML-Content" id="A3.2.p2.79.8.m8.1b"><cn id="A3.2.p2.79.8.m8.1.1.cmml" type="integer" xref="A3.2.p2.79.8.m8.1.1">0</cn></annotation-xml><annotation encoding="application/x-tex" id="A3.2.p2.79.8.m8.1c">\bm{0}</annotation><annotation encoding="application/x-llamapun" id="A3.2.p2.79.8.m8.1d">bold_0</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="A3.3.p3"> <p class="ltx_p" id="A3.3.p3.2">3) The proof of closed-loop exponential stability under IE is similar to that under partial IE, so we omit some similar steps. Applying the Lyapunov function <math alttext="V" class="ltx_Math" display="inline" id="A3.3.p3.1.m1.1"><semantics id="A3.3.p3.1.m1.1a"><mi id="A3.3.p3.1.m1.1.1" xref="A3.3.p3.1.m1.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="A3.3.p3.1.m1.1b"><ci id="A3.3.p3.1.m1.1.1.cmml" xref="A3.3.p3.1.m1.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.3.p3.1.m1.1c">V</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.1.m1.1d">italic_V</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A3.E39" title="In Proof. ‣ Appendix C The proof of Theorem 2 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">39</span></a>) and the IE condition <math alttext="\Psi(T_{\rm e})\geq\sigma I" class="ltx_Math" display="inline" id="A3.3.p3.2.m2.1"><semantics id="A3.3.p3.2.m2.1a"><mrow id="A3.3.p3.2.m2.1.1" xref="A3.3.p3.2.m2.1.1.cmml"><mrow id="A3.3.p3.2.m2.1.1.1" xref="A3.3.p3.2.m2.1.1.1.cmml"><mi id="A3.3.p3.2.m2.1.1.1.3" mathvariant="normal" xref="A3.3.p3.2.m2.1.1.1.3.cmml">Ψ</mi><mo id="A3.3.p3.2.m2.1.1.1.2" xref="A3.3.p3.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="A3.3.p3.2.m2.1.1.1.1.1" xref="A3.3.p3.2.m2.1.1.1.1.1.1.cmml"><mo id="A3.3.p3.2.m2.1.1.1.1.1.2" stretchy="false" xref="A3.3.p3.2.m2.1.1.1.1.1.1.cmml">(</mo><msub id="A3.3.p3.2.m2.1.1.1.1.1.1" xref="A3.3.p3.2.m2.1.1.1.1.1.1.cmml"><mi id="A3.3.p3.2.m2.1.1.1.1.1.1.2" xref="A3.3.p3.2.m2.1.1.1.1.1.1.2.cmml">T</mi><mi id="A3.3.p3.2.m2.1.1.1.1.1.1.3" mathvariant="normal" xref="A3.3.p3.2.m2.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="A3.3.p3.2.m2.1.1.1.1.1.3" stretchy="false" xref="A3.3.p3.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A3.3.p3.2.m2.1.1.2" xref="A3.3.p3.2.m2.1.1.2.cmml">≥</mo><mrow id="A3.3.p3.2.m2.1.1.3" xref="A3.3.p3.2.m2.1.1.3.cmml"><mi id="A3.3.p3.2.m2.1.1.3.2" xref="A3.3.p3.2.m2.1.1.3.2.cmml">σ</mi><mo id="A3.3.p3.2.m2.1.1.3.1" xref="A3.3.p3.2.m2.1.1.3.1.cmml">⁢</mo><mi id="A3.3.p3.2.m2.1.1.3.3" xref="A3.3.p3.2.m2.1.1.3.3.cmml">I</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.3.p3.2.m2.1b"><apply id="A3.3.p3.2.m2.1.1.cmml" xref="A3.3.p3.2.m2.1.1"><geq id="A3.3.p3.2.m2.1.1.2.cmml" xref="A3.3.p3.2.m2.1.1.2"></geq><apply id="A3.3.p3.2.m2.1.1.1.cmml" xref="A3.3.p3.2.m2.1.1.1"><times id="A3.3.p3.2.m2.1.1.1.2.cmml" xref="A3.3.p3.2.m2.1.1.1.2"></times><ci id="A3.3.p3.2.m2.1.1.1.3.cmml" xref="A3.3.p3.2.m2.1.1.1.3">Ψ</ci><apply id="A3.3.p3.2.m2.1.1.1.1.1.1.cmml" xref="A3.3.p3.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.3.p3.2.m2.1.1.1.1.1.1.1.cmml" xref="A3.3.p3.2.m2.1.1.1.1.1">subscript</csymbol><ci id="A3.3.p3.2.m2.1.1.1.1.1.1.2.cmml" xref="A3.3.p3.2.m2.1.1.1.1.1.1.2">𝑇</ci><ci id="A3.3.p3.2.m2.1.1.1.1.1.1.3.cmml" xref="A3.3.p3.2.m2.1.1.1.1.1.1.3">e</ci></apply></apply><apply id="A3.3.p3.2.m2.1.1.3.cmml" xref="A3.3.p3.2.m2.1.1.3"><times id="A3.3.p3.2.m2.1.1.3.1.cmml" xref="A3.3.p3.2.m2.1.1.3.1"></times><ci id="A3.3.p3.2.m2.1.1.3.2.cmml" xref="A3.3.p3.2.m2.1.1.3.2">𝜎</ci><ci id="A3.3.p3.2.m2.1.1.3.3.cmml" xref="A3.3.p3.2.m2.1.1.3.3">𝐼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.3.p3.2.m2.1c">\Psi(T_{\rm e})\geq\sigma I</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.2.m2.1d">roman_Ψ ( italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) ≥ italic_σ italic_I</annotation></semantics></math> yield</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx74"> <tbody id="A3.Ex61"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}\leq" class="ltx_Math" display="inline" id="A3.Ex61.m1.1"><semantics id="A3.Ex61.m1.1a"><mrow id="A3.Ex61.m1.1.1" xref="A3.Ex61.m1.1.1.cmml"><mover accent="true" id="A3.Ex61.m1.1.1.2" xref="A3.Ex61.m1.1.1.2.cmml"><mi id="A3.Ex61.m1.1.1.2.2" xref="A3.Ex61.m1.1.1.2.2.cmml">V</mi><mo id="A3.Ex61.m1.1.1.2.1" xref="A3.Ex61.m1.1.1.2.1.cmml">˙</mo></mover><mo id="A3.Ex61.m1.1.1.1" xref="A3.Ex61.m1.1.1.1.cmml">≤</mo><mi id="A3.Ex61.m1.1.1.3" xref="A3.Ex61.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex61.m1.1b"><apply id="A3.Ex61.m1.1.1.cmml" xref="A3.Ex61.m1.1.1"><leq id="A3.Ex61.m1.1.1.1.cmml" xref="A3.Ex61.m1.1.1.1"></leq><apply id="A3.Ex61.m1.1.1.2.cmml" xref="A3.Ex61.m1.1.1.2"><ci id="A3.Ex61.m1.1.1.2.1.cmml" xref="A3.Ex61.m1.1.1.2.1">˙</ci><ci id="A3.Ex61.m1.1.1.2.2.cmml" xref="A3.Ex61.m1.1.1.2.2">𝑉</ci></apply><csymbol cd="latexml" id="A3.Ex61.m1.1.1.3.cmml" xref="A3.Ex61.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex61.m1.1c">\displaystyle\dot{V}\leq</annotation><annotation encoding="application/x-llamapun" id="A3.Ex61.m1.1d">over˙ start_ARG italic_V end_ARG ≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-k_{\rm c}\bm{e}^{T}\bm{e}-\kappa\sigma^{*}\lambda_{\min}(\Gamma)% \tilde{\bm{\theta}}^{T}\Gamma^{-1}\tilde{\bm{\theta}}" class="ltx_Math" display="inline" id="A3.Ex61.m2.1"><semantics id="A3.Ex61.m2.1a"><mrow id="A3.Ex61.m2.1.2" xref="A3.Ex61.m2.1.2.cmml"><mrow id="A3.Ex61.m2.1.2.2" xref="A3.Ex61.m2.1.2.2.cmml"><mo id="A3.Ex61.m2.1.2.2a" xref="A3.Ex61.m2.1.2.2.cmml">−</mo><mrow id="A3.Ex61.m2.1.2.2.2" xref="A3.Ex61.m2.1.2.2.2.cmml"><msub id="A3.Ex61.m2.1.2.2.2.2" xref="A3.Ex61.m2.1.2.2.2.2.cmml"><mi id="A3.Ex61.m2.1.2.2.2.2.2" xref="A3.Ex61.m2.1.2.2.2.2.2.cmml">k</mi><mi id="A3.Ex61.m2.1.2.2.2.2.3" mathvariant="normal" xref="A3.Ex61.m2.1.2.2.2.2.3.cmml">c</mi></msub><mo id="A3.Ex61.m2.1.2.2.2.1" xref="A3.Ex61.m2.1.2.2.2.1.cmml">⁢</mo><msup id="A3.Ex61.m2.1.2.2.2.3" xref="A3.Ex61.m2.1.2.2.2.3.cmml"><mi id="A3.Ex61.m2.1.2.2.2.3.2" xref="A3.Ex61.m2.1.2.2.2.3.2.cmml">𝒆</mi><mi id="A3.Ex61.m2.1.2.2.2.3.3" xref="A3.Ex61.m2.1.2.2.2.3.3.cmml">T</mi></msup><mo id="A3.Ex61.m2.1.2.2.2.1a" xref="A3.Ex61.m2.1.2.2.2.1.cmml">⁢</mo><mi id="A3.Ex61.m2.1.2.2.2.4" xref="A3.Ex61.m2.1.2.2.2.4.cmml">𝒆</mi></mrow></mrow><mo id="A3.Ex61.m2.1.2.1" xref="A3.Ex61.m2.1.2.1.cmml">−</mo><mrow id="A3.Ex61.m2.1.2.3" xref="A3.Ex61.m2.1.2.3.cmml"><mi id="A3.Ex61.m2.1.2.3.2" xref="A3.Ex61.m2.1.2.3.2.cmml">κ</mi><mo id="A3.Ex61.m2.1.2.3.1" xref="A3.Ex61.m2.1.2.3.1.cmml">⁢</mo><msup id="A3.Ex61.m2.1.2.3.3" xref="A3.Ex61.m2.1.2.3.3.cmml"><mi id="A3.Ex61.m2.1.2.3.3.2" xref="A3.Ex61.m2.1.2.3.3.2.cmml">σ</mi><mo id="A3.Ex61.m2.1.2.3.3.3" xref="A3.Ex61.m2.1.2.3.3.3.cmml">∗</mo></msup><mo id="A3.Ex61.m2.1.2.3.1a" xref="A3.Ex61.m2.1.2.3.1.cmml">⁢</mo><msub id="A3.Ex61.m2.1.2.3.4" xref="A3.Ex61.m2.1.2.3.4.cmml"><mi id="A3.Ex61.m2.1.2.3.4.2" xref="A3.Ex61.m2.1.2.3.4.2.cmml">λ</mi><mi id="A3.Ex61.m2.1.2.3.4.3" xref="A3.Ex61.m2.1.2.3.4.3.cmml">min</mi></msub><mo id="A3.Ex61.m2.1.2.3.1b" xref="A3.Ex61.m2.1.2.3.1.cmml">⁢</mo><mrow id="A3.Ex61.m2.1.2.3.5.2" xref="A3.Ex61.m2.1.2.3.cmml"><mo id="A3.Ex61.m2.1.2.3.5.2.1" stretchy="false" xref="A3.Ex61.m2.1.2.3.cmml">(</mo><mi id="A3.Ex61.m2.1.1" mathvariant="normal" xref="A3.Ex61.m2.1.1.cmml">Γ</mi><mo id="A3.Ex61.m2.1.2.3.5.2.2" stretchy="false" xref="A3.Ex61.m2.1.2.3.cmml">)</mo></mrow><mo id="A3.Ex61.m2.1.2.3.1c" xref="A3.Ex61.m2.1.2.3.1.cmml">⁢</mo><msup id="A3.Ex61.m2.1.2.3.6" xref="A3.Ex61.m2.1.2.3.6.cmml"><mover accent="true" id="A3.Ex61.m2.1.2.3.6.2" xref="A3.Ex61.m2.1.2.3.6.2.cmml"><mi id="A3.Ex61.m2.1.2.3.6.2.2" xref="A3.Ex61.m2.1.2.3.6.2.2.cmml">𝜽</mi><mo id="A3.Ex61.m2.1.2.3.6.2.1" xref="A3.Ex61.m2.1.2.3.6.2.1.cmml">~</mo></mover><mi id="A3.Ex61.m2.1.2.3.6.3" xref="A3.Ex61.m2.1.2.3.6.3.cmml">T</mi></msup><mo id="A3.Ex61.m2.1.2.3.1d" xref="A3.Ex61.m2.1.2.3.1.cmml">⁢</mo><msup id="A3.Ex61.m2.1.2.3.7" xref="A3.Ex61.m2.1.2.3.7.cmml"><mi id="A3.Ex61.m2.1.2.3.7.2" mathvariant="normal" xref="A3.Ex61.m2.1.2.3.7.2.cmml">Γ</mi><mrow id="A3.Ex61.m2.1.2.3.7.3" xref="A3.Ex61.m2.1.2.3.7.3.cmml"><mo id="A3.Ex61.m2.1.2.3.7.3a" xref="A3.Ex61.m2.1.2.3.7.3.cmml">−</mo><mn id="A3.Ex61.m2.1.2.3.7.3.2" xref="A3.Ex61.m2.1.2.3.7.3.2.cmml">1</mn></mrow></msup><mo id="A3.Ex61.m2.1.2.3.1e" xref="A3.Ex61.m2.1.2.3.1.cmml">⁢</mo><mover accent="true" id="A3.Ex61.m2.1.2.3.8" xref="A3.Ex61.m2.1.2.3.8.cmml"><mi id="A3.Ex61.m2.1.2.3.8.2" xref="A3.Ex61.m2.1.2.3.8.2.cmml">𝜽</mi><mo id="A3.Ex61.m2.1.2.3.8.1" xref="A3.Ex61.m2.1.2.3.8.1.cmml">~</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex61.m2.1b"><apply id="A3.Ex61.m2.1.2.cmml" xref="A3.Ex61.m2.1.2"><minus id="A3.Ex61.m2.1.2.1.cmml" xref="A3.Ex61.m2.1.2.1"></minus><apply id="A3.Ex61.m2.1.2.2.cmml" xref="A3.Ex61.m2.1.2.2"><minus id="A3.Ex61.m2.1.2.2.1.cmml" xref="A3.Ex61.m2.1.2.2"></minus><apply id="A3.Ex61.m2.1.2.2.2.cmml" xref="A3.Ex61.m2.1.2.2.2"><times id="A3.Ex61.m2.1.2.2.2.1.cmml" xref="A3.Ex61.m2.1.2.2.2.1"></times><apply id="A3.Ex61.m2.1.2.2.2.2.cmml" xref="A3.Ex61.m2.1.2.2.2.2"><csymbol cd="ambiguous" id="A3.Ex61.m2.1.2.2.2.2.1.cmml" xref="A3.Ex61.m2.1.2.2.2.2">subscript</csymbol><ci id="A3.Ex61.m2.1.2.2.2.2.2.cmml" xref="A3.Ex61.m2.1.2.2.2.2.2">𝑘</ci><ci id="A3.Ex61.m2.1.2.2.2.2.3.cmml" xref="A3.Ex61.m2.1.2.2.2.2.3">c</ci></apply><apply id="A3.Ex61.m2.1.2.2.2.3.cmml" xref="A3.Ex61.m2.1.2.2.2.3"><csymbol cd="ambiguous" id="A3.Ex61.m2.1.2.2.2.3.1.cmml" xref="A3.Ex61.m2.1.2.2.2.3">superscript</csymbol><ci id="A3.Ex61.m2.1.2.2.2.3.2.cmml" xref="A3.Ex61.m2.1.2.2.2.3.2">𝒆</ci><ci id="A3.Ex61.m2.1.2.2.2.3.3.cmml" xref="A3.Ex61.m2.1.2.2.2.3.3">𝑇</ci></apply><ci id="A3.Ex61.m2.1.2.2.2.4.cmml" xref="A3.Ex61.m2.1.2.2.2.4">𝒆</ci></apply></apply><apply id="A3.Ex61.m2.1.2.3.cmml" xref="A3.Ex61.m2.1.2.3"><times id="A3.Ex61.m2.1.2.3.1.cmml" xref="A3.Ex61.m2.1.2.3.1"></times><ci id="A3.Ex61.m2.1.2.3.2.cmml" xref="A3.Ex61.m2.1.2.3.2">𝜅</ci><apply id="A3.Ex61.m2.1.2.3.3.cmml" xref="A3.Ex61.m2.1.2.3.3"><csymbol cd="ambiguous" id="A3.Ex61.m2.1.2.3.3.1.cmml" xref="A3.Ex61.m2.1.2.3.3">superscript</csymbol><ci id="A3.Ex61.m2.1.2.3.3.2.cmml" xref="A3.Ex61.m2.1.2.3.3.2">𝜎</ci><times id="A3.Ex61.m2.1.2.3.3.3.cmml" xref="A3.Ex61.m2.1.2.3.3.3"></times></apply><apply id="A3.Ex61.m2.1.2.3.4.cmml" xref="A3.Ex61.m2.1.2.3.4"><csymbol cd="ambiguous" id="A3.Ex61.m2.1.2.3.4.1.cmml" xref="A3.Ex61.m2.1.2.3.4">subscript</csymbol><ci id="A3.Ex61.m2.1.2.3.4.2.cmml" xref="A3.Ex61.m2.1.2.3.4.2">𝜆</ci><min id="A3.Ex61.m2.1.2.3.4.3.cmml" xref="A3.Ex61.m2.1.2.3.4.3"></min></apply><ci id="A3.Ex61.m2.1.1.cmml" xref="A3.Ex61.m2.1.1">Γ</ci><apply id="A3.Ex61.m2.1.2.3.6.cmml" xref="A3.Ex61.m2.1.2.3.6"><csymbol cd="ambiguous" id="A3.Ex61.m2.1.2.3.6.1.cmml" xref="A3.Ex61.m2.1.2.3.6">superscript</csymbol><apply id="A3.Ex61.m2.1.2.3.6.2.cmml" xref="A3.Ex61.m2.1.2.3.6.2"><ci id="A3.Ex61.m2.1.2.3.6.2.1.cmml" xref="A3.Ex61.m2.1.2.3.6.2.1">~</ci><ci id="A3.Ex61.m2.1.2.3.6.2.2.cmml" xref="A3.Ex61.m2.1.2.3.6.2.2">𝜽</ci></apply><ci id="A3.Ex61.m2.1.2.3.6.3.cmml" xref="A3.Ex61.m2.1.2.3.6.3">𝑇</ci></apply><apply id="A3.Ex61.m2.1.2.3.7.cmml" xref="A3.Ex61.m2.1.2.3.7"><csymbol cd="ambiguous" id="A3.Ex61.m2.1.2.3.7.1.cmml" xref="A3.Ex61.m2.1.2.3.7">superscript</csymbol><ci id="A3.Ex61.m2.1.2.3.7.2.cmml" xref="A3.Ex61.m2.1.2.3.7.2">Γ</ci><apply id="A3.Ex61.m2.1.2.3.7.3.cmml" xref="A3.Ex61.m2.1.2.3.7.3"><minus id="A3.Ex61.m2.1.2.3.7.3.1.cmml" xref="A3.Ex61.m2.1.2.3.7.3"></minus><cn id="A3.Ex61.m2.1.2.3.7.3.2.cmml" type="integer" xref="A3.Ex61.m2.1.2.3.7.3.2">1</cn></apply></apply><apply id="A3.Ex61.m2.1.2.3.8.cmml" xref="A3.Ex61.m2.1.2.3.8"><ci id="A3.Ex61.m2.1.2.3.8.1.cmml" xref="A3.Ex61.m2.1.2.3.8.1">~</ci><ci id="A3.Ex61.m2.1.2.3.8.2.cmml" xref="A3.Ex61.m2.1.2.3.8.2">𝜽</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex61.m2.1c">\displaystyle-k_{\rm c}\bm{e}^{T}\bm{e}-\kappa\sigma^{*}\lambda_{\min}(\Gamma)% \tilde{\bm{\theta}}^{T}\Gamma^{-1}\tilde{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex61.m2.1d">- italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_e - italic_κ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Γ ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Γ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex62"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\leq" class="ltx_Math" display="inline" id="A3.Ex62.m1.1"><semantics id="A3.Ex62.m1.1a"><mo id="A3.Ex62.m1.1.1" xref="A3.Ex62.m1.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="A3.Ex62.m1.1b"><leq id="A3.Ex62.m1.1.1.cmml" xref="A3.Ex62.m1.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex62.m1.1c">\displaystyle\leq</annotation><annotation encoding="application/x-llamapun" id="A3.Ex62.m1.1d">≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-k_{\rm e}V,t\geq T_{\rm e}" class="ltx_Math" display="inline" id="A3.Ex62.m2.2"><semantics id="A3.Ex62.m2.2a"><mrow id="A3.Ex62.m2.2.2" xref="A3.Ex62.m2.2.2.cmml"><mrow id="A3.Ex62.m2.2.2.1.1" xref="A3.Ex62.m2.2.2.1.2.cmml"><mrow id="A3.Ex62.m2.2.2.1.1.1" xref="A3.Ex62.m2.2.2.1.1.1.cmml"><mo id="A3.Ex62.m2.2.2.1.1.1a" xref="A3.Ex62.m2.2.2.1.1.1.cmml">−</mo><mrow id="A3.Ex62.m2.2.2.1.1.1.2" xref="A3.Ex62.m2.2.2.1.1.1.2.cmml"><msub id="A3.Ex62.m2.2.2.1.1.1.2.2" xref="A3.Ex62.m2.2.2.1.1.1.2.2.cmml"><mi id="A3.Ex62.m2.2.2.1.1.1.2.2.2" xref="A3.Ex62.m2.2.2.1.1.1.2.2.2.cmml">k</mi><mi id="A3.Ex62.m2.2.2.1.1.1.2.2.3" mathvariant="normal" xref="A3.Ex62.m2.2.2.1.1.1.2.2.3.cmml">e</mi></msub><mo id="A3.Ex62.m2.2.2.1.1.1.2.1" xref="A3.Ex62.m2.2.2.1.1.1.2.1.cmml">⁢</mo><mi id="A3.Ex62.m2.2.2.1.1.1.2.3" xref="A3.Ex62.m2.2.2.1.1.1.2.3.cmml">V</mi></mrow></mrow><mo id="A3.Ex62.m2.2.2.1.1.2" xref="A3.Ex62.m2.2.2.1.2.cmml">,</mo><mi id="A3.Ex62.m2.1.1" xref="A3.Ex62.m2.1.1.cmml">t</mi></mrow><mo id="A3.Ex62.m2.2.2.2" xref="A3.Ex62.m2.2.2.2.cmml">≥</mo><msub id="A3.Ex62.m2.2.2.3" xref="A3.Ex62.m2.2.2.3.cmml"><mi id="A3.Ex62.m2.2.2.3.2" xref="A3.Ex62.m2.2.2.3.2.cmml">T</mi><mi id="A3.Ex62.m2.2.2.3.3" mathvariant="normal" xref="A3.Ex62.m2.2.2.3.3.cmml">e</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex62.m2.2b"><apply id="A3.Ex62.m2.2.2.cmml" xref="A3.Ex62.m2.2.2"><geq id="A3.Ex62.m2.2.2.2.cmml" xref="A3.Ex62.m2.2.2.2"></geq><list id="A3.Ex62.m2.2.2.1.2.cmml" xref="A3.Ex62.m2.2.2.1.1"><apply id="A3.Ex62.m2.2.2.1.1.1.cmml" xref="A3.Ex62.m2.2.2.1.1.1"><minus id="A3.Ex62.m2.2.2.1.1.1.1.cmml" xref="A3.Ex62.m2.2.2.1.1.1"></minus><apply id="A3.Ex62.m2.2.2.1.1.1.2.cmml" xref="A3.Ex62.m2.2.2.1.1.1.2"><times id="A3.Ex62.m2.2.2.1.1.1.2.1.cmml" xref="A3.Ex62.m2.2.2.1.1.1.2.1"></times><apply id="A3.Ex62.m2.2.2.1.1.1.2.2.cmml" xref="A3.Ex62.m2.2.2.1.1.1.2.2"><csymbol cd="ambiguous" id="A3.Ex62.m2.2.2.1.1.1.2.2.1.cmml" xref="A3.Ex62.m2.2.2.1.1.1.2.2">subscript</csymbol><ci id="A3.Ex62.m2.2.2.1.1.1.2.2.2.cmml" xref="A3.Ex62.m2.2.2.1.1.1.2.2.2">𝑘</ci><ci id="A3.Ex62.m2.2.2.1.1.1.2.2.3.cmml" xref="A3.Ex62.m2.2.2.1.1.1.2.2.3">e</ci></apply><ci id="A3.Ex62.m2.2.2.1.1.1.2.3.cmml" xref="A3.Ex62.m2.2.2.1.1.1.2.3">𝑉</ci></apply></apply><ci id="A3.Ex62.m2.1.1.cmml" xref="A3.Ex62.m2.1.1">𝑡</ci></list><apply id="A3.Ex62.m2.2.2.3.cmml" xref="A3.Ex62.m2.2.2.3"><csymbol cd="ambiguous" id="A3.Ex62.m2.2.2.3.1.cmml" xref="A3.Ex62.m2.2.2.3">subscript</csymbol><ci id="A3.Ex62.m2.2.2.3.2.cmml" xref="A3.Ex62.m2.2.2.3.2">𝑇</ci><ci id="A3.Ex62.m2.2.2.3.3.cmml" xref="A3.Ex62.m2.2.2.3.3">e</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex62.m2.2c">\displaystyle-k_{\rm e}V,t\geq T_{\rm e}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex62.m2.2d">- italic_k start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT italic_V , italic_t ≥ italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.3.p3.25">in which <math alttext="k_{\rm e}" class="ltx_Math" display="inline" id="A3.3.p3.3.m1.1"><semantics id="A3.3.p3.3.m1.1a"><msub id="A3.3.p3.3.m1.1.1" xref="A3.3.p3.3.m1.1.1.cmml"><mi id="A3.3.p3.3.m1.1.1.2" xref="A3.3.p3.3.m1.1.1.2.cmml">k</mi><mi id="A3.3.p3.3.m1.1.1.3" mathvariant="normal" xref="A3.3.p3.3.m1.1.1.3.cmml">e</mi></msub><annotation-xml encoding="MathML-Content" id="A3.3.p3.3.m1.1b"><apply id="A3.3.p3.3.m1.1.1.cmml" xref="A3.3.p3.3.m1.1.1"><csymbol cd="ambiguous" id="A3.3.p3.3.m1.1.1.1.cmml" xref="A3.3.p3.3.m1.1.1">subscript</csymbol><ci id="A3.3.p3.3.m1.1.1.2.cmml" xref="A3.3.p3.3.m1.1.1.2">𝑘</ci><ci id="A3.3.p3.3.m1.1.1.3.cmml" xref="A3.3.p3.3.m1.1.1.3">e</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.3.p3.3.m1.1c">k_{\rm e}</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.3.m1.1d">italic_k start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="A3.3.p3.4.m2.1"><semantics id="A3.3.p3.4.m2.1a"><mo id="A3.3.p3.4.m2.1.1" xref="A3.3.p3.4.m2.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="A3.3.p3.4.m2.1b"><csymbol cd="latexml" id="A3.3.p3.4.m2.1.1.cmml" xref="A3.3.p3.4.m2.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A3.3.p3.4.m2.1c">:=</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.4.m2.1d">:=</annotation></semantics></math> <math alttext="\min\{k_{\rm c},2\kappa\sigma^{*}\lambda_{\min}(\Gamma)/(1+p)\}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A3.3.p3.5.m3.4"><semantics id="A3.3.p3.5.m3.4a"><mrow id="A3.3.p3.5.m3.4.4" xref="A3.3.p3.5.m3.4.4.cmml"><mrow id="A3.3.p3.5.m3.4.4.2.2" xref="A3.3.p3.5.m3.4.4.2.3.cmml"><mi id="A3.3.p3.5.m3.2.2" xref="A3.3.p3.5.m3.2.2.cmml">min</mi><mo id="A3.3.p3.5.m3.4.4.2.2a" xref="A3.3.p3.5.m3.4.4.2.3.cmml">⁡</mo><mrow id="A3.3.p3.5.m3.4.4.2.2.2" xref="A3.3.p3.5.m3.4.4.2.3.cmml"><mo id="A3.3.p3.5.m3.4.4.2.2.2.3" stretchy="false" xref="A3.3.p3.5.m3.4.4.2.3.cmml">{</mo><msub id="A3.3.p3.5.m3.3.3.1.1.1.1" xref="A3.3.p3.5.m3.3.3.1.1.1.1.cmml"><mi id="A3.3.p3.5.m3.3.3.1.1.1.1.2" xref="A3.3.p3.5.m3.3.3.1.1.1.1.2.cmml">k</mi><mi id="A3.3.p3.5.m3.3.3.1.1.1.1.3" mathvariant="normal" xref="A3.3.p3.5.m3.3.3.1.1.1.1.3.cmml">c</mi></msub><mo id="A3.3.p3.5.m3.4.4.2.2.2.4" xref="A3.3.p3.5.m3.4.4.2.3.cmml">,</mo><mrow id="A3.3.p3.5.m3.4.4.2.2.2.2" xref="A3.3.p3.5.m3.4.4.2.2.2.2.cmml"><mrow id="A3.3.p3.5.m3.4.4.2.2.2.2.3" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.cmml"><mn id="A3.3.p3.5.m3.4.4.2.2.2.2.3.2" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.2.cmml">2</mn><mo id="A3.3.p3.5.m3.4.4.2.2.2.2.3.1" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.1.cmml">⁢</mo><mi id="A3.3.p3.5.m3.4.4.2.2.2.2.3.3" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.3.cmml">κ</mi><mo id="A3.3.p3.5.m3.4.4.2.2.2.2.3.1a" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.1.cmml">⁢</mo><msup id="A3.3.p3.5.m3.4.4.2.2.2.2.3.4" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.4.cmml"><mi id="A3.3.p3.5.m3.4.4.2.2.2.2.3.4.2" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.4.2.cmml">σ</mi><mo id="A3.3.p3.5.m3.4.4.2.2.2.2.3.4.3" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.4.3.cmml">∗</mo></msup><mo id="A3.3.p3.5.m3.4.4.2.2.2.2.3.1b" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.1.cmml">⁢</mo><msub id="A3.3.p3.5.m3.4.4.2.2.2.2.3.5" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.5.cmml"><mi id="A3.3.p3.5.m3.4.4.2.2.2.2.3.5.2" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.5.2.cmml">λ</mi><mi id="A3.3.p3.5.m3.4.4.2.2.2.2.3.5.3" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.5.3.cmml">min</mi></msub><mo id="A3.3.p3.5.m3.4.4.2.2.2.2.3.1c" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.1.cmml">⁢</mo><mrow id="A3.3.p3.5.m3.4.4.2.2.2.2.3.6.2" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.cmml"><mo id="A3.3.p3.5.m3.4.4.2.2.2.2.3.6.2.1" stretchy="false" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.cmml">(</mo><mi id="A3.3.p3.5.m3.1.1" mathvariant="normal" xref="A3.3.p3.5.m3.1.1.cmml">Γ</mi><mo id="A3.3.p3.5.m3.4.4.2.2.2.2.3.6.2.2" stretchy="false" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="A3.3.p3.5.m3.4.4.2.2.2.2.2" xref="A3.3.p3.5.m3.4.4.2.2.2.2.2.cmml">/</mo><mrow id="A3.3.p3.5.m3.4.4.2.2.2.2.1.1" xref="A3.3.p3.5.m3.4.4.2.2.2.2.1.1.1.cmml"><mo id="A3.3.p3.5.m3.4.4.2.2.2.2.1.1.2" stretchy="false" xref="A3.3.p3.5.m3.4.4.2.2.2.2.1.1.1.cmml">(</mo><mrow id="A3.3.p3.5.m3.4.4.2.2.2.2.1.1.1" xref="A3.3.p3.5.m3.4.4.2.2.2.2.1.1.1.cmml"><mn id="A3.3.p3.5.m3.4.4.2.2.2.2.1.1.1.2" xref="A3.3.p3.5.m3.4.4.2.2.2.2.1.1.1.2.cmml">1</mn><mo id="A3.3.p3.5.m3.4.4.2.2.2.2.1.1.1.1" xref="A3.3.p3.5.m3.4.4.2.2.2.2.1.1.1.1.cmml">+</mo><mi id="A3.3.p3.5.m3.4.4.2.2.2.2.1.1.1.3" xref="A3.3.p3.5.m3.4.4.2.2.2.2.1.1.1.3.cmml">p</mi></mrow><mo id="A3.3.p3.5.m3.4.4.2.2.2.2.1.1.3" stretchy="false" xref="A3.3.p3.5.m3.4.4.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="A3.3.p3.5.m3.4.4.2.2.2.5" stretchy="false" xref="A3.3.p3.5.m3.4.4.2.3.cmml">}</mo></mrow></mrow><mo id="A3.3.p3.5.m3.4.4.3" xref="A3.3.p3.5.m3.4.4.3.cmml">∈</mo><msup id="A3.3.p3.5.m3.4.4.4" xref="A3.3.p3.5.m3.4.4.4.cmml"><mi id="A3.3.p3.5.m3.4.4.4.2" xref="A3.3.p3.5.m3.4.4.4.2.cmml">ℝ</mi><mo id="A3.3.p3.5.m3.4.4.4.3" xref="A3.3.p3.5.m3.4.4.4.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.3.p3.5.m3.4b"><apply id="A3.3.p3.5.m3.4.4.cmml" xref="A3.3.p3.5.m3.4.4"><in id="A3.3.p3.5.m3.4.4.3.cmml" xref="A3.3.p3.5.m3.4.4.3"></in><apply id="A3.3.p3.5.m3.4.4.2.3.cmml" xref="A3.3.p3.5.m3.4.4.2.2"><min id="A3.3.p3.5.m3.2.2.cmml" xref="A3.3.p3.5.m3.2.2"></min><apply id="A3.3.p3.5.m3.3.3.1.1.1.1.cmml" xref="A3.3.p3.5.m3.3.3.1.1.1.1"><csymbol cd="ambiguous" id="A3.3.p3.5.m3.3.3.1.1.1.1.1.cmml" xref="A3.3.p3.5.m3.3.3.1.1.1.1">subscript</csymbol><ci id="A3.3.p3.5.m3.3.3.1.1.1.1.2.cmml" xref="A3.3.p3.5.m3.3.3.1.1.1.1.2">𝑘</ci><ci id="A3.3.p3.5.m3.3.3.1.1.1.1.3.cmml" xref="A3.3.p3.5.m3.3.3.1.1.1.1.3">c</ci></apply><apply id="A3.3.p3.5.m3.4.4.2.2.2.2.cmml" xref="A3.3.p3.5.m3.4.4.2.2.2.2"><divide id="A3.3.p3.5.m3.4.4.2.2.2.2.2.cmml" xref="A3.3.p3.5.m3.4.4.2.2.2.2.2"></divide><apply id="A3.3.p3.5.m3.4.4.2.2.2.2.3.cmml" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3"><times id="A3.3.p3.5.m3.4.4.2.2.2.2.3.1.cmml" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.1"></times><cn id="A3.3.p3.5.m3.4.4.2.2.2.2.3.2.cmml" type="integer" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.2">2</cn><ci id="A3.3.p3.5.m3.4.4.2.2.2.2.3.3.cmml" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.3">𝜅</ci><apply id="A3.3.p3.5.m3.4.4.2.2.2.2.3.4.cmml" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.4"><csymbol cd="ambiguous" id="A3.3.p3.5.m3.4.4.2.2.2.2.3.4.1.cmml" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.4">superscript</csymbol><ci id="A3.3.p3.5.m3.4.4.2.2.2.2.3.4.2.cmml" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.4.2">𝜎</ci><times id="A3.3.p3.5.m3.4.4.2.2.2.2.3.4.3.cmml" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.4.3"></times></apply><apply id="A3.3.p3.5.m3.4.4.2.2.2.2.3.5.cmml" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.5"><csymbol cd="ambiguous" id="A3.3.p3.5.m3.4.4.2.2.2.2.3.5.1.cmml" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.5">subscript</csymbol><ci id="A3.3.p3.5.m3.4.4.2.2.2.2.3.5.2.cmml" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.5.2">𝜆</ci><min id="A3.3.p3.5.m3.4.4.2.2.2.2.3.5.3.cmml" xref="A3.3.p3.5.m3.4.4.2.2.2.2.3.5.3"></min></apply><ci id="A3.3.p3.5.m3.1.1.cmml" xref="A3.3.p3.5.m3.1.1">Γ</ci></apply><apply id="A3.3.p3.5.m3.4.4.2.2.2.2.1.1.1.cmml" xref="A3.3.p3.5.m3.4.4.2.2.2.2.1.1"><plus id="A3.3.p3.5.m3.4.4.2.2.2.2.1.1.1.1.cmml" xref="A3.3.p3.5.m3.4.4.2.2.2.2.1.1.1.1"></plus><cn id="A3.3.p3.5.m3.4.4.2.2.2.2.1.1.1.2.cmml" type="integer" xref="A3.3.p3.5.m3.4.4.2.2.2.2.1.1.1.2">1</cn><ci id="A3.3.p3.5.m3.4.4.2.2.2.2.1.1.1.3.cmml" xref="A3.3.p3.5.m3.4.4.2.2.2.2.1.1.1.3">𝑝</ci></apply></apply></apply><apply id="A3.3.p3.5.m3.4.4.4.cmml" xref="A3.3.p3.5.m3.4.4.4"><csymbol cd="ambiguous" id="A3.3.p3.5.m3.4.4.4.1.cmml" xref="A3.3.p3.5.m3.4.4.4">superscript</csymbol><ci id="A3.3.p3.5.m3.4.4.4.2.cmml" xref="A3.3.p3.5.m3.4.4.4.2">ℝ</ci><plus id="A3.3.p3.5.m3.4.4.4.3.cmml" xref="A3.3.p3.5.m3.4.4.4.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.3.p3.5.m3.4c">\min\{k_{\rm c},2\kappa\sigma^{*}\lambda_{\min}(\Gamma)/(1+p)\}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.5.m3.4d">roman_min { italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT , 2 italic_κ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Γ ) / ( 1 + italic_p ) } ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="p" class="ltx_Math" display="inline" id="A3.3.p3.6.m4.1"><semantics id="A3.3.p3.6.m4.1a"><mi id="A3.3.p3.6.m4.1.1" xref="A3.3.p3.6.m4.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="A3.3.p3.6.m4.1b"><ci id="A3.3.p3.6.m4.1.1.cmml" xref="A3.3.p3.6.m4.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.3.p3.6.m4.1c">p</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.6.m4.1d">italic_p</annotation></semantics></math> <math alttext="=" class="ltx_Math" display="inline" id="A3.3.p3.7.m5.1"><semantics id="A3.3.p3.7.m5.1a"><mo id="A3.3.p3.7.m5.1.1" xref="A3.3.p3.7.m5.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="A3.3.p3.7.m5.1b"><eq id="A3.3.p3.7.m5.1.1.cmml" xref="A3.3.p3.7.m5.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="A3.3.p3.7.m5.1c">=</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.7.m5.1d">=</annotation></semantics></math> <math alttext="\delta^{2}/" class="ltx_math_unparsed" display="inline" id="A3.3.p3.8.m6.1"><semantics id="A3.3.p3.8.m6.1a"><mrow id="A3.3.p3.8.m6.1b"><msup id="A3.3.p3.8.m6.1.1"><mi id="A3.3.p3.8.m6.1.1.2">δ</mi><mn id="A3.3.p3.8.m6.1.1.3">2</mn></msup><mo id="A3.3.p3.8.m6.1.2">/</mo></mrow><annotation encoding="application/x-tex" id="A3.3.p3.8.m6.1c">\delta^{2}/</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.8.m6.1d">italic_δ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT /</annotation></semantics></math> <math alttext="(2k_{\rm c}\kappa\sigma^{*})" class="ltx_Math" display="inline" id="A3.3.p3.9.m7.1"><semantics id="A3.3.p3.9.m7.1a"><mrow id="A3.3.p3.9.m7.1.1.1" xref="A3.3.p3.9.m7.1.1.1.1.cmml"><mo id="A3.3.p3.9.m7.1.1.1.2" stretchy="false" xref="A3.3.p3.9.m7.1.1.1.1.cmml">(</mo><mrow id="A3.3.p3.9.m7.1.1.1.1" xref="A3.3.p3.9.m7.1.1.1.1.cmml"><mn id="A3.3.p3.9.m7.1.1.1.1.2" xref="A3.3.p3.9.m7.1.1.1.1.2.cmml">2</mn><mo id="A3.3.p3.9.m7.1.1.1.1.1" xref="A3.3.p3.9.m7.1.1.1.1.1.cmml">⁢</mo><msub id="A3.3.p3.9.m7.1.1.1.1.3" xref="A3.3.p3.9.m7.1.1.1.1.3.cmml"><mi id="A3.3.p3.9.m7.1.1.1.1.3.2" xref="A3.3.p3.9.m7.1.1.1.1.3.2.cmml">k</mi><mi id="A3.3.p3.9.m7.1.1.1.1.3.3" mathvariant="normal" xref="A3.3.p3.9.m7.1.1.1.1.3.3.cmml">c</mi></msub><mo id="A3.3.p3.9.m7.1.1.1.1.1a" xref="A3.3.p3.9.m7.1.1.1.1.1.cmml">⁢</mo><mi id="A3.3.p3.9.m7.1.1.1.1.4" xref="A3.3.p3.9.m7.1.1.1.1.4.cmml">κ</mi><mo id="A3.3.p3.9.m7.1.1.1.1.1b" xref="A3.3.p3.9.m7.1.1.1.1.1.cmml">⁢</mo><msup id="A3.3.p3.9.m7.1.1.1.1.5" xref="A3.3.p3.9.m7.1.1.1.1.5.cmml"><mi id="A3.3.p3.9.m7.1.1.1.1.5.2" xref="A3.3.p3.9.m7.1.1.1.1.5.2.cmml">σ</mi><mo id="A3.3.p3.9.m7.1.1.1.1.5.3" xref="A3.3.p3.9.m7.1.1.1.1.5.3.cmml">∗</mo></msup></mrow><mo id="A3.3.p3.9.m7.1.1.1.3" stretchy="false" xref="A3.3.p3.9.m7.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.3.p3.9.m7.1b"><apply id="A3.3.p3.9.m7.1.1.1.1.cmml" xref="A3.3.p3.9.m7.1.1.1"><times id="A3.3.p3.9.m7.1.1.1.1.1.cmml" xref="A3.3.p3.9.m7.1.1.1.1.1"></times><cn id="A3.3.p3.9.m7.1.1.1.1.2.cmml" type="integer" xref="A3.3.p3.9.m7.1.1.1.1.2">2</cn><apply id="A3.3.p3.9.m7.1.1.1.1.3.cmml" xref="A3.3.p3.9.m7.1.1.1.1.3"><csymbol cd="ambiguous" id="A3.3.p3.9.m7.1.1.1.1.3.1.cmml" xref="A3.3.p3.9.m7.1.1.1.1.3">subscript</csymbol><ci id="A3.3.p3.9.m7.1.1.1.1.3.2.cmml" xref="A3.3.p3.9.m7.1.1.1.1.3.2">𝑘</ci><ci id="A3.3.p3.9.m7.1.1.1.1.3.3.cmml" xref="A3.3.p3.9.m7.1.1.1.1.3.3">c</ci></apply><ci id="A3.3.p3.9.m7.1.1.1.1.4.cmml" xref="A3.3.p3.9.m7.1.1.1.1.4">𝜅</ci><apply id="A3.3.p3.9.m7.1.1.1.1.5.cmml" xref="A3.3.p3.9.m7.1.1.1.1.5"><csymbol cd="ambiguous" id="A3.3.p3.9.m7.1.1.1.1.5.1.cmml" xref="A3.3.p3.9.m7.1.1.1.1.5">superscript</csymbol><ci id="A3.3.p3.9.m7.1.1.1.1.5.2.cmml" xref="A3.3.p3.9.m7.1.1.1.1.5.2">𝜎</ci><times id="A3.3.p3.9.m7.1.1.1.1.5.3.cmml" xref="A3.3.p3.9.m7.1.1.1.1.5.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.3.p3.9.m7.1c">(2k_{\rm c}\kappa\sigma^{*})</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.9.m7.1d">( 2 italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT italic_κ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math> <math alttext="\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A3.3.p3.10.m8.1"><semantics id="A3.3.p3.10.m8.1a"><mrow id="A3.3.p3.10.m8.1.1" xref="A3.3.p3.10.m8.1.1.cmml"><mi id="A3.3.p3.10.m8.1.1.2" xref="A3.3.p3.10.m8.1.1.2.cmml"></mi><mo id="A3.3.p3.10.m8.1.1.1" xref="A3.3.p3.10.m8.1.1.1.cmml">∈</mo><msup id="A3.3.p3.10.m8.1.1.3" xref="A3.3.p3.10.m8.1.1.3.cmml"><mi id="A3.3.p3.10.m8.1.1.3.2" xref="A3.3.p3.10.m8.1.1.3.2.cmml">ℝ</mi><mo id="A3.3.p3.10.m8.1.1.3.3" xref="A3.3.p3.10.m8.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.3.p3.10.m8.1b"><apply id="A3.3.p3.10.m8.1.1.cmml" xref="A3.3.p3.10.m8.1.1"><in id="A3.3.p3.10.m8.1.1.1.cmml" xref="A3.3.p3.10.m8.1.1.1"></in><csymbol cd="latexml" id="A3.3.p3.10.m8.1.1.2.cmml" xref="A3.3.p3.10.m8.1.1.2">absent</csymbol><apply id="A3.3.p3.10.m8.1.1.3.cmml" xref="A3.3.p3.10.m8.1.1.3"><csymbol cd="ambiguous" id="A3.3.p3.10.m8.1.1.3.1.cmml" xref="A3.3.p3.10.m8.1.1.3">superscript</csymbol><ci id="A3.3.p3.10.m8.1.1.3.2.cmml" xref="A3.3.p3.10.m8.1.1.3.2">ℝ</ci><plus id="A3.3.p3.10.m8.1.1.3.3.cmml" xref="A3.3.p3.10.m8.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.3.p3.10.m8.1c">\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.10.m8.1d">∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\delta\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A3.3.p3.11.m9.1"><semantics id="A3.3.p3.11.m9.1a"><mrow id="A3.3.p3.11.m9.1.1" xref="A3.3.p3.11.m9.1.1.cmml"><mi id="A3.3.p3.11.m9.1.1.2" xref="A3.3.p3.11.m9.1.1.2.cmml">δ</mi><mo id="A3.3.p3.11.m9.1.1.1" xref="A3.3.p3.11.m9.1.1.1.cmml">∈</mo><msup id="A3.3.p3.11.m9.1.1.3" xref="A3.3.p3.11.m9.1.1.3.cmml"><mi id="A3.3.p3.11.m9.1.1.3.2" xref="A3.3.p3.11.m9.1.1.3.2.cmml">ℝ</mi><mo id="A3.3.p3.11.m9.1.1.3.3" xref="A3.3.p3.11.m9.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.3.p3.11.m9.1b"><apply id="A3.3.p3.11.m9.1.1.cmml" xref="A3.3.p3.11.m9.1.1"><in id="A3.3.p3.11.m9.1.1.1.cmml" xref="A3.3.p3.11.m9.1.1.1"></in><ci id="A3.3.p3.11.m9.1.1.2.cmml" xref="A3.3.p3.11.m9.1.1.2">𝛿</ci><apply id="A3.3.p3.11.m9.1.1.3.cmml" xref="A3.3.p3.11.m9.1.1.3"><csymbol cd="ambiguous" id="A3.3.p3.11.m9.1.1.3.1.cmml" xref="A3.3.p3.11.m9.1.1.3">superscript</csymbol><ci id="A3.3.p3.11.m9.1.1.3.2.cmml" xref="A3.3.p3.11.m9.1.1.3.2">ℝ</ci><plus id="A3.3.p3.11.m9.1.1.3.3.cmml" xref="A3.3.p3.11.m9.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.3.p3.11.m9.1c">\delta\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.11.m9.1d">italic_δ ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> satisfies <math alttext="\|\Phi-\Phi_{\rm f}\|" class="ltx_Math" display="inline" id="A3.3.p3.12.m10.1"><semantics id="A3.3.p3.12.m10.1a"><mrow id="A3.3.p3.12.m10.1.1.1" xref="A3.3.p3.12.m10.1.1.2.cmml"><mo id="A3.3.p3.12.m10.1.1.1.2" stretchy="false" xref="A3.3.p3.12.m10.1.1.2.1.cmml">‖</mo><mrow id="A3.3.p3.12.m10.1.1.1.1" xref="A3.3.p3.12.m10.1.1.1.1.cmml"><mi id="A3.3.p3.12.m10.1.1.1.1.2" mathvariant="normal" xref="A3.3.p3.12.m10.1.1.1.1.2.cmml">Φ</mi><mo id="A3.3.p3.12.m10.1.1.1.1.1" xref="A3.3.p3.12.m10.1.1.1.1.1.cmml">−</mo><msub id="A3.3.p3.12.m10.1.1.1.1.3" xref="A3.3.p3.12.m10.1.1.1.1.3.cmml"><mi id="A3.3.p3.12.m10.1.1.1.1.3.2" mathvariant="normal" xref="A3.3.p3.12.m10.1.1.1.1.3.2.cmml">Φ</mi><mi id="A3.3.p3.12.m10.1.1.1.1.3.3" mathvariant="normal" xref="A3.3.p3.12.m10.1.1.1.1.3.3.cmml">f</mi></msub></mrow><mo id="A3.3.p3.12.m10.1.1.1.3" stretchy="false" xref="A3.3.p3.12.m10.1.1.2.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.3.p3.12.m10.1b"><apply id="A3.3.p3.12.m10.1.1.2.cmml" xref="A3.3.p3.12.m10.1.1.1"><csymbol cd="latexml" id="A3.3.p3.12.m10.1.1.2.1.cmml" xref="A3.3.p3.12.m10.1.1.1.2">norm</csymbol><apply id="A3.3.p3.12.m10.1.1.1.1.cmml" xref="A3.3.p3.12.m10.1.1.1.1"><minus id="A3.3.p3.12.m10.1.1.1.1.1.cmml" xref="A3.3.p3.12.m10.1.1.1.1.1"></minus><ci id="A3.3.p3.12.m10.1.1.1.1.2.cmml" xref="A3.3.p3.12.m10.1.1.1.1.2">Φ</ci><apply id="A3.3.p3.12.m10.1.1.1.1.3.cmml" xref="A3.3.p3.12.m10.1.1.1.1.3"><csymbol cd="ambiguous" id="A3.3.p3.12.m10.1.1.1.1.3.1.cmml" xref="A3.3.p3.12.m10.1.1.1.1.3">subscript</csymbol><ci id="A3.3.p3.12.m10.1.1.1.1.3.2.cmml" xref="A3.3.p3.12.m10.1.1.1.1.3.2">Φ</ci><ci id="A3.3.p3.12.m10.1.1.1.1.3.3.cmml" xref="A3.3.p3.12.m10.1.1.1.1.3.3">f</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.3.p3.12.m10.1c">\|\Phi-\Phi_{\rm f}\|</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.12.m10.1d">∥ roman_Φ - roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ∥</annotation></semantics></math> <math alttext="\leq" class="ltx_Math" display="inline" id="A3.3.p3.13.m11.1"><semantics id="A3.3.p3.13.m11.1a"><mo id="A3.3.p3.13.m11.1.1" xref="A3.3.p3.13.m11.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="A3.3.p3.13.m11.1b"><leq id="A3.3.p3.13.m11.1.1.cmml" xref="A3.3.p3.13.m11.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="A3.3.p3.13.m11.1c">\leq</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.13.m11.1d">≤</annotation></semantics></math> <math alttext="\delta" class="ltx_Math" display="inline" id="A3.3.p3.14.m12.1"><semantics id="A3.3.p3.14.m12.1a"><mi id="A3.3.p3.14.m12.1.1" xref="A3.3.p3.14.m12.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="A3.3.p3.14.m12.1b"><ci id="A3.3.p3.14.m12.1.1.cmml" xref="A3.3.p3.14.m12.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.3.p3.14.m12.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.14.m12.1d">italic_δ</annotation></semantics></math>, <math alttext="\forall t\geq 0" class="ltx_Math" display="inline" id="A3.3.p3.15.m13.1"><semantics id="A3.3.p3.15.m13.1a"><mrow id="A3.3.p3.15.m13.1.1" xref="A3.3.p3.15.m13.1.1.cmml"><mrow id="A3.3.p3.15.m13.1.1.2" xref="A3.3.p3.15.m13.1.1.2.cmml"><mo id="A3.3.p3.15.m13.1.1.2.1" rspace="0.167em" xref="A3.3.p3.15.m13.1.1.2.1.cmml">∀</mo><mi id="A3.3.p3.15.m13.1.1.2.2" xref="A3.3.p3.15.m13.1.1.2.2.cmml">t</mi></mrow><mo id="A3.3.p3.15.m13.1.1.1" xref="A3.3.p3.15.m13.1.1.1.cmml">≥</mo><mn id="A3.3.p3.15.m13.1.1.3" xref="A3.3.p3.15.m13.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.3.p3.15.m13.1b"><apply id="A3.3.p3.15.m13.1.1.cmml" xref="A3.3.p3.15.m13.1.1"><geq id="A3.3.p3.15.m13.1.1.1.cmml" xref="A3.3.p3.15.m13.1.1.1"></geq><apply id="A3.3.p3.15.m13.1.1.2.cmml" xref="A3.3.p3.15.m13.1.1.2"><csymbol cd="latexml" id="A3.3.p3.15.m13.1.1.2.1.cmml" xref="A3.3.p3.15.m13.1.1.2.1">for-all</csymbol><ci id="A3.3.p3.15.m13.1.1.2.2.cmml" xref="A3.3.p3.15.m13.1.1.2.2">𝑡</ci></apply><cn id="A3.3.p3.15.m13.1.1.3.cmml" type="integer" xref="A3.3.p3.15.m13.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.3.p3.15.m13.1c">\forall t\geq 0</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.15.m13.1d">∀ italic_t ≥ 0</annotation></semantics></math> and <math alttext="\sigma^{*}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A3.3.p3.16.m14.1"><semantics id="A3.3.p3.16.m14.1a"><mrow id="A3.3.p3.16.m14.1.1" xref="A3.3.p3.16.m14.1.1.cmml"><msup id="A3.3.p3.16.m14.1.1.2" xref="A3.3.p3.16.m14.1.1.2.cmml"><mi id="A3.3.p3.16.m14.1.1.2.2" xref="A3.3.p3.16.m14.1.1.2.2.cmml">σ</mi><mo id="A3.3.p3.16.m14.1.1.2.3" xref="A3.3.p3.16.m14.1.1.2.3.cmml">∗</mo></msup><mo id="A3.3.p3.16.m14.1.1.1" xref="A3.3.p3.16.m14.1.1.1.cmml">∈</mo><msup id="A3.3.p3.16.m14.1.1.3" xref="A3.3.p3.16.m14.1.1.3.cmml"><mi id="A3.3.p3.16.m14.1.1.3.2" xref="A3.3.p3.16.m14.1.1.3.2.cmml">ℝ</mi><mo id="A3.3.p3.16.m14.1.1.3.3" xref="A3.3.p3.16.m14.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.3.p3.16.m14.1b"><apply id="A3.3.p3.16.m14.1.1.cmml" xref="A3.3.p3.16.m14.1.1"><in id="A3.3.p3.16.m14.1.1.1.cmml" xref="A3.3.p3.16.m14.1.1.1"></in><apply id="A3.3.p3.16.m14.1.1.2.cmml" xref="A3.3.p3.16.m14.1.1.2"><csymbol cd="ambiguous" id="A3.3.p3.16.m14.1.1.2.1.cmml" xref="A3.3.p3.16.m14.1.1.2">superscript</csymbol><ci id="A3.3.p3.16.m14.1.1.2.2.cmml" xref="A3.3.p3.16.m14.1.1.2.2">𝜎</ci><times id="A3.3.p3.16.m14.1.1.2.3.cmml" xref="A3.3.p3.16.m14.1.1.2.3"></times></apply><apply id="A3.3.p3.16.m14.1.1.3.cmml" xref="A3.3.p3.16.m14.1.1.3"><csymbol cd="ambiguous" id="A3.3.p3.16.m14.1.1.3.1.cmml" xref="A3.3.p3.16.m14.1.1.3">superscript</csymbol><ci id="A3.3.p3.16.m14.1.1.3.2.cmml" xref="A3.3.p3.16.m14.1.1.3.2">ℝ</ci><plus id="A3.3.p3.16.m14.1.1.3.3.cmml" xref="A3.3.p3.16.m14.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.3.p3.16.m14.1c">\sigma^{*}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.16.m14.1d">italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="\sigma^{*}\leq\sigma" class="ltx_Math" display="inline" id="A3.3.p3.17.m15.1"><semantics id="A3.3.p3.17.m15.1a"><mrow id="A3.3.p3.17.m15.1.1" xref="A3.3.p3.17.m15.1.1.cmml"><msup id="A3.3.p3.17.m15.1.1.2" xref="A3.3.p3.17.m15.1.1.2.cmml"><mi id="A3.3.p3.17.m15.1.1.2.2" xref="A3.3.p3.17.m15.1.1.2.2.cmml">σ</mi><mo id="A3.3.p3.17.m15.1.1.2.3" xref="A3.3.p3.17.m15.1.1.2.3.cmml">∗</mo></msup><mo id="A3.3.p3.17.m15.1.1.1" xref="A3.3.p3.17.m15.1.1.1.cmml">≤</mo><mi id="A3.3.p3.17.m15.1.1.3" xref="A3.3.p3.17.m15.1.1.3.cmml">σ</mi></mrow><annotation-xml encoding="MathML-Content" id="A3.3.p3.17.m15.1b"><apply id="A3.3.p3.17.m15.1.1.cmml" xref="A3.3.p3.17.m15.1.1"><leq id="A3.3.p3.17.m15.1.1.1.cmml" xref="A3.3.p3.17.m15.1.1.1"></leq><apply id="A3.3.p3.17.m15.1.1.2.cmml" xref="A3.3.p3.17.m15.1.1.2"><csymbol cd="ambiguous" id="A3.3.p3.17.m15.1.1.2.1.cmml" xref="A3.3.p3.17.m15.1.1.2">superscript</csymbol><ci id="A3.3.p3.17.m15.1.1.2.2.cmml" xref="A3.3.p3.17.m15.1.1.2.2">𝜎</ci><times id="A3.3.p3.17.m15.1.1.2.3.cmml" xref="A3.3.p3.17.m15.1.1.2.3"></times></apply><ci id="A3.3.p3.17.m15.1.1.3.cmml" xref="A3.3.p3.17.m15.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.3.p3.17.m15.1c">\sigma^{*}\leq\sigma</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.17.m15.1d">italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ≤ italic_σ</annotation></semantics></math> satisfies (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S5.E31" title="In V-C Robustness Results ‣ V Theoretical Guarantees ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">31</span></a>). <span class="ltx_text" id="A3.3.p3.25.8" style="color:#000099;">Thus, the equilibrium point <math alttext="(\bm{e}" class="ltx_math_unparsed" display="inline" id="A3.3.p3.18.1.m1.1"><semantics id="A3.3.p3.18.1.m1.1a"><mrow id="A3.3.p3.18.1.m1.1b"><mo id="A3.3.p3.18.1.m1.1.1" mathcolor="#000099" stretchy="false">(</mo><mi id="A3.3.p3.18.1.m1.1.2" mathcolor="#000099">𝒆</mi></mrow><annotation encoding="application/x-tex" id="A3.3.p3.18.1.m1.1c">(\bm{e}</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.18.1.m1.1d">( bold_italic_e</annotation></semantics></math>, <math alttext="\tilde{\bm{\theta}})=\bm{0}" class="ltx_math_unparsed" display="inline" id="A3.3.p3.19.2.m2.1"><semantics id="A3.3.p3.19.2.m2.1a"><mrow id="A3.3.p3.19.2.m2.1b"><mover accent="true" id="A3.3.p3.19.2.m2.1.1"><mi id="A3.3.p3.19.2.m2.1.1.2" mathcolor="#000099">𝜽</mi><mo id="A3.3.p3.19.2.m2.1.1.1" mathcolor="#000099">~</mo></mover><mo id="A3.3.p3.19.2.m2.1.2" mathcolor="#000099" stretchy="false">)</mo><mo id="A3.3.p3.19.2.m2.1.3" mathcolor="#000099">=</mo><mn id="A3.3.p3.19.2.m2.1.4" mathcolor="#000099">𝟎</mn></mrow><annotation encoding="application/x-tex" id="A3.3.p3.19.2.m2.1c">\tilde{\bm{\theta}})=\bm{0}</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.19.2.m2.1d">over~ start_ARG bold_italic_θ end_ARG ) = bold_0</annotation></semantics></math> of the closed-loop system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S3.E7" title="In III Modular Backstepping Control Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">7</span></a>) with (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>) is exponential stable on <math alttext="t" class="ltx_Math" display="inline" id="A3.3.p3.20.3.m3.1"><semantics id="A3.3.p3.20.3.m3.1a"><mi id="A3.3.p3.20.3.m3.1.1" mathcolor="#000099" xref="A3.3.p3.20.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A3.3.p3.20.3.m3.1b"><ci id="A3.3.p3.20.3.m3.1.1.cmml" xref="A3.3.p3.20.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.3.p3.20.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.20.3.m3.1d">italic_t</annotation></semantics></math> <math alttext="\in" class="ltx_Math" display="inline" id="A3.3.p3.21.4.m4.1"><semantics id="A3.3.p3.21.4.m4.1a"><mo id="A3.3.p3.21.4.m4.1.1" mathcolor="#000099" xref="A3.3.p3.21.4.m4.1.1.cmml">∈</mo><annotation-xml encoding="MathML-Content" id="A3.3.p3.21.4.m4.1b"><in id="A3.3.p3.21.4.m4.1.1.cmml" xref="A3.3.p3.21.4.m4.1.1"></in></annotation-xml><annotation encoding="application/x-tex" id="A3.3.p3.21.4.m4.1c">\in</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.21.4.m4.1d">∈</annotation></semantics></math> <math alttext="[T_{\rm e},\infty)" class="ltx_Math" display="inline" id="A3.3.p3.22.5.m5.2"><semantics id="A3.3.p3.22.5.m5.2a"><mrow id="A3.3.p3.22.5.m5.2.2.1" xref="A3.3.p3.22.5.m5.2.2.2.cmml"><mo id="A3.3.p3.22.5.m5.2.2.1.2" mathcolor="#000099" stretchy="false" xref="A3.3.p3.22.5.m5.2.2.2.cmml">[</mo><msub id="A3.3.p3.22.5.m5.2.2.1.1" xref="A3.3.p3.22.5.m5.2.2.1.1.cmml"><mi id="A3.3.p3.22.5.m5.2.2.1.1.2" mathcolor="#000099" xref="A3.3.p3.22.5.m5.2.2.1.1.2.cmml">T</mi><mi id="A3.3.p3.22.5.m5.2.2.1.1.3" mathcolor="#000099" mathvariant="normal" xref="A3.3.p3.22.5.m5.2.2.1.1.3.cmml">e</mi></msub><mo id="A3.3.p3.22.5.m5.2.2.1.3" mathcolor="#000099" xref="A3.3.p3.22.5.m5.2.2.2.cmml">,</mo><mi id="A3.3.p3.22.5.m5.1.1" mathcolor="#000099" mathvariant="normal" xref="A3.3.p3.22.5.m5.1.1.cmml">∞</mi><mo id="A3.3.p3.22.5.m5.2.2.1.4" mathcolor="#000099" stretchy="false" xref="A3.3.p3.22.5.m5.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.3.p3.22.5.m5.2b"><interval closure="closed-open" id="A3.3.p3.22.5.m5.2.2.2.cmml" xref="A3.3.p3.22.5.m5.2.2.1"><apply id="A3.3.p3.22.5.m5.2.2.1.1.cmml" xref="A3.3.p3.22.5.m5.2.2.1.1"><csymbol cd="ambiguous" id="A3.3.p3.22.5.m5.2.2.1.1.1.cmml" xref="A3.3.p3.22.5.m5.2.2.1.1">subscript</csymbol><ci id="A3.3.p3.22.5.m5.2.2.1.1.2.cmml" xref="A3.3.p3.22.5.m5.2.2.1.1.2">𝑇</ci><ci id="A3.3.p3.22.5.m5.2.2.1.1.3.cmml" xref="A3.3.p3.22.5.m5.2.2.1.1.3">e</ci></apply><infinity id="A3.3.p3.22.5.m5.1.1.cmml" xref="A3.3.p3.22.5.m5.1.1"></infinity></interval></annotation-xml><annotation encoding="application/x-tex" id="A3.3.p3.22.5.m5.2c">[T_{\rm e},\infty)</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.22.5.m5.2d">[ italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT , ∞ )</annotation></semantics></math>, where the tracking error <math alttext="\bm{e}(t)" class="ltx_Math" display="inline" id="A3.3.p3.23.6.m6.1"><semantics id="A3.3.p3.23.6.m6.1a"><mrow id="A3.3.p3.23.6.m6.1.2" xref="A3.3.p3.23.6.m6.1.2.cmml"><mi id="A3.3.p3.23.6.m6.1.2.2" mathcolor="#000099" xref="A3.3.p3.23.6.m6.1.2.2.cmml">𝒆</mi><mo id="A3.3.p3.23.6.m6.1.2.1" xref="A3.3.p3.23.6.m6.1.2.1.cmml">⁢</mo><mrow id="A3.3.p3.23.6.m6.1.2.3.2" xref="A3.3.p3.23.6.m6.1.2.cmml"><mo id="A3.3.p3.23.6.m6.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="A3.3.p3.23.6.m6.1.2.cmml">(</mo><mi id="A3.3.p3.23.6.m6.1.1" mathcolor="#000099" xref="A3.3.p3.23.6.m6.1.1.cmml">t</mi><mo id="A3.3.p3.23.6.m6.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="A3.3.p3.23.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.3.p3.23.6.m6.1b"><apply id="A3.3.p3.23.6.m6.1.2.cmml" xref="A3.3.p3.23.6.m6.1.2"><times id="A3.3.p3.23.6.m6.1.2.1.cmml" xref="A3.3.p3.23.6.m6.1.2.1"></times><ci id="A3.3.p3.23.6.m6.1.2.2.cmml" xref="A3.3.p3.23.6.m6.1.2.2">𝒆</ci><ci id="A3.3.p3.23.6.m6.1.1.cmml" xref="A3.3.p3.23.6.m6.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.3.p3.23.6.m6.1c">\bm{e}(t)</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.23.6.m6.1d">bold_italic_e ( italic_t )</annotation></semantics></math> and the parameter estimation error <math alttext="\tilde{\bm{\theta}}(t)" class="ltx_Math" display="inline" id="A3.3.p3.24.7.m7.1"><semantics id="A3.3.p3.24.7.m7.1a"><mrow id="A3.3.p3.24.7.m7.1.2" xref="A3.3.p3.24.7.m7.1.2.cmml"><mover accent="true" id="A3.3.p3.24.7.m7.1.2.2" xref="A3.3.p3.24.7.m7.1.2.2.cmml"><mi id="A3.3.p3.24.7.m7.1.2.2.2" mathcolor="#000099" xref="A3.3.p3.24.7.m7.1.2.2.2.cmml">𝜽</mi><mo id="A3.3.p3.24.7.m7.1.2.2.1" mathcolor="#000099" xref="A3.3.p3.24.7.m7.1.2.2.1.cmml">~</mo></mover><mo id="A3.3.p3.24.7.m7.1.2.1" xref="A3.3.p3.24.7.m7.1.2.1.cmml">⁢</mo><mrow id="A3.3.p3.24.7.m7.1.2.3.2" xref="A3.3.p3.24.7.m7.1.2.cmml"><mo id="A3.3.p3.24.7.m7.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="A3.3.p3.24.7.m7.1.2.cmml">(</mo><mi id="A3.3.p3.24.7.m7.1.1" mathcolor="#000099" xref="A3.3.p3.24.7.m7.1.1.cmml">t</mi><mo id="A3.3.p3.24.7.m7.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="A3.3.p3.24.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.3.p3.24.7.m7.1b"><apply id="A3.3.p3.24.7.m7.1.2.cmml" xref="A3.3.p3.24.7.m7.1.2"><times id="A3.3.p3.24.7.m7.1.2.1.cmml" xref="A3.3.p3.24.7.m7.1.2.1"></times><apply id="A3.3.p3.24.7.m7.1.2.2.cmml" xref="A3.3.p3.24.7.m7.1.2.2"><ci id="A3.3.p3.24.7.m7.1.2.2.1.cmml" xref="A3.3.p3.24.7.m7.1.2.2.1">~</ci><ci id="A3.3.p3.24.7.m7.1.2.2.2.cmml" xref="A3.3.p3.24.7.m7.1.2.2.2">𝜽</ci></apply><ci id="A3.3.p3.24.7.m7.1.1.cmml" xref="A3.3.p3.24.7.m7.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.3.p3.24.7.m7.1c">\tilde{\bm{\theta}}(t)</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.24.7.m7.1d">over~ start_ARG bold_italic_θ end_ARG ( italic_t )</annotation></semantics></math> exponentially converge to <math alttext="\bm{0}" class="ltx_Math" display="inline" id="A3.3.p3.25.8.m8.1"><semantics id="A3.3.p3.25.8.m8.1a"><mn id="A3.3.p3.25.8.m8.1.1" mathcolor="#000099" xref="A3.3.p3.25.8.m8.1.1.cmml">𝟎</mn><annotation-xml encoding="MathML-Content" id="A3.3.p3.25.8.m8.1b"><cn id="A3.3.p3.25.8.m8.1.1.cmml" type="integer" xref="A3.3.p3.25.8.m8.1.1">0</cn></annotation-xml><annotation encoding="application/x-tex" id="A3.3.p3.25.8.m8.1c">\bm{0}</annotation><annotation encoding="application/x-llamapun" id="A3.3.p3.25.8.m8.1d">bold_0</annotation></semantics></math></span> ∎</p> </div> </div> </section> <section class="ltx_appendix" id="A4"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix D </span>The proof of Theorem 3</h2> <div class="ltx_proof" id="A4.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A4.1.p1"> <p class="ltx_p" id="A4.1.p1.96">1) The proof of Item 1 follows the results in the proof of Item 1 in Theorem 2. Using (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S5.E26" title="In V-C Robustness Results ‣ V Theoretical Guarantees ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">26</span></a>), (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S5.E30" title="In V-C Robustness Results ‣ V Theoretical Guarantees ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">30</span></a>), and (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S5.E31" title="In V-C Robustness Results ‣ V Theoretical Guarantees ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">31</span></a>), the equality (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A3.E44" title="In Proof. ‣ Appendix C The proof of Theorem 2 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">44</span></a>) becomes</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx75"> <tbody id="A4.Ex63"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}=" class="ltx_Math" display="inline" id="A4.Ex63.m1.1"><semantics id="A4.Ex63.m1.1a"><mrow id="A4.Ex63.m1.1.1" xref="A4.Ex63.m1.1.1.cmml"><mover accent="true" id="A4.Ex63.m1.1.1.2" xref="A4.Ex63.m1.1.1.2.cmml"><mi id="A4.Ex63.m1.1.1.2.2" xref="A4.Ex63.m1.1.1.2.2.cmml">V</mi><mo id="A4.Ex63.m1.1.1.2.1" xref="A4.Ex63.m1.1.1.2.1.cmml">˙</mo></mover><mo id="A4.Ex63.m1.1.1.1" xref="A4.Ex63.m1.1.1.1.cmml">=</mo><mi id="A4.Ex63.m1.1.1.3" xref="A4.Ex63.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex63.m1.1b"><apply id="A4.Ex63.m1.1.1.cmml" xref="A4.Ex63.m1.1.1"><eq id="A4.Ex63.m1.1.1.1.cmml" xref="A4.Ex63.m1.1.1.1"></eq><apply id="A4.Ex63.m1.1.1.2.cmml" xref="A4.Ex63.m1.1.1.2"><ci id="A4.Ex63.m1.1.1.2.1.cmml" xref="A4.Ex63.m1.1.1.2.1">˙</ci><ci id="A4.Ex63.m1.1.1.2.2.cmml" xref="A4.Ex63.m1.1.1.2.2">𝑉</ci></apply><csymbol cd="latexml" id="A4.Ex63.m1.1.1.3.cmml" xref="A4.Ex63.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex63.m1.1c">\displaystyle\dot{V}=</annotation><annotation encoding="application/x-llamapun" id="A4.Ex63.m1.1d">over˙ start_ARG italic_V end_ARG =</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\bm{e}^{T}(-K\bm{e}+\Phi^{T}\tilde{\bm{\theta}})+\bm{e}^{T}\bm{d}% -(1+p)\tilde{\bm{\theta}}^{T}\Phi_{{\rm f}}(\Phi_{{\rm f}}^{T}\tilde{\bm{% \theta}}+{\bm{d}}_{\rm f})" class="ltx_Math" display="inline" id="A4.Ex63.m2.3"><semantics id="A4.Ex63.m2.3a"><mrow id="A4.Ex63.m2.3.3" xref="A4.Ex63.m2.3.3.cmml"><mrow id="A4.Ex63.m2.1.1.1" xref="A4.Ex63.m2.1.1.1.cmml"><mrow id="A4.Ex63.m2.1.1.1.1" xref="A4.Ex63.m2.1.1.1.1.cmml"><msup id="A4.Ex63.m2.1.1.1.1.3" xref="A4.Ex63.m2.1.1.1.1.3.cmml"><mi id="A4.Ex63.m2.1.1.1.1.3.2" xref="A4.Ex63.m2.1.1.1.1.3.2.cmml">𝒆</mi><mi id="A4.Ex63.m2.1.1.1.1.3.3" xref="A4.Ex63.m2.1.1.1.1.3.3.cmml">T</mi></msup><mo id="A4.Ex63.m2.1.1.1.1.2" xref="A4.Ex63.m2.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex63.m2.1.1.1.1.1.1" xref="A4.Ex63.m2.1.1.1.1.1.1.1.cmml"><mo id="A4.Ex63.m2.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex63.m2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex63.m2.1.1.1.1.1.1.1" xref="A4.Ex63.m2.1.1.1.1.1.1.1.cmml"><mrow id="A4.Ex63.m2.1.1.1.1.1.1.1.2" xref="A4.Ex63.m2.1.1.1.1.1.1.1.2.cmml"><mo id="A4.Ex63.m2.1.1.1.1.1.1.1.2a" xref="A4.Ex63.m2.1.1.1.1.1.1.1.2.cmml">−</mo><mrow id="A4.Ex63.m2.1.1.1.1.1.1.1.2.2" xref="A4.Ex63.m2.1.1.1.1.1.1.1.2.2.cmml"><mi id="A4.Ex63.m2.1.1.1.1.1.1.1.2.2.2" xref="A4.Ex63.m2.1.1.1.1.1.1.1.2.2.2.cmml">K</mi><mo id="A4.Ex63.m2.1.1.1.1.1.1.1.2.2.1" xref="A4.Ex63.m2.1.1.1.1.1.1.1.2.2.1.cmml">⁢</mo><mi id="A4.Ex63.m2.1.1.1.1.1.1.1.2.2.3" xref="A4.Ex63.m2.1.1.1.1.1.1.1.2.2.3.cmml">𝒆</mi></mrow></mrow><mo id="A4.Ex63.m2.1.1.1.1.1.1.1.1" xref="A4.Ex63.m2.1.1.1.1.1.1.1.1.cmml">+</mo><mrow id="A4.Ex63.m2.1.1.1.1.1.1.1.3" xref="A4.Ex63.m2.1.1.1.1.1.1.1.3.cmml"><msup id="A4.Ex63.m2.1.1.1.1.1.1.1.3.2" xref="A4.Ex63.m2.1.1.1.1.1.1.1.3.2.cmml"><mi id="A4.Ex63.m2.1.1.1.1.1.1.1.3.2.2" mathvariant="normal" xref="A4.Ex63.m2.1.1.1.1.1.1.1.3.2.2.cmml">Φ</mi><mi id="A4.Ex63.m2.1.1.1.1.1.1.1.3.2.3" xref="A4.Ex63.m2.1.1.1.1.1.1.1.3.2.3.cmml">T</mi></msup><mo id="A4.Ex63.m2.1.1.1.1.1.1.1.3.1" xref="A4.Ex63.m2.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mover accent="true" id="A4.Ex63.m2.1.1.1.1.1.1.1.3.3" xref="A4.Ex63.m2.1.1.1.1.1.1.1.3.3.cmml"><mi id="A4.Ex63.m2.1.1.1.1.1.1.1.3.3.2" xref="A4.Ex63.m2.1.1.1.1.1.1.1.3.3.2.cmml">𝜽</mi><mo id="A4.Ex63.m2.1.1.1.1.1.1.1.3.3.1" xref="A4.Ex63.m2.1.1.1.1.1.1.1.3.3.1.cmml">~</mo></mover></mrow></mrow><mo id="A4.Ex63.m2.1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex63.m2.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A4.Ex63.m2.1.1.1.2" xref="A4.Ex63.m2.1.1.1.2.cmml">+</mo><mrow id="A4.Ex63.m2.1.1.1.3" xref="A4.Ex63.m2.1.1.1.3.cmml"><msup id="A4.Ex63.m2.1.1.1.3.2" xref="A4.Ex63.m2.1.1.1.3.2.cmml"><mi id="A4.Ex63.m2.1.1.1.3.2.2" xref="A4.Ex63.m2.1.1.1.3.2.2.cmml">𝒆</mi><mi id="A4.Ex63.m2.1.1.1.3.2.3" xref="A4.Ex63.m2.1.1.1.3.2.3.cmml">T</mi></msup><mo id="A4.Ex63.m2.1.1.1.3.1" xref="A4.Ex63.m2.1.1.1.3.1.cmml">⁢</mo><mi id="A4.Ex63.m2.1.1.1.3.3" xref="A4.Ex63.m2.1.1.1.3.3.cmml">𝒅</mi></mrow></mrow><mo id="A4.Ex63.m2.3.3.4" xref="A4.Ex63.m2.3.3.4.cmml">−</mo><mrow id="A4.Ex63.m2.3.3.3" xref="A4.Ex63.m2.3.3.3.cmml"><mrow id="A4.Ex63.m2.2.2.2.1.1" xref="A4.Ex63.m2.2.2.2.1.1.1.cmml"><mo id="A4.Ex63.m2.2.2.2.1.1.2" stretchy="false" xref="A4.Ex63.m2.2.2.2.1.1.1.cmml">(</mo><mrow id="A4.Ex63.m2.2.2.2.1.1.1" xref="A4.Ex63.m2.2.2.2.1.1.1.cmml"><mn id="A4.Ex63.m2.2.2.2.1.1.1.2" xref="A4.Ex63.m2.2.2.2.1.1.1.2.cmml">1</mn><mo id="A4.Ex63.m2.2.2.2.1.1.1.1" xref="A4.Ex63.m2.2.2.2.1.1.1.1.cmml">+</mo><mi id="A4.Ex63.m2.2.2.2.1.1.1.3" xref="A4.Ex63.m2.2.2.2.1.1.1.3.cmml">p</mi></mrow><mo id="A4.Ex63.m2.2.2.2.1.1.3" stretchy="false" xref="A4.Ex63.m2.2.2.2.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex63.m2.3.3.3.3" xref="A4.Ex63.m2.3.3.3.3.cmml">⁢</mo><msup id="A4.Ex63.m2.3.3.3.4" xref="A4.Ex63.m2.3.3.3.4.cmml"><mover accent="true" id="A4.Ex63.m2.3.3.3.4.2" xref="A4.Ex63.m2.3.3.3.4.2.cmml"><mi id="A4.Ex63.m2.3.3.3.4.2.2" xref="A4.Ex63.m2.3.3.3.4.2.2.cmml">𝜽</mi><mo id="A4.Ex63.m2.3.3.3.4.2.1" xref="A4.Ex63.m2.3.3.3.4.2.1.cmml">~</mo></mover><mi id="A4.Ex63.m2.3.3.3.4.3" xref="A4.Ex63.m2.3.3.3.4.3.cmml">T</mi></msup><mo id="A4.Ex63.m2.3.3.3.3a" xref="A4.Ex63.m2.3.3.3.3.cmml">⁢</mo><msub id="A4.Ex63.m2.3.3.3.5" xref="A4.Ex63.m2.3.3.3.5.cmml"><mi id="A4.Ex63.m2.3.3.3.5.2" mathvariant="normal" xref="A4.Ex63.m2.3.3.3.5.2.cmml">Φ</mi><mi id="A4.Ex63.m2.3.3.3.5.3" mathvariant="normal" xref="A4.Ex63.m2.3.3.3.5.3.cmml">f</mi></msub><mo id="A4.Ex63.m2.3.3.3.3b" xref="A4.Ex63.m2.3.3.3.3.cmml">⁢</mo><mrow id="A4.Ex63.m2.3.3.3.2.1" xref="A4.Ex63.m2.3.3.3.2.1.1.cmml"><mo id="A4.Ex63.m2.3.3.3.2.1.2" stretchy="false" xref="A4.Ex63.m2.3.3.3.2.1.1.cmml">(</mo><mrow id="A4.Ex63.m2.3.3.3.2.1.1" xref="A4.Ex63.m2.3.3.3.2.1.1.cmml"><mrow id="A4.Ex63.m2.3.3.3.2.1.1.2" xref="A4.Ex63.m2.3.3.3.2.1.1.2.cmml"><msubsup id="A4.Ex63.m2.3.3.3.2.1.1.2.2" xref="A4.Ex63.m2.3.3.3.2.1.1.2.2.cmml"><mi id="A4.Ex63.m2.3.3.3.2.1.1.2.2.2.2" mathvariant="normal" xref="A4.Ex63.m2.3.3.3.2.1.1.2.2.2.2.cmml">Φ</mi><mi id="A4.Ex63.m2.3.3.3.2.1.1.2.2.2.3" mathvariant="normal" xref="A4.Ex63.m2.3.3.3.2.1.1.2.2.2.3.cmml">f</mi><mi id="A4.Ex63.m2.3.3.3.2.1.1.2.2.3" xref="A4.Ex63.m2.3.3.3.2.1.1.2.2.3.cmml">T</mi></msubsup><mo id="A4.Ex63.m2.3.3.3.2.1.1.2.1" xref="A4.Ex63.m2.3.3.3.2.1.1.2.1.cmml">⁢</mo><mover accent="true" id="A4.Ex63.m2.3.3.3.2.1.1.2.3" xref="A4.Ex63.m2.3.3.3.2.1.1.2.3.cmml"><mi id="A4.Ex63.m2.3.3.3.2.1.1.2.3.2" xref="A4.Ex63.m2.3.3.3.2.1.1.2.3.2.cmml">𝜽</mi><mo id="A4.Ex63.m2.3.3.3.2.1.1.2.3.1" xref="A4.Ex63.m2.3.3.3.2.1.1.2.3.1.cmml">~</mo></mover></mrow><mo id="A4.Ex63.m2.3.3.3.2.1.1.1" xref="A4.Ex63.m2.3.3.3.2.1.1.1.cmml">+</mo><msub id="A4.Ex63.m2.3.3.3.2.1.1.3" xref="A4.Ex63.m2.3.3.3.2.1.1.3.cmml"><mi id="A4.Ex63.m2.3.3.3.2.1.1.3.2" xref="A4.Ex63.m2.3.3.3.2.1.1.3.2.cmml">𝒅</mi><mi id="A4.Ex63.m2.3.3.3.2.1.1.3.3" mathvariant="normal" xref="A4.Ex63.m2.3.3.3.2.1.1.3.3.cmml">f</mi></msub></mrow><mo id="A4.Ex63.m2.3.3.3.2.1.3" stretchy="false" xref="A4.Ex63.m2.3.3.3.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex63.m2.3b"><apply id="A4.Ex63.m2.3.3.cmml" xref="A4.Ex63.m2.3.3"><minus id="A4.Ex63.m2.3.3.4.cmml" xref="A4.Ex63.m2.3.3.4"></minus><apply id="A4.Ex63.m2.1.1.1.cmml" xref="A4.Ex63.m2.1.1.1"><plus id="A4.Ex63.m2.1.1.1.2.cmml" xref="A4.Ex63.m2.1.1.1.2"></plus><apply id="A4.Ex63.m2.1.1.1.1.cmml" xref="A4.Ex63.m2.1.1.1.1"><times id="A4.Ex63.m2.1.1.1.1.2.cmml" xref="A4.Ex63.m2.1.1.1.1.2"></times><apply id="A4.Ex63.m2.1.1.1.1.3.cmml" xref="A4.Ex63.m2.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.Ex63.m2.1.1.1.1.3.1.cmml" xref="A4.Ex63.m2.1.1.1.1.3">superscript</csymbol><ci id="A4.Ex63.m2.1.1.1.1.3.2.cmml" xref="A4.Ex63.m2.1.1.1.1.3.2">𝒆</ci><ci id="A4.Ex63.m2.1.1.1.1.3.3.cmml" xref="A4.Ex63.m2.1.1.1.1.3.3">𝑇</ci></apply><apply id="A4.Ex63.m2.1.1.1.1.1.1.1.cmml" xref="A4.Ex63.m2.1.1.1.1.1.1"><plus id="A4.Ex63.m2.1.1.1.1.1.1.1.1.cmml" xref="A4.Ex63.m2.1.1.1.1.1.1.1.1"></plus><apply id="A4.Ex63.m2.1.1.1.1.1.1.1.2.cmml" xref="A4.Ex63.m2.1.1.1.1.1.1.1.2"><minus id="A4.Ex63.m2.1.1.1.1.1.1.1.2.1.cmml" xref="A4.Ex63.m2.1.1.1.1.1.1.1.2"></minus><apply id="A4.Ex63.m2.1.1.1.1.1.1.1.2.2.cmml" xref="A4.Ex63.m2.1.1.1.1.1.1.1.2.2"><times id="A4.Ex63.m2.1.1.1.1.1.1.1.2.2.1.cmml" xref="A4.Ex63.m2.1.1.1.1.1.1.1.2.2.1"></times><ci id="A4.Ex63.m2.1.1.1.1.1.1.1.2.2.2.cmml" xref="A4.Ex63.m2.1.1.1.1.1.1.1.2.2.2">𝐾</ci><ci id="A4.Ex63.m2.1.1.1.1.1.1.1.2.2.3.cmml" xref="A4.Ex63.m2.1.1.1.1.1.1.1.2.2.3">𝒆</ci></apply></apply><apply id="A4.Ex63.m2.1.1.1.1.1.1.1.3.cmml" xref="A4.Ex63.m2.1.1.1.1.1.1.1.3"><times id="A4.Ex63.m2.1.1.1.1.1.1.1.3.1.cmml" xref="A4.Ex63.m2.1.1.1.1.1.1.1.3.1"></times><apply id="A4.Ex63.m2.1.1.1.1.1.1.1.3.2.cmml" xref="A4.Ex63.m2.1.1.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A4.Ex63.m2.1.1.1.1.1.1.1.3.2.1.cmml" xref="A4.Ex63.m2.1.1.1.1.1.1.1.3.2">superscript</csymbol><ci id="A4.Ex63.m2.1.1.1.1.1.1.1.3.2.2.cmml" xref="A4.Ex63.m2.1.1.1.1.1.1.1.3.2.2">Φ</ci><ci id="A4.Ex63.m2.1.1.1.1.1.1.1.3.2.3.cmml" xref="A4.Ex63.m2.1.1.1.1.1.1.1.3.2.3">𝑇</ci></apply><apply id="A4.Ex63.m2.1.1.1.1.1.1.1.3.3.cmml" xref="A4.Ex63.m2.1.1.1.1.1.1.1.3.3"><ci id="A4.Ex63.m2.1.1.1.1.1.1.1.3.3.1.cmml" xref="A4.Ex63.m2.1.1.1.1.1.1.1.3.3.1">~</ci><ci id="A4.Ex63.m2.1.1.1.1.1.1.1.3.3.2.cmml" xref="A4.Ex63.m2.1.1.1.1.1.1.1.3.3.2">𝜽</ci></apply></apply></apply></apply><apply id="A4.Ex63.m2.1.1.1.3.cmml" xref="A4.Ex63.m2.1.1.1.3"><times id="A4.Ex63.m2.1.1.1.3.1.cmml" xref="A4.Ex63.m2.1.1.1.3.1"></times><apply id="A4.Ex63.m2.1.1.1.3.2.cmml" xref="A4.Ex63.m2.1.1.1.3.2"><csymbol cd="ambiguous" id="A4.Ex63.m2.1.1.1.3.2.1.cmml" xref="A4.Ex63.m2.1.1.1.3.2">superscript</csymbol><ci id="A4.Ex63.m2.1.1.1.3.2.2.cmml" xref="A4.Ex63.m2.1.1.1.3.2.2">𝒆</ci><ci id="A4.Ex63.m2.1.1.1.3.2.3.cmml" xref="A4.Ex63.m2.1.1.1.3.2.3">𝑇</ci></apply><ci id="A4.Ex63.m2.1.1.1.3.3.cmml" xref="A4.Ex63.m2.1.1.1.3.3">𝒅</ci></apply></apply><apply id="A4.Ex63.m2.3.3.3.cmml" xref="A4.Ex63.m2.3.3.3"><times id="A4.Ex63.m2.3.3.3.3.cmml" xref="A4.Ex63.m2.3.3.3.3"></times><apply id="A4.Ex63.m2.2.2.2.1.1.1.cmml" xref="A4.Ex63.m2.2.2.2.1.1"><plus id="A4.Ex63.m2.2.2.2.1.1.1.1.cmml" xref="A4.Ex63.m2.2.2.2.1.1.1.1"></plus><cn id="A4.Ex63.m2.2.2.2.1.1.1.2.cmml" type="integer" xref="A4.Ex63.m2.2.2.2.1.1.1.2">1</cn><ci id="A4.Ex63.m2.2.2.2.1.1.1.3.cmml" xref="A4.Ex63.m2.2.2.2.1.1.1.3">𝑝</ci></apply><apply id="A4.Ex63.m2.3.3.3.4.cmml" xref="A4.Ex63.m2.3.3.3.4"><csymbol cd="ambiguous" id="A4.Ex63.m2.3.3.3.4.1.cmml" xref="A4.Ex63.m2.3.3.3.4">superscript</csymbol><apply id="A4.Ex63.m2.3.3.3.4.2.cmml" xref="A4.Ex63.m2.3.3.3.4.2"><ci id="A4.Ex63.m2.3.3.3.4.2.1.cmml" xref="A4.Ex63.m2.3.3.3.4.2.1">~</ci><ci id="A4.Ex63.m2.3.3.3.4.2.2.cmml" xref="A4.Ex63.m2.3.3.3.4.2.2">𝜽</ci></apply><ci id="A4.Ex63.m2.3.3.3.4.3.cmml" xref="A4.Ex63.m2.3.3.3.4.3">𝑇</ci></apply><apply id="A4.Ex63.m2.3.3.3.5.cmml" xref="A4.Ex63.m2.3.3.3.5"><csymbol cd="ambiguous" id="A4.Ex63.m2.3.3.3.5.1.cmml" xref="A4.Ex63.m2.3.3.3.5">subscript</csymbol><ci id="A4.Ex63.m2.3.3.3.5.2.cmml" xref="A4.Ex63.m2.3.3.3.5.2">Φ</ci><ci id="A4.Ex63.m2.3.3.3.5.3.cmml" xref="A4.Ex63.m2.3.3.3.5.3">f</ci></apply><apply id="A4.Ex63.m2.3.3.3.2.1.1.cmml" xref="A4.Ex63.m2.3.3.3.2.1"><plus id="A4.Ex63.m2.3.3.3.2.1.1.1.cmml" xref="A4.Ex63.m2.3.3.3.2.1.1.1"></plus><apply id="A4.Ex63.m2.3.3.3.2.1.1.2.cmml" xref="A4.Ex63.m2.3.3.3.2.1.1.2"><times id="A4.Ex63.m2.3.3.3.2.1.1.2.1.cmml" xref="A4.Ex63.m2.3.3.3.2.1.1.2.1"></times><apply id="A4.Ex63.m2.3.3.3.2.1.1.2.2.cmml" xref="A4.Ex63.m2.3.3.3.2.1.1.2.2"><csymbol cd="ambiguous" id="A4.Ex63.m2.3.3.3.2.1.1.2.2.1.cmml" xref="A4.Ex63.m2.3.3.3.2.1.1.2.2">superscript</csymbol><apply id="A4.Ex63.m2.3.3.3.2.1.1.2.2.2.cmml" xref="A4.Ex63.m2.3.3.3.2.1.1.2.2"><csymbol cd="ambiguous" id="A4.Ex63.m2.3.3.3.2.1.1.2.2.2.1.cmml" xref="A4.Ex63.m2.3.3.3.2.1.1.2.2">subscript</csymbol><ci id="A4.Ex63.m2.3.3.3.2.1.1.2.2.2.2.cmml" xref="A4.Ex63.m2.3.3.3.2.1.1.2.2.2.2">Φ</ci><ci id="A4.Ex63.m2.3.3.3.2.1.1.2.2.2.3.cmml" xref="A4.Ex63.m2.3.3.3.2.1.1.2.2.2.3">f</ci></apply><ci id="A4.Ex63.m2.3.3.3.2.1.1.2.2.3.cmml" xref="A4.Ex63.m2.3.3.3.2.1.1.2.2.3">𝑇</ci></apply><apply id="A4.Ex63.m2.3.3.3.2.1.1.2.3.cmml" xref="A4.Ex63.m2.3.3.3.2.1.1.2.3"><ci id="A4.Ex63.m2.3.3.3.2.1.1.2.3.1.cmml" xref="A4.Ex63.m2.3.3.3.2.1.1.2.3.1">~</ci><ci id="A4.Ex63.m2.3.3.3.2.1.1.2.3.2.cmml" xref="A4.Ex63.m2.3.3.3.2.1.1.2.3.2">𝜽</ci></apply></apply><apply id="A4.Ex63.m2.3.3.3.2.1.1.3.cmml" xref="A4.Ex63.m2.3.3.3.2.1.1.3"><csymbol cd="ambiguous" id="A4.Ex63.m2.3.3.3.2.1.1.3.1.cmml" xref="A4.Ex63.m2.3.3.3.2.1.1.3">subscript</csymbol><ci id="A4.Ex63.m2.3.3.3.2.1.1.3.2.cmml" xref="A4.Ex63.m2.3.3.3.2.1.1.3.2">𝒅</ci><ci id="A4.Ex63.m2.3.3.3.2.1.1.3.3.cmml" xref="A4.Ex63.m2.3.3.3.2.1.1.3.3">f</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex63.m2.3c">\displaystyle\bm{e}^{T}(-K\bm{e}+\Phi^{T}\tilde{\bm{\theta}})+\bm{e}^{T}\bm{d}% -(1+p)\tilde{\bm{\theta}}^{T}\Phi_{{\rm f}}(\Phi_{{\rm f}}^{T}\tilde{\bm{% \theta}}+{\bm{d}}_{\rm f})</annotation><annotation encoding="application/x-llamapun" id="A4.Ex63.m2.3d">bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( - italic_K bold_italic_e + roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG ) + bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_d - ( 1 + italic_p ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ( roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG + bold_italic_d start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex64"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-\kappa(1+p)\tilde{\bm{\theta}}^{T}(Q(t,t_{\rm e}){\tilde{\bm{% \theta}}}+{\bm{d}}_{\rm g})." class="ltx_Math" display="inline" id="A4.Ex64.m1.2"><semantics id="A4.Ex64.m1.2a"><mrow id="A4.Ex64.m1.2.2.1" xref="A4.Ex64.m1.2.2.1.1.cmml"><mrow id="A4.Ex64.m1.2.2.1.1" xref="A4.Ex64.m1.2.2.1.1.cmml"><mo id="A4.Ex64.m1.2.2.1.1a" xref="A4.Ex64.m1.2.2.1.1.cmml">−</mo><mrow id="A4.Ex64.m1.2.2.1.1.2" xref="A4.Ex64.m1.2.2.1.1.2.cmml"><mi id="A4.Ex64.m1.2.2.1.1.2.4" xref="A4.Ex64.m1.2.2.1.1.2.4.cmml">κ</mi><mo id="A4.Ex64.m1.2.2.1.1.2.3" xref="A4.Ex64.m1.2.2.1.1.2.3.cmml">⁢</mo><mrow id="A4.Ex64.m1.2.2.1.1.1.1.1" xref="A4.Ex64.m1.2.2.1.1.1.1.1.1.cmml"><mo id="A4.Ex64.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="A4.Ex64.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex64.m1.2.2.1.1.1.1.1.1" xref="A4.Ex64.m1.2.2.1.1.1.1.1.1.cmml"><mn id="A4.Ex64.m1.2.2.1.1.1.1.1.1.2" xref="A4.Ex64.m1.2.2.1.1.1.1.1.1.2.cmml">1</mn><mo id="A4.Ex64.m1.2.2.1.1.1.1.1.1.1" xref="A4.Ex64.m1.2.2.1.1.1.1.1.1.1.cmml">+</mo><mi id="A4.Ex64.m1.2.2.1.1.1.1.1.1.3" xref="A4.Ex64.m1.2.2.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A4.Ex64.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="A4.Ex64.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex64.m1.2.2.1.1.2.3a" xref="A4.Ex64.m1.2.2.1.1.2.3.cmml">⁢</mo><msup id="A4.Ex64.m1.2.2.1.1.2.5" xref="A4.Ex64.m1.2.2.1.1.2.5.cmml"><mover accent="true" id="A4.Ex64.m1.2.2.1.1.2.5.2" xref="A4.Ex64.m1.2.2.1.1.2.5.2.cmml"><mi id="A4.Ex64.m1.2.2.1.1.2.5.2.2" xref="A4.Ex64.m1.2.2.1.1.2.5.2.2.cmml">𝜽</mi><mo id="A4.Ex64.m1.2.2.1.1.2.5.2.1" xref="A4.Ex64.m1.2.2.1.1.2.5.2.1.cmml">~</mo></mover><mi id="A4.Ex64.m1.2.2.1.1.2.5.3" xref="A4.Ex64.m1.2.2.1.1.2.5.3.cmml">T</mi></msup><mo id="A4.Ex64.m1.2.2.1.1.2.3b" xref="A4.Ex64.m1.2.2.1.1.2.3.cmml">⁢</mo><mrow id="A4.Ex64.m1.2.2.1.1.2.2.1" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.cmml"><mo id="A4.Ex64.m1.2.2.1.1.2.2.1.2" stretchy="false" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.cmml">(</mo><mrow id="A4.Ex64.m1.2.2.1.1.2.2.1.1" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.cmml"><mrow id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.cmml"><mi id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.3" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.3.cmml">Q</mi><mo id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.2" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.1.1" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.1.2.cmml"><mo id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.1.1.2" stretchy="false" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.1.2.cmml">(</mo><mi id="A4.Ex64.m1.1.1" xref="A4.Ex64.m1.1.1.cmml">t</mi><mo id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.1.1.3" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.1.2.cmml">,</mo><msub id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.1.1.1" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.1.1.1.cmml"><mi id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.1.1.1.2" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.1.1.1.2.cmml">t</mi><mi id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.1.1.1.3" mathvariant="normal" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.1.1.4" stretchy="false" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.1.2.cmml">)</mo></mrow><mo id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.2a" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.2.cmml">⁢</mo><mover accent="true" id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.4" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.4.cmml"><mi id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.4.2" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.4.2.cmml">𝜽</mi><mo id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.4.1" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.4.1.cmml">~</mo></mover></mrow><mo id="A4.Ex64.m1.2.2.1.1.2.2.1.1.2" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.2.cmml">+</mo><msub id="A4.Ex64.m1.2.2.1.1.2.2.1.1.3" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.3.cmml"><mi id="A4.Ex64.m1.2.2.1.1.2.2.1.1.3.2" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.3.2.cmml">𝒅</mi><mi id="A4.Ex64.m1.2.2.1.1.2.2.1.1.3.3" mathvariant="normal" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.3.3.cmml">g</mi></msub></mrow><mo id="A4.Ex64.m1.2.2.1.1.2.2.1.3" stretchy="false" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A4.Ex64.m1.2.2.1.2" lspace="0em" xref="A4.Ex64.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex64.m1.2b"><apply id="A4.Ex64.m1.2.2.1.1.cmml" xref="A4.Ex64.m1.2.2.1"><minus id="A4.Ex64.m1.2.2.1.1.3.cmml" xref="A4.Ex64.m1.2.2.1"></minus><apply id="A4.Ex64.m1.2.2.1.1.2.cmml" xref="A4.Ex64.m1.2.2.1.1.2"><times id="A4.Ex64.m1.2.2.1.1.2.3.cmml" xref="A4.Ex64.m1.2.2.1.1.2.3"></times><ci id="A4.Ex64.m1.2.2.1.1.2.4.cmml" xref="A4.Ex64.m1.2.2.1.1.2.4">𝜅</ci><apply id="A4.Ex64.m1.2.2.1.1.1.1.1.1.cmml" xref="A4.Ex64.m1.2.2.1.1.1.1.1"><plus id="A4.Ex64.m1.2.2.1.1.1.1.1.1.1.cmml" xref="A4.Ex64.m1.2.2.1.1.1.1.1.1.1"></plus><cn id="A4.Ex64.m1.2.2.1.1.1.1.1.1.2.cmml" type="integer" xref="A4.Ex64.m1.2.2.1.1.1.1.1.1.2">1</cn><ci id="A4.Ex64.m1.2.2.1.1.1.1.1.1.3.cmml" xref="A4.Ex64.m1.2.2.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="A4.Ex64.m1.2.2.1.1.2.5.cmml" xref="A4.Ex64.m1.2.2.1.1.2.5"><csymbol cd="ambiguous" id="A4.Ex64.m1.2.2.1.1.2.5.1.cmml" xref="A4.Ex64.m1.2.2.1.1.2.5">superscript</csymbol><apply id="A4.Ex64.m1.2.2.1.1.2.5.2.cmml" xref="A4.Ex64.m1.2.2.1.1.2.5.2"><ci id="A4.Ex64.m1.2.2.1.1.2.5.2.1.cmml" xref="A4.Ex64.m1.2.2.1.1.2.5.2.1">~</ci><ci id="A4.Ex64.m1.2.2.1.1.2.5.2.2.cmml" xref="A4.Ex64.m1.2.2.1.1.2.5.2.2">𝜽</ci></apply><ci id="A4.Ex64.m1.2.2.1.1.2.5.3.cmml" xref="A4.Ex64.m1.2.2.1.1.2.5.3">𝑇</ci></apply><apply id="A4.Ex64.m1.2.2.1.1.2.2.1.1.cmml" xref="A4.Ex64.m1.2.2.1.1.2.2.1"><plus id="A4.Ex64.m1.2.2.1.1.2.2.1.1.2.cmml" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.2"></plus><apply id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.cmml" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1"><times id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.2.cmml" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.2"></times><ci id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.3.cmml" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.3">𝑄</ci><interval closure="open" id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.1.2.cmml" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.1.1"><ci id="A4.Ex64.m1.1.1.cmml" xref="A4.Ex64.m1.1.1">𝑡</ci><apply id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.1.1.1.cmml" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.1.1.1.1.cmml" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.1.1.1.2.cmml" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.1.1.1.2">𝑡</ci><ci id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.1.1.1.3.cmml" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.1.1.1.3">e</ci></apply></interval><apply id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.4.cmml" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.4"><ci id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.4.1.cmml" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.4.1">~</ci><ci id="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.4.2.cmml" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.1.4.2">𝜽</ci></apply></apply><apply id="A4.Ex64.m1.2.2.1.1.2.2.1.1.3.cmml" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.3"><csymbol cd="ambiguous" id="A4.Ex64.m1.2.2.1.1.2.2.1.1.3.1.cmml" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.3">subscript</csymbol><ci id="A4.Ex64.m1.2.2.1.1.2.2.1.1.3.2.cmml" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.3.2">𝒅</ci><ci id="A4.Ex64.m1.2.2.1.1.2.2.1.1.3.3.cmml" xref="A4.Ex64.m1.2.2.1.1.2.2.1.1.3.3">g</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex64.m1.2c">\displaystyle-\kappa(1+p)\tilde{\bm{\theta}}^{T}(Q(t,t_{\rm e}){\tilde{\bm{% \theta}}}+{\bm{d}}_{\rm g}).