Document Zbl 1203.94022

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O.</a>; <a href="/authors/ferreira.rui-a-c" title="Author Profile">Ferreira, Rui A. C.</a>; <a href="/authors/torres.delfim-f-m" title="Author Profile">Torres, Delfim F. M.</a></div> <h2 class="title"> <strong>Discrete-time fractional variational problems.</strong> <i>(English)</i> <a class="label nowrap" href="/1203.94022">Zbl 1203.94022</a> </h2> <div class="source"> <a href="/serials/1303" title="Journal Profile">Signal Process.</a> <a href="/?q=in%3A270524" title="Articles in this Issue">91, No. 3, 513-524 (2011)</a>. </div> <div class="abstract">Summary: We introduce a discrete-time fractional calculus of variations on the time scale $(h \mathbb Z)_a, a \in \mathbb R, h > 0$. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler-Lagrange and Legendre type conditions are given. They show that solutions to the considered fractional problems become the classical discrete-time solutions when the fractional order of the discrete-derivatives are integer values, and that they converge to the fractional continuous-time solutions when $h$ tends to zero. Our Legendre type condition is useful to eliminate false candidates identified via the Euler-Lagrange fractional equation.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5830278">Cited in <strong>131</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A94A12" title="MSC2020">94A12</a> </td> <td class="space"> Signal theory (characterization, reconstruction, filtering, etc.) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A26A33" title="MSC2020">26A33</a> </td> <td class="space"> Fractional derivatives and integrals </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62M10" title="MSC2020">62M10</a> </td> <td class="space"> Time series, auto-correlation, regression, etc. in statistics (GARCH) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A49J99" title="MSC2020">49J99</a> </td> <td class="space"> Existence theories in calculus of variations and optimal control </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Afractional+difference+calculus">fractional difference calculus</a>; <a href="/?q=ut%3Acalculus+of+variations">calculus of variations</a>; <a href="/?q=ut%3Afractional+summation+by+parts">fractional summation by parts</a>; <a href="/?q=ut%3AEuler-Lagrange+equation">Euler-Lagrange equation</a>; <a href="/?q=ut%3Anatural+boundary+conditions">natural boundary conditions</a>; <a href="/?q=ut%3ALegendre+necessary+condition">Legendre necessary condition</a>; <a href="/?q=ut%3Atime+scale+%5C%28h+%5Cmathbb+Z%5C%29">time scale $h \mathbb Z$</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 1203.94022" data-ciurl="/ci/05830278" data-biburl="/bibtex/05830278.bib" data-amsurl="/amsrefs/05830278.bib" data-xmlurl="/xml/05830278.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05830278.pdf" title="Zbl 1203.94022 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/j.sigpro.2010.05.001" aria-label="DOI for “Discrete-time fractional variational problems”" title="10.1016/j.sigpro.2010.05.001">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1005.0252"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Agrawal, O. 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