Nonsmooth Analysis in Systems and Control Theory
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Nonsmooth Analysis in Systems and Control Theory Francis Clarke Institut universitaire de France et Universite de Lyon [January 2008. To appear in the Encyclopedia of Complexity and System Science, Springer.] Article Outline Glossary I. Definition of the Subject and Its Importance II. Introduction III. Elements of Nonsmooth Analysis IV. Necessary Conditions in Optimal Control V. Verification Functions VI. Dynamic Programming and Viscosity Solutions VII. Lyapunov Functions VIII. Stabilizing Feedback IX. Future Directions X. Bibliography Glossary Generalized gradients and subgradients These terms refer to various set- valued replacements for the usual derivative which are used in developing differential calculus for functions which are not differentiable in the classi- cal sense. The subject itself is known as nonsmooth analysis. One of the best-known theories of this type is that of generalized gradients. Another basic construct is the subgradient, of which there are several variants. The approach also features generalized tangent and normal vectors which apply to sets which are not classical manifolds. A short summary of the essential definitions is given in III. Pontryagin Maximum Principle The main theorem on necessary conditions in optimal control was developed in the 1950s by the Russian mathemati- cian L. Pontryagin and his associates. The Maximum Principle unifies and extends to the control setting the classical necessary conditions of Euler and 1
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Nonsmooth Analysis in Systems and Control Theory Francis Clarke Institut universitaire de France et Universite de Lyon [January 2008. To appear in the Encyclopedia of Complexity and System Science, Springer.] Article Outline Glossary I. Definition of the Subject and Its Importance II. Introduction III. Elements of Nonsmooth Analysis IV. Necessary Conditions in Optimal Control V. Verification Functions VI. Dynamic Programming and Viscosity Solutions VII. Lyapunov Functions VIII. Stabilizing Feedback IX. Future Directions X. Bibliography Glossary Generalized gradients and subgradients These terms refer to various set- valued replacements for the usual derivative which are used in developing differential calculus for functions which are not differentiable in the classi- cal sense. The subject itself is known as nonsmooth analysis. One of the best-known theories of this type is that of generalized gradients. Another basic construct is the subgradient, of which there are several variants. The approach also features generalized tangent and normal vectors which apply to sets which are not classical manifolds. A short summary of the essential definitions is given in III. Pontryagin Maximum Principle The main theorem on necessary conditions in optimal control was developed in the 1950s by the Russian mathemati- cian L. Pontryagin and his associates. The Maximum Principle unifies and extends to the control setting the classical necessary conditions of Euler and 1
- stabilizing feedback
- optimal control
- transversality con- ditions
- has been
- problems involving
- lyapunov function
- theory has
- ordinary differential
- control theory
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NonsmoothAnalysisinSystemsandControlTheoryFrancisClarkeInstitutuniversitairedeFranceetUniversite´deLyon[January2008.ToappearintheEncyclopediaofComplexityandSystemScience,Springer.]ArticleOutlineGlossaryI.DefinitionoftheSubjectandItsImportanceII.IntroductionIII.ElementsofNonsmoothAnalysisIV.NecessaryConditionsinOptimalControlV.VerificationFunctionsVI.DynamicProgrammingandViscositySolutionsVII.LyapunovFunctionsVIII.StabilizingFeedbackIX.FutureDirectionsX.BibliographyGlossaryGeneralizedgradientsandsubgradientsThesetermsrefertovariousset-valuedreplacementsfortheusualderivativewhichareusedindevelopingdifferentialcalculusforfunctionswhicharenotdifferentiableintheclassi-calsense.Thesubjectitselfisknownasnonsmoothanalysis.Oneofthebest-knowntheoriesofthistypeisthatofgeneralizedgradients.Anotherbasicconstructisthesubgradient,ofwhichthereareseveralvariants.Theapproachalsofeaturesgeneralizedtangentandnormalvectorswhichapplytosetswhicharenotclassicalmanifolds.Ashortsummaryoftheessentialdefinitionsisgivenin§III.PontryaginMaximumPrincipleThemaintheoremonnecessaryconditionsinoptimalcontrolwasdevelopedinthe1950sbytheRussianmathemati-cianL.Pontryaginandhisassociates.TheMaximumPrincipleunifiesandextendstothecontrolsettingtheclassicalnecessaryconditionsofEulerand1
