Controllability and Stabilization of Parabolic Equations (Progress in Nonlinear Differential Equations and Their Applications (90)) [1st ed. 2018] 9783319766652, 3319766651

Table of contents :
Preface
Contents
Acronyms
1 Preliminaries
1.1 Notations
1.2 The Nonlinear Cauchy Problem in Banach Spaces
1.3 Semilinear Parabolic Equations
1.4 Navier–Stokes Equations
1.5 Infinite Dimensional Linear Control Systems
2 The Carleman Inequality for Linear Parabolic Equations
2.1 The Carleman and Observability Inequality
2.2 Notes on Chapter 2
3 Exact Controllability of Parabolic Equations
3.1 Exact Controllability of Linear Parabolic Equations
3.2 Controllability of Semilinear Parabolic Equations
3.3 Approximate Controllability
3.4 Local Controllability of Semilinear Parabolic Equations
3.5 Controllability of the Kolmogorov Equation
3.6 Exact Controllability of Stochastic Parabolic Equations
3.7 Approximate Controllability of Stochastic Parabolic Equation
3.8 Notes on Chapter 3
4 Internal Controllability of Parabolic Equations with Inputs in Coefficients
4.1 The Exact Controllability via Self-Organized Criticality
4.2 Exact Controllability via Fast Diffusion Equation
4.3 Exact Controllability via Total Variation Flow
4.4 Exact Null Controllability in Rd
4.5 Exact Controllability of Linear Stochastic Parabolic Equations
4.6 Notes on Chapter 4
5 Feedback Stabilization of Semilinear Parabolic Equations
5.1 Riccati-based Internal Stabilization
5.2 Boundary Stabilization of Parabolic Equations
5.3 Stabilization of Semilinear Equations
5.4 Internal Stabilization of Stochastic Parabolic Equations
5.5 Stabilization of Navier–Stokes Equations Driven by Linear Multiplicative Noise
5.6 Notes on Chapter 5
6 Boundary Stabilization of Navier–Stokes Equations
6.1 The Main Stabilization Results
6.2 Proof of Theorem 6.1
6.3 Proof of Theorem 6.2
6.4 Real Stabilizing Feedback Controllers
6.5 An Example to Stabilization of a Periodic Flow in a 2D Channel
6.6 Notes on Chapter 6
References
Index

Citation preview

Progress in Nonlinear Differential Equations and Their Applications Subseries in Control 90

Viorel Barbu

Controllability and Stabilization of Parabolic Equations

Progress in Nonlinear Differential Equations and Their Applications: Subseries in Control Volume 90

Editor Jean-Michel Coron, Université Pierre et Marie Curie, Paris, France Editorial Board Viorel Barbu, Facultatea de Matematic˘a, Universitatea “Alexandru Ioan Cuza” din Ia¸si, Romania Piermarco Cannarsa, Department of Mathematics, University of Rome “Tor Vergata”, Italy Karl Kunisch, Institute of Mathematics and Scientific Computing, University of Graz, Austria Gilles Lebeau, Laboratoire J.A. Dieudonné, Université de Nice Sophia-Antipolis, France Tatsien Li, School of Mathematical Sciences, Fudan University, China Shige Peng, Institute of Mathematics, Shandong University, China Eduardo Sontag, Department of Mathematics, Rutgers University, USA Enrique Zuazua, Departamento de Matemáticas, Universidad Autónoma de Madrid, Spain

More information about this series at https://www.springer.com/series/15137

Viorel Barbu

Controllability and Stabilization of Parabolic Equations

Viorel Barbu A1. I CUZA UNIVERSITY IASI, Romania

ISSN 1421-1750 ISSN 2374-0280 (electronic) Progress in Nonlinear Differential Equations and Their Applications ISBN 978-3-319-76665-2 ISBN 978-3-319-76666-9 (eBook) https://doi.org/10.1007/978-3-319-76666-9 Library of Congress Control Number: 2018935387 Mathematics Subject Classification (2010): 93D15, 93C10, 49N35, 35K55, 35K58 © Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This book is published under the trade name Birkhäuser, www.birkhauser-science.com by the registered company Springer International Publishing AG part of Springer Nature. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book is based on the author’s works and lectures on controllability and stabilization of parabolic equations and part of it was used for a graduate course at the University of Ia¸si, Romania. In the inaugural lecture delivered at Warwick on the 7th of October 1970, Lawrence Markus has briefly defined the objectives of the mathematical control theory as: “. . . the modification of differential equations, within prescribed limitations so that the solutions behave in some desired manner. The common feature of all these problems is that we prescribe the desired behaviour of the solutions and then we seek to modify the coefficients of the dynamical equations so as to induce this behaviour. In classical mathematical physics, we know all about the physical laws guiding the development of the phenomenon we may be studying – we do not know all the rules of the game – and we want to predict the outcome. In control theory, we know the rules, in fact we can change them within certain limitations, but we do know exactly how we want the game to end. Thus, the mathematical problems of control theory are inverse to the usual problems of mathematical physics.”

The controllability and stabilization are without any doubt two fundamental problems of mathematical control theory which, for parabolic-like systems, have a special significance and difficulty due to the time irreversibility of dynamics generated by these systems. The first part of the book is devoted to the internal null controllability of parabolic equations based essentially on the influential linear controllability result of G. Lebeau and L. Robbiano (1995), developed later on by A.V. Fursikov and O.Yu. Imanuvilov (1996), via a Carleman-type inequality for linear parabolic equations. Compared with controllability, internal and boundary stabilization are apparently weaker properties but, since they are usually realized by feedback controllers, they are structurally stable and so more convenient for applications and numerical simulations. The first stabilization method developed here is based on the spectral decomposition technique, which is used to represent the linear parabolic control systems in a convenient product space as a null controllable finite dimensional unstable system and a stable infinite dimensional one. Such a system is stabilizable v

vi

Preface

by an open loop controller, and so a stabilizable linear feedback controller can be obtained by standard optimal control arguments from an algebraic Riccati equation. As a matter of fact, this stabilization method applies to general semilinear control systems of the form dy dt + Ay + F y = Bu in a Hilbert space H , where the operator −A is the infinitesimal generator of a C0 -analytic semigroup in H and has a compact resolvent. This is the class of so-called parabolic-like systems which, besides the standard heat and diffusion equations, includes Navier–Stokes and other related systems such as Boussinesq and magnetohydrodynamic equations. The second method mostly applicable to boundary control systems consists in the explicit construction of a stabilizing feedback controller by using a finite number of unstable modes. Both methods provide stabilizable feedback controllers with finite dimensional structure and stabilization. The exact controllability of stochastic parabolic equations with linear multiplicative noise is also briefly treated in this book, though so far only partial results were obtained in this direction. It should be mentioned, however, that none of the above topics was exhaustively treated in this book, which may be viewed only as a survey on controllability and stabilization techniques for parabolic boundary value problems. Many of these results, exact controllability in particular, keep their original strength and flavor, even though they were established at the end of the 1990s. There are several important topics related to the content of this book which were not treated here and most notable is perhaps the controllability of Navier–Stokes equations extensively studied in the last two decades. We wish to thank J.M. Coron for the invitation to write this book and publish it with the Control/PNLDE series. Thanks are due also to my colleagues Gabriela Marinoschi, C˘at˘alin-George Lefter and Ionu¸t Munteanu who read and criticized various chapters. Thanks are also due to the anonymous reviewers for their suggestions. I am also indebted to Mrs. Elena Mocanu for typing and processing this book. Iasi, Romania January 1st , 2018

Viorel Barbu

Contents

1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Nonlinear Cauchy Problem in Banach Spaces. . . . . . . . . . . . . . . . . . . . 1.3 Semilinear Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Navier–Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Infinite Dimensional Linear Control Systems . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 5 13 19

2

The Carleman Inequality for Linear Parabolic Equations . . . . . . . . . . . . . . 2.1 The Carleman and Observability Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Notes on Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 27 42

3

Exact Controllability of Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Exact Controllability of Linear Parabolic Equations . . . . . . . . . . . . . . . . . . 3.2 Controllability of Semilinear Parabolic Equations . . . . . . . . . . . . . . . . . . . . 3.3 Approximate Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Local Controllability of Semilinear Parabolic Equations . . . . . . . . . . . . . 3.5 Controllability of the Kolmogorov Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Exact Controllability of Stochastic Parabolic Equations . . . . . . . . . . . . . . 3.7 Approximate Controllability of Stochastic Parabolic Equation. . . . . . . 3.8 Notes on Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 43 59 69 70 81 89 95 99

4

Internal Controllability of Parabolic Equations with Inputs in Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Exact Controllability via Self-Organized Criticality . . . . . . . . . . . . . 4.2 Exact Controllability via Fast Diffusion Equation. . . . . . . . . . . . . . . . . . . . . 4.3 Exact Controllability via Total Variation Flow . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Exact Null Controllability in Rd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Exact Controllability of Linear Stochastic Parabolic Equations . . . . . . 4.6 Notes on Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103 103 114 122 124 125 126

5

Feedback Stabilization of Semilinear Parabolic Equations . . . . . . . . . . . . . 129 5.1 Riccati-based Internal Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.2 Boundary Stabilization of Parabolic Equations. . . . . . . . . . . . . . . . . . . . . . . . 175 vii

viii

6

Contents

5.3 Stabilization of Semilinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Internal Stabilization of Stochastic Parabolic Equations . . . . . . . . . . . . . . 5.5 Stabilization of Navier–Stokes Equations Driven by Linear Multiplicative Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Notes on Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

182 186

Boundary Stabilization of Navier–Stokes Equations. . . . . . . . . . . . . . . . . . . . . 6.1 The Main Stabilization Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Proof of Theorem 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Proof of Theorem 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Real Stabilizing Feedback Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 An Example to Stabilization of a Periodic Flow in a 2D Channel . . . 6.6 Notes on Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

197 197 204 207 210 214 218

191 194

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

Acronyms

pK recc(K) C N R Rd R+ Rd+ O ∂O Q Σ  · X X∗ (·, ·)H x·y L(X, Y ) ∇f ∂f B∗ C int C conv C D(A) R(A) IC sign C k (O)

the Minkowski functional of the set K the recession cone of K the set of all complex numbers the set of all natural numbers the real line (−∞, ∞) the d-dimensional Euclidean space = (0, +∞) = {(x1 , . . . , xd ); xd > 0} an open subset of Rd the boundary of O = O × (0, T ) = ∂O × (0, T ), where 0 < T < ∞ the norm of the linear normed space X the dual of the space X the scalar product of the Hilbert space H the scalar product of the vectors x, y ∈ Rd the space of linear continuous operators from X to Y the gradient of the function f the subdifferential of the function f the adjoint of the operator B the closure of the set C the interior of the set C the convex hull of the set C the domain of the operator A the range of the operator A the indicator function of the set C the signum function on X : sign x = x/xX if x = 0, sign 0 = {x; xX ≤ 1} the space of real-valued functions on O that are continuously differentiable up to order k, k ≤ ∞

ix

x

Acronyms

C0k (O) D(O) dku , u(k) dt k

D (O)

C(O) Lp (O)

p

Lm (O) W m,p (O) m,p

W0 (O) W −m,q (O) H k (O), H0k (O) Lp (a, b; X) C([a, b]; X) AC([a, b]; X) W 1,p ([a, b]; X) ν, n ∂u ∂u ∂ν , ∂n

the subspace of functions in C k (O) with compact support in O the space C0∞ (O) the derivative of order k of the function u : [a, b] → X the dual of D(O) (i.e., the space of distributions on O) the space of continuous functions on O the space of p-summable functions u : O → R 1  endowed with the norm |u|p = ( O |u(x)|p dx) p , 1≤p 1 the spaces W k,2 (O) and W0k,2 (O), respectively the space of p-summable functions from (a, b) to X, 1 ≤ p ≤ ∞, −∞ ≤ a < b ≤ ∞ the space of X-valued continuous functions on [a, b] the space of absolutely [a, b] to X  continuous functions from  p the Sobolev space u ∈ AC([a, b]; X); du dt ∈ L ((a, b); X) the outward normal to O the normal derivative of the function u : O → R

Chapter 1

Preliminaries

Here we survey for later use some basic existence results for the infinite dimensional Cauchy problem, semilinear parabolic-like boundary value problems, and infinite dimensional control systems.

1.1 Notations Given an open subset O of Rd , d ∈ N, we denote by O its closure and ∂O its boundary. By Lp (O), 1 ≤ p ≤ ∞, we denote the standard space of Lp -Lebesgue integrable functions on O with the norm denoted | · |p or  · Lp (O) . By D (O), we denote the space of distributions on O, that is, the dual of C0∞ (O). Denote by W 1,p (O), 1 ≤ p ≤ ∞, the Sobolev space 

where

∂ ∂xi

u ∈ Lp (O),

 ∂u ∈ Lp (O), i = 1, . . . , d , ∂xi

is taken in D (O). We denote by C0∞ (O) = D(O) the space of infinitely 1,p

differentiable functions with compact support in O and by W0 (O) the closure of C0∞ (O) in W 1,p (O). For p = 2, we set W 1,2 (O) = H 1 (O), W01,2 (O) = H01 (O),   ∂ 2u 2 2 2 H (O) = u ∈ L (O); ∈ L (O), i, j = 1, . . . , d . ∂xi ∂xj

The dual space of W0 (O) is denoted by W −1,p (O), p1 + p1 = 1, and set W −1,2 (O) = H −1 (O). Given a Banach space X, denote by C([0, T ]; X) the 1,p

© Springer International Publishing AG, part of Springer Nature 2018 V. Barbu, Controllability and Stabilization of Parabolic Equations, Progress in Nonlinear Differential Equations and Their Applications 90, https://doi.org/10.1007/978-3-319-76666-9_1

1

2

1 Preliminaries

space of X-valued continuous functions on [0, T ] ⊂ [0, ∞) and by Lp (0, T ; X) the space of X-valued Lp -Bochner integrable functions on (0, T ). We denote by Cw ([0, T ]; X) the space of weakly continuous X-valued functions on 1,p we denote [0, T ]. pBy W ([0,duT ]; X),  the infinite dimensional Sobolev space u ∈ L (0, T ; X); dt ∈ Lp (0, T ; X) , where du dt is taken in sense of X-valued vectorial distributions on (0, T ). If X, Y are Banach spaces, L(X, Y ) is the space of linear continuous operators from X to Y with the operatorial norm  · L(X,Y ) . The norm of X will be denoted  · X or simply  · .

1.2 The Nonlinear Cauchy Problem in Banach Spaces Let X be a real Banach space with the dual space X∗ and let A : D(A) ⊂ X → 2X be a nonlinear multivalued operator. Here D(A) is the domain of A, that is, D(A) = {u ∈ X; Au = ∅}. The multivalued mapping A is very often identified with a subset of X × X by A = {(v, u) ∈ X × X; v ∈ Au}. The operator A is said to be accretive if X (v1

− v2 , η)X∗ ≥ 0 for η ∈ J (u1 − u2 ), ∀ vi ∈ Aui , i = 1, 2,

where J : X → X∗ is the duality map of the space X and ·, ·X∗ is the duality function on X × X∗ . Equivalently, (I + λA)−1 f − (I + λA)−1 gX ≤ f − gX , for all f, g ∈ R(I + λA) and λ > 0. Here R(I + λA) is the range of the operator I +λA. The operator A is said to be m-accretive if A is accretive and R(I +λA) = X for all λ > 0 (equivalently, for some λ > 0). The operator is said to be quasi-maccretive if A + αI is m-accretive for some α > 0 and quasi-accretive if A + αI is accretive. If X is a real Hilbert space with the scalar product (·, ·), then A is m-accretive if and only if it is maximal monotone in X × X, that is, (v1 − v2 , u1 − u2 ) ≥ 0 for all vi ∈ Aui , i = 1, 2, and A has no proper monotone extension on X × X (see [17], p. 39, [18, 26]). If X is a Hilbert space and ϕ : X →] − ∞, +∞] is a convex and lower semicontinuous function, then its subdifferential ∂ϕ : X → 2X is defined by ∂ϕ(x) = {z ∈ X; ϕ(x) ≤ ϕ(y) + (z, x − y), ∀ y ∈ X}. The operator A = ∂ϕ is m-accretive (equivalently, maximal monotone) (see, [26, 54], p.47).

1.2 The Nonlinear Cauchy Problem in Banach Spaces

3

Consider the Cauchy problem dy + Ay  0 in (0, T ), dt y(0) = y0 .

(1.1)

The function y : [0, T ] → X is said to be a strong solution to (1.1) if it is absolutely continuous on [0, T ] and there is η ∈ L1 (0, T ; X) such that η(t) ∈ Ay(t), t ∈ (0, T ), and dy (t) + η(t) = 0, a.e. t ∈ (0, T ), dt y(0) = y0 .

(1.2)

The function y ∈ C([0, T ]; X) is said to be a mild solution to (1.2) if y(t) = lim yh (t), ∀ t ∈ [0, T ], h→0

(1.3)

where yh : [0, T ] → H is the step function, yh (t) = yih for t ∈ [ih, (i + 1)h), i = 0, 1, . . . , N − 1,

(1.4)

h + hAy h  y h . yi+1 i i+1

(1.5)

y0h = y0 and N h = T . In other words, a mild solution is a continuous X-valued function on [0, T ] which is the limit of the solution yh to the finite difference scheme (1.4)–(1.5) associated with problem (1.1). Equivalently,

−n t I+ A y0 , ∀ t ∈ [0, T ]. h→∞ n

y(t) = lim

(1.6)

We have the following fundamental result (see [26], p. 138). Theorem 1.1 (Crandall & Liggett). Let A be a quasi-m-accretive operator in a real Banach space X. Then, for each y0 ∈ D(A), there is a unique mild solution y to the Cauchy problem (1.1). If the space X is reflexive and y0 ∈ D(A), then the mild ∞ 1,∞ ([0, T ]; X)). solution y is strong solution and dy dt ∈ L (0, T ; X) (that is, y ∈ W If X is a Hilbert space and A = ∂ϕ, then, for y0√∈ D(ϕ) = {ϕ; ϕ(y) < ∞}, 2 y ∈ W 1,2 ([0, T ]; X), ϕ(y) ∈ L∞ (0, T ; X), and t dy dt ∈ L (0, T ; X), ϕ(y) ∈ L1 (0, T ; X) for y0 ∈ D(A) = D(ϕ). Here D(A) is the closure of D(A) in X. We note also that the map y0 → y(t) defines a continuous semigroup of quasi-contractions on D(A), S(t) : D(A) → D(A) also denoted by e−tA . We have also the following more general result (see [26], p. 130).

4

1 Preliminaries

Theorem 1.2. Let A be quasi-accretive in a Banach space X and let C be a closed convex cone of X such that  R(1 + λA) D(A) ⊂ C ⊂ 0 0 is the Yosida approximation of A. Example 1.1. Let X = L1 (O), where O is a bounded and open domain of Rd and let A : D(A) ⊂ L1 (O) → L1 (O) be the operator Ay = {−Δη; η ∈ β(y), a.e. in O}, ∀ y ∈ D(A), D(A) = {y ∈ L1 (O); ∃ η ∈ W01,1 (O), η ∈ β(y), a.e. in O}, where β : D(β) ⊂ R → 2R , β(0)  0, is a maximal monotone graph in R × R, that is, β is monotone and R(I + β) = R. Then (see [26], p. 117) the operator A is m-accretive in L1 (O) and, by Theorem 1.1, we have Theorem 1.3. For each y0 ∈ L1 (O), such that y0 (x) ∈ D(β), a.e., x ∈ O, there is a unique mild solution to the equation ∂y − Δβ(y)  0 in (0, T ) × O, ∂t y(0, x) = y0 (x), x ∈ O, β(y) = 0 on (0, T ) × ∂O.

(1.8)

It turns out that the mild solution y to (1.8) is just a solution in sense of distributions on (0, T ) × O. Equation (1.8) can be also studied in a different functional setting, namely in the Sobolev space H −1 (O). In fact, the operator = {−Δη; η ∈ β(y), η ∈ H01 (O)} Ay is m-accretive in H −1 (O) (see, e.g., [26], p. 237) and so, again by Theorem 1.1, a unique strong solution y ∈ C([0, T ]; H −1 (O)) equation (1.8) has, for y0 ∈ D(A), dy ∞ −1 such that dt ∈ L (0, T ; H (O)), β(y) ∈ L∞ (0, T ; H01 (O)).

1.3 Semilinear Parabolic Equations

5

1.3 Semilinear Parabolic Equations We shall review here the main existence result for the parabolic boundary value

∂ ∂y (x, t) − ∂t ∂xi d

i,j =1



∂y aij (x) (x, t) +f (x, y(x, t))=f0 (x, t) ∂xj in Q = O×(0, T ),

∂y + α2 y = 0 α1 ∂ν y(x, 0) = y0 (x)

(1.9)

on Σ = ∂O×(0, T ), in O.

Here O is an open and bounded set of Rd with the boundary ∂O of class C 2 and f0 ∈ L2 (Q), y0 ∈ L2 (O) are given functions. We assume that αi ≥ 0, α1 + α2 > 0, αi ∈ C(∂O), i = 1, 2, . . . , d, aij ∈ C 1 (O), aij = a j i ,

d  i,j =1

aij ξi ξj ≥ ω|ξ |2 , ∀ ξ ∈ Rd ,

(1.10) (1.11)

where ω > 0, ξ = (ξ1 , . . . , ξd ). Finally, f : O×R → R is continuous function satisfying the following assumptions f ∈ C 1 (O×R), fy (x, y) ≥ −γ , ∀ (x, y) ∈ O×R, fx ∈ where fy = ∇y f, fx = ∇x f. By

L∞ (Q),

∂ ∂νA

(1.12) (1.13)

we have denoted the outward conormal   d  ∂ ∂ derivative corresponding to the elliptic operator ∂xi aij ∂xj , that is, i,j =1

d

∂y ∂y = aij νj , ∂νA ∂xj i,j =1

where ν = (ν1 , . . . , νd ) is the outward normal to O. In the special case aij = δij (Kronecker’s symbol), ∂y ∂y = ∇y · ν on ∂O = ∂νA ∂ν is the outward normal derivative of y. We have

6

1 Preliminaries

Theorem 1.4. Assume that y0 ∈ H 2 (O), α1

∂y0 + α2 y0 = 0 on ∂O, ∂ν

f0 ∈ W 1,1 ([0, T ]; L2 (O)).

(1.14) (1.15)

Then there is a unique strong solution y to (1.9) satisfying y ∈ C([0, T ]; H 1 (O)) ∩ L∞ (0, T ; H 2 (O)) ∩ W 1,∞ ([0, T ]; L2 (O)).

(1.16)

If one merely assumes that y0 ∈ H 1 (O) (y0 ∈ H01 (O) if α1 =0), f0 ∈ L2 (Q), j (x, y0 ) ∈ L1 (O),

(1.17)

then y ∈ W 1,2 ([0, T ]; L2 (O)) ∩ L2 (0, T ; H 2 (O)) ∩ Cw ([0, T ]; H 1 (O)).

(1.18)

Finally, if y0 ∈ L2 (O) and f0 ∈ L2 (Q), then y ∈ C([0, T ]; L2 (O)) ∩ L2 (0, T ; H 1 (O)), √ t yt ∈ L2 (Q).

√ t y ∈ L2 (0, T ; H 2 (O)), (1.19) (1.20)

r Here j (x, r) ≡ 0 f (x, s)ds. By strong solution to (1.9) we mean a continuous function y : [0, T ] → L2 (O) which is a.e. differentiable and satisfies a.e. the equation on Q. By yt we mean of 1,2 ([0, T ]; L2 (O)). course dy dt , the strong derivative of y ∈ W A convenient way to treat problem (1.9) is to write it as a Cauchy problem of the form (1.1) in the space H = L2 (O). Namely, we view y = y(x, t) as an H -valued function y : [0, T ] → H and rewrite (1.9) as dy (t) + Ay(t) = f0 (t), t ∈ (0, T ), dt y(0) = y0 ,

(1.21)

where A : H → H is the operator

d

∂ ∂y Ay = − aij + f (x, y), ∀ y ∈ D(A), ∂xi ∂xj i,j =1

with the domain   ∂y D(A) = y ∈ H 2 (O); α1 + α2 y = 0 on ∂O; f (x, y) ∈ L2 (O) . ∂ν We have

(1.22)

1.3 Semilinear Parabolic Equations

7

Lemma 1.1. The operator A is quasi-m-accretive in H ×H . Proof. As seen earlier, this means that (λ(y − y) ¯ + Ay − Ay, ¯ y − y) ¯ ≥ 0, ∀ y, y¯ ∈ D(A), λ > λ0 ,

(1.23)

and R(λI + A) = L2 (O), ∀ λ > λ0 ,

(1.24)

where R(λI + A) is the range of λI + A and (·, ·) is the scalar product in L2 (O). Inequality (1.23) is an immediate consequence of (1.10)–(1.12). To prove (1.24), we fix g ∈ L2 (O) and consider the equation λy + Ay = g.

(1.25)

We set 1 (r − (1 + εf (x, ·))−1 r) ∀ r ∈ R, ε > 0, ε

d

∂ ∂y aij , A0 y = − ∂xi ∂xj  i=1  ∂y D(A0 ) = y ∈ H 2 (O); α1 + α2 y = 0 on ∂O . ∂νA

fε (x, r) =

This is the Yosida approximation of f (x, ·) and it is easily seen that for each x ∈ O, fε (x, ·) is Lipschitzian, monotonically increasing and 1 (r − Jε (r)), ∀ r ∈ R, ε Jε (r) = (1 + εf (x, ·))−1 (r), lim fε (x, r) = f (x, r), ∀ x ∈ O, r ∈ R, fε (x, r) = f (x, Jε (r)) =

(1.26)

ε↓0

|Jε (r) − Jε (¯r )| ≤ (1 − εγ )−1 |r − r¯ |, ∀ r, r¯ ∈ R. Consider the approximating equation λy + A0 y + fε (x, y) = g, x ∈ O, ε > 0, and note that it has a unique solution yε ∈ H 2 (O). Indeed, for each z ∈ L2 (O), arbitrary but fixed, the equation



d  1 1 ∂ ∂y λ+ y− aij = g + Jε (z) in O, ε ∂xj ε i,j =1 ∂xi ∂y α1 on ∂O, + α2 y = 0 ∂ν

(1.27)

8

1 Preliminaries

has, by the Lax-Milgram lemma and elliptic regularity theorem, a unique solution y = F (z) ∈ H 2 (O) (see, e.g., [55]). Moreover, by an elementary calculation involving Green’s formula, we see by (1.26) that F z − F z¯ L2 (O) ≤ ρz − z¯ L2 (O) , 0 < ρ < 1, for λ ≥ λ0 . Then, by Banach’s fixed point theorem, there is a solution yε ∈ H 2 (O). A Priori Estimates Assume here α1 > 0, but the case α1 = 0 can be similarly treated. Multiplying (1.27) by yε and taking into account that, by (1.26) and (1.12), fε (x, r)r = f (x, Jε (r)Jε (r)) + εfε2 (x, r) ≥ −γ |yε (r)|2 + f (x, 0)yε (r) + εfε2 (r) ≥ −2γ |r 2 |, ∀ r ∈ R, we obtain via Green’s formula that λyε 2L2 (O) + ω∇yε 2L2 (O) ≤ C(yε 2L2 (O) + 1). Hence, for λ ≥ λ0 large enough, yε H 1 (O) ≤ C, ∀ ε > 0.

(1.28)

(We shall use the same symbol C to denote several positive constants independent of ε.) Next, multiply (1.27) by fε (x, yε ) and integrate on O. Noticing that −

d 

i,j =1





d 

∂fε ∂ ∂yε ∂yε ∂yε ∂fε aij fε (x, yε )dx = dx + aij ∂xj ∂xj ∂xi ∂y ∂xi O ∂xi O i,j =1  α2 + fε (x, yε )yε dσ, ∂O α1

we obtain, via the trace theorem, the estimate

  d  

 ∂ ∂yε 2  a dx + fε2 (x, yε )dx ≤ C, ∀ ε > 0, ij   ∂x ∂x i j O O

i,j =1

and, therefore, we have yε H 2 (O) + fε (·, yε )L2 (O) ≤ C, ∀ ε > 0.

(1.29)

By (1.11) and (1.27), we obtain also λyε − yμ 2L2 (O)+ ω∇(yε − yμ )2L2 (O)  + (fε (x, yε (x)) − fμ (x, yε (x)))(yε (x) − yμ (x))dx ≤ 0. O

1.3 Semilinear Parabolic Equations

9

Taking into account that, by (1.26), (fε (·, yε ) − fμ (·, yμ ))(yε − yμ ) = (f (·, Jε (yε )) − f (·, Jμ (yμ )))(Jε (yε ) − Jμ (yμ )) +(fε (x, yε ) − fμ (x, yμ ))(εfε (x, yε ) − μfμ (x, yμ ))

(1.30)

and that y → f (x, y) + γ y is monotonically increasing, we obtain by (1.29) and (1.30) that λyε − yμ 2L2 (O) + ωyε − yμ 2H 1 (O) ≤ C(ε + λ), ∀ ε, λ > 0. Hence {yε } is strongly convergent in H 1 (O) for ε → 0. We set y = lim yε in H 1 (O).

(1.31)

ε↓0

Moreover, by (1.29) it follows that {A0 yε } and {fε (·, yε )} are weakly compact in L2 (O). Hence, for ε → 0, we have A0 yε −→ A0 y weakly in L2 (O), fε (·, yε ) −→ η yε −→ y

weakly in L2 (O), 2

weakly in H (O).

(1.32) (1.33) (1.34)

It remains to be shown that η(x) = f (x, y(x)), a.e. x ∈ O.

(1.35)

To this end, we note that since |yε − Jε (x, yε )| = ε|fε (x, yε )|, we have Jε (x, yε ) −→ y strongly in L2 (O) and, therefore, fε (x, yε (x)) −→ f (x, y(x)), a.e. x ∈ O because fε (x, yε ) = f (x, Jε (yε )) and y → f (x, y) is continuous. Then, by (1.33), we get (1.35), as claimed. This completes the proof of Lemma 1.1. Proof of Theorem 1.4 (Continued). Assume first that (1.14) and (1.15) hold. Hence y0 ∈ D(A) and so, according to Theorem 1.1, there is a unique function y ∈ W 1,∞ ([0, T ]; H ) ∩ L∞ (0, T ; D(A))

10

1 Preliminaries

which satisfies a.e. on (0, T ) equation (1.21). This implies that y satisfies (1.16). Moreover, multiplying (1.9) by dy dt and integrating on O, we obtain      dy 2 dy 1 d d 2   ∇y(t) (t) ω dx. + + j (x, y(t))dx ≤ f0 2 (O)  2  dt L 2 dt dt O dt O L (O) (1.36) Integrating on (0, t), we get the estimate 2  t    dy 2   (s) ds + ω∇y(t)L2 (O) + j (x, y(x, t))dx  O  0 dt  L2 (O) (1.37)

 t 2 |∇y0 (x)| dx + j (x, y0 (x))dx + f0 (s)2L2 (O) ds . ≤C O

If multiply (1.9) by t  0

t

O

dy dt

0

and integrate on (0, t), we obtain

2     dy 2   (s) ds + ωt∇y(t)L2 (O) + t j (x, y(x, t))dx s O dt    

≤C

t



0 t

≤C 0

∇y(s)2L2 (O) ds+

t

0

O

f02 (x, s)dx

t

ds+ 0

O



j (x, y(x, s))dx ds

f0 (s)2L2 (O) ds + y0 2L2 (O) , ∀ t ∈ (0, T )

(1.38)

because, as easily seen by (1.8)–(1.10), we have  t  t 1 y(t)2L2 (O) + ω ∇y(s)2L2 (O) ds + j (y(x, s))dx ds 2 0  0 O t t 1 ≤ y0 2L2 (O) + γ0 y(s)2L2 (O) ds + f0 (s)L2 (O) y(s)L2 (O) ds 2 0 0 and so, by Gronwall’s lemma,  t  t y(t)2L2 (O)+ ω ∇y(s)2L2 (O) ds + j (y(x, s))dx ds 0 O

0  t 2 2 ≤ C y0 L2 (O) + f0 (s)L2 (O) ds . 0

By (1.29) and (1.34), it follows that yH 2 (O) ≤ C. Then, by (1.37), we obtain 

T 0

    dy 2 2 2   + y(t)H 2 (O) + f (t, y(t))L2 (O) dt  dt (t) 2 L (O)

 T f0 2L2 (O) dt +y(t)2H 1 (O) ≤ C y0 2H 1 (O) + 0

(1.39)

1.3 Semilinear Parabolic Equations

11

while (1.38) yields 

T 0

    dy 2 2 2   (t) + y(t)H 2 (O) + f (t, y(t))L2 (O) dt t  dt

 T 2 2 f0 (t)L2 (O) dt . ≤ C y0 L2 (O) +

(1.40)

0

Assume now that (1.17) holds. Then, choose the sequences {y0k }⊂D(A), such that

{f0k }⊂W 1,1 ([0, T ]; L2 (O))

y0k −→ y0 strongly in H01 (O), f0k −→ f0 strongly in L2 (0, T ; H ) = L2 (Q). The corresponding solution yk to (1.9) satisfies (1.16) and estimate (1.39). Moreover, subtracting equation for yk and ym and multiplying by yk − ym , we see that  yk (t) − ym (t)2L2 (O)

+ω

t

∇(yn − ym )(s)2L2 (O) ds 0 

t 2 2 ≤ C y0k − y0m L2 (O) + f0k (s) − f0m (s)L2 (O) ds . 0

Hence there is y ∈ C([0, T ]; H ) ∩ L2 (0, T ; H 2 (O)) such that for k → ∞ yk −→ y strongly in C(0, T ]; H ) ∩ L2 (0, T ; H 1 (O)) and, by estimate (1.37), it follows that dy dyk −→ weakly in L2 (0, T ; H ), dt dt A0 yk −→ A0 y weakly in L2 (0, T ; H ), f (t, yk ) −→ f (t, y) weakly in L2 (0, T ; H ). Hence y ∈ W 1,2 ([0, T ]; H ) ∩ L2 (0, T ; D(A)) is a solution to (1.9) (obviously unique) and satisfies (1.18). By a similar argument, it follows by estimate (1.40) that, if y0 ∈ L2 (O), f0 ∈ L2 (Q), there is a solution y which satisfies (1.19), (1.20). This completes the proof of Theorem 1.4. A similar existence result follows for the equation ∂y − Δy + f (x, y) + F (y) = 0 in Q, ∂t y(0) = y0 in O, y=0 on Σ,

(1.41)

12

1 Preliminaries

where f satisfies conditions (1.12)–(1.13), and the operator F : L2 (0, T ; H01 (O)) → L2 (0, T ; L2 (O)) is Lipschitzian. Namely, we have Theorem 1.5. Let y0 ∈ L2 (O). Then equation (1.41) has a unique solution y ∈ C([0, T ]; L2 (O)) ∩ L2 (0, T ; H01 (O)), dy ∈ L2 (0, T ; H −1 (O)). dt

(1.42)

Moreover, √ dy √ ∈ L2 (0, T ; L2 (O)) t y ∈ L2 (0, T ; H 2 (O)), t dt

(1.43)

and, if y0 ∈ H01 (O), then y ∈ L2 (0, T ; H 2 (O)) ∩ L∞ (0, T ; H01 (O)),

dy ∈ L2 (0, T ; L2 (O)). dt

(1.44)

Proof. It is easily seen via fixed point arguments that the operator Ay = −Δy + F (y), D(A) = H01 (O) ∩ H 2 (O) is quasi-m-accretive in H = L2 (O). Then, by Theorem 1.1, for y0 ∈ D(A), equation (1.41) has a unique solution y ∈ L∞ (0, T ; H01 (O) ∩ H 2 (O)) with dy ∞ 2 dt ∈ L (0, T ; L (O)). Moreover, arguing as in the proof of Theorem 1.4, one finds (1.42)–(1.44). It is useful to note that the solution y to (1.41) is given by y = lim yε in C([0, T ]; H ), ε→0

(1.45)

where yε ∈ L2 (0, T ; H (O)) ∩ C([0, T ]; H ) is the solution to the approximating equation ∂yε − Δyε + fε (x, yε ) + F (yε ) = 0, ∂t yε (x, 0) = y0 (x) in O, yε = 0 on Σ.

(1.46)

Indeed, arguing as above, it follows that, for each g ∈ H and λ ≥ λ0 sufficiently large, the solution zε to the equation λzε − Δzε + fε (x, zε ) + F (zε ) = g in O, zε ∈ H01 (O) ∩ H 2 (O),

1.4 Navier–Stokes Equations

13

is strongly convergent to the solution z to the elliptic boundary value problem λz − Δz + f (x, z) + F (z) = g in O, z ∈ H01 (O) ∩ H 2 (O). (This simply follows by subtracting the equations, multiplying by zε − zε , where ε, ε > 0, and integrating on O.) Then, by the Trotter–Kato theorem for nonlinear semigroups (see [26], p. 168), it follows (1.45), as claimed. On the other hand, by multiplying (1.46) by yε , Δyε and tΔyε , respectively, and integrating on (0, t) × O, we get as above the estimates (see (1.38)–(1.40))  |yε (t)|22 + yε (t)2

H01 (O)

t

0

+ 0

tyε 2

|∇yε (s)|22 ds ≤ C(|y0 |22 + 1), ∀ t ∈ [0, T ], t

|Δyε (s)|22 ds ≤ C(y0 2H 1 (O) + 1), t ∈ [0, T ), 0  t 2 + s|Δyε |2 ds ≤ C(|y0 |22 + 1).

H01 (O)

0

Hence, for y0 ∈ L2 (O), we have also, for ε → 0, y −→ y weakly in L2 (0, T ; H01 (O)), strongly in C([0, T ]; L2 (O)), √ √ ε t yε −→ t y weakly in L2 (0, T ; H 2 (O)) ∩ L∞ (0, T ; H01 (O)). (1.47)

1.4 Navier–Stokes Equations Let O be an open subset of Rd with a smooth boundary ∂O. The Navier–Stokes system yt (x, t) = ν0 Δy(x, t)− (y · ∇)y(x, t) + f (x, t) + ∇p(x, t), x ∈ O, t ∈ (0, T ), (∇ · y)(x, t) = 0, ∀ (x, t) ∈ O×(0, T ), (1.48) y = 0 on ∂O ∈ (0, T ), y(x, 0) = y0 (x), x ∈ O, describes the nonslip motion of a viscous, incompressible, Newtonian fluid in an open domain O⊂Rd , d = 2, 3. Here y = (y1 , y2 , . . . , yd ) is the fluid velocity field, p = p(t, x) is the pressure, f is the density of an external force, and ν0 > 0 is the kinematic viscosity. Equations (1.48) are obtained by Newton’s second law, while the condition ∇ · u = 0 represents the incompressibility constraint. We have used the following notation:

14

1 Preliminaries

∇·y = div y =

d

i=1

 d d

∂ Di yi , Di = , i = 1, . . . , d, (y·∇)y = yi Di yj ∂xi i=1

.

j =1

If the force f = fe is independent of t, then the motion of the fluid is governed by the stationary Navier–Stokes equation −ν0 Δy(x) + (y · ∇)y(x) = fe (x) + ∇p(x), x ∈ O, ∇ ·y =0 in O, y=0 on ∂O.

(1.49)

The linear equation yt − ν0 Δy = f + ∇p ∇ ·y =0 y=0 y(x, 0) = y0 (x)

in O×(0, T ), in O × (0, T ), in O, in O,

(1.50)

is called the Stokes equation. A convenient and standard way to treat the boundary value problem (1.48) is to represent it as an infinite dimensional Cauchy problem in an appropriate function space on O. To this end, we shall introduce the following spaces H = {y ∈ (L2 (O))d ; ∇ · y = 0, y · ν = 0 on ∂O},

(1.51)

V = {y ∈ (H01 (O))d ; ∇ · y = 0}.

(1.52)

Here ν is the outward normal to ∂O. The space H is a closed subspace of (L2 (O))d and it is a Hilbert space with the scalar product  (y, z) =

O

y · z dx

and the norm  |y| =

1/2 O

|y|2 dx

.

(1.53)

(We shall denote by the same symbol | · | the norm in Rd , (L2 (O))d and H , respectively.)

1.4 Navier–Stokes Equations

15

The norm of the space V will be denoted by  · , i.e.,  y =

1/2 |∇y(x)| dx 2

O

.

We shall denote by Π : (L2 (O))d → H the orthogonal projection of (L2 (O))d onto H (the Leray projector) and set  a(y, z) =

O

∇y · ∇z dx, ∀ y, z ∈ V .

A = −Π Δ, D(A) = (H 2 (O))d ∩ V . Equivalently, (Ay, z) = a(y, z), ∀ y, z ∈ V . The Stokes operator A is self-adjoint in H , A ∈ L(V , V ) (V is the dual of V ) and (Ay, y) = y2 , ∀ y ∈ V . Finally, consider the trilinear functional  b(y, z, w) =

d

O i,j =1

yi Di zj wj dx, ∀ y, z, w ∈ V ,

and denote by B : V → V the operator defined by By = Π ((y · ∇)y) or, equivalently, (By, w) = b(y, y, w), ∀ w ∈ V . Then, taking in account that Π (∇p) = 0, problem (1.48) can be written as dy (t) + ν0 Ay(t) + By(t) = Πf (t), t ∈ (0, T ), dt y(0) = y0 . (We have assumed of course that y0 ∈ H.) Similarly, equation (1.49) can be rewritten as ν0 Ay + By = Πfe .

(1.54)

16

1 Preliminaries

Let f ∈ L2 (0, T ; V ) and y0 ∈ H. The function y : [0, T ] → H is said to be a weak solution to equation (1.48) if y ∈ L2 (0, T ; V ) ∩ Cw ([0, T ]; H ) ∩ W 1,1 ([0, T ]; V ), d (y(t), ψ)+ν0 a(y(t), ψ)+b(y(t), y(t), ψ)=(f (t), ψ), a.e. t∈(0, T ), dt y(0) = y0 , ∀ ψ∈V . (Here (·, ·) is, as usual, the pairing between V , V and the scalar product of H .) This equation can be, equivalently, written as dy (t) + ν0 Ay(t) + By(t) = f (t), a.e. t ∈ (0, T ), dt y(0) = y0 ,

(1.55)

where dy dt is the strong derivative of function y : [0, T ] → V . The function y is said to be strong solution to (1.9) if y ∈ W 1,1 ([0, T ]; H ) ∩ 1 L2 (0, T ; D(A)) and (1.55) holds with dy dt ∈ L (0, T ; H ) the strong derivative of function y : [0, T ] → H. The existence theory for the Navier–Stokes equations is based on the following estimate on the trilinear functional b,

|b(y, z, w)| ≤ Cym1 zm2 +1 wm3 , where ym1 = y(H m1 (O))d = |A

m1 2

(1.56)

y|2 and (see, [113], [26], p. 252)

d d if mi = , ∀ i = 1, 2, 3, 2 2 d d if m1 = for some i. m1 + m 2 + m 3 > 2 2

m1 + m2 + m3 ≥

A fundamental question so far only partially solved is whether problem (1.48) (equivalently, (1.55)) is well posed, that is, if the solutions to the Navier–Stokes equations exist for all time t > 0 and are sufficiently smooth to imply uniqueness. The answer is positive if d = 2 or if the initial data y0 is sufficiently small in the space V , but the problem is open for d = 3. We pause briefly to discuss this problem via the abstract existence theory presented in Section 1.3 (see [26]). Theorem 1.6. Let d = 2, 3 and f ∈ W 1,1 ([0, T ]; H ), y0 ∈ D(A) where 0 < T < ∞. Then there is a unique function y ∈ W 1,∞ ([0, T ∗ ]; H ) ∩ L∞ (0, T ∗ ; D(A)) ∩ C([0, T ∗ ]; V )

1.4 Navier–Stokes Equations

17

such that dy(t) + ν0 Ay(t) + By(t) = f (t), a.e. t ∈ (0, T ∗ ), dt y(0) = y0 ,

(1.57)

for some T ∗ = T ∗ (y0 , f ) ≤ T . If d = 2, then T ∗ = T . Moreover, y is right differentiable and d+ y(t) + ν0 Ay(t) + By(t) = f (t), ∀ t ∈ [0, T ∗ ). dt

(1.58)

1

Finally, if d = 3, f ≡ 0, and y0  ≤ C ∗ (ν0 ) 2 , where C ∗ is independent of y0 , then T ∗ = T . Proof. One considers the operator BN : V → V , defined by ⎧ if y ≤ N, ⎨ By BN y = N2 ⎩ By if y > N, y2 and the equation dyN + ν0 AyN + BN yN = 0, t ∈ (0, T ), dt yN (0) = y0 .

(1.59)

Taking into account that By − BzV ≤ Cy − z(y + z), ∀ y, z ∈ V , it follows that ν0 A + BN is quasi-m-accretive in H × H and so, by Theorem 1.1, the Cauchy problem (1.59) has a unique strong solution yN = e−t (ν0 A+BN ) y0 . By (1.57) and (1.56), one gets for yN the Leray energetical estimate 



t

|yN (t)| + ν0 2

t

yN (t) ≤ |x| +

|f (t)|2 , ∀ t ∈ (0, T ).

(1.60)



 t |AyN (s)|2 ds ≤ C y0 2 + f (s)|2 ds ,

(1.61)

2

2

0

0

Moreover, for d = 2, we get the estimate  yN (t)2 + ν0 0

t

0

∀ t ∈ [0, T ], y0 ∈ V ,

and 1 d yN (t)2 + ν0 |AyN (t)|2 ≤ C(yN (t)6 + fT2 ), 2 dt for d = 3. (Here C is a positive constant independent of N and fT2 =

(1.62) T 0

|f (t)|2 dt.)

18

1 Preliminaries

√ By (1.61) we see that, if d = 2, then, for N ≥ Cy0 , BN (yN ) ≡ ByN , and so for such an N, yN is the solution to (1.57). If d = 3, it follows by (1.62) that yN (t)2 ≤ ϕ(t), ∀ t ∈ [0, T ], where ϕ is the solution to the equation (α > 0) ϕ (t) + αν0 ϕ(t) = C(ϕ 3 (t) + fT2 ), t ∈ [0, T ], ϕ(0) = ϕ0 = y0 2 .

(1.63)

Since equation (1.63) has a local solution only, there is T ∗ = T ∗ (y0 , f ) such that 1

yN (t) ≤ ϕ 2 (t) ≤ C, ∀ t ∈ [0, T ∗ ], and so, yN (t) ≤ N, ∀ t ∈ [0, T ∗ ] for N sufficiently large. Assume now that f ≡ 0. Solving equation (1.63) yields  yN (t) ≤ ϕ(t) ≤ e 2

−αν0 t

C 1 + α (e−2αν0 t − 1) ν0 ϕ02

− 1 2

, ∀ t ∈ [0, T ],

where C > 0 is independent of N and ϕ0 . If Cϕ02 ≤ αν0 , that is, if y0  ≤

 αν  1 0

C

2

,

(1.64)

then yN 2 ≤ C1 , ∀ t ∈ [0, T ], and so, for N large enough, yN = y is a global solution to (1.57). If condition (1.64) does not hold, then as seen earlier yN (t) ≤ C1 on some interval [0, T ] and, for N large, yN = y satisfies (1.55). This completes the proof. We have also (see [26], p. 260) Theorem 1.7. Let y0 ∈ H, f ∈ L2 (0, T ; H ), T > 0 and d = 2. Then there is a unique solution y ∈ C([0, T ]; V ) ∩ Cw ([0, T ]; H ) ∩ L2 (0, T ; V ), t 1/2 y ∈ L2 (0, T ; D(A)) ∩ L∞ (0, T ; V ), 2 dy dy ∈ L2 (0, T ; H ), ∈ L 1+ε (0, T ; V ) t 1/2 dt dt to equation (1.57), i.e., dy (t) + ν0 Ay(t) + By(t) = f (t), a.e. t ∈ (0, T ) dt y(0) = y0 . If y0 ∈ V , then y ∈ L∞ (0, T ; V ) ∩ L2 (0, T ; D(A)).

(1.65)

1.5 Infinite Dimensional Linear Control Systems

19

For N → ∞, it turns out that {yN } is weakly convergent in L2 (0, T ; V ) ∩ 4 W 1, 3 (0, T ; V ) to a weak solution y to equation (1.55) (see [26], p. 264). Due to its low regularity, one cannot prove its uniqueness except on the interval [0, T ∗ ] mentioned above, or if y0  is sufficiently small. From the above construction of the solution y, it follows, however, that it coincides globally if d = 2 or for y0  small, and locally if d = 3 with a smooth solution yN to equation (1.65). Taking into account the Leray estimate (1.60), we have, for all N ∈ N,  ν0 m{t ∈ [0, T ]; yN (t) ≥ N } ≤ N −2 |x|2 +

T

|f (t)|2 ds .

0

Hence in 3D the weak solution y satisfies everywhere equation (1.57), excepting a measurable subset E0 ⊂ (0, T ) of an arbitrary small Lebesgue measure. Remark 1.1. Theorem 1.6 extends to more general equations of the form ∂y − ν0 Δy + (y · ∇)a1 + (a2 · ∇)y + (y · ∇)y = ∇p, ∂t ∇ · y = 0, y = 0 on [0, T ] × ∂O, where ai ∈ C 2 (O), i = 1, 2. The details are omitted.

1.5 Infinite Dimensional Linear Control Systems Let H be a real Hilbert space with scalar product (·, ·) and norm denoted | · |H . Consider the Cauchy problem dy (t) + A(t)y(t) = 0, t ∈ (0, T ), dt y(0) = y0 ,

(1.66)

where {A(t); t ∈ [0, T ]} is a family of closed and densely defined operators form H to itself. We say that the Cauchy problem (1.66) is well posed if there exists a function S(t, s) : Δ = {0 ≤ s ≤ t ≤ T } → L(H, H ) such that 1◦ 2◦ 3◦ 4◦

For each y0 ∈ H , the function (t, s) → S(t, s)y0 is continuous on Δ. S(s, s) = I, ∀ s ∈ [0, T ]. S(t, s)S(s, τ ) = S(t, τ ) for all 0 ≤ τ ≤ s ≤ t ≤ T . For each s ∈ [0, T ], there is a densely linear space E(s) ⊂ H such that, for each y0 ∈ E(s), the function t → S(t, s)y0 is continuously differentiable on [S, T ] and

20

1 Preliminaries

d S(t, s)y0 = A(t)S(t, s)y0 , ∀ t ∈ [S, T ], dt S(t, s)L(H,H ) ≤ C, ∀ (s, t) ∈ Δ.

(1.67) (1.68)

If conditions 1◦ –4◦ hold, we say that {A(t); t ∈ [0, T ]} generates the evolution S(t, s). If problem (1.66) is well posed and f ∈ L1 (0, T ; H ), then by mild solution to the nonhomogeneous Cauchy problem dy (t) + A(t)y(t) = f (t), t ∈ (0, T ), dt y(0) = y0 , we mean the continuous function y : [0, T ] → H given by  t y(t) = S(t, 0)y0 + S(t, s)f (s)ds, ∀ t ∈ [0, T ].

(1.69)

(1.70)

0

Examples (i) If A(t) ≡ A is independent of t, then problem (1.66) is well posed if and only if −A is the infinitesimal generator of a C0 -semigroup {S(t) = e−tA } of linear continuous operators on H . By the Hille–Yosida theorem (see, e.g., [104]) this happens if there are M, ω ∈ R, such that (λI + A)−n L(H,H ) ≤ M(λ − ω)−n , ∀ λ > ω, n = 1, 2, . . .

(1.71)

(ii) Let V be a Hilbert space such that V ⊂ H with dense and continuous embedding and let V be the dual space of V with the space pivot H , that is V ⊂ H ⊂ V with dense and continuous embeddings. Let A : [0, T ] → L(V , V ) be such that, for each y0 ∈ V , the function t → A(t)y0 is V -valued measurable on (0, T ) and A(t)y0 V ≤ Cy0 V , ∀ t ∈ (0, T ), y0 ∈ V , 2 V (A(t)y, y)V + α1 |y|H

≥

α2 y2V ,

∀ y ∈ V,

(1.72) (1.73)

where α2 > 0 and α1 ∈ R. Here V (·, ·)V is the duality pairing between V and V

which coincides with the scalar product (·, ·) of H on H × H . Then AH (t)u = A(t)u ∩ H, ∀ u ∈ V , generates an evolution on H . Moreover, we have (see, e.g., [26], p. 177) Proposition 1.1. For each y0 ∈ H and f ∈ L2 (0, T ; V ), the Cauchy problem dy + A(t)y = f (t), t ∈ (0, T ), dt y(0) = y0 ,

(1.74)

1.5 Infinite Dimensional Linear Control Systems

21

has a unique strong solution y ∈ L2 (0, T ; V ) ∩ C([0, T ]; H ),

dy ∈ L2 (0, T ; V ). dt

(1.75)

This means that y : [0, T ] → V is absolutely continuous and (1.74) holds a.e. on (0, T ). (See [26], p. 25.) This result applies neatly to the liner parabolic equation yt (x, t) − Δy(x, t) + b(x, t) · ∇y(x, t) + a(x, t)y(x, t) = f0 (x, t), ∀ (x, t) ∈ Q = O × (0, T ), ∂y (x, t) + α2 y(x, t) = 0, ∀ (x, t) ∈ Σ = ∂O × (0, T ), α1 ∂ν y(x, 0) = y0 (x), x ∈ O,

(1.76)

d where yt = ∂y ∂t , O is an open, bounded set of R with smooth boundary ∂O, a, b ∈ ∞ 2 L (Q), f0 ∈ L (Q) and α1 , α2 are as in (1.10). We take H = L2 (O), V = H 1 (O) (or V = H01 (O) if α1 = 0) and A(t) : V → V defined by  (∇y(x) · ∇z(x) + a(x, t)y(x)z(x) + b(x, t)∇y(x) · z(x))dx V (A(t)y, z)V = O α2 + y(x)z(x)dσ. ∂O α1

Then, by Proposition 1.1, we infer that for y0 ∈ L2 (O) there is a unique solution y to (1.76) satisfying y ∈ L2 (0, T ; H 1 (O)) ∩ C([0, T ]; L2 (O)),

dy ∈ L2 (0, T ; (H 1 (O)) ), dt

(1.77)

and, by Theorem 1.4, part (1.19)–(1.20), √ √ t y ∈ L2 (0, T ; H 2 (O)), t yt ∈ L2 (Q).

(1.78)

Consider now the linear control system dy + A(t)y = B(t)u, t ∈ (0, T ), dt y(0) = y0 ,

(1.79)

where A(t) generates the evolution S(t, s) and B : [0, T ] → L(U, H ) is strongly measurable and B(t)L(U,H ) ≤ C < ∞, a.e. t ∈ (0, T ). Here U is a real Hilbert space with the norm  · U .

(1.80)

22

1 Preliminaries

For each u ∈ L1 (0, T ; U ), (1.79) has a unique mild solution y ∈ C([0, T ]; H ) given by  t y(t) = S(t, 0)y0 + S(t, s)B(s)u(s)ds, t ∈ [0, T ]. (1.81) 0

The input function u is called the controller and the corresponding solution y = y u is called the state of the control system (1.79). The control system (1.79) is said to be exact null controllable if, for each y0 ∈ H , there is u ∈ L2 (0, T ; H ) such that y u (T ) = 0 and uL2 (0,T ;U ) ≤ CT |y0 |2H . We associate with (1.79) the dual-backward system dp − A∗ (t)p = 0, t ∈ (0, T ), dt

(1.82)

where A∗ (t) is the adjoint of A(t), that is, (A∗ (t)y, z) = (y, A(t)z), ∀ y ∈ D(A∗ (t)), z ∈ D(A(t)). We have Theorem 1.8. The dual system (1.82) is observable, that is, there is CT > 0 such that  |p(0)|2H

≤ CT 0

T

B ∗ (t)p(t)2U dt

(1.83)

for every mild solution p to (1.82), if and only if the control system (1.79) is exactly null controllable. Proof. Assume that (1.83) holds for each mild solution to (1.82) and consider for fixed y0 ∈ H and ε > 0 the minimization problem  Minimize 0

T

u(t)2U dt +

1 |y(T )|2H 2ε

 subject to (1.79).

(1.84)

By a standard argument, it follows that problem (1.84) has a unique solution (yε , uε ) given by (the maximum principle) dyε + A(t)yε = B(t)uε , t ∈ (0, T ), dt yε (0) = y0 , uε = B ∗ (t)pε , a.e. t ∈ (0, T ), dpε − A∗ (t)pε = 0, t ∈ (0, T ), dt 1 pε (T ) = − yε (T ). ε

(1.85)

(1.86)

1.5 Infinite Dimensional Linear Control Systems

23

By (1.85)–(1.86), we obtain that  T 1 B ∗ (t)pε (t)2U dt + |yε (T )|2H = (y0 , pε (0))H ε 0 and so, by (1.83), it follows that  T 1 |yε (T )|2H ≤ CT |y0 |2H , ∀ ε > 0. uε (t)2U dt + 2ε 0

(1.87)

Hence {uε } is bounded in L2 (0, T ; U ) and |yε (T )|2H ≤ 2εCT |y0 |2H , ∀ ε > 0. On a subsequence, again denoted {ε}, we have uε −→ u∗ weakly in L2 (0, T ; U ), strongly in H, yε (T ) −→ 0 ∗ yε (t) −→ y (t) weakly in H , ∀ t ∈ (0, T ). Clearly, (y ∗ , u∗ ) satisfies (1.79) and y ∗ (T ) = 0, as desired. We note also that, by (1.87), we have u∗ L2 (0,T ;U ) ≤



CT |y0 |H ,

and, therefore, the observability inequality (1.83) implies the uniform exact null controllability of system (1.79). Conversely, if system (1.79) is exactly null controllable, then, for each y0 with |y0 |H ≤ 1, there is ρ > 0 such that 



T

− S(T , 0)y0 ∈ K =

S(T , s)B(s)u(s)ds; uL2 (0,T ;U ) ≤ ρ .

0

(1.88)

Hence, for each ξ ∈ H ,  −(S(T , 0)y0 , ξ ) ≤ sup u

T

0

 (S(T , s)B(s)u(s), ξ )ds, uL2 (0,T ;U ) ≤ ρ .

This yields 

T

−(S(T , 0)y0 , ξ ) ≤ ρ

∗

∗

|B (s)S (T , s)ξ | ds 2

12

, ∀ ξ ∈ H,

0

and so |S ∗ (T , 0)ξ |2 ≤ ρ 2



T

|B ∗ (s)S ∗ (T , s)ξ |2 ds, ∀ ξ ∈ H,

0

which implies (1.83) with p(0) = S ∗ (T , 0)ξ.

24

1 Preliminaries

We note that, if (B ∗ (t))−1 ∈ L(H, U ), ∀ t ∈ [0, T ], and the function t → is in L2 (0, T ; L(H, H )), then (1.83) trivially holds because

(B ∗ (t))−1

 |p(0)|2H ≤ C

T 0

|p(t)|2H dt.

