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1、Chapter12DifferentialGaines,DistributedSystems,andImpulseControlInpreviouschapters,weweremainlyconcernedwiththeoptimalcontrolproblemsformulatedinChapters3and4andtheirapplicationstovari•ousfunctionalareasofmanagementandtosomeproblemsineconomics.Theseproblemsweredescribedbyasingleobject
2、ivefunction(orasingledecisionmaker)andasetofordinarydifferentialequations,calledthestateequations,definedinadeterministicframework.Inthischapter,wedealwithgeneralizationsofthe(ordinary)deter•ministicoptimalcontrolproblemsthatcanbemadeinoneormoreofthefollowingdirections.SeeChapter13for
3、stochasticoptimalcontrolproblems.Theremaybemorethanonedecisionmaker,eachhavingseparateobjectivefunctionswhicheachistryingtomaximize,subjecttoasetofdifferentialequations.Thisextensionoftheoptimalcontroltheoryisreferredtoasthetheoryofdifferentialgames.Section12.1containsabriefintroducti
4、ontodifferentialgamesalongwithanapplication.Anotherextensionreplacesthesystemofordinarydifferentialequa^tionsbyasetofpartialdifferentialequations.ThesecomeundertheclassificationofdistributedparametersystemsandaretreatedinSection12.2.FinallyinSection12.3,wetreatthetheoryofim,pulsecontr
5、ol.Thiscontrolisdevelopedtodealwithsystemswhich,inadditiontoconven•tionalcontrols,aUowacontrollertomakediscretechangesinthestate30812.DifferentialGames,DistributedSystems,andImpulseControlvariablesatselectedinstantsoftimeinanoptimalfashion.Thistheoryisespeciallyusefulindealingwithinve
6、ntoryproblemssuchasdynamiclotsizeproblems.Italsoallowasetupcostassociatedwiththediscretechangesinthestatevariablestobechargedtosuchdiscreteactionsassettingupamachinetoolfrommakingoneproducttomakinganother,openingawarehouse,etc.Extensionsofoptimalcontrolproblemswhereuncertaintiesarepre
7、sentwillbediscussedinthenextchapter.12.1DifferentialGainesThestudyofdifferentialgameswasinitiatedbyIsaacs(1965).AfterthedevelopmentofPontryagin'smaximumprinciple,itbecameclearthattherewasaconnectionbetweendifferentialgamesandoptimalcontroltheory.Infact,diflFerentialgameproblemsreprese
8、ntage
