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1、HJBEquationsfortheOptimalControlofDifferentialEquationswithDelaysandStateConstraints,I:RegularityandApplications.∗SalvatoreFederico†BenGoldys‡FaustoGozzi§July9,2009AbstractWestudyaclassofoptimalcontrolproblemswithstateconstraintswherethestateequationisadifferentialequationwi
2、thdelays.Thisclassincludessomeproblemsarisingineconomics,inparticulartheso-calledmodelswithtimetobuild,see[1,2,25].WeembedtheprobleminasuitableHilbertspaceHandconsidertheassociatedHamilton-Jacobi-Bellman(HJB)equation.Thiskindofinfinite-dimensionalHJBequationhasnotbeenprevio
3、uslystudiedandisdifficultduetothepresenceofstateconstraintsandthelackofsmoothingpropertiesofthestateequation.OurmainresultontheregularityofsolutionstosuchaHJBequationseemstobecompletelynew.MorepreciselyweprovethatthevaluefunctioniscontinuousinasufficientlybigopensetofH,thatits
4、olvesintheviscositysensetheassociatedHJBequationandithascontinuousclassicalderivativeinthedirectionofthe“present”.Thisregularityresultisthestartingpointtodefineafeedbackmapinclassicalsense,whichgivesrisetoacandidateoptimalfeedbackstrategyfortheproblem.Thestudyofverificationt
5、heoremsandoftheclosedloopequationwillbethesubjectoftheforthcomingpaper[20].Keywords:Hamilton-Jacobi-Bellmanequation,optimalcontrol,delayequations,viscosityso-lutions,regularity.A.M.S.SubjectClassification:34K35,49L25,49K25.arXiv:0905.1825v2[math.OC]9Jul2009∗Thisworkwasparti
6、allysupportedbyanAustralianResearchCouncilDiscoveryProject†SalvatoreFederico,DipartimentodiScienzeEconomicheedAziendali,Facolt`adiEconomia,LiberaUniversit`ainternazionaledeglistudisociali“GuidoCarli”,vialeRomania32,00197Roma,Italy.Email:sfedericoatluiss.it.‡BenGoldys,Schoo
7、loofMathematicsandStatistics,UniversityofNewSouthWales,Sydney,Australia.Email:B.Goldysatunsw.edu.au.§FaustoGozzi(correspondingauthor),DipartimentodiScienzeEconomicheedAziendali,Facolt`adiEconomia,LiberaUniversit`ainternazionaledeglistudisociali“GuidoCarli”,vialeRomania32,0
8、0197Roma,Italy.Email:fgozziatluiss.it.1Contents1Introduction22Setupofthecontrolproblemand
