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Uniquenessoflowersemicontinuousviscosity solutionsfortheminimumtimeproblem OlivierAlvarez UPRESA SiteColbert Universit edeRouen Mont Saint AignanCedex France ShigeakiKoikeandIsaoNakayama DepartmentofMathematics SaitamaUniversity Shimo Okubo Urawa Saitama Japan Abstract Weobtaintheuniquenessoflowersemicontinuous lscforshort viscositysolutionsofthetransformedminimumtimeproblemassum ingthattheyconvergetozeroona reachable partofthetargetin appropriatedirections Wepresentacounter examplewhichshows thattheuniquenessdoesnotholdwithoutthisconvergenceassump tion ItwasshownbySoraviathattheuniquenessoflscviscositysolu tionshavinga subsolutionproperty onthetargetholds Inorder toverifythissubsolutionproperty weshowthattheDynamicPro grammingPrinciple DPPforshort holdsinsideforanylscviscosity solutions InordertoobtaintheDPP weprepareappropriateapproximate PDEsderivedthroughBarles inf convolutionanditsvariant Introduction Inthismanuscript wediscusstheminimumtimeproblemofdeterministic optimalcontrol whichhasbeenstudiedviaviscositysolutionapproachby manyauthors Asthe rstresult werefertoBardi Ba Seealso EJ BF BS and BSo whichalsotreatedtheminimumtimeproblemofdi erential games Inthoseworks theycharacterizedthevaluefunctionoftheminimumtime problemtoreachagiventargetastheuniqueviscositysolutionofa rst orderPDE However theyonlytreatedthecasewhentheresultingvalue functionsarecontinuoussincethoseuniquenessresultsimplythecontinuity ofsolutions Wenotethatthereoftenappeardiscontinuousvaluefunctions forpracticalminimumtimeproblems Thebreak throughtotreatsemicontinuoussolutionsfor rst orderPDEs wasdonebyBarronandJensen BJ Indeed theyintroducedanewdef initionofsemicontinuousviscositysolutionsforCauchyproblemswithcon vexHamiltonians whicharisewhenwedealwithoptimalcontrolproblems Undertheirsetting itwasshownin BJ thatthesemicontinuousvalue functionistheuniquesolutionoftheassociatedPDE Wenotethat ifwe restrictourselvestotreatcontinuousviscositysolutions thentheirde nition isequivalenttothatofthestandardone Afterwards Barles B discussedsemicontinuoussolutionsforstationary problemsutilizing Barles convolution Withthisidea Soravia S stud iedtheDirichlettypeproblems Moreprecisely heimposeda subsolution propertyontheboundaryofthetarget underwhichtheuniquenessoflscvis cositysolutionsforthe transformed minimumtimeproblemwasobtained Seealso K and BL forrelatedtopics Recently C arj aetal in CMP seealso C studiedlscviscositysolutionsoftheminimumtimeproblem assumingthattheyconvergetotheDirichletdatafrominside Fortheviscositysolutiontheoryof rst orderHamilton Jacobiequations werefertoanewbookbyBardiandCapuzzoDolcetta BC Ontheotherhand innon smoothanalysis lscsolutionshavebeenstud iedinoptimalcontroltheory Forthe rstresult werefertoFrankowska F Morerecently WolenskiandZhuang WZ haveprovedtheuniquenessoflsc solutionsoftheminimumtimeproblemassumingthesubsolutionproperty onthetargetasin S whichthevaluefunctionsatis es Wenotethattheir de nitionofsolutionsisslightlydi erentfromthatofviscositysolutions It isworthmentioningthat in WZ toshowtheuniqueness theycompared theotherlscsolution ifitexists withthevaluefunctionbytheso called invariancetheorywhile intheliteratureoftheviscositysolutiontheory wehaveshownitviacomparisonprincipleforaboundaryvalueproblemof PDEs Ouraimhereistoobtainauniquenessresultwithoutassumingthesub solutionpropertyonthetarget Infact wewillderivesuchapropertyfrom thede nitionofsolutionsundersomecontinuityassumptionona reachable partoftheboundary Then wewillbeabletoapplySoravia sargumentin S togettheuniqueness Moreover wewillmentionthatourcontinuityconditionisequivalent toSoravia sone Infact toshowthattheSoravia sconditionimpliesthe continuityone wegiveadirectproofthoughwecanproveitusingthe uniquenessofsolutions Inanexample wewillseethatthiscontinuityassumptionisnecessary toobtaintheuniquenessresult Here weshallrecalltheoriginalminimumtimeproblem Considerthe stateequationassociatedwithcontrols A f Ameasurableg whereAisacompactsetinR m forsomem N Forx R n dX dt t g X t t fort X x whereg R n A R n isagivenfunctionandx