Boundary Stabilization of Parabolic Equations (Progress in Nonlinear Differential Equations and Their Applications) [1st ed. 2019] 3030110982, 9783030110987

Table of contents :
Preface
Contents
Acronyms
1 Preliminaries
1.1 Notation and Theoretical Results
2 Stabilization of Abstract Parabolic Equations
2.1 Presentation of the Abstract Model
2.2 The Design of the Boundary Stabilizer
2.2.1 The Case of Mutually Distinct Unstable Eigenvalues
2.2.2 The Semisimple Eigenvalues Case
2.3 A Numerical Example
2.4 Comments
3 Stabilization of Periodic Flows in a Channel
3.1 Presentation of the Problem
3.2 The Stabilization Result
3.2.1 The Feedback Law and the Stability of the System
3.3 Design of a Riccati-Based Feedback
3.4 Comments
4 Stabilization of the Magnetohydro- dynamics Equations in a Channel
4.1 The Magnetohydrodynamics Equations of an Incompressible Fluid
4.2 The Stabilizing Proportional Feedback
4.3 Comments
5 Stabilization of the Cahn–Hilliard System
5.1 Presentation of the Problem
5.1.1 Stabilization of the Linearized System
5.2 Comments
6 Stabilization of Equations with Delays
6.1 Presentation of the Problem
6.2 Stability of the Linearized System
6.3 Feedback Stabilization of the Nonlinear System (6.1)
6.4 Comments
7 Stabilization of Stochastic Equations
7.1 Robustness in the Presence of Noise Perturbation of the Boundary Feedback
7.2 Stabilization of the Stochastic Heat Equation on a Rod
7.2.1 Mild Formulation of the Solution and Proof of the Main Result
7.3 Stabilization of the Stochastic Burgers Equation
7.4 Stabilization by Discrete-Time Feedback Control
7.5 Comments
8 Stabilization of Unsteady States
8.1 Presentation of the Problem
8.2 The Stabilization Result and Applications
8.2.1 Observer Design
8.2.2 Applications
8.3 Comments
9 Internal Stabilization of Abstract Parabolic Systems
9.1 Presentation of the Problem
9.2 Stabilization of the Full Nonlinear Equation (9.9)
9.3 The Design of a Real Stabilizing Feedback Controller
9.4 Comments
References
Index

Citation preview

Progress in Nonlinear Differential Equations and Their Applications Subseries in Control 93

Ionut¸ Munteanu

Boundary Stabilization of Parabolic Equations

Progress in Nonlinear Differential Equations and Their Applications PNLDE Subseries in Control Volume 93

Editors Jean-Michel Coron, Université Pierre et Marie Curie, Paris, France Editorial Board Viorel Barbu, Facultatea de Matematică, Universitatea “Alexandru Ioan Cuza” din, Iaşi, Romania Piermarco Cannarsa, Department of Mathematics, University of Rome “Tor Vergata”, Roma, Italy Karl Kunisch, Institute of Mathematics and Scientific Computing, University of Graz, Graz, Austria Gilles Lebeau, Laboratoire J.A. Dieudonné, Université de Nice Sophia-Antipolis, Nice, France Tatsien Li, School of Mathematical Sciences, Fudan University, Shanghai, China Shige Peng, Institute of Mathematics, Shandong University, Jinan, China Eduardo Sontag, Department of Electrical and Computer Engineering, Northeastern University, Boston, MA, USA Enrique Zuazua, Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain

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

Ionuţ Munteanu

Boundary Stabilization of Parabolic Equations

Ionuţ Munteanu Faculty of Mathematics Alexandru Ioan Cuza University Iaşi, Romania

ISSN 1421-1750 ISSN 2374-0280 (electronic) Progress in Nonlinear Differential Equations and Their Applications PNLDE Subseries in Control ISBN 978-3-030-11098-7 ISBN 978-3-030-11099-4 (eBook) https://doi.org/10.1007/978-3-030-11099-4 Library of Congress Control Number: 2018966441 Mathematics Subject Classification (2010): 35K05, 93D15, 93B52, 93C20, 47F05, 60H15, 35R09, 35Q30, 35Q92 © Springer Nature Switzerland AG 2019 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. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To my beloved daughter, Anastasia

Preface

In recent years, many researchers have been working on designing stabilizers in different technological areas such as surface design for controlling aircraft, voltage regulators in electronics, camera stabilizers, chemical substances to prevent unwanted change in the state of another substance, different types of food preservatives, medical processes for preventing shock in sick or injured people, and mood stabilizers, among many others. In this book, we will treat this subject from a mathematical point of view. More exactly, we will consider different models from different fields such as fluid flows modeled by the Navier–Stokes equations; electrically conducted fluid flows modeled by the magnetohydrodynamic equations; phase separation modeled by the Cahn– Hilliard equations; different cases of semilinear heat equations arising from biology, chemistry, or population dynamics as well as their stochastic versions. Then we will address the problem of boundary stabilization associated with these models. All these models can be combined under the rubric of abstract parabolic-like equations, namely equations whose linear parts are generated by analytic C0 -semigroups. That is why, in Chap. 2, we consider the boundary stabilization problem associated to abstract parabolic-like equations and develop an algorithm to design proportionaltype boundary feedback stabilizers, of finite-dimensional structure, expressed in a very simple form, that are easy to manipulate in numerical simulations. It should be emphasized that no rigorous stabilization theory is possible without a unique continuation theory for the eigenfunctions of the linear operator obtained from the linearization of the equation around the target solution. So, once a model, such as those above, can be formulated in a parabolic abstract form, the boundary stabilizing control design method can be applied, provided that a unique continuation property of the eigenfunctions is established. This provides the power of this control design technique; namely, it can be applied to a wide range of models. But it requires that we prove a priori a unique continuation result that relies on some advanced results and techniques involving both the theory of parabolic-like equations and functional analysis.

vii

viii

Preface

We mention that in the literature, there are also other notable results concerning the boundary stabilization of parabolic equations, and though we mention some basic references and offer a brief presentation of other significant works in the field, we have not presented them in detail. We confine ourselves to the proportional-type feedback design only, which is based on the spectral decomposition of a linearized system in stable and unstable systems, thereby omitting other important results in the literature. This book was written with the goal of presenting in detail new results related to an algorithm for the design of proportional-type feedback forms, which enabled us to obtain some of the first results in areas such as boundary stabilization of the Cahn–Hilliard system, and trajectories for the semilinear heat equation and even for stochastic partial differential equations. These ideas are still being developed, and one might expect in the future to obtain other spectacular achievements. Besides stabilization, the robustness of stabilizable feedback under stochastic perturbations is also discussed. The form of the feedback is based on the eigenfunctions of the linear operator, and we have tried to use a minimal set of them. The reader is assumed to have a basic knowledge of linear functional analysis, linear algebra, probability theory, and the general theory of elliptic, parabolic, and stochastic equations. Most of this is reviewed in Chap. 1. The material included in this book (excepting the comments on the references) represents the original contribution of the author and his coworkers. The author is indebted to Prof. Viorel Barbu for suggesting to us, five years ago, that we develop some of his own earlier ideas on constructing proportional-type feedback forms, which led to the conception of this entire book. We are indebted to him as well for encouraging us to write this book and for useful discussions, pertinent observations and suggestions, and unstinting support and guidance in the writing of this book. Many thanks go to Hanbing Liu, and special thanks to my parents for their love and support. Also, the author is indebted to Mrs. Elena Mocanu, from the Institute of Mathematics Iaşi, who assisted in the typesetting of this text. Iaşi, Romania August 2018

Ionuţ Munteanu

Contents

1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Notation and Theoretical Results . . . . . . . . . . . . . . . . . . . . . . . . .

1 1

2 Stabilization of Abstract Parabolic Equations . . . . . . . . . . . . . . 2.1 Presentation of the Abstract Model . . . . . . . . . . . . . . . . . . . 2.2 The Design of the Boundary Stabilizer . . . . . . . . . . . . . . . . 2.2.1 The Case of Mutually Distinct Unstable Eigenvalues . 2.2.2 The Semisimple Eigenvalues Case . . . . . . . . . . . . . . 2.3 A Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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19 19 25 26 37 41 45

3 Stabilization of Periodic Flows in a Channel . . . . . . . 3.1 Presentation of the Problem . . . . . . . . . . . . . . . . . 3.2 The Stabilization Result . . . . . . . . . . . . . . . . . . . 3.2.1 The Feedback Law and the Stability of the 3.3 Design of a Riccati-Based Feedback . . . . . . . . . . 3.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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49 49 51 63 71 75

....... ....... ....... System . ....... .......

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4 Stabilization of the Magnetohydrodynamics Equations in a Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Magnetohydrodynamics Equations of an Incompressible Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Stabilizing Proportional Feedback . . . . . . . . . . . . . . . . . 4.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

....

77

.... .... ....

77 86 91

5 Stabilization of the Cahn–Hilliard System . . . . . 5.1 Presentation of the Problem . . . . . . . . . . . . . . 5.1.1 Stabilization of the Linearized System 5.2 Comments . . . . . . . . . . . . . . . . . . . . . . . . . .

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. 93 . 93 . 96 . 106

ix

x

Contents

6 Stabilization of Equations with Delays . . . . . . . . . 6.1 Presentation of the Problem . . . . . . . . . . . . . . . 6.2 Stability of the Linearized System . . . . . . . . . . 6.3 Feedback Stabilization of the Nonlinear System 6.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . .

....... ....... ....... (6.1) . . . .......

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109 109 113 120 125

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7 Stabilization of Stochastic Equations . . . . . . . . . . . . . . . . . 7.1 Robustness in the Presence of Noise Perturbation of the Boundary Feedback . . . . . . . . . . . . . . . . . . . . . 7.2 Stabilization of the Stochastic Heat Equation on a Rod . 7.2.1 Mild Formulation of the Solution and Proof of the Main Result . . . . . . . . . . . . . . . . . . . . . . 7.3 Stabilization of the Stochastic Burgers Equation . . . . . . 7.4 Stabilization by Discrete-Time Feedback Control . . . . . 7.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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140 150 162 165

8 Stabilization of Unsteady States . . . . . . . . . . 8.1 Presentation of the Problem . . . . . . . . . . . 8.2 The Stabilization Result and Applications 8.2.1 Observer Design . . . . . . . . . . . . . 8.2.2 Applications . . . . . . . . . . . . . . . . 8.3 Comments . . . . . . . . . . . . . . . . . . . . . . .

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171 171 172 176 180 183

9 Internal Stabilization of Abstract Parabolic Systems . . . 9.1 Presentation of the Problem . . . . . . . . . . . . . . . . . . . . 9.2 Stabilization of the Full Nonlinear Equation (9.9) . . . . 9.3 The Design of a Real Stabilizing Feedback Controller 9.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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187 187 195 203 205

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

Acronyms

C N R Rd O @O Q R k : kX X0 ð; ÞH hx; yid k kd LðX; YÞ B DðAÞ RðAÞ 1C sign C k ðOÞ C0k ðOÞ DðOÞ dk u ðkÞ dtk ; u D 0 ðOÞ CðOÞ Lp ðOÞ

The set of all complex numbers The set of all natural numbers The real line ð1; 1Þ The d-dimensional Euclidean space An open subset of Rd The boundary of O ¼ O  ð0; TÞ ¼ @O  ð0; TÞ, where 0\T\1 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 2 Rd The Euclidean norm in Rd The space of linear continuous operators from X to Y The adjoint of the operator B 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 ¼ x=k xkX if x 6¼ 0, sign 0 ¼ fx; k xkX  1g The space of real-valued functions on O that are continuously differentiable up to order k, k  1 The subspace of functions in C k ðOÞ with compact support in O The space C01 ðOÞ The derivative of order k of the function u : ½a; b ! X The dual of DðOÞ (i.e., the space of distributions on O) The space of continuous functions on O The space of p-summable functions u : O ! R endowed with the R 1 norm jujp ¼ ð O juðxÞjp dxÞp ; 1  p\1

xi

xii

W m; p ðOÞ W0m; p ðOÞ H k ðOÞ; H0k ðOÞ Lp ða; b; XÞ Cð½a; b; XÞ W 1; p ð½a; b; XÞ n @u @n

Acronyms

The Sobolev space fu 2 Lp ðOÞ; Da u 2 Lp ðOÞ; jaj  m; 1  p  1g The closure of C01 ðOÞ in the norm of W m; p ðOÞ The spaces W k; 2 ðOÞ and W0k; 2 ðOÞ, respectively The space of p-summable functions from ða; bÞ to X, 1  p  1; 1  a\b  1 The space of X-valued continuous functions on ½a; b p The Sobolev space fu 2 ACð½a; b; XÞ; du dt 2 L ðða; bÞ; XÞg The outward normal to O the normal derivative of the function u : O ! R

Chapter 1

Preliminaries

For easy reference, we collect here some standard notation and results in functional analysis and partial differential equations that will be used throughout this work.

1.1 Notation and Theoretical Results Functional Spaces Here O will stand for a bounded domain in Rd , d ∈ N \ {0} , i.e., a nontrivial connected open subset of Rd such that    O ⊂ x ∈ Rd : x12 + · · · + xd2 ≤ R , for some R > 0. We denote by O its closure and by ∂O its boundary. We always assume that the boundary of O is piecewise smooth (e.g., the boundary of a polygon or a sphere). In many cases, the boundary of O will be split into two parts as ∂O = Γ1 ∪ Γ2 , where Γ1 has nonzero surface measure. Consider H to be a Hilbert space over C, with the inner product ·, ·. Then the system {x1 , x2 , . . . , xN } ⊂ H is linearly independent in H if and only if the Gram matrix N  G := xi , xj  i,j=1 is invertible. It is known that for every system in H , the corresponding Gramian is a positive semidefinite Hermitian matrix, i.e., G = G T and Gc, cN ≥ 0, ∀c ∈ CN , © Springer Nature Switzerland AG 2019 I. Munteanu, Boundary Stabilization of Parabolic Equations, Progress in Nonlinear Differential Equations and Their Applications 93, https://doi.org/10.1007/978-3-030-11099-4_1

1

2

1 Preliminaries

where G T denotes the transpose of the matrix G, ·, ·N denotes the scalar product in CN , and z denotes the complex conjugate of z ∈ C (for more details on Gramians, see, e.g., [69]). Now consider a normed vector space (X , · ) over R. A sequence (un )n ⊂ X is called a Cauchy sequence if for all ε > 0, there exists Nε ∈ N such that

un − um < ε, ∀n, m ≥ Nε . We say that X is a Banach space if it is a complete normed vector space, i.e., a normed space in which every Cauchy sequence converges to a limit point in X . Next let X be a nonempty set. A set F of subsets of X is a σ -algebra if: (a) the empty set belongs to F ; / F} belongs to F for all F ∈ F ; (b) the complement F c := {x ∈ X : x ∈  Fj belongs to F for Fj ∈ F , j ∈ N. (c) the union j∈N

A measure μ on a measurable space (X , F ) is a mapping from F to R+ ∪ {∞} such that: (a) the empty has measure zero;  set  Fj = μ(Fj ) if Fj ∈ F are mutually disjoint. (b) μ j∈N

j∈N

Together, the pair (X , F , μ) forms a measure space. A set F ∈ F is called a null set if μ(F) = 0, and a measure space is said to be complete if all subsets of null sets belong to F . (Every measure space (X , F , μ) can be extended to a complete measure space by including all subsets of null sets). The Borel σ -algebra of a topological space Y , denoted by B(Y ), is the smallest σ -algebra containing all open subsets of Y . The usual notion of volume of subsets of Rd gives rise to Lebesgue measure, thereby introducing the notion of Lebesgue measure space. A function u : X → Y is F -measurable if the pullback set u−1 (G) is in F for every G ∈ B(Y ). A function s : X → Y is simple if there exist sj ∈ Y and Fj ∈ F for j = 1, 2, . . . , N with μ(Fj ) < ∞ such that s(x) =

N

sj 1Fj (x), x ∈ X ,

j=1

where 1F is the indicator function, i.e.,  1F (x) :=

1, x ∈ F, 0, x ∈ / F.

The integral of a simple function s with respect to the measure space (X , F , μ) is

s(x)d μ(x) := X

N j=1

sj μ(Fj ).

1.1 Notation and Theoretical Results

3

We say that a measurable function u is integrable with respect to μ if there exist simple functions {un }n∈N such that un (x) → u(x) as n → ∞ for almost all x ∈ X , and un is a Cauchy sequence in the sense that for all ε > 0,

un (x) − um (x) Y d μ(x) < ε X

for all n, m sufficiently large. If u is integrable, we define



u(x)d μ(x) := lim

n→∞ X

X

un (x)d μ(x).

If F ∈ F , we define



u(x)d μ(x) := F

u(x)1F (x)d μ(x). X

The above definition can be extended to C-valued functions as follows: letting u = u1 + iu2 , with u1,2 R-valued functions, we say that u is measurable (integrable) if both u1 , u2 are measurable (integrable), and define





u(x)d μ(x) := X

u1 (x)d μ(x) + i X

u2 (x)d μ(x). X

Theorem 1.1 (Dominated convergence) Consider a sequence of measurable functions un : X → Y such that un (x) → u(x) in Y as n → ∞ for almost all x ∈ X . If there is a real-valued integrable function U such that un (x) Y ≤ |U (x)|, ∀n ∈ N and almost all x ∈ X , then



lim un (x)d μ(x) = lim un (x)d μ(x) = u(x)d μ(x). n→∞ X

X n→∞

X

Let (Xk , Fk , μk ), k = 1, 2, be two measure spaces. We set F1 × F2 for the smallest σ -algebra containing all sets F1 × F2 for Fk ∈ Fk . The product measure μ1 × μ2 on F1 × F2 is defined by (μ1 × μ2 )(F1 × F2 ) := μ1 (F1 )μ2 (F2 ). Theorem 1.2 (Fubini) Suppose that (Xk , Fk , μk ), k = 1, 2, are σ -finite measure spaces and consider a measurable function u : X1 × X2 → Y . If



X2



u(x1 , x2 ) Y d μ1 (x1 ) d μ2 (x2 ) < ∞, X1

then u is integrable with respect to the product measure μ1 × μ2 and

4

1 Preliminaries





u(x1 , x2 ) Y d μ1 (x1 ) d μ2 (x2 ) X X

2  1

u(x1 , x2 ) Y d μ2 (x2 ) d μ1 (x1 ). =

X1 ×X2

u(x1 , x2 )d (μ1 × μ2 )(x1 , x2 ) =

X1

X2

Next, by Lp (O), 1 ≤ p ≤ ∞, we denote the standard space of Lebesgue integrable L functions on O endowed with the norm p



f

Lp (O )

:=

|f (x)| dx p

O

p1

.

One can easily see that Lq (O) ⊂ Lp (O) for 1 ≤ p ≤ q. For the particular case p = 2, L2 (O) is a Hilbert space, with the inner product

f , gL2 (O ) =

O

f (x)g(x)dx.

Next, we set D(O) = C0∞ (O) for the space of infinitely differentiable functions with compact support defined in O. Then denote by D (O) its dual, known as the space of distributions on O. Based on this, one can introduce the Sobolev spaces W 1,p (O), 1 ≤ p ≤ ∞, defined as   ∂u p p u ∈ L (O) : ∈ L (O), i = 1, 2, . . . , d . ∂xi Here the partial derivatives, like ∂x∂ i above, are taken in the sense of distributions, i.e., given a multi-index α ∈ Nd of order |α| = α1 + α2 + · · · + αd , we use the notation Dα u := and define



α

O

|α|

D uφdx = (−1)

∂ |α| u . . . ∂xdαd

∂x1α1

O

uDα φdx, ∀φ ∈ D(O).

Further, we set W0 (O) for the closure of C0∞ (O) in W 1,p (O), i.e., 1,p

W0 (O) := {u ∈ W 1,p (O) : ∃(un )n∈N ⊂ C0∞ (O) such that un −→ u in W 1,p (O)}. 1,p

1,p

In other words, W0 (O) consists of functions from W 1,p (O) that can be approximated by smooth functions with compact support. For the particular case p = 2, we define W 1,2 (O) = H 1 (O) and W01,2 (O) = H01 (O).

1.1 Notation and Theoretical Results

5

In general, we may introduce for each m ∈ N the space

 H m (O) := u ∈ L2 (O) : Dα u ∈ L2 (O), for each multi-index α with |α| ≤ m .

p The dual of the space W0 (O) is denoted by W −1,p (O), for p = p−1 . In partic−1,2 −1 ular, W (O) = H (O). The connection between the Lebesgue and Sobolev spaces is given by the Sobolev we have W 1,p (O) ⊂ embeddings. Namely, assume u ∈ W 1,p (O). Then for q = dpd −p Lq (O); i.e., the identity map from W 1,p (O) to Lq (O) is bounded. 1,p

Proposition 1.1 (Poincaré’s inequality) There exists a constant CO (depending only on the domain O) such that for all u ∈ H01 (O), we have

u L2 (O ) ≤ CO ∇u L2 (O ) . Proposition 1.2 (Hölder’s inequality) Let p, q ∈ [1, ∞] with all measurable functions u, v, we have

1 p

+

1 q

= 1. Then for

uv L1 (O ) ≤ u Lp (O ) v Lq (O ) . We continue by defining the space of X -valued continuous functions on [0, T ] ⊂ [0, ∞), denoted by C([0, T ]; X ), where X is some Banach space. The space C([0, T ]; X ) is endowed with the sup norm

u C([0,T ];X ) := sup u(t) X , ∀u ∈ C([0, T ]; X ). t∈[0,T ]

Similarly, we introduce the space of continuous differentiable X -valued functions C 1 ([0, T ]; X ), with the norm    d 

u C 1 ([0,T ];X ) := sup u(t) X + sup  u(t)  . t∈[0,T ] t∈[0,T ] dt X Also, Lp (0, T ; X ) stands for the space of X -valued Bochner integrable Lp functions on (0, T ) with the norm 

u Lp (0,T ;X ) := 0

T

p

u(t) X dt

p1

.

By W 1,p ([0, T ]; X ) we denote the Sobolev space   d p p u ∈ L (0, T ; X ) : u ∈ L (0, T ; X ) , dt where

d u dt

is taken in the sense of X -valued vectorial distributions on (0, T ).

6

1 Preliminaries

Fourier Series Let f : [x0 , x0 + P] → C be an integrable function on the interval [x0 , x0 + P], where x0 , P are real numbers. If f (x0 ) = f (x0 + P), it follows that f can be represented on the whole real line (f is extended by periodicity to the whole line) as f (x) =



fk e i

2πkx P

, x ∈ R,

k∈Z

where fk :=

1 P



x0 +P

f (x)e−i

2πkx P

dx, k ∈ Z.

x0

The coefficients fk are called the Fourier modes of the function f . In particular, if x0 = 0 and P = 2π , we get that f can be represented as f (x) =



fk eikx , x ∈ R,

(1.1)

k∈Z

 2π 1 −ikx with fk := 2π dx, k ∈ Z. 0 f (x)e One can immediately deduce that f is real-valued if and only if fk = f−k , ∀k ∈ Z, that is, if fk and f−k are complex conjugates. Introduce the space L2per (0, 2π ) consisting of all locally square-integrable functions on R that are 2π -periodic. The norm in L2per (0, 2π ) is defined as

f 2L2per (0,2π) = 2π



|fk |2 ,

k∈Z

where fk are the corresponding Fourier modes introduced above (also known as Parseval’s identity). Next, define

 1 Hper (0, 2π ) := f ∈ H 1 (0, 2π ) : f (0) = f (2π ) , with the norm

f 2Hper 1 (0,2π) := 2π



(1 + k 2 )|fk |2 .

k∈Z

We can also define the Fourier series for functions of two variables x and y in the square [0, 2π ] × [0, 2π ]: f (x, y) = fkl eikx eily , k,l∈Z

where fkl :=

1 4π 2

 2π  2π 0

0

f (x, y)e−ikx e−ily dxdy.

1.1 Notation and Theoretical Results

7

Linear Operators If X , Y are Banach spaces, then L(X , Y ) is the space of all linear continuous operators from X to Y with the operatorial norm 

A L(X ,Y )



Ax Y := sup , ∀x ∈ X , x = 0 .

x X

If X and Y are two Hilbert spaces, we introduce the Hilbert–Schmidt norm of an operator A : X → Y as

Aei 2Y ,

A 2HS := i∈I

where {ei : i ∈ I } is an orthonormal basis in X , i.e., ei , ej  = δij , ∀i, j, where δij is the Kronecker delta δij = 1 if i = j, and δij = 0 otherwise. We call A a Hilbert–Schmidt operator if A HS < ∞. Letting A ∈ L(X , X ), for each λ ∈ C we denote by (λI − A)−1 the resolvent of A, by ρ(A) the resolvent set 

ρ(A) = λ ∈ C : (λI − A)−1 ∈ L(X , X ) , and by σ (A) = C \ ρ(A) the spectrum of A. We say that λ ∈ C is an eigenvalue of the linear operator A if there exists x ∈ D(A), x = 0, such that Ax = λx. Such an x is called an eigenvector of A corresponding to the eigenvalue λ. If λ is an eigenvalue for A, then the dimension of the linear eigenvector space Ker(λI − A) := {x ∈ X : Ax = λx} is called the geometric multiplicity of λ. The vector x is called a generalized eigenvector corresponding to the eigenvalue λ if (λI − A)m x = 0, for some m ∈ N. The dimension of the space of generalized eigenvectors is called the algebraic multiplicity of the eigenvalue λ. An eigenvalue λ of the operator A is called semisimple if its algebraic multiplicity coincides with its geometric multiplicity. We say that the linear operator A : D(A) ⊂ X → Y is closed if its graph is closed, that is, if xn → x in X and yn = Axn → y in Y implies y = Ax. The operator A is said to be densely defined if its domain D(A) is dense in X .

8

1 Preliminaries

An operator is called compact if it maps bounded sets into relatively compact sets. Concerning the eigenvalues of an operator, we have the following result, known as the Riesz–Schauder–Fredholm theorem (see [128], p. 283). Theorem 1.3 Let A be a closed and densely defined operator in X with compact resolvent (λI − A) −1, for some λ ∈ ρ(A). Then the spectrum σ (A) consists of isolated eigenvalues λj j∈N each of finite (algebraic) multiplicity. If X is a Banach space, we denote by X its dual space endowed with the dual norm



x∗ X := sup X (x, x∗ )X : x X = 1 , where by X (x, x∗ )X , we mean the value of x∗ computed in x. We say that A : D(A) ⊂ X → X is symmetric if (Ay, z) = (y, Az), ∀y, z ∈ D(A). We note that a symmetric operator has only positive semisimple eigenvalues. Let A ∈ L(X , Y ) be a closed and densely defined operator. Then the adjoint, A∗ : Y → X of A is defined by ∗ ∗ X (A y , x)X

=Y (y∗ , Ax)Y , ∀x ∈ D(A),

 with D(A∗ ) = y∗ ∈ Y : ∃C > 0, |Y (y∗ , Ax)Y | ≤ C x X , ∀x ∈ D(A) . If λ is an eigenvalue of A, then λ is an eigenvalue of A∗ , of the same multiplicity. Assume now that X is a Hilbert space with the scalar product ·, ·X and the induced norm · X . Also assume that there exists λ0 ∈ ρ(A). Then define the space (D(A)) as the completion of X in the norm

x (D (A)) = (λ0 I − A)−1 x X , ∀x ∈ X . Then one has

D(A) ⊂ X ⊂ (D(A)) ,

algebraically and topologically. Moreover, the operator A has an extension, denoted ˜ : X → (D(A∗ )) and defined by by A ˜

(D (A∗ )) Ax, yD (A∗ )

= x, A∗ y, ∀y ∈ D(A∗ ).

(1.2)

˜ ∈ L(X , (D(A∗ )) ). Moreover, one has By the closed graph theorem, one has that A ˜ coincide with those of A. that the spectrum and the eigenvalues of A Positive Semidefinite Operators A linear operator L ∈ L (H ) on a Hilbert space H is called positive semidefinite if

1.1 Notation and Theoretical Results

9

u, Lu ≥ 0, ∀u ∈ H , and positive definite if u, Lu > 0, ∀u ∈ H \ {0} . In the particular case in which H = Cd and L = A, with A a d × d complex symmetric matrix (also known as a Hermitian matrix, i.e., AT = A), we recover the definitions of positive semidefinite and positive definite matrices, respectively. We say that a function k : D × D → R is positive semidefinite if for all N ∈ N, xj ∈ D, and aj ∈ R, for j = 1, 2, . . . , N , we have N

ai aj k(xi , xj ) ≥ 0.

i,j=1

For a domain D, if k ∈ C(D × D) is a positive semidefinite function, then the integral operator L on H = L2 (D) is positive semidefinite, i.e.,

u(x)k(x, y)u(y)dxdy ≥ 0, ∀u ∈ L2 (D).

u, Lu = D×D

Powers of a Linear Operator Let A be a linear operator

from D(A) ⊂ H to a Hilbert space H with an orthonormal basis of eigenfunctions ϕj : j ∈ N∗ and eigenvalues λj > 0, ordered so that λj+1 ≥ λj . Such operators are self-adjoint, i.e., they satisfy Au, v = u, Av, ∀u, v ∈ D(A).  ∗ We see that given u ∈ H , we have u = ∞ j=1 uj ϕj , where uj = u, ϕj , j ∈ N . Consequently, ∞ ∞ Au = A uj ϕj = λj uj ϕj . j=1

j=1

We may define the fractional power α ∈ R of A as Aα u :=

∞

λαj uj ϕj , ∀u ∈ H ,

(1.3)

j=1

 α α and let D(Aα ) be the set of all u = ∞ j=1 uj ϕj such that A u ∈ H . The domain D(A ) is a Hilbert space with inner product

10

1 Preliminaries

u, vα :=

∞

Aα u, Aα v

j=1

and corresponding induced norm u α := Aα u H . In addition, we have 1

1

• u, v 12 = A 2 u, A 2 v = Au, v, ∀u, v ∈ D(A); • for α > 0, u α ≥ λα1 u , u ∈ D(Aα ). Elliptic Operators Let A be a scalar linear partial differential operator A = a(x, D) =



aα (x)Dα

|α|≤2l

on O of even order 2l, with coefficients infinitely differentiable on O. (Here Dα stands for the partial derivative operator introduced above.) Its symbol is the polynomial a(x, ξ ) obtained from a(x, D) by replacing all Dj by real numbers ξj . The principal symbol a0 (x, ξ ) is the leading homogeneous part of the symbol: a0 (x, ξ ) =



aα (x)ξ α .

|α|=2l

Suppose we are also given a boundary operator B = b(x, D) =



bβ (x)Dβ ,

|β|≤r

on Γ = ∂O, of nonnegative order r, with infinitely differentiable coefficients. The boundary value problem that we will consider in this section is 

a(x, D)u(x) = 0 in O, b(x, D)u(x) = g on Γ.

(1.4)

Now let us give the definition of ellipticity for this problem: (a) The operator A is elliptic on O, that is, for its main symbol a0 (x, ξ ), we have a0 (x, ξ ) = 0, ∀x ∈ O, ∀ξ ∈ Rd \ {0} . (b) Moreover, the operator A is regular elliptic, i.e., the equation a0 (x, ξ , ζ ) = 0 with ξ = 0 has the same number of roots ζ in the upper and lower half-planes (and this number equals l). (c) The boundary operator B obeys the Lopatinskii condition. In order to have a simple statement of this, we transfer the origin to a point x0 and rotate the coordinate system so that the t = xd axis is directed along the inner normal

1.1 Notation and Theoretical Results

11

to the boundary at this point. Suppose that the operators of the problem are rewritten in this coordinate system. Consider the following problem on the ray R+ = {t : t > 0} for fixed ξ = ξ0 = 0: 

a0 (x0 , ξ0 , Dt )v(t) = 0, t > 0, b(x0 , ξ0 , Dt )v|t=0 = h,

(1.5)

where b0 is the principal symbol of the operator b, b0 (x, ξ ) =



bβ (x)ξ β .

|β|=r

Problem (1.5) is required to have precisely one solution in L2 (R+ ) for every ξ0 = 0 and every number h. (More about the Lopatinskii condition can be found in [90].) Problem (1.5) is obtained from the original problem by freezing the coefficients at the point x0 , removing the lower-order terms, and applying the formal Fourier transform with respect to the tangent variables. If all three conditions stated above hold, then the problem is said to be elliptic. Let us assume that A can be written in divergence form as A=



(−1)|α| ∂ α (aα,β ∂ β ),

|α|≤l,|β|≤l

such that the strong ellipticity condition 



aα,β (x)ξ α+β ≥ C|ξ |2

|α|=l,|β|=l

holds. Then, via the Lax–Milgram theorem, one may show that an elliptic problem such as (1.5) has a unique solution. For more details on this subject, see [2]. The Cauchy Problem Let X be a real Banach space with the dual denoted by X , and let A : D(A) ⊂ X → X be a linear unbounded operator. The operator A is said to be accretive if (Au, η)X ≥ 0 for η ∈ J (u), ∀u ∈ D(A), where J : X → X is the duality map of the space X . Equivalently,

(I + λA)−1 f X ≤ f X , λ > 0, for all f , g in the range R(I + λA), of the operator I + λA. The operator A is said to be m-accretive if it is accretive and R(I + λA) = X for all λ > 0 (equivalently, for some λ > 0).

12

1 Preliminaries

A semilinear evolution equation on the Banach space X is a differential equation of the form d u = −Au + f (u), u(0) = u0 . (1.6) dt We start with f ≡ 0. We have Theorem 1.4 (Hille–Yosida) Let A be m-accretive. Then given uo ∈ D(A), there exists a unique function u ∈ C 1 ([0, ∞); X ) ∩ C([0, ∞); D(A)) satisfying 

+ Au = 0, t ≥ 0, u(0) = uo . d u dt

Moreover,    d 

u(t) X ≤ uo X and  u(t)  = Au(t) X ≤ Auo X , t ≥ 0. dt X The map uo −→ u(t) extended by continuity to all of X is denoted by e−At . It is a continuous (C0 -)semigroup of contractions on X , and −A is called its infinitesimal generator. Proof For a proof, see [35, Chap. 7]. We say that f : X → X is Lipschitz if

f (u1 ) − f (u2 ) X ≤ Cf u1 − u2 X , ∀u1 , u2 ∈ X , for some constant Cf > 0. If A is m-accretive, then the variation of constants formula in (1.6) gives the following reformulation of Eq. (1.6): u(t) = e

−tA

uo +

t

e−(t−s)A f (u(s))ds.

0

Such a function u is called a mild solution of Eq. (1.6). We have the following results (for proofs, see [7]). Theorem 1.5 Suppose that A is an m-accretive operator and f is Lipschitz. Then there exists a mild solution u(t) to (1.6) for t ≥ 0. Further, for T > 0, there exists CT > 0 such that for all uo ∈ X ,

u(t) X ≤ CT (1 + uo X ), 0 ≤ t ≤ T . Usually, the evolution Eq. (1.6) is understood in the weak sense. Proposition 1.3 Let X be a Hilbert space, let A be a self-adjoint m-accretive operator and f a Lipschitz function. Then the mild solution u(t) of (1.6) given by Theorem 1 1 1.5 belongs to D(A 2 ) and dtd u(t) ∈ D(A− 2 ) for all t > 0.

1.1 Notation and Theoretical Results

13

The above proposition does not provide enough smoothness of u(t) to interpret (1.6) directly, since we do not know whether u(t) ∈ D(A). We can, however, develop 1 the weak form using test functions v ∈ D(A 2 ). Taking the inner product of (1.6) with v gives   d u, v = −Au, v + f (u), v, dt or equivalently,

    1 d 1 1 1 A− 2 u, A 2 v = A 2 u, A 2 v + f (u), v. dt

And so, due to the result in the above proposition, the expression is well defined for 1 t > 0. This leads to the definition of a weak solution for Eq. (1.6): Let V = D(A 2 ) 1 and V = D(A− 2 ), which is the dual of V . We say that u : [0, T ] → V is a weak solution of (1.6) if for almost all s ∈ [0, T ], we have dtd u(s) ∈ V and 

   1 d 1 u(s), v = −a(u(s), v) + f (u(s)), v, ∀v ∈ V, where a(u, v) := A 2 u, A 2 v . dt

Stochastic Processes For this section, we refer for further details to [113]. A triple (Ω, F , P) is called a probability space, where Ω is the set of possible outcomes, F is a σ -algebra of subsets of Ω, called the set of events, and P : F → [0, 1] is a probability measure, which assigns “probabilities” to the outcomes of Ω with P(Ω) = 1. An F -measurable function X : Ω → H , i.e., a function such that X −1 (B) ⊂ F for each Borel set B of H , where H is a Banach space, is called a random variable. A family {X (t) : t ≥ 0} of random variables is called a stochastic process. The quantity

E(X ) :=

Ω

Xd P

is called the expectation, or the average, of the random variable X . Let V ⊂ F a sub-σ -algebra of F . The random variable X : Ω → H is usu ally not V -measurable. Thus, the integral V Xd P|V , where V ∈ V cannot be well defined in general. However, the local averages V Xd P can be recovered in (Ω, V , P|V ) via the conditional expectation. A conditional expectation of X given V , denoted by E(X |V ), is a V -measurable function E(X |V ) : Ω → H that satisfies



E(X |V )d P = V

Xd P for each V ∈ V . V

Let X (t) be a stochastic process such that E(|X (t)|) < ∞ for all t ≥ 0. Then X (t) is called a martingale if

14

1 Preliminaries

X (s) = E(X (t)|U (s)), P-a.s., for all t ≥ s > 0, where U (s) := σ (X (τ ) : 0 ≤ τ ≤ s) is the σ -algebra generated by the random variables X (τ ) for 0 ≤ τ ≤ s. The quadratic variation of X (t) is the process, written [X ](t), defined bas [X ](t) = lim

Δ →0

N

(X (tk ) − X (tk−1 ))2 ,

k=1

where Δ ranges over partitions of the interval [0, t] and the norm of the partition Δ is the mesh. A collection of σ -algebras Ft , t ≥ 0, satisfying Fs ⊂ Ft for all 0 ≤ s ≤ t is called a filtration, and a stochastic process X (t) is said to be adapted to the filtration Ft if for each t ≥ 0, X (t) is Ft -measurable. A random variable τ : Ω → [0, ∞) is an Ft -stopping time if {τ ≤ t} ⊂ Ft , ∀t ≥ 0. The stochastic process X (t) is said to be a local martingale if there is a sequence of stopping times (τn )n such that τn → ∞, P-a.s., as n → ∞, and for each n, X (min {t, τn }) is a martingale. The stochastic process X (t) is an Ft -semimartingale if X = M + Y , where M is a local martingale with respect to Ft and Y is an Ft -adapted finite variation process, that is, for each t > 0, sup



|Y (ti+1 ) − Y (ti )| < ∞,

i

where the supremum is taken over all the partitions of [0, t]. The following result, which is related to the martingale convergence theorem, is important for obtaining convergence in probability of stochastic processes. Its proof can be found in [85]. Lemma 1.1 Let I and I1 be nondecreasing adapted processes, V a nonnegative semimartingale, and M a local martingale such that E(V (t)) < ∞, ∀t ≥ 0, I1 (∞) < ∞, P-a.s., and V (t) + I (t) = V (0) + I1 (t) + M (t), ∀t ≥ 0. Then there exists lim V (t) < ∞, P-a.s.,

and I (∞) < ∞, P-a.s.

t→∞

A real-valued random variable Y is said to be N (0, q)-Gaussian distributed if 1 P[a ≤ Y ≤ b] = √ 2π q



b

x2

e− 2q dx, for all − ∞ < a < b < ∞.

a

A real-valued stochastic process β(·) is called a Brownian motion or a Wiener process if

1.1 Notation and Theoretical Results

15

(a) β(0) = 0, P-a.s., (b) β(t) − β(s) is N (0, t − s) Gaussian distributed for all 0 ≤ s ≤ t, (c) for all times 0 < t1 < t2 < · · · < tn , the random variables β(t1 ), β(t2 ) − β(t1 ), . . . , β(tn ) − β(tn−1 ) are independent. We have the following bounds (for a proof, see [10, Lemma 4.6]). Lemma 1.2 Let β(t), t ≥ 0, be a real Brownian motion in some probability space (Ω, F , P). Then for each λ > 0, we have 

P sup e

β(t)−λt

t>0





  ≥ r = P esupt>0 {β(t)−λt} ≥ r  = P sup {β(t) − λt} ≥ log r t>0

= r −2λ . Let U be another Hilbert space. A simple process is a stochastic process X (t) of the form N Φ(t) = Φi−1 1[ti−1 ,ti ) (t), i=1

where 0 = t0 < t1 < · · · < tn = 1 is a partition of [0, 1] and Φi−1 is an L(U, H )valued Fti−1 -measurable random variable, and 1[ti−1 ,ti ) is the characteristic function of the interval [ti−1 , ti ). Then one defines the stochastic integral

t

N

Φ(s)d β(s) :=

0

Φi−1 [β(min {ti , t}) − β(min {ti−1 , t})].

i=1

This definition can be extended, in a standard way (see [113], pp. 1–4), to adapted processes Φ : [0, T ] → L(U, H ) such that 

t

P 0



Φ(s) 2HS ds < ∞, t ≥ 0 = 1,

where · HS is the Hilbert–Schmidt norm in L(U, H ). Then we have (a) Itô’s isometry: 

t

E

2 Φ(s)d β(s)



=E

t 0

Φ (s)ds , 2

0

(b)

t 0

Φ(s)d β(s) is a martingale, in particular, 

E 0

t

Φ(s)d β(s) = 0,

16

1 Preliminaries

(c) the solution X = X (t) of the stochastic differential equation dX (t) = Φ(t)d β(t) + f (t)dt, t ∈ (0, T ), X (0) = x, is defined as the process given by

t

X (t) := x +

f (s)ds +

0

t

Φ(s)d β(s), t ∈ [0, T ],

0

(d) Stochastic calculus: let X (t) = (X (t)1 , . . . , X (t)n )T be a random vector such that dX (t) = μ(t)dt + G(t)d β(t), for a vector μ(t) and a matrix G(t); then Itô’s lemma states that  1 ∂φ T T + (∇X φ) μ(t) + Tr[G (t)(HX φ)G(t)] dt d φ(t, X (t)) = ∂t 2 

+ (∇X φ)T G(t)d β(t), where φ : [0, ∞) × Rn → R is once differentiable in the first variable and twice differentiable in the second one, ∇X φ is the gradient of φ with respect to X , HX φ is the Hessian matrix of φ with respect to X , and Tr is the trace operator. Some important inequalities: 1. Chebyshev’s inequality. If X is a random variable and 1 ≤ p < ∞, then P(|X | ≥ λ) ≤

1 E(|X |p ), ∀λ > 0. λp

2. Borel–Cantelli Lemma application. If Xk → X in probability, then there exists a subsequence

∞ Xkj j=1 ⊂ {Xk }∞ k=1 such that Xkj (ω) → X (ω) for almost every ω. 3. Burkholder–Davis–Gundy inequality. If X is a martingale with X (0) = 0, and 1 < p < ∞, then there exist cp and Cp such that      p p cp E [X ] 2 (t) ≤ E max |X (s)|p ≤ Cp E [X ] 2 (t) . 0≤s≤t

Let H be a separable Hilbert space and consider the stochastic differential equation

1.1 Notation and Theoretical Results

17

dX (t) + AX (t)dt = f (t)dt + B(X (t))d β(t), X (0) = x,

(1.7)

where −A is the infinitesimal generator of a C0 -semigroup e−tA on H , and B is a continuous operator. The adapted process X (t) is said to be a mild solution of (1.7) if X (t) = e−tA x +



t

e−(t−s)A f (s)ds +



0

t

e−(t−s)A B(X (s))d β(s), t ∈ [0, T ].

0

We have (see [113, p. 67]) the following theorem. Theorem 1.6 Assume that f ∈ L2 (0, T ; H ) and

e−tA Bx − e−tA By HS ≤ γ x − y H , ∀t ∈ [0, T ], x, y ∈ H . Then (1.7) has a unique mild solution X ∈ C([0, T ]; L2 (Ω, F , P, H )), which is the space of all continuous functions [0, T ] → L2 (Ω, F , P, H ) that are adapted to the filtration Ft . This result extends to nonlinear differential equations of the form dX + AXdt + F(X )dt = fdt + B(X )d β, for Lipschitz mappings F : H → H . A stochastic differential equation of the form dX (t) + AX (t)dt = f (t)dt + h(t)X (t)d β(t), t ≥ 0, can be transformed into the equivalent random deterministic equation ∂t y + e−h(t)β(t) A(eh(t)β(t) y) = f (t) −



1 d h(t)β(t) + h2 y, P-a.s., t ≥ 0, dt 2

via the rescaling y(t) := e−h(t)β(t) X (t) (for more details, see [22]).

Chapter 2

Stabilization of Abstract Parabolic Equations

In this chapter, we present a technique to design asymptotically exponentially stabilizing boundary proportional-type feedback controllers for nonlinear parabolic-like equations, namely equations for which their linear parts are generated by analytic C0 semigroups. In what follows, we will simply refer to them as parabolic equations, in concordance with the title of this book. The feedback law’s main features are that it is expressed in an explicit simple form and has a finite-dimensional structure involving only the eigenfunctions of the linear operator obtained from the linearized equation. As we will see, these features will enable us to obtain the first results to appear in the literature regarding the stabilization of different equations, such as the stochastic heat equation, the Chan–Hilliard equations, and for boundary stabilization to nonsteady states for parabolic-type equations.

2.1 Presentation of the Abstract Model Let O be an open bounded domain in Rd , d ∈ N∗ , with smooth boundary ∂O, split into two parts ∂O = Γ1 ∪ Γ2 , such that Γ1 has nonzero surface measure. We set n for the outward unit normal to the boundary ∂O. Let A be a closed and densely defined linear differential operator on L 2 (O), with domain D(A), and let F0 : D(F0 ) ⊂ D(A) → L 2 (O) be a nonlinear (differential) operator. We assume that (A1) −A generates a C0 -analytic semigroup on L 2 (O). (A2) For all y, yˆ ∈ D(A), there exists the limit  1 F0 ( yˆ + λy) − F0 ( yˆ ) λ→0 λ

F0 ( yˆ )(y) := lim

in L 2 (O). Moreover, F0 (0) = 0, and for some α ∈ (0, 1) and C > 0, we have © Springer Nature Switzerland AG 2019 I. Munteanu, Boundary Stabilization of Parabolic Equations, Progress in Nonlinear Differential Equations and Their Applications 93, https://doi.org/10.1007/978-3-030-11099-4_2

19

20

2 Stabilization of Abstract Parabolic Equations

F0 ( yˆ )y ≤ αAy + Cy, ∀y ∈ D(A).

(2.1)

Now fix some yˆ ∈ D(A), and introduce the linear operator A := A + F0 ( yˆ ), D(A) = D(A).

(2.2)

It is easy to see that A is closed and densely defined and that −A generates a C0 semigroup on L 2 (O). The operator A can be viewed as the linearization of A + F0 around yˆ . In addition to (A1) and (A2), we assume that (A3) the resolvent (λI + A)−1 of A is compact in L 2 (O). Hypothesis (A3) implies, via the Fredholm–Riesz theory (see Theorem 1.3), that the operator A has a countable set of eigenvalues λ j , j ∈ N∗ (repeated accordingly to their multiplicities), and corresponding eigenfunctions ϕ j , j ∈ N∗ , i.e., Aϕ j = λ j ϕ j , j ∈ N∗ . Besides this, given ρ > 0, there is a finite number N of eigenvalues such that λ j < ρ, j = 1, 2, . . . , N , while λ j ≥ ρ, j = N + 1, N + 2, . . . .

(2.3)

The first N eigenvalues are usually called the unstable eigenvalues. Recall that if the algebraic multiplicity of an eigenvalue coincides with the geometric multiplicity, then that eigenvalue is called semisimple (see Chap. 1). We add to the above context the following assumption: (A4) Each unstable eigenvalue λ j , j = 1, 2, . . . , N , is semisimple. Example 2.1 A classical example is A = −Δ, with 

 ∂y | =0 , D(A) = y ∈ H (O) : y|Γ1 = 0, ∂n Γ2 2

F0 (y) = f (y), where f is some C 1 -nonlinear function in y, and yˆ ∈ L ∞ (O). In this case, A = −Δ + f  ( yˆ ). Taking into account that A is self-adjoint, one can easily check that hypotheses (A1)–(A4) hold for this case. Denote by ·, · and by  ·  the scalar product and the corresponding norm in L 2 (O), respectively. Since the spectrum of A might contain some complex eigenvalues, it will be convenient in the sequel to view A as a linear operator (still denoted by A) in the complexified space L 2 (O) + iL 2 (O) (which will still be denoted by L 2 (O)). We denote by ·, · and by  ·  the corresponding scalar product and the induced norm of the complexified L 2 (O), respectively.

2.1 Presentation of the Abstract Model

21

 N It is easily seen that the finite part of the spectrum λ j j=1 can be separated from the rest of the spectrum by a rectifiable curve Γ N in the complex space C. Set Xu to  N be the linear space generated by the eigenfunctions ϕ j j=1 , that is,  N Xu := lin span ϕ j j=1 . Then the operator PN : L 2 (O) → Xu defined by PN :=

1 2π i

 ΓN

(λI − A)−1 dλ

(2.4)

is known as the algebraic projection of L 2 (O) onto Xu . It is easy to see that the operator (2.5) Au := PN A  N maps the space Xu into itself and σ (Au ) = λ j j=1 . More exactly, Au : Xu → Xu is finite-dimensional and can be represented by an N × N matrix. (σ (Au ) stands for the spectrum of the operator Au ; see Chap. 1.)   If A∗ is the adjoint operator of A, then its eigenvalues are precisely λ j j∈N∗ , with the corresponding eigenfunctions A∗ ϕ ∗j = λ j ϕ ∗j , j ∈ N∗ . The adjoint PN∗ of PN is given by PN∗ N while X N∗ = lin span ϕ ∗j

1 = 2πi

j=1

 ΓN

(λI − A∗ )−1 dλ,

 = PN∗ L 2 (O) .

Via the Schmidt orthogonalizationprocedure, it follows by hypothesis (A4) that N   N ∗ one can find a biorthogonal system ϕ j j=1 , ϕ j of eigenfunctions of A j=1

and A∗ , respectively, i.e.,

ϕ j , ϕ ∗j  = δi j , i, j = 1, 2, . . . , N ,

(2.6)

Aϕ j = λ j ϕ j , A∗ ϕ ∗j = λ j ϕ ∗j .

(2.7)

Here δi j stands for the Kronecker symbol, namely  δi j =

1, i = j 0, i = j.

22

2 Stabilization of Abstract Parabolic Equations

If y ∈ L 2 (O) but y ∈ / D(A), we will understand by Ay a differential form involving y rather than the operator A acting on y. In this light, consider the problem ⎧ ∂y ⎨ ∂t + Ay + F0 (y) = 0 in (0, ∞) × O, B.C. (y, 0) on (0, ∞) × ∂O, ⎩ y(0) = yo in O.

(2.8)

Here B.C. (y, 0) denotes some appropriate boundary conditions for the unknown function y, and yo ∈ D(A) is the initial data. The operator ∂t∂ + A + F0 is called the abstract parabolic differential operator. Example 2.2 With respect to Example 2.1, Eq. (2.8) reads ⎧ ∂y ⎨ ∂t − Δy + f (y) = 0 in (0, ∞) × O, ∂y y = 0 on (0, ∞) × Γ1 , ∂n = 0, on (0, ∞) × Γ2 , ⎩ y(0) = yo in O.

(2.9)

Under appropriate conditions on A, F0 , and B.C. (y, 0), problem (2.8) is well posed. Here we do not give any further details about the well-posedness, since these were discussed in Chap. 1, Theorem 1.5. We simply assume that (2.8) with B.C. (y, 0) generates a semiflow y = y(t, yo ), t ≥ 0. An equilibrium (steady-state or stationary) solution yˆ to system (2.8) is a solution to the stationary equation (if there exists one) A yˆ + F0 ( yˆ ) = 0. The equilibrium yˆ is said to be locally asymptotically exponentially stable if lim ect (y(t, yo ) − yˆ ) = 0

t→∞

in L 2 (O) for all yo in a neighborhood of yˆ . Here c > 0 is some constant. By defining the fluctuation variable z := y − yˆ , the stability of yˆ can be equivalently expressed as the stability of the null solution to the equation

where

⎧∂ ⎨ ∂t z + Az + G(z) = 0 in (0, ∞) × O, B.C. (z + yˆ , 0) on (0, ∞) × ∂O, ⎩ z(0) = z o := yo − yˆ in O,

(2.10)

G(z) := F0 (z + yˆ ) − F0 ( yˆ ) − F0 ( yˆ )(z).

(2.11)

If the steady state yˆ is not stable, a way to stabilize it is to plug into (2.8) a controller function u : [0, ∞) → U that takes values in another space U (which is assumed Hilbert), obtaining thereby the following boundary controlled problem:

2.1 Presentation of the Abstract Model

⎧ ∂y ⎨ ∂t + Ay + F0 (y) = 0, in (0, ∞) × O, B.C. (y, u) on (0, ∞) × ∂O, ⎩ y(0) = yo in O,

23

(2.12)

or equivalently, by (2.10), ⎧∂ ⎨ ∂t z + Az + G(z) = 0 in (0, ∞) × O, B.C. (z, v) on (0, ∞) × ∂O, ⎩ z(0) = z o in O,

(2.13)

Example 2.3 The boundary control problems (2.12) and (2.13) associated with (2.9) from Example 2.2 look like ⎧ ∂y ⎨ ∂t − Δy + f (y) = 0 in (0, ∞) × O, ∂y y = u, on (0, ∞) × Γ1 , ∂n = 0 on (0, ∞) × Γ2 , ⎩ y(0) = yo in O,

(2.14)

and ⎧ ∂z ⎨ ∂t − Δz + f  ( yˆ )z + G(z) = 0 in (0, ∞) × O, ∂z z = v := u − yˆ on (0, ∞) × Γ1 , ∂n = 0 on (0, ∞) × Γ2 , ⎩ z(0) = z o in O, respectively. Here

(2.15)

G(z) := f (z + yˆ ) − f ( yˆ ) − f  ( yˆ )(z).

Hence the asymptotic exponential stabilization problem consists in finding a controller u ∈ L 2 (0, ∞; U ) such that once it is inserted into Eq. (2.12), the corresponding solution y = y(t, yo , u) to Eq. (2.12) has the property that lim ect (y(t, yo , u) − yˆ ) = 0

t→∞

in L 2 (O) for some constant c > 0, for all yo in a neighborhood of yˆ . Throughout this book, we will use the shortened terminology stabilization (stability, stabilizability) in referring to the asymptotic exponential stabilization (asymptotic exponential stability, stabilizability, respectively) of some system. If one can find such a controller, then the equation is said to be stabilizable from the boundary. If the controller is in feedback form, i.e., u(t) = K (y(t)), t ≥ 0, where K is a given operator from L 2 (O) to U , then Eq. (2.12) is said to be a closedloop equation.. In practice, a controller given in feedback form is the most desirable, since it ensures better performance. Roughly speaking, if at time t ∗ , the solution of Eq.

24

2 Stabilization of Abstract Parabolic Equations

(2.12) gets away from yˆ , then at the very same moment t ∗ , the feedback controller u(t ∗ ) = K (y(t ∗ )) reacts and brings back the trajectory close to the steady state. One can equivalently express the stabilization problem for Eq. (2.13). More precisely, the problem consists in finding a feedback control v such that once it is inserted into Eq. (2.13), the corresponding solution z to the closed-loop equation (2.13) satisfies lim ect z(t) = 0 in L 2 (O). t→∞

There is another type of control largely used in stabilization theory of parameter distributed systems, namely the internal control. More precisely, in this case, the controlled system (2.12) is of the form ⎧ ∂y ⎨ ∂t + Ay + F0 (y) = 1O 0 u in (0, ∞) × O, B.C. (y, 0) on (0, ∞) × ∂O, ⎩ y(0) = yo in O, where 1O 0 is the characteristic function of a subdomain O0 ⊂ O. While in (2.12) the controller’s actuation and sensing are applied only through the boundary conditions, in this case the controller actuation penetrates the domain of the PDE system or is evenly distributed everywhere in the domain. Boundary control is generally considered to be physically more realistic, because actuation and sensing are nonintrusive, but it turns out to be a harder problem than the internal control one. This is the main reason why throughout this book, we will confine ourselves mostly to the boundary control case (except in the last chapter). One may consider as well the problem of (boundary or internal) stabilization to non-steady states associated with (2.8). That means a nonstationary yˆ , i.e., yˆ = yˆ (t), solution to ∂ yˆ + A yˆ + F0 ( yˆ ) = 0. ∂t The main difficulty in this case is that the corresponding linear operator A has timedependent spectrum, making useless all the considerations and results from the stationary case. Therefore, this is a challenging subject, and in a subsequent chapter, we will pose and solve this problem for a special case. The theory of control and stabilization uses a number of tools, many of them developed in the 1960 by Kalman with his theory of filtering and algebraic approach to control systems, then by Pontryagin with his maximum principle, which is a generalization of Lagrange multipliers, and by Bellman with his principle of dynamic programming, or by Lyapunov and his Lyapunov functions, among others. In the present book, we will rely mainly on unique continuation results and construct a Lyapunov function for the system. More exactly, the form of the feedback controller is given a priori (guaranteed by a unique continuation result), and then, once it has been plugged into the system, one proves the stability of the closed-loop system by finding a Lyapunov function.