Inequality (1.83) is also called the observability property of the dual backward system (1.79) and, in specific control systems governed by linear partial differential equations and, in particular, for linear parabolic systems of the form (1.9), it is implied by a Carleman-type inequality to be discussed later on. For instance, if H = L2 (O), U = L2 (O), A = −Δ, D(A) = H01 (O) ∩ H 2 (O) and Bu = 1ω u, where ω is an open subset of O and 1ω is its characteristic function, system (1.79) reduces to the controlled heat equation ∂y − Δy = 1ω u in (0, T ) × O, ∂t y=0 on (0, T ) × ∂O, y(x, 0) = y0 (x), x ∈ O,

(1.89)

and the observability inequality (1.84) is  |p(0)|22

T

≤ CT 0

 p2 (x, t)dx, dt, ω

for all the solutions p to the backward equation ∂p + Δp = 0 in (0, T ) × O, ∂t p=0 on (0, T ) × ∂O. If H = Rn , U = Rm and A(t) ≡ A, B(t) ≡ B, where A and B are time-independent n × n and, respectively, n × m matrices, then the observability condition (1.83) reduces to Kalman’s rank condition rankB, AB, . . . , An−1 B = n,

(1.90)

which is equivalent with the exact null controllability of the finite-dimensional system (1.79). Indeed, in this case, the exact null controllability is equivalent with the observability condition ∗

B ∗ e−tA p0 = 0, ∀ t ∈ [0, T ] ⇒ p0 = 0, which, in turn, is equivalent with (1.90).

1.5 Infinite Dimensional Linear Control Systems

25

By (1.83) and (1.90), we see that the exact controllability property is sensitive (unstable) with respect to the structure of system (1.79) and this is the reason that it has a limited impact in automatic system theory. A simple example is just system (1.89), which can be approximated by the hyperbolic equation ∂y ∂ 2y − ε 2 − Δy = 1ω u in (0, T ) × O, ∂t ∂t ∂y y = 0 on (0, T ) × O, y(x, 0) = y0 (x), (x, 0) = y1 (x). ∂t However, due to the finite speed of propagation, the latter is not exactly controllable. From this perspective, a more convenient concept is that of stabilization. System (1.79) defined on (0, ∞) is said to be stabilizable if there is a controller u : [0, ∞) → U such that the corresponding solution y satisfies |y(t)|H ≤ C exp(−γ t)|y0 |H , ∀ t ≥ 0, y0 ∈ H,

(1.91)

for some positive constants C and γ . Such a controller u is called a stabilizing open loop controller. System (1.79) is said to be feedback stabilizable if there is a mapping Φ : [0, ∞) × H → U called feedback controller, such that the solution y to the closed loop system dy + A(t)y + BΦ(t, y) = 0, t ≥ 0, dt y(0) = y0 , exists and satisfies (1.91). These definitions extend mutatis-mutandis to nonlinear control systems of the form (1.79) for y0 in a neighborhood of the origin. It should be mentioned that, if A, B are time-independent, then the exact controllability or, more generally, the existence of a stabilizing open loop controller u ∈ L2 (0, T ; U ) implies the feedback stabilization by a special device involving algebraic infinite dimensional Riccati equations (see, e.g., [19], p. 210 and [51]). More precisely, the stabilizing feedback controller u in this case is given by u = −B ∗ Ry, where R ∈ L(H, H ), R = R ∗ , R ≥ 0 is the solution to the algebraic Riccati equation A∗ R + RA + RBB ∗ R = C ∗ C, where C is a linear densely defined closed operator in H such that |Cy|2H ≥ γ0 |y|2H , ∀ y ∈ D(C).

(1.92)

26

1 Preliminaries

The solution R to (1.92) is given by (Ry0 , y0 ) =

 1 inf 2

 0

∞

dy + Ay = Bu, dt  t ≥ 0, y(0) = y0 .

(|C(y(t)|2H + u(t)2U )dt;

(1.93)

Remark 1.2. We refer to the survey [109] and to the books [51, 116] for an abstract theory of controllability of linear differential systems in Banach spaces and applications to partial differential equations of parabolic and hyperbolic type.

Chapter 2

The Carleman Inequality for Linear Parabolic Equations

This chapter is concerned with the Carleman estimates for the backward linear parabolic equations on smooth and bounded domains of Rd which implies observability that, as seen earlier, is the main tool to investigate the exact controllability of the forward parabolic controlled system.

2.1 The Carleman and Observability Inequality Consider here the controlled linear parabolic equation yt (x, t) − Δy(x, t) + b(x, t) · ∇y(x, t) + a(x, t)y(x, t) = m(x)u(x, t) + F (x, t), ∀ (x, t) ∈ Q = O×(0, T ), ∂y α1 (x, t) + α2 y(x, t) = 0, ∀ (x, t) ∈ Σ = ∂O×(0, T ), ∂ν y(x, 0) = y0 (x), x ∈ O,

(2.1)

where O is an open, bounded set of Rd with smooth boundary ∂O (of class C 2 , for instance), yt = ∂y ∂t , and m is the characteristic function of an open set ω⊂O. ∂ Here ∂ν is the outward normal derivative, α1 ∈ C(∂O), α1 ≥ 0 is constant, α2 ∈ C 1 (∂O), α2 ≥ 0, α1 + α2 > 0. Moreover, a ∈ L∞ (Q), b ∈ C 1 (Q; Rd ) and F ∈ L2 (Q) are given functions while u ∈ L2 (Q) is a control input. This is a parabolic equation of the form (1.76) which, as seen earlier, is well posed for y0 ∈ L2 (O) and u ∈ L2 (Q).

© Springer International Publishing AG, part of Springer Nature 2018 V. Barbu, Controllability and Stabilization of Parabolic Equations, Progress in Nonlinear Differential Equations and Their Applications 90, https://doi.org/10.1007/978-3-319-76666-9_2

27

28

2 The Carleman Inequality for Linear Parabolic Equations

We shall associate with (2.1) the backward dual parabolic problem pt (x, t) + Δp(x, t) + divx (b(x, t)p(x, t))− a(x, t)p(x, t)= g(x, t), (x, t) ∈ Q, α1

∂p + (α2 + α1 b · ν)p = 0 on Σ, p(T ) ∈ L2 (O), ∂ν

(2.2)

where g ∈ L2 (Q) is a given function, ∂p ∂ν = ∇p · ν and ν is the outward normal to ∂O. Recalling (1.77)–(1.78), it follows that, for p(T ) ∈ L2 (O), the solution p to (2.2) satisfies (see Section 1.3) p ∈ C([0, T ]; L2 (O)) ∩ L2 (0, T ; H 1 (O)), (T − t)1/2 pt ∈ L2 (Q) and (T − t)1/2 p ∈ L2 (0, T ; H 2 (O)). If we denote by A(t) : D(A(t)) ⊂ L2 (O) → L2 (O) the linear operator A(t)y = −Δy + b(t) · ∇y + a(t)y with the domain 

 ∂y D(A(t)) = y ∈ H (O); α1 + α2 y = 0 on ∂O , ∂ν 2

then equation (2.1) can be written as dy + A(t)y = F, t ∈ (0, T ), dt y(0) = y0 ,

(2.3)

and (2.2) is the backward equation dp − A∗ (t)p = g, t ∈ (0, T ), dt

(2.4)

where A∗ (t) is the dual operator A∗ (t)p = −Δp − div(b(t)p) + a(t)p, p ∈ D(A(t)), with the domain   ∂p D(A∗ (t)) = p ∈ H 2 (O0 ); α1 + (α2 + α1 b(t).ν)p = 0 on ∂O0 . ∂ν Taking into account the relevance of the observability of the dual system (2.4) for the exact null controllability of (2.3), this section is entirely devoted to the proof of the observability inequality (1.79) in this special case. This will be derived from a more precise estimate for solutions to equation (2.2), known in literature as the Carleman inequality.

2.1 The Carleman and Observability Inequality

29

Let ω be an open subset of O and let ω0 be an open subset of ω such that ω0 ⊂ω. An essential ingredient of the proof is the following technical lemma. Lemma 2.1. There is ψ ∈ C 2 (O) such that ψ(x) > 0, ∀ x ∈ O; ψ = 0 on ∂O |∇ψ(x)| > 0, ∀ x ∈ O0 = O\ω0 .

(2.5)

Proof. The existence of such a function ψ is easily seen in 1 − D or for rectangular domains O, but we shall give below a simple proof in the case where O is an open convex set of Rd and refer to [76] and [116], p. 439, for the general case. We consider the solution ψ ∈ C 2 (O) to the Dirichlet problem Δψ = −1 in O, ψ =0 on ∂O, ∂ψ By the maximum principle, √we know that ψ > 0 on O and ∂ν < 0 on ∂O. We recall (see [91]) that u = ψ is concave and the Hessian matrix of u is negative definite. This means that every critical point (x0 , y0 ) ∈ O is not degenerate and so, by the Morse theory, u has a unique local maximum (x0 , y0 ) in O. If (x0 , y0 ) ∈ ω0 , then clearly the function ψ satisfies conditions (2.5). Otherwise, we take

(x, y) = ψ(F −1 (x, y)), ∀ (x, y) ∈ O, ψ where F is a diffeomorphism of O into itself such that F (x0 , y0 ) ∈ ω0 . It is clear (x, y) = 0} ⊂ ω0 , and so ψ satisfies (2.5). As regards the that {(x, y) ∈ O; ∇ ψ mapping F , it can be constructed as follows. We fix (x0∗ , y0∗ ) ∈ ω0 and consider the operator A0 : I → Rd A0 (s(x0∗ , y0∗ ) + (1 − s)(x0 , y0 )) = (x0 − x0∗ , y0 − y0∗ ), ∀ s ∈ (0, 1) defined on the segment I = [s(x0∗ , y0∗ ) + (1 − s)(x0 , y0 ); s ∈ [0, 1] ⊂ O. The operator A0 has a maximal monotone extension A : D(A) ⊂ O → Rd , such that A0 ⊂ A and D(A) ⊂ O (see [17], p. 41, Theorem 1.4). Let S(t) : D(A) → O be the semigroup generated by −A, that is,

30

2 The Carleman Inequality for Linear Parabolic Equations

d (S(t)u0 ) + A(S(t)u0 )  0, a.e. t > 0, dt S(0)u0 = u0 . Since A0 ⊂ A, we have S(t)(x0 , y0 ) = t (x0∗ , y0∗ ) + (1 − t)(x0 , y0 ), ∀ t ∈ [0, 1]. Now, we consider a smooth invertible approximation Aλ ∈ C 1 (O; R2 ) of A such that (Aλ )−1 ∈ Lip(R2 , Rd ) and, for each α > 0, (I + αAλ )−1 → (I + αA)−1 as λ → 0. (Such an operator Aλ might be λI + Aλ ∗ ρλ , where Aλ is the Yosida approximation of A and ρλ is a mollifier.) Then, by the Kato-Trotter theorem (see [26], p. 168) for λ → 0, Sλ (t)u0 → S(t)u0 , ∀ u0 ∈ O, t ≥ 0, where Sλ (t) is the semigroup generated by −Aλ . Clearly, Sλ (1) is a diffeomorphism of O into itself and, for λ small enough, Sλ (1)(x0 , y0 ) ∈ ω0 . Hence, F = Sλ (1) is the desired mapping. For λ > 0 and a function ψ which satisfies (2.5) we set α(x, t) =

eλψ(x) − e2λψC(O) eλψ(x) , ϕ(x, t) = , ∀ (x, t) ∈ Q, (t (T − t))k (t (T − t))k

(2.6)

where k ≥ 1 is a natural number. However, for the purposes of this chapter one might choose k = 1. Theorem 2.1 below (the Carleman inequality) is the main result of this section. Theorem 2.1. Let a ∈ L∞ (Q) and b ∈ C 1 (Q; Rd ). Then there are λ0 > 0 and s0 > 0 such that, for λ ≥ λ0 and s ≥ s0 ,  Q

e2sα (s 3 λ4 ϕ 3 p2 + sλ2 ϕ|∇p|2 + (sϕ)−1 (pt2 + |Δp|2 ))dx dt

  e2sα ϕ 3 p2 dx dt + e2sα g 2 dx dt ≤ Cλ s 3 λ 4 Qω

(2.7)

Q

for all solutions p to equation (2.3). Here Qω = ω×(0, T ) and C is a positive constant independent of s, λ and p. It should be observed that this inequality is symmetric with respect to 0 and T . Thus, by the change of variable t → T − t, we see that (2.1) remains true and has the same form for the forward equation (2.1). As mentioned earlier, Theorem 2.1 is motivated by the exact controllability of equation (2.1) but it has, however, an

2.1 The Carleman and Observability Inequality

31

intrinsic interest. We note also that Theorem 2.1 remains true if −Δ is replaced by   the linear elliptic operator L = − ∂x∂ i aij (x, t) ∂x∂ j with smooth coefficients. We recall that the classical generic form for the Carleman inequality on a domain Q ⊂ Rn+1 for a linear differential operator P (D) is τ eτ θ u ≤ Ceτ θ P (D)u, ∀τ > τ0 , u ∈ C0∞ (Q), τ > τ0 . Here θ is a C ∞ function such that |∇θ | > 0 in Q0 and  ·  is a Sobolev space norm. In the present situation θ = α + log ϕ and Q0 = Qω . Theorem 2.1 implies the observability of system (2.1). Corollary 2.1. Under the assumptions of Theorem 2.1, there are λ0 >0, s0 and C, μ, independent of p such that, for λ ≥ λ0 , s ≥ s0 , the following inequality holds 

 p (x, 0)dx ≤ Ce 2

μs

Ω



 e

ϕ p dx dt +

2sα 3 2

Qω

2

(2.8)

g dx dt Q

for all solutions p to equation (2.3). Proof. We multiply equation (2.3) by p and integrate on O. After some calculation involving Green’s formula, we get 1 2



d |p(t)|22 − dt

 O

|∇p(x, t)|2 dx ≥ − C(|p(t)|22 + |g(t)|22 ), a.e. t∈(0, T ),

where | · |2 is the L2 (O) norm. Then, by the Gronwall lemma, we obtain

 t |p(τ )|22 ≤ C |p(t)|22 + |g(θ )|22 dθ , 0 ≤ τ ≤ t. τ

Hence |p(0)|22

  ≤ C γ (t) e2sα(x,t) ϕ 3 (x, t)p2 (x, t)dx + O

T 0

|g(t)|22 dt

,

μs

where γ (t) = sup{e−2sα(x,t) ϕ −3 (x, t); x ∈ O} ≤ Ce (t (T −t))k and μ = 2e2λψC(O) . Now, we integrate the latter inequality on the interval (t1 , t2 ) to get  |p(0)|22

t2

e t1

−

μs ((T −t)t)k

 dt ≤ C

g 2 dx dt ,

 e2sα ϕ 3 p2 dx dt +

Q

Q

for 0 < t1 < t2 < T . Then, using estimate (2.7), we obtain (2.8), where Cλ is suitable chosen. This completes the proof.

32

2 The Carleman Inequality for Linear Parabolic Equations

Proof of Theorem 2.1. We set z = esα p and note that z satisfies the boundary value problem zt + Δz + (λ2 s 2 ϕ 2 |∇ψ|2 − λ2 sϕ|∇ψ|2 )z − (2sλϕ∇ψ − b) · ∇z sα −(a − div b + sλϕ(b · ∇ψ) + (sα

t + λsϕΔψ)z = ge in Q, ∂z ∂ψ α1 + α2 + α1 b · ν − α1 sλϕ z = 0 on Σ, ∂ν ∂ν z(x, 0) = z(x, T ) = 0 in O.

(2.9)

Without no loss of generality, we may assume that z ∈ H 2,1 (Q). We set X(t)z =−2(sλ2 ϕ|∇ψ|2 z + sλϕ∇z · ∇ψ), B(t)z =−Δz − (λ2 s 2 ϕ 2 |∇ψ|2 + sλ2 ϕ|∇ψ|2 )z + sαt z, and rewrite equation (2.9) as zt + X(t)z − B(t)z = esα g + Z(t)z in Q, where Z(t)z = −(−sλϕΔψ + div b − sλϕ(b · ∇ψ) − a)z − b · ∇z. This yields d dt



 (B(t)z)(x, t)z(x, t)dx =

O

((B(t)z)(x, t)zt (x, t)

O

+(B(t)zt )(x, t)z(x, t))dx + (Bt (t)z)(x, t)z(x, t)dx  O = 2 (B(t)z)(x, t)(B(t)z(x, t) − X(t)z(x, t) + Z(t)z(x, t) + esα g) dx

  O ∂zt ∂z z − zt dσ dt. + (Bt z)(x, t)z(x, t)dx − ∂ν ∂ν O ∂O Integrating by parts with respect to t on the surface integral on Σ, we get after some calculations that   2 (B(t)z(x, t))2 dx dt + 2 (B(t)z)(x, t)(Z(t)z + esα g)(x, t)dx dt + 2Y Q Q   ∂z ≤ − (Bt (t)z)(x, t)z(x, t)dx dt − 2 zt dσ dt, ∂ν Q Σ (2.10) where  Y = −2 (sλ2 ϕ|∇ψ|2 z+sλϕ∇z · ∇ψ) (2.11) Q (Δz+(λ2 s 2 ϕ 2 |∇ψ|2 +sλ2 ϕ|∇ψ|2 − sαt )zdx dt.

2.1 The Carleman and Observability Inequality

33

We note that 

 ∂z Bt (t)z z dx dt + 2 zt dσ dt ∂ν Q Σ  = − z2 (2λ2 s 2 ϕϕt |∇ψ|2 + sλ2 ϕt |∇ψ|2 − sαtt )dx dt Q

 ∂ψ −1 dσ dt. zt z α2 + α1 b · ν − α1 sλϕ − 2α1 ∂ν Σ

We set γ (λ) = e2λψC(O) . We have −1

|αt | ≤ γ (λ)|ϕt | ≤ Cγ (λ)ϕ 1+k , −1 |αtt | ≤ γ (λ)|ϕtt | ≤ Cγ (λ)ϕ 1+2k . Taking into account the boundary condition in (2.9), we get after some integration by parts that       zt ∂z dσ dt  ≤ C (1 + sλϕ 2 )z2 dσ dt.   Σ ∂ν Σ On the other hand, by the trace theorem and the interpolation inequality, we have  1 1 −ε +ε z2 dσ dt ≤ Cz2 ≤ CzL2 2 (0,T ;L2 (O)) zL2 2 (0,T ;H 1 (O)) . 1 +ε L2 (0,T ;H 2

Σ

This yields 

(O))

 ϕ z dσ dt ≤ C

Σ



 ϕ z dxdt +

2 2

3 2

Q

2

ϕ|∇z| dxdt

(2.12)

Q

and, therefore, for s ≥ 1, we have       (Bt (t)z)z dx dt + 2 zt ∂z dσ dt    ∂ν Q Σ ≤ C(λ2 s 2 + sγ (λ)) ϕ 3 z2 dxdt Q  +Csλ ϕ|∇z|2 dx dt. Q

(Here and everywhere in the sequel, C is a positive constant independent of s, λ, z and g.) Note also that      2 (B(t)z)(Z(t)z + esα g) dx dt    Q

≤ B(t)z2L2 (Q) + esα g2L2 (Q) +C(s 2 λ2 ϕz2L2 (Q) + ∇z2L2 (Q) )

34

2 The Carleman Inequality for Linear Parabolic Equations

and so, by (2.10), we see that 



Y ≤ C(s 2 λ2 + sγ (λ))

ϕ 3 z2 dx dt + C Q

(sλϕ|∇z|2 + e2sα g 2 )dx dt,

(2.13)

Q

while, by (2.11), we obtain that  Y ≥ −2s

(λ2 ϕ|∇ψ|2 z + λϕ∇ψ · ∇z) Q

2 2 2 2 2 2 (Δz+(λ  s ϕ |∇ψ| +sλ ϕ|∇ψ| )z)dx dt +s 2 λ z2 ϕαt ∇ψ · νdσ dt − CD1 (s, λ, z)

(2.14)

Σ

for s, λ ≥ λ0 sufficiently large, where 

 D1 (s, λ, z) = s γ (λ)λ 2

ϕ z dx dt + sλ

2

3 2

Q

Since ψ = 0 on ∂Ω and ψ ≥ 0 in Ω, it follows that we see that

ϕ|∇z|2 dxdt.

(2.15)

Q ∂ψ ∂ν

≤ 0 on Σ and so, by (2.9),

∂z z ≤ −b · νz2 on Σ. ∂ν

(2.16)

Then, by Green’s formula, we obtain that   2 − zΔzϕ|∇ψ| dx dt = ϕ|∇z|2 |∇ψ|2 dx dt Q Q   ∂z dσ dt + z∇z · ∇(ϕ|∇ψ|2 )dx dt − ϕ|∇ψ|2 z ∂ν Σ Q  3 2 ≥ ϕ|∇ψ| |∇z|2 dx dt − C(λ2 + 1) ϕ 2 z2 dx dt. 4 Q Q

(2.17)

Here, to estimate the surface integral, we have used the boundary conditions in (2.9) along with the above interpolation inequality given before which, taking into account (2.12), (2.16), allows to estimate the above integral by    1 2 2 2 2 − ϕ|∇ψ| z dσ dt ≥ −C ϕ |z| dxdt − ϕ|∇ψ|2 |∇z|2 dt. 8 Q Σ Q Note also that

2.1 The Carleman and Observability Inequality

35

 −2sλ

ϕ∇z · ∇ψ(λ2 s 2 ϕ 2 |∇ψ|2 + sλ2 ϕ|∇ψ|2 )z dx dt Q  = −sλ ∇z2 · ∇ψ(λ2 s 2 ϕ 3 + sλ2 ϕ 2 )|∇ψ|2 dx dt Q = −sλ div (z2 ∇ψ(λ2 s 2 ϕ 3 + sλ2 ϕ 2 )|∇ψ|2 )dx dt Q +sλ z2 div (∇ψ(λ2 s 2 ϕ 3 + sλ2 ϕ 2 )|∇ψ|2 )dx dt Q ∂ψ dσ dt ≥ −sλ z2 (λ2 s 2 ϕ 3 + sλ2 ϕ 2 )|∇ψ|2 ∂ν  Σ + (3s 3 λ4 ϕ 3 + 2s 2 λ4 ϕ 2 )|∇ψ|4 z2 dx dt Q − C (λ3 s 3 ϕ 3 + s 2 λ3 ϕ 2 )z2 dx dt.

(2.18)

Q

By (2.14), (2.15), (2.16), and (2.17), we get  3 Y ≥ (s 3 λ4 ϕ 3 |∇ψ|4 z2 + sλ2 ϕ|∇ψ|2 |∇z|2 )dxdt − I0 (z) 2 Q   (2.19) 2 2 +λs z ϕαt ∇ψ · νdσ dt − 2sλ ϕ(∇z · ∇ψ)Δz dx dt − CD(s, λ, z), Σ

where

Q

 ∂ψ I0 (z) = sλ z2 (λ2 s 2 ϕ 3 + sλ2 ϕ 2 )|∇ψ|2 dσ dt ≤ 0 ∂ν  Σ D(s, λ, z) = ((s 3 λ3 ϕ 3 + s 2 λ2 γ (λ)ϕ 2 )z2 + sλϕ|∇z|2 )dx dt. Q

Next, by the Green formula, it follows after some calculation that 

 ϕ(∇z · ∇ψ)Δz dx dt = −2sλ ϕ(∇ψ · ∇z)(∇z · ν)dσ dt Q Σ  +2sλ ∇z · ∇(ϕ∇z · ∇ψ)dx dt = −2sλ ϕ(∇ψ · ∇z)(∇z · ν)dσ dt Σ  Q  +sλ ϕ|∇z|2 (∇ψ · ν)dσ dt − sλ2 ϕ|∇ψ|2 |∇z|2 dx dt ⎛ Q ⎛Σ ⎞⎞  d

+ ⎝2sλ2 ϕ(∇z · ∇ψ)2 − sλϕ ⎝|∇z|2 Δψ − zxi zxj ψxi xj ⎠⎠ dx dt,

−2sλ

Q

i,j =1

where ν is the outward normal to ∂O. Since, as seen earlier, ∂ψ ∂ν ≤ 0 on ∂O and ψ = 0 on ∂O, we have ν=−

∇ψ on ∂O, (∇ψ · ∇z)(∇z · ν) = −(∇ψ · ∇z)2 |∇ψ|−1 on ∂O. |∇ψ|

36

2 The Carleman Inequality for Linear Parabolic Equations

We have, therefore,   −2sλ ϕ∇z · ∇ψΔz dx dt ≥ 2sλ ϕ(∇ψ · ∇z)2 |∇ψ|−1 dσ dt Q  Σ 2 2 −sλ ϕ|∇z| |∇ψ|dσ dt − sλ ϕ|∇ψ|2 |∇z|2 dx dt − CD(s, λ, z). Σ

Q

Then, by (2.13), (2.14) and (2.18), we obtain that   s 3 λ4 ϕ 3 |∇ψ|4 z2 dx dt + sλ2 ϕ|∇ψ|2 |∇z|2 dx dt Q Q

 2 2sα ≤ C D(s, λ, z) + g e dx dt − I (z) + I0 (z),

(2.20)

Q

where  I (z) = sλ

(2(∇ψ · ∇z)2 − |∇z|2 |∇ψ|2 )ϕ|∇ψ|−1 + sz2 ϕαt ∇ψ · νdσ dt.

Σ ∇z We note that, if α1 = 0, then ν = ± |∇z| on Σ and so |∇z · ∇ψ| = |∇z||∇ψ| on Σ. This implies that I (z) ≥ 0 and, since I0 (z) ≤ 0, we infer that



 ϕ |∇ψ| z dx dt + sλ ϕ|∇ψ|2 |∇z|2 dx dt Q

 2sα 2 ≤ C D(s, λ, z) + e g dx dt . 3

s 3 λ4 Q

4 2

2

(2.21)

Q

We shall prove now that the latter inequality still remains true if α1 > 0. To this purpose, we set z¯ = es α¯ p, where α¯ =

e−λψ e−λψ − e2λψC(Ω) , ϕ= · k (t (T − t)) (t (T − t))k

Clearly, α¯ ≤ α, ϕ ≤ ϕ in Q and α¯ = α, ϕ = ϕ on Σ. We note that z¯ satisfies equation (2.9), where ϕ is replaced by ϕ and λ by −λ. We have, therefore, z¯ t + X(t)¯z − B(t)¯z = esα g + Z(t)¯z in Q, where X(t)¯z = −2(sλ2 ϕ|∇ψ|2 − sλϕ∇ z¯ · ∇ψ), B(t)¯z = −Δ¯z − (λ2 s 2 ϕ 2 |∇ψ|2 + sλ2 ϕ|∇ψ|2 )¯z + sα t z¯ , Z(t)¯z = (a − sλϕΔψ − div b + sλϕ(b · ∇ψ)¯z − b · ∇ z¯ . We obtain as above (see (2.13), (2.14))  −2s (λ2 ϕ|∇ψ|2 z¯ −λϕ∇ψ·∇ z¯ )(Δ¯z+(λ2 s 2 ϕ 2 |∇ψ|2 +sλ2 ϕ|∇ψ|2 )¯z)dx dt Q

  2s α¯ 2 2 2 ≤ C D(s, λ, z¯ ) + e g dx dt + s λ z¯ ϕα t ∇ψ · νdσ dt Q

Σ

2.1 The Carleman and Observability Inequality

37

because ϕ ≤ ϕ and α ≤ α. Arguing as above, we obtain also that (see (2.16))    3 − z¯ Δ¯zϕ|∇ψ|2 dx dt ≥ ϕ|∇ψ|2 |∇ z¯ |2 dx dt − C(λ2 + 1) ϕ 2 z¯ 2 dx dt 4 Q Q Q (2.22) and (see (2.17))  2sλ ϕ∇ z¯ · ∇ψ(λ2 s 2 ϕ 2 |∇ψ|2 + sλ2 ϕ|∇ψ|2 )¯zdx dt Q  (2.23) ≥ (3s 3 λ4 ϕ 3 + 2s 2 λ4 ϕ 2 )¯z2 dx dt + I0 (¯z) − CD(s, λ, z¯ ). Q

(Here we must take into account that ∇ϕ = −λ∇ψ · ϕ.) Arguing as in the previous case, we obtain also that 

 2sλ

ϕ(∇ z¯ · ∇ψ)Δ¯zdx dt ≥ −2sλ ϕ(∇ψ · ∇ z¯ )2 |∇ψ|−1 dσ dt Σ  2 2 +sλ ϕ|∇ z¯ | |∇ψ|dσ dt + sλ ϕ|∇ψ|2 |∇z|2 dx dt − CD(s, λ, z¯ ).

Q

Σ

Q

Substituting the latter inequality along with (2.22), (2.23) into (2.20), we obtain  (s 3 λ4 ϕ 3 |∇ψ|4 z¯ 2 + sλ2 ϕ|∇ψ|2 |∇ z¯ |2 )dx dt

 ≤ C D(s, λ, z¯ ) + e2sα g 2 dx dt + I (¯z) − I0 (¯z) .

Q

(2.24)

Q

By (2.19) and (2.24), we see that  (s 3 λ4 ϕ 3 |∇ψ|4 z2 + sλ2 ϕ|∇ψ|2 |∇z|2 )dx dt  + (s 3 λ4 ϕ 3 |∇ψ|4 z¯ 2 + sλ2 ϕ|∇ψ|2 |∇ z¯ |2 )dx dt Q

 ≤ C D(s, λ, z) + e2sα g 2 dx dt ,

Q

Q ¯ ¯ because ϕ ≤ ϕ and z¯ = es(α−α) z, ∇ z¯ = ∇zes(α−α) + s∇(α¯ − α)z. Indeed, the latter imply that z = z¯ , ∇z = ∇ z¯ on Σ

|∇ z¯ | ≤ |∇z| + sλ|∇ψ|ϕ|z| in Q and we have, therefore, I0 (z) = I0 (¯z), I (z) = I (¯z), D(s, λ, z¯ ) ≤ CD(s, λ, z).

(2.25)

38

2 The Carleman Inequality for Linear Parabolic Equations

Thus, by (2.25), we get the desired estimate (2.21), that is,  (s 3 λ4 ϕ 3 |∇ψ|4 z2 + sλ2 ϕ|∇ψ|2 |∇z|2 )dx dt

 ≤ C D(s, λ, z) + e2sα g 2 dx dt .

Q

(2.26)

Q

On the other hand, by (2.9), we see that (sλ)−1



 (zt + Δz)2 ϕ −1 dx dt ≤ C D(s, λ, z) + g 2 e2sα dx dt .



Q

Q

This yields (sλ)−1



(zt + |Δz|2 )ϕ −1 dx dt

  1 2 2sα −1 zt2 ϕ −1 dx dt ≤ C D(s, λ, z) + g e dx dt + (sλ) 2 Q Q Q

because z(0) = z(T ) = 0,    ∂z zt dσ dt zt Δzϕ −1 dxdt = − ∇z · ∇(zt ϕ −1 )dx dt + ϕ −1 ∂ν Q Q Σ while, by the boundary conditions in (2.9),



 ∂ψ α2 ∂z 1 sλ zt dσ dt = − + b·ν ϕ −1 (z)2t dσ dt 2 Σ ∂ν

α1 Σ  ∂ν α1 α2 2 −1 1 + b·ν ϕ z dσ dt ≤ C (z2 + |∇z|2 )dx dt = 2α1 Σ α2 Q t



ϕ −1

(2.27)

by the trace theorem. We obtain, therefore, that for s, λ ≥ λ0 (sλ)

−1

 (zt2 Q

+ |Δz| )ϕ 2

−1



 2 2sα dx dt ≤ C D(s, λ, z) + g e dx dt

(2.28)

Q

and recall also that, by (2.26), we have that  s 3 λ4

 ϕ 3 |∇ψ|4 z2 dx dt + sλ2 ϕ|∇ψ|2 |∇z|2 dx dt Q Q

 ≤ C D(s, λ, z) + e2sα g 2 dx dt . Q

(2.29)

2.1 The Carleman and Observability Inequality

39

Recalling the definition of D(s, λ, z) and the fact that |∇ψ(x)| ≥ ρ > 0, ∀ x ∈ O0 = O\ω0 , ω0 ⊂⊂ω, it follows by (2.29) that   ϕ 3 z2 dx dt + sλ2 ρ 2 ϕ|∇z|2 dx dt s 3 λ4 ρ 4 Q Q 0

0    ≤ C (s 2 λ2 γ (λ) + s 3 λ3 ) ϕ 3 z2 dx dt + sλ ϕ|∇z|2 dx dt + e2sα g 2 dx dt , Q

Q

Q

where Q0 = O0 ×(0, T ). Hence there are λ0 > 0 and s0 = s0 (λ) such that, for λ ≥ λ0 and s ≥ s0 (λ), we have  (s 3 λ4 ϕ 3 z2 + sλ2 ϕ|∇z|2 )dx dt Q0    ≤C (s 3 λ4 ϕ 3 z2 + sλϕ|∇z|2 )dx dt + e2sα g 2 dx dt . Qω0

Q

Finally,  (s 3 λ4 ϕ 3 z2 + sλ2 ϕ|∇z|2 )dx dt    3 4 3 2 2 2sα 2 ≤ Cλ (s λ ϕ z + sλϕ|∇z| )dx dt + e g dx dt .

Q

Qω0

(2.30)

Q

Coming back to p, we get  e2sα (s 3 λ4 ϕ 3 p2 + sλ2 ϕ|sλϕp∇ψ + ∇p|2 )dx dt      2sα 3 4 3 2 2 2 2sα 2 s λ ϕ p +sλ ϕ|sλϕp∇ψ +∇p| dx dt + e g dx dt . ≤Cλ e Q

Qω0

Q

(2.31)

We note that, for any δ ∈ (0, 1), we have  e2sα ϕ 2 p∇ψ · ∇pdx dt   ≥ −δ −1 s 3 λ4 e2sα ϕ 3 |∇ψ|2 p2 dxdt − δsλ2 e2sα ϕ|∇p|2 dxdt.

2s 2 λ3

Q

Q

(2.32)

Q

Then, by (2.31), it follows that, for δ −1 ∇ψ2 < 1 and for λ ≥ λ0 large enough, C(O) s ≥ s0 (λ), we have  e2sα (s 3 λ4 ϕ 3 p2 + sλ2 ϕ|∇p|2 )dxdt Q    e2sα (s 3 λ4 ϕ 3 p2 + sλ2 ϕ|∇p|2 )dxdt + e2sα g 2 dxdt . ≤ Cλ Qω0

Q

40

2 The Carleman Inequality for Linear Parabolic Equations

Next, we choose χ ∈ C0∞ (O) such that χ = 1 in ω0 and χ = 0 in O\ω. If multiply equation (2.3) by χ ϕe2sα p and integrate on Q, we get after some calculation involving Green’s formula that 

 e

2sα



ϕ χ |∇p|2 dx dt ≤ Cλ s 2 λ2

Q

e

ϕ p dx dt + Cλ

2sα 3 2

Qω

e2sα g 2 dx dt. Q

Substituting into (2.32), we get  Q

e2sα (s 3 λ4 ϕ 3 p2 + sλ2 ϕ|∇p|2 )dx dt

  2sα 3 2 2sα 2 3 4 e ϕ p dx dt + e g dx dt . ≤ Cλ s λ Qω

Q

Hence  Q

e2sα (s 3 ϕ 3 p2 + sϕ|∇p|2 )dx dt 

 e2sα ϕ 3 p2 dx dt + e2sα g 2 dx dt ≤ Cλ s 3 Qω

(2.33)

Q

for all λ ≥ λ0 sufficiently large and s ≥ s0 (λ). Finally, by (2.28) and (2.33), we see that s −1



e2sα (pt2 + |Δp|2 )ϕ −1 dx dt 

 2sα 3 3 2 2 2 2sα e (s ϕ p + sϕ|∇p| )dx dt + g e dx dt ≤ Cλ Q

Q  2sα 3 2 2sα 2 3 e ϕ p dx dt + e g dx dt ≤ Cλ s

Q

Q

Q

for λ ≥ λ0 , s ≥ s0 (λ). Along with (2.33), the latter implies (2.3), thereby completing the proof. Remark 2.1. The constant Cλ arising in Carleman’s inequality depends on |a|∞ and |div b|∞ , |b|∞ only. As a function of λ, Cλ ≤ Cγ (λ) for λ ≥ λ0 . Notice also that Theorem 2.1 extends to general second order parabolic equations of the form pt +

n

(aij pxi )xj + div (bp) − ap = g

i,j =1

with smooth coefficients aij , b. It turns out, however, that a similar result remains true if ai,j ∈ W 1,∞ (Q), b ∈ L∞ (0, T ; Lr (O)), r > 2n and −μ a ∈ L∞ (0, T ; Wr1 (Ω) for 0 < μ < 12 and r1 sufficiently large. (See also [21].)

2.1 The Carleman and Observability Inequality

41

Theorem 2.1 remains true for the solutions p ∈ H 2 (O) to the elliptic boundaryvalue problem Δp(x, t) + divx (b(x, t)p(x, t)) + c(x, t) · ∇p(x, t) −a(x, t)p(x, t) = g(x, t), ∀(x, t) ∈ Q, ∂p + (α2 + α1 b · ν)p = 0 on Σ, α1 ∂ν

(2.34)

where t ∈ [0, T ] is a parameter. Namely, one has Theorem 2.2. Under the assumptions of Theorem 2.1, there are λ0 > 0 and s0 such that, for λ ≥ λ0 and s ≥ s0 ,  e2sα (s 3 ϕ 3 p2 + sϕ|∇p|2 + (sϕ)−1 |Δp|2 )dx O 

 (2.35) ≤ Cλ s 3 e2sα ϕ 3 p2 dx + e2sα g 2 dx , a.e. t ∈ (0, T ) O

ω

Proof. We shall argue as in the proof of Theorem 2.1. As a matter of fact, the proof is identic, but we shall forget the terms pt , zt and αt , ϕt from the previous calculation. Namely, we set z = esα p and obtain the elliptic equation (see (2.9)) Δz + (λ2 s 2 ϕ 2 |∇ψ|2 − λ2 sϕ|∇ψ|2 )z −2sλϕ∇ψ · ∇z + (b + c) · ∇z − (a + sλϕ∇ψ · (b + c) −div b − λsϕΔψ)z = gesα ,

(2.36)

that is, X(t)z − B0 (t)z = esα g + Z(t)z,

(2.37)

where X(t)z = −2(sλ2 ϕ 2 |∇ψ|2 z + sλϕ∇z · ∇ψ), B0 (t)z = −Δz − (λ2 s 2 ϕ 2 |∇ψ|2 + sλ2 ϕ|∇ψ|2 )z, Z(t)z = (a − div b − sλϕΔψ + sλϕ(b + c) · ∇ψ)z − (b + c) · ∇z. This yields  O

B0 (t)z(B0 (t)z − X(t)z + Z(t)z + esα g)dx = 0.

For simplicity, we shall assume that α1 = 0. Then the same calculation as in the proof of Theorem 2.1 leads us to (see (2.20))  s 3 λ4

 ϕ |∇ψ| z dx + sλ ϕ|∇ψ|2 |∇z|2 dx O 

2sα 2 ≤ C D(s, λ, z)(t) + e g dx , 3

O

4 2

2

O

(2.38)

42

2 The Carleman Inequality for Linear Parabolic Equations

where  D(s, λ, z)(t) =

O

((s 3 λ3 ϕ 3 + s 2 λ4 ϕ 2 )z2 + sλϕ|∇z|2 )dx, t ∈ (0, T ).

On the other hand, by (2.36), we see that (sλ)−1

 O



 |Δz|2 ϕ −1 dx ≤ C D(s, λ, z)(t) + e2sα g 2 dx . O

Then, by (2.38), we obtain that  s 3 λ4

  ϕ 3 |∇ψ|4 z2 dx + sλ2 ϕ|∇ψ|2 |∇z|2 dx + (sλ)−1 |Δz|2 ϕ −1 dx O O O

 ≤ C D(s, λ, z)(t) + e2sα g 2 dx . O

Then, arguing as in the proof of Theorem 2.1 (see (2.10), (2.29)), one obtains inequality (2.30) on O and, consequently, (2.31), which leads as before to the desired inequality (2.35). This completes the proof.

2.2 Notes on Chapter 2 Theorem 2.1 was established by A.V. Fursikov and O.Yu. Imanuvilov [76]. The proof given here closely follows [20, 22]. As shown in [81], it extends to more general linear second order parabolic equations with nonregular lower order coefficients and the right-hand side in a Sobolev space of negative order. Sharper Carleman estimates for linear parabolic equations were recently obtained by J. Le Rousseau and G. Lebeau [87] and J. Le Rousseau and L. Robbiano [88]. More precisely, the results of [88] refer to a Carleman inequality of the form (2.7) for solutions of the equation yt − div(a(t, x)∇y) = f in (0, T )×O, where a is piecewise smooth in a spatial variable and discontinuous across a smooth interface.

Chapter 3

Exact Controllability of Parabolic Equations

This chapter is concerned with the presentation of some basic results on the exact internal and boundary controllability of linear and semilinear parabolic equations on smooth domains of Rd . The exact controllability of linear stochastic parabolic equations with linear multiplicative Gaussian noise is also briefly studied. The main ingredient to exact controllability is the observability inequality for the dual parabolic equations established in Chapter 2.

3.1 Exact Controllability of Linear Parabolic Equations We come back to the controlled linear parabolic equation (2.1), that is, yt (x, t) − Δy(x, t) + b(x, t) · ∇y(x, t) + a(x, t)y(x, t), = m(x)u(x, t) + F (x, t), ∀ (x, t) ∈ Q = O×(0, T ), ∂y α1 (x, t) + α2 y(x, t) = 0, ∀ (x, t) ∈ Σ = ∂O×(0, T ), ∂ν y(x, 0) = y0 (x), x ∈ O,

(3.1)

where O is an open, bounded, and smooth set of Rd (of class C 2 for instance) and m = 1ω is the characteristic function of an open set ω⊂O. Let y0 ∈ L2 (O) be arbitrary but fixed. If F (x, T ) ≡ 0, then system (3.1) is said to be exactly null controllable if there is u ∈ L2 (Q) such that y u (T ) ≡ 0. We have denoted by y u ∈ C([0, T ]; L2 (O)) ∩ L2 (0, T ; H 1 (O)) the solution to (3.1) and recall (see (1.76)–(1.77)) that √ √ u tyt ∈ L2 (Q), ty u ∈ L2 (0, T ; H 2 (O))

© Springer International Publishing AG, part of Springer Nature 2018 V. Barbu, Controllability and Stabilization of Parabolic Equations, Progress in Nonlinear Differential Equations and Their Applications 90, https://doi.org/10.1007/978-3-319-76666-9_3

43

44

3 Exact Controllability of Parabolic Equations

and if y0 ∈ H 1 (O) (y0 ∈ H01 (O) if α1 = 0) then y u ∈ Cw ([0, T ]; H 1 (O)) ∩ L2 (0, T ; H 2 (O)), ytu ∈ L2 (Q). Let a ∈ L∞ (O), b ∈ C 1 (O; Rd ), α2 ∈ C 1 (∂O), F ∈ L2 (O) and let ye ∈ be a steady-state (equilibrium) solution to system (3.1), that is, a solution to elliptic boundary value problem H 2 (O)

−Δye + b · ∇ye + aye = F in O, ∂ye + α2 ye = 0 on ∂O. α1 ∂ν

(3.2)

The steady-state solution ye is said to be exactly controllable if there is u ∈ L2 (Q) such that y u (T ) ≡ ye . Subtracting equations (3.1) and (3.2), we may reduce the exact controllability of the steady-state solution ye to the exact null controllability of system (3.1) with modified coefficients a, b. We may view v = mu as an internal controller, i.e., as a distributed controller with the support in ω×(0, T ) and for this reason we also refer to this property as the internal exact controllability. As it is well known, (3.1) describes the heat propagation and also the density fluctuation y = y(x, t) of a diffusing material at x ∈ O at time t. The term F +mu− ay in equation (3.1) represents sources of substance, composed of a fixed source and another one which is proportional with the concentration y. If the concentration of the substance y is specified on Σ, one has α1 = 0, while, if α2 = 0, one assumes to have a zero flow of substance through boundary (impermeable boundary). The exact null controllability problem formulated above is whether one can steers in time T the concentration y to equilibrium state ye by applying a source mu of material active on some subdomain ω of O. A similar problem if the source u is applied on the boundary ∂O or on some parts of it. In the absence of exterior sources, the solution of diffusion systems naturally decreases to an equilibrium state, but the role of the controller u introduced here is to achieve this goal in a finite specified time T . For practical purposes, it is desirable to find such a control u in an arbitrary small domain ω ∈ Ω or on the boundary. If equation (3.1) reduces to the heat equation ∂y − Δy = mu, in (0, T × O, ∂t y=0 on (0, T ) × ∂O, y(x, 0) = y0 (x) on O ⊂ R3 ,

(3.3)

the exact null controllability in time T means that one wants to drive the temperature y in zero by applying a heating (cooling) source u(t, x) on a subdomain ω of conductor Ω.

3.1 Exact Controllability of Linear Parabolic Equations

45

Let us illustrate the problem with the following simple example pertaining the exact null controllability of the 1 − D heat equation (3.3) on a finite interval of R. Problem 3.1. Given y0 ∈ L2 (0, π ) and an interval (a, b) ∈ (0, π ), find a controller v ∈ L2 (0, T ) such that the solution y to the equation ∂y ∂ 2y − 2 = v(t)m on (0, T ) × (0, π ), ∂t ∂x y(x, 0) = y0 (x), y(0, t) = y(π, t) = 0, ∀ t ∈ (0, T ), satisfies the condition y(x, T ) = 0, ∀ x ∈ (0, π ). Here, m is the characteristic function 1[a,b] of the interval [a, b] ⊂ (0, π ). This problem can be solved directly by invoking Theorem 1.8 and Corollary 2.1, that is, showing that the solution p to the backward dual equation ∂p ∂ 2 p + 2 =0 in (0, T ) × (0, π ), ∂t ∂x p(0, t) = p(π, t) = 0 ∀ t ∈ (0, T ),

(3.4)

satisfies the observability inequality 



T

T

p (x, 0)dx ≤ C 2



2

b

p(x, t)dx

0

0

dt.

(3.5)

a

(We note that, in this case, H = L2 (0, π ), U = R, B : R → L2 (0, π ) is given π b by Bu = mu and B ∗ p = 0 mp dx = a p(x)dx.) To prove (3.5), we express the solution p to equation (3.4) with the final value p(x, T ) = pT (x) as the Fourier series p(x, t) =

∞

pj e−j

2 (T −t)

sin(j x),

j =1

where pj = reduces to

√ π √ 2

π 0

pT sin(j x)dx. Then, by Parseval’s identity, inequality (3.5)

∞

pj2 e

−2j 2 T

j =1

where hj =

b a

sin(j x)dx.



T

≤C 0

⎛ ⎞2 ∞

2 ⎝ pj e−j t hj ⎠ dt, j =1

(3.6)

46

3 Exact Controllability of Parabolic Equations ∞ 2 2 Given the system {e−j t }∞ j =0 ⊂ L (0, T ), consider {ϕj }j =1 ⊂ L (0, T ) such that 2



T

ϕj (t)e−k t dt = δj k , ∀ j, k. 2

(3.7)

0

Such a system {ϕj }∞ j =1 is given by , ϕj (t) = (e−j t − qj (t))e−j t − qj −2 L2 (0,T ) 2

where

2



T

qj = arg min

|e−j t − q(t)|2 dt; q ∈ Λj 2



0

and Λj is the linear closed space of L2 (0, T ) spanned by {e−k t ; k = j }. By (3.7), we have 2

e−j T = hj 2

2 e−j T p

j

e−j T hj 2

=

−j 2 T

e ≤ |hj |



T

ϕj (t) 0



T

ϕj (t)

∞

k=1 ∞

0

⎛  ⎝

T



0

 hj e

−k 2 t

pk e−k t hk dt 2

k=1 ∞

pk e

pk dt 

2 −k 2 t

⎞1 2

ds ⎠ ϕj L2 (0,T ) .

hk

k=1

This yields ∞

j =1

pj2 e

−2j 2 T

≤

∞ −2j 2 T

e j =1

|hj |2

 ϕj 2L2 (0,T )

T 0

∞

2 pk e

−k 2 t

hk

dt.

k=1

We see, therefore, that a sufficient condition to have (3.6) is that |hj | ≥ M, for all j . We note that this happens and so the exact controllability problem has a solution if b−a and b+a are not rational numbers. Of course, even if this happens, we can find a smaller interval (a , b ) ⊂ (a, b) with this property. Hence, for each subinterval ω, there is a controller u(t, x) ≡ u(t)m(x) with support in ω such that y u (T ) = 0. This approach, which was extended to more general linear parabolic equations (see D. Russell [108, 109]) based on sharp analysis of Fourier series corresponding to eigenfunctions of the Laplace operator, was the first successful attempt to solve the controllability problem for parabolic equations. (We note here also the pioneering works of V. Mizel and T. Seidman [96], H. Fattorini and D. Russell [69] and refer to the monograph [116] by M. Tucsnack and G. Weiss for other recent results in this area.) However, it is limited to time independent parabolic operators and in more dimensions it involves sophisticated results of harmonic analysis.

3.1 Exact Controllability of Linear Parabolic Equations

47

Here we shall derive an exact controllability result for equation (3.1) via Carleman’s inequality (2.8). To this end, we shall associate with (3.1) the dual backward system (2.3), that is, pt + Δp + divx (bp) − ap = 0 in Q ∂p α1 + (α2 + α1 b · ν)p = 0 on Σ ∂ν

(3.8)

already considered in Section 2.1. Keeping in mind Theorem 1.1 and the observability inequality (2.8), we have at this stage all ingredients for studying the exact null controllability of equation (3.1). Throughout in the sequel, the functions α and ϕ are defined by equations (2.6), where k = 1. The main result is Theorem 3.1 below. Theorem 3.1. Let O be a C 2 -open and bounded domain of L∞ (Q), b ∈ C 1 (Q; Rd ). Let F0 ∈ L2 (Q) be such that

Rd and a ∈

|F (x, t)| ≤ |F0 (x, t)|esα(x,t) ϕ 3/2 (x, t), a.e. (x, t) ∈ Q

(3.9)

where s ≥ s0 , λ ≥ λ0 , are as in Theorem 2.1. Then, for each y0 ∈ L2 (O), there is u ∈ L2 (Q) such that y u (x, T ) ≡ 0 and uL2 (Q) ≤ C(|y0 |2 + F0 L2 (Q) ). Here y u ∈ C([0, T ]; L2 (O)) ∩ L2 (0, T ; H 1 (O)) is the solution to (3.1) and C = C(a∞ , ∇b∞ ). By |·|p , 1 ≤ p ≤ ∞, we denote as usually the norm of the space Lp (O). Theorem 3.1 amounts to saying that system (3.1) is exactly null controllable uniformly with respect to y0 and F. Proof. We note that, in the special case F ≡ 0, Theorem 3.1 is a direct consequence of Theorem 1.8. For the more general case considered here, we shall apply the argument used in the proof of Theorem 1.8. Namely, consider the optimal control problem:  u2 dx dt +

Minimize Q

1 ε

 O

y 2 (x, T )dx subject to (3.1).

(3.10)

By a standard argument, it follows that, for each ε > 0, problem (3.10) has a unique solution (yε , uε ). Moreover, by the maximum principle (see (1.85)–(1.86)), we have uε = mpε , a.e. in Q

(3.11)

where pε ∈ C([0, T ]; L2 (O)) ∩ L2 (0, T ; H01 (O)) is the solution to the backward dual system (3.8), i.e.,

48

3 Exact Controllability of Parabolic Equations

(pε )t + Δpε + divx (bpε ) − apε = 0 in Q ∂pε + (α2 + α1 b · ν)pε = 0 on Σ α1 ∂ν 1 pε (T ) = − yε (T ) in O. ε

(3.12)

Now, we multiply (3.12) by yε , equation (3.1) (where y = yε ) by pε and integrate on Q. We get, after summing the two identities, that  Qω

pε2 dx dt +

1 ε

 O

  yε2 (x, T )dx = − y0 (x)pε (x, 0)dx − Fpε dx dt. O

Q

(3.13)

By Corollary 2.1, we have       y0 (x)pε (x, 0)dx  ≤ C   O

Qω

1/2 pε2 dx

dt

|y0 |2

whilst the Carleman estimate (2.7) and condition (3.9) imply that   

1/2 

1/2   2sα 3 2 2  Fpε dx dt  ≤ C e ϕ p dx dt F (x, t)dx dt ε 0   Q Q Q 

1 2 ≤C e2sα ϕ 3 pε2 dx dt F0 L2 (Q) Qω

for s ≥ s0 and λ ≥ λ0 . (Here and everywhere in the sequel, we shall denote by the same symbol C a positive constant independent of ε and y0 .) Putting the latter inequality into (3.13), we get the estimate  Q

u2ε dx dt +

1 ε

 O

yε2 (x, T )dx ≤ C(|y0 |22 +F0 2L2 (Q) ), ∀ ε > 0.

This means that on a subsequence, again denoted ε, we have uε → u∗ weakly in L2 (Q) and |yε (T )|22 → 0 as ε → 0. By (3.1) it follows that {yε } is bounded in L2 (0, T ; H 1 (O)) ∩ C([0, T ]; L2 (O)), √ { t(yε )t } is bounded in L2 (Q), √ { tyε } is bounded in L2 (0, T ; H 2 (O)) ∩ L∞ (0, T ; H 1 (O)). Hence, selecting further subsequence, if necessary, we may assume that

(3.14)

3.1 Exact Controllability of Linear Parabolic Equations

√

49

yε −→ y ∗ weakly in L2 (0, T ; H 1 (O)) and weak–star in L2 (Q), √ ∗ t(yε )t −→ tyt weakly in L2 (Q), yε (t) −→ y ∗ (t) strongly in L2 (O) and uniformly on each [δ, T ], ∗

where 0 < δ < T . Clearly, y ∗ = y u and y ∗ (T ) = 0. Moreover, letting ε tend to zero into (3.14), we see that u∗ satisfies the desired estimate. This completes the proof. Theorem 3.1 has several important consequences. In particular, it implies the exact controllability of the steady-state solutions ye to equation (3.1). Corollary 3.1. Under assumptions of Theorem 3.1, let b, a, F be independent of t and let ye be a steady state solution to (3.1). Then for each y0 ∈ L2 (O) there is u ∈ L2 (Q) such that y u (T ) ≡ ye . Proof. One applies Theorem 3.1 to y¯ = y − ye which satisfies the homogeneous equation (3.1) with the initial value y¯0 = y0 − ye . Note also that Theorem 3.1 implies the exact boundary controllability of the boundary control system yt (x, t) − Δy(x, t) + b(x, t) · ∇y(x, t) + a(x, t) = F (x, t) in Q, ∂y + α2 y = u on Γ1 ×(0, T ) = Σ1 , α1 ∂ν ∂y + α2 y = 0 on Γ2 ×(0, T ) = Σ2 , α1 ∂ν y(x, 0) = y0 (x),

(3.15)

where ∂O = Γ1 ∪ Γ2 and Γ1 ∩ Γ2 = ∅. Corollary 3.2. Under assumptions of Theorem 3.1, for each y0 ∈ H 1 (O), there are u ∈ L2 (Σ1 ) and y ∈ C([0, T ]; H 1 (O)) ∩ L2 (0, T ; H 2 (O)), yt ∈ L2 (Q), which satisfy (3.15) and y(T ) ≡ 0. Proof. The idea is to “inflate” a little bit the domain O and apply Theorem 3.1 on be an open bounded subset such that O ⊃ O and this new domain. Namely, let O = Γ2 ∪ Γ 3 . ∂O We set ω = O\O extend a, b to smooth functions on O × (0, T ) and apply with the Dirichlet homogeneous boundary conTheorem 3.1 to equation (3.1) on O ditions and the initial value conditions y(x, 0) = y0 (x) on O where y0 is an H01 –extension of y0 to O. In fact, as the boundary is of class C 1 the function y0 can be extended by the usual method to a function y 0 ∈ H 1 (Rd ) (i.e., by symmetry if O is flat and by a

50

3 Exact Controllability of Parabolic Equations

similar argument using a partition of unity on a finite covering of O in the general case.) We set y0 = ρy 0 where ρ ∈ C0∞ (Rd ) and ρ = 1 in a neighborhood of O. We take F = 0 on O\O). Consequently, by virtue of Theorem 3.1 there are 1 and × (0, T )) satisfying (3.1) y ∈ L2 (O y ∈ C([0, T ]; H0 (O)) ∩ L2 (0, T ; H 2 (O)) on O×(0, T ) and such that y (T ) = 0. y y + α2 ∂ Let u be the trace of α1 ∂ν to Σ1 = Γ1 × (0, T ). (It should be recalled that by the trace theorem u is well defined and belongs to L2 (Σ1 ).) Clearly, the restriction y of y to O×(0, T ) satisfies all requirements of Corollary 3.2. We note also that, if y0 ∈ L2 (O), then by the regularity theory of parabolic equations and by the trace theorem√ it follows that Corollary 3.2 remains true with a controller u ∈ L2loc (0, T ; L2 (Γ1 )), tu ∈ L2 (Σ1 ). Coming back to Theorem 3.1 it should be observed that the Carleman inequality (2.7) leads to a sharper controllability result than that expressed in Theorem 3.1. Theorem 3.2. Under the assumptions of Theorem 3.1, there are u ∈ L2 (Q) and y ∈ C([0, T ]; L2 (O)) ∩ L2 (0, T ; H 1 (O)), which satisfy (3.1) and such that    e−2sα ϕ −3 u2 dx dt + s 3 e−2sα y 2 dx dt ≤ C e−2sα ϕ −3 F 2 dx dt. (3.16) Q

Q

Q

Proof. Let αε =

eλψ − e2λψC(O) , (t + ε)(T + ε − t)

where ε is positive and sufficiently small. Consider the optimal control problem   −2sα −3 2 3 ϕ u dx dt + s e−2sαε y 2 dx dt subject to (3.1). Minimize e Q

Q

Let (yε , uε ) be an optimal pair. Then, by the maximum principle, we have uε = me2sα ϕ 3 pε , a.e. in Q, where pε is the solution to (pε )t + Δpε + div (bpε ) − apε = s 3 yε e−2sαε in Q, ∂pε + (α2 + b · ν)pε = 0 on Σ, α1 ∂ν pε (T ) = 0 in O. By the transversality condition in the maximum principle in the above optimal control problem, we have also pε (0) = 0. Then, multiplying the latter by yε and (3.1) by pε , we get as above (see (3.13)) 

 Qω

e2sα ϕ 2 pε2 dx

dt + s

3 Q

e−2sαε yε2 dx

 dt = − pε F dx dt. Q

3.1 Exact Controllability of Linear Parabolic Equations

51

By (3.9), this yields 

 Qω

e2sα ϕ 3 pε2 dx dt + s 3

Q

e−2sαε yε2 dx dt ≤ C



e−2sα ϕ −3 F 2 dx dt Q

and, therefore,    1 e−2sα ϕ −3 u2ε dx dt + s 3 e−2sαε yε2 dx dt ≤ C e−2sα ϕ −3 F 2 dx dt. 2 Qω Q Q Letting ε tend to zero, we get uε −→ u weakly in L2 (Q) yε −→ y weakly in L2 (Q) and conclude as above that (y, u) satisfy system (3.1) along with estimate (3.16). We shall study now the exact null controllability of equation (3.1) with L∞ – controllers u. Theorem 3.3. Let F =f e2sα ϕ 3 , y0 ∈ L1 (O) and a ∈ L∞ (Q), b ∈ C 1 (Q; Rd ). Then there are λ0 > 0, μ = μ(λ) and s0 = s0 (λ) such that for λ ≥ λ0 and s ≥ s0 (λ) ∗ there is u∗ ∈ L∞ (Q) such that y u (T ) = 0 and for all δ > 0, I (u∗ )



2

e−2sα ϕ −3 |u∗ |2 dx dt + e(−s(1−η(λ))+δ(1+η(λ)))α u∗ L∞ (Q) Q

 2sα 3 2 2 2μs 6 ≤ C e |y0 |2 + s e ϕ f dx dt

=

(3.17)

Q

if y0 ∈ L2 (O) and

 I (u∗ ) ≤ Ce2μs |y0 |21 + e2sα ϕ 3 f 2 dx dt

(3.18)

Q

if y0 ∈ L1 (O) and α1 = 0. Here η(λ) = e−λψC(O) and C = Cλδ . Proof. We shall assume first that y0 ∈ L2 (O) and consider the optimal control problem  Minimize Q

e−2sα ϕ −3 u2 dx dt + ε−1

 O

y 2 (x, T )dx subject to (3.1).

