Startseite Evolution Equations: Long Time Behavior and Control

Evolution Equations: Long Time Behavior and Control

,
This volume constitutes the proceedings of the summer school MIS 2015,
“Mathematics In Savoie 2015,” whose theme was: “Evolution Equations:
long time behavior and control.”
This summer school was held at the University Savoie Mont Blanc,
Chambéry in the period June 15–18, 2015 (see http://lama.univ-savoie.fr/
MIS2015 for details). It was organized by Kaïs Ammari, UR Analysis and
Control of PDE, University of Monastir, Tunisia, and Stéphane Gerbi,
Laboratoire de Mathématiques, University Savoie Mont Blanc, France.
The summer school consisted of two mini-courses in the morning while
the afternoons were devoted to various contributions on the theme.
The 䱌rst mini-course was held by Farid Ammar-Khodja, University of
Franche-Comté, France. The topic was: “Controllability of parabolic sys-
tems: the moment method.” This recent point of view on the controllability
of parabolic systems permits to overview the moment method for parabolic
equations. This course constitutes the 䱌rst part of this volume.
The second part of this volume is devoted to the second mini-course
which was held by Emmanuel Trélat, UPMC, Paris. The topic was
“Stabilization of semilinear PDEs, and uniform decay under discretiza-
tion.” This course was devoted to the numerical stabilization and control
of partial differential equations and more speci䱌cally it addresses the prob-
lem of the construction of numerical feedback control that will preserve the
theoretical rate of decay.
Categories: Mathematics
Jahr: 2018
Verlag: Cambridge University Press
Sprache: english
Seiten: 206
ISBN 13: 978-1-108-41230-8
Series: LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES 439
File: PDF, 5.61 MB
Download (pdf, 5.61 MB)
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LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES
Managing Editor: Professor M. Reid, Mathematics Institute,
University of Warwick, Coventry CV4 7AL, United Kingdom
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Model theory with applications to algebra and analysis I, Z. CHATZIDAKIS, D. MACPHERSON,
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Epidemics and rumours in complex networks, M. DRAIEF & L. MASSOULIÉ
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Probability and mathematical genetics, N. H. BINGHAM & C. M. GOLDIE (eds)
Finite and algorithmic model theory, J. ESPARZA, C. MICHAUX & C. STEINHORN (eds)
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Symmetries and integrability of difference equations, D. LEVI, P. OLVER, Z. THOMOVA & P. WINTERNITZ (eds)
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383 Motivic integration and its interactions with model theory and non-Archimedean geometry I,
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384 Motivic integration and its interactions with model theory and non-Archimedean geometry II,
R. CLUCKERS, J. NICAISE & J. SEBAG (eds)
385 Entropy of hidden Markov processes and connections to dynamical systems, B. MARCUS,
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386 Independence-friendly logic, A.L. MANN, G. SANDU & M. SEVENSTER
387 Groups St Andrews 2009 in Bath I, C.M. CAMPBELL et al (eds)
388 Groups St Andrews 2009 in Bath II, C.M. CAMPBELL et al (eds)
389 Random fields on the sphere, D. MARINUCCI & G. PECCATI
390 Localization in periodic potentials, D.E. PELINOVSKY
391 Fusion systems in algebra and topology, M. ASCHBACHER, R. KESSAR & B. OLIVER
392 Surveys in combinatorics 2011, R. CHAPMAN (ed)
393 Non-abelian fundamental groups and Iwasawa theory, J. COATES et al (eds)
394 Variational problems in differential geometry, R. BIELAWSKI, K. HOUSTON & M. SPEIGHT (eds)
395 How groups grow, A. MANN
396 Arithmetic differential operators over the p-adic integers, C.C. RALPH & S.R. SIMANCA
397 Hyperbolic geometry and applications in quantum chaos and cosmology, J. BOLTE & F. STEINER (eds)
398 Mathematical models in contact mechanics, M. SOFONEA & A. MATEI
399 Circuit double cover of graphs, C.-Q. ZHANG
400 Dense sphere packings: a blueprint for formal proofs, T. HALES
401 A double Hall algebra approach to affine quantum Schur–Weyl theory, B. DENG, J. DU & Q. FU
402 Mathematical aspects of fluid mechanics, J.C. ROBINSON, J.L. RODRIGO & W. SADOWSKI (eds)
403 Foundations of computational mathematics, Budapest 2011, F. CUCKER, T. KRICK,
A. PINKUS & A. SZANTO (eds)
404 Operator methods for boundary value problems, S. HASSI, H.S.V. DE SNOO & F.H. SZAFRANIEC (eds)
405 Torsors, étale homotopy and applications to rational points, A.N. SKOROBOGATOV (ed)
406 Appalachian set theory, J. CUMMINGS & E. SCHIMMERLING (eds)
407 The maximal subgroups of the low-dimensional finite classical groups, J.N. BRAY, D.F. HOLT &
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408 Complexity science: the Warwick master’s course, R. BALL, V. KOLOKOLTSOV & R.S. MACKAY (eds)
409 Surveys in combinatorics 2013, S.R. BLACKBURN, S. GERKE & M. WILDON (eds)
410 Representation theory and harmonic analysis of wreath products of finite groups, T. CECCHERINI-SILBERSTEIN,
F. SCARABOTTI & F. TOLLI
411 Moduli spaces, L. BRAMBILA-PAZ, O. GARCÍA-PRADA, P. NEWSTEAD & R.P. THOMAS (eds)
412 Automorphisms and equivalence relations in topological dynamics, D.B. ELLIS & R. ELLIS
413 Optimal transportation, Y. OLLIVIER, H. PAJOT & C. VILLANI (eds)
414 Automorphic forms and Galois representations I, F. DIAMOND, P.L. KASSAEI & M. KIM (eds)
415 Automorphic forms and Galois representations II, F. DIAMOND, P.L. KASSAEI & M. KIM (eds)
416 Reversibility in dynamics and group theory, A.G. O’FARRELL & I. SHORT
417 Recent advances in algebraic geometry, C.D. HACON, M. MUSTAŢĂ & M. POPA (eds)
418 The Bloch–Kato conjecture for the Riemann zeta function, J. COATES, A. RAGHURAM, A. SAIKIA &
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419 The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations, J.C. MEYER &
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420 Arithmetic and geometry, L. DIEULEFAIT et al (eds)
421 O-minimality and Diophantine geometry, G.O. JONES & A.J. WILKIE (eds)
422 Groups St Andrews 2013, C.M. CAMPBELL et al (eds)
423 Inequalities for graph eigenvalues, Z. STANIĆ
424 Surveys in combinatorics 2015, A. CZUMAJ et al (eds)
425 Geometry, topology and dynamics in negative curvature, C.S. ARAVINDA, F.T. FARRELL & J.-F. LAFONT (eds)
426 Lectures on the theory of water waves, T. BRIDGES, M. GROVES & D. NICHOLLS (eds)
427 Recent advances in Hodge theory, M. KERR & G. PEARLSTEIN (eds)
428 Geometry in a Fréchet context, C. T. J. DODSON, G. GALANIS & E. VASSILIOU
429 Sheaves and functions modulo p, L. TAELMAN
430 Recent progress in the theory of the Euler and Navier-Stokes equations, J.C. ROBINSON,
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431 Harmonic and subharmonic function theory on the real hyperbolic ball, M. STOLL
432 Topics in graph automorphisms and reconstruction (2nd Edition), J. LAURI & R. SCAPELLATO
433 Regular and irregular holonomic D-modules, M. KASHIWARA & P. SCHAPIRA
434 Analytic semigroups and semilinear initial boundary value problems (2nd Edition), K. TAIRA
435 Graded rings and graded Grothendieck groups, R. HAZRAT
436 Groups, graphs and random walks, T. CECCHERINI-SILBERSTEIN, M. SALVATORI &
E. SAVA-HUSS (eds)
437 Dynamics and analytic number theory, D. BADZIAHIN, A. GORODNIK & N. PEYERIMHOFF (eds)
438 Random walks and heat kernels on graphs, M.T. BARLOW
439 Evolution equations, K. AMMARI & S. GERBI (eds)
440 Surveys in combinatorics 2017, A. CLAESSON et al (eds)
441 Polynomials and the mod 2 Steenrod algebra I, G. WALKER & R.M.W. WOOD
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443 Asymptotic analysis in general relativity, T. DAUDÉ, D. HÄFNER & J.-P. NICOLAS (eds)
444 Geometric and cohomological group theory, P.H. KROPHOLLER, I.J. LEARY, C. MARTÍNEZ-PÉREZ &
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London Mathematical Society Lecture Note Series: 439

Evolution Equations:
Long Time Behavior and Control
Edited by

KA Ï S A M M A RI
University of Monsatir, Tunisia
S T É P H A N E G E RBI
University Savoie Mont Blanc, Chambéry, France

University Printing House, Cambridge CB2 8BS, United Kingdom
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Information on this title: www.cambridge.org/9781108412308
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© Cambridge University Press 2018
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no reproduction of any part may take place without the written
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accurate or appropriate.

Contents

Preface
List of Contributors Present at the Summer School
1

Controllability of Parabolic Systems: The Moment Method

page vii
ix
1

FARID AMMAR-KHODJA

2

Stabilization of Semilinear PDEs, and Uniform Decay under
Discretization

31

EMMANUEL TRÉLAT

3

A Null-Controllability Result for the Linear System of
Thermoelastic Plates with a Single Control

77

CARLOS CASTRO AND LUZ DE TERESA

4

Doubly Connected V-States for the Generalized Surface
Quasi-geostrophic Equations

90

FRANCISCO DE LA HOZ, ZINEB HASSAINIA, AND
TAOUFIK HMIDI

5

About Least-Squares Type Approach to Address Direct
and Controllability Problems

118

ARNAUD MÜNCH AND PABLO PEDREGAL

6

A Note on the Asymptotic Stability of Wave-Type Equations
with Switching Time-Delay

137

SERGE NICAISE AND CRISTINA PIGNOTTI

7

Ill-Posedness of Coupled Systems with Delay
LISA FISCHER AND REINHARD RACKE

v

151

vi

Contents

8

Controllability of Parabolic Equations by the Flatness Approach

161

PHILIPPE MARTIN, LIONEL ROSIER, AND
PIERRE ROUCHON

9

Mixing for the Burgers Equation Driven by a Localized
Two-Dimensional Stochastic Forcing
ARMEN SHIRIKYAN

179

Preface

This volume constitutes the proceedings of the summer school MIS 2015,
“Mathematics In Savoie 2015,” whose theme was: “Evolution Equations:
long time behavior and control.”
This summer school was held at the University Savoie Mont Blanc,
Chambéry in the period June 15–18, 2015 (see http://lama.univ-savoie.fr/
MIS2015 for details). It was organized by Kaïs Ammari, UR Analysis and
Control of PDE, University of Monastir, Tunisia, and Stéphane Gerbi,
Laboratoire de Mathématiques, University Savoie Mont Blanc, France.
The summer school consisted of two mini-courses in the morning while
the afternoons were devoted to various contributions on the theme.
The first mini-course was held by Farid Ammar-Khodja, University of
Franche-Comté, France. The topic was: “Controllability of parabolic systems: the moment method.” This recent point of view on the controllability
of parabolic systems permits to overview the moment method for parabolic
equations. This course constitutes the first part of this volume.
The second part of this volume is devoted to the second mini-course
which was held by Emmanuel Trélat, UPMC, Paris. The topic was
“Stabilization of semilinear PDEs, and uniform decay under discretization.” This course was devoted to the numerical stabilization and control
of partial differential equations and more specifically it addresses the problem of the construction of numerical feedback control that will preserve the
theoretical rate of decay.
Several of the speakers agreed to write review papers related to their contributions to the summer school, while others have written more traditional
research papers, which constitute the last part of this volume.
We believe that this volume therefore provides an accessible summary of
a wide range of active research topics, along with some exciting new results,

vii

viii

Preface

and we hope that it will prove a useful resource for both graduate students
new to the area and to more established researchers.
The summer school brought together internationally leading researchers
from the community of control theory and young researchers who came
from all around the world. The organizers’ intention was to provide a
wide angle snapshot of this exciting and fast moving area and facilitate the
exchange of ideas on recent advances in its various aspects. The numerous
formal, informal, and sometimes lively discussions that resulted from this
interaction were for us a sign that we achieved something in the direction
of fulfilling this aim.
Our second aim was to ensure that the diffusion of these recent results
was not limited to established researchers in the area who were present at
the summer school, but also available to newcomers and more junior members of the research community. This was reflected by the presence of many
unfamiliar and/or young faces in the audience. The present proceedings
should hopefully complete the fulfillment of our second aim.
This summer school would not have materialized without the help and
support of the following institutions.
We are very grateful to the CNRS (Centre National de la Recherche
Scientifique), the University Savoie Mont Blanc; La Région AuvergneRhône-Alpes; the GDRI LEM2I: “Laboratoire Euro-Maghrébin de
Mathématiques et leurs Interactions;” the GDR MACS: “Modelisation,
Analyse et Conduite des Systèmes dynamiques;” the GDR EDP: “Equations aux dérivées partielles;” the GDRE CONEDP: “Control of Partial
Differential Equations;” the MaiMoSine: “Maison de la Modélisation et
de la Simulation, Nanosciences et Environnement;” and the PERSYVALlab: “PERvasive SYstems and ALgorithms” for their financial support
without which this summer school would not be accessible without fees.
Finally we would like to thank all the participants of the summer school
who have made this event a success, the contributors to these proceedings,
and the reviewers for their hard work.
Kaïs Ammari and Stéphane Gerbi
Chambéry, July 07, 2017

List of Contributors
Present at the Summer School

Farid Ammar Khodja
University and ESPE of Franche-Comté
16, Route de Gray, 25030 Besançon Cedex, France
fammarkh@univ-fcomte.fr
Carlos Castro
Department of Mathematics and Information
ETSI Roads, Canals, and Ports
Technical University of Madrid
Ciudad Universitaria
28040 Madrid, Spain
carlos.castro@upm.es
Taoufik Hmidi
University of Rennes1
Campus de Beaulieu, IRMAR
263, Avenue du Général Leclerc
35042 Rennes, France
thmidi@univ-rennes1.fr
Arnaud Münch
Blaise Pascal University
Laboratoire de Mathématiques, UMR CNRS 6620
Clermont-Ferrand, France
arnaud.munch@math.univ-bpclermont.fr

ix

x

List of Contributors Present at the Summer School

Serge Nicaise
University of Valenciennes and of Hainaut Cambrésis
Le Mont Houy
59313 Valenciennes Cedex 9, France
snicaise@univ-valenciennes.fr
Cristina Pignotti
Department of Engineering and Computer Science and Mathematics
Via Vetoio, Loc. Coppito
67010 L’Aquila, Italy
pignotti@univaq.it
Reinhard Racke
Department of Mathematics and Statistics
University of Konstanz
Fach D 187, 78457 Konstanz, Germany
reinhard.racke@uni-konstanz.de
Lionel Rosier
Automatic Control and Systems Center
MINES ParisTech
60 Bd Saint-Michel, 75272 Paris Cedex, France
lionel.rosier@mines-paristech.fr
Armen Shirikyan
Department of Mathematics
Université de Cergy-Pontoise
Site de Saint Martin
2, Avenue Adolphe Chauvin
95302 Cergy-Pontoise Cedex, France
Armen.Shirikyan@u-cergy.fr
Emmanuel Trélat
University of Pierre et Marie Curie (Paris 6)
Laboratoire Jacques-Louis Lions
CNRS, UMR 7598
4 Place Jussieu, BC 187
75252 Paris Cedex 05, France
emmanuel.trelat@upmc.fr

1
Controllability of Parabolic
Systems: The Moment Method
FARI D AMMAR-KHO D J A

Abstract
We give some recent controllability results of linear hyperbolic systems and
we will apply them to solve some nonlinear control problems.
Mathematics Subject Classification 2010. 93B05, 93B07, 93C20, 93C05,
35K40
Key words and phrases. Parabolic systems, null controllability, moment
method

Contents
1.1.
1.2.
1.3.
1.4.
1.5.

