Stabilization of Navier–Stokes Flows PDF
Preview Stabilization of Navier–Stokes Flows
Communications and Control Engineering Forothertitlespublishedinthisseries,goto www.springer.com/series/61 SeriesEditors (cid:2) (cid:2) (cid:2) (cid:2) A.Isidori J.H.vanSchuppen E.D.Sontag M.Thoma M.Krstic Publishedtitlesinclude: StabilityandStabilizationofInfiniteDimensional SwitchedLinearSystems SystemswithApplications ZhendongSunandShuzhiS.Ge Zheng-HuaLuo,Bao-ZhuGuoandOmerMorgul SubspaceMethodsforSystemIdentification NonsmoothMechanics(Secondedition) TohruKatayama BernardBrogliato DigitalControlSystems NonlinearControlSystemsII IoanD.LandauandGianlucaZito AlbertoIsidori MultivariableComputer-controlledSystems L2-GainandPassivityTechniquesinNonlinearControl EfimN.RosenwasserandBernhardP.Lampe ArjanvanderSchaft DissipativeSystemsAnalysisandControl ControlofLinearSystemswithRegulationandInput (Secondedition) Constraints BernardBrogliato,RogelioLozano,BernhardMaschke AliSaberi,AntonA.StoorvogelandPeddapullaiah andOlavEgeland Sannuti AlgebraicMethodsforNonlinearControlSystems RobustandH∞Control GiuseppeConte,ClaudeH.MoogandAnnaM.Perdon BenM.Chen PolynomialandRationalMatrices ComputerControlledSystems TadeuszKaczorek EfimN.RosenwasserandBernhardP.Lampe Simulation-basedAlgorithmsforMarkovDecision ControlofComplexandUncertainSystems Processes StanislavV.EmelyanovandSergeyK.Korovin HyeongSooChang,MichaelC.Fu,JiaqiaoHuand StevenI.Marcus RobustControlDesignUsingH∞Methods IanR.Petersen,ValeryA.Ugrinovskiand IterativeLearningControl AndreyV.Savkin Hyo-SungAhn,KevinL.MooreandYangQuanChen ModelReductionforControlSystemDesign DistributedConsensusinMulti-vehicleCooperative GoroObinataandBrianD.O.Anderson Control WeiRenandRandalW.Beard ControlTheoryforLinearSystems HarryL.Trentelman,AntonStoorvogelandMaloHautus ControlofSingularSystemswithRandomAbrupt Changes FunctionalAdaptiveControl El-KébirBoukas SimonG.FabriandVisakanKadirkamanathan NonlinearandAdaptiveControlwithApplications Positive1Dand2DSystems AlessandroAstolfi,DimitriosKaragiannisandRomeo TadeuszKaczorek Ortega IdentificationandControlUsingVolterraModels Stabilization,OptimalandRobustControl FrancisJ.DoyleIII,RonaldK.PearsonandBabatunde AzizBelmiloudi A.Ogunnaike ControlofNonlinearDynamicalSystems Non-linearControlforUnderactuatedMechanical FelixL.Chernous’ko,IgorM.AnanievskiandSergey Systems A.Reshmin IsabelleFantoniandRogelioLozano PeriodicSystems RobustControl(Secondedition) SergioBittantiandPatrizioColaneri JürgenAckermann DiscontinuousSystems FlowControlbyFeedback YuryV.Orlov OleMortenAamoandMiroslavKrstic ConstructionsofStrictLyapunovFunctions LearningandGeneralization(Secondedition) MichaelMalisoffandFrédéricMazenc MathukumalliVidyasagar ControllingChaos ConstrainedControlandEstimation HuaguangZhang,DerongLiuandZhiliangWang GrahamC.Goodwin,MariaM.Seronand JoséA.DeDoná ControlofComplexSystems AleksandarZecˇevic´andDragoslavD.Šiljak RandomizedAlgorithmsforAnalysisandControl ofUncertainSystems RobertoTempo,GiuseppeCalafioreandFabrizio Dabbene Viorel Barbu Stabilization of Navier–Stokes Flows Prof.ViorelBarbu Fac.Mathematics Al.I.CuzaUniversity Blvd.CarolI11 700506Ias¸i Romania [email protected] ISSN0178-5354 ISBN978-0-85729-042-7 e-ISBN978-0-85729-043-4 DOI10.1007/978-0-85729-043-4 SpringerLondonDordrechtHeidelbergNewYork BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ©Springer-VerlagLondonLimited2011 MATLAB® and Simulink® are registered trademarks of The MathWorks, Inc., 3 Apple Hill Drive, Natick,MA01760-2098,U.S.A.http://www.mathworks.com Apartfromanyfairdealingforthepurposesofresearchorprivatestudy,orcriticismorreview,asper- mittedundertheCopyright,DesignsandPatentsAct1988,thispublicationmayonlybereproduced, storedortransmitted,inanyformorbyanymeans,withthepriorpermissioninwritingofthepublish- ers,orinthecaseofreprographicreproductioninaccordancewiththetermsoflicensesissuedbythe