Control of Partial Differential Equations: Cetraro, Italy 2010, Editors: Piermarco Cannarsa, Jean-Michel Coron PDF
Preview Control of Partial Differential Equations: Cetraro, Italy 2010, Editors: Piermarco Cannarsa, Jean-Michel Coron
Lecture Notes in Mathematics 2048 Editors: J.-M.Morel,Cachan B.Teissier,Paris Forfurthervolumes: http://www.springer.com/series/304 FondazioneC.I.M.E.,Firenze C.I.M.E. stands for Centro Internazionale Matematico Estivo, that is, International MathematicalSummerCentre.Conceivedintheearlyfifties,itwasbornin1954inFlorence, Italy,andwelcomedbytheworldmathematical community:itcontinues successfully,year foryear,tothisday. Many mathematicians from all over the world have been involved in a way oranother in C.I.M.E.’sactivitiesovertheyears.ThemainpurposeandmodeoffunctioningoftheCentre maybesummarisedasfollows:everyyear,duringthesummer,sessionsondifferentthemes from pure and applied mathematics are offered byapplication to mathematicians from all countries. ASessionis generally basedonthree orfourmaincourses given byspecialists ofinternationalrenown,plusacertainnumberofseminars,andisheldinanattractiverural locationinItaly. TheaimofaC.I.M.E.sessionistobringtotheattentionofyoungerresearcherstheorigins, development, and perspectives of some very active branch of mathematical research. The topicsofthecoursesaregenerallyofinternationalresonance.Thefullimmersionatmosphere ofthecoursesandthedailyexchangeamongparticipantsarethusaninitiationtointernational collaborationinmathematicalresearch. C.I.M.E.Director C.I.M.E.Secretary PietroZECCA ElviraMASCOLO DipartimentodiEnergetica“S.Stecco” DipartimentodiMatematica“U.Dini” Universita`diFirenze Universita`diFirenze ViaS.Marta,3 vialeG.B.Morgagni67/A 50139Florence 50134Florence Italy Italy e-mail:zecca@unifi.it e-mail:[email protected]fi.it FormoreinformationseeCIME’shomepage:http://www.cime.unifi.it CIMEactivityiscarriedoutwiththecollaborationandfinancialsupportof: -INdAM(IstitutoNazionalediAltaMatematica) -MIUR(Ministerodell’Universita’edellaRicerca) ThiscourseispartiallysupportedbyGDR-GDREonCONTROLEDESEQUATIONSAUX DERIVEESPARTIELLES(CONEDP). Fatiha Alabau-Boussouira Roger Brockett (cid:2) Olivier Glass Je´roˆme Le Rousseau (cid:2) Enrique Zuazua Control of Partial Differential Equations Cetraro, Italy 2010 Editors: Piermarco Cannarsa Jean-Michel Coron 123 FatihaAlabau-Boussouira Je´roˆmeLeRousseau Universite´PaulVerlaine-Metz Universite´d’Orle´ans LMAM LaboratoireMAPMO IleduSaulcy Orle´ans Metz France France EnriqueZuazua RogerBrockett BasqueCenterforAppliedMathematics HarvardUniversity BizkaiaTechnologyPark EngineeringandAppliedSciences Derio OxfordSt.33 Spain CambridgeMassachusetts MaxwellDworkin USA OlivierGlass Universite´Paris-Dauphine CEREMADE PlaceduMare´chaldeLattrede Tassigny Paris France ISBN978-3-642-27892-1 e-ISBN978-3-642-27893-8 DOI10.1007/978-3-642-27893-8 SpringerHeidelbergDordrechtLondonNewYork LectureNotesinMathematicsISSNprintedition:0075-8434 ISSNelectronicedition:1617-9692 LibraryofCongressControlNumber:2012934256 MathematicsSubjectClassification(2010):35B35,93B05,93B07,93B52,93C20,93D15,76B75, 65M10,65M12 (cid:2)c Springer-VerlagBerlinHeidelberg2012 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations areliabletoprosecutionundertheGermanCopyrightLaw. Theuseofgeneral descriptive names,registered names, trademarks, etc. inthis publication does not imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotective lawsandregulationsandthereforefreeforgeneraluse. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface Oneofthefindingsofthe1988ReportbythePanelonFutureDirectionsinControl Theory,chairedbyWendellH.Fleming,was: Manyfundamentaltheoreticalissues,suchascontrolofnonlinearmultivariablesystems,or controlofnonlinearpartialdifferentialequations,arenotyetunderstood. Nowadays,morethan20yearslater,webelievewecansaythatalotoffundamental issues concerning the latter topic have definitely been understood, thanks to the effortsof many researchers who produceda large body of results and techniques. And yet, this process has led to an enormous amount of open questions that will needtobeaddressedbynewgenerationsofscientists.Surveyingthemostimportant advancesof the lasttwo decadesandoutliningfutureresearchdirectionswere the main motivations that led us to organize the CIME Course on Control of Partial Differential Equations that took place in Cetraro (CS, Italy), July 19–23, 2010. We hope thisvolume,which is one of the outcomesof that event,will providean ultimate formative step for those who attended the course, and will represent an authoritativereferenceforthosewhowereunabletodoso. Thecourseconsistedoffiveseriesoflectures,whichare nowthe sourceofthe chaptersofthismonograph.Specifically,thefollowingtopicswerecovered: • Stabilization of evolution equations (by Fatiha Alabau-Boussouira): these lec- turesdiscussedrecentadvances,aswellasclassicalmethods,forthestabilization of wave-like equations. Special attention was paid to nonlinear problems, memory-damping,andindirectstabilizationofcoupledPDEs.Alltheproblems were treated by a unified methodology based on energy estimates. It was shown how the introduction of optimal-weight convexity methods leads to easy computable upper energy decay estimates, and how these results can be completedbylowerenergyestimatesforseveralexamples. • ControloftheLiouvilleequation(byRogerBrockett):theseequationsdescribe the evolution of an initial density of points that move according to a given differential equation, and may depend on a control which can be chosen in ordertosatisfysomeprescribedgoals.Thisframeworkalsoallowstoovercome v vi Preface limitations of the classical theory:for example, the expense required to imple- mentcontrollaws.Severalresults(e.g.,onensemblecontrol:controlling,witha singlecontrol,afinitebutoftenlargenumberofcopiesofagivensystem)aswell asopenproblemswerepresented. • Controlinfluidmechanics(byOlivierGlass):thelecturestreatedvariousissues related to the controllability of two well-known equations in fluid mechanics, namelytheEulerequationforperfectincompressiblefluidsinbothEulerianand Lagrangian coordinates, and the one-dimensionalisentropic Euler equation for compressiblefluidsintheframeworkofentropysolutions.Specialemphasiswas putontheaspectsofthe theorythatareconnectedwith thenonlinearnatureof the problem: linearization around an equilibrium gives here no information on thecontrollabilityofthenonlinearsystem. • Carleman estimates for elliptic and parabolic equations, with application to control (by Je´roˆme Le Rousseau): these are weighted energy estimates for solutionsofpartialdifferentialequationswithweightsofexponentialtype.The lectures derived Carleman estimates for elliptic and parabolic operators using several methods: a microlocal approach where the main tool is the Ga˚rding inequality, and a more computational direct approach. It was also shown how Carleman estimates can be used to provide unique continuation properties, as wellasapproximateandnullcontrollabilityresults. • Controlandnumericsforthewaveequation(byEnriqueZuazua):theselectures provided a self-contained presentation of the theory that has been developed recentlyforthenumericalanalysisofthecontrollabilitypropertiesofwaveprop- agationphenomena.The methodologyadoptedthe so-calleddiscrete approach, whichconsistsinanalyzingwhetherthesemidiscreteorfullydiscretedynamics arising when discretizing the wave equation by means of the most classical schemeofnumericalanalysissharethepropertyofbeingcontrollable,uniformly with respect to