Document Zbl 0959.49021

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L.</a></div> <h2 class="title"> <strong>Solving Hamilton-Jacobi-Bellman equations by a modified method of characteristics.</strong> <i>(English)</i> <a class="label nowrap" href="/0959.49021">Zbl 0959.49021</a> </h2> <div class="source"> <a href="/serials/605" title="Journal Profile">Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods</a> <a href="/?q=in%3A62976" title="Articles in this Issue">40, No. 1-8, 279-293 (2000)</a>. </div> <div class="abstract">Continuing previous work, the authors propose yet another numerical procedure for solving fixed time-interval optimal control problems (without end-point constraints) which consist in minimizing cost functionals of the form: \[ C(u(.))=h(x(1))+\int_{0}^{1}L(t,x(t),u(t)) dt \] subject to: \[ x'(t)=f(t,x(t),u(t)), \;u(t)\in U \subset R^m, \text{ a.e. }([0,1]), \;x(0)=y\in \Omega \subset R^n . \] Mentioning certain existing results in the theory of viscosity solutions in [<span class="zbmathjax-textit">M. Bardi</span> and <span class="zbmathjax-textit">I. Capuzzo-Dolcetta</span>, “Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations” (1997; <a href="/0890.49011">Zbl 0890.49011</a>)], the authors propose a numerical procedure for solving the “viscosity approximating equation”: \[ \varepsilon \nabla^2v-{{\partial v}\over {\partial t}}+H(t,x,-\nabla_x v)=0, \;v(1,x)=h(x), \;x\in \Omega, (1) \] defined by the Hamiltonian of the problem: \[ H(t,x,p) =\sup_{u\in U}{\mathcal H}(t,x,p,u), \;{\mathcal H}(t,x,p,u)=\langle p,f(t,x,u)\rangle- L(t,x,u), \]<br class="zbmathjax-paragraph">\[ u^*(t,x,p) = \text{argmax}_{u\in U} {\mathcal H}(t,x,p,u) \] and to which one adds certain “artificial” Dirichlet and Neumann boundary conditions. <br class="zbmathjax-paragraph">The numerical algorithm proposed by the authors uses an adaptation of the “modified method of characteristics” in [<span class="zbmathjax-textit">J. Douglas jun.</span> and <span class="zbmathjax-textit">T. F. Russell</span>, SIAM J. Numer. Anal. 19, 871-885 (1982; <a href="/0492.65051">Zbl 0492.65051</a>)] for the “convection-diffusion” equation in (1) in which the “marginal multifunction” $u^*(.,.,.)$ seems to be considered a (single-valued) function. <br class="zbmathjax-paragraph">The largest part of the paper (some 8 pages out of 15) is dedicated to the reports (Tables and Figures) of numerical experiments on 5 examples.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A1105">Stefan Mirica (Bucureşti)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A1472219">Cited in <strong>28</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td class="mono"> 49M30 </td> <td class="space"> Other numerical methods in calculus of variations (MSC2010) </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><tr> <td> <a class="mono" href="/classification/?q=cc%3A49L25" title="MSC2020">49L25</a> </td> <td class="space"> Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65K10" title="MSC2020">65K10</a> </td> <td class="space"> Numerical optimization and variational techniques </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aoptimal+control">optimal control</a>; <a href="/?q=ut%3AHamilton-Jacobi-Bellman+equation">Hamilton-Jacobi-Bellman equation</a>; <a href="/?q=ut%3Aviscosity+solution">viscosity solution</a>; <a href="/?q=ut%3Aviscosity+approximation">viscosity approximation</a>; <a href="/?q=ut%3Aconvection-diffusion+equation">convection-diffusion equation</a>; <a href="/?q=ut%3Anumerical+method">numerical method</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0890.49011">Zbl 0890.49011</a>; <a href="/0492.65051">Zbl 0492.65051</a></div> <div class="software"> <h3>Software:</h3><a href="/software/5154">FFSQP(f77)</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 0959.49021" data-ciurl="/ci/01472219" data-biburl="/bibtex/01472219.bib" data-amsurl="/amsrefs/01472219.bib" data-xmlurl="/xml/01472219.