Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach (Encyclopedia of Mathematics and its Applications (No. 117))

471 Pages·2008·10.99 MB·English

by  Roland Glowinski

Exact and Approximate Controllability for
Distributed Parameter Systems
Thebehaviorofsystemsoccurringinreallifeisoftenmodeledbypartial
differentialequations.Thisbookinvestigateshowauserorobservercaninfluence
thebehaviorofsuchsystemsmathematicallyandcomputationally.Athorough
mathematicalanalysisofcontrollabilityproblemsiscombinedwithadetailed
investigationofmethodsusedtosolvethemnumerically,thesemethodsbeing
validatedbytheresultsofnumericalexperiments.InPartIofthebook,theauthors
discussthemathematicsandnumericsrelatingtothecontrollabilityofsystems
modeledbylinearandnonlineardiffusionequations;PartIIisdedicatedtothe
controllabilityofvibratingsystems,typicalonesbeingthosemodeledbylinear
waveequations;finally,PartIIIcoversflowcontrolforsystemsgovernedbythe
Navier–Stokesequationsmodelingincompressibleviscousflow.Thebookis
accessibletograduatestudentsinappliedandcomputationalmathematics,
engineering,andphysics;itwillalsobeofusetomoreadvancedpractitioners.
EncyclopediaofMathematicsanditsApplications
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62 H.O.FattoriniInfiniteDimensionalOptimizationandControlTheory
63 A.C. ThompsonMinkowskiGeometry
64 R.B.BapatandT.E.S.RaghavanNonnegativeMatriceswithApplications
65 K.EngelSpernerTheory
66 D.Cvetkovic,P.Rowlinson,andS.SimicEigenspacesofGraphs
67 F.Bergeron,G.Labelle,andP.LerouxCombinationalSpeciesandTree-LikeStructures
68 R.GoodmanandN.WallachRepresentationsandInvariantsoftheClassicalGroups
69 T.Beth,D.Jungnickel,andH.LenzDesignTheory1,2ndedn
70 A.PietschandJ.WenzelOrthonormalSystemsforBanachSpaceGeometry
71 G.E.Andrews,R.Askey,andR.RoySpecialFunctions
72 R.TicciatiQuantumFieldTheoryforMathematicians
73 M.SternSemimodularLattices
74 I.LasieckaandR.TriggianiControlTheoryforPartialDifferentialEquationsI
75 I.LasieckaandR.TriggianiControlTheoryforPartialDifferentialEquationsII
76 A.A.IvanovGeometryofSporadicGroupsI
77 A.SchinzelPolynomialswithSpecialRegardtoReducibility
78 H.Lenz,T.Beth,andD.JungnickelDesignTheoryII,2ndedn
∗
79 T.PalmerBanachAlgebrasandtheGeneralTheoryof -AlbegrasII
80 O.StormarkLie’sStructuralApproachtoPDESystems
81 C.F.DunklandY.XuOrthogonalPolynomialsofSeveralVariables
82 J.P.MayberryTheFoundationsofMathematicsintheTheoryofSets
83 C.Foias,O.Manley,R.Rosa,andR.TemamNavier–StokesEquationsandTurbulence
84 B.PolsterandG.SteinkeGeometriesonSurfaces
85 R.B.ParisandD.KaminskiAsymptoticsandMellin–BarnesIntegrals
86 R.McElieceTheTheoryofInformationandCoding,2ndedn
87 B.MagurnAlgebraicIntroductiontoK-Theory
88 T.MoraSolvingPolynomialEquationSystemsI
89 K.BichtelerStochasticIntegrationwithJumps
90 M.LothaireAlgebraicCombinatoricsonWords
91 A.A.IvanovandS.V.ShpectorovGeometryofSporadicGroupsII
92 P.McMullenandE.SchulteAbstractRegularPolytopes
93 G.Gierzetal.ContinuousLatticesandDomains
94 S.FinchMathematicalConstants
95 Y.JabriTheMountainPassTheorem
96 G.GasperandM.RahmanBasicHypergeometricSeries,2ndedn
97 M.C.PedicchioandW.Tholen(eds.)CategoricalFoundations
98 M.E.H.IsmailClassicalandQuantumOrthogonalPolynomialsinOneVariable