(3.19)

Let (yε , uε ) be optimal in (3.19). Then, once again by the maximum principle, we have uε = mpε e2sα ϕ 3 a.e. in Q

(3.20)

52

3 Exact Controllability of Parabolic Equations

where (pε )t + Δpε + div (bpε ) − apε = 0 in Q, ∂pε α1 + (α2 + b · ν)pε = 0 on Σ, ∂ν 1 pε (T ) = − yε (T ) in O. ε

(3.21)

By (3.9) and (3.20), (3.21) it follows that (see, e.g., (3.13)) 



Qω

pε2 e2sα ϕ 3 dx dt +

f e2sα ϕ 3 pε dx dt + ε−1



Q

O

 yε2 (x, T )dx = −

O

y0 (x)pε (x, 0)dx.

This yields 1 2

 Qω

pε2 e2sα ϕ 3 dx

dt + ε

−1

 O

yε2 (x, T )dx

1 ≤ 2

 Q

f 2 e2sα ϕ 3 dx dt + |y0 |2 |pε (0)|2

and so, by Corollary 3.1, we obtain 

 pε2 e2sα ϕ 3 dx dt + ε−1 yε2 (x, T )dx Qω

O 2 2μs ≤ C e |y0 |2 + e2sα ϕ 3 f 2 dx dt

(3.22)

Q

for λ ≥ λ0 , s ≥ s0 (λ). Equivalently, 

 e−2sα ϕ −3 u2ε dx dt + ε−1 yε2 (x, T )dx Q

O 2 2μs ≤ C e |y0 |2 + e2sα ϕ 3 f 2 dx dt .

(3.23)

Q

We shall use in the following a bootstrap argument to improve estimate (3.23). Let δ ∈ (s0 , s) be arbitrary but fixed. We set α0 (t) =

1 − e2λψC(O) 1 , ϕ0 (t) = · t (T − t) t (T − t)

Clearly, we have α0 ≤ α ≤

eλψC(O) 1+e

λψC(O)

α0 , ϕ0 ≤ ϕ ≤ eλψC(O) ϕ0 .

(3.24)

3.1 Exact Controllability of Linear Parabolic Equations

53

Let δ > 0 be arbitrarily small but fixed and let {δj } be an increasing sequence such that 0 < δj < δ for all j . For each j , we set vj (x, t) = e(s+δj )α0 (t) ϕ03 (t)pε (x, t). We have (vj )t + Δvj + div (bvj ) − avj = gj in Q, ∂vj α1 on Σ, + (α2 + b · ν)vj = 0 ∂ν vj (x, 0) = vj (x, T ) = 0 in O,

(3.25)

where gj = pε (e(s+δj )α0 ϕ03 )t . By (3.9) and by (3.24), we see that  ((g1 )2t + g12 + |∇g1 |2 )dx dt  ≤ C e2(s+δ1 )α ϕ 7 ((s + δ1 )4 (pε2 + |∇pε |2 ) + (s + δ1 )2 (pε )2t )dx dt Q ≤ C1 s 3 e2sα (s 3 ϕ 3 pε2 + sϕ|∇pε |2 + (sϕ)−1 (pε )2t )dx dt Q 6 ≤ C1 s e2sα ϕ 3 pε2 dx dt.

Q

(3.26)

Qω

Then (3.22) yields

 g1 2H 1 (Q) ≤ C e2μs |y0 |22 + s 6 e2sα ϕ 3 f 2 dx dt .

(3.27)

Q

Recalling that, by the Sobolev embedding theorem, H 1 (Q) ⊂ Lp1 (Q) for p1 = 2(d+1) d−1 , we obtain by (3.27) that 

1/2 



|g1 |Lp1 (Q) ≤ C e |y0 |2 + s μs

3

e

2sα 3 2

ϕ f dx dt

(3.28)

.

Q

(Here C is a generic positive constant independent of ε, s, pi and f.) Then, by the parabolic regularity (see [83], p. 341), we have  v1 W 2,1 (Q) ≤ C e |y0 |2 + s p1

1/2 

 μs

3

e

2sα 3 2

,

(3.29)

1 1 2 − ≤ . p q d +2

(3.30)

ϕ f dx dt

Q

Wp2,1 (Q) ⊂ Lq (Q) for

54

3 Exact Controllability of Parabolic Equations

Then we obtain by (3.29) that  v1 Lp2 (Q) ≤ C e |y0 |2 + s where p2 = p1 + implies that

dp12 d+2−dp1

1/2 

 μs

3

e

2sα 3 2

ϕ f dx dt

,

Q

∞ (if d + 2 − 2p1 ≤ 0, then Wp2,1 1 (Q) ⊂ L (Q)). This



1/2 



g2 Lp2 (Q) ≤ C eμs |y0 |2 + s 3

e2sα ϕ 3 f 2 dx dt Q

p3 and, therefore, v2 ∈ Wp2,1 2 (Q) ⊂ L (Q) satisfies the estimate



1/2 



v2 W 2,1 (Q) ≤ C e |y0 |2 + s μs

3

e

p2

2sα 3 2

ϕ f dx dt

.

Q

In general, it follows that 

1/2 



vj W 2,1 (Q) ≤ C e |y0 |2 + s μs

pj

3

e

2sα 3 2

ϕ f dx dt

(3.31)

,

Q

dp2

j −1 where pj = pj −1 + d+2−dp · j −1 Thus, there is N such that d +2−2pN ≤ 0 and for such an pN we have, therefore, ∞ Wp2,1 N (Q) ⊂ L (Q). Moreover, by (3.31), we have



1/2 



vN L∞ (Q) ≤ C eμs |y0 |2 + s 3

e2sα ϕ 3 f 2 dx dt Q

and, therefore, 

1/2 



e(s+δ)α0 ϕ0 pε L∞ (Q) ≤ C eμs |y0 |2 + s 3

e2sα ϕ 3 f 2 dx dt

.

Q

By (3.24), we obtain that  e(s+δ)(1−η(λ))α ϕ 3 pε L∞ (Q) ≤ Cλ eμs |y0 |2 + s 3

1/2 

 e2sα ϕ 3 f 2 dx dt Q

3.1 Exact Controllability of Linear Parabolic Equations

55

and, by (3.17), it follows that e(−s(1−η(λ))+δ(1+η(λ)))α uε L∞ (Q)  

1/2  2sα 3 2 μs 3 ≤ Cλ e |y0 |2 + s e ϕ f dx dt

(3.32)

Q

for s ≥ s0 , λ ≥ λ0 and s0 ≤ δ < s. By estimates (3.23) and (3.32), it follows that on a subsequence, for simplicity again denoted {ε} → 0, we have uε −→ u∗ y −→ y ∗ √ √ ε tyε −→ ty ∗ √ √ t(yε )t −→ tyt∗ yε (t) −→ y ∗ (t)

weak star in L∞ (Q), weakly in L2 (0, T ; H 1 (O)), weak star in L∞ (0, T ; L2 (O)), weakly in L2 (0, T ; H 2 (O)), weakly in L2 (Q), uniformly in L2 (O) on every compact interval.

Clearly, y ∗ = u∗ , y ∗ (T ) = 0 and u∗ satisfies all the requirements of Theorem 3.2 (i.e., estimate (3.17)). Let us assume now that α1 = 0. Let S(t) be the C0 semigroup generated on Lp (O) by the Laplace operator with the Dirichlet homogeneous boundary value conditions (see Section 1.5). Then the solution y to (3.1) can be represented by the variation of the constant formula  t y(t) = S(t)y0 + S(t − s)(F (x, s) − b(x, s) · ∇y(x, s) − a(x, s)y(x, s))ds. 0

We recall that d

|S(t)z|p ≤ Ct − 2 (q

−1 −p −1 )

|z|q , ∀ t > 0,

(3.33)

for all 1 ≤ q ≤ p ≤ ∞. Moreover, we recall that (see, e.g., [18]) t|∇S(t)(z)|22 ≤ C|z|22 , ∀ z ∈ L2 (O), t > 0. For p = 2 and q = 1, we get by (3.33)

 t − d4 − 21 |y(t)|2 ≤ C t |y0 |1 + (|t − s| |y(s)|2 + |y(s)|2 + |F (s)|2 )ds 0

and, therefore, for η > 0 and sufficiently small, we have

 t d |y(t)|2 ≤ C t − 2 |y0 |1 + |F (s)|2 ds , ∀t ∈ (0, η). 0

56

3 Exact Controllability of Parabolic Equations

By density, this implies that equation (3.1) has, for each y0 ∈ L1 (O), a unique solution y ∈ C([0, T ]; L1 (O)) ∩ C(]0, T ]; H01 (O)) which satisfies the estimate  |y(η)|2 ≤ C |y0 |1 +

η

|f e ϕ

sα 3/2

0

|2 dt .

Let η ∈ (0, T ) be arbitrary but fixed. According to the first part of the proof, there are ( y, u) which satisfy equation (3.1) on O×(η, T ), y (η) = y(η), y (T ) = 0 and  T

2

e−2sα ϕ −3 ( u)2 dx dt + e(−s(1−η(λ))+δ(1+η(λ)))α uL∞ (Q) O



  ≤ C e2μs |y(η)|22 + e2sα ϕ 3 f 2 dx dt ≤ Ce2μs |y0 |21 + e2sα ϕ 3 f 2 dx dt .

η

Q

Q

The function u∗ (t) =



0 if 0 < t ≤ η u∗ (t) if η < t ≤ T

clearly satisfies the estimate (3.17) and steers y0 into origin in the time T . This completes the proof of Theorem 3.3. From the proof of Theorems 3.1 and 3.2, one might suspect that there is an equivalence between Carleman’s inequality (2.2) for the dual equation and the exact null controllability of the linear equation (3.1) with F = e2sα ϕ 3 f. We shall see below that this is, indeed, the case in a certain precise sense. Theorem 3.4. The Carleman inequality (2.2) holds for (3.8) if and only if for each f ∈ L2 (Q) there are u ∈ L2 (Q) and y ∈ C([0, T ]; L2 (O)) ∩ L2 (0, T ; H 1 (O)) satisfying (3.1) where y0 = 0, F = e2sα ϕ 3 f , and such that 

e−2sα ϕ −3 u2 dx dt + s 3

Q



e−2sα y 2 dx dt ≤ C

Q

 f 2 ϕ 3 e2sα dx dt, Q

(3.34) for s ≥ s0 (λ), λ ≥ λ0 . Proof. The only if part was proved in Theorem 3.2, so we shall confine to prove the if part. To this end, consider u and y satisfying (3.1) and (3.34). Multiplying (3.1) by p and (3.8) by y, we get 





mu p dxdt + Q



gy dx dt + Q

for all f ∈ L2 (Q) and y0 ∈ L2 (O).

3 2sα

fϕ e Q

p dxdt +

O

y0 (x)p(x, 0)dx = 0, (3.35)

3.1 Exact Controllability of Linear Parabolic Equations

57

For y0 = 0 and f = p, it follows by (3.35) that 

1/2 

1/2  2sα 3 2 2sα 3 2 2 −2sα −3 e ϕ p dx dt ≤ e ϕ p dx dt u e ϕ dx dt Q Qω 

1/2  Q

1/2 + e−2sα y 2 dx dt e2sα g 2 dx dt . Q

Q

Then, by (3.34), it follows that 

  e2sα ϕ 3 p2 dx dt ≤ C e2sα ϕ 3 p2 dx dt + s −3 e2sα g 2 dx dt . Q

Qω

(3.36) (3.37)

Q

Next, we multiply equation (3.8) by ϕe2sα p and integrate on Q. After some calculation involving Green’s formula, we get the estimate    2sα 2 2 2sα 3 2 −1 e ϕ|∇p| dx dt ≤ Cs e ϕ p dx dt + Cs e2sα g 2 dx dt. (3.38) Q

Q

Q

Finally, we multiply (3.8) by e2sα ϕ −1 (Δp + pt ) and integrate on Q. Again using Green’s formula, we obtain that  e2sα ϕ −1 (|Δp|2 + |pt |2 )dx dt Q   2sα ≤ Cs e |pt ||∇p|dx dt + C e2sα (p2 + |∇p|2 + g 2 )dx dt

Q Q 1 2sα 2 2sα −1 2 e ϕ |pt | dx dt ≤ C s e ϕ|∇p| dx dt +  2 Q  Q e2sα ϕ 3 p2 dx dt + C

+Cs 4

(3.39)

e2sα g 2 dx dt.

Q

Q

By (3.37), (3.38), and (3.39), we get inequality (3.34), as claimed. By formula (3.35), we see that more regular is the control u sharper is the Carleman inequality for the dual equation (3.8). In particular, by Theorem 3.4, we get the following Carleman inequality in L1 (O). Corollary 3.3. Let g ≡ 0. Then, under the assumptions of Theorem 3.2, for λ ≥ λ0 , s ≥ s0 (λ), the following inequality holds 

1/2 e2sα ϕ 3 p2 dx dt

Q

 ≤ Cδλ s 3

e(s(1−η(λ))−δ(1+η(λ)))α |p|dx dt

(3.40)

Qω

and, if α1 = 0,  |p(0)|∞ ≤

Cδλ eμs

e(s(1−η(λ))−δ(1+η(λ)))α |p|dx dt Qω

for 0 < δ < s0 (λ) and all weak solutions p ∈ L1 (Q) to (3.8).

(3.41)

58

3 Exact Controllability of Parabolic Equations

Proof. In (3.35), we take f = p and y0 = 0. Then, by (3.17), we see that 

 Q

e2sα ϕ 3 p2 dx dt ≤ Cδλ s 3

e(s(1−η(λ))−δ(1+η(λ))α |p|dx dt. Qω

Similarly, for f = 0, estimates (3.35) and (3.18) imply that  |p(0)|∞ ≤

Cδλ eμs

e(s(1−η(λ))−δ(1+η(λ))α |p|dx dt, Qω

as claimed. In particular, we deduce by Corollary 3.3 that the homogeneous system (3.8) is L1 − L∞ observable. We have a similar result for the nonhomogeneous equation (3.8). Corollary 3.4. Let g ∈ L1 (Q). Then  |p(0)|1 ≤ Cs



 e

(s(1−η(λ))−δ(1+η(λ)))α

|p|dxdt +

Qω

|g|dxdt

(3.42)

Q

for 0 < δ < s0 and all solutions p to (3.8). Proof. We write p = p1 + p2 where p1 is the solution to the homogeneous equation (3.8) and p2 is the solution to (3.8) with the final value p2 (T )=0. Then we apply inequality (3.40) to p1 and use the obvious inequality  |p2 (t)|1 ≤ C

T

|g(θ )|dθ

t

to obtain the following estimate 

 e2sα ϕ 3 |p|dx dt ≤ Cs Q

|g|dx dt ,

 es(1−η(λ))−δ(1+η(λ)))α |p|dx dt +

Qω

Q

which clearly implies (3.42), as claimed. Remark 3.1. By inequality (3.40), it follows via the bootstrap argument developed earlier in the proof of Theorem 3.4 the following sharper Carleman inequality  esα0 pL∞ (Q) ≤ C

|p|esδα0 dx dt Qω

for some μ suitable chosen. This implies an inequality of the form (3.41) in L∞ norm for the left-hand side. Remark 3.2. One might ask if the above controllability results remain true if int ω is empty. The following example shows that in general the answer is negative. For

3.2 Controllability of Semilinear Parabolic Equations

59

instance let us take ω to be a smooth boundary of a simple connected subdomain Oω of O. Consider the equation yt + Ay = μ in Q = O × (0, T ), where μ ∈ H −1 (O) is defined by  μ(ϕ) = ω

uϕdx, ∀ϕ ∈ H01 (O)

and  (Ay, ϕ) =

O

∇y · ∇ϕdx, ∀ϕ ∈ H01 (O).

Here u is a given L2 function on ω. This is just system (3.1), where b = 0, a = 0, F = 0 and with a distributed control u with the support in ω. We may, equivalently, write it as yt − Δy = 0 in Q, ∂ −y ∂ +y − = u on ω × (0, T ), y = 0 on ∂O × (0, T ). ∂ν ∂ν As noticed earlier, the exact null controllability of the above system is equivalent with the observability inequality  |p(0)|22

≤C

p2 dxdt ω×(0,T )

for all the solutions to the equation pt + Δp = 0 in Q; p = 0 on Σ, which, obviously, is false.

3.2 Controllability of Semilinear Parabolic Equations We shall study in this section the exact null controllability of the equation yt − Δy + b · ∇y + f (x, t, y) = mu + F (x, t) in Q, y(x, 0) = y0 (x) in O, ∂y + α2 y = 0 on Σ, α1 ∂ν

(3.43)

60

3 Exact Controllability of Parabolic Equations

where α1 , α2 ≥ 0 are nonnegative constants such that α1 + α2 > 0, m = 1ω is the characteristic function of some open subset ω⊂O and f : O×(0, T )×R → R is continuous in y, measurable in (x, t) and satisfies the following conditions |f (x, t, r)| ≤ L|r|(η(|r|) + 1), a.e. (x, t) ∈ Q, r ∈ R,

(3.44)

f (x, t, r)r ≥ −γ0 r , ∀ (x, t, r) ∈ Q×R,

(3.45)

2

3

|F (x, t)| ≤ |F0 (x, t)|esα ϕ 2 , ∀(x, t) ∈ Q.

(3.46)

Here L > 0, γ0 ≥ 0, F0 ∈ L2 (Q), η is a nonnegative, continuous, and increasing function and α, λ ≥ λ0 , s ≥ s0 are fixed as in Theorem 2.1 and α, ϕ are defined by equations (2.6), where k = 1. It should be said that the sign condition (3.45) precludes the blow up of solutions and so it implies the existence of a global solution for the Cauchy problem (3.43) by a standard continuation argument. Note also that, if for some γ the function y → f (x, t, y) + γ y is monotonically increasing, then by Theorem 1.4 equation (3.43) has a unique global solution. We shall see here that the exact null controllability is possible for nonlinear function r → f (·, r) with mild growth to +∞. To begin with, we shall consider first the case where f is sublinear as a function of r. Theorem 3.5. Assume that b ∈ C 1 (Q; Rd ) and |f (t, x, r)| ≤ L|r|, a.e. (x, t) ∈ Q, ∀ r ∈ R. Then for each y0 ∈ L√2 (O) there are √u ∈ L2 (Q) and y ∈ C([0, T ]; L2 (O)) ∩ L2 (0, T ; H01 (O)), tyt ∈ L2 (Q), ty ∈ L2 (0, T ; H 2 (O)) satisfying (3.43) and such that y(T ) = 0 in O. Proof. The argument is standard for this type of controllability result and will be frequently used in the sequel. Namely, consider the set K = {z ∈ L2 (Q); zL2 (Q) ≤ ρ} and the linear system yt − Δy + b · ∇y + g(x, t, z)y = mu + F in Q, in O, y(x, 0) = y0 (x) ∂y + α2 y = 0 on ∂O, α1 ∂ν

(3.47)

where ⎧ f (x, t, r) ⎪ ⎨ if |r| > 0, r g(x, t, r) = f (x, t, θ ) ⎪ ⎩ lim if r = 0. θ→0 θ

(3.48)

3.2 Controllability of Semilinear Parabolic Equations

61

(Without loss of generality we may assume that the previous limit exists; otherwise we approximate g by a family of smooth functions gε and let ε → 0 in the corresponding controllability result.) By Theorem 3.1, for each z ∈ K, there is at least one controller u ∈ L2 (Q) such that y u (T ) = 0 and  u2 dx dt ≤ γ (|y0 |22 + F0 2L2 (Q) ), (3.49) Q

where γ is independent of z (because |g(x, t, z)| ≤ L). Here yzu is the solution to (3.47). 2 Define the mapping Φ : K → 2L (Q) by Φ(z) = {y u ; yzu (T ) = 0; u satisfies (3.49)}. By (3.47) and (3.48), we see that yzu satisfies the estimates 

 T 2 u 2 2 2 2 u |∇yz (x, t)| dx dt ≤ C1 (u + F )dx dt + |y0 |2 |yz (t)|2 + 0 O Q

2 2 ≤ C2 F dx dt + |y0 |2 , t ∈ [0, T ] Q

and, therefore, by (3.47) we have (yzu )t L2 (0,T ;(H 1 (O)) ) ≤ C3 (|y0 |2 + 1), ∀ z ∈ K.

(3.50) (3.51)

This implies that, for ρ sufficiently large Φ maps K into itself and for each z ∈ K, Φ(z) is a convex and compact subset of L2 (Q). It is also easily seen that this multivalued map is upper semicontinuous on L2 (Q), i.e., if zk → z strongly in L2 (Q) and yk ∈ Φ(zk ) then y ∈ Φ(z). Indeed, yk = yzukk where (see (3.49)) uk L2 (Q) ≤ C. By estimates (3.50) and (3.51), it follows via the Aubin–Lions compactness theorem (see, e.g., [89], Theorem 5.1) that {yk } is compact in L2 (Q) and, therefore, selecting a subsequence uk −→ u weakly in L2 (Q) yk −→ y strongly in L2 (Q) and weakly in L2 (0, T ; H 1 (O)) Moreover, since g(zk )yk → g(z)y, a.e. on Q and {g(zk )yk } is bounded in L2 (Q), we infer that g(zk )yk −→ g(z)y weakly in L2 (Q). Hence (yk )t − Δyk + b · ∇yk −→ yt − Δy + b · ∇y weakly in L2 (Q) and, therefore, y = yzu as claimed.

62

3 Exact Controllability of Parabolic Equations

By the infinite dimensional Kakutani theorem (see, e.g., [18]) and [65], p. 310, we may conclude that there is y ∈ K such that y ∈ Φ(y). In other words, there is u ∈ L2 (Q) such that y u (T ) = 0, as claimed. This completes the proof. In spite of the restrictive condition (3.45), Theorem 3.5 is, however, applicable to a large class of physical processes. For instance, this is the case with the thermostat control model or with the kinetics of enzymatic reactions (the Michaelis–Menten model). In the latter case, f (y) = a1 y(a2 y + a3 )−1 , ∀y ∈ R. Analyzing the proof of Theorem 3.5, one sees that the sublinearity condition was essential to proving that Φ(K)⊂K. However, it turns out that the conclusions of Theorem 3.5 still remain true for nonlinearities f with mild growth to +∞. Namely, one has Theorem 3.6. Assume that the functions f, F satisfy conditions (3.44), (3.45), and (3.46) where F0 ∈ L∞ (Q), |η(r)| ≤ μ(r)((log r + 1)3/2 + 1), ∀ r ∈ R+ ,

(3.52)

and lim μ(r) = 0. Then, for each y0 ∈ L2 (O), there are u ∈ L∞ (Q) and y ∈ r→∞

C([0, T ]; L2 (O)) ∪ L2 (0, T ; H 1 (O)) which satisfy (3.43) and such that y(T ) ≡ 0, uL∞ (Q) ≤ C(|y0 |2 + F0 L∞ (Q) ). Proof. We set K∞ = {z ∈ L∞ (Q); zL∞ (Q) ≤ ρ}

(3.53)

and, for each z ∈ K∞ , denote by yzu the solution to equation (3.47). Lemma 3.1 below is the main ingredient of the proof. Lemma 3.1. Under the assumptions of Theorem 3.6, for each z ∈ K∞ there are u ∈ L∞ (Q) and y = yzu such that yzu (T ) = 0 and uL∞ (Q) ≤ C(eμ0 η

2 3 (ρ)

|y0 |2 + F0 ∞ ),

(3.54)

where C and μ0 are independent of ρ and z and F0 ∞ is the norm of F0 in L∞ (Q). Proof. Arguing as in the proof of Theorem 3.2, consider the optimal control problem  Minimize

e Q

1 u dx dt + ε

−2sα −3 2

ϕ

 O

y 2 (x, T )dx subject to (3.47).

We have uε = mpε e2sα ϕ 3 , a.e. in Q,

(3.55)

3.2 Controllability of Semilinear Parabolic Equations

63

where pε is the solution to linear system (pε )t + Δpε − g(x, t, z)pε + div (b · pε ) = 0 in Q, ∂pε + (α2 + b · ν)pε = 0 on Σ, α1 ∂ν 1 pε (x, T ) = − yε (x, T ) in O. ε

(3.56)

Multiplying (3.56) by yε , (3.47) by pε , and integrating on Q, we obtain  Qω

e2sα ϕ 3 pε2

+ε

−1

 O

  = − y0 (x)pε (x, 0)dx − Fpε dx dt O  Q 2sα 3 2 ≤ |y0 |2 |pε (0)|2 + δ e ϕ pε dx dt + Cδ F0 2∞

yε2 (x, T )dx

Q

(3.57) for each δ > 0. Next, applying the Carleman inequality (2.7) into equation (3.57) (i.e., g is replaced by g(x, t, z)pε ) and using condition (3.44), we see that, for λ = λ0 sufficiently large and s ≥ s0 , we have  s3 Q

 e2sα ϕ 3 pε2 dx dt ≤ C s 3

 Qω

e2sα ϕ 3 pε2 dx dt + η2 (ρ)

Q

e2sα pε2 dx dt .

2

This means that, for s ≥ C0 η 3 (ρ) + s0 , we have   2sα 3 2 e ϕ pε dx dt ≤ C e2sα ϕ 3 pε2 dx dt, Q

(3.58)

Qω

where C is independent of z and ε. On the other hand, multiplying equation (3.56) by pε and recalling that, by condition (3.45), g(x, t, z) ≥ −γ0 , ∀ (x, t) ∈ Q, z ∈ K, we obtain after some calculation involving Green’s formula that d |pε (t)|22 ≥ −C1 |pε (t)|22 , a.e. t ∈ (0, T ), dt where C1 is independent of z and ρ. This yields μs

|pε (0)|2 2 ≤ C3 e t (T −t)

 O

e2sα ϕ 3 |pε (x, t)|2 dx, ∀ t ∈ (t1 , t2 ),

64

3 Exact Controllability of Parabolic Equations

where C3 is independent of ρ, z and s. From now on, we shall argue as in the proof of Corollary 2.1. Integrating the latter on (t1 , t2 ) and using estimate (3.58), we obtain, for some C and μ0 > 0 independent of ρ, |pε (0)|2 ≤ Ce

μ0 η2/3 (ρ)



1/2 Qω

e2sα ϕ 3 pε2 dx

dt

2

for s = Cη 3 (ρ) + s0 and so (3.57) yields  Qω

e2sα ϕ 3 pε2 dx dt + ε−1



yε2 (x, T )dx O  2/3 ≤ C|y0 |22 e2μ0 η (ρ) + δ e2sα ϕ 3 pε2 dx dt + Cδ F0 2∞ .

(3.59)

Q

By the Carleman inequality (3.9) applied to equation (3.56), it follows also that  Q

e2sα (s 3 ϕ 2 pε2 + sϕ|∇pε |2 + (sϕ)−1 (|Δpε |2 + (pε )2t )) dxdt

  e2sα ϕ 3 pε2 dx dt + η2 (ρ) e2sα ϕ 3 pε2 dx dt . ≤ C s3 Qω

Q

Then, by (3.59), we get for δ suitable chosen that 

 e2sα (ϕ 3 pε2 + ϕ|∇pε |2 + ϕ −1 (pε )2t + |Δpε |2 )dx dt + ε−1 yε2 (x, T )dx Q O  2 2sα 3 2 2μ0 η 3 (ρ) e ϕ pε dx dt ≤ C(e |y0 |22 + F0 2∞ ) ≤C Qω

(3.60) 2 for s = Cη 3 (ρ) + s0 and λ = λ0 large enough. We shall continue the proof with a bootstrap argument already used in the proof of Theorem 3.3. Namely, we fix 0 < δ < s = Cη2/3 (ρ) + s0 and consider an increasing sequence 0 < δj < δ, j = 1, . . . . We set vj = e(s+δj )α0 ϕ03 pε , gj = pj (e(s+δj )α0 ϕ03 )t , j = 1, . . . , where α0 and ϕ0 are defined as in the proof of Theorem 3.3. We recall that, by (3.26), we have the estimate  e2sα ϕ 3 pε2 dx dt. g1 2H 1 (Q) ≤ Cs 6 Qω

Then, by (3.60), we obtain g1 2H 1 (Q) ≤ Cs 6 (e2μ0 η

2/3 (ρ)

|y0 |22 + F0 ∞ )

3.2 Controllability of Semilinear Parabolic Equations

65

for s = Cη2/3 (ρ) + s0 and λ = λ0 fixed but large enough. This implies that (see (3.28)) 2 3

g1 Lp1 (Q) ≤ C(eμ0 η |y0 |2 + F0 ∞ ) for p1 = 2(d+1) d−1 and, therefore, by the parabolic regularity and by (3.29), we obtain the estimate v1 W 2,1 (Q) ≤ C(eμ0 η

2/3 (ρ)

p1

|y0 |2 + F0 ∞ )

for p1 = 2(d+1) d−1 · In general, we have (see (3.31)) vj W 2,1 (Q) ≤ C(eμ0 η

2/3 (ρ)

pj

where pj = pj −1 +

|y0 |2 + F0 ∞ ),

dpj2−1 d+2−dpj −1 ·

For j = N sufficiently large but finite, we have, therefore, vN L∞ (Q) ≤ C(eμ0 η

2/3 (ρ)

|y0 |2 + F0 ∞ ),

where C is independent of ε. Hence e(s+δ)α0 ϕ03 pε L∞ (Q) ≤ C(eμ0 η

2/3 (ρ)

|y0 |2 + F0 ∞ )

and, therefore, (see ((3.32)) uε L∞ (Q) ≤ e(−s(1−η(λ))+δ(1+η(λ)))α uε ∞ ≤ C(eμ0 η

2/3 (ρ)

|y0 |2 + F0 ∞ ), (3.61)

for s = Cη2/3 (ρ) + s0 and 0 < δ < s. Here C is independent of ε and, redefining μ0 if necessary, we may assume also that it is independent of ρ, too. Now, multiplying equation (3.47), where u = uε and y = yε , by yε and integrating on O×(0, t), it follows by (3.45) that (see (3.50)) |yε (t)|22

+

 T 0

   2 2 |∇yε | dx dt ≤ C F0 ∞ + |y0 |2 + C u2ε dx dt. 2

O

(3.62)

Q

Similarly, multiplying by t (yε )t and integrating on O×(0, t), we get after some calculation that

  t (yε2 + |Δyε |2 )dx dt + tyε (t)2H 1 (O) ≤ C F0 2∞ + |y0 |22 + u2ε dx dt . Q

Q

(3.63)

66

3 Exact Controllability of Parabolic Equations

In particular, it follows by estimate (3.62) that

 (yε )t 2L2 (0,T ;(H 1 (O)) ) ≤ C F0 2∞ + |y0 |22 + |uε |2 dx dt .

(3.64)

Q

(The constant C = C(ρ) arising in (3.63), (3.64) is independent of ε.) Then we conclude by (3.61)–(3.64) that {uε } is weak-star compact in L∞ (Q) and {yε } is compact in L2 (Q) and weakly compact in L2 (0, T ; H 1 (O)) ∩ L∞ (0, T ; L2 (O)). Thus, selecting a subsequence if necessary, we have uε −→ u weak star in L∞ (Q) yε −→ y weakly in L2 (0, T ; H 1 (O)) and weak star in L∞ (0, T ; L2 (O)). Moreover, by estimate (3.63), it follows that yε −→ y weak star in L∞ (η, T ; H 1 (O)) ∩ L2 (0, T ; H 2 (O)), (yε )t −→ yt weakly in L2 (η, T ; L2 (O)), yε (t) −→ y(t) strongly in L2 (O), uniformly on [η, T ] for each 0 < η < T . Then, letting ε tend to zero into equation (3.43), where u = uε , we infer that y = yzu , and by (3.61) it follows that uL∞ (Q) ≤ C(eμ0 η

2/3 (ρ)

|y0 |2 + F0 ∞ ),

where C is independent of ρ and z. This is precisely the desired estimate (3.54). We note also for later use that, by (3.62) and (3.64), it follows that yzu satisfies the estimates  |yzu (t)|22 +

T 0

 yzu (t)2H 1 (O) dt +

≤ C(eμ0 η

2/3 (ρ)

T 0

(yzu )t (t)2(H 1 (O)) dt

(3.65)

|y0 |22 + F0 2∞ ),

where C is independent of z and ρ. This completes the proof of Lemma 3.1. Proof of Theorem 3.6 (Continued). For each z ∈ K∞ defined above denote by Φ(z)⊂L2 (Q) the set of all yzu ∈ L2 (Q) such that yzu (T ) = 0, uL∞ (Q) ≤ C(eμ0 η

2 3 (ρ)

|y0 |2 + F0 ∞ ),

where C and μ0 are as in Lemma 3.1. Hence Φ(z) = ∅, ∀ z ∈ K∞ . It is also easily seen by estimates (3.65) that Φ(z) is a closed, convex, and compact subset of L2 (Q). Moreover, arguing as in the proof of Theorem 3.5, it follows that Φ is upper

3.2 Controllability of Semilinear Parabolic Equations

67

semicontinuous on L2 (Q)×L2 (Q). To conclude the proof, it remains to be shown that Φ(K∞ )⊂K∞ and to apply the Kakutani fixed point theorem as in the previous theorem. We shall assume first that y0 ∈ L∞ (Q). We set M0 = |y0 |∞ , M1 = muL∞ (Q) + F L∞ (Q) . Let w = yzu − M0 − Mt, where M will be made precise below. By (3.48) and (3.44), we have |g(z)| ≤ L(η(ρ) + 1), a.e. in Q. We chose T sufficiently small such that L(η(ρ) + 1)T ≤ 1.

(3.66)

Now, we choose M = 2(M1 + L(η(ρ) + 1)M0 ). We have wt −Δw+b·∇w+g(z)w=mu + F − M − g(z)(M0 + Mt)≤0, ∂w + α2 w = −α2 (M0 + Mt) ≤ 0 on Σ, α1 ∂ν w(x, 0) = y0 − M0 ≤ 0.

(3.67)

If we multiply (3.67) by w + and integrate on Q, we see via Gronwall’s lemma that w + = 0 and, therefore, w ≤ 0 in Q. Hence yzu ≤ M0 + MT , a.e. on Q. Similarly, it follows that yzu ≥ −M0 − MT . Recalling estimate (3.54), we obtain, therefore, that yzu L∞ (Q) ≤ 2(M1 + L(η(ρ) + 1)M0 )T + M0 2

≤ 2C(eμ0 η 3 (ρ) |y0 |∞ + L|y0 |∞ (η(ρ) + 1) + F0 ∞ ), ∀ z ∈ K∞ . Then, by (3.52), it follows that for ρ large enough and for T chosen as in (3.66) we have yzu ∈ K∞ for all z ∈ K∞ . This completes the proof in the case y0 ∈ L∞ (O) and T sufficiently small. Obviously, this implies the controllability for any T > 0 along with the conclusions of Theorem 3.6. Now, let y0 ∈ L2 (O) and let y be the solution to the uncontrolled equation yt − Δy + b · ∇y + f (x, t, y) = F in Q, in O, y(x, 0) = y0 (x) ∂y + α2 y = 0 on Σ. α1 ∂ν As noticed earlier, problem (3.68) has at least one solution y ∈ C([0, T ]; L2 (O)) ∩ C(]0, T [; H 1 (O)) ∩ L2 (η, T ; H 2 (O))

(3.68)

68

3 Exact Controllability of Parabolic Equations 2d

for each 0 < η < T . By the Sobolev embedding theorem, H 1 (Oω ) ⊂ L d−2 (Oω ) and, by condition (3.52), it follows that b · ∇y + f (x, t, y) ∈ Lp1 (O×(η, T )), p1
2, then for each y0 ∈ L2 (O) such that y0 > 0 on an open subset of O there is T1 > 0 such that the system is not null controllable in T for each T > T1 . Previously, weaker versions of this result were established by A. Fursikov and O.Yu. Imanuvilov [76]. (See also [71].) This result suggests that one cannot expect null controllability for f which grows to +∞ faster than log2 (|s| + 1) though it does not preclude the exact null controllability for some special classes of nonlinearities f. Anyway, it seems that, so far, the case 32 < p < 2 remained open. Remark 3.4. By Theorem 3.6, under conditions (3.44)–(3.46) and (3.52), it follows that system (3.43) is exactly null boundary controllable, i.e., for each y0 ∈ H 1 (O)

3.3 Approximate Controllability

69

there are v ∈ L2 (Σ) and y ∈ C([0, T ; L2 (O)) ∩ L2 (0, T ; H 1 (O)) which satisfy system (3.43) with u = 0 and y(T ) = 0, y = v on Σ. The proof is exactly the same as that given in Corollary 3.2, i.e., one inflates a little bit the domain O and one applies Theorem 3.6 on the new domain. The details are left to the reader. Note also that, if f (x, t, y) ≡ f (x, y) and if the function r −→ f (x, r + ye ) − f (x, ye ) satisfies conditions (3.43)–(3.45) and (3.52), then one might derive by Theorem 3.6 the exact controllability of the steady-state solutions ye to system (3.43). In fact, by the substitution z = y − ye , one reduces the exact controllability of ye to the null controllability of a system of form (3.43) and one might apply Theorem 3.6 to obtain the desired result.

3.3 Approximate Controllability System (3.43) is said to be approximately controllable if for all y0 , y1 ∈ L2 (O) there are u ∈ L2 (Q) and y ∈ L2 (Q) satisfying (3.43) and such that |y(T ) − y1 |2 ≤ ε.

(3.69)

It should be recalled that, for linear systems, the approximate controllability is implied by exact controllability and it is an immediate consequence of a unique continuation property for the dual equation, but this does not happen for the nonlinear systems considered here. As a matter of fact, this weaker concept of controllability has the same degree of difficulty as the exact controllability, though it can be deduced by similar arguments and, essentially, under the same conditions. We shall illustrate this for the case described in Theorem 3.6 only, though it extends mutatis–mutandis to all the situations considered above. (See also [70], [121], [122] for other results of this type.) Theorem 3.7. Assume that F ≡ 0 and that, for each y1 ∈ L∞ (O), the function (x, t, r) → f1 (x, t, r) = f (x, t, r + y1 ) − f (x, t, y1 ) satisfies conditions (3.44), (3.45). Then system (3.43) is approximately controllable. Proof. By density, it suffices to prove (3.69) for each y1 ∈ W 2,∞ (O) ∩ H01 (O). We set w = y − y1 and note that w satisfies a system of form (3.43), i.e., wt − Δw + b · ∇w + f1 (w) = mu + F1 in Q, w(x, 0) = y0 (x) − y1 (x) in O; w = 0 on Σ,

(3.70)

70

3 Exact Controllability of Parabolic Equations

where f1 (x, t, w) = f (x, t, w + y1 ) − f (x, t, y1 ), F1 = Δy1 + b · ∇y1 − f (x, t, y1 ). We set Fε (x, t, w) = e2sα (ε + e2sα )−1 F1 (x, t), where ε ≥ 0 and s, α are chosen as in Theorem 3.6. Hence, there are uε , wε such that (wε )t − Δwε + b · ∇wε + f1 (x, t, wε ) = muε + Fε in Q wε (x, 0) = y0 − y1 , wε (x, T ) = 0 in O, wε = 0 on Σ.

(3.71)

Let wε be a solution to equation (3.70), where u = uε . We set θε = w ε − wε and notice that (θε )t −Δθε +b·∇θε +f1 (x, t, θε +wε )−f1 (x, t, wε )=(ε+e2sα )−1 F1 in Q, θε (x, 0) = 0 in O, θε = 0 on Σ. Since the right-hand side is a.e. convergent to 0 on Q and is bounded by |F1 |∞ , we infer by virtue of conditions (3.44), (3.45) that |θε (T )|2 −→ 0 as ε → 0 and, therefore, w ε (T ) −→ 0 strongly in L2 (O) as ε → 0. This implies (3.69) for a suitable chosen controller u, thereby completing the proof.

3.4 Local Controllability of Semilinear Parabolic Equations For general systems of form (3.43), the best one can expect is the local null controllability i.e., the exact null controllability for the initial data in a neighborhood of the origin. Theorem 3.8. Let F ≡ 0, b ∈ C 1 (Q; R) , and let f : Q×R → R satisfy conditions (3.44), (3.45). Then there is ρ0 > 0 such that for all y0 ∈ L∞ (O), |y0 |∞ ≤ ρ0 there are y ∈ C([0, T ]; L2 (O)) ∩ L2 (0, T ; H 1 (O)) ∩ L∞ (Q) and u ∈ L∞ (Q) which satisfy (3.43) and such that y(T ) ≡ 0.

3.4 Local Controllability of Semilinear Parabolic Equations

71

Proof. Consider the set K = {z ∈ L2 (Q); zL∞ (Q) ≤ ρ} 2

and define the mapping Φ : K → 2L (Q)   2 u 2 u μ0 η 3 (ρ) Φ(z) = yz ∈ L (Q); yz (T ) = 0, uL∞ (Q) ≤ Ce |y0 |2 , where yzu is the solution to (3.47) and C, μ0 are as in Lemma 3.1. We have (recall that F ≡ 0) yzu L∞ (Q) ≤ C(ρ)(|y0 |∞ + uL∞ (Q) ) ≤ C1 (ρ)|y0 |∞ , ∀ z ∈ K, and so Φ(K)⊂K for |y0 |∞ ≤ ρ0 sufficiently small. We recall also the estimate

 t  t 2 u 2 u 2 2 2 u |yz (t)|2 + yz (s)H 1 (O) ds ≤ C |y0 |2 + η(ρ) |yz (s)| ds + uL2 (Q) , 0

0

which implies that |yzu (t)|22 + |yzu |2L2 (0,T ;H 1 (O)) + (yzu )t 2 L2 (0,T ;(H 1 (O)) ≤ C(ρ)|y0 |22 . Hence, Φ(K) is compact in L2 (Q) and, arguing as in the previous cases, we see also that Φ is upper semicontinuous on L2 (Q). Thus, there is y ∈ K such that y ∈ Φ(y) and this completes the proof. In particular, we find by Theorem 3.8 the following local boundary controllability result. Corollary 3.5. Let f satisfy conditions (3.44) and (3.45). Then there is ρ0 > 0 such that, for all y0 ∈ L∞ (O), |y0 |∞ ≤ ρ0 , there are y ∈ C([0, T ]; L2 (O)) ∩ L2 (0, T ; H 1 (O)) ∩ L∞ (Q) and u ∈ L2 (Σ) such that yt − Δy + b · ∇y + f (x, t, y) = 0 in Q, y = u on Σ, y(x, 0) = y0 (x), y(x, T ) = 0 in O. The proof is the same as that used in the previous cases (see Corollary 3.1), i.e., one extends the domain O near ∂O and one applies Theorem 3.8 on Q1 = O1 × (0, T ) where O1 is the extension of O. If f (x, t, y) ≡ f (x, y), b ≡ b(x) and ye ∈ H 1 (O) ∩ L∞ (O) is a steady-state solution to system (3.43), that is, −Δye + b · ∇ye + f (x, ye ) = 0 in O ∂ye on Σ, α1 + α2 ye = 0 ∂ν

72

3 Exact Controllability of Parabolic Equations

then, by applying Theorem 3.8 to the equation obtained by the substitution y → y − ye , we are led to the following controllability result. Corollary 3.6. Let f satisfy the assumptions of Theorem 3.8. Then there is ρ0 > 0 such that, for all y0 ∈ L∞ (O) such that |y0 − ye |∞ ≤ ρ0 , there are (y, u) which satisfy (3.43) and y(T ) = ye . In other words, the steady-state solution is locally exactly controllable. One of the main consequences of this corollary is that each equilibrium solution y0 ∈ H 1 (O) ∩ L∞ (O) to equation (3.43) (which, in general, is unstable) is stabilizable by an open loop internal distributed controller with the support in an arbitrary open subset ω of O (or by boundary controllers). More generally, we may replace in Corollary 3.6 the steady-state solution ye by the final value y ∗ (T ) of a solution y ∗ to the equation yt∗ − Δy ∗ + b · ∇y ∗ + f (x, t, y ∗ ) = 0 in Q , ∂y on Σ. α1 + α2 y = 0 ∂ν Then, it follows as above that, if y ∗ is regular enough, then for |y0 − y ∗ (0)|∞ sufficiently small there are (y, u), which satisfy equation (3.43) with the righthand side mu + f0 and y(T ) = y ∗ (T ). In particular, this means that the nonlinear diffusion equation (3.43) excited by a distributed control with the support in Qω is locally reversible. It should be also said that, as in the case of Theorem 3.6, condition y0 ∈ L∞ (O) can be relaxed to y0 ∈ L2 (O) due to smoothing effect of solutions to parabolic equations on initial data. The previous treatment suggests a general strategy to obtain the local (global) controllability of nonlinear distributed systems of the form yt + Ly + f (y) = mu in Q, in O, y(0) = y0 where L is a linear partial differential operator with homogeneous boundary value conditions. Namely, one check first (via Carleman inequality) the (global) exact null controllability of the linearized system yt + Ly + g(z)y = mu in Q, in O, y(0) = y0 where f (y) = g(y)y and one applies the Kakutani fixed point theorem to the map z → yzu on a suitable chosen set K⊂L2 (Q). A quite different approach, systematically used in the works of Fursikov and Imanuvilov (see, e.g., [76]), is to invoke the implicit function theorem in a convenient function space but we did not pursue this way here.

3.4 Local Controllability of Semilinear Parabolic Equations

73

Now, we shall study with the above methods the local controllability of the phase field system ut (x, t) + ϕt (x, t) − kΔu(x, t) = m(x)w(x, t) + f1 (x), ϕt (x, t) − αΔϕ(x, t) − β(ϕ(x, t) − ϕ 3 (x, t)) + γ u(x, t) = m(x)v(x, t) + f2 (x), (x, t) ∈ Q = O×(0, T ), u(x, 0) = u0 (x), ϕ(x, 0) = ϕ0 (x), u(x, t) = u(x), ¯ ϕ(x, t) = ϕ(x), ∀ (x, t) ∈ Σ = ∂O×(0, T ),

(3.72)

where w, v are input controllers and α, β, γ , , k are positive constants, m is the characteristic function of an open set ω⊂O, f1 , f2 ∈ L2 (O), and u0 , ϕ0 ∈ H01 (O), u, ¯ ϕ are given functions. Let (u∗ , ϕ ∗ ) ∈ H 2 (O)×H 2 (O) be a steady-state solution to (3.72), that is, the solution of the Landau–Ginzburg equations ϕ ∗ − kΔu∗ = f1 in O −αΔϕ ∗ − β(ϕ ∗ − (ϕ ∗ )3 ) + γ u∗ = f2 in O ¯ ϕ∗ = ϕ on ∂O. u∗ = u,

(3.73)

It is well known that, for each w and v in L2 (Q) such that u0 − u∗ , ϕ0 − ϕ ∗ ∈ H01 (O), problem (3.72) has a unique solution (u, ϕ) ∈ (H 2,1 (Q))2 . (See, e.g., [26], p 235.) We recall that H 2,1 (Q) = {y ∈ L2 (Q); H01 (O) ∩ H 2 (O), yt ∈ L2 (Q)}. System (3.72) models the phase transition of several physical processes including the melting and solidification. The two phase Stefan problem as well as other classical mathematical models of phase transition are limiting cases of system (3.72). (See [57].) Here u is the temperature while ϕ is the phase field function defining the liquid or the solid phase. As a matter of fact, in phase transition models ϕ = ϕ(t, x) is an order parameter which takes values is a specific interval [ϕ0 , ϕ1 ], describing the phase evolution. For instance, if [(t, x); ϕ(t, x) = ϕ0 ] and [(t, x); ϕ(t, x) = ϕ1 ] in the first and, respectively, second phase, then the domain {(t, x); ϕ(t, x) = ϕ ∗ }, where ϕ ∗ ∈ (ϕ0 , ϕ1 ) may define the interface. It should be said, however, that the value of the temperature u does not determine the phase and so, if wants to steer the phase system in a certain steady state, one must act not only on the temperature u, but also on the phase state ϕ. For f2 ≡ 0, the second equation in (3.72) is derived from the kinetic equation ∂ϕ ∂t + ∇ϕ E(u, ϕ) = 0, where E is the Landau energy functional



 1 1 4 2 2 E(u, ϕ) = + (γ u − mv)ϕ dx. α|∇ϕ| + β ϕ −ϕ 2 O 2 So, the controller v, likes w in the first equation, is acting on temperature in order to restore (u, ϕ) to the equilibrium state (u∗ , ϕ ∗ ). Throughout in the sequel, α0 (d) is any real number which satisfies the following conditions: α0 (d) >

d +2 if d = 2, 3; α0 (d) = 2 if d = 1. 2

(3.74)

74

3 Exact Controllability of Parabolic Equations

We set uα0 (d) = u

2 1−

W0

1 ,α0 (d) α0 (d)

. (O)

Theorem 3.9 below is the main result. Theorem 3.9. Let (u∗ , ϕ ∗ ) ∈ H 2 (O)×H 2 (O) be any steady-state solution to (3.72). Then there is δ > 0 such that, for all (u0 , ϕ0 ) ∈ H 1 (O)×H 1 (O) satisfying the conditions u∗ − u0 α0 (d) + ϕ ∗ − ϕ0 H 1 (O) ≤ δ, 0

(3.75)

there is (w, v) ∈ L2 (Q)×L2 (Q) such that uw,v (T ) = u∗ , ϕ w,v (T ) = ϕ ∗ . Here (uw,v , ϕ w,v ) ∈ H 2,1 (Q)×H 2,1 (Q) is the solution to (3.72). A similar result remains true for the boundary control system ut + ϕt − kΔu = f1 ϕt − αΔϕ − β(ϕ − ϕ 3 ) + γ u = f2 u(0) = u0 , ϕ(0) = ϕ0 u = u, ¯ ϕ=ϕ

in Q, in Q, in O, on Σ.

(3.76)

Namely, we have Corollary 3.7. Let (u∗ , ϕ ∗ ) ∈ H 2 (O)×H 2 (O) be any steady-state solution to (3.72). Then there is δ > 0 such that, for all u0 , ϕ0 ∈ H 1 (O) satisfying (3.75), there are u, ¯ ϕ ∈ L2 (Σ)×L2 (Σ) such that ∗ ∗ uu,ϕ ¯ (T ) = u , ϕu,ϕ ¯ (T ) = ϕ . 2,1 (Q)×H 2,1 (Q) is the solution to (3.76). Here (uu, ¯ ϕ¯ , ϕu,ϕ ¯ )∈H As noted earlier, the function ϕ determines the phase of physical system (liquid or solid, for instance) and the above result amounts to saying that the system can be steered in equilibrium state by prescribing a certain values of temperature and phase functions on the boundary.

Proof of Theorem 3.9. Let y = ϕ − ϕ ∗ , z = u − u∗ . Then by redefining the controllers v, w, and the state (y, z), after some elementary calculation, Theorem 3.9 reduces to the local null controllability of the system yt − Δy + ay + by 2 + cy 3 + dz = mw zt − Δz + Δy + a1 y + b1 y 2 + c1 y 3 + d1 z = mv y(0) = y0 , z(0) = z0 y = 0, z = 0 where a, b, c, d, a1 , b2 , c1 , d1 ∈ L∞ (O).

in Q, in Q, in O, on Σ,

(3.77)

3.4 Local Controllability of Semilinear Parabolic Equations

75

According to the general scheme presented above, we shall consider first the linearized system yt − Δy + ξy + dz = mw zt − Δz + Δy + ηy + d1 z = mv y(0) = y0 , z(0) = z0 y = 0, z = 0

in Q, in Q, in O, on Σ,

(3.78)

where ξ, η ∈ L∞ (Q) are given functions. We have, Lemma 3.2. There are (w, v) ∈ Lα0 (d) (Q)×Lα0 (d) (Q) such that wLα0 (d) (Q) + vLα0 (d) (Q) ≤ μ(|ξ |∞ , |η|∞ )(|y0 |2 + |z0 |2 ) and y(T ) = 0, z(T ) = 0. Here μ : R → R is a positive function which is bounded on bounded subsets. To prove this lemma we shall consider the corresponding dual backward system pt + (Δ − ξ )p − (Δ + η)q = 0 in Q, in Q, qt + (Δ − d1 )q − dp = 0 p=q=0 on Σ,

(3.79)

and prove a Carleman estimate for (3.79). Lemma 3.3. There are s, λ > 0 such that   e2sα ϕ(p2 + ϕ 2 q 2 )dx dt + e2sα (ϕ −1 |∇p|2 + ϕ|∇q|2 )dx dt Q Q   + e2sα ϕ −1 |Δq|2 dx dt ≤ C(|ξ |∞ , |η|∞ ) ϕe2sα (p2 + ϕ 2 q 2 )dx dt. Q

Qω

Proof. We shall apply the Carleman estimate (2.2) in the second equation of (3.79). We obtain  e2sα ((sϕ)−1 |Δq|2 + sϕ|∇q|2 + s 3 ϕ 3 q 2 )dx dt Q 

 (3.80) 2sα 2 3 3 2sα 2 e p dx dt + s ϕ e q dx dt ≤ C(λ, |ξ |∞ ) Q

Qω

for λ ≥ λ0 and s ≥ s0 (λ).

76

3 Exact Controllability of Parabolic Equations

On the other hand, we have (t (T −t)p)t + (Δ − ξ )(t (T − t)p) = t (T − t)(Δ + η)q + (t (T − t))t p in Q, t (T −t)p = 0 on Σ. Then, again applying the Carleman inequality in the latter equation, we get   ϕp2 e2sα dx dt + s ϕ −1 |∇p|2 e2sα ≤ Cs 3 ϕp2 e2sα dx dt Q Q Q  ω +C ϕ −2 ((Δ + η)q)2 e2sα dx dt + C e2sα p2 dx dt.

 s3

Q

Q

Thus, for λ = λ0 and s ≥ s0 large enough, we have  s3

 ϕp2 e2sα dx dt + s ϕ −1 |∇p|2 e2sα dx dt Q Q   2 2sα 3 ≤ Cs ϕp e dx dt + C ϕ −2 ((Δ + d)q)2 e2sα dx dt Qω

Q

and so, by (3.80), we obtain 

 2 2sα

s3

ϕp e Q

dx dt +s

−1



ϕ |∇p| e dx dt ≤ Cs ϕp2 e2sα dx dt Qω

  2sα 2 4 3 2sα 2 ϕ e q dx dt, + C s e p dx dt + s 2 2sα

3

Q

Q

Qω

where C = C(|ξ |∞ , |η|∞ ). Therefore, for s large enough, we have 



ϕ −1 |∇p|2 e2sα dx dt  ≤ C(|ξ |∞ , |η|∞ ) ϕe2sα (p2 + ϕ 2 q 2 )dx dt

ϕp2 e2sα dx dt + Q

Q

Qω

and 

e2sα (ϕ|∇q|2 + ϕ −1 |Δq|2 + ϕ 3 q 2 )dx dt Q  ≤ C(|ξ |∞ , |η|∞ ) ϕe2sα (p2 + ϕ 2 q 2 )dx dt, Qω

as claimed. (Here and everywhere in the following, C(r1 , r2 ) is a positive function bounded on bounded subsets.) This completes the proof of Lemma 3.3.

3.4 Local Controllability of Semilinear Parabolic Equations

77

By Lemma 3.3, we get the following observability inequality for the solution (p, q) to system (3.79)  ϕe2sα (p2 + ϕ 2 q 2 )dx dt. (3.81) |p(0)|22 + |q(0)|22 ≤ C(|ξ |∞ , |η|∞ ) Qω

In fact, multiplying the first equation of (3.79) by ρp, where ρ > 0 is suitable chosen, the second by q and integrating on O, we see after some calculation involving Green’s formula that (|p(t)|22 + |q(t)|22 )t ≥ −C(|ξ |∞ , |η|∞ )(|p(t)|22 + |q(t)|22 . This yields 2 2 |p(0)|22 + |q(0)|22 ≤ C(|ξ |∞ , |η|∞ )(|p(t)|  2 + |q(t)|2 )

≤ C(|ξ |∞ , |η|∞ )sup (e−2sα ϕ −1 )

e2sα ϕ(p2 + ϕ 2 q 2 )dx, ∀t ∈ [0, T ]. Q

Then, integrating on some subinterval (t1 , t2 ) and using the Carleman inequality given in Lemma 3.3, we obtain (3.81), as claimed. Proof of Lemma 3.2. As in the previous situations (see (1.84)), we shall consider the optimal control problem  Min

e Q

−2sα

(ϕ

−1

w +ϕ 2

1 v )dx dt + ε

−3 2



 (y (x, T ) + z (x, T ))dx 2

2

O

subject to (3.78). Let (yε , zε ), (wε , vε ) be an optimal solution. Then, by the maximum principle, we have wε = me2sα ϕpε , vε = me2sα ϕ 2 qε , a.e. in Q, where (pε , qε ) is the solution to (3.79) with the final conditions 1 1 pε (T ) = − yε (T ), qε (T ) = − zε (T ). ε ε Thus, taking into account (3.81), we obtain that 

 1 e2sα (ϕpε2 + ϕ 3 qε2 )dx dt + (yε2 (x, T ) + zε2 (x, T ))dx ε Qω O  = − (y0 (x)pε (x, 0) + z0 (x)qε (x, 0))dx O  1 2 2 ≤ C(|ξ |∞ , |η|∞ )(|y0 |2 + |z0 |2 ) + 2 ϕe2sα (pε2 + ϕ 2 qε2 )dx dt. Qω

78

3 Exact Controllability of Parabolic Equations

Hence  e

−2sα

Q

 1 (ϕ dt+ (y 2 (x, T )+zε2 (x, T ))dx ε O ε ≤ C(|ξ |∞ , |η|∞ )(|y0 |22 + |z0 |22 ), ∀ ε > 0. −1

wε2 +ϕ −3 vε2 )dx

(3.82)

Thus, letting ε tend to 0, we infer that ∃(w, v) ∈ L2 (Q)×L2 (Q) such that ϕ −1/2 e−sα w ∈ L2 (Q), ϕ −3/2 e−sα v ∈ L2 (Q) and y(T ) = 0, z(T ) = 0. By (3.82), it follows that  Qω

e2sα ϕ(pε2 dx dt + ϕ 2 qε2 )dx dt ≤ C(|ξ |∞ , |η|∞ )(|y0 |22 + |z0 |22 )

(3.83)

and so, by Lemma 3.3, 

 e2sα ϕ(pε2 + ϕ 2 qε2 )dx dt + e2sα ϕ −1 |Δqε |2 dx dt Q Q  2sα −1 2 + e ϕ |∇pε | dx dt ≤ C(|ξ |∞ , |η|∞ )(|y0 |22 + |z0 |22 ).

