Exact Controllability and Stabilization: The Multiplier Method

167 Pages·1995·5.924 MB·English

by  V. Komornik

EXACT CONTROLLABILITY AND
STABILIZATION
THE MULTIPLIER METHOD
OTHER TITLES IN THE SAME SERIES
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Li Ta-tsien. 1994, 328 pages.
Recent Advances in Partial Differential Equations, by M.-A. Herrero and
E. ZuAZUA. 1994,160 pages.
Exponential Attractors for Dissipative Evolution Equations, by A. Eden,
C. Foias, B. Темам. 1994,192 pages.
D A Research in Applied Mathematics
K M fV I Series Editors : P.G. CIARLET and J.-L. LIONS
EXACT CONTROLLABILITY AND
STABILIZATION
THE MULTIPLIER METHOD
V. Komornik
Université Louis Pasteur, Strasbourg, France
JOHN WILEY & SONS
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Icdnaky Timinek es ZsoUnak
PREFACE
This book grew out of a series of lectures given over the past four years
in FVance, Hungary and the USA.
In the first part exact boimdary controllability problems are studied by
the Hilbert Uniqueness Method. This approach, introduced by Lions [3] in
1986, is based on uniqueness theorems leading to the construction of suitable
Hilbert spaces of the controllable spaces. It is closely related to a duality
theory of Dolecki and Russell [1]. Following Ho [1] and Lions [2], these
spaces may often be characterized by using the multiplier method. In chapters
2 to 4 we reproduce some results of Lions [4], [5] with certain changes :
- some compactness-uniqueness arguments are replaced by constructive
proofs;
- equations containing lower-order terms are also considered;
- more general boundary conditions are used which in fact simplify the
theory.
The results of chapters 5 and 6 were obtained after the publication of
Lions’ monography. In chapter 5 we develop a general and constructive
approach to improve the usual estimates of the exact controllability time. It
was inspired by a new estimation method of Haraux [3]. Using this approach,
in chapter 6 we improve most of the results obtained in chapters 3 and 4. We
also give elementary and constructive proofs for certain results of Zuazua
[1], obtained earlier by indirect arguments.
The second part of the book is devoted to stabilizability. In chapters 8
and 9 strong and uniform boundary stabilization theorems are proved. Our
method is a modified and simplified version of a Liapunov type approach
introduced in Komornik and Zuazua [1]. We also present here a classical
principle of Russell [2] connecting the exact controllability to the stabiliz­
ability, and some recent results of Conrad and Rao [1].
For the sake of brevity in the first nine chapters we consider only the
wave equation. Maxwell’s equations and very simple plate models. In the
last chapter we consider the internal stabilization of the Korteweg-de Vries
equation : we prove a special case of a theorem in Komornik, Russell and
Zhang [2].
The multiplier method, applied systematically in this book, is remarkably
elementary and efficient. In the bibliography we have included some refer­
ences which use other approaches : see in particular the work of Bardos et
VI
al., Littmann, Russell, Joo and their references. We have also included some
material concerning other equations.
I wish to express my gratitude to
- J.-L. Lions for his advice and numerous suggestions concerning the
subject of this book, for his valuable remarks on a preliminary version of
these notes and for his proposal to publish it in the collection RMA;
- P. G. Ciarlet who also proposed the publication of this book in this
collection and who gave me useful advice concerning the presentation of the
material;
- F. Conrad, A. Haraux, J. Lagnese, M. Pierre, B. Rao, D. L. Russell and
E. Zuazua for many fruitful discussions;
- the Mathematical Departments of Virginia Tech, the Universities of
Strasbourg I, Nancy I, the Eotvos University of Budapest and the Institute
of Mathematics and its Applications at the University of Minnesota, where
my lectures were given and/or part of this book was written;
- the students and colleagues following my lectures, in particular S.
Kouemou-Patcheu who read the manuscript and made a number of useful
remarks.
Strasbourg, April 12, 1994
Vll
TABLE OF CONTENTS
P reface..............................................................................................................................v
0. Introduction. Vibrating strings....................................................................1
1. Linear evolutionary problems..........................................................................7
1.1. The diagram V CH = H^ C V *..................................................................7
1.2. The equation u" + Au = 0 ........................................................................10
1.3. The wave equation....................................................................................11
1.4. A Petrovsky system....................................................................................13
1.5. Another Petrovsky system........................................................................15
2. Hidden regularity. Weak solutions ................................................18
2.1. A special vector field .............................................................................18
2.2. The wave equation. The multiplier method.........................................19
2.3. The first Petrovsky system.....................................................................24