99 T.MoraSolvingPolynomialEquationSystemsII
100 E.OlivierandM.EuláliáVaresLargeDeviationsandMetastability
101 A.Kushner,V.Lychagin,andV.RubtsovContactGeometryandNonlinear
DifferentialEquations
102 L.W.Beineke,R.J.Wilson,andP.J.Cameron.(eds.)TopicsinAlgebraicGraphTheory
103 O.StaffansWell-PosedLinearSystems
104 J.M.Lewis,S.Lakshmivarahan,andS.DhallDynamicDataAssimilation
105 M.LothaireAppliedCombinatoricsonWords
106 A.MarkoeAnalyticTomography
107 P.A.MartinMultipleScattering
108 R.A.BrualdiCombinatorialMatrixClasses
110 M.-J.LaiandL.L.SchumakerSplineFunctionsonTriangulations
111 R.T.CurtisSymmetricGenerationofGroups
112 H.Salzmann,T.Grundhöfer,H.Hähl,andR.LöwenTheClassicalFields
113 S.PeszatandJ.ZabczykStochasticPartialDifferentialEquationswithLévyNoise
114 J.BeckCombinatorialGames
115 L.BarreiraandY.PesinNonuniformHyperbolicity
116 D.Z.ArovandH.DymJ-ContractiveMatrixValuedFunctionsandRelatedTopics
117 R.Glowinski,J.-L.Lions,andJ.HeExactandApproximateControllabilityfor
DistributedParameterSystems
Exact and Approximate Controllability for
Distributed Parameter Systems
ANumericalApproach
ROLAND GLOWINSKI
UniversityofHouston
JACQUES-LOUIS LIONS
CollegedeFrance,Paris
JIWEN HE
UniversityofHouston
cambridgeuniversitypress
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©R.Glowinski,J.-L.LionsandJ.He2008
Thispublicationisincopyright.Subjecttostatutoryexception
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thewrittenpermissionofCambridgeUniversityPress.
Firstpublished2008
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
LibraryofCongressCataloginginPublicationdata
Glowinski,R.
Exactandapproximatecontrollabilityfordistributedparametersystems:anumerical
approach/RolandGlowinski,Jacques-LouisLions,JiwenHe.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-521-88572-0(hardback:alk.paper)
1.Controltheory.2.Distributedparametersystems.3.Differentialequations,
Partial–Numericalsolutions.I.Lions,JacquesLouis.II.He,Jiwen.III.Title.
QA402.3.G562008
(cid:2)
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LENSLARQUE-homonyms,withdefinitions.
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JackVance,TheFace.InTheDemonPrinces,VolumeII,
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ThemostchallengingcourseItookinhighschoolwascalculus.
BillClinton,MyLife,Knopf,NewYork,NY,2004
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Contents
Preface pagexi
Introduction 1
I.1 Whatitisallabout? 1
I.2 Motivation 2
I.3 Topologiesandnumericalmethods 3
I.4 Choiceofthecontrol 4
I.5 Relaxationofthecontrollabilitynotion 4
I.6 Variousremarks 5
PartI DiffusionModels
1 Distributedandpointwisecontrolforlineardiffusionequations 9
1.1 Firstexample 9
1.2 Approximatecontrollability 12
1.3 Formulationoftheapproximatecontrollabilityproblem 14
1.4 Dualproblem 15
1.5 Directsolutiontothedualproblem 17
1.6 Penaltyarguments 19
1.7 L∞costfunctionsandbang-bangcontrols 22
1.8 Numericalmethods 28
1.9 Relaxationofcontrollability 57
1.10 Pointwisecontrol 62
1.11 Furtherremarks(I):Additionalconstraintsonthestatefunction 96
1.12 Furtherremarks(II):Abisectionbasedmemorysavingmethodfor
thesolutionoftimedependentcontrolproblemsbyadjointequation
basedmethodologies 112
1.13 Furtherremarks(III):AbriefintroductiontoRiccatiequations
