an introduction to mathematical optimal control theory version (evans)

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1、AnIntroductiontoMathematicalOptimalControlTheoryVersion0.2ByLawrenceC.EvansDepartmentofMathematicsUniversityofCalifornia,BerkeleyChapter1:IntroductionChapter2:Controllability,bang-bangprincipleChapter3:Lineartime-optimalcontrolChapter4:ThePontryaginMaximumPrincipleCha

2、pter5:DynamicprogrammingChapter6:GametheoryChapter7:IntroductiontostochasticcontroltheoryAppendix:ProofsofthePontryaginMaximumPrincipleExercisesReferences1PREFACEThesenotesbuilduponacourseItaughtattheUniversityofMarylandduringthefallof1983.MygreatthanksgotoMartinoBard

3、i,whotookcarefulnotes,savedthemalltheseyearsandrecentlymailedthemtome.FayeYeagertypeduphisnotesintoafirstdraftoftheselecturesastheynowappear.Ihaveradicallymodifiedmuchofthenotation(tobeconsistentwithmyotherwritings),updatedthereferences,addedseveralnewexamples,andprovid

4、edaproofofthePontryaginMaximumPrinciple.Asthisisacourseforundergraduates,Ihavedispensedincertainproofswithvariousmeasurabilityandcontinuityissues,andascompensationhaveaddedvariouscritiquesastothelackoftotalrigor.ScottArmstrongreadoverthenotesandsuggestedmanyimprovemen

5、ts:thanks.AndStephenMoyeoftheAmericanMathSocietyhelpedmealotwithAMSTeXversusLaTeXissues.Thiscurrentversionofthenotesisnotyetcomplete,butmeetsIthinktheusualhighstandardsformaterialpostedontheinternet.IhaveaddedforVersion0.2aroughbutinformativecalculationforvariationsin

6、thenewSectionA.1,addedmorereferencesandfixedvarioustypos.Pleaseemailmeatevans@math.berkeley.eduwithanycorrectionsorcomments.2CHAPTER1:INTRODUCTION1.1.Thebasicproblem1.2.Someexamples1.3.Ageometricsolution1.4.Overview1.1THEBASICPROBLEM.DYNAMICS.Weopenourdiscussionbyconsi

7、deringanordinarydifferentialequation(ODE)havingtheformx˙(t)=f(x(t))(t>0)(1.1)0x(0)=x.Weareheregiventheinitialpointx0∈Rnandthefunctionf:Rn→Rn.Theun-knownisthecurvex:[0,∞)→Rn,whichweinterpretasthedynamicalevolutionofthestateofsome“system”.CONTROLLEDDYNAMICS.Wegeneralize

8、abitandsupposenowthatfdependsalsouponsome“control”parametersbelongingtoasetA⊂Rm;sothatf:Rn×A→Rn.Thenifweselectsomevaluea∈Aan