1.6.

Introduction
Parabolic Systems and Controllability Concepts
Controllability Results for the Scalar Case: The
Carleman Inequality
First Application to a Parabolic System
The Moment Method
1.5.1. Presentation: Example 1
1.5.2. Generalization of the Moment Problem
1.5.3. Going Back to the Heat Equation
1.5.4. Example 2: A Minimal Time of Control for a 2 × 2
Parabolic System due to the Coupling Function
1.5.5. Example 3: A Minimal Time of Control Due to the
Condensation of the Eigenvalues of the System
The Index of Condensation
1.6.1. Definition
1.6.2. Optimal Condensation Grouping
1.6.3. Interpolating Function
1

2
2
4
7
8
8
10
14
15
21
24
24
25
27

2

Farid Ammar-Khodja

1.6.4.
1.6.5.
References

An Interpolating Formula of Jensen
Going Back to the Boundary Control Problem

28
28
29

1.1 Introduction
The main goal of these notes is to give a review of results relating to controllability issues for some parabolic systems obtained via the moment method.
We will follow Fattorini and Russell who, in the 1970s, solved controllability problems for scalar parabolic equations (see [10, 11]). This method
is very efficient in the one-dimensional space setting. But it has also been
used to prove the boundary null-controllability of the heat equation for
particular geometries of the space domain (disks, parallelipepidons, etc.).
At the beginning of the 1990s, Fursikov and Imanuvilov [12] solved the
null-controllability problem for a general second-order parabolic equation.
They did this by proving a global Carleman inequality for solutions of quite
general parabolic equations. This Carleman inequality implies observability inequality and thus controllability of the corresponding parabolic equation when the control function acts on an arbitrary open subset of the space
domain or on an arbitrary relatively open subset of its boundary. At the
same time, Lebeau and Robbiano [16] also proved the null-controllability
of the heat equation with constant coefficients. Their method of proof is
less general than that of Fursikov–Imanuvilov when dealing with parabolic
equations but it generalizes to abstract diagonal systems.
Since then, a huge literature has been devoted to solving control problems by a systematic use of Carleman estimates: Stokes and Navier–Stokes
equations, Burger’s equations, etc. But as usual in mathematics, any powerful tool or method has its limitations. These appeared in particular when
dealing with parabolic systems. It is one of the goals of these notes to
explain these limits.

1.2 Parabolic Systems and Controllability Concepts
Consider the following system:

 (∂t − D∆ − A)y = Bu1ω ,
y = Cv1Γ0 ,

y(0, ·) = y0 ,

QT := (0, T) × Ω,
ΣT := (0, T) × ∂Ω,
Ω,

(1.1)

where
• Ω ⊂ RN is a smooth bounded domain, ω ⊂ Ω is an open set, Γ0 ⊂ ∂Ω is a
relatively open subset;

Controllability of Parabolic Systems: The Moment Method

3

• D = diag(d1 , ..., dn ), A = (aij )1≤i,j≤N ∈ L∞ (QT ; L (Rn )),
• B = (bij ), C = (cij ) ∈ L∞ (QT ; L (Rm , Rn )): control matrices.
Definition 1.1 System
( (1.1)
) is approximately controllable at time T > 0 if
for all ε > 0, for all y0 , y1 ∈ X × X, there exists (u, v) ∈ L2 (QT ) × L2 (ΣT )
such that y(T) − y1 X ≤ ε.
System (1.1) is null-controllable at time T > 0 if for all y0 ∈ X, there exists
(u, v) ∈ L2 (QT ) × L2 (ΣT ) such that y(T) = 0 in Ω.
Here X is a space where the system (1.1) is well-posed. For example, when
2
n
C = 0 (distributed control), it is enough to work with
( 0 X )= L (Ω; R2 ). In this
case, variational methods should prove that for y , u ∈ X × L (QT ; Rm ),
system (1.1) admits a unique solution
(
)
y ∈ C([0, T]; X) ∩ L2 0, T; H10 (Ω, Rn ) .
When B = 0 and C ̸= 0 (boundary control), a suitable space
( 0 )is X =
−1
n
H (Ω; R ). The transposition method proves that for y , u ∈ X ×
L2 (ΣT ; Rm ), system (1.1) admits a unique solution
y ∈ C([0, T]; X) ∩ L2 (QT ; Rn ).
The previous two controllability concepts have dual equivalent concepts.
Introduce the backward adjoint system:

∗
 (∂t + D∆ + A )φ = 0, in QT ,
(1.2)
φ = 0,
on ΣT ,

0
φ(T) = φ ,
in Ω.
If φ0 ∈ L2 (Ω, Rn ) (resp. φ0 ∈ H10 (Ω, Rn )) then there exists a unique solution
φ to (1.2) such that:
(
)
(
)
φ ∈ C 0, T; L2 (Ω, Rn ) ∩ L2 0, T; H10 (Ω, Rn ) ,
(

(
)
(
))
resp. φ ∈ C 0, T; H10 (Ω, Rn ) ∩ L2 0, T; H2 ∩ H10 (Ω, Rn ) .

The following characterizations have been known for a long time and their
proof can be found in [9] for instance.
Proposition 1.2
• Assume that C = 0 (distributed control)
System (1.1) is approximately controllable if, and only if, for any φ0 ∈
L2 (Ω, Rn ) the associated solution to (1.2) satisfies the property:
B∗ φ = 0

in (0, T) × ω ⇒ φ = 0 in QT .

(1.3)

4

Farid Ammar-Khodja

System (1.1) is null-controllable if, and only if, there exists C = CT > 0
such that for any solution to (1.2)
ˆ Tˆ
2
2
∥φ(0)∥L2(Ω,Rn ) ≤ C
|B∗ φ| dxdt.
(1.4)
0

ω

• Assume B = 0 (boundary control)
System (1.1) is approximately controllable if, and only if, for any φ0 ∈
1
H0 (Ω, Rn ) the associated solution to (1.2) satisfies the property:
C∗

∂φ
=0
∂ν

in (0, T) × Γ0 ⇒ φ = 0 in QT .

(1.5)

System (1.1) is null-controllable if, and only if, there exists C = CT > 0
such that for any solution to (1.2)
ˆ
2
∥φ(0)∥H1(Ω,Rn )
0

≤C
0

T

ˆ
Γ0

C∗

∂φ
∂ν

2

dxdt.

1.3 Controllability Results for the Scalar Case: The
Carleman Inequality
We describe in this section known controllability results for the scalar
parabolic equation and give (without proof) the general form of the
Carleman inequality proved in [12].
Theorem 1.3 The problem

 (∂t − ∆ − a) y = u1ω ,
y = 0,

y(0, ·) = y0 ,

QT := (0, T) × Ω,
ΣT := (0, T) × ∂Ω,
Ω,

(1.6)

is null and approximately controllable in X = L2 (Ω) for any open set ω ⊂ Ω,
provided that a ∈ L∞ (QT ).
As a consequence, the problem

QT := (0, T) × Ω,
 (∂t − ∆ − a)y = 0,
(1.7)
y = v1Γ0 ,
ΣT := (0, T) × ∂Ω,

y(0, ·) = y0 ,
Ω,
is null and approximately controllable in X = H−1 (Ω) for any relatively open
set Γ0 ⊂ ∂Ω.

Controllability of Parabolic Systems: The Moment Method

5

( )
To prove this result, let β0 ∈ C 2 Ω and s ∈ R a parameter. Introduce the
functions
β0 (x)
η(t, x) := s
,
(t, x) ∈ QT ,
t(T − t)
s
ρ(t) :=
, (t, x) ∈ QT
t(T − t)
and the functional
ˆ
(
)
2
2
2
2
I(τ, φ) =
ρτ −1 e−2η |φt | + |∆φ| + ρ2 |∇φ| + ρ4 |φ| .
QT

Theorem 1.4 (Carleman inequality) There exist a positive function β0 ∈
C 2 (Ω), s0 > 0 and C > 0 such that ∀s ≥ s0 and ∀τ ∈ R:
(ˆ
)
ˆ Tˆ
2
2
τ −2η
τ +3 −2η
ρ e |φ| ,
I(τ, φ) ≤ C
ρ e |φt ± c∆φ| +
(1.8)
QT

0

ω

for any function φ satisfying φ = 0 on ΣT and for which the right-hand side is
defined.
More detailed information about the function β0 can be found in [12].
Let us see how this inequality is applied to prove null and approximate
controllability of a system (1.6). Consider the associated backward adjoint
system:

QT := (0, T) × Ω,
 (∂t + ∆ + a) φ = 0,
(1.9)
φ = 0,
ΣT := (0, T) × ∂Ω,

0
φ(T, ·) = φ ,
Ω.
From Theorem 1.4, for any φ0 ∈ L2 (Ω), the solution of (1.9) satisfies (1.8)
which, in particular gives the estimate:
(ˆ
)
ˆ
ˆ Tˆ
2
2
2
ρτ +3 e−2η |φ| ≤ C
ρτ e−2η |aφ| +
ρτ +3 e−2η |φ| .
QT

QT

Since

ˆ

τ −2η

ρ e

0

ˆ
|aφ|

QT

2

2
≤ ∥a∥∞

it appears that
ˆ
ˆ
(
)
2
2
ρτ ρ3 − ∥a∥∞ e−2η |φ| ≤ C
QT

3 3

ω

ρτ e−2η |φ|

2

QT

T
0

ˆ

ρτ +3 e−2η |φ| .
2

(1.10)

ω
2

2/3

But, for s > 0, we have ρ3 ≥ 4Ts6 and taking s ≥ 2T5/3 ∥a∥∞ , we see that ρ3 −
2
∥a∥∞ > 0 on (0, T). With this choice of the parameter s, the approximate
controllability property is readily implied by (1.10).

6

Farid Ammar-Khodja

To prove the null-controllability property, something more has to be
done. According to (1.4), we have to deduce from (1.10) that
ˆ

ˆ

T

2

|φ(0, x)| dx ≤ CT

ˆ
2

|φ| ,

0

Ω

ω

for any solution of (1.9). After noting that e−2η ≥ e−2sβ0 ρ (here β0 =
maxΩ β0 ), the other argument
´ is that there exists α = α(∥a∥∞ ) such that
the function t 7→ E (t) := eαt Ω φ2 is increasing on (0, T) (this is quite easy:
it suffices to compute E′ (t), to use the equation satisfied by φ and to choose
α in such a way that E′ (t) ≤ 0 for t ∈ (0, T)). Using this, we get
ˆ
QT

ˆ
(
)
ρτ ρ3 − ∥a∥2∞ e−2η |φ|2 ≥

T

0

ˆ
≥

( ˆ
)
(
)
ρτ ρ3 − ∥a∥2∞ e−2sβ0 ρ−αt eαt |φ|2 dx dt

T

0

ρτ
ˆ

(

Ω

ˆ
)
|φ(0, x)|2 dx
ρ3 − ∥a∥2∞ e−2sβ0 ρ−αt dt
Ω

2

≥ mT

|φ(0, x)| dx.
Ω

On the other hand, there exists cT > 0 such that
ˆ

T

ˆ

0

We arrive to: Ω

ρτ +3 e−2η |φ| ≤ cT

ˆ

ˆ

0

ω

ˆ

ˆ

T

2

|φ(0, x)| dx ≤ CT
Ω

T

2

0

2

|φ| .
ω

ˆ
2

|φ| ,
ω

which is exactly the observability inequality (1.4). This proves the distributed null-controllability.
Due to this distributed null-controllability property holding true for any
open subset ω ⊂ Ω, it allows to deduce the boundary controllability result
for an arbitrary relatively open subset Γ0 ⊂ ∂Ω. Here is the (heuristic) proof.
Let Ω′ ⊃ Ω another smooth bounded domain such that Ω′ = Ω ∪ Ω0 with
Ω ∩ Ω0 = ∅ and Ω ∩ Ω0 ⊂ Γ0 . By the previous result, the problem (1.6) is
null-controllable on Q′T = (0, T) × Ω′ with any ω ⊂ Ω0 . The restriction to
QT = (0, T) × Ω of a controlled solution on Q′T is a controlled solution
of system (1.7) (and the control function is just the Dirichlet trace of this
controlled solution to (0, T) × Γ0 .
Remark 1.5 Note that the Carleman inequality (1.8) allows to prove both
null and approximate controllability.

Controllability of Parabolic Systems: The Moment Method

7

1.4 First Application to a Parabolic System
Consider the 2 × 2 parabolic system:

(∂t − ∆)y1 = a11 y1 + a12 y2



(∂t − d∆) y2 = a21 y1 + a22 y2 + u1ω ,
 y = (y1 , y2 ) = 0,


y(0, ·) = y0 ,

QT ,
ΣT ,
Ω,

(1.11)

where aij ∈ L∞ (QT ). The following result is proved in [2] and in a most
general version in [13].
Theorem 1.6 If there exists ω0 ⊂ ω such that a12 ≥ σ > 0 on (0, T) × ω0 then
system (1.11) is null and approximately controllable for any d > 0.
The proof of this result uses Carleman inequalities for scalar parabolic
equations (see Theorem 1.4) applied to each equation of the backward
adjoint system:

−(∂t + ∆)φ1 = a11 φ1 + a21 φ2


QT ,

−(∂t + ∆)φ2 = a12 φ1 + a22 φ2 ,
(1.12)

φ = (φ1 , φ2 ) = 0,
ΣT ,


φ(0, ·) = φ0 ,
Ω.
The assumption a12 ≥ σ > 0 on (0, T) × ω0 is used to get an estimate of the
L2 − norm of φ1 on (0, T) × ω0 using the second equation in (1.12). For
more precise details, see [2, 13].
Natural questions arise at this level:
• What happens if
supp(a12 ) ∩ ω = ∅?
The technique of proof used for the previous theorem cannot be extended
to this case. It seems that Carleman estimates cannot treat this kind of
situation.
• What happens for the boundary control system:

(∂ − ∆)y1 = a11 y1 + a12 y2

 t
QT ,


(∂
d∆)y = a y + a22 y2 ,
 t−
( ) 2 ( 21) 1
y1
0

y=
=
1Γ0 v,
ΣT ,


y2
1


y(0, ·) = y0 ,
Ω,
where Γ0 is a relatively open subset of ∂Ω?

8

Farid Ammar-Khodja

There exist only partial answers to these two questions: even in the onedimensional space case. In any space dimension, the single result is the
one proved by Alabau-Boussouira and Léautaud in [1]. They considered
the special system

(∂t − ∆)y1 = ay1 + by2


QT ,

(∂t − d∆)y2 = δby1 + ay2 + u1ω ,
(1.13)

y = (y1 , y2 ) = 0,
ΣT ,


y(0, ·) = y0 ,
Ω,
and proved.
Theorem 1.7 [1] Let b ≥ 0 on Ω. Assume that there exists b0 > 0 and
ωb :=supp(b) ⊂ Ω satisfying the Geometric Control Condition (GCC)
(see [6]) with b ≥ b0 in ω
√b . Assume that ω also satisfies GCC. Then there exists
δ0 > 0 such that if 0 < δ∥b∥L∞(Ω) ≤ δ0 , System (1.13) is null controllable at
any positive time T.
Carleman’s inequalities are not used in the proof of this result. It is
obtained as a consequence of the controllability of the corresponding
hyperbolic system of two wave equations and the transmutation method.
In the forthcoming sections, we will study the one-dimensional version
of system (1.11) by means of the moment method.

