Document Zbl 0873.93008

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Calc. Var.</a> <a href="/?q=in%3A40209" title="Articles in this Issue">2, 33-55 (1997)</a>. </div> <div class="abstract">The author proves a controllability result for the nonlinear Korteweg-de Vries equation on bounded domains for sufficiently small initial and final states. He uses the Banach contraction fixed point theorem and the corresponding controllability result for the linear equation. For the linear case, the author uses the Hilbert Uniqueness Method and the multiplier method.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A4944">O.Cârjá (Iaşi)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A997470">Cited in <strong>3</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A997470">Cited in <strong>180</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A93B05" title="MSC2020">93B05</a> </td> <td class="space"> Controllability </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C20" title="MSC2020">93C20</a> </td> <td class="space"> Control/observation systems governed by partial differential equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Acontrollability">controllability</a>; <a href="/?q=ut%3AHilbert+uniqueness+method">Hilbert uniqueness method</a>; <a href="/?q=ut%3Aboundary+control">boundary control</a>; <a href="/?q=ut%3Anonlinear+Korteweg-de+Vries+equation">nonlinear Korteweg-de Vries equation</a>; <a href="/?q=ut%3Abounded+domains">bounded domains</a>; <a href="/?q=ut%3Afixed+point+theorem">fixed point theorem</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 0873.93008" data-ciurl="/ci/00997470" data-biburl="/bibtex/00997470.bib" data-amsurl="/amsrefs/00997470.bib" data-xmlurl="/xml/00997470.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/00997470.pdf" title="Zbl 0873.93008 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/10.1051/cocv:1997102" aria-label="DOI for “Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain”" title="10.1051/cocv:1997102">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://eudml.org/doc/90511" title="Open Access at EuDML">EuDML</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">C. Bardos, G. Lebeau and J. Rauch: Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim., 30, 1992, 1024-1065. Zbl0786.93009 MR1178650 · <a href="/0786.93009" class="nowrap">Zbl 0786.93009</a> · <a href="https://doi.org/10.1137/0330055" class="nowrap">doi:10.1137/0330055</a></td> </tr><tr> <td>[2]</td> <td class="space">J. Bona and R. Winther: The Korteweg-de Vries equation, posed in a quarter-plane, SIAM J. Math Anal., 14, 1983, 1056-1106. Zbl0529.35069 MR718811 · <a href="/0529.35069" class="nowrap">Zbl 0529.35069</a> · <a href="https://doi.org/10.1137/0514085" class="nowrap">doi:10.1137/0514085</a></td> </tr><tr> <td>[3]</td> <td class="space">J.M. Coron: Contrôlabilité exacte frontière de l’équation d’Euler des fluides parfaits incompressibles bidimensionnels, C. R. Acad. Sci. Paris, t. 317, Série I, 1993, 271-276. 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Zbl0014.21503 MR1545625 · <a href="/0014.21503" class="nowrap">Zbl 0014.21503</a> · <a href="https://doi.org/10.1007/BF01180426" class="nowrap">doi:10.1007/BF01180426</a></td> </tr><tr> <td>[7]</td> <td class="space">V. Komornik: Exact controllability and stabilization, the multiplier method, R.A.M. 36, John Wiley-Masson, 1994. Zbl0937.93003 MR1359765 · <a href="/0937.93003" class="nowrap">Zbl 0937.93003</a></td> </tr><tr> <td>[8]</td> <td class="space">V. Komornik, D.L. Russel and B.-Y. Zhang: Control and stabilization of the Korteweg-de Vries equation on a periodic domain, submitted to J. Differential Equations. · <a href="/1213.93015" class="nowrap">Zbl 1213.93015</a> · <a href="https://doi.org/10.1080/03605300903585336" class="nowrap">doi:10.1080/03605300903585336</a></td> </tr><tr> <td>[9]</td> <td class="space">D.J. Korteweg and G. de Vries: On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. 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Zhang: Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain, SIAM J. Control Optim., 31, 1993, 659-673. Zbl0771.93073 MR1214759 · <a href="/0771.93073" class="nowrap">Zbl 0771.93073</a> · <a href="https://doi.org/10.1137/0331030" class="nowrap">doi:10.1137/0331030</a></td> </tr><tr> <td>[17]</td> <td class="space">J.C. Saut and R. Temam: Remarks on the Korteweg-De Vries equation, Israel. Math., 24, 1976, 78-87. Zbl0334.35062 MR454425 · <a href="/0334.35062" class="nowrap">Zbl 0334.35062</a> · <a href="https://doi.org/10.1007/BF02761431" class="nowrap">doi:10.1007/BF02761431</a></td> </tr><tr> <td>[18]</td> <td class="space">J. Simon: Compact sets in the space Lp(0,T, B), Annali di Matematica pura ed applicata (IV), vol. CXLVI, 1987, 65-96. 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