Systems with Persistent Memory: Controllability, Stability, Identification (Interdisciplinary Applied Mathematics, 54) 3030802809, 9783030802806

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Interdisciplinary Applied Mathematics 54

Luciano Pandolfi

Systems with Persistent Memory Controllability, Stability, Identification

Interdisciplinary Applied Mathematics Volume 54 Series Editors Anthony Bloch, University of Michigan, Ann Arbor, MI, USA Charles L. Epstein, University of Pennsylvania, Philadelphia, PA, USA Alain Goriely, University of Oxford, Oxford, UK Leslie Greengard, New York University, New York, USA Advisory Editors Rick Durrett, Duke University, Durham, NC, USA Andrew Fowler, University of Oxford, Oxford, UK L. Glass, McGill University, Montreal, QC, Canada R. Kohn, New York University, New York, NY, USA P.S. Krishnaprasad, University of Maryland, College Park, MD, USA C. Peskin, New York University, New York, NY, USA S.S. Sastry, University of California, Berkeley, CA, USA J. Sneyd, University of Auckland, Auckland, New Zealand

Problems in engineering, computational science, and the physical and biological sciences are using increasingly sophisticated mathematical techniques. Thus, the bridge between the mathematical sciences and other disciplines is heavily traveled. The correspondingly increased dialog between the disciplines has led to the establishment of the series: Interdisciplinary Applied Mathematics. The purpose of this series is to meet the current and future needs for the interaction between various science and technology areas on the one hand and mathematics on the other. This is done, firstly, by encouraging the ways that mathematics may be applied in traditional areas, as well as point towards new and innovative areas of applications; and secondly, by encouraging other scientific disciplines to engage in a dialog with mathematicians outlining their problems to both access new methods as well as to suggest innovative developments within mathematics itself. The series will consist of monographs and high-level texts from researchers working on the interplay between mathematics and other fields of science and technology.

More information about this series at http://www.springer.com/series/1390

Luciano Pandolfi

Systems with Persistent Memory Controllability, Stability, Identification

Luciano Pandolfi Department of Mathematics “G.L. Lagrange” Politecnico di Torino (retired) Torino, Italy

ISSN 0939-6047 ISSN 2196-9973 (electronic) Interdisciplinary Applied Mathematics ISBN 978-3-030-80280-6 ISBN 978-3-030-80281-3 (eBook) https://doi.org/10.1007/978-3-030-80281-3 Mathematics Subject Classification: 35K20, 35L20, 35R09, 35R30, 45K05, 45Q05, 93C20 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed 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, expressed 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 Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The goal of this book is to introduce the readers to the rich field of systems with persistent memory in Hilbert spaces. Applications of this class of systems are mostly in viscoelasticity, thermodynamics with memory and in the study of diffusion processes in the presence of complex molecular structures, as for example in the study of diffusion in polymers. Clearly this case presents itself in biology. Most of these applications have been introduced from the second half of the nineteenth century, as shortly described in Chap. 1. Systems with persistent memory are an active field of research now, and their stability properties, identification of the parameters and controllability are widely studied. In order to make contact with this research field, we confine ourselves to the study of linear systems and to two archetypal classes of systems, which, respectively, reduce to the heat equation and to the wave equation when there is no memory. The techniques we encounter can be used for the study of larger classes of systems, as can be seen in the final sections of the chapters. The chapters of the book can be read independently, apart from Chap. 2, whose material is of general use. The content of the book can be described as follows. Chapter 1 studies a simple example of a system with memory in order to get a feeling of the properties to be expected and to contrast the properties of the heat equation with memory with those of the wave equation with memory. Some of the results proved in this chapter are then used in the study of an identification problem in Sect. 6.3 of Chap. 6. These results are explicitly recalled at the moment of their use so that Chap. 6 is self-contained. Chapter 2 presents the elements of functional analysis which are used to describe evolution systems with nonhomogeneous boundary conditions: we present the extension by transposition of an operator; a cursory description of the Laplace equation with Dirichlet and Neumann nonhomogeneous boundary conditions; and the use of semigroups and cosine operators in the study of evolution systems with nonhomogeneous boundary conditions.

v

vi

Preface

Chapter 3 studies the heat equation with memory. We study the existence and properties of the solutions and the controllability properties. The goal is to introduce the readers to the use of semigroup techniques and of “frequency domain methods”, i.e. methods based on the Laplace transformation. We achieve this goal by studying the equation and its controllability with both the approaches. We note that the frequency domain methods presented in this chapter can be used to study systems with fractional integrals and derivatives. Chapter 4 studies the wave equation with memory. Here, the goal is to make contact with the use of cosine operators and to understand that different boundary conditions make a great deal of difference since (similar to the memoryless wave equation) the system is “admissible” when the boundary condition is of Dirichlet type and is not when the boundary condition is of Neumann type. The goal of the chapter is achieved by studying in parallel the properties of the solutions and the control properties in the two cases when the boundary conditions are of Dirichlet or of Neumann type. Chapter 5 is an introduction to the study of stability, a very active field of research now. Stability results are derived with the use of three different techniques: frequency domain techniques, energy inequalities and Dafermos semigroup representation of the system in the “memory space”. These techniques are separately studied in this chapter, but of course they can be combined, and, for example, energy estimates can be used together with the semigroup approach, as can be seen from the references listed in the concluding section of the chapter. Chapter 6 introduces the reader to the mathematical techniques used to identify the physical parameters of the equation with memory. We consider dynamical methods for the identification of the memory kernel or for the identification of elastic coefficients which describe the interaction with the external environment. The “quasistatic” methods mostly used in the engineering literature are shortly commented. The arguments in Chaps. 3–6 are described in detail, but do not exhaust the field. Chapter 7 hints in a sketchy and informal way to some of the important topics which are not covered in this book. The goal is to direct the reader to search the literature on the several facets of the theory of systems with persistent memory. The book is intended for both mathematicians and research engineers and it could be used in doctorate courses. For this reason Chap. 2 presents certain arguments of functional analysis with which young researchers in particular may need to familiarize. These arguments are ubiquitous in the study of boundary control systems with or without memory.

Acknowledgements The author is a retired professor from the Dipartimento di Scienze Matematiche “G. L. Lagrange” of the Politecnico di Torino. The author wishes to thank the Politecnico and the Department for the opportunity to continue his research activity.

Torino, Italy December 2020

Luciano Pandolfi

Contents

1

2

Preliminary Considerations and Examples . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Systems with Persistent Memory Studied in This Book . . . . . . . 1.2 Heuristics on the Wave and Heat Equations with and Without Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Archetypal Model: The String Equation with Memory . . . . . . . . 1.3.1 The String Equation Without Memory on a Half Line . . . . . . 1.3.1.1 A Control Problem for the String Equation Without Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The String with Persistent Memory on a Half Line . . . . . . . . . . . . . . . 1.4.1 Finite Propagation Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 A Formula for the Solutions and a Control Problem . . . . . . . 1.4.2.1 From the Regularity of the Input to the Interior Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2.2 The Response Operator . . . . . . . . . . . . . . . . . . . . . . . 1.4.2.3 From the Regularity of the Target to the Regularity of the Control . . . . . . . . . . . . . . . . . . . . . . 1.5 Diffusion Processes and Viscoelasticity: the Derivation of the Equations with Persistent Memory . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Thermodynamics with Memory and Non-Fickian Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 A Problem of Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.4 Physical Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1

26 28 32 33 33

Operators and Semigroups for Systems with Boundary Inputs . . . . . . 2.1 Preliminaries on Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Continuous Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 The Complexification of Real Banach or Hilbert Spaces . . . . 2.1.3 Operators and Resolvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Closed and Closable Operators . . . . . . . . . . . . . . . . . . . . . . . . .

37 37 37 39 40 41

3 7 7 11 12 15 17 19 25 25 26

vii

viii

Contents

2.1.5

The Transpose and the Adjoint Operators . . . . . . . . . . . . . . . . 41 2.1.5.1 The Adjoint, the Transpose and the Image of an Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.1.6 Compact Operators in Hilbert Spaces . . . . . . . . . . . . . . . . . . . 43 2.2 Integration, Volterra Integral Equations and Convolutions . . . . . . . . . 45 2.3 Laplace Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.3.1 Holomorphic Functions in Banach Spaces . . . . . . . . . . . . . . . 49 2.3.2 Definition and Properties of the Laplace Transformation . . . 50 2.3.3 The Hardy Space H 2 (Π+ ; H) and the Laplace Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.4 Graph Norm, Dual Spaces, and the Riesz Map of Hilbert spaces . . . 55 2.4.1 Relations of the Adjoint and the Transpose Operators . . . . . . 58 2.5 Extension by Transposition and the Extrapolation Space . . . . . . . . . . 60 2.5.1 Selfadjoint Operators with Compact Resolvent . . . . . . . . . . . . 64 2.5.1.1 Fractional Powers of Positive Operators with Compact Resolvent . . . . . . . . . . . . . . . . . . . . . . 66 2.6 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.6.1 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.6.1.1 Sobolev Spaces of Hilbert Space Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.6.1.2 Sobolev Spaces of Any Real Order . . . . . . . . . . . . . 75 2.7 The Laplace Operator and the Laplace Equation . . . . . . . . . . . . . . . . . 75 2.7.1 The Laplace Equation with Nonhomogeneous Dirichlet Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.7.2 The Laplace Equation with Nonhomogeneous Neumann Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.8 Semigroups of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.8.1 Holomorphic Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.9 Cosine Operators and Differential Equations of the Second Order . . 92 2.10 Extensions by Transposition and Semigroups . . . . . . . . . . . . . . . . . . . 96 2.10.1 Semigroups, Cosine Operators, and Boundary Inputs . . . . . . 97 2.11 On the Terminology and a Final Observation . . . . . . . . . . . . . . . . . . . . 101 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3

The Heat Equation with Memory and Its Controllability . . . . . . . . . . . 103 3.1 The Abstract Heat Equation with Memory . . . . . . . . . . . . . . . . . . . . . . 103 3.2 Preliminaries on the Associated Memoryless System . . . . . . . . . . . . . 105 3.2.1 Controllability of the Heat Equation (Without Memory) . . . . 111 3.3 Systems with Memory, Semigroups, and Volterra Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 3.3.1 Projection on the Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . 117 3.4 The Definitions of Controllability for the Heat Equation with Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.5 Memory Kernels of Class H 1 : Controllability via Semigroups . . . . . 122

Contents

ix

3.5.1

Approximate Controllability Is Inherited by the System with Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.5.2 Controllability to the Target 0 Is Not Preserved . . . . . . . . . . . 126 3.6 Frequency Domain Methods for Systems with Memory . . . . . . . . . . . 130 3.6.1 Well Posedness via Laplace Transformation . . . . . . . . . . . . . . 137 3.7 Controllability via Laplace Transformation . . . . . . . . . . . . . . . . . . . . . 150 3.7.1 Approximate Controllability Is Inherited by the System with Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 3.7.2 Controllability to the Target 0 Is Not Preserved . . . . . . . . . . . 155 3.8 Final Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 4

The Wave Equation with Memory and Its Controllability . . . . . . . . . . . 163 4.1 The Equations with and Without Memory . . . . . . . . . . . . . . . . . . . . . . 163 4.1.1 Admissibility and the Direct Inequality for the System Without Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 4.2 The Solution of the Wave Equation with Memory . . . . . . . . . . . . . . . . 172 4.2.1 Admissibility and the Direct Inequality for the Wave Equation with Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 4.2.1.1 Admissibility and Fourier Expansion . . . . . . . . . . . . 176 4.2.2 Finite Speed of Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 4.2.3 Memory Kernel of Class H 2 and Compactness . . . . . . . . . . . 180 4.3 The Definitions of Controllability and Their Consequences . . . . . . . . 182 4.3.1 Controllability of the Wave equation (Without Memory) with Dirichlet Boundary Controls . . . . . . . . . . . . . . . . . . . . . . 184 4.3.1.1 Fourier Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 185 4.3.2 Controllability of the Wave Equation (Without Memory) with Neumann Boundary Controls . . . . . . . . . . . . . . . . . . . . . . 186 4.3.2.1 Fourier Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 192 4.3.3 Controllability of the Wave Equation and Eigenvectors . . . . . 193 4.4 Controllability of the Wave Equation with Memory: The Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 ∗ ∗ and VˆN,T . . . . . . . . . . . . . . . . . . . . . . . . . 197 4.4.1 Computation of VˆD,T 4.5 Wave Equation with Memory: the Proof of Controllability . . . . . . . . 199 4.6 Final Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

5

The Stability of the Wave Equation with Persistent Memory . . . . . . . . 223 5.1 Introduction to Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 5.2 The Memory Kernel When the System Is Stable . . . . . . . . . . . . . . . . . 230 5.2.1 Consequent Properties of the Memory Kernel M(t) . . . . . . . . 234 ˆ 5.2.1.1 The Real and the Imaginary Parts of M(λ) . . . . . . . 236 5.2.1.2 The Resolvent Kernel R(t) of −M(t) . . . . . . . . . . . . 238 5.2.2 Positive Real (Transfer) Functions . . . . . . . . . . . . . . . . . . . . . . 240 5.3 L 2 -Stability via Laplace Transform and Frequency Domain Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

x

Contents

5.3.1 L 2 -Stability When w˜ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 5.3.2 The Memory Prior to the Time 0 . . . . . . . . . . . . . . . . . . . . . . . 251 5.4 Stability via Energy Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 5.5 Stability via the Semigroup Approach of Dafermos . . . . . . . . . . . . . . 263 5.5.1 Generation of the Semigroup in the History Space . . . . . . . . . 271 5.5.2 Exponential Stability via Semigroups . . . . . . . . . . . . . . . . . . . 274 5.6 Final Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 6

Dynamical Algorithms for Identification Problems . . . . . . . . . . . . . . . . . 283 6.1 Introduction to Identification Problems . . . . . . . . . . . . . . . . . . . . . . . . . 283 6.1.1 Deconvolution and Numerical Computation of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 6.2 Dynamical Algorithms for Kernel Identification . . . . . . . . . . . . . . . . . 288 6.2.1 A Linear Algorithm with Two Independent Measurements: Reduction to a Deconvolution Problem . . . . . . . . . . . . . . . . . . 289 6.2.2 One Measurement: A Nonlinear Identification Algorithm . . . 296 6.2.3 Quasi Static Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 6.3 Dynamical Identification of an Elastic Coefficient . . . . . . . . . . . . . . . 307 6.3.1 From the Connecting Operator to the Identification of q(x) . 311 6.3.2 The Blagoveshchenskiˇı Equation and the Computation of the Connecting Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 6.3.2.1 Well Posedness of the Blagoveshchenskiˇı Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 6.4 Final Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

7

Final Miscellaneous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 7.1 Solutions of a Nonlinear System with Memory: Galerkin’s Method . 331 7.2 Asymptotics of Linear Heat Equations with Memory Perturbed by a Sector Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 7.2.1 Dafermos Method for the Heat Equation with Memory . . . . . 338 7.2.2 Asymptotic Properties of the Perturbed Equation . . . . . . . . . . 340 7.3 Controllability and Small Perturbations . . . . . . . . . . . . . . . . . . . . . . . . 342 7.3.1 The Belleni-Morante Method and Controllability . . . . . . . . . . 344 7.4 Memory on the Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 7.5 A Glimpse to Numerical Methods for Systems with Persistent Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

Chapter 1

Preliminary Considerations and Examples

1.1 The Systems with Persistent Memory Studied in This Book A model for systems with persistent memory is the equation ∫ t w (t) = αAw(t) + N(t − s)Aw(s) ds + h(t) , t > 0,

w(0) = w0 . (1.1)

0

The operator A is an operator in a Hilbert space, and in archetypal cases A is the laplacian in a region Ω with suitable boundary conditions (possibly non-homogeneous, for example when the system is controlled on the boundary). In this section, we get a feeling of the properties of Eq. (1.1) and, at the end, we outline some of the physical problems that are modeled by this equation. The final result of this chapter, Theorem 1.15, will be recalled in Chap. 6 and used to solve an identification problem. In applications, (1.1) represents the evolution in time of the measure of a certain “quantity” (as the temperature, concentration, deformation, and so on). This fact imposes restrictions on α and on N(t). Let us consider the case that A = Δ (for example with Dirichlet boundary conditions). The first restriction is α ≥ 0. In fact, if α < 0 and N(t) = 0, then Eq. (1.1) is the backward heat equation, which is not solvable, unless w0 has special regularity properties. If α = 0 and N(t) is smooth, then it must be N(0)  0. In fact, for example ∫ t w (t) = (t − s)Δ(s) ds 0

is the integrated version of

w  = Δw .

This problem, for example on (0, π) with Dirichlet homogeneous boundary conditions, is not well posed, see [23, p. 99]. When N(t) and h(t) are differentiable, α = 0 and N(0) > 0, Eq. (1.1) can be written as © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 L. Pandolfi, Systems with Persistent Memory, Interdisciplinary Applied Mathematics 54, https://doi.org/10.1007/978-3-030-80281-3_1

1

2

1 Preliminary Considerations and Examples

w (t) = N(0)Aw(t) +

∫

t

N (t − s)Aw(s) ds + h (t) ,

0



w(0) = w0 , w (0) = w1 .

(1.2)

We prefer to rewrite the equation as w (t) = Aw(t) −

∫

t

M(t − s)Aw(s) ds + h(t)

(1.3)

0

with N(0) absorbed in the operator A, M(t) = −N (t)/N(0) and h (t) renamed to h(t). The reason is that physical considerations impose M(t) = −N (t)/N(0) ≥ 0 (see Sect. 1.5.4) although this condition is not used in this book with the exception of Chap. 5 where it has a crucial role to ensure stability of the system. Systems of the form (1.1) have been introduced in the second half of the nineteenth century by several authors and for different reasons. One of the earliest instances is a paper by J.C. Maxwell, which uses (1.1) to model a viscoelastic fluid (see [41]). In that paper J.C. Maxwell introduced a device, which then became standard in the engineering description of viscoelastic materials: any element of a viscoelastic fluid is described as a “cell,” which contains both a spring and a damper in series, so that the stress solves 1 σ (x, t) = − σ(x, t) − κ∇w(x, t) τ

τ > 0, κ > 0 .

Combining this material law with conservation of momentum (see Sect. 1.5.2), we get (1.3) with a memory kernel which is a decreasing exponential, βe−ηt . If instead the element is modeled as a spring and damper in parallel, then we get Voigt’s model, which is more suited when studying solids. Equation (1.1) with α > 0 is a generalization of Voigt’s model. As first proved in [38], glasses are viscoelastic materials. At the time, glasses were used as dielectrics in capacitors (Leiden jars). Following a suggestion of J.C. Maxwell, the viscoelastic properties of glasses were exploited in [31, 32] to explain the relaxation in time of the potential of a Leiden  jar. In these papers, J. Hopkinson realized that a kernel of the form N(t) = βk e−ηk t can be used to satisfactorily fit experimental data and this fact was interpreted as follows: a glass constituted by pure silica would produce a discharge law whose relaxation in time is represented by one exponential, N(t) = βe−ηt . Real glasses are a mixture, each component being responsible for a relaxation of the potential with its own exponential βe−ηt : these different exponentials have to be linearly combined in order to have an accurate description of the process. This justifies the use of the “relaxation kernel”  N(t) = βk e−ηk t , βk > 0 , ηk ≥ 0, which is called a Prony sum in the engineering literature. Hopkinson’s research is an early instance of the solution of the inverse problem we study in Sect. 6.2. Few years later, in 1889, M.J. Curie in his studies on piezoelectricity noted that the discharge of capacitors with certain crystals as dielectric is best described by

1.2 Heuristics on the Wave and Heat Equations with and Without Memory

3

N(t) = (const)/tγ (see [14] and [63]). This leads to a fractional integral law for the current released by a capacitor with crystalline dielectric. Fractional integrals are now much studied, and we make contact with the techniques used in this kind of study in Sects. 3.6 and 3.7. The memory effect in viscoelasticity was studied in general by L. Boltzmann (see [7, 8]) and then by V. Volterra (see [59–62]), while the study of the memory effect of ferromagnetic materials was initiated by D. Graffi in [27, 28]. Note that when N(t) = βe−ηt and α = 0, then (1.1) (with A = Δ and F(t) = 0) is the integrated version of the telegrapher’s equation w  = −ηw  + βΔw

(β > 0, η > 0),

(1.4)

and it is known that solutions of (1.4) have a wave front, i.e. a discontinuity of the solutions, in the direction of the propagation, but not backward (unless η = 0 when the equation reduces to the wave equation). This “forward wave front” is observed in the diffusion of solutes in solvents with complex molecular structure, and for this reason, (1.4) and more in general (1.1) are used also to model non-Fickian diffusion (see [12, 16]), for example in biology. Equation (1.4) was first introduced in thermodynamics in [10] as a “hyperbolic” model for heat diffusion with a finite propagation speed. Finally, the authors, in [13, 29], introduced a general equation with memory (which can be reduced to (1.1)) to model heat processes. Stimulated by so many applications, distributed systems with persistent memory described by (1.1), with suitable initial and boundary conditions, became a subject of intense study (see for example the books [1, 12, 22, 49, 50, 54]) and [39, Chap. 5]. A rich overview on the history of (1.1) is in [36, 37], and a deep study of the existence and properties of the solutions for a class of systems far more general than those considered in this book is in [56].

1.2 Heuristics on the Wave and Heat Equations with and Without Memory Equation (1.1) (with A = Δ) is the heat equation if α > 0 and N(t) = 0; it is the wave equation if α = 0 and N(t) = 1. So, Eq. (1.1) is expected to have properties that are intermediate between those of the wave and of the heat equations. The heuristic considerations in this section aim at an illustration of this intuition. We use the following shorthand notation, when A = Δ: • if α > 0, then Eq. (1.1) is the heat equation with memory (shortly, Heat-M); • if α = 0 and N(t) differentiable with N(0) > 0, then Eq. (1.1) (often written in the form (1.3)) is the wave equation with memory (shortly, Weave-M); • the solution of the equation with memory is denoted w, while u denotes the solution of the heat or wave equation without memory.

4

1 Preliminary Considerations and Examples

In this section, we informally compare the properties of the Weave-M, the Heat-M, and the wave and heat equations (without memory). We consider ∫ t N(t − s)wxx (s)ds , x ∈ (0, π) (1.5) w  = wxx + 0

and

 w(0, t) = w(π, t) = 0 ,

w(x, 0) =

1 if x ∈ ( (19π/20), (21π/20) ) 0 otherwise.

The memory kernel is the Prony sum N(t) = 3e−t − 3e−2t + e−4t

(note N(0) = 1) .

Remark 1.1. (Almost) vertical segments in the graphs correspond to jumps of the functions. The segments are almost vertical since the values of the functions are computed at finitely many points and then linearly interpolated. Figure 1.1 compares the graphs of the solutions of the heat equation with and without memory (left) and those of the wave equation with and without memory (right). The figures present the graphs of the solutions1 at the time π/4, i.e. u(x, π/4) (green) and of w(x, π/4) (red) and the graph of the initial condition (yellow). The solution of Heat-M looks quite similar to that of the heat equation without memory. In particular, both the graphs are never zero on an entire subinterval. Instead, the solution of Weave-M displays a significant similarity and also a significant difference with respect to that of the wave equation. So, we concentrate on this case. w w

x

x

Fig. 1.1 The solution at the time T = π/4 of Heat-M compared with the solution of the heat equation (left) and that of Weave-M compared with the solution of the wave equation (right). The yellow graph is that of the initial condition (as explained in Remark 1.1, vertical segments of the graphs indicate jumps of the functions) 1

We recall that u is the solution of the system without memory, while w is the solution of the system with memory. The graphs are obtained from truncated Fourier series.

1.2 Heuristics on the Wave and Heat Equations with and Without Memory

5

w

x

Fig. 1.2 The solutions of the wave and Weave-M equations after the reflection. Note that the solution of Weave-M is not zero on any interval

The plot shows that the solution of Weave-M did not “move too much” in space: the initial “wave” propagates with  finite speed, like in the case of the wave equation: the velocity of the “wave” is N(0) (which is 1 in the example) so that at time t = π/4 the wave did not hit the ends of the interval and it displays the same wave front in the direction of the motion as the wave equation, but it does not became zero backward: the solution of the wave equation leaves no memory of itself after it has passed over a point P, while the memory of Weave-M persists. This is confirmed by Fig. 1.2, which shows the solutions of Weave-M and of the wave equation (without memory) soon after the time T = π/2, i.e. soon after the signal has been reflected back at the ends of the interval (0, π). A further difference between the wave equations with and without memory is that the plot of the solution of Weave-M is significantly “lower” than that of the equation without memory: it looks as if the memory would dissipate energy. This property is examined in Chap. 5. The plots in Fig. 1.3 decompose the solutions of the wave equations with and without memory computed at T = π/4 (the ones displayed in Fig. 1.1) in the contributions of the low and high frequencies. The figure presents the graphs of the truncation of the Fourier series up to the thirtieth eigenvalue (left) and the sum of the components of higher frequency (right). These figures display the well known fact that at high frequency a viscoelastic material behaves much like a purely elastic one, and the discrepancy is mostly due to the low frequency components. The graphs discussed up to now suggest that the Weave-M equation should approximate the wave equation. This is so when N(t) is “constant enough” as seen from Fig. 1.4 (left), which superimposes the solutions of (1.5) with the kernel N(t) = e−kt for several values of k, k = 1, 1/2, 1/3, 1/4 (the initial condition is a step function centered at t = π/2, as in the previous figures, but it is not represented in order to make the graphs of the solutions more “separated”). The plots represent the (trun-

6

1 Preliminary Considerations and Examples

w

w

x x

Fig. 1.3 Low (left) and high (right) frequency content of the solutions at the time T = π/4 of the wave equation (green) and Weave-M equation (red) w w x

x

Fig. 1.4 The solutions of Weave-M compared with the solutions of the wave equation when k(t) is “flat” (left) and with the solution of the heat equation when k(t) is “steep” (right) (ripples are due to the truncation of the Fourier series). Note that the solutions of Weave-M have jumps, while the solution of the heat equation is smooth

cation of the Fourier series2 of the) solution at time 7π/10, soon after the reflection of the waves. It is seen that as the memory kernel becomes more and more flat, the solution of Weave-M gives a good approximation of the solution of the wave equation (black). The graphs on the right (computed at the time 7π/10 also in this case) are the solutions when N(t) = ke−kt , k = 5, 10, 30, 40. It is seen that when N(t) becomes more and more steep, then the solutions of Weave-M approximate3 that of the heath equation (black graph). This last observation explains the reason for using Weave-M (with a suitable kernel) as the equation of certain thermal processes. 2

Truncation is responsible for the ripple in the graphs. But the solution of the heat equation is far smoother than that of the Weave-M, a fact that can be seen in the plots only when k is small. Note the jumps—represented as vertical segments—in the blue plot.

3

1.3 The Archetypal Model: The String Equation with Memory

7

See [11] for a precise analysis of this phenomenon for general memory kernels. Finally, we cite few papers that study the velocity of propagation of the Weave-M. Among the relevant papers, we cite [18, 25, 30, 42] and in particular [48]. It is proved in this last paper that signals propagate with a finite speed if the memory kernel M(t) in (1.3) is bounded and in this case the initial conditions are not smoothed at large time. If instead M(t) ∼ 1/tγ , γ ∈ (0, 1), like in Curie’s law, then the velocity of propagation is still finite as in the wave equation, but the solution x → w(x, t) is smoothed when t increases.

1.3 The Archetypal Model: The String Equation with Memory In this section, we study the string equation with memory, i.e. Eq. (1.1) with (Aw)(x) = wxx (x) + q(x)w(x) and α = 0. As explained in Remark 1.18, the term q(x)w(x) describes the elastic interaction of the viscoelastic string with the environment. By using MacCamy’s trick described in Chap. 4, when M(t) ∈ C 2 , the string equation with memory can be rewritten in the form4 ∫ t w (x, t) = wxx (x, t)+q(x)w(x, t)+ K(t −s)w(x, s) ds , x > 0 , t > 0 . (1.6) 0

This is the equation we study now, when K(t) is continuous (a consequence of M ∈ C 2 ) and when q(x) is continuous. We want to understand something of the solutions and to study a control problem when the string is controlled on its left. So, we impose the initial and the boundary conditions w(0, t) = f (t) . (1.7) w(x, 0) = 0 , w (x, 0) = 0 , First we present few information on the string equation without memory.

1.3.1 The String Equation Without Memory on a Half Line We use the letter u instead of w when K = 0. We consider the string equation   t > 0, x > 0 u = u xx + F(x, t) , (1.8) u(x, 0) = u0 (x) , u (x, 0) = v0 (x) , u(0, t) = f (t) (note that we did not insert q(x) in this equation). As seen in [53, Chap. II Sect. 2.9], the solution is the sum of three functions:

4

The coefficient of w(x) in (1.6) is q(x) + b with b ∈ R, renamed to q(x); see (4.35) in Sect. 4.2.3.

8

1 Preliminary Considerations and Examples

u(x, t) = u(x, t; u0, v0 ) + u f (x, t) + u(x, t; F),

(1.9)

where • u(x, t; u0, v0 ) is the contribution of the initial conditions, ∫ 1 1 x+t ψ(s) ds . u(x, t; u0, v0 ) = [φ(x + t) + φ(x − t)] + 2 2 x−t

(1.10a)

The functions φ(x) and ψ(x) are the odd extensions to R of u0 (x) and v0 (x). • u f (t) is the contribution of the boundary input f , u f (x, t) = f (t − x)H(t − x),

(1.10b)

where H(t) is the Heaviside function H(t) = 1 if t ≥ 0 and H(t) = 0 if t < 0. This is an abuse of notation, since f is defined only if its argument is nonnegative. The notation wants to indicate that we consider the extension of f (t) which is zero for t < 0. • u(x, t; F) is the contribution of the distributed affine term F, ∫ ∫ ∫ 1 t x+t−τ 1 F(ξ, τ) dξ dτ = F(ξ, τ) dξ dτ . (1.10c) u(x, t; F) = 2 D(x,t) 2 0 |x−t+τ | In order to illustrate the structure of the domain of integration D(x, t), we represent the space variable ξ on the horizontal axis and the time variable τ on the vertical axis (Fig. 1.5). In the plane (ξ, τ), we fix a point (x0, t0 ) and we represent the domain of integration D(x0, t0 ), which is identified by the inequalities 0 ≤ τ ≤ t0 ,

|(x0 − t0 ) + τ| ≤ ξ ≤ x0 + t0 − τ .

τ τ (x0,t0)

t0−x0

D(x ,t ) 0 0

(x ,t ) 0 0

D(x ,t ) 0 0

x0−t0

x0+t0

ξ

t0−x0

t +x 0

0

ξ

Fig. 1.5 The domains D(x0, t0 ). Note that in order to compute u(x0, t0 ), the function F(x, t) has to be known with t up to t0 and x up to t0 + x0 > x0 . (This figure reproduces Fig. 1.1 in [43].)

1.3 The Archetypal Model: The String Equation with Memory

9

We have two cases: if x0 ≥ t0 ≥ 0, then D(x0, t0 ) is the triangle of the plane (ξ, τ) identified by the characteristics τ − t0 = ξ − x0 ,

τ − t0 = −(ξ − x0 )

(1.11)

(Fig. 1.5, left), and otherwise, if 0 ≤ x0 ≤ t0 , then D(x0, t0 ) is the quadrilateral in Fig. 1.5, right. A classical solution of (1.8) is a function u(x, t), which is of class C 2 in (0, +∞) × (0, +∞), continuous with continuous derivatives on [0, +∞) × [0, +∞) and which satisfies both the equation and the prescribed initial and boundary conditions. It is easily seen that classical solutions rarely exist. Let for example u0 (x) = 0, v0 (x) = 0, and F(x, t) = 0. The solution (1.10b) is  f (t − x) if x < t u f (x, t) = (1.12) 0 otherwise . The function u f (x, t) is not continuous on the line x = t, not even if f ∈ C 2 . In this case, we have continuity if f (0) = 0, we have u ∈ C 1 if we also have f (0) = 0, and in order to have u ∈ C 2 (hence a classical solution), we must also require f (0) = 0. In particular, we have the following lemma.5 Lemma 1.2. If f ∈ D(0, +∞), then the function u f (x, t) in (1.12) is a classical solution of (1.8) with u0 = v0 = 0, F = 0. Let us consider the restriction of the solutions of (1.8) to a rectangle R(X, T) = {(x, t)

0 ≤ x ≤ X,

0 ≤ t ≤ T}

and a sequence { fn } of D(0, T) functions. Let u fn (x, t) be the classical solution of (1.8) with u0 = v0 = 0, F = 0, and the boundary input fn . The function u fn (x, t) is given by (1.12) with f = fn . The function u fn (x, t) is a classical solution in the rectangle R(X, T). Let f ∈ L 2 (0, T). There exists a sequence { fn } in D(0, T) such that fn → f in L 2 (0, T). Then u fn → u f in L 2 (R(X, T)) for every X > 0, T > 0: the function in (1.12) is not even continuous in general, but it is the L 2 -limit of classical solutions. We call it a mild solution. In order to state more properties of the solutions u f , we need the Lebesgue Theorem on continuity of the shift (see [19, Example 5.4 p. 39]). If f = f (t) ∈ L 2 (0, T) and s > 0, we consider the function t → f (·; t) from [0, T] to L 2 (0, X) (any X > 0) where f (s; t) = f (t − s)H(t − s) . We have the following lemma. Lemma 1.3. Let f (t) be defined for t > 0, and let f ∈ L 2 (0, T) for every T > 0. The transformation t → f (·; t) is continuous from [0, +∞) to L 2 (0, X) for every X > 0. We recall the notation for 0 < T ≤ +∞: f ∈ D(0, T ) when f ∈ C ∞ (0, T ) with compact support in (0, T ).

5

10

1 Preliminary Considerations and Examples

Lemma 1.3 can be reformulated as follows: Theorem 1.4. Let f ∈ L 2 (0, T), and let us consider the restriction of u f (x, t) to R(X, T). The function t → u f (·, t) belongs to C([0, T]; L 2 (0, X)) and the transformation f → u f is continuous from f ∈ L 2 (0, T) to C([0, T]; L 2 (0, X)). Similar considerations can be repeated when F and the initial conditions are not zero (we do not make an effort to be precise about that now), and these considerations lead to the following definition: Definition 1.5. The function u in (1.9) with u(x, t; u0, v0 ), u f (x, t), and u(x, t; F) given by (1.10a)–(1.10c) is the mild solution (shortly, the solution) of (1.8). The previous formulas clearly display the finite propagation speed of the string equation: • let F = 0 and f = 0. If u0 (x) = 0, v0 (x) = 0 for x > X > 0, then u(x, t; u0, v0 ) = 0 if x > X +t: signals due to the initial conditions propagate with finite speed (equal to 1, since 1 is the coefficient in front of u xx ). • Let u0 = 0, v0 = 0, and F = 0. Then, – u(x, t) = 0 for x > t: also the signal “injected” from the boundary travels with velocity 1. – let f (t)  0 for t ∈ [a, b] and f (t) = 0 for t ∈ [0, a) (when a > 0) and for t > b. Then, u(x, t) = 0 when t < x + a and when t > x + b. Let us look at the solution by “sitting” at a fixed position x0 . We see that the signal acts on the position x0 during the interval of time [x0 + a, x0 + b]. In particular, the fact that u(x0, t) is zero for t > x0 + b can be expressed in a picturesque way by saying that the “wave” u excited by f passes over the point x0 without leaving any memory of itself in the future. An important consequence of Lemma 1.3 concerns the derivative σf (x, t) = ∂u f (x, t)/∂ x, which is interpreted as the traction in position x at the time t, see Sect. 1.5.2. We wonder under which condition on f the traction exerted by the string on the support of the left end x = 0 is square integrable and it is the “trace” on the boundary of the traction in the interior points. In order to clarify this point, we introduce the space  ∫t H−1 (0, T) = f (t) = 0 g(s) ds , g ∈ L 2 (0, T) , (1.13) f H−1 (0,T ) = g L 2 (0,T ) . It turns out that f ∈ H−1 (0, T) is continuous and a.e. differentiable, with f (t) = g(t) a.e. and that H−1 (0, T) is a Hilbert space, in fact a closed subspace of the space H 1 (0, T) introduced in Sect. 2.6.1. Readers not yet familiar with this notion may assume f ∈ C 1 with f (0) = 0 and consider the warning in footnote 6.

1.3 The Archetypal Model: The String Equation with Memory

11

If f ∈ H−1 (0, T), then u f (x, t) ∈ C([0, X] × [0, T]) for every T > 0 and X > 0; that is, u f (x, t) is continuous also on the diagonal x = t (and of course it is zero for x > t). Note that the condition f (0) = 0 is crucial to have continuity of x → u x (x, t) at the point x = t and that for x  t σf (x, t) =

∂ ∂ u f (x, t) = [ f (t − x)H(t − x)] = − f (t − x)H(t − x). ∂x ∂x

We rename to x the variable s in Lemma 1.3 and we interchange the roles of t and x = s. Lemma 1.3 shows that, when considered as an L 2 (0, T)-valued function of the variable x, the map x → σf (x, ·) is an L 2 (0, T)-valued continuous function6 and so Theorem 1.6. Let T > 0. The map f → σf (0, t) = (∂/∂ x)u f (0, t) ∈ L 2 (0, T) is defined for every f ∈ H−1 (0, T) and lim σf (0, t) − σf (x, t) L 2 (0,T ) = 0 .

x→0+

In this sense, we say that the traction on the left end x = 0 is the “trace” on the boundary of the trace at interior points.

1.3.1.1 A Control Problem for the String Equation Without Memory We consider the wave equation (1.8) with u0 = 0, v0 = 0, and F = 0. We fix L > 0 and a target U ∈ L 2 (0, L). We intend that f ∈ L 2 (0, T) is a control which should steer the zero initial condition to hit the target U, i.e. we want u(x, T) = u f (x, T) = f (T − x)H(T − x) = U(x) if x ∈ (0, L) . Nothing is required if x > L. It is clear that • such control f does not exist if T < L since in this case u = 0, possibly different from U(x), if x ∈ (T, L); • if T = L, then the steering control7 exists and it is unique (for t ≤ L), given by f (t) = U(T − t) ;

(1.14)

• if T > L, then the control f exists, given by (1.14) for t ∈ (T − L, T) and otherwise arbitrary. These observations show the properties we state now. Before stating these properties, it is convenient to introduce a useful notation. the conditions f ∈ C 1 ([0, T ]) and f (0) = 0 do not imply that x → u f (x, t) is of class C 1 since the derivative does not exist at x = t, unless f  (0) = 0. For this reason, we prefer H 1 to C 1 . 7 That is, a control such that u (x, T ) = U(x). f 6

12

1 Preliminary Considerations and Examples

Notation Let a time T ≥ 0 be given. The notation TX denotes the number T when it is used to identify a point of the space axis. So (0, T) and (0, TX ) both denote intervals of length T whose left hand is 0, but (0, T) is in the time axis while (0, TX ) is in the space axis.

With this notation, we can state the following theorem. Theorem 1.7. Let T > 0. The following properties hold: 1. if U ∈ L 2 (0, TX ), then there exists a unique steering control fU ∈ L 2 (0, T) and the transformation U → fU is continuous from L 2 (0, TX ) to L 2 (0, T); 2. the function fU has the same regularity as U. In particular, a. if U(x) is continuous on [0, TX ], then fU (t) is continuous on [0, T] and U(TX ) = lim− u(x, T) = lim+ fU (t) = fU (0+ ) ; x→TX

t→0

b. let p ≥ 0 be an integer. If U ∈ C p ([0, TX ]), then fU ∈ C p ([0, T]); if U ∈ H p (0, TX ), then fU ∈ H p (0, T) (the definition of the Sobolev space H p is in Sect. 2.6.1), and the transformation U → fU is continuous from H p (0, TX ) to H p (0, T). Remark 1.8. These results can be interpreted as follows: in time T, the “waves” issued from the left hand x = 0 fill the space interval (0, TX ), and every target U(x) ∈ L 2 (0, TX ) can be hit by using a suitable input f . This property is specific of dimension 1. As proved in [2], it cannot be extended if x ∈ Rd with d > 1.

1.4 The String with Persistent Memory on a Half Line We confine ourselves to study the equation with zero initial condition. We note that Eq. (1.8) is the string equation associated with the equation with memory (1.6) and Eq. (1.6) is the same as (1.8), the function F(x, t) being now ∫ t F(x, t) = q(x)w(x, t) + K(t − s)w(x, s) ds . 0

So we have the following representation formula for the solution when the initial conditions are zero:

1.4 The String with Persistent Memory on a Half Line

13

w(x, t) = w f (x, t) = f (t − x)H(t − x)

  1 + 2

u f (x,t)



∫

∫

q(ξ)w(ξ, τ) +

τ

 K(τ − s)w(ξ, s) ds dξ dτ .

(1.15)

0

D(x,t)

This formula is obtained by considering the equation with memory as a perturbation of the memoryless string equation8 . We note that • the integral on the right hand side of (1.15) depends only on w(ξ, s) with s ≤ t, as expected by inspecting Eq. (1.6). Instead, ξ runs up to x + t; • the properties of the string equation without memory suggest w ∈ C([0, T]; L 2 (0, X)) for every T > 0 and X > 0. Now we prove that (1.15) is well posed: we prove that Eq. (1.15) admits a solution 2 (0, +∞; L 2 (0, +∞)), which “depends continuously” on f (in the sense w(x, t) ∈ Lloc loc specified in Theorem 1.9) and that the solution is unique. We fix any T > 0 and we note that t ≤ T , x ≤ 2T − t =⇒ D(x, t) ⊆ TT , where TT is the trapezium in Fig. 1.6 (left). We study w(x, t) in TT for every fixed T > 0 (note that we are not asserting that w is defined or it is not zero only in TT ). We fix a parameter γ > 0 (to be specified later on) and we introduce y(x, t) = e−γt w(x, t) ,

ψ(x, t) = e−γt f (t − x)H(t − x) = e−γt u f (x, t) .

(1.16)

We rewrite Eq. (1.15) in terms of y:

τ

τ

(T,T)

(0,T)

(2T,T)

(T,T)

(0,T)

(x,t)

(x,t)

D(x,t)

D(x,t)

(x,t)

(x,t)

D(x,t)

D(x,t) (0,0)

(2T,0)

ξ

(0,0)

(2T,0)

ξ

Fig. 1.6 The trapezium TT (left) and the corresponding rectangle RT = (0, 2T ) × (0, T ) (right). (Compare Fig. 1.3 in [43].) 8

This idea, of using (1.10c) to represent the solutions of a system with memory, has been exploited to study controllability in [3, 40] .

14

1 Preliminary Considerations and Examples

∫

1 e−γ(t−τ) q(ξ)y(ξ, τ) dξ dτ 2 D(x,t)  ∫ τ   e−γ(τ−s) K(τ − s) y(ξ, s) ds dξ dτ . e−γ(t−τ)

y(x, t) = ψ(x, t) + +

1 2

∫

(1.17)

0

D(x,t)

A function w solves (1.15) if and only if y = e−γt w solves (1.17). So, it is enough that we study Eq. (1.17): we must study the restriction of y(x, t) to TT . We prove that the transformation ∫ 1 y → L y = e−γ(t−τ) q(ξ)y(ξ, τ) dξ dτ 2 D(x,t)

  1 + 2

∫

e−γ(t−τ)

D(x,t)

∫

τ



=Lq y

 e−γ(τ−s) K(τ − s) y(ξ, s) ds dξ dτ

(1.18)

0

  =Lmem y

is linear and continuous on L 2 (TT ) to itself and that it is a contraction, i.e. its norm is strictly less than 1, when γ is sufficiently large. This implies that Eq. (1.17) admits a solution in L 2 (TT ), and the solution is unique. Furthermore, the solution depends continuously on ψ ∈ L 2 (TT ), see Theorem 2.2 and von Neumann’s formula (2.4). Hence, it depends continuously on f ∈ L 2 (0, T). In order to compute the norm in L 2 (TT ), we extend y(x, t) with 0 to the rectangle RT = (0, 2T) × (0, T) (see Fig. 1.6, right). Let y˜ (x, t) be this extension, and let M be a number such that |K(t)| < M for t ∈ [0, T] and |q(x)| < M for x ∈ [0, 2T]. We give an estimate for Lmem y. We have ∫   

2  e e K(τ − s) y(ξ, s) ds dξ dτ  ≤ D(x,t) 0 2 ∫ ∫ τ    e−γ(τ−s) |K(τ − s)| | y˜ (ξ, s)| ds dξ dτ  ≤ e−γ(t−τ) ≤  0 R  ∫ ∫ τ  ∫ τ   ∫ T −2γt −2γs 2 2 e dξ dτ e M ds y˜ (ξ, s) ds dξ dτ ≤ ≤ −γ(t−τ)

RT

M 2T 2 ≤ 2γ 2

∫ RT

∫

τ





−γ(τ−s)

RT

0

0



M 2T 2 M 2T 2 2 y˜ 2 (ξ, s) dξ ds = y ˜ = y L2 2 (T ) . 2 L (RT ) T 2γ 2 2γ 2

We integrate both the sides on TT and we see that the norm of the transformation is strictly less than 1/2 if γ is sufficiently large. A similar computation shows that for γ large enough we also have Lq L(TT )
0, and let f (t) ∈ L 2 (0, T). Then, Eq. (1.17) admits a unique solution y(x, t) ∈ L 2 (TT ). The transformation f → y from L 2 (0, T) to L 2 (TT ) is continuous. The regularity of the solutions in Theorem 1.9 can be improved since the integral operator (1.18) is a linear continuous transformation from L 2 (TT ) to C(TT ) and the integral is a.e. differentiable. So, w(x, t) is the sum of the L 2 function u f (x, t) and of a continuous a.e. differentiable function which depends continuously on u f . Hence, Theorems 1.4 and 1.6 have the following counterpart. Theorem 1.10. Let T > 0. Then, 1. if f ∈ L 2 (0, T), then w f ∈ C([0, T]; L 2 (0, TX )), and the transformation f → w f is continuous in the indicated spaces; 2. the transformation f → (∂/∂ x)w f is continuous from the space H−1 (0, T) to the space C([0, T]; L 2 (0, TX )) (the space H−1 is defined in (1.13)).