1.5 The Moment Method
1.5.1 Presentation: Example 1
We present in this section the moment method through the study of the null
controllability issue for the scalar one-dimensional heat equation:
 ′
QT = (0, T) × (0, π)
 y − yxx = f (x) u(t),
(1.14)
y|x=0,π = 0,
(0, T)

y|t=0 = y0
(0, π) .
Here the constraint is that the control has separate variables: f ∈ L2 (0, π)
and u ∈ L2 (0,√
T).

If φk (x) = π2 sin(kx), then {φk }k≥1 is an orthonormal basis of L2 (0, π).
We look for a solution in the form
∑
y(t, x) =
yk (t)φk (x) .
k≥1

Controllability of Parabolic Systems: The Moment Method

Set

∑

f (x) =

fk sin(kx),

y0 =

k≥1

y0k sin(kx).

k≥1

Then y is a solution if, and only if,
{ ′
yk = −k2 yk + fk u(t),
yk|t=0 = y0k ,
i.e.
2
yk (t) = e−k t y0k

∑

9

ˆ

t

+ fk

(0, T)

, ∀k ≥ 1,

e−k (t−s) u(s) ds,
2

∀k ≥ 1.

0

Therefore, there exists a control function u ∈ L2 (0, T) such that the solution
satisfies y(T, x) = 0 for any x ∈ (0, π) if, and only if, there exists u ∈ L2 (0, T)
such that:
ˆ T
2
2
fk
e−k (T−s) u(s) ds = −e−k T y0k , ∀k ≥ 1.
0

After a change of variable in the integral, we arrive to the reduction of the
null-controllability issue to the problem (v(t) = u(T − t))


Find v ∈ L2 (0, T):
´ T −k2 t
(1.15)
2
 fk 0 e
v(t) dt = −e−k T y0k , k ≥ 1.
This is a moment problem in L2 (0, T) with respect to the family
2
{e−k t }k≥1 .
A necessary condition for the existence of a solution for any y0 ∈
2
L (0, π) is:
fk ̸= 0,

k ≥ 1.

If {e−k t }k≥1 admits a biorthogonal family {qk }k≥1 in L2 (0, T), i.e. a
family {qk }k≥1 such that
2

ˆ

T

e−k t qℓ (t) dt = δkℓ ,
2

k, ℓ ≥ 1,

0

then a formal solution is
v(t) = −

∑ e−k2 T
k≥1

The question is then: v ∈ L2 (0, T)?

fk

y0k qk .

10

Farid Ammar-Khodja

The next subsection is devoted to proving the existence of this biorthogonal family {qk }k≥1 ⊂ L2 (0, T) and to the estimate of ∥qk ∥L2(0,T) as k tends
to ∞ (in order to prove that v ∈ L2 (0, T)).

1.5.2 Generalization of the Moment Problem
Let {λk } ⊂ R such that
0 < λ1 < λ2 < · · · < λk < · · ·,

lim λk = ∞.

k→∞

Let {mk }k≥1 ∈ ℓ2 and consider the moment problem:


Find v ∈ L2 (0, T):
´
 0T e−λk t v(t) dt = mk , k ≥ 1.
To solve this problem, we need to answer the following two questions:
{
}
1. Does the family e−λk t k≥1 admit a biorthogonal family {qk }k≥1 in
L2 (0, T)?
2. If a biorthogonal family {qk }k≥1 exists, is it possible to estimate
∥qk ∥L2(0,T) as k → ∞?
{
}
As a first step, consider e−λk t k≥1 in L2 (0, ∞). Then following Schwartz
[18], we have:
{
}
Theorem 1.8 The family e−λk t k≥1 is
∑
1. complete in L2 (0, ∞) if k≥1 1/λk = ∞ and in this case, it is not minimal;
∑
2. minimal in L2 (0, ∞) if k≥1 1/λk < ∞ and in this case, it is not complete.

Recall that a family {xk }k≥1 is complete in a Hilbert space H if
span{xk , k ≥ 1} = H; it is minimal if for any n ≥ 1, xn ∈
/ span{xk , k ≥ 1, k̸= n}.
The proof is based on classical properties of the Laplace transform and
zeros of holomorphic functions.
Let f ∈ L2 (0, ∞) and its Laplace transform F given by:
ˆ ∞
F (λ) =
e−λt f (t) dt, ℜ(λ) > 0.
0

The main properties we will use are the following (see for instance [18]):
1. F ∈ H(C+ ), the space of holomorphic functions on C+ := {λ ∈ C :
ℜ (λ) > 0}.

Controllability of Parabolic Systems: The Moment Method

11

2. For any ε > 0, F ∈ H∞ (Cε ) the space of bounded holomorphic functions
on Cε = {λ ∈ C : ℜ(λ) > ε}, and moreover
lim

|λ|→∞,λ∈Cε

F (λ) = 0.

{
}
´
2
3. The space H2 (C+ ) = F ∈ H(C+ ) : R |F(σ + iτ )| dτ < ∞, ∀σ > 0 is
(´
)1/2
2
a Hilbert space with norm ∥F∥H2(C+ ) = R |F(iτ )| dτ
and the
Laplace transform is an isometry from L2 (0, ∞) in H2 (C+ ):
∥F∥H2(C+ ) = ∥ f ∥L2(0,∞) ,

f ∈ L2 (0, ∞).

For bounded holomorphic functions we also have the following properties (see [14]):
Theorem 1.9 If f ∈ H∞ (C+ ) is a nontrivial function and if Λ = {zk }k≥1 is the
sequence of its zeros in C+ , then
∑ R(zk )
< ∞.
(1.16)
2
k≥1 1 + |zk |
If (1.16) is satisfied for a sequence Λ = {zk }k≥1 ⊂ C+ , then the infinite
product
∏ 1 − λ/zk
W (λ) =
, λ ∈ C+
(1.17)
1 + λ/zk
k≥1

converges absolutely in C+ and defines a function W ∈ H∞ (C+ ) whose set of
zeroes is the sequence Λ.
∑
Proof of Theorem 1.8 Assume that n≥1 1/λn = ∞ and let φ ∈ L2 (0, ∞)
such that:
ˆ ∞
∀n ≥ 1,
e−λn t φ(t) dt = 0.
0

Let J : C+ → C be the Laplace transform of φ:
ˆ ∞
J (λ) =
e−λt φ(t) dt.
0

J is holomorphic on C+ and uniformly bounded on Cε = {λ ∈ C : R(λ) > ε}
for all ε > 0. Moreover, by the assumption on φ, J(λn ) = 0 for all n ≥ 1.
If φ was nontrivial, from Theorem 1.9 it should follow that
∑ λn
< ∞.
1 + λ2n
n≥1

12

Farid Ammar-Khodja

∑
But this condition is equivalent to n≥1 1/λn < ∞ and contradicts the
starting assumption:
{
} thus φ = 0.
Therefore, e−λn t n≥1 is complete in L2 (0, ∞) since we have proved that
{
}⊥
span e−λn t , n ≥ 1
= {0}.
However, under the same assumption, it is not minimal since for any
k ≥ 1, the sequence (λn )n≥1 still has the same properties.
∑ n̸=k
Assume now that n≥1 1/λn < ∞. Set
W (λ) =

∏ 1 − λ/λk
,
1 + λ/λk

λ ∈ C+ .

(1.18)

k≥1

If a function J is defined by
J (λ) =

W (λ)
(1 + λ)

2

,

λ ∈ C+

(1.19)

then J ∈ H2 (C+ ). From the properties of the Laplace transform, there exists
a nontrivial function φ ∈ L2 (0, ∞) such that
ˆ ∞
J (λ) =
e−λt φ(t) dt, λ ∈ C+ ,
ˆ

∞

0

ˆ
2

|φ(t)| dt =

+∞

−∞

0

2

|J(iτ )| dτ.

{
}
The λn ’s are the zeros of J and thus it follows that the family e−λn t n≥1 is
⊥

−λn t , n ≥ 1⟩ . To prove
not complete
( −λ t ) since φ is nontrivial and belongs to ⟨e
n
that e
is minimal, a biorthogonal family will be explicitly built
n≥1
for which an estimate of the asymptotic behavior of the norm of its
elements will be given. Set for k ≥ 1,

Jk (λ) =

J′ (λ

J(λ)
,
k )(λ − λk )

λ ∈ C+ ,

where J is the previously defined function. It can easily be proved that
Jk ∈ H2 (C+ ). Again, from the Laplace transform properties, there exists
χk ∈ L2 (0, ∞) such that:
ˆ ∞
Jk (λ) =
e−λt χk (t) dt, λ ∈ C+ ,
ˆ

∞
0

0

ˆ
2

|χk (t)| dt =

+∞

−∞

2

|Jk (iτ )| dτ.

Since,
{ −λ t }by definition, Jk (λn ) = δkn , the biorthogonality of the families
e n n≥1 and {χn }n≥1 follows.

Controllability of Parabolic Systems: The Moment Method

13

To estimate the norm of χk , we have from the previous considerations:
ˆ

∞

ˆ
2

|χk (t)| dt =

0

+∞

−∞

J(iτ )
J′ (λk )(iτ − λk )

2

dτ,

Since, |W (iτ )| = 1 for all τ ∈ R, it appears that:
ˆ ∞
ˆ ∞
2
dτ
2
(
|χk (t)| dt =
2
′
|λk J (λk )| 0 (1 + τ 2 )2 τ
0
λk

k ≥ 1.

2

).
+1

By the Lebesgue dominated convergence,
ˆ ∞
ˆ ∞
dτ
π
dτ
) →
(
= .
2
2
2
k→∞
4
2
(1 + τ )
0
0
τ
+1
(1 + τ 2 )
λk
This leads immediately to:
ˆ ∞
2
|χk (t)| dt ∼
0

k→∞

π
2 |λk J′ (λk )|

2

.

The second step in Schwartz’s work is the following: consider the closed
subspace of L2 (0, T) defined by:
L2(0,T)

A(Λ; T) = Span{e−λk t , k ≥ 1}
, 0 < T ≤ ∞.
∑
Theorem 1.10 Assume that k≥1 1/λk < ∞. The restriction operator
RT : A(Λ, ∞) →
φ
7→

A(Λ, T)
φ|(0,T)

is an isomorphism. In particular, there exists CT > 0 such that:
∥ f ∥L2(0,∞) ≤ CT ∥ f ∥L2(0,T) ,

∀f ∈ A(Λ, ∞).

This result is admitted: see [18] for a proof.
With this result in hand, if ek (t) = e−λk t for t ≥ 0 and k ≥ 1, remark that
RT ek = ek|(0,T) . Thus,
δkj = ⟨ek , χj ⟩L2(0,∞)
⟨
⟩
= R−1
R
e
,
χ
T
j
k
T
L2(0,∞)
⟨
⟩
∗
= ek , (R−1
.
T ) χj
2
L (0,T)

14

Farid Ammar-Khodja

∗
Therefore,
the family {qk }k≥1 = {(R−1
T ) χk }k≥1 is biorthogonal to
{
}
−λk t
2
e
in L (0, T) and we have the estimate
k≥1
C1
|λk J′(λk )|

2
≤ ∥qk ∥L2(0,T) ≤ |λk JC′(λ
,
k )|

k ≥ 1.

(1.20)

1.5.3 Going Back to the Heat Equation
Anything amounts to estimate:
2

1
|λk J′ (λk )|

=∏

2(1 + λk )

1−λk /λn
n≥1 1+λk /λn
n̸=k

if we want to solve problem (1.15). Here, the function J is defined in (1.19).
As a consequence of the results of Fattorini and Russell proved in [10, 11],
we have in particular that:
Theorem 1.11 If λk = k2 , then
lim

k→∞

ln |J′(λ1 k )|
λk

= 0.

In other words, for all ε > 0, there exists Cε > 0 such that
∥qk ∥L2(0,T) ≤

C
≤ Cε eελk ,
|λk J′ (λk )|

∀k ≥ 1.

The control problem (1.14) was reduced to solving the moment problem:


Find v ∈ L2 (0, T):
´ T −k2 t
2
 fk 0 e
v(t) dt = −e−k T y0k , k ≥ 1.
If fk ̸= 0 for all k, a formal solution is
v=−

∑ e−k2 T
k≥1

fk

y0k qk

where {qk }k≥1 ⊂ L (0, T) is the biorthogonal family previously constructed.
The function f can be chosen such that
2

C
⇒ ∀ε > 0,
k→∞ kp

fk ∼

2
1
= o(eεk ).
| fk |

Controllability of Parabolic Systems: The Moment Method

15

But, in view of Theorem 1.11, for any ε > 0:
2
2
2
e−k T 0
yk ∥qk ∥L2 (0,T) ≤ Cε e−k T e2εk = Cε e−k (T−2ε) .
| fk |
2

Thus

∑
k

e−k (T−2ε) < ∞ for any ε < T/2. This allows to conclude that
2

v=−

∑ e−k2 T
k≥1

fk

y0k qk ∈ L2 (0, T)

and therefore, that the scalar heat equation (1.14) is null-controllable.
Note that the function f could be chosen such that Supp( f) b (0, π).

1.5.4 Example 2: A Minimal Time of Control for a 2 × 2 Parabolic
System due to the Coupling Function
Consider the 2 × 2 distributed control system:

(∂t − ∂xx )y1 + q(x)y2 = 0 in QT ,



(∂t − ∂xx )y2 = v1ω
 y(0, ·) = 0, y(π, ·) = 0
on (0, T ),


y(·, 0) = y0
in (0, π),

(1.21)

where
• q ∈ L∞ (0, π) is a given function, y0 is the initial datum and v ∈ L2 (QT ) is
the control function.
• ω = (a, b) ⊂ (0, π).
The system possesses a unique solution which satisfies
y ∈ L2 (0, T; H10 (0, π; R2 )) ∩ C 0 ([0, T]; L2 (0, π; R2 )).
Assume that q satisfies
supp(q) ∩ ω = ∅ (⇔ supp(q) ⊂ [0, a] ∪ [b, π]).

(1.22)

For any k ≥ 1, we associate with the function q ∈ L∞ (0, π) the sequences
{Ik (q)}k≥1 and {Ii,k (q)}k≥1 , i = 1, 2, given by
ˆ a
ˆ π


2

I
(q)
:=
q(x)|φ
(x)|
dx,
I
(q)
:=
q(x)|φk (x)|2 dx,
k
2,k
 1,k
0
b
ˆ π

2

 Ik (q) := I1,k (q) + I2,k (q) =
q(x)|φk (x)| dx,
0

16

Farid Ammar-Khodja

where

√
φk (x) =

2
sin(kx),
π

∀x ∈ (0, π),

k ≥ 1.

We will outline the proof of the following result whose details can be
found in [5].
Theorem 1.12 With the previous notations, assume that q ∈ L∞ (0, π) satisfies
(1.22).
1. The system is approximately controllable at time T > 0 if and only if
|Ik (q)| + |I1,k (q)| ̸= 0

∀k ≥ 1.

2. Define
T0 (q) := lim

k→∞

min{−log|I1,k (q)|, −log|Ik (q)|}
.
k2

Then:
1. If T > T0 (q), the system is null-controllable at time T.
2. If T < T0 (q), the system is not null-controllable at time T.
Remark 1.13
• The first point of this theorem has been proved by Boyer and Olive [8].
We will sketch the proof of the second point.
• Note that in [5], the authors show that for any δ ∈ [0, ∞], there exists q ∈
L∞ (0, π) satisfying (1.22) such that T0 (q) = δ. In particular, depending
on q, the system may be always approximately controllable and, at the
same time, never null-controllable.
• The boundary control problem

(∂t − ∂xx )y1 + q(x)y2 = 0
in QT ,


 (∂ − ∂ )y = 0

 t
xx 2
( )
0
 y(0, ·) =
v(t), y(π, ·) = 0 on (0, T),


1


y(·, 0) = y0
in (0, π),
with v ∈ L2 (0, T), is also studied in [5]. It appears that the minimal time
of control is given by:
−log|Ik (q)|
.
k→∞
k2

Tb (q) := lim

From this, it follows that boundary and distributed controllability may
occur independently.

