Measure-Valued Solutions for Nonlinear Evolution Equations on Banach Spaces and Their Optimal Control [1 ed.] 9783031372599, 9783031372605

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N. U. Ahmed Shian Wang

Measure-Valued Solutions for Nonlinear Evolution Equations on Banach Spaces and Their Optimal Control

Measure-Valued Solutions for Nonlinear Evolution Equations on Banach Spaces and Their Optimal Control

N. U. Ahmed • Shian Wang

Measure-Valued Solutions for Nonlinear Evolution Equations on Banach Spaces and Their Optimal Control

N. U. Ahmed University of Ottawa Ottawa, ON, Canada

Shian Wang The University of Texas at El Paso El Paso, TX, USA

ISBN 978-3-031-37259-9 ISBN 978-3-031-37260-5 https://doi.org/10.1007/978-3-031-37260-5

(eBook)

Mathematics Subject Classification: 35A01, 35A02, 35B44, 35K57, 35Q30, 37N35, 46G10, 46E27, 49J20, 49J27, 49J55, 49K20, 49K27, 78A02, 93E03, 93E20 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 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 Paper in this product is recyclable.

In memory of my beloved mother, father, uncles, wife, aunties, sisters, brother and the last but not the least, all my teachers who gave so much. Dedicated to my sons: Jordan and Shockley; daughters: Pamela, Rebeka, Mona, Lisa; and my grandchildren: Reynah-Sofia, Maximus, Achilles, Eliza, Pearl, Austin, Rio, Kira and Jazzmine. Dedicated to my parents Guihua Xu and Shiwu Wang, and my sister Man Wang.

Preface

In this monograph, we consider measure-valued solutions for nonlinear deterministic and stochastic evolution equations on infinite dimensional Banach spaces. It is known that, in finite dimensional spaces, if the vector field of a nonlinear differential equation is merely continuous, the system has continuous solution locally and may blow up in finite time. In infinite dimensional spaces, this is no longer true; the vector field may be continuous, but the system has no solution. There are counterexamples as seen later in the monograph [67, 80]. Consequently, in most of the literature on differential equations, assumptions like Lipschitz or more generally local Lipschitz continuity and at most linear growth, are widely used to prove existence and uniqueness of solutions in both finite and infinite dimensional spaces [39, 74, 100]. In the study of abstract differential equations on infinite dimensional Banach spaces, there are several notions of solutions, such as classical, strong, mild and weak solutions. For detailed exposition, see [5, Chapter 5, p. 143]. The notions of classical and strong solutions are too restrictive as they require the solution to be differentiable and to belong to the domain of the semigroup generator which is an unbounded operator. This limits the class of systems to those with infinitesimal generators of analytic or differentiable semigroups. Further, for semilinear problems, the vector fields are also limited. The notion of weak solutions is very powerful for linear problems and some special classes of nonlinear problems. In the case of nonlinear systems, determined by monotone or maximal monotone operators which are also hemicontinuous, the notion of weak solutions is used. There is a class of nonlinear parabolic and hyperbolic partial differential equations satisfying such special properties, and in such cases, one can prove existence (and uniqueness) of weak solutions, e.g., see [4, Chapter 4, p. 94]. The notion of mild solutions is generally very powerful for semilinear problems provided the nonlinear vector fields satisfy local Lipschitz continuity and (at most) linear growth. Thus, generalizing the very notion of solutions to measure-valued solutions, one can substantially expand the theory to cover the broader class of abstract evolution equations with vector fields which are merely continuous and bounded on bounded sets and, more generally, Borel measurable vector fields which are bounded on vii

viii

Preface

bounded sets. Hence, the notions of classical, strong, weak and mild solutions are special cases of the notion of measure-valued solutions. In fact, these path-wise solutions, obtained under standard assumptions, can be considered as degenerate measure-valued solutions and hence can be embedded in the class of measurevalued solutions. In this monograph, we present the existence of measure solutions for differential equations having no solutions in the usual sense (classical, strong, mild and weak). Moreover, this monograph covers measure-valued solutions of differential equations on infinite dimensional Banach spaces and considers standard and nonstandard optimal control problems. The published books [39, 74, 85, 100, 116, 128] are primarily on nonlinear evolution equations with classical and restrictive assumptions. In contrast, this monograph is much more focused on the development of measurevalued solutions for nonlinear evolution equations on infinite dimensional spaces under much relaxed assumptions. Further, the optimal control problems considered cover existence of solutions, necessary conditions of optimality, among others, which are not considered in the published books mentioned. This monograph covers a broad range of topics, such as evolution equations with continuous/discontinuous vector fields, neutral evolution equations subject to impulsive forces, stochastic evolution equations and optimal control of evolution equations on the Banach space of regular bounded finitely additive measures. It is well beyond the topics on dynamic systems subject to regular controls determined only by measurable functions. This monograph contains seven chapters. In Chap. 1, we present some fundamental results from functional analysis required to read the monograph smoothly. The contents of this chapter will also serve as a quick reference for readers both familiar and unfamiliar with the subject. It contains many relevant and powerful results from the theory of vector measures and functional analysis used extensively in the text. Chapter 2 focuses on measure solutions for evolution equations with continuous vector fields, and under relaxed hypothesis, with Borel measurable vector fields, covering existence of solutions and regularity properties thereof. Chapter 3 studies evolution equations subject to impulsive forces, considering the questions of existence and uniqueness of measure-valued solutions and regularity properties thereof. Chapter 4 deals with measure solutions for stochastic systems, with extension to measurable vector fields. Neutral evolution equations are studied in Chap. 5, covering both deterministic and stochastic second order neutral systems. In Chap. 6, optimal control problems are studied for deterministic systems, impulsive systems, stochastic systems and neutral systems. These systems are considered as evolution equations on the Banach space of regular bounded finitely additive measures, with a focus on the question of existence of optimal controls. In Chap. 7, a few examples from physical sciences with applications to engineering and medicine are presented to demonstrate the applicability of the theories developed in this monograph. We hope that the notion of measure solutions considered in this monograph will inspire further research in the field and advance theory and applications in many areas which are not approachable by classical notions.

Preface

ix

Finally, we would like to appreciate Dr. Remi Lodh, the Senior Editor of Mathematics with Springer Verlag, for his continued support and excellent cooperation throughout the publishing process. Ottawa, ON, Canada El Paso, TX, USA

N. U. Ahmed Shian Wang

Contents

1

Background Materials From Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Vector-Valued Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Multi Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 4 9 11

2

Measure Solutions for Deterministic Evolution Equations . . . . . . . . . . . . . . 2.1 Evolution Equations with Continuous Vector Fields . . . . . . . . . . . . . . . . . . 2.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Evolution Equations Under Relaxed Hypothesis . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Competing Notions of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Quasilinear Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Evolution Equations with Measurable Vector Fields . . . . . . . . . . . . . . . . . . 2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Existence of Measure Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Differential Equations on the Space of Measures . . . . . . . . . . . . . 2.4 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 15 15 16 26 30 32 36 37 38 46 47

3

Measure Solutions for Impulsive Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Spaces of Measure-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Measure-Valued Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Existence of Measure Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Measure Solutions vs. Pathwise Solutions . . . . . . . . . . . . . . . . . . . . 3.4 Differential Equations on the Space of Measures . . . . . . . . . . . . . . . . . . . . . 3.5 Differential Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Classical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 General Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 49 50 51 55 66 67 69 69 71 74

xi

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Contents

4

Measure Solutions for Stochastic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2 Existence of Measure Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.2.1 Martingale vs. Generalized Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2.2 Some Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.3 Stochastic Systems Driven by Martingale Measures . . . . . . . . . . . . . . . . . . 90 4.3.1 Special Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.3.2 Some Basic Properties of the Martingale Measure M . . . . . . . . 91 4.3.3 Basic Formulation of the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.3.4 Existence of Measure Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.4 Extension to Measurable Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.5 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5

Measure Solutions for Neutral Evolution Equations . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Basic Background Materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Existence of Measure Solutions and Their Regularity . . . . . . . . . . . . . . . . 5.4 Stochastic Neutral Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Basic Background Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Existence of Measure Solutions and Their Regularity . . . . . . . . 5.5 Second Order Neutral Differential Equations. . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Some Basic Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 System Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4 System Models Generating C0 -Group . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.5 Existence and Regularity of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Stochastic Second Order Neutral Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103 103 104 104 111 112 113 122 122 122 123 124 125 128 134

6

Optimal Control of Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Optimal Control of Deterministic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Optimal Control of Impulsive Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Optimal Control of Stochastic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Optimal Control of Neutral Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Deterministic Neutral Systems (DNS) . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Stochastic Neutral Systems (SNS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135 135 150 161 168 168 175 180

7

Examples From Physical Sciences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Nonlinear Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Basic Formulation of the System Model . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Existence and Uniqueness of Solutions. . . . . . . . . . . . . . . . . . . . . . . .