</annotation><annotation encoding="application/x-llamapun" id="A4.Ex64.m1.2d">- italic_κ ( 1 + italic_p ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_Q ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG + bold_italic_d start_POSTSUBSCRIPT roman_g end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.1.p1.97">Following the steps from (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A3.E44" title="In Proof. ‣ Appendix C The proof of Theorem 2 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">44</span></a>) to (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A3.E45" title="In Proof. ‣ Appendix C The proof of Theorem 2 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">45</span></a>), it is straightforward to get</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx76"> <tbody id="A4.Ex65"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}\leq" class="ltx_Math" display="inline" id="A4.Ex65.m1.1"><semantics id="A4.Ex65.m1.1a"><mrow id="A4.Ex65.m1.1.1" xref="A4.Ex65.m1.1.1.cmml"><mover accent="true" id="A4.Ex65.m1.1.1.2" xref="A4.Ex65.m1.1.1.2.cmml"><mi id="A4.Ex65.m1.1.1.2.2" xref="A4.Ex65.m1.1.1.2.2.cmml">V</mi><mo id="A4.Ex65.m1.1.1.2.1" xref="A4.Ex65.m1.1.1.2.1.cmml">˙</mo></mover><mo id="A4.Ex65.m1.1.1.1" xref="A4.Ex65.m1.1.1.1.cmml">≤</mo><mi id="A4.Ex65.m1.1.1.3" xref="A4.Ex65.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex65.m1.1b"><apply id="A4.Ex65.m1.1.1.cmml" xref="A4.Ex65.m1.1.1"><leq id="A4.Ex65.m1.1.1.1.cmml" xref="A4.Ex65.m1.1.1.1"></leq><apply id="A4.Ex65.m1.1.1.2.cmml" xref="A4.Ex65.m1.1.1.2"><ci id="A4.Ex65.m1.1.1.2.1.cmml" xref="A4.Ex65.m1.1.1.2.1">˙</ci><ci id="A4.Ex65.m1.1.1.2.2.cmml" xref="A4.Ex65.m1.1.1.2.2">𝑉</ci></apply><csymbol cd="latexml" id="A4.Ex65.m1.1.1.3.cmml" xref="A4.Ex65.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex65.m1.1c">\displaystyle\dot{V}\leq</annotation><annotation encoding="application/x-llamapun" id="A4.Ex65.m1.1d">over˙ start_ARG italic_V end_ARG ≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-\bm{e}^{T}(K-I/4)\bm{e}-\kappa\tilde{\bm{\theta}}^{T}Q(t,t_{\rm e% }){\tilde{\bm{\theta}}}+\bm{e}^{T}(\Phi-\Phi_{\rm f})\tilde{\bm{\theta}}" class="ltx_Math" display="inline" id="A4.Ex65.m2.4"><semantics id="A4.Ex65.m2.4a"><mrow id="A4.Ex65.m2.4.4" xref="A4.Ex65.m2.4.4.cmml"><mrow id="A4.Ex65.m2.3.3.2" xref="A4.Ex65.m2.3.3.2.cmml"><mrow id="A4.Ex65.m2.2.2.1.1" xref="A4.Ex65.m2.2.2.1.1.cmml"><mo id="A4.Ex65.m2.2.2.1.1a" xref="A4.Ex65.m2.2.2.1.1.cmml">−</mo><mrow id="A4.Ex65.m2.2.2.1.1.1" xref="A4.Ex65.m2.2.2.1.1.1.cmml"><msup id="A4.Ex65.m2.2.2.1.1.1.3" xref="A4.Ex65.m2.2.2.1.1.1.3.cmml"><mi id="A4.Ex65.m2.2.2.1.1.1.3.2" xref="A4.Ex65.m2.2.2.1.1.1.3.2.cmml">𝒆</mi><mi id="A4.Ex65.m2.2.2.1.1.1.3.3" xref="A4.Ex65.m2.2.2.1.1.1.3.3.cmml">T</mi></msup><mo id="A4.Ex65.m2.2.2.1.1.1.2" xref="A4.Ex65.m2.2.2.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex65.m2.2.2.1.1.1.1.1" xref="A4.Ex65.m2.2.2.1.1.1.1.1.1.cmml"><mo id="A4.Ex65.m2.2.2.1.1.1.1.1.2" stretchy="false" xref="A4.Ex65.m2.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex65.m2.2.2.1.1.1.1.1.1" xref="A4.Ex65.m2.2.2.1.1.1.1.1.1.cmml"><mi id="A4.Ex65.m2.2.2.1.1.1.1.1.1.2" xref="A4.Ex65.m2.2.2.1.1.1.1.1.1.2.cmml">K</mi><mo id="A4.Ex65.m2.2.2.1.1.1.1.1.1.1" xref="A4.Ex65.m2.2.2.1.1.1.1.1.1.1.cmml">−</mo><mrow id="A4.Ex65.m2.2.2.1.1.1.1.1.1.3" xref="A4.Ex65.m2.2.2.1.1.1.1.1.1.3.cmml"><mi id="A4.Ex65.m2.2.2.1.1.1.1.1.1.3.2" xref="A4.Ex65.m2.2.2.1.1.1.1.1.1.3.2.cmml">I</mi><mo id="A4.Ex65.m2.2.2.1.1.1.1.1.1.3.1" xref="A4.Ex65.m2.2.2.1.1.1.1.1.1.3.1.cmml">/</mo><mn id="A4.Ex65.m2.2.2.1.1.1.1.1.1.3.3" xref="A4.Ex65.m2.2.2.1.1.1.1.1.1.3.3.cmml">4</mn></mrow></mrow><mo id="A4.Ex65.m2.2.2.1.1.1.1.1.3" stretchy="false" xref="A4.Ex65.m2.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex65.m2.2.2.1.1.1.2a" xref="A4.Ex65.m2.2.2.1.1.1.2.cmml">⁢</mo><mi id="A4.Ex65.m2.2.2.1.1.1.4" xref="A4.Ex65.m2.2.2.1.1.1.4.cmml">𝒆</mi></mrow></mrow><mo id="A4.Ex65.m2.3.3.2.3" xref="A4.Ex65.m2.3.3.2.3.cmml">−</mo><mrow id="A4.Ex65.m2.3.3.2.2" xref="A4.Ex65.m2.3.3.2.2.cmml"><mi id="A4.Ex65.m2.3.3.2.2.3" xref="A4.Ex65.m2.3.3.2.2.3.cmml">κ</mi><mo id="A4.Ex65.m2.3.3.2.2.2" xref="A4.Ex65.m2.3.3.2.2.2.cmml">⁢</mo><msup id="A4.Ex65.m2.3.3.2.2.4" xref="A4.Ex65.m2.3.3.2.2.4.cmml"><mover accent="true" id="A4.Ex65.m2.3.3.2.2.4.2" xref="A4.Ex65.m2.3.3.2.2.4.2.cmml"><mi id="A4.Ex65.m2.3.3.2.2.4.2.2" xref="A4.Ex65.m2.3.3.2.2.4.2.2.cmml">𝜽</mi><mo id="A4.Ex65.m2.3.3.2.2.4.2.1" xref="A4.Ex65.m2.3.3.2.2.4.2.1.cmml">~</mo></mover><mi id="A4.Ex65.m2.3.3.2.2.4.3" xref="A4.Ex65.m2.3.3.2.2.4.3.cmml">T</mi></msup><mo id="A4.Ex65.m2.3.3.2.2.2a" xref="A4.Ex65.m2.3.3.2.2.2.cmml">⁢</mo><mi id="A4.Ex65.m2.3.3.2.2.5" xref="A4.Ex65.m2.3.3.2.2.5.cmml">Q</mi><mo id="A4.Ex65.m2.3.3.2.2.2b" xref="A4.Ex65.m2.3.3.2.2.2.cmml">⁢</mo><mrow id="A4.Ex65.m2.3.3.2.2.1.1" xref="A4.Ex65.m2.3.3.2.2.1.2.cmml"><mo id="A4.Ex65.m2.3.3.2.2.1.1.2" stretchy="false" xref="A4.Ex65.m2.3.3.2.2.1.2.cmml">(</mo><mi id="A4.Ex65.m2.1.1" xref="A4.Ex65.m2.1.1.cmml">t</mi><mo id="A4.Ex65.m2.3.3.2.2.1.1.3" xref="A4.Ex65.m2.3.3.2.2.1.2.cmml">,</mo><msub id="A4.Ex65.m2.3.3.2.2.1.1.1" xref="A4.Ex65.m2.3.3.2.2.1.1.1.cmml"><mi id="A4.Ex65.m2.3.3.2.2.1.1.1.2" xref="A4.Ex65.m2.3.3.2.2.1.1.1.2.cmml">t</mi><mi id="A4.Ex65.m2.3.3.2.2.1.1.1.3" mathvariant="normal" xref="A4.Ex65.m2.3.3.2.2.1.1.1.3.cmml">e</mi></msub><mo id="A4.Ex65.m2.3.3.2.2.1.1.4" stretchy="false" xref="A4.Ex65.m2.3.3.2.2.1.2.cmml">)</mo></mrow><mo id="A4.Ex65.m2.3.3.2.2.2c" xref="A4.Ex65.m2.3.3.2.2.2.cmml">⁢</mo><mover accent="true" id="A4.Ex65.m2.3.3.2.2.6" xref="A4.Ex65.m2.3.3.2.2.6.cmml"><mi id="A4.Ex65.m2.3.3.2.2.6.2" xref="A4.Ex65.m2.3.3.2.2.6.2.cmml">𝜽</mi><mo id="A4.Ex65.m2.3.3.2.2.6.1" xref="A4.Ex65.m2.3.3.2.2.6.1.cmml">~</mo></mover></mrow></mrow><mo id="A4.Ex65.m2.4.4.4" xref="A4.Ex65.m2.4.4.4.cmml">+</mo><mrow id="A4.Ex65.m2.4.4.3" xref="A4.Ex65.m2.4.4.3.cmml"><msup id="A4.Ex65.m2.4.4.3.3" xref="A4.Ex65.m2.4.4.3.3.cmml"><mi id="A4.Ex65.m2.4.4.3.3.2" xref="A4.Ex65.m2.4.4.3.3.2.cmml">𝒆</mi><mi id="A4.Ex65.m2.4.4.3.3.3" xref="A4.Ex65.m2.4.4.3.3.3.cmml">T</mi></msup><mo id="A4.Ex65.m2.4.4.3.2" xref="A4.Ex65.m2.4.4.3.2.cmml">⁢</mo><mrow id="A4.Ex65.m2.4.4.3.1.1" xref="A4.Ex65.m2.4.4.3.1.1.1.cmml"><mo id="A4.Ex65.m2.4.4.3.1.1.2" stretchy="false" xref="A4.Ex65.m2.4.4.3.1.1.1.cmml">(</mo><mrow id="A4.Ex65.m2.4.4.3.1.1.1" xref="A4.Ex65.m2.4.4.3.1.1.1.cmml"><mi id="A4.Ex65.m2.4.4.3.1.1.1.2" mathvariant="normal" xref="A4.Ex65.m2.4.4.3.1.1.1.2.cmml">Φ</mi><mo id="A4.Ex65.m2.4.4.3.1.1.1.1" xref="A4.Ex65.m2.4.4.3.1.1.1.1.cmml">−</mo><msub id="A4.Ex65.m2.4.4.3.1.1.1.3" xref="A4.Ex65.m2.4.4.3.1.1.1.3.cmml"><mi id="A4.Ex65.m2.4.4.3.1.1.1.3.2" mathvariant="normal" xref="A4.Ex65.m2.4.4.3.1.1.1.3.2.cmml">Φ</mi><mi id="A4.Ex65.m2.4.4.3.1.1.1.3.3" mathvariant="normal" xref="A4.Ex65.m2.4.4.3.1.1.1.3.3.cmml">f</mi></msub></mrow><mo id="A4.Ex65.m2.4.4.3.1.1.3" stretchy="false" xref="A4.Ex65.m2.4.4.3.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex65.m2.4.4.3.2a" xref="A4.Ex65.m2.4.4.3.2.cmml">⁢</mo><mover accent="true" id="A4.Ex65.m2.4.4.3.4" xref="A4.Ex65.m2.4.4.3.4.cmml"><mi id="A4.Ex65.m2.4.4.3.4.2" xref="A4.Ex65.m2.4.4.3.4.2.cmml">𝜽</mi><mo id="A4.Ex65.m2.4.4.3.4.1" xref="A4.Ex65.m2.4.4.3.4.1.cmml">~</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex65.m2.4b"><apply id="A4.Ex65.m2.4.4.cmml" xref="A4.Ex65.m2.4.4"><plus id="A4.Ex65.m2.4.4.4.cmml" xref="A4.Ex65.m2.4.4.4"></plus><apply id="A4.Ex65.m2.3.3.2.cmml" xref="A4.Ex65.m2.3.3.2"><minus id="A4.Ex65.m2.3.3.2.3.cmml" xref="A4.Ex65.m2.3.3.2.3"></minus><apply id="A4.Ex65.m2.2.2.1.1.cmml" xref="A4.Ex65.m2.2.2.1.1"><minus id="A4.Ex65.m2.2.2.1.1.2.cmml" xref="A4.Ex65.m2.2.2.1.1"></minus><apply id="A4.Ex65.m2.2.2.1.1.1.cmml" xref="A4.Ex65.m2.2.2.1.1.1"><times id="A4.Ex65.m2.2.2.1.1.1.2.cmml" xref="A4.Ex65.m2.2.2.1.1.1.2"></times><apply id="A4.Ex65.m2.2.2.1.1.1.3.cmml" xref="A4.Ex65.m2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="A4.Ex65.m2.2.2.1.1.1.3.1.cmml" xref="A4.Ex65.m2.2.2.1.1.1.3">superscript</csymbol><ci id="A4.Ex65.m2.2.2.1.1.1.3.2.cmml" xref="A4.Ex65.m2.2.2.1.1.1.3.2">𝒆</ci><ci id="A4.Ex65.m2.2.2.1.1.1.3.3.cmml" xref="A4.Ex65.m2.2.2.1.1.1.3.3">𝑇</ci></apply><apply id="A4.Ex65.m2.2.2.1.1.1.1.1.1.cmml" xref="A4.Ex65.m2.2.2.1.1.1.1.1"><minus id="A4.Ex65.m2.2.2.1.1.1.1.1.1.1.cmml" xref="A4.Ex65.m2.2.2.1.1.1.1.1.1.1"></minus><ci id="A4.Ex65.m2.2.2.1.1.1.1.1.1.2.cmml" xref="A4.Ex65.m2.2.2.1.1.1.1.1.1.2">𝐾</ci><apply id="A4.Ex65.m2.2.2.1.1.1.1.1.1.3.cmml" xref="A4.Ex65.m2.2.2.1.1.1.1.1.1.3"><divide id="A4.Ex65.m2.2.2.1.1.1.1.1.1.3.1.cmml" xref="A4.Ex65.m2.2.2.1.1.1.1.1.1.3.1"></divide><ci id="A4.Ex65.m2.2.2.1.1.1.1.1.1.3.2.cmml" xref="A4.Ex65.m2.2.2.1.1.1.1.1.1.3.2">𝐼</ci><cn id="A4.Ex65.m2.2.2.1.1.1.1.1.1.3.3.cmml" type="integer" xref="A4.Ex65.m2.2.2.1.1.1.1.1.1.3.3">4</cn></apply></apply><ci id="A4.Ex65.m2.2.2.1.1.1.4.cmml" xref="A4.Ex65.m2.2.2.1.1.1.4">𝒆</ci></apply></apply><apply id="A4.Ex65.m2.3.3.2.2.cmml" xref="A4.Ex65.m2.3.3.2.2"><times id="A4.Ex65.m2.3.3.2.2.2.cmml" xref="A4.Ex65.m2.3.3.2.2.2"></times><ci id="A4.Ex65.m2.3.3.2.2.3.cmml" xref="A4.Ex65.m2.3.3.2.2.3">𝜅</ci><apply id="A4.Ex65.m2.3.3.2.2.4.cmml" xref="A4.Ex65.m2.3.3.2.2.4"><csymbol cd="ambiguous" id="A4.Ex65.m2.3.3.2.2.4.1.cmml" xref="A4.Ex65.m2.3.3.2.2.4">superscript</csymbol><apply id="A4.Ex65.m2.3.3.2.2.4.2.cmml" xref="A4.Ex65.m2.3.3.2.2.4.2"><ci id="A4.Ex65.m2.3.3.2.2.4.2.1.cmml" xref="A4.Ex65.m2.3.3.2.2.4.2.1">~</ci><ci id="A4.Ex65.m2.3.3.2.2.4.2.2.cmml" xref="A4.Ex65.m2.3.3.2.2.4.2.2">𝜽</ci></apply><ci id="A4.Ex65.m2.3.3.2.2.4.3.cmml" xref="A4.Ex65.m2.3.3.2.2.4.3">𝑇</ci></apply><ci id="A4.Ex65.m2.3.3.2.2.5.cmml" xref="A4.Ex65.m2.3.3.2.2.5">𝑄</ci><interval closure="open" id="A4.Ex65.m2.3.3.2.2.1.2.cmml" xref="A4.Ex65.m2.3.3.2.2.1.1"><ci id="A4.Ex65.m2.1.1.cmml" xref="A4.Ex65.m2.1.1">𝑡</ci><apply id="A4.Ex65.m2.3.3.2.2.1.1.1.cmml" xref="A4.Ex65.m2.3.3.2.2.1.1.1"><csymbol cd="ambiguous" id="A4.Ex65.m2.3.3.2.2.1.1.1.1.cmml" xref="A4.Ex65.m2.3.3.2.2.1.1.1">subscript</csymbol><ci id="A4.Ex65.m2.3.3.2.2.1.1.1.2.cmml" xref="A4.Ex65.m2.3.3.2.2.1.1.1.2">𝑡</ci><ci id="A4.Ex65.m2.3.3.2.2.1.1.1.3.cmml" xref="A4.Ex65.m2.3.3.2.2.1.1.1.3">e</ci></apply></interval><apply id="A4.Ex65.m2.3.3.2.2.6.cmml" xref="A4.Ex65.m2.3.3.2.2.6"><ci id="A4.Ex65.m2.3.3.2.2.6.1.cmml" xref="A4.Ex65.m2.3.3.2.2.6.1">~</ci><ci id="A4.Ex65.m2.3.3.2.2.6.2.cmml" xref="A4.Ex65.m2.3.3.2.2.6.2">𝜽</ci></apply></apply></apply><apply id="A4.Ex65.m2.4.4.3.cmml" xref="A4.Ex65.m2.4.4.3"><times id="A4.Ex65.m2.4.4.3.2.cmml" xref="A4.Ex65.m2.4.4.3.2"></times><apply id="A4.Ex65.m2.4.4.3.3.cmml" xref="A4.Ex65.m2.4.4.3.3"><csymbol cd="ambiguous" id="A4.Ex65.m2.4.4.3.3.1.cmml" xref="A4.Ex65.m2.4.4.3.3">superscript</csymbol><ci id="A4.Ex65.m2.4.4.3.3.2.cmml" xref="A4.Ex65.m2.4.4.3.3.2">𝒆</ci><ci id="A4.Ex65.m2.4.4.3.3.3.cmml" xref="A4.Ex65.m2.4.4.3.3.3">𝑇</ci></apply><apply id="A4.Ex65.m2.4.4.3.1.1.1.cmml" xref="A4.Ex65.m2.4.4.3.1.1"><minus id="A4.Ex65.m2.4.4.3.1.1.1.1.cmml" xref="A4.Ex65.m2.4.4.3.1.1.1.1"></minus><ci id="A4.Ex65.m2.4.4.3.1.1.1.2.cmml" xref="A4.Ex65.m2.4.4.3.1.1.1.2">Φ</ci><apply id="A4.Ex65.m2.4.4.3.1.1.1.3.cmml" xref="A4.Ex65.m2.4.4.3.1.1.1.3"><csymbol cd="ambiguous" id="A4.Ex65.m2.4.4.3.1.1.1.3.1.cmml" xref="A4.Ex65.m2.4.4.3.1.1.1.3">subscript</csymbol><ci id="A4.Ex65.m2.4.4.3.1.1.1.3.2.cmml" xref="A4.Ex65.m2.4.4.3.1.1.1.3.2">Φ</ci><ci id="A4.Ex65.m2.4.4.3.1.1.1.3.3.cmml" xref="A4.Ex65.m2.4.4.3.1.1.1.3.3">f</ci></apply></apply><apply id="A4.Ex65.m2.4.4.3.4.cmml" xref="A4.Ex65.m2.4.4.3.4"><ci id="A4.Ex65.m2.4.4.3.4.1.cmml" xref="A4.Ex65.m2.4.4.3.4.1">~</ci><ci id="A4.Ex65.m2.4.4.3.4.2.cmml" xref="A4.Ex65.m2.4.4.3.4.2">𝜽</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex65.m2.4c">\displaystyle-\bm{e}^{T}(K-I/4)\bm{e}-\kappa\tilde{\bm{\theta}}^{T}Q(t,t_{\rm e% }){\tilde{\bm{\theta}}}+\bm{e}^{T}(\Phi-\Phi_{\rm f})\tilde{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="A4.Ex65.m2.4d">- bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_K - italic_I / 4 ) bold_italic_e - italic_κ over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT italic_Q ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG + bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( roman_Φ - roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex66"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle+\bm{e}^{T}\bm{d}-(1+p)\tilde{\bm{\theta}}^{T}\Phi_{{\rm f}}{\bm{% d}}_{\rm f}-\kappa(1+p)\tilde{\bm{\theta}}^{T}{\bm{d}}_{\rm g}." class="ltx_Math" display="inline" id="A4.Ex66.m1.1"><semantics id="A4.Ex66.m1.1a"><mrow id="A4.Ex66.m1.1.1.1" xref="A4.Ex66.m1.1.1.1.1.cmml"><mrow id="A4.Ex66.m1.1.1.1.1" xref="A4.Ex66.m1.1.1.1.1.cmml"><mrow id="A4.Ex66.m1.1.1.1.1.4" xref="A4.Ex66.m1.1.1.1.1.4.cmml"><mo id="A4.Ex66.m1.1.1.1.1.4a" xref="A4.Ex66.m1.1.1.1.1.4.cmml">+</mo><mrow id="A4.Ex66.m1.1.1.1.1.4.2" xref="A4.Ex66.m1.1.1.1.1.4.2.cmml"><msup id="A4.Ex66.m1.1.1.1.1.4.2.2" xref="A4.Ex66.m1.1.1.1.1.4.2.2.cmml"><mi id="A4.Ex66.m1.1.1.1.1.4.2.2.2" xref="A4.Ex66.m1.1.1.1.1.4.2.2.2.cmml">𝒆</mi><mi id="A4.Ex66.m1.1.1.1.1.4.2.2.3" xref="A4.Ex66.m1.1.1.1.1.4.2.2.3.cmml">T</mi></msup><mo id="A4.Ex66.m1.1.1.1.1.4.2.1" xref="A4.Ex66.m1.1.1.1.1.4.2.1.cmml">⁢</mo><mi id="A4.Ex66.m1.1.1.1.1.4.2.3" xref="A4.Ex66.m1.1.1.1.1.4.2.3.cmml">𝒅</mi></mrow></mrow><mo id="A4.Ex66.m1.1.1.1.1.3" xref="A4.Ex66.m1.1.1.1.1.3.cmml">−</mo><mrow id="A4.Ex66.m1.1.1.1.1.1" xref="A4.Ex66.m1.1.1.1.1.1.cmml"><mrow id="A4.Ex66.m1.1.1.1.1.1.1.1" xref="A4.Ex66.m1.1.1.1.1.1.1.1.1.cmml"><mo id="A4.Ex66.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex66.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex66.m1.1.1.1.1.1.1.1.1" xref="A4.Ex66.m1.1.1.1.1.1.1.1.1.cmml"><mn id="A4.Ex66.m1.1.1.1.1.1.1.1.1.2" xref="A4.Ex66.m1.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A4.Ex66.m1.1.1.1.1.1.1.1.1.1" xref="A4.Ex66.m1.1.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="A4.Ex66.m1.1.1.1.1.1.1.1.1.3" xref="A4.Ex66.m1.1.1.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A4.Ex66.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex66.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex66.m1.1.1.1.1.1.2" xref="A4.Ex66.m1.1.1.1.1.1.2.cmml">⁢</mo><msup id="A4.Ex66.m1.1.1.1.1.1.3" xref="A4.Ex66.m1.1.1.1.1.1.3.cmml"><mover accent="true" id="A4.Ex66.m1.1.1.1.1.1.3.2" xref="A4.Ex66.m1.1.1.1.1.1.3.2.cmml"><mi id="A4.Ex66.m1.1.1.1.1.1.3.2.2" xref="A4.Ex66.m1.1.1.1.1.1.3.2.2.cmml">𝜽</mi><mo id="A4.Ex66.m1.1.1.1.1.1.3.2.1" xref="A4.Ex66.m1.1.1.1.1.1.3.2.1.cmml">~</mo></mover><mi id="A4.Ex66.m1.1.1.1.1.1.3.3" xref="A4.Ex66.m1.1.1.1.1.1.3.3.cmml">T</mi></msup><mo id="A4.Ex66.m1.1.1.1.1.1.2a" xref="A4.Ex66.m1.1.1.1.1.1.2.cmml">⁢</mo><msub id="A4.Ex66.m1.1.1.1.1.1.4" xref="A4.Ex66.m1.1.1.1.1.1.4.cmml"><mi id="A4.Ex66.m1.1.1.1.1.1.4.2" mathvariant="normal" xref="A4.Ex66.m1.1.1.1.1.1.4.2.cmml">Φ</mi><mi id="A4.Ex66.m1.1.1.1.1.1.4.3" mathvariant="normal" xref="A4.Ex66.m1.1.1.1.1.1.4.3.cmml">f</mi></msub><mo id="A4.Ex66.m1.1.1.1.1.1.2b" xref="A4.Ex66.m1.1.1.1.1.1.2.cmml">⁢</mo><msub id="A4.Ex66.m1.1.1.1.1.1.5" xref="A4.Ex66.m1.1.1.1.1.1.5.cmml"><mi id="A4.Ex66.m1.1.1.1.1.1.5.2" xref="A4.Ex66.m1.1.1.1.1.1.5.2.cmml">𝒅</mi><mi id="A4.Ex66.m1.1.1.1.1.1.5.3" mathvariant="normal" xref="A4.Ex66.m1.1.1.1.1.1.5.3.cmml">f</mi></msub></mrow><mo id="A4.Ex66.m1.1.1.1.1.3a" xref="A4.Ex66.m1.1.1.1.1.3.cmml">−</mo><mrow id="A4.Ex66.m1.1.1.1.1.2" xref="A4.Ex66.m1.1.1.1.1.2.cmml"><mi id="A4.Ex66.m1.1.1.1.1.2.3" xref="A4.Ex66.m1.1.1.1.1.2.3.cmml">κ</mi><mo id="A4.Ex66.m1.1.1.1.1.2.2" xref="A4.Ex66.m1.1.1.1.1.2.2.cmml">⁢</mo><mrow id="A4.Ex66.m1.1.1.1.1.2.1.1" xref="A4.Ex66.m1.1.1.1.1.2.1.1.1.cmml"><mo id="A4.Ex66.m1.1.1.1.1.2.1.1.2" stretchy="false" xref="A4.Ex66.m1.1.1.1.1.2.1.1.1.cmml">(</mo><mrow id="A4.Ex66.m1.1.1.1.1.2.1.1.1" xref="A4.Ex66.m1.1.1.1.1.2.1.1.1.cmml"><mn id="A4.Ex66.m1.1.1.1.1.2.1.1.1.2" xref="A4.Ex66.m1.1.1.1.1.2.1.1.1.2.cmml">1</mn><mo id="A4.Ex66.m1.1.1.1.1.2.1.1.1.1" xref="A4.Ex66.m1.1.1.1.1.2.1.1.1.1.cmml">+</mo><mi id="A4.Ex66.m1.1.1.1.1.2.1.1.1.3" xref="A4.Ex66.m1.1.1.1.1.2.1.1.1.3.cmml">p</mi></mrow><mo id="A4.Ex66.m1.1.1.1.1.2.1.1.3" stretchy="false" xref="A4.Ex66.m1.1.1.1.1.2.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex66.m1.1.1.1.1.2.2a" xref="A4.Ex66.m1.1.1.1.1.2.2.cmml">⁢</mo><msup id="A4.Ex66.m1.1.1.1.1.2.4" xref="A4.Ex66.m1.1.1.1.1.2.4.cmml"><mover accent="true" id="A4.Ex66.m1.1.1.1.1.2.4.2" xref="A4.Ex66.m1.1.1.1.1.2.4.2.cmml"><mi id="A4.Ex66.m1.1.1.1.1.2.4.2.2" xref="A4.Ex66.m1.1.1.1.1.2.4.2.2.cmml">𝜽</mi><mo id="A4.Ex66.m1.1.1.1.1.2.4.2.1" xref="A4.Ex66.m1.1.1.1.1.2.4.2.1.cmml">~</mo></mover><mi id="A4.Ex66.m1.1.1.1.1.2.4.3" xref="A4.Ex66.m1.1.1.1.1.2.4.3.cmml">T</mi></msup><mo id="A4.Ex66.m1.1.1.1.1.2.2b" xref="A4.Ex66.m1.1.1.1.1.2.2.cmml">⁢</mo><msub id="A4.Ex66.m1.1.1.1.1.2.5" xref="A4.Ex66.m1.1.1.1.1.2.5.cmml"><mi id="A4.Ex66.m1.1.1.1.1.2.5.2" xref="A4.Ex66.m1.1.1.1.1.2.5.2.cmml">𝒅</mi><mi id="A4.Ex66.m1.1.1.1.1.2.5.3" mathvariant="normal" xref="A4.Ex66.m1.1.1.1.1.2.5.3.cmml">g</mi></msub></mrow></mrow><mo id="A4.Ex66.m1.1.1.1.2" lspace="0em" xref="A4.Ex66.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex66.m1.1b"><apply id="A4.Ex66.m1.1.1.1.1.cmml" xref="A4.Ex66.m1.1.1.1"><minus id="A4.Ex66.m1.1.1.1.1.3.cmml" xref="A4.Ex66.m1.1.1.1.1.3"></minus><apply id="A4.Ex66.m1.1.1.1.1.4.cmml" xref="A4.Ex66.m1.1.1.1.1.4"><plus id="A4.Ex66.m1.1.1.1.1.4.1.cmml" xref="A4.Ex66.m1.1.1.1.1.4"></plus><apply id="A4.Ex66.m1.1.1.1.1.4.2.cmml" xref="A4.Ex66.m1.1.1.1.1.4.2"><times id="A4.Ex66.m1.1.1.1.1.4.2.1.cmml" xref="A4.Ex66.m1.1.1.1.1.4.2.1"></times><apply id="A4.Ex66.m1.1.1.1.1.4.2.2.cmml" xref="A4.Ex66.m1.1.1.1.1.4.2.2"><csymbol cd="ambiguous" id="A4.Ex66.m1.1.1.1.1.4.2.2.1.cmml" xref="A4.Ex66.m1.1.1.1.1.4.2.2">superscript</csymbol><ci id="A4.Ex66.m1.1.1.1.1.4.2.2.2.cmml" xref="A4.Ex66.m1.1.1.1.1.4.2.2.2">𝒆</ci><ci id="A4.Ex66.m1.1.1.1.1.4.2.2.3.cmml" xref="A4.Ex66.m1.1.1.1.1.4.2.2.3">𝑇</ci></apply><ci id="A4.Ex66.m1.1.1.1.1.4.2.3.cmml" xref="A4.Ex66.m1.1.1.1.1.4.2.3">𝒅</ci></apply></apply><apply id="A4.Ex66.m1.1.1.1.1.1.cmml" xref="A4.Ex66.m1.1.1.1.1.1"><times id="A4.Ex66.m1.1.1.1.1.1.2.cmml" xref="A4.Ex66.m1.1.1.1.1.1.2"></times><apply id="A4.Ex66.m1.1.1.1.1.1.1.1.1.cmml" xref="A4.Ex66.m1.1.1.1.1.1.1.1"><plus id="A4.Ex66.m1.1.1.1.1.1.1.1.1.1.cmml" xref="A4.Ex66.m1.1.1.1.1.1.1.1.1.1"></plus><cn id="A4.Ex66.m1.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A4.Ex66.m1.1.1.1.1.1.1.1.1.2">1</cn><ci id="A4.Ex66.m1.1.1.1.1.1.1.1.1.3.cmml" xref="A4.Ex66.m1.1.1.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="A4.Ex66.m1.1.1.1.1.1.3.cmml" xref="A4.Ex66.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.Ex66.m1.1.1.1.1.1.3.1.cmml" xref="A4.Ex66.m1.1.1.1.1.1.3">superscript</csymbol><apply id="A4.Ex66.m1.1.1.1.1.1.3.2.cmml" xref="A4.Ex66.m1.1.1.1.1.1.3.2"><ci id="A4.Ex66.m1.1.1.1.1.1.3.2.1.cmml" xref="A4.Ex66.m1.1.1.1.1.1.3.2.1">~</ci><ci id="A4.Ex66.m1.1.1.1.1.1.3.2.2.cmml" xref="A4.Ex66.m1.1.1.1.1.1.3.2.2">𝜽</ci></apply><ci id="A4.Ex66.m1.1.1.1.1.1.3.3.cmml" xref="A4.Ex66.m1.1.1.1.1.1.3.3">𝑇</ci></apply><apply id="A4.Ex66.m1.1.1.1.1.1.4.cmml" xref="A4.Ex66.m1.1.1.1.1.1.4"><csymbol cd="ambiguous" id="A4.Ex66.m1.1.1.1.1.1.4.1.cmml" xref="A4.Ex66.m1.1.1.1.1.1.4">subscript</csymbol><ci id="A4.Ex66.m1.1.1.1.1.1.4.2.cmml" xref="A4.Ex66.m1.1.1.1.1.1.4.2">Φ</ci><ci id="A4.Ex66.m1.1.1.1.1.1.4.3.cmml" xref="A4.Ex66.m1.1.1.1.1.1.4.3">f</ci></apply><apply id="A4.Ex66.m1.1.1.1.1.1.5.cmml" xref="A4.Ex66.m1.1.1.1.1.1.5"><csymbol cd="ambiguous" id="A4.Ex66.m1.1.1.1.1.1.5.1.cmml" xref="A4.Ex66.m1.1.1.1.1.1.5">subscript</csymbol><ci id="A4.Ex66.m1.1.1.1.1.1.5.2.cmml" xref="A4.Ex66.m1.1.1.1.1.1.5.2">𝒅</ci><ci id="A4.Ex66.m1.1.1.1.1.1.5.3.cmml" xref="A4.Ex66.m1.1.1.1.1.1.5.3">f</ci></apply></apply><apply id="A4.Ex66.m1.1.1.1.1.2.cmml" xref="A4.Ex66.m1.1.1.1.1.2"><times id="A4.Ex66.m1.1.1.1.1.2.2.cmml" xref="A4.Ex66.m1.1.1.1.1.2.2"></times><ci id="A4.Ex66.m1.1.1.1.1.2.3.cmml" xref="A4.Ex66.m1.1.1.1.1.2.3">𝜅</ci><apply id="A4.Ex66.m1.1.1.1.1.2.1.1.1.cmml" xref="A4.Ex66.m1.1.1.1.1.2.1.1"><plus id="A4.Ex66.m1.1.1.1.1.2.1.1.1.1.cmml" xref="A4.Ex66.m1.1.1.1.1.2.1.1.1.1"></plus><cn id="A4.Ex66.m1.1.1.1.1.2.1.1.1.2.cmml" type="integer" xref="A4.Ex66.m1.1.1.1.1.2.1.1.1.2">1</cn><ci id="A4.Ex66.m1.1.1.1.1.2.1.1.1.3.cmml" xref="A4.Ex66.m1.1.1.1.1.2.1.1.1.3">𝑝</ci></apply><apply id="A4.Ex66.m1.1.1.1.1.2.4.cmml" xref="A4.Ex66.m1.1.1.1.1.2.4"><csymbol cd="ambiguous" id="A4.Ex66.m1.1.1.1.1.2.4.1.cmml" xref="A4.Ex66.m1.1.1.1.1.2.4">superscript</csymbol><apply id="A4.Ex66.m1.1.1.1.1.2.4.2.cmml" xref="A4.Ex66.m1.1.1.1.1.2.4.2"><ci id="A4.Ex66.m1.1.1.1.1.2.4.2.1.cmml" xref="A4.Ex66.m1.1.1.1.1.2.4.2.1">~</ci><ci id="A4.Ex66.m1.1.1.1.1.2.4.2.2.cmml" xref="A4.Ex66.m1.1.1.1.1.2.4.2.2">𝜽</ci></apply><ci id="A4.Ex66.m1.1.1.1.1.2.4.3.cmml" xref="A4.Ex66.m1.1.1.1.1.2.4.3">𝑇</ci></apply><apply id="A4.Ex66.m1.1.1.1.1.2.5.cmml" xref="A4.Ex66.m1.1.1.1.1.2.5"><csymbol cd="ambiguous" id="A4.Ex66.m1.1.1.1.1.2.5.1.cmml" xref="A4.Ex66.m1.1.1.1.1.2.5">subscript</csymbol><ci id="A4.Ex66.m1.1.1.1.1.2.5.2.cmml" xref="A4.Ex66.m1.1.1.1.1.2.5.2">𝒅</ci><ci id="A4.Ex66.m1.1.1.1.1.2.5.3.cmml" xref="A4.Ex66.m1.1.1.1.1.2.5.3">g</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex66.m1.1c">\displaystyle+\bm{e}^{T}\bm{d}-(1+p)\tilde{\bm{\theta}}^{T}\Phi_{{\rm f}}{\bm{% d}}_{\rm f}-\kappa(1+p)\tilde{\bm{\theta}}^{T}{\bm{d}}_{\rm g}.</annotation><annotation encoding="application/x-llamapun" id="A4.Ex66.m1.1d">+ bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_d - ( 1 + italic_p ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT bold_italic_d start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT - italic_κ ( 1 + italic_p ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_d start_POSTSUBSCRIPT roman_g end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.1.p1.12">Noting that the partial IE condition holds at the beginning and some moments later and using Theorem 1 and (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A3.E52" title="In Proof. ‣ Appendix C The proof of Theorem 2 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">52</span></a>), the robustness of the estimation error <math alttext="\tilde{\bm{\theta}}" class="ltx_Math" display="inline" id="A4.1.p1.1.m1.1"><semantics id="A4.1.p1.1.m1.1a"><mover accent="true" id="A4.1.p1.1.m1.1.1" xref="A4.1.p1.1.m1.1.1.cmml"><mi id="A4.1.p1.1.m1.1.1.2" xref="A4.1.p1.1.m1.1.1.2.cmml">𝜽</mi><mo id="A4.1.p1.1.m1.1.1.1" xref="A4.1.p1.1.m1.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="A4.1.p1.1.m1.1b"><apply id="A4.1.p1.1.m1.1.1.cmml" xref="A4.1.p1.1.m1.1.1"><ci id="A4.1.p1.1.m1.1.1.1.cmml" xref="A4.1.p1.1.m1.1.1.1">~</ci><ci id="A4.1.p1.1.m1.1.1.2.cmml" xref="A4.1.p1.1.m1.1.1.2">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.1.m1.1c">\tilde{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.1.m1.1d">over~ start_ARG bold_italic_θ end_ARG</annotation></semantics></math> is ensured, thereby guaranteeing that 1) <math alttext="\tilde{\bm{\theta}}(t)\in L_{\infty}" class="ltx_Math" display="inline" id="A4.1.p1.2.m2.1"><semantics id="A4.1.p1.2.m2.1a"><mrow id="A4.1.p1.2.m2.1.2" xref="A4.1.p1.2.m2.1.2.cmml"><mrow id="A4.1.p1.2.m2.1.2.2" xref="A4.1.p1.2.m2.1.2.2.cmml"><mover accent="true" id="A4.1.p1.2.m2.1.2.2.2" xref="A4.1.p1.2.m2.1.2.2.2.cmml"><mi id="A4.1.p1.2.m2.1.2.2.2.2" xref="A4.1.p1.2.m2.1.2.2.2.2.cmml">𝜽</mi><mo id="A4.1.p1.2.m2.1.2.2.2.1" xref="A4.1.p1.2.m2.1.2.2.2.1.cmml">~</mo></mover><mo id="A4.1.p1.2.m2.1.2.2.1" xref="A4.1.p1.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="A4.1.p1.2.m2.1.2.2.3.2" xref="A4.1.p1.2.m2.1.2.2.cmml"><mo id="A4.1.p1.2.m2.1.2.2.3.2.1" stretchy="false" xref="A4.1.p1.2.m2.1.2.2.cmml">(</mo><mi id="A4.1.p1.2.m2.1.1" xref="A4.1.p1.2.m2.1.1.cmml">t</mi><mo id="A4.1.p1.2.m2.1.2.2.3.2.2" stretchy="false" xref="A4.1.p1.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="A4.1.p1.2.m2.1.2.1" xref="A4.1.p1.2.m2.1.2.1.cmml">∈</mo><msub id="A4.1.p1.2.m2.1.2.3" xref="A4.1.p1.2.m2.1.2.3.cmml"><mi id="A4.1.p1.2.m2.1.2.3.2" xref="A4.1.p1.2.m2.1.2.3.2.cmml">L</mi><mi id="A4.1.p1.2.m2.1.2.3.3" mathvariant="normal" xref="A4.1.p1.2.m2.1.2.3.3.cmml">∞</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.2.m2.1b"><apply id="A4.1.p1.2.m2.1.2.cmml" xref="A4.1.p1.2.m2.1.2"><in id="A4.1.p1.2.m2.1.2.1.cmml" xref="A4.1.p1.2.m2.1.2.1"></in><apply id="A4.1.p1.2.m2.1.2.2.cmml" xref="A4.1.p1.2.m2.1.2.2"><times id="A4.1.p1.2.m2.1.2.2.1.cmml" xref="A4.1.p1.2.m2.1.2.2.1"></times><apply id="A4.1.p1.2.m2.1.2.2.2.cmml" xref="A4.1.p1.2.m2.1.2.2.2"><ci id="A4.1.p1.2.m2.1.2.2.2.1.cmml" xref="A4.1.p1.2.m2.1.2.2.2.1">~</ci><ci id="A4.1.p1.2.m2.1.2.2.2.2.cmml" xref="A4.1.p1.2.m2.1.2.2.2.2">𝜽</ci></apply><ci id="A4.1.p1.2.m2.1.1.cmml" xref="A4.1.p1.2.m2.1.1">𝑡</ci></apply><apply id="A4.1.p1.2.m2.1.2.3.cmml" xref="A4.1.p1.2.m2.1.2.3"><csymbol cd="ambiguous" id="A4.1.p1.2.m2.1.2.3.1.cmml" xref="A4.1.p1.2.m2.1.2.3">subscript</csymbol><ci id="A4.1.p1.2.m2.1.2.3.2.cmml" xref="A4.1.p1.2.m2.1.2.3.2">𝐿</ci><infinity id="A4.1.p1.2.m2.1.2.3.3.cmml" xref="A4.1.p1.2.m2.1.2.3.3"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.2.m2.1c">\tilde{\bm{\theta}}(t)\in L_{\infty}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.2.m2.1d">over~ start_ARG bold_italic_θ end_ARG ( italic_t ) ∈ italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\forall t\in[0,\infty)" class="ltx_Math" display="inline" id="A4.1.p1.3.m3.2"><semantics id="A4.1.p1.3.m3.2a"><mrow id="A4.1.p1.3.m3.2.3" xref="A4.1.p1.3.m3.2.3.cmml"><mrow id="A4.1.p1.3.m3.2.3.2" xref="A4.1.p1.3.m3.2.3.2.cmml"><mo id="A4.1.p1.3.m3.2.3.2.1" rspace="0.167em" xref="A4.1.p1.3.m3.2.3.2.1.cmml">∀</mo><mi id="A4.1.p1.3.m3.2.3.2.2" xref="A4.1.p1.3.m3.2.3.2.2.cmml">t</mi></mrow><mo id="A4.1.p1.3.m3.2.3.1" xref="A4.1.p1.3.m3.2.3.1.cmml">∈</mo><mrow id="A4.1.p1.3.m3.2.3.3.2" xref="A4.1.p1.3.m3.2.3.3.1.cmml"><mo id="A4.1.p1.3.m3.2.3.3.2.1" stretchy="false" xref="A4.1.p1.3.m3.2.3.3.1.cmml">[</mo><mn id="A4.1.p1.3.m3.1.1" xref="A4.1.p1.3.m3.1.1.cmml">0</mn><mo id="A4.1.p1.3.m3.2.3.3.2.2" xref="A4.1.p1.3.m3.2.3.3.1.cmml">,</mo><mi id="A4.1.p1.3.m3.2.2" mathvariant="normal" xref="A4.1.p1.3.m3.2.2.cmml">∞</mi><mo id="A4.1.p1.3.m3.2.3.3.2.3" stretchy="false" xref="A4.1.p1.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.3.m3.2b"><apply id="A4.1.p1.3.m3.2.3.cmml" xref="A4.1.p1.3.m3.2.3"><in id="A4.1.p1.3.m3.2.3.1.cmml" xref="A4.1.p1.3.m3.2.3.1"></in><apply id="A4.1.p1.3.m3.2.3.2.cmml" xref="A4.1.p1.3.m3.2.3.2"><csymbol cd="latexml" id="A4.1.p1.3.m3.2.3.2.1.cmml" xref="A4.1.p1.3.m3.2.3.2.1">for-all</csymbol><ci id="A4.1.p1.3.m3.2.3.2.2.cmml" xref="A4.1.p1.3.m3.2.3.2.2">𝑡</ci></apply><interval closure="closed-open" id="A4.1.p1.3.m3.2.3.3.1.cmml" xref="A4.1.p1.3.m3.2.3.3.2"><cn id="A4.1.p1.3.m3.1.1.cmml" type="integer" xref="A4.1.p1.3.m3.1.1">0</cn><infinity id="A4.1.p1.3.m3.2.2.cmml" xref="A4.1.p1.3.m3.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.3.m3.2c">\forall t\in[0,\infty)</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.3.m3.2d">∀ italic_t ∈ [ 0 , ∞ )</annotation></semantics></math> such that there exists a constant <math alttext="c_{\theta}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A4.1.p1.4.m4.1"><semantics id="A4.1.p1.4.m4.1a"><mrow id="A4.1.p1.4.m4.1.1" xref="A4.1.p1.4.m4.1.1.cmml"><msub id="A4.1.p1.4.m4.1.1.2" xref="A4.1.p1.4.m4.1.1.2.cmml"><mi id="A4.1.p1.4.m4.1.1.2.2" xref="A4.1.p1.4.m4.1.1.2.2.cmml">c</mi><mi id="A4.1.p1.4.m4.1.1.2.3" xref="A4.1.p1.4.m4.1.1.2.3.cmml">θ</mi></msub><mo id="A4.1.p1.4.m4.1.1.1" xref="A4.1.p1.4.m4.1.1.1.cmml">∈</mo><msup id="A4.1.p1.4.m4.1.1.3" xref="A4.1.p1.4.m4.1.1.3.cmml"><mi id="A4.1.p1.4.m4.1.1.3.2" xref="A4.1.p1.4.m4.1.1.3.2.cmml">ℝ</mi><mo id="A4.1.p1.4.m4.1.1.3.3" xref="A4.1.p1.4.m4.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.4.m4.1b"><apply id="A4.1.p1.4.m4.1.1.cmml" xref="A4.1.p1.4.m4.1.1"><in id="A4.1.p1.4.m4.1.1.1.cmml" xref="A4.1.p1.4.m4.1.1.1"></in><apply id="A4.1.p1.4.m4.1.1.2.cmml" xref="A4.1.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="A4.1.p1.4.m4.1.1.2.1.cmml" xref="A4.1.p1.4.m4.1.1.2">subscript</csymbol><ci id="A4.1.p1.4.m4.1.1.2.2.cmml" xref="A4.1.p1.4.m4.1.1.2.2">𝑐</ci><ci id="A4.1.p1.4.m4.1.1.2.3.cmml" xref="A4.1.p1.4.m4.1.1.2.3">𝜃</ci></apply><apply id="A4.1.p1.4.m4.1.1.3.cmml" xref="A4.1.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.4.m4.1.1.3.1.cmml" xref="A4.1.p1.4.m4.1.1.3">superscript</csymbol><ci id="A4.1.p1.4.m4.1.1.3.2.cmml" xref="A4.1.p1.4.m4.1.1.3.2">ℝ</ci><plus id="A4.1.p1.4.m4.1.1.3.3.cmml" xref="A4.1.p1.4.m4.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.4.m4.1c">c_{\theta}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.4.m4.1d">italic_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> that satisfies <math alttext="\|\tilde{\bm{\theta}}(t)\|\leq c_{\theta}" class="ltx_Math" display="inline" id="A4.1.p1.5.m5.2"><semantics id="A4.1.p1.5.m5.2a"><mrow id="A4.1.p1.5.m5.2.2" xref="A4.1.p1.5.m5.2.2.cmml"><mrow id="A4.1.p1.5.m5.2.2.1.1" xref="A4.1.p1.5.m5.2.2.1.2.cmml"><mo id="A4.1.p1.5.m5.2.2.1.1.2" stretchy="false" xref="A4.1.p1.5.m5.2.2.1.2.1.cmml">‖</mo><mrow id="A4.1.p1.5.m5.2.2.1.1.1" xref="A4.1.p1.5.m5.2.2.1.1.1.cmml"><mover accent="true" id="A4.1.p1.5.m5.2.2.1.1.1.2" xref="A4.1.p1.5.m5.2.2.1.1.1.2.cmml"><mi id="A4.1.p1.5.m5.2.2.1.1.1.2.2" xref="A4.1.p1.5.m5.2.2.1.1.1.2.2.cmml">𝜽</mi><mo id="A4.1.p1.5.m5.2.2.1.1.1.2.1" xref="A4.1.p1.5.m5.2.2.1.1.1.2.1.cmml">~</mo></mover><mo id="A4.1.p1.5.m5.2.2.1.1.1.1" xref="A4.1.p1.5.m5.2.2.1.1.1.1.cmml">⁢</mo><mrow id="A4.1.p1.5.m5.2.2.1.1.1.3.2" xref="A4.1.p1.5.m5.2.2.1.1.1.cmml"><mo id="A4.1.p1.5.m5.2.2.1.1.1.3.2.1" stretchy="false" xref="A4.1.p1.5.m5.2.2.1.1.1.cmml">(</mo><mi id="A4.1.p1.5.m5.1.1" xref="A4.1.p1.5.m5.1.1.cmml">t</mi><mo id="A4.1.p1.5.m5.2.2.1.1.1.3.2.2" stretchy="false" xref="A4.1.p1.5.m5.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="A4.1.p1.5.m5.2.2.1.1.3" stretchy="false" xref="A4.1.p1.5.m5.2.2.1.2.1.cmml">‖</mo></mrow><mo id="A4.1.p1.5.m5.2.2.2" xref="A4.1.p1.5.m5.2.2.2.cmml">≤</mo><msub id="A4.1.p1.5.m5.2.2.3" xref="A4.1.p1.5.m5.2.2.3.cmml"><mi id="A4.1.p1.5.m5.2.2.3.2" xref="A4.1.p1.5.m5.2.2.3.2.cmml">c</mi><mi id="A4.1.p1.5.m5.2.2.3.3" xref="A4.1.p1.5.m5.2.2.3.3.cmml">θ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.5.m5.2b"><apply id="A4.1.p1.5.m5.2.2.cmml" xref="A4.1.p1.5.m5.2.2"><leq id="A4.1.p1.5.m5.2.2.2.cmml" xref="A4.1.p1.5.m5.2.2.2"></leq><apply id="A4.1.p1.5.m5.2.2.1.2.cmml" xref="A4.1.p1.5.m5.2.2.1.1"><csymbol cd="latexml" id="A4.1.p1.5.m5.2.2.1.2.1.cmml" xref="A4.1.p1.5.m5.2.2.1.1.2">norm</csymbol><apply id="A4.1.p1.5.m5.2.2.1.1.1.cmml" xref="A4.1.p1.5.m5.2.2.1.1.1"><times id="A4.1.p1.5.m5.2.2.1.1.1.1.cmml" xref="A4.1.p1.5.m5.2.2.1.1.1.1"></times><apply id="A4.1.p1.5.m5.2.2.1.1.1.2.cmml" xref="A4.1.p1.5.m5.2.2.1.1.1.2"><ci id="A4.1.p1.5.m5.2.2.1.1.1.2.1.cmml" xref="A4.1.p1.5.m5.2.2.1.1.1.2.1">~</ci><ci id="A4.1.p1.5.m5.2.2.1.1.1.2.2.cmml" xref="A4.1.p1.5.m5.2.2.1.1.1.2.2">𝜽</ci></apply><ci id="A4.1.p1.5.m5.1.1.cmml" xref="A4.1.p1.5.m5.1.1">𝑡</ci></apply></apply><apply id="A4.1.p1.5.m5.2.2.3.cmml" xref="A4.1.p1.5.m5.2.2.3"><csymbol cd="ambiguous" id="A4.1.p1.5.m5.2.2.3.1.cmml" xref="A4.1.p1.5.m5.2.2.3">subscript</csymbol><ci id="A4.1.p1.5.m5.2.2.3.2.cmml" xref="A4.1.p1.5.m5.2.2.3.2">𝑐</ci><ci id="A4.1.p1.5.m5.2.2.3.3.cmml" xref="A4.1.p1.5.m5.2.2.3.3">𝜃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.5.m5.2c">\|\tilde{\bm{\theta}}(t)\|\leq c_{\theta}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.5.m5.2d">∥ over~ start_ARG bold_italic_θ end_ARG ( italic_t ) ∥ ≤ italic_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\forall t\in[0,\infty)" class="ltx_Math" display="inline" id="A4.1.p1.6.m6.2"><semantics id="A4.1.p1.6.m6.2a"><mrow id="A4.1.p1.6.m6.2.3" xref="A4.1.p1.6.m6.2.3.cmml"><mrow id="A4.1.p1.6.m6.2.3.2" xref="A4.1.p1.6.m6.2.3.2.cmml"><mo id="A4.1.p1.6.m6.2.3.2.1" rspace="0.167em" xref="A4.1.p1.6.m6.2.3.2.1.cmml">∀</mo><mi id="A4.1.p1.6.m6.2.3.2.2" xref="A4.1.p1.6.m6.2.3.2.2.cmml">t</mi></mrow><mo id="A4.1.p1.6.m6.2.3.1" xref="A4.1.p1.6.m6.2.3.1.cmml">∈</mo><mrow id="A4.1.p1.6.m6.2.3.3.2" xref="A4.1.p1.6.m6.2.3.3.1.cmml"><mo id="A4.1.p1.6.m6.2.3.3.2.1" stretchy="false" xref="A4.1.p1.6.m6.2.3.3.1.cmml">[</mo><mn id="A4.1.p1.6.m6.1.1" xref="A4.1.p1.6.m6.1.1.cmml">0</mn><mo id="A4.1.p1.6.m6.2.3.3.2.2" xref="A4.1.p1.6.m6.2.3.3.1.cmml">,</mo><mi id="A4.1.p1.6.m6.2.2" mathvariant="normal" xref="A4.1.p1.6.m6.2.2.cmml">∞</mi><mo id="A4.1.p1.6.m6.2.3.3.2.3" stretchy="false" xref="A4.1.p1.6.m6.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.6.m6.2b"><apply id="A4.1.p1.6.m6.2.3.cmml" xref="A4.1.p1.6.m6.2.3"><in id="A4.1.p1.6.m6.2.3.1.cmml" xref="A4.1.p1.6.m6.2.3.1"></in><apply id="A4.1.p1.6.m6.2.3.2.cmml" xref="A4.1.p1.6.m6.2.3.2"><csymbol cd="latexml" id="A4.1.p1.6.m6.2.3.2.1.cmml" xref="A4.1.p1.6.m6.2.3.2.1">for-all</csymbol><ci id="A4.1.p1.6.m6.2.3.2.2.cmml" xref="A4.1.p1.6.m6.2.3.2.2">𝑡</ci></apply><interval closure="closed-open" id="A4.1.p1.6.m6.2.3.3.1.cmml" xref="A4.1.p1.6.m6.2.3.3.2"><cn id="A4.1.p1.6.m6.1.1.cmml" type="integer" xref="A4.1.p1.6.m6.1.1">0</cn><infinity id="A4.1.p1.6.m6.2.2.cmml" xref="A4.1.p1.6.m6.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.6.m6.2c">\forall t\in[0,\infty)</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.6.m6.2d">∀ italic_t ∈ [ 0 , ∞ )</annotation></semantics></math>, and 2) <math alttext="\bm{\epsilon}\in L_{\infty}" class="ltx_Math" display="inline" id="A4.1.p1.7.m7.1"><semantics id="A4.1.p1.7.m7.1a"><mrow id="A4.1.p1.7.m7.1.1" xref="A4.1.p1.7.m7.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="A4.1.p1.7.m7.1.1.2" mathvariant="bold-italic" xref="A4.1.p1.7.m7.1.1.2.cmml">ϵ</mi><mo id="A4.1.p1.7.m7.1.1.1" xref="A4.1.p1.7.m7.1.1.1.cmml">∈</mo><msub id="A4.1.p1.7.m7.1.1.3" xref="A4.1.p1.7.m7.1.1.3.cmml"><mi id="A4.1.p1.7.m7.1.1.3.2" xref="A4.1.p1.7.m7.1.1.3.2.cmml">L</mi><mi id="A4.1.p1.7.m7.1.1.3.3" mathvariant="normal" xref="A4.1.p1.7.m7.1.1.3.3.cmml">∞</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.7.m7.1b"><apply id="A4.1.p1.7.m7.1.1.cmml" xref="A4.1.p1.7.m7.1.1"><in id="A4.1.p1.7.m7.1.1.1.cmml" xref="A4.1.p1.7.m7.1.1.1"></in><ci id="A4.1.p1.7.m7.1.1.2.cmml" xref="A4.1.p1.7.m7.1.1.2">bold-italic-ϵ</ci><apply id="A4.1.p1.7.m7.1.1.3.cmml" xref="A4.1.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.7.m7.1.1.3.1.cmml" xref="A4.1.p1.7.m7.1.1.3">subscript</csymbol><ci id="A4.1.p1.7.m7.1.1.3.2.cmml" xref="A4.1.p1.7.m7.1.1.3.2">𝐿</ci><infinity id="A4.1.p1.7.m7.1.1.3.3.cmml" xref="A4.1.p1.7.m7.1.1.3.3"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.7.m7.1c">\bm{\epsilon}\in L_{\infty}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.7.m7.1d">bold_italic_ϵ ∈ italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\forall t\in[0,t_{\rm f})" class="ltx_Math" display="inline" id="A4.1.p1.8.m8.2"><semantics id="A4.1.p1.8.m8.2a"><mrow id="A4.1.p1.8.m8.2.2" xref="A4.1.p1.8.m8.2.2.cmml"><mrow id="A4.1.p1.8.m8.2.2.3" xref="A4.1.p1.8.m8.2.2.3.cmml"><mo id="A4.1.p1.8.m8.2.2.3.1" rspace="0.167em" xref="A4.1.p1.8.m8.2.2.3.1.cmml">∀</mo><mi id="A4.1.p1.8.m8.2.2.3.2" xref="A4.1.p1.8.m8.2.2.3.2.cmml">t</mi></mrow><mo id="A4.1.p1.8.m8.2.2.2" xref="A4.1.p1.8.m8.2.2.2.cmml">∈</mo><mrow id="A4.1.p1.8.m8.2.2.1.1" xref="A4.1.p1.8.m8.2.2.1.2.cmml"><mo id="A4.1.p1.8.m8.2.2.1.1.2" stretchy="false" xref="A4.1.p1.8.m8.2.2.1.2.cmml">[</mo><mn id="A4.1.p1.8.m8.1.1" xref="A4.1.p1.8.m8.1.1.cmml">0</mn><mo id="A4.1.p1.8.m8.2.2.1.1.3" xref="A4.1.p1.8.m8.2.2.1.2.cmml">,</mo><msub id="A4.1.p1.8.m8.2.2.1.1.1" xref="A4.1.p1.8.m8.2.2.1.1.1.cmml"><mi id="A4.1.p1.8.m8.2.2.1.1.1.2" xref="A4.1.p1.8.m8.2.2.1.1.1.2.cmml">t</mi><mi id="A4.1.p1.8.m8.2.2.1.1.1.3" mathvariant="normal" xref="A4.1.p1.8.m8.2.2.1.1.1.3.cmml">f</mi></msub><mo id="A4.1.p1.8.m8.2.2.1.1.4" stretchy="false" xref="A4.1.p1.8.m8.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.8.m8.2b"><apply id="A4.1.p1.8.m8.2.2.cmml" xref="A4.1.p1.8.m8.2.2"><in id="A4.1.p1.8.m8.2.2.2.cmml" xref="A4.1.p1.8.m8.2.2.2"></in><apply id="A4.1.p1.8.m8.2.2.3.cmml" xref="A4.1.p1.8.m8.2.2.3"><csymbol cd="latexml" id="A4.1.p1.8.m8.2.2.3.1.cmml" xref="A4.1.p1.8.m8.2.2.3.1">for-all</csymbol><ci id="A4.1.p1.8.m8.2.2.3.2.cmml" xref="A4.1.p1.8.m8.2.2.3.2">𝑡</ci></apply><interval closure="closed-open" id="A4.1.p1.8.m8.2.2.1.2.cmml" xref="A4.1.p1.8.m8.2.2.1.1"><cn id="A4.1.p1.8.m8.1.1.cmml" type="integer" xref="A4.1.p1.8.m8.1.1">0</cn><apply id="A4.1.p1.8.m8.2.2.1.1.1.cmml" xref="A4.1.p1.8.m8.2.2.1.1.1"><csymbol cd="ambiguous" id="A4.1.p1.8.m8.2.2.1.1.1.1.cmml" xref="A4.1.p1.8.m8.2.2.1.1.1">subscript</csymbol><ci id="A4.1.p1.8.m8.2.2.1.1.1.2.cmml" xref="A4.1.p1.8.m8.2.2.1.1.1.2">𝑡</ci><ci id="A4.1.p1.8.m8.2.2.1.1.1.3.cmml" xref="A4.1.p1.8.m8.2.2.1.1.1.3">f</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.8.m8.2c">\forall t\in[0,t_{\rm f})</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.8.m8.2d">∀ italic_t ∈ [ 0 , italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT )</annotation></semantics></math> such that there exists a constant <math alttext="\bar{\epsilon}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A4.1.p1.9.m9.1"><semantics id="A4.1.p1.9.m9.1a"><mrow id="A4.1.p1.9.m9.1.1" xref="A4.1.p1.9.m9.1.1.cmml"><mover accent="true" id="A4.1.p1.9.m9.1.1.2" xref="A4.1.p1.9.m9.1.1.2.cmml"><mi id="A4.1.p1.9.m9.1.1.2.2" xref="A4.1.p1.9.m9.1.1.2.2.cmml">ϵ</mi><mo id="A4.1.p1.9.m9.1.1.2.1" xref="A4.1.p1.9.m9.1.1.2.1.cmml">¯</mo></mover><mo id="A4.1.p1.9.m9.1.1.1" xref="A4.1.p1.9.m9.1.1.1.cmml">∈</mo><msup id="A4.1.p1.9.m9.1.1.3" xref="A4.1.p1.9.m9.1.1.3.cmml"><mi id="A4.1.p1.9.m9.1.1.3.2" xref="A4.1.p1.9.m9.1.1.3.2.cmml">ℝ</mi><mo id="A4.1.p1.9.m9.1.1.3.3" xref="A4.1.p1.9.m9.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.9.m9.1b"><apply id="A4.1.p1.9.m9.1.1.cmml" xref="A4.1.p1.9.m9.1.1"><in id="A4.1.p1.9.m9.1.1.1.cmml" xref="A4.1.p1.9.m9.1.1.1"></in><apply id="A4.1.p1.9.m9.1.1.2.cmml" xref="A4.1.p1.9.m9.1.1.2"><ci id="A4.1.p1.9.m9.1.1.2.1.cmml" xref="A4.1.p1.9.m9.1.1.2.1">¯</ci><ci id="A4.1.p1.9.m9.1.1.2.2.cmml" xref="A4.1.p1.9.m9.1.1.2.2">italic-ϵ</ci></apply><apply id="A4.1.p1.9.m9.1.1.3.cmml" xref="A4.1.p1.9.m9.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.9.m9.1.1.3.1.cmml" xref="A4.1.p1.9.m9.1.1.3">superscript</csymbol><ci id="A4.1.p1.9.m9.1.1.3.2.cmml" xref="A4.1.p1.9.m9.1.1.3.2">ℝ</ci><plus id="A4.1.p1.9.m9.1.1.3.3.cmml" xref="A4.1.p1.9.m9.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.9.m9.1c">\bar{\epsilon}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.9.m9.1d">over¯ start_ARG italic_ϵ end_ARG ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> that satisfies <math alttext="\|\bm{\epsilon}\|\leq\bar{\epsilon}" class="ltx_Math" display="inline" id="A4.1.p1.10.m10.1"><semantics id="A4.1.p1.10.m10.1a"><mrow id="A4.1.p1.10.m10.1.2" xref="A4.1.p1.10.m10.1.2.cmml"><mrow id="A4.1.p1.10.m10.1.2.2.2" xref="A4.1.p1.10.m10.1.2.2.1.cmml"><mo id="A4.1.p1.10.m10.1.2.2.2.1" stretchy="false" xref="A4.1.p1.10.m10.1.2.2.1.1.cmml">‖</mo><mi class="ltx_mathvariant_bold-italic" id="A4.1.p1.10.m10.1.1" mathvariant="bold-italic" xref="A4.1.p1.10.m10.1.1.cmml">ϵ</mi><mo id="A4.1.p1.10.m10.1.2.2.2.2" stretchy="false" xref="A4.1.p1.10.m10.1.2.2.1.1.cmml">‖</mo></mrow><mo id="A4.1.p1.10.m10.1.2.1" xref="A4.1.p1.10.m10.1.2.1.cmml">≤</mo><mover accent="true" id="A4.1.p1.10.m10.1.2.3" xref="A4.1.p1.10.m10.1.2.3.cmml"><mi id="A4.1.p1.10.m10.1.2.3.2" xref="A4.1.p1.10.m10.1.2.3.2.cmml">ϵ</mi><mo id="A4.1.p1.10.m10.1.2.3.1" xref="A4.1.p1.10.m10.1.2.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.10.m10.1b"><apply id="A4.1.p1.10.m10.1.2.cmml" xref="A4.1.p1.10.m10.1.2"><leq id="A4.1.p1.10.m10.1.2.1.cmml" xref="A4.1.p1.10.m10.1.2.1"></leq><apply id="A4.1.p1.10.m10.1.2.2.1.cmml" xref="A4.1.p1.10.m10.1.2.2.2"><csymbol cd="latexml" id="A4.1.p1.10.m10.1.2.2.1.1.cmml" xref="A4.1.p1.10.m10.1.2.2.2.1">norm</csymbol><ci id="A4.1.p1.10.m10.1.1.cmml" xref="A4.1.p1.10.m10.1.1">bold-italic-ϵ</ci></apply><apply id="A4.1.p1.10.m10.1.2.3.cmml" xref="A4.1.p1.10.m10.1.2.3"><ci id="A4.1.p1.10.m10.1.2.3.1.cmml" xref="A4.1.p1.10.m10.1.2.3.1">¯</ci><ci id="A4.1.p1.10.m10.1.2.3.2.cmml" xref="A4.1.p1.10.m10.1.2.3.2">italic-ϵ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.10.m10.1c">\|\bm{\epsilon}\|\leq\bar{\epsilon}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.10.m10.1d">∥ bold_italic_ϵ ∥ ≤ over¯ start_ARG italic_ϵ end_ARG</annotation></semantics></math>, <math alttext="\forall t\in[0,t_{\rm f})" class="ltx_Math" display="inline" id="A4.1.p1.11.m11.2"><semantics id="A4.1.p1.11.m11.2a"><mrow id="A4.1.p1.11.m11.2.2" xref="A4.1.p1.11.m11.2.2.cmml"><mrow id="A4.1.p1.11.m11.2.2.3" xref="A4.1.p1.11.m11.2.2.3.cmml"><mo id="A4.1.p1.11.m11.2.2.3.1" rspace="0.167em" xref="A4.1.p1.11.m11.2.2.3.1.cmml">∀</mo><mi id="A4.1.p1.11.m11.2.2.3.2" xref="A4.1.p1.11.m11.2.2.3.2.cmml">t</mi></mrow><mo id="A4.1.p1.11.m11.2.2.2" xref="A4.1.p1.11.m11.2.2.2.cmml">∈</mo><mrow id="A4.1.p1.11.m11.2.2.1.1" xref="A4.1.p1.11.m11.2.2.1.2.cmml"><mo id="A4.1.p1.11.m11.2.2.1.1.2" stretchy="false" xref="A4.1.p1.11.m11.2.2.1.2.cmml">[</mo><mn id="A4.1.p1.11.m11.1.1" xref="A4.1.p1.11.m11.1.1.cmml">0</mn><mo id="A4.1.p1.11.m11.2.2.1.1.3" xref="A4.1.p1.11.m11.2.2.1.2.cmml">,</mo><msub id="A4.1.p1.11.m11.2.2.1.1.1" xref="A4.1.p1.11.m11.2.2.1.1.1.cmml"><mi id="A4.1.p1.11.m11.2.2.1.1.1.2" xref="A4.1.p1.11.m11.2.2.1.1.1.2.cmml">t</mi><mi id="A4.1.p1.11.m11.2.2.1.1.1.3" mathvariant="normal" xref="A4.1.p1.11.m11.2.2.1.1.1.3.cmml">f</mi></msub><mo id="A4.1.p1.11.m11.2.2.1.1.4" stretchy="false" xref="A4.1.p1.11.m11.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.11.m11.2b"><apply id="A4.1.p1.11.m11.2.2.cmml" xref="A4.1.p1.11.m11.2.2"><in id="A4.1.p1.11.m11.2.2.2.cmml" xref="A4.1.p1.11.m11.2.2.2"></in><apply id="A4.1.p1.11.m11.2.2.3.cmml" xref="A4.1.p1.11.m11.2.2.3"><csymbol cd="latexml" id="A4.1.p1.11.m11.2.2.3.1.cmml" xref="A4.1.p1.11.m11.2.2.3.1">for-all</csymbol><ci id="A4.1.p1.11.m11.2.2.3.2.cmml" xref="A4.1.p1.11.m11.2.2.3.2">𝑡</ci></apply><interval closure="closed-open" id="A4.1.p1.11.m11.2.2.1.2.cmml" xref="A4.1.p1.11.m11.2.2.1.1"><cn id="A4.1.p1.11.m11.1.1.cmml" type="integer" xref="A4.1.p1.11.m11.1.1">0</cn><apply id="A4.1.p1.11.m11.2.2.1.1.1.cmml" xref="A4.1.p1.11.m11.2.2.1.1.1"><csymbol cd="ambiguous" id="A4.1.p1.11.m11.2.2.1.1.1.1.cmml" xref="A4.1.p1.11.m11.2.2.1.1.1">subscript</csymbol><ci id="A4.1.p1.11.m11.2.2.1.1.1.2.cmml" xref="A4.1.p1.11.m11.2.2.1.1.1.2">𝑡</ci><ci id="A4.1.p1.11.m11.2.2.1.1.1.3.cmml" xref="A4.1.p1.11.m11.2.2.1.1.1.3">f</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.11.m11.2c">\forall t\in[0,t_{\rm f})</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.11.m11.2d">∀ italic_t ∈ [ 0 , italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT )</annotation></semantics></math>. Applying these results to the above inequality and letting <math alttext="k_{{\rm c}i}>1/4" class="ltx_Math" display="inline" id="A4.1.p1.12.m12.1"><semantics id="A4.1.p1.12.m12.1a"><mrow id="A4.1.p1.12.m12.1.1" xref="A4.1.p1.12.m12.1.1.cmml"><msub id="A4.1.p1.12.m12.1.1.2" xref="A4.1.p1.12.m12.1.1.2.cmml"><mi id="A4.1.p1.12.m12.1.1.2.2" xref="A4.1.p1.12.m12.1.1.2.2.cmml">k</mi><mrow id="A4.1.p1.12.m12.1.1.2.3" xref="A4.1.p1.12.m12.1.1.2.3.cmml"><mi id="A4.1.p1.12.m12.1.1.2.3.2" mathvariant="normal" xref="A4.1.p1.12.m12.1.1.2.3.2.cmml">c</mi><mo id="A4.1.p1.12.m12.1.1.2.3.1" xref="A4.1.p1.12.m12.1.1.2.3.1.cmml">⁢</mo><mi id="A4.1.p1.12.m12.1.1.2.3.3" xref="A4.1.p1.12.m12.1.1.2.3.3.cmml">i</mi></mrow></msub><mo id="A4.1.p1.12.m12.1.1.1" xref="A4.1.p1.12.m12.1.1.1.cmml">></mo><mrow id="A4.1.p1.12.m12.1.1.3" xref="A4.1.p1.12.m12.1.1.3.cmml"><mn id="A4.1.p1.12.m12.1.1.3.2" xref="A4.1.p1.12.m12.1.1.3.2.cmml">1</mn><mo id="A4.1.p1.12.m12.1.1.3.1" xref="A4.1.p1.12.m12.1.1.3.1.cmml">/</mo><mn id="A4.1.p1.12.m12.1.1.3.3" xref="A4.1.p1.12.m12.1.1.3.3.cmml">4</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.12.m12.1b"><apply id="A4.1.p1.12.m12.1.1.cmml" xref="A4.1.p1.12.m12.1.1"><gt id="A4.1.p1.12.m12.1.1.1.cmml" xref="A4.1.p1.12.m12.1.1.1"></gt><apply id="A4.1.p1.12.m12.1.1.2.cmml" xref="A4.1.p1.12.m12.1.1.2"><csymbol cd="ambiguous" id="A4.1.p1.12.m12.1.1.2.1.cmml" xref="A4.1.p1.12.m12.1.1.2">subscript</csymbol><ci id="A4.1.p1.12.m12.1.1.2.2.cmml" xref="A4.1.p1.12.m12.1.1.2.2">𝑘</ci><apply id="A4.1.p1.12.m12.1.1.2.3.cmml" xref="A4.1.p1.12.m12.1.1.2.3"><times id="A4.1.p1.12.m12.1.1.2.3.1.cmml" xref="A4.1.p1.12.m12.1.1.2.3.1"></times><ci id="A4.1.p1.12.m12.1.1.2.3.2.cmml" xref="A4.1.p1.12.m12.1.1.2.3.2">c</ci><ci id="A4.1.p1.12.m12.1.1.2.3.3.cmml" xref="A4.1.p1.12.m12.1.1.2.3.3">𝑖</ci></apply></apply><apply id="A4.1.p1.12.m12.1.1.3.cmml" xref="A4.1.p1.12.m12.1.1.3"><divide id="A4.1.p1.12.m12.1.1.3.1.cmml" xref="A4.1.p1.12.m12.1.1.3.1"></divide><cn id="A4.1.p1.12.m12.1.1.3.2.cmml" type="integer" xref="A4.1.p1.12.m12.1.1.3.2">1</cn><cn id="A4.1.p1.12.m12.1.1.3.3.cmml" type="integer" xref="A4.1.p1.12.m12.1.1.3.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.12.m12.1c">k_{{\rm c}i}>1/4</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.12.m12.1d">italic_k start_POSTSUBSCRIPT roman_c italic_i end_POSTSUBSCRIPT > 1 / 4</annotation></semantics></math> yields</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx77"> <tbody id="A4.Ex67"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}\leq" class="ltx_Math" display="inline" id="A4.Ex67.m1.1"><semantics id="A4.Ex67.m1.1a"><mrow id="A4.Ex67.m1.1.1" xref="A4.Ex67.m1.1.1.cmml"><mover accent="true" id="A4.Ex67.m1.1.1.2" xref="A4.Ex67.m1.1.1.2.cmml"><mi id="A4.Ex67.m1.1.1.2.2" xref="A4.Ex67.m1.1.1.2.2.cmml">V</mi><mo id="A4.Ex67.m1.1.1.2.1" xref="A4.Ex67.m1.1.1.2.1.cmml">˙</mo></mover><mo id="A4.Ex67.m1.1.1.1" xref="A4.Ex67.m1.1.1.1.cmml">≤</mo><mi id="A4.Ex67.m1.1.1.3" xref="A4.Ex67.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex67.m1.1b"><apply id="A4.Ex67.m1.1.1.cmml" xref="A4.Ex67.m1.1.1"><leq id="A4.Ex67.m1.1.1.1.cmml" xref="A4.Ex67.m1.1.1.1"></leq><apply id="A4.Ex67.m1.1.1.2.cmml" xref="A4.Ex67.m1.1.1.2"><ci id="A4.Ex67.m1.1.1.2.1.cmml" xref="A4.Ex67.m1.1.1.2.1">˙</ci><ci id="A4.Ex67.m1.1.1.2.2.cmml" xref="A4.Ex67.m1.1.1.2.2">𝑉</ci></apply><csymbol cd="latexml" id="A4.Ex67.m1.1.1.3.cmml" xref="A4.Ex67.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex67.m1.1c">\displaystyle\dot{V}\leq</annotation><annotation encoding="application/x-llamapun" id="A4.Ex67.m1.1d">over˙ start_ARG italic_V end_ARG ≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-k_{\rm c}\|\bm{e}\|^{2}+\delta c_{\theta}\|\bm{e}\|+\bar{% \epsilon}(1+p)\|{\bm{d}}_{\rm f}\|" class="ltx_Math" display="inline" id="A4.Ex67.m2.4"><semantics id="A4.Ex67.m2.4a"><mrow id="A4.Ex67.m2.4.4" xref="A4.Ex67.m2.4.4.cmml"><mrow id="A4.Ex67.m2.4.4.4" xref="A4.Ex67.m2.4.4.4.cmml"><mo id="A4.Ex67.m2.4.4.4a" xref="A4.Ex67.m2.4.4.4.cmml">−</mo><mrow id="A4.Ex67.m2.4.4.4.2" xref="A4.Ex67.m2.4.4.4.2.cmml"><msub id="A4.Ex67.m2.4.4.4.2.2" xref="A4.Ex67.m2.4.4.4.2.2.cmml"><mi id="A4.Ex67.m2.4.4.4.2.2.2" xref="A4.Ex67.m2.4.4.4.2.2.2.cmml">k</mi><mi id="A4.Ex67.m2.4.4.4.2.2.3" mathvariant="normal" xref="A4.Ex67.m2.4.4.4.2.2.3.cmml">c</mi></msub><mo id="A4.Ex67.m2.4.4.4.2.1" xref="A4.Ex67.m2.4.4.4.2.1.cmml">⁢</mo><msup id="A4.Ex67.m2.4.4.4.2.3" xref="A4.Ex67.m2.4.4.4.2.3.cmml"><mrow id="A4.Ex67.m2.4.4.4.2.3.2.2" xref="A4.Ex67.m2.4.4.4.2.3.2.1.cmml"><mo id="A4.Ex67.m2.4.4.4.2.3.2.2.1" stretchy="false" xref="A4.Ex67.m2.4.4.4.2.3.2.1.1.cmml">‖</mo><mi id="A4.Ex67.m2.1.1" xref="A4.Ex67.m2.1.1.cmml">𝒆</mi><mo id="A4.Ex67.m2.4.4.4.2.3.2.2.2" stretchy="false" xref="A4.Ex67.m2.4.4.4.2.3.2.1.1.cmml">‖</mo></mrow><mn id="A4.Ex67.m2.4.4.4.2.3.3" xref="A4.Ex67.m2.4.4.4.2.3.3.cmml">2</mn></msup></mrow></mrow><mo id="A4.Ex67.m2.4.4.3" xref="A4.Ex67.m2.4.4.3.cmml">+</mo><mrow id="A4.Ex67.m2.4.4.5" xref="A4.Ex67.m2.4.4.5.cmml"><mi id="A4.Ex67.m2.4.4.5.2" xref="A4.Ex67.m2.4.4.5.2.cmml">δ</mi><mo id="A4.Ex67.m2.4.4.5.1" xref="A4.Ex67.m2.4.4.5.1.cmml">⁢</mo><msub id="A4.Ex67.m2.4.4.5.3" xref="A4.Ex67.m2.4.4.5.3.cmml"><mi id="A4.Ex67.m2.4.4.5.3.2" xref="A4.Ex67.m2.4.4.5.3.2.cmml">c</mi><mi id="A4.Ex67.m2.4.4.5.3.3" xref="A4.Ex67.m2.4.4.5.3.3.cmml">θ</mi></msub><mo id="A4.Ex67.m2.4.4.5.1a" xref="A4.Ex67.m2.4.4.5.1.cmml">⁢</mo><mrow id="A4.Ex67.m2.4.4.5.4.2" xref="A4.Ex67.m2.4.4.5.4.1.cmml"><mo id="A4.Ex67.m2.4.4.5.4.2.1" stretchy="false" xref="A4.Ex67.m2.4.4.5.4.1.1.cmml">‖</mo><mi id="A4.Ex67.m2.2.2" xref="A4.Ex67.m2.2.2.cmml">𝒆</mi><mo id="A4.Ex67.m2.4.4.5.4.2.2" stretchy="false" xref="A4.Ex67.m2.4.4.5.4.1.1.cmml">‖</mo></mrow></mrow><mo id="A4.Ex67.m2.4.4.3a" xref="A4.Ex67.m2.4.4.3.cmml">+</mo><mrow id="A4.Ex67.m2.4.4.2" xref="A4.Ex67.m2.4.4.2.cmml"><mover accent="true" id="A4.Ex67.m2.4.4.2.4" xref="A4.Ex67.m2.4.4.2.4.cmml"><mi id="A4.Ex67.m2.4.4.2.4.2" xref="A4.Ex67.m2.4.4.2.4.2.cmml">ϵ</mi><mo id="A4.Ex67.m2.4.4.2.4.1" xref="A4.Ex67.m2.4.4.2.4.1.cmml">¯</mo></mover><mo id="A4.Ex67.m2.4.4.2.3" xref="A4.Ex67.m2.4.4.2.3.cmml">⁢</mo><mrow id="A4.Ex67.m2.3.3.1.1.1" xref="A4.Ex67.m2.3.3.1.1.1.1.cmml"><mo id="A4.Ex67.m2.3.3.1.1.1.2" stretchy="false" xref="A4.Ex67.m2.3.3.1.1.1.1.cmml">(</mo><mrow id="A4.Ex67.m2.3.3.1.1.1.1" xref="A4.Ex67.m2.3.3.1.1.1.1.cmml"><mn id="A4.Ex67.m2.3.3.1.1.1.1.2" xref="A4.Ex67.m2.3.3.1.1.1.1.2.cmml">1</mn><mo id="A4.Ex67.m2.3.3.1.1.1.1.1" xref="A4.Ex67.m2.3.3.1.1.1.1.1.cmml">+</mo><mi id="A4.Ex67.m2.3.3.1.1.1.1.3" xref="A4.Ex67.m2.3.3.1.1.1.1.3.cmml">p</mi></mrow><mo id="A4.Ex67.m2.3.3.1.1.1.3" stretchy="false" xref="A4.Ex67.m2.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex67.m2.4.4.2.3a" xref="A4.Ex67.m2.4.4.2.3.cmml">⁢</mo><mrow id="A4.Ex67.m2.4.4.2.2.1" xref="A4.Ex67.m2.4.4.2.2.2.cmml"><mo id="A4.Ex67.m2.4.4.2.2.1.2" stretchy="false" xref="A4.Ex67.m2.4.4.2.2.2.1.cmml">‖</mo><msub id="A4.Ex67.m2.4.4.2.2.1.1" xref="A4.Ex67.m2.4.4.2.2.1.1.cmml"><mi id="A4.Ex67.m2.4.4.2.2.1.1.2" xref="A4.Ex67.m2.4.4.2.2.1.1.2.cmml">𝒅</mi><mi id="A4.Ex67.m2.4.4.2.2.1.1.3" mathvariant="normal" xref="A4.Ex67.m2.4.4.2.2.1.1.3.cmml">f</mi></msub><mo id="A4.Ex67.m2.4.4.2.2.1.3" stretchy="false" xref="A4.Ex67.m2.4.4.2.2.2.1.cmml">‖</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex67.m2.4b"><apply id="A4.Ex67.m2.4.4.cmml" xref="A4.Ex67.m2.4.4"><plus id="A4.Ex67.m2.4.4.3.cmml" xref="A4.Ex67.m2.4.4.3"></plus><apply id="A4.Ex67.m2.4.4.4.cmml" xref="A4.Ex67.m2.4.4.4"><minus id="A4.Ex67.m2.4.4.4.1.cmml" xref="A4.Ex67.m2.4.4.4"></minus><apply id="A4.Ex67.m2.4.4.4.2.cmml" xref="A4.Ex67.m2.4.4.4.2"><times id="A4.Ex67.m2.4.4.4.2.1.cmml" xref="A4.Ex67.m2.4.4.4.2.1"></times><apply id="A4.Ex67.m2.4.4.4.2.2.cmml" xref="A4.Ex67.m2.4.4.4.2.2"><csymbol cd="ambiguous" id="A4.Ex67.m2.4.4.4.2.2.1.cmml" xref="A4.Ex67.m2.4.4.4.2.2">subscript</csymbol><ci id="A4.Ex67.m2.4.4.4.2.2.2.cmml" xref="A4.Ex67.m2.4.4.4.2.2.2">𝑘</ci><ci id="A4.Ex67.m2.4.4.4.2.2.3.cmml" xref="A4.Ex67.m2.4.4.4.2.2.3">c</ci></apply><apply id="A4.Ex67.m2.4.4.4.2.3.cmml" xref="A4.Ex67.m2.4.4.4.2.3"><csymbol cd="ambiguous" id="A4.Ex67.m2.4.4.4.2.3.1.cmml" xref="A4.Ex67.m2.4.4.4.2.3">superscript</csymbol><apply id="A4.Ex67.m2.4.4.4.2.3.2.1.cmml" xref="A4.Ex67.m2.4.4.4.2.3.2.2"><csymbol cd="latexml" id="A4.Ex67.m2.4.4.4.2.3.2.1.1.cmml" xref="A4.Ex67.m2.4.4.4.2.3.2.2.1">norm</csymbol><ci id="A4.Ex67.m2.1.1.cmml" xref="A4.Ex67.m2.1.1">𝒆</ci></apply><cn id="A4.Ex67.m2.4.4.4.2.3.3.cmml" type="integer" xref="A4.Ex67.m2.4.4.4.2.3.3">2</cn></apply></apply></apply><apply id="A4.Ex67.m2.4.4.5.cmml" xref="A4.Ex67.m2.4.4.5"><times id="A4.Ex67.m2.4.4.5.1.cmml" xref="A4.Ex67.m2.4.4.5.1"></times><ci id="A4.Ex67.m2.4.4.5.2.cmml" xref="A4.Ex67.m2.4.4.5.2">𝛿</ci><apply id="A4.Ex67.m2.4.4.5.3.cmml" xref="A4.Ex67.m2.4.4.5.3"><csymbol cd="ambiguous" id="A4.Ex67.m2.4.4.5.3.1.cmml" xref="A4.Ex67.m2.4.4.5.3">subscript</csymbol><ci id="A4.Ex67.m2.4.4.5.3.2.cmml" xref="A4.Ex67.m2.4.4.5.3.2">𝑐</ci><ci id="A4.Ex67.m2.4.4.5.3.3.cmml" xref="A4.Ex67.m2.4.4.5.3.3">𝜃</ci></apply><apply id="A4.Ex67.m2.4.4.5.4.1.cmml" xref="A4.Ex67.m2.4.4.5.4.2"><csymbol cd="latexml" id="A4.Ex67.m2.4.4.5.4.1.1.cmml" xref="A4.Ex67.m2.4.4.5.4.2.1">norm</csymbol><ci id="A4.Ex67.m2.2.2.cmml" xref="A4.Ex67.m2.2.2">𝒆</ci></apply></apply><apply id="A4.Ex67.m2.4.4.2.cmml" xref="A4.Ex67.m2.4.4.2"><times id="A4.Ex67.m2.4.4.2.3.cmml" xref="A4.Ex67.m2.4.4.2.3"></times><apply id="A4.Ex67.m2.4.4.2.4.cmml" xref="A4.Ex67.m2.4.4.2.4"><ci id="A4.Ex67.m2.4.4.2.4.1.cmml" xref="A4.Ex67.m2.4.4.2.4.1">¯</ci><ci id="A4.Ex67.m2.4.4.2.4.2.cmml" xref="A4.Ex67.m2.4.4.2.4.2">italic-ϵ</ci></apply><apply id="A4.Ex67.m2.3.3.1.1.1.1.cmml" xref="A4.Ex67.m2.3.3.1.1.1"><plus id="A4.Ex67.m2.3.3.1.1.1.1.1.cmml" xref="A4.Ex67.m2.3.3.1.1.1.1.1"></plus><cn id="A4.Ex67.m2.3.3.1.1.1.1.2.cmml" type="integer" xref="A4.Ex67.m2.3.3.1.1.1.1.2">1</cn><ci id="A4.Ex67.m2.3.3.1.1.1.1.3.cmml" xref="A4.Ex67.m2.3.3.1.1.1.1.3">𝑝</ci></apply><apply id="A4.Ex67.m2.4.4.2.2.2.cmml" xref="A4.Ex67.m2.4.4.2.2.1"><csymbol cd="latexml" id="A4.Ex67.m2.4.4.2.2.2.1.cmml" xref="A4.Ex67.m2.4.4.2.2.1.2">norm</csymbol><apply id="A4.Ex67.m2.4.4.2.2.1.1.cmml" xref="A4.Ex67.m2.4.4.2.2.1.1"><csymbol cd="ambiguous" id="A4.Ex67.m2.4.4.2.2.1.1.1.cmml" xref="A4.Ex67.m2.4.4.2.2.1.1">subscript</csymbol><ci id="A4.Ex67.m2.4.4.2.2.1.1.2.cmml" xref="A4.Ex67.m2.4.4.2.2.1.1.2">𝒅</ci><ci id="A4.Ex67.m2.4.4.2.2.1.1.3.cmml" xref="A4.Ex67.m2.4.4.2.2.1.1.3">f</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex67.m2.4c">\displaystyle-k_{\rm c}\|\bm{e}\|^{2}+\delta c_{\theta}\|\bm{e}\|+\bar{% \epsilon}(1+p)\|{\bm{d}}_{\rm f}\|</annotation><annotation encoding="application/x-llamapun" id="A4.Ex67.m2.4d">- italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ∥ bold_italic_e ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_δ italic_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ∥ bold_italic_e ∥ + over¯ start_ARG italic_ϵ end_ARG ( 1 + italic_p ) ∥ bold_italic_d start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ∥</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex68"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle+\bm{e}^{T}\bm{d}-\kappa(1+p)\tilde{\bm{\theta}}^{T}{\bm{d}}_{\rm g}" class="ltx_Math" display="inline" id="A4.Ex68.m1.1"><semantics id="A4.Ex68.m1.1a"><mrow id="A4.Ex68.m1.1.1" xref="A4.Ex68.m1.1.1.cmml"><mrow id="A4.Ex68.m1.1.1.3" xref="A4.Ex68.m1.1.1.3.cmml"><mo id="A4.Ex68.m1.1.1.3a" xref="A4.Ex68.m1.1.1.3.cmml">+</mo><mrow id="A4.Ex68.m1.1.1.3.2" xref="A4.Ex68.m1.1.1.3.2.cmml"><msup id="A4.Ex68.m1.1.1.3.2.2" xref="A4.Ex68.m1.1.1.3.2.2.cmml"><mi id="A4.Ex68.m1.1.1.3.2.2.2" xref="A4.Ex68.m1.1.1.3.2.2.2.cmml">𝒆</mi><mi id="A4.Ex68.m1.1.1.3.2.2.3" xref="A4.Ex68.m1.1.1.3.2.2.3.cmml">T</mi></msup><mo id="A4.Ex68.m1.1.1.3.2.1" xref="A4.Ex68.m1.1.1.3.2.1.cmml">⁢</mo><mi id="A4.Ex68.m1.1.1.3.2.3" xref="A4.Ex68.m1.1.1.3.2.3.cmml">𝒅</mi></mrow></mrow><mo id="A4.Ex68.m1.1.1.2" xref="A4.Ex68.m1.1.1.2.cmml">−</mo><mrow id="A4.Ex68.m1.1.1.1" xref="A4.Ex68.m1.1.1.1.cmml"><mi id="A4.Ex68.m1.1.1.1.3" xref="A4.Ex68.m1.1.1.1.3.cmml">κ</mi><mo id="A4.Ex68.m1.1.1.1.2" xref="A4.Ex68.m1.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex68.m1.1.1.1.1.1" xref="A4.Ex68.m1.1.1.1.1.1.1.cmml"><mo id="A4.Ex68.m1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex68.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex68.m1.1.1.1.1.1.1" xref="A4.Ex68.m1.1.1.1.1.1.1.cmml"><mn id="A4.Ex68.m1.1.1.1.1.1.1.2" xref="A4.Ex68.m1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A4.Ex68.m1.1.1.1.1.1.1.1" xref="A4.Ex68.m1.1.1.1.1.1.1.1.cmml">+</mo><mi id="A4.Ex68.m1.1.1.1.1.1.1.3" xref="A4.Ex68.m1.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A4.Ex68.m1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex68.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex68.m1.1.1.1.2a" xref="A4.Ex68.m1.1.1.1.2.cmml">⁢</mo><msup id="A4.Ex68.m1.1.1.1.4" xref="A4.Ex68.m1.1.1.1.4.cmml"><mover accent="true" id="A4.Ex68.m1.1.1.1.4.2" xref="A4.Ex68.m1.1.1.1.4.2.cmml"><mi id="A4.Ex68.m1.1.1.1.4.2.2" xref="A4.Ex68.m1.1.1.1.4.2.2.cmml">𝜽</mi><mo id="A4.Ex68.m1.1.1.1.4.2.1" xref="A4.Ex68.m1.1.1.1.4.2.1.cmml">~</mo></mover><mi id="A4.Ex68.m1.1.1.1.4.3" xref="A4.Ex68.m1.1.1.1.4.3.cmml">T</mi></msup><mo id="A4.Ex68.m1.1.1.1.2b" xref="A4.Ex68.m1.1.1.1.2.cmml">⁢</mo><msub id="A4.Ex68.m1.1.1.1.5" xref="A4.Ex68.m1.1.1.1.5.cmml"><mi id="A4.Ex68.m1.1.1.1.5.2" xref="A4.Ex68.m1.1.1.1.5.2.cmml">𝒅</mi><mi id="A4.Ex68.m1.1.1.1.5.3" mathvariant="normal" xref="A4.Ex68.m1.1.1.1.5.3.cmml">g</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex68.m1.1b"><apply id="A4.Ex68.m1.1.1.cmml" xref="A4.Ex68.m1.1.1"><minus id="A4.Ex68.m1.1.1.2.cmml" xref="A4.Ex68.m1.1.1.2"></minus><apply id="A4.Ex68.m1.1.1.3.cmml" xref="A4.Ex68.m1.1.1.3"><plus id="A4.Ex68.m1.1.1.3.1.cmml" xref="A4.Ex68.m1.1.1.3"></plus><apply id="A4.Ex68.m1.1.1.3.2.cmml" xref="A4.Ex68.m1.1.1.3.2"><times id="A4.Ex68.m1.1.1.3.2.1.cmml" xref="A4.Ex68.m1.1.1.3.2.1"></times><apply id="A4.Ex68.m1.1.1.3.2.2.cmml" xref="A4.Ex68.m1.1.1.3.2.2"><csymbol cd="ambiguous" id="A4.Ex68.m1.1.1.3.2.2.1.cmml" xref="A4.Ex68.m1.1.1.3.2.2">superscript</csymbol><ci id="A4.Ex68.m1.1.1.3.2.2.2.cmml" xref="A4.Ex68.m1.1.1.3.2.2.2">𝒆</ci><ci id="A4.Ex68.m1.1.1.3.2.2.3.cmml" xref="A4.Ex68.m1.1.1.3.2.2.3">𝑇</ci></apply><ci id="A4.Ex68.m1.1.1.3.2.3.cmml" xref="A4.Ex68.m1.1.1.3.2.3">𝒅</ci></apply></apply><apply id="A4.Ex68.m1.1.1.1.cmml" xref="A4.Ex68.m1.1.1.1"><times id="A4.Ex68.m1.1.1.1.2.cmml" xref="A4.Ex68.m1.1.1.1.2"></times><ci id="A4.Ex68.m1.1.1.1.3.cmml" xref="A4.Ex68.m1.1.1.1.3">𝜅</ci><apply id="A4.Ex68.m1.1.1.1.1.1.1.cmml" xref="A4.Ex68.m1.1.1.1.1.1"><plus id="A4.Ex68.m1.1.1.1.1.1.1.1.cmml" xref="A4.Ex68.m1.1.1.1.1.1.1.1"></plus><cn id="A4.Ex68.m1.1.1.1.1.1.1.2.cmml" type="integer" xref="A4.Ex68.m1.1.1.1.1.1.1.2">1</cn><ci id="A4.Ex68.m1.1.1.1.1.1.1.3.cmml" xref="A4.Ex68.m1.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="A4.Ex68.m1.1.1.1.4.cmml" xref="A4.Ex68.m1.1.1.1.4"><csymbol cd="ambiguous" id="A4.Ex68.m1.1.1.1.4.1.cmml" xref="A4.Ex68.m1.1.1.1.4">superscript</csymbol><apply id="A4.Ex68.m1.1.1.1.4.2.cmml" xref="A4.Ex68.m1.1.1.1.4.2"><ci id="A4.Ex68.m1.1.1.1.4.2.1.cmml" xref="A4.Ex68.m1.1.1.1.4.2.1">~</ci><ci id="A4.Ex68.m1.1.1.1.4.2.2.cmml" xref="A4.Ex68.m1.1.1.1.4.2.2">𝜽</ci></apply><ci id="A4.Ex68.m1.1.1.1.4.3.cmml" xref="A4.Ex68.m1.1.1.1.4.3">𝑇</ci></apply><apply id="A4.Ex68.m1.1.1.1.5.cmml" xref="A4.Ex68.m1.1.1.1.5"><csymbol cd="ambiguous" id="A4.Ex68.m1.1.1.1.5.1.cmml" xref="A4.Ex68.m1.1.1.1.5">subscript</csymbol><ci id="A4.Ex68.m1.1.1.1.5.2.cmml" xref="A4.Ex68.m1.1.1.1.5.2">𝒅</ci><ci id="A4.Ex68.m1.1.1.1.5.3.cmml" xref="A4.Ex68.m1.1.1.1.5.3">g</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex68.m1.1c">\displaystyle+\bm{e}^{T}\bm{d}-\kappa(1+p)\tilde{\bm{\theta}}^{T}{\bm{d}}_{\rm g}</annotation><annotation encoding="application/x-llamapun" id="A4.Ex68.m1.1d">+ bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_d - italic_κ ( 1 + italic_p ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_d start_POSTSUBSCRIPT roman_g end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.1.p1.20">with <math alttext="k_{\rm c}" class="ltx_Math" display="inline" id="A4.1.p1.13.m1.1"><semantics id="A4.1.p1.13.m1.1a"><msub id="A4.1.p1.13.m1.1.1" xref="A4.1.p1.13.m1.1.1.cmml"><mi id="A4.1.p1.13.m1.1.1.2" xref="A4.1.p1.13.m1.1.1.2.cmml">k</mi><mi id="A4.1.p1.13.m1.1.1.3" mathvariant="normal" xref="A4.1.p1.13.m1.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="A4.1.p1.13.m1.1b"><apply id="A4.1.p1.13.m1.1.1.cmml" xref="A4.1.p1.13.m1.1.1"><csymbol cd="ambiguous" id="A4.1.p1.13.m1.1.1.1.cmml" xref="A4.1.p1.13.m1.1.1">subscript</csymbol><ci id="A4.1.p1.13.m1.1.1.2.cmml" xref="A4.1.p1.13.m1.1.1.2">𝑘</ci><ci id="A4.1.p1.13.m1.1.1.3.cmml" xref="A4.1.p1.13.m1.1.1.3">c</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.13.m1.1c">k_{\rm c}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.13.m1.1d">italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="A4.1.p1.14.m2.1"><semantics id="A4.1.p1.14.m2.1a"><mo id="A4.1.p1.14.m2.1.1" xref="A4.1.p1.14.m2.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="A4.1.p1.14.m2.1b"><csymbol cd="latexml" id="A4.1.p1.14.m2.1.1.cmml" xref="A4.1.p1.14.m2.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.14.m2.1c">:=</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.14.m2.1d">:=</annotation></semantics></math> <math alttext="\min_{1\leq i\leq n}\{k_{{\rm c}i}\}-1/4\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A4.1.p1.15.m3.2"><semantics id="A4.1.p1.15.m3.2a"><mrow id="A4.1.p1.15.m3.2.2" xref="A4.1.p1.15.m3.2.2.cmml"><mrow id="A4.1.p1.15.m3.2.2.2" xref="A4.1.p1.15.m3.2.2.2.cmml"><mrow id="A4.1.p1.15.m3.2.2.2.2.2" xref="A4.1.p1.15.m3.2.2.2.2.3.cmml"><msub id="A4.1.p1.15.m3.1.1.1.1.1.1" xref="A4.1.p1.15.m3.1.1.1.1.1.1.cmml"><mi id="A4.1.p1.15.m3.1.1.1.1.1.1.2" xref="A4.1.p1.15.m3.1.1.1.1.1.1.2.cmml">min</mi><mrow id="A4.1.p1.15.m3.1.1.1.1.1.1.3" xref="A4.1.p1.15.m3.1.1.1.1.1.1.3.cmml"><mn id="A4.1.p1.15.m3.1.1.1.1.1.1.3.2" xref="A4.1.p1.15.m3.1.1.1.1.1.1.3.2.cmml">1</mn><mo id="A4.1.p1.15.m3.1.1.1.1.1.1.3.3" xref="A4.1.p1.15.m3.1.1.1.1.1.1.3.3.cmml">≤</mo><mi id="A4.1.p1.15.m3.1.1.1.1.1.1.3.4" xref="A4.1.p1.15.m3.1.1.1.1.1.1.3.4.cmml">i</mi><mo id="A4.1.p1.15.m3.1.1.1.1.1.1.3.5" xref="A4.1.p1.15.m3.1.1.1.1.1.1.3.5.cmml">≤</mo><mi id="A4.1.p1.15.m3.1.1.1.1.1.1.3.6" xref="A4.1.p1.15.m3.1.1.1.1.1.1.3.6.cmml">n</mi></mrow></msub><mo id="A4.1.p1.15.m3.2.2.2.2.2a" xref="A4.1.p1.15.m3.2.2.2.2.3.cmml">⁡</mo><mrow id="A4.1.p1.15.m3.2.2.2.2.2.2" xref="A4.1.p1.15.m3.2.2.2.2.3.cmml"><mo id="A4.1.p1.15.m3.2.2.2.2.2.2.2" stretchy="false" xref="A4.1.p1.15.m3.2.2.2.2.3.cmml">{</mo><msub id="A4.1.p1.15.m3.2.2.2.2.2.2.1" xref="A4.1.p1.15.m3.2.2.2.2.2.2.1.cmml"><mi id="A4.1.p1.15.m3.2.2.2.2.2.2.1.2" xref="A4.1.p1.15.m3.2.2.2.2.2.2.1.2.cmml">k</mi><mrow id="A4.1.p1.15.m3.2.2.2.2.2.2.1.3" xref="A4.1.p1.15.m3.2.2.2.2.2.2.1.3.cmml"><mi id="A4.1.p1.15.m3.2.2.2.2.2.2.1.3.2" mathvariant="normal" xref="A4.1.p1.15.m3.2.2.2.2.2.2.1.3.2.cmml">c</mi><mo id="A4.1.p1.15.m3.2.2.2.2.2.2.1.3.1" xref="A4.1.p1.15.m3.2.2.2.2.2.2.1.3.1.cmml">⁢</mo><mi id="A4.1.p1.15.m3.2.2.2.2.2.2.1.3.3" xref="A4.1.p1.15.m3.2.2.2.2.2.2.1.3.3.cmml">i</mi></mrow></msub><mo id="A4.1.p1.15.m3.2.2.2.2.2.2.3" stretchy="false" xref="A4.1.p1.15.m3.2.2.2.2.3.cmml">}</mo></mrow></mrow><mo id="A4.1.p1.15.m3.2.2.2.3" xref="A4.1.p1.15.m3.2.2.2.3.cmml">−</mo><mrow id="A4.1.p1.15.m3.2.2.2.4" xref="A4.1.p1.15.m3.2.2.2.4.cmml"><mn id="A4.1.p1.15.m3.2.2.2.4.2" xref="A4.1.p1.15.m3.2.2.2.4.2.cmml">1</mn><mo id="A4.1.p1.15.m3.2.2.2.4.1" xref="A4.1.p1.15.m3.2.2.2.4.1.cmml">/</mo><mn id="A4.1.p1.15.m3.2.2.2.4.3" xref="A4.1.p1.15.m3.2.2.2.4.3.cmml">4</mn></mrow></mrow><mo id="A4.1.p1.15.m3.2.2.3" xref="A4.1.p1.15.m3.2.2.3.cmml">∈</mo><msup id="A4.1.p1.15.m3.2.2.4" xref="A4.1.p1.15.m3.2.2.4.cmml"><mi id="A4.1.p1.15.m3.2.2.4.2" xref="A4.1.p1.15.m3.2.2.4.2.cmml">ℝ</mi><mo id="A4.1.p1.15.m3.2.2.4.3" xref="A4.1.p1.15.m3.2.2.4.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.15.m3.2b"><apply id="A4.1.p1.15.m3.2.2.cmml" xref="A4.1.p1.15.m3.2.2"><in id="A4.1.p1.15.m3.2.2.3.cmml" xref="A4.1.p1.15.m3.2.2.3"></in><apply id="A4.1.p1.15.m3.2.2.2.cmml" xref="A4.1.p1.15.m3.2.2.2"><minus id="A4.1.p1.15.m3.2.2.2.3.cmml" xref="A4.1.p1.15.m3.2.2.2.3"></minus><apply id="A4.1.p1.15.m3.2.2.2.2.3.cmml" xref="A4.1.p1.15.m3.2.2.2.2.2"><apply id="A4.1.p1.15.m3.1.1.1.1.1.1.cmml" xref="A4.1.p1.15.m3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.1.p1.15.m3.1.1.1.1.1.1.1.cmml" xref="A4.1.p1.15.m3.1.1.1.1.1.1">subscript</csymbol><min id="A4.1.p1.15.m3.1.1.1.1.1.1.2.cmml" xref="A4.1.p1.15.m3.1.1.1.1.1.1.2"></min><apply id="A4.1.p1.15.m3.1.1.1.1.1.1.3.cmml" xref="A4.1.p1.15.m3.1.1.1.1.1.1.3"><and id="A4.1.p1.15.m3.1.1.1.1.1.1.3a.cmml" xref="A4.1.p1.15.m3.1.1.1.1.1.1.3"></and><apply id="A4.1.p1.15.m3.1.1.1.1.1.1.3b.cmml" xref="A4.1.p1.15.m3.1.1.1.1.1.1.3"><leq id="A4.1.p1.15.m3.1.1.1.1.1.1.3.3.cmml" xref="A4.1.p1.15.m3.1.1.1.1.1.1.3.3"></leq><cn id="A4.1.p1.15.m3.1.1.1.1.1.1.3.2.cmml" type="integer" xref="A4.1.p1.15.m3.1.1.1.1.1.1.3.2">1</cn><ci id="A4.1.p1.15.m3.1.1.1.1.1.1.3.4.cmml" xref="A4.1.p1.15.m3.1.1.1.1.1.1.3.4">𝑖</ci></apply><apply id="A4.1.p1.15.m3.1.1.1.1.1.1.3c.cmml" xref="A4.1.p1.15.m3.1.1.1.1.1.1.3"><leq id="A4.1.p1.15.m3.1.1.1.1.1.1.3.5.cmml" xref="A4.1.p1.15.m3.1.1.1.1.1.1.3.5"></leq><share href="https://arxiv.org/html/2401.10785v2#A4.1.p1.15.m3.1.1.1.1.1.1.3.4.cmml" id="A4.1.p1.15.m3.1.1.1.1.1.1.3d.cmml" xref="A4.1.p1.15.m3.1.1.1.1.1.1.3"></share><ci id="A4.1.p1.15.m3.1.1.1.1.1.1.3.6.cmml" xref="A4.1.p1.15.m3.1.1.1.1.1.1.3.6">𝑛</ci></apply></apply></apply><apply id="A4.1.p1.15.m3.2.2.2.2.2.2.1.cmml" xref="A4.1.p1.15.m3.2.2.2.2.2.2.1"><csymbol cd="ambiguous" id="A4.1.p1.15.m3.2.2.2.2.2.2.1.1.cmml" xref="A4.1.p1.15.m3.2.2.2.2.2.2.1">subscript</csymbol><ci id="A4.1.p1.15.m3.2.2.2.2.2.2.1.2.cmml" xref="A4.1.p1.15.m3.2.2.2.2.2.2.1.2">𝑘</ci><apply id="A4.1.p1.15.m3.2.2.2.2.2.2.1.3.cmml" xref="A4.1.p1.15.m3.2.2.2.2.2.2.1.3"><times id="A4.1.p1.15.m3.2.2.2.2.2.2.1.3.1.cmml" xref="A4.1.p1.15.m3.2.2.2.2.2.2.1.3.1"></times><ci id="A4.1.p1.15.m3.2.2.2.2.2.2.1.3.2.cmml" xref="A4.1.p1.15.m3.2.2.2.2.2.2.1.3.2">c</ci><ci id="A4.1.p1.15.m3.2.2.2.2.2.2.1.3.3.cmml" xref="A4.1.p1.15.m3.2.2.2.2.2.2.1.3.3">𝑖</ci></apply></apply></apply><apply id="A4.1.p1.15.m3.2.2.2.4.cmml" xref="A4.1.p1.15.m3.2.2.2.4"><divide id="A4.1.p1.15.m3.2.2.2.4.1.cmml" xref="A4.1.p1.15.m3.2.2.2.4.1"></divide><cn id="A4.1.p1.15.m3.2.2.2.4.2.cmml" type="integer" xref="A4.1.p1.15.m3.2.2.2.4.2">1</cn><cn id="A4.1.p1.15.m3.2.2.2.4.3.cmml" type="integer" xref="A4.1.p1.15.m3.2.2.2.4.3">4</cn></apply></apply><apply id="A4.1.p1.15.m3.2.2.4.cmml" xref="A4.1.p1.15.m3.2.2.4"><csymbol cd="ambiguous" id="A4.1.p1.15.m3.2.2.4.1.cmml" xref="A4.1.p1.15.m3.2.2.4">superscript</csymbol><ci id="A4.1.p1.15.m3.2.2.4.2.cmml" xref="A4.1.p1.15.m3.2.2.4.2">ℝ</ci><plus id="A4.1.p1.15.m3.2.2.4.3.cmml" xref="A4.1.p1.15.m3.2.2.4.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.15.m3.2c">\min_{1\leq i\leq n}\{k_{{\rm c}i}\}-1/4\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.15.m3.2d">roman_min start_POSTSUBSCRIPT 1 ≤ italic_i ≤ italic_n end_POSTSUBSCRIPT { italic_k start_POSTSUBSCRIPT roman_c italic_i end_POSTSUBSCRIPT } - 1 / 4 ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>. Noting <math alttext="\|\bm{d}\|" class="ltx_Math" display="inline" id="A4.1.p1.16.m4.1"><semantics id="A4.1.p1.16.m4.1a"><mrow id="A4.1.p1.16.m4.1.2.2" xref="A4.1.p1.16.m4.1.2.1.cmml"><mo id="A4.1.p1.16.m4.1.2.2.1" stretchy="false" xref="A4.1.p1.16.m4.1.2.1.1.cmml">‖</mo><mi id="A4.1.p1.16.m4.1.1" xref="A4.1.p1.16.m4.1.1.cmml">𝒅</mi><mo id="A4.1.p1.16.m4.1.2.2.2" stretchy="false" xref="A4.1.p1.16.m4.1.2.1.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.16.m4.1b"><apply id="A4.1.p1.16.m4.1.2.1.cmml" xref="A4.1.p1.16.m4.1.2.2"><csymbol cd="latexml" id="A4.1.p1.16.m4.1.2.1.1.cmml" xref="A4.1.p1.16.m4.1.2.2.1">norm</csymbol><ci id="A4.1.p1.16.m4.1.1.cmml" xref="A4.1.p1.16.m4.1.1">𝒅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.16.m4.1c">\|\bm{d}\|</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.16.m4.1d">∥ bold_italic_d ∥</annotation></semantics></math>, <math alttext="\|\bm{d}_{\rm f}\|\leq\bar{d}" class="ltx_Math" display="inline" id="A4.1.p1.17.m5.1"><semantics id="A4.1.p1.17.m5.1a"><mrow id="A4.1.p1.17.m5.1.1" xref="A4.1.p1.17.m5.1.1.cmml"><mrow id="A4.1.p1.17.m5.1.1.1.1" xref="A4.1.p1.17.m5.1.1.1.2.cmml"><mo id="A4.1.p1.17.m5.1.1.1.1.2" stretchy="false" xref="A4.1.p1.17.m5.1.1.1.2.1.cmml">‖</mo><msub id="A4.1.p1.17.m5.1.1.1.1.1" xref="A4.1.p1.17.m5.1.1.1.1.1.cmml"><mi id="A4.1.p1.17.m5.1.1.1.1.1.2" xref="A4.1.p1.17.m5.1.1.1.1.1.2.cmml">𝒅</mi><mi id="A4.1.p1.17.m5.1.1.1.1.1.3" mathvariant="normal" xref="A4.1.p1.17.m5.1.1.1.1.1.3.cmml">f</mi></msub><mo id="A4.1.p1.17.m5.1.1.1.1.3" stretchy="false" xref="A4.1.p1.17.m5.1.1.1.2.1.cmml">‖</mo></mrow><mo id="A4.1.p1.17.m5.1.1.2" xref="A4.1.p1.17.m5.1.1.2.cmml">≤</mo><mover accent="true" id="A4.1.p1.17.m5.1.1.3" xref="A4.1.p1.17.m5.1.1.3.cmml"><mi id="A4.1.p1.17.m5.1.1.3.2" xref="A4.1.p1.17.m5.1.1.3.2.cmml">d</mi><mo id="A4.1.p1.17.m5.1.1.3.1" xref="A4.1.p1.17.m5.1.1.