Document Zbl 1281.34002

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Berlin: Springer (ISBN 978-3-642-34927-0/pbk; 978-3-642-34928-7/ebook). Differential-Algebraic Equations Forum, 63-95 (2013). </div> <div class="abstract">The paper summarizes investigations on robust stability of linear differential-algebraic equations (DAEs). The goal is to give bounds for perturbations such that the perturbed system keeps the same properties as the unperturbed one.<br class="zbmathjax-paragraph">Here, the exponential and the asymptotic stability are considered and stability radii are derived. Implicitly, also changes of the index or the nilpotency structure have to be considered here.<br class="zbmathjax-paragraph">The stability concepts for ODEs are generalized for regular DAEs. The investigation of the time-invariant case uses mainly the Weierstrass-Kronecker form of the DAE, whereas in the time-varying case the tractability and the strangeness index concept are applied.<br class="zbmathjax-paragraph">The step from ODEs to DAEs needs to consider structured perturbations, i.e. those related to the structure of the DAE. In the time-varying case the investigations are restricted to index-1 DAEs.<br class="zbmathjax-paragraph">The derivation of the different stability radii (e.g. in the real or in the complex world) is well presented and illustratively discussed.<br class="zbmathjax-paragraph">It is shown that under structured perturbations (and additional assumptions) stability radii of DAEs exist and can be computed.<br class="zbmathjax-paragraph">For the entire collection see [<a href="/1261.65003">Zbl 1261.65003</a>].<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A7287">René Lamour (Berlin)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6177008">Cited in <strong>17</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A34-02" title="MSC2020">34-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to ordinary differential equations </td> </tr><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%3A34D10" title="MSC2020">34D10</a> </td> <td class="space"> Perturbations of ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93B35" title="MSC2020">93B35</a> </td> <td class="space"> Sensitivity (robustness) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93D09" title="MSC2020">93D09</a> </td> <td class="space"> Robust stability </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%3A34A30" title="MSC2020">34A30</a> </td> <td class="space"> Linear ordinary differential equations and systems </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Adifferential-algebraic+equation">differential-algebraic equation</a>; <a href="/?q=ut%3Arestricted+perturbation">restricted perturbation</a>; <a href="/?q=ut%3Arobust+stability">robust stability</a>; <a href="/?q=ut%3Astability+radius">stability radius</a>; <a href="/?q=ut%3Aspectrum">spectrum</a>; <a href="/?q=ut%3Aindex">index</a></div> <div class="software"> <h3>Software:</h3><a href="/software/28496">PSAPSR</a>; <a href="/software/7020">Eigtool</a>; <a href="/software/4112">RODAS</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 1281.34002" data-ciurl="/ci/06177008" data-biburl="/bibtex/06177008.bib" data-amsurl="/amsrefs/06177008.bib" data-xmlurl="/xml/06177008.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06177008.pdf" title="Zbl 1281.34002 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/10.1007/978-3-642-34928-7_2" aria-label="DOI for “Robust stability of differential-algebraic equations”" title="10.1007/978-3-642-34928-7_2">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://depositonce.tu-berlin.de/handle/11303/15733" title="Full Text Link">Link</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Adrianova, L. 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