A new method for optimal control of Volterra integ

AnewmethodforoptimalcontrolofVolterraintegralequationsS.A.BelbasMathematicsDepartmentUniversityofAlabamaTuscaloosa,AL.35487-0350.USA.e-mail:[e-mail]wmethodforsolvingoptimalcontrolproblemsforsystemsgovernedbyVolterraintegralequations.OurmethodutilizesdiscretizationoftheoriginalVolterracontrolledsystemandanoveltypeofdynamicprogrammingjnwhichtheHamilton-Jacobifunctionisparametrizedbythecontrolfunction(ratherthanthestate,asinthecaseofordinarydynamicprogramming).Wealsoderiveestimatesforthecomputationalcostofourmethod.Keywords:Optimalcontrol,Volterraintegralequation,discreteapproximation.AMSSubjectClassification:49K22,49M25.1.Introduction.Theclassicaltheoryofoptimalcontrolwasoriginallydevelopedtodealwithsystemsofcontrolledordinarydifferentialequations.Ithasbeenunderstoodthatmanyphysical,technological,biological,andsocio-economicproblemscannotbeadequatelydescribedbyordinarydifferentialequations,andothermathematicalmodels,includingsystemswithmemory,distributedsystems,andothertypesofsystems,havebeenaddedtothearsenalofthetheoryofoptimalcontrol.AbroadcategoryofsystemscanbedescribedbyVolterraintegralequations.ThesimplestformofacontrolledVolterraintegralequationisds))s(u),s(x,s,t(f)t(x)t(xt00∫+=---(1.1)Inthissystem,x(t)isthen-dimensionalstatefunction,andu(t)isthem-dimensionalcontrolfunction.Forthepurposesofthisexposition,wepostulatethatfbecontinuouswithrespecttoallvariablesanduniformlyLipschitzwithrespecttox.Forthepurposesofdescribingthenecessaryconditionsthatarebrieflyreviewedinthissection,theadmissiblecontrolfunctionsarecontinuousfunctionswithvaluesinacompactsetU,nR⊆U.IncertainpartsoftheoveralltheoryofoptimalcontrolforVolterraintegralequations,theclassofadmissiblecontrolfunctionscanbemoregeneral,forexampleitmayconsistofboundedmeasurableorp-integrablefunctions.Ontheotherhand,asitwillbeexplainedbelow,forsomeoftheresultsofthepresentpaperitisnecessarytofurtherrestricttheclassofadmissiblecontrolsandpostulateLipschitzcontinuity.Volterraintegralequationsariseinawidevarietyofapplications.Infact,itseemsthat,withtheexceptionofthesimplestphysicalproblems,practicallyeverysituationthatcanbemodelledbyordinarydiffrentialequationscanbeextendedtoamodelwithVolterraintegralequations.Forexample,ageneralODEsystemofinteractingbiologicalpopulations,oftheform0,iijijjkjijkk,jix)0(x;)t(xb)t(x)t(xadt)t(dx=+=∑∑canbeextendedtoanintegro-differentialsystem0,iijijjkjijkk,jt0jijjkjijkk,jix)0(x;ds)s(x)s,t(B)s(x)s(x)s,t(A)t(xb)t(x)t(xadt)t(dx=++++=∑∑∫∑∑Indeed,somerelatedextensionshavealreadybeenconsideredin[CU]andinotherworks.Inturn,everyintegrodifferentialsystemoftheform0t0x)0(x;ds))s(x,s,t(f),t(x,tgdt)t(dx==∫canbereducedtoasystemofVolterraintegralequationsbysettingds)s(x,s,t(f)t(yt0∫=,thentheintegrodifferentialequationbecomesds))s(y),s(x,s(gx)t(xt00∫+=,sothatwegetasystemofVolterraintegralequationsintheunknowns(x(t),y(t)).ProblemsinmathematicaleconomicsalsoleadtoVolterraintegralequations.Therelationshipsamongdifferentquantities,forexamplebetweencapitalandinvestment,includememoryeffects(e.g.thepresentstockofcapitaldependsonthehistoryofinvestmentstrategiesoveraperiodoftime,cf.[KM]),andthesimplestwaytodescribesuchmemoryeffectsisthroughVolterraintegraloperators.Nowwereturntothegeneralmodelofstatedynamics(1.1).Anoptimalcontrolproblemfor(1.1)concernstheminimizationofacostfunctionaldt))t(u),t(x,t(F))T(x(F:JT00∫+=---(1.2)Thetheoryofoptimalcontrolofordinarydifferentialequationshastwomainmethods:theextremum(usuallycalledmaximum)principleofPontryaginandhiscoworkers,andthemethodofdynamicprogramming.Themetodofdynamicprogrammingisparticularlyusefulasitprovidessufficientconditionsforoptimality.However,thenatureofcontrolledVolterraequationsisnot,atfirstglance,conducivetotheapplicationofdynamicprogrammingmethods.Ifthestatex(t)isknownatsomeparticulartimet,andacontrolfunctionisspecifiedoveraninterval]tt,t(δ+,thesetwobitsofinformationarenotenoughforthedeterminationofthesolutionof(1.1)overtheinterval]tt,t(δ+.Bycontrast,forordinarydifferentialequations,itisalwaystruethat,givenx(t)andacontrolfunctionover]tt,t(δ+,thetrajectoryover]tt,t(δ+canbedeterminedbysolvinganinitialvalueproblemforanordinarydifferentialequationwithinitialtimet.Forthesereasons,optimalcontrolproblemsforVolterraintegralequationshavebeentraditionallytreatedbyextensionsofPontryagin'sextremumprinciple.Therelatedresultsarefoundinanumberofpapers,including[M,S,V,NW];anapproachbasedondirectvariationalmethods,butstillutilizingnecessaryconditionsforoptimality,maybefoundin[B].Theco-state)t(ψfortheproblemconsistingof(1.1)and(1.2)satisfiesthefollowingadjointequation,whichisthecounterpartofHamiltonianequations(see,e.g.,[S,V]):()ds))t(u),t(x,t,s(f)s())t(u),t(x,t,T(f))T(x(F))t(u),t(x,t(F)t(xTtx0xx∇ψ++∇∇+∇=ψ∫---(1.3)Thestatex(.)isann-dimensionalcolumnvector;thefunctionftakesvaluesthataren-dimensionalcolumnvectors;theco-stateψ(.)isann-dimensionalrowvector.Thegradient,withrespecttox,ofascalar-valuedfunctionisann-dimensionalrowvector;fx∇isamatrixwithelementsjixf∂∂(eac