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.17.m5.1b"><apply id="A4.1.p1.17.m5.1.1.cmml" xref="A4.1.p1.17.m5.1.1"><leq id="A4.1.p1.17.m5.1.1.2.cmml" xref="A4.1.p1.17.m5.1.1.2"></leq><apply id="A4.1.p1.17.m5.1.1.1.2.cmml" xref="A4.1.p1.17.m5.1.1.1.1"><csymbol cd="latexml" id="A4.1.p1.17.m5.1.1.1.2.1.cmml" xref="A4.1.p1.17.m5.1.1.1.1.2">norm</csymbol><apply id="A4.1.p1.17.m5.1.1.1.1.1.cmml" xref="A4.1.p1.17.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.1.p1.17.m5.1.1.1.1.1.1.cmml" xref="A4.1.p1.17.m5.1.1.1.1.1">subscript</csymbol><ci id="A4.1.p1.17.m5.1.1.1.1.1.2.cmml" xref="A4.1.p1.17.m5.1.1.1.1.1.2">𝒅</ci><ci id="A4.1.p1.17.m5.1.1.1.1.1.3.cmml" xref="A4.1.p1.17.m5.1.1.1.1.1.3">f</ci></apply></apply><apply id="A4.1.p1.17.m5.1.1.3.cmml" xref="A4.1.p1.17.m5.1.1.3"><ci id="A4.1.p1.17.m5.1.1.3.1.cmml" xref="A4.1.p1.17.m5.1.1.3.1">¯</ci><ci id="A4.1.p1.17.m5.1.1.3.2.cmml" xref="A4.1.p1.17.m5.1.1.3.2">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.17.m5.1c">\|\bm{d}_{\rm f}\|\leq\bar{d}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.17.m5.1d">∥ bold_italic_d start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ∥ ≤ over¯ start_ARG italic_d end_ARG</annotation></semantics></math>, and <math alttext="\|{\bm{d}}_{\rm g}(t)\|" class="ltx_Math" display="inline" id="A4.1.p1.18.m6.2"><semantics id="A4.1.p1.18.m6.2a"><mrow id="A4.1.p1.18.m6.2.2.1" xref="A4.1.p1.18.m6.2.2.2.cmml"><mo id="A4.1.p1.18.m6.2.2.1.2" stretchy="false" xref="A4.1.p1.18.m6.2.2.2.1.cmml">‖</mo><mrow id="A4.1.p1.18.m6.2.2.1.1" xref="A4.1.p1.18.m6.2.2.1.1.cmml"><msub id="A4.1.p1.18.m6.2.2.1.1.2" xref="A4.1.p1.18.m6.2.2.1.1.2.cmml"><mi id="A4.1.p1.18.m6.2.2.1.1.2.2" xref="A4.1.p1.18.m6.2.2.1.1.2.2.cmml">𝒅</mi><mi id="A4.1.p1.18.m6.2.2.1.1.2.3" mathvariant="normal" xref="A4.1.p1.18.m6.2.2.1.1.2.3.cmml">g</mi></msub><mo id="A4.1.p1.18.m6.2.2.1.1.1" xref="A4.1.p1.18.m6.2.2.1.1.1.cmml">⁢</mo><mrow id="A4.1.p1.18.m6.2.2.1.1.3.2" xref="A4.1.p1.18.m6.2.2.1.1.cmml"><mo id="A4.1.p1.18.m6.2.2.1.1.3.2.1" stretchy="false" xref="A4.1.p1.18.m6.2.2.1.1.cmml">(</mo><mi id="A4.1.p1.18.m6.1.1" xref="A4.1.p1.18.m6.1.1.cmml">t</mi><mo id="A4.1.p1.18.m6.2.2.1.1.3.2.2" stretchy="false" xref="A4.1.p1.18.m6.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="A4.1.p1.18.m6.2.2.1.3" stretchy="false" xref="A4.1.p1.18.m6.2.2.2.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.18.m6.2b"><apply id="A4.1.p1.18.m6.2.2.2.cmml" xref="A4.1.p1.18.m6.2.2.1"><csymbol cd="latexml" id="A4.1.p1.18.m6.2.2.2.1.cmml" xref="A4.1.p1.18.m6.2.2.1.2">norm</csymbol><apply id="A4.1.p1.18.m6.2.2.1.1.cmml" xref="A4.1.p1.18.m6.2.2.1.1"><times id="A4.1.p1.18.m6.2.2.1.1.1.cmml" xref="A4.1.p1.18.m6.2.2.1.1.1"></times><apply id="A4.1.p1.18.m6.2.2.1.1.2.cmml" xref="A4.1.p1.18.m6.2.2.1.1.2"><csymbol cd="ambiguous" id="A4.1.p1.18.m6.2.2.1.1.2.1.cmml" xref="A4.1.p1.18.m6.2.2.1.1.2">subscript</csymbol><ci id="A4.1.p1.18.m6.2.2.1.1.2.2.cmml" xref="A4.1.p1.18.m6.2.2.1.1.2.2">𝒅</ci><ci id="A4.1.p1.18.m6.2.2.1.1.2.3.cmml" xref="A4.1.p1.18.m6.2.2.1.1.2.3">g</ci></apply><ci id="A4.1.p1.18.m6.1.1.cmml" xref="A4.1.p1.18.m6.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.18.m6.2c">\|{\bm{d}}_{\rm g}(t)\|</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.18.m6.2d">∥ bold_italic_d start_POSTSUBSCRIPT roman_g end_POSTSUBSCRIPT ( italic_t ) ∥</annotation></semantics></math> <math alttext="\leq" class="ltx_Math" display="inline" id="A4.1.p1.19.m7.1"><semantics id="A4.1.p1.19.m7.1a"><mo id="A4.1.p1.19.m7.1.1" xref="A4.1.p1.19.m7.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="A4.1.p1.19.m7.1b"><leq id="A4.1.p1.19.m7.1.1.cmml" xref="A4.1.p1.19.m7.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.19.m7.1c">\leq</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.19.m7.1d">≤</annotation></semantics></math> <math alttext="{\bar{d}}_{\rm g}" class="ltx_Math" display="inline" id="A4.1.p1.20.m8.1"><semantics id="A4.1.p1.20.m8.1a"><msub id="A4.1.p1.20.m8.1.1" xref="A4.1.p1.20.m8.1.1.cmml"><mover accent="true" id="A4.1.p1.20.m8.1.1.2" xref="A4.1.p1.20.m8.1.1.2.cmml"><mi id="A4.1.p1.20.m8.1.1.2.2" xref="A4.1.p1.20.m8.1.1.2.2.cmml">d</mi><mo id="A4.1.p1.20.m8.1.1.2.1" xref="A4.1.p1.20.m8.1.1.2.1.cmml">¯</mo></mover><mi id="A4.1.p1.20.m8.1.1.3" mathvariant="normal" xref="A4.1.p1.20.m8.1.1.3.cmml">g</mi></msub><annotation-xml encoding="MathML-Content" id="A4.1.p1.20.m8.1b"><apply id="A4.1.p1.20.m8.1.1.cmml" xref="A4.1.p1.20.m8.1.1"><csymbol cd="ambiguous" id="A4.1.p1.20.m8.1.1.1.cmml" xref="A4.1.p1.20.m8.1.1">subscript</csymbol><apply id="A4.1.p1.20.m8.1.1.2.cmml" xref="A4.1.p1.20.m8.1.1.2"><ci id="A4.1.p1.20.m8.1.1.2.1.cmml" xref="A4.1.p1.20.m8.1.1.2.1">¯</ci><ci id="A4.1.p1.20.m8.1.1.2.2.cmml" xref="A4.1.p1.20.m8.1.1.2.2">𝑑</ci></apply><ci id="A4.1.p1.20.m8.1.1.3.cmml" xref="A4.1.p1.20.m8.1.1.3">g</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.20.m8.1c">{\bar{d}}_{\rm g}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.20.m8.1d">over¯ start_ARG italic_d end_ARG start_POSTSUBSCRIPT roman_g end_POSTSUBSCRIPT</annotation></semantics></math>, one obtains</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx78"> <tbody id="A4.Ex69"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}\leq" class="ltx_Math" display="inline" id="A4.Ex69.m1.1"><semantics id="A4.Ex69.m1.1a"><mrow id="A4.Ex69.m1.1.1" xref="A4.Ex69.m1.1.1.cmml"><mover accent="true" id="A4.Ex69.m1.1.1.2" xref="A4.Ex69.m1.1.1.2.cmml"><mi id="A4.Ex69.m1.1.1.2.2" xref="A4.Ex69.m1.1.1.2.2.cmml">V</mi><mo id="A4.Ex69.m1.1.1.2.1" xref="A4.Ex69.m1.1.1.2.1.cmml">˙</mo></mover><mo id="A4.Ex69.m1.1.1.1" xref="A4.Ex69.m1.1.1.1.cmml">≤</mo><mi id="A4.Ex69.m1.1.1.3" xref="A4.Ex69.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex69.m1.1b"><apply id="A4.Ex69.m1.1.1.cmml" xref="A4.Ex69.m1.1.1"><leq id="A4.Ex69.m1.1.1.1.cmml" xref="A4.Ex69.m1.1.1.1"></leq><apply id="A4.Ex69.m1.1.1.2.cmml" xref="A4.Ex69.m1.1.1.2"><ci id="A4.Ex69.m1.1.1.2.1.cmml" xref="A4.Ex69.m1.1.1.2.1">˙</ci><ci id="A4.Ex69.m1.1.1.2.2.cmml" xref="A4.Ex69.m1.1.1.2.2">𝑉</ci></apply><csymbol cd="latexml" id="A4.Ex69.m1.1.1.3.cmml" xref="A4.Ex69.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex69.m1.1c">\displaystyle\dot{V}\leq</annotation><annotation encoding="application/x-llamapun" id="A4.Ex69.m1.1d">over˙ start_ARG italic_V end_ARG ≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-k_{\rm c}\|\bm{e}\|^{2}+\delta c_{\theta}\|\bm{e}\|+\bar{d}\|\bm% {e}\|+(1+p)(\bar{\epsilon}\bar{d}+\kappa c_{\theta}{\bar{d}}_{\rm g})." class="ltx_Math" display="inline" id="A4.Ex69.m2.4"><semantics id="A4.Ex69.m2.4a"><mrow id="A4.Ex69.m2.4.4.1" xref="A4.Ex69.m2.4.4.1.1.cmml"><mrow id="A4.Ex69.m2.4.4.1.1" xref="A4.Ex69.m2.4.4.1.1.cmml"><mrow id="A4.Ex69.m2.4.4.1.1.4" xref="A4.Ex69.m2.4.4.1.1.4.cmml"><mo id="A4.Ex69.m2.4.4.1.1.4a" xref="A4.Ex69.m2.4.4.1.1.4.cmml">−</mo><mrow id="A4.Ex69.m2.4.4.1.1.4.2" xref="A4.Ex69.m2.4.4.1.1.4.2.cmml"><msub id="A4.Ex69.m2.4.4.1.1.4.2.2" xref="A4.Ex69.m2.4.4.1.1.4.2.2.cmml"><mi id="A4.Ex69.m2.4.4.1.1.4.2.2.2" xref="A4.Ex69.m2.4.4.1.1.4.2.2.2.cmml">k</mi><mi id="A4.Ex69.m2.4.4.1.1.4.2.2.3" mathvariant="normal" xref="A4.Ex69.m2.4.4.1.1.4.2.2.3.cmml">c</mi></msub><mo id="A4.Ex69.m2.4.4.1.1.4.2.1" xref="A4.Ex69.m2.4.4.1.1.4.2.1.cmml">⁢</mo><msup id="A4.Ex69.m2.4.4.1.1.4.2.3" xref="A4.Ex69.m2.4.4.1.1.4.2.3.cmml"><mrow id="A4.Ex69.m2.4.4.1.1.4.2.3.2.2" xref="A4.Ex69.m2.4.4.1.1.4.2.3.2.1.cmml"><mo id="A4.Ex69.m2.4.4.1.1.4.2.3.2.2.1" stretchy="false" xref="A4.Ex69.m2.4.4.1.1.4.2.3.2.1.1.cmml">‖</mo><mi id="A4.Ex69.m2.1.1" xref="A4.Ex69.m2.1.1.cmml">𝒆</mi><mo id="A4.Ex69.m2.4.4.1.1.4.2.3.2.2.2" stretchy="false" xref="A4.Ex69.m2.4.4.1.1.4.2.3.2.1.1.cmml">‖</mo></mrow><mn id="A4.Ex69.m2.4.4.1.1.4.2.3.3" xref="A4.Ex69.m2.4.4.1.1.4.2.3.3.cmml">2</mn></msup></mrow></mrow><mo id="A4.Ex69.m2.4.4.1.1.3" xref="A4.Ex69.m2.4.4.1.1.3.cmml">+</mo><mrow id="A4.Ex69.m2.4.4.1.1.5" xref="A4.Ex69.m2.4.4.1.1.5.cmml"><mi id="A4.Ex69.m2.4.4.1.1.5.2" xref="A4.Ex69.m2.4.4.1.1.5.2.cmml">δ</mi><mo id="A4.Ex69.m2.4.4.1.1.5.1" xref="A4.Ex69.m2.4.4.1.1.5.1.cmml">⁢</mo><msub id="A4.Ex69.m2.4.4.1.1.5.3" xref="A4.Ex69.m2.4.4.1.1.5.3.cmml"><mi id="A4.Ex69.m2.4.4.1.1.5.3.2" xref="A4.Ex69.m2.4.4.1.1.5.3.2.cmml">c</mi><mi id="A4.Ex69.m2.4.4.1.1.5.3.3" xref="A4.Ex69.m2.4.4.1.1.5.3.3.cmml">θ</mi></msub><mo id="A4.Ex69.m2.4.4.1.1.5.1a" xref="A4.Ex69.m2.4.4.1.1.5.1.cmml">⁢</mo><mrow id="A4.Ex69.m2.4.4.1.1.5.4.2" xref="A4.Ex69.m2.4.4.1.1.5.4.1.cmml"><mo id="A4.Ex69.m2.4.4.1.1.5.4.2.1" stretchy="false" xref="A4.Ex69.m2.4.4.1.1.5.4.1.1.cmml">‖</mo><mi id="A4.Ex69.m2.2.2" xref="A4.Ex69.m2.2.2.cmml">𝒆</mi><mo id="A4.Ex69.m2.4.4.1.1.5.4.2.2" stretchy="false" xref="A4.Ex69.m2.4.4.1.1.5.4.1.1.cmml">‖</mo></mrow></mrow><mo id="A4.Ex69.m2.4.4.1.1.3a" xref="A4.Ex69.m2.4.4.1.1.3.cmml">+</mo><mrow id="A4.Ex69.m2.4.4.1.1.6" xref="A4.Ex69.m2.4.4.1.1.6.cmml"><mover accent="true" id="A4.Ex69.m2.4.4.1.1.6.2" xref="A4.Ex69.m2.4.4.1.1.6.2.cmml"><mi id="A4.Ex69.m2.4.4.1.1.6.2.2" xref="A4.Ex69.m2.4.4.1.1.6.2.2.cmml">d</mi><mo id="A4.Ex69.m2.4.4.1.1.6.2.1" xref="A4.Ex69.m2.4.4.1.1.6.2.1.cmml">¯</mo></mover><mo id="A4.Ex69.m2.4.4.1.1.6.1" xref="A4.Ex69.m2.4.4.1.1.6.1.cmml">⁢</mo><mrow id="A4.Ex69.m2.4.4.1.1.6.3.2" xref="A4.Ex69.m2.4.4.1.1.6.3.1.cmml"><mo id="A4.Ex69.m2.4.4.1.1.6.3.2.1" stretchy="false" xref="A4.Ex69.m2.4.4.1.1.6.3.1.1.cmml">‖</mo><mi id="A4.Ex69.m2.3.3" xref="A4.Ex69.m2.3.3.cmml">𝒆</mi><mo id="A4.Ex69.m2.4.4.1.1.6.3.2.2" stretchy="false" xref="A4.Ex69.m2.4.4.1.1.6.3.1.1.cmml">‖</mo></mrow></mrow><mo id="A4.Ex69.m2.4.4.1.1.3b" xref="A4.Ex69.m2.4.4.1.1.3.cmml">+</mo><mrow id="A4.Ex69.m2.4.4.1.1.2" xref="A4.Ex69.m2.4.4.1.1.2.cmml"><mrow id="A4.Ex69.m2.4.4.1.1.1.1.1" xref="A4.Ex69.m2.4.4.1.1.1.1.1.1.cmml"><mo id="A4.Ex69.m2.4.4.1.1.1.1.1.2" stretchy="false" xref="A4.Ex69.m2.4.4.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex69.m2.4.4.1.1.1.1.1.1" xref="A4.Ex69.m2.4.4.1.1.1.1.1.1.cmml"><mn id="A4.Ex69.m2.4.4.1.1.1.1.1.1.2" xref="A4.Ex69.m2.4.4.1.1.1.1.1.1.2.cmml">1</mn><mo id="A4.Ex69.m2.4.4.1.1.1.1.1.1.1" xref="A4.Ex69.m2.4.4.1.1.1.1.1.1.1.cmml">+</mo><mi id="A4.Ex69.m2.4.4.1.1.1.1.1.1.3" xref="A4.Ex69.m2.4.4.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A4.Ex69.m2.4.4.1.1.1.1.1.3" stretchy="false" xref="A4.Ex69.m2.4.4.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex69.m2.4.4.1.1.2.3" xref="A4.Ex69.m2.4.4.1.1.2.3.cmml">⁢</mo><mrow id="A4.Ex69.m2.4.4.1.1.2.2.1" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.cmml"><mo id="A4.Ex69.m2.4.4.1.1.2.2.1.2" stretchy="false" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.cmml">(</mo><mrow id="A4.Ex69.m2.4.4.1.1.2.2.1.1" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.cmml"><mrow id="A4.Ex69.m2.4.4.1.1.2.2.1.1.2" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.cmml"><mover accent="true" id="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.2" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.2.cmml"><mi id="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.2.2" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.2.2.cmml">ϵ</mi><mo id="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.2.1" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.2.1.cmml">¯</mo></mover><mo id="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.1" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.1.cmml">⁢</mo><mover accent="true" id="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.3" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.3.cmml"><mi id="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.3.2" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.3.2.cmml">d</mi><mo id="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.3.1" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.3.1.cmml">¯</mo></mover></mrow><mo id="A4.Ex69.m2.4.4.1.1.2.2.1.1.1" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.1.cmml">+</mo><mrow id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.cmml"><mi id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.2" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.2.cmml">κ</mi><mo id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.1" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.1.cmml">⁢</mo><msub id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.3" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.3.cmml"><mi id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.3.2" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.3.2.cmml">c</mi><mi id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.3.3" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.3.3.cmml">θ</mi></msub><mo id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.1a" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.1.cmml">⁢</mo><msub id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.4" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.4.cmml"><mover accent="true" id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.4.2" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.4.2.cmml"><mi id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.4.2.2" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.4.2.2.cmml">d</mi><mo id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.4.2.1" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.4.2.1.cmml">¯</mo></mover><mi id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.4.3" mathvariant="normal" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.4.3.cmml">g</mi></msub></mrow></mrow><mo id="A4.Ex69.m2.4.4.1.1.2.2.1.3" stretchy="false" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A4.Ex69.m2.4.4.1.2" lspace="0em" xref="A4.Ex69.m2.4.4.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex69.m2.4b"><apply id="A4.Ex69.m2.4.4.1.1.cmml" xref="A4.Ex69.m2.4.4.1"><plus id="A4.Ex69.m2.4.4.1.1.3.cmml" xref="A4.Ex69.m2.4.4.1.1.3"></plus><apply id="A4.Ex69.m2.4.4.1.1.4.cmml" xref="A4.Ex69.m2.4.4.1.1.4"><minus id="A4.Ex69.m2.4.4.1.1.4.1.cmml" xref="A4.Ex69.m2.4.4.1.1.4"></minus><apply id="A4.Ex69.m2.4.4.1.1.4.2.cmml" xref="A4.Ex69.m2.4.4.1.1.4.2"><times id="A4.Ex69.m2.4.4.1.1.4.2.1.cmml" xref="A4.Ex69.m2.4.4.1.1.4.2.1"></times><apply id="A4.Ex69.m2.4.4.1.1.4.2.2.cmml" xref="A4.Ex69.m2.4.4.1.1.4.2.2"><csymbol cd="ambiguous" id="A4.Ex69.m2.4.4.1.1.4.2.2.1.cmml" xref="A4.Ex69.m2.4.4.1.1.4.2.2">subscript</csymbol><ci id="A4.Ex69.m2.4.4.1.1.4.2.2.2.cmml" xref="A4.Ex69.m2.4.4.1.1.4.2.2.2">𝑘</ci><ci id="A4.Ex69.m2.4.4.1.1.4.2.2.3.cmml" xref="A4.Ex69.m2.4.4.1.1.4.2.2.3">c</ci></apply><apply id="A4.Ex69.m2.4.4.1.1.4.2.3.cmml" xref="A4.Ex69.m2.4.4.1.1.4.2.3"><csymbol cd="ambiguous" id="A4.Ex69.m2.4.4.1.1.4.2.3.1.cmml" xref="A4.Ex69.m2.4.4.1.1.4.2.3">superscript</csymbol><apply id="A4.Ex69.m2.4.4.1.1.4.2.3.2.1.cmml" xref="A4.Ex69.m2.4.4.1.1.4.2.3.2.2"><csymbol cd="latexml" id="A4.Ex69.m2.4.4.1.1.4.2.3.2.1.1.cmml" xref="A4.Ex69.m2.4.4.1.1.4.2.3.2.2.1">norm</csymbol><ci id="A4.Ex69.m2.1.1.cmml" xref="A4.Ex69.m2.1.1">𝒆</ci></apply><cn id="A4.Ex69.m2.4.4.1.1.4.2.3.3.cmml" type="integer" xref="A4.Ex69.m2.4.4.1.1.4.2.3.3">2</cn></apply></apply></apply><apply id="A4.Ex69.m2.4.4.1.1.5.cmml" xref="A4.Ex69.m2.4.4.1.1.5"><times id="A4.Ex69.m2.4.4.1.1.5.1.cmml" xref="A4.Ex69.m2.4.4.1.1.5.1"></times><ci id="A4.Ex69.m2.4.4.1.1.5.2.cmml" xref="A4.Ex69.m2.4.4.1.1.5.2">𝛿</ci><apply id="A4.Ex69.m2.4.4.1.1.5.3.cmml" xref="A4.Ex69.m2.4.4.1.1.5.3"><csymbol cd="ambiguous" id="A4.Ex69.m2.4.4.1.1.5.3.1.cmml" xref="A4.Ex69.m2.4.4.1.1.5.3">subscript</csymbol><ci id="A4.Ex69.m2.4.4.1.1.5.3.2.cmml" xref="A4.Ex69.m2.4.4.1.1.5.3.2">𝑐</ci><ci id="A4.Ex69.m2.4.4.1.1.5.3.3.cmml" xref="A4.Ex69.m2.4.4.1.1.5.3.3">𝜃</ci></apply><apply id="A4.Ex69.m2.4.4.1.1.5.4.1.cmml" xref="A4.Ex69.m2.4.4.1.1.5.4.2"><csymbol cd="latexml" id="A4.Ex69.m2.4.4.1.1.5.4.1.1.cmml" xref="A4.Ex69.m2.4.4.1.1.5.4.2.1">norm</csymbol><ci id="A4.Ex69.m2.2.2.cmml" xref="A4.Ex69.m2.2.2">𝒆</ci></apply></apply><apply id="A4.Ex69.m2.4.4.1.1.6.cmml" xref="A4.Ex69.m2.4.4.1.1.6"><times id="A4.Ex69.m2.4.4.1.1.6.1.cmml" xref="A4.Ex69.m2.4.4.1.1.6.1"></times><apply id="A4.Ex69.m2.4.4.1.1.6.2.cmml" xref="A4.Ex69.m2.4.4.1.1.6.2"><ci id="A4.Ex69.m2.4.4.1.1.6.2.1.cmml" xref="A4.Ex69.m2.4.4.1.1.6.2.1">¯</ci><ci id="A4.Ex69.m2.4.4.1.1.6.2.2.cmml" xref="A4.Ex69.m2.4.4.1.1.6.2.2">𝑑</ci></apply><apply id="A4.Ex69.m2.4.4.1.1.6.3.1.cmml" xref="A4.Ex69.m2.4.4.1.1.6.3.2"><csymbol cd="latexml" id="A4.Ex69.m2.4.4.1.1.6.3.1.1.cmml" xref="A4.Ex69.m2.4.4.1.1.6.3.2.1">norm</csymbol><ci id="A4.Ex69.m2.3.3.cmml" xref="A4.Ex69.m2.3.3">𝒆</ci></apply></apply><apply id="A4.Ex69.m2.4.4.1.1.2.cmml" xref="A4.Ex69.m2.4.4.1.1.2"><times id="A4.Ex69.m2.4.4.1.1.2.3.cmml" xref="A4.Ex69.m2.4.4.1.1.2.3"></times><apply id="A4.Ex69.m2.4.4.1.1.1.1.1.1.cmml" xref="A4.Ex69.m2.4.4.1.1.1.1.1"><plus id="A4.Ex69.m2.4.4.1.1.1.1.1.1.1.cmml" xref="A4.Ex69.m2.4.4.1.1.1.1.1.1.1"></plus><cn id="A4.Ex69.m2.4.4.1.1.1.1.1.1.2.cmml" type="integer" xref="A4.Ex69.m2.4.4.1.1.1.1.1.1.2">1</cn><ci id="A4.Ex69.m2.4.4.1.1.1.1.1.1.3.cmml" xref="A4.Ex69.m2.4.4.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="A4.Ex69.m2.4.4.1.1.2.2.1.1.cmml" xref="A4.Ex69.m2.4.4.1.1.2.2.1"><plus id="A4.Ex69.m2.4.4.1.1.2.2.1.1.1.cmml" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.1"></plus><apply id="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.cmml" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.2"><times id="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.1.cmml" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.1"></times><apply id="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.2.cmml" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.2"><ci id="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.2.1.cmml" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.2.1">¯</ci><ci id="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.2.2.cmml" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.2.2">italic-ϵ</ci></apply><apply id="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.3.cmml" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.3"><ci id="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.3.1.cmml" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.3.1">¯</ci><ci id="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.3.2.cmml" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.2.3.2">𝑑</ci></apply></apply><apply id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.cmml" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3"><times id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.1.cmml" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.1"></times><ci id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.2.cmml" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.2">𝜅</ci><apply id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.3.cmml" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.3"><csymbol cd="ambiguous" id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.3.1.cmml" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.3">subscript</csymbol><ci id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.3.2.cmml" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.3.2">𝑐</ci><ci id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.3.3.cmml" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.3.3">𝜃</ci></apply><apply id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.4.cmml" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.4"><csymbol cd="ambiguous" id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.4.1.cmml" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.4">subscript</csymbol><apply id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.4.2.cmml" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.4.2"><ci id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.4.2.1.cmml" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.4.2.1">¯</ci><ci id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.4.2.2.cmml" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.4.2.2">𝑑</ci></apply><ci id="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.4.3.cmml" xref="A4.Ex69.m2.4.4.1.1.2.2.1.1.3.4.3">g</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex69.m2.4c">\displaystyle-k_{\rm c}\|\bm{e}\|^{2}+\delta c_{\theta}\|\bm{e}\|+\bar{d}\|\bm% {e}\|+(1+p)(\bar{\epsilon}\bar{d}+\kappa c_{\theta}{\bar{d}}_{\rm g}).</annotation><annotation encoding="application/x-llamapun" id="A4.Ex69.m2.4d">- italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ∥ bold_italic_e ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_δ italic_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ∥ bold_italic_e ∥ + over¯ start_ARG italic_d end_ARG ∥ bold_italic_e ∥ + ( 1 + italic_p ) ( over¯ start_ARG italic_ϵ end_ARG over¯ start_ARG italic_d end_ARG + italic_κ italic_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT over¯ start_ARG italic_d end_ARG start_POSTSUBSCRIPT roman_g end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.1.p1.23">Applying Young’s inequality <math alttext="ab\leq a^{2}/2+b^{2}/2" class="ltx_Math" display="inline" id="A4.1.p1.21.m1.1"><semantics id="A4.1.p1.21.m1.1a"><mrow id="A4.1.p1.21.m1.1.1" xref="A4.1.p1.21.m1.1.1.cmml"><mrow id="A4.1.p1.21.m1.1.1.2" xref="A4.1.p1.21.m1.1.1.2.cmml"><mi id="A4.1.p1.21.m1.1.1.2.2" xref="A4.1.p1.21.m1.1.1.2.2.cmml">a</mi><mo id="A4.1.p1.21.m1.1.1.2.1" xref="A4.1.p1.21.m1.1.1.2.1.cmml">⁢</mo><mi id="A4.1.p1.21.m1.1.1.2.3" xref="A4.1.p1.21.m1.1.1.2.3.cmml">b</mi></mrow><mo id="A4.1.p1.21.m1.1.1.1" xref="A4.1.p1.21.m1.1.1.1.cmml">≤</mo><mrow id="A4.1.p1.21.m1.1.1.3" xref="A4.1.p1.21.m1.1.1.3.cmml"><mrow id="A4.1.p1.21.m1.1.1.3.2" xref="A4.1.p1.21.m1.1.1.3.2.cmml"><msup id="A4.1.p1.21.m1.1.1.3.2.2" xref="A4.1.p1.21.m1.1.1.3.2.2.cmml"><mi id="A4.1.p1.21.m1.1.1.3.2.2.2" xref="A4.1.p1.21.m1.1.1.3.2.2.2.cmml">a</mi><mn id="A4.1.p1.21.m1.1.1.3.2.2.3" xref="A4.1.p1.21.m1.1.1.3.2.2.3.cmml">2</mn></msup><mo id="A4.1.p1.21.m1.1.1.3.2.1" xref="A4.1.p1.21.m1.1.1.3.2.1.cmml">/</mo><mn id="A4.1.p1.21.m1.1.1.3.2.3" xref="A4.1.p1.21.m1.1.1.3.2.3.cmml">2</mn></mrow><mo id="A4.1.p1.21.m1.1.1.3.1" xref="A4.1.p1.21.m1.1.1.3.1.cmml">+</mo><mrow id="A4.1.p1.21.m1.1.1.3.3" xref="A4.1.p1.21.m1.1.1.3.3.cmml"><msup id="A4.1.p1.21.m1.1.1.3.3.2" xref="A4.1.p1.21.m1.1.1.3.3.2.cmml"><mi id="A4.1.p1.21.m1.1.1.3.3.2.2" xref="A4.1.p1.21.m1.1.1.3.3.2.2.cmml">b</mi><mn id="A4.1.p1.21.m1.1.1.3.3.2.3" xref="A4.1.p1.21.m1.1.1.3.3.2.3.cmml">2</mn></msup><mo id="A4.1.p1.21.m1.1.1.3.3.1" xref="A4.1.p1.21.m1.1.1.3.3.1.cmml">/</mo><mn id="A4.1.p1.21.m1.1.1.3.3.3" xref="A4.1.p1.21.m1.1.1.3.3.3.cmml">2</mn></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.21.m1.1b"><apply id="A4.1.p1.21.m1.1.1.cmml" xref="A4.1.p1.21.m1.1.1"><leq id="A4.1.p1.21.m1.1.1.1.cmml" xref="A4.1.p1.21.m1.1.1.1"></leq><apply id="A4.1.p1.21.m1.1.1.2.cmml" xref="A4.1.p1.21.m1.1.1.2"><times id="A4.1.p1.21.m1.1.1.2.1.cmml" xref="A4.1.p1.21.m1.1.1.2.1"></times><ci id="A4.1.p1.21.m1.1.1.2.2.cmml" xref="A4.1.p1.21.m1.1.1.2.2">𝑎</ci><ci id="A4.1.p1.21.m1.1.1.2.3.cmml" xref="A4.1.p1.21.m1.1.1.2.3">𝑏</ci></apply><apply id="A4.1.p1.21.m1.1.1.3.cmml" xref="A4.1.p1.21.m1.1.1.3"><plus id="A4.1.p1.21.m1.1.1.3.1.cmml" xref="A4.1.p1.21.m1.1.1.3.1"></plus><apply id="A4.1.p1.21.m1.1.1.3.2.cmml" xref="A4.1.p1.21.m1.1.1.3.2"><divide id="A4.1.p1.21.m1.1.1.3.2.1.cmml" xref="A4.1.p1.21.m1.1.1.3.2.1"></divide><apply id="A4.1.p1.21.m1.1.1.3.2.2.cmml" xref="A4.1.p1.21.m1.1.1.3.2.2"><csymbol cd="ambiguous" id="A4.1.p1.21.m1.1.1.3.2.2.1.cmml" xref="A4.1.p1.21.m1.1.1.3.2.2">superscript</csymbol><ci id="A4.1.p1.21.m1.1.1.3.2.2.2.cmml" xref="A4.1.p1.21.m1.1.1.3.2.2.2">𝑎</ci><cn id="A4.1.p1.21.m1.1.1.3.2.2.3.cmml" type="integer" xref="A4.1.p1.21.m1.1.1.3.2.2.3">2</cn></apply><cn id="A4.1.p1.21.m1.1.1.3.2.3.cmml" type="integer" xref="A4.1.p1.21.m1.1.1.3.2.3">2</cn></apply><apply id="A4.1.p1.21.m1.1.1.3.3.cmml" xref="A4.1.p1.21.m1.1.1.3.3"><divide id="A4.1.p1.21.m1.1.1.3.3.1.cmml" xref="A4.1.p1.21.m1.1.1.3.3.1"></divide><apply id="A4.1.p1.21.m1.1.1.3.3.2.cmml" xref="A4.1.p1.21.m1.1.1.3.3.2"><csymbol cd="ambiguous" id="A4.1.p1.21.m1.1.1.3.3.2.1.cmml" xref="A4.1.p1.21.m1.1.1.3.3.2">superscript</csymbol><ci id="A4.1.p1.21.m1.1.1.3.3.2.2.cmml" xref="A4.1.p1.21.m1.1.1.3.3.2.2">𝑏</ci><cn id="A4.1.p1.21.m1.1.1.3.3.2.3.cmml" type="integer" xref="A4.1.p1.21.m1.1.1.3.3.2.3">2</cn></apply><cn id="A4.1.p1.21.m1.1.1.3.3.3.cmml" type="integer" xref="A4.1.p1.21.m1.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.21.m1.1c">ab\leq a^{2}/2+b^{2}/2</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.21.m1.1d">italic_a italic_b ≤ italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 + italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2</annotation></semantics></math> with <math alttext="a=\sqrt{k_{\rm c}}\|\bm{e}\|" class="ltx_Math" display="inline" id="A4.1.p1.22.m2.1"><semantics id="A4.1.p1.22.m2.1a"><mrow id="A4.1.p1.22.m2.1.2" xref="A4.1.p1.22.m2.1.2.cmml"><mi id="A4.1.p1.22.m2.1.2.2" xref="A4.1.p1.22.m2.1.2.2.cmml">a</mi><mo id="A4.1.p1.22.m2.1.2.1" xref="A4.1.p1.22.m2.1.2.1.cmml">=</mo><mrow id="A4.1.p1.22.m2.1.2.3" xref="A4.1.p1.22.m2.1.2.3.cmml"><msqrt id="A4.1.p1.22.m2.1.2.3.2" xref="A4.1.p1.22.m2.1.2.3.2.cmml"><msub id="A4.1.p1.22.m2.1.2.3.2.2" xref="A4.1.p1.22.m2.1.2.3.2.2.cmml"><mi id="A4.1.p1.22.m2.1.2.3.2.2.2" xref="A4.1.p1.22.m2.1.2.3.2.2.2.cmml">k</mi><mi id="A4.1.p1.22.m2.1.2.3.2.2.3" mathvariant="normal" xref="A4.1.p1.22.m2.1.2.3.2.2.3.cmml">c</mi></msub></msqrt><mo id="A4.1.p1.22.m2.1.2.3.1" xref="A4.1.p1.22.m2.1.2.3.1.cmml">⁢</mo><mrow id="A4.1.p1.22.m2.1.2.3.3.2" xref="A4.1.p1.22.m2.1.2.3.3.1.cmml"><mo id="A4.1.p1.22.m2.1.2.3.3.2.1" stretchy="false" xref="A4.1.p1.22.m2.1.2.3.3.1.1.cmml">‖</mo><mi id="A4.1.p1.22.m2.1.1" xref="A4.1.p1.22.m2.1.1.cmml">𝒆</mi><mo id="A4.1.p1.22.m2.1.2.3.3.2.2" stretchy="false" xref="A4.1.p1.22.m2.1.2.3.3.1.1.cmml">‖</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.22.m2.1b"><apply id="A4.1.p1.22.m2.1.2.cmml" xref="A4.1.p1.22.m2.1.2"><eq id="A4.1.p1.22.m2.1.2.1.cmml" xref="A4.1.p1.22.m2.1.2.1"></eq><ci id="A4.1.p1.22.m2.1.2.2.cmml" xref="A4.1.p1.22.m2.1.2.2">𝑎</ci><apply id="A4.1.p1.22.m2.1.2.3.cmml" xref="A4.1.p1.22.m2.1.2.3"><times id="A4.1.p1.22.m2.1.2.3.1.cmml" xref="A4.1.p1.22.m2.1.2.3.1"></times><apply id="A4.1.p1.22.m2.1.2.3.2.cmml" xref="A4.1.p1.22.m2.1.2.3.2"><root id="A4.1.p1.22.m2.1.2.3.2a.cmml" xref="A4.1.p1.22.m2.1.2.3.2"></root><apply id="A4.1.p1.22.m2.1.2.3.2.2.cmml" xref="A4.1.p1.22.m2.1.2.3.2.2"><csymbol cd="ambiguous" id="A4.1.p1.22.m2.1.2.3.2.2.1.cmml" xref="A4.1.p1.22.m2.1.2.3.2.2">subscript</csymbol><ci id="A4.1.p1.22.m2.1.2.3.2.2.2.cmml" xref="A4.1.p1.22.m2.1.2.3.2.2.2">𝑘</ci><ci id="A4.1.p1.22.m2.1.2.3.2.2.3.cmml" xref="A4.1.p1.22.m2.1.2.3.2.2.3">c</ci></apply></apply><apply id="A4.1.p1.22.m2.1.2.3.3.1.cmml" xref="A4.1.p1.22.m2.1.2.3.3.2"><csymbol cd="latexml" id="A4.1.p1.22.m2.1.2.3.3.1.1.cmml" xref="A4.1.p1.22.m2.1.2.3.3.2.1">norm</csymbol><ci id="A4.1.p1.22.m2.1.1.cmml" xref="A4.1.p1.22.m2.1.1">𝒆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.22.m2.1c">a=\sqrt{k_{\rm c}}\|\bm{e}\|</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.22.m2.1d">italic_a = square-root start_ARG italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT end_ARG ∥ bold_italic_e ∥</annotation></semantics></math> and <math alttext="b=(\delta c_{\theta}+\bar{d})/\sqrt{k_{\rm c}}" class="ltx_Math" display="inline" id="A4.1.p1.23.m3.1"><semantics id="A4.1.p1.23.m3.1a"><mrow id="A4.1.p1.23.m3.1.1" xref="A4.1.p1.23.m3.1.1.cmml"><mi id="A4.1.p1.23.m3.1.1.3" xref="A4.1.p1.23.m3.1.1.3.cmml">b</mi><mo id="A4.1.p1.23.m3.1.1.2" xref="A4.1.p1.23.m3.1.1.2.cmml">=</mo><mrow id="A4.1.p1.23.m3.1.1.1" xref="A4.1.p1.23.m3.1.1.1.cmml"><mrow id="A4.1.p1.23.m3.1.1.1.1.1" xref="A4.1.p1.23.m3.1.1.1.1.1.1.cmml"><mo id="A4.1.p1.23.m3.1.1.1.1.1.2" stretchy="false" xref="A4.1.p1.23.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.1.p1.23.m3.1.1.1.1.1.1" xref="A4.1.p1.23.m3.1.1.1.1.1.1.cmml"><mrow id="A4.1.p1.23.m3.1.1.1.1.1.1.2" xref="A4.1.p1.23.m3.1.1.1.1.1.1.2.cmml"><mi id="A4.1.p1.23.m3.1.1.1.1.1.1.2.2" xref="A4.1.p1.23.m3.1.1.1.1.1.1.2.2.cmml">δ</mi><mo id="A4.1.p1.23.m3.1.1.1.1.1.1.2.1" xref="A4.1.p1.23.m3.1.1.1.1.1.1.2.1.cmml">⁢</mo><msub id="A4.1.p1.23.m3.1.1.1.1.1.1.2.3" xref="A4.1.p1.23.m3.1.1.1.1.1.1.2.3.cmml"><mi id="A4.1.p1.23.m3.1.1.1.1.1.1.2.3.2" xref="A4.1.p1.23.m3.1.1.1.1.1.1.2.3.2.cmml">c</mi><mi id="A4.1.p1.23.m3.1.1.1.1.1.1.2.3.3" xref="A4.1.p1.23.m3.1.1.1.1.1.1.2.3.3.cmml">θ</mi></msub></mrow><mo id="A4.1.p1.23.m3.1.1.1.1.1.1.1" xref="A4.1.p1.23.m3.1.1.1.1.1.1.1.cmml">+</mo><mover accent="true" id="A4.1.p1.23.m3.1.1.1.1.1.1.3" xref="A4.1.p1.23.m3.1.1.1.1.1.1.3.cmml"><mi id="A4.1.p1.23.m3.1.1.1.1.1.1.3.2" xref="A4.1.p1.23.m3.1.1.1.1.1.1.3.2.cmml">d</mi><mo id="A4.1.p1.23.m3.1.1.1.1.1.1.3.1" xref="A4.1.p1.23.m3.1.1.1.1.1.1.3.1.cmml">¯</mo></mover></mrow><mo id="A4.1.p1.23.m3.1.1.1.1.1.3" stretchy="false" xref="A4.1.p1.23.m3.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.1.p1.23.m3.1.1.1.2" xref="A4.1.p1.23.m3.1.1.1.2.cmml">/</mo><msqrt id="A4.1.p1.23.m3.1.1.1.3" xref="A4.1.p1.23.m3.1.1.1.3.cmml"><msub id="A4.1.p1.23.m3.1.1.1.3.2" xref="A4.1.p1.23.m3.1.1.1.3.2.cmml"><mi id="A4.1.p1.23.m3.1.1.1.3.2.2" xref="A4.1.p1.23.m3.1.1.1.3.2.2.cmml">k</mi><mi id="A4.1.p1.23.m3.1.1.1.3.2.3" mathvariant="normal" xref="A4.1.p1.23.m3.1.1.1.3.2.3.cmml">c</mi></msub></msqrt></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.23.m3.1b"><apply id="A4.1.p1.23.m3.1.1.cmml" xref="A4.1.p1.23.m3.1.1"><eq id="A4.1.p1.23.m3.1.1.2.cmml" xref="A4.1.p1.23.m3.1.1.2"></eq><ci id="A4.1.p1.23.m3.1.1.3.cmml" xref="A4.1.p1.23.m3.1.1.3">𝑏</ci><apply id="A4.1.p1.23.m3.1.1.1.cmml" xref="A4.1.p1.23.m3.1.1.1"><divide id="A4.1.p1.23.m3.1.1.1.2.cmml" xref="A4.1.p1.23.m3.1.1.1.2"></divide><apply id="A4.1.p1.23.m3.1.1.1.1.1.1.cmml" xref="A4.1.p1.23.m3.1.1.1.1.1"><plus id="A4.1.p1.23.m3.1.1.1.1.1.1.1.cmml" xref="A4.1.p1.23.m3.1.1.1.1.1.1.1"></plus><apply id="A4.1.p1.23.m3.1.1.1.1.1.1.2.cmml" xref="A4.1.p1.23.m3.1.1.1.1.1.1.2"><times id="A4.1.p1.23.m3.1.1.1.1.1.1.2.1.cmml" xref="A4.1.p1.23.m3.1.1.1.1.1.1.2.1"></times><ci id="A4.1.p1.23.m3.1.1.1.1.1.1.2.2.cmml" xref="A4.1.p1.23.m3.1.1.1.1.1.1.2.2">𝛿</ci><apply id="A4.1.p1.23.m3.1.1.1.1.1.1.2.3.cmml" xref="A4.1.p1.23.m3.1.1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="A4.1.p1.23.m3.1.1.1.1.1.1.2.3.1.cmml" xref="A4.1.p1.23.m3.1.1.1.1.1.1.2.3">subscript</csymbol><ci id="A4.1.p1.23.m3.1.1.1.1.1.1.2.3.2.cmml" xref="A4.1.p1.23.m3.1.1.1.1.1.1.2.3.2">𝑐</ci><ci id="A4.1.p1.23.m3.1.1.1.1.1.1.2.3.3.cmml" xref="A4.1.p1.23.m3.1.1.1.1.1.1.2.3.3">𝜃</ci></apply></apply><apply id="A4.1.p1.23.m3.1.1.1.1.1.1.3.cmml" xref="A4.1.p1.23.m3.1.1.1.1.1.1.3"><ci id="A4.1.p1.23.m3.1.1.1.1.1.1.3.1.cmml" xref="A4.1.p1.23.m3.1.1.1.1.1.1.3.1">¯</ci><ci id="A4.1.p1.23.m3.1.1.1.1.1.1.3.2.cmml" xref="A4.1.p1.23.m3.1.1.1.1.1.1.3.2">𝑑</ci></apply></apply><apply id="A4.1.p1.23.m3.1.1.1.3.cmml" xref="A4.1.p1.23.m3.1.1.1.3"><root id="A4.1.p1.23.m3.1.1.1.3a.cmml" xref="A4.1.p1.23.m3.1.1.1.3"></root><apply id="A4.1.p1.23.m3.1.1.1.3.2.cmml" xref="A4.1.p1.23.m3.1.1.1.3.2"><csymbol cd="ambiguous" id="A4.1.p1.23.m3.1.1.1.3.2.1.cmml" xref="A4.1.p1.23.m3.1.1.1.3.2">subscript</csymbol><ci id="A4.1.p1.23.m3.1.1.1.3.2.2.cmml" xref="A4.1.p1.23.m3.1.1.1.3.2.2">𝑘</ci><ci id="A4.1.p1.23.m3.1.1.1.3.2.3.cmml" xref="A4.1.p1.23.m3.1.1.1.3.2.3">c</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.23.m3.1c">b=(\delta c_{\theta}+\bar{d})/\sqrt{k_{\rm c}}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.23.m3.1d">italic_b = ( italic_δ italic_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT + over¯ start_ARG italic_d end_ARG ) / square-root start_ARG italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT end_ARG</annotation></semantics></math> to the above inequality yields</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx79"> <tbody id="A4.Ex70"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}\leq-k_{\rm c}\|\bm{e}\|^{2}/2+\rho(k_{\rm c},\kappa,\bar{% d})" class="ltx_Math" display="inline" id="A4.Ex70.m1.4"><semantics id="A4.Ex70.m1.4a"><mrow id="A4.Ex70.m1.4.4" xref="A4.Ex70.m1.4.4.cmml"><mover accent="true" id="A4.Ex70.m1.4.4.3" xref="A4.Ex70.m1.4.4.3.cmml"><mi id="A4.Ex70.m1.4.4.3.2" xref="A4.Ex70.m1.4.4.3.2.cmml">V</mi><mo id="A4.Ex70.m1.4.4.3.1" xref="A4.Ex70.m1.4.4.3.1.cmml">˙</mo></mover><mo id="A4.Ex70.m1.4.4.2" xref="A4.Ex70.m1.4.4.2.cmml">≤</mo><mrow id="A4.Ex70.m1.4.4.1" xref="A4.Ex70.m1.4.4.1.cmml"><mrow id="A4.Ex70.m1.4.4.1.3" xref="A4.Ex70.m1.4.4.1.3.cmml"><mo id="A4.Ex70.m1.4.4.1.3a" xref="A4.Ex70.m1.4.4.1.3.cmml">−</mo><mrow id="A4.Ex70.m1.4.4.1.3.2" xref="A4.Ex70.m1.4.4.1.3.2.cmml"><mrow id="A4.Ex70.m1.4.4.1.3.2.2" xref="A4.Ex70.m1.4.4.1.3.2.2.cmml"><msub id="A4.Ex70.m1.4.4.1.3.2.2.2" xref="A4.Ex70.m1.4.4.1.3.2.2.2.cmml"><mi id="A4.Ex70.m1.4.4.1.3.2.2.2.2" xref="A4.Ex70.m1.4.4.1.3.2.2.2.2.cmml">k</mi><mi id="A4.Ex70.m1.4.4.1.3.2.2.2.3" mathvariant="normal" xref="A4.Ex70.m1.4.4.1.3.2.2.2.3.cmml">c</mi></msub><mo id="A4.Ex70.m1.4.4.1.3.2.2.1" xref="A4.Ex70.m1.4.4.1.3.2.2.1.cmml">⁢</mo><msup id="A4.Ex70.m1.4.4.1.3.2.2.3" xref="A4.Ex70.m1.4.4.1.3.2.2.3.cmml"><mrow id="A4.Ex70.m1.4.4.1.3.2.2.3.2.2" xref="A4.Ex70.m1.4.4.1.3.2.2.3.2.1.cmml"><mo id="A4.Ex70.m1.4.4.1.3.2.2.3.2.2.1" stretchy="false" xref="A4.Ex70.m1.4.4.1.3.2.2.3.2.1.1.cmml">‖</mo><mi id="A4.Ex70.m1.1.1" xref="A4.Ex70.m1.1.1.cmml">𝒆</mi><mo id="A4.Ex70.m1.4.4.1.3.2.2.3.2.2.2" stretchy="false" xref="A4.Ex70.m1.4.4.1.3.2.2.3.2.1.1.cmml">‖</mo></mrow><mn id="A4.Ex70.m1.4.4.1.3.2.2.3.3" xref="A4.Ex70.m1.4.4.1.3.2.2.3.3.cmml">2</mn></msup></mrow><mo id="A4.Ex70.m1.4.4.1.3.2.1" xref="A4.Ex70.m1.4.4.1.3.2.1.cmml">/</mo><mn id="A4.Ex70.m1.4.4.1.3.2.3" xref="A4.Ex70.m1.4.4.1.3.2.3.cmml">2</mn></mrow></mrow><mo id="A4.Ex70.m1.4.4.1.2" xref="A4.Ex70.m1.4.4.1.2.cmml">+</mo><mrow id="A4.Ex70.m1.4.4.1.1" xref="A4.Ex70.m1.4.4.1.1.cmml"><mi id="A4.Ex70.m1.4.4.1.1.3" xref="A4.Ex70.m1.4.4.1.1.3.cmml">ρ</mi><mo id="A4.Ex70.m1.4.4.1.1.2" xref="A4.Ex70.m1.4.4.1.1.2.cmml">⁢</mo><mrow id="A4.Ex70.m1.4.4.1.1.1.1" xref="A4.Ex70.m1.4.4.1.1.1.2.cmml"><mo id="A4.Ex70.m1.4.4.1.1.1.1.2" stretchy="false" xref="A4.Ex70.m1.4.4.1.1.1.2.cmml">(</mo><msub id="A4.Ex70.m1.4.4.1.1.1.1.1" xref="A4.Ex70.m1.4.4.1.1.1.1.1.cmml"><mi id="A4.Ex70.m1.4.4.1.1.1.1.1.2" xref="A4.Ex70.m1.4.4.1.1.1.1.1.2.cmml">k</mi><mi id="A4.Ex70.m1.4.4.1.1.1.1.1.3" mathvariant="normal" xref="A4.Ex70.m1.4.4.1.1.1.1.1.3.cmml">c</mi></msub><mo id="A4.Ex70.m1.4.4.1.1.1.1.3" xref="A4.Ex70.m1.4.4.1.1.1.2.cmml">,</mo><mi id="A4.Ex70.m1.2.2" xref="A4.Ex70.m1.2.2.cmml">κ</mi><mo id="A4.Ex70.m1.4.4.1.1.1.1.4" xref="A4.Ex70.m1.4.4.1.1.1.2.cmml">,</mo><mover accent="true" id="A4.Ex70.m1.3.3" xref="A4.Ex70.m1.3.3.cmml"><mi id="A4.Ex70.m1.3.3.2" xref="A4.Ex70.m1.3.3.2.cmml">d</mi><mo id="A4.Ex70.m1.3.3.1" xref="A4.Ex70.m1.3.3.1.cmml">¯</mo></mover><mo id="A4.Ex70.m1.4.4.1.1.1.1.5" stretchy="false" xref="A4.Ex70.m1.4.4.1.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex70.m1.4b"><apply id="A4.Ex70.m1.4.4.cmml" xref="A4.Ex70.m1.4.4"><leq id="A4.Ex70.m1.4.4.2.cmml" xref="A4.Ex70.m1.4.4.2"></leq><apply id="A4.Ex70.m1.4.4.3.cmml" xref="A4.Ex70.m1.4.4.3"><ci id="A4.Ex70.m1.4.4.3.1.cmml" xref="A4.Ex70.m1.4.4.3.1">˙</ci><ci id="A4.Ex70.m1.4.4.3.2.cmml" xref="A4.Ex70.m1.4.4.3.2">𝑉</ci></apply><apply id="A4.Ex70.m1.4.4.1.cmml" xref="A4.Ex70.m1.4.4.1"><plus id="A4.Ex70.m1.4.4.1.2.cmml" xref="A4.Ex70.m1.4.4.1.2"></plus><apply id="A4.Ex70.m1.4.4.1.3.cmml" xref="A4.Ex70.m1.4.4.1.3"><minus id="A4.Ex70.m1.4.4.1.3.1.cmml" xref="A4.Ex70.m1.4.4.1.3"></minus><apply id="A4.Ex70.m1.4.4.1.3.2.cmml" xref="A4.Ex70.m1.4.4.1.3.2"><divide id="A4.Ex70.m1.4.4.1.3.2.1.cmml" xref="A4.Ex70.m1.4.4.1.3.2.1"></divide><apply id="A4.Ex70.m1.4.4.1.3.2.2.cmml" xref="A4.Ex70.m1.4.4.1.3.2.2"><times id="A4.Ex70.m1.4.4.1.3.2.2.1.cmml" xref="A4.Ex70.m1.4.4.1.3.2.2.1"></times><apply id="A4.Ex70.m1.4.4.1.3.2.2.2.cmml" xref="A4.Ex70.m1.4.4.1.3.2.2.2"><csymbol cd="ambiguous" id="A4.Ex70.m1.4.4.1.3.2.2.2.1.cmml" xref="A4.Ex70.m1.4.4.1.3.2.2.2">subscript</csymbol><ci id="A4.Ex70.m1.4.4.1.3.2.2.2.2.cmml" xref="A4.Ex70.m1.4.4.1.3.2.2.2.2">𝑘</ci><ci id="A4.Ex70.m1.4.4.1.3.2.2.2.3.cmml" xref="A4.Ex70.m1.4.4.1.3.2.2.2.3">c</ci></apply><apply id="A4.Ex70.m1.4.4.1.3.2.2.3.cmml" xref="A4.Ex70.m1.4.4.1.3.2.2.3"><csymbol cd="ambiguous" id="A4.Ex70.m1.4.4.1.3.2.2.3.1.cmml" xref="A4.Ex70.m1.4.4.1.3.2.2.3">superscript</csymbol><apply id="A4.Ex70.m1.4.4.1.3.2.2.3.2.1.cmml" xref="A4.Ex70.m1.4.4.1.3.2.2.3.2.2"><csymbol cd="latexml" id="A4.Ex70.m1.4.4.1.3.2.2.3.2.1.1.cmml" xref="A4.Ex70.m1.4.4.1.3.2.2.3.2.2.1">norm</csymbol><ci id="A4.Ex70.m1.1.1.cmml" xref="A4.Ex70.m1.1.1">𝒆</ci></apply><cn id="A4.Ex70.m1.4.4.1.3.2.2.3.3.cmml" type="integer" xref="A4.Ex70.m1.4.4.1.3.2.2.3.3">2</cn></apply></apply><cn id="A4.Ex70.m1.4.4.1.3.2.3.cmml" type="integer" xref="A4.Ex70.m1.4.4.1.3.2.3">2</cn></apply></apply><apply id="A4.Ex70.m1.4.4.1.1.cmml" xref="A4.Ex70.m1.4.4.1.1"><times id="A4.Ex70.m1.4.4.1.1.2.cmml" xref="A4.Ex70.m1.4.4.1.1.2"></times><ci id="A4.Ex70.m1.4.4.1.1.3.cmml" xref="A4.Ex70.m1.4.4.1.1.3">𝜌</ci><vector id="A4.Ex70.m1.4.4.1.1.1.2.cmml" xref="A4.Ex70.m1.4.4.1.1.1.1"><apply id="A4.Ex70.m1.4.4.1.1.1.1.1.cmml" xref="A4.Ex70.m1.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.Ex70.m1.4.4.1.1.1.1.1.1.cmml" xref="A4.Ex70.m1.4.4.1.1.1.1.1">subscript</csymbol><ci id="A4.Ex70.m1.4.4.1.1.1.1.1.2.cmml" xref="A4.Ex70.m1.4.4.1.1.1.1.1.2">𝑘</ci><ci id="A4.Ex70.m1.4.4.1.1.1.1.1.3.cmml" xref="A4.Ex70.m1.4.4.1.1.1.1.1.3">c</ci></apply><ci id="A4.Ex70.m1.2.2.cmml" xref="A4.Ex70.m1.2.2">𝜅</ci><apply id="A4.Ex70.m1.3.3.cmml" xref="A4.Ex70.m1.3.3"><ci id="A4.Ex70.m1.3.3.1.cmml" xref="A4.Ex70.m1.3.3.1">¯</ci><ci id="A4.Ex70.m1.3.3.2.cmml" xref="A4.Ex70.m1.3.3.2">𝑑</ci></apply></vector></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex70.m1.4c">\displaystyle\dot{V}\leq-k_{\rm c}\|\bm{e}\|^{2}/2+\rho(k_{\rm c},\kappa,\bar{% d})</annotation><annotation encoding="application/x-llamapun" id="A4.Ex70.m1.4d">over˙ start_ARG italic_V end_ARG ≤ - italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ∥ bold_italic_e ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 + italic_ρ ( italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT , italic_κ , over¯ start_ARG italic_d end_ARG )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.1.p1.32">with <math alttext="\rho" class="ltx_Math" display="inline" id="A4.1.p1.24.m1.1"><semantics id="A4.1.p1.24.m1.1a"><mi id="A4.1.p1.24.m1.1.1" xref="A4.1.p1.24.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="A4.1.p1.24.m1.1b"><ci id="A4.1.p1.24.m1.1.1.cmml" xref="A4.1.p1.24.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.24.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.24.m1.1d">italic_ρ</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="A4.1.p1.25.m2.1"><semantics id="A4.1.p1.25.m2.1a"><mo id="A4.1.p1.25.m2.1.1" xref="A4.1.p1.25.m2.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="A4.1.p1.25.m2.1b"><csymbol cd="latexml" id="A4.1.p1.25.m2.1.1.cmml" xref="A4.1.p1.25.m2.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.25.m2.1c">:=</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.25.m2.1d">:=</annotation></semantics></math> <math alttext="(\delta c_{\theta}+\bar{d})^{2}/(2k_{\rm c})+(1+p)(\bar{\epsilon}\bar{d}+% \kappa c_{\theta}{\bar{d}}_{\rm g})\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A4.1.p1.26.m3.4"><semantics id="A4.1.p1.26.m3.4a"><mrow id="A4.1.p1.26.m3.4.4" xref="A4.1.p1.26.m3.4.4.cmml"><mrow id="A4.1.p1.26.m3.4.4.4" xref="A4.1.p1.26.m3.4.4.4.cmml"><mrow id="A4.1.p1.26.m3.2.2.2.2" xref="A4.1.p1.26.m3.2.2.2.2.cmml"><msup id="A4.1.p1.26.m3.1.1.1.1.1" xref="A4.1.p1.26.m3.1.1.1.1.1.cmml"><mrow id="A4.1.p1.26.m3.1.1.1.1.1.1.1" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.cmml"><mo id="A4.1.p1.26.m3.1.1.1.1.1.1.1.2" stretchy="false" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.1.p1.26.m3.1.1.1.1.1.1.1.1" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.cmml"><mrow id="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2.cmml"><mi id="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2.2" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2.2.cmml">δ</mi><mo id="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2.1" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2.1.cmml">⁢</mo><msub id="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2.3" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2.3.cmml"><mi id="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2.3.2" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2.3.2.cmml">c</mi><mi id="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2.3.3" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2.3.3.cmml">θ</mi></msub></mrow><mo id="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.1" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.1.cmml">+</mo><mover accent="true" id="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.3" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.3.cmml"><mi id="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.3.2" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.3.2.cmml">d</mi><mo id="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.3.1" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.3.1.cmml">¯</mo></mover></mrow><mo id="A4.1.p1.26.m3.1.1.1.1.1.1.1.3" stretchy="false" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="A4.1.p1.26.m3.1.1.1.1.1.3" xref="A4.1.p1.26.m3.1.1.1.1.1.3.cmml">2</mn></msup><mo id="A4.1.p1.26.m3.2.2.2.2.3" xref="A4.1.p1.26.m3.2.2.2.2.3.cmml">/</mo><mrow id="A4.1.p1.26.m3.2.2.2.2.2.1" xref="A4.1.p1.26.m3.2.2.2.2.2.1.1.cmml"><mo id="A4.1.p1.26.m3.2.2.2.2.2.1.2" stretchy="false" xref="A4.1.p1.26.m3.2.2.2.2.2.1.1.cmml">(</mo><mrow id="A4.1.p1.26.m3.2.2.2.2.2.1.1" xref="A4.1.p1.26.m3.2.2.2.2.2.1.1.cmml"><mn id="A4.1.p1.26.m3.2.2.2.2.2.1.1.2" xref="A4.1.p1.26.m3.2.2.2.2.2.1.1.2.cmml">2</mn><mo id="A4.1.p1.26.m3.2.2.2.2.2.1.1.1" xref="A4.1.p1.26.m3.2.2.2.2.2.1.1.1.cmml">⁢</mo><msub id="A4.1.p1.26.m3.2.2.2.2.2.1.1.3" xref="A4.1.p1.26.m3.2.2.2.2.2.1.1.3.cmml"><mi id="A4.1.p1.26.m3.2.2.2.2.2.1.1.3.2" xref="A4.1.p1.26.m3.2.2.2.2.2.1.1.3.2.cmml">k</mi><mi id="A4.1.p1.26.m3.2.2.2.2.2.1.1.3.3" mathvariant="normal" xref="A4.1.p1.26.m3.2.2.2.2.2.1.1.3.3.cmml">c</mi></msub></mrow><mo id="A4.1.p1.26.m3.2.2.2.2.2.1.3" stretchy="false" xref="A4.1.p1.26.m3.2.2.2.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="A4.1.p1.26.m3.4.4.4.5" xref="A4.1.p1.26.m3.4.4.4.5.cmml">+</mo><mrow id="A4.1.p1.26.m3.4.4.4.4" xref="A4.1.p1.26.m3.4.4.4.4.cmml"><mrow id="A4.1.p1.26.m3.3.3.3.3.1.1" xref="A4.1.p1.26.m3.3.3.3.3.1.1.1.cmml"><mo id="A4.1.p1.26.m3.3.3.3.3.1.1.2" stretchy="false" xref="A4.1.p1.26.m3.3.3.3.3.1.1.1.cmml">(</mo><mrow id="A4.1.p1.26.m3.3.3.3.3.1.1.1" xref="A4.1.p1.26.m3.3.3.3.3.1.1.1.cmml"><mn id="A4.1.p1.26.m3.3.3.3.3.1.1.1.2" xref="A4.1.p1.26.m3.3.3.3.3.1.1.1.2.cmml">1</mn><mo id="A4.1.p1.26.m3.3.3.3.3.1.1.1.1" xref="A4.1.p1.26.m3.3.3.3.3.1.1.1.1.cmml">+</mo><mi id="A4.1.p1.26.m3.3.3.3.3.1.1.1.3" xref="A4.1.p1.26.m3.3.3.3.3.1.1.1.3.cmml">p</mi></mrow><mo id="A4.1.p1.26.m3.3.3.3.3.1.1.3" stretchy="false" xref="A4.1.p1.26.m3.3.3.3.3.1.1.1.cmml">)</mo></mrow><mo id="A4.1.p1.26.m3.4.4.4.4.3" xref="A4.1.p1.26.m3.4.4.4.4.3.cmml">⁢</mo><mrow id="A4.1.p1.26.m3.4.4.4.4.2.1" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.cmml"><mo id="A4.1.p1.26.m3.4.4.4.4.2.1.2" stretchy="false" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.cmml">(</mo><mrow id="A4.1.p1.26.m3.4.4.4.4.2.1.1" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.cmml"><mrow id="A4.1.p1.26.m3.4.4.4.4.2.1.1.2" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.cmml"><mover accent="true" id="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.2" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.2.cmml"><mi id="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.2.2" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.2.2.cmml">ϵ</mi><mo id="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.2.1" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.2.1.cmml">¯</mo></mover><mo id="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.1" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.1.cmml">⁢</mo><mover accent="true" id="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.3" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.3.cmml"><mi id="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.3.2" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.3.2.cmml">d</mi><mo id="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.3.1" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.3.1.cmml">¯</mo></mover></mrow><mo id="A4.1.p1.26.m3.4.4.4.4.2.1.1.1" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.1.cmml">+</mo><mrow id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.cmml"><mi id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.2" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.2.cmml">κ</mi><mo id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.1" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.1.cmml">⁢</mo><msub id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.3" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.3.cmml"><mi id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.3.2" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.3.2.cmml">c</mi><mi id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.3.3" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.3.3.cmml">θ</mi></msub><mo id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.1a" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.1.cmml">⁢</mo><msub id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.4" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.4.cmml"><mover accent="true" id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.4.2" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.4.2.cmml"><mi id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.4.2.2" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.4.2.2.cmml">d</mi><mo id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.4.2.1" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.4.2.1.cmml">¯</mo></mover><mi id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.4.3" mathvariant="normal" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.4.3.cmml">g</mi></msub></mrow></mrow><mo id="A4.1.p1.26.m3.4.4.4.4.2.1.3" stretchy="false" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A4.1.p1.26.m3.4.4.5" xref="A4.1.p1.26.m3.4.4.5.cmml">∈</mo><msup id="A4.1.p1.26.m3.4.4.6" xref="A4.1.p1.26.m3.4.4.6.cmml"><mi id="A4.1.p1.26.m3.4.4.6.2" xref="A4.1.p1.26.m3.4.4.6.2.cmml">ℝ</mi><mo id="A4.1.p1.26.m3.4.4.6.3" xref="A4.1.p1.26.m3.4.4.6.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.26.m3.4b"><apply id="A4.1.p1.26.m3.4.4.cmml" xref="A4.1.p1.26.m3.4.4"><in id="A4.1.p1.26.m3.4.4.5.cmml" xref="A4.1.p1.26.m3.4.4.5"></in><apply id="A4.1.p1.26.m3.4.4.4.cmml" xref="A4.1.p1.26.m3.4.4.4"><plus id="A4.1.p1.26.m3.4.4.4.5.cmml" xref="A4.1.p1.26.m3.4.4.4.5"></plus><apply id="A4.1.p1.26.m3.2.2.2.2.cmml" xref="A4.1.p1.26.m3.2.2.2.2"><divide id="A4.1.p1.26.m3.2.2.2.2.3.cmml" xref="A4.1.p1.26.m3.2.2.2.2.3"></divide><apply id="A4.1.p1.26.m3.1.1.1.1.1.cmml" xref="A4.1.p1.26.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.1.p1.26.m3.1.1.1.1.1.2.cmml" xref="A4.1.p1.26.m3.1.1.1.1.1">superscript</csymbol><apply id="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.cmml" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1"><plus id="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.1.cmml" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.1"></plus><apply id="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2.cmml" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2"><times id="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2.1.cmml" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2.1"></times><ci id="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2.2.cmml" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2.2">𝛿</ci><apply id="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2.3.cmml" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2.3.1.cmml" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2.3">subscript</csymbol><ci id="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2.3.2.cmml" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2.3.2">𝑐</ci><ci id="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2.3.3.cmml" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.2.3.3">𝜃</ci></apply></apply><apply id="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.3.cmml" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.3"><ci id="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.3.1.cmml" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.3.1">¯</ci><ci id="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.3.2.cmml" xref="A4.1.p1.26.m3.1.1.1.1.1.1.1.1.3.2">𝑑</ci></apply></apply><cn id="A4.1.p1.26.m3.1.1.1.1.1.3.cmml" type="integer" xref="A4.1.p1.26.m3.1.1.1.1.1.3">2</cn></apply><apply id="A4.1.p1.26.m3.2.2.2.2.2.1.1.cmml" xref="A4.1.p1.26.m3.2.2.2.2.2.1"><times id="A4.1.p1.26.m3.2.2.2.2.2.1.1.1.cmml" xref="A4.1.p1.26.m3.2.2.2.2.2.1.1.1"></times><cn id="A4.1.p1.26.m3.2.2.2.2.2.1.1.2.cmml" type="integer" xref="A4.1.p1.26.m3.2.2.2.2.2.1.1.2">2</cn><apply id="A4.1.p1.26.m3.2.2.2.2.2.1.1.3.cmml" xref="A4.1.p1.26.m3.2.2.2.2.2.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.26.m3.2.2.2.2.2.1.1.3.1.cmml" xref="A4.1.p1.26.m3.2.2.2.2.2.1.1.3">subscript</csymbol><ci id="A4.1.p1.26.m3.2.2.2.2.2.1.1.3.2.cmml" xref="A4.1.p1.26.m3.2.2.2.2.2.1.1.3.2">𝑘</ci><ci id="A4.1.p1.26.m3.2.2.2.2.2.1.1.3.3.cmml" xref="A4.1.p1.26.m3.2.2.2.2.2.1.1.3.3">c</ci></apply></apply></apply><apply id="A4.1.p1.26.m3.4.4.4.4.cmml" xref="A4.1.p1.26.m3.4.4.4.4"><times id="A4.1.p1.26.m3.4.4.4.4.3.cmml" xref="A4.1.p1.26.m3.4.4.4.4.3"></times><apply id="A4.1.p1.26.m3.3.3.3.3.1.1.1.cmml" xref="A4.1.p1.26.m3.3.3.3.3.1.1"><plus id="A4.1.p1.26.m3.3.3.3.3.1.1.1.1.cmml" xref="A4.1.p1.26.m3.3.3.3.3.1.1.1.1"></plus><cn id="A4.1.p1.26.m3.3.3.3.3.1.1.1.2.cmml" type="integer" xref="A4.1.p1.26.m3.3.3.3.3.1.1.1.2">1</cn><ci id="A4.1.p1.26.m3.3.3.3.3.1.1.1.3.cmml" xref="A4.1.p1.26.m3.3.3.3.3.1.1.1.3">𝑝</ci></apply><apply id="A4.1.p1.26.m3.4.4.4.4.2.1.1.cmml" xref="A4.1.p1.26.m3.4.4.4.4.2.1"><plus id="A4.1.p1.26.m3.4.4.4.4.2.1.1.1.cmml" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.1"></plus><apply id="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.cmml" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.2"><times id="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.1.cmml" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.1"></times><apply id="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.2.cmml" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.2"><ci id="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.2.1.cmml" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.2.1">¯</ci><ci id="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.2.2.cmml" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.2.2">italic-ϵ</ci></apply><apply id="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.3.cmml" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.3"><ci id="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.3.1.cmml" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.3.1">¯</ci><ci id="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.3.2.cmml" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.2.3.2">𝑑</ci></apply></apply><apply id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.cmml" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3"><times id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.1.cmml" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.1"></times><ci id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.2.cmml" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.2">𝜅</ci><apply id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.3.cmml" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.3"><csymbol cd="ambiguous" id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.3.1.cmml" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.3">subscript</csymbol><ci id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.3.2.cmml" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.3.2">𝑐</ci><ci id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.3.3.cmml" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.3.3">𝜃</ci></apply><apply id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.4.cmml" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.4"><csymbol cd="ambiguous" id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.4.1.cmml" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.4">subscript</csymbol><apply id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.4.2.cmml" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.4.2"><ci id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.4.2.1.cmml" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.4.2.1">¯</ci><ci id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.4.2.2.cmml" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.4.2.2">𝑑</ci></apply><ci id="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.4.3.cmml" xref="A4.1.p1.26.m3.4.4.4.4.2.1.1.3.4.3">g</ci></apply></apply></apply></apply></apply><apply id="A4.1.p1.26.m3.4.4.6.cmml" xref="A4.1.p1.26.m3.4.4.6"><csymbol cd="ambiguous" id="A4.1.p1.26.m3.4.4.6.1.cmml" xref="A4.1.p1.26.m3.4.4.6">superscript</csymbol><ci id="A4.1.p1.26.m3.4.4.6.2.cmml" xref="A4.1.p1.26.m3.4.4.6.2">ℝ</ci><plus id="A4.1.p1.26.m3.4.4.6.3.cmml" xref="A4.1.p1.26.m3.4.4.6.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.26.m3.4c">(\delta c_{\theta}+\bar{d})^{2}/(2k_{\rm c})+(1+p)(\bar{\epsilon}\bar{d}+% \kappa c_{\theta}{\bar{d}}_{\rm g})\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.26.m3.4d">( italic_δ italic_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT + over¯ start_ARG italic_d end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( 2 italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ) + ( 1 + italic_p ) ( over¯ start_ARG italic_ϵ end_ARG over¯ start_ARG italic_d end_ARG + italic_κ italic_c start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT over¯ start_ARG italic_d end_ARG start_POSTSUBSCRIPT roman_g end_POSTSUBSCRIPT ) ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, which is valid on <math alttext="\bm{x}\in\Omega_{{\rm c}_{x}}" class="ltx_Math" display="inline" id="A4.1.p1.27.m4.1"><semantics id="A4.1.p1.27.m4.1a"><mrow id="A4.1.p1.27.m4.1.1" xref="A4.1.p1.27.m4.1.1.cmml"><mi id="A4.1.p1.27.m4.1.1.2" xref="A4.1.p1.27.m4.1.1.2.cmml">𝒙</mi><mo id="A4.1.p1.27.m4.1.1.1" xref="A4.1.p1.27.m4.1.1.1.cmml">∈</mo><msub id="A4.1.p1.27.m4.1.1.3" xref="A4.1.p1.27.m4.1.1.3.cmml"><mi id="A4.1.p1.27.m4.1.1.3.2" mathvariant="normal" xref="A4.1.p1.27.m4.1.1.3.2.cmml">Ω</mi><msub id="A4.1.p1.27.m4.1.1.3.3" xref="A4.1.p1.27.m4.1.1.3.3.cmml"><mi id="A4.1.p1.27.m4.1.1.3.3.2" mathvariant="normal" xref="A4.1.p1.27.m4.1.1.3.3.2.cmml">c</mi><mi id="A4.1.p1.27.m4.1.1.3.3.3" xref="A4.1.p1.27.m4.1.1.3.3.3.cmml">x</mi></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.27.m4.1b"><apply id="A4.1.p1.27.m4.1.1.cmml" xref="A4.1.p1.27.m4.1.1"><in id="A4.1.p1.27.m4.1.1.1.cmml" xref="A4.1.p1.27.m4.1.1.1"></in><ci id="A4.1.p1.27.m4.1.1.2.cmml" xref="A4.1.p1.27.m4.1.1.2">𝒙</ci><apply id="A4.1.p1.27.m4.1.1.3.cmml" xref="A4.1.p1.27.m4.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.27.m4.1.1.3.1.cmml" xref="A4.1.p1.27.m4.1.1.3">subscript</csymbol><ci id="A4.1.p1.27.m4.1.1.3.2.cmml" xref="A4.1.p1.27.m4.1.1.3.2">Ω</ci><apply id="A4.1.p1.27.m4.1.1.3.3.cmml" xref="A4.1.p1.27.m4.1.1.3.3"><csymbol cd="ambiguous" id="A4.1.p1.27.m4.1.1.3.3.1.cmml" xref="A4.1.p1.27.m4.1.1.3.3">subscript</csymbol><ci id="A4.1.p1.27.m4.1.1.3.3.2.cmml" xref="A4.1.p1.27.m4.1.1.3.3.2">c</ci><ci id="A4.1.p1.27.m4.1.1.3.3.3.cmml" xref="A4.1.p1.27.m4.1.1.3.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.27.m4.1c">\bm{x}\in\Omega_{{\rm c}_{x}}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.27.m4.1d">bold_italic_x ∈ roman_Ω start_POSTSUBSCRIPT roman_c start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="t\in[0,t_{\rm f})" class="ltx_Math" display="inline" id="A4.1.p1.28.m5.2"><semantics id="A4.1.p1.28.m5.2a"><mrow id="A4.1.p1.28.m5.2.2" xref="A4.1.p1.28.m5.2.2.cmml"><mi id="A4.1.p1.28.m5.2.2.3" xref="A4.1.p1.28.m5.2.2.3.cmml">t</mi><mo id="A4.1.p1.28.m5.2.2.2" xref="A4.1.p1.28.m5.2.2.2.cmml">∈</mo><mrow id="A4.1.p1.28.m5.2.2.1.1" xref="A4.1.p1.28.m5.2.2.1.2.cmml"><mo id="A4.1.p1.28.m5.2.2.1.1.2" stretchy="false" xref="A4.1.p1.28.m5.2.2.1.2.cmml">[</mo><mn id="A4.1.p1.28.m5.1.1" xref="A4.1.p1.28.m5.1.1.cmml">0</mn><mo id="A4.1.p1.28.m5.2.2.1.1.3" xref="A4.1.p1.28.m5.2.2.1.2.cmml">,</mo><msub id="A4.1.p1.28.m5.2.2.1.1.1" xref="A4.1.p1.28.m5.2.2.1.1.1.cmml"><mi id="A4.1.p1.28.m5.2.2.1.1.1.2" xref="A4.1.p1.28.m5.2.2.1.1.1.2.cmml">t</mi><mi id="A4.1.p1.28.m5.2.2.1.1.1.3" mathvariant="normal" xref="A4.1.p1.28.m5.2.2.1.1.1.3.cmml">f</mi></msub><mo id="A4.1.p1.28.m5.2.2.1.1.4" stretchy="false" xref="A4.1.p1.28.m5.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.28.m5.2b"><apply id="A4.1.p1.28.m5.2.2.cmml" xref="A4.1.p1.28.m5.2.2"><in id="A4.1.p1.28.m5.2.2.2.cmml" xref="A4.1.p1.28.m5.2.2.2"></in><ci id="A4.1.p1.28.m5.2.2.3.cmml" xref="A4.1.p1.28.m5.2.2.3">𝑡</ci><interval closure="closed-open" id="A4.1.p1.28.m5.2.2.1.2.cmml" xref="A4.1.p1.28.m5.2.2.1.1"><cn id="A4.1.p1.28.m5.1.1.cmml" type="integer" xref="A4.1.p1.28.m5.1.1">0</cn><apply id="A4.1.p1.28.m5.2.2.1.1.1.cmml" xref="A4.1.p1.28.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="A4.1.p1.28.m5.2.2.1.1.1.1.cmml" xref="A4.1.p1.28.m5.2.2.1.1.1">subscript</csymbol><ci id="A4.1.p1.28.m5.2.2.1.1.1.2.cmml" xref="A4.1.p1.28.m5.2.2.1.1.1.2">𝑡</ci><ci id="A4.1.p1.28.m5.2.2.1.1.1.3.cmml" xref="A4.1.p1.28.m5.2.2.1.1.1.3">f</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.28.m5.2c">t\in[0,t_{\rm f})</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.28.m5.2d">italic_t ∈ [ 0 , italic_t start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT )</annotation></semantics></math>. The remaining derivation is similar to the steps from (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A3.E46" title="In Proof. ‣ Appendix C The proof of Theorem 2 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">46</span></a>) to the end of the proof of Item 1 in Theorem 2, so that it is omitted here for saving space. <br class="ltx_break"/>2) Consider the control problem under partial IE on <math alttext="t" class="ltx_Math" display="inline" id="A4.1.p1.29.m6.1"><semantics id="A4.1.p1.29.m6.1a"><mi id="A4.1.p1.29.m6.1.1" xref="A4.1.p1.29.m6.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A4.1.p1.29.m6.1b"><ci id="A4.1.p1.29.m6.1.1.cmml" xref="A4.1.p1.29.m6.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.29.m6.1c">t</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.29.m6.1d">italic_t</annotation></semantics></math> <math alttext="\in" class="ltx_Math" display="inline" id="A4.1.p1.30.m7.1"><semantics id="A4.1.p1.30.m7.1a"><mo id="A4.1.p1.30.m7.1.1" xref="A4.1.p1.30.m7.1.1.cmml">∈</mo><annotation-xml encoding="MathML-Content" id="A4.1.p1.30.m7.1b"><in id="A4.1.p1.30.m7.1.1.cmml" xref="A4.1.p1.30.m7.1.1"></in></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.30.m7.1c">\in</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.30.m7.1d">∈</annotation></semantics></math> <math alttext="[T_{\rm a}" class="ltx_math_unparsed" display="inline" id="A4.1.p1.31.m8.1"><semantics id="A4.1.p1.31.m8.1a"><mrow id="A4.1.p1.31.m8.1b"><mo id="A4.1.p1.31.m8.1.1" stretchy="false">[</mo><msub id="A4.1.p1.31.m8.1.2"><mi id="A4.1.p1.31.m8.1.2.2">T</mi><mi id="A4.1.p1.31.m8.1.2.3" mathvariant="normal">a</mi></msub></mrow><annotation encoding="application/x-tex" id="A4.1.p1.31.m8.1c">[T_{\rm a}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.31.m8.1d">[ italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\infty)" class="ltx_math_unparsed" display="inline" id="A4.1.p1.32.m9.1"><semantics id="A4.1.p1.32.m9.1a"><mrow id="A4.1.p1.32.m9.1b"><mi id="A4.1.p1.32.m9.1.1" mathvariant="normal">∞</mi><mo id="A4.1.p1.32.m9.1.2" stretchy="false">)</mo></mrow><annotation encoding="application/x-tex" id="A4.1.p1.32.m9.1c">\infty)</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.32.m9.1d">∞ )</annotation></semantics></math>. Using (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S5.E26" title="In V-C Robustness Results ‣ V Theoretical Guarantees ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">26</span></a>), (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S5.E30" title="In V-C Robustness Results ‣ V Theoretical Guarantees ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">30</span></a>), and (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S5.E31" title="In V-C Robustness Results ‣ V Theoretical Guarantees ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">31</span></a>), the equality (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A3.Ex53" title="Proof. ‣ Appendix C The proof of Theorem 2 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">C</span></a>) becomes</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx80"> <tbody id="A4.Ex71"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle{\dot{V}}_{\zeta}=" class="ltx_Math" display="inline" id="A4.Ex71.m1.1"><semantics id="A4.Ex71.m1.1a"><mrow id="A4.Ex71.m1.1.1" xref="A4.Ex71.m1.1.1.cmml"><msub id="A4.Ex71.m1.1.1.2" xref="A4.Ex71.m1.1.1.2.cmml"><mover accent="true" id="A4.Ex71.m1.1.1.2.2" xref="A4.Ex71.m1.1.1.2.2.cmml"><mi id="A4.Ex71.m1.1.1.2.2.2" xref="A4.Ex71.m1.1.1.2.2.2.cmml">V</mi><mo id="A4.Ex71.m1.1.1.2.2.1" xref="A4.Ex71.m1.1.1.2.2.1.cmml">˙</mo></mover><mi id="A4.Ex71.m1.1.1.2.3" xref="A4.Ex71.m1.1.1.2.3.cmml">ζ</mi></msub><mo id="A4.Ex71.m1.1.1.1" xref="A4.Ex71.m1.1.1.1.cmml">=</mo><mi id="A4.Ex71.m1.1.1.3" xref="A4.Ex71.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex71.m1.1b"><apply id="A4.Ex71.m1.1.1.cmml" xref="A4.Ex71.m1.1.1"><eq id="A4.Ex71.m1.1.1.1.cmml" xref="A4.Ex71.m1.1.1.1"></eq><apply id="A4.Ex71.m1.1.1.2.cmml" xref="A4.Ex71.m1.1.1.2"><csymbol cd="ambiguous" id="A4.Ex71.m1.1.1.2.1.cmml" xref="A4.Ex71.m1.1.1.2">subscript</csymbol><apply id="A4.Ex71.m1.1.1.2.2.cmml" xref="A4.Ex71.m1.1.1.2.2"><ci id="A4.Ex71.m1.1.1.2.2.1.cmml" xref="A4.Ex71.m1.1.1.2.2.1">˙</ci><ci id="A4.Ex71.m1.1.1.2.2.2.cmml" xref="A4.Ex71.m1.1.1.2.2.2">𝑉</ci></apply><ci id="A4.Ex71.m1.1.1.2.3.cmml" xref="A4.Ex71.m1.1.1.2.3">𝜁</ci></apply><csymbol cd="latexml" id="A4.Ex71.m1.1.1.3.cmml" xref="A4.Ex71.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex71.m1.1c">\displaystyle{\dot{V}}_{\zeta}=</annotation><annotation encoding="application/x-llamapun" id="A4.Ex71.m1.1d">over˙ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT =</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\bm{e}^{T}(-K\bm{e}+\Phi^{T}_{\zeta}{\tilde{\bm{\theta}}}_{\zeta}% )+\bm{e}^{T}\bm{d}-(1+p){\tilde{\bm{\theta}}}_{\zeta}^{T}\Phi_{{\rm f},\zeta}(% \Phi_{{\rm f},\zeta}^{T}\tilde{\bm{\theta}}_{\zeta}+{\bm{d}}_{\rm f})" class="ltx_Math" display="inline" id="A4.Ex71.m2.7"><semantics id="A4.Ex71.m2.7a"><mrow id="A4.Ex71.m2.7.7" xref="A4.Ex71.m2.7.7.cmml"><mrow id="A4.Ex71.m2.5.5.1" xref="A4.Ex71.m2.5.5.1.cmml"><mrow id="A4.Ex71.m2.5.5.1.1" xref="A4.Ex71.m2.5.5.1.1.cmml"><msup id="A4.Ex71.m2.5.5.1.1.3" xref="A4.Ex71.m2.5.5.1.1.3.cmml"><mi id="A4.Ex71.m2.5.5.1.1.3.2" xref="A4.Ex71.m2.5.5.1.1.3.2.cmml">𝒆</mi><mi id="A4.Ex71.m2.5.5.1.1.3.3" xref="A4.Ex71.m2.5.5.1.1.3.3.cmml">T</mi></msup><mo id="A4.Ex71.m2.5.5.1.1.2" xref="A4.Ex71.m2.5.5.1.1.2.cmml">⁢</mo><mrow id="A4.Ex71.m2.5.5.1.1.1.1" xref="A4.Ex71.m2.5.5.1.1.1.1.1.cmml"><mo id="A4.Ex71.m2.5.5.1.1.1.1.2" stretchy="false" xref="A4.Ex71.m2.5.5.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex71.m2.5.5.1.1.1.1.1" xref="A4.Ex71.m2.5.5.1.1.1.1.1.cmml"><mrow id="A4.Ex71.m2.5.5.1.1.1.1.1.2" xref="A4.Ex71.m2.5.5.1.1.1.1.1.2.cmml"><mo id="A4.Ex71.m2.5.5.1.1.1.1.1.2a" xref="A4.Ex71.m2.5.5.1.1.1.1.1.2.cmml">−</mo><mrow id="A4.Ex71.m2.5.5.1.1.1.1.1.2.2" xref="A4.Ex71.m2.5.5.1.1.1.1.1.2.2.cmml"><mi id="A4.Ex71.m2.5.5.1.1.1.1.1.2.2.2" xref="A4.Ex71.m2.5.5.1.1.1.1.1.2.2.2.cmml">K</mi><mo id="A4.Ex71.m2.5.5.1.1.1.1.1.2.2.1" xref="A4.Ex71.m2.5.5.1.1.1.1.1.2.2.1.cmml">⁢</mo><mi id="A4.Ex71.m2.5.5.1.1.1.1.1.2.2.3" xref="A4.Ex71.m2.5.5.1.1.1.1.1.2.2.3.cmml">𝒆</mi></mrow></mrow><mo id="A4.Ex71.m2.5.5.1.1.1.1.1.1" xref="A4.Ex71.m2.5.5.1.1.1.1.1.1.cmml">+</mo><mrow id="A4.Ex71.m2.5.5.1.1.1.1.1.3" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.cmml"><msubsup id="A4.Ex71.m2.5.5.1.1.1.1.1.3.2" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.2.cmml"><mi id="A4.Ex71.m2.5.5.1.1.1.1.1.3.2.2.2" mathvariant="normal" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.2.2.2.cmml">Φ</mi><mi id="A4.Ex71.m2.5.5.1.1.1.1.1.3.2.3" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.2.3.cmml">ζ</mi><mi id="A4.Ex71.m2.5.5.1.1.1.1.1.3.2.2.3" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.2.2.3.cmml">T</mi></msubsup><mo id="A4.Ex71.m2.5.5.1.1.1.1.1.3.1" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.1.cmml">⁢</mo><msub id="A4.Ex71.m2.5.5.1.1.1.1.1.3.3" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.3.cmml"><mover accent="true" id="A4.Ex71.m2.5.5.1.1.1.1.1.3.3.2" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.3.2.cmml"><mi id="A4.Ex71.m2.5.5.1.1.1.1.1.3.3.2.2" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.3.2.2.cmml">𝜽</mi><mo id="A4.Ex71.m2.5.5.1.1.1.1.1.3.3.2.1" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.3.2.1.cmml">~</mo></mover><mi id="A4.Ex71.m2.5.5.1.1.1.1.1.3.3.3" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.3.3.cmml">ζ</mi></msub></mrow></mrow><mo id="A4.Ex71.m2.5.5.1.1.1.1.3" stretchy="false" xref="A4.Ex71.m2.5.5.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A4.Ex71.m2.5.5.1.2" xref="A4.Ex71.m2.5.5.1.2.cmml">+</mo><mrow id="A4.Ex71.m2.5.5.1.3" xref="A4.Ex71.m2.5.5.1.3.cmml"><msup id="A4.Ex71.m2.5.5.1.3.2" xref="A4.Ex71.m2.5.5.1.3.2.cmml"><mi id="A4.Ex71.m2.5.5.1.3.2.2" xref="A4.Ex71.m2.5.5.1.3.2.2.cmml">𝒆</mi><mi id="A4.Ex71.m2.5.5.1.3.2.3" xref="A4.Ex71.m2.5.5.1.3.2.3.cmml">T</mi></msup><mo id="A4.Ex71.m2.5.5.1.3.1" xref="A4.Ex71.m2.5.5.1.3.1.cmml">⁢</mo><mi id="A4.Ex71.m2.5.5.1.3.3" xref="A4.Ex71.m2.5.5.1.3.3.cmml">𝒅</mi></mrow></mrow><mo id="A4.Ex71.m2.7.7.4" xref="A4.Ex71.m2.7.7.4.cmml">−</mo><mrow id="A4.Ex71.m2.7.7.3" xref="A4.Ex71.m2.7.7.3.cmml"><mrow id="A4.Ex71.m2.6.6.2.1.1" xref="A4.Ex71.m2.6.6.2.1.1.1.cmml"><mo id="A4.Ex71.m2.6.6.2.1.1.2" stretchy="false" xref="A4.Ex71.m2.6.6.2.1.1.1.cmml">(</mo><mrow id="A4.Ex71.m2.6.6.2.1.1.1" xref="A4.Ex71.m2.6.6.2.1.1.1.cmml"><mn id="A4.Ex71.m2.6.6.2.1.1.1.2" xref="A4.Ex71.m2.6.6.2.1.1.1.2.cmml">1</mn><mo id="A4.Ex71.m2.6.6.2.1.1.1.1" xref="A4.Ex71.m2.6.6.2.1.1.1.1.cmml">+</mo><mi id="A4.Ex71.m2.6.6.2.1.1.1.3" xref="A4.Ex71.m2.6.6.2.1.1.1.3.cmml">p</mi></mrow><mo id="A4.Ex71.m2.6.6.2.1.1.3" stretchy="false" xref="A4.Ex71.m2.6.6.2.