R n is xed Weshalldenote underappropriatehypotheses byX x the unique solutionof WewillalsodenotebyX x theuniquesolutionfor avector eld W R n R n dX dt t X t fort X x Forsimplicity weshallsupposethat A T R n iscompact Withthesenotations werecallthevaluefunctionoftheminimumtime problem V x inf A T x whereT x infft jX t x Tg SinceV x mightbein nityinasubregionof R n nT wewillhave tostudythefreeboundaryproblem max a A f hg x a DV x i g inR fx jV x g SincewecannotexpectthatRisopeningeneralaswewillsee wemeet somedi cultyifwetreat directly Therefore inthispaper follow ingthepreviousworks weshallconsiderthetransformedvaluefunctionby Kruzkovtransformation u x inf A e T x Then wecanexpectutobeasolutionof u x max a A f hg x a Du x ig in Thus onceweverifythatuistheuniquesolutionof wewillbeableto derivethereachablesetbyR fx ju x g Thispaperisorganizedasfollows Section isdevotedtogiveourde nitionoftheminimumtimeproblemandtheDPPwhichimpliesthesubso lutionproperty WepresentouruniquenessresultandexamplesinSection Also wediscussabouttheequivalenceofboundaryconditionsinSection Inthe nalsection weprovetheDPPinSection Acknowledgement WewishtothankProfessorE N Barronforin formingusthemanuscript WZ WealsowishtothankProfessorM Bardi forlettingusknowaninterestingexample seeExample below dueto P Soravia We nallywishtothanktherefereesfortheirsuggestionstothe rst draft ThesecondauthorSKwassupportedbyGrant in AidforScienti cRe searchNo andNo theMinistryofEducation Science andCultureinJapan DynamicProgrammingPrinciple Ourhypothesisontheregularityofgivenfunctionsisasfollows A g C R n A R n andsup a A kg a k W R n R n Forlaterconvenience weshallconsiderthefollowinggeneral rst order PDEinaset R n u x max a A f hg x a Du x i f x a g in wheref R n A Risagivencontinuousfunction Forsimplicity weshallusethenotation H x r p r max a A f hg x a pi f x a g Wewillsupposethefollowingreguralityongivenfunctionsin A g C R n A R n f C R n A R and sup a A n kg a k W R n R n kf a k W R n R o Following BJ also B wepresentourde nitionofsolutionsof De nition Forafunctionu R wecallitasubsolution resp supersolution of ifuislscin and H x u x p resp forx andp D u x whereD u x denotesthestandardsubdi erentialofuatx D u x fp R n ju y u x hp y xi o jy xj asy xg Forafunctionu R wealsocallitasolutionof ifuisbotha sub andsupersolutionof Wecharacterizethesetofreachablecontrolsinthefollowingway For x T A x a A Thereexistst suchthat X s x g a fors t Weshallderivethe subsolution propertieson Tforsolutionsthrough thefollowingpropositions The rstoneis Lemma Assumethat A holds Letubeasolutionof Assumealsothatu onT Then forx Tandp D u x wehave hg x a pi provideda AnA x Proof Choose C suchthatu attainsitsminimumoverR n at x T u x x andD x p SetX X x g a Sincea AnA x thereexistsft k g k suchthatlim k t k and X t k T k Hence X t k x u X t k Therefore dividingt k andthen sendingk weconcludetheassertion QED Forsimplicity weshallsupposethat A isopen and iscompact For wede neanopensubset fx jdist x g Also foranopensubsetO andx O weusethenotation x O infft jX t x Og WepresenttheDPPfor whoseproofwillbegiveninthe nalsection sinceitisrathercomplicated Theorem cf L Assumethat A and A hold Letu Rbeaboundedsolutionof H x u Du in Then for andx u x inf A Z x e s f X s x s ds e x u X x x Corollary Assumethat A and A hold Fixx T Let u Rbeaboundedsolutionof u x max a A f hg x a Du x ig in Assumealsothatu onTand foranyx Tanda A x liminf s t u X s x t a u x holds wherex t X t x g a fort Then u x hg x a pi forp D u x ProofofCorollary Letx T p D u x anda A x inthe hypothesis Wethensetx t X t x g a forsmallt Choose C suchthatu x x u inR n andD x p Fixsmallt andchoose t suchthatx t for t ByTheorem wehave u x t e x t a e x t a u X x t a x t a whereastandsfortheconstantcontrol a Wenotethatlim x t a t Wealsonotethattheuniquenessofsolutionsof yieldsX x t a x t a x t x t a Takethelimitin mum as togetherwiththeseintheabove toget x t e t x u x t e t u x e t Dividingt andthen sendingt intheabove weconcludetheproof QED Mainresults Inordertoobtaintheuniquenessresult wewillsupposethefollowing continuityassumption Lettingubealscfunctionin