Controllability of Parabolic Systems: The Moment Method
(
Set A0 =

0
0

L := −

1
0

17

)
and consider the vectorial operator:

d2
+ q(x)A0 : D(L) ⊂ L2 (0, π; R2 ) −→ L2 (0, π; R2 )
dx2

with domain D(L) = H 2 (0, π; R2 ) ∩ H10 (0, π; R2 ) and also its adjoint L∗ .
We summarize some properties of the eigenspaces and generalized
eigenspaces of these operators in the following proposition:
Proposition 1.14
• The spectra of L and L∗ are given by σ(L) = σ(L∗ ) = {k2 : k ≥ 1}.
• Given k ≥ 1, let ψk be the unique solution of the nonhomogeneous
Sturm–Liouville problem:

−ψxx − k2 ψ = [Ik (q) − q(x)] φk , in (0, π),




ψ(0) = 0, ψ(π) = 0,
ˆ π




ψ(x)φk (x) dx = 0.
0

{

(

• The family B = Φ1,k =

)
φk
,
0

(
Φ2,k =

ψk
φk

)}
satisfies
k≥1

(L − k2 Id )Φ1,k = 0 and (L − k2 Id )Φ2,k = Ik (q)Φ1,k .
{
( )
( )}
φk
0
• The family B∗ = Φ∗1,k :=
, Φ∗2,k :=
is biorthogonal
ψk
φk
k≥1
to B and
( ∗
)
( ∗
)
L − k2 Id Φ∗2,k = 0.
L − k2 Id Φ∗1,k = Ik (q)Φ∗2,k and
• In particular, if Ik ̸= 0 then k2 is a simple eigenvalue and Φ1,k and Φ2,k
(resp., Φ∗2,k and Φ∗1,k ) are, respectively, an eigenfunction and a generalized eigenfunction of the operator L (resp., L∗ ) associated with k2 ,
while if Ik = 0 then Φ1,k and Φ2,k are both eigenfunctions of L (resp.,
L∗ ) associated with k2 .
• B and B∗ are Riesz bases in L2 (0, π; R2 ) and for any y0 ∈ L2 (0, π; R2 )
∑ {⟨
⟩
}
⟩
⟨
y0 =
y0 , Φ∗1,k Φ1,k + y0 , Φ∗2,k Φ2,k
k≥1

∑{
}
=
y01,k Φ1,k + y02,k Φ2,k .
k≥1

18

Farid Ammar-Khodja

If we look for the solution of System (1.21) in the form:
∑
y(t) =
{y1,k (t)Φ1,k + y2,k (t)Φ2,k }
k≥1

we readily get the system that {yi,k , i = 1, 2; k ≥ 1} must solve the sequence
of 2 × 2 differential systems:

⟨
⟩

y′1,k + k2 y1,k + Ik (q)y2,k = Bv1ω , Φ∗1,k



⟨
⟩
y′2,k + k2 y2,k = Bv1ω , Φ∗2,k

(
)


0
0
 (y1,k , y2,k )
|t=0 = y1,k , y2,k
( )
0
.
with B =
1

⟨
⟩
Solving this system, we get by setting vi,k (t) = Bv1ω , Φ∗i,k for i = 1, 2:
)
2 (
y1,k (T) = e−k T y01,k − TIk y02,k
ˆ T
2
+
e−k (T−t) [v1,k (t) − (T − t)Ik v2,k (t)]dt,
0

y2,k (T) = e−k T y02,k +
2

ˆ

T

e−k (T−t) v2,k (t) dt.
2

0

Then, y(T) = 0 if and only if
(
)
{ ´T
−k2(T−t)
−k2 T
0
0
e
[v
(t)
−
(T
−
t)I
v
(t)]dt
=
−e
y
+
TI
y
1,k
k
2,k
k
1,k
2,k ,
0
´ T −k2(T−t)
−k2 T 0
e
v2,k (t) dt = −e
y2,k .
0
If we look for v in the form:
v(x, t) = f1 (x)v1 (T − t) + f2 (x)v2 (T − t),

(x, t) ∈ QT ,

where v1 , v2 ∈ L2 (0, T) are new controls, only depending on t, and f1 , f2 ∈
L2 (0, π) are appropriate functions satisfying the condition supp( f1 ),
supp(f2 ) ⊆ ω = (a, b), then we get the system:

ˆ T
ˆ T
⟩
2
2 ⟨

−k2 t

f1,k
v1 (t)e
dt + f2,k
v2 (t)e−k t dt = −e−k T y0 , Φ∗2,k




ˆ0 T
ˆ0 T


2
−k2 t
ef1,k
e
v1 (t)e
dt + f2,k
v2 (t)e−k t dt
0
0


 −I (q)f ´ T v (t)te−k2 t dt − I (q)f ´ T v (t)te−k2 t dt

2
1
k
1,k
k ⟨ 2,k 0 ⟩)

0(⟨
⟩



−k2 T
∗
∗
= −e
y0 , Φ1,k − TIk (q) y0 , Φ2,k ,
(1.23)

Controllability of Parabolic Systems: The Moment Method
where, for k ≥ 1, ef1,k , ef2,k are given by
ˆ π
ˆ
efi,k :=
fi (x)ψk (x) dx, fi,k :=
0

19

π

fi (x)φk (x) dx,

i = 1, 2.

0

Remark that, if we fix k ≥ 1, it is a linear system of two equations and
four unknown quantities:
ˆ T
ˆ T
2
−k2 t
vi (t)e
dt,
vi (t)te−k t dt, i = 1, 2.
0

0

Working on system (1.23), it is possible to prove the following result:
Lemma 1.15 The moment problem (1.23) has the form
 ˆ
T

2
2
(k)


vi (t)e−k t dt = e−k T M1,i (y0 ),

0
ˆ T

2
2

(k)

vi (t)te−k t dt = e−k T M2,i (y0 ),

0

(k)

where the quantities Mi,j (y0 ) ∈ R, with k ≥ 1 and 1 ≤ i, j ≤ 2, satisfy the
following property: for any ε > 0 there exists a positive constant Cε (only
depending on ε) such that
2

(k)

Mi,j (y0 ) ≤ Cε ek (T0 (q)+2ε) ∥y0 ∥L2 (0,π;R2 ) ,

∀k ≥ 1,

1 ≤ i, j ≤ 2.

The conclusion giving the positive null controllability result is based on
the following two observations (proved in [3]):
• The family {e−k t , te−k t }k≥1 is minimal in L2 (0, T).
• There exists a biorthogonal family {q1,k , q2,k }k≥1 in L2 (0, T) such that
2

2

2

2

∥qi,k ∥L2(0,T) ≤ Cε eεk ,

i = 1, 2; k ≥ 1.

(1.24)

The formal solution of the moment problem is then given by
}
∑{ 2
2
(k)
(k)
vi (t) =
e−k T M1,i (y0 )q1,k + e−k T M2,i (y0 ) , i = 1, 2.
k≥1

With the previous estimates in Lemma 1.15 and (1.24), it can be checked
exactly as in Section 1.5.3, that vi ∈ L2 (0, T) for T > T0 (q) and leads to the
first point of the theorem.
Let now assume that T < T0 (q) and consider the adjoint problem:

∗

 −θt − θxx + q(x)A0 θ = 0 in QT ,
θ(0, ·) = 0, θ(π, ·) = 0
on (0, T ),


0
θ(·, T) = θ
in (0, π).

20

Farid Ammar-Khodja

( )
The idea, is to find a sequence of initial data θk0 k≥1 for which
´´
|B∗ θk (x, t)|2 dx dt
ω×(0,T )
→ 0
k→∞
∥θk (·, 0)∥2L2 (0,π;R2 )
where θk is the solution of the adjoint problem associated with θk0 .
In this way, the observability inequality
ˆ
2
∥θ(·, 0)∥L2 (0,π;R2 ) ≤ C
|B∗ θ(x, t)|2 dx dt
ω×(0,T )

fails.
For θk0 = ak Φ∗1,k + bk Φ∗2,k , with (ak , bk ) ∈ R2 , the solution of the adjoint
problem is given by:
)
(
2
2
θk (t, x) = ak e−k (T−t) Φ∗1,k − (T − t)Ik (q)Φ∗2,k + bk e−k (T−t) Φ∗2,k .
Computing, we get
∥θk (·, 0)∥2L2 (0,π;R2 )
{
[
]}
2
2
2
= e−2k T |ak |2 + |ak |2 | ∥ψk ∥L2(0,π) + (bk − Tak Ik (q))
≥ e−2k T |ak |2
2

and
ˆ Tˆ
0

|B∗ θ(x, t)|2 =

ˆ Tˆ
0

ω

e−2k t |ak ψk (x) + (bk − tak Ik (q))φk (x)| dx.
2

2

ω

It can be proved that for any x ∈ ω
ψk (x) = τk φk (x) − Ik (q)gk (x) −

√

π1
I1,k (q) cos(kx).
2k

Thus:
ˆ Tˆ
ˆ Tˆ
2
|B∗ θ(x, t)|2 =
e−2k t (ak τk + bk )φk (x)
0

0

ω

ω

√
2
π ak
I1,k (q) cos(kx) dxdt.
− ak Ik (q)(gk (x) + tφk (x)) −
2 k

Choosing ak = 1 and bk = −τk gives the inequality:
ˆ Tˆ
0

(
)
2
2
|B∗ θ(x, t)|2 ≤ C |I1,k (q)| + |Ik (q)|
ω

Controllability of Parabolic Systems: The Moment Method

21

and to summarize:
(
)
2
2
2
e−k T ≤ C |I1,k (q)| + |Ik (q)|
≤ Ce−2k [ k2
2

1

min(−log|I1,k (q)|,− log|Ik (q)|)]

.

The contradiction follows with a suitable choice of a subsequence {kn } in
connection with the definition of T0 (q).

1.5.5 Example 3: A Minimal Time of Control Due to the
Condensation of the Eigenvalues of the System
Consider the system

∂ 2 y1


y′1 =



∂x2






2

 y′ = d ∂ y2
2
∂x2






yi (t, 0) = bi v(t),







yi (0, ·) = y0i

QT = (0, T) × (0, π)
QT
yi (t, π) = 0

i = 1, 2
i = 1, 2

where 0 < d(< 1, b)i ∈ R (i =
1, 2) and
v ∈ L2 (0, T).
(
)
For y0 = y01 , y02 ∈ H−1 0, π; R2 , there exists a unique solution
(
(
))
y ∈ C [0, T] ; H−1 0, π; R2 ∩ L2 (QT ) .
Indeed, the solution is given by:
yi =

∑

yi,k φk

k≥1

√
where φk (x) =

2
π

sin(kx) and
√

ˆ t
2
2
e−k (t−s) v(s) ds
+
kb1
π
0
√
ˆ t
2
2
2
y2,k (t) = e−dk t y02,k +
kb2
e−dk (t−s) v(s) ds.
π
0
2
y1,k (t) = e−k t y01,k

(1.25)

22

Farid Ammar-Khodja
Thus the null controllability issue reduces to: find v ∈ L2 (0, T) such that
 √
´ T −k2 t
2
2

v(t − t) ds = −e−k T y01,k


π kb1 0 e
, k ≥ 1.
(1.26)
√

´

2
2
T

2
kb
e−dk t v(t − t) ds = −e−dk T y0
π

2 0

2,k

Exercise 1.1
1. A first necessary condition for solvability of (1.26) for any initial data is
bi ̸= 0,

i = 1, 2.

2. A second necessary condition for solvability of (1.26) for any initial
data is
dk2 ̸= ℓ2 ,

∀k, ℓ ≥ 1.
√
/ Q.)
(This last condition is equivalent to d ∈
With these two necessary conditions, we now have to solve:
 ´
0
T −k2 t
−k2 T √y1,k


e
v(t
−
t)
ds
=
−e

2
 0
π kb1
, k ≥ 1.

´ T −dk2 t

2
y02,k

−dk
T
√

e
v(t − t) ds = −e
0

2
π kb2

So, this time, we are dealing with the family {e−k t , e−dk t }. Set
2

λ2k = dk2 ,

λ2k+1 = k2 ,

2

k ≥ 1.

{
}
Then clearly n≥1 1/λn < ∞ and it follows from Theorem 1.8 that e−λn t
is minimal in L2 (0, T) and a biorthogonal family {qn } can be found such
that
C1
C2
2
≤ ∥qn ∥L2(0,T) ≤
, n≥1
(1.27)
′
|λn J (λn )|
|λn J′ (λn )|
∑

with, let us recall (see (1.19)):
2

2(1 + λn )
1
=
.
1−λn /λℓ
|λn J′ (λn )| ∏
ℓ≥1 1+λn /λℓ
n̸=ℓ

Again, a formal solution is given by:
∑
e−λn T mn qn
v(t − t) =
n≥1

(1.28)

Controllability of Parabolic Systems: The Moment Method

23

and as in the Fattorini–Russell example (Sections 1.5.1 and 1.5.3), we need
an estimate of 1/ |λn J′ (λn )| or, in other words, to compute
lim log

k→∞

1
.
|λn J′ (λn )|

Remember that if λn = n2 , we had
lim log

k→∞

1
= 0.
|n2 J′ (n2 )|

Indeed, it has been proven (see [17] for instance) that for any ordered real
sequence
|λn − λm | ≥ α |n − m| =⇒ lim log
k→∞

1
|λn J′ (λn )|

= 0.

}
{
2
2
But for the sequence λ2k = dk
√ , λ2k+1 = k , there does not exist a
positive real number such that d ∈
/ Q and for which this separability
condition satisfied. Indeed, it is proven in [4] that:
√
Proposition
1.16
For
any
c
∈
[0,
∞],
there
exists
d∈
/ Q such that for
{
}
2
2
λ2k = dk , λ2k+1 = k
lim

k→∞

log |λn J′1(λn )|
λn

= c.

Actually, the number
C(Λ) := lim

k→∞

log |λn J′1(λn )|
λn

(1.29)

associated with the sequence Λ = {λn } has a name: it is the index of
condensation of the sequence {λn }.
Before saying more about this index, let us see to what kind of conclusion
leads the introduction of this number:
Theorem 1.17 Assume that
√
d∈
/ Q.
{
}
Let c(Λ) be the condensation index associated with Λ= λ2k =dk2 , λ2k+1 =k2 .
Then:
bi ̸= 0 (i = 1, 2)

and

1. If T > c, then the system is null-controllable.
2. If T < c, then the system is not null-controllable.

24

Farid Ammar-Khodja

To prove the first point, it suffices to use the definition of C(Λ) in (1.29)
and the estimate (1.27 to prove that the function v defined in 1.28 belongs to
L2 (0, T). The second point is more tricky: we need some of the intermediate
results given in the forthcoming section.

1.6 The Index of Condensation
1.6.1 Definition
Definition 1.18 Let Λ = (λk )k≥1 be an increasing sequence of real numbers.
A condensation grouping of Λ is any sequence of sets G =(Gk )k≥1 satisfying
the following properties:
1. Λ ∩ Gk ̸= ∅ for all k ≥ 1 and Λ = ∪k≥1 (Λ ∩ Gk ).
2. If
Λ ∩ Gk = {λnk , λnk +1 , . . . , λnk +pk },

k≥1

then
lim

pk

k→∞ λnk

lim

k→∞

=0

λnk +pk
= 1.
λnk

The second item of this definition characterizes what is meant by
condensation.
Let G = (Gk )k≥1 a condensation grouping of Λ.
• For all k ≥ 1, the index of condensation of Gk = {λnk , . . . , λnk +pk } is the
number






pk !