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex71.m2.7.7.3.3" xref="A4.Ex71.m2.7.7.3.3.cmml">⁢</mo><msubsup id="A4.Ex71.m2.7.7.3.4" xref="A4.Ex71.m2.7.7.3.4.cmml"><mover accent="true" id="A4.Ex71.m2.7.7.3.4.2.2" xref="A4.Ex71.m2.7.7.3.4.2.2.cmml"><mi id="A4.Ex71.m2.7.7.3.4.2.2.2" xref="A4.Ex71.m2.7.7.3.4.2.2.2.cmml">𝜽</mi><mo id="A4.Ex71.m2.7.7.3.4.2.2.1" xref="A4.Ex71.m2.7.7.3.4.2.2.1.cmml">~</mo></mover><mi id="A4.Ex71.m2.7.7.3.4.2.3" xref="A4.Ex71.m2.7.7.3.4.2.3.cmml">ζ</mi><mi id="A4.Ex71.m2.7.7.3.4.3" xref="A4.Ex71.m2.7.7.3.4.3.cmml">T</mi></msubsup><mo id="A4.Ex71.m2.7.7.3.3a" xref="A4.Ex71.m2.7.7.3.3.cmml">⁢</mo><msub id="A4.Ex71.m2.7.7.3.5" xref="A4.Ex71.m2.7.7.3.5.cmml"><mi id="A4.Ex71.m2.7.7.3.5.2" mathvariant="normal" xref="A4.Ex71.m2.7.7.3.5.2.cmml">Φ</mi><mrow id="A4.Ex71.m2.2.2.2.4" xref="A4.Ex71.m2.2.2.2.3.cmml"><mi id="A4.Ex71.m2.1.1.1.1" mathvariant="normal" xref="A4.Ex71.m2.1.1.1.1.cmml">f</mi><mo id="A4.Ex71.m2.2.2.2.4.1" xref="A4.Ex71.m2.2.2.2.3.cmml">,</mo><mi id="A4.Ex71.m2.2.2.2.2" xref="A4.Ex71.m2.2.2.2.2.cmml">ζ</mi></mrow></msub><mo id="A4.Ex71.m2.7.7.3.3b" xref="A4.Ex71.m2.7.7.3.3.cmml">⁢</mo><mrow id="A4.Ex71.m2.7.7.3.2.1" xref="A4.Ex71.m2.7.7.3.2.1.1.cmml"><mo id="A4.Ex71.m2.7.7.3.2.1.2" stretchy="false" xref="A4.Ex71.m2.7.7.3.2.1.1.cmml">(</mo><mrow id="A4.Ex71.m2.7.7.3.2.1.1" xref="A4.Ex71.m2.7.7.3.2.1.1.cmml"><mrow id="A4.Ex71.m2.7.7.3.2.1.1.2" xref="A4.Ex71.m2.7.7.3.2.1.1.2.cmml"><msubsup id="A4.Ex71.m2.7.7.3.2.1.1.2.2" xref="A4.Ex71.m2.7.7.3.2.1.1.2.2.cmml"><mi id="A4.Ex71.m2.7.7.3.2.1.1.2.2.2.2" mathvariant="normal" xref="A4.Ex71.m2.7.7.3.2.1.1.2.2.2.2.cmml">Φ</mi><mrow id="A4.Ex71.m2.4.4.2.4" xref="A4.Ex71.m2.4.4.2.3.cmml"><mi id="A4.Ex71.m2.3.3.1.1" mathvariant="normal" xref="A4.Ex71.m2.3.3.1.1.cmml">f</mi><mo id="A4.Ex71.m2.4.4.2.4.1" xref="A4.Ex71.m2.4.4.2.3.cmml">,</mo><mi id="A4.Ex71.m2.4.4.2.2" xref="A4.Ex71.m2.4.4.2.2.cmml">ζ</mi></mrow><mi id="A4.Ex71.m2.7.7.3.2.1.1.2.2.3" xref="A4.Ex71.m2.7.7.3.2.1.1.2.2.3.cmml">T</mi></msubsup><mo id="A4.Ex71.m2.7.7.3.2.1.1.2.1" xref="A4.Ex71.m2.7.7.3.2.1.1.2.1.cmml">⁢</mo><msub id="A4.Ex71.m2.7.7.3.2.1.1.2.3" xref="A4.Ex71.m2.7.7.3.2.1.1.2.3.cmml"><mover accent="true" id="A4.Ex71.m2.7.7.3.2.1.1.2.3.2" xref="A4.Ex71.m2.7.7.3.2.1.1.2.3.2.cmml"><mi id="A4.Ex71.m2.7.7.3.2.1.1.2.3.2.2" xref="A4.Ex71.m2.7.7.3.2.1.1.2.3.2.2.cmml">𝜽</mi><mo id="A4.Ex71.m2.7.7.3.2.1.1.2.3.2.1" xref="A4.Ex71.m2.7.7.3.2.1.1.2.3.2.1.cmml">~</mo></mover><mi id="A4.Ex71.m2.7.7.3.2.1.1.2.3.3" xref="A4.Ex71.m2.7.7.3.2.1.1.2.3.3.cmml">ζ</mi></msub></mrow><mo id="A4.Ex71.m2.7.7.3.2.1.1.1" xref="A4.Ex71.m2.7.7.3.2.1.1.1.cmml">+</mo><msub id="A4.Ex71.m2.7.7.3.2.1.1.3" xref="A4.Ex71.m2.7.7.3.2.1.1.3.cmml"><mi id="A4.Ex71.m2.7.7.3.2.1.1.3.2" xref="A4.Ex71.m2.7.7.3.2.1.1.3.2.cmml">𝒅</mi><mi id="A4.Ex71.m2.7.7.3.2.1.1.3.3" mathvariant="normal" xref="A4.Ex71.m2.7.7.3.2.1.1.3.3.cmml">f</mi></msub></mrow><mo id="A4.Ex71.m2.7.7.3.2.1.3" stretchy="false" xref="A4.Ex71.m2.7.7.3.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex71.m2.7b"><apply id="A4.Ex71.m2.7.7.cmml" xref="A4.Ex71.m2.7.7"><minus id="A4.Ex71.m2.7.7.4.cmml" xref="A4.Ex71.m2.7.7.4"></minus><apply id="A4.Ex71.m2.5.5.1.cmml" xref="A4.Ex71.m2.5.5.1"><plus id="A4.Ex71.m2.5.5.1.2.cmml" xref="A4.Ex71.m2.5.5.1.2"></plus><apply id="A4.Ex71.m2.5.5.1.1.cmml" xref="A4.Ex71.m2.5.5.1.1"><times id="A4.Ex71.m2.5.5.1.1.2.cmml" xref="A4.Ex71.m2.5.5.1.1.2"></times><apply id="A4.Ex71.m2.5.5.1.1.3.cmml" xref="A4.Ex71.m2.5.5.1.1.3"><csymbol cd="ambiguous" id="A4.Ex71.m2.5.5.1.1.3.1.cmml" xref="A4.Ex71.m2.5.5.1.1.3">superscript</csymbol><ci id="A4.Ex71.m2.5.5.1.1.3.2.cmml" xref="A4.Ex71.m2.5.5.1.1.3.2">𝒆</ci><ci id="A4.Ex71.m2.5.5.1.1.3.3.cmml" xref="A4.Ex71.m2.5.5.1.1.3.3">𝑇</ci></apply><apply id="A4.Ex71.m2.5.5.1.1.1.1.1.cmml" xref="A4.Ex71.m2.5.5.1.1.1.1"><plus id="A4.Ex71.m2.5.5.1.1.1.1.1.1.cmml" xref="A4.Ex71.m2.5.5.1.1.1.1.1.1"></plus><apply id="A4.Ex71.m2.5.5.1.1.1.1.1.2.cmml" xref="A4.Ex71.m2.5.5.1.1.1.1.1.2"><minus id="A4.Ex71.m2.5.5.1.1.1.1.1.2.1.cmml" xref="A4.Ex71.m2.5.5.1.1.1.1.1.2"></minus><apply id="A4.Ex71.m2.5.5.1.1.1.1.1.2.2.cmml" xref="A4.Ex71.m2.5.5.1.1.1.1.1.2.2"><times id="A4.Ex71.m2.5.5.1.1.1.1.1.2.2.1.cmml" xref="A4.Ex71.m2.5.5.1.1.1.1.1.2.2.1"></times><ci id="A4.Ex71.m2.5.5.1.1.1.1.1.2.2.2.cmml" xref="A4.Ex71.m2.5.5.1.1.1.1.1.2.2.2">𝐾</ci><ci id="A4.Ex71.m2.5.5.1.1.1.1.1.2.2.3.cmml" xref="A4.Ex71.m2.5.5.1.1.1.1.1.2.2.3">𝒆</ci></apply></apply><apply id="A4.Ex71.m2.5.5.1.1.1.1.1.3.cmml" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3"><times id="A4.Ex71.m2.5.5.1.1.1.1.1.3.1.cmml" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.1"></times><apply id="A4.Ex71.m2.5.5.1.1.1.1.1.3.2.cmml" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A4.Ex71.m2.5.5.1.1.1.1.1.3.2.1.cmml" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.2">subscript</csymbol><apply id="A4.Ex71.m2.5.5.1.1.1.1.1.3.2.2.cmml" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A4.Ex71.m2.5.5.1.1.1.1.1.3.2.2.1.cmml" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.2">superscript</csymbol><ci id="A4.Ex71.m2.5.5.1.1.1.1.1.3.2.2.2.cmml" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.2.2.2">Φ</ci><ci id="A4.Ex71.m2.5.5.1.1.1.1.1.3.2.2.3.cmml" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.2.2.3">𝑇</ci></apply><ci id="A4.Ex71.m2.5.5.1.1.1.1.1.3.2.3.cmml" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.2.3">𝜁</ci></apply><apply id="A4.Ex71.m2.5.5.1.1.1.1.1.3.3.cmml" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="A4.Ex71.m2.5.5.1.1.1.1.1.3.3.1.cmml" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.3">subscript</csymbol><apply id="A4.Ex71.m2.5.5.1.1.1.1.1.3.3.2.cmml" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.3.2"><ci id="A4.Ex71.m2.5.5.1.1.1.1.1.3.3.2.1.cmml" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.3.2.1">~</ci><ci id="A4.Ex71.m2.5.5.1.1.1.1.1.3.3.2.2.cmml" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.3.2.2">𝜽</ci></apply><ci id="A4.Ex71.m2.5.5.1.1.1.1.1.3.3.3.cmml" xref="A4.Ex71.m2.5.5.1.1.1.1.1.3.3.3">𝜁</ci></apply></apply></apply></apply><apply id="A4.Ex71.m2.5.5.1.3.cmml" xref="A4.Ex71.m2.5.5.1.3"><times id="A4.Ex71.m2.5.5.1.3.1.cmml" xref="A4.Ex71.m2.5.5.1.3.1"></times><apply id="A4.Ex71.m2.5.5.1.3.2.cmml" xref="A4.Ex71.m2.5.5.1.3.2"><csymbol cd="ambiguous" id="A4.Ex71.m2.5.5.1.3.2.1.cmml" xref="A4.Ex71.m2.5.5.1.3.2">superscript</csymbol><ci id="A4.Ex71.m2.5.5.1.3.2.2.cmml" xref="A4.Ex71.m2.5.5.1.3.2.2">𝒆</ci><ci id="A4.Ex71.m2.5.5.1.3.2.3.cmml" xref="A4.Ex71.m2.5.5.1.3.2.3">𝑇</ci></apply><ci id="A4.Ex71.m2.5.5.1.3.3.cmml" xref="A4.Ex71.m2.5.5.1.3.3">𝒅</ci></apply></apply><apply id="A4.Ex71.m2.7.7.3.cmml" xref="A4.Ex71.m2.7.7.3"><times id="A4.Ex71.m2.7.7.3.3.cmml" xref="A4.Ex71.m2.7.7.3.3"></times><apply id="A4.Ex71.m2.6.6.2.1.1.1.cmml" xref="A4.Ex71.m2.6.6.2.1.1"><plus id="A4.Ex71.m2.6.6.2.1.1.1.1.cmml" xref="A4.Ex71.m2.6.6.2.1.1.1.1"></plus><cn id="A4.Ex71.m2.6.6.2.1.1.1.2.cmml" type="integer" xref="A4.Ex71.m2.6.6.2.1.1.1.2">1</cn><ci id="A4.Ex71.m2.6.6.2.1.1.1.3.cmml" xref="A4.Ex71.m2.6.6.2.1.1.1.3">𝑝</ci></apply><apply id="A4.Ex71.m2.7.7.3.4.cmml" xref="A4.Ex71.m2.7.7.3.4"><csymbol cd="ambiguous" id="A4.Ex71.m2.7.7.3.4.1.cmml" xref="A4.Ex71.m2.7.7.3.4">superscript</csymbol><apply id="A4.Ex71.m2.7.7.3.4.2.cmml" xref="A4.Ex71.m2.7.7.3.4"><csymbol cd="ambiguous" id="A4.Ex71.m2.7.7.3.4.2.1.cmml" xref="A4.Ex71.m2.7.7.3.4">subscript</csymbol><apply id="A4.Ex71.m2.7.7.3.4.2.2.cmml" xref="A4.Ex71.m2.7.7.3.4.2.2"><ci id="A4.Ex71.m2.7.7.3.4.2.2.1.cmml" xref="A4.Ex71.m2.7.7.3.4.2.2.1">~</ci><ci id="A4.Ex71.m2.7.7.3.4.2.2.2.cmml" xref="A4.Ex71.m2.7.7.3.4.2.2.2">𝜽</ci></apply><ci id="A4.Ex71.m2.7.7.3.4.2.3.cmml" xref="A4.Ex71.m2.7.7.3.4.2.3">𝜁</ci></apply><ci id="A4.Ex71.m2.7.7.3.4.3.cmml" xref="A4.Ex71.m2.7.7.3.4.3">𝑇</ci></apply><apply id="A4.Ex71.m2.7.7.3.5.cmml" xref="A4.Ex71.m2.7.7.3.5"><csymbol cd="ambiguous" id="A4.Ex71.m2.7.7.3.5.1.cmml" xref="A4.Ex71.m2.7.7.3.5">subscript</csymbol><ci id="A4.Ex71.m2.7.7.3.5.2.cmml" xref="A4.Ex71.m2.7.7.3.5.2">Φ</ci><list id="A4.Ex71.m2.2.2.2.3.cmml" xref="A4.Ex71.m2.2.2.2.4"><ci id="A4.Ex71.m2.1.1.1.1.cmml" xref="A4.Ex71.m2.1.1.1.1">f</ci><ci id="A4.Ex71.m2.2.2.2.2.cmml" xref="A4.Ex71.m2.2.2.2.2">𝜁</ci></list></apply><apply id="A4.Ex71.m2.7.7.3.2.1.1.cmml" xref="A4.Ex71.m2.7.7.3.2.1"><plus id="A4.Ex71.m2.7.7.3.2.1.1.1.cmml" xref="A4.Ex71.m2.7.7.3.2.1.1.1"></plus><apply id="A4.Ex71.m2.7.7.3.2.1.1.2.cmml" xref="A4.Ex71.m2.7.7.3.2.1.1.2"><times id="A4.Ex71.m2.7.7.3.2.1.1.2.1.cmml" xref="A4.Ex71.m2.7.7.3.2.1.1.2.1"></times><apply id="A4.Ex71.m2.7.7.3.2.1.1.2.2.cmml" xref="A4.Ex71.m2.7.7.3.2.1.1.2.2"><csymbol cd="ambiguous" id="A4.Ex71.m2.7.7.3.2.1.1.2.2.1.cmml" xref="A4.Ex71.m2.7.7.3.2.1.1.2.2">superscript</csymbol><apply id="A4.Ex71.m2.7.7.3.2.1.1.2.2.2.cmml" xref="A4.Ex71.m2.7.7.3.2.1.1.2.2"><csymbol cd="ambiguous" id="A4.Ex71.m2.7.7.3.2.1.1.2.2.2.1.cmml" xref="A4.Ex71.m2.7.7.3.2.1.1.2.2">subscript</csymbol><ci id="A4.Ex71.m2.7.7.3.2.1.1.2.2.2.2.cmml" xref="A4.Ex71.m2.7.7.3.2.1.1.2.2.2.2">Φ</ci><list id="A4.Ex71.m2.4.4.2.3.cmml" xref="A4.Ex71.m2.4.4.2.4"><ci id="A4.Ex71.m2.3.3.1.1.cmml" xref="A4.Ex71.m2.3.3.1.1">f</ci><ci id="A4.Ex71.m2.4.4.2.2.cmml" xref="A4.Ex71.m2.4.4.2.2">𝜁</ci></list></apply><ci id="A4.Ex71.m2.7.7.3.2.1.1.2.2.3.cmml" xref="A4.Ex71.m2.7.7.3.2.1.1.2.2.3">𝑇</ci></apply><apply id="A4.Ex71.m2.7.7.3.2.1.1.2.3.cmml" xref="A4.Ex71.m2.7.7.3.2.1.1.2.3"><csymbol cd="ambiguous" id="A4.Ex71.m2.7.7.3.2.1.1.2.3.1.cmml" xref="A4.Ex71.m2.7.7.3.2.1.1.2.3">subscript</csymbol><apply id="A4.Ex71.m2.7.7.3.2.1.1.2.3.2.cmml" xref="A4.Ex71.m2.7.7.3.2.1.1.2.3.2"><ci id="A4.Ex71.m2.7.7.3.2.1.1.2.3.2.1.cmml" xref="A4.Ex71.m2.7.7.3.2.1.1.2.3.2.1">~</ci><ci id="A4.Ex71.m2.7.7.3.2.1.1.2.3.2.2.cmml" xref="A4.Ex71.m2.7.7.3.2.1.1.2.3.2.2">𝜽</ci></apply><ci id="A4.Ex71.m2.7.7.3.2.1.1.2.3.3.cmml" xref="A4.Ex71.m2.7.7.3.2.1.1.2.3.3">𝜁</ci></apply></apply><apply id="A4.Ex71.m2.7.7.3.2.1.1.3.cmml" xref="A4.Ex71.m2.7.7.3.2.1.1.3"><csymbol cd="ambiguous" id="A4.Ex71.m2.7.7.3.2.1.1.3.1.cmml" xref="A4.Ex71.m2.7.7.3.2.1.1.3">subscript</csymbol><ci id="A4.Ex71.m2.7.7.3.2.1.1.3.2.cmml" xref="A4.Ex71.m2.7.7.3.2.1.1.3.2">𝒅</ci><ci id="A4.Ex71.m2.7.7.3.2.1.1.3.3.cmml" xref="A4.Ex71.m2.7.7.3.2.1.1.3.3">f</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex71.m2.7c">\displaystyle\bm{e}^{T}(-K\bm{e}+\Phi^{T}_{\zeta}{\tilde{\bm{\theta}}}_{\zeta}% )+\bm{e}^{T}\bm{d}-(1+p){\tilde{\bm{\theta}}}_{\zeta}^{T}\Phi_{{\rm f},\zeta}(% \Phi_{{\rm f},\zeta}^{T}\tilde{\bm{\theta}}_{\zeta}+{\bm{d}}_{\rm f})</annotation><annotation encoding="application/x-llamapun" id="A4.Ex71.m2.7d">bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( - italic_K bold_italic_e + roman_Φ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ) + bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_d - ( 1 + italic_p ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f , italic_ζ end_POSTSUBSCRIPT ( roman_Φ start_POSTSUBSCRIPT roman_f , italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT + bold_italic_d start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex72"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-\kappa(1+p)\tilde{\bm{\theta}}^{T}_{\zeta}(Q_{\zeta}(t,t_{\rm e}% ){\tilde{\bm{\theta}}}_{\zeta}+{\bm{d}}_{\rm g})." class="ltx_Math" display="inline" id="A4.Ex72.m1.2"><semantics id="A4.Ex72.m1.2a"><mrow id="A4.Ex72.m1.2.2.1" xref="A4.Ex72.m1.2.2.1.1.cmml"><mrow id="A4.Ex72.m1.2.2.1.1" xref="A4.Ex72.m1.2.2.1.1.cmml"><mo id="A4.Ex72.m1.2.2.1.1a" xref="A4.Ex72.m1.2.2.1.1.cmml">−</mo><mrow id="A4.Ex72.m1.2.2.1.1.2" xref="A4.Ex72.m1.2.2.1.1.2.cmml"><mi id="A4.Ex72.m1.2.2.1.1.2.4" xref="A4.Ex72.m1.2.2.1.1.2.4.cmml">κ</mi><mo id="A4.Ex72.m1.2.2.1.1.2.3" xref="A4.Ex72.m1.2.2.1.1.2.3.cmml">⁢</mo><mrow id="A4.Ex72.m1.2.2.1.1.1.1.1" xref="A4.Ex72.m1.2.2.1.1.1.1.1.1.cmml"><mo id="A4.Ex72.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="A4.Ex72.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex72.m1.2.2.1.1.1.1.1.1" xref="A4.Ex72.m1.2.2.1.1.1.1.1.1.cmml"><mn id="A4.Ex72.m1.2.2.1.1.1.1.1.1.2" xref="A4.Ex72.m1.2.2.1.1.1.1.1.1.2.cmml">1</mn><mo id="A4.Ex72.m1.2.2.1.1.1.1.1.1.1" xref="A4.Ex72.m1.2.2.1.1.1.1.1.1.1.cmml">+</mo><mi id="A4.Ex72.m1.2.2.1.1.1.1.1.1.3" xref="A4.Ex72.m1.2.2.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A4.Ex72.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="A4.Ex72.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex72.m1.2.2.1.1.2.3a" xref="A4.Ex72.m1.2.2.1.1.2.3.cmml">⁢</mo><msubsup id="A4.Ex72.m1.2.2.1.1.2.5" xref="A4.Ex72.m1.2.2.1.1.2.5.cmml"><mover accent="true" id="A4.Ex72.m1.2.2.1.1.2.5.2.2" xref="A4.Ex72.m1.2.2.1.1.2.5.2.2.cmml"><mi id="A4.Ex72.m1.2.2.1.1.2.5.2.2.2" xref="A4.Ex72.m1.2.2.1.1.2.5.2.2.2.cmml">𝜽</mi><mo id="A4.Ex72.m1.2.2.1.1.2.5.2.2.1" xref="A4.Ex72.m1.2.2.1.1.2.5.2.2.1.cmml">~</mo></mover><mi id="A4.Ex72.m1.2.2.1.1.2.5.3" xref="A4.Ex72.m1.2.2.1.1.2.5.3.cmml">ζ</mi><mi id="A4.Ex72.m1.2.2.1.1.2.5.2.3" xref="A4.Ex72.m1.2.2.1.1.2.5.2.3.cmml">T</mi></msubsup><mo id="A4.Ex72.m1.2.2.1.1.2.3b" xref="A4.Ex72.m1.2.2.1.1.2.3.cmml">⁢</mo><mrow id="A4.Ex72.m1.2.2.1.1.2.2.1" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.cmml"><mo id="A4.Ex72.m1.2.2.1.1.2.2.1.2" stretchy="false" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.cmml">(</mo><mrow id="A4.Ex72.m1.2.2.1.1.2.2.1.1" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.cmml"><mrow id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.cmml"><msub id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.3" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.3.cmml"><mi id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.3.2" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.3.2.cmml">Q</mi><mi id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.3.3" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.3.3.cmml">ζ</mi></msub><mo id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.2" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.1.1" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.1.2.cmml"><mo id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.1.1.2" stretchy="false" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.1.2.cmml">(</mo><mi id="A4.Ex72.m1.1.1" xref="A4.Ex72.m1.1.1.cmml">t</mi><mo id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.1.1.3" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.1.2.cmml">,</mo><msub id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.1.1.1" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.1.1.1.cmml"><mi id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.1.1.1.2" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.1.1.1.2.cmml">t</mi><mi id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.1.1.1.3" mathvariant="normal" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.1.1.4" stretchy="false" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.1.2.cmml">)</mo></mrow><mo id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.2a" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.2.cmml">⁢</mo><msub id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.4" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.4.cmml"><mover accent="true" id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.4.2" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.4.2.cmml"><mi id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.4.2.2" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.4.2.2.cmml">𝜽</mi><mo id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.4.2.1" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.4.2.1.cmml">~</mo></mover><mi id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.4.3" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.4.3.cmml">ζ</mi></msub></mrow><mo id="A4.Ex72.m1.2.2.1.1.2.2.1.1.2" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.2.cmml">+</mo><msub id="A4.Ex72.m1.2.2.1.1.2.2.1.1.3" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.3.cmml"><mi id="A4.Ex72.m1.2.2.1.1.2.2.1.1.3.2" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.3.2.cmml">𝒅</mi><mi id="A4.Ex72.m1.2.2.1.1.2.2.1.1.3.3" mathvariant="normal" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.3.3.cmml">g</mi></msub></mrow><mo id="A4.Ex72.m1.2.2.1.1.2.2.1.3" stretchy="false" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A4.Ex72.m1.2.2.1.2" lspace="0em" xref="A4.Ex72.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex72.m1.2b"><apply id="A4.Ex72.m1.2.2.1.1.cmml" xref="A4.Ex72.m1.2.2.1"><minus id="A4.Ex72.m1.2.2.1.1.3.cmml" xref="A4.Ex72.m1.2.2.1"></minus><apply id="A4.Ex72.m1.2.2.1.1.2.cmml" xref="A4.Ex72.m1.2.2.1.1.2"><times id="A4.Ex72.m1.2.2.1.1.2.3.cmml" xref="A4.Ex72.m1.2.2.1.1.2.3"></times><ci id="A4.Ex72.m1.2.2.1.1.2.4.cmml" xref="A4.Ex72.m1.2.2.1.1.2.4">𝜅</ci><apply id="A4.Ex72.m1.2.2.1.1.1.1.1.1.cmml" xref="A4.Ex72.m1.2.2.1.1.1.1.1"><plus id="A4.Ex72.m1.2.2.1.1.1.1.1.1.1.cmml" xref="A4.Ex72.m1.2.2.1.1.1.1.1.1.1"></plus><cn id="A4.Ex72.m1.2.2.1.1.1.1.1.1.2.cmml" type="integer" xref="A4.Ex72.m1.2.2.1.1.1.1.1.1.2">1</cn><ci id="A4.Ex72.m1.2.2.1.1.1.1.1.1.3.cmml" xref="A4.Ex72.m1.2.2.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="A4.Ex72.m1.2.2.1.1.2.5.cmml" xref="A4.Ex72.m1.2.2.1.1.2.5"><csymbol cd="ambiguous" id="A4.Ex72.m1.2.2.1.1.2.5.1.cmml" xref="A4.Ex72.m1.2.2.1.1.2.5">subscript</csymbol><apply id="A4.Ex72.m1.2.2.1.1.2.5.2.cmml" xref="A4.Ex72.m1.2.2.1.1.2.5"><csymbol cd="ambiguous" id="A4.Ex72.m1.2.2.1.1.2.5.2.1.cmml" xref="A4.Ex72.m1.2.2.1.1.2.5">superscript</csymbol><apply id="A4.Ex72.m1.2.2.1.1.2.5.2.2.cmml" xref="A4.Ex72.m1.2.2.1.1.2.5.2.2"><ci id="A4.Ex72.m1.2.2.1.1.2.5.2.2.1.cmml" xref="A4.Ex72.m1.2.2.1.1.2.5.2.2.1">~</ci><ci id="A4.Ex72.m1.2.2.1.1.2.5.2.2.2.cmml" xref="A4.Ex72.m1.2.2.1.1.2.5.2.2.2">𝜽</ci></apply><ci id="A4.Ex72.m1.2.2.1.1.2.5.2.3.cmml" xref="A4.Ex72.m1.2.2.1.1.2.5.2.3">𝑇</ci></apply><ci id="A4.Ex72.m1.2.2.1.1.2.5.3.cmml" xref="A4.Ex72.m1.2.2.1.1.2.5.3">𝜁</ci></apply><apply id="A4.Ex72.m1.2.2.1.1.2.2.1.1.cmml" xref="A4.Ex72.m1.2.2.1.1.2.2.1"><plus id="A4.Ex72.m1.2.2.1.1.2.2.1.1.2.cmml" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.2"></plus><apply id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.cmml" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1"><times id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.2.cmml" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.2"></times><apply id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.3.cmml" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.3"><csymbol cd="ambiguous" id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.3.1.cmml" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.3">subscript</csymbol><ci id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.3.2.cmml" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.3.2">𝑄</ci><ci id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.3.3.cmml" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.3.3">𝜁</ci></apply><interval closure="open" id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.1.2.cmml" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.1.1"><ci id="A4.Ex72.m1.1.1.cmml" xref="A4.Ex72.m1.1.1">𝑡</ci><apply id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.1.1.1.cmml" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.1.1.1.1.cmml" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.1.1.1.2.cmml" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.1.1.1.2">𝑡</ci><ci id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.1.1.1.3.cmml" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.1.1.1.3">e</ci></apply></interval><apply id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.4.cmml" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.4"><csymbol cd="ambiguous" id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.4.1.cmml" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.4">subscript</csymbol><apply id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.4.2.cmml" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.4.2"><ci id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.4.2.1.cmml" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.4.2.1">~</ci><ci id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.4.2.2.cmml" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.4.2.2">𝜽</ci></apply><ci id="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.4.3.cmml" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.1.4.3">𝜁</ci></apply></apply><apply id="A4.Ex72.m1.2.2.1.1.2.2.1.1.3.cmml" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.3"><csymbol cd="ambiguous" id="A4.Ex72.m1.2.2.1.1.2.2.1.1.3.1.cmml" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.3">subscript</csymbol><ci id="A4.Ex72.m1.2.2.1.1.2.2.1.1.3.2.cmml" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.3.2">𝒅</ci><ci id="A4.Ex72.m1.2.2.1.1.2.2.1.1.3.3.cmml" xref="A4.Ex72.m1.2.2.1.1.2.2.1.1.3.3">g</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex72.m1.2c">\displaystyle-\kappa(1+p)\tilde{\bm{\theta}}^{T}_{\zeta}(Q_{\zeta}(t,t_{\rm e}% ){\tilde{\bm{\theta}}}_{\zeta}+{\bm{d}}_{\rm g}).</annotation><annotation encoding="application/x-llamapun" id="A4.Ex72.m1.2d">- italic_κ ( 1 + italic_p ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_Q start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t , italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT + bold_italic_d start_POSTSUBSCRIPT roman_g end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.1.p1.35">As the partial IE condition <math alttext="\Psi_{\zeta}(t_{\rm e})\geq\sigma_{\rm c}(T_{\rm a})I\geq\sigma I" class="ltx_Math" display="inline" id="A4.1.p1.33.m1.2"><semantics id="A4.1.p1.33.m1.2a"><mrow id="A4.1.p1.33.m1.2.2" xref="A4.1.p1.33.m1.2.2.cmml"><mrow id="A4.1.p1.33.m1.1.1.1" xref="A4.1.p1.33.m1.1.1.1.cmml"><msub id="A4.1.p1.33.m1.1.1.1.3" xref="A4.1.p1.33.m1.1.1.1.3.cmml"><mi id="A4.1.p1.33.m1.1.1.1.3.2" mathvariant="normal" xref="A4.1.p1.33.m1.1.1.1.3.2.cmml">Ψ</mi><mi id="A4.1.p1.33.m1.1.1.1.3.3" xref="A4.1.p1.33.m1.1.1.1.3.3.cmml">ζ</mi></msub><mo id="A4.1.p1.33.m1.1.1.1.2" xref="A4.1.p1.33.m1.1.1.1.2.cmml">⁢</mo><mrow id="A4.1.p1.33.m1.1.1.1.1.1" xref="A4.1.p1.33.m1.1.1.1.1.1.1.cmml"><mo id="A4.1.p1.33.m1.1.1.1.1.1.2" stretchy="false" xref="A4.1.p1.33.m1.1.1.1.1.1.1.cmml">(</mo><msub id="A4.1.p1.33.m1.1.1.1.1.1.1" xref="A4.1.p1.33.m1.1.1.1.1.1.1.cmml"><mi id="A4.1.p1.33.m1.1.1.1.1.1.1.2" xref="A4.1.p1.33.m1.1.1.1.1.1.1.2.cmml">t</mi><mi id="A4.1.p1.33.m1.1.1.1.1.1.1.3" mathvariant="normal" xref="A4.1.p1.33.m1.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="A4.1.p1.33.m1.1.1.1.1.1.3" stretchy="false" xref="A4.1.p1.33.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A4.1.p1.33.m1.2.2.4" xref="A4.1.p1.33.m1.2.2.4.cmml">≥</mo><mrow id="A4.1.p1.33.m1.2.2.2" xref="A4.1.p1.33.m1.2.2.2.cmml"><msub id="A4.1.p1.33.m1.2.2.2.3" xref="A4.1.p1.33.m1.2.2.2.3.cmml"><mi id="A4.1.p1.33.m1.2.2.2.3.2" xref="A4.1.p1.33.m1.2.2.2.3.2.cmml">σ</mi><mi id="A4.1.p1.33.m1.2.2.2.3.3" mathvariant="normal" xref="A4.1.p1.33.m1.2.2.2.3.3.cmml">c</mi></msub><mo id="A4.1.p1.33.m1.2.2.2.2" xref="A4.1.p1.33.m1.2.2.2.2.cmml">⁢</mo><mrow id="A4.1.p1.33.m1.2.2.2.1.1" xref="A4.1.p1.33.m1.2.2.2.1.1.1.cmml"><mo id="A4.1.p1.33.m1.2.2.2.1.1.2" stretchy="false" xref="A4.1.p1.33.m1.2.2.2.1.1.1.cmml">(</mo><msub id="A4.1.p1.33.m1.2.2.2.1.1.1" xref="A4.1.p1.33.m1.2.2.2.1.1.1.cmml"><mi id="A4.1.p1.33.m1.2.2.2.1.1.1.2" xref="A4.1.p1.33.m1.2.2.2.1.1.1.2.cmml">T</mi><mi id="A4.1.p1.33.m1.2.2.2.1.1.1.3" mathvariant="normal" xref="A4.1.p1.33.m1.2.2.2.1.1.1.3.cmml">a</mi></msub><mo id="A4.1.p1.33.m1.2.2.2.1.1.3" stretchy="false" xref="A4.1.p1.33.m1.2.2.2.1.1.1.cmml">)</mo></mrow><mo id="A4.1.p1.33.m1.2.2.2.2a" xref="A4.1.p1.33.m1.2.2.2.2.cmml">⁢</mo><mi id="A4.1.p1.33.m1.2.2.2.4" xref="A4.1.p1.33.m1.2.2.2.4.cmml">I</mi></mrow><mo id="A4.1.p1.33.m1.2.2.5" xref="A4.1.p1.33.m1.2.2.5.cmml">≥</mo><mrow id="A4.1.p1.33.m1.2.2.6" xref="A4.1.p1.33.m1.2.2.6.cmml"><mi id="A4.1.p1.33.m1.2.2.6.2" xref="A4.1.p1.33.m1.2.2.6.2.cmml">σ</mi><mo id="A4.1.p1.33.m1.2.2.6.1" xref="A4.1.p1.33.m1.2.2.6.1.cmml">⁢</mo><mi id="A4.1.p1.33.m1.2.2.6.3" xref="A4.1.p1.33.m1.2.2.6.3.cmml">I</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.33.m1.2b"><apply id="A4.1.p1.33.m1.2.2.cmml" xref="A4.1.p1.33.m1.2.2"><and id="A4.1.p1.33.m1.2.2a.cmml" xref="A4.1.p1.33.m1.2.2"></and><apply id="A4.1.p1.33.m1.2.2b.cmml" xref="A4.1.p1.33.m1.2.2"><geq id="A4.1.p1.33.m1.2.2.4.cmml" xref="A4.1.p1.33.m1.2.2.4"></geq><apply id="A4.1.p1.33.m1.1.1.1.cmml" xref="A4.1.p1.33.m1.1.1.1"><times id="A4.1.p1.33.m1.1.1.1.2.cmml" xref="A4.1.p1.33.m1.1.1.1.2"></times><apply id="A4.1.p1.33.m1.1.1.1.3.cmml" xref="A4.1.p1.33.m1.1.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.33.m1.1.1.1.3.1.cmml" xref="A4.1.p1.33.m1.1.1.1.3">subscript</csymbol><ci id="A4.1.p1.33.m1.1.1.1.3.2.cmml" xref="A4.1.p1.33.m1.1.1.1.3.2">Ψ</ci><ci id="A4.1.p1.33.m1.1.1.1.3.3.cmml" xref="A4.1.p1.33.m1.1.1.1.3.3">𝜁</ci></apply><apply id="A4.1.p1.33.m1.1.1.1.1.1.1.cmml" xref="A4.1.p1.33.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.1.p1.33.m1.1.1.1.1.1.1.1.cmml" xref="A4.1.p1.33.m1.1.1.1.1.1">subscript</csymbol><ci id="A4.1.p1.33.m1.1.1.1.1.1.1.2.cmml" xref="A4.1.p1.33.m1.1.1.1.1.1.1.2">𝑡</ci><ci id="A4.1.p1.33.m1.1.1.1.1.1.1.3.cmml" xref="A4.1.p1.33.m1.1.1.1.1.1.1.3">e</ci></apply></apply><apply id="A4.1.p1.33.m1.2.2.2.cmml" xref="A4.1.p1.33.m1.2.2.2"><times id="A4.1.p1.33.m1.2.2.2.2.cmml" xref="A4.1.p1.33.m1.2.2.2.2"></times><apply id="A4.1.p1.33.m1.2.2.2.3.cmml" xref="A4.1.p1.33.m1.2.2.2.3"><csymbol cd="ambiguous" id="A4.1.p1.33.m1.2.2.2.3.1.cmml" xref="A4.1.p1.33.m1.2.2.2.3">subscript</csymbol><ci id="A4.1.p1.33.m1.2.2.2.3.2.cmml" xref="A4.1.p1.33.m1.2.2.2.3.2">𝜎</ci><ci id="A4.1.p1.33.m1.2.2.2.3.3.cmml" xref="A4.1.p1.33.m1.2.2.2.3.3">c</ci></apply><apply id="A4.1.p1.33.m1.2.2.2.1.1.1.cmml" xref="A4.1.p1.33.m1.2.2.2.1.1"><csymbol cd="ambiguous" id="A4.1.p1.33.m1.2.2.2.1.1.1.1.cmml" xref="A4.1.p1.33.m1.2.2.2.1.1">subscript</csymbol><ci id="A4.1.p1.33.m1.2.2.2.1.1.1.2.cmml" xref="A4.1.p1.33.m1.2.2.2.1.1.1.2">𝑇</ci><ci id="A4.1.p1.33.m1.2.2.2.1.1.1.3.cmml" xref="A4.1.p1.33.m1.2.2.2.1.1.1.3">a</ci></apply><ci id="A4.1.p1.33.m1.2.2.2.4.cmml" xref="A4.1.p1.33.m1.2.2.2.4">𝐼</ci></apply></apply><apply id="A4.1.p1.33.m1.2.2c.cmml" xref="A4.1.p1.33.m1.2.2"><geq id="A4.1.p1.33.m1.2.2.5.cmml" xref="A4.1.p1.33.m1.2.2.5"></geq><share href="https://arxiv.org/html/2401.10785v2#A4.1.p1.33.m1.2.2.2.cmml" id="A4.1.p1.33.m1.2.2d.cmml" xref="A4.1.p1.33.m1.2.2"></share><apply id="A4.1.p1.33.m1.2.2.6.cmml" xref="A4.1.p1.33.m1.2.2.6"><times id="A4.1.p1.33.m1.2.2.6.1.cmml" xref="A4.1.p1.33.m1.2.2.6.1"></times><ci id="A4.1.p1.33.m1.2.2.6.2.cmml" xref="A4.1.p1.33.m1.2.2.6.2">𝜎</ci><ci id="A4.1.p1.33.m1.2.2.6.3.cmml" xref="A4.1.p1.33.m1.2.2.6.3">𝐼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.33.m1.2c">\Psi_{\zeta}(t_{\rm e})\geq\sigma_{\rm c}(T_{\rm a})I\geq\sigma I</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.33.m1.2d">roman_Ψ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) ≥ italic_σ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ( italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT ) italic_I ≥ italic_σ italic_I</annotation></semantics></math> is satisfied with some constants <math alttext="T_{\rm a}" class="ltx_Math" display="inline" id="A4.1.p1.34.m2.1"><semantics id="A4.1.p1.34.m2.1a"><msub id="A4.1.p1.34.m2.1.1" xref="A4.1.p1.34.m2.1.1.cmml"><mi id="A4.1.p1.34.m2.1.1.2" xref="A4.1.p1.34.m2.1.1.2.cmml">T</mi><mi id="A4.1.p1.34.m2.1.1.3" mathvariant="normal" xref="A4.1.p1.34.m2.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="A4.1.p1.34.m2.1b"><apply id="A4.1.p1.34.m2.1.1.cmml" xref="A4.1.p1.34.m2.1.1"><csymbol cd="ambiguous" id="A4.1.p1.34.m2.1.1.1.cmml" xref="A4.1.p1.34.m2.1.1">subscript</csymbol><ci id="A4.1.p1.34.m2.1.1.2.cmml" xref="A4.1.p1.34.m2.1.1.2">𝑇</ci><ci id="A4.1.p1.34.m2.1.1.3.cmml" xref="A4.1.p1.34.m2.1.1.3">a</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.34.m2.1c">T_{\rm a}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.34.m2.1d">italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\sigma\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A4.1.p1.35.m3.1"><semantics id="A4.1.p1.35.m3.1a"><mrow id="A4.1.p1.35.m3.1.1" xref="A4.1.p1.35.m3.1.1.cmml"><mi id="A4.1.p1.35.m3.1.1.2" xref="A4.1.p1.35.m3.1.1.2.cmml">σ</mi><mo id="A4.1.p1.35.m3.1.1.1" xref="A4.1.p1.35.m3.1.1.1.cmml">∈</mo><msup id="A4.1.p1.35.m3.1.1.3" xref="A4.1.p1.35.m3.1.1.3.cmml"><mi id="A4.1.p1.35.m3.1.1.3.2" xref="A4.1.p1.35.m3.1.1.3.2.cmml">ℝ</mi><mo id="A4.1.p1.35.m3.1.1.3.3" xref="A4.1.p1.35.m3.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.35.m3.1b"><apply id="A4.1.p1.35.m3.1.1.cmml" xref="A4.1.p1.35.m3.1.1"><in id="A4.1.p1.35.m3.1.1.1.cmml" xref="A4.1.p1.35.m3.1.1.1"></in><ci id="A4.1.p1.35.m3.1.1.2.cmml" xref="A4.1.p1.35.m3.1.1.2">𝜎</ci><apply id="A4.1.p1.35.m3.1.1.3.cmml" xref="A4.1.p1.35.m3.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.35.m3.1.1.3.1.cmml" xref="A4.1.p1.35.m3.1.1.3">superscript</csymbol><ci id="A4.1.p1.35.m3.1.1.3.2.cmml" xref="A4.1.p1.35.m3.1.1.3.2">ℝ</ci><plus id="A4.1.p1.35.m3.1.1.3.3.cmml" xref="A4.1.p1.35.m3.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.35.m3.1c">\sigma\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.35.m3.1d">italic_σ ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, noting the proof of Item 2 in Theorem 2, the above result leads to</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx81"> <tbody id="A4.Ex73"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}_{\zeta}\leq" class="ltx_Math" display="inline" id="A4.Ex73.m1.1"><semantics id="A4.Ex73.m1.1a"><mrow id="A4.Ex73.m1.1.1" xref="A4.Ex73.m1.1.1.cmml"><msub id="A4.Ex73.m1.1.1.2" xref="A4.Ex73.m1.1.1.2.cmml"><mover accent="true" id="A4.Ex73.m1.1.1.2.2" xref="A4.Ex73.m1.1.1.2.2.cmml"><mi id="A4.Ex73.m1.1.1.2.2.2" xref="A4.Ex73.m1.1.1.2.2.2.cmml">V</mi><mo id="A4.Ex73.m1.1.1.2.2.1" xref="A4.Ex73.m1.1.1.2.2.1.cmml">˙</mo></mover><mi id="A4.Ex73.m1.1.1.2.3" xref="A4.Ex73.m1.1.1.2.3.cmml">ζ</mi></msub><mo id="A4.Ex73.m1.1.1.1" xref="A4.Ex73.m1.1.1.1.cmml">≤</mo><mi id="A4.Ex73.m1.1.1.3" xref="A4.Ex73.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex73.m1.1b"><apply id="A4.Ex73.m1.1.1.cmml" xref="A4.Ex73.m1.1.1"><leq id="A4.Ex73.m1.1.1.1.cmml" xref="A4.Ex73.m1.1.1.1"></leq><apply id="A4.Ex73.m1.1.1.2.cmml" xref="A4.Ex73.m1.1.1.2"><csymbol cd="ambiguous" id="A4.Ex73.m1.1.1.2.1.cmml" xref="A4.Ex73.m1.1.1.2">subscript</csymbol><apply id="A4.Ex73.m1.1.1.2.2.cmml" xref="A4.Ex73.m1.1.1.2.2"><ci id="A4.Ex73.m1.1.1.2.2.1.cmml" xref="A4.Ex73.m1.1.1.2.2.1">˙</ci><ci id="A4.Ex73.m1.1.1.2.2.2.cmml" xref="A4.Ex73.m1.1.1.2.2.2">𝑉</ci></apply><ci id="A4.Ex73.m1.1.1.2.3.cmml" xref="A4.Ex73.m1.1.1.2.3">𝜁</ci></apply><csymbol cd="latexml" id="A4.Ex73.m1.1.1.3.cmml" xref="A4.Ex73.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex73.m1.1c">\displaystyle\dot{V}_{\zeta}\leq</annotation><annotation encoding="application/x-llamapun" id="A4.Ex73.m1.1d">over˙ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-k_{\rm c}\bm{e}^{T}\bm{e}-\kappa(1+p)\sigma^{*}\tilde{\bm{\theta% }}^{T}_{\zeta}\tilde{\bm{\theta}}_{\zeta}+\delta\|\bm{e}\|\|\tilde{\bm{\theta}% }_{\zeta}\|" class="ltx_Math" display="inline" id="A4.Ex73.m2.3"><semantics id="A4.Ex73.m2.3a"><mrow id="A4.Ex73.m2.3.3" xref="A4.Ex73.m2.3.3.cmml"><mrow id="A4.Ex73.m2.2.2.1" xref="A4.Ex73.m2.2.2.1.cmml"><mrow id="A4.Ex73.m2.2.2.1.3" xref="A4.Ex73.m2.2.2.1.3.cmml"><mo id="A4.Ex73.m2.2.2.1.3a" xref="A4.Ex73.m2.2.2.1.3.cmml">−</mo><mrow id="A4.Ex73.m2.2.2.1.3.2" xref="A4.Ex73.m2.2.2.1.3.2.cmml"><msub id="A4.Ex73.m2.2.2.1.3.2.2" xref="A4.Ex73.m2.2.2.1.3.2.2.cmml"><mi id="A4.Ex73.m2.2.2.1.3.2.2.2" xref="A4.Ex73.m2.2.2.1.3.2.2.2.cmml">k</mi><mi id="A4.Ex73.m2.2.2.1.3.2.2.3" mathvariant="normal" xref="A4.Ex73.m2.2.2.1.3.2.2.3.cmml">c</mi></msub><mo id="A4.Ex73.m2.2.2.1.3.2.1" xref="A4.Ex73.m2.2.2.1.3.2.1.cmml">⁢</mo><msup id="A4.Ex73.m2.2.2.1.3.2.3" xref="A4.Ex73.m2.2.2.1.3.2.3.cmml"><mi id="A4.Ex73.m2.2.2.1.3.2.3.2" xref="A4.Ex73.m2.2.2.1.3.2.3.2.cmml">𝒆</mi><mi id="A4.Ex73.m2.2.2.1.3.2.3.3" xref="A4.Ex73.m2.2.2.1.3.2.3.3.cmml">T</mi></msup><mo id="A4.Ex73.m2.2.2.1.3.2.1a" xref="A4.Ex73.m2.2.2.1.3.2.1.cmml">⁢</mo><mi id="A4.Ex73.m2.2.2.1.3.2.4" xref="A4.Ex73.m2.2.2.1.3.2.4.cmml">𝒆</mi></mrow></mrow><mo id="A4.Ex73.m2.2.2.1.2" xref="A4.Ex73.m2.2.2.1.2.cmml">−</mo><mrow id="A4.Ex73.m2.2.2.1.1" xref="A4.Ex73.m2.2.2.1.1.cmml"><mi id="A4.Ex73.m2.2.2.1.1.3" xref="A4.Ex73.m2.2.2.1.1.3.cmml">κ</mi><mo id="A4.Ex73.m2.2.2.1.1.2" xref="A4.Ex73.m2.2.2.1.1.2.cmml">⁢</mo><mrow id="A4.Ex73.m2.2.2.1.1.1.1" xref="A4.Ex73.m2.2.2.1.1.1.1.1.cmml"><mo id="A4.Ex73.m2.2.2.1.1.1.1.2" stretchy="false" xref="A4.Ex73.m2.2.2.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex73.m2.2.2.1.1.1.1.1" xref="A4.Ex73.m2.2.2.1.1.1.1.1.cmml"><mn id="A4.Ex73.m2.2.2.1.1.1.1.1.2" xref="A4.Ex73.m2.2.2.1.1.1.1.1.2.cmml">1</mn><mo id="A4.Ex73.m2.2.2.1.1.1.1.1.1" xref="A4.Ex73.m2.2.2.1.1.1.1.1.1.cmml">+</mo><mi id="A4.Ex73.m2.2.2.1.1.1.1.1.3" xref="A4.Ex73.m2.2.2.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A4.Ex73.m2.2.2.1.1.1.1.3" stretchy="false" xref="A4.Ex73.m2.2.2.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex73.m2.2.2.1.1.2a" xref="A4.Ex73.m2.2.2.1.1.2.cmml">⁢</mo><msup id="A4.Ex73.m2.2.2.1.1.4" xref="A4.Ex73.m2.2.2.1.1.4.cmml"><mi id="A4.Ex73.m2.2.2.1.1.4.2" xref="A4.Ex73.m2.2.2.1.1.4.2.cmml">σ</mi><mo id="A4.Ex73.m2.2.2.1.1.4.3" xref="A4.Ex73.m2.2.2.1.1.4.3.cmml">∗</mo></msup><mo id="A4.Ex73.m2.2.2.1.1.2b" xref="A4.Ex73.m2.2.2.1.1.2.cmml">⁢</mo><msubsup id="A4.Ex73.m2.2.2.1.1.5" xref="A4.Ex73.m2.2.2.1.1.5.cmml"><mover accent="true" id="A4.Ex73.m2.2.2.1.1.5.2.2" xref="A4.Ex73.m2.2.2.1.1.5.2.2.cmml"><mi id="A4.Ex73.m2.2.2.1.1.5.2.2.2" xref="A4.Ex73.m2.2.2.1.1.5.2.2.2.cmml">𝜽</mi><mo id="A4.Ex73.m2.2.2.1.1.5.2.2.1" xref="A4.Ex73.m2.2.2.1.1.5.2.2.1.cmml">~</mo></mover><mi id="A4.Ex73.m2.2.2.1.1.5.3" xref="A4.Ex73.m2.2.2.1.1.5.3.cmml">ζ</mi><mi id="A4.Ex73.m2.2.2.1.1.5.2.3" xref="A4.Ex73.m2.2.2.1.1.5.2.3.cmml">T</mi></msubsup><mo id="A4.Ex73.m2.2.2.1.1.2c" xref="A4.Ex73.m2.2.2.1.1.2.cmml">⁢</mo><msub id="A4.Ex73.m2.2.2.1.1.6" xref="A4.Ex73.m2.2.2.1.1.6.cmml"><mover accent="true" id="A4.Ex73.m2.2.2.1.1.6.2" xref="A4.Ex73.m2.2.2.1.1.6.2.cmml"><mi id="A4.Ex73.m2.2.2.1.1.6.2.2" xref="A4.Ex73.m2.2.2.1.1.6.2.2.cmml">𝜽</mi><mo id="A4.Ex73.m2.2.2.1.1.6.2.1" xref="A4.Ex73.m2.2.2.1.1.6.2.1.cmml">~</mo></mover><mi id="A4.Ex73.m2.2.2.1.1.6.3" xref="A4.Ex73.m2.2.2.1.1.6.3.cmml">ζ</mi></msub></mrow></mrow><mo id="A4.Ex73.m2.3.3.3" xref="A4.Ex73.m2.3.3.3.cmml">+</mo><mrow id="A4.Ex73.m2.3.3.2" xref="A4.Ex73.m2.3.3.2.cmml"><mi id="A4.Ex73.m2.3.3.2.3" xref="A4.Ex73.m2.3.3.2.3.cmml">δ</mi><mo id="A4.Ex73.m2.3.3.2.2" xref="A4.Ex73.m2.3.3.2.2.cmml">⁢</mo><mrow id="A4.Ex73.m2.3.3.2.4.2" xref="A4.Ex73.m2.3.3.2.4.1.cmml"><mo id="A4.Ex73.m2.3.3.2.4.2.1" stretchy="false" xref="A4.Ex73.m2.3.3.2.4.1.1.cmml">‖</mo><mi id="A4.Ex73.m2.1.1" xref="A4.Ex73.m2.1.1.cmml">𝒆</mi><mo id="A4.Ex73.m2.3.3.2.4.2.2" stretchy="false" xref="A4.Ex73.m2.3.3.2.4.1.1.cmml">‖</mo></mrow><mo id="A4.Ex73.m2.3.3.2.2a" xref="A4.Ex73.m2.3.3.2.2.cmml">⁢</mo><mrow id="A4.Ex73.m2.3.3.2.1.1" xref="A4.Ex73.m2.3.3.2.1.2.cmml"><mo id="A4.Ex73.m2.3.3.2.1.1.2" stretchy="false" xref="A4.Ex73.m2.3.3.2.1.2.1.cmml">‖</mo><msub id="A4.Ex73.m2.3.3.2.1.1.1" xref="A4.Ex73.m2.3.3.2.1.1.1.cmml"><mover accent="true" id="A4.Ex73.m2.3.3.2.1.1.1.2" xref="A4.Ex73.m2.3.3.2.1.1.1.2.cmml"><mi id="A4.Ex73.m2.3.3.2.1.1.1.2.2" xref="A4.Ex73.m2.3.3.2.1.1.1.2.2.cmml">𝜽</mi><mo id="A4.Ex73.m2.3.3.2.1.1.1.2.1" xref="A4.Ex73.m2.3.3.2.1.1.1.2.1.cmml">~</mo></mover><mi id="A4.Ex73.m2.3.3.2.1.1.1.3" xref="A4.Ex73.m2.3.3.2.1.1.1.3.cmml">ζ</mi></msub><mo id="A4.Ex73.m2.3.3.2.1.1.3" stretchy="false" xref="A4.Ex73.m2.3.3.2.1.2.1.cmml">‖</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex73.m2.3b"><apply id="A4.Ex73.m2.3.3.cmml" xref="A4.Ex73.m2.3.3"><plus id="A4.Ex73.m2.3.3.3.cmml" xref="A4.Ex73.m2.3.3.3"></plus><apply id="A4.Ex73.m2.2.2.1.cmml" xref="A4.Ex73.m2.2.2.1"><minus id="A4.Ex73.m2.2.2.1.2.cmml" xref="A4.Ex73.m2.2.2.1.2"></minus><apply id="A4.Ex73.m2.2.2.1.3.cmml" xref="A4.Ex73.m2.2.2.1.3"><minus id="A4.Ex73.m2.2.2.1.3.1.cmml" xref="A4.Ex73.m2.2.2.1.3"></minus><apply id="A4.Ex73.m2.2.2.1.3.2.cmml" xref="A4.Ex73.m2.2.2.1.3.2"><times id="A4.Ex73.m2.2.2.1.3.2.1.cmml" xref="A4.Ex73.m2.2.2.1.3.2.1"></times><apply id="A4.Ex73.m2.2.2.1.3.2.2.cmml" xref="A4.Ex73.m2.2.2.1.3.2.2"><csymbol cd="ambiguous" id="A4.Ex73.m2.2.2.1.3.2.2.1.cmml" xref="A4.Ex73.m2.2.2.1.3.2.2">subscript</csymbol><ci id="A4.Ex73.m2.2.2.1.3.2.2.2.cmml" xref="A4.Ex73.m2.2.2.1.3.2.2.2">𝑘</ci><ci id="A4.Ex73.m2.2.2.1.3.2.2.3.cmml" xref="A4.Ex73.m2.2.2.1.3.2.2.3">c</ci></apply><apply id="A4.Ex73.m2.2.2.1.3.2.3.cmml" xref="A4.Ex73.m2.2.2.1.3.2.3"><csymbol cd="ambiguous" id="A4.Ex73.m2.2.2.1.3.2.3.1.cmml" xref="A4.Ex73.m2.2.2.1.3.2.3">superscript</csymbol><ci id="A4.Ex73.m2.2.2.1.3.2.3.2.cmml" xref="A4.Ex73.m2.2.2.1.3.2.3.2">𝒆</ci><ci id="A4.Ex73.m2.2.2.1.3.2.3.3.cmml" xref="A4.Ex73.m2.2.2.1.3.2.3.3">𝑇</ci></apply><ci id="A4.Ex73.m2.2.2.1.3.2.4.cmml" xref="A4.Ex73.m2.2.2.1.3.2.4">𝒆</ci></apply></apply><apply id="A4.Ex73.m2.2.2.1.1.cmml" xref="A4.Ex73.m2.2.2.1.1"><times id="A4.Ex73.m2.2.2.1.1.2.cmml" xref="A4.Ex73.m2.2.2.1.1.2"></times><ci id="A4.Ex73.m2.2.2.1.1.3.cmml" xref="A4.Ex73.m2.2.2.1.1.3">𝜅</ci><apply id="A4.Ex73.m2.2.2.1.1.1.1.1.cmml" xref="A4.Ex73.m2.2.2.1.1.1.1"><plus id="A4.Ex73.m2.2.2.1.1.1.1.1.1.cmml" xref="A4.Ex73.m2.2.2.1.1.1.1.1.1"></plus><cn id="A4.Ex73.m2.2.2.1.1.1.1.1.2.cmml" type="integer" xref="A4.Ex73.m2.2.2.1.1.1.1.1.2">1</cn><ci id="A4.Ex73.m2.2.2.1.1.1.1.1.3.cmml" xref="A4.Ex73.m2.2.2.1.1.1.1.1.3">𝑝</ci></apply><apply id="A4.Ex73.m2.2.2.1.1.4.cmml" xref="A4.Ex73.m2.2.2.1.1.4"><csymbol cd="ambiguous" id="A4.Ex73.m2.2.2.1.1.4.1.cmml" xref="A4.Ex73.m2.2.2.1.1.4">superscript</csymbol><ci id="A4.Ex73.m2.2.2.1.1.4.2.cmml" xref="A4.Ex73.m2.2.2.1.1.4.2">𝜎</ci><times id="A4.Ex73.m2.2.2.1.1.4.3.cmml" xref="A4.Ex73.m2.2.2.1.1.4.3"></times></apply><apply id="A4.Ex73.m2.2.2.1.1.5.cmml" xref="A4.Ex73.m2.2.2.1.1.5"><csymbol cd="ambiguous" id="A4.Ex73.m2.2.2.1.1.5.1.cmml" xref="A4.Ex73.m2.2.2.1.1.5">subscript</csymbol><apply id="A4.Ex73.m2.2.2.1.1.5.2.cmml" xref="A4.Ex73.m2.2.2.1.1.5"><csymbol cd="ambiguous" id="A4.Ex73.m2.2.2.1.1.5.2.1.cmml" xref="A4.Ex73.m2.2.2.1.1.5">superscript</csymbol><apply id="A4.Ex73.m2.2.2.1.1.5.2.2.cmml" xref="A4.Ex73.m2.2.2.1.1.5.2.2"><ci id="A4.Ex73.m2.2.2.1.1.5.2.2.1.cmml" xref="A4.Ex73.m2.2.2.1.1.5.2.2.1">~</ci><ci id="A4.Ex73.m2.2.2.1.1.5.2.2.2.cmml" xref="A4.Ex73.m2.2.2.1.1.5.2.2.2">𝜽</ci></apply><ci id="A4.Ex73.m2.2.2.1.1.5.2.3.cmml" xref="A4.Ex73.m2.2.2.1.1.5.2.3">𝑇</ci></apply><ci id="A4.Ex73.m2.2.2.1.1.5.3.cmml" xref="A4.Ex73.m2.2.2.1.1.5.3">𝜁</ci></apply><apply id="A4.Ex73.m2.2.2.1.1.6.cmml" xref="A4.Ex73.m2.2.2.1.1.6"><csymbol cd="ambiguous" id="A4.Ex73.m2.2.2.1.1.6.1.cmml" xref="A4.Ex73.m2.2.2.1.1.6">subscript</csymbol><apply id="A4.Ex73.m2.2.2.1.1.6.2.cmml" xref="A4.Ex73.m2.2.2.1.1.6.2"><ci id="A4.Ex73.m2.2.2.1.1.6.2.1.cmml" xref="A4.Ex73.m2.2.2.1.1.6.2.1">~</ci><ci id="A4.Ex73.m2.2.2.1.1.6.2.2.cmml" xref="A4.Ex73.m2.2.2.1.1.6.2.2">𝜽</ci></apply><ci id="A4.Ex73.m2.2.2.1.1.6.3.cmml" xref="A4.Ex73.m2.2.2.1.1.6.3">𝜁</ci></apply></apply></apply><apply id="A4.Ex73.m2.3.3.2.cmml" xref="A4.Ex73.m2.3.3.2"><times id="A4.Ex73.m2.3.3.2.2.cmml" xref="A4.Ex73.m2.3.3.2.2"></times><ci id="A4.Ex73.m2.3.3.2.3.cmml" xref="A4.Ex73.m2.3.3.2.3">𝛿</ci><apply id="A4.Ex73.m2.3.3.2.4.1.cmml" xref="A4.Ex73.m2.3.3.2.4.2"><csymbol cd="latexml" id="A4.Ex73.m2.3.3.2.4.1.1.cmml" xref="A4.Ex73.m2.3.3.2.4.2.1">norm</csymbol><ci id="A4.Ex73.m2.1.1.cmml" xref="A4.Ex73.m2.1.1">𝒆</ci></apply><apply id="A4.Ex73.m2.3.3.2.1.2.cmml" xref="A4.Ex73.m2.3.3.2.1.1"><csymbol cd="latexml" id="A4.Ex73.m2.3.3.2.1.2.1.cmml" xref="A4.Ex73.m2.3.3.2.1.1.2">norm</csymbol><apply id="A4.Ex73.m2.3.3.2.1.1.1.cmml" xref="A4.Ex73.m2.3.3.2.1.1.1"><csymbol cd="ambiguous" id="A4.Ex73.m2.3.3.2.1.1.1.1.cmml" xref="A4.Ex73.m2.3.3.2.1.1.1">subscript</csymbol><apply id="A4.Ex73.m2.3.3.2.1.1.1.2.cmml" xref="A4.Ex73.m2.3.3.2.1.1.1.2"><ci id="A4.Ex73.m2.3.3.2.1.1.1.2.1.cmml" xref="A4.Ex73.m2.3.3.2.1.1.1.2.1">~</ci><ci id="A4.Ex73.m2.3.3.2.1.1.1.2.2.cmml" xref="A4.Ex73.m2.3.3.2.1.1.1.2.2">𝜽</ci></apply><ci id="A4.Ex73.m2.3.3.2.1.1.1.3.cmml" xref="A4.Ex73.m2.3.3.2.1.1.1.3">𝜁</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex73.m2.3c">\displaystyle-k_{\rm c}\bm{e}^{T}\bm{e}-\kappa(1+p)\sigma^{*}\tilde{\bm{\theta% }}^{T}_{\zeta}\tilde{\bm{\theta}}_{\zeta}+\delta\|\bm{e}\|\|\tilde{\bm{\theta}% }_{\zeta}\|</annotation><annotation encoding="application/x-llamapun" id="A4.Ex73.m2.3d">- italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_e - italic_κ ( 1 + italic_p ) italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT + italic_δ ∥ bold_italic_e ∥ ∥ over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ∥</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex74"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle+\bm{e}^{T}\bm{d}-(1+p){\tilde{\bm{\theta}}}_{\zeta}^{T}\Phi_{{% \rm f},\zeta}{\bm{d}}_{\rm f}-\kappa(1+p){\tilde{\bm{\theta}}}_{\zeta}^{T}{\bm% {d}}_{\rm g}." class="ltx_Math" display="inline" id="A4.Ex74.m1.3"><semantics id="A4.Ex74.m1.3a"><mrow id="A4.Ex74.m1.3.3.1" xref="A4.Ex74.m1.3.3.1.1.cmml"><mrow id="A4.Ex74.m1.3.3.1.1" xref="A4.Ex74.m1.3.3.1.1.cmml"><mrow id="A4.Ex74.m1.3.3.1.1.4" xref="A4.Ex74.m1.3.3.1.1.4.cmml"><mo id="A4.Ex74.m1.3.3.1.1.4a" xref="A4.Ex74.m1.3.3.1.1.4.cmml">+</mo><mrow id="A4.Ex74.m1.3.3.1.1.4.2" xref="A4.Ex74.m1.3.3.1.1.4.2.cmml"><msup id="A4.Ex74.m1.3.3.1.1.4.2.2" xref="A4.Ex74.m1.3.3.1.1.4.2.2.cmml"><mi id="A4.Ex74.m1.3.3.1.1.4.2.2.2" xref="A4.Ex74.m1.3.3.1.1.4.2.2.2.cmml">𝒆</mi><mi id="A4.Ex74.m1.3.3.1.1.4.2.2.3" xref="A4.Ex74.m1.3.3.1.1.4.2.2.3.cmml">T</mi></msup><mo id="A4.Ex74.m1.3.3.1.1.4.2.1" xref="A4.Ex74.m1.3.3.1.1.4.2.1.cmml">⁢</mo><mi id="A4.Ex74.m1.3.3.1.1.4.2.3" xref="A4.Ex74.m1.3.3.1.1.4.2.3.cmml">𝒅</mi></mrow></mrow><mo id="A4.Ex74.m1.3.3.1.1.3" xref="A4.Ex74.m1.3.3.1.1.3.cmml">−</mo><mrow id="A4.Ex74.m1.3.3.1.1.1" xref="A4.Ex74.m1.3.3.1.1.1.cmml"><mrow id="A4.Ex74.m1.3.3.1.1.1.1.1" xref="A4.Ex74.m1.3.3.1.1.1.1.1.1.cmml"><mo id="A4.Ex74.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="A4.Ex74.m1.3.3.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex74.m1.3.3.1.1.1.1.1.1" xref="A4.Ex74.m1.3.3.1.1.1.1.1.1.cmml"><mn id="A4.Ex74.m1.3.3.1.1.1.1.1.1.2" xref="A4.Ex74.m1.3.3.1.1.1.1.1.1.2.cmml">1</mn><mo id="A4.Ex74.m1.3.3.1.1.1.1.1.1.1" xref="A4.Ex74.m1.3.3.1.1.1.1.1.1.1.cmml">+</mo><mi id="A4.Ex74.m1.3.3.1.1.1.1.1.1.3" xref="A4.Ex74.m1.3.3.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A4.Ex74.m1.3.3.1.1.1.1.1.3" stretchy="false" xref="A4.Ex74.m1.3.3.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex74.m1.3.3.1.1.1.2" xref="A4.Ex74.m1.3.3.1.1.1.2.cmml">⁢</mo><msubsup id="A4.Ex74.m1.3.3.1.1.1.3" xref="A4.Ex74.m1.3.3.1.1.1.3.cmml"><mover accent="true" id="A4.Ex74.m1.3.3.1.1.1.3.2.2" xref="A4.Ex74.m1.3.3.1.1.1.3.2.2.cmml"><mi id="A4.Ex74.m1.3.3.1.1.1.3.2.2.2" xref="A4.Ex74.m1.3.3.1.1.1.3.2.2.2.cmml">𝜽</mi><mo id="A4.Ex74.m1.3.3.1.1.1.3.2.2.1" xref="A4.Ex74.m1.3.3.1.1.1.3.2.2.1.cmml">~</mo></mover><mi id="A4.Ex74.m1.3.3.1.1.1.3.2.3" xref="A4.Ex74.m1.3.3.1.1.1.3.2.3.cmml">ζ</mi><mi id="A4.Ex74.m1.3.3.1.1.1.3.3" xref="A4.Ex74.m1.3.3.1.1.1.3.3.cmml">T</mi></msubsup><mo id="A4.Ex74.m1.3.3.1.1.1.2a" xref="A4.Ex74.m1.3.3.1.1.1.2.cmml">⁢</mo><msub id="A4.Ex74.m1.3.3.1.1.1.4" xref="A4.Ex74.m1.3.3.1.1.1.4.cmml"><mi id="A4.Ex74.m1.3.3.1.1.1.4.2" mathvariant="normal" xref="A4.Ex74.m1.3.3.1.1.1.4.2.cmml">Φ</mi><mrow id="A4.Ex74.m1.2.2.2.4" xref="A4.Ex74.m1.2.2.2.3.cmml"><mi id="A4.Ex74.m1.1.1.1.1" mathvariant="normal" xref="A4.Ex74.m1.1.1.1.1.cmml">f</mi><mo id="A4.Ex74.m1.2.2.2.4.1" xref="A4.Ex74.m1.2.2.2.3.cmml">,</mo><mi id="A4.Ex74.m1.2.2.2.2" xref="A4.Ex74.m1.2.2.2.2.cmml">ζ</mi></mrow></msub><mo id="A4.Ex74.m1.3.3.1.1.1.2b" xref="A4.Ex74.m1.3.3.1.1.1.2.cmml">⁢</mo><msub id="A4.Ex74.m1.3.3.1.1.1.5" xref="A4.Ex74.m1.3.3.1.1.1.5.cmml"><mi id="A4.Ex74.m1.3.3.1.1.1.5.2" xref="A4.Ex74.m1.3.3.1.1.1.5.2.cmml">𝒅</mi><mi id="A4.Ex74.m1.3.3.1.1.1.5.3" mathvariant="normal" xref="A4.Ex74.m1.3.3.1.1.1.5.3.cmml">f</mi></msub></mrow><mo id="A4.Ex74.m1.3.3.1.1.3a" xref="A4.Ex74.m1.3.3.1.1.3.cmml">−</mo><mrow id="A4.Ex74.m1.3.3.1.1.2" xref="A4.Ex74.m1.3.3.1.1.2.cmml"><mi id="A4.Ex74.m1.3.3.1.1.2.3" xref="A4.Ex74.m1.3.3.1.1.2.3.cmml">κ</mi><mo id="A4.Ex74.m1.3.3.1.1.2.2" xref="A4.Ex74.m1.3.3.1.1.2.2.cmml">⁢</mo><mrow id="A4.Ex74.m1.3.3.1.1.2.1.1" xref="A4.Ex74.m1.3.3.1.1.2.1.1.1.cmml"><mo id="A4.Ex74.m1.3.3.1.1.2.1.1.2" stretchy="false" xref="A4.Ex74.m1.3.3.1.1.2.1.1.1.cmml">(</mo><mrow id="A4.Ex74.m1.3.3.1.1.2.1.1.1" xref="A4.Ex74.m1.3.3.1.1.2.1.1.1.cmml"><mn id="A4.Ex74.m1.3.3.1.1.2.1.1.1.2" xref="A4.Ex74.m1.3.3.1.1.2.1.1.1.2.cmml">1</mn><mo id="A4.Ex74.m1.3.3.1.1.2.1.1.1.1" xref="A4.Ex74.m1.3.3.1.1.2.1.1.1.1.cmml">+</mo><mi id="A4.Ex74.m1.3.3.1.1.2.1.1.1.3" xref="A4.Ex74.m1.3.3.1.1.2.1.1.1.3.cmml">p</mi></mrow><mo id="A4.Ex74.m1.3.3.1.1.2.1.1.3" stretchy="false" xref="A4.Ex74.m1.3.3.1.1.2.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex74.m1.3.3.1.1.2.2a" xref="A4.Ex74.m1.3.3.1.1.2.2.cmml">⁢</mo><msubsup id="A4.Ex74.m1.3.3.1.1.2.4" xref="A4.Ex74.m1.3.3.1.1.2.4.cmml"><mover accent="true" id="A4.Ex74.m1.3.3.1.1.2.4.2.2" xref="A4.Ex74.m1.3.3.1.1.2.4.2.2.cmml"><mi id="A4.Ex74.m1.3.3.1.1.2.4.2.2.2" xref="A4.Ex74.m1.3.3.1.1.2.4.2.2.2.cmml">𝜽</mi><mo id="A4.Ex74.m1.3.3.1.1.2.4.2.2.1" xref="A4.Ex74.m1.3.3.1.1.2.4.2.2.1.cmml">~</mo></mover><mi id="A4.Ex74.m1.3.3.1.1.2.4.2.3" xref="A4.Ex74.m1.3.3.1.1.2.4.2.3.cmml">ζ</mi><mi id="A4.Ex74.m1.3.3.1.1.2.4.3" xref="A4.Ex74.m1.3.3.1.1.2.4.3.cmml">T</mi></msubsup><mo id="A4.Ex74.m1.3.3.1.1.2.2b" xref="A4.Ex74.m1.3.3.1.1.2.2.cmml">⁢</mo><msub id="A4.Ex74.m1.3.3.1.1.2.5" xref="A4.Ex74.m1.3.3.1.1.2.5.cmml"><mi id="A4.Ex74.m1.3.3.1.1.2.5.2" xref="A4.Ex74.m1.3.3.1.1.2.5.2.cmml">𝒅</mi><mi id="A4.Ex74.m1.3.3.1.1.2.5.3" mathvariant="normal" xref="A4.Ex74.m1.3.3.1.1.2.5.3.cmml">g</mi></msub></mrow></mrow><mo id="A4.Ex74.m1.3.3.1.2" lspace="0em" xref="A4.Ex74.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex74.m1.3b"><apply id="A4.Ex74.m1.3.3.1.1.cmml" xref="A4.Ex74.m1.3.3.1"><minus id="A4.Ex74.m1.3.3.1.1.3.cmml" xref="A4.Ex74.m1.3.3.1.1.3"></minus><apply id="A4.Ex74.m1.3.3.1.1.4.cmml" xref="A4.Ex74.m1.3.3.1.1.4"><plus id="A4.Ex74.m1.3.3.1.1.4.1.cmml" xref="A4.Ex74.m1.3.3.1.1.4"></plus><apply id="A4.Ex74.m1.3.3.1.1.4.2.cmml" xref="A4.Ex74.m1.3.3.1.1.4.2"><times id="A4.Ex74.m1.3.3.1.1.4.2.1.cmml" xref="A4.Ex74.m1.3.3.1.1.4.2.1"></times><apply id="A4.Ex74.m1.3.3.1.1.4.2.2.cmml" xref="A4.Ex74.m1.3.3.1.1.4.2.2"><csymbol cd="ambiguous" id="A4.Ex74.m1.3.3.1.1.4.2.2.1.cmml" xref="A4.Ex74.m1.3.3.1.1.4.2.2">superscript</csymbol><ci id="A4.Ex74.m1.3.3.1.1.4.2.2.2.cmml" xref="A4.Ex74.m1.3.3.1.1.4.2.2.2">𝒆</ci><ci id="A4.Ex74.m1.3.3.1.1.4.2.2.3.cmml" xref="A4.Ex74.m1.3.3.1.1.4.2.2.3">𝑇</ci></apply><ci id="A4.Ex74.m1.3.3.1.1.4.2.3.cmml" xref="A4.Ex74.m1.3.3.1.1.4.2.3">𝒅</ci></apply></apply><apply id="A4.Ex74.m1.3.3.1.1.1.cmml" xref="A4.Ex74.m1.3.3.1.1.1"><times id="A4.Ex74.m1.3.3.1.1.1.2.cmml" xref="A4.Ex74.m1.3.3.1.1.1.2"></times><apply id="A4.Ex74.m1.3.3.1.1.1.1.1.1.cmml" xref="A4.Ex74.m1.3.3.1.1.1.1.1"><plus id="A4.Ex74.m1.3.3.1.1.1.1.1.1.1.cmml" xref="A4.Ex74.m1.3.3.1.1.1.1.1.1.1"></plus><cn id="A4.Ex74.m1.3.3.1.1.1.1.1.1.2.cmml" type="integer" xref="A4.Ex74.m1.3.3.1.1.1.1.1.1.2">1</cn><ci id="A4.Ex74.m1.3.3.1.1.1.1.1.1.3.cmml" xref="A4.Ex74.m1.3.3.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="A4.Ex74.m1.3.3.1.1.1.3.cmml" xref="A4.Ex74.m1.3.3.1.1.1.3"><csymbol cd="ambiguous" id="A4.Ex74.m1.3.3.1.1.1.3.1.cmml" xref="A4.Ex74.m1.3.3.1.1.1.3">superscript</csymbol><apply id="A4.Ex74.m1.3.3.1.1.1.3.2.cmml" xref="A4.Ex74.m1.3.3.1.1.1.3"><csymbol cd="ambiguous" id="A4.Ex74.m1.3.3.1.1.1.3.2.1.cmml" xref="A4.Ex74.m1.3.3.1.1.1.3">subscript</csymbol><apply id="A4.Ex74.m1.3.3.1.1.1.3.2.2.cmml" xref="A4.Ex74.m1.3.3.1.1.1.3.2.2"><ci id="A4.Ex74.m1.3.3.1.1.1.3.2.2.1.cmml" xref="A4.Ex74.m1.3.3.1.1.1.3.2.2.1">~</ci><ci id="A4.Ex74.m1.3.3.1.1.1.3.2.2.2.cmml" xref="A4.Ex74.m1.3.3.1.1.1.3.2.2.2">𝜽</ci></apply><ci id="A4.Ex74.m1.3.3.1.1.1.3.2.3.cmml" xref="A4.Ex74.m1.3.3.1.1.1.3.2.3">𝜁</ci></apply><ci id="A4.Ex74.m1.3.3.1.1.1.3.3.cmml" xref="A4.Ex74.m1.3.3.1.1.1.3.3">𝑇</ci></apply><apply id="A4.Ex74.m1.3.3.1.1.1.4.cmml" xref="A4.Ex74.m1.3.3.1.1.1.4"><csymbol cd="ambiguous" id="A4.Ex74.m1.3.3.1.1.1.4.1.cmml" xref="A4.Ex74.m1.3.3.1.1.1.4">subscript</csymbol><ci id="A4.Ex74.m1.3.3.1.1.1.4.2.cmml" xref="A4.Ex74.m1.3.3.1.1.1.4.2">Φ</ci><list id="A4.Ex74.m1.2.2.2.3.cmml" xref="A4.Ex74.m1.2.2.2.4"><ci id="A4.Ex74.m1.1.1.1.1.cmml" xref="A4.Ex74.m1.1.1.1.1">f</ci><ci id="A4.Ex74.m1.2.2.2.2.cmml" xref="A4.Ex74.m1.2.2.2.2">𝜁</ci></list></apply><apply id="A4.Ex74.m1.3.3.1.1.1.5.cmml" xref="A4.Ex74.m1.3.3.1.1.1.5"><csymbol cd="ambiguous" id="A4.Ex74.m1.3.3.1.1.1.5.1.cmml" xref="A4.Ex74.m1.3.3.1.1.1.5">subscript</csymbol><ci id="A4.Ex74.m1.3.3.1.1.1.5.2.cmml" xref="A4.Ex74.m1.3.3.1.1.1.5.2">𝒅</ci><ci id="A4.Ex74.m1.3.3.1.1.1.5.3.cmml" xref="A4.Ex74.m1.3.3.1.1.1.5.3">f</ci></apply></apply><apply id="A4.Ex74.m1.3.3.1.1.2.cmml" xref="A4.Ex74.m1.3.3.1.1.2"><times id="A4.Ex74.m1.3.3.1.1.2.2.cmml" xref="A4.Ex74.m1.3.3.1.1.2.2"></times><ci id="A4.Ex74.m1.3.3.1.1.2.3.cmml" xref="A4.Ex74.m1.3.3.1.1.2.3">𝜅</ci><apply id="A4.Ex74.m1.3.3.1.1.2.1.1.1.cmml" xref="A4.Ex74.m1.3.3.1.1.2.1.1"><plus id="A4.Ex74.m1.3.3.1.1.2.1.1.1.1.cmml" xref="A4.Ex74.m1.3.3.1.1.2.1.1.1.1"></plus><cn id="A4.Ex74.m1.3.3.1.1.2.1.1.1.2.cmml" type="integer" xref="A4.Ex74.m1.3.3.1.1.2.1.1.1.2">1</cn><ci id="A4.Ex74.m1.3.3.1.1.2.1.1.1.3.cmml" xref="A4.Ex74.m1.3.3.1.1.2.1.1.1.3">𝑝</ci></apply><apply id="A4.Ex74.m1.3.3.1.1.2.4.cmml" xref="A4.Ex74.m1.3.3.1.1.2.4"><csymbol cd="ambiguous" id="A4.Ex74.m1.3.3.1.1.2.4.1.cmml" xref="A4.Ex74.m1.3.3.1.1.2.4">superscript</csymbol><apply id="A4.Ex74.m1.3.3.1.1.2.4.2.cmml" xref="A4.Ex74.m1.3.3.1.1.2.4"><csymbol cd="ambiguous" id="A4.Ex74.m1.3.3.1.1.2.4.2.1.cmml" xref="A4.Ex74.m1.3.3.1.1.2.4">subscript</csymbol><apply id="A4.Ex74.m1.3.3.1.1.2.4.2.2.cmml" xref="A4.Ex74.m1.3.3.1.1.2.4.2.2"><ci id="A4.Ex74.m1.3.3.1.1.2.4.2.2.1.cmml" xref="A4.Ex74.m1.3.3.1.1.2.4.2.2.1">~</ci><ci id="A4.Ex74.m1.3.3.1.1.2.4.2.2.2.cmml" xref="A4.Ex74.m1.3.3.1.1.2.4.2.2.2">𝜽</ci></apply><ci id="A4.Ex74.m1.3.3.1.1.2.4.2.3.cmml" xref="A4.Ex74.m1.3.3.1.1.2.4.2.3">𝜁</ci></apply><ci id="A4.Ex74.m1.3.3.1.1.2.4.3.cmml" xref="A4.Ex74.m1.3.3.1.1.2.4.3">𝑇</ci></apply><apply id="A4.Ex74.m1.3.3.1.1.2.5.cmml" xref="A4.Ex74.m1.3.3.1.1.2.5"><csymbol cd="ambiguous" id="A4.Ex74.m1.3.3.1.1.2.5.1.cmml" xref="A4.Ex74.m1.3.3.1.1.2.5">subscript</csymbol><ci id="A4.Ex74.m1.3.3.1.1.2.5.2.cmml" xref="A4.Ex74.m1.3.3.1.1.2.5.2">𝒅</ci><ci id="A4.Ex74.m1.3.3.1.1.2.5.3.cmml" xref="A4.Ex74.m1.3.3.1.1.2.5.3">g</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex74.m1.3c">\displaystyle+\bm{e}^{T}\bm{d}-(1+p){\tilde{\bm{\theta}}}_{\zeta}^{T}\Phi_{{% \rm f},\zeta}{\bm{d}}_{\rm f}-\kappa(1+p){\tilde{\bm{\theta}}}_{\zeta}^{T}{\bm% {d}}_{\rm g}.</annotation><annotation encoding="application/x-llamapun" id="A4.Ex74.m1.3d">+ bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_d - ( 1 + italic_p ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_f , italic_ζ end_POSTSUBSCRIPT bold_italic_d start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT - italic_κ ( 1 + italic_p ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_d start_POSTSUBSCRIPT roman_g end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.1.p1.40">Noting <math alttext="\|\bm{d}\|" class="ltx_Math" display="inline" id="A4.1.p1.36.m1.1"><semantics id="A4.1.p1.36.m1.1a"><mrow id="A4.1.p1.36.m1.1.2.2" xref="A4.1.p1.36.m1.1.2.1.cmml"><mo id="A4.1.p1.36.m1.1.2.2.1" stretchy="false" xref="A4.1.p1.36.m1.1.2.1.1.cmml">‖</mo><mi id="A4.1.p1.36.m1.1.1" xref="A4.1.p1.36.m1.1.1.cmml">𝒅</mi><mo id="A4.1.p1.36.m1.1.2.2.2" stretchy="false" xref="A4.1.p1.36.m1.1.2.1.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.36.m1.1b"><apply id="A4.1.p1.36.m1.1.2.1.cmml" xref="A4.1.p1.36.m1.1.2.2"><csymbol cd="latexml" id="A4.1.p1.36.m1.1.2.1.1.cmml" xref="A4.1.p1.36.m1.1.2.2.1">norm</csymbol><ci id="A4.1.p1.36.m1.1.1.cmml" xref="A4.1.p1.36.m1.1.1">𝒅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.36.m1.1c">\|\bm{d}\|</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.36.m1.1d">∥ bold_italic_d ∥</annotation></semantics></math>, <math alttext="\|\bm{d}_{\rm f}\|\leq\bar{d}" class="ltx_Math" display="inline" id="A4.1.p1.37.m2.1"><semantics id="A4.1.p1.37.m2.1a"><mrow id="A4.1.p1.37.m2.1.1" xref="A4.1.p1.37.m2.1.1.cmml"><mrow id="A4.1.p1.37.m2.1.1.1.1" xref="A4.1.p1.37.m2.1.1.1.2.cmml"><mo id="A4.1.p1.37.m2.1.1.1.1.2" stretchy="false" xref="A4.1.p1.37.m2.1.1.1.2.1.cmml">‖</mo><msub id="A4.1.p1.37.m2.1.1.1.1.1" xref="A4.1.p1.37.m2.1.1.1.1.1.cmml"><mi id="A4.1.p1.37.m2.1.1.1.1.1.2" xref="A4.1.p1.37.m2.1.1.1.1.1.2.cmml">𝒅</mi><mi id="A4.1.p1.37.m2.1.1.1.1.1.3" mathvariant="normal" xref="A4.1.p1.37.m2.1.1.1.1.1.3.cmml">f</mi></msub><mo id="A4.1.p1.37.m2.1.1.1.1.3" stretchy="false" xref="A4.1.p1.37.m2.1.1.1.2.1.cmml">‖</mo></mrow><mo id="A4.1.p1.37.m2.1.1.2" xref="A4.1.p1.37.m2.1.1.2.cmml">≤</mo><mover accent="true" id="A4.1.p1.37.m2.1.1.3" xref="A4.1.p1.37.m2.1.1.3.cmml"><mi id="A4.1.p1.37.m2.1.1.3.2" xref="A4.1.p1.37.m2.1.1.3.2.cmml">d</mi><mo id="A4.1.p1.37.m2.1.1.3.1" xref="A4.1.p1.37.m2.1.1.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.37.m2.1b"><apply id="A4.1.p1.37.m2.1.1.cmml" xref="A4.1.p1.37.m2.1.1"><leq id="A4.1.p1.37.m2.1.1.2.cmml" xref="A4.1.p1.37.m2.1.1.2"></leq><apply id="A4.1.p1.37.m2.1.1.1.2.cmml" xref="A4.1.p1.37.m2.1.1.1.1"><csymbol cd="latexml" id="A4.1.p1.37.m2.1.1.1.2.1.cmml" xref="A4.1.p1.37.m2.1.1.1.1.2">norm</csymbol><apply id="A4.1.p1.37.m2.1.1.1.1.1.cmml" xref="A4.1.p1.37.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.1.p1.37.m2.1.1.1.1.1.1.cmml" xref="A4.1.p1.37.m2.1.1.1.1.1">subscript</csymbol><ci id="A4.1.p1.37.m2.1.1.1.1.1.2.cmml" xref="A4.1.p1.37.m2.1.1.1.1.1.2">𝒅</ci><ci id="A4.1.p1.37.m2.1.1.1.1.1.3.cmml" xref="A4.1.p1.37.m2.1.1.1.1.1.3">f</ci></apply></apply><apply id="A4.1.p1.37.m2.1.1.3.cmml" xref="A4.1.p1.37.m2.1.1.3"><ci id="A4.1.p1.37.m2.1.1.3.1.cmml" xref="A4.1.p1.37.m2.1.1.3.1">¯</ci><ci id="A4.1.p1.37.m2.1.1.3.2.cmml" xref="A4.1.p1.37.m2.1.1.3.2">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.37.m2.1c">\|\bm{d}_{\rm f}\|\leq\bar{d}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.37.m2.1d">∥ bold_italic_d start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ∥ ≤ over¯ start_ARG italic_d end_ARG</annotation></semantics></math>, and <math alttext="\|{\bm{d}}_{\rm g}(t)\|" class="ltx_Math" display="inline" id="A4.1.p1.38.m3.2"><semantics id="A4.1.p1.38.m3.2a"><mrow id="A4.1.p1.38.m3.2.2.1" xref="A4.1.p1.38.m3.2.2.2.cmml"><mo id="A4.1.p1.38.m3.2.2.1.2" stretchy="false" xref="A4.1.p1.38.m3.2.2.2.1.cmml">‖</mo><mrow id="A4.1.p1.38.m3.2.2.1.1" xref="A4.1.p1.38.m3.2.2.1.1.cmml"><msub id="A4.1.p1.38.m3.2.2.1.1.2" xref="A4.1.p1.38.m3.2.2.1.1.2.cmml"><mi id="A4.1.p1.38.m3.2.2.1.1.2.2" xref="A4.1.p1.38.m3.2.2.1.1.2.2.cmml">𝒅</mi><mi id="A4.1.p1.38.m3.2.2.1.1.2.3" mathvariant="normal" xref="A4.1.p1.38.m3.2.2.1.1.2.3.cmml">g</mi></msub><mo id="A4.1.p1.38.m3.2.2.1.1.1" xref="A4.1.p1.38.m3.2.2.1.1.1.cmml">⁢</mo><mrow id="A4.1.p1.38.m3.2.2.1.1.3.2" xref="A4.1.p1.38.m3.2.2.1.1.cmml"><mo id="A4.1.p1.38.m3.2.2.1.1.3.2.1" stretchy="false" xref="A4.1.p1.38.m3.2.2.1.1.cmml">(</mo><mi id="A4.1.p1.38.m3.1.1" xref="A4.1.p1.38.m3.1.1.cmml">t</mi><mo id="A4.1.p1.38.m3.2.2.1.1.3.2.2" stretchy="false" xref="A4.1.p1.38.m3.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="A4.1.p1.38.m3.2.2.1.3" stretchy="false" xref="A4.1.p1.38.m3.2.2.2.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.38.m3.2b"><apply id="A4.1.p1.38.m3.2.2.2.cmml" xref="A4.1.p1.38.m3.2.2.1"><csymbol cd="latexml" id="A4.1.p1.38.m3.2.2.2.1.cmml" xref="A4.1.p1.38.m3.2.2.1.2">norm</csymbol><apply id="A4.1.p1.38.m3.2.2.1.1.cmml" xref="A4.1.p1.38.m3.2.2.1.1"><times id="A4.1.p1.38.m3.2.2.1.1.1.cmml" xref="A4.1.p1.38.m3.2.2.1.1.1"></times><apply id="A4.1.p1.38.m3.2.2.1.1.2.cmml" xref="A4.1.p1.38.m3.2.2.1.1.2"><csymbol cd="ambiguous" id="A4.1.p1.38.m3.2.2.1.1.2.1.cmml" xref="A4.1.p1.38.m3.2.2.1.1.2">subscript</csymbol><ci id="A4.1.p1.38.m3.2.2.1.1.2.2.cmml" xref="A4.1.p1.38.m3.2.2.1.1.2.2">𝒅</ci><ci id="A4.1.p1.38.m3.2.2.1.1.2.3.cmml" xref="A4.1.p1.38.m3.2.2.1.1.2.3">g</ci></apply><ci id="A4.1.p1.38.m3.1.1.cmml" xref="A4.1.p1.38.m3.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.38.m3.2c">\|{\bm{d}}_{\rm g}(t)\|</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.38.m3.2d">∥ bold_italic_d start_POSTSUBSCRIPT roman_g end_POSTSUBSCRIPT ( italic_t ) ∥</annotation></semantics></math> <math alttext="\leq" class="ltx_Math" display="inline" id="A4.1.p1.39.m4.1"><semantics id="A4.1.p1.39.m4.1a"><mo id="A4.1.p1.39.m4.1.1" xref="A4.1.p1.39.m4.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="A4.1.p1.39.m4.1b"><leq id="A4.1.p1.39.m4.1.1.cmml" xref="A4.1.p1.39.m4.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.39.m4.1c">\leq</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.39.m4.1d">≤</annotation></semantics></math> <math alttext="{\bar{d}}_{\rm g}" class="ltx_Math" display="inline" id="A4.1.p1.40.m5.1"><semantics id="A4.1.p1.40.m5.1a"><msub id="A4.1.p1.40.m5.1.1" xref="A4.1.p1.40.m5.1.1.cmml"><mover accent="true" id="A4.1.p1.40.m5.1.1.2" xref="A4.1.p1.40.m5.1.1.2.cmml"><mi id="A4.1.p1.40.m5.1.1.2.2" xref="A4.1.p1.40.m5.1.1.2.2.cmml">d</mi><mo id="A4.1.p1.40.m5.1.1.2.1" xref="A4.1.p1.40.m5.1.1.2.1.cmml">¯</mo></mover><mi id="A4.1.p1.40.m5.1.1.3" mathvariant="normal" xref="A4.1.p1.40.m5.1.1.3.cmml">g</mi></msub><annotation-xml encoding="MathML-Content" id="A4.1.p1.40.m5.1b"><apply id="A4.1.p1.40.m5.1.1.cmml" xref="A4.1.p1.40.m5.1.1"><csymbol cd="ambiguous" id="A4.1.p1.40.m5.1.1.1.cmml" xref="A4.1.p1.40.m5.1.1">subscript</csymbol><apply id="A4.1.p1.40.m5.1.1.2.cmml" xref="A4.1.p1.40.m5.1.1.2"><ci id="A4.1.p1.40.m5.1.1.2.1.cmml" xref="A4.1.p1.40.m5.1.1.2.1">¯</ci><ci id="A4.1.p1.40.m5.1.1.2.2.cmml" xref="A4.1.p1.40.m5.1.1.2.2">𝑑</ci></apply><ci id="A4.1.p1.40.m5.1.1.3.cmml" xref="A4.1.p1.40.m5.1.1.3">g</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.40.m5.1c">{\bar{d}}_{\rm g}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.40.m5.1d">over¯ start_ARG italic_d end_ARG start_POSTSUBSCRIPT roman_g end_POSTSUBSCRIPT</annotation></semantics></math>, one obtains</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx82"> <tbody id="A4.Ex75"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}_{\zeta}\leq" class="ltx_Math" display="inline" id="A4.Ex75.m1.1"><semantics id="A4.Ex75.m1.1a"><mrow id="A4.Ex75.m1.1.1" xref="A4.Ex75.m1.1.1.cmml"><msub id="A4.Ex75.m1.1.1.2" xref="A4.Ex75.m1.1.1.2.cmml"><mover accent="true" id="A4.Ex75.m1.1.1.2.2" xref="A4.Ex75.m1.1.1.2.2.cmml"><mi id="A4.Ex75.m1.1.1.2.2.2" xref="A4.Ex75.m1.1.1.2.2.2.cmml">V</mi><mo id="A4.Ex75.m1.1.1.2.2.1" xref="A4.Ex75.m1.1.1.2.2.1.cmml">˙</mo></mover><mi id="A4.Ex75.m1.1.1.2.3" xref="A4.Ex75.m1.1.1.2.3.cmml">ζ</mi></msub><mo id="A4.Ex75.m1.1.1.1" xref="A4.Ex75.m1.1.1.1.cmml">≤</mo><mi id="A4.Ex75.m1.1.1.3" xref="A4.Ex75.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex75.m1.1b"><apply id="A4.Ex75.m1.1.1.cmml" xref="A4.Ex75.m1.1.1"><leq id="A4.Ex75.m1.1.1.1.cmml" xref="A4.Ex75.m1.1.1.1"></leq><apply id="A4.Ex75.m1.1.1.2.cmml" xref="A4.Ex75.m1.1.1.2"><csymbol cd="ambiguous" id="A4.Ex75.m1.1.1.2.1.cmml" xref="A4.Ex75.m1.1.1.2">subscript</csymbol><apply id="A4.Ex75.m1.1.1.2.2.cmml" xref="A4.Ex75.m1.1.1.2.2"><ci id="A4.Ex75.m1.1.1.2.2.1.cmml" xref="A4.Ex75.m1.1.1.2.2.1">˙</ci><ci id="A4.Ex75.m1.1.1.2.2.2.cmml" xref="A4.Ex75.m1.1.1.2.2.2">𝑉</ci></apply><ci id="A4.Ex75.m1.1.1.2.3.cmml" xref="A4.Ex75.m1.1.1.2.3">𝜁</ci></apply><csymbol cd="latexml" id="A4.Ex75.m1.1.1.3.cmml" xref="A4.Ex75.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex75.m1.1c">\displaystyle\dot{V}_{\zeta}\leq</annotation><annotation encoding="application/x-llamapun" id="A4.Ex75.m1.1d">over˙ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-k_{\rm c}\|\bm{e}\|^{2}-\kappa(1+p)\sigma^{*}\|\tilde{\bm{\theta% }}_{\zeta}\|^{2}+\delta\|\bm{e}\|\|\tilde{\bm{\theta}}_{\zeta}\|" class="ltx_Math" display="inline" id="A4.Ex75.m2.5"><semantics id="A4.Ex75.m2.5a"><mrow id="A4.Ex75.m2.5.5" xref="A4.Ex75.m2.5.5.cmml"><mrow id="A4.Ex75.m2.4.4.2" xref="A4.Ex75.m2.4.4.2.cmml"><mrow id="A4.Ex75.m2.4.4.2.4" xref="A4.Ex75.m2.4.4.2.4.cmml"><mo id="A4.Ex75.m2.4.4.2.4a" xref="A4.Ex75.m2.4.4.2.4.cmml">−</mo><mrow id="A4.Ex75.m2.4.4.2.4.2" xref="A4.Ex75.m2.4.4.2.4.2.cmml"><msub id="A4.Ex75.m2.4.4.2.4.2.2" xref="A4.Ex75.m2.4.4.2.4.2.2.cmml"><mi id="A4.Ex75.m2.4.4.2.4.2.2.2" xref="A4.Ex75.m2.4.4.2.4.2.2.2.cmml">k</mi><mi id="A4.Ex75.m2.4.4.2.4.2.2.3" mathvariant="normal" xref="A4.Ex75.m2.4.4.2.4.2.2.3.cmml">c</mi></msub><mo id="A4.Ex75.m2.4.4.2.4.2.1" xref="A4.Ex75.m2.4.4.2.4.2.1.cmml">⁢</mo><msup id="A4.Ex75.m2.4.4.2.4.2.3" xref="A4.Ex75.m2.4.4.2.4.2.3.cmml"><mrow id="A4.Ex75.m2.4.4.2.4.2.3.2.2" xref="A4.Ex75.m2.4.4.2.4.2.3.2.1.cmml"><mo id="A4.Ex75.m2.4.4.2.4.2.3.2.2.1" stretchy="false" xref="A4.Ex75.m2.4.4.2.4.2.3.2.1.1.cmml">‖</mo><mi id="A4.Ex75.m2.1.1" xref="A4.Ex75.m2.1.1.cmml">𝒆</mi><mo id="A4.Ex75.m2.4.4.2.4.2.3.2.2.2" stretchy="false" xref="A4.Ex75.m2.4.4.2.4.2.3.2.1.1.cmml">‖</mo></mrow><mn id="A4.Ex75.m2.4.4.2.4.2.3.3" xref="A4.Ex75.m2.4.4.2.4.2.3.3.cmml">2</mn></msup></mrow></mrow><mo id="A4.Ex75.m2.4.4.2.3" xref="A4.Ex75.m2.4.4.2.3.cmml">−</mo><mrow id="A4.Ex75.m2.4.4.2.2" xref="A4.Ex75.m2.4.4.2.2.cmml"><mi id="A4.Ex75.m2.4.4.2.2.4" xref="A4.Ex75.m2.4.4.2.2.4.cmml">κ</mi><mo id="A4.Ex75.m2.4.4.2.2.3" xref="A4.Ex75.m2.4.4.2.2.3.cmml">⁢</mo><mrow id="A4.Ex75.m2.3.3.1.1.1.1" xref="A4.Ex75.m2.3.3.1.1.1.1.1.cmml"><mo id="A4.Ex75.m2.3.3.1.1.1.1.2" stretchy="false" xref="A4.Ex75.m2.3.3.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex75.m2.3.3.1.1.1.1.1" xref="A4.Ex75.m2.3.3.1.1.1.1.1.cmml"><mn id="A4.Ex75.m2.3.3.1.1.1.1.1.2" xref="A4.Ex75.m2.3.3.1.1.1.1.1.2.cmml">1</mn><mo id="A4.Ex75.m2.3.3.1.1.1.1.1.1" xref="A4.Ex75.m2.3.3.1.1.1.1.1.1.cmml">+</mo><mi id="A4.Ex75.m2.3.3.1.1.1.1.1.3" xref="A4.Ex75.m2.3.3.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A4.Ex75.m2.3.3.1.1.1.1.3" stretchy="false" xref="A4.Ex75.m2.3.3.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex75.m2.4.4.2.2.3a" xref="A4.Ex75.m2.4.4.2.2.3.cmml">⁢</mo><msup id="A4.Ex75.m2.4.4.2.2.5" xref="A4.Ex75.m2.4.4.2.2.5.cmml"><mi id="A4.Ex75.m2.4.4.2.2.5.2" xref="A4.Ex75.m2.4.4.2.2.5.2.cmml">σ</mi><mo id="A4.Ex75.m2.4.4.2.2.5.3" xref="A4.Ex75.m2.4.4.2.2.5.3.cmml">∗</mo></msup><mo id="A4.Ex75.m2.4.4.2.2.3b" xref="A4.Ex75.m2.4.4.2.2.3.cmml">⁢</mo><msup id="A4.Ex75.m2.4.4.2.2.2" xref="A4.Ex75.m2.4.4.2.2.2.cmml"><mrow id="A4.Ex75.m2.4.4.2.2.2.1.1" xref="A4.Ex75.m2.4.4.2.2.2.1.2.cmml"><mo id="A4.Ex75.m2.4.4.2.2.2.1.1.2" stretchy="false" xref="A4.Ex75.m2.4.4.2.2.2.1.2.1.cmml">‖</mo><msub id="A4.Ex75.m2.4.4.2.2.2.1.1.1" xref="A4.Ex75.m2.4.4.2.2.2.1.1.1.cmml"><mover accent="true" id="A4.Ex75.m2.4.4.2.2.2.1.1.1.2" xref="A4.Ex75.m2.4.4.2.2.2.1.1.1.2.cmml"><mi id="A4.Ex75.m2.4.4.2.2.2.1.1.1.2.2" xref="A4.Ex75.m2.4.4.2.2.2.1.1.1.2.2.cmml">𝜽</mi><mo id="A4.Ex75.m2.4.4.2.2.2.1.1.1.2.1" xref="A4.Ex75.m2.4.4.2.2.2.1.1.1.2.1.cmml">~</mo></mover><mi id="A4.Ex75.m2.4.4.2.2.2.1.1.1.3" xref="A4.Ex75.m2.4.4.2.2.2.1.1.1.3.cmml">ζ</mi></msub><mo id="A4.Ex75.m2.4.4.2.2.2.1.1.3" stretchy="false" xref="A4.Ex75.m2.4.4.2.2.2.1.2.1.cmml">‖</mo></mrow><mn id="A4.Ex75.m2.4.4.2.2.2.3" xref="A4.Ex75.m2.4.4.2.2.2.3.cmml">2</mn></msup></mrow></mrow><mo id="A4.Ex75.m2.5.5.4" xref="A4.Ex75.m2.5.5.4.cmml">+</mo><mrow id="A4.Ex75.m2.5.5.3" xref="A4.Ex75.m2.5.5.3.cmml"><mi id="A4.Ex75.m2.5.5.3.3" xref="A4.Ex75.m2.5.5.3.3.cmml">δ</mi><mo id="A4.Ex75.m2.5.5.3.2" xref="A4.Ex75.m2.5.5.3.2.cmml">⁢</mo><mrow id="A4.Ex75.m2.5.5.3.4.2" xref="A4.Ex75.m2.5.5.3.4.1.cmml"><mo id="A4.Ex75.m2.5.5.3.4.2.1" stretchy="false" xref="A4.Ex75.m2.5.5.3.4.1.1.cmml">‖</mo><mi id="A4.Ex75.m2.2.2" xref="A4.Ex75.m2.2.2.cmml">𝒆</mi><mo id="A4.Ex75.m2.5.5.3.4.2.2" stretchy="false" xref="A4.Ex75.m2.5.5.3.4.1.1.cmml">‖</mo></mrow><mo id="A4.Ex75.m2.5.5.3.2a" xref="A4.Ex75.m2.5.5.3.2.cmml">⁢</mo><mrow id="A4.Ex75.m2.5.5.3.1.1" xref="A4.Ex75.m2.5.5.3.1.2.cmml"><mo id="A4.Ex75.m2.5.5.3.1.1.2" stretchy="false" xref="A4.Ex75.m2.5.5.3.1.2.1.cmml">‖</mo><msub id="A4.Ex75.m2.5.5.3.1.1.1" xref="A4.Ex75.m2.5.5.3.1.1.1.cmml"><mover accent="true" id="A4.Ex75.m2.5.5.3.1.1.1.2" xref="A4.Ex75.m2.5.5.3.1.1.1.2.cmml"><mi id="A4.Ex75.m2.5.5.3.1.1.1.2.2" xref="A4.Ex75.m2.5.5.3.1.1.1.2.2.cmml">𝜽</mi><mo id="A4.Ex75.m2.5.5.3.1.1.1.2.1" xref="A4.Ex75.m2.5.5.3.1.1.1.2.1.cmml">~</mo></mover><mi id="A4.Ex75.m2.5.5.3.1.1.1.3" xref="A4.Ex75.m2.5.5.3.1.1.1.3.cmml">ζ</mi></msub><mo id="A4.Ex75.m2.5.5.3.1.1.3" stretchy="false" xref="A4.Ex75.m2.5.5.3.1.2.1.cmml">‖</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex75.m2.5b"><apply id="A4.Ex75.m2.5.5.cmml" xref="A4.Ex75.m2.5.5"><plus id="A4.Ex75.m2.5.5.4.cmml" xref="A4.Ex75.m2.5.5.4"></plus><apply id="A4.Ex75.m2.4.4.2.cmml" xref="A4.Ex75.m2.4.4.2"><minus id="A4.Ex75.m2.4.4.2.3.cmml" xref="A4.Ex75.m2.4.4.2.3"></minus><apply id="A4.Ex75.m2.4.4.2.4.cmml" xref="A4.Ex75.m2.4.4.2.4"><minus id="A4.Ex75.m2.4.4.2.4.1.cmml" xref="A4.Ex75.m2.4.4.2.4"></minus><apply id="A4.Ex75.m2.4.4.2.4.2.cmml" xref="A4.Ex75.m2.4.4.2.4.2"><times id="A4.Ex75.m2.4.4.2.4.2.1.cmml" xref="A4.Ex75.m2.4.4.2.4.2.1"></times><apply id="A4.Ex75.m2.4.4.2.4.2.2.cmml" xref="A4.Ex75.m2.4.4.2.4.2.2"><csymbol cd="ambiguous" id="A4.Ex75.m2.4.4.2.4.2.2.1.cmml" xref="A4.Ex75.m2.4.4.2.4.2.2">subscript</csymbol><ci id="A4.Ex75.m2.4.4.2.4.2.2.2.cmml" xref="A4.Ex75.m2.4.4.2.4.2.2.2">𝑘</ci><ci id="A4.Ex75.m2.4.4.2.4.2.2.3.cmml" xref="A4.Ex75.m2.4.4.2.4.2.2.3">c</ci></apply><apply id="A4.Ex75.m2.4.4.2.4.2.3.cmml" xref="A4.Ex75.m2.4.4.2.4.2.3"><csymbol cd="ambiguous" id="A4.Ex75.m2.4.4.2.4.2.3.1.cmml" xref="A4.Ex75.m2.4.4.2.4.2.3">superscript</csymbol><apply id="A4.Ex75.m2.4.4.2.4.2.3.2.1.cmml" xref="A4.Ex75.m2.4.4.2.4.2.3.2.2"><csymbol cd="latexml" id="A4.Ex75.m2.4.4.2.4.2.3.2.1.1.cmml" xref="A4.Ex75.m2.4.4.2.4.2.3.2.2.1">norm</csymbol><ci id="A4.Ex75.m2.1.1.cmml" xref="A4.Ex75.m2.1.1">𝒆</ci></apply><cn id="A4.Ex75.m2.4.4.2.4.2.3.3.cmml" type="integer" xref="A4.Ex75.m2.4.4.2.4.2.3.3">2</cn></apply></apply></apply><apply id="A4.Ex75.m2.4.4.2.2.cmml" xref="A4.Ex75.m2.4.4.2.2"><times id="A4.Ex75.m2.4.4.2.2.3.cmml" xref="A4.Ex75.m2.4.4.2.2.3"></times><ci id="A4.Ex75.m2.4.4.2.2.4.cmml" xref="A4.Ex75.m2.4.4.2.2.4">𝜅</ci><apply id="A4.Ex75.m2.3.3.1.1.1.1.1.cmml" xref="A4.Ex75.m2.3.3.1.1.1.1"><plus id="A4.Ex75.m2.3.3.1.1.1.1.1.1.cmml" xref="A4.Ex75.m2.3.3.1.1.1.1.1.1"></plus><cn id="A4.Ex75.m2.3.3.1.1.1.1.1.2.cmml" type="integer" xref="A4.Ex75.m2.3.3.1.1.1.1.1.2">1</cn><ci id="A4.Ex75.m2.3.3.1.1.1.1.1.3.cmml" xref="A4.Ex75.m2.3.3.1.1.1.1.1.3">𝑝</ci></apply><apply id="A4.Ex75.m2.4.4.2.2.5.cmml" xref="A4.Ex75.m2.4.4.2.2.5"><csymbol cd="ambiguous" id="A4.Ex75.m2.4.4.2.2.5.1.cmml" xref="A4.Ex75.m2.4.4.2.2.5">superscript</csymbol><ci id="A4.Ex75.m2.4.4.2.2.5.2.cmml" xref="A4.Ex75.m2.4.4.2.2.5.2">𝜎</ci><times id="A4.Ex75.m2.4.4.2.2.5.3.cmml" xref="A4.Ex75.m2.4.4.2.2.5.3"></times></apply><apply id="A4.Ex75.m2.4.4.2.2.2.cmml" xref="A4.Ex75.m2.4.4.2.2.2"><csymbol cd="ambiguous" id="A4.Ex75.m2.4.4.2.2.2.2.cmml" xref="A4.Ex75.m2.4.4.2.2.2">superscript</csymbol><apply id="A4.Ex75.m2.4.4.2.2.2.1.2.cmml" xref="A4.Ex75.m2.4.4.2.2.2.1.1"><csymbol cd="latexml" id="A4.Ex75.m2.4.4.2.2.2.1.2.1.cmml" xref="A4.Ex75.m2.4.4.2.2.2.1.1.2">norm</csymbol><apply id="A4.Ex75.m2.4.4.2.2.2.1.1.1.cmml" xref="A4.Ex75.m2.4.4.2.2.2.1.1.1"><csymbol cd="ambiguous" id="A4.Ex75.m2.4.4.2.2.2.1.1.1.1.cmml" xref="A4.Ex75.m2.4.4.2.2.2.1.1.1">subscript</csymbol><apply id="A4.Ex75.m2.4.4.2.2.2.1.1.1.2.cmml" xref="A4.Ex75.m2.4.4.2.2.2.1.1.1.2"><ci id="A4.Ex75.m2.4.4.2.2.2.1.1.1.2.1.cmml" xref="A4.Ex75.m2.4.4.2.2.2.1.1.1.2.1">~</ci><ci id="A4.Ex75.m2.4.4.2.2.2.1.1.1.2.2.cmml" xref="A4.Ex75.m2.4.4.2.2.2.1.1.1.2.2">𝜽</ci></apply><ci id="A4.Ex75.m2.4.4.2.2.2.1.1.1.3.cmml" xref="A4.Ex75.m2.4.4.2.2.2.1.1.1.3">𝜁</ci></apply></apply><cn id="A4.Ex75.m2.4.4.2.2.2.3.cmml" type="integer" xref="A4.Ex75.m2.4.4.2.2.2.3">2</cn></apply></apply></apply><apply id="A4.Ex75.m2.5.5.3.cmml" xref="A4.Ex75.m2.5.5.3"><times id="A4.Ex75.m2.5.5.3.2.cmml" xref="A4.Ex75.m2.5.5.3.2"></times><ci id="A4.Ex75.m2.5.5.3.3.cmml" xref="A4.Ex75.m2.5.5.3.3">𝛿</ci><apply id="A4.Ex75.m2.5.5.3.4.1.cmml" xref="A4.Ex75.m2.5.5.3.4.2"><csymbol cd="latexml" id="A4.Ex75.m2.5.5.3.4.1.1.cmml" xref="A4.Ex75.m2.5.5.3.4.2.1">norm</csymbol><ci id="A4.Ex75.m2.2.2.cmml" xref="A4.Ex75.m2.2.2">𝒆</ci></apply><apply id="A4.Ex75.m2.5.5.3.1.2.cmml" xref="A4.Ex75.m2.5.5.3.1.1"><csymbol cd="latexml" id="A4.Ex75.m2.5.5.3.1.2.1.cmml" xref="A4.Ex75.m2.5.5.3.1.1.2">norm</csymbol><apply id="A4.Ex75.m2.5.5.3.1.1.1.cmml" xref="A4.Ex75.m2.5.5.3.1.1.1"><csymbol cd="ambiguous" id="A4.Ex75.m2.5.5.3.1.1.1.1.cmml" xref="A4.Ex75.m2.5.5.3.1.1.1">subscript</csymbol><apply id="A4.Ex75.m2.5.5.3.1.1.1.2.cmml" xref="A4.Ex75.m2.5.5.3.1.1.1.2"><ci id="A4.Ex75.m2.5.5.3.1.1.1.2.1.cmml" xref="A4.Ex75.m2.5.5.3.1.1.1.2.1">~</ci><ci id="A4.Ex75.m2.5.5.3.1.1.1.2.2.cmml" xref="A4.Ex75.m2.5.5.3.1.1.1.2.2">𝜽</ci></apply><ci id="A4.Ex75.m2.5.5.3.1.1.1.3.cmml" xref="A4.Ex75.m2.5.5.3.1.1.1.3">𝜁</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex75.m2.5c">\displaystyle-k_{\rm c}\|\bm{e}\|^{2}-\kappa(1+p)\sigma^{*}\|\tilde{\bm{\theta% }}_{\zeta}\|^{2}+\delta\|\bm{e}\|\|\tilde{\bm{\theta}}_{\zeta}\|</annotation><annotation encoding="application/x-llamapun" id="A4.Ex75.m2.5d">- italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ∥ bold_italic_e ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_κ ( 1 + italic_p ) italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_δ ∥ bold_italic_e ∥ ∥ over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ∥</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex76"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle+\bar{d}\|\bm{e}\|+(1+p)(\bar{\epsilon}\bar{d}+\kappa{\bar{d}}_{% \rm g}\|\tilde{\bm{\theta}}_{\zeta}\|)." class="ltx_Math" display="inline" id="A4.Ex76.m1.2"><semantics id="A4.Ex76.m1.2a"><mrow id="A4.Ex76.m1.2.2.1" xref="A4.Ex76.m1.2.2.1.1.cmml"><mrow id="A4.Ex76.m1.2.2.1.1" xref="A4.Ex76.m1.2.2.1.1.cmml"><mrow id="A4.Ex76.m1.2.2.1.1.4" xref="A4.Ex76.m1.2.2.1.1.4.cmml"><mo id="A4.Ex76.m1.2.2.1.1.4a" xref="A4.Ex76.m1.2.2.1.1.4.cmml">+</mo><mrow id="A4.Ex76.m1.2.2.1.1.4.2" xref="A4.Ex76.m1.2.2.1.1.4.2.cmml"><mover accent="true" id="A4.Ex76.m1.2.2.1.1.4.2.2" xref="A4.Ex76.m1.2.2.1.1.4.2.2.cmml"><mi id="A4.Ex76.m1.2.2.1.1.4.2.2.2" xref="A4.Ex76.m1.2.2.1.1.4.2.2.2.cmml">d</mi><mo id="A4.Ex76.m1.2.2.1.1.4.2.2.1" xref="A4.Ex76.m1.2.2.1.1.4.2.2.1.cmml">¯</mo></mover><mo id="A4.Ex76.m1.2.2.1.1.4.2.1" xref="A4.Ex76.m1.2.2.1.1.4.2.1.cmml">⁢</mo><mrow id="A4.Ex76.m1.2.2.1.1.4.2.3.2" xref="A4.Ex76.m1.2.2.1.1.4.2.3.1.cmml"><mo id="A4.Ex76.m1.2.2.1.1.4.2.3.2.1" stretchy="false" xref="A4.Ex76.m1.2.2.1.1.4.2.3.1.1.cmml">‖</mo><mi id="A4.Ex76.m1.1.1" xref="A4.Ex76.m1.1.1.cmml">𝒆</mi><mo id="A4.Ex76.m1.2.2.1.1.4.2.3.2.2" stretchy="false" xref="A4.Ex76.m1.2.2.1.1.4.2.3.1.1.cmml">‖</mo></mrow></mrow></mrow><mo id="A4.Ex76.m1.2.2.1.1.3" xref="A4.Ex76.m1.2.2.1.1.3.cmml">+</mo><mrow id="A4.Ex76.m1.2.2.1.1.2" xref="A4.Ex76.m1.2.2.1.1.2.cmml"><mrow id="A4.Ex76.m1.2.2.1.1.1.1.1" xref="A4.Ex76.m1.2.2.1.1.1.1.1.1.cmml"><mo id="A4.Ex76.