An Introduction to the Theory of Viscosity Solutions for First-Order Hamilton–Jacobi Equations and Applications
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Auteurs : Guy BarlesSource :
- Lecture Notes in Mathematics [ 0075-8434 ] ; NaN.
Abstract
Abstract: In this course, we first present an elementary introduction to the concept of viscosity solutions for first-order Hamilton–Jacobi Equations: definition, stability and comparison results (in the continuous and discontinuous frameworks), boundary conditions in the viscosity sense, Perron’s method, Barron–Jensen solutions … etc. We use a running example on exit time control problems to illustrate the different notions and results. In a second part, we consider the large time behavior of periodic solutions of Hamilton–Jacobi Equations: we describe recents results obtained by using partial differential equations type arguments. This part is complementary of the course of H. Ishii which presents the dynamical system approach (“weak KAM approach”).
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DOI: 10.1007/978-3-642-36433-4_2
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Karetnyi per. 19/1</Street><City>Moscow</City><Postcode>127994</Postcode><Country>Russia</Country></OrgAddress></Affiliation></AuthorGroup></BookHeader><Chapter ID="b978-3-642-36433-4_2" Language="En"><ChapterInfo ChapterType="OriginalPaper" ContainsESM="No" Language="En" NumberingDepth="3" NumberingStyle="ContentOnly" OutputMedium="All" TocLevels="0"><ChapterID>2</ChapterID><ChapterDOI>10.1007/978-3-642-36433-4_2</ChapterDOI><ChapterSequenceNumber>2</ChapterSequenceNumber><ChapterTitle Language="En">An Introduction to the Theory of Viscosity Solutions for First-Order Hamilton–Jacobi Equations and Applications</ChapterTitle><ChapterFirstPage>49</ChapterFirstPage><ChapterLastPage>109</ChapterLastPage><ChapterCopyright><CopyrightHolderName>Springer-Verlag Berlin Heidelberg</CopyrightHolderName><CopyrightYear>2013</CopyrightYear></ChapterCopyright><ChapterHistory><RegistrationDate><Year>2013</Year><Month>1</Month><Day>28</Day></RegistrationDate><OnlineDate><Year>2013</Year><Month>3</Month><Day>20</Day></OnlineDate></ChapterHistory><ChapterContext><SeriesID>304</SeriesID><SubSeriesID>3114</SubSeriesID><BookID>978-3-642-36433-4</BookID><BookTitle>Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications</BookTitle></ChapterContext></ChapterInfo><ChapterHeader><AuthorGroup><Author AffiliationIDS="Aff1" CorrespondingAffiliationID="Aff1"><AuthorName DisplayOrder="Western"><GivenName>Guy</GivenName><FamilyName>Barles</FamilyName></AuthorName><Contact><Email>barles@lmpt.univ-tours.fr</Email></Contact></Author><Affiliation ID="Aff1"><OrgDivision>Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 7350)</OrgDivision><OrgName>Fédération Denis Poisson (FR CNRS 2964), Université de Tours</OrgName><OrgAddress><Street>Parc de Grandmont</Street><Postcode>37200</Postcode><City>Tours</City><Country>France</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En" OutputMedium="All"><Heading>Abstract</Heading><Para>In this course, we first present an elementary introduction to the concept of viscosity solutions for first-order Hamilton–Jacobi Equations: definition, stability and comparison results (in the continuous and discontinuous frameworks), boundary conditions in the viscosity sense, Perron’s method, Barron–Jensen solutions <Emphasis Type="Italic">…</Emphasis> etc. We use a running example on exit time control problems to illustrate the different notions and results. In a second part, we consider the large time behavior of periodic solutions of Hamilton–Jacobi Equations: we describe recents results obtained by using partial differential equations type arguments. This part is complementary of the course of H. Ishii which presents the dynamical system approach (“weak KAM approach”).</Para></Abstract></ChapterHeader><NoBody></NoBody></Chapter></Book></SubSeries></Series></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>An Introduction to the Theory of Viscosity Solutions for First-Order Hamilton–Jacobi Equations and Applications</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>An Introduction to the Theory of Viscosity Solutions for First-Order Hamilton–Jacobi Equations and Applications</title></titleInfo><name type="personal" displayLabel="corresp"><namePart type="given">Guy</namePart><namePart type="family">Barles</namePart><affiliation>Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 7350), Fédération Denis Poisson (FR CNRS 2964), Université de Tours, Parc de Grandmont, 37200, Tours, France</affiliation><affiliation>E-mail: barles@lmpt.univ-tours.fr</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="chapter" displayLabel="chapter" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-CGT4WMJM-6"></genre><originInfo><publisher>Springer Berlin Heidelberg</publisher><place><placeTerm type="text">Berlin, Heidelberg</placeTerm></place><dateIssued encoding="w3cdtf">2013-03-20</dateIssued><copyrightDate encoding="w3cdtf">2013</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><abstract lang="en">Abstract: In this course, we first present an elementary introduction to the concept of viscosity solutions for first-order Hamilton–Jacobi Equations: definition, stability and comparison results (in the continuous and discontinuous frameworks), boundary conditions in the viscosity sense, Perron’s method, Barron–Jensen solutions … etc. We use a running example on exit time control problems to illustrate the different notions and results. In a second part, we consider the large time behavior of periodic solutions of Hamilton–Jacobi Equations: we describe recents results obtained by using partial differential equations type arguments. This part is complementary of the course of H. Ishii which presents the dynamical system approach (“weak KAM approach”).</abstract><relatedItem type="host"><titleInfo><title>Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications</title><subTitle>Cetraro, Italy 2011, Editors: Paola Loreti, Nicoletta Anna Tchou</subTitle></titleInfo><name type="personal"><namePart type="given">Yves</namePart><namePart type="family">Achdou</namePart><affiliation>UMR 7598, Lab. Jacques-Louis Lions - UPMC, Bât. Sophie Germain-Univ. Paris Diderot, avenue de France, 75013, Paris, France</affiliation><affiliation>E-mail: achdou@ljll-univ-parisdiderot.fr</affiliation></name><name type="personal"><namePart type="given">Guy</namePart><namePart type="family">Barles</namePart><affiliation>et Physique Théorique, UMR 6083, Laboratoire de Mathématiques, Parc de Grandmont, 37200, TOURS, France</affiliation><affiliation>E-mail: Guy.Barles@lmpt.univ-tours.fr</affiliation></name><name type="personal"><namePart type="given">Hitoshi</namePart><namePart type="family">Ishii</namePart><affiliation>Department of Mathematics, Waseda University, Nishi-Waseda 1-6-1, 169-8050, Shinjuku-ku, Tokyo, Japan</affiliation><affiliation>E-mail: hitoshi.ishii@waseda.jp</affiliation></name><name type="personal"><namePart type="given">Grigory L.</namePart><namePart type="family">Litvinov</namePart><affiliation>Institution, for Information Transmission Problems, Russsian Academy of Sciences, B. 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