183 183 184 185

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xiii

7.2 Stochastic Navier-Stokes Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 7.3 Reaction Diffusion Equation (Biomedical Application) . . . . . . . . . . . . . . 211 7.4 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

Chapter 1

Background Materials From Analysis

1.1 Introduction In this chapter we present some basic and crucial results from functional analysis which are useful for the development and understanding of the subject matters of this monograph. We first introduce some function spaces. Some Special Function Spaces Definition 1.1.1 ([69]) Space .B(S): Let .S denote an arbitrary set and .B(S) denote the space of real valued bounded functions defined on .S. This is endowed with the sup norm topology, .

 f ≡ sup {|f (s)|, s ∈ S} for f ∈ B(S).

Theorem 1.1.2 With respect to the sup norm topology as defined above, .B(S) is a complete normed linear vector space and hence a Banach space. The elements of this space may not be even measurable. For this, we must introduce an algebra (or field) of sets in .S. Let . denote a field of subsets of the set .S. Recall that a field . is closed under operations of finite union and intersections and that .{∅, S} are in .. Definition 1.1.3 Space .B(S, ): Let .B(S, ) denote the space of bounded scalar valued functions defined on .S which are the uniform limits of finite linear combinations of characteristic functions of sets in .. An element .f ∈ B(S, ) is said to be measurable if for every Borel set .D ∈ R, .f −1 (D) ∈ . With respect to the norm topology, .

 f ≡ sup {|f (s)|, s ∈ S} ,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. U. Ahmed, S. Wang, Measure-Valued Solutions for Nonlinear Evolution Equations on Banach Spaces and Their Optimal Control, https://doi.org/10.1007/978-3-031-37260-5_1

1

2

1 Background Materials From Analysis

B(S, ) is also a Banach space. This follows from the fact that .B(S, ) is a closed linear subspace of the Banach space .B(S).

.

Definition 1.1.4 Space .BC(S): For a topological space .S, let .BC(S) denote the space of real valued bounded continuous functions defined on .S. Furnished with the sup norm topology this is a Banach space. Space of Set Functions We are now ready to introduce set functions. There are several classes of set functions. Here we present three different classes of such functions intimately related to the function spaces described above. Let . denote a field of subsets of the set .S and .Mba (S, ) the space of scalar valued functions defined on ., called set functions. For any .μ ∈ Mba (S, ) and any . ∈ , the variation of .μ on . is given by |μ|() ≡ sup



.

|μ(E)|,

 E∈

where the summation is taken over the elements of any partition . of the set . by a family of finite number of pairwise disjoint . measurable sets. The supremum is taken over all such finite partitions. The norm of .μ is given by .|μ| = |μ|(S). Let .B ∗ (S, ) denote the space of continuous linear functionals on .B(S, ). Then, we have the following result on duality relating .B ∗ (S, ) with .Mba (S, ). Theorem 1.1.5 Space .Mba (S, ): Endowed with the total variation norm .|μ| ≡ |μ|(S), .Mba (S, ) is a Banach space. Further, it is the topological dual of the Banach space .B(S, ) in the sense that, for every continuous linear functional ∗ . ∈ B (S, ), there exists a unique .μ ∈ Mba (S, ) such that  (f ) =

.

S

f (s)μ(ds),

(1.1)

and conversely, every .μ ∈ Mba (S, ) defines a unique continuous linear functional  on .B(S, ) through the integration (1.1) satisfying .  = |μ|, where

.

.

  = sup {|(f )|, f ∈ B(S, ),  f ≤ 1} .

This is an isometric isomorphism indicated by the expression Mba (S, ) ∼ = B ∗ (S, ).

.

Proof See [69, Theorem IV.5.1, p. 258].



Next we consider the Banach space .B(S) and its topological dual (conjugate space). Let . denote the family of all subsets of the space .S. Clearly, .B(S) = B(S, ). Therefore, the following result follows as a corollary of Theorem 1.1.5.

1.1 Introduction

3

Corollary 1.1.6 Let .B ∗ (S) denote the space of continuous linear functionals on .B(S) and .Mba (S) the space of all scalar valued set functions defined on .S. Then, ∗ ∗ .B (S) is isometrically isomorphic to .Mba (S). That is, .B (S) ∼ = Mba (S). In other ∗ words, for every . ∈ B (S) there exists a unique .μ ∈ Mba (S) such that  (f ) =

.

S

f (s)μ(ds), f ∈ B(S),

and conversely, every such .μ defines a continuous linear functional on .B(S) satisfying .|μ| = f . So far we have considered an arbitrary set .S and defined set functions on a field  of subsets of the set .S. No topology was assumed. Next, we consider .S to be a topological space and assume that . is a field of sets generated by closed sets. Let .BC(S) denote the linear space of bounded continuous real valued functions defined on .S. Endowed with sup norm topology, .BC(S) is a Banach space. .

Definition 1.1.7 A set function (or a measure) .ν is said to be regular if for each set E ∈  and .ε > 0, there exists a closed set .F ∈  contained in E, and an open set .G ∈  containing E, such that .|ν|(G \ F ) < ε. .

Recall that a topological space .Z is said to be Hausdorff if, and only if, for any pair of distinct points .x, y ∈ Z, there exist disjoint open sets .U ⊂ Z and .V ⊂ Z such that .x ∈ U and .y ∈ V . A topological space .Z is said to be regular if, and only if, for any closed set .C ⊂ Z and a point z not in C, there exist disjoint open sets .U ⊂ Z and .V ⊂ Z such that .C ⊂ U and .z ∈ V . A topological space .Z is said to be normal if, and only if, for any pair of disjoint closed sets .C1 , C2 ⊂ Z there exists a pair of disjoint open sets .O1 , O2 ⊂ Z such that .C1 ⊂ O1 and .C2 ⊂ O2 . Let .S be a normal topological space and . a field of subsets of the set .S. Let .Mrba (S, ) denote the class of regular bounded finitely additive set functions defined on .. Theorem 1.1.8 Suppose .S is a normal topological space. Let .(BC(S))∗ denote the space of (bounded) continuous linear functionals on the Banach space .BC(S). Then, the space .(BC(S))∗ is isometrically isomorphic to the space of regular bounded finitely additive measures .Mrba (S, ) in the sense that for any . ∈ (BC(S))∗ there exists a unique element .μ ∈ Mrba (S, ) such that  (g) =

.

S

g(s)μ(ds), for g ∈ BC(S).

(1.2)

and .  = |μ|. Proof See [69, Theorem IV.6.2, p. 262].



4

1 Background Materials From Analysis

In case .S is a compact Hausdorff space, we have the well-known Riesz representation theorem. Let .S be a compact Hausdorff space and denote by .BC(S) the linear space of bounded continuous functions endowed with the sup norm topology and .BC ∗ (S) the space of continuous linear functionals on .BC(S). Let .Mrca (S) denote the space of regular countably additive set functions defined on .S. Then, we have the following Riesz representation theorem. Theorem 1.1.9 For every . ∈ BC ∗ (S), there exists a unique .μ ∈ Mrca (S) such that  .(f ) = f (s)μ(ds), for f ∈ BC(S), (1.3) S

and conversely. Further, this is an isometric isomorphism expressed as .BC ∗ (S) ∼ = Mrca (S).