1.4.1 Finite Propagation Speed It is important to understand that also in the case of Eq. (1.6) signals propagate with finite speed (as suggested by the discussion in Sect. 1.2). We use the notations y and ψ as in Sect. 1.4 and γ is chosen in such a way that the map L in (1.18) is a contraction. We prove that the input f does not affect a point in position x if x > t in the sense that x > t =⇒ y(x, t) = 0 . We fix any X > 0 and we consider the domain (a triangle) TrX = {(ξ, τ) : 0 < τ < ξ < X − τ}, which is represented in Fig. 1.7, left. Note that D(x, t) ⊆ TrX for every (x, t) ∈ TrX . Hence, the integral operator in (1.17) is also a transformation in L 2 (TrX ) (and it is a contraction for γ large). On TrX we have ψ = 0. The integral operator (1.17) being a contraction in L 2 (TrX ), the equation has a unique solution. The fact that ψ = 0 on TrX implies that y = 0. This proves that signals propagate with finite velocity also when the equation is (1.6). The velocity of the signals of the associated (memoryless) system is equal 1, and the previous argument shows that in the case of Eq. (1.6) the velocity is at most 1. We shall see in Theorem 1.11 that neither the term q(x)w nor the memory changes the velocity of wave propagation, which is precisely equal to 1 also for the system (1.6). Now we consider the effect of a forcing term f , which is zero for 0 ≤ t ≤ a and for t > b.

16

1 Preliminary Considerations and Examples

We fix any X > b and this time we consider the two parallels to the line τ = ξ, from (0, a) and from (0, b), and their intersection with the line ξ = X − τ. We get two quadrilaterals, respectively, Q(X, a) and Q(X, b). The quadrilateral Q(X, a) is the one below the line τ = ξ + a and Q(X, b) is below the line τ = ξ + b. Let S(X, a, b) = Q(X, b) \ Q(X, a), i.e. the strip between the lines τ = ξ + a and τ = ξ + b in Fig. 1.7, right. The strip S(X, a, b) contains the support of ψ(x, t). It is clear that if (x, t) ∈ Q(X, a), then D(x, t) ⊆ Q(X, a). The same argument as above shows that y(x, t) = 0 in Q(X, a). Instead, if (x, t) ∈ Q(X, b) \ Q(X, a), i.e. if (x, t) is in the strip, in general it will be y(x, t)  0. In conclusion, if f (t) = 0 for t < a, then y(x, t) = 0 in Q(X, a) for every X > 0 and it is nonzero in the strip S(X, a, b). This is the same as for the memoryless string equation. But, let us consider now a point (x, t), which is above the line τ = ξ + b, as in Fig. 1.7 (right). It is clear that D(x, t) will intersect the strip S(x, t), and then we can represent ∫ τ ∫ ds dξ dτ = D(x,t) 0 ∫ ∫ τ ∫ τ ∫ ds dξ dτ + ds dξ dτ . D(x,t)∩S(X,a,b)

0

D(x,t)\S(X,a,b)

0

In general, these integrals are not zero: the effect of the operator L in (1.18) is to introduce a “persistent affine term” in the right hand side of (1.17), and we cannot expect y to become zero after the effect of the boundary term f has been “forgotten” by the solution u f of the memoryless string equation: once the signal reaches a point of abscissa x, we have to expect that the memory of the signal at that position will persist forever in the future. It is indeed so, as seen in the plots of Fig. 1.1 (right) and 1.2.

τ

τ= ξ (X/2,X/2)

(x,t)

Tr

ξ=X−τ

b S(X,a,b)

X

a

X

ξ

D(x,t)

X

Fig. 1.7 The domain Tr X (left) and the domains Q(X, b), Q(X, a), and S(X, a, b) (right). (This figure reproduces Fig. 1.4 in [43]).

1.4 The String with Persistent Memory on a Half Line

17

1.4.2 A Formula for the Solutions and a Control Problem We give a representation for the solutions of (1.17) (i.e. of (1.6)–(1.7)) and we use it to solve a control problem. The initial conditions are zero and we recall finite propagation speed: y(x, t) = 0 when x > t. We fix a time T, and we confine ourselves to study the solutions in the square9 Q = {(x, t) , 0 < t < T , 0 < x < TX } . Since we know that y(x, t) = 0 when x > t, the integral operator in (1.18) can be considered as a linear and continuous transformation in L 2 (Q). We already proved that the norm of the transformation can be taken as small as we wish, by a proper choice of γ. We fix γ so to have the norm less than 1/4 (and note that the integral has a factor 1/2 in front). We elaborate the integral operator. Let ˜ τ, s) = e−γ(t−s) K(τ − s) ⎧ K(t, ⎪ ∫τ ⎪ ⎨ ⎪ ˜ τ, s)y(ξ, s) ds F(t, τ, ξ) = e−γ(t−τ) q(ξ)y(ξ, t) + 0 K(t, ⎪ ∫ ⎪ t ⎪ ˜ τ, s)y(ξ, s) ds . = e−γ(t−τ) q(ξ)y(ξ, t) + 0 H(τ − s)K(t, ⎩ Then, using (1.10c), the integral operator in (1.18) can be written as ∫ t ∫ x+t−τ ∫ F(t, τ, ξ) dξ dτ = H(ξ − |x − t + τ|)F(t, τ, ξ) dξ dτ D(x,t) ∫ t ∫ x+t

=

0

=

0

0

H(x + t − τ − ξ)H(ξ − |x − t + τ|)F(t, τ, ξ) dξ dτ

0

∫ t∫ 0

x+t

G1 (t, x, s, ξ)y(ξ, s) dξ ds,

(1.19)

0

where G1 (t, x, s, ξ) = H(x + t − s − ξ)H(ξ − |x + t − s|)e−γ(t−τ) q(ξ) ∫ t ˜ τ, s) dτ . + H(x + t − τ − ξ)H(ξ − |x − t + τ|)H(τ − s)K(t, 0

Now we invoke again finite propagation speed so that y(ξ, s) = 0 for ξ > s, in particular for ξ > t, and we get the following form for the Eq. (1.17) of y(t): ∫ ∫ 1 t t 1 y(x, t) = ψ(x, t) + G1 (t, x, s, ξ)y(ξ, s) dξ ds = ψ + Gy, 2 0 0 2 where G is the integral transformation in L 2 (Q):

We recall the notation TX from Sect. 1.3.1.1: for every T , the number TX is equal to T , but [0, T ] denotes an interval of the time axis, while [0, TX ] denotes the interval from 0 to T of the space axis.

9

18

1 Preliminary Considerations and Examples

(Gy) (t) =

∫ t∫ 0

t

G1 (t, x, s, ξ)y(ξ, s) dξ ds .

0

The choice of γ is such that G
t and it is continuous in any rectangle [0, L] × [0, T]. Proof. The first statement is a consequence of finite propagation speed, while the second statement is an immediate consequence of the absolute continuity of the integral which is the following property: let Φ(x) be a summable10 function on Rn . For every ε > 0, there exists δε > 0 such that (λ(D) is the Lebesgue measure of D): ∫     λ(D) < δε =⇒  Φ(x) dx  ≤ ε . D

This result can be applied when Φ(x, t) = F(x, t) since statement 1 of Theorem 1.10 in particular implies that w(x, t) is square integrable on every bounded rectangle. So, continuity of v(x, t) follows from a simple look at Fig. 1.8 in which two integration domains Dx,t and Dx1,t1 are compared.   Now we insert w(x, t) = (1/2)v(x, t) + f (t − x)H(t − x) in the expression of F(ξ, τ). We have, for ξ ≥ 0 and τ ≥ 0, 10

That is, integrable with finite integral. In the following, when using the more common term integrable, we intend that the integral is finite.

20

1 Preliminary Considerations and Examples

(x 1,t 1)

(x,t) (t,t)

(x,t)

Fig. 1.8 Continuity of v(x, t) via absolute continuity of the integral

F(ξ, τ) = q(ξ) f (τ − ξ)H(τ − ξ) + V(ξ, τ) , ∫ 1 τ 1 V(ξ, τ) = q(ξ)v(ξ, τ) + K(τ − s)v(ξ, s) ds 2 2 0 ∫ τ−ξ + K(τ − ξ − r) f (r)H(r) dr . −ξ

Absolute continuity of the integral and continuity of K(t) show that V(ξ, τ) ∈ C([0, L] × [0, T]) for every L > 0, T > 0 (equal zero for ξ ≥ τ) while q(ξ) f (τ − ξ) H(τ − ξ) ∈ L 2 ((0, L) × (0, T)). Remark 1.13. we note that • we reconsider the two Fig. 1.8. Thanks to finite propagation speed (and because the initial conditions are zero), we have w(x, t) = 0 if11 x > tX . Hence, only the part of the integration domains that are above the bisectrix gives a contribution to the integral on Dx,t in (1.22). • Fig. 1.8 is a bit particular in the sense that it presents the set D(x, t) in the case (t + x)/2 < x i.e. t < 3x. For the following computations, it is useful to keep in mind also the case t > 3x. The sets Dx,t ∩ {(t, x) , t > x} in the two cases are represented in Fig. 1.9. As we already know w(x, t) = v(x, t) = 0 if x > t, we study the property of the restrictions of w and v to the triangles  T ;− = T ∩ {t ≤ 3x} , T = {(x, t) ⊆ R2 0 ≤ t ≤ T , 0 ≤ x ≤ tX } , T ;+ = T ∩ {t ≥ 3x} . In the triangle T , we have H(τ − ξ) = 1 and we do not indicate this factor, which is then reintroduced when we pass to the properties on [0, L] × [0, T]. Now we investigate the properties of v and of w. We use the partition of the integration domain in Fig. 1.9 and we proceed in four steps. 11

We recall that tX denotes the point of the horizontal axis at distance t from the origin.

1.4 The String with Persistent Memory on a Half Line

21

(x,t)

(t-x)/2

x

(x,t)

(t+x)/2

x

(t-x)/2

(t+x)/2

Fig. 1.9 The domain of integration when t < 3x so that (t − x)/2 < x (left) and when t > 3x so that (t − x)/2 > x (right). Finite propagation speed accounts for the fact that the domains of integration are above the bisectrix

Step 1: the function v and its first derivatives in T ;− When (x, t) ∈ T ;− , we have ∫ v(x, t) = ∫

0

+ x

∫ = ∫ + +

(t−x)/2 ∫ ξ−x+t t−x−ξ (t+x)/2 ∫ x+t−ξ ξ

(t−x)/2 ∫ ξ−x+t

0 x

∫

+

x

∫

t−x−ξ ξ−x+t

ξ

x

(t−x)/2

∫

ξ−x+t

ξ

F(ξ, τ) dτ dξ

F(ξ, τ) dτ dξ V(ξ, τ) dτ dξ ∫

(t+x)/2 ∫ x+t−ξ

x

ξ

V(ξ, τ) dτ dξ

q(ξ) f (τ − ξ) dτ dξ

q(ξ) f (τ − ξ) dτ dξ

(t−x)/2 ξ ∫ (t+x)/2 ∫ x+t−ξ x

F(ξ, τ) dτ dξ +

V(ξ, τ) dτ dξ +

(t−x)/2 ξ ∫ (t−x)/2 ∫ ξ−x+t 0

+

∫

t−x−ξ ξ−x+t

∫

q(ξ) f (τ − ξ) dτ dξ .

(1.23)

This expressions confirms that (as we already know from Lemma 1.12) v(tX− , t) = lim v(x, t) = 0 . x→t X

x 0 and T > 0, while q(ξ) f (τ − ξ)H(ξ − τ) is square integrable. We deduce the following properties of the terms that appear in the expression of the derivatives: • the integrals that contain V belong to C 1 ([0, L] × [0, T]) for every L > 0 and T > 0 (note that the notation stresses the fact that the derivatives have continuous extension to the closed rectangle). • the integrals in which both q and f are integrated are continuous functions of (x, t) in [0, L] × [0, T]. In order to see this, we reintroduce H and as an example we consider the following integral: ∫

(t−x)/2

∫ q(ξ) f (t − x − 2ξ) dξ =

0

t−x

(t−x)/2

q((t − x − r)/2) f (r) dr .

Continuity follows from absolute continuity of the integral and the assumed continuity of the function q(x). • the function  ∫ x q(ξ) dξ f (t − x)H(t − x) (1.25) 0

belongs to

L 2 ((0,

L) × (0, T)) for every L > 0, T > 0.

The previous properties and Lemma 1.3 applied to (1.25) give the following lemma. Lemma 1.14. Let f ∈ L 2 (0, T). Then the functions vx (x, t) and vt (x, t) are square integrable on [0, L] × [0, T] for every L > 0 and T > 0. Furthermore, x → vx (x, t) is continuous from x > 0 to L 2 (0, T) and lim vx (x, t)

x→0+

exists in L 2 (0, T) .

Looking again at (1.25), we see that the derivatives vx and vt are continuous both in T ;− and in T ;+ if and only if f ∈ C([0, T]). In this case we have more: the derivatives have continuous extension to T as it is easily seen by considering the limits of the derivatives for (x, t) → (x0, 3x0 ). We use the equality w(x, t) =

1 v(x, t) + f (t − x)H(t − x) 2

(1.26)

24

1 Preliminary Considerations and Examples

to lift the properties of v to the function w. We collect the properties that we are going to use: if f ∈L 2 (0,T )

  

v(x, t) ∈ H 1 (T ) v(0, t) = v(tX− , t) = 0

w(x, t) ∈ L 2 (T ) (1.27)

w(x, t) ∈ C(T ) , w(tX− , t) = f (0) (wx (x, t) ∈ L 2 (T ) if f ∈ H 1 (0, T))

  v(x, t) ∈ C 1 (T )

if f ∈C([0,T ])

The noticeable facts in (1.27) are that v is smoother than w and f and the fact that if f ∈ C([0, T]), then w(tX− , t) ≡ f (0). Step 4: the derivative wxx (x, t) We are interested in the properties of wxx (x, t), and so we confine ourselves to examine the first and second derivatives with respect to the space variable x of v and w. The similar analysis for the time derivatives and the mixed derivatives vxt and wxt is left to the reader. In order to compute the second derivatives, we need the derivative of the forcing term f (t). When f ∈ H 1 (0, T), then in particular f ∈ C([0, T]). It follows from (1.27) that v ∈ C 1 (T ), and from (1.26), we see that w ∈ C(T ) and wx ∈ L 2 (T ). In fact, the expressions of vx in T ;− and in T ;+ show that v is even more regular: vxx ∈ L 2 (T ) and vxx ∈ C(T ) if f ∈ C 1 ([0, T]), but this property cannot be used to assert the existence of wxx , unless f admits a second derivative. If f ∈ H 2 (0, T), then we have wxx ∈ L 2 (T ) and we have the following slightly more precise statements: if f ∈H 1 (0,T )

  vxx ∈ L 2 (T ) w ∈ C(T ), wx ∈ L 2 (T ) (vxx ∈ C(T ) if f ∈ C 1 ([0, T])) (w ∈ C 1 (T ) if f ∈ C 1 ([0, T])) w ∈ L 2 (T ) vxx ∈ C(T ) xx (wxx ∈ C(T ) if f ∈ C 2 ([0, T]))

 

(1.28)

if f ∈H 2 (0,T )

Note that if f ∈ H 1 or f ∈ H 2 , then v is smoother than w and f , while in the case f ∈ C 2 the regularity of v cannot be improved since we cannot compute the third derivative of v unless q is differentiable.

1.4 The String with Persistent Memory on a Half Line

25

1.4.2.2 The Response Operator Let T > 0. The response operator is the map RT from L 2 (0, T) to itself: (RT f )(t) = wx (0, t) . As described in Sect. 1.5.2, the response operator is related to the traction exerted by the string on the support of its left end. The linear operator RT is densely defined in L 2 (0, T), since its domain contains D(0, T), but it is unbounded. We want to assign a domain to RT in such a way that wx on the boundary is related to its interior values, i.e. we want lim (RT f )(t) − wx (x, t) L 2 (0,T ) = 0 .

(1.29)

x→0+

In this formula, wx (x, t) is computed for x ∈ (0, L), where L > 0 is fixed. We noted in Lemma 1.14 that for every f ∈ L 2 ([0, T]), we have vx (x, t) ∈ L 2 (0, T) for every x ∈ (0, L), and the L 2 (0, T)-limit for x → 0+ exists in L 2 (0, T). So, in order to have (1.29), we must require the analogous properties to u f (x, t) = f (t − x)H(t − x). This problem has already been studied in Sect. 1.3.1, and from Theorem 1.6 we see that, as for the string equation without memory, dom RT = H−1 = { f ∈ H 1 (0, T) ,

f (0) = 0} .

1.4.2.3 From the Regularity of the Target to the Regularity of the Control Let T > 0. We proved that any W ∈ L 2 (0, TX ) can be reached (from the zero initial conditions) by using a unique steering control12 f ∈ L 2 (0, T). The results in Sect. 1.4.2.1 can be used to see that smoother targets can be reached by using smoother controls. The results are consequences of the representation (1.26) and of the results listed in (1.27) and in (1.28) and are as follows: Theorem 1.15. Let f ∈ L 2 (0, T) be the steering control to W ∈ L 2 (0, TX ). the following properties hold: 1. if W ∈ C([0, TX ]), then f ∈ C([0, T]); 2. if W ∈ C k ([0, TX ]) and k = 1 or k = 2, then f ∈ C k ([0, T]); 3. if W ∈ H k (0, TX ) and k = 1 or k = 2, then f ∈ H k (0, T). Proof. The proof is based on a bootstrap argument. We sketch the proof that if W ∈ C 2 ([0, TX ]), then the steering control f belongs to C 2 ([0, T]), granted the previously listed properties. W ∈ C 2 ([0, TX ]) =⇒ W ∈ C 1 ([0, TX ])

=⇒ f ∈ C 1 ([0, T]) =⇒ vxx ∈ C(T ) .



 previous property

12

That is, a control such that w f (T, x) = U(x).

from (1.28)

26

1 Preliminary Considerations and Examples

Now we use the equality 1 1 W(x) − v(x, T) = w(x, T) − v(x, T) = f (T − x) 2 2

x ∈ [0, TX ] .

The left hand side is of class C 2 so that the steering control f is of class C 2 as well.   Remark 1.16. The analogous problem in any spatial dimension has been studied in [44], where analogous results have been proved. It is interesting to note that the results can be extended to any k > 2 and any spatial dimension for the system without memory (see [20]), but they cannot be extended to k > 2 for systems with memory, not even in dimension 1 and q = 0 (see [44]).

1.5 Diffusion Processes and Viscoelasticity: the Derivation of the Equations with Persistent Memory In this section, we give a short account of the derivation of the equations with persistent memory in thermodynamics (and non-Fickian diffusion), in viscoelasticity and in a problem of filtration. See [15] for a detailed analysis.

1.5.1 Thermodynamics with Memory and Non-Fickian Diffusion We begin by recalling the derivation of the memoryless heat equation in a bar, which follows from two fundamental physical facts: conservation of energy and the fact that the temperature is a measure of energy, i.e. (the prime denotes time derivative) e (x, t) = −qx (x, t) ,

θ (x, t) = γe (x, t) ,

γ>0.

Here q is the (density of the) flux of heat, e is the (density of the) energy, and θ is the temperature13 . So we have θ (x, t) = −γqx (x, t) . If there is a distributed source of heat, we have θ (x, t) = −γqx (x, t) + g(x, t) . We combine with a “constitutive law.” Fourier’s law assumes that the flux responds immediately to changes in temperature: ∫b The minus sign because the total internal energy a e(x, t) dx of a segment (a, b) decreases when the flux of heat is directed to the exterior of the segment.

13

1.5 Derivation of the Equations with Memory

27

q(x, t) = −αθ x (x, t)

(1.30)

(α > 0 and the minus sign because the flux is toward parts which have lower temperature). By combining these equalities, we get the memoryless heat equation θ (x, t) = (αγ)θ xx (x, t) + g(x, t) .

(1.31)

The heat equation with memory is obtained when we take into account the fact that the transmission of heat is not immediate, and Fourier’s law is modified as in [13, 29] (after the special case in [10]. An analysis of [10] is in [45, 52]): ∫ t q(x, t) = −αθ x (x, t) − N(t − s)θ x (x, s) ds, (1.32) −∞

which gives θ (x, t) = αθ xx (x, t) +

∫

t

−∞

N(t − s)θ xx (x, s) ds + g(x, t) .

(1.33)

The memory kernel N(t) describes the way the flux “relaxes” in the course of the time and it is called the relaxation kernel. Let the system be subject to the action of an external control f , for example, let us regulate the temperature at one end: θ(0, t) = f (t). Then the control acts after a certain initial time t0 , say t0 = 0, and we get the problem14 ∫ t  θ = αθ xx + N(t − s)θ xx (s) ds + h , θ(0, t) = f (t). (1.34) 0

The distributed affine term h(t) = h(x, t) takes into account the previous, uncontrolled, history of the system and possibly a distributed source represented by the function g: ∫ 0 h(x, t) = N(t − s)θ xx (x, s) ds + g(x, t) . −∞

A similar equation is obtained when θ represents the concentration of a solute in a solvent and q represents the flux across the position x at time t. Then, the constitutive law (1.30) is the Fick law, which leads to (1.31) as the law for the variation in time and space of the concentration (denoted θ), but it is clear that the assumption that the flux of matter reacts immediately to the variation of concentration is even less acceptable, and, in the presence of complex molecular structures, the law (1.32) is preferred, leading to (1.33) as the equation for the concentration, see [16]. 14 Note the use of the symbols: θ or θ(t) or θ(x, t) and analogous notations for h and f , as most convenient.

28

1 Preliminary Considerations and Examples

We note that H. Jeffreys introduced the special case α > 0 and N(t) = βe−ηt in order to take into account viscosity of the ocean water (see [35]). In fact, Eq. (1.1) with α > 0 and N(t) = e−ηt is the Moore–Gibson–Thompson equation of acoustics, see [9, 17]. We mention that especially in the study of diffusion a relaxation kernel N(t), which is singular at t = 0, as for example N(t) ∼ 1/t α , is often used. We study systems with singular kernels in Sect.s 3.6 and 3.7.

1.5.2 Viscoelasticity We consider the simple case of a string (of constant density  > 0). We assume that when it is not subject to “vertical” forces, it lays in equilibrium on the segment [a, b] of the horizontal axis. After the application of a force (and under certain “smallness” conditions in the sense specified below), the point in position (x, 0) will be found in position (w(x), 0) (so, we assume negligible horizontal motion, a first “smallness” constraint). The function x → w(x, t) is the displacement of the string at time t, while the function t → w(x, t) (with values in a suitable space of x-valued function) represents the evolution in time of the string and it is called its motion. A displacement by itself does not produce an elastic traction: for example, a vertical translation w(x, t) = h, the same for every x, does not produce any traction in the string. An elastic traction appears when neighboring points undergo different displacements. In this case we call deformation the displacement of the string. Let us fix a point x. ˆ A segment ( x, ˆ xˆ + δ) on the right of xˆ exerts a traction on ˆ t)  0 and elasticity assumes that this traction [a, x] ˆ in the case that w( xˆ + δ, t) − w( x, is k [w( xˆ + δ, t) − w( x, ˆ t) + o (w( xˆ + δ, t) − w( x, ˆ t))]. It is an experimental fact that k = k(δ) and, with an acceptable error, k=

k0 δ

(k0 does not depend on δ)

(in order to produce the same deformation in a longer string a smaller force is ˆ t) > 0, required). The coefficient15 k0 is positive since if δ > 0 and w( xˆ + δ, t) − w( x, then the traction on [a, x] ˆ exerted by the part of the segment x > xˆ points upward. We assume the conditions of linear elasticity, which are new “smallness” condition. The first one is that the effect of o (w( xˆ + δ, t) − w( x, ˆ t)) is negligibly small, and we ignore it. So, the traction exerted at xˆ from the nearby segment ( x, ˆ xˆ + δ) is approximated by k0 [w( xˆ + δ, t) − w( x, ˆ t)] = k0 wx ( x, ˆ t) + o(1), δ In general k0 = k0 (x). For simplicity we assume that the string is homogeneous so that k0 does not depend on x.

15

1.5 Derivation of the Equations with Memory

29

which in its turn is approximated by σ( x, ˆ t) = k0 wx ( x, ˆ t) (a new “smallness” assumption), which is the traction or stress16 exerted on ˆ t) is the deformation the end xˆ of the segment (a, x) ˆ from the right, while wx ( x, gradient (at time t and position x). ˆ Now we consider a segment (x0, x1 ) of the string. The resultant of the traction exerted at time t on the segment by the adjacent parts (x0 − δ, x0 ) and (x1, x1 + δ) is the difference of the tractions at the ends: k0 [wx (x1, t) − wx (x0, t)] .

(1.35)

This is not the sole force that acts on the segment. There will also be a force due to the interaction of the segment with the environment. This force is often a body force, i.e. a force given by ∫ x1

−

P(ξ, t) dξ .

(1.36)

x0

For example, P(ξ, t) = g is due to gravity (g is the acceleration of gravity); P(ξ, t) = q(ξ)w(ξ, t) is an elastic force, which appears when the string is bound to an elastic bed. We balance the momentum on the segment (x0, x1 ) and the forces acting on this segment. We have   ∫ x1 ∫ x1 d  w (x, t) dx = − P(ξ, t) dξ + k0 [wx (x1, t) − wx (x0, t)] . (1.37)  dt x0 x0 Now we divide both the sides with x1 − x0 and we pass (formally!) to the limit for x1 → x0 . As x0 is a generic abscissa of points of the string, we rename it as x and we get the string equation w (x, t) = −P(x, t) + k0 wxx (x, t) .

(1.38)

Remark 1.17. It is important to interpret the derivatives that appear in (1.35): w(x1 + h, t) − w(x1, t) , h w(x0 − h, t) − w(x0, t) −wx (x0, t) = lim . h→0 h wx (x1, t) = lim

h→0

These expressions represent the derivatives computed by moving toward the exterior of the segment (x0, x1 ). We denote by γ1 the exterior derivative, and with this notation, (1.35) takes the form k0 [γ1 w(x1, t) + γ1 w(x0, t)] . 16

In a three dimensional body, the stress is the force for unit area of the surface on which it acts.

30

1 Preliminary Considerations and Examples

The string is located on the right of its left support. So, the traction exerted by the string on its left support is k0 wx (a, t) = −k0 γ1 w(a, t), while k0 γ1 w(a, t) is the traction exerted by the left support on the string. A hidden assumption in the previous derivation is that the elastic traction appears at the same time as the deformation and that it adjusts itself instantaneously to the variation of the deformation. In a sense, this is a common experience: a rubber cord shortens abruptly when released. But, if the cord has been in the freezer for a while, it will shorten slowly: the effect of the traction fades slowly with the time. This is taken into account by the Boltzmann superposition principle introduced in [7, 8]. L. Boltzmann assumed that if at a fixed position x, ˆ the deformation gradient changes ˆ τ + h) − wx ( x, ˆ τ)  0, then with time and at time τ and time τ + h we have wx ( x, also this discrepancy in time will produce a traction, approximately equal to hNwx ( x, ˆ τ) (N is a positive coefficient, and we recall that the prime denotes derivative with respect to the time τ). Also the properties that produce body forces may change with time, and in this case the argument applies to the body force as well: the traction on a segment (x0, x1 ) is   ∫ x1 d P(ξ, τ) dξ . (1.39) hN (wx (x1, τ) − wx (x0, τ)) − dτ x0 The traction (1.39), which originates at time τ, is sensed at every future time t with an “intensity,” which decreases with t − τ: this traction originated at time τ is sensed at time t > τ as   ∫ x1 d hN(t − τ) P(ξ, τ) dξ (wx (x1, τ) − wx (x0, τ)) − dτ x0 (we recall that h is the length of an interval of time). The Boltzmann superposition principle is the property that at time t the tractions originated at every previous intervals of time are “summed” so that the total traction to be inserted at the right side of the balance equation (1.37) is   ∫ x1 ∫ t d N(t − τ) P(ξ, τ) dξ dτ, (wx (x1, τ) − wx (x0, τ)) − dτ −∞ x0 and the balance equation takes the form   ∫ x1 d  w (x, t) dx  dt x0   ∫ t ∫ x1 d = N(t − τ) P(ξ, τ) dξ dτ . (wx (x1, τ) − wx (x0, τ)) − dτ −∞ x0 Now we divide both the sides with (x1 − x0 ) and we pass to the limit. We get

1.5 Derivation of the Equations with Memory

w (x, t) dx =

∫

t

−∞

31

N(t − τ)

d [wxx (x, τ) − P(x, τ)] dτ . dτ

(1.40)

Finally, if it happens that N(t) is differentiable, then we can integrate by parts. We get ∫ t  w (x, t) dx = N(0) [wxx (x, t) − P(x, t)] − M(t − s) [wxx (x, s) − P(x, s)] ds −∞

where M(t) = −N (t) ≥ 0

− lim N(t − τ) [wxx (x, τ) − P(x, τ)] τ→−∞

(note that M(t) ≥ 0 since, as we stated, N(t) is decreasing). Physical considerations suggest lim N(t − τ) [wxx (x, τ) − P(x, τ)] = 0,

τ→−∞

and we get the wave equation with memory w (x, t) dx = N(0) [wxx (x, t) − P(x, t)] ∫ t M(t − s) [wxx (x, s) − P(x, s)] ds . − −∞

(1.41)

It is interesting to note that, when there is no body force, an integration of both the sides gives ∫ w (t) =

t

−∞

N(t − s)wxx (x, s)

(provided that limτ→−∞ w (τ) = 0). This is the same equation as (1.33) with α = 0. For this reason, Eq. (1.33) with α = 0 is also called the hyperbolic heat equation. Remark 1.18. We note that • the traction exerted at x by the part of the string at its right is (when N(t) is differentiable)   ∫ t σ(x, t) = N(0)wx (x, t) + N (t − s)wx (x, s) ds −∞ ∫ t M(t − s)wx (x, s) ds . (1.42) = N(0)wx (x, t) − −∞

This expression with x = a is the traction exerted by the string on its left support. • an implicit assumption in the previous derivation: the traction acting on the extremum x1 of the segment [x0, x1 ] is approximated with k0 wx (x1, s): we disregard the traction which may diffuse from distant parts of the body. Models in which the traction depends on distant points in space (but without the effect of the memory) have been studied for example in [24, 51]. The use of fractional laplacian accounts also for a similar phenomenon. The paper [5] takes into account both nonlocal phenomena in the space and the time memory. See also the abstract treatment in the papers [46, 57, 58].

32

1 Preliminary Considerations and Examples

The derivations of models which are nonlocal in space and time can be found in [47]. • an interesting case of the body force in viscoelasticity is when a viscoelastic body is connected to an elastic bed whose vibrations can be ignored since its mass is far greater, as in the case of an underground viscoelastic pipe connected to the surrounding soil, see17 [4, 6, 55]. In this case, the body force entering the integral in (1.40) is P(ξ, τ) = q(ξ)w(ξ, τ),

where q(ξ) is the elastic coefficient .

1.5.3 A Problem of Filtration When a fluid is filtered through a granular material, the flux depends on the gradient of the pressure and it is directed toward regions of lower pressure. A phenomenological law that accounts for this property is q = −c∇p,

(1.43)

where q is the flux and p is the pressure. Of course, the constant c > 0 depends on the properties of the fluid and on those of the filter since it is clear that large pores are more permeable than small ones. The density of the flux is higher in regions of higher density. We take into account this observation and the continuity equation, i.e. conservation of the material, and we get the equations (1.44) p = γ , ∇ · q = −, where  is the density, γ > 0, and the prime denotes time derivative. We combine (1.43) and(1.44), and we find  = γcΔ .

(1.45)

This derivation does not take into account for example the fact that the fluid transports impurities which in the course of the time reduce the size of the pores. So, in practice the coefficient c is not constant but depends on the past history of the process. In practice, it may be ∫ t  c = c0 + G N(t − s)F(∇(s)) ds , 0

where the positive functions F and G have certain monotonicity properties and N is positive since the obstruction of the pores increases with the amount of the

17

These papers introduce a Cattaneo type equation, and so the systems in these papers can be put in the form (1.40) with a memory kernel which is a decaying exponential.

References

33

fluid which transverses the filter. These observations show that problem of filtration naturally leads to study nonlinear equations with memory. See [34], where it is shown that a similar nonlinear problem is encountered in the study of the magnetic field in materials whose physical properties depend on the changes of the temperature due to the dissipation of energy, a consequence of the time varying magnetic field. The paper [33] models filtration of water through a sand bed by assuming constant coefficients, hence a linear equation. The accumulation of impurities is accounted by replacing the first derivative  with a fractional derivative. This way, the authors obtain a linear equation with memory, the memory being due to the fractional derivative, but with a memory kernel which is singular at the time t = 0. In this book, we confine ourselves to study linear problems. A cursory account of certain nonlinear problems is in the final Chap. 7, mostly to stimulate the reader to investigate also this class of important problems.

1.5.4 Physical Constraints As noted in Sect. 1.5.3, physical considerations impose severe restrictions to the memory kernel, which do not have a role in this book, a part when studying stability in Chap. 5. So, we confine ourselves to few observations. First of all, it is clear that in practice temperature or deformation of a (viscoelastic) body will have little influence after sufficient time has elapsed. And it is easily understood that a past effect, when “almost forgotten” will not appear again in the future. These observations suggest that the memory kernels are decreasing functions of time. In particular, if N(t) or M(t) is differentiable, their derivative will be negative. This is the first reason why we preferred to write Eq. (1.3) in that form, i.e. with M(t) = −N (t) ≥ 0 in order to better visualize the sign. As noted in [60], when a deformation is applied to a viscoelastic material, the resulting stress is minor than that of a purely elastic material with similar elastic constants. This is a second reason for the negative signs in front of the integral in (1.3) (with the condition that M(t) ≥ 0). A far more refined analysis of the constraints imposed by physical principles to the memory kernels can be found for example in [1, 21, 26].

References 1. Amendola, G., Fabrizio, M., Golden, J.: Thermodynamics of materials with memory: Theory and applications. Springer, New York (2012) 2. Avdonin, S.A., Belishev, M.I., Ivanov, S.A.: Controllability in a captured domain for a multidimensional wave equation with a singular boundary control. J. Math. Sci. 83(2), 165–174 (1997)

34

1 Preliminary Considerations and Examples

3. Barbu, V., Iannelli, M.: Controllability of the heat equation with memory. Differential Integral Equations 13(10-12), 1393–1412 (2000) 4. Basu, D., Kameswara Rao, N.: Analytical solution for Euler-Bernoulli beam on visco-elastic foundation subject to moving load. Internat. J. Numer. Anal. Methods Geomech. 37, 945–960 (2013) 5. Biccari, U., Warma, M.: Null-controllability properties of a fractional wave equation with a memory term. Evol. Equ. Control Theory 9(2), 399–430 (2020) 6. Bogacz, R.: Response of beams on viscoelastic foundation to moving distributed loads. J. Theoret. Appl. Mech. 46, 763–775 (2008) 7. Boltzmann, L.: Zur theorie der elastischen nachwirkung. Wien. Ber. 70, 275–306 (1874) 8. Boltzmann, L.: Zur theorie der elastischen nachwirkung. Wied. Ann. 5, 430–432 (1878) 9. Bucci, F., Pandolfi, L.: On the regularity of solutions to the Moore-Gibson-Thompson equation: a perspective via wave equations with memory. J. Evol. Equ. 20(3), 837–867 (2020) 10. Cattaneo, C.: Sulla conduzione del calore. Atti Sem. Mat. Fis. Univ. Modena 3, 83–101 (1949) 11. Conti, M., Pata, V., Squassina, M.: Singular limit of differential systems with memory. Indiana Univ. Math. J. 55(1), 169–215 (2006) 12. Christensen, R.: Theory of viscoelasticity, an introduction. Academic Press, New York (1982) 13. Coleman, B.D., Gurtin, M.E.: Equipresence and constitutive equations for rigid heat conductors. Z. Angew. Math. Phys. 18, 199–208 (1967) 14. Curie, M.: Recherches sur la conductibilité des corps cristallines. Ann. Chim. Phys. 18, 203–269 (1889) 15. Dautray, R., Lions, J.L.: Mathematical analysis and numerical methods for science and technology. Vol. 1: Physical origins and classical methods. Springer-Verlag, Berlin (1990) 16. De Kee, D., Liu, Q., Hinestroza, J.: (Non-fickian) diffusion. The Canada J. of chemical engineering 83, 913–929 (2005) 17. Dell’Oro, F., Pata, V.: On the Moore-Gibson-Thompson equation and its relation to linear viscoelasticity. Appl. Math. Optim. 76(3), 641–655 (2017) 18. Desch, W., Grimmer, R.: Invariance and wave propagation for nonlinear integro-differential equations in Banach spaces. J. Integral Equations 8(2), 137–164 (1985) 19. Engel, K.J., Nagel, R.: One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194. Springer-Verlag, New York (2000) 20. Ervedoza, S., Zuazua, E.: A systematic method for building smooth controls for smooth data. Discrete Contin. Dyn. Syst. Ser. B, 14(4), 1375–1401 (2010) 21. Fabrizio, M., Lazzari, B.: On asymptotic stability for linear viscoelastic fluids. Differential Integral Equations 6(3), 491–505 (1993) 22. Fabrizio, M., Morro, A.: Mathematical problems in linear viscoelasticity, SIAM Studies in Applied Mathematics, vol. 12. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1992) 23. Fattorini, H.O.: The Cauchy problem, Encyclopedia of Mathematics and its Applications, vol. 18. Addison-Wesley Publishing Co., Reading, Mass. (1983) 24. Fernández-Cara, E., Lü, Q., Zuazua, E.: Null controllability of linear heat and wave equations with nonlocal spatial terms. SIAM J. Control Optim. 54(4), 2009–2019 (2016) 25. Fisher, G.M.C., Gurtin, M.E.: Wave propagation in the linear theory of viscoelasticity. Quart. Appl. Math. 23, 257–263 (1965) 26. Giorgi, C., Gentili, G.: Thermodynamic properties and stability for the heat flux equation with linear memory. Quart. Appl. Math. 51(2), 343–362 (1993) 27. Graffi, D.: Sui problemi della eredità lineare. Nuovo Cimento 5, 53–71 (1928) 28. Graffi, D.: Sopra alcuni fenomeni ereditari dell’elettrologia. Rend. Istit. Lombardo Sci. Lett 68-69, 124–139 (1936) 29. Gurtin, M.E., Pipkin, A.C.: A general theory of heat conduction with finite wave speeds. Arch. Rational Mech. Anal. 31(2), 113–126 (1968) 30. Herrera, I., Gurtin, M.E.: A correspondence principle for viscoelastic wave propagation. Quart. Appl. Math. 22, 360–364 (1965) 31. Hopkinson, J.: The residual charge of the Leiden jar. Phil. Trans. Roy. Soc. Lond. 166, 489–494 (1876)

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32. Hopkinson, J.: Residual charge of the Leiden jar—dielectric properties of different glasses. Phil. Trans. Roy. Soc. Lond. 167, 569–626 (1877) 33. Iaffaldano, G., Caputo, M., Martino, S.: Experimental and theoretical memory diffusion of water in sand. Hydrology and Earth System Science Discussion. European Geoscience Union. 2(4), 1329–1357, hal-00298687 (2005) 34. Jangveladze, T.: Investigation and numerical solution of nonlinear partial differential and integro-differential models based on system of Maxwell equations. Mem. Differ. Equ. Math. Phys 76, 1–118 (2019) 35. Jeffreys, H.: The Earth. Cambridge University Press, Cambridge (1924) 36. Joseph, D.D., Preziosi, L.: Heat waves. Rev. Modern Phys. 61(1), 41–73 (1989) 37. Joseph, D.D., Preziosi, L.: Addendum to the paper: “Heat waves” [Rev. Modern Phys. 61 (1989), no. 1, 41–73. Rev. Modern Phys. 62(2), 375–391 (1990) 38. Kohlrausch, F.: Über die elastische Nakwirkung bei der Torsion. Ann. Phys. Chem. (Pogg. Ann.) 119(3), 337–368 (1863) 39. Kolsky, H.: Stress waves in solids. Dover Publications, Inc., New York (1963) 40. Leugering, G.: Time optimal boundary controllability of a simple linear viscoelastic liquid. Math. Methods Appl. Sci. 9(3), 413–430 (1987) 41. Maxwell, J.C.: On the dynamical theory of gases. Phil. Trans. Roy. Soc. London 157, 49–88 (1867) 42. Narain, A., Joseph, D.D.: Linearized dynamics of shearing deformation perturbing rest in viscoelastic materials. In: Equadiff 82 (Würzburg, 1982), Lecture Notes in Math., vol. 1017, pp. 476–507. Springer, Berlin (1983) 43. Pandolfi, L.: Distributed systems with persistent memory. Control and moment problems. SpringerBriefs in Control, Automation and Robotics, Springer, Cham (2014) 44. Pandolfi, L.: Controllability and lack of controllability with smooth controls in viscoelasticity via moment methods. Integral Equations Operator Theory 90(3), Paper No. 33, 18 (2018) 45. Pandolfi, L.: On the fourth order Cattaneo equation of heat conduction with memory. arXiv:2005.02383 (2020) 46. Perez Ortiz, R., Vlasov, V. V.: Correct solvability of Volterra integrodifferential Electron. J. Qual. Theory Differ. Equ. Paper No. 31, 17 (2016) 47. Povstenko, Y. Z.: From the chemical potential tensor and concentration tensor to nonlocal continuum theories. J. Math. Sci. (N.Y.) 249(3), 389–403 (2020) 48. Renardy, M.: Some remarks on the propagation and nonpropagation of discontinuities in linearly viscoelastic liquids. Rheol. Acta 21(3), 251–254 (1982) 49. Renardy, M.: Mathematical analysis of viscoelastic flows, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 73. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2000) 50. Renardy, M., Hrusa, W.J., Nohel, J.A.: Mathematical problems in viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 35. Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York (1987) 51. Skubachevskii, A.L.: Elliptic functional-differential equations and applications, Operator Theory: Advances and Applications, vol. 91. Birkhäuser Verlag, Basel (1997) 52. Spigler, R.: More around Cattaneo equation to describe heat transfer processes. Math. Meth. Appl. Sci. 43(9), 5953–5962 (2020) 53. Tikhonov, A.N., Samarski˘ı, A.A.: Equations of mathematical physics. Dover Publications, Inc., New York (1990) 54. Tschoegl, N.: The phenomenological theory of linear viscoelastic behavior. An introduction. Springer-Verlag, Berlin (1989) 55. Vassilev, V., Djondiorov, P.: Dynamic stability of viscoelastic pipes on elastic foundations of variable modulus. J. Sound Vibration 297, 414–419 (2006) 56. Vlasov, V. V., Medvedev, D. A.: Functional-differential equations in Sobolev spaces and related problems in spectral theory. J. Math. Sci. (N.Y.) 164(5), 659–841 (2010) 57. Vlasov, V. V., Rautian, N. A.: Well-posed solvability of Volterra integro-differential equations in Hilbert space. Differ. Equ. 52(9), 1123–1132 (2016)

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1 Preliminary Considerations and Examples

58. Vlasov, V. V., Rautian, N. A.: Well-posed solvability and the representation of solutions of integro-differential equations arising in viscoelasticity. Differ. Equ. 55(4), 561–574 (2019) 59. Volterra, V.: Sulle equazioni integro-differenziali. Rend. Acad. Naz. Lincei, Ser. 5 XVIII, 167–174 (1909) 60. Volterra, V.: Sur les équations intégro-différentielles et leurs applications. Acta Math. 35, 295–356 (1912) 61. Volterra, V.: Leçons sur les fonctions de lignes. Gauthier-Villars, Paris (1913) 62. Volterra, V.: La teoria dei funzionali applicata ai fenomeni ereditary. In: Atti del congresso internazionale dei matematici. Bologna, 1928 (VI), vol. Tomo 1: rendicontto del congresso. Conferenze., pp. 215–232. Nicola Zanichelli Editore, Bologna (1929 (VII)) 63. Westerlund, S., Ekstam, L.: Capacitors theory. IEEE Trans. Diel. Electr. Insul. 1(5), 826–839 (1994)

Chapter 2

Operators and Semigroups for Systems with Boundary Inputs

2.1 Preliminaries on Functional Analysis In this section, mainly to fix the notations, we collect few notions concerning bounded and unbounded operators in Banach and Hilbert spaces. We refer to standard books, as, for example, [2, 4, 7].

2.1.1 Continuous Operators The Banach spaces we use in this book are always separable i.e. they contain a dense denumerable subset. A Hilbert space is separable when it contains a sequence {ϕn } which is an orthonormal basis, i.e. ϕn = 1 ,

ϕn, ϕk  = 0 if n  k ,

cl span{ϕn } = H .