2.2 The Design of the Boundary Stabilizer

25

2.2 The Design of the Boundary Stabilizer We begin to develop the ideas on how to construct the boundary proportional-type controller v (equivalently, u) for the stationary case of yˆ . In order to represent (2.13) as an abstract Cauchy problem, we will lift the B.C. into the equations. This will be done via the Dirichlet map, D, which will now be introduced. Let β ∈ L 2 (∂O). We denote by Dγ β the solution z˜ to the equation ⎧ ⎨ ⎩

A(˜z ) − 2

N  k=1

λk ˜z , ϕk∗ ϕk + γ z˜ = 0 in O,

(2.16)

B.C. (˜z , β) on ∂O.

Under appropriate boundary conditions B.C. (˜z , β) and γ > 0 sufficiently large, there exists a unique solution to (2.16), defined such that the operator Dγ belongs to 1 L(L 2 (∂O), H 2 (O)) (for details, see [19] or [80], and also (1.4). For later use, we need to compute the scalar product Dγ β, ϕ ∗j , j = 1, 2, . . . , N . To this end, scalar multiplying (2.16) by ϕ ∗j and taking into account relations (2.6) and (2.7) yields, via Green’s formula, that Dγ β, ϕ ∗j  = − Here

1 β, D ϕ ∗j 0 , j = 1, 2, . . . , N . γ − λj

(2.17)

D ϕ ∗j := −(γ − λ j )D∗γ ϕ j , j = 1, 2, . . . , N ,

where D∗γ denotes the adjoint operator of Dγ . And ·, ·0 stands for the scalar product in L 2 (∂O). As we shall see below, the algorithm requires a new hypothesis. More exactly, (A5) None of the functions D ϕ ∗j , j = 1, 2, . . . , N , is identically zero on the boundary ∂O. This assumption is related to the unique continuation property of the eigenfunctions of the adjoint A∗ of the linear operator A. It arises naturally in the context of boundary control problems. In the existing literature on this subject, instead of hypothesis (A5), a stronger one is assumed, namely linear independence of the traces of the eigenfunctions on the boundary (see more in the “Comments” section below). For such a hypothesis, it is usually hard to check its validity in practical examples, and there are many simple cases of domains O for which it fails to hold. For all the examples in this book, the weaker assumption (A5) is satisfied, while the one related to linear independence is not. As a matter of fact, validation of assumption (A5), for different models, will involve the major effort of this book, since once one has (A5) satisfied (together with (A1)–(A4)), the control design algorithm may be applied similarly, for all the models, as described in this chapter. Consequently, roughly speaking, every evolution equation governed by an elliptic operator can be stabilized

26

2 Stabilization of Abstract Parabolic Equations

from the boundary by a proportional-type feedback of the form (2.26) below, once a unique continuation-type result such as (A5) is provided. Example 2.4 For Example 2.3, the corresponding map Dγ looks like Dγ := z˜ , where z˜ satisfies ⎧ N  ⎨ −Δ˜z + f  ( yˆ )˜z − 2 λk ˜z , ϕk ϕk + γ z˜ = 0 in O, (2.18) k=1 ⎩ ∂ z˜ z˜ = β on Γ1 , ∂n = 0 on Γ2 . Since the linear operator is self-adjoint, we have that ϕ ∗j = ϕ j , j ∈ N∗ . Then simple ∂ computations show that D = ∂n |Γ1 , and that  Dγ β, ϕ j  =

1 − γ −λ β, j 1 − γ +λ j

β,

∂ϕ j  , ∂n 0 ∂ϕ j  , ∂n 0

j = 1, 2, . . . , N , j = N + 1, N + 2, . . . .

(2.19)

This time, ·, ·0 stands for the scalar product in L 2 (Γ1 ). Assumption (A5), in this ∂ϕ case, says that each trace of ∂nj , j = 1, 2, . . . , N , cannot identically vanish on Γ1 . Equivalently, if ϕ j satisfies −Δϕ j + f  ( yˆ )ϕ j = λ j ϕ j in O, ϕ j = 0 on Γ1

and

∂ϕ j = 0 on ∂O, ∂n

then necessarily ϕ j ≡ 0 in O. Since the boundary ∂O is assumed to be smooth and the Lebesgue measure of Γ1 is assumed to be nonzero, it is known that for the elliptic operator −Δ + f  ( yˆ ), the above condition holds. Therefore, for this case, assumption (A5) is valid. Next, for the convenience of the reader, we will split the presentation into two parts: first, we strengthen the hypothesis (A4) by (A4.1) below, assuming that the unstable eigenvalues are distinct. Then in the second part, we slightly adjust the feedback law to show that it still achieves stability in the more general framework given by hypothesis (A4).

2.2.1 The Case of Mutually Distinct Unstable Eigenvalues For the moment, let us assume that (A4.1) The unstable eigenvalues λi , i = 1, . . . , N , are simple, or, equivalently, are mutually distinct. We choose ρ < γ1 < γ2 < · · · < γ N to be N real constants such that Eq. (2.16) is well posed for each of them, and denote by Dγi , i = 1, . . . , N , the corresponding solutions.

2.2 The Design of the Boundary Stabilizer

27

N Next let us denote by B the Gram matrix of the system D ϕ ∗j

j=1

space L 2 (∂O). That is,

in the Hilbert

⎞ D ϕ1∗ , D ϕ1∗ 0 D ϕ1∗ , D ϕ2∗ 0 . . . D ϕ1∗ , D ϕ N∗ 0 ⎜ D ϕ ∗ , D ϕ ∗ 0 D ϕ ∗ , D ϕ ∗ 0 . . . D ϕ ∗ , D ϕ ∗ 0 ⎟ 2 1 2 2 2 N ⎟ B := ⎜ ⎝ .............................................................................. ⎠ . D ϕ N∗ , D ϕ1∗ 0 D ϕ N∗ , D ϕ2∗ 0 . . . D ϕ N∗ , D ϕ N∗ 0 ⎛

(2.20)

Further, we introduce the matrices ⎞ ... 0 1 ⎜ 0 ... 0 ⎟ γk −λ2 ⎟ Λγk := ⎜ ⎝ ................................. ⎠ , k = 1, . . . , N , 1 0 0 . . . γk −λ N ⎛

1 γk −λ1

0

(2.21)

and Bk := Λγk BΛγk , k = 1, . . . , N .

(2.22)

We have the following result. Proposition 2.1 The sum of the Bk , i.e., B1 + B2 + · · · + B N , is an invertible matrix. Before we prove this, let us note that for each k = 1, 2, . . . , N , Bk is invertible if and only if the Gramian B is invertible, and this happens only when the system

N D ϕ ∗j is linearly independent in the Hilbert space L 2 (∂O). This usually is not j=1

the case, even for simple domains O. Moreover, when the dimension d of the space equals 1, D ϕ ∗j , j = 1, 2, . . . , N , are just some complex numbers, and therefore, it makes no sense to talk about linear independence unless N equals 1. But usually N = 1 is not enough to ensure (2.3). To conclude, it is highly possible that each Bk , k = 1, 2, . . . , N , is singular, but it turns out that their sum is always nonsingular. The ideas in this chapter are based on those from the work [13] of Barbu, which requires the linear independence assumption. Here, we drop it by defining the feedback law based on the invertible matrix B1 + B2 + · · · + B N , rather than only one Bk as is done in [13]. The reason why we need the invertibility of B1 + · · · + B N becomes clearer in the computations (2.35)–(2.37) below, which allow us to obtain in a simple form the stable finite-dimensional differential system (2.39). Proof of Proposition ⎛ ⎞ 2.1. Arguing by contradiction, let us assume that there exists a z1 ⎜ z2 ⎟ N ⎟ nonzero z = ⎜ ⎝ . . . ⎠ ∈ C such that (B1 + · · · + B N )z = 0. Scalar multiplying this zN relation by z in C N yields

28

2 Stabilization of Abstract Parabolic Equations N   k=1

∂O

2    N   1  ∗  d x = 0.  z D ϕ (x) j j     j=1 γk − λ j

We deduce from the above that N  j=1

zj

1 D ϕ ∗j (x) = 0, a.e. on ∂O, γk − λ j

for all k = 1, . . . , N . This gives an N × N linear homogeneous system, with the unknowns z j , j = 1, . . . , N , for almost all x ∈ ∂O. The determinant of the matrix of the corresponding system is  1  1 1  γ1 −λ1 D ϕ1∗ (x) γ1 −λ D ϕ2∗ (x) . . . γ1 −λ D ϕ N∗ (x)  2 N  1  1 1  ∗  ∗  ∗    γ2 −λ1 D ϕ1 (x) γ2 −λ2 D ϕ2 (x) . . . γ2 −λ N D ϕ N (x)   ........................................................................     1 D ϕ ∗ (x) 1 D ϕ ∗ (x) . . . 1 D ϕ ∗ (x)  1 2 N γ N −λ1 γ N −λ2 γ N −λ N   1 1 1  . . . γ1 −λ ⎛ ⎞  γ1 −λ γ1 −λ2 1 N   N 1 1 1   . . . γ2 −λ N  = ⎝ D ϕ ∗j (x)⎠  γ2 −λ1 γ2 −λ2   ...................................  j=1 1 1  1 . . . γ N −λ N  γ N −λ1 γ N −λ2 ⎛ ⎞   j−1 N N j−1    (λ − λ )(γ − γ ) (−1) j k j k = ⎝ D ϕ ∗j (x)⎠ = 0, γ j − λ j k=1 (γk − λ j )(γ j − λk ) j=1 j=2

(2.23)

N  at least for some x ∈ ∂O, by virtue of the unique continuation property of D ϕi∗ i=1  N assumed in (A5), the fact that the set λ j j=1 contains distinct elements, and the inequality ρ < γ1 < γ 2 < · · · < γ N . This implies that the above homogeneous system has a unique solution, namely the trivial one z = 0. This is in contradiction to our assumption. Hence we conclude that  the sum B1 + · · · + B N is indeed an invertible matrix. Proposition 2.1 enables us to define the matrix A := (B1 + B2 + · · · + B N )−1 . Now let us introduce the following proportional-type feedback laws:

(2.24)

2.2 The Design of the Boundary Stabilizer

29

⎞ ⎛ 1 ⎞ D ϕ1∗ (x)  z(t), ϕ1∗  γk −λ1 ⎜ z(t), ϕ ∗  ⎟ ⎜ 1 D ϕ2∗ (x) ⎟ 2 ⎟ ⎜ γk −λ2 ⎟ vk (z(t))(= vk (t, x)) := A ⎜ ⎝ .............. ⎠ , ⎝ ....................... ⎠ , 1 z(t), ϕ N∗  D ϕ N∗ (x) γk −λ N N t ≥ 0, x ∈ ∂O, for k = 1, 2, . . . , N . 

⎛

(2.25)

We recall that ·, · N denotes the standard scalar product in C N . Then we define v as v(z) := v1 (z) + v2 (z) + · · · + v N (z), which in condensed form can be written as ⎞ ⎛  ∗⎞ ⎛ D ϕ1  z(t), ϕ1∗   N  ⎜ z(t), ϕ ∗  ⎟ ⎜ D ϕ ∗ ⎟ 2 2 ⎟ ⎟ ⎜ ⎜ , , where Λ := Λγk . v = ΛS A ⎝ S ............ ⎠ ⎝ ........ ⎠ k=1 D ϕ N∗ z(t), ϕ N∗  N

(2.26)

Next, we lift the boundary control v into Eq. (2.13). To achieve this, by (2.13) and N  (2.16), setting η := z − Dγk vk , we have k=1 N   ∂Dγk vk ∂η + Aη + G(z) = − − ADγk vk ∂t ∂t k=1

=−

N N N   ∂  Dγk vk − 2 λ j Dγk vk , ϕ ∗j ϕ j + γk Dγk vk . ∂t k=1 k, j=1 k=1

(2.27) Thus N N  ∂  ∂η + Aη = − Dγ vk + G (z), η(0) = z o − Dγk vk (z(0)), ∂t ∂t k=1 k k=1

where G (z) := −G(z) − 2

N  k, j=1

λ j Dγk vk (z), ϕ ∗j ϕ j

+

N 

γk Dγk vk (z).

k=1

Recall that −A generates a C0 -semigroup. Then the variation of constants formula yields

30

2 Stabilization of Abstract Parabolic Equations

η(t) = e−tA η(0) −



t

e−(t−s)A

0

 t N ∂  Dγk vk ds + e−(t−s)A G (z(s))ds. ∂s k=1 0

Some integration by parts performed on the first integral term leads us to z(t) = e

−tA

 zo +

t

e

−(t−s)A

˜ A

0

N 



t

Dγk vk ds +

k=1

e−(t−s)A G (z(s))ds,

(2.28)

0

˜ stands for the extension of the operator A to the whole space L 2 (O) (see where A (1.2) in Chap. 1). For the sake of notational simplicity, in the sequel we will omit writing the symbol ˜, but keep in mind that by A we refer, in fact, to the extended operator of A. Hence, we finally get that (2.13) is equivalent to N  dz + Az + G(z) = (γk I + A) Dγk vk (z) dt k=1 N 

−2

λ j Dγk vk (z), ϕ ∗j ϕ j ,

(2.29) t ≥ 0; z(0) = z o .

k, j=1

In order to show the stability of (2.29), we first consider only the linear part of it, and obtain the following result. Theorem 2.1 Under (A1)–(A5) and (A4.1), the unique solution to the linear equation N  dz + Az = (γk I + A) Dγk vk (z) dt k=1 (2.30) N  ∗ −2 λ j Dγk vk (z), ϕ j ϕ j , t ≥ 0; z(0) = z o , k, j=1

satisfies

z(t)2 ≤ Ce−ρt z o 2 , ∀t ≥ 0,

for some positive constant C. Proof We represent the solution z to Eq. (2.30) as z = z u + z s , where z u = PN z and z s = (I − PN )z, with PN the projector defined by (2.4). In this way, (2.30) can be split as

(2.31)

2.2 The Design of the Boundary Stabilizer

31

 dz u + Au z u = (γk I + Au ) Dγk vk (z u ) dt k=1 N

on Xu :

−2

N 

λ j Dγk vk (z u ), ϕ ∗j ϕ j ,

(2.32)

t ≥ 0; z u (0) = PN z o ,

k, j=1

and  dz s + As z s = (γk I + As ) Dγk vk (z u ), t ≥ 0; z s (0) = (I − PN )z o . dt k=1 (2.33) N

on Xs : Here

Au := PN A and As := (I − PN )A. Recall that the spaces Xu := PN L 2 (O) and Xs := (I − PN )L 2 (O) are invariant with respect to A, and we have that  N  ∞ σ (Au ) = λ j j=1 and σ (As ) = λ j j=N +1 . Note that by the definition of vk given by (2.25), we have vk = vk (z u ). That is why in (2.32) and (2.33) we wrote vk (z u ). We see that (2.32) is a finite-dimensional system, while (2.33) is infinitedimensional. However, −As generates a C0 -analytic semigroup in Xs , and together with σ (As ) ⊂ {λ ∈ C : λ ≥ ρ} , this implies that

e−tAs  L(L 2 (O ),L 2 (O )) ≤ Ce−ρt , ∀t ≥ 0,

(2.34)

which tells us that the infinite-dimensional system (2.33) is governed by an exponentially asymptotically stable operator. Consequently, one may expect that (2.33) is stable. Hence, the main effort will be to prove the stability of the remaining system (2.32). For later purposes, we show that ⎛

⎞ ⎞ ⎛ Dγk vk , ϕ1∗  z(t), ϕ1∗  ⎜ Dγk vk , ϕ ∗  ⎟ ⎜ z(t), ϕ ∗  ⎟ 2 ⎟ 2 ⎟ ⎜ ⎜ ⎝ .................. ⎠ = −Bk A ⎝ ................ ⎠ , Dγk vk , ϕ N∗  z(t), ϕ N∗ 

(2.35)

where the Bk were introduced in (2.22) above, for k = 1, . . . , N . This is indeed so. We have by (2.25) that

32

2 Stabilization of Abstract Parabolic Equations

⎞ ⎛ ⎞ 1 Dγk D ϕ1∗ , ϕ ∗j   z(t), ϕ1∗  γk −λ1 1 ∗  ∗ ⎜ z(t), ϕ ∗  ⎟ ⎜ ⎟ 2 ⎟ , ⎜ γk −λ2 Dγk D ϕ2 , φ j  ⎟ Dγk vk , ϕ ∗j  = A ⎜ ⎝ ............... ⎠ ⎝ ................................. ⎠ , j = 1, . . . , N . 1 z(t), ϕ N∗  Dγk D ϕ N∗ , ϕ ∗j  γ −λ ⎛



k

N

N

Then by relation (2.17), it follows that ⎞ ⎞ ⎛ − 1 D ϕ1∗ , D ϕ ∗j 0  z(t), ϕ1∗  (γk −λ1 )(γk −λ j ) 1 ⎜ z(t), ϕ ∗  ⎟ ⎜ − (γk −λ2 )(γ D ϕ2∗ , D ϕ ∗j 0 ⎟ ⎟ 2 ⎟,⎜ k −λ j ) Dγk vk , ϕ ∗j  = A ⎜ , ⎜ ⎝ .............. ⎠ ⎝ ............................................. ⎟ (2.36) ⎠ 1  ∗  ∗ z(t), ϕ N∗  D ϕ , D ϕ  − (γk −λ N )(γ 0 N j k −λ j ) N j = 1, . . . , N , 

⎛

from which we immediately obtain (2.35). In particular, this yields that ⎛

⎞ ⎞ ⎛ Dγk vk , ϕ1∗  z(t), ϕ1∗  N N   ⎜ Dγk vk , ϕ ∗  ⎟ ⎜ z(t), ϕ ∗  ⎟ 2 ⎟=− 2 ⎟ ⎜ ⎜ B A k ⎝ ................ ⎠ ⎝ ............... ⎠ k=1 k=1 Dγk vk , ϕ N∗  z(t), ϕ N∗  ⎞ ⎛ z(t), ϕ1∗   N   ⎜ z(t), ϕ ∗  ⎟ 2 ⎟ =− Bk A ⎜ ⎝ ............... ⎠ k=1 z(t), ϕ N∗  ⎞ ⎛ z(t), ϕ1∗  ⎜ z(t), ϕ ∗  ⎟ 2 ⎟ = −⎜ ⎝ ................. ⎠ , z(t), ϕ N∗ 

(2.37)

if we recall that A is the inverse of the sum of the Bk . Returning to (2.32) and representing the solution z u as zu =

N 

z j (t)ϕ j ,

j=1

where z j = z, ϕ ∗j , j = 1, 2, . . . , N , by (2.6), (2.7), (2.35), and (2.37), we may rewrite (2.32) as 1 1 γk Bk AZ , t > 0; Z (0) = Zo , Zt + ΛZ − 2 2 k=1 N

Zt + ΛZ = or equivalently,

(2.38)

2.2 The Design of the Boundary Stabilizer

Zt = −γ1 Z +

N 

33

(γ1 − γk )Bk AZ , t > 0; Z (0) = Zo .

(2.39)

k=2

Here

⎞ z(t), ϕ1∗  ⎜ z(t), ϕ ∗  ⎟ 2 ⎟ Z := ⎜ ⎝ .............. ⎠ and Λ := diag(λ1 , λ2 , . . . , λ N ). z(t), ϕ N∗  ⎛

Recall that Bk , k = 1, . . . , N , are positive semidefinite Hermitian matrices (by the definition of Bk , Λγk and the fact that B is a Gram matrix). Therefore, Bk q, q N ≥ 0, ∀q ∈ C N , k = 1, . . . , N . Consequently, A = (B1 + · · · + B N )−1 is a positive definite Hermitian matrix (see Chap. 1, for details). Thus one can define another positive definite Hermitian matrix, 1 denoted by A 2 , such that 1

Aq, q N = A 2 q2N , ∀q ∈ C N . 1

(A 2 is the square root of A; for details, see (1.3) or [34].) Now let us scalar multiply Eq. (2.39) by AZ to get N  1 d 1 1 A 2 Z (t)2N = −γ1 A 2 Z (t)2N + (γ1 − γk )Bk AZ (t), AZ (t) N , 2 dt k=2 (2.40) which leads to 1 d 1 1 A 2 Z (t)2N ≤ −γ1 A 2 Z (t)2N , t ≥ 0, 2 dt

since γ1 − γk < 0, k = 2, . . . , N . Here  ·  N stands for the Euclidean norm in C N . 1 The above relation implies the exponential decay of Z in the A 2 ·  N -norm, i.e., A 2 Z (t)2N ≤ e−2γ1 t A 2 Zo 2N , t ≥ 0, 1

1

1

where using the fact that A 2 is a positive definite Hermitian matrix, we finally arrive at Z (t)2N ≤ Ce−2γ1 t Zo 2N , t ≥ 0,

(2.41)

for some positive constant C. (We note that A·, · N is a Lyapunov function for the differential system (2.32).) Returning from C N to Xu , relation (2.41) yields

34

2 Stabilization of Abstract Parabolic Equations

z u (t)2 = Z (t)2N ≤ Ce−2γ1 t PN z o 2 , t ≥ 0.

(2.42)

Finally, (2.33), (2.34), and (2.42) imply that z s (t)2 ≤ Ce−ρt (I − PN )z o 2 , ∀t ≥ 0.

(2.43)

Hence the conclusion of the theorem follows immediately by (2.42) and (2.43),  since z = z u + z s . Now let us return to the full nonlinear system (2.29). In order to be able to show its stability, we need to strengthen the assumptions on the nonlinear part F0 to (A6) |F0 (y)| ≤ C(|y|m + 1), ∀y ∈ R, where 0 < m < ∞ for d = 1, 2 and m = 3 for d = 3. Then we have the following result. Theorem 2.2 Let 1 ≤ d ≤ 3. Assume that (A1)–(A6) hold together with (A4.1). Then for each z o ∈ Uo , there exists a unique solution z to the equation N N   dz + Az + G(z) = λ j Dγk vk (z), ϕ ∗j ϕ j , (γk I + A) Dγk vk (z) − 2 (2.44) dt k=1 k, j=1

t ≥ 0; z(0) = z o ,

satisfying, for some constants C, c > 0, z(t)2 ≤ Ce−ct z o 2 , ∀t ≥ 0. Here

  Uo := z o ∈ L 2 (O); z o  ≤ σ ,

for some sufficiently small σ > 0. Proof Let us define A z := Az −

N  k=1

(γk I + A) Dγk vk (z) + 2

N 

λ j Dγk vk (z), ϕ ∗j ϕ j , z ∈ D(A ),

k, j=1

with D(A ) = D(A). By Theorem 2.1, we know that −A generates a C0 -analytic semigroup in L 2 (O) that satisfies e−tA 2L(L 2 (O ),L 2 (O )) ≤ Ce−ρt , ∀t ≥ 0.

(2.45)

2.2 The Design of the Boundary Stabilizer

35

We may equivalently rewrite (2.44) as z(t) = e

−tA

 zo +

t

e−(t−s)A G(z(s))ds, t ≥ 0.

(2.46)

0

We are going to show that for z o  ≤ σ sufficiently small, Eq. (2.46) has a unique solution z ∈ L m+1 (0, ∞; H 1 (O)). To this end, we will proceed as in [19]. More precisely, we consider the map  : L m+1 (0, ∞; H 1 (O)) → L m+1 (0, ∞; H 1 (O)) defined by  t

z := e−tA z o +

e−(t−s)A G(z(s))ds,

0

and we shall show that for r sufficiently small, it maps the ball   B(0, r ) := z ∈ L m+1 (0, ∞; H 1 (O)) : z L m+1 (0,∞;H 1 (O )) ≤ r into itself and is a contraction on B(0, r ), for z o  sufficiently small and r suitably chosen. By (A6), the definition of G, and the Sobolev embedding theorem (for dimension 1 ≤ d ≤ 3), we have that G(z 1 ) − G(z 2 ) ≤ Cz 1 − z 2  H 1 (O ) (z 1 mH 1 (O ) + z 2 mH 1 (O ) ), while G(z) ≤ Czm+1 H 1 (O ) . Then arguing as in the proof of [10, Theorem 3.5], one may show that  is a contraction on B(0, r ). Hence, via the contraction mapping theorem, for z o  ≤ σ sufficiently small, Eq. (2.46) has a unique solution z ∈ L m+1 (0, ∞; H 1 (O)). By a standard argument, such as the one in [10, Proposition 5.9], this implies also that for some constants C, c > 0, z(t) ≤ Ce−ct z o 2 , ∀t ≥ 0, thereby completing the proof.



To conclude this section, we recall the notation z := y − yˆ and see that Theorems 2.1 and 2.2 imply the following stabilization result for the original system (2.12). Theorem 2.3 Under (A1)–(A6) and (A4.1), for 1 ≤ d ≤ 3, we have that for each yo ∈ Uo , there exists a unique solution y to the equation  ∂y

+ Ay + F0 (y) = 0 in (0, ∞) × O; ∂t B.C.(y + yˆ , u(y)) on (0, ∞) × ∂O; y(0) = yo in O,

satisfying, for some constants C, c > 0,

(2.47)

36

2 Stabilization of Abstract Parabolic Equations

y(t) − yˆ 2 ≤ Ce−ct yo − yˆ 2 , ∀t ≥ 0. Here

⎞ ⎛  ∗⎞ D ϕ1  y(t) − yˆ , ϕ1∗  ⎜ y(t) − yˆ , ϕ ∗  ⎟ ⎜ D ϕ ∗ ⎟ 2 2 ⎟ ⎟ ⎜ ⎜ , u := Λ S A ⎝ ................... ⎠ ⎝ ......... ⎠ D ϕ N∗ y(t) − yˆ , ϕ N∗  N

and

  Uo := yo ∈ L 2 (O); yo − yˆ  ≤ σ ,

⎛



for some sufficiently small σ > 0. Example 2.5 The corresponding stabilization results in Theorems 2.1–2.3, expressed for Eq. (2.9) in Example 2.2, are as follows. First, for the linearized case, we have the following theorem. Theorem 2.4 Assume that hypothesis (A4.1) holds and that f is a C 1 function such that f  ∈ C(R). Then the solution y to the equation ⎧ yt (t, x) − Δy(t, x) + f  ( yˆ (x))y(t, x) = 0, t > 0, x ∈ O, ⎪ ⎪ ⎪ ⎞ ⎛ ∂ϕ1 ⎞ ⎛ ⎪ ⎪ y(t), ϕ1  ⎪   ∂n ⎪ ⎪ ⎜ y(t), ϕ2  ⎟ ⎜ ∂ϕ2 ⎟ ⎪ ⎨ ⎜ ⎟ ⎟ ⎜ ∂n , , t > 0, x ∈ Γ1 , y(t, x) = Λ S A ⎝ .............. ⎠ ⎝ ...... ⎠ ⎪ ∂ϕ N ⎪ y(t), ϕ N  ⎪ ⎪ ∂n N ⎪ ⎪ ∂ ⎪ y(t, x) = 0, t > 0, x ∈ Γ2 , ⎪ ∂n ⎪ ⎩ y(0, x) = yo , x ∈ Ω,

(2.48)

satisfies the exponential decay y(t)2 ≤ Ce−ρt yo 2 , t ≥ 0,

(2.49)

for a prescribed ρ > 0 and a constant C > 0. Here Λ S := Λγ1 + · · · + Λγ N with ⎞ ... 0 ⎜ 0 ... 0 ⎟ ⎟ Λγk := ⎜ ⎝ ................................ ⎠ , k = 1, · · · , N . 1 0 0 . . . γk −λ N ⎛

1 γk −λ1

0

1 γk −λ2

Moreover, A := (B1 + B2 + · · · + B N )−1 , where Bk := Λγk BΛγk , k = 1, . . . , N , with B the Gram matrix

2.2 The Design of the Boundary Stabilizer

37

⎞ 1 1 1 N , ∂ϕ1   ∂ϕ , ∂ϕ2  . . .  ∂ϕ , ∂ϕ   ∂ϕ ∂n ∂n 0 ∂n ∂n 0 ∂n ∂n 0 ⎟ ⎜  ∂ϕ2 , ∂ϕ1   ∂ϕ2 , ∂ϕ2  . . . 2 N  ∂ϕ , ∂ϕ  ⎟ ∂n ∂n 0 ∂n ∂n 0 ∂n ∂n 0 B := ⎜ ⎝ .................................................................. ⎠ , N 1 N 2 N N , ∂ϕ   ∂ϕ , ∂ϕ  ...  ∂ϕ , ∂ϕ   ∂ϕ ∂n ∂n 0 ∂n ∂n 0 ∂n ∂n 0 ⎛

 ∞  ∞ where ·, ·0 stands for the scalar product in L 2 (Γ1 ). Finally, λ j j=1 , ϕ j j=1 denote the eigenvalues and eigenfunctions of the linear operator −Δ + f  ( yˆ ), respectively; and γ1 , . . . , γ N are some real positive numbers. And for the full nonlinear case, we have the following. Theorem 2.5 Assume 1 ≤ d ≤ 3 and that hypothesis (A4.1) holds. In addition, assume also that there exist C1 > 0, q ∈ N, αi > 0, i = 1, . . . , q, when d = 1, 2, and 0 < αi ≤ 1, i = 1, . . . , q, when d = 3, such that 

| f (y)| ≤ C1

 q 

 αi

|y| + 1 , ∀y ∈ R.

i=1

Then the solution to the closed-loop nonlinear equation ⎧ yt (t, x) − Δy(t, x) + f (y) = 0, t > 0, x ∈ O, ⎪ ⎪ ⎪ ⎞ ⎛ ∂ϕ1 ⎞ ⎛ ⎪ ⎪ y(t) − yˆ , ϕ1  ⎪   ∂n ⎪ ⎪ ⎜ y(t) − yˆ , ϕ2  ⎟ ⎜ ∂ϕ2 ⎟ ⎪ ⎪ ⎟ ⎜ ∂n ⎟ ⎪ ⎨ y(t, x) = Λ S A ⎜ ⎝ .................... ⎠ , ⎝ ...... ⎠ + yˆ (x), ∂ϕ N y(t) − yˆ , ϕ N  ⎪ ⎪ ∂n N ⎪ ⎪ ⎪ t > 0, x ∈ Γ1 , ⎪ ⎪ ⎪ ∂ ⎪ y(t, x) = 0, t > 0, x ∈ Γ , ⎪ 2 ∂n ⎪ ⎩ y(0, x) = yo , x ∈ Ω,

(2.50)

satisfies the exponential decay y(t) − yˆ 2 ≤ Ce−ρt yo − yˆ 2 , t ≥ 0, for a prescribed ρ > 0 and a constant C > 0, provided that yo − yˆ  is small enough. For the notation A, Λs , ϕ j , . . ., we refer to Theorem 2.4.

2.2.2 The Semisimple Eigenvalues Case In this section, we drop the hypothesis (A4.1) but keep the more general one (A4). In this case, the result in Proposition 2.1 may fail to hold, since the determinant given by (2.46) may be zero. In other words, in this case, the sum B1 + · · · + B N may be a singular matrix. To overcome this problem, we slightly perturb the spectrum of the

38

2 Stabilization of Abstract Parabolic Equations

linear operator A. To illustrate our approach, let us assume, for instance, that λ1 = λ2 and λ j = λk , ∀ j, k = 2, 3, . . . , N , j = k (other cases can be treated in a similar manner). This time, the operator introduced in (2.16) is given as follows: for β ∈ L 2 (∂O), we define Dγ β := z˜ , where z˜ is a solution to ⎧ ⎪ ⎨ ⎪ ⎩

A˜z − 2

N 

λk ˜z , ϕk∗ ϕk (x) − δ˜z , ϕ1∗ ϕ1 + γ z˜ = 0 in O,

(2.51)

k=1

B.C.(˜z , β) on ∂O,

for some δ > 0. We choose ρ < γ1 < γ2 := γ1 + N 1−1 < γ3 := γ1 + N 1−2 · · · < γ N := γ1 + 1, with γ1 large enough that for δ = γ14 , Eq. (2.51) is well posed for each γi , 1 i = 1, . . . , N , and such that (λ1 + δ), λ2 , . . . , λ N are distinct. One may easily show that  1 β, D ϕ ∗j 0 , j = 1, − γ −δ−λ ∗ 1 Dγ β, ϕ j  = (2.52) 1 − γ −λ β, D ϕ ∗j 0 , j = 2, . . . , N . j Now the feedback has the form ⎞ ⎛  ∗⎞ ⎛ D ϕ1  z(t), ϕ1∗   ⎜ z(t), ϕ ∗  ⎟ ⎜ D ϕ ∗ ⎟ 2 2 ⎟ ⎟ ⎜ v = ΛS A ⎜ ⎝ ............ ⎠ , ⎝ ....... ⎠ , D ϕ N∗ z(t), ϕ N∗  N where Λ S :=

N 

(2.53)

Λγk , with Λγk given this time as

k=1

 Λγk := diag

1 1 1 , ,..., γk − δ − λ1 γk − λ 2 γk − λ N

,

(2.54)

while the matrix A is given similarly as in (2.24) (we mention that since (λ1 + δ), λ2 , . . . , λ N are distinct, a result similar to that in Proposition 2.1 can be proved, showing in this way that A is well defined). Computations similar to those in (2.27)– (2.29) yield that system (2.13) is equivalent to N N   dz + Az = λ j Dγk vk (z), ϕ ∗j ϕ j (γk I + A) Dγk vk (z) − 2 dt k=1 k, j=1

−δ

N  Dγk vk , ϕ1∗ ϕ1 , t ≥ 0; z(0) = z o . k=1

2.2 The Design of the Boundary Stabilizer

39

The main results in the present context are similar to those above. First, concerning the linearized system, we have the following theorem. Theorem 2.6 Under (A1)–(A5), the unique solution to the equation N N   dz λ j Dγk vk (z), ϕ ∗j ϕ j + Az = (γk I + A) Dγk vk (z) − 2 dt k=1 k, j=1 N  −δ Dγk vk , ϕ1∗ ϕ1 , t ≥ 0; z(0) = z o ,

(2.55)

k=1

satisfies

z(t)2 ≤ Ce−ρt z o 2 , ∀t ≥ 0,

(2.56)

for some positive constant C. Proof We argue as in the proof of Theorem 2.1. We get this time that the finitedimensional unstable part of the linear system (2.55) (which corresponds to (2.32)), has the equivalent form (see all the computations between (2.35) and (2.39)) Zt = −γ1 Z +

N  δ (γ1 − γk )Bk AZ − O; Z (0) = Zo , 2 k=2

⎞ z(t), ϕ1∗  ⎟ ⎜ 0 ⎟ O := ⎜ ⎝ .............. ⎠ . 0 ⎛

where

Then scalar multiplying the above equation by AZ yields d 1 1 A 2 Z (t)2N ≤ −2γ1 A 2 Z (t)2N + δAZ (t)2N , t ≥ 0, dt

(2.57)

where A stands for the classical Euclidean induced norm of the matrix A. Denote by λ1 (A) > 0 the first eigenvalue of A and integrate over time in (2.57). This yields λ1 (A)Z (t)2N ≤ e−2γ1 t A 2 Z0 2N + 1

 0

t

e−2γ1 (t−s) δAZ (s)2N ds,

(2.58)

where making use of Grönwall’s lemma gives us Z

(t)2N

1 1 A 2 Z0 2N exp ≤ λ1 (A)

"

# δA − 2γ1 t , t ≥ 0. λ1 (A)

(2.59)

40

2 Stabilization of Abstract Parabolic Equations

Let us denote by bi j , i, j = 1, . . . , N , the entries of the matrix B1 + B2 + · · · + B N . By the definition of Bi (see (2.22)) and the constants γi , i = 1, . . . , N and δ (see after (2.51)), we have that lim γ 2 |bi j | γ1 →∞ 1

∈ R+ , ∀i, j = 1, . . . , N .

Let bi∗j denote the entries of the adjoint of the matrix B1 + · · · + B N . By virtue of the definition of the adjoint matrix and the above observation, we deduce that lim γ 2(N −1) |bi∗j | γ1 →∞ 1

∈ R+ , ∀i, j = 1, . . . , N .

Besides this, the above observation also implies that lim γ 2(N +1) |det (B1 γ1 →∞ 1

+ · · · + B N )| = +∞.

In other words, we have |bi∗j | ≤ ci j

    1 2(N +1)   , i, j = 1, . . . , N , and ,  det (B + · · · + B )  ≤ cγ1 2(N −1) 1 N γ1 1

for some positive constants ci j , c, i, j = 1, . . . , N , independent of γ1 , for γ1 large enough. This yields, if we denote by ai j , i, j = 1, . . . , N , the entries of the matrix A = (B1 + · · · + B N )−1 , that there exists some constant C > 0, independent of γ1 , such that |ai j | ≤ Cγ14 , i, j = 1, . . . , N . Consequently, A ≤ Cγ14 ,

(2.60)

for γ1 large enough. In conclusion, for γ1 large enough, there exists some μ > 0 such that 1 δA A − 2γ1 = 4 − 2γ1 ≤ −μ, − 2γ1 ≤ C N 2 λ1 (A) λ1 (A) γ1 λ1 (A) since

1 λ1 (A)

→ 0 for γ1 → ∞. This, together with (2.59), implies Z (t)2N ≤

1 1 e−μt A 2 Z0 2N , t ≥ 0, λ1 (A)

which represents the exponential decay of the first N modes of z. The rest of the proof mimics the proof of Theorem 2.1, and it is therefore omitted.  Finally, based on the above result, one can immediately deduce the following counterpart of Theorem 2.3. Theorem 2.7 Under (A1)–(A6), for 1 ≤ d ≤ 3, we have that for each yo ∈ Uo , there exists a unique solution y to the equation

2.2 The Design of the Boundary Stabilizer

41

 ∂y

+ Ay + F0 (y) = 0 in (0, ∞) × O; ∂t B.C.(y + yˆ , u(y)) on (0, ∞) × ∂O; y(0) = yo in O,

(2.61)

satisfying, for some constants C, c > 0, y(t) − yˆ 2 ≤ Ce−ct yo − yˆ 2 , ∀t ≥ 0. Here

⎞ ⎛  ∗⎞ D ϕ1  y(t) − yˆ , ϕ1∗  ⎜ y(t) − yˆ , ϕ ∗  ⎟ ⎜ D ϕ ∗ ⎟ 2 ⎟ ⎜ 2 ⎟ u := Λ S A ⎜ ⎝ .................. ⎠ , ⎝ ........ ⎠ , D ϕ N∗ y(t) − yˆ , ϕ N∗  N 

⎛

with Λ S given by (2.54) and A by (2.24); and   Uo := yo ∈ L 2 (O); yo − yˆ  ≤ σ , for some sufficiently small σ > 0.

2.3 A Numerical Example In this section, we further particularize the model in Example 2.2 (see also Example 2.3), in the sense that we take the space dimension to be equal to one, and f of the form f (y) := −αy + βy 2 , obtaining thereby the 1-dimensional Fischer model, which reads as ⎧ ⎨ yt (t, x) − yx x (t, x) − αy(t, x) + βy 2 (t, x) = 0, (t, x) ∈ (0, ∞) × (0, 1), y(t, 0) = 0, y(t, 1) = u(t), t ∈ (0, ∞), ⎩ y(0, x) = y0 (x), x ∈ (0, 1), (2.62) where α, β are positive constants. Fisher’s equation is a nonlinear parabolic equation first proposed by Fisher to model the advance of a mutant gene in an infinite onedimensional habitat [54]. Moreover, Fisher’s equation has been used as a basis for a wide variety of models for the spatial spread of genes in a population, chemical wave propagation, flame propagation, branching Brownian motion processes, and even nuclear reactor theory [36, 108]. It is well known that the uncontrolled Fisher’s equation is unstable. The goal is to stabilize the null solution via the proportionalfeedback law designed in the previous section. To this end, it is clear that, arguing as before, one is able to obtain similar results as in Theorems 2.4 and 2.5, in Example 2.5. More precisely, we have the following result.

42

2 Stabilization of Abstract Parabolic Equations

Theorem 2.8 The solution y to the equation ⎧ yt (t, x) − yx x (t, x) − αy(t, x) = 0, t > 0, x ∈ (0, 1), ⎪ ⎪ ⎪ ⎪ y(t, 0)) = 0, ⎪ ⎪ ⎞ ⎛ ⎞ ⎛ ⎪ ⎪ (ϕ1 )x (1)  y(t), ϕ1  ⎨  ⎜ y(t), ϕ2  ⎟ ⎜ (ϕ2 )x (1) ⎟ ⎟ ⎜ ⎟ y(t, 1) = Λ S A ⎜ ⎪ ⎪ ⎝ .............. ⎠ , ⎝ .............. ⎠ , t > 0, ⎪ ⎪ ⎪ ⎪ y(t), ϕ N  (ϕ N )x (1) ⎪ ⎪ N ⎩ y(0, x) = yo , x ∈ (0, 1),

(2.63)

satisfies the exponential decay y(t)2L 2 (0,1) ≤ Ce−ρt yo 2L 2 (0,1) , t ≥ 0,

(2.64)

for a prescribed ρ > 0 and a constant C > 0. Here Λ S := Λγ1 + · · · + Λγ N with ⎞ ... 0 ⎜ 0 ... 0 ⎟ ⎟ Λγk := ⎜ ⎝ ................................ ⎠ , k = 1, . . . , N . 1 0 0 . . . γk −λ N ⎛

1 γk −λ1

0

1 γk −λ2

Moreover, A := (B1 + B2 + · · · + B N )−1 , where Bk := Λγk BΛγk , k = 1, . . . , N , with B being the Gram matrix ⎞ ⎛ (ϕ1 )x (1)(ϕ1 )x (1) (ϕ1 )x (1)(ϕ2 )x (1) . . . (ϕ1 )x (1)(ϕ N )x (1) ⎜ (ϕ2 )x (1)(ϕ1 )x (1) (ϕ2 )x (1)(ϕ2 )x (1) . . . (ϕ2 )x (1)(ϕ N )x (1) ⎟ ⎟ B := ⎜ ⎝ ..................................................................................... ⎠ , (ϕ N )x (1)(ϕ1 )x (1) (ϕ N )x (1)(ϕ2 )x (1) . . . (ϕ N )x (1)(ϕ N )x (1)  ∞  ∞ where λ j j=1 , ϕ j j=1 denote the eigenvalues and eigenfunctions of the linear operator −∂x x − α, respectively; and γ1 , . . . , γ N are some positive numbers. And for the full nonlinear case, we have this theorem: Theorem 2.9 The solution to the closed-loop nonlinear equation ⎧ yt (t, x) − yx x (t, x) − αy(t, x) + βy 2 (t, x) = 0, t > 0, x ∈ (0, 1), ⎪ ⎪ ⎪ ⎪ y(t, 0) = 0, ⎪ ⎪ ⎞ ⎛ ⎞ ⎛ ⎪ ⎪ (ϕ1 )x (1)  y(t), ϕ1  ⎨  ⎜ y(t), ϕ2  ⎟ ⎜ (ϕ2 )x (1) ⎟ ⎟ ⎜ ⎟ y(t, 1) = Λ S A ⎜ ⎪ ⎪ ⎝ .............. ⎠ , ⎝ ............ ⎠ , t > 0, ⎪ ⎪ ⎪ ⎪ y(t), ϕ N  (ϕ N )x (1) ⎪ ⎪ N ⎩ y(0, x) = yo , x ∈ (0, 1),

(2.65)

2.3 A Numerical Example

43

satisfies the exponential decay y(t)2L 2 (0,1) ≤ Ce−ρt yo 2L 2 (0,1) , t ≥ 0, for a prescribed ρ > 0 and a constant C > 0, provided that yo  L 2 (0,1) is small enough. For the notation A, Λs , ϕ j , . . ., we refer to Theorem 2.8. Now let us study the problem numerically. It is easy to see that when α > (2π )2 , there is more than one unstable eigenvalue. We take α = 50, β = 0.30, and set the initial profile to be u 0 (x) = 5xe x , x ∈ [0, 1]. In this case, for decay rate ρ ∈ (0, (3π )2 − 50], a two-dimensional feedback controller can be designed to stabilize the system. For a larger rate ρ ∈ ((3π )2 − 50, (4π )2 − 50], we need to design a three-dimensional feedback controller. If one wants to archive an even bigger decay rate, a feedback controller of larger dimension can be accordingly designed. We can observe from Fig. 2.1 that the state is unstable without control. First, we take ρ = 10. In this case, we need to take γ1 > α such that the Dirichlet operator, given by (2.18), is well defined. Taking γ1 = 60, γ2 = 70, and simulating the controlled system with the above parameters, we get Fig. 2.2. Second, we take ρ = 50. In this case, we need to take γ1 > α + λ3 such that Eq. (2.18) has a unique solution. Taking γ1 = 90, γ2 = 100, γ3 = 110, and simulating the controlled system with the above parameters, we get Fig. 2.3. It is obvious that the state with 3-D control decays faster than that with 2-D control. We want to stress that the decay rate of the controlled system may be larger than the value of ρ we set. For example, in the case ρ = 10, the actual decay rate is greater than 10 but does not exceed min{λ3 , γ1 }. On the other hand, if we fix a decay rate

Fig. 2.1 State of Fisher’s equation without control

44

2 Stabilization of Abstract Parabolic Equations

Fig. 2.2 State of Fisher’s equation with 2-D control

Fig. 2.3 State of Fisher’s equation with 3-D control

ρ, then as the parameter α increases, we may need to design a higher-dimensional feedback controller. The analysis is quite similar to that in the above discussion with respect to ρ for fixed α, and we will not go into details. We take γ1 = 15, γ2 = 20, and then  B1 = π

2

B2 = π

2



1 (45−π 2 )2 −2 (45−π 2 )(45−(2π)2 )

−2 (45−π 2 )(45−(2π)2 ) 4 (45−(2π)2 )2

1 (50−π 2 )2 −2 (50−π 2 )(50−(2π)2 )

−2 (50−π 2 )(50−(2π)2 ) 4 (50−(2π)2 )2

 ,  .

By Theorem 2.9, we know that the control of feedback form  u(t) = F(y)(t) := T A

 $1 $ 10 0

   1 , , 1 y sin 2π xd x y sin π xd x

2

(2.66)

2.3 A Numerical Example

45

exponentially stabilizes Fisher’s equation (as shown in Fig. 2.2). Here  T :=

−π 45−π 2 −π 50−π 2

2π 45−(2π)2 2π 50−(2π)2

 , A = (B1 + B2 )−1 .

(2.67)

As we may realize, it is more practical to use only part of the information about the state. One can take a modified feedback law such as   $b    1 a y sin π xd x u(t) = F(y)(t) := T A $ b , , (2.68) 1 a y sin 2π xd x 2 where [a, b] is a proper subset of [0, 1]. If we fix b = 1 and simulate the closed-loop system with different values for a, we find that for all a ≤ 0.24, the system governed by Fisher’s equation can still be stabilized using the feedback law (2.68). However, the value of a cannot be too small. When a = 0.25, it seems that the system cannot be stabilized with a control of the form (2.68). Some further computations show that the feedback controller u(t) can stabilize the solution of Fisher’s equation not only in L 2 (0, 1), but also in H 1 (0, 1) when the initial data satisfy some compatibility condition. Similarly as above, we take a control of the form (2.68), which uses only part of the state information. Simulating the closed-loop system with b = 1, and with different values for a, we find that when a ≤ 0.24, the solution is exponentially stable in H 1 (0, 1). It is reasonable that when we use less information about the state for the feedback control, the feedback controller is less effective in stabilizing the solution of the closed-loop system.

2.4 Comments The problem of boundary stabilization of the heat equation was first solved in the pioneering work of Triggiani [116]. His approach was based on spectral decomposition, similar to what we do here and to what has been done in many papers on this subject. Then several other methods were proposed for deriving new types of controls. One of the most fruitful is the so-called backstepping method, developed by Krstic and coworkers; see, for example, [1, 27, 29, 38, 87, 114, 124, 127] or the book [75]. Let us briefly present it here, since it can be related to the results of a subsequent chapter. Let us consider the reaction–diffusion equation on (0, 1): 

yt (t, x) = yx x (t, x) + λy(t, x), y(t, 0) = 0, y(t, 1) = u(t).

(2.69)

46

2 Stabilization of Abstract Parabolic Equations

The uncontrolled equation (2.69) (i.e., u ≡ 0) is unstable when λ > 0 is sufficiently large. In other words, the term λy is a source of instability in (2.69), and the natural objective for a boundary feedback is to eliminate it. To this end, one considers the state transformation  x k(x, ξ )y(t, ξ )dξ, w(t, x) = y(t, x) − 0

with the kernel k such that w satisfies the target stable equation 

wt (t, x) = wx x (t, x), w(t, 0) = 0, w(t, 1) = 0.

(2.70)

The control u is given by 

1

u(t) =

k(1, ξ )y(t, ξ )dξ.

0

Thus, the whole problem reduces to finding a kernel k that ensures this passage. Plugging the form of w into Eq. (2.70), one easily deduces that k must obey 

k x x (x, ξ ) − kξ ξ (x, ξ ) = λk(x, ξ ), x, ξ ∈ (0, 1), k(x, 0) = 0, k(x, x) = − λ2 x, x ∈ (0, 1).

(2.71)

These form a well-posed PDE of hyperbolic type in the Goursat form. Moreover, one can obtain explicitly the form of the kernel k, and consequently the form of the feedback u. A more direct method is the so-called design of proportional type feedback, which we use here as well. We begin by mentioning the significant results obtained by Barbu in [12, 13]; see also the monograph [14]. As mentioned before, the design algorithm we developed here is based on the ideas in [13]. So in order to have a clear comparison between what we have presented here and [13], let us briefly describe what is stated and proved in that work. Consider the parabolic equation (see also Examples 2.1, 2.2) 

yt (t, x) = Δy(t, x) + f (x, y(t, x)), in (0, ∞) × O, ∂y y = u on Γ1 , ∂n = 0 on Γ2 .

Under the assumption that the traces the feedback u=η

∂ϕ j ∂n

N  j=1

N j=1

(2.72)

are linearly independent in L 2 (Γ1 ),

μ j y, ϕ j  j

2.4 Comments

47

achieves stability in (2.72). Here η and μ j , j = 1, 2, . . . , N , are positive real param N ∂ϕ eters, while  j are linear combinations of the traces ∂nj such that j=1

% & ∂ϕl j, = δ jl , j, l = 1, 2, . . . , N . ∂n 0 It is clear that such  j can be constructed if and only if the above hypothesis on linear independence holds. Here ·, ·0 stands for the scalar product in L 2 (Γ1 ). Note that this u can be equivalently written as ⎞ ⎛ ∂ϕ1 y, ϕ1  ∂n ⎜ ⎟ ⎜ ∂ϕ2 −1 ⎜ y, ϕ2  ⎟ ⎜ ∂n , u = η Λ1 B ⎝ .......... ⎠ ⎝ ... ∂ϕ N y, ϕ N  

⎛

∂n

⎞  ⎟ ⎟ , ⎠ N

N ∂ where Λ1 = diag(μ1 , . . . , μ N ) and B is the Gram matrix of the system ∂n ϕ j |Γ1 j=1 in L 2 (Γ1 ). One can clearly see the similarity between this u and the control v in (2.26). In the same proportional-type feedback context, we mention as well the recent work of Lasiecka and Triggiani [82], in which the hypothesis of semisimple eigenvalues is dropped, but instead an additional internal controller is inserted into the equations. Other stabilization results are obtained for specific models, and they will be mentioned in the relevant subsequent chapters. The results in this chapter, carried out for the particular case presented in Examples 2.1, 2.2, appeared in the work Munteanu [105], while their formulation here in the general framework for an abstract parabolic differential operator of the type ∂ + A + F0 , obeying assumptions (A1)–(A6), is new. The need to consider the ∂t general abstract context is to emphasize that in all that follows in the chapters below, we are presenting just some particular cases. The whole effort is to show that (A1)– (A6) (especially (A5)) are satisfied for the considered examples. Consequently, the present controller design algorithm is not confined to the considered models, but can be applied by those working on this subject to a larger spectrum of models, generically named parabolic-like equations. The feedback designed here has many advantages: it is linear and of finite-dimensional structure, expressed in a very simple form involving only the eigenfunctions of the linear operator derived from the linearized equations, and is therefore easy to manipulate from the numerical point of view. We mention that one can easily adapt the present feedback control design technique to systems of the form 

yt = Δy + f (x, y), in (0, ∞) × O, ∂y y = u on Γ1 , ∂n = 0 on Γ2 ,

(2.73)

48

where this time, y is a vector

2 Stabilization of Abstract Parabolic Equations

⎛

⎞ y1 ⎜ y2 ⎟ ⎟ y=⎜ ⎝ ... ⎠ . yM

However, we will not go into details about this problem here, since later, we will treat different types of systems such as the Navier–Stokes equations, the magnetohydrodynamics equations, and the phase field equations. Concerning the internal stabilization of (2.73), one may consult the work [25]. Other important topics related to the control of parabolic-like equations, for instance exact and approximate controllability and optimal control, are beyond the scope of this presentation, and we refer to Coron’s book [50] for significant recent results in this direction. The numerical examples were published in the work [86] of of Liu et al.

Chapter 3

Stabilization of Periodic Flows in a Channel

Here we apply the control design algorithm from Chap. 2 to the Navier–Stokes equations, placed in a particular geometry, namely a semi-infinite channel. The high instability of the Navier–Stokes equations is well known as is the fact that the principal way to suppress the turbulence occurring in the dynamics of a fluid is to plug in a stabilizing feedback control. In addition, a Riccati-based robust controller is also constructed.

3.1 Presentation of the Problem The 3-D boundary controlled incompressible Navier–Stokes equations in the geometry of a 2π -periodic channel in the x and z coordinates are given by (x, y, z) ∈ (−∞, ∞) × [0, 1] × (−∞, ∞), and ⎧ u t − νΔu + u ∂∂ux + v ∂u + w ∂u = − ∂∂ px , ⎪ ∂y ∂z ⎪ ⎪ ⎪ ∂p ∂v ∂v ∂v ⎪ ⎪ ⎪ vt − νΔv + u ∂ x + v ∂ y + w ∂z = − ∂ y , ⎪ ⎪ ⎪ ⎪ + v ∂w + w ∂w = − ∂∂zp , w − νΔw + u ∂w ⎪ ∂x ∂y ∂z ⎨ t ∂u + ∂∂vy + ∂w = 0, ∀t ≥ 0, x, z ∈ R, y ∈ (0, 1), ∂x ∂z ⎪ ⎪ ⎪ (u, v, w, p)(t, x + 2π, y, z + 2π ) = (u, v, w, p)(t, x, y, z), ⎪ ⎪ ⎪ ⎪ ∀t ≥ 0, x, z ∈ R, y ∈ (0, 1), ⎪ ⎪ ⎪ ⎪ (u, w)(t, x, 0, z) = (u, w)(t, x, 1, z) = 0, ⎪ ⎩ v(t, x, 0, z) = 0, v(t, x, 1, z) = Ψ (t, x, z), ∀t ≥ 0, x, z ∈ R, y ∈ (0, 1), (3.1) and the initial data (u, v, w)(0, x, y, z) = (u o , vo , wo )(x, y, z), x, z ∈ R, y ∈ (0, 1), © Springer Nature Switzerland AG 2019 I. Munteanu, Boundary Stabilization of Parabolic Equations, Progress in Nonlinear Differential Equations and Their Applications 93, https://doi.org/10.1007/978-3-030-11099-4_3

(3.2) 49

50

3 Stabilization of Periodic Flows in a Channel

where the standard notation includes u for the streamwise velocity, v for the wallnormal velocity, w for the spanwise velocity, p for the pressure, ν for the viscosity coefficient of the fluid, while Ψ is the control. The incompressibility of the fluid is described by the divergence-free condition. In order not to deal with infinite domains, we have assumed that both the velocity field and the pressure are 2π -periodic in the first and last spatial coordinates (for more details on this, one may consult the work [33]). Instead of 2π , one could take any L > 0, and all the results below still hold. However, we keep the particular period 2π for the sake of simplicity of notation. The parabolic Poiseuille profile, denoted here by [(U (y), 0, 0) , −ax] , will be the target of our controlled problem. Here a U (y) = − (y 2 − y), y ∈ (0, 1), 2ν

(3.3)

for some a ∈ R+ . The Poiseuille flow is viewed as a pressure-induced flow in a long duct, being a laminar flow of an incompressible Newtonian fluid of viscosity ν. It is obtained from the stationary uncontrolled equations (3.1) by taking both the wall-normal velocity and the spanwise velocity equal to zero. Besides the obvious use of these equations, namely to model the evolution of fluids in a pipe, they are often and successfully used to model the circulation of blood through the vessels. It is known that for high values of the viscosity coefficient, we have laminar flow (no turbulence), but when ν is low, turbulence occurs. Thus one seeks to plug some control into the system to force the corresponding solution to behave like the steady parabolic Poiseuille profile. We aim here to construct such a control, namely Ψ , using ideas from Chap. 2. Note that the control is with actuation only on the upper wall, and only on the normal component of the velocity field. There is no action, however, in y = 0, on the streamwise and spanwise components or inside the channel. This is of great importance from the practical point of view, since it makes the applicability of the control highly feasible. Next, following the approach from the second chapter, we introduce the linearization of (3.1), around the equilibrium profile (3.3), given by ⎧ u t − νΔu + U ∂∂ux + v ∂U = − ∂∂ px , ⎪ ⎪ ∂y ⎪ ⎪ ∂p ∂v ⎪ ⎪ ⎪ vt − νΔv + U ∂ x = − ∂ y , ⎪ ⎪ ⎪ ⎪ wt − νΔw + U ∂w = − ∂ p , ⎨ ∂x ∂z ∂u ∂v ∂w + + = 0, ∀t ≥ 0, x, z ∈ R, y ∈ (0, 1), ∂x ∂y ∂z ⎪ ⎪ ⎪ ⎪ (u, v, w, p)(t, x + 2π, y, z + 2π ) = (u, v, w, p)(t, x, y, z), ⎪ ⎪ ⎪ ⎪ ⎪ (u, w)(t, x, 0, z) = (u, w)(t, x, 1, z) = 0, ⎪ ⎪ ⎩ v(t, x, 0, z) = 0, v(t, x, 1, z) = Ψ (t, x, z), x, z ∈ R, y ∈ (0, 1),

(3.4)

3.1 Presentation of the Problem

51

and initial data u(0) = u 0 := u o − U, v(0) = v0 := vo , w(0) = w0 := wo . System (3.4) is too complex for applying the argument from Chap. 2 directly, mainly because it contains four unknowns and only three equations. To overcome this defect, we will take advantage of the 2π -periodicity assumption. More exactly, we decompose the system into Fourier modes (for details, see Chap. 1), then reduce the pressure from it.