1.2 Vector-Valued Measures Let .(, , μ) be a finite positive measure space and X a real Banach space. Let m be an X valued measure defined on . in the sense that, for every .E ∈ , .m(E) ∈ X. The measure m is countably additive if for every (countable) family of disjoint . measurable sets .{En } with .En ⊂ ,  m

∞ 

.

 En

n=1

=

∞ 

m(En ).

n=1

It is said to be finitely additive if the identity holds only for finitely many disjoint sets. Note that the identity is understood in the sense that   k       En − . lim m m(En ) = 0.  k→∞  n=1

X

The measure m is said to be of bounded variation if .|m| = |m|() < ∞, where the variation is defined in the same way as in the scalar case. For any . measurable set .A ⊂ , the variation of m over A is given by |m|(A) ≡ sup



.

 m(E) X ,

 E∈

where . is any partition of A by a finite family of disjoint . measurable sets .{E} ⊂ . The summation is taken over . and the supremum is taken over all such finite

1.2 Vector-Valued Measures

5

partitions .{}. There is another notion of variation, known as semi-variation. This is defined as follows .

  m  (A) = sup |x ∗ (m)|(A), x ∗ ∈ B1 (X∗ ) ,

where .|x ∗ (m)| is the variation of the scalar measure .x ∗ (m) and .B1 (X∗ ) is the closed unit ball in the dual .X∗ of the Banach space X. Since we rarely use semi-variation in this monograph, we will not duel on this topic any further. The domain of vector measures is either the field or the sigma field .. Thus, it is reasonable to express this by simply writing .M(, X) in place of .M(, ; X). Again in the vector-valued case, we are interested in several different classes of X valued measures such as .Mba (, X) and .Mca (, X); and in case . is also a topological space, we have the spaces .Mrba (, X) and .Mrca (, X) of regular vector-valued measures. These are also Banach spaces with respect to total variation norm. It is clear that the space .Mrba (, X) ⊂ Mba (, X) and .Mca (, X) ⊂ Mba (, X) and hence .Mrca (, X) = Mca (, X) ∩ Mrba (, X). Let .μ be a nonnegative finite measure and consider the finite measure space .(, , μ). Definition 1.2.1 A vector measure m is said to be .μ continuous if and only if m vanishes on .μ null sets, that is, for every set .σ ∈ , .m(σ ) = 0 whenever .μ(σ ) = 0, and hence .limμ(σ )→0 |m|(σ ) = 0. The notion of differentiability of any scalar valued measure m with respect to another, possibly, positive measure .μ defined on the same field (or .σ field) is guaranteed if m is absolutely continuous with respect to .μ as expressed by .m  μ. It is well-known that if .m  μ then there exists a measurable scalar valued function .g ∈ L1 (μ, R/C) such that  m(D) =

g(ω)μ(dω)

.

(1.4)

D

for every .D ∈ . This can be expressed by simply writing .dm = gdμ, where g is called the Radon-Nikodym derivative (RND) of m with respect to .μ. Clearly, this is a nontrivial extension of the well-known fact that an absolutely continuous function defined on any interval I is differentiable almost everywhere. In this case one thinks of differentiation with respect to the Lebesgue measure on I . In the case of vector-valued measures, similar but more intricate notions, do exist. For this, we need to introduce the concept of Radon-Nikodym property (RNP). First let us recall the notion of Bochner integrals. Let .(, , μ) be a finite (positive) measure space and f be an X valued strongly measurable function defined on .. The function f is Bochner integrable if the real valued function, .  ω −→ f (ω) X , is Lebesgue measurable and .μ-integrable in the sense that   f (ω) X μ(dω) < ∞.

.



6

1 Background Materials From Analysis

Now we are ready to introduce the notion of Radon-Nikodym property for Banach spaces. Definition 1.2.2 (RNP) Let .(, , μ) be any finite positive measure space and X a Banach space. The space X is said to have the Radon-Nikodym property (RNP) if, and only if, for every X valued .μ continuous vector measure m of bounded variation, there exists a strongly measurable Bochner integrable function g such that for every set .E ∈ ,  .m(E) = g(s)μ(ds). E

Examples of frequently used Banach spaces having the RNP are Hilbert spaces, reflexive Banach spaces, separable duals of Banach spaces, and many more. Some examples of Banach spaces not possessing the RNP include .L1 (μ) with .μ not purely atomic, .c0 , .∞ , .L∞ (I ) and .C() for any (infinite) compact Hausdorff space (.). For proof and many more examples see [66]. There is another more general notion of RNP, called weak RNP related to Pettis integrable X valued weakly measurable functions. If we need the weak RNP at any stage of the development, we will present it where it is necessary. If X is any finite dimensional Banach space, for example .R n , it is well-known that the dual of .Lp (μ, X) is given by .Lq (μ, X∗ ) for .1 ≤ p < ∞, where .1/p + 1/q = 1. In the case of infinite dimensional Banach spaces this is no longer true. Instead, we have the following result. Theorem 1.2.3 Let .(, , μ) be a finite positive measure space and X an infinite dimensional Banach space with dual .X∗ . Then, the dual of the Banach space ∗ .Lp (μ, X) is .Lq (μ, X ) for .1 ≤ p < ∞, with .1/p + 1/q = 1, if and only if ∗ .X has the RNP with respect to the measure .μ. Proof See [66, Theorem IV.1.1, p. 98].



Clearly, it follows from the above theorem that the dual of .L1 (μ, X) is not L∞ (μ, X∗ ) unless .X∗ has the RNP. However, if X is a reflexive Banach space then the dual of .L1 (μ, X) is always .L∞ (μ, X∗ ). This is due to the fact that if X is reflexive so also is .X∗ and hence it has the RNP. The property of a Banach space having the RNP has a substantial bearing, not only on characterization of the dual of the space .Lp (μ, X), but also on the characterization of compact sets therein. Let .(, , μ) be a finite positive measure space and consider the space .L1 (μ, R n ). It is well-known from the Dunford-Pettis theorem that a set .K ⊂ L1 (μ, R n ) is weakly relatively compact if and only if (i) K is bounded, and (ii) K is uniformly integrable with respect to the measure .μ. Under an additional condition, a similar result holds for the Banach space of Bochner integrable functions .L1 (μ, X). This is stated in the following theorem originally due to Dunford.

.

1.2 Vector-Valued Measures

7

Theorem 1.2.4 (Dunford) Let .(, , μ) be a finite measure space and X a Banach space with both X and its dual .X∗ having the RNP. Then, a set .K ⊂ L1 (μ, X) is weakly relatively compact if the following conditions hold: (i) K is bounded; (ii) K is uniformly .μ integrable;  (iii) For each .E ∈ , the set . E f (s)μ(ds), f ∈ K is a relatively weakly compact subset of X. Proof See [66, Theorem IV.2.1, p. 101].



The RNP is crucial. Even if all the three conditions are satisfied but X lacks the RNP, then the conclusion of the theorem is false. For an excellent proof of this fact see [66, Theorem IV.2.3, p. 103]. The influence of the RNP is far-reaching. It also plays a fundamental role in the characterization of compact sets in the space of vector measures. This is presented in the following theorem due to Bartle, Dunford and Schwartz (BDS). Let .Mca (, X) denote the space of countably additive X valued vector measures having bounded variations. Theorem 1.2.5 (Bartle-Dunford-Schwartz (BDS)) Suppose both X and its dual X∗ have the RNP and let . ⊂ Mca (, X) be a set satisfying the following conditions:

.

(i) The set . is bounded (in variation norm); (ii) There exists a countably additive nonnegative real valued measure .ν such that . is uniformly .ν continuous in the sense that .

lim |m|(D) = 0, D ∈ ,

ν(D)→0

uniformly with respect to .m ∈ ; (iii) For every .E ∈ , the set .{m(E) : m ∈ } is a relatively weakly compact subset of X. Then, the set . is a relatively weakly compact subset of .Mca (, X). Proof See [66, Theorem IV.2.2, p. 105].