CopyrightLicensingAgency.Enquiriesconcerningreproductionoutsidethosetermsshouldbesentto thepublishers. Theuseofregisterednames,trademarks,etc.,inthispublicationdoesnotimply,evenintheabsenceofa specificstatement,thatsuchnamesareexemptfromtherelevantlawsandregulationsandthereforefree forgeneraluse. Thepublishermakesnorepresentation,expressorimplied,withregardtotheaccuracyoftheinformation containedinthisbookandcannotacceptanylegalresponsibilityorliabilityforanyerrorsoromissions thatmaybemade. Coverdesign:eStudioCalamarS.L. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface In the last years, notable progresses were obtained in mathematical theory of sta- bilization of equilibrium solution to Newtonian fluid flows as a principal tool to eliminateorattenuatetheturbulence.Oneofthemainresultsobtainedinthisdirec- tionisthattheequilibriumsolutionstoNavier–Stokesequationsareexponentially stabilizable by finite-dimensional feedback controllers with support in the interior ofthedomainorontheboundary.Thisbookwascompletedintheideatopresent these new results and techniques which are in our opinion the core of a discipline stillindevelopmentandfromwhichonemightexpectinthefuturesomespectacular achievements. Beside internal and boundarystabilizationof Navier–Stokesequations, the sto- chasticstabilizationandrobustnessofstabilizablefeedbacksarealsodiscussed.We hadinmindarigorousmathematicaltreatmentofthestabilizationproblem,which relies on some advanced results and techniques involving the theory of Navier– Stokesequationsandfunctionalanalysisaswell.Wetriedtoanswertothefollowing questions:whichisthestructureofthestabilizingfeedbackcontrollerandhowcan be designed by using a minimal set of eigenfunctions of the Stokes–Oseen opera- tor. Though most of the feedback controllers constructed here are conceptual and their practical implementation requires a computational effort which still remains to be done, the analysis developed here provides a rigorous pattern for the design ofefficientstabilizablefeedbackcontrollersinmostspecificproblemsofpractical interest. To this purpose, the exposition is in mathematical style: definitions, hy- potheses, theorems, proof. It should be emphasized that no rigorous stabilization theorywithinternalorboundarycontrollersispossiblewithoutuniquecontinuation theoryforthesolutionstoStokes–Oseenequations. Byincludingapreparatorychapteroninfinite-dimensionaldifferentialequations andafewappendicespertaininguniquecontinuationpropertiesofeigenfunctionsof theStokes–Oseenoperatorandstochasticprocesses,wehaveattemptedtomakethis work essentially self-contained and so, accessible to a broad spectrum of readers. What is assumed of the reader is a knowledge of basic results in linear functional analysis,linearalgebra,probabilitytheoryandgeneralvariationaltheoryofelliptic, parabolic and Navier–Stokes equations, most of these being reviewed in Chap. 1 and in Sects. 3.8 and 4.5. An important part of the material included in this book v vi Preface representthepersonalcontributionoftheauthorandhiscoworkersand,thoughwe mentioned the basic references and a brief presentation of other significant works in this field, we did not present them, however, in details. In fact, the presentation wasconfinedtothestabilizationtechniquesbasedonthespectraldecompositionof thelinearizedsysteminstableandunstablesystemsandsowehaveomittedother importantresultsinliterature. The author is indebted to Ca˘ta˘lin Lefter who made pertinent observations and suggestions. I also thank Irena Lasiecka and Roberto Triggiani for useful discus- sions on several results presented in this book. Also, I am indebted to Mrs. Elena MocanufromInstituteofMathematicsinIas¸iwhopreparedthistextforprinting. I wish to express my thanks to Professor Miroslav Krstic, from University of California, San Diego, for the invitation to write this book for the Springer series CommunicationandControlEngineeringheiscoordinating. SpecialthanksareduetoMr.OliverJackson,EditorofEngineeringatSpringer, forunderstandingandassistanceintheelaborationofthiswork. Contents 1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 BanachSpacesandLinearOperators . . . . . . . . . . . . . . . . 