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1、Chapter8TheMaximumPrinciple:DiscreteTimeFormanypurposesitisconvenienttoassumethattimeisrepresentedbyadiscretevariable,fc=0,1,2,...,T,ratherthanbyacontinuousvariabletG[0,T].Thisisparticularlytruewhenwewishtosolvealargecontroltheoryproblembymeansofacomputer.Itisalsodesirable,evenwhensol
2、vingsmallproblemswhichhavestateoradjointdifferentialequationswhosesolutionscannotbeexpressedinclosedform,toformulatethemasdiscreteproblems,andletthecomputersolvetheminastepwisemanner.Wewillseethatthemaximumprinciple,whichistobederivedinthischapter,isnotvalidforthediscrete-timeproblemi
3、naswideasenseasforthecontinuous-timeproblem.Infactwewillreduceittoanonlinearprogrammingproblemandstatenecessaryconditionsforitssolutionbyusingthewell-knownKuhn-Tuckertheorem.Inordertofollowthisprocedure,wehavetomakesomesimplifyingassimiptionsandhencewillobtainonlyarestrictedformofthed
4、iscretemaximumprinciple.InSection8.2.5westatewithoutproofamoregeneralformofthediscretemaximumprinciple.8.1NonlinearProgrammingProblemsWebeginbystatingageneralformofanonlinearprogrammingproblem.Letybeann-componentcol"anmvector,abeanr-componentcolumnvector,andban5-componentcolumnvector.
5、Leth:E'^-^E^^g:E'^-^E'^,andw:E'^-^E^hegivenfunctions.Weassumefimctionsgandwtobecolimrmvectorswithcomponentsrands,respectively.We2188,TheMaximumPrincipal:DiscreteTimeconsiderthenonlinearprogrammingproblem:maxh{y)(8.1)subjectto9{y)=a,(8.2)w{y)>h.(8.3)Wewilldevelopnecessaryconditions,cal
6、ledtheKnhn-Tuckerconditions,whichasolutiony*tothisproblemmustsatisfy.Westartwithsimplerproblemsandworkuptothestatementoftheseconditionsforthegen•eralprobleminaheuristicfashion.Referencesaregivenforrigorousdevelopmentsoftheseresults.Inthischapter,wheneverwetakederivativesoffunctions,we
7、assimiethatthosederivativesexistandarecontinuous.ItwouldbealsohelpfultorecallthenotationdevelopedinSection1.4.8.1.1LagrangeMultipliersSupposewewanttosolve(8.1)withoutimposingconstraints(8.2)or(8.3).Theproblemisnowtheclassicalunconstrainedmaximizationproblemofcalculus,andthefirst-order
8、neces