(3.84)

Q

3s

We set θε = e 2 α ϕ −1 pε and notice that, by (3.79), we have 3s

3s

(θε )t + (Δ − ξ )θε = ϕ −1 e 2 α (Δ + η)qε + pε (e 2 α ϕ −1 ) 3s 3s + pε Δ(e 2 α ϕ −1 ) + 2∇pε · ∇(e 2 α ϕ −1 ). Then, by parabolic regularity and by estimate (3.84), we get θε H 2,1 (Q) ≤ C(|ξ |∞ , |η|∞ )(|y0 |2 + |z0 |2 ). Since H 2,1 (Q)⊂Lα0 (d) (Q) (we recall that, by the Sobolev embedding theorem, H 2,1 (Q)⊂L

2(d+2) d−2

(Q)), we infer that

pε ϕe2sα Lα0 (d) (Q) ≤ C(|ξ |∞ , |η|∞ )(|y0 |2 + |z0 |2 ). Hence wε Lα0 (d) (Q) ≤ C(|ξ |∞ , |η|∞ )(|y0 |2 + |z0 |2 ).

(3.85)

Similarly, it follows that vε Lα0 (d) (Q) ≤ C(|ξ |∞ , |η|∞ )(|y0 |2 + |z0 |2 ).

(3.86)

3.4 Local Controllability of Semilinear Parabolic Equations

79

Letting ε tend to zero into (3.85), (3.86), we conclude that (w, v) satisfy the conditions of Lemma 3.2. Proof of Theorem 3.9 (Continued). We set K = {ζ ∈ L∞ (Q); ζ L∞ (Q) ≤ R}

(3.87)

and let Φ : K → L2 (Q) be defined by Φ(ζ ) = {y ∈ L2 (Q); y(T ) = 0, z(T ) = 0; wLα0 (d) (Q) +vLα0 (d) )Q) ≤M}, (3.88) where (y, z) ∈ H 2,1 (Q)×H 2,1 (Q) is the solution to the linearized system yt − Δy + (a + bζ + cζ 2 )y + dz = mv zt − Δz + Δy + (a1 + b1 ζ + c1 ζ 2 )y + d1 z = mv y(0) = y0 , z(0) = z0 y=z=0

in Q, in Q, in O, on Σ.

(3.89)

We shall assume that  2 1− α

y0 ∈ W0

1 0 (d)

 ,α0 (d)

(O), z0 ∈ H01 (O).

We note first that, by Lemma 3.2, Φ(ζ ) = ∅ for ζ ∈ K if M ≤ μ(R, R)(|y0 |2 + |z0 |2 ).

(3.90)

Let us show now that for y0 α0 (d) +z0 H 1 (O) sufficiently small Φ(K)⊂K. 0 Indeed, by (3.89), we have the standard parabolic estimates yH 2,1 (Q) + zH 2,1 (Q) ≤ C(M + y0 H 1 (O) + z0 H 1 (O) ). 0

0

(3.91)

Next, by the first equation in (3.89) and the Lp − theory of parabolic boundary value problems, we have yW 2,1

α0 (d) (Q)

≤ C(R)(zH 2,1 (Q) + y0 α0 (d) ) ≤ C(R)(M + y0 α0 (d) + z0 H 1 (O) ).

(3.92)

0

(Here C = C(r) is a positive and continuous and monotonically increasing function.) The last estimate follows by the following argument. By the first equation, we have y2H 2,1 (Q) ≤ C(R)(M + z2L2 (Q) + y0 2H 1 (O) ) 0

80

3 Exact Controllability of Parabolic Equations

and, by the second equation, z2H 2,1 (Q) ≤ C(R)(M + y2H 2,1 (Q) + z0 2H 1 (O) ). 0

Next, we multiply the first equation by y and the second by δz, where δ > 0 is sufficiently small. Then, after some calculation involving Gronwall’s lemma, we get the desired estimate. Since α0 (d) > d+2 2 , α0 (d) = d + 2 for d = 1, 2, 3, we infer by (3.92) that yL∞ (Q) ≤ C(r)(M + y0 α0 (d) + z0 H 1 (O) ).

(3.93)

0

Hence, by (3.90) and (3.93), we obtain that yL∞ (Q) ≤ C(R)(μ(R, R) + 1)(y0 α0 (d) + z0 H 1 (O) ) ≤ R 0

if y0 α0 (d) + z0 H 1 (O) is sufficiently small. 0

Moreover, it follows by (3.91) via standard compactness results in L2 (Q) that Φ(K) is a compact subset of L2 (Q). It is also easily seen that Φ(ζ ) is a closed convex subset for each ζ ∈ K and that the map ζ → Φ(ζ ) is upper semicontinuous from L2 (O) to itself. Then, once again, by the Kakutani fixed point theorem, it follows that Φ has a fixed point y, i.e., there is (w, v) ∈ L2 (Q)×L2 (Q) such that y(T ) = 0, z(T ) = 0. This completes the proof of Theorem 3.9. Proof of Corollary 3.7. By the substitution y = ϕ − ϕ ∗ , z = u − u∗ the problem reduces to that of boundary controllability of system (3.77). Then the conclusions of Corollary 3.7 follow from Theorem 3.9, as in the previous situations. Namely, let O d be an open smooth subset of R such that O⊂O. We extend, by standard procedure, to = O×[0, T ) and y0 , z0 on O the functions a, b, c, d, a1 , b1 , c1 , d1 on Q  2 1− α

( y0 , z0 ) ∈ W0

1 0 (d)

 1 (O)×H 0 (O).

z0 H 1 (O) Then, by Theorem 3.9 and its proof, we know that for  y0 α0 (d) +  0 2 2 sufficiently small there are ( u, w ) ∈ L (Q)×L (Q) such that y (T ) = 0, z(T ) = 0, where ( y , z) is the solution to (3.77). z|Q is the solution to system (3.77), where v = w = 0 Clearly, y = y |Q , z = and with the boundary conditions y |Σ , z|Σ = y |Σ y|Σ = and y(T ) = z(T ) = 0. This completes the proof.

3.5 Controllability of the Kolmogorov Equation

81

Remark 3.5. Theorem 3.9 is, in particular, important because the Stefan two phase problem (see, e.g., [18, 26]), yt − Δβ(y) = mu in Q, which describes the dynamic of melting processes, is so far beyond the controllability approach developed in the previous section. (See, however, [50] for a controllability result in 1 − D.) Remark 3.6. Theorem 3.9, which was established in [23], was meantime extended to the controllability of phase-field system (3.72) using an internal control only (see [3, 10, 77]).

3.5 Controllability of the Kolmogorov Equation On unbounded domains O a controllability result similar to Theorem 3.1 is not true in general. (See examples in [93, 94].) We shall present, however, in the following such a result for the linear parabolic equation with a drift term, 1 ∂y − Δy + F (x) · ∇y = 1O0 u in (0, T ) × O, ∂t 2 ∂y = 0 on (0, T ) × ∂O, ∂ν y(0) = y0 (x), x ∈ O,

(3.94)

where O is an open and convex set in Rd (eventually unbounded), d ≥ 1, O0 is an open subset of O, and F : Rd → Rd is a C 1 -continuous, coercive, and bounded mapping of gradient type. The linear parabolic equation (3.94) is known in literature as the Kolmogorov equation and arises in the theory of stochastic processes on the domain O. In the special case where div F = 0, (3.94) reduces to the Fokker-Planck equation on O. Given a closed convex set K ⊂ Rd , the recession cone of K is defined by recc(K) = {y ∈ Rd ; x + λy ∈ K, ∀ x ∈ K, ∀ λ ≥ 0} or, equivalently, recc(K) =



λ(K − y), ∀ y ∈ K.

λ>0

If K is bounded, then recc(K) = {0}, but otherwise recc(K) is an unbounded set (cone). Denote by pK the Minkowski functional (gauge) associated with the closed convex set K, that is,   1 pK (x) = inf λ ≥ 0; x ∈ K , ∀ x ∈ Rd . (3.95) λ

82

3 Exact Controllability of Parabolic Equations ◦

We recall that pK is subadditive, positively homogeneous and, if K = ∅, then ◦

K = {x ∈ Rd ; pK (x) < 1}, ∂K = {x ∈ Rd ; pK (x) = 1}.

(3.96)

◦

(Here, K is the interior of K and ∂K is its boundary.) ◦

If 0 ∈ K, then recc(K) = {x ∈ K; pK (x) = 0}.

(3.97)

We shall assume in the following that (i) O is an open convex set of Rd with C 2 -boundary ∂O, 0 ∈ O. (ii) F = ∇g, where g ∈ C 2 (Rd ) is convex and sup{|F (x)|d + DF (x); x ∈ Rd } < ∞, g(x) ≥ α1 |x|d + α2 , ∀ x ∈

Rd ,

(3.98) (3.99)

where α1 > 0 and α2 ∈ R. Here, DF stands for the differential of F : Rd → Rd and  ·  is the norm in L(Rd , Rd ). (iii) O0 is an open subset of O which contains an open subset O1 such that O 1 ⊂ O0 and inf{|∇pO (x)|d ; x ∈ O \ O1 } = γ > 0.

(3.100)

We note that pO ∈ C 2 (O \ recc(O)) and so (3.100) makes sense. Taking into account (3.97), we see by (iii) that recc(O) ⊂ O1 ⊂ O0 . A convenient functional space to treat equation (3.94) is L2 (O; μ), where μ is a Gaussian measure with the density ρ : Rd → [0, ∞) defined by  ρ(x) =

! "−1 exp(−2g(x)) O exp(−2g(x))dx , x ∈ O, 0, x ∈ Rd \ O.

(3.101)

A simple calculation shows that, if g ∈ C 2 (Rd ), then ρ is a solution to the Neumann problem 1 Δρ + div (Fρ) = 0 in O, 2 ∂ρ + (F · ν)ρ = 0 on ∂O. ∂ν

(3.102)

(Otherwise, (3.102) holds in the weak distributional sense.) We consider the probability measure μ defined by dμ = ρ dx

(3.103)

3.5 Controllability of the Kolmogorov Equation

83

and consider in the space L2 (O; dμ) = L2 (O; μ) the operator 1 Ny = − Δy + F · ∇y, y ∈ D(N ),  2 ∂y 2,2 = 0 on ∂O . D(N) = y ∈ W (O; μ); ∂ν

(3.104)

In fact, N is the Kolmogorov operator associated with the stochastic reflection equation dX + F (X)dt + NK (X)dt  dWt , X(0) = x,

(3.105)

where NK is the normal cone to K and Wt is a Wiener d-dimensional process. As shown in [38], the operator N is m-accretive in L2 (O; μ) and so it generates a C0 semigroup of contractions e−tN in L2 (O; μ), given by (e−tN y0 )(x) = Ey0 (X(x, t)), x ∈ O, ∀ y0 ∈ L2 (O; dμ), where X(x, t) is the solution to (3.105). In particular, this implies that problem (3.94), or equivalently dy + Ny = 1O0 u, t ∈ [0, T ], y(0) = y0 , dt

(3.106)

has, for each y0 ∈ L2 (O; μ) and all u ∈ L2 (0, T ; L2 (O; μ)), a unique mild solution y u ∈ C([0, T ]; L2 (O; dμ)), that is, y u (t) = e−tN y0 +

 0

t

e−(t−s)N (1O0 u)(s)ds, t ∈ [0, T ].

(3.107)

Now, we can formulate the main controllability result. Theorem 3.10. Under assumptions (i)–(iii), for each 0 < T < ∞ and all y0 ∈ L2 (O; μ) there is at least one controller u ∈ L2 (0, T ; L2 (O; μ)) such that y u (T ) ≡ 0. Theorem 3.10 will be proved as Theorem 3.1 via the Carleman inequality for the backward dual equation associated with (3.106) similar to Theorem 2.1. The main difference is that here the Lebesgue measure will be replaced by the probability measure μ. It should be said that in equation (3.94) (respectively (3.104)) the coefficient 12 in front of Δ was taken for the sake of symmetry only. Of course, one can replace it by any constant a > 0. As a matter of fact, the operator 12 Δ can be replaced by any second order elliptic operator with constant coefficients.

84

3 Exact Controllability of Parabolic Equations

In the following, we discuss the form of O0 arising in (iii) in some special cases. Assume that O = {(x , xd ) ∈ Rd ; xd > φ(x ) − b},

(3.108)

where b > 0 and φ ∈ C 2 (Rd−1 ) is a convex function satisfying φ(0) = 0 and d φ(u) ≥ a|u|m d−1 , ∀ u ∈ R ,

(3.109)

where a > 0 and 1 ≤ m < ∞. (We have always such a local representation of O.) It is readily seen that recc(O) = {(0, xd ); xd ≥ 0} for m > 1, recc(O) = {(x , xd ); xd ≥ a|x |d−1 } for m = 1. (Here, x = x1 , . . . , xd−1 ).) We have Proposition 3.1. Let η > 0 be the solution to the equation xd = ηφ

x

η

− bη.

(3.110)

Then, η = pO and any set O0 = O \ Gεα , Gεα := {(x , xd ) ∈ O; |x |d−1 ≤ αpO (x) − ε}, α > 0, 0 < ε < b, satisfies (iii). Proof. By (3.95), we see that, for each x ∈ O, pO (x) = η(x) is the unique positive solution to (3.110). We have ∂η η     = x ∂x2 ηφ η − ∇φ xη · x − b and, since φ and ∇φ are bounded on bounded sets, we infer that, for each α > 0,    ∂η 

   ∂x  ≥ γ (α) > 0, for |x |d−1 ≤ αη, 2 which implies that inf{|∇pO (x)|; x ∈ O \ O1 } > 0, where O1 = {(x , xd ); |x |d−1 > αpO (x)} ⊂ O0 .

3.5 Controllability of the Kolmogorov Equation

85

Example 3.1. Let φ(u) = a|u|m d−1 , where a > 0 and m ≥ 2. Then (3.110) reduces to xd = aη1−m |x |m d−1 − bη.

(3.111)

Equivalently, by m + where y = α > 0,

xd |x |d−1 ·

xd m−1

|x |d−1

y m−1 − a = 0,

(3.112)

A simple analysis of equation (3.112) reveals that, for each y ≥ α if 0 < xd ≤ ζ (α)|x |dm−1 .

Then, by Proposition (3.1), it follows that, for each γ > 0, 0 < ε < b, and Gεγ = {x xd ) ∈ O; xd ≤ γ |x |dm−1 − ε},

(3.113)

the set O0 ⊂ O \ Gεγ satisfies (iii). Then, Theorem 3.10 implies that Corollary 3.8. Let O = {(x , xd ); xd > a|x |m d − b} for a, b > 0, 2 ≤ m < ∞. Then, we may take O0 , any set of the form {(x , xd ); xd > γ |x |dm−1 − ε}, where γ > 0 and 0 < ε < b.

(3.114)

In particular, it follows by Theorem 3.10 that (3.94) is exactly null controllable with controllers v = 1O0 u in any set O0 of the form (3.114). At finite distance, this set can be taken as close as we want of the recession cone {(0, xd ); xd ≥ 0}. Remark 3.7. The conclusion of Corollary 3.8 remains true if O is of the form (3.108) away from origin, that is, φ(u) = a|u|m d−1 for |u|d−1 ≥ λ > a, 1 < m < ∞, a > 0, b > 0.

(3.115)

Indeed, the calculation in Example 3.1 shows that (3.113) holds because only the values |x |d−1 +xd large enough are relevant. This extends to the case m = 1, where pO (x) = b−1 (a|x |d−1 − xd )− for |x |d−1 ≥ λ > 0, and so O0 = {(x , xd ); xd − a|x |d−1 ≥ −ε}, where ε > 0 is arbitrarily small.

(3.116)

86

3 Exact Controllability of Parabolic Equations

Proof of Theorem 3.10. Denote by N ∗ the dual operator of N in the space L2 (O; μ), that is, #

N ∗ p, y

$ L2 (O;μ)

= p, NyL2 (O;μ) ,

for all y ∈ D(N ) and p ∈ D(N ∗ ). A simple calculation involving (3.102) and (3.104) shows that 1 N ∗ p = − Δp − F · ∇p − ∇(log ρ) · ∇p,  2 ∂p ∗ 2,2 = 0 on ∂O . D(N ) = p ∈ W (O); ∂ν

(3.117)

Moreover, taking into account that F = ∇g = − 12 ∇(log ρ), we see by (3.117) that N ∗ = N. As it is well known, for the exact controllability of (3.106) we need the observability inequality  p(0)2L2 (O;μ) ≤ C



T

dt 0

O0

|p(x, t)|2 dμ = C

 T 0

O0

ρ(x)|p(x, t)|2 dx dt, (3.118)

for any solution p to the backward equation dp − N ∗ p = 0, t ∈ (0, T ), dt or, equivalently, ∂p 1 + Δp − F · ∇p = 0 in (0, T ) × O, ∂t 2 ∂p =0 on (0, T ) × ∂O. ∂ν

(3.119)

To get (3.118), we prove first a Carleman-type inequality for solutions p to equation (3.119). To this end, proceeding as in Section 2.1, we consider an open set O1 , O 1 ⊂ O0 as in assumption (iii), and set α(x, t) =

e−λψ(x) e−λψ(x) − e2λψC(O) , ϕ(x, t) = , t (T − t) t (T − t)

x ∈ O, t ∈ (0, T ),

where ψ is the function given by Lemma 3.4 below. Lemma 3.4. There is ψ ∈ C 2 (O) such that ψ(x) > 0, ∀ x ∈ O, ψ(x) = 0, ∀ x ∈ ∂O,

(3.120)

3.5 Controllability of the Kolmogorov Equation

87

|∇ψ(x)|d ≥ γ > 0, ∀ x ∈ O \ O1 ,

(3.121)

sup{|∇ψ(x)|d + |Dx2i xj ψ(x)|; i, j = 1, . . . , d} < ∞.

(3.122)

Proof. Let O2 be an open subset of O1 such that O 2 ⊂ O1 and dist(∂O2 , ∂O1 ) > 0 is sufficiently small. Then, consider a function X ∈ Cb∞ (Rd ) such that 0 ≤ X ≤ 1, X = 0 on O \O1 and X = 1 in O 2 . (This function can be constructed in a standard way via mollifiers technique.) Then, we set ψ = 1−(1−X )pO . Taking into account assumption (iii), we see that ψ satisfies (3.120), (3.121) (because ∇ψ = −∇pO on O \ O1 ). Moreover, by Lemma 3.5 below, (3.122) follows, too. The following Carleman inequality is exactly of the same form as that given in Theorem 2.1, so we give it without proof (see [31] for the proof). Proposition 3.2. There are λ0 > 0 and a function s0 : R+ → R+ such that, for λ ≥ λ0 and s ≥ s0 ,  T  T e2sα (s 3 ϕ 3 p2 + sϕ|∇p|2 dμ dt ≤ Cλ s 3 e2sα ϕ 3 p2 dμ dt, O

0

0

O0

(3.123)

for all the solutions p to (3.119). By (3.123), we obtain estimate (3.118). Namely,

Corollary 3.9. The observability inequality (3.118) holds for all the solutions p to (3.119). Proof. We note first that p, NpL2 (O;μ) =

1 2

 O

|∇p|2 dμ,

which yields d dt



 p (x, t)dμ − 2

O

O

|∇p(x, t)|2 dμ = 0, ∀ t ≥ 0.

We have, for all 0 ≤ τ ≤ t < ∞,  O

p2 (x, τ )dμ +

 t

 O

τ

|∇p(x, θ )|2 dμ dθ =

O

p2 (x, t)dμ,

and, therefore,    p2 (x, 0)dμ ≤ p2 (x, t)dμ ≤ γ (t) e2sα ϕ 3 p2 dμ, O

O

O

where γ (t) = sup{e−2sα ϕ −3 (x, t); x ∈ O} ≤ C exp



μs t (T −t)



(3.124)

(3.125)

, μ = 2e2λ ψC(O) .

88

3 Exact Controllability of Parabolic Equations

Integrating on (t1 , t2 ) ⊂ (0, T ), we obtain by (3.123) the desired inequality (3.118) with a suitable constant C. We note also that (3.124) implies that p ∈ W 1,2 (0, T ; L2 (O; μ)). Without loss of generality, we may assume in the following that p ∈ L2 (0, T ; W 2,2 (O; μ)). (Taking into account the structure of the domain D(N) this happens if p(T ) ∈ D(N ) which, without any loss of generality, we may assume.) Also, for the sake of simplicity, we shall prove Proposition 3.2 for the equation ∂p + Δp − F · ∇p = 0 in (0, T ) × O, ∂t ∂p =0 on (0, T ) × ∂O, ∂ν

(3.126)

because (3.119) is obtained from (3.126) by rescaling the time t. Lemma 3.5. Let O be an open and convex set with C 2 -boundary ∂O and 0 ∈ O. Let pO be the gauge of O. Then pO ∈ C 2 (O \ recc(O)),

(3.127)

sup{|∇pO (x)|d ; x ∈ O \ recc(O)} < ∞,

(3.128)

sup{|Dx2i xj pO (x)|; x ∈ O \ recc(O)} < ∞, i, j = 0, 1, 2 . . . , d.

(3.129)

Proof. We set η = p−1 on O \ recc(O). Then, η(x)x ∈ ∂O and, since ∂O is of O class C 2 , it is locally represented as xd = φ(x ), where φ is convex and of class C 2 . We have, therefore, xd η = φ(ηx ), ∀ x ∈ O \ recc(O), and, since ∇φ is monotone and of class C 1 in Rd−1 , αI + ∇φ, is invertible for all α > 0. So, we conclude that η is of class C 2 on O \ recc(O). Since pO is positively homogeneous, subadditive, and 0 ∈ O, we have, for all y ∈ O with |y|d ≤ ε sufficiently small, ∇pO (x) · y ≤ pO (y), ∀ x ∈ O \ recc(O), and this, clearly, implies (3.128). Now, if we differentiate two times with respect to x the equation pO (λx) = λpO (x), we obtain λDij2 pO (λx) = Dij2 pO (x), ∀ x ∈ O \ recc(O). This yields Dij2 pO (y) =

y  1 2 Dij pO , y ∈ O \ recc(O), λ λ

3.6 Exact Controllability of Stochastic Parabolic Equations

89

and, for λ = |y|d /ε, we get Dij2 pO (y) =



y ε , ∀ y ∈ O \ recc(O), D 2 pO ε |y|d |y|d

which, clearly, implies (3.129), as desired.

3.6 Exact Controllability of Stochastic Parabolic Equations The linear parabolic stochastic equation is a model for linear diffusion dynamic perturbed by a Gaussian random driving force. In most situations, this force can be viewed as a linear multiplicative Gaussian noise and so the dynamic can be represented as the stochastic parabolic equation dt X(ξ, t) − Δξ X(ξ, t)dt + a(ξ, t)X(ξ, t)dt + (b(ξ, t) · ∇ξ X(ξ, t))dt N

μk (ξ, t)X(ξ, t)dβk (t) in (0, T )×O, = 1O0 u(ξ, t)dt + X(ξ, 0) = x(ξ ), ξ ∈ O, X(ξ, t) = 0 on (0, T )×∂O,

k=1

(3.130) {βk }N k=1

where is a linear independent system of Brownian motion in a probability space {O, F , P}, O is a bounded and open subset of Rd , d ≥ 1, with smooth boundary ∂O and O0 is any open subset of O. By 1O0 we have denoted, as usually, the characteristic function of O0 . As regard the functions a, b and μk , we assume that the following hypothesis holds: (i) a ∈ L∞ (O×(0, T )), b ∈ C 1 (O×[0, T ]; Rd ), μk ∈ C 2 (O×[0, T ]), k = 1, 2, . . . , N. Here and everywhere in the following, the controller u : O×(0, T )×Ω → R, u = u(ξ, t), is taken in the class of all adapted L2 (O)-valued processes with respect to the natural filtration {Ft }t≥0 induced by the Brownian motions {βk }N k=1 . It should be said that also in this case, the exact null controllability is related and, in a certain sense, which will be made precise later on, equivalent with the observability of the dual backward stochastic equation dt p(ξ, t) + Δξ (ξ, t)dt − a(ξ, t)p(ξ, t)dt + div(b(ξ, t)p(ξ, t))dt N N  

μk (ξ, t)qk (ξ, t) dt = qk (ξ, t)dβk (t), (ξ, t) ∈ O×(0, T ), + k=1

k=1

p(ξ, t) = 0, (ξ, t) ∈ ∂O×(0, T ).

(3.131)

90

3 Exact Controllability of Parabolic Equations

Given the stochastic basis {Ω, F , P, {Ft }t≥0 } and H a Hilbert space with the norm | · |H and 1 ≤ p < ∞, we denote by MP2 (0, T ; H ) the space of all H -valued progressively measurable processes X : Ω × (0, T ) → H , such that  T 2 |X(t)|2H dt < ∞, XM 2 (0,T ;H ) = E P

0

where E is the expectation. Denote by CP ([0, T ]; H ) the space of all Ft -adapted processes X ∈ MP2 (0, T ; H ) which have a modification in C([0, T ]; L2 (Ω)). We denote by L2ad (Ω; C([0, T ]; L2 (O))) the space of all Ft -adapted processes X : O ×[0, T ] → L2 (O), X ∈ C([0, T ]; L2 (O)), P-a.s., and EX2C([0,T ];L2 (O)) < ∞. We denote by L2T (Ω, O) the space of all random variables η : Ω → L2 (O) such that Z is FT measurable and  E |Z(ξ )|2 dξ < ∞. O

By solution to (3.130), we mean a process X ∈ L2ad (Ω; C([0, T ]; L2 (O))) which satisfies P-a.s. the equation  t  t X(ξ, t) = x + Δξ X(ξ, s)ds − (a(s)X(ξ, s) + b(s) · ∇ξ X(ξ, s))ds 0 0  t N  t

+ 1O0 u(ξ, s)ds + μk (ξ, s)X(ξ, s)dβk (s), ∀ t ∈ (0, T ), ξ ∈ O. 0

k=1 0

(3.132) Here the integral arising in the right-hand side of (3.132) is taken in sense of Itô with values in H −1 (O). (We refer to [64] for existence and uniqueness of such a solution.) The solution X to (3.130) is denoted by Xu . In order to study the exact null controllability of (3.130), we shall reduce it by a rescaling procedure presented below to a random parabolic equation and apply to this equation the controllability results established in Section 3.1. Namely, we set W (ξ, t) = μ(ξ, t) =

N

μk (ξ, t)βk (t), (ξ, t) ∈ (0, T )×O,

k=1 N

k=1

∂μk 1 2 (t, ξ )βk (t) + μk (ξ, t) , (ξ, t) ∈ (0, T )×O. ∂t 2

By the substitution X(ξ, t) = eW (ξ,t) y(ξ, t),

(3.133)

3.6 Exact Controllability of Stochastic Parabolic Equations

91

equation (3.130) reduces to the random parabolic equation ∂y − e−W Δ(eW y) + (a + μ)y + e−W b · ∇(eW y) = 1O0 v in O×(0, T ), ∂t (3.134) y(ξ, 0) = x(ξ ) in O, y = 0 on ∂O×(0, T ), where v = e−W u. Indeed, if y is a regular solution to (3.134) (for instance, absolutely continuous in t), which is progressively measurable in (t, ω) in the probability space {Ω, P, F , Ft } and y ∈ MP2 (0, T ; H01 (O) ∩ H 2 (O)), then, by Itô’s formula in (0, T ) × Ω × O, we have dX = yd(eW ) + eW

∂y dt ∂t

in (0, T )×O

and d(eW ) = eW dW +

N N N N

1 W 2 1 ∂μk μk dt = eW μk dβk + eW μ2k dt + eW e βk dt. 2 2 ∂t k=1

k=1

k=1

k=1

Then, we obtain for y the random equation (3.134), as claimed. On the other hand, any Ft -adapted solution t → y(t) to equation (3.134) leads via transformation (3.133) to a solution X to (3.130) in the sense of the above definition. We write (3.134) as ∂y − Δy + F y + G · ∇y = 1O0 v in (0, T )×O ∂t y(0) = x in O, y = 0 in (0, T ) × ∂O,

(3.135)

where F = a + μ + b · ∇W − |∇W |2 − ΔW,

G = −2∇W + b.

(3.136)

We may, equivalently, rewrite (3.135) as 

t

y(t) = U (t, 0)x + 0

U (t, s)1O0 v(s)ds, t ∈ [0, T ],

(3.137)

where U (t, s) : [0, T ] → L(L2 (O), L2 (O)) is the evolution generated by the operator z → −Δz+F z+G·∇z with the domain H01 (O)∩H 2 (O) (see Section 1.5). It is easily seen that the operator function t → U (t, s) is Ft -adapted on [0, T ]. We note also that, if μk (t, ξ ) ≡ μk (t), then (3.135) is a deterministic linear parabolic system which, by virtue of Theorem 3.1, is exactly null controllable.

92

3 Exact Controllability of Parabolic Equations

Now, we define the controllability target set ST for system (3.130) as   ST = η = 0

T

U (T , t)v(t)dt, v ∈ MP2 (0, T ; L2 (O)),  E[v(t)|Ft ] = 0, ∀ t ∈ [0, T ] .

(3.138)

(For each σ -algebra G ⊂ F0 , E(Y |G ) denotes the conditional expectation of the random variable Y with respect to G .) We note that ST is a linear closed subspace of L2T (Ω; O). Theorem 3.11 below is the main controllability result. Theorem 3.11. Let T > 0 and let O0 be an arbitrary open subset of O. Then, under hypothesis (i), for each x ∈ L2 (O), there is u∗ ∈ MP2 (0, T ; L2 (O)) such that  E

Q0

|u∗ |2 dt dξ ≤ C|x|22 ,

∗

Xu (T , ξ ) ∈ ST .

(3.139) (3.140)

Here C is independent of x. Theorem 3.11 amounts to saying that system (3.130) is exactly ST controllable on an interval [0, T ] by an internal MP2 -controller u∗ with support in O0 . Denote by ST⊥ the orthogonal of ST in L2T (Ω; O). We consider, now, the boundary control stochastic system dt X − ΔX dt + aX dt + (b · ∇X)dt =

N

Xμk dβk in (0, T )×O,

k=1

X = u on (0, T )×∂O, P-a.s. X(ξ, 0) = x(ξ ), ∀ ξ ∈ O.

(3.141)

We have Theorem 3.12. Under assumption (i), for each T > 0 and x ∈ L2 (O), system (3.141) is exactly ST -controllable, that is, there are u ∈ MP2 (0, T ; L2 (∂O)) and X ∈ CP ([0, T ]; L2 (O)), which satisfy (3.141) and  X(T ) ∈ ST , E Σ

|u|2 dt dξ ≤ C|x|22 .

(3.142)

Theorem 3.12 follows by Theorem 3.11 by the following simple argument, already used in the controllability of deterministic systems (see Corollary 3.2). ⊃ O and for O0 = O \ O Namely, one applies Theorem 3.11 on the domain O to equation (3.130), where a, b, μk are replaced by a : O×(0, T ) → R, b : O×(0, T ) → Rd , μk : (0, T ) → R which are extensions of a, b, μk to

3.6 Exact Controllability of Stochastic Parabolic Equations

93

Then, O×(0, T ). The initial value x ∈ L2 (O, F0 , P) is an extension of x to O. and X ∈ CP ([0, T ], L2 (O)) ∩ by Theorem 3.11 there is u ∈ MP2 (0, T ; L2 (O)) which satisfy equation (3.130) (respectively, (3.131)) on MP2 (0, T ; H 1 (O)), ) ∈ ST . (0, T )×O and such that X(T If we take u = X|∂O , we see by the trace theorem that u ∈ MP2 (0, T ; L2 (∂O)) ∈ CP ([0, T ]; L2 (O)) satisfy the conditions of Theorem 3.12. and X = X| O (Estimate (3.142) follows by (3.139) via trace theorem.) In this case, too, the solution X = Xu satisfies (3.141). Remark 3.8. A similar result remains true for the boundary control u on some segment Γ ⊂ ∂O of the boundary with nonempty interior. By virtue of the duality relation between exact controllability and observability, by Theorem 3.11 we derive the following internal observability result for the backward stochastic equation (3.131). Theorem 3.13. Under assumption (i), there is C independent of p such that  |p(0)|2 ≤ CE

1 2

O0

p dt dξ

2

,

Q0 = (0, T )×O0 ,

(3.143)

for all the solutions {p ∈ CP ([0, T ]; L2 (O)) ∩ MP2 (0, T ; H01 (O)), qk ∈ MP2 (0, T ; L2 (O)), k = 1, . . . , N } to the dual backward equation (3.131) such that p(T ) ∈ ST⊥ . (We note that, since the filtration {Ft }t≥0 is natural, that is, it is induced by the system {βj }N j =1 , the process p(0) is deterministic.) We have also the following boundary observability inequality. Theorem 3.14. There is C independent of p such that    ∂p  |p(0)|2 ≤ CE  Σ ∂ν

1 2 2   dt dξ , 

(3.144)

for all solutions p ∈ MP2 (0, T ; H 2 (O)) to the dual backward stochastic equation (3.131) with p(T ) ∈ ST⊥ . Proof of Theorem 3.11. By Theorem 3.2, for each ω ∈ Ω, equation (3.135) is exactly null controllable, that is, there are a controller v ∈ C([0, T ]; L2 (O)) and y v ∈ C([0, T ]; L2 (O)) ∩ L2 (0, T ; H01 (O)) which satisfy (3.135) and such that y(ξ, T ) = 0, a.e. in O. However, this does not complete the proof since a priori is not known that such a controller v = v(ξ, t, ω) is an Ft -adapted process. (In fact, N  this happens if W (t, ξ ) ≡ μk (t)βk (t) because, in this case, equation (3.135) is k=1

deterministic.) We set v ∗ (t) = E[v(t)|Ft ], t ∈ [0, T ],

(3.145)

94

3 Exact Controllability of Parabolic Equations

and consider the solution y ∗ to equation (3.135) where v is replaced by v ∗ , that is, ∂y ∗ − Δy ∗ + F y ∗ + G · ∇y ∗ = 1O0 v ∗ in (0, T ) × O, ∂t y ∗ (0) = x in O, y ∗ = 0 on (0, T ) × ∂O.

(3.146)

We set z = y ∗ − y v and obtain ∂z − Δz + F z + G · ∇z = 1O0 (v ∗ − v) in (0, T ) × O, ∂t z(0) = 0 in O, z∗ = 0 on (0, T ) × ∂O.

(3.147)

We have 

T

z(T ) = 0

U (T , t)(1O0 (v ∗ (t) − v(t)))dt =



T

ζ (t)dt, P-a.s.

0

By (3.145), we see that E[ζ (t)|Ft ] = 0, ∀ t ∈ [0, T ], because E[v(t)−v ∗ (t)|Ft ] = 0 and t → U (T , t) is Ft -adapted. On the other hand, since y v (T ) = 0, this yields 

∗

y (T ) =

T

U (T , t)ζ (t)dt. 0

Hence y ∗ (T ) ∈ ST , as claimed. Moreover, by (3.16), we have  0

T

 O

(u∗ )2 dt dξ ≤ CT |x|22 , P-a.s.,

where CT = CT (ω) is estimated by CT ≤ C1 sup

m

t∈[0,T ] k=1

 for some natural m. This implies that E

T

0 C|x|22 .

therefore, E sup{|y ∗ (t)|22 ; t ∈ [0, T ]} ≤ This completes the proof of Theorem 3.11.

|βk (t)|m ,  O

|v ∗ (t)|2 dt dξ ≤ CT∗ |x|22 and,

3.7 Approximate Controllability of Stochastic Parabolic Equation

95

Proof of Theorem 3.13. If p is the solution to the dual backward equation (3.131), with the final condition p(T ) = pT in O, then by Itô’s formula we have % &   pu dt dξ , (3.148) E Xu (ξ, T )pT (ξ )dξ − x(ξ )p(ξ, 0)dξ = E O

O

Q0

for all u ∈ MP2 (0, T ; L2 (O)). In (3.148), we take u = u∗ to obtain that, since the filtration {Ft }t≥0 is the natural one, p(0) is P-a.s. deterministic),       x(ξ )p(ξ, 0)dξ ≤ E  

1 

1 2 2 2 E (u ) dt dξ (1O0 p) dt dξ Q0 Q 

1 0 2 ≤ C|x|22 E p2 dt dξ ,

O

∗ 2

Q0

  1 2 because pT ∈ ST⊥ . This yields |p(0)|2 ≤ C E Q0 p2 dt dξ , as claimed. Proof of Theorem 3.14. Estimate (3.144) follows by (3.131) and (3.141) which   ∂p u dt dξ, where u imply via Itô’s formula the equality O x(ξ )p(ξ, 0)dξ = E Σ ∂ν is a boundary controller provided by Theorem 3.13.

3.7 Approximate Controllability of Stochastic Parabolic Equation We shall prove here an approximate controllability result for (3.130) obtained from exact controllability of the stochastic n-dimensional equation dY + A(t)Y dt = B(t)u(t)dt +

N

σk (t, Y )dβk ,

(3.149)

k=1

where A and B are matrices. Theorem 3.15. Assume that hypothesis (i) holds. Then, for every ε > 0, there is an (Ft )t≥0 -adapted controller uε ∈ MP2 (0, T ; L2 (O)) such that P[Xuε (t)L2 (O) ≤ ε, ∀ t ≥ T ] ≥ 1 − ε.

(3.150)

Proof. For n ∈ N, we set n = X

n

i=1

Xin ei , un =

n

i=1

uni ei ,

(3.151)

96

3 Exact Controllability of Parabolic Equations

2 where {ei }∞ i=1 is the orthonormal basis in L (O) defined

−Δei = λi ei in O; ei = 0 on ∂O. We approximate equation (3.130) by the finite dimensional control system dXn + An Xn dt + Dn (t)Xn dt = Xn (0) = xn = {

#

$

x, ej 2 }nj=1 .

N

σk (t, Xn )dβk + Bn un dt,

k=1

(3.152)

Here Xn = {Xin }ni=1 , un = {uni }ni=1 , n An = diagλ i i=1 ,  n  Bn =  O0 ei ej dξ i,j =1 ,

Dn (t) =  a(t)ek + b(t) · ∇ek , ei 2 ni,k=1 ,  n # $ n  σk (t, Xn ) = σki (t, Xn ) = Xjn μk (t)ej , ei 2 j =1

i=1

and ·, ·2 is the scalar product of L2 (O). We have Lemma 3.6. Let n arbitrary but fixed. For each ε > 0, there is an (Ft )t≥0 -adapted controller unε ∈ L∞ ((0, T ) × Ω; Rn ) such that, if τn = inf{t ≥ 0; |Xn (t)|n = 0},

(3.153)

P[τn ≤ T ) ≥ 1 − ε.

(3.154)

then

Proof. We consider in equation (3.152) the feedback controller un = −ρ sign(Bn Xn ),

(3.155)

where sign v = |v|v n if v = 0, sign 0 = {v ∈ Rn ; |v|n ≤ 1}. By a standard device based on the approximation of the m-accretive mapping y → sign(Bn y) by a Lipschitzian monotone mapping (Yosida approximation) (see, e.g., (1.7)), it follows that the corresponding closed loop system dXn +An Xn dt+Dn (t)Xn dt= Xn (0)

= xn ,

N

σk (t, Xn )dβk −ρBn sign(Bn Xn )dt, k=1

(3.156)

3.7 Approximate Controllability of Stochastic Parabolic Equation

97

has a unique strong solution Xn ∈ L2 (Ω; C([0, T ]; Rn )). (For a detailed proof, see [42].) As solution to (3.156), Xn as well as un are (Ft )t≥0 -adapted Rn -valued processes. Let ϕε ∈ C 2 (R+ ) be such that ϕε (r) = εr , ∀ r ∈ [0, t], ϕε (r) = 1 + ε, ∀ r ≥ 2ε,

|ϕε (r)| ≤ εc , ∀ r ∈ R. We set φε (y) = ϕε (|y|n ), ∀ y ∈ Rn , and note that ∇φε (y) = ϕε (|y|n )sign y, ∇ 2 φε (y) = 0 for |y|n > 2ε, c |∇ 2 φε (y)|n ≤ , ∀ y ∈ Rn . ε

(3.157)

We apply in (3.156) Itô’s formula to the function t → φε (Xn (t)). We obtain dφε (Xn (t)) + An Xn (t) + Dn (t)Xn (t), ∇φε (Xn (t))n dt + ρ Bn sign(Bn Xn (t)), ∇φε (Xn (t))n dt n N

$ # 1 n αij (t)(∇ 2 φε (Xn (t)))ij dt + σj (t, Xn (t))dβj (t), ∇φε (Xn (t)) n , = 2 i,j =1

j =1

(3.158) where αijn =

m  =1

j

σi (t, Xn (t))σ (t, Xn (t)).

Here ·, ·n is the scalar product in Rn , | · |n is the Euclidean norm. We have |αijn (t)| ≤ Ln |Xn (t)|2n , i, j = 1, . . . , n.

(3.159)

On the other hand, by the unique continuation property of the eigenfunctions ej (see Theorem 2.2), we have det Bn = 0, and so, |Bn X|n ≥ γn |X|n , ∀ X ∈ Rn .

(3.160)

Integrating (3.158) on (s, t), 0 < s < t < T , and using (3.158)–(3.160), we get P-a.s., 

#

$ sign(Bn Xn (θ )), Bn ∇φε (Xn (θ )) n dθ s  t n |Xn (θ )|n ∇ 2 φε (Xn (θ ))L(Rn ,Rn ) dθ ≤ φε (X (s)) + Cn s  t N # $ + σj (θ, Xn (θ ))dβj (θ ), ∇φε (Xn (θ )) n .

φε (Xn (t)) + ρ

t

(3.161)

s j =1

Taking into account (3.160) and that, for ε → 0, ∇φε (y) → η ∈ sign y, ∀ y ∈ Rn , by letting ε → 0 in (3.161), we get

98

3 Exact Controllability of Parabolic Equations

 |Xn (t)|

t

+ ργn

1[Xn =0] dθ  t N # $ ≤ |Xn (s)|n + σj (θ, Xn (θ ))dβj (θ ), sign Xn (θ ) n , n

s

s j =1

and so, by the stochastic Gronwall’s lemma, 

t

|Xn (t)|n + ργn eCn (t−θ) 1[Xn >0] dθ ≤ eCn (t−s) |Xn (s)|n s  t N

# $ σj (θ, Xn (θ ))dβj (θ ), Xn (θ ) n , + eCn (t−θ) s

(3.162)

j =1

0 ≤ s < t < T . In particular, it follows by (3.162) that t → e−Cn t |Xn (t)|n is a nonnegative supermartingale, that is, ( ' E |Xn (t)|n e−Cn t | Fs = |Xn (s)|n e−Cn s , ∀ t ≥ s. Hence, if τn is the stopping time (3.153), we have |Xn (t)|n e−Cn t ≤ |Xn (τn )|n e−Cn τn and so the extinction of Xn (t) occurs at the time τn . Moreover, taking the expectation in (3.162) with s = 0 yields  E|Xn (t)|n + ργn

t

e−Cn θ P[τn > θ ]dθ ≤ |xn |n

0

and, therefore, ργn (1 − e−Cn t )P[τn > t] ≤ |xn |n , ∀ t ∈ [0, T ]. Cn If we take ρ = ρn = Cn (γn ε|xn |n )−1 (1 − e−Cn T )−1 , we get (3.154), as claimed. Proof of Theorem 3.15 (Continued). We fix ε > 0 and take n sufficiently large such un to (3.130), where un = that x − x n L2 (O) ≤ ε, and consider the solution X n  n n , ui ei and {uni }ni=1 = un is given by (3.155). Subtracting the equation in (X un )

i=1

given by (3.150), we get un − X un n n d(X ⎛ ) − Δ(X − X )dt ⎞ ⎛ ⎞ n N n

n u n un + b · ∇ X un − ⎝μk X n − + ⎝aX Dn (t)Xi ei ⎠ dt = σk (t, X )ei ⎠ dβk , i=1 un − X n )(0) = x − xn . (X

k=1

i=1

3.8 Notes on Chapter 3

99

By applying Itô’s formula, we get after some calculation involving Gronwall’s lemma that ( '  un n (t)2 2 ; t ∈ [0, T ] ≤ Cx − x n L2 (O) ≤ Cε, (t) − X E sup X L (O) where C is independent of n. Taking into account (3.154), we obtain un P[X (t)L2 (O) ≤ (1 + C)ε, ∀ t ≥ T ] ≥ 1 − 2ε.

Then, redefining ε, we see that the controller un = uε satisfies (3.150), as claimed.

3.8 Notes on Chapter 3 Theorem 3.1 was first established by A.V. Fursikov and O.Yu. Imanuvilov [76]. (See also O.Yu. Imanuvilov and Iamamoto [81].) However, for the heat equation the internal exact controllability was previously proved by G. Lebeau and L. Robbiano [85] via the Carleman inequality for elliptic equations. The boundary controllability of the heat equation with zero potential was first proved by D.L. Russell [108] (see also the survey [109]), via harmonic analysis. (We refer also to [96] and [69] for previous results in 1 − D.) Theorems 3.2, 3.3 and Corollaries 3.3, 3.4 closely follow the author’s work [22]. In the special case of nonlinearities f satisfying the growth condition lim f (|r|)/|r|(log |r| + 1) = 0,

r→∞

Theorem 3.2 was first proved by E. Fernandez-Cara [71]. For the more general nonlinearity of form (3.52), this theorem was proved by V. Barbu [20] in the case 1 ≤ n ≤ 5. Later, E. Fernandez-Cara and E. Zuazua [72] extended this result to the arbitrary n and to blowing up systems, that is, in the absence of sign condition (3.45). The proofs of Theorem 3.6 and Lemma 3.1 closely follow the treatment of [22]. The case of parabolic equations with discontinuous coefficients was studied in [67] and, more recently, by J. Le Rousseau and L. Robbiano [88] for more general systems. In [7], it is studied the exact controllability of the nonlinear heat (diffusion) processes with nonlinear flux of the form q = −∇y + b(y), that is, yt − Δy + div(b(y)) = mu in Q, y = 0 on Σ; y(0) = y0 on O, where b satisfies the following growth conditions: |b (r)| ≤ ϕ(r)(1 + ln(|r| + 1), |b

(r)| ≤ ϕ0 (r)(1 + lnδ (|r| + 1)(|r| + 1)−1 ,

100

3 Exact Controllability of Parabolic Equations

δ ∈ (0, 12 ) and ϕ(r), ϕ0 (r) → 0 as |r| → ∞. It is shown that if d = 2, 3, then the above system is exactly null controllable for all y0 ∈ H01 (O) ∩ H 2 (O). For an extension of this result we refer to [68]. The exact controllability of the equation with impulse controllers was studied in [105]. We note also the works [110, 119] for sharper results concerning the exact and feedback controllability. During the last decade or so, some important work was done toward the exact controllability of degenerate parabolic equations of the form yt − div(a(x)yx ) = mu, and we refer to the works [58, 59] of P. Cannarsa et al. Such an equation is relevant in financial mathematics (the Black-Scholes equation). As regards the approximate controllability, we cite the works [70, 72]. It should be said that Theorem 3.8 extends to systems with nonlinear flux boundary conditions of the form yt − Δy = mu in Q,

∂ y + β(y) = 0 on ∂O, ∂ν

where β is a smooth monotonically nondecreasing function. In the case where β is Lipschitzian the controllability was studied by J.I. Diaz et al. [66]. The exact controllability of low diffusion porous media equations was studied in 1 − D in the works [50] and [63], but the general 3 − D case remains open. In [34], it is studied the controllability of the heat equation with memory, that is,  t yt (x, t) − γ0 Δy(x, t) − a(t − s)Δy(x, s)ds = m(x)u(x, t) in Q = O × (0, T ), 0

where γ0 ≥ 0 and a is a completely monotone kernel. If γ0 > 0, then this equation is of parabolic type and is approximately controllable ([34]). In this context, we mention also the works [102, 103] of L. Pandolfi. However, as recently shown by S. Guerrero and O. Imanuvilov [78], an equation of this type is not boundary controllable for all the initial data in L2 (O). Moreover, for γ0 = 0 the above equation is of hyperbolic type, and so the internal null controllability is not possible for all the intervals (0, T ). Local controllability of semilinear parabolic equations was first studied in [76] via the implicit function theorem. However, Theorem 3.8 seems to be new in this context. Theorem 3.9 was taken from the author’s work [23]. The local controllability results presented above were extended in several directions and we want to mention a few of them. For instance, in [8] it is studied the local internal controllability of the steady-state (equilibrium) solution to the reaction–diffusion system yt − k1 Δy + ayz = f + mu in Q,

zt − k2 Δz + byz = g + mv in Q.

Sharper results concerning one control controllability were also given in [73, 77]. It should be said, however, that the controllability of 2-D or 3-D parabolic systems by one control force is a difficult problem and it is only partially solved. For

3.8 Notes on Chapter 3

101

two and three heat equations coupled with cubic nonlinearities, this problem was solved by J.M. Coron, S. Guerrero, and L. Rossier [62] and in the work [61] by J.M. Coron and J.Ph. Guilleron by the so-called return method introduced by J.M. Coron in the controllability of hydrodynamic equations. (See also [60].) The results of Section 3.5 on exact controllability on unbounded domains are taken from author’s work [31]. For related results, we mention the works [94, 95]. The exact controllability of the stochastic parabolic equation (3.130) remains so far an open problem, the results given in Sections 3.6 and 3.8 being only partial results in this direction. Theorem 3.11 was established in a slight different form in [43] along with a Carleman-type inequality for stochastic parabolic equations. For other results in this direction, we mention [90, 112] and [91]. (See, also, Zhang’s survey [120].) As regards Theorem 3.14, it was established in [42] along with other results pertaining controllability of finite dimensional stochastic equations of the form (3.149), under Kalman’s rank assumption.

Chapter 4

Internal Controllability of Parabolic Equations with Inputs in Coefficients

Very often, the input control arises in the coefficients of a parabolic equation and the exact controllability of initial data to origin or to a given stationary state is a delicate problem which cannot be treated by the linearization method developed in the previous chapter. However, in some situations, one can construct explicit feedback controllers which steer initial data to a given stationary state. In general, such a controller is nonlinear, eventually multivalued mapping, and its controllability effect is based on the property of solutions to certain nonlinear partial differential equations to have extinction in finite time. Here we shall study a few examples of this type.

4.1 The Exact Controllability via Self-Organized Criticality Consider the control system ∂y − Δ(uy) = 0 in (0, T ) × O, ∂t y(x, 0) = y0 (x), x ∈ O, y(x, t) = 0, ∀ (t, x) ∈ (0, T ) × ∂O,

(4.1)

where O is a bounded and open set of Rd , d = 1, 2, 3, with smooth boundary ∂O, u ∈ U , where the controller set U is defined by U = {u ∈ L∞ ((0, T ) × O); u ≥ 0, a.e. in (0, T ) × O}.

© Springer International Publishing AG, part of Springer Nature 2018 V. Barbu, Controllability and Stabilization of Parabolic Equations, Progress in Nonlinear Differential Equations and Their Applications 90, https://doi.org/10.1007/978-3-319-76666-9_4

103

104

4 Internal Controllability of Parabolic Equations with Inputs in Coefficients

As mentioned earlier, equation (4.1) models a large variety of nonisotropic diffusion processes (for instance, gas flow dynamics through a porous medium) and the control parameter u describes the physical properties of the medium. The problem to be studied here is that of the structural controllability of the system via nonlinear feedback controller. Problem 4.1. Let y1 ∈ L∞ (O), y1 ≥ 0, a.e. in O, be given. Find a feedback controller u = Φ(y, x) which steers y0 in y1 in time T . In other words, one should find the map y → Φ(y, x) such that the solution y ∈ C([0, T ]; L2 (O)) to the closed loop system ∂ y(x, t) − Δ(Φ(y(x, t), x)y(x, t)) = 0 in (0, T ) × O, ∂t y(x, 0) = y0 (x) in O; y = 0 on (0, T ) × ∂O,

(4.2)

satisfies y(T , x) = y1 (x), a.e. x ∈ O.

(4.3)

We shall see below that such a function Φ : R × O → R can be taken of the form Φ(y, x) = αH (y − y1 (x)), ∀ y ∈ R, x ∈ O,

(4.4)

where α is a positive constant and H is the Heaviside (multivalued) function ⎧ ⎨ 1 if r > 0, H (r) = [0, 1] if r = 0, (4.5) ⎩ 0 if r < 0. More precisely, we shall prove that, for each T > 0, problem (4.2) is well posed in a weak sense to be made precise below and that, for α suitable chosen, the controllability condition (4.3) holds. The feedback law u = Φ(y(x, t), x) has a special interest because it is a relay, bang-bang controller. On the other hand, the closed loop system (4.2) is physically motivated. In fact, the parabolic boundary value problem (4.2), (4.3) is a nonlinear diffusion equation with phase transition which models the self-organized criticality of diffusion processes. More precisely, for d = 2, (4.2) describes the dynamic of the sandpile model of self-organized criticality which we briefly present below (see, e.g., [14, 27, 56] for details). If ρ = ρ(t, x1 , x2 ) is the energy (or mass density) assigned to the site x = (x1 , x2 ) of a square lattice at time t, then, if ρ(t, x1 , x2 ) exceeds the critical value ρc (x1 , x2 ), the site becomes unstable and an avalanche develops according to the following evolution ρ(t + 1, x1 , x2 ± 1) = ρ(t, x1 , x2 ± 1) +

1 ρ(t, x1 , x2 ), 4

ρ(t + 1, x1 ± 1, x2 ) = ρ(t, x1 ± 1, x2 ) +

1 ρ(t, x1 , x2 ). 4

4.1 The Exact Controllability via Self-Organized Criticality

105

If ρ(t, j ) is the energy of the j -th cell at time t, this dynamic can be described by the following discrete system of equations: ρ(t + 1, k) = ρ(t, k) − ρ(t, k)H (ρ(t, k) − ρc (k)) 1 ρ(t, j )H (ρ(t, j ) − ρc (j )). + 4

(4.6)

j =k

The continuous version of (4.6) is just equation (4.2) with Φ given by (4.4), α = 1 and y(t, x) = ρ(t, x), ρc = y1 . In cellular automata model presented above, the set {(t, x); ρ(t, x) = ρc (x)} is the critical region while {(t, x); ρ(t, x) > ρc (x)} is the supercritical region which is unstable and absorbed in finite time by the critical region (see [33]) and this is essentially the self-organized criticality mechanism. It should be emphasized that the diffusion dynamic induced by the avalanche process described above is not apparently generated by the Fick’s first law, but seems to be the materialization of a more complex physical process. By weak solution to problem (4.2) on (0, T ) × O, we mean a function y ∈ C([0, T ]; L1 (O)) which satisfies equation (4.2) in sense of distributions. An equivalent definition can be done in terms of mild solutions to nonlinear m-accretive Cauchy problem in the space L1 (O). Namely, setting y − y1 = z, we reduce problem (4.2) to ∂z − αΔ((z + y1 )H (z))  0 in (0, T ) × O, ∂t z(x, 0) = z0 (x) = y0 − y1 in O, H (z)(z + y1 ) = 0 on (0, T ) × ∂O.

(4.7)

In the following, we shall assume that z0 = y0 − y1 ≥ 0, a.e. in O. We may write problem (4.7) as an infinite-dimensional Cauchy problem in the space L1 (O) (see Section 1.1) dz + Az  0 for t ≥ 0, dt z(0) = z0 ,

(4.8)

where the operator A : D(A) ⊂ L1 (O) → L1 (O) is defined by Az = {−Δη; η ∈ W01,1 (O), Δη ∈ L1 (O), η(x) ∈ α(z(x) + y1 (x))H (z(x)), a.e. x ∈ O}, ∀ z ∈ D(A),

(4.9)

where D(A) consists of all z ∈ L1 (O), z ≥ 0, a.e. in O, for which there is η ∈ W01,1 (O) such that η ∈ α(z + y1 )H (z), a.e. in O. We have Lemma 4.1. The operator A is accretive in L1 (O) and {z ∈ L1 (O); z ≥ 0, a.e. in O} ⊂ R(I + λA), ∀ λ > 0.

(4.10)

106

4 Internal Controllability of Parabolic Equations with Inputs in Coefficients

Proof. Taking into account that the multivalued operator J : L1 (O) → L∞ (O), J (z)(x) = {ζ ∈ L∞ (O); ζ (x) ∈ sign z(x) a.e. x ∈ O}, z for z = 0, sign 0 = [−1, 1], is the duality mapping of the space where sign z = |z| 1 L (O) (see, e.g., [26], p. 4), for the accretivity of A it suffices to show that

 −

O

Δ(H (z)(z + y1 ) − H (¯z)(¯z + y1 ))ζ dx ≥ 0, ∀ z1 , z¯ 1 ∈ D(A),

for some ζ ∈ L∞ (O), ζ ∈ sign(z − z¯ ), a.e. in O. Taking into account that sign(z − z¯ ) = sign(H (z)(z+y1 )−H (¯z)(¯z+y1 )), a.e. in O, the latter follows by the accretivity of the elliptic operator A0 y = −αΔy, with the domain {y ∈ W01,1 (O); Δy ∈ 1,q d L1 (O)} ⊂ W0 (O) ⊂ L2 (O), 1 ≤ q < d−1 . (See, e.g., [26], p. 110.) It remains 1 to prove (4.10), for each f ∈ L (O), f ≥ 0, that is, the existence of a solution z ∈ D(A) to the equation z − αΔ((z + y1 )H (z))  f in D (O).

(4.11)

Assume first that f ∈ L2 (O) and approximate (4.11) by an appropriate convenient equation. Namely, we note that, for ε > 0, the operator (A0 + εI )−1 is accretive and continuous in L1 (O), while Bz = α(z + y1 )H (z) is m-accretive. Hence, (A0 + εI )−1 + B is m-accretive (see [26], p. 104). This means that, for each ε > 0, there is zε ∈ L1 (O) such that εzε + (A0 + εI )−1 zε + α(zε + y1 )H (zε )  (A0 + εI )−1 f.

(4.12)

Equivalently, (1 + ε2 )zε − εΔzε − αΔ((zε + y1 )H (zε )) + αε(zε + y1 )H (zε )  f in D (O), (4.13) where εzε + α(zε + y1 )H (zε ) ⊂ D(A0 ) ⊂ L2 (O). Hence zε ∈ L2 (O) and, since f ∈ L2 (O), by the elliptic regularity (see, e.g., [55], p. 297) it follows that εzε + α(zε + y1 )H (zε ) = gε ∈ H01 (O) ∩ H 2 (O). Since zε = (ε + α)−1 (gε − αy1 )+ , this implies that zε ∈ H01 (O). By (4.13), we see via the maximum principle that, for f ≥ 0, we have zε ≥ 0, a.e. in O. Moreover, multiplying (4.13) by sign zε and integrating on O, we get by the accretivity of A0 that 

 O

|zε |dx ≤

O

|f |dx, ∀ ε > 0.