1 1.2 SobolevSpacesandEllipticBoundaryValueProblems . . . . . . . 3 1.2.1 VariationalTheoryofEllipticBoundaryValueProblems . . 7 1.2.2 Infinite-dimensionalSobolevSpaces . . . . . . . . . . . . 10 1.3 TheSemigroupsofClassC . . . . . . . . . . . . . . . . . . . . . 12 0 1.4 TheNonlinearCauchyProblem . . . . . . . . . . . . . . . . . . . 14 1.5 StrongSolutionstoNavier–StokesEquations . . . . . . . . . . . . 17 2 StabilizationofAbstractParabolicSystems . . . . . . . . . . . . . . 25 2.1 NonlinearParabolic-likeSystems . . . . . . . . . . . . . . . . . . 25 2.2 InternalStabilizationofLinearizedSystem . . . . . . . . . . . . . 30 2.2.1 TheCaseofNotSemisimpleEigenvalues . . . . . . . . . . 37 2.2.2 DirectProportionalStabilizationofUnstableModes . . . . 40 2.3 BoundaryStabilizationofLinearizedSystem . . . . . . . . . . . . 42 2.4 StabilizationbyNoiseoftheLinearizedSystems . . . . . . . . . . 45 2.4.1 TheBoundaryStabilizationbyNoise . . . . . . . . . . . . 50 2.5 InternalStabilizationofNonlinearParabolic-likeSystems . . . . . 52 2.5.1 High-gainRiccati-basedStabilizableFeedback . . . . . . . 54 2.5.2 Low-gainRiccati-basedStabilizableFeedback . . . . . . . 57 2.5.3 InternalStabilizationofNonlinearSystemviaHigh-gain Riccati-basedFeedback . . . . . . . . . . . . . . . . . . . 58 2.5.4 InternalStabilizationofNonlinearSystemviaLow-gain Riccati-basedFeedback . . . . . . . . . . . . . . . . . . . 61 2.5.5 High-gain Feedback Controller Versus Low-gain ControllerandRobustness . . . . . . . . . . . . . . . . . . 63 2.6 StabilizationofTime-periodicFlows . . . . . . . . . . . . . . . . 66 2.6.1 TheFunctionalSetting. . . . . . . . . . . . . . . . . . . . 66 2.6.2 StabilizationoftheLinearizedTime-periodicSystem . . . 68 2.6.3 TheStabilizingRiccatiEquation . . . . . . . . . . . . . . 74 2.6.4 StabilizationofNonlinearSystem(2.148) . . . . . . . . . 79 2.7 CommentstoChap.2 . . . . . . . . . . . . . . . . . . . . . . . . 85 vii viii Contents 3 StabilizationofNavier–StokesFlows . . . . . . . . . . . . . . . . . . 87 3.1 TheNavier–StokesEquationsofIncompressibleFluidFlows . . . 87 3.2 TheSpectralPropertiesoftheStokes–OseenOperator . . . . . . . 91 3.3 InternalStabilizationviaSpectralDecomposition . . . . . . . . . 93 3.3.1 TheInternalStabilizationoftheStokes–OseenSystem . . . 94 3.3.2 TheStabilizationofStokes–OseenSystembyProportional FeedbackController . . . . . . . . . . . . . . . . . . . . . 100 3.3.3 InternalStabilizationviaFeedbackController;High-gain Riccati-basedFeedback . . . . . . . . . . . . . . . . . . . 103 3.3.4 InternalStabilization;Low-gainRiccati-basedFeedback . . 112 3.4 TheTangentialBoundaryStabilizationofNavier–StokesEquations 119 3.4.1 The Tangential Boundary Stabilization of the Stokes–OseenEquation . . . . . . . . . . . . . . . . . . . 120 3.4.2 StabilizableBoundaryFeedbackControllersviaLow-gain RiccatiEquation . . . . . . . . . . . . . . . . . . . . . . . 130 3.4.3 TheBoundaryFeedbackStabilizationofNavier–Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 135 3.5 NormalStabilizationofaPlane-periodicChannelFlow . . . . . . 142 3.5.1 FeedbackStabilization . . . . . . . . . . . . . . . . . . . . 149 3.6 InternalStabilizationofTime-periodicFlows . . . . . . . . . . . . 156 3.7 TheNumericalImplementationofStabilizingFeedback . . . . . . 159 3.8 UniqueContinuationandGenericPropertiesofEigenfunctions . . 164 3.9 CommentsonChap.3 . . . . . . . . . . . . . . . . . . . . . . . . 