Document Zbl 0949.93022

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The GHJB equation gives the cost of an arbitrary control law and can be used to improve the performance of this control. The GHJB equation can also be used to successively approximate the Hamilton-Jacobi-Bellman equation. We state sufficient conditions that guarantee that the Galerkin approximation converges to the solution of the GHJB equation and that the resulting approximate control is stabilizing on the same region as the initial control. The method is demonstrated on a simple nonlinear system and is compared to a result obtained by using exact feedback linearization in conjunction with the LQR design method.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A1161172">Cited in <strong>96</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td class="mono"> 93B40 </td> <td class="space"> Computational methods in systems theory (MSC2010) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A49M15" title="MSC2020">49M15</a> </td> <td class="space"> Newton-type methods </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A49L20" title="MSC2020">49L20</a> </td> <td class="space"> Dynamic programming in optimal control and differential games </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AGalerkin+approximation">Galerkin approximation</a>; <a href="/?q=ut%3AHamilton-Jacobi-Bellman">Hamilton-Jacobi-Bellman</a>; <a href="/?q=ut%3Afeedback+linearization">feedback linearization</a>; <a href="/?q=ut%3ALQR+design+method">LQR design method</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 0949.93022" data-ciurl="/ci/01161172" data-biburl="/bibtex/01161172.bib" data-amsurl="/amsrefs/01161172.bib" data-xmlurl="/xml/01161172.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/01161172.pdf" title="Zbl 0949.93022 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/S0005-1098(97)00128-3" aria-label="DOI for “Galerkin approximations of the generalized Hamilton-Jacobi-Bellman equation”" title="10.1016/S0005-1098(97)00128-3">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">Aeyels, D., Stabilization of a class of nonlinear systems by a smooth feedback control, System Control Lett., 5, 289-294 (1985) · <a href="/0569.93056" class="nowrap">Zbl 0569.93056</a></td> </tr><tr> <td>[2]</td> <td class="space">Aeyels, D., Local and global stabilizability for nonlinear systems, (Byrnes, C. 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