.
∏
ln
δ(Gk ) = sup
|(λnk +l − λnk +j )| 
0≤l≤pk |λnk +l |


1

0≤ j≤pk
j̸=l

• The index of condensation of G = (Gk )k≥1 is the number: δ(G) =
limk→∞ δ(Gk ).
Definition 1.19 The index of condensation of Λ = (λk )k≥1 is the number
δ(Λ) defined to be the supremum of the set {δ(G)} where G may be any
condensation grouping.

Controllability of Parabolic Systems: The Moment Method

25

Example 1.20 Let Λ = (λn ) and set Gn = {λn }.
G = (Gn ) is a condensation grouping with pn = 0 and
δ(Gn ) = 0,

n ≥ 1.

Conclusion: δ(Λ) ≥ 0 for any Λ.
Example 1.21 Let α ≥ 1 and β > 0 and set Λ = {λ2n = nα , λ2n+1 = nα + e−n ,
n ≥ 1}.
]
[
Define: Gn = λ2n − 21 ; λ2n + 21 , n ≥ 1. G = (Gn ) is a condensation
grouping with pn = 1 and
{
}
1!
1
1!
1
δ(Gn ) = max α ln −nβ , α
ln
n
e
n + e−nβ
e−nβ
1
1
= α ln −nβ = nβ−α .
n
e
β

Thus:


 0, 0 < β < α,
δ(G) = lim sup δ(Gn ) =
.
1,
β = α,

n→∞
∞,
β > α.

1.6.2 Optimal Condensation Grouping
Definition 1.22 Let Λ = (λn )n≥1 be a real increasing sequence. A sequence
n
= D. The
Λ is measurable if there exists D ∈ [0, ∞[ such that limn→∞
λn
number D when it exists is the density of Λ.
Example 1.23 Let α ≥ 1 and β > 0 and set Λ = {λ2n = nα , λ2n+1 = nα + e−n ,
n ≥ 1}. Then

if α > 1
 0,
D=
1,
if α = 1

∞,
if α < 1.
β

For any subset E ⊂ R, NΛ (E) = card(Λ ∩ E) is the counting function of
E. The following intermediate result is due to Shackell [17]:
Lemma 1.24 Let Λ = (λk )k≥1 be an increasing sequence of real numbers,
with finite density D, and let 0 < q < 1/(2D + η) where η > 0 is some arbitrary fixed number. For all λ ∈ Λ and any integer r ≥ 1, let I(λ, rq) =
]λ − rq, λ + rq[.

26

Farid Ammar-Khodja

Then there exists a greatest integer p(λ) such that
NΛ (I(λ, p(λ) q)) ≥ p(λ).
With the previous lemma in hand, we have the following result:
Theorem 1.25 [17] Let Λ = (λn )n≥1 be an increasing sequence of positive
real numbers, with density D and 0 < q < 1/(2D + η) where η > 0 is a fixed
arbitrary number.
Then there exists a condensation grouping G = (Gk )k≥1 such that:
1. δ(G) = δ(Λ).
2. Fix k ≥ 1. For all λ ∈ Λ ∩ Gk and (µn )1≤n≤m ⊂ Λ\(Λ ∩ Gk ), we have
m
∏

|λ − µn | ≥ qm m!.

n=1

Proof [Sketch of the proof] By successive applications of the previous
Lemma, define a sequence of intervals in the following way. Let:
G1 = I(λ1 , p1 q), n1 = 1, p1 = p(λn1 ).
Denote by λn2 the smallest element of Λ not belonging to I1 (q) and set
G2 = I(λn2 , p2 q),

p2 = p(λn2 ).

Let k ≥ 1 and suppose constructed the intervals (Gj )1≤ j≤k . Denote by λnk+1
the smallest element of Λ not belonging to ∪1≤ j≤k Gj . We then define
(
)
(
)
Gk+1 = I λnk+1 , pk+1 q ,
pk+1 = p λnk+1 .
The sequence thus defined satisfies the conclusion of the theorem as it can
be easily checked.
Example 1.26 Let α ≥ 1 and β > 0 and set Λ = {λ2n = nα , λ2n+1 = nα + e−n ,
n ≥ 1}.
]
[
Define: Gn = λ2n − 21 ; λ2n + 12 , n ≥ 1.
G = (Gn ) is a condensation grouping with pn = 1 and
}
{
1
1!
1
1!
ln −nβ
δ(Gn ) = max α ln −nβ , α
n
e
n + e−nβ
e
1
1
= α ln −nβ = nβ−α .
n
e
β

Controllability of Parabolic Systems: The Moment Method

Thus:

27


 0, 0 < β < α,
δ(G) = lim sup δ(Gn ) =
.
1,
β = α,

n→∞
∞,
β > α.

With this previous construction, it can be checked that G is optimal and
thus: δ(G) = δ(Λ).

1.6.3 Interpolating Function
To the sequence Λ = (λn )n≥1 with density D ≥ 0, is associated the interpolating function
)
∏(
λ2
C(λ) =
1 − 2 , λ ∈ C.
λn
n≥1

The following result is due to Bernstein [7] for real sequences and
Shackell [17] for complex sequences.
Theorem 1.27 Let Λ = (λn )n≥1 be an increasing real sequence of positive
numbers, measurable with finite density D ≥ 0. Then its index of condensation
δ(Λ) is given by:
δ(Λ) = lim

1
ln |C′(λ
k )|

k→∞

λk

.

The property that links with the boundary control problem (1.25) of the
previous section is the following:
Theorem 1.28 If Λ = (λn )n≥1 is an increasing sequence of positive numbers
∑
such that n≥1 1/λn < ∞, then
δ(Λ) = lim

k→∞

where

ln |J′(λ1 k )|
λk

∏
|J′ (λk )| =

1−λk /λn
n≥1 1+λk /λn
n̸=k
2
2λk (1 + λk )

.

Remark 1.29 Note that the condensation index is defined even for
∑
sequences which do not satisfy the condition n≥1 1/λn < ∞.

28

Farid Ammar-Khodja

Exercise 1.2 Define Λ = (λn )n≥1 by
λk2 +l = k2 + le−k ,
β

and prove that:

k ≥ 1,

0 ≤ l ≤ 2k, (β > 0)


 ∞, β > 1
δ(Λ) =
.
2,
β =1

0, 0 < β < 1

1.6.4 An Interpolating Formula of Jensen
An interpolation formula due to Jensen [15] assures that if f is a holomorphic function on a convex domain Ω ⊂ C and A = {aj }0≤ j≤q ⊂ Ω is a set of
distinct points, then there exists θ ∈ [−1, 1] and ξ ∈ Conv(A), the convex
hull of A, such that
q
∑
f(aj )
θ dq f
=
(ξ),
′
PA(aj ) q! dzq
j=0

where for any finite set F the function PF is defined by:
∏
PF (λ) =
(λ − µ).
µ∈F

As a consequence of this formula, we have:
Theorem 1.30 Let Λ = {λk }k≥1 be an increasing sequence of real numbers
and G = {Gk }k≥1 any condensation grouping associated with Λ. Then:
ˆ

∑

∞

lim

k→∞

0

λn ∈Gk

2

pk !
e−λn t
P′Gk(λn )

dt = 0.

This result will prove decisive to establish noncontrollability results.

1.6.5 Going Back to the Boundary Control Problem
The associated observability inequality with the boundary control problem
(1.25) is of the form:
ˆ

T
0

∑
k≥1

2

ck e

−λk t

dt ≥ CT

∑
k≥1

e−λk T c2k ,

∀c ∈ ℓ2 .

Controllability of Parabolic Systems: The Moment Method

29

Let G = (Gk ) be the optimal condensation grouping given by Theorem
1.25. Fix k ≥ 1 and set

pk !

, if λn ∈ Gk ,
′
k
P (λn )
cn =
(1.30)
 Gk
0
otherwise.
Then, from Theorem 1.30:
ˆ
0

T

∑

2

ckn e−λn t dt =

ˆ

T

0

n≥1

∑
λn ∈Gk

2

pk !
e−λn t
P′Gk(λn )

dt → 0.
k→∞

On the other hand, if T < δ(Λ)
∑
n≥1

2
ckn e−λn T

=

∑
λn ∈Gn

pk !
e−λn T
P′Gk(λn )

2

2

≥ eλnk (δ(Λ)−ε−T) → ∞.
This implies that the observability inequality does not hold and concludes the proof of the Theorem 1.17.

References
[1] F. Alabau-Boussouira, M. Léautaud, Indirect controllability of locally coupled wave-type systems and applications, J. Math. Pures Appl. (9) 99 (2013),
no. 5, 544–76.
[2] F. Ammar Khodja, A. Benabdallah, C. Dupaix, Null-controllability of some
reaction-diffusion systems with one control force, J. Math. Anal. Appl. 320
(2006), no. 2, 928–43.
[3] F. Ammar-Khodja, A. Benabdallah, M. González-Burgos, L. de Teresa, The
Kalman condition for the boundary controllability of coupled parabolic
systems. Bounds on biorthogonal families to complex matrix exponentials,
J. Math. Pures Appl. 96 (2011), no. 6, 555–90.
[4] F. Ammar Khodja, A. Benabdallah, M. Gonzalez-Burgos, L. de Teresa. Minimal time for the null controllability of parabolic systems: The effect of the
condensation index of complex sequences. J. Funct. Anal. 267 (2014), no. 7,
2077–151.
[5] F. Ammar Khodja, A. Benabdallah, M. González-Burgos, L. de Teresa, New
phenomena for the null controllability of parabolic systems: Minimal time and
geometrical dependence. https://hal.archives-ouvertes.fr/hal-01165713 (2015).
[6] C. Bardos, G. Lebeau, J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control
Optim. 30 (1992), no. 5, 1024–65.

30

Farid Ammar-Khodja

[7] V. Bernstein, Leçons sur les Progrès Récents de la Théorie des Séries de
Dirichlet, Gauthier-Villars, Paris, 1933.
[8] F. Boyer, G. Olive, Approximate controllability conditions for some linear
1D parabolic systems with space-dependent coefficients, Math. Control Relat.
Fields 4 (2014), no. 3, 263–87.
[9] J.-M. Coron. Control and Nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007.
[10] H.O. Fattorini, D. L. Russell, Exact controllability theorems for linear
parabolic equations in one space dimension, Arch. Rational Mech. Anal. 43
(1971), 272–92.
[11] H.O. Fattorini, D. L. Russell, Uniform bounds on biorthogonal functions
for real exponentials with an application to the control theory of parabolic
equations, Quart. Appl. Math. 32 (1974/5), 45–69.
[12] A. V. Fursikov, O. Yu. Imanuvilov, Controllability of Evolution Equations,
Lecture Notes Series, 34. Seoul National University, Research Institute of
Mathematics, Global Analysis Research Center, Seoul, 1996.
[13] M. González-Burgos, L. de Teresa, Controllability results for cascade systems
of m coupled parabolic PDEs by one control force, Port. Math. 67 (2010), no. 1,
91–113.
[14] E. Hille. Analytic Function Theory. Vol. II. Second edition. AMS Chelsea
Publishing, Boston, MA, 2000.
[15] J.L.W.V. Jensen, Sur une expression simple du reste dans la formule
d’interpolation de Newton, Bull. Acad. Roy. Danemark 1894 (1894), 246–52.
[16] G. Lebeau, L. Robbiano, Contrôle exact de l’équation de la chaleur, Comm.
Partial Diff. Eq. 20 (1995), no. 1–2, 335–56.
[17] J. R. Shackell, Overconvergence of Dirichlet series with complex exponents,
J. Analyse Math. 22 (1969), 135–70.
[18] L. Schwartz. Étude des Sommes d’Exponentielles Réelles. Hermann, Paris,
France, 1943.

LABORATOIRE DE MATHÉMATIQUES DE BESANÇON
E-mail address: farid.ammar-khodja@univ-fcomte.fr

2
Stabilization of Semilinear PDEs, and
Uniform Decay under Discretization
EMMAN UEL TRÉL A T

Abstract
These notes are issued from a short course given by the author in a summer
school in Chambéry in June 2015.
We consider general semilinear PDEs and we address the following two
questions:
1. How to design an efficient feedback control locally stabilizing the
equation asymptotically to 0?
2. How to construct such a stabilizing feedback from approximation
schemes?
To address these issues, we distinguish between parabolic and hyperbolic
semilinear PDEs. By parabolic, we mean that the linear operator underlying
the system generates an analytic semigroup. By hyperbolic, we mean that
this operator is skew-adjoint.
We first recall general results allowing one to consider the nonlinear term
as a perturbation that can be absorbed when one is able to construct a
Lyapunov function for the linear part. We recall in particular some known
results borrowed from the Riccati theory.
However, since the numerical implementation of Riccati operators is
computationally demanding, we focus then on the question of being able to
design “simple” feedbacks. For parabolic equations, we describe a method
consisting of designing a stabilizing feedback, based on a small finitedimensional (spectral) approximation of the whole system. For hyperbolic equations, we focus on simple linear or nonlinear feedbacks and we
investigate the question of obtaining sharp decay results.
When considering discretization schemes, the decay obtained in the continuous model cannot in general be preserved for the discrete model, and
31

32

Emmanuel Trélat

we address the question of adding appropriate viscosity terms in the numerical scheme, in order to recover a uniform decay. We consider space, time,
and then full discretizations and we report in particular on the most recent
results obtained in the literature.
Finally, we describe several open problems and issues.
Mathematics Subject Classification 2010. 35B37,74B05,93B05
Key words and phrases. Null controllability, thermoelastic plate system,
single distributed control

Contents
2.1.

Introduction and General Results
2.1.1. General Setting
2.1.2. In Finite Dimension
2.1.3. In Infinite Dimension
2.1.4. Existing Results for Discretizations
2.1.5. Conclusion
2.2. Parabolic PDEs
2.3. Hyperbolic PDEs
2.3.1. The Continuous Setting
2.3.2. Space Semidiscretizations
2.3.3. Time Semidiscretizations
2.3.4. Full Discretizations
2.3.5. Conclusion and Open Problems
References

32
32
33
34
37
38
38
43
44
53
66
69
71
73

2.1 Introduction and General Results
2.1.1 General Setting
Let X and U be Hilbert spaces. We consider the semilinear control system
ẏ(t) = Ay(t) + F(y(t)) + Bu(t),

(2.1)

where A : D(A) → X is an operator on X, generating a C0 semigroup, B ∈
L(U, D(A∗ )′ ), and F : X → X is a (nonlinear) mapping of class C 1 , assumed
to be Lipschitz on the bounded sets of X, with F(0) = 0. We refer to
[10, 18, 43, 44] for well-posedness of such systems (existence, uniqueness,
appropriate functional spaces, etc.).
We focus on the following two questions:
1. How to design an efficient feedback control u = Ky, with K ∈ L(X, U),
locally stabilizing (2.1) asymptotically to 0?
2. How to construct such a stabilizing feedback from approximation
schemes?

Stabilization of Semilinear PDEs

33

Moreover, we want the feedback to be as simple as possible, in order to promote a simple implementation. Given the fact that the decay obtained in the
continuous setting may not be preserved under discretization, we are also
interested in the way one should design a numerical scheme, in order to get
a uniform decay for the solutions of the approximate system, i.e., in order to
guarantee uniform properties with respect to the discretization parameters
△x and/or △t. This is the objective of these notes, to address those issues.
Concerning the first point, note that, in general, stabilization cannot be
global because there may exist other steady states than 0, i.e., ȳ ∈ X such
that Aȳ + F(ȳ) = 0. This is why we speak of local stabilization. Without
loss of generality we focus on the steady-state 0 (otherwise, just design a
feedback of the kind u = K(y − ȳ)).
We are first going to recall well-known results on how to obtain local
stabilization results, first in finite dimension, and then in infinite dimension. As a preliminary remark, we note that, replacing if necessary A with
A + dF(0), we can always assume, without loss of generality, that dF(0) =
0, and thus,
∥F(y)∥ = o(∥y∥)

near y = 0.

