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1、cvoc-formattedAugust24,20117x10CalculusofVariationsandOptimalControlTheorycvoc-formattedAugust24,20117x10cvoc-formattedAugust24,20117x10CalculusofVariationsandOptimalControlTheoryAConciseIntroductionDanielLiberzonPRINCETONUNIVERSITYPRESSPRINCETONANDOXFORDCopyright©2012byPrincetonUniversityPressP
2、ublishedbyPrincetonUniversityPress,41WilliamStreet,Princeton,NewJersey08540IntheUnitedKingdom:PrincetonUniversityPress,6OxfordStreet,Woodstock,OxfordshireOX201TWAllRightsReservedISBN:978-0-691-15187-8LibraryofCongressControlNumber:2011935625BritishLibraryCataloging-in-PublicationDataisavailableT
3、hisbookhasbeencomposedinLATEXThepublisherwouldliketoacknowledgetheauthorofthisvolumeforprovidingthedigitalfilesfromwhichthisbookwasprintedPrintedonacid-freepaper∞press.princeton.eduPrintedintheUnitedStatesofAmerica10987654321cvoc-formattedAugust24,20117x10Sincethebuildingoftheuniverseisperfectan
4、discreatedbythewisdomcreator,nothingarisesintheuniverseinwhichonecannotseethesenseofsomemaximumorminimum.
5、LeonhardEulerThewordscontroltheory"are,ofcourse,ofrecentorigin,butthesubjectitselfismucholder,sinceitcontainstheclassicalcalculusofvariationsasaspecialcase,andthe rstcalculusofvariationspro
6、blemsgobacktoclassicalGreece.
7、HectorJ.Sussmanncvoc-formattedAugust24,20117x10cvoc-formattedAugust24,20117x10ContentsPrefacexiii1Introduction11.1Optimalcontrolproblem11.2Somebackgroundon nite-dimensionaloptimization31.2.1Unconstrainedoptimization...............41.2.2Constrainedoptimization.......
8、.........111.3Previewofin nite-dimensionaloptimization171.3.1Functionspaces,norms,andlocalminima.......181.3.2Firstvariationand rst-ordernecessarycondition...191.3.3Secondvariationandsecond-orderconditions.....211.3.4Globalminimaandconvexproblems..........231.4NotesandreferencesforChapter1242Cal
9、culusofVariations262.1Examplesofvariationalproblems262.1.1Dido'sisoperimetricproblem..............262.1.2Lightre
ectionandrefraction..............272.1.3Catenary.........................282.1.4Brachistochrone................