If we multiply (4.12) by (zε + y1 )H (zε ) and integrate on O, we obtain via Green’s formula

4.1 The Exact Controllability via Self-Organized Criticality

107

  (1 + ε2 ) zε (zε + y1 )dx + ε ∇(zε ) · ∇((zε + y1 )H (zε ))dx O O   |∇((zε + y1 )H (zε ))|2 dx + αε (zε + y1 )2 H (zε )dx +α  O O = f (zε + y1 )H (zε )dx. O

This yields    |zε |2 dx + |∇((zε + y1 )H (zε ))|2 dx ≤ C (f 2 + y12 )dx, ∀ ε > 0. O

O

O

Hence, on a subsequence {ε} → 0 we have (zε + y1 )H (zε ) −→ (z + y1 )H (z) weakly in H01 (O) and strongly in L2 (O), strongly in L2 (O), zε −→ z Δ(zε + y1 )H (zε ) −→ Δ(z + y1 )H (z) strongly in H −1 (O), where z is the solution to (4.11). Now, if f ∈ L1 (O), we choose a sequence {fn } ⊂ L2 (O) such that fn → f strongly in L1 (O). If zn ∈ L2 (O) are corresponding solutions to (4.11), we see once again by the accretivity of A0 that   |zn − zm |dx ≤ |fn − fm |dx, ∀ n, m ∈ N. O

O

If z = lim zn in L1 (O), it follows that z is the solution to (4.11), as claimed. n→∞

By Theorem 1.2, applied with X = L1 (O) and C = {z ∈ L1 (O); z ≥ 0, a.e. in O}, it follows that, for each T ∈ (0, ∞) and z0 ∈ D(A) = {z0 ∈ L1 (O), z0 ≥ 0, a.e. in O}, equation (4.8) has a unique mild solution z ∈ C([0, T ); L1 (O)) given by Crandall & Liggett exponential formula (1.6), that is,

−n t z(t) = lim I + A z0 strongly in L1 (O) n→∞ n uniformly on compact intervals. Equivalently, z(t) = lim zh (t) strongly in L1 (O), h→0

(4.14)

uniformly in t on every compact interval [0, T ], where zh is the solution to the finite difference scheme % & T h h (4.15) z (t) = zi , t ∈ [ih, (i + 1)h), i = 0, 1, . . . , N = h h h zi+1 + hAzi+1  zih , i = 0, 1, . . . , N, z0h = z0 .

This existence result can be completed as follows.

(4.16)

108

4 Internal Controllability of Parabolic Equations with Inputs in Coefficients

Theorem 4.1. Equation (4.7) (equivalently (4.8)) has a unique mild solution z ∈ C([0, T ]; L1 (O)). Moreover, if z0 ∈ L2 (O), then z ∈ C([0, T ]; L1 (O)) ∩ W 1,1 ([0, T ]; H −1 (O)), z ≥ 0, a.e.in (0, T ) × O and z is a strong solution to (4.7), that is, dz (t) − Δη(t) = 0, a.e. t ∈ (0, T ), dt η(x, t) ∈ α(z(x, t) + y1 (x))H (z(x, t)), a.e. (t, x) ∈ (0, T ) × O, η ∈ L2 (0, T ; H01 (O)).

(4.17)

(4.18)

If z0 ∈ L∞ (O), then z ∈ L∞ ((0, T ) × O). Proof. If z0 ∈ L2 (O), then, by (4.15)–(4.16), we have h − hΔηh = zh in O, ηh ∈ H 1 (O), zi+1 i i+1 i+1 0

(4.19) h ∈ α(zh + y )H (zh ), a.e. in O. ηi+1 1 i+1 i+1

We set (x, r) = α(r + y1 (x))H (r), ∀ r ∈ R, H and j : O × R → R be defined by α j (x, r) =

2

r 2 + αry1 (x) if r ≥ 0, 0

if r < 0.

(x, r), ∀ x ∈ O, r ∈ R, and so, by the convexity of j , We have ∇r j (x, r) = H we have h (x, zh )(zh − z), ∀ z ∈ R. ) ≤ j (x, z) + H j (x, zi+1 i+1 i+1

Then, (4.19) yields, via Green’s formula, that 

 O

h j (x, zi+1 (x))dx + h

O

 h |∇ηi+1 (x)|2 dx ≤

O

j (x, zih (x))dx,

(4.20)

∀ i = 0, 1, . . . , N.

Hence  O

h j (x, zi+1 )dx + h

i 

k=0 O

 |∇ηkh (x)|2 dx ≤

O

j (x, z0 (x))dx, i = 0, 1, . . . , N,

4.1 The Exact Controllability via Self-Organized Criticality

109

and, therefore,  O

j (x, zh (t, x))dx +

 t



O

0

|∇ηh (x, s)|2 ds dx ≤

O

j (x, z0 (x))ds,

(4.21)

where ηh (x, t) = ηih (x) for t ∈ [ih, (i + 1)h). Hence, for h → 0, zh −→ z strongly in L∞ (0, T ; L1 (O)), weak star in L∞ (0, T ; L2 (O)), ηh −→ η weakly in L2 (0, T ; H01 (O)). (4.22) By (4.15), (4.16), and (4.19), we have 

T

) zh (t),

0

dϕ (t) dt

*



T

dt + 0

2

 O

∇ηh (t) · ∇ϕ(t)dx dt = 0, ∀ h > 0,

for all ϕ ∈ C 1 ([0, T ]; L1 (O)) ∩ C([0, T ]; H01 (O)), ϕ(0) = ϕ(T ) = 0. Then, by (4.22), we get, for h → 0, 

T

)

0

dϕ z(t), (t) dt

*



T

dt + 2

0

η(t), Δϕ(t)2 dt = 0,

and this yields z ∈ W 1,2 ([0, T ]; H −1 (O)), that is,

dz dt

∈ L2 (0, T ; H −1 (O)), and

dz − Δη = 0 in D (0, T ; H −1 (O)). dt

(4.23)

Here ·, ·2 is the duality functional on H01 (O) × H −1 (O) (respectively, the scalar 2 −1 #product−1on L$ (O)), and ·, ·−1 is the scalar product of H (O), that1 is, u, v2−1 = (−Δ) u, v , where −Δ is the Laplace operator with the domain H0 (O) ∩ H (O), and D (0, T ; H −1 (O)) is the space of H −1 (O)-valued distributions on (0, T ). It (x, z), a.e. in (0, T )×O. Because the map z → H (x, z) remains to show that η ∈ H is maximal monotone in L2 ((0, T ) × O) × L2 ((0, T ) × O), to this end it suffices to show that 

T

lim sup 0

h→0



 O

zh ηh dx dt ≤ 0

T

 O

z η dx dt.

(See [26], p. 41.) By (4.19), we have 

T

lim sup h→0

0

 O

zh ηh dx dt +

1 (|z(T )|2−1 − |z0 |2−1 ) ≤ 0, 2

(4.24)

110

4 Internal Controllability of Parabolic Equations with Inputs in Coefficients

whereas (4.23) yields 

T

 O

0

ηz dx dt +

1 (|z(T )|2−1 − |z0 |2−1 ) = 0, 2

2 −1 (O)). (Here | · | −1 (O).) because dz −1 is, as usually, the norm of H dt ∈ L (0, T ; H (x, z), a.e. in (0, T ) × O, as claimed. This implies η ∈ H Hence, z is a strong solution to (4.7) in the senseof (4.17), (4.18). Moreover, by equation (4.23), it follows that the function ϕ(t) = O j (x, z(x, t))dx is absolutely continuous on [0, T ] and (see [26], p. 158)

) * dz d ϕ(t) = η(t), (t) , a.e. t ∈ (0, T ). dt dt −1 Assume now that z0 ∈ L∞ (O). To show that z ∈ L∞ ((0, T ) × O), it suffices to check that zh L∞ ((0,T )×O) ≤ C < ∞, ∀ h > 0. More precisely, we shall prove that the solution zh to (4.15)–(4.16) satisfies the estimate h |zi+1 |∞ ≤ |zih |∞ ≤ |z0 |∞ , ∀ i = 1, 2, . . . , N, h > 0.

(4.25)

It is readily seen that, for each i, zih = lim (zλh )i strongly in H (O), where λ > 0, and λ→0

(zh )i+1 + λ(zh )i+1 )  zh in O, (zλh )i+1 − hΔ(H λ λ i (4.26) (zλh )i+1

∈

H01 (O).

Now, by (4.26), we see that, for Mi = ess sup zih , we have by Green’s formula 

 (zλh )i+1 + λ(zλh )i+1 ) · ∇((zλh )i+1 − Mi )+ dx (((zλh )i+1 − Mi )+ )2 dx + h ∇(H  O O = ((zλh )i − Mi )((zh )i+1 − Mi )+ dx ≤ 0. O

h This yields lim λi ((zλh )i+1 − Mi )+ = 0, a.e. in O and, therefore, zi+1 ≤ Mi , λ→0

a.e. in O. Similarly, we obtain, for Mi∗ = ess sup {−zih }, that (zλh + Mi∗ )− = 0, a.e. in O. O

h ≥ −Mi∗ , a.e. in O and so (4.25) holds. Hence, zi+1

4.1 The Exact Controllability via Self-Organized Criticality

111

h Finally, if z0 ≥ 0, a.e. in O, we obtain in a similar way that zi+1 ≥ 0, a.e. in O, and so, by (4.22), we conclude that z ≥ 0, a.e. in (0, T ) × O. This completes the proof.

Now, we come back to the controllability problem and assume that y0 , y1 ∈ L∞ (O), y0 ≥ y1 ≥ ρ > 0, a.e. on O,

(4.27)

where ρ is a positive constant. We shall prove that, for α suitable chosen (dependent of T ), the solution z to equation (4.7) satisfies z(T , x) = 0, a.e. x ∈ O.

(4.28)

In other words, the feedback controller u defined by (4.4) steers in the time T the initial data y0 in the state y1 . 2d if d ≥ 3 and p∗ > 2 if d = 1, 2. Then, by the Sobolev To this end, let p∗ = d−2 ∗ 1 embedding theorem, H0 (O) ⊂ Lp (O) (see, e.g., [55], p. 278) and so γ = sup{uLp∗ (O) ; uH 1 (O) = 1} < ∞.

(4.29)

0

We come back to (4.19) and note that, by (4.29), we have  O

∗ h |ηi+1 |p dx





1 p∗

≤γ

O

1 2

h |∇ηi+1 |2 dx

, ∀ i,

(4.30)

while, by (4.20) and (4.30), it follows that  O

h j (x, zi+1 (x))dx

+ hγ

−2

 O

∗ h |ηi+1 (x)|p dx

Recalling that, by (4.19),



2 p∗

 ≤

O

j (x, zih (x))dx, ∀ i = 0, 1, . . . (4.31)

h (zh (x)) = α(zh + y1 (x))H (zh (x)), a.e. x ∈ O, ηi+1 (x) ∈ H i+1 i+1 i+1

we get 

p∗

O

h |ηi+1 | dx



2 p∗



 = α2

h >0] [zi+1

p∗

2 p∗

h |zi+1 (x) + y1 (x)| dx

and so, (4.31) yields  O

h j (x, zi+1 (x))dx + α 2 γ −2

j =k

 ≤

 i+1 

O

[zjh (x)>0]

j (x, zkh (x))dx, ∀ k ≤ i + 1.

 p∗

|zjh (x) + y1 (x)| dx

2 p∗

(4.32)

112

4 Internal Controllability of Parabolic Equations with Inputs in Coefficients

Then, by (4.27), we have  O

h j (x, zi+1 (x))dx + α 2 γ −2 ρ 2

j =k

 ≤

i+1

2 h (m(x ∈ O; zi+1 (x) > 0)) p∗

O

j (x, zkh (x))dx,

where m is the Lebesgue measure. Taking into account that by (4.25)  h h zi+1 (x)dx ≤ |z0 |∞ m(x ∈ O; zi+1 (x) > 0), O

we get, for 1 ≤ k ≤ i + 1,  O

2 ∗

−

h j (x, zi+1 )dx + α 2 γ −2 ρ 2 |z0 |∞p

O

O

j =k

 ≤

i+1 

2 p∗

zj (x)dx

j (x, zk (x))dx.

Letting h → 0, we obtain that 

2 −2

O

− 2∗ |z0 |∞p ρ 2



 t 

j (x, z(x, t))dx + α γ s  ≤ j (x, z(x, s))dx, 0 ≤ s ≤ t ≤ T .

O

(z(τ, x))dx

2 p∗

dτ

O

We set

 ϕ(t) =

O

j (x, z(x, t))dx, t ∈ [0, T ],

and obtain 2 −2

ϕ(t) + α γ

− 2∗ |z0 |∞p ρ 2

 t  s

O

z(τ, x)dx

2 p∗

dτ ≤ ϕ(s), 0 ≤ s ≤ t ≤ T . (4.33)

Since ϕ is absolutely continuous, we get by (4.33) the differential inequality 

2∗ p − 2∗ ϕ (t) + α 2 γ −2 |z0 |∞p ρ 2 z(x, t)dx ≤ 0, a.e. t > 0, O  ϕ(0) = ϕ0 = j (x, z0 (x))dx. O

Taking into account that j (x, z) ≥ αρz, ∀ z ≥ 0, we have that  O

z(x, t)dx ≥

1 ϕ(t), ∀ t > 0. αρ

(4.34)

4.1 The Exact Controllability via Self-Organized Criticality

113

We set −

2 ∗

λ0 = α 2 γ −2 ρ 2 |z0 |∞p . Then, by (4.34), we get 2

ϕ (t) + λ0 (ϕ(t)) p∗ ≤ 0, a.e. t > 0, and, therefore, ϕ(T ) = 0 if p∗ λ0 T = ∗ p −2

 O

1− j (x, z0 (x))dx

2 p∗

p∗ = ∗ p −2

 O

1− j (x, y0 (x)−y1 (x))dx

2 p∗

. (4.35)

We have proved, therefore, Theorem 4.2. Let condition (4.27) hold. If α is chosen in such a way that (4.35) holds, then (4.28) follows. Now, coming back to equation (4.2), we have by Theorem 4.2 Theorem 4.3. Let y0 and y1 be such that y0 , y1 ∈ L∞ (O), y0 ≥ y1 ≥ ρ > 0, a.e. in O. Then, the feedback controller u(t) = αH (y(t) − y1 ), t ∈ [0, T ],

(4.36)

steers y0 into y1 in time T if α > 0 is chosen as in equation (4.35). In other words, equation (4.1) is exactly controllable in y1 on [0, T ] by the feedback controller (4.36). It should be mentioned also that, if the solution y to the closed loop system (4.2) reaches y1 in t = T , then, by the uniqueness of the solution z to (4.7), it follows that y(t) = y1 for all t ≥ T . If we interpret y1 as a critical state of diffusion system, the physical significance of Theorem 4.3 is that the supercritical region {x; y(x, t) > y1 (x)} is absorbed in time T in the critical region {x; y(x, t) = y1 (x)}. In other words, the feedback controller (4.4) induces a nonlinear diffusion dynamic which steers the initial supercritical state y0 in the critical state y1 . Remark 4.1. The above exact controllability approach applies as well to the nonlinear parabolic equation ∂y − Δ(uy) + f (y) = 0 in (0, T ) × O, ∂t y(x, 0) = y0 (x) in O, y = 0 on (0, T ) × ∂O, where f : R → R is a continuous and monotonically nondecreasing function such that f (0) = 0. We note that also in this case the operator z → −αΔ(z + y1 )H (z) +

114

4 Internal Controllability of Parabolic Equations with Inputs in Coefficients

f (z + y1 ) is accretive in L1 (O) and satisfies condition (4.10). Then the feedback law (4.4) has a similar exact controllability effect. Note also that the above results extend to parabolic equations of the form d

∂y ∂2 aij (uy) = 0 in (0, ∞) × O, − ∂t ∂xi ∂xj i,j =1

where aij = aj i and

d  i,j =1

aij ξi ξj ≥ γ |ξ |2d , ∀ ξ ∈ Rd , γ > 0. We omit the details.

4.2 Exact Controllability via Fast Diffusion Equation We consider here system (4.1) with the feedback controller u(t) = α|y|m−1 sign y,

(4.37)

where α > 0 and % m∈

d −2 , 1 , d ≥ 2, d +2

(4.38)

and m = 0 if d = 1.Here sign y =

⎧ ⎨

1 for y > 0, [−1, 1] for y = 0, ⎩ −1 for y < 0.

Theorem 4.4. Let y0 ∈ H −1 (O) ∩ L1 (O). Then, equation (4.1) with the feedback controller (4.37) has a unique strong solution y ∈ C([0, T ]; L1 (O) ∩ H −1 (O)). Moreover, for 1 y0 1−m (γ m+1 T )−1 , d ≥ 2, H −1 (O) 1−m γ = sup{yH −1 (O) ; uLm+1 (O) = 1},

(4.39)

y(T ) = 0.

(4.40)

α=

one has

If d = 1 and m = 0, then y(x, T ) = 0 for α = y0 H −1 (O) (γ T )−1 .

4.2 Exact Controllability via Fast Diffusion Equation

115

Proof. By Theorem 5.3 in [26] (see, also, Theorem 1.3), it follows the existence of a strong solution u ∈ C([0, T ]; H −1 (O) ∩ L1 (O)) to the multivalued porous media equation α ∂y − Δ(|y|m η)  0 in (0, T ) × O, ∂t m y(x, 0) = y0 (x), x ∈ O, y = 0 on (0, T ) × O, η ∈ sign y, a.e. in (0, T ) × O.

(4.41)

Moreover, one has 1

t2

1 d y ∈ L2 (0, T ; H −1 (O)), t 2 (|y|m η) ∈ L2 (0, T ; H01 (O)). dt

If we multiply scalarly in H −1 (O) equation (4.41) by y, we get 1 d y(t)2H −1 (O) + α 2 dt

 O

|y(x, t)|m+1 dx = 0, ∀ t > 0,

and, since by the Sobolev embedding theorem, γ yH −1 (O) ≤ yLm+1 (O) for m ≥

d −2 , d +2

we obtain that 1 d m+1 y(t)2H −1 (O) + αγ m+1 y(t)H −1 (O) ≤ 0. 2 dt This yields d y(t)H −1 (O) + αγ m+1 y(t)m ≤ 0, t > 0, H −1 (O) dt and so, by (4.39), (4.40) follows. The case d = 1, m = 0 follows in a similar way from the equation 1 d y(t)2−1 + αy(t)L1 (O) ≤ 0, a.e. t > 0, 2 dt taking into account that yL1 (O) ≥ γ yH −1 (O) . This completes the proof. The case m = 1 in (4.37), which was ruled out by condition (4.38), leads to the feedback relay controller u(t) = α sign y(t), t ∈ [0, T ].

(4.42)

116

4 Internal Controllability of Parabolic Equations with Inputs in Coefficients

The corresponding closed loop system is ∂y − αΔ(sign y)  0 in (0, T ) × O, ∂t y(x, 0) = y0 (x), x ∈ O, y=0 on (0, T ) × ∂O.

(4.43)

As in the previous case, it follows that, for a suitable constant α, the controller (4.42) steers the initial data y0 in origin in the time T . Namely, one has Theorem 4.5. Let y0 ∈ L1 (O) ∩ L∞ (O) be such that y0 ≥ 0 and T > 0 be given. Then there is α = α(y0 , T ) > 0 such that the solution y ∈ C([0, T ]; L1 (O)) to equation (4.43) satisfies y(T ) = 0. Proof. Likewise all porous media equations of the form ∂y ∂t − Δψ(y)  0 with the R maximal monotone function ψ : R → 2 , problem (4.43) can be represented as the Cauchy problem in the space L1 (O), dy + Ay(t)  0, t ≥ 0, dt y(0) = y0 ,

(4.44)

where the operator A : D(A) ⊂ L1 (O) → L1 (O) is defined by Ay = {−αΔη; η ∈ W01,1 (O), η(ξ ) ∈ sign(y(ξ )), a.e. ξ ∈ O},

(4.45)

and D(A) consists of all y ∈ L1 (O) for which such a section η ∈ W01,1 (O) of sign y(x) exists. We note that the multivalued operator A is m-accretive in L1 (O) (see Theorem 1.3 in Chapter 1) and so, for each y0 ∈ D(A) = L1 (O) and T > 0, the Cauchy problem (4.44) has a unique mild solution y ∈ C([0, T ]; L1 (O)) given by the exponential formula

−n t I+ A y0 strongly in L1 (O), t ≥ 0. n→∞ n

(4.46)

y(t) = lim yh (t) uniformly in t,

(4.47)

y(t) = lim Equivalently,

h→0

where yh : [0, T ] → L1 (O) is the step function yh (t) = yih for t ∈ [ih, (i + 1)h), h + hAy h yi+1 i+1  yi , i = 0, 1, 2, . . . , N, N ∈ y0h = y0 .

Moreover, we have

(4.48) +T , h

,

(4.49)

4.2 Exact Controllability via Fast Diffusion Equation

117

Proposition 4.1. Assume that y0 ∈ L2 (O). Then (4.43) (equivalently, (4.44)) has a unique solution y satisfying y(t) ∈ C([0, T ]; L1 (O) ∩ H −1 (O)) ∩ W 1,1 ([0, T ]; H −1 (O)), dy (t) − αΔη(t) = 0, a.e. t ∈ [0, T ], dt η(ξ, t) ∈ sign(y(ξ, t)), a.e. (ξ, t) ∈ (0, T ) × O,

(4.50)

 where η ∈ L1 (0, T ; H01 (O)). Moreover, t → O |y(ξ, t)|dξ is absolutely continuous. If y0 ∈ L∞ (O), then y ∈ L∞ ((0, T ) × O). If y0 ≥ 0, a.e. in O, then y ≥ 0, a.e. in (0, T ) × O. Proof. The proof is much similar to that of Theorem 4.1 and so it will be outlined only. Namely, we fix h > 0. By (4.46) and (4.48), we see that h h yi+1 − hαΔηi+1 = yih in O, i = 0, 1, . . . , N,

(4.51)

h ∈ sign y h , a.e. in O. This yields where ηi+1 i+1



 O

h j (yi+1 (ξ ))dξ

+ hα

 O

|∇(ηih (ξ ))|2 dξ

≤

O

j (yih (ξ ))dξ, i = 0, 1, . . . , N,

where j (r) = |r|. Hence  O

|yh (ξ, t)|dξ + α

 t

 O

0

|∇ηh (s, ξ )|2 ds dξ ≤

O

|y0 (ξ )|dξ,

(4.52)

t ∈ (0, T ), ∀ h > 0,

where ηh (ξ, t) = ηih (ξ ) for t ∈ (ih, (i + 1)h), h ∈ O. We also have, by (4.49), that 

 O

h 2 |yi+1 | dξ ≤

and, therefore,  O

O

|yih |2 dξ, ∀ i, h > 0,

 |yih (t)|2 dξ ≤

O

|y0 |2 dξ, ∀ t ∈ (0, T ), h > 0.

(4.53)

Hence, we have by (4.46), (4.52), and (4.53) that, for h → 0, yh → y strongly on L∞ (0, T ; H −1 (O)), weak-star in L∞ (0, T ; L2 (O)), h η → η weakly in L2 (0, T ; H01 (O)).

(4.54)

118

4 Internal Controllability of Parabolic Equations with Inputs in Coefficients

For any test function ϕ ∈ C 1 ([0, T ]; H −1 (O))∩C([0, T ]; H01 (O)), ϕ(0) = ϕ(T ) = 0, we have by (4.51) that  h

T

)

dϕ yh (t), (t) dt

*

 −1

T

dt + α



0

O

ηh (t)ϕ(t)dξ dt = 0, ∀ h > 0,

and this yields y ∈ W 1,2 ([0, T ]; H −1 (O)), that is,

dy dt

∈ L2 (0, T ; H −1 (O)), and

dy − αΔη = 0 in D (0, T ). dt

(4.55)

It remains to show that η ∈ sign(y), a.e. in (0, T ) × O. This will be shown as above by checking that 

T

lim sup 0

h→0



 O

yh ηh dξ dt ≤ 0

T

 O

y η dξ dt.

(4.56)

In fact, by (4.51), (4.54), we have 

T

α lim sup

O

0

h→0

 yh ηh dξ dt +

1 (|y(T )|2−1 − |y0 |2−1 ) ≤ 0, 2

while (4.55) yields 

T

α 0

 O

ηy dξ dt +

1 (|y(T )|2−1 − |y0 |2−1 ) = 0, 2

2 −1 (O)). This implies (4.56), as claimed. because dy dt ∈ L (0, T ; H Hence, y is strong solution to (4.44) in sense of (4.50). Moreover, by (4.50) it follows that the function ϕ(t) = O |y(ξ, t)|dξ is absolutely continuous on [0, T ] and, as seen above, ) * dy d ϕ(t) = η(t), (t) , a.e. t ∈ (0, T ). dt dt −1

Assume now that y0 ∈ L∞ (O). We have yh L∞ ((0,T )×O) ≤ C < ∞, ∀ h > 0,

(4.57)

as a consequence of the estimate h |∞ ≤ |yi |h∞ ≤ |y0 |∞ , ∀ i = 1, 2, . . . , N, h > 0. |yi+1

(4.58)

4.2 Exact Controllability via Fast Diffusion Equation

119

h Indeed, as in the previous case, we have, for each i, yi+1 = lim yλh strongly in λ→∞

H 1 (O), where λ > 0, and yλh − εαΔ(sign(yλh ) + λyλh )  yih in O,

(4.59)

yλh ∈ H01 (O). Now, by (4.59), we see that, for Mi = ess supO yiε , we have via Green’s formula  O

((yλh



(yλh ) + λyλh ) · ∇(yλh − Mi )+ dξ − Mi ) ) dξ + hα ∇H  O = (yih − Mi )(yλh − Mi )+ dξ ≤ 0. + 2

O

h ≤ M , a.e. in O. Similarly, This yields (yλh −Mi )+ = 0, a.e. in O and, therefore, yi+1 i h h ∗ we obtain, for Mi = ess supO {−yi } that (yλ + Mi∗ )− = 0, a.e. in O. Hence, h ≥ −M ∗ , a.e. in O, and so (4.58) holds. yi+1 i h ≥ 0, a.e. in Finally, if y0 ≥ 0, a.e. in O, we obtain in a similar way that yi+1 O, and so by (4.54) we conclude that y ≥ 0, a.e. in (0, T ) × O. This completes the proof of Proposition 4.1.

To complete the proof of Theorem 4.5, we shall prove that y(x, t) ≡ 0, a.e. x ∈ O, ∀ t ∈ T ∗ ,

(4.60)

where 2 p∗ p∗ γ 2 |y0 |∞ T = ∗ p −α

∗

Here, p∗ =

2d d−2

 O

t−

2 p∗

y0 dξ

.

 for d ≥ 3, p∗ > 2 for d = 1, 2, and γ = sup |u|p∗ u−11



H0 (O)

.

We start with the difference scheme (4.49) (or (4.51)), that is, h − hαΔηh = y h in O, i = 0, 1, . . . ., N, N = yi+1 i i+1 h h ∈ sign(y h ), a.e. in O. ηi+1 ∈ H01 (O), ηi+1 i+1

+T , N

,

(4.61)

 $ # h h − yh ≥ h , yi+1 Taking into account that yih ≥ 0, a.e. in O, and ηi+1 i O (yi+1 − h yi )dξ, we obtain as above that 

 O

h yi+1 (ξ )dξ

+ hα

 O

h |∇ηi+1 (ξ )|2 dξ

≤

O

yih (ξ )dξ, ∀ i = 0, 1.

(4.62)

120

4 Internal Controllability of Parabolic Equations with Inputs in Coefficients

On the other hand, by the Sobolev embedding theorem, we have (see (4.30)) 



p∗

O

h |ηi+1 | dξ

1 p∗

 ≤γ

p∗ 2

h |∇ηi+1 |2 dξ

O

, ∀ i = 0, 1, . . .

(4.63)

yih dξ, ∀ i = 0, 1, . . .

(4.64)

Then, by (4.62) we obtain  O

h yi+1 dξ + hαγ −2

 O



∗

h |ηi+1 |p dξ

2 p∗

 ≤

O

h ∈ sign(y h ), a.e. in O, we have and recalling that ηi+1 i+1

 O

∗ h |ηi+1 |p dξ



2 p∗

2

h ≥ (m[ξ ∈ O; yi+1 (ξ ) > 0]) p∗ ,

(4.65)

where m stands for the Lebesgue measure. On the other hand, we have by (4.58) h 0 ≤ yi+1 (ξ ) ≤ |y0 |∞ , a.e. ξ ∈ O, i = 1, . . . ,

and so, we obtain that  h (ξ ) > 0] ≥ |y0 |∞ m[ξ ∈ O; yi+1

O

h yi+1 (ξ )dξ, ∀ i.

(4.66)

By (4.62)–(4.63), we have that  O

h yi+1 (ξ )dξ

+ hαγ

−2

i+1 

O

j =k

∗ |ηjh+1 |p dξ



2 p∗

 ≤

O

ykh (ξ )dξ,

for all i > k. Summing up and letting ε tend to zero and keeping in mind (4.48), we obtain by virtue of (4.44) that, for all 0 ≤ s < t < ∞, we have  O

y(ξ, t)ξ + αγ

−2

− 2∗ |y0 |∞p

 t 



O

s

y(τ, ξ )dξ

2 p∗

 dτ ≤

O

y(s, ξ )dξ.

We set  ϕ(t) =

O

y(ξ, t)dξ, t ≥ 0,

and rewrite (4.67) as −

2 ∗

ϕ(t) + αγ −2 |y0 |∞p



t s

2

(ϕ(τ )) p∗ dτ ≤ ϕ(s), 0 ≤ s < ∞.

(4.67)

4.2 Exact Controllability via Fast Diffusion Equation

121

Recalling that ϕ is absolutely continuous on [0, T ] and p∗ > 2, we obtain that − 2∗ d p∗ 1− 2 (ϕ(t)) p∗ + αγ −2 |y0 |∞p ≤ 0, a.e. t > 0, − 2 dt

p∗ and, therefore,

2 p∗ p∗ α −1 γ 2 |y0 |∞ ϕ(t) = 0 for t > T = ∗ (p − 2)

∗

1−

 O

y0 dξ

2 p∗

.

Hence, y(ξ, t) = 0, a.e. ξ ∈ O for t ≥ T ∗ , and taking 2 p∗ p∗ α= ∗ γ 2 |y0 |∞ (p − 2)T

 O

1− y0 dx

2 p∗

,

(4.68)

we get T = T ∗ , as claimed. If the control parameter u is taken the diffusivity of medium, then by the Fick diffusion law we have for the concentration y the equation ∂y ∂t

− div(u∇y) = 0 in (0, T ) × O, y(x, 0) = y0 (x).

(4.69)

α The feedback controller u(t) = m |y(t)|m−1 , t ≥ 0, inserted in (4.69) yields a closed loop system of the form (4.41), that is,

α ∂y − Δ(|y|m sign y) = 0 in (0, T ) × O, ∂t m y(x, 0) = y0 (x) in O, y(x, t) = 0, on (0, T ) × ∂O,

(4.70)

while, for m = 0, one gets a system of the form'(4.43).Then, by Theorem 4.4 and Theorem 4.7, it follows that y(T ) = 0 if m ∈ d−2 d+2 , 1 and a condition of the form (4.39) (respectively, (4.68)) holds. Moreover, as easily seen, if y0 ≥ 0, a.e. in O, then y ≥ 0 on (0, T ) × O. Remark 4.2. It is easily seen that the linear system ∂y − Δy + u = 0 in (0, ∞) × O, ∂t y(x, 0) = y0 (x), x ∈ O, y = 0 on (0, ∞) × ∂O,

122

4 Internal Controllability of Parabolic Equations with Inputs in Coefficients

can be controlled to y1 in a finite time via the feedback controller u = −αH (y −y1 ) if y1 ≥ ρ > 0 in O. However, it should be mentioned that in this case the physical significance of controller (4.4) (as well as of (4.37)) is different since it intervenes within the flow of the diffusion system.

4.3 Exact Controllability via Total Variation Flow For the linear parabolic equation (4.69), it turns out that a controller u of the form u(x, t) = α|∇y(x, t)|−1 d , t ≥ 0, x ∈ O,

(4.71)

steers y0 into origin in time T for a suitable positive constant α = α(y0 , T ). Indeed, substituting (4.71) in (4.69), we get the nonlinear parabolic equation

∇y ∂y = 0, t ≥ 0, x ∈ O, − α div ∂t |∇y|d y(x, 0) = y0 (x), x ∈ O, y(x, t) = 0, x ∈ ∂O.

(4.72)

It should be said, however, that problem (4.72) (as well as controller (4.71)) is not well posed in the Sobolev space W01,1 (O), but in the space BV (O) of functions with bounded variations on O. Namely, it has a natural formulation as the infinite dimensional Cauchy problem dy + Ay  0 on (0, T ), dt y(0) = y0 ,

(4.73)

where A is the subdifferential ∂ϕ : L2 (O) → L2 (O) of the convex function  ϕ(u) = α|Du| + α γ0 (u)dH d−1 , u ∈ BV (O), ∂O

where |Du| is the total variation of u ∈ BV (O), that is,  |Du| = sup

O

 u div ϕ dx; ϕ ∈ C0∞ (O), |ϕ|∞ ≤ 1 .

γ0 (u) is the trace of u to ∂O and dH d−1 is the d−1 dimensional Hausdorff measure on ∂O (see, e.g., [4, 41]). By Theorem 1.1, the Cauchy problem (4.73) in the space X = L2 (O) has, for each y0 ∈ L2 (O), a unique strong solution y ∈ C([0, T ]; L2 (O)) ∩

4.3 Exact Controllability via Total Variation Flow

123

L1 (0, T ; BV (O)) and y ∈ L∞ (0, T ; BV (O)) if y0 ∈ BV (O). As a matter of fact, t → y(t) is L2 (O)-absolutely continuous if y0 ∈ D(∂ϕ). In this sense, problem (4.72) is well posed for each y0 ∈ L2 (O). For simplicity, we assume 1 ≤ d ≤ 2 and so BV (O) ⊂ L2 (O), but this condition can be removed by redefining ϕ(u) = +∞ for u ∈ L2 (O \ BV (O). We also have Theorem 4.6. Assume that O is bounded and convex, and let y0 ∈ L2 (O) and T > 0 be arbitrary but fixed. Then, there is α > 0 such that y(T ) = 0 and y(t) = 0 for all t > T . Proof. To fix the idea, we give first a formal argument which may work if y(t) is a strong solution to (4.72) in the Sobolev space W01,1 (O). If we multiply equation (4.72) by y and integrate on (0, t) × O, we get 1 |y(t)|22 + α 2

 t 0

O

|∇y(x, s)|d dx ds ≤

1 |y0 |22 , ∀ t ≥ 0. 2

On the other hand, by the Sobolev embedding theorem, we have  O

|∇y(x, s)|d dx ≥ γ |y(s)|2 , ∀ s ≥ 0.

This yields |y(t)|2 + αγ t ≤ |y0 |2 , ∀ t ∈ (0, T ),

(4.74)

and, therefore, y(t) = 0 for t ≥ T = (αγ )−1 |y0 |2 . This argument can be made rigorous if we take into account that equation (4.73) can be approximated by

∇y dy = 0 in (0, T ) × O, − εΔy − α div ∂t |∇y|d y(x, 0) = y0 (x), x ∈ O, y(x, t) = 0, x ∈ ∂O.

(4.75)

This follows via the Kato-Trotter theorem (see[26], p. 168) because, under our ∇u with the domain D(Aε ) = assumptions, the operator Aε u = −εΔu − α div |∇u| d H01 (O) ∩ H 2 (O) is m-accretive in L2 (O) and, for each f ∈ L2 , the solution uε ∈ H01 (O) ∩ H 2 (O) to the equation

∇uε uε − εΔuε − α div |∇uε |

= f in O,

(4.76)

is strongly convergent, for ε → 0, to (I −∂ϕ)−1 f. To get the m-accretivity of Aε and so the existence in (4.76), we approximate in this equation the mapping u → |u|ud

124

4 Internal Controllability of Parabolic Equations with Inputs in Coefficients

by ψλ (u) = λ1 u for |u|d ≤ λ, ψλ (u) = |u|ud for |u|d ≥ λ, and let λ → 0 in the resulting equation, by taking into account that (see Corollary (8.2) in [41])  O

Δy · div ψλ (∇y)dx ≥ 0, ∀ y ∈ H01 (O) ∩ H 2 (O), λ > 0.

Then, by Theorem 4.1, equation (4.75) has a unique solution yε ∈ L2 (0, T ; H01 (O)) ∩ C([0, T ]; L2 (O)),

√ ∂yε ∈ L2 (0, T ; L2 (O)), t ∂t

and, arguing as above, we get the estimate 1 |yε (t)|22 + 2

 t 0

O

(ε|∇yε |2 + |∇yε |)ds dx =

1 |y0 |22 , ∀ t ∈ (0, T ), ∀ ε > 0, 2

which implies that |yε (t)|2 + αγ t ≤ |y0 |2 , ∀ t ∈ [0, T ]. Letting ε → 0, we get, for the solution y = lim yε to (4.72), estimate (4.74), as claimed. ε→0

The physical significance of Theorem 4.6 is obvious: the concentration y of the diffusion system (4.69) can be driven at zero in a finite time by a diffusion controller y proportional with the inverse |∇y(t)|−1 d of flux magnitude.

4.4 Exact Null Controllability in Rd Consider here the control system ∂y + (λ − Δ)(uy) = 0 in (0, ∞) × Rd , ∂t y(x, 0) = y0 (x),

(4.77)

where λ > 0 and u is the feedback controller (4.37). The corresponding closed loop equation ∂y + α(λ − Δ)(y m sign y) = 0 in (0, ∞) × Rd , ∂t y(x, 0) = y0 (x), has, for y0 ∈ L1 (Rd ) ∩ H −1 (Rn ), a unique strong solution y ∈ C([0, T ]; L1 (Rd ) ∩ H −1 (Rd )), ∀ T > 0. (see, e.g., [26], p. 233.) We have

(4.78)

4.5 Exact Controllability of Linear Stochastic Parabolic Equations

125

Theorem 4.7. Let y0 ∈ L1 (Rd ) ∩ H −1 (Rd ), d ≥ 3, and m = solution y to (4.78) satisfies y(T ) = 0 for α =

d−2 d+2 .

1 y0 1−m , H −1 (Rd ) γ m+1 T (1 − m)

Then the

(4.79)

where γ = sup{yH −1 (Rd ) ; yLm+1 (Rd ) = 1}. Here, the space H −1 (Rd ) is endowed with the norm  yH −1 (Rd ) =

Rd

((I − Δ)

−1

1 2

y, y) dx

.

Proof. By multiplying (4.78) scalarly in H −1 (Rd ) with y and integrating on Rd , we get  1 d 2 |y(x, t)|m+1 dx = 0, a.e. t > 0. y(t)H −1 (Rd ) + α 2 dt Rd By the Sobolev embedding theorem, we have yLm+1 (Rd ) ≥ γ yH −1 (Rd ) for m = d−2 d+2 (see, e.g., [55], p. 278). This yields d y(t)H −1 (Rd ) + αγ m+1 y(t)m ≤ 0, a.e. t > 0. H −1 (Rd ) dt We set ϕ(t) = y(t)H −1 (Rd ) and get the differential inequality ϕ (t) + αγ m+1 ≤ 0, a.e. t > 0. (ϕ(t))m This yields 1 1 (ϕ(t))1−m + αγ m+1 t ≤ y0 1−m . H −1 (Rd ) 1−m 1−m Hence, ϕ(T ) = 0 for α =

1 T γ m+1 (1−m)

y0 1−m , as claimed. H −1 (Rd )

4.5 Exact Controllability of Linear Stochastic Parabolic Equations Consider the linear stochastic equation dX − Δ(uX)dt = XdW in (0, T ) × O, X(x, 0) = X0 (x), X(x, t) = 0, (x, t) ∈ [0, T ] × ∂O,

(4.80)

126

4 Internal Controllability of Parabolic Equations with Inputs in Coefficients

where O ⊂ Rd is bounded and open, while W is a cylindrical Gaussian process in a ∞  probability space (Ω, F , P). More precisely, W is given by W (t) = μj ej βj (t), j =1

∞ −1 (O) is an where {βj }∞ j =1 are independent Brownian motions, {ej }j =1 ⊂ H orthonormal basis in H −1 (O), and μj are suitable chosen. (See Section 3.6.) The controllability problem considered here is to design a feedback controller u = Φ(X) which steers X0 in origin in time T . The controller is of the form (4.37), where m ∈ [0, 1) and the corresponding closed loop stochastic differential system is

dX − αΔ(|X|m sign X)dt = X dW in (0, ∞) × O, X(0) = X0 in O, X = 0 on (0, ∞) × ∂O.

(4.81)

−1 (O). Then Theorem 4.8. Assume that m ∈ [0, 1), 1 ≤ d < 2(m+1) 1−m , and X0 ∈ H 2 −1 equation (4.81) has a unique solution X ∈ L (Ω; C([0, T ]; H (O)), ∀ T > 0, and, if τ = τ (ω) is the stopping time τ (ω) = inf{t > 0; X(t, ω) = 0}, ω ∈ Ω, then we have

P(τ > t) ≤ Cα −1 X0 1−m , H −1 (O)

(4.82)

where C > 0 is independent of ω. Theorem 4.4 amounts to saying that P-a.s. system (4.80) is exactly controlled in origin in a random time T = T (ω) with a probability P estimated by formula (4.82). The existence of the solution X as well as estimate (4.82) for the stopping time τ was proved in [48], p. 68 (Theorem 3.73) and we omit here the details. We only note that (4.82) follows by (4.77) via Itô’s formula applied to (4.81) with the Lyapunov function ϕ(X) = X1−m , which yields after some calculation H −1 (O) X(t)1−m H −1 (O)

1−m +α 1+m



t

−m−1 m+1 X(s)H −1 (O) X(s)Lm+1 (O) 1[X(s)H −1 (O) >0] ds r  t ≤ X(r)1−m + C X(s)1−m ds H −1 (O) H −1 (O) r

for 0 < r < t < ∞. This implies (4.82) via the stochastic Gronwall’s lemma.

4.6 Notes on Chapter 4 The results of this chapter are new in the controllability theory context. The feedback controllers constructed here transform the original equation in a nonlinear closed loop system of phase transition type which is at the origin of the exact controllability mechanism. Roughly speaking, this approach relies on the finite time extinction

4.6 Notes on Chapter 4

127

property of solutions to nonlinear fast diffusion equations (the main references about this mechanism are the works [27, 33, 40, 46, 48]). As a matter of fact, this property is true for an entire class of nonlinear infinite dimensional equations and is expressed in the following abstract result: Let X be a Banach space, J : X → X∗ the duality mapping of X and A : D(A) ⊂ X → X be an m-accretive (multivalued) operator such that X (v, J (u))X∗ ≥ ρuX , ∀ u ∈ D(A), v ∈ Au, where ρ > 0. Then every mild solution u ∈ C([0, T ]; X) to the Cauchy problem du dt + Au  0, t ≥ 0, u(0) = u0 , has the property u(t) = 0 for t ≥ ρ −1 u0 X . The proof is quite immediate if u is a strong solution and follows by the convergence of the finite difference scheme ui+1 + hAui+1  ui to the mild solution u in the general case. One might expect to find such nonlinear feedback controllers for more general parabolic equations and also for controllers with internal or boundary support. A nice feature of this method is that it provides exact controllability by feedback and not by open loop controllers as usually happens by the linearization technique presented in Chapter 3.

Chapter 5

Feedback Stabilization of Semilinear Parabolic Equations

We shall discuss here the internal and boundary feedback stabilization of equilibrium solutions to semilinear parabolic equations. The main conclusion is that such an equation is stabilizable by a feedback controller with finite dimensional structure dependent of the unstable spectrum of the corresponding linearized system around the equilibrium solution.

5.1 Riccati-based Internal Stabilization An important objective of the control theory is to design a feedback controller that stabilizes the differential system around a given orbit which in most situations is a steady-state or periodic orbit of the system. Here we shall study this problem for semilinear parabolic equations of the form (1.9). As in the exact controllability problem, it is desirable to design stabilizable feedback controllers which are supported by accessible small subdomains or on the boundary. Compared with the exact null controllability property discussed in Chapter 3, the stabilization which is a weaker one has two important advantages very appreciated in engineering applications: it is stable to structural modifications of the system and may be achieved by feedback controllers, that is, in real time. Let O ∈ Rd , d = 1, 2, 3, be an open and bounded subset with smooth boundary ∂O, ω⊂O be an open subset and let m = 1ω be the characteristic function of ω. Let Q = O×(0, ∞) and Σ = ∂O×(0, ∞). Consider the quasilinear controlled reaction-diffusion equation yt (x, t) − Δy(x, t) + f (x, y(x, t), ∇y(x, t)) = m(x)u(x, t) in Q, in O, y(x, 0) = y0 (x), y(x, t) = 0, on Σ, © Springer International Publishing AG, part of Springer Nature 2018 V. Barbu, Controllability and Stabilization of Parabolic Equations, Progress in Nonlinear Differential Equations and Their Applications 90, https://doi.org/10.1007/978-3-319-76666-9_5

(5.1)

129

130

5 Feedback Stabilization of Semilinear Parabolic Equations

and a steady-state (equilibrium) solution ye to (5.1), that is, −Δye (x) + f (x, ye (x), ∇ye (x)) = 0, in O, on ∂O. ye (x) = 0,

(5.2)

Let H = L2 (O) with the norm | · |2 and V = H01 (O) with the norm  · . We shall denote by | · |s the norm of Ls (O) and by  · r the norm of H r (O). We use (·, ·) to denote the inner product in H and the paring between V and V and between H 2 (O) and (H 2 (O)) , respectively. Use | · |ω and (·, ·)ω to denote the norm and the inner product of L2 (ω), respectively. The same notations will be used for the complex space L2 (O) and L2 (ω), respectively. We shall consider now a set of technical conditions on the steady-state solution ye and the nonlinear function f . (H1 ) ye ∈ C(O), ∇ye ∈ (C(O))d . (H2 ) f (x, y, θ ) ≡ g(y) + a(x)y + b(x) · θ , where g : R → R is of class C 2 , a ∈ C(O), b ∈ C(O; Rd ) and g (r) ≥ −γ ,

∀ r ∈ R,

|g

(r)| ≤ γ1 (|r|η−2 + 1),

(5.3) ∀ r ∈ R,

(5.4)

where γ ∈ R, γ1 ≥ 0, and 2≤η≤

d +3 · d −1

(5.5)

The standard example is the polynomial function g(r) ≡ a1 |r|η−1 r + a2 |r|r, where ai , a2 ≥ 0 and η satisfies condition (5.5). Here, we shall not discuss the existence of a solution y to (5.1) under assumptions (H1 )–(H2 ) though this problem can be treated as in the case of Theorem 1.4. The problem to be addressed here is that to design a feedback controller u = Φ(t, y) such that the solution y to (5.1) where u = Φ(t, y) satisfies lim y(t) = ye strongly in L2 (O)

t→∞

for all y0 in a suitable chosen neighborhood of ye . By the substitution y → ye + y, we reduce the problem of stabilization of ye to that of stabilization of the null solution to the equation yt − Δy + f (x, y + ye , ∇y + ∇ye ) − f (x, ye , ∇ye ) = mu in Q, in O, y(x, 0) = y 0 (x) = y0 (x) − ye (x) y=0 on Σ.

(5.6)

5.1 Riccati-based Internal Stabilization

131

We consider the corresponding linearized equation about zero solution yt − Δy + g (ye )y + ay + b · ∇y = mu in Q, in O, y(x, 0) = y 0 (x) y=0 on Σ.

(5.7)

It should be said that, as seen earlier in Chapter 3, under the above assumptions for each T there is an open loop controller u∗ : [0, T ] → L2 (O) such that ∗ ∗ the corresponding solution y u to (5.1) satisfies y u (T ) = ye . If we extend this ∗ controller with 0 on (T , ∞), we may view u as a stabilizable controller for ye . However, being obtained by fixed point theoretical arguments the above controller u∗ is not in feedback form and its structure is somehow unclear. We shall develop in the following a constructive procedure to obtain a feedback stabilizable controller u based on Riccati-based stabilization of linear system (5.7) and which is expressed in function of a finite system of unstable modes of the system. Let A = −Δ with the domain D(A) = H 2 (O) ∩ H01 (O), and A0 y = g (ye )y + a(x)y + b(x) · ∇y. 1

We denote by As , s ∈ (0, 1) the fractional power of operator A. Let W = D(A 4 ) 1 with the graph norm yW = |A 4 y|2 . Recall that (see [84], pp. 66), 1

1

1

D(A 4 ) = {y ∈ H 2 (O); (dist(x, ∂O))− 2 y ∈ L2 (O)}. In terms of A and A0 , equation (5.7) can be rewritten as y + Ay + A0 y = mu, t ∈ (0, T ) y(0) = y0 .

(5.8)

Consider the operator F0 = −(A + A0 ) with the domain D(F0 ) = D(A) = H01 (O) ∩ H 2 (O) and denote by "F the extension of F0 on the complexified space H = L2 (O) ⊕ iL2 (O). This means that F (y1 + iy2 ) = F0 y1 + iF0 y2 , ∀ y1 , y2 ∈ D(A). We shall denote again (·, ·) the scalar product in the space H . It is readily seen that F has a compact resolvent and generates an analytic C0 -semigroup on H = L2 (O). Then, by the Fredholm–Riesz theorem, the operator F has a countable set of complex eigenvalues λj and corresponding proper eigenvectors ϕj , that is, F ϕj = λj ϕj . For each λj there is a finite number mj of linear independent vectors {ϕij ; i = 1, . . . , mj } such that (F − λj ) ϕij = 0, 1 ≤  ≤ mj . These vectors are called generalized eigenvectors or, simply, eigenvectors (eigenfunctions). The algebraic multiplicity j of λj is the number of generalized eigenvectors, while the geometric multiplicity of λj is the number of proper vectors corresponding to λj . We note

132

5 Feedback Stabilization of Semilinear Parabolic Equations

also that the operator F has a finite number N of eigenvalues λj with Re λj ≥ 0 (unstable eigenvalues). In the following, the eigenvalues are repeated according to their algebraic multiplicity j . An eigenvalue λj is said to be simple if mj = 1 and semisimple if the algebraic and geometric multiplicity coincide. This means that λj is a pole of order mj . Let {ϕj }N j =1 be the corresponding system of eigenfunctions, ϕj = ϕj1 + iϕj2 and let YNm = span{ϕjm }N j =1 , m = 1, 2.

(5.9)

Denote by M the number of distinct unstable eigenvalues, that is, 1 + 2 + · · · + M = N, and let K = max{j }M j =1 .

(5.10)

We shall prove first the following stabilization result for equation (5.7) or, equivalently, (5.8). 1 Theorem 5.1. Under assumption (H1 ), there are the functions {ψi }K i=1 ⊂YN , 2K 2 {ψi }i=K+1 ⊂YN and a linear self-adjoint operator G : D(G)⊂H → H such that for some 0 < γ1 < γ2 , C1 > 0, 1

1

1

γ1 |A 4 y|22 ≤ (Gy, y) ≤ γ2 |A 4 y|22 , ∀ y ∈ D(A 4 ), |Gy|2 ≤ C1 y, ∀ y ∈ V , 1 1 3 (ψi , Gy)2ω = |A 4 y|22 , ∀ y ∈ D(A). 2 2

(5.11) (5.12)

2K

(F y, Gy) +

(5.13)

i=1

Moreover, the feedback controller 2K

u = − (Gy, ψi )ω ψi

(5.14)

i=1

stabilizes exponentially system (5.8). Equation (5.13) can be, equivalently, written as the algebraic operational Riccati equation 3

GF + F ∗ G + N = A 2 , where the linear operator N : H → H is defined by Ny=

2K 

i=1

(Gy)(ξ )ψi (ξ )dξ G(mψi ), ∀ y ∈ H. ω

(5.15)

5.1 Riccati-based Internal Stabilization

133

It turns out that the stabilizing feedback controller (5.14) also stabilizes the semilinear system (5.6). Namely, we have Theorem 5.2. Under assumptions (H1 ), (H2 ), the feedback controller 2K

u = − (G(y − ye ), ψi )ω ψi

(5.16)

i=1

exponentially stabilizes the steady-state solution ye to (5.1) in a neighborhood 1

Uρ = {y0 ∈ D(A 4 ); y0 − ye W < ρ} of ye for a suitable ρ > 0. More precisely, if ρ > 0 is sufficiently small, then for each y0 ∈ Uρ there is a solution y ∈ C(R+ ; H ) ∩ L2loc (R+ ; V ) to the closed loop system yt − Δy + f (x, y, ∇y) + y(x, 0) = y0 (x)

2K

mψi (G(y − ye ), ψi )ω = 0 in Q, i=1

y=0

in O,

(5.17)

on Σ,

such that 

∞

0

3

|A 4 (y(t) − ye )|22 dt ≤ Cy0 − ye 2W |y(t) − ye |2 ≤ Ce−γ0 t y0 − ye W , ∀ t ≥ 0,

(5.18) (5.19)

for some γ0 > 0. Remark 5.1. In particular, it follows by Theorem 5.2 that, if all unstable eigenvalues λj are simple, that is, K = 1, then the feedback controller (5.16) is of the form u = −(G(y − ye ), ψ)ω , where ψ ∈ span{Re ϕj , Im ϕj }N j =1 is arbitrary. Remark 5.2. As mentioned earlier, by controllability theorems established in Chapter 3, we know that equation (5.7) (or (5.8)) is exactly null controllable in every finite time which implies by a Riccati-based device, mentioned in Section 1.5, the feedback stabilization (see (1.92)–(1.93)). The novelty of Theorem 5.1 and 5.2 is that the stabilizable controllers (5.14), (5.16) found here have a simple finite dimensional structure which is not the case with the controllers arising in Chapter 3. As seen in Theorems 3.1–3.3 for exact null controllability, it was necessary a Carleman-type inequality for the solutions to dual parabolic equations, while as

134

5 Feedback Stabilization of Semilinear Parabolic Equations

seen later on, for stabilization it is sufficient, and in a certain sense necessary, a weaker property, i.e., the unique continuation property of eigenfunctions the operator A + A0 . As in the case of the exact controllability, the stabilization result given by Theorem 5.2 is local and works only for nonlinear functions f with mild polynomial growth. This can be explained by the fact that a large nondissipative reaction term f is a source of instability, which cannot be compensated by the linear dissipative operator −Δ. It should be mentioned also that the stabilization property, obtained via this approach, is only local for initial data in a neighborhood of an equilibrium solution. Proof of Theorem 5.1. We shall proceed as in [37] (see, also, [45]), via the pole location technique. In a few words, the idea is to decompose the system in an unstable finite-dimensional system generated by unstable eigenvectors and an infinite-dimensional exponentially stable system. Then the finite-dimensional unstable system is stabilized via the standard pole allocation technique. We shall consider the extension of system (5.8) on the complexified space L2 (O) ⊕ iL2 (O) = H , that is, dz − F z = mv, t > 0, dt z(0) = z0 = y 0 + i y0,

(5.20)

y 0 . As noted below, the operator where z = y + i y , v = u + i u, z0 = y 0 + i F has a compact resolvent in H , and generates an analytic C0 -semigroup on H , so that only a finite number of eigenvalues of F are in the right complex half-plane {λ ∈ C; Re λ ≥ 0}. We have already denoted by λ1 , λ2 , . . . , λN these (unstable) eigenvalues repeated according to their algebraic multiplicity i so that Re λN +1 < 0 ≤ Re λN ≤ · · · ≤ Re λ1 . Let Γj be a closed curve enclosing λj but no other point of the spectrum of F . Define the eigenprojection PN,j , PN,j

1 = 2π i



(λI − F )−1 dλ, j = 1, . . . , M. Γj

The space ZN,j = R(PN,j ) (the range of PN,j ) is called the algebraic eigenspace for the eigenvalue λj and j = dim ZN,j is just the algebraic multiplicity for λj . We recall (see [82], Theorem 6.17) that ZN,j = {z ∈ H ; (λj − F )j z = 0}, j = 1, . . . , M.

5.1 Riccati-based Internal Stabilization

135

We now denote by {ϕij }ji=1 the (normalized) linearly independent system of generalized eigenfunctions corresponding to each unstable distinct eigenvalue λi of F . Then the space H can be decomposed as (see [82], p. 178) u s u u ⊕ ZN ; ZN = span{ϕij , i = 1, . . . , M; j = 1, . . . , i }; dim ZN = N, H = ZN u and Z s is invariant under F . where each of the spaces ZN N

Denote by PN the projection, explicitly identified as a contour integral, PN =  (λI − F )−1 dλ where Γ separates the unstable spectrum from the stable one Γ  and similarly define PN∗ = 2π1 i Γ (λI − F ∗ )−1 dλ. We set 1 2π i

FNu = PN F |ZNu ; FNs = (I − PN )F = F |ZNs . u and Z s coincide with {λ }N We then have that the spectra of F on ZN j j =1 and N ∞ {λj }j =N +1 , respectively: s ∞ σ (FNu ) = {λj }N j =1 ; σ (FN ) = {λj }j =N +1 .

Moreover, since F generates a C0 -analytic semigroup on H , then its restriction s generates likewise a C -analytic semigroup on Z s . This implies that FNs to ZN 0 N s (see [114]), and so it satisfies the spectrum determined growth condition on ZN we have eFN t L (H ;H ) ≤ Cγ0 e−γ0 t , ∀ t ≥ 0, s

(5.21)

for any γ0 > |Re λN +1 |, where Cγ0 depends on γ0 . u , then the eigenvalues of (F u )∗ If (FNu )∗ is the adjoint operator of FNu on ZN N are just the complex conjugate λj of eigenvalues λj of FN , j = 1, . . . , N. The eigenvalue λj of FNu is called semisimple if the algebraic and geometric multiplicity of λj coincide. This means that FNu ϕj k = λj ϕj k , j = 1, . . . , M, k = 1, . . . , j , and we have also (FNu )∗ ϕj∗k = λj ϕj∗k for an eigenvalue λj of (FNu )∗ . If all the eigenvalues of FNu are semisimple, that is, the operator FNu is diagonlizable M



M



j j (see [82], p. 41), then the systems {ϕj k }j =1 k=1 , {ϕj∗k }j =1 k=1 can be chosen in such a way to form a bi-orthogonal system, that is,

∗ ) = 0 if j = 1, j = i, k = m, (ϕj k , ϕim

and (ϕj k , ϕj∗k ) = 0 for k = 1, . . . , j , j = 1, . . . , M. (See [42] for the proof.)

136

5 Feedback Stabilization of Semilinear Parabolic Equations

Then system (5.20) can be, accordingly, decomposed as z = zN + ζN ,

zn = PN z,

ζN = (I − PN )z,

(5.22)

u

on ZN : zN − FNu zN = PN (mv), zN (0) = PN z0 ,

(5.23)

s on ZN : ζN − FNs ζN = (I − PN )(mv), ζN (0) = (I − PN )z0 .