Let H and K be Banach spaces (the norm is · ). We assume that H and K are linear spaces on the same scalar field (in practice either R or C). A linear operator L from H to K is continuous if and only it is bounded, i.e. if and only if L < +∞ where (2.1) L = sup L x . x =1

We denote L(H, K) the linear space of the linear bounded operators with values in K and whose domain is the entire space H. It turns out that L → L is a norm on L(H, K) and that the space L(H, K) with this norm is a Banach space too. For the sake of clarity, it is sometime useful to add an index to the norms as follows: L L(H,K) = sup L x K . x H =1

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 L. Pandolfi, Systems with Persistent Memory, Interdisciplinary Applied Mathematics 54, https://doi.org/10.1007/978-3-030-80281-3_2

37

38

2 Operators and Semigroups for Systems with Boundary Inputs

When H = K we write L(H) instead of L(H, H). When K is the scalar field, in practice either R or C, the space L(H, K) is denoted H  (the dual space of H). If χ ∈ H , then the value taken by χ on the element h ∈ H is denoted1 h, χ = h, χ H, H 

and ·, · is the pairing of the space and its dual .

Remark 2.1. Many authors use the notation  χ, h, with the element of the dual in the first position since this notation is more readable when h is expressed by a long formula. We usually write the element of the dual in the second position but this is not always convenient. If needed, we use indexes to specify the position of the element of the space and that of the element of the dual. So, we write either

h, h  H, H 

or

h , h H , H

as most convenient (note also that the elements of the dual are often identified by a prime). It is known that the dual of a Hilbert space can be identified with the Hilbert space itself, H = H . This is Riesz Theorem discussed in Sect. 2.4. We shall use the following terms: • a map t → L(t) from (a, b) to L(H, K) is strongly continuous when t → L(t)h is a K-valued continuous function for every h ∈ H. • let2 A: H → K be a linear possibly not continuous operator and let u(t) be an Hvalued function which takes values in dom A. We say that u ∈ C k ([0, T]; dom A) when Au(t) ∈ C k ([0, T]; K). The operators we are going to use are for the most part linear. Concerning nonlinear operators, we recall Banach fixed point theorem. Let H be a Banach space and let T be a possibly nonlinear operator in H (i.e. an operator from H to H). A fixed point of T is a point such that T h = h. Let there exist a ball B which is invariant under T: B = {h ∈ H such that h − h0 ≤ R}

and T h ∈ B ∀h ∈ B .

It can be R = +∞. In this case B = H. The operator T is a contraction in B if there exists α ∈ [0, 1) such that T h − T h1 ≤ α h − h1

∀h , h1 ∈ B .

(2.2)

We stress the strict inequality: α < 1. We shall introduce the space D(Ω) which is not a Banach space. Also in this case the value on φ ∈ D(Ω) of an element χ of the dual D  (Ω) is denoted φ, χ. 2 The reader may wonder the reason why here we shift the notation from L to A. The reason is that we use the notation A when we present considerations which in the book are most often applied to the infinitesimal generator of a semigroup (introduced in Sect. 2.8) since A is a standard notation for the infinitesimal generators. 1

2.1 Preliminaries on Functional Analysis

39

We have: Theorem 2.2 (Banach fixed point theorem). If T is a contraction in B, then it has one and only one fixed point in B. Let T = (I + L) with L ∈ L(H) and let us consider the equation h + Lh = f .

(2.3)

A solution of (2.3) is a fixed point of the operator h → f − Lh. This transformation is a contraction if and only if L < 1 (and then it is a contraction on the entire space H). Hence, if L < 1, then Eq. (2.3) admits a solution and this solution is unique in H. It turns out that the (unique) solution is a linear and continuous function of f and it is given by the von Neumann formula: h=

+∞ 

(−1)n L n f = f −

n=0

+∞ 

(−1)n−1 L n f .

(2.4)

n=1

Formula (2.4) shows that when L < 1 the solution of the equation h + Lh = f depends continuously on f .

2.1.2 The Complexification of Real Banach or Hilbert Spaces The complexification of a real Banach space H is the complex Banach space HC whose elements are h1 + ih2 where both h1 and h2 belong to H. If (a + ib) ∈ C, then by definition (a + ib)(h1 + ih2 ) = ah1 − bh2 + i(ah2 + bh1 ) . The norm in HC is h1 + ih2 HC =

2 + h 2 . h1 H 2 H

(2.5)

The elements of the dual are χ  = χ1 + i χ2 (with χ1 and χ2 in H ) and by definition the action of χ1 + i χ2 on h1 + ih2 is h1 + ih2, χ1 + i χ2  HC, HC = h1, χ1  H, H  − h2, χ2  H, H 

+i(h2, χ1  H, H  + h1, χ2  H, H  ) .

The pairing ·, · HC, HC is linear in both its arguments. If H is a real Hilbert space with inner product ·, · H , the inner product in HC is h1 + ih2, h˜ 1 + i h˜ 2  HC = h1, h˜ 1  H + h2, h˜ 2  H + i[h2, h˜ 1  H − h1, h˜ 2  H ] . (2.6)

40

2 Operators and Semigroups for Systems with Boundary Inputs

The inner product ·, · HC is linear in its first argument and antilinear in the second one. The space HC with this inner product is a complex Hilbert space. Using the fact that in the real Hilbert space H we have h, k H = k, h H , we see that 2 . h1 + ih2, h1 + ih2  HC = h1 + ih2 H C Remark 2.3 (Linear Operator and Conjugation). Let HC be the complexification of the real Hilbert (or Banach) space H. The element h1 − ih2 is the conjugate of h1 + ih2 and it is denoted h1 + ih2 . It is important to note that conjugation does not commute with any L ∈ L(HC, KC ). For example, if KC = HC and L(h1 + ih2 ) = i(h1 + ih2 ) = −h2 + ih1 , then   L h1 + ih2  L(h1 + ih2 ) . A necessary and sufficient condition that conjugation commutes with L is L(h + i0) = k + i0 ∀h ∈ H (where k ∈ K) (so that we have also L(0 + ih) = 0 + ik)

(2.7)

(we recall that H and K are real linear spaces). The necessity is obvious. Conversely, if condition (2.7) holds, then for every h1 and h2 in H we have L(h1 + ih2 ) = L ((h1 + i0) + (0 + ih2 )) = L(h1 + i0) + L(0 + ih2 ) = k1 + ik2   L h1 + ih2 = L ((h1 + i0) − (0 + ih2 )) = L(h1 + i0) − L(0 + ih2 ) = k1 − ik2 = L(h1 + ih2 ) . Finally we note the common practice of dropping the index C , unless really needed for clarity, and to denote the complexification HC of H as H.

2.1.3 Operators and Resolvents The resolvent set (A) of the operator A acting from H to itself is the set of those complex numbers3 λ ∈ C such that (I denotes the identity operator, I h = h): • (λI − A)h = 0 if and only if h = 0, and so (λI − A)−1 exists, defined on im(λI − A). • the image of (λI − A) is dense in H and (λI − A)−1 is continuous. If (λI − A)h = 0 has a non zero solution h, then λ is an eigenvalue of A and h is an eigenvector The operator (λI − A)−1 , defined for λ ∈ (A), is the resolvent operator of A. and the set C \ (A) is the spectrum of A. 3

So that when using the resolvent we work in a complex space, possibly the complexification of a real space.

2.1 Preliminaries on Functional Analysis

41

2.1.4 Closed and Closable Operators Let A be a linear operator from a Hilbert space H to a Hilbert space K. The graph of A is G(A) = {(h, Ah), h ∈ dom A}. The set G(A) is a linear subspace of H × K. If it is a closed subspace, then the operator A is a closed operator. The explicit test to verify that an operator is closed is as follows: we consider a sequence {(hn, Ahn )} ∈ G(A). If this sequence does not converge, then noting is required. But, if it converges it must converge to an element of the graph: if hn → h and if Ahn → k, then it must be h ∈ dom A and Ah = k. Important properties are: Theorem 2.4. Let A be a closed operator in H. The following properties hold: • • • •

if A is invertible, then its inverse is closed too. if A is everywhere defined on H, then it is continuous. if A is surjective and invertible, then its inverse A−1 is continuous. let D ∈ L(U, H). If im D ⊆ dom A, then AD ∈ L(U, H).

Some (densely defined in the cases of our interest) operators A which are not closed may be closable. This means that the closure of the graph is still a graph (i.e. the graph of an operator which by definition is the closure of the operator A). A test for closability is simple: if {(hn, Ahn )} and {(xn, Axn )} both converge in H × K, then: if lim hn = lim xn

, then it must be lim Ahn = lim Axn .

An equivalent test is: Lemma 2.5. The operator A is closable if any sequence {hn } such that hn → 0 and { Ahn } is convergent has the additional property that Ahn → 0. An operator which is densely defined, if closable, has a unique closure.

2.1.5 The Transpose and the Adjoint Operators Let H and K be Banach spaces and let A ∈ L(H, K). We recall that H  and K  are the dual spaces and we denote their elements with a prime: for example, h denotes an element of H and h  an element of H . The functional h → Ah, k  K,K 

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2 Operators and Semigroups for Systems with Boundary Inputs

is linear and continuous on H, i.e. it is an element h  ∈ H  which depends on k : h  = h (k ) and

Ah, k  K,K  = h, h (k ) H, H 

∀h ∈ H .

The transformation k  → h (k ) from K  to H  is called the transpose of the operator A and it is denoted4 A. So we have Ah, k K,K  = h, A k  H, H 

∀h ∈ H ,

k ∈ K .

The properties of the transpose operator are: 1. 2. 3. 4.

the operator A is linear and continuous from K  to H . the norm is preserved: A L(H,K) = A L(K , H  ) . we have (A) = A. the transformation A → A is linear and continuous from L(H, K) to L(K , H ).

Of course, this definition can be applied in the special case that H and K are Hilbert spaces, but in this special case we can define also the adjoint operator A∗ acting from K to H. The operator A has been defined only when A is bounded (extension is possible, but not needed in this book). Instead, we need the definition of the adjoint also of unbounded operators. So, the definition is more delicate. Let A be a linear operator from a Hilbert space H to a Hilbert space K, possibly unbounded but with dense domain. Then it is possible to define its (Hilbert space) adjoint operator A∗ , acting from K to H, as follows: A∗ k is defined when there exists z ∈ H such that the following equalities hold: Ah, kK = h, z H

∀h ∈ dom A .

(2.8)

Density of the domain of A implies that z, if it exists, it is unique. By definition, A∗ k = z so that

Ah, kK = h, A∗ k H

∀h ∈ dom A , ∀k ∈ dom A∗ .

The operator A is selfadjoint when it acts from H to itself and A = A∗ (note that this implies dom A = dom A∗ ). The properties of the adjoint operators are: • A∗ is defined on a subspace of K (possibly not dense and even reduced to {0}) and it is linear from K to H. • any adjoint operator is closed. So, in particular, a selfadjoint operator is closed. • if the operator A is closed with dense domain, then A∗ (is closed and) has dense domain too. So, the operator (A∗ )∗ , shortly denoted A∗∗ , can be defined. • the operator A is continuous if and only if A∗ is continuous. In this case A L(H,K) = A∗ L(K, H) . Different notations are used and quite often the transpose is denoted A∗ but we avoid this notation in order to distinguish the transpose operator A from the adjoint operator A∗ in a Hilbert space defined below.

4

2.1 Preliminaries on Functional Analysis

43

• if dom A∗ is dense, then A is closable and the operator A∗∗ is the closed extension of A (in particular, if A has dense domain and it is continuous, then A∗∗ is the extension of A to H by continuity). • let us consider the transformation A → A∗ from L(H, K) to L(K, H). The transformation is continuous and antilinear, i.e. (A+ A1 )∗ = A∗ + A∗1 and (αA)∗ = αA∗ . Note that in order to define (A + A1 )∗ we must require that dom A ∩ dom A1 is dense in H. The fact that A → A is linear while A → A∗ is antilinearity is not such a deep difference, but it is a bit annoying and it has to be taken into account when both the transpose and the adjoint of an operator are used. The precise relation between the adjoint and the transpose is discussed in Sect. 2.4.

2.1.5.1 The Adjoint, the Transpose and the Image of an Operator It is clear that if A is a (possibly unbounded) operator from H to K (both Hilbert spaces), then ξ ⊥ im A if and only if ξ, Ah = 0 for every h ∈ dom A and so the map h → ξ, Ah = 0 is continuous. Hence if dom A is dense, then ξ ⊥ im A if and only if ξ ∈ ker A∗ ⊆ dom A∗ . By elaborating on this observation, it is possible to recast in a “dual” way the property of having dense image or of being surjective. The next result is proved, for example, in [7, Chap. II.7] and in [13, Sects. II.3 and II.4]. We state it in the more delicate case of unbounded operators in Hilbert spaces but the same statement holds when A is a linear bounded operator between Banach spaces (in this case A∗ has to be replaced with A and when needed by the formulas the dual spaces have to replace H and K). Theorem 2.6. Let A be a linear closed operator with dense domain in a Hilbert space H and image in a Hilbert space K. Then we have: 1. the operator A is invertible with bounded inverse if and only if there exists m > 0 such that m x H ≤ Ax K for every x ∈ dom A (when this inequality holds and A is selfadjoint we say that A is coercive). 2. if A∗ is surjective, then A is invertible and A−1 is bounded and conversely. 3. the operator A is surjective if and only if (A∗ )−1 is continuous i.e. if and only if there exists m1 > 0 such that m1 k K ≤ A∗ k H (when this inequality holds and A∗ is selfadjoint the operator A∗ is coercive). 4. A is continuous and boundedly invertible if and only if its adjoint is continuous and boundedly invertible.

2.1.6 Compact Operators in Hilbert Spaces Norm convergence of a sequence {xn } to x0 in a Hilbert space H, denoted xn → x0 , can be formulated as follows: for every ε > 0 there exists Nε such that

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2 Operators and Semigroups for Systems with Boundary Inputs

n > Nε and h ≤ 1 =⇒ |xn, h − x0, h| < ε . We say that {xn } converges weakly to x0 in a Hilbert space H when for every h ∈ H we have limxn, h = x0, h, i.e. for every ε > 0 and every h we can find Nε,h such that n > Nε,h =⇒ |xn, h − x0, h| < ε . We stress the dependence of Nε,h on h even if h ≤ 1 in the case of weak convergence. The notation for the weak convergence is either w − lim xn = x0 or xn  x0 . The properties we need of the weak convergence are: • any weakly convergent sequence is bounded. • if L ∈ L(H, K) and if xn  x0 in H, then L xn  L x0 in K. The previous properties hold in every Banach space. Now we assume that H and K are Hilbert spaces. A property which holds in Hilbert spaces is the following one: • any bounded sequence in a Hilbert space has weakly convergent subsequences. A K-valued operator L defined on H with H and K both Hilbert spaces is a compact operator5 when it transforms every weakly convergent sequence in H to a norm convergent sequence in K. Of course, in order to prove that L is compact we can prove hn  0 implies Lhn → 0 (norm convergence) . It is easily seen that every compact operator is continuous and that the composition of a continuous operator with a compact operator is compact. If it happens that H = K, a Hilbert space, then we can study the spectrum of the compact operator L. We state the following properties: • every λ  0 which is not an eigenvalue belongs to the resolvent set. • the number of the linearly independent eigenvectors which correspond to a non zero eigenvalue is finite. • Let A be a linear possibly unbounded operator with dense domain in H (we recall: H is separable) and with values in H. If it is selfadjoint and if its resolvent (λI − A)−1 is a compact operator,6 then the operator A has a sequence of normalized (i.e. of norm equal 1) eigenvectors which is an orthonormal basis of H.

5 6

The definition of compact operator as given here is specific of Hilbert spaces. For one value of λ. But this implies that (λI − A)−1 is compact for every λ ∈ (A).

2.2 Integration, Volterra Integral Equations and Convolutions

45

2.2 Integration, Volterra Integral Equations and Convolutions Let f (t) be a function of a real variable t with values in a Banach space. Then f (t0 ) = lim

t→t0

f (t) − f (t0 ) t − t0

and the partial derivatives defined pointwise7 of functions from Rd to a Banach space are defined similarly. The definition of the integral instead is more delicate and it has several facets. We assume a working knowledge of (Bochner) integration of Banach space valued functions defined on a region Ω of Rd , d ≥ 1. A detailed treatment can be found in [1, Sect. 1.1] and in [18, Chap. 3]. The idea is to define the simple functions as those functions which are constant on measurable sets and have finitely many values, and the measurable functions as those functions which are a.e. limit of sequences of simple functions. The integral of a simple function is defined in the obvious way. A measurable function f on a region Ω is (Bochner) integrable when there exists a sequence {gn } such that  gn is∫ simple, gn (x) → f (x) a.e. on Ω lim Ω f (x) − gn (x) dx = 0 . In this case the Bochner integral exists, defined by ∫ ∫ f (x) dx = lim gn (x) dx Ω

Ω

(it is possible to prove that the limit does not depend on the approximating sequence {gn } of simple functions). The important result is that f is Bochner integrable if and only if it is measurable and the real valued function f is integrable.8 The usual properties of the integral hold, in particular ∫  ∫    f (x) dx  ≤ f (x) dx .  Ω

Ω

The spaces L p (Ω) are defined in the usual way, as the spaces of (equivalent classes) of functions such that ∫ f L p (Ω) =

7

 1/p f (x) dx p

Ω

if p ∈ [1, +∞),

In general different from the derivatives in the sense of the distributions defined in Sect. 2.6. See Remark 2.31. 8 In the case of real valued functions, some authors use integrable when the limit which defines the integral exists, possibly not finite and summable when the limit, i.e. the integral, is finite. Unless the contrary is explicitly stated, in this book a real valued function is “integrable” when its integral is finite.

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2 Operators and Semigroups for Systems with Boundary Inputs

while L ∞ (Ω) is the space of the essentially bounded functions, with the usual L ∞ norm: f L ∞ (Ω) = ess supx ∈Ω f (x) . These definitions apply in particular when f takes values in L(H, K). Let L(t, s) ∈ L(H) (H is a Hilbert space in our applications, but the same results hold in Banach spaces too). The equation ∫ t w(t) + L(t, s)w(s) ds = f (t) (2.9) 0

is a linear Volterra integral equation of the second kind on H. For brevity, we call it simply a “Volterra integral equation.” Equation (2.9) can be written in operator form as follows: we introduce the linear operator L ∫ t L(t, s)w(s) ds so that Eq. (2.9) is w + Lw = f . (Lw)(t) = 0

With this notation, the Volterra integral equation takes the form (2.3) and so, at least if L < 1, the solution is expressed by the von Neumann series (2.4), i.e.: w= f−

+∞  (−1)n−1 Ln f .

(2.10)

n=1

The operators Ln have a very special structure. For example, ∫ t ∫ s   L(t, s) L(s, r)w(r) dr ds . L2 w (t) = 0

0

By exploiting the special form of these operators it is possible to prove: Theorem 2.7. Let TT be the triangle 0 ≤ s ≤ t ≤ T. Let L(t, s) ∈ L(H) a.e. 0 ≤ s ≤ t and let L(t, s) ∈ L 2 (TT , L(H)). The series in (2.10) converges in L 2 ([0, T]; H) and furthermore for every T > 0 formula (2.10) defines a linear continuous transformation from L 2 (0, T; H) to itself. If L(t, s) is strongly continuous, then the series converges in C([0, T]; H) and defines a linear continuous transformation from L 2 (0, T; H) to C([0, T]; H) (and so also a linear continuous transformation from C([0, T]; H) to itself). The important observation is that, thanks to the special form of the operator L, Theorem 2.7 asserts convergence of the series (2.10) even if L is not a contraction. Formula (2.10) when applied to a Volterra integral equation is called the Picard formula. Note the way the Picard formula is obtained: Eq. (2.9) gives ∫ t w(t) = f (t) − L(t, s)w(s) ds 0   ∫ s ∫ t L(t, s) f (s) − L(s, r)w(r) dr ds = f (t) − 0

0

2.2 Integration, Volterra Integral Equations and Convolutions

47

= f − L f + L2 w = f − L f + L2 [ f − Lw] = · · · In the context of Volterra integral equation, this method is the Picard iteration. This kind of general (linear) Volterra integral equations are encountered in Sect. 3.3.1 (while in Chap. 6 we encounter Volterra integral equations of the first kind and nonlinear Volterra integral equations). In most of the cases we are interested in Volterra integral equations of convolution type which we introduce now. Let H and K be Banach spaces and let F(t) and h(t) be measurable functions on t ≥ 0 with values, respectively, in L(H, K) and in H. Their convolution F ∗ h is defined as follows: ∫ t F(t − s)h(s) ds . k(t) = (F ∗ g)(t) = 0

Of course in general the integral does not exist. A result which ensures existence of the convolution and which specifies its properties goes under the name of Young theorem while (2.11) and (2.12) are the Young inequalities (see [1] for more details). Theorem 2.8. Let H and K be Banach spaces. We have: • if F ∈ L p (0, +∞; L(H, K)) and h ∈ L q (0, T; H), then k = F ∗ h ∈ L r (0, +∞; K) where r, p, and q are related by 1 1 1 = + −1 r p q

(if

1 p

+

1 q

− 1 = 0, then r = ∞)

and k L r (0,+∞;K) ≤ F L p (0,+∞;L(H,K)) h L q (0,+∞;H) .

(2.11)

• if p1 + q1 − 1 = 0 and furthermore p ∈ (1, +∞), q ∈ (1, +∞), then k = F ∗ h ∈ C([0, +∞); K) and for every T > 0 we have: k C([0,T ];K) ≤ F L p (0,T ;L(H,K)) h L q (0,T ;H) .

(2.12)

A Volterra integral equation (of the second kind) of convolution type in a Hilbert space H is ∫ t w(t) + L(t − s)w(s) ds = f (t) . (2.13) 0

The function L(t) is the kernel of the Volterra integral equation. We assume that: 1. L(t) takes values in L(H) for a.e. t > 0; 2. t → L(t) is either strongly continuous or integrable; 3. the function f (t) belong to L 2 (0, T; H) or to C ([0, T]; H). The notation L ∗k denotes iterated convolution of L with itself: L ∗1 = L ,

L ∗(k+1) = L ∗ L ∗k .

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Then we have: Lemma 2.9. Equation (2.13) admits a unique solution w(t) in L 2 (0, T; H), which depends continuously on f ∈ L 2 (0, T; H). The solution w is given by the Picard formula +∞  (−1)k−1 L ∗k ∗ f (2.14) w= f− k=1

L 2 (0, T; H)).

(the series converges in If f (t) ∈ C ([0, T]; H) and L(t) ∈ L p (0, T; L(H)) with p > 1, then the series (2.14) converges in C([0, T]; H) so that w ∈ C ([0, T]; H) and the transformation f → w is continuous in C ([0, T]; H). Let R=

+∞  (−1)k−1 L ∗k . k=1

Then we can write ∫ w = f − R ∗ f = f (t) −

t

∫ R(t − s) f (s) ds = f (t) −

0

t

R(s) f (t − s) ds . (2.15)

0

The function R(t) is called the resolvent kernel of (2.13) and it is identified by the fact that it is the unique solution of the Volterra integral equation in L(H) R+R∗L = L . The resolvent kernel R(t) is (square) integrable when L(t) is (square) integrable and it is strongly continuous when L(t) is strongly continuous. An observation we shall use is as follows: Lemma 2.10. Let L(t) be strongly continuous in 0 ≤ s ≤ t. If f (t) ∈ C 1 ([0, T]; H), then the solution w of (2.9) belongs to C 1 ([0, T]; H). In fact, the right hand side of (2.15) is differentiable, so that w(t) is differentiable and ∫ t   w (t) = f (t) + R(t) f (0) + R(s) f (t − s) ds 0 ∫ t R(t − s) f (s) ds . = f (t) + R(t) f (0) + 0

When dealing with Volterra integral equations, Gronwall inequality is a useful device. Lemma 2.11 (Gronwall Inequality). Let u(t) be a continuous real valued function defined on [0, T], T ≤ +∞. If ∫ t u(s) ds, (2.16) 0 ≤ u(t) ≤ a + b 0

2.3 Laplace Transformation

49

then we have also 0 ≤ u(t) ≤ aebt

∀t ∈ [0, T] .

(2.17)

Now we consider the following Volterra integro-differential equation (where N, f , and w are scalar valued, N(t) is continuous, and f (t) square integrable): ∫ t w  = 2αw + N(t − s)w(s) ds + f (t) , w(0) = w0 . 0

The solution exists and it is unique since integrating both the sides we see that w(t) solves a Volterra integral equation. We are interested in a formula for the solution. Let z(t) solve ∫ t  z (t) = 2αz(t) + N(t − s)z(s) ds , z(0) = 1 . 0

Then w(t) is given by the following variation of constants formula: ∫ t w(t) = z(t)w0 + z(t − s) f (s) ds .

(2.18)

0

2.3 Laplace Transformation Frequency domain techniques, i.e. techniques based on the use of the Laplace transformation, are a powerful instrument in the analysis of linear systems. We collect here the fundamental properties we shall use. The functions we consider take values in a Banach space, say Y . We cannot confine ourselves to functions in Hilbert spaces since we shall use also the Laplace transform of operator valued functions but some of the properties are specific of functions with values in Hilbert spaces and they do not hold in general Banach spaces. In order to stress the difference, Banach spaces are denoted Y , U, V while H is used to denote a Hilbert space.

2.3.1 Holomorphic Functions in Banach Spaces Let Y be any Banach space and let f (λ) be a function from a region D ⊆ C to Y . The function f is holomorphic on D when its derivative f (λ0 ) = lim

λ→λ0

1 [ f (λ) − f (λ0 )] λ − λ0

exists for every λ0 ∈ D. The limit must be computed in the norm of Y but the following result is important (see [18, Chap. III.2] for a study of vector valued holomorphic functions):

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Theorem 2.12. Let f (λ) be defined on a region D ⊆ C with values in the Banach space Y . The following properties hold: 1. f (λ) is holomorphic if and only if the complex valued functions  f (λ), y Y,Y  are holomorphic for every y ∈ Y . If Y = L(U, V) (U and V are Banach spaces), then f (λ) is holomorphic if and only if the complex valued functions  f (λ)u, v V,V  are holomorphic for every u ∈ U and v  ∈ V . 2. Let f (λ) be bounded on D, f (λ) Y ≤ M for all λ ∈ D. Then, f (λ) is holomorphic on D if and only if the function  f (λ), y Y,Y  is holomorphic for every y  in a dense subspace of Y ; if f (λ) takes values in L(U, V) (U and V are Banach spaces) and f (λ) L(U,V ) is bounded on D, then f (λ) is holomorphic on D if and only if  f (λ)u, v V,V  is holomorphic for any u and v  in dense subspaces of U and of V . Statement 2 is [33, Remark 1.38 p. 139]. Thanks to these results, it is easy to prove that the elementary properties of holomorphic Y -valued functions are the same as in the case Y = C. In particular, it is easily seen that Cauchy theorem and Cauchy integral formula, Morera theorem, the residue theorem and Weierstrass convergence theorem hold. We recall Weierstrass theorem: Theorem 2.13. Let { f N (λ)} be a sequence of holomorphic functions in a region D ⊆ C which converges to f (λ). If the convergence is uniform on every compact set K ⊆ D, then f (λ) is holomorphic on D.

2.3.2 Definition and Properties of the Laplace Transformation A study of the Laplace transform of Banach space valued functions can be found in [18, Chap. VI]. Let f (t) be defined (a.e.) on t ≥ 0 with values in a Banach space Y . This function is Laplace transformable when the following conditions hold:9 1. the function is integrable on (0, T) for every T > 0; 2. there exists ω such that e−ωt f (t) is integrable on (0, +∞). Under these conditions, the function ∫ +∞ ˆf (λ) = e−λt f (t) dt

(2.19)

0

is defined (i.e. the integral converges) in the half plane e λ > ω and fˆ(λ) is holomorphic in this halfplane. The function fˆ(λ) is the Laplace transform of f (t). 9

These conditions can be weakened and a more general definition of the Laplace transformation can be given. For example, Theorem 2.19 gives an extended definition.

2.3 Laplace Transformation

51

Remark 2.14. The function f (t) might be real valued but fˆ(λ) is complex valued. If f (t) takes values in a real Banach space, then fˆ(λ) takes values in the complexification of the space. The Laplace transform fˆ(λ) defined by the integral in (2.19) may have a holomorphic extension to a region larger than the half plane in which the integral converges, as in the case f (t) = 1. In this case fˆ(λ) = 1/λ. The integral converges for e λ > 0 but fˆ(λ) admits a holomorphic continuation to C \ {0}. The Laplace transform of a function is denoted also L( f )(λ): L( f )(λ) = fˆ(λ). This notation is convenient when f has a long expression, as in the following equalities which hold in the common domains of the Laplace transforms of the functions involved: ˆ , (L(α f + βg)) (λ) = α fˆ(λ) + βg(λ) (L ( f )) (λ) = λ fˆ(λ) − f (0) , ˆ g(λ) ˆ . (L (F ∗ g)) (λ) = F(λ) In the last equality we intend that F(t) takes values in L(U, V) (U and V both Banach spaces), g(t) takes values in U and ∗ denotes the convolution as defined in Sect. 2.2: ∫ t F(t − s)g(s) ds . (F ∗ g) (t) = 0

The following lemma is easily proved: Lemma 2.15. We have: 1. if f ∈ L 1 (0, +∞; Y ), then fˆ(λ) exists and it is bounded and uniformly continuous in e λ ≥ 0. 2. if f ∈ C 1 ([0, +∞); Y ) and both f (t) and f (t) are integrable, then lim fˆ(λ) = 0 . e λ≥0

(2.20)

|λ|→+∞

3. if f ∈ D(0, +∞; Y ) (i.e. if it is of class C ∞ with compact support), then for every k ∈ N we have lim λ k fˆ(λ) = 0 . (2.21) |λ|→+∞ e λ≥0 Proof. The integral fˆ(λ) converges in e λ ≥ 0 since f (t) is integrable and so fˆ(λ) is holomorphic and bounded in e λ > 0. We prove uniform continuity in the closed half plane. Let e λ ≥ 0, e λ  ≥ 0. We have  ∫ +∞      −λt −λ t  ˆ ˆ e −e f (t) dt  f (λ) − f (λ ) =  ∫

≤ 0

0

T

∫    −λt −λ t  e − e  f (t) dt + 2

T

+∞

f (t) dt .

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2 Operators and Semigroups for Systems with Boundary Inputs

Given ε > 0, the second integral is less than ε/2 if T ≥ Tε , for Tε sufficiently large. With this value of T = Tε now fixed, there exists δε > 0 such that the first integral is less than ε/2 if |λ − λ  | < δε . In order to prove the statement 2 we note that integrability of f  implies the existence of limT →+∞ f (T) and integrability of f then implies that the limit is 0. So we have 1 fˆ(λ) = − lim λ T →+∞

∫ 0

T

 ∫ +∞  1   f (t) d e−λt = e−λt f (t) dt . f (0) + λ 0

(2.22)

Then we have lim

|λ|→+∞

fˆ(λ) = 0

since integrability of f  implies boundedness of its Laplace transformation. This argument can be iterated if the assumption of statement 3 holds and (2.21) follows.   In fact, integrable functions with values in Hilbert spaces have a far deeper property, known as Riemann–Lebesgue Lemma: (see [1, Theorem 1.8.1]): Theorem 2.16. Let f ∈ L 1 (0, +∞; H) (H is a Hilbert space). Then lim fˆ(λ) = 0 . e λ≥0 |λ|→+∞

Note that the limit is zero in particular if it is computed along the imaginary axis. When the function takes values in a (real or complex) Hilbert space we can consider a “convolution” inside the inner product. Let f and g both belong to L 2 (0, +∞; H) where H is a Hilbert space. From Theorem 2.8 we have ∫ t h(t) =  f (t − s), g(s) H ds ∈ L ∞ (0, +∞) ∩ C([0, +∞)) 0

and the Laplace transform of h(t) is ˆ ¯ H. h(λ) =  fˆ(λ), g( ˆ λ) This equality is seen as follows:  ∫ t ∫ +∞ −λt ˆh(λ) = e  f (t − s), g(s) H ds dt 0 0   ∫ +∞ ∫ +∞ −λt = e  f (t − s), g(s) dt ds 0 s ∫ +∞ ∫ +∞ ¯ ¯ . = e−λr f (r), e−λs g(s) dr ds =  f (λ), g(λ) 0

0

2.3 Laplace Transformation

53

Finally we consider explicitly the case that the functions take values in a real Hilbert space. Let λ = ξ + iω. Then ∫ +∞ ∫ +∞ e−ξt f (t) cos ωt dt − i e−ξt f (t) sin ωt dt . fˆ(ξ + iω) = 0

0

Hence

⎧ ⎪ ¯ , fˆ(λ) = fˆ(λ) ⎪ ⎪ ∫ +∞  ⎪ ⎨ ⎪ f (t) cos ωt dt =0 lim |ω |→+∞ 0 (2.23) if f is real, then ⎪ ∫ +∞  ⎪ ⎪ ⎪ ⎪ lim |ω |→+∞ 0 f (t) sin ωt dt = 0 . ⎩ The Laplace transform fˆ(λ) uniquely identifies (the equivalent class of) f (t), in the following sense: Theorem 2.17. There exists c such that fˆ(λ) = g(λ) ˆ in the half plane e λ > c if and only if f (t) = g(t) a.e. t > 0. The problem of recovering f (t) from fˆ(λ) is delicate. We do not need to dig into this question, but we note that if it happens that fˆ(λ) is holomorphic for e λ > c0 and if it is infinitesimal (for |λ| → +∞ in e λ > c0 ) of the order 1 + γ respect to 1/(Im λ) (with γ > 0), then for a.e. t > 0 and every c > c0 we have ∫ +∞ ∫ 1 1 (c+is)t ˆ f (t) = e ezt fˆ(z) dz . (2.24) f (c + is) ds = 2π −∞ 2πi rc

  rc : z=(c+is) s ∈R

We stress the fact that, under the stated conditions, the value of the integral does not depend on c > c0 .

2.3.3 The Hardy Space H 2 (Π+ ; H) and the Laplace Transformation In this section the space Y is renamed H in order to stress the fact that the results hold for functions with values in Hilbert spaces:10 H is a complex (separable) Hilbert space. We use Π+ to denote the right half plane, Π+ = {e λ > 0}. The Hardy space H 2 (Π+ ; H) is the space of the H-valued functions F(λ) which are holomorphic in e λ > 0 and such that ∫ +∞ 2 2 F H F(x + iy) H dy < +∞ . (2.25) 2 (Π ;H) = sup +

x>0

−∞

If there is no risk of confusion or if H = C, then we use the notation H 2 (Π+ ). Functions of H 2 (Π+ ; H) have strong properties. In particular: It is a fact that the results can be suitably adapted for functions which take values in L(H) or in L(H, K) when both H and K are Hilbert spaces.

10

54

2 Operators and Semigroups for Systems with Boundary Inputs

Theorem 2.18. Let F ∈ H 2 (Π+ ; H). Then: • limx→0+ F(x + iy) = F(iy) exists a.e. and it is square integrable on the imaginary axis. • the limit F(iy) exists in L 2 (iR; H): limx→0+ F(x + iy) − F(iy) L 2 (−∞,+∞;H) = 0. • if F and G belong to H 2 (Π+ ; H) and F  G, then F(iy)  G(iy) a.e. y ∈ R. • the following equality holds: ∫ +∞ 2 2 F H 2 (Π ;H) = F(iy) H dy . (2.26) +

−∞

Our interest in the Hardy spaces stems from the following Paley–Wiener theorem which shows the relation with the Laplace transformation: Theorem 2.19. Let f ∈ L 2 (0, +∞; H). We have: ∫T • 0 e−λt f (t) dt belongs to H 2 (Π+ ; H) for every T > 0 and ∫

T

fˆ(λ) = lim

T →+∞

e−λt f (t) dt

0

exists in H 2 (Π+ ; H) .

(2.27)

By definition, (2.27) is the Laplace transform of f ∈ L 2 (0, +∞; H). 1 2 fˆ H i.e. • we have f L2 2 (0,+∞;H)) = 2π 2 (π ;H)) +

∫

1 2π ∫ =

0

fˆ(iy) Y2 dy =

R +∞

  ∫ +∞   1  fˆ(x + iy)2 dy sup 2π x>0 0

f (t) Y2 dt .

(2.28)

• conversely, any element of H 2 (Π+ ; H) is the Laplace transform of a (unique) element of L 2 (0, +∞; H). • when f ∈ L 2 (0, +∞; H) (which is equivalent to fˆ ∈ H 2 (Π+ ; H)) then we have ∫ +∞ 1 f (t) = eiωt fˆ(iω) dω (2.29) 2π −∞ in the sense that 1 f (t) = lim T →+∞ 2π

∫

+T

−T

eiωt fˆ(iω) dω

in L 2 (0, +∞; H) .

(2.30)

If it happens that fˆ(iω) ∈ L 1 (−∞, +∞; H), then f (t) is continuous and the limit in (2.30) is uniform on compact subsets of [0, +∞). When f ∈ L 2 (0, +∞; H), its Laplace transform is shortly denoted as in (2.19): ∫ +∞ ˆf (λ) = e−λt f (t) dt 0

2.4 Graph Norm, Dual Spaces, and the Riesz Map of Hilbert spaces

55

but now the integral has to be intended as the limit (2.27) in the space H 2 (Π+ ; H). Finally we note that H 2 (Π+ ; H) endowed with the norm in (2.25) is a (separable) Hilbert space too with the inner product ! ∫ +∞ F(x + iy), G(x + iy) H dy F, G H 2 (Π+ ;H) = sup x>0 −∞ ∫ +∞ = F(iy), G(iy) H dy . −∞

It follows that ∫

+∞

 f (t), g(t) H dt =

0

1 2π

∫

+∞

−∞

 fˆ(iω), g(iω) ˆ H dω

(2.31)

for every pair of functions f and g in L 2 (0, +∞; H). This equality is the PlancherelParseval identity.

2.4 Graph Norm, Dual Spaces, and the Riesz Map of Hilbert spaces We recall that Banach in particular Hilbert spaces used in this book are separable. Let A be a unbounded densely defined and closed operator in a Hilbert space H, A :

H → H .

Its domain is denoted X so that X is a dense subspace which is properly contained in H. The subspace X when endowed with · H is not closed, hence it is not a Hilbert space. It is possible to define a second norm on X, related to the operator A, under which it is a Hilbert space. This norm is 2 2 x 2A = (x, Ax) 2G(A) = x H + Ax H

(the graph norm of X) .

A sequence {xn } converges in X, with the graph norm, if and only if {(xn, Axn )} converges in the graph of A. The graph of A is closed in the Hilbert space H × H by assumption and so with this norm X is complete and it is a Hilbert space: the norm · A is the norm of a Hilbert space with inner product x1, x2  A = x1, x2  H + Ax1, Ax2  H . Note that if 0 ∈ (A), then an equivalent norm is x A = Ax H

(2.32)

and with this norm A is an isometry between the Hilbert spaces X and H. In fact, the assumption 0 ∈ (A) is not restrictive, provided that (A)  ∅. In fact, let c0 ∈ (A).

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2 Operators and Semigroups for Systems with Boundary Inputs

Then X = dom A = dom (A − c0 I) and G(A − c0 I) = {(x, y) : x ∈ dom A and y = Ax − c0 x} . Hence, a sequence {(xn, Axn )} converges in G(A) if and only if {(xn, (A − c0 I)xn )} converges in G(A − c0 I). The norm (2.32) is the one we shall use when 0 ∈ (A). Remark 2.20 (On the Notations). There are two norms on the space X = dom A. When needed for clarity, we refer to the space either as (X, · H ) or as (X, · A). And sometimes for clarity it may be convenient to specify (H, · H ). The operator A is a linear operator from (X, · A) to (H, · H ) which is continuous: Ax H ≤ x A . A second linear transformation which is linear and continuous among these same spaces is the inclusion of X in H, defined as follows: if x ∈ X, then11 ix is that same element but considered as an element of H. The map i: (X, · A) → (H, · H ) is continuous since 2 2 2 2 ix H = x H ≤ x H + Ax H = x 2A .

Note that x, h H = ix, h H

∀x ∈ X

but the two maps x → x, h H and x → ix, h H are different. In fact: • the map x → x, h H defined for x ∈ X is a linear functional on H, whose domain is the dense subspace X of H. This map is continuous, and it admits a continuous extension to H since |x, h| ≤ x H h H . • the map x → ix, h H is a linear functional on (X, · A). It is continuous on (X, · A), i.e. it is an element of its dual for each h ∈ H since |ix, h H | ≤ x A h H . See remark 2.25 for a more precise statement. Remark 2.21 (On the Notation). The dual X  of X is always intended as the dual of (X, · A). Note that the dual of (X, · H ) is H . The norm in X  is denoted · X  . Let χ ∈ X . By definition, χ X  = sup x, χX,X  = x A =1

11

sup x 2H + Ax 2H =1

x, χX,X  .

(2.33)

The notation i for the inclusion has not to be confused with the imaginary unity denoted i.

2.4 Graph Norm, Dual Spaces, and the Riesz Map of Hilbert spaces

57

A Hilbert space can be identified with its dual, thanks to the Riesz theorem. We state Riesz Theorem for a generic Hilbert space K, in general a complex Hilbert space. This is recalled by the index C of the Riesz map: Theorem 2.22 (Riesz Theorem). Let K be a Hilbert space and let K  be its dual. There exists a surjective invertible operator RC from K  to K such that k, k  K,K  = k, RC k K .

(2.34)

The map RC is the Riesz map and: RC k  K = k  K  ∀k  ∈ K  (RC is an isometry) ,  RC (k  + k  ) = RC k  + RC k  ∀k  , k  ∈ K  , α ∈ C (RC is antilinear) . RC (αk ) = αR ¯ Ck Antilinearity is seen as follows. For every k ∈ K and k , k  in K  we have k, RC (αk  + βk  )K = k, [αk  + βk  ]K,K  = αk, k  K,K  + βk, k  K,K  ¯ C k ]K = αk, RC k K + βk, RC k  K = k, [αR ¯ C k  + βR     ¯ Ck . so that RC (αk + βk ) = αR ¯ C k + βR When K = KC is the complexification of a real Hilbert space KR , the fact that RC is antilinear is sometime inconvenient. In this case, we replace the map RC with a new map Rlin which is a linear isometry of KC onto KC as follows. We recall that the elements k of KC have the form k = k1 + ik2 ,

k1 ∈ KR ,

k2 ∈ KR .

In this sense we write KC = KR + iKR . Then if RC k  = k1 + ik2

k1 ∈ KR ,

k2 ∈ KR

we define Rlin k  = RC k  = k1 + ik2 = k1 − ik2 .

(2.35)

It is clear that Rlin is a linear isometry of K  onto K. With an abuse of language also the map Rlin is called the Riesz map. It is useful to note that Rlin commutes with the operator of conjugation. First we note (2.36) Rlin (KR + i0) ⊆ KR + i0 . This is seen as follows: let k  + i0 ∈ KC and Rlin (k  + i0) = RC (k  + i0) = k1 + ik2 = k1 − ik2 . Then, for every h ∈ KR we have

58

2 Operators and Semigroups for Systems with Boundary Inputs ∈R

=

h + i0, k  + i0KC,KC =

  h, k KR,KR + i0

h + i0, k1 + ik2 KC = h, h1 KR − ih, k2 KR and so h, k2 KR = 0 for every h ∈ KR . Hence k2 = 0 and (2.36) holds. Now, as in Remark 2.3 we see that Rlin commutes with the conjugate:   Rlin (k1 + ik2 ) = Rlin (k1 + i0) + i(k2 + i0) = Rlin (k1 + i0) + iRlin (k2 + i0)   (2.37) = Rlin (k1 + i0) − iRlin (k2 + i0) = Rlin (k1 − ik2 ) = Rlin k1 + ik2 . We combine (2.35) and (2.37). We replace k  = RC−1 k in (2.35) and we get, for every k ∈ K = KC , k¯ = Rlin RC−1 k and we have also

so that, from (2.37): −1 Rlin k = RC−1 k

k = Rlin RC−1 k

∀k ∈ K .

(2.38)

Remark 2.23. Note that the use of the map Rlin instead of the “true” Riesz map RC ¯ H instead of the inner product in the is like using the bilinear form (h, k) → h, k definition of the Riesz map.

2.4.1 Relations of the Adjoint and the Transpose Operators Let K and H be two (complex) Hilbert spaces. We use the notations RK,C and RK,lin , respectively, RH,C and RH,lin , for the corresponding Riesz operators. We recall that when K and H are Hilbert spaces and12 B ∈ L(K, H) then we can define both the adjoint B∗ ∈ L(H, K), which depends on the Hilbert structure of the spaces, and the transpose operator B  ∈ L(H , K ), whose definition uses only the Banach space structure of the spaces. In this section we investigate the relations between B∗ and B  in terms of RK,C and RH,C and we recast the relation in terms of the linear operators RK,lin and RH,lin . Then we shall see an interesting consequence. We recall the definitions: Bk, h H = k, B∗ hK ,

Bk, h  H, H  = k, B  h K,K  .

Using the antilinear operators RK,C and RH,C we have 12 In this section it is convenient not to follow the alphabetic order and to choose that the domain of B is the space K. Furthermore, the operator is now denoted B since the result we find here will be applied to the operator B = A∗ in Sect. 2.5.

2.4 Graph Norm, Dual Spaces, and the Riesz Map of Hilbert spaces

59

=

=

−1 −1 Bk, h H = Bk, RH,C h H, H  = k, B  RH,C hK,K 

k, B∗ hK

−1 k, RK,C B  RH,C hK

so that ∗

B h=

−1 RK,C B  RH,C h

 ∀h ∈ H

i.e.

−1 B∗ = RK,C B  RH,C  −1 ∗ B = RK,C B RH,C .