2Weierstrassfromthecalculusofvariations,aswellasthetransversalitycon-ditions.Therehavebeennumerousextensionssincethen,astheneedtoconsidernewtypesofproblemscontinuestoarise,andasdiscussedin§IV.VerificationfunctionsInattemptingtoprovethatacertaincontrolisindeedthesolutiontoagivenoptimalcontrolproblem,oneimportantap-proachhingesuponexhibitingafunctionhavingcertainpropertiesimplyingtheoptimalityofthegivencontrol.Suchafunctionistermedaverificationfunction(see§V).Theapproachbecomeswidelyapplicableifoneallowsnonsmoothverificationfunctions.DynamicprogrammingAwell-knowntechniqueindynamicproblemsofoptimizationistosolve(inadiscretecontext)abackwardsrecursionforacertainvaluefunctionrelatedtotheproblem.Thistechnique,whichwasdevelopednotablybyBellman,canbeappliedinparticulartooptimalcon-trolproblems.Inthecontinuoussetting,therecursioncorrespondstotheHamilton-Jacobiequation.Thispartialdifferentialequationdoesnotgener-allyadmitsmoothclassicalsolutions.Thetheoryofviscositysolutionsusessubgradientstodefinegeneralizedsolutions,andobtainstheirexistenceanduniqueness(see§VI).LyapunovfunctionIntheclassicaltheoryofordinarydifferentialequations,globalasymptoticstabilityismostoftenverifiedbyexhibitingaLyapunovfunction,afunctionwhichdecreasesalongtrajectories.Inthatsetting,theexistenceofasmoothLyapunovfunctionisbothnecessaryandsufficientforstability.TheLyapunovfunctionconceptcanbeextendedtocontrolsystems,butinthatcaseitturnsoutthatnonsmoothfunctionsareessential.ThesegeneralizedcontrolLyapunovfunctionsplayanimportantroleindesigningoptimalorstabilizingfeedback(see§§VII,VIII).I.DefinitionoftheSubjectandItsImportanceThetermnonsmoothanalysisreferstothebodyoftheorywhichdevelopsdifferentialcalculusforfunctionswhicharenotdifferentiableintheusualsense,andforsetswhicharenotclassicalsmoothmanifolds.Thereareseveraldifferent(butrelated)approachestodoingthis.Amongthebetter-knownconstructsofthetheoryarethefollowing:generalizedgradientsandJacobians,proximalsubgradients,subdifferentials,generalizeddirectional(orDini)derivates,togetherwithvariousassociatedtangentandnormalcones.Nonsmoothanalysisisasubjectinitself,withinthelargermathe-maticalfieldofdifferential(variational)analysisorfunctionalanalysis,butithasalsoplayedanincreasinglyimportantroleinseveralareasofapplica-tion,notablyinoptimization,calculusofvariations,differentialequations,mechanics,andcontroltheory.Amongthosewhohaveparticipatedinitsdevelopment(inadditiontotheauthor)areJ.Borwein,A.D.Ioffe,B.
3Mordukhovich,R.T.Rockafellar,andR.B.Vinter,butmanymorehavecontributedaswell.Inthecaseofcontroltheory,theneedfornonsmoothanalysisfirstcametolightinconnectionwithfindingproofsofnecessaryconditionsforopti-malcontrol,notablyinconnectionwiththePontryaginMaximumPrinci-ple.Thisnecessityholdsevenforproblemswhichareexpressedentirelyintermsofsmoothdata.Subsequently,itbecameclearthatproblemswithintrinsicallynonsmoothdataarisenaturallyinavarietyofoptimalcontrolsettings.Generally,nonsmoothanalysisentersthepictureassoonasweconsiderproblemswhicharetrulynonlinearornonlinearizable,whetherforderivingorexpressingnecessaryconditions,inapplyingsufficientconditions,orinstudyingthesensitivityoftheproblem.Theneedtoconsidernonsmoothnessinthecaseofstabilizing(asopposedtooptimal)controlhascometolightmorerecently.Itappearsinparticularthatintheanalysisoftrulynonlinearcontrolsystems,theconsiderationofnonsmoothLyapunovfunctionsanddiscontinuousfeedbacksbecomesun-avoidable.II.IntroductionThebasicobjectinthecontroltheoryofordinarydifferentialequationsisthesystemx˙(t)=f(x(t),u(t))a.e.,0≤t≤T,(1)wherethe(measurable)controlfunctionu(∙)ischosensubjecttothecon-straintu(t)∈Ua.e.(2)(Inthisarticle,UisagivensetinaEuclideanspace.)Theensuingstatex(∙)(afunctionwithvaluesinRn)issubjecttocertainconditions,includingmostoftenaninitialoneoftheformx(0)=x0,andperhapsotherconstraints,eitherthroughouttheinterval(pointwise)orattheterminaltime.Acontrolfunctionu(∙)ofthistypeisreferredtoasanopenloopcontrol.Thisindirectcontrolofx(∙)viathechoiceofu(∙)istobeexercisedforapurpose,ofwhichtherearetwoprincipalsorts:•positional:x(t)istoremaininagivensetinRn,orapproachthatset;•optimal:x(∙),togetherwithu(∙),istominimizeagivenfunctional.Thesecondofthesecriteriafollowsdirectlyinthetraditionofthecalculusofvariations,andgivesrisetothesubjectofoptimalcontrol,inwhichthedominantissuesarethoseofoptimization:necessaryconditionsforoptimal-ity,sufficientconditions,regularityoftheoptimalcontrol,sensitivity.We