(5.24)

We shall prove now that the finite dimensional unstable system (5.23) is stabilizable by a controller v with finite structure (Lemma 5.1) and show afterwards that the infinite dimensional system (5.24) is stable. Lemma 5.1. Given γ1 > 0 arbitrarily large, there is a controller v = vN = K  i (t)ψ , ψ ∈ Z u , such that once inserted in (5.23), yields the estimate vN i i N i=1

|zN (t)| + |vN (t)| ≤ Cγ1 e−γ1 t |PN z0 |, t ≥ 0.

(5.25)

u , λ , i = 1, . . . , M, be the distinct unstable eigenProof. Let w1 , w2 , . . . , wk ∈ ZN i i M value of F , and {{ϕij }j =1 }i=1 = {ϕj }N j =1 its corresponding system of generalized eigenfunctions linearly, independent on L2 (O). By the classical procedure, we may u choose the system {{ϕij }ji=1 }M i=1 such that the matrix Λ corresponding to FN in this basis has the Jordan canonical form. More precisely,

FNu {ϕij }ji=1 = Ji {ϕij }ji=1 , ∀ i = 1, . . . , M,

(5.26)

where Ji is the Jordan block for λi . N In the following, we shall simply write the system {ϕij }ji=1 M i=1 as {ϕk }k=1 . (We recall that 1 + 2 + M = N.) u N Consider the orthonormal system {φj }N j =1 ∈ ZN obtained from {ϕk }k=1 by Schmidt’s orthogonalization procedure. Let us represent φj as φj =

N

aj k ϕk , j = 1, 2, . . . , N k=1

and set χ =

N N

bk ϕk , bk = aj k aj  , k,  = 1, . . . , N. k=1

j =1

We shall regroup the vectors χ  from χ 1 to χ 1 , then from χ 1 +1 to χ 1 +2 , and so on. We rename them as χ j = χ 1j , j = 1, . . . , 1 , χ 1 +j = χ 2j , j = 1, . . . ., 2 , ..................................... χ 1 +...+M−1 +j = χ Mj , j = 1, . . . ., M .

5.1 Riccati-based Internal Stabilization

137

u , consider the  ×k matrix D For k = K and any fixed vector {ψ1 , . . . , ψk } ∈ ZN i i defined by

   (ψ1 , P ∗ χ i )ω (ψ2 , P ∗ χ i )ω · · · (ψk , P ∗ χ i )ω    N 1 N 1 N 1     ∗ ∗ ∗ χ χ χ  (ψ , P ) (ψ , P ) · · · (ψ , P ) 2 k Di =  1 N i2 ω N i2 ω N i2 ω    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..     (ψ1 , P ∗ χ i )ω (ψ2 , P ∗ χ i )ω · · · (ψk , P ∗ χ i )ω  1 1 1 N N N

(5.27)

for i = 1, 2, . . . , M. Then we shall use the following finite-dimensional feedback stabilization result which will play a key role in the proof of Lemma 5.1. Lemma 5.2. Let K = max{i ; i = 1, . . . , M} and assume that rank Di = i , ∀ i = 1, . . . , M.

(5.28)

Then there is a controller v = vN such that (5.25) holds. Proof of Lemma 5.2. In order to illustrate better the proof technique, we shall assume first that all eigenvalues λj , j = 1, . . . , N , are simple. Hence, K = i = 1 for all i = 1, . . . , N = M and ϕij ≡ φi , i = 1, . . . , N, j = 1, the matrix Λ is diagonal, Ji = {ϕi } and φj = ϕj , bj k = aj k = δj k (Kronecker’s u , the matrix (5.27) is symbol) χ  = ϕ , ∀  = 1, . . . , N. Then, for ψ1 ∈ ZN Di = (ψ1 , PN∗ ϕi )ω , i = 1, . . . , N, and condition (5.28) reduces to (ψ1 , PN∗ ϕj )ω = 0, j = 1, . . . , N.

(5.29)

Let us show that, under this assumptions, system (5.23), that is,

− FNu zN = PN (mψ1 )v zN

is exponentially stable, that is, there is v ∈ C([0, T ]) such that (5.25) holds. To prove this, we set zN = N

j =1

j (zN ) ϕj

=

N  j =1 N

j =1

j

zN ϕj and so rewrite (5.30) as

j

FNu (ϕj )zN (t) + v1 (t)PN (mψ1 ).

(5.30)

138

5 Feedback Stabilization of Semilinear Parabolic Equations

Equivalently, N

j j ((zN ) (t) − λj zN (t))ϕj = v1 (t)PN (mψ1 ). j =1

By using the Schmidt orthogonalization technique, we may replace the system {ϕj }N j =1 by an orthonormal one in H , and so we get (zN ) − λj zN = (PN (mψ1 ), ϕj )v1 = (w1 , PN∗ ϕj )ω v1 = Dj v1 , j = 1, . . . , N. (5.31) Recall that, by the Kalman controllability theorem (see condition (1.90) in Section 1.5), the linear system (5.31) is exactly null controllable if and only if j

j

rankD, DA, . . . , DAN −1  = N, N where A = diagλj N j =1 , D = colDj j =1 . Equivalently, N1 N N det col(Dj )N j =1 , λ1 col(Dj )j =1 , . . . , λ1 col(Dj )j =1  = 0.

By condition (5.29), this condition is obviously satisfied for system (5.31). The above argument extends to the case where all unstable eigenvalues λj are semisimple. That means that the Jordan block Ji is a diagonal matrix of dimension i ×i , that is,    λi 0     Ji =  . . .  , i = 1, 2, . . . , M.   0 λi 

(5.32)

Indeed, assume that {ψi } satisfy condition (5.28) and consider the controlled system

zN

− FNu zN

= PN

K

 i vN (t)mψi

i=1

=

K

i vN (t)PN (mψi )

(5.33)

i=1

obtained from (5.23), with the controller v=

K

i vN (t)wi ,

i=1 i }K ⊂ CK . In this case, we have by (5.27) where vN = {vN i=1

FNu (ϕij = λi ϕij , i = 1, . . . , M; j = 1, . . . , i , (FNu )∗ (ϕij∗ ) = λi ϕij∗ , i = 1, . . . ., M; j = 1, . . . , i .

(5.34)

5.1 Riccati-based Internal Stabilization

139

i N ∗ M i ∗ N If we rename the sequences {ϕij }M i=1,j =1 , {ϕij }i=1,j =1 as {φi }i−1 and {φi }j =1 , respectively, we may assume that

(φi , φj∗ ) = δij , i, j = 1, . . . , N. If we insert, as above, zN = N

N  j =1

(zN ) φj = j

j =1

(5.35)

j

zN φj in system (5.33), we get

N

j =1

j

FNu (φj )zN +

K

i vN PN (mψi ).

i=1

Taking into account (5.35) and that FNu φj = λj , we get (zN ) = λj zN + j

j

K

i (ψi , PN∗ (φj∗ ))ω vN , j = 1, . . . , N.

i=1

Equivalently,

− Λ zN = DvN , zN

(5.36)

K,N ∗ ∗ where zN = colzN N j =1 , D = dij i,j =1 , dij = (ψi , PN (φj ))ω and Λ is the N×N matrix j

   J1 0      Λ =  ... ,   0 JM 

(5.37)

where Ji , i = 1, . . . , M, are given by (5.32). It should be noted that D is just the χ matrix colDi M i=1 , where  = φ ,  = 1, . . . , N . Taking into account that  m  J 0   1    .. Λm =   , ∀ m ∈ N, .   m  0 JM  it is easily seen by assumption (5.28) that, also in this case, the Kalman condition rankD, DA, . . . , DAN −1  = N holds, which implies the exact null controllability of system (5.36), as desired. The general case of not semisimple unstable eigenvalues λj follows similarly, but with some technical details. Namely, also in this case, system (5.32) reduces to

140

5 Feedback Stabilization of Semilinear Parabolic Equations

(5.35), but the matrices Λ and D are given by (5.37) and, respectively, ⎡ D = (wi , PN∗ χ j )ω K i=1

N j =1

⎢ ⎢ =⎢ ⎣

⎤

D1 D2 .. .

⎥ ⎥ ⎥. ⎦

DM Here Ji is the Jordan block of dimension i corresponding to the eigenvalue λi of F , that is, of FNu . We shall test the controllability of the pair {Λ, D} via Kalman’s theorem recalled above. To this end, we introduce for n = 1, 2, . . . the matrices ⎡ ⎢ ⎢ Λn D = ⎢ ⎣

J1n

⎤ ⎡ J2n

..

⎥ ⎥ ⎥ ⎦

. n JM

⎢ ⎢ ⎢ ⎣

D1 D2 .. .

⎤

⎡

⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎦ ⎣

DM

J1n D1 J2n D2 .. .

⎤ ⎥ ⎥ ⎥ ⎦

nD JM M

The Kalman controllability matrix ⎡ ⎢ ⎢ ⎢ [D, ΛD, Λ2 D, . . . , ΛN −1 D] = ⎢ ⎢ ⎣

D1

J1 D1

D2 .. .

J2 D2 .. .

DM JM DM

λ21 D1 · · · J1N −1 D1

⎤

⎥ ⎥ λ22 D1 · · · J2N −1 D2 ⎥ ⎥ .. .. .. ⎥ ⎦ . . . N −1 λ2M D1 · · · JM DM

u , is of full rank that satisfies Kalman’s controllability of size N × KN, N = dim ZN condition, if and only if rank Wi = i , i = 1, . . . , M, which is true by assumption (5.28). Thus, there is a controller vN such that vN (T ) = 0. u such that the In conclusion, given any γ1 > 0, there exist {p1 , . . . , pK } ⊂ ZN solution zN to (5.30) satisfies the estimate  u    |vN (t)| + |zN (t)| = eFN t zN (0) ≤ Cγ1 e−γ1 t |zN (0)|.

Now, to complete the proof of Lemma 5.1, we need to check condition (5.28). The proof relies on the following basic properties of the generalized eigenfunctions i N u {ϕij }M i=1 j =1 = {ϕj }j =1 , of the operator FN . 1◦ The eigenfunction system {ϕij , 1 ≤ i ≤ M, 1 ≤ j ≤ i } is linearly independent in L2 (ω). ◦ 2 The system {PN∗ ϕij , 1 ≤ i ≤ M, 1 ≤ j ≤ i } is linearly independent in L2 (O) and also in L2 (ω). u and P ∗ ϕ = 0 on ω then ϕ ≡ 0. ◦ 3 If ϕ ∈ ZN N

5.1 Riccati-based Internal Stabilization

141

We shall see that all these properties are consequences of the unique continuation property for the elliptic equation −Δϕ + fy (x, ye , ∇ye )ϕ + fθ (x, ye , ∇ye ) · ∇ϕ = λϕ ϕ = 0 on ∂O

(5.38)

and for the corresponding dual equation. More precisely, if ϕ = 0 on ω, then ϕ ≡ 0. To begin with, we treat first where the eigenvalues {λj }, j = 1, . . . , M, are semisimple, that is, the case of proper eigenvectors (eigenfunctions) ϕj . 1◦ . We note first that, if {ϕj , 1 ≤ j ≤ μ} is a system of eigenfunctions corresponding to the same eigenvalue λj , it is linearly independent on O, and so, by the unique continuation property for solutions ϕij to (5.38), it is linearly independent on ω, too. Consider now the case of two distinct eigenvalues λ1 , λ2 . Assume by contradiction that the corresponding system of eigenfunctions {ϕ1j , ϕ2k ; j = 1, 2, . . . , 1 , k = 1, 2, . . . , 2 } is linearly dependent on ω. We may assume, therefore, that ϕ2,2 =

1

αj ϕ1j +

j =1

where

1  j =1

 2 −1

βk ϕ2k on ω,

k=1

αj > 0. Taking into account that F ϕ1j = λ1 ϕ1j , F ϕ2k = λ2 ϕ2k and

F ϕ22 = λ2 ϕ22 on O, it follows after some calculation that (λ2 − λ1 )

1

αj ϕ1j = 0 on ω.

j =1

Since, by the first step, the system {ϕ1j } is linearly independent on ω, and so αj = 0 for all j , which is a contradiction. By induction with respect to number q of distinct eigenvalues λj , 1◦ follows in general. Indeed, assume that 1◦ is true for q − 1 and consider a system {ϕij , 1 ≤ i ≤ q, 1 ≤ j ≤ i }. Assume that this system is linearly dependent on ω and we shall argue from this to a contradiction. As above, we may assume that ϕqq =

1

q −1

αj 1 ϕj 1 + . . . +

j =1

where

q−1 k   j =1 k=1

αj q ϕj q on ω,

j =1

αj k > 0. Taking in account that F ϕij = λi ϕij on O, and

consequently on ω too, this implies that (λ1 (λq )−1 − 1)

1

j =1

q−1

αj 1 ϕj 1 + . . . + (λq−1 (λq )−1 − 1)

j =1

αj (q−1) ϕj (q−1) = 0 on ω.

142

5 Feedback Stabilization of Semilinear Parabolic Equations

By the inductive hypothesis, the latter implies that all αj k are zero. The contradiction we arrived at completes the proof of 1◦ . We note that property 1◦ remains true, by the same argument, for the eigenfunctions ϕij∗ , i = 1, . . . , M, j = 1, . . . , i of the dual operator FNu ∗ . In particular, it u is zero on ω, then it is zero on all of O. follows by 1◦ that, if ϕ ∈ ZN M, i αij PN∗ ϕij = 0 on O and we will show that To prove 2◦ , we assume that i=1,j =1

αij = 0. Let h ∈ H be arbitrary but fixed. We have ⎛

⎞

M,

i

⎝h,

⎛

M,

i

αij PN∗ ϕij ⎠ = ⎝PN h,

i=1,j =1

⎞ αij ϕij ⎠ = 0.

i=1,j =1

u , we conclude that Taking into account that, as h runs H , PN h fills all of ZN  αij ϕij = 0 on O. Since the system {ϕij } is linearly independent, we conclude that αij = 0, as claimed. Now, assume that

ϕ=

M,

i

αij PN∗ ϕij = 0 on ω.

i=1,j =1 N 

We may write ϕ =

k=1

βk ϕk∗ , where ϕk∗ are eigenfunctions of the dual operator FN∗ .

By property 1◦ applied to the dual operator FN∗ , we infer that system {ϕj∗ } is linearly independent on ω and so all βk are zero. Hence ϕ = 0 on O, as claimed. As regards 3◦ , it is an immediate consequence of 1◦ and 2◦ . Indeed, if PN∗ ϕ = 0 N  on ω and ϕ = αj ϕj , then by 2◦ it follows that all αj are zero. Hence, by 1◦ , ϕ j =1

is identically zero, which is absurd. Now, let us prove properties 1◦ , 2◦ , and 3◦ in the general case of generalized eigenfunctions. We note first that, if ϕ ∈ ZN,j , that is, (λj − F )j ϕ = 0 and ϕ = 0 on ω, then according to first step (λj − F )j −1 ϕ ≡ 0 (because (λj − F )j ϕ = 0 on ω and, finally, it follows that ϕ ≡ 0. Assume, now, that ϕ1 ∈ ZN,1 , ϕ2 ∈ ZN,2 are such that ϕ1 = αϕ2 on ω. We may take α = 1 and assume 1 < 2 . We have (λj I − F )k ϕj = 0, j = 1, 2, where k   k     j j j j j F k−j ϕ = j k−j ϕ = 0 on ω. k = 1 , and so λ (−1) 1 1 1 k k λ2 (−1) F j =1

j =1

This yields (λ1 − F )k ϕ1 = (λ2 − F )k ϕ1 on ω and so we have, for k = 1 < 2 , k

j j =1

k

(λ1 )j (−1)k−j F k−j ϕ1 =

k

j j =1

k

(λ2 )j (−1)k−j F k−j ϕ1 .

5.1 Riccati-based Internal Stabilization

143

This yields k

j j =1

k

j

j

(λ1 − λ2 )(−1)k−j F k−j ϕ1 = 0 on ω.

Keeping in mind that (λ1 − F )1 ϕ1 = 0, we get, for z=

k

j j =1

k

j

j

(λ1 − λ2 )(−1)k−j F k−j ϕ1 ,

(λ1 − F )1 z = 0, z = 0 on ω. Hence z ≡ 0 on O and this implies that (λ1 − F )1 ϕ1 = (λ2 − F )1 ϕ1 = 0 on O. Therefore, ϕ1 is the eigenfunction of F for the eigenvalue λ2 . The contradiction we arrived at shows that ϕ1 and ϕ2 are linearly independent on ω. By induction with respect to m = 1, . . . , N , it follows that system {ϕj }M j =1 is linearly independent on ω, as claimed. Similarly, it follows that {PN∗ ϕj }N j =1 is linearly independent on ω. Taking into account that the orthogonalization matrix {aj k } as well as {bj k } are nonsingular, it follows that properties 1◦ , 2◦ , and 3◦ remain true for the systems χ N {φj }N j =1 and { j }j =1 as well. Now, let us turn to the proof of existence of {wi }K i=1 satisfying (5.28). For simplicity, we shall assume first that λj , j = 1, . . . , N, are all distinct eigenvalues. u such that (w, P ∗ ϕ ) = 0 for all i = Then we must prove that there is w ∈ ZN N i ω N

αj ϕj . Taking into account (5.27) and 1, . . . , N . We choose w of the form w = j =1

the fact that the matrix {bk } is nonsingular (it is symmetric and positive definite), we conclude that, for existence of w, it suffices to show that det (ϕj , PN∗ ϕi )ω  = 0. N  But the latter is obviously true because otherwise, there is η = βi PN∗ ϕi such that i=1

(η, ϕj )ω = 0 for all j = 1, . . . , N. Since, as follows by 1◦ and 2◦ , both systems {ϕj } and {PN∗ ϕj } are linearly independent on (L2 (ω))d , we arrived to a contradiction. It should be mentioned in this context also that there are infinitely many solutions w1 , w2 , . . . , wK to problem (5.28). Let us consider now the general case of multiple eigenvalues. i−1 +i u )i , i = 1, . . . , M and D 0 = [w , . . . , w ] ∈ We set Φi = {χ k }k= ⊂ (ZN 1 K i−1 +1 u )K . Choose the K ×  matrices D such that rank D =  and consider the (ZN i i i i system (PN∗ Φi , D 0 )ω = Di , i = 1, . . . , M.

(5.39)

144

5 Feedback Stabilization of Semilinear Parabolic Equations

Here PN∗ Φi = col[PN∗ χ i−1 +1 , . . . PN∗ χ i−1 +i ]. We set = col[D1 , . . . , DM ] D We and denote by dkj , k = 1, . . . , N, j = 1, . . . , K the elements of the matrix D. look for wk of the form wk =

N

αkj ϕj , k = 1, . . . , K.

j =1

Then system (5.39) reduces to N

αj (ϕj , PN∗ χ i )ω = di , i = 1, . . . , N,  = 1, . . . , k.

j =1

Thus system (5.39) has a solution if det(ϕj , PN∗ χ i )ω is not zero. But the letter is an immediate consequence of properties 1◦ , 2◦ . This completes the proof of Lemma 5.1. Now, we shall return back to system (5.8) and prove that Lemma 5.3. For each y 0 ∈ H = L2 (O), there is a real valued controller u = uN (x, t), uN (x, t) =

2K

uiN (t)ψi (x),

(5.40)

i=1

where ψi ∈ YN1 for 1 ≤ i ≤ K, ψi ∈ YN2 for K + 1 ≤ i ≤ 2K, such that y ∈ L∞ (0, T ; L2 (O)) ∩ L2 (0, T ; H01 (O)) ∩ L2 (δ, T ; H 2 (O)), ∀ 0 < δ < T , (5.41) |y(t)|2 + |uiN (t)| ≤ Ce−γ0 t |y 0 |2 , ∀ t ≥ 0,

(5.42)

where 0 < γ0 < |Re λN +1 |. Proof. If we substitute the feedback controller vN given by Lemma 5.1 in system (5.24), we get by (5.21) that |ζN (t)| ≤ Ce−γ1 t |(I − PN )z0 | and, therefore, the solution z to system (5.20) satisfies |z(t)| ≤ Ce−γ1 t |z0 |, ∀ t ≥ 0.

(5.43)

5.1 Riccati-based Internal Stabilization

145

u such that the Hence there is a controller v = vN of the form (5.34) with wi ∈ ZN solution z to (5.20) satisfies estimate (5.43). Hence

vN =

K

(vj1 + ivj2 )(φj1 + iφj2 ),

j =1

where vj1 (t), vj2 , φj1 , φj2 are real valued and φj1 ∈ YN1 , φj2 ∈ YN , for j = 1, . . . , K. Coming to system (5.8), we infer that there is a controller uN of the form (5.40) such that (5.41), (5.42) hold. This completes the proof. Remark 5.3. The constant 2K arising in Lemma 5.3 and defined by (5.10) is the minimal dimension of the stabilizing controller u. As mentioned earlier, if all unstable eigenvalues are simple, then K = 1 and, if all unstable eigenvalues are real and simple, then the dimension of the controller u is 1. However, the proof of Lemma 5.3, that is, the existence of a controller u which stabilizes the linear system (2.8) becomes much simpler for K = N . In fact, if in system (5.23) we look N  for a stabilizing controller v of the form vN = vjN wj , then the corresponding j =1

u system is exactly null controllable because one can find {wj }N j =1 ⊂ ZN such that det D = 0. The details are omitted.

Proof of Theorem 5.1. The idea of the proof was already outlined in Section 1.5 (see (1.92), (1.93)). Namely, consider the optimal control problem   ∞   3 1 2 2 ϕ(y ) = Min |A 4 y(t)|2 + |u(t)|2K dt 2 0 0

(5.44)

subject to u ∈ L2 (0, ∞; R2K ) and to the state system y + Ay + A0 y = m

2K 

ui ψi

i=1

y(0) =  Here |u|2K =

2K 

i=1

y0,

(5.45)

t ∈ (0, ∞).

1/2 u2i

and the functions ψi are as in Lemma 5.3.

We set D(ϕ) = {y 0 ∈ H ; ϕ(y 0 ) < ∞}. By Lemma 5.3, we know that for each there is u ∈ L2 (0, ∞; R 2K ) such that |y(t)|2 ≤ Ce−γ t , ∀ t > 0. If we multiply 1 equation (5.45) by A 2 y, we obtain

y0

1 3 1 1 1 d |A 4 y(t)|22 + |A 4 y(t)|32 ≤ |(A0 y, A 2 y)| + C|u(t)|2K |A 2 y(t)|2 . 2 dt

146

5 Feedback Stabilization of Semilinear Parabolic Equations 1

1

3

1

1

Taking into account that |A 2 y|22 ≤ |A 4 y| |A 4 y|2 , |(A0 y, A 2 y)|2 ≤ C|A 2 y|22 and 3

1

|y|2 ∈ L∞ (0, ∞), we see that A 4 y ∈ L2 (0, ∞; H ) if y 0 ∈ D(A 4 ). Hence 1

1

1

1

ϕ(y 0 ) ≤ C|A 4 y 0 |22 , ∀ y 0 ∈ D(A 4 ).

(5.46)

On the other hand, we have ϕ(y 0 ) ≥ C|A 4 y 0 |22 , ∀ y 0 ∈ D(A 4 ).

(5.47)

Indeed, it is easy to see that for each y 0 ∈ D(ϕ), problem (5.44) has a unique solu3 1 tion (y ∗ , u∗ ) ∈ L2 (R+ ; D(A 4 ))×L2 (R+ ; R2K ). Moreover, y ∗ ∈ C(R+ ; D(A 4 )). 1 If we multiply the equation (5.45), where y = y ∗ , u = u∗ , by A 2 y ∗ and integrate it on (0, ∞), we obtain 1 1 02 |A 4 y |2 ≤ 2



∞

0

1

1

1

((Ay ∗ , A 2 y ∗ ) + (A0 y ∗ , A 2 y ∗ ) + |u∗ |2K |A 2 y ∗ |2 )dt ∞

≤C 0

3

(|A 4 y ∗ |22 + |u|22K )dt = Cϕ(y 0 )

because 1

1

1

|(A0 y ∗ , A 2 y ∗ )| ≤ Cg (ye )C(O) (|y ∗ |2 |A 2 y ∗ |2 + |A 2 y ∗ |22 ) ≤ Cy ∗ 2 .  1 This implies that D(ϕ) = D A 4 = W. We note that, by (5.44), ϕ is of the form ϕ(y 0 ) =

1 2

 0

∞

3

(|A 4 L1 (y 0 )|22 + |L2 (y 0 )|22K )dt,

(5.48)

3

where L1 : W → D(A 2 ), L2 : W → R2k , are linear continuous operators. Hence, there exists a linear self-adjoint positive operator G : H → H with domain D(G)⊂W such that 1 (Gy 0 , y 0 ) = ϕ(y 0 ), ∀ y 0 ∈ D(G). 2

(5.49)

Moreover, the operator G extends to all of W and G ∈ L(W, W ). 3

Lemma 5.4. Let (y ∗ , u∗ ) ∈ L2 (0, ∞; D(A 4 ))×L2 (0, ∞; R2K ) be the optimal 1 pair for problem (5.44) corresponding to y 0 ∈ D(A 4 ). Then u∗ (t) = −{(Gy ∗ (t), ψi )ω }2K i=1 , ∀ t ≥ 0.

(5.50)

5.1 Riccati-based Internal Stabilization

147

Moreover, V ⊂D(G), that is, |Gy|2 ≤ Cy, ∀ y ∈ V

(5.51)

and there exist Ci > 0, i = 1, 2, such that C1 y2W ≤ (Gy, y) ≤ C2 y2W , ∀ y ∈ W.

(5.52)

The operator G is the solution to the algebraic Riccati equation 1 1 3 (Gy(t), ψi )2ω = |A 4 y|22 , ∀ y ∈ D(A). 2 2 2K

(Ay + A0 y, Gy) +

(5.53)

i=1

Proof. Estimate (5.52) follows immediately from (5.46) and (5.47). By the dynamic programming principle, it follows that, for each T > 0, (y ∗ , u∗ ) is the solution to optimal control problem min

   T 3 1 (|A 4 y(t)|2 + |u(t)|22K )dt + ϕ(y(T )); (y, u) subject to (5.45) . 2 0

By the maximum principle, we obtain that u∗ (t) = {(q T (t), ψi )ω }2K i=1 ,

(5.54)

where q T is the solution to adjoint equation 3 d T q − (A + A0 )∗ q T = A 2 y ∗ , t ∈ (0, T ), dt q T (T ) = −Gy ∗ (T ).

(5.55)

Since q T (t) ∈ W ⊂V , it follows from the standard existence theory for linear evolution equations that q T ∈ L2 (0, T ; H ) ∩ C([0, T ], V ). Moreover, if y 0 ∈ V , we have q T ∈ C([0, T ]; H ). Indeed, y ∗ ∈ L2 (0, T ; D(A)) if y 0 ∈ V . If we set 1 z = A− 2 q T , it follows from (5.47) that 1

1

z − Az − A− 2 A∗0 A 2 z = Ay ∗ . It is easy to check that the estimate 1 d z(t)2 ≥ |Az(t)|22 − |Ay ∗ (t)|2 |Az(t)|2 − C|Az(t)|2 z(t), a.e. t ∈ (0, T ), 2 dt which, together with (5.55), shows that z ∈ C([0, T ]; V ). Hence q T ∈ C([0, T ]; H ), as claimed.

148

5 Feedback Stabilization of Semilinear Parabolic Equations

By (5.55) and by the unique continuous property for the backward linear parabolic equation qt − (A + A0 )∗ q = 0 in Q, q = 0 on ∂O,

it follows that q T = q T on (0, T ) for 0 < T < T . Hence q T = q is independent of T and so (5.54) and (5.55) extend to all T of R+ . Moreover, we have Gy 0 = −q T (0).

(5.56)

 1 Here is the argument. For all z0 ∈ D A 4 , we have  ϕ(y 0 ) − ϕ(z0 ) ≤ 0

T

3

3

((A 4 y ∗ (t), A 4 (y ∗ (t) − z∗ (t)) + (u∗ (t), u∗ (t) − v ∗ (t))2K )dt +(Gy ∗ (T ), y ∗ (T ) − z∗ (T )),

where (z∗ , v ∗ ) is the optimal pair of problem (5.45) corresponding to z0 . On the other hand, it follows from (5.55) that 3 3 d T (q (t), y ∗ (t)−z∗ (t)) = (A 4 y ∗ (t), A 4 (y ∗ (t)−z∗ (t)))+(u∗ (t), u∗ (t)−v ∗ (t))2K . dt

Integrating it on (0, T ) and then substituting the result into the previous inequality, we obtain that ϕ(y 0 ) − ϕ(z0 ) ≤ −(q T (0), y 0 − z0 ) which yields (5.56), as desired. By (5.55) and (5.56), we infer that q(t) = −Gy ∗ (t) for all t ≥ 0, which shows (5.50), as desired. Now, let y 0 ∈ V . Then, by the previous argument, we see that q ∈ C([0, T ]; H ) which, together with (5.60), show that G : V → H. By the closed graph theorem, one obtains (5.51), as desired. Next, we have 1 ϕ(y (t)) = 2 ∗

 t

∞

3

(|A 4 y ∗ (s)|22 + |u∗ (s)|22K )ds, ∀ t ≥ 0

and, therefore,

2K d ∗ 1 3 1 Gy ∗ (t), y (t) + |A 4 y ∗ (t)|22 + (Gy ∗ (t), ψi )2ω = 0. dt 2 2 i=1

5.1 Riccati-based Internal Stabilization

149

Hence we have, for all t ≥ 0, −(Gy ∗ (t), Ay ∗ (t) + A0 y ∗ (t)) −

1 1 3 (Gy ∗ (t), ψi )2ω + |A 4 y ∗ (t)|22 = 0 2 2 2K

i=1

which implies (5.53), thereby completing the proof. Proof of Theorem 5.2 (Continued). Let G be the operator defined by (5.49). Then, putting the feedback controller (5.50) in system (5.45), we are led to the closed loop system yt + Ay + f (x, y, ∇y) + y(0) = y0 .

2K

m(G(y − ye ), ψi )ω ψi = 0, t > 0,

i=1

(5.57)

By Theorem 1.5, it follows that, under hypotheses (H1 ) and (H2 ), for y0 ∈ L2 (O), equation (5.57) has a unique solution y ∈ L2 (0, T ; V ) ∩ C([0, T ]; H ). 1

Moreover, t 2 y ∈ L2 (0, T ; D(A)) ∩ W 1,2 ([0, T ]; H ). Also, for ε → 0, we have yε → y strongly in C([0, T ]; L2 (O)), weakly in L2 (0, T ; V ),

(5.58)

where yε is the solution to the approximating equation (see (1.46)) (yε )t + Ayε + gε (yε ) + ayε + b · ∇yε +

2K 

m(G(yε − ye )ψi )ω ψi = 0

i=1

in (0, ∞) × O, yε (0) = y0 in O,

(5.59) where gε = g((1 + εg)−1 ) is the Yosida approximation of g. We shall show that, if y0 ∈ Uρ for ρ sufficiently small, then, for t → ∞, the solution y to (5.57) goes exponentially to ye . To this end, we substitute y by y + ye into (5.57) and reduce the problem to that of stability of the null solution to equation yt + Ay + A0 y + R(y) + y(0) = y 0 ≡ y0 − ye ,

2K

i=1

m(Gy, ψi )ω ψi = 0, t > 0,

(5.60)

where R(y) = f (x, y+ye , ∇(y+ye ))−f (x, ye , ∇ye )−A0 y ≡ g(y+ye )−g(ye )−g (ye )y.

150

5 Feedback Stabilization of Semilinear Parabolic Equations

As seen above, we may approximate (5.60) by (see (5.59)) (yε )t + Ayε + A0 yε + Rε (yε ) + yε (0) =

y0

2K

m(Gyε , ψi )ω ψi = 0,

i=1

= y0 − ye ,

(5.61)

where Rε (y) ≡ gε (y + ye ) − gε (ye ) − gε (ye )y. Taking into account that g (J (y)) , J (y) = (I + εg)−1 (y), 1 + εg (J (y)) g

(J (y)) gε

(y) ≡ , y ∈ R, ε > 0, (1 + εg (J (y)))3

gε (y) ≡

by (5.4) and (5.61), we see that |Rε (y)| ≤ C(|y|η + |y|2 ), ∀ y ∈ R, ε > 0,

(5.62)

where C is independent of ε. We multiply (5.61) by Gy and use (5.51) and (5.53) to get after some calculation that

3 d (Gyε , ψi )2ω + |A 4 yε (t)|22 ≤ 2|(Gyε (t), Rε yε (t))|. (Gyε (t), yε (t)) + dt i=1 (5.63) We shall show that there exist C > 0 independent of ε, such that 2K

1

1

η−1

|(Gy, R(y))| ≤ C(|A 4 y|2 + |A 4 y|2

3

3

)|A 4 y|22 , ∀ y ∈ D(A 4 ).

(5.64)

To this end, by virtue of (5.62), we prove that 1

η−1

|(Gy, |y|η )| ≤ C|A 4 y|2

3

3

|A 4 y|22 , ∀ y ∈ D(A 4 )

(5.65)

and 1

3

3

|(G(y), |y|2 )| ≤ C|A 4 y|2 |A 4 y|22 , ∀ y ∈ D(A 4 ). 1

1

(5.66)

We recall that D(As ) = H02s (O) for s > 14 and D(A 4 )⊂H 2 (O). Thus, the norm | · |D(As ) and  · 2s are equivalent for s > 14 · By (5.51) and by the interpolation inequality 3

1

1

1

3

y ≤ |A 4 y|22 |A 4 y|22 , ∀ y ∈ D(A 4 ),

5.1 Riccati-based Internal Stabilization

151

it follows that 1

1

η

1

3

η

|(Gy, |y|η )| ≤ Cy|y|2η ≤ C|A 4 y|22 |A 4 y|22 |y|2η

(5.67)

while, by Sobolev’s embedding theorem (see, e.g., [2], p 217), |y|2η ≤ Cyα , for α =

d(η−1) 2η

(5.68)

· Then, again by the interpolation inequality, we obtain that 3

1

yα ≤ C|A 4 y|22

−α

α− 12

3

|A 4 y|2

, α=

d(η − 1) · 2η

This, together with (5.67), implies that 1

1

1

3

|(Gy, |y|η )| ≤ C|A 4 y| 2 (3η+1−(η−1)d) |A 4 y|22

(η−1)(d−1)

1

η−1

≤ C|A 4 y|2

3

|A 4 y|22 , (5.69)

and so, (5.67) follows. Similarly, we have 1

1

1

3

|(G(y), |y|2 )| ≤ Cy |y|24 ≤ C|A 4 y|22 |A 4 y|22 y2d

4

1 4

1 2 (7−d)

≤ C|A y|2

3 4

|A y|

d−1 2

1 4

3

≤ C|A y|2 |A 4 y|22 ,

and (5.66) follows. Now, we come back to (5.60). Then, by (5.52)and (5.63)–(5.64), we see that 2 3 d 1 3 η−1 (Gyε (t), yε (t)) + |A 4 yε (t)|2 ≤ C(yε (t)W + yε (t)W )|A 4 yε (t)|22 dt 2

3 1 3 1 2 4 |A 4 yε (t)|22 , ≤ |A yε (t)|2 + C ϕ(yε (t)W ) − 4 4C (5.70) where ϕ(r) = r η−1 + r. Taking into account (5.50), we get by (5.70) that d dt

3

(Gyε (t), yε (t)) + 12 A 4 yε (t)|22  1

(Gyε (t),yε (t)) 2 − ≤C ϕ C1

1 2C

3

(5.71)

|A 4 yε (t)|22 ,

where C, C1 are independent of ε. 1 If we multiply (5.61) by A 2 yε and take into account that, since gε is Lipschitz, |Rε (y)| ≤ Cε (|y| + 1), ∀ y ∈ R,

152

5 Feedback Stabilization of Semilinear Parabolic Equations

we get 1 d 1 3 yε (t)2W + A 4 yε (t)|2 ≤ Cε1 yε (t)W (1 + yε (t)), a.e. t > 0. 2 dt 2 This yields yε (t)W ≤ y 0 W + Cε1 t, ∀ t > 0.

(5.72)

We denote by (0, Tε ) the maximal interval with the property that  ϕ

(Gyε (t), yε (t)) C1

1  2

−

1 ≤ 0, t ∈ [0, Tε ). 2C

By (5.72), it follows that, for y 0 W ≤ ρ sufficiently small, Tε > 0. On the other hand, by (5.71) we have d 1 3 (Gyε (t), yε (t)) + |A 4 yε (t)|2 ≤ 0, a.e. t ∈ (0, Tε ). dt 2 Since ϕ is monotonically decreasing and t → (Gyε (t), yε (t)) is decreasing on (0, Tε ), we infer that Tε = ∞ and d 1 3 (Gyε (t), yε (t)) + |A 4 yε (t)|22 ≤ 0, a.e. t > 0. dt 2 Letting ε → 0, this yields, for some γ > 0, d (Gy(t), y(t)) + γ (Gy(t), y(t)) ≤ 0, a.e. t > 0 dt

(5.73)

and  0

∞

3

2

|A 4 y(t)|2 dt ≤ 2(Gy 0 , y 0 ).

Moreover, by (5.50) and (5.73), we have |y(t)|2 ≤ y(t)W ≤ C|y 0 |e−γ0 t , ∀ t ≥ 0. This completes the proof. Remark 5.4. It should be mentioned that the above argument shows that the feedback controller (5.16) is robust with respect to smooth perturbations. More precisely, if Qε ∈ L(V , H ) is such that |Qε y| ≤ δ(ε)|Gy|, ∀ y ∈ V , where

5.1 Riccati-based Internal Stabilization

153

δ(ε) → 0 as ε → 0, then, for all ε > 0 sufficiently small, the feedback controller 2K

u = − ((G + Qε )(y − ye ), ψi )ω ψi i=1

still exponentially stabilizes ye in the neighborhood Uρ of ye . This implies that, if GN is a finite dimensional approximation of the Riccati equation (5.13) (equivalently, (5.15)), that is 3

GN FN + FN∗ GN + NN = (A 2 )N , then, for N large enough, the corresponding feedback controller (5.16) is stabilizable. Example 5.1. The steady-state solution ye = 0 to the nonlinear heat equation yt − yxx − y − y 3 = mu in (0, π )×(0, ∞), y(0, t) = y(π, t) = 0

in (0, ∞),

y(x, 0) = y0 (x)

in (0, π )

where m = 1[0, π ] is unstable. Clearly, A = −Δ, D(A) = H01 (0, π ) ∩ H 2 (0, π ), 2 A0 y = −y and ψj (x) = sin(j x), λj = 1 − j 2 . The stabilizable feedback controller  π  (5.16) is, in this case, u = − 02 G(y)(x) sin x dx sin x, where G : L2 (0, π ) → L2 (0, π ) is the solution to the Riccati equation (5.13) (or (5.15)), 

π 2

−G(Δy + y) − (Δ + 1)G(y) + G(y)(ξ ) sin ξ dξ G(m(x) sin x) 0 

∞ π

3 = j2 y(ξ ) sin(j ξ )dξ sin j x.

(5.74)

0

j =1

If we define G (z) = N

N

i,j =1

gij zi sin j x, z(x) =

N

zi sin ix,

i=1

we approximate (5.74) by the finite dimensional Riccati equation ∗ N 2GN AN + GN BN BN G = FN,

(5.75)

154

5 Feedback Stabilization of Semilinear Parabolic Equations

where 3

F N = diagj 2 N j =1 and  π N  2    sin x sin j x dx  BN = column   0 

j =1

AN = diagj 2 − 1N j =1 , 3

, F N = columnj 2 N j =1 .

If GN = gijN N i,j =1 is the solution to (5.75), then the feedback controller u(x, t) = −

N

gijN yi (t)bj sin x,

i,j =1



π 2

bj =



π

sin j x sin x dx, yi (t) =

y(x, t) sin ix dx, 0

0

is stabilizable for N sufficiently large. Remark 5.5. Theorem 5.1 extends along the above lines to the boundary stabilization of the semilinear heat equation yt − Δy + ay + g(y) = 0 in Q, y=u

on Σ,

y(x, 0) = y0 (x)

in O,

where g ∈ C 2 (R) satisfies conditions (5.3) and (5.4) and a ∈ R. Remark 5.6. Theorems 5.1 and 5.2 extend to more general parabolic equations (5.1) where Δ is replaced by the second order elliptic operator

n

∂ ∂ Ly ≡ aij (x) , ∂xi ∂xj i,j =1

where aij satisfy condition (1.11). Moreover, the Dirichlet homogeneous condition ∂y = 0. The proofs in (5.1) can be replaced by the Neumann boundary condition ∂ν are exactly the same. It should be said also that, taking into account the Sobolev embedding theorem, condition 1 ≤ d ≤ 3 can be relaxed.

5.1 Riccati-based Internal Stabilization

155

Stabilization of Semilinear Parabolic Systems Consider here the reaction-diffusion system yt (x, t) − Δy(x, t) + f (y(x, t), z(x, t)) = m(x)u(x, t) in Q ≡ O×R + , zt (x, t) − αΔz(x, t) + g(y(x, t), z(x, t)) = 0, in Q, y(x, t) = z(x, t) = 0 on Σ ≡ ∂O×R+ ,

(5.76)

y(x, 0) = y0 (x), z(x, 0) = z0 (x) in O, where α is a positive constant and m is the characteristic function of an open subset ω ⊂ O ⊂ Rd , d = 1, 2, 3. This reaction diffusion system is relevant in mathematical description of several physical processes including chemical reactions, semiconductor theory, nuclear reactor dynamics, and population dynamics (see, e.g., [80] and references given there). Let (ye , ze ) be a steady-state (equilibrium) solution to (5.76), that is, −Δye (x) + f (ye (x), ze (x)) = 0 in O, −αΔze (x) + g(ye (x), ze (x)) = 0 in O,

(5.77)

ye (x) = ze (x) = 0 on ∂O. We shall prove here with the methods developed above that, if f and g are of quadratic growth in (y, z) (the exact conditions will be specified later), then the steady-state solution (ye , ze ) to (5.76) is locally exponentially stabilizable by a feedback controller provided by a linear quadratic LQ control problem associated with the linearized systems. In the present situation it seems that system (5.76) is not locally controllable (anyway this is still an open problem) and so the stabilization cannot be established via local controllability. Then, proceeding as in the previous case, we first prove the exponential stabilization of the linearized system of (5.76) in a direct way and then introduce an appropriate infinite horizon LQ problem associated with the linearized system with unbounded cost functional, from which we find a solution R to an algebraic Riccati equation associated with the LQ problem, such that the feedback controller provided by R locally stabilizes the steady-state solution (ye , ze ). In a similar way will be treated the local stabilization of the phase-field system with one internal controller on the temperature field. The following assumptions will be in effect everywhere in the following: (K1 ) The steady-state solution ye is in C(O) with ∇ye ∈ (C(O))d . (K2 ) f, g ∈ C 1 (R×R) satisfy the growth condition ∇f (y, z)Lip + ∇g(y, z)Lip ≤ C, ∀ y, z ∈ R,

(5.78)

where  · Lip is a Lipschitz norm and (f (y, z) − f (y, ¯ z¯ ))(y − y) ¯ + (g(y, z) − g(y, ¯ z¯ ))(z − z¯ ) 2 ≥ −γ ((y − y) ¯ + (z − z¯ )2 ), for some γ ∈ R.

(5.79)

156

5 Feedback Stabilization of Semilinear Parabolic Equations

(K3 ) fz (ye , ze ) ≡ b, gy (ye , ze ) ≡ b1 , gz (ye , ze ) − αfy (ye , ze ) ≡ c0 , where c0 , b, and b1 are constants, b1 = 0 and α is a positive constant. Let A = −Δ and A1 = −Δ + aI with domain D(A) = D(A1 ) = H 2 (O) ∩ where I is the identity operator on H = L2 (O), a = fy (ye , ze ) ∈ C(O), and let A , A0 : H ×H → H ×H be defined by

H01 (O),

& % & % & fy (ye , ze ) fz (ye , ze ) , aI bI A 0 = A = , A0 = gy (ye , ze ) gz (ye , ze ) b1 I (aα + c0 )I 0 αA %

respectively, with the domain D(A ) = D(A)×D(A). It is clear that A is linear positive and self-adjoint operator on H ×H. We denote by A s and As , s ∈ (0, 1), 1 the fractional powers of operator the A and A, respectively. If W = D(A 4 ) with 1 1 the graph norm yW = |A 4 y|2 , then W ×W = (D(A 4 ))2 . Let B : H → H ×H be the operator defined by % B=

& mI , 0

and let B ∗ : H ×H → H be the adjoint operator of B, that is,

p B = mp, ∀ (p, q) ∈ H × H. q ∗

We have A (y, z) ≡ A



y = (Ay, αAz), ∀ (y, z) ∈ D(A ). z

In the following, the vectors

f g

and



y z

will be also denoted by (f, g) and (y, z), respectively. We denote by ·, · and ·, · the scalar products of H and H × H , respectively. Let λj and ϕj be the eigenvalues and corresponding orthonormal system of eigenfunctions of A1 , that is, A1 ϕj = λj ϕj , j = 1, . . . , ∞. (We note that A1 is self-adjoint and so all the eigenvalues λj are real.) From now on, we shall omit all x, t in the functions of x, t if there is no ambiguity, and we shall use the same symbol C to denote several positive constants.

5.1 Riccati-based Internal Stabilization

157

Theorem 5.3. Suppose that (K1 ), (K2 ) and (K3 ) hold. Then there exist N and a linear positive self-adjoint operator R : D(R) ⊂ H ×H → H ×H such that the feedback controller N

u = − (R11 (y − ye ) + R12 (z − ze ), ϕi )ω ϕi i=1

exponentially stabilizes (ye , ze ) in a neighborhood Eρ = {(y0 , z0 ) ∈ W ×W ; y0 − ye W +z0 − ze W < ρ} of (ye , ze ). More precisely, for each pair (y0 , z0 ) ∈ Eρ there is a solution (y, z) ∈ C(R+ ; H ) ∩ L2loc (R+ ; V )×C(R+ ; H ) ∩ L2loc (R + ; V ) to closed loop system (yt , zt ) + A (y, z) + (f (y, z), g(y, z)) N 

(R11 (y − ye ) + R12 (z − ze ), ϕi )ω ϕi = 0, t > 0, +B

(5.80)

i=1

(y(0), z(0)) = (y0 , z0 ), such that  ∞ 3 3 |A 4 (y(t), z(t)) − A 4 (ye , ze )|2H ×H dt ≤ C(y0 − ye 2W + z0 − ze 2W ) 0

(5.81)

and |y(t) − ye |2 + |z(t) − ze |2 ≤ Ce−γ t (y0 − ye W + z0 − ze W ), ∀ t > 0, (5.82) for some γ > 0, where (·, ·)ω denotes the inner product in L2 (ω) and ⎡ R=⎣

R11 R12

⎤ ⎦,

R12 R22 where Rij ∈ L(H ; H ), i, j = 1, 2. Moreover, R is the solution to the algebraic Riccati equation R(y, z), (A + A0 )(y, z) 3 1 1 (R11 (y − ye ) + R12 (z − ze ), ϕi )2ω = |A 4 (y, z)|2H ×H 2 2

N

+

i=1

for all (y, z) ∈ D(A ). Here {ϕi } are eigenvectors of A1 corresponding to λi .

(5.83)

158

5 Feedback Stabilization of Semilinear Parabolic Equations

Theorem 5.3 can be understood better in the framework of stability theory for nonlinear parabolic systems. If the spectrum σ (L) of the linearized operator % & & % −Δ 0 f (y , z ) f (y , z ) L= + y e e z e e 0 −αΔ gy (ye , ze ) gz (ye , ze ) has a nonempty intersection with {λ; Re λ < 0}, then the equilibrium solution (ye , ze ) is unstable. However, under assumptions (K1 ), (K2 ) the nonlinear system (5.76) is locally stabilizable by a linear feedback controller u with the support in an arbitrary open subset ω of O. An Example The parabolic system yt − yxx + ay + bz + f (y, z) = 0, t > 0, x ∈ (0, π ), zt − zxx + cy + dz + g(y, z) = 0, t > 0, x ∈ (0, π ),

(5.84)

y(0, t) = y(π, t) = 0, z(0, t) = z(π, t) = 0 arises in chemical reactor theory. (See, e.g., [11].) Here f, g are smooth functions such that f (0, 0) = g(0, 0) = 0, ∇f (0, 0) = ∇g(0, 0) = 0 and a, b, c, d are constants. It is easily seen that the eigenvalues λ = λj of the corresponding linearized system should satisfy the equation λ2 + λ(d + 2j 2 + a) + (a + j 2 )(d + j 2 ) − bc = 0. ∞  (ϕjn , ψjn ) sin nx.) (To get this, we look for eigenfunctions (ϕj , ψj ) of the form n=1

This implies that, for bc − (a + j 2 )(d + j 2 ) > 0, there are positive eigenvalues λj , that is, the zero solution is unstable. However, according to Theorem 5.3, for each open subset ω⊂(0, π ) there is a controller u with the support in ω×(0, ∞) which stabilizes the system yt − yxx + ay + bz + f (y, z) = mu, t > 0, x ∈ (0, π ), zt − zxx + cy + dz + g(y, z) = 0, t > 0, x ∈ (0, π ), y(0, t) = y(π, t) = 0, z(0, t) = z(π, t) = 0. The above example reveals another interesting phenomenon which has deep implications in biology and the dynamics of chemical reactions. We have seen that the constant solution (0, 0) to system (5.84) is unstable for a, b, c, d chosen as above but, as easily seen, for corresponding ordinary differential system dy + ay + bz + f (y, z) = 0, dt dz + cy + dz + g(y, z) = 0, dt

5.1 Riccati-based Internal Stabilization

159

the solution (0, 0) is asymptotically stable because the eigenvalues of this system satisfy λ2 + (a + d)λ + ad − bc = 0 and so, for a +d > 0, (a +d)2 −4(ad −bc) < 0, all have negative real parts. This is the so-called Turing’s instability (see Murray [101]), which roughly speaking means that the diffusion may destabilize a 2D ordinary differential system. More precisely, the stable constant solutions to ordinary differential systems could become unstable in the presence of diffusions and this is at the origin of spatial pattern which can be experimentally seen in chemical reaction or biological evolution. Proof of Theorem 5.3. Proceeding as in Theorem 5.2, we shall linearize system (5.76) in (ye , ze ). Namely, yt + A1 y + bz = mu, t > 0, zt + αA1 z + c0 z + b1 y = 0, t > 0,

(5.85)

y(0) = y 0 ≡ y0 − ye , z(0) = z0 ≡ z0 − ze , where A1 = A + aI. We shall prove first Lemma 5.5. There exist N and uj ∈ L∞ (R+ ), j = 1, . . . , N , such that the controller u(x, t) =

N

uj (t)ϕi (x) i=1

stabilizes exponentially system (5.85). More precisely, we have |y(t)| + |z(t)| +

N

|uj (t)| ≤ Ce−γ t (|y 0 | + |z0 |), ∀ t > 0, j =1

for some γ , C > 0, where (y, z) is the solution to (5.85) corresponding to u = N  uj (t)ϕj (x).

j =1

Proof. Let {ϕi }∞ i=1 be an orthonormal basis formed with the eigenfunctions of A1 and let {λi }∞ be the corresponding eigenvalues of A1 . Since by hypotheses (K1 ) i=1 and (K2 ), a = fy (ye , ze ) ∈ C(O), it follows that λi → ∞ as i → ∞. Let XN = 2 span{ϕi }N i=1 , PN : L (O) → XN be the orthonormal projection on XN and QN = yN = QN y and zN = QN z. Applying PN I − PN . Set yN = PN y, zN = PN z, and QN to system (5.85), respectively, and taking into account that y = yN + yN , zN , we obtain that z = zN +

160

5 Feedback Stabilization of Semilinear Parabolic Equations i dyN i i + bzN = PN (mu, ϕi ), t > 0, i = 1, . . . , N, + λi yN dt i dzN i i i + λi αzN + c0 zN + b1 yN = 0, t > 0, i = 1, . . . , N, dt i (0) = (P y 0 )i , zi (0) = (P z0 )i , yN N N N

(5.86)

1 , . . . , y N ), z = (z1 , . . . , zN ), P y 0 = ((P y 0 )1 , . . . , (P y 0 )N ) where yN = (yN N N N N N N N 0 and PN z = ((PN z0 )1 , . . . , (PN z0 )N ), and

d yN + b zN = QN (mu) t > 0, yN + QN A1 dt d zN + c0 zN + b1 yN = 0 t > 0, zN + αQN A1 dt zN (0) = QN z0 . yN (0) = QN y 0 ,

(5.87)

We prove first for N large enough the exact null controllability of the finite– dimensional system (5.86), where u is given as above, that is, N

d i i i + bzN = uj (t)(ϕj , ϕi )ω , t > 0, i = 1, . . . , N, yN + λi yN dt j =1 d i i i i z + λi αzN + c0 zN + b1 yN = 0, t > 0, i = 1, . . . , N, dt N yN (0) = PN y 0 , zN (0) = PN z0 .

(5.88)

The backward dual system corresponding to (5.88) is the following: d i i i p − λ i pN − b1 qN = 0, t > 0, i = 1, . . . , N, dt N d i i i i q − λi αqN − c0 qN − bpN = 0, t > 0, i = 1, . . . , N. dt N

(5.89)

We set BN = (ϕj , ϕi )ω N i,j =1 ,

(5.90)

where (ϕj , ϕi )ω N i,j =1 denotes the N ×N matrix whose components are (ϕj , ϕi )ω , j, i = 1, . . . , N. We note that the right-hand side of system (5.105) is Bu = ∗ N N → RN of B : RN → (BN u, 0), u = {uj }N j =1 and the adjoint B : R ×R RN ×RN is given by B ∗ (p, q) = BN p, ∀ (p, q) ∈ RN ×RN , i = 1, . . . , N.

5.1 Riccati-based Internal Stabilization

161

Recall that (see Section 1.5) system (5.88) is exactly null controllable on [0, T ], T > 0, if and only if BN (pN (t)) = 0, ∀ t ∈ (0, T )

(5.91)

implies that pN (t) ≡ 0 and qN (t) ≡ 0, ∀ t ∈ (0, T ), where     1 N 1 N (t) . . . pN (t) , qN (t) = qN (t) . . . qN (t) . pN (t) = pN

(5.92)

By (5.91) and (5.92), it follows that N

i (ϕj , ϕi )ω pN (t) ≡ 0, t ∈ (0, T ), j = 1, . . . , N.

(5.93)

i=1

On the other hand, we have det (ϕj , ϕi )ω N i,j =1 = 0.

(5.94)

To prove this, we give an argument by reduction to absurdity. If (5.94) does not hold, 2 then the system {ϕj }N j =1 is linearly dependent in L (ω), and so, we might assume without loss of generality that ϕN =

N −1

γi ϕi on ω, where γi = 0, for some i ∈ {1, . . . , N − 1}.

i=1

Then, we have (−Δ + a)ϕN =

N −1

γi (−Δ + a)ϕi =

i=1

N −1

N −1

i=1

i=1

γi λi ϕi = λN ϕN =

λN γi ϕi on ω,

which implies that N −1

(λN − λi )γi ϕi = 0 on ω.

i=1

Since, by hypotheses (K1 ) and (K2 ), a = fy (ye , ze ) ∈ C(O), it follows that λi → ∞ as i → ∞. So we may take N large enough such that λN > λi for i = 1, . . . , N − 1. Thus in this way one arrives to conclusion that there is at least one ϕi such that ϕi = 0 on ω. By virtue of the unique continuation property of the solutions to elliptic equations, this implies that ϕi = 0 on O. This is a contradiction and so we obtain (5.94).

162

5 Feedback Stabilization of Semilinear Parabolic Equations

It follows that pN (x, t) = 0, ∀ t ≥ 0. Then we obtain that qN (x, t) = 0, ∀ t ≥ 0, as claimed. Thus system (5.88) is exactly null controllable and this implies that there are {uj (t)}N j =1 (given in feedback form) such that system (5.86) is exponentially stable with arbitrary exponent γ0 . More precisely, we have i i (t)| + |zN (t)| + |ui (t)| ≤ Cγ0 e−γ0 t (|y 0 | + |z0 |), i = 1, . . . , N, |yN

(5.95)

where Cγ0 is a positive constant independent of i and t but dependent on γ0 . Then, by (5.87), we obtain d i 2 i 2 i i i 2 zN ) ) + 2λi ( yN ) + (b + b1 ) yN zN + 2(αλi + c0 )( zN ) (( y i )2 + ( dt N ⎛ ⎞ N

i , i = 1 + N, . . . , = ⎝ (QN (muj ), ϕi )⎠ yN j =1

which together with (5.95) implies that, for N large enough, we have i (t)|2 + | i (t)|2≤ Ce−γN t (|y 0 |2 + |z0 |2 ) + zN | yN



t

e−γN (t−s)

0

N

|uj (s)|2 ds

j =1

≤ Ceγ t (|y 0 |2 + |z0 |2 ), ∀ t ≥ 0, i = N + 1, . . . , where γ , γN are positive constants. This completes the proof of the lemma. Proof of Theorem 5.3 (Continued). Now, we rewrite system (5.85) as (yt , zt ) + A (y, z) + A0 (y, z) = Bu, t > 0 (y(0), z(0)) = (y 0 , z0 ),

(5.96)

and consider the LQ (linear quadratic) optimal control problem Ψ (y 0 , z0 ) = Min

  ∞  3 1 (|A 4 (y, z)|2H ×H + |u|22 )dt subject to (5.96) . 2 0

(5.97)

Let D(Ψ ) be the set of all (y 0 , z0 ) ∈ H ×H such that Ψ (y 0 , z0 ) < ∞. We 1 observe first that for each (y 0 , z0 ) ∈ D(A 4 ), there exists u ∈ L2 (R+ , H ) such 3 that the system (5.96) has a unique solution (y, z) ∈ L2 (R + , D(A 4 )). Indeed, 1 it follows from Lemma 5.5 that, for each pair (y 0 , z0 ) ∈ D(A 4 ), there exists a u ∈ L2 (R+ ; H ) such that the system (5.96) has a solution (y, z) satisfying   2 2 |(y(t), z(t)|2H ×H ≤ Ce−γ t |y 0 |2 + |z0 |2 , ∀ t > 0.

5.1 Riccati-based Internal Stabilization

163 1

Multiplying scalarly in H ×H equation (5.96) by A 2 (y, z), we obtain after some calculation that 2 2 1 3 1 d |A 4 (y(t), z(t))|H ×H + |A 4 (y(t), z(t))|H ×H 2 dt 33 44 33 44 1 1     ≤  Bu, A 2 (y, x)  +  A0 (y, z), A 2 (y, z)  ! " 1 ≤ C |mu|2 + |y|2 + |z|2 |A 2 (y, z)| ! " 3 ≤ C |mu|2 + |y|2 + |z|2 |A 4 (y, z)|H ×H

2 3 1 ≤ C |mu|22 + |y|22 + |z|22 + |A 4 (y, z)|H ×H 2

which implies that 2 2 1 3 d |A 4 (y(t), z(t))|H ×H + |A 4 (y(t), z(t))|H ×H dt   ≤ C |y(t)|22 + |z(t)|22 + |mu(t)|22     2 2 ≤ C e−2γ t |y 0 |2 + |z0 |2 + |mu(t)|22 .