3.2 The Stabilization Result Let u = u(t, x, y, z) be 2π -periodic in x and z, with u(t, x, y, z) =



u kl (t, y)eikx eilz , u kl = u −k−l ,

k,l∈Z

such that



1

|u kl (y)|2 dy < ∞.

k,l∈Z 0

The corresponding L 2 -norm can be expressed in terms of the Fourier modes as ⎛ u(t) := ⎝

 k,l∈Z

 2π

1

⎞ 21 |u kl (t, y)|2 dy ⎠ .

0

Moreover, for a function f : [0, 1] → C, f = f (y), y ∈ [0, 1], we denote by f  := ∂∂y f the derivative with respect to the y-coordinate. We consider H to be the complexified space of L 2 (0, 1). We denote also by  ·  the norm in H and by < ·, · > the scalar product. Now we return to system (3.4) and rewrite it in terms of Fourier coefficients as ⎧  2 2  ⎪ ⎪ (u kl )t − ν[−(k2 + l2 )u kl + ukl ] + ikU u kl + U v kl = −ikpkl , a.e. in (0, 1), ⎪ ⎪ ⎨ (vkl )t − ν[−(k + l )vkl + vkl ] + ikU vkl = − pkl , a.e. in (0, 1),  ] + ikU wkl = −ilpkl , a.e. in (0, 1), (wkl )t − ν[−(k 2 + l 2 )wkl + wkl ⎪  ⎪ + v + ilw = 0, a.e. in (0, 1), iku ⎪ kl kl kl ⎪ ⎩ (u kl , wkl )(0) = (u kl , wkl )(1) = 0, vkl (0) = 0, vkl (1) = ψkl , ∀t ≥ 0, (3.5) 0 0 , wkl . with initial data u 0kl , vkl It is clear that the stabilizability of (3.4) is equivalent to that of (3.5) at each level (k, l) ∈ Z × Z. That is why in what follows, we will stabilize (3.5) for each pair (k, l) separately. Then we will conclude with the main stabilization result. Firstly, let us take care of the cases k = l = 0 and k = 0, l = 0. We insert ψ00 ≡ ψ0l ≡ 0.

52

3 Stabilization of Periodic Flows in a Channel

After some straightforward computations involving the free divergence condition, one can easily arrive at the conclusion that 0 2 0 2  + w00  ), (3.6) u 00 (t)2 + v00 (t)2 + w00 (t)2 ≤ C1 e−νt (u 000 2 + v00

t ≥ 0, and that 0 2 0 2  + w0l  ), (3.7) u 0l (t)2 + v0l (t)2 + w0l (t)2 ≤ C1 e−2νl t (u 00l 2 + v0l 2

t ≥ 0, for some constant C1 > 0. From now on, we consider only the cases k = 0 and k 2 + l 2 = 0. The result below is an important one. It makes the connection between the behavior of the Fourier modes vkl and u kl , wkl . More exactly, it says that it is enough to have that vkl is exponentially stable in order to have that system (3.5) is stable. In other words, the normal component of the velocity field is the leading one in measuring the laminar or turbulent character of the flow. Lemma 3.1 It suffices to stabilize vkl exponentially in the norm  (t) + (k 2 + l 2 )vkl (t)  − vkl

in order to have that (3.5) is exponentially stable. Proof Assume that we have found some control ψkl such as d |ψkl (t)| + |ψkl (t)| ≤ Ce−μt , t ≥ 0, dt

(3.8)

for some positive constants C, μ, such that once plugged into (3.5), we have   − vkl (t) + (k 2 + l 2 )vkl (t)2 ≤ Ce−μt , t ≥ 0.

(3.9)

Introduce the function Vkl = Vkl (t, y), given by Vkl (t, y) = vkl (t, y) + (2y 3 − 3y 2 )ψkl (t), t ≥ 0, y ∈ (0, 1). By (3.5), we see that Vkl satisfies the equation ⎧ [−Vkl + (k 2 + l 2 )Vkl ]t + νVkliv − [2ν(k 2 + l 2 ) + ikU ]Vkl ⎪ ⎪ ⎪ ⎪ +[ν(k 2 + l 2 )2 + ik(k 2 + l 2 )U + ikU  ]Vkl ⎪ ⎪ ⎪ ⎨ = [−(12y − 6)ψkl + (k 2 + l 2 )(2y 3 − 3y 2 )ψkl ]t ⎪ −[2ν(k 2 + l 2 ) + ikU ](12y − 6)ψkl ⎪ ⎪ ⎪ ⎪ +[ν(k 2 + l 2 )2 + ik(k 2 + l 2 )U + ikU  ](2y 3 − 3y 2 )ψkl , y ∈ (0, 1), ⎪ ⎪ ⎩  Vkl (0) = Vkl (1) = 0, Vkl (0) = 0, Vkl (1) = 0. (3.10)

3.2 The Stabilization Result

53

We scalar multiply Eq. (3.10) by Vkl , take the real part of the result, and obtain 1 d (Vkl 2 + (k 2 + l 2 )Vkl 2 ) + νVkl 2 + 2ν(k 2 + l 2 )Vkl 2 2 dt  1

+ ν(k 2 + l 2 )2 Vkl 2 = ik U  Vkl Vkl dy 0  1

+ [−(12y − 6)ψkl + (k 2 + l 2 )(2y 3 − 3y 2 )ψkl ]t Vkl dy 0  1

2 2 − ((2ν(k + l ) + ikU )(12y − 6)ψkl )Vkl dy 

0 1

+



(3.11)

(ν(k + l ) + ik(k + l )U + ikU )(2y − 3y )ψkl )Vkl dy , 2

2 2

2

2

3

2

0

from which we get that 1 d (Vkl 2 + (k 2 + l 2 )Vkl 2 ) + ν(k 2 + l 2 )(Vkl 2 + (k 2 + l 2 )Vkl 2 ) 2 dt   2  (3.12)  d 2 2 2 ≤ Ckl Vkl  +  ψkl  + |ψkl | + Vkl  , t ≥ 0, dt for some Ckl > 0. By (3.8), (3.9), and the definition of Vkl , it follows that 0 2  , t ≥ 0, Vkl (t)2 ≤ Ckl e−μt vkl

(3.13)

for some Ckl > 0. By (3.12), together with (3.13) and (3.8), we obtain 0 2  , t ≥ 0. Vkl (t)2 + (k 2 + l 2 )Vkl 2 ≤ Ckl e−μt vkl

Therefore,

 0 2 (t)2 ≤ Ce−μt vkl  , t ≥ 0. vkl

(3.14)

Now, taking into account that  , iku kl + ilwkl = −vkl

relation (3.14) implies that 0 2  , t ≥ 0. ku kl (t) + lwkl (t)2 ≤ Ce−μt vkl

(3.15)

Multiplying the first equation of system (3.5) by il and the third by −ik and summing them, we get that

54

3 Stabilization of Periodic Flows in a Channel

(ilu kl − ikwkl )t − ν[−(k 2 + l 2 )(ilu kl − ikwkl ) + (ilu kl − ikwkl ) ] +ikU (ilu kl − ikwkl ) + ilU  vkl = 0.

(3.16)

Scalar multiplying Eq. (3.16) by (ilu kl − ikwkl ) and taking the real part of the result, we obtain that 1 d ilu kl − ikwkl 2 + ν(k 2 + l 2 )ilu kl − ikwkl 2 + ν(ilu kl − ikwkl ) 2 2 dt    1 U  vkl ilu kl − ikwkl dy .

= −il

0

Hence  1    1 d U  vkl ilu kl − ikwkl dy  ilu kl − ikwkl 2 + ν(k 2 + l 2 )ilu kl − ikwkl 2 |l| ≤  2 dt 0

 a 2 |l| 1 ν(k 2 + l 2 ) 2 . v  ≤ |l| ilu kl − ikwkl 2 + kl 2 |l| ν(k 2 + l 2 ) 2ν This yields

 a 2 d vkl 2 . ilu kl − ikwkl 2 + ν(k 2 + l 2 )ilu kl − ikwkl 2 ≤ dt 2ν

From the above estimate and relation (3.9), one can easily obtain that 0 2 0 2  + wkl  ), ∀t ≥ 0, lu kl (t) − kwkl (t)2 ≤ Ce−μt (u 0kl 2 + vkl

(3.17)

for some C > 0. Now, taking into account that ku kl + lwkl 2 + lu kl − kwkl 2 = (k 2 + l 2 )(u kl 2 + wkl 2 ), relations (3.15) and (3.17) imply that 0 2  , t ≥ 0. u kl 2 + wkl 2 ≤ Ce−μt vkl

(3.18)

Finally, relations (3.9) and (3.18) imply that   0 2 0 2  + wkl  , (3.19) u kl (t)2 + vkl (t)2 + wkl (t)2 ≤ Ce−μt u 0kl 2 + vkl for some Ckl > 0, t ≥ 0, k, l = 0.



We intend to write the system (3.5) in an abstract form, aiming to apply the control design for abstract parabolic-like equations from Chap. 2. To this end, we define the following operators: for each 0 = k ∈ Z, we define Lk : D(Lk ) ⊂ H → H and Fk : D(Fk ) ⊂ H → H , by

3.2 The Stabilization Result

55



Lk v := −v + k 2 v, D(Lk ) = H 2 (0, 1) ∩ H01 (0, 1), 



(3.20) 

Fk v := νv − (2νk + ikU )v + k(νk + ik U + iU )v, D(Fk ) = H 4 (0, 1) ∩ H02 (0, 1), 2

3

2

(3.21)

Next, we define Ak : D(Ak ) ⊂ H → H   −1 Ak := Fk L−1 k , D(Ak ) = v ∈ H : Lk v ∈ D(Fk ) .

(3.22)

Furthermore, for each k, l ∈ Z, k, l = 0, we denote by Lkl : D(Lkl ) ⊂ H → H and Fkl : D(Fkl ) ⊂ H → H the operators 

Lkl v := −v + (k 2 + l 2 )v, D (Lkl ) = H01 (0, 1) ∩ H 2 (0, 1), Fkl v := νv







− [2ν(k 2 + l 2 ) + ikU ]v + [ν(k 2 + l 2 )2 + ik(k 2 + l 2 )U + ikU ]v,

(3.23) (3.24)

D (Fkl ) = H 4 (0, 1) ∩ H02 (0, 1),

which introduce the operators Akl : D(Akl ) ⊂ H → H , defined by   −1 Akl := Fkl L−1 kl , D(Akl ) = v ∈ H : Lkl v ∈ D(Fkl ) .

(3.25)

Finally, we set the differential forms 



Lk v := −v + k 2 v, Lkl v := −v + (k 2 + l 2 )v, Fk v := νv Fkl v := νv

 





− (2νk 2 + ikU )v + k(νk 3 + ik 2 U + iU )v, 



− [2ν(k 2 + l 2 ) + ikU ]v + [ν(k 2 + l 2 )2 + ik(k 2 + l 2 )U + ikU ]v.

(3.26) (3.27) (3.28)

System (3.5) can be equivalently rewritten as abstract equations, governed by the operators Ak and Akl , respectively (see (3.63) below). Therefore, it is necessary to study these operators in detail. In the two lemmas below, we gather their properties. First, we show that they generate C0 -analytic semigroups, which, for k and l sufficiently large, are exponentially asymptotically stable. Lemma 3.2 The operators −Ak , k ∈ Z∗ , and −Akl , k, l ∈ Z∗ , generate C0 analytic semigroups on H , and for each λ ∈ ρ(−Ak ), (λI + Ak )−1 is compact; also, for each λ ∈ ρ(−Akl ), (λI + Akl )−1 is compact. Moreover, one has, for each γ > 0, σ (−Ak ) ⊂ {λ ∈ C : λ ≤ −γ } , ∀|k| > S, and

 σ (−Akl ) ⊂ {λ ∈ C : λ ≤ −γ } , ∀ k 2 + l 2 > S,

where 1 S := √ 2ν



21 a 1+γ + √ . 2ν

(3.29)

56

3 Stabilization of Periodic Flows in a Channel

Here σ (−A) is the spectrum of the operator −A and ρ(−A) is the resolvent set of −A. Proof We will consider only the more complex case of −Akl (for −Ak , one may argue likewise, because of their similar forms). So for λ ∈ C and f ∈ H , consider the equation λg + Akl g = f, or equivalently, λLkl v + Fkl v = f.

(3.30)

Scalar multiplying this equation by v and taking into account (3.23) and (3.24), yields  1 (|v |2 + (k 2 + l 2 )|v|2 )dy + ν |v |2 dy 0 0  1  1 + 2ν(k 2 + l 2 ) |v |2 dy + ν(k 2 + l 2 )2 |v|2 dy 0 0  1 U  ( v v − v v)dy =  f, v +k

 λ

1

(3.31)

0

and

 1 (|v |2 + (k 2 + l 2 )|v|2 )dy + k |v |2 dy 0 0

 1 1  2 2 2 +k (k + l )U + U |v| =  f, v. 2 0

 λ

1

(3.32)

Then, via Poincaré’s inequality, we see by (3.31) and (3.32) that for some r > 0, |(λI + Akl )−1 f | ≤

C | f | for |λ| > r, |λ| − r

which implies, via the Hille–Yosida theorem (see Chap. 1), that −Akl is the infinitesimal generator of a C0 -analytic semigroup, denoted by e−Akl t , t ≥ 0, on H . Moreover, by (3.31), (3.32) we see that (λI + Akl )−1 is compact on H , and it follows also that all the eigenvalues λ of −Akl satisfy the estimates  λ

1

 2



(|v | + (k + l )|v| )dy + 2ν(k + l ) 0  1  1  2 2 2 2 +ν |v | dy + ν(k + l ) |v|2 dy 0 0  1 U  ( v v − v v)dy ≤ −k 0

2

2

2

2

1

2

0

|v |2 dy

3.2 The Stabilization Result



57

 1 1 ≤ 2νk |v | dy + |U  |2 |v|2 dy 2ν 0 0  1  1 a2 2 2  2 |v | + 3 |v|2 dy, ≤ 2ν(k + l ) 8ν 0 0 1

2

 2

where Av = −λv. By the above estimate we see that for γ > 0 arbitrary but fixed, we have

 a 1 2 2 1+γ + √ , λ ≤ −γ if k + l ≥ √ 2ν 2ν  √ In particular, it follows by Lemma 3.2 that for |k| ≥ S and k 2 + l 2 ≥ S, we have

which completes the proof.

e−Ak t  L(H,H ) ≤ Ce−γ t and e−Akl t  L(H,H ) ≤ Ce−γ t ,

(3.33)

for all t ≥ 0. This √ implies that for the stabilization of (3.4), it suffices to stabilize system (3.5) for k 2 + l 2 ≤ S only. Moreover, Lemma 3.2 ensures that the hypotheses (A1)–(A3) from Chap. 2 hold in the present case. Furthermore,    again by Lemma 3.2, −Ak has a countable set of eigenvalues, denoted by λkj

∞

j=1

, and the same holds for −Akl , and we denote by λklj

∞

j=1

the corresponding eigenvalues. Besides this, there is only a finite number Nk of eigenvalues λkj with λkj ≥ 0, the unstable eigenvalues; and a finite number Nkl of   eigenvalues with λklj ≥ 0, j = 1, . . . , Nkl . We denote by ϕ kj , j = 1, 2, . . . , and   ϕ k∗ , j = 1, 2, . . . , the corresponding eigenfunctions for −Ak and its adjoint −A∗k , j     respectively; and by ϕ klj , j = 1, 2, . . . and ϕ kl∗ j , j = 1, 2, . . . the corresponding eigenfunctions for −Akl and its adjoint −A∗kl , respectively. Next, we show that the unique continuation-type hypothesis (A5) from Chap. 2 holds in the present case. The Dirichlet map D that lifts the boundary conditions into the equations is introduced in (3.54) below, similarly as in (2.16). Then one easily deduces that in the present case, 

D ϕ = ϕ (1), where D was defined in (2.17). Hence, validation of hypothesis (A5) is equivalent to showing that for the solution ϕ ∗ to ⎧ ∗  2 2 ∗   ∗  ⎪ ⎨ ν(ϕ ) − (2ν(k + l ) − ikU + λ¯ )(ϕ ) + 2ikU (ϕ ) + ((k 2 + l 2 )λ¯ + ν(k 2 + l 2 )2 − ik(k 2 + l 2 )U )ϕ ∗ = 0, ⎪ ⎩ ϕ ∗ (0) = ϕ ∗ (1) = 0, (ϕ ∗ ) (0) = (ϕ ∗ ) (1) = 0,

(3.34)

58

3 Stabilization of Periodic Flows in a Channel

if it satisfies in addition that (ϕ ∗ ) (1) = 0, then necessarily ϕ ∗ ≡ 0. This is in contradiction to the fact that ϕ ∗ is an eigenfunction of the adjoint operator −A∗kl of −Akl . Thus necessarily,  D ϕ ∗ = (ϕ ∗ ) (1) = 0. As announced in Chap. 2, this task is not an easy one. Even if the fourth-order equation (3.34) has five null boundary conditions, we cannot deduce immediately that the only solution is the trivial one, since the two boundary conditions in y = 0 may be linearly dependent with those given in y = 1. To overcome this problem, we take into account the special form of the equation, more precisely its symmetric nature. Indeed, by the form of U in (3.3), one can easily see that U (y) = U (1 − y), ∀y ∈ (0, 1). This implies that if ϕ ∗ (y) is a solution to (3.34), then ϕ ∗ (1 − y) is also a solution to (3.34). This enables us to assume that the eigenfunction is either symmetric or antisymmetric, i.e., ϕ ∗ (y) = ±ϕ ∗ (1 − y), ∀y ∈ [0, 1]. By assuming this, we gain another null boundary condition, namely (ϕ ∗ ) (0) = ±(ϕ ∗ ) (1) = 0, and it turns out that these six null boundary conditions are enough to establish our claim. Lemma 3.3 Let λkj for some 0 < |k| ≤ S, and let j ∈ {1, . . . , Nk } be an unstable eigenvalue of −A∗k . Then we can choose the corresponding adjoint eigenfunction ϕ k∗ j  such that (ϕ k∗ ) (1) > 0. j √ Moreover, let λklj for some 0 < k 2 + l 2 ≤ S, and let j ∈ {1, . . . , Nkl } be an unstable eigenvalue of −A∗kl . Then we can choose the corresponding adjoint eigenkl∗  function ϕ kl∗ j such that (ϕ j ) (1) > 0. Proof We will consider only the more complex case, the eigenvalues and eigenfunctions of −A∗kl , while for −A∗k one may construct a similar argument. We aim to show kl∗  that we can choose the eigenfunction ϕ kl∗ j such that (ϕ j ) (1) = 0. Then if needed, kl∗  kl∗ replacing ϕ kl∗ j by (ϕ j ) (1)ϕ j , we obtain the desired result. The proof follows in three steps. Step 1. For a function f : [0, 1] → C, let us denote by fˇ : [0, 1] → C the function

fˇ(y) := f (1 − y), ∀y ∈ [0, 1]. We say that the function f : [0, 1] → C is symmetric if f (y) = fˇ(y), ∀y ∈ [0, 1], and antisymmetric if f (y) = − fˇ(y), ∀y ∈ [0, 1]. In this step, we show that we can choose a basis of the adjoint eigenfunction space consisting of symmetric or antisymmetric functions.

3.2 The Stabilization Result

59

Let us denote by λ := λklj the unstable eigenvalue. If ϕ ∗ := ϕ kl∗ j is an eigenfunction ∗ ¯ corresponding to λ, then ϕ satisfies the boundary value problem ⎧ ∗  2 2 ¯ ∗ ) + 2ikU  (ϕ ∗ ) ⎪ ⎨ ν(ϕ ) − (2ν(k + l ) − ikU + λ)(ϕ + ((k 2 + l 2 )λ¯ + ν(k 2 + l 2 )2 − ik(k 2 + l 2 )U )ϕ ∗ = 0, ⎪ ⎩ ϕ ∗ (0) = ϕ ∗ (1) = 0, (ϕ ∗ ) (0) = (ϕ ∗ ) (1) = 0.

(3.35)

Let us observe that if ϕ ∗ is a solution to (3.35), then ϕˇ∗ is also a solution to (3.35), because of the symmetric form of the equation. Let us denote by H the fourth-dimensional linear space of the solutions to the fourth-order linear homogeneous differential equation ν(ϕ ∗ ) − (2ν(k 2 + l 2 ) − ikU + λ¯ )(ϕ ∗ ) + 2ikU  (ϕ ∗ ) + ((k 2 + l 2 )λ¯ + ν(k 2 + l 2 )2 − ik(k 2 + l 2 )U )ϕ ∗ = 0, a.e. in (0, 1). 



Then the eigenfunction space can be written as the linear space E , defined as   E := ϕ ∈ H : ϕ(0) = ϕ(1) = 0, (ϕ) (0) = (ϕ) (1) = 0 . It is easy to see that the dimension of E is ≤ 2. We claim that we can find a basis for this linear space consisting of symmetric functions or antisymmetric functions. Indeed, let us assume that there exists ϕ ∈ E that is neither symmetric nor antisymmetric. Then the two functions ϕ1 := ϕ + ϕˇ and ϕ2 := ϕ − ϕˇ are both in E . Moreover, ϕ1 = 0, ϕ2 = 0, and the system {ϕ1 , ϕ2 } is linearly independent, because ϕ1 is symmetric and ϕ2 is antisymmetric. This, together with the fact that the dimension of E is ≤ 2, proves our claim. Hence we can assume that the corresponding eigenfunction ϕ ∗ is either symmetric or antisymmetric. Assume, for instance, that ϕ ∗ is symmetric. The other case can be treated similarly. We want to show that we have (ϕ ∗ ) (1) = 0. Let us assume, for the sake of a contradiction, that (ϕ ∗ ) (1) = 0. From symmetry, we get also that (ϕ ∗ ) (0) = 0. It is easy to see that ϕ ∗ satisfies F∗kl ϕ ∗ = 0, where F∗kl is the adjoint operator of Fkl given by (3.24). We have 

1

0= 0

F∗kl ϕ ∗ ϕdy

 = 0

1

ϕ ∗ Fkl ϕdy + ν((ϕ ∗ ) (1)ϕ  (1) − (ϕ ∗ ) (0)ϕ  (0)), ∀ϕ ∈ D(Fkl ),

(3.36)

60

3 Stabilization of Periodic Flows in a Channel

by taking into account the boundary conditions for ϕ ∗ , i.e., ϕ ∗ (0) = ϕ ∗ (1) = 0, (ϕ ∗ ) (0) = (ϕ ∗ ) (1) = 0, (ϕ ∗ ) (0) = (ϕ ∗ ) (1) = 0. From (3.36), we get that



1

|ϕ ∗ |2 dy = 0,

(3.37)

0

provided that ϕ satisfies



Fkl ϕ = ϕ ∗ , in (0, 1), ϕ  (0) = ϕ  (1) = 0,

(3.38)

where Fkl is the differential form given in (3.26). Relation (3.37) implies that ϕ ∗ ≡ 0, which is in contradiction to the fact that ϕ ∗ is an eigenfunction. This implies that the assumption (ϕ ∗ ) (1) = 0 is false, which leads to our desired result. So in order to complete the proof, it remains to show that there exists a solution ϕ to the Eq. (3.38). Step 2. We claim that there exists a function ϕ1 such that 

Fkl ϕ1 = 0, y ∈ (0, 1) , ϕ1 (0) − ϕ1 (1) = 0.

(3.39)

The proof of this claim will be given in the last step of the proof. In this step, we will prove that under the above claim, there exists a solution to the Eq. (3.38), and so we obtain the desired result. Let us construct the function ϕ2 := ϕ1 + ϕˇ1 . As seen before, the equation Fkl ϕ1 = 0 is symmetric, and this implies that if ϕ1 is a solution, then also ϕˇ1 is. Hence we have ⎧ ⎨ Fkl ϕ2 = 0, ϕ  (0) = ϕ1 (0) − ϕ1 (1) = 0, (3.40) ⎩ 2 ϕ2 is symmetric. Let ϕ3 be a solution to the equation Fkl ϕ3 = ϕ ∗ such that ϕ3 is symmetric. There exists a symmetric solution to the above equation because ϕ ∗ is symmetric. Indeed, let ϕ4 be a solution to the equation Fkl ϕ4 = 21 ϕ ∗ . If we take ϕ3 := ϕ4 + ϕˇ4 , we have Fkl ϕ3 = 21 ϕ ∗ + 21 ϕ ∗ = ϕ ∗ , and ϕ3 is symmetric, as desired. ϕ  (0)

Now we define ϕ5 := − ϕ3 (0) ϕ2 + ϕ3 . We have that ϕ5 is well defined, and more2 over, ϕ5 satisfies ⎧ ⎨ Fkl ϕ5 = ϕ ∗ , ϕ5 is symmetric, (3.41) ⎩  ϕ5 (0) = 0 and, because of the symmetry, ϕ5 (1) = 0. So we can take ϕ := ϕ5 .

3.2 The Stabilization Result

61

Step 3. In the last step we show that there exists a function ϕ1 such that 

Fkl ϕ1 = 0, y ∈ (0, 1) , ϕ1 (0) − ϕ1 (1) = 0.

(3.42)

We assume, for the sake of a contradiction, that this is not true. Hence for every solution ψ to the equation Fkl ψ = 0, we have ψ  (0) − ψ  (1) = 0. Let us denote by H1 the linear space of the solutions to the equation Fkl ψ = 0, and by E1 the linear subspace of H1 defined by   E1 := ψ ∈ H1 : ψ  (0) − ψ  (1) = 0 . ˇ We have Let ψ ∈ E1 . Define Ψ := ψ + ψ. Ψ  (0) = ψ  (0) − ψ  (1) = 0, Ψ  (1) = ψ  (1) − ψ  (0) = 0, since ψ ∈ H1 . Also, Ψ  (0) = ψ  (0) − ψ  (1) = 0, Ψ  (1) = ψ  (1) − ψ  (0) = 0, since ψ ∈ E1 . Let us set Φ := Ψ  − (k 2 + l 2 )Ψ . The equation Fkl Ψ = 0 can be rewritten in the form (3.43) νΦ  − (ν(k 2 + l 2 ) + ikU + λ)Φ + ikU  Ψ = 0, and

Ψ  − (k 2 + l 2 )Ψ = Φ.

(3.44)

Observe that since Ψ  (0) = Ψ  (1) = 0 and Ψ  (0) = Ψ  (1) = 0, we have Φ  (0) = Φ  (1) = 0. If we scalar multiply Eq. (3.43) by Φ and Eq. (3.44) by Ψ , we get  −ν

1

 2

 2

0

and



1

− 0

1

|Φ | dy − (ν(k + l ) + λ) |Φ|2 dy 0  1 0  1 U |Φ|2 dy + ikU  Ψ Φdy = 0, −ik 2

|Ψ  |2 dy − (k 2 + l 2 )

(3.45)

0



1 0



1

|Ψ |2 dy = 0

ΦΨ dy.

(3.46)

62

3 Stabilization of Periodic Flows in a Channel

1 From (3.46) we see that 0 Ψ Φdy is a real number. Using this and taking the real part of (3.45), we obtain that  −ν

1

 2



1

|Φ | dy − (ν(k + l ) + λ) 2

2

0

|Φ|2 dy = 0.

0

Since λ is an unstable eigenvalue, we have that λ > 0. So the relation above yields Φ ≡ 0. ˇ Hence It is easy to see that Φ ≡ 0 implies Ψ ≡ 0, which implies that ψ = −ψ. ˇ ψ ∈ E1 ⇒ ψ = −ψ.

(3.47)

Let us consider the following subspaces of H1 :   S := ψ ∈ H1 : ψ = ψˇ ,   A S := ψ ∈ H1 : ψ = −ψˇ , which are the symmetric subspace of H1 and the antisymmetric subspace of H1 , respectively. It is easy to see that   1   1 =ψ =0 S = ψ ∈ H1 : ψ 2 2   1 1  =ψ =0 . A S = ψ ∈ H1 : ψ 2 2 This implies that dimC S = dimC A S = 2. Relation (3.47) implies that E1 ⊂ A S . Let us define the subspace of H1   F1 := ψ ∈ H1 : ψ  (0) = 0 . We have dimC F1 = 3. Since dimC S = 2, dimC H1 = 4, and F1 , S ⊂ H1 , we have that F1 ∩ S = {0} . Hence there exists 0 = ψ ∈ F1 ∩ S . That ψ ∈ F1 implies that ψ  (0) = 0, and from symmetry (because ψ ∈ S ), ψ  (1) = 0. These yield that ψ  (0) − ψ  (1) = 0 and ψ ∈ H1 . We conclude that ψ ∈ E1 ⊂ A S . Finally, we have ψ ∈ S ∩ A S , which implies ψ ≡ 0, which is absurd. Hence the assumption made is not true, which means that there exists a function ϕ1 such that

3.2 The Stabilization Result

63



Fkl ϕ1 = ϕ ∗ , y ∈ (0, 1), ϕ1 (0) − ϕ1 (1) = 0.

As we mentioned earlier, this completes the proof.

(3.48) 

3.2.1 The Feedback Law and the Stability of the System We saw earlier that for k = l = 0, k = 0, and l = 0, the stability of the system is guaranteed without any boundary control. We continue with the case k = 0 and l = 0. System (3.5) reduces to ⎧ (u k0 )t − ν[−k 2 u k0 + u k0 ] + ikU u k0 + U  vk0 = −ikpk0 , a.e. in (0, 1), ⎪ ⎪ ⎪   ⎪ (vk0 )t − ν[−k 2 vk0 + vk0 ] + ikU vk0 = − pk0 , a.e. in (0, 1), ⎪ ⎪ ⎪ ⎨  iku k0 + vk0 = 0, a.e. in (0, 1),  ⎪ (wk0 )t − ν[−k 2 wk0 + wk0 ] + ikU wk0 = 0, a.e. in (0, 1), ⎪ ⎪ ⎪ ⎪ u (0) = u (1) = 0, v (0) = 0, vk0 (1) = ψk0 , wk0 (0) = wk0 (1) = 0, ⎪ k0 k0 k0 ⎪ ⎩ ∀t ≥ 0.

(3.49)

First, we consider separately only the equations for u k0 and vk0 , that is, ⎧ (u k0 )t − ν[−k 2 u k0 + u k0 ] + ikU u k0 + U  vk0 = −ikpk0 , a.e. in (0, 1), ⎪ ⎪ ⎪ ⎨ (v ) − ν[−k 2 v + v ] + ikU v = − p  , a.e. in (0, 1), k0 t k0 k0 k0 k0  ⎪ iku + v = 0, a.e. in (0, 1), k0 ⎪ k0 ⎪ ⎩ u k0 (0) = u k0 (1) = 0, vk0 (0) = 0, vk0 (1) = ψk0 , ∀t ≥ 0.

(3.50)

We reduce the pressure from the system and use the free divergence condition to get that vk0 satisfies the equation ⎧    2 2 ⎪ ⎨ (−vk0 + k vk0 )t + νvk0 − (2νk + ikU )vk0 + k(νk 3 + ik 2 U + iU  )vk0 = 0, t ≥ 0, y ∈ (0, 1), ⎪ ⎩ v (0) = v (1) = 0, v (0) = 0, v (1) = ψ (t), k0 k0 k0 k0 k0

(3.51)

which can be equivalently rewritten as 

(Lk vk0 )t + Fk vk0 = 0, t ≥ 0, y ∈ (0, 1),   vk0 (0) = vk0 (1) = 0, vk0 (0) = 0, vk0 (1) = ψk0 (t),

(3.52)

where Lk and Fk are given in (3.26). Note that formally, Eq. (3.52) may be rewritten as (z k )t + Ak z k = 0

64

3 Stabilization of Periodic Flows in a Channel

by setting z k := Lk vk0 , where we recall the operators −Ak with their eigenvalues {λkj } j and their eigenfunctions {ϕ kj } j , described in the previous section. For the sake of simplicity, we assume that the unstable eigenvalues λkj , j = 1, . . . , Nk , are simple,

(3.53)

which is the counterpart of hypothesis (A4.1) from Chap. 2. The present algorithm works equally well in the case of semisimple eigenvalues by doing tricks similar to those in Chap. 2. However, we will not develop this subject here since the presentation may get too hard to follow. Using (if necessary) the Gram–Schmidt procedure, we may assume that the sysNk k and {ϕ k∗ tems {ϕ kj } Nj=1 j } j=1 are biorthonormal, that is, ϕik , ϕ k∗ j  = δi j , i, j = 1, . . . , Nk , where δi j is the Kronecker symbol. Next we will lift the boundary conditions into the Eq. (3.52) via the Dirichlet operator, defined as (see also (2.16)) follows: let α ∈ C. We denote by Dγ α := ω the solution to the equation ⎧ ⎪ ⎨ ⎪ ⎩

Fk ω + 2

Nk  j=1

k λkj Lk ω, φ k∗ j φ j + γ Lk ω = 0, y ∈ (0, 1), 

(3.54)



ω(1) = α, ω(0) = ω (0) = ω (1) = 0.

(It is known that for γ > 0 large enough, the above equation has a unique solution 1 in H 2 (0, 1).) Next, let us compute Lk Dγ α, φmk∗ , for some 1 ≤ m ≤ Nk . To this end, we have from (3.54) scalar multiplied by φmk∗ and by the biorthogonality of the eigenfunctions systems that 0 = Fk ω, φmk∗  + 2λkm Lk ω, φmk∗  + γ Lk ω, φmk∗  = −α(φmk∗ ) (1) + ω, F∗k φmk∗  + (γ + 2λkm )Lk ω, φmk∗ (by Lemma 3.3)) = −α(φmk∗ ) (1) + Lk ω, A∗k φmk∗  + (γ + 2λkm )Lk ω, φmk∗ . This yields that Lk Dγ α, φmk∗  = Note that in this case,

α (φ k∗ ) (1), 1 ≤ m ≤ Nk . γ + λkm m 

D ϕ = ϕ (1),

where D was introduced in (2.17).

(3.55)

3.2 The Stabilization Result

65

We choose Nk constants 0 < γ1k < γ2k < · · · < γ Nk k large enough that Eq. (3.54), corresponding to each γik , i = 1, . . . , Nk , has a solution, and denote by Dγik , i = 1, . . . , Nk , the corresponding solutions. Now for each 0 < |k| ≤ S, we introduce the feedback ψk0 as (see the proportional feedback defined in (2.6) in Chap. 2) " ⎞ ⎛ k∗  ⎛! ⎞ k∗ (ϕ1 ) (1) $ !Lk vk0 (t), ϕ1k∗ " ⎜ Lk vk0 (t), ϕ ⎟ ⎜ (ϕ k∗ ) (1) ⎟ 2 2 ⎟ ⎜ ⎟ , ψk0 (t) := − Λksum Ak ⎜ ⎝ ...................... ⎠ , ⎝ ............... ⎠ ! " k∗  k∗ (ϕ Nk ) (1) Lk vk0 (t), ϕ Nk N 

(3.56)

k

with Λksum := Λkγ k + · · · + Λkγ k , for 1

Nk

⎛

1 γik +λk1

0

...

⎞

0

1 ⎜ 0 ... 0 ⎟ ⎟ ⎜ γik +λk2 Λkγ k := ⎜ ⎟ , i = 1, . . . , Nk . i .................................. ⎠ ⎝ 1 0 0 . . . γ k +λk i

Moreover,

(3.57)

Nk

Ak := (B1k + B2k + · · · + B Nk k )−1

(3.58)

(the counterpart of the matrix A introduced in (2.24), Chap. 2), where Bik := Λkγ k Bk Λkγ k , i = 1, . . . , Nk i

(3.59)

i

(see also (2.22)), with Bk the matrix ⎞ [ϕ1k∗ ) (1)k ]2 (ϕ1k∗ ) (1)(ϕ2k∗ ) (1) . . . (ϕ1k∗ ) (1)(ϕ Nk∗k ) (1) ⎜(ϕ k∗ ) (1)(ϕ k∗ ) (1) [(ϕ2k∗ ) (1)]2 . . . (ϕ2k∗ ) (1)(ϕ Nk∗k ) (1)⎟ 2 1 ⎟ Bk :=⎜ ⎝.................................................................................................⎠ (3.60) (ϕ Nk∗k ) (1)(ϕ1k∗ ) (1) (ϕ Nk∗k ) (1)(ϕ2k∗ ) (1) . . . [(ϕ Nk∗k ) (1)]2 ⎛

(he counterpart of the Gram matrix B introduced in (2.20) in Chap. 2). Here ·, · N stands for the classical scalar product in C N . By Lemma 3.3, we know that (ϕik∗ ) (1) = 0, i = 1, . . . , Nk , and therefore, hypothesis (A5) is verified for the present case. Hence just as in Proposition 2.1, one can show that the above matrices Ak are well defined, and consequently, the feedback ψk0 is well defined. We plug the above ψk0 into (3.52) and show that it ensures its stability. Similarly to (2.25), we decompose ψk0 as ψk0 = v1k + · · · + vkNk ,

66

3 Stabilization of Periodic Flows in a Channel

where " ⎞ ⎛ 1 (φ k∗ ) (1) ⎞ ⎛! 1 Lk vk0 (t), ϕ1k∗ γik +λk1 $ " ! k∗ ⎟ ⎜ k 1 k (φ k∗ ) (1) ⎟ ⎜ ⎟ 2 k k ⎜ Lk vk0 (t), ϕ2 ⎟ ⎜ γi +λ2 vi (t) := − A ⎝ ⎟ ⎠,⎜ ....................... ⎝ ....................... ⎠ ! " 1 Lk vk0 (t), ϕ Nk∗k (φ Nk∗k ) (1) γ k +λk 

i

Nk

, t ≥ 0,

(3.61)

Nk

for i = 1, 2, . . . , Nk . Likewise in (2.35), we have that &⎞ ⎛% "⎞ ⎛! Lk Dγik vik , ϕ1k∗ Lk vk0 , ϕ1k∗ " ⎜% &⎟ ! ⎜ ⎟ ⎜ Lk vk0 , ϕ k∗ ⎟ ⎜ Lk Dγik vik , ϕ2k∗ ⎟ 2 ⎟ ⎜ ⎟ = −Bik Ak ⎜ ⎝ .................. ⎠ , for i = 1, . . . , Nk . ⎜ ....................... ⎟ ! " ⎝% &⎠ Lk vk0 , ϕ Nk∗k k k∗ Lk Dγik vi , ϕ Nk

(3.62)

In the next lines, the approach will slightly differ from that in Chap. 2, in the sense that we will not consider the equivalent reformulation of Eq. (3.52) via the variation of constants formula and the extension operators. Instead, we perform computations directly in Eq. (3.52). The reason is that in this case, we do not care about the nonlinear equation, but only the linearized one. Returning to the linear equation (3.52), we define z k := Lk [vk0 − Dγ1k v1k − · · · − Dγ Nk vkNk ]. k

We see that z k belongs to D(−Ak ). Subtracting (3.52) and (3.54), corresponding to Dγik vik , i = 1, . . . , Nk , we arrive at (z )t = −Ak z + 2 k

k

 − Lk

Nk 

k λkj Lk Dγik vik , ϕ k∗ j ϕ j

i, j=1 Nk  i=1

+



Dγik vik

Nk 

γik Lk Dγik vik

i=1

(3.63)

. t

In terms of the new variable z k , the feedbacks vik , i = 1, .., Nk , have the form " ⎞ ⎛ k 1 k (ϕ k∗ ) (1) ⎞ ! k k∗ 1 γi +λ1 (t), ϕ z $ 1 " ! k 1 k∗  ⎜ ⎟ k∗ ⎜ ⎟ (ϕ ) (1) 1 k k z (t), ϕ ⎜ ⎟ 2 γi +λ2 2 ⎟ vik (t) = − Ak ⎜ ⎟ ⎝ ....................... ⎠ , ⎜ ⎝ ....................... ⎠ 2 ! k " k∗ 1 k∗  z (t), ϕ Nk (ϕ Nk ) (1) γ k +λk 

⎛

i

Nk

. Nk

(3.64)

3.2 The Stabilization Result

67

To see this, we do the following straightforward computations: & ⎞ ⎛ ⎜  ⎜ % & ⎟ ⎟ ⎜ ⎜ 1 ⎜ z k (t), ϕ2k∗ ⎟ ⎜ k A ⎜ ⎟,⎜ ⎜ ....................... ⎟ ⎜ 2 ⎝ % & ⎠ ⎜ ⎝ z k (t), ϕ k∗ N ⎛ %

z k (t), ϕ1k∗

k

1 (ϕ k∗ ) (1) γik +λk1 1 1 (ϕ k∗ ) (1) γ k +λk 2

⎞

⎟$ ⎟ ⎟ ⎟ 2 i ⎟ ....................... ⎟ ⎠ 1 k∗  k k (ϕ N ) (1)

γi +λ N k

⎞ " ⎞ ⎛ 1 ⎛ ! (ϕ k∗ ) (1) Lk vk0 , ϕ1k∗ γik +λk1 1 " ⎟ ⎜ ⎟$ 1 ⎜ ! k∗ (ϕ k∗ ) (1) ⎟ 1 ⎟ ⎜ k ⎜ Lk vk0 , ϕ2 γik +λk2 2 ⎜ ⎟ = A ⎜ ....................... ⎟ , ⎜ ⎟ ⎝ % 2 & ⎠ ⎝ ....................... ⎠ 1 k∗ ) (1) Lk vk0 , ϕ k∗ (ϕ k k Nk Nk

k

Nk



⎛%

&⎞

γi +λ N

k

Nk

⎞ ⎜ ⎟ & % ⎟$ ⎜ Nk ⎜ k k∗ ⎟ ⎜ ⎟ 1 k ⎜ Lk D γ k v j , ϕ 2 ⎟ ⎜ ⎟ A ⎜ − ⎟, j ⎟ ⎜ ....................... ⎟ ⎜ 2 ....................... ⎠ ⎝ j=1 ⎝% ⎠ & 1 k∗  k +λk (ϕ Nk ) (1) γ Lk Dγ k v kj , ϕ k∗ i N Nk k 

Lk Dγ k v kj , ϕ1k∗ j

⎛

1 (ϕ k∗ ) (1) γik +λk1 1 1 (ϕ k∗ ) (1) γik +λk2 2

j

, Nk

(taking into account relation (3.62))

⎞ 1 " ⎞ ⎛ ⎛ ! (ϕ k∗ ) (1) k∗ γik +λk1 1  !Lk vk0 , ϕ1k∗ " ⎟$ 1 ⎟ ⎜ ( ⎜ Lk vk0 , ϕ (ϕ k∗ ) (1) ⎟ 1 ' 2 ⎟ ⎜ k k k k⎜ γik +λk2 2 ⎟ ⎜ I + A (B1 + · · · + B Nk ) A ⎜ ....................... ⎟ , ⎜ = ⎝ % 2 & ⎠ ⎝ ....................... ⎟ ⎠ 1 k∗  Lk vk0 , ϕ k∗ Nk k k (ϕ Nk ) (1) γi +λ N

k

Nk

= −vik ,

since Ak = (B1k + · · · + B Nk k )−1 . Moreover, as in (3.62), we have now ⎛%

&⎞

"⎞ ⎛! k z (t), ϕ1k∗ " ⎜% &⎟ ! ⎜ ⎟ ⎜ z k (t), ϕ k∗ ⎟ 1 ⎜ Lk Dγik vik , ϕ2k∗ ⎟ 2 ⎟ ⎜ ⎟ = − Bik Ak ⎜ ⎝ ................. ⎠ , i = 1, . . . , Nk . ⎜ ....................... ⎟ 2 ! " ⎝% &⎠ z k (t), ϕ Nk∗k k k∗ Lk Dγik vi , ϕ Nk Lk Dγik vik , ϕ1k∗

(3.65)

Next, we decompose system (3.63) into its stable and unstable parts. Recall the projections PNk , and its adjoint PN∗k , defined by PNk

1 := 2π i

 Γ

−1

(λI + Ak ) dλ;

PN∗k

1 := 2π i

 Γ¯

(λI + A∗k )−1 dλ,

where Γ (its conjugate Γ¯ , respectively) separates the unstable spectrum from the stable one of −Ak (−A∗k , respectively). We set

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3 Stabilization of Periodic Flows in a Channel

− AuNk := PNk (−Ak ), −AsNk := (I − PNk )(−Ak ),

(3.66)

for the restrictions of −Ak , respectively. The system (3.63) can accordingly be decomposed as z k = z Nk + ζ Nk , z Nk := PNk z k , ζ Nk := (I − PNk )z k , where applying PNk and (I − PNk ) to (3.63), we obtain d z N + AuNk z Nk dt k ⎡ ⎞⎤ ⎛ Nk Nk Nk    k k k∗ k k k k = PNk ⎣2 λ j Lk Dγ k vi , ϕ j ϕ j + γi Lk Dγ k vi − ⎝Lk Dγ k vi ⎠ ⎦ i

i, j=1

i=1

i

i=1

i

t

d ζ N + AsNk ζ Nk dt k ⎡ ⎛ ⎞⎤ Nk Nk Nk    k = (I −PNk )⎣2 λkj Lk Dγ k vik , ϕ k∗ γik Lk Dγ k vik − ⎝Lk Dγ k vik⎠ ⎦ j ϕ j + i

i, j=1

i=1

i

i=1

(3.67)

i

(3.68)

t

respectively. Let us decompose z Nk as z Nk (t, y) =

Nk 

k z k (t), ϕ k∗ j ϕ j (y).

j=1

We introduce this z Nk in Eq. (3.67). Then we scalar multiply (3.67) successively by ϕ k∗ j , j = 1, . . . , Nk , take account the biorthogonality of the systems of eigenfunck∗ ∗ tions, notice that we may assume that PN∗k ϕ k∗ j = ϕ j (since PNk is idempotent), and take advantage of relation (3.65) to get that Zt k = Λk Z k −

Nk 

k k 1 1 γik Bik Ak Z k + B k Ak Zt k , t ≥ 0, 2 i=1 2 i=1 i

N

Λk Bik Ak Z k −

i=1

N

⎞ z k (t), ϕ1k∗  ⎜ z k (t), ϕ k∗  ⎟ k k k k 2 ⎟ where Z k := ⎜ ⎝ ................. ⎠ and Λ := diag (λ1 , λ2 , . . . , λ Nk ). Recalling that z k (t), ϕ Nk∗k  Ak = (B1k + · · · + B Nk k )−1 , we see that the above relation yields ⎛

Zt k = −γ1k Z k +

Nk  (γ1k − γik )Bik Ak Z k , t ≥ 0, i=2

(3.69)

3.2 The Stabilization Result

69

which is the counterpart of Eq. (2.39) from Chap. 2. Thus continuing with arguments similar to those in (2.39)–(2.41), we conclude that Z k (t)2Nk ≤ Ce−2γ1 t Zok 2Nk , t ≥ 0,

(3.70)

for some positive constants C, γ1 > 0 independent of k. The above relation says that the unstable part of the system (3.63) is, in fact, stable. Further, one can argue as at the end of the proof of Theorem 2.1 in order to deduce that the system (3.50) is stabilized by ψk0 defined in (3.56). The details are omitted. Therefore, by virtue of Lemma 3.2 and (3.70), we may conclude that on plugging the feedback "⎞ ⎛ k∗  ⎞ ⎛! k∗ (ϕ1 ) (1) $  !Lk vk0 (t), ϕ1k∗ " k∗  ⎜ ⎟ ⎟ ⎜ L v (t), ϕ k k0 2 ⎟,⎜ (ϕ2 ) (1) ⎟ ψk0 = − Λksum Ak ⎜ ⎝......................⎠ ⎝ .............. ⎠ , 0 < |k| ≤ S, (3.71) ! " (ϕ Nk∗k ) (1) Lk vk0 (t), ϕ Nk∗k Nk ψk0 ≡ 0, |k| > S, into (3.49), we obtain that 0 2 0 2  + wk0  ), u k0 (t)2 + vk0 (t)2 + wk0 (t)2 ≤ C3 e−μ3 t (u 0k0 2 + vk0 (3.72) ∀t ≥ 0, ∀|k| > 0, for some constants C3 , μ3 > 0, independent of k. The case k = 0 and l = 0 can be treated similarly to that above, obtaining that the feedback " ⎞ ⎛ kl∗  ⎛ ! ⎞ kl∗ (ϕ1 ) (1) $  !Lkl vkl (t), ϕ1kl∗ " ⎜ ⎟ ⎜ (ϕ kl∗ ) (1) ⎟ kl ⎜ Lkl vkl (t), ϕ2 ⎟ ⎟ ⎜ 2 ψkl = − Λkl sum A ⎝ .......................... ⎠ , ⎝ ................. ⎠ ! "  Lkl vkl (t), ϕ Nkl∗kl (ϕ Nkl∗kl ) (1) (3.73) Nkl  for 0 < k 2 + l 2 ≤ S,  ψkl ≡ 0 for k 2 + l 2 > S kl kl ensures the stability. Here Λkl sum := Λγ kl + · · · + Λγ kl , for 1

⎛

1 γikl +λkl 1

0

Nkl

...

⎞

0

1 ⎜ 0 ... 0 ⎜ γikl +λkl 2 Λkl kl := ⎜ γi ⎝ .................................... 1 0 0 . . . γ kl +λ kl i

⎟ ⎟ ⎟ , i = 1, . . . , Nkl , ⎠

(3.74)

Nkl

for some 0 < γ1kl < · · · < γ Nklkl , Nkl real constants sufficiently large. Moreover,

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3 Stabilization of Periodic Flows in a Channel

Akl := (B1kl + B2kl + · · · + B Nklkl )−1 ,

(3.75)

Bkl Λkl , i = 1, . . . , Nkl , Bikl := Λkl γ kl γ kl

(3.76)

where i

i

Bkl being the matrix ⎛



[(ϕ1kl∗ ) (1)]2









(ϕ1kl∗ ) (1)(ϕ2kl∗ ) (1) . . . (ϕ1kl∗ ) (1)(ϕ kl∗ Nkl ) (1)

⎞

⎟ ⎜ kl∗     kl∗  ⎜ [(ϕ2kl∗ ) (1)]2 . . . (ϕ2kl∗ ) (1)(ϕ kl∗ Nkl ) (1) ⎟ Bkl := ⎜ (ϕ2 ) (1)(ϕ1 ) (1) ⎟. ⎝ ....................................................................................................... ⎠      kl∗ kl∗ kl∗ 2 [(ϕ kl∗ (ϕ kl∗ Nkl ) (1)(ϕ1 ) (1) (ϕ Nkl ) (1)(ϕ2 ) (1) . . . Nkl ) (1)]

(3.77)

We may deduce then that 0 2 0 2 u kl (t)2 + vkl (t)2 + wkl (t)2 ≤ C4 e−μ4 t (u 0kl 2 + vkl  + wkl  ), (3.78)

∀t ≥ 0, for all k, l ∈ Z∗ , for some constants C4 , μ4 > 0, independent of k, l. Collecting all the above results, we conclude with the following theorem: Theorem 3.1 Given initial data u o , vo , wo in L 22π (O), define the feedback Ψ as Ψ (t, x, z) =

 0 0 sufficiently large there is a solution. Then doing computations similar to those in (2.27)–(2.29), we obtain that (3.83) is equivalent to d 0 z k + Ak z k = Ak Dγ k ψk0 , t ≥ 0; z k (0) = z k0 := Lk vk0 , dt

(3.85)

where z k := Lk vk0 . We associate to (3.85) the following linear quadratic control problem: φ(z k0 ) := min

1 2



∞ 0

2 2 (L−1 k z k (t) + |ψk0 (t)| )dt,

(3.86)

subject to ψk0 ∈ L 2 (0, ∞; X ) and z k satisfying (3.85). Here   X := H 2 (0, 1) ∩ H01 (0, 1) is the dual of the space H 2 (0, 1) ∩ H01 (0, 1). First let us show that the optimization problem is well posed on the state space X , i.e., φ(z k0 ) < ∞, ∀z k0 ∈ X . In other words, we must show that with z k0 ∈ X arbitrary, there exists some control ψk0 ∈ L 2 (0, ∞; X ) such that the corresponding solution z k of (3.85) satisfies z k ∈ L 2 (0, ∞; X ). In fact, such a feedback ψk0 is provided in the above section. (It is the proportional controller given by (3.56) that provides the exponential decay (3.72).) From the exponential stability and the form of the feedback, we deduce also that there exists some constant a2 > 0 such that 0 2 0 φ(z k0 ) ≤ a2 ||L−1 k z k || , ∀z k ∈ X .