It is interesting to note that this theorem has an extension to the space of finitely additive measures originally due to Brooks and Dinculeanu. Let . be a field of subsets of the set . and let .Mba (, X) denote the space of bounded finitely additive measures with values in the Banach space X.

8

1 Background Materials From Analysis

Theorem 1.2.6 (Brooks-Dinculeanu) Suppose both X and its dual .X∗ have the RNP and let . ⊂ Mba (, X) be a set satisfying the following conditions: (i) The set . is bounded (in variation norm); (ii) There exists a finitely additive nonnegative real valued measure .ν such that .

lim |m|(D) = 0

ν(D)→0

uniformly with respect to .m ∈ ; (iii) For every .E ∈ , the set .{m(E) : m ∈ } is a relatively weakly compact subset of X. Then, the set . is a relatively weakly compact subset of .Mba (, X). Proof See [66, Theorem IV.2.3, p. 106].



In fact the proof is quite similar to that of the previous theorem. This requires the Boolean isomorphism of an algebra to a sigma algebra, and an isometric isomorphism that maps the space of finitely additive measures onto the space of countably additive measures. Under this isomorphism weak compactness is preserved and the conclusion of the theorem follows from that of the previous theorem. We have seen in Theorem 1.2.3 that if the dual .X∗ of the Banach space X has the RNP, then for any finite positive measure space .(, , μ), we have the isometric isomorphism (Lp (μ, X))∗ ∼ = Lq (μ, X∗ )

.

for .1 ≤ p < ∞ satisfying .1/p + 1/q = 1. This certainly limits the class of Banach spaces only to those having duals which satisfy the RNP. In applications, we have in mind, we need Banach spaces whose duals do not possess the RNP. In order to cover such classes we need the theory of lifting due to Tulcea and Tulcea [121]. ∗ ∗ Let .Lw ∞ (μ, X ) denote the space of weak star .(w ) .μ-measurable functions ∗ w defined on . and taking values in .X . For any .g ∈ L∞ (μ, X∗ ), its norm is given by .

  g ∞ ≡ inf α ≥ 0 : |g(ω), xX∗ ,X | ≤ α, ∀ x ∈ B1 (X), μ a.e. ω ∈  .

Then, by virtue of the theory of lifting [121] we have the following result. Theorem 1.2.7 ([121]) Let .(, , μ) be a finite measure space and X any Banach space and . any continuous linear functional on the Banach space .L1 (μ, X). Then, ∗ there exists a unique .g ∈ Lw ∞ (μ, X ) such that  (f ) =

g(s), f (s)X∗ ,X μ(ds) for every f ∈ L1 (μ, X),

.

 ∗ and .  = g ∞ . Hence, .(L1 (μ, X))∗ ∼ = Lw ∞ (μ, X ).

1.3 Multi Valued Functions

9

Proof See [121, Theorem 7, pp. 94–95] and its corollary.



The following well-known result is also often used in this monograph. Theorem 1.2.8 (Alaoglu) Let X be a Banach space with the dual .X∗ and suppose that .X∗ is endowed with the weak star topology (also called the X topology of .X∗ ). Then, the closed unit ball .B1 (X∗ ) is weak star (.w ∗ ) compact.



Proof See [69, Theorem V.2.4, p. 424].

Corollary 1.2.9 Let X be a Banach space with the dual .X∗ . Then, any subset . ⊂ X∗ is weak star compact if, and only if, (1) it is bounded in norm and (2) it is weak star closed. Corollary 1.2.10 Let .(D, ≤) be a directed set and let X be a Banach

space with the dual .X∗ , the latter endowed with the weak star topology, and let . xα∗ ∈ X∗ , α ∈ D be a net (or a generalized sequence) such that there exists a finite positive number r such that .

 sup |xα∗ |X∗ , α ∈ D ≤ r.

 Then, there exists a subnet . xβ∗ , β ∈ D (or a generalized subsequence) of the net ∗

∗  ∗ w ∗ ∗ ∗ . xα , α ∈ D and an .xo ∈ X such that .x −→ xo . β Remark 1.2.11 ([69]) In case the Banach space X is separable, the weak star topology on .X∗ is metrizable with the metric d(x ∗ , y ∗ ) =

.



(1/2n )

|(x ∗ − y ∗ )(xn )| , 1 + |(x ∗ − y ∗ )(xn )|

where .{xn } is a dense subset of X. Hence, for any .r ∈ (0, ∞), the closed ball Br (X∗ ) is weak star .(w ∗ ) compact.

.

1.3 Multi Valued Functions In this section we briefly present some prominent results from the theory of multi functions also called set-valued maps. These results are used later in the study of differential inclusions. Let X be a metric space, Y a Banach space, .2Y the set of all subsets of Y called the power set of Y , and .∅ the empty set. A map F from X to Y .2 \ ∅ is called a multi function. A multi function .F : X → 2Y \ ∅ is said to be upper semicontinuous at .x0 ∈ X, if for every open set .𝒪 ⊂ Y containing the set .F (x0 ), there exists an .ε > 0 such that .F (x) ⊂ 𝒪 for all .x ∈ Nε (x0 ), where .Nε (x0 ) is the .ε neighbourhood of .x0 . The multi function F is said to be upper semicontinuous if it is so for every .x ∈ X.

10

1 Background Materials From Analysis

A multi function F is said to be lower semicontinuous at the point .x0 ∈ X, if for every open set .𝒪 ⊂ Y for which .F (x0 ) ∩ 𝒪 = ∅, there exists an .ε > 0 such that .F (x) ∩ 𝒪 = ∅ for all .x ∈ Nε (x0 ). The multi function F is said to be lower semicontinuous if it is so for every .x ∈ X. Let .(, ) be a measurable space and .Z a topological space and .F :  → 2Z \ ∅. The multi function F is said to be measurable if for every open set .𝒪 ⊂ Z the set .F − (𝒪) = {ω ∈  : F (ω) ⊂ 𝒪} ∈ . In the study of differential inclusions, which can be used as the dynamic model for uncertain systems and variational inequalities, we need selection theorems. Let X, Y be two topological spaces, and let .c(Y ), .cc(Y ), and .cbc(Y ) denote the class of nonempty closed, closed convex, closed bounded convex, subsets of Y , respectively. Let .F : X → cc(Y ) be a multi function. A single valued function .f : X → Y is called a selection of the multi function F , if .f (x) ∈ F (x) for all .x ∈ X. It is obvious that given any nonempty set-valued function .F (x), there is always a point .s(x) ∈ F (x) for every .x ∈ X. The question is: does the function .s = s(x), x ∈ I , so constructed have the property that we may need, for example, continuity, measurability, etc.? This is where selection theorems play a significant role by introducing additional properties that guarantee the required regularity properties. For our purpose, in some problems we need continuous selections and for some others we need merely measurable selections. There are two well-known results in the literature, one known as Michael’s selection theorem proving existence of continuous selection, and the other, known as the Kuratowski and Ryll-Nardzewski selection theorem proving existence of measurable selections. Here we present both the results. First, let us recall the notion of paracompact spaces. A topological space X is called a paracompact space if every open cover has a locally finite refinement. The following theorem is known as Michael’s selection theorem originally proved by Michael [90]. Theorem 1.3.1 (Michael’s Selection Theorem) Let X be a paracompact space and Y a Banach space, and .F : X → cc(Y ) is a lower semicontinuous multi function. Then, F admits a continuous selection, that is, there exists a continuous function .f : X → Y such that .f (x) ∈ F (x) for all .x ∈ X. Proof See Hu and Papageorgiou [90, Theorem 4.6, p. 92].



