Table Of ContentSystems & Control: Foundations & Applications
SeriesEditor
TamerBas¸ar,UniversityofIllinoisatUrbana-Champaign
EditorialBoard
KarlJohanA˚stro¨m,LundUniversityofTechnology,Lund,Sweden
Han-FuChen,AcademiaSinica,Beijing
WilliamHelton,UniversityofCalifornia,SanDiego
AlbertoIsidori,UniversityofRome(Italy)and
WashingtonUniversity,St.Louis
PetarV.Kokotovic´,UniversityofCalifornia,SantaBarbara
AlexanderKurzhanski,RussianAcademyofSciences,Moscow
andUniversityofCalifornia,Berkeley
H.VincentPoor,PrincetonUniversity
MeteSoner,Koc¸ University,Istanbul
Alain Bensoussan
Giuseppe Da Prato
Michel C. Delfour
Sanjoy K. Mitter
Representation and Control of
Infinite Dimensional Systems
Second Edition
Birkha¨user
Boston • Basel • Berlin
AlainBensoussan GiuseppeDaPrato
SchoolofManagement ScuolaNormaleSuperiore
UniversityofTexasatDallas PalazzodellaCarovana
Office3.211 Piano3,studio56
Richardson,TX75083-0688 Pisa56126
USA Italy
SanjoyK.Mitter
MichelC.Delfour
DepartmentofElectricalEngineering
Centrederecherchesmathe´matiques
andComputerScience
and
LaboratoryforInformationand
De´partementdemathe´matiquesetdestatistique
DecisionSystems
Universite´deMontre´al
MassachusettsInstituteofTechnology
C.P.6128,succ.Centre-ville
77MassachusettsAvenue
Montre´al(QC),CanadaH3C3J7
Cambridge,MA02139
USA
MathematicsSubjectClassification:34H05,47D03,47N70,49-02,49J27,49K27,93C25,93C95
LibraryofCongressControlNumber:2006936851
ISBN-10:0-8176-4461-X e-ISBN-10:0-8176-4581-0
ISBN-13:978-0-8176-4461-1 e-ISBN-13:978-0-8176-4581-6
Printedonacid-freepaper.
(cid:1)c2007Birkha¨userBoston
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www.birkhauser.com (SB)
In memory of Andr´ee
Preface to the Second Edition
A new edition in a single volume
Over the past decade, more and more sophisticaced mathematical tools and
approaches have been incorporated in the field of Control of infinite dimen-
sional systems. This was motivated by a whole range of challenging applica-
tionsarisingfromnew phenomenologicalstudies,technologicaldevelopments,
and more stringent design requirements. At the same time, researchers and
advanced engineers have been steadily using an impressive amount of very
sophisticatedmathematics in their analysis,synthesis,anddesign ofsystems.
Whatwasregardedastooabstract,specialized,ortheoreticalin1990hasnow
become a standard part of the toolkit.
The decision to produce a second edition of the original 1992–1993 two-
volume edition is further motivated by several other factors. Over the years
thebookhasbeenrecognizedasakeyreferenceinthefield,andarevisedand
corrected edition was desirable. Even if some good books on the control of
infinite dimensional linear systems have appearedsince then, we felt that the
originalmaterialhas not agedtoo muchand thatthe breadthof its presenta-
tionisstillattractiveandverycompetitive.Theresultisacompletelyrevised
and corrected second edition in a single convenient volume with integrated
bibliography and index.
The book has been restructured into five parts and each part into several
chapters.ThemostsignificantchangesoccurredinPartI,whichnowprovides
a verybroadaccountoffinite dimensional linear systems.Itservesasa back-
ground and a motivation for the other parts. The scope of this part has been
expandedbyaddingtopicssuchasdissipative systems inChapter1andintro-
ducing a new Chapter 2 on linear quadratic two-person zero-sum differential
games thatprovidesanexampleofasolutiontothematrixRiccatidifferential
equation that is not necessarily positive semi-definite and some connections
with the H∞-theory and dissipative systems of Chapter 1.
viii Preface to theSecond Edition
Description of the five parts
PartIonthecontroloffinitedimensional linear dynamical systems andlinear
quadratic two-person zero-sum differential games presents a broad review of
thefield.Itisanintroductionandamotivationforthebook,sothatthereader
understands the background from which the infinite dimensional theory is
developed and also obtains an appreciationof the substantialconceptual and
technical difficulties that had to be overcometo give a satisfactory treatment
ofthesubjectintheinfinitedimensionalcontext.Chapter1isconcernedwith
the theory of controllability and observability of linear systems, and the role
theseconceptsplayinthestudyofthequadraticcostproblemoveraninfinite
time horizon. It has been expanded to include, in addition to the updated
section on the H∞-theory, a new section on the theory of dissipative systems
of J. C. Willems [2].
