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Forfurthervolumes: http://www.springer.com/series/10030 BCAM SpringerBriefs EditorialBoard EnriqueZuazua BCAM-BasqueCenterforAppliedMathematics &Ikerbasque Bilbao,BasqueCountry,Spain IreneFonseca CenterforNonlinearAnalysis DepartmentofMathematicalSciences CarnegieMellonUniversity Pittsburgh,USA JuanJ.Manfredi DepartmentofMathematics UniversityofPittsburgh Pittsburgh,USA EmmanuelTre´lat LaboratoireJacques-LouisLions InstitutUniversitairedeFrance Universite´ PierreetMarieCurie CNRS,UMR,Paris XuZhang SchoolofMathematics SichuanUniversity Chengdu,China BCAM SpringerBriefs aims to publishcontributionsin the followingdisciplines: Applied Mathematics, Finance, Statistics and Computer Science. BCAM has appointedanEditorialBoardthatwillevaluateandreviewproposals. 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Please submit your proposal to the Editorial Board or to Francesca Bonadei, Executive Editor Mathematics, Statistics, and Engineering: francesca.bonadei@ springer.com Sylvain Ervedoza • Enrique Zuazua Numerical Approximation of Exact Controls for Waves 123 SylvainErvedoza EnriqueZuazua InstitutdeMathe´matiques BCAM-BasqueCenterforApplied deToulouse&CNRS Mathematics Universite´PaulSabatier Ikerbasque,Bilbao Toulouse,France BasqueCountry,Spain ISSN2191-8198 ISSN2191-8201(electronic) ISBN978-1-4614-5807-4 ISBN978-1-4614-5808-1(eBook) DOI10.1007/978-1-4614-5808-1 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2012956551 MathematicsSubjectClassification(2010):35L05,93B05,93B07,34H05,35L90,49M25 (cid:2)c SylvainErvedozaandEnriqueZuazua2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface In this book, we fully develop and compare two approaches for the numerical approximationof exactcontrolsfor wave propagationphenomena:the continuous one, based on a thorough analysis of the continuousmodel, and the discrete one, whichreliesupontheanalysisofthediscretemodelsunderconsideration.Wedoit intheabstractfunctionalsettingofconservativesemigroups. The main results of this book end up unifying,to a large extent, these two ap- proachesyieldingsimilaralgorithmsandconvergencerates.Thediscreteapproach, however,hastheaddedadvantageofyieldingnotonlyefficientnumericalapproxi- mationsofthecontinuouscontrolsbutalsoensuringthepartialcontrollabilityofthe finite-dimensionalapproximateddynamics,i.e.,thefactthatasubstantialprojection oftheapproximatedynamicsis controlled.Italso leadsto iterativeapproximation processes that converge without a limiting threshold in the number of iterations. Sucha thresholdhastobetakenintoaccount,necessarily,formethodsderivedby the continuousapproach, and it is hard to compute and estimate in practice. This isadrawbackofthemethodsemanatingfromthecontinuousapproachthatexhibit divergence phenomena when the number of iterations in the algorithms aimed to yieldaccurateapproximationsofthecontrolgoesbeyondthisthreshold. Weshallalsobrieflyexplainhowtheseideascanbeappliedfordataassimilation problems. Though our results apply in a wide functionalsetting, our approach requires a fineanalysisinthecaseofunboundedcontroloperators,e.g.,inthecaseofboundary controls.Wewillthereforeshowhowthiscanbedoneinasimplecase,namelythe 1−d wave equation approximatedby finite-difference methods. In particular, we presentseveralnewresultsontheratesofconvergenceforthesolutionofthewave equationwithnonhomogeneousDirichletboundarydata. Toulouse,France SylvainErvedoza Bilbao,BasqueCountry,Spain EnriqueZuazua v Acknowledgments Sylvain Ervedozais partially supportedby the Agence Nationale de la Recherche (ANR,France),ProjectC-QUIDnumberBLAN-3-139579,ProjectCISIFSnumber NT09-437023,theAOPICANofUniversityPaulSabatier(Toulouse3),andgrant MTM2011-29306of the MICINN, Spain. Part of this work has been done while hewasvisitingtheBCAM—BasqueCenterforAppliedMathematics,asavisiting fellow. Enrique Zuazua is supported by the ERC Advanced Grant FP7-246775 NU- MERIWAVES,theGrantPI2010-04oftheBasqueGovernment,theESFResearch NetworkingProgramOPTPDE,andGrantMTM2011-29306oftheMICINN,Spain. vii Contents 1 NumericalApproximationofExactControlsforWaves............. 1 1.1 Introduction ............................................... 1 1.1.1 AnAbstractFunctionalSetting......................... 1 1.1.2 ContentsofChap.1 .................................. 4 1.2 MainResults .............................................. 6 1.2.1 An“Algorithm”inanInfinite-DimensionalSetting ........ 6 1.2.2 TheContinuousApproach............................. 9 1.2.3 TheDiscreteApproach ............................... 11 1.2.4 OutlineofChap.1 ................................... 13 1.3 ProofoftheMainResultontheContinuousSetting.............. 13 1.3.1 ClassicalConvergenceResults ......................... 13 1.3.2 ConvergenceRatesinX .............................. 14 s 1.4 TheContinuousApproach ................................... 16 1.4.1 ProofofTheorem1.2................................. 17 1.4.2 ProofofTheorem1.3................................. 18 1.5 ImprovedConvergenceRates:TheDiscreteApproach............ 18 1.5.1 ProofofTheorem1.4................................. 19 1.5.2 ProofofTheorem1.5................................. 20 1.6 AdvantagesoftheDiscreteApproach.......................... 20 1.6.1 TheNumberofIterations ............................. 20 1.6.2 ControllingNon-smoothData.......................... 21 1.6.3 OtherMinimizationAlgorithms ........................ 23 1.7 ApplicationtotheWaveEquation............................. 24 1.7.1 BoundaryControl.................................... 24 1.7.2 DistributedControl................................... 40 1.8 ADataAssimilationProblem ................................ 44 1.8.1 TheSetting ......................................... 44 1.8.2 NumericalApproximationMethods..................... 46 ix