Progress in Nonlinear Differential Equations and Their Applications Volume58 Editor HaimBrezis Universite´ PierreetMarieCurie Paris and RutgersUniversity NewBrunswick,N.J. EditorialBoard AntonioAmbrosetti,ScuolaNormaleSuperiore,Pisa A.Bahri,RutgersUniversity,NewBrunswick FelixBrowder,RutgersUniversity,NewBrunswick LuisCaffarelli,CourantInstituteofMathematics,NewYork LawrenceC.Evans,UniversityofCalifornia,Berkeley MarianoGiaquinta,UniversityofPisa DavidKinderlehrer,Carnegie-MellonUniversity,Pittsburgh SergiuKlainerman,PrincetonUniversity RobertKohn,NewYorkUniversity P.L.Lions,UniversityofParisIX JeanMawhin,Universite´ CatholiquedeLouvain LouisNirenberg,NewYorkUniversity LambertusPeletier,UniversityofLeiden PaulRabinowitz,UniversityofWisconsin,Madison JohnToland,UniversityofBath Piermarco Cannarsa Carlo Sinestrari Semiconcave Functions, Hamilton–Jacobi Equations, and Optimal Control Birkha¨user Boston • Basel • Berlin PiermarcoCannarsa CarloSinestrari Universita`diRoma“TorVergata” Universita`diRoma“TorVergata” DipartimentodiMatematica DipartimentodiMatematica 00133Roma 00133Roma Italy Italy LibraryofCongressCataloging-in-PublicationData Cannarsa,Piermarco,1957- Semiconcavefunctions,Hamilton–Jacobiequations,andoptimalcontrol/Piermarco Cannarsa,CarloSinestrari. p.cm.–(Progressinnonlineardifferentialequationsandtheirapplications;v.58) Includesbibliographicalreferencesandindex. ISBN 0-8176-4084-3 (alk. paper) 1.Concavefunctions.2.Hamilton–Jacobiequations.3.Controltheory.4.Mathematical optimization. I.Sinestrari,Carlo,1970-II.Title.III.Series. QA353.C64C362004 515’.355–dc22 2004043695 CIP AMS Subject Classifications: Primary: 35F20, 49J52, 49-XX, 49-01; Secondary: 35D10, 26B25, 49L25,35A21,49J15,49K15,49Lxx,49L20 ISBN 0-8176-4336-2 Printed on acid-free paper. (cid:2)c2004Birkha¨userBoston Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(Birkha¨userBoston,c/oSpringer-VerlagNewYork,Inc.,175Fifth Avenue,NewYork,NY10010,USA),exceptforbriefexcerptsinconnectionwithreviewsor scholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronic adaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafter developedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,evenifthey arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare subjecttopropertyrights. PrintedintheUnitedStatesofAmerica. (TXQ/HP) 987654321 SPIN 10982358 www.birkhauser.com ToFrancesca Preface A gifted British crime novelist1 once wrote that mathematics is “like one of those languages that is simple, straightforward and logical in the early stages, but which rapidlyspiralsoutofcontrolinafrenzyofidioms,oddities,idiosyncrasiesandex- ceptionstotherulewhichevennativespeakerscannotalwaysgetright,nevermind explain.” In fact, providing evidence to contradict such a statement has been one ofourguidesinwritingthismonograph.Itmaythenberecommendedtodescribe, rightfromthebeginning,theessentialobjectofourinterest,thatis,semiconcavity,a propertythatplaysacentralroleinoptimization. There are various possible ways to introduce semiconcavity. For instance, one cansaythatafunctionu issemiconcaveifitcanberepresented,locally,asthesum ofaconcavefunctionplusasmoothone.Thus,semiconcavefunctionssharemany regularity properties with concave functions, but include several other significant examples. Roughly speaking, semiconcave functions can be obtained as envelopes of smooth functions, in the same way as concave functions are envelopes of linear functions.Typicalexamplesofsemiconcavefunctionsarethedistancefunctionfrom aclosedsetS ⊂Rn,theleasteigenvalueofasymmetricmatrixdependingsmoothly on parameters, and the so-called “inf-convolutions.” Another class of examples we are particularly interested in are viscosity solutions of Hamilton–Jacobi–Bellman equations. Atthispoint,thereadermaywonderwhyweconsidersemiconcavityratherthan the symmetric—yet more usual—notion of semiconvexity. The thing is that as far as optimization