Then, a first possibility to stabilize (2.1) locally at 0 is to consider the nonlinear term F(y) as a perturbation, that we are going to absorb with a linear
feedback. Let us now recall standard results and methods.

2.1.2 In Finite Dimension
In this section, we assume that X = Rn . In that case, A is a square matrix of
size n, and B is a matrix of size n × m. Then, as it is well known, stabilization
is doable under the Kalman condition
rank(B, AB, . . . , An−1 B) = n,
and there are several ways to do it (see, e.g., [42] for a reference on what
follows).
2.1.2.1 First Possible Way: By Pole-Shifting
According to the pole-shifting theorem, there exists K (matrix of size m × n)
such that A + BK is Hurwitz, that is, all its eigenvalues have negative real
part. Besides, according to the Lyapunov lemma, there exists a positive
definite symmetric matrix P of size n, such that
P(A + BK) + (A + BK)⊤ P = −In .

34

Emmanuel Trélat

It easily follows that the function V defined on Rn by
V(y) = y⊤ Py
is a Lyapunov function for the closed-loop system ẏ(t) = (A + BK)y(t).
Now, for the semilinear system (2.1) in closed loop with u(t) = Ky(t),
we have
d
V(y(t)) = −∥y(t)∥2 + y(t)⊤ PF(y(t)) ≤ −C1 ∥y(t)∥2 ≤ −C2 V(y(t)),
dt
under an a priori assumption on y(t), for some positive constants C1 and
C2 , whence the local asymptotic stability.
2.1.2.2 Second Possible Way: By Riccati Procedure
The Riccati procedure consists of computing the unique negative definite
solution of the algebraic Riccati equation
A⊤ E + EA + EBB⊤ E = In .
Then, we set u(t) = B⊤ Ey(t).
´ +∞
Note that this is the control minimizing 0 (∥y(t)∥2 + ∥u(t)∥2 )dt for
the control system ẏ(t) = Ay(t) + Bu(t). This is one of the best well-known
results of the linear quadratic theory in optimal control.
Then the function
V(y) = −y∗ Ey
is a Lyapunov function, as before.
Now, for the semilinear system (2.1) in closed loop with u(t) = B⊤ Ey(t),
we have
d
V(y(t)) = −y(t)⊤ (In + EBB⊤ E)y(t) − y(t)⊤ EF(y(t))
dt
≤ −C1 ∥y(t)∥2 ≤ −C2 V(y(t)),
under an a priori assumption on y(t), and we easily infer the local asymptotic stability property.

2.1.3 In Infinite Dimension
2.1.3.1 Several Reminders
When the Hilbert space X is infinite-dimensional, several difficulties occur
with respect to the finite-dimensional setting. To explain them, we consider
the uncontrolled linear system
ẏ(t) = Ay(t),
with A : D(A) → X generating a C0 semigroup S(t).

(2.2)

Stabilization of Semilinear PDEs

35

The first difficulty is that none of the following three properties are
equivalent:
1. S(t) is exponentially stable, i.e., there exist C > 0 and δ > 0 such that
∥S(t)∥ ≤ Ce−δt , for every t ≥ 0.
2. The spectral abscissa is negative, i.e., sup{Re(λ) | λ ∈ σ(A)} < 0.
3. All solutions of (2.2) converge to 0 in X, i.e., S(t)y0 −→ 0, for every
t→+∞

y0 ∈ X.

For example, if we consider the linear wave equation with local damping
ytt − △y + χω y = 0,
in some domain Ω of R , with Dirichlet boundary conditions, and with ω an
open subset of Ω, then it is always true that any solution (y, yt ) converges to
(0, 0) in H10 (Ω) × L2 (Ω) (see [16]). Besides, it is known that we have exponential stability if and only if ω satisfies the geometric control condition
(GCC). This condition says, roughly, that any generalized ray of geometric
optics must meet ω in finite time. Hence, if for instance Ω is a square, and
ω is a small ball in Ω, then GCC does not hold and hence the exponential
stability fails, whereas convergence of solutions to the equilibrium is valid.
In general, we always have
n

sup{Re(λ) | λ ∈ σ(A)} ≤ inf{µ ∈ R | ∃C > 0, ∥S(t)∥ ≤ Ceµt

∀t ≥ 0},

in other words the spectral abscissa is always less than or equal to the best
exponential decay rate. The inequality may be strict, and the equality is
referred to as “spectral growth condition.”
Let us go ahead by recalling the following known results:
• Datko theorem: S(t) is exponentially stable if and only if, for every y0 ∈ X,
S(t)y0 converges exponentially to 0.
• Arendt–Batty theorem: If there exists M > 0 such that ∥S(t)∥ ≤ M for
every t ≥ 0, and if i R ⊂ ρ(A) (where ρ(A) is the resolvent set of A), then
S(t)y0 −→ 0 for every y0 ∈ X.
t→+∞

• Huang–Prüss theorem: Assume that there exists M > 0 such that ∥S(t)∥ ≤
M for every t ≥ 0. Then S(t) is exponentially stable if and only if i R ⊂
ρ(A) and supβ∈R ∥(iβid − A)−1 ∥ < +∞.
Finally, we recall that:
• Exactly null-controllable implies exponentially stabilizable, meaning that
there exists K ∈ L(X, U) such that A + BK generates an exponentially
stable C0 semigroup.
• Approximately controllable does not imply exponentially stabilizable.

36

Emmanuel Trélat

For all reminders done here, we refer to [15, 18, 31, 32, 45].
2.1.3.2 Stabilization
Let us now consider the linear control system
ẏ = Ay + Bu
and let us first assume that B ∈ L(U, X), that is, the control operator B is
bounded. We assume that the pair (A, B) is exponentially stabilizable.
Riccati procedure. As before, the Riccati procedure consists of finding the
unique negative definite solution E ∈ L(X) of the algebraic Riccati equation
A∗ E + EA + EBB∗ E = id,
in the sense of ⟨(2EA + EBB∗ E − id)y, y⟩ = 0, for every y ∈ D(A), and
then
u(t) = B∗ Ey(t). Note that this is the control minimizing
´ +∞ of setting
2
(∥y(t)∥ + ∥u(t)∥2 ) dt for the control system ẏ = Ay + Bu.
0
Then, as before, the function V(y) = −⟨y, Ey⟩ is a Lyapunov function for
the system in closed loop ẏ = (A + BK)y.
Now, for the semilinear system (2.1) in closed loop with u(t) = B∗ Ey(t),
we have
d
V(y(t)) = −⟨y(t), (id + EBB∗ E)y(t)⟩ − ⟨y(t), EF(y(t))⟩
dt
≤ −C1 ∥y(t)∥2 ≤ −C2 V(y(t))
and we infer local asymptotic stability.
For B ∈ L(U, D(A∗ )′ ) unbounded, things are more complicated.
Roughly, the theory is complete in the parabolic case (i.e., when A generates an analytic semigroup), but is incomplete in the hyperbolic case (see
[29, 30] for details).
Rapid stabilization. An alternative method exists in the case where A generates a group S(t), and B ∈ L(U, D(A∗ )′ ) an admissible control operator
(see [44] for the notion of admissibility). An example covered by this setting
is the wave equation with Dirichlet control.
The strategy developed in [27] consists of setting
∗

u = −B

C−1
λ y

ˆ
with Cλ =
0

T+1/2λ

fλ (t)S(−t)BB∗ S(−t)∗ dt,

Stabilization of Semilinear PDEs

37

with λ > 0 arbitrary, fλ (t) = e−2λt if t ∈ [0, T] and fλ (t) = 2λe−2λT (T +
1/2λ − t) if t ∈ [T, T + 1/2λ]. Besides, the function
V(y) = ⟨y, C−1
λ y⟩
is a Lyapunov function (as noticed in [11]). Actually, this feedback
yields exponential stability with rate −λ for the closed-loop system ẏ =
(A − BB∗ C−1
λ )y, whence the wording “rapid stabilization” since λ > 0 is
arbitrary.
Then, thanks to that Lyapunov function, the above rapid stabilization
procedure applies as well to the semilinear control system (2.1), yielding a
local stabilization result (with any exponential rate).

2.1.4 Existing Results for Discretizations
We recall hereafter existing convergence results for space semidiscretizations of the Riccati procedure. We denote by EN the approximate Riccati
solution, expecting that EN → E as N → +∞.
One can find in [7, 8, 22, 26, 32] a general result showing convergence
of the approximations EN of the Riccati operator E, under assumptions of
uniform exponential stabilizability, and of uniform boundedness of EN :
∥SAN +BN B⊤
(t)∥ ≤ Ce−δt ,
N EN

∥EN ∥ ≤ M,

for every t ≥ 0 and every N ∈ N∗ , with uniform positive constants C, δ
and M.
In [7, 29, 32], the convergence of EN to E is proved in the general
parabolic case, for unbounded control operators, that is, when A : D(A) →
X generates an analytic semigroup, B ∈ L(U, D(A∗ )′ ), and (A, B) is exponentially stabilizable.
The situation is therefore definitive in the parabolic setting. In contrast,
if the semigroup S(t) is not analytic, the theory is not complete. Uniform
exponential stability is proved under uniform Huang-Prüss conditions in
[32]. More precisely, it is proved that, given a sequence (Sn (·)) of C0 semigroups on Xn , of generators An , (Sn (·)) is uniformly exponentially stable if
and only if iR ⊂ ρ(An ) for every n ∈ N and
sup

∥(iβid − An )−1 ∥ < +∞.

β∈R,n∈N

This result is used, e.g., in [37], to prove uniform stability of second-order
equations with (bounded) damping and with viscosity term, under uniform
gap condition on the eigenvalues.

38

Emmanuel Trélat

This result also allows to obtain convergence of the Riccati operators,
for second-order systems
ÿ + Ay = Bu,
with A : D(A) → X positive self-adjoint, with compact inverse, and B ∈
L(U, X) (bounded control operator).
But the approximation theory for general LQR problems remains incomplete in the general hyperbolic case with unbounded control operators, for
instance it is not done for wave equations with Dirichlet boundary control.

2.1.5 Conclusion
Concerning implementation issues, solving Riccati equations (or computing Gramians, in the case of rapid stabilization) in large dimension is
computationally demanding. In what follows, we would like to find other
ways to proceed.
Our objective is therefore to design simple feedbacks with efficient
approximation procedures.
In the sequel, we are going to investigate two situations:

1. Parabolic case (A generates an analytic semigroup): we are going to show
how to design feedbacks based on a small number of spectral modes.
2. Hyperbolic case, i.e., A = −A∗ in (2.1): we are going to consider two
“simple” feedbacks:
• Linear feedback u = −B∗ y: in that case, if F = 0 then 21 dtd ∥y(t)∥2 =
−∥B∗ y(t)∥2 . We will investigate the question of how to ensure uniform
exponential decay for approximations.
• Nonlinear feedback u = B∗ G(y): we will also investigate the question
of how to ensure uniform (sharp) decay for approximations.

2.2 Parabolic PDEs
In this section, we assume that the operator A in (2.1) generates an analytic
semigroup. Our objective is to design stabilizing feedbacks based on a small
number of spectral modes.
To simplify the exposition, we consider a 1D semilinear heat equation,
and we will comment further on extensions.

Stabilization of Semilinear PDEs

39

Let L > 0 be fixed and let f : R → R be a function of class C 2 such that
f(0) = 0. Following [12], We consider the 1D semilinear heat equation
yt = yxx + f(y),

y(t, 0) = 0,

y(t, L) = u(t),

(2.3)

where the state is y(t, ·) : [0, L] → R and the control is u(t) ∈ R.
We want to design a feedback control locally stabilizing (2.3) asymptotically to 0. Note that this cannot be global, because we can have other steady
states (a steady state is a function y ∈ C 2 (0, L) such that y ′′ (x) + f(y(x)) = 0
on (0, L) and y(0) = 0). By the way, here, without loss of generality we
consider the steady-state 0.
Let us first note that, for every T > 0, (2.3) is well posed in the Banach
space
YT = L2 (0, T; H 2 (0, L)) ∩ H1 (0, T; L2 (0, L)),
which is continuously embedded in L∞ ((0, T) × (0, L)).1
First of all, in order to end up with a Dirichlet problem, we set
x
z(t, x) = y(t, x) − u(t).
L
Assuming (for the moment) that u is differentiable, we set v(t) = u′ (t), and
we consider in the sequel v as a control. We also assume that u(0) = 0. Then
we have
x
x
zt = zxx + f ′ (0)z + f ′ (0)u − v + r(t, x),
z(t, 0) = z(t, L) = 0,
L
L
(2.4)
with z(0, x) = y(0, x) and
(
)2 ˆ 1
(
)
x
x
r(t, x) = z(t, x) + u(t)
(1 − s)f ′′ sz(s, x) + s u(s) ds.
L
L
0
Note that, given B > 0 arbitrary, there exist positive constants C1 and C2
such that, if |u(t)| ≤ B and ∥z(t, ·)∥L∞ (0,L) ≤ B, then
∥r(t, ·)∥L∞ (0,L) ≤ C1 (u(t)2 + ∥z(t, ·)∥2L∞ (0,L) ) ≤ C2 (u(t)2 + ∥z(t, ·)∥2H1 (0,L) ).
0

1

v ∈ L2 (0, T; H 2 (0, L))

Indeed, considering
∑
v = j,k cjk eijt eikx , we have
∑
j,k


∑
|cjk | ≤ 
j,k

with vt

1/2 
1
1 + j 2 + k4





∈ H1 (0, T; L2 (0, L)),

∑

writing

1/2
2

4

2

(1 + j + k )|cjk |

,

j,k

and these series converge, whence the embedding, allowing to give a sense to f(y).
Now, if y1 and y2 are solutions of (2.3) on [0, T ], then y1 = y2 . Indeed, v = y1 − y2 is a
solution of vt = vxx + av, v(t, 0) = v(t, L) = 0, v(0, x) = 0, with
a(t, x) = g(y1 (t, x), y2 (t, x)) where g is a function of class C 1 . We infer that v = 0.

40

Emmanuel Trélat

In the sequel, r(t, x) will be considered as a remainder.
We define the operator A = △ + f ′ (0)id on D(A) = H 2 (0, L) ∩ H10 (0, L),
so that (2.4) is written as
u̇ = v,

zt = Az + au + bv + r,

z(t, 0) = z(t, L) = 0,

(2.5)

with a(x) = Lx f ′ (0) and b(x) = − Lx .
Since A is self-adjoint and has a compact resolvent, there exists
a Hilbert basis (ej )j≥1 of L2 (0, L), consisting of eigenfunctions ej ∈
H10 (0, L) ∩ C 2 ([0, L]) of A, associated with eigenvalues (λj )j≥1 such that
−∞ < · · · < λn < · · · < λ1 and λn → −∞ as n → +∞.
Any solution z(t, ·) ∈ H 2 (0, L) ∩ H10 (0, L) of (2.4), as long as it is well
defined, can be expanded as a series
z(t, ·) =

∞
∑

zj (t)ej (·)

j=1

(converging in H10 (0, L)), and then we have, for every j ≥ 1,
żj (t) = λj zj (t) + aj u(t) + bj v(t) + rj (t),
with
aj =

f ′ (0)
L

ˆ

L

xej (x) dx, bj = −

0

1
L

Setting, for every n ∈ N∗ ,



u(t)
0 0
z1 (t)
a1 λ1



Xn (t)=  . , An =  .
..
 .. 
 ..
.
zn (t)
an 0

ˆ

ˆ

L

L

xej (x) dx, rj (t) =
0

r(t, x)ej (x) dx.
0

···
···
..
.