(2.39)

Now we recast this equality in terms of the linear Riesz maps RK,lin and RH,lin . We use the fact that RK,lin and RH,lin commute with the conjugation and we rewrite (2.39) as follows:    ⎧ ⎪   −1 ∗  ⎪ B h = RK,lin B RH,lin h ⎪ ⎨ ⎪   (2.40)  ⎪ ∗ ⎪ −1  ⎪ ⎪ B h = RK,lin B RH,lin h . ⎩ Remark 2.24 (On the Notations). A common practice is that the Riesz maps are not explicitly indicated. Then, the equalities (2.39) and (2.40) (in real spaces) are subsumed in the formal equality (2.41) B  = B∗ but it is clear that strictly speaking this equality does not make any sense since the operators act among different spaces. When the Riesz maps are not explicitly indicated we say that a Hilbert space is identified with its dual. Now we consider the special case13 that K = X is a subspace of H. Let B = i be the inclusion of X in H. We assume • a norm · X is defined on X and (X, · X ) is a Hilbert space; • the inclusion i is linear and continuous from (X, · X ) to (H, · H ). • iX is dense in H. We use (2.39) and we see RH,C = (i ∗ )−1 RX,Ci  . In fact, i: (X, · X ) → (H, · H ) is injective, hence invertible from its image which is dense in H, but the inverse is not continuous. So also the maps i  and i ∗ have unbounded inverses. It follows that the operators RH,C and RX,C (and so also RH,lin and RX,lin ) are not related by “bounded transformation of coordinates.” For this reason, it is usually not convenient to invoke Riesz theorem in order to represent both the dual spaces of X and H with the spaces themselves. It is convenient to do the identification for only one of the spaces, often the larger space H. The space 13 In practice it will be X = dom A with the graph norm but we do not need this special structure for the observations below.

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2 Operators and Semigroups for Systems with Boundary Inputs

which is identified with its dual is called the pivot space. This is the “interesting consequence” we wanted to note. Remark 2.25. We noted already that the map x → ix, h H belongs to X  for every h ∈ H. We need to be precise. We note the following chain of equalities: −1 −1 ix, h H = ix, RH,C h H, H  = x, i  RH,C hX,X  .

(2.42)

−1 h ∈ X  and the equality (2.42) This equality shows that h identifies the element i  RH,C  −1  is expressed by saying that i restricts RH,C h ∈ H to the subspace X. Similar to the Riesz map, in practice the inclusion operator and its transpose are not explicitly indicated and loose statement as follows can be found in the literature: if X is a dense subspace of H with continuous injection, then H ⊆ X  and, when h ∈ H then the following equality holds: x, h H = x, hX,X  . The precise meaning of this equality is (2.42).

The equalities (2.42) show also: −1 h, h ∈ H} of X  is dense in X  . This fact Theorem 2.26. The subspace {i  RH,C is often informally stated as follows: the Hilbert space H is dense in the Hilbert space X . −1 h In fact, if x0, i  RH,C X,X  = 0 for all h ∈ H, then we have also ix0, h H = 0 for all h and so x0 = 0.

2.5 Extension by Transposition and the Extrapolation Space As in Sect. 2.4, the linear operator A: H → H is unbounded, closed and with dense domain in H so that A∗ : H → H is linear, unbounded, closed and with dense domain too. Furthermore, (A∗ )∗ = A. So now we have two operators to consider and we introduce the following notations: dom A = (dom A, · A) ,

dom A∗ = (dom A∗, · A∗ )

while the inclusions are denoted, respectively14 i : dom A → H ,

i∗ : dom A∗ → H .

We recall the definition of the domain of A∗ : k ∈ dom A∗ when h → Ah, k H admits a continuous extension to H and analogously, using (A∗ )∗ = A, Note that i∗ is neither the adjoint nor the transpose of i. In fact, i ∗ ∈ L(H, (dom A, · A )) while i ∈ L(H , (dom A, · A ) ).

14

2.5 Extension by Transposition and the Extrapolation Space

61

h ∈ dom A when k → A∗ k, h H admits a continuous extension to H . Furthermore: k, Ah H = A∗ k, h H

∀h ∈ dom A, k ∈ dom A∗ .

Now we use the fact that A∗ ∈ L(dom A∗, H). In order to be precise, we note that is defined on H with a dense domain, but we want to consider the corresponding operator defined on (dom A∗, · A∗ ) and with values in H so that strictly speaking A∗

A∗ k, h H

is

A∗ (i∗ k), h H .

We consider the following chains 1 and 2 of equalities which hold when k belongs to the Hilbert space (dom A∗, · A∗ ) (so that i∗ k belongs to dom A∗ considered as a subspace of H) and every h ∈ H (the chain 1 is (2.42) in this context): 1) i∗ k, h H = i∗ k, RC−1 h H, H  = k, (i∗ )  RC−1 hdom A∗,(dom A∗ ) ∗

∗

2) A (i∗ k), h H = A (i∗ k),





RC−1 h H, H 

∗

= (A i∗ )k,

(2.43a)

RC−1 h H, H 

= k, (A∗ i∗ ) RC−1 hdom A∗,(dom A∗ )

(2.43b)

(note that A∗ i∗ ∈ L((dom A∗, · A∗ ), H)). The equalities (2.43a) show that the injective antilinear map (i∗ ) RC−1 is continuous from H in (dom A∗ ) and furthermore its image is dense. This is the restatement of Theorem 2.26 in this context. Now we use (2.43b) in the special case that h ∈ dom A so that A∗ (i∗ k), h H = i∗ k, Ah H (and, we recall, i∗ k ∈ (dom A∗, · H ) ). We have:

=

i∗ k, RC−1 Ah H, H  k, (i∗ RC−1 A)hdom A∗,(dom A∗ )

A∗ (i∗ k), RC−1 h H, H  =

= A∗ (i∗ k), h H =

(A∗i∗ )k, RC−1 h H, H  =

=

i∗ k, Ah H

k, (A∗ i∗ ) RC−1 hdom A∗,(dom A∗ ) .

These equalities hold for every k ∈ dom A∗ and every h ∈ dom A so that for every h ∈ dom A we have (2.44) i∗ RC−1 Ah = (A∗ i∗ ) RC−1 h . We note that: • the equality holds for every h ∈ dom A; • (A∗ i∗ ) RC−1 is defined on the entire space H and it is a continuous (dom A∗ )-valued operator; • the operators in both the sides of (2.44) are antilinear. In order to interpret (2.44) first we consider the case that H is a real Hilbert spaces. In this case RC = Rlin is a linear operator and the operators in both the sides of (2.44) are linear operators. We introduce

62

2 Operators and Semigroups for Systems with Boundary Inputs −1 Ae = (A∗ i∗ ) RC−1 = (A∗ i∗ ) Rlin ∈ L(H, (dom, A∗ )) .

(2.45)

We compare with (2.44) and we see that the operator Ae extends the (dom A∗ )−1 )A, defined on dom A, to a linear continuous operator valued linear operator (i∗ Rlin defined on the entire space H. As done in the identification of H with a dense subspace of (dom A∗ ), the −1 usually are not explicitly indicated,15 and it is informally stated operators i∗ and Rlin that Ae extends A to a (dom A∗ )-valued linear continuous operator defined on the entire space H; and, consistent with the suppression of the inclusion and the Riesz maps, the operator Ae is simply denoted (A∗ ). Even more, it is usual to write A instead of Ae leaving the interpretation of the notation to the reader. For the sake of clarity, in this chapter we retain the notation Ae to distinguish between the original operator A and its extension but in the next chapters the index e will be dropped. Now we interpret formula (2.44) when A is originally defined in a real space H which is then complexified. We denote AC and i∗C the operators in the complexified space16 HC = H + iH , so that

AC (h1 + ih2 ) = Ah1 + iAh2 ,

  AC (h1 + ih2 ) = AC h1 + ih2 ,

i∗C (h1 + ih2 ) = i∗ h1 + i i∗ h2

  i∗C (h1 + ih2 ) = i∗C h1 + ih2 .

It is a simple verification that A∗C (h1 + ih2 ) = A∗ h1 + iA∗ h2 , so that

  A∗C (h1 + ih2 ) = A∗C h1 + ih2 ,

 i∗,C (h1 + ih2 ) = i∗ h1 + i i∗ h2

   (h  + ih  ) = i   + ih  . i∗,C h ∗,C 1 2 1 2

We consider (2.44) with A = AC and we compute the conjugate of both the sides. −1 k = R−1 k. We get We use (2.38) i.e. Rlin C −1 −1 i∗ Rlin AC h = (A∗Ci∗ ) Rlin h = Ae h −1 h . since also in this case we define Ae h = (A∗Ci∗ ) Rlin

(2.46)

Hence, also in this case the linear operator Ae is interpreted as the extension by continuity of AC to the entire space H, with values in (dom A∗ ). 15 16

So in particular H is identified with its dual. Keep in mind: i is the inclusion while i is the imaginary unity.

2.5 Extension by Transposition and the Extrapolation Space

63

The usual terminology is that the operator Ae is the extension by transposition of the operator A and the space (dom A∗ ) is the extrapolation space of the operator A since it is usual to drop the index C and A is used to denote AC . We repeat once more that in practice, neither the inclusion nor the Riesz map are explicitly indicated and the extension of A is written as ⎧ Ae = (A∗ ) and more commonly as Ae = A ⎪ ⎪ ⎪ ⎪ while the definition of Ae is written as ⎪ ⎨ ⎪ k, Ahdom A∗,(dom A∗ ) = k, Ah H ⎪ ⎪ ⎪

  ⎪ ⎪ ⎪ = k, A h e ⎩ dom A∗ ,(dom A∗ )

(2.47)

leaving to interpretation to the reader. Extension by transposition of unbounded operators is a key ingredient in the Hilbert space theory of nonhomogeneous Dirichlet or Neumann problems (see [23] for a full account and [6] for a concise exposition). The following example shows the reason why values of functions on the boundary of a domain show up when computing the extension by transposition of the second derivative. Example 2.27. Let H be the real space L 2 (0, π) and let us consider the dense subspace X of those functions φ which are of class C 1 with a square integrable second derivative.17 We consider the following operators on H: dom A0 = {φ ∈ X , φ(0) = φ(π) = φ (0) = φ (π) = 0} dom A1 = {φ ∈ X , φ(0) = φ(π) = 0} dom A2 = {φ ∈ X , φ(0) = φ (π) = 0}

A0 φ = φ  , A1 φ = φ  , A2 φ = φ  .

We are going to use formula (2.47) in order to compute Ae u in the three cases, when u ∈ L 2 (0, π). Using the definition by transposition, Ae u is that element of (dom A) such that ∫ φ, (Ae u)X,X  = Aφ, u L 2 (0,π) =

π

φ (x)u(x) dx

∀φ ∈ dom A .

0

This is the definition of Ae u for every u ∈ L 2 (0, π). Let us consider the special case u ∈ C 2 ([0, π]). Then we can integrate twice by parts and we get ∫ π φ, A0,e u = u (s)φ(s) ds , (2.48a) 0 ∫ π φ, A1,e u = −φ (π)u(π) + φ (0)u(0) + u(s)φ (s) ds , (2.48b) 0

17

Implicitly we require that the first derivative is a primitive of the second derivative. The standard notation for this space is H 2 (0, π), see Sect. 2.6.1.

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2 Operators and Semigroups for Systems with Boundary Inputs

φ, A2,e u = φ(π)u (π) + φ (0)u(0) +

∫

π

u(s)φ (s) ds .

(2.48c)

0

The first equality shows that A0,e u = u  is a square integrable function while neither A1,e u nor A2,e are represented by the L 2 -inner product of φ with a square integrable function: the action of these operators on φ involves the values of φ on the end points 0 and π of the interval. This is the reason why the extension Ae of A is used to describe the solutions of boundary value problems.

2.5.1 Selfadjoint Operators with Compact Resolvent The case that A is selfadjoint with compact resolvent in a separable Hilbert space is particularly important and it is explicitly considered here. The cases encountered most often are the cases that A is positive, i.e. x, Ax ≥ 0 or positive defined. This is the case when there exists c > 0 such that x, Ax ≥ c x 2 . We say that the operator A is negative or negative defined when −A is either positive or positive defined. We consider the case that A is positive defined so that 0 ∈ (A) and we use the equivalent norm x A = Ax H

in dom A .

The results we find are adapted to positive operators in Remark 2.28. The fact that A is positive defined and with compact resolvent implies that the spectrum is a sequence of positive eigenvalues which we denote λn2 . Furthermore, the space H has an orthonormal basis {ϕn } of eigenvectors of A, Aϕn = λn2 ϕn . Note that different eigenvectors may correspond to the same eigenvalue but every eigenvalue has finite multiplicity. We assume that the eigenvectors ϕn have been ordered in such a way that the corresponding sequence {λn2 } is increasing (in general not strictly increasing). Any h ∈ H has the representation h=

+∞ 

hn ϕn

{hn } ∈ l 2

yn ϕn ,

xn yn = 2 λn

n=1

and

" −1

dom A = im A

= y=

+∞  n=1

# with

{xn } ∈ l

2

.

2.5 Extension by Transposition and the Extrapolation Space

65

This is seen because continuity of A−1 gives  +∞  +∞ +∞    1 y = A−1 x = A−1 xn ϕn = xn A−1 ϕn = xn 2 ϕn . λn n=1 n=1 n=1 The important consequence of this formula is that Ay = x =

+∞ 

xn ϕn =

+∞ 

n=1

yn Aϕn :

(2.49)

n=1

• Eq. (2.49) shows that the operator A can be exchanged with the series as if it were a continuous operator; +∞  2 x • if x = +∞ n=1 n ϕn = n=1 (hn /λn )ϕn ∈ dom A, then x 2A =

+∞ 

|hn2 | .

n=1

• if x =

+∞

2 n=1 (hn /λn )ϕn ,

y=

+∞

2 n=1 (k n /λn )ϕn

x, y A =

+∞ 

are two elements of dom A, then

k¯ n hn .

n=1

• the sequence {ϕn /λn2 } is an orthonormal basis of (domA, · A). It is now possible to find a “concrete” representation of X  = (dom A). The space X = domA (with the graph norm) is isometrically isomorphic to the weighted l 2 -space % $  2 {hn /λn2 } with the norm {hn /λn2 } 2 = +∞ n=1 |hn | and so X  is isometrically isomorphic to the dual space $ 2 %  2 . {λn χn } with the norm {λn2 χn } 2 = +∞ n=1 | χn | It is a common practice to represent the elements of X  with the notation   +∞ +∞ +∞       2 2 λn χn ϕn , λ χ ϕ = | χn | 2 .  n n n    n=1

n=1

X

(2.50)

n=1

For every n the element λn2 ϕn belongs to X , it has norm equal 1 and the sequence is an orthonormal basis of X . It follows that the extension Ae ∈ L(H, X ) is given by & +∞ ' +∞   Ae hn ϕn = λn2 hn ϕn .

{λn2 ϕn }

n=1

n=1

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2 Operators and Semigroups for Systems with Boundary Inputs

Remark 2.28. The previous arguments can be extended to the case that  H0 = ker A H = H0 ⊕ H1 , H0 ⊥ H1 and A1 = A | H1 is positive defined . Let X1 = dom A1 . Then dom A = H0 ⊕ X1 and (dom A) = R0−1 H0 ⊕ X1 where R0 is the Riesz map of H0 (usually not indicated).

2.5.1.1 Fractional Powers of Positive Operators with Compact Resolvent Also in this section, A is a positive defined unbounded operator with compact resolvent in a Hilbert space H. The arguments are then extended as in Remark 2.28 to positive operators. We note " #   +∞ +∞ +∞   ϕn Aϕn  dom A = x = xn 2 , {xn } ∈ l 2 xn 2 = xn ϕn , Ax = λn λn n=1 n=1 n=1 " #   +∞ +∞ +∞   ϕn A2 ϕn  2 2 dom A = y = yn 4 , {yn } ∈ l yn 4 = yn ϕn . A2 y = λn λn n=1 n=1 n=1 The spaces dom A with x 2A =

+∞

n=1 |xn |

2

,

dom A2 with y 2A2 =

+∞

n=1 |yn |

2

are Hilbert spaces. These observations suggest the introduction of the fractional powers18 of A as follows. Let γ > 0. We define " #  +∞  +∞ +∞   ϕn  ϕn γ 2 γ yn 2γ with {yn } ∈ l , A yn 2γ = yn ϕn . dom A = y = λn λn n=1 n=1 n=1 γ  (2.51) It is easy to see that Aγ is boundedly invertible in H and that (Aγ )−1 = A−1 . The operator (Aγ )−1 is denoted A−γ . The space dom Aγ is a Hilbert space when endowed with the norm   +∞ +∞   ϕn   2 y dom Aγ =  yn 2γ  = |yn | 2 . (2.52)   λn dom Aγ n=1 n=1 It follows that the sequence {ϕn /λn } is an orthonormal basis of dom Aγ with the graph norm. It is possible to give a “concrete” representation of the extrapolation spaces (domAγ ) in terms of the eigenfunctions of A. In fact, the operator Aγ are closed densely defined on H so that, by simply replacing Aγ in the place of A in (2.50), we 2γ

18

In fact, the exponent is any positive real number. See below for negative exponents.

2.6 Distributions

find

67

   +∞ 2γ ⎧ γ  2 ⎪ ⎨ (domA ) = χ = n=1 χn λn ϕn , { χn } ∈ l , ⎪    2    +∞ 2γ 2 2 ⎪ λ = χ ϕ = +∞ ⎪ χ (dom   n n γ  n n=1 n=1 | χn | . A ) ⎩ (domAγ )

(2.53)

These considerations are easily extended when A is as in Remark 2.28. The spaces dom Aγ and their duals are of great utility in the study of linear systems in Hilbert spaces. We introduce special notations as follows.

Notations • When A is selfadjoint positive defined and with compact resolvent: – for γ ≥ 0 we define   Xγ = dom (A)γ/2 , X−γ = Xγ = dom (A)γ/2 . The norm in these spaces is those in (2.52) and (2.53) (note that X0 = X0 = H). – the special case γ = 1 is particularly important and we reserve special notations:  V = X1 = dom A 1/2 A = (A) and we define V  = X−1 = (dom A) . • when −A is selfadjoint positive defined and with compact resolvent: – we define the operators A = (−A)1/2 ,

A = i(−A)1/2 = iA

(in order to define A we complexify the space). – the spaces Xγ and V are those of −A. • So: if A is positive, then

A2

= A; if −A is positive, then

2.6 Distributions Let d > 0 and ni ≥ be integers. We use the notations



A2 = A A2 = −A .

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2 Operators and Semigroups for Systems with Boundary Inputs

α = (n1 . . . nd ) , and Dα =

∂ n1 +···nd , ∂ x1n1 ∂ x2n2 . . . ∂ xdnd

|α| = n1 + · · · + nd .

The number |α| is the “length” of the string α. The special case of the first partial derivative respect to xr will be denoted as ∂φ = Dr φ = φ xr = φ,r . ∂ xr The last notation is especially convenient when a partial derivative is under the integral sign and we change the variables of integration. Let Ω ∈ Rd be a bounded region. The symbol D(Ω) denotes the linear space of the functions φ ∈ C ∞ (Ω) with compact support in Ω. The space D(Ω) is sometimes called the linear spaces of the test functions. It is possible to introduce a topology in D(Ω). We confine ourselves to define the convergence of sequences. A sequence {φn } converges to φ0 in D(Ω) when both the following conditions hold: • there exists a compact set K ⊆ Ω such that for every n we have φn (x) = 0 if x ∈ Ω \ K. • for every α, Dα φn → Dα φ0 uniformly on Ω (and so we have φ0 (x) = 0 in Ω \ K). The dual space D (Ω) is the space of the linear functionals χ on D(Ω) which are continuous in the sense that19 φn → φ0 =⇒ φn, χ → φ0, χ . By definition, χn → χ in D (Ω) when

limφ, χn  = φ, χ

∀φ ∈ D(Ω) .

The elements of D (Ω) are called distributions. The operators Dα can be defined also on D (Ω), as follows: φ, Dα χ = (−1) |α | Dα φ, χ . It is clear that Dα from D(Ω) to itself and also from D (Ω) to itself is continuous. We call regular distributions those distributions χ f defined by ∫ φ, χ f  = f (x)φ(x) dx, (2.54) Ω

1 (Ω) (unicity of the representation is easily seen). Distributions which where f ∈ Lloc are not regular exist. An example is the Dirac delta at x0 ∈ Ω defined as follows: φ → φ, δx0  = φ(x0 ) (the Dirac delta at x0 = 0 is simply denoted δ).

As already noted, the pairing of D  (Ω) and D(Ω) is denoted φ, χ: χ(φ) = φ, χ. When needed for clarity, φ, χD(Ω), D  (Ω) .

19

2.6 Distributions

69

1 (Ω) we can compute D α χ . In general, D α χ is not a regular For every f ∈ Lloc f f distribution (for example,20 DH = δ). If it is a regular distribution, Dα χ f = χg , then the function g is called the derivative in the sense of the distributions of the function f and it is shortly denoted Dα f . Finally, we say that χ = 0 on a subregion Ω of Ω when φ, χ = 0 for every φ ∈ D(Ω) whose support is contained in Ω. The complement of the largest region over which χ = 0 is the support of χ. It is clear that the support of Dα χ is contained in that of χ. It is a fact that if K ⊆ Ω and K is compact, then there exist (infinitely many) functions f ∈ D(Ω) such that f (x) = 1 on K. Hence, if φ ∈ C ∞ (Ω) (possibly not zero on the boundary), then f (x)φ(x) ∈ D(Ω) so that we can compute (with χ ∈ D (Ω))  f φ, χ . (2.55)

It is a fact: Lemma 2.29. If χ has compact support K ⊆ Ω and φ ∈ C ∞ (Ω), then the number (2.55) does not depend on the function f ∈ D(Ω) which is equal 1 on K and it is simply denoted φ, χ. In this sense, distributions with compact support can be applied to every C ∞ function, and not only to the functions of D(Ω) and furthermore we have: if φ ∈ C ∞ (Ω), χ with compact support, then

Dα φ, χ = (−1) |α | φ, Dα χ . (2.56)

2.6.1 Sobolev Spaces We shall work in a bounded region Ω ⊆ Rd and we assume that every point of ∂Ω has a neighborhood D which is transformed to the interior of a “cube” Q ⊆ Rd by a transformation F which has the following properties, illustrated in Fig. 2.1 (in this figure m is a plane which splits the “cube” into two parallelepipeds Q+ and Q− ): • The function F is of class C 2 , and its Jacobian J is not zero. • the image of the open set D ∩ Ω is contained Q+ , say the part above the “middle plane” m ∩ Q of the “cube” Q. • the set D ∩ ∂Ω is transformed in the “middle plane” m ∩ Q. Using J  0 we can represent ∂Ω ∩ D as ξ = ψ(η) (if D is “small”) and: – (ξ, η) belongs to D ∩ Ω if and only if ξ > ψ(η). We express this property by saying that “Ω lays on one side of its boundary.” A region with these properties is a region of class C 2 (which lays on one side of its boundary). If the function F is of class C k , k ≤ +∞, then we say that ∂Ω is of class C k . 20

We recall that H is the Heaviside function, H(x) = 0 if x < 0 and H(x) = 1 if x ≥ 0.

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2 Operators and Semigroups for Systems with Boundary Inputs

D Q m

Fig. 2.1 The definition of “region with C 2 boundary”

We define H m (Ω) as the spaces of those L 2 (Ω) functions whose derivatives in the sense of distributions up to the order m included are regular distributions, identified by square integrable functions. The spaces H m (Ω) are Hilbert spaces with the norm  ∫ 2 φ H m (Ω) = |Dα φ(x)| 2 dx . (2.57) |α | ≤m

Ω

The space C ∞ (Ω) is dense in H m (Ω) for every m (smoothness of ∂Ω is used here). Furthermore, we define H0m (Ω) as the closure of D(Ω) in H m (Ω) (in the case we are interested in, that Ω  Rd , we have H0m (Ω)  H m (Ω)). The dual space of H0m (Ω) is denoted H −m (Ω). When Ω is bounded, the following two norms on H01 (Ω) are equivalent: &∫ Ω

|φ(x)| 2 dx +

d ∫  k=1

Ω

' 1/2 |Dk φ(x)| 2 dx

& ,

d ∫  k=1

Ω

' 1/2 |Dk φ(x)| 2 dx

The equivalence of these norms and in particular the inequality & d ∫ ' ∫  2 2 |φ(x)| dx ≤ M |Dk φ(x)| dx ∀φ ∈ H01 (Ω) Ω

k=1

Ω

.

(2.58)

is called Poincaré inequality. The norm in H −m (Ω) is χ H −m (Ω) = sup{φ, χ ,

φ H0m (Ω) = 1}

(2.59)

so that for every χ ∈ H −m (Ω) and φ ∈ H0m (Ω) we have |φ, χ| ≤ χ H −m (Ω) φ H0m (Ω) .

(2.60)

It is clear that D(Ω) ⊆ H −m (Ω) in the sense that any φ ∈ D(Ω) identifies a regular distribution which belongs to H −m (Ω). An important property is that D(Ω) is dense in H −m (Ω) too.

2.6 Distributions

71

By its very definition, D(Ω) is dense in H0m (Ω) and it is clear that when φn → φ in D(Ω) then we have also φn → φ in H0m (Ω). So, any element of H −m (Ω) is continuous on D(Ω): $H −m%(Ω) ⊆ D (Ω). Let m = 1 and let χ fn be a sequence of regular distributions in H −1 (Ω), such that χ fn → χ0 in H −1 (Ω). Then we have ∫ φ, χ0  = limφ, χ fn  = lim fn (x)φ(x) dx ∀φ ∈ H01 (Ω) . (2.61) Ω

Shortly, we shall write fn → χ0 in H −1 (Ω) instead of χ fn → χ0 .

(2.62)

Density of D(Ω) in H −1 (Ω) implies that for every χ0 ∈ H −1 (Ω) there exists a sequence of regular distributions χ fn such that fn ∈ D(Ω) and fn → χ in the sense that χ fn → χ in H −1 (Ω). Remark 2.30. We stress the following fact: the inclusion H −m (Ω) ⊆ D (Ω) holds because D(Ω) is dense in of H0m (Ω) by the very definition of this space. Instead, D(Ω) is not dense in H m (Ω) (unless Ω = Rd ), not even in H 1 (Ω) and so the dual of H m (Ω) is not a space of distributions. In particular, the dual of the space H 2 (Ω) ∩ H01 (Ω) (equipped with the norm of H 2 (Ω)) is not a space of distributions. This is clearly seen from the computation of A1,e in Example 2.27: φ, A1,e u H 2 (Ω)∩H 1 (Ω),(H 2 (Ω)∩H 1 (Ω)) 0

0

involves φ (0) and φ (π). This observation has a role in the study of the Laplace equation with nonhomogeneous boundary conditions (see Sect. 2.7). Let us consider the special case Ω = (0, T), an interval. Our definition is that the elements of H 1 (0, T) are equivalent classes of square integrable functions. It is a fact that every equivalent class of H 1 (0, T) contains a (unique) continuous element (which is used to identify its class), which furthermore is a.e. differentiable with square integrable derivative and this element φ(t) can be recovered from its derivative, using Lebesgue integration: ∫ t  2 φ (s) ds (2.63) φ ∈ L (0, T) and φ(t) = φ(a) + a

for every a and t in [0, T]. It is useful to note that (when T < +∞) an equivalent norm in H 1 (0, T) is ( ∫ T 2 |φ(0)| + |φ (s)| 2 ds . 0

The properties of higher order spaces, (H 2 (0, T), H 3 (0, T) . . . ) are easily deduced.

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2 Operators and Semigroups for Systems with Boundary Inputs

Remark 2.31 (An Important Caveat). Equality (2.63) shows that the two definitions of the derivative either in the sense of distribution of φ ∈ H 1 (0, T) or of the derivative computed a.e. do not coincide, not even when the derivative exists a.e. and it is bounded. An example is the Cantor ternary function whose derivative on [0, 1] is equal 0 a.e. while the function transforms [0, 1] onto [0, 1] so that equality (2.63) does not hold for the Cantor function. Equality (2.63) holds if φ ∈ H 1 (Ω) ⊆ Rd with d = 1. If instead dim Ω > 1, an element φ ∈ H 1 (Ω) needs not be represented by a smooth function, not even a continuous function. In spite of this, we would like to define the trace of elements of H k (Ω) and of their derivatives on ∂Ω. If φ ∈ C ∞ (Ω) (with continuous derivatives up to ∂Ω), then γ0 φ = φ |∂Ω while γ1 φ = ∂φ/∂ν is the (exterior) normal derivative on ∂Ω. We have (smoothness of the boundary is crucial here, see [31, Sect. 13.6]): Theorem 2.32. The linear transformation γ0 with values in L 2 (∂Ω) admits a linear continuous extension to H 1 (Ω) whose kernel is H01 (Ω); the linear transformation γ1 with values in L 2 (∂Ω) admits a linear continuous extension to H 2 (Ω). If u ∈ H k (Ω) for every k, then the equivalent class u contains a (unique) C ∞ function. If ∂Ω is of class C ∞ , then this function and its derivatives admit continuous extensions to cl Ω. The image of γ0 (extended to H 1 (Ω)) is not the entire space L 2 (∂Ω) but it is a dense subspace which is, by definition, the space H 1/2 (∂Ω). Of course, different H 1 functions may have the same trace. So, we define, for g ∈ H 1/2 (∂Ω): $ % g H 1/2 (∂Ω) = inf u H 1 (Ω) : γ0 u = g . With this norm, the linear space H 1/2 (∂Ω) is a Hilbert space and it is clear by the definition of the norm that the map γ0 is continuous from H 1 (Ω) to H 1/2 (∂Ω). Analogously, H k+1/2 (∂Ω) is the image of γ0 extended to H k+1 (Ω) (the obvious norm makes it a Hilbert space and the trace is continuous). We note that H k+1/2 (∂Ω) is also the image of γ1 extended to H k+2 (Ω). In particular, H 1/2 (∂Ω) is the image of γ1 extended to H 2 (Ω). Due to this definition, the Hilbert spaces H k+1/2 (∂Ω) are called trace spaces on ∂Ω. Theorem 2.32 can be extended to less regular regions, in particular to cylinders Ω × (0, T). If f ∈ H k+1 (Ω × (0, T)) the space of its traces f (x, 0) are the spaces H k+1/2 (Ω). This extension is usually formulated in terms of the spaces W which we introduce in (2.66) below (see [23, p. 13]). A final observation is as follows: we stated that any f ∈ H 1 (a, b) is an equivalent class which contains a function f ∈ C([a, b]), a fact which does not hold for functions in H 1 (Ω) when dim Ω > 1. The following result is true: Theorem 2.33. Let Ω be a region in Rd , with C 2 -boundary. There exists an exponent N which depends solely on d and not on Ω such that if f ∈ H n (Ω) with n > N, then ¯ the class f contains a (unique) function f ∈ C 2 (Ω) ∩ C(Ω).

2.6 Distributions

73

This statement can be reformulated by replacing C 2 with any C k and there exists an explicit formula for N, which we do not need (see [7]). The complete statement of this result is known as the Sobolev embedding theorem.

2.6.1.1 Sobolev Spaces of Hilbert Space Valued Functions We need few pieces of information about Sobolev spaces of functions with values in a Hilbert space H. We confine ourselves to the case that H is a real Hilbert space. Extension to complex spaces is obvious. For our goals, it is sufficient that we consider functions of one variable t ∈ (0, T) with values in a Hilbert space H. The definition of the derivative computed pointwise is as usual f (t + h) − f (t) , f (t) = lim h→0 h where the limit is computed in the norm of H. So, we can define the space D(0, T; H) of the C ∞ (0, T; H) functions with compact support. The convergence in this space, continuity, duality, etc. are defined precisely as in the case of H = R, with obvious modification. For example, f ∈ L p (0, T; H) defines the regular distribution χ f defined by ∫ φ, χ f  =

T

φ(t), f (t) H dt,

0

where the inner product in H replaces the multiplication in R. The derivative in the sense of the distributions is defined as in the case H = R: (2.64) φ, f (n)  = (−1)n φ(n), f , where now φ is a H-valued C ∞ function with compact support in (0, T). A simple observation which is sometime useful is (see [2]): Theorem 2.34. Let f ∈ L 2 (0, T; H). Its derivative f  in the sense of the distribution is a regular distribution and f  ∈ L 2 (0, T; H) if and only if there exists a constant C such that for every φ ∈ D(0, T; H) we have ∫ t      f (t), φ (t) H dt  ≤ C φ L 2 (0,T ;H) .   0

The derivatives in the sense of distributions are denoted with the standard notations: f  is the first derivative. Definition (2.64) is the basis for the definition of the spaces H k (0, T; H). For example, H 1 (0, T; H) is the space of H-valued square integrable functions which admit first derivative in H in the sense of the distributions which is square integrable too, endowed with the norm 2 2  2 2  2 f H 1 (0,T ) = f L 2 (0,T ;H) + f L 2 (0,T ;H) equivalent to f (0) H + f L 2 (0,T ;H)

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2 Operators and Semigroups for Systems with Boundary Inputs

(the two norms are equivalent when T < +∞). Formula (2.63) holds also in the present case: if f ∈ H 1 (0, T; H), then (its derivative in the sense of the distributions exists, it is square integrable and) for every a and t in [0, T] we have ∫ t f (t) = f (a) + f (s) ds, (2.65) a

where now the integral is the Bochner integral of H-valued functions21 (see Sect. 2.2 and [1, Sect. 1.1], [18, Chap. 3] for more details). Of course the caveat in Remark 2.31 has been kept in mind also in this case. Remark 2.35. Sobolev spaces of functions with values in Sobolev spaces are a large and interesting class. For example, let22 R = (0, T) × (0, L). The space H 1 (0, T; H 1 (0, L)) is neither H 1 (R) nor H 2 (R). In fact, the norm of H 1 (0, T; H 1 (0, L)) is: ∫ T ∫ T 2 2 2 f H 1 (0,T ;H 1 (0, L)) = f (·, t) H 1 (0, L) dt + ft (·, t) H 1 (0, L) dt 0 0  ∫ T ∫ L ∫ L = | f (x, t)| 2 dx + | fx (x, t)| 2 dx dt 0 0 0  ∫ L ∫ T ∫ L 2 2 | ft (x, t)| dx + | ft,x (x, t)| dx dt . + 0

0

0

Furthermore, equality (2.65) holds with f (0) ∈ H 1 (0, L) and ft (·, t) ∈ H 1 (0, T) a.e. t ∈ (0, T) so that " ∫x f (0) = f (x, 0) = f (0, 0) + f (ξ, 0) dξ , 0 x ∫x ft (x, s) = ft (0, s) + 0 ft x (ξ, s) dξ a.e. s ∈ (0, T) . Hence ∫ f (x, t) = f (0, 0) + 0

x

∫ fx (ξ, 0) dξ + 0

t

∫ ft (0, s) ds + 0

t

∫

x

 ft x (ξ, s) dξ ds .

0

In particular, f (x, t) is continuous in the closed rectangle, a property which is not enjoyed by every element of H 1 (R): the function f (x, y) = log(x 2 + y 2 ) is of class H 1 (R) but it does not belong to H 1 (0, T; H 1 (0, L)) since it does not admit a continuous extension to the closure of the rectangle. Using the Sobolev spaces of functions of one variable with values in a Hilbert space it is possible to define a more general class of spaces H s (Ω) as trace spaces. We introduce

21

The equality holds for every t when f is the unique continuous element of the equivalence class. Note that the boundary of R is not of class C 2 but this property is not required for the sole definition of the Sobolev spaces.

22

2.7 The Laplace Operator and the Laplace Equation

 W = f ∈ L 2 (0, T; H k (Ω)) , f (m) ∈ L 2 (0, T; L 2 (Ω)) .

75

(2.66)

The norm in the space W is 2 = f L2 2 (0,T ;H k (Ω)) + f (m) L2 2 (0,T ;L 2 (Ω)) . f W

It is easy to see that W, with this norm, is a Hilbert space. k 2 we have also f ∈ The  inclusion  H (Ω) ⊆ L (Ω) implies that if f ∈ W, then 1 2 H 0, T; L (Ω) so that f is (represented by) a continuous L 2 (Ω)-valued  function f and f (0) ∈ L 2 (Ω) but, due to the restriction f ∈ L 2 0, T; H k (Ω) , the space { f (0) , f ∈ W } is a proper (dense) subspace of L 2 (Ω), which is denoted H s (Ω) with s = k(1 − 1/2m). This linear space23 becomes a Hilbert space when endowed with the norm u H s (Ω) = inf { f W : f (0) = u} . f ∈W

2.6.1.2 Sobolev Spaces of Any Real Order Up to now we defined H s (Ω) when s is either an integer number or it has a special rational form. At few places we shall need to know that it is possible to define the spaces H s (Ω) for every real s. We refer to [23] for the definitions. We confine ourselves to state the following properties: 1. the sequence of the spaces H s (Ω) is decreasing, in the sense that s1 > s2 implies H s1 (Ω) ⊆ H s2 (Ω). 2. if s1 > s2 , then H s1 (Ω) is dense in H s2 (Ω) and the inclusion is continuous and compact. The reason why we are interested in these spaces is the relation they have with the fractional powers of the Neumann or the Dirichlet Laplacian. These relations will be described when needed.

2.7 The Laplace Operator and the Laplace Equation The Laplace operator or Laplacian in a region Ω ⊆ Rd is defined by Δφ = φ x1 x1 + · · · + φ x d x d ∈ D (Ω), where the derivative is defined in the sense of distributions. When Δφ is a regular distribution we have ∫ ∫ Δφ = G ⇐⇒ φ(x)Δψ(x) dx = G(x)ψ(x) dx ∀ψ ∈ D(Ω) . (2.67) Ω

Ω

23 Which does not depend on T and not on the fact that we computed f (0). In fact, f ∈ C ([0, T ]; H s (Ω)).

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2 Operators and Semigroups for Systems with Boundary Inputs

Formula (2.67) is a formula of integration by parts which does not involve the boundary values of φ since ψ ∈ D(Ω). It is possible to prove that the usual formula ∫ ∫ ∫ (Δφ)ψ dx = [(γ1 φ)γ0 ψ − (γ0 φ)γ1 ψ] dΓ + φΔψ dx (2.68) Ω

Γ

Ω

holds for every φ and ψ of class H 2 (Ω). The Laplace equation is the equation Δu = G which clearly does not have a unique solution since any constant function solves Δu = 0. In order to have a unique solution we assign boundary conditions on ∂Ω. We consider the following problems: • the Dirichlet problem is the problem Δu = G ,

γ0 u = f .

(2.69)

The Dirichlet Laplacian is the operator A in L 2 (Ω) defined by dom A = H 2 (Ω) ∩ H01 (Ω)

Aφ = Δφ .

• Let ∂Ω = Γ ∪ (∂Ω \ Γ) where Γ is nonempty and relatively open in ∂Ω. Let us consider the problem  γ1 u = f on Γ Δu = G , γ0 u = 0 on ∂Ω \ Γ . We call this problem the Neumann problem and the Neumann laplacian is the operator A = Δ in L 2 (Ω) with domain  γ1 u = 0 on Γ 2 φ ∈ H (Ω) , γ0 u = 0 on ∂Ω \ Γ . We explicitly assume that ∂Ω \ Γ contains a nonempty subset which is relatively open in ∂Ω. The consequence is ker A = 0. Remark 2.36. In the usual formulation of the Neumann problem, Γ = ∂Ω is not excluded. In this case ker A  {0} and the Neumann problem is generally not solvable, ∫ not even ∫ when G = 0 since Gauss theorem asserts that if a solution exists, then γ f dΓ = Ω G(x) dx. If Γ = ∂Ω, all we are going to see can be reformulated by replacing A with the operator A − I (we recall that I is the identity operator). The assumption on ∂Ω\Γ implies ker A = 0 and it is introduced to avoid bothering with this caveat. The Dirichlet and the Neumann operators are similar in certain respect, but have also important differences. We state the properties we shall need.

2.7 The Laplace Operator and the Laplace Equation

77

• (Both in the Dirichlet and the Neumann cases) the operator A is selfadjoint (hence it is closed) and 0 ∈ (A), hence A is surjective and A−1 ∈ L(L 2 (Ω)). Furthermore, the operator A from its domain endowed with the norm of H 2 (Ω) to L 2 (Ω) is bounded and boundedly invertible. In fact, the graph norm on dom A is equivalent to that induced by H 2 (Ω). • (Both in the Dirichlet and the Neumann cases) the operator A has compact resolvent and −A is positive defined (boundedness of Ω and, in the Neumann case, the assumption on ∂Ω \ Γ are used here). Hence, L 2 (Ω) has an orthonormal basis of eigenvectors of the operator A. We denote {ϕn } such a basis and: – it is possible to choose the functions ϕn (x) to be real valued. – We denote −λn2 the eigenvalue of ϕn, Aϕn = −λn2 ϕn . We have λn2 > 0 for every n and lim λn2 → +∞. We put λn = λn2 (a positive number). – the eigenvalues are not distinct, but each one of them has finite multiplicity. We intend that the eigenvectors are listed in such a way that λn ≤ λn+1 . Needless to say, the Dirichlet and the Neumann Laplacians have different eigenvalues and different eigenvectors! • Asymptotic estimates (we recall that Ω ⊆ Rd is bounded): Dirichlet laplacian: there exist two constants m0 > 0 and M0 such that m0 n1/d ≤ λn ≤ M0 n1/d ,

m0 ≤

1 γ1 ϕn L 2 (Γ) ≤ M0 . λn

(2.70)

(see [17] and [25, Chap. IV Sect. 1.5]). Neumann laplacian: there exists m0 > 0, C > 0 such that (see [5, 19])  γ0 ϕn L 2 (Γ) ≤ C 3 λn . (2.71) λn > m0 n1/d , • Thanks to the properties listed above, the construction of the fractional powers in Sect. 2.5.1.1 can be performed both for the Dirichlet and for the Neumann laplacian (both with sign changed so to have positive defined operators) and in both the cases the operators can be extended by transposition to (dom A). • In both the cases we can give an explicit description of V = X1 = dom (−A)1/2 (see [12, 15, 20]): – in the case of the Dirichlet laplacian: V = dom (−A)1/2 = H01 (Ω) . – in the case of the Neumann laplacian: $ V = dom (−A)1/2 = φ ∈ H 1 (Ω) ,

% γ0 φ = 0 on ∂Ω \ Γ .

(2.72)

(2.73)

In Sects. 2.7.1 and 2.7.2 we present few results on the existence of solutions of the Laplace equation with Dirichlet or Neumann boundary conditions. We shall use the  following theorem. In this theorem we use f C 2 (∂Ω) = |α | ≤2 supx ∈∂Ω |Dα f (x)|.

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2 Operators and Semigroups for Systems with Boundary Inputs

The derivatives are computed by representing ∂Ω as a finite union of C 2 graphs. The values of the constants C in Theorem 2.37 depend on the choice of the partition of ∂Ω as a union of graphs and on their parametrization. Theorem 2.37. Let Ω be a bounded region with C 2 boundary. 1. Let f ∈ D(∂Ω). There exists F ∈ H 2 (Ω) such that γ0 F = f

on ∂Ω and

F H 2 (Ω) ≤ C f C 2 (Ω) .

The constant C does not depend on f . 2. Let Γ ⊆ ∂Ω be a relatively open subset and let f ∈ D(Γ). There exists F ∈ H 2 (Ω) such that  γ1 F = f on Γ and F H 2 (Ω) ≤ C f C 2 (∂Ω) . γ0 F = 0 on ∂Ω \ Γ The constant C does not depend on f . The proof of item 1 is in [25, Theorem 2 in Chap. 3 Sect. 4.2]. The second statement easily follows from the first statement if Γ is a graph. Otherwise it follows by using a partition of unity in Ω. See [23, Theorem 8.3 p. 44] for a more general statement. Of course, the function F in the theorem is not unique.

2.7.1 The Laplace Equation with Nonhomogeneous Dirichlet Boundary Conditions We study the problem24 Δu = G ∈ L 2 (Ω)

γ0 u = f ∈ L 2 (Γ) where Γ = ∂Ω: L 2 (Γ) = L 2 (∂Ω)

(2.74)

(of course, it might be f = 0 on a part of the boundary). Due to the linearity of the problem, we can study separately the case G  0 and f = 0 and the case G = 0 and f  0. The case f = 0 is simple, because 0 ∈ (A) where A is the Dirichlet laplacian: the solution is u = A−1 G ∈ dom A = H 2 (Ω) ∩ H01 (Ω) and the problem is well posed in L 2 (Ω) in the sense that: • it admits a solution u ∈ L 2 (Ω) for every G ∈ L 2 (Ω). • the solution is unique. • the transformation G → u is continuous in L 2 (Ω). We need to distinguish between the affine term in L 2 (Ω) and that acting on the boundary. So, we use an upper case letter for the first function and a lower case letter for the boundary term.

24

2.7 The Laplace Operator and the Laplace Equation

79

Let us consider now the problem with G = 0 and f  0. First we consider the case f ∈ D(Γ). Let F be the function in the statement 1 of Theorem 2.37 and let v = u − F. Then, u is a solution of problem (2.74) with G = 0 if Δv = −ΔF ∈ L 2 (Ω) , and so

v = −A−1 ΔF

γ0 v = 0 on Γ = ∂Ω

hence u = F − A−1 ΔF .