(3.87)

It is easy to see that the map φ(z) → z ∈ X is continuous, and thus, ||L−1 k z|| ≤ cφ(z). This, together with relation (3.87), shows that there exist constants a1 and a2 such that −1 0 2 0 2 0 0 (3.88) a1 L−1 k z k  ≤ φ(z k ) ≤ a2 Lk z k  , ∀z k ∈ X . Thus by (3.88), there is a linear nonnegative self-adjoint operator Rk : X → X associated with the linear symmetric form φ(·) such that φ(z k0 ) =

1 Rk z k0 , z k0 X , ∀z k0 ∈ X , 2

(3.89)

where Rk ∈ L(X , X ). By the dynamic programming principle, for each 0 < t < T , the optimal solution (ψk∗ , z k∗ ) to (3.86) and (3.85) is also the solution to the optimization problem ⎧  T ⎫ ⎨1 ⎬ 2 2 (||L−1 z (s)|| + |ψ (s)| )ds + φ(z (T )), k k0 k k min 2 t , ⎩ ⎭ subject to (3.85), z k (t) = z k∗ (t)

(3.90)

3.3 Design of a Riccati-Based Feedback

73

z k∗ (t) ∈ X as initial condition, where z k∗ (T ) ∈ X as well. By the maximum principle, we obtain that ∗  ψk∗ (t) = Ak Dγ k qT = νqT (1), a.e. t ∈ (0, T ), ∗ L−2 k Rk z k (t) = −q T (t), ∀t ∈ [0, T ], ∗  ∗ ψk∗ (t) = Ak Dγ k (L−2 k Rk z k (t)), ∀t ≥ 0,

(3.91)

where qT is the solution to the dual equation 0

∗ − A∗k qT = L−2 k z k , ∀t ∈ (0, T ), −2 ∗ qT (T ) = −Lk Rk z k (T ). d q dt T

(3.92)

Let us show that Rk : H → H . More precisely, we show that if z k∗ is the optimal solution to (3.85)–(3.86) (which is also optimal to (3.90)) with z k∗ (0) ∈ H , then ∗ 2 4 Rk z k∗ (0) ∈ H . We have that L−2 k z k ∈ L (0, T ; H (0, 1)). From Eq. (3.92), since −A∗k generates an analytic C0 -semigroup, we know that (T − t) 2 qT ∈ C([0, T ); D((−A∗k ) 2 )) 1

3

(for more details, see, for example, [80]). This yields qT (0) ∈ D((−A∗k ) 2 ) ⊂ H 6 (0, 1), 3

because D(−A∗k ) ⊂ H 4 (0, 1). So Rk z k∗ (0) = −L2k qT (0) ∈ H 2 (0, 1) ⊂ H, as claimed. Finally, we show that Rk is a solution to a Riccati-type equation. To this end, we first notice that again by the dynamic programming principle and (3.89), we have 1 1 Rz k∗ (t), z k∗ (t)X = φ(z k∗ (t)) = 2 2



∞ t

∗ 2 ∗ 2 (L−1 k z k (s) + |ψk0 (s)| )ds,

(3.93)

∀t ≥ 0. Differentiating (3.93) in t and using the self-adjointness of Rk on X and Eq. (3.85), we obtain 1 2 −2 1 −1 ∗ −1 ∗ ∗ ∗  2 2 L−1 k Ak z k (t), Lk Rk z k (t) + ν |(Lk Rk z k (t)) (1)| = Lk z k (t) , (3.94) 2 2 t ≥ 0, which implies, by setting t = 0, that Rk satisfies the following Riccati equation: 1 2 −2 1 −1 0 2 −1 0 0 0  2 0 L−1 k Ak z k , Lk Rk z k  + ν |(Lk Rk z k ) (1)| = Lk z k  , ∀z k ∈ H. 2 2

(3.95)

74

3 Stabilization of Periodic Flows in a Channel

Let us notice that we have proved that Rk z k0 ∈ H for all z k0 ∈ H ; hence 0 4 L−2 k Rk z k ∈ H (0, 1). 0 Thus the third derivative of L−2 k Rk z k in the Riccati algebraic equation (3.95) makes sense. In the end, using the classical Datko’s theorem, we finally obtain the exponential decay of the solution, once the feedback

  ψk0 = −ν L−2 (1) for 0 < |k| ≤ S k Rk Lk vk0 and ψk0 ≡ 0 for |k| > S are plugged into the equations. For the case k = 0 and l = 0, we can easily argue as in the previous case and rely on the results in the above section to deduce that once plugged into the feedback    (1) for 0 < k 2 + l 2 ≤ S ψkl = −ν L−2 kl Rkl Lkl vkl and ψkl ≡ 0 for



k 2 + l 2 > S,

the corresponding solution to the Eq. (3.5) with k = 0 and l = 0 is exponentially decaying. Here Rkl is a self-adjoint linear operator satisfying a Riccati equation similar to (3.95). Putting together all we have obtained above, we deduce the following Riccatibased stabilization result for our problem: Theorem 3.2 Once the feedback Ψ (t, x, z) :=



ψkl (t)eikx eilz ,

k,l∈Z

where ⎧  −ν(L−2 ⎪ k Rk Lk vk0 (t)) (1) ⎪ ⎪ ⎪ 0 ⎨ 0 ψkl (t) := ⎪ −2  ⎪ −ν(L R L ⎪ kl kl vkl (t)) (1) kl ⎪ ⎩ 0

for 0 < |k| ≤ S, l = 0, for |k| > S, l = 0, for k = 0, l ∈ Z,√ for k, l = 0 and √k 2 + l 2 ≤ S, for k, l = 0 and k 2 + l 2 > S,

is plugged into system (3.4), this yields its exponential stability. Here Rk , Rkl : X → X are linear, self-adjoint operators satisfying Riccati-type equations of the form

3.3 Design of a Riccati-Based Feedback

75

1 2 −2 1 −1 0 2 −1 0 0 0  2 0 L−1 k Ak z k , Lk Rk z k  + ν |(Lk Rk z k ) (1)| = Lk z k  , ∀z k ∈ H, 2 2 and 1 2 −2 1 −1 0 2 −1 0 0 0  2 0 L−1 kl Akl z kl , Lkl Rkl z kl  + ν |(Lkl Rkl z kl ) (1)| = Lkl z kl  , ∀z kl ∈ H, 2 2 respectively, where X is the dual of the space H 2 (0, 1) ∩ H01 (0, 1).

3.4 Comments The local stabilization theory for the Navier–Stokes equations by feedback control supported on the boundary of a domain filled with liquid was created in Fursikov [59, 60]. In particular, the feedback theory was developed in Fursikov [58]. The idea to construct boundary controllers was based on previous results on stabilization via internal distributed feedbacks. Roughly speaking, it consists in extending the domain O by a thin strip around the boundary, obtaining thereby the new domain O ∪ Oε , and considering Oε to be the support of the internal feedback. Once the internal feedback is constructed for the new extended system, one may let ε go to zero. The boundary controller for the former problem is the trace of the solution to the latter problem. Another method to deal with boundary actuators is to lift them into the equations via an auxiliary operator acting on functions defined on the boundary with values on the whole domain. (This method is used as well in the results presented in this book.) Then, via the Riccati-based method, boundary stabilizing actuators were constructed in Barbu et al. [19]. Other results on this subject were obtained in Raymond [118, 119]. In [12], Barbu designed an explicit feedback law of proportional type, called oblique, that acts almost normal to the boundary. It has the following form:

∂Φ j + α(x)n(x) , x ∈ ∂O, μ j y, ϕ j  u=η ∂n j=1

N 

where α is an arbitrary continuous function with zero circulation on ∂O, that is,  O

α(x)d x = 0.

One can easily see that u(t, x) · n(x) = α(x), ∀x ∈ ∂O,

76

3 Stabilization of Periodic Flows in a Channel

and | cosu(t, x), n(x)| ≥ 1 −

C , ∀x ∈ ∂O, C + |α(x)|

where C > 0 is independent of α. This means that the stabilizable boundary controller u can be chosen almost normal to ∂O. However, for technical reasons the limit case |α| = +∞, that is, u normal, is excluded from the discussion. Moreover, again the feedback is under the requirement of linear independence of the system of eigenfunctions. The general domain O is replaced by the particular infinite channel form, and via the backstepping technique, stabilizing feedbacks for the Poiseuille profile were designed by Krstic and his coworkers in [1, 29, 124]. In all these works, in order to achieve stability, all the components of the velocity field are controlled on the boundary. Other results are obtained by Triggiani in [117]. From the practical point of view, to implement a tangential control into the system is quite demanding, both from the technological point of view and the cost. The most feasible case is that in which the control acts only on the normal component of the velocity field, the so-called wall-normal controller. Results in this direction were obtained by Barbu in [9, 12] and for the stochastic case in Barbu [11]. More results on the stabilization of the Navier–Stokes flows can be found in the book Barbu [10]. The results presented in this chapter provide as well normal boundary stabilizers, and concerning the construction of a proportional type feedback stabilizer, they appeared in Munteanu [103], and concerning the Riccati-based approach, in the author’s work [95, 96]. We mention that we were not able to deduce the local stability of the full nonlinear Navier–Stokes system, because of the normal boundary conditions. More precisely, in trying to reduce the pressure from the nonlinear system, the usual trick is to apply the Leray projector. However, due to the nontangential conditions, this cannot be done.

Chapter 4

Stabilization of the Magnetohydrodynamics Equations in a Channel

Here we consider again a channel flow. But in addition to the assumptions of the previous chapter, we assume that the incompressible fluid is electrically conducting and affected by a constant transverse magnetic field. This kind of flow was first investigated both experimentally and theoretically by Hartmann [67]. The governing equations are the magnetohydrodynamics equations (MHD, for short), which are a coupling between the Navier–Stokes equations and the Maxwell equations.

4.1 The Magnetohydrodynamics Equations of an Incompressible Fluid The 2-D MHD equations are ⎧ ρ(ut − νΔu + uux + vvy ) + CCx − CBy = −px , ⎪ ⎪ ⎪ ⎪ ⎨ ρ(vt − νΔv + uvx + vvy ) + BBy − BCx = −py , 1 ΔB + uBx + vBy − Bux − Cuy = 0, Bt − μσ ⎪ 1 ⎪ ⎪ Ct − μσ ΔC + uCx + vCy − Bvx − Cvy = 0, ⎪ ⎩ ux + vy = 0, Bx + Cy = 0, t ≥ 0, (x, y) ∈ R × (−L, L).

(4.1)

Here (u, v) is the velocity field, p is the scalar pressure, and (B, C) is the magnetic field. The positive constants ρ, ν, μ, and σ represent the fluid mass density, the kinematic viscosity, the magnetic permeability, and the electrical conductivity, respectively; 2L is the distance between the walls. These equations are of huge importance, and they are used in the study of magnetofluids such as plasmas, liquid metals, salt water, and electrolytes.

© Springer Nature Switzerland AG 2019 I. Munteanu, Boundary Stabilization of Parabolic Equations, Progress in Nonlinear Differential Equations and Their Applications 93, https://doi.org/10.1007/978-3-030-11099-4_4

77

78

4 Stabilization of the Magnetohydrodynamics Equations in a Channel

The fully developed steady state of (4.1), the Hartmann–Poiseuille profile, which we are going to stabilize, is given by   1 cosh(Hay∗ ) 1 1− , vˆ ≡ 0 uˆ (y ) = Ha tanh(Ha) cosh(Ha) y∗ 1 sinh(Hay∗ ) ˆ Bˆ = − + , C ≡ B0 , Ha Ha sinh(Ha) ∗

(4.2)

 σ where y∗ := Ly , Ha := B0 L ρν . For later purposes, we notice that ∗ 1 e−Hay ˆ ∗ ≤ 2, y∗ ∈ [−1, 1]. + |(ˆu + B)y | = − Ha sinh(Ha)

(4.3)

Here B0 is the constant external applied magnetic field. As one can easily see, the MHD equations, in comparison to the Navier–Stokes equations, are far more complex. Thus, the whole effort from now on is to reduce, in various ways, their complexity. A first simplification is to define the dimensionless variables x∗ :=

x 1 v0 t 1 , (u∗ , v∗ ) := (u, v), t ∗ := , (B∗ , C ∗ ) := (B, C), L v0 L b0

with v0 := −

L2 pˆ x and b0 := −μL2 ρν



σ pˆ y , ρν

(ˆp is the pressure corresponding to the equilibrium solution (4.2)). For the sake of simplicity we drop the star notation. However, we keep in mind that now we are dealing with the above variables. Again we assume 2π -periodicity with respect to the x-coordinate of the velocity field, the magnetic field, and the pressure. In addition, we impose that the magnetic Prandtl number of the fluid, i.e., Prm := νμσ , be equal to one. Such a periodic MHD channel flow does not directly correspond to a specific laboratory fluid. It is, however, often studied as an approximation to torus devices of plasma-controlled fusion, such as the Tokamak and the reversed field pinch. Numerical simulations have shown that turbulence may appear in the movement of this kind of flow; that is, the flow may become unstable. It is easily seen that Prm = 1 implies N = R = Rm = 1, after rescaling as necessary. So the linearization of system (4.1) around the equilibrium profile (4.2), supplemented with the boundary conditions, has the form

4.1 The Magnetohydrodynamics Equations of an Incompressible Fluid

⎧ ut − Δu + uˆ ux + vuˆ y + B0 Cx − B0 By − Bˆ y C = px , ⎪ ⎪ ⎪ ⎪ ˆ + BB ˆ y − BC ˆ x = py , vt − Δv + uˆ vx + BB ⎪ ⎪ ⎪ ⎪ ˆ ˆ ⎪ Bt − ΔB + uˆ Bx + By v − Bux − B0 uy − uˆ y C = 0, ⎪ ⎪ ⎪ ˆ x − B0 vy = 0, ⎪ Ct − ΔC + uˆ Cx − Bv ⎪ ⎪ ⎪ u + v = 0, B + C = 0, ⎨ x

y

x

y

u(t, x + 2π, y) = u(t, x, y), v(t, x + 2π, y) = v(t, x, y), ⎪ ⎪ ⎪ B(t, x + 2π, y) = B(t, x, y), C(t, x + 2π, y) = C(t, x, y), ⎪ ⎪ ⎪ ⎪ p(t, x + 2π, y) = p(t, x, y), ⎪ ⎪ ⎪ ⎪ u(t, x, −1) = u(t, x, 1) = v(t, x, −1) = 0, v(t, x, 1) = Ψ (t, x), ⎪ ⎪ ⎪ ⎪ B(t, x, −1) = B(t, x, 1) = C(t, x, −1) = 0, C(t, x, 1) = Ξ (t, x), ⎪ ⎩ t ≥ 0, x ∈ R, y ∈ (−1, 1),

79

(4.4)

and initial data u0 , v0 , B0 , C 0 . Here Ψ and Ξ are the boundary controllers, which means that both the normal components of the velocity field and the magnetic field are controlled on the upper wall. Of course, from the practical point of view, it would have been more convenient to control only the wall-normal velocity. Unfortunately, this is not possible with the algorithm from Chap. 2, because, in trying to show the unique continuation property (see Lemma 4.2 below) related to a vector-valued operator, one cannot prove that both components of the unstable eigenvector are nonzero. Rather, a weaker result is available, saying that both components cannot vanish simultaneously. Consequently, both the velocity and the magnetic field must be controlled. As in the previous chapter, we take advantage of the 2π -periodicity and decompose (4.4) into Fourier modes. We get the following infinite system, indexed by k ∈ Z: ⎧ (uk )t − (−k 2 uk + uk ) + ik uˆ uk + uˆ  vk + ikB0 ck − B0 bk − Bˆ  ck = ikpk , ⎪ ⎪ ⎪ ⎪ ˆ  − ik Bc ˆ k = p , (vk )t − (−k 2 vk + vk ) + ik uˆ vk + Bˆ  bk + Bb ⎪ k k ⎪ ⎪ ⎪ (b ) − (−k 2 b + b ) + ik uˆ b + Bˆ  v − ik Bu ˆ k − B0 u − uˆ  ck = 0, ⎨ k t k k k k k ˆ k − B0 v = 0, (ck )t − (−k 2 ck + ck ) + ik uˆ ck − ik Bv k ⎪ ⎪   ⎪ ikuk + vk = 0, ikbk + ck = 0, t ≥ 0, y ∈ (−1, 1), ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ bk (−1) = bk (1) = ck (−1) = 0, uk (−1) = uk (1) = vk (−1) = 0, vk (1) = ψk , ck (1) = ξk ,

(4.5)

with initial data uk0 , vk0 , b0k , dk0 . ∂ . Here  denotes the derivative with respect to y, i.e., ∂y Stabilization of (4.4) is equivalent to stabilization of (4.5) at each level k ∈ Z. When k = 0, we put ψ0 ≡ ξ0 ≡ 0, and after some straightforward computations, we deduce that u0 (t)2 + v0 (t)2 + b0 (t)2 + c0 (t)2 ≤ Ce−αt (u00 2 + v00 2 + b00 2 + c00 2 ), t ≥ 0, for some positive constants C, α.

(4.6)

80

4 Stabilization of the Magnetohydrodynamics Equations in a Channel

Since we have taken care of k = 0, from now on we will consider only k = 0. System (4.5) still looks complicated, since it involves five unknowns (with only four equations). Of course, one may try to reduce the pressure in the same manner as we did in the previous chapter. We will do this, but before that, we aim to reduce the number of the field’s unknowns as well. To this end, let us set S1k := uk + bk , S2k := vk + ck , D1k := uk − bk , D2k := vk − ck , and 0 0 0 0 S1k := uk0 + b0k , S2k := vk0 + ck0 , D1k := uk0 − b0k , D2k := vk0 − ck0 .

Then we add the first equation to the third one of (4.5), and the second equation to the fourth one of (4.5). In this way, we obtain the two-equation system    +u ˆ k ) = ikpk , ˆ  D2k + Bˆ  D2k + ik(B0 ck − Bu (S1k )t − (−k 2 S1k + S1k ) + ik uˆ S1k − B0 S1k     2  ˆ ˆ ˆ (S2k )t − (−k S2k + S2k ) + ik uˆ S2k + ik BS2k − B0 vk + B bk + Bbk = pk .

Then we reduce the pressure from it and use the divergence-free conditions to find that     ˆ 2k ˆ 2k (−S2k + k 2 S2k )t + S2k + BS − [2k 2 + ik D]S

 ˆ  + k 2 B0 ]S2k ˆ 2k + ik[(Sˆ  D2k ] = 0. − [ik D + [k 4 + ik 3 D]S

(4.7)

Here and in the following, ˆ := uˆ − B. ˆ Sˆ := uˆ + Bˆ and D We do the same for the differences. More precisely, we subtract the third equation from the first one of (4.5), the fourth equation form the second one of (4.5), and reduce the pressure as before to arrive at    ˆ 2k + k 2 D2k )t + D2k − B0 D2k − [2k 2 + ik S]D − [ik sˆ  − k 2 B0 ]D2k (−D2k 



ˆ 2k + ik[D ˆ  S2k ] = 0. + [k 4 + ik 3 S]D

(4.8)

Hence by (4.7) and (4.8), we get that (4.5) is equivalent to ⎧     ˆ 2k ˆ 2k (−S2k + k 2 S2k )t + S2k + BS − [2k 2 + ik D]S ⎪ ⎪ ⎪ ⎪  ⎪ ˆ  + k 2 B0 ]S2k ˆ 2k + ik[(Sˆ  D2k ] = 0, ⎪ − [ik D + [k 4 + ik 3 D]S ⎪ ⎪ ⎨      ˆ 2k + k 2 D2k )t + D2k − B0 D2k − [2k 2 + ik S]D − [ik sˆ  − k 2 B0 ]D2k (−D2k ⎪ ⎪ ˆ 2k + ik[D ˆ  S2k ] = 0, ⎪ + [k 4 + ik 3 S]D ⎪ ⎪     ⎪ ⎪ S2k (−1) = S2k (1) = S2k (−1) = D2k (−1) = D2k (1) = D2k (−1) = 0, ⎪ ⎩ S D S2k (1) = ψk := ψk + ξk , D2k (1) = ψk := ψk − ξk , (4.9) 0 0 := vk0 + ck0 , D2k := vk0 − ck0 . Thus we have reduced the fiveand the initial data S2k unknown problem (4.5) to the two-unknown problem (4.9).

4.1 The Magnetohydrodynamics Equations of an Incompressible Fluid

81

To write the above system in an abstract form, we introduce the linear operators Lk : D(Lk ) ⊂ H × H → H × H and Fk : D(Fk ) ⊂ H × H → H × H , defined as Lk (S D)T :=

−S  + k 2 S −D + k 2 D

 2 , D(Lk ) = H 2 (−1, 1) ∩ H01 (−1, 1)

(4.10)

(here (· ·)T means the transpose matrix) and Fk (S D)T (4.11)     ˆ  − [ik D ˆ  + k 2 B0 ]S  + [(k 4 + ik 3 D]S ˆ + ik[Sˆ  D] S + B0 S − [2k 2 + ik D]S , :=   ˆ  − [ik Sˆ  − k 2 B0 ]D + [(k 4 + ik 3 S]D ˆ + ik[D ˆ  S] D − B0 D − [2k 2 + ik S]D  2 D (Fk ) = H 4 (−1, 1) ∩ H02 (−1, 1) ,

respectively. We will denote by Lk and by Fk the differential forms of the operators Lk and Fk , respectively (that is, we do not take into account the domain of definition, but only the form). Moreover, we define the operator   −1 T T Ak := Fk L−1 k , D(Ak ) = (S D) : Lk (S D) ∈ D(Fk ) . Regarding the operator −Ak , we may prove the following lemma, arguing similarly as in the proof of Lemma 3.2. Lemma 4.1 The operator −Ak generates a C0 -analytic semigroup on H × H , and for each λ ∈ ρ(−Ak ) (the resolvent set of −Ak ), (λI + Ak )−1 is compact. Moreover, there exists M > 0 such that σ (−Ak ) ⊂ {λ ∈ C : λ < 0} , ∀|k| > M . Here σ (−Ak ) is the spectrum of −Ak . Lemma 4.1 says that for all |k| > M , we may take ψkS ≡ ψkD ≡ 0, since at these levels the system is stable. Therefore, it remains to stabilize the system (4.9) for 0 < |k| ≤ M only. Besides this, Lemma 4.1 guarantees that the operator −Ak has a countable set of eigenvalues, denoted by {λkj }∞ j=1 (each repeated according to its multiplicity); and there is only a finite number Nk of eigenvalues for which λkj ≥ 0, j = 1, . . . , Nk , the unstable eigenvalues. Finally, let 

ϕjk := (ϕ1jk ϕ2jk )T

∞ j=1

∞  and ϕjk∗ := (ϕ1jk∗ ϕ2jk∗ )T

j=1

denote the corresponding eigenvectors of the operator −Ak and its adjoint −A∗k , respectively. For the sake of simplicity, we assume that the unstable eigenvalues are simple. Hence, we may suppose that they are arranged such that

λk1 < λk2 < · · · < λkNk .

82

4 Stabilization of the Magnetohydrodynamics Equations in a Channel

k k This assumption implies also that the systems {ϕjk }Nj=1 and {ϕjk∗ }Nj=1 may be chosen to be biorthogonal, that is,

ϕik , ϕjk∗  = δij , i, j = 1, . . . , Nk , with δij the Kronecker symbol. However, one may apply the present stabilizing algorithm to the semisimple case of eigenvalues as well (see Chap. 2), but since it might be difficult to follow the computations, we will not develop this problem here. Furthermore, as in Lemma 3.3, a “unique continuation”-type result for the eigenvectors of the dual operator −A∗k of −Ak can be obtained, though there is a major difference between the operator −Ak introduced in (3.22) and the present −Ak . Namely, the latter acts on vectors. Let us consider the counterpart of (3.34), which reads as  (λLk + F∗k )(ϕ1 ϕ2 )T = 0, t > 0, (4.12) (ϕ1 ϕ2 )T (−1) = (ϕ1 ϕ2 )T (1) = (ϕ1 ϕ2 )T (−1) = (ϕ1 ϕ2 )T (1) = 0. Now our goal is to show that (ϕ1 ϕ2 )T (1) = (0 0)T . In other words, ϕ1 (1) and ϕ2 (1) cannot vanish simultaneously for every eigenvector corresponding to an unstable eigenvalue of the adjoint operator. This is equivalent to the fact that there exists μk ∈ C such that (ϕ1 ) (1) + μk (ϕ2 ) (1) = 0, for all the eigenvectors corresponding to the unstable eigenvalues. This is exactly what we prove below. Then, with the help of this μk , we will construct our controller (see (4.19) below). Similarly as in Lemma 3.3, we will show that if (ϕ1 ϕ2 )T solves (4.12) plus   T (ϕ1 ϕ2 ) (1) = (0 0)T , then necessarily (ϕ1 ϕ2 )T ≡ (0 0)T , which is in contradiction to the fact that (ϕ1 ϕ2 )T is an eigenvector. The symmetric nature of Eq. (4.12) will be decisive again. However, this time, the symmetry is more curious, namely if (ϕ1 ϕ2 )T (y) solves (4.12), then (ϕ2 ϕ1 )T (−y) solves (4.12) as well. Although this is much weaker than what we had in Lemma 3.3, it is enough to prove our claim. Replacing the eigenvector as necessary, we may gain an additional null boundary condition, that is, (ϕ1 ϕ2 )T (−1) = (0 0)T . Taking into account Eqs. (4.26)–(4.27) below, we see that in this case, the corresponding D (given in (2.17)) reads as

4.1 The Magnetohydrodynamics Equations of an Incompressible Fluid

83

D (ϕ1 ϕ2 )T = (ϕ1 ) (1) + μk (ϕ2 ) (1). The lemma below says nothing but the fact that assumption (A5) from Chap. 2 holds in the present case. Lemma 4.2 Let 0 < |k| ≤ M . Then there exists μk ∈ C such that (ϕ1jk∗ ) (1) + μk (ϕ2jk∗ ) (1) > 0, j = 1, . . . , Nk . Proof Below, we will understand by ∧ and by ∨ the logical symbols for “and” and “or,” respectively. Fix k ∈ Z such that 0 < |k| ≤ M . For the sake of simplicity of notation, let us set λ := λkj and ϕ := ϕjk∗ . First, consider the case in which ϕ ∗ is a classical eigenvector corresponding to the eigenvalue λ, i.e., −A∗k ϕ ∗ = λϕ ∗ . ∗ Hence ϕ := L−1 k ϕ solves

(λLk + F∗k )ϕ = 0,

(4.13)

where F∗k is the dual of the operator Fk defined by (4.11). Given a function f : [−1, 1] → C, we denote by fˇ : [−1, 1] → C the function fˇ (y) := f (−y), y ∈ [−1, 1]. ˇ ˆ and that It is easy to check that Sˆ = D, (λLk + F∗k )(ϕ1 ϕ2 )T = 0 ⇒ (λLk + F∗k )(ϕˇ 2 ϕˇ1 )T = 0. Set (ψ1 ψ2 )T := (ϕ1 + ϕˇ2 ϕ2 + ϕˇ1 )T . From above, we deduce that (λLk + F∗k )(ψ1 ψ2 )T = 0. 



(4.14) 



We have two cases: either (ψ1 (1) = 0) ∨ (ψ2 (1) = 0) or ψ1 (1) = ψ2 (1) = 0.   Assume first that ψ1 (1) = ψ2 (1) = 0. Since ψˇ 1 = ψ2 , we also have that  ψ1 (−1) = 0. Defining  Ψ := −ψ1 + k 2 ψ1 , we have from (4.14) that

84

4 Stabilization of the Magnetohydrodynamics Equations in a Channel



ˆ + λ)Ψ + ik D ˆ  (ψ1 + ψˇ 1 ) = 0 in (−1, 1), −Ψ + B0 Ψ  + (k 2 − ik D   Ψ (−1) = Ψ (1) = 0. 

(4.15)

Scalar multiplying (4.15) by Ψ and taking the real part of the result, we get  Ψ  + (k + λ)Ψ  + ik  2

2

1

2

−1

  ˆ ˇ D (ψ1 + ψ1 ) Ψ dy = 0.

(4.16)

Simple computations show that 



Ψ  2 = ψ1 2 + 2k 2 ψ1 2 + k 4 ψ1 2 . It follows from (4.16), via Poincaré’s inequality π 2 v2 ≤ v 2 , ∀v ∈ H 2 (−1, 1) ∩ H01 (−1, 1), and relation (4.3) that 

ψ1 2 + (k 4 + 2k 2 π 2 )ψ1 2 + (k 2 + λ)Ψ 2  1  1 ˆ  (ψˇ 1 + ψ1 ) Ψˇ dy ˆ  (ψ1 + ψˇ 1 ) Ψ dy = ik S ≤ ik D ≤

−1 4ψ1 2

(4.17)

−1

+ k 2 Ψ 2 .

Recall that λ is an unstable eigenvalue. Consequently, λ ≥ 0. Moreover, k 4 + 2π 2 k 2 − 4 > 0, ∀k ∈ Z∗ . Thus relation (4.17) implies that ψ1 ≡ ψ2 ≡ 0. Therefore, we see that ϕ1 = −ϕˇ2 . With this in hand, we claim that 



(ϕ1 (1) = 0) ∨ (ϕ2 (1) = 0). Indeed, assume for the sake of contradiction that 



ϕ (1) = ϕ2 (1) = 0. 

Since ϕ1 = −ϕˇ2 , we get as well that ϕ1 (−1) = 0.  Hence setting Φ := −ϕ1 + k 2 ϕ1 , we obtain by (4.13) that 

 ˆ + λ)Φ + ik D ˆ  (ϕ1 + ϕˇ1 ) = 0, −Φ + B0 Φ  + (k 2 − ik D   Φ (−1) = Φ (1) = 0.

(4.18)

Similarly as above, scalar multiplying (4.18) by Φ and taking the real part of the result, we obtain that

4.1 The Magnetohydrodynamics Equations of an Incompressible Fluid

85

ϕ1 = ϕ2 = 0. This is in contradiction to the fact that ϕ = (ϕ1 ϕ2 )T is an eigenvector. We conclude   that in the case ψ1 (1) = ψ2 (1) = 0, we necessarily have that 



(ϕ1 (1) = 0) ∨ (ϕ2 (1) = 0). Now if we take (χ1 χ2 )T := (ϕ1 − ϕˇ1 ϕ2 − ϕˇ2 )T 



and argue as before, we get that in the case χ1 (1) = χ2 (1) = 0, we necessarily have that   (ϕ1 (1) = 0) ∨ (ϕ2 (1) = 0). From the above, we get the following cases: (1) [ψ1 (1) = ψ2 (1) = 0] ∧ [χ1 (1) = χ2 (1) = 0]. This implies that 



(ϕ1 (1) = 0) ∨ (ϕ2 (1) = 0). 

ϕ (1)



This means that for all θ ∈ C∗ such that θ = − ϕ1 (1) (in the case that ϕ2 (1) = 0, 2 otherwise for all θ ∈ C∗ ), we have 



ϕ1 (1) + θ ϕ2 (1) = 0. (2) [ψ1 (1) = ψ2 (1) = 0] ∧ [(χ1 (1) = 0) ∨ (χ2 (1) = 0)]. Again the first one implies that   (ϕ1 (1) = 0) ∨ (ϕ2 (1) = 0), 

ϕ (1)

which means, as before, that for all θ ∈ C∗ such that θ = − ϕ1 (1) (in the case 

that ϕ2 (1) = 0, otherwise for all θ ∈ C∗ ), we have 

2



ϕ1 (1) + θ ϕ2 (1) = 0. (3) [(ψ1 (1) = 0) ∨ (ψ2 (1) = 0)] ∧ [χ1 (1) = χ2 (1) = 0]. The second one implies 

ϕ (1)



as before that for all θ ∈ C∗ such that θ = − ϕ1 (1) (in the case that ϕ2 (1) = 0, 2 otherwise for all θ ∈ C∗ ), we have 



ϕ1 (1) + θ ϕ2 (1) = 0. (4) [(ψ1 (1) = 0) ∨ (ψ2 (1) = 0)] ∧ [(χ1 (1) = 0) ∨ (χ2 (1) = 0)]. By the fact that (ψ1 + χ1 ψ2 + χ2 )T = 2(ϕ1 ϕ2 )T , we get as before that there exists infinitely many θ ∈ C such that

86

4 Stabilization of the Magnetohydrodynamics Equations in a Channel 



ϕ1 (1) + θ ϕ2 (1) = 0. We conclude that in every case, there exists some μk ∈ C such that 



ϕ1 (1) + μk ϕ2 (1) = 0, and the conclusion follows immediately for this case. Now we treat the general case of generalized eigenvectors. Let us consider the chain (ϕ11 ϕ21 )T , . . . , (ϕ1J ϕ2J )T for some J ∈ N such that (λ + A∗k )(ϕ11 ϕ21 )T = 0 and

(λ + A∗k )j (ϕ1j ϕ2j )T = 0, j = 2, 3, . . . , J .

Concerning (ϕ11 ϕ21 )T , we may show, as in the above lines, that there exists some μ such that   ϕ11 (1) + μϕ21 (1) = 0. Then if needed, replacing (ϕ1j ϕ2j )T , j = 2, 3, . . . , J , by (ϕ1j ϕ2j )T + qj (ϕ11 ϕ21 )T , with qj > 0 properly chosen such that 







[ϕ1j (1) + qj ϕ11 (1)] + μ[ϕ2j (1) + qj ϕ21 (1)] = 0, 

we easily complete the proof.

4.2 The Stabilizing Proportional Feedback The main theorem of this section amounts to saying that the following feedback laws, once plugged into the system (4.4) yield its stability. Let us define 1  (1 + μk )U k (t)eikx , 2 0 0, α0

(5.7)

with α0 > 0 chosen such that

that is,  α0 =

γ0 . l0

(5.8)

Writing the system (5.1)–(5.3) in the variables ϕ and σ and using (5.7) and the notation l := γ0 l0 , we obtain the equivalent nonlinear system

(5.9)

5.1 Presentation of the Problem

95

⎧ 2  ⎪ ⎪ ϕt + νΔ ϕ − ΔF (ϕ) − lΔϕ + γ Δσ = 0, in (0, ∞) × O, ⎪ ⎪ ⎪ σt − Δσ + γ Δϕ = 0, in (0, ∞) × O, ⎪ ⎪ ⎪ ⎪ ⎪ ∂ϕ ⎪ ⎨ = 0, on (0, ∞) × Γ, ∂n ⎪ ∂Δϕ γ0 ∂σ ⎪ ⎪ ⎪ = − u, = α0 u, on (0, ∞) × Γ1 , ⎪ ⎪ ∂n ν ∂n ⎪ ⎪ ⎪ ⎪ ∂σ ∂Δϕ ⎪ ⎩ = = 0, on (0, ∞) × Γ2 . ∂n ∂n

(5.10)

Then we continue with the classical step, namely reducing the problem to the null stabilization, via the fluctuation variables y := ϕ − ϕ, ˆ z := σ − σˆ , ˆ z o := σo − σˆ , yo := ϕo − ϕ,

(5.11) (5.12)

ˆ and σo := α0 (θo + l0 ϕo ). And so system (5.1)–(5.3) where, clearly, σˆ := α0 (θˆ + l0 ϕ) transforms into the equivalent null boundary stabilization problem ⎧ yt + νΔ2 y − Δ[F  (y + ϕ) ˆ − F  (ϕ)] ˆ − lΔy + γ Δz = 0, in (0, ∞) × O, ⎪ ⎪ ⎪ ⎪ ⎪ z − Δz + γ Δy = 0, in (0, ∞) × O, ⎪ ⎪ t ⎪ ⎪ ⎪ ∂y ⎪ ⎪ ⎪ ⎨ ∂n = 0, in (0, ∞) × Γ, ∂Δy γ0 ∂z ⎪ = − u, = α0 u, on (0, ∞) × Γ1 , ⎪ ⎪ ⎪ ∂n ν ∂n ⎪ ⎪ ⎪ ∂z ∂Δy ⎪ ⎪ = = 0, on (0, ∞) × Γ2 , ⎪ ⎪ ⎪ ∂n ∂n ⎪ ⎩ y(0) = yo , z(0) = z o . (5.13) This is a fourth-order differential system due to the presence of Δ2 . We have met fourth-order differential equations before, in Chap. 3, regarding the Navier–Stokes equations in a channel, and in Chap. 4 for the magnetohydrodynamics equations in a channel. The main difference between those equations and (5.13) is that now we are dealing with nonlinearities under the Laplace operator. Consequently, the linearized system will not be subtracted from (5.13) in the classical way. More precisely, we set  + l, (5.14) Fl := F∞ where  := F∞

1 mO

 O

F  (ϕ(ξ ˆ ))dξ,

with m O the Lebesgue measure of O, and we introduce the linear system

(5.15)

96

5 Stabilization of the Cahn–Hilliard System

⎧ yt + νΔ2 y − Fl Δy + γ Δz = 0 in (0, ∞) × O, ⎪ ⎪ ⎪ ⎪ ⎪ z t − Δz + γ Δy = 0 in (0, ∞) × O, ⎪ ⎪ ⎪ ⎪ ⎪ ∂y ⎪ ⎪ ⎪ ⎨ ∂n = 0 in (0, ∞) × Γ, ∂Δy γ0 ∂z ⎪ = − u, = α0 u on (0, ∞) × Γ1 , ⎪ ⎪ ⎪ ∂n ν ∂n ⎪ ⎪ ⎪ ∂z ∂Δy ⎪ ⎪ = = 0 on (0, ∞) × Γ2 , ⎪ ⎪ ⎪ ∂n ∂n ⎪ ⎩ y(0) = yo , z(0) = z o .

(5.16)

We remark that the above system is not the linearization of (5.13), since the replacement of the nonlinear term is different from the usual one.

5.1.1 Stabilization of the Linearized System Set A : D(A) ⊂ L 2 (O) × L 2 (O) → L 2 (O) × L 2 (O),  A :=

νΔ2 − Fl Δ γ Δ , γΔ −Δ

(5.17)

having the domain

 ∂y ∂Δy ∂z D(A) = (y z)T ∈ L 2 (O ) × L 2 (O ) : A (y z)T ∈ L 2 (O ) × L 2 (O ), = = = 0 on Γ ∂n ∂n ∂n

endowed with its graph norm. By the regularity of O, it follows that D(A) ⊂ H 4 (O) × H 2 (O); also we notice that A is self-adjoint. Proposition 5.1 The operator A is quasi-m-accretive on L 2 × L 2 , that is, λI + A is m-accretive for some λ > 0, and its resolvent is compact. 1

Proof We set V = D(A) × D(A 2 ). We compute the scalar product  A(y z)T , (φ ψ)T =

(νΔy · Δφ + Fl ∇ y · ∇φ − γ ∇z · ∇φ)d x  + (∇z · ∇ψ − γ ∇ y · ∇ψ)d x, O

O

for all (φ ψ)T ∈ V. We see easily that A is bounded from V to V  , the dual of V . Indeed, we have

5.1 Presentation of the Problem

A(y z)T V  =

97

sup (φ ψ)T ∈V,(φ ψ)T )V ≤1

A(y z)T , (φ ψ)T ≤ C(y z)T V .

Moreover,  A(y z)T , (y z)T =

O

(ν|Δy|2 + Fl |∇ y|2 − 2γ ∇ y · ∇z + |∇z|2 )d x

1 ≥ νΔy2 − (|Fl | + 2γ 2 )∇ y2 + ∇z2 2 1 1 = νΔy2 + z2H 1 (O ) − νy2 − z2 − a0 ∇ y2 , 2 2 with a0 := |Fl | + 2γ 2 . Next, by the interpolation a0 ∇ y2 ≤ CΔyy ≤

ν C2 Δy2 + y2 , 2 2ν

we deduce from the above that A(y z)T , (y z)T ≥ C1 (y z)T 2V − C2 (y z)T 2 , for all (y z)T ∈ V.

(5.18)

The above relations lead to the fact that A is quasi-m-accretive, which means that A + C2 I : V → V  is coercive, thus surjective. Consequently, (λI + A)−1 is well defined for λ ≥ C2 . Let ( f 1 f 2 )T ∈ L 2 × L 2 , and define (λI + A)−1 ( f 1 f 2 )T = (y z)T . It is readily seen that (5.18) implies (y z)T 2V ≤ C( f 1 f 2 )T 2 , for λ ≥ C2 , and some C > 0, whence it follows that (λI + A)−1 (E) is relatively compact when ever E is bounded in L 2 × L 2 . (For more details, see [17, Proposition 2.1].)

∞ Therefore, A has a countable set λ j j=1 of real eigenvalues and a complete set of corresponding eigenvectors. Moreover, all the eigenspaces are finite-dimensional, and by repeating each eigenvalue according to its multiplicity, we have that λ1 ≤ λ2 ≤ λ3 ≤ . . . and lim λ j = +∞. j→∞

(5.19)

 We note that zero is an eigenvalue, and it is of multiplicity 2, since 2m1O (1 1)T  and 2m1O (−1 1)T are eigenvectors for it. By (5.19), the number of nonpositive eigenvalues is finite, i.e., for some N ∈ N, we have that

98

5 Stabilization of the Cahn–Hilliard System

λ j < 0, j = 1, 2, . . . , N − 2, λ N −1 = λ N = 0 and λ j > 0 for j > N .

(5.20)

For the sake of simplicity, we assume that (H1 ) : Each negative eigenvalue is simple.

(5.21)

(Of course, one can consider the general case as well, namely the semisimple case, arguing similarly ∞last part of Chap. 2, and still obtain a stabilization result.)

as in the Denote by (ϕ j ψ j )T j=1 the corresponding eigenvectors, that is, ⎧ 2 νΔ ϕ j − Fl Δϕ j + γ Δψ j = λ j ϕ j , in O, ⎪ ⎪ ⎨ γ Δϕ j − Δψ j = λ j ψ j , in O, ⎪ ⎪ ⎩ ∂ϕ j = ∂Δϕ j = ∂ψ j = 0, on Γ, ∂n ∂n ∂n

(5.22)

for all j = 1, 2, . . . . ∞

By the self-adjointness of A, we may assume that the system (ϕ j ψ j )T j=1 forms an orthonormal basis in L 2 (O) × L 2 (O) that is orthogonal in D(A). The control design procedure developed in Chap. 2 requires further knowledge about the eigenvectors of the linear operator A. We refer to the validation of the decisive hypothesis (A5) regarding the unique continuation of the eigenvectors. It is clear by the form of the operator A, which involves the Laplace operator, that the eigenvectors (ϕ j ψ j )T can be associated with the eigenfunctions of the Neumann–

∞ Laplacian (this is indeed true; see (5.23) below). In this light, let us denote by μ j j=1

∞ and by e j j=1 the eigenvalues and the normalized eigenfunctions of the Neumann– Laplacian, respectively, i.e., Δe j = μ j e j in O and

∂e j = 0 on Γ, ∂n

to which we simply refer as the Laplace operator Δ in the sequel. ∞ know that

We μ j ≤ 0 for all j = 1, 2, . . . , μ j → −∞ for j → ∞. Moreover, e j j=1 forms an orthonormal basis in L 2 (O) that is orthogonal in H 1 (O). We have enough experience (from the previous chapters) to realize that the Neumann boundary conditions yield that hypothesis (A5) reads as follows: the trace of the eigenvector (ϕ j ψ j )T , j = 1, 2, . . . , N , is not identically zero on Γ1 . In any case, due to the boundary conditions in (5.16) and the definition of the Neumann map Dη in (5.27) below, in the present case we have that D (ϕ ψ)T = ψ; see (5.28) below. Hence our task is considerably simplified, since we must show the nonvanishing of the second component of the eigenvector only. More exactly, it

5.1 Presentation of the Problem

99

is enough to show that only ψ j , j = 1, 2, . . . , N , does not vanish on Γ1 . To this end, we will adopt the idea from Chap. 3 (or Chap. 4), namely a priori we choose the eigenvectors (ϕ j ψ j )T , j = 1, 2, . . . , N , such that ψ j = 0 on Γ1 . For j = N − 1, N , there is nothing to prove, since we have already set (see after (5.19))  (ϕ N −1 ψ N −1 )T =

 1 (1 1)T and (ϕ N ψ N )T = 2m O

1 (−1 1)T . 2m O

Therefore, the result below concerns only the negative eigenvalues. Lemma 5.1 For all j = 1, 2, . . . , N − 2, there exists an eigenfunction ek of the 2 Laplace operator corresponding to the eigenvalue μk that satisfies 0 > μk ≥ Fl −γ ν such that γ μk ek . ψj ≡  2 (γ μk ) + (λ j + μk )2

(5.23)

In this case, the eigenvalue λ j is a root of the second-degree polynomial X 2 + [(Fl + 1)μk − νμ2k ]X − νμ3k + (Fl − γ 2 )μ2k . In particular, we deduce that necessarily Fl − γ 2 ≤ 0, in order to have unstable negative eigenvalues for the operator A; and ψ j ≡ 0 on Γ1 , ∀ j = 1, 2, . . . , N .

(5.24)

Proof Let j ∈ {1, 2, . . . , N − 2} . For the sake of simplicity of notation, we drop the indices j, that is, we use the notation (ϕ ψ)T = (ϕ j ψ j )T and λ = λ j . We have that (ϕ ψ)T satisfies ⎧ 2 νΔ ϕ − Fl Δϕ + γ Δψ = λϕ, in , ⎪ ⎪ ⎨ γ Δϕ − Δψ = λψ, in , (5.25) ⎪ ⎪ ⎩ ∂ϕ = ∂Δϕ = ∂ψ = 0, on Γ. ∂n ∂n ∂n

∞ Let us decompose ϕ and ψ in the basis e j j=1 of the eigenfunctions of the Neumann Laplacian as ϕ=

∞  j=1

ϕ jej, ψ =

∞  j=1

ψ jej.

100

5 Stabilization of the Cahn–Hilliard System

Successively scalar multiplying equation (5.25) by e j , j = 1, 2, . . ., using the boundary conditions and Green’s formula, we deduce that 

(νμ2j − Fl μ j − λ)ϕ j + γ μ j ψ j = 0, γ μ j ϕ j − (λ + μ j )ψ j = 0, ∀ j ∈ N∗ .

For all j, this is a second-order linear homogeneous system, with the unknowns ϕ j , ψ j . Computing the determinant of the matrix of the system, we get that μ j must satisfy − ν(μ j )3 + (Fl − γ 2 − νλ)(μ j )2 + λ(Fl + 1)μ j + λ2 = 0,

(5.26)

in order not to have ϕ j = ψ j = 0. The polynomial −ν X 3 + (Fl − γ 2 − νλ)X 2 + λ(Fl + 1)X + λ2 , may have at most two distinct negative roots (since the free term is λ2 > 0). Denote them by X 1 < 0, X 2 < 0. Assume that we have μk = μk+1 = · · · = μk+M = X 1 , and μs = μs+1 = · · · = μs+L = X 2 , i.e., X 1 is an eigenvalue of the Laplace operator of multiplicity M + 1, and X 2 is an eigenvalue of the Laplace operator of multiplicity L + 1. Of course, it may happen that only one of X 1 , X 2 is an eigenvalue of the Laplace operator (we see that at least one is an eigenvalue in order not to have ϕ ≡ ψ ≡ 0, which is in contradiction to the fact that (ϕ ψ)T is an eigenvector). In that case, all the discussion below can be easily revised, and one would arrive at similar conclusions. However, we will consider the “worst-case scenario,” namely that X 1 , X 2 are both eigenvalues of the Lapalcian. One can easily check that each vector from the systems ⎧ T ⎫ ⎪ ⎪ ⎪ ⎪ λ + X γ X 1 1 ⎨  eq  eq ,⎬ 2 2 2 2 , X1 := (γ X 1 ) + (λ + X 1 ) (γ X 1 ) + (λ + X 1 ) ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ q = k, k + 1, . . . , k + M T ⎫ ⎪ ⎪  eq  eq ,⎬ 2 2 2 2 X2 := , (γ X 2 ) + (λ + X 2 ) (γ X 2 ) + (λ + X 2 ) ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ q = s, s + 1, . . . , s + L ⎧ ⎪ ⎪ ⎨

λ + X2

γ X2

has unit norm and satisfies equation (5.25), i.e., it is an eigenvector for A corresponding to the eigenvalue λ. Notice that X1 ∪ X2 contains M + L + 2 orthogonal unit vectors, which in particular are linearly independent.

5.1 Presentation of the Problem

101

˜ T satisfy (5.25). Then necessarily Furthermore, arguing as above, let (ϕ˜ ψ) ϕ˜ = ϕ˜ k ek + ϕ˜ k+1 ek+1 + · · · + ϕ˜ k+M ek+M + ϕ˜ s es + ϕ˜ s+1 es+1 + · · · + ϕ˜ s+L es+L , ψ˜ = ψ˜ k ek + ψ˜ k+1 ek+1 + · · · + ψ˜ k+M ek+M + ψ˜ s es + ψ˜ s+1 es+1 + · · · + ψ˜ s+L es+L ,

that is, ϕ˜ and ψ˜ are linear combinations of the eigenfunctions {ek , ek+1 , . . . , ek+M , es , es+1 , . . . , es+L } . Taking into account that the system {ek , ek+1 , . . . , ek+M , es , es+1 , . . . , es+L } is linearly independent, plugging the above ϕ˜ and ψ˜ into relations (5.25), and recalling that μk = · · · = μk+M = X 1 , μs = μs+1 = · · · = μs+L = X 2 , we deduce that λ + X1 q ψ˜ , q = k, k + 1, . . . , k + M, γ X1 λ + X2 q ϕ˜ q = ψ˜ , q = s, s + 1, . . . , s + L . γ X2

ϕ˜ q =

Hence ˜ T = ψ˜ k (ϕ˜ ψ)



+ψ˜ s

λ+X 1 e γ X1 k



λ+X 2 e γ X2 s

T ek

T

es

+ · · · + ψ˜ k+M + · · · + ψ˜ s+L

 

λ+X 1 e γ X 1 k+M λ+X 2 e γ X 2 s+L

T ek+M T es+L .

Or equivalently,  T √ 2 2 γ X1 ˜ T = (λ+X 1 ) +(γ X 1 ) ψ˜ k √ λ+X 1 √ (ϕ˜ ψ) e e + ··· k k γ X1 (λ+X 1 )2 +(γ X 1 )2 (λ+X 1 )2 +(γ X 1 )2  T √ (λ+X 1 )2 +(γ X 1 )2 k+M γ X1 λ+X 1 ˜ √ √ ψ + ek+M ek+M γ X1 2 2 2 2 (λ+X 1 ) +(γ X 1 )

√

(λ+X 2 )2 +(γ X 2 )2 s ψ˜ γ X2



(λ+X 1 ) +(γ X 1 )

γ X2 √ λ+X2 2 es √ es (λ+X 2 ) +(γ X 2 )2 (λ+X 2 )2 +(γ X 2 )2  √ (λ+X 2 )2 +(γ X 2 )2 s+L γ X2 √ λ+X2 2 ψ˜ + es+L √ γX 2 2

+

2

(λ+X 2 ) +(γ X 2 )

T + ··· T es+L . 2

(λ+X 2 ) +(γ X 2 )

˜ T may be written as a linear combination of the Thus we obtain that the above (ϕ˜ ψ) vectors from X1 ∪ X2 . In other words, X1 ∪ X2 forms a system of generators for the subspace of the eigenvectors of the operator A corresponding to the eigenvalue λ. Recalling that X1 ∪ X2 is linearly independent, we conclude that in fact, X1 ∪ X2 represents an orthonormal basis of this subspace. Consequently, we may choose the

102

5 Stabilization of the Cahn–Hilliard System

eigenvector (ϕ ψ)T to be one of the elements from X1 ∪ X2 , in particular such that ψ is of the form (5.23). Let us consider now the identity (5.26) as one of unknown λ. So λ must be a root of the second-degree polynomial X 2 + [(Fl + 1)μk − νμ2k ]X − νμ3k + (Fl − γ 2 )μ2k . We observe that the above polynomial has a negative root, namely λ, and a nonnegative one. Indeed, assume for the sake of a contradiction that both roots are negative. Then by Viète’s relations, we deduce that −νμ3k + (Fl − γ 2 )μ2k > 0 and (Fl + 1)μk − νμ2k > 0. Adding the second relation multiplied by −μk ≥ 0 to the first we deduce −μ2k (1 + γ 2 ) > 0, which is absurd. Therefore, again by Viète’s relations, we get that necessarily −νμ3k + (Fl − γ 2 )μ2k ≤ 0, and therefore, μk must satisfy μk ≥

Fl −γ 2 . ν



We continue following the algorithm in Chap. 2. We transform the boundary control problem into an internal-type one by lifting the boundary conditions into equations. In order to do this, let us define the so-called Neumann operator as follows: given a ∈ L 2 (Γ1 ) and η > 0, we set Dη a := (y z)T , the solution to the system (recall (2.51) and the fact that λ N −1 = λ N = 0) ⎧ N  ⎪ ⎪ 2 ⎪ νΔ y − F Δy + γ Δz − 2 λ j (y, z), (ϕ j , ψ j ) ϕ j ⎪ l ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ − δ(y, z), (ϕ N , ψ N ) ϕ N + ηy = 0 in O, ⎪ ⎪ ⎪ ⎪ ⎪ N ⎪  ⎪ ⎪ ⎪ − Δz + γ Δy − 2 λ j (y, z), (ϕ j , ψ j ) ψ j ⎪ ⎪ ⎨ j=1 ⎪ − δ(y, z), (ϕ N , ψ N ) ϕ N + ηz = 0 in O, ⎪ ⎪ ⎪ ⎪ ⎪ ∂ y ⎪ ⎪ = 0 in (0, ∞) × Γ, ⎪ ⎪ ∂n ⎪ ⎪ ⎪ ⎪ γ0 ∂z ∂Δy ⎪ ⎪ = − a, = α0 a on (0, ∞) × Γ1 , ⎪ ⎪ ⎪ ∂n ν ∂n ⎪ ⎪ ⎪ ⎪ ⎩ ∂Δy = ∂z = 0 on (0, ∞) × Γ2 , ∂n ∂n

(5.27)

5.1 Presentation of the Problem

103

where δ > 0 is such that λ1 , . . . , λ N −1 , λ N + δ are distinct and η is sufficiently large to ensure the existence of a unique solution to (5.27), defining thereby the map Dη ∈ L(L 2 (Γ1 ), H 1 (O) × H 1/2 (O)). Easy computations lead to α0 a, ψ j 0 , j = 1, 2, . . . , N − 1, η − λj α0 Dη a, (ϕ N ψ N )T = a, ψ N 0 , η − λN − δ

Dη a, (ϕ j ψ j )T =

(5.28)

where  · , · 0 stands for the classical scalar product in L 2 (Γ1 ). Let 0 < η1 < η2 < · · · < η N be N constants sufficiently large such that (5.27) is well posed for each of them. For future use, we set Dηi , i = 1, 2, . . . , N , the corresponding solutions of (5.27).

(5.29)

Further, set 

1 1 1 1 , ,..., , ηk := diag ηk − λ1 ηk − λ2 ηk − λ N −1 ηk − λ N − δ k = 1, 2, . . . , N , and  S :=

N 

 ,

(5.30)

ηk .

k=1

Moreover, define Bk := ηk Bηk , k = 1, 2, . . . , N ,

(5.31)

N

where B is the Gram matrix of the system ψ j |Γ1 j=1 , in L 2 (Γ1 ), i.e., ⎞ ψ1 , ψ1 0 ψ1 , ψ2 0 . . . ψ1 , ψ N 0 ⎜ ψ2 , ψ1 0 ψ2 , ψ2 0 . . . ψ2 , ψ N 0 ⎟ ⎟ B := ⎜ ⎝ ................................................... ⎠ . ψ N , ψ1 0 ψ N , ψ2 0 . . . ψ N , ψ N 0

(5.32)

(B1 + B2 + · · · + B N )−1 =: A.

(5.33)

⎛

Set

Then A is well defined. Indeed, this can be shown by arguing similarly as in Proposition 2.1 and making use of Lemma 5.1, from which we know that for all j = 1, 2, . . . , N , the traces of ψ j are not identically zero on Γ1 .

104

5 Stabilization of the Cahn–Hilliard System

Now, plugging the feedback # u(t) =

⎞ ⎛ ⎞ ψ1 (x) $ (y(t) z(t))T , (ϕ1 ψ1 )T ⎜ (y(t) z(t))T , (ϕ2 ψ2 )T ⎟ ⎜ ψ2 (x) ⎟ ⎟ ⎜ ⎟ S A ⎜ ⎝ .................................... ⎠ , ⎝ ......... ⎠ , (y(t) z(t))T , (ϕ N ψ N )T ψ N (x) N ⎛

(5.34)

into equations (5.16), one may show, similarly as in Theorems 2.6, 2.7, that it achieves its exponential stability. More exactly, we have the following result, which is commented on in the forthcoming Remark 5.1. Its proof is omitted, since it is similar to the proof of Theorem 2.6, and it has been repeated several times in previous chapters. Proposition 5.2 The solution (y, z) to the system ⎧ yt + νΔ2 y − Fl Δy + γ Δz = 0, in (0, ∞) × O, ⎪ ⎪ ⎪ ⎪ ⎪ z t − Δz + γ Δy = 0, in (0, ∞) × O, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ y = 0, on (0, ∞) × Γ, ⎪ ⎪ ⎪ ∂n ⎪ ⎞ ⎞ ⎛ ⎛ ⎪ ⎪ ⎪ ψ1 $ (y z)T , (ϕ1 ψ1 )T # ⎪ ⎪ ⎪ ⎜ (y z)T , (ϕ2 ψ2 )T ⎟ ⎜ ψ2 ⎟ ⎪ γ0 ∂Δy ⎪ ⎟ ⎟ ⎜ ⎜ ⎪ = − A  S ⎪ ⎝ ....................... ⎠ , ⎝ .... ⎠ , ⎪ ∂n ν ⎪ ⎪ ⎪ ⎪ ψN (y z)T , (ϕ N ψ N )T ⎪ N ⎪ ⎪ ⎨ on (0, ∞) × Γ1 ⎪ ⎞ ⎞ ⎛ ⎛ ⎪ T T ⎪ ⎪ , (ϕ ψ ) ψ1 $ (y z) # 1 1 ⎪ ⎪ ⎪ ⎟ ⎜ ψ2 ⎟ ⎜ ∂z ⎪ (y z)T , (ϕ2 ψ2 )T ⎪ ⎟ ⎟,⎜ ⎪ = α0  S A ⎜ ⎪ ⎠ ⎝ ..... ⎠ , ⎝ ⎪ ...................................... ∂n ⎪ ⎪ ⎪ ⎪ ψN (y z)T , (ϕ N ψ N )T ⎪ N ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ on (0, ∞) × Γ1 , ⎪ ⎪ ⎪ ⎪ ⎪ ∂Δy ∂z ⎪ ⎪ = = 0, on (0, ∞) × Γ2 , ⎪ ⎪ ∂n ∂n ⎪ ⎪ ⎩ y(0) = yo , z(0) = z o ,

(5.35)

satisfies the exponential decay (y(t) z(t))T 2 ≤ C1 e−C2 t (yo z o )T 2 , t ≥ 0,

(5.36)

for some constants C1 , C2 > 0. Remark 5.1 From the practical point of view, it is important to describe how one can compute the first N eigenvectors of the operator A, since the boundary feedback law

5.1 Presentation of the Problem

105

is expressed only in terms of those eigenvectors. In order to do this, we will mainly rely on the results from the proof of Lemma 5.1. First of all, one should compute the first, let us say K , eigenvalues and eigenfunctions of the Neumann Laplace operator, i.e., Δe j = μ j e j , in O; for which μj ≥

∂e j = 0, on Γ ; j = 1, 2, . . . , K , ∂n

Fl − γ 2 , j = 1, 2, . . . , K . ν

We have that μ1 = 0 and μi = 0 for i = 2, 3, . . . , K . Then for each j = 1, 2, . . . , K one should check whether the polynomial X 2 + [(Fl + 1)μ j − νμ2j ]X − νμ3j + (Fl − γ 2 )μ2j has a nonpositive root. If it does, we denote it by λ, and this is in fact a nonpositive eigenvalue of the operator A. Let us assume that we have found N such nonpositive roots, and denote them by λi , i = 1, 2, . . . , N . Hence for each i = 1, 2, . . . , N , either λi = 0, in which case one can take  (ϕi ψi )T =

 1 2m O

1 2m O

T

 



1 or (ϕi ψi )T = − 2m O

1 2m O

T ,

or there exists some j ∈ {2, . . . , K } such that the eigenvalue λi can be computed as a root of the following second-degree polynomial: X 2 + [(Fl + 1)μ j − νμ2j ]X − νμ3j + (Fl − γ 2 )μ2j . Then the corresponding eigenvector is given by  (ϕi ψi ) = T



λi + μ j (γ μ j )2 + (λi + μ j )2

ej 

γ μj (γ μ j )2 + (λi + μ j )2

T ej

.

