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1、Chapter3SomeQuestionsofControlinFluidMechanicsOlivierGlassAbstractThegoaloftheselecturenotesistopresentsometechniquesofnon-linearcontrolofPDEs,inthecontextoffluidmechanics.Wewillconsidertheproblemofcontrollabilityoftwodifferentmodels,namelytheEulerequatio
2、nforperfectincompressiblefluids,andtheone-dimensionalisentropicEulerequationforcompressiblefluids.ThestandardtechniquesusedtodealwiththeCauchyproblemforthesetwomodelsareofratherdifferentnature,despitethefactthatthemodelsareclose.Aswewillsee,thisdifferencew
3、illalsoappearwhenconstructingsolutionsofthecontrollabilityproblem;howeveracommontechnique(orpointofview)willbeusedinbothcases.Thistechnique,introducedbyJ.-M.Coronasthereturnmethod,isawaytoexploitthenonlinearityoftheequationforcontrolpurposes.Hencewewills
4、eeitsapplicationintworatherdifferenttypesofPDEs.Theplanofthesenotesisthefollowing.Inafirstpart,werecallinaverybasicwaysometypesofquestionsthatcanberaisedinPDEcontrol(inanon-exhaustiveway).Inasecondpart,weexposeresultsconcerningthecontrollabilityoftheincom
5、pressibleEulerequation.Inathirdpart,weshowhowthetechniquesusedtoprovethecontrollabilityoftheincompressibleEulerequationcanbeusedtoprovesomeothercontrollabilitypropertiesforthisequation,namelytheso-calledLagrangiancontrollability.Inafourthandlastpart,weco
6、nsiderthecontrollabilityoftheisentropicEulerequation.O.Glass()Ceremade,CNRSUMR7534,UniversiteParis-Dauphine,PlaceduMar´echaldeLattrede´Tassigny,75775ParisCedex16,Francee-mail:glass@ceremade.dauphine.frF.Alabau-Boussouiraetal.,ControlofPartialDifferentia
7、lEquations,131LectureNotesinMathematics2048,DOI10.1007/978-3-642-27893-83,©Springer-VerlagBerlinHeidelberg2012132O.Glass3.1IntroductionInthisfirstsection,wegiveashortandelementarypresentationofsomequestionsincontroltheoryasageneralintroductionbeforegettin
8、gtosomespecificcontrolproblemsinfluidmechanics.3.1.1ControlSystemsWestartwiththedefinitionofthebasicobjectstudiedincontroltheory.Definition3.1.1.Acontrolsystemisanevolutionequation(anODEoraPDE)dependingonaparameteru,thatwewill