(2.75)

When f ∈ D(Γ), the function u is a classical solution of (2.74) with G = 0. Let φ ∈ dom A. Integrating by parts (see (2.68)) and taking into account the condition γ0 φ = 0 we get ∫ ∫ ∫ 0= Δu(x)φ(x) dx = − f (x)γ1 φ(x) dΓ + u(x)Δφ(x) dx Γ Ω ∫Ω f (x)γ1 φ(x) dΓ + u, Aφ L 2 (Ω) . (2.76) =− Γ

  We know from Lemma 2.32 operator γ1 belongs to L H 2(Ω), L 2 (Γ)   2that the trace and so it belongs also to L H (Ω) ∩ H01 (Ω), L 2 (Γ) = L dom A, L 2 (Γ) . Then, for every f ∈ L 2 (Γ) the map25 ∫ φ → f γ1 φ dΓ = φ, Σ f  Γ

is a linear and continuous functional on dom A: Σ f ∈ (dom A). We recall the definition of the extension by transposition Ae of A to (dom A) in (2.47). For every φ ∈ dom A we have A∗ φ, u L 2 (Ω)

=

=

= Aφ, u L 2 (Ω)

φ, Ae udom A,(dom

A)

φ, Σ f dom A,(dom

(2.77) A)

.

This equality shows that Ae u = Σ f

in

(dom A)

and it suggests that we define the function u and the operator D as u = A−1 e Σf = D f .

(2.78)

By definition, u is the mild solution of (2.74) (with G = 0). So, we have D ∈ L(L 2 (Γ), L 2 (Ω)). In fact the operator D has a stronger regularity property: Theorem 2.38. The operator D is linear and continuous from L 2 (Γ) to H 1/2 (Ω): The map Σ f is the double layer of strength f , see [30, vol. 2 p. 12] (the notation Σ f for the double layer is not standard).

25

80

2 Operators and Semigroups for Systems with Boundary Inputs

D ∈ L(L 2 (Γ), H 1/2 (Ω)) . It is important to know that, using the notation Xγ introduced in Sect. 2.5.1.1, H 1/2 (Ω) ⊆ dom(−A)1/4−ε = X1/2−2ε

(2.79)

for every ε > 0 (the inclusion does not hold if ε = 0, see [22, Remark 3.1.4 p. 186]). By definition, the function u = A−1 G + D f is the mild solution of the Dirichlet problem (2.69). In particular, the transformation (G, f ) → u is linear and continuous from L 2 (Ω) × L 2 (Γ) to L 2 (Ω). So, the Dirichlet problem is well posed in these spaces in the sense that it is uniquely solvable for every (G, f ) ∈ L 2 (Ω) × L 2 (Γ) and the solution u ∈ L 2 (Ω) depends continuously on the data. Let us look again at the formula (2.76): ∫ f γ1 φ dΓ = u, Aφ L 2 (Ω) = D f , Aφ L 2 (Ω) =  f , D∗ Aφ L 2 (Γ) . (2.80) Γ

This equality holds for every fixed φ ∈ dom A and every f ∈ L 2 (Γ). Hence:26 Theorem 2.39. For every φ ∈ dom A we have D∗ A ∈ L(dom A, L 2 (Γ)) ,

D∗ Aφ = γ1 φ .

(2.81)

If we choose φ = ϕn (an eigenvector of A) in (2.80), then we have D ∗ ϕn = −

1 γ1 ϕn . λn2

(2.82)

Now we combine (2.76) and (2.81). We consider the equality (2.76) with any function u of class H 2 (Ω) whose derivatives can be extended by continuity to the boundary. The function u needs not solve the Laplace equation so that “= 0” on the left has to be ignored. The equality gives Δu, φ L 2 (Ω) = −γ0 u, D∗ Aφ L 2 (Γ) + u, Aφ L 2 (Γ) = −D(γ0 u), Aφ L 2 (Ω) + Ae u, φ(domA),dom A = Ae u, φ(domA),dom A − Ae D(γ0 u), φ(domA),dom A . This equality holds for every φ ∈ dom A. It shows that Δ can be extended to any u ∈ H 2 (Ω) with value in the extrapolation space (dom A) as follows: Δu = Ae u − Ae D(γ0 u) . 26

See also [22, p. 181] and [31, Prop. 10.6.1] and note that our operator A is −A0 in [31].

(2.83)

2.7 The Laplace Operator and the Laplace Equation

81

This equality has to be compared with (2.48b). Remark 2.40. In the previous arguments f ∈ L 2 (∂Ω) might be zero in ∂Ω \ Γ where now Γ is a fixed relatively open in ∂Ω. It may be that we are interested solely in those square integrable f whose support is contained in Γ (the same as we described for the Neumann problem). In this case, we can consider f ∈ L 2 (Γ) ⊆ L 2 (∂Ω) so that D ∈ L(L 2 (Γ), L 2 (Ω)) and the same computations as above give D∗ A ∈ L(dom A, L 2 (Γ))

D∗ Aφ = (γ1 φ) |Γ ∈ L 2 (Γ) .

(2.84)

2.7.2 The Laplace Equation with Nonhomogeneous Neumann Boundary Conditions A similar treatment leads to the solutions of the Neumann problem  γ1 u = f ∈ L 2 (Γ) Δu = G ∈ L 2 (Ω) γ0 u = 0 on ∂Ω \ Γ .

(2.85)

Thanks to the linearity of the problem, we study first the case f = 0 and G  0. Here we use in a crucial way the consequence of the assumption that ∂Ω \ Γ has nonempty relative interior: the consequence is that 0 ∈ (A) so that u = A−1 G ∈ dom A

(A is the Neumann laplacian) .

(2.86)

Then we solve the problem when G = 0 and f ∈ D(Γ). We rely on the statement 2 of Theorem 2.37. Let F be the H 2 (Ω) function such that γ1 F = f on Γ and γ0 F = 0 on ∂Ω \ Γ and let v = u − F. Then u solves (2.85) if and only if v solves  γ1 v = 0 ∈ L 2 (Γ) Δv = −ΔF , i.e. v = −A−1 ΔF, γ0 v = 0 on ∂Ω \ Γ where now A is the Neumann Laplacian. The function u = A−1 G + F − A−1 ΔF is the classical solution of the Neumann problem (2.85): classical solutions exist for a dense set of functions f ∈ L 2 (Γ). Now we take the inner product of both the sides of (2.85) with φ ∈ dom A and we integrate by parts. Any classical solution u (with G = 0) satisfies ∫ f γ0 φ dΓ + u, Aφ L 2 (Ω) 0 = Δu, φ L 2 (Ω) = Γ

∫

and so u, Aφ = −

Γ

f γ0 φ dΓ = φ, Σ f  ,

Σ f ∈ (dom A),

(2.87)

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2 Operators and Semigroups for Systems with Boundary Inputs

where now27

∫ φ, Σ f  = −

Γ

f γ0 φ dΓ .

We note again the equality u, Aφ L 2 (Ω) = Ae u, φ(dom A),dom A and this suggests the definition of the mild solution of the Neumann problem (with G = 0) D ∈ L(L 2 (Γ); L 2 (Ω)) . (2.88) u = A−1 e Σf = D f , The following result can be found in [15, 23]: Theorem 2.41. We have D ∈ L(L 2 (Γ), H 3/2 (Ω)) and γ0 D = 0 on ∂Ω \ Γ. In particular we have im D ⊆ dom (−A)3/4−ε any ε > 0

im D ⊆ V = X1 .

As in the case of the Dirichlet problem, we elaborate on formula (2.87). In this case we get D∗ A ∈ L(dom A, L 2 (Γ))

D∗ Aφ = −(γ0 φ) |Γ ∈ L 2 (Γ) .

(2.89)

Finally, as in the case of the Dirichlet laplacian, we get the following formulas: if Aϕn = −λn2 ϕn , then D∗ ϕn =

1 γ ϕ λ2n 0 n

(compare with (2.82)) ,

Δu = Ae u − Ae D(γ1 u) (compare with (2.83) and (2.48c)) .

(2.90) (2.91)

As in the Dirichlet case, formula (2.91) holds even if u is not a solution of the Laplace equation and it has to be intended as follows: for any u ∈ L 2 (Ω) the right side gives an extension of the Neumann laplacian to a (dom A)-valued function defined on L 2 (Ω). As in the case of the Dirichlet problem, the Neumann problem is well posed in the sense that for every (G, f ) ∈ L 2 (Ω) × L 2 (Γ) it admits a unique mild solution u ∈ L 2 (Ω) and the transformation (G, f ) → u is continuous between these spaces: we combine (2.86) and (2.88) and we get u = A−1 G + D f ,

D f = A−1 Σ f .

Remark 2.42. We stress that in spite of the same letters used for the operators, the operators A and D in this section correspond to the Neumann problem, so they are different from the operators A and D in Sect. 2.7.1 which are the operators of the Dirichlet problem. And so, the eigenvalues −λn2 and the eigenfunctions ϕn in the formulas (2.82) and (2.90) are not the same: the Dirichlet eigenvalues and eigenvectors Σ f ∈ (dom A) is the simple layer of strength f , see [30, vol. 2 p. 6]. The notation Σ f for the simple layer is not standard.

27

2.8 Semigroups of Operators

83

appear in formula (2.82) while the Neumann eigenvalues and eigenvectors appear in formula (2.90). We used the same notations so that in the rest of the book we can study both the problems in parallel as much as possible. This way, the important differences are highlighted.

2.8 Semigroups of Operators We refer, for example, to [4, 8, 28] for this subject. If x ∈ Rn and if A is a constant n × n matrix, it is known that the Cauchy problem x  = Ax + h(t) ,

x(t0 ) = x0

(2.92)

has the unique solution x(t; t0, x0, h) = e

A(t−t0 )

∫ x0 +

t

e A(t−s) h(s) ds ,

t0

e At =

+∞  1 n n A t . n! n=0

(2.93)

The exponential e At has the characteristic properties of the (scalar) exponential:  n e A0 = I , e A(t+τ) = e At e Aτ so that e At = en At but note: e A+B  e AeB

unless A and B commute .

In terms of the solutions of the differential equation, the semigroup property rewritten as

e At = e A(t−t0 ) e At0

x(t; 0, x0 ) = x (t; t0 ; x(t0 ; 0, x0 ))

(we intend h = 0)

e A(t+τ) = e At e Aτ is the property

and this is the unicity of solutions of the Cauchy problem. It is easily seen that precisely the same properties hold if x ∈ H and A ∈ L(H). Here H is any Banach space. Unfortunately, a differential equation as (2.92) in an infinite dimensional Banach space and A a bounded operator is rarely encountered. In most of the cases, the operator A is unbounded. In this case dom A  H and the right hand side Ax(t) + h(t) even does not make sense if, for example, x0  dom A. So, we go the opposite way and we define first what should be the “exponential” without making reference to the operator. Definition 2.43. Let H be a Banach or Hilbert space. A function t → E(t) from [0, +∞) to L(H) is a C0 -semigroup when 1. the following semigroup property holds:

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2 Operators and Semigroups for Systems with Boundary Inputs

• E(0) = I. • E(t + τ) = E(t)E(τ) for all t > 0, τ > 0 (so we have E(t)E(τ) = E(τ)E(t)). 2. the function t → E(t) is strongly continuous i.e. E(t)h ∈ C([0, +∞); H) for every h ∈ H. 3. if E(t) is defined and strongly continuous on R, E(0) = I and E(t + τ) = E(t)E(τ) for every t, τ in R, then E(t) is a C0 -group. Thanks to the property 2, a C0 -semigroup is also called a strongly continuous semigroup (or group). The infinitesimal generator A (shortly, the “generator”) of the C0 -semigroup E(t) is defined as follows: Ah = lim+ t→0

E(t)h − h , t

dom A = {h ∈ H such that the limit exists}

(2.94)

(we intend that the limit is computed in the norm of H). It turns out that: Theorem 2.44. Let E(t) be a C0 -semigroup and let A be its infinitesimal generator. Then: 1. the operator A is closed and densely defined. 2. there is ω such that σ(A) ⊆ {λ : e λ < ω}. 3. we have28 ∫ t

E(s)x0 ds = A−1 [E(t)x0 − E(t0 )x0 ]

t0

∫

so that

t

x0 → A

H → C([t0, T]; H)

E(s)x0 ds : t0

is continuous for every T > ∫0. t 4. the transformation h → t E(t − s)h(s) ds is linear and continuous from 0 L p (t0, T; H) to C([t0, T]; H) for every p ≥ 1 and every T > t0 . 5. if h ∈ C 1 (t0, +∞; H), then for every t ≥ t0 we have ∫ t E(t − s)h(s) ds ∈ C([t0, +∞); dom A) t0

and29 d dt

∫

t

∫ E(t − s)h(s) ds = A

t0

∫

= E(t)h(0) +

t

E(t − s)h(s) ds + h(t)

t0 t

E(t − s)h (s) ds .

0 28

The integration is in the sense of Bochner as outlined in Sect. 2.2. See [1, Sect. 1.1] and [18, Chap. 3] for more details. 29 Note that this equality can be interpreted as integration by parts.

2.8 Semigroups of Operators

85

6. if h ∈ C 1 ([t0, +∞); H) and if x0 ∈ dom A, then the function ∫ t x(t) = x(t; t0, x0, h) = E(t − t0 )x0 + E(t − s)h(s) ds

(2.95)

t0

belongs to C 1 ([t0, +∞); H), take values in dom A and verifies the following equalities: x(t0 ) = x0

and

x (t) = Ax(t) + h(t)

∀t ≥ t0 .

(2.96)

7. if two functions x(t) and x1 (t) of C([0, T]; H) both verify the equalities in (2.96), then they coincide, x(t) ≡ x1 (t). Now we define: Definition 2.45. The function x(t) is a classical solution on [t0, T] of x  = Ax + h ,

x(t0 ) = x0

(2.97)

when x ∈ C 1 ([t0, T]; H) ∩ C([t0, T]; dom A) and the equalities (2.96) hold for t ∈ [t0, T]. When x0 ∈ H and h ∈ L p (t0, T; H) (any p ≥ 1) the function ∫ t x(t) = x(t; t0, x0, h) = E(t − t0 )x0 + E(t − s)h(s) ds t ≥ t0 (2.98) t0

is a mild solution on [t0, T] (shortly, a solution) of the problem (2.97). Theorem 2.44 shows that (2.98) is a classical solution when x0 ∈ dom A and h ∈ C 1 ([0, T]; H) and it shows that any mild solution is the limit of a sequence of classical solutions (and this justifies the definition of the mild solutions).

Notations 1. The notation x(·; t0, x0, h) denotes the mild solution of Eq. (2.97). If one of the values t0 , x0 or h is zero, then it is not indicated but we use the simpler notation x(·) when no confusion can arise. So, for example, x(·; x0, h) is used when t0 = 0 and x(·; h) is the solution when t0 = 0 and x0 = 0 (but we can use also the simpler notation x(·)). 2. In order to stress that A is the infinitesimal generator of the semigroup E(t), we use the notations E(t) = e At in spite of the fact that the exponential series does not even make sense if A is unbounded.

From now, unless explicitly stated, we simplify the notations by taking the initial time t0 equal to 0, t0 = 0.

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2 Operators and Semigroups for Systems with Boundary Inputs

In practice, Eq. (2.97) is given and we wish to know whether (mild) solutions exist, i.e. whether the operator A generates a C0 -semigroup. This is not always the case. Example 2.46. Let H = l 2 and let A have the diagonal representation ) * A = diag 1 2 3 4 . . . , dom A = {h ∈ l 2 : Ah ∈ l 2 } . The operator A is closed and densely defined but it is not a generator of a C0 semigroup. In fact, the candidate semigroup would be * ) e At = diag et e2t e3t e4t . . . but this is not a C0 -semigroup since e At is not a bounded operator on l 2 if t > 0 (a different way to see that A is not a generator is to note that its spectrum is unbounded above in contrast to the property 2 of Theorem 2.44). Instead, the operator −A is the generator of the C0 -semigroup * ) e−At = diag e−t e−2t e−3t e−4t . . . . This semigroup is not “reversible,” i.e. e−At is defined for t ≥ 0 but not for t < 0. In order to give sufficient conditions in order that an operator A be the infinitesimal generator of a C0 -semigroup, first we define: Definition 2.47. Let H be a Hilbert space (in general, a complex Hilbert space). We define: • The operator A in H is dissipative when its domain is dense in H and Ah, h + h, Ah = 2e Ah, h ≤ 0

∀h ∈ dom A .   • A C0 -semigroup on H is a semigroup of contractions when e At  ≤ 1 for every t ≥ 0. Remark 2.48. Note a discrepancy between the definition (2.2) of contraction operator and that of a semigroup of contractions: when L is linear then it is a contraction operator if L ≤ α < 1 and the inequality α < 1 is strict. Instead, E(t) is a contraction semigroup when E(t) ≤ 1. The inequality cannot be strict: surely it is not strict when t = 0 and it might even be E(t) ≡ 1 for every t. The relation between dissipative operators and semigroups of contractions is specified by the Lumer–Phillips Theorem: Theorem 2.49 (Lumer–Phillips). Let H be a Hilbert space. The following properties hold: 1. the infinitesimal generator of a contraction semigroup on H is dissipative. 2. let A be a closed densely defined linear operator in the Hilbert space H. We assume:

2.8 Semigroups of Operators

87

either: the operator A is dissipative and there exists λ > 0 such that (λI − A) is surjective; or: the operator A is closed and both the operators A and A∗ are dissipative. If one of these sets of conditions holds, then A is the generator of a C0 semigroup e At and furthermore the semigroup is a semigroup of contractions. A special and important case is when the operator A is skew adjoint, i.e. A∗ = −A: Corollary 2.50. Let H be a Hilbert space. If A∗ = −A, then A generates a C0 -group and e At = 1. In particular, if A0 is selfadjoint, then A = iA0 is the infinitesimal generator of a C0 -group of operator and eiA0 t = 1 for all t ∈ R. This statement goes under the name of Stone Theorem. Remark 2.51 (A Warning). In this book we shall always use C0 -semigroups in Hilbert spaces but in this chapter we specify that the semigroup is in a Hilbert or Banach space since semigroups in a Hilbert space have few special properties. The previous results are two instances (in fact, Lumer–Phillips theorem can be reformulated in Banach spaces too). Moreover we observe: if A is the infinitesimal generator of a C0 semigroup E(t) in a Hilbert space, then E ∗ (t) is a C0 -semigroup whose generator is A∗ . A similar property does not hold when H is a general Banach space.30 Another important property which holds in Hilbert spaces but generally not in Banach spaces is Theorem 5.47 which is used in the study of stability. Finally we introduce a definition: Definition 2.52. The semigroup e At is exponentially stable when there exist M > 0 and ω0 > 0 such that e At ≤ Me−ω0 t . It is a fact that in this case the spectrum of A is in the half plane e λ ≤ −ω0 < 0. It is important to know that the converse implication holds for the holomorphic semigroups introduced in Sect. 2.8.1, but it does not hold in general, see [8, Counterexample 3.4 p. 273]. We conclude this section with an example which is used in the study of exponential stability of systems with memory (see Chap. 5). Example 2.53 (The Right Shift Semigroup). We denote η = η(x) a function of x > 0 with values in a Hilbert space denoted V and we consider the family of the operators S(t) defined as follows (when t ≥ 0):  0 if x < t S(t)η = (S(t)η(·))(x) = (2.99) η(x − t) if x > t . 30

See [18, Chap. XIV] for the Banach space case.

88

2 Operators and Semigroups for Systems with Boundary Inputs

It is easily seen that S(0) = I and S(t + τ) = S(t)S(τ) for all t ≥ 0, τ ≥ 0. For this reason, S(t) is called the right translation or right shift semigroup. The semigroup S(t) is usually studied in the space L 2 (0, +∞; V) (see [11, p. 148]). For a reason we shall see at the end of this example, the L 2 space used in Sect 5.5 is a weighted L 2 space with an integrable weight M(t) ≥ 0. This space will be denoted M and by definition  M = η(x) ∈ V a.e. x > 0

∫ such that 0

+∞

! M(x) η(x) V2 dx < +∞ .

The semigroup S(t) in the space M is studied in [14] (see also [18, p. 536]) where it is proved that it is a semigroup of class C0 and that its infinitesimal generator is the following operator AS : "

 dom AS = η(x) ∈ M ,

η(x) =

∫x 0

η (r) dr ,

η (x) ∈ M



(AS η)(x) = −η (x) .

(2.100)

We solve the equation η (t) = AS η + h

η(0) = η0 ∈ M .

(2.101)

The functions η(t) and h(t) denote functions in M so that η(t) = η(t, x), h(t) = h(t, x) and η0 = η0 (x), x > 0. We apply formula (2.98) to the Eq. (2.101): ∫ t η(t) = S(t)η0 + S(t − s)h(s) ds . 0

Formula (2.99) gives (S(t)η0 )(x) and when applied to (S(t − s)h(s)) (x) we have " h (s, x − (t − s)) if x − (t − s) > 0 S(t − s)h(s) = (S(t − s)h(s, ·)) (x) = 0 if x − (t − s) < 0 . So,

∫ 0

t

S(t − s)h(s) ds =

⎧ ⎨ ⎪ ⎪ ⎩

∫t 0

∫t

h(s, x − t + s) ds if x > t

t−x

h(s, x − t + s) ds if x < t .

Hence, the explicit expression for the mild solution η(x, t) is ∫ t ⎧ ⎪ ⎪ ⎪ 0 + h(s, x − t + s) ds x ∈ (0, t) ⎨ ⎪ x−t ∫ t η(x, t) = ⎪ ⎪ ⎪ h(s, x − t + s) ds x>t. ⎪ η0 (x − t) + 0 ⎩

(2.102)

2.8 Semigroups of Operators

89

We consider the special case used in Sect. 5.5. We have two V-valued functions: the function w ∈ H 1 (0, T; V) for every T > 0 and the V-valued function w˜ defined on (−∞, 0) and such that w(−s) ˜ ∈ M. We consider Eq. (2.101) when h(x, t) = −w (t)1(x) ,

η0 (x) = w(−x) ˜ − w(0)1(x)

where 1(x) = 1 for x ≥ 0 .

Note that 1(x) = 1 is not square integrable on (0, +∞) but h(·, t) (for every t) and η0 belong to M since the weight M(t) is integrable: The integrable weight is introduced in order to have this property. Formula (2.102) gives ⎧ if x > t: ⎪ ⎪ ∫t ⎪ ⎪ ⎪ − 0 w (s)1(x − t + s) ds ⎪ ⎪ ∫ t ⎪ ⎨ ⎪ S(t − s) (w(s)1) (x) ds = if x ∈ (0, t): ⎪ 0 ⎪ ⎪ ⎪ ⎪ − ∫ t w (s)1(x − t + s) ds ⎪ ⎪ ⎪ ⎪ t−x ⎩

∫t = − 0 w (s) ds = −w(t) + w(0) ∫t = − t−x w (s) ds = −w(t) + w(t − x) .

We insert the contribution of the initial condition and we get  w(t ˜ − x) − w(t) if x > t η(t, x) = w(t − x) − w(t) x ∈ (0, t) .

(2.103)

2.8.1 Holomorphic Semigroups It turns out that if e At is a contraction semigroup, then the following estimate holds: (λI − A)−1 ≤

1 λ

if

λ>0.

This estimate does not hold if we replace λ > 0 with e λ > 0 and 1/λ with 1/|λ|. This observation suggests that we study the special class of those generators such that (λI − A)−1 ≤ 1/|λ| in e λ > 0. The following result holds (see [8, p. 96] and [28, p. 61]): Theorem 2.54. Let A be the generator of a bounded C0 -semigroup on a Banach or Hilbert space H. The properties 1 and 2 are equivalent: ˜ there 1. there exists θ˜ ∈ (0, π/2) with the following property: for every θ 0 ∈ (0, θ) exists a constant C0 > 0 such that   (λI − A)−1  ≤ C0 |λ|

for | arg λ| < θ 0 + π/2 and λ  0 .

2. there exists a sector Σ1 = {λ : | arg λ| < θ 1 }

(2.104)

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2 Operators and Semigroups for Systems with Boundary Inputs

and a function E(λ) ∈ L(H) defined on Σ1 which is holomorphic in this sector and such that • if | arg λ| < θ 1 and | arg μ| < θ 1 , then E(λ + μ) = E(λ)E(μ). • let θ ∈ (0, θ 1 ). For all x0 ∈ H we have lim λ→0 E(λ)x0 = x0 . | arg λ| 0 is e At : E(t + i0) = e At . For this reason, E(λ) is called a holomorphic semigroup and it is denoted e Aλ . Furthermore we have: • let λ ∈ Σ1 . The operator E(λ) is given by ∫ 1 eλζ (ζ I − A)−1 dζ E(λ) = 2πi Gε

(2.105)

where the path of integration G ε (see Fig. 2.2) is composed by the following two half lines and circular arc:  ζ = s [cos α ± i sin α] s ∈ [ε, +∞) , α ∈ (π/2, θ + π/2) , G± : ζ = εeis , −α < s < α . • let h ∈ dom A. The limit in (2.94) exists when t is replaced by λ and the limit is computed for λ → 0 in the sector | arg λ| ≤ θ < θ 1 (θ 1 is in property 2). • if 0 ∈ (A), then there exists θ˜ > 0 such that inequality (2.104) holds if and only if there exists C > 0 such that   (λI − A)−1  ≤ C |λ|

for e λ > 0 .

If (A − cI) satisfy the inequality (2.104) and Ec (λ) is given by the integral (2.105) with A replaced by A−cI, then we say that A generates the holomorphic semigroup e Aλ = ecλ Ec (λ) . Holomorphic semigroups have strong regularity properties in time of course, but they have strong regularity properties also in the space H since:

G+

G-

Fig. 2.2 The path of integration in (2.105)

2.8 Semigroups of Operators

91

Theorem 2.55. Let E(t) be a holomorphic semigroup and let t > 0. Then, im E(t) ⊆ dom Ak for every k. An important case of operators which generate holomorphic semigroups is the case of selfadjoint operators with compact resolvent in a separable Hilbert space H, when the spectrum is a sequence {λn } such that lim λn = −∞ .

(2.106)

Let {ϕn } be an orthonormal basis in H of eigenvectors of A Aϕn = λn ϕn . Condition (2.106) implies that A generates the holomorphic semigroup & +∞ ' +∞   At e xn ϕn = eλn t xn ϕn . n=1

(2.107)

n=1

This semigroup is exponentially stable if and only if sup{λn } < 0. Both the Dirichlet and the Neumann laplacians have these properties. The corresponding semigroups are the heat semigroups. When −A is selfadjoint positive definite and with compact resolvent we can define the fractional powers (−A)γ (see Sect 2.5.1.1). It is a fact that fractional powers (−A)γ can be defined more in general when A is the generator of an exponentially stable holomorphic semigroup31 (see [8, 28]). When −A is selfadjoint positive the two definitions coincide. In general, even if A is not selfadjoint, the following properties hold (see [8, 28]): Theorem 2.56. Let A generate an exponentially stable holomorphic semigroup on a Hilbert or Banach space H. The fractional powers (−A)γ are densely defined and closed operators in H and there exist constants ω0 > 0, m0 > 0 and θ 0 ∈ (0, π/2) such that   0t ⎧ (−A)γ e At  ≤ m0 e−ω ∀γ ∈ [0, 1]) , ⎪ ⎪ tγ ⎪ ⎪ ⎪ ⎪ " m ⎪ 0  ⎨ ⎪  |λ+ω0 | −1  (2.108) ≤ if |arg λ| < π2 + θ 0 , − A)  (λI m 0 ⎪ ⎪ ⎪ |λ| ⎪ ⎪ ⎪ ⎪ ⎪ (−A)γ x ≤ m x 1−γ Ax γ ∀x ∈ dom A . 0 ⎩ We are going to use the following consequence not explicitly stated in this form in standard reference books: Lemma 2.57. Let A generate an exponentially stable holomorphic semigroup on H. Let D ∈ L(U, H) (where U is a second Banach or Hilbert space) and let there exists σ0 > 0 such that im D ⊆ (−A)σ0 . Then, there exist M > 0 and ω0 > 0 such that the 31

And even for larger classes of operators, see [16], but this is not needed in this book.

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2 Operators and Semigroups for Systems with Boundary Inputs

following inequality holds for |arg λ| < as in Theorem 2.56).    A(λI − A)−1 D f 

π 2

+ θ 0 (the numbers ω0 and θ 0 are the same

H

≤

M f U . |λ + ω0 | σ0

(2.109)

Proof. The facts that: im D ⊆ dom (−A)σ0 ; D is continuous; (−A)σ0 is closed imply that (−A)σ0 D ∈ L(U, H). In order to prove (2.109) we replace x = (λI − A)−1 y (any y ∈ H) in the last inequality in (2.108). We get :   (−A)γ (λI − A)−1 y 

H

 1−γ  γ ≤ m0 (λI − A)−1 y  H  A(λI − A)−1 y  H

m0 M y H |λ + ω0 | 1−γ   since it is easily seen that  A(λI − A)−1  L(H) < M when |arg λ| < θ 0 + π/2 . Now we observe that ≤

A (λI − A)−1 D f = (−A)1−σ (λI − A)−1 y

where

y = (−A)σ0 D f

so that    A(λI − A)−1 D f  ≤

H

  = (−A)1−σ0 (λI − A)−1 [(−A)σ0 D f ] H

m0 M (−A)σ0 D f H . |λ + ω0 | σ0

The result follows since the operator (−A)σ0 D is continuous.

 

Remark 2.58. If A ∈ L(H) it is easily computed that the Laplace transform of e At is (λI − A)−1 and this observation can be extended to any C0 -semigroup but formula (2.24) cannot be used to recover the semigroup from (λI − A)−1 since this function does not decays fast enough on vertical lines. In the special case of the holomorphic semigroups, formula (2.105) can be interpreted as an inversion formula for the Laplace transformation which replaces (2.24).

2.9 Cosine Operators and Differential Equations of the Second Order we consider the equation of the forced harmonic oscillator u  = −ω2 u + h ,

u(0) = u0 , u (0) = v0 ,

We introduce v = u  and we rewrite the equation as

u∈R.

2.9 Cosine Operators and Differential Equations of the Second Order

93

       d u 0 1 u 0 + h(t) . = 1 −ω2 0 v dt v

  Aω

It is known and easily checked that the solution is ∫ 1 1 t u(t) = u0 cos ωt + v0 sin ωt + sin ω(t − s)h(s) ds ω ω 0 so that, when h = 0,     u(t) A ω t u0 =e u (t) v0



where

e

Aω t

cos ωt (1/ω) sin ωt = −ω sin ωt cos ωt

 .

(2.110)

Similar formulas exist, at least in certain cases, when the equation is a second order equation in a Banach or Hilbert spaces. We confine ourselves to the case that H is a Hilbert space. Let A be selfadjoint, negative defined, with compact resolvent (in a separable Hilbert space H). So, its spectrum is a divergent sequence of negative numbers. Let {ϕn } be an orthonormal basis of H whose elements are eigenvalues of A: Aϕn = −λn2 ϕn . We consider the abstract wave equation u  = Au + h . We introduce v = u  and we rewrite this second order equation as         d u u 0 0 I Å Å = + h(t) where = . v 1 A 0 dt v

(2.111)

(2.112)

This is similar to the scalar case, but now we must find a suitable functional framework where the operator Å generates a C0 -semigroup or group. We recall the notations A = (−A)1/2 , A = iA and V = X1 = dom A = dom A. We introduce the product Hilbert space V × H, whose elements we denote U, and dom Å = {U = (u, v) :

u ∈ dom A, v ∈ V } .

If U and U˜ belong to dom Å , then we have ˜ V ×H = v, u ˜ V ×H ÅU, U Å ˜ V + Au, v˜  H = Av, Au ˜ H − Au, A˜v  H = −U, ÅU Å and in fact it is possible to prove equality: Å ∗ = −Å Å. It follows and so Å ∗ extends −Å from Corollary 2.50 that Å generates a C0 -group on V × H. We wish to represent this group with a formula similar to (2.110).

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2 Operators and Semigroups for Systems with Boundary Inputs

We use again Corollary 2.50 and we see that A generates a C0 -group e At on H. In terms of the expansion of A in (2.107) we have "  +∞ +∞  , λ α ϕ A y = i n n n n=1 (2.113) αn ϕn ∈ L 2 (Ω) , then if y =  iλ n t α ϕ . e At y = +∞ n n n=1 n=1 e The (strongly continuous) operator valued function R+ (t) =

* 1 ) At e + e−At 2

t∈R

is the cosine operator generated by A. Its key property is the cosine formula R+ (t)R+ (τ) =

1 (R+ (t) + R+ (τ)) 2

∀t , τ ∈ R .

(2.114)

It is convenient to introduce the operators R− (t) =

* 1 ) At e − e−At , 2

S(t) = A −1 R− (t) ,

t ∈ R.

The operator S(t) is the sine operator (generated by A). Formulas (2.113) give32  +∞  +∞   R+ (t) αn ϕn = (αn cos λn t) ϕn (x) , n=1 n=1  +∞    +∞   R− (t) αn ϕn = i (αn sin λn t) ϕn (x) , n=1  +∞    +∞n=1    sin λn t  S(t) αn ϕn = αn ϕn (x) . λn n=1 n=1

(2.115)

Remark 2.59. It is a standard and unfortunate terminology that the sine operator generated by A is not the operator (1/i)R− (t) but the operator S(t) = A −1 R− (t). The reason is that the operator A −1 R− (t) enters the representation formula of the solutions of (2.111). The following properties are easily seen from the formulas (2.115): • S(t) takes values in dom A. ∫ t • for all h ∈ H we have S(t)h = 0 R+ (r)h dr. α • the spaces dom (−A) (any α ≥ 0) are invariant for R+ (t), R− (t) and S(t) (any t ∈ R) and the operators R+ (t), R− (t), S(t) can be extended to strongly continuous functions on the spaces (dom (−A)α ). The cosine formula holds for the restriction/extension of R+ (t). 32

Note that the Dirichlet and the Neumann Laplacians generate cosine operators in L 2 (Ω).

2.9 Cosine Operators and Differential Equations of the Second Order

95

• for every h ∈ dom A we have d R+ (t)h = AR− (t)h = AS(t)h , dt

d R− (t)h = AR+ (t)h . dt

• if h ∈ dom Ak for every k, then e At h, e At h, R+ (t)h, R− (t)h and S(t)h are of class C ∞ . • the following integration by parts formulas hold for h(t) ∈ C 1 ([0, T]; H): (see [26]): ∫ t R+ (t − s)h (s) ds = h(t) − R+ (t)h(0) 0 ∫ t R− (t − s)h(s) ds , (2.116a) +A 0 ∫ t R− (t − s)h (s) ds = −R− (t)h(0) 0 ∫ t R+ (t − s)h(s) ds . (2.116b) +A 0

With this information we can easily see that the unique mild solution of the problem u(0) = u0 , u (0) = v0 u  = Au + h , written in the form (2.112) is given by U(t) = (u(t), u (t)) where ∫ t −1 −1 u(t) = R+ (t)u0 + A R− (t)v0 + A R− (t − s)h(s) ds 0 ∫ t R+ (t − s)h(s) ds . u (t) = AR− (t)u0 + R+ (t)v0 +

(2.117a) (2.117b)

0



It follows eÅ t =

   R+ (t) S(t) R+ (t) A −1 R− (t) . = AS(t) R+ (t) AR− (t) R+ (t)

(2.118)

The following result is an obvious adaptation of Theorem 2.44: Theorem 2.60. Let u be the function in (2.117a). For every α we have: 1. Let h = 0. The transformation (u0, u1 ) → u is linear and continuous from Xα × Xα−1 to C ([0, T]; Xα ) ∩ C 1 ([0, T]; Xα−1 ). 2. Let u0 = 0, u1 = 0. The transformation h → u is linear and continuous from L 2 (0, T; Xα ) to C ([0, T]; Xα+1 ) ∩ C 1 ([0, T]; Xα ) ∩ C 2 ([0, T]; Xα−1 ). 3. If h ∈ C 1 ([0, T]; Xα ), then u ∈ C ([0, T]; Xα+2 ) ∩ C 1 ([0, T]; Xα+1 ) ∩ C 2 ([0, T]; Xα ). Finally, we recall from Sect. 2.8: Theorem 2.61. The function U(t) = (u(t), u (t)) given by (2.117a)–(2.117b) is a classical solution of Eq. (2.112) when u0 ∈ dom A, v0 ∈ dom A and h ∈ C 1 ([0, +∞); H).

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2 Operators and Semigroups for Systems with Boundary Inputs

In this case by definition the function u is a classical solution of the abstract wave equation (2.111). Remark 2.62. We call the attention on the following facts (we use that A is selfadjoint): • • • • • • •

A∗ = A while A ∗ = −A. R−∗ (t) = R− (−t) = −R− (t) while R+∗ (t) = R+ (t) = R+ (−t). (AR− (t))∗ = R− (t)A = AR− (t), (AR+ (t))∗ = −AR+ (t). (AR+ (t))∗ = R+ (t)A = AR+ (t), (AR− (t))∗ = −AR− (t). R+ (t)R+ (τ) = R+ (τ)R+ (t), R− (t)R− (τ) = R− (τ)R− (t). R+ (t)R− (τ) = R− (τ)R+ (t). Aα R+ (t)u = R+ (t)Aα u, Aα R− (t)u = R− (t)Aα u for all u ∈ dom Aα .

The theory of the cosine operators and their applications to the solutions of differential equations of second order can be found in [10, 29].

2.10 Extensions by Transposition and Semigroups Let E(t) = e At be a C0 -semigroup in a Hilbert space H, whose generator is A. As noted in Remark 2.51, E ∗ (t) is a C0 -semigroup too, whose generator is A∗ . We proved that H ⊆ (dom A∗ ) and the operator A admits an extension Ae ∈ L(H, (dom A∗ )): the operator Ae is an (unbounded) operator in (dom A∗ ) whose domain is H. See Sect. 2.5 for rigorous statements (Sect. 2.5.1 for selfadjoint operators). The following result holds (more details are in [8]): Theorem 2.63. Let A be the infinitesimal generator of a C0 -semigroup (or group) on H. The operator Ae (in (dom A∗ ) with domain H) is the infinitesimal generator of a C0 -semigroup (or group) on (dom A∗ ). If e At is a holomorphic semigroup in H, then e Ae t is a holomorphic semigroup in (dom A∗ ). Moreover, if h ∈ H ⊆ (dom A∗ ), then e Ae t h = e At h. Note a consequence of the last statement: if h ∈ H = dom Ae , then t → e At h = is differentiable in the norm of (dom A∗ ) for every t ≥ 0 but not in the norm of H (unless h ∈ dom A). If the operator −A is selfadjoint positive with compact resolvent, then e At is a holomorphic semigroup and it is possible to define the fractional powers of −A, their domains Xγ and the duals X−γ (see Sect. 2.5.1.1) and using an orthonormal basis of eigenvectors of A the following results are easily seen:33 e Ae t h

Theorem 2.64. Let the operator A in H be selfadjoint with compact resolvent and let −A be positive definite. Let e At be the (holomorphic) semigroup generated by A. Then we have: 33

We noted that the fractional powers and so the spaces Xγ can be defined also when A generates a holomorphic exponentially stable semigroup. The statement holds also in this case but the proofs are not easy.

2.10 Extensions by Transposition and Semigroups

97

• let γ ≥ 0. Then e At Xγ ⊆ Xγ and the restriction of e At to Xγ is a holomorphic semigroup. Its generator is the part of A in Xγ i.e. it is the operator Ah ∈ Xγ

defined on

{h ∈ Xγ : Ah ∈ Xγ } .

• let γ ≥ 0. The spaces X−γ ⊆ (dom A) are invariant for e Ae t (defined on (dom A) ) and the restriction of e Ae t to X−γ is a holomorphic semigroup whose generator is the part of Ae in X−γ . The fractional powers (−A)γ are positive and in general are not infinitesimal generators of semigroups. Instead: Theorem 2.65. If A is selfadjoint negative defined, then −(−A)γ is the infinitesimal generator of a C0 semigroup on H (and so the semigroup admits an extension to (dom (−A)γ )). Remark 2.66. Using the expansions in eigenfunctions it is easily seen that the semiγ group e[−(−A) ]t is not the restriction of e At to dom (−A)γ . Finally, slightly adapting Theorem 2.65 we have: Theorem 2.67. The semigroup eiAt and the operators R± (t) admits extensions from H to V  = (dom A) = (dom A).

2.10.1 Semigroups, Cosine Operators, and Boundary Inputs In this section we outline a method first introduced in control theory by Fattorini in [9] (see also [3]), which transforms a boundary control system to a linear affine system (of the form x  = Ax + B f ) in a larger space. More details can be found in the appendix of the paper [27]. Let H and U be Hilbert spaces and let L be a linear (unbounded) operator in H with dense domain. Let T be a linear (unbounded) operator from H to U. We assume: • dom L ⊆ dom T ; • we introduce the following (unbounded) operator A in H: dom A = (dom L) ∩ ker T ,

Ah = Lh

∀h ∈ dom A .

We assume that A is the infinitesimal generator of a C0 -semigroup on H (and so, implicitly, we assume also that (dom L) ∩ ker T is dense in H) and we assume ker A = 0. • there exists D ∈ L(U, H) with the following properties: there exists a dense subspace U0 of U such that if f ∈ U0 , then D f ∈ ker L and T D f = f .

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2 Operators and Semigroups for Systems with Boundary Inputs

Now we define: Definition 2.68. When f ∈ U0 , u = D f is a classical solution of the “stationary” problem  Lu = 0 (2.119) Tu = f and, for every f ∈ U, u = D f is a mild solution of the problem (2.119). The definition of mild solution is justified by the fact that any mild solution is the limit of a sequence of classical solutions. This follows because U0 is dense in U and D ∈ L(U, H). Remark 2.69. In practice, L is a differential operator on a region Ω and T accounts for the boundary conditions. For example, if H = L 2 (Ω), L = Δ and T = γ0 , then problem (2.119) is the Dirichlet problem. We called the problem (2.119) “stationary” since now we adapt the ideas to solve the following evolutionary boundary control problem: u  = Lu + h

u(0) = u0 ∈ H ,

T u = f ∈ L 2 (0, T; U) .

(2.120)

The idea to recast this problem in a semigroup setting is as follows: let f ∈ C 1 ([0, T]; U), with values in U0 and let us assume that a solution u (“regular” enough) exists. We introduce y(t) = u(t) − D f (t) so that T y(t) = 0 and, by using LD f = 0 for every f ∈ U0 y  = (u − D f ) = Lu + h − D f  = L(u − D f ) + h − D f  = Ay + h − D f  . By definition, the equation y  = Ay + h − D f 

y(0) = u0 − D f (0)

(2.121)

is the abstract version of the boundary control problem (2.120) (when f ∈ C 1 and takes values in U0 ). Of course, the previous arguments are solely a heuristic justification of this definition. Now we solve (2.121) and we get ∫ t At u(t) = y(t) + D f (t) = e (u0 − D f (0)) − e A(t−s) D f (s) ds + D f (t) 0 ∫ t e A(t−s) Dh(s) ds . (2.122) + 0

2.10 Extensions by Transposition and Semigroups

99

By adapting to this case the statement 6 in Theorem 2.44 we see that if u0 − D f (0) ∈ dom A, h ∈ C 1 ([0, T]; H) and if f ∈ C 2 ([0, T]; U), then y(t) = u(t) − D f (t) is a classical solution of the problem (2.121). So, we define: Definition 2.70. The function u(t) is a classical solution on the interval [0, T] of the boundary control problem (2.120) when it belongs to C 1 ([0, T]; H), y(t) = u(t)− D f (t) ∈ C([0, T]; dom A) and the following equalities hold for every t ∈ [0, T]: u (t) = Ay(t) + h(t) ,

y(t) = u(t) − D f (t) ,

u(0) = u0 .

We recapitulate: Theorem 2.71. If u(0) − D f (0) ∈ dom A, h ∈ C 1 ([0, T]; H) and f ∈ C 2 ([0, T]; U), then the function u(t) in (2.122) is a classical solution of the boundary control problem (2.120). We would like to extend formula (2.122) to the case that f is solely of class L 2 . The obvious device is an integration by parts but unfortunately this cannot be done in the space H unless im D ⊆ dom A, a property that does not hold in practical cases. We invoke the operator Ae , the extension by transposition of A defined in Sect. 2.5 and applied to semigroups in Sect. 2.10. If u(t) is a classical solution, then we have also u(t) ∈ C([0, T]; H) ⊆ C([0, T]; (dom A∗ )) and ∫ t ∫ t e Ae (t−s) D f (s) ds + D f (t) + e Ae (t−s) h(s) ds . u(t) = e Ae t (u0 − D f (0)) − 0

0

From im D ⊆ H ⊆ (dom A∗ ) and H = dom Ae we see that we can integrate by parts in the space (dom A∗ ). We get ∫ t ∫ t Ae t A e (t−s) u(t) = e (u0 − D f (0)) + e h(s) ds − e Ae (t−s) D f (s) ds + D f (t) 0 0 ∫ t Ae t e Ae (t−s) h(s) ds = e (u0 − D f (0)) + D f (t) + 0   ∫ t e Ae (t−s) D f (s) ds ∈ C ([0, T]; (dom A∗ )) . − D f (t) − e Ae t D f (0) + Ae 0

After cancellations of terms, this formula is usually written without the index e : ∫ t ∫ t u(t) = e At u0 + A e A(t−s) D f (s) ds + e A(t−s) h(s) ds 0 0 ∫ t ∫ t e A(t−s) AD f (s) ds + e A(t−s) h(s) ds (2.123) = e At u0 + 0

0

(the operator A and the integral can be exchanged because A = Ae is a closed operator). Definition 2.72. The function u ∈ C([0, T]; (dom A∗ )) in (2.123) is the mild solution of the boundary control problem (2.120).