This selection theorem is used later in the study of measure solutions for differential inclusions, where the multi functions are lower semicontinuous. In case the multi functions are only measurable we need selection theorems for measurable multi functions. Definition 1.3.2 (1): A topological space X is said to be a Polish space if it is metrizable and with respect to the metric it is a complete separable metric space. (2): A topological space is Souslin (Lusin) if it is a continuous (and bijective) image of a Polish space.

1.4 Bibliographical Notes

11

The following theorem is due to Kuratowski and Ryll-Nardzewski [90]. Theorem 1.3.3 (Kuratowski-Ryll-Nardzewski Selection Theorem) Let .(, ) be a measurable space and X a Polish space, and .F :  → c(X) a measurable multi function. Then, F admits a measurable selection, in the sense that there exists a . measurable function f such that .f (ω) ∈ F (ω), ∀ ω ∈ . Proof See Hu and Papageorgiou [90, Theorem 2.1, p. 154].



Another measurable selection theorem of significant interest used in this book is known as Yankov-von Neumann-Aumann selection theorem. Theorem 1.3.4 (Yankov-von Neumann-Aumann Selection Theorem) If .(, ) is a complete measurable space and X is a Souslin space with .B(X) denoting the sigma algebra of Borel sets in X, and .F :  → 2X \ ∅ is a multi function such that its graph, given by Gr(F ) = {(ω, z) : z ∈ F (ω)},

.

is . × B(X) measurable, then F admits a measurable selection, that is, there exists a measurable function .f :  −→ X such that .f (ω) ∈ F (ω) for all .ω ∈ . Proof See Hu and Papageorgiou [90, Theorem 2.14,p158].



Remark 1.3.5 For detailed study of selection theorems asserting existence of continuous selections, measurable selections, .Lp (p ≥ 1) selections, see the excellent book of Hu and Papageorgiou [90].

1.4 Bibliographical Notes In general, a nonlinear continuous function of a weakly convergent (or weak star convergent) sequence in any Banach space may not converge to the function of the weak limit. This was observed by L. C. Young in his study of calculus of variations that led to the development of Young measure [125, 126]. For further development of Young measure and its refinement see Ball [48]. Later with the development of control theory the notion of Bang-Bang control (and chattering control) was introduced by LaSalle leading to the so called LaSalle bang-bang principle [88]. Chattering control was also developed by Gamkrelidze [77]. These are given by sums of Dirac measures. Let .(S, S, λ) be a finite measure space and let .{zk } be a sequence converging weakly to .z0 in .Lp (S, R n ). Then, there exists a probability measure .μs ∈ P(R n ) such that for every . ∈ C b (R n ), along a subsequence, relabeled as the original sequence, .(zk (s)) −→ R n (ξ )μs (dξ ) for .λ almost

12

1 Background Materials From Analysis

n all .s ∈ S. More generally, there exists a .μ ∈ Lw q (S, P(R )) such that for every n .(s, ξ ) ∈ Lp (S, Cb (R )),



 .

lim

k→∞ S

(s, zk (s))dλ =

S×R n

(s, ξ )μs (dξ )dλ,

where .1 < q ≤ ∞ and .1 ≤ p < ∞, satisfying .1/p + 1/q = 1. The measure .μ is known as the Young measure. Theorem 1.2.7, proved for infinite dimensional spaces, is far reaching covering the Young measure and much more. It was independently proved by Tulcea and Tulcea [121] entirely based on functional analysis and topology. This theorem is used extensively in this monograph.

Chapter 2

Measure Solutions for Deterministic Evolution Equations

There are several notions of solutions for differential equations on infinite dimensional spaces. Putting them in the increasing order of generality, they are (1) classical solutions (2) strong solutions (3) mild solutions (4) weak solutions, and finally (5) measure-valued solutions. For detailed explanation on these various notions of solutions, see [5]. For a brief illustration, let X be a Banach space and consider the differential equation x˙ = Ax + f (t), x(0) = x0 , t ∈ I = [0, T ],

.

on the space X with .x0 ∈ X and .f ∈ L1 (I, X). Here A is an unbounded, closed densely defined linear operator generating a .C0 -semigroup .S(t), t ∈ [0, T ] which are bounded linear operators in X. For details on semigroup theory and its applications to abstract differential equations and control theory, see [4, 5, 39, 114]. These notions of solutions can be described informally as follows: 1. Classical solutions are X valued functions on I satisfying (i) .x ∈ C((0, T ], X) ∩ C 1 ((0, T ), X), (ii) .x(t) ∈ D(A) for .t ∈ (0, T ), and (iii) x satisfies the above differential equation for all .t ∈ (0, T ). 2. Strong solutions are also continuous functions of time taking values in the Banach space X and belonging to the domain of the differential operator A almost everywhere on .(0, T ), and that both .x˙ and Ax belong to .L1 ([0, T ], X). 3. Mild solutions are also continuous functions of time taking values in the Banach space X, but generally neither differentiable nor contained in the domain of A unless the semigroup generated by A is analytic. It is given by the expression  x(t) = S(t)x0 +

.

t

S(t − s)f (s)ds, t ∈ I.

0

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. U. Ahmed, S. Wang, Measure-Valued Solutions for Nonlinear Evolution Equations on Banach Spaces and Their Optimal Control, https://doi.org/10.1007/978-3-031-37260-5_2

13

14

2 Measure Solutions for Deterministic Evolution Equations

4. Weak solutions are also functions of time, not necessarily differentiable, nor do they belong to the domain of the operator A. It is given by any function .x ∈ C([0, T ], X) that satisfies the following identity ∗



x0 , y (0) .

X,X∗

T

+

f (t), y ∗ (t)X,X∗ dt

0



T

=−

x(t), (y˙ ∗ (t) + A∗ y ∗ (t))X,X∗ dt,

0

for every .y ∗ ∈ C([0, T ], X∗ ) satisfying .y ∗ (T ) = 0 and .y˙ ∗ + A∗ y ∗ ∈ L1 (I, X∗ ). The notion of weak solutions works very well for linear problems, and some special nonlinear problems admitting specific nonlinear operators A satisfying some regularity properties. 5. Measure solutions are functions of time taking values not in X but in the space of measures .M(X) on X, and certainly cannot belong to the domain of A. This is the class of most general (relaxed) solutions. The first four classes of solutions are well understood as pathwise solutions traversing through X with time. For existence of solutions in the sense of any of these four classes, strong conditions are required for the generator A and the function f including the initial state .x0 . For the class of measure-valued solutions, these requirements are significantly relaxed to be seen throughout this book. There is another notion of solutions called “relaxed solutions” used in control theory. In case the vector field f is a function of state x and control u taking values in a metric space U , it is often the case that if U is not convex, optimal controls may not exist. In this case the vector field f is convexified by use of relaxed controls giving fˆ(t, x, ut ) ≡

 f (t, x, ξ )ut (dξ )

.

U

where u is a strongly measurable function on I taking values in the space of probability measures on the metric space U . This notion of relaxed controls is not new. It has been extensively used by Warga [122], Cesari [58], Ahmed and Teo [39], Papageorgiou [112], Frankowska [76], and Borkar [53], among others. Relaxed controls have been used also in many of the papers of the authors cited in this book. Some authors describe the solutions of the convexified system x˙ = Ax + fˆ(t, x, ut ), x(0) = x0 , t ∈ I,

.

as “relaxed solutions”. It is worth noting that they are functions of time taking values in the Banach space X, rather than in the space of measures. So the terminology may be misleading. To avoid the possibility of confusion we have consistently used the term measure-valued solutions instead of relaxed solutions.

2.1 Evolution Equations with Continuous Vector Fields

15

In this chapter we consider deterministic evolution equations on Banach spaces. Under standard assumptions, such as uniform or local Lipschitz properties, one can prove existence of pathwise solutions, that is solutions taking values in the state space such as X. As we will see shortly, in infinite dimensional spaces, even under the assumption of continuity of the vector fields such as f with .A = 0, the evolution equation does not have a solution. We admit nonlinear vector fields which do not possess the standard conditions and hence pathwise solutions do not exist. However, under fairly relaxed assumptions we prove existence of measure-valued solutions. In Sect. 2.1, we consider evolution equations with continuous vector fields; in Sect. 2.2, we consider more relaxed assumptions; and in Sect. 2.3, we consider measurable vector fields.