In conclusion, the problem reduces to finding the first K eigenvalues and eigenfunctions of the Neumann Laplace operator and computing the roots of some third-degree polynomials. Recalling the notation (5.11), The following result concerning the linearized system of (5.10) follows immediately by Proposition 5.2. Theorem 5.1 The unique solution to the linear system

106

5 Stabilization of the Cahn–Hilliard System

⎧ (θ + l0 ϕ)t − Δθ = 0, in (0, ∞) × O, ⎪ ⎪ ⎪ ⎪ ⎪ ϕt − Δμ = 0, in (0, ∞) × O, ⎪ ⎪ ⎪ ⎪ ⎪ μ = −νΔϕ − ϕ − γ0 θ, in (0, ∞) × O, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂ϕ = ∂μ = 0, on (0, ∞) × Γ, ∂n ∂n ⎪ ∂θ ⎪ ⎪ ⎪ = u =  S AO, J N , on (0, ∞) × Γ1 , ⎪ ⎪ ∂n ⎪ ⎪ ⎪ ⎪ ∂θ ⎪ ⎪ = 0, on (0, ∞) × Γ2 , ⎪ ⎪ ∂n ⎪ ⎪ ⎩ θ (0) = θo , ϕ(0) = ϕo , in O,

(5.37)

satisfies the exponential decay ˆ T 2 , ∀t ≥ 0, (ϕ(t) θ (t))T − (ϕˆ θˆ )T 2 ≤ C5 e−C6 t (ϕo θo )T − (ϕˆ θ) for some positive constants C5 , C6 . Here O = O(θ, ϕ) ⎛ ⎞ ϕ, ϕ1 + α0 (θ + l0 ϕ), ψ1 − ϕ∞ , ϕ1 − α0 (θ∞ − l0 ϕ∞ ), ψ1 ⎜ ϕ, ϕ2 + α0 (θ + l0 ϕ), ψ2 − ϕ∞ , ϕ2 − α0 (θ∞ − l0 ϕ∞ ), ψ2 ⎟ ⎟ := ⎜ ⎝ .......................................................................................... ⎠ ϕ, ϕ N + α0 (θ + l0 ϕ), ψ N − ϕ∞ , ϕ N − α0 (θ∞ − l0 ϕ∞ ), ψ N (5.38) and ⎞ ⎛ ψ1 ⎜ ψ2 ⎟ ⎟ J := ⎜ ⎝ ... ⎠ . ψN Via a fixed-point argument similar to what we discussed in the previous chapters, one can deduce from Theorem 5.1 the local boundary stabilization for the nonlinear system as well.

5.2 Comments Instead of the Cahn-Hilliard system, it is usually studied its simpler form, the socalled phase-field system, which reads as %

θt − kΔθ + laΔϕ + lb(ϕ − ϕ 3 ) − ldθ = 0, ϕt − aΔϕ − b(ϕ − ϕ 3 ) + dθ = 0, in R+ × O.

(5.39)

5.2 Comments

107

The problem of stabilization of the phase field system has been intensively studied in the literature using various methods. The Riccati-based approach is used in Barbu [8], where a stabilizing finite-dimensional feedback controller with compact support acting only on one component of the system is constructed. The boundary stabilization problem was studied, for example, in Chen [45] using the time optimal control technique, while in Munteanu [99], a proportional boundary feedback is designed under the constraint that the eigenfunctions are linearly independent. Concerning the problem of stabilization of systems of Cahn-Hilliard type (5.1)– (5.3), we mention the work Barbu et al. [17], which constructs an internal stabilizing feedback, while concerning the boundary stabilization case, the result of this chapter represents the first result in this direction, and it is based on the ideas in [17]. Other results related to the control problem associated with the Cahn-Hilliard system are the sliding mode controls in Barbu et al. [16] and Colli et al. [49], while in Marinoschi [93], the singular potential case is investigated.

Chapter 6

Stabilization of Equations with Delays

In this chapter, we consider equations with delays. Namely, the derivative of an unknown function at a certain time depends on the values of the function at previous times. More exactly, we consider in the model aftereffect phenomena by adding a memory term. Engineers conclude that actuators, sensors that are involved in feedback control, introduce, in addition, delays into the system. That is why from the control engineering point of view it is of great interest to consider control problems associated with equations with delays. Furthermore, special kinds of substances, such as viscoelastic fluids, may also impose such delays. We will prove here that the proportional feedback, designed in Chap. 2, still ensures stability for this kind of system.

6.1 Presentation of the Problem The subject of this chapter is the Dirichlet boundary control problem of the following evolution integro–partial differential equation:  t  t ⎧ ⎪ ⎪ ∂ y(t, x) = Δy(t, x) + k(t − s)Δy(s, x) + μ k(t − s)y(s, x)ds t ⎪ ⎪ ⎪ −∞ −∞ ⎪ ⎨ + f (y(t, x)), (t, x) ∈ Q := (0, ∞) × O, ⎪ y(t, x) = u(t, x) on Σ 1 := (0, ∞) × Γ1 , ⎪ ⎪ ∂ ⎪ ⎪ y = 0 on Σ := (0, ∞) × Γ2 , 2 ⎪ ⎩ ∂n y(t, x) = yo (t, x), (t, x) ∈ (−∞, 0] × O. (6.1) The effects of the memory are expressed in the linear time convolution of the functions Δy(·, ·), respectively y(·, ·), and the memory kernel k(·). The Dirichlet controller u is applied on Γ1 , while Γ2 is insulated. © Springer Nature Switzerland AG 2019 I. Munteanu, Boundary Stabilization of Parabolic Equations, Progress in Nonlinear Differential Equations and Their Applications 93, https://doi.org/10.1007/978-3-030-11099-4_6

109

110

6 Stabilization of Equations with Delays

As can be seen, in the initial condition, the fourth equation in (6.1), it is assumed that the function y(t, x) is known for all t ≤ 0. However, y(t, x) does not necessarily satisfy the equation for negative t. We shall assume for the nonlinear function f that f (0) = 0 and f  (0) > 0, and choose from the following two hypotheses, which we have met before: (i) f ∈ C 1 (R); (ii) f ∈ C 2 (R), and there exist C1 > 0, q ∈ N, αi > 0, i = 1, . . . , q, when d = 1, 2, and 0 < αi ≤ 1, i = 1, . . . , q, when d = 3, such that | f  (y)| ≤ C1

 q 

|y|αi + 1 , ∀y ∈ R.

i=1 d (Here f  stands for the derivative dy f.) Finally, μ is a nonnegative constant. The instability of the null solution in (6.1) may occur because of the presence of the nonlinear term f . But since the kernel k will be assumed to be positive, the instability may also be caused by the presence of the memory term containing the positive constant μ. Therefore, it makes sense to deal with the stabilization of the null solution in (6.1). We set A : D(A) ⊂ L 2 (O) → L 2 (O), defined by

Ay := −Δy, ∀y ∈ D(A), 

∂ 2 y = 0 on Γ2 . D(A) = y ∈ H (O) : y = 0 on Γ1 and ∂n ∞ ∞ Let {ϕi }i=1 be an orthonormal basis of eigenfunctions of A. By {λi }i=1 we denote the corresponding eigenvalues, repeated according to their multiplicity. It is easy to see that we can rearrange the set of eigenvalues as

0 < λ1 ≤ λ2 ≤ · · · ≤ λi ≤ · · · ., with λi → ∞ when i → ∞. For the rest of this chapter, we let · stand for the norm in L 2 (O). In addition to the above context, we assume as well that (k) there exists some δ > 0 such that the nonnegative memory kernel k ∈ C([0, ∞), R+ ) ∩ C 2 ((0, ∞), R+ ) satisfies

k  (t) + 2δk(t) ≤ 0 and k  (t) + 2δk  (t) ≥ 0, ∀t ≥ 0,

and eρδ· k· = const., for all 0 ≤ ρ ≤ 1. Straightforward computations show that this implies that

(6.2)

6.1 Presentation of the Problem

111

k(t) ≤ k(0)e−2ρδt , 0 ≤ ρ ≤ 1, t ≥ 0, and moreover, (−1)m

(6.3)

d m ρδt e k(t) ≥ 0, m = 0, 1, 2, m dt

for all t > 0 and 0 ≤ ρ ≤ 1. Hence by [7, Proposition 4.1], we have that eρδ· k(·) is a positive kernel, i.e., 

t



τ

w(τ )

e

0

ρδ(τ −s)

 k(τ − s)w(s)ds dτ ≥ 0, ∀w ∈ L 2 (0, t; R), t ≥ 0,

0

(6.4) for all 0 ≤ ρ ≤ 1. It should be noted that such kernels k satisfying (6.2) are often considered in the literature in studying the stability of heat equations with memory (see, for example, [46, 88]). In fact, the exponential decay of k reflects the fading of the far history in the model. Besides this, a simple example of a kernel that obeys M  bi e−ai t , t ≥ 0, for some ai , bi > 0, i = 1, . . . , M. (6.2) is k(t) = i=1

(o) The initial data yo belongs to the space L 2 (−∞, 0; H 2 (O)). On setting  η(t, x) :=



0

−∞

k(t − s)Δyo (s, x)ds + μ

0 −∞

k(t − s)yo (s, x)ds, (t, x) ∈ Q, (6.5)

assumption (o) leads to the following estimates:  η(t) 2 ≤ 2

0

−∞

2 k(t − s) Δyo (s) ds

  +2 μ

0 −∞

2 k(t − s) yo (s) ds

(using Schwarz’s inequality and relation (6.3), with ρ = 1)  0   0  0 e−4δ(t−s) ds Δyo (s) 2 ds + μ2 yo (s) 2 ds ≤2 −∞ −∞ −∞   2 max 1, μ −4δt e yo L 2 (−∞,0;H 2 (O )) , t ≥ 0. ≤ 2δ (6.6) Therefore, there exists a constant C > 0 such that η(t) 2 ≤ Ce−δt yo 2L 2 (−∞,0;H 2 (O )) , t ≥ 0, and  ∞ e2δt η(t) 2 dt ≤ C yo 2L 2 (−∞,0;H 2 (O )) . 0

(N) N ∈ N is a constant sufficiently large that

(6.7)

112

6 Stabilization of Equations with Delays

− λi + f  (0) +

δ < 0 and − λi + μ < 0 for i = N + 1, N + 2, . . . . (6.8) 4

As we have done already in this book, in order to simplify the presentation, we assume that the first N eigenvalues are simple. Of course, one may argue as in the second part of Chap. 2 to deal with the general semisimple case. Accordingly, changing the proportional feedback law provided in Chap. 2 to suit  ∂ N ϕi i=1 in the Hilbert our case, we have that B is the Gram matrix of the system ∂n space L 2 (Γ1 ), with the standard scalar product 

g, h0 :=

Γ1

f (x)g(x)dσ

(σ being the corresponding Lebesgue measure on the boundary Γ1 ). That is, ⎛

⎞ ∂ ∂ ∂ ∂ ∂ ∂

∂n φ1 , ∂n φ1 0 ∂n φ1 , ∂n φ2 0 . . . ∂n φ1 , ∂n φ N 0 ⎜ ∂ φ , ∂ φ  ∂ φ , ∂ φ  ... ∂ φ , ∂ φ  ⎟ ∂n 2 ∂n 1 0 ∂n 2 ∂n 2 0 ∂n 2 ∂n N 0 ⎟ B := ⎜ ⎝ ...................................................................... ⎠ . ∂ ∂ ∂ ∂ ∂ ∂

∂n φ N , ∂n φ1 0 ∂n φ N , ∂n φ2 0 . . . ∂n φ N , ∂n φ N 0

(6.9)

Then we define the matrices ⎞ ... 0 1 ⎜ 0 ... 0 ⎟ γk −λ2 ⎟ Λγk := ⎜ ⎝ ............................ ⎠ , k = 1, . . . , N , 1 0 0 . . . γk −λ N ⎛

1 γk −λ1

0

Λ S :=

N 

Λγk

(6.10)

(6.11)

k=1

and

A = (B1 + B2 + · · · + B N )−1 ,

(6.12)

Bk := Λγk BΛγk , k = 1, . . . , N .

(6.13)

where (Recall that by virtue of Example 2.4 and Proposition 2.1, the sum B1 + · · · + B N is invertible.) Finally, the feedback laws are ⎞ ⎞ ⎛ 1 ∂ ϕ (x)

y(t), ϕ1  γk −λ1 ∂n 1 ⎟ 1 ∂ ⎜ y(t), ϕ2  ⎟ ⎜ ⎜ γk −λ2 ∂n ϕ2 (x) ⎟ ⎟ ⎜ ,⎜ u k (t, x) = A ⎝ ⎟ , t ≥ 0, x ∈ Γ1 , .............. ⎠ ⎝ ...................... ⎠ N 1 ∂

y(t), ϕ N  ϕ (x) γ −λ ∂n N 

⎛

k

N

(6.14)

6.1 Presentation of the Problem

113

for k = 1, 2, . . . , N . Here ·, · denotes the standard scalar product in L 2 (O), while

·, · N denotes the standard scalar product in R N . Then we take u to be the sum u = u1 + u2 + · · · + u N , which in a condensed form, can be rewritten as ⎞ ⎞ ⎛ ∂ ϕ

y(t), ϕ1  ∂n 1  ⎜ y(t), ϕ2  ⎟ ⎜ ∂ ϕ ⎟ 2 ⎜ ⎟ ⎟ ⎜ ∂n , , t ≥ 0. u = ΛS A ⎝ .............. ⎠ ⎝ ....... ⎠ N ∂

y(t), ϕ N  ϕN ⎛



(6.15)

∂n

Remark 6.1 For negative time t, the controller u is defined as ⎞ ⎞ ⎛ ∂ ϕ

yo (t), ϕ1  ∂n 1  ⎜ yo (t), ϕ2  ⎟ ⎜ ∂ ϕ ⎟ ⎟ , ⎜ ∂n 2 ⎟ . u(t, x) := Λ S A ⎜ ⎝ ................ ⎠ ⎝ ........ ⎠ N ∂

yo (t), ϕ N  ϕ ∂n N 

⎛

(6.16)

Since yo is known, we deduce that for negative time, u is in fact a known function.

6.2 Stability of the Linearized System The following result amounts to saying that the feedback u given by (6.15) globally exponentially stabilizes the first-order approximation of (6.1). Theorem 6.1 Let N ∈ N as in (6.8). Under hypothesis (k), (o), (i), for each yo ∈ C([0, ∞); L 2 (O)) ∩ L 2 (−∞, 0; H 2 (O)), there exists a unique solution

y ∈ C [0, ∞); L 2 (O) ∩ L 2 (0, ∞; H 1 (O)) to the closed-loop system  t  t ⎧ ⎪ ⎪ ∂ y(t, x) = Δy(t, x) + k(t − s)Δy(s, x)ds + μ k(t − s)y(s, x)ds t ⎪ ⎪ ⎪ 0 0 ⎪ ⎨ + f  (0)y(t, x) + η(t, x), (t, x) ∈ Q, ⎪ y(t, x) = u(y(t)), on Σ1 ⎪ ⎪ ∂ ⎪ ⎪ y = 0 on Σ2 , ⎪ ⎩ ∂n y(0, x) = yo (0, x), x ∈ O, (6.17) which satisfies the exponential decay δ

y(t) 2 ≤ Ce− 2 t ( yo (0) 2 + yo 2L 2 (−∞,0;H 2 (O )) ), ∀t ≥ 0,

(6.18)

114

6 Stabilization of Equations with Delays

for some constant C > 0. Here η is as introduced in (6.5), δ > 0 is as introduced in (6.2), and the feedback u is given in (6.15). Besides this, there exists c > 0 such that 

∞ 0

  y(t) 2H 1 (O ) ≤ c yo (0) 2 + yo 2L 2 (−∞,0;H 2 (O )) .

(6.19)

Proof First, we equivalently rewrite Eq. (6.17) as one with null boundary conditions. To this end, we introduce, similarly as in (2.16) and (2.18), the map D, as follows: given β ∈ L 1 (Γ1 ), we denote by Dγ β := y the solution to the equation ⎧ ⎨ ⎩

−Δy − 2

N  k=1

y = β on Γ1 ,

λk y, ϕk ϕk + γ y = 0 in O, ∂ y ∂n

(6.20)

= 0 on Γ2 .

Doing similar computations as in (2.19), we get 

Dγ β, ϕ j  =

1 − γ −λ

β, j 1 − γ +λ j

β,

∂ϕ j  , ∂n 0 ∂ϕ j  , ∂n 0

j = 1, 2, . . . , N , j = N + 1, N + 2, . . . .

(6.21)

We let γ1 , . . . , γ N be N positive sufficiently large constants, and set Dγ1 , . . . , Dγ N for the corresponding solutions to (6.20). Similarly as in (2.35), we have that ⎛

⎞ ⎞ ⎛

Dγk u k , ϕ1 

y(t), ϕ1  ⎜ Dγk u k , ϕ2  ⎟ ⎜ y(t), ϕ2  ⎟ ⎜ ⎟ ⎟ ⎜ ⎝ ................. ⎠ = −Bk A ⎝ .............. ⎠ ,

Dγk u k , ϕ N 

y(t), ϕ N 

(6.22)

where the Bk were introduced in (6.13) above, for k = 1, . . . , N . Let us define z(t, x) := y(t, x) −

N 

Dγk u k (t, x), (t, x) ∈ Q,

k=1

and z o (x) := yo (0, x) −

N 

Dγk u k (0, x), x ∈ O,

k=1

(u k is given by (6.14)). Then with similar arguments as in (2.36)–(2.37), we have that the feedback u may be expressed in terms of z only as

6.2 Stability of the Linearized System

115

⎞ ⎞ ⎛ 1 ∂ ϕ1

z(t), ϕ  γ −λ ∂n 1 k 1  ⎜ 1 ∂ ϕ ⎟ 1 ⎜ z(t), ϕ2  ⎟ 2 ⎟ ⎟,⎜ u k (t, x) = A ⎜ ⎜ γk −λ2 ∂n ⎟ . 2 ⎝ ................. ⎠ ⎝ ................. ⎠ N 1 ∂

z(t), ϕ N  ϕ γk −λ N ∂n N ⎛

(6.23)

Finally, as in (6.22), we have now ⎛

⎞ ⎞ ⎛

Dγk u k , ϕ1 

z(t), ϕ1  ⎜ Dγk u k , ϕ2  ⎟ ⎜ z(t), ϕ2  ⎟ 1 ⎜ ⎟ ⎟ ⎜ ⎝ ................. ⎠ = − 2 Bk A ⎝ ................ ⎠ , k = 1, . . . , N ,

Dγk u k , ϕ N 

z(t), ϕ N 

(6.24)

and Eq. (6.17) may be rewritten in terms of z as follows: ⎧  t  t ⎪ ⎪ z (t, x) = −Az(t, x) − k(t − s)Az(s)ds + μ k(t − s)z(s)ds ⎪ t ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ N  t ⎪  ⎪ ⎪ ⎪ ⎪ +μ k(t − s)Dγk u k (s)ds ⎪ ⎪ ⎪ ⎨ k=1 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

+ (R1 + R2 )( z(t), ϕ1 , . . . , z(t), ϕ N ) + f  (0)z + f  (0)  + z(0, x) =

N 

Dγk u k

k=1 t

k(t − s)R2 ( z(s), ϕ1 , . . . , z(s), ϕ N ) + η(t), t > 0, x ∈ O,

0 z o (x),

x ∈ O, (6.25)

where  R1 ( z, ϕ1 , . . . , z, ϕ N ) := −

N  i=1

R2 ( z, ϕ1 , . . . , z, ϕ N ) := −2

N  i, j=1

Dγi u i t

λ j Dγi u i , ϕ j ϕ j +

N 

(6.26) γi Dγi u i .

i=1

(One may show that given an initial datum yo ∈ C((−∞, 0]; L 2 (O)) ∩ L 2 (0, ∞; H 1 (O)), there exists a unique solution z ∈ C([0, ∞); L 2 (O)) ∩ L 2 (0, ∞; H 1 (O)) to the system (6.25), proving thereby the well-posedness of (6.25) and consequently that of (6.17). See, for instance, [46, Theorem 2.1].) By virtue of (6.24), using the fact that A is the inverse of the sum of Bk ’s, k = 1, . . . , N , we immediately see that

116

6 Stabilization of Equations with Delays

⎛

⎞

R1 , ϕ1  N ⎜ R1 , ϕ2  ⎟ 1  1 ⎜ ⎟= Bk AZt = Zt ⎝ .......... ⎠ 2 2 k=1

R1 , ϕ N  ⎛ ⎞

R2 , ϕ1  N N  ⎜ R2 , ϕ2  ⎟ 1 ⎜ ⎟=Λ B AZ − γk Bk AZ k ⎝ ........... ⎠ 2 k=1 k=1

R2 , ϕ N 

(6.27)

1 1 = ΛZ − γ1 Z + (γ1 − γk )Bk AZ , 2 2 k=2 N

⎞ ⎞ ⎛ λ1 0 . . . 0

z(t), ϕ1  ⎜ 0 λ2 . . . 0 ⎟ ⎜ z(t), ϕ2  ⎟ ⎟ ⎟ ⎜ where we have set Z (t) := ⎜ ⎝ ........... ⎠, t ≥ 0, and Λ := ⎝ ............... ⎠ .

z(t), ϕ N  0 0 . . . λN Taking into account the above relations and projecting the Eq. (6.25) into the space  N Xu := lin span ϕ j j=1 , it follows that ⎛

 t  t d Z (t) = −ΛZ (t) − k(t − s)ΛZ (s)ds + μ k(t − s)Z (s)ds dt 0 0  1 d μ t Z (t) + ΛZ (t) k(t − s)Z (s)ds + − 2 0 2 dt N 1 1 − γ1 Z (t) + (γ1 − γk )Bk AZ (t) 2 2 k=2 + f  (0)Z (t) −   t

+ 0

f  (0) Z (t) 2

 N 1 1 k(t − s) ΛZ (s) − γ1 Z (s) + (γ1 − γk )Bk AZ (s) ds + L(t), 2 2 k=2

⎞

η(t), ϕ1  ⎜ η(t), ϕ2  ⎟ ⎟ for t > 0, where L(t) := ⎜ ⎝ .............. ⎠ .

η(t), ϕ N  Equivalently, ⎛

! d Z (t) = −γ1 + f  (0) Z (t) + (μ − γ1 ) dt N  + (γ1 − γk )Bk AZ (t) k=2



t 0

k(t − s)Z (s)ds

6.2 Stability of the Linearized System

 +

t

117



 N  k(t − s) (γ1 − γk )Bk AZ (s) ds + 2L(t), t > 0. (6.28)

0

k=2

Now let us scalar multiply (in R N ) Eq. (6.28) by AZ (t), to arrive at (see (2.39)– (2.40)) "2 1 d " " 12 " "A Z (t)" N 2 dt

   t " 1 "2 1 1 " " ≤ [−γ1 + f  (0)] "A 2 Z (t)" + (μ − γ1 ) A 2 Z (t), k(t − s)A 2 Z (s)ds N 0 N ⎡ ⎤ ( #  t N  1 1 1 1 + A 2 Z (t), k(t − s) ⎣ (γ1 − γk )A 2 Bk A 2 A 2 Z (s)⎦ ds 0

k=2

N

+ 2 AZ (t), L(t) N , t > 0.

After multiplying the above equation by e2δt and changing t to τ , we get " 2 " 2  " 1 d  δτ " " " " 1 " ≤ 2[−γ1 + f  (0) + δ] eδτ "A 2 Z (τ )" e "A 2 Z (τ )" N N dτ  τ δτ 21 δ(τ −s) δs 21 + 2(μ − γ1 ) e A Z (τ ), e k(τ − s)e A Z (s)ds N +2

N ) 

0

(γ1 − γk )

k=2

   1 1 1 1 2 A 2 Z (τ ), × eδτ A 2 Bk A 2

τ

 *  1 1 1 1 2 eδ(τ −s) k(τ − s)eδs A 2 Bk A 2 A 2 Z (s)ds

0

N

  1 + 4e A N L(τ ) N eδτ A 2 Z (τ ) N , τ > 0. 1 2

δτ

1

Here we used in the last term the Cauchy–Schwarz inequality and set A 2 N for the 1 induced Euclidean norm of the matrix A 2 . Then integrating the above equation with respect to τ over (0, t), we deduce that  t " 2 " 1 1 1 " " eδτ "A 2 Z (τ )" e2δt A 2 Z (t) 2N ≤ A 2 Z (0) 2N + 2[−γ1 + f  (0) + δ] dτ N 0  t  τ 1 1

eδτ A 2 Z (τ ), eδ(τ −s) k(τ − s)eδs A 2 Z (s)ds N dτ + 2(μ − γ1 ) +2

0

N ) 

0

(γ1 − γk )

k=2

 t# 0

+

 τ  1 1 1  1 1 1 1 2 1 2 A 2 Z (τ ), eδ(τ −s) k(τ − s)eδs A 2 Bk A 2 A 2 Z (s)ds eδτ A 2 Bk A 2

1 4 A 2 N

 t 0

0

eδτ L(τ ) N



1 eδτ A 2 Z



(τ ) N dτ

(

 dτ N

118

6 Stabilization of Equations with Delays

(using in the third and the fourth term relation (6.4) with ρ = 1, and the fact that μ − γ1 < 0 and γ1 − γk < 0, k = 2, . . . , N )  t " 2 " 1 1 " " eδτ "A 2 Z (τ )" dτ ≤ A 2 Z (0) 2N + 2[−γ1 + f  (0) + δ] N 0  t   1 1 eδτ L(τ ) N eδτ A 2 Z (τ ) N dτ + 4 A 2 N 0

(using, in the last term, Young’s inequality and the fact that − γ1 + f  (0) + δ < 0) 1  t 8 A 2 2N 1 2 2 ≤ A Z (0) N + e2δτ L(τ ) 2N dτ, t > 0. −γ1 + f  (0) + δ 0

It follows that 1

A 2 Z (t) N  ≤e

−2δt

1

1 2

A Z

(0) 2N



8 A 2 2N + −γ1 + f  (0) + δ





t

e

2δτ

η(τ ) dτ 2

0

1 8 A 2 2N 2 C yo L 2 (−∞,0;H 2 (O )) , t ≥ 0, ≤e + A Z −γ1 + f  (0) + δ (6.29) 1 using (6.7). Hence recalling that A 2 is symmetric and positive definite, we see that (6.29) yields the existence of a constant C > 0 such that −2δt

1 2

(0) 2N

  Z (t) N ≤ Ce−2δt Z (0) 2N + yo 2L 2 (−∞,0;H 2 (O )) , t ≥ 0.

(6.30)

Now taking the norm in (6.28) and using (6.30), (6.3) with ρ = 21 and (6.7), we deduce that " " " "d " Z (t)" " " dt N    t  −2δt −δ(t−s) −2δs −δt Z (0) 2N + yo 2L 2 (−∞,0;H 2 (O )) ≤C e + e e ds + e 0   −δt Z (0) 2N + yo 2L 2 (−∞,0;H 2 (O )) , t ≥ 0, ≤ Ce (6.31) for some constant C > 0. Next, let us take care of the remaining modes of z, namely z j := z, ϕ j , j = N + 1, N + 2, . . . . To this end, scalar multiplying Eq. (6.25) by ϕ j , j = N + 1, N + 2, . . . , we get

6.2 Stability of the Linearized System

119

 t d z j (t) = −λ j z j (t) + (−λ j + μ + f  (0)) k(t − s)z j (s) dt 0 N  t  +μ k(t − s) Dγi u i (s), ϕ j ds 0

i=1

+ R1 (z 1 , . . . , z N ), ϕ j  + +

N  t  i=1

N 

(6.32) [γi + f  (0)] Dγi u i (t), ϕ j 

i=1

k(t − s)

0

N 

γi Dγi u i (s), ϕ j ds + η(t), ϕ j , t > 0.

k=1

By (6.23), we see that u i , i = 1, 2, . . . , N , depend only on the first N modes of z. Consequently, by (6.30) and (6.3) with ρ = 21 , we get that μ

N  

t

k(t − s) Dγi u i (s), ϕ j ds +

0

i=1

+

N  [γi + f  (0)] Dγi u i (t), ϕ j  i=1

N  

t

i=1

0

≤ Ce

−δt

k(t − s)



N 

γi Dγi u i (s), ϕ j ds

(6.33)

k=1

Z

(0) 2N

 + yo 2L 2 (−∞,0;H 2 (O )) , t ≥ 0.

On the other hand, R1 depends on the time derivatives of the modes z j , j = 1, 2, . . . , N . So by (6.31), we also have that  

R1 (z 1 , . . . , z N ), ϕ j  ≤ Ce−δt Z (0) 2N + yo 2L 2 (−∞,0;H 2 (O )) , t ≥ 0. (6.34) In conclusion, arguing similarly as above, i.e., multiplying (6.32) by e2δt z j (t), setting τ instead of t, then integrating the result with respect to τ over (0, t) and taking advantage of (6.4), (6.7), (6.33), and (6.34), we obtain that ∞  j=N +1

  |z j (t)|2 ≤ Ce−δt z(0) 2 + yo 2L 2 (−∞,0;H 2 (O )) , t ≥ 0.

This together with (6.30) yields that z(t) 2 =

∞ 

  |z j (t)|2 ≤ Ce−δt z(0) 2 + yo 2L 2 (−∞,0;H 2 (O )) , t ≥ 0. (6.35)

j=1

Recalling that y = z +

N i=1

Dγi u i , we immediately obtain (6.18), as desired.

120

6 Stabilization of Equations with Delays

To get the H 1 -norm estimate in (6.19), we scalar multiply Eq. (6.25) by z. After some straightforward computations, using relations (6.35), (6.33), and (6.7), we get that  t ∇z(τ ) 2 dτ z(t) 2 ≤ z o 2 − 2 0  τ  *  ) t ∇z(τ, x) k(τ − s)∇z(s, x)ds dτ d x −2 0 0 O   τ  t z(τ ) k(τ − s) z(s) ds dτ +μ 0 0  t   (6.36)  2 −δt 2 2 z(0) + yo L 2 (−∞,0;H 2 (O )) z(τ ) dτ + Ce + f (0) 0  t ≤ z o 2 − 2 ∇z(τ ) 2 dτ 0  t   z(τ ) 2 dτ + Ce−δt z(0) 2 + yo 2L 2 (−∞,0;H 2 (O )) , + f  (0) 0

t ≥ 0, using relation (6.4), with ρ = 0. Hence we deduce the existence of a constant c > 0 such that  ∞ z(t) 2H 1 (O ) dt ≤ c( z o 2 + yo 2L 2 (−∞,0;H 2 (O )) ). (6.37) 0

Recalling that y = z +

N i=1

Dγi u i , we get immediately that (6.19) holds, as claimed. 

6.3 Feedback Stabilization of the Nonlinear System (6.1) Here we plug the feedback u given by (6.15) into the nonlinear system (6.1) and show that it locally stabilizes it. More precisely, we have the following theorem. Theorem 6.2 Let N ∈ N as in (6.8). Under hypotheses (k), (o), (H), (ii), the feedback controller u given by (6.15) locally exponentially stabilizes the nonlinear system (6.1). More exactly, there exists ρ > 0 sufficiently small that for all yo ∈ L 2 (−∞, 0; H 2 (O)) with yo (0) 2 + yo 2L 2 (−∞,0;H 2 (O )) ≤ ρ, there exists a unique solution y ∈ C([0, ∞); L 2 (O)) ∩ L 2 (0, ∞; H 1 (O)) to the equation

6.3 Feedback Stabilization of the Nonlinear System (6.1)

121

 t  t ⎧ ⎪ ⎪ ∂ y(t, x) =Δy(t, x) + k(t − s)Δy(s, x)ds + μ k(t − s)y(s, x)ds t ⎪ ⎪ ⎨ −∞ −∞ + f (y(t, x)), (t, x) ∈ (0, ∞) × O, ⎪ ⎪ ∂ ⎪ y(t, x) = u(y(t)) on Γ1 and ∂n y = 0 on Γ2 , ⎪ ⎩ y(t, x) = yo (t, x), (t, x) ∈ (−∞, 0] × O, (6.38) which is L 2 -exponentially decaying. Proof As in the proof of Theorem 6.1 (see (6.25)), we rewrite (6.38) via the operators Dγk , k = 1, 2, . . . , N , as ⎧  t ⎪ ⎪ z (t, x) = −Az(t, x) − k(t − s)Az(s)ds ⎪ t ⎪ ⎪ 0 ⎪ ⎪  t  t ⎪ ⎪ ⎪ ⎪ ⎪ k(t − s)z(s)ds + μ k(t − s)Du(s)ds + μ ⎪ ⎪ ⎪ 0 0 ⎨ + (R1 + R2 )( z(t), ϕ1 , . . . , z(t), ϕ N ) + f  (0)z + f  (0)Du ⎪  t ⎪ ⎪ ⎪ ⎪ k(t − s)R2 ( z(s), ϕ1 , . . . , z(s), ϕ N ) + η(t) + ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ + f (z + Du(z)) − f  (0)(z + Du(z)), t > 0, x ∈ O, ⎪ ⎪ ⎪ ⎩ z(0, x) = z o (x), x ∈ O, where we have set Du =

N 

(6.39)

Dγi u i .

i=1

In the sequel, we will consider an approximation of (6.39), to which we will apply a fixed-point argument in order to show its well-posedness. To this end, consider, for each M > 0, the following truncation functions JM : H 1 (O) → L 2 (O), defined as  JM (ξ ) :=

 f

f (ξ + Du(ξ )) − f  (0)(ξ + Du(ξ )),

M (ξ ξ 1

+ Du(ξ )) − f



(0) ξM 1 (ξ

ξ 1 ≤ M,

+ Du(ξ )), ξ 1 > M.

We claim that JM (ξ ) ≤ C M ξ 2H 1 (O ) , ∀ξ ∈ H 1 (O),

(6.40)

for some C M > 0 depending only on M. Indeed, let ξ H 1 (O ) ≤ M. Then since f satisfies estimate (ii), we have that ⎞ ⎛ q  C 1 ⎝ | f (ξ + Du(ξ )) − f  (0)(ξ + Du(ξ ))| ≤ |ξ + Du(ξ )|αi + 1⎠ |ξ + Du(ξ )|2 . 2 i=1

It follows from the above relation, via Hölder’s inequality, that

122

6 Stabilization of Equations with Delays

"2 2 " q " " " " JM (ξ ) 2 ≤ |ξ + Du(ξ )|αi + 1" ξ + Du(ξ ) 4L 6 (O ) " " " 6 i=1 L (O ) 2  2  q C1 αi ≤ |ξ + Du(ξ )| L 6 (O ) + 1 L 6 (O ) ξ + Du(ξ ) 4L 6 (O ) 2 i=1 2  2  q C1 αi = ξ + Du(ξ ) L 6αi (O ) + σ (O) ξ + Du(ξ ) 4L 6 (O ) , 2 i=1 

C1 2

where using the Sobolev embedding theorem (i.e., L p (O) → H 1 (O), ∀0 < p ≤ 6), we obtain 

C1 JM (ξ ) ≤ 2 2

2 q

2 ξ +

Du(ξ ) αHi 1 (O )

+ σ (O) ξ + Du(ξ ) 4H 1 (O ) . (6.41)

i=1

Since ξ + Du(ξ ) H 1 (O ) ≤ (1 + C D ) ξ H 1 (O ) ,

(6.42)

it follows, by (6.41), (6.42), and the fact that ξ H 1 (O ) ≤ M, that  JM (ξ ) ≤ 2

C1 (1 + C D )2 2

Hence taking C M =

C1 (1+C D )2 2

2 2  q αi αi (1 + C D ) M + σ (O) ξ 4H 1 (O ) . i=1



 (1 + C D ) M + σ (O) , we get (6.40), as

q 

αi

αi

i=1

claimed. Likewise for ξ H 1 (O ) > M. We have, as before, that +  +  + + M M  +f (ξ + Du(ξ )) − f (0) (ξ + Du(ξ ))++ + ξ H 1 (O ) ξ H 1 (O ) +  q + +αi +2  + + + + M M C1 + + + ≤ (ξ + Du(ξ ))+ + 1 + (ξ + Du(ξ ))++ . + 1 1 2 ξ ξ i=1

H (O )

H (O )

Then as before, applying Hölder’s inequality and Sobolev embeddings, we obtain that JM (ξ ) 2 q " 2  " M ≤ C21 " ξ 1

H (O )

i=1

" " Since " ξ M1

H (O )

"αi " (ξ + Du(ξ ))" 1

" " (ξ + Du(ξ ))"

H (O )

H 1 (O )

+ σ (O)

2 " " M " ξ 1

H (O )

"4 " (ξ + Du(ξ ))" 1

≤ (1 + C D )M (see (6.42)), we get

H (O )

.

6.3 Feedback Stabilization of the Nonlinear System (6.1)

 JM (ξ ) 2 ≤

C1 2

123

2 2  q ((1 + C D )M)αi + σ (O) (1 + C D )4 ξ 4H 1 (O ) . i=1

Taking C M as before, we conclude that (6.40) holds in this case, as well. With similar arguments, one may also obtain the estimate ! JM (ξ1 ) − JM (ξ2 ) ≤ C M ξ1 H 1 (O ) + ξ2 H 1 (O ) ξ1 − ξ2 H 1 (O ) ,

(6.43)

for all ξ1 , ξ2 ∈ H 1 (O). Now let us consider the approximation problem  t ⎧ ⎪ ⎪ (z ) (t, x) = −Az (t, x) − k(t − s)Az M (s)ds M t M ⎪ ⎪ ⎪ 0 ⎪  t  t ⎪ ⎪ ⎪ ⎪ ⎪ k(t − s)z (s)ds + μ k(t − s)Du M (s)ds + μ M ⎪ ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ ⎨ + (R1 + R2 )( z M (t), ϕ1 , . . . , z M (t), ϕ N ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

+ f  (0)z M + f  (0)Du M  t + k(t − s)R2 ( z M (s), ϕ1 , . . . , z M (s), ϕ N )

(6.44)

0

+ η(t) + JM (z M ), t > 0, x ∈ Ω, z(0, x) = z o (x), x ∈ Ω,

where u M = u(z M ). Let us denote by {S(t) : t ≥ 0} the semigroup generated by the evolution Eq. (6.25), guaranteed by Theorem 6.1, defined as follows: for each initial datum z o ∈ L 2 (O), we denote by S(t)z o , t ≥ 0, the solution to (6.25). In the proof of Theorem 6.1, we have actually shown that the semigroup {S(t) : t ≥ 0} is L 2 exponentially stable and satisfies  ∞ S(t)g 2H 1 (O ) dt < c( g 2 + yo 2L 2 (−∞,0;H 2 (O )) ), ∀g ∈ D(L) 0

(see relations (6.18) and (6.19)).

  Next, for ξ ∈ S(0, r M ) := g ∈ L 2 (0, ∞; H 1 (O)) : g L 2 (0,∞;H 1 (O )) ≤ r M , introduce the map  (Λξ )(t) := S(t)z o + (N ξ )(t); (N ξ )(t) :=

t

S(t − τ )JM (ξ )(τ )dτ.

0

Since all the hypotheses from [19] are satisfied in the present case, we may apply the same fixed-point argument for Λ on S(0, r M ) as in the proof of [19, Theorem 5.1], in order to deduce that for each M > 0, there exist r M > 0 and ρ M > 0

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6 Stabilization of Equations with Delays

sufficiently small such that for each z o , z o 2 + yo 2L 2 (−∞,0;H 2 (O )) ≤ ρ M , there exists a unique solution z M ∈ C([0, ∞; L 2 (O)) ∩ L 2 (0, ∞; H 1 (O)) to (6.44), with z M L 2 -exponentially decaying. More precisely, z M (t) 2 ≤ C (M, r M )e−γ (M,r M )t ρ M , t ≥ 0.

(6.45)

The details are omitted. Here C , γ : R+ × R+ → R∗+ are some continuous functions depending only on M and r M . To conclude with the proof, it remains to show that there exists C > 0, independent of M, such that (6.46) z M (t) H 1 (O ) ≤ C, ∀t ≥ 0. Then, taking M sufficiently large, we will have JM (z M (t)) = f (z M (t) + Du(z M (t))) − f  (0)(z M (t) + Du(z M )(t)), ∀t ≥ 0, which immediately will lead to the conclusion that the exponentially decaying z M is, in fact, a solution to the system (6.39). Then, recalling that y = z + Du, the result stated in the theorem has been proved. To show relation (6.46), we scalar multiply Eq. (6.44) by Az M , and use (6.3), (6.45) and relations (6.7), (6.40). We deduce that  z M (t) 2H 1 (O )

≤ C1 (M, r M )ρ M + C M 0

t

z M (τ ) 4H 1 (O ) dτ, t ≥ 0,

for some positive continuous function C1 : R+ × R+ → R+ . Then by Grönwall’s lemma, we get z M (t) 2H 1 (O ) ≤ C1 (M, r M )ρ M e

CM

,∞ 0

z M (τ ) 2H 1 (O ) dτ

≤ C1 (M, r M )ρ M eC M C 2 (M,r M )ρ M ,

for some positive continuous function C2 : R+ × R+ → R+ , since z M ∈ L 2 (0, ∞; H 1 (O)). Now it is clear that given a constant C > 0, for each M > 0 we may take ρ M so small that C1 (M, r M )ρ M eC M C 2 (M,r M )ρ M < C, which implies that z M (t) H 1 (O ) < C, ∀t ≥ 0, ∀M > 0, thereby completing the proof.



Remark 6.2 It is easy to see that if in Eq. (6.1) we change the null boundary Neumann condition to a null Dirichlet boundary condition, the result, stated in Theorem 6.2,  N still holds for a feedback u of similar form (with the eigenfunctions ϕ j j=1 changed accordingly).

6.3 Feedback Stabilization of the Nonlinear System (6.1)

125

Remark 6.3 It should be noted that the same stabilizing method developed here may be also applied to the the following type of heat equation with memory: ,t ,t ⎧ ⎨∂t y(t, x) = Δy(t, x) + −∞ k1 (t − s)Δy(s, x)ds + μ −∞ k2 (t − s)y(s, x)ds + f (y(t, x)), (t, x) ∈ Q, ⎩ ∂ y = 0 on Σ , y(t, x) = y (t, x), (t, x) ∈ (−∞, 0]×O, y(t, x) = u(t, x) on Σ1 , ∂n o 2

(6.47) with k1 , k2 two different positive kernels satisfying hypothesis (k). A similar result to Theorem 6.2 can be obtained in this case. The details are omitted. Remark 6.4 If there exists a constant a ∈ R, a = 0, such that f (a) = 0, then similar results to those in Theorem 6.2 concerning the local stabilization of the steady-state solution a in the nonlinear system (6.1) with μ = 0 can be obtained, following the algorithm developed above. Indeed, setting y := y − a, we reduce the problem to the null stabilization of the equivalent system ⎧ ,t ∂t y(t, x) = Δy(t, x) + −∞ k(t − s)Δy(s, x)ds + f˜(y(t, x)), ⎪ ⎪ ⎨ (t, x) ∈ Q := (0, ∞) × O, ∂ ⎪ y(t, x) = u(t, x) on Σ1 := (0, ∞) × Γ1 , ∂n y = 0 on Σ2 := (0, ∞) × Γ2 , ⎪ ⎩ y(t, x) = yo (t, x), (t, x) ∈ (−∞, 0] × O, (6.48) where f˜(y) := f (y + a), y ∈ R satisfies similar assumptions ( f 1 ), ( f 2 ) to those that f does. Then it is clear that the algorithm can be applied. The details are omitted.

6.4 Comments The model (6.1) was introduced in [66], and it describes the heat flow in a rigid isotropic homogeneous heat conductor with memory. It is derived in the framework of the theory of heat flows with memory established in [48]. Moreover, a system of first-order hyperbolic PDEs can be transformed to a system described by retarded functional differential equations like (6.1) (for details, see [71]). These equations serve as a model for physical phenomena such as traffic flows, chemical reactors, and heat exchangers. Similar equations have been considered in different papers, but the problem of the behavior of solutions and stability was directly addressed in [46, 62]. There, the main ingredient used is the so-called history space setting, which consists in considering some past history variables as additional components of the phase space corresponding to the equation under study (this idea is due to Dafermos [51]), whereas concerning the first-order hyperbolic equations, the backstepping method is implemented by Krstic et al. [76]. The boundary stabilization problem associated with (6.1) with k ≡ 0 was studied in Chap. 2. When the model incorporates memory terms, this problem is far from being solved and well understood. The character of Eq. (6.1) is determined by the nature of the kernel k, and in some situations, this equation might be of hyperbolic

126

6 Stabilization of Equations with Delays

type (that is, with finite speed propagation). This is the case, for instance, if k(t) = e−εt . By virtue of relation (6.3), we clearly see the hyperbolic nature of Eq. (6.1). In any case, there are cases of kernels k for which the equation is of parabolic type, namely k(t) = a0 t −ε , a0 > 0, 0 < ε < 1. But we see that such a kernel cannot satisfy hypothesis (6.2). In the parabolic case, the controllability problem is solved by relying on similar arguments to those in the free memory case; see, for instance, the work of Barbu and Iannelli [26] or Pandolfi [109]. Since our stabilization method requires an exponential decay of the kernel (see hypothesis (k)) it is clear that the parabolic case is left outside, while the hyperbolic case can be treated similarly to the free memory case. This is an interesting difference between the stabilization and controllability problem associated with equations with memory. The results presented in this chapter are new, and are based on those obtained in Munteanu [101]. While in [101] an additional hypothesis of linear independence of the traces of the normal derivatives of the eigenfunctions on Γ1 is imposed, here we drop it by using the control design in Chap. 2. Other results concerning the Navier– Stokes equations with memory were obtained in the author’s work [100]. For more results on the controllability problem associated with (6.1), see [18], for example, and for the optimal control problem, see [40], for instance. For more details about heat equations with memory, one may consult the book [3]. Finally, we call the reader’s attention to the result on the present subject in [64], as well as [15], concerning the stochastic version of the problem.

Chapter 7

Stabilization of Stochastic Equations

In this chapter, we consider stochastic PDEs. We address the boundary stabilization problem, which will be of two types: pathwise stabilization and stabilization in mean. Stochastic differential equations can be viewed as a generalization of dynamical systems theory to models with noise. This generalization arises naturally due to the fact that real systems cannot be completely isolated from their environments and for this reason always experience external stochastic influence. Clearly, noise perturbation complicates the problem considerably. However, it turns out that similar proportional-type deterministic feedback, designed in Chap. 2, ensures stability, though only for some special cases of stochastic equations. Depending on the equation, the technique that we use is to start with an argument similar to that in Chap. 2, then improve it by writing the solution in an integral form, and finally improve the latter by adding a fixed-point argument to the procedure.

7.1 Robustness in the Presence of Noise Perturbation of the Boundary Feedback This section answers to the following question: in the case in which the stabilizing feedback designed in Chap. 2 is perturbed by a noise, will it still ensure the stability of the system? This situation directly corresponds to practice. More precisely, measuring instruments may present some malfunctions, and therefore, the accuracy of the collected data may be randomly negatively affected. Thus, in order to have a more realistic model, it makes sense to add a noise perturbation to the controller. We confine ourselves to the one-dimensional case, with Neumann boundary conditions in which the derivative of the unknown is equal to the sum of the control and a white © Springer Nature Switzerland AG 2019 I. Munteanu, Boundary Stabilization of Parabolic Equations, Progress in Nonlinear Differential Equations and Their Applications 93, https://doi.org/10.1007/978-3-030-11099-4_7

127

128

7 Stabilization of Stochastic Equations

noise in time. For higher dimensions it is not even known whether this problem is well posed. The governing equations are ⎧ ⎨ ∂t Y (t, x) = Yx x (t, x) + f (x, Y (t, x)), t > 0, x ∈ (0, L), ˙ Yx (t, L) = 0, t > 0, Y (t, 0) = u(t) + e−δt β(t), ⎩ x Y (0, x) = Yo (x), x ∈ (0, L).

(7.1)

Here {β = β(t), t ≥ 0} is a standard real Brownian motion in the probability space (Ω, P, F ); the unknown Y = Y (t, x, ω) is a real-valued process; the initial data Yo belongs to L 2 (0, L); f is a nonlinear function; and u = u(t) is the control (see Chap. 1 for details). The target solution, which we aim to stabilize, is any Yˆ ∈ C 2 ([0, L]) satisfying the equation Yˆx x (x) + f (x, Yˆ (x)) = 0 in (0, L); Yˆx (L) = 0. Once we define the regulation error Y − Yˆ → Z , we translate the problem to the origin, by equivalently rewriting (7.1) as’ ⎧ ⎨ ∂t Z (t, x) = Z x x (t, x) + f (x, Z (t, x) + Yˆ (x)) − f (x, Yˆ (x)), t > 0, x ∈ (0, L), ˙ Z x (t, L) = 0, t > 0, Z (t, 0) = v + e−δt β, ⎩ x o Z (0, x) = Z (x) := Yo (x) − Yˆ (x), x ∈ (0, L),

(7.2) where v = u − yˆ x (0). Recall the classical assumptions on f , which we have met before: (i)

f, f  ∈ C([0, L] × R),

where f  = f y . Besides this, when needed, we will strengthen assumption (i) to (ii)

| f  (x, y)| ≤ C(|y|m + 1), ∀x ∈ [0, L], y ∈ R,

where 0 < m < ∞. The corresponding linear operator given in (2.2) in Chap. 2, obtained from the linearization around the steady state Yˆ , is defined here as Ay := yx x + f  (x, Yˆ )y, ∀y ∈ D(A),   D(A) = y ∈ H 2 (0, L) : yx (0) = yx (L) = 0 .

(7.3)

It is easy to check that A is self-adjoint in L 2 (0, L), and satisfies assumptions (A1)– (A4) denoted by  ∞  ∞from Chap. 2. Hence −A has a countable set of real eigenvalues, λ j j=1 , with the corresponding eigenfunctions denoted by ϕ j j=1 , that is, −Aϕ j = λ j ϕ j , j = 1, 2, 3, . . . .

7.1 Robustness in the Presence of Noise Perturbation of the Boundary Feedback

129

  The system ϕ j j∈N∗ may be chosen to be orthonormal. Besides this, given ρ > 0, there exists N ∈ N such that λ j ≤ ρ for j = 1, 2, . . . , N and λ j > ρ j ≥ N + 1.

(7.4)

We fix ρ (and so we fix N as well) such that 2ρ − δ < 0. In order to simplify our presentation, we will assume further that the first N eigenvalues are distinct. The general case can be also considered and treated as in Chap. 2, Sect. 2.2.2. Recalling the notation in (2.20)–(2.26), we introduce the feedback law ⎞ ⎛ ⎞

Z (t), ϕ1  ϕ1 (0)  ⎜ Z (t), ϕ2  ⎟ ⎜ ϕ2 (0) ⎟ ⎟ ⎜ ⎟ v(t) := Λ S A ⎜ ⎝ .............. ⎠ , ⎝ .......... ⎠

Z (t), ϕ N  ϕ N (0) N ⎛



(7.5)

and rewrite it as the sum v = v1 + v2 + · · · + v N , where ⎞ ⎛ 1 ⎞ ϕ (0) 

Z (t), ϕ1  γk −λ1 1 ⎜ Z (t), ϕ2  ⎟ ⎜ 1 ϕ2 (0) ⎟ ⎟ ⎜ γk −λ2 ⎟ vk (t) := A ⎜ ⎝ ............ ⎠ , ⎝ ................. ⎠ , t ≥ 0, k = 1, . . . , N . 1

Z (t), ϕ N  ϕ N (0) γ −λ 

⎛

k

N

(7.6)

N

In this case, the Gram matrix B has the form ⎞ ⎛ ϕ1 (0)ϕ2 (0) . . . ϕ1 (0)ϕ N (0) (ϕ1 (0))2 ⎜ ϕ2 (0)ϕ1 (0) (ϕ2 (0))2 . . . ϕ2 (0)ϕ N (0) ⎟ ⎟ B := ⎜ ⎝ ............................................................... ⎠ . ϕ N (0)ϕ1 (0) ϕ N (0)ϕ2 (0) . . . (ϕ N (0))2

(7.7)

It is important to emphasize that ϕi (0) = 0, i = 1, . . . , N , since otherwise, we would have ϕi ≡ 0, i = 1, . . . , N , which is absurd. That is, hypothesis (A5) also holds for this case. For a Hilbert space H , we denote by MP2 (0, T ; H ) the space of all H −valued progressively measurable processes X : Ω × (0, T ) → H such that



T

E 0

X (t)2H dt < ∞,

130

7 Stabilization of Stochastic Equations

where E is the expectation. Denote by CP ([0, T ]; H ) the space of all Ft -adapted processes X ∈ MP2 (0, T ; H ) that have a modification in C([0, T ]; L 2 (Ω)). Our goal is to show the following robustness in the presence of noise perturbation result for the proportional feedback we introduced in Chap. 2. Theorem 7.1 Under assumptions (i) and (ii), the closed-loop equation ⎧ ∂t Y (t, x) =Yx x (t, x) + f (x, Y (t, x)), t > 0, x ∈ (0, L), ⎪ ⎪ ⎪ ⎞ ⎛ ⎞ ⎛ ⎪ ⎪

Y (t) − Yˆ , ϕ1  ⎪ ϕ1 (0)   ⎪ ⎪ ⎪ ⎟ ⎜ ⎟ ⎜ ⎪ ⎨ Yx (t, 0) = Λ S A ⎜ Y (t) − Yˆ , ϕ2  ⎟ , ⎜ ϕ2 (0) ⎟ ⎝ .................... ⎠ ⎝ ........ ⎠ ⎪ ⎪ ϕ N (0)

Y (t) − Yˆ , ϕ N  ⎪ N ⎪ ⎪ ⎪ ⎪ ˆx (0) + e−δt β(t), ˙ ⎪ Y + Y (t, L) = 0, t ≥ 0, ⎪ x ⎪ ⎩ Y (0, x) =Yo (x), x ∈ (0, L),

(7.8)

has a unique solution Y ∈ CP ([0, T ]; L 2 (0, L)) that satisfies ρ

lim e 2 t Y (t) − Yˆ 2L 2 (0,L) < ∞, P − a.s.,

t→∞

provided that Yo − Yˆ  L 2 (0,L) ≤ θ for some θ > 0 sufficiently small. Following the approach in Chap. 2, we first plug the proposed feedback law (7.5) into the first-order approximation of Eq. (7.2) and derive the following result (the counterpart of the result in Theorem 2.1). Theorem 7.2 Under assumption (i), the closed-loop equation ⎧ ⎪ ∂t Z (t, x) = Z x x (t, x) + f  (x, Yˆ )Z (t, x), t > 0, x ∈ (0, L), ⎪ ⎪ ⎞ ⎛ ⎞ ⎛ ⎪ ⎪ ⎪

Z (t), ϕ1  ϕ1 (0)   ⎪ ⎪ ⎪ ⎟ ⎜ ⎟ ⎪ ⎨ Z (t, 0) = Λ A ⎜ ⎜ Z (t), ϕ2  ⎟ , ⎜ ϕ2 (0) ⎟ + e−δt β(t), ˙ x S ⎝ ............ ⎠ ⎝ ........ ⎠ ⎪ ⎪

Z (t), ϕ N  ϕ N (0) ⎪ N ⎪ ⎪ ⎪ ⎪ Z (t, L) = 0, t ≥ 0, ⎪ x ⎪ ⎪ ⎩ Z (0, x) = Z o (x), x ∈ (0, L), has a unique solution Z ∈ CP ([0, T ]; L 2 (0, L)) that satisfies ρ

lim e 2 t Z (t)2L 2 (0,L) < ∞, P − a.s.

t→∞

Proof In order to lift the boundary control into the equations, for some δ < γ1 < · · · < γ N

(7.9)

7.1 Robustness in the Presence of Noise Perturbation of the Boundary Feedback

131

we introduce as in (2.16) the Neumann operators Dγk . The noise is lifted as well via the map D = D(x), x ∈ (0, L), which is the solution to the equation 

−Dx x (x) − f  (x, yˆ )D(x) + γ D(x) = 0, x ∈ (0, L), Dx (0) = 1, Dx (L) = 0,

(7.10)

for some sufficiently large γ > 0. Then arguing as in (2.27)–(2.29), it follows that Eq. (7.9) may be equivalently rewritten as ⎛ ⎧  N  N ⎪   ⎪ ⎪ ⎪ d Z (t) = ⎝AZ (t) − 2 λj Dγk vk (Z (t)), ϕ j ϕ j (x) ⎪ ⎪ ⎪ ⎪ j=1 k=1 ⎨  N  ⎪ ⎪ (γk − A)Dγk (x)vk (Z (t)) dt + (γ − A)De−δt dβ, t > 0, + ⎪ ⎪ ⎪ ⎪ k=1 ⎪ ⎪ ⎩ o Z (0) = Z . (7.11) Equation (7.11) is formal. The precise meaning of the state equation is as follows: we say that a continuous L 2 (0, L) predictable process Z is a solution to the state equation if P − a.s. Z (t) = etA Z o − 2

N   j=1

+

N   k=1 t

 +

t

0

t

 e(t−τ )A λ j

N 

 Dγk vk (Z (τ )), ϕ j ϕ j (x)dτ

k=1

e(t−τ )A (γk − A)Dγk (x)vk (Z (τ ))dτ

(7.12)

0

e(t−τ )A (γ − A)De−δτ dβ(τ ).

