(J.P.Puel)Carleman Estimates-Course__ Fudan-01-2013

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1、ApplicationsofglobalCarlemaninequalitiestocontrollabilityandinverseproblemsJean-PierrePuel1May20081J.-P.Puel:(jppuel@cmapx.polytechnique.fr)LaboratoiredeMathematiquesdeVer-sailes,UniversitedeVersaillesSaint-Quentin,45avenuedesEtatsUnis,78035VersaillesChapter1IntroductionThesenotesco

2、ntainthreemainpartscorrespondingtothreeapplicationsofglobalCarlemaninequalities.Therstpartwilldescribewhatwecallexactcontrollabilitytotrajectories.Thesetermswillbedenedpreciselylateronbutforlinearsystems,itcorrespondstonull(orzero)controllabilityandroughlyspeakingitmeansthatwelookfo

3、ranexactcontrollabilitywhenthetargetstobereachedarepointsontrajectoriesofthesameoperator.ThesenotionsarerelevantfordissipativesystemssuchasheatequationsorNavierStokesequationsorrelatedsystems.Indeed,fortheseequations,onecannotexpecttoobtainexactcontrollabilityintheusualsense(seeforexa

4、mple[2]or[5])asforwave-typeequations.Thenotionofapproximatecontrollability,whichhasbeenextensivelystudied(seeforexample[11])andwhichwillbequiteusefulinarststephere,isnotreallysatisfactoryforrealpurposes.Exactcontrollabilitytotrajectoriescorrespondstorealobjectivesand,whenapplicable,i

5、sapowerfultool.Asalreadymentionned,themainmathematicaltooltoobtainexactcontrolabilitytotrajectoriesistheuseofglobalCarlemaninequalities.ThemethodalongtheselineshasbeengivenintheworksofFursikov-Imanuvilov(e.g.[24])andImanuvilov([26],[27])andwewilluseessentialideascomingfromtheseworks,e

6、venifwerevisitedthequestioninordertoclarifythepresentation.InparticularwewillcompletelyprovethebasicCarlemaninequalitiesfordiusion-convectionequations.Thesecondpartwillbedevotedtoadataassimilationproblem.Thisquestionisofgreatimportanceinmeteorology(wheatherprediction),climatology,stu

7、dyofoceanmovementsforexample.Atthemoment,thewheatherpredictionmodelsmakeuseofwhatiscalledvariationaldataassimilation(see[8]foraninterestingpresenta-tion),basedonoptimalcontrolmethods.Togiveanideaoftheimportanceofthisquestiononehastoknowthatitrequiresmorethanahalfofthecomputationaltime

8、eect