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This definition is justified by the following result: Theorem 2.73. The following properties hold: • the transformation H × L 2 (0, T; H) × L 2 (0, T; U)  (u0, h, f ) → u ∈ C([0, T]; (dom A∗ )) is linear and continuous. • every mild solution of the problem (2.120) is the limit of a sequence of classical solutions (the limit is in the norm of in the space C([0, T]; (dom A∗ )) ). The mild solutions might take values in the space H itself, usually at the expenses of the time regularity. We shall describe results of this type in Chaps. 3 and 4. Similar considerations can be done for second order equations. The problem u  = A(u − D f ) + h

u(0) = u0 , u (0) = u1

(2.124)

is the abstract version of u  = Lu + h ,

u(0) = u0 , u (0) = u1 ,

Tu = f .

If A generates a cosine operator, then     ∫ t   u(t) u 0 U(t) =  = eÅ t 0 + eÅ (t−s) ds u (t) u1 h(s) 0   ∫ t D f (s) eÅ (t−s)Å − ds . 0 0

(2.125)

The operator Å is the (extension by transposition of the) operator defined in (2.112). Note in fact that     D f (s) 0 Å = 0 AD f (s) (where AD is in fact Ae D). The first component u(t) of U(t) is the mild solution of (2.124). Definition 2.45 of classical solution when applied to the problem (2.124) gives: the function u(t) is a classical solution when: 1) u(t) − D f (t) ∈ C([0, T]; dom A) ∩ C 1 ([0, T]; dom A) . 2) u(t) = C 1 ([0, T]; dom A) ∩ C 2 ([0, T]; H) . Theorem 2.71 can be reformulated for the semigroup generated by Å and we get Theorem 2.74. Let h ∈ C 1 ([0, T]; H), f ∈ C 3 ([0, T]; H), u0 − D f (0) ∈ dom A, u1 − D f (0) ∈ V = dom A. The first component u(t) of U(t) in (2.125) is a classical solution of problem (2.124). Hence, every mild solution is the limit of a sequence of classical solutions.

References

101

Using the explicit formula (2.118) for the semigroup eÅ t we can rewrite (2.125) as follows: ∫t u(t) = R+ (t)u0 + A −1 R− (t)u1 − A 0 R− (t − s)D f (s) ds ∫t +A −1 0 R− (t − s)h(s) ds , ∫t (2.126) u (t) = AR− (t)u0 + R+ (t)u1 − A 0 R+ (t − s)D f (s) ds ∫t + 0 R+ (t − s)h(s) ds . Finally we note that the cosine operator approach to systems with boundary inputs has been introduced in [21].

2.11 On the Terminology and a Final Observation The approach we take in this book is mostly based on semigroup theory. Different approaches to the solutions of evolutionary systems are possible. The reader can see, for example, the nice presentation in [24]. An approach based on Laplace transformation is in Sect. 3.6. The reader should also consider that we distinguish between “classical solutions” and “mild solutions.” This is not a standard terminology. Mild solutions when the boundary conditions are zero are often called weak solutions (equivalent to strong solutions in semigroup theory) while the solutions introduced in Sect. 2.10.1 when the boundary condition is nonhomogeneous are the ultraweak solutions. We do not need to enter in such details.

References 1. Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics, vol. 96, second edn. Birkhäuser/Springer Basel AG, Basel (2011) 2. Aubin, J.P.: Applied functional analysis. Wiley-Interscience, New York (2000) 3. Balakrishnan, A.V.: Boundary control of parabolic equations: L − Q − R theory. In: Theory of nonlinear operators (Proc. Fifth Internat. Summer School, Central Inst. Math. Mech. Acad. Sci. GDR, Berlin, 1977), Abh. Akad. Wiss. DDR, Abt. Math. Naturwiss. Tech., 1978, vol. 6, pp. 11–23. Akademie-Verlag, Berlin (1978) 4. Balakrishnan, A.V.: Applied functional analysis. Springer-Verlag, New York-Berlin (1981) 5. Barnett, A., Hassell, A.: Estimates on Neumann eigenfunctions at the boundary, and the “method of particular solutions” for computing them. In: AMS Proceedings of symposia in pure mathematics, vol. 84, pp. 195–208. Birkhäuser, Basel (2012) 6. Barros-Neto, J.: Problèmes aux limites non homogènes. Séminaire de Mathématiques Supérieures, No. 17 (Été, 1965). Les Presses de l’Université de Montréal, Montreal, Que. (1966) 7. Brezis, H.: Functional analysis, Sobolev spaces and partial differential equations. Springer, New York (2011) 8. Engel, K.J., Nagel, R.: One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194. Springer-Verlag, New York (2000)

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9. Fattorini, H.O.: Boundary control systems. SIAM J. Control 6, 349–385 (1968) 10. Fattorini, H.O.: Second order linear differential equations in Banach spaces. North-Holland Publishing Co., Amsterdam (1985) 11. Fuhrmann, P.A.: Linear systems and operators in Hilbert space. McGraw-Hill International Book Co., New York (1981) 12. Fujiwara, D.: Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order. Proc. Japan Acad. 43, 82–86 (1967) 13. Goldberg, S.: Unbounded linear operators: Theory and applications. McGraw-Hill Book Co., New York-Toronto, Ont.-London (1966) 14. Grasselli, M., Pata, V.: Uniform attractors of nonautonomous dynamical systems with memory. In: Evolution equations, semigroups and functional analysis (Milano, 2000), Progr. Nonlinear Differential Equations Appl., vol. 50, pp. 155–178. Birkhäuser, Basel (2002) 15. Grisvard, P.: Caractérisation de quelques espaces d’interpolation. Arch. Rational Mech. Anal. 25, 40–63 (1967) 16. Haase, M.: The functional calculus for sectorial operators, Operator Theory: Advances and Applications, vol. 169. Birkhäuser Verlag, Basel (2006) 17. Hassell, A., Tao, T.: Erratum for “Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions”. Math. Res. Lett. 17(4), 793–794 (2010) 18. Hille, E., Phillips, R.S.: Functional analysis and semi-groups. American Mathematical Society Colloquium Publications, vol. 31. American Mathematical Society, Providence, R. I. (1957) 19. Jakšić, V., Molčanov, S., Simon, B.: Eigenvalue asymptotics of the Neumann Laplacian of regions and manifolds with cusps. J. Funct. Anal. 106(1), 59–79 (1992) 20. Lasiecka, I.: Unified theory for abstract parabolic boundary problems—a semigroup approach. Appl. Math. Optim. 6(4), 287–333 (1980) 21. Lasiecka, I., Triggiani, R.: A cosine operator approach to modeling L2 (0, T ; L2 (Γ))— boundary input hyperbolic equations. Appl. Math. Optim. 7(1), 35–93 (1981) 22. Lasiecka, I., Triggiani, R.: Control theory for partial differential equations: continuous and approximation theories. I Abstract parabolic systems, Encyclopedia of Mathematics and its Applications, vol. 74. Cambridge University Press, Cambridge (2000) 23. Lions, J.L., Magenes, E.: Non-homogeneous boundary value problems and applications. Vol. I. Springer-Verlag, New York-Heidelberg (1972). Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181 24. Medeiros, L. A., Miranda M. M., Lourêdo, A.T.: Introduction to exact control theory. Method HUM. Eduepb, Editora da Universidade Estadual de Paraíba, Campina Grande-PB (2013) (it can be downloaded from http://dspace.bc.uepb.edu.br/jspui/handle/123456789/13358) 25. Mikha˘ılov, V.P.: Partial differential equations. “Mir”, Moscow (1978) 26. Pandolfi, L.: The controllability of the Gurtin-Pipkin equation: A cosine operator approach. Applied Mathematics and Optimization 52(2), 143–165 (2005) 27. Pandolfi, L., Priola, E., Zabczyk, J.: Linear operator inequality and null controllability with vanishing energy for unbounded control systems. SIAM J. Control Optim. 51(1), 629–659 (2013) 28. Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York (1983) 29. Sova, M.: Cosine operator functions. Rozprawy Mat. 49, 47 (1966) 30. Stakgold, I.: Boundary value problems of mathematical physics. Vol. I, II, Classics in Applied Mathematics, vol. 29. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2000) 31. Tucsnak, M., Weiss, G.: Observation and control for operator semigroups. Birkhäuser Verlag, Basel (2009)

Chapter 3

The Heat Equation with Memory and Its Controllability

3.1 The Abstract Heat Equation with Memory The operators A and D have the properties listed below. The abstract version of the heat equation with memory studied in this chapter is1 ∫ t w  = A(w − D f ) + N(t − s)A(w(s) − D f (s)) ds + h(t) , w(0) = w0 . (3.1) 0

The (associated) memoryless system is u  = A(u − D f ) + h = Au − AD f + h ,

w(0) = w0 .

(3.2)

The prototype of system (3.1), introduced in [12], is the concrete case when A is either the Dirichlet or the Neumann laplacian, and the interpretation of (3.1) and (3.2) as abstract models for a system with boundary input is in Sect. 2.10.1. Remark 3.1 (On the Notations). We shall be consistent to denote w the solution of the system with memory, i.e. when N(t)  0, while u is the solution when N(t) = 0 but note that u(0) = w(0) = w0 .

1

The equation studied in Sect. 3.6 is slightly more general.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 L. Pandolfi, Systems with Persistent Memory, Interdisciplinary Applied Mathematics 54, https://doi.org/10.1007/978-3-030-80281-3_3

103

104

3 The Heat Equation with Memory and Its Controllability

Standing Assumptions and Notations of This Chapter 1. H and U are separable Hilbert spaces and D ∈ L(U, H). The operator A is selfadjoint with compact resolvent and −A is positive definite. 2. We use the notations A = (−A)1/2 and V = dom A. 3. There exists σ0 ∈ (0, 1) such that im D ⊆ dom (−A)σ0 . 4. As usual, A denotes both the operator in Assumption 1 and its extension by transposition Ae ∈ L(H, (dom A)) described in Sect. 2.5 and, strictly speaking, AD has to be intended as Ae D ∈ L(U, (dom (−A)1−σ0 )) ⊆ L(U, (dom A)). 5. h ∈ L 2 (0, T; H) and f ∈ L 2 (0, T; U) for every T > 0. f is the control and U is the control space, while h is the distributed affine term. 6. Let θ ∈ (0, π/2). We denote Σθ the sector of the complex plane:  π . (3.3) Σθ = λ , λ  0 and |arg λ| < θ + 2 7. When the dependence of the data have to be specified, we write w f (·; w0, h) respectively u f (·; w0, h). When one of the arguments is not indicated (while at least one of the others is explicitly indicated), we intend that it is zero. For example, w f (·) denotes the solution with w0 = 0 and h = 0. The notations w(·) and u(·) are used when there is no risk of confusion.

As seen in Sect. 2.8.1, the assumptions have the following consequences: Theorem 3.2. The following properties hold: • the operator A is the infinitesimal generator of a holomorphic exponentially stable semigroup e At in H. • there exists θ 0 ∈ (0, π/2) such that  π ∪ {0} ⊆ (A) . Σθ0 ∪ {0} = λ , |arg λ| < θ 0 + 2 • there exist positive numbers ω0 , m0 , and M such that   ⎧ m0 e−ω0 t ⎪ γ At ⎪ ∀γ ∈ [0, 1] ⎨ (−A) e  L(H) ≤ ⎪ tγ   m0  ⎪ −1  ⎪ ∀λ ∈ Σθ0 ∪ {0} ≤ ⎪ (λI − A)  L(H) |λ + ω0 | ⎩

(3.4)

and 1 ⎧ At p ⎪ ⎪ ⎨ Ae D L(U, H) ∈ L (0, T) ∀p ∈ [1, 1 − σ ) ⎪ 0   M ⎪ −1   ⎪ ∀λ ∈ Σθ0 ∪ {0} . ⎪ A(λI − A) D L(U, H) ≤ |λ + ω0 | σ0 ⎩

(3.5)

3.2 Preliminaries on the Associated Memoryless System

105

The first problem we study in this chapter is to understand whether it is possible to define a solution of problem (3.1) in some extrapolation space (dom (−A)σ ), possibly in H, and then we study the control properties. Equation (3.1) can be studied with different methods. In this chapter we present two of them: the use of semigroup theory to reduce Eq. (3.1) to a Volterra integral equation of the second kind and frequency domain methods based on the use of the Laplace transformation. The first method (in Sect. 3.3) is more suited when the memory kernel N(t) is smooth, while Laplace transform methods (deeply studied in [49]) can be used under less stringent conditions. This is the reason for not listing the assumptions on the memory kernel N(t) here: the assumptions used in the semigroup approach are in Sect. 3.3, while those used in the frequency domain approach are in Sect. 3.6. Remark 3.3. The condition that A is selfadjoint with compact resolvent is used in the study of controllability via semigroup methods in Sect. 3.5, based on Sect. 3.3.1. It is not used in the proofs concerning the existence of the solutions, provided that the inequalities (3.4) and (3.5) hold. These inequalities, suitably adapted, hold also if the holomorphic semigroup is not even bounded. As an example of the changes needed in this case, we note that the condition im D ⊆ dom (−A)σ0 has to be replaced with im D ⊆ dom (cI − A)σ0 , where c is a suitable positive constant.

3.2 Preliminaries on the Associated Memoryless System In this section first we recall (from Sect. 2.10.1) the solutions of (3.2) when A, D, f , and h satisfy the general conditions listed in the previous section. Then we consider the specific properties when A is either the Dirichlet or the Neumann laplacian, i.e. when (3.2) is the associated heat equation. The mild solutions of Eq. (3.2) are given by the following semigroup formula (see Sects. 2.8 and 2.10.1): ∫ t ∫ t At A(t−s) e h(s) ds − Ae A(t−s) D f (s) ds . (3.6) u(t) = e w0 + 0

0

Thanks to the standing assumption 3, for every (w0, h, f ) ∈ H × L 2 (0, T; H) × the function u in (3.6) belongs to C([0, T]; (dom (−A)1−σ0 )) ⊆ C([0, T]; (dom A)) for every T > 0, and the transformation (w0, h, f ) → u is continuous among the indicated spaces. By definition, the function u in (3.6) is a classical solution when y = u − D f ∈ C([0, T]; dom A), u ∈ C 1 ([0, T]; H), and the following equalities hold: L 2 (0, T; U),

u (t) = Ay(t) + h(t)

∀t ≥ 0 ,

u(0) = w0 .

106

3 The Heat Equation with Memory and Its Controllability

The first inequality in (3.5) shows that Ae At D L(U, H) ∈ L 1 (0, T) for every T > 0 (and in fact it belongs to L p (0, T) for suitable values of p > 1). The fact that f ∈ L 2 (0, T; U) implies that ∫ t Ae A(t−s) D f (s) ds 0

belongs to L 2 (0, T; H) for every T > 0 (use Young inequalities in Sect. 2.2). More precisely, we have (see Sect. 2.10.1) Lemma 3.4. The following properties hold for every T > 0: ∫

1. let u(t) = e At w0 +

t

e A(t−s) h(s) ds .

0

The transformation (w0, h) → u is linear and continuous from H × L 2 ([0, T]; H) to C([0, T]; H). If h ∈ C 1 ([0, T]; H) and w0 ∈ dom A, then u(t) is a classical solution of (3.2) with f = 0. 2. let ∫ t ∫ t A(t−s) Ae D f (s) ds = (−A)1−σ0 e A(t−s) ((−A)σ0 D) f (s) ds . (3.7) u(t) = − 0

0

The H-valued function u(t) in (3.7) has the following properties: a. if f ∈ L 2 (0, T; L 2 (Γ)), then u(t) is square integrable and f → u is continuous from L 2 (0, T; U) to L 2 (0, T; H). b. if f ∈ C 1 ([0, T]; U), then u(t) ∈ C([0, T]; H) and u(0) = 0. c. if f ∈ C 2 ([0, T]; U) and f (0) = 0, then u(t) is a classical solution of (3.2) with h = 0 and w0 = 0. 3. if h ∈ C 1 ([0, T]; H), f ∈ C 2 ([0, T]; H), and w0 − D f (0) ∈ dom A, then the function u(t) = u f (t; w0, h) in (3.6) is a classical solution of (3.2). The lemma can be adapted to the case f and h in L p , but we are not interested to do that. Now we consider the concrete case that A is either the Dirichlet or the Neumann laplacian so that the memoryless system is the heat equation. We describe the relevant properties of the associated heat equation.

3.2 Preliminaries on the Associated Memoryless System

107

On the Heat Equation 1. H = L 2 (Ω) (Ω ⊆ Rd is a bounded region with C 2 boundary). 2. U = L 2 (Γ), where Γ ⊆ ∂Ω is relatively open in ∂Ω. 3. The operator A in L 2 (Ω) is A = Δ on the domain dom A = H 2 (Ω) ∩ H01 (Ω) ; !   γ1 φ = 0 on Γ , 2 . Neumann laplacian: dom A = φ ∈ H (Ω) , γ0 φ = 0 on ∂Ω \ Γ Dirichlet laplacian:

4. If A is the Neumann laplacian, we assume that ∂Ω \ Γ contains a nonempty relatively open subset so that 0 ∈ (A) (in the case of the Dirichlet laplacian, we have 0 ∈ (A) even if Γ = ∂Ω). 5. The space V = dom A = dom (−A)1/2 is Dirichlet laplacian: V = H01 (Ω) Neumann laplacian: V = {φ ∈ H 1 (Ω) such that γ0 φ = 0 on ∂Ω \ Γ} . 6. The operator D is defined as follows: u = D f is the mild solution of the “stationary” problem (2.119), which now is the Laplace equation   T u = f on Γ γ (Dirichlet) Δu = 0 where T = 0 (3.8) =0 on ∂Ω \ Γ, γ1 (Neumann) (in the Dirichlet case, Γ = ∂Ω is not excluded. In this case the condition on ∂Ω \ Γ has to be disregarded.). An important known fact is the existence of σ0 ∈ (0, 1) such that im D ⊆ dom (−A)

σ0



any σ0 < 1/4 in the Dirichlet case any σ0 < 3/4 in the Neumann case .

(3.9)

Under these conditions, the associated memoryless system is the heat equation  ⎧ either γ0 u = f on Γ ⊆ ∂Ω ⎪ ⎨ ⎪ on Γ ⊆ ∂Ω or γ1 u = f (3.10) u  = Δu + h , u(0) = w0 , ⎪ ⎪ γ0 u = 0 on ∂Ω \ Γ . ⎩ The reason for the assumption on ∂Ω \ Γ is explained in Sect. 2.7.2. It can be removed as describe there. It implies that −A is positive defined in the Neumann case too, and it is assumed solely to avoid technicalities (actually minor technicalities in this chapter). We take into account the values of the exponent σ0 in (3.9) and the first inequality in (3.4). We get

108

3 The Heat Equation with Memory and Its Controllability

• (−A)1−σ0 e At is square integrable on (0, T) (any T > 0) in the case T = γ1 , • (−A)1−σ0 e At integrable2 on (0, T) (any T > 0) in the case T = γ0 , while ((−A)σ0 D) f (t) is square integrable, since (−A)σ0 D is a continuous operator. Using these observations, we give a more precise version of Statement 2a of Lemma 3.4. Theorem 3.5. Let A be either the Dirichlet or the Neumann laplacian, and let f ∈ L 2 (0, T; U) = L 2 (0, T; L 2 (Γ)). The function ∫ t u(t) = Ae A(t−s) D f (s) ds 0

in (3.7) has the following properties: 1. if T = γ1 (Neumann laplacian, i.e. when f is the boundary flux), then u ∈ C([0, +∞); L 2 (Ω)) and the transformation f → u is linear and continuous from L 2 ([0, T]; L 2 (Γ)) to C([0, T]; L 2 (Ω)) for every T > 0. 2. if T = γ0 (Dirichlet laplacian, i.e. when f is the boundary temperature), then u ∈ L 2 ([0, T]; L 2 (Ω)) for every T > 0 and the transformation f → u is linear and continuous from L 2 ([0, T]; L 2 (Γ)) to L 2 ([0, T]; L 2 (Ω)) for every T > 0. Finally, we state the following theorem: Theorem 3.6. Let w0 , h, and f be of class C ∞ with compact support, respectively, in Ω, in Ω × (0, T), and in Γ × (0, T). Then u(x, t) has continuous first derivative in t and second derivatives in the space variable x, and the following properties hold: • the equality ut (x, t) = Δu(x, t) + h(x, t) holds for every t > 0, x ∈ Ω. • for every t > 0, the equality T u(x, t) = f (x, t) holds for every x ∈ Γ and γ0 u(x, t) = 0 holds for x ∈ ∂Ω \ Γ. • we have limt→0+ u(x, t) = w0 (x) for every x ∈ Ω. More precise properties are in [21, Theorem 7 p. 65 and Theorem 2 p. 144] and [36, pp. 187, 194]. It is important to know that in general, the solution of the Dirichlet problem for the heat equation is not an L 2 (Ω)-valued continuous function for t > 0, not even when Ω ⊆ R, as the following example (taken from [39, p. 217]) shows: Example 3.7. We consider Ω = (0, π) and u  = u xx ,

u(x, 0) = 0

u(0, t) = 0 , u(π, t) = f (t) . (3.11)  We expand the solution u in series of ϕn (x) = 2/π sin nx, which is an orthonormal basis of L 2 (0, π) whose elements are eigenvectors of the Dirichlet laplacian. We have

It belongs to L p with 1 ≤ p < 1/(1 − σ0 ) < 2, but this does not help much to improve the regularity of the solutions.

2

3.2 Preliminaries on the Associated Memoryless System

u(x, t) =

+∞ 

109

∫ ϕn (x)un (t) ,

un (t) =

π

u(x, t)ϕn (x) dx .

0

n=1

We multiply both the sides in (3.11) with ϕn (x), and we integrate on (0, π). We integrate by parts twice. We find  un (0) = 0 . un = −n2 un + n(−1)n+1 2/π f (t) , So, we have

+ un (t) = (−1)

n+1

2 n π

∫

t

e−n

2 (t−s)

f (s) ds .

0

We fix T > 0, and we choose 1 f (t) = √4 ∈ L 2 (0, T) . T −t It is clear that u ∈ C([0, T); L 2 (0, π)) since f ∈ C 1 (0, T ) for every T  < T. In particular, u(t) can be computed for t < T. We prove that lim u(·, t) does not exist in L 2 (0, π).

(3.12)

t→T −

In fact, we have + (−1)

n+1

∫ =n 0

n2 T

∫ T ∫ T 2 2 π 1 1 un (T) = n e−n (T −s) √4 e−n s √4 ds ds = n 2 s 0 0 T−s   ∫ T /2 1 √ 1 1 1 c e−r √4 n 2 dr ≥ √ e−r √4 dr = √ . n n 0 n r r

It follows that {un (T)}  l 2 . We conclude that the limit in (3.12) does not exist thanks to the following observations: • we prove below that if the limit in (3.12) exists, then limt→T − {un (t)} exists in l 2 . • for every n, we have limt→T − un (t) = un (T). • it follows that if limt→T − {un (t)} exists in l 2 , then we have also {un (T)} ∈ l 2 and, as we already noted, this is false. We conclude that the limit in (3.12) does not exist. So, in order to complete our argument, we must prove that if the limit in (3.12) exists, then also limt→T − {un (t)} exists in l 2 . This follows because if the limit (3.12) exists, then for every ε > 0 there exists Tε such that if Tε < t < T , Tε < t  < T then ε > u(t) − u(t



) L2 2 (0,π)

 2 +∞      =  ϕn (x)[un (t) − un (t )]   2 n=1

L (0,π)

110

3 The Heat Equation with Memory and Its Controllability

=

+∞ 

[un (t) − un (t )]2

so that {un (t)} converges in l 2 for t → T − .

n=1

We sum up that the limit (3.12) does not exist in L 2 (0, π) and t → u(t) does not belong to C([0, T]; L 2 (0, π)). In spite of Example 3.7, even when the control acts in the Dirichlet boundary condition, the solutions of the heat equation have strong regularity properties which we describe now (see [16, p. 258]). We use the multiindex notations for the partial derivatives with respect to the components of x ∈ Ω ⊆ Rd . Let α be a sequence of d nonnegative integers, α = (α1 , α2 , . . . , αd ). Then |α| = α1 + α2 + · · · + αd,

Dα =

∂ |α | . ∂xα11 ∂xα22 · · · ∂xαdd

˜ where Ω ˜ ⊆ Ω ⊆ Rd is Theorem 3.8. Let us assume that h(x, t) = 0 for every x ∈ Ω, 2 open and nonempty. We fix any t0 > 0 and  so small that t0 − 16 > 0. ˜ and let B denote the ball of radius  and center x0 . If needed, we Let x0 ∈ Ω,   ˜ (so that cl B4 ∩ Γ = ∅). reduce the value of  so to have cl B4 ⊆ Ω We introduce the cylinders C = B × (t0 − 2, t0 ) ,

Cˆ = B4 × (t0 − 162, t0 ) .

˜ for every k and Let u(x, t) solve the heat equation. Then ∂ k u/∂t k and Dα u exist in Ω α. Furthermore, there exist constants a and C, which depends only on the dimension d of Ω, such that the following inequalities hold (μ is the volume of Cˆ ): sup |Dα u(x, t)| ≤ a

(x,t)∈C 

C |α | |α|!  |α | μ

∫ Cˆ 

|u(ξ, τ)| dξ dτ

 k  ∫ 2k  ∂ u(x, t)   ≤ a C (2k)! |u(ξ, τ)| dξ dτ . sup  ∂t k  2k μ Cˆ  (x,t)∈C  So, we have also ' 1/2 & ⎧ ⎪ |α | |α|! ∫ ⎪ C ⎪ ⎪ sup |Dα u(x, t)| ≤ a |α | √ |u(ξ, τ)| 2 dξ dτ ⎪ ⎪ ⎨ (x,t)∈C  ⎪  μ Cˆ  ' 1/2 &  k  ⎪ ⎪ ˆ 2k (2k)! ∫  ∂ u(x, t)  C ⎪ 2 ⎪ ≤a ⎪ sup  |u(ξ, τ)| dξ dτ . √ ⎪ ⎪ (x,t)∈C   ∂t k  2k μ Cˆ  ⎩

(3.13)

Remark 3.9. The previous inequalities have been stated in [16, p. 258] for every ˜ which satisfy function u ∈ H (Ω), u (x, t) = Δu(x, t)

˜. ∀t > 0 , ∀x ∈ Ω

(3.14)

3.2 Preliminaries on the Associated Memoryless System

111

˜ is the space of those functions u(x, t) such that ut (x, t) and The space H (Ω) ˜ × (0, +∞). This is the case if w0 , h, and f are of u x j x j (x, t) are continuous on Ω ∞ ˜ of the class C with compact support. As stated in Theorem 3.6, the restriction to Ω ˜ and function in (3.6) is the L 2 -limit of a sequence of solutions that belong to H (Ω) satisfy the equality (3.14). ˜ to It follows that the inequalities (3.13) can be lifted from solutions in H (Ω) every mild solution given by (3.6). Indeed, we see from (3.13) that L 2 (0, T; L 2 (Ω))convergence of a sequence of solutions implies that the sequence of partial derivatives of any order is uniformly convergent on C . It follows that the partial derivatives of the mild solution exist in C . We shall use the following consequence. Let Ω1 ⊆ Ω be an open set such that Ω \ cl Ω1  ∅, and let h be supported in Ω1 . Thanks to Theorem 3.6, the result in Theorem 3.8 can be applied to the restriction to Ω \ cl Ω1 of every (mild) solution. In particular: Corollary 3.10. Let Ω1 ⊆ Ω be such that Ω \ cl Ω1  ∅. Let h ∈ L 2 (0, T; L 2 (Ω)) be zero in Ω \ Ω1 , f ∈ L 2 (0, T; L 2 (Γ)), w0 ∈ L 2 (Ω). For every t > 0, the solution u(x, t) of the heat equation is of class C ∞ (Ω \ cl Ω1 ).

3.2.1 Controllability of the Heat Equation (Without Memory) The reachable states at time T are the values u f (T). Example 3.7 shows that u f (T) cannot be defined in the Dirichlet case, unless the control f has some particular property: in order to define u f (T), we must require the existence of the limit limt→T − u f (t) in L 2 (Ω). Definition 3.11. A boundary control f is admissible for the heat equation at time T when limt→T − u f (t) exists in H. In this case, u f (T) is by definition limt→T − u f (t). Hence, every L 2 -control is admissible when the boundary condition is of Neumann type, but there exist square integrable boundary controls that are not admissible when they act in the Dirichlet boundary condition. Using Theorem 3.6, we have the following lemma: Lemma 3.12. The admissible boundary controls f are dense in L 2 (0, T; L 2 (Γ)) for every T > 0. We introduce the following definition: Definition 3.13. Let Ω1 ⊆ Ω be a nonempty region and Γ ⊆ ∂Ω be relatively open and nonempty in ∂Ω. We say that the equation is subject to controls localized in Ω1 or in Γ when it is imposed that h is supported in Ω1 , respectively, and f is supported in Γ. In this case we say that Ω1 or Γ are the active parts of Ω or of ∂Ω. Now we recall the known controllability properties of the heat equation, from [24, 38] (see also [42, 43] for a different approach).

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3 The Heat Equation with Memory and Its Controllability

Theorem 3.14. Let the active part Ω1 of Ω or the active part Γ of ∂Ω be fixed (and so we impose, respectively, h(x, t) = 0 if x  Ω1 and f (x, t) = 0 if x ∈ ∂Ω \ Γ). The following properties hold: 1. approximate controllability (in any time T > 0): let u(0) = 0, and let T > 0. For every ξ ∈ L 2 (Ω) and every ε > 0, there exist controls, either distributed controls h localized in Ω1 or admissible boundary controls f localized in Γ, such that u(T) − ξ L 2 (Ω) < ε. Even more, these controls can be taken to be of class D(Ω1 × (0, T)), and D(Γ × (0, T)). 2. controllability to the target zero (in any time T > 0): let u(0) = w0 ∈ L 2 (Ω), and let T > 0. There exist controls (either distributed control h localized in Ω1 or admissible boundary controls f localized in Γ) such that u(T) = 0. Even more, these controls can be taken to be of class D(Ω1 × (0, T)), and D(Γ × (0, T)). Note that once u(T) = 0, by putting the control to zero for t > T, we get u(t) ≡ 0 for t > T. That is, the heat equation is controllable to the rest. Controllability to the rest is also called null controllability. We cannot expect controllability to the rest for a system with persistent memory (a part exceptional cases [31]). But, we can wonder whether approximate controllability holds or whether every initial condition w0 can be steered to hit the target 0 at a certain time T > 0. A guess on these matters is not easy, since • systems whose solutions have similar analytic properties may have different controllability properties. See [35] for an equation whose solutions have properties similar to those of the heat equation, but with different controllability properties. • approximate controllability is a weak property, which is not preserved under perturbations (examples are easily found, see [46]), but here we have the particular perturbation that consists in the addition of a memory. • Controllability to zero seems to be a stronger property, but even this property is not preserved under perturbations, not even in the concrete case of systems with finite memory in Rn (see [46]). In this chapter, we are going to prove that approximate controllability is preserved by the memory, while controllability to the target zero is not preserved. It turns out that approximate controllability can be proved under quite general conditions, in the abstract setup described in Sect. 3.1 or 3.6, while the proof of lack of controllability to the target zero uses specific properties either of the heat equation (in Sect. 3.5.2) or of the Laplace equation (in Sect. 3.7.2), and so in this case we are forced to work with the concrete case that A is the laplacian.

3.3 Systems with Memory, Semigroups, and Volterra Integral Equations In this section we show an approach to the study of the solutions and controllability of system (3.1), which is best suited when the memory kernel N(t) is smooth.

3.3 Systems with Memory, Semigroups, and Volterra Integral Equations

113

The Assumption on N(t) Used in This Section and in Sect. 3.5 The real memory kernel N(t) is of class H 1 (0, T) for every T > 0; i.e. there exists an L 2 (0, T) function N (t) such that ∫ t N (s) ds . (3.15) N(t) = N(0) + 0

We consider Eq. (3.1) as a Volterra integral equation in the “unknown” A(w − D f ) in the space (dom A∗ ) : ∫ t A(w(t) − D f (t)) + N(t − s)A(w(s) − D f (s)) ds = w (t) − h(t) . (3.16) 0

We “solve” this Volterra integral equation. Note that the existence of the solution w(t) has still to be proved, and so the computations are formal. Let R(t) be the resolvent kernel of N(t): ∫ t N(t − s)R(s) ds = N(t) (the function R(t) is of class H 1 as N(t)) . R(t) + 0

Equation (3.16) when solved in the “unknown” A(w(t) − D f ) gives ∫ t R(t − s)w (s) ds + R(t − s)h(s) ds 0 0 ∫ t R (t − s)w(s) ds + R(t)w0 − G(t) = w (t) − R(0)w(t) − 0   ∫ t R(t − s)h(s) ds . where G(t) = h(t) −

A(w(t) − D f (t)) = w (t) − h(t) −

∫

t

0

Hence, we get ∫



w (t) = A(w(t) − D f (t)) + R(0)w(t) +

t

R (t − s)w(s) ds − R(t)w0 + G(t) ,

0

w(0) = w0 . A last manipulation is as follows: we replace w(t) = eR(0)t v(t). The left hand side is transformed to R(0)eR(0)t v(t) + eR(0)t v (t), and the first addendum cancels the term R(0)w(t) on the right. Then we multiply both the sides with e−R(0)t , and we get   ∫ t v (t) = A v(t) − De−R(0)t f (t) + e−R(0)(t−s) R (t − s)v(s) ds 0 −R(0)t

−e

−R(0)t

R(t)w0 + e

G(t) ,

v(0) = w0 .

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3 The Heat Equation with Memory and Its Controllability

The functions e−R(0)t G(t) and eR(0)t f (t) are as arbitrary as h(t) and f (t) are, respectively, in the spaces L 2 (0, T; H) and L 2 (0, T; U) (furthermore, when H = L 2 (Ω) and U = L 2 (Γ), they are localized in the same sets as, respectively, h(x, t) and f (x, t)). So, we rename f (t) and G(t) as the functions e−R(0)t G(t) and e−R(0)t f (t). We rename R(t) as the function e−R(0)t R(t), and we introduce the notation L(t) = e−R(0)t R (t) ∈ L 2 (0, T) .

(3.17)

Finally, the function v is renamed as w. Then, w(0) = w0 , and we get the equation ∫ t  w (t) = A(w(t) − D f (t)) + L(t − s)w(s) ds − R(t)w0 + G(t) . (3.18) 0

The associated memoryless equation is still (3.2), i.e. u  = A(u − D f ) + G(t) ,

u(0) = w0 .

(3.19)

Note that we do not insert the term −R(t)w0 in the right side of (3.19). Remark 3.15. This formal procedure is known as MacCamy trick. We note that • the bonus of MacCamy trick is that the unbounded operator is removed from the memory; • the initial condition w0 appears also in the right hand side of the equation. As we shall see, this fact does not make any difficulty in the definition and in the study of the solutions, but it has an important consequence on controllability. We apply formula (3.6)–(3.18), and we find u f (t;w0,G)

  ∫ t  ∫ t w(t) = e At w0 + e A(t−s) G(s) ds − e A(t−s) AD f (s) ds 0 0  ∫ t ∫ t−s ∫ t e A(t−s) R(s)w0 ds + e A(t−s−r) L(r)w(s) dr ds . − 0

0

(3.20)

0

The first integral at the second line is a linear continuous transformation from H to C([0, T]; H) for every T > 0, and u f (t; w0, G) is the solution of the associated memoryless system.

3.3 Systems with Memory, Semigroups, and Volterra Integral Equations

115

In terms of w, Eq. (3.20) is a Volterra integral equation of the second kind in H, with strongly continuous kernel. The properties of the associated memoryless system and of the Volterra integral equations (see Sect. 2.2) imply the following: Theorem 3.16. Equation (3.20) admits a unique solution w = w f (·; w0, G) ∈ L 2 (0, T; H), and 1. let f = 0. The transformation (w0, G) → w(·; w0, G) is linear and continuous from H × L 2 (0, T; H) to C([0, T]; H). 2. let w0 = 0 and G = 0. The transformation f → w f (t) is linear and continuous from L 2 (0, T; U) to L 2 (0, T; H). So, we give the following definition: Definition 3.17. The solution w(t) ∈ L 2 (0, T; H) of the Volterra integral equation (3.20) is the mild solution (shortly, “solution”) w f (t; w0, h) of (3.1). The solution w = w f (t; w0, h) is a classical solution if w(0) = w0 , y = w − D f belongs to C([0, T]; dom A), w ∈ C 1 ([0, T]; H), and furthermore the following equality holds in H for every t > 0: ∫ t w (t) = Ay(t) + L(t − s)w(s) ds − R(t)w0 + G(t) . (3.21) 0

A control f is admissible for the problem (3.1) at time T if the corresponding mild solution w admits limit in H for t → T − . The limit is denoted w(T). Note that every addendum in the right hand side of (3.20) is a H-valued continuous function, with the possible exception of u f (t), which is square integrable. Hence, we have the following: Theorem 3.18. Equation (3.1) admits a unique mild solution w f (t; w0, h) for every w0 ∈ H, h ∈ L 2 (0, T; H), and f ∈ L 2 (0, T; U). The properties of the mild solution have been specified in Theorem 3.16. A boundary control f is admissible for (3.1) if and only if it is admissible for the associated memoryless system. As in the case of the heat equation, the definition of the (mild) solutions is justified by the fact that every mild solution is the L 2 (0, T; H)-limit of a sequence of classical solutions. In fact, we have the following theorem: Theorem 3.19. Let T > 0, and let w0 ∈ dom A, h ∈ D(0, T; H) and f ∈ D(0, T; U). The mild solution w of (3.1) is a classical solution. Proof. Linearity of the problem shows that we can study separately w0 , h, and f . We confine ourselves to the case w0 = 0, h = 0, and f  0 leaving the contribution of w0 and h to the reader. The mild solution of (3.1) is the solution of the Volterra integral equation (3.20). We know already that w(t) ∈ L 2 (0, T; H) so that the last integral in (3.20) is continuous and equal to zero for t = 0.

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3 The Heat Equation with Memory and Its Controllability

The assumption that f (t) is of class C ∞ with f (0) = 0 shows that ∫ t ∫ t − e A(t−s) AD f (s) ds = D f (t) − e A(t−s) D f (s) ds 0

0

so that the solution w(t) of (3.20) verifies w(0) = 0. Moreover, y(t) = w(t) − D f (t) ∫ t ∫ t−s ∫ t = e A(t−s−r) L(r)w(s) dr ds − e A(t−s) D f (s) ds 0

0

(3.22)

0

so that y(t) =

∫ t∫ 0

∫ +

t−s

0

∫

t

e A(t−r) 0

e A(t−s−r) L(r) [w(s) − D f (s)] dr ds

  y(s) r

∫

L(r − s)D f (s) ds dr −

0

t

 e A(t−r) D f (r) dr .

(3.23)

0

Thanks to the regularity of f (t), both the addenda in the last square bracket of (3.23) belong to C([0, T]; dom A) ∩ C 1 ([0, T]; H) (see Theorem 2.44, Statement 5). Furthermore, using that L(t) is a scalar function, the map ∫ t ∫ t−s y → e A(t−s−r) L(r)y(s) dr ds 0

0

belongs to L(C([0, T]; dom A)) (dom A with the graph norm). Hence, (3.23) is a Volterra integral equation for y(t) in the Hilbert space dom A so that its solution y verifies y(t) ∈ C([0, T]; dom A) . Even more, the affine term of the Volterra integral equation, i.e. the square bracket in (3.23), is of class C 1 ([0, T]; dom A). In fact, ∫ r ∫ t e A(t−r) L(s)D f (r − s) ds dr 0 0  ∫ t ∫ t ∫ r e A(t−r) L(s)D f (r − s) ds dr − L(s)D f (t − s) ds , = A−1 0 0 0  ∫ t ∫ t A(t−r)  −1 A(t−r)  e D f (r) dr = A e D f (r) dr − D f (t) . 0

0

Hence, from Lemma 2.10,  y(t) ∈ C ([0, T]; dom A) =⇒ 1

w(t) = y(t) + D f (t) ∈ C 1 ([0, T]; H) w(0) = 0,

(3.24)

3.3 Systems with Memory, Semigroups, and Volterra Integral Equations

117

and both the integrals on the line (3.23) take values in dom A (see Statement 5 of Theorem 2.44). In order to complete the proof that w(t) is a classical solution, we must prove equality (3.21) for every t. We consider the equality (3.20), which is an equality in L 2 (0, T; H). We apply the operator A−1 to both the sides. We get the following equality in L 2 (0, T; H): ∫ t ∫ t ∫ t−s −1 −1 A(t−s−r) A w(t) = A e L(r)w(s) dr ds − e A(t−s) D f (s) ds . (3.25) 0

0

0

Both the sides are continuous H-valued functions, and the equality holds for every t ≥ 0. Now we observe ∫ t ∫ t−s e A(t−s−r) L(r)w(s) dr ds 0 0   ∫ t ∫ t−s −1 A(t−s−r) e L(r)w(s) dr ds A =A 0 0   ∫ t ∫ t−s ∫ t d −1 A(t−s−r) e L(r)w(s) dr ds − L(t − s)w(s) ds . A dt 0 0 0 Using this equality, (3.22) gives  ∫ t ∫ t−s ∫ t d −1 A(t−s−r) −1 y(t) = A e L(r)w(s) dr ds − A L(t − s)w(s) ds dt 0 0 0 ∫ t e A(t−s) D f (s) ds . − 0

We use (3.25), and we get   ∫ t ∫ t d y(t) = e As D f (t − s) ds − A−1 L(t − s)w(s) ds A−1 w(t) + dt 0 0 ∫ t e A(t−s) D f (s) ds (use w ∈ C 1 ([0, T]; H), see (3.24)) − 0 ∫ t L(t − s)w(s) ds = A−1 w (t) − A−1 0

(note the use of f (0) = 0 in the computation of the derivative of the integral). Then we see that the equality (3.21) holds for every t ≥ 0, as wanted.  

3.3.1 Projection on the Eigenfunctions Let {ϕn } be an orthonormal basis of eigenfunctions of A, Aϕn = −λn2 ϕn . Then

118

3 The Heat Equation with Memory and Its Controllability ∞ ⎧  ⎪ ⎪ ⎨ w(t) = ⎪ ϕn wn (t) ,

wn (t) = w(t), ϕn  ,

n=1 ⎪ ⎪ ⎪ G n (t) = G(t), ϕn  , ⎩

w0,n = w0, ϕn 

fn (t) = D f (t), Aϕn  =

(the crochet denotes the inner product in H). We see from (3.20),  ∫ t ∫ t−s 2 wn (t) − L(t − s − r)e−λn r dr wn (s) ds 0 0   ∫ t 2 −λ2n t − e−λn (t−s) R(s) ds w0,n = e 0 ∫ t ∫ t 2 −λ2n (t−s) + e G n (s) ds − e−λn (t−s) fn (s) ds . 0

(3.26)

−λn2  f (t), D∗ ϕn 

(3.27)

0

This is a Volterra integral equation for wn (t), with kernel ∫ t 2 Zn (t) = − L(t − s)e−λn s ds .

(3.28)

0

Let Hn (t) be the resolvent of Zn (t): ∫ Hn (t) = Zn (t) −

t

Zn (t − s)Hn (s) ds .

(3.29)

0

Then we have ∫

∫ t t 2 2 wn (t) = e w0,n − e−λn (t−s) R(s) ds + Hn (t − s)e−λn s ds 0 0 ∫ t−s   ∫ t 2 R(s) Hn (τ)e−λn (t−s−τ) dτ ds w0,n − 0 0  ∫ t ∫ t−s 2 2 + Hn (τ)e−λn (t−s−τ) dτ (G n (s) − fn (s)) ds . (3.30) e−λn (t−s) − −λ2n t

0

0

We prove the following lemma: Lemma 3.20. The functions Zn (t) are continuous, and for every T > 0, there exists MT , which does not depend on n, such that the following inequalities hold on [0, T]:   MT Hn (t − τ)e dτ  ≤ 3 . λn 0 (3.31) If N (t) is bounded on [0, T], then the right hand sides of the previous inequalities improve to (MT /λn2 ) (inequalities (a) and (b)) and to (MT /λn4 ) (the last one). MT (a) |Zn (t)| ≤ , λn

MT (b) |Hn (t)| ≤ , λn

∫  c) 

t

−λ2n τ

3.3 Systems with Memory, Semigroups, and Volterra Integral Equations

119

Proof. The function L(t) is defined in (3.17):  L(t) ∈ L 2 (0, T) since N (t) ∈ L 2 (0, T) −R(0)t  R (t), hence L(t) = e if N (t) is bounded, then L(t) is bounded too . Furthermore, Zn (t) is continuous since it is the convolution of L(t) with the contin2 uous function e−λn t . We have ∫ t    2 L(t − s)e−λn s ds |Zn (t)| =  0 ∫ T  1/2 ∫ T  1/2 1 2 −2λ2n s ≤ L (s) ds e ds ≤ MT . λn 0 0 Let N (t) be bounded. Boundedness of N (t) and L(t) implies ∫  |Zn (t)| = 

t

−λ2n s

L(t − s)e

0

∫     ds ≤ sup |L(t)| [0,T ]

T

e−λn s ds ≤ MT 2

0

1 . λn2

This is inequality (a) when N (t) is bounded, and the inequality (b) in both the cases follows from the Gronwall inequality applied to (3.29). Inequality (c) follows since ∫     ∫ t T   2τ 2t 1 −λ −λ  Hn (t − τ)e n dτ  ≤ sup |Hn (t)| e n dt ≤ 2 sup |Hn (t)| .  λn t ∈[0,T ] 0 0 t ∈[0,T ]  The proof is finished by taking into account the already found estimate of Hn (t).  The resolvent Hn (t) has the following property first noted in [27] (see also [28]). Theorem 3.21. There exists a function J(t, s) (which does not depend on n) such that ∫ t ∫ t 2 2 Hn (t − s)e−λn s ds = J(t, s)e−λn s ds . (3.32) 0

0

Moreover, 1. the function J(t, s) is square integrable on the triangle T = {(t, s) 0 ≤ s ≤ t ≤ T }, every T > 0. 2. if N ∈ C 1 , then J(t, s) is continuous on the triangle T. 3. let us extend J(t, s) = 0 for t < s < T. Both the functions t → J(t, ·) and s → J(·, s) belong to C([0, T]; L 2 (0, T)). Proof. We note that N(t) need not be defined for t < 0. So, Hn (t − s) is defined in the triangle T. We recall the notation for the iterated convolution: L ∗1 = L ,

L ∗2 = L ∗ L ,

L ∗(k+1) = L ∗ L ∗k .