2.1 Evolution Equations with Continuous Vector Fields 2.1.1 Motivation For motivation let us consider the evolution equation x˙ = Ax + f (x), t ≥ 0, x(0) = ξ

.

(2.1)

in a Banach space E, where A is the infinitesimal generator of a .C0 semigroup, S(t), .t ≥ 0, on E and .f : E −→ E is a continuous map. It is well-known that if E is finite dimensional, mere continuity of f is sufficient to prove the existence of local solutions with possibly a finite blowup time [3]. If E is an infinite dimensional Banach space this is no longer true unless the semigroup .S(t), .t > 0, is compact, for example, see [5, Theorem 5.3.6]. An interesting example was given by Dieudonne [67], proving nonexistence of solutions of an infinite dimensional system with f merely continuous. To support this statement we present the following example due to Dieudonne [67]. Let .E = c0 be the sequence space, .E = (xn , n ≥ 1) = (x1 , x2 , · · · ) with norm .|x| ≡ supn∈N |xn | satisfying .limn→∞ xn = 0. It is well-known that with respect to this norm, it is a Banach space. Define the map .

   f (x) ≡ fn (xn ) = |xn | + 1/(1 + n), xn ∈ R, n = 1, 2, · · · .

.

It is clear that f is a continuous map from E to E. Now consider the evolution equation on E, x˙ = f (x), x(0) = 0, t ≥ 0.

.

16

2 Measure Solutions for Deterministic Evolution Equations

For each .n ∈ N, it is easy to verify that the following scalar equation x˙n = fn (xn ), xn (0) = 0

.

(2.2)

has a continuous solution and that .|xn (t)| ≥ t 2 /4 for all .n ∈ N , hence .x(t) = (xn (t), n ≥ 1) ∈ / E for any .t > 0. Another example on a Hilbert space, due to Godunov [80], also demonstrates the nonexistence of a solution even though f is continuous. This led to the notion of measure-valued solutions seen in studies like [9, 10, 12, 14, 19, 20, 22, 23, 73]. Thus, generalizing the notion of solutions beyond the notions of classical, strong, weak, and the so-called mild solutions, it has been possible to prove existence of measure-valued solutions for a very broad class of systems on Banach spaces as seen throughout this monograph.

2.1.2 Introduction In the seminal article [72], Fattorini introduced the concept of relaxed solutions, more precisely, measure-valued solutions. One of the authors of this monograph further relaxed this notion and showed that under much milder hypothesis on f Eq. (2.1) has generalized (or measure-valued) solutions [9]. For this purpose, we need the characterization of the dual of the Banach space .L1 (I, X), where .I ≡ [0, T ] is a finite interval of the real line and X is a Banach space. Let .X∗ denote the topological (or continuous) dual of X, and .·, · the duality pairing of .X∗ and X. We have seen in Chap. 1 that an .X∗ valued function h is .w ∗ measurable if and only ∗ if .h(·), x is a scalar valued measurable function for each .x ∈ X. Let .Lw ∞ (I, X ) ∗ denote the class of all .w measurable functions h, for which there exists a finite number .α > 0 such that for every .x ∈ X, .|h(t), x| ≤ αh |x|X , for almost all (possibly depending on x) .t ∈ I . The space is furnished with the norm .|h|Lw∞ (I,X∗ ) = αh , where .αh is the smallest number .α for which the inequality is satisfied. Clearly, this result follows from Theorem 1.2.7 of Chap. 1. Let Z denote any normal topological space and .BC(Z) the space of bounded continuous functions on Z with the topology of sup norm, and let .Mrba (Z) denote the space of regular bounded finitely additive set functions on . endowed with total variation norm, where . denotes the algebra generated by the closed subsets of Z. With respect to these topologies, these are Banach spaces and the dual of .BC(Z) is .Mrba (Z) (see [69, Theorem 2, p. 262]). Note that if Z is a compact Hausdorff space, .Mrba (Z) = Mrca (Z), with the latter space being the space of regular countably additive measures. Let .rba (Z) ⊂ Mrba (Z) denote the class of regular finitely additive probability measures furnished with the relative topology. It follows from the characterization result discussed above that the dual of .L1 (I, BC(Z)) is given by .Lw ∞ (I, Mrba (Z)) which is furnished with the weak star topology. First we consider the following Cauchy problem in the Banach space X, x˙ = Ax + f (t, x), t ∈ I, x(0) = x0 ∈ X,

.

(2.3)

2.1 Evolution Equations with Continuous Vector Fields

17

where A is the generator of a .C0 semigroup .S(t), .t ≥ 0, in X, and f is measurable in t on I for each .x ∈ X, and continuous on X for almost all .t ∈ I, and it satisfies a linear growth condition such as |f (t, x)|X ≤ K(t)(1 + |x|X ), x ∈ X

.

with .K ∈ L+ 1 (I ). This is a sufficient condition (almost close to necessary conditions) guaranteeing a priori bounds for arbitrary but finite intervals of time. If f were also Lipschitz (or locally Lipschitz), the problem would have a global (or local) mild solution given by the solution of the following integral equation 

t

x(t) = S(t)x0 +

.

S(t − s)f (s, x(s))ds, t ∈ I.

(2.4)

0

In any case, one can use this equation to determine the a priori bounds. Using the facts, that .S(t), .t ≥ 0, is a semigroup, f satisfies the linear growth as indicated above, and .I ≡ [0, T ] is a finite interval, one can easily deduce from Grönwall’s inequality that there exists a ball .Br ⊂ X of radius .r < ∞ centered at the origin such that .x(t) ∈ Br for all .t ∈ I . Following Fattorini’s notion of generalized solutions we have the following definition. Definition 2.1.1 An element .λ ∈ Lw ∞ (I, Mrba (Br )) is said to be a generalized solution of the evolution equation (2.1) if for each .e∗ ∈ X∗ , the dual of X, the following identity holds 

e∗ , ξ λt (dξ ) = S ∗ (t)e∗ , x0 

.

Br





t

+

ds 0

S ∗ (t − s)e∗ , f (s, ξ )λs (dξ ), t ∈ I.

(2.5)

Br

Assuming X to be a reflexive Banach space and using an approximation theory [72, 73], Fattorini proved the existence of a measure-valued solution for the problem (2.1). Here we shall present a direct proof. Theorem 2.1.2 Let X be a Banach space with the dual .X∗ . Consider the system (2.3) and suppose A is the generator of a .Co semigroup .{S(t), t ≥ 0} on X and f is a Borel measurable map from .I × X to X, Bochner integrable on I for each .x ∈ X and continuous in .x ∈ X for almost all .t ∈ I and satisfies the growth condition as stated above. Let .Br ⊂ X denote the closed ball of radius r centered at the origin. Then, for a sufficiently large .r > 0, the system has at least one measurevalued solution .λ ∈ Lw ∞ (I, Mrba (Br )) that satisfies the integral equation (2.5). Further, .t → λt is .w ∗ continuous on I . Proof Using the expression (2.4), the growth condition of f , and Grönwall’s inequality, one can easily verify the existence of a closed ball .Br ⊂ X, of radius .r > 0 centered at the origin, such that if the system has any solution at all, it must

18

2 Measure Solutions for Deterministic Evolution Equations

be contained in .Br . Define the operator L on .Lw ∞ (I, Mrba (Br )) as follows,  .

e∗ , xμt (dx) −

(Lμ)(t) ≡



t

S ∗ (t − s)e∗ , f (s, x)μs (dx), t ∈ I,

ds 0

Br ∗



Br

∗

ξ(t) ≡ S (t)e , x0 , t ∈ I. Then, the functional equation (2.5) can be written compactly as Lμ = ξ.

