Document Zbl 1162.34004

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Differ. Equations</a> <a href="/?q=in%3A232486" title="Articles in this Issue">245, No. 8, 2078-2102 (2008)</a>. </div> <div class="abstract">The authors consider time-varying linear implicit differential equations <br class="zbmathjax-paragraph">\[ E(t)\dot{x}(t)=A(t)x(t),\quad t\geq 0, \]<br class="zbmathjax-paragraph">which are strongly index one, i.e.<span class="zbmathjax-space"></span>the equation can be decoupled into an explicite differential equation and an algebraic equation, furthermore, certain transformation matrices are assumed to be bounded. For this class of equations the stability radius with respect to structured dynamic perturbations of the matrix $A$ and the Bohl exponent are defined and studied. It follows that the zero solution of the above equation is exponentially stable if, and only if, the Bohl exponent is negative and it is shown that exponential stability is equivalent to the property that the inhomogeneous equation <br class="zbmathjax-paragraph">\[ E(t)\dot{x}(t)=A(t)x(t)+f(t),\quad t\geq 0, \]<br class="zbmathjax-paragraph">has a “bounded-input-bounded-output” property. Finally, the dependency of the stability radius and of the Bohl exponent on additive changes in the matrix $A$ are studied. The authors obtain “continuity”-results, i.e.<span class="zbmathjax-space"></span>sufficiently small disturbances in the matrix $A$ will not change the stability radius and the Bohl exponent significantly.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A11138">Stephan Trenn (Ilmenau)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5354957">Cited in <strong>23</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A34A09" title="MSC2020">34A09</a> </td> <td class="space"> Implicit ordinary differential equations, differential-algebraic equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34D20" title="MSC2020">34D20</a> </td> <td class="space"> Stability of solutions to ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34D10" title="MSC2020">34D10</a> </td> <td class="space"> Perturbations of ordinary differential equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aexponential+stability">exponential stability</a>; <a href="/?q=ut%3ABohl+exponent">Bohl exponent</a>; <a href="/?q=ut%3Astability+radii">stability radii</a>; <a href="/?q=ut%3Adifferential-algebraic+equations">differential-algebraic equations</a>; <a href="/?q=ut%3Adata-dependence">data-dependence</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1162.34004" data-ciurl="/ci/05354957" data-biburl="/bibtex/05354957.bib" data-amsurl="/amsrefs/05354957.bib" data-xmlurl="/xml/05354957.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05354957.pdf" title="Zbl 1162.34004 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.jde.2008.07.016" aria-label="DOI for “On data-dependence of exponential stability and stability radii for linear time-varying differential-algebraic systems”" title="10.1016/j.jde.2008.07.016">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Ascher, U. M.; Petzold, L. R., Stability of computation for constrained dynamical systems, SIAM J. Sci. Statist. Comput., 14, 95-120 (1993) · <a href="/0773.65044" class="nowrap">Zbl 0773.65044</a></td> </tr><tr> <td>[2]</td> <td class="space">Balla, K.; März, R., Linear differential algebraic equations of index 1 and their adjoint equations, Results Math., 37, 1-2, 13-35 (2000) · <a href="/0963.34001" class="nowrap">Zbl 0963.34001</a></td> </tr><tr> <td>[3]</td> <td class="space">Beckner, W., Inequalities in Fourier analysis, Ann. of Math. (2), 102, 1, 159-182 (1975) · <a href="/0338.42017" class="nowrap">Zbl 0338.42017</a></td> </tr><tr> <td>[4]</td> <td class="space">M. Bracke, On stability radii of parametrized linear differential-algebraic systems, PhD thesis, University of Kaiserslautern, 2000; M. Bracke, On stability radii of parametrized linear differential-algebraic systems, PhD thesis, University of Kaiserslautern, 2000</td> </tr><tr> <td>[5]</td> <td class="space">Brenan, K. E.; Campbell, S. L.; Petzold, L. R., Numerical