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3 The Heat Equation with Memory and Its Controllability

We choose any natural number n, and we keep it fixed. We introduce the notations

e0 =

e0∗1

−λ2n t

=e

2 tk ek (t) = e−λn t , k!

,

 so that

e0 ∗ ek = ek+1 , e0∗k = ek−1 for k ≥ 1 .

We repeat that e0 = e−λn t for a fixed index n, and we recall ∫ t Zn = − L(t − s)e0 (s) ds = −L ∗ e0 , so that Zn∗k = (−1)k L ∗k ∗ ek−1 . 2

0

We search for a representation of Hn ∗ e0 . We recall that Zn (t) is continuous. Hence, Eq. (3.29) is a Volterra integral equation in C(0, T) (the unknown is Hn (t)). Picard iteration applied to (3.29) gives the following uniformly convergent series for Hn (t): Hn (t) =

+∞ +∞   (−1)k−1 Zn∗k = − L ∗k ∗ ek−1 . n=1

(3.33)

n=1

We compute Hn ∗ e0 :

Hn ∗ e0 = −e0 ∗ ∫ =− 0

where

t

 +∞ 

L ∗k ∗ ek−1 = −

k=1

L ∗k (t − s)

k=1 +∞ 

+∞ 

sk k!

+∞  k=1



e−λn s ds = 2

L ∗k ∗ ek ∫

t

J(t, s)e−λn s ds, 2

0

+∞ k  s ∗k sk L (t − s) J(t, s) = L (t − s) = sL(t − s) + k! k! k=1 k=2 ∗k

(3.34)

does not depend on n. We recall L(t) ∈ L 2 (0, T). Young inequalities show that the iterated convolutions ∗k L are continuous and L ∗k C([0,T ]) ≤ L Lk 2 (0,T ) . It follows that the series with index k ≥ 2 is a uniformly convergent series of continuous functions, and its sum is continuous, while the first addendum sL(t − s) is square integrable if N ∈ H 1 (it is continuous if N ∈ C 1 ). These observations prove Statements 1 and 2. Statement 3 follows from Lebesgue’s lemma on continuity of the shift3 applied to sL(t − s). This lemma states that if L(s) ∈ L 2 (0, +∞), then the functions t → L(s − t) and s → L(s − t)  (equal zero if t −s < 0) are continuous L 2 (0, T)-valued functions for every T > 0. 

3

That is, Lemma 1.3 used also in Example 2.53.

3.4 The Definitions of Controllability for the Heat Equation with Memory

121

We insert (3.32) in the last integral of (3.30). We get ∫ t  ∫ t 2 −λ2n t −λ2n (t−s) wn (t) = e w0,n − e R(s) ds w0,n − Hn (t − s)e−λn s ds 0 0 ∫ t−s   ∫ t 2 R(s) Hn (t − s − τ)e−λn τ dτ ds w0,n − 0 0  ∫ t ∫ t−s −λ2n (t−s) −λ2n τ + − J(t − s, τ)e dτ G n (s) ds e 0 0  ∫ t−s ∫ t 2 2 +λn2 J(t − s, τ)e−λn τ dτ fn (s) ds . (3.35) e−λn (t−s) − 0

0

3.4 The Definitions of Controllability for the Heat Equation with Memory These definitions are given here, but they will be used also when the memory kernel N(t) satisfies the assumptions in Sect. 3.6. We recall the definition of admissible (boundary) control: a control f is admissible for the problem (3.1) at time T if w f (T) = limt→T − w f (t) exists in H. We define and then study controllability either under the action of distributed controls h, and in this case we assume f = 0, or under the action of the boundary controls f , and in this case we assume h = 0. Definition 3.22. An initial condition w0 can be steered or controlled to hit the target ξ at time T when there exists either a distributed control h or an admissible boundary control f such that either w(T; w0, h) = ξ or w f (T; w0 ) = ξ. In this case, ξ is a reachable vector4 at time T (from w0 ). Approximate controllability is the property that the set of the reachable vectors w(T) at time T from w0 = 0 (either under the action of distributed controls or under the action of admissible boundary controls) is dense in H. The system is controllable to the target 0 in time T when 0 is reachable at time T from any w0 , i.e. when for every w0 ∈ H, there exists either a distributed control h such that w(T; w0, h) = 0 or a distributed admissible boundary control f such that w f (T; w0 ) = 0.

4

The analogous term we used for the memoryless heat equation is “reachable state.” We use different terms since, as we shall see in Sect. 5.5, u f (T ; w0 ) is a state of the heat equation without memory, while w f (T ; w0 ) is not a state of the system with memory.

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3 The Heat Equation with Memory and Its Controllability

Facts to be Kept in Mind in the Study of Controllability 1. Density in H of the set of the reachable vectors from w0 does not depend on w0 , and so when studying approximate controllability we assume w0 = 0. 2. We are pedantic, and we say “w0 can be steered to hit the target zero” and “controllable to the target 0” since even if the target 0 is reached, then w(t) will evolve, and it will be w(t)  0 at later times, unless N(t) = 0. So, this property is the property of hitting a certain target, which happens to be 0, without remaining at the rest in the future. 3. We recall that in the concrete case of the heat equation, approximate controllability and null controllability hold also when the controls are localized in Ω1 ⊆ Ω or in Γ ⊆ ∂Ω. 4. Controllability to the target zero is studied in the concrete case of the heat equation with memory, i.e. A is the laplacian, and either admissible boundary controls localized in a relatively open subset Γ ⊆ ∂Ω (equality is not excluded) or distributed controls localized in nonempty subregion Ω1 of Ω with the property that Ω \ Ω1 contains an open nonempty set. 5. When studying approximate controllability of the system with memory, we assume that the associated memoryless system is approximately controllable. We do not put ourselves in the concrete case that A is the laplacian.

3.5 Memory Kernels of Class H 1 : Controllability via Semigroups We recall equality (3.15) for N(t). We see from (3.35),  2 • let ξ ∈ H, ξ(x) = +∞ n=1 ξn ϕn (x) with {ξn } ∈ l . The vector ξ is reachable at time T (from w0 = 0) – by using distributed controls: when there exists G ∈ L 2 (0, T; H) such that ∫

T



−λ2n (T −s)

e 0

∫ − 0

T −s

−λ2n τ

J(T − s, τ)e

 dτ G(s), ϕn  ds = ξn .

 

(3.36)

G n (s)

– by using boundary controls: when there exists f ∈ L 2 (0, T; U) which is admissible and such that

3.5 Memory Kernels of Class H 1 : Controllability via Semigroups

∫

T



∫

e−λn (T −s) − 2

0

T −s

J(T − s, τ)e−λn τ dτ 2

0

123



 λn2 D f (s), ϕn  ds = ξn .

  fn (s)

(3.37)

+∞

• An initial condition w0 =

n=1 w0,n ϕn

can be steered to hit the target 0

– by distributed controls: when there exists G ∈ L 2 (0, T; H) such that ∫

T



e−λn (T −s) − 2

∫

0

T −s

 2 J(T − s, τ)e−λn τ dτ G n (s) ds = −ξn (w0 ),

(3.38)

0

where −λ2n T

∫

T

−λ2n (T −s)



w0,n − e R(s) ds w0,n ξn (w0 ) = e 0 ∫ T 2 − Hn (T − s)e−λn s ds 0 ∫ T −s   ∫ T 2 R(s) Hn (T − s − τ)e−λn τ dτ ds w0,n . − 0

(3.39)

0

Of course, the minus sign on the right hand side of (3.38) has no effect and can be ignored (i.e. by renaming −G as G). – a similar condition characterizes controllability to the target zero with boundary controls, but in this case we must require also that the control is admissible: ∫

T



−λ2n (T −s)

e

∫ −

0

T −s

−λ2n τ

J(T − s, τ)e

 dτ fn (s) ds = ξn (w0 )

(3.40)

0

(ξn (w0 ) is in (3.39)). When N = 0, R = 0, and J = 0, we get the corresponding controllability conditions for the associated memoryless system.

3.5.1 Approximate Controllability Is Inherited by the System with Memory In this section we prove that the property of approximate controllability is inherited by the system with memory; i.e. we prove the following theorem: Theorem 3.23. Let the associated memoryless system be approximately controllable at time T with either distributed or boundary controls. The system with memory (3.1) either with distributed control or with boundary control is approximately controllable at the same time T > 0, and the reachable targets (from the initial condition

124

3 The Heat Equation with Memory and Its Controllability

equal to zero) are the reachable targets (at the same time T) of the associated memoryless system. In the concrete case of the heat equation with memory, approximate controllability is preserved also if the controls are localized in Ω1 or in Γ. Proof. We recall w0 = 0. We assume f = 0, and we study approximate controllability under the action of the distributed control h, i.e. G (in the concrete case of the heat equation with memory, localized in the subregion Ω1 ⊆ Ω). We exchange the order of integration in (3.36), and we see that ξ ∈ L 2 (Ω) is reachable when there exists G ∈ L 2 (0, T; U), which solves ∫ τ ∫ T 2 e−λn (T −τ)  G(τ) − J(T − s, T − τ)G(s) ds, ϕn  dτ = ξn ∀n . 0 0

  G1 (τ)

Let ξ be reachable for the associated memoryless system. Then, there exists a function G1 , which solves these equalities for every value of n. The equation ∫ τ G(τ) − J(T − s, T − τ)G(s) ds = G1 (τ) (3.41) 0

is a Volterra integral equation of the second kind, and so for every G1 ∈ L 2 (0, T; U), it has a (unique) solution G ∈ L 2 (0, T; U). This function G (i.e. the corresponding function h) steers the initial condition 0 of the system with memory to the target ξ. This proves the statement concerning controllability with distributed controls. We examine the special case H = L 2 (Ω). In this case, ∫ τ G1 (τ) = G1 (x, τ) = h(x, τ) − R(τ − s)h(x, s) ds, 0

where R(τ) is a real function: this equality holds for every x. Hence, if h(x, τ) is localized in Ω1 , then G1 (x, τ) is localized in Ω1 , and for an analogous reason, G(x, τ) is localized in Ω1 as well. In fact, J(τ, r) is a real function so that, for every fixed x ∈ Ω, Eq. (3.41) is a Volterra integral equation for the function τ → G(x, τ). The proof of controllability under the action of boundary controls is similar, but we must pay attention to the fact that the boundary control has to be admissible. The proof goes as follows: a target ξ is reachable when there exists f ∈ L 2 (0, T; U), which is admissible and solves the following equalities for every n: ∫ − λn2

0

T

∫ τ 2 e−λn (T −τ)  f (τ) + J(T − s, T − τ) f (s) ds, D∗ ϕn  dτ = ξn . 0

  f1 (τ)

(3.42)

3.5 Memory Kernels of Class H 1 : Controllability via Semigroups

125

When ξ is reachable with boundary controls for the associated memoryless system, there exists a square integrable boundary control f1 , which solves (3.42) for every n and which is admissible at time T for the memoryless system. Hence, there exists a square integrable boundary control f such that the equalities (3.42) for the equation with memory (3.1) are verified for every n: f is the solution of the Volterra integral equation (in the Hilbert space U): ∫ τ f (τ) = f1 (τ) − J(T − s, T − τ) f (s) ds . (3.43) 0

In order to complete the proof, we prove that this control f is admissible, i.e. that limt→T − w f (t) exists in L 2 (Ω). We state this result as a lemma since it is used also in Sect. 3.5.2. Lemma 3.24. The function f1 ∈ L 2 (0, T; U) is admissible for the associated memoryless system if and only if the solution f of the Volterra integral equation (3.43) is admissible for the equation with memory. Proof. We replace the right hand side of (3.43) in the Volterra integral equation (3.20) for w(t), with w0 = 0 and h = 0. We get 2’

  ∫ τ ∫ t ∫ t e A(t−s) AD f1 (s) ds + e A(t−τ) AD J(T − s, T − τ) f (s) ds dτ w f (t) = − 0 0 0

    ∫ +

u f1 (t)

∫

t

e A(t−r) 0

r

2

 L(r − s)w f (s) ds dr .

0

  3

In order to prove that w f (T − ) exists if and only if u f1 (T − ) exists, we prove continuity of 2 and 3 . We consider 3 . We know that w f (t) is square integrable and that L(t) = −R(0)t e R (t) is square integrable as well. Hence, from Young inequalities (i.e. Theorem 2.8), the convolution L ∗ w is continuous and 3 is continuous as well. A similar argument holds for the addendum 2 since, form (3.5), there exists p > 1 such that e At AD ∈ L p (0, T; H) and 2’ is bounded as we can see from Schwarz inequality: the following function of τ ∫   

0

τ

 ∫  J(T − s, T − τ) f1 (s) ds ≤

0

τ

 1/2 |J(T − s, T − τ)| ds 2

f1 L 2 (0,T ;U)

is bounded since τ → J(T − ·, T − τ) L 2 (0,T ) is continuous on [0, T], see Statement 2 of Theorem 3.21.

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3 The Heat Equation with Memory and Its Controllability

Young inequalities show that 2 is continuous.

 

Finally, we note that in the concrete case of the heat equation with memory, the control f is localized in Γ if and only if f1 is localized in Γ. The argument to show this fact is similar to the one for distributed controls. The proof of approximate controllability is now complete.  

3.5.2 Controllability to the Target 0 Is Not Preserved We put ourselves in the following concrete setup:

The Setup for the Study of Controllability to the Target 0 1. H = L 2 (Ω), U = L 2 (Γ), and A is either the Dirichlet or the Neumann laplacian so that the solution u of the associated heat equation has the properties stated in Theorem 3.8. 2. When working with distributed controls, we fix a region Ω1 ⊆ Ω such that Ω \ Ω1 contains a nonempty open set, and we assume that the distributed control is localized in Ω1 . Instead, we do not exclude that the boundary control is distributed on the entire ∂Ω. 3. N(t) ∈ H 1 (0, T) is not identically zero. Hence, its resolvent kernel R(t) is of class H 1 (0, T) and not identically zero.

In fact, in the case of boundary controls, the arguments in this section can be adapted to the general equation with memory (3.1). The reason for putting ourselves in the concrete case is that when studying distributed controls the fact that the control is localized in a proper subregion Ω1 ⊆ Ω has its role and an attempt to recast the argument in an abstract form, although possible, looks artificial. We recall that every initial condition w0 of the associated heat equation can be steered to hit the target 0 at every time t0 > 0. We are going to prove that this is not the case for the heat equation with memory: it is clear that if N(t) ≡ 0 for t ∈ (0, τ), then on this interval the heat equation with memory reduces to the standard heat equation, and every w0 can be controlled to hit the target zero in time less than τ. The effect of the memory is that w(t) will depart from zero as soon as the memory kernel becomes nonzero. So, we consider what is going on at a time t0 when the memory affects the system. This fact is best expressed in terms of the resolvent kernel. We prove the following theorem: Theorem 3.25. We fix a time instant t0 such that R(t0 )  0. There exist initial conditions which cannot be steered to hit the target 0 at time t0 either by distributed controls localized in Ω1 or by boundary controls.

3.5 Memory Kernels of Class H 1 : Controllability via Semigroups

127

We prove the theorem. Using formula (3.35), we see that w0 is controllable to hit the target 0 when ∫

t0



−λ2n (t0 −s)

e

∫ −

t0 −s

−λ2n τ

J(t0 − s, τ)e

0

0

∫ =

t0

−λ2n (t0 −s)

e 0



∫ χn (s) −

s

⎤ ⎡ ⎥ ⎢ ⎥ ⎢ ⎢ dτ ⎢G n (s) − fn (s)⎥⎥ ds ⎢   ⎥ ⎥ ⎢ χn (s) ⎦ ⎣ 

J(t0 − r, t0 − s) χn (r) dr ds

0

  χ1 n (s)

= −ξn (w0 )

(either G = 0 or f = 0),

(3.44)

where ∫ t0 2 2 ξn (w0 ) = e−λn t0 w0,n − e−λn (t0 −s) R(s)w0,n ds 0  ∫ t0 ∫ t0 ∫ s −λ2n s −λ2n τ Hn (t0 − s)e ds − R(t0 − s) Hn (s − τ)e dτ ds w0,n . − 0

0

0

(3.45) A similar problem corresponds to the controllability to the target zero of the heat equation. In this case, R = 0 and Hn = 0 and so J = 0: ∫ t0 2 2 e−λn (t0 −s) [G n (s) − fn (s)] ds = −e−λn t0 w0,n . (3.46)

  0 χn (s)

We combine (3.45) and (3.46), and we see the following fact: let ∫t either χ(t) = G(t) − 0 J(t0 − r, t0 − t)G(r) dr ∫t or χ(t) = f (t) − 0 J(t0 − r, t0 − t) f (r) dr, and let u χ (x, t) be the solution of the associated heat equation with null initial condition and control5 χ(t). If equality (3.44) holds (under the action of the distributed or the boundary control), then u χ (x, t0 ) = ξw0 (x) =

+∞ 

φn (x)ξn (w0 ),

(3.47)

n=1

i.e. if w0 can be steered to the target zero then ξw0 is reachable from 0 for the memoryless heat equation. We prove the existence of w0 such that ξw0 is not reachable and hence such that the equality (3.44) does not hold. The construction of such vector w0 is suggested 5 Note that χ is a boundary control when G = 0, but it is a distributed control if f = 0. In spite of this, here we are going to use the same notation u χ in both the cases.

128

3 The Heat Equation with Memory and Its Controllability

by the following observation: the crucial difference between problem (3.46) and −λ2n t0 w } tends to zero exponentially fast, problem (3.44) is that 0,n the sequence {e and so e−λn t0 w0,n is the sequence of the Fourier coefficients of an element of 2

dom Ak for every k, while the sequence {ξn (w0 )} in (3.45) does not decay so fast. In fact: Lemma 3.26. Let R(t0 )  0. There exists a natural number N0 such that for every {cn } ∈ l 2 , there exists w0 such that the sequence {ξn (w0 )} given by (3.45) satisfies ξn (w0 ) =

cn λn2

∀n ≥ N0 .

(3.48)

Proof. We use the fact that R(t), as N(t), is of class H 1 , and we integrate by parts the first integral on the right hand side of (3.45). We get −1 ξn (w0 ) = 2 R(t0 )w0,n λn    ∫ t0 2 2 2 1 + e−λn t0 + 2 e−λn t0 R(0) + e−λn (t0 −s) R (s) ds w0,n λn 0 ∫ t0  ∫ t0 ∫ s 2 −λ2n s − Hn (t0 − s)e ds − R(t0 − s) Hn (s − τ)e−λn τ dτ ds w0,n . 0

0

0

We use the inequality (c) in (3.31), and we see that the bracket in the second line is 2 dominated by M/λn3 . The same holds for the bracket at the first line. In fact, e−λn t0 decays exponentially fast, while the integral in parenthesis is estimated as follows: ∫   

t0

−λ2n (t0 −s)

e

0

 ∫  R (s) ds ≤ 

t0

0

e−2λn s ds 2

 1/2

R  L 2 (0,t0 ) = M

1 . λn

So, there exists a bounded sequence {Mn } such that problem (3.48) takes the form ξn (w0 ) = R(t0 )

1 Mn w0,n + 3 w0,n 2 λn λn

{Mn } bounded: |Mn | < M .

(3.49)

2 2 In order to$ prove % the lemma, we prove the existence, for every {cn } ∈ l , of an l sequence w0,n such that the equalities (3.48) hold at least for n > N0 , and we prove that the number N0 does not depend on {cn }. Using (3.49), problem (3.48) can be written as $ % $ % in l 2 (N0, +∞), (−I + M) w0,n = R(t0 )−1 cn

where l 2 (N0, +∞) is the Hilbert space of the square integrable sequences indexed by n ≥ N0 (N0 still to be identified). The operator I is the identity, while  ! Mn R−1 (t0 ) M{w0,n } = w0,n , λn

3.5 Memory Kernels of Class H 1 : Controllability via Semigroups

129

so that (we recall that {λn } is an increasing sequence)   M{w0,n } 2

l (N0,+∞)

& ≤ M |R(t0 )| −1

= |R(t0 )|

 1/2 +∞  1 2 2  M w    λk2 k 0,k  k=N

−1 

+∞  1 |w0,k | 2 2 λ k k=N

' 1/2

0

≤ M |R(t0 )| −1

0

 1  {w0,k }l2 (N0,+∞) . λ N0

The number M is in (3.49) and does not depend on N0 . We recall that |λn | → +∞, and we see that if N0 is sufficiently large, then M is a contraction in l 2 (N0, +∞). Solvability of (3.48) in l 2 (N0, +∞) follows from the Banach fixed point theorem (see Theorem 2.2).   It follows that, when R(t0 )  0, the sequence {ξn } in (3.44) is indeed the sequence of the Fourier coefficients of a function ξ ∈ dom A but not too regular. In fact, ξ  dom (−A)1+ε if ε > 0. For example, ξ  H 3 (Ω) ∩ dom A (unless {w0,n } decays fast enough). Now we complete the proof of Theorem 3.25. We consider first the case of boundary controls, hence G = 0. We fix any ball B ⊆ Ω at positive distance from ∂Ω. Let c(x) ∈ dom A with support in B be such that c(x)  H 3 (B ). We expand in series of {ϕn }: c(x) =

+∞ 

(cn /λn2 )ϕn .

(3.50)

n=1

 We use Lemma 3.26. Let w0 (x) = +∞ n=1 wn ϕn be any initial condition such that (3.48) holds when {cn } is the sequence in (3.50). Note that noting we are asserting on the finite sequence {ξn (w0 )}n 0 when studying controllability), but the following fact is far more important: the assumption that N(t) is of class H 1 (0, T) used in Sect. 3.3 and in the study of controllability with the methods presented in Sect. 3.5 excludes unbounded kernels since H 1 (0, T) kernels are represented by functions which are continuous on [0, T]. Instead, as we shall see in Remark 3.28, frequency domain methods do not exclude certain classes of unbounded kernels. In particular, unbounded kernels which can be studied with the methods we are going to develop in the rest of the chapter include N(t) = 1/t σ , any σ ∈ (0, 1). These kernels correspond to system with fractional integration, a class of systems that are encountered in numerous applications.

The companion paper [40] proves a similar result, for a general kernel N (t), for the system studied in Chap. 4.

6

3.6 Frequency Domain Methods for Systems with Memory

131

Assumptions on the Memory Kernel N(t) and Notations When Using Frequency Domain Methods We recall the sector Σθ0 in Theorem 3.2. We assume: 1. α ≥ 0 and the kernel N(t) is real valued and Laplace transformable according ∫to the definition in Sect. 2.3.2, with ω = 0. Hence, the integral +∞ ˆ N(λ) = 0 e−λt N(t) dt converges for e λ > 0. ˆ 2. there exists θˆ ∈ $(0, π/2) such that N(λ) admits a%bounded holomorphic extension to Σˆ = λ : λ  0 and |arg λ| < θˆ + π/2 , and we assume ˆ lim N(λ) =0.

|λ|→+∞

(3.52)

λ∈Σˆ

ˆ we can have θˆ < θ 0 , i.e. Σˆ ⊆ Σθ0 . By reducing the value of θ, ˆ then J(λ)  0, where 3. if λ ∈ Σ, ˆ J(λ) = α + N(λ) . ˆ then 4. if λ ∈ Σ, 5. we have

λ J(λ)

=

λ α+ Nˆ (λ)

(3.53)

∈ Σθ0 . lim

λ→0, λ∈Σˆ

λJ(λ) = 0 .

(3.54)

6. there exist M > 0, ω ≥ 0, and γ ∈ [0, 1) such that |J(λ)| ≥

M |λ + ω|γ

if λ ∈ Σˆ .

(3.55)

Finally, we introduce the path of integration in Fig. 3.1 (left), defined as ˆ and a (small) number ε > 0. The parametrization is follows: we fix θ˜ ∈ (0, θ) ) * λ = s − sin θ˜ ± i cos θ˜ , s ≥ ε λ = εeiτ − θ˜ ≤ τ ≤ θ˜ . (3.56) G ε is composed of two straight lines G± in the left half plane and of an arc of circumference which intersects the right half plane.

Remark 3.28. We note: • the assumptions are restrictive, and in certain applications the properties do not hold on the entire sector Σˆ but on Σˆ ∩ {λ : |λ| > r > 0}. The proofs can be adapted to this case. We are assuming r = 0 for the sake of simplicity. • physical considerations impose strong properties to N(t), which automatically imply some of the stated conditions. • the sector Σˆ need not be the largest sector in which the assumptions hold.

132

3 The Heat Equation with Memory and Its Controllability G

G

+

+

y

y

x

x

G

−

G

−

Fig. 3.1 The paths of integration G ε (left). The paths G ε and, dotted, Gt ε (right)

• the assumptions can be satisfied even if α = 0 and N(t) is singular. For example, 1 with σ ∈ (0, 1) since in this case they are satisfied when α = 0 and N(t) = t 1−σ 1 ˆ J(λ) = N(λ) = Γ(σ) σ λ

• •

• •

(Γ is the Euler Gamma function) .

Kernels of this type, which correspond to “fractional integration,” are often encountered in applications (see for example [2, 8, 23]). Assumption 6 follows from the previous ones if α > 0 (in this case we have γ = 0). in spite of the previous observation, it is important to note that Assumption 4 is not satisfied when α = 0 and N(t) ∈ H 1 with N(0) > 0 since in this case ˆ λ/ N(λ) ∼ λ2 /N(0) for large λ. In this case the system has deeply different properties, as seen in Chap. 4. ˆ ˆ In the Laplace transform N(λ) might have branch points, which belong to C \ Σ. ˆ this case also the branch cuts are chosen in C \ Σ. Assumption (3.52) implies ˆ J(λ) bounded on  {λ  ∈ Σ , |λ| > r > 0} ∀r > 0  λ  lim |λ|→+∞, λ∈Σˆ  J(λ)  = +∞ .

(3.57)

• by combining the properties of A and of N(t) and using the inequalities (3.4) ˆ and (3.5), we get the following inequalities in Σ: λ 1 M ( I − A)−1 ≤ , |J(λ)| J(λ) |λ + ωJ(λ)| M (λI − J(λ)A)−1 D ≤ . 1−σ 0 |J(λ)| |λ − ω0 J(λ)| σ0 (λI − J(λ)A)−1 =

(3.58)

We stress that Assumption 4 is crucial for these inequalities to hold. We shall meet functions that are holomorphic in a “small” sector of the right half plane, centered on the real axis. The angle of the sector is not crucial, but in fact it is any angle less than the angle θ˜ in (3.56). In order to prove that these functions are

3.6 Frequency Domain Methods for Systems with Memory

133

holomorphic, we need an estimate of |eλz | when z belongs to the sector and λ ∈ G± . ˜ and any ε˜ ∈ (0, θ). We give an estimate in To find this estimate, we fix any θ ∈ (0, θ) the sector Σ = {λ : λ  0 and |arg λ| < θ − ε} ˜

0 < ε˜ < θ < θ˜ .

(3.59)

Any z ∈ Σ is z = eic with |c| < θ − ε. ˜ Of course,  = (z) and c = c(z). In order to give an estimate for |eλz | when λ ∈ G± and z ∈ Σ, we note if λ ∈ G+ , then e λz = −s  sin(θ˜ + c) . if λ ∈ G− , then e λz = −s  sin(θ˜ − c) . The important fact is that θ˜ ± c ∈ (ε, ˜ π − ε) ˜ so that sin(θ˜ ± c) > sin ε. ˜ Hence, in λz both the cases, e decays exponentially fast when |λ| → +∞ and λ ∈ G± : |eλz | ≤ e−s sin ε˜ .

(3.60)

Using (3.60), we easily get the following lemma: Lemma 3.29. Let Σ be the sector in (3.59), and let z0 ∈ Σ. Let B(z0, r) = {z : |z − z0 | < r }

r small so that

cl B(z0, r) ⊆ Σ .

There exists a > 0, which depends on z0 and r, such that z ∈ B(z0, r) =⇒ |eλz | ≤ e−as

∀λ ∈ G± .

(3.61)

The proofs of Lemma 3.30 below and of item 1 of Lemma 3.31 use standard devices in the analysis of C-valued holomorphic functions. The properties in Sect. 2.3.1 justify the use of these same methods for holomorphic functions in Banach spaces. Here we give the details of these standard proofs. Later on, similar arguments will be repeatedly used, without giving every detail. Lemma 3.30. The path of integration G ε is that in (3.56) and represented in Fig. 3.1 (left). If J(λ) is constant, i.e. if N(t) = 0, then for every z ∈ Σ, we have ∫ 1 dλ = 0 . ezλ J(λ) Gε ˆ Proof. If J(λ) is constant, J(λ) = J0 , then N(λ) = 0. Assumption 3 implies that α = J0  0. So, the integral exists for every z ∈ Σ thanks to the fast decay of the exponential, see (3.60). The definition of the improper integral is ∫ ∫ 1 zλ 1 e dλ = lim ezλ dλ . N →+∞ J J 0 0 Gε G ε ∩{λ, |λ| 0, we denote G ε, N the path composed by G ε ∩ {λ , |λ| < N }, closed by the arc CN of the circle of radius N in e λ < 0, as in Fig. 3.2 (left).

134

3 The Heat Equation with Memory and Its Controllability

The Cauchy integral theorem gives ∫ ∫ zλ 1 0= e dλ = J0 G ε, N G ε ∩{λ,

|λ| 0 and N(t) ≡ 0. Proof. We prove that Ψ(z) is holomorphic in Σ ∩ {z , |z| > ε}. It will turn out that the function is holomorphic in the angle |arg z| < θ˜ since the arguments below can ˜ every ε˜ ∈ (0, θ), and every radius ε of the circular arc be repeated for every θ < θ, of the integration path. We fix any N > 0, and we note that ∫ 1 dλ, Ψ(z) = lim ΨN (z) , ΨN (z) = eλz N →+∞ J(λ) G ε, N where G ε, N is as in Lemma 3.30 (see Fig. 3.2, left). It is clear that every ΨN (z) is a holomorphic function. We fix any z0 = 0 eic0 ∈ Σ, and we prove uniform convergence of the sequence {ΨN (z)} in the disk B(z0, r) with center z0 and such that cl B(z0, r) ⊆ {z , |z| > ε}∩Σ. Clearly, ∫ 1 dλ . eλz Ψ(z) − ΨN (z) = J(λ) G ε , |λ|>N Note, from the inequality (3.55) in Assumption 6, 1 |λ + ω| < . |J(λ)| M We combine this estimate with (3.61), and we see convergence to zero of the difference |Ψ(z) − ΨN (z)|, uniform on the ball B(z0, r). Hence, Ψ(z) is holomorphic in B(z0, r) thanks to the Weierstrass convergence theorem (i.e. Theorem 2.13 and [19, p. 228]). The fact that Ψ(z) does not depend on ε provided that ε < |z| follows from the Cauchy integral theorem. In fact, let ε  < ε < |z|, and let G ε and G ε be the corresponding paths. Then ∫ ∫ 1 1 dλ − dλ = 0 eλz eλz J(λ) J(λ) Gε Gε since this difference is the integral on a path which does not enclose singularities. We prove item 2. If J(λ) is constant, then the integral is zero, see Lemma 3.30. Conversely, let Ψ(z) = 0. We prove that J(λ) is a constant, and then we see that the constant is positive. As in [49, p. 59], we introduce the function ∫ 1 1 1 Φ(z) = dλ . eλz 2 2πi Gε λ J(λ) Thanks to the exponential decay of the integrand, Φ(z) is holomorphic and the derivatives can be exchanged with the integral. Hence,

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3 The Heat Equation with Memory and Its Controllability

1 2πi

Φ(z) =

∫

eλz

Gε

1 dλ = 0 (equality to 0 is the assumption) . J(λ)

It follows that Φ(z) is a polynomial of the first degree at most, Φ(z) = Φ0 + Φ1 z . In particular this equality holds when z = t > 0: ∫ 1 1 1 Φ(t) = dλ = Φ0 + Φ1 t . eλt 2 2πi Gε λ J(λ) We fix any c0 > 0, and we note that for every fixed t ≥ 0 , ⎧ ⎪ ⎨ λ = c + is with ⎪ c ∈ (0, c0 ] ⎪ ⎪ s ∈ R, ⎩

if

then

⎧ |e(c+is)t | < M ⎪ ⎪ ⎨ ⎪   ⎪   ⎪ ⎪  (c+is)21J(c+is)  ≤ ⎩

M s 2−γ

.

Note that 2 − γ > 1. Hence, the following integral converges: 1 2πi

∫

eλt

rc

1 λ2 J(λ)

dλ ,

rc : λ = (c + is)

and furthermore we have ∫ ∫ 1 1 1 1 λt 1 dλ = dλ , e 2 eλt 2 2πi Gε 2πi rc λ J(λ) λ J(λ)

s ∈ R (c > 0 fixed),

rc : λ = (c + is)

s∈R.

This equality is seen by using the ideas we illustrated in Lemma 3.30: we cut the lines of integration at distance R from the origin, and we close the path of integration with the two arcs of radius R (see Fig. 3.2, right). The integral on the resulting closed path is zero. We pass to the limit for R → +∞. The lengths of the arcs increase of the order R, while the integrand function tends to zero of higher order. Hence, the contributions of the arcs to the integral on the closed path tend to zero, and the previous equality follows (take into account that when computing the integral on the closed path, the part of the closed path contained in G ε is described downward). Comparing with the formula (2.24) for the inverse Laplace transform, we see that ˆ Φ(λ) =

Φ0 Φ1 1 + 2 = λ λ2 J(λ) λ

so that

λ = Φ0 λ2 + Φ1 λ . J(λ)

λ We recall Assumption 4, i.e. the fact that J(λ) transforms Σˆ inside the sector Σθ0 and this is not true if Φ0  0. So, it must be Φ0 = 0 and, being also J(λ)  0,

ˆ J(λ) = α + N(λ) =

1 = const. Φ1

and so N(t) = 0 ,

J(λ) ≡ α ≥ 0 .

3.6 Frequency Domain Methods for Systems with Memory

137

Finally, we invoke Assumption 3 and we see that J(λ) ≡ α > 0 (strict inequality) .

 

Remark 3.32. The properties of the holomorphic functions imply that if Ψ(t) = 0 on a segment of the real axis, then it is zero on its domain.

3.6.1 Well Posedness via Laplace Transformation In order to define the solutions of (3.51), we formally compute the Laplace transformations of both the sides. If a Laplace transformable solution w exists, it should be ˆ given by (we recall J(λ) = α + N(λ))  ) * ˆ − ADg(λ) ˆ w(λ) ˆ = (λ − J(λ)A)−1 w0 + h(λ) (3.63) ˆ g(λ) ˆ = α fˆ(λ) + N(λ) fˆ(λ) = J(λ) fˆ(λ) . We are going to prove that the function w(λ) ˆ is the Laplace transform of a certain function w(t) in a suitable space. This function w(t) is by definition the (mild) solution of problem (3.51). The computations are modeled on those in [14, 49] and are similar to those used in the study of holomorphic semigroups, when J(λ) = 1, see for example [48, Theorem 7.7]. An important difference, which has consequences on controllability, is Theorem 3.38. Thanks to the linearity of the problem, we study separately the contributions of the initial condition w0 , of h, and of f . For later use, we note the following inequalities, which follow from (3.58) and (3.60):    λt ⎧ −1  1 −st sin θ˜ ⎪ if λ ∈ G± , ⎨ e (λI − J(λ)A)  ≤ M s e ⎪   (3.64) ⎪ ⎪ eλt (λI − J(λ)A)−1  ≤ M ε1 etε on the circular arc . ⎩ Dependence on the Initial Condition: The Case h = 0, f = 0, and w0  0 The idea is to recover a function w(t) from the Laplace transformation w(λ) ˆ in (3.63) with h = 0, g = 0. Unfortunately, the integral (2.24) of the standard formula for the inverse Laplace transform does not converge since w(λ) ˆ does not decay fast enough for |λ| → +∞ on vertical lines. So, we must rely on a different formula. We get a convergent integral if the path of integration, instead of a vertical line, is the path of integration G ε contained in Σˆ described in (3.56) and represented in Fig. 3.1 (left). We let t > 0, and we consider the operator E(t) ∫ 1 E(t) = eλt (λI − J(λ)A)−1 dλ 2πi Gε

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3 The Heat Equation with Memory and Its Controllability

1 = lim R→+∞ 2πi

∫

eλt (λI − J(λ)A)−1 dλ .

(3.65)

G ε, R

The integral on G ε,R exists since the integrand is a continuous function of λ and the limit exists, in the uniform operator topology, thanks to an argument similar to that in Lemmas 3.30 and 3.31. In particular, as in Lemma 3.31, we see that E(t) does not depend on ε. So, E(t) ∈ L(H) for every t > 0 and E(t) is bounded for t ≥ t0 , any t0 > 0. We prove the following alternative formula for E(t): Lemma 3.33. Let E(t) be the operator in (3.65). We have E(t) =

1 2πi

∫ Gε

  −1 1 ζ e (ζ/t)I − J (ζ/t) A dζ . t

(3.66)

The operator E(t) does not depend on ε > 0 in the sense that neither the integral in (3.65) nor the integral in (3.66) depends on the choice of ε > 0 provided that ε < t. Proof. We transform the variable of the integral (3.65): λt = ζ . Then, 1 E(t) = 2πi

∫ Gt ε

  −1 1 ζ e (ζ/t)I − J (ζ/t) A dζ . t

(3.67)

The path of integration Gtε (dotted path in Fig. 3.1, right) is composed of the two half lines with the same angles ±(θ˜ + π/2) as those of G ε , but the arc of circumference joining them has radius tε. Hence, G ε and Gtε differ by a closed curve in the region in which the integrand is a holomorphic function. This implies that the two integrals in (3.66) and in (3.67) coincide, and so (3.67) converges to the same operator as the integral (3.65). In particular, the common value of the integrals (3.65), (3.66), and (3.67) does not depend on the choice of ε ∈ (0, t).   Analogous to the inequalities (3.64), we have   1 ζ ˜ ⎧ −1  ⎪ e − J A) ((ζ/t)I (ζ/t)   ≤ M ε1 e−s sin θ if ζ ∈ G± , ⎨ t ⎪   ⎪ ⎪  1t eζ ((ζ/t)I − J (ζ/t) A)−1  ≤ M ε1 eε on the arc . ⎩

(3.68)

Theorem 3.34. Let t > 0, and let E(t) ∈ L(H) be the operator in (3.65). Let w(t; w0 ) = E(t)w0 . We have: 1. there exists a constant M such that for every t > 0, we have E(t) L(H) < M

∀t > 0 .

(3.69)

3.6 Frequency Domain Methods for Systems with Memory

139

2. the function t → w(t; w0 ) is bounded on (0, +∞) and w(t; w0 ) ≤ M w0

∀t > 0 .

(3.70)

3. the function t → E(t) is a continuous L(H)-valued function of t > 0 . 4. for every w0 ∈ H, we have lim w(t; w0 ) = w0 .

t→0+

(3.71)

5. let E(0)w0 = w0 . for every T > 0, the transformation  w0 → E(t)w0 =

w(t; w0 ) if t > 0 if t = 0 w0

(3.72)

is linear and continuous from H to C([0, T]; H). 6. the Laplace transform of the function w(t; w0 ) is w(λ; ˆ w0 ) = (λI − J(λ)A)−1 w0 . Proof. We noted already that t → E(t) L(H) is bounded on [t0, +∞) (any t0 > 0), while boundedness on (0, t0 ] follows from the representation (3.66) of E(t) and the inequalities (3.68). Hence, Statement 1 holds and Statement 2 is an immediate consequence. Now we see time continuity for t > 0 still using the representation (3.66) of E(t). First we consider the integral on the closed path G ε,R in Fig. 3.2, left. For every fixed R, the transformation t →

1 2πi

∫ G ε, R

  −1 1 ζ e (ζ/t)I − J (ζ/t) A dζ t

is continuous. In order to see continuity of E(t) for t > 0, we fix any interval [a, b] with a > 0, and we represent E(t) by using any ε ∈ (0, a). Continuity follows since inequalities (3.68) show that the limit for R → +∞ exists uniformly on [a, b]. Now we prove the property in Statement 4. Thanks to the boundedness in item 2, it is sufficient to prove that the limit in (3.71) exists and it is equal to w0 for every w0 in a dense subset of H. We use w0 ∈ dom A, i.e. w0 = A−1 y, so that 1 w(t; A y) = E(t)A y = 2πi −1

−1

∫ Gε

 −1  ζ 1 ζ e I−A A−1 y dζ . t J (ζ/t) t J (ζ/t)

We invoke the first formula of the resolvent, i.e. the equality: (λI − A)−1 (μI − A)−1 =

* 1 ) (μI − A)−1 − (λI − A)−1 λ−μ

(3.73)

140

3 The Heat Equation with Memory and Its Controllability

so that 

ζ I−A t J (ζ/t)

 −1

&  −1 '  ζ t J (ζ/t) −1 I−A A y= , A + ζ t J (ζ/t) −1

  and the integral of w t; A−1 y splits into the sum of two integrals: 1 w(t; A y) = 2πi −1

∫ Gε

1 1 e A−1 y dζ + ζ 2πi ζ

∫ Gε

 −1  ζ 1 I−A e y dζ . ζ t J (ζ/t) ζ

The integrand eζ (1/ζ) has the simple pole ζ = 0, with residue equal to 1 inside the region bounded by G ε . So, the first integral is8 A−1 y: ∫ 1 1 eζ A−1 y dζ = A−1 y . 2πi Gε ζ The second integral splits into the sum of three integrals: the integrals on G+ and G− and the integral on the circular arc of radius ε. These integrals tend to zero for t → 0+ as it is seen from the inequalities (3.68). In fact, on G± and on the circular arc, the integrands are dominated by, respectively,     1 −s sin θ˜ 1 ε M e t and M e t. ε ε The property in 5 combines those in 3 and 4. Finally, we prove Statement 6, which justifies taking w(t; w0 ) as the mild solution of Eq. (3.51) when h = 0 and f = 0 (see Definition 3.36 below). We use the representation (3.65) of E(t). We denote z a second complex variable, and we compute w(z; ˆ w0 ) when e z > 0:   ∫ +∞ ∫ 1 w(z; ˆ w0 ) = e−zt eλt (λI − J(λ)A)−1 w0 dλ dt 2πi Gε 0  ∫ ∫ +∞ 1 = e−(z−λ)t dt (λI − J(λ)A)−1 w0 dλ 2πi Gε 0 ∫ 1 −1 = (λI − J(λ)A)−1 w0 dλ = 2πi Gε λ − z ∫ 1 −1 = (λI − J(λ)A)−1 w0 dλ (rc : λ = (c + is) 0 < c < e z, s ∈ R) 2πi rc λ − z = (zI − J(z)A)−1 w0 .

As seen with the usual trick, integrate on a bounded curve, composed of G ε ∩ {λ |λ| < R } closed on the left side by an arc of circumference of radius R. Then pass to the limit for R → +∞, and note that the contribution of the arc to the integral converges to 0.