is concerned, in this book we focus our attention on minimization ratherthanmaximization.Thismakessemiconcavitythenaturalpropertytolookat. Interestinsemiconcavefunctionswasinitiallymotivatedbyresearchonnonlin- ear partial differential equations. In fact, it was exactly in classes of semiconcave functions that the first global existence and uniqueness results were obtained for Hamilton–Jacobi–Bellmanequations,seeDouglis[69]andKruzhkov[99,100,102]. Afterwards,morepowerfuluniquenesstheories,suchasviscositysolutionsandmin- imax solutions, were developed. Nevertheless, semiconcavity maintains its impor- 1M.Dibdin,Bloodrain,FaberandFaber,London,1999. viii Preface tanceeveninmodernPDEtheory,beingthemaximaltypeofregularitythatcanbe expectedforcertainnonlinearproblems.Assuch,ithasbeeninvestigatedinmodern textbooks on Hamilton–Jacobi equations such as Lions [110], Bardi and Capuzzo- Dolcetta [20], Fleming and Soner [81], and Li and Yong [109]. In the context of nonsmoothanalysisandoptimization,semiconcavefunctionshavealsoreceivedat- tentionunderthenameoflowerCk functions,see,e.g.,Rockafellar[123]. Comparedtotheabovereferences,theperspectiveofthisbookisdifferent.First, in Chapters 2, 3 and 4, we develop the theory of semiconcave functions without aimingatonespecificapplication,butasatopicinnonsmoothanalysisofinterestin its own right. The exposition ranges from well-known properties for the experts— analyzed here for the first time in a comprehensive way—to recent results, such as the latest developments in the analysis of singularities. Then, in Chapters 5, 6, 7 and 8, we discuss contexts in which semiconcavity plays an important role, such as Hamilton–Jacobi equations and control theory. Moreover, the book opens with an introductory chapter studying a model problem from the calculus of variations: this allows us to present, in a simple situation, some of the main ideas that will be developedintherestofthebook.Amoredetaileddescriptionofthecontentsofthis workcanbefoundintheintroductionatthebeginningofeachchapter. Inouropinion,anattractivefeatureofthepresentexpositionisthatitrequires,on thereader’spart,littlemorethanastandardbackgroundinrealanalysisandPDEs. Althoughwedoemploynotionsandtechniquesfromdifferentfields,wehavenever- thelessmadeanefforttokeepthisbookasself-containedaspossible.Intheappendix wehavecollectedallthedefinitionsweneeded,andmostproofsofthebasicresults. For the more advanced ones—not too many indeed—we have given precise refer- encesintheliterature. Weareconfidentthatthisbookwillbeusefulfordifferentkindsofreaders.Re- searchersinoptimalcontroltheoryandHamilton–Jacobiequationswillherefindthe recentprogressofthistheoryaswellasasystematiccollectionofclassicalresults— for which a precise citation may be hard to recover. On the other hand, for readers at the graduate level, learning the basic properties of semiconcave functions could also be an occasion to become familiar with important fields of modern analysis, suchascontroltheory,nonsmoothanalysis,geometricmeasuretheoryandviscosity solutions. We will now sketch some shortcuts for readers with specific interests. As we mentioned before, Chapter 1 is introductory to the whole text; it can also be used on its own to teach a short course on calculus of variations. The first section of Chapter2andmostofChapter3areessentialforthecomprehensionofanythingthat follows.Onthecontrary,Chapter4,devotedtosingularities,couldbeomittedona first reading. The PDE-oriented reader could move on to Chapter 5 on Hamilton– Jacobi equations, and then to Chapter 6 on the calculus of variations, where sharp regularity results are obtained for solutions to suitable classes of equations. On the otherhand,thereaderwhowishestofollowadirectpathtodynamicoptimization, withoutincludingtheclassicalcalculusofvariations,couldgodirectlyfromChapter 3toChapters7and8wherefinitehorizonoptimalcontrolproblemsandoptimalexit