Solution of Initial Value Problems in Differential Algebraic Equations (1996), SIAM: SIAM Philadelphia · <a href="/0844.65058" class="nowrap">Zbl 0844.65058</a></td> </tr><tr> <td>[6]</td> <td class="space">Byers, R.; Nichols, N., On the stability radius of a generalized state-space system, Linear Algebra Appl., 188/189, 113-134 (1993) · <a href="/0783.65056" class="nowrap">Zbl 0783.65056</a></td> </tr><tr> <td>[7]</td> <td class="space">Cong, N. D.; Nam, H., Lyapunov inequality for differential-algebraic equations, Acta Math. Vietnam., 28, 1, 73-88 (2003) · <a href="/1048.34089" class="nowrap">Zbl 1048.34089</a></td> </tr><tr> <td>[8]</td> <td class="space">Cong, N. D.; Nam, H., Lyapunov’s regularity for linear differential-algebraic equations of index-1, Acta Math. Vietnam., 29, 1, 1-21 (2004) · <a href="/1080.34039" class="nowrap">Zbl 1080.34039</a></td> </tr><tr> <td>[9]</td> <td class="space">Daleckii, J. L.; Krein, M. G., Stability of Solutions of Differential Equations in Banach Spaces (1974), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · <a href="/0286.34094" class="nowrap">Zbl 0286.34094</a></td> </tr><tr> <td>[10]</td> <td class="space">Du, N. H., Stability radii for differential-algebraic equations, Vietnam J. Math., 27, 379-382 (1999) · <a href="/0971.34002" class="nowrap">Zbl 0971.34002</a></td> </tr><tr> <td>[11]</td> <td class="space">Du, N. H.; Linh, V. H., Robust stability of implicit linear systems containing a small parameter in the leading term, IMA J. Math. Control Inform., 23, 67-84 (2006) · <a href="/1106.93044" class="nowrap">Zbl 1106.93044</a></td> </tr><tr> <td>[12]</td> <td class="space">Du, N. H.; Linh, V. H., Stability radii for linear time-varying differential-algebraic equations with respect to dynamic perturbation, J. Differential Equations, 230, 579-599 (2006) · <a href="/1113.34002" class="nowrap">Zbl 1113.34002</a></td> </tr><tr> <td>[13]</td> <td class="space">Fridman, E., Stability of linear descriptor systems with delay: A Lyapunov-based approach, J. Math. Anal. Appl., 273, 24-44 (2002) · <a href="/1032.34069" class="nowrap">Zbl 1032.34069</a></td> </tr><tr> <td>[14]</td> <td class="space">Griepentrog, E.; März, R., Differential-Algebraic Equations and Their Numerical Treatment (1986), Teubner-Texte zur Mathematik: Teubner-Texte zur Mathematik Leipzig · <a href="/0629.65080" class="nowrap">Zbl 0629.65080</a></td> </tr><tr> <td>[15]</td> <td class="space">Higueras, I.; März, R.; Tischendorf, C., Stability preserving integration of index-1 DAEs, Appl. Numer. Math., 45, 175-200 (2003) · <a href="/1041.65065" class="nowrap">Zbl 1041.65065</a></td> </tr><tr> <td>[16]</td> <td class="space">Hinrichsen, D.; Ilchmann, A.; Pritchard, A. J., Robustness of stability of time-varying linear systems, J. Differential Equations, 82, 219-250 (1989) · <a href="/0702.34048" class="nowrap">Zbl 0702.34048</a></td> </tr><tr> <td>[17]</td> <td class="space">Hinrichsen, D.; Pritchard, A. J., Stability radius for structured perturbations and the algebraic Riccati equation, Systems Control Lett., 8, 105-113 (1986) · <a href="/0626.93054" class="nowrap">Zbl 0626.93054</a></td> </tr><tr> <td>[18]</td> <td class="space">Hinrichsen, D.; Pritchard, A. J., A note on some difference between real and complex stability radii, Systems Control Lett., 14, 401-408 (1990) · <a href="/0698.93067" class="nowrap">Zbl 0698.93067</a></td> </tr><tr> <td>[19]</td> <td class="space">Hinrichsen, D.; Pritchard, A. J., Destabilization by output feedback, Differential Integral Equations, 5, 2, 357-386 (1992) · <a href="/0763.93068" class="nowrap">Zbl 0763.93068</a></td> </tr><tr> <td>[20]</td> <td class="space">Ilchmann, A.; Mareels, I. M.Y., On stability radii of slowly time-varying systems, (Advances in Mathematical System Theory (2001), Birkhäuser: Birkhäuser Boston), 55-75</td> </tr><tr> <td>[21]</td> <td class="space">Jacob, B., A formula for the stability radius of time-varying systems, J. Differential Equations, 142, 167-187 (1998) · <a href="/0902.34039" class="nowrap">Zbl 0902.34039</a></td> </tr><tr> <td>[22]</td> <td