Document Zbl 1139.58011

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Program.</a> <a href="/?q=in%3A216466" title="Articles in this Issue">113, No. 2 (A), 345-424 (2008)</a>. </div> <div class="abstract">Summary: This paper introduces and studies the class of differential variational inequalities (DVIs) in a finite-dimensional Euclidean space. The DVI provides a powerful modeling paradigm for many applied problems in which dynamics, inequalities, and discontinuities are present; examples of such problems include constrained time-dependent physical systems with unilateral constraints, differential Nash games, and hybrid engineering systems with variable structures. The DVI unifies several mathematical problem classes that include ordinary differential equations (ODEs) with smooth and discontinuous right-hand sides, differential algebraic equations (DAEs), dynamic complementarity systems, and evolutionary variational inequalities. Conditions are presented under which the DVI can be converted, either locally or globally, to an equivalent ODE with a Lipschitz continuous right-hand function. For DVIs that cannot be so converted, we consider their numerical resolution via an Euler time-stepping procedure, which involves the solution of a sequence of finite-dimensional variational inequalities. Borrowing results from differential inclusions (DIs) with upper semicontinuous, closed and convex valued multifunctions, we establish the convergence of such a procedure for solving initial-value DVIs. We also present a class of DVIs for which the theory of DIs is not directly applicable, and yet similar convergence can be established. Finally, we extend the method to a boundary-value DVI and provide conditions for the convergence of the method. The results in this paper pertain exclusively to systems with “index” not exceeding two and which have absolutely continuous solutions.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A5248725">Cited in <strong>4</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A5248725">Cited in <strong>227</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A58E35" title="MSC2020">58E35</a> </td> <td class="space"> Variational inequalities (global problems) in infinite-dimensional spaces </td> </tr></table> </div> <div class="software"> <h3>Software:</h3><a href="/software/1112">Meschach</a>; <a href="/software/6242">PATH Solver</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 1139.58011" data-ciurl="/ci/05248725" data-biburl="/bibtex/05248725.bib" data-amsurl="/amsrefs/05248725.bib" data-xmlurl="/xml/05248725.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05248725.pdf" title="Zbl 1139.58011 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/s10107-006-0052-x" aria-label="DOI for “Differential variational inequalities”" title="10.1007/s10107-006-0052-x">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://hal.science/hal-01366027" title="Open Access at HAL">HAL </a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Acary, V., Brogliato, B., Goeleven, D.: Higher order Moreau’s sweeping process: Mathematical formulation and numerical simulation. 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