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Numerical Approximation of Exact Controls for Waves PDF

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SpringerBriefs in Mathematics SeriesEditors KrishnaswamiAlladi NicolaBellomo MicheleBenzi TatsienLi MatthiasNeufang OtmarScherzer DierkSchleicher BenjaminSteinberg VladasSidoravicius YuriTschinkel LoringW.Tu G. GeorgeYin PingZhang SpringerBriefs in Mathematics showcases expositions in all areas of mathematics and applied mathematics. Manuscripts presenting new resultsorasinglenewresultinaclassicalfield,newfield,oranemerging topic,applications,orbridgesbetweennewresultsandalreadypublished works, are encouraged. The series is intended for mathematicians and appliedmathematicians. 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. Typical topics include: a timely report of state-of-the-art analytical techniques, bridgebetweennew researchresultspublishedin journalarticlesand a contextual literaturereview,asnapshotofahotoremergingtopic,apresentationofcorecon- ceptsthatstudentsmustunderstandinordertomakeindependentcontributions. 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. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. 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

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