 


0
1
0
b1 
r1 (t)
0

 


.. , Bn =  .. , Rn (t)=  .. ,




.
.
. 

···

λn

bn

rn (t)

we have, then,
Ẋn (t) = An Xn (t) + Bn v(t) + Rn (t).
Lemma 2.1 The pair (An , Bn ) satisfies the Kalman condition.
Proof We compute
det(Bn , An Bn , . . . , Ann Bn ) =

n
∏
(aj + λj bj )VdM(λ1 , . . . , λn ),

(2.6)

j=1

where VdM(λ1 , . . . , λn ) is a Van der Monde determinant, and thus is never
equal to zero since the λi , i = 1, . . . , n, are pairwise distinct. On the other

Stabilization of Semilinear PDEs

41

part, using the fact that each ej is an eigenfunction of A and belongs to
H10 (0, L), we compute
aj + λj bj =

1
L

ˆ

L

x( f ′ (0) − λj )ej (x) dx = −

0

1
L

ˆ

L
0

xe′′j (x) dx = −e′j (L),

and this quantity is never equal to zero since ej (L) = 0 and ej is a nontrivial
solution of a linear second-order scalar differential equation. Therefore the
determinant (2.6) is never equal to zero.
By the pole-shifting theorem, there exists Kn = (k0 , . . . , kn ) such that the
matrix An + Bn Kn has −1 as an eigenvalue of multiplicity n + 1. Moreover,
by the Lyapunov lemma, there exists a symmetric positive definite matrix
Pn of size n + 1 such that
⊤

Pn (An + Bn Kn ) + (An + Bn Kn ) Pn = −In+1 .
Therefore, the function defined by Vn (X) = X⊤ Pn X for any X ∈ Rn+1 is a
Lyapunov function for the closed-loop system Ẋn (t) = (An + Bn Kn )Xn (t).
Let γ > 0 and n ∈ N∗ to be chosen later. For every u ∈ R and every z ∈
H 2 (0, L) ∩ H10 (0, L), we set
V(u, z) = γ

X⊤
n Pn Xn

∞

1
1∑ 2
− ⟨z, Az⟩L2 (0,L) = γ X⊤
λj zj ,
n Pn Xn −
2
2

(2.7)

j=1

where Xn ∈ Rn+1 is defined by Xn = (u, z1 , . . . , zn )⊤ and zj = ⟨z(·), ei (·)⟩L2 (0,L)
for every j.
Using that λn → −∞ as n → +∞, it is clear that, choosing γ > 0 and
n ∈ N∗ large enough, we have V(u, z) > 0 for all (u, z) ∈ R × (H 2 (0, L) ∩
H10 (0, L)) \ {(0, 0)}. More precisely, there exist positive constants C3 , C4 ,
C5 and C6 such that
(
)
(
)
C3 u2 + ∥z∥2H1 (0,L) ≤ V(u, z) ≤ C4 u2 + ∥z∥2H1 (0,L) ,
0
0
(
)
V(u, z) ≤ C5 ∥Xn ∥22 + ∥Az∥2L2 (0,L) ,
γC6 ∥Xn ∥22 ≤ V(u, z),
for all (u, z) ∈ R × (H 2 (0, L) ∩ H10 (0, L)). Here, ∥ ∥2 designates the
Euclidean norm of Rn+1 .
Our objective is now to prove that V is a Lyapunov function for the
system (2.5) in closed loop with the control v = Kn Xn .

42

Emmanuel Trélat

In what follows, we thus take v = Kn Xn and u defined by u̇ = v and
u(0) = 0. We compute
d
V(u(t), z(t)) = −γ ∥Xn (t)∥22 − ∥Az(t, ·)∥2L2 − ⟨Az(t, ·), a(·)⟩L2 u(t)
dt
−⟨Az(t, ·), b(·)⟩L2 Kn Xn (t) − ⟨Az(t, ·), r(t, ·)⟩L2
(
)
(2.8)
+γ Rn (t)⊤ Pn Xn (t) + Xn (t)⊤ Pn Rn (t) .
Let us estimate the terms at the right-hand side of (2.8). Under the a priori
estimates |u(t)| ≤ B and ∥z(t, ·)∥L∞ (0,L) ≤ B, there exist positive constants
C7 , C8 and C9 such that
1
|⟨Az, a⟩L2 u| + |⟨Az, b⟩L2 Kn Xn | ≤ ∥Az∥2L2 + C7 ∥Xn ∥22 ,
4
1
C2
V,
|⟨Az, r⟩L2 | ≤ ∥Az∥2L2 + C8 V2 ,
∥Rn ∥∞ ≤
4
C3
(
)
C2 √ 3/2
⊤
√
|γ R⊤
γV .
n Pn Xn + Xn Pn Rn | ≤
C3 C6
We infer that, if γ > 0 is large enough, then there exist positive constants
C10 and C11 such that dtd V ≤ −C10 V + C11 V 3/2 . We easily conclude the
local asymptotic stability of the system (2.5) in closed loop with the control
v = Kn Xn .
Remark 2.2 Of course, the above local asymptotic stability may be achieved
with other procedures, for instance, by using the Riccati theory. However,
the procedure developed here is much more efficient because it consists of
stabilizing a finite-dimensional part of the system, mainly, the part that is
not naturally stable. We refer to [12] for examples and for more details.
Actually, we have proved in that reference that, thanks to such a strategy,
we can pass from any steady-state to any other one, provided that the two
steady states belong to a same connected component of the set of steady
states: this is a partially global exact controllability result.
The main idea used above is the following fact, already used in the
remarkable early paper [38]. Considering the linearized system with no
control, we have an infinite-dimensional linear system that can be aligned,
through a spectral decomposition, in two parts: the first part is finitedimensional, and consists of all spectral modes that are unstable (meaning
that the corresponding eigenvalues have nonnegative real part); the second part is infinite-dimensional, and consists of all spectral modes that are
asymptotically stable (meaning that the corresponding eigenvalues have
negative real part). The idea used here then consists of focusing on the

Stabilization of Semilinear PDEs

43

finite-dimensional unstable part of the system, and to design a feedback
control in order to stabilize that part. Then, we plug this control in the
infinite-dimensional system, and we have to check that this feedback indeed
stabilizes the whole system (in the sense that it does not destabilize the
other infinite-dimensional part). This is the role of the Lyapunov function
V defined by (2.7).
The extension to general systems (2.1) is quite immediate, at least in
the parabolic setting under appropriate spectral assumptions (see [39] for
Couette flows and [14] for Navier–Stokes equations).
But it is interesting to note that it does not work only for parabolic
equations: this idea has been as well used in [13] for the 1D semilinear
equation
ytt = yxx + f(y),

y(t, 0) = 0, yx (t, L) = u(t),

with the same assumptions on f as before. We first note that, if f(y) = cy is
linear (with c ∈ L∞ (0, L)), then, setting u(t)´ = −αyt (t, L) with α > 0 yields
L
an exponentially decrease of the energy 0 (yt (t, x)2 + yx (t, x)2 ) dt, and
moreover, the eigenvalues of the corresponding operator have a real part
tending to −∞ as α tends to 1. Therefore, in the general case, if α is sufficiently close to 1 then at most a finite number of eigenvalues may have
a nonnegative real part. Using a Riesz spectral expansion, the same kind
of method as the one developed above can therefore be applied, and yields
a feedback based on a finite number of modes, that stabilizes locally the
semilinear wave equation, asymptotically to equilibrium.

2.3 Hyperbolic PDEs
In this section, we assume that the operator A in (2.1) is skew-adjoint,
that is,
A∗ = −A,

D(A∗ ) = D(A).

Let us start with a simple remark. If F = 0 (linear case), then, choosing the very simple linear feedback u = −B∗ y and setting V(y) = 12 dtd ∥y∥2X ,
we have
d
V(y(t)) = −∥B∗ y(t)∥2X ≤ 0,
dt
and then we expect that, under reasonable assumptions, we have exponential asymptotic stability (and this will be the case under observability
assumptions, as we are going to see).

44

Emmanuel Trélat

Now, if we choose a nonlinear feedback u = B∗ G(y), we ask the same
question: what are sufficient conditions ensuring asymptotic stability, and
if so, with which sharp decay?
Besides, we will investigate the following important question: how to
ensure uniform properties when discretizing?

2.3.1 The Continuous Setting
2.3.1.1 Linear Case
In this section, we assume that F = 0 (linear case), and we assume that B
is bounded. Taking the linear feedback u = −B∗ y as said above, we have
the closed-loop system ẏ = Ay − BB∗ y. For convenience, in what follows
we rather write this equation in the form (more standard in the literature)
ẏ(t) + Ay(t) + By(t) = 0,

(2.9)

where A is a densely defined skew-adjoint operator on X and B is a bounded
nonnegative self-adjoint operator on X (we have just replaced A with −A
and BB∗ with B).
We start hereafter with the question of the exponential stability of
solutions of (2.9).
Equivalence between Observability and Exponential Stability. The following result is a generalization of the main result of [23].
Theorem 2.3 Let X be a Hilbert space, let A : D(A) → X be a densely defined
skew-adjoint operator, let B be a bounded self-adjoint nonnegative operator
on X. We have equivalence of:
1. There exist T > 0 and C > 0 such that every solution of the conservative
equation
ϕ̇(t) + Aϕ(t) = 0

(2.10)

satisfies the observability inequality
ˆ T0
∥ϕ(0)∥2X ≤ C
∥B1/2 ϕ(t)∥2X dt.
0

2. There exist C1 > 0 and δ > 0 such that every solution of the damped
equation
ẏ(t) + Ay(t) + By(t) = 0

(2.11)

Stabilization of Semilinear PDEs

45

satisfies
Ey (t) ≤ C1 Ey (0)e−δt ,
where Ey (t) = 12 ∥y(t)∥2X .
Proof Let us first prove that the first property implies the second one: we
want to prove that every solution of (2.11) satisfies
1
1
Ey (t) = ∥y(t)∥2X ≤ Ey (0)e−δt = ∥y(0)∥2X e−δt .
2
2
Consider ϕ solution of (2.10) with ϕ(0) = y(0). Setting θ = y − ϕ, we have
θ̇ + Aθ + By = 0,

θ(0) = 0.

Then, taking the scalar product with θ, since A is skew-adjoint, we get
⟨θ̇ + By, θ⟩X = 0. But, setting Eθ (t) = 12 ∥θ(t)∥2X , we have Ėθ = −⟨By, θ⟩X .
Then, integrating a first time over [0, t], and then a second time over [0, T],
since Eθ (0) = 0, we get
ˆ T
ˆ Tˆ t
Eθ (t) dt = −
⟨By(s), θ(s)⟩X ds dt
0

0

ˆ
=−

0
T

(T − t)⟨B1/2 y(t), B1/2 θ(t)⟩X dt,

0

where we have used the Fubini theorem. Hence, thanks to the Young
1 2
inequality ab ≤ α2 a2 + 2α
b with α = 2, we infer that
1
2

ˆ
0

T

ˆ
∥θ(t)∥2X dt ≤ T∥B1/2 ∥

T

∥B1/2 y(t)∥X ∥θ(t)∥X dt

0

ˆ

≤ T ∥B
2

∥

1/2 2

T

∥B

1/2

0

and therefore,
ˆ

T
0

y(t)∥2X

ˆ
∥θ(t)∥2X dt ≤ 4T 2 ∥B1/2 ∥2X

T
0

1
dt +
4

0

0

T

∥θ(t)∥2X dt,

∥B1/2 y(t)∥2X dt.

Now, since ϕ = y − θ, it follows that
ˆ T
ˆ T
ˆ
1/2
2
1/2
2
∥B ϕ(t)∥X dt ≤ 2
∥B y(t)∥X dt + 2
0

ˆ

0

ˆ

T

≤ (2 + 8T 2 ∥B1/2 ∥4 )
0

T

∥B1/2 θ(t)∥2X dt

∥B1/2 y(t)∥2X dt.

46

Emmanuel Trélat

Finally, since
1
C
Ey (0) = Eϕ (0) = ∥ϕ(0)∥2X ≤
2
2

ˆ

T

0

∥B1/2 ϕ(t)∥2X dt

it follows that
ˆ
Ey (0) ≤ C(1 + 4T 2 ∥B1/2 ∥4 )
0

T

∥B1/2 y(t)∥2X dt.

Besides, one has E′y (t) = −∥B1/2 y(t)∥2X , and then
Ey (0) − Ey (T). Therefore

´T
0

∥B1/2 y(t)∥2X dt =

Ey (0) ≤ C(1 + 4T 2 ∥B1/2 ∥4 )(Ey (0) − Ey (T)) = C1 (Ey (0) − Ey (T))
and hence
Ey (T) ≤

C1 − 1
Ey (0) = C2 Ey (0),
C1

with C2 < 1.
Actually this can be done on every interval [kT, (k + 1)T], and it yields
Ey ((k + 1)T) ≤ C2 Ey (kT ) for every k ∈ N, and hence
≤ Ey (0)C2k .
[tE
] y (kT)
t
For every t ∈ [kT, (k + 1)T), noting that k = T > T − 1, and that
ln C12 > 0, it follows that
C2k

(
)
− ln C12
1
1
= exp(k ln C2 ) = exp(−k ln
)≤
exp
t
C2
C2
T

and hence Ey (t) ≤ Ey (kT) ≤ δEy (0) exp(−δt) for some δ > 0.
Let us now prove the converse: assume the exponential decrease of
solutions of (2.11), and let us prove the observability property for solutions
of (2.10).
From the exponential decrease inequality, one has
ˆ T
∥B1/2 y(t)∥2X dt = Ey (0) − Ey (T ) ≥ (1 − C1 e−δT )Ey (0) = C2 Ey (0),
0

−δT

(2.12)

and for T > 0 large enough there holds C2 = 1 − C1 e
> 0.
Then we make the same proof as before, starting from (2.10), that we
write in the form
ϕ̇ + Aϕ + Bϕ = Bϕ,

Stabilization of Semilinear PDEs

47

and considering the solution of (2.11) with y(0) = ϕ(0). Setting θ = ϕ − y,
we have
θ̇ + Aθ + Bθ = Bϕ,

θ(0) = 0.

Taking the scalar product with θ, since A is skew-adjoint, we get ⟨θ̇ +
Bθ, θ⟩X = ⟨Bϕ, θ⟩X , and therefore
Ėθ + ⟨Bθ, θ⟩X = ⟨Bϕ, θ⟩X .
1/2
Since ⟨Bθ,
θ∥X ≥ 0, it follows that Ėθ ≤ ⟨Bϕ, θ⟩X . As before we
X = ∥B
´ T ´ θ⟩
t
apply 0 0 and hence, since Eθ (0) = 0,

ˆ

T

Eθ (t) dt ≤

ˆ Tˆ

0

0

ˆ

t

T

⟨Bϕ(s), θ(s)⟩X ds dt=

0

(T − t)⟨B1/2 ϕ(t), B1/2 θ(t)⟩X dt.

0

Thanks to the Young inequality, we get, exactly as before,
ˆ
ˆ T
1 T
∥θ(t)∥2X dt ≤ T ∥B1/2 ∥
∥B1/2 ϕ(t)∥X ∥θ(t)∥X dt
2 0
0
ˆ T
ˆ
1 T
2
1/2 2
≤ T ∥B ∥X
∥B1/2 ϕ(t)∥2X dt +
∥θ(t)∥2X dt,
4
0
0
and finally,
ˆ
0

T

ˆ
∥θ(t)∥2X

dt ≤ 4T

2

∥B1/2 ∥2X

T

∥B1/2 ϕ(t)∥2X dt.