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="A4.Ex76.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex76.m1.2.2.1.1.1.1.1.1" xref="A4.Ex76.m1.2.2.1.1.1.1.1.1.cmml"><mn id="A4.Ex76.m1.2.2.1.1.1.1.1.1.2" xref="A4.Ex76.m1.2.2.1.1.1.1.1.1.2.cmml">1</mn><mo id="A4.Ex76.m1.2.2.1.1.1.1.1.1.1" xref="A4.Ex76.m1.2.2.1.1.1.1.1.1.1.cmml">+</mo><mi id="A4.Ex76.m1.2.2.1.1.1.1.1.1.3" xref="A4.Ex76.m1.2.2.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A4.Ex76.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="A4.Ex76.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex76.m1.2.2.1.1.2.3" xref="A4.Ex76.m1.2.2.1.1.2.3.cmml">⁢</mo><mrow id="A4.Ex76.m1.2.2.1.1.2.2.1" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.cmml"><mo id="A4.Ex76.m1.2.2.1.1.2.2.1.2" stretchy="false" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.cmml">(</mo><mrow id="A4.Ex76.m1.2.2.1.1.2.2.1.1" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.cmml"><mrow id="A4.Ex76.m1.2.2.1.1.2.2.1.1.3" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.cmml"><mover accent="true" id="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.2" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.2.cmml"><mi id="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.2.2" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.2.2.cmml">ϵ</mi><mo id="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.2.1" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.2.1.cmml">¯</mo></mover><mo id="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.1" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.1.cmml">⁢</mo><mover accent="true" id="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.3" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.3.cmml"><mi id="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.3.2" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.3.2.cmml">d</mi><mo id="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.3.1" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.3.1.cmml">¯</mo></mover></mrow><mo id="A4.Ex76.m1.2.2.1.1.2.2.1.1.2" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.2.cmml">+</mo><mrow id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.cmml"><mi id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.3" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.3.cmml">κ</mi><mo id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.2" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.2.cmml">⁢</mo><msub id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.4" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.4.cmml"><mover accent="true" id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.4.2" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.4.2.cmml"><mi id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.4.2.2" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.4.2.2.cmml">d</mi><mo id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.4.2.1" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.4.2.1.cmml">¯</mo></mover><mi id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.4.3" mathvariant="normal" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.4.3.cmml">g</mi></msub><mo id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.2a" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.2.cmml"><mo id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.2" stretchy="false" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.2.1.cmml">‖</mo><msub id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.1" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.1.cmml"><mover accent="true" id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.1.2" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.1.2.cmml"><mi id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.1.2.2" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.1.2.2.cmml">𝜽</mi><mo id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.1.2.1" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.1.2.1.cmml">~</mo></mover><mi id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.1.3" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.1.3.cmml">ζ</mi></msub><mo id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.3" stretchy="false" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.2.1.cmml">‖</mo></mrow></mrow></mrow><mo id="A4.Ex76.m1.2.2.1.1.2.2.1.3" stretchy="false" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A4.Ex76.m1.2.2.1.2" lspace="0em" xref="A4.Ex76.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex76.m1.2b"><apply id="A4.Ex76.m1.2.2.1.1.cmml" xref="A4.Ex76.m1.2.2.1"><plus id="A4.Ex76.m1.2.2.1.1.3.cmml" xref="A4.Ex76.m1.2.2.1.1.3"></plus><apply id="A4.Ex76.m1.2.2.1.1.4.cmml" xref="A4.Ex76.m1.2.2.1.1.4"><plus id="A4.Ex76.m1.2.2.1.1.4.1.cmml" xref="A4.Ex76.m1.2.2.1.1.4"></plus><apply id="A4.Ex76.m1.2.2.1.1.4.2.cmml" xref="A4.Ex76.m1.2.2.1.1.4.2"><times id="A4.Ex76.m1.2.2.1.1.4.2.1.cmml" xref="A4.Ex76.m1.2.2.1.1.4.2.1"></times><apply id="A4.Ex76.m1.2.2.1.1.4.2.2.cmml" xref="A4.Ex76.m1.2.2.1.1.4.2.2"><ci id="A4.Ex76.m1.2.2.1.1.4.2.2.1.cmml" xref="A4.Ex76.m1.2.2.1.1.4.2.2.1">¯</ci><ci id="A4.Ex76.m1.2.2.1.1.4.2.2.2.cmml" xref="A4.Ex76.m1.2.2.1.1.4.2.2.2">𝑑</ci></apply><apply id="A4.Ex76.m1.2.2.1.1.4.2.3.1.cmml" xref="A4.Ex76.m1.2.2.1.1.4.2.3.2"><csymbol cd="latexml" id="A4.Ex76.m1.2.2.1.1.4.2.3.1.1.cmml" xref="A4.Ex76.m1.2.2.1.1.4.2.3.2.1">norm</csymbol><ci id="A4.Ex76.m1.1.1.cmml" xref="A4.Ex76.m1.1.1">𝒆</ci></apply></apply></apply><apply id="A4.Ex76.m1.2.2.1.1.2.cmml" xref="A4.Ex76.m1.2.2.1.1.2"><times id="A4.Ex76.m1.2.2.1.1.2.3.cmml" xref="A4.Ex76.m1.2.2.1.1.2.3"></times><apply id="A4.Ex76.m1.2.2.1.1.1.1.1.1.cmml" xref="A4.Ex76.m1.2.2.1.1.1.1.1"><plus id="A4.Ex76.m1.2.2.1.1.1.1.1.1.1.cmml" xref="A4.Ex76.m1.2.2.1.1.1.1.1.1.1"></plus><cn id="A4.Ex76.m1.2.2.1.1.1.1.1.1.2.cmml" type="integer" xref="A4.Ex76.m1.2.2.1.1.1.1.1.1.2">1</cn><ci id="A4.Ex76.m1.2.2.1.1.1.1.1.1.3.cmml" xref="A4.Ex76.m1.2.2.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="A4.Ex76.m1.2.2.1.1.2.2.1.1.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1"><plus id="A4.Ex76.m1.2.2.1.1.2.2.1.1.2.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.2"></plus><apply id="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.3"><times id="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.1.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.1"></times><apply id="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.2.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.2"><ci id="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.2.1.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.2.1">¯</ci><ci id="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.2.2.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.2.2">italic-ϵ</ci></apply><apply id="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.3.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.3"><ci id="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.3.1.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.3.1">¯</ci><ci id="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.3.2.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.3.3.2">𝑑</ci></apply></apply><apply id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1"><times id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.2.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.2"></times><ci id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.3.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.3">𝜅</ci><apply id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.4.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.4"><csymbol cd="ambiguous" id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.4.1.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.4">subscript</csymbol><apply id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.4.2.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.4.2"><ci id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.4.2.1.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.4.2.1">¯</ci><ci id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.4.2.2.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.4.2.2">𝑑</ci></apply><ci id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.4.3.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.4.3">g</ci></apply><apply id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.2.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1"><csymbol cd="latexml" id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.2.1.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.2">norm</csymbol><apply id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.1.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.1.1.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.1">subscript</csymbol><apply id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.1.2.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.1.2"><ci id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.1.2.1.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.1.2.1">~</ci><ci id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.1.2.2.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.1.2.2">𝜽</ci></apply><ci id="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.1.3.cmml" xref="A4.Ex76.m1.2.2.1.1.2.2.1.1.1.1.1.1.3">𝜁</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex76.m1.2c">\displaystyle+\bar{d}\|\bm{e}\|+(1+p)(\bar{\epsilon}\bar{d}+\kappa{\bar{d}}_{% \rm g}\|\tilde{\bm{\theta}}_{\zeta}\|).</annotation><annotation encoding="application/x-llamapun" id="A4.Ex76.m1.2d">+ over¯ start_ARG italic_d end_ARG ∥ bold_italic_e ∥ + ( 1 + italic_p ) ( over¯ start_ARG italic_ϵ end_ARG over¯ start_ARG italic_d end_ARG + italic_κ over¯ start_ARG italic_d end_ARG start_POSTSUBSCRIPT roman_g end_POSTSUBSCRIPT ∥ over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ∥ ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.1.p1.98">Following the proof of Item 2 in Theorem 2, one obtains</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx83"> <tbody id="A4.Ex77"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}_{\zeta}\leq" class="ltx_Math" display="inline" id="A4.Ex77.m1.1"><semantics id="A4.Ex77.m1.1a"><mrow id="A4.Ex77.m1.1.1" xref="A4.Ex77.m1.1.1.cmml"><msub id="A4.Ex77.m1.1.1.2" xref="A4.Ex77.m1.1.1.2.cmml"><mover accent="true" id="A4.Ex77.m1.1.1.2.2" xref="A4.Ex77.m1.1.1.2.2.cmml"><mi id="A4.Ex77.m1.1.1.2.2.2" xref="A4.Ex77.m1.1.1.2.2.2.cmml">V</mi><mo id="A4.Ex77.m1.1.1.2.2.1" xref="A4.Ex77.m1.1.1.2.2.1.cmml">˙</mo></mover><mi id="A4.Ex77.m1.1.1.2.3" xref="A4.Ex77.m1.1.1.2.3.cmml">ζ</mi></msub><mo id="A4.Ex77.m1.1.1.1" xref="A4.Ex77.m1.1.1.1.cmml">≤</mo><mi id="A4.Ex77.m1.1.1.3" xref="A4.Ex77.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex77.m1.1b"><apply id="A4.Ex77.m1.1.1.cmml" xref="A4.Ex77.m1.1.1"><leq id="A4.Ex77.m1.1.1.1.cmml" xref="A4.Ex77.m1.1.1.1"></leq><apply id="A4.Ex77.m1.1.1.2.cmml" xref="A4.Ex77.m1.1.1.2"><csymbol cd="ambiguous" id="A4.Ex77.m1.1.1.2.1.cmml" xref="A4.Ex77.m1.1.1.2">subscript</csymbol><apply id="A4.Ex77.m1.1.1.2.2.cmml" xref="A4.Ex77.m1.1.1.2.2"><ci id="A4.Ex77.m1.1.1.2.2.1.cmml" xref="A4.Ex77.m1.1.1.2.2.1">˙</ci><ci id="A4.Ex77.m1.1.1.2.2.2.cmml" xref="A4.Ex77.m1.1.1.2.2.2">𝑉</ci></apply><ci id="A4.Ex77.m1.1.1.2.3.cmml" xref="A4.Ex77.m1.1.1.2.3">𝜁</ci></apply><csymbol cd="latexml" id="A4.Ex77.m1.1.1.3.cmml" xref="A4.Ex77.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex77.m1.1c">\displaystyle\dot{V}_{\zeta}\leq</annotation><annotation encoding="application/x-llamapun" id="A4.Ex77.m1.1d">over˙ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-k_{\rm c}\|\bm{e}\|^{2}/2+\bar{d}\|\bm{e}\|-\kappa\sigma^{*}\|% \tilde{\bm{\theta}}_{\zeta}\|^{2}" class="ltx_Math" display="inline" id="A4.Ex77.m2.3"><semantics id="A4.Ex77.m2.3a"><mrow id="A4.Ex77.m2.3.3" xref="A4.Ex77.m2.3.3.cmml"><mrow id="A4.Ex77.m2.3.3.3" xref="A4.Ex77.m2.3.3.3.cmml"><mrow id="A4.Ex77.m2.3.3.3.2" xref="A4.Ex77.m2.3.3.3.2.cmml"><mo id="A4.Ex77.m2.3.3.3.2a" xref="A4.Ex77.m2.3.3.3.2.cmml">−</mo><mrow id="A4.Ex77.m2.3.3.3.2.2" xref="A4.Ex77.m2.3.3.3.2.2.cmml"><mrow id="A4.Ex77.m2.3.3.3.2.2.2" xref="A4.Ex77.m2.3.3.3.2.2.2.cmml"><msub id="A4.Ex77.m2.3.3.3.2.2.2.2" xref="A4.Ex77.m2.3.3.3.2.2.2.2.cmml"><mi id="A4.Ex77.m2.3.3.3.2.2.2.2.2" xref="A4.Ex77.m2.3.3.3.2.2.2.2.2.cmml">k</mi><mi id="A4.Ex77.m2.3.3.3.2.2.2.2.3" mathvariant="normal" xref="A4.Ex77.m2.3.3.3.2.2.2.2.3.cmml">c</mi></msub><mo id="A4.Ex77.m2.3.3.3.2.2.2.1" xref="A4.Ex77.m2.3.3.3.2.2.2.1.cmml">⁢</mo><msup id="A4.Ex77.m2.3.3.3.2.2.2.3" xref="A4.Ex77.m2.3.3.3.2.2.2.3.cmml"><mrow id="A4.Ex77.m2.3.3.3.2.2.2.3.2.2" xref="A4.Ex77.m2.3.3.3.2.2.2.3.2.1.cmml"><mo id="A4.Ex77.m2.3.3.3.2.2.2.3.2.2.1" stretchy="false" xref="A4.Ex77.m2.3.3.3.2.2.2.3.2.1.1.cmml">‖</mo><mi id="A4.Ex77.m2.1.1" xref="A4.Ex77.m2.1.1.cmml">𝒆</mi><mo id="A4.Ex77.m2.3.3.3.2.2.2.3.2.2.2" stretchy="false" xref="A4.Ex77.m2.3.3.3.2.2.2.3.2.1.1.cmml">‖</mo></mrow><mn id="A4.Ex77.m2.3.3.3.2.2.2.3.3" xref="A4.Ex77.m2.3.3.3.2.2.2.3.3.cmml">2</mn></msup></mrow><mo id="A4.Ex77.m2.3.3.3.2.2.1" xref="A4.Ex77.m2.3.3.3.2.2.1.cmml">/</mo><mn id="A4.Ex77.m2.3.3.3.2.2.3" xref="A4.Ex77.m2.3.3.3.2.2.3.cmml">2</mn></mrow></mrow><mo id="A4.Ex77.m2.3.3.3.1" xref="A4.Ex77.m2.3.3.3.1.cmml">+</mo><mrow id="A4.Ex77.m2.3.3.3.3" xref="A4.Ex77.m2.3.3.3.3.cmml"><mover accent="true" id="A4.Ex77.m2.3.3.3.3.2" xref="A4.Ex77.m2.3.3.3.3.2.cmml"><mi id="A4.Ex77.m2.3.3.3.3.2.2" xref="A4.Ex77.m2.3.3.3.3.2.2.cmml">d</mi><mo id="A4.Ex77.m2.3.3.3.3.2.1" xref="A4.Ex77.m2.3.3.3.3.2.1.cmml">¯</mo></mover><mo id="A4.Ex77.m2.3.3.3.3.1" xref="A4.Ex77.m2.3.3.3.3.1.cmml">⁢</mo><mrow id="A4.Ex77.m2.3.3.3.3.3.2" xref="A4.Ex77.m2.3.3.3.3.3.1.cmml"><mo id="A4.Ex77.m2.3.3.3.3.3.2.1" stretchy="false" xref="A4.Ex77.m2.3.3.3.3.3.1.1.cmml">‖</mo><mi id="A4.Ex77.m2.2.2" xref="A4.Ex77.m2.2.2.cmml">𝒆</mi><mo id="A4.Ex77.m2.3.3.3.3.3.2.2" stretchy="false" xref="A4.Ex77.m2.3.3.3.3.3.1.1.cmml">‖</mo></mrow></mrow></mrow><mo id="A4.Ex77.m2.3.3.2" xref="A4.Ex77.m2.3.3.2.cmml">−</mo><mrow id="A4.Ex77.m2.3.3.1" xref="A4.Ex77.m2.3.3.1.cmml"><mi id="A4.Ex77.m2.3.3.1.3" xref="A4.Ex77.m2.3.3.1.3.cmml">κ</mi><mo id="A4.Ex77.m2.3.3.1.2" xref="A4.Ex77.m2.3.3.1.2.cmml">⁢</mo><msup id="A4.Ex77.m2.3.3.1.4" xref="A4.Ex77.m2.3.3.1.4.cmml"><mi id="A4.Ex77.m2.3.3.1.4.2" xref="A4.Ex77.m2.3.3.1.4.2.cmml">σ</mi><mo id="A4.Ex77.m2.3.3.1.4.3" xref="A4.Ex77.m2.3.3.1.4.3.cmml">∗</mo></msup><mo id="A4.Ex77.m2.3.3.1.2a" xref="A4.Ex77.m2.3.3.1.2.cmml">⁢</mo><msup id="A4.Ex77.m2.3.3.1.1" xref="A4.Ex77.m2.3.3.1.1.cmml"><mrow id="A4.Ex77.m2.3.3.1.1.1.1" xref="A4.Ex77.m2.3.3.1.1.1.2.cmml"><mo id="A4.Ex77.m2.3.3.1.1.1.1.2" stretchy="false" xref="A4.Ex77.m2.3.3.1.1.1.2.1.cmml">‖</mo><msub id="A4.Ex77.m2.3.3.1.1.1.1.1" xref="A4.Ex77.m2.3.3.1.1.1.1.1.cmml"><mover accent="true" id="A4.Ex77.m2.3.3.1.1.1.1.1.2" xref="A4.Ex77.m2.3.3.1.1.1.1.1.2.cmml"><mi id="A4.Ex77.m2.3.3.1.1.1.1.1.2.2" xref="A4.Ex77.m2.3.3.1.1.1.1.1.2.2.cmml">𝜽</mi><mo id="A4.Ex77.m2.3.3.1.1.1.1.1.2.1" xref="A4.Ex77.m2.3.3.1.1.1.1.1.2.1.cmml">~</mo></mover><mi id="A4.Ex77.m2.3.3.1.1.1.1.1.3" xref="A4.Ex77.m2.3.3.1.1.1.1.1.3.cmml">ζ</mi></msub><mo id="A4.Ex77.m2.3.3.1.1.1.1.3" stretchy="false" xref="A4.Ex77.m2.3.3.1.1.1.2.1.cmml">‖</mo></mrow><mn id="A4.Ex77.m2.3.3.1.1.3" xref="A4.Ex77.m2.3.3.1.1.3.cmml">2</mn></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex77.m2.3b"><apply id="A4.Ex77.m2.3.3.cmml" xref="A4.Ex77.m2.3.3"><minus id="A4.Ex77.m2.3.3.2.cmml" xref="A4.Ex77.m2.3.3.2"></minus><apply id="A4.Ex77.m2.3.3.3.cmml" xref="A4.Ex77.m2.3.3.3"><plus id="A4.Ex77.m2.3.3.3.1.cmml" xref="A4.Ex77.m2.3.3.3.1"></plus><apply id="A4.Ex77.m2.3.3.3.2.cmml" xref="A4.Ex77.m2.3.3.3.2"><minus id="A4.Ex77.m2.3.3.3.2.1.cmml" xref="A4.Ex77.m2.3.3.3.2"></minus><apply id="A4.Ex77.m2.3.3.3.2.2.cmml" xref="A4.Ex77.m2.3.3.3.2.2"><divide id="A4.Ex77.m2.3.3.3.2.2.1.cmml" xref="A4.Ex77.m2.3.3.3.2.2.1"></divide><apply id="A4.Ex77.m2.3.3.3.2.2.2.cmml" xref="A4.Ex77.m2.3.3.3.2.2.2"><times id="A4.Ex77.m2.3.3.3.2.2.2.1.cmml" xref="A4.Ex77.m2.3.3.3.2.2.2.1"></times><apply id="A4.Ex77.m2.3.3.3.2.2.2.2.cmml" xref="A4.Ex77.m2.3.3.3.2.2.2.2"><csymbol cd="ambiguous" id="A4.Ex77.m2.3.3.3.2.2.2.2.1.cmml" xref="A4.Ex77.m2.3.3.3.2.2.2.2">subscript</csymbol><ci id="A4.Ex77.m2.3.3.3.2.2.2.2.2.cmml" xref="A4.Ex77.m2.3.3.3.2.2.2.2.2">𝑘</ci><ci id="A4.Ex77.m2.3.3.3.2.2.2.2.3.cmml" xref="A4.Ex77.m2.3.3.3.2.2.2.2.3">c</ci></apply><apply id="A4.Ex77.m2.3.3.3.2.2.2.3.cmml" xref="A4.Ex77.m2.3.3.3.2.2.2.3"><csymbol cd="ambiguous" id="A4.Ex77.m2.3.3.3.2.2.2.3.1.cmml" xref="A4.Ex77.m2.3.3.3.2.2.2.3">superscript</csymbol><apply id="A4.Ex77.m2.3.3.3.2.2.2.3.2.1.cmml" xref="A4.Ex77.m2.3.3.3.2.2.2.3.2.2"><csymbol cd="latexml" id="A4.Ex77.m2.3.3.3.2.2.2.3.2.1.1.cmml" xref="A4.Ex77.m2.3.3.3.2.2.2.3.2.2.1">norm</csymbol><ci id="A4.Ex77.m2.1.1.cmml" xref="A4.Ex77.m2.1.1">𝒆</ci></apply><cn id="A4.Ex77.m2.3.3.3.2.2.2.3.3.cmml" type="integer" xref="A4.Ex77.m2.3.3.3.2.2.2.3.3">2</cn></apply></apply><cn id="A4.Ex77.m2.3.3.3.2.2.3.cmml" type="integer" xref="A4.Ex77.m2.3.3.3.2.2.3">2</cn></apply></apply><apply id="A4.Ex77.m2.3.3.3.3.cmml" xref="A4.Ex77.m2.3.3.3.3"><times id="A4.Ex77.m2.3.3.3.3.1.cmml" xref="A4.Ex77.m2.3.3.3.3.1"></times><apply id="A4.Ex77.m2.3.3.3.3.2.cmml" xref="A4.Ex77.m2.3.3.3.3.2"><ci id="A4.Ex77.m2.3.3.3.3.2.1.cmml" xref="A4.Ex77.m2.3.3.3.3.2.1">¯</ci><ci id="A4.Ex77.m2.3.3.3.3.2.2.cmml" xref="A4.Ex77.m2.3.3.3.3.2.2">𝑑</ci></apply><apply id="A4.Ex77.m2.3.3.3.3.3.1.cmml" xref="A4.Ex77.m2.3.3.3.3.3.2"><csymbol cd="latexml" id="A4.Ex77.m2.3.3.3.3.3.1.1.cmml" xref="A4.Ex77.m2.3.3.3.3.3.2.1">norm</csymbol><ci id="A4.Ex77.m2.2.2.cmml" xref="A4.Ex77.m2.2.2">𝒆</ci></apply></apply></apply><apply id="A4.Ex77.m2.3.3.1.cmml" xref="A4.Ex77.m2.3.3.1"><times id="A4.Ex77.m2.3.3.1.2.cmml" xref="A4.Ex77.m2.3.3.1.2"></times><ci id="A4.Ex77.m2.3.3.1.3.cmml" xref="A4.Ex77.m2.3.3.1.3">𝜅</ci><apply id="A4.Ex77.m2.3.3.1.4.cmml" xref="A4.Ex77.m2.3.3.1.4"><csymbol cd="ambiguous" id="A4.Ex77.m2.3.3.1.4.1.cmml" xref="A4.Ex77.m2.3.3.1.4">superscript</csymbol><ci id="A4.Ex77.m2.3.3.1.4.2.cmml" xref="A4.Ex77.m2.3.3.1.4.2">𝜎</ci><times id="A4.Ex77.m2.3.3.1.4.3.cmml" xref="A4.Ex77.m2.3.3.1.4.3"></times></apply><apply id="A4.Ex77.m2.3.3.1.1.cmml" xref="A4.Ex77.m2.3.3.1.1"><csymbol cd="ambiguous" id="A4.Ex77.m2.3.3.1.1.2.cmml" xref="A4.Ex77.m2.3.3.1.1">superscript</csymbol><apply id="A4.Ex77.m2.3.3.1.1.1.2.cmml" xref="A4.Ex77.m2.3.3.1.1.1.1"><csymbol cd="latexml" id="A4.Ex77.m2.3.3.1.1.1.2.1.cmml" xref="A4.Ex77.m2.3.3.1.1.1.1.2">norm</csymbol><apply id="A4.Ex77.m2.3.3.1.1.1.1.1.cmml" xref="A4.Ex77.m2.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.Ex77.m2.3.3.1.1.1.1.1.1.cmml" xref="A4.Ex77.m2.3.3.1.1.1.1.1">subscript</csymbol><apply id="A4.Ex77.m2.3.3.1.1.1.1.1.2.cmml" xref="A4.Ex77.m2.3.3.1.1.1.1.1.2"><ci id="A4.Ex77.m2.3.3.1.1.1.1.1.2.1.cmml" xref="A4.Ex77.m2.3.3.1.1.1.1.1.2.1">~</ci><ci id="A4.Ex77.m2.3.3.1.1.1.1.1.2.2.cmml" xref="A4.Ex77.m2.3.3.1.1.1.1.1.2.2">𝜽</ci></apply><ci id="A4.Ex77.m2.3.3.1.1.1.1.1.3.cmml" xref="A4.Ex77.m2.3.3.1.1.1.1.1.3">𝜁</ci></apply></apply><cn id="A4.Ex77.m2.3.3.1.1.3.cmml" type="integer" xref="A4.Ex77.m2.3.3.1.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex77.m2.3c">\displaystyle-k_{\rm c}\|\bm{e}\|^{2}/2+\bar{d}\|\bm{e}\|-\kappa\sigma^{*}\|% \tilde{\bm{\theta}}_{\zeta}\|^{2}</annotation><annotation encoding="application/x-llamapun" id="A4.Ex77.m2.3d">- italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ∥ bold_italic_e ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 + over¯ start_ARG italic_d end_ARG ∥ bold_italic_e ∥ - italic_κ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex78"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle+(1+p)\kappa{\bar{d}}_{\rm g}\|\tilde{\bm{\theta}}_{\zeta}\|+(1+p% )\bar{\epsilon}\bar{d}" class="ltx_Math" display="inline" id="A4.Ex78.m1.3"><semantics id="A4.Ex78.m1.3a"><mrow id="A4.Ex78.m1.3.3" xref="A4.Ex78.m1.3.3.cmml"><mrow id="A4.Ex78.m1.2.2.2" xref="A4.Ex78.m1.2.2.2.cmml"><mo id="A4.Ex78.m1.2.2.2a" xref="A4.Ex78.m1.2.2.2.cmml">+</mo><mrow id="A4.Ex78.m1.2.2.2.2" xref="A4.Ex78.m1.2.2.2.2.cmml"><mrow id="A4.Ex78.m1.1.1.1.1.1.1" xref="A4.Ex78.m1.1.1.1.1.1.1.1.cmml"><mo id="A4.Ex78.m1.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex78.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex78.m1.1.1.1.1.1.1.1" xref="A4.Ex78.m1.1.1.1.1.1.1.1.cmml"><mn id="A4.Ex78.m1.1.1.1.1.1.1.1.2" xref="A4.Ex78.m1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A4.Ex78.m1.1.1.1.1.1.1.1.1" xref="A4.Ex78.m1.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="A4.Ex78.m1.1.1.1.1.1.1.1.3" xref="A4.Ex78.m1.1.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A4.Ex78.m1.1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex78.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex78.m1.2.2.2.2.3" xref="A4.Ex78.m1.2.2.2.2.3.cmml">⁢</mo><mi id="A4.Ex78.m1.2.2.2.2.4" xref="A4.Ex78.m1.2.2.2.2.4.cmml">κ</mi><mo id="A4.Ex78.m1.2.2.2.2.3a" xref="A4.Ex78.m1.2.2.2.2.3.cmml">⁢</mo><msub id="A4.Ex78.m1.2.2.2.2.5" xref="A4.Ex78.m1.2.2.2.2.5.cmml"><mover accent="true" id="A4.Ex78.m1.2.2.2.2.5.2" xref="A4.Ex78.m1.2.2.2.2.5.2.cmml"><mi id="A4.Ex78.m1.2.2.2.2.5.2.2" xref="A4.Ex78.m1.2.2.2.2.5.2.2.cmml">d</mi><mo id="A4.Ex78.m1.2.2.2.2.5.2.1" xref="A4.Ex78.m1.2.2.2.2.5.2.1.cmml">¯</mo></mover><mi id="A4.Ex78.m1.2.2.2.2.5.3" mathvariant="normal" xref="A4.Ex78.m1.2.2.2.2.5.3.cmml">g</mi></msub><mo id="A4.Ex78.m1.2.2.2.2.3b" xref="A4.Ex78.m1.2.2.2.2.3.cmml">⁢</mo><mrow id="A4.Ex78.m1.2.2.2.2.2.1" xref="A4.Ex78.m1.2.2.2.2.2.2.cmml"><mo id="A4.Ex78.m1.2.2.2.2.2.1.2" stretchy="false" xref="A4.Ex78.m1.2.2.2.2.2.2.1.cmml">‖</mo><msub id="A4.Ex78.m1.2.2.2.2.2.1.1" xref="A4.Ex78.m1.2.2.2.2.2.1.1.cmml"><mover accent="true" id="A4.Ex78.m1.2.2.2.2.2.1.1.2" xref="A4.Ex78.m1.2.2.2.2.2.1.1.2.cmml"><mi id="A4.Ex78.m1.2.2.2.2.2.1.1.2.2" xref="A4.Ex78.m1.2.2.2.2.2.1.1.2.2.cmml">𝜽</mi><mo id="A4.Ex78.m1.2.2.2.2.2.1.1.2.1" xref="A4.Ex78.m1.2.2.2.2.2.1.1.2.1.cmml">~</mo></mover><mi id="A4.Ex78.m1.2.2.2.2.2.1.1.3" xref="A4.Ex78.m1.2.2.2.2.2.1.1.3.cmml">ζ</mi></msub><mo id="A4.Ex78.m1.2.2.2.2.2.1.3" stretchy="false" xref="A4.Ex78.m1.2.2.2.2.2.2.1.cmml">‖</mo></mrow></mrow></mrow><mo id="A4.Ex78.m1.3.3.4" xref="A4.Ex78.m1.3.3.4.cmml">+</mo><mrow id="A4.Ex78.m1.3.3.3" xref="A4.Ex78.m1.3.3.3.cmml"><mrow id="A4.Ex78.m1.3.3.3.1.1" xref="A4.Ex78.m1.3.3.3.1.1.1.cmml"><mo id="A4.Ex78.m1.3.3.3.1.1.2" stretchy="false" xref="A4.Ex78.m1.3.3.3.1.1.1.cmml">(</mo><mrow id="A4.Ex78.m1.3.3.3.1.1.1" xref="A4.Ex78.m1.3.3.3.1.1.1.cmml"><mn id="A4.Ex78.m1.3.3.3.1.1.1.2" xref="A4.Ex78.m1.3.3.3.1.1.1.2.cmml">1</mn><mo id="A4.Ex78.m1.3.3.3.1.1.1.1" xref="A4.Ex78.m1.3.3.3.1.1.1.1.cmml">+</mo><mi id="A4.Ex78.m1.3.3.3.1.1.1.3" xref="A4.Ex78.m1.3.3.3.1.1.1.3.cmml">p</mi></mrow><mo id="A4.Ex78.m1.3.3.3.1.1.3" stretchy="false" xref="A4.Ex78.m1.3.3.3.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex78.m1.3.3.3.2" xref="A4.Ex78.m1.3.3.3.2.cmml">⁢</mo><mover accent="true" id="A4.Ex78.m1.3.3.3.3" xref="A4.Ex78.m1.3.3.3.3.cmml"><mi id="A4.Ex78.m1.3.3.3.3.2" xref="A4.Ex78.m1.3.3.3.3.2.cmml">ϵ</mi><mo id="A4.Ex78.m1.3.3.3.3.1" xref="A4.Ex78.m1.3.3.3.3.1.cmml">¯</mo></mover><mo id="A4.Ex78.m1.3.3.3.2a" xref="A4.Ex78.m1.3.3.3.2.cmml">⁢</mo><mover accent="true" id="A4.Ex78.m1.3.3.3.4" xref="A4.Ex78.m1.3.3.3.4.cmml"><mi id="A4.Ex78.m1.3.3.3.4.2" xref="A4.Ex78.m1.3.3.3.4.2.cmml">d</mi><mo id="A4.Ex78.m1.3.3.3.4.1" xref="A4.Ex78.m1.3.3.3.4.1.cmml">¯</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex78.m1.3b"><apply id="A4.Ex78.m1.3.3.cmml" xref="A4.Ex78.m1.3.3"><plus id="A4.Ex78.m1.3.3.4.cmml" xref="A4.Ex78.m1.3.3.4"></plus><apply id="A4.Ex78.m1.2.2.2.cmml" xref="A4.Ex78.m1.2.2.2"><plus id="A4.Ex78.m1.2.2.2.3.cmml" xref="A4.Ex78.m1.2.2.2"></plus><apply id="A4.Ex78.m1.2.2.2.2.cmml" xref="A4.Ex78.m1.2.2.2.2"><times id="A4.Ex78.m1.2.2.2.2.3.cmml" xref="A4.Ex78.m1.2.2.2.2.3"></times><apply id="A4.Ex78.m1.1.1.1.1.1.1.1.cmml" xref="A4.Ex78.m1.1.1.1.1.1.1"><plus id="A4.Ex78.m1.1.1.1.1.1.1.1.1.cmml" xref="A4.Ex78.m1.1.1.1.1.1.1.1.1"></plus><cn id="A4.Ex78.m1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A4.Ex78.m1.1.1.1.1.1.1.1.2">1</cn><ci id="A4.Ex78.m1.1.1.1.1.1.1.1.3.cmml" xref="A4.Ex78.m1.1.1.1.1.1.1.1.3">𝑝</ci></apply><ci id="A4.Ex78.m1.2.2.2.2.4.cmml" xref="A4.Ex78.m1.2.2.2.2.4">𝜅</ci><apply id="A4.Ex78.m1.2.2.2.2.5.cmml" xref="A4.Ex78.m1.2.2.2.2.5"><csymbol cd="ambiguous" id="A4.Ex78.m1.2.2.2.2.5.1.cmml" xref="A4.Ex78.m1.2.2.2.2.5">subscript</csymbol><apply id="A4.Ex78.m1.2.2.2.2.5.2.cmml" xref="A4.Ex78.m1.2.2.2.2.5.2"><ci id="A4.Ex78.m1.2.2.2.2.5.2.1.cmml" xref="A4.Ex78.m1.2.2.2.2.5.2.1">¯</ci><ci id="A4.Ex78.m1.2.2.2.2.5.2.2.cmml" xref="A4.Ex78.m1.2.2.2.2.5.2.2">𝑑</ci></apply><ci id="A4.Ex78.m1.2.2.2.2.5.3.cmml" xref="A4.Ex78.m1.2.2.2.2.5.3">g</ci></apply><apply id="A4.Ex78.m1.2.2.2.2.2.2.cmml" xref="A4.Ex78.m1.2.2.2.2.2.1"><csymbol cd="latexml" id="A4.Ex78.m1.2.2.2.2.2.2.1.cmml" xref="A4.Ex78.m1.2.2.2.2.2.1.2">norm</csymbol><apply id="A4.Ex78.m1.2.2.2.2.2.1.1.cmml" xref="A4.Ex78.m1.2.2.2.2.2.1.1"><csymbol cd="ambiguous" id="A4.Ex78.m1.2.2.2.2.2.1.1.1.cmml" xref="A4.Ex78.m1.2.2.2.2.2.1.1">subscript</csymbol><apply id="A4.Ex78.m1.2.2.2.2.2.1.1.2.cmml" xref="A4.Ex78.m1.2.2.2.2.2.1.1.2"><ci id="A4.Ex78.m1.2.2.2.2.2.1.1.2.1.cmml" xref="A4.Ex78.m1.2.2.2.2.2.1.1.2.1">~</ci><ci id="A4.Ex78.m1.2.2.2.2.2.1.1.2.2.cmml" xref="A4.Ex78.m1.2.2.2.2.2.1.1.2.2">𝜽</ci></apply><ci id="A4.Ex78.m1.2.2.2.2.2.1.1.3.cmml" xref="A4.Ex78.m1.2.2.2.2.2.1.1.3">𝜁</ci></apply></apply></apply></apply><apply id="A4.Ex78.m1.3.3.3.cmml" xref="A4.Ex78.m1.3.3.3"><times id="A4.Ex78.m1.3.3.3.2.cmml" xref="A4.Ex78.m1.3.3.3.2"></times><apply id="A4.Ex78.m1.3.3.3.1.1.1.cmml" xref="A4.Ex78.m1.3.3.3.1.1"><plus id="A4.Ex78.m1.3.3.3.1.1.1.1.cmml" xref="A4.Ex78.m1.3.3.3.1.1.1.1"></plus><cn id="A4.Ex78.m1.3.3.3.1.1.1.2.cmml" type="integer" xref="A4.Ex78.m1.3.3.3.1.1.1.2">1</cn><ci id="A4.Ex78.m1.3.3.3.1.1.1.3.cmml" xref="A4.Ex78.m1.3.3.3.1.1.1.3">𝑝</ci></apply><apply id="A4.Ex78.m1.3.3.3.3.cmml" xref="A4.Ex78.m1.3.3.3.3"><ci id="A4.Ex78.m1.3.3.3.3.1.cmml" xref="A4.Ex78.m1.3.3.3.3.1">¯</ci><ci id="A4.Ex78.m1.3.3.3.3.2.cmml" xref="A4.Ex78.m1.3.3.3.3.2">italic-ϵ</ci></apply><apply id="A4.Ex78.m1.3.3.3.4.cmml" xref="A4.Ex78.m1.3.3.3.4"><ci id="A4.Ex78.m1.3.3.3.4.1.cmml" xref="A4.Ex78.m1.3.3.3.4.1">¯</ci><ci id="A4.Ex78.m1.3.3.3.4.2.cmml" xref="A4.Ex78.m1.3.3.3.4.2">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex78.m1.3c">\displaystyle+(1+p)\kappa{\bar{d}}_{\rm g}\|\tilde{\bm{\theta}}_{\zeta}\|+(1+p% )\bar{\epsilon}\bar{d}</annotation><annotation encoding="application/x-llamapun" id="A4.Ex78.m1.3d">+ ( 1 + italic_p ) italic_κ over¯ start_ARG italic_d end_ARG start_POSTSUBSCRIPT roman_g end_POSTSUBSCRIPT ∥ over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ∥ + ( 1 + italic_p ) over¯ start_ARG italic_ϵ end_ARG over¯ start_ARG italic_d end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.1.p1.43">with <math alttext="p" class="ltx_Math" display="inline" id="A4.1.p1.41.m1.1"><semantics id="A4.1.p1.41.m1.1a"><mi id="A4.1.p1.41.m1.1.1" xref="A4.1.p1.41.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="A4.1.p1.41.m1.1b"><ci id="A4.1.p1.41.m1.1.1.cmml" xref="A4.1.p1.41.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.41.m1.1c">p</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.41.m1.1d">italic_p</annotation></semantics></math> <math alttext="=" class="ltx_Math" display="inline" id="A4.1.p1.42.m2.1"><semantics id="A4.1.p1.42.m2.1a"><mo id="A4.1.p1.42.m2.1.1" xref="A4.1.p1.42.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="A4.1.p1.42.m2.1b"><eq id="A4.1.p1.42.m2.1.1.cmml" xref="A4.1.p1.42.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.42.m2.1c">=</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.42.m2.1d">=</annotation></semantics></math> <math alttext="\delta^{2}/(2k_{\rm c}\kappa\sigma^{*})\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A4.1.p1.43.m3.1"><semantics id="A4.1.p1.43.m3.1a"><mrow id="A4.1.p1.43.m3.1.1" xref="A4.1.p1.43.m3.1.1.cmml"><mrow id="A4.1.p1.43.m3.1.1.1" xref="A4.1.p1.43.m3.1.1.1.cmml"><msup id="A4.1.p1.43.m3.1.1.1.3" xref="A4.1.p1.43.m3.1.1.1.3.cmml"><mi id="A4.1.p1.43.m3.1.1.1.3.2" xref="A4.1.p1.43.m3.1.1.1.3.2.cmml">δ</mi><mn id="A4.1.p1.43.m3.1.1.1.3.3" xref="A4.1.p1.43.m3.1.1.1.3.3.cmml">2</mn></msup><mo id="A4.1.p1.43.m3.1.1.1.2" xref="A4.1.p1.43.m3.1.1.1.2.cmml">/</mo><mrow id="A4.1.p1.43.m3.1.1.1.1.1" xref="A4.1.p1.43.m3.1.1.1.1.1.1.cmml"><mo id="A4.1.p1.43.m3.1.1.1.1.1.2" stretchy="false" xref="A4.1.p1.43.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.1.p1.43.m3.1.1.1.1.1.1" xref="A4.1.p1.43.m3.1.1.1.1.1.1.cmml"><mn id="A4.1.p1.43.m3.1.1.1.1.1.1.2" xref="A4.1.p1.43.m3.1.1.1.1.1.1.2.cmml">2</mn><mo id="A4.1.p1.43.m3.1.1.1.1.1.1.1" xref="A4.1.p1.43.m3.1.1.1.1.1.1.1.cmml">⁢</mo><msub id="A4.1.p1.43.m3.1.1.1.1.1.1.3" xref="A4.1.p1.43.m3.1.1.1.1.1.1.3.cmml"><mi id="A4.1.p1.43.m3.1.1.1.1.1.1.3.2" xref="A4.1.p1.43.m3.1.1.1.1.1.1.3.2.cmml">k</mi><mi id="A4.1.p1.43.m3.1.1.1.1.1.1.3.3" mathvariant="normal" xref="A4.1.p1.43.m3.1.1.1.1.1.1.3.3.cmml">c</mi></msub><mo id="A4.1.p1.43.m3.1.1.1.1.1.1.1a" xref="A4.1.p1.43.m3.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="A4.1.p1.43.m3.1.1.1.1.1.1.4" xref="A4.1.p1.43.m3.1.1.1.1.1.1.4.cmml">κ</mi><mo id="A4.1.p1.43.m3.1.1.1.1.1.1.1b" xref="A4.1.p1.43.m3.1.1.1.1.1.1.1.cmml">⁢</mo><msup id="A4.1.p1.43.m3.1.1.1.1.1.1.5" xref="A4.1.p1.43.m3.1.1.1.1.1.1.5.cmml"><mi id="A4.1.p1.43.m3.1.1.1.1.1.1.5.2" xref="A4.1.p1.43.m3.1.1.1.1.1.1.5.2.cmml">σ</mi><mo id="A4.1.p1.43.m3.1.1.1.1.1.1.5.3" xref="A4.1.p1.43.m3.1.1.1.1.1.1.5.3.cmml">∗</mo></msup></mrow><mo id="A4.1.p1.43.m3.1.1.1.1.1.3" stretchy="false" xref="A4.1.p1.43.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A4.1.p1.43.m3.1.1.2" xref="A4.1.p1.43.m3.1.1.2.cmml">∈</mo><msup id="A4.1.p1.43.m3.1.1.3" xref="A4.1.p1.43.m3.1.1.3.cmml"><mi id="A4.1.p1.43.m3.1.1.3.2" xref="A4.1.p1.43.m3.1.1.3.2.cmml">ℝ</mi><mo id="A4.1.p1.43.m3.1.1.3.3" xref="A4.1.p1.43.m3.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.43.m3.1b"><apply id="A4.1.p1.43.m3.1.1.cmml" xref="A4.1.p1.43.m3.1.1"><in id="A4.1.p1.43.m3.1.1.2.cmml" xref="A4.1.p1.43.m3.1.1.2"></in><apply id="A4.1.p1.43.m3.1.1.1.cmml" xref="A4.1.p1.43.m3.1.1.1"><divide id="A4.1.p1.43.m3.1.1.1.2.cmml" xref="A4.1.p1.43.m3.1.1.1.2"></divide><apply id="A4.1.p1.43.m3.1.1.1.3.cmml" xref="A4.1.p1.43.m3.1.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.43.m3.1.1.1.3.1.cmml" xref="A4.1.p1.43.m3.1.1.1.3">superscript</csymbol><ci id="A4.1.p1.43.m3.1.1.1.3.2.cmml" xref="A4.1.p1.43.m3.1.1.1.3.2">𝛿</ci><cn id="A4.1.p1.43.m3.1.1.1.3.3.cmml" type="integer" xref="A4.1.p1.43.m3.1.1.1.3.3">2</cn></apply><apply id="A4.1.p1.43.m3.1.1.1.1.1.1.cmml" xref="A4.1.p1.43.m3.1.1.1.1.1"><times id="A4.1.p1.43.m3.1.1.1.1.1.1.1.cmml" xref="A4.1.p1.43.m3.1.1.1.1.1.1.1"></times><cn id="A4.1.p1.43.m3.1.1.1.1.1.1.2.cmml" type="integer" xref="A4.1.p1.43.m3.1.1.1.1.1.1.2">2</cn><apply id="A4.1.p1.43.m3.1.1.1.1.1.1.3.cmml" xref="A4.1.p1.43.m3.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.43.m3.1.1.1.1.1.1.3.1.cmml" xref="A4.1.p1.43.m3.1.1.1.1.1.1.3">subscript</csymbol><ci id="A4.1.p1.43.m3.1.1.1.1.1.1.3.2.cmml" xref="A4.1.p1.43.m3.1.1.1.1.1.1.3.2">𝑘</ci><ci id="A4.1.p1.43.m3.1.1.1.1.1.1.3.3.cmml" xref="A4.1.p1.43.m3.1.1.1.1.1.1.3.3">c</ci></apply><ci id="A4.1.p1.43.m3.1.1.1.1.1.1.4.cmml" xref="A4.1.p1.43.m3.1.1.1.1.1.1.4">𝜅</ci><apply id="A4.1.p1.43.m3.1.1.1.1.1.1.5.cmml" xref="A4.1.p1.43.m3.1.1.1.1.1.1.5"><csymbol cd="ambiguous" id="A4.1.p1.43.m3.1.1.1.1.1.1.5.1.cmml" xref="A4.1.p1.43.m3.1.1.1.1.1.1.5">superscript</csymbol><ci id="A4.1.p1.43.m3.1.1.1.1.1.1.5.2.cmml" xref="A4.1.p1.43.m3.1.1.1.1.1.1.5.2">𝜎</ci><times id="A4.1.p1.43.m3.1.1.1.1.1.1.5.3.cmml" xref="A4.1.p1.43.m3.1.1.1.1.1.1.5.3"></times></apply></apply></apply><apply id="A4.1.p1.43.m3.1.1.3.cmml" xref="A4.1.p1.43.m3.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.43.m3.1.1.3.1.cmml" xref="A4.1.p1.43.m3.1.1.3">superscript</csymbol><ci id="A4.1.p1.43.m3.1.1.3.2.cmml" xref="A4.1.p1.43.m3.1.1.3.2">ℝ</ci><plus id="A4.1.p1.43.m3.1.1.3.3.cmml" xref="A4.1.p1.43.m3.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.43.m3.1c">\delta^{2}/(2k_{\rm c}\kappa\sigma^{*})\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.43.m3.1d">italic_δ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( 2 italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT italic_κ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, which can be rewritten into</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx84"> <tbody id="A4.Ex79"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}_{\zeta}\leq" class="ltx_Math" display="inline" id="A4.Ex79.m1.1"><semantics id="A4.Ex79.m1.1a"><mrow id="A4.Ex79.m1.1.1" xref="A4.Ex79.m1.1.1.cmml"><msub id="A4.Ex79.m1.1.1.2" xref="A4.Ex79.m1.1.1.2.cmml"><mover accent="true" id="A4.Ex79.m1.1.1.2.2" xref="A4.Ex79.m1.1.1.2.2.cmml"><mi id="A4.Ex79.m1.1.1.2.2.2" xref="A4.Ex79.m1.1.1.2.2.2.cmml">V</mi><mo id="A4.Ex79.m1.1.1.2.2.1" xref="A4.Ex79.m1.1.1.2.2.1.cmml">˙</mo></mover><mi id="A4.Ex79.m1.1.1.2.3" xref="A4.Ex79.m1.1.1.2.3.cmml">ζ</mi></msub><mo id="A4.Ex79.m1.1.1.1" xref="A4.Ex79.m1.1.1.1.cmml">≤</mo><mi id="A4.Ex79.m1.1.1.3" xref="A4.Ex79.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex79.m1.1b"><apply id="A4.Ex79.m1.1.1.cmml" xref="A4.Ex79.m1.1.1"><leq id="A4.Ex79.m1.1.1.1.cmml" xref="A4.Ex79.m1.1.1.1"></leq><apply id="A4.Ex79.m1.1.1.2.cmml" xref="A4.Ex79.m1.1.1.2"><csymbol cd="ambiguous" id="A4.Ex79.m1.1.1.2.1.cmml" xref="A4.Ex79.m1.1.1.2">subscript</csymbol><apply id="A4.Ex79.m1.1.1.2.2.cmml" xref="A4.Ex79.m1.1.1.2.2"><ci id="A4.Ex79.m1.1.1.2.2.1.cmml" xref="A4.Ex79.m1.1.1.2.2.1">˙</ci><ci id="A4.Ex79.m1.1.1.2.2.2.cmml" xref="A4.Ex79.m1.1.1.2.2.2">𝑉</ci></apply><ci id="A4.Ex79.m1.1.1.2.3.cmml" xref="A4.Ex79.m1.1.1.2.3">𝜁</ci></apply><csymbol cd="latexml" id="A4.Ex79.m1.1.1.3.cmml" xref="A4.Ex79.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex79.m1.1c">\displaystyle\dot{V}_{\zeta}\leq</annotation><annotation encoding="application/x-llamapun" id="A4.Ex79.m1.1d">over˙ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-k_{\rm c}(1-\mu)\|\bm{e}\|^{2}/2-k_{\rm c}\mu(\|\bm{e}\|^{2}-2% \bar{d}/(k_{\rm c}\mu)\|\bm{e}\|)/2" class="ltx_Math" display="inline" id="A4.Ex79.m2.5"><semantics id="A4.Ex79.m2.5a"><mrow id="A4.Ex79.m2.5.5" xref="A4.Ex79.m2.5.5.cmml"><mrow id="A4.Ex79.m2.4.4.1" xref="A4.Ex79.m2.4.4.1.cmml"><mo id="A4.Ex79.m2.4.4.1a" xref="A4.Ex79.m2.4.4.1.cmml">−</mo><mrow id="A4.Ex79.m2.4.4.1.1" xref="A4.Ex79.m2.4.4.1.1.cmml"><mrow id="A4.Ex79.m2.4.4.1.1.1" xref="A4.Ex79.m2.4.4.1.1.1.cmml"><msub id="A4.Ex79.m2.4.4.1.1.1.3" xref="A4.Ex79.m2.4.4.1.1.1.3.cmml"><mi id="A4.Ex79.m2.4.4.1.1.1.3.2" xref="A4.Ex79.m2.4.4.1.1.1.3.2.cmml">k</mi><mi id="A4.Ex79.m2.4.4.1.1.1.3.3" mathvariant="normal" xref="A4.Ex79.m2.4.4.1.1.1.3.3.cmml">c</mi></msub><mo id="A4.Ex79.m2.4.4.1.1.1.2" xref="A4.Ex79.m2.4.4.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex79.m2.4.4.1.1.1.1.1" xref="A4.Ex79.m2.4.4.1.1.1.1.1.1.cmml"><mo id="A4.Ex79.m2.4.4.1.1.1.1.1.2" stretchy="false" xref="A4.Ex79.m2.4.4.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex79.m2.4.4.1.1.1.1.1.1" xref="A4.Ex79.m2.4.4.1.1.1.1.1.1.cmml"><mn id="A4.Ex79.m2.4.4.1.1.1.1.1.1.2" xref="A4.Ex79.m2.4.4.1.1.1.1.1.1.2.cmml">1</mn><mo id="A4.Ex79.m2.4.4.1.1.1.1.1.1.1" xref="A4.Ex79.m2.4.4.1.1.1.1.1.1.1.cmml">−</mo><mi id="A4.Ex79.m2.4.4.1.1.1.1.1.1.3" xref="A4.Ex79.m2.4.4.1.1.1.1.1.1.3.cmml">μ</mi></mrow><mo id="A4.Ex79.m2.4.4.1.1.1.1.1.3" stretchy="false" xref="A4.Ex79.m2.4.4.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex79.m2.4.4.1.1.1.2a" xref="A4.Ex79.m2.4.4.1.1.1.2.cmml">⁢</mo><msup id="A4.Ex79.m2.4.4.1.1.1.4" xref="A4.Ex79.m2.4.4.1.1.1.4.cmml"><mrow id="A4.Ex79.m2.4.4.1.1.1.4.2.2" xref="A4.Ex79.m2.4.4.1.1.1.4.2.1.cmml"><mo id="A4.Ex79.m2.4.4.1.1.1.4.2.2.1" stretchy="false" xref="A4.Ex79.m2.4.4.1.1.1.4.2.1.1.cmml">‖</mo><mi id="A4.Ex79.m2.1.1" xref="A4.Ex79.m2.1.1.cmml">𝒆</mi><mo id="A4.Ex79.m2.4.4.1.1.1.4.2.2.2" stretchy="false" xref="A4.Ex79.m2.4.4.1.1.1.4.2.1.1.cmml">‖</mo></mrow><mn id="A4.Ex79.m2.4.4.1.1.1.4.3" xref="A4.Ex79.m2.4.4.1.1.1.4.3.cmml">2</mn></msup></mrow><mo id="A4.Ex79.m2.4.4.1.1.2" xref="A4.Ex79.m2.4.4.1.1.2.cmml">/</mo><mn id="A4.Ex79.m2.4.4.1.1.3" xref="A4.Ex79.m2.4.4.1.1.3.cmml">2</mn></mrow></mrow><mo id="A4.Ex79.m2.5.5.3" xref="A4.Ex79.m2.5.5.3.cmml">−</mo><mrow id="A4.Ex79.m2.5.5.2" xref="A4.Ex79.m2.5.5.2.cmml"><mrow id="A4.Ex79.m2.5.5.2.1" xref="A4.Ex79.m2.5.5.2.1.cmml"><msub id="A4.Ex79.m2.5.5.2.1.3" xref="A4.Ex79.m2.5.5.2.1.3.cmml"><mi id="A4.Ex79.m2.5.5.2.1.3.2" xref="A4.Ex79.m2.5.5.2.1.3.2.cmml">k</mi><mi id="A4.Ex79.m2.5.5.2.1.3.3" mathvariant="normal" xref="A4.Ex79.m2.5.5.2.1.3.3.cmml">c</mi></msub><mo id="A4.Ex79.m2.5.5.2.1.2" xref="A4.Ex79.m2.5.5.2.1.2.cmml">⁢</mo><mi id="A4.Ex79.m2.5.5.2.1.4" xref="A4.Ex79.m2.5.5.2.1.4.cmml">μ</mi><mo id="A4.Ex79.m2.5.5.2.1.2a" xref="A4.Ex79.m2.5.5.2.1.2.cmml">⁢</mo><mrow id="A4.Ex79.m2.5.5.2.1.1.1" xref="A4.Ex79.m2.5.5.2.1.1.1.1.cmml"><mo id="A4.Ex79.m2.5.5.2.1.1.1.2" stretchy="false" xref="A4.Ex79.m2.5.5.2.1.1.1.1.cmml">(</mo><mrow id="A4.Ex79.m2.5.5.2.1.1.1.1" xref="A4.Ex79.m2.5.5.2.1.1.1.1.cmml"><msup id="A4.Ex79.m2.5.5.2.1.1.1.1.3" xref="A4.Ex79.m2.5.5.2.1.1.1.1.3.cmml"><mrow id="A4.Ex79.m2.5.5.2.1.1.1.1.3.2.2" xref="A4.Ex79.m2.5.5.2.1.1.1.1.3.2.1.cmml"><mo id="A4.Ex79.m2.5.5.2.1.1.1.1.3.2.2.1" stretchy="false" xref="A4.Ex79.m2.5.5.2.1.1.1.1.3.2.1.1.cmml">‖</mo><mi id="A4.Ex79.m2.2.2" xref="A4.Ex79.m2.2.2.cmml">𝒆</mi><mo id="A4.Ex79.m2.5.5.2.1.1.1.1.3.2.2.2" stretchy="false" xref="A4.Ex79.m2.5.5.2.1.1.1.1.3.2.1.1.cmml">‖</mo></mrow><mn id="A4.Ex79.m2.5.5.2.1.1.1.1.3.3" xref="A4.Ex79.m2.5.5.2.1.1.1.1.3.3.cmml">2</mn></msup><mo id="A4.Ex79.m2.5.5.2.1.1.1.1.2" xref="A4.Ex79.m2.5.5.2.1.1.1.1.2.cmml">−</mo><mrow id="A4.Ex79.m2.5.5.2.1.1.1.1.1" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.cmml"><mrow id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.cmml"><mrow id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.3" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.3.cmml"><mn id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.3.2" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.3.2.cmml">2</mn><mo id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.3.1" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.3.1.cmml">⁢</mo><mover accent="true" id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.3.3" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.3.3.cmml"><mi id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.3.3.2" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.3.3.2.cmml">d</mi><mo id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.3.3.1" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.3.3.1.cmml">¯</mo></mover></mrow><mo id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.2" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.2.cmml">/</mo><mrow id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.cmml"><mo id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.cmml"><msub id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.2" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.2.2" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.2.2.cmml">k</mi><mi id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.2.3" mathvariant="normal" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.2.3.cmml">c</mi></msub><mo id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.1" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.3" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.3.cmml">μ</mi></mrow><mo id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A4.Ex79.m2.5.5.2.1.1.1.1.1.2" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex79.m2.5.5.2.1.1.1.1.1.3.2" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.3.1.cmml"><mo id="A4.Ex79.m2.5.5.2.1.1.1.1.1.3.2.1" stretchy="false" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.3.1.1.cmml">‖</mo><mi id="A4.Ex79.m2.3.3" xref="A4.Ex79.m2.3.3.cmml">𝒆</mi><mo id="A4.Ex79.m2.5.5.2.1.1.1.1.1.3.2.2" stretchy="false" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.3.1.1.cmml">‖</mo></mrow></mrow></mrow><mo id="A4.Ex79.m2.5.5.2.1.1.1.3" stretchy="false" xref="A4.Ex79.m2.5.5.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A4.Ex79.m2.5.5.2.2" xref="A4.Ex79.m2.5.5.2.2.cmml">/</mo><mn id="A4.Ex79.m2.5.5.2.3" xref="A4.Ex79.m2.5.5.2.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex79.m2.5b"><apply id="A4.Ex79.m2.5.5.cmml" xref="A4.Ex79.m2.5.5"><minus id="A4.Ex79.m2.5.5.3.cmml" xref="A4.Ex79.m2.5.5.3"></minus><apply id="A4.Ex79.m2.4.4.1.cmml" xref="A4.Ex79.m2.4.4.1"><minus id="A4.Ex79.m2.4.4.1.2.cmml" xref="A4.Ex79.m2.4.4.1"></minus><apply id="A4.Ex79.m2.4.4.1.1.cmml" xref="A4.Ex79.m2.4.4.1.1"><divide id="A4.Ex79.m2.4.4.1.1.2.cmml" xref="A4.Ex79.m2.4.4.1.1.2"></divide><apply id="A4.Ex79.m2.4.4.1.1.1.cmml" xref="A4.Ex79.m2.4.4.1.1.1"><times id="A4.Ex79.m2.4.4.1.1.1.2.cmml" xref="A4.Ex79.m2.4.4.1.1.1.2"></times><apply id="A4.Ex79.m2.4.4.1.1.1.3.cmml" xref="A4.Ex79.m2.4.4.1.1.1.3"><csymbol cd="ambiguous" id="A4.Ex79.m2.4.4.1.1.1.3.1.cmml" xref="A4.Ex79.m2.4.4.1.1.1.3">subscript</csymbol><ci id="A4.Ex79.m2.4.4.1.1.1.3.2.cmml" xref="A4.Ex79.m2.4.4.1.1.1.3.2">𝑘</ci><ci id="A4.Ex79.m2.4.4.1.1.1.3.3.cmml" xref="A4.Ex79.m2.4.4.1.1.1.3.3">c</ci></apply><apply id="A4.Ex79.m2.4.4.1.1.1.1.1.1.cmml" xref="A4.Ex79.m2.4.4.1.1.1.1.1"><minus id="A4.Ex79.m2.4.4.1.1.1.1.1.1.1.cmml" xref="A4.Ex79.m2.4.4.1.1.1.1.1.1.1"></minus><cn id="A4.Ex79.m2.4.4.1.1.1.1.1.1.2.cmml" type="integer" xref="A4.Ex79.m2.4.4.1.1.1.1.1.1.2">1</cn><ci id="A4.Ex79.m2.4.4.1.1.1.1.1.1.3.cmml" xref="A4.Ex79.m2.4.4.1.1.1.1.1.1.3">𝜇</ci></apply><apply id="A4.Ex79.m2.4.4.1.1.1.4.cmml" xref="A4.Ex79.m2.4.4.1.1.1.4"><csymbol cd="ambiguous" id="A4.Ex79.m2.4.4.1.1.1.4.1.cmml" xref="A4.Ex79.m2.4.4.1.1.1.4">superscript</csymbol><apply id="A4.Ex79.m2.4.4.1.1.1.4.2.1.cmml" xref="A4.Ex79.m2.4.4.1.1.1.4.2.2"><csymbol cd="latexml" id="A4.Ex79.m2.4.4.1.1.1.4.2.1.1.cmml" xref="A4.Ex79.m2.4.4.1.1.1.4.2.2.1">norm</csymbol><ci id="A4.Ex79.m2.1.1.cmml" xref="A4.Ex79.m2.1.1">𝒆</ci></apply><cn id="A4.Ex79.m2.4.4.1.1.1.4.3.cmml" type="integer" xref="A4.Ex79.m2.4.4.1.1.1.4.3">2</cn></apply></apply><cn id="A4.Ex79.m2.4.4.1.1.3.cmml" type="integer" xref="A4.Ex79.m2.4.4.1.1.3">2</cn></apply></apply><apply id="A4.Ex79.m2.5.5.2.cmml" xref="A4.Ex79.m2.5.5.2"><divide id="A4.Ex79.m2.5.5.2.2.cmml" xref="A4.Ex79.m2.5.5.2.2"></divide><apply id="A4.Ex79.m2.5.5.2.1.cmml" xref="A4.Ex79.m2.5.5.2.1"><times id="A4.Ex79.m2.5.5.2.1.2.cmml" xref="A4.Ex79.m2.5.5.2.1.2"></times><apply id="A4.Ex79.m2.5.5.2.1.3.cmml" xref="A4.Ex79.m2.5.5.2.1.3"><csymbol cd="ambiguous" id="A4.Ex79.m2.5.5.2.1.3.1.cmml" xref="A4.Ex79.m2.5.5.2.1.3">subscript</csymbol><ci id="A4.Ex79.m2.5.5.2.1.3.2.cmml" xref="A4.Ex79.m2.5.5.2.1.3.2">𝑘</ci><ci id="A4.Ex79.m2.5.5.2.1.3.3.cmml" xref="A4.Ex79.m2.5.5.2.1.3.3">c</ci></apply><ci id="A4.Ex79.m2.5.5.2.1.4.cmml" xref="A4.Ex79.m2.5.5.2.1.4">𝜇</ci><apply id="A4.Ex79.m2.5.5.2.1.1.1.1.cmml" xref="A4.Ex79.m2.5.5.2.1.1.1"><minus id="A4.Ex79.m2.5.5.2.1.1.1.1.2.cmml" xref="A4.Ex79.m2.5.5.2.1.1.1.1.2"></minus><apply id="A4.Ex79.m2.5.5.2.1.1.1.1.3.cmml" xref="A4.Ex79.m2.5.5.2.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.Ex79.m2.5.5.2.1.1.1.1.3.1.cmml" xref="A4.Ex79.m2.5.5.2.1.1.1.1.3">superscript</csymbol><apply id="A4.Ex79.m2.5.5.2.1.1.1.1.3.2.1.cmml" xref="A4.Ex79.m2.5.5.2.1.1.1.1.3.2.2"><csymbol cd="latexml" id="A4.Ex79.m2.5.5.2.1.1.1.1.3.2.1.1.cmml" xref="A4.Ex79.m2.5.5.2.1.1.1.1.3.2.2.1">norm</csymbol><ci id="A4.Ex79.m2.2.2.cmml" xref="A4.Ex79.m2.2.2">𝒆</ci></apply><cn id="A4.Ex79.m2.5.5.2.1.1.1.1.3.3.cmml" type="integer" xref="A4.Ex79.m2.5.5.2.1.1.1.1.3.3">2</cn></apply><apply id="A4.Ex79.m2.5.5.2.1.1.1.1.1.cmml" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1"><times id="A4.Ex79.m2.5.5.2.1.1.1.1.1.2.cmml" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.2"></times><apply id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.cmml" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1"><divide id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.2.cmml" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.2"></divide><apply id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.3.cmml" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.3"><times id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.3.1.cmml" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.3.1"></times><cn id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.3.2.cmml" type="integer" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.3.2">2</cn><apply id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.3.3.cmml" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.3.3"><ci id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.3.3.1.cmml" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.3.3.1">¯</ci><ci id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.3.3.2.cmml" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.3.3.2">𝑑</ci></apply></apply><apply id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.cmml" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1"><times id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.1.cmml" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.1"></times><apply id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.2.cmml" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.2">subscript</csymbol><ci id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.2.2">𝑘</ci><ci id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.2.3.cmml" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.2.3">c</ci></apply><ci id="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.3.cmml" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.1.1.1.1.3">𝜇</ci></apply></apply><apply id="A4.Ex79.m2.5.5.2.1.1.1.1.1.3.1.cmml" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.3.2"><csymbol cd="latexml" id="A4.Ex79.m2.5.5.2.1.1.1.1.1.3.1.1.cmml" xref="A4.Ex79.m2.5.5.2.1.1.1.1.1.3.2.1">norm</csymbol><ci id="A4.Ex79.m2.3.3.cmml" xref="A4.Ex79.m2.3.3">𝒆</ci></apply></apply></apply></apply><cn id="A4.Ex79.m2.5.5.2.3.cmml" type="integer" xref="A4.Ex79.m2.5.5.2.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex79.m2.5c">\displaystyle-k_{\rm c}(1-\mu)\|\bm{e}\|^{2}/2-k_{\rm c}\mu(\|\bm{e}\|^{2}-2% \bar{d}/(k_{\rm c}\mu)\|\bm{e}\|)/2</annotation><annotation encoding="application/x-llamapun" id="A4.Ex79.m2.5d">- italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ( 1 - italic_μ ) ∥ bold_italic_e ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 - italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT italic_μ ( ∥ bold_italic_e ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 over¯ start_ARG italic_d end_ARG / ( italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT italic_μ ) ∥ bold_italic_e ∥ ) / 2</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex80"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-\kappa\sigma^{*}(1-\mu)\|\tilde{\bm{\theta}}_{\zeta}\|^{2}-% \kappa\sigma^{*}\mu(\|\tilde{\bm{\theta}}_{\zeta}\|^{2}-(1+p){\bar{d}}_{\rm g}% /(\sigma^{*}\mu)\|\tilde{\bm{\theta}}_{\zeta}\|)" class="ltx_Math" display="inline" id="A4.Ex80.m1.3"><semantics id="A4.Ex80.m1.3a"><mrow id="A4.Ex80.m1.3.3" xref="A4.Ex80.m1.3.3.cmml"><mrow id="A4.Ex80.m1.2.2.2" xref="A4.Ex80.m1.2.2.2.cmml"><mo id="A4.Ex80.m1.2.2.2a" xref="A4.Ex80.m1.2.2.2.cmml">−</mo><mrow id="A4.Ex80.m1.2.2.2.2" xref="A4.Ex80.m1.2.2.2.2.cmml"><mi id="A4.Ex80.m1.2.2.2.2.4" xref="A4.Ex80.m1.2.2.2.2.4.cmml">κ</mi><mo id="A4.Ex80.m1.2.2.2.2.3" xref="A4.Ex80.m1.2.2.2.2.3.cmml">⁢</mo><msup id="A4.Ex80.m1.2.2.2.2.5" xref="A4.Ex80.m1.2.2.2.2.5.cmml"><mi id="A4.Ex80.m1.2.2.2.2.5.2" xref="A4.Ex80.m1.2.2.2.2.5.2.cmml">σ</mi><mo id="A4.Ex80.m1.2.2.2.2.5.3" xref="A4.Ex80.m1.2.2.2.2.5.3.cmml">∗</mo></msup><mo id="A4.Ex80.m1.2.2.2.2.3a" xref="A4.Ex80.m1.2.2.2.2.3.cmml">⁢</mo><mrow id="A4.Ex80.m1.1.1.1.1.1.1" xref="A4.Ex80.m1.1.1.1.1.1.1.1.cmml"><mo id="A4.Ex80.m1.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex80.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex80.m1.1.1.1.1.1.1.1" xref="A4.Ex80.m1.1.1.1.1.1.1.1.cmml"><mn id="A4.Ex80.m1.1.1.1.1.1.1.1.2" xref="A4.Ex80.m1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A4.Ex80.m1.1.1.1.1.1.1.1.1" xref="A4.Ex80.m1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="A4.Ex80.m1.1.1.1.1.1.1.1.3" xref="A4.Ex80.m1.1.1.1.1.1.1.1.3.cmml">μ</mi></mrow><mo id="A4.Ex80.m1.1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex80.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex80.m1.2.2.2.2.3b" xref="A4.Ex80.m1.2.2.2.2.3.cmml">⁢</mo><msup id="A4.Ex80.m1.2.2.2.2.2" xref="A4.Ex80.m1.2.2.2.2.2.cmml"><mrow id="A4.Ex80.m1.2.2.2.2.2.1.1" xref="A4.Ex80.m1.2.2.2.2.2.1.2.cmml"><mo id="A4.Ex80.m1.2.2.2.2.2.1.1.2" stretchy="false" xref="A4.Ex80.m1.2.2.2.2.2.1.2.1.cmml">‖</mo><msub id="A4.Ex80.m1.2.2.2.2.2.1.1.1" xref="A4.Ex80.m1.2.2.2.2.2.1.1.1.cmml"><mover accent="true" id="A4.Ex80.m1.2.2.2.2.2.1.1.1.2" xref="A4.Ex80.m1.2.2.2.2.2.1.1.1.2.cmml"><mi id="A4.Ex80.m1.2.2.2.2.2.1.1.1.2.2" xref="A4.Ex80.m1.2.2.2.2.2.1.1.1.2.2.cmml">𝜽</mi><mo id="A4.Ex80.m1.2.2.2.2.2.1.1.1.2.1" xref="A4.Ex80.m1.2.2.2.2.2.1.1.1.2.1.cmml">~</mo></mover><mi id="A4.Ex80.m1.2.2.2.2.2.1.1.1.3" xref="A4.Ex80.m1.2.2.2.2.2.1.1.1.3.cmml">ζ</mi></msub><mo id="A4.Ex80.m1.2.2.2.2.2.1.1.3" stretchy="false" xref="A4.Ex80.m1.2.2.2.2.2.1.2.1.cmml">‖</mo></mrow><mn id="A4.Ex80.m1.2.2.2.2.2.3" xref="A4.Ex80.m1.2.2.2.2.2.3.cmml">2</mn></msup></mrow></mrow><mo id="A4.Ex80.m1.3.3.4" xref="A4.Ex80.m1.3.3.4.cmml">−</mo><mrow id="A4.Ex80.m1.3.3.3" xref="A4.Ex80.m1.3.3.3.cmml"><mi id="A4.Ex80.m1.3.3.3.3" xref="A4.Ex80.m1.3.3.3.3.cmml">κ</mi><mo id="A4.Ex80.m1.3.3.3.2" xref="A4.Ex80.m1.3.3.3.2.cmml">⁢</mo><msup id="A4.Ex80.m1.3.3.3.4" xref="A4.Ex80.m1.3.3.3.4.cmml"><mi id="A4.Ex80.m1.3.3.3.4.2" xref="A4.Ex80.m1.3.3.3.4.2.cmml">σ</mi><mo id="A4.Ex80.m1.3.3.3.4.3" xref="A4.Ex80.m1.3.3.3.4.3.cmml">∗</mo></msup><mo id="A4.Ex80.m1.3.3.3.2a" xref="A4.Ex80.m1.3.3.3.2.cmml">⁢</mo><mi id="A4.Ex80.m1.3.3.3.5" xref="A4.Ex80.m1.3.3.3.5.cmml">μ</mi><mo id="A4.Ex80.m1.3.3.3.2b" xref="A4.Ex80.m1.3.3.3.2.cmml">⁢</mo><mrow id="A4.Ex80.m1.3.3.3.1.1" xref="A4.Ex80.m1.3.3.3.1.1.1.cmml"><mo id="A4.Ex80.m1.3.3.3.1.1.2" stretchy="false" xref="A4.Ex80.m1.3.3.3.1.1.1.cmml">(</mo><mrow id="A4.Ex80.m1.3.3.3.1.1.1" xref="A4.Ex80.m1.3.3.3.1.1.1.cmml"><msup id="A4.Ex80.m1.3.3.3.1.1.1.1" xref="A4.Ex80.m1.3.3.3.1.1.1.1.cmml"><mrow id="A4.Ex80.m1.3.3.3.1.1.1.1.1.1" xref="A4.Ex80.m1.3.3.3.1.1.1.1.1.2.cmml"><mo id="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex80.m1.3.3.3.1.1.1.1.1.2.1.cmml">‖</mo><msub id="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.1" xref="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.1.cmml"><mover accent="true" id="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.1.2" xref="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.1.2.cmml"><mi id="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.1.2.2" xref="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.1.2.2.cmml">𝜽</mi><mo id="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.1.2.1" xref="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.1.2.1.cmml">~</mo></mover><mi id="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.1.3" xref="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.1.3.cmml">ζ</mi></msub><mo id="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex80.m1.3.3.3.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mn id="A4.Ex80.m1.3.3.3.1.1.1.1.3" xref="A4.Ex80.m1.3.3.3.1.1.1.1.3.cmml">2</mn></msup><mo id="A4.Ex80.m1.3.3.3.1.1.1.5" xref="A4.Ex80.m1.3.3.3.1.1.1.5.cmml">−</mo><mrow id="A4.Ex80.m1.3.3.3.1.1.1.4" xref="A4.Ex80.m1.3.3.3.1.1.1.4.cmml"><mrow id="A4.Ex80.m1.3.3.3.1.1.1.3.2" xref="A4.Ex80.m1.3.3.3.1.1.1.3.2.cmml"><mrow id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.cmml"><mrow id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.1.1" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.1.1.1.cmml"><mo id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.1.1.2" stretchy="false" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.1.1.1" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.1.1.1.cmml"><mn id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.1.1.1.2" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.1.1.1.2.cmml">1</mn><mo id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.1.1.1.1" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.1.1.1.1.cmml">+</mo><mi id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.1.1.1.3" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.1.1.3" stretchy="false" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.2" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.2.cmml">⁢</mo><msub id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.3" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.3.cmml"><mover accent="true" id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.3.2" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.3.2.cmml"><mi id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.3.2.2" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.3.2.2.cmml">d</mi><mo id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.3.2.1" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.3.2.1.cmml">¯</mo></mover><mi id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.3.3" mathvariant="normal" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.3.3.cmml">g</mi></msub></mrow><mo id="A4.Ex80.m1.3.3.3.1.1.1.3.2.3" xref="A4.Ex80.m1.3.3.3.1.1.1.3.2.3.cmml">/</mo><mrow id="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1" xref="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.cmml"><mo id="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.2" stretchy="false" xref="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.cmml">(</mo><mrow id="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1" xref="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.cmml"><msup id="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.2" xref="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.2.cmml"><mi id="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.2.2" xref="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.2.2.cmml">σ</mi><mo id="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.2.3" xref="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.2.3.cmml">∗</mo></msup><mo id="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.1" xref="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.1.cmml">⁢</mo><mi id="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.3" xref="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.3.cmml">μ</mi></mrow><mo id="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.3" stretchy="false" xref="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="A4.Ex80.m1.3.3.3.1.1.1.4.4" xref="A4.Ex80.m1.3.3.3.1.1.1.4.4.cmml">⁢</mo><mrow id="A4.Ex80.m1.3.3.3.1.1.1.4.3.1" xref="A4.Ex80.m1.3.3.3.1.1.1.4.3.2.cmml"><mo id="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.2" stretchy="false" xref="A4.Ex80.m1.3.3.3.1.1.1.4.3.2.1.cmml">‖</mo><msub id="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.1" xref="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.1.cmml"><mover accent="true" id="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.1.2" xref="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.1.2.cmml"><mi id="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.1.2.2" xref="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.1.2.2.cmml">𝜽</mi><mo id="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.1.2.1" xref="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.1.2.1.cmml">~</mo></mover><mi id="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.1.3" xref="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.1.3.cmml">ζ</mi></msub><mo id="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.3" stretchy="false" xref="A4.Ex80.m1.3.3.3.1.1.1.4.3.2.1.cmml">‖</mo></mrow></mrow></mrow><mo id="A4.Ex80.m1.3.3.3.1.1.3" stretchy="false" xref="A4.Ex80.m1.3.3.3.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex80.m1.3b"><apply id="A4.Ex80.m1.3.3.cmml" xref="A4.Ex80.m1.3.3"><minus id="A4.Ex80.m1.3.3.4.cmml" xref="A4.Ex80.m1.3.3.4"></minus><apply id="A4.Ex80.m1.2.2.2.cmml" xref="A4.Ex80.m1.2.2.2"><minus id="A4.Ex80.m1.2.2.2.3.cmml" xref="A4.Ex80.m1.2.2.2"></minus><apply id="A4.Ex80.m1.2.2.2.2.cmml" xref="A4.Ex80.m1.2.2.2.2"><times id="A4.Ex80.m1.2.2.2.2.3.cmml" xref="A4.Ex80.m1.2.2.2.2.3"></times><ci id="A4.Ex80.m1.2.2.2.2.4.cmml" xref="A4.Ex80.m1.2.2.2.2.4">𝜅</ci><apply id="A4.Ex80.m1.2.2.2.2.5.cmml" xref="A4.Ex80.m1.2.2.2.2.5"><csymbol cd="ambiguous" id="A4.Ex80.m1.2.2.2.2.5.1.cmml" xref="A4.Ex80.m1.2.2.2.2.5">superscript</csymbol><ci id="A4.Ex80.m1.2.2.2.2.5.2.cmml" xref="A4.Ex80.m1.2.2.2.2.5.2">𝜎</ci><times id="A4.Ex80.m1.2.2.2.2.5.3.cmml" xref="A4.Ex80.m1.2.2.2.2.5.3"></times></apply><apply id="A4.Ex80.m1.1.1.1.1.1.1.1.cmml" xref="A4.Ex80.m1.1.1.1.1.1.1"><minus id="A4.Ex80.m1.1.1.1.1.1.1.1.1.cmml" xref="A4.Ex80.m1.1.1.1.1.1.1.1.1"></minus><cn id="A4.Ex80.m1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A4.Ex80.m1.1.1.1.1.1.1.1.2">1</cn><ci id="A4.Ex80.m1.1.1.1.1.1.1.1.3.cmml" xref="A4.Ex80.m1.1.1.1.1.1.1.1.3">𝜇</ci></apply><apply id="A4.Ex80.m1.2.2.2.2.2.cmml" xref="A4.Ex80.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="A4.Ex80.m1.2.2.2.2.2.2.cmml" xref="A4.Ex80.m1.2.2.2.2.2">superscript</csymbol><apply id="A4.Ex80.m1.2.2.2.2.2.1.2.cmml" xref="A4.Ex80.m1.2.2.2.2.2.1.1"><csymbol cd="latexml" id="A4.Ex80.m1.2.2.2.2.2.1.2.1.cmml" xref="A4.Ex80.m1.2.2.2.2.2.1.1.2">norm</csymbol><apply id="A4.Ex80.m1.2.2.2.2.2.1.1.1.cmml" xref="A4.Ex80.m1.2.2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="A4.Ex80.m1.2.2.2.2.2.1.1.1.1.cmml" xref="A4.Ex80.m1.2.2.2.2.2.1.1.1">subscript</csymbol><apply id="A4.Ex80.m1.2.2.2.2.2.1.1.1.2.cmml" xref="A4.Ex80.m1.2.2.2.2.2.1.1.1.2"><ci id="A4.Ex80.m1.2.2.2.2.2.1.1.1.2.1.cmml" xref="A4.Ex80.m1.2.2.2.2.2.1.1.1.2.1">~</ci><ci id="A4.Ex80.m1.2.2.2.2.2.1.1.1.2.2.cmml" xref="A4.Ex80.m1.2.2.2.2.2.1.1.1.2.2">𝜽</ci></apply><ci id="A4.Ex80.m1.2.2.2.2.2.1.1.1.3.cmml" xref="A4.Ex80.m1.2.2.2.2.2.1.1.1.3">𝜁</ci></apply></apply><cn id="A4.Ex80.m1.2.2.2.2.2.3.cmml" type="integer" xref="A4.Ex80.m1.2.2.2.2.2.3">2</cn></apply></apply></apply><apply id="A4.Ex80.m1.3.3.3.cmml" xref="A4.Ex80.m1.3.3.3"><times id="A4.Ex80.m1.3.3.3.2.cmml" xref="A4.Ex80.m1.3.3.3.2"></times><ci id="A4.Ex80.m1.3.3.3.3.cmml" xref="A4.Ex80.m1.3.3.3.3">𝜅</ci><apply id="A4.Ex80.m1.3.3.3.4.cmml" xref="A4.Ex80.m1.3.3.3.4"><csymbol cd="ambiguous" id="A4.Ex80.m1.3.3.3.4.1.cmml" xref="A4.Ex80.m1.3.3.3.4">superscript</csymbol><ci id="A4.Ex80.m1.3.3.3.4.2.cmml" xref="A4.Ex80.m1.3.3.3.4.2">𝜎</ci><times id="A4.Ex80.m1.3.3.3.4.3.cmml" xref="A4.Ex80.m1.3.3.3.4.3"></times></apply><ci id="A4.Ex80.m1.3.3.3.5.cmml" xref="A4.Ex80.m1.3.3.3.5">𝜇</ci><apply id="A4.Ex80.m1.3.3.3.1.1.1.cmml" xref="A4.Ex80.m1.3.3.3.1.1"><minus id="A4.Ex80.m1.3.3.3.1.1.1.5.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.5"></minus><apply id="A4.Ex80.m1.3.3.3.1.1.1.1.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.1"><csymbol cd="ambiguous" id="A4.Ex80.m1.3.3.3.1.1.1.1.2.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.1">superscript</csymbol><apply id="A4.Ex80.m1.3.3.3.1.1.1.1.1.2.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.1.1.1"><csymbol cd="latexml" id="A4.Ex80.m1.3.3.3.1.1.1.1.1.2.1.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.2">norm</csymbol><apply id="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.1.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.1.1.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.1">subscript</csymbol><apply id="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.1.2.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.1.2"><ci id="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.1.2.1.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.1.2.1">~</ci><ci id="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.1.2.2.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.1.2.2">𝜽</ci></apply><ci id="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.1.3.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.1.1.1.1.3">𝜁</ci></apply></apply><cn id="A4.Ex80.m1.3.3.3.1.1.1.1.3.cmml" type="integer" xref="A4.Ex80.m1.3.3.3.1.1.1.1.3">2</cn></apply><apply id="A4.Ex80.m1.3.3.3.1.1.1.4.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.4"><times id="A4.Ex80.m1.3.3.3.1.1.1.4.4.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.4.4"></times><apply id="A4.Ex80.m1.3.3.3.1.1.1.3.2.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.3.2"><divide id="A4.Ex80.m1.3.3.3.1.1.1.3.2.3.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.3.2.3"></divide><apply id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1"><times id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.2.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.2"></times><apply id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.1.1.1.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.1.1"><plus id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.1.1.1.1.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.1.1.1.1"></plus><cn id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.1.1.1.2.cmml" type="integer" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.1.1.1.2">1</cn><ci id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.1.1.1.3.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.1.1.1.3">𝑝</ci></apply><apply id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.3.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.3"><csymbol cd="ambiguous" id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.3.1.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.3">subscript</csymbol><apply id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.3.2.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.3.2"><ci id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.3.2.1.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.3.2.1">¯</ci><ci id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.3.2.2.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.3.2.2">𝑑</ci></apply><ci id="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.3.3.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.2.1.1.3.3">g</ci></apply></apply><apply id="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1"><times id="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.1.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.1"></times><apply id="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.2.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.2"><csymbol cd="ambiguous" id="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.2.1.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.2">superscript</csymbol><ci id="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.2.2.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.2.2">𝜎</ci><times id="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.2.3.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.2.3"></times></apply><ci id="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.3.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.3.2.2.1.1.3">𝜇</ci></apply></apply><apply id="A4.Ex80.m1.3.3.3.1.1.1.4.3.2.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.4.3.1"><csymbol cd="latexml" id="A4.Ex80.m1.3.3.3.1.1.1.4.3.2.1.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.2">norm</csymbol><apply id="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.1.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.1"><csymbol cd="ambiguous" id="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.1.1.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.1">subscript</csymbol><apply id="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.1.2.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.1.2"><ci id="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.1.2.1.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.1.2.1">~</ci><ci id="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.1.2.2.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.1.2.2">𝜽</ci></apply><ci id="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.1.3.cmml" xref="A4.Ex80.m1.3.3.3.1.1.1.4.3.1.1.3">𝜁</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex80.m1.3c">\displaystyle-\kappa\sigma^{*}(1-\mu)\|\tilde{\bm{\theta}}_{\zeta}\|^{2}-% \kappa\sigma^{*}\mu(\|\tilde{\bm{\theta}}_{\zeta}\|^{2}-(1+p){\bar{d}}_{\rm g}% /(\sigma^{*}\mu)\|\tilde{\bm{\theta}}_{\zeta}\|)</annotation><annotation encoding="application/x-llamapun" id="A4.Ex80.m1.3d">- italic_κ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( 1 - italic_μ ) ∥ over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_κ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_μ ( ∥ over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ( 1 + italic_p ) over¯ start_ARG italic_d end_ARG start_POSTSUBSCRIPT roman_g end_POSTSUBSCRIPT / ( italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_μ ) ∥ over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ∥ )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex81"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle+(1+p)\bar{\epsilon}\bar{d}" class="ltx_Math" display="inline" id="A4.Ex81.m1.1"><semantics id="A4.Ex81.m1.1a"><mrow id="A4.Ex81.m1.1.1" xref="A4.Ex81.m1.1.1.cmml"><mo id="A4.Ex81.m1.1.1a" xref="A4.Ex81.m1.1.1.cmml">+</mo><mrow id="A4.Ex81.m1.1.1.1" xref="A4.Ex81.m1.1.1.1.cmml"><mrow id="A4.Ex81.m1.1.1.1.1.1" xref="A4.Ex81.m1.1.1.1.1.1.1.cmml"><mo id="A4.Ex81.m1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex81.