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1、SIAMJ.NUMER.ANAL.c2010SocietyforIndustrialandAppliedMathematicsVol.48,No.4,pp.1492–1517INEXACTRESTORATIONFORRUNGE–KUTTADISCRETIZATIONOFOPTIMALCONTROLPROBLEMS∗C.YALC¸INKAYA†Abstract.AnumericalmethodispresentedforRunge–Kuttadiscretizationofunconstrainedoptimalcontrolproble
2、ms.First,generalRunge–Kuttadiscretizationiscarriedouttoobtainafinite-dimensionalapproximationofthecontinous-timeoptimalcontrolproblem.Thenarecentoptimiza-tiontechnique,theinexactrestoration(IR)method,duetoMart´ınezandcoworkers[E.G.BirginandJ.M.Mart´ınez,J.Optim.TheoryAppl.
3、,127(2005),pp.229–247;J.M.Mart´ınezandE.A.Pilotta,J.Optim.TheoryAppl.,104(2000),pp.135–163;J.M.Mart´ınez,J.Optim.TheoryAppl.,111(2001),pp.39–58],isappliedtothediscretizedproblemtofindanapproximatesolution.Itisprovedthat,foroptimalcontrolproblems,akeysufficiencyconditionforco
4、nvergenceoftheIRmethodisreadilysatisfied.Underreasonableassumptions,theIRmethodforoptimalcontrolprob-lemsisshowntoconvergetoasolutionofthediscretizedproblem.Convergenceofasolutionofthediscretizedproblemtoasolutionofthecontinuous-timeproblemisalsoshown.Itturnsoutthatoptimal
5、ityphaseequationsoftheIRmethodemanatefromanassociatedHamiltoniansystem,andsogeneralRunge–KuttadiscretizationinducesasymplecticpartitionedRunge–Kuttascheme.Acomputationalalgorithmisdescribed,andnumericalexperimentsaremadetodemonstratetheworkingofthemethodforoptimalcontrolo
6、fthevanderPolsystem,employingthethree-stage(order6)Gauss–Legendrediscretization.Keywords.optimalcontrol,inexactrestoration,Runge–Kuttadiscretization,symplecticpar-titionedRunge–Kuttascheme,Lagrangemultiplierupdate,costateupdate,vanderPolsystemAMSsubjectclassifications.49K1
7、5,49M05,49M25,65K10,65L06DOI.10.1137/0907666681.Introduction.Itiscommonpracticetodiscretizeacontinuous-timeoptimalcontrolprobleminordertoperformcomputationsandtofindatleastanapproximatesolution.Onetypicallydiscretizes(i.e.,partitions)thetimehorizonoftheoptimalcontrolproble
8、mandcomputesoptimalvaluesofthestatesandcontrolsatthediscretetimepointsbyapplyingastandardfinite-d