class="space">Kunkel, P.; Mehrmann, V., Differential-Algebraic Equations. Analysis and Numerical Solution (2006), EMS Publishing House: EMS Publishing House Zürich, Switzerland · <a href="/0707.65043" class="nowrap">Zbl 0707.65043</a></td> </tr><tr> <td>[23]</td> <td class="space">Kunkel, P.; Mehrmann, V., Stability properties of differential-algebraic equations and spin-stabilized discretization, Electron. Trans. Numer. Anal., 26, 385-420 (2007) · <a href="/1171.65418" class="nowrap">Zbl 1171.65418</a></td> </tr><tr> <td>[24]</td> <td class="space">Lamour, R.; März, R.; Winkler, R., How Floquet theory applies to differential-algebraic equations, J. Math. Anal. Appl., 217, 372-394 (1998) · <a href="/0903.34002" class="nowrap">Zbl 0903.34002</a></td> </tr><tr> <td>[25]</td> <td class="space">V.H. Linh, V. Mehrmann, Spectral intervals for differential algebraic equations and their numerical approximations, preprint 402, DFG Research Center <span class="zbmathjax-textsc">Matheon</span>http://www.matheon.de/; V.H. Linh, V. Mehrmann, Spectral intervals for differential algebraic equations and their numerical approximations, preprint 402, DFG Research Center <span class="zbmathjax-textsc">Matheon</span>http://www.matheon.de/</td> </tr><tr> <td>[26]</td> <td class="space">März, R., Numerical methods for differential-algebraic equations, Acta Numer., 141-198 (1992) · <a href="/0769.65044" class="nowrap">Zbl 0769.65044</a></td> </tr><tr> <td>[27]</td> <td class="space">März, R., Practical Lyapunov stability criteria for differential algebraic equations, Banach Center Publ., 29, 245-266 (1994) · <a href="/0801.34004" class="nowrap">Zbl 0801.34004</a></td> </tr><tr> <td>[28]</td> <td class="space">März, R., Criteria for the trivial solution of differential algebraic equations with small nonlinearities to be asymptotically stable, J. Math. Anal. Appl., 225, 587-607 (1998) · <a href="/0955.34003" class="nowrap">Zbl 0955.34003</a></td> </tr><tr> <td>[29]</td> <td class="space">Qiu, L.; Benhardson, B.; Rantzer, A.; Davison, E. J.; Young, P. M.; Doyle, J. C., A formula for computation of the real stability radius, Automatica, 31, 879-890 (1995) · <a href="/0839.93039" class="nowrap">Zbl 0839.93039</a></td> </tr><tr> <td>[30]</td> <td class="space">Qiu, L.; Davison, E. J., The stability robustness of generalized eigenvalues, IEEE Trans. Automat. Control, 37, 886-891 (1992) · <a href="/0775.93181" class="nowrap">Zbl 0775.93181</a></td> </tr><tr> <td>[31]</td> <td class="space">Stykel, T., On criteria for asymptotic stability of differential-algebraic equations, Z. Angew. Math. Mech., 92, 147-158 (2002) · <a href="/1014.34037" class="nowrap">Zbl 1014.34037</a></td> </tr><tr> <td>[32]</td> <td class="space">Tischendorf, C., On stability of solutions of autonomous index-1 tractable and quasilinear index-2 tractable DAE’s, Circuits Systems Signal Process., 13, 139-154 (1994) · <a href="/0801.34005" class="nowrap">Zbl 0801.34005</a></td> </tr><tr> <td>[33]</td> <td class="space">Wirth, F.; Hinrichsen, D., On stability radii of infinite-dimensional time-varying discrete-time systems, IMA J. Math. Control Inform., 11, 3, 253-276 (1994) · <a href="/0812.93055" class="nowrap">Zbl 0812.93055</a></td> </tr><tr> <td>[34]</td> <td class="space">Xu, S.; Van Dooren, P.; Radu, S.; Lam, J., Robust stability and stabilization for singular systems with state delay and parameter uncertainty, IEEE Trans. Automat. Control, 47, 1122-1128 (2002) · <a href="/1364.93723" class="nowrap">Zbl 1364.93723</a></td> </tr><tr> <td>[35]</td> <td class="space">Zhu, W.; Petzold, L., Asymptotic stability of linear delay differential algebraic equations and numerical methods, Appl. Numer. Math., 24, 247-264 (1997) · <a href="/0879.65060" class="nowrap">Zbl 0879.65060</a></td> </tr><tr> <td>[36]</td> <td class="space">Zhu, W.; Petzold, L., Asymptotic stability of Hessenberg delay differential-algebraic equations of retarded or neutral type, Appl. Numer. Math., 27, 309-325 (1998) · <a href="/0937.34059" class="nowrap">Zbl 0937.34059</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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