0

Here the integral arising on the right-hand side of (7.12) is taken in the sense of Itô with values in H −1 (0, L). (We refer to [52, Proposition 2.4] or [113] for the existence and uniqueness of such a solution.) We continue with the argument in Chap. 2 by projecting the system on Xu : =  N  ∞ linspan ϕ j j=1 and Xs := linspan ϕ j j=N +1 (see (2.32) and (2.33)). The so-called unstable part in this case reads as (see as well (2.39)) ⎧ ⎪ ⎨ ⎪ ⎩

 dZ = −γ1 Z + Z (0) = Zo .

N  k=2

 (γ1 − γk )Bk AZ

dt + Φe−δt dβ, t > 0,

(7.13)

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7 Stabilization of Stochastic Equations

⎞ −ϕ1 (0) ⎜ −ϕ2 (0) ⎟ ⎟ Φ := ⎜ ⎝ .......... ⎠ . −ϕ N (0) ⎛

Here

Applying Itô’s formula to eδt AZ , Z  N in (7.13) yields (see (2.39)–(2.41) for the notation and computations) eδt A 2 Z 2N = A 2 Zo 2N + 1

1

 t 0

(δ − 2γ1 )eδs A 2 Z 2N 1

 N  +2e (γ1 − γk ) Bk AZ , AZ  N ds δs

 +

k=2 t

e−δs AΦ, Φ N ds + 2

0



t

AΦ, Z  N dβ(s).

0

(7.14) We see that (recall the positive semidefiniteness of the matrices Bk and the fact that the sequence (γk )k=1,N is increasing) 2eδs

N 

(γ1 − γk ) Bk AZ , AZ  N ≤ 0, s ≥ 0.

k=2

Also, recall that γ1 was taken such that γ1 > δ, which implies that δ − 2γ1 < 0. Finally, notice that since A is positive definite, we have AΦ, Φ ≥ 0. Hence taking the expectation in (7.14) yields   1 1 1 E eδt A 2 Z 2N ≤ A 2 Zo 2N + AΦ, Φ N < ∞, ∀t ≥ 0. δ

(7.15)

Now let us define V (t) := eδt A 2 Z 2N ,  t e−δs ds, I1 (t) := AΦ, Φ N 0  t

AΦ, Z  N dβ, M(t) := 2 0   1

t

I (t) := − 0

δs

1 2

(δ − 2γ1 )e A Z

2N

 N  + 2e (γ1 − γk ) Bk AZ , AZ  N ds. δs

k=2

Taking into account that M is a local martingale and I, I1 are nondecreasing, adapted, and with finite variation processes, we conclude that

7.1 Robustness in the Presence of Noise Perturbation of the Boundary Feedback

133

V (t) = Z (0) + I1 (t) − I (t) + M(t) is a semimartingale. By (7.15) we are able to apply Lemma 1.1 to V, I, I1 and M (noticing also the obvious fact that I1 (∞) < ∞) to obtain that there exists the limit   1 lim eδt A 2 Z 2N < ∞, P − a.s.

t→∞ 1

Using that A 2 is an invertible positive definite symmetric matrix, it follows that lim eδt Z 2N < ∞, P−a.s. This implies that

t→∞

lim eδt z u (t)2L 2 (0,L) < ∞, P − a.s.,

t→∞

since z u (t)2L 2 (0,L) =

(7.16)

N    | Z (t), ϕ j |2 = Z (t)2N . j=1

Concerning the stable part, since the spectrum of the operator As consists of 

−λ j

∞ j=N +1

with − λ j < −ρ, j ≥ N + 1,

by Lyapunov’s theorem, there exists Q ∈ L(Xs , Xs ), Q = Q ∗ ≥ 0, such that − Qz, As z + ρz =

1 z2X s , ∀z ∈ Xs . 2

(7.17)

The stable part of the system can be written as 

dz s (t) = (As z s (t) + F(v(t))) dt + e−δt G(x)dβ, z s (0) = z so ,

where F(v(t)) :=

N 

(7.18)

(γk − As )Dγk (x)vk (t)

k=1

and G(x) := (γ − As )D(x). ˜ s (to recall this, see We point out that by As , we understand, in fact, its extension A (2.28), while for the extension operator, see Chap. 1). So N  k=1

(γk − As )Dγk ∈ (D(As )) .

134

7 Stabilization of Stochastic Equations

By (7.16) and the definition of vk in (7.6), we easily see from the definition of F that we have  N      −δt

g, F(v(t)) ≤ C1 e g L 2  (γk − As )Dγk    (7.19) k=1

≤ C2 e

− 21 δt

(D (As ))

g L 2 , t ≥ 0, P-a.s.,

for all g ∈ D(As ). Applying Itô’s formula in (7.18) to the function e2ρt Qz, z, we get via (7.17) that 1

1

e2ρt Q 2 z s 2L 2 (0,L) = Q 2 z so 2L 2 (0,L)  t  −e2ρτ z s 2L 2 (0,L) + 2e2ρτ Qz s , F(v(τ )) + e2(ρ−δ)τ QG, G dτ (7.20) + 0  t + e(2ρ−δ)τ Qz s , G dβ. 0

First, we show that     1 E Q 2 z s (t)2L 2 (0,L) ≤ Ce−ρt and E z s (t)2L 2 (0,L) ≤ Ce−ρt ,

(7.21)

∀t ≥ 0, for some positive constant C. To this end, taking the expectation in (7.20) and recalling that 2ρ − δ < 0, we deduce that  t   1 1    1 e2ρτ Q 2 Q 2 z s , F(v(τ )) dτ , E e2ρt Q 2 z s 2L 2 (0,L) ≤ C + 2E

(7.22)

0

∀t ≥ 0, where 1

1

C = Q 2 z s0 2L 2 (0,L) + Q 2

1 G2 2 . 2(δ − ρ) L (0,L)

This implies via the Schwarz inequality, the stochastic Fubini’s theorem, and the estimate (7.19) that  t     1 1 1 E e2ρt Q 2 z s 2L 2 (0,L) ≤ C + 2C e2ρτ E Q 2 z s  L 2 (0,L) e− 2 δτ dτ 0  t  t   1 E e2ρτ Q 2 z s 2L 2 (0,L) dτ + C e(2ρ−δ)τ dτ ≤ C + ρC 0

(recall that 2ρ − δ < 0)  t   1 ρE e2ρτ Q 2 z s 2L 2 (0,L) dτ. ≤C +C 0

0

7.1 Robustness in the Presence of Noise Perturbation of the Boundary Feedback

135

1

Then via Grönwall’s inequality and the fact that Q 2 is a symmetric positive definite operator, (7.21) follows immediately. Next, from (7.20), we have 

 P

 1 εp  ≤ P Q 2 z s ( p)2L 2 (0,L) ≥ 5 t∈[ p, p+1] !  p+1   εp 1 2ρQ 2 z s 2L 2 (0,L) + z s 2L 2 (0,L) dτ ≥ +P 5 p   " t  "  " " εp 1 1 Q 2 Q 2 z s , F(v(τ )) dτ "" ≥ + P 2 sup "" 5 t∈[ p, p+1] p   " " t " εp " +P sup "" e−2δτ QG, G dτ "" ≥ 5 t∈[ p, p+1] p   " t "  1  " ε " 1 p −δτ " " Q 2 z s , Q 2 G dβ " ≥ +P sup " e 5 t∈[ p, p+1] p sup

Q

1 2

z s (t)2L 2 (0,L)

≥ εp

(using the Cebyshev inequality and the Burkholder–Davis–Gundy inequality)  p+1    5 5  1 1 ≤ E Q 2 z s ( p)2L 2 (0,L) + E 2ρQ 2 z s 2L 2 (0,L) + z s 2L 2 (0,L) dτ εp εp p  p+1 " " 10 " 1 1 " + E " Q 2 Q 2 z s , F(v(τ )) " dτ εp p  p+1 $ # 5 + E e−2δτ | QG, G| dτ εp p  p+1 " 1 "2 ! 5 1 " " + E e−2δτ " Q 2 z s , Q 2 G " dτ εp p (making use of the estimates (7.19) and (7.21)) 1 ≤ Ce−ρp , εp for some ε p > 0. Taking ε p = e− 2 ρp , we get from the above that 1

 P

 sup t∈[ p, p+1]

Q

1 2

z s (t)2L 2 (0,L)

≥e

− 21 ρp

≤ Ce− 2 ρp , ∀ p ∈ N∗ . 1

(7.23)

The Borel–Cantelli lemma now implies that there exists p(ω) such that if p > p(ω), then 1 1 sup Q 2 z s (t)2L 2 (0,L) ≤ Ce− 2 ρp , t∈[ p, p+1]

which implies that

136

7 Stabilization of Stochastic Equations

lim e 2 ρt z s (t)2L 2 (0,L) < ∞, P − a.s. 1

t→∞

(7.24)

Recalling that z = z u + z s and invoking (7.16) and (7.24), we are led to the conclusion of the theorem, thereby completing the proof.  Theorem 7.2 provides a global asymptotic exponential stabilization result of the linearization of Eq. (7.2) under the action of the feedback (7.5). Then arguing as in Theorem 2.2, one may deduce the following local stabilization result of the nonlinear system (7.2). Its proof is omitted. Theorem 7.3 Under assumptions (i) and (ii), the closed-loop equation ⎧ ⎪ ∂t Z (t, x) =Z x x (t, x) + f (x, Z (t, x) + Yˆ (x)) − f (x, Yˆ (x)), t > 0, x ∈ (0, L), ⎪ ⎪ ⎞ ⎛ ⎞ ⎛ ⎪ ⎪ ⎪

Z (t), ϕ1  ϕ1 (0)   ⎪ ⎪ ⎪ ⎟ ⎜ ⎟ ⎪ ⎨ Z (t, 0) = Λ A ⎜ ⎜ Z (t), ϕ2  ⎟ , ⎜ ϕ2 (0) ⎟ + e−δt β(t), ˙ x S ⎝ ............ ⎠ ⎝ ......... ⎠ ⎪ ⎪

Z (t), ϕ N  ϕ N (0) ⎪ N ⎪ ⎪ ⎪ ⎪ Z x (t, L) =0, t ≥ 0, ⎪ ⎪ ⎪ ⎩ Z (0, x) =Z o (x), x ∈ (0, L), (7.25) has a unique solution Z ∈ CP ([0, T ]; L 2 (0, L)), which satisfies ρ

lim e 2 t Z (t)2L 2 (0,L) < ∞, P − a.s.,

t→∞

provided that Z o  ≤ θ for some θ > 0 sufficiently small. To end this section, returning to the initial variable y, Theorem 7.3 immediately implies Theorem 7.1.

7.2 Stabilization of the Stochastic Heat Equation on a Rod The subject of this section is represented by the heat equation on (0, L), L > 0, perturbed by an internal multiplicative noise, i.e., ⎧ ⎪ ⎨ ∂t Y (t, x) = Yx x (t, x) + λσ (x, Y (t, x))dβ(t, x), 0 < x < L , t > 0, Yx (t, 0) = u(t), Yx (t, L) = 0, t ≥ 0, ⎪ ⎩ Y (0, x) = Yo (x), for x ∈ [0, L].

(7.26)

Here dβ denotes a Gaussian space-time noise on [0, ∞) × [0, L] that is usually understood as the distribution derivative of the Brownian sheet β(t, x) in t and x; see Chap. 1 or [125]. And σ is a uniformly globally Lipschitz function with σ (x, 0) = 0 , ∀x ∈ (0, L). In particular, there exists L σ > 0 such that

7.2 Stabilization of the Stochastic Heat Equation on a Rod

|σ (x, y)| ≤ L σ |y|, ∀x ∈ [0, L], ∀y ∈ R,

137

(7.27)

where λ is a positive number, usually refereed as the level of the noise; u is the boundary control. It was shown by Foondun and Nualart in [56] that in the absence of a control, no matter how small (or large) λ is, the corresponding solution y to (7.26) is exponentially unstable in the expectation. More precisely, it is shown in [56, Theorem 1.5] that the solution y to (7.26) without the boundary control (i.e., u ≡ 0) satisfies 0 < lim inf t→∞

1 log E|Y (t, x)|2 < ∞, ∀x ∈ (0, L). t

Hence it makes sense to search for a stabilizing feedback u for (7.26) in the sense that once inserted into the equation, the corresponding solution of the closed-loop equation satisfies lim inf t→∞

1 log E|Y (t, x)|2 < −γ , ∀x ∈ (0, L), t

for some γ > 0. Our goal is to show that a feedback similar to the one described in Chap. 2 ensures that the corresponding solution of the closed-loop equation (7.26) goes exponentially fast to zero in a certain sense (see Theorem 7.4 and relation (7.35) below). Since the stochastic force is of multiplicative type, one may guess that the method, presented in Chap. 2, that was successfully applied in the previous section (in the case of additive noise) may fail to work now. Indeed, the spectral decomposition method is useless due to the presence of the nonlinearity σ (y)dβ, unless it is considered separately. That is why, in comparison with the previous section, we will change the approach in the following way: we consider separately the linear equation and show that after we lift the control, the corresponding obtained linear operator generates a C0 −semigroup that can be expressed in a mild formulation, via a kernel. Then returning to the full nonlinear equation, we write its solution in an integral formulation, a fact that allows one to obtain the desired exponential decay. All our effort will be concentrated in showing that the kernel has “good properties”; see Lemma 7.1 below. In any case, let us further explain the approach we will follow. Say that we are dealing with the following heat equation: 

∂t y(t, x) − y(t, x) − ay(t, x) = 0 in (0, ∞) × (0, L), y(t, 0) = y(t, L) = 0, ∀t ≥ 0, y(0) = yo ,

(7.28)

 ∞  ∞ where a and L are some positive constants. Let us denote by λ j j=1 and by ϕ j j=1 the system of eigenvalues and the system of eigenfunctions that diagonalizes the Dirichlet Laplacian in L 2 (0, L), respectively. It is known that they are given by

138

7 Stabilization of Stochastic Equations

λj =

jπ L

!2 and ϕ j =

2 L

! 21

jπ x L

sin

! , j = 1, 2, . . . .

Then denoting by ∞ 

p(t, x, ξ ) :=

e−λ j t ϕ j (x)ϕ j (ξ ), t ≥ 0, x, ξ ∈ O,

(7.29)

j=1

the so-called Dirichlet heat kernel, it is easy to see that p(t, x, ξ ) ≤ pG (t, x, y), ∀t > 0, where pG (t, x, ξ ) := √

1 4π t

e−

(7.30)

|x−ξ |2 4t

is the Gaussian kernel. Next, for t0 > 0 fixed, there is a constant c > 0 such that p(t, x, ξ ) ≤ ce−λ1 t , ∀t > t0 .

(7.31)

It follows that there exists some constant c1 > 0 such that for all η ∈ (0, λ1 ), we have 

∞

eηt p(t, x, x)dt ≤ c1

0

! 1 1 . √ + η λ1 − η

(7.32)

This is indeed so. We use the bounds (7.30) and (7.31). We therefore write 

∞

eηt p(t, x, x)dt =



0

t0

eηt p(t, x, x)dt +



∞

eηt p(t, x, x)dt =: I1 + I2 .

t0

0

For I1 , using (7.30), we have 

t0

I1 ≤



ηt

t0

e pG (t, x, x)dt ≤ c2

0

0

eηt √ dt. t

As for I2 , by (7.31), we have  I2 ≤ t0

∞

eηt p(t, x, x)dt ≤ c



∞

eηt e−λ1 t dt.

t0

Combining these two results yields the estimate (7.32).

7.2 Stabilization of the Stochastic Heat Equation on a Rod

139

Let us first consider the case a = 0. We can represent the corresponding solution to (7.28) as  L p(t, x, ξ )yo (ξ )dξ. y(t, x) = 0

Hence for each η ∈ (0, 2λ1 ), we have 

∞

ηt



∞

e y(t) dt = 2

0

e 

0

%

∞

=

ηt

e 0



ηt

!

L

y (t, x)d x dt 2



0



L

0



∞

≤ yo 2 0

 = yo 

0

eηt 

L

L

2



L

!2 p(t, x, ξ )yo (ξ )dξ

d x dt

! p(2t, x, x)d x dt

0 ∞



ηt

!

e p(2t, x, x)dt d x 0

0

(using estimate (7.32)) ≤ Cyo 2 , for some positive constant C. It follows by the semigroup property that the solution y is exponentially decaying in the L 2 −norm  · . Now consider the case a = 0 such that λ1 − a < 0. Set μ j := λ j − a, for j = 1, 2, .., and ∞  e−μ j t ϕ j (x)ϕ j (ξ ). p(t, ˜ x, ξ ) := j=1

Then the solution to (7.28) can be written again in integral form as  y(t, x) =

L

p(t, ˜ x, ξ )yo (ξ )dξ.

(7.33)

0

It is clear that in trying to show an L 2 −norm exponential decay for this y, by following the above ideas, an estimate like (7.32) fails to hold for the kernel p, ˜ because the eigenvalue μ1 = λ1 − a is negative, and so (7.31) can no longer be obtained. In order to achieve the stability in (7.28), a boundary feedback control may be inserted. To achieve this goal, we look for a special feedback law that allows us to write the corresponding solution of the closed-loop equation in an integral form similar to (7.33). It is clear that in order for us to be able to do this, the feedback law must be explicitly given in a simple form. It turns out that the proportional feedback law designed in Chap. 2 is able to do this job.

140

7 Stabilization of Stochastic Equations

7.2.1 Mild Formulation of the Solution and Proof of the Main Result The goal of this section is to prove the following result. Theorem 7.4 Let ρ > 0 be arbitrary but fixed. For N ∈ N large enough, the closedloop equation ⎧ ∂t Y (t, x) = Yx x (t, x) + λσ (x, Y (t, x))dβ(t, x), ⎪ ⎪ ⎪ ⎨ for 0 < x < L and t > 0, ⎪ Yx (t, 0) = u(Y (t)), Yx (t, L) = 0 for t > 0, ⎪ ⎪ ⎩ Y (0, x) = Yo (x) for x ∈ [0, L],

(7.34)

has a unique solution Y ∈ CP ([0, T ]; L 2 (0, L)), which satisfies − ∞ < lim sup log E|Y (t, x)|2 < −ρ, ∀x ∈ (0, L).

(7.35)

t→∞

Here the feedback u is defined by (7.42) below. In contrast to the previous discussions, here we will also estimate the magnitude, with respect to the parameter N , of the controller u. More precisely, in what follows, we will estimate the norm of the matrices involved in the definition of the controller u, with respect to the parameter N . To this end, we define below what we understand by a quantity to be of order some power of N . Let k > 0, and consider a function f that depends on the parameter N , i.e., f = f (N ). We say that f is of order N k , and denote this by O(N k ), if f (N ) lim N →∞ N k exists and is finite and nonzero. Now let us construct the stabilizing feedback law, using the ideas in Chap. 2. The governing operator of Eq. (7.34) is the Neumann–Laplace operator on (0, L), given by (7.36) Ay = −yx x , ∀y ∈ D(A),   D(A) = y ∈ H 2 (0, L) : yx (0) = yx (L) = 0 . It is well known that it has a countable set of eigenvalues, namely μj =

( j − 1)2 π 2 , j = 1, 2, . . . , L2

with the corresponding eigenfunctions

7.2 Stabilization of the Stochastic Heat Equation on a Rod

141

⎧& ⎨ 1 , j =1 L   ϕ j (x) = & ( j−1)π x 2 ⎩ , j = 2, 3, . . . . cos L L which form an orthonormal basis in L 2 (0, 1). Let us fix some large enough N ∈ N. In the previous chapters, the choice of N was correlated with the magnitude of the eigenvalue μ N +1 . Namely, N was chosen to be large enough that μ N +1 was greater than some given constant. This time, besides the magnitude of μ N +1 , here we will take into account also the magnitude of N itself. In other words, we will study how the controller u depends on N . It turns out that it can be estimated to be of order O(N η ), for some good enough η to imply the desired stability (see (7.48) below). In this light, this time, the constants γ1 , . . . , γ N will not be arbitrary, but of a precise form. More exactly, γk := μ N + with

7 4

k + N α , k = 1, 2, . . . , N , N

(7.37)

< α < 2. Note that we have lim

N →∞

γk γk − μ N π2 = and lim = 1. N →∞ N2 L2 Nα

Thus we stress that

and

γk , k = 1, 2, . . . , N , are of order O(N 2 ),

(7.38)

γk − μ N , k = 1, 2, . . . , N , are of order O(N α ).

(7.39)

This time, the counterpart of the Gram matrix B introduced in (2.20) is given by √ √ ⎞ 2 ... 2 √1 1 ⎜ 2 2 ... 2 ⎟ ⎟. B := ⎜ .................... ⎠ L ⎝√ 2 2 ... 2 ⎛

(7.40)

And the corresponding form of the feedback law (2.26), in the present case, reads as follows: for each k = 1, . . . , N , we set ⎞ ⎛ ϕ1 (0) ⎞

y, ϕ1  γk −μ1  ϕ2 (0) ⎟ ⎜ y, ϕ2  ⎟ ⎜ ⎟ ⎜ γk −μ2 ⎟ ⎟ , u k (y) := A ⎜ ⎝ ........... ⎠ , ⎜ ⎝ ......... ⎠ ϕ (0) N

y, ϕ N  

⎛

γk −μ N

N

(7.41)

142

7 Stabilization of Stochastic Equations

then introduce u as u(y) := u 1 (y) + · · · + u N (y).

(7.42)

For the definition of A, see (2.24). Recall the definition of the Bk from (2.22). Their sum is precisely the following: N  k=1

1 Bk = L

 N  k=1

bi j (γk − μi )(γk − μ j )

N ,

(7.43)

i, j=1

√

where b11 = 1 and bi j = 2, (i, j) = (1, 1). We want to estimate the magnitude of u k , k = 1, 2, . . . , N . This reduces to estimating the first and last eigenvalues of the matrix A, or equivalently, by the definition of A, to estimating the last and first eigenvalues of the sum matrix B1 + · · · + B N . Let us denote by r1 , . . . , r N (arranged as an increasing sequence) the positive eigenvalues of the latter matrix. By virtue of (7.43), we have that r1 + · · · + r N =

N bii 1  . L i,k=1 (γk − μi )2

By (7.38) and (7.39), we deduce that C1

1 1 ≤ r1 + r2 + · · · + r N ≤ C2 2α−2 , N2 N

(7.44)

for some positive constants C1 , C2 , independent of N (but for N large enough). ' NNext by the Gershgorin circle theorem, we know that the eigenvalues of the matrix k=1 Bk cannot be far from its diagonal entries. More precisely, we know that there exists some j ∈ {1, 2, . . . , N } such that " " √ √ N N "  "  2 2 " " . "≤ "r N − 2" " (γ − μ ) (γ − μ )(γ k j k j k − μl ) k=1 k,l=1 Since in the above inequality the  right-hand side tends  to zero as N → ∞, we deduce √ N  2 that r N is maximal of order O . Taking into account relations (γk − μ N )2 k=1 (7.38) and (7.39), and that N  k=1

√ √ 2 2N ≤ , (γk − μ N )2 (γ1 − μ N )2

7.2 Stabilization of the Stochastic Heat Equation on a Rod

it follows that r N is maximal of order O ri ≤ C 3

1 N 2α−1

#

1 N 2α−1

143

$ . In other words, we obtain that

, i = 1, 2, . . . , N ,

(7.45)

for some constant C3 > 0, independent of N . Finally, taking α very close to 2 as necessary, we see by (7.44) and (7.45) that necessarily, 1 (7.46) ri ≥ C4 3 , i = 1, 2, . . . , N , N N$ . Hence we conclude that the orders for some positive constant C#4 , independent $ # of 1 . of r1 , . . . , r N lie between O N13 and O N 2α−1 Recalling that A = (B1 + B2 + · · · + B N )−1 , and denoting by λ1 (A) and λ N (A) the first and last eigenvalues of the matrix A, respectively, we get that λ1 (A) and λ N (A) = A have order between O(N 2α−1 ) and O(N 3 ).

(7.47)

Here A denotes the classical Euclidean norm of the matrix A. By (7.41), (7.39), and (7.47), we deduce then ⎛ ⎞  y, ϕ1     √ ⎜ y, ϕ2  ⎟ 3−α ⎜   ⎟ N ⎝ |u k (Y )| ≤ C N ⎠ , ∀t ≥ 0.  .........   y, ϕ N   N

(7.48)

Proof of Theorem 7.4. First of all, we note that the stochastic equation (7.34) is well posed, since both σ and the Neumann boundary conditions are Lipschitz, and thus one can argue for the existence and uniqueness as in [113]. We lift the boundary conditions into the Eq. (7.34) by arguing similarly as in (2.27)–(2.29), obtaining thereby the internal control-type problem ∂t Y (t) = − AY (t) +

N 

˜ + γi )Dγi − 2 u i (Y (t))(A

i=1

N 

  μ j u i (Y (t))Dγi , ϕ j ϕ j

i, j=1

+ λσ (Y (t))dβ; Y (0) = Yo . (7.49) (One can check the section above or [19, Sect. 1] for additional explanations on the precise definition of a solution to (7.49).) Next, the idea is to forget, for a while, about the stochastic perturbation, and express the solution z to the linear equation

144

7 Stabilization of Stochastic Equations

∂t z(t) = − Az(t) +

N 

˜ + γi )Dγi u i (z(t))(A

i=1 N 

−2

  μ j u i (z(t))Dγi , ϕ j ϕ j , t > 0,

(7.50)

i, j=1

z(0) = z o , in an integral form. This enables one to have a mild formulation for the solution y to (7.49). One can prove the following results. Lemma 7.1 The solution z of N 

∂t z(t) = − Az(t) +

˜ + γi )Dγi − 2 u i (z(t))(A

i=1

N 

  μ j u i (z(s))Dγi , ϕ j ϕ j ;

i, j=1

z(0) = z o , (7.51) can be written in a mild formulation as 

L

z(t, x) =

p(t, x, ξ )z o (ξ )dξ.

0

Moreover, we have that  0

∞



L

e N t p 2 (t, x, ξ )dξ dt ≤ C

0

1 , ∀x ∈ (0, L), Nθ

(7.52)

for some positive θ and C > 0 independent of N . Proof We will decompose z as z(t) =

∞ 

z j (t)ϕ j (x),

j=1

  where z j (t) = z(t), ϕ j , j = 1, 2, . . . . From now on, all our effort will be devoted to writing, for each j ∈ N∗ , z j in the form z j (t) =

∞ 

f i j (t) z o , ϕi  ,

(7.53)

i=1

with | f i j (t)| ≤ Ci j e−ci j t , t ≥ 0, for some Ci j , ci j > 0 of order some powers of N . Once we do this, then immediately we may write the kernel p as

7.2 Stabilization of the Stochastic Heat Equation on a Rod

p(t, x, ξ ) =

∞ 

145

f i j (t)ϕi (x)ϕ j (ξ )

i, j=1

and play with the estimates (in terms of N ) of Ci j , ci j in order to deduce (7.52) as well. We emphasize that in the case of a general feedback law, after a lift of the boundary conditions into the equations, it is not easy (or even possible) to get a relation like (7.53). However, the simple explicit form of our feedback law allows us to do this. Scalar multiplying equation (7.51) by ϕ j , j = 1, . . . , N , and arguing as in (2.38)– (2.41), we get that the first N modes of the solution z satisfy  d (γ1 − γk )Bk AZ , t > 0. Z = −γ1 Z + dt k=2 N

(7.54)

The solution Z can be expressed as Z (t) = e

  'N t −γ1 I + k=2 (γ1 −γk )Bk A

Zo .

 N This yields that there exist continuous functions qi j : [0, ∞) → R i, j=1 such that z i (t) =

N 

  qi j (t) z o , ϕ j , i = 1, . . . , N .

(7.55)

j=1

Furthermore, using the results in (2.40)–(2.41), we conclude that |qi j (t)|2 ≤

C e−γ1 t , ∀t ≥ 0, ∀i, j = 1, . . . , N . λ1 (A)

(7.56)

Making use of (7.38) and (7.47), relation (7.56) implies |qi j (t)| ≤ C

1 N

α− 21

e−cN t , ∀t ≥ 0, 2

(7.57)

for some positive constants C, c, independent of N . Clearly, we may take c < πL 2 (which we shall do for later purposes). Since by (7.41), the feedbacks u i , i = 1, . . . , N , are some linear combinations of the modes z 1 , . . . , z N , we get from (7.55) that there exist continuous functions  N ri j : [0, ∞) → R i, j=1 such that 2

u i = u i (t) =

N  j=1

  ri j (t) z o , ϕ j , i = 1, . . . , N ,

(7.58)

146

7 Stabilization of Stochastic Equations

where by (7.41), (7.47), (7.39), and (7.57), we get that there exists C > 0 such that √

√ N 1 N 2 3 −cN 2 t |ri j (t)| ≤ AZ (t) N ≤ CN = C N 4−2α e−cN t , 1 e α α− γ1 − μ N N N 2 (7.59) ∀t ≥ 0, ∀i, j = 1, . . . , N . We move on to the modes z j , j > N . Scalar multiplying equation (7.51) by ϕ j , j > N , we get d z j = −μ j z j − dt

(

N 2 u i , t > 0. L i=1

Then the variation of constants formula gives z j (t) =e

−μ j t





(

zo , ϕ j −

We write

2 L

( j wi (t)

:= −



t

e−μ j (t−s)

0

N 

u i (s)ds, t ≥ 0.

(7.60)

i=1

N  2  t −μ j (t−s) e rki (s)ds. L k=1 0

Involving (7.58), the above relation (7.60) reads as N    j z j (t) = e−μ j t z o , ϕ j + wi (t) z o , ϕi  , t ≥ 0.

(7.61)

i=1

Simple computations, taking advantage of the estimates (7.59), yield j

|wi (t)| ≤ C

1 2 N 5−2α e−cN t , ∀t ≥ 0, μ j − cN 2

(7.62)

for all i = 1, 2, . . . , N and j = N + 1, N + 2, . . .. Above, we have used the fact 2 that c was chosen such that c < πL 2 . We may now conclude by (7.55) and (7.61) that the solution z to (7.51) may be written as  L p(t, x, ξ )z o (ξ )dξ, z(t, x) = 0

where the kernel p is given as p(t, x, ξ ) := p1 (t, x, ξ ) + p2 (t, x, ξ ) + p3 (t, x, ξ ), for t ≥ 0, x, ξ ∈ (0, L), where

(7.63)

7.2 Stabilization of the Stochastic Heat Equation on a Rod

147

⎞ ⎛ N N   ⎝ p1 (t, x, ξ ) := q ji (t)ϕ j (x)⎠ ϕi (ξ ), i=1

j=1

∞ 

p2 (t, x, ξ ) :=

e−μi t ϕi (x)ϕi (ξ ), and

i=N +1

p3 (t, x, ξ ) :=

N 

⎛ ⎝

i=1

∞ 

⎞ wi (t)ϕ j (x)⎠ ϕi (ξ ). j

j=N +1

Now we want to estimate the quantity 

∞ 0



L

e N t p 2 (t, x, ξ )dξ dt.

0

Taking into account the form of p, by Parseval’s identity, we have that 

L

p 2 (t, x, ξ )dξ

(7.64)

0

⎞2 ⎛ N N ∞ ∞     j ⎝ = q ji (t)ϕ j (x) + wi (t)ϕ j (x)⎠ + e−2μi t ϕi2 (x) i=1

j=N +1

j=1

i=N +1

⎞2 N N N ∞ ∞      j ⎝ ⎝ q ji (t)ϕ j (x)⎠ + 2 wi (t)ϕ j (x)⎠ + e−2μi t ϕi2 (x). ≤2 i=1

Thus

⎞2

⎛

j=1

⎛

i=1



∞



0

L

j=N +1

i=N +1

e N t p 2 (t, x, ξ )dξ dt ≤ I1 + I2 + I3 ,

(7.65)

0

where

 I1 :=2

∞

⎞2 ⎛ N N   ⎝ eNt q ji (t)ϕ j (x)⎠ dt

0

 ≤C 0

i=1 ∞

j=1

e N t N 3 max |q ji (t)|2 dt i, j=1,N

(using (7.57) and taking N large enough)  ∞ 1 2 ≤ C N3 e N t 2α−1 e−2cN t dt N 0 1 ≤ C 2α−2 , ∀x ∈ (0, L). N

(7.66)

148

7 Stabilization of Stochastic Equations

Next,

 I2 := 2

∞

eNt

0

N 

⎛ ⎝

⎡

wi (t)ϕ j (x)⎠ dt j

j=N +1

i=1

(using (7.62))

⎞2

∞ 

⎤2 ! 1 2 ≤C eNt N ⎣ N 5−2α e−cN t ⎦ dt 2 μ − cN j 0 j=N +1 ⎛ ⎞2  ∞ ∞  1 2 ⎠ e N t e−2cN t dt. = C N 11−4α ⎝ 2 μ − cN j 0 j=N +1 

∞

∞ 

(7.67)

Note that ∞ 

1 ≤ μ − cN 2 j j=N +1

1 π2

−c  ∞

L2

=C

N −1

∞  j=N +1

1 ≤C j2



N N −1

1 dx + x2



N +1 N

1 dx + · · · x2

1 1 . dx = C x2 N −1

It follows by (7.67) that I2 ≤ C N 11−4α

1 1 1 ≤ C 4α−7 , 2 2 N 2cN − N N

(7.68)

for N large enough. Finally, notice that the kernel p2 has a similar structure to that of the heat kernel p defined by (7.29), except that in the present case, the first eigenvalue of the infinite 2 summation is μ N +1 = πL 2 N 2 > 0. Hence we can bound the kernel p2 similarly as in (7.30)–(7.32), by taking η = N . In this way, we obtain that 



∞

I3 :=

e

Nt

0



∞ 

e

i=N +1

−2μi t

ϕi2 (x)



∞

dt = 0

1 e N t p2 (2t, x, x)dt ≤ C √ , N (7.69)

for some C > 0. Again considering all the above estimates, namely (7.65), (7.66), (7.68), and (7.69), we deduce that  0

∞



L 0

e N t p 2 (t, x, ξ )dξ dt ≤ C

1 , ∀x ∈ (0, L), Nθ

where θ > 0 is defined as (recall that α was chosen such that α > 47 )

(7.70)

7.2 Stabilization of the Stochastic Heat Equation on a Rod

  1 , θ := min 2α − 2; 4α − 7; 2

149

(7.71) 

thereby completing the proof.

Proof of Theorem 7.4 (continued). Next, the idea is to get rid of the Brownian motion. This is usually done by taking the second moment into the equation and using Itô’s isometry. Before doing that, let us introduce Y 2,N := essupt>0 essupx∈(0,L) e N t E|Y (t, x)|2 . Then, by virtue of Lemma 7.1, we write the solution of (7.71) in a mild formulation via the kernel p, i.e.,  Y (t, x) =

L

p(t, x, ξ )Yo (ξ )dξ  t L p(t − s, x, ξ )σ (x, Y (s, ξ )))β(ds, dξ ). +λ 0

0

(7.72)

0

Notice that the kernel p, defined by (7.63), has similar structure, with similar properties, to the classical heat kernel. Consequently, one can easily argue as in [125, Ex. 3.4] or [44, Theorem 13] in order to deduce the unique existence of a solution Y to (7.72). Taking the second moment in (7.72) and using Itô’s isometry and relation (7.27), we obtain that 

L

E|Y (t, x)|2 ≤ 2L 0

+ 2Lλ2 L 2σ

p2 (t, x, ξ )yo (ξ )dξ  t 0

L

p2 (t − s, x, ξ )E|Y (s, ξ )|2 dξ ds

0

(using (7.52)) ≤ Ce−N t + 2Lλ2 L 2σ = Ce−N t + 2Lλ2 L 2σ

 t 0

L

 t 0

p2 (t − s, x, ξ )E|Y (s, ξ )|2 dξ ds

0 L

e N (t−s) p2 (t − s, x, ξ )e−N (t−s) E|Y (s, ξ )|2 dξ ds

0

≤ Ce−N t + Y 2,N 2Lλ2 L 2σ e−N t

 t 0

L

e N (t−s) p2 (t − s, x, ξ )dξ ds

0

(again using relation (7.52) ) ≤ Ce−N t 1 + Lλ2 L 2σ

! 1 . Y  2,N Nθ

The above relation implies that Y 2,N ≤ C + Cλ2 L L 2σ

1 Y 2,N . Nθ

150

7 Stabilization of Stochastic Equations

Therefore, if we choose N large enough that Cλ2 L L 2σ N1θ < 1 and N > ρ, we obtain that Y 2,ρ ≤ Y 2,N < ∞, 

thereby completing the proof.

7.3 Stabilization of the Stochastic Burgers Equation Here we propose to further develop the ideas from the previous section regarding the equivalent rewrite of the solution in an integral form. We will consider again a nonlinear stochastic equation, namely the stochastic Burgers equation, and stabilize its null solution from the boundary. This equation reads as ⎧ dY (t, x) = νYx x (t, x)dt + b(t, x)Y (t, x)Yx (t, x)dt +θ Y (t, x)dβ(t), ⎪ ⎪ ⎨ t > 0, x ∈ (0, L), (t, 0) = v(t), Y (t, L) = 0, t > 0, Y ⎪ x x ⎪ ⎩ Y (0, x) = yo (x), x ∈ (0, L). (7.73) It is clear that the second-order nonlinearity Y Yx significantly complicates the context, which is left outside by the previous approach, applied to the stochastic heat equation. In any case, this time, the stochastic perturbation is very simple, θ Y dβ, with θ a positive constant. This allows one to do a rescaling in order to transform (7.73) into a deterministic random PDE. Concerning the obtained random deterministic equation, one does not even know whether it well posed. So in fact, we have to solve three problems at once, namely existence, uniqueness, and stabilization. This can be achieved via a fixed-point argument. More exactly, we will consider an auxiliary functional space, namely  Z = y = y(t, x) : sup e (y(t) + t yx (t)) < ∞ , 

Nt

ρ

t>0

for some positive ρ, write again the solution in a mild formulation, then show that the corresponding nonlinear functional leaves the ball   Br (0) = e N t (y(t) + t ρ yx (t)) ≤ r invariant and that it is a contraction on it, for r and initial data small enough. From this, via the contraction mapping theorem, the three problems are solved. Let us give some details about the functions and parameters that constitute (7.73). The function b is such that there exist Cb > 0 and 0 ≤ m 1 ≤ m 2 ≤ · · · ≤ m S , for some S ∈ N, for which

7.3 Stabilization of the Stochastic Burgers Equation

sup |b(t, x)| ≤ Cb x∈(0,L)

 S 

151

 t

mk

+ 1 , ∀t > 0.

(7.74)

k=1

Moreover, we assume that m S and θ are such that θ can be split as 1 2 1 θ = m S + + θ1 , 2 4

(7.75)

where θ1 > 0. In (7.73), consider the substitution Y (t) = Γ (t)y(t), t ∈ [0, ∞),

(7.76)

where Γ (t) : L 2 (0, L) → L 2 (0, L) is the linear continuous operator defined by the equations dΓ (t) = θ Γ (t)dβ(t), t ≥ 0, Γ (0) = 1, which can be equivalently expressed as Γ (t) = eθβ(t)− 2 θ , t ≥ 0. t

2

(7.77)

By the transformation (7.76), Eq. (7.73) reduces to the random parabolic equation ⎧ ∂ ⎪ ⎪ y(t) = νΓ −1 (t)(Γ (t)y(t))x x + Γ −1 (t)b(t)(Γ (t)y(t)) (Γ (t)y(t))x , ⎪ ⎨ ∂t t ∈ [0, ∞), ⎪ ⎪ yx (t, 0) = Γ −1 (t)v(t), yx (t, 1) = 0, t ∈ [0, ∞), ⎪ ⎩ y(0) = yo . (7.78) Indeed, if y is a regular solution to (7.78) (for instance absolutely continuous in t) that is progressively measurable in (t, ω) in the probability space {Ω, P, F , Ft } and  T

E 0

y(t)2H 2 (O ) dt < ∞,

then by Itô’s formula in (0, T ) × Ω × O, we have dY = ydΓ (t) + Γ (t)

∂y dt in (0, T ) × O. ∂t

Then we obtain for y the random Eq. (7.78), as claimed. On the other hand, an Ft adapted solution t → y(t) to Eq. (7.78) leads via transformation (7.76) to a solution Y to (7.73) in the sense of the above definition. We equivalently write (7.78) as

152

7 Stabilization of Stochastic Equations

⎧ ∂ ⎪ ⎪ y(t) = νθ [βx x (t)y(t) + (βx (t))2 y(t) + 2βx (t)yx (t) + yx x (t)] ⎪ ⎪ ∂t ⎪ ⎨ + b(t)Γ (t)y(t)[θβx (t)y(t) + yx (t)], t ∈ [0, ∞), ⎪ ⎪ ⎪ yx (t, 0) = Γ −1 (t)v(t), yx (t, 1) = 0, t ∈ [0, ∞), ⎪ ⎪ ⎩ y(0) = yo .

(7.79)

In order to simplify the problem, we assume that the Brownian motion β is only time-dependent. Hence it follows by (7.79) that in fact, y satisfies the equation ⎧∂ ⎨ ∂t y(t) = νyx x (t) + Γ (t)b(t)y(t)yx (t), t ∈ [0, ∞), y (t, 0) = u(t) := Γ −1 (t)v(t), yx (t, 1) = 0, t ∈ [0, ∞), ⎩ x y(0) = yo .

(7.80)

Of course, one may think to consider the more general case of β = β(t, x) and apply the argument that follows to (7.79). Unfortunately, this is not a trivial task. We will explain later what other difficulties appear in this general case and what additional hypotheses should be added. Below, we will frequently use the following obvious but useful inequality: e−at ≤ t −a , ∀t > 0, a ≥ 0. Before moving on, let us see that by the law of the iterated logarithm, arguing as in Lemma 3.4 in [21], it follows that there exists a constant CΓ > 0 such that Γ (t) = eθβ(t)−θ1 t e−(m S + 4 )t ≤ CΓ e−(m S + 4 )t , ∀t > 0, P-a.s., 1

1

where we have used that 21 θ 2 = m S +

1 4

 Γ (t) sup |b(t, x)| ≤ CΓ Cb x∈(0,L)

(7.81)

+ θ1 . Then by (7.74), we have that

S 

 m k −(m S + 41 )t

t e

+e

−(m S + 41 )t

k=1

(since 0 ≤ m 1 ≤ m 2 ≤ · · · ≤ m S )  S   m k −(m k + 41 )t − 14 t ≤C t e +e  ≤C

k=1 S 

(7.82)

 m k −(m k + 41 )

t t

+t

− 41

k=1

≤ (S + 1)Ct − 4 , ∀t > 0. 1

Next, we recall the Neumann–Laplace operator A, and its spectrum {μk }∞ k=1 and its eigenfunction system {ϕk }∞ k=1 ; the Gram matrix B, the matrices Λk and Bk , k = 1, . . . , N , and A; the Neumann operators Dγk , k = 1, 2, . . . , N , introduced in relations (7.36)–(7.43) above, respectively. Also, we recall that based on them, we

7.3 Stabilization of the Stochastic Burgers Equation

153

have introduced the feedback laws ⎞ ⎛ ϕ1 (0) ⎞

y, ϕ1  γk −μ1  ϕ (0) ⎟ ⎜ y, ϕ2  ⎟ ⎜ ⎜ γk2−μ2 ⎟ ⎟ ⎜ ,⎜ u k (y) := A ⎝ ⎟ , ......... ⎠ ⎝ ........ ⎠ ϕ N (0)

y, ϕ N  

⎛

γk −μ N

(7.83)

N

and u as u(y) := u 1 (y) + · · · + u N (y).

(7.84)

Next, arguing similarly as in (7.49), we equivalently rewrite (7.80) as an internal control-type problem: ∂t y(t) = − Ay(t) +

N 

u i (y(t))(A + γi )Dγi − 2

i=1

N 

  μ j u i (y(t))Dγi , ϕ j ϕ j

i, j=1

+ b(t)Γ (t)y(t)yx (t); y(0) = yo . (7.85) Finally, as in Lemma 7.1, one may show that the solution z of ∂t z(t) = −Az(t) +

N 

u i (z(t))(A + γi )Dγi − 2

i=1

N 

  μ j u i (z(s))Dγi , ϕ j ϕ j ,

i, j=1

z(0) = z o , (7.86) can be written in a mild formulation as  L p(t, x, ξ )z o (ξ )dξ, z(t, x) = 0

where p(t, x, ξ ) := p1 (t, x, ξ ) + p2 (t, x, ξ ) + p3 (t, x, ξ ), for t ≥ 0, x, ξ ∈ (0, L). Here ⎞ ⎛ N N   ⎝ p1 (t, x, ξ ) := q ji (t)ϕ j (x)⎠ ϕi (ξ ), i=1

p2 (t, x, ξ ) :=

j=1

∞ 

e−μi t ϕi (x)ϕi (ξ ), and

i=N +1

p3 (t, x, ξ ) :=

N  i=1

⎛ ⎝

∞  j=N +1

⎞ wi (t)ϕ j (x)⎠ ϕi (ξ ). j

(7.87)

154

7 Stabilization of Stochastic Equations j

The quantities q ji (t) and wi (t) involved in the definition of p satisfy the following estimates: for some Cq > 0, depending on N , |q ji (t)| ≤ Cq e−cN t , ∀t ≥ 0, 2

(7.88)

for all i, j = 1, 2, . . . , N , and for some Cw > 0, depending on N , j

|wi (t)| ≤ Cw

1 2 e−cN t , ∀t ≥ 0, 2 μ j − cN

(7.89)

for all i = 1, 2, . . . , N and j = N + 1, N + 2, . . .. Moreover, for all z o ∈ L 2 (0, L), we have that ⎧ ∞ ⎨  ⎩

μj

j=N +1

% N 

j wi (t) z o , ϕi 

2 ⎫ 21 ⎬

i=1

⎭

≤ Ce−cN

2

t

sup

| z o , ϕl  |, ∀t ≥ 0.

l=1,2,...,N

(7.90) We will not comment on (7.87)–(7.89) since they can be directly obtained from the proof of Lemma 7.1. In any case, relation (7.90) is a new property of the kernel p, and we prove it below. Note that the kernel p is the same as in the framework of the stochastic heat equation from the previous section. Yet we succeed in finding new properties of it, namely relation (7.90). This suggests that the kernel p may have further important properties that can be developed and used for stabilization of other more complex problems. In other words, we are sure that in the future, one may solve the stabilization problem of other diverse complicated stochastic equations using these proportional-type feedback laws. The need for relation (7.90) becomes clearer if we recall that we are now dealing 1 with the derivative 0 yx as1well. Therefore, estimates of the H -norm will be needed. It is known that

1 √ μj

ϕj

∞

j=1

forms a basis in H 1 ; hence it is easy to see that relation

(7.90) is an estimate of an H 1 -norm. Now let us prove it. Setting B(u)(t) :=

N 

u i (z(t))(A + γi )Dγi − 2

i=1

and

N 

  μ j u i (z(s))Dγi , ϕ j ϕ j ,

i, j=1

  (B(u)(t)) j := B(u)(t), ϕ j , j = 1, 2, . . . ,

we have, via (7.60)–(7.61), that N  i=1

 j wi (t) z o , ϕi 

t

= 0

e−μ j (t−s) (B(u)(s)) j ds,

7.3 Stabilization of the Stochastic Burgers Equation

155

which yields that ⎧ ∞ ⎨ 

μj ⎩ j=N +1 ⎛ 4

t

≤

e 0



i=1

⎭

=

5

−μ N +1 (t−s)

t

≤C

2 ⎫ 21 ⎬

j wi (t) z o , ϕi 

1 √ ϕj μj

⎝since 

% N 

⎧ ∞ ⎨  ⎩

2

t

μj

j=N +1

e

−μ j (t−s)

(B (u)(s)) j ds

⎫1 32 ⎬ 2

0

⎭

⎞

is an orthogonal basis in H 1 (0, L)⎠ j

 [B (u)(s)]x ds

e−μ N +1 (t−s) e−cN s ds 2

0

sup

| z o , ϕl  | ≤ Ce−cN

2t

l=1,2,...,N

sup

| z o , ϕl  |, ∀t ≥ 0,

l=1,2,...,N

where we have used the form of B(u)) and the fact that |u i (t)| ≤ Ce−cN

2

t

sup

| z o , ϕl  |, ∀t ≥ 0, i = 1, . . . , N .

l=1,2,...,N

That is exactly what we have claimed. The goal of the present section is stated in the theorem below. Theorem 7.5 Let η > 0, depending on ω and sufficiently small, and let N ∈ N be sufficiently large. Then for each yo ∈ L 2 (0, L) with y0  < η, there exists a unique solution y to the random deterministic Eq. (7.85) belonging to the space Y , 

   1 Nt 2 Y := y ∈ Cb ((0, ∞), H (0, L)) : sup e (y(t) + t yx (t)) < ∞ . 1

t≥0

In particular, the stochastic Burgers equation ⎧ dY (t, x) = νYx x (t, x)dt + b(t, x)Y (t, x)Yx (t, x)dt + θ Y (t, x)dβ(t), ⎪ ⎪ ⎪ ⎪ ⎪ t > 0, x ∈ (0, L), ⎪ ⎪ ⎪ ⎛ ϕ1 (0) ⎞ ⎪ ⎞ ⎛ ⎪ ⎪

Y (t), ϕ1  ⎪ γk −μ1   ⎪ N ⎨ ϕ2 (0) ⎟  ⎜ Y (t), ϕ2  ⎟ ⎜ ⎜ ⎟ γk −μ2 ⎟ ⎟ Yx (t, 0) = , A⎜ ⎝ ............ ⎠ , ⎜ ⎪ ⎝ ........ ⎠ ⎪ k=1 ⎪ ϕ N (0) ⎪

Y (t), ϕ N  ⎪ ⎪ γk −μ N ⎪ N ⎪ ⎪ ⎪ ⎪ Y (t, L) = 0, t > 0, x ⎪ ⎪ ⎩ Y (0, x) = yo (x), x ∈ (0, L),

(7.91)

has a unique solution Y = Γ y that pathwise almost surely is exponentially decaying in the L 2 -norm.

156

7 Stabilization of Stochastic Equations

Proof The norm in Y is given as   1 |y|Y := sup e N t (y(t) + t 2 yx (t)) . t≥0

It is clear that for all y ∈ Y , we have e N t y(t) ≤ |y|Y and e N t yx (t) ≤ t − 2 |y|Y , ∀t > 0. 1

(7.92)

We set Br (0) := {y ∈ Y : |y|Y ≤ r } . Based on what we have discussed above, we may rewrite (7.85) in a mild formulation as  y(t, x) =

L

p(t, x, ξ )yo (ξ )dξ +

0

 t 0

L 0

p(t − s, x, ξ )b(s, ξ )Γ (s)y(s, ξ )yξ (s, ξ )dξ ds,

where p is defined in (7.87). Thus the existence of a solution y is equivalent to the fact that the map G : Y → Y , defined as  G y :=

L

p(t, x, ξ )y(0, ξ )dξ + F y,

0

where (F y) (t) :=

 t 0

L

p(t − s, x, ξ )b(s, ξ )Γ (s)y(s, ξ )yξ (s, ξ )dξ ds,

0

has a fixed point. In what follows, we aim to show that G is a contraction on Br (0) that maps the ball Br (0) into itself, for r > 0 properly chosen. Then via the contraction mappings theorem, we will deduce that G has a unique fixed point y ∈ Br (0) that is in fact the mild solution to the Eq. (7.85). Then one easily arrives at the conclusion claimed by the theorem. We will need to estimate the norm | · |Y of G y. So in particular, we will need to estimate the | · |Y -norm of F y, for y ∈ Y . We begin with the L 2 -norm of F y. We propose to use Parseval’s identity, so in order to do this, based on the kernel’s form (7.87), we conveniently rewrite the term F y as  F y(t) = 0

t

(F1 (y(s)) + F2 (y(s)) + F3 (y(s))) ds,

(7.93)

7.3 Stabilization of the Stochastic Burgers Equation

157

where F1 (y)(t, s, x) % N  N   := q ji (t − s)Γ (s) j=1

L

b(s, ξ )y(s, ξ )yξ (s, ξ )ϕi (ξ )dξ ϕ j (x),

0

i=1

F2 (y)(t, s, x)  ∞ 2  −μ j (t−s) e := Γ (s)

L

3 b(s, ξ )y(s, ξ )yξ (s, ξ )ϕ j (ξ )dξ ϕ j (x),

0

j=N +1

F3 (y)(t, s, x) % N  ∞   j := wi (t − s)Γ (s) j=N +1



i=1

L

 b(s, ξ )y(s, ξ )yξ (s, ξ )ϕi (ξ )dξ ϕ j (x).

0

(7.94) It follows via Parseval’s identity that F1 (y) ⎧ 2 ⎫ 21 % N  L N ⎨ ⎬  q ji (t − s)Γ (s) b(s, ξ )y(s, ξ )yξ (s, ξ )ϕi (ξ )dξ = ⎩ ⎭ 0 j=1

i=1

(using the uniform boundedness of eigenfunctions and (7.82))  L N  − 41 ≤C |q ji (t − s)|s |y(s, ξ )||yξ (s, ξ )|dξ 0

i, j=1

(involving relation (7.88) and Schwarz’s inequality) ≤ Ce−cN

2

(t−s) − 41

s

y(s)yξ (s)

−2N t (−cN 2 +2N + 14 )(t−s) − 41 (t−s) − 41 N s

e y(s)e N s yξ (s) 1 (by (7.92) and the fact that − cN 2 + 2N + < 0, N large) 4 1 1 1 ≤ Ce−N t (t − s)− 4 s − 4 s − 2 |y|2Y = Ce

e

e

s

= Ce−N t (t − s)− 4 s − 4 |y|2Y , ∀0 < s < t. 1

3

We continue with F2 (y) ⎧ ⎫1 32 ⎬ 2  L ∞ 2 ⎨  e−μ j (t−s) Γ (s) b(s, ξ )y(s, ξ )yξ (s, ξ )ϕ j (ξ )dξ = ⎩ ⎭ 0 j=N +1

(7.95)

158

7 Stabilization of Stochastic Equations

= e−2N t

⎧ ∞ 2 ⎨ 



L

e−(μ j −2N )(t−s) Γ (s)e2N s b(s, ξ )y(s, ξ )yξ (s, ξ )ϕ j (ξ )dξ ⎩ 0 j=N +1 ⎧⎡ ∞  L ⎨  # −(μ j −2N )(t−s) N s $ −N t ⎣ e ≤ Ce e y(s, ξ )ϕ j (ξ ) ⎩ 0

⎫1 32 ⎬ 2 ⎭

j=N +1

$ 62 1 21 # × Γ (s)b(s, ξ )e N s yξ (s, ξ ) dξ (by Schwarz’s inequality) ⎧ ∞  L ⎨  ≤ Ce−N t e−2(μ j −2N )(t−s) ϕ 2j (ξ )e2N s y 2 (s, ξ )dξ × ⎩ 0 j=N +1



L

2

0

(s, ξ )e2N s yξ2 (s, ξ )dξ

 21

Γ (s)b ⎧ ⎤ ⎡ ∞ ⎨ L  ⎣ = Ce−N t e−2(μ j −2N )(t−s) ϕ 2j (ξ )⎦ e2N s y 2 (s, ξ )dξ × ⎩ 0 2

j=N +1



L

Γ (s)b 2

0

2

(s, ξ )e2N s yξ2 (s, ξ )dξ

 21

(use inequality between the heat and the Gaussian kernel (7.30))  L  21  L −N t − 21 2N s 2 2 2 2N s 2 (t − s) e y (s, ξ )dξ Γ (s)b (s, ξ )e yξ (s, ξ )dξ ≤ Ce 0

0

(by (7.82)) ≤ Ce−N t (t − s)− 4 s − 4 e N s y(s)e N s yξ (s) (using (7.92)) 1

1

≤ Ce−N t (t − s)− 4 s − 4 |y|2Y , ∀0 < s < t. 1

3

(7.96)

Finally, we deal with F3 (y) ⎧ 2 ⎫ 21 % N  L ∞ ⎨  ⎬  j wi (t − s)Γ (s) b(s, ξ )y(s, ξ )yξ (s, ξ )ϕi (ξ )dξ = ⎩ ⎭ 0 j=N +1

i=1

(by (7.82), (7.89) and the uniform boundedness of the eigenfunctions) ⎛ ⎞ ∞  1 ⎠ e−cN 2 (t−s) s − 14 y(s)yξ (s) ≤C⎝ 2 μ − cN j j=N +1

7.3 Stabilization of the Stochastic Burgers Equation

159

(the series converge; see (7.67)–(7.68)) ≤ Ce−2N t e(−cN

2

+2N + 41 )(t−s) − 14 (t−s) − 14 N s

e y(s)e N s yξ (s) 1 (by (7.92) and the fact that − cN 2 + 2N + < 0 for N large enough) 4 1 3 ≤ Ce−N t (t − s)− 4 s − 4 |y|2Y , ∀0 < s < t. (7.97) e

s

We conclude that (7.95)–(7.97) imply that F (y)(t) ≤ Ce

−N t



t

s

− 43

(t − s)

− 41

0

ds|y|2Y

=e

−N t

CB

! 1 3 , |y|2Y , (7.98) 4 4

∀t ≥ 0, where B(x, y) is the classical beta function. By the exponential semigroup property, we have as well that    

L 0

  −N t p(t, x, ξ )yo (ξ )dξ   ≤ Ce yo .