.

(2.6)

We prove the existence of a solution of this equation. Clearly, for each .e∗ ∈ X∗ , we have .ξ ∈ L∞ (I ). By virtue of weak star continuity of the adjoint semigroup .S ∗ (t), .t ≥ 0, the function .ξ is also continuous. In any case, it is clear that L : Lw ∞ (I, Mrba (Br )) −→ L∞ (I )

.

and that it is a continuous linear map. One can easily verify that the dual .L∗ of this map is given by (L∗ η)(t, x) = e∗ , xη(t) −



T

.

ds η(s)S ∗ (s − t)e∗ , f (t, x), t ∈ I,

t

for .η ∈ L1 (I ). It is clear from the above expression and continuity of f in x that L∗ : L1 (I ) −→ L1 (I, BC(Br ))

.

and that it is also a bounded linear map. Thus, there exists a constant .C > 0 such that |L∗ η|L1 (I,BC(Br )) ≤ C|η|L1 (I ) , η ∈ L1 (I ).

.

Define   Y ≡ y ∈ L1 (I, BC(Br )) : y = L∗ η for some η ∈ L1 (I ) .

.

Clearly, this is a linear vector space and is the image of .L1 (I ) under the map .L∗ . We introduce a norm on this space as follows. For .y ∈ Y , define   |y|Y ≡ sup |η|L1 (I ) , η ∈ L1 (I ) : L∗ η = y .

.

It is not difficult to verify that this defines a norm on the vector space Y and that, with respect to this norm topology, Y is a Banach space. Further, for each .η ∈ L1 (I ) we have .|L∗ η|Y ≥ |η|L1 (I ) . Thus, .L∗ is an isomorphism of .L1 (I ) onto Y denoted

2.1 Evolution Equations with Continuous Vector Fields

19

by .L∗ ∈ Iso(L1 (I ), Y ). Consider Eq. (2.6) and note that, for the given .ξ ∈ L∞ (I ) as shown above this equation, ξ (η) ≡ (ξ, η)L∞ (I ),L1 (I )

.

defines a continuous linear functional on .L1 (I ). Since .L∗ ∈ Iso(L1 (I ), Y ), it is clear that the composition map .ξ o(L∗ )−1 given by (ξ o(L∗ )−1 )(y) ≡ ξ ((L∗ )−1 y)

.

is a continuous linear functional on Y . Hence, there exists a .μo ∈ Y ∗ , not necessarily unique, such that (ξ o(L∗ )−1 )(y) = μo , yY ∗ ,Y ,

.

for all .y ∈ Y ⊂ L1 (I, BC(Br )). Therefore, by virtue of the Hahn-Banach extension theorem (see [97, Theorem 4.2.2, p. 87]), there exists a .λo ∈ (L1 (I, BC(Br )))∗ = o Lw ∞ (I, Mrba (Br )) such that its restriction to Y coincides with .μ , while preserving norm. Hence, we have (ξ o(L∗ )−1 )(y) = λo , yLw∞ (I,Mrba (Br )),L1 (I,BC(Br )) , y ∈ Y.

.

Again, by virtue of the isomorphism, this is equivalent to ξ (η) = λo , L∗ η, η ∈ L1 (I ).

.

Thus, we have .ξ (η) = (Lλo , η)L∞ (I ),L1 (I ) , for all .η ∈ L1 (I ). This implies that o o .Lλ = ξ and hence .λ is a solution of Eq. (2.6). The last statement of the theorem follows from the fact that both the terms on the right-hand side of Eq. (2.5) are real valued continuous functions in .t ∈ I and hence the left-hand member is also 

continuous in .t ∈ I . Since .e∗ ∈ E ∗ is arbitrary this proves the theorem. Remark 2.1.3 The technique used by Fattorini is known as the principle of transposition. It is a very powerful technique and has been extensively used in the proof of existence of solutions of general linear partial differential equations with general boundary conditions [101]. It has been used by one of the authors in the study of electro-magnetic communication channels based on Maxwell equations [45]. In this monograph we use a different technique which applies to a much broader class of both deterministic and stochastic evolution equations on Banach spaces without requiring any assumptions on apriori bounds. Since we are dealing with finitely additive measures and continuous functions on infinite dimensional Banach spaces X (which are not even locally compact), it is convenient to use a suitable compactification .X+ of X, so that .X+ is Hausdorff and X is a dense subspace of .X+ . In our case X is a Banach space and hence a complete metric space. It is known from general topology that every metric space is

20

2 Measure Solutions for Deterministic Evolution Equations

ˆ a Tychonoff space and that Stone-Cech compactification of a Tychonoff space is a ˆ compact Hausdorff space. Hence, the Stone-Cech compactification of X, denoted by .X+ ≡ βX, is a compact Hausdorff space. Since homeomorphic spaces are topologically equivalent, we can write .X ⊂ X+ . Interested reader may see [16, 123] ˆ for detailed steps of Stone-Cech compactification. We introduce a more general notion of generalized solutions which is applicable to systems containing nonlinearities having polynomial growth without requiring a priori hypothesis on boundedness of solutions. Let us consider the system x˙ = Ax + f (t, x), x(0) = x0 ∈ X,

.

(2.7)

with f being a general nonlinear map in X, which is continuous and bounded on bounded sets. Let .Dφ denote the Fréchet derivative of .φ ∈ BC(X) and introduce the class of functions .F given by   F ≡ φ ∈ BC(X) : Dφ exists, Dφ ∈ BC(X, X∗ ) and φ has bounded support .

.

This is used as the class of test functions. Define the operator .A with its domain given by   D(A) ≡ φ ∈ F : A(φ) ∈ BC(X+ ) ,

.

where .(Aφ)(ξ ) = (A∗ Dφ(ξ ), ξ )X∗ ,X + (Dφ(ξ ), f (t, ξ ))X∗ ,X , for .φ ∈ D(A). Note that .D(A) = ∅, for example, for .ψ ∈ F, the function .φ given by .φ(x) ≡ ψ(λR(λ, A)x), belongs to .D(A) for each .λ ∈ ρ(A), the resolvent set of A. + Definition 2.1.4 A measure-valued function .μ ∈ Lw ∞ (I, Mrba (X )) is said to be a generalized solution of Eq. (2.7) if, for every .φ ∈ D(A) with .φ and hence .Dφ having bounded support, the following equality holds

 μt (φ) = φ(x0 ) +

.

t

μs (Aφ)ds, t ∈ I,

0

where  μt (ψ) ≡

ψ(ξ )μt (dξ ), t ∈ I.

.

E

Note that the operator .A is time-dependent though not explicitly indicated. This is chosen to indicate that the domain is time-invariant. In other words, the domain remains fixed. Let .rba (X+ ) denote set of all regular finitely additive probabilw (I,  + ity measures on .X+ . For simplicity of notation, we use .M∞ rba (X )) ⊂ w + L∞ (I, Mrba (X )) to denote the set of measure-valued functions defined on I and

2.1 Evolution Equations with Continuous Vector Fields

21

+ taking values in the set .rba (X+ ). Note that .Lw ∞ (I, Mrba (X )) is a linear vector w + space while .M∞ (I, rba (X )) is a (special) bounded subset of this space.

Theorem 2.1.5 Let A be the infinitesimal generator of a .C0 semigroup in X and suppose .f : I × X −→ X is strongly measurable and integrable in t on I for each .x ∈ X; and, for almost all .t ∈ I , it is continuous and bounded in x on bounded subsets of X satisfying the following approximation properties: (ai): there exists a sequence .{fn } such that .fn (t, x) ∈ D(A) for .(t, x) ∈ I × X, and further, for almost all .t ∈ I , .fn (t, x) −→ f (t, x) in X for each .x ∈ X. (aii): there exists a pair of sequences .{αn , βn } ∈ L+ 1 (I ), possibly .αn , βn → ∞ as .n → ∞, such that .

 fn (t, x) − fn (t, y) X ≤ αn (t)  x − y X ,  fn (t, x) X ≤ βn (t)(1 + |x|), for all x, y ∈ X.