A new Chapter 2 on linear quadratic two-person zero-sum differential
games has been included to broaden the perspective. The pioneering work
in that area has been done in the paper of P. Bernhard [2] in 1979 and
the seminalbook ofT. Ba¸sar andP. Bernhard [1] in 1991and 1995.This
chapterisself-containedwithcompletemathematicalproofs.Itfocusesonthe
openloopcase.Newresultsusingtheinvariantembeddingapproach ofR.Bell-
maninthestyleofJ. L. Lions[3]areincludedtoprovidefurtherinsightinto
the associated inf sup (open loop upper value) and sup inf (open loop lower
value)problems.ThisiscombinedwithdevelopmentsbyP.Zhang[1]in2005
whoestablishedthatinthelinearquadraticcasethedualitygapiseitherzero
orinfinite.This meansthatonlythreecasescanoccur:(a)sup inf=−∞and
inf sup finite; (b) sup inf finite and inf sup = +∞; and (c) sup inf and inf
sup both finite; in which case, there is equality and the existence of a saddle
point. In particular it illustrates the occurrence of symmetrical solutions to
thematrixRiccatidifferentialequationthatarenotnecessarilypositivesemi-
definite. It also connects with the glimpse of H∞-theory and the new section
on the theory of dissipative systems at the end of Chapter 1.
Part II deals with the representation of infinite dimensional systems. It
develops semigroup theory and variational methods for the representation of
infinite dimensional systems such as dynamical partial differential equations
and delay differential systems. Chapter 1 gives a unique presentation of the
theory of semigroups of linear operators integrated with interpolation theory.
It brings together advanced concepts and techniques that are usually treated
independently.Chapter2providesabasicintroductiontothevariational the-
ory of parabolic systems.It nicely describes the explicitand illuminating con-
nectionbetweentheearlyworkofT.KatoandJ.-L.Lionsandthelaterresults
of P. Auscher, S. Hofmann, J. L. Lewis, and Ph. Tchamitchian [1] in
2001 on the Kato’s conjecture on the domain of the square root operators.
Chapter3containsthebasicconstructionstoeffectivelydealwithunbounded
control and observation operators via semigroup methods.
Preface to theSecond Edition ix
Chapter4onthemodeling ofdifferentialsystemswithdelaysinthe state,
control, and observation is self-contained. To our knowledge, it is possibly
the only book where the state space theory is completely developed using
the fundamental structural operators introduced by Delfour and Manitius in
1976–1977and further extended to the neutral case by M. C. Delfour and
J. Karrakchou [1,2] in 1987.
Part III is devoted to the generic qualitative properties of controlled sys-
tems. It studies the controllability for an infinite dimensional abstract linear
dynamicalsystem,whichcanbespecializedtoobtainresultsforcontrollability
ofparabolicandhyperbolicpartialdifferentialequationsbothwhencontrolis
exercised in the interior of the domain and when controlis exercisedthrough
the boundary. The important problem of exact controllability of hyperbolic
equations in appropriate spaces, which leads to stabilizability properties for
these systems (and hence verification of the finite cost condition for infinite
time problems) is discussed in detail in this chapter. The systematic use of
eigenvalues and eigenfunctions of appropriate differential operators to obtain
the results is a somewhat novel aspect of this chapter.
PartsIVandVpresentadynamicalprogrammingapproachtotheoptimal
linear quadratic control problem over a finite and an infinite time horizon of
certain classes of infinite dimensional systems. Dynamic programming is a
deep conceptual idea. Its applications are manifold. For the philosophically
minded reader, we have provided a quotation from Soren Kierkegaard on
pagexviofthePreface toVolume II of the First Edition asanindicationthat
philosophy might often anticipate developments in science.
Part IV is devoted to the quadratic cost optimal control problem over a
finite time horizon. We develop the theory for an abstract dynamical model
satisfyingcertainsemigroup(group)assumptions,andthenwetreatconcrete
situationsbyverifyingtheseassumptionsusingdifferentialequations(partial,
functional-differential)methods.This chapterpresentsareasonablycomplete
treatment of the subject, which includes both boundary control and bound-
ary observation(not necessarilysimultaneously)for parabolic andhyperbolic
systems. Technically, these are the more difficult parts of the book because
they involve unbounded control and observation operators. Many of the re-
sults presented here appeared in book form for the first time in 1993. The
approachweadoptisdynamic programming, whichleadsto asynthesisofthe
optimalcontrolinfeedbackformviaastudyofanoperatorRiccatidifferential
equation.The systematic useofdynamic programming givesa unifiedview of
this topic.
