Table Of ContentLecture Notes in Mathematics 2048
Editors:
J.-M.Morel,Cachan
B.Teissier,Paris
Forfurthervolumes:
http://www.springer.com/series/304
FondazioneC.I.M.E.,Firenze
C.I.M.E. stands for Centro Internazionale Matematico Estivo, that is, International
MathematicalSummerCentre.Conceivedintheearlyfifties,itwasbornin1954inFlorence,
Italy,andwelcomedbytheworldmathematical community:itcontinues successfully,year
foryear,tothisday.
Many mathematicians from all over the world have been involved in a way oranother in
C.I.M.E.’sactivitiesovertheyears.ThemainpurposeandmodeoffunctioningoftheCentre
maybesummarisedasfollows:everyyear,duringthesummer,sessionsondifferentthemes
from pure and applied mathematics are offered byapplication to mathematicians from all
countries. ASessionis generally basedonthree orfourmaincourses given byspecialists
ofinternationalrenown,plusacertainnumberofseminars,andisheldinanattractiverural
locationinItaly.
TheaimofaC.I.M.E.sessionistobringtotheattentionofyoungerresearcherstheorigins,
development, and perspectives of some very active branch of mathematical research. The
topicsofthecoursesaregenerallyofinternationalresonance.Thefullimmersionatmosphere
ofthecoursesandthedailyexchangeamongparticipantsarethusaninitiationtointernational
collaborationinmathematicalresearch.
C.I.M.E.Director C.I.M.E.Secretary
PietroZECCA ElviraMASCOLO
DipartimentodiEnergetica“S.Stecco” DipartimentodiMatematica“U.Dini”
Universita`diFirenze Universita`diFirenze
ViaS.Marta,3 vialeG.B.Morgagni67/A
50139Florence 50134Florence
Italy Italy
e-mail:zecca@unifi.it e-mail:[email protected]fi.it
FormoreinformationseeCIME’shomepage:http://www.cime.unifi.it
CIMEactivityiscarriedoutwiththecollaborationandfinancialsupportof:
-INdAM(IstitutoNazionalediAltaMatematica)
-MIUR(Ministerodell’Universita’edellaRicerca)
ThiscourseispartiallysupportedbyGDR-GDREonCONTROLEDESEQUATIONSAUX
DERIVEESPARTIELLES(CONEDP).
Fatiha Alabau-Boussouira Roger Brockett
(cid:2)
Olivier Glass Je´roˆme Le Rousseau
(cid:2)
Enrique Zuazua
Control of Partial Differential
Equations
Cetraro, Italy 2010
Editors:
Piermarco Cannarsa
Jean-Michel Coron
123
FatihaAlabau-Boussouira Je´roˆmeLeRousseau
Universite´PaulVerlaine-Metz Universite´d’Orle´ans
LMAM LaboratoireMAPMO
IleduSaulcy Orle´ans
Metz France
France
EnriqueZuazua
RogerBrockett BasqueCenterforAppliedMathematics
HarvardUniversity BizkaiaTechnologyPark
EngineeringandAppliedSciences Derio
OxfordSt.33 Spain
CambridgeMassachusetts
MaxwellDworkin
USA
OlivierGlass
Universite´Paris-Dauphine
CEREMADE
PlaceduMare´chaldeLattrede
Tassigny
Paris
France
ISBN978-3-642-27892-1 e-ISBN978-3-642-27893-8
DOI10.1007/978-3-642-27893-8
SpringerHeidelbergDordrechtLondonNewYork
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2012934256
MathematicsSubjectClassification(2010):35B35,93B05,93B07,93B52,93C20,93D15,76B75,
65M10,65M12
(cid:2)c Springer-VerlagBerlinHeidelberg2012
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Preface
Oneofthefindingsofthe1988ReportbythePanelonFutureDirectionsinControl
Theory,chairedbyWendellH.Fleming,was:
Manyfundamentaltheoreticalissues,suchascontrolofnonlinearmultivariablesystems,or
controlofnonlinearpartialdifferentialequations,arenotyetunderstood.
Nowadays,morethan20yearslater,webelievewecansaythatalotoffundamental
issues concerning the latter topic have definitely been understood, thanks to the
effortsof many researchers who produceda large body of results and techniques.
And yet, this process has led to an enormous amount of open questions that will
needtobeaddressedbynewgenerationsofscientists.Surveyingthemostimportant
advancesof the lasttwo decadesandoutliningfutureresearchdirectionswere the
main motivations that led us to organize the CIME Course on Control of Partial
Differential Equations that took place in Cetraro (CS, Italy), July 19–23, 2010.
We hope thisvolume,which is one of the outcomesof that event,will providean
ultimate formative step for those who attended the course, and will represent an
authoritativereferenceforthosewhowereunabletodoso.
Thecourseconsistedoffiveseriesoflectures,whichare nowthe sourceofthe
chaptersofthismonograph.Specifically,thefollowingtopicswerecovered:
• Stabilization of evolution equations (by Fatiha Alabau-Boussouira): these lec-
turesdiscussedrecentadvances,aswellasclassicalmethods,forthestabilization
of wave-like equations. Special attention was paid to nonlinear problems,
memory-damping,andindirectstabilizationofcoupledPDEs.Alltheproblems
were treated by a unified methodology based on energy estimates. It was
shown how the introduction of optimal-weight convexity methods leads to
easy computable upper energy decay estimates, and how these results can be
completedbylowerenergyestimatesforseveralexamples.