wewillsuppose that foranyx Tanda A x A liminf s t u X s x t a forsmallt wherex t X t x g a NoticethatwedonotsupposethatA x inthishypothesis Ouruniquenessresultfor isasfollows Theorem Assumethat A and A hold Letuandv R n R beboundedsolutionsof andsatisfy A Assumealsothatu v onT Then u vinR n ProofofTheorem InviewofLemma andCorollary wesee that u x max a A f hg x a pig forx Tandp D u x ThispropertyenablesustoapplySoravia sresultTheorem in S to concludetheproof QED Thankstotheabovetheorem itiseasytoshowthattherelaxedvalue functionistheuniqueboundedviscositysolutionof satisfying A Tothisend letusintroducetheuniquesolution X x oftheassociated stateequation X t x Z t Z A g X s a d s a ds wheres s M A ismeasurable Here M A isthesetof allRadonprobabilitymeasuresonA Weshalldenoteby Athesetofsuch maps Therelaxedvaluefunctionisasfollows V x inf A e T x where T x infft j X t x Tg Theorem Assumethat A and A hold Then Vistheunique boundedsolutionof satisfying A ProofofTheorem Followingtheargumentin BJ weseethat Vis lscandsatis es inoursensein Tocheckthat A holdsfor V weobservethat forx Tanda A x V X s X t x g a a e t s Hence since V x sendings t weobtain A for V Weremark herethatthenonnegativityof Vindeedyields lim s t V X s X t x g a a Therefore Theorem immediatelyimpliestheassertion QED Remark Ifwehavealscsolutionuof satisfying then u x V x inR n Hence A holdstrueforubySoravia sargumentin S since Valsosatis es Thus throughtheabovetheorem thecondition isequivalentto A See WZ forthesameargument Now weshallshowthatcondition impliesabitstrongerassertion than A Theorem Assumethat A and A hold Letu be aboundedsubsolutionof satisfying andu onT Then for eachx T a A x andsmallt wehave lim s t u X s X t x g a a ProofofTheorem Fixx Tanda A x Asusual wemay supposex andg a e n wheree n Furthermore wemaysupposethatg a e n neartheorigin Indeed settingv y v y y n u X y n y y n g a wehave v y v y n y u X hg X a Du X i De neQ h fx x x n j x n h jx j gforsmallh and x x n x n jx j Sincemin Q h u u the minimumpoint x Q h canbeattainedat x x h Indeed otherwise we have possibilities Incasewhen x n weimmediatelysee u x u Incasewhenj x j holds wealsohave u x u Incasewhen x x x n Q h n thereis suchthat u x e n u x Intheabovethreecases wegetacontradictiontothechoiceofthe minimumpoint x Theremainingcaseisasfollows Incasewhen x Q h thede nitionofsolutionsyields u x x n x whichisacontradiction Therefore taking alongasubsequenceifnecessarily bythelower semicontituityofu we ndu h h whichconcludestheasser tion QED Remark Recently P Soraviakindlyletusknowthatwecaneasilyobtain theaboveassertionusingtheoptimalityprinciplein S Now wegiveanexampleduetoP Soravia Example ForT considerthePDE u u x in RnT Itiseasytoshowthattheunique continuous solutionisgivenby V x e jxj forjxj forjxj Ontheotherhand weobservethatthefollowingfunctionsatis es in V x e x forx forjxj forx Noticethat Vdoesnotsatisfy A atx Thus thisexampleindicates thatitisnecessaryfortheuniquenessresulttosuppose A Wealsonotethat byTheorem Vistheuniquelscsolutionof u u x in Wenextgiveanexample inwhichthereachablesetisnotopenandthe discontinuityappearsin Example ForT fx x R j x g u max u x a x u x in R nT where a x forx x forx forx WeeasilyverifythatthereachablesetR fx jV x gisgiven by f x x j x g Moreover itisnothardtocalculatethevaluefunction V x x forx orx e jx j forx e jx j x forx x e jx j forx NoticethatthediscontinuityofVoccursat x ProofofTheorem ThebasicideaofourproofwasobtainedbyP L Lionsin L forsecond orderPDEs Wealsoreferto EI and BSo But intheirargument we needsomeregularityofsolutions Hence wewilladaptsomeapproximation techniques Letubeasolutionof Weshallextendu withthesamenotation tothewholespacebysettingu x forx We xanyT We rstapproximateubylocallyLipschitzcontinuousfunctions For and x t R n T wede ne u x t inf y R n u y e t jx yj and u x t inf y R n u y e t jx yj Here we x max a A kDg a k Noticethatthe rstoneis Barles convolutionbutthesecondonehasanoppositesignofthepoweron theexponential Itisimmediatetoseethatu u inR n T Wecaneasilyshowtheproperties jx yj e t kuk ifu x t u y e t jx yj jx yj e t kuk ifu