Part V, the final part of the book, is concerned with the quadratic cost
optimal control problem over an infinite time horizon. Here the concepts of
stabilizabilityanddetectabilityplayanessentialrole.Thepropertiesofstabi-
lizability and detectability have to be verified, in some sense directly, for the
paraboliccase(whichhasafinite dimensionalunstable part)andviathe the-
oryofexactcontrollability(forexample)inthehyperboliccase.Thus,thereis
acloserelationbetweenPartVandPartsIandIIIofthebook.Theapproach
x Preface to theSecond Edition
here is again via dynamic programming and a study of an algebraic matrix
Riccati equation. We also prove the stability of the closed loop system. In
many (but not all) situations, the theory of the infinite time quadratic cost
problemintheinfinitedimensionalcaseisascompleteasforthefinitedimen-
sional situation. Again, many results here appearedfor the first time in book
form in 1993.
Acknowledgments
It is a pleasant task to acknowledge the support of the Centre de recherches
math´ematiques of the Universit´e de Montr´eal and the Natural Sciences and
EngineeringCouncilthroughDiscoveryGrant–8730inthe“re-engineering”of
the first edition of the book into this new second edition.
We are very grateful to Thomas Grasso of Birkh¨auser Boston Inc. for his
cooperation and efforts since the 2003 IEEE CDC Meeting to promote the
book and for his kind invitation to consider a second edition. It is also a
pleasure to thank Tamer Ba¸sar, Editor-in-Chief, for accepting it in the book
series Systems & Control: Foundations & Applications.
Most sincere thanks to Louise Letendre of the Centre de Recherches
Math´ematiques for her colossal, patient, and truly professional work of com-
pletely reorganizing and upgrading the old LaTeX files of the original two-
volume book to the latest AMS LaTex standards in a single volume. Very
special thanks also go to Andr´e Montpetit of the Centre de Recherches
Math´ematiques who provided his technical support, experience, and talent
to adapt, upgrade, and ajust the TeX files and the special format of the bib-
liography to the latest Birkha¨user/Springermacros.
Additional earlier acknowledgments can be found in the Prefaces to Vol-
ume I (pages xii to xiii) and II (pages xvi to xvii) of the first edition.
October, 2006 Alain Bensoussan, Richardson, USA
Giuseppe Da Prato, Pisa, Italia
Michel C. Delfour, Montreal, Canada
Sanjoy K. Mitter, Cambridge, USA.
Preface to Volume I of the First Edition
Attheendofthe1960s,thestatespacetheoryoflinearsystemsbothfortime-
invariantandtime-varyingsystemshadessentiallybeenworkedout.Thebasic
concepts of controllability, observability, and their relationship to the theory
of minimal realizations received almost a complete treatment in the work
of Kalman and others. The feedback solution of the quadratic cost optimal
controlproblem,bothinafiniteandaninfinitetimeintervalwerewellunder-
stood. The concepts of controllability and observability and the weaker no-
tionsofstabilizabilityanddetectabilityplayanessentialrolethroughstability
ideas in the solution of the infinite time quadratic cost optimal control prob-
lem. The ideas of optimal control from the Hamilton–Jacobi–Caratheodory,
dynamic programming, calculus of variations (and in its modern form the
McShane–Pontryagin maximum principle) points of view had received defin-
itive treatments. As far as deterministic control problems for finite dimen-
sional linear systems are concerned, attention shifted away from optimality
to synthesis problems,suchas decoupling,regulator,and trackingwith inter-
nal stability, and to a unification of the time and frequency domain points of
view.Aqualitativetheoryofnonlinearcontroloffinitedimensionaldynamical
systems began its development at the same time.
It is a fact that most lumped parameter systems are approximations of
distributed parameter systems, and hence, the study of infinite dimensional
systems such as control of systems governed by partial differential equations
and functional differential equations are both of intrinsic interest and poten-
tiallyimportantforapplicationareassuchaschemicalprocesscontrol,control
ofelasticstructures,andevenforchallengingissuessuchasthestabilizationof
plasma instabilities. It was, in some sense, natural to concentrate on optimal
control problems for infinite dimensional systems so that the well-developed
theory of calculus of variations and Hamilton–Jacobi theory could be suit-
ably generalized and used. Clearly the first step in such a development is a
reasonablycompleteunderstanding ofthelinear quadratic problembothover
a finite and an infinite time interval. This book attempts to do this with a
reasonable degree of completeness.
