the maximum principle mixed inequality constraints

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1、Chapter3TheMsLximumPrinciple:MixedInequalityConstraintsTheproblemstowhichthemaximumprinciplederivedinthepreviouschapterwasapphcablehadconstraintsinvolvingonlythecontrolvari•ables.Weshallseethatinmanyappliedmodelsitisnecessarytoimposeconstraintsinvolvingbothcontrolandstatevariables.I

2、nequalitycon•straintsinvolvingcontrolandpossiblystatevariablesarecalledmixedinequalityconstraints,Inthesolutionspacesofproblemswithmixedconstraints,theremayberegionsinwhichoneormoreoftheconstraintsistight.Whenthishappens,thesystemmustbecontrolledinsuchawaythatthetightconstraintsaren

3、otviolated.Asaresult,themaximumprincipleofChapter2mustberevisedsothattheHamiltonianismaximizedsubjecttotheconstraints.ThisisdonebyappendingtheHamiltonianwiththemixedconstraintsandtheassociatedLagrangemultipMerstoformaLagrangian,andthensettingthederivativesoftheresultingLagrangianwit

4、hrespecttothecontrolvariablestozero.InSection3.1,aLagrangianformofthemaximumprincipleisdis•cussedformodelsinwhichtherearesomeconstraintswhichinvolveonlycontrolvariables,andotherswhichinvolvebothstateandcontrolvari•ablessimultaneously.Problemshavingpiirestatevariableconstraints,i.e.,

5、thoseinvolvingstatevariablesbutnocontrolvariables,wiUbedealtwithinChapter4.InSection3.2,westateconditionsimderwhichtheLagrangianmax-583.TheMaximumPrinciple:MixedInequalityConstraintsimumprincipleisalsosufficientforoptimality.Economistsfrequentlyanalyzecontrolmodelshavinganinfiniteho

6、ri•zontogetherwithacontinuousdiscountrate.BycombiningthediscountfactorwiththeadjointvariablesandtheLagrangemultipliersandmak•ingsuitablechangesinthedefinitionsoftheHamiltonianandLagrangianfunctions,itispossibletoderivethecurrent-valueformulationofthemaximumprincipleasdescribedinSect

7、ion3.3.TerminalconditionsforeachoftheabovemodelsarediscussedinSection3.4,andthemodelswithinfinitehorizonsandtheirstationaryequilibriumsolutionsarecoveredinSection3.5.Section3.6presentsaclassificationofanumberofthemostimportantandcommonlyusedkindsofoptimalcontrolmodels,togetherwithab

8、riefdescriptionofth