Principal
Evolution Equations: Long Time Behavior and Control
Evolution Equations: Long Time Behavior and Control
Kaïs Ammari, Stéphane Gerbi
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.univsavoie.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 minicourses in the morning while
the afternoons were devoted to various contributions on the theme.
The 䱌rst minicourse was held by Farid AmmarKhodja, University of
FrancheComté, 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 minicourse
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.
“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.univsavoie.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 minicourses in the morning while
the afternoons were devoted to various contributions on the theme.
The 䱌rst minicourse was held by Farid AmmarKhodja, University of
FrancheComté, 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 minicourse
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
Año:
2018
Editorial:
Cambridge University Press
Idioma:
english
Páginas:
206
ISBN 13:
9781108412308
Series:
LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES 439
File:
PDF, 5.61 MB
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NUCINKIS (eds) 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 One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 4843/24, 2nd Floor, Ansari Road, Daryaganj, Delhi – 110002, India 79 Anson Road, #0604/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108412308 DOI: 10.1017/9781108412308 © Cambridge University Press 2018 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2018 Printed in the United Kingdom by Clays, St Ives plc A catalogue record for this publication is available from the British Library. ISBN 9781108412308 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or thirdparty internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, 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 AMMARKHODJA 2 Stabilization of Semilinear PDEs, and Uniform Decay under Discretization 31 EMMANUEL TRÉLAT 3 A NullControllability Result for the Linear System of Thermoelastic Plates with a Single Control 77 CARLOS CASTRO AND LUZ DE TERESA 4 Doubly Connected VStates for the Generalized Surface Quasigeostrophic Equations 90 FRANCISCO DE LA HOZ, ZINEB HASSAINIA, AND TAOUFIK HMIDI 5 About LeastSquares Type Approach to Address Direct and Controllability Problems 118 ARNAUD MÜNCH AND PABLO PEDREGAL 6 A Note on the Asymptotic Stability of WaveType Equations with Switching TimeDelay 137 SERGE NICAISE AND CRISTINA PIGNOTTI 7 IllPosedness 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 TwoDimensional 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.univsavoie.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 minicourses in the morning while the afternoons were devoted to various contributions on the theme. The first minicourse was held by Farid AmmarKhodja, University of FrancheComté, 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 minicourse 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 AuvergneRhôneAlpes; the GDRI LEM2I: “Laboratoire EuroMaghrébin 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 FrancheComté 16, Route de Gray, 25030 Besançon Cedex, France fammarkh@univfcomte.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@univrennes1.fr Arnaud Münch Blaise Pascal University Laboratoire de Mathématiques, UMR CNRS 6620 ClermontFerrand, France arnaud.munch@math.univbpclermont.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@univvalenciennes.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@unikonstanz.de Lionel Rosier Automatic Control and Systems Center MINES ParisTech 60 Bd SaintMichel, 75272 Paris Cedex, France lionel.rosier@minesparistech.fr Armen Shirikyan Department of Mathematics Université de CergyPontoise Site de Saint Martin 2, Avenue Adolphe Chauvin 95302 CergyPontoise Cedex, France Armen.Shirikyan@ucergy.fr Emmanuel Trélat University of Pierre et Marie Curie (Paris 6) Laboratoire JacquesLouis 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 AMMARKHO 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 AmmarKhodja 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 onedimensional space setting. But it has also been used to prove the boundary nullcontrollability 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 nullcontrollability problem for a general secondorder 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 nullcontrollability 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 nullcontrollable 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 wellposed. 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 AmmarKhodja System (1.1) is nullcontrollable 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 nullcontrollable 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 righthand 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 AmmarKhodja To prove the nullcontrollability 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 φ2 ≥ T 0 ˆ ≥ ( ˆ ) ( ) ρτ ρ3 − ∥a∥2∞ e−2sβ0 ρ−αt eαt φ2 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 nullcontrollability. Due to this distributed nullcontrollability 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 nullcontrollable 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 AmmarKhodja 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 AlabauBoussouira 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 onedimensional 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 onedimensional heat equation: ′ QT = (0, T) × (0, π) y − yxx = f (x) u(t), (1.14) yx=0,π = 0, (0, T) yt=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), ykt=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 nullcontrollability 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 AmmarKhodja 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 AmmarKhodja ∑ 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 AmmarKhodja ∗ 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 nullcontrollable. 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 AmmarKhodja 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{−logI1,k (q), −logIk (q)} . k2 Then: 1. If T > T0 (q), the system is nullcontrollable at time T. 2. If T < T0 (q), the system is not nullcontrollable 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 nullcontrollable. • 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: −logIk (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 AmmarKhodja 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 AmmarKhodja ( ) 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(−logI1,k (q),− logIk (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 AmmarKhodja 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 nullcontrollable. 2. If T < c, then the system is not nullcontrollable. 24 Farid AmmarKhodja 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 AmmarKhodja 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 AmmarKhodja 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. AlabauBoussouira, M. Léautaud, Indirect controllability of locally coupled wavetype systems and applications, J. Math. Pures Appl. (9) 99 (2013), no. 5, 544–76. [2] F. 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LABORATOIRE DE MATHÉMATIQUES DE BESANÇON Email address: farid.ammarkhodja@univfcomte.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 skewadjoint. 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 wellposedness 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 steadystate 0 (otherwise, just design a feedback of the kind u = K(y − ȳ)). We are first going to recall wellknown 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 PoleShifting According to the poleshifting 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 closedloop 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 wellknown 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 infinitedimensional, several difficulties occur with respect to the finitedimensional 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 nullcontrollable 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 closedloop 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 HuangPrüss 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 secondorder 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 secondorder systems ÿ + Ay = Bu, with A : D(A) → X positive selfadjoint, 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 steadystate 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 selfadjoint 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 secondorder scalar differential equation. Therefore the determinant (2.6) is never equal to zero. By the poleshifting 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 closedloop 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 righthand 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 finitedimensional 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 steadystate 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 infinitedimensional 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 infinitedimensional, 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 finitedimensional unstable part of the system, and to design a feedback control in order to stabilize that part. Then, we plug this control in the infinitedimensional system, and we have to check that this feedback indeed stabilizes the whole system (in the sense that it does not destabilize the other infinitedimensional 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 skewadjoint, 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 closedloop 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 skewadjoint operator on X and B is a bounded nonnegative selfadjoint 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 skewadjoint operator, let B be a bounded selfadjoint 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 skewadjoint, 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 skewadjoint, 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 secondorder equations, but the proof works exactly in the same 48 Emmanuel Trélat way for more general firstorder systems, as shown here. More precisely, the statement done in [23] for secondorder 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 secondorder 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 skewadjoint operator, B : X → X a nontrivial bounded selfadjoint 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 welldefined 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 selfadjoint operator B implies that B is unitarily equivalent to a multiplication: there exist a probability space (Ω, µ), a realvalued 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 finitedimensional 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 skewsymmetric 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