8

3.6 Frequency Domain Methods for Systems with Memory

141

The previous computation has to be clarified: in order to compute the Laplace transform for e z > 0, we used boundedness of w(t; w0 ) for t > 0. The integrals can be exchanged thanks to the fast decay of the integrands. More important is the following observation, which explains the integral on the vertical line rc . We cannot use the Cauchy integral formula applied to the path in Fig. 3.2 (left) since this path encloses eigenvalues of A. Rather, we use the same argument as in Lemma 3.31: the integrand at the third line decays of order larger than 1 (with respect to 1/λ for |λ| → +∞), and it gives the same value when integrated along G ε and on the vertical line rc = c + is as in Fig. 3.2 (right) with 0 < c < e z. This observation proves the equality of the third and fourth lines. Finally, we compute the integral on rc by integrating on the closed circuit composed of the segment c + is with |s| ≤ R closed by an arc of circumference in the right half plane. We apply the Cauchy integral formula (we note that the segment is described upward, and so the path is described clockwise), and we compute the limit for R → +∞.   By definition, E(t) is the resolvent of Eq. (3.51). We study its regularity. Theorem 3.35. The operator E(t) has the following properties: 1. it transforms dom A into itself and E(t)A−1 y = A−1 E(t)y for every y ∈ H. 2. if w0 = A−1 y ∈ dom A, then w(t; w0 ) = E(t)w0 is of class C 1 , w (t; w0 ) is bounded for t ≥ 0, and the equality (3.51) (with h = 0, f = 0) holds for every t > 0. Proof. The first property follows since A−1 is bounded and can be exchanged with the integral. We replace w0 = A−1 y in (3.65), and we get E(t)A−1 y = A−1 E(t)y ∈ dom A. In order to prove differentiability, we show that the derivative with respect to the variable t can be exchanged with the integral. We use dominated convergence theorem thanks to the fact that the norm of the incremental quotient is dominated by an integrable function. In fact, the incremental quotient is * eλh − 1 ) λt λe (λI − J(λ)A)−1 A−1 y , λh and the square bracket is integrable on G ε , while the factor (eλh −1)/(λh) is bounded on G ε , as it is seen by examining the real and the imaginary parts. Dominated convergence theorem implies that the derivative of w(t; A−1 y) exists, equal to ∫ 1 λeλt (λI − J(λ)A)−1 A−1 y dλ ∈ C([0, T]; H) (3.74) w (t; A−1 y) = 2πi Gε (continuity at t = 0 is seen by using the first formula of the resolvent as in the proof of Statement 4 of Theorem 3.34). Furthermore, the previous equality shows boundedness of w (t) on [0, +∞), and hence its Laplace transform converges for e λ > 0. Now we prove that the equality (3.51) holds for every t > 0. We recall that two continuous functions are equal everywhere when they have the same Laplace

142

3 The Heat Equation with Memory and Its Controllability

transform. So, we prove the equality of the Laplace transform of the left and the right hand side of (3.51) when w0 = A−1 y. When w0 = A−1 y, the function t → (αAEw0 + N ∗ AEw0 )(t) is continuous and bounded on [0, +∞) from the first statement. The fact that w (t) ∈ C 1 and w(t), w (t) both bounded on [0, +∞) implies that the Laplace transform of both the sides of (3.51) exists in e λ > 0. The Laplace transform of the right side is J(λ)Aw(λ), ˆ while that of the left side is λ w(λ) ˆ − w0 , and we know w(λ) ˆ = (λI − J(λ)A)−1 w0 . We replace this expression in L(w )(λ). We have L(w )(λ) = λ w(λ) ˆ − w0 = (λI − J(λ)A)−1 w0 − w0 = J(λ)Aw(λ) ˆ  

as wanted. These results justify the following definition analogous to Definition 3.17:

Definition 3.36. When h = 0 and f = 0, a classical solution with initial condition w0 is a function w(t; w0 ) ∈ C 1 ([0, T]; H) ∩ C([0, T]; dom A) for every T > 0 such that the equality (3.51) (with h = 0 and f = 0) holds for t ≥ 0. ˆ w0 ) = If w0 ∈ H, a mild solution is a continuous function w(t; w0 ) such that w(λ; (λI − J(λ)A)−1 w0 . The previous results prove that any classical solution is a mild solution too and that the mild solution with initial condition w0 is unique since two continuous functions with the same Laplace transform coincide. Hence, the mild solution is w(t; w0 ) = E(t)w0 . Theorems 3.34 and 3.35 state that every mild solution is the limit in C([0, T]; H) of classical solutions. This observation justifies the definition of the mild solutions. The properties of the resolvent that we stated in Theorem 3.35, sufficient to justify the definition of mild solutions, can be strengthened, and we shall see the interest of this fact in the study of controllability. First we extend E(t) to a sector of the complex plane. The extension is denoted E(z), while we reserve λ for the variable of integration. Theorem 3.37. The following properties hold: 1. the resolvent E(t) is the restriction to the axis t > 0 of an operator valued function E(z), which is holomorphic in a sector Σ1 = {z : e z > 0 ,

|arg z| < θ 1 }

θ1 > 0 .

(3.75)

2. for every w0 ∈ H and every fixed z ∈ Σ1 , we have E(z)w0 ∈ dom A, and AE(z), originally defined on dom A, admits a continuous extension to H. This extension, still denoted AE(z), is holomorphic on Σ1 . 3. the function AE(t) is the restriction to the axis t > 0 of a function, which is holomorphic on Σ1 (and so, in particular, im E(t) ⊆ dom A for every t > 0).

3.6 Frequency Domain Methods for Systems with Memory

143

Proof. As expected, we define (G ε, N is the path in Fig. 3.2 (left)) ∫ 1 E(z) = eλz (λI − J(λ)A)−1 dλ = lim E N (z) , N →+∞ 2πi Gε ∫ 1 eλz (λI − J(λ)A)−1 dλ . E N (z) = 2πi Gε, N

(3.76)

The function E N (z) is holomorphic in e z > 0 since it is differentiable there: its derivative is ∫ λeλz (λI − J(λ)A)−1 dλ . G ε, N

We prove the existence of a sector Σ1 such that E N (z) → E(z) uniformly in any disk contained in Σ1 so that E(z) is holomorphic thanks to the Weierstrass theorem (Theorem 2.13). The proof is similar to the proof of the first statement of Lemma 3.31. We choose as the sector Σ1 any sector as in (3.59). We fix z0 ∈ Σ1 and a disk D(z0, r) of center z0 whose closure is contained in Σ1 . In particular, D(z0, r) is at positive distance from 0 and (notations as in (3.59)) z = eic ∈ D(z0, r) =⇒  > 0 > 0

and

|c| < θ − ε˜ < θ˜ − ε˜ .

In order to prove that the limit (3.76) is uniform on D(z0, r), we apply Lemma 3.29 to z ∈ D(z0, r). We see that λz −s 0 sin ε˜ , ⎧ ⎪ ⎨ |e | ≤ e ⎪   (3.77)   ⎪ e−s 0 sin ε˜ . ⎪ eλz (λI − J(λ)A)−1  ≤ M s ⎩ We use (3.77) in order to give an estimate for the difference E(z) − E N (z) : ∫   1  −1 λz e (λI − J(λ)A) dλ E(z) − E N (z) ≤  2π G+ ∩{λ : |λ|>N } ∫   1  −1 λz +  e (λI − J(λ)A) dλ . 2π G− ∩{λ : |λ|>N }

We prove that both these integrals tend to zero uniformly on D(z0, r). We consider the integral on G+ (the integral on G− is similar): ∫  ∫ +∞   1 −1 λz   e (λI − J(λ)A) dλ ≤ M e−s 0 sin ε˜ ds → 0  s G+ ∩{λ : |λ|>N } N (for N → +∞). This proves Statement 1 . We prove Statement 2. So, let z ∈ Σ1 be fixed. We prove that AE(z), originally defined on dom A, admits a continuous extension to H. We see from (3.76) that

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3 The Heat Equation with Memory and Its Controllability

∫

2πiAE(z)w0 = eλz A (λI − J(λ)A)−1 w0 dλ Gε ∫ ∫ λ λz 1 w0 dλ + =− e ezλ (λI − J(λ)A)−1 w0 dλ . J(λ) J(λ) Gε Gε

(3.78)

Convergence of the first integral on the second line is seen from the first inequality in (3.77). The second integral converges since, from (3.55) and (3.77), we have     zλ λ M −s 0 sin ε˜ −1  e ≤ Ne−s 0 sin ε˜ |λ + ω|γ .  J(λ) (λI − J(λ)A)  ≤ |J(λ)| e This proves Statement 2. Finally, we prove that AE(z) is holomorphic. We use Statement 2 of Theorem 2.12: AE(z)w, v = E(z)w, Av is a holomorphic (complex valued) function on Σ1 for every w ∈ H and every v ∈ dom A (which is dense in H). Hence, AE(z) is holomorphic.   To interpret the following result, we recall that if N(t) = 0, then E(t) = e At and e At is a holomorphic semigroup, so that when t > 0 we have im e At ⊆ dom Ak for every k (see Theorem 2.55). The regularity of e At has to be contrasted with the following statement. Theorem 3.38. Let Σ1 be the sector in Theorem 3.37, and let σ > 0. Let w0  dom (−A)1+σ . If E(z)w0 ∈ dom (−A)1+σ for every z in a subset of Σ1 , which contains one of its accumulation points (in particular, in a segment of the real axis), then N(t) = 0 and E(t) = eα At is a holomorphic semigroup. Proof. Computations similar to those used in the proof of Statement 2 of Theorem 3.37 show that the second integral in the right hand side of (3.78) takes values in dom A, and so AE(z)w0 ∈ dom (−A)1+σ for every w0 if and only if the first integral is equal to zero. Lemma 3.31 shows that the integral is zero if and only if α > 0 and   N = 0, i.e. if E(t) = eα At is a holomorphic semigroup. Dependence on h: The Case f = 0, w0 = 0, and h  0 Now it is easy to study the solution when w0 = 0 and f = 0. In this case, ∫ t −1 ˆ w(λ; ˆ h) = (λI − J(λ)A) h(λ) so that w(t; h) = E(t − s)h(s) ds

(3.79)

0

as it is seen by using the theorem on the Laplace transform of a convolution. Note that if h is solely square integrable, then w need not be of class C 1 , i.e. it is not a classical solution. The following theorem shows its properties and justifies taking (3.79) as the mild solution in this case.

3.6 Frequency Domain Methods for Systems with Memory

145

Theorem 3.39. The function w(t; h) in (3.79) has the following properties: 1. the transformation h → w(·; h) is linear and continuous from L 2 (0, T; H) to C([0, T]; H) for every T > 0. 2. if h is of class C 1 ([0, T]; H) and takes values in dom A, then w(t; h) ∈ C 1 ([0, T]; H) and takes values in dom A. Moreover, equality (3.51) (with w0 = 0, f = 0) holds for every t > 0. 3. every h ∈ L 2 (0, T; H) is the L 2 (0, T; H)-limit of a sequence {hn } such that hn ∈ C([0, T]; dom A) ∩ C 1 ([0, T]; H), and so lim w(t; hn ) = w(t; h) in C([0, T]; H). Proof. The first statement follows from Young inequalities, according to which ∫ t w(t; h) = E(t − s)h(s) ds 0

is a continuous function of t, and for every t ∈ [0, T], we have w(t; h) H ≤ E L 2 (0,T ;L(H)) h L 2 (0,T ;H) . The proof of differentiability when h(t) ∈ C 1 is standard: ∫ t ∫  w(t; h) = E(s)h(t − s) ds so that w (t; h) = E(t)h(0) + 0

t

E(s)h (t − s) ds .

0

We prove that if it happens that h ∈ C 1 ([0, T]; H) and takes values in dom A, then w(t; h) solves (3.51) (with w0 = 0, f = 0). A proof that uses the Laplace transformation, similar to that of the second statement of Theorem 3.35, is possible. We present a different idea. We use the second statement in Theorem 3.35: when h(s) ∈ dom A (here s is fixed), the function y(τ) = E(τ)h(s) is differentiable and verifies ∫ τ d E(τ)h(s) = αAE(τ)h(s) + N(τ − r)AE(r)h(s) dr . dτ 0 So, when τ = t − s with t > s, we have d E(t − s)h(s) = αAE(t − s)h(s) + dt

∫

t−s

N(t − s − r)AE(r)h(s) dr .

0

It follows that9 w (t; h) = h(t) +

∫ 0

9

t



 d E(t − s)h(s) ds dt

Note that the operator A is unbounded but closed, since it generates a C0 semigroup. Furthermore, E(t −s)h(s) ∈ dom A and AE(t −s)h(s) is continuous, hence ∫integrable from the second statement ∫t t in Theorem 3.35. It follows from [30, Theorem 3.7.12] that 0 AE(t − s)h(s) ds = A 0 E(t − s)h(s) ds.

146

3 The Heat Equation with Memory and Its Controllability

! ∫ t−s N(t − s − r)A [E(r)h(s)] dr ds αA [E(t − s)h(s)] + 0 0 ∫ r  ∫ t N(t − r)A E(r − s)h(s) ds dr + h(t) = αAw(t) + 0 0 ∫ t N(t − r)Aw(r; h) dr + h(t) . = αAw(t) + ∫

= h(t) +

t



0

Finally, we prove the third statement, i.e. we prove the existence of an approximating sequence with the required properties for every h ∈ L 2 (0, T; H). This approximating sequence is constructed in two steps: • we approximate h(t) with a sequence {hn (t)} in C 1 ([0, T]; H), ∫ t hn (t) = hn (0) + hn (s) ds . 0

This is possible since C 1 functions are dense in L 2 (0, T; H). The sequence {hn }, being convergent, is bounded in L 2 (0, T; H). • we construct a sequence hn,k ∈ C([0, T] dom A) ∩ C 1 ([0, T]; H) such that lim hn − hn,k L 2 (0,T ;H) = 0 .

k→+∞

We use the following fact (see Statement 3 of Theorem 2.44 or [48, Theorem 2.4 p. 4]): ∫ 1/k ⎧ ⎪ ⎪ ⎨ hn,k (t) = k ⎪ e As hn (t) ds = k[e A/k − I]A−1 hn (t) ∈ C([0, T]; dom A) , 0 ⎪ ⎪ lim hn,k (t) = hn (t) in H for all t ≥ 0 . ⎪ ⎩ k→0 Note that e At is bounded on t ≥ 0, e At < M so that we have hn,k (t) H ≤ M hn (t) H

∀t ∈ [0, T]

(3.80)

and, for a different constant M, hn,k L 2 (0,T ;H) < M

since {hn } is bounded in L 2 (0, T; H) .

Furthermore, ∫ hn,k (t) = k ∫ =k

1/k

e As hn (t) ds

0 1/k

=k 0

1/k

t

e hn (0) ds + k

0

∫

∫

&∫

As

∫ e hn (0) ds + As

0

0 t

0

1/k

' e As hn (r) ds

dr

k[e A/k − I]A−1 hn (r) dr

3.6 Frequency Domain Methods for Systems with Memory

147

is a C 1 function. So, we constructed hn,k of class C 1 with values in dom A and such that – limk→+∞ [hn,k (t) − hn (t)] = 0 for every t ∈ [0, T]; 2 ∈ L 1 (0, T) for – inequality (3.80) gives hn,k (t) − hn (t) 2 ≤ (1 + M)2 hn (t) H every fixed n. Dominated convergence theorem implies lim hn,k = hn in L 2 (0, T; H)

k→0

for every n .

For every r ∈ N, we choose hn(r) such that hn(r) − h L 2 (0,T ;H) < 1/(2r), and then we choose k(r) such that hn(r),k(r) − hn(r) L 2 (0,T ;H) 0, real on the real axis and with positive real part in the half plane e λ > 0 is called a positive real (transfer) function. This property will not be used in the proof of stability but it is important in systems theory and identification problems (see Sect. 6.1.1). We give two criteria for a function to be positive real which are easily derived along the lines we are using in this section. We recall the definitions of κ(t) and N(t) in (5.7): ∫ t ∫ +∞ M(s) ds , N(t) = 1 − M(s) ds = 1 − κ0 + μ(t) . κ(t) = t

0

Lemma 5.22. Let M(t) ≥ 0 be an integrable function such that also t M(t) is integrable on (0, +∞) and such that   ˆ κ0 = κ(0) ∈ [0, 1) , ω !m M(iω) ≤0. Then

e κ(λ) ˆ ≥ 0 if ˆ e N(λ) > 0 if

e λ ≥ 0 , e λ ≥ 0 and λ  0 .

(5.38)

ˆ If furthermore κ(λ) ˆ has an analytic extension to a disk |λ| < ε, then N(λ) has a simple pole λ = 0 and the residue is positive. Proof. The result is obvious if M(t) ≡ 0. We assume that M(t) is not identically zero so that κ0 > 0. First we prove that κ(t) is integrable, so that its Laplace transform κ(λ) ˆ is continuous in e λ ≥ 0. For every T > 0 we have

5.2 The Memory Kernel When the System Is Stable

241

∫ T ∫ +∞ κ(t) dt = M(s) ds dt 0 0 t ∫ s  ∫ ∫ T ∫ +∞ = M(s) 1 dt ds + M(s)

∫

T

0

∫

0 T

=

T

∫

+∞

sM(s) ds + T

0

∫

M(s) ds ≤

T

0 +∞

T

 1 dt ds

sM(s) ds < +∞ .

0

So, κ(t) is integrable and κ(λ) ˆ is continuous in e λ ≥ 0 (see statement 1 of Lemma 2.15). We compute κ(iω) ˆ and we prove e κ(iω) ˆ > 0 for every ω. If ω = 0 we have ∫ +∞ ∫ +∞ ∫ +∞ M(s) ds dt = sM(s) ds > 0 . κ(0) ˆ = 0

If ω  0 we have ∫ κ(iω) ˆ = ∫ =

0 +∞

0

+∞

t

e−iωt

∫

0

+∞

∫ M(s) ds dt =

t

∫

+∞

M(s) 0

s

 e−iωt dt ds

0

1 − e−iωs ds . M(s) iω

So, for λ = iω  0 we have ∫  1  1 +∞ ˆ !m M(iω) ≥0 M(t) sin ωt dt = − e κ(iω) = ω 0 ω as we wished to prove. Property 5.15 shows that e κ(λ) ˆ > 0 for e λ ≥ 0. ˆ The statement concerning N(λ) follows from the assumption 1 − κ0 > 0 and the equality 1 − κ0 ˆ .   N(λ) = κ(λ) ˆ + λ The next important result requires additional conditions on the memory kernel, not consequences of the standing assumptions of this chapter. Theorem 5.23. We assume that the function M(t) has the following properties: 1. 2. 3. 4.

it is of class H 2 (0, T) for every T > 0. M(t) ≥ 0 and M ∈ L 1 (0, +∞). M (t) ≤ 0 and M  ∈ L 1 (0, +∞). M (t) ≥ 0 a.e. t > 0.

Then we have ˆ e M(λ) ≥0

if

e λ ≥ 0 .

ˆ Proof. The function M(λ) is continuous in e λ ≥ 0 since M(t) is integrable. So, it is sufficient that we prove the inequality on the imaginary axis. The conclusion then follows from Property 5.15.

242

5 The Stability of the Wave Equation with Persistent Memory

ˆ If ω = 0 then M(0) = κ(0) ≥ 0. If ω  0 we compute as follows: ∫ +∞ ∫ 1 +∞ ˆ e M(iω) = M(s) cos ωs ds = M(s) d sin ωs ω 0 0 ∫ T ∫ 1 +∞  1 =− M (s) sin ωs ds = 2 lim M (s) d cos ωs ω 0 ω T →+∞ 0   ∫ T 1 = 2 lim M (T) cos ωT − M (0) − M (s) cos ωs ds ω T →+∞ 0   ∫ T 1 = 2 lim M (s) (1 − cos ωs) ds ≥ 0 ω T →+∞ 0 since integrability and monotonicity of M (t) imply that limT →+∞ M (T) = 0. So the required inequality holds on the imaginary axis and Property 5.15 implies that it holds in e λ > 0 too.   Note also that under the conditions of this theorem the integral of M (t) converges. ˆ Remark 5.24. In spite of the fact that the sign of e M(λ) is not used in the study of stability, the condition that certain kernel is positive real is imposed by the second principle of thermodynamics. See, for example, [20, 22].

5.3 L 2 -Stability via Laplace Transform and Frequency Domain Techniques Time domain and frequency domain properties are connected by the Paley–Wiener Theorem 2.19 and so the natural property to be studied by frequency domain methods, i.e. by studying the properties of the Laplace transform of the solutions, is L 2 -stability. In this section we show that a result concerning L 2 -stability can indeed be achieved in this way. First we consider the case w˜ = 0, hence h = 0. The effect of the history for t < 0 is then studied in Sect. 5.3.2. So, we begin our study with the system ∫ t w  = Aw − M(t − s)Aw(s) ds , w(0) = w0 ∈ dom A , w (0) = w1 ∈ H . 0

(5.39) The standing assumptions of this chapter are in force and the Hilbert space H is complexified (since we use the Laplace transformation). The main result we prove is: Theorem 5.25. Let w˜ = 0 so that EV (0) is given by (5.12). Under the standing assumptions of this chapter, there exists C > 0 such that ∫ +∞   )  * w (t) 2 + Aw(t) 2 dt ≤ C w1 2 + Aw0 2 = CEV (0) . (5.40) 0

This result is the key to study L 2 -stability when w˜  0 belongs to the space described in Sect. 5.3.2.

5.3 L 2 -Stability via Laplace Transform and Frequency Domain Techniques

243

5.3.1 L 2 -Stability When w˜ = 0 The proof is in two steps: first we prove the inequality (5.40) for the classical solutions of Eq. (5.39). Then the inequality is extended by continuity to every mild solution. We compute the Laplace transform of both the sides of Eq. (5.39). The computation makes sense when w(t) is a classical solution and provided that e λ > 0. In ˆ fact, M(λ) exists for e λ > −ξ while the Laplace transform of Aw(t) and of w (t) exist in e λ > 0 since the energy EV (t) is (decreasing and) bounded as shown by the equation of the energy (5.10). The computation of the Laplace transform gives   ˆ λ2 I − J(λ)A w(λ) ˆ = λw0 + w1 , J(λ) = 1 − M(λ) . Hence  −1  w(λ) ˆ = λ2 I − J(λ)A [λw0 + w1 ] =

 −1  2 λ 1 I − A [λw0 + w1 ] . (5.41) J(λ) J(λ)

This equality makes sense in the set !  λ2  σ(A) . λ : e λ > 0 , J(λ)  0 and J(λ)

(5.42)

Paley–Wiener Theorem 2.19 shows that:3 Theorem 5.26. Theorem 5.25 holds if and only if Aw(λ) ˆ and L(w )(λ) both belong 2 to H (Π+ ; H) and there exists C > 0 such that   2 2 2 , Aw ˆ H 2 (Π ;H) ≤ C w1 H + Aw0 H +   (5.43) 2  2 2 2 . + Aw0 H L(w )(λ) H 2 (Π+ ;H) = λ wˆ − w0 H 2 (Π ;H) ≤ C w1 H +

ˆ The inequalities (5.43) are proved by relaying on the properties of M(λ) described in Sect. 5.2.1. The first observation is: ˆ Lemma 5.27. We recall J(λ) = 1 − M(λ). We have: 1. The function 1/J(λ) is continuous and bounded in e λ ≥ 0, holomorphic in e λ > 0. 2. Let e λ ≥ 0. The number λ2 /J(λ) is not an eigenvalue of A since, as we are going to see, it is not a negative number. So: a. the function (λ2 I − J(λ)A)−1 is holomorphic in e λ > 0 and continuous in e λ ≥ 0; 3

The space H 2 (Π+ ; H) in the statement of the theorem is the Hardy space described in Sect. 2.3.3.

244

5 The Stability of the Wave Equation with Persistent Memory

b. for every −λn2 ∈ σ(A), the functions 1/(λ2 + λn2 J(λ)) are holomorphic in e λ > 0 and continuous in e λ ≥ 0. ˆ Proof. Statement 1 holds since in the half plane e λ ≥ 0 we have | M(λ)| ≤ κ0 < 1 ˆ and lim |λ|→+∞ M(λ) = 0. In order to prove the statement 2, we consider three cases: ∫ +∞ ˆ • the case λ = σ ≥ 0. In this case M(σ) = 0 e−σt M(s) ds ∈ (0, 1). So, λ2 /J(λ) ≥ 0, in particular, it does not belong to σ(A) . ˆ • the case that e λ ≥ 0 and !m λ > 0. In this case !m M(λ) is negative and so   2      ˆ arg J(λ) = arg 1 − M(λ) ∈ (0, π/2) hence arg λ  < π J(λ)

so that

λ2 /J(λ)

is not negative, in particular,

λ2 J(λ)

 σ(A) .

• a similar argument holds in the quarter plane e λ ≥ 0 and !m λ < 0.

 

A consequence: Corollary 5.28. The function w(λ) ˆ admits a continuous extension to e λ ≥ 0. Now we prove Theorem 5.26. Thanks to the linearity of the problem, we can study separately the contributions of w1 and that of w0 and their contributions separately to the norms of Aw(t) and of w (t). So, the proof is in two steps, each one consisting of two substeps, plus a preliminary step which outlines the general idea.

Step 0: An Outline of the General Idea In this preliminary step we illustrate the idea of the proof of the following facts:4 2 −1 y ∈ H 2 (Π ; H) , 1) (λ +  I2 − J(λ)A) −1  2) (λ I − J(λ)A) y  H 2 (Π+ ;H) ≤ C y H .

We expand any y ∈ H in series of the eigenfunctions ϕn of A:

y=

+∞  n=1

yn ϕn

−1

so that (λ I − J(λ)A) y = 2

+∞  n=1

 ϕn

 1 yn λ2 + λn2 J(λ)

and If it would be possible to apply the inequality (μI − A)−1 ≤ C/(|μ | + 1), then the proof would be immediate. But, this inequality holds if | arg μ | < π − ε where ε is strictly positive while the inequality should be used with μ = λ2 /J(λ). Our assumptions imply that λ2 /J(λ) is not a negative number but they do not imply the existence of ε > 0 such that | arg(λ2 /J(λ))| < π − ε.

4

5.3 L 2 -Stability via Laplace Transform and Frequency Domain Techniques

 2  (λ I − J(λ)A)−1 y 2 2 H (Π+ ;H) In order to have

2 +∞     1   =  λ2 + λ2 J(λ) 

H 2 (Π+ )

n

n=1

 2  (λ I − J(λ)A)−1 y 2

H 2 (Π+ ;H)

245

|yn | 2 .

2 ≤ C 2 y H

we must prove   sup 

1 ∈ H 2 (Π+ ) and λ2 + λn2 J(λ)

n

2  1  < C < +∞ . 2 2 λ + λn J(λ)  H 2 (Π+ )

(5.44)

We recall from Lemma 5.27 that the function 1/(λ2 + λn2 J(λ)) is holomorphic in ˆ e λ > 0 and continuous in e λ ≥ 0 while, from Lemma 5.11, M(λ) is uniformly continuous in e λ ≥ 0 and lim J(λ) = 1 . e λ≥0

ˆ lim M(λ) =0, e λ≥0 |λ|→+∞

|λ|→+∞

So, there exists cn > 0 such that 2    1 cn    λ2 + λ2 J(λ)  ≤ |λ| 2 + 1 in e λ > 0 hence n

λ2

1 ∈ H 2 (Π+ ) . + λn2 J(λ)

From (2.28) we see that the inequality (5.44) has the following explicit form: there exists C > 0 such that for every n we have  2   1    λ2 + λ2 J(λ)  n

H 2 (Π+ )

∫ =

+∞

−∞

|−

ω2

1 dω < C . + λn2 J(iω)| 2

(5.45)

Remark 5.29. Thanks to the fact that each integral in (5.45) converges, it is sufficient that we prove the inequality for large values of n, n > N0 . We manipulate the integrands as in [29] (see also [1]), using the results in Sect. 5.2.1.2. In particular, we use boundedness of 1/|J(iω)| in e λ ≥ 0: |−

ω2

1 1 1 1 = ≤M 2 . 2 2 2 2 2 2 2 |J(iω)| + λn J(iω)| |λn − ω /J(iω)| |λn − ω /J(iω)| 2

We recall from (5.35) and (5.37) 1 ⎧ ⎪ ˆ ⎪ ⎨ J(iω) = 1 − R(iω) , ⎪ M(0) 1 ⎪ ˆ ⎪ =i + [p1 (ω) + ip2 (ω)] , ⎪ R(iω) ω ω ⎩ Then, we must evaluate

lim

|ω |→+∞

pi (ω) = 0 .

246

5 The Stability of the Wave Equation with Persistent Memory

∫

+∞

−∞

∫

=2

1

dω + ω2 [M(0) + p2 (ω)]2 1 dω . 2 2 2 [−ω + ωp1 (ω) + λn ] + ω2 [M(0) + p2 (ω)]2

[−ω2 +∞

0

+ ωp1 (ω) +

λn2 ]2

(5.46)

In fact, p1 (ω) is an odd function while p2 (ω) is even (see (5.35)). In conclusion, we must evaluate from below the denominator of the integrand in (5.46). In order to evaluate from below the square of a function, we need to know the sign of that function. We recall that we can confine ourselves to give an estimate when n is large. The estimates are in Table 5.1 where we use the notation ∫ +∞ |R (s)| ds . D = max |p1 (ω)| ≤ 0

Table 5.1 suggests splitting the integral as follows: ∫

+∞

0

∫ =

λ n −D−1

∫ +

∫

2λ n

λ n −D−1

0

  here we use (5.47)

+

+∞

= I1 + I2 + I3 .

(5.49)

2λ n

 here we use (5.48)

After this preliminary illustration of the ideas, we embark on the proof of the inequality (5.43), hence of the inequality (5.40).

Table 5.1 The sign of −ω 2 + ωp1 (ω) + λ2n and the estimate for the denominator of (5.46) λ2n − ω 2 + ωp1 (ω)

ω 2 − ωp1 (ω) − λ2n

− ω − Dω > 0  if 0 ≤ ω ≤ 12 ( D 2 + 4λ2n − D) .

≥ ω 2 − Dω − λ2n > 0  if ω > 12 (D + D 2 + 4λ2n ) .

≥

λ2n

2

For large n we have: λn − D > 0 (λ n − D)
0 .

(5.47)

For large n we have:   1 D + D 2 + 4λ2n 2 < λ n + D < 2λ n . So, to be on the safe side, we put ourselves on the half line ω ≥ 2λ n where for large n we have ω 2 − ωp1 (ω) − λ2n ≥ ω 2 − Dω − λ2n > 0 .

(5.48)

5.3 L 2 -Stability via Laplace Transform and Frequency Domain Techniques

247

Step 1: The Inequalities (5.43) in the Case w0 = 0, w1  0 In this step we shall use [17, formula 142.2]: ∫ a + x 1 x x2   log − dx =  +c . a−x (a2 − x 2 )2 2(a2 − x 2 ) 4a We expand w1 = y =

+∞ 

(5.50)

yn ϕn

n=1

so that  +∞  −1   L (w ) (iω) = iω −ω2 I − J(iω)A w1 = ϕn

 iω yn −ω2 + λn2 J(iω) n=1   +∞   −1  λn 2 Aw(iω) ˆ = A −ω I − J(iω)A w1 = ϕn yn . −ω2 + λn2 J(iω) n=1 We adapt the arguments in Step 0: we must prove the existence of C ∈ R such that ∫ +∞ ω2 sup dω < C (5.51) [−ω2 + ωp1 (ω) + λn2 ]2 + ω2 [M(0) + p2 (ω)]2 n 0 ∫ +∞ λn2 sup dω < C (5.52) [−ω2 + ωp1 (ω) + λn2 ]2 + ω2 [M(0) + p2 (ω)]2 n 0 and, as noted in Remark 5.29, we can confine ourselves to evaluate the supremum for large n, n > N1 . Step 1A: The Proof of (5.51) We split the integral as in (5.49). We consider I1 . We use the inequality in (5.47) and we note that λn2 − Dλn > 0 and that the denominator is not zero if n is large so that we can use formula (5.50). We see ∫

λ n −D−1

λn − D − 1 * ) 2 (2 + D)λn − (D + 1)2 − Dλn −    λ2 − Dλ + λ − D − 1  1 1  n  n n (for n → +∞) log   −  →  λ2 − Dλn − λn + D + 1  2D + 4 4 λn2 − Dλn n I1 ≤

0

)

ω2

λn2



*2 ω2

dω =

so that sup I1 < C . n

248

5 The Stability of the Wave Equation with Persistent Memory

The estimate of I3 is similar (we use (5.48)): ∫

+∞

ω2 dω 2 2 2 2λ n (ω − Dω − λn ) ∫ +∞ ∫ +∞ ω2 (s + D/2)2 = dω = ds, 2 2 2 2 2 2 2 2λ n [(ω − D/2) − (λn + D /4)] 2λ n −D/2 (s − a ) I3 ≤

where a=

λn2 + D2 /4 .

We note that when n is large the inequality s > 2λn − D/2 implies (s + D/2)2 < 4s2 so that ∫ +∞ 4s2 2λn − D/2 ds ≤ −2 2 I3 ≤ 2 2 2 2 (λn + D /4) − (2λn − D/2)2 2λ n −D/2 (s − a )    λ2 + D2 /4 + 2λ − D/2  1  n  n (5.53) log   − →0 2 2  2 2 λ + D /4 λ + D /4 − 2λn + D/2  n

n

(for n → +∞). Hence there exists C such that I3 < C . The integral I2 is split as follows: ∫ I2 = I2a + I2b =

λ n +D+1

λ n −D−1

∫ +

2λ n

λ n +D+1

.

The integral I2a is easily estimated: for n large we have |p2 (ω)| < M(0)/2 so that ∫ I2a ≤

λ n +D+1

λ n −D−1

4ω2 8(d + 1) dω = . 2 2 M(0) ω M(0)2

We consider I2b . We recall limω→+∞ p1 (ω) = 0. So, there exists ω0 such that ω > ω0 =⇒ |p1 (ω)| < ε
ω0 . Then we have ω2 − ωp1 (ω) − λn2 > ω2 − 2ελn − λn2 > 0 . Positivity follows since ω > λn + D + 1 so that ω2 − 2ελn − λn2 ≥ (λn + D + 1)2 − 2ελn − λn2 ≥ 2λn (D + 1 − ε) + (D + 1)2 . So on [λn + D + 1, 2λn ] we have for large n

5.3 L 2 -Stability via Laplace Transform and Frequency Domain Techniques

249

(ω2 − ωp1 (ω) − λn2 )2 > (ω2 − 2ελn − λn2 )2 and ∫ I2b ≤

2λ n

λ n +D+1

ω2 dω ≤ [ω2 + ωp1 (ω) − λn2 ]2

1 → 4(D + 1 − ε)

∫

2λ n

λ n +D+1

ω2 dω [ω2 − (λn2 + 2ελn )]2

(for n → +∞)

as it is easily seen by a direct computation of the integral, using formula (5.50). This ends the proof of inequality (5.51). Step 1B: The Proof of (5.52) In this step we use [17, formula 140.2]: ∫ a + x 1 1 x   + dx = log +c .  a−x (a2 − x 2 )2 2a2 (a2 − x 2 ) 4a3

(5.54)

The integral in (5.52) is splitted as in (5.49) and we note that if ω > 2λn then λn2 < ω2 so that the integral I3 of this substep is dominated by the integral in (5.53) and there exists C such that I3 ≤ C. We give an estimate for I1 . We use the inequality in (5.47) and formula (5.54). ∫ I1 ≤

λ n −D−1

λn2

* 2 dω λn2 − Dλn − ω2 λn − D − 1 = λn2 2 2(λn − Dλn )[(D + 2)λn − (D − 1)2 ]    λ2 − Dλ + λ − D − 1  λn2 1  n  n n . + log   → 2 3/2 2   4 + 2D 4(λn − Dλn ) λn − Dλn − λn + D + 1 0

)

The bound of I2 follows from the corresponding bound found in the substep 1A since ∫ 2λn  2  λn ω2 I2 = dω 2 [−ω2 + ωp (ω) + λ 2 ]2 + ω2 [M(0) + p (ω)]2 λ n −D−1 ω 1 2 n and

λn2 λn2 ≤ < 2 (for large n) . ω2 (λn − D − 1)2

This ends the proof of inequality (5.52).

250

5 The Stability of the Wave Equation with Persistent Memory

Step 2: The Inequalities (5.43) in the Case w0  0, w1 = 0 We recall that w0 ∈ dom A and we note that, from Eq. (5.41):   −1 Aw(λ) ˆ = λ λ2 I − J(λ)A (Aw0 ) ,   −1 L (w ) (λ) = λ w(λ) ˆ − w0 = J(λ)A λ2 I − J(λ)A w0    −1  2 = J(λ)A λ I − J(λ)A (Aw0 ) .

(5.55)

We must prove 2 2 Aw(λ) ˆ H2 (Π+ ;H) ≤ C Aw0 H ,

2 2 L (w ) (λ)) H ≤ C Aw0 H . 2 (Π+ ;H)

So, we expand y = Aw0 ∈ H in series of eigenfunctions and we see that we must prove ! ∫ +∞ ω2 sup dω < C < +∞ (5.56) | − ω2 + λn2 J(iω)| 2 n 0 and ∫ sup n

0

+∞

! λn2 |J(iω)| dω < C < +∞ . | − ω2 + λn2 J(iω)| 2 2

(5.57)

Inequality (5.56) has been proved in the Substep 1A while the inequality (5.57) follows from the one in Substep 1B because J(iω) is bounded. We sum up: there exists C such that every classical solution of Eq. (5.39) satisfies the required inequality (5.40). In order to complete the proof of Theorem 5.25, we prove that also the mild solutions5 satisfy the inequality (5.40). The proof follows from a simple continuity argument. We proceed as follows. For every classical solution and every T > 0 we have ∫ T   )  2 * (5.58) w (t)| + Aw(t) 2 dt ≤ C Aw0 2 + w1 2 . 0

The constant C does not depend either on the initial conditions or on T ≥ 0. It is proved in Sect. 4.2 that classical solutions correspond to a set of initial conditions which is dense in (domA) × H; and the transformation (w0, w1 ) → (w(t), w (t)) is linear and continuous from (dom A) × H to C ([0, T]; dom A) × C ([0, T]; H) (for every T > 0). Hence, inequality (5.58) holds for every initial condition, hence for 5 If (as in Sect. 3.6) mild solutions are defined as inverse Laplace transformations, then the proof is finished here but this is not the definition in Sect. 4.2.

5.3 L 2 -Stability via Laplace Transform and Frequency Domain Techniques

251

every mild solution, and every T. We pass to the limit for T → +∞ and we get inequality (5.40) for every mild solution. The proof of Theorem 5.25 is finished.

5.3.2 The Memory Prior to the Time 0 We consider the system (5.1) which we rewrite as follows: w  = Aw −

∫

t

∫ M(t − s)Aw(s) ds + A

0

0

M(t − s)Aw(s) ˜ ds .

  −∞

(5.59)

h(t)=hw˜ (t)

Note the definition of the function h = hw˜ in this formula, which is different from the function h in (5.1).

Notations and the Assumptions on w(t) ˜ We assume that w(t) ˜ satisfies the following conditions (compare with the definition of the space M in Sect. 5.5): • w(t) ˜ ∈ dom A a.e. t < 0 and Aw(t) ˜ ∈ L 2 (−S, 0; H) for every S > 0. ∫0 2 = 2 ds < +∞. M(−s) Aw(s) ˜ • w ˜ M −∞

The first result that we prove is: Theorem 5.30. Let the stated assumptions on M(t) and on w(t) ˜ hold. There exists C > 0 such that every solution of (5.59) satisfies the following inequality: ∫ +∞   )  * 2 (5.60) w (t) 2 + Aw(t) 2 dt ≤ C w1 2 + Aw0 2 + w ˜ M 0

and, with EV (0) now the energy function in (5.14), we have also ∫ +∞ )  * w (t) 2 + Aw(t) 2 dt ≤ C1 EV (0) .

(5.61)

0

Then we prove L 2 -stability: Theorem 5.31. Under the stated assumptions, the system (5.59) is L 2 -stable. We expand w(t) ˜ and the function hw˜ (t) in (5.59) in series of the eigenfunctions ϕn of A and we note:

252

5 The Stability of the Wave Equation with Persistent Memory

(a) w(t) ˜ =

+∞ 

ϕn w˜ n (t) so that

n=1 +∞ 

(b) hw˜ (t) =

hw˜ L2 2 (0,+∞;H)

(d) hˆ w˜ (λ) =

+∞ 

=

ϕn L

+∞ 

hn L2 2 (0,+∞)

n=1 ∫ 0

−∞

n=1

+∞ 

=

∫n=1 0

ϕn hn (t) where hn (t) =

n=1

(c)

2 w ˜ M

−∞

∫ λn2

0

−∞

 M(−s)| w˜ n (s)| ds , 2

M(t − s)[λn w˜ n (s)] ds , (5.62)

, 

M(t − s)[λn w˜ n (s)] ds (λ) .

We need two preliminary lemmas. Lemma 5.32. Under the stated assumptions the function ∫ t → hw˜ (t) =

0

−∞

M(t − s)Aw(s) ˜ ds

belongs to L 1 (0, +∞; H)∩ L 2 (0, +∞; H). In particular, its Laplace transform belongs to H 2 (Π+ ; H) and it is continuous in e λ ≥ 0. Furthermore, there exists C > 0 such that 2 2 ˜ M . (5.63) hw˜ L2 2 (0,+∞;H) = (1/2π) hˆ w˜ H 2 (Π ;H) ≤ C w +

Proof. First we prove integrability. The property M (t) ≤ −ξ M(t) implies (see (5.31)) M(t + τ) ≤ e−ξt M(τ) ∀t ≥ 0 , ∀τ ≥ 0 . Hence   

+∞ ∫ 0

∫ 0

∫

≤

+∞

−∞

  M(t − s)Aw(s) ˜ ds dt

e−ξt

∫

0

0



−∞

 √  κ0 w ˜ M. M(−s) M(−s) Aw(s) ˜ ds dt ≤ ξ

In a similar way, ∫ hw˜ L2 2 (0,+∞;H) ∫ ≤ 0

+∞

=

  

+∞ ∫ 0

0

 ∫  −2ξt e dt 

0

−∞



−∞

2  M(t − s)Aw(s) ˜ ds dt

2   κ0 2 w ˜ M M(−s) M(−s) Aw(s) ˜ ds ≤ . 2ξ

 

Lemma 5.33. Let y be a scalar function such that ∫

0

−∞

M(s)|y(s)| 2 ds < +∞

and let e λ ≥ 0. The following inequality holds in the closed halfplane e λ ≥ 0:

5.3 L 2 -Stability via Laplace Transform and Frequency Domain Techniques

 ∫  |λ| L

0

−∞

 ∫  √  M(t − s)y(s) ds (λ) ≤ 2 κ0 

0

−∞

253

 1/2 .

M(s)|y(s)| 2 ds

Proof. The inequality for λ = 0 is obvious. If λ  0 we compute as follows: ∫ 0  ∫ +∞ ∫ 0 L M(t − s)y(s) ds (λ) = e−λt M(t − s)y(s) ds dt −∞

1 λ ∫

=− 1 = λ

∫

0

−∞ 0

∫

0 +∞

de−λt

0

dt

M(t − s) dt y(s) ds ∫

1 M(−s)y(s) ds + λ −∞

∫

0

−∞

+∞

e

=

0

∫

0

−∞

0

∫

−λt





M (t − s) dt y(s) ds .

0

We recall |M (t)| = −M (t) so that: ∫ 0 ∫ +∞  ∫    −λt   e M (t − s) dt y(s) ds ≤  −∞

−∞



+∞





−M (t − s) dt |y(s)| ds

0

M(−s)|y(s)| ds .

−∞

The required inequality follows since ∫



0

−∞

∫  √ M(−s) M(−s)|y(s)| ds ≤ κ0

0

−∞

 1/2 M(−s)|y(s)| 2 ds

.

 

After these preliminaries, we can prove the theorems.

The Proof of Theorem 5.30 First we note that (5.60) implies (5.61). In fact, the expression (5.14) of EV (0) shows 2 . We have that we must evaluate the integral term w ˜ M ∫

∫

+∞

0

2 M(s) Aw(−s) ˜ ds =

∫

+∞

M(s) (Aw(−s) ˜ − Aw0 ) + Aw0 2 ds

0

+∞

M(s) Aw(−s) ˜ − Aw0 2 ds + 2κ0 Aw0 2   ∫ +∞ ≤ 2 Aw0 2 + w1 2 + M(s) Aw(−s) ˜ − Aw0 2 ds = 2EV (0) . ≤2

0

0

So in order to prove Theorem 5.30 it is sufficient that we prove the inequality (5.60). Thanks to the linearity of the problem, it is sufficient to prove the inequality with ˜ w0 = 0 and w1 = 0 and we prove the result when w˜ is a smooth function, say Aw(t) of class C ∞ with compact support. The inequality is then extended by continuity to every w˜ with values in dom A and such that w ˜ M < +∞. We have

254

5 The Stability of the Wave Equation with Persistent Memory

  −1 ∫ w(λ) ˆ = A λ2 I − J(λ)A L

0

−∞

 M(t − s)Aw(s) ˜ ds (λ) .

  hˆ w˜ (λ)

We expand w(t), y(t), and hˆ w˜ (t) in series of the eigenfunctions ϕn of A as in (5.62): ∫ 0  λn ˆ ˆ M(t − s) [λn w˜ n (s)] ds (λ) , wˆ n (λ) = 2 hn (λ) , hn (λ) = L λ + λn2 J(λ) −∞ 2 +∞ ∫ +∞     iωλn 2   dω , ˆ λ w(λ) ˆ = (iω) h n 2   H (Π+ ;H) −ω2 + λn2 J(iω) n=1 −∞ 2 +∞ ∫ +∞     λn2 2   ˆ Aw(λ) ˆ =  −ω2 + λ2 J(iω) hn (iω) dω . H 2 (Π+ ;H) −∞ n n=1 We use Lemma 5.33 with y(t) = λn w˜ n (t). We have |ω hˆ n (iω)| 2 ≤ c (s)| 2 ds where c = 4κ0 . Then we have for the velocity: 2 λ w(λ) ˆ H 2 (Π+ ;H)

≤ ≤

≤

+∞ ∫ 

+∞

−∞

λn2 |iω hˆ n (iω)| 2 dω | − ω2 + λn2 J(iω)| 2  ∫ 0  2 dω c M(s)|λn w˜ n (s)| ds 2

−∞ n=1 2 λn 2 2 −∞ | − ω + λn J(iω)| n=1 ∫ +∞ λn2 c sup 2 2 2 n −∞ | − ω + λn J(iω)|

+∞ ∫ 

∫0

+∞

M(s)|λn w˜ n



−∞

dω

! ∫0

 

−∞

M(s)

+∞

˜n n=1 |λn w

(s)| 2 ds



(5.64)