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex81.m1.1.1.1.1.1.1" xref="A4.Ex81.m1.1.1.1.1.1.1.cmml"><mn id="A4.Ex81.m1.1.1.1.1.1.1.2" xref="A4.Ex81.m1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A4.Ex81.m1.1.1.1.1.1.1.1" xref="A4.Ex81.m1.1.1.1.1.1.1.1.cmml">+</mo><mi id="A4.Ex81.m1.1.1.1.1.1.1.3" xref="A4.Ex81.m1.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A4.Ex81.m1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex81.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex81.m1.1.1.1.2" xref="A4.Ex81.m1.1.1.1.2.cmml">⁢</mo><mover accent="true" id="A4.Ex81.m1.1.1.1.3" xref="A4.Ex81.m1.1.1.1.3.cmml"><mi id="A4.Ex81.m1.1.1.1.3.2" xref="A4.Ex81.m1.1.1.1.3.2.cmml">ϵ</mi><mo id="A4.Ex81.m1.1.1.1.3.1" xref="A4.Ex81.m1.1.1.1.3.1.cmml">¯</mo></mover><mo id="A4.Ex81.m1.1.1.1.2a" xref="A4.Ex81.m1.1.1.1.2.cmml">⁢</mo><mover accent="true" id="A4.Ex81.m1.1.1.1.4" xref="A4.Ex81.m1.1.1.1.4.cmml"><mi id="A4.Ex81.m1.1.1.1.4.2" xref="A4.Ex81.m1.1.1.1.4.2.cmml">d</mi><mo id="A4.Ex81.m1.1.1.1.4.1" xref="A4.Ex81.m1.1.1.1.4.1.cmml">¯</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex81.m1.1b"><apply id="A4.Ex81.m1.1.1.cmml" xref="A4.Ex81.m1.1.1"><plus id="A4.Ex81.m1.1.1.2.cmml" xref="A4.Ex81.m1.1.1"></plus><apply id="A4.Ex81.m1.1.1.1.cmml" xref="A4.Ex81.m1.1.1.1"><times id="A4.Ex81.m1.1.1.1.2.cmml" xref="A4.Ex81.m1.1.1.1.2"></times><apply id="A4.Ex81.m1.1.1.1.1.1.1.cmml" xref="A4.Ex81.m1.1.1.1.1.1"><plus id="A4.Ex81.m1.1.1.1.1.1.1.1.cmml" xref="A4.Ex81.m1.1.1.1.1.1.1.1"></plus><cn id="A4.Ex81.m1.1.1.1.1.1.1.2.cmml" type="integer" xref="A4.Ex81.m1.1.1.1.1.1.1.2">1</cn><ci id="A4.Ex81.m1.1.1.1.1.1.1.3.cmml" xref="A4.Ex81.m1.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="A4.Ex81.m1.1.1.1.3.cmml" xref="A4.Ex81.m1.1.1.1.3"><ci id="A4.Ex81.m1.1.1.1.3.1.cmml" xref="A4.Ex81.m1.1.1.1.3.1">¯</ci><ci id="A4.Ex81.m1.1.1.1.3.2.cmml" xref="A4.Ex81.m1.1.1.1.3.2">italic-ϵ</ci></apply><apply id="A4.Ex81.m1.1.1.1.4.cmml" xref="A4.Ex81.m1.1.1.1.4"><ci id="A4.Ex81.m1.1.1.1.4.1.cmml" xref="A4.Ex81.m1.1.1.1.4.1">¯</ci><ci id="A4.Ex81.m1.1.1.1.4.2.cmml" xref="A4.Ex81.m1.1.1.1.4.2">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex81.m1.1c">\displaystyle+(1+p)\bar{\epsilon}\bar{d}</annotation><annotation encoding="application/x-llamapun" id="A4.Ex81.m1.1d">+ ( 1 + italic_p ) over¯ start_ARG italic_ϵ end_ARG over¯ start_ARG italic_d end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.1.p1.49">with <math alttext="\mu\in(0,1)" class="ltx_Math" display="inline" id="A4.1.p1.44.m1.2"><semantics id="A4.1.p1.44.m1.2a"><mrow id="A4.1.p1.44.m1.2.3" xref="A4.1.p1.44.m1.2.3.cmml"><mi id="A4.1.p1.44.m1.2.3.2" xref="A4.1.p1.44.m1.2.3.2.cmml">μ</mi><mo id="A4.1.p1.44.m1.2.3.1" xref="A4.1.p1.44.m1.2.3.1.cmml">∈</mo><mrow id="A4.1.p1.44.m1.2.3.3.2" xref="A4.1.p1.44.m1.2.3.3.1.cmml"><mo id="A4.1.p1.44.m1.2.3.3.2.1" stretchy="false" xref="A4.1.p1.44.m1.2.3.3.1.cmml">(</mo><mn id="A4.1.p1.44.m1.1.1" xref="A4.1.p1.44.m1.1.1.cmml">0</mn><mo id="A4.1.p1.44.m1.2.3.3.2.2" xref="A4.1.p1.44.m1.2.3.3.1.cmml">,</mo><mn id="A4.1.p1.44.m1.2.2" xref="A4.1.p1.44.m1.2.2.cmml">1</mn><mo id="A4.1.p1.44.m1.2.3.3.2.3" stretchy="false" xref="A4.1.p1.44.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.44.m1.2b"><apply id="A4.1.p1.44.m1.2.3.cmml" xref="A4.1.p1.44.m1.2.3"><in id="A4.1.p1.44.m1.2.3.1.cmml" xref="A4.1.p1.44.m1.2.3.1"></in><ci id="A4.1.p1.44.m1.2.3.2.cmml" xref="A4.1.p1.44.m1.2.3.2">𝜇</ci><interval closure="open" id="A4.1.p1.44.m1.2.3.3.1.cmml" xref="A4.1.p1.44.m1.2.3.3.2"><cn id="A4.1.p1.44.m1.1.1.cmml" type="integer" xref="A4.1.p1.44.m1.1.1">0</cn><cn id="A4.1.p1.44.m1.2.2.cmml" type="integer" xref="A4.1.p1.44.m1.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.44.m1.2c">\mu\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.44.m1.2d">italic_μ ∈ ( 0 , 1 )</annotation></semantics></math>. Applying Young’s inequality <math alttext="2ab-a^{2}\leq b^{2}" class="ltx_Math" display="inline" id="A4.1.p1.45.m2.1"><semantics id="A4.1.p1.45.m2.1a"><mrow id="A4.1.p1.45.m2.1.1" xref="A4.1.p1.45.m2.1.1.cmml"><mrow id="A4.1.p1.45.m2.1.1.2" xref="A4.1.p1.45.m2.1.1.2.cmml"><mrow id="A4.1.p1.45.m2.1.1.2.2" xref="A4.1.p1.45.m2.1.1.2.2.cmml"><mn id="A4.1.p1.45.m2.1.1.2.2.2" xref="A4.1.p1.45.m2.1.1.2.2.2.cmml">2</mn><mo id="A4.1.p1.45.m2.1.1.2.2.1" xref="A4.1.p1.45.m2.1.1.2.2.1.cmml">⁢</mo><mi id="A4.1.p1.45.m2.1.1.2.2.3" xref="A4.1.p1.45.m2.1.1.2.2.3.cmml">a</mi><mo id="A4.1.p1.45.m2.1.1.2.2.1a" xref="A4.1.p1.45.m2.1.1.2.2.1.cmml">⁢</mo><mi id="A4.1.p1.45.m2.1.1.2.2.4" xref="A4.1.p1.45.m2.1.1.2.2.4.cmml">b</mi></mrow><mo id="A4.1.p1.45.m2.1.1.2.1" xref="A4.1.p1.45.m2.1.1.2.1.cmml">−</mo><msup id="A4.1.p1.45.m2.1.1.2.3" xref="A4.1.p1.45.m2.1.1.2.3.cmml"><mi id="A4.1.p1.45.m2.1.1.2.3.2" xref="A4.1.p1.45.m2.1.1.2.3.2.cmml">a</mi><mn id="A4.1.p1.45.m2.1.1.2.3.3" xref="A4.1.p1.45.m2.1.1.2.3.3.cmml">2</mn></msup></mrow><mo id="A4.1.p1.45.m2.1.1.1" xref="A4.1.p1.45.m2.1.1.1.cmml">≤</mo><msup id="A4.1.p1.45.m2.1.1.3" xref="A4.1.p1.45.m2.1.1.3.cmml"><mi id="A4.1.p1.45.m2.1.1.3.2" xref="A4.1.p1.45.m2.1.1.3.2.cmml">b</mi><mn id="A4.1.p1.45.m2.1.1.3.3" xref="A4.1.p1.45.m2.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.45.m2.1b"><apply id="A4.1.p1.45.m2.1.1.cmml" xref="A4.1.p1.45.m2.1.1"><leq id="A4.1.p1.45.m2.1.1.1.cmml" xref="A4.1.p1.45.m2.1.1.1"></leq><apply id="A4.1.p1.45.m2.1.1.2.cmml" xref="A4.1.p1.45.m2.1.1.2"><minus id="A4.1.p1.45.m2.1.1.2.1.cmml" xref="A4.1.p1.45.m2.1.1.2.1"></minus><apply id="A4.1.p1.45.m2.1.1.2.2.cmml" xref="A4.1.p1.45.m2.1.1.2.2"><times id="A4.1.p1.45.m2.1.1.2.2.1.cmml" xref="A4.1.p1.45.m2.1.1.2.2.1"></times><cn id="A4.1.p1.45.m2.1.1.2.2.2.cmml" type="integer" xref="A4.1.p1.45.m2.1.1.2.2.2">2</cn><ci id="A4.1.p1.45.m2.1.1.2.2.3.cmml" xref="A4.1.p1.45.m2.1.1.2.2.3">𝑎</ci><ci id="A4.1.p1.45.m2.1.1.2.2.4.cmml" xref="A4.1.p1.45.m2.1.1.2.2.4">𝑏</ci></apply><apply id="A4.1.p1.45.m2.1.1.2.3.cmml" xref="A4.1.p1.45.m2.1.1.2.3"><csymbol cd="ambiguous" id="A4.1.p1.45.m2.1.1.2.3.1.cmml" xref="A4.1.p1.45.m2.1.1.2.3">superscript</csymbol><ci id="A4.1.p1.45.m2.1.1.2.3.2.cmml" xref="A4.1.p1.45.m2.1.1.2.3.2">𝑎</ci><cn id="A4.1.p1.45.m2.1.1.2.3.3.cmml" type="integer" xref="A4.1.p1.45.m2.1.1.2.3.3">2</cn></apply></apply><apply id="A4.1.p1.45.m2.1.1.3.cmml" xref="A4.1.p1.45.m2.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.45.m2.1.1.3.1.cmml" xref="A4.1.p1.45.m2.1.1.3">superscript</csymbol><ci id="A4.1.p1.45.m2.1.1.3.2.cmml" xref="A4.1.p1.45.m2.1.1.3.2">𝑏</ci><cn id="A4.1.p1.45.m2.1.1.3.3.cmml" type="integer" xref="A4.1.p1.45.m2.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.45.m2.1c">2ab-a^{2}\leq b^{2}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.45.m2.1d">2 italic_a italic_b - italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="a=\|\bm{e}\|" class="ltx_Math" display="inline" id="A4.1.p1.46.m3.1"><semantics id="A4.1.p1.46.m3.1a"><mrow id="A4.1.p1.46.m3.1.2" xref="A4.1.p1.46.m3.1.2.cmml"><mi id="A4.1.p1.46.m3.1.2.2" xref="A4.1.p1.46.m3.1.2.2.cmml">a</mi><mo id="A4.1.p1.46.m3.1.2.1" xref="A4.1.p1.46.m3.1.2.1.cmml">=</mo><mrow id="A4.1.p1.46.m3.1.2.3.2" xref="A4.1.p1.46.m3.1.2.3.1.cmml"><mo id="A4.1.p1.46.m3.1.2.3.2.1" stretchy="false" xref="A4.1.p1.46.m3.1.2.3.1.1.cmml">‖</mo><mi id="A4.1.p1.46.m3.1.1" xref="A4.1.p1.46.m3.1.1.cmml">𝒆</mi><mo id="A4.1.p1.46.m3.1.2.3.2.2" stretchy="false" xref="A4.1.p1.46.m3.1.2.3.1.1.cmml">‖</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.46.m3.1b"><apply id="A4.1.p1.46.m3.1.2.cmml" xref="A4.1.p1.46.m3.1.2"><eq id="A4.1.p1.46.m3.1.2.1.cmml" xref="A4.1.p1.46.m3.1.2.1"></eq><ci id="A4.1.p1.46.m3.1.2.2.cmml" xref="A4.1.p1.46.m3.1.2.2">𝑎</ci><apply id="A4.1.p1.46.m3.1.2.3.1.cmml" xref="A4.1.p1.46.m3.1.2.3.2"><csymbol cd="latexml" id="A4.1.p1.46.m3.1.2.3.1.1.cmml" xref="A4.1.p1.46.m3.1.2.3.2.1">norm</csymbol><ci id="A4.1.p1.46.m3.1.1.cmml" xref="A4.1.p1.46.m3.1.1">𝒆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.46.m3.1c">a=\|\bm{e}\|</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.46.m3.1d">italic_a = ∥ bold_italic_e ∥</annotation></semantics></math> or <math alttext="\|\tilde{\bm{\theta}}_{\zeta}\|" class="ltx_Math" display="inline" id="A4.1.p1.47.m4.1"><semantics id="A4.1.p1.47.m4.1a"><mrow id="A4.1.p1.47.m4.1.1.1" xref="A4.1.p1.47.m4.1.1.2.cmml"><mo id="A4.1.p1.47.m4.1.1.1.2" stretchy="false" xref="A4.1.p1.47.m4.1.1.2.1.cmml">‖</mo><msub id="A4.1.p1.47.m4.1.1.1.1" xref="A4.1.p1.47.m4.1.1.1.1.cmml"><mover accent="true" id="A4.1.p1.47.m4.1.1.1.1.2" xref="A4.1.p1.47.m4.1.1.1.1.2.cmml"><mi id="A4.1.p1.47.m4.1.1.1.1.2.2" xref="A4.1.p1.47.m4.1.1.1.1.2.2.cmml">𝜽</mi><mo id="A4.1.p1.47.m4.1.1.1.1.2.1" xref="A4.1.p1.47.m4.1.1.1.1.2.1.cmml">~</mo></mover><mi id="A4.1.p1.47.m4.1.1.1.1.3" xref="A4.1.p1.47.m4.1.1.1.1.3.cmml">ζ</mi></msub><mo id="A4.1.p1.47.m4.1.1.1.3" stretchy="false" xref="A4.1.p1.47.m4.1.1.2.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.47.m4.1b"><apply id="A4.1.p1.47.m4.1.1.2.cmml" xref="A4.1.p1.47.m4.1.1.1"><csymbol cd="latexml" id="A4.1.p1.47.m4.1.1.2.1.cmml" xref="A4.1.p1.47.m4.1.1.1.2">norm</csymbol><apply id="A4.1.p1.47.m4.1.1.1.1.cmml" xref="A4.1.p1.47.m4.1.1.1.1"><csymbol cd="ambiguous" id="A4.1.p1.47.m4.1.1.1.1.1.cmml" xref="A4.1.p1.47.m4.1.1.1.1">subscript</csymbol><apply id="A4.1.p1.47.m4.1.1.1.1.2.cmml" xref="A4.1.p1.47.m4.1.1.1.1.2"><ci id="A4.1.p1.47.m4.1.1.1.1.2.1.cmml" xref="A4.1.p1.47.m4.1.1.1.1.2.1">~</ci><ci id="A4.1.p1.47.m4.1.1.1.1.2.2.cmml" xref="A4.1.p1.47.m4.1.1.1.1.2.2">𝜽</ci></apply><ci id="A4.1.p1.47.m4.1.1.1.1.3.cmml" xref="A4.1.p1.47.m4.1.1.1.1.3">𝜁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.47.m4.1c">\|\tilde{\bm{\theta}}_{\zeta}\|</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.47.m4.1d">∥ over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ∥</annotation></semantics></math> and <math alttext="b=\bar{d}/(k_{\rm c}\mu)" class="ltx_Math" display="inline" id="A4.1.p1.48.m5.1"><semantics id="A4.1.p1.48.m5.1a"><mrow id="A4.1.p1.48.m5.1.1" xref="A4.1.p1.48.m5.1.1.cmml"><mi id="A4.1.p1.48.m5.1.1.3" xref="A4.1.p1.48.m5.1.1.3.cmml">b</mi><mo id="A4.1.p1.48.m5.1.1.2" xref="A4.1.p1.48.m5.1.1.2.cmml">=</mo><mrow id="A4.1.p1.48.m5.1.1.1" xref="A4.1.p1.48.m5.1.1.1.cmml"><mover accent="true" id="A4.1.p1.48.m5.1.1.1.3" xref="A4.1.p1.48.m5.1.1.1.3.cmml"><mi id="A4.1.p1.48.m5.1.1.1.3.2" xref="A4.1.p1.48.m5.1.1.1.3.2.cmml">d</mi><mo id="A4.1.p1.48.m5.1.1.1.3.1" xref="A4.1.p1.48.m5.1.1.1.3.1.cmml">¯</mo></mover><mo id="A4.1.p1.48.m5.1.1.1.2" xref="A4.1.p1.48.m5.1.1.1.2.cmml">/</mo><mrow id="A4.1.p1.48.m5.1.1.1.1.1" xref="A4.1.p1.48.m5.1.1.1.1.1.1.cmml"><mo id="A4.1.p1.48.m5.1.1.1.1.1.2" stretchy="false" xref="A4.1.p1.48.m5.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.1.p1.48.m5.1.1.1.1.1.1" xref="A4.1.p1.48.m5.1.1.1.1.1.1.cmml"><msub id="A4.1.p1.48.m5.1.1.1.1.1.1.2" xref="A4.1.p1.48.m5.1.1.1.1.1.1.2.cmml"><mi id="A4.1.p1.48.m5.1.1.1.1.1.1.2.2" xref="A4.1.p1.48.m5.1.1.1.1.1.1.2.2.cmml">k</mi><mi id="A4.1.p1.48.m5.1.1.1.1.1.1.2.3" mathvariant="normal" xref="A4.1.p1.48.m5.1.1.1.1.1.1.2.3.cmml">c</mi></msub><mo id="A4.1.p1.48.m5.1.1.1.1.1.1.1" xref="A4.1.p1.48.m5.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="A4.1.p1.48.m5.1.1.1.1.1.1.3" xref="A4.1.p1.48.m5.1.1.1.1.1.1.3.cmml">μ</mi></mrow><mo id="A4.1.p1.48.m5.1.1.1.1.1.3" stretchy="false" xref="A4.1.p1.48.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.48.m5.1b"><apply id="A4.1.p1.48.m5.1.1.cmml" xref="A4.1.p1.48.m5.1.1"><eq id="A4.1.p1.48.m5.1.1.2.cmml" xref="A4.1.p1.48.m5.1.1.2"></eq><ci id="A4.1.p1.48.m5.1.1.3.cmml" xref="A4.1.p1.48.m5.1.1.3">𝑏</ci><apply id="A4.1.p1.48.m5.1.1.1.cmml" xref="A4.1.p1.48.m5.1.1.1"><divide id="A4.1.p1.48.m5.1.1.1.2.cmml" xref="A4.1.p1.48.m5.1.1.1.2"></divide><apply id="A4.1.p1.48.m5.1.1.1.3.cmml" xref="A4.1.p1.48.m5.1.1.1.3"><ci id="A4.1.p1.48.m5.1.1.1.3.1.cmml" xref="A4.1.p1.48.m5.1.1.1.3.1">¯</ci><ci id="A4.1.p1.48.m5.1.1.1.3.2.cmml" xref="A4.1.p1.48.m5.1.1.1.3.2">𝑑</ci></apply><apply id="A4.1.p1.48.m5.1.1.1.1.1.1.cmml" xref="A4.1.p1.48.m5.1.1.1.1.1"><times id="A4.1.p1.48.m5.1.1.1.1.1.1.1.cmml" xref="A4.1.p1.48.m5.1.1.1.1.1.1.1"></times><apply id="A4.1.p1.48.m5.1.1.1.1.1.1.2.cmml" xref="A4.1.p1.48.m5.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="A4.1.p1.48.m5.1.1.1.1.1.1.2.1.cmml" xref="A4.1.p1.48.m5.1.1.1.1.1.1.2">subscript</csymbol><ci id="A4.1.p1.48.m5.1.1.1.1.1.1.2.2.cmml" xref="A4.1.p1.48.m5.1.1.1.1.1.1.2.2">𝑘</ci><ci id="A4.1.p1.48.m5.1.1.1.1.1.1.2.3.cmml" xref="A4.1.p1.48.m5.1.1.1.1.1.1.2.3">c</ci></apply><ci id="A4.1.p1.48.m5.1.1.1.1.1.1.3.cmml" xref="A4.1.p1.48.m5.1.1.1.1.1.1.3">𝜇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.48.m5.1c">b=\bar{d}/(k_{\rm c}\mu)</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.48.m5.1d">italic_b = over¯ start_ARG italic_d end_ARG / ( italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT italic_μ )</annotation></semantics></math> or <math alttext="(1+p){\bar{d}}_{\rm g}/(2\sigma^{*}\mu)" class="ltx_Math" display="inline" id="A4.1.p1.49.m6.2"><semantics id="A4.1.p1.49.m6.2a"><mrow id="A4.1.p1.49.m6.2.2" xref="A4.1.p1.49.m6.2.2.cmml"><mrow id="A4.1.p1.49.m6.1.1.1" xref="A4.1.p1.49.m6.1.1.1.cmml"><mrow id="A4.1.p1.49.m6.1.1.1.1.1" xref="A4.1.p1.49.m6.1.1.1.1.1.1.cmml"><mo id="A4.1.p1.49.m6.1.1.1.1.1.2" stretchy="false" xref="A4.1.p1.49.m6.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.1.p1.49.m6.1.1.1.1.1.1" xref="A4.1.p1.49.m6.1.1.1.1.1.1.cmml"><mn id="A4.1.p1.49.m6.1.1.1.1.1.1.2" xref="A4.1.p1.49.m6.1.1.1.1.1.1.2.cmml">1</mn><mo id="A4.1.p1.49.m6.1.1.1.1.1.1.1" xref="A4.1.p1.49.m6.1.1.1.1.1.1.1.cmml">+</mo><mi id="A4.1.p1.49.m6.1.1.1.1.1.1.3" xref="A4.1.p1.49.m6.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A4.1.p1.49.m6.1.1.1.1.1.3" stretchy="false" xref="A4.1.p1.49.m6.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.1.p1.49.m6.1.1.1.2" xref="A4.1.p1.49.m6.1.1.1.2.cmml">⁢</mo><msub id="A4.1.p1.49.m6.1.1.1.3" xref="A4.1.p1.49.m6.1.1.1.3.cmml"><mover accent="true" id="A4.1.p1.49.m6.1.1.1.3.2" xref="A4.1.p1.49.m6.1.1.1.3.2.cmml"><mi id="A4.1.p1.49.m6.1.1.1.3.2.2" xref="A4.1.p1.49.m6.1.1.1.3.2.2.cmml">d</mi><mo id="A4.1.p1.49.m6.1.1.1.3.2.1" xref="A4.1.p1.49.m6.1.1.1.3.2.1.cmml">¯</mo></mover><mi id="A4.1.p1.49.m6.1.1.1.3.3" mathvariant="normal" xref="A4.1.p1.49.m6.1.1.1.3.3.cmml">g</mi></msub></mrow><mo id="A4.1.p1.49.m6.2.2.3" xref="A4.1.p1.49.m6.2.2.3.cmml">/</mo><mrow id="A4.1.p1.49.m6.2.2.2.1" xref="A4.1.p1.49.m6.2.2.2.1.1.cmml"><mo id="A4.1.p1.49.m6.2.2.2.1.2" stretchy="false" xref="A4.1.p1.49.m6.2.2.2.1.1.cmml">(</mo><mrow id="A4.1.p1.49.m6.2.2.2.1.1" xref="A4.1.p1.49.m6.2.2.2.1.1.cmml"><mn id="A4.1.p1.49.m6.2.2.2.1.1.2" xref="A4.1.p1.49.m6.2.2.2.1.1.2.cmml">2</mn><mo id="A4.1.p1.49.m6.2.2.2.1.1.1" xref="A4.1.p1.49.m6.2.2.2.1.1.1.cmml">⁢</mo><msup id="A4.1.p1.49.m6.2.2.2.1.1.3" xref="A4.1.p1.49.m6.2.2.2.1.1.3.cmml"><mi id="A4.1.p1.49.m6.2.2.2.1.1.3.2" xref="A4.1.p1.49.m6.2.2.2.1.1.3.2.cmml">σ</mi><mo id="A4.1.p1.49.m6.2.2.2.1.1.3.3" xref="A4.1.p1.49.m6.2.2.2.1.1.3.3.cmml">∗</mo></msup><mo id="A4.1.p1.49.m6.2.2.2.1.1.1a" xref="A4.1.p1.49.m6.2.2.2.1.1.1.cmml">⁢</mo><mi id="A4.1.p1.49.m6.2.2.2.1.1.4" xref="A4.1.p1.49.m6.2.2.2.1.1.4.cmml">μ</mi></mrow><mo id="A4.1.p1.49.m6.2.2.2.1.3" stretchy="false" xref="A4.1.p1.49.m6.2.2.2.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.49.m6.2b"><apply id="A4.1.p1.49.m6.2.2.cmml" xref="A4.1.p1.49.m6.2.2"><divide id="A4.1.p1.49.m6.2.2.3.cmml" xref="A4.1.p1.49.m6.2.2.3"></divide><apply id="A4.1.p1.49.m6.1.1.1.cmml" xref="A4.1.p1.49.m6.1.1.1"><times id="A4.1.p1.49.m6.1.1.1.2.cmml" xref="A4.1.p1.49.m6.1.1.1.2"></times><apply id="A4.1.p1.49.m6.1.1.1.1.1.1.cmml" xref="A4.1.p1.49.m6.1.1.1.1.1"><plus id="A4.1.p1.49.m6.1.1.1.1.1.1.1.cmml" xref="A4.1.p1.49.m6.1.1.1.1.1.1.1"></plus><cn id="A4.1.p1.49.m6.1.1.1.1.1.1.2.cmml" type="integer" xref="A4.1.p1.49.m6.1.1.1.1.1.1.2">1</cn><ci id="A4.1.p1.49.m6.1.1.1.1.1.1.3.cmml" xref="A4.1.p1.49.m6.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="A4.1.p1.49.m6.1.1.1.3.cmml" xref="A4.1.p1.49.m6.1.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.49.m6.1.1.1.3.1.cmml" xref="A4.1.p1.49.m6.1.1.1.3">subscript</csymbol><apply id="A4.1.p1.49.m6.1.1.1.3.2.cmml" xref="A4.1.p1.49.m6.1.1.1.3.2"><ci id="A4.1.p1.49.m6.1.1.1.3.2.1.cmml" xref="A4.1.p1.49.m6.1.1.1.3.2.1">¯</ci><ci id="A4.1.p1.49.m6.1.1.1.3.2.2.cmml" xref="A4.1.p1.49.m6.1.1.1.3.2.2">𝑑</ci></apply><ci id="A4.1.p1.49.m6.1.1.1.3.3.cmml" xref="A4.1.p1.49.m6.1.1.1.3.3">g</ci></apply></apply><apply id="A4.1.p1.49.m6.2.2.2.1.1.cmml" xref="A4.1.p1.49.m6.2.2.2.1"><times id="A4.1.p1.49.m6.2.2.2.1.1.1.cmml" xref="A4.1.p1.49.m6.2.2.2.1.1.1"></times><cn id="A4.1.p1.49.m6.2.2.2.1.1.2.cmml" type="integer" xref="A4.1.p1.49.m6.2.2.2.1.1.2">2</cn><apply id="A4.1.p1.49.m6.2.2.2.1.1.3.cmml" xref="A4.1.p1.49.m6.2.2.2.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.49.m6.2.2.2.1.1.3.1.cmml" xref="A4.1.p1.49.m6.2.2.2.1.1.3">superscript</csymbol><ci id="A4.1.p1.49.m6.2.2.2.1.1.3.2.cmml" xref="A4.1.p1.49.m6.2.2.2.1.1.3.2">𝜎</ci><times id="A4.1.p1.49.m6.2.2.2.1.1.3.3.cmml" xref="A4.1.p1.49.m6.2.2.2.1.1.3.3"></times></apply><ci id="A4.1.p1.49.m6.2.2.2.1.1.4.cmml" xref="A4.1.p1.49.m6.2.2.2.1.1.4">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.49.m6.2c">(1+p){\bar{d}}_{\rm g}/(2\sigma^{*}\mu)</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.49.m6.2d">( 1 + italic_p ) over¯ start_ARG italic_d end_ARG start_POSTSUBSCRIPT roman_g end_POSTSUBSCRIPT / ( 2 italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_μ )</annotation></semantics></math> to the above expression, one has</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx85"> <tbody id="A4.Ex82"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}_{\zeta}\leq" class="ltx_Math" display="inline" id="A4.Ex82.m1.1"><semantics id="A4.Ex82.m1.1a"><mrow id="A4.Ex82.m1.1.1" xref="A4.Ex82.m1.1.1.cmml"><msub id="A4.Ex82.m1.1.1.2" xref="A4.Ex82.m1.1.1.2.cmml"><mover accent="true" id="A4.Ex82.m1.1.1.2.2" xref="A4.Ex82.m1.1.1.2.2.cmml"><mi id="A4.Ex82.m1.1.1.2.2.2" xref="A4.Ex82.m1.1.1.2.2.2.cmml">V</mi><mo id="A4.Ex82.m1.1.1.2.2.1" xref="A4.Ex82.m1.1.1.2.2.1.cmml">˙</mo></mover><mi id="A4.Ex82.m1.1.1.2.3" xref="A4.Ex82.m1.1.1.2.3.cmml">ζ</mi></msub><mo id="A4.Ex82.m1.1.1.1" xref="A4.Ex82.m1.1.1.1.cmml">≤</mo><mi id="A4.Ex82.m1.1.1.3" xref="A4.Ex82.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex82.m1.1b"><apply id="A4.Ex82.m1.1.1.cmml" xref="A4.Ex82.m1.1.1"><leq id="A4.Ex82.m1.1.1.1.cmml" xref="A4.Ex82.m1.1.1.1"></leq><apply id="A4.Ex82.m1.1.1.2.cmml" xref="A4.Ex82.m1.1.1.2"><csymbol cd="ambiguous" id="A4.Ex82.m1.1.1.2.1.cmml" xref="A4.Ex82.m1.1.1.2">subscript</csymbol><apply id="A4.Ex82.m1.1.1.2.2.cmml" xref="A4.Ex82.m1.1.1.2.2"><ci id="A4.Ex82.m1.1.1.2.2.1.cmml" xref="A4.Ex82.m1.1.1.2.2.1">˙</ci><ci id="A4.Ex82.m1.1.1.2.2.2.cmml" xref="A4.Ex82.m1.1.1.2.2.2">𝑉</ci></apply><ci id="A4.Ex82.m1.1.1.2.3.cmml" xref="A4.Ex82.m1.1.1.2.3">𝜁</ci></apply><csymbol cd="latexml" id="A4.Ex82.m1.1.1.3.cmml" xref="A4.Ex82.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex82.m1.1c">\displaystyle\dot{V}_{\zeta}\leq</annotation><annotation encoding="application/x-llamapun" id="A4.Ex82.m1.1d">over˙ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-k_{\rm c}(1-\mu)\|\bm{e}\|^{2}/2-\kappa\sigma^{*}(1-\mu)\|\tilde% {\bm{\theta}}_{\zeta}\|^{2}+\rho_{2}(k_{\rm c},\kappa,\bar{d})" class="ltx_Math" display="inline" id="A4.Ex82.m2.7"><semantics id="A4.Ex82.m2.7a"><mrow id="A4.Ex82.m2.7.7" xref="A4.Ex82.m2.7.7.cmml"><mrow id="A4.Ex82.m2.6.6.3" xref="A4.Ex82.m2.6.6.3.cmml"><mrow id="A4.Ex82.m2.4.4.1.1" xref="A4.Ex82.m2.4.4.1.1.cmml"><mo id="A4.Ex82.m2.4.4.1.1a" xref="A4.Ex82.m2.4.4.1.1.cmml">−</mo><mrow id="A4.Ex82.m2.4.4.1.1.1" xref="A4.Ex82.m2.4.4.1.1.1.cmml"><mrow id="A4.Ex82.m2.4.4.1.1.1.1" xref="A4.Ex82.m2.4.4.1.1.1.1.cmml"><msub id="A4.Ex82.m2.4.4.1.1.1.1.3" xref="A4.Ex82.m2.4.4.1.1.1.1.3.cmml"><mi id="A4.Ex82.m2.4.4.1.1.1.1.3.2" xref="A4.Ex82.m2.4.4.1.1.1.1.3.2.cmml">k</mi><mi id="A4.Ex82.m2.4.4.1.1.1.1.3.3" mathvariant="normal" xref="A4.Ex82.m2.4.4.1.1.1.1.3.3.cmml">c</mi></msub><mo id="A4.Ex82.m2.4.4.1.1.1.1.2" xref="A4.Ex82.m2.4.4.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex82.m2.4.4.1.1.1.1.1.1" xref="A4.Ex82.m2.4.4.1.1.1.1.1.1.1.cmml"><mo id="A4.Ex82.m2.4.4.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex82.m2.4.4.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex82.m2.4.4.1.1.1.1.1.1.1" xref="A4.Ex82.m2.4.4.1.1.1.1.1.1.1.cmml"><mn id="A4.Ex82.m2.4.4.1.1.1.1.1.1.1.2" xref="A4.Ex82.m2.4.4.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A4.Ex82.m2.4.4.1.1.1.1.1.1.1.1" xref="A4.Ex82.m2.4.4.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="A4.Ex82.m2.4.4.1.1.1.1.1.1.1.3" xref="A4.Ex82.m2.4.4.1.1.1.1.1.1.1.3.cmml">μ</mi></mrow><mo id="A4.Ex82.m2.4.4.1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex82.m2.4.4.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex82.m2.4.4.1.1.1.1.2a" xref="A4.Ex82.m2.4.4.1.1.1.1.2.cmml">⁢</mo><msup id="A4.Ex82.m2.4.4.1.1.1.1.4" xref="A4.Ex82.m2.4.4.1.1.1.1.4.cmml"><mrow id="A4.Ex82.m2.4.4.1.1.1.1.4.2.2" xref="A4.Ex82.m2.4.4.1.1.1.1.4.2.1.cmml"><mo id="A4.Ex82.m2.4.4.1.1.1.1.4.2.2.1" stretchy="false" xref="A4.Ex82.m2.4.4.1.1.1.1.4.2.1.1.cmml">‖</mo><mi id="A4.Ex82.m2.1.1" xref="A4.Ex82.m2.1.1.cmml">𝒆</mi><mo id="A4.Ex82.m2.4.4.1.1.1.1.4.2.2.2" stretchy="false" xref="A4.Ex82.m2.4.4.1.1.1.1.4.2.1.1.cmml">‖</mo></mrow><mn id="A4.Ex82.m2.4.4.1.1.1.1.4.3" xref="A4.Ex82.m2.4.4.1.1.1.1.4.3.cmml">2</mn></msup></mrow><mo id="A4.Ex82.m2.4.4.1.1.1.2" xref="A4.Ex82.m2.4.4.1.1.1.2.cmml">/</mo><mn id="A4.Ex82.m2.4.4.1.1.1.3" xref="A4.Ex82.m2.4.4.1.1.1.3.cmml">2</mn></mrow></mrow><mo id="A4.Ex82.m2.6.6.3.4" xref="A4.Ex82.m2.6.6.3.4.cmml">−</mo><mrow id="A4.Ex82.m2.6.6.3.3" xref="A4.Ex82.m2.6.6.3.3.cmml"><mi id="A4.Ex82.m2.6.6.3.3.4" xref="A4.Ex82.m2.6.6.3.3.4.cmml">κ</mi><mo id="A4.Ex82.m2.6.6.3.3.3" xref="A4.Ex82.m2.6.6.3.3.3.cmml">⁢</mo><msup id="A4.Ex82.m2.6.6.3.3.5" xref="A4.Ex82.m2.6.6.3.3.5.cmml"><mi id="A4.Ex82.m2.6.6.3.3.5.2" xref="A4.Ex82.m2.6.6.3.3.5.2.cmml">σ</mi><mo id="A4.Ex82.m2.6.6.3.3.5.3" xref="A4.Ex82.m2.6.6.3.3.5.3.cmml">∗</mo></msup><mo id="A4.Ex82.m2.6.6.3.3.3a" xref="A4.Ex82.m2.6.6.3.3.3.cmml">⁢</mo><mrow id="A4.Ex82.m2.5.5.2.2.1.1" xref="A4.Ex82.m2.5.5.2.2.1.1.1.cmml"><mo id="A4.Ex82.m2.5.5.2.2.1.1.2" stretchy="false" xref="A4.Ex82.m2.5.5.2.2.1.1.1.cmml">(</mo><mrow id="A4.Ex82.m2.5.5.2.2.1.1.1" xref="A4.Ex82.m2.5.5.2.2.1.1.1.cmml"><mn id="A4.Ex82.m2.5.5.2.2.1.1.1.2" xref="A4.Ex82.m2.5.5.2.2.1.1.1.2.cmml">1</mn><mo id="A4.Ex82.m2.5.5.2.2.1.1.1.1" xref="A4.Ex82.m2.5.5.2.2.1.1.1.1.cmml">−</mo><mi id="A4.Ex82.m2.5.5.2.2.1.1.1.3" xref="A4.Ex82.m2.5.5.2.2.1.1.1.3.cmml">μ</mi></mrow><mo id="A4.Ex82.m2.5.5.2.2.1.1.3" stretchy="false" xref="A4.Ex82.m2.5.5.2.2.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex82.m2.6.6.3.3.3b" xref="A4.Ex82.m2.6.6.3.3.3.cmml">⁢</mo><msup id="A4.Ex82.m2.6.6.3.3.2" xref="A4.Ex82.m2.6.6.3.3.2.cmml"><mrow id="A4.Ex82.m2.6.6.3.3.2.1.1" xref="A4.Ex82.m2.6.6.3.3.2.1.2.cmml"><mo id="A4.Ex82.m2.6.6.3.3.2.1.1.2" stretchy="false" xref="A4.Ex82.m2.6.6.3.3.2.1.2.1.cmml">‖</mo><msub id="A4.Ex82.m2.6.6.3.3.2.1.1.1" xref="A4.Ex82.m2.6.6.3.3.2.1.1.1.cmml"><mover accent="true" id="A4.Ex82.m2.6.6.3.3.2.1.1.1.2" xref="A4.Ex82.m2.6.6.3.3.2.1.1.1.2.cmml"><mi id="A4.Ex82.m2.6.6.3.3.2.1.1.1.2.2" xref="A4.Ex82.m2.6.6.3.3.2.1.1.1.2.2.cmml">𝜽</mi><mo id="A4.Ex82.m2.6.6.3.3.2.1.1.1.2.1" xref="A4.Ex82.m2.6.6.3.3.2.1.1.1.2.1.cmml">~</mo></mover><mi id="A4.Ex82.m2.6.6.3.3.2.1.1.1.3" xref="A4.Ex82.m2.6.6.3.3.2.1.1.1.3.cmml">ζ</mi></msub><mo id="A4.Ex82.m2.6.6.3.3.2.1.1.3" stretchy="false" xref="A4.Ex82.m2.6.6.3.3.2.1.2.1.cmml">‖</mo></mrow><mn id="A4.Ex82.m2.6.6.3.3.2.3" xref="A4.Ex82.m2.6.6.3.3.2.3.cmml">2</mn></msup></mrow></mrow><mo id="A4.Ex82.m2.7.7.5" xref="A4.Ex82.m2.7.7.5.cmml">+</mo><mrow id="A4.Ex82.m2.7.7.4" xref="A4.Ex82.m2.7.7.4.cmml"><msub id="A4.Ex82.m2.7.7.4.3" xref="A4.Ex82.m2.7.7.4.3.cmml"><mi id="A4.Ex82.m2.7.7.4.3.2" xref="A4.Ex82.m2.7.7.4.3.2.cmml">ρ</mi><mn id="A4.Ex82.m2.7.7.4.3.3" xref="A4.Ex82.m2.7.7.4.3.3.cmml">2</mn></msub><mo id="A4.Ex82.m2.7.7.4.2" xref="A4.Ex82.m2.7.7.4.2.cmml">⁢</mo><mrow id="A4.Ex82.m2.7.7.4.1.1" xref="A4.Ex82.m2.7.7.4.1.2.cmml"><mo id="A4.Ex82.m2.7.7.4.1.1.2" stretchy="false" xref="A4.Ex82.m2.7.7.4.1.2.cmml">(</mo><msub id="A4.Ex82.m2.7.7.4.1.1.1" xref="A4.Ex82.m2.7.7.4.1.1.1.cmml"><mi id="A4.Ex82.m2.7.7.4.1.1.1.2" xref="A4.Ex82.m2.7.7.4.1.1.1.2.cmml">k</mi><mi id="A4.Ex82.m2.7.7.4.1.1.1.3" mathvariant="normal" xref="A4.Ex82.m2.7.7.4.1.1.1.3.cmml">c</mi></msub><mo id="A4.Ex82.m2.7.7.4.1.1.3" xref="A4.Ex82.m2.7.7.4.1.2.cmml">,</mo><mi id="A4.Ex82.m2.2.2" xref="A4.Ex82.m2.2.2.cmml">κ</mi><mo id="A4.Ex82.m2.7.7.4.1.1.4" xref="A4.Ex82.m2.7.7.4.1.2.cmml">,</mo><mover accent="true" id="A4.Ex82.m2.3.3" xref="A4.Ex82.m2.3.3.cmml"><mi id="A4.Ex82.m2.3.3.2" xref="A4.Ex82.m2.3.3.2.cmml">d</mi><mo id="A4.Ex82.m2.3.3.1" xref="A4.Ex82.m2.3.3.1.cmml">¯</mo></mover><mo id="A4.Ex82.m2.7.7.4.1.1.5" stretchy="false" xref="A4.Ex82.m2.7.7.4.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex82.m2.7b"><apply id="A4.Ex82.m2.7.7.cmml" xref="A4.Ex82.m2.7.7"><plus id="A4.Ex82.m2.7.7.5.cmml" xref="A4.Ex82.m2.7.7.5"></plus><apply id="A4.Ex82.m2.6.6.3.cmml" xref="A4.Ex82.m2.6.6.3"><minus id="A4.Ex82.m2.6.6.3.4.cmml" xref="A4.Ex82.m2.6.6.3.4"></minus><apply id="A4.Ex82.m2.4.4.1.1.cmml" xref="A4.Ex82.m2.4.4.1.1"><minus id="A4.Ex82.m2.4.4.1.1.2.cmml" xref="A4.Ex82.m2.4.4.1.1"></minus><apply id="A4.Ex82.m2.4.4.1.1.1.cmml" xref="A4.Ex82.m2.4.4.1.1.1"><divide id="A4.Ex82.m2.4.4.1.1.1.2.cmml" xref="A4.Ex82.m2.4.4.1.1.1.2"></divide><apply id="A4.Ex82.m2.4.4.1.1.1.1.cmml" xref="A4.Ex82.m2.4.4.1.1.1.1"><times id="A4.Ex82.m2.4.4.1.1.1.1.2.cmml" xref="A4.Ex82.m2.4.4.1.1.1.1.2"></times><apply id="A4.Ex82.m2.4.4.1.1.1.1.3.cmml" xref="A4.Ex82.m2.4.4.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.Ex82.m2.4.4.1.1.1.1.3.1.cmml" xref="A4.Ex82.m2.4.4.1.1.1.1.3">subscript</csymbol><ci id="A4.Ex82.m2.4.4.1.1.1.1.3.2.cmml" xref="A4.Ex82.m2.4.4.1.1.1.1.3.2">𝑘</ci><ci id="A4.Ex82.m2.4.4.1.1.1.1.3.3.cmml" xref="A4.Ex82.m2.4.4.1.1.1.1.3.3">c</ci></apply><apply id="A4.Ex82.m2.4.4.1.1.1.1.1.1.1.cmml" xref="A4.Ex82.m2.4.4.1.1.1.1.1.1"><minus id="A4.Ex82.m2.4.4.1.1.1.1.1.1.1.1.cmml" xref="A4.Ex82.m2.4.4.1.1.1.1.1.1.1.1"></minus><cn id="A4.Ex82.m2.4.4.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A4.Ex82.m2.4.4.1.1.1.1.1.1.1.2">1</cn><ci id="A4.Ex82.m2.4.4.1.1.1.1.1.1.1.3.cmml" xref="A4.Ex82.m2.4.4.1.1.1.1.1.1.1.3">𝜇</ci></apply><apply id="A4.Ex82.m2.4.4.1.1.1.1.4.cmml" xref="A4.Ex82.m2.4.4.1.1.1.1.4"><csymbol cd="ambiguous" id="A4.Ex82.m2.4.4.1.1.1.1.4.1.cmml" xref="A4.Ex82.m2.4.4.1.1.1.1.4">superscript</csymbol><apply id="A4.Ex82.m2.4.4.1.1.1.1.4.2.1.cmml" xref="A4.Ex82.m2.4.4.1.1.1.1.4.2.2"><csymbol cd="latexml" id="A4.Ex82.m2.4.4.1.1.1.1.4.2.1.1.cmml" xref="A4.Ex82.m2.4.4.1.1.1.1.4.2.2.1">norm</csymbol><ci id="A4.Ex82.m2.1.1.cmml" xref="A4.Ex82.m2.1.1">𝒆</ci></apply><cn id="A4.Ex82.m2.4.4.1.1.1.1.4.3.cmml" type="integer" xref="A4.Ex82.m2.4.4.1.1.1.1.4.3">2</cn></apply></apply><cn id="A4.Ex82.m2.4.4.1.1.1.3.cmml" type="integer" xref="A4.Ex82.m2.4.4.1.1.1.3">2</cn></apply></apply><apply id="A4.Ex82.m2.6.6.3.3.cmml" xref="A4.Ex82.m2.6.6.3.3"><times id="A4.Ex82.m2.6.6.3.3.3.cmml" xref="A4.Ex82.m2.6.6.3.3.3"></times><ci id="A4.Ex82.m2.6.6.3.3.4.cmml" xref="A4.Ex82.m2.6.6.3.3.4">𝜅</ci><apply id="A4.Ex82.m2.6.6.3.3.5.cmml" xref="A4.Ex82.m2.6.6.3.3.5"><csymbol cd="ambiguous" id="A4.Ex82.m2.6.6.3.3.5.1.cmml" xref="A4.Ex82.m2.6.6.3.3.5">superscript</csymbol><ci id="A4.Ex82.m2.6.6.3.3.5.2.cmml" xref="A4.Ex82.m2.6.6.3.3.5.2">𝜎</ci><times id="A4.Ex82.m2.6.6.3.3.5.3.cmml" xref="A4.Ex82.m2.6.6.3.3.5.3"></times></apply><apply id="A4.Ex82.m2.5.5.2.2.1.1.1.cmml" xref="A4.Ex82.m2.5.5.2.2.1.1"><minus id="A4.Ex82.m2.5.5.2.2.1.1.1.1.cmml" xref="A4.Ex82.m2.5.5.2.2.1.1.1.1"></minus><cn id="A4.Ex82.m2.5.5.2.2.1.1.1.2.cmml" type="integer" xref="A4.Ex82.m2.5.5.2.2.1.1.1.2">1</cn><ci id="A4.Ex82.m2.5.5.2.2.1.1.1.3.cmml" xref="A4.Ex82.m2.5.5.2.2.1.1.1.3">𝜇</ci></apply><apply id="A4.Ex82.m2.6.6.3.3.2.cmml" xref="A4.Ex82.m2.6.6.3.3.2"><csymbol cd="ambiguous" id="A4.Ex82.m2.6.6.3.3.2.2.cmml" xref="A4.Ex82.m2.6.6.3.3.2">superscript</csymbol><apply id="A4.Ex82.m2.6.6.3.3.2.1.2.cmml" xref="A4.Ex82.m2.6.6.3.3.2.1.1"><csymbol cd="latexml" id="A4.Ex82.m2.6.6.3.3.2.1.2.1.cmml" xref="A4.Ex82.m2.6.6.3.3.2.1.1.2">norm</csymbol><apply id="A4.Ex82.m2.6.6.3.3.2.1.1.1.cmml" xref="A4.Ex82.m2.6.6.3.3.2.1.1.1"><csymbol cd="ambiguous" id="A4.Ex82.m2.6.6.3.3.2.1.1.1.1.cmml" xref="A4.Ex82.m2.6.6.3.3.2.1.1.1">subscript</csymbol><apply id="A4.Ex82.m2.6.6.3.3.2.1.1.1.2.cmml" xref="A4.Ex82.m2.6.6.3.3.2.1.1.1.2"><ci id="A4.Ex82.m2.6.6.3.3.2.1.1.1.2.1.cmml" xref="A4.Ex82.m2.6.6.3.3.2.1.1.1.2.1">~</ci><ci id="A4.Ex82.m2.6.6.3.3.2.1.1.1.2.2.cmml" xref="A4.Ex82.m2.6.6.3.3.2.1.1.1.2.2">𝜽</ci></apply><ci id="A4.Ex82.m2.6.6.3.3.2.1.1.1.3.cmml" xref="A4.Ex82.m2.6.6.3.3.2.1.1.1.3">𝜁</ci></apply></apply><cn id="A4.Ex82.m2.6.6.3.3.2.3.cmml" type="integer" xref="A4.Ex82.m2.6.6.3.3.2.3">2</cn></apply></apply></apply><apply id="A4.Ex82.m2.7.7.4.cmml" xref="A4.Ex82.m2.7.7.4"><times id="A4.Ex82.m2.7.7.4.2.cmml" xref="A4.Ex82.m2.7.7.4.2"></times><apply id="A4.Ex82.m2.7.7.4.3.cmml" xref="A4.Ex82.m2.7.7.4.3"><csymbol cd="ambiguous" id="A4.Ex82.m2.7.7.4.3.1.cmml" xref="A4.Ex82.m2.7.7.4.3">subscript</csymbol><ci id="A4.Ex82.m2.7.7.4.3.2.cmml" xref="A4.Ex82.m2.7.7.4.3.2">𝜌</ci><cn id="A4.Ex82.m2.7.7.4.3.3.cmml" type="integer" xref="A4.Ex82.m2.7.7.4.3.3">2</cn></apply><vector id="A4.Ex82.m2.7.7.4.1.2.cmml" xref="A4.Ex82.m2.7.7.4.1.1"><apply id="A4.Ex82.m2.7.7.4.1.1.1.cmml" xref="A4.Ex82.m2.7.7.4.1.1.1"><csymbol cd="ambiguous" id="A4.Ex82.m2.7.7.4.1.1.1.1.cmml" xref="A4.Ex82.m2.7.7.4.1.1.1">subscript</csymbol><ci id="A4.Ex82.m2.7.7.4.1.1.1.2.cmml" xref="A4.Ex82.m2.7.7.4.1.1.1.2">𝑘</ci><ci id="A4.Ex82.m2.7.7.4.1.1.1.3.cmml" xref="A4.Ex82.m2.7.7.4.1.1.1.3">c</ci></apply><ci id="A4.Ex82.m2.2.2.cmml" xref="A4.Ex82.m2.2.2">𝜅</ci><apply id="A4.Ex82.m2.3.3.cmml" xref="A4.Ex82.m2.3.3"><ci id="A4.Ex82.m2.3.3.1.cmml" xref="A4.Ex82.m2.3.3.1">¯</ci><ci id="A4.Ex82.m2.3.3.2.cmml" xref="A4.Ex82.m2.3.3.2">𝑑</ci></apply></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex82.m2.7c">\displaystyle-k_{\rm c}(1-\mu)\|\bm{e}\|^{2}/2-\kappa\sigma^{*}(1-\mu)\|\tilde% {\bm{\theta}}_{\zeta}\|^{2}+\rho_{2}(k_{\rm c},\kappa,\bar{d})</annotation><annotation encoding="application/x-llamapun" id="A4.Ex82.m2.7d">- italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ( 1 - italic_μ ) ∥ bold_italic_e ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 - italic_κ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( 1 - italic_μ ) ∥ over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_ρ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT , italic_κ , over¯ start_ARG italic_d end_ARG )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.1.p1.52">with <math alttext="\rho_{2}(k_{\rm c},\kappa,\bar{d})" class="ltx_Math" display="inline" id="A4.1.p1.50.m1.3"><semantics id="A4.1.p1.50.m1.3a"><mrow id="A4.1.p1.50.m1.3.3" xref="A4.1.p1.50.m1.3.3.cmml"><msub id="A4.1.p1.50.m1.3.3.3" xref="A4.1.p1.50.m1.3.3.3.cmml"><mi id="A4.1.p1.50.m1.3.3.3.2" xref="A4.1.p1.50.m1.3.3.3.2.cmml">ρ</mi><mn id="A4.1.p1.50.m1.3.3.3.3" xref="A4.1.p1.50.m1.3.3.3.3.cmml">2</mn></msub><mo id="A4.1.p1.50.m1.3.3.2" xref="A4.1.p1.50.m1.3.3.2.cmml">⁢</mo><mrow id="A4.1.p1.50.m1.3.3.1.1" xref="A4.1.p1.50.m1.3.3.1.2.cmml"><mo id="A4.1.p1.50.m1.3.3.1.1.2" stretchy="false" xref="A4.1.p1.50.m1.3.3.1.2.cmml">(</mo><msub id="A4.1.p1.50.m1.3.3.1.1.1" xref="A4.1.p1.50.m1.3.3.1.1.1.cmml"><mi id="A4.1.p1.50.m1.3.3.1.1.1.2" xref="A4.1.p1.50.m1.3.3.1.1.1.2.cmml">k</mi><mi id="A4.1.p1.50.m1.3.3.1.1.1.3" mathvariant="normal" xref="A4.1.p1.50.m1.3.3.1.1.1.3.cmml">c</mi></msub><mo id="A4.1.p1.50.m1.3.3.1.1.3" xref="A4.1.p1.50.m1.3.3.1.2.cmml">,</mo><mi id="A4.1.p1.50.m1.1.1" xref="A4.1.p1.50.m1.1.1.cmml">κ</mi><mo id="A4.1.p1.50.m1.3.3.1.1.4" xref="A4.1.p1.50.m1.3.3.1.2.cmml">,</mo><mover accent="true" id="A4.1.p1.50.m1.2.2" xref="A4.1.p1.50.m1.2.2.cmml"><mi id="A4.1.p1.50.m1.2.2.2" xref="A4.1.p1.50.m1.2.2.2.cmml">d</mi><mo id="A4.1.p1.50.m1.2.2.1" xref="A4.1.p1.50.m1.2.2.1.cmml">¯</mo></mover><mo id="A4.1.p1.50.m1.3.3.1.1.5" stretchy="false" xref="A4.1.p1.50.m1.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.50.m1.3b"><apply id="A4.1.p1.50.m1.3.3.cmml" xref="A4.1.p1.50.m1.3.3"><times id="A4.1.p1.50.m1.3.3.2.cmml" xref="A4.1.p1.50.m1.3.3.2"></times><apply id="A4.1.p1.50.m1.3.3.3.cmml" xref="A4.1.p1.50.m1.3.3.3"><csymbol cd="ambiguous" id="A4.1.p1.50.m1.3.3.3.1.cmml" xref="A4.1.p1.50.m1.3.3.3">subscript</csymbol><ci id="A4.1.p1.50.m1.3.3.3.2.cmml" xref="A4.1.p1.50.m1.3.3.3.2">𝜌</ci><cn id="A4.1.p1.50.m1.3.3.3.3.cmml" type="integer" xref="A4.1.p1.50.m1.3.3.3.3">2</cn></apply><vector id="A4.1.p1.50.m1.3.3.1.2.cmml" xref="A4.1.p1.50.m1.3.3.1.1"><apply id="A4.1.p1.50.m1.3.3.1.1.1.cmml" xref="A4.1.p1.50.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="A4.1.p1.50.m1.3.3.1.1.1.1.cmml" xref="A4.1.p1.50.m1.3.3.1.1.1">subscript</csymbol><ci id="A4.1.p1.50.m1.3.3.1.1.1.2.cmml" xref="A4.1.p1.50.m1.3.3.1.1.1.2">𝑘</ci><ci id="A4.1.p1.50.m1.3.3.1.1.1.3.cmml" xref="A4.1.p1.50.m1.3.3.1.1.1.3">c</ci></apply><ci id="A4.1.p1.50.m1.1.1.cmml" xref="A4.1.p1.50.m1.1.1">𝜅</ci><apply id="A4.1.p1.50.m1.2.2.cmml" xref="A4.1.p1.50.m1.2.2"><ci id="A4.1.p1.50.m1.2.2.1.cmml" xref="A4.1.p1.50.m1.2.2.1">¯</ci><ci id="A4.1.p1.50.m1.2.2.2.cmml" xref="A4.1.p1.50.m1.2.2.2">𝑑</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.50.m1.3c">\rho_{2}(k_{\rm c},\kappa,\bar{d})</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.50.m1.3d">italic_ρ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT , italic_κ , over¯ start_ARG italic_d end_ARG )</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="A4.1.p1.51.m2.1"><semantics id="A4.1.p1.51.m2.1a"><mo id="A4.1.p1.51.m2.1.1" xref="A4.1.p1.51.m2.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="A4.1.p1.51.m2.1b"><csymbol cd="latexml" id="A4.1.p1.51.m2.1.1.cmml" xref="A4.1.p1.51.m2.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.51.m2.1c">:=</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.51.m2.1d">:=</annotation></semantics></math> <math alttext="(1+p)\bar{\epsilon}\bar{d}+\bar{d}^{2}/(2k_{\rm c}\mu)+\kappa((1+p){\bar{d}}_{% \rm g})^{2}/(4\sigma^{*}\mu)\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A4.1.p1.52.m3.4"><semantics id="A4.1.p1.52.m3.4a"><mrow id="A4.1.p1.52.m3.4.4" xref="A4.1.p1.52.m3.4.4.cmml"><mrow id="A4.1.p1.52.m3.4.4.4" xref="A4.1.p1.52.m3.4.4.4.cmml"><mrow id="A4.1.p1.52.m3.1.1.1.1" xref="A4.1.p1.52.m3.1.1.1.1.cmml"><mrow id="A4.1.p1.52.m3.1.1.1.1.1.1" xref="A4.1.p1.52.m3.1.1.1.1.1.1.1.cmml"><mo id="A4.1.p1.52.m3.1.1.1.1.1.1.2" stretchy="false" xref="A4.1.p1.52.m3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.1.p1.52.m3.1.1.1.1.1.1.1" xref="A4.1.p1.52.m3.1.1.1.1.1.1.1.cmml"><mn id="A4.1.p1.52.m3.1.1.1.1.1.1.1.2" xref="A4.1.p1.52.m3.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A4.1.p1.52.m3.1.1.1.1.1.1.1.1" xref="A4.1.p1.52.m3.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="A4.1.p1.52.m3.1.1.1.1.1.1.1.3" xref="A4.1.p1.52.m3.1.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A4.1.p1.52.m3.1.1.1.1.1.1.3" stretchy="false" xref="A4.1.p1.52.m3.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.1.p1.52.m3.1.1.1.1.2" xref="A4.1.p1.52.m3.1.1.1.1.2.cmml">⁢</mo><mover accent="true" id="A4.1.p1.52.m3.1.1.1.1.3" xref="A4.1.p1.52.m3.1.1.1.1.3.cmml"><mi id="A4.1.p1.52.m3.1.1.1.1.3.2" xref="A4.1.p1.52.m3.1.1.1.1.3.2.cmml">ϵ</mi><mo id="A4.1.p1.52.m3.1.1.1.1.3.1" xref="A4.1.p1.52.m3.1.1.1.1.3.1.cmml">¯</mo></mover><mo id="A4.1.p1.52.m3.1.1.1.1.2a" xref="A4.1.p1.52.m3.1.1.1.1.2.cmml">⁢</mo><mover accent="true" id="A4.1.p1.52.m3.1.1.1.1.4" xref="A4.1.p1.52.m3.1.1.1.1.4.cmml"><mi id="A4.1.p1.52.m3.1.1.1.1.4.2" xref="A4.1.p1.52.m3.1.1.1.1.4.2.cmml">d</mi><mo id="A4.1.p1.52.m3.1.1.1.1.4.1" xref="A4.1.p1.52.m3.1.1.1.1.4.1.cmml">¯</mo></mover></mrow><mo id="A4.1.p1.52.m3.4.4.4.5" xref="A4.1.p1.52.m3.4.4.4.5.cmml">+</mo><mrow id="A4.1.p1.52.m3.2.2.2.2" xref="A4.1.p1.52.m3.2.2.2.2.cmml"><msup id="A4.1.p1.52.m3.2.2.2.2.3" xref="A4.1.p1.52.m3.2.2.2.2.3.cmml"><mover accent="true" id="A4.1.p1.52.m3.2.2.2.2.3.2" xref="A4.1.p1.52.m3.2.2.2.2.3.2.cmml"><mi id="A4.1.p1.52.m3.2.2.2.2.3.2.2" xref="A4.1.p1.52.m3.2.2.2.2.3.2.2.cmml">d</mi><mo id="A4.1.p1.52.m3.2.2.2.2.3.2.1" xref="A4.1.p1.52.m3.2.2.2.2.3.2.1.cmml">¯</mo></mover><mn id="A4.1.p1.52.m3.2.2.2.2.3.3" xref="A4.1.p1.52.m3.2.2.2.2.3.3.cmml">2</mn></msup><mo id="A4.1.p1.52.m3.2.2.2.2.2" xref="A4.1.p1.52.m3.2.2.2.2.2.cmml">/</mo><mrow id="A4.1.p1.52.m3.2.2.2.2.1.1" xref="A4.1.p1.52.m3.2.2.2.2.1.1.1.cmml"><mo id="A4.1.p1.52.m3.2.2.2.2.1.1.2" stretchy="false" xref="A4.1.p1.52.m3.2.2.2.2.1.1.1.cmml">(</mo><mrow id="A4.1.p1.52.m3.2.2.2.2.1.1.1" xref="A4.1.p1.52.m3.2.2.2.2.1.1.1.cmml"><mn id="A4.1.p1.52.m3.2.2.2.2.1.1.1.2" xref="A4.1.p1.52.m3.2.2.2.2.1.1.1.2.cmml">2</mn><mo id="A4.1.p1.52.m3.2.2.2.2.1.1.1.1" xref="A4.1.p1.52.m3.2.2.2.2.1.1.1.1.cmml">⁢</mo><msub id="A4.1.p1.52.m3.2.2.2.2.1.1.1.3" xref="A4.1.p1.52.m3.2.2.2.2.1.1.1.3.cmml"><mi id="A4.1.p1.52.m3.2.2.2.2.1.1.1.3.2" xref="A4.1.p1.52.m3.2.2.2.2.1.1.1.3.2.cmml">k</mi><mi id="A4.1.p1.52.m3.2.2.2.2.1.1.1.3.3" mathvariant="normal" xref="A4.1.p1.52.m3.2.2.2.2.1.1.1.3.3.cmml">c</mi></msub><mo id="A4.1.p1.52.m3.2.2.2.2.1.1.1.1a" xref="A4.1.p1.52.m3.2.2.2.2.1.1.1.1.cmml">⁢</mo><mi id="A4.1.p1.52.m3.2.2.2.2.1.1.1.4" xref="A4.1.p1.52.m3.2.2.2.2.1.1.1.4.cmml">μ</mi></mrow><mo id="A4.1.p1.52.m3.2.2.2.2.1.1.3" stretchy="false" xref="A4.1.p1.52.m3.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="A4.1.p1.52.m3.4.4.4.5a" xref="A4.1.p1.52.m3.4.4.4.5.cmml">+</mo><mrow id="A4.1.p1.52.m3.4.4.4.4" xref="A4.1.p1.52.m3.4.4.4.4.cmml"><mrow id="A4.1.p1.52.m3.3.3.3.3.1" xref="A4.1.p1.52.m3.3.3.3.3.1.cmml"><mi id="A4.1.p1.52.m3.3.3.3.3.1.3" xref="A4.1.p1.52.m3.3.3.3.3.1.3.cmml">κ</mi><mo id="A4.1.p1.52.m3.3.3.3.3.1.2" xref="A4.1.p1.52.m3.3.3.3.3.1.2.cmml">⁢</mo><msup id="A4.1.p1.52.m3.3.3.3.3.1.1" xref="A4.1.p1.52.m3.3.3.3.3.1.1.cmml"><mrow id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.cmml"><mo id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.2" stretchy="false" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.cmml">(</mo><mrow id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.cmml"><mrow id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.1.1" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.1.1.1.cmml"><mo id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.1.1.2" stretchy="false" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.1.1.1" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.1.1.1.cmml"><mn id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.1.1.1.2" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.1.1.1.1" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.1.1.1.3" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.1.1.3" stretchy="false" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.2" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.2.cmml">⁢</mo><msub id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.3" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.3.cmml"><mover accent="true" id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.3.2" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.3.2.cmml"><mi id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.3.2.2" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.3.2.2.cmml">d</mi><mo id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.3.2.1" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.3.2.1.cmml">¯</mo></mover><mi id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.3.3" mathvariant="normal" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.3.3.cmml">g</mi></msub></mrow><mo id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.3" stretchy="false" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.cmml">)</mo></mrow><mn id="A4.1.p1.52.m3.3.3.3.3.1.1.3" xref="A4.1.p1.52.m3.3.3.3.3.1.1.3.cmml">2</mn></msup></mrow><mo id="A4.1.p1.52.m3.4.4.4.4.3" xref="A4.1.p1.52.m3.4.4.4.4.3.cmml">/</mo><mrow id="A4.1.p1.52.m3.4.4.4.4.2.1" xref="A4.1.p1.52.m3.4.4.4.4.2.1.1.cmml"><mo id="A4.1.p1.52.m3.4.4.4.4.2.1.2" stretchy="false" xref="A4.1.p1.52.m3.4.4.4.4.2.1.1.cmml">(</mo><mrow id="A4.1.p1.52.m3.4.4.4.4.2.1.1" xref="A4.1.p1.52.m3.4.4.4.4.2.1.1.cmml"><mn id="A4.1.p1.52.m3.4.4.4.4.2.1.1.2" xref="A4.1.p1.52.m3.4.4.4.4.2.1.1.2.cmml">4</mn><mo id="A4.1.p1.52.m3.4.4.4.4.2.1.1.1" xref="A4.1.p1.52.m3.4.4.4.4.2.1.1.1.cmml">⁢</mo><msup id="A4.1.p1.52.m3.4.4.4.4.2.1.1.3" xref="A4.1.p1.52.m3.4.4.4.4.2.1.1.3.cmml"><mi id="A4.1.p1.52.m3.4.4.4.4.2.1.1.3.2" xref="A4.1.p1.52.m3.4.4.4.4.2.1.1.3.2.cmml">σ</mi><mo id="A4.1.p1.52.m3.4.4.4.4.2.1.1.3.3" xref="A4.1.p1.52.m3.4.4.4.4.2.1.1.3.3.cmml">∗</mo></msup><mo id="A4.1.p1.52.m3.4.4.4.4.2.1.1.1a" xref="A4.1.p1.52.m3.4.4.4.4.2.1.1.1.cmml">⁢</mo><mi id="A4.1.p1.52.m3.4.4.4.4.2.1.1.4" xref="A4.1.p1.52.m3.4.4.4.4.2.1.1.4.cmml">μ</mi></mrow><mo id="A4.1.p1.52.m3.4.4.4.4.2.1.3" stretchy="false" xref="A4.1.p1.52.m3.4.4.4.4.2.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A4.1.p1.52.m3.4.4.5" xref="A4.1.p1.52.m3.4.4.5.cmml">∈</mo><msup id="A4.1.p1.52.m3.4.4.6" xref="A4.1.p1.52.m3.4.4.6.cmml"><mi id="A4.1.p1.52.m3.4.4.6.2" xref="A4.1.p1.52.m3.4.4.6.2.cmml">ℝ</mi><mo id="A4.1.p1.52.m3.4.4.6.3" xref="A4.1.p1.52.m3.4.4.6.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.52.m3.4b"><apply id="A4.1.p1.52.m3.4.4.cmml" xref="A4.1.p1.52.m3.4.4"><in id="A4.1.p1.52.m3.4.4.5.cmml" xref="A4.1.p1.52.m3.4.4.5"></in><apply id="A4.1.p1.52.m3.4.4.4.cmml" xref="A4.1.p1.52.m3.4.4.4"><plus id="A4.1.p1.52.m3.4.4.4.5.cmml" xref="A4.1.p1.52.m3.4.4.4.5"></plus><apply id="A4.1.p1.52.m3.1.1.1.1.cmml" xref="A4.1.p1.52.m3.1.1.1.1"><times id="A4.1.p1.52.m3.1.1.1.1.2.cmml" xref="A4.1.p1.52.m3.1.1.1.1.2"></times><apply id="A4.1.p1.52.m3.1.1.1.1.1.1.1.cmml" xref="A4.1.p1.52.m3.1.1.1.1.1.1"><plus id="A4.1.p1.52.m3.1.1.1.1.1.1.1.1.cmml" xref="A4.1.p1.52.m3.1.1.1.1.1.1.1.1"></plus><cn id="A4.1.p1.52.m3.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A4.1.p1.52.m3.1.1.1.1.1.1.1.2">1</cn><ci id="A4.1.p1.52.m3.1.1.1.1.1.1.1.3.cmml" xref="A4.1.p1.52.m3.1.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="A4.1.p1.52.m3.1.1.1.1.3.cmml" xref="A4.1.p1.52.m3.1.1.1.1.3"><ci id="A4.1.p1.52.m3.1.1.1.1.3.1.cmml" xref="A4.1.p1.52.m3.1.1.1.1.3.1">¯</ci><ci id="A4.1.p1.52.m3.1.1.1.1.3.2.cmml" xref="A4.1.p1.52.m3.1.1.1.1.3.2">italic-ϵ</ci></apply><apply id="A4.1.p1.52.m3.1.1.1.1.4.cmml" xref="A4.1.p1.52.m3.1.1.1.1.4"><ci id="A4.1.p1.52.m3.1.1.1.1.4.1.cmml" xref="A4.1.p1.52.m3.1.1.1.1.4.1">¯</ci><ci id="A4.1.p1.52.m3.1.1.1.1.4.2.cmml" xref="A4.1.p1.52.m3.1.1.1.1.4.2">𝑑</ci></apply></apply><apply id="A4.1.p1.52.m3.2.2.2.2.cmml" xref="A4.1.p1.52.m3.2.2.2.2"><divide id="A4.1.p1.52.m3.2.2.2.2.2.cmml" xref="A4.1.p1.52.m3.2.2.2.2.2"></divide><apply id="A4.1.p1.52.m3.2.2.2.2.3.cmml" xref="A4.1.p1.52.m3.2.2.2.2.3"><csymbol cd="ambiguous" id="A4.1.p1.52.m3.2.2.2.2.3.1.cmml" xref="A4.1.p1.52.m3.2.2.2.2.3">superscript</csymbol><apply id="A4.1.p1.52.m3.2.2.2.2.3.2.cmml" xref="A4.1.p1.52.m3.2.2.2.2.3.2"><ci id="A4.1.p1.52.m3.2.2.2.2.3.2.1.cmml" xref="A4.1.p1.52.m3.2.2.2.2.3.2.1">¯</ci><ci id="A4.1.p1.52.m3.2.2.2.2.3.2.2.cmml" xref="A4.1.p1.52.m3.2.2.2.2.3.2.2">𝑑</ci></apply><cn id="A4.1.p1.52.m3.2.2.2.2.3.3.cmml" type="integer" xref="A4.1.p1.52.m3.2.2.2.2.3.3">2</cn></apply><apply id="A4.1.p1.52.m3.2.2.2.2.1.1.1.cmml" xref="A4.1.p1.52.m3.2.2.2.2.1.1"><times id="A4.1.p1.52.m3.2.2.2.2.1.1.1.1.cmml" xref="A4.1.p1.52.m3.2.2.2.2.1.1.1.1"></times><cn id="A4.1.p1.52.m3.2.2.2.2.1.1.1.2.cmml" type="integer" xref="A4.1.p1.52.m3.2.2.2.2.1.1.1.2">2</cn><apply id="A4.1.p1.52.m3.2.2.2.2.1.1.1.3.cmml" xref="A4.1.p1.52.m3.2.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.52.m3.2.2.2.2.1.1.1.3.1.cmml" xref="A4.1.p1.52.m3.2.2.2.2.1.1.1.3">subscript</csymbol><ci id="A4.1.p1.52.m3.2.2.2.2.1.1.1.3.2.cmml" xref="A4.1.p1.52.m3.2.2.2.2.1.1.1.3.2">𝑘</ci><ci id="A4.1.p1.52.m3.2.2.2.2.1.1.1.3.3.cmml" xref="A4.1.p1.52.m3.2.2.2.2.1.1.1.3.3">c</ci></apply><ci id="A4.1.p1.52.m3.2.2.2.2.1.1.1.4.cmml" xref="A4.1.p1.52.m3.2.2.2.2.1.1.1.4">𝜇</ci></apply></apply><apply id="A4.1.p1.52.m3.4.4.4.4.cmml" xref="A4.1.p1.52.m3.4.4.4.4"><divide id="A4.1.p1.52.m3.4.4.4.4.3.cmml" xref="A4.1.p1.52.m3.4.4.4.4.3"></divide><apply id="A4.1.p1.52.m3.3.3.3.3.1.cmml" xref="A4.1.p1.52.m3.3.3.3.3.1"><times id="A4.1.p1.52.m3.3.3.3.3.1.2.cmml" xref="A4.1.p1.52.m3.3.3.3.3.1.2"></times><ci id="A4.1.p1.52.m3.3.3.3.3.1.3.cmml" xref="A4.1.p1.52.m3.3.3.3.3.1.3">𝜅</ci><apply id="A4.1.p1.52.m3.3.3.3.3.1.1.cmml" xref="A4.1.p1.52.m3.3.3.3.3.1.1"><csymbol cd="ambiguous" id="A4.1.p1.52.m3.3.3.3.3.1.1.2.cmml" xref="A4.1.p1.52.m3.3.3.3.3.1.1">superscript</csymbol><apply id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.cmml" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1"><times id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.2.cmml" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.2"></times><apply id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.1.1.1.cmml" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.1.1"><plus id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.1.1.1.1.cmml" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.1.1.1.1"></plus><cn id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.1.1.1.2">1</cn><ci id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.1.1.1.3.cmml" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.3.cmml" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.3.1.cmml" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.3">subscript</csymbol><apply id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.3.2.cmml" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.3.2"><ci id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.3.2.1.cmml" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.3.2.1">¯</ci><ci id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.3.2.2.cmml" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.3.2.2">𝑑</ci></apply><ci id="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.3.3.cmml" xref="A4.1.p1.52.m3.3.3.3.3.1.1.1.1.1.3.3">g</ci></apply></apply><cn id="A4.1.p1.52.m3.3.3.3.3.1.1.3.cmml" type="integer" xref="A4.1.p1.52.m3.3.3.3.3.1.1.3">2</cn></apply></apply><apply id="A4.1.p1.52.m3.4.4.4.4.2.1.1.cmml" xref="A4.1.p1.52.m3.4.4.4.4.2.1"><times id="A4.1.p1.52.m3.4.4.4.4.2.1.1.1.cmml" xref="A4.1.p1.52.m3.4.4.4.4.2.1.1.1"></times><cn id="A4.1.p1.52.m3.4.4.4.4.2.1.1.2.cmml" type="integer" xref="A4.1.p1.52.m3.4.4.4.4.2.1.1.2">4</cn><apply id="A4.1.p1.52.m3.4.4.4.4.2.1.1.3.cmml" xref="A4.1.p1.52.m3.4.4.4.4.2.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.52.m3.4.4.4.4.2.1.1.3.1.cmml" xref="A4.1.p1.52.m3.4.4.4.4.2.1.1.3">superscript</csymbol><ci id="A4.1.p1.52.m3.4.4.4.4.2.1.1.3.2.cmml" xref="A4.1.p1.52.m3.4.4.4.4.2.1.1.3.2">𝜎</ci><times id="A4.1.p1.52.m3.4.4.4.4.2.1.1.3.3.cmml" xref="A4.1.p1.52.m3.4.4.4.4.2.1.1.3.3"></times></apply><ci id="A4.1.p1.52.m3.4.4.4.4.2.1.1.4.cmml" xref="A4.1.p1.52.m3.4.4.4.4.2.1.1.4">𝜇</ci></apply></apply></apply><apply id="A4.1.p1.52.m3.4.4.6.cmml" xref="A4.1.p1.52.m3.4.4.6"><csymbol cd="ambiguous" id="A4.1.p1.52.m3.4.4.6.1.cmml" xref="A4.1.p1.52.m3.4.4.6">superscript</csymbol><ci id="A4.1.p1.52.m3.4.4.6.2.cmml" xref="A4.1.p1.52.m3.4.4.6.2">ℝ</ci><plus id="A4.1.p1.52.m3.4.4.6.3.cmml" xref="A4.1.p1.52.m3.4.4.6.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.52.m3.4c">(1+p)\bar{\epsilon}\bar{d}+\bar{d}^{2}/(2k_{\rm c}\mu)+\kappa((1+p){\bar{d}}_{% \rm g})^{2}/(4\sigma^{*}\mu)\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.52.m3.4d">( 1 + italic_p ) over¯ start_ARG italic_ϵ end_ARG over¯ start_ARG italic_d end_ARG + over¯ start_ARG italic_d end_ARG start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( 2 italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT italic_μ ) + italic_κ ( ( 1 + italic_p ) over¯ start_ARG italic_d end_ARG start_POSTSUBSCRIPT roman_g end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( 4 italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_μ ) ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>. It follows from the above inequality that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx86"> <tbody id="A4.Ex83"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}_{\zeta}\leq" class="ltx_Math" display="inline" id="A4.Ex83.m1.1"><semantics id="A4.Ex83.m1.1a"><mrow id="A4.Ex83.m1.1.1" xref="A4.Ex83.m1.1.1.cmml"><msub id="A4.Ex83.m1.1.1.2" xref="A4.Ex83.m1.1.1.2.cmml"><mover accent="true" id="A4.Ex83.m1.1.1.2.2" xref="A4.Ex83.m1.1.1.2.2.cmml"><mi id="A4.Ex83.m1.1.1.2.2.2" xref="A4.Ex83.m1.1.1.2.2.2.cmml">V</mi><mo id="A4.Ex83.m1.1.1.2.2.1" xref="A4.Ex83.m1.1.1.2.2.1.cmml">˙</mo></mover><mi id="A4.Ex83.m1.1.1.2.3" xref="A4.Ex83.m1.1.1.2.3.cmml">ζ</mi></msub><mo id="A4.Ex83.m1.1.1.1" xref="A4.Ex83.m1.1.1.1.cmml">≤</mo><mi id="A4.Ex83.m1.1.1.3" xref="A4.Ex83.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex83.m1.1b"><apply id="A4.Ex83.m1.1.1.cmml" xref="A4.Ex83.m1.1.1"><leq id="A4.Ex83.m1.1.1.1.cmml" xref="A4.Ex83.m1.1.1.1"></leq><apply id="A4.Ex83.m1.1.1.2.cmml" xref="A4.Ex83.m1.1.1.2"><csymbol cd="ambiguous" id="A4.Ex83.m1.1.1.2.1.cmml" xref="A4.Ex83.m1.1.1.2">subscript</csymbol><apply id="A4.Ex83.m1.1.1.2.2.cmml" xref="A4.Ex83.m1.1.1.2.2"><ci id="A4.Ex83.m1.1.1.2.2.1.cmml" xref="A4.Ex83.m1.1.1.2.2.1">˙</ci><ci id="A4.Ex83.m1.1.1.2.2.2.cmml" xref="A4.Ex83.m1.1.1.2.2.2">𝑉</ci></apply><ci id="A4.Ex83.m1.1.1.2.3.cmml" xref="A4.Ex83.m1.1.1.2.3">𝜁</ci></apply><csymbol cd="latexml" id="A4.Ex83.m1.1.1.3.cmml" xref="A4.Ex83.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex83.m1.1c">\displaystyle\dot{V}_{\zeta}\leq</annotation><annotation encoding="application/x-llamapun" id="A4.Ex83.m1.1d">over˙ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-k_{\rm c}(1-\mu)\bm{e}^{T}\bm{e}/2-\kappa\sigma^{*}\lambda_{\min% }(\Gamma_{\zeta})(1-\mu)\tilde{\bm{\theta}}_{\zeta}^{T}\Gamma^{-1}_{\zeta}% \tilde{\bm{\theta}}_{\zeta}" class="ltx_Math" display="inline" id="A4.Ex83.m2.3"><semantics id="A4.Ex83.m2.3a"><mrow id="A4.Ex83.m2.3.3" xref="A4.Ex83.m2.3.3.cmml"><mrow id="A4.Ex83.m2.1.1.1" xref="A4.Ex83.m2.1.1.1.cmml"><mo id="A4.Ex83.m2.1.1.1a" xref="A4.Ex83.m2.1.1.1.cmml">−</mo><mrow id="A4.Ex83.m2.1.1.1.1" xref="A4.Ex83.m2.1.1.1.1.cmml"><mrow id="A4.Ex83.m2.1.1.1.1.1" xref="A4.Ex83.m2.1.1.1.1.1.cmml"><msub id="A4.Ex83.m2.1.1.1.1.1.3" xref="A4.Ex83.m2.1.1.1.1.1.3.cmml"><mi id="A4.Ex83.m2.1.1.1.1.1.3.2" xref="A4.Ex83.m2.1.1.1.1.1.3.2.cmml">k</mi><mi id="A4.Ex83.m2.1.1.1.1.1.3.3" mathvariant="normal" xref="A4.Ex83.m2.1.1.1.1.1.3.3.cmml">c</mi></msub><mo id="A4.Ex83.m2.1.1.1.1.1.2" xref="A4.Ex83.m2.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex83.m2.1.1.1.1.1.1.1" xref="A4.Ex83.m2.1.1.1.1.1.1.1.1.cmml"><mo id="A4.Ex83.m2.1.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex83.m2.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex83.m2.1.1.1.1.1.1.1.1" xref="A4.Ex83.m2.1.1.1.1.1.1.1.1.cmml"><mn id="A4.Ex83.m2.1.1.1.1.1.1.1.1.2" xref="A4.Ex83.m2.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A4.Ex83.m2.1.1.1.1.1.1.1.1.1" xref="A4.Ex83.m2.1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="A4.Ex83.m2.1.1.1.1.1.1.1.1.3" xref="A4.Ex83.m2.1.1.1.1.1.1.1.1.3.cmml">μ</mi></mrow><mo id="A4.Ex83.m2.1.1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex83.m2.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex83.m2.1.1.1.1.1.2a" xref="A4.Ex83.m2.1.1.1.1.1.2.cmml">⁢</mo><msup id="A4.Ex83.m2.1.1.1.1.1.4" xref="A4.Ex83.m2.1.1.1.1.1.4.cmml"><mi id="A4.Ex83.m2.1.1.1.1.1.4.2" xref="A4.Ex83.m2.1.1.1.1.1.4.2.cmml">𝒆</mi><mi id="A4.Ex83.m2.1.1.1.1.1.4.3" xref="A4.Ex83.m2.1.1.1.1.1.4.3.cmml">T</mi></msup><mo id="A4.Ex83.m2.1.1.1.1.1.2b" xref="A4.Ex83.m2.1.1.1.1.1.2.cmml">⁢</mo><mi id="A4.Ex83.m2.1.1.1.1.1.5" xref="A4.Ex83.m2.1.1.1.1.1.5.cmml">𝒆</mi></mrow><mo id="A4.Ex83.m2.1.1.1.1.2" xref="A4.Ex83.m2.1.1.1.1.2.cmml">/</mo><mn id="A4.Ex83.m2.1.1.1.1.3" xref="A4.Ex83.m2.1.1.1.1.3.cmml">2</mn></mrow></mrow><mo id="A4.Ex83.m2.3.3.4" xref="A4.Ex83.m2.3.3.4.cmml">−</mo><mrow id="A4.Ex83.m2.3.3.3" xref="A4.Ex83.m2.3.3.3.cmml"><mi id="A4.Ex83.m2.3.3.3.4" xref="A4.Ex83.m2.3.3.3.4.cmml">κ</mi><mo id="A4.Ex83.m2.3.3.3.3" xref="A4.Ex83.m2.3.3.3.3.cmml">⁢</mo><msup id="A4.Ex83.m2.3.3.3.5" xref="A4.Ex83.m2.3.3.3.5.cmml"><mi id="A4.Ex83.m2.3.3.3.5.2" xref="A4.Ex83.m2.3.3.3.5.2.cmml">σ</mi><mo id="A4.Ex83.m2.3.3.3.5.3" xref="A4.Ex83.m2.3.3.3.5.3.cmml">∗</mo></msup><mo id="A4.Ex83.m2.3.3.3.3a" xref="A4.Ex83.m2.3.3.3.3.cmml">⁢</mo><msub id="A4.Ex83.m2.3.3.3.6" xref="A4.Ex83.m2.3.3.3.6.cmml"><mi id="A4.Ex83.m2.3.3.3.6.2" xref="A4.Ex83.m2.3.3.3.6.2.cmml">λ</mi><mi id="A4.Ex83.m2.3.3.3.6.3" xref="A4.Ex83.m2.3.3.3.6.3.cmml">min</mi></msub><mo id="A4.Ex83.m2.3.3.3.3b" xref="A4.Ex83.m2.3.3.3.3.cmml">⁢</mo><mrow id="A4.Ex83.m2.2.2.2.1.1" xref="A4.Ex83.m2.2.2.2.1.1.1.cmml"><mo id="A4.Ex83.m2.2.2.2.1.1.2" stretchy="false" xref="A4.Ex83.m2.2.2.2.1.1.1.cmml">(</mo><msub id="A4.Ex83.m2.2.2.2.1.1.1" xref="A4.Ex83.m2.2.2.2.1.1.1.cmml"><mi id="A4.Ex83.m2.2.2.2.1.1.1.2" mathvariant="normal" xref="A4.Ex83.m2.2.2.2.1.1.1.2.cmml">Γ</mi><mi id="A4.Ex83.m2.2.2.2.1.1.1.3" xref="A4.Ex83.m2.2.2.2.1.1.1.3.cmml">ζ</mi></msub><mo id="A4.Ex83.m2.2.2.2.1.1.3" stretchy="false" xref="A4.Ex83.m2.2.2.2.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex83.m2.3.3.3.3c" xref="A4.Ex83.m2.3.3.3.3.cmml">⁢</mo><mrow id="A4.Ex83.m2.3.3.3.2.1" xref="A4.Ex83.m2.3.3.3.2.1.1.cmml"><mo id="A4.Ex83.m2.3.3.3.2.1.2" stretchy="false" xref="A4.Ex83.m2.3.3.3.2.1.1.cmml">(</mo><mrow id="A4.Ex83.m2.3.3.3.2.1.1" xref="A4.Ex83.m2.3.3.3.2.1.1.cmml"><mn id="A4.Ex83.m2.3.3.3.2.1.1.2" xref="A4.Ex83.m2.3.3.3.2.1.1.2.cmml">1</mn><mo id="A4.Ex83.m2.3.3.3.2.1.1.1" xref="A4.Ex83.m2.3.3.3.2.1.1.1.cmml">−</mo><mi id="A4.Ex83.m2.3.3.3.2.1.1.3" xref="A4.Ex83.m2.3.3.3.2.1.1.3.cmml">μ</mi></mrow><mo id="A4.Ex83.m2.3.3.3.2.1.3" stretchy="false" xref="A4.Ex83.m2.3.3.3.2.1.1.cmml">)</mo></mrow><mo id="A4.Ex83.m2.3.3.3.3d" xref="A4.Ex83.m2.3.3.3.3.cmml">⁢</mo><msubsup id="A4.Ex83.m2.3.3.3.7" xref="A4.Ex83.m2.3.3.3.7.cmml"><mover accent="true" id="A4.Ex83.m2.3.3.3.7.2.2" xref="A4.Ex83.m2.3.3.3.7.2.2.cmml"><mi id="A4.Ex83.m2.3.3.3.7.2.2.2" xref="A4.Ex83.m2.3.3.3.7.2.2.2.cmml">𝜽</mi><mo id="A4.Ex83.m2.3.3.3.7.2.2.1" xref="A4.Ex83.m2.3.3.3.7.2.2.1.cmml">~</mo></mover><mi id="A4.Ex83.m2.3.3.3.7.2.3" xref="A4.Ex83.m2.3.3.3.7.2.3.cmml">ζ</mi><mi id="A4.Ex83.m2.3.3.3.7.3" xref="A4.Ex83.m2.3.3.3.7.3.cmml">T</mi></msubsup><mo id="A4.Ex83.m2.3.3.3.3e" xref="A4.Ex83.m2.3.3.3.3.cmml">⁢</mo><msubsup id="A4.Ex83.m2.3.3.3.8" xref="A4.Ex83.m2.3.3.3.8.cmml"><mi id="A4.Ex83.m2.3.3.3.8.2.2" mathvariant="normal" xref="A4.Ex83.m2.3.3.3.8.2.2.cmml">Γ</mi><mi id="A4.Ex83.m2.3.3.3.8.3" xref="A4.Ex83.m2.3.3.3.8.3.cmml">ζ</mi><mrow id="A4.Ex83.m2.3.3.3.8.2.3" xref="A4.Ex83.m2.3.3.3.8.2.3.cmml"><mo id="A4.Ex83.m2.3.3.3.8.2.3a" xref="A4.Ex83.m2.3.3.3.8.2.3.cmml">−</mo><mn id="A4.Ex83.m2.3.3.3.8.2.3.2" xref="A4.Ex83.m2.3.3.3.8.2.3.2.cmml">1</mn></mrow></msubsup><mo id="A4.Ex83.m2.3.3.3.3f" xref="A4.Ex83.m2.3.3.3.3.cmml">⁢</mo><msub id="A4.Ex83.m2.3.3.3.9" xref="A4.Ex83.m2.3.3.3.9.cmml"><mover accent="true" id="A4.Ex83.m2.3.3.3.9.2" xref="A4.Ex83.m2.3.3.3.9.2.cmml"><mi id="A4.Ex83.m2.3.3.3.9.2.2" xref="A4.Ex83.m2.3.3.3.9.2.2.cmml">𝜽</mi><mo id="A4.Ex83.m2.3.3.3.9.2.1" xref="A4.Ex83.m2.3.3.3.9.2.1.cmml">~</mo></mover><mi id="A4.Ex83.m2.3.3.3.9.3" xref="A4.Ex83.m2.3.3.3.9.3.cmml">ζ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex83.m2.3b"><apply id="A4.Ex83.m2.3.3.cmml" xref="A4.Ex83.m2.3.3"><minus id="A4.Ex83.m2.3.3.4.cmml" xref="A4.Ex83.m2.3.3.4"></minus><apply id="A4.Ex83.m2.1.1.1.cmml" xref="A4.Ex83.m2.1.1.1"><minus id="A4.Ex83.m2.1.1.1.2.cmml" xref="A4.Ex83.m2.1.1.1"></minus><apply id="A4.Ex83.m2.1.1.1.1.cmml" xref="A4.Ex83.m2.1.1.1.1"><divide id="A4.Ex83.m2.1.1.1.1.2.cmml" xref="A4.Ex83.m2.1.1.1.1.2"></divide><apply id="A4.Ex83.m2.1.1.1.1.1.cmml" xref="A4.Ex83.m2.1.1.1.1.1"><times id="A4.Ex83.m2.1.1.1.1.1.2.cmml" xref="A4.Ex83.m2.1.1.1.1.1.2"></times><apply id="A4.Ex83.m2.1.1.1.1.1.3.cmml" xref="A4.Ex83.m2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.Ex83.m2.1.1.1.1.1.3.1.cmml" xref="A4.Ex83.m2.1.1.1.1.1.3">subscript</csymbol><ci id="A4.Ex83.m2.1.1.1.1.1.3.2.cmml" xref="A4.Ex83.m2.1.1.1.1.1.3.2">𝑘</ci><ci id="A4.Ex83.m2.1.1.1.1.1.3.3.cmml" xref="A4.Ex83.m2.1.1.1.1.1.3.3">c</ci></apply><apply id="A4.Ex83.m2.1.1.1.1.1.1.1.1.cmml" xref="A4.Ex83.m2.1.1.1.1.1.1.1"><minus id="A4.Ex83.m2.1.1.1.1.1.1.1.1.1.cmml" xref="A4.Ex83.m2.1.1.1.1.1.1.1.1.1"></minus><cn id="A4.Ex83.m2.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A4.Ex83.m2.1.1.1.1.1.1.1.1.2">1</cn><ci id="A4.Ex83.m2.1.1.1.1.1.1.1.1.3.cmml" xref="A4.Ex83.m2.1.1.1.1.1.1.1.1.3">𝜇</ci></apply><apply id="A4.Ex83.m2.1.1.1.1.1.4.cmml" xref="A4.Ex83.m2.1.1.1.1.1.4"><csymbol cd="ambiguous" id="A4.Ex83.m2.1.1.1.1.1.4.1.cmml" xref="A4.Ex83.m2.1.1.1.1.1.4">superscript</csymbol><ci id="A4.Ex83.m2.1.1.1.1.1.4.2.cmml" xref="A4.Ex83.m2.1.1.1.1.1.4.2">𝒆</ci><ci id="A4.Ex83.m2.1.1.1.1.1.4.3.cmml" xref="A4.Ex83.m2.1.1.1.1.1.4.3">𝑇</ci></apply><ci id="A4.Ex83.m2.1.1.1.1.1.5.cmml" xref="A4.Ex83.m2.1.1.1.1.1.5">𝒆</ci></apply><cn id="A4.Ex83.m2.1.1.1.1.3.cmml" type="integer" xref="A4.Ex83.m2.1.1.1.1.3">2</cn></apply></apply><apply id="A4.Ex83.m2.3.3.3.cmml" xref="A4.Ex83.m2.3.3.3"><times id="A4.Ex83.m2.3.3.3.3.cmml" xref="A4.Ex83.m2.3.3.3.3"></times><ci id="A4.Ex83.m2.3.3.3.4.cmml" xref="A4.Ex83.m2.3.3.3.4">𝜅</ci><apply id="A4.Ex83.m2.3.3.3.5.cmml" xref="A4.Ex83.m2.3.3.3.5"><csymbol cd="ambiguous" id="A4.Ex83.m2.3.3.3.5.1.cmml" xref="A4.Ex83.m2.3.3.3.5">superscript</csymbol><ci id="A4.Ex83.m2.3.3.3.5.2.cmml" xref="A4.Ex83.m2.3.3.3.5.2">𝜎</ci><times id="A4.Ex83.m2.3.3.3.5.3.cmml" xref="A4.Ex83.m2.3.3.3.5.3"></times></apply><apply id="A4.Ex83.m2.3.3.3.6.cmml" xref="A4.Ex83.m2.3.3.3.6"><csymbol cd="ambiguous" id="A4.Ex83.m2.3.3.3.6.1.cmml" xref="A4.Ex83.m2.3.3.3.6">subscript</csymbol><ci id="A4.Ex83.m2.3.3.3.6.2.cmml" xref="A4.Ex83.m2.3.3.3.6.2">𝜆</ci><min id="A4.Ex83.m2.3.3.3.6.3.cmml" xref="A4.Ex83.m2.3.3.3.6.3"></min></apply><apply id="A4.Ex83.m2.2.2.2.1.1.1.cmml" xref="A4.Ex83.m2.2.2.2.1.1"><csymbol cd="ambiguous" id="A4.Ex83.m2.2.2.2.1.1.1.1.cmml" xref="A4.Ex83.m2.2.2.2.1.1">subscript</csymbol><ci id="A4.Ex83.m2.2.2.2.1.1.1.2.cmml" xref="A4.Ex83.m2.2.2.2.1.1.1.2">Γ</ci><ci id="A4.Ex83.m2.2.2.2.1.1.1.3.cmml" xref="A4.Ex83.m2.2.2.2.1.1.1.3">𝜁</ci></apply><apply id="A4.Ex83.m2.3.3.3.2.1.1.cmml" xref="A4.Ex83.m2.3.3.3.2.1"><minus id="A4.Ex83.m2.3.3.3.2.1.1.1.cmml" xref="A4.Ex83.m2.3.3.3.2.1.1.1"></minus><cn id="A4.Ex83.m2.3.3.3.2.1.1.2.cmml" type="integer" xref="A4.Ex83.m2.3.3.3.2.1.1.2">1</cn><ci id="A4.Ex83.m2.3.3.3.2.1.1.3.cmml" xref="A4.Ex83.m2.3.3.3.2.1.1.3">𝜇</ci></apply><apply id="A4.Ex83.m2.3.3.3.7.cmml" xref="A4.Ex83.m2.3.3.3.7"><csymbol cd="ambiguous" id="A4.Ex83.m2.3.3.3.7.1.cmml" xref="A4.Ex83.m2.3.3.3.7">superscript</csymbol><apply id="A4.Ex83.m2.3.3.3.7.2.cmml" xref="A4.Ex83.m2.3.3.3.7"><csymbol cd="ambiguous" id="A4.Ex83.m2.3.3.3.7.2.1.cmml" xref="A4.Ex83.m2.3.3.3.7">subscript</csymbol><apply id="A4.Ex83.m2.3.3.3.7.2.2.cmml" xref="A4.Ex83.m2.3.3.3.7.2.2"><ci id="A4.Ex83.m2.3.3.3.7.2.2.1.cmml" xref="A4.Ex83.m2.3.3.3.7.2.2.1">~</ci><ci id="A4.Ex83.m2.3.3.3.7.2.2.2.cmml" xref="A4.Ex83.m2.3.3.3.7.2.2.2">𝜽</ci></apply><ci id="A4.Ex83.m2.3.3.3.7.2.3.cmml" xref="A4.Ex83.m2.3.3.3.7.2.3">𝜁</ci></apply><ci id="A4.Ex83.m2.3.3.3.7.3.cmml" xref="A4.Ex83.m2.3.3.3.7.3">𝑇</ci></apply><apply id="A4.Ex83.m2.3.3.3.8.cmml" xref="A4.Ex83.m2.3.3.3.8"><csymbol cd="ambiguous" id="A4.Ex83.m2.3.3.3.8.1.cmml" xref="A4.Ex83.m2.3.3.3.8">subscript</csymbol><apply id="A4.Ex83.m2.3.3.3.8.2.cmml" xref="A4.Ex83.m2.3.3.3.8"><csymbol cd="ambiguous" id="A4.Ex83.m2.3.3.3.8.2.1.cmml" xref="A4.Ex83.m2.3.3.3.8">superscript</csymbol><ci id="A4.Ex83.m2.3.3.3.8.2.2.cmml" xref="A4.Ex83.m2.3.3.3.8.2.2">Γ</ci><apply id="A4.Ex83.m2.3.3.3.8.2.3.cmml" xref="A4.Ex83.m2.3.3.3.8.2.3"><minus id="A4.Ex83.m2.3.3.3.8.2.3.1.cmml" xref="A4.Ex83.m2.3.3.3.8.2.3"></minus><cn id="A4.Ex83.m2.3.3.3.8.2.3.2.cmml" type="integer" xref="A4.Ex83.m2.3.3.3.8.2.3.2">1</cn></apply></apply><ci id="A4.Ex83.m2.3.3.3.8.3.cmml" xref="A4.Ex83.m2.3.3.3.8.3">𝜁</ci></apply><apply id="A4.Ex83.m2.3.3.3.9.cmml" xref="A4.Ex83.m2.3.3.3.9"><csymbol cd="ambiguous" id="A4.Ex83.m2.3.3.3.9.1.cmml" xref="A4.Ex83.m2.3.3.3.9">subscript</csymbol><apply id="A4.Ex83.m2.3.3.3.9.2.cmml" xref="A4.Ex83.m2.3.3.3.9.2"><ci id="A4.Ex83.m2.3.3.3.9.2.1.cmml" xref="A4.Ex83.m2.3.3.3.9.2.1">~</ci><ci id="A4.Ex83.m2.3.3.3.9.2.2.cmml" xref="A4.Ex83.m2.3.3.3.9.2.2">𝜽</ci></apply><ci id="A4.Ex83.m2.3.3.3.9.3.cmml" xref="A4.Ex83.m2.3.3.3.9.3">𝜁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex83.m2.3c">\displaystyle-k_{\rm c}(1-\mu)\bm{e}^{T}\bm{e}/2-\kappa\sigma^{*}\lambda_{\min% }(\Gamma_{\zeta})(1-\mu)\tilde{\bm{\theta}}_{\zeta}^{T}\Gamma^{-1}_{\zeta}% \tilde{\bm{\theta}}_{\zeta}</annotation><annotation encoding="application/x-llamapun" id="A4.Ex83.m2.3d">- italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ( 1 - italic_μ ) bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_e / 2 - italic_κ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Γ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ) ( 1 - italic_μ ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Γ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex84"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle+\rho_{2}(k_{\rm c},\kappa,\bar{d})." class="ltx_Math" display="inline" id="A4.Ex84.m1.3"><semantics id="A4.Ex84.m1.3a"><mrow id="A4.Ex84.m1.3.3.1" xref="A4.Ex84.m1.3.3.1.1.cmml"><mrow id="A4.Ex84.m1.3.3.1.1" xref="A4.Ex84.m1.3.3.1.1.cmml"><mo id="A4.Ex84.m1.3.3.1.1a" xref="A4.Ex84.m1.3.3.1.1.cmml">+</mo><mrow id="A4.Ex84.m1.3.3.1.1.1" xref="A4.Ex84.m1.3.3.1.1.1.cmml"><msub id="A4.Ex84.m1.3.3.1.1.1.3" xref="A4.Ex84.m1.3.3.1.1.1.3.cmml"><mi id="A4.Ex84.m1.3.3.1.1.1.3.2" xref="A4.Ex84.m1.3.3.1.1.1.3.2.cmml">ρ</mi><mn id="A4.Ex84.m1.3.3.1.1.1.3.3" xref="A4.Ex84.m1.3.3.1.1.1.3.3.cmml">2</mn></msub><mo id="A4.Ex84.m1.3.3.1.1.1.2" xref="A4.Ex84.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex84.m1.3.3.1.1.1.1.1" xref="A4.Ex84.m1.3.3.1.1.1.1.2.cmml"><mo id="A4.Ex84.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="A4.Ex84.m1.3.3.1.1.1.1.2.cmml">(</mo><msub id="A4.Ex84.m1.3.3.1.1.1.1.1.1" xref="A4.Ex84.m1.3.3.1.1.1.1.1.1.cmml"><mi id="A4.Ex84.m1.3.3.1.1.1.1.1.1.2" xref="A4.Ex84.m1.3.3.1.1.1.1.1.1.2.cmml">k</mi><mi id="A4.Ex84.m1.3.3.1.1.1.1.1.1.3" mathvariant="normal" xref="A4.Ex84.m1.3.3.1.1.1.1.1.1.3.cmml">c</mi></msub><mo id="A4.Ex84.m1.3.3.1.1.1.1.1.3" xref="A4.Ex84.m1.3.3.1.1.1.1.2.cmml">,</mo><mi id="A4.Ex84.m1.1.1" xref="A4.Ex84.m1.1.1.cmml">κ</mi><mo id="A4.Ex84.m1.3.3.1.1.1.1.1.4" xref="A4.Ex84.m1.3.3.1.1.1.1.2.cmml">,</mo><mover accent="true" id="A4.Ex84.m1.2.2" xref="A4.Ex84.m1.2.2.cmml"><mi id="A4.Ex84.m1.2.2.2" xref="A4.Ex84.m1.2.2.2.cmml">d</mi><mo id="A4.Ex84.m1.2.2.1" xref="A4.Ex84.m1.2.2.1.cmml">¯</mo></mover><mo id="A4.Ex84.m1.3.3.1.1.1.1.1.5" stretchy="false" xref="A4.Ex84.m1.3.3.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="A4.Ex84.m1.3.3.1.2" lspace="0em" xref="A4.Ex84.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex84.m1.3b"><apply id="A4.Ex84.m1.3.3.1.1.cmml" xref="A4.Ex84.m1.3.3.1"><plus id="A4.Ex84.m1.3.3.1.1.2.cmml" xref="A4.Ex84.m1.3.3.1"></plus><apply id="A4.Ex84.m1.3.3.1.1.1.cmml" xref="A4.Ex84.m1.3.3.1.1.1"><times id="A4.Ex84.m1.3.3.1.1.1.2.cmml" xref="A4.Ex84.m1.3.3.1.1.1.2"></times><apply id="A4.Ex84.m1.3.3.1.1.1.3.cmml" xref="A4.Ex84.m1.3.3.1.1.1.3"><csymbol cd="ambiguous" id="A4.Ex84.m1.3.3.1.1.1.3.1.cmml" xref="A4.Ex84.m1.3.3.1.1.1.3">subscript</csymbol><ci id="A4.Ex84.m1.3.3.1.1.1.3.2.cmml" xref="A4.Ex84.m1.3.3.1.1.1.3.2">𝜌</ci><cn id="A4.Ex84.m1.3.3.1.1.1.3.3.cmml" type="integer" xref="A4.Ex84.m1.3.3.1.1.1.3.3">2</cn></apply><vector id="A4.Ex84.m1.3.3.1.1.1.1.2.cmml" xref="A4.Ex84.m1.3.3.1.1.1.1.1"><apply id="A4.Ex84.m1.3.3.1.1.1.1.1.1.cmml" xref="A4.Ex84.m1.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.Ex84.m1.3.3.1.1.1.1.1.1.1.cmml" xref="A4.Ex84.m1.3.3.1.1.1.1.1.1">subscript</csymbol><ci id="A4.Ex84.m1.3.3.1.1.1.1.1.1.2.cmml" xref="A4.Ex84.m1.3.3.1.1.1.1.1.1.2">𝑘</ci><ci id="A4.Ex84.m1.3.3.1.1.1.1.1.1.3.cmml" xref="A4.Ex84.m1.3.3.1.1.1.1.1.1.3">c</ci></apply><ci id="A4.Ex84.m1.1.1.cmml" xref="A4.Ex84.m1.1.1">𝜅</ci><apply id="A4.Ex84.m1.2.2.cmml" xref="A4.Ex84.m1.2.2"><ci id="A4.Ex84.m1.2.2.1.cmml" xref="A4.Ex84.m1.2.2.1">¯</ci><ci id="A4.Ex84.m1.2.2.2.cmml" xref="A4.Ex84.m1.2.2.2">𝑑</ci></apply></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex84.m1.3c">\displaystyle+\rho_{2}(k_{\rm c},\kappa,\bar{d}).</annotation><annotation encoding="application/x-llamapun" id="A4.Ex84.m1.3d">+ italic_ρ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT , italic_κ , over¯ start_ARG italic_d end_ARG ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.1.p1.99">Consequently, one immediately gets</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx87"> <tbody id="A4.Ex85"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}_{\zeta}\leq" class="ltx_Math" display="inline" id="A4.Ex85.m1.1"><semantics id="A4.Ex85.m1.1a"><mrow id="A4.Ex85.m1.1.1" xref="A4.Ex85.m1.1.1.cmml"><msub id="A4.Ex85.m1.1.1.2" xref="A4.Ex85.m1.1.1.2.cmml"><mover accent="true" id="A4.Ex85.m1.1.1.2.2" xref="A4.Ex85.m1.1.1.2.2.cmml"><mi id="A4.Ex85.m1.1.1.2.2.2" xref="A4.Ex85.m1.1.1.2.2.2.cmml">V</mi><mo id="A4.Ex85.m1.1.1.2.2.1" xref="A4.Ex85.m1.1.1.2.2.1.cmml">˙</mo></mover><mi id="A4.Ex85.m1.1.1.2.3" xref="A4.Ex85.m1.1.1.2.3.cmml">ζ</mi></msub><mo id="A4.Ex85.m1.1.1.1" xref="A4.Ex85.m1.1.1.1.cmml">≤</mo><mi id="A4.Ex85.m1.1.1.3" xref="A4.Ex85.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex85.m1.1b"><apply id="A4.Ex85.m1.1.1.cmml" xref="A4.Ex85.m1.1.1"><leq id="A4.Ex85.m1.1.1.1.cmml" xref="A4.Ex85.m1.1.1.1"></leq><apply id="A4.Ex85.m1.1.1.2.cmml" xref="A4.Ex85.m1.1.1.2"><csymbol cd="ambiguous" id="A4.Ex85.m1.1.1.2.1.cmml" xref="A4.Ex85.m1.1.1.2">subscript</csymbol><apply id="A4.Ex85.m1.1.1.2.2.cmml" xref="A4.Ex85.m1.1.1.2.2"><ci id="A4.Ex85.m1.1.1.2.2.1.cmml" xref="A4.Ex85.m1.1.1.2.2.1">˙</ci><ci id="A4.Ex85.m1.1.1.2.2.2.cmml" xref="A4.Ex85.m1.1.1.2.2.2">𝑉</ci></apply><ci id="A4.Ex85.m1.1.1.2.3.cmml" xref="A4.Ex85.m1.1.1.2.3">𝜁</ci></apply><csymbol cd="latexml" id="A4.Ex85.m1.1.1.3.cmml" xref="A4.Ex85.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex85.m1.1c">\displaystyle\dot{V}_{\zeta}\leq</annotation><annotation encoding="application/x-llamapun" id="A4.Ex85.m1.1d">over˙ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-k_{\rm r}V_{\zeta}+\rho_{2}(k_{\rm c},\kappa,\bar{d}),\forall t% \geq T_{\rm a}" class="ltx_Math" display="inline" id="A4.Ex85.m2.4"><semantics id="A4.Ex85.m2.4a"><mrow id="A4.Ex85.m2.4.4" xref="A4.Ex85.m2.4.4.cmml"><mrow id="A4.Ex85.m2.4.4.2.2" xref="A4.Ex85.m2.4.4.2.3.cmml"><mrow id="A4.Ex85.m2.3.3.1.1.1" xref="A4.Ex85.m2.3.3.1.1.1.cmml"><mrow id="A4.Ex85.m2.3.3.1.1.1.3" xref="A4.Ex85.m2.3.3.1.1.1.3.cmml"><mo id="A4.Ex85.m2.3.3.1.1.1.3a" xref="A4.Ex85.m2.3.3.1.1.1.3.cmml">−</mo><mrow id="A4.Ex85.m2.3.3.1.1.1.3.2" xref="A4.Ex85.m2.3.3.1.1.1.3.2.cmml"><msub id="A4.Ex85.m2.3.3.1.1.1.3.2.2" xref="A4.Ex85.m2.3.3.1.1.1.3.2.2.cmml"><mi id="A4.Ex85.m2.3.3.1.1.1.3.2.2.2" xref="A4.Ex85.m2.3.3.1.1.1.3.2.2.2.cmml">k</mi><mi id="A4.Ex85.m2.3.3.1.1.1.3.2.2.3" mathvariant="normal" xref="A4.Ex85.m2.3.3.1.1.1.3.2.2.3.cmml">r</mi></msub><mo id="A4.Ex85.m2.3.3.1.1.1.3.2.1" xref="A4.Ex85.m2.3.3.1.1.1.3.2.1.cmml">⁢</mo><msub id="A4.Ex85.m2.3.3.1.1.1.3.2.3" xref="A4.Ex85.m2.3.3.1.1.1.3.2.3.cmml"><mi id="A4.Ex85.m2.3.3.1.1.1.3.2.3.2" xref="A4.Ex85.m2.3.3.1.1.1.3.2.3.2.cmml">V</mi><mi id="A4.Ex85.m2.3.3.1.1.1.3.2.3.3" xref="A4.Ex85.m2.3.3.1.1.1.3.2.3.3.cmml">ζ</mi></msub></mrow></mrow><mo id="A4.Ex85.m2.3.3.1.1.1.2" xref="A4.Ex85.m2.3.3.1.1.1.2.cmml">+</mo><mrow id="A4.Ex85.m2.3.3.1.1.1.1" xref="A4.Ex85.m2.3.3.1.1.1.1.cmml"><msub id="A4.Ex85.m2.3.3.1.1.1.1.3" xref="A4.Ex85.m2.3.3.1.1.1.1.3.cmml"><mi id="A4.Ex85.m2.3.3.1.1.1.1.3.2" xref="A4.Ex85.m2.3.3.1.1.1.1.3.2.cmml">ρ</mi><mn id="A4.Ex85.m2.3.3.1.1.1.1.3.3" xref="A4.Ex85.m2.3.3.1.1.1.1.3.3.cmml">2</mn></msub><mo id="A4.Ex85.m2.3.3.1.1.1.1.2" xref="A4.Ex85.m2.