Document Zbl 1502.93010

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I: Frequency-sweeping stability tests and applications.</strong> <i>(English)</i> <a class="label nowrap" href="/1502.93010">Zbl 1502.93010</a> </h2> <div class="source"> Breda, Dimitri (ed.), Controlling delayed dynamics. Advances in theory, methods and applications. Cham: Springer. CISM Courses Lect. 604, 117-155 (2023). </div> <div class="abstract">Summary: This chapter addresses the stability analysis of linear dynamical systems represented by delay differential equations with a focus on the effects induced by the delay, seen as a <span class="zbmathjax-textit">parameter</span>, on the dynamical behavior. More precisely, we propose a <span class="zbmathjax-textit">frequency-sweeping framework</span> for treating the problem, and the stability problem is reformulated in terms of properties of frequency-sweeping curves. The presentation is teaching-oriented and focuses more on discussing the main ideas of the method and their illustration through appropriate examples and less on explicit proofs of the results. Some applications from Life Sciences complete the presentation. <br class="zbmathjax-paragraph">For Part II, see [<span class="zbmathjax-textit">S.-I. Niculescu</span> and <span class="zbmathjax-textit">I. Boussaada</span>, CISM Courses Lect. 604, 157–193 (2023; <a href="/1504.93290">Zbl 1504.93290</a>)].<br class="zbmathjax-paragraph">For the entire collection see [<a href="/1496.93003">Zbl 1496.93003</a>].</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7626440">Cited in <strong>3</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A93D05" title="MSC2020">93D05</a> </td> <td class="space"> Lyapunov and other classical stabilities (Lagrange, Poisson, $L^p, l^p$, etc.) in control theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C23" title="MSC2020">93C23</a> </td> <td class="space"> Control/observation systems governed by functional-differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C80" title="MSC2020">93C80</a> </td> <td class="space"> Frequency-response methods in control theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C05" title="MSC2020">93C05</a> </td> <td class="space"> Linear systems in control theory </td> </tr></table> </div><div class="keywords"> <h3>Citations:</h3><a href="/1504.93290">Zbl 1504.93290</a></div> <div class="software"> <h3>Software:</h3><a href="/software/2396">DDE-BIFTOOL</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 1502.93010" data-ciurl="/ci/07626440" data-biburl="/bibtex/07626440.bib" data-amsurl="/amsrefs/07626440.bib" data-xmlurl="/xml/07626440.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07626440.pdf" title="Zbl 1502.93010 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/10.1007/978-3-031-01129-0_5" aria-label="DOI for “Counting characteristic roots of linear delay differential equations. I: Frequency-sweeping stability tests and applications”" title="10.1007/978-3-031-01129-0_5">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">Ahlfors, L. V. (1979). Complex analysis. McGraw-Hill. · <a href="/0395.30001" class="nowrap">Zbl 0395.30001</a></td> </tr><tr> <td>[2]</td> <td class="space">Bellman, R.; Cooke, KL, Differential-difference equations (1963), New York: Academic Press, New York · <a href="/0105.06402" class="nowrap">Zbl 0105.06402</a></td> </tr><tr> <td>[3]</td> <td class="space">Casas-Alvero, E. (2000). Singularities of plane curves. London Mathematical Society Lecture Note Series. 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