Table Of ContentProgress 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
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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
