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1、Chapter2TheMaximumPrinciple:ContinuousTimeThemainpurposeofthischapteristointroducethemaximumprincipleasanecessaryconditionthatmustbesatisfiedbyanyoptimalcontrolforthebasicproblemspecifiedinSection2.1.Althoughvectornotationisused,thereadercanthinkoftheproblemashavingonlyasinglestatevariab
2、leandasinglecontrolvariableonthefirstreading.InSection2.2,themethodofdynamicprogrammingisusedtoderivethemaximumprinciple.Weusethismethodbecauseofthesimplicityandfamiliarityofthedynamicprogrammingconcept.Thederivationalsoyieldssignifi•canteconomicinterpretations.(InAppendixC,themaximumpri
3、ncipleisalsoderivedbyusingamoregeneralmethodsimilartothatofPontrya-ginetal.(1962),butwithcertainsimplifications.)InSection2.4,themaximumprincipleisshowntobesufficientforoptimalcontrolimderanappropriateconcavitycondition,whichholdsinmanymanagementscienceapphcations.2.1StatementoftheProble
4、mOptimalcontroltheorydealswiththeproblemofoptimizationofdy•namicsystems.Theproblemmustbewellposedbeforeanysolutioncanbeattempted.Thisrequiresaclearmathematicaldescriptionofthesystemtobeoptimized,theconstraintsimposedonthesystem,andtheobjectivefunctiontobemaximized(orminimized).242.TheMaj
5、dmumPrinciple:ContinuousTime2.1.1TheMathematicalModelAnimportantpartofanycontrolproblemistheprocessofmodelingthedynamicsystem,physical,business,orotherwise,underconsideration.Theaimistoarriveatamathematicaldescriptionwhichissimpleenoughtodealwith,andrealisticenoughtobeabletopredicttheres
6、ponseofthesystemtoanygiveninput.Themodelofthesystemforourpurposeisrestrictedtosystemsthatcanbecharacterizedbyasystemofordinarydifferentialequations(or,ordinarydifferenceequationsinthediscrete-timecasetreatedinChapter8).Thus,giventheinitialstateXQofthesystemandcontrolhistoryu(t),tG[0,T],o
7、ftheprocess,theevolutionofthesystemmaybedescribedbythefirst-orderdifferentialequation,knownalsoasthestateequation,x{t)=f{x{t),u{t),t),:r(0)=XQ,(2.1)wherethevectorofstatevariables,x(t)€E'^,thevectorofcontrolvariables,u{t)€E"^,and/:E"^xE"^xE^-^E"^.Furthermore,thefunction/is