• ControloftheLiouvilleequation(byRogerBrockett):theseequationsdescribe
the evolution of an initial density of points that move according to a given
differential equation, and may depend on a control which can be chosen in
ordertosatisfysomeprescribedgoals.Thisframeworkalsoallowstoovercome
v
vi Preface
limitations of the classical theory:for example, the expense required to imple-
mentcontrollaws.Severalresults(e.g.,onensemblecontrol:controlling,witha
singlecontrol,afinitebutoftenlargenumberofcopiesofagivensystem)aswell
asopenproblemswerepresented.
• Controlinfluidmechanics(byOlivierGlass):thelecturestreatedvariousissues
related to the controllability of two well-known equations in fluid mechanics,
namelytheEulerequationforperfectincompressiblefluidsinbothEulerianand
Lagrangian coordinates, and the one-dimensionalisentropic Euler equation for
compressiblefluidsintheframeworkofentropysolutions.Specialemphasiswas
putontheaspectsofthe theorythatareconnectedwith thenonlinearnatureof
the problem: linearization around an equilibrium gives here no information on
thecontrollabilityofthenonlinearsystem.
• Carleman estimates for elliptic and parabolic equations, with application to
control (by Je´roˆme Le Rousseau): these are weighted energy estimates for
solutionsofpartialdifferentialequationswithweightsofexponentialtype.The
lectures derived Carleman estimates for elliptic and parabolic operators using
several methods: a microlocal approach where the main tool is the Ga˚rding
inequality, and a more computational direct approach. It was also shown how
Carleman estimates can be used to provide unique continuation properties, as
wellasapproximateandnullcontrollabilityresults.
• Controlandnumericsforthewaveequation(byEnriqueZuazua):theselectures
provided a self-contained presentation of the theory that has been developed
recentlyforthenumericalanalysisofthecontrollabilitypropertiesofwaveprop-
agationphenomena.The methodologyadoptedthe so-calleddiscrete approach,
whichconsistsinanalyzingwhetherthesemidiscreteorfullydiscretedynamics
arising when discretizing the wave equation by means of the most classical
schemeofnumericalanalysissharethepropertyofbeingcontrollable,uniformly
with respect to the mesh-size parameters, and the corresponding controls con-
vergetothecontinuousonesasthemeshsizetendstozero.Alltheresultswere
illustratedbymeansofseveralnumericalexperiments.
Besidestheabovelectures,therewerethreeseminars,givenbyKarineBeauchard
(Some controllability results for the 2D Kolmogorovequation),Sylvain Ervedoza
(Regularity of HUM controls for conservative systems and convergence rates for
discrete controls), and Lionel Rosier (Control of some dispersive equations for
waterwaves).TherewerealsofourcommunicationsgivenbyIdoBright(Periodic
optimizationsufficesforinfinitehorizonplanaroptimalcontrol),KhaiTienNguyen
(Theregularityoftheminimumtimefunctionvianonsmoothanalysisandgeometric
measure theory), Camille Laurent (On stabilization and control for the critical
Klein-Gordonequationona3-Dcompactmanifold),andVincentPerrollaz(Exact
controllabilityofentropicsolutionsofscalarconservationlawswiththreecontrols).
Seminarsandcommunicationsarenotreproducedinthesenotes.
Oneimportantpoint,containedinthe1988Reportwementionedabove,isthat
advances in the control field are made through a combination of mathematics,
modeling,computation,andexperimentation.Hopingthereaderwillfindthepresent
Preface vii
exposition in accord with such a basic principle, we wish to thank the lecturers
and authors who designed their contributions in a detailed-yet-focussedform, for
helpingusrealizethisproject.Overall,weareverygratefultoallthe57participants
in the CIME course, for their enthusiasm that created a friendly and stimulating
atmosphereinCetraro.Finally,specialgratitudeisduetotheGDRECONEDP,for
providingtheessentialsupportthatallowedustoreceiveandacceptalargenumber
ofapplications,andtotheC.I.M.E.Foundation,formakingthiseventpossibleand
foritsveryhelpfulassistancebeforeandallalongthelectures.
RomeandParis PiermarcoCannarsa
Jean-MichelCoron
•
Contents
1 OnSomeRecentAdvancesonStabilization
forHyperbolicEquations ................................................... 1
FatihaAlabau-Boussouira
1.1 Introduction............................................................ 2
1.1.1 OnNonlinearandMemoryStabilization..................... 4
1.1.2 OnIndirectStabilizationforCoupledSystems.............. 7
1.2 Notation................................................................ 8
1.3 StrongStabilization.................................................... 8
1.3.1 Dafermos’StrongStabilizationResult....................... 9
1.4 LinearStabilization.................................................... 11
1.4.1 Introduction.................................................... 12
1.4.2 GeometricalAspects .......................................... 12
1.4.3 ExponentialDecayforLinearFeedbacks.................... 18
1.4.4 TheCompactness–UniquenessMethod...................... 20
1.5 NonlinearStabilizationinFiniteDimensions........................ 25
1.5.1 NonlinearGronwallInequalities.............................. 25
1.5.2 AComparisonLemma........................................ 33
1.5.3 EnergyDecayRates Characterization:The
ScalarCase .................................................... 36
1.5.4 TheVectorialCaseandSemi-discretizedPDE’s............. 42
1.5.5 ExamplesofFeedbacksandOptimality...................... 45
1.6 PolynomialFeedbacksinInfiniteDimensions....................... 46
1.7 TheOptimal-WeightConvexityMethod ............................. 48
1.7.1 IntroductionandScope........................................ 48
1.7.2 DominantKineticEnergyEstimates ......................... 52
1.7.3 WeightFunctionAsanOptimalUnknown................... 55
1.7.4 SimplificationoftheEnergyDecayRates ................... 59
1.7.5 GeneralizationtoOpticGeometricConditions:
TheIndirectOptimal-WeightConvexityMethod............ 60
ix
