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1、StochasticOptimalControlProblemswithaBoundedMemory∗Mou-HsiungChang†TaoPang‡MoustaphaPemy§June2,2006AbstractThispapertreatsafinitetimehorizonoptimalcontrolprobleminwhichthecontrolledstatedynamicsisgovernedbyageneralsystemofstochasticfunctionaldifferentialequat
2、ionswithaboundedmemory.Aninfinite-dimensionalHJBequationisderivedusingaBellman-typedynamicprogrammingprinciple.ItisshownthatthevaluefunctionistheuniqueviscositysolutionoftheHJBequation.Inaddition,thecomputationissuesarealsostudied.Moreparticularly,afinitediffe
3、renceschemeisobtainedtoapproximatetheviscositysolutionoftheinfinitedimensionalHJBequation.TheconvergenceoftheschemeisprovedusingtheBanachfixedpointtheorem.Thecomputationalalgorithmisalsoprovidedbasedontheschemeobtained.Keywords:Stochasticcontrol,stochasticfun
4、ctionaldifferentialequations,viscositysolutions,finitedifferenceapproximation.AMS2000subjectclassifications:primary93E20,60H35;secondary34K50,49L25∗TheresearchofthispaperispartiallysupportedbyagrantW911NF-04-D-0003fromtheU.S.ArmyResearchOffice†MathematicsDivision
5、,U.S.ArmyResearchOffice,P.O.Box12211,RTP,NC27709,USA,mouhsiung.chang@us.army.mil‡DepartmentofMathematicsandCenterforResearchinScientificComputation,NorthCarolinaStateUniversity,Raleigh,NC27695USA,tpang@unity.ncsu.edu.Correspondingauthor.§DepartmentofMathematic
6、sandCenterforResearchinScientificComputation,NorthCarolinaStateUniversity/SAMSI,Raleigh,NC27695USA,mnpemy@unity.ncsu.edu11IntroductionThetheoryofstochasticfunctionaldifferentialequationshasbeenwidelyusedtodescribethestochasticsystemswhoseevolutiondependonthep
7、asthistoryofthestate.Ithasmanyapplicationsinrealworldapplications(seeMohammed[20],[21]andKolmanovskiiandShaikhet[13]forbasictheoryandsomeapplications).Thelinear-quadraticregulatoryprobleminvolvingstochasticdelayequationswasfirststudiedinKolmanovskiiandMaizen
8、-berg[12],andoptimalcontrolproblemsforaclassofnonlinearstochasticequationsthatinvolveacontinuousdelayofthefollowingtypedX(s)=α(s,X(s),Y(s),u(s))ds+β(s,X(s),Y(s),u(s))dW(s),s∈[t,T],(1)havebeenstudiedinr