timeproblemsareconsidered. Preface ix Wewouldliketoexpressourgratitudeforalltheassistancewehavereceivedfor the realization of this project. The first author is indebted to Sergio Campanato for inspiring his interest in regularity theory, to Giuseppe Da Prato for communicating his taste for functional analysis, to Wendell Fleming and Craig Evans for opening powerful views on optimal control and viscosity solutions, and to his friend Mete Sonerforsharingwithhimtheinitialenthusiasmforsemiconcavityandvariational problems. The subsequent collaboration with Halina Frankowska acquainted him withset-valuedanalysis.LuigiAmbrosiorevealedtohimenlighteningconnections with geometric measure theory. The second author is grateful to Alberto Tesei and RobertoNatalini,whofirstintroducedhiminthestudyofnonlinearfirstorderequa- tions. He is also indebted to Constantine Dafermos and Alberto Bressan for their inspiringteachingsabouthyperbolicconservationlawsandcontroltheory. A significant part of the topics of the book was conceived or refined in the framework of the Graduate School in Mathematics of the University of Rome Tor Vergata, as material developed in graduate courses, doctoral theses, and research papers.Wewishtothankalltheoneswhoparticipatedintheseactivities,inpartic- ular Paolo Albano, Cristina Pignotti, and Elena Giorgieri. Special thanks are due to our friends Italo Capuzzo-Dolcetta and Francis Clarke who read parts of the manuscriptimprovingitwiththeircomments.Helpfulsuggestionswerealsooffered bymanyotherfriendsandcolleagues,suchasGiovanniAlberti,MartinoBardi,Nick Barron, Pierre Cardaliaguet, Giovanni Colombo, Alessandra Cutr`ı, Robert Jensen, VilmosKomornik,AndreaMennucci,RobertoPeirone.Finally,wewishtoexpress ourwarmestthankstoMarianoGiaquinta,whoseinterestgaveusessentialencour- agement in starting this book, and to Ann Kostant, who followed us with patience duringthewritingofthiswork. PiermarcoCannarsa CarloSinestrari Thepowerofdoinganythingwithquicknessisalwaysmuchprizedbythepossessor, andoftenwithoutanyattentiontotheimperfectionoftheperformance. —JANEAUSTEN,PrideandPrejudice Contents Preface ......................................................... vii 1 AModelProblem ............................................ 1 1.1 Semiconcavefunctions...................................... 2 1.2 Aprobleminthecalculusofvariations ........................ 4 1.3 TheHopfformula .......................................... 6 1.4 Hamilton–Jacobiequations .................................. 9 1.5 Methodofcharacteristics .................................... 11 1.6 SemiconcavityofHopf’ssolution............................. 18 1.7 Semiconcavityandentropysolutions .......................... 25 2 SemiconcaveFunctions ....................................... 29 2.1 Definitionandbasicproperties ............................... 29 2.2 Examples ................................................. 38 2.3 SpecialpropertiesofSCL(A) ................................ 41 2.4 AdifferentialHarnackinequality ............................. 43 2.5 Ageneralizedsemiconcavityestimate ......................... 45 3 GeneralizedGradientsandSemiconcavity ....................... 49 3.1 Generalizeddifferentials..................................... 50 3.2 Directionalderivatives ...................................... 55 3.3 Superdifferentialofasemiconcavefunction .................... 56 3.4 Marginalfunctions ......................................... 65 3.5 Inf-convolutions............................................ 68 3.6 Proximalanalysisandsemiconcavity .......................... 73 4 SingularitiesofSemiconcaveFunctions .......................... 77 4.1 Rectifiabilityofthesingularsets .............................. 77 4.2 PropagationalongLipschitzarcs.............................. 84 4.3 Singularsetsofhigherdimension ............................. 88 4.4 Applicationtothedistancefunction ........................... 94