Table Of ContentCommunications 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
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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
