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1、DefinitionsSelectionsDifferentialInclusionsIntroductiontoDifferentialInclusionsIntroductiontoDifferentialInclusionsDefinitionsSelectionsDifferentialInclusionsSeptember26,2007RickBarnardStudentSeminaronControlTheoryandOptimizationFall2007IntroductiontoDifferentialInclusionsDefinitionsSelecti
2、onsDifferentialInclusionsIntroductiontoDifferentialInclusionsDefinitionsSelections1DefinitionsDifferentialInclusions2Selections3DifferentialInclusionsIntroductiontoDifferentialInclusionsDefinitionsSelectionsDifferentialInclusionsReferencesIntroductiontoDifferentialInclusionsDefinitionsAubin,J.
3、P.andCellina,A.DifferentialInclusions.SelectionsDifferentialSpringer-Verlag,Berlin,1984.InclusionsClarke,F.H.OptimizationandNonsmoorthAnalysis.CentredeRecherchesMath´ematiques,Montr´eal,1989.Smirnov,G.S.IntroductiontotheTheoryofDifferentialInclusions.AMS,Providence,2002.C.Cai,R.Goebel,
4、R.Sanfelice,A.Teel.LectureNotestoWorkshoponRobustHybridSystems:TheoryandApplications.CDc’06.IntroductiontoDifferentialInclusionsDefinitionsSelectionsDifferentialInclusionsMultifunctionsIntroductiontoDifferentialInclusionsDefinitionsSelectionsAmultifunctionF:Rm→RnisamapfromRmtotheDifferent
5、ialInclusionssubsetsofRn,thatisforeveryx∈Rm,weassociatea(potentiallyempty)setF(x).Itsgraph,denotedGr(F)isdefinedbyGr(F)={(x,y)
6、y∈F(x)}.IntroductiontoDifferentialInclusionsDefinitionsSelectionsDifferentialInclusionsMeasurabilityIntroductiontoDifferentialInclusionsDefinitionsSelectionsDiffer
7、entialnInclusionsAmultifunctionF:S→Rismeasurableifforeveryopen(closed)C⊆Rn,{x∈S:F(x)∩C6=∅}isLebesguemeasurable.IntroductiontoDifferentialInclusionsDefinitionsSelectionsDifferentialInclusionsContinuityIntroductiontoDifferentialInclusionsDefinitionsAmultifunctionFiscalleduppersemi-continuo
8、usatx0SelectionsifforanyopenMcontainingF(x0)thereisaDifferentialInclusionsneighborhoodΩofx0sothatF(Ω)⊂M.IntroductiontoDifferentialInclusionsDefinitionsSelectionsDifferentialInclusionsContinuityIntroductiontoDifferentialInclusionsDefinitionsAmultifunctionFiscalleduppersemi-continuousatx0Se
9、lectionsifforanyope
