Document Zbl 1100.93048

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Control Lett.</a> <a href="/?q=in%3A185914" title="Articles in this Issue">55, No. 10, 826-834 (2006)</a>. </div> <div class="abstract">Summary: Neutral stochastic differential delay equations (NSDDEs) have recently been studied intensively (see e.g. <span class="zbmathjax-textit">V.B. Kolmanovskii, V.R. Nosov</span> [Stability and periodic regimes of control systems with aftereffect, Nauka, Moscow (1981; <a href="/0457.93002">Zbl 0457.93002</a>)] and <span class="zbmathjax-textit">X. Mao</span> [Syst. Control Lett. 26, 245–251 (1995; <a href="/0877.93133">Zbl 0877.93133</a> ); SIAM J. Math. Anal. 28, No. 2, 389–401 (1997; <a href="/0876.60047">Zbl 0876.60047</a>); Stochastic differential equations and their applications, Horwood Publishing, Chichester (1997; <a href="/0892.60057">Zbl 0892.60057</a>)]). More recently, <span class="zbmathjax-textit">X. Mao</span> [Asymptotic properties of neutral stochastic differential delay equations, Stochastics Stochastics Rep. 68, No. 3-4, 273–295 (2000; <a href="/0964.60066">Zbl 0964.60066</a>)] provided with some useful criteria on the exponential stability for NSDDEs. However, the criteria there require not only the coefficients of the NSDDEs to obey the linear growth condition but also the time delay to be a constant. One of our aims in this paper is to remove these two restrictive conditions. Moreover, the key condition on the diffusion operator associated with the underlying NSDDE will take a much more general form. Our new stability criteria not only cover many highly non-linear NSDDEs with variable time delays but they can also be verified much more easily than the known criteria.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5068281">Cited in <strong>76</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A93E15" title="MSC2020">93E15</a> </td> <td class="space"> Stochastic stability in control theory </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3ABrownian+motion">Brownian motion</a>; <a href="/?q=ut%3AIt%C3%B4%27s+formula">Itô’s formula</a>; <a href="/?q=ut%3Amartingale">martingale</a>; <a href="/?q=ut%3Aexponential+stability">exponential stability</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0457.93002">Zbl 0457.93002</a>; <a href="/0877.93133">Zbl 0877.93133</a>; <a href="/0964.60066">Zbl 0964.60066</a>; <a href="/0876.60047">Zbl 0876.60047</a>; <a href="/0892.60057">Zbl 0892.60057</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 1100.93048" data-ciurl="/ci/05068281" data-biburl="/bibtex/05068281.bib" data-amsurl="/amsrefs/05068281.bib" data-xmlurl="/xml/05068281.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05068281.pdf" title="Zbl 1100.93048 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.sysconle.2006.04.005" aria-label="DOI for “New criteria on exponential stability of neutral stochastic differential delay equations”" title="10.1016/j.sysconle.2006.04.005">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">Brayton, R., Nonlinear oscillations in a distributed network, Quart. Appl. Math., 24, 289-301 (1976) · <a href="/0166.35102" class="nowrap">Zbl 0166.35102</a></td> </tr><tr> <td>[2]</td> <td class="space">Hale, J. K.; Lunel, S. M.V., Introduction to Functional Differential Equations (1993), Springer: Springer Berlin · <a href="/0787.34002" class="nowrap">Zbl 0787.34002</a></td> </tr><tr> <td>[3]</td> <td class="space">Has’minskii, R. Z., Stochastic Stability of Differential Equations (1981), Sijthoff and Noordhoff · <a href="/0276.60059" class="nowrap">Zbl 0276.60059</a></td> </tr><tr> <td>[4]</td> <td class="space">Kolmanovskii, V. B.; Nosov, V. R., Stability and Periodic Modes of Control Systems with Aftereffect (1981), Nauka: Nauka Moscow · <a href="/0457.93002" class="nowrap">Zbl 0457.93002</a></td> </tr><tr> <td>[5]</td> <td class="space">Ladde, G. S.; Lakshmikantham, V., Random Differential Inequalities (1980), Academic Press: Academic Press New York · <a href="/0466.60002" class="nowrap">Zbl 0466.60002</a></td> </tr><tr> <td>[6]</td> <td class="space">Lipster, R. Sh.; Shiryayev, A. N., Theory of Martingales (1989), Kluwer Academic Publisher: Kluwer Academic Publisher Dordrecht · <a href="/0728.60048" class="nowrap">Zbl 0728.60048</a></td> </tr><tr> <td>[7]</td> <td class="space">Mao, X., Stability of Stochastic Differential Equations with Respect to Semimartingales (1991), Longman Scientific and Technical · <a href="/0724.60059" class="nowrap">Zbl 0724.60059</a></td> </tr><tr> <td>[8]</td> <td class="space">Mao, X., Exponential Stability of Stochastic Differential Equations (1994), Marcel Dekker: Marcel Dekker New York · <a href="/0851.93074" class="nowrap">Zbl 0851.93074</a></td> </tr><tr> <td>[9]</td> <td class="space">Mao, X., Exponential stability in mean square of neutral stochastic differential functional equations, Systems Control Lett., 26, 245-251 (1995) · <a href="/0877.93133" class="nowrap">Zbl 0877.93133</a></td> </tr><tr> <td>[10]</td> <td class="space">Mao, X., Razumikhin type theorems on exponential stability of neutral stochastic functional differential equations, SIAM J. Math. Anal., 28, 2, 389-401 (1997) · <a href="/0876.60047" class="nowrap">Zbl 0876.60047</a></td> </tr><tr> <td>[11]</td> <td class="space">Mao, X., Stochastic Differential Equations and Their Applications (1997), Horwood Publishing: Horwood Publishing Chichester · <a href="/0892.60057" class="nowrap">Zbl 0892.60057</a></td> </tr><tr> <td>[12]</td> <td class="space">Mao, X., Asymptotic properties of neutral stochastic differential delay equations, Stochastics and Stochastics Rep., 68, 273-295 (2000) · <a href="/0964.60066" class="nowrap">Zbl 0964.60066</a></td> </tr><tr> <td>[13]</td> <td class="space">Mohammed, S.-E. A., Stochastic Functional Differential Equations (1986), Longman Scientific and Technical · <a href="/0584.60066" class="nowrap">Zbl 0584.60066</a></td> </tr><tr> <td>[14]</td> <td class="space">Rubanik, V. P., Oscillations of Quasilinear Systems with Retardation (1969), Nauka: Nauka Moscow · <a href="/0403.34061" class="nowrap">Zbl 0403.34061</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. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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