Integrating it on (0, ∞), we obtain that  0

∞

2

3

|A 4 (y(t), z(t))|H ×H dt ≤ C < ∞

as desired.

 3 Thus, for each pair (y 0 , z0 ) ∈ D A 4 , Ψ (y 0 , z0 ) < ∞ and, therefore,  1 2 1 Ψ (y 0 , z0 ) ≤ C|A 4 (y 0 , z0 )|H ×H , ∀ (y 0 , z0 ) ∈ D A 4 .

(5.98)

On the other hand, we have  1 2 1 Ψ (y 0 , z0 ) ≥ C|A 4 (y 0 , z0 )|H ×H , ∀ (y 0 , z0 ) ∈ D A 4 .

(5.99)

1

Indeed, it is readily seen for each pair (y 0 , z0 ) ∈ D(A 4 ), problem (5.97) has a unique optimal solution 3

3

(y ∗ , z∗ , u∗ ) ∈ L2 (R+ ; D(A 4 ))×L2 (R+ ; D(A 4 ))×L2 (R+ ; L2 (O)). 1

Multiplying, once again, equation (5.96) by A 2 (y ∗ , z∗ ) and integrating on R+ , we obtain that

164

5 Feedback Stabilization of Semilinear Parabolic Equations

 ∞ 33 44 1 1  1 0 0 2   4 ≤ A (y , z )  A (y ∗ , z∗ ), A 2 (y ∗ , z∗ )  H ×H 2 0 33  1   44 1     +  A0 (y ∗ , u∗ ), A 2 (y ∗ , u∗ )  + |mu∗ |2 A 2 y ∗  dt 2

 ∞  2 3  4 ∗ ∗  ≤C + |u∗ |22 dt = CΨ (y 0 , z0 ), A (y , u ) H ×H

0

because

33 3 4 44 1 1      A0 (y ∗ , z∗ ), A 2 (y ∗ , z∗ )  ≤ C  (1 + a)y ∗ , A 2 y ∗   3 3 2 4 3 4 1 1       +  y ∗ , A 2 z∗  +  (1 + a)z∗ , A 2 z∗  ≤ C A 4 (y ∗ , z∗ )

H ×H

.

By (5.98) and (5.99), we see that 1

D(Ψ ) = D(A 4 ) = W ×W. Since Ψ likewise ϕ (see (5.48)) is a quadratic functional, there exists a linear positive and self-adjoint operator R : H ×H → H with the domain D(R)⊂W ×W such that 44 1 33 R(y 0 , z0 ), (y 0 , z0 ) = Ψ (y 0 , z0 ), ∀ (y 0 , z0 ) ∈ D(Ψ ). 2

(5.100)

Moreover, R extends to all W ×W and R ∈ L(W ×W ; W ×W ). Lemma 5.6. Let (y ∗ , z∗ , u∗ ) be optimal for problem (5.97) corresponding to (y 0 , z0 ) ∈ W ×W. Then N

u∗ (t) = − (R11 y ∗ (t) + R12 z∗ (t), ϕi )ω ϕi , ∀ t > 0,

(5.101)

i=1

Moreover, V ×V ⊂D(R), that is, |R(y, z)|2H ×H ≤ C(y, z)2V ×V , ∀ (y, z) ∈ V ×V ,

(5.102)

and there are constants c1 > 0, c2 > 0 such that c1 (y, z)2W ×W ≤ R(y, z), (y, z) ≤ c2 (y, z)2W ×W .

(5.103)

The operator R is the solution to algebraic Riccati equation 2 3 1 1 (R11 y + R12 z, ϕi )2ω = |A 4 (y, z)|H ×H . 2 2 i=1 (5.104) N

(A + A0 )(y, z), R(y, z) +

5.1 Riccati-based Internal Stabilization

165

Proof. The proof follows closely that of Lemma 5.4. In fact, estimate (5.103) follows from the previous estimates. By the dynamic programming principle, it follows that ∀ T > 0, (y ∗ , z∗ , u∗ ) is the solution to optimal control problem

   T 2 3 1 2 4 |A (y, z)|H ×H + |u| dt + Ψ (y(T ), z(T )), subject to (5.96) . Min 2 0 By the maximum principle it follows that 4 3 u∗ (t) = B ∗ (pT (t), q T (t)), u∗i (t) = pT (t), ϕi , ∀ t ∈ [0, T ), ω

(5.105)

where (pT , q T ) is the solution to the adjoint system 3 d (pT , q T ) − (A + A0∗ )(pT , q T ) = A 2 (y ∗ , z∗ ), t > 0, dt (pT (T ), q T (T )) = −R(y ∗ (T ), z∗ (T )).

(5.106)

Since R(y ∗ (T ), z∗ (T )) ∈ W ×W ⊂V ×V , it follows from the standard existence theory for linear evolution equations that (pT , q T ) ∈ L2 (0, T ; H ×H ) ∩ C([0, T ]; V ×V ). Moreover, if (y 0 , z0 ) ∈ V ×V , then we have that (pT , q T ) ∈ C([0, T ]; H ×H ). Indeed, we have that (y ∗ , z∗ ) ∈ L2 (0, T ; D(A )) if (y 0 , z0 ) ∈ 1 V ×V . Let ( p, q ) = A − 2 (pT , q T ). It follows from (5.106) that 1 1 d ( p, q ) − A ( p, q ) − A − 2 A0∗ A 2 ( p, q ) = A (y ∗ , z∗ ). dt

One can check easily that 33 44 1 1   p, q )|V ×V p, q )  ≤ C|(  A − 2 A0∗ A 2 ( which, combined with (5.106), implies that ( p, q ) ∈ C([0, T ]; V ×V ). Hence (pT , q T ) ∈ C([0, T ]; H ×H ). By (5.105) and by the unique continuation property for the linear parabolic system d (p, q) − (A + A0 )∗ (p, q) = 0, dt



it follows that (pT , q T ) = (pT , q T ) on (0, T ) for 0 < T < T . Hence (pT , q T ) = (p, q) is independent of T and so (5.105) and (5.106) extends to all of R+ . Moreover, we have R(y 0 , z0 ) = −(p(0), q(0)).

(5.107)

166

5 Feedback Stabilization of Semilinear Parabolic Equations

 1 Indeed, for all (y 1 , z1 ) ∈ D A 4 , we have Ψ (y 0 , z0 ) − Ψ (y 1 , z1 )  T 33 44 # $ 3 3 A 4 (y ∗ , z∗ ), A 4 (y ∗ − y1∗ , z∗ − z1∗ ) + u∗ , u∗ − u∗1 dt ≤ $$ ##0 + R(y ∗ (T ), z∗ (T )), (y ∗ (T ) − y1∗ (T ) − z1∗ (T )) ,

(5.108)

where (y1∗ , z1∗ , u∗1 ) is the optimal solution to problem (5.96) corresponding to (y 1 , z1 ). On the other hand, by (5.107), we have 44 d 33 T (p (t), q T (t)), (y ∗ (t) − y1∗ (t), z∗ (t) − z1∗ (t)) dt 44 # 33 3 $ 3 = A 4 (y ∗ , z∗ ), A 4 (y ∗ − y1∗ , z∗ − z1∗ ) + u∗ , u∗ − u∗1 . Integrating it over (0, T ) and then substituting the result into (5.108), we obtain that 33 44 Ψ (y 0 , z0 ) − Ψ (y 1 , z1 ) ≤ − (p(0), q(0)), (y 0 − y 1 , z0 − z1 ) which implies (5.107), as desired. By (5.106) and (5.107), we conclude that (p(t), q(t)) = −R(y ∗ (t), z∗ (t)), ∀ t ≥ 0, which along with (5.105) implies (5.107), as desired. Now, let (y 0 , z0 ) ∈ V ×V . Then, it follows from the previous argument that (p, q) ∈ C([0, T ]; H ×H ) which together with (5.107) implies that R : V ×V → H ×H . By the closed graph theorem, we obtain (5.102), as desired. Next, we observe that

 2 3 1 ∞ |A 4 (y ∗ (s), z∗ (s))|H ×H + |u∗ (s)|22 ds, ∀ t ≥ 0, Ψ (y ∗ (t), z∗ (t)) = 2 t and, therefore, )) ** 2 d ∗ 1  3  ∗ ∗ ∗ R(y (t), z (t)), (y (t), z (t)) + A 4 (y ∗ (t), z∗ (t)) H ×H dt 2 1 ∗ + |B R(y ∗ (t), z∗ (t))|2H = 0, a.e. t > 0. 2

(5.109)

Since |B ∗ R(y, z)|H ≤ C|(y, z)|V ×V , ∀ (y, z) ∈ V ×V , we see that the operator A + A0 + BB ∗ R with the domain D(A ) generates a C0 -semigroup in H ×H . This

5.1 Riccati-based Internal Stabilization

167

implies that A (y ∗ , z∗ ), A0 (y ∗ , z∗ ), BB ∗ R(y ∗ , z∗ ) ∈ C([0, ∞); H ×H ). Then it follows from (5.109) that, for (y 0 , z0 ) ∈ D(A ), 1 − R(y ∗ (t), z∗ (t)), (A + A0 )(y ∗ (t), z∗ (t)) − |B ∗ R(y ∗ (t), z∗ (t))|2 2 2 1  3 ∗  ∗ + A 4 (y (t), z (t)) = 0, ∀ t ≥ 0, H ×H 2 which implies (5.104), thereby completing the proof of Lemma 5.6. Now, let F (y, z) = (f (y, z), g(y, z)) and consider the closed loop system (yt , zt ) + A (y, z) + F (y, z) + B

N

(R11 (y − ye ) + R12 (z − ze ), ϕi )ω ϕ i = 0, i=1

t > 0, y(0), z(0) = (y0 , z0 ). (5.110) It follows that, under hypotheses (K1 ) and (K2 ), for each pair (y0 , z0 ) ∈ W ×W , systems (5.110) have a unique solution (y, z) ∈ L2 (0, T ; V ×V )∩C([0, T ]; H ×H ) 1 for all T > 0. Moreover, t 2 (y, z) ∈ L2 (0, T ; D(A )) ∩ W 1,2 ([0, T ]; H ×H ). (This follows as in Theorem 1.5 by proving first that the operator (y, z) → A (y, z) + F (y, z) + B

N

(R11 (y − ye ) + R12 (z − ze ), ϕi )ω ϕi i=1

is quasi-m-accretive in the space H ×H , but we omit the details. We shall prove that if (y0 , z0 ) ∈ Eρ for ρ small enough, then this solution exponentially stabilizes (ye , ze ). To this end, we substitute (y, z) by (y + ye , z + ze ) into equation (5.110) and reduce thus the problem to that of stability of the null solution to system (yt , zt ) + A (y, z) + A0 (y, z) + Φ(y, z) +B

N

(R11 y + R12 z, ϕi )ω ϕi = 0, t > 0,

(5.111)

i=1

(y(0), z(0)) = (y 0 , z0 ) ≡ (y0 − ye , z0 − ze ), where Φ(y, z) ≡ (Φ 1 (y, z), Φ 2 (y, z)) = (f (y + ye , z + ze ) − f (ye , ze ), g(y + ye , z + ze ) − g(ye , ze )) − A0 (y, z). By hypotheses (K1 ) and (K2 ), we see that |Φ(y, z)| ≤ C(|y|2 + |z|2 ).

(5.112)

168

5 Feedback Stabilization of Semilinear Parabolic Equations

Multiplying (5.111) by R(y, z) and using equation (5.104), we obtain after some calculation that 2 3 d R(y, z), (y, z) + |B ∗ R(y, z)|2H ×H + |A 4 (y, z)|2 dt ≤ 2|R(y, z), Φ(y, z)|.

(5.113)

We claim that there exist C > 0 independent of y and z, such that |R(y, z), Φ(y, z)| 3

2

2

3

1

1

(5.114)

3

≤ C(|A 4 y|2 + |A 4 z|2 )(|A 4 y|2 + |A 4 z|2 ), ∀ (y, z) ∈ D(A 4 ). It is sufficient to show that 3

2

3

2

1

1

|(R i (y, z), |y|r + |z|r )| ≤ C(|A 4 y|2 + |A 4 z|2 )(|A 4 y|2 + |A 4 z|2 ), i = 1, 2, (5.115) where r = 2 and R 1 (y, z) = R11 y + R12 z, R 2 (y, z) = R12 y + R22 z. 1 1 3 By (5.102) and by the interpolation relation D(A 2 ) = (D(A 4 ), D(A 4 )) 1 , we 2 obtain that |(R i (y, z), |y|r )| ≤ C (y + z) |y|r2r 1

1

1

3

1

1

3

1

≤ C(|A 4 y|22 |A 4 y|22 + |A 4 z|22 |A 4 z|22 )|y|r2r , i = 1, 2, (5.116) while, by Sobolev’s embedding theorem (see (5.68)), |y|2r ≤ Cyα , for 1 ≤ r ≤

d , d > 2α. d − 2α

(5.117)

Then, again by the interpolation inequality, we obtain that 1

3

yα ≤ C|A 4 y|22

−α

3

α− 21

|A 4 y|2

,

which together with (5.116) implies that      1 3 1 1 # i $ 1 3 2 + 2 −α r 2 + α− 2 r r | R (y, z), |y| | ≤ C |A 4 y|2 |A 4 y|2  1 4

1 2

1 4

+|A z|2 |A y|2

3 2 −α

 r

3 4

1 2

3 4

   α− 12 r

|A z|2 |A y|2

(5.118) .

Now, (5.117) holds if we choose α = 54 , and so, by (5.118), we obtain that 4 3 2 2 3 3 1 1 | R i (y, z), |y|r |≤C(|A 4 y|2 +|A 4 z|2 )(|A 4 y|2 +|A 4 z|2 ), i=1, 2.

(5.119)

5.1 Riccati-based Internal Stabilization

169

Similarly, we may obtain that 4 3 2 2 3 3 1 1 | R i (y, z), |z|r | ≤ C(|A 4 y|2 + |A 4 z|2 )(|A 4 y|2 + |A 4 z|2 ), and so, we get as above estimate (5.114), as claimed. By (5.103), we see that, for ρ small enough, if (y0 , z0 ) ∈ Eρ , we have 2 1 3 d R(y(t), z(t)), (y(t), z(t)) + |A 4 (y(t), z(t))|H ×H ≤ 0, a.e. t ∈ (0, T ∗ ), dt 2

where (0, T ∗ ) is the maximum interval of existence of the solution (y, z) with the property that R(y(t), z(t)), (y(t), z(t)) ≤ C1

(5.120)

for some constant C1 > 0. From this, it follows that d R(y(t), z(t)), (y(t), z(t)) + γ R(y(t), z(t)), (y(t), z(t)) ≤ 0, a.e. t > 0, dt which implies, as in Theorem 5.2, that  0

∞

33 44 2 3 |A 4 (y(t), z(t))|H ×H dt ≤ 2 R(y 0 , z0 ), (y 0 , z0 )

and |(y(t), z(t))|2H ×H ≤ (y(t), z(t))2W ×W ≤ C(y 0 , z0 )W ×W e−γ t , ∀ t ≥ 0, as claimed. It should be said, however, that the existence of the interval (0, T ∗ ) with property (5.120) or, equivalently, (y(t), z(ε))W ×W ≤ C2 , t ∈ (0, T ∗ ), for some C2 > 0, is a delicate problem which can be solved as in the proof of Theorem 5.2. Namely, one approximates (5.111) by a similar system where f, g are replaced by fε , gε , the Yosida approximations of f and g, and get for the corresponding solution (yε , zε ) the estimates (5.119)–(5.120). Stabilization of the Phase-Field System In this section, we shall use a similar method to study the stabilization of the zero equilibrium solution to the phase-field system yt + ϕt − kΔy = mu in Q = O×R+ , ϕt − aΔϕ − b(ϕ − ϕ 3 ) + dy = 0, in Q, y = 0, ϕ = 0 on Σ = ∂O×R+ , y(x, 0) = y0 (x), ϕ(x, 0) = ϕ0 (x) in O.

(5.121)

170

5 Feedback Stabilization of Semilinear Parabolic Equations

Equivalently, yt − kΔy + (aΔϕ + b(ϕ − ϕ 3 ) − dy) = mu in Q, ϕt − aΔϕ − b(ϕ − ϕ 3 ) + dy = 0 in Q, y = 0, ϕij = 0 on Σ, y(x, 0) = y0 (x), ϕij (x, 0) = ϕ0 (x) in O.

(5.122)

Here O ∈ Rd , d = 1, 2, 3, is an open and bounded subset, , k, a, b are nonnegative constants, and m is, as usually, the characteristic function of an open subset ω⊂O. The local controllability of steady-state solution to (5.121) with internal controllers on both equations was established in Section 3.3 via Carleman estimates for the linearized system. Here we shall assume that (K4 ) a, b, k,  > 0 and d = 0. We notice that hypothesis (K4 ) alone does not imply the stability of the zero steadystate solution to the phase-field system. We shall denote, as above, by | · |2 the norm of H = L2 (O) and, by ·, · and ·, · the scalar products of H and H ×H , respectively. Throughout this section, we set A = −Δ with D(A) = H 2 (O) ∩ H01 (O) and let {ψi }∞ i=1 be an orthonormal basis of eigenfunctions of A. By {λi }∞ we denote the corresponding system of i=1 eigenvalues. Consider the linearized system yt + kAy − aAϕ + bϕ − dy = mu, t > 0, ϕt + aAϕ − bϕ + dy = 0, t > 0,

(5.123)

y(0) = y0 , ϕ(0) = ϕ0 . 2 Let XN = span{ψi }N i=1 , PN : L (O) → XN be the orthonormal projection on XN and QN = I − PN . Applying PN and QN to systems (5.123), we obtain that

d i i i y + (kλi − d)yN − (aλi − b)ϕN = mu, ψi  , t > 0, i = 1, . . . , N, dt N d i i i ϕ + (aλi − b)ϕN + dyN = 0, t > 0, i = 1, . . . , N, dt N i (0) = y i , ϕ i (0) = ϕ i yN N 0N 0N (5.124) d yN − aQN A ϕN − d yN + b ϕN = QN (mu), t > 0, yN + kQN A dt d (5.125) ϕN + aQN A ϕN − b ϕN + d yN = 0, t>0, dt N (0) = QN ϕ0 , yN (0) = QN y0 , Q i and ϕ i are the i-th where yN = PN y, ϕN = PN ϕ, yN = QN y, ϕN = QN ϕ, yN N components of yN and ϕN , respectively.

5.1 Riccati-based Internal Stabilization

171

The backward dual system of (5.124) is given by d i p − (kλi − d)pi − (aλi − b)q i = 0, i = 1, . . . , N, dt d i q − (aλi − b)q i − dpi = 0, i = 1, . . . , N. dt

(5.126)

Lemma 5.7. There are uj ∈ L∞ (R+ ), j = 1, . . . , N, such that for N large enough the controller u(x, t) =

N

(5.127)

ui (t)ψi (x)

i=1

stabilizes exponentially system (5.123), that is, |y(t)|2 + |ϕ(t)|2 +

N

|ui (t)| ≤ Ce−γ t (|y0 |2 + |ϕ0 |2 ), ∀ t ≥ 0

i=1

for some γ > 0 and C > 0. Proof. By (K4 ) and by the same argument as that used in the proof of Lemma 5.6, it follows that system (5.124) is exactly null controllable and so there are ui (t), N  i = 1, . . . , N , such that system (5.124), where u = ui ψi , is exponentially stable i=1

with arbitrary exponent γ0 > 0, that is, j

j

|yN (t)| + |ϕN (t)| +

N

|ui (t)| ≤ Cγ0 e−γ0 t (|y0 | + |ϕ|), ∀ t ≥ 0.

i=1

Multiplying the first equation of (5.125) by yN , where u is given by (5.127) and the second equation of (5.125) by β ϕN , where β > 0 will be made precise later, we get 1 d j j j j j (( y )2 + β( ϕN )2 ) + (kλj − d)(z yN )2 − (aλj + b) yN ϕN 2 dt N N



 j j j j yN , +βd yN ϕN + β(aλj − b)( ϕN )2 = QN m ui ψi , ψj i=1

j = N + 1, . . . (5.128)

then for β suitable chosen and N large enough we have that j | yN (t)|2

j j + β| ϕN (t)|2 ≤ e−γN t (| yN (0)|2

j + | ϕN (0)|2 ) +



t 0

e−γN (t−s)

N

|ui (s)|2 ds i=1

≤ Ce−γ t (|y0 |22 + |ϕ0 |22 ), ∀ t ≥ 0, j = N + 1, . . . , where γ > 0. This completes the proof of Lemma 5.7.

172

5 Feedback Stabilization of Semilinear Parabolic Equations

Consider now the LQ optimal control problem  Ψ (y0 , ϕ0 ) = Min

∞ 0

3

2

3

2

(|A 4 y|2 + |A 4 ϕ|2 + |u|22 )dt;

u=

N 

ui (t)ψi (t), (y, ϕ, u) satisfies (5.123) . i=1

(5.129) First, we claim that, ∀ (y0 , ϕ0 ) ∈ D(A )×D(A ), Ψ (y0 , z0 ) < ∞. Indeed, by Lemma 5.7, there exists a u ∈ L2 (Q) such that the corresponding solution (y, ϕ) to systems (5.123) satisfies 1 4

1 4

|u(t)|2 + |y(t)|22 + |ϕ(t)|22 ≤ Ce−γ t , ∀ t > 0, where γ > 0 and C > 0 are constants. We get after some calculation that 1 d 2 dt



2 2 2 1 1 3 |A 4 y(t)|2 + |A 4 ϕ(t)|2 + k|A 4 y(t)|2 3

3

2

3

−a|A 4 y(t)|2 |A 4 ϕ(t)|2 + a|A 4 ϕ(t)|2  " 3 ! 3 ≤ C |ϕ(t)|2 + |y(t)|2 + |mu(t)|2 |A 4 y(t)|2 + |A 4 ϕ(t)|2 .

(5.130)

Then it follows that ∞

 0

|A

3 4

2 y(t)|2

+ |A

3 4

2 ϕ(t)|2

dt < ∞,

as claimed. Hence

2 2 1 1 4 4 Ψ (y0 , ϕ0 ) ≤ C |A y0 |2 + |A ϕ0 |2 . Then, by the similar arguments to those in the proof of Lemma 5.6, we obtain that there exists a linear,  positive,  1  and self-adjoint operator R : H ×H → H ×H , 1 4 with D(R) = D A ×D A 4 = W ×W , such that Ψ (y0 , ϕ0 ) = R(y0 , ϕ0 ), (y0 , ϕ0 ) . We set as above ⎡ R=⎣

R11 R12 R12 R22

We have, therefore,

⎤ ⎦.

5.1 Riccati-based Internal Stabilization

173

Lemma 5.8. Let (y ∗ , ϕ ∗ , u∗ ) be optimal for problem (5.129) corresponding to (y0 , ϕ0 ) ∈ W ×W. Then N

u∗ (t) = − (R11 y ∗ (t) + R12 ϕ ∗ (t), ψi )ω ψi , ∀ y, ϕ(t) > 0,

(5.131)

i=1

and there are constants ci > 0, i = 1, 2, such that ! " ! " c1 y2W + ϕ2W ≤ R(y, ϕ), (y, ϕ) ≤ c2 y2W + ϕ2W , ∀ y, ϕ ∈ W.

(5.132)

The operator R satisfies the Riccati equation kAy − aAϕ − bdy + bϕ, R11 y + R12 ϕ

N 1 |(R11 y + R12 ϕ, ψi )ω |2 + aAϕ + dy − bϕ, R12 y + R22 ϕ + 2 i=1

2 2 3 3 1 |A 4 y|2 + |A 4 ϕ|2 , ∀ y, ϕ ∈ D(A). = 2 (5.133)

Now, we insert in (5.121) (equivalently, (5.122)) the feedback controller (5.131) and get the closed loop system yt + kAy − aAϕ − dy + bϕ − bϕ 3 N

+ m(R11 y + R12 ϕ, ψi )ω ψi = 0, t > 0,

(5.134)

i=1

ϕt + aAϕ − b(ϕ − ϕ 3 ) + dy = 0, t > 0,

(5.135)

y(0) = y0 , ϕ(0) = ϕ0 . By the substitution z = y + ϕ, we rewrite system (5.134) as zt + bAz − bAϕ + ϕt + aAϕ − b(ϕ

N

m(R11 (z − ϕ) + R12 ϕ, ψi )ω ψi i=1 − ϕ 3 ) + d(z − ϕ) = 0,

= 0, (5.136)

z(0) = z0 − y0 − ϕ0 , ϕ(0) = ϕ0 . Equivalently, d (z, ϕ) + G (z, ϕ) = 0, t ≥ 0, dt (z, ϕ)(0) = (z0 , ϕ),

(5.137)

174

5 Feedback Stabilization of Semilinear Parabolic Equations

where G : H ×H is given by G (z, ϕ) = (bAz − bAϕ + +d(z − ϕ),

N

m(R11 (z − ϕ) + R12 ϕ, ψi )ω ψi , aAϕ − b(ϕ − ϕ 3 )

i=1

with D(G ) = (H01 (O) ∩ H 2 (O))2 . If we provide the space H ×H with a suitable equivalent scalar product, we see that the operator G is quasi-m-accretive in H ×H (see, e.g., [26], p. 237). Then, by Theorem 1.1, it follows that (see also Theorem 1.5) for all (y0 , ϕ0 ) ∈ H ×H and T > 0, problem (5.136) is well posed in H ×H , and so, system (5.134)–(5.135) has a unique solution (y, ϕ) ∈ (L2 (0, T ; V ))2 such that 1

t 2 (y, ϕ) ∈ (L2 (0, T ; D(A)) ∩ W 1,2 ([0, T ]; H ))2 . Multiplying (5.134) by R11 y + R12 ϕ and (5.135) by R12 y + R22 ϕ, respectively, we get after some calculation that

4 3 2 2 3 3 d 2 R(y, ϕ), (y, ϕ) + |A 4 y|2 + |A 4 ϕ|2 + |mR 1 (y, ϕ)|2 ≤ 2b| R 2 (y, ϕ), ϕ 3 |, dt ∀ t ∈ (0, T ),

where R 1 (y, ϕ) = R11 y + R12 ϕ, R 2 (y, ϕ) = R12 y + R22 ϕ. We have # $ | R 1 (y, ϕ), ϕ 3 | ≤ C (y + ϕ) |ϕ|36

1 1 1 1 1 3 1 3 2 2 2 2 3 3 4 4 4 4 ≤ C |A y|2 |A y|2 |ϕ|6 + |A ϕ|2 |A ϕ|2 |ϕ|6

1 3 2 2 2 2 1 3 1 1 3 3 2 2 4 4 4 4 4 4 ≤ C |A y|2 |A y|2 + |A ϕ|2 |A y|2 |A y|2 + |A ϕ|2

1 3 2 2 2 1 1 1 3 3 2 2 |A 4 y|2 + |A 4 ϕ|2 , ≤ C |A 4 y|2 + |A 4 ϕ|2 |A 4 y|2 3

because ϕ|6 ≤ C|A 4 ϕ|2 . This yields 1 3 d 1 3 R(y(t), ϕ(t)), (y(t), ϕ(t)) + |A 4 y(t)|22 + |A 4 ϕ(t)|22 dt 2 2   1 1 1 ≤ C |A 4 y(t)|2 + |A 4 ϕ(t)|2 − 2C

(5.138)

≤ C1 (R(y(t), ϕ(t)), (y(t), ϕ(t)) − C2 , where C1 , C2 ≥ 0. If we replace in system (5.110) and the above computation ϕ 3 by its Yosida approximation (ϕ 3 )ε , we get, as in the proof of Theorem 5.2, that there is a maximal

5.2 Boundary Stabilization of Parabolic Equations

175

interval (0, T ∗ ), T ∗ > 0, where R(y(t)), (y(t), ϕ(t)) ≤ C2 . Then, by (5.138), it follows that

2 2 3 3 1 d 4 4 R(y, ϕ), (y, ϕ) + |A y|2 + |A ϕ|2 ≤ 0, ∀ t > 0 dt 2 which, by virtue of (5.132), implies that y(t)W + ϕ(t)W ≤ C(y0 W + ϕ0 W )e−γ t , ∀ t > 0 and ∞



|A

0

3 4

2 y|2

+ |A

3 4

2 ϕ|2

dt ≤ C.

We have obtained, therefore, the following stabilization result. Theorem 5.4. Suppose that (K4 ) holds. Then there exists a linear positive and selfadjoint operator R : H ×H → H ×H with V ×V ⊂D(A)⊂W ×W , such that the feedback controller N

u = − (R11 y + R12 ϕ, ψi )ω ψi i=1

stabilizes exponentially the zero solution of the phase-field system (5.121) for (y0 , ϕ0 ) ∈ Eρ . Moreover, the operator R is the solution to the algebraic Riccati equation (5.133).

5.2 Boundary Stabilization of Parabolic Equations Since in applications the boundary ∂O is more accessible than the interior of O, the design of a stabilizable feedback controller supported by a boundary is an important objective in engineering control practice. The boundary stabilization of equation (5.1) can be treated by a Riccati-based technique similar to that developed above for the internal stabilization. However, it should be said that, in spite of its theoretical simplicity and robustness, the Riccatibased approach to the boundary feedback stabilization might be impractical. One of its major drawbacks is that it involves a great computational complexity in the numerical treatment of specific problems. We shall present here a direct stabilization technique for parabolic equations which has an interest in itself and is applicable to a wider class of parabolic-like control systems and, in particular, to Navier–Stokes systems (see Chapter 6). For the sake of simplicity, we confine to the stabilization of the equilibrium solutions ye to the parabolic equation

176

5 Feedback Stabilization of Semilinear Parabolic Equations

∂y = Δy + f (x, y) in (0, ∞) × O, ∂t y(0, x) = y0 (x) in O, ∂y = 0 on (0, ∞) × Γ2 , y = u on (0, ∞) × Γ1 , ∂n

(5.139)

where O is a bounded and open domain of Rd with a smooth boundary ∂O = Γ1 ∪ Γ2 , Γ1 , Γ2 being connected parts of ∂O. Here and everywhere in the sequel, ∂y we shall denote by ∂n the outward normal derivative of y. ∂ is the The Dirichlet controller u is applied on Γ1 while Γ2 is insulated. Here, ∂n 2 normal derivative and ye ∈ C (O) is any solution to the equation Δye + f (x, ye ) = 0 in O,

∂ye = 0 on Γ2 . ∂n

Translating ye into zero via substitution y − ye → y we can rewrite (5.139) as ∂y = Δy + f (x, y + ye ) − f (x, ye ) in (0, ∞) × O, ∂t y(0, x) = y0∗ (x) = y0 (x) − ye (x) in O, ∂y = 0 on (0, ∞) × Γ2 , y = u − ye on (0, ∞) × Γ1 ; ∂n

(5.140)

and the stabilization problem reduces to design a feedback controller u = F (y) such that the solution to the corresponding closed loop system satisfies, for some γ > 0,  |y(t, x)| dx ≤ Ce 2

O

−γ t

 O

|y(0, x)|2 dx, ∀ t ≥ 0,

(5.141)

for all y0∗ in a L2 (O)–neighborhood of the origin. The first step toward this goal is the stabilization of the linearized system associated with (5.140), that is, ∂y = Δy + fy (x, ye )y in (0, ∞) × O ∂t ∂y = 0 on (0, ∞) × Γ2 , y = v on (0, ∞) × Γ1 ; ∂n

(5.142)

where fy (x, ye ) = ∂f ∂y (x, ye ). The stabilizing feedback controller v = F (y) for (5.142) will be used afterwards to stabilize locally system (5.140), and implicitly the equilibrium solution ye . Everywhere in the following, we shall assume that (i) f, fy ∈ C(O × R). In particular, this implies that x → fy (x, ye (x)) is continuous on O.

5.2 Boundary Stabilization of Parabolic Equations

177

We consider the linear self-adjoint operator in H = L2 (O), Ly = Δy+ fy (x, ye )y, ∀ y ∈ D(L), D(L) = y ∈

H 2 (O);

 ∂y = 0 on Γ2 . y = 0 on Γ1 , ∂n

The operator −L has a countable set of real eigenvalues λj with corresponding eigenfunctions ϕj , that is, −Lϕj = λj ϕj , j = 1, 2, . . .. Each eigenvalue λj is repeated here according to its multiplicity and let N ∈ N be such that λj ≤ 0 for j = 1, . . . , N ; λN +1 > 0. (Since the resolvent of L is compact and Ly, y2 ≤ −|∇y|22 + C|y|22 , it is clear that N is finite.) ∂ϕ

Here ∂nj is the normal derivative of ϕj to ∂O. By the unique continuation property of eigenfunctions ϕj , we know that, for all ∂ϕj j , ∂n ≡ 0 on Γ1 . In the following, we shall assume that   ∂ϕ (ii) The system ∂nj , 1 ≤ j ≤ N is linearly independent on Γ1 . We note that (ii) always holds if N = 1 and, for d = 1, only in this case. For d > 1, there are, however, significant situations where (ii) holds (see Example 5.3 below). Consider the feedback controller v=η

N

j =1

# $ μj y, ϕj 2 φj on (0, ∞) × Γ1 ,

(5.143)

where η, k > 0 are parameters to be made precise later on, and μj = φj =

k + λj , j = 1, . . . , N, k + λj − η N

=1

N

)

aj 

(5.144)

∂ϕ , j = 1, . . . , N, on Γ1 , ∂n

∂ϕ ∂ϕi , ∂n ∂n

(5.145)

* = δij , i, j = 1, . . . , N.

(5.146)

3 4    ∂ϕi N We note that, by assumption (ii), the Gram matrix  ∂ϕ , ∂n ∂n 

is nonsingular.

=1

aj

0

0 i,=1

Hence, aj and φj are well defined. Theorem 5.5. Let k and η be positive and sufficiently large such that ((k + λj − η)λj + kη)(k + λj − 2η)−1 ≥ γ0 > 0, for j = 1, . . . , N.

(5.147)

178

5 Feedback Stabilization of Semilinear Parabolic Equations

Then, the feedback controller (5.143) stabilizes exponentially system (5.142). More precisely, the solution y to the closed loop system ∂y = Δy + fy (x, ye )y in (0, ∞) × O, ∂t N

# $ y=η μj y, ϕj 2 φj in (0, ∞) × Γ1 ;

(5.148)

j =1

∂y = 0 on (0, ∞) × Γ2 , ∂n satisfies, for γ = inf{γ0 , λN +1 }, the estimate |y(t)|2 ≤ C exp(−γ t)|y(0)|2 , ∀ t ≥ 0.

(5.149)

It should be remarked that Theorem 5.5 provides a simple algorithm for the stabilization of the linear system (5.142) as well as for the nonlinear system (5.139) which, in a few words, can be described as follows. Determine first the unstable eigenvalues {λj }N j =1 and the corresponding eigenfunctions ϕj of the operator L and construct afterwards the feedback controller (5.143), where μj , φj are given (5.144)–(5.146) and k, η satisfy condition (5.147). In this setting, the controller u is coordinated with the measurements of boundary values of the state system and this enhances the stability of motion. Of course, in specific situations, the eigenvalues λj and ϕj cannot be computed exactly and so, instead of (5.143), we must consider an approximating feedback controller vh of the form (5.143) corresponding to the approximations λhj and ϕjh of λj and ϕj , respectively. However, the approximating controller vh is still stabilizing in problem (5.142) by the robustness of the stabilizer controller. Let us illustrate the method on a few examples. Example 5.2. We consider the boundary stabilization problem yt = yxx + λy, x ∈ (0, 1), t ≥ 0, yx (0, t) = 0, y(1, t) = v(t), t > 0,

(5.150)

with Dirichlet actuation in x = 1, where λ > 0 is a constant parameter. The eigenvalues of the operator −Ly = −y

− λy with the domain D(L) = {y ∈ H 2 (0, 1); 2π 2 y (0), y(1) = 0} are λj = (2j −1) − λ with eigenfunctions ϕj = cos (2j −1)π x. 4 2  2 3π Then, for λ < 2 , we have N = 1 and so Theorem 5.5 is applicable with the feedback controller (5.143) of the form 3 π 4 v = ημ1 y, cos x φ1 , 2 2 where μ1 is given by (5.144), and φ1 = 1.

(5.151)

5.2 Boundary Stabilization of Parabolic Equations

179

By Theorem 5.5, for(λ1 (k +λ1 −η)+k)(λ1 +k −2η)−1 > γ0 > 0, this feedback controller stabilizes (5.150) with the exponent decay 



3π γ = inf γ0 , 2

2

 −λ .

In [52], a stabilizing feedback controller was first designed by the backstepping 2 method for λ < 3π4 and later on, this condition was removed in [15] by a sharpening of the method. It should be said, however, that the feedback controller (5.151) is  2 · simpler than that constructed via the backstepping method for λ < 3π 2 Example 5.3. Consider the boundary stabilization of the heat equation in Rd yt = Δy + λy, x ∈ (0, π )d , t ≥ 0,

(5.152)

y(t, x) = v(t, x), x ∈ ∂O, t ≥ 0,

where x = (x1 , x2 , . . . , xd ) and O = (0, π )d . The eigenvalues of the operator −Ly = −Δy − λy with the domain {y ∈ H01 (O); Δy ∈ L2 (O)} are given λk = |k|2 − λ, |k|2 = k12 + · · · + kd2 , (k1 , . . . , kd ) ∈ Nd , with the corresponding eigenfunctions ϕk (x) = sin k1 x1 . . . sin kd xd . Then, N ∈ N is determined by the condition |k|2 < λ and, since ∂ϕk (x) = −k1 sin k2 x2 . . . sin kd xd for x1 = 0, (x2 , . . . , xd ) ∈ (0, π )d−1 , ∂n it turns out that assumption (ii) holds on ∂O (as a matter of fact on each Γj = {0} × (0, π )d−1 ). Hence, Theorem 5.5 is applicable in the present situation and so there is a feedback controller v of the form (5.143)–(5.145), which stabilizes system (5.152). Remark 5.7. The previous equation might suggest that, for d ≥ 1, assumption (ii) is always satisfied, but the following example shows that, in general, this is not true if ∂O = Γ1 ∪ Γ2 , where Γ2 = ∅. Take, for instance, equation (5.152) with ∂y λ = 11, O = (0, π )2 and boundary conditions: y = u on Γ1 , ∂ν = 0 on Γ2 , where Γ1 = {x1 = 0} ∪{x1 = π } × (0, π ), Γ2 = (0, π ) × {x2 = 0} ∪ {x2 = π }. Then, ϕ1 = sin x1 cos x2 is an eigenfunction for λ1 = −9 and ϕ2 = sin 3x1 cosx2 for ∂ϕ2  1 are λ2 = −1 (both unstable eigenvalues). However, as easily seen, ∂ϕ ∂x1  , ∂x1  linearly dependent.

Γ1

Γ1

180

5 Feedback Stabilization of Semilinear Parabolic Equations

Remark 5.8. Numerical tests for the computation of the stabilizing controller (5.143) were performed in [79] for (5.152) on O = (0, π ) × (0, π ), and λ = 3, 7. Also, the case O = (0, 1)×(0, 1), Γ1 = {1}×{0, 1), Γ2 = ∂O \Γ2 was numerically treated. Proof of Theorem 5.5. Consider the map y = Dv defined by −Δy − fy (x, ye )y + ky = 0 in O, ∂y y = v on Γ1 , = 0 on Γ2 ∂n

(5.153)

For k sufficiently large, the Dirichlet map D is well defined and D ∈ L(L2 (Γ1 ), 1 H 2 (O)) (see, e.g., [84]). Moreover, we have Dv 1 ≤ 2

C v||L2 (Γ1 ) , ∀ v ∈ L2 (Γ1 ). k ∂ ∂t ,

Since D commutes with the operator and D as

(5.154)

we may rewrite (5.142) in terms of L

d y = L(y − Dv) + kDv, t ≥ 0. dt Equivalently, dv dz = Lz − D + kDv, t ≥ 0, dt dt

(5.155)

where z = y − Dv. Moreover, for later purpose, it is convenient to express the feedback controller (5.143) in terms of z as N

#

v=η

j =1

$ z, ϕj 2 φj .

(5.156)

Indeed, taking z = y − Dv in (5.143), we obtain that v=η

N

# j =1

N

$ # $ y, ϕj 2 φj − η v, D ∗ ϕj 0 φj on Γ1 ,

(5.157)

j =1

where D ∗ ∈ L(L2 (O), L2 (Γ1 )) is the adjoint of D. Now, multiplying equation (5.153), where y = Dφj , by ϕi and recalling that Lϕi = −λi ϕi , we obtain via Green’s formula that  O

Dφj ϕi dx = −

1 λi + k

 φj Γ1

∂ϕi dx, i, j = 1, . . . , N, ∂n

5.2 Boundary Stabilization of Parabolic Equations

and so, by (5.144) and (5.146), we have  # ∗ $ D ϕi , φj 0 = Dφj ϕi dx= − O

This yields #

v, D ∗ ϕi

$ 0

=−

1 λi +k

181

 φj Γ1

δij ∂ϕi dx= − · ∂n λi +k

η y, ϕi 2 , i = 1, . . . , N, k + λi − η

(5.158)

and, substituting into (5.157), we obtain (5.156), as claimed. Substituting (5.156) into (5.155), we obtain * N )

dz dz = Lz − η − kz, ϕj Dφj . dt dt 2

(5.159)

j =1

2 1 Let X1 = lin span{ϕj }N j =1 , PN the algebraic projection of L (O) on X and set z1 = PN z, z2 = (I − PN )z. Then, we may decompose system (5.159) as follows:

* N )

dz1 d 1 z = L1 z1 − ηPN − kz1 , ϕj Dφj , dt dt 2 j =1 * N ) 1

d 2 dz z = L2 z2 − η(I − PN ) − kz1 , ϕj Dφj , dt dt 2

(5.160)

j =1

where L1 = PN L, L2 = (I − PN )L and z = z1 + z2 . N

zj ϕj , we see by (5.158) that If we represent z1 as j =1

zj + λj zj =

η (z − kzj ), k + λj − η j

or, equivalently, zj +

(k + λj − η)λj + kη zj = 0, j = 1, . . . , N, k + λj − 2η

(5.161)

and, by condition (5.147), we have that |zj (t)| + |zj (t)| ≤ Ce−γ0 t |zj (0)|, j = 1, . . . , N.

(5.162)

On the other hand, taking into account that the spectrum of L2 , σ (L2 ) = {λj }∞ j =N +1 , we have |etL2 z02 |2 ≤ Ce−λN+1 t |z02 |2 , ∀ t ≥ 0, z02 ∈ X2 ,

182

5 Feedback Stabilization of Semilinear Parabolic Equations

because the C0 -semigroup etL2 generated by the operator L2 is analytic in L2 (O). Taking into account the second equation in (5.160) and (5.162), we obtain that |z(t)|2 ≤ Ce−γ t |z(0)|2 , ∀ t ≥ 0,

(5.163)

where γ = inf{γ0 , λN +1 }. Keeping in mind that y = z +Du, by (5.163) and (5.156) we see that (5.149) holds. Remark 5.9. The above design of a stabilizable feedback controller applies as well to equation (5.139) with homogeneous Dirichlet condition on Γ2 and Dirichlet actuation on Γ1 , that is, y = u on Γ1 ; y = 0 on Γ2 , or to the Neumann boundary ∂y ∂y control ∂n = u on Γ1 ; ∂n = 0 on Γ2 , but we omit the details. We also note that Theorem 5.5 and the above stabilization construction extend word by word to the controlled parabolic linear equation ∂y − div(a(x)∇y) + b(x)y = 0 in (0, T ) × O, ∂t y = v on (0, ∞) × Γ1 , a∇y · n = 0 on (0, T ) × Γ2 , where a, b ∈ C(O), a(x) > 0, ∀ x ∈ O.

5.3 Stabilization of Semilinear Equations We strengthen assumption (i) to |fy (x, y)| ≤ C(|y|m + 1), ∀ x ∈ O, y ∈ R,

(5.164)

d where 0 < m < ∞ for d = 1, 2, m = d−2 for d ≥ 3. 2 If ye ∈ C (O) is an equilibrium solution to (5.139), we consider the feedback controller

u = F (y) = η

N

j =1

# $ μj y − ye , ϕj 2 φj + ye on Γ1 .

(5.165)

where μj , φj are given by (5.144)–(5.146). Theorem 5.6. Let 1 ≤ d ≤ 3. Then, under assumptions (i), (ii), (5.164), and (5.147), the feedback controller (5.165) stabilizes exponentially the solutions ye to system (5.139). More precisely, the solution y to the closed loop system

5.3 Stabilization of Semilinear Equations

183

∂y = Δy + f (x, y) in (0, ∞) × O, ∂t y = F (y) on (0, ∞) × Γ1 ; ∂y = 0 on (0, ∞) × Γ2 , ∂n y(0, x) = y0 (x), x ∈ O,

(5.166)

satisfies for |y0 − ye |2 ≤ ρ and ρ sufficiently small, the estimate |y(t) − ye |2 ≤ C exp(−γ t)|y0 − ye |2 , ∀ t ≥ 0,

(5.167)

where γ > 0. Proof. By substitution y − ye → y, we reduce the problem to the stability of the null solution to system (5.140) with the boundary controller (5.143), that is, ∂y = Ly + g(x, y) in (0, ∞) × O, ∂t y = G(y) on (0, ∞) × Γ1 ; ∂y = 0 on (0, ∞) × Γ2 , ∂n

(5.168)

where g(x, y) = f (x, y + ye ) − f (x, ye ) − fy (x, ye )y, G(y) = η

N

j =1

# $ μj y, ϕj 2 φj .

Arguing as in the previous case, we write (5.168) as dy = L(y − DG(y)) + g(x, y) + kDG(y), t ≥ 0, dt

(5.169)

and setting z = y − DG(y), we obtain that * N )  

dz dz = Lz + g x, (I − DG)−1 z − η − kz, ϕj Dφj . dt dt 2 j =1

We set as above z = z1 + z2 , z1 ∈ X1 , z2 ∈ X2 andz =

N

j =1

zj ϕj .

(5.170)

184

5 Feedback Stabilization of Semilinear Parabolic Equations

This yields zj +

3 4 (k + λj − η)λj + kη k + λj g(x, (I − DG)−1 z, ϕj ) zj + 2 k + λj − 2η k + λj − 2η

and zj − kzj = Kj (z) =−

3 4 1 ((k+λj )2 zj +(k+λj )) g(x, (I −DG)−1 , ϕj ) . 2 k+λj −2η

We get N 3 4

z+λj dz1 1 z1 + =L g(x, (I −DG)−1 z), ϕj ϕj = 0 2 dt k+λj −2η

(5.171)

j =1

dz2 = L2 z2 + (I −PN )g(x, (I −DG)−1 z)−η(I −PN ) Kj (z)Dφj , (5.172) dt N

j =1

N

(k + λj − η)λj + kη 1 z1= L zj ϕj . k + λj − 2η

(5.173)

j =1

1 and L2 are exponentially stable on X1 , By virtue of (5.163), both operators L 2 1 z1 +L2 z2 , z = z1 +z2 respectively X , and therefore so is the operator z → Bz = L 2 on the space L (O). We set G (z) = −

N

j =1

4 k + λj 3 g(x, (I − DG)−1 z), ϕj ϕj 2 k + λj + η

−(I −PN )g(x, (I −DG)−1 z))−η(I −PN )

N

Kj (z)Dj φ,

j =1

and rewrite (5.171), (5.172) as dz = Bz + G (z), t ≥ 0; z(0) = z0 = y0 − DGy0 . dt Equivalently,  z(t) = etB z0 + 0

t

e(t−s)B G (z(s))ds, t ≥ 0.

(5.174)

5.3 Stabilization of Semilinear Equations

185

We are going to show that, for |z0 |2 ≤ ρ sufficiently small, equation (5.174) has a unique solution z ∈ Lm+1 (0, ∞; H 1 (O)). To this end, we proceed as in [15]. Namely, consider the map Λ : Lm+1 (0, ∞; H 1 (O)) → Lm+1 (0, ∞; H 1 (O)) defined by  Λz(t) = etB z0 +

t

e(t−s)B G (z(s))ds

(5.175)

0

and show that, for r sufficiently small, it maps the ball {z ∈ Lm+1 (0, ∞; H 1 (O)); zLm+1 (0,∞;H 1 (O)) ≤ r} = S(0, r) into itself and is a contraction on S(0, r) for |z0 |2 sufficiently small and r suitable chosen. Indeed, by assumption (5.164), we have |g(x, (I − DG)−1 z) − g(x, (I − DG)−1 z¯ )| ≤ C|(I − DG)−1 z − (I − DG)−1 z¯ |(|(I − DG)−1 z|m + |(I − DG)−1 z¯ |m + 1). Taking into account Sobolev’s embedding theorem (recall that 1 ≤ d ≤ 3) this yields |g(x, (I − DG)−1 z) − g(x, (I − DG)−1 z¯ )|2 ≤ C(I − DG)−1 z−(I −DG)−1 z¯ L4 (O) ((I −DG)−1 zm +(I −DG)−1 z¯ m ) L2m (O) L2m (O) ≤ C1 (I −DG)−1 z−(I −DG)−1 z¯ H 1 (O) ((I −DG)−1 zm +(I −DG)−1 z¯ m +1) H 1 (O) H 1 (O) + ¯zm 1 ≤ C2 z − z¯ H 1 (O) (zm H 1 (O)

H (O)

), ∀ z, z¯ ∈ H 1 (O).

Hence, |G (z) − G (¯z)|2 ≤ C2 z − z¯ H 1 (O) (z|m + ¯zm ), H 1 (O) H 1 (O)

(5.176)

while m+1 |G (z)|2 ≤ C3 zH 1 (O) .

(5.177)

Taking into account that, by (5.171)–(5.172), etB z0 H 1 (O) ≤ Ce−γ t |z0 |2 , ∀ t ≥ 0, we see by (5.175) that Λ maps S(0, r) into itself for r suitable chosen and |z0 |2 sufficiently small. Moreover, by (5.176), it follows that Λ is a contraction on S(0, r).

186

5 Feedback Stabilization of Semilinear Parabolic Equations

Hence, equation (5.174) has, for |y0 |2 ≤ ρ sufficiently small, a unique solution z ∈ Lm+1 (0, ∞; H 1 (O)). By a standard argument (see, e.g., Proposition 5.9 in [28]), this implies also that |z(t)|2 ≤ Ce−γ t |z(0)|2 , ∀ t ≥ 0, and the latter extends to the solution y to (5.168). Then (5.167) follows. Example 5.4. Consider the classical Fitzhugh–Nagumo equation which models the dynamics of electrical impulses across a cell membrane of an excited neuron. Namely, in its simplified version, it has the form yt = yxx + y(1 − y)(y − a), 0 < x < , t ≥ 0, yx (t, ) = 0, y(t, 0) = u(t), t ≥ 0, where 0 < a
0.

5.4 Internal Stabilization of Stochastic Parabolic Equations Consider the stochastic nonlinear controlled parabolic equation dX(t) − ΔX(t)dt + a(t, ξ )X(t)dt + b(t, ξ ) · ∇ξ X(t)dt +f (X(t))dt = X(t)dW (t) + 1O0 u(t)dt in (0, ∞) × O, X = 0 on (0, ∞) × ∂O, X(0) = x in O.

(5.180)

Here, O is a bounded and open domain of Rd , d ≥ 1, with smooth boundary ∂O and W (t) is a Wiener cylindrical process of the form W (t) =

∞

k=1

μk ek (ξ )βk (t), t ≥ 0, ξ ∈ O,

(5.181)

5.4 Internal Stabilization of Stochastic Parabolic Equations

187

where μk are real numbers, {ek } ⊂ C 2 (O) is an orthonormal system in L2 (O) and {βk }∞ k=1 are independent Brownian motions in a stochastic basis {Ω, F , Ft , P}. We assume that ∞

μ2k |ek |2∞ < ∞,

(5.182)

k=1

where | · |∞ denotes the L∞ (O)-norm. The functions a : [0, ∞) × O → R, b : [0, ∞) × O → Rd and f : R → R are assumed to satisfy a ∈ C([0, ∞) × O), b ∈ C 1 ([0, ∞) × O)

(5.183)

sup{|a(t)|∞ + |∇b(t)|∞ ; t ≥ 0} < ∞

(5.184)

f ∈ Lip(R), f (0) = 0.

(5.185)

Finally, O0 is an open subdomain of O with smooth boundary, 1O0 is its characteristic function, and u = u(t, ξ ) is an adapted controller with respect to the natural filtration {Ft }. The main problem we address here is the design of a feedback controller u = F (X) such that the corresponding closed loop system (5.180) is asymptotically stable in probability, that is, lim X(t) = 0, P-a.s.

t→∞

(As seen in Section 3.6, a stronger property, the exact null controllability of (5.180) in finite time, is in general an open problem.) Given an Ft -adapted process u ∈ L2 (0, T ; L2 (O, L2 (O))), a continuous Ft adapted process X : [0, T ] → L2 (O) is said to be a solution to (5.180) if X ∈ L2 (O; L∞ (0, T ; L2 (O))) ∩ C([0, T ]; L2 (O; L2 (O)))

(5.186)

and  t X(t, ξ ) = (ΔX(s, ξ ) − a(t, ξ )X(s, ξ ) − b(s, ξ ) · ∇ξ X(s, ξ ) 0  t  t 1O0 u(s, ξ )ds + X(s, ξ )dW (s), −f (X(s, ξ )))ds + 0

(5.187)

0

ξ ∈ O, t ∈ (0, T ), P-a.s. Taking into account assumptions (5.182)–(5.185), we may conclude (see, e.g., [64], p. 208) that (5.180) has a unique solution X satisfying (5.186), (5.187).

188

5 Feedback Stabilization of Semilinear Parabolic Equations

Let O0 be an open subset of O. We set O1 = O \ O 0 and denote by A1 : D(A1 )⊂L2 (O1 ) → L2 (O1 ) defined by A1 y = −Δy, y ∈ D(A1 ) = H01 (O1 ) ∩ H 2 (O1 ),

(5.188)

or, equivalently,  A1 y, z1 =

O1

∇y · ∇z dξ, ∀ y, z ∈ H01 (O1 ),

(5.189)

where ·, ·1 is the duality on H01 (O1 )×H −1 (O1 ) induced by L2 (O1 ) as pivot space. Denote by λ∗1 (O1 ) the first eigenvalue of the operator A1 , that is, λ∗1 (O1 ) = inf



 O1

|∇y|2 dξ ; y ∈ H01 (O1 ),

O1

 y 2 dξ = 1 .

(5.190)

Consider in (5.180) the feedback controller u = −ηX, η ∈ R+ ,

(5.191)

and the corresponding closed loop system dX − Xdt + aXdt + b · ∇Xdt + f (X)dt = Xdt − η1O0 Xdt in (0, ∞) × O, X(0) = x in O, X = 0 on (0, ∞) × ∂O. (5.192) Theorem 5.7 is the main result. Theorem 5.7. Assume that ∞

1 2 μj |ej |2∞ − f Lip 2 j =1   1 − sup −a(t, ξ ) + divξ b(t, ξ ); (t, ξ ) ∈ R+ × O > 0. 2

λ∗1 (O1 ) −

(5.193)

Then, for each x ∈ L2 (O) and for η sufficiently large (independent of x), the feedback controller (5.191) exponentially stabilizes in probability equation (5.180). More precisely, there is γ > 0 such that the solution X to (5.192) satisfies lim eγ t |X(t)|22 = 0, P-a.s.  ∞ eγ t E|X(t)|22 + E eγ t |X(t)|22 dt ≤ C|x|22 . t→∞

0

(5.194) (5.195)

5.4 Internal Stabilization of Stochastic Parabolic Equations

189

We recall that, by the classical Rayleigh–Faber–Krahn perimetric inequality in dimension d ≥ 2, we have λ∗1 (O1 )

≥

ωd |O1 |

2 d

J d −1,1 , 2

(5.196)

! "" d ! where |O1 | = Vol(O1 ), ωd = π 2 / Γ d2 + 1 , and Jm,1 is the first positive zero of the Bessel function Im (r). In particular, by Theorem 5.7, we conclude that, if |O1 | is sufficiently small, then the feedback controller (5.191) is exponentially stabilizable. More precisely, we have Corollary 5.1. Assume under hypotheses (5.182)–(5.185) that ⎛ |O1 | ≤ ωd J

d 2 d 2 −1,1

⎞− d 2   ∞

1 1 2 2 ⎝ ⎠ μj |ej |∞ + sup −a + divξ b + f Lip . 2 2 R+ ×O Lip

(5.197) Then, for each x ∈ L2 (O), the feedback controller (5.191) exponentially stabilizes system (5.180) in sense of (5.194), (5.195). An Example The stochastic equation dX − Xξ ξ dt + (aX + bXξ )dt = μXdβ + V dt, 0 < ξ < 1, X(t, 0) = X(t, 1) = 0, t ≥ 0, where β is a Brownian motion and μ ∈ R, a ∈ C([0, T ] × R), b ∈ C 1 ([0, 1] × R), is exponentially stabilizable in probability by any feedback controller V = −η1[a1 ,a2 ] X, where η > 0 is sufficiently large and 0 < a1 < a2 < 1 are such that 

1 1 , π inf a1 1 − a2



  μ2 1 + sup > −a(t, ξ ) + bξ (t, ξ ) . 2 2 (t,ξ )∈R+ ×(0,1)

Proof of Theorem 5.7. The main ingredient of the proof is the following lemma. Lemma 5.9. For each ε > 0 there is η0 = η0 (ε) such that   |∇y(ξ )|2 dξ + η y 2 (ξ )dξ ≥ (λ∗1 (O1 ) − ε)|y|22 , O

O0

∀ y ∈ H01 (O), η ≥ η0 .

Proof. Denote by ν1 the first eigenvalue of the self-adjoint operator Aη y = Ay + η1O0 y, ∀ y ∈ D(Aη ) = H01 (O) ∩ H 2 (O), where A = −Δ, D(A) = H01 (O) ∩ H 2 (O), and η ∈ R+ .

