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1、ANINTRODUCTIONTOMATHEMATICALOPTIMALCONTROLTHEORYVERSION0.1ByLawrenceC.EvansDepartmentofMathematicsUniversityofCalifornia,BerkeleyChapter1:IntroductionChapter2:Controllability,bang-bangprincipleChapter3:Lineartime-optimalcontrolChapter4:ThePontryaginMaximumPrincipleChapter5:Dynamicprogrammi
2、ngChapter6:GametheoryChapter7:IntroductiontostochasticcontroltheoryAppendix:ProofsofthePontryaginMaximumPrincipleExercisesReferences1PREFACEThesenotesbuilduponacourseItaughtattheUniversityofMarylandduringthefallof1983.MygreatthanksgotoMartinoBardi,whotookcarefulnotes,savedthemalltheseyears
3、andrecentlymailedthemtome.FayeYeagertypeduphisnotesintoa?rstdraftoftheselecturesastheynowappear.Ihaveradicallymodi?edmuchofthenotation(tobeconsistentwithmyotherwrit-ings),updatedthereferences,addedseveralnewexamples,andprovidedaproofofthePontryaginMaximumPrinciple.Asthisisacourseforundergr
4、aduates,Ihavedispensedincertainproofswithvariousmeasurabilityandcontinuityissues,andascompensationhaveaddedvariouscritiquesastothelackoftotalrigor.ScottArmstrongreadoverthenotesandsuggestedmanyimprovements:thanks.Thiscurrentversionofthenotesisnotyetcomplete,butmeetsIthinktheusualhighstanda
5、rdsformaterialpostedontheinternet.Pleaseemailmeatevans@math.berkeley.eduwithanycorrectionsorcomments.2CHAPTER1:INTRODUCTION1.1.Thebasicproblem1.2.Someexamples1.3.Ageometricsolution1.4.Overview1.1THEBASICPROBLEM.DYNAMICS.Weopenourdiscussionbyconsideringanordinarydi?erentialequation(ODE)havi
6、ngtheformx˙(t)=f(x(t))(t>0)(1.1)0x(0)=x.Weareheregiventheinitialpointx0∈Rnandthefunctionf:Rn→Rn.Theunknownisthecurvex:[0,∞)→Rn,whichweinterpretasthedynamicalevolutionofthestateofsome“system”.CONTROLLEDDYNAMICS.Wegeneralizeabitandsupposenowthatfdependsalsouponsome“control”parametersbelongi
7、ngtoasetA?Rm;sothatf:Rn×A→Rn.Thenifweselectsomevaluea∈Aandconsiderthecorrespondingdynamics:x˙(t)=f(x(t),a)(t>0)x(0)=x0,weobtaintheevolutionofoursystemwhentheparameterisconstantlysettothevaluea.Thenextpossibilityisthatwechangethevalueoftheparameterasthesysteme