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/01472219.pdf" title="Zbl 0959.49021 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/S0362-546X(00)85016-6" aria-label="DOI for “Solving Hamilton-Jacobi-Bellman equations by a modified method of characteristics”" title="10.1016/S0362-546X(00)85016-6">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">Bardi, M.; Capuzzo-Dolcetta, I., Optimal Control and Viscosity Solutions of Hamilton—Jacobi—Bellman Equations (1997), Birkhäuser: Birkhäuser Boston · <a href="/0890.49011" class="nowrap">Zbl 0890.49011</a></td> </tr><tr> <td>[2]</td> <td class="space">Crandall, M. G.; Lions, P. L., Viscosity solutions of Hamilton—Jacobi equations, Trans. AMS, 277, 1, 1-42 (1983) · <a href="/0599.35024" class="nowrap">Zbl 0599.35024</a></td> </tr><tr> <td>[3]</td> <td class="space">Crandall, M. G.; Evans, L. C.; Lions, P. L., Some properties of viscosity solutions of Hamilton—Jacobi equations, Trans. AMS, 282, 2, 487-502 (1984) · <a href="/0543.35011" class="nowrap">Zbl 0543.35011</a></td> </tr><tr> <td>[4]</td> <td class="space">Craven, B. D., Control and Optimization (1995), Chapman & Hall: Chapman & Hall London · <a href="/0915.49002" class="nowrap">Zbl 0915.49002</a></td> </tr><tr> <td>[5]</td> <td class="space">Lions, P. L., Generalised solutions of Hamilton—Jacobi Equations, (Pitman Research Notes in Mathematics, Vol. 69 (1982), Longman Scientific and Technical: Longman Scientific and Technical Harbour) · <a href="/0659.35015" class="nowrap">Zbl 0659.35015</a></td> </tr><tr> <td>[6]</td> <td class="space">Douglas, J.; Ewing, R. E.; Wheeler, M. F., A time-discretization Procedure for a mixed finite element approximation of miscible displacement in porous media, RAIRO Anal. Numér., 17, 249-265 (1983) · <a href="/0526.76094" class="nowrap">Zbl 0526.76094</a></td> </tr><tr> <td>[7]</td> <td class="space">Douglas, J.; Russell, T. F., Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal., 19, 871-885 (1982) · <a href="/0492.65051" class="nowrap">Zbl 0492.65051</a></td> </tr><tr> <td>[8]</td> <td class="space">Maslov, V. P., On a new principle of superposition for optimization problems, Russian Maths. Surveys, 42, 3, 43-84 (1987) · <a href="/0707.35138" class="nowrap">Zbl 0707.35138</a></td> </tr><tr> <td>[9]</td> <td class="space">Teo, K. L.; Goh, C. J.; Wong, K. H., A Unified Computational Approach to Optimal Control Problems (1991), Longman Scientific and Technical: Longman Scientific and Technical Essex · <a href="/0747.49005" class="nowrap">Zbl 0747.49005</a></td> </tr><tr> <td>[10]</td> <td class="space">S. Wang, F. Gao, K.L. Teo, An upwind finite difference method for the approximation of viscosity solutions to Hamilton—Jacobi—Bellman equations, IMA J. Math. Control Appl., to appear.; S. Wang, F. Gao, K.L. Teo, An upwind finite difference method for the approximation of viscosity solutions to Hamilton—Jacobi—Bellman equations, IMA J. Math. Control Appl., to appear. · <a href="/0952.49025" class="nowrap">Zbl 0952.49025</a></td> </tr><tr> <td>[11]</td> <td class="space">Zhou, J. L.; Tits, A. L.; Lawrence, C. T., User’s Guide for FFSQP Version 3.7: A Fortran Code for solving Constrained Optimization Problems, Generating Iterates Satisfying All Inequality and Linear Constraints. User’s Guide for FFSQP Version 3.7: A Fortran Code for solving Constrained Optimization Problems, Generating Iterates Satisfying All Inequality and Linear Constraints, Technical Report TR92-107-r2 (1997), System Research Center, University of Maryland</td> </tr><tr> <td>[12]</td> <td class="space">Zhou, X. Y., Verification theorems within the framework of viscosity solutions, J. Math. Anal. Appl., 177, 208-225 (1993) · <a href="/0780.49023" class="nowrap">Zbl 0780.49023</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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