the mesh-size parameters, and the corresponding controls con- vergetothecontinuousonesasthemeshsizetendstozero.Alltheresultswere illustratedbymeansofseveralnumericalexperiments. Besidestheabovelectures,therewerethreeseminars,givenbyKarineBeauchard (Some controllability results for the 2D Kolmogorovequation),Sylvain Ervedoza (Regularity of HUM controls for conservative systems and convergence rates for discrete controls), and Lionel Rosier (Control of some dispersive equations for waterwaves).TherewerealsofourcommunicationsgivenbyIdoBright(Periodic optimizationsufficesforinfinitehorizonplanaroptimalcontrol),KhaiTienNguyen (Theregularityoftheminimumtimefunctionvianonsmoothanalysisandgeometric measure theory), Camille Laurent (On stabilization and control for the critical Klein-Gordonequationona3-Dcompactmanifold),andVincentPerrollaz(Exact controllabilityofentropicsolutionsofscalarconservationlawswiththreecontrols). Seminarsandcommunicationsarenotreproducedinthesenotes. Oneimportantpoint,containedinthe1988Reportwementionedabove,isthat advances in the control field are made through a combination of mathematics, modeling,computation,andexperimentation.Hopingthereaderwillfindthepresent Preface vii exposition in accord with such a basic principle, we wish to thank the lecturers and authors who designed their contributions in a detailed-yet-focussedform, for helpingusrealizethisproject.Overall,weareverygratefultoallthe57participants in the CIME course, for their enthusiasm that created a friendly and stimulating atmosphereinCetraro.Finally,specialgratitudeisduetotheGDRECONEDP,for providingtheessentialsupportthatallowedustoreceiveandacceptalargenumber ofapplications,andtotheC.I.M.E.Foundation,formakingthiseventpossibleand foritsveryhelpfulassistancebeforeandallalongthelectures. RomeandParis PiermarcoCannarsa Jean-MichelCoron • Contents 1 OnSomeRecentAdvancesonStabilization forHyperbolicEquations ................................................... 1 FatihaAlabau-Boussouira 1.1 Introduction............................................................ 2 1.1.1 OnNonlinearandMemoryStabilization..................... 4 1.1.2 OnIndirectStabilizationforCoupledSystems.............. 7 1.2 Notation................................................................ 8 1.3 StrongStabilization.................................................... 8 1.3.1 Dafermos’StrongStabilizationResult....................... 9 1.4 LinearStabilization.................................................... 11 1.4.1 Introduction.................................................... 12 1.4.2 GeometricalAspects .......................................... 12 1.4.3 ExponentialDecayforLinearFeedbacks.................... 18 1.4.4 TheCompactness–UniquenessMethod...................... 20 1.5 NonlinearStabilizationinFiniteDimensions........................ 25 1.5.1 NonlinearGronwallInequalities.............................. 25 1.5.2 AComparisonLemma........................................ 33 1.5.3 EnergyDecayRates Characterization:The ScalarCase .................................................... 36 1.5.4 TheVectorialCaseandSemi-discretizedPDE’s............. 42 1.5.5 ExamplesofFeedbacksandOptimality...................... 45 1.6 PolynomialFeedbacksinInfiniteDimensions....................... 46 1.7 TheOptimal-WeightConvexityMethod ............................. 48 1.7.1 IntroductionandScope........................................ 48 1.7.2 DominantKineticEnergyEstimates ......................... 52 1.7.3 WeightFunctionAsanOptimalUnknown................... 55 1.7.4 SimplificationoftheEnergyDecayRates ................... 59 1.7.5 GeneralizationtoOpticGeometricConditions: TheIndirectOptimal-WeightConvexityMethod............ 60 ix
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