0

Now, since y = ϕ − θ, it follows that
ˆ T
ˆ T
ˆ
∥B1/2 y(t)∥2X dt ≤ 2
∥B1/2 ϕ(t)∥2X dt + 2
0

0

ˆ

≤ (2 + 8T 2 ∥B1/2 ∥4 )
0

0

T

T

∥B1/2 θ(t)∥2X dt

∥B1/2 ϕ(t)∥2X dt.

Now, using (2.12) and noting that Ey (0) = Eϕ (0), we infer that
ˆ T
2
1/2 4
C2 Eϕ (0) ≤ (2 + 8T ∥B ∥ )
∥B1/2 ϕ(t)∥2X dt.
0

This is the desired observability inequality.
Remark 2.4 This result says that the observability property for the linear
conservative equation (2.10) is equivalent to the exponential stability property for the linear damped equation (2.11). This result has been written in
[23] for second-order equations, but the proof works exactly in the same

48

Emmanuel Trélat

way for more general first-order systems, as shown here. More precisely,
the statement done in [23] for second-order equations looks as follows:
We have equivalence of:
1. There exist T > 0 and C > 0 such that every solution of
ϕ̈(t) + Aϕ(t) = 0

(conservative equation)

satisfies

ˆ
∥A1/2 ϕ(0)∥2X + ∥ϕ̇(0)∥2X ≤ C

0

T0

∥B1/2 ϕ̇(t)∥2X dt.

2. There exist C1 > 0 and δ > 0 such that every solution of
ÿ(t) + Ay(t) + Bẏ(t) = 0

(damped equation)

satisfies
Ey (t) ≤ C1 Ey (0)e−δt ,
where

)
1 ( 1/2
∥A y(t)∥2X + ∥ẏ(t)∥2X .
2
Remark 2.5 A second remark is that the proof uses in a crucial way the
fact that the operator B is bounded. We refer to [5] for a generalization
for unbounded operators with degree of unboundedness ≤ 1/2 (i.e., B ∈
L(U, D(A1/2 )′ )), and only for second-order equations, with a proof using
Laplace transforms, and under a condition on the unboundedness of B that
is not easy to check (related to “hidden regularity” results), namely,
Ey (t) =

∀β > 0

sup ∥B∗ λ(λ2 I + A)−1 B∥L(U) < +∞.
Re(λ)=β

For instance this works for waves with a nonlocal operator B corresponding
to a Dirichlet condition, in the state space L2 × H−1 , but not for the usual
Neumann one, in the state space H1 × L2 (except in 1D).
2.3.1.2 Semilinear Case
In the case with a nonlinear feedback, still in order to be in agreement with
standard notations used in the existing literature, we rather write the equation in the form u̇ + Au + F(u) = 0, where u now designates the solution
(and not the control).
Therefore, from now on and throughout the rest of this chapter, we
consider the differential system
u ′ (t) + Au(t) + BF(u(t)) = 0,

(2.13)

Stabilization of Semilinear PDEs

49

with A : D(A) → X a densely defined skew-adjoint operator, B : X → X a
nontrivial bounded self-adjoint nonnegative operator, and F : X → X a
(nonlinear) mapping assumed to be Lipschitz continuous on bounded
subsets of X. These are the framework and notations adopted in [4].
If F = 0 then the system (2.13) is purely conservative, and ∥u(t)∥X =
∥u(0)∥X for every t ≥ 0. If F ̸= 0 then the system (2.13) is expected to be
dissipative if the nonlinearity F has “the good sign.” Along any solution
of (2.13) (while it is well defined), the derivative with respect to time of the
energy Eu (t) = 21 ∥u(t)∥2X is
Eu′ (t) = −⟨u(t), BF(u(t))⟩X = −⟨B1/2 u(t), B1/2 F(u(t))⟩X .
In the sequel, we will make appropriate assumptions on B and on F
ensuring that E′u (t) ≤ 0. It is then expected that the solutions are globally
well-defined and that their energy decays asymptotically to 0 as t → +∞.
We make the following assumptions.
• For every u ∈ X
⟨u, BF(u)⟩X ≥ 0.
This assumption implies that Eu′ (t) ≤ 0.
The spectral theorem applied to the bounded nonnegative self-adjoint
operator B implies that B is unitarily equivalent to a multiplication: there
exist a probability space (Ω, µ), a real-valued bounded nonnegative measurable function b defined on X, and an isometry U from L2 (Ω, µ) into X,
such that U−1 BUf = bf for every f ∈ L2 (Ω, µ).
Now, we define the (nonlinear) mapping ρ : L2 (Ω, µ) → L2 (Ω, µ) by
ρ( f ) = U−1 F(Uf).
We make the following assumptions on ρ:
• ρ(0) = 0 and fρ( f) ≥ 0 for every f ∈ L2 (Ω, µ).
• There exist c1 > 0 and c2 > 0 such that, for every f ∈ L∞(Ω, µ),
c1 g(| f(x)|) ≤ |ρ( f )(x)| ≤ c2 g−1 (| f(x)|) for almost every x ∈ Ω such that | f(x)| ≤ 1,
c1 | f(x)| ≤ |ρ( f )(x)| ≤ c2 | f(x)|

for almost every x ∈ Ω such that | f(x)| ≥ 1,

where g is an increasing odd function of class C 1 such that g(0) = g′ (0) =
0, sg′ (s)2 /g(s) → 0 as s → 0, and such that the function H defined by
√ √
H(s) = sg( s), for every s ∈ [0, 1], is strictly convex on [0, s20 ] for some
s0 ∈ (0, 1].

50

Emmanuel Trélat

This assumption is issued from [2] where the optimal weight method
has been developed.
Examples of such functions g are given by
g(s) = s/ ln p (1/s),

s p,

e−1/s ,
2

s p lnq (1/s),

e− ln

p

(1/s)

.

b on R by H(s)
b = H(s) for every s ∈ [0, s2 ] and
We define the function H
0
b = +∞ otherwise. We define the function L on [0, +∞) by L(0) = 0
by H(s)
and, for r > 0, by
L(r) =

(
)
b ∗ (r) 1
H
b
= sup rs − H(s)
,
r
r s∈R

b ∗ is the convex conjugate of H.
b By construction, the function L :
where H
2
[0, +∞) → [0, s0 ) is continuous and increasing.
We define ΛH : (0, s20 ] → (0, +∞) by ΛH (s) = H(s)/sH ′ (s), and we set
∀s ≥ 1/H

′

(s20 )

1
ψ(s) = ′ 2 +
H (s0 )

ˆ

H ′(s20 )
1/s

v2 (1

1
dv.
− ΛH ((H ′ )−1 (v)))

The function ψ : [1/H ′ (s20 ), +∞) → [0, +∞) is continuous and increasing.
Hereafter, we use the notations . and ≃ in the estimates, with the following meaning. Let S be a set, and let F and G be nonnegative functions
defined on R × Ω × S. The notation F . G (equivalently, G & F) means that
there exists a constant C > 0, only depending on the function g or on the
mapping ρ, such that F(t, x, λ) ≤ CG(t, x, λ) for all (t, x, λ) ∈ R × Ω × S.
The notation F1 ≃ F2 means that F1 . F2 and F1 & F2 .
In the sequel, we choose S = X, or equivalently, using the isometry U, we
choose S = L2 (Ω, µ), so that the notation . designates an estimate in which
the constant does not depend on u ∈ X, or on f ∈ L2 (Ω, µ), but depends only
on the mapping ρ. We will use these notations to provide estimates on the
solutions u(·) of (2.13), meaning that the constants in the estimates do not
depend on the solutions.
Theorem 2.6 [4] In addition to the above assumptions, we assume that there
exist T > 0 and CT > 0 such that
ˆ T
CT ∥ϕ(0)∥2X ≤
∥B1/2 ϕ(t)∥2X dt,
0

′

for every solution of ϕ (t) + Aϕ(t) = 0 (observability inequality for the linear
conservative equation).

Stabilization of Semilinear PDEs

51

Table 2.1. Examples
g(s)

ΛH (s)

decay of E(t)

s/ lnp (1/s), p > 0

lim sup ΛH (s) = 1

e−t

1/(p+1)

/t1/(p+1)

x↘0

ΛH (s) ≡

s p on [0, s20 ], p > 1
2

e−1/s

p

(1/s)

t−2/(p−1)

<1

lim ΛH (s) = 0

1/ ln(t)

s↘0

s p lnq (1/s), p > 1, q > 0
e−ln

2
p+1

, p>1

lim ΛH (s) =

s↘0

2
p+1

t−2/(p−1) ln−2q/(p−1) (t)

<1

e−2 ln

lim ΛH (s) = 0

s↘0

1/p

(t)

Then, for every u0 ∈ X, there exists a unique solution u(·) ∈ C0 (0, +∞; X) ∩
C (0, +∞; D(A)′ ) of (2.13) such that u(0) = u0 .2 Moreover, the energy of
any solution satisfies
(
)
1
Eu (t) . T max(γ1 , Eu (0))L
,
ψ −1 (γ2 t)
1

for every time t ≥ 0, with γ1 ≃ ∥B∥/γ2 and γ2 ≃ CT /T(T 2 ∥B1/2 ∥4 + 1). If
moreover
lim sup ΛH (s) < 1,

(2.14)

s↘0

then we have the simplified decay rate
Eu (t) . T max(γ1 , Eu (0)) (H ′ )−1

(γ )
3

t

,

for every time t ≥ 0, for some positive constant γ3 ≃ 1.
Note the important fact that this result gives sharp decay rates (see
Table 2.1 for examples).
Theorem 2.6 improves and generalizes to a wide class of equations the
main result of [3], in which the authors dealt with locally damped wave
equations. Examples of applications are given in [4], that we mention here
without giving the precise framework, assumptions, and comments:
• Schrödinger equation with nonlinear damping (nonlinear absorption):
i∂t u(t, x) + △u(t, x) + ib(x)u(t, x)ρ(x, |u(t, x)|) = 0.
2

Here, the solution is understood in the weak sense (see [10, 18]), and D(A)′ is the dual of
D(A) with respect to the pivot space X. If u0 ∈ D(A), then
u(·) ∈ C 0 (0, +∞; D(A)) ∩ C 1 (0, +∞; X).

52

Emmanuel Trélat

• Wave equation with nonlinear damping:
∂tt u(t, x) − △u(t, x) + b(x)ρ(x, ∂t u(t, x)) = 0.
• Plate equation with nonlinear damping:
∂tt u(t, x) + △2 u(t, x) + b(x)ρ(x, ∂t u(t, x)) = 0.
• Transport equations with nonlinear damping:
∂t u(t, x) + div(v(x)u(t, x)) + b(x)ρ(x, u(t, x)) = 0,

x ∈ Tn ,

with div(v) = 0.
• Dissipative equations with nonlocal terms:
∂t f + v · ∇x f = ρ(f),
with kernels ρ satisfying the sign assumption fρ(f ) ≥ 0.
Proof It is interesting to quickly give the main steps of the proof.
• Step 1. Comparison of the nonlinear equation with the linear damped
model:
Prove that the solutions of
u′ (t) + Au(t) + BF(u(t)) = 0,
z′ (t) + Az(t) + Bz(t) = 0,
satisfy
ˆ
0

T

ˆ
∥B1/2 z(t)∥2X dt ≤ 2

T
0

(

z(0) = u(0),

)
∥B1/2 u(t)∥2X + ∥B1/2 F(u(t))∥2X dt.

• Step 2. Comparison of the linear damped equation with the conservative
linear equation:
Prove that the solutions of
z′ (t) + Az(t) + Bz(t) = 0,
ϕ′ (t) + Aϕ(t) = 0,
satisfy

ˆ

T
0

ϕ(0) = u(0),
ˆ

∥B

1/2

ϕ(t)∥2X

with kT = 8T 2 ∥B1/2 ∥4 + 2.

z(0) = u(0),

T

dt ≤ kT
0

∥B1/2 z(t)∥2X dt,

Stabilization of Semilinear PDEs

53

• Step 3. Following the optimal weight
( ) method introduced by F. Alabau
−1 s
(see, e.g., [2]), we set w(s) = L
β with β appropriately chosen. Nonlinear energy estimate:
Prove that
ˆ T
(
)
w(Eϕ (0)) ∥B1/2 u(t)∥2X + ∥B1/2 F(u(t))∥2X dt
0

. T∥B∥H∗ (w(Eϕ (0))) + (w(Eϕ (0)) + 1)

ˆ

T

⟨Bu(t), F(u(t))⟩X dt.

0

• Step 4. End of the proof:
Using the results of the three steps above, we have
ˆ T
∗
T∥B∥H (w(Eϕ (0))) + (w(Eϕ (0)) + 1)
⟨Bu(t), F(u(t))⟩X dt
ˆ

T

&
ˆ

0

0

)
w(Eϕ (0)) ∥B1/2 u(t)∥2X + ∥B1/2 F(u(t))∥2X dt

(Step 3)

w(Eϕ (0))∥B1/2 z(t)∥2X dt

(Step 1)

w(Eϕ (0))∥B1/2 ϕ(t)∥2X dt

(Step 2)

T

&
ˆ

(

0
T

&
0

& Cst w(Eϕ (0))Eϕ (0)

(uniform observability inequality)

from which we infer that

))
(
(
Eu (0)
,
Eu (T) ≤ Eu (0) 1 − ρT L−1
β

and then the exponential decrease is finally established.

2.3.2 Space Semidiscretizations
In this section, we define a general space semidiscrete version of (2.13),
with the objective of obtaining a theorem similar to Theorem 2.6, but in
this semidiscrete setting, with estimates that are uniform with respect to the
mesh parameter. As we are going to see, uniformity is not true in general,
and in order to recover it we add an appropriate extra numerical viscosity
term in the numerical scheme.
2.3.2.1 Space Semidiscretization Setting
We denote by △x > 0 the space discretization parameter (typically, step size
of the mesh), with 0 < △x < △x0 , for some fixed △x0 > 0. We follow [28, 29]

54

Emmanuel Trélat

for the setting. Let (X△x )0<△x<△x0 be a family of finite-dimensional vector
spaces (X△x ∼ RN(△x) with N(△x) ∈ N).
We use the notations . and ≃ as before, also meaning that the involved
constants are uniform with respect to △x.
Let β ∈ ρ(A) (resolvent of A). Following [18], we define X1/2 = (βidX −
A)−1/2 (X), endowed with the norm ∥u∥X1/2 = ∥(βidX − A)1/2 u∥X (for
instance, if A1/2 is well defined, then X1/2 = D(A1/2 )), and we define
′
X−1/2 = X1/2
(dual with respect to X).
The general semidiscretization setting is the following. We assume
that, for every △x ∈ (0, △x0 ), there exist linear mappings P△x : X−1/2 →
e △x : X△x → X1/2 such that P△x P
e △x = idX△x , and such that
X△x and P
∗
e . We assume that the scheme is convergent, that is, ∥(I −
P△x = P
△x
e
P△x P△x )u∥X → 0 as △x → 0, for every u ∈ X. Here, we have implicitly used
the canonical injections D(A) ,→ X1/2 ,→ X ,→ X−1/2 (see [18]).
For every △x ∈ (0, △x0 ):
e △x u△x ∥X , for
• X△x is endowed with the Euclidean norm ∥u△x ∥△x = ∥P
u△x ∈ X△x . The corresponding scalar product is denoted by ⟨·, ·⟩△x .3
• We set4
e △x ,
e △x .
A△x = P△x AP
B△x = P△x BP
e ∗ , A△x is skew-symmetric and B△x is symmetric
Since P△x = P
△x
nonnegative
• We define F△x : X△x → X△x by
∀u△x ∈ X△x

e △x u△x ).
F△x (u△x ) = P△x F(P

Note that B△x is uniformly bounded with respect to △x, and F△x is
Lipschitz continuous on bounded subsets of X△x , uniformly with respect
to △x.
Now, a priori we consider the space semidiscrete approximation of (2.13)
given by
u′△x (t) + A△x