(5.198)

190

5 Feedback Stabilization of Semilinear Parabolic Equations

We have by the Rayleigh formula    η ν1 = inf |∇y|2 dξ + η |y|2 dξ ; |y|2 = 1 ≤ λ∗1 (O1 ) O

(5.199)

O0

because any function y ∈ H01 (O1 ) can be extended by zero to H01 (O) across the η smooth boundary ∂O1 = ∂O0 . Let ϕ1 ∈ H01 (O) ∩ H 2 (O) be such that η

η η

η

Aη ϕ1 = ν1 ϕ1 , |ϕ1 |2 = 1. We have by (5.199) that  O

η |∇ϕ1 |2 dξ

 +η

|ϕ1 |2 dξ = ν1 ≤ λ∗1 (O), ∀ η > 0. η

O0

η

Then, on a subsequence, again denoted η, we have for η → ∞, that ν1 → ν ∗ and η

 η

O0

η

ϕ1 −→ ϕ1 weakly in H01 (O), strongly in L2 (O) η

|ϕ1 |2 dξ −→ 0. η

η

We have, therefore, ϕ1 ∈ H01 (O1 ), |ϕ1 |2 = 0 and, since Aη ϕ1 |O1 = A1 ϕ1 , we have also that A1 ϕ1 = ν ∗ ϕ1 . Moreover, by (5.199) we see that ν ∗ ≤ λ∗1 (O1 ). Since λ∗1 (O1 ) is the first eigenvalue of A1 , we have that ν ∗ = λ∗1 (O1 ) and 

 |∇y| dξ + η 2

lim inf

η→∞

O

 y dξ ; |y|1 = 1 = λ∗1 (O1 ). 2

O0

This yields (5.198), as claimed. Proof of Theorem 5.7 (Continued). By applying Itô’s formula in (5.192), which by virtue of (5.186) is possible, we obtain that   1 d(eγ t |X(t)|22 ) + eγ t |∇X(t, ξ )|2 dξ + (a(t, ξ ) − γ 2 O O  ∞

1 1 − divξ b(t, ξ ))eγ t X2 (t, ξ )dξ = eγ t |(Xek )(t, ξ )|2 dξ 2 2 O k=1  γt − e f (X(t, ξ ))X(t, ξ )dξ O



−η

O0

 eγ t |X(t, ξ )|2 dξ +

O

eγ t

∞

(Xek )(t, ξ )X(t, ξ )dβk (t), P-a.s., t ≥ 0.

k=1

Equivalently, 1 γt e |X(t)|22 + 2



t 0

K(s)ds =

1 |x|22 + M(t), t ≥ 0, P-a.s., 2

(5.200)

5.5 Stabilization of Navier–Stokes Equations Driven by Linear Multiplicative. . .

where K(t) = ∞

 O

191

1 divξ b(t, ξ ))|X(t, ξ )|2 2  γt 2 e |X(t, ξ )| dξ + eγ t f (X(t, ξ ))X(t, ξ )dξ,

eγ t (|∇X(t, ξ )|2 + (a(t, ξ ) − γ −

1 2 − X (t, ξ )ek2 (ξ ))dξ + η 2 k=1

M(t) =

 t ∞ 0

O k=1

 O0

O

(5.201) eγ s X2 (s, ξ )ek (s, ξ )dβk (s), t ≥ 0.

(5.202)

By (5.199) and (5.193), we see that, for η ≥ η0 sufficiently large and 0 < γ ≤ γ0 sufficiently small, we have  eγ t |X(t, ξ )|2 dξ, ∀ t > 0, P-a.s., (5.203) K(t) ≥ ε0 O

where ε0 > 0. Taking expectation into (5.200), we obtain that  t 1 1 eγ s E|X(s)|22 ds ≤ |x|22 , ∀ t ≥ 0. (5.204) E[eγ t |X(t)|22 ] + ε0 2 2 0 t Since t → 0 K(s) is an a.s. nondecreasing stochastic process and t → M(t) is a continuous local martingale, we infer by (5.200), (5.204) that there exist lim (eγ t |X(t)|22 ) < ∞, K(∞) < ∞,

t→∞

which imply (5.194), (5.195), as claimed. Remark 5.10. By the proof, it is clear that Theorem 5.7 extends to more general nonlinear functions f = f (t, ξ, x), as well as to smooth functions μk = μk (t, ξ ). Also, the Lipschitz condition (5.185) can be weakened to f continuous, monotonically increasing and with polynomial growth. Moreover, Δ can be replaced by any strongly elliptic linear operator in O. The details are omitted.

5.5 Stabilization of Navier–Stokes Equations Driven by Linear Multiplicative Noise We consider here the stochastic Navier–Stokes equation dX(t) − ν0 ΔX(t)dt + (a(t) · ∇)X(t)dt + (X(t) · ∇)b(t)dt + (X(t) · ∇)X(t)dt = X(t)dW (t) + ∇p(t)dt + 1O0 u(t)dt in (0, ∞) × O, ∇ · X(t) = 0 in (0, ∞) × O, X(t) = 0 on (0, ∞) × ∂O, X(0) = x in O,

(5.205)

192

5 Feedback Stabilization of Semilinear Parabolic Equations

where ν > 0, a, b ∈ (C 1 ((0, ∞) × O))2 , ∇ · a = ∇ · b = 0, a · n = b · n = 0 on ∂O. Here O is a bounded and open domain of R2 and O0 is an open subset of O. The boundaries ∂O and ∂O0 are assumed to be smooth. We set H = {y ∈ (L2 (O))2 ; ∇ · y = 0, y · n = 0 on ∂O}, where n is the normal to ∂O. We denote by ·, ·H the scalar product of H and by | · |H the norm. The Wiener process W (t) is of the form (5.181), where {ek } is the orthonormal system in L2 (O) given by −Δek = λk ek in O, ek = 0 on ∂O, and μk ∈ R. As in the previous case, the main objective here is the design of a stabilizable feedback controller u for equation (5.205). We use the standard notations (see Section 1.4) H = {y ∈ (L2 (O))d ; ∇ · y = 0 in O, y · n = 0 on ∂O}, V = {y ∈ (H01 (O))d ; ∇ · y = 0 in O}, A = −ν0 Π Δ, D(A) = (H 2 (O))d ∩ V , where Π is the Leray projector on H . Consider the Stokes operator A1 on O1 = O \ O0 , that is, A1 y, ϕ = ν0

d 

i=1

O1

∇yi · ∇ϕi dξ, ∀ ϕ ∈ V1 ,

where V1 = {y ∈ (H01 (O1 ))d ; ∇ · y = 0 in O1 }. Denote again by λ∗1 (O1 ) the first eigenvalue of A1 , that is,  λ∗1 (O1 )

= inf ν0

i=1





d 

|∇ϕi | dξ, ϕ ∈ V1 , 2

O

|ϕ| dξ = 1 . 2

O1

(5.206)

Also, in this case, we have (see Lemma 1 in [35]), for η ≥ η0 (ε) and ε > 0, # $ Ay, yH + η Π (1O0 y), y H ≥ (λ∗1 (O1 ) − ε)|y|2H , ∀ y ∈ V .

(5.207)

We consider in system (5.205) the linear feedback controller u = −ηX, η > 0.

(5.208)

  We set γ ∗ (t) = sup O |yi Di bj yj dξ |; |y|H = 1 < ∞, where b = {b1 , b2 }. The closed loop system (5.205) with the feedback controller (5.208) has a unique strong solution in the sense of (5.186), (5.187). (See, e.g., [64], p. 281.) We have Theorem 5.8. Assume that λ∗1 (O1 ) >

∞

1 2 μj |ej |2∞ + sup γ ∗ (t). 2 t∈R+ j =1

(5.209)

5.5 Stabilization of Navier–Stokes Equations Driven by Linear Multiplicative. . .

193

Then, for each x ∈ H and η sufficiently large independent of x, the solution X to the closed loop system (5.205) with the feedback controller (5.208) satisfies  E[eγ t |X(t)|2H ] +

∞ 0

eγ t E|X(t)|2H dt < C|x|2H ,

lim eγ t |X(t)|2H = 0, P-a.s.,

t→∞

(5.210) (5.211)

for some γ > 0. The proof is essentially the same as that of Theorem 5.7, and so it will be sketched only. Taking into account that (X · ∇)X, XH + (a(t) · ∇)X, XH = 0, t > 0, P-a.s., we obtain by (5.205), (5.208), via Itô’s formula, that  t  1 γt e |X(t)|2H + eγ s AX(s), X(s)H + X(s) · ∇b(s), X(s)H 2 0 ∞

# $  1 − |X(s)ej |2H + η 1O0 X(s), X(s) H ds 2 j =1  t ∞

$ # 1 = |x|2H + eγ s X(s)ej , X(s) H dβj (s), t ≥ 0. 2 0

(5.212)

j =1

Then, by virtue of (5.207) and (5.209), we have, by (5.212), that 1 γt 1 e |X(t)|2H + I (t) = |x|2H + M ∗ (t), t ≥ 0, P-a.s., 2 2 where I (t) is a nondecreasing process, which satisfies 

t

E[I (t)] ≥ ε0 0

eγ s E|X(s)|2H ds, ∀ t ≥ 0,

for η sufficiently large, and ∗



M (t) = 0

t

eγ s

∞

# j =1

$ X(s)ej , X(s) H dβj (s)

is a continuous local martingale. As in the previous case, this implies that lim eγ t |X(t)|2H exists P-a.s. and, t→∞ therefore, (5.210) and (5.211) hold.

194

5 Feedback Stabilization of Semilinear Parabolic Equations

Remark 5.11. By (5.206), we see that λ∗1 (O1 ) > νμ1 , where μ1 is the first eigenvalue on O of the Stokes operator −P Δ and we have also that  λ∗1 (O1 ) ≥ Cν

−2 sup dist(x, ∂O)

.

x∈O1

5.6 Notes on Chapter 5 Section 5.1 is based on the stabilization method developed in [37] (see also [28, 39, 44]) for stabilization of Navier–Stokes equations. Earlier results were given in [36]. In fact, the stabilization of a linearized parabolic system closely follows the corresponding treatment in [37] for the Oseen–Stokes equation. The spectral decomposition method was first used for the stabilization of parabolic equations by R. Triggiani in [115] and it is confined to parabolic-like systems that is to infinitedimensional systems with compact resolvents and with spectrum determined growth property [114]. For other resent contributions in this direction, we mention the work of H. Badra and T. Takahashi [13]. It should be said that this method leaves out the stabilization of nonsteadystate trajectories of time-dependent systems. However, the latter case can be also treated by the method developed in [47] for the Navier–Stokes equations. A different approach to the stabilization of time-dependent parabolic equations based on the cost of approximate controllability was developed by C. Lefter [86]. In the works [49, 92] there are designed – by the above spectral decomposition method combined with a fixed point technique on suitable functions space – stabilizable feedback controllers for Cahn–Hillard type systems, while in [53] is studied, by similar methods, the internal stabilization of a FitzHugh–Nagumo model with diffusion. Section 5.2 is based on the author work [30]. An extension of these results to more general systems are given by I. Munteanu in [98, 99]. A different stabilization approach, called backstepping, was developed by A. Balough et al. in [15, 16] (see also [111]). The results of Section 5.4 are taken from the author work [32]. Earlier results for the diffusion equation were given in [5, 6]. Another internal and boundary stabilization technique not developed here, but which was treated in some detail in [28], is the stabilization by Gaussian noises. For system (5.1), such a stabilizing stochastic controller u is of the form u(t) =

N

(y(t), ϕi∗ )2 φi β˙i (t),

i=1 2 where {ϕi }i=1 , {φi }N i=1 ⊂ L (O) are suitable chosen and {βi } is a system of independent Brownian motions on a probability space. The corresponding closed loop system is a stochastic parabolic equation with linear multiplicative noise.

5.6 Notes on Chapter 5

195

As mentioned earlier, a general procedure to stabilize the nonlinear systems, also called the linearization technique, is to use the stabilizable feedback controllers for the linearized equations. The use of this linear feedback law in the nonlinear system is advantageous for its simplicity and elegance, but it might be not efficient enough to stabilize systems with high order nonlinearity. One might suspect that, choosing in Theorem 5.2 a feedback controller u of the form  2k 

(G(y − ye ), ψi )ω ψi , u=ψ i=1

the stabilization effect is effective for functions f of higher order. However, this remains to be done.

Chapter 6

Boundary Stabilization of Navier–Stokes Equations

The stabilization of fluid flows and, in particular, of Navier–Stokes equations was extensively studied via the Riccati-based approach in the last decade and the main references are the works [24, 28, 44, 45, 74, 107]. Here, following [29], we shall briefly present a few results concerning the stabilization with oblique feedback controllers and, more precisely, with almost normal boundary feedback controllers.

6.1 The Main Stabilization Results It is well known that the steady-state solutions to the Navier–Stokes equation might be unstable for a large Reynold number (that is, for small viscosity ν0 ). The objective here is to design a stabilizable feedback controller defined on the boundary of the domain guaranteeing the stability of the fluid flow for large Reynolds numbers. The Navier–Stokes system considered here is of the form (see (1.48)) ∂y − ν0 Δy + (y · ∇)y = ∇p + fe ∂t ∇· y=0 y = v, y · n = 0 y(0) = y0

in (0, ∞) × O, in (0, ∞) × O, on (0, ∞) × ∂O, in O,

(6.1)

in a bounded open domain O ⊂ Rd , d = 2, 3, with the boundary ∂O which is assumed to be a finite union of d − 1 dimensional C 2 – connected manifolds. Here ν0 > 0, fe is a given smooth function and v is a boundary input. If ye is an equilibrium (steady-state) solution to (6.1), then (6.1) can be, equivalently, written as

© Springer International Publishing AG, part of Springer Nature 2018 V. Barbu, Controllability and Stabilization of Parabolic Equations, Progress in Nonlinear Differential Equations and Their Applications 90, https://doi.org/10.1007/978-3-319-76666-9_6

197

198

6 Boundary Stabilization of Navier–Stokes Equations

∂y − ν0 Δy + (ye · ∇)y + (y · ∇)ye + (y · ∇)y = ∇p ∂t ∇· y=0 y = u, y · n = 0 y(0) = y0 − ye

in (0, ∞) × O, in (0, ∞) × O, on (0, ∞) × ∂O, in O. (6.2)

As seen in Section 1.5 (Remark 1.1), in 3 − D, equation (6.2) ha a local solution which is global if ν0−1 y0 − ye 2 is sufficiently small. However, in general, the null solution to (6.2) is not asymptotically stable. As seen below, this can be achieved, however, by choosing a suitable feedback controller supported by the boundary ∂O. Let Γ be a connected component of ∂O. Our main concern here is the design of an oblique boundary feedback controller with support in Γ which stabilizes exponentially the equilibrium state ye , or, equivalently, the zero solution to (6.2). The main step toward this end is the stabilization of the linear system corresponding to (6.2) or, more generally, of the Oseen–Stokes system ∂y − ν0 Δy + (y · ∇)a1 + (a2 · ∇)y = ∇p in (0, ∞) × O, ∂t ∇· y=0 in (0, ∞) × O, y = u, y · n = 0 on (0, ∞) × ∂O,

(6.3)

where a1 , a2 ∈ (C 2 (O))d , ∇ · a1 = ∇ · a2 = 0 in O. Besides its significance as first order linear approximation of (6.2), this system models the dynamics of a Stokes flow with inclusion of a convection acceleration (a2 · ∇)y and also the disturbance flow induced by a moving body in a Stokes fluid flow. In its complex form, the main result Theorem 6.1 below, amounts to saying that, if the unstable eigenvalues of system (6.3) are semi-simple and a certain unique continuation type property for eigenfunctions of the dual linearized system holds, then there is a boundary feedback controller of the form u(t, x) = η1Γ

N

 μj

j =1

O

y(t, x)ϕ ∗j (x)dx

(φj (x) + α(x)n(x)),

(6.4)

t ≥ 0, x ∈ ∂O,

which stabilizes exponentially system (6.1). Here, 1Γ is the characteristic function of Γ as subset of ∂O, φj ⊂ (C 2 (Γ ))d are suitably chosen functions and {ϕj∗ }N j =1 is an eigenfunction system for the adjoint L ∗ of the Stokes–Oseen operator L ϕ = −ν0 Δϕ + (a · ∇)ϕ + (ϕ · ∇)b, ϕ ∈ D(L ), D(L ) = {ϕ ∈ (H 2 (O))d ∩ (H01 (O))d ; ∇ · ϕ = 0 in O},

(6.5)

that is, (L ∗ ψ)j = −ν0 Δψj −

N

(Di (ai ψj ) − ψi Dj bi ), j = 1, .., d. i=1

(6.6)

6.1 The Main Stabilization Results

199

It turns out (see Theorem 6.2) that this feedback controller also stabilizes the Navier–Stokes system (6.2) in a neighborhood of the origin. In (6.4), N is the number of the eigenvalue λj of L with Re λj ≤ 0 and α ∈ C 2 (∂O) is an arbitrary function with zero circulation on Γ , that is,  α(x)dx = 0.

(6.7)

Γ

If α is identically zero, then the controller (6.4) is tangential but in general it is oblique to domain O which makes it more effective for control actuation. As a matter of fact, we shall see below (see Corollary 6.1) that, with exception of a set of Lebesgue measure arbitrarily small, the controller u given by (6.4) can be chosen in a direction close to n, that is, almost normal. It should be mentioned that, in literature, only in a few situations normal stabilizing controllers for equation (6.1) were designed, and this happened mostly for periodic flows in 2−D channels only (see, e.g., [1, 16, 25, 117, 118]). However, there is a large body of results on boundary stabilization of system (6.1) by tangential or not made precise boundary feedback controllers and here the works [12, 74, 75, 106, 107] should be cited. The Riccati-based approach used in these works is essentially the same, as that developed in Chapter 5 for the internal stabilization of parabolic equations and can be described in a few words as follows: one decomposes system (6.1) in a finite-dimensional unstable part which is exactly controllable and an infinitedimensional part which is exponentially stable and proves so its stabilization by an open loop boundary controller with finite-dimensional structure. Then, one designs in a standard way a stabilizing feedback controller via the algebraic Riccati equation associated with an infinite horizon quadratic optimal control problem. As mentioned earlier, a major drawback of this method is that it involves a large amount of computation. Here we shall construct an explicit feedback stabilizing controller using the same method as in Section 5.3. Everywhere in the following, O is a bounded and open domain of Rd , d = 2, 3, its boundary ∂O is a finite union of d − 1 dimensional C 2 – connected manifolds and Γ is a connected component of ∂O. We set H = {y ∈ (L2 (O))d ; ∇ · y = 0 in O, y · n = 0 on ∂O} and denote by Π : (L2 (O))d → H the Leray projector on H . We consider the operator A : D(A)⊂H → H , A : D(A )⊂H → H , Ay = −ν0 Π (Δy), ∀ y ∈ D(A) = (H01 (O))d ∩ (H 2 (O))d ∩ H, A y = Π (−ν0 Δy + (y · ∇)a + (b · ∇)y)

(6.8) (6.9)

= Ay + Π ((y · ∇)a + (b · ∇)y), ∀ y ∈ D(A ) = D(A). the complexified space H = H + iH and consider the We denote, as usually, by H , that is, A (y + iz) = A y + iA z for all y, z ∈ D(A ). extension A of A to H

200

6 Boundary Stabilization of Navier–Stokes Equations

are denoted by ·, · and ·, ·H , respectively. The scalar product of H and of H The corresponding norms are denoted by | · |H and | · |H , respectively. For simplicity, we denote in the following again by A the operator A and the difference will be clear from the content. The operator A has a compact resolvent (λI − A )−1 . Consequently, A has a countable number of eigenvalues {λj }∞ j =1 with corresponding eigenfunctions ϕj each with finite algebraic multiplicity mj . In the following, each eigenvalue λj is repeated according to its algebraic multiplicity j . Note also that there is a finite number of eigenvalues {λj }N j =1 with Re λj ≤0 and N are invariant with that the spaces Xu = lin span{ϕj }j =1 = PN H , Xs = (I − PN )H on Xu and is defined by respect to A . Here, PN is the algebraic projection of H  1 PN = (λI − A )−1 dλ, 2π i Γ0 where Γ0 is a closed curve which contains in interior the eigenvalues {λj }N j =1 . If we set Au = A |Xu , As = A |Xs , then we have σ (Au ) = {λj : Re λj ≤ 0}, σ (As ) = {λj : Re λj > 0}.

(6.10)

We recall that the eigenvalue λj is called semi-simple if its algebraic multiplicity g mj coincides with its geometric multiplicity mj . In particular, this happens if λj is simple and it turns out that the property of the eigenvalues λj to be all simple is generic (see [28], p. 164). The dual operator A ∗ has the eigenvalues λj with the eigenfunctions ϕj∗ , j = 1, . . . . For the time being, the following hypotheses will be assumed. (L1 ) The eigenvalues λj , j = 1, . . . , N, are semi-simple. This implies that A ϕj = λj ϕj , A ∗ ϕj∗ = λj ϕj∗ , j = 1, . . . , N,

(6.11)

or, equivalently, L ϕj = λj ϕj + ∇pj , L ∗ ϕj∗ = λj ϕj∗ + ∇pj∗ , j = 1, . . . , N,

(6.12)

and so we can choose systems {ϕj }, {ϕj∗ } in such a way that #

ϕj , ϕk∗

$

H

= δj k , j, k = 1, . . . , N.

(6.13)

The next hypothesis is a unique continuation type assumption on the normal ∂ϕj∗ , j = 1, . . . , N. derivatives ∂n  ∗ N ∂ϕj (L2 ) The system is linearly independent on Γ . ∂n j =1

6.1 The Main Stabilization Results

201

In the special case where the unstable spectrum A has only one distinct eigenvalue λ1 (eventually multivalued), hypothesis (L2 ) is implied by the following weaker assumption (L2 )

∂ϕ ∗ ∂n

is not identically zero on Γ,

where ϕ ∗ is any eigenfunction corresponding to the unstable eigenvalue λ¯ 1 . Since any linear combination of this system of eigenfunctions is again an eigenfunction corresponding to λ¯ 1 , it is clear that in this case (L2 ) is implied by (L2 ) . It is not known whether (L2 ) is always satisfied, but likewise hypothesis (L1 ), if Γ = ∂O, it holds, however, for “almost all a, b” in the generic sense (see [28, 39]). In Section 6.5, it is presented, however, a significant example, where (L2 ) holds. As regards hypothesis (L1 ), it is necessary here for the existence of the biorthogonal system {ϕj }, {ϕj∗ }, whose existence simplifies the construction of the stabilizable controller. Consider the feedback boundary controller u = η1Γ

N

3 4 μj PN y, ϕj∗ (φj + αn), H

j =1

(6.14)

where μj =

φj =

k + λj j = 1, . . . , N, k + λj − ν0 η N

=1

aj 

∂ϕ∗ , j = 1, . . . , N, ∂n

(6.15)

(6.16)

and aj ∈ C are chosen in such a way that N

=1

 aj Γ

$ ∂ϕ∗ ∂ϕ ∗i 1 # dx = δij − α, pi∗ 0 for i, j = 1, . . . , N. ∂n ∂n ν0

(6.17)

By virtue of hypothesis (L2 ), such a system {aj }N ,j =1 exists because the Gram matrix N   ∂ϕ∗ ∂ϕ ∗i    = Z0 ,   Γ ∂n ∂n i,=1 is not singular. By (6.13), (6.17), we have  $ ∂ϕ ∗i 1 # dx = δij − φj α, pi∗ 0 , i, j = 1, . . . , N. ∂n ν0 Γ Here {pi∗ } are given by (6.12) and ·, ·0 is the scalar product in L2 (Γ ). The first stabilization result refers to the Oseen-Stokes system (6.3).

(6.18)

202

6 Boundary Stabilization of Navier–Stokes Equations

Theorem 6.1. Assume that d = 2, 3, (L1 ), (L2 ), (6.7) hold, and that Re λj ≤ 0 for j = 1, . . . , N , Re λj > 0 for j > N. Let k > 0 sufficiently large and η > 0 be such that Re λj +ην0 +η2 ν02 (Re λj +k−ην0 )|k+λj −η|−2 > 0 for j = 1, . . . , N.

(6.19)

Then the feedback controller (6.14) stabilizes exponentially system (6.3), that is, the solution y to the closed loop system ∂y − ν0 Δy + (y · ∇)a + (b · ∇)y = ∇p in (0, ∞) × O, ∂t ∇ ·y =0 in (0, ∞) × O, N 3 4

y = η1Γ μj PN y, ϕj∗ (φj + αn) on (0, ∞) × ∂O,

(6.20)

H

j =1

satisfies for some γ > 0 the estimate |y(t)|H ≤ Ce−γ t |y(0)|H , ∀ t ≥ 0.

(6.21)

It is easily seen that (6.19) holds for k sufficiently large and η > 0 such that Re λj + ην0 > 0, ∀ j = 1, . . . , N . If λj are complex, then the controller (6.14) is complex valued too and plugged into system (6.3) leads to a real closed loop system in the state variables (Re y, Im y). In order to circumvent such a situation, we shall construct in Section 6.4 a real stabilizing feedback controller of the form (6.14) which has a similar stabilization effect. (See Theorem 6.3.) A problem of major interest is whether the controller u can be chosen almost normal, that is, its normal component un is close to the normal n. We have Corollary 6.1. There is a stabilizing controller u of the form (6.14), (6.16), with −1 ∗ ∗ 2 aij N i,j =1 = Z0 and α = λα , where λ ∈ C is arbitrary and α ∈ C (Γ ) satisfies (6.7) and |α ∗ (x)| = 0, a.e. x ∈ Γ.

(6.22)

The exact significance of this result is that, for each ε > 0, there is a Lebesgue measurable subset Γε such that m(Γ \ Γε ) ≤ ε and, on Γε , the normal component λα ∗ n of the controller u is = 0 and arbitrarily large with respect to the tangential component represented by φj . (Here m is the Lebesgue measure on Γ .) Proof of Corollary 6.1. We set    2 X = ψ ∈ L (Γ ); ψ dx = 0 , Γ ⎧ ⎫

 N ⎨ ⎬ 1 Y = γj pj∗ − pj∗ dx ; γj ∈ C , Y1 = X ∩ Y ⊥ . ⎩ ⎭ m(Γ ) Γ j =1

6.1 The Main Stabilization Results

203

4 3 Then Y1 = {ψ ∈ X; X, pj∗ = 0, ∀ j = 1, . . . , N}, L2 (Γ ) = Y ⊕ Y1 ⊕ C, and 0

so, any ψ ∈ C 2 (Γ ) can be written as ψ = α+

N

γj pj∗ −

1 m(Γ )

j =1

 Γ

pj∗ dx

+ γ0 ,

(6.23)

for some α ∈ Y1 and γj ∈ C, j = 0, 1, . . . , N. We note that there are ψ ∗ ∈ C 2 (Γ ) ∗ and γj ∈ C such that ∗

|ψ (x) −

N

γj∗

pj∗ (x) −

j =1

1 m(Γ )

 Γ

pj∗ dx



− γ0∗ | > 0, a.e. x ∈ Γ.

(6.24)

∈ C 2 (Γ ) and { Otherwise, for each ψ γj } N j =0 ⊂ C, there is a Lebesgue measurable subset Γ ⊂ Γ such that m(Γ ) > 0 and − ψ

N

j =1

γj pj∗ −

1 m(Γ )

 Γ

pj∗ dx



− γ0 = 0, in Γ ,

and γj ∈ C, which, by virtue of arbitrarity of ψ γj , is absurd. Indeed, it suffices to fix ∈ C 2 (Γ ) in such a way that j = 0, 1, . . . , N, and take ψ    

 N  1 ∗ ∗  | > sup inf |ψ γ j pj − pj dx − γ0   Γ m(Γ ) Γ Γ j =1  ∗ to arrive to a contradiction. Then, for the corresponding α in (6.23), denoted $ α , we # ∗ ∗ have (6.22). Now, we see that, for each λ ∈ C, α = λα , we have α, pi 0 = 0, −1 ∀ i = 1, . . . , N , and so, by (6.17) we have aij N i,j =1 = Z0 , as claimed.

Remark 6.1. The idea of the proof already used in the previous works mentioned in a finite differential system above is to decompose system (6.20) or the space H corresponding to unstable eigenvalue {λj }N and an infinite and stable differential j =1 system. For this, Hypothesis (L1 ) is not absolutely necessary (see [97]) and it was assumed only for convenience in order to get a simple diagonal form for the finitedimensional unstable system and implicitly for the stabilizing feedback. As regards (L2 ), one might suspect that it can be replaced by the weaker assumption (L2 )

eventually modifying the form of stabilizing controller. Remark 6.2. As easily follows from the proof in (6.14), the function α can be replaced by a system of functions {αj }N j =1 satisfying (6.7) with the corresponding modification of (6.17).

204

6 Boundary Stabilization of Navier–Stokes Equations

The above construction works also in more general case where Γ is a smooth part (not necessarily connected) of ∂O but in this case 1Γ should be replaced by a C 2 – function on ∂O with compact support in Γ and in condition (6.7) α should be replaced by 1Γ α. In the boundary stabilization of Navier–Stokes equation (6.3) with finite dimensional controllers due to compatibility of the boundary trace of state y with the boundary control there are two feasible regularity levels for the solution y, namely 1 1 (H 2 −ε (O))d for d = 2 and (H 2 +ε (O))d for d = 3. (See [28, 84].) However, in 1 3−D the high topological level (H 2 +ε (O))d is not appropriate for some technical reasons related to properties of the inertial term (y · ∇)y and so, contrary to the linear case, the treatment should be confined to d = 2. 1 , Z = (H 32 −ε (O))2 ∩ H ), Consider the Sobolev spaces W = (H 2 −ε (O))2 ∩ H 1 where 0 < ε < 2 , with the norms denoted by || · ||W , || · ||Z . The main stabilization result for the Navier–Stokes system (6.1) is Theorem 6.2 below. Theorem 6.2. Let d = 2 and a = b = ye . Then, under the assumptions of Theorem 6.1, the feedback boundary controller (6.14) stabilizes exponentially system (6.2) in a neighborhood W = {y0 ∈ W ; y0 W < ρ}. More precisely, the solution y ∈ C([0, ∞); W ) ∩ L2 (0, ∞; Z) to the closed loop system ∂y − ν0 Δy + (y · ∇)ye + (ye · ∇)y + (y · ∇)y = ∇p in (0, ∞) × O, ∂t N 3 4

y = η1Γ μj PN y, ϕj∗ (φj + αn) on (0, ∞) × ∂O,

(6.25)

H

j =1

satisfies for y(0) ∈ W and ρ sufficiently small y(t)W ≤ Ce−γ t y(0)W , ∀ t ≥ 0,

(6.26)

for some γ > 0. In particular, it follows that the boundary feedback controller u=η

N

3 4 μj PN (y − ye ), ϕj∗ (φj + αn) H

j =1

(6.27)

stabilizes exponentially the equilibrium solution ye to (6.1) in a neighborhood {y0 ∈ W ; y0 − ye W < ρ}.

6.2 Proof of Theorem 6.1 We set

  U = u ∈ (L2 (∂O))d ;

 u(x) · n(x)dx = 0 .

0

∂O

6.2 Proof of Theorem 6.1

205 1

Then, for k > 0 sufficiently large, there is a unique solution y ∈ (H 2 (O))d to the equation −ν0 Δy + (y · ∇)a + (b · ∇)y + ky = ∇p in O, ∇ · y = 0 in O, y = u on ∂O. We set y = Du and note that 1 1 D ∈ L((H s (∂O))d ∩ U 0 ; (H s+ 2 )O))d ), for s ≥ − · 2

In terms of A (see (6.9)) and of the Dirichlet map D, system (6.3) can be written as d y(t) + A (y(t) − Du(t)) = kΠ Du, t ≥ 0, dt y(0) = y0 .

Π

(6.28)

Equivalently,

d du z(t) + A z(t) = −Π D (t) − kDu(t) , t ≥ 0, dt dt z(0) = y0 − Du(0), z(t) = y(t) − Du(t), t ≥ 0.

(6.29)

(6.30)

In the following, we fix k > 0 sufficiently large and η > 0 such that (6.19) holds. In particular, for this choice of k and η, we also have λi + k − ν0 η = 0 for i = 1, 2, . . . , N.

(6.31)

We note first that in terms of z the controller (6.14) can be, equivalently, expressed as u(t) = η1Γ

N 3

PN z(t), ϕj∗

4 H

j =1

(φj + αn).

(6.32)

Indeed, by (6.30) and (6.32), we have u(t) = η1Γ

N 3

PN y(t), ϕj∗

j =1 N 3

j =1

(φj + αn)

H

u(t), D ∗ ϕj∗

−η1Γ where D ∗ is the adjoint of D.

4 4

0

(6.33) (φj + αn),

206

6 Boundary Stabilization of Navier–Stokes Equations

On the other hand, if we set ψ = D1Γ (φj + αn) and recall that L ∗ ϕi∗ − λi ϕi∗ = ∇pi∗ in O, ϕi∗ = 0 on ∂O, ∇ · ϕj∗ = 0, L ψ + kψ = ∇ p in O, ψ = 1Γ (φj + αn) on ∂O, ∇ · ψ = 0, we get by (6.18) via Green’s formula #

because n ·



 ∂ϕ ∗ ν0 ψ · ϕ ∗i dx = − (φj +αn)· i dx λi +k Γ ∂n O # $ ν0 1 ∗ α, pi 0 = − δij , ∀ i, j = 1, . . . , N, − k + λi λi +k

$ φj +αn, D ∗ ϕi∗ 0 =

(6.34)

∂ϕi∗ = 0, a.e. on ∂O (see [44]. Then, by (6.33), (6.34), we see that ∂n #

u(t), D ∗ ϕi∗

$ 0

=

# $ −ην0 PN y, ϕi∗ H k + λi − ν0 η

(6.35)

and, substituting into (6.33), we get (6.14) as claimed. Now, by (6.32) and (6.29), we obtain that *

N )

d dz ∗ PN +A z = −η z(t)−kz(t) , ϕj Π D(1Γ (φj +αn)), dt dt H j =1

(6.36)

z(0) = z0 = y0 − Du(0). It is convenient to decompose system (6.36) into a finite dimensional part corresponding to the unstable spectrum {λj , j = 1, . . . N} of A and an infinite dimensional one which corresponds to the stable spectrum {λj , j > N }. Namely, we write (6.36) as *

N )

dzu dz ∗ PN + Au zu = −ηPN − kz , ϕj Π D(1Γ (φj + αn)), dt dt H j =1

dzs + As zs = −η(I − PN ) dt

N )

PN

j =1



dz − kz , ϕj∗ dt

(6.37)

* H

Π D(1Γ (φj + αn)),

(6.38) where z = zu + zs , zu ∈ Xu , zs ∈ Xs and PN is the algebraic projection on Xu defined in Section 2.1. If we represent zu as zu =

N

j =1

zj ϕj ,

6.3 Proof of Theorem 6.2

207

and recall (6.35), we can rewrite (6.37) as zj + λj zj =

ην0 (z − kzj ), t ≥ 0. k + λj j

(6.39)

Equivalently, zj +

(k + λj )λj + kην0 zj = 0, j = 1, . . . , N. k + λj − ην0

(6.40)

By (6.19) we have Re

(k + λj )λj + kην0 > 0 for j = 1, . . . , N. k + λj − ην0

Then, by (6.39) there is γ0 > 0 such that |zj (t)| ≤ e−γ0 t |zj (0)|, j = 1, . . . , N.

(6.41)

On the other hand, by (6.39) we have

dzs + As zs = −η(I − PN ) (zj − kzj )Π D(1G (φj + αn)), dt N

(6.42)

j =1

and since e−As t L(H ,H ) ≤ Ce−γ1 t , ∀ t ≥ 0, for some γ1 > 0, we see by (6.40), (6.42) that |zs (t)|H ≤ C exp(−γ0 t)|zs (0)|H , ∀ t ≥ 0,

(6.43)

which together with (6.41) yields |z(t)|H ≤ C exp(−γ0 t)|z(0)|H , ∀ t ≥ 0. Now, recalling (6.30) and (6.32), we obtain the estimate (6.21), thereby completing the proof.

6.3 Proof of Theorem 6.2 System (6.2) with the feedback controller u = F y = η1Γ

N

j =1

3 4 μj PN y, ϕj∗ (φj + αn) H

208

6 Boundary Stabilization of Navier–Stokes Equations

can be written as (see(6.28)) dy + A (y − DF y) + By = kΠ DF y, t > 0, dt y(0) = y0 ,

Π

where By = Π ((y · ∇)y). Setting z = y − DF y we rewrite it as (see (6.36)) dz + A z + B((I − DF )−1 z) dt *

N )

d ∗ = −η Π D(1Γ (φj + αn)). PN z(t) − kz(t) , ϕj dt H j =1

We set, as in previous case, z = zu + zs , zu ∈ Xu , zs ∈ Xs and zu =

N

zj ϕj .

j =1

Recalling (6.39), (6.40), and (6.42), we get for j = 1, . . . , N zj +

3 4 (k + λj )λj + kην0 k + λj B((I − DF )−1 (z), ϕj∗ ) = 0 zj + H k + λj − ην0 k + λj − ην0

and zj − kzj = Kj (z), where Kj (z) = −

1 k + λj − ην0

 3 4  (k + λj )2 zj + (k + λj ) B((I − DF )−1 z, ϕj∗ ) . H

Then, we may write the above system as N 3 4

k + λj dzu + A u zu + B((I − DF )−1 (z), ϕj ) PN ϕj = 0, H dt k + λj − ην0

(6.44)

j =1

dzs + As zs + (I − PN )B((I − DF )−1 )z dt N

= −η(I − PN ) Kj (z)Π D(1Γ (φj + αn)). j =1

Here A u ∈ L(Xu , Xu ) is the operator defined by A u zu =

N

(k + λj )λj + kην0 zj ϕj . k + λj − ην0 j =1

(6.45)

6.3 Proof of Theorem 6.2

209

By virtue of (6.41), (6.43), both operators A u , As are exponentially stable on spaces Xu , respectively Xs and, therefore, so is the operator C (z) = A u zu + As zs , z = zu + zs . We set on the space H B(z) =

N

j =1

k + λj B((I − DF )−1 z), ϕj )H PN ϕj k + λj + ην0

+(I − PN )B((I − DF )−1 z) + η(I − PN )

N

Kj (z)Π D(1Γ (φj + αn))

j =1

and rewrite (6.44), (6.45) as dz + C z + B(z) = 0, t ≥ 0, z(0) = z0 = y0 − DF y0 . dt Equivalently, z(t) = e−tC z0 −



t

e−(t−s)C B(z(s))ds, t ≥ 0.

(6.46)

0

We recall that (see Section 1.3) B(z1 ) − B(z2 )W ≤ C(z1 Z + z2 Z )z1 − z2 Z , ∀z1 , z2 ∈ Z.

(6.47)

On the other hand, we have |Dz|s+ 1 ≤ 2

C 1 |z|s , ∀ z ∈ (H s (∂O))d , s ≥ − , k−c 2

where c, C are independent of k, and this yields, for s = 1 − ε, DF yZ ≤

Cη yZ , ∀ y ∈ Z. k−c

This implies that, for k large enough and η as in condition (6.19), the operator (I − DF )−1 is Lipschitz on the space Z and this implies that the local Lipschitz property (6.47) extends to the operator B. Then arguing exactly as in the proof of Theorem 5.1 in [28] (see, also, [44] and [45]) we conclude that the integral equation (6.46) has for ||z0 ||W ≤ ρ sufficiently small, a unique solution z ∈ C([0, ∞); W ) ∩ L2 (0, ∞; Z) which has the exponential decay ||z(t)||W ≤ Me−γ t ||z0 ||W , ∀t > 0 which completes the proof.

210

6 Boundary Stabilization of Navier–Stokes Equations

6.4 Real Stabilizing Feedback Controllers We shall construct here a real stabilizing feedback controller of the form (6.14). To do this we replace the system of functions {ϕj } by that obtained taking the real and imaginary parts of this one. Namely, we consider the following system of functions in the space H : ψ2j −1 = Re ϕj , ψ2j = Im ϕj , and similarly for the adjoint system ∗ ∗ ∗ ∗ ψ2j −1 = Re ϕj , ψ2j = Im ϕj .

For the sake of simplicity, we assume that all unstable eigenvalues λj , j = 1, . . . , N are simple mentioning, however, that the general case can be treated completely similar. u = lin span{ψj ; j = 1., ..N }. It should be mentioned that the We set X dimension of this space is still N and denote again by PN the algebraic projection u ⊕ X s and note that the real of H on X˜ u . Then we decompose the space as H = X s and X u and since X s + i X s = Xs we operator A leaves invariant both spaces X infer that the operator A s∗ = A |X s generates an exponential stable semigroup on s ⊂H . We set also A u∗ = A |X X u . We have A ψ2j −1 = Re λ2j −1 ψ2j −1 − Im λ2j −1 ψ2j , A ψ2j = Im λ2j −1 ψ2j −1 + Re λ2j −1 ψ2j ,

(6.48)

and, similarly for ψj∗ , i.e., ∗ ∗ ∗ A ∗ ψ2j −1 = Re λ2j −1 ψ2j −1 − Im λ2j −1 ψ2j , ∗ ∗ ∗ . ∗ A ψ2j = Im λ2j −1 ψ2j −1 + Re λ2j −1 ψ2j

(6.49)

Equivalently, ∗ ∗ ∗ ∗ L ∗ ψ2j −1 = Re λ2j −1 ψ2j −1 − Im λ2j −1 ψ2j + ∇p2j −1 , ∗ ∗ ∗ ∗ . ∗ L ψ2j = Im λ2j −1 ψ2j −1 + Re λ2j −1 ψ2j + ∇p2j

(6.50)

Under assumption (L2 ), the following real version of this hypothesis holds. (L2 )∗ The system {

∂ψj∗ ∂n ,

, j = 1, .., N} is linearly independent on Γ .

6.4 Real Stabilizing Feedback Controllers

211

Then, following (6.14) consider the real feedback controller  N 3

∗

PN y, ψj∗

u = η1Γ

4

−

j =1

N



#

Kj  PN y, ψ∗

$

(φj∗ + αn),

(6.51)

=1

where Kj  are made precise later on, φj∗ is of the form φj∗ =

N

aij∗

i=1

∂ψi∗ , j = 1, . . . , N, ∂n

(6.52)

and aij∗ are chosen in a such a way that (see (6.18)), )

∂ψi∗ ∗ , φj ∂n

*

1 # ∗ $ p , α 0 + δij , i, j = 1, . . . , N. ν0 i

=− 0

(6.53)

(As seen earlier, this choice is possible by virtue of (L2 )∗ .) Now, proceeding as in Section 5.1, we show that for Kj  suitably chosen the feedback controller (6.51) can be put in the form N 3

4 PN z, ψj∗ (φj∗ + αn),

u = η1Γ

(6.54)

j =1

where z is given by (6.30). Indeed, in terms of y, (6.54) can be written as (see (6.33)) u = η1Γ

N 3

N 3 4 4

PN y, ψj∗ (φj∗ + αn) − η1Γ u, D ∗ ψj∗ (φj∗ + αn).

j =1

0

j =1

(6.55)

This yields u, D ∗ ψi 0 = η −η

N 3

43 4 PN y, ψj∗ φj∗ + αn, D ∗ ψi∗

j =1

N 3

u, D

j =1

∗

ψj∗

0

4 3 0

φj∗

+ αn, D

∗

ψi∗

On the other hand, by (6.50), (6.54), we see that

4 0

(6.56) , i = 1, . . . , N.

212

6 Boundary Stabilization of Navier–Stokes Equations

3 4 ∗ (Re λ2i−1 + k) ψ2i−1 , D(1Γ (φj∗ + αn)) 3 4 ∗ , D(1 (φ ∗ + αn)) −Im λ2i−1 ψ2i Γ j ) ∗ * # ∗ $ ∂ψ2i−1 ∗ , φj + αn − p2i−1 , α 0 = −ν0 δ2i−1j , = −ν0 0 3∂n 4 ∗ , D(1 (φ ∗ + α n)) (Re λ2i−1 + k) ψ2i Γ k j 3 4 ∗ −Im λ2i−1 ψ2i−1 , D(1Γ (φj∗ + αn)) ) ∗ * # ∗ $ ∂ψ2i ∗ , φj + αn − p2i = −ν0 , α 0 = −ν0 δ2ij . ∂n 0

(6.57)

We set 3

4 ψi∗ , D(1Γ (φj∗ + αn)) = ηij ,

(6.58)

γi = ((Re λi + k)2 + (Im λi )2 )−1 , i, j = 1, . . . , N. This yields (Re λ2i−1 + k)η2i−1j + Im λ2i−1 η2ij = −ν0 δ2i−1j (Re λ2i−1 + k)η2ij − Im λ2i−1 η2i−1j = −ν0 δ2ij .

(6.59)

Then, by (6.56), (6.58), we obtain #

u, D

∗

$

ψi∗ 0

=η

N 3

PN y, ψj∗

4

ηij − η

j =1

N 3

4 u, D ∗ ψj∗ ηij .

j =1

0

Equivalently, N

j =1

N 3 4

$ # PN y, ψj∗ ηij . (δij + ηηij ) u, D ∗ ψj 0 = η

(6.60)

j =1

By (6.58), (6.59) we have η2i−1j = −ν0 γ2i−1 (Re λ2i−1 + k)δ2i−1j − Im λ2i−1 δ2ij ) η2ij = −ν0 γ2i−1 (Im λ2i−1 δ2i−1j + (Re λ2i−1 + k)δ2i ).

(6.61)

We set K = Kj  N j,=1 , where −1 K = η(δij + ν0 ηηij N × ||ηij ||. i,j =1 )

(6.62)

6.4 Real Stabilizing Feedback Controllers

213

By (6.61) we see that, for k sufficiently large, K is well defined. By (6.60), we have #

u, D ∗ ψj

$ 0

=

N

# $ Kj  PN y, ψ∗ , j = 1, . . . , N.

=1

Substituting the latter into (6.56), we see that u is of the form (6.51), where Kj  is given by (6.62). Now, we rewrite system (6.28) as (see (6.36), (6.54)) *

N )

dz dz PN + A z = −η − kz , ψj∗ Π D(1Γ (φj∗ + αn)) dt dt

(6.63)

j =1

with the corresponding projection on Xu∗ (see (6.37)) *

N )

dzu dz PN + A u∗ zu = −ηPN − kz , ψj∗ Π D(1Γ (φj∗ + αn)). dt dt

(6.64)

j =1

We set zu =

N

zi ψi and so, we may write (6.58) as

i=1 N

(bi zi + ai zi ) = −η

N N

bij (zi − kzi )ηj ,

(6.65)

j =1 i=1

i=1

$ $ # # where bi = ψi , ψ∗ , ai = A ψi , ψ∗ and ηj are given by (6.61). N N We set B = bi N i,=1 , A0 = ai i,=1 , E = ηj ,j =1 and rewrite (6.64) as Bz + A0 z + ηEB(z − kz) = 0, t ≥ 0,

(6.66)

where z = {zi }N j =1 . To study the stability of system (6.66) it is convenient to consider the limit case k = ∞. Taking into account (6.61), we see that for k → ∞ ηij k → δij and so system (6.66) reduces to Bz + A0 z + ηBz = 0, which is exponentially stable if η > 0 is sufficiently large because |Bz(t)| ≤ e

−ηt



t

|Bz(0)| + 0

e−η(t−s) |A0 z(s)|ds

214

6 Boundary Stabilization of Navier–Stokes Equations

and B is invertible as consequence of the independence of the systems {ψj }N j =1 and N ∗ {ψj }j =1 . This implies that system (6.66) and, consequently, (6.63) is exponentially stable for k and η sufficiently large. Then, we have the following real version of Theorem 6.1. Theorem 6.3. Under assumptions (L1 ), (L2 ) and (6.7) for k and η sufficiently large, there is a boundary feedback controller u∗ of the form (6.62) which stabilizes exponentially system (6.3).

6.5 An Example to Stabilization of a Periodic Flow in a 2D Channel The previous results remain true for the Navier–Stokes system in a 2D channel O = {(x, y) ∈ R × (0, 1)} with periodic condition in direction x. We illustrate this on the standard problem of laminar flows in a two-dimensional channel with the walls located at y = 0, 1. (See, e.g., [1, 16, 25, 28].) We assume that the velocity field (u(t, x, y), v(t, x, y)) and the pressure p(t, x, y) are 2π periodic in x. Then, the dynamic of flow is governed by the system ut − ν0 Δu + uux + vuy = px , x ∈ R, y ∈ (0, 1), vt − ν0 Δv + uvx + vvy = py , x ∈ R, y ∈ (0, 1), ux + vy = 0, u(t, x + 2π, y) ≡ u(t, x, y), v(t, x + 2π, y) ≡ v(t, x, y), y ∈ (0, 1). (6.67) Consider a steady-state flow with zero vertical velocity component, that is, (U (x, y), 0). We have U (x, y) = U (y) = c(y 2 − y), ∀ y ∈ (0, 1), and take c = − 2νa0 , a ∈ R+ . The linearization of (6.67) around the steady-state flow (U (y), 0) leads to the following system ut − ν0 Δu + ux U + vU = px , y ∈ (0, 1), x, t ∈ R, vt − ν0 Δv + vx U = py , ux + vy = 0, u(t, x + 2π, y) ≡ u(t, x, y), v(t, x + 2π, y) ≡ v(t, x, y).

(6.68)

A convenient way to treat this system is to represent u, v as Fourier series. Let us briefly recall this standard procedure. Denote by L2π (Q), Q = (0, 2π ) × (0, 1) the space of all the functions u ∈ L2loc (R × (0, 1)) which are 2π -periodic in x. These functions are characterized by their Fourier series u(x, y) = a0 (y) +

k=0

ak (y)e

ikx

, ak = a¯ −k ,



1

|ak |2 dy < ∞.

k∈Z 0

Similarly, there are defined the Sobolev spaces Hπ1 (Q), Hπ2 (Q). We set H = {(u, v) ∈ (L2π (Q))2 ; ux + vy = 0, v(x, 0) = v(x, 1) = 0}.

6.5 An Example to Stabilization of a Periodic Flow in a 2D Channel

215

If ux + vy = 0, then the trace of v at y = 0, 1 is well defined as an element of H −1 (0, 2π ) × H −1 (0, 2π ). We also set V = {(u, v) ∈ H ∩ Hπ1 (Q); u(x, 0) = u(x, 1) = v(x, 0) = v(x, 1) = 0}. The space H can be defined equally as 

H = u= uk (y)eikx , v = vk (y)eikx , vk (0) = vk (1) = 0,



k∈Z 1

k∈Z 0

k∈Z

 (|uk | + |vk | )dy < ∞, ikuk (y) + vk (y) = 0, a.e. y ∈ (0, 1), k ∈ Z . 2

2

Let Π : L2π (Q) → H be the Leray projector and A : D(A )⊂H → H the operator A (u, v) = Π {−ν0 Δu + ux U + vU , −ν0 Δv + vx U }, ∀ (u, v) ∈ D(A ) = (H 2 ((0, 2π ) × (0, 1)) ∩ V .

(6.69)

We associate with (6.69) the boundary value conditions u(t, x, 0) = u0 (t, x), u(t, x, 1) = u1 (t, x), t ≥ 0, x ∈ R, v(t, x, 0) = v 0 (t, x), v(t, x, 1) = v 1 (t, x), t ≥ 0, x ∈ R,

(6.70)

and, for k ∗ > 0 sufficiently large, we consider the Dirichlet map D : X → L2π (Q) defined by D(u∗ , v ∗ ) = ( u, v ), −ν0 Δ u + ux U + v U + k ∗ u = px , x ∈ R, y ∈ (0, 1), v + vx U + k ∗ v = py , x ∈ R, y ∈ (0, 1), −ν0 Δ vy = 0, u(x + 2π, y) = u(x, y), v (x + 2π, y) = v (x, y), ux + v (x, y) = v ∗ (x, y), y = 0, 1. u(x, y) = u∗ (x, y),

(6.71)

 Here X = (u∗ , v ∗ ) ∈ L2 ((0, 2π ) × ∂(0, 1)); u∗ (x + 2π, y) = u∗ (x, y), v ∗ (x +  2π  2π  2π, y) = v ∗ (x, y), 0 v ∗ (x, 0)dx = 0 v ∗ (x, 1)dx . Then system (6.69), (6.70) can be written as d y(t) + A (y(t) − DU ∗ (t)) = k ∗ Π DU ∗ (t), t ≥ 0, dt y(0) = (u0 , v0 ),

Π

(6.72)

where y = (u, v), U ∗ = (u∗ , v ∗ ).We denote again by A the extension of A and by λj , ϕj the eigenvalues and corresponding on the complexified space H

216

6 Boundary Stabilization of Navier–Stokes Equations

eigenvectors of the operator A . By ϕj∗ , we denote the eigenvector to the dual operator A ∗ . Written into this form, which is exactly (6.28), it is clear that one can apply Theorem 6.1 provided hypotheses (L1 ), (L2 ) are satisfied for A . We check below the unique continuation hypothesis (L2 ) which has also an interest in itself. Lemma 6.1. Assume that all eigenvalues λj , j = 1, 2, . . . , N, are semi-simple. Then we have ∂ϕj (x, y) ≡ 0, x ∈ (0, 2π ), y = 0, 1, ∂n ∂ϕj∗ (x, y) ≡ 0, x ∈ (0, 2π ), y = 0, 1. ∂n

(6.73) (6.74)

Proof. If we represent ϕj = (uj , v j ), then (6.73) reduces to     ∂ j  ∂ j   +  > 0, x ∈ (0, 2π ), y = 0, 1. v u (x, y) (x, y)  ∂y   ∂y 

(6.75)

We set λ = λj and ϕj = (u, v). This means that, if λ is semisimple, then −ν0 Δu + ux U + vU = λu + px , x ∈ R, y ∈ (0, 1), −ν0 Δv + vx U = λv + py , x ∈ R, y ∈ (0, 1), ux + vy = 0, u(x + 2π, y) = u(x, y), v(x + 2π, y) = v(x, y).

(6.76)

If we represent u, v, p as Fourier series with coefficients uk , vk , pk we reduce (6.76) to the system −ν0 u

k + (ν0 k 2 + ikU )uk + U vk = ikpk + λuk , y ∈ (0, 1), −ν0 vk

+ (ν0 k 2 + ikU )vk = pk + λvk , ikuk + vk = 0 in (0, 1), uk (0) = uk (1) = 0, vk (0) = vk (1) = 0. Equivalently, −ν0 vkiv +(2ν0 k 2 +ikU )vk

−k(ν0 k 3 +ik 2 U +iU

)vk −λ(vk

− k 2 vk ) = 0, y ∈ (0, 1), vk (0) = vk (1) = 0, vk (0) = vk (1) = 0, ∀ k = 0. (6.77) Now, let us check (6.74) or, equivalently, (6.76). We have

eikx ∂ u(x, y) = −i v

(y), ∀ x, y ∈ 0, 1, ∂n k k k=0

6.5 An Example to Stabilization of a Periodic Flow in a 2D Channel

217

and so (6.76) reduces to |vk

(0)| + |vk

(1)| > 0 for all k.

(6.78)

Assume that vk

(0) = vk

(1) = 0 for all k and lead from this to a contradiction. To this end we set Wk = vk

− k 2 vk and rewrite (6.78) as −ν0 Wk

+ (ν0 k 2 + ikU − λ)Wk = ikU

vk in (0, 1), Wk (0) = Wk (1) = 0.

(6.79)

If we multiply (6.79) by W k , integrate on (0, 1), and take the real part, we obtain that 

1

0

(ν0 |Wk |2 + (ν0 k 2 − Re λ)|Wk |2 )dy = 0, ∀ k

and since Re λ = Re λj ≤ 0 for all j = 1, . . . , N , we get Wk ≡ 0, and so vk ≡ 0. The contradiction we arrived at proves (6.78) and (6.73). To prove (6.74) one proceeds similarly with dual system of eigenfunction but since the proof is more delicate we refer to [97]. 

We suspect that the same argument applies to prove hypothesis (L2 ), that is, N is linearly independent on ∂O but this remains to be done. Of course,

∂ϕj∗ ∂n

j =1

if all ϕj∗ , j = 1, . . . , N , are eigenvectors corresponding to the same eigenvalue, the independence follows by Lemma 6.1. Then, following the general case (6.14), we can design a feedback controller ( u, v ) for system (6.68). (In terms of (6.70), u = (u0 , u1 ), v = (v 0 , v 1 ).) We set ϕj∗ = (u∗j , vj∗ ), j = 1, . . . , N, where ϕj∗ are eigenvectors of the dual operator A ∗ with corresponding eigenvalues λj and Re λj < 0 for j = 1, . . . , N. (Recall that the eigenvalues λj are repeated according to their multiplicity.) We consider the boundary feedback controller u(t, x, y) = η

N

μj vj (t)φj1 (x, y), x ∈ R, y = 0, 1,

j =1

v (t, x, y) = η

N

μj vj (t)(φj2 (x, y) + aH (y)), x ∈ R, y = 0, 1,

=1  j2π vj (t) = (u(t, x, y)u¯ ∗j (x, y) + v(t, x, y)v¯j∗ (x, y))dx dy 0

= (uk (t, y)(u¯ ∗j )k (y) + vk (t, y)(v¯k∗ )k (y)). k=0

(6.80)

218

6 Boundary Stabilization of Navier–Stokes Equations

Here a is an arbitrary constant, H is a smooth function such that H (0) = −1, H (1) = 1, μj are defined as in (6.15), φji , i = 1, 2, are of the form (see (6.16)) N N   ∂ϕ ∗ aij χ 1i , φj2 = aij χ 2i , where (χ 1i , χ 2i ) = χ i , χ i (x, 0) = − ∂yi (x, 0), φj1 = j =1

χ i (x, 1) =

∂ϕi∗ ∂y (x, 1)

i=1

and aij are chosen as in (6.17). By Theorem 6.1, we have

Corollary 6.2. If there is at most one unstable semi-simple eigenvalue for (6.76) (eventually multiple), then, for each a ∈ R and η > 0 suitably chosen, the feedback boundary controller (6.80) stabilizes exponentially system (6.68). We note also that, by Theorem 6.2, the feedback controller (6.80) stabilizes the Navier–Stokes equation (6.67).

6.6 Notes on Chapter 6 This chapter is based on the author work [29]. The method developed here was extended by I. Munteanu [100] to the boundary stabilization of 2-D periodic MHD equations. It should be said that the design of a normal stabilizable feedback controller for general domains is of crucial importance in fluid flow control and the results presented here only partially respond to this demanding objective.

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Index

Symbols C0 -semigroup, 20 m-accretive, 2

eigenvector, 131 exact null controllability, 59 exactly null controllable, 43

A accretive, 2 algebraic multiplicity, 134 algebraic Riccati equation, 25, 147 approximate controllability, 69, 95

F feedback controller, 25, 157 feedback stabilizable, 25

B boundary controller, 183 Brownian motion, 89

C Carleman inequality, 24 Cauchy problem, 1, 3 closed loop system, 25, 178, 182 controller, 22

D Dirichlet map, 180, 205

E eigenfunction, 131 eigenvalue, 131 semisimple, 132, 135 simple, 132

G Gaussian noise, 89

I infinitesimal generator, 20 internal controller, 44 Itô formula, 91

K Kakutani theorem, 62 Kalman’s rank condition, 101 Kolmogorov operator, 83

L Leray projector, 15, 199, 215 local null controllability, 70

© Springer International Publishing AG, part of Springer Nature 2018 V. Barbu, Controllability and Stabilization of Parabolic Equations, Progress in Nonlinear Differential Equations and Their Applications 90, https://doi.org/10.1007/978-3-319-76666-9

225

226 M maximal monotone, 4 mild solution, 3, 20, 22 Minkowski functional (gauge), 81

N normal derivative, 5 null controllable, 22

O observability inequality, 27, 43

P phase field system, 73, 169

Q quasi-m-accretive, 2 quasi-accretive, 2

Index R reaction–diffusion system, 100 recession cone, 81 Riccati equation, 25

S self-organized criticality, 104 semi-simple eigenvalue, 216 Sobolev embedding theorem, 53, 68, 111 stabilizable, 72 stabilizing open loop controller, 25 stochastic equation, 89, 186 Stokes equation, 14 Stokes–Oseen operator, 198 strong solution, 3, 16

W Wiener process, 83, 186, 192

Y Yosida approximation, 4, 7, 149