Numerical Approximation of Exact Controls for Waves

139 Pages·2013·1.366 MB·English

by  Sylvain Ervedoza, Enrique Zuazua (auth.)

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BCAM-BasqueCenterforAppliedMathematics
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CenterforNonlinearAnalysis
DepartmentofMathematicalSciences
CarnegieMellonUniversity
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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
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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
x Contents
2 Observabilityforthe1−dFinite-DifferenceWaveEquation ........ 49
2.1 Objectives................................................. 49
2.2 SpectralDecompositionoftheDiscreteLaplacian ............... 50
2.3 UniformAdmissibilityofDiscreteWaves ...................... 51
2.3.1 TheMultiplierIdentity................................ 51
2.3.2 ProofoftheUniformHiddenRegularityResult ........... 52
2.4 AnObservabilityResult ..................................... 53
2.4.1 EquipartitionoftheEnergy............................ 53
2.4.2 TheMultiplierIdentityRevisited ....................... 54
2.4.3 UniformObservabilityforFilteredSolutions ............. 55
2.4.4 ProofofTheorem2.3................................. 58
3 ConvergenceoftheFinite-DifferenceMethodforthe1−dWave
EquationwithHomogeneousDirichletBoundaryConditions........ 59
3.1 Objectives................................................. 59
3.2 ExtensionOperators ........................................ 59
3.2.1 TheFourierExtension ................................ 60
3.2.2 OtherExtensionOperators ............................ 60
3.3 OrdersofConvergenceforSmoothInitialData.................. 64
3.4 FurtherConvergenceResults ................................. 70
3.4.1 StronglyConvergentInitialData ....................... 70
3.4.2 SmoothInitialData .................................. 71
3.4.3 GeneralInitialData .................................. 73
3.4.4 ConvergenceRatesinWeakerNorms ................... 74
3.5 Numerics ................................................. 75
4 ConvergencewithNonhomogeneousBoundaryConditions.......... 79
4.1 TheSetting................................................ 79
4.2 TheLaplaceOperator....................................... 81
4.2.1 NaturalFunctionalSpaces............................. 81
4.2.2 StrongerNorms ..................................... 86
4.2.3 NumericalResults ................................... 88
4.3 UniformBoundsony ...................................... 89
h
4.3.1 EstimatesinC([0,T];L2(0,1))......................... 90
4.3.2 Estimateson∂y .................................... 93
t h
4.4 ConvergenceRatesforSmoothData........................... 98
4.4.1 MainConvergenceResult ............................. 98
4.4.2 Convergenceofy ...................................100
h
4.4.3 Convergenceof∂y ..................................104
t h
4.4.4 MoreRegularData...................................106
4.5 FurtherConvergenceResults .................................110
4.6 NumericalResults..........................................111