(7.99)

We go on with the estimates in the H 1 -norm. Using the above notation, we have (F1 (y))x  ⎧ 2 ⎫ 21 % N  L N ⎨ ⎬  μj q ji (t − s)Γ (s) b(s, ξ )y(s, ξ )yξ (s, ξ )ϕ j (ξ )dξ = ⎩ ⎭ 0 j=1

i=1

(arguing as in (7.95)) ≤ Ce−cN

2

(t−s) − 41

s

= Ce−2N t e(−cN

2

y(s)yξ (s)

+2N + 34 )(t−s) − 43 (t−s) − 14 N s

e y(s)e N s yξ (s) 3 (by (7.92) and the fact that − cN 2 + 2N + < 0 for N large enough) 4 3 3 ≤ Ce−N t (t − s)− 4 s − 4 |y|2Y , ∀0 < s < t. (7.100) Next, e

s

(F2 (y))x  ⎧ ⎫1 2 32 ⎬ 2  L ∞ ⎨  μ j e−μ j (t−s) Γ (s) b(s, ξ )y(s, ξ )yξ (s, ξ )ϕ j (ξ )dξ = ⎩ ⎭ 0 j=N +1

= (t − s)− 2

1

160

7 Stabilization of Stochastic Equations

⎧ ⎫1 32 ⎬ 2  L ∞ 2 ⎨  1 1 (t − s) 2 μ j2 e−μ j (t−s) Γ (s) × b(s, ξ )y(s, ξ )yξ (s, ξ )ϕ j (ξ )dξ ⎩ ⎭ 0 j=N +1

(using the obvious inequality [(t − s)μ j ] 2 ≤ e 2 μ j (t−s) ) 1

1

≤ (t − s)− 2 ⎧ ⎫1 32 ⎬ 2  L ∞ 2 ⎨  1 e− 2 μ j (t−s) Γ (s) × b(s, ξ )y(s, ξ )yξ (s, ξ )ϕ j (ξ )dξ ⎩ ⎭ 0 1

j=N +1

(arguing as in (7.96)) ≤ C(t − s)− 2 e−N t (t − s)− 4 s − 4 |y|2Y 1

1

3

= Ce−N t (t − s)− 4 s − 4 |y|2Y , ∀0 < s < t. 3

3

(7.101)

Finally, (F3 (y))x  ⎧ 2 ⎫ 21 % N  L ∞ ⎨  ⎬  j μj wi (t − s)Γ (s) b(s, ξ )y(s, ξ )yξ (s, ξ )ϕi (ξ )dξ = ⎩ ⎭ 0 j=N +1

i=1

(by (7.90)) ≤ Ce−cN

2

(t−s)

" " " Γ (s)b(s, ·)y(s, ·)yξ (s, ·), ϕl (·) "

sup l=1,2,...,N

(with similar arguments as before) ≤ Ce−N t (t − s)− 4 s − 4 |y|2Y , ∀0 < s < t. 3

3

(7.102)

Therefore, (7.100)–(7.102) imply that 

(F (y)(t))x  ≤ Ce

t

−N t

(t − s)

0

− 43 − 34

s

ds|y|2Y

=e

−N t − 21

t

CB

! 1 1 , |y|2Y , 4 4 (7.103)

∀t > 0. Heading toward the end of the proof, we note that 

∞

Nt

e t 0

1 2

%

L 0

 21

!2

∂p (t, x, ξ ) ∂x

dξ

dt < ∞,

since the presence of the μ j in the infinite sum is controlled as in (7.101) by the 1 presence of t 2 . Consequently, via the semigroup property, we deduce that

7.3 Stabilization of the Stochastic Burgers Equation

   

0

L

  ∂p −N t − 21 (t, x, ξ )dξ   ≤ Ce t yo . ∂x

161

(7.104)

Now, gathering together the relations (7.97), (7.99), (7.103), and (7.104), we arrive at the fact that there exists a constant C1 > 0 such that |G y|Y ≤ C1 (yo  + |y|2Y ),

(7.105)

for all y ∈ Y . It is easily seen that arguments similar to those above lead as well to |G y − G y|Y ≤ C2 (|y|Y + |y|Y )|y − y|Y , ∀y, y ∈ Y ,

(7.106)

for some constant C2 > 0. Recall that yo  < η. It then follows that if η is small enough that  η < min

 1 1 , , 4C12 4C1 C2

then taking r = 2C1 η, we get from (7.106) that G is a contraction and by (7.105) that G maps the ball Br (0) into itself, as claimed. Note that C1 , C2 depend on ω, since in the above, ω-estimates for Γ were used (CΓ is ω-dependent). Thus η should depend on ω too. This means that in fact, yo must depend on ω.  Remark 7.1 Let us return to Eq. (7.79). If one assumes that β depends on the space variable as well, then in trying to apply the above approach, one has to estimate terms like βx . The law of the iterated logarithm should work again. In any case, this time, we will no longer have |y|2Y on the right-hand side, because of the terms like βx y, βx yx . This implies that in applying the fixed-point argument, at some point one should find an r > 0 sufficiently small that for some constants c1 , c2 , c1 r + c2 r 2 < r. This is possible if and only if c1 < 1, but no one can guarantee this. In the above case, we had c2 r 2 < r, and this is possible, provided r is sufficiently close to zero. Hence, we believe that the above argument surely fails to work for the case of a space-dependent Brownian motion β, unless some additional hypothesis on the coefficients of the equation, such as small enough, are imposed.

162

7 Stabilization of Stochastic Equations

7.4 Stabilization by Discrete-Time Feedback Control In the previous sections we discussed boundary stabilization by continuous-time (regular) actuators, for different types of stochastic equations. Such a continuous-time feedback control requires continuous observation of the state y(t), for all time t ≥ 0. However, it is more realistic and costs less in practice if the state is observed only at discrete times, say 0, τ, 2τ, . . . , where τ > 0 is the duration between two consecutive observations. Accordingly, the feedback control should be designed based on these7 discrete-time observations, namely the feedback control should be of the form 6 7t 6 t τ ), t), where is the integer part of τt . Thus, the idea of this section is u(y( ˜ τ τ to design a discrete-time proportional-type feedback, plug it into the equations, and show that it still ensures the stability of the system. We will use as a guide the work of Mao [92], which treats the case of stochastic differential equations. The equation under study reads as ⎧ ⎨ d y˜ (t) = y˜ x x (t)dt + λσ ( y˜ (t))dβ, t > 0, x ∈ (0, L), ˜ y˜ x (t, L) = 0, t ≥ 0, y˜ x (t, 0) = u(t), ⎩ y˜ (0) = yo .

(7.107)

Recall  we have  denoted by A the Neumann–Laplace operator, see (7.36),  that and by μ j j and ϕ j j its system of eigenvalues and system of eigenfunctions, respectively. Let N ∈ N be sufficiently large that Theorem 7.4 holds, and assume that this time, σ is a Lipschitz function of y˜ that depends only on the first N modes of y˜ , namely y˜ , ϕ1  , . . . , y˜ , ϕ N . Using the notation in (7.37)–(7.42), we introduce the feedback form u˜ as

where

u( ˜ y˜ ) := u˜ 1 ( y˜ ) + u˜ 2 ( y˜ ) + · · · + u˜ N ( y˜ ),

(7.108)

 ⎞ ⎛ ϕ1 (0) ⎞ ⎛  #7 t 6 $ y˜ τ τ , ϕ1 γk −μ1    #7 6 $ ϕ2 (0) ⎟ ⎜ y˜ t τ , ϕ2 ⎟ ⎜ ⎜ ⎟ γ −μ τ ⎟ k 2 ⎟ u˜ k ( y˜ ) := A ⎜ . ⎝ .................... ⎠ , ⎜ ⎝ ........ ⎠   #7 t 6 $ ϕ N (0) y˜ τ τ , ϕ N

(7.109)



γk −μ N

N

We observe that the feedback control u˜ is designed based on the discrete-time state observations y˜ (0), y˜ (τ ), y˜ (2τ ), . . .. Next, we lift the boundary conditions into Eq. (7.107) by arguing similarly as in (2.27)–(2.29), obtaining thereby the internal control-type problem

7.4 Stabilization by Discrete-Time Feedback Control

4 d y˜ (t) = −A y˜ (t) + +

⎧ ⎨ ⎩

N 

u˜ i

i=1

−2

N 

8 μ j u˜ i

i, j=1

163

5 2 3 !! t y˜ τ (A + γi )Dγi dt τ ⎫ 2 3 !! 9 ⎬ t y˜ τ )Dγi , ϕ j ϕ j dt ⎭ τ

(7.110)

+ λσ ( y˜ (t))dβ; y˜ (0) = yo . We note that Eq. (7.110) is in fact a stochastic PDE with delays, with a bounded variable delay. Indeed, if we define the bounded variable ζ : [0, ∞) → [0, τ ] by ζ (t) := t − kτ for kτ ≤ t < (k + 1)τ, for k ∈ N, then Eq. (7.110) can be equivalently rewritten as 4 d y˜ (t) = −A y˜ (t) +

N 

5 u˜ i ( y˜ (t − ζ (t))) (A + γi )Dγi

dt

i=1

⎫ ⎧ N ⎨    ⎬ μ j u˜ i ( y˜ (t − ζ (t))) Dγi , ϕ j ϕ j dt + −2 ⎭ ⎩

(7.111)

i, j=1

+ λσ ( y˜ (t))dβ; y˜ (0) = yo . Hence, the classical existence theory for delay SPDEs can be applied in order to ensure that (7.111) (and implicitly (7.110)) is well posed. To address the mean-square exponential stability of the controlled Eq. (7.110), we will relate it to its continuous-time controlled version (also refereed to as the auxiliary problem) 4 dy(t) = −Ay(t) +

N 

u i (y(t))(A + γi )Dγi

i=1

−2

N 

  μ j u i (y(s))Dγi , ϕ j ϕ j

i, j=1

⎫ ⎬ ⎭

dt + λσ (y(t))dβ; y(0) = yo ,

(7.112) which is mean exponential stable by virtue of Theorem 7.4. More precisely, we have obtained that − ∞ < lim sup log E|y(t, x)|2 < −ρ, ∀x ∈ (0, L).

(7.113)

t→∞

We aim to prove a relation similar to (7.113) for the solution y˜ to (7.110). To this end, we will compare y˜ with y, and show that they are close enough (in a proper sense),

164

7 Stabilization of Stochastic Equations

provided that τ is sufficiently small (namely, we make state observations frequently enough). As in (7.54) (see also (7.13) and (2.38)), we decompose both (7.110) and (7.112) into an eigenbasis, and get that the first N modes satisfy 4

2 3 !  2 3 !5 N t t τ + τ d Y˜ = −ΛY˜ (t) + ΛY˜ Bk AY˜ dt τ τ k=1   + λσ˜ Y˜ (t) dβ(t)

(7.114)

and 4 dY = −ΛY (t) + ΛY (t) +

N 

5 Bk AY (t) dt + λσ˜ (Y (t)) dβ(t),

(7.115)

k=1

respectively. Here ⎞ ⎛ ⎞

y, ϕ1 

y˜ , ϕ1  ⎜ ⎟ ⎜ y˜ , ϕ2  ⎟ ⎟ , Y := ⎜ y, ϕ2  ⎟ Y˜ := ⎜ ⎝ ....... ⎠ ⎝ ......... ⎠

y˜ , ϕ N 

y, ϕ N  ⎛

and Λ := diag(μ1 μ2 . . . μ N ). Finally, σ˜ is a function depending only on the first N modes, since σ was assumed to be like that, ⎞ ⎛

σ (Y ), ϕ1  ⎜ σ (Y ), ϕ2  ⎟ ⎟ σ˜ := ⎜ ⎝ .............. ⎠ .

σ (Y ), ϕ N  Note that with the two Eqs. (7.114) and (7.115) we are placed in the context of finite-dimensional stochastic differential equations from [92]. And since the sublinear assumptions from [92] are satisfied for the present case (see [92, Assumption 2.1], because here we are dealing with matrices and σ is Lipschitz), we may deduce similar results to those in [92, Lemma 3.2, Theorem 3.1]. More precisely, one may show that "2 " " " E "Y˜ (t) − Y (t)" ≤ C1 e−C2 t E|Yo |2 , ∀t ≥ 0, (7.116) for some positive constants C1 and C2 . And taking advantage of the exponential decay (7.113), we finally get that

7.4 Stabilization by Discrete-Time Feedback Control

165

"2 " " " E "Y˜ (t)" ≤ Ce−ρt E|Yo |2 , ∀t ≥ 0,

(7.117)

for some constants C, ρ > 0. Now let us scalar multiply (7.111) by ϕi , i > N . We get ⎫ ⎧ 2 3 !! N ⎨       ⎬ t (A + γk )Dγk , ϕi dt + λ σ Y˜ (t) , ϕi dβ(t), τ d y˜i = −μi y˜i + u˜ k Y˜ ⎭ ⎩ τ k=1

where obviously, y˜i = y˜ , ϕi . Of course, this will be compared with its continuous version, obtained from (7.112), 4 dyi = −μi yi +

N 



u k (Y (t)) (A + γk )Dγk , ϕi

5



dt + λ σ (Y (t)) , ϕi  dβ(t).

k=1

Similarly as in (7.117), one may deduce with arguments from [92, Lemma 3.2, Theorem 3.1] that E| y˜i (t)|2 ≤ Ce−ρt E| y˜i (0)|2 , ∀t ≥ 0, for all i = N + 1, N + 2, . . .. We conclude that by virtue of the above relation and (7.117), we have that E y˜ (t)2 ≤ Ce−ρt Eyo 2 , t ≥ 0.

(7.118)

7.5 Comments The problem of boundary stabilization of stochastic parabolic-type equations has been discussed in many papers; see, for instance, [70], [10, Sect. 2.4.1], and the references therein. In all those works the equation always contains noise as a forcing term, which makes the problem easier, since the presence of enough noise guaranties the stability of the system. However, in [52] the limit is considered as well as the more difficult situation of a noise acting only at the boundary, corresponding to a realistic situation in which the control itself is perturbed by a noise dβ. In the first section of this chapter, we assumed a fading-type noise, namely e−δt dβ, for some δ > 0 (the noise perturbation is not permanently of the same intensity but vanishes exponentially fast). The exponential decay of the noise requirement is mandatory, since we are studying here the problem of stabilization. In [52] is addressed the optimal control problem associated with Eq. (7.1). Those results were improved in [63], while in [129] the authors considered in addition some delays in Eq. (7.1). The results presented in this section were published in Munteanu [104]. Of course, another interesting case is that in which a Dirichlet boundary controller is perturbed by noise. The main difficulties that appear in that case are related to the

166

7 Stabilization of Stochastic Equations

fact that the solution is no longer L 2 -valued. More precisely, the solution lies in a negative Sobolev space H α , α < − 14 . The reason is that the smoothing properties of the heat equation are not strong enough to regularize a rough term such as a white noise. However, one may suggest reconsidering the problem in the new framework proposed in [53], namely in weighted L 2 -spaces. The difficulty is that in order to apply the reduction method, one should consider the eigenbasis in the weighted L 2 space of the weighted Laplacian. Another idea is to consider the solution in the space of distributions as in [37]. Besides this, even in the case of Neumann boundary conditions, it would be interesting to consider a space dimension higher than one. The main difficulty in that case is that the solution D of (7.10) should satisfy noise boundary conditions of the type ∂ D(x) = e−δt dβ(t, x), x ∈ Γ, ∂n where Γ is a part of the boundary of the domain in which the equation is considered, while n is its outward unit normal. To define such a D, one may rely on the existing results in [112], after imposing some additional conditions. Then the control design algorithm may be applied. The problem of stabilization of the stochastic versions of the deterministic models has arisen naturally in the scientific community. First, the finite-dimensional case was considered, and we mention Mao and his coworkers for notable results in this direction; see the book [91], for example. There, it is mainly the Lyapunov stability technique that is used, which consists in finding proper Lyapunov functions for the equation under discussion. Then these ideas were reconsidered in the infinitedimensional case, and we refer to the joint work of Caraballo et al. [42], which treats a similar problem to the one we presented above. Let us give some details on how the Lyapunov functions are used. Let A denote the Dirichlet–Laplace operator on (0, L), and σ = σ (y) a Lipschitz function such that σ (0) = 0, and consider the problem 

dy(t) = Ay(t)dt + λσ (y(t))dβ, t > 0, y(0) = yo .

(7.119)

Assume that V (t, y) : R+ × L 2 (0, L) → R+ is a C 1,2 -positive functional such that for all y ∈ H01 (0, L), t ∈ R+ , Vy (t, y) ∈ H01 (0, L). Define the operators L and Q (which are, in fact, the deterministic part and the stochastic part of Itô’s formula applied to V (t, y) in (7.119), respectively): for y ∈ H01 (0, L) and t ∈ R+ ,   1    L V (t, y) = Vt (t, y) + Vy (t, y), Ay + Vyy (t, y)λσ (y), λσ (y) 2 and

  QV (t, y) = (Vy (t, y))2 , λσ (y) .

7.5 Comments

167

Assume that the solution to (7.119) satisfies |y(t)| = 0 for all t ≥ 0 a.s., provided |yo | = 0 a.s., and that there exists a function V (t, y) ∈ C 1 (R+ , R+ ) × C 2 (L 2 (0, L); R+ ), and ψ1 (t), ψ2 (t) ≥ 0 are two functions for which there exist constants p > 0, γ ≥ 0, and θ ∈ R such that (1) y p ≤ V (t, y), ∀y ∈ H 1 (0, L); (2) L V (t, y) ≤ ψ1 (t)V (t, y), ∀y ∈ H01 (0, L), ∀t ∈ R+ ; (3) QV (t, y) ≥ ψ2 (t)V 2 (t, y), ∀x: ∈ H01 (0, L), ∀t ∈ R+ ; : (4) lim sup t→∞

t 0

ψ1 (s)ds t

t 0

≤ θ, lim inf t→∞

ψ2 (s)ds t

≥ 2γ .

Then the strong solution of equation (7.119) satisfies lim sup t→∞

γ −θ log |y(t)| ≤− a.s., t p

since V is a Lyapunov function for the system. Then the problem of internal stabilization by noise was proposed in the same work [42]. More precisely, the problem 

dy(t) = Ay(t)dt + λσ (y(t))dβ1 + h(t, y(t)dβ2 , t > 0, y(0) = yo

(7.120)

was considered. Here β1 and β2 are two independent Brownian motions. If h is a Lipschitz function with h(t, 0) = 0 for which there exist λ(·), ρ(·) t ≥ 0 and ν0 , ρ0 ∈ R such that h(t, y)2 ≤ λ(t)y2 , t ≥ 0, y ∈ L 2 (0, L),

y, h(t, y) ≥ ρ(t)y4 , t ≥ 0, y ∈ L 2 (0, L), where

1 lim sup t→∞ t



t

0

1 λ(s)ds ≤ λ0 and lim inf t→∞ t



t

ρ(s)ds ≥ ρ0 ,

0

then under the assumption that 4

λσ (y), y2 ≥ ρ(t)y ˜ , ∀y ∈ L 2 (0, L),

where ρ(·) ˜ is a nonnegative continuous function such that lim inf t→∞

1 t



T

ρ(s)ds ˜ ≥ ρ˜0 , ρ˜0 ∈ R+ ,

0

the solution of (7.120) satisfies lim sup t→∞

1 log y(t)2 ≤ −(2(ρ0 + ρ˜0 ) − λ0 ), P − a.s. t

168

7 Stabilization of Stochastic Equations

A more direct and simple control is proposed in [14, Sect. 5.5], where the internal stabilization of the Navier–Stokes equations driven by linear multiplicative noise is treated. Provided that the first eigenvalue of the Oseen operator is large enough, the feedback u = −η1O 0 y once inserted into the equations ensures the stability of the closed-loop system, P−a.s.. Proportional-type feedback laws (both internal and from the boundary) were proposed, in the context of noise stabilization of deterministic equations, by Barbu; see the book [10, Chap. 4]. Regarding the boundary case, in [10] the equation under consideration is the Navier–Stokes equation, but the ideas can be easily reformulated for general parabolic-like equations. The control law involves a family of independent  N Brownian motions β j j=1 and is given as u=η

N 

  μ j y, ϕ j Φ j dβ j .

j=1

The boundary conditions are lifted into the equations, and then the system is decomposed into its unstable and stable parts. The solution of the unstable part is given explicitly and it is shown that if it is stable, then via a Lyapunov function and Itô’s formula it is shown that the stable part is stable as well. This allows us to conclude that the corresponding solution of the closed-loop equation satisfies 

∞

e2γ t y(t)2 dt < ∞ P − a.s.

0

The proof of the stability of the system is almost identical to the proof of the main result of Sect. 7.1, except that the solution of the unstable part is given explicitly. This is possible due to the imposed hypothesis of linear independence of the traces of the normal derivatives of the dual eigenfunctions on the boundary (as described in the Comments of Chap. 2). Of course, following the ideas in Chap. 2, one may define another stabilizing noise control, where this kind of hypothesis is dropped. The results presented in the second section of this chapter were published in the author’s work [106]. On the other hand, concerning the internal stabilization by noise of deterministic equations, there are substantially more results. We refer first to the early work of Arnold [5], which provides an example of an unstable system stabilized by a random parameter noise, followed by the work on linear systems Arnold et al. [6]. Other stabilization results are provided via Lyapunov exponents in Kwiecinska [78, 79]. Let us briefly describe the ideas behind those works. The equation d X (t) = AX (t) dt is considered, where A generates a C0 -semigroup in a Hilbert space. It is denoted by

7.5 Comments

169

λdet := lim sup t→∞

1 log X det (t), t

the Lyapunov exponent of the deterministic equation. Here X det is the solution of the deterministic equation. Then the equation is perturbed by d X = AX dt + σ

N 

Bk X dβk ,

k=1

where the Bk are linear continuous operators satisfying some diagonalizable and commutation properties. Similarly, a Lyapunov coefficient of the stochastic equation is introduced: 1 λst := lim sup log X st (t), t→∞ t where X st is the solution of the stochastic equation. The author proves that the stochastic Lyapunov exponents turn out to be smaller, almost surely, than their deterministic counterparts. This means that the deterministic system is made more stable by adding a term with white noise. Moreover, there exists σ0 such that for σ ≥ σ0 , all the stochastic Lyapunov exponents are strictly smaller than zero with probability one. For a collection of more results on this subject, one may see [41]. Regarding the third section of this chapter, Burgers’s equation is often referred to as a one-dimensional “cartoon” of the Navier–Stokes equation because it does not exhibit turbulence. In contrast, it turns out that its stochastic version, (7.73), models turbulence; for details, one can see [47, 111]. In the literature there are plenty of results concerning the stabilization of the deterministic Burgers equation; for example, we refer to [74], which provides a global stabilization result, with some consequences on the stabilizability of the stochastic version. The results of Sect. 7.3 are new. The ideas are based on the mild formulation, described above, plus a fixed-point argument. The idea to use fixed-point arguments in order to prove the stability of deterministic or stochastic equations has been previously used in papers such as [89]. Finally, the result asserting that Eq. (7.107) (see also (7.110)) is stabilizable by a proportional-type feedback law involving only time-discrete measurements of the state, see (7.108) and (7.109), can be viewed as a completion of the result in (2.68), where via some numerical simulations, we observed that there is no need for fullstate knowledge, but only on a part of the domain. So we may conclude that the proportional-type feedback law designed in Chap. 2 and used through out this book to stabilize different types of deterministic or stochastic PDEs can be improved in order to involve only time-discrete measurements of the state on only a part of the domain where the phenomena (modeled by the PDE) are evolving. From the practical point of view and the costs, this is a very important feature. The results in Sect. 7.4 are new.

Chapter 8

Stabilization of Unsteady States

In this chapter, we address the problem of stabilization of unsteady-state trajectories of time-dependent systems. In this case, the linear operator obtained from the linearization of the equation around the trajectory is time-dependent, so its spectrum is time-dependent as well. This means that the spectral method leaves out this case. We will follow the approach from Sect. 7.2, Chap. 7. Namely, we will write the solution of the nonlinear equation in a mild formulation via a kernel and prove its stability.

8.1 Presentation of the Problem The subject of this chapter is the following boundary-controlled parabolic-type equation on (0, L), L > 0: ⎧ ∂t y(t, x) = yx x (t, x) + a(t, x)yx (t, x) + b(t, x)y(t, x) + σ (t, x, y(t, x)), ⎪ ⎪ ⎪ ⎨ 0 < x < L , t > 0, ⎪ yx (t, 0) = u(t), yx (t, L) = 0, t ≥ 0, ⎪ ⎪ ⎩ y(0, x) = yo (x) for x ∈ [0, L]. (8.1) Now let us give details. Functions a, b : R+ × R → R are uniformly bounded with a of class C 1 in both t and x, in the sense that we may find some positive constant c1 such that ess supt>0 ess supx∈(0,L) (|∂t a(t, x)| + |ax (t, x)| + |a(t, x)| + |b(t, x)|) ≤ c1 . (8.2) In addition, we assume that a(t, 0) = a(t, L) = 0, ∀t ≥ 0. © Springer Nature Switzerland AG 2019 I. Munteanu, Boundary Stabilization of Parabolic Equations, Progress in Nonlinear Differential Equations and Their Applications 93, https://doi.org/10.1007/978-3-030-11099-4_8

171

172

8 Stabilization of Unsteady States

Further, σ is a uniformly globally Lipschitz nonlinear function, i.e., there exists L σ > 0 such that |σ (t, x, y) − σ (t, x, y¯ )| ≤ L σ |y − y¯ |, ∀t ≥ 0, x ∈ [0, L], ∀y, y¯ ∈ R,

(8.3)

and σ (t, x, 0) = 0. Now let yˆ be some trajectory of the uncontrolled (8.1). More precisely, yˆ = yˆ (t, x) satisfies ⎧ ∂t yˆ (t, x) = yˆ x x (t, x) + a(t, x) yˆ x (t, x) + b(t, x) yˆ (t, x) + σ (t, x, yˆ (t, x)), ⎪ ⎪ ⎪ ⎨ 0 < x < L , t > 0, ⎪ yˆ x (t, L) = 0, t ≥ 0, ⎪ ⎪ ⎩ yˆ (0, x) = yˆo (x) for x ∈ [0, L]. (8.4) Then define the fluctuation variable z := y − yˆ , which by virtue of (8.1) and (8.4) satisfies the equation ⎧ ∂t z(t, x) = z x x (t, x) + a(t, x)z x (t, x) + b(t, x)z(t, x) ⎪ ⎪ ⎪ ⎨ + σ (t, x, z(t, x) + yˆ (t, x)) − σ (t, x, yˆ (t, x)), for 0 < x < L , t > 0, ⎪ z x (t, 0) = U (t) := u(t) − yˆ x (t, 0), z x (t, L) = 0, t ≥ 0, ⎪ ⎪ ⎩ z(0, x) = z o (x) := yo (x) − yˆo (x) for x ∈ [0, L]. (8.5)

8.2 The Stabilization Result and Applications We will use all the notation from Chap. 7, Sect. 7.2, except that this time, we take the γk to be γk := N α +

k , k = 1, 2, . . . , N , N

(8.6)

with α > 2. We emphasize that γk , γk − μ1 , γk − μ N are of order O(N α ),

(8.7)

for all k = 1, 2, . . . , N . The given a priori feedback law v is the same as in (7.41)–(7.42); namely, for w ∈ L 2 (0, L), we set v(w) := v1 (w) + · · · + v N (w),

(8.8)

8.2 The Stabilization Result and Applications

where

173

⎞ ⎛ ϕ1 (0) ⎞ γk −μ1 w, ϕ1 

ϕ2 (0) ⎟ ⎜ w, ϕ2  ⎟ ⎜ ⎟ ⎜ ⎟ γk −μ2 ⎟ vk (w) := A ⎜ . ⎝ ........ ⎠ , ⎜ ⎝ ........ ⎠ ϕ N (0) w, ϕ N  ⎛



γk −μ N

(8.9)

N

The main result of stabilization concerning the Eq. (8.1) is stated and proved below. Theorem 8.1 Let ρ > 0 be arbitrary but fixed. For N ∈ N large enough, the solution y to the equation ⎧ ∂t y(t, x) = yx x (t, x) + a(t, x)yx (t, x) + b(t, x)y(t, x) + σ (t, x, y(t, x)), ⎪ ⎪ ⎪ ⎪ ⎪ for 0 < x < L and t > 0, ⎪ ⎨  1 x a(t,ξ )dξ (y(t) − yˆ (t))) + yˆ x (t, 0), t > 0 yx (t, 0) = v(e 2 0 ⎪ ⎪ ⎪ ⎪ yx (t, L) = 0 for t > 0, ⎪ ⎪ ⎩ y(0, x) = yo (x) for x ∈ [0, L], (8.10) satisfies lim sup eρt ess sup |y(t, x) − yˆ (t, x)| < ∞. t→∞

(8.11)

x∈(0,L)

Here the feedback v is defined by (8.8). Proof As mentioned above, stabilization of (8.10) is equivalent to the null stabilization of the translated Eq. (8.5). That is why in the following, we will consider only the latter situation. Carrying out the transformation 1

w(t, x) := e 2

x 0

a(t,ξ )dξ

z(t, x), t > 0, x ∈ [0, L],

in (8.5), we get that w satisfies ⎧ ∂ w(t, x) = wx x (t, x) + d(t, x)w(t, x) + σ˜ (w(t)) for 0 < x < L , t > 0, ⎪ ⎨ t wx (t, 0) = v(w(t)), wx (t, L) = 0, t ≥ 0, ⎪  1 x ⎩ w(0, x) = wo (x) := e 2 0 a(0,ξ )dξ (yo (x) − yˆo (x)) for x ∈ [0, L], (8.12) where we have used that a(t, 0) = a(t, L) = 0, ∀t > 0. Here 1 d(t, x) := 2

 0

x

1 1 ∂t a(t, ξ )dξ − ax (t, x) − a 2 (t, x) + b(t, x), 2 4

(8.13)

174

and

8 Stabilization of Unsteady States x

1

σ˜ (w(t)) := e 2

−e

0 1 2

a(t,ξ )dξ

x 0

σ (t, x, e− 2

a(t,ξ )dξ

1

x 0

a(t,ξ )dξ

w(t, x) + yˆ (t, x))

σ (t, x, yˆ (t, x)).

(8.14)

It is easily seen from (8.2) and (8.3) that we may find some constant c2 > 0 such that ess sup ess sup |d(t, x)| ≤ c2 t>0

x∈(0,L)

|σ˜ (w(t, x))| ≤ c2 L σ |w(t, x)|, ∀t > 0, x ∈ (0, L).

(8.15)

Now we lift the boundary conditions into the Eq. (8.12), obtaining thereby an internal control-type problem. As in (7.49), we find that (8.12) is equivalent to ∂t w(t) = − Aw(t) +

N 

N 

˜ + γi )Dγi − 2 vi (w(t))(A

i=1

μ j vi (w(t))Dγi , ϕ j ϕ j

i, j=1

+ Γ (w(t)); w(0) = wo , (8.16) where Γ (w) := dw + σ˜ (w). It is clear that one can find some constant c3 such that |Γ (w)| ≤ c3 (1 + L σ )|w|.

(8.17)

Similarly as in Lemma 7.1, we may prove the following result. Lemma 8.1 The solution z of ∂t z(t) = − Az(t) +

N 

˜ + γi )Dγi vi (z(t))(A

i=1

−2

N 

(8.18)

μ j vi (z(s))Dγi , ϕ j ϕ j ; z(0) = z o ,

i, j=1

can be written in a mild formulation as  z(t, x) =

L

p(t, x, ξ )z o (ξ )dξ.

0

Moreover, we have that 

∞ 0

 0

L

e N t | p(t, x, ξ )|dξ dt ≤ C

1 , ∀x ∈ (0, L), Nθ

(8.19)

8.2 The Stabilization Result and Applications

175

for some positive θ and C > 0, independent of N . Proof of Theorem 8.1 (continued). Now we are ready to conclude the proof of Theorem 8.1. Let us define

w 1,N := ess sup ess sup e N t |w(t, x)|. t>0

x∈(0,L)

Then by Lemma 8.1, we write the solution of (8.16) in a mild formulation via the kernel p, i.e.,  w(t, x) =

L

p(t, x, ξ )wo (ξ )dξ +

 t

0

Then we have  |w(t, x)| ≤

0

L

| p(t, x, ξ )||wo (ξ )|dξ +

0

L

p(t − s, x, ξ )Γ (w(s))ds.

(8.20)

0

 t 0

L

| p(t − s, x, ξ )||Γ (w(s))|ds

0

(using (8.19) in the first term and (8.17) in the second one) ≤ Ce−N t ess sup |wo (x)|  t

x∈(0,L) L

e N (t−s) | p(t − s, x, ξ )|e−N (t−s) c3 (1 + L σ )|w(s)|ds  ∞ L −N t −N t e N t | p(t, x, ξ )|dξ dt ≤ Ce ess sup |wo (x)| + c3 (1 + L σ ) w 1,N e

+

0

0

x∈(0,L)

0

0

(again using (8.19) in the second term) ≤ Ce−N t ess sup |wo (x)| + Ce−N t c3 (1 + L σ ) w 1,N x∈(0,L)

1 , ∀t > 0, x ∈ (0, L). Nθ

The above relation implies that

w 1,N ≤ C ess sup |wo (x)| + Cc3 (1 + L σ ) w 1,N x∈(0,L)

1 . Nθ

Therefore, if we choose N large enough that Cc3 (1 + L σ ) N1θ < 1 and N > ρ, we obtain that (8.21)

w 1,ρ ≤ w 1,N < ∞. Keeping in mind that we have defined y(t, x) − yˆ (t, x) = e− 2

1

x 0

a(t,ξ )dξ

we see that relation (8.21) yields the desired result.

w(t, x), 

176

8 Stabilization of Unsteady States

Remark 8.1 In comparison with the proof of Theorem 7.4, here we did not estimate the L 2 -norm of the solution (using Parseval’s identity), but instead we estimated the L ∞ -norm. That is why in comparison to Sect. 7.2, here we had to change the values of γk s, k = 1, 2, . . . , N , in order to obtain relation (8.19). This suggests that the various ways of choosing the γk also play an important role, allowing one to solve stabilization problems for different equations in different frameworks.

8.2.1 Observer Design Let us reconsider Eq. (8.1), but this time with σ ≡ 0, i.e., ⎧ ⎨ ∂t y(t, x) = yx x (t, x) + a(t, x)yx (t, x) + b(t, x)y(t, x), 0 < x < L , t > 0, yx (t, 0) = u(t), yx (t, L) = 0, t ≥ 0, ⎩ y(0, x) = yo (x), 0 ≤ x ≤ L . (8.22) Again, let yˆ stand for a particular time-dependent solution of the uncontrolled (8.22) (that is, yˆ satisfies (8.4) with σ ≡ 0). Then as was done in (8.5), we write the equation satisfied by the fluctuation variable Y := y − yˆ . This, via the transformation 1

w := e 2

x 0

a(t,ξ )dξ

Y (t, x),

may be equivalently rewritten as (see also (8.12)) ⎧ ⎨ ∂t w(t, x) = wx x (t, x) + d(t, x)w(t, x), 0 < x < L , t > 0, wx (t, 0) = v(w(t)), wx (t, L) = 0, t ≥ 0, ⎩ w(0, x) = w0 (x), 0 ≤ x ≤ L .

(8.23)

Here (see (8.9) and (8.8)) v(w) := v1 (w) + · · · + v N (w), where

(8.24)

⎞ ⎛ ϕ1 (0) ⎞ w, ϕ1  γk −μ1

ϕ2 (0) ⎟ ⎜ w, ϕ2  ⎟ ⎜ ⎜ ⎟ γk −μ2 ⎟ ⎟ . vk (w) := A ⎜ ⎝ ........ ⎠ , ⎜ ⎝ ......... ⎠ ϕ (0) N w, ϕ N  

⎛

γk −μ N

(8.25)

N

In the proof of Theorem 8.1, we have shown that the feedback v given by (8.24) ensures the exponential stability of (8.23). When trying to apply this theoretical result in practice, things may become difficult to implement, since the feedback law (8.24) requires full state knowledge, while in practice, only measurements at the end x = L are available.

8.2 The Stabilization Result and Applications

177

Based on the ideas from Krstic [75], we propose the following observer for system (8.23)–(8.25): ⎧ ∂t w(t, ˆ x) = wˆ x x (t, x) + d(t, x)w(t, ˆ x) + K 1 (t, x)[w(t, L) − w(t, ˆ L)], ⎪ ⎪ ⎨ 0 < x < L , t > 0, ˆ wˆ x (t, L) = K 10 (t)[w(t, L) − w(t, ˆ L)], t ≥ 0, wˆ x (t, 0) = v(w(t)), ⎪ ⎪ ⎩ w(0, ˆ x) = w0 (x), 0 ≤ x ≤ L . (8.26) Here K 1 , K 10 are output injection functions. Observer (8.26) is in the standard form of a copy of the system plus injection of the output estimation error. This form is usually used for the finite-dimensional case, in which observers of the form d ˆ X = A Xˆ + Bu + L(Y − C Xˆ ) dt are constructed for plants d X = AX + Bu, Y = C X. dt This standard form allows us to pursue duality between the observer and the controller design, that is, to find the observer gain function using the solution to the stabilization problem we studied in the previous section. This can be put in connection with the way duality is used to find the gains of a Luenberger observer based on the pole placement control algorithm, or the way duality is used to construct Kalman filters based on the LQR design. In practice, things go as follows: once the estimates of measurements in x = L are available, one inserts them into the observer equation (8.26) and numerically computes the solution wˆ and, at the same time, v(w). ˆ With this v(w) ˆ plugged into the plant equation (8.23), instead of v(w), one expects that the corresponding solution of (8.23) will go exponentially fast to zero. Keeping in mind that in (8.23), v(w) is replaced by v(w), ˆ we deduce that the observer error w(t, ˜ x) := w(t, x) − w(t, ˆ x) satisfies the following PDE: 

˜ x) = w˜ x x (t, x) + d(t, x)w(t, ˜ x) − K 1 (t, x)w(t, ˜ L), 0 < x < L , t > 0, ∂t w(t, ˜ , t), t ≥ 0. w˜ x (t, 0) = 0, w˜ x (t, L) = −K 10 (t)w(L (8.27)

Now the functions K 1 (t, x) and K 10 (t) must be determined such that (8.27) is exponentially stable to zero. To this end, we look for a backstepping-like coordinate transformation  L

w(t, ˜ x) := z˜ (t, x) − x

K (t, x, ξ )˜z (t, ξ )dξ

178

8 Stabilization of Unsteady States

that transforms (8.27) into the exponentially stable (for c > 0) system 

∂t z˜ (t, x) = z˜ x x (t, x) − c˜z (t, x), x ∈ (0, L), t > 0, z˜ x (0) = z˜ x (L) = 0.

(8.28)

The free parameter c can be used to set the desired observer convergence speed. Straightforward computations involving (8.28) give 

L

˜ x) = ∂t z˜ (t, x) − ∂t w(t, 



x

∂t K (t, x, ξ )˜z (t, ξ )dξ

x



L

−



L

K (t, x, ξ )˜z ξ ξ (t, ξ )dξ + c

x

K (t, x, ξ )∂t z˜ (t, ξ )dξ

x L

= ∂t z˜ (t, x) −

L

∂t K (t, x, ξ )˜z (t, ξ )dξ −

K (t, x, ξ )˜z (t, ξ )dξ

x



L

= ∂t z˜ (t, x) −

∂t K (t, x, ξ )˜z (t, ξ )dξ + K (t, x, x)˜z x (t, x)

x



L

− K ξ (t, x, x)˜z (t, x) + K ξ (t, x, L)˜z (t, L) −

K ξ ξ (t, x, ξ )˜z (t, ξ )dξ

x



L

+c

K (t, x, ξ )˜z (t, ξ )dξ.

x

(8.29)

Likewise, we have w˜ x x (t, x) = z˜ x x (t, x) + K x (t, x, x)˜z (t, x) + K (t, x, x)˜z x (t, x)  x d K (t, x, x)˜z (t, x) − + K x x (t, x, ξ )˜z (t, ξ )dξ. dx 0

(8.30)

Subtracting (8.30) from (8.29), we get that   d ˜ x) − w˜ x x (t, x) = −2 K (t, x, x) − c z˜ (t, x) ∂t w(t, dx  L   + −∂t K (t, x, ξ ) + K x x (t, x, ξ ) − K ξ ξ (t, x, ξ ) z˜ (t, ξ )dξ

(8.31)

x

+ K ξ (t, x, L)˜z (t, L). It follows by (8.27) and (8.31) that one should have 



L



K (t, x, ξ )˜z (t, ξ )dξ − K 1 (t, x)w(t, ˜ L) d(t, x) z˜ (t, x) − x   d = −2 K (t, x, x) − c z˜ (t, x) dx

(8.32)

8.2 The Stabilization Result and Applications



L

+

179

  −∂t K (t, x, ξ ) + K x x (t, x, ξ ) − K ξ ξ (t, x, ξ ) z˜ (t, ξ )dξ

x

+ K ξ (t, x, L)˜z (t, L). In order for (8.32) to hold, we must have ⎧ ⎨ −∂t K (t, x, ξ ) + K x x (t, x, ξ ) − K ξ ξ (t, x, ξ ) = −d(t, x)K (t, x, ξ ), −2 ddx K (t, x, x) − c = d(t, x), ⎩ K ξ (t, x, L) = −K 1 (t, x).

(8.33)

Recall that 

L

w˜ x (t, x) = z˜ x (t, x) + K (t, x, x)˜z (t, x) − x

d K (t, x, ξ )˜z (t, ξ )dξ. dx

Putting x = 0 and assuming that K (t, 0, 0) = 0, we get d K (t, 0, ξ ) = 0, dx where we have used that w˜ x (t, 0) = z˜ x (t, 0) = 0. Now put x = L and recall that w˜ x (t, L) = −K 10 (t)w(t, L) and that z˜ x (t, L) = 0. We arrive at K (t, L , L) = −K 10 (t). Hence, we have to solve the equation − ∂t K (t, x, ξ ) + K x x (t, x, ξ ) − K ξ ξ (t, x, ξ ) = −d(t, x)K (t, x, ξ )

(8.34)

with boundary conditions ⎧ ⎪ ⎪ ⎨

K (t, x, x) = −

⎪ d ⎪ ⎩ K (t, 0, ξ ) = 0. dx

1 2



x

[c + d(t, ξ )]dξ,

0

We introduce the standard change of variables τ = x + ξ η = x − ξ, and define

  τ +η τ −η , , G(t, τ, η) := K (t, x, ξ ) = K t, 2 2

(8.35)

180

8 Stabilization of Unsteady States

thereby transforming the problem into the following PDE:   τ +η G(t, τ, η), (τ, η) ∈ O1 , (8.36) − ∂t G(t, τ, η) + 4G τ η (t, τ, η) = −d t, 2 with boundary conditions ⎧  τ ⎪ ⎨ G(t, τ, 0) = 1 2 [c + d(t, ξ )]dξ, 2 0 ⎪ ⎩ G (t, τ, −τ ) − G (t, τ, −τ ) = 0. τ η

(8.37)

Here the domain is O1 := {(τ, η) : 0 < τ < 2L , 0 < η < min(τ, 2L − τ )} . If d is such that (8.36)–(8.37) has a solution G, then we can recover K as K (t, x, y) = G(t, x + y, x − y). So we are able to say that Eqs. (8.28) and (8.27) are equivalent. This implies the asymptotic exponential decay of the error w, ˜ from which we conclude that the observer (8.26) asymptotically exponentially approximates the plant equation (8.23).

8.2.2 Applications Now we will consider a stabilization problem associated with the following SPDE: ⎧ ⎪ ⎪ dY (t, x) = Yx x (t, x)dt + f (t, x)Yx (t, x)dt + h(t)Y (t, x)dβ(t, x), ⎪ ⎪ ⎪ for 0 < x < L and t > 0, ⎪ ⎨  h(t)β(t)  h(t)β(t,0) v e Y (t) , t > 0 Yx (t, 0) = u(t) := e ⎪ ⎪ ⎪ ⎪ Yx (t, L) = 0 for t > 0, ⎪ ⎪ ⎩ Y (0, x) = Yo (x) for x ∈ [0, L].

(8.38)

Here f = −2hβx , where β is a Brownian motion in time and colored in space such that βx (t, 0) = βx (t, L) = 0; and h is such that 1 |h(t)| ≤ C √ , t > 0. t (For the precise formulation of the solution to (8.38), see Chap. 7.)

(8.39)

8.2 The Stabilization Result and Applications

181

A fair question would be why this stochastic PDE is related to the deterministic PDE (8.1), studied above in this chapter. The reason is that in order to study the boundary stabilization of (8.1), we will reduce it by a rescaling procedure (similarly as in the third section of Chap. 7) to a random parabolic equation and apply to this equation the stabilization result established in Theorem 8.1. Namely, by the substitution w(t) := e−h(t)β(t) Y (t), doing similar computations as in [22], we obtain that w is the solution to the following random deterministic equation: ⎧ ∂t w(t, x) = e−h(t)β(t,x) (eh(t)β(t,x) w(t, x))x x ⎪ ⎪ ⎪ ⎪ ⎪ + f (t, x)e−h(t)β(t,x) (eh(t)β(t,x) w(t, x))x ⎪ ⎪ ⎪   ⎨ 1 2 d h(t)β(t, x) + h (t) w(t, x), P-a.s.,t > 0, x ∈ (0, L), − ⎪ dt 2 ⎪ ⎪ ⎪ ⎪ ⎪ wx (t, 0) = v(w(t)), wx (t, L) = 0, P − a.s., t > 0, ⎪ ⎪ ⎩ w(0) = yo , (8.40) where in order to recover the boundary conditions, we used that βx (t, 0) = βx (t, L) = 0. Or equivalently, ⎧ ⎪ ⎨ ∂t w(t, x) = wx x (t, x) + q(t, x)w(t, x), P-a.s., for t > 0, x ∈ (0, L), wx (t, 0) = v(w(t)), wx (t, L) = 0, P-a.s., t > 0, ⎪ ⎩ w(0) = yo ,

(8.41)

where we used that f = −2hβx and set q(t, x) := h(t)βx x (t, x) + (h(t)βx (t, x))2 −

1 d h(t)β(t, x) − h 2 (t) − 2h(t)(βx (t, x))2 . dt 2

It is clear that Eq. (8.41) is a particular case of Eq. (8.1). In fact, in applying a rescaling argument to reduce a stochastic PDE to a deterministic one, the latter will usually have time-dependent coefficients (as (8.41) does). Consequently, the problem of stabilization of an SPDE can be solved via the stabilization to trajectories for some deterministic PDE. However, for the moment, this is not the best approach for this problem, since in the literature, there are very few results on unsteady-state stabilization. On the other hand, since we have obtained in Theorem 8.1 a result concerning the stabilization of trajectories for a semilinear heat equation, it is clear that we may immediately obtain a stabilization result for its stochastic version as well. In fact, we can prove the following result. Theorem 8.2 Let ρ > 0 be arbitrary but fixed. For N ∈ N large enough, the solution Y to the equation

182

8 Stabilization of Unsteady States

⎧ dY (t, x) = Yx x (t, x)dt + f (t, x)Yx (t, x)dt + h(t)Y (t, x)dβ(t, x), ⎪ ⎪ ⎪ ⎪ ⎪ for 0 < x < L and t > 0, ⎪ ⎨ h(t)β(t,0) h(t)β(t) v(e Y (t)), t > 0 Yx (t, 0) = u(t) := e ⎪ ⎪ ⎪ ⎪ Y (t, L) = 0 for t > 0, x ⎪ ⎪ ⎩ Y (0, x) = Yo (x) for x ∈ [0, L],

(8.42)

satisfies −∞ < lim sup log |Y (t, x)| < −ρ, ∀x ∈ (0, L), P-a.e.

(8.43)

t→∞

Here v is given by (8.8). Proof As mentioned earlier, Eq. (8.42) is equivalent to ⎧ ⎪ ⎨ ∂t w(t, x) = wx x (t, x) + q(t, x)w(t, x), P-a.s., for t > 0, x ∈ (0, L), wx (t, 0) = v(w(t)), wx (t, L) = 0, P-a.s., t > 0, ⎪ ⎩ w(0) = yo .

(8.44)

Making use of (8.39), we get, for some constant C > 0, that  |q(t)| ≤ C

 |β(t)| |βx x (t)| (βx (t))2 + √ + 1 , t > 0. + √ t t t

By the law of the iterated logarithm, it follows that 

|βx x (t)| (βx (t))2 |β(t)| + √ sup + √ t t t t≥0 whence for

 < ∞, P-a.e.,

    |βx x (t)| (βx (t))2 |β(t)| ≤r , Ωr := sup + √ + √ t t t t≥0

we have P(Ωrc ) → 0 as r → ∞ (for details, see Lemma 3.4 in [21]). Hence we may conclude that ess sup ess sup |q(t, x)| ≤ c4 , P − a.e., t>0

(8.45)

x∈(0,L)

for some constant c4 > 0. To conclude the proof, we remark that Eq. (8.44) is of the form (8.1) with a ≡ 0, b ≡ q, and σ ≡ 0. Notice that (8.45) ensures that in this case, b satisfies (8.2), as needed. Therefore, one may argue as in the previous section in order to obtain the null exponential stability of the random deterministic equation (8.44), and consequently relation (8.42). The details are omitted. 

8.3 Comments

183

8.3 Comments As already mentioned and discussed, regarding the nonstationary case, there are very few results, most of them treating the internal stabilization problem only, see [4, 23, 77], while Rodrigues [122] deals with the boundary case. The reason for this impoverished literature is that all the techniques developed for the stationary case seem not to work for stabilizing trajectories. In the work Barbu et al. [23], the Foias–Prodi property for parabolic PDEs is used. Roughly speaking, this property means that if the projections of two solutions to the unstable modes converge to each other as time goes to infinity, then the difference between these solutions goes to zero. However, it turns out that the conclusion remains true if the projections are close to each other at times proportional to a fixed constant. So the main idea in [23] was to design a control that ensures equality at integer times for the projections of two solutions to the unstable modes. More precisely, assume that for a sufficiently large integer N , one manages to construct an (internal) control such that once plugged into (8.5), the corresponding solution to the closed-loop equation (8.5) satisfies PN Y (1) = 0, where PN is the projection of L 2 on the space spanned by the first N eigenfunctions of the Laplacian in (0, L). Then using Poincaré’s inequality and the regularizing property of the resolving operator for (8.5), one gets −1

Y (1) = (I − PN )Y (1) ≤ C1 μ N 2 Y (1) H 1 (0,L)  −1  −1 ≤ C2 μ N 2 Yo + U L 2 (0,1);X ) ≤ C3 μ N 2 Yo ,

(8.46)

  where μ j j denotes the increasing sequence of eigenvalues of the Laplace operator and Ci , i = 1, 2, 3, are some constants not depending on N . It is clear that the fact that C3 is independent of N is of great importance, and in [23], this is shown based on a truncated observability inequality. It follows from (8.46) that provided that N is sufficiently large, one has

Y (1) ≤ e−μ Y0 . Iterating this procedure, one gets an exponentially decaying solution. Then via the dynamic programming principle, a Riccati feedback stabilizing controller is designed. As noticed above, the Riccati-based controls are not the best ones, from a practical point of view, since the algebraic Riccati equations require a large number of hard computations. Here we propose simple proportional-type ones. The results in this chapter were published in the author’s work [107], except those concerning the observer design, which are new. Of course, one may try to stabilize Eq. (8.1) by a Dirichlet boundary control, in the same manner in which we did so for the Neumann boundary case. Namely, consider the following problem:

184

8 Stabilization of Unsteady States

⎧ ∂t y(t, x) = yx x (t, x) + a(t, x)yx (t, x) + b(t, x)y(t, x) + σ (t, x, y(t, x)), ⎪ ⎪ ⎪ ⎨ 0 < x < L , t > 0, ⎪ ⎪ ⎪ ⎩

y(t, 0) = u(t), y(t, L) = 0, t ≥ 0, y(0, x) = yo (x) for x ∈ [0, L].

(8.47) This situation corresponds to Examples 2.3–2.5. So the feedback v, in this case, should look like (8.48) v(w) := v1 (w) + · · · + v N (w), ⎞ ⎛ (ϕ1 )x (0) ⎞ w, ϕ1  γk −μ1

(ϕ2 )x (0) ⎟ ⎜ w, ϕ2  ⎟ ⎜ ⎜ ⎟ γk −μ2 ⎟ ⎟ . vk (w) := A ⎜ ⎝ ......... ⎠ , ⎜ ⎝ ......... ⎠ (ϕ N )x (0) w, ϕ N 

where

⎛



γk −μ N

(8.49)

N

In this case, j 2π 2 μj = and ϕ j = L2



2 sin L



jπ x L

 , j = 1, 2, 3, . . . ,

are the eigenvalues and eigenfunctions of the Dirichlet Laplacian Ay = −yx x , y ∈ D(A) = H 2 (0, L) ∩ H01 (0, L), respectively. Now let us compare the form of the vk in the Neumann case given in (8.9) with the Dirichlet case from (8.49), the first of which involves the quantities  ϕ j (0) =

   ( j − 1)π 0 2 2 cos = , L L L

whereas the latter one involves  (ϕ j )x (0) =

2 j cos LL



jπ 0 L



 =

2√ μj. L

√ That is, now an additional μ j appears in the form of the feedback law. Consequently, the analogue of the estimate (7.48) reads, in this case, as ⎛ ⎞  w, ϕ1     √ ⎜ w, ϕ2  ⎟ 4−α ⎜   ⎟ N ⎝ |u k (w)| ≤ C N ⎠ .  .........   w, ϕ N  

(8.50)

8.3 Comments

185

So it gains an extra N . This is bad. Moreover, if we look at the Gram matrix B, given now as ⎛√ ⎞ √ √ μ1 μ1 μ1 μ2 ... μ1 μ N √ √ √ 2 ⎜ μ2 μ1 μ2 μ2 ... μ2 μ N ⎟ ⎟, B := ⎜ ⎝ ⎠ ........................................................ L √ √ √ μ N μ1 . . . μ N μ N −1 μ N μ N and argue as in (7.43)–(7.47), we may show that the best we can get is 1 N2 = B1 + · · · + B N ≤ C 2α−1 . λ1 (A) N This has bad repercussions for estimating |qi j | in (7.56)–(7.57), |qi j (t)| ≤ C

1 N

α− 25

e−cN t , t ≥ 0. 2

That is, we gain an extra N 2 in the estimates. Overall, in the final estimates we will have an extra N 3 . This is too much to be able to manipulate the powers of N to obtain relations such as (7.70)–(7.71). In conclusion, it is not straightforward to pass from the Neumann case to the Dirichlet one. Further subtle estimates of the quantities and properties of the feedback law must be deduced.

Chapter 9

Internal Stabilization of Abstract Parabolic Systems

In this chapter, we will reconsider the abstract parabolic equation framework from Chap. 2. This time, we will design an internal stabilizing proportional-type actuator. As in the boundary case, the feedback laws are of finite-dimensional nature, given in a simple form, and easy to manipulate from the computational point of view. And since we formulate the results in an abstract form, it is clear that for different types of precise models satisfying the imposed abstract hypotheses, these can be applied to the stabilization problem.

9.1 Presentation of the Problem For the reader’s convenience, we will restate the abstract formulation from Chap. 2. Let O be an open bounded domain in Rd , d ∈ N∗ , with smooth boundary ∂O. We denote by  ·  the norm in L2 (O). Let A be a closed and densely defined linear differential operator on L2 (O), with domain D(A); and let F0 : D(F0 ) ⊂ D(A) → L2 (O) be a nonlinear differential operator. We assume that (1) −A generates a C0 -analytic semigroup on L2 (O). (2) For all y, yˆ ∈ D(A), there exists the limit  1 F0 (ˆy + λy) − F0 (ˆy) , λ→0 λ

F0 (ˆy)(y) := lim

in L2 (O). Moreover, F0 (0) = 0, and for for some α ∈ (0, 1) and C > 0, we have F0 (ˆy)y ≤ αAy + Cy, ∀y ∈ D(A).

