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1、ChapterOptimalControlTheoryAttheendofChap.13,werefern;dtodynamicoptimizationasatypeofproblemwev.'erenotreadytotacklebecausewedidnotyethavethetoolsofdynamicanalysissuchasdifferentialequations.Nowthatwehaveacquiredsuchtools,wecanfinallytryatasteofdynamicoptimization.Theclassicalap
2、proachtodynamicoptimizationiscalledthecalculus0/variations.Inthelaterdevelopmentofthismethodology,however,amorepowerfulapproachknO'Wllasoptimal('ol1lroltheoryhas.forthemostpart.supplantedthecalculusofvariations.Forthisreason,weshall,inthischapter,contineourattentiontooptimal(;on
3、tl'Oltheory,explainingitsbasicnature,introducingthemajorsolutiontoolcalledthemaximumprlndple,andtillustratingitsuseinsomeelementaryeconomicmodels.20.1TheNatureofOptimalControlInstaticoptimization,thetaskistofindasinglevalueforeachchoicevariable,suchthatastatedobjectivefunctionwi
4、llbemaximizedorminimized,asthecasemaybc.Suchaprob•lemisdevoidofatimedimension.Incontrast,timeentersexpllcitlyandprominentlyinadynamicoptimizationproblem.Insuchaproblem,wevillalwayshaveinmindaplanningperiod,sayfromaninitialtimct=0toatenninaltimet:;;;;;;T,andtrytofindthebestcours
5、eofactiontotakeduringthatentireperiod.Thusthesolutionforanyvariablewilltaketheformofnotasinglevalue,butacompletetimepath.Supposetheproblemisoneofprofitmaximir.ationOVl;ratimeperiod.AtanypointoftimeI,wehavetoChOOSlCthevalueofsomecontrolvariahle,u(t),which",illthenaffectthevalueo
6、fsomcstatevariable.y(t),viaaso~calledequation1-{motion.Inturn,y(r)willdeterminetheprofitJ!'(t).Sinceourobjectiveistomaximizetheprofitovertheentirepe~riod,theobjectivefunctionshouldtaketheformofadefiniteintegralofJ!'from(=0tot::;;T.Tobecomplete,theproblemalsospecifiestheinitialva
7、lueofthestatevariabley.tForamorecompletetreatmentofQptimi;llcontroltheory(a~wellas"calculusofvariation;"),thestudentisreferredtoElement5ofDynamiCOptimizationbyAlphaC.Chiang,McGraw.Hill,NewYork,1992,nowpublishedbyWavelandPress,Inc.,ProspectHeights,Illinoi5.Thischapterdrawsheavily
8、frommaterialinthiscitedbook.631632PartFiveDynrm