2.4. The second Petrovsky system....................................................................29
3. Uniqueness theorem s.......................................................................................35
3.1. The wave equation. Dirichlet condition.................................................35
3.2. The first Petrovsky system........................................................................40
3.3. The second Petrovsky system....................................................................44
3.4. The wave equation. Mixed boimdary conditions...................................47
4. Exact controllability. Hilbert uniqueness method . . . . 53
4.1. The wave equation. Dirichlet control.......................................................53
4.2. The first Petrovsky system........................................................................57
4.3. The wave equation. Neumann or Robin control.................................59
5. Norm inequalities.................................................................................................63
5.1. Riesz sequences............................................................................................63
5.2. Formulation of the results................................................ 65
5.3. Proof of theorem 5.3....................................................................................69
5.4. Proof of theorem 5.5..................................................................................73
6. New uniqueness and exact controllability results . . . . 75
6.1. A unique continuation theorem................................................................75
6.2. The wave equation. Dirichlet condition...................................................77
6.3. The first Petrovsky system........................................................................79
vni
6.4. The second Petrovsky system. Uniqueness theorems...........................81
6.5. The second Petrovsky system. Exact controllability...........................83
6.6. The wave equation. Neumann or Robin condition...............................85
7. Dissipative evolutionary system s..........................................................90
7.1. Maximal monotone operators................................ 90
7.2. The wave equation....................................................................................91
7.3. KirchhoiF plates............................................................................................97
8* Linear stabilization.........................................................................................103
8.1. An integral inequality..............................................................................103
8.2. Uniform stabilization of the wave equation I .................................104
8.3. Links with the exact controllability. Russell’s principle................108
8.4. Uniform stabilization of the wave equation I I .................................Ill
8.5. Strong stabilization. LaSalle’s principle ..............................................114
8.6. Uniform stabilization of the wave equation III ............................116
8.7. Uniform stabilization of Maxwell’s equations.....................................118
9. Nonlinear stabilization ...............................................................................124
9.1. A nonlinear integral inequality..............................................................124
9.2. Uniform stabilization of the wave equation I .....................................126
9.3. Uniform stabilization of the wave equation I I .....................................134
9.4. Uniform stabilization of KirchhoiF plates..............................................135
10. Internal stabilization of the Korteweg~de Vries equation 144
10.1. Well-posedness and conservation laws..................................................144
10.2. Uniform stabilization by linear feedbacks..........................................146
References....................................................................................................................152
0. Introduction. Vibrating strings
Let I = (a, 6) be a bounded interval, T a positive number and consider
the following problem, modelling among other things the small transversal
vibrations of a string :
{uit - u^x){x, t) = 0, (x, t)elx (0, T), (1)
u{a^t) = Va{t) and u{b^t) = Vb{t)y t€[0,T], (2)
u(a:,0) = u®(a:) and ut(x,0) = u^(a;), x £ I. (3)
The problem (l)-(3) is said to be exactly controllable if for ’’arbitrarily”
given initial ’’state” there exist suitable ’’control” fimctions Va and
Vb such that the solution of (l)-(3) satisfies
u(xjT) = ut{x,T) = 0, X £ I, (4)
We say that the controls Va and vj drive the system to rest in time T.
Naturally, we have to specify the functional spaces of the initial states and
of the controls; the results depend on these choices.
The solution of (l)-(3) is by definition a function
u e C{[0,T];H\l))nC\[0,T]-,L^(I)) (5)
satisfying (1) in the distributional sense, the equalities (2) pointwise, and the
equalities (3) almost everywhere. (As for the usual properties of the Sobolev
spaces applied in this book we refer e.g. to Lions and Magenes [1].)
We have the following result :
Theorem 0.1. — LetT = b - a and lei (г¿°,г¿^) £ H^{I) x L^(/) be such
that
u^{a) + u^{b) -f J ds = 0. (6)
(0 Then there is a unique choice of functions
cO
c
<u
Va,Vb£ (7)
QO.
8
such that the solution o/(l)-(3) satisfies (4).
2 Moreover^ Va and are given by the formulae
a-f t
/
w^(s) ds, (8)