© Springer Nature Switzerland AG 2019 I. Munteanu, Boundary Stabilization of Parabolic Equations, Progress in Nonlinear Differential Equations and Their Applications 93, https://doi.org/10.1007/978-3-030-11099-4_9

(9.1)

187

188

9 Internal Stabilization of Abstract Parabolic Systems

Now fix some yˆ ∈ D(A) and introduce the linear operator A := A + F0 (ˆy), D(A) = D(A).

(9.2)

It is easy to see that A is closed, densely defined, and −A generates a C0 -semigroup on L2 (O). In addition, we assume that (3) The resolvent (λI − A)−1 of A is compact in L2 (O). Hypothesis (3) implies that the operator A has a countable set of eigenvalues λj , j ∈ N∗ (repeated according to their multiplicity), and corresponding eigenvectors ϕj , j ∈ N∗ , i.e., Aϕj = λj ϕj , j ∈ N∗ . Besides this, given ρ > 0, there is a finite number N of eigenvalues such that

λj < ρ, j = 1, 2, . . . , N , while λj ≥ ρ, j = N + 1, N + 2, . . .

(9.3)

We assume that (4) Each unstable eigenvalue λj , j = 1, 2, . . . , N , is semisimple.  N It is easily seen that the finite part of the spectrum λj j=1 can be separated from the rest of the spectrum by a rectifiable curve ΓN in the complex space C. Set Xu to  N be the linear space generated by the eigenfunctions ϕj j=1 , that is,  N Xu := lin span ϕj j=1 . Then the operator PN : L2 (O) → Xu defined by PN :=

1 2π i

 ΓN

(λI − A)−1 d λ

(9.4)

is known as the algebraic projection of L2 (O) onto Xu . It is easy to see that the operator Au := PN A  N maps the space Xu into itself and σ (Au ) = λj j=1 . More precisely, Au : Xu → Xu is finite-dimensional and can be represented by an N × N matrix. If A∗ is the dual operator of A, then its eigenvalues are precisely {λj }j∈N∗ , and the corresponding eigenfunctions are A∗ ϕj∗ = λj ϕj∗ , j ∈ N∗ .

9.1 Presentation of the Problem

189

The dual PN∗ of PN is given by PN∗

1 = 2π i

 ΓN

(λI − A∗ )−1 d λ,

  while XN∗ = lin span{ϕj∗ }Nj=1 = PN∗ L2 (O) . Via the Schmidt orthogonalization follows by hypothesis (4) that procedure, it

N ∗ N one can find a biorthogonal system {ϕj }j=1 , {ϕj }j=1 of eigenfunctions of A corresponding to the first eigenvalues {λj }Nj=1 , i.e., ϕj , ϕj∗ = δij , i, j = 1, 2, . . . , N .

(9.5)

Consider the Cauchy problem dy + Ay + F0 (y) = 0, t > 0; y(0) = yo , dt

(9.6)

and let yˆ be an equilibrium solution to the system (9.6), i.e., yˆ satisfies A(ˆy) + F0 (ˆy) = 0. By defining the fluctuation variable z := y − yˆ , the stability of yˆ can be equivalently reduced to the stability of the null solution to the equation

where

⎧ ⎨ ∂ z + Az + G(z) = 0 in (0, ∞) × O, ∂t ⎩ z(0) = zo := yo − yˆ in O,

(9.7)

G(z) := F0 (z + yˆ ) − F0 (ˆy) − F0 (ˆy)(z).

(9.8)

We consider an open subdomain O0 ⊂ O, and associate with (9.7) the control system ⎧ ⎨ ∂ z + Az + G(z) = 1 u in (0, ∞) × O, O0 ∂t (9.9) ⎩ z(0) = zo in O, where 1O 0 is the characteristic function of the set O0 . We will construct two stabilizing feedback laws for (9.9). The first approach is from Barbu [10], and it is directly related to the proportional controllers. More precisely, u is given as N  u(t) = −η z(t), ϕj∗ Φj , (9.10) j=1

190

9 Internal Stabilization of Abstract Parabolic Systems

  where Φj is a system of functions such that Φi , 1O 0 ϕj∗ = δij , i, j = 1, 2, . . . , N . Such a system can be found in the form Φi =

N 

αki ϕk∗ , i = 1, 2, . . . , N ,

k=1

where αki ∈ C are chosen to form the system N 

αki ϕk∗ , 1O 0 ϕj∗ = δij , i, j = 1, . . . , N .

k=1

By the unique continuation of the eigenfunctions, it follows immediately that the above system has a unique solution (a fact that was not true for the boundary case). Assuming that η ≥ γ − λj , j = 1, 2, . . . , N , one may show that once u, given by (9.10), is plugged into the Eq. (9.9), we obtain its stability. The method of proof for this is classical, and it has been used frequently throughout this book. Briefly, one first considers the linearization part and splits the system in two: the stable and unstable parts. Concerning the finite-dimensional unstable part, simple computations lead to d zi + λi zi = −ηzi , i = 1, 2, . . . , N , dt where zi = z, ϕi , i = 1, 2, . . . , N . Then the stability of the linearized part follows immediately (for details, see [10, Theorem 2.3]). Then via a fixed-point argument, local stability can be deduced as well (see [10, Sect. 2.5]). The second feedback law, which we propose for stabilization, involves the sign function, and it reads as u(t) := −η

N 

sign( PN z(t), ϕj∗ )PN Φj ,

(9.11)

j=1

where sign is the multivalued function on C defined by  sign(z) :=

z , |z|

if z = 0, {w ∈ C : |w| ≤ 1} , if z = 0,

(9.12)

9.1 Presentation of the Problem

191

and Φj ∈ L2 are defined by Φj :=

N 

αjk ϕk∗ , j = 1, . . . , N ,

k=1

with

N 

αik ϕk∗ , ϕj∗ 0 = δij , i, j = 1, . . . , N .

(9.13)

k=1

As a matter of fact, the only difference between the feedback laws (9.10) and (9.11) is the presence of the sign function. The reason to introduce this function is, as we will see below, that it allows us to obtain a stronger result, namely, that in finite time, the solution z belongs to the stable space Xs . Theorem 9.1 below amounts to saying that for η sufficiently large, the feedback controller (9.11) is exponentially stabilizing, with exponent −γ , in the linearized system, and it steers zo into Xs in a finite time T > 0. Theorem 9.1 Let ρ > 0 and zo ∈ L2 (O) be such that zo  ≤ ρ. Then the closedloop system ⎧ N ⎪ ⎨ dz + Az + η  sign( P z, ϕ ∗ )P (mΦ ) = 0, t ≥ 0, N N j j dt j=1 ⎪ ⎩ z(0) = zo ,

(9.14)

where m = 1O 0 has a unique solution z ∈ L∞ (0, t; L2 (O)) ∩ L2 (δ, t; D(A)), ∀0 < δ < t, such that for T > 0 arbitrary but fixed and η such that  η ≥ ρ max

1≤j≤N



λj , e λj T − 1

(9.15)

we have

and

PN z(t) = 0, ∀t ≥ T ,

(9.16)

z(t) ≤ Ce−γ t z0 , ∀t ≥ T ,

(9.17)

for some C > 0. If for some j ∈ {1, . . . , N }, we have λj = 0, then in (9.15) we take

λj to be T1 . e λj T −1 Remark 9.1 In Theorem 9.1, N can be taken arbitrarily large. In fact, for each N there exists γ such that max λj ≤ γ , and for η satisfying (9.15), relations (9.16), 1≤j≤N

192

9 Internal Stabilization of Abstract Parabolic Systems

(9.17) hold. Roughly speaking, this means that for each N , there is a controller of the form (9.11) that steers Yo into Xs . Proof We apply the projector PN to the system (9.14) and obtain that ⎧ N  ⎪ ⎨ dzu sign( zu , ϕi∗ )PN (mΦi ) = 0, t ≥ 0, + Au zu + η dt i=1 ⎪ ⎩ zu (0) = PN zo , where zu := PN z. If we decompose zu as zu =

N 

(9.18)

zj ϕj , introduce it into (9.18), and

j=1

scalar multiply by ϕj∗ the Eq. (9.18), we get that

⎧ ⎨ dzj + λ z + ηsignz = 0, ∀t ≥ 0, j j j dt ⎩ zj (0) = zjo ,

(9.19)

for all j = 1, . . . , N . Here we have used the relations (9.5) and (9.13). It should be said that the multivalued ordinary differential system (9.19) is well posed, because the multivalued function z → signz is maximal monotone on C.  N Hence there is a unique absolutely continuous solution zj j=1 to the system (9.19). Moreover, if we take account the relation signz · z = |z|, ∀z ∈ C, we have by (9.19) that 1d |zj (t)|2 + λj |zj (t)|2 + η|zj (t)| = 0, t ≥ 0. 2 dt This yields d |zj (t)| + λj |zj (t)| + η = 0, t ≥ 0, dt and therefore e λj t |zj (t)| − |zj (0)| +

 η  λj t e − 1 = 0, ∀t ≥ 0.

λj

It is easy to see that since η satisfies (9.15), we have  η  λj t e − 1 − |zj (0)| ≥ 0, ∀t ≥ T ,

λj which together with (9.20), implies immediately relation (9.16).

(9.20)

9.1 Presentation of the Problem

193

Next, we apply to the system (9.14) the projector I − PN , and get that ⎧ ⎨ dzs + A z = 0, t ≥ 0, s s dt ⎩ zs (0) = (I − PN )zo ,

(9.21)

where zs := (I − PN )z. We have that zs (t) ≤ Ce−γ t (I − PN )zo  ≤ Ce−γ t zo , ∀t ≥ 0. This, together with relation (9.16), implies (9.17), as desired.



Remark 9.2 As mentioned, the results obtained above hold as well in the case in which the eigenvalues are not necessarily semisimple. Indeed, let us assume, for example, that the matrix  Aϕi , ϕj Ni,j=1 has the form ⎛

 Aϕi , ϕj Ni,j=1

⎞ λ1 0 0 0 . . . 0 ⎜ 1 λ1 0 0 . . . 0 ⎟ ⎜ ⎟ ⎟ =⎜ ⎜ 0 0 λ3 0 . . . 0 ⎟ . ⎝ ....................... ⎠ 0 0 0 0 . . . λN

Thus in this case, (9.19) has the form ⎧ dz1 ⎪ ⎪ + λ1 z1 + ηsignz1 = 0, t ≥ 0; z1 (0) = z1o , ⎪ ⎪ ⎪ dt ⎪ ⎨ dz2 + z1 + λ1 z2 + ηsignz2 = 0, t ≥ 0; z2 (0) = z2o , ⎪ dt ⎪ ⎪ ⎪ ⎪ dzj ⎪ ⎩ + λj zj + ηsignzj = 0, t ≥ 0; zj (0) = zjo , j = 3, . . . , N . dt

(9.22)

(9.23)

Just as in the proof of Theorem 9.1, one can obtain that η λj t (e − 1) = 0, t ≥ 0, j = 1 and j = 3, 4, . . . , N .

λj (9.24) It follows, in particular, that e λj t |zj (t)| − |zj (0)| +

e λ1 t |z1 (t)| ≤ |z1 (0)| ≤ ρ.

(9.25)

We multiply the second equation of (9.23) by z¯2 and take the real part of the result to obtain that 1d |z2 |2 + λ1 |z2 |2 + η|z2 | = − (z1 z¯2 ). 2 dt

194

9 Internal Stabilization of Abstract Parabolic Systems

This yields e

λ1 t

e λ1 t − 1 |z2 (t)| − |z2 (0)| + η ≤

λ1



t

e λ1 τ |z1 (τ )|d τ.

(9.26)

0

Using relations (9.24), (9.25), and (9.26), it is easy to see that zj (t) = 0, t ≥ T , j = 1, . . . , N , 



λj . if η ≥ ρ(1 + T ) max 1≤j≤N e λj T − 1 Now let us treat another case. Let us assume that ⎛ ⎞ λ1 1 0 0 . . . 0 ⎜ 1 λ1 0 0 . . . 0 ⎟ ⎜ ⎟ ⎟  Aϕi , ϕj Ni,j=1 = ⎜ ⎜ 0 0 λ3 0 . . . 0 ⎟ . ⎝ ....................... ⎠ 0 0 0 0 . . . λN

(9.27)

We get immediately that ⎧d ⎪ |z1 | + λ1 |z1 | + η ≤ |z2 |, t ≥ 0, ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎨ d |z2 | + λ1 |z2 | + η ≤ |z1 |, t ≥ 0, ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ d |z | + λ |z | + η = 0, t ≥ 0, j = 3, . . . , N . j j j dt

(9.28)

We sum the first two equations of (9.28) to see that d (|z1 | + |z2 |) + ( λ1 − 1)(|z1 | + |z2 |) + 2η ≤ 0. dt It is easy to observe that if  

λj 1 λ1 − 1 ; max , η ≥ ρ max 2 e( λ1 −1)T − 1 3≤j≤N e λj T − 1 

then via (9.28) and (9.29), we have zj (t) = 0, t ≥ T , j = 1, . . . , N , as desired.

(9.29)

9.1 Presentation of the Problem

195

We conclude that when the unstable eigenvalues are not necessarily semisimple, one can choose η > 0 in an appropriate way, sufficiently large, to obtain the same results as in Theorem 9.1.

9.2 Stabilization of the Full Nonlinear Equation (9.9) In what follows, we will consider γ to be 0, and so N is such that Xs is generated by the eigenfunctions corresponding to the stable eigenvalues, i.e., λj > 0, j = N +1, N + 2, . . .. Hence in this case, λj ≤ 0, for j = 1, . . . , N . We define β := min λj , j = N + 1, . . . . Thus in this case, we have the following estimate for the operator As : (9.30) e−As t L(L2 (O ),L2 (O )) ≤ Ce−βt , ∀t ≥ 0. The main result of this section is the next theorem, which amounts to saying that the feedback controller (9.11) exponentially stabilizes the nonlinear system (9.9), and just as for the linear equation, it steers zo into Xs in finite time T > 0. Theorem 9.2 Let T , ρ > 0 be sufficiently small. For each zo ∈ W such that zo W ≤ ρ, the problem ⎧ N ⎪ ⎨ dz + Az + η  sign( P z, ϕ ∗ )P (mΦ ) + Gz = 0, t ≥ 0, N N j j dt j=1 ⎪ ⎩ z(0) = zo ,

(9.31)

is well posed on W with unique solution z ∈ C([0, ∞); W ) ∩ L2 (0, ∞; Z ), if η is such that  ⎫ ⎧ ⎨ λj kϕj∗  + ρ ⎬ η ≥ max . (9.32) 1≤j≤N ⎩ ⎭ e λj T − 1 Moreover, these solutions satisfy

and

PN z(t) = 0, ∀t ≥ T ,

(9.33)

z(t) ≤ Ce−βt z0 , ∀t ≥ T ,

(9.34)

for some C > 0. Here k is given by relation (9.39) below, and W = L2 (O) and Z = H 1 (O). If

λj to be T1 . for some j ∈ {1, . . . , N }, we have λj = 0, then in (9.32) we take e λj T −1

196

9 Internal Stabilization of Abstract Parabolic Systems

Proof For r ≤ 1, let us introduce the ball of radius r centered at the origin of the space L2 (0, ∞; Z ): 



S(0, r) := f ∈ L (0, ∞; Z ) : f L2 (0,∞;Z ) = 2

∞

f

0

(t)2Z

1 2

dt

! ≤r .

For all Z ∈ S(0, r), let us consider the system ⎧ N ⎪ ⎨ dz + Az + η  sign( P z, ϕ ∗ )P (mΦ ) = −GZ, t ≥ 0, N N j j dt j=1 ⎪ ⎩ z(0) = zo .

(9.35)

The idea of the proof is as follows: we show that for all Z ∈ S(0, r), problem (9.35) has a solution zZ ∈ S(0, r) for T , ρ, and r sufficiently small. Moreover, we show that PN zZ (t) = 0, ∀t ≥ T . Then we denote by Γ the operator that associates Z to the solution zZ . In doing so, we get that Γ : S(0, r) → S(0, r) is a contraction on S(0, r), for T , r sufficiently small. It follows that there exists a unique solution z ∈ S(0, r) for (9.31). Next, we show that z ∈ C([0, ∞); W ) and z(t) ∈ B(0, b) := {f ∈ W : f W ≤ b} , t ≥ T , for some b > 0, which will imply the claimed exponential decay (9.34). By Theorem 9.1, one can easily deduce that e−As t L(W ) ≤ Ce−δt , t ≥ 0, and

 0

∞

e−As t W 2Z ≤ cW 2W , ∀W ∈ W ,

for some c > 0. Next, we define

 (N Z)(t) :=

t

e−As (t−τ ) (GZ)(τ )d τ.

(9.36)

(9.37)

(9.38)

0

About the nonlinearity G, we will assume that it is of second-order type, namely GZW ≤ kZ2Z , ∀Z ∈ Z ,

(9.39)

for some k > 0, and GZ1 − GZ2 W ≤ k (Z1 Z + Z2 Z ) Z1 − Z2 Z , ∀Z1 , Z2 ∈ S(0, r). (9.40) By duality, we have that for Z ∈ L2 (0, ∞; Z  ), Z  is the dual of Z ,

9.2 Stabilization of the Full Nonlinear Equation (9.9)

197

#  ∞ " t N Z(t), ζ (t) dt = e−As (t−τ ) (GZ)(τ )d τ, ζ (t) dt 0 0 0  ∞ t $ $ −A (t−τ ) $e s ≤ (GZ)(τ )$Z ζ (t)Z  d τ dt 0 0   ∞  ∞ $ $ −A (t−τ ) $e s (GZ)(τ )$Z ζ (t)Z  dt d τ = 0 τ & 21  ∞ % ∞ $2 $ −A (t−τ ) $e s ≤ (GZ)(τ )$ dt



∞

% × 0

Z

τ

0 ∞

&1 !

ζ (t)2Z 

2

 = ζ (t)L2 (0,∞;Z  )

∞

%



≤ Cζ (t)L2 (0,∞;Z  )

∞ τ

0

(by (9.37))

(9.41)

dτ

∞

$2 $ −A (t−τ ) $e s (GZ)(τ )$Z dt

& 21

dτ

GZ(τ )W d τ

0

(by (9.39)) ≤ Cζ (t)L2 (0,∞;Z  ) Z(t)2L2 (0,∞;Z ) . Therefore, (N Z)(t)L2 (0,∞;Z ) ≤ Cr 2 ,

(9.42)

for all Z ∈ S(0, r). In a similar manner, one can show as well that N Z1 − N Z2 2L2 (0,∞;Z ) ≤ 4ck 2 r 2 Z1 − Z2 2L2 (0,∞;Z ) , ∀Z1 , Z2 ∈ S(0, r). (9.43) Finally, we define (ΛZ)(t) := e−As t (I − PN )zo + (N Z)(t).

(9.44)

By (9.36) and (9.42), we deduce that '



ΛZ2L2 (0,∞;Z ) ≤ C (I − PN )zo 2W +

0

∞

2

Z(t)2Z

( (9.45)

≤ C(ρ + r ), ∀Z ∈ S(0, r). 2

4

With these key results in hand, we can proceed with the proof. To prove that there exists a solution to the equation (9.35), one can argue as in the proof of the Theorem 9.1, using the fact that the function sign is maximal monotone on C. Next, one may show that this solution remains in S(0, r), for r sufficiently

198

9 Internal Stabilization of Abstract Parabolic Systems

small, and it satisfies (9.33) and (9.34). Then one applies the projector PN to (9.35) and gets that   1d (9.46) |zj (t)|2 + λj |zj (t)|2 + η|zj (t)| = − GZ, ϕj∗ ¯zj , t ≥ 0, 2 dt N  for all j = 1, . . . , N , where PN z = zj ϕj . Next, using the Schwarz inequality, it j=1

follows that )) )  )  ) ) ) ) − GZ, ϕj∗ ¯zj ≤ ) GZ, ϕj∗ ) )zj ) ≤ GZϕj∗  )zj ) . This, together with (9.46), yields d |zj (t)| + λj |zj (t)| + η ≤ GZϕj∗ , ∀t ≥ 0, dt

(9.47)

for all j = 1, . . . , N . Multiplying (9.47) by e λj τ and integrating over (0, t), we obtain that  t  t

λj t

λj τ e |zj (t)| − |zj (0)| + ηe dτ ≤ e λj τ GZ(τ )ϕj∗ d τ, t ≥ 0, (9.48) 0

0

for all j = 1, . . . , N . Now using estimate (9.39), we get that  t e λj τ GZ(τ )ϕj∗ d τ ≤ e λj τ GZ(τ )W ϕj∗ d τ (9.49) 0 0  t  ∞ ≤ kϕj∗  e λj τ Z(τ )2Z d τ ≤ kϕj∗  Z(τ )2Z ≤ kr 2 ϕj∗ , (9.50) 

t

0

0

since Z ∈ S(0, r). Hence (9.49)–(9.50) and (9.46) yield e λj t |zj (t)| + η

 e λj t − 1  2 ∗ − kr ϕj  + |zj (0)| ≤ 0, ∀t ≥ 0,

λj

(9.51)

for all j = 1, . . . , N . It is easy to see that if η satisfies relation (9.32), we get that |zj (t)| = 0, ∀t ≥ T , for all j = 1, . . . , N . Moreover, we get also from (9.51) that   |zj (t)| ≤ e− λj T kr 2 ϕj∗  + ρ , 0 ≤ t ≤ T .

(9.52)

We choose T > 0 sufficiently small that hT

N   2  r 2  |λj |e−2 λj T kr 2 ϕj∗  + ρ ≤ , 4 j=1

(9.53)

9.2 Stabilization of the Full Nonlinear Equation (9.9)

199

where h > 0 is given by the following relation between the norms: 1

 · H 1 (O ) ≤ hA 2 · .

(9.54)

Thus one can obtain via (9.52), (9.54), and (9.53) that 

∞ 0

 PN z(t)2Z

dt =



T

PN z(t)2Z

0

T

dt ≤ h 0

N 

|λj ||zj (t)|2 dt

(9.55)

j=1

N   2  |λj |e−2 λj T kr 2 ϕj∗  + ρ ≤ hT

≤

j=1

r2 . (9.56) 4

Now applying the projector I − PN to (9.35), we get that d zs + As zs + (I − PN )GZ = 0, t ≥ 0; zs (0) = (I − PN )zo , dt

(9.57)

where zs = (I − PN )z. Using the variation of constants formula, we have that 

zs (t) = e−As t (I − PN )zo +

t

e−As (t−τ ) (I − PN )GZ(τ )d τ, t ≥ 0.

(9.58)

0

It is easy to see that by making use of the relations (9.58), (9.38), and (9.44), we have the equality zs (t) = (ΛZ)(t), ∀t ≥ 0. Thus from (9.45), we obtain that zs 2L2 (0,∞;Z ) ≤ C(ρ 2 + r 4 ).

(9.59)

Taking ρ and r sufficiently small that C(ρ 2 + r 4 ) ≤

r2 , 4

(9.60)

we get from (9.59) that (I − PN )z2L2 (0,∞;Z ) = zs 2L2 (0,∞;Z ) ≤

r2 . 4

(9.61)

Finally, we conclude that if T , ρ, and r are small enough that they satisfy relations (9.53) and (9.60), we have

200

9 Internal Stabilization of Abstract Parabolic Systems

 z2L2 (0,∞;Z ) = ≤2

∞

0



 z(t)2Z dt ≤ 2 2

2

r r + 4 4

0

∞



 PN z(t)2Z + (I − PN )z(t)2Z dt

= r2,

if we take account of the relations (9.55)–(9.56) and (9.61). This means that the solution z remains in the ball S(0, r). Hence if we denote by Γ the operator that associates Z to the corresponding solution z to the system (9.35), we have that Γ maps the ball S(0, r) into itself. Therefore, in order to complete the proof, it is enough to show that Γ is a contraction on S(0, r). To this end, we have the following. Let Z1 , Z2 be two functions in S(0, r), and z1 , z2 ∈ S(0, r) the corresponding solutions to the system (9.35). Consequently, z1 and z2 satisfy ⎧ N ⎪ ⎨ dz1 + Az + η  sign( P z , ϕ ∗ )P (mΦ ) = −GZ , t ≥ 0, 1 N 1 N j 1 j dt j=1 ⎪ ⎩ z1 (0) = zo

(9.62)

⎧ N ⎪ ⎨ dz2 + Az + η  sign( P z , ϕ ∗ )P (mΦ ) = −GZ , t ≥ 0, 2 N 2 N j 2 j dt j=1 ⎪ ⎩ z2 (0) = zo .

(9.63)

and

Applying, N as before, the projector N PN to (9.62) and (9.63), and decomposing PN z1 = j=1 z1j ϕj and PN z2 = j=1 z2j ϕj , we get that

and

⎧ ⎨ d z + λ z + ηsign(z ) = − GZ , ϕ ∗ , t ≥ 0, 1j j 1j 1j 1 j dt ⎩ z1j (0) = zjo ,

(9.64)

⎧ ⎨ d z + λ z + ηsign(z ) = − GZ , ϕ ∗ , t ≥ 0, 2j j 2j 2j 2 j dt ⎩ o z2j (0) = zj ,

(9.65)

for all j = 1, . . . , N . Subtracting (9.64) and (9.65), we obtain ⎧      d  ⎪ ⎪ ⎪ ⎨ dt z1j − z2j + λj z1j − z2j + η sign(z1j ) − sign(z2j ) = − GZ1 − GZ2 , ϕj∗ , t ≥ 0, ⎪ ⎪ ⎪  ⎩ z1j − z2j (0) = 0,

(9.66)

9.2 Stabilization of the Full Nonlinear Equation (9.9)

201

for all j = 1, . . . , N . Taking into account that sign is a maximal operator, we get from (9.66) multiplied by z¯1j − z¯2j , that ⎧ ) ⎨ d )z − z )) + λ ))z − z )) ≤ | GZ − GZ , ϕ ∗ |, t ≥ 0, 1j 2j j 1j 2j 1 2 j dt  ⎩ z1j − z2j (0) = 0,

(9.67)

for all j = 1, . . . , N . Hence   e λj t | z1j − z2j (t)| ≤



t

e λj τ | (GZ1 − GZ2 ) (τ ), ϕj∗ |d τ  t ≤ ϕj∗  GZ1 − GZ2 W 0

0

(using (9.40))  t {Z1 Z + Z2 Z } Z1 − Z2 Z dt ≤ kϕj∗ 

(9.68)

0

≤ 2krϕj∗ Z1 − Z2 L2 (0,∞;Z ) , for all j = 1, . . . , N . Hence   | z1j − z2j (t)| ≤ e− λj T 2krϕj∗ Z1 − Z2 L2 (0,∞;Z ) , 0 ≤ t < T and

(9.69)

  | z1j − z2j (t)| = 0, ∀t ≥ T ,

for all j = 1, . . . , N . In the same manner as in relation (9.55), we obtain, via relation (9.69), that  0

∞

PN (z1 − z2 )(t)2Z dt ≤ hT

N %  2 &  |λj |e−2 λj T 2krϕj∗  Z1 − Z2 2L2 (0,∞;Z ) .

(9.70)

j=1

To obtain estimates for (I − PN )(z1 − z2 ), we apply the projector (I − PN ) to (9.62) and (9.63), use the variation of constants formula as above, and get that 

t

(I − PN )(z1 − z2 )(t) =

e−As (t−τ ) (I − PN )(GZ1 − GZ2 )(τ )d τ, t ≥ 0. (9.71)

0

Using (9.43), we obtain that (I − PN )(z1 − z2 )2L2 (0,∞;Z ) ≤ 4ck 2 r 2 Z1 − Z2 2L2 (0,∞;Z ) .

(9.72)

202

9 Internal Stabilization of Abstract Parabolic Systems

Now (9.70) and (9.72) together yield z1 − z2 2L2 (0,∞;Z ) ⎧ ⎫ N  ⎨  ⎬  2 (9.73) |λj |e−2 λj T 2krϕj∗  + 4ck 2 r 2 Z1 − Z2 2L2 (0,∞;Z ) . ≤ 2 hT ⎩ ⎭ j=1

Thus if we take T and r sufficiently small that ⎧ ⎫ N  ⎨  ⎬  2 2 hT |λj |e−2 λj T 2krϕj∗  + 4ck 2 r 2 < μ2 < 1, ⎩ ⎭

(9.74)

j=1

we get that Γ Z1 − Γ Z2 L2 (0,∞;Z ) ≤ μZ1 − Z2 L2 (0,∞;Z ) , ∀Z1 , Z2 ∈ S(0, r), with μ < 1. Hence Γ is a contraction on S(0, r), as desired. We conclude that if T , r, ρ are small enough that they satisfy relations (9.53), (9.60), and (9.74), then via the contraction mapping principle, there exists a unique solution z ∈ S(0, r) ⊂ L2 (0, ∞; Z ) to equation (9.31) that satisfies relation (9.33). Next, we want to show that this solution z is in C([0, ∞); W ) as well. To this end, we take into account that zj ∈ C([0, T ], C) and zj (t) = 0, t ≥ T , for all j = 1, . . . , N . Hence PN z ∈ C([0, ∞); W ). Moreover, since e−As t is an analytic semigroup on W , uniformly stable there, by (9.36), it follows by convolution of the definition of the operator N that (I − PN )z ∈ C([0, ∞); W ). Thus z ∈ C([0, ∞); W ), as claimed. Finally, by (9.45), we have for all t ≥ T , taking into account that z = (I − PN )z, z(t)W ≤ Cρ + Cr 2 .

(9.75)

Thus z(t) ∈ S(0, b) for all t ≥ T , where b := Cρ + Cr 2 . This then yields that  ∞

T

z(t)2Z dt ≤ Kzo 2W ,

from which, using the classical strategy for nonlinear autonomous systems [27, p. 178], we get the claimed exponential decay (9.34). 

9.3 The Design of a Real Stabilizing Feedback Controller

203

9.3 The Design of a Real Stabilizing Feedback Controller In applications, it is convenient to design a real stabilizing controller of the form (9.11). To do this, we consider again γ ≥ 0 to be arbitrary but fixed. We set {ψj }Nj=1 =   N2

ϕj , ϕj j=1 (we assume, for simplicity, that all λj , 1 ≤ j ≤ N , are complex, and so N is even). We set Xˆu = linspan{ψj }Nj=1 , and denote by Pˆ N : H → Xˆu the algebraic projection on Xˆu . We set also Xˆs = (I − PN )H , and ˆ s = A| ˆ . ˆ u = A| ˆ , A A Xu Xs ˆ u = Au and A ˆ s = As . Moreover, we can orthogonalize We have, of course, A {ψj }Nj=1 , via the Gram–Schmidt procedure, and get thereby ψj , ψi = δij , i, j = 1, . . . , N .

(9.76)

Now we consider the feedback controller u = −η

N 

sign( PN z, ψj PN Ψj ,

(9.77)

j=1

where Ψj =

N 

αjk ψk , j = 1, . . . , N ,

(9.78)

k=1

and

N 

αjk ψk , ψi 0 = δji , i, j = 1, . . . , N .

(9.79)

k=1

(We can choose αjk in this way because the system {ψj }Nj=1 is linearly independent d  in L2 (O0 ) .) Then substituting u into the linearized system, we have ⎧ N ⎪ ⎨ d z + Az = −η  sign( P , ψ )P (m(Ψ )), t ≥ 0, N j N j dt j=1 ⎪ ⎩ z(0) = zo . Arguing as in the proof of Theorem 9.1 and taking account of the fact that

(9.80)

204

9 Internal Stabilization of Abstract Parabolic Systems

Ψj , ψi = δij , i, j = 1, . . . , N , and

ˆ

e−As t L(H ,H ) ≤ Ce−γ t , ∀t ≥ 0,

we deduce the following result. Theorem 9.3 Let T , ρ > 0, and zo be such that zo  ≤ ρ. For 0 < η = η(T , ρ) sufficiently large, we have for the solution z to the closed-loop system (9.80),

and

PN z(t) = 0, for all t ≥ T ,

(9.81)

z(t) ≤ Ce−γ t zo , ∀t ≥ T .

(9.82)

Proof For simplicity, let us assume that N = 4. The other cases can be treated similarly. We have ˜ 1 + iψ2 ) = Aψ1 + iAψ2 . ˜ 1 ) = A(ψ A(ϕ On the other hand, we have ˜ 1 ) = λ1 ϕ1 = λ1 (ψ1 + iψ2 ). A(ϕ Hence Aψ1 = λ1 ψ1 − λ1 ψ2 and Aψ2 = λ1 ψ2 + λ1 ψ1 .

(9.83)

In the same manner, we get also that Aψ3 = λ2 ψ3 − λ2 ψ4 and Aψ4 = λ2 ψ4 + λ2 ψ3 .

(9.84)

Thus in this case, the finite-dimensional system  d ˆ u zu = −η zu + A sign( zu , ψj )PN (m(Ψj )) dt j=1 4

reads as

⎧ d ⎪ ⎪ z1 + λ1 z1 + λ1 z2 ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪d ⎪ ⎪ ⎨ z2 + λ1 z2 − λ1 z1 dt d ⎪ ⎪ ⎪ z3 + λ2 z3 + λ2 z4 ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ d ⎪ ⎩ z4 + λ2 z4 − λ2 z3 dt

= −ηsign(z1 ), = −ηsign(z2 ), (9.85) = −ηsign(z3 ), = −ηsign(z4 ), ∀t ≥ 0.

9.3 The Design of a Real Stabilizing Feedback Controller

205

Multiplying the first equation of (9.85) by z1 , the second by z2 , and summing them, we get 1d (|z1 |2 + |z2 |2 ) + λ1 (|z1 |2 + |z2 |2 ) + η(|z1 | + |z2 |) = 0, ∀t ≥ 0. 2 dt Hence 1 1d (|z1 | + |z2 |)2 + λ1 (|z1 | + |z2 |)2 + η(|z1 | + |z2 |) ≤ 0, ∀t ≥ 0. 4 dt 2 The same result can be obtained for the coefficients z3 and z4 . Now arguing as in the proof of Theorem 9.1, one can obtain the desired result. The details are omitted.  In the same manner, following the ideas in the proof of Theorem 9.2, one can obtain for the nonlinear system  d z + Az + Gz = −η sign( PN , ψj )PN (m(Ψj )), t ≥ 0; z(0) = zo , dt j=1 N

(9.86)

the following theorem. Theorem 9.4 Let T , ρ > 0 be sufficiently small. For each zo ∈ W such that zo W ≤ ρ, the problem (9.86) is well posed on W with the unique solution z ∈ C([0, ∞); W ) ∩ L2 (0, ∞; Z ) if η = η(T , ρ) is large enough. Moreover, these solutions satisfy (9.87) PN z(t) = 0, ∀t ≥ T , and

z(t) ≤ Ce−βt z0 , ∀t ≥ T ,

(9.88)

for some C, β > 0.

9.4 Comments The stabilization problems presented above have been studied extensively over the last six or seven years, and we refer to the works [11–13, 19, 30, 31, 50, 118, 119, 121], as well as to the book [10], for significant results in this direction. We have presented here an internal stabilizing control design associated with abstract parabolic-like equations. The proportional feedback law is similar to the one in Chap. 2, but reconsidered for the internal case. A similar result was published in

206

9 Internal Stabilization of Abstract Parabolic Systems

Barbu and Munteanu [20]. However, the abstract setting discussed here is new. It should be mentioned that these results are connected with those in [123], where the exact controllability in projections for the Navier–Stokes equations is obtained. However, there is no overlap, and the technique used here is completely different. The proof of the stability of the nonlinear system is based mainly on the ideas in [19].

References

1. Aamo OM, Krstic M, Bewley TR (2003) Control of mixing by boundary feedback in a 2Dchannel. Automatica 39:1597–1606 2. Agranovich MS (2015) Sobolev spaces, their generalizations and elliptic problems in smooth and lipschitz domains. Springer, New York 3. Amendola G, Fabrizio M, Golden JM (2012) Thermodynamics of materials with memory: theory and applications. Springer, New York 4. Ammari K, Duyckaerts T, Shirikyan A (2016) Local feedback stabilisation to a nonstationary solution for a damped non-linear wave equation. Math Control Relat Fields 6(1):1–5 5. Arnold L (1979) A new example of an unstable system being stabilized by random parameter noise. Inform Comm Math Chem 133–140 6. Arnold L, Crauel H, Wihstutz V (1983) Stabilization of linear systems by noise. SIAM J Control Optim 21:451–461 7. Barbu V (1975) Nonlinear semigroups and differential equations in Banach spaces. Noordhoff, Leyden 8. Barbu V (2003) Internal stabilization of the phase filed system. Adv Autom Control 754:1–8 9. Barbu V (2007) Stabilization of a plane channel flow by wall normal controllers. Nonlin Anal Theory-Methods Appl 67(9):2573–2588 10. Barbu V (2010) Stabilization of Navier-Stokes flows. Springer, New York 11. Barbu V (2010) Stabilization of a plane channel flow by noise wall normal controllers. Syst Control Lett 59(10):608–614. Barbu V (2010) Optimal stabilizable feedback controller for Navier-Stokes equations. Contemp Math 513:43–53 12. Barbu V (2012) Stabilization of Navier-Stokes equations by oblique boundary feedback controllers. SIAM J Control Optim 50(4):2288–2307 13. Barbu V (2013) Boundary stabilization of equilibrium solutions to parabolic equations. IEEE Trans Autom Control 58:2416–2420 14. Barbu V (2018) Controllability and stabilization of parabolic equations. Birkhäuser Basel 15. Barbu V, Bonaccorsi S, Tubaro L (2014) Existence and asymptotic behavior for hereditary stochastic evolution equations. Appl Math Optim 69:273–314 16. Barbu V, Colli P, Gilardi G, Marinoschi G, Rocca E (2017) Sliding mode control for a nonlinear phase-field system. SIAM J Control Optim 55(3):2108–2133 17. Barbu V, Colli P, Gilardi G, Marinoschi G (2017) Feedback stabilization of the Cahn-Hilliard type system for phase separation. J Diff Eqs 262:2286–2334 © Springer Nature Switzerland AG 2019 I. Munteanu, Boundary Stabilization of Parabolic Equations, Progress in Nonlinear Differential Equations and Their Applications 93, https://doi.org/10.1007/978-3-030-11099-4

207

208

References

18. Barbu V, Iannelli M (2000) Controllability of the heat equation with memory. Diff Int Eqs 13:1393–1412 19. Barbu V, Lasiecka I, Triggiani R (2006) Abstract settings for tangential boundary stabilization of Navier-Stokes equations by high- and low-gain feedback controllers. Nonlin Anal 64:2704– 2746 20. Barbu V, Munteanu I (2012) Internal stabilization of Navier-Stokes equation with exact controllability on spaces with finite codimension. Evol Eqs Control Theory 1(1):1–16 21. Barbu V, Da Prato G (2012) Internal stabilization by noise of the Navier-Stokes equation. SIAM J Control Optim 49(1):1–20 22. Barbu V, Röckner M (2015) An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise. J Eur Math Soc 17:1789–1815 23. Barbu V, Rodrigues SS, Shirikyan A (2011) Internal exponential stabilization to a nonstationary solution for 3D Navier-Stokes equations. SIAM J Control Optim 49(4):1454–1478 24. Barbu V, Triggiani R (2004) Internal stabilization of Navier-Stokes equations with finite dimensional controllers. Indiana Univ Math J 53:1443–1469 25. Barbu V, Wang G (2003) Internal stabilization of semilinear parabolic systems. J Math Anal Appl 285:387–407 26. Barbu V, Iannelli M (2000) Controllability of the heat equation with memory. Differ Integr Eqs 13:1393–1412 27. Balogh A, Krstic M (2002) Infinite dimensional backstepping-style feedback transformations for a heat equation with an arbitrary level of instability. Eur J Control 8:165–176 28. Badra M, Takahashi T (2011) Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers: application to the Navier-Stokes system. SIAM J Control Optim 49(2):420–463 29. Balogh A, Liu W-J, Krstic M (2001) Stability enhancement by boundary control in 2D channel flow. IEEE Trans Autom Control 46:1696–1711 30. Bedra M (2009) Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system. ESAIM COCV 15:934–968 31. Bedra M (2009) Lyapunov functions and local feedback stabilization of the Navier-Stokes equations. SIAM J Control Optim 48:1797–1830 32. Bertini L, Giacomin G (1997) Stochastic Burgers and KPZ equations from particle systems. Commun Math Phys 183:571–607 33. Bewley TR (2001) Flow control: new challenges for new renaissance. Prog Aerospace Sci 37:21–58 34. Bourbaki N (2007) Théories spectrales. Springer, Berlin 35. Brezis H (2011) Functional analysis. Sobolev spaces and partial differential equations. Springer, New York 36. Britton NF (1986) Reaction-diffusion equations and their applications to biology. Academic Press, New York 37. Brzezniak Z, Goldys B, Peszat S, Russo F (2013) Second order PDEs with Dirichlet white noise boundary conditions. J Evol Eqs 15(1):1–26 38. Boskovic DM, Krstic M, Liu W (2001) Boundary control of an unstable heat equation via measurement of domain-averaged temperature. IEEE Trans Autom Control 46(12):2028– 2028 39. Caginalp G (1988) Conserved-phase field system: implications for kinetic undercooling. Phys Rev B 38:789–791 40. Cannarsa P, Frankowska H, Marchini EM (2013) Optimal control for evolution equations with memory. J Evol Eqs 13:197–227 41. Carballo T (2006) Recent results on stabilization of PDEs by noise. Bol Soc Esp Mat Apl 37:47–70 42. Caraballo T, Liu K, Mao X (2001) On stabilization of partial equations by noise. Nagoya Math J 161:155–170 43. Cochran J, Vasquez R, Krstic M (2006) Backstepping boundary control of Navier-Stokes channel flow: a 3D extension, 25th edn. In: American control conference

References

209

44. Dalang RC (1999) Extending the martingale measure stochastic integral with applications to spatially homogeneous S.P.D.E’s. Electron J Probab 4(6) (online) 45. Chen Z (1994) Optimal boundary controls for a phase field model. IMA J Math Control Inform 10(2):157–176 46. Chepyzhov V, Miranville A (2006) On trajectory and global attractors for semilinear heat equations with fading memory. Ind Univ Math J 55(1):119–167 47. Choi H, Temam R, Moin P, Kim J (1993) Feedback control for unsteady flow and its application to the stochastic Burgers equation. J Fluid Mech 253:509–543 48. Coleman B, Gurtin M (1967) Equipresence and constitutive equations for rigid heat conductors. Z Angew Math Phys 18:199–208 49. Colli P, Gilardi G, Marinoschi G, Rocca E (2017) Sliding mode control for a phase field system related to tumor growth. Appl Math Optim 1–24 50. Coron JM (2007) Control and nonlinearity. AMS, Providence, RI 51. Dafermos CM (1970) Asymptotic stability in viscoelasticity. Arch Rat Mech Anal 37:297–308 52. Debussche A, Fuhrman M, Tessitore G (2007) Optimal control of a stochastic heat equation with boundary-noise and boundary-control. ESAIM COCV 13:178–205 53. Fabbri G, Goldys B (2009) An LQ problem for the heat equation on the halfline with Dirichlet boundary control and noise. SICON 48(3):1473–1488 54. Fisher RA (1937) The wave of advance of advantageous genes. Ann Eugen 7:353–369 55. Fitzhugh A (1961) Impulses and physiological states in theoretical phenomena in the nerve membrane. Biophys J 1:445–466 56. Foondun M, Nualart E (2015) On the behaviour of stochastic heat equations on bounded domains. Lat Am J Probab Math Stat 12(2):551–571 57. Funaki T, Quastel J (2015) KPZ equation, its renormalization and invariant measures. Stoch Partial Diff Eqs Anal Comput 3(2):159-220 58. Fursikov A (2002) Real process corresponding to 3D Navier-Stokes system and its feedback stabilization form boundary. Am Math Soc Transl 206(2):95–123 59. Fursikov AV (2001) Stabilizability of quasi-linear parabolic equations by feedback boundary control. Sbornik Math 192:593–639 60. Fursikov AV (2004) Stabilization for the 3D Navier-Stokes system by feedback boundary control. Discret Contin Dyn Syst 10(1):289–314 61. Gilbarg D, Trudinger N (2001) Elliptic partial differential equations of second order. Springer, Berlin 62. Giorgi C, Pata V, Marzocchi A (1998) Asymptotic behavior of a semilinear problem in heat conduction with memory. NoDEA 5:333–354 63. Guatteri G, Masiero F (2013) On the existence of optimal controls for SPDEs with boundary noise and boundary control. SIAM J Control Optim 51(13):1909–1939 64. Guerrero S, Imanuvilov OY (2013) Remarks on non controllability of the heat equation with memory. ESAIM Control Optim Calc Var 19(1):288–300 65. Gyongy I, Nualart D (1999) On the stochastic Burgers equation in the real line. Ann Probab 27(2):782–802 66. Gurtin M, Pipkin A (1968) A general theory of heat conduction with finite wave speed. Arch Rational Mech Anal 31:113–126 67. Hartmann J (1937) Theory of the laminar flow of an electrically conductive liquid in a homogeneous magnetic field. Det Kgl. Danske Videnskabernes Selskab Mathematisk-fysiske Meddelelser XV 6:1–27 68. Haussmann UG (1978) Asymptotic stability of the linear Ito equation in infinite dimensions. J Math Anal Appl 65:219–235 69. Hazewinkel M (2001) [1994] Gram matrix. Encyclopedia of mathematics. Springer Science+Business Media B.V./Kluwer Academic Publishers 70. Ichikawa A (1985) Stability of parabolic equations with boundary and pointwise noise. Lecture notes in control and information sciences 69. Springer, Berlin 71. Karafyllis I, Krstic M (2014) On the relation of delay equations to first-order hyperbolic partial differential equations. ESAIM: COCV 20:894–923

210

References

72. Kato T (1966) Perturbation theory for linear operators. Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer, New York 73. Komornik V (1994) Exact controllability and stabilization-the multiplier method. Masson 74. Krstic M (1999) On global stabilization of Burgers’ equation by boundary control. Syst Control Lett 37:123–141 75. Krstic M, Smyshlyaev A (2008) Boundary control of PDEs: a course on backstepping designs. SIAM, Philadelphia 76. Krstic M, Smyshlyaev A (2008) Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays. Syst Control Lett 57(9):750– 758 77. Kroner A, Rodrigues SS (2015) Remarks on the internal exponential stabilization to a nonstationary solution for 1D Burgers equations. SIAM J Control Optim 53(2):1020–1055 78. Kwieci´nska A (1999) Stabilization of partial differential equations by noise. Stoch Process Appl 79:179–184 79. Kwieci´nska A (2002) Stabilization of evolution equations by noise. Proc Am Math Soc 130(10):3067–3074 80. Lasiecka I, Triggiani R (1991) Differential and algebraic Riccati equations with application to boundary/point control problems: continuous theory and approximation theory. Lecture notes in control and information sciences. Springer, Berlin, p 164 81. Lasiecka I, Triggiani R (2000) Control theory for partial differential equations: continuous and approximation theories. Cambridge University Press, Cambridge 82. Lasiecka I, Triggiani R (2015) Stabilization to an equilibrium of the Navier-Stokes equations with tangential action of feedback controllers. Nonlin Anal Theory Meth Appl 121:424–446 83. Lee D, Choi H (2001) Magnetohydrodynamic turbulent flow in a channel at low magnetic Reynolds number. J Fluid Mech 439:367–394 84. Lefter C (2010) On a unique continuation property related to the boundary stabilization of Magnetohydrodynamic equations. Annals of Uiv AL.I CUZA Math, LVI 85. Lipster R, Shiraev AN (1989) Theory of martingales. Kluwer, Dordrecht 86. Liu HB, Hub P, Munteanu I (2016) Boundary feedback stabilization of Fisher’s equations. Syst Control Lett 97:55–60 87. Liu WJ (2003) Boundary feedback stabilization of an unstable heat equation. SIAM J Control Optim 42(3):1033–1043 88. Liu Z, Zheng S (1999) Semigroups associated with dissipative systems. Chapman & Hall/CRC research notes in mathematics 398. Chapman & Hall/CRC, Boca Raton, FL 89. Luo J (2008) Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays. J Math Anal Appl 342:753–760 90. Lopatinskij YaB (1953) A method of reduction of boundary-value problems for systems of differential equations of elliptic type to a system of regular integral equations. Ukrain Mat Zh 5:123–151 91. Mao X (1994) Exponential stability of stochastic. Differential equations. Pure and applied mathematics. Chapman & Hall, New York 92. Mao X (2013) Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control. Automatica 49(12):3677–3681 93. Marionschi G (2017) A note on the feedback stabilization of a Cahn Hilliard type system with a singular logarithmic potential. Solvability, regularity, and optimal control of boundary value problems for PDEs. Springer, Berlin, pp 357–377 94. Munteanu I (2011) Tangential feedback stabilization of periodic flows in a 2-D channel. Differ Integr Eqs 24(5–6):469–494 95. Munteanu I (2012) Normal feedback stabilization of periodic flows in a two-dimensional channel. J Optim Theory Appl 152(2):413–443 96. Munteanu I (2012) Normal feedback stabilization of periodic flows in a three-dimensional channel. Numer Funct Anal Optim 33(6):611–637 97. Munteanu I (2013) Normal feedback stabilization for linearized periodic MHD channel flow, at low magnetic Reynolds number. Syst Control Lett 62:55–62

References

211

98. Munteanu I (2013) Boundary feedback stabilization of periodic fluid flows in a magnetohydrodynamic channel. IEEE Trans Autom Control 58(8):2119–2125 99. Munteanu I (2014) Boundary stabilization of the phase field system by finite-dimensional feedback controllers. J Math Anal Appl 412:964–975 100. Munteanu I (2015) Boundary stabilization of the Navier-Stokes equation with fading memory. Int J Control 88(3):531–542 101. Munteanu I (2015) Stabilization of semilinear heat equations, with fading memory, by boundary feedbacks. J Differ Eqs 259:454–472 102. Munteanu I (2017) Boundary stabilization of a 2-D periodic MHD channel flow, by proportional feedbacks. ESAIM COCV 23(4):1253–1266 103. Munteanu I (2017) Stabilization of a 3-D periodic channel flow by explicit normal boundary feedbacks. J Dyn Control Syst 23(2):387–403 104. Munteanu I (2017) Stabilization of stochastic parabolic equations with boundary-noise and boundary-control. J Math Anal Appl 449(1):829–842 105. Munteanu I (2017) Stabilisation of parabolic semilinear equations. Int J Control 90(5):1063– 1076 106. Munteanu I (2018) Boundary stabilization of the stochastic heat equation by proportional feedbacks. Automatica 87:152–158 107. Munteanu I (2018) Boundary stabilization to non-stationary solutions for deterministic and stochastic parabolic-type equations. J Control Int. https://doi.org/10.1080/00207179.2017. 1407878 108. Murray JD (1993) Mathematical biology. Springer, Berlin 109. Pandolfi L (2013) Boundary controllability and source reconstruction in a viscoelastic string under external traction. J Math Anal Appl 407:464–479 110. Partington JR (2004) Linear operators and linear systems. London mathematical society student texts (60). Cambridge University Press, Cambridge 111. Da Prato G, Debussche A (1999) Control of the stochastic Burgers model of turbulence. SIAM J Control Optim 37(4):1123–1149 112. Da Prato G, Zabczyk J (1993) Evolution equations with white-noise boundary conditions. Stoch Rep 42:167–182 113. Da Prato G, Zabczyk J (2013) Stochastic equations in infinite dimensions. Cambridge University Press, Cambridge 114. Schuster E, Luo L, Krstic M (2008) MHD channel flow control in 2D: mixing enhancement by boundary feedback. Automatica 44:2498–2507 115. Takashima M (1996) The stability of the modified plane Poiseuille flow in the presence of a transverse magnetic field. Fluid Dyn Res 17:293–310 116. Triggiani R (1980) Boundary feedback stabilization of parabolic equations. Appl Math Optim 6:201–220 117. Triggiani R (2007) Stability enhancement of a 2-D linear Navier-Stokes channel flow by a 2-D wall normal boundary controller. Discret Contin Dyn Syst SB 8(2):279–314 118. Raymond JP (2006) Feedback boundary stabilization of the two-dimensional Navier-Stokes equations. SIAM J Control Optim 45:790–828 119. Raymond JP (2007) Feedback boundary stabilization of the three dimensional incompressible Navier-Stokes equations. Math Pures et Appl 87:627–669 120. Smyshlaev A, Krstic M (2005) On control design for PDEs with space-dependent diffusivity or time-dependent reactivity. Automatica 41:1601–1608 121. Ravindran SS (2000) Reduced-order adaptive controllers for fluid flows using POD. J Sci Comput 15:457–478 122. Rodrigues SS (2018) Feedback boundary stabilization to trajectories for 3D Navier-Stokes equations. Math Optim Appl. https://doi.org/10.1007/s00245-017-9474-5 123. Shirikyan A (2007) Exact controllability in projections for three-dimensional Navier-Stokes equations. Ann I. H. Poincaré 24:521–537 124. Vasquez R, Krstic M (2007) A closed-form feedback controller for stabilization of the linearized 2-D Navier-Stokes Poisseuille system. IEEE Trans Autom Control 52:2298–2312

212

References

125. Walsh JB (1986) An introduction to stochastic partial differential equations. Lecture notes in mathematical 1180. Springer, Berlin, pp 265–439 126. Xie B (2016) Some effects of the noise intensity upon non-linear stochastic heat equations on [0, 1]. Stoch Proc Appl 126:1184–1205 127. Xu C, Schuster E, Vazquez R, Krstic M (2008) Stabilization of linearized 2D magnetohydrodynamic channel flow by backstepping boundary control. Syst Control Lett 57:805–812 128. Yosida K (1980) Functional analysis. Springer, Berlin 129. Zhou J (2014) Optimal control of a stochastic delay heat equation with boundary-noise and boundary-control. Int J Control 87(9):1808–1821

Index

A Abstract parabolic, 22 Accretive operator, 11 Adapted process, 14 Adjoint operator, 8 Algebraic multiplicity, 7 B Banach space, 2 Bochner integrable, 5 Borel–Cantelli lemma, 16 Borel σ -algebra, 2 Boundary control problem, 22 Boundary value problem, 10 Brownian motion, 14 Burkholder–Davis–Gundy inequality, 16 C Cauchy problem, 11 Cauchy sequence, 2 Closed and densely defined operator, 7 Closed-loop equation, 23 Compact operator, 8 Complete measure space, 2 Conditional expectation, 13 Controlled Cahn–Hilliard system, 93 Controlled heat equation with delays, 109 Controlled magnetohydrodynamics equations, 78 Controlled Navier–Stokes equations, 49 Controlled stochastic equations, 128 C0 -semigroup, 12 C([0, T ]; X ), 5 C 1 ([0, T ]; X ), 5

D Dirichlet map, 25 Distributions space, 4 Dominated convergence theorem, 3 Dual space, 8 E Eigenvalue, 7 Eigenvector, 7 Ellipticity condition, 10 Elliptic operators, 10 Equilibrium solution, 22 Expectation E, 13 Extension operator, 8 F Feedback control, 23 Filtration, 14 Fourier series, 6 Fractional power, 9 Fubini’s theorem, 3 G Gaussian distribution, 14 Generalized eigenvector, 7 Geometric multiplicity, 7 Gram matrix, 1 H Hartmann–Poiseuille profile, 78 Hilbert–Schmidt norm, 7 Hille–Yosida theorem, 12 Hölder’s inequality, 5

© Springer Nature Switzerland AG 2019 I. Munteanu, Boundary Stabilization of Parabolic Equations, Progress in Nonlinear Differential Equations and Their Applications 93, https://doi.org/10.1007/978-3-030-11099-4

213

214 I Integrable function, 3 Internal control, 24 Itô’s isometry, 15

L Linearization, 20 Linear operators, 7 Lipschitz function, 12 Local martingale, 14 Lopatinskii condition, 10 L p (O), 4

M m-accretive operator, 11 Martingale, 13 Measurable function, 2 Measure, 2 Measure space, 2 Mild solution, 12 Modes, 6

N Null set, 2

O Operatorial norm, 7 Orthonormal basis, 7

P Parabolic Poiseuille profile, 50 Parseval’s identity, 6 Poincaré’s inequality, 5 Positive definite function, 9 Positive definite operators, 8 Powers of a linear operator, 9 Probability space, 13 Product measure, 3 Proportional feedback, 28

Index Q Quadratic variation, 14

R Random deterministic equation, 17 Random variable, 13 Rescaling, 17 Resolvent operator, 7 Resolvent set, 7 Riesz–Schauder–Fredholm theorem, 8

S Self-adjoint operator, 9 Semilinear evolution equation, 12 Semimartingale, 14 Semisimple eigenvalue, 7 σ -algebra, 2 Sobolev embeddings, 5 Sobolev space, 4 Spectrum, 7 Stabilizable controller, 23 Stabilization problem, 23 Stabilization to non-steady states, 24 Stabilization to trajectories, 171 Stable equilibrium, 22 Stochastic Chebyshev inequality, 16 Stochastic claculus, 16 Stochastic integral, 15 Stochastic processes, 13 Stopping time, 14 Symmetric operator, 8

U Unstable eigenvalues, 20

V Variation of constants formula, 12

W Weak solution, 13