basedcontrolmethods 117
viii Contents
2 Boundarycontrol 124
2.1 Dirichletcontrol(I):Formulationofthecontrolproblem 124
2.2 Dirichletcontrol(II):Optimalityconditionsanddualformulations 126
2.3 Dirichletcontrol(III):Iterativesolutionofthecontrolproblems 128
2.4 Dirichletcontrol(IV):Approximationofthecontrolproblems 133
2.5 Dirichletcontrol(V):Iterativesolutionofthefullydiscrete
dualproblem(2.124) 143
2.6 Dirichletcontrol(VI):Numericalexperiments 146
2.7 Neumanncontrol(I):Formulationofthecontrolproblems
andsynopsis 155
2.8 Neumanncontrol(II):Optimalityconditionsanddualformulations 163
2.9 Neumanncontrol(III):Conjugategradientsolutionofthe
dualproblem(2.192) 176
2.10 Neumanncontrol(IV):Iterativesolutionofthe
dualproblem(2.208),(2.209) 178
2.11 Neumanncontrolofunstableparabolicsystems:
anumericalapproach 178
2.12 Closed-loopNeumanncontrolofunstableparabolicsystems
viatheRiccatiequationapproach 223
3 ControloftheStokessystem 231
3.1 Generalities.Synopsis 231
3.2 FormulationoftheStokessystem.Afundamental
controllabilityresult 231
3.3 Twoapproximatecontrollabilityproblems 234
3.4 Optimalityconditionsanddualproblems 234
3.5 Iterativesolutionofthecontrolproblem(3.19) 236
3.6 Timediscretizationofthecontrolproblem(3.19) 238
3.7 Numericalexperiments 239
4 Controlofnonlineardiffusionsystems 243
4.1 Generalities.Synopsis 243
4.2 Exampleofanoncontrollablenonlinearsystem 243
4.3 PointwisecontroloftheviscousBurgersequation 245
4.4 Onthecontrollabilityandthestabilizationofthe
Kuramoto-Sivashinskyequationinonespacedimension 259
5 Dynamicprogrammingforlineardiffusionequations 277
5.1 Introduction.Synopsis 277
5.2 DerivationoftheHamilton–Jacobi–Bellmanequation 278
5.3 Someremarks 279
Contents ix
PartII WaveModels
6 Waveequations 283
6.1 Waveequations:Dirichletboundarycontrol 283
6.2 Approximatecontrollability 285
6.3 Formulationoftheapproximatecontrollabilityproblem 286
6.4 Dualproblems 287
6.5 Directsolutionofthedualproblem 288
6.6 Exactcontrollabilityandnewfunctionalspaces 289
6.7 OnthestructureofspaceE 291
6.8 NumericalmethodsfortheDirichletboundarycontrollabilityofthe
waveequation 291
6.9 ExperimentalvalidationofthefilteringprocedureofSection6.8.7
viathesolutionofthetestproblemofSection6.8.5 315
6.10 Somereferencesonalternativeapproximationmethods 319
6.11 Otherboundarycontrols 320
6.12 Distributedcontrolsforwaveequations 328
6.13 Dynamicprogramming 329
7 Ontheapplicationofcontrollabilitymethodstothesolutionofthe
Helmholtzequationatlargewavenumbers 332
7.1 Introduction 332
7.2 TheHelmholtzequationanditsequivalentwaveproblem 332
7.3 Exactcontrollabilitymethodsforthecalculationoftime-periodic
solutionstothewaveequation 334
7.4 Least-squaresformulationoftheproblem(7.8)–(7.11) 334
7.5 CalculationofJ(cid:2) 336
7.6 Conjugategradientsolutionoftheleast-squaresproblem(7.14) 337
7.7 Afiniteelement–finitedifferenceimplementation 340
7.8 Numericalexperiments 341
7.9 Furthercomments.Descriptionofamixedformulation
basedvariantofthecontrollabilitymethod 349
7.10 Afinalcomment 355
8 Otherwaveandvibrationproblems.Coupledsystems 356
8.1 Generalitiesandfurtherreferences 356
8.2 CoupledSystems(I):aproblemfromthermo-elasticity 359
8.3 Coupledsystems(II):Othersystems 367