x t u y e t jx yj Inviewofthesefacts wede netheconstant c c T e T kuk Weclaimthatthefollowingpropertieshold ForsomeC andC independentof and u x t q max a A f hg x a pi f x a g C e t for x t c T and p q D u x t and u x t q max a A f hg x a pi f x a g C e t for x t c T and p q D u x t Here D u x t D u x t Wenotethat holdsinalargersetthan c butthisissu cientto concludetheproof Althoughitisnothardtoshow and bytheargumentin B togetherwith wegiveabriefproofforthereader sconvenience Since canbeobtainedeasilybyremarkingthesignofthepoweron e weshallonlyshow Seealsoourprooffor below LetusrecalltheBarron Jensenlemma whichwillbeneededalsofor checkingthesignofqin Lemma See BJ or K Fix x t R n T and p q D u x t Forany thereexist x k t k R n T p k q k D u x k t k fork f n g withsomen N x t R n T C andf k g n k suchthat i lim jx k x j ii lim x t k x t k n iii jp k j C k n iv n X k k v lim n X k k p k q k p q For x t c T and p q D u x t in weshallchoose x k t k etc inLemma Sincewemaysupposex k c forsmall inviewof wecan choosey k suchthat u x k t k u y k e t k jx k y k j Sincep k D u y k thede nitionyields u y k max a A f hg y k a p k i f y k a g Notingp k e t k x k y k wecalculateinthefollowingway u y k max a A f hg x k a p k i f y k a g max a A kDg a k e t k jx k y k j u y k max a A f hg x a p k i f x k a g max a A kDg a k e t k jx k y k j jx k x jjp k j max a A kDf a k jx k y k j Sincewemayalsosuppose e t k jx k y k j q k forsmall by and iii of wecan ndC suchthat u x k t k q k max a A f hg x a p k i f x k a g C max a A kDg a k jx k x j C e t k max a A kDg a k e t k jx k y k j Fromthechoiceof weseethatthelasttermintherighthandsideof theaboveisnonnegative Takingtheconvexcombinationwithf k g n k andthen sending with i ii iv v of intheabove wehave u x t q max a A f hg x a pi f x a g C e t Now for wechoose C R n suchthat inR n in and in c Wesetthefunctions g x a x g x a f x t a x f x a C e t x u x t f x t a x f x a C e t x u x t Wethenconsidertheproblems For x t p q R n T R n R u u t H x t Du and u u t H x t Du where H x t p max a A f hg x a pi f x t a g H x t p max a A f hg x a pi f x t a g Inwhatfollows wesupposethat c Weclaimthatu andu respectively arethestandardviscositysubsolu tionandsupersolutionofu u t H andu u t H inR n T For x t R n T u x t q H x t p provided p q D u x t and u x t q H x t p provided p q D u x t Indeed itisimmediatetocheckthatu andu respectively satisfythat for x t R n T u x t x q H x t p provided p q D u x t and u x t x q H x t p provided p q D u x t Here wehaveusedthefact x forx c We rstshow Since p q D u x t fromthede nition wehaveq e t jx yj forsomey Hence weconcludeourclaimbecause Thus for itissu centtoshowthatq provided p q D u x t Thisisnotstraightforwardunlikethatfor However inviewof iv and v ofLemma qcanbeapproximatedby P n k k q k as for p k q k D u x k t k withappropriate x k t k Hence wecanseethatq k e t k jx k yj forsomey Therefore q Now weshallgivethevaluefunctionsu andu respectively for and withinitialconditionu andu u x t inf A Z t e s f X s x t s s ds e t u X t x and u x t inf A Z t e s f X s x t s s ds e t u X t x Sincef x t f x t C e t C e t andu u thereexists C suchthat u x t u x t C e t inR n T Wealsoremarkthatu andu areboundedandcontinuous Hence thestandardcomparisonprincipleyieldsthat u x t u x t andu x t u x t inR n T Fixx andchoose sothatx Then theDPPforu at x T with and impliesthat u x T inf A e x T u X x T x T x Z x T e s f X s x s ds inf A e x T u X x T x T x Z x T e s f X s x s ds C e T u x T C e T Wenotethat foreach A and imply u X x T x lim u X x T x T x lim u X x T x T x Therefore sending with in wehave u x inf A e x T u X x T x Z x T e s f X s x s ds Finally sendingT weconcludetheproof QED References Ba M Bardi Aboundaryvalueproblemfortheminimumtime function SIAMJ Control BC M Bardi I CapuzzoDolcetta OptimalControland ViscositySolutionsofHamilton Jacobi BellmanEquations Birkh auser BF M Bardi M Falcone Anapproximationschemeforthe minimumtimefunction SIAMJ ControlOptim BS M Bardi V Staicu TheBellmanequationfortime optimalcontrolofnoncontrollablenonlinearsystems Acta ApplicandaMathematicae BSo