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex85.m2.3.3.1.1.1.1.1.1" xref="A4.Ex85.m2.3.3.1.1.1.1.1.2.cmml"><mo id="A4.Ex85.m2.3.3.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex85.m2.3.3.1.1.1.1.1.2.cmml">(</mo><msub id="A4.Ex85.m2.3.3.1.1.1.1.1.1.1" xref="A4.Ex85.m2.3.3.1.1.1.1.1.1.1.cmml"><mi id="A4.Ex85.m2.3.3.1.1.1.1.1.1.1.2" xref="A4.Ex85.m2.3.3.1.1.1.1.1.1.1.2.cmml">k</mi><mi id="A4.Ex85.m2.3.3.1.1.1.1.1.1.1.3" mathvariant="normal" xref="A4.Ex85.m2.3.3.1.1.1.1.1.1.1.3.cmml">c</mi></msub><mo id="A4.Ex85.m2.3.3.1.1.1.1.1.1.3" xref="A4.Ex85.m2.3.3.1.1.1.1.1.2.cmml">,</mo><mi id="A4.Ex85.m2.1.1" xref="A4.Ex85.m2.1.1.cmml">κ</mi><mo id="A4.Ex85.m2.3.3.1.1.1.1.1.1.4" xref="A4.Ex85.m2.3.3.1.1.1.1.1.2.cmml">,</mo><mover accent="true" id="A4.Ex85.m2.2.2" xref="A4.Ex85.m2.2.2.cmml"><mi id="A4.Ex85.m2.2.2.2" xref="A4.Ex85.m2.2.2.2.cmml">d</mi><mo id="A4.Ex85.m2.2.2.1" xref="A4.Ex85.m2.2.2.1.cmml">¯</mo></mover><mo id="A4.Ex85.m2.3.3.1.1.1.1.1.1.5" stretchy="false" xref="A4.Ex85.m2.3.3.1.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="A4.Ex85.m2.4.4.2.2.3" xref="A4.Ex85.m2.4.4.2.3.cmml">,</mo><mrow id="A4.Ex85.m2.4.4.2.2.2" xref="A4.Ex85.m2.4.4.2.2.2.cmml"><mo id="A4.Ex85.m2.4.4.2.2.2.1" rspace="0.167em" xref="A4.Ex85.m2.4.4.2.2.2.1.cmml">∀</mo><mi id="A4.Ex85.m2.4.4.2.2.2.2" xref="A4.Ex85.m2.4.4.2.2.2.2.cmml">t</mi></mrow></mrow><mo id="A4.Ex85.m2.4.4.3" xref="A4.Ex85.m2.4.4.3.cmml">≥</mo><msub id="A4.Ex85.m2.4.4.4" xref="A4.Ex85.m2.4.4.4.cmml"><mi id="A4.Ex85.m2.4.4.4.2" xref="A4.Ex85.m2.4.4.4.2.cmml">T</mi><mi id="A4.Ex85.m2.4.4.4.3" mathvariant="normal" xref="A4.Ex85.m2.4.4.4.3.cmml">a</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex85.m2.4b"><apply id="A4.Ex85.m2.4.4.cmml" xref="A4.Ex85.m2.4.4"><geq id="A4.Ex85.m2.4.4.3.cmml" xref="A4.Ex85.m2.4.4.3"></geq><list id="A4.Ex85.m2.4.4.2.3.cmml" xref="A4.Ex85.m2.4.4.2.2"><apply id="A4.Ex85.m2.3.3.1.1.1.cmml" xref="A4.Ex85.m2.3.3.1.1.1"><plus id="A4.Ex85.m2.3.3.1.1.1.2.cmml" xref="A4.Ex85.m2.3.3.1.1.1.2"></plus><apply id="A4.Ex85.m2.3.3.1.1.1.3.cmml" xref="A4.Ex85.m2.3.3.1.1.1.3"><minus id="A4.Ex85.m2.3.3.1.1.1.3.1.cmml" xref="A4.Ex85.m2.3.3.1.1.1.3"></minus><apply id="A4.Ex85.m2.3.3.1.1.1.3.2.cmml" xref="A4.Ex85.m2.3.3.1.1.1.3.2"><times id="A4.Ex85.m2.3.3.1.1.1.3.2.1.cmml" xref="A4.Ex85.m2.3.3.1.1.1.3.2.1"></times><apply id="A4.Ex85.m2.3.3.1.1.1.3.2.2.cmml" xref="A4.Ex85.m2.3.3.1.1.1.3.2.2"><csymbol cd="ambiguous" id="A4.Ex85.m2.3.3.1.1.1.3.2.2.1.cmml" xref="A4.Ex85.m2.3.3.1.1.1.3.2.2">subscript</csymbol><ci id="A4.Ex85.m2.3.3.1.1.1.3.2.2.2.cmml" xref="A4.Ex85.m2.3.3.1.1.1.3.2.2.2">𝑘</ci><ci id="A4.Ex85.m2.3.3.1.1.1.3.2.2.3.cmml" xref="A4.Ex85.m2.3.3.1.1.1.3.2.2.3">r</ci></apply><apply id="A4.Ex85.m2.3.3.1.1.1.3.2.3.cmml" xref="A4.Ex85.m2.3.3.1.1.1.3.2.3"><csymbol cd="ambiguous" id="A4.Ex85.m2.3.3.1.1.1.3.2.3.1.cmml" xref="A4.Ex85.m2.3.3.1.1.1.3.2.3">subscript</csymbol><ci id="A4.Ex85.m2.3.3.1.1.1.3.2.3.2.cmml" xref="A4.Ex85.m2.3.3.1.1.1.3.2.3.2">𝑉</ci><ci id="A4.Ex85.m2.3.3.1.1.1.3.2.3.3.cmml" xref="A4.Ex85.m2.3.3.1.1.1.3.2.3.3">𝜁</ci></apply></apply></apply><apply id="A4.Ex85.m2.3.3.1.1.1.1.cmml" xref="A4.Ex85.m2.3.3.1.1.1.1"><times id="A4.Ex85.m2.3.3.1.1.1.1.2.cmml" xref="A4.Ex85.m2.3.3.1.1.1.1.2"></times><apply id="A4.Ex85.m2.3.3.1.1.1.1.3.cmml" xref="A4.Ex85.m2.3.3.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.Ex85.m2.3.3.1.1.1.1.3.1.cmml" xref="A4.Ex85.m2.3.3.1.1.1.1.3">subscript</csymbol><ci id="A4.Ex85.m2.3.3.1.1.1.1.3.2.cmml" xref="A4.Ex85.m2.3.3.1.1.1.1.3.2">𝜌</ci><cn id="A4.Ex85.m2.3.3.1.1.1.1.3.3.cmml" type="integer" xref="A4.Ex85.m2.3.3.1.1.1.1.3.3">2</cn></apply><vector id="A4.Ex85.m2.3.3.1.1.1.1.1.2.cmml" xref="A4.Ex85.m2.3.3.1.1.1.1.1.1"><apply id="A4.Ex85.m2.3.3.1.1.1.1.1.1.1.cmml" xref="A4.Ex85.m2.3.3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.Ex85.m2.3.3.1.1.1.1.1.1.1.1.cmml" xref="A4.Ex85.m2.3.3.1.1.1.1.1.1.1">subscript</csymbol><ci id="A4.Ex85.m2.3.3.1.1.1.1.1.1.1.2.cmml" xref="A4.Ex85.m2.3.3.1.1.1.1.1.1.1.2">𝑘</ci><ci id="A4.Ex85.m2.3.3.1.1.1.1.1.1.1.3.cmml" xref="A4.Ex85.m2.3.3.1.1.1.1.1.1.1.3">c</ci></apply><ci id="A4.Ex85.m2.1.1.cmml" xref="A4.Ex85.m2.1.1">𝜅</ci><apply id="A4.Ex85.m2.2.2.cmml" xref="A4.Ex85.m2.2.2"><ci id="A4.Ex85.m2.2.2.1.cmml" xref="A4.Ex85.m2.2.2.1">¯</ci><ci id="A4.Ex85.m2.2.2.2.cmml" xref="A4.Ex85.m2.2.2.2">𝑑</ci></apply></vector></apply></apply><apply id="A4.Ex85.m2.4.4.2.2.2.cmml" xref="A4.Ex85.m2.4.4.2.2.2"><csymbol cd="latexml" id="A4.Ex85.m2.4.4.2.2.2.1.cmml" xref="A4.Ex85.m2.4.4.2.2.2.1">for-all</csymbol><ci id="A4.Ex85.m2.4.4.2.2.2.2.cmml" xref="A4.Ex85.m2.4.4.2.2.2.2">𝑡</ci></apply></list><apply id="A4.Ex85.m2.4.4.4.cmml" xref="A4.Ex85.m2.4.4.4"><csymbol cd="ambiguous" id="A4.Ex85.m2.4.4.4.1.cmml" xref="A4.Ex85.m2.4.4.4">subscript</csymbol><ci id="A4.Ex85.m2.4.4.4.2.cmml" xref="A4.Ex85.m2.4.4.4.2">𝑇</ci><ci id="A4.Ex85.m2.4.4.4.3.cmml" xref="A4.Ex85.m2.4.4.4.3">a</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex85.m2.4c">\displaystyle-k_{\rm r}V_{\zeta}+\rho_{2}(k_{\rm c},\kappa,\bar{d}),\forall t% \geq T_{\rm a}</annotation><annotation encoding="application/x-llamapun" id="A4.Ex85.m2.4d">- italic_k start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT italic_V start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT + italic_ρ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT , italic_κ , over¯ start_ARG italic_d end_ARG ) , ∀ italic_t ≥ italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.1.p1.55">with <math alttext="k_{\rm r}" class="ltx_Math" display="inline" id="A4.1.p1.53.m1.1"><semantics id="A4.1.p1.53.m1.1a"><msub id="A4.1.p1.53.m1.1.1" xref="A4.1.p1.53.m1.1.1.cmml"><mi id="A4.1.p1.53.m1.1.1.2" xref="A4.1.p1.53.m1.1.1.2.cmml">k</mi><mi id="A4.1.p1.53.m1.1.1.3" mathvariant="normal" xref="A4.1.p1.53.m1.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="A4.1.p1.53.m1.1b"><apply id="A4.1.p1.53.m1.1.1.cmml" xref="A4.1.p1.53.m1.1.1"><csymbol cd="ambiguous" id="A4.1.p1.53.m1.1.1.1.cmml" xref="A4.1.p1.53.m1.1.1">subscript</csymbol><ci id="A4.1.p1.53.m1.1.1.2.cmml" xref="A4.1.p1.53.m1.1.1.2">𝑘</ci><ci id="A4.1.p1.53.m1.1.1.3.cmml" xref="A4.1.p1.53.m1.1.1.3">r</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.53.m1.1c">k_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.53.m1.1d">italic_k start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="A4.1.p1.54.m2.1"><semantics id="A4.1.p1.54.m2.1a"><mo id="A4.1.p1.54.m2.1.1" xref="A4.1.p1.54.m2.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="A4.1.p1.54.m2.1b"><csymbol cd="latexml" id="A4.1.p1.54.m2.1.1.cmml" xref="A4.1.p1.54.m2.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.54.m2.1c">:=</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.54.m2.1d">:=</annotation></semantics></math> <math alttext="\min\{k_{\rm c}(1-\mu),2\kappa\sigma^{*}\lambda_{\min}(\Gamma_{\zeta})(1-\mu)/% (1+p)\}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A4.1.p1.55.m3.3"><semantics id="A4.1.p1.55.m3.3a"><mrow id="A4.1.p1.55.m3.3.3" xref="A4.1.p1.55.m3.3.3.cmml"><mrow id="A4.1.p1.55.m3.3.3.2.2" xref="A4.1.p1.55.m3.3.3.2.3.cmml"><mi id="A4.1.p1.55.m3.1.1" xref="A4.1.p1.55.m3.1.1.cmml">min</mi><mo id="A4.1.p1.55.m3.3.3.2.2a" xref="A4.1.p1.55.m3.3.3.2.3.cmml">⁡</mo><mrow id="A4.1.p1.55.m3.3.3.2.2.2" xref="A4.1.p1.55.m3.3.3.2.3.cmml"><mo id="A4.1.p1.55.m3.3.3.2.2.2.3" stretchy="false" xref="A4.1.p1.55.m3.3.3.2.3.cmml">{</mo><mrow id="A4.1.p1.55.m3.2.2.1.1.1.1" xref="A4.1.p1.55.m3.2.2.1.1.1.1.cmml"><msub id="A4.1.p1.55.m3.2.2.1.1.1.1.3" xref="A4.1.p1.55.m3.2.2.1.1.1.1.3.cmml"><mi id="A4.1.p1.55.m3.2.2.1.1.1.1.3.2" xref="A4.1.p1.55.m3.2.2.1.1.1.1.3.2.cmml">k</mi><mi id="A4.1.p1.55.m3.2.2.1.1.1.1.3.3" mathvariant="normal" xref="A4.1.p1.55.m3.2.2.1.1.1.1.3.3.cmml">c</mi></msub><mo id="A4.1.p1.55.m3.2.2.1.1.1.1.2" xref="A4.1.p1.55.m3.2.2.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.1.p1.55.m3.2.2.1.1.1.1.1.1" xref="A4.1.p1.55.m3.2.2.1.1.1.1.1.1.1.cmml"><mo id="A4.1.p1.55.m3.2.2.1.1.1.1.1.1.2" stretchy="false" xref="A4.1.p1.55.m3.2.2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.1.p1.55.m3.2.2.1.1.1.1.1.1.1" xref="A4.1.p1.55.m3.2.2.1.1.1.1.1.1.1.cmml"><mn id="A4.1.p1.55.m3.2.2.1.1.1.1.1.1.1.2" xref="A4.1.p1.55.m3.2.2.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A4.1.p1.55.m3.2.2.1.1.1.1.1.1.1.1" xref="A4.1.p1.55.m3.2.2.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="A4.1.p1.55.m3.2.2.1.1.1.1.1.1.1.3" xref="A4.1.p1.55.m3.2.2.1.1.1.1.1.1.1.3.cmml">μ</mi></mrow><mo id="A4.1.p1.55.m3.2.2.1.1.1.1.1.1.3" stretchy="false" xref="A4.1.p1.55.m3.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A4.1.p1.55.m3.3.3.2.2.2.4" xref="A4.1.p1.55.m3.3.3.2.3.cmml">,</mo><mrow id="A4.1.p1.55.m3.3.3.2.2.2.2" xref="A4.1.p1.55.m3.3.3.2.2.2.2.cmml"><mrow id="A4.1.p1.55.m3.3.3.2.2.2.2.2" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.cmml"><mn id="A4.1.p1.55.m3.3.3.2.2.2.2.2.4" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.4.cmml">2</mn><mo id="A4.1.p1.55.m3.3.3.2.2.2.2.2.3" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.3.cmml">⁢</mo><mi id="A4.1.p1.55.m3.3.3.2.2.2.2.2.5" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.5.cmml">κ</mi><mo id="A4.1.p1.55.m3.3.3.2.2.2.2.2.3a" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.3.cmml">⁢</mo><msup id="A4.1.p1.55.m3.3.3.2.2.2.2.2.6" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.6.cmml"><mi id="A4.1.p1.55.m3.3.3.2.2.2.2.2.6.2" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.6.2.cmml">σ</mi><mo id="A4.1.p1.55.m3.3.3.2.2.2.2.2.6.3" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.6.3.cmml">∗</mo></msup><mo id="A4.1.p1.55.m3.3.3.2.2.2.2.2.3b" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.3.cmml">⁢</mo><msub id="A4.1.p1.55.m3.3.3.2.2.2.2.2.7" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.7.cmml"><mi id="A4.1.p1.55.m3.3.3.2.2.2.2.2.7.2" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.7.2.cmml">λ</mi><mi id="A4.1.p1.55.m3.3.3.2.2.2.2.2.7.3" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.7.3.cmml">min</mi></msub><mo id="A4.1.p1.55.m3.3.3.2.2.2.2.2.3c" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.3.cmml">⁢</mo><mrow id="A4.1.p1.55.m3.3.3.2.2.2.2.1.1.1" xref="A4.1.p1.55.m3.3.3.2.2.2.2.1.1.1.1.cmml"><mo id="A4.1.p1.55.m3.3.3.2.2.2.2.1.1.1.2" stretchy="false" xref="A4.1.p1.55.m3.3.3.2.2.2.2.1.1.1.1.cmml">(</mo><msub id="A4.1.p1.55.m3.3.3.2.2.2.2.1.1.1.1" xref="A4.1.p1.55.m3.3.3.2.2.2.2.1.1.1.1.cmml"><mi id="A4.1.p1.55.m3.3.3.2.2.2.2.1.1.1.1.2" mathvariant="normal" xref="A4.1.p1.55.m3.3.3.2.2.2.2.1.1.1.1.2.cmml">Γ</mi><mi id="A4.1.p1.55.m3.3.3.2.2.2.2.1.1.1.1.3" xref="A4.1.p1.55.m3.3.3.2.2.2.2.1.1.1.1.3.cmml">ζ</mi></msub><mo id="A4.1.p1.55.m3.3.3.2.2.2.2.1.1.1.3" stretchy="false" xref="A4.1.p1.55.m3.3.3.2.2.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="A4.1.p1.55.m3.3.3.2.2.2.2.2.3d" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.3.cmml">⁢</mo><mrow id="A4.1.p1.55.m3.3.3.2.2.2.2.2.2.1" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.2.1.1.cmml"><mo id="A4.1.p1.55.m3.3.3.2.2.2.2.2.2.1.2" stretchy="false" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.2.1.1.cmml">(</mo><mrow id="A4.1.p1.55.m3.3.3.2.2.2.2.2.2.1.1" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.2.1.1.cmml"><mn id="A4.1.p1.55.m3.3.3.2.2.2.2.2.2.1.1.2" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.2.1.1.2.cmml">1</mn><mo id="A4.1.p1.55.m3.3.3.2.2.2.2.2.2.1.1.1" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.2.1.1.1.cmml">−</mo><mi id="A4.1.p1.55.m3.3.3.2.2.2.2.2.2.1.1.3" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.2.1.1.3.cmml">μ</mi></mrow><mo id="A4.1.p1.55.m3.3.3.2.2.2.2.2.2.1.3" stretchy="false" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="A4.1.p1.55.m3.3.3.2.2.2.2.4" xref="A4.1.p1.55.m3.3.3.2.2.2.2.4.cmml">/</mo><mrow id="A4.1.p1.55.m3.3.3.2.2.2.2.3.1" xref="A4.1.p1.55.m3.3.3.2.2.2.2.3.1.1.cmml"><mo id="A4.1.p1.55.m3.3.3.2.2.2.2.3.1.2" stretchy="false" xref="A4.1.p1.55.m3.3.3.2.2.2.2.3.1.1.cmml">(</mo><mrow id="A4.1.p1.55.m3.3.3.2.2.2.2.3.1.1" xref="A4.1.p1.55.m3.3.3.2.2.2.2.3.1.1.cmml"><mn id="A4.1.p1.55.m3.3.3.2.2.2.2.3.1.1.2" xref="A4.1.p1.55.m3.3.3.2.2.2.2.3.1.1.2.cmml">1</mn><mo id="A4.1.p1.55.m3.3.3.2.2.2.2.3.1.1.1" xref="A4.1.p1.55.m3.3.3.2.2.2.2.3.1.1.1.cmml">+</mo><mi id="A4.1.p1.55.m3.3.3.2.2.2.2.3.1.1.3" xref="A4.1.p1.55.m3.3.3.2.2.2.2.3.1.1.3.cmml">p</mi></mrow><mo id="A4.1.p1.55.m3.3.3.2.2.2.2.3.1.3" stretchy="false" xref="A4.1.p1.55.m3.3.3.2.2.2.2.3.1.1.cmml">)</mo></mrow></mrow><mo id="A4.1.p1.55.m3.3.3.2.2.2.5" stretchy="false" xref="A4.1.p1.55.m3.3.3.2.3.cmml">}</mo></mrow></mrow><mo id="A4.1.p1.55.m3.3.3.3" xref="A4.1.p1.55.m3.3.3.3.cmml">∈</mo><msup id="A4.1.p1.55.m3.3.3.4" xref="A4.1.p1.55.m3.3.3.4.cmml"><mi id="A4.1.p1.55.m3.3.3.4.2" xref="A4.1.p1.55.m3.3.3.4.2.cmml">ℝ</mi><mo id="A4.1.p1.55.m3.3.3.4.3" xref="A4.1.p1.55.m3.3.3.4.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.55.m3.3b"><apply id="A4.1.p1.55.m3.3.3.cmml" xref="A4.1.p1.55.m3.3.3"><in id="A4.1.p1.55.m3.3.3.3.cmml" xref="A4.1.p1.55.m3.3.3.3"></in><apply id="A4.1.p1.55.m3.3.3.2.3.cmml" xref="A4.1.p1.55.m3.3.3.2.2"><min id="A4.1.p1.55.m3.1.1.cmml" xref="A4.1.p1.55.m3.1.1"></min><apply id="A4.1.p1.55.m3.2.2.1.1.1.1.cmml" xref="A4.1.p1.55.m3.2.2.1.1.1.1"><times id="A4.1.p1.55.m3.2.2.1.1.1.1.2.cmml" xref="A4.1.p1.55.m3.2.2.1.1.1.1.2"></times><apply id="A4.1.p1.55.m3.2.2.1.1.1.1.3.cmml" xref="A4.1.p1.55.m3.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.55.m3.2.2.1.1.1.1.3.1.cmml" xref="A4.1.p1.55.m3.2.2.1.1.1.1.3">subscript</csymbol><ci id="A4.1.p1.55.m3.2.2.1.1.1.1.3.2.cmml" xref="A4.1.p1.55.m3.2.2.1.1.1.1.3.2">𝑘</ci><ci id="A4.1.p1.55.m3.2.2.1.1.1.1.3.3.cmml" xref="A4.1.p1.55.m3.2.2.1.1.1.1.3.3">c</ci></apply><apply id="A4.1.p1.55.m3.2.2.1.1.1.1.1.1.1.cmml" xref="A4.1.p1.55.m3.2.2.1.1.1.1.1.1"><minus id="A4.1.p1.55.m3.2.2.1.1.1.1.1.1.1.1.cmml" xref="A4.1.p1.55.m3.2.2.1.1.1.1.1.1.1.1"></minus><cn id="A4.1.p1.55.m3.2.2.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A4.1.p1.55.m3.2.2.1.1.1.1.1.1.1.2">1</cn><ci id="A4.1.p1.55.m3.2.2.1.1.1.1.1.1.1.3.cmml" xref="A4.1.p1.55.m3.2.2.1.1.1.1.1.1.1.3">𝜇</ci></apply></apply><apply id="A4.1.p1.55.m3.3.3.2.2.2.2.cmml" xref="A4.1.p1.55.m3.3.3.2.2.2.2"><divide id="A4.1.p1.55.m3.3.3.2.2.2.2.4.cmml" xref="A4.1.p1.55.m3.3.3.2.2.2.2.4"></divide><apply id="A4.1.p1.55.m3.3.3.2.2.2.2.2.cmml" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2"><times id="A4.1.p1.55.m3.3.3.2.2.2.2.2.3.cmml" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.3"></times><cn id="A4.1.p1.55.m3.3.3.2.2.2.2.2.4.cmml" type="integer" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.4">2</cn><ci id="A4.1.p1.55.m3.3.3.2.2.2.2.2.5.cmml" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.5">𝜅</ci><apply id="A4.1.p1.55.m3.3.3.2.2.2.2.2.6.cmml" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.6"><csymbol cd="ambiguous" id="A4.1.p1.55.m3.3.3.2.2.2.2.2.6.1.cmml" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.6">superscript</csymbol><ci id="A4.1.p1.55.m3.3.3.2.2.2.2.2.6.2.cmml" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.6.2">𝜎</ci><times id="A4.1.p1.55.m3.3.3.2.2.2.2.2.6.3.cmml" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.6.3"></times></apply><apply id="A4.1.p1.55.m3.3.3.2.2.2.2.2.7.cmml" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.7"><csymbol cd="ambiguous" id="A4.1.p1.55.m3.3.3.2.2.2.2.2.7.1.cmml" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.7">subscript</csymbol><ci id="A4.1.p1.55.m3.3.3.2.2.2.2.2.7.2.cmml" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.7.2">𝜆</ci><min id="A4.1.p1.55.m3.3.3.2.2.2.2.2.7.3.cmml" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.7.3"></min></apply><apply id="A4.1.p1.55.m3.3.3.2.2.2.2.1.1.1.1.cmml" xref="A4.1.p1.55.m3.3.3.2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="A4.1.p1.55.m3.3.3.2.2.2.2.1.1.1.1.1.cmml" xref="A4.1.p1.55.m3.3.3.2.2.2.2.1.1.1">subscript</csymbol><ci id="A4.1.p1.55.m3.3.3.2.2.2.2.1.1.1.1.2.cmml" xref="A4.1.p1.55.m3.3.3.2.2.2.2.1.1.1.1.2">Γ</ci><ci id="A4.1.p1.55.m3.3.3.2.2.2.2.1.1.1.1.3.cmml" xref="A4.1.p1.55.m3.3.3.2.2.2.2.1.1.1.1.3">𝜁</ci></apply><apply id="A4.1.p1.55.m3.3.3.2.2.2.2.2.2.1.1.cmml" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.2.1"><minus id="A4.1.p1.55.m3.3.3.2.2.2.2.2.2.1.1.1.cmml" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.2.1.1.1"></minus><cn id="A4.1.p1.55.m3.3.3.2.2.2.2.2.2.1.1.2.cmml" type="integer" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.2.1.1.2">1</cn><ci id="A4.1.p1.55.m3.3.3.2.2.2.2.2.2.1.1.3.cmml" xref="A4.1.p1.55.m3.3.3.2.2.2.2.2.2.1.1.3">𝜇</ci></apply></apply><apply id="A4.1.p1.55.m3.3.3.2.2.2.2.3.1.1.cmml" xref="A4.1.p1.55.m3.3.3.2.2.2.2.3.1"><plus id="A4.1.p1.55.m3.3.3.2.2.2.2.3.1.1.1.cmml" xref="A4.1.p1.55.m3.3.3.2.2.2.2.3.1.1.1"></plus><cn id="A4.1.p1.55.m3.3.3.2.2.2.2.3.1.1.2.cmml" type="integer" xref="A4.1.p1.55.m3.3.3.2.2.2.2.3.1.1.2">1</cn><ci id="A4.1.p1.55.m3.3.3.2.2.2.2.3.1.1.3.cmml" xref="A4.1.p1.55.m3.3.3.2.2.2.2.3.1.1.3">𝑝</ci></apply></apply></apply><apply id="A4.1.p1.55.m3.3.3.4.cmml" xref="A4.1.p1.55.m3.3.3.4"><csymbol cd="ambiguous" id="A4.1.p1.55.m3.3.3.4.1.cmml" xref="A4.1.p1.55.m3.3.3.4">superscript</csymbol><ci id="A4.1.p1.55.m3.3.3.4.2.cmml" xref="A4.1.p1.55.m3.3.3.4.2">ℝ</ci><plus id="A4.1.p1.55.m3.3.3.4.3.cmml" xref="A4.1.p1.55.m3.3.3.4.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.55.m3.3c">\min\{k_{\rm c}(1-\mu),2\kappa\sigma^{*}\lambda_{\min}(\Gamma_{\zeta})(1-\mu)/% (1+p)\}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.55.m3.3d">roman_min { italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ( 1 - italic_μ ) , 2 italic_κ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Γ start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ) ( 1 - italic_μ ) / ( 1 + italic_p ) } ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>. Applying <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#bib.bib40" title="">40</a>, Lemma A.3.2]</cite> to solve the above inequality, one has</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx88"> <tbody id="A4.Ex86"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle V_{\zeta}(t)\leq(V_{\zeta}(0)-\rho_{2}(k_{\rm c},\kappa,\bar{d})% )e^{-k_{\rm r}t}+\rho_{2}(k_{\rm c},\kappa,\bar{d}),\forall t\geq T_{\rm a}." class="ltx_Math" display="inline" id="A4.Ex86.m1.7"><semantics id="A4.Ex86.m1.7a"><mrow id="A4.Ex86.m1.7.7.1"><mrow id="A4.Ex86.m1.7.7.1.1.2" xref="A4.Ex86.m1.7.7.1.1.3.cmml"><mrow id="A4.Ex86.m1.7.7.1.1.1.1" xref="A4.Ex86.m1.7.7.1.1.1.1.cmml"><mrow id="A4.Ex86.m1.7.7.1.1.1.1.4" xref="A4.Ex86.m1.7.7.1.1.1.1.4.cmml"><msub id="A4.Ex86.m1.7.7.1.1.1.1.4.2" xref="A4.Ex86.m1.7.7.1.1.1.1.4.2.cmml"><mi id="A4.Ex86.m1.7.7.1.1.1.1.4.2.2" xref="A4.Ex86.m1.7.7.1.1.1.1.4.2.2.cmml">V</mi><mi id="A4.Ex86.m1.7.7.1.1.1.1.4.2.3" xref="A4.Ex86.m1.7.7.1.1.1.1.4.2.3.cmml">ζ</mi></msub><mo id="A4.Ex86.m1.7.7.1.1.1.1.4.1" xref="A4.Ex86.m1.7.7.1.1.1.1.4.1.cmml">⁢</mo><mrow id="A4.Ex86.m1.7.7.1.1.1.1.4.3.2" xref="A4.Ex86.m1.7.7.1.1.1.1.4.cmml"><mo id="A4.Ex86.m1.7.7.1.1.1.1.4.3.2.1" stretchy="false" xref="A4.Ex86.m1.7.7.1.1.1.1.4.cmml">(</mo><mi id="A4.Ex86.m1.1.1" xref="A4.Ex86.m1.1.1.cmml">t</mi><mo id="A4.Ex86.m1.7.7.1.1.1.1.4.3.2.2" stretchy="false" xref="A4.Ex86.m1.7.7.1.1.1.1.4.cmml">)</mo></mrow></mrow><mo id="A4.Ex86.m1.7.7.1.1.1.1.3" xref="A4.Ex86.m1.7.7.1.1.1.1.3.cmml">≤</mo><mrow id="A4.Ex86.m1.7.7.1.1.1.1.2" xref="A4.Ex86.m1.7.7.1.1.1.1.2.cmml"><mrow id="A4.Ex86.m1.7.7.1.1.1.1.1.1" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.cmml"><mrow id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.cmml"><mo id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.cmml"><mrow id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.cmml"><msub id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.2" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.2.cmml"><mi id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.2.2" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.2.2.cmml">V</mi><mi id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.2.3" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.2.3.cmml">ζ</mi></msub><mo id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.1" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.3.2" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.cmml"><mo id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.3.2.1" stretchy="false" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.cmml">(</mo><mn id="A4.Ex86.m1.2.2" xref="A4.Ex86.m1.2.2.cmml">0</mn><mo id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.3.2.2" stretchy="false" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.cmml">)</mo></mrow></mrow><mo id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.2" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.2.cmml">−</mo><mrow id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.cmml"><msub id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.3" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.3.2" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.3.2.cmml">ρ</mi><mn id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.3.3" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.3.3.cmml">2</mn></msub><mo id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.2" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.2.cmml"><mo id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.2.cmml">(</mo><msub id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.2" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">k</mi><mi id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.3" mathvariant="normal" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml">c</mi></msub><mo id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.3" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.2.cmml">,</mo><mi id="A4.Ex86.m1.3.3" xref="A4.Ex86.m1.3.3.cmml">κ</mi><mo id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.4" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.2.cmml">,</mo><mover accent="true" id="A4.Ex86.m1.4.4" xref="A4.Ex86.m1.4.4.cmml"><mi id="A4.Ex86.m1.4.4.2" xref="A4.Ex86.m1.4.4.2.cmml">d</mi><mo id="A4.Ex86.m1.4.4.1" xref="A4.Ex86.m1.4.4.1.cmml">¯</mo></mover><mo id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.5" stretchy="false" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex86.m1.7.7.1.1.1.1.1.1.2" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.2.cmml">⁢</mo><msup id="A4.Ex86.m1.7.7.1.1.1.1.1.1.3" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.cmml"><mi id="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.2" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.2.cmml">e</mi><mrow id="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.cmml"><mo id="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3a" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.cmml">−</mo><mrow id="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2.cmml"><msub id="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2.2" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2.2.cmml"><mi id="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2.2.2" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2.2.2.cmml">k</mi><mi id="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2.2.3" mathvariant="normal" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2.2.3.cmml">r</mi></msub><mo id="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2.1" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2.1.cmml">⁢</mo><mi id="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2.3" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2.3.cmml">t</mi></mrow></mrow></msup></mrow><mo id="A4.Ex86.m1.7.7.1.1.1.1.2.3" xref="A4.Ex86.m1.7.7.1.1.1.1.2.3.cmml">+</mo><mrow id="A4.Ex86.m1.7.7.1.1.1.1.2.2" xref="A4.Ex86.m1.7.7.1.1.1.1.2.2.cmml"><msub id="A4.Ex86.m1.7.7.1.1.1.1.2.2.3" xref="A4.Ex86.m1.7.7.1.1.1.1.2.2.3.cmml"><mi id="A4.Ex86.m1.7.7.1.1.1.1.2.2.3.2" xref="A4.Ex86.m1.7.7.1.1.1.1.2.2.3.2.cmml">ρ</mi><mn id="A4.Ex86.m1.7.7.1.1.1.1.2.2.3.3" xref="A4.Ex86.m1.7.7.1.1.1.1.2.2.3.3.cmml">2</mn></msub><mo id="A4.Ex86.m1.7.7.1.1.1.1.2.2.2" xref="A4.Ex86.m1.7.7.1.1.1.1.2.2.2.cmml">⁢</mo><mrow id="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.1" xref="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.2.cmml"><mo id="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.1.2" stretchy="false" xref="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.2.cmml">(</mo><msub id="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.1.1" xref="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.1.1.cmml"><mi id="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.1.1.2" xref="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.1.1.2.cmml">k</mi><mi id="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.1.1.3" mathvariant="normal" xref="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.1.1.3.cmml">c</mi></msub><mo id="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.1.3" xref="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.2.cmml">,</mo><mi id="A4.Ex86.m1.5.5" xref="A4.Ex86.m1.5.5.cmml">κ</mi><mo id="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.1.4" xref="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.2.cmml">,</mo><mover accent="true" id="A4.Ex86.m1.6.6" xref="A4.Ex86.m1.6.6.cmml"><mi id="A4.Ex86.m1.6.6.2" xref="A4.Ex86.m1.6.6.2.cmml">d</mi><mo id="A4.Ex86.m1.6.6.1" xref="A4.Ex86.m1.6.6.1.cmml">¯</mo></mover><mo id="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.1.5" stretchy="false" xref="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="A4.Ex86.m1.7.7.1.1.2.3" xref="A4.Ex86.m1.7.7.1.1.3a.cmml">,</mo><mrow id="A4.Ex86.m1.7.7.1.1.2.2" xref="A4.Ex86.m1.7.7.1.1.2.2.cmml"><mrow id="A4.Ex86.m1.7.7.1.1.2.2.2" xref="A4.Ex86.m1.7.7.1.1.2.2.2.cmml"><mo id="A4.Ex86.m1.7.7.1.1.2.2.2.1" rspace="0.167em" xref="A4.Ex86.m1.7.7.1.1.2.2.2.1.cmml">∀</mo><mi id="A4.Ex86.m1.7.7.1.1.2.2.2.2" xref="A4.Ex86.m1.7.7.1.1.2.2.2.2.cmml">t</mi></mrow><mo id="A4.Ex86.m1.7.7.1.1.2.2.1" xref="A4.Ex86.m1.7.7.1.1.2.2.1.cmml">≥</mo><msub id="A4.Ex86.m1.7.7.1.1.2.2.3" xref="A4.Ex86.m1.7.7.1.1.2.2.3.cmml"><mi id="A4.Ex86.m1.7.7.1.1.2.2.3.2" xref="A4.Ex86.m1.7.7.1.1.2.2.3.2.cmml">T</mi><mi id="A4.Ex86.m1.7.7.1.1.2.2.3.3" mathvariant="normal" xref="A4.Ex86.m1.7.7.1.1.2.2.3.3.cmml">a</mi></msub></mrow></mrow><mo id="A4.Ex86.m1.7.7.1.2" lspace="0em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex86.m1.7b"><apply id="A4.Ex86.m1.7.7.1.1.3.cmml" xref="A4.Ex86.m1.7.7.1.1.2"><csymbol cd="ambiguous" id="A4.Ex86.m1.7.7.1.1.3a.cmml" xref="A4.Ex86.m1.7.7.1.1.2.3">formulae-sequence</csymbol><apply id="A4.Ex86.m1.7.7.1.1.1.1.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1"><leq id="A4.Ex86.m1.7.7.1.1.1.1.3.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.3"></leq><apply id="A4.Ex86.m1.7.7.1.1.1.1.4.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.4"><times id="A4.Ex86.m1.7.7.1.1.1.1.4.1.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.4.1"></times><apply id="A4.Ex86.m1.7.7.1.1.1.1.4.2.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.4.2"><csymbol cd="ambiguous" id="A4.Ex86.m1.7.7.1.1.1.1.4.2.1.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.4.2">subscript</csymbol><ci id="A4.Ex86.m1.7.7.1.1.1.1.4.2.2.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.4.2.2">𝑉</ci><ci id="A4.Ex86.m1.7.7.1.1.1.1.4.2.3.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.4.2.3">𝜁</ci></apply><ci id="A4.Ex86.m1.1.1.cmml" xref="A4.Ex86.m1.1.1">𝑡</ci></apply><apply id="A4.Ex86.m1.7.7.1.1.1.1.2.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.2"><plus id="A4.Ex86.m1.7.7.1.1.1.1.2.3.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.2.3"></plus><apply id="A4.Ex86.m1.7.7.1.1.1.1.1.1.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1"><times id="A4.Ex86.m1.7.7.1.1.1.1.1.1.2.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.2"></times><apply id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1"><minus id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.2.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.2"></minus><apply id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3"><times id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.1"></times><apply id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.2.1.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.2">subscript</csymbol><ci id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.2.2.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.2.2">𝑉</ci><ci id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.2.3.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.3.2.3">𝜁</ci></apply><cn id="A4.Ex86.m1.2.2.cmml" type="integer" xref="A4.Ex86.m1.2.2">0</cn></apply><apply id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1"><times id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.2"></times><apply id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.3.2">𝜌</ci><cn id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.3.3">2</cn></apply><vector id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1"><apply id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.2">𝑘</ci><ci id="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.3">c</ci></apply><ci id="A4.Ex86.m1.3.3.cmml" xref="A4.Ex86.m1.3.3">𝜅</ci><apply id="A4.Ex86.m1.4.4.cmml" xref="A4.Ex86.m1.4.4"><ci id="A4.Ex86.m1.4.4.1.cmml" xref="A4.Ex86.m1.4.4.1">¯</ci><ci id="A4.Ex86.m1.4.4.2.cmml" xref="A4.Ex86.m1.4.4.2">𝑑</ci></apply></vector></apply></apply><apply id="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.1.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.3">superscript</csymbol><ci id="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.2.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.2">𝑒</ci><apply id="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3"><minus id="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.1.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3"></minus><apply id="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2"><times id="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2.1.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2.1"></times><apply id="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2.2.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2.2"><csymbol cd="ambiguous" id="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2.2.1.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2.2">subscript</csymbol><ci id="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2.2.2.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2.2.2">𝑘</ci><ci id="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2.2.3.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2.2.3">r</ci></apply><ci id="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2.3.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.1.1.3.3.2.3">𝑡</ci></apply></apply></apply></apply><apply id="A4.Ex86.m1.7.7.1.1.1.1.2.2.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.2.2"><times id="A4.Ex86.m1.7.7.1.1.1.1.2.2.2.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.2.2.2"></times><apply id="A4.Ex86.m1.7.7.1.1.1.1.2.2.3.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.2.2.3"><csymbol cd="ambiguous" id="A4.Ex86.m1.7.7.1.1.1.1.2.2.3.1.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.2.2.3">subscript</csymbol><ci id="A4.Ex86.m1.7.7.1.1.1.1.2.2.3.2.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.2.2.3.2">𝜌</ci><cn id="A4.Ex86.m1.7.7.1.1.1.1.2.2.3.3.cmml" type="integer" xref="A4.Ex86.m1.7.7.1.1.1.1.2.2.3.3">2</cn></apply><vector id="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.2.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.1"><apply id="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.1.1.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.1.1"><csymbol cd="ambiguous" id="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.1.1.1.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.1.1">subscript</csymbol><ci id="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.1.1.2.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.1.1.2">𝑘</ci><ci id="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.1.1.3.cmml" xref="A4.Ex86.m1.7.7.1.1.1.1.2.2.1.1.1.3">c</ci></apply><ci id="A4.Ex86.m1.5.5.cmml" xref="A4.Ex86.m1.5.5">𝜅</ci><apply id="A4.Ex86.m1.6.6.cmml" xref="A4.Ex86.m1.6.6"><ci id="A4.Ex86.m1.6.6.1.cmml" xref="A4.Ex86.m1.6.6.1">¯</ci><ci id="A4.Ex86.m1.6.6.2.cmml" xref="A4.Ex86.m1.6.6.2">𝑑</ci></apply></vector></apply></apply></apply><apply id="A4.Ex86.m1.7.7.1.1.2.2.cmml" xref="A4.Ex86.m1.7.7.1.1.2.2"><geq id="A4.Ex86.m1.7.7.1.1.2.2.1.cmml" xref="A4.Ex86.m1.7.7.1.1.2.2.1"></geq><apply id="A4.Ex86.m1.7.7.1.1.2.2.2.cmml" xref="A4.Ex86.m1.7.7.1.1.2.2.2"><csymbol cd="latexml" id="A4.Ex86.m1.7.7.1.1.2.2.2.1.cmml" xref="A4.Ex86.m1.7.7.1.1.2.2.2.1">for-all</csymbol><ci id="A4.Ex86.m1.7.7.1.1.2.2.2.2.cmml" xref="A4.Ex86.m1.7.7.1.1.2.2.2.2">𝑡</ci></apply><apply id="A4.Ex86.m1.7.7.1.1.2.2.3.cmml" xref="A4.Ex86.m1.7.7.1.1.2.2.3"><csymbol cd="ambiguous" id="A4.Ex86.m1.7.7.1.1.2.2.3.1.cmml" xref="A4.Ex86.m1.7.7.1.1.2.2.3">subscript</csymbol><ci id="A4.Ex86.m1.7.7.1.1.2.2.3.2.cmml" xref="A4.Ex86.m1.7.7.1.1.2.2.3.2">𝑇</ci><ci id="A4.Ex86.m1.7.7.1.1.2.2.3.3.cmml" xref="A4.Ex86.m1.7.7.1.1.2.2.3.3">a</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex86.m1.7c">\displaystyle V_{\zeta}(t)\leq(V_{\zeta}(0)-\rho_{2}(k_{\rm c},\kappa,\bar{d})% )e^{-k_{\rm r}t}+\rho_{2}(k_{\rm c},\kappa,\bar{d}),\forall t\geq T_{\rm a}.</annotation><annotation encoding="application/x-llamapun" id="A4.Ex86.m1.7d">italic_V start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t ) ≤ ( italic_V start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( 0 ) - italic_ρ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT , italic_κ , over¯ start_ARG italic_d end_ARG ) ) italic_e start_POSTSUPERSCRIPT - italic_k start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT + italic_ρ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT , italic_κ , over¯ start_ARG italic_d end_ARG ) , ∀ italic_t ≥ italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.1.p1.68"><span class="ltx_text" id="A4.1.p1.66.11" style="color:#000099;">Thus, the equilibrium point <math alttext="(\bm{e}" class="ltx_math_unparsed" display="inline" id="A4.1.p1.56.1.m1.1"><semantics id="A4.1.p1.56.1.m1.1a"><mrow id="A4.1.p1.56.1.m1.1b"><mo id="A4.1.p1.56.1.m1.1.1" mathcolor="#000099" stretchy="false">(</mo><mi id="A4.1.p1.56.1.m1.1.2" mathcolor="#000099">𝒆</mi></mrow><annotation encoding="application/x-tex" id="A4.1.p1.56.1.m1.1c">(\bm{e}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.56.1.m1.1d">( bold_italic_e</annotation></semantics></math>, <math alttext="\tilde{\bm{\theta}})=\bm{0}" class="ltx_math_unparsed" display="inline" id="A4.1.p1.57.2.m2.1"><semantics id="A4.1.p1.57.2.m2.1a"><mrow id="A4.1.p1.57.2.m2.1b"><mover accent="true" id="A4.1.p1.57.2.m2.1.1"><mi id="A4.1.p1.57.2.m2.1.1.2" mathcolor="#000099">𝜽</mi><mo id="A4.1.p1.57.2.m2.1.1.1" mathcolor="#000099">~</mo></mover><mo id="A4.1.p1.57.2.m2.1.2" mathcolor="#000099" stretchy="false">)</mo><mo id="A4.1.p1.57.2.m2.1.3" mathcolor="#000099">=</mo><mn id="A4.1.p1.57.2.m2.1.4" mathcolor="#000099">𝟎</mn></mrow><annotation encoding="application/x-tex" id="A4.1.p1.57.2.m2.1c">\tilde{\bm{\theta}})=\bm{0}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.57.2.m2.1d">over~ start_ARG bold_italic_θ end_ARG ) = bold_0</annotation></semantics></math> of the closed-loop system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S5.E26" title="In V-C Robustness Results ‣ V Theoretical Guarantees ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">26</span></a>) with (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>) is partial practical exponential stable on <math alttext="t" class="ltx_Math" display="inline" id="A4.1.p1.58.3.m3.1"><semantics id="A4.1.p1.58.3.m3.1a"><mi id="A4.1.p1.58.3.m3.1.1" mathcolor="#000099" xref="A4.1.p1.58.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A4.1.p1.58.3.m3.1b"><ci id="A4.1.p1.58.3.m3.1.1.cmml" xref="A4.1.p1.58.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.58.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.58.3.m3.1d">italic_t</annotation></semantics></math> <math alttext="\in" class="ltx_Math" display="inline" id="A4.1.p1.59.4.m4.1"><semantics id="A4.1.p1.59.4.m4.1a"><mo id="A4.1.p1.59.4.m4.1.1" mathcolor="#000099" xref="A4.1.p1.59.4.m4.1.1.cmml">∈</mo><annotation-xml encoding="MathML-Content" id="A4.1.p1.59.4.m4.1b"><in id="A4.1.p1.59.4.m4.1.1.cmml" xref="A4.1.p1.59.4.m4.1.1"></in></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.59.4.m4.1c">\in</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.59.4.m4.1d">∈</annotation></semantics></math> <math alttext="[T_{\rm a},\infty)" class="ltx_Math" display="inline" id="A4.1.p1.60.5.m5.2"><semantics id="A4.1.p1.60.5.m5.2a"><mrow id="A4.1.p1.60.5.m5.2.2.1" xref="A4.1.p1.60.5.m5.2.2.2.cmml"><mo id="A4.1.p1.60.5.m5.2.2.1.2" mathcolor="#000099" stretchy="false" xref="A4.1.p1.60.5.m5.2.2.2.cmml">[</mo><msub id="A4.1.p1.60.5.m5.2.2.1.1" xref="A4.1.p1.60.5.m5.2.2.1.1.cmml"><mi id="A4.1.p1.60.5.m5.2.2.1.1.2" mathcolor="#000099" xref="A4.1.p1.60.5.m5.2.2.1.1.2.cmml">T</mi><mi id="A4.1.p1.60.5.m5.2.2.1.1.3" mathcolor="#000099" mathvariant="normal" xref="A4.1.p1.60.5.m5.2.2.1.1.3.cmml">a</mi></msub><mo id="A4.1.p1.60.5.m5.2.2.1.3" mathcolor="#000099" xref="A4.1.p1.60.5.m5.2.2.2.cmml">,</mo><mi id="A4.1.p1.60.5.m5.1.1" mathcolor="#000099" mathvariant="normal" xref="A4.1.p1.60.5.m5.1.1.cmml">∞</mi><mo id="A4.1.p1.60.5.m5.2.2.1.4" mathcolor="#000099" stretchy="false" xref="A4.1.p1.60.5.m5.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.60.5.m5.2b"><interval closure="closed-open" id="A4.1.p1.60.5.m5.2.2.2.cmml" xref="A4.1.p1.60.5.m5.2.2.1"><apply id="A4.1.p1.60.5.m5.2.2.1.1.cmml" xref="A4.1.p1.60.5.m5.2.2.1.1"><csymbol cd="ambiguous" id="A4.1.p1.60.5.m5.2.2.1.1.1.cmml" xref="A4.1.p1.60.5.m5.2.2.1.1">subscript</csymbol><ci id="A4.1.p1.60.5.m5.2.2.1.1.2.cmml" xref="A4.1.p1.60.5.m5.2.2.1.1.2">𝑇</ci><ci id="A4.1.p1.60.5.m5.2.2.1.1.3.cmml" xref="A4.1.p1.60.5.m5.2.2.1.1.3">a</ci></apply><infinity id="A4.1.p1.60.5.m5.1.1.cmml" xref="A4.1.p1.60.5.m5.1.1"></infinity></interval></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.60.5.m5.2c">[T_{\rm a},\infty)</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.60.5.m5.2d">[ italic_T start_POSTSUBSCRIPT roman_a end_POSTSUBSCRIPT , ∞ )</annotation></semantics></math>, where the tracking error <math alttext="\bm{e}(t)" class="ltx_Math" display="inline" id="A4.1.p1.61.6.m6.1"><semantics id="A4.1.p1.61.6.m6.1a"><mrow id="A4.1.p1.61.6.m6.1.2" xref="A4.1.p1.61.6.m6.1.2.cmml"><mi id="A4.1.p1.61.6.m6.1.2.2" mathcolor="#000099" xref="A4.1.p1.61.6.m6.1.2.2.cmml">𝒆</mi><mo id="A4.1.p1.61.6.m6.1.2.1" xref="A4.1.p1.61.6.m6.1.2.1.cmml">⁢</mo><mrow id="A4.1.p1.61.6.m6.1.2.3.2" xref="A4.1.p1.61.6.m6.1.2.cmml"><mo id="A4.1.p1.61.6.m6.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="A4.1.p1.61.6.m6.1.2.cmml">(</mo><mi id="A4.1.p1.61.6.m6.1.1" mathcolor="#000099" xref="A4.1.p1.61.6.m6.1.1.cmml">t</mi><mo id="A4.1.p1.61.6.m6.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="A4.1.p1.61.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.61.6.m6.1b"><apply id="A4.1.p1.61.6.m6.1.2.cmml" xref="A4.1.p1.61.6.m6.1.2"><times id="A4.1.p1.61.6.m6.1.2.1.cmml" xref="A4.1.p1.61.6.m6.1.2.1"></times><ci id="A4.1.p1.61.6.m6.1.2.2.cmml" xref="A4.1.p1.61.6.m6.1.2.2">𝒆</ci><ci id="A4.1.p1.61.6.m6.1.1.cmml" xref="A4.1.p1.61.6.m6.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.61.6.m6.1c">\bm{e}(t)</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.61.6.m6.1d">bold_italic_e ( italic_t )</annotation></semantics></math> and the estimation error <math alttext="\tilde{\bm{\theta}}_{\zeta}(t)" class="ltx_Math" display="inline" id="A4.1.p1.62.7.m7.1"><semantics id="A4.1.p1.62.7.m7.1a"><mrow id="A4.1.p1.62.7.m7.1.2" xref="A4.1.p1.62.7.m7.1.2.cmml"><msub id="A4.1.p1.62.7.m7.1.2.2" xref="A4.1.p1.62.7.m7.1.2.2.cmml"><mover accent="true" id="A4.1.p1.62.7.m7.1.2.2.2" xref="A4.1.p1.62.7.m7.1.2.2.2.cmml"><mi id="A4.1.p1.62.7.m7.1.2.2.2.2" mathcolor="#000099" xref="A4.1.p1.62.7.m7.1.2.2.2.2.cmml">𝜽</mi><mo id="A4.1.p1.62.7.m7.1.2.2.2.1" mathcolor="#000099" xref="A4.1.p1.62.7.m7.1.2.2.2.1.cmml">~</mo></mover><mi id="A4.1.p1.62.7.m7.1.2.2.3" mathcolor="#000099" xref="A4.1.p1.62.7.m7.1.2.2.3.cmml">ζ</mi></msub><mo id="A4.1.p1.62.7.m7.1.2.1" xref="A4.1.p1.62.7.m7.1.2.1.cmml">⁢</mo><mrow id="A4.1.p1.62.7.m7.1.2.3.2" xref="A4.1.p1.62.7.m7.1.2.cmml"><mo id="A4.1.p1.62.7.m7.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="A4.1.p1.62.7.m7.1.2.cmml">(</mo><mi id="A4.1.p1.62.7.m7.1.1" mathcolor="#000099" xref="A4.1.p1.62.7.m7.1.1.cmml">t</mi><mo id="A4.1.p1.62.7.m7.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="A4.1.p1.62.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.62.7.m7.1b"><apply id="A4.1.p1.62.7.m7.1.2.cmml" xref="A4.1.p1.62.7.m7.1.2"><times id="A4.1.p1.62.7.m7.1.2.1.cmml" xref="A4.1.p1.62.7.m7.1.2.1"></times><apply id="A4.1.p1.62.7.m7.1.2.2.cmml" xref="A4.1.p1.62.7.m7.1.2.2"><csymbol cd="ambiguous" id="A4.1.p1.62.7.m7.1.2.2.1.cmml" xref="A4.1.p1.62.7.m7.1.2.2">subscript</csymbol><apply id="A4.1.p1.62.7.m7.1.2.2.2.cmml" xref="A4.1.p1.62.7.m7.1.2.2.2"><ci id="A4.1.p1.62.7.m7.1.2.2.2.1.cmml" xref="A4.1.p1.62.7.m7.1.2.2.2.1">~</ci><ci id="A4.1.p1.62.7.m7.1.2.2.2.2.cmml" xref="A4.1.p1.62.7.m7.1.2.2.2.2">𝜽</ci></apply><ci id="A4.1.p1.62.7.m7.1.2.2.3.cmml" xref="A4.1.p1.62.7.m7.1.2.2.3">𝜁</ci></apply><ci id="A4.1.p1.62.7.m7.1.1.cmml" xref="A4.1.p1.62.7.m7.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.62.7.m7.1c">\tilde{\bm{\theta}}_{\zeta}(t)</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.62.7.m7.1d">over~ start_ARG bold_italic_θ end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> exponentially converge to a small neighborhood of <math alttext="\bm{0}" class="ltx_Math" display="inline" id="A4.1.p1.63.8.m8.1"><semantics id="A4.1.p1.63.8.m8.1a"><mn id="A4.1.p1.63.8.m8.1.1" mathcolor="#000099" xref="A4.1.p1.63.8.m8.1.1.cmml">𝟎</mn><annotation-xml encoding="MathML-Content" id="A4.1.p1.63.8.m8.1b"><cn id="A4.1.p1.63.8.m8.1.1.cmml" type="integer" xref="A4.1.p1.63.8.m8.1.1">0</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.63.8.m8.1c">\bm{0}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.63.8.m8.1d">bold_0</annotation></semantics></math> dominated by <math alttext="k_{{\rm c}i}" class="ltx_Math" display="inline" id="A4.1.p1.64.9.m9.1"><semantics id="A4.1.p1.64.9.m9.1a"><msub id="A4.1.p1.64.9.m9.1.1" xref="A4.1.p1.64.9.m9.1.1.cmml"><mi id="A4.1.p1.64.9.m9.1.1.2" mathcolor="#000099" xref="A4.1.p1.64.9.m9.1.1.2.cmml">k</mi><mrow id="A4.1.p1.64.9.m9.1.1.3" xref="A4.1.p1.64.9.m9.1.1.3.cmml"><mi id="A4.1.p1.64.9.m9.1.1.3.2" mathcolor="#000099" mathvariant="normal" xref="A4.1.p1.64.9.m9.1.1.3.2.cmml">c</mi><mo id="A4.1.p1.64.9.m9.1.1.3.1" xref="A4.1.p1.64.9.m9.1.1.3.1.cmml">⁢</mo><mi id="A4.1.p1.64.9.m9.1.1.3.3" mathcolor="#000099" xref="A4.1.p1.64.9.m9.1.1.3.3.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A4.1.p1.64.9.m9.1b"><apply id="A4.1.p1.64.9.m9.1.1.cmml" xref="A4.1.p1.64.9.m9.1.1"><csymbol cd="ambiguous" id="A4.1.p1.64.9.m9.1.1.1.cmml" xref="A4.1.p1.64.9.m9.1.1">subscript</csymbol><ci id="A4.1.p1.64.9.m9.1.1.2.cmml" xref="A4.1.p1.64.9.m9.1.1.2">𝑘</ci><apply id="A4.1.p1.64.9.m9.1.1.3.cmml" xref="A4.1.p1.64.9.m9.1.1.3"><times id="A4.1.p1.64.9.m9.1.1.3.1.cmml" xref="A4.1.p1.64.9.m9.1.1.3.1"></times><ci id="A4.1.p1.64.9.m9.1.1.3.2.cmml" xref="A4.1.p1.64.9.m9.1.1.3.2">c</ci><ci id="A4.1.p1.64.9.m9.1.1.3.3.cmml" xref="A4.1.p1.64.9.m9.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.64.9.m9.1c">k_{{\rm c}i}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.64.9.m9.1d">italic_k start_POSTSUBSCRIPT roman_c italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\kappa" class="ltx_Math" display="inline" id="A4.1.p1.65.10.m10.1"><semantics id="A4.1.p1.65.10.m10.1a"><mi id="A4.1.p1.65.10.m10.1.1" mathcolor="#000099" xref="A4.1.p1.65.10.m10.1.1.cmml">κ</mi><annotation-xml encoding="MathML-Content" id="A4.1.p1.65.10.m10.1b"><ci id="A4.1.p1.65.10.m10.1.1.cmml" xref="A4.1.p1.65.10.m10.1.1">𝜅</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.65.10.m10.1c">\kappa</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.65.10.m10.1d">italic_κ</annotation></semantics></math>, and <math alttext="\bar{d}" class="ltx_Math" display="inline" id="A4.1.p1.66.11.m11.1"><semantics id="A4.1.p1.66.11.m11.1a"><mover accent="true" id="A4.1.p1.66.11.m11.1.1" xref="A4.1.p1.66.11.m11.1.1.cmml"><mi id="A4.1.p1.66.11.m11.1.1.2" mathcolor="#000099" xref="A4.1.p1.66.11.m11.1.1.2.cmml">d</mi><mo id="A4.1.p1.66.11.m11.1.1.1" mathcolor="#000099" xref="A4.1.p1.66.11.m11.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="A4.1.p1.66.11.m11.1b"><apply id="A4.1.p1.66.11.m11.1.1.cmml" xref="A4.1.p1.66.11.m11.1.1"><ci id="A4.1.p1.66.11.m11.1.1.1.cmml" xref="A4.1.p1.66.11.m11.1.1.1">¯</ci><ci id="A4.1.p1.66.11.m11.1.1.2.cmml" xref="A4.1.p1.66.11.m11.1.1.2">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.66.11.m11.1c">\bar{d}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.66.11.m11.1d">over¯ start_ARG italic_d end_ARG</annotation></semantics></math>.</span> <br class="ltx_break"/>3) The proof of robustness under IE is similar to that under partial IE, so we omit some similar steps. Applying the Lyapunov function <math alttext="V" class="ltx_Math" display="inline" id="A4.1.p1.67.m1.1"><semantics id="A4.1.p1.67.m1.1a"><mi id="A4.1.p1.67.m1.1.1" xref="A4.1.p1.67.m1.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="A4.1.p1.67.m1.1b"><ci id="A4.1.p1.67.m1.1.1.cmml" xref="A4.1.p1.67.m1.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.67.m1.1c">V</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.67.m1.1d">italic_V</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#A3.E39" title="In Proof. ‣ Appendix C The proof of Theorem 2 ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">39</span></a>) and the IE condition <math alttext="\Psi(T_{\rm e})\geq\sigma I" class="ltx_Math" display="inline" id="A4.1.p1.68.m2.1"><semantics id="A4.1.p1.68.m2.1a"><mrow id="A4.1.p1.68.m2.1.1" xref="A4.1.p1.68.m2.1.1.cmml"><mrow id="A4.1.p1.68.m2.1.1.1" xref="A4.1.p1.68.m2.1.1.1.cmml"><mi id="A4.1.p1.68.m2.1.1.1.3" mathvariant="normal" xref="A4.1.p1.68.m2.1.1.1.3.cmml">Ψ</mi><mo id="A4.1.p1.68.m2.1.1.1.2" xref="A4.1.p1.68.m2.1.1.1.2.cmml">⁢</mo><mrow id="A4.1.p1.68.m2.1.1.1.1.1" xref="A4.1.p1.68.m2.1.1.1.1.1.1.cmml"><mo id="A4.1.p1.68.m2.1.1.1.1.1.2" stretchy="false" xref="A4.1.p1.68.m2.1.1.1.1.1.1.cmml">(</mo><msub id="A4.1.p1.68.m2.1.1.1.1.1.1" xref="A4.1.p1.68.m2.1.1.1.1.1.1.cmml"><mi id="A4.1.p1.68.m2.1.1.1.1.1.1.2" xref="A4.1.p1.68.m2.1.1.1.1.1.1.2.cmml">T</mi><mi id="A4.1.p1.68.m2.1.1.1.1.1.1.3" mathvariant="normal" xref="A4.1.p1.68.m2.1.1.1.1.1.1.3.cmml">e</mi></msub><mo id="A4.1.p1.68.m2.1.1.1.1.1.3" stretchy="false" xref="A4.1.p1.68.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A4.1.p1.68.m2.1.1.2" xref="A4.1.p1.68.m2.1.1.2.cmml">≥</mo><mrow id="A4.1.p1.68.m2.1.1.3" xref="A4.1.p1.68.m2.1.1.3.cmml"><mi id="A4.1.p1.68.m2.1.1.3.2" xref="A4.1.p1.68.m2.1.1.3.2.cmml">σ</mi><mo id="A4.1.p1.68.m2.1.1.3.1" xref="A4.1.p1.68.m2.1.1.3.1.cmml">⁢</mo><mi id="A4.1.p1.68.m2.1.1.3.3" xref="A4.1.p1.68.m2.1.1.3.3.cmml">I</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.68.m2.1b"><apply id="A4.1.p1.68.m2.1.1.cmml" xref="A4.1.p1.68.m2.1.1"><geq id="A4.1.p1.68.m2.1.1.2.cmml" xref="A4.1.p1.68.m2.1.1.2"></geq><apply id="A4.1.p1.68.m2.1.1.1.cmml" xref="A4.1.p1.68.m2.1.1.1"><times id="A4.1.p1.68.m2.1.1.1.2.cmml" xref="A4.1.p1.68.m2.1.1.1.2"></times><ci id="A4.1.p1.68.m2.1.1.1.3.cmml" xref="A4.1.p1.68.m2.1.1.1.3">Ψ</ci><apply id="A4.1.p1.68.m2.1.1.1.1.1.1.cmml" xref="A4.1.p1.68.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.1.p1.68.m2.1.1.1.1.1.1.1.cmml" xref="A4.1.p1.68.m2.1.1.1.1.1">subscript</csymbol><ci id="A4.1.p1.68.m2.1.1.1.1.1.1.2.cmml" xref="A4.1.p1.68.m2.1.1.1.1.1.1.2">𝑇</ci><ci id="A4.1.p1.68.m2.1.1.1.1.1.1.3.cmml" xref="A4.1.p1.68.m2.1.1.1.1.1.1.3">e</ci></apply></apply><apply id="A4.1.p1.68.m2.1.1.3.cmml" xref="A4.1.p1.68.m2.1.1.3"><times id="A4.1.p1.68.m2.1.1.3.1.cmml" xref="A4.1.p1.68.m2.1.1.3.1"></times><ci id="A4.1.p1.68.m2.1.1.3.2.cmml" xref="A4.1.p1.68.m2.1.1.3.2">𝜎</ci><ci id="A4.1.p1.68.m2.1.1.3.3.cmml" xref="A4.1.p1.68.m2.1.1.3.3">𝐼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.68.m2.1c">\Psi(T_{\rm e})\geq\sigma I</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.68.m2.1d">roman_Ψ ( italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT ) ≥ italic_σ italic_I</annotation></semantics></math>, one obtains</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx89"> <tbody id="A4.Ex87"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\dot{V}\leq" class="ltx_Math" display="inline" id="A4.Ex87.m1.1"><semantics id="A4.Ex87.m1.1a"><mrow id="A4.Ex87.m1.1.1" xref="A4.Ex87.m1.1.1.cmml"><mover accent="true" id="A4.Ex87.m1.1.1.2" xref="A4.Ex87.m1.1.1.2.cmml"><mi id="A4.Ex87.m1.1.1.2.2" xref="A4.Ex87.m1.1.1.2.2.cmml">V</mi><mo id="A4.Ex87.m1.1.1.2.1" xref="A4.Ex87.m1.1.1.2.1.cmml">˙</mo></mover><mo id="A4.Ex87.m1.1.1.1" xref="A4.Ex87.m1.1.1.1.cmml">≤</mo><mi id="A4.Ex87.m1.1.1.3" xref="A4.Ex87.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex87.m1.1b"><apply id="A4.Ex87.m1.1.1.cmml" xref="A4.Ex87.m1.1.1"><leq id="A4.Ex87.m1.1.1.1.cmml" xref="A4.Ex87.m1.1.1.1"></leq><apply id="A4.Ex87.m1.1.1.2.cmml" xref="A4.Ex87.m1.1.1.2"><ci id="A4.Ex87.m1.1.1.2.1.cmml" xref="A4.Ex87.m1.1.1.2.1">˙</ci><ci id="A4.Ex87.m1.1.1.2.2.cmml" xref="A4.Ex87.m1.1.1.2.2">𝑉</ci></apply><csymbol cd="latexml" id="A4.Ex87.m1.1.1.3.cmml" xref="A4.Ex87.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex87.m1.1c">\displaystyle\dot{V}\leq</annotation><annotation encoding="application/x-llamapun" id="A4.Ex87.m1.1d">over˙ start_ARG italic_V end_ARG ≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-k_{\rm c}(1-\mu)\bm{e}^{T}\bm{e}/2-\kappa\sigma^{*}\lambda_{\min% }(\Gamma)(1-\mu)\tilde{\bm{\theta}}^{T}\Gamma^{-1}\tilde{\bm{\theta}}+\rho_{2}" class="ltx_Math" display="inline" id="A4.Ex87.m2.3"><semantics id="A4.Ex87.m2.3a"><mrow id="A4.Ex87.m2.3.3" xref="A4.Ex87.m2.3.3.cmml"><mrow id="A4.Ex87.m2.3.3.2" xref="A4.Ex87.m2.3.3.2.cmml"><mrow id="A4.Ex87.m2.2.2.1.1" xref="A4.Ex87.m2.2.2.1.1.cmml"><mo id="A4.Ex87.m2.2.2.1.1a" xref="A4.Ex87.m2.2.2.1.1.cmml">−</mo><mrow id="A4.Ex87.m2.2.2.1.1.1" xref="A4.Ex87.m2.2.2.1.1.1.cmml"><mrow id="A4.Ex87.m2.2.2.1.1.1.1" xref="A4.Ex87.m2.2.2.1.1.1.1.cmml"><msub id="A4.Ex87.m2.2.2.1.1.1.1.3" xref="A4.Ex87.m2.2.2.1.1.1.1.3.cmml"><mi id="A4.Ex87.m2.2.2.1.1.1.1.3.2" xref="A4.Ex87.m2.2.2.1.1.1.1.3.2.cmml">k</mi><mi id="A4.Ex87.m2.2.2.1.1.1.1.3.3" mathvariant="normal" xref="A4.Ex87.m2.2.2.1.1.1.1.3.3.cmml">c</mi></msub><mo id="A4.Ex87.m2.2.2.1.1.1.1.2" xref="A4.Ex87.m2.2.2.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex87.m2.2.2.1.1.1.1.1.1" xref="A4.Ex87.m2.2.2.1.1.1.1.1.1.1.cmml"><mo id="A4.Ex87.m2.2.2.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex87.m2.2.2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex87.m2.2.2.1.1.1.1.1.1.1" xref="A4.Ex87.m2.2.2.1.1.1.1.1.1.1.cmml"><mn id="A4.Ex87.m2.2.2.1.1.1.1.1.1.1.2" xref="A4.Ex87.m2.2.2.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A4.Ex87.m2.2.2.1.1.1.1.1.1.1.1" xref="A4.Ex87.m2.2.2.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="A4.Ex87.m2.2.2.1.1.1.1.1.1.1.3" xref="A4.Ex87.m2.2.2.1.1.1.1.1.1.1.3.cmml">μ</mi></mrow><mo id="A4.Ex87.m2.2.2.1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex87.m2.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex87.m2.2.2.1.1.1.1.2a" xref="A4.Ex87.m2.2.2.1.1.1.1.2.cmml">⁢</mo><msup id="A4.Ex87.m2.2.2.1.1.1.1.4" xref="A4.Ex87.m2.2.2.1.1.1.1.4.cmml"><mi id="A4.Ex87.m2.2.2.1.1.1.1.4.2" xref="A4.Ex87.m2.2.2.1.1.1.1.4.2.cmml">𝒆</mi><mi id="A4.Ex87.m2.2.2.1.1.1.1.4.3" xref="A4.Ex87.m2.2.2.1.1.1.1.4.3.cmml">T</mi></msup><mo id="A4.Ex87.m2.2.2.1.1.1.1.2b" xref="A4.Ex87.m2.2.2.1.1.1.1.2.cmml">⁢</mo><mi id="A4.Ex87.m2.2.2.1.1.1.1.5" xref="A4.Ex87.m2.2.2.1.1.1.1.5.cmml">𝒆</mi></mrow><mo id="A4.Ex87.m2.2.2.1.1.1.2" xref="A4.Ex87.m2.2.2.1.1.1.2.cmml">/</mo><mn id="A4.Ex87.m2.2.2.1.1.1.3" xref="A4.Ex87.m2.2.2.1.1.1.3.cmml">2</mn></mrow></mrow><mo id="A4.Ex87.m2.3.3.2.3" xref="A4.Ex87.m2.3.3.2.3.cmml">−</mo><mrow id="A4.Ex87.m2.3.3.2.2" xref="A4.Ex87.m2.3.3.2.2.cmml"><mi id="A4.Ex87.m2.3.3.2.2.3" xref="A4.Ex87.m2.3.3.2.2.3.cmml">κ</mi><mo id="A4.Ex87.m2.3.3.2.2.2" xref="A4.Ex87.m2.3.3.2.2.2.cmml">⁢</mo><msup id="A4.Ex87.m2.3.3.2.2.4" xref="A4.Ex87.m2.3.3.2.2.4.cmml"><mi id="A4.Ex87.m2.3.3.2.2.4.2" xref="A4.Ex87.m2.3.3.2.2.4.2.cmml">σ</mi><mo id="A4.Ex87.m2.3.3.2.2.4.3" xref="A4.Ex87.m2.3.3.2.2.4.3.cmml">∗</mo></msup><mo id="A4.Ex87.m2.3.3.2.2.2a" xref="A4.Ex87.m2.3.3.2.2.2.cmml">⁢</mo><msub id="A4.Ex87.m2.3.3.2.2.5" xref="A4.Ex87.m2.3.3.2.2.5.cmml"><mi id="A4.Ex87.m2.3.3.2.2.5.2" xref="A4.Ex87.m2.3.3.2.2.5.2.cmml">λ</mi><mi id="A4.Ex87.m2.3.3.2.2.5.3" xref="A4.Ex87.m2.3.3.2.2.5.3.cmml">min</mi></msub><mo id="A4.Ex87.m2.3.3.2.2.2b" xref="A4.Ex87.m2.3.3.2.2.2.cmml">⁢</mo><mrow id="A4.Ex87.m2.3.3.2.2.6.2" xref="A4.Ex87.m2.3.3.2.2.cmml"><mo id="A4.Ex87.m2.3.3.2.2.6.2.1" stretchy="false" xref="A4.Ex87.m2.3.3.2.2.cmml">(</mo><mi id="A4.Ex87.m2.1.1" mathvariant="normal" xref="A4.Ex87.m2.1.1.cmml">Γ</mi><mo id="A4.Ex87.m2.3.3.2.2.6.2.2" stretchy="false" xref="A4.Ex87.m2.3.3.2.2.cmml">)</mo></mrow><mo id="A4.Ex87.m2.3.3.2.2.2c" xref="A4.Ex87.m2.3.3.2.2.2.cmml">⁢</mo><mrow id="A4.Ex87.m2.3.3.2.2.1.1" xref="A4.Ex87.m2.3.3.2.2.1.1.1.cmml"><mo id="A4.Ex87.m2.3.3.2.2.1.1.2" stretchy="false" xref="A4.Ex87.m2.3.3.2.2.1.1.1.cmml">(</mo><mrow id="A4.Ex87.m2.3.3.2.2.1.1.1" xref="A4.Ex87.m2.3.3.2.2.1.1.1.cmml"><mn id="A4.Ex87.m2.3.3.2.2.1.1.1.2" xref="A4.Ex87.m2.3.3.2.2.1.1.1.2.cmml">1</mn><mo id="A4.Ex87.m2.3.3.2.2.1.1.1.1" xref="A4.Ex87.m2.3.3.2.2.1.1.1.1.cmml">−</mo><mi id="A4.Ex87.m2.3.3.2.2.1.1.1.3" xref="A4.Ex87.m2.3.3.2.2.1.1.1.3.cmml">μ</mi></mrow><mo id="A4.Ex87.m2.3.3.2.2.1.1.3" stretchy="false" xref="A4.Ex87.m2.3.3.2.2.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex87.m2.3.3.2.2.2d" xref="A4.Ex87.m2.3.3.2.2.2.cmml">⁢</mo><msup id="A4.Ex87.m2.3.3.2.2.7" xref="A4.Ex87.m2.3.3.2.2.7.cmml"><mover accent="true" id="A4.Ex87.m2.3.3.2.2.7.2" xref="A4.Ex87.m2.3.3.2.2.7.2.cmml"><mi id="A4.Ex87.m2.3.3.2.2.7.2.2" xref="A4.Ex87.m2.3.3.2.2.7.2.2.cmml">𝜽</mi><mo id="A4.Ex87.m2.3.3.2.2.7.2.1" xref="A4.Ex87.m2.3.3.2.2.7.2.1.cmml">~</mo></mover><mi id="A4.Ex87.m2.3.3.2.2.7.3" xref="A4.Ex87.m2.3.3.2.2.7.3.cmml">T</mi></msup><mo id="A4.Ex87.m2.3.3.2.2.2e" xref="A4.Ex87.m2.3.3.2.2.2.cmml">⁢</mo><msup id="A4.Ex87.m2.3.3.2.2.8" xref="A4.Ex87.m2.3.3.2.2.8.cmml"><mi id="A4.Ex87.m2.3.3.2.2.8.2" mathvariant="normal" xref="A4.Ex87.m2.3.3.2.2.8.2.cmml">Γ</mi><mrow id="A4.Ex87.m2.3.3.2.2.8.3" xref="A4.Ex87.m2.3.3.2.2.8.3.cmml"><mo id="A4.Ex87.m2.3.3.2.2.8.3a" xref="A4.Ex87.m2.3.3.2.2.8.3.cmml">−</mo><mn id="A4.Ex87.m2.3.3.2.2.8.3.2" xref="A4.Ex87.m2.3.3.2.2.8.3.2.cmml">1</mn></mrow></msup><mo id="A4.Ex87.m2.3.3.2.2.2f" xref="A4.Ex87.m2.3.3.2.2.2.cmml">⁢</mo><mover accent="true" id="A4.Ex87.m2.3.3.2.2.9" xref="A4.Ex87.m2.3.3.2.2.9.cmml"><mi id="A4.Ex87.m2.3.3.2.2.9.2" xref="A4.Ex87.m2.3.3.2.2.9.2.cmml">𝜽</mi><mo id="A4.Ex87.m2.3.3.2.2.9.1" xref="A4.Ex87.m2.3.3.2.2.9.1.cmml">~</mo></mover></mrow></mrow><mo id="A4.Ex87.m2.3.3.3" xref="A4.Ex87.m2.3.3.3.cmml">+</mo><msub id="A4.Ex87.m2.3.3.4" xref="A4.Ex87.m2.3.3.4.cmml"><mi id="A4.Ex87.m2.3.3.4.2" xref="A4.Ex87.m2.3.3.4.2.cmml">ρ</mi><mn id="A4.Ex87.m2.3.3.4.3" xref="A4.Ex87.m2.3.3.4.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex87.m2.3b"><apply id="A4.Ex87.m2.3.3.cmml" xref="A4.Ex87.m2.3.3"><plus id="A4.Ex87.m2.3.3.3.cmml" xref="A4.Ex87.m2.3.3.3"></plus><apply id="A4.Ex87.m2.3.3.2.cmml" xref="A4.Ex87.m2.3.3.2"><minus id="A4.Ex87.m2.3.3.2.3.cmml" xref="A4.Ex87.m2.3.3.2.3"></minus><apply id="A4.Ex87.m2.2.2.1.1.cmml" xref="A4.Ex87.m2.2.2.1.1"><minus id="A4.Ex87.m2.2.2.1.1.2.cmml" xref="A4.Ex87.m2.2.2.1.1"></minus><apply id="A4.Ex87.m2.2.2.1.1.1.cmml" xref="A4.Ex87.m2.2.2.1.1.1"><divide id="A4.Ex87.m2.2.2.1.1.1.2.cmml" xref="A4.Ex87.m2.2.2.1.1.1.2"></divide><apply id="A4.Ex87.m2.2.2.1.1.1.1.cmml" xref="A4.Ex87.m2.2.2.1.1.1.1"><times id="A4.Ex87.m2.2.2.1.1.1.1.2.cmml" xref="A4.Ex87.m2.2.2.1.1.1.1.2"></times><apply id="A4.Ex87.m2.2.2.1.1.1.1.3.cmml" xref="A4.Ex87.m2.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.Ex87.m2.2.2.1.1.1.1.3.1.cmml" xref="A4.Ex87.m2.2.2.1.1.1.1.3">subscript</csymbol><ci id="A4.Ex87.m2.2.2.1.1.1.1.3.2.cmml" xref="A4.Ex87.m2.2.2.1.1.1.1.3.2">𝑘</ci><ci id="A4.Ex87.m2.2.2.1.1.1.1.3.3.cmml" xref="A4.Ex87.m2.2.2.1.1.1.1.3.3">c</ci></apply><apply id="A4.Ex87.m2.2.2.1.1.1.1.1.1.1.cmml" xref="A4.Ex87.m2.2.2.1.1.1.1.1.1"><minus id="A4.Ex87.m2.2.2.1.1.1.1.1.1.1.1.cmml" xref="A4.Ex87.m2.2.2.1.1.1.1.1.1.1.1"></minus><cn id="A4.Ex87.m2.2.2.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A4.Ex87.m2.2.2.1.1.1.1.1.1.1.2">1</cn><ci id="A4.Ex87.m2.2.2.1.1.1.1.1.1.1.3.cmml" xref="A4.Ex87.m2.2.2.1.1.1.1.1.1.1.3">𝜇</ci></apply><apply id="A4.Ex87.m2.2.2.1.1.1.1.4.cmml" xref="A4.Ex87.m2.2.2.1.1.1.1.4"><csymbol cd="ambiguous" id="A4.Ex87.m2.2.2.1.1.1.1.4.1.cmml" xref="A4.Ex87.m2.2.2.1.1.1.1.4">superscript</csymbol><ci id="A4.Ex87.m2.2.2.1.1.1.1.4.2.cmml" xref="A4.Ex87.m2.2.2.1.1.1.1.4.2">𝒆</ci><ci id="A4.Ex87.m2.2.2.1.1.1.1.4.3.cmml" xref="A4.Ex87.m2.2.2.1.1.1.1.4.3">𝑇</ci></apply><ci id="A4.Ex87.m2.2.2.1.1.1.1.5.cmml" xref="A4.Ex87.m2.2.2.1.1.1.1.5">𝒆</ci></apply><cn id="A4.Ex87.m2.2.2.1.1.1.3.cmml" type="integer" xref="A4.Ex87.m2.2.2.1.1.1.3">2</cn></apply></apply><apply id="A4.Ex87.m2.3.3.2.2.cmml" xref="A4.Ex87.m2.3.3.2.2"><times id="A4.Ex87.m2.3.3.2.2.2.cmml" xref="A4.Ex87.m2.3.3.2.2.2"></times><ci id="A4.Ex87.m2.3.3.2.2.3.cmml" xref="A4.Ex87.m2.3.3.2.2.3">𝜅</ci><apply id="A4.Ex87.m2.3.3.2.2.4.cmml" xref="A4.Ex87.m2.3.3.2.2.4"><csymbol cd="ambiguous" id="A4.Ex87.m2.3.3.2.2.4.1.cmml" xref="A4.Ex87.m2.3.3.2.2.4">superscript</csymbol><ci id="A4.Ex87.m2.3.3.2.2.4.2.cmml" xref="A4.Ex87.m2.3.3.2.2.4.2">𝜎</ci><times id="A4.Ex87.m2.3.3.2.2.4.3.cmml" xref="A4.Ex87.m2.3.3.2.2.4.3"></times></apply><apply id="A4.Ex87.m2.3.3.2.2.5.cmml" xref="A4.Ex87.m2.3.3.2.2.5"><csymbol cd="ambiguous" id="A4.Ex87.m2.3.3.2.2.5.1.cmml" xref="A4.Ex87.m2.3.3.2.2.5">subscript</csymbol><ci id="A4.Ex87.m2.3.3.2.2.5.2.cmml" xref="A4.Ex87.m2.3.3.2.2.5.2">𝜆</ci><min id="A4.Ex87.m2.3.3.2.2.5.3.cmml" xref="A4.Ex87.m2.3.3.2.2.5.3"></min></apply><ci id="A4.Ex87.m2.1.1.cmml" xref="A4.Ex87.m2.1.1">Γ</ci><apply id="A4.Ex87.m2.3.3.2.2.1.1.1.cmml" xref="A4.Ex87.m2.3.3.2.2.1.1"><minus id="A4.Ex87.m2.3.3.2.2.1.1.1.1.cmml" xref="A4.Ex87.m2.3.3.2.2.1.1.1.1"></minus><cn id="A4.Ex87.m2.3.3.2.2.1.1.1.2.cmml" type="integer" xref="A4.Ex87.m2.3.3.2.2.1.1.1.2">1</cn><ci id="A4.Ex87.m2.3.3.2.2.1.1.1.3.cmml" xref="A4.Ex87.m2.3.3.2.2.1.1.1.3">𝜇</ci></apply><apply id="A4.Ex87.m2.3.3.2.2.7.cmml" xref="A4.Ex87.m2.3.3.2.2.7"><csymbol cd="ambiguous" id="A4.Ex87.m2.3.3.2.2.7.1.cmml" xref="A4.Ex87.m2.3.3.2.2.7">superscript</csymbol><apply id="A4.Ex87.m2.3.3.2.2.7.2.cmml" xref="A4.Ex87.m2.3.3.2.2.7.2"><ci id="A4.Ex87.m2.3.3.2.2.7.2.1.cmml" xref="A4.Ex87.m2.3.3.2.2.7.2.1">~</ci><ci id="A4.Ex87.m2.3.3.2.2.7.2.2.cmml" xref="A4.Ex87.m2.3.3.2.2.7.2.2">𝜽</ci></apply><ci id="A4.Ex87.m2.3.3.2.2.7.3.cmml" xref="A4.Ex87.m2.3.3.2.2.7.3">𝑇</ci></apply><apply id="A4.Ex87.m2.3.3.2.2.8.cmml" xref="A4.Ex87.m2.3.3.2.2.8"><csymbol cd="ambiguous" id="A4.Ex87.m2.3.3.2.2.8.1.cmml" xref="A4.Ex87.m2.3.3.2.2.8">superscript</csymbol><ci id="A4.Ex87.m2.3.3.2.2.8.2.cmml" xref="A4.Ex87.m2.3.3.2.2.8.2">Γ</ci><apply id="A4.Ex87.m2.3.3.2.2.8.3.cmml" xref="A4.Ex87.m2.3.3.2.2.8.3"><minus id="A4.Ex87.m2.3.3.2.2.8.3.1.cmml" xref="A4.Ex87.m2.3.3.2.2.8.3"></minus><cn id="A4.Ex87.m2.3.3.2.2.8.3.2.cmml" type="integer" xref="A4.Ex87.m2.3.3.2.2.8.3.2">1</cn></apply></apply><apply id="A4.Ex87.m2.3.3.2.2.9.cmml" xref="A4.Ex87.m2.3.3.2.2.9"><ci id="A4.Ex87.m2.3.3.2.2.9.1.cmml" xref="A4.Ex87.m2.3.3.2.2.9.1">~</ci><ci id="A4.Ex87.m2.3.3.2.2.9.2.cmml" xref="A4.Ex87.m2.3.3.2.2.9.2">𝜽</ci></apply></apply></apply><apply id="A4.Ex87.m2.3.3.4.cmml" xref="A4.Ex87.m2.3.3.4"><csymbol cd="ambiguous" id="A4.Ex87.m2.3.3.4.1.cmml" xref="A4.Ex87.m2.3.3.4">subscript</csymbol><ci id="A4.Ex87.m2.3.3.4.2.cmml" xref="A4.Ex87.m2.3.3.4.2">𝜌</ci><cn id="A4.Ex87.m2.3.3.4.3.cmml" type="integer" xref="A4.Ex87.m2.3.3.4.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex87.m2.3c">\displaystyle-k_{\rm c}(1-\mu)\bm{e}^{T}\bm{e}/2-\kappa\sigma^{*}\lambda_{\min% }(\Gamma)(1-\mu)\tilde{\bm{\theta}}^{T}\Gamma^{-1}\tilde{\bm{\theta}}+\rho_{2}</annotation><annotation encoding="application/x-llamapun" id="A4.Ex87.m2.3d">- italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ( 1 - italic_μ ) bold_italic_e start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_e / 2 - italic_κ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Γ ) ( 1 - italic_μ ) over~ start_ARG bold_italic_θ end_ARG start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT roman_Γ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT over~ start_ARG bold_italic_θ end_ARG + italic_ρ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex88"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\leq" class="ltx_Math" display="inline" id="A4.Ex88.m1.1"><semantics id="A4.Ex88.m1.1a"><mo id="A4.Ex88.m1.1.1" xref="A4.Ex88.m1.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="A4.Ex88.m1.1b"><leq id="A4.Ex88.m1.1.1.cmml" xref="A4.Ex88.m1.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex88.m1.1c">\displaystyle\leq</annotation><annotation encoding="application/x-llamapun" id="A4.Ex88.m1.1d">≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-k_{\rm r}V+\rho_{2}(k_{\rm c},\kappa,\bar{d}),\forall t\geq T_{% \rm e}" class="ltx_Math" display="inline" id="A4.Ex88.m2.4"><semantics id="A4.Ex88.m2.4a"><mrow id="A4.Ex88.m2.4.4" xref="A4.Ex88.m2.4.4.cmml"><mrow id="A4.Ex88.m2.4.4.2.2" xref="A4.Ex88.m2.4.4.2.3.cmml"><mrow id="A4.Ex88.m2.3.3.1.1.1" xref="A4.Ex88.m2.3.3.1.1.1.cmml"><mrow id="A4.Ex88.m2.3.3.1.1.1.3" xref="A4.Ex88.m2.3.3.1.1.1.3.cmml"><mo id="A4.Ex88.m2.3.3.1.1.1.3a" xref="A4.Ex88.m2.3.3.1.1.1.3.cmml">−</mo><mrow id="A4.Ex88.m2.3.3.1.1.1.3.2" xref="A4.Ex88.m2.3.3.1.1.1.3.2.cmml"><msub id="A4.Ex88.m2.3.3.1.1.1.3.2.2" xref="A4.Ex88.m2.3.3.1.1.1.3.2.2.cmml"><mi id="A4.Ex88.m2.3.3.1.1.1.3.2.2.2" xref="A4.Ex88.m2.3.3.1.1.1.3.2.2.2.cmml">k</mi><mi id="A4.Ex88.m2.3.3.1.1.1.3.2.2.3" mathvariant="normal" xref="A4.Ex88.m2.3.3.1.1.1.3.2.2.3.cmml">r</mi></msub><mo id="A4.Ex88.m2.3.3.1.1.1.3.2.1" xref="A4.Ex88.m2.3.3.1.1.1.3.2.1.cmml">⁢</mo><mi id="A4.Ex88.m2.3.3.1.1.1.3.2.3" xref="A4.Ex88.m2.3.3.1.1.1.3.2.3.cmml">V</mi></mrow></mrow><mo id="A4.Ex88.m2.3.3.1.1.1.2" xref="A4.Ex88.m2.3.3.1.1.1.2.cmml">+</mo><mrow id="A4.Ex88.m2.3.3.1.1.1.1" xref="A4.Ex88.m2.3.3.1.1.1.1.cmml"><msub id="A4.Ex88.m2.3.3.1.1.1.1.3" xref="A4.Ex88.m2.3.3.1.1.1.1.3.cmml"><mi id="A4.Ex88.m2.3.3.1.1.1.1.3.2" xref="A4.Ex88.m2.3.3.1.1.1.1.3.2.cmml">ρ</mi><mn id="A4.Ex88.m2.3.3.1.1.1.1.3.3" xref="A4.Ex88.m2.3.3.1.1.1.1.3.3.cmml">2</mn></msub><mo id="A4.Ex88.m2.3.3.1.1.1.1.2" xref="A4.Ex88.m2.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex88.m2.3.3.1.1.1.1.1.1" xref="A4.Ex88.m2.3.3.1.1.1.1.1.2.cmml"><mo id="A4.Ex88.m2.3.3.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex88.m2.3.3.1.1.1.1.1.2.cmml">(</mo><msub id="A4.Ex88.m2.3.3.1.1.1.1.1.1.1" xref="A4.Ex88.m2.3.3.1.1.1.1.1.1.1.cmml"><mi id="A4.Ex88.m2.3.3.1.1.1.1.1.1.1.2" xref="A4.Ex88.m2.3.3.1.1.1.1.1.1.1.2.cmml">k</mi><mi id="A4.Ex88.m2.3.3.1.1.1.1.1.1.1.3" mathvariant="normal" xref="A4.Ex88.m2.3.3.1.1.1.1.1.1.1.3.cmml">c</mi></msub><mo id="A4.Ex88.m2.3.3.1.1.1.1.1.1.3" xref="A4.Ex88.m2.3.3.1.1.1.1.1.2.cmml">,</mo><mi id="A4.Ex88.m2.1.1" xref="A4.Ex88.m2.1.1.cmml">κ</mi><mo id="A4.Ex88.m2.3.3.1.1.1.1.1.1.4" xref="A4.Ex88.m2.3.3.1.1.1.1.1.2.cmml">,</mo><mover accent="true" id="A4.Ex88.m2.2.2" xref="A4.Ex88.m2.2.2.cmml"><mi id="A4.Ex88.m2.2.2.2" xref="A4.Ex88.m2.2.2.2.cmml">d</mi><mo id="A4.Ex88.m2.2.2.1" xref="A4.Ex88.m2.2.2.1.cmml">¯</mo></mover><mo id="A4.Ex88.m2.3.3.1.1.1.1.1.1.5" stretchy="false" xref="A4.Ex88.m2.3.3.1.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="A4.Ex88.m2.4.4.2.2.3" xref="A4.Ex88.m2.4.4.2.3.cmml">,</mo><mrow id="A4.Ex88.m2.4.4.2.2.2" xref="A4.Ex88.m2.4.4.2.2.2.cmml"><mo id="A4.Ex88.m2.4.4.2.2.2.1" rspace="0.167em" xref="A4.Ex88.m2.4.4.2.2.2.1.cmml">∀</mo><mi id="A4.Ex88.m2.4.4.2.2.2.2" xref="A4.Ex88.m2.4.4.2.2.2.2.cmml">t</mi></mrow></mrow><mo id="A4.Ex88.m2.4.4.3" xref="A4.Ex88.m2.4.4.3.cmml">≥</mo><msub id="A4.Ex88.m2.4.4.4" xref="A4.Ex88.m2.4.4.4.cmml"><mi id="A4.Ex88.m2.4.4.4.2" xref="A4.Ex88.m2.4.4.4.2.cmml">T</mi><mi id="A4.Ex88.m2.4.4.4.3" mathvariant="normal" xref="A4.Ex88.m2.4.4.4.3.cmml">e</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex88.m2.4b"><apply id="A4.Ex88.m2.4.4.cmml" xref="A4.Ex88.m2.4.4"><geq id="A4.Ex88.m2.4.4.3.cmml" xref="A4.Ex88.m2.4.4.3"></geq><list id="A4.Ex88.m2.4.4.2.3.cmml" xref="A4.Ex88.m2.4.4.2.2"><apply id="A4.Ex88.m2.3.3.1.1.1.cmml" xref="A4.Ex88.m2.3.3.1.1.1"><plus id="A4.Ex88.m2.3.3.1.1.1.2.cmml" xref="A4.Ex88.m2.3.3.1.1.1.2"></plus><apply id="A4.Ex88.m2.3.3.1.1.1.3.cmml" xref="A4.Ex88.m2.3.3.1.1.1.3"><minus id="A4.Ex88.m2.3.3.1.1.1.3.1.cmml" xref="A4.Ex88.m2.3.3.1.1.1.3"></minus><apply id="A4.Ex88.m2.3.3.1.1.1.3.2.cmml" xref="A4.Ex88.m2.3.3.1.1.1.3.2"><times id="A4.Ex88.m2.3.3.1.1.1.3.2.1.cmml" xref="A4.Ex88.m2.3.3.1.1.1.3.2.1"></times><apply id="A4.Ex88.m2.3.3.1.1.1.3.2.2.cmml" xref="A4.Ex88.m2.3.3.1.1.1.3.2.2"><csymbol cd="ambiguous" id="A4.Ex88.m2.3.3.1.1.1.3.2.2.1.cmml" xref="A4.Ex88.m2.3.3.1.1.1.3.2.2">subscript</csymbol><ci id="A4.Ex88.m2.3.3.1.1.1.3.2.2.2.cmml" xref="A4.Ex88.m2.3.3.1.1.1.3.2.2.2">𝑘</ci><ci id="A4.Ex88.m2.3.3.1.1.1.3.2.2.3.cmml" xref="A4.Ex88.m2.3.3.1.1.1.3.2.2.3">r</ci></apply><ci id="A4.Ex88.m2.3.3.1.1.1.3.2.3.cmml" xref="A4.Ex88.m2.3.3.1.1.1.3.2.3">𝑉</ci></apply></apply><apply id="A4.Ex88.m2.3.3.1.1.1.1.cmml" xref="A4.Ex88.m2.3.3.1.1.1.1"><times id="A4.Ex88.m2.3.3.1.1.1.1.2.cmml" xref="A4.Ex88.m2.3.3.1.1.1.1.2"></times><apply id="A4.Ex88.m2.3.3.1.1.1.1.3.cmml" xref="A4.Ex88.m2.3.3.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.Ex88.m2.3.3.1.1.1.1.3.1.cmml" xref="A4.Ex88.m2.3.3.1.1.1.1.3">subscript</csymbol><ci id="A4.Ex88.m2.3.3.1.1.1.1.3.2.cmml" xref="A4.Ex88.m2.3.3.1.1.1.1.3.2">𝜌</ci><cn id="A4.Ex88.m2.3.3.1.1.1.1.3.3.cmml" type="integer" xref="A4.Ex88.m2.3.3.1.1.1.1.3.3">2</cn></apply><vector id="A4.Ex88.m2.3.3.1.1.1.1.1.2.cmml" xref="A4.Ex88.m2.3.3.1.1.1.1.1.1"><apply id="A4.Ex88.m2.3.3.1.1.1.1.1.1.1.cmml" xref="A4.Ex88.m2.3.3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.Ex88.m2.3.3.1.1.1.1.1.1.1.1.cmml" xref="A4.Ex88.m2.3.3.1.1.1.1.1.1.1">subscript</csymbol><ci id="A4.Ex88.m2.3.3.1.1.1.1.1.1.1.2.cmml" xref="A4.Ex88.m2.3.3.1.1.1.1.1.1.1.2">𝑘</ci><ci id="A4.Ex88.m2.3.3.1.1.1.1.1.1.1.3.cmml" xref="A4.Ex88.m2.3.3.1.1.1.1.1.1.1.3">c</ci></apply><ci id="A4.Ex88.m2.1.1.cmml" xref="A4.Ex88.m2.1.1">𝜅</ci><apply id="A4.Ex88.m2.2.2.cmml" xref="A4.Ex88.m2.2.2"><ci id="A4.Ex88.m2.2.2.1.cmml" xref="A4.Ex88.m2.2.2.1">¯</ci><ci id="A4.Ex88.m2.2.2.2.cmml" xref="A4.Ex88.m2.2.2.2">𝑑</ci></apply></vector></apply></apply><apply id="A4.Ex88.m2.4.4.2.2.2.cmml" xref="A4.Ex88.m2.4.4.2.2.2"><csymbol cd="latexml" id="A4.Ex88.m2.4.4.2.2.2.1.cmml" xref="A4.Ex88.m2.4.4.2.2.2.1">for-all</csymbol><ci id="A4.Ex88.m2.4.4.2.2.2.2.cmml" xref="A4.Ex88.m2.4.4.2.2.2.2">𝑡</ci></apply></list><apply id="A4.Ex88.m2.4.4.4.cmml" xref="A4.Ex88.m2.4.4.4"><csymbol cd="ambiguous" id="A4.Ex88.m2.4.4.4.1.cmml" xref="A4.Ex88.m2.4.4.4">subscript</csymbol><ci id="A4.Ex88.m2.4.4.4.2.cmml" xref="A4.Ex88.m2.4.4.4.2">𝑇</ci><ci id="A4.Ex88.m2.4.4.4.3.cmml" xref="A4.Ex88.m2.4.4.4.3">e</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex88.m2.4c">\displaystyle-k_{\rm r}V+\rho_{2}(k_{\rm c},\kappa,\bar{d}),\forall t\geq T_{% \rm e}</annotation><annotation encoding="application/x-llamapun" id="A4.Ex88.m2.4d">- italic_k start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT italic_V + italic_ρ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT , italic_κ , over¯ start_ARG italic_d end_ARG ) , ∀ italic_t ≥ italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.1.p1.95">with <math alttext="k_{\rm r}" class="ltx_Math" display="inline" id="A4.1.p1.69.m1.1"><semantics id="A4.1.p1.69.m1.1a"><msub id="A4.1.p1.69.m1.1.1" xref="A4.1.p1.69.m1.1.1.cmml"><mi id="A4.1.p1.69.m1.1.1.2" xref="A4.1.p1.69.m1.1.1.2.cmml">k</mi><mi id="A4.1.p1.69.m1.1.1.3" mathvariant="normal" xref="A4.1.p1.69.m1.