174 4 StabilizationbyNoiseofNavier–StokesEquations . . . . . . . . . . 177 4.1 InternalStabilizationbyNoise. . . . . . . . . . . . . . . . . . . . 177 4.1.1 StabilizationbyNoiseoftheLinearizedNavier–Stokes System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 4.1.2 ProofofTheorem4.1 . . . . . . . . . . . . . . . . . . . . 181 4.1.3 StabilizationbyNoiseofNavier–StokesEquations . . . . . 186 4.1.4 ProofofTheorem4.2 . . . . . . . . . . . . . . . . . . . . 188 4.1.5 StochasticStabilizationVersusDeterministicStabilization . 199 4.2 StabilizationoftheStokes–OseenEquationbyImpulseFeedback NoiseControllers . . . . . . . . . . . . . . . . . . . . . . . . . . 201 4.2.1 ProofofTheorem4.3 . . . . . . . . . . . . . . . . . . . . 204 4.2.2 ProofofTheorem4.4 . . . . . . . . . . . . . . . . . . . . 209 4.2.3 TheRobustnessoftheNoiseFeedbackController . . . . . 209 4.2.4 DeterministicImpulseController . . . . . . . . . . . . . . 211 4.3 TheTangentialBoundaryStabilizationbyNoise . . . . . . . . . . 211 4.4 StochasticStabilizationofPeriodicChannelFlowsbyNoiseWall NormalControllers . . . . . . . . . . . . . . . . . . . . . . . . . 217 4.4.1 FeedbackStabilization . . . . . . . . . . . . . . . . . . . . 221 4.4.2 ProofofTheorem4.6 . . . . . . . . . . . . . . . . . . . . 223 4.5 StochasticProcesses . . . . . . . . . . . . . . . . . . . . . . . . . 230 4.6 CommentsonChap.4 . . . . . . . . . . . . . . . . . . . . . . . . 236 Contents ix 5 Robust Stabilization of the Navier–Stokes Equation ∞ viatheH -ControlTheory . . . . . . . . . . . . . . . . . . . . . . . 237 ∞ 5.1 TheState-spaceFormulationoftheH -ControlProblem . . . . . 237 ∞ 5.2 TheH -ControlProblemfortheStokes–OseenSystem . . . . . . 248 5.2.1 InternalRobustStabilizationwithRegulationofTurbulent KineticEnergy . . . . . . . . . . . . . . . . . . . . . . . . 249 5.2.2 RobustInternalStabilizationwiththeRegulationofFluid Enstrophy . . . . . . . . . . . . . . . . . . . . . . . . . . 251 ∞ 5.3 TheH -ControlProblemfortheNavier–StokesEquations . . . . 252 5.3.1 ProofofTheorem5.7 . . . . . . . . . . . . . . . . . . . . 254 ∞ 5.4 TheH -ControlProblemforBoundaryControlProblem . . . . . 267 5.4.1 TheAbstractFormulation . . . . . . . . . . . . . . . . . . 268 ∞ 5.4.2 TheH -BoundaryControlProblemfortheLinearized Navier–StokesEquation . . . . . . . . . . . . . . . . . . . 269 5.5 CommentsonChap.5 . . . . . . . . . . . . . . . . . . . . . . . . 270 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Symbols and Notation Rd thed-dimensionalEuclideanspace R therealline(−∞,+∞) R+ thehalfline[0,+∞) C thecomplexspace Cd thed-dimensionalcomplexspace x·y thedotproductofvectorsx,y∈Rd |·| , (cid:4)·(cid:4) thenormofalinearnormedspaceX X X D y ∂y, i=1,...,d ∇fi t∂hxei gradientofthemapf :X→Y ∇·f thedivergenceofvectorfieldf :O→Rd ⊂Rd L(X,Y) thespaceoflinearcontinuousoperatorsfromXtoY (cid:4)·(cid:4) thenormofL(X,Y) L(X,Y) ∗ (cid:9) X , X thedualofthespaceX (x,y), (x,y) thescalarproductofthevectorsx,y∈H (aHilbertspace).If H x∈X, y∈X∗,thisisthevalueofy atx Aα thefractionalpoweroforderα∈(0,1)oftheoperator A:D(A)⊂H →H D(A) thedomainoftheoperatorA ∗ (cid:9) A , A theadjointoftheoperatorA A−1 theinverseoftheoperatorA {Ω,F,P} theprobabilityspace Lp(0,T;X) (XBanachspace)thespaceofallp-sumablefunctionsfrom [0,T]toX Lp (0,∞;X) thespaceofmeasurablefunctionsy:(0,∞)→Xwhichare loc p-integrableoneachinterval(a,b)⊂(0,∞) y(cid:9)(t),dy(t) thederivativeofthefunctiony:[0,T]→X dt AC([0,T];X) thespaceofabsolutelycontinuousfunctionsfrom[0,T]toX W1,p([0,T];X) thespace{y∈AC([0,T]; y(cid:9)∈Lp(0,T;X)} C([0,T];X) thespaceofallcontinuousfunctionsfrom[0,T]toX C ([0,T];X) thespaceofallweaklycontinuousfunctionsfrom[0,T]toX w eAt theC -semigroupgeneratedbyA 0 Hk(O) theSobolevspaceoforderkonO⊂Rd xi
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