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="A4.1.p1.69.m1.1b"><apply id="A4.1.p1.69.m1.1.1.cmml" xref="A4.1.p1.69.m1.1.1"><csymbol cd="ambiguous" id="A4.1.p1.69.m1.1.1.1.cmml" xref="A4.1.p1.69.m1.1.1">subscript</csymbol><ci id="A4.1.p1.69.m1.1.1.2.cmml" xref="A4.1.p1.69.m1.1.1.2">𝑘</ci><ci id="A4.1.p1.69.m1.1.1.3.cmml" xref="A4.1.p1.69.m1.1.1.3">r</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.69.m1.1c">k_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.69.m1.1d">italic_k start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="A4.1.p1.70.m2.1"><semantics id="A4.1.p1.70.m2.1a"><mo id="A4.1.p1.70.m2.1.1" xref="A4.1.p1.70.m2.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="A4.1.p1.70.m2.1b"><csymbol cd="latexml" id="A4.1.p1.70.m2.1.1.cmml" xref="A4.1.p1.70.m2.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.70.m2.1c">:=</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.70.m2.1d">:=</annotation></semantics></math> <math alttext="\min\{k_{\rm c}(1-\mu),2\kappa\sigma^{*}\lambda_{\min}(\Gamma)(1-\mu)/(1+p)\}% \in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A4.1.p1.71.m3.4"><semantics id="A4.1.p1.71.m3.4a"><mrow id="A4.1.p1.71.m3.4.4" xref="A4.1.p1.71.m3.4.4.cmml"><mrow id="A4.1.p1.71.m3.4.4.2.2" xref="A4.1.p1.71.m3.4.4.2.3.cmml"><mi id="A4.1.p1.71.m3.2.2" xref="A4.1.p1.71.m3.2.2.cmml">min</mi><mo id="A4.1.p1.71.m3.4.4.2.2a" xref="A4.1.p1.71.m3.4.4.2.3.cmml">⁡</mo><mrow id="A4.1.p1.71.m3.4.4.2.2.2" xref="A4.1.p1.71.m3.4.4.2.3.cmml"><mo id="A4.1.p1.71.m3.4.4.2.2.2.3" stretchy="false" xref="A4.1.p1.71.m3.4.4.2.3.cmml">{</mo><mrow id="A4.1.p1.71.m3.3.3.1.1.1.1" xref="A4.1.p1.71.m3.3.3.1.1.1.1.cmml"><msub id="A4.1.p1.71.m3.3.3.1.1.1.1.3" xref="A4.1.p1.71.m3.3.3.1.1.1.1.3.cmml"><mi id="A4.1.p1.71.m3.3.3.1.1.1.1.3.2" xref="A4.1.p1.71.m3.3.3.1.1.1.1.3.2.cmml">k</mi><mi id="A4.1.p1.71.m3.3.3.1.1.1.1.3.3" mathvariant="normal" xref="A4.1.p1.71.m3.3.3.1.1.1.1.3.3.cmml">c</mi></msub><mo id="A4.1.p1.71.m3.3.3.1.1.1.1.2" xref="A4.1.p1.71.m3.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.1.p1.71.m3.3.3.1.1.1.1.1.1" xref="A4.1.p1.71.m3.3.3.1.1.1.1.1.1.1.cmml"><mo id="A4.1.p1.71.m3.3.3.1.1.1.1.1.1.2" stretchy="false" xref="A4.1.p1.71.m3.3.3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.1.p1.71.m3.3.3.1.1.1.1.1.1.1" xref="A4.1.p1.71.m3.3.3.1.1.1.1.1.1.1.cmml"><mn id="A4.1.p1.71.m3.3.3.1.1.1.1.1.1.1.2" xref="A4.1.p1.71.m3.3.3.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A4.1.p1.71.m3.3.3.1.1.1.1.1.1.1.1" xref="A4.1.p1.71.m3.3.3.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="A4.1.p1.71.m3.3.3.1.1.1.1.1.1.1.3" xref="A4.1.p1.71.m3.3.3.1.1.1.1.1.1.1.3.cmml">μ</mi></mrow><mo id="A4.1.p1.71.m3.3.3.1.1.1.1.1.1.3" stretchy="false" xref="A4.1.p1.71.m3.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A4.1.p1.71.m3.4.4.2.2.2.4" xref="A4.1.p1.71.m3.4.4.2.3.cmml">,</mo><mrow id="A4.1.p1.71.m3.4.4.2.2.2.2" xref="A4.1.p1.71.m3.4.4.2.2.2.2.cmml"><mrow id="A4.1.p1.71.m3.4.4.2.2.2.2.1" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.cmml"><mn id="A4.1.p1.71.m3.4.4.2.2.2.2.1.3" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.3.cmml">2</mn><mo id="A4.1.p1.71.m3.4.4.2.2.2.2.1.2" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.2.cmml">⁢</mo><mi id="A4.1.p1.71.m3.4.4.2.2.2.2.1.4" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.4.cmml">κ</mi><mo id="A4.1.p1.71.m3.4.4.2.2.2.2.1.2a" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.2.cmml">⁢</mo><msup id="A4.1.p1.71.m3.4.4.2.2.2.2.1.5" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.5.cmml"><mi id="A4.1.p1.71.m3.4.4.2.2.2.2.1.5.2" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.5.2.cmml">σ</mi><mo id="A4.1.p1.71.m3.4.4.2.2.2.2.1.5.3" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.5.3.cmml">∗</mo></msup><mo id="A4.1.p1.71.m3.4.4.2.2.2.2.1.2b" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.2.cmml">⁢</mo><msub id="A4.1.p1.71.m3.4.4.2.2.2.2.1.6" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.6.cmml"><mi id="A4.1.p1.71.m3.4.4.2.2.2.2.1.6.2" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.6.2.cmml">λ</mi><mi id="A4.1.p1.71.m3.4.4.2.2.2.2.1.6.3" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.6.3.cmml">min</mi></msub><mo id="A4.1.p1.71.m3.4.4.2.2.2.2.1.2c" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.2.cmml">⁢</mo><mrow id="A4.1.p1.71.m3.4.4.2.2.2.2.1.7.2" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.cmml"><mo id="A4.1.p1.71.m3.4.4.2.2.2.2.1.7.2.1" stretchy="false" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.cmml">(</mo><mi id="A4.1.p1.71.m3.1.1" mathvariant="normal" xref="A4.1.p1.71.m3.1.1.cmml">Γ</mi><mo id="A4.1.p1.71.m3.4.4.2.2.2.2.1.7.2.2" stretchy="false" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.cmml">)</mo></mrow><mo id="A4.1.p1.71.m3.4.4.2.2.2.2.1.2d" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.2.cmml">⁢</mo><mrow id="A4.1.p1.71.m3.4.4.2.2.2.2.1.1.1" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.1.1.1.cmml"><mo id="A4.1.p1.71.m3.4.4.2.2.2.2.1.1.1.2" stretchy="false" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.1.1.1.cmml">(</mo><mrow id="A4.1.p1.71.m3.4.4.2.2.2.2.1.1.1.1" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.1.1.1.cmml"><mn id="A4.1.p1.71.m3.4.4.2.2.2.2.1.1.1.1.2" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.1.1.1.2.cmml">1</mn><mo id="A4.1.p1.71.m3.4.4.2.2.2.2.1.1.1.1.1" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.1.1.1.1.cmml">−</mo><mi id="A4.1.p1.71.m3.4.4.2.2.2.2.1.1.1.1.3" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.1.1.1.3.cmml">μ</mi></mrow><mo id="A4.1.p1.71.m3.4.4.2.2.2.2.1.1.1.3" stretchy="false" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A4.1.p1.71.m3.4.4.2.2.2.2.3" xref="A4.1.p1.71.m3.4.4.2.2.2.2.3.cmml">/</mo><mrow id="A4.1.p1.71.m3.4.4.2.2.2.2.2.1" xref="A4.1.p1.71.m3.4.4.2.2.2.2.2.1.1.cmml"><mo id="A4.1.p1.71.m3.4.4.2.2.2.2.2.1.2" stretchy="false" xref="A4.1.p1.71.m3.4.4.2.2.2.2.2.1.1.cmml">(</mo><mrow id="A4.1.p1.71.m3.4.4.2.2.2.2.2.1.1" xref="A4.1.p1.71.m3.4.4.2.2.2.2.2.1.1.cmml"><mn id="A4.1.p1.71.m3.4.4.2.2.2.2.2.1.1.2" xref="A4.1.p1.71.m3.4.4.2.2.2.2.2.1.1.2.cmml">1</mn><mo id="A4.1.p1.71.m3.4.4.2.2.2.2.2.1.1.1" xref="A4.1.p1.71.m3.4.4.2.2.2.2.2.1.1.1.cmml">+</mo><mi id="A4.1.p1.71.m3.4.4.2.2.2.2.2.1.1.3" xref="A4.1.p1.71.m3.4.4.2.2.2.2.2.1.1.3.cmml">p</mi></mrow><mo id="A4.1.p1.71.m3.4.4.2.2.2.2.2.1.3" stretchy="false" xref="A4.1.p1.71.m3.4.4.2.2.2.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="A4.1.p1.71.m3.4.4.2.2.2.5" stretchy="false" xref="A4.1.p1.71.m3.4.4.2.3.cmml">}</mo></mrow></mrow><mo id="A4.1.p1.71.m3.4.4.3" xref="A4.1.p1.71.m3.4.4.3.cmml">∈</mo><msup id="A4.1.p1.71.m3.4.4.4" xref="A4.1.p1.71.m3.4.4.4.cmml"><mi id="A4.1.p1.71.m3.4.4.4.2" xref="A4.1.p1.71.m3.4.4.4.2.cmml">ℝ</mi><mo id="A4.1.p1.71.m3.4.4.4.3" xref="A4.1.p1.71.m3.4.4.4.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.71.m3.4b"><apply id="A4.1.p1.71.m3.4.4.cmml" xref="A4.1.p1.71.m3.4.4"><in id="A4.1.p1.71.m3.4.4.3.cmml" xref="A4.1.p1.71.m3.4.4.3"></in><apply id="A4.1.p1.71.m3.4.4.2.3.cmml" xref="A4.1.p1.71.m3.4.4.2.2"><min id="A4.1.p1.71.m3.2.2.cmml" xref="A4.1.p1.71.m3.2.2"></min><apply id="A4.1.p1.71.m3.3.3.1.1.1.1.cmml" xref="A4.1.p1.71.m3.3.3.1.1.1.1"><times id="A4.1.p1.71.m3.3.3.1.1.1.1.2.cmml" xref="A4.1.p1.71.m3.3.3.1.1.1.1.2"></times><apply id="A4.1.p1.71.m3.3.3.1.1.1.1.3.cmml" xref="A4.1.p1.71.m3.3.3.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.71.m3.3.3.1.1.1.1.3.1.cmml" xref="A4.1.p1.71.m3.3.3.1.1.1.1.3">subscript</csymbol><ci id="A4.1.p1.71.m3.3.3.1.1.1.1.3.2.cmml" xref="A4.1.p1.71.m3.3.3.1.1.1.1.3.2">𝑘</ci><ci id="A4.1.p1.71.m3.3.3.1.1.1.1.3.3.cmml" xref="A4.1.p1.71.m3.3.3.1.1.1.1.3.3">c</ci></apply><apply id="A4.1.p1.71.m3.3.3.1.1.1.1.1.1.1.cmml" xref="A4.1.p1.71.m3.3.3.1.1.1.1.1.1"><minus id="A4.1.p1.71.m3.3.3.1.1.1.1.1.1.1.1.cmml" xref="A4.1.p1.71.m3.3.3.1.1.1.1.1.1.1.1"></minus><cn id="A4.1.p1.71.m3.3.3.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A4.1.p1.71.m3.3.3.1.1.1.1.1.1.1.2">1</cn><ci id="A4.1.p1.71.m3.3.3.1.1.1.1.1.1.1.3.cmml" xref="A4.1.p1.71.m3.3.3.1.1.1.1.1.1.1.3">𝜇</ci></apply></apply><apply id="A4.1.p1.71.m3.4.4.2.2.2.2.cmml" xref="A4.1.p1.71.m3.4.4.2.2.2.2"><divide id="A4.1.p1.71.m3.4.4.2.2.2.2.3.cmml" xref="A4.1.p1.71.m3.4.4.2.2.2.2.3"></divide><apply id="A4.1.p1.71.m3.4.4.2.2.2.2.1.cmml" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1"><times id="A4.1.p1.71.m3.4.4.2.2.2.2.1.2.cmml" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.2"></times><cn id="A4.1.p1.71.m3.4.4.2.2.2.2.1.3.cmml" type="integer" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.3">2</cn><ci id="A4.1.p1.71.m3.4.4.2.2.2.2.1.4.cmml" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.4">𝜅</ci><apply id="A4.1.p1.71.m3.4.4.2.2.2.2.1.5.cmml" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.5"><csymbol cd="ambiguous" id="A4.1.p1.71.m3.4.4.2.2.2.2.1.5.1.cmml" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.5">superscript</csymbol><ci id="A4.1.p1.71.m3.4.4.2.2.2.2.1.5.2.cmml" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.5.2">𝜎</ci><times id="A4.1.p1.71.m3.4.4.2.2.2.2.1.5.3.cmml" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.5.3"></times></apply><apply id="A4.1.p1.71.m3.4.4.2.2.2.2.1.6.cmml" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.6"><csymbol cd="ambiguous" id="A4.1.p1.71.m3.4.4.2.2.2.2.1.6.1.cmml" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.6">subscript</csymbol><ci id="A4.1.p1.71.m3.4.4.2.2.2.2.1.6.2.cmml" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.6.2">𝜆</ci><min id="A4.1.p1.71.m3.4.4.2.2.2.2.1.6.3.cmml" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.6.3"></min></apply><ci id="A4.1.p1.71.m3.1.1.cmml" xref="A4.1.p1.71.m3.1.1">Γ</ci><apply id="A4.1.p1.71.m3.4.4.2.2.2.2.1.1.1.1.cmml" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.1.1"><minus id="A4.1.p1.71.m3.4.4.2.2.2.2.1.1.1.1.1.cmml" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.1.1.1.1"></minus><cn id="A4.1.p1.71.m3.4.4.2.2.2.2.1.1.1.1.2.cmml" type="integer" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.1.1.1.2">1</cn><ci id="A4.1.p1.71.m3.4.4.2.2.2.2.1.1.1.1.3.cmml" xref="A4.1.p1.71.m3.4.4.2.2.2.2.1.1.1.1.3">𝜇</ci></apply></apply><apply id="A4.1.p1.71.m3.4.4.2.2.2.2.2.1.1.cmml" xref="A4.1.p1.71.m3.4.4.2.2.2.2.2.1"><plus id="A4.1.p1.71.m3.4.4.2.2.2.2.2.1.1.1.cmml" xref="A4.1.p1.71.m3.4.4.2.2.2.2.2.1.1.1"></plus><cn id="A4.1.p1.71.m3.4.4.2.2.2.2.2.1.1.2.cmml" type="integer" xref="A4.1.p1.71.m3.4.4.2.2.2.2.2.1.1.2">1</cn><ci id="A4.1.p1.71.m3.4.4.2.2.2.2.2.1.1.3.cmml" xref="A4.1.p1.71.m3.4.4.2.2.2.2.2.1.1.3">𝑝</ci></apply></apply></apply><apply id="A4.1.p1.71.m3.4.4.4.cmml" xref="A4.1.p1.71.m3.4.4.4"><csymbol cd="ambiguous" id="A4.1.p1.71.m3.4.4.4.1.cmml" xref="A4.1.p1.71.m3.4.4.4">superscript</csymbol><ci id="A4.1.p1.71.m3.4.4.4.2.cmml" xref="A4.1.p1.71.m3.4.4.4.2">ℝ</ci><plus id="A4.1.p1.71.m3.4.4.4.3.cmml" xref="A4.1.p1.71.m3.4.4.4.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.71.m3.4c">\min\{k_{\rm c}(1-\mu),2\kappa\sigma^{*}\lambda_{\min}(\Gamma)(1-\mu)/(1+p)\}% \in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.71.m3.4d">roman_min { italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ( 1 - italic_μ ) , 2 italic_κ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( roman_Γ ) ( 1 - italic_μ ) / ( 1 + italic_p ) } ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\rho_{2}" class="ltx_Math" display="inline" id="A4.1.p1.72.m4.1"><semantics id="A4.1.p1.72.m4.1a"><msub id="A4.1.p1.72.m4.1.1" xref="A4.1.p1.72.m4.1.1.cmml"><mi id="A4.1.p1.72.m4.1.1.2" xref="A4.1.p1.72.m4.1.1.2.cmml">ρ</mi><mn id="A4.1.p1.72.m4.1.1.3" xref="A4.1.p1.72.m4.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="A4.1.p1.72.m4.1b"><apply id="A4.1.p1.72.m4.1.1.cmml" xref="A4.1.p1.72.m4.1.1"><csymbol cd="ambiguous" id="A4.1.p1.72.m4.1.1.1.cmml" xref="A4.1.p1.72.m4.1.1">subscript</csymbol><ci id="A4.1.p1.72.m4.1.1.2.cmml" xref="A4.1.p1.72.m4.1.1.2">𝜌</ci><cn id="A4.1.p1.72.m4.1.1.3.cmml" type="integer" xref="A4.1.p1.72.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.72.m4.1c">\rho_{2}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.72.m4.1d">italic_ρ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext=":=" class="ltx_Math" display="inline" id="A4.1.p1.73.m5.1"><semantics id="A4.1.p1.73.m5.1a"><mo id="A4.1.p1.73.m5.1.1" xref="A4.1.p1.73.m5.1.1.cmml">:=</mo><annotation-xml encoding="MathML-Content" id="A4.1.p1.73.m5.1b"><csymbol cd="latexml" id="A4.1.p1.73.m5.1.1.cmml" xref="A4.1.p1.73.m5.1.1">assign</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.73.m5.1c">:=</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.73.m5.1d">:=</annotation></semantics></math> <math alttext="(1+p)\bar{\epsilon}\bar{d}+\bar{d}^{2}/(2k_{\rm c}\mu)+\kappa((1+p){\bar{d}}_{% \rm g})^{2}/(4\sigma^{*}\mu)\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A4.1.p1.74.m6.4"><semantics id="A4.1.p1.74.m6.4a"><mrow id="A4.1.p1.74.m6.4.4" xref="A4.1.p1.74.m6.4.4.cmml"><mrow id="A4.1.p1.74.m6.4.4.4" xref="A4.1.p1.74.m6.4.4.4.cmml"><mrow id="A4.1.p1.74.m6.1.1.1.1" xref="A4.1.p1.74.m6.1.1.1.1.cmml"><mrow id="A4.1.p1.74.m6.1.1.1.1.1.1" xref="A4.1.p1.74.m6.1.1.1.1.1.1.1.cmml"><mo id="A4.1.p1.74.m6.1.1.1.1.1.1.2" stretchy="false" xref="A4.1.p1.74.m6.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.1.p1.74.m6.1.1.1.1.1.1.1" xref="A4.1.p1.74.m6.1.1.1.1.1.1.1.cmml"><mn id="A4.1.p1.74.m6.1.1.1.1.1.1.1.2" xref="A4.1.p1.74.m6.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A4.1.p1.74.m6.1.1.1.1.1.1.1.1" xref="A4.1.p1.74.m6.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="A4.1.p1.74.m6.1.1.1.1.1.1.1.3" xref="A4.1.p1.74.m6.1.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A4.1.p1.74.m6.1.1.1.1.1.1.3" stretchy="false" xref="A4.1.p1.74.m6.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.1.p1.74.m6.1.1.1.1.2" xref="A4.1.p1.74.m6.1.1.1.1.2.cmml">⁢</mo><mover accent="true" id="A4.1.p1.74.m6.1.1.1.1.3" xref="A4.1.p1.74.m6.1.1.1.1.3.cmml"><mi id="A4.1.p1.74.m6.1.1.1.1.3.2" xref="A4.1.p1.74.m6.1.1.1.1.3.2.cmml">ϵ</mi><mo id="A4.1.p1.74.m6.1.1.1.1.3.1" xref="A4.1.p1.74.m6.1.1.1.1.3.1.cmml">¯</mo></mover><mo id="A4.1.p1.74.m6.1.1.1.1.2a" xref="A4.1.p1.74.m6.1.1.1.1.2.cmml">⁢</mo><mover accent="true" id="A4.1.p1.74.m6.1.1.1.1.4" xref="A4.1.p1.74.m6.1.1.1.1.4.cmml"><mi id="A4.1.p1.74.m6.1.1.1.1.4.2" xref="A4.1.p1.74.m6.1.1.1.1.4.2.cmml">d</mi><mo id="A4.1.p1.74.m6.1.1.1.1.4.1" xref="A4.1.p1.74.m6.1.1.1.1.4.1.cmml">¯</mo></mover></mrow><mo id="A4.1.p1.74.m6.4.4.4.5" xref="A4.1.p1.74.m6.4.4.4.5.cmml">+</mo><mrow id="A4.1.p1.74.m6.2.2.2.2" xref="A4.1.p1.74.m6.2.2.2.2.cmml"><msup id="A4.1.p1.74.m6.2.2.2.2.3" xref="A4.1.p1.74.m6.2.2.2.2.3.cmml"><mover accent="true" id="A4.1.p1.74.m6.2.2.2.2.3.2" xref="A4.1.p1.74.m6.2.2.2.2.3.2.cmml"><mi id="A4.1.p1.74.m6.2.2.2.2.3.2.2" xref="A4.1.p1.74.m6.2.2.2.2.3.2.2.cmml">d</mi><mo id="A4.1.p1.74.m6.2.2.2.2.3.2.1" xref="A4.1.p1.74.m6.2.2.2.2.3.2.1.cmml">¯</mo></mover><mn id="A4.1.p1.74.m6.2.2.2.2.3.3" xref="A4.1.p1.74.m6.2.2.2.2.3.3.cmml">2</mn></msup><mo id="A4.1.p1.74.m6.2.2.2.2.2" xref="A4.1.p1.74.m6.2.2.2.2.2.cmml">/</mo><mrow id="A4.1.p1.74.m6.2.2.2.2.1.1" xref="A4.1.p1.74.m6.2.2.2.2.1.1.1.cmml"><mo id="A4.1.p1.74.m6.2.2.2.2.1.1.2" stretchy="false" xref="A4.1.p1.74.m6.2.2.2.2.1.1.1.cmml">(</mo><mrow id="A4.1.p1.74.m6.2.2.2.2.1.1.1" xref="A4.1.p1.74.m6.2.2.2.2.1.1.1.cmml"><mn id="A4.1.p1.74.m6.2.2.2.2.1.1.1.2" xref="A4.1.p1.74.m6.2.2.2.2.1.1.1.2.cmml">2</mn><mo id="A4.1.p1.74.m6.2.2.2.2.1.1.1.1" xref="A4.1.p1.74.m6.2.2.2.2.1.1.1.1.cmml">⁢</mo><msub id="A4.1.p1.74.m6.2.2.2.2.1.1.1.3" xref="A4.1.p1.74.m6.2.2.2.2.1.1.1.3.cmml"><mi id="A4.1.p1.74.m6.2.2.2.2.1.1.1.3.2" xref="A4.1.p1.74.m6.2.2.2.2.1.1.1.3.2.cmml">k</mi><mi id="A4.1.p1.74.m6.2.2.2.2.1.1.1.3.3" mathvariant="normal" xref="A4.1.p1.74.m6.2.2.2.2.1.1.1.3.3.cmml">c</mi></msub><mo id="A4.1.p1.74.m6.2.2.2.2.1.1.1.1a" xref="A4.1.p1.74.m6.2.2.2.2.1.1.1.1.cmml">⁢</mo><mi id="A4.1.p1.74.m6.2.2.2.2.1.1.1.4" xref="A4.1.p1.74.m6.2.2.2.2.1.1.1.4.cmml">μ</mi></mrow><mo id="A4.1.p1.74.m6.2.2.2.2.1.1.3" stretchy="false" xref="A4.1.p1.74.m6.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="A4.1.p1.74.m6.4.4.4.5a" xref="A4.1.p1.74.m6.4.4.4.5.cmml">+</mo><mrow id="A4.1.p1.74.m6.4.4.4.4" xref="A4.1.p1.74.m6.4.4.4.4.cmml"><mrow id="A4.1.p1.74.m6.3.3.3.3.1" xref="A4.1.p1.74.m6.3.3.3.3.1.cmml"><mi id="A4.1.p1.74.m6.3.3.3.3.1.3" xref="A4.1.p1.74.m6.3.3.3.3.1.3.cmml">κ</mi><mo id="A4.1.p1.74.m6.3.3.3.3.1.2" xref="A4.1.p1.74.m6.3.3.3.3.1.2.cmml">⁢</mo><msup id="A4.1.p1.74.m6.3.3.3.3.1.1" xref="A4.1.p1.74.m6.3.3.3.3.1.1.cmml"><mrow id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.cmml"><mo id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.2" stretchy="false" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.cmml">(</mo><mrow id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.cmml"><mrow id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.1.1" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.1.1.1.cmml"><mo id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.1.1.2" stretchy="false" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.1.1.1" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.1.1.1.cmml"><mn id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.1.1.1.2" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.1.1.1.1" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.1.1.1.3" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.1.1.3" stretchy="false" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.2" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.2.cmml">⁢</mo><msub id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.3" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.3.cmml"><mover accent="true" id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.3.2" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.3.2.cmml"><mi id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.3.2.2" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.3.2.2.cmml">d</mi><mo id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.3.2.1" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.3.2.1.cmml">¯</mo></mover><mi id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.3.3" mathvariant="normal" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.3.3.cmml">g</mi></msub></mrow><mo id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.3" stretchy="false" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.cmml">)</mo></mrow><mn id="A4.1.p1.74.m6.3.3.3.3.1.1.3" xref="A4.1.p1.74.m6.3.3.3.3.1.1.3.cmml">2</mn></msup></mrow><mo id="A4.1.p1.74.m6.4.4.4.4.3" xref="A4.1.p1.74.m6.4.4.4.4.3.cmml">/</mo><mrow id="A4.1.p1.74.m6.4.4.4.4.2.1" xref="A4.1.p1.74.m6.4.4.4.4.2.1.1.cmml"><mo id="A4.1.p1.74.m6.4.4.4.4.2.1.2" stretchy="false" xref="A4.1.p1.74.m6.4.4.4.4.2.1.1.cmml">(</mo><mrow id="A4.1.p1.74.m6.4.4.4.4.2.1.1" xref="A4.1.p1.74.m6.4.4.4.4.2.1.1.cmml"><mn id="A4.1.p1.74.m6.4.4.4.4.2.1.1.2" xref="A4.1.p1.74.m6.4.4.4.4.2.1.1.2.cmml">4</mn><mo id="A4.1.p1.74.m6.4.4.4.4.2.1.1.1" xref="A4.1.p1.74.m6.4.4.4.4.2.1.1.1.cmml">⁢</mo><msup id="A4.1.p1.74.m6.4.4.4.4.2.1.1.3" xref="A4.1.p1.74.m6.4.4.4.4.2.1.1.3.cmml"><mi id="A4.1.p1.74.m6.4.4.4.4.2.1.1.3.2" xref="A4.1.p1.74.m6.4.4.4.4.2.1.1.3.2.cmml">σ</mi><mo id="A4.1.p1.74.m6.4.4.4.4.2.1.1.3.3" xref="A4.1.p1.74.m6.4.4.4.4.2.1.1.3.3.cmml">∗</mo></msup><mo id="A4.1.p1.74.m6.4.4.4.4.2.1.1.1a" xref="A4.1.p1.74.m6.4.4.4.4.2.1.1.1.cmml">⁢</mo><mi id="A4.1.p1.74.m6.4.4.4.4.2.1.1.4" xref="A4.1.p1.74.m6.4.4.4.4.2.1.1.4.cmml">μ</mi></mrow><mo id="A4.1.p1.74.m6.4.4.4.4.2.1.3" stretchy="false" xref="A4.1.p1.74.m6.4.4.4.4.2.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A4.1.p1.74.m6.4.4.5" xref="A4.1.p1.74.m6.4.4.5.cmml">∈</mo><msup id="A4.1.p1.74.m6.4.4.6" xref="A4.1.p1.74.m6.4.4.6.cmml"><mi id="A4.1.p1.74.m6.4.4.6.2" xref="A4.1.p1.74.m6.4.4.6.2.cmml">ℝ</mi><mo id="A4.1.p1.74.m6.4.4.6.3" xref="A4.1.p1.74.m6.4.4.6.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.74.m6.4b"><apply id="A4.1.p1.74.m6.4.4.cmml" xref="A4.1.p1.74.m6.4.4"><in id="A4.1.p1.74.m6.4.4.5.cmml" xref="A4.1.p1.74.m6.4.4.5"></in><apply id="A4.1.p1.74.m6.4.4.4.cmml" xref="A4.1.p1.74.m6.4.4.4"><plus id="A4.1.p1.74.m6.4.4.4.5.cmml" xref="A4.1.p1.74.m6.4.4.4.5"></plus><apply id="A4.1.p1.74.m6.1.1.1.1.cmml" xref="A4.1.p1.74.m6.1.1.1.1"><times id="A4.1.p1.74.m6.1.1.1.1.2.cmml" xref="A4.1.p1.74.m6.1.1.1.1.2"></times><apply id="A4.1.p1.74.m6.1.1.1.1.1.1.1.cmml" xref="A4.1.p1.74.m6.1.1.1.1.1.1"><plus id="A4.1.p1.74.m6.1.1.1.1.1.1.1.1.cmml" xref="A4.1.p1.74.m6.1.1.1.1.1.1.1.1"></plus><cn id="A4.1.p1.74.m6.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A4.1.p1.74.m6.1.1.1.1.1.1.1.2">1</cn><ci id="A4.1.p1.74.m6.1.1.1.1.1.1.1.3.cmml" xref="A4.1.p1.74.m6.1.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="A4.1.p1.74.m6.1.1.1.1.3.cmml" xref="A4.1.p1.74.m6.1.1.1.1.3"><ci id="A4.1.p1.74.m6.1.1.1.1.3.1.cmml" xref="A4.1.p1.74.m6.1.1.1.1.3.1">¯</ci><ci id="A4.1.p1.74.m6.1.1.1.1.3.2.cmml" xref="A4.1.p1.74.m6.1.1.1.1.3.2">italic-ϵ</ci></apply><apply id="A4.1.p1.74.m6.1.1.1.1.4.cmml" xref="A4.1.p1.74.m6.1.1.1.1.4"><ci id="A4.1.p1.74.m6.1.1.1.1.4.1.cmml" xref="A4.1.p1.74.m6.1.1.1.1.4.1">¯</ci><ci id="A4.1.p1.74.m6.1.1.1.1.4.2.cmml" xref="A4.1.p1.74.m6.1.1.1.1.4.2">𝑑</ci></apply></apply><apply id="A4.1.p1.74.m6.2.2.2.2.cmml" xref="A4.1.p1.74.m6.2.2.2.2"><divide id="A4.1.p1.74.m6.2.2.2.2.2.cmml" xref="A4.1.p1.74.m6.2.2.2.2.2"></divide><apply id="A4.1.p1.74.m6.2.2.2.2.3.cmml" xref="A4.1.p1.74.m6.2.2.2.2.3"><csymbol cd="ambiguous" id="A4.1.p1.74.m6.2.2.2.2.3.1.cmml" xref="A4.1.p1.74.m6.2.2.2.2.3">superscript</csymbol><apply id="A4.1.p1.74.m6.2.2.2.2.3.2.cmml" xref="A4.1.p1.74.m6.2.2.2.2.3.2"><ci id="A4.1.p1.74.m6.2.2.2.2.3.2.1.cmml" xref="A4.1.p1.74.m6.2.2.2.2.3.2.1">¯</ci><ci id="A4.1.p1.74.m6.2.2.2.2.3.2.2.cmml" xref="A4.1.p1.74.m6.2.2.2.2.3.2.2">𝑑</ci></apply><cn id="A4.1.p1.74.m6.2.2.2.2.3.3.cmml" type="integer" xref="A4.1.p1.74.m6.2.2.2.2.3.3">2</cn></apply><apply id="A4.1.p1.74.m6.2.2.2.2.1.1.1.cmml" xref="A4.1.p1.74.m6.2.2.2.2.1.1"><times id="A4.1.p1.74.m6.2.2.2.2.1.1.1.1.cmml" xref="A4.1.p1.74.m6.2.2.2.2.1.1.1.1"></times><cn id="A4.1.p1.74.m6.2.2.2.2.1.1.1.2.cmml" type="integer" xref="A4.1.p1.74.m6.2.2.2.2.1.1.1.2">2</cn><apply id="A4.1.p1.74.m6.2.2.2.2.1.1.1.3.cmml" xref="A4.1.p1.74.m6.2.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.74.m6.2.2.2.2.1.1.1.3.1.cmml" xref="A4.1.p1.74.m6.2.2.2.2.1.1.1.3">subscript</csymbol><ci id="A4.1.p1.74.m6.2.2.2.2.1.1.1.3.2.cmml" xref="A4.1.p1.74.m6.2.2.2.2.1.1.1.3.2">𝑘</ci><ci id="A4.1.p1.74.m6.2.2.2.2.1.1.1.3.3.cmml" xref="A4.1.p1.74.m6.2.2.2.2.1.1.1.3.3">c</ci></apply><ci id="A4.1.p1.74.m6.2.2.2.2.1.1.1.4.cmml" xref="A4.1.p1.74.m6.2.2.2.2.1.1.1.4">𝜇</ci></apply></apply><apply id="A4.1.p1.74.m6.4.4.4.4.cmml" xref="A4.1.p1.74.m6.4.4.4.4"><divide id="A4.1.p1.74.m6.4.4.4.4.3.cmml" xref="A4.1.p1.74.m6.4.4.4.4.3"></divide><apply id="A4.1.p1.74.m6.3.3.3.3.1.cmml" xref="A4.1.p1.74.m6.3.3.3.3.1"><times id="A4.1.p1.74.m6.3.3.3.3.1.2.cmml" xref="A4.1.p1.74.m6.3.3.3.3.1.2"></times><ci id="A4.1.p1.74.m6.3.3.3.3.1.3.cmml" xref="A4.1.p1.74.m6.3.3.3.3.1.3">𝜅</ci><apply id="A4.1.p1.74.m6.3.3.3.3.1.1.cmml" xref="A4.1.p1.74.m6.3.3.3.3.1.1"><csymbol cd="ambiguous" id="A4.1.p1.74.m6.3.3.3.3.1.1.2.cmml" xref="A4.1.p1.74.m6.3.3.3.3.1.1">superscript</csymbol><apply id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.cmml" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1"><times id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.2.cmml" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.2"></times><apply id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.1.1.1.cmml" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.1.1"><plus id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.1.1.1.1.cmml" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.1.1.1.1"></plus><cn id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.1.1.1.2">1</cn><ci id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.1.1.1.3.cmml" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.3.cmml" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.3.1.cmml" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.3">subscript</csymbol><apply id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.3.2.cmml" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.3.2"><ci id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.3.2.1.cmml" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.3.2.1">¯</ci><ci id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.3.2.2.cmml" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.3.2.2">𝑑</ci></apply><ci id="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.3.3.cmml" xref="A4.1.p1.74.m6.3.3.3.3.1.1.1.1.1.3.3">g</ci></apply></apply><cn id="A4.1.p1.74.m6.3.3.3.3.1.1.3.cmml" type="integer" xref="A4.1.p1.74.m6.3.3.3.3.1.1.3">2</cn></apply></apply><apply id="A4.1.p1.74.m6.4.4.4.4.2.1.1.cmml" xref="A4.1.p1.74.m6.4.4.4.4.2.1"><times id="A4.1.p1.74.m6.4.4.4.4.2.1.1.1.cmml" xref="A4.1.p1.74.m6.4.4.4.4.2.1.1.1"></times><cn id="A4.1.p1.74.m6.4.4.4.4.2.1.1.2.cmml" type="integer" xref="A4.1.p1.74.m6.4.4.4.4.2.1.1.2">4</cn><apply id="A4.1.p1.74.m6.4.4.4.4.2.1.1.3.cmml" xref="A4.1.p1.74.m6.4.4.4.4.2.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.74.m6.4.4.4.4.2.1.1.3.1.cmml" xref="A4.1.p1.74.m6.4.4.4.4.2.1.1.3">superscript</csymbol><ci id="A4.1.p1.74.m6.4.4.4.4.2.1.1.3.2.cmml" xref="A4.1.p1.74.m6.4.4.4.4.2.1.1.3.2">𝜎</ci><times id="A4.1.p1.74.m6.4.4.4.4.2.1.1.3.3.cmml" xref="A4.1.p1.74.m6.4.4.4.4.2.1.1.3.3"></times></apply><ci id="A4.1.p1.74.m6.4.4.4.4.2.1.1.4.cmml" xref="A4.1.p1.74.m6.4.4.4.4.2.1.1.4">𝜇</ci></apply></apply></apply><apply id="A4.1.p1.74.m6.4.4.6.cmml" xref="A4.1.p1.74.m6.4.4.6"><csymbol cd="ambiguous" id="A4.1.p1.74.m6.4.4.6.1.cmml" xref="A4.1.p1.74.m6.4.4.6">superscript</csymbol><ci id="A4.1.p1.74.m6.4.4.6.2.cmml" xref="A4.1.p1.74.m6.4.4.6.2">ℝ</ci><plus id="A4.1.p1.74.m6.4.4.6.3.cmml" xref="A4.1.p1.74.m6.4.4.6.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.74.m6.4c">(1+p)\bar{\epsilon}\bar{d}+\bar{d}^{2}/(2k_{\rm c}\mu)+\kappa((1+p){\bar{d}}_{% \rm g})^{2}/(4\sigma^{*}\mu)\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.74.m6.4d">( 1 + italic_p ) over¯ start_ARG italic_ϵ end_ARG over¯ start_ARG italic_d end_ARG + over¯ start_ARG italic_d end_ARG start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( 2 italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT italic_μ ) + italic_κ ( ( 1 + italic_p ) over¯ start_ARG italic_d end_ARG start_POSTSUBSCRIPT roman_g end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( 4 italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_μ ) ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="p" class="ltx_Math" display="inline" id="A4.1.p1.75.m7.1"><semantics id="A4.1.p1.75.m7.1a"><mi id="A4.1.p1.75.m7.1.1" xref="A4.1.p1.75.m7.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="A4.1.p1.75.m7.1b"><ci id="A4.1.p1.75.m7.1.1.cmml" xref="A4.1.p1.75.m7.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.75.m7.1c">p</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.75.m7.1d">italic_p</annotation></semantics></math> <math alttext="=" class="ltx_Math" display="inline" id="A4.1.p1.76.m8.1"><semantics id="A4.1.p1.76.m8.1a"><mo id="A4.1.p1.76.m8.1.1" xref="A4.1.p1.76.m8.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="A4.1.p1.76.m8.1b"><eq id="A4.1.p1.76.m8.1.1.cmml" xref="A4.1.p1.76.m8.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.76.m8.1c">=</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.76.m8.1d">=</annotation></semantics></math> <math alttext="\delta^{2}/(2k_{\rm c}\kappa\sigma^{*})\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A4.1.p1.77.m9.1"><semantics id="A4.1.p1.77.m9.1a"><mrow id="A4.1.p1.77.m9.1.1" xref="A4.1.p1.77.m9.1.1.cmml"><mrow id="A4.1.p1.77.m9.1.1.1" xref="A4.1.p1.77.m9.1.1.1.cmml"><msup id="A4.1.p1.77.m9.1.1.1.3" xref="A4.1.p1.77.m9.1.1.1.3.cmml"><mi id="A4.1.p1.77.m9.1.1.1.3.2" xref="A4.1.p1.77.m9.1.1.1.3.2.cmml">δ</mi><mn id="A4.1.p1.77.m9.1.1.1.3.3" xref="A4.1.p1.77.m9.1.1.1.3.3.cmml">2</mn></msup><mo id="A4.1.p1.77.m9.1.1.1.2" xref="A4.1.p1.77.m9.1.1.1.2.cmml">/</mo><mrow id="A4.1.p1.77.m9.1.1.1.1.1" xref="A4.1.p1.77.m9.1.1.1.1.1.1.cmml"><mo id="A4.1.p1.77.m9.1.1.1.1.1.2" stretchy="false" xref="A4.1.p1.77.m9.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.1.p1.77.m9.1.1.1.1.1.1" xref="A4.1.p1.77.m9.1.1.1.1.1.1.cmml"><mn id="A4.1.p1.77.m9.1.1.1.1.1.1.2" xref="A4.1.p1.77.m9.1.1.1.1.1.1.2.cmml">2</mn><mo id="A4.1.p1.77.m9.1.1.1.1.1.1.1" xref="A4.1.p1.77.m9.1.1.1.1.1.1.1.cmml">⁢</mo><msub id="A4.1.p1.77.m9.1.1.1.1.1.1.3" xref="A4.1.p1.77.m9.1.1.1.1.1.1.3.cmml"><mi id="A4.1.p1.77.m9.1.1.1.1.1.1.3.2" xref="A4.1.p1.77.m9.1.1.1.1.1.1.3.2.cmml">k</mi><mi id="A4.1.p1.77.m9.1.1.1.1.1.1.3.3" mathvariant="normal" xref="A4.1.p1.77.m9.1.1.1.1.1.1.3.3.cmml">c</mi></msub><mo id="A4.1.p1.77.m9.1.1.1.1.1.1.1a" xref="A4.1.p1.77.m9.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="A4.1.p1.77.m9.1.1.1.1.1.1.4" xref="A4.1.p1.77.m9.1.1.1.1.1.1.4.cmml">κ</mi><mo id="A4.1.p1.77.m9.1.1.1.1.1.1.1b" xref="A4.1.p1.77.m9.1.1.1.1.1.1.1.cmml">⁢</mo><msup id="A4.1.p1.77.m9.1.1.1.1.1.1.5" xref="A4.1.p1.77.m9.1.1.1.1.1.1.5.cmml"><mi id="A4.1.p1.77.m9.1.1.1.1.1.1.5.2" xref="A4.1.p1.77.m9.1.1.1.1.1.1.5.2.cmml">σ</mi><mo id="A4.1.p1.77.m9.1.1.1.1.1.1.5.3" xref="A4.1.p1.77.m9.1.1.1.1.1.1.5.3.cmml">∗</mo></msup></mrow><mo id="A4.1.p1.77.m9.1.1.1.1.1.3" stretchy="false" xref="A4.1.p1.77.m9.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A4.1.p1.77.m9.1.1.2" xref="A4.1.p1.77.m9.1.1.2.cmml">∈</mo><msup id="A4.1.p1.77.m9.1.1.3" xref="A4.1.p1.77.m9.1.1.3.cmml"><mi id="A4.1.p1.77.m9.1.1.3.2" xref="A4.1.p1.77.m9.1.1.3.2.cmml">ℝ</mi><mo id="A4.1.p1.77.m9.1.1.3.3" xref="A4.1.p1.77.m9.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.77.m9.1b"><apply id="A4.1.p1.77.m9.1.1.cmml" xref="A4.1.p1.77.m9.1.1"><in id="A4.1.p1.77.m9.1.1.2.cmml" xref="A4.1.p1.77.m9.1.1.2"></in><apply id="A4.1.p1.77.m9.1.1.1.cmml" xref="A4.1.p1.77.m9.1.1.1"><divide id="A4.1.p1.77.m9.1.1.1.2.cmml" xref="A4.1.p1.77.m9.1.1.1.2"></divide><apply id="A4.1.p1.77.m9.1.1.1.3.cmml" xref="A4.1.p1.77.m9.1.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.77.m9.1.1.1.3.1.cmml" xref="A4.1.p1.77.m9.1.1.1.3">superscript</csymbol><ci id="A4.1.p1.77.m9.1.1.1.3.2.cmml" xref="A4.1.p1.77.m9.1.1.1.3.2">𝛿</ci><cn id="A4.1.p1.77.m9.1.1.1.3.3.cmml" type="integer" xref="A4.1.p1.77.m9.1.1.1.3.3">2</cn></apply><apply id="A4.1.p1.77.m9.1.1.1.1.1.1.cmml" xref="A4.1.p1.77.m9.1.1.1.1.1"><times id="A4.1.p1.77.m9.1.1.1.1.1.1.1.cmml" xref="A4.1.p1.77.m9.1.1.1.1.1.1.1"></times><cn id="A4.1.p1.77.m9.1.1.1.1.1.1.2.cmml" type="integer" xref="A4.1.p1.77.m9.1.1.1.1.1.1.2">2</cn><apply id="A4.1.p1.77.m9.1.1.1.1.1.1.3.cmml" xref="A4.1.p1.77.m9.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.77.m9.1.1.1.1.1.1.3.1.cmml" xref="A4.1.p1.77.m9.1.1.1.1.1.1.3">subscript</csymbol><ci id="A4.1.p1.77.m9.1.1.1.1.1.1.3.2.cmml" xref="A4.1.p1.77.m9.1.1.1.1.1.1.3.2">𝑘</ci><ci id="A4.1.p1.77.m9.1.1.1.1.1.1.3.3.cmml" xref="A4.1.p1.77.m9.1.1.1.1.1.1.3.3">c</ci></apply><ci id="A4.1.p1.77.m9.1.1.1.1.1.1.4.cmml" xref="A4.1.p1.77.m9.1.1.1.1.1.1.4">𝜅</ci><apply id="A4.1.p1.77.m9.1.1.1.1.1.1.5.cmml" xref="A4.1.p1.77.m9.1.1.1.1.1.1.5"><csymbol cd="ambiguous" id="A4.1.p1.77.m9.1.1.1.1.1.1.5.1.cmml" xref="A4.1.p1.77.m9.1.1.1.1.1.1.5">superscript</csymbol><ci id="A4.1.p1.77.m9.1.1.1.1.1.1.5.2.cmml" xref="A4.1.p1.77.m9.1.1.1.1.1.1.5.2">𝜎</ci><times id="A4.1.p1.77.m9.1.1.1.1.1.1.5.3.cmml" xref="A4.1.p1.77.m9.1.1.1.1.1.1.5.3"></times></apply></apply></apply><apply id="A4.1.p1.77.m9.1.1.3.cmml" xref="A4.1.p1.77.m9.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.77.m9.1.1.3.1.cmml" xref="A4.1.p1.77.m9.1.1.3">superscript</csymbol><ci id="A4.1.p1.77.m9.1.1.3.2.cmml" xref="A4.1.p1.77.m9.1.1.3.2">ℝ</ci><plus id="A4.1.p1.77.m9.1.1.3.3.cmml" xref="A4.1.p1.77.m9.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.77.m9.1c">\delta^{2}/(2k_{\rm c}\kappa\sigma^{*})\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.77.m9.1d">italic_δ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( 2 italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT italic_κ italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="\delta\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A4.1.p1.78.m10.1"><semantics id="A4.1.p1.78.m10.1a"><mrow id="A4.1.p1.78.m10.1.1" xref="A4.1.p1.78.m10.1.1.cmml"><mi id="A4.1.p1.78.m10.1.1.2" xref="A4.1.p1.78.m10.1.1.2.cmml">δ</mi><mo id="A4.1.p1.78.m10.1.1.1" xref="A4.1.p1.78.m10.1.1.1.cmml">∈</mo><msup id="A4.1.p1.78.m10.1.1.3" xref="A4.1.p1.78.m10.1.1.3.cmml"><mi id="A4.1.p1.78.m10.1.1.3.2" xref="A4.1.p1.78.m10.1.1.3.2.cmml">ℝ</mi><mo id="A4.1.p1.78.m10.1.1.3.3" xref="A4.1.p1.78.m10.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.78.m10.1b"><apply id="A4.1.p1.78.m10.1.1.cmml" xref="A4.1.p1.78.m10.1.1"><in id="A4.1.p1.78.m10.1.1.1.cmml" xref="A4.1.p1.78.m10.1.1.1"></in><ci id="A4.1.p1.78.m10.1.1.2.cmml" xref="A4.1.p1.78.m10.1.1.2">𝛿</ci><apply id="A4.1.p1.78.m10.1.1.3.cmml" xref="A4.1.p1.78.m10.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.78.m10.1.1.3.1.cmml" xref="A4.1.p1.78.m10.1.1.3">superscript</csymbol><ci id="A4.1.p1.78.m10.1.1.3.2.cmml" xref="A4.1.p1.78.m10.1.1.3.2">ℝ</ci><plus id="A4.1.p1.78.m10.1.1.3.3.cmml" xref="A4.1.p1.78.m10.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.78.m10.1c">\delta\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.78.m10.1d">italic_δ ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> satisfies <math alttext="\|\Phi-\Phi_{\rm f}\|" class="ltx_Math" display="inline" id="A4.1.p1.79.m11.1"><semantics id="A4.1.p1.79.m11.1a"><mrow id="A4.1.p1.79.m11.1.1.1" xref="A4.1.p1.79.m11.1.1.2.cmml"><mo id="A4.1.p1.79.m11.1.1.1.2" stretchy="false" xref="A4.1.p1.79.m11.1.1.2.1.cmml">‖</mo><mrow id="A4.1.p1.79.m11.1.1.1.1" xref="A4.1.p1.79.m11.1.1.1.1.cmml"><mi id="A4.1.p1.79.m11.1.1.1.1.2" mathvariant="normal" xref="A4.1.p1.79.m11.1.1.1.1.2.cmml">Φ</mi><mo id="A4.1.p1.79.m11.1.1.1.1.1" xref="A4.1.p1.79.m11.1.1.1.1.1.cmml">−</mo><msub id="A4.1.p1.79.m11.1.1.1.1.3" xref="A4.1.p1.79.m11.1.1.1.1.3.cmml"><mi id="A4.1.p1.79.m11.1.1.1.1.3.2" mathvariant="normal" xref="A4.1.p1.79.m11.1.1.1.1.3.2.cmml">Φ</mi><mi id="A4.1.p1.79.m11.1.1.1.1.3.3" mathvariant="normal" xref="A4.1.p1.79.m11.1.1.1.1.3.3.cmml">f</mi></msub></mrow><mo id="A4.1.p1.79.m11.1.1.1.3" stretchy="false" xref="A4.1.p1.79.m11.1.1.2.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.79.m11.1b"><apply id="A4.1.p1.79.m11.1.1.2.cmml" xref="A4.1.p1.79.m11.1.1.1"><csymbol cd="latexml" id="A4.1.p1.79.m11.1.1.2.1.cmml" xref="A4.1.p1.79.m11.1.1.1.2">norm</csymbol><apply id="A4.1.p1.79.m11.1.1.1.1.cmml" xref="A4.1.p1.79.m11.1.1.1.1"><minus id="A4.1.p1.79.m11.1.1.1.1.1.cmml" xref="A4.1.p1.79.m11.1.1.1.1.1"></minus><ci id="A4.1.p1.79.m11.1.1.1.1.2.cmml" xref="A4.1.p1.79.m11.1.1.1.1.2">Φ</ci><apply id="A4.1.p1.79.m11.1.1.1.1.3.cmml" xref="A4.1.p1.79.m11.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.79.m11.1.1.1.1.3.1.cmml" xref="A4.1.p1.79.m11.1.1.1.1.3">subscript</csymbol><ci id="A4.1.p1.79.m11.1.1.1.1.3.2.cmml" xref="A4.1.p1.79.m11.1.1.1.1.3.2">Φ</ci><ci id="A4.1.p1.79.m11.1.1.1.1.3.3.cmml" xref="A4.1.p1.79.m11.1.1.1.1.3.3">f</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.79.m11.1c">\|\Phi-\Phi_{\rm f}\|</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.79.m11.1d">∥ roman_Φ - roman_Φ start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT ∥</annotation></semantics></math> <math alttext="\leq" class="ltx_Math" display="inline" id="A4.1.p1.80.m12.1"><semantics id="A4.1.p1.80.m12.1a"><mo id="A4.1.p1.80.m12.1.1" xref="A4.1.p1.80.m12.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="A4.1.p1.80.m12.1b"><leq id="A4.1.p1.80.m12.1.1.cmml" xref="A4.1.p1.80.m12.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.80.m12.1c">\leq</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.80.m12.1d">≤</annotation></semantics></math> <math alttext="\delta" class="ltx_Math" display="inline" id="A4.1.p1.81.m13.1"><semantics id="A4.1.p1.81.m13.1a"><mi id="A4.1.p1.81.m13.1.1" xref="A4.1.p1.81.m13.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="A4.1.p1.81.m13.1b"><ci id="A4.1.p1.81.m13.1.1.cmml" xref="A4.1.p1.81.m13.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.81.m13.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.81.m13.1d">italic_δ</annotation></semantics></math>, <math alttext="\forall t\geq 0" class="ltx_Math" display="inline" id="A4.1.p1.82.m14.1"><semantics id="A4.1.p1.82.m14.1a"><mrow id="A4.1.p1.82.m14.1.1" xref="A4.1.p1.82.m14.1.1.cmml"><mrow id="A4.1.p1.82.m14.1.1.2" xref="A4.1.p1.82.m14.1.1.2.cmml"><mo id="A4.1.p1.82.m14.1.1.2.1" rspace="0.167em" xref="A4.1.p1.82.m14.1.1.2.1.cmml">∀</mo><mi id="A4.1.p1.82.m14.1.1.2.2" xref="A4.1.p1.82.m14.1.1.2.2.cmml">t</mi></mrow><mo id="A4.1.p1.82.m14.1.1.1" xref="A4.1.p1.82.m14.1.1.1.cmml">≥</mo><mn id="A4.1.p1.82.m14.1.1.3" xref="A4.1.p1.82.m14.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.82.m14.1b"><apply id="A4.1.p1.82.m14.1.1.cmml" xref="A4.1.p1.82.m14.1.1"><geq id="A4.1.p1.82.m14.1.1.1.cmml" xref="A4.1.p1.82.m14.1.1.1"></geq><apply id="A4.1.p1.82.m14.1.1.2.cmml" xref="A4.1.p1.82.m14.1.1.2"><csymbol cd="latexml" id="A4.1.p1.82.m14.1.1.2.1.cmml" xref="A4.1.p1.82.m14.1.1.2.1">for-all</csymbol><ci id="A4.1.p1.82.m14.1.1.2.2.cmml" xref="A4.1.p1.82.m14.1.1.2.2">𝑡</ci></apply><cn id="A4.1.p1.82.m14.1.1.3.cmml" type="integer" xref="A4.1.p1.82.m14.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.82.m14.1c">\forall t\geq 0</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.82.m14.1d">∀ italic_t ≥ 0</annotation></semantics></math>, and <math alttext="\sigma^{*}\in\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A4.1.p1.83.m15.1"><semantics id="A4.1.p1.83.m15.1a"><mrow id="A4.1.p1.83.m15.1.1" xref="A4.1.p1.83.m15.1.1.cmml"><msup id="A4.1.p1.83.m15.1.1.2" xref="A4.1.p1.83.m15.1.1.2.cmml"><mi id="A4.1.p1.83.m15.1.1.2.2" xref="A4.1.p1.83.m15.1.1.2.2.cmml">σ</mi><mo id="A4.1.p1.83.m15.1.1.2.3" xref="A4.1.p1.83.m15.1.1.2.3.cmml">∗</mo></msup><mo id="A4.1.p1.83.m15.1.1.1" xref="A4.1.p1.83.m15.1.1.1.cmml">∈</mo><msup id="A4.1.p1.83.m15.1.1.3" xref="A4.1.p1.83.m15.1.1.3.cmml"><mi id="A4.1.p1.83.m15.1.1.3.2" xref="A4.1.p1.83.m15.1.1.3.2.cmml">ℝ</mi><mo id="A4.1.p1.83.m15.1.1.3.3" xref="A4.1.p1.83.m15.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.83.m15.1b"><apply id="A4.1.p1.83.m15.1.1.cmml" xref="A4.1.p1.83.m15.1.1"><in id="A4.1.p1.83.m15.1.1.1.cmml" xref="A4.1.p1.83.m15.1.1.1"></in><apply id="A4.1.p1.83.m15.1.1.2.cmml" xref="A4.1.p1.83.m15.1.1.2"><csymbol cd="ambiguous" id="A4.1.p1.83.m15.1.1.2.1.cmml" xref="A4.1.p1.83.m15.1.1.2">superscript</csymbol><ci id="A4.1.p1.83.m15.1.1.2.2.cmml" xref="A4.1.p1.83.m15.1.1.2.2">𝜎</ci><times id="A4.1.p1.83.m15.1.1.2.3.cmml" xref="A4.1.p1.83.m15.1.1.2.3"></times></apply><apply id="A4.1.p1.83.m15.1.1.3.cmml" xref="A4.1.p1.83.m15.1.1.3"><csymbol cd="ambiguous" id="A4.1.p1.83.m15.1.1.3.1.cmml" xref="A4.1.p1.83.m15.1.1.3">superscript</csymbol><ci id="A4.1.p1.83.m15.1.1.3.2.cmml" xref="A4.1.p1.83.m15.1.1.3.2">ℝ</ci><plus id="A4.1.p1.83.m15.1.1.3.3.cmml" xref="A4.1.p1.83.m15.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.83.m15.1c">\sigma^{*}\in\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.83.m15.1d">italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="\sigma^{*}\leq\sigma" class="ltx_Math" display="inline" id="A4.1.p1.84.m16.1"><semantics id="A4.1.p1.84.m16.1a"><mrow id="A4.1.p1.84.m16.1.1" xref="A4.1.p1.84.m16.1.1.cmml"><msup id="A4.1.p1.84.m16.1.1.2" xref="A4.1.p1.84.m16.1.1.2.cmml"><mi id="A4.1.p1.84.m16.1.1.2.2" xref="A4.1.p1.84.m16.1.1.2.2.cmml">σ</mi><mo id="A4.1.p1.84.m16.1.1.2.3" xref="A4.1.p1.84.m16.1.1.2.3.cmml">∗</mo></msup><mo id="A4.1.p1.84.m16.1.1.1" xref="A4.1.p1.84.m16.1.1.1.cmml">≤</mo><mi id="A4.1.p1.84.m16.1.1.3" xref="A4.1.p1.84.m16.1.1.3.cmml">σ</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.84.m16.1b"><apply id="A4.1.p1.84.m16.1.1.cmml" xref="A4.1.p1.84.m16.1.1"><leq id="A4.1.p1.84.m16.1.1.1.cmml" xref="A4.1.p1.84.m16.1.1.1"></leq><apply id="A4.1.p1.84.m16.1.1.2.cmml" xref="A4.1.p1.84.m16.1.1.2"><csymbol cd="ambiguous" id="A4.1.p1.84.m16.1.1.2.1.cmml" xref="A4.1.p1.84.m16.1.1.2">superscript</csymbol><ci id="A4.1.p1.84.m16.1.1.2.2.cmml" xref="A4.1.p1.84.m16.1.1.2.2">𝜎</ci><times id="A4.1.p1.84.m16.1.1.2.3.cmml" xref="A4.1.p1.84.m16.1.1.2.3"></times></apply><ci id="A4.1.p1.84.m16.1.1.3.cmml" xref="A4.1.p1.84.m16.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.84.m16.1c">\sigma^{*}\leq\sigma</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.84.m16.1d">italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ≤ italic_σ</annotation></semantics></math> satisfies (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S5.E31" title="In V-C Robustness Results ‣ V Theoretical Guarantees ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">31</span></a>). <span class="ltx_text" id="A4.1.p1.95.11" style="color:#000099;">Thus, the equilibrium point <math alttext="(\bm{e}" class="ltx_math_unparsed" display="inline" id="A4.1.p1.85.1.m1.1"><semantics id="A4.1.p1.85.1.m1.1a"><mrow id="A4.1.p1.85.1.m1.1b"><mo id="A4.1.p1.85.1.m1.1.1" mathcolor="#000099" stretchy="false">(</mo><mi id="A4.1.p1.85.1.m1.1.2" mathcolor="#000099">𝒆</mi></mrow><annotation encoding="application/x-tex" id="A4.1.p1.85.1.m1.1c">(\bm{e}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.85.1.m1.1d">( bold_italic_e</annotation></semantics></math>, <math alttext="\tilde{\bm{\theta}})=\bm{0}" class="ltx_math_unparsed" display="inline" id="A4.1.p1.86.2.m2.1"><semantics id="A4.1.p1.86.2.m2.1a"><mrow id="A4.1.p1.86.2.m2.1b"><mover accent="true" id="A4.1.p1.86.2.m2.1.1"><mi id="A4.1.p1.86.2.m2.1.1.2" mathcolor="#000099">𝜽</mi><mo id="A4.1.p1.86.2.m2.1.1.1" mathcolor="#000099">~</mo></mover><mo id="A4.1.p1.86.2.m2.1.2" mathcolor="#000099" stretchy="false">)</mo><mo id="A4.1.p1.86.2.m2.1.3" mathcolor="#000099">=</mo><mn id="A4.1.p1.86.2.m2.1.4" mathcolor="#000099">𝟎</mn></mrow><annotation encoding="application/x-tex" id="A4.1.p1.86.2.m2.1c">\tilde{\bm{\theta}})=\bm{0}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.86.2.m2.1d">over~ start_ARG bold_italic_θ end_ARG ) = bold_0</annotation></semantics></math> of the closed-loop system (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S5.E26" title="In V-C Robustness Results ‣ V Theoretical Guarantees ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">26</span></a>) with (<a class="ltx_ref" href="https://arxiv.org/html/2401.10785v2#S4.E24" title="In IV-A Composite Learning High-Order Tuner ‣ IV Composite Learning Design ‣ Composite Learning Backstepping Control With Guaranteed Exponential Stability and Robustness"><span class="ltx_text ltx_ref_tag">24</span></a>) is practical exponential stable on <math alttext="t" class="ltx_Math" display="inline" id="A4.1.p1.87.3.m3.1"><semantics id="A4.1.p1.87.3.m3.1a"><mi id="A4.1.p1.87.3.m3.1.1" mathcolor="#000099" xref="A4.1.p1.87.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A4.1.p1.87.3.m3.1b"><ci id="A4.1.p1.87.3.m3.1.1.cmml" xref="A4.1.p1.87.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.87.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.87.3.m3.1d">italic_t</annotation></semantics></math> <math alttext="\in" class="ltx_Math" display="inline" id="A4.1.p1.88.4.m4.1"><semantics id="A4.1.p1.88.4.m4.1a"><mo id="A4.1.p1.88.4.m4.1.1" mathcolor="#000099" xref="A4.1.p1.88.4.m4.1.1.cmml">∈</mo><annotation-xml encoding="MathML-Content" id="A4.1.p1.88.4.m4.1b"><in id="A4.1.p1.88.4.m4.1.1.cmml" xref="A4.1.p1.88.4.m4.1.1"></in></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.88.4.m4.1c">\in</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.88.4.m4.1d">∈</annotation></semantics></math> <math alttext="[T_{\rm e},\infty)" class="ltx_Math" display="inline" id="A4.1.p1.89.5.m5.2"><semantics id="A4.1.p1.89.5.m5.2a"><mrow id="A4.1.p1.89.5.m5.2.2.1" xref="A4.1.p1.89.5.m5.2.2.2.cmml"><mo id="A4.1.p1.89.5.m5.2.2.1.2" mathcolor="#000099" stretchy="false" xref="A4.1.p1.89.5.m5.2.2.2.cmml">[</mo><msub id="A4.1.p1.89.5.m5.2.2.1.1" xref="A4.1.p1.89.5.m5.2.2.1.1.cmml"><mi id="A4.1.p1.89.5.m5.2.2.1.1.2" mathcolor="#000099" xref="A4.1.p1.89.5.m5.2.2.1.1.2.cmml">T</mi><mi id="A4.1.p1.89.5.m5.2.2.1.1.3" mathcolor="#000099" mathvariant="normal" xref="A4.1.p1.89.5.m5.2.2.1.1.3.cmml">e</mi></msub><mo id="A4.1.p1.89.5.m5.2.2.1.3" mathcolor="#000099" xref="A4.1.p1.89.5.m5.2.2.2.cmml">,</mo><mi id="A4.1.p1.89.5.m5.1.1" mathcolor="#000099" mathvariant="normal" xref="A4.1.p1.89.5.m5.1.1.cmml">∞</mi><mo id="A4.1.p1.89.5.m5.2.2.1.4" mathcolor="#000099" stretchy="false" xref="A4.1.p1.89.5.m5.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.89.5.m5.2b"><interval closure="closed-open" id="A4.1.p1.89.5.m5.2.2.2.cmml" xref="A4.1.p1.89.5.m5.2.2.1"><apply id="A4.1.p1.89.5.m5.2.2.1.1.cmml" xref="A4.1.p1.89.5.m5.2.2.1.1"><csymbol cd="ambiguous" id="A4.1.p1.89.5.m5.2.2.1.1.1.cmml" xref="A4.1.p1.89.5.m5.2.2.1.1">subscript</csymbol><ci id="A4.1.p1.89.5.m5.2.2.1.1.2.cmml" xref="A4.1.p1.89.5.m5.2.2.1.1.2">𝑇</ci><ci id="A4.1.p1.89.5.m5.2.2.1.1.3.cmml" xref="A4.1.p1.89.5.m5.2.2.1.1.3">e</ci></apply><infinity id="A4.1.p1.89.5.m5.1.1.cmml" xref="A4.1.p1.89.5.m5.1.1"></infinity></interval></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.89.5.m5.2c">[T_{\rm e},\infty)</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.89.5.m5.2d">[ italic_T start_POSTSUBSCRIPT roman_e end_POSTSUBSCRIPT , ∞ )</annotation></semantics></math>, where the tracking error <math alttext="\bm{e}(t)" class="ltx_Math" display="inline" id="A4.1.p1.90.6.m6.1"><semantics id="A4.1.p1.90.6.m6.1a"><mrow id="A4.1.p1.90.6.m6.1.2" xref="A4.1.p1.90.6.m6.1.2.cmml"><mi id="A4.1.p1.90.6.m6.1.2.2" mathcolor="#000099" xref="A4.1.p1.90.6.m6.1.2.2.cmml">𝒆</mi><mo id="A4.1.p1.90.6.m6.1.2.1" xref="A4.1.p1.90.6.m6.1.2.1.cmml">⁢</mo><mrow id="A4.1.p1.90.6.m6.1.2.3.2" xref="A4.1.p1.90.6.m6.1.2.cmml"><mo id="A4.1.p1.90.6.m6.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="A4.1.p1.90.6.m6.1.2.cmml">(</mo><mi id="A4.1.p1.90.6.m6.1.1" mathcolor="#000099" xref="A4.1.p1.90.6.m6.1.1.cmml">t</mi><mo id="A4.1.p1.90.6.m6.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="A4.1.p1.90.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.90.6.m6.1b"><apply id="A4.1.p1.90.6.m6.1.2.cmml" xref="A4.1.p1.90.6.m6.1.2"><times id="A4.1.p1.90.6.m6.1.2.1.cmml" xref="A4.1.p1.90.6.m6.1.2.1"></times><ci id="A4.1.p1.90.6.m6.1.2.2.cmml" xref="A4.1.p1.90.6.m6.1.2.2">𝒆</ci><ci id="A4.1.p1.90.6.m6.1.1.cmml" xref="A4.1.p1.90.6.m6.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.90.6.m6.1c">\bm{e}(t)</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.90.6.m6.1d">bold_italic_e ( italic_t )</annotation></semantics></math> and the estimation error <math alttext="\tilde{\bm{\theta}}(t)" class="ltx_Math" display="inline" id="A4.1.p1.91.7.m7.1"><semantics id="A4.1.p1.91.7.m7.1a"><mrow id="A4.1.p1.91.7.m7.1.2" xref="A4.1.p1.91.7.m7.1.2.cmml"><mover accent="true" id="A4.1.p1.91.7.m7.1.2.2" xref="A4.1.p1.91.7.m7.1.2.2.cmml"><mi id="A4.1.p1.91.7.m7.1.2.2.2" mathcolor="#000099" xref="A4.1.p1.91.7.m7.1.2.2.2.cmml">𝜽</mi><mo id="A4.1.p1.91.7.m7.1.2.2.1" mathcolor="#000099" xref="A4.1.p1.91.7.m7.1.2.2.1.cmml">~</mo></mover><mo id="A4.1.p1.91.7.m7.1.2.1" xref="A4.1.p1.91.7.m7.1.2.1.cmml">⁢</mo><mrow id="A4.1.p1.91.7.m7.1.2.3.2" xref="A4.1.p1.91.7.m7.1.2.cmml"><mo id="A4.1.p1.91.7.m7.1.2.3.2.1" mathcolor="#000099" stretchy="false" xref="A4.1.p1.91.7.m7.1.2.cmml">(</mo><mi id="A4.1.p1.91.7.m7.1.1" mathcolor="#000099" xref="A4.1.p1.91.7.m7.1.1.cmml">t</mi><mo id="A4.1.p1.91.7.m7.1.2.3.2.2" mathcolor="#000099" stretchy="false" xref="A4.1.p1.91.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.1.p1.91.7.m7.1b"><apply id="A4.1.p1.91.7.m7.1.2.cmml" xref="A4.1.p1.91.7.m7.1.2"><times id="A4.1.p1.91.7.m7.1.2.1.cmml" xref="A4.1.p1.91.7.m7.1.2.1"></times><apply id="A4.1.p1.91.7.m7.1.2.2.cmml" xref="A4.1.p1.91.7.m7.1.2.2"><ci id="A4.1.p1.91.7.m7.1.2.2.1.cmml" xref="A4.1.p1.91.7.m7.1.2.2.1">~</ci><ci id="A4.1.p1.91.7.m7.1.2.2.2.cmml" xref="A4.1.p1.91.7.m7.1.2.2.2">𝜽</ci></apply><ci id="A4.1.p1.91.7.m7.1.1.cmml" xref="A4.1.p1.91.7.m7.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.91.7.m7.1c">\tilde{\bm{\theta}}(t)</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.91.7.m7.1d">over~ start_ARG bold_italic_θ end_ARG ( italic_t )</annotation></semantics></math> exponentially converge to a small neighborhood of <math alttext="\bm{0}" class="ltx_Math" display="inline" id="A4.1.p1.92.8.m8.1"><semantics id="A4.1.p1.92.8.m8.1a"><mn id="A4.1.p1.92.8.m8.1.1" mathcolor="#000099" xref="A4.1.p1.92.8.m8.1.1.cmml">𝟎</mn><annotation-xml encoding="MathML-Content" id="A4.1.p1.92.8.m8.1b"><cn id="A4.1.p1.92.8.m8.1.1.cmml" type="integer" xref="A4.1.p1.92.8.m8.1.1">0</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.92.8.m8.1c">\bm{0}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.92.8.m8.1d">bold_0</annotation></semantics></math> dominated by <math alttext="k_{{\rm c}i}" class="ltx_Math" display="inline" id="A4.1.p1.93.9.m9.1"><semantics id="A4.1.p1.93.9.m9.1a"><msub id="A4.1.p1.93.9.m9.1.1" xref="A4.1.p1.93.9.m9.1.1.cmml"><mi id="A4.1.p1.93.9.m9.1.1.2" mathcolor="#000099" xref="A4.1.p1.93.9.m9.1.1.2.cmml">k</mi><mrow id="A4.1.p1.93.9.m9.1.1.3" xref="A4.1.p1.93.9.m9.1.1.3.cmml"><mi id="A4.1.p1.93.9.m9.1.1.3.2" mathcolor="#000099" mathvariant="normal" xref="A4.1.p1.93.9.m9.1.1.3.2.cmml">c</mi><mo id="A4.1.p1.93.9.m9.1.1.3.1" xref="A4.1.p1.93.9.m9.1.1.3.1.cmml">⁢</mo><mi id="A4.1.p1.93.9.m9.1.1.3.3" mathcolor="#000099" xref="A4.1.p1.93.9.m9.1.1.3.3.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A4.1.p1.93.9.m9.1b"><apply id="A4.1.p1.93.9.m9.1.1.cmml" xref="A4.1.p1.93.9.m9.1.1"><csymbol cd="ambiguous" id="A4.1.p1.93.9.m9.1.1.1.cmml" xref="A4.1.p1.93.9.m9.1.1">subscript</csymbol><ci id="A4.1.p1.93.9.m9.1.1.2.cmml" xref="A4.1.p1.93.9.m9.1.1.2">𝑘</ci><apply id="A4.1.p1.93.9.m9.1.1.3.cmml" xref="A4.1.p1.93.9.m9.1.1.3"><times id="A4.1.p1.93.9.m9.1.1.3.1.cmml" xref="A4.1.p1.93.9.m9.1.1.3.1"></times><ci id="A4.1.p1.93.9.m9.1.1.3.2.cmml" xref="A4.1.p1.93.9.m9.1.1.3.2">c</ci><ci id="A4.1.p1.93.9.m9.1.1.3.3.cmml" xref="A4.1.p1.93.9.m9.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.93.9.m9.1c">k_{{\rm c}i}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.93.9.m9.1d">italic_k start_POSTSUBSCRIPT roman_c italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\kappa" class="ltx_Math" display="inline" id="A4.1.p1.94.10.m10.1"><semantics id="A4.1.p1.94.10.m10.1a"><mi id="A4.1.p1.94.10.m10.1.1" mathcolor="#000099" xref="A4.1.p1.94.10.m10.1.1.cmml">κ</mi><annotation-xml encoding="MathML-Content" id="A4.1.p1.94.10.m10.1b"><ci id="A4.1.p1.94.10.m10.1.1.cmml" xref="A4.1.p1.94.10.m10.1.1">𝜅</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.94.10.m10.1c">\kappa</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.94.10.m10.1d">italic_κ</annotation></semantics></math>, and <math alttext="\bar{d}" class="ltx_Math" display="inline" id="A4.1.p1.95.11.m11.1"><semantics id="A4.1.p1.95.11.m11.1a"><mover accent="true" id="A4.1.p1.95.11.m11.1.1" xref="A4.1.p1.95.11.m11.1.1.cmml"><mi id="A4.1.p1.95.11.m11.1.1.2" mathcolor="#000099" xref="A4.1.p1.95.11.m11.1.1.2.cmml">d</mi><mo id="A4.1.p1.95.11.m11.1.1.1" mathcolor="#000099" xref="A4.1.p1.95.11.m11.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="A4.1.p1.95.11.m11.1b"><apply id="A4.1.p1.95.11.m11.1.1.cmml" xref="A4.1.p1.95.11.m11.1.1"><ci id="A4.1.p1.95.11.m11.1.1.1.cmml" xref="A4.1.p1.95.11.m11.1.1.1">¯</ci><ci id="A4.1.p1.95.11.m11.1.1.2.cmml" xref="A4.1.p1.95.11.m11.1.1.2">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.1.p1.95.11.m11.1c">\bar{d}</annotation><annotation encoding="application/x-llamapun" id="A4.1.p1.95.11.m11.1d">over¯ start_ARG italic_d end_ARG</annotation></semantics></math>.</span> ∎</p> </div> </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_tag_bibitem">[1]</span> <span class="ltx_bibblock"> G. Tao, “Multivariable adaptive control: A survey,” <em class="ltx_emph ltx_font_italic" id="bib.bib1.1.1">Automatica</em>, vol. 50, no. 11, pp. 2737–2764, Nov. 2014. </span> </li> <li class="ltx_bibitem" id="bib.bib2"> <span class="ltx_tag ltx_tag_bibitem">[2]</span> <span class="ltx_bibblock"> L. Chan, F. Naghdy, and D. Stirling, “Application of adaptive controllers in teleoperation systems: A survey,” <em class="ltx_emph ltx_font_italic" id="bib.bib2.1.1">IEEE Trans. Human-Mach. Syst.</em>, vol. 44, no. 3, pp. 337–352, Jun. 2014. </span> </li> <li class="ltx_bibitem" id="bib.bib3"> <span class="ltx_tag ltx_tag_bibitem">[3]</span> <span class="ltx_bibblock"> S. P. Nageshrao, G. A. D. Lopes, D. Jeltsema, and R. Babuška, “Port-hamiltonian systems in adaptive and learning control: A survey,” <em class="ltx_emph ltx_font_italic" id="bib.bib3.1.1">IEEE Trans. Autom. Control</em>, vol. 61, no. 5, pp. 1223–1238, May 2016. </span> </li> <li class="ltx_bibitem" id="bib.bib4"> <span class="ltx_tag ltx_tag_bibitem">[4]</span> <span class="ltx_bibblock"> A. M. Annaswamy and A. L. Fradkov, “A historical perspective of adaptive control and learning,” <em class="ltx_emph ltx_font_italic" id="bib.bib4.1.1">Annu. Rev. Control</em>, vol. 52, pp. 18–41, Dec. 2021. </span> </li> <li class="ltx_bibitem" id="bib.bib5"> <span class="ltx_tag ltx_tag_bibitem">[5]</span> <span class="ltx_bibblock"> R. Ortega, V. Nikiforov, and D. Gerasimov, “On modified parameter estimators for identification and adaptive control. A unified framework and some new schemes,” <em class="ltx_emph ltx_font_italic" id="bib.bib5.1.1">Annu. Rev. Control</em>, vol. 50, pp. 278–293, 2020. </span> </li> <li class="ltx_bibitem" id="bib.bib6"> <span class="ltx_tag ltx_tag_bibitem">[6]</span> <span class="ltx_bibblock"> K. Guo and Y. Pan, “Composite adaptation and learning for robot control: A survey,” <em class="ltx_emph ltx_font_italic" id="bib.bib6.1.1">Annu. Rev. Control</em>, vol. 55, pp. 279–290, Dec. 2023. </span> </li> <li class="ltx_bibitem" id="bib.bib7"> <span class="ltx_tag ltx_tag_bibitem">[7]</span> <span class="ltx_bibblock"> I. Kanellakopoulos, P. V. Kokotović, and A. S. Morse, “Systematic design of adaptive controllers for feedback linearizable systems,” <em class="ltx_emph ltx_font_italic" id="bib.bib7.1.1">IEEE Trans. Autom. Control</em>, vol. 36, no. 11, pp. 1241–1253, Nov. 1991. </span> </li> <li class="ltx_bibitem" id="bib.bib8"> <span class="ltx_tag ltx_tag_bibitem">[8]</span> <span class="ltx_bibblock"> M. Krstić, I. Kanellakopoulos, and P. V. Kokotović, <em class="ltx_emph ltx_font_italic" id="bib.bib8.1.1">Nonlilnear and Adaptive Control Design</em>. New York, NY: Wiley, 1995. </span> </li> <li class="ltx_bibitem" id="bib.bib9"> <span class="ltx_tag ltx_tag_bibitem">[9]</span> <span class="ltx_bibblock"> G. Tao and L. Wen, “Higher-order tracking properties of adaptive backstepping control systems,” <em class="ltx_emph ltx_font_italic" id="bib.bib9.1.1">Automatica</em>, vol. 153, Art. No. 111019, Jul. 2023. </span> </li> <li class="ltx_bibitem" id="bib.bib10"> <span class="ltx_tag ltx_tag_bibitem">[10]</span> <span class="ltx_bibblock"> Y. Pan, T. Sun, and H. Yu, “Composite adaptive dynamic surface control using online recorded data,” <em class="ltx_emph ltx_font_italic" id="bib.bib10.1.1">Int. J. Robust Nonlinear Control</em>, vol. 26, no. 18, pp. 3921–3936, Dec. 2016. </span> </li> <li class="ltx_bibitem" id="bib.bib11"> <span class="ltx_tag ltx_tag_bibitem">[11]</span> <span class="ltx_bibblock"> D. Wang, “Neural network-based adaptive dynamic surface control of uncertain nonlinear pure-feedback systems,” <em class="ltx_emph ltx_font_italic" id="bib.bib11.1.1">Int. J. Robust Nonlinear Control</em>, vol. 21, no. 5, pp. 527–541, Mar. 2011. </span> </li> <li class="ltx_bibitem" id="bib.bib12"> <span class="ltx_tag ltx_tag_bibitem">[12]</span> <span class="ltx_bibblock"> Z. Zhang, G. Duan, and M. Hou, “An improved adaptive dynamic surface control approach for uncertain nonlinear systems,” <em class="ltx_emph ltx_font_italic" id="bib.bib12.1.1">Int. J. Adapt. Control Signal Process.</em>, vol. 32, no. 5, pp. 713–728, May 2018. </span> </li> <li class="ltx_bibitem" id="bib.bib13"> <span class="ltx_tag ltx_tag_bibitem">[13]</span> <span class="ltx_bibblock"> S. Ling, H. Wang, and P. X. Liu, “Adaptive tracking control of high-order nonlinear systems under asymmetric output constraint,” <em class="ltx_emph ltx_font_italic" id="bib.bib13.1.1">Automatica</em>, vol. 122, Art. No. 109281, Dec. 2020. </span> </li> <li class="ltx_bibitem" id="bib.bib14"> <span class="ltx_tag ltx_tag_bibitem">[14]</span> <span class="ltx_bibblock"> K. Zhao, Y. Song, and Z. Zhang, “Tracking control of MIMO nonlinear systems under full state constraints: A single-parameter adaptation approach free from feasibility conditions,” <em class="ltx_emph ltx_font_italic" id="bib.bib14.1.1">Automatica</em>, vol. 107, pp. 52–60, Sep. 2019. </span> </li> <li class="ltx_bibitem" id="bib.bib15"> <span class="ltx_tag ltx_tag_bibitem">[15]</span> <span class="ltx_bibblock"> C. Bechlioulis and G. Rovithakis, “Reinforcing robustness of adaptive dynamic surface control,” <em class="ltx_emph ltx_font_italic" id="bib.bib15.1.1">Int. J. Adapt. Control Signal Process.</em>, vol. 27, no. 4, pp. 323–339, Apr. 2013. </span> </li> <li class="ltx_bibitem" id="bib.bib16"> <span class="ltx_tag ltx_tag_bibitem">[16]</span> <span class="ltx_bibblock"> J. A. Farrell, M. Polycarpou, M. Sharma, and W. Dong, “Command filtered backstepping,” <em class="ltx_emph ltx_font_italic" id="bib.bib16.1.1">IEEE Trans. Autom. Control</em>, vol. 54, no. 6, pp. 1391–1395, Jun. 2009. </span> </li> <li class="ltx_bibitem" id="bib.bib17"> <span class="ltx_tag ltx_tag_bibitem">[17]</span> <span class="ltx_bibblock"> J. Farrell, M. Sharma, and M. Polycarpou, “Backstepping-based flight control with adaptive function approximation,” <em class="ltx_emph ltx_font_italic" id="bib.bib17.1.1">J. Guid., Control, Dyn.</em>, vol. 28, no. 6, pp. 1089–1102, Nov.-Dec. 2005. </span> </li> <li class="ltx_bibitem" id="bib.bib18"> <span class="ltx_tag ltx_tag_bibitem">[18]</span> <span class="ltx_bibblock"> Y. Pan, Y. Liu, and H. Yu, “Online data-driven composite adaptive backstepping control with exact differentiators,” <em class="ltx_emph ltx_font_italic" id="bib.bib18.1.1">Int. J. Adapt. Control Signal Process.</em>, vol. 30, no. 5, pp. 779–789, May 2016. </span> </li> <li class="ltx_bibitem" id="bib.bib19"> <span class="ltx_tag ltx_tag_bibitem">[19]</span> <span class="ltx_bibblock"> J. Hu and H. Zhang, “Immersion and invariance based command-filtered adaptive backstepping control of VTOL vehicles,” <em class="ltx_emph ltx_font_italic" id="bib.bib19.1.1">Automatica</em>, vol. 49, no. 7, pp. 2160–2167, Jul. 2013. </span> </li> <li class="ltx_bibitem" id="bib.bib20"> <span class="ltx_tag ltx_tag_bibitem">[20]</span> <span class="ltx_bibblock"> Y. Pan and H. Yu, “Composite learning from adaptive dynamic surface control,” <em class="ltx_emph ltx_font_italic" id="bib.bib20.1.1">IEEE Trans. Autom. Control</em>, vol. 61, no. 9, pp. 2603–2609, Sep. 2016. </span> </li> <li class="ltx_bibitem" id="bib.bib21"> <span class="ltx_tag ltx_tag_bibitem">[21]</span> <span class="ltx_bibblock"> W. Wang and C. Wen, “Adaptive compensation for infinite number of actuator failures or faults,” <em class="ltx_emph ltx_font_italic" id="bib.bib21.1.1">Automatica</em>, vol. 47, no. 10, pp. 2197–2210, Oct. 2011. </span> </li> <li class="ltx_bibitem" id="bib.bib22"> <span class="ltx_tag ltx_tag_bibitem">[22]</span> <span class="ltx_bibblock"> A. S. Morse, “High-order parameter tuners for the adaptive control of linear and nonlinear systems,” in <em class="ltx_emph ltx_font_italic" id="bib.bib22.1.1">Systems, Models and Feedback: Theory and Applications</em>, A. Isidori and T.-J. Tarn, Eds., vol. 12. Boston, MA: Birkhäuser, 1992, pp. 339–364. </span> </li> <li class="ltx_bibitem" id="bib.bib23"> <span class="ltx_tag ltx_tag_bibitem">[23]</span> <span class="ltx_bibblock"> V. O. Nikiforov and K. V. Voronov, “Adaptive backstepping with a high-order tuner,” <em class="ltx_emph ltx_font_italic" id="bib.bib23.1.1">Automatica</em>, vol. 37, no. 12, pp. 1953–1960, Dec. 2001. </span> </li> <li class="ltx_bibitem" id="bib.bib24"> <span class="ltx_tag ltx_tag_bibitem">[24]</span> <span class="ltx_bibblock"> V. Nikiforov, D. Gerasimov, and A. Pashenko, “Modular adaptive backstepping design with a high-order tuner,” <em class="ltx_emph ltx_font_italic" id="bib.bib24.1.1">IEEE Trans. Autom. Control</em>, vol. 67, no. 5, pp. 2663–2668, May 2022. </span> </li> <li class="ltx_bibitem" id="bib.bib25"> <span class="ltx_tag ltx_tag_bibitem">[25]</span> <span class="ltx_bibblock"> S. Sastry and M. Bodson, <em class="ltx_emph ltx_font_italic" id="bib.bib25.1.1">Adaptive Control: Stability, Convergence and Robustness</em>. Englewood Cliffs, NJ, USA: Prentice Hall, 1989. </span> </li> <li class="ltx_bibitem" id="bib.bib26"> <span class="ltx_tag ltx_tag_bibitem">[26]</span> <span class="ltx_bibblock"> G. Kreisselmeier and G. Rietze-Augst, “Richness and excitation on an interval-with application to continuous-time adaptive control,” <em class="ltx_emph ltx_font_italic" id="bib.bib26.1.1">IEEE Trans. Autom. Control</em>, vol. 35, no. 2, pp. 165–171, Feb. 1990. </span> </li> <li class="ltx_bibitem" id="bib.bib27"> <span class="ltx_tag ltx_tag_bibitem">[27]</span> <span class="ltx_bibblock"> H. K. Khalil, <em class="ltx_emph ltx_font_italic" id="bib.bib27.1.1">Nonlinear Systems</em>, 3rd ed. Upper Saddle River, NJ, USA: Prentice Hall, 2002. </span> </li> <li class="ltx_bibitem" id="bib.bib28"> <span class="ltx_tag ltx_tag_bibitem">[28]</span> <span class="ltx_bibblock"> D. N. Gerasimov and V. O. Nikiforov, “On key properties of the Lion’s and Kreisselmeier’s adaptation algorithms,” <em class="ltx_emph ltx_font_italic" id="bib.bib28.1.1">IFAC-PapersOnLine</em>, vol. 53, no. 2, pp. 3773–3778, 2020. </span> </li> <li class="ltx_bibitem" id="bib.bib29"> <span class="ltx_tag ltx_tag_bibitem">[29]</span> <span class="ltx_bibblock"> L. Wang, R. Ortega, A. Bobtsov, J. G. Romero, and B. Yi, “Identifiability implies robust, globally exponentially convergent on-line parameter estimation,” <em class="ltx_emph ltx_font_italic" id="bib.bib29.1.1">Int. J. Control</em>, early access, Aug. 17, 2023, doi: 10.1080/00207179.2023.2246595. </span> </li> <li class="ltx_bibitem" id="bib.bib30"> <span class="ltx_tag ltx_tag_bibitem">[30]</span> <span class="ltx_bibblock"> Y. Pan and T. Shi, “Adaptive estimation and control with online data memory: A historical perspective,” <em class="ltx_emph ltx_font_italic" id="bib.bib30.1.1">IEEE Control Syst. Lett.</em>, vol. 8, pp. 267–278, 2024. </span> </li> <li class="ltx_bibitem" id="bib.bib31"> <span class="ltx_tag ltx_tag_bibitem">[31]</span> <span class="ltx_bibblock"> W. Dong, J. A. Farrell, M. M. Polycarpou, V. Djapic, and M. Sharma, “Command filtered adaptive backstepping,” <em class="ltx_emph ltx_font_italic" id="bib.bib31.1.1">IEEE Trans. Control Syst. Technol.</em>, vol. 20, no. 3, pp. 566–580, May 2012. </span> </li> <li class="ltx_bibitem" id="bib.bib32"> <span class="ltx_tag ltx_tag_bibitem">[32]</span> <span class="ltx_bibblock"> B. Lai, Z. Li, W. Li, C. Yang, and Y. Pan, “Homography-based visual servoing of eye-in-hand robots with exact depth estimation,” <em class="ltx_emph ltx_font_italic" id="bib.bib32.1.1">IEEE Trans. Ind. Electron.</em>, vol. 71, no. 4, pp. 3832–3841, Apr. 2024. </span> </li> <li class="ltx_bibitem" id="bib.bib33"> <span class="ltx_tag ltx_tag_bibitem">[33]</span> <span class="ltx_bibblock"> Y. Pan, Z. Li, T. Shi, and C. Wen, “Composite learning variable impedance robot control with stability and passivity guarantees,” <em class="ltx_emph ltx_font_italic" id="bib.bib33.1.1">IEEE Robot. Autom. Lett.</em>, vol. 9, no. 1, pp. 119–126, Jan. 2024. </span> </li> <li class="ltx_bibitem" id="bib.bib34"> <span class="ltx_tag ltx_tag_bibitem">[34]</span> <span class="ltx_bibblock"> Y. Pan, K. Guo, A. Bobtsov, C. Yang, and H. Yu, “Composite error learning robot control using discontinuous lyapunov analysis,” <em class="ltx_emph ltx_font_italic" id="bib.bib34.1.1">IEEE Trans. Autom. Control</em>, vol. 69, no. 3, pp. 1705–1712, 2024. </span> </li> <li class="ltx_bibitem" id="bib.bib35"> <span class="ltx_tag ltx_tag_bibitem">[35]</span> <span class="ltx_bibblock"> Z. Li, B. Lai, H. Wang, and Y. Pan, “Homography-based visual servoing of robot pose under an uncalibrated eye-to-hand camera,” <em class="ltx_emph ltx_font_italic" id="bib.bib35.1.1">IEEE/ASME Trans. Mechatron.</em>, vol. 29, no. 3, pp. 1891–1902, Jun. 2024. </span> </li> <li class="ltx_bibitem" id="bib.bib36"> <span class="ltx_tag ltx_tag_bibitem">[36]</span> <span class="ltx_bibblock"> Z. Li, B. Lai, and Y. Pan, “Image-based composite learning robot visual servoing with an uncalibrated eye-to-hand camera,” <em class="ltx_emph ltx_font_italic" id="bib.bib36.1.1">IEEE/ASME Trans. Mechatron.</em>, vol. 29, no. 4, pp. 2499–2509, Aug. 2024. </span> </li> <li class="ltx_bibitem" id="bib.bib37"> <span class="ltx_tag ltx_tag_bibitem">[37]</span> <span class="ltx_bibblock"> P. A. Ioannou and J. Sun, <em class="ltx_emph ltx_font_italic" id="bib.bib37.1.1">Robust Adaptive Control</em>. Englewood Cliffs, NJ, USA: Prentice Hall, 1996. </span> </li> <li class="ltx_bibitem" id="bib.bib38"> <span class="ltx_tag ltx_tag_bibitem">[38]</span> <span class="ltx_bibblock"> Y. Pan and H. Yu, “Composite learning robot control with guaranteed parameter convergence,” <em class="ltx_emph ltx_font_italic" id="bib.bib38.1.1">Automatica</em>, vol. 89, pp. 398–406, 2018. </span> </li> <li class="ltx_bibitem" id="bib.bib39"> <span class="ltx_tag ltx_tag_bibitem">[39]</span> <span class="ltx_bibblock"> V. Adetola and M. Guay, “Performance improvement in adaptive control of linearly parameterized nonlinear systems,” <em class="ltx_emph ltx_font_italic" id="bib.bib39.1.1">IEEE Trans. Autom. Control</em>, vol. 55, no. 9, pp. 2182–2186, Sep. 2010. </span> </li> <li class="ltx_bibitem" id="bib.bib40"> <span class="ltx_tag ltx_tag_bibitem">[40]</span> <span class="ltx_bibblock"> J. A. Farrel and M. M. Polycarpou, <em class="ltx_emph ltx_font_italic" id="bib.bib40.1.1">Adaptive Approximation Based Control: Unifying Neural, Fuzzy and Traditional Adaptive Approximation Approaches</em>. Hoboken, NJ, USA: Wiley, 2006. </span> </li> <li class="ltx_bibitem" id="bib.bib41"> <span class="ltx_tag ltx_tag_bibitem">[41]</span> <span class="ltx_bibblock"> H. K. Khalil, <em class="ltx_emph ltx_font_italic" id="bib.bib41.1.1">Nonlinear Control</em>. Upper Saddle River, NJ, USA: Prentice Hall, 2015. </span> </li> </ul> </section> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Tue Mar 18 17:40:33 2025 by <a class="ltx_LaTeXML_logo" href="http://dlmf.nist.gov/LaTeXML/"><span style="letter-spacing:-0.2em; margin-right:0.1em;">L<span class="ltx_font_smallcaps" style="position:relative; bottom:2.2pt;">a</span>T<span class="ltx_font_smallcaps" style="font-size:120%;position:relative; bottom:-0.2ex;">e</span></span><span style="font-size:90%; position:relative; bottom:-0.2ex;">XML</span><img alt="Mascot Sammy" src="data:image/png;base64,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"/></a> </div></footer> </div> </body> </html>