Then, for every .x0 ∈ X, the evolution equation (2.7) has at least one generalized + solution .μ ∈ Lw ∞ (I, Mrba (X )) in the sense of Definition 2.1.4. Further, .μ ∈ w + M∞ (I, rba (X )) and it is .w ∗ continuous on I . Proof Since .D(A) is dense in X and .x0 ∈ X, there exists a sequence .{x0,n } such s that .x0,n −→ x0 . Consider the Cauchy problem: x˙ = Ax + fn (t, x), x(0) = x0,n ∈ X.

.

Since for almost all .t ∈ I , .fn (t, x) ∈ D(A) for each .n ∈ N, it follows from the assumption .(aii) that this problem has a unique strong solution .xn = {xn (t), t ∈ I } and hence for almost all .t ∈ I , .xn is differentiable and .xn (t) ∈ D(A). Now let .φ ∈ D(A) with .φ, and hence .Dφ, having compact support. Then, for each .t ∈ I ,  φ(xn (t)) = φ(x0,n ) +

.

t

Dφ(xn (s)), Axn (s) + fn (s, xn (s))X∗ ,X ds. (2.8)

0

Letting .δe (dξ ) denote the Dirac measure concentrated at the point .e ∈ X, and defining .λnt (dξ ) ≡ δxn (t) (dξ ), .t ∈ I , using the notation of Definition 2.1.4, we can rewrite Eq. (2.8) as  n .λt (φ)

= φ(x0,n ) + 0

t

λns (An (s)φ)ds, t ∈ I,

(2.9)

where for all .φ ∈ D(A), (An (t)φ)(ξ ) = A∗ Dφ(ξ ), ξ X∗ ,X + Dφ(ξ ), fn (t, ξ )X∗ ,X .

.

w (I,  For each integer n, .λn ∈ M∞ rba (X)) and, as seen below, the set of measuren w (I,  + w + valued functions .{λ } is contained in .M∞ rba (X )) ⊂ L∞ (I, Mrba (X )).

22

2 Measure Solutions for Deterministic Evolution Equations

Indeed, the functional .n , given by  n (ψ) ≡

.

I ×X+

ψ(t, ξ )λnt (dξ )dt,

is well-defined for each .ψ ∈ L1 (I, BC(X+ )) and .|n (ψ)| ≤ |ψ|L1 (I,BC(X+ )) , for all .n ∈ N . Thus, the sequence .{n } is contained in a bounded subset of the space + ∗ + .(L1 (I, BC(X ))) (the space of continuous linear functionals on .L1 (I, BC(X ))), n and hence the corresponding sequence of measure-valued functions .{λ } is confined + n in a bounded subset of .Lw ∞ (I, Mrba (X )). In fact, .{λ } is contained in the set w + w + .M∞ (I, rba (X )) ⊂ L∞ (I, Mrba (X )). Thus, by Alaoglu’s theorem, there exists a generalized subsequence of the sequence .{λn }, relabeled as such, and an element w∗

+ n o λ o ∈ Lw ∞ (I, Mrba (X )), so that .λ −→ λ . Defining

.

(Fn (t)φ)(ξ ) ≡ Dφ(ξ ), fn (t, ξ ) − f (t, ξ )X∗ ,X ,

.

we can rewrite Eq. (2.9) as  λnt (φ) = φ(x0,n ) +

t

.

0

 λns (A(s)φ)ds +

0

t

λns (Fn (s)φ)ds, t ∈ I.

(2.10)

s

Since .x0,n −→ x0 in X and .φ is continuous, .φ(x0,n ) −→ φ(x0 ) as .n → ∞. For + .φ ∈ D(A), we have .Aφ ∈ L1 (I, BC(X )). Hence, it follows from weak star n o convergence of .λ to .λ that 

t

.

0

 λns (A(s)φ)ds

−→ 0

t

λos (A(s)φ)ds, t ∈ I.

Let .Fn (φ) denote the .BC(X) valued function as introduced above. Since .φ and hence .Dφ has compact support, it follows from integrability of .f (·, x) on I for each .x ∈ X and the assumption .(ai) that .Fn (φ) ∈ L1 (I, BC(X)) and that .Fn (φ) −→ 0 w∗

strongly in this space as .n → ∞. Using this and the fact that .λn −→ λo , we conclude that, as .n → ∞, 

t

λns (Fn (s)φ)ds −→ 0, for each t ∈ I.

.

0

Thus, the expression on the right-hand side of Eq. (2.10) converges to 

t

φ(x0 ) +

.

0

λos (A(s)φ)ds, t ∈ I,

for each .φ ∈ D(A) having compact support. Clearly, it follows from similar argument that, for each .t ∈ I , the expression on the left-hand side of Eq. (2.10)

2.1 Evolution Equations with Continuous Vector Fields

23

converges to .λot (φ). This shows that for each .t ∈ I , we have  λot (φ) = φ(x0 ) +

.

0

t

λos (A(s)φ)ds, t ∈ I for all φ ∈ D(A).

(2.11)

Thus, .λo is a generalized solution of Eq. (2.7) in the sense of Definition 2.1.4. The proof of the last assertion follows from the fact that the approximating sequence is a sequence of Dirac measures and that positivity is preserved under weak star limit. w (I,  + o Thus, .λo ∈ M∞ rba (X )). The weak star continuity, .t −→ λt , follows readily from the expression (2.11). This completes the proof. 

Remark 2.1.6 The class of functions .{f } that satisfies the hypothesis of Theorem 2.1.5 is fairly broad. In case the operator A is the infinitesimal generator of an analytic semigroup, the assumptions on the nonlinear operator f can be further relaxed. Remark 2.1.7 In Definition 2.1.4, as well as in Theorem 2.1.5, the initial state was assumed to be a given point .x0 ∈ X or equivalently the Dirac measure .δx0 concentrated at .x0 . In fact, there is no difficulty considering the initial state as a measure .λ0 ∈ rba (X+ ) having possibly a bounded support in X. Remark 2.1.8 Note that for each .t ∈ I , the measure solution .λt ∈ Mrba (X) ∩ Mrca (X+ ). Thus, the measure solutions restricted to X are only finitely additive while, with respect to the compact Hausdorff space .X+ , they are countably additive. This is understood without further notice. Example E1 For a simple illustration of Theorem 2.1.5, let us consider the system (2.7) on a Hilbert space X with A being dissipative and the vector field f given by .f (x) = x p−1 x for .p ≥ 1. Then, one can easily verify that .

d   x(t) 2 = 2(x(t), ˙ x(t)) dt = 2(Ax, x)+2(f (x(t)), x(t)) ≤ 2  x(t) 1+p , t ≥ 0.

(2.12)

Hence, by virtue of Grönwall’s inequality, we have .

 −1  x(t) p−1 ≤ x0 p−1 1 − t (p − 1)  x0 p−1

−1

for all .0 ≤ t < (p − 1)  x0 p−1 . This indicates that the solution may blow 

−1 up at .τ ≡ (p − 1)  x0 p−1 . For .γ > 0, define gγ (x) = ( x p−1 /(1 + γ  x p−1 ))x, γ > 0.

.

24

2 Measure Solutions for Deterministic Evolution Equations

It is easy to verify that  gγ (x) ≤ (1/γ )  x , .

 gγ (x) − gγ (y) ≤ (p/γ )  x − y , γ > 0.

Then, define fn (x) = g1/n (nR(n, A)x), n ∈ N ∩ ρ(A),

.

where .R(λ, A) is the resolvent of the operator A corresponding to .λ ∈ ρ(A), the resolvent set of A. One can easily verify that  fn (x) ≤ n  nR(n, A)x ≤ αn (1+  x ), .

 fn (x) − fn (y) ≤ np  nR(n, A)x − nR(n, A)y ≤ βn  x − y  .

Clearly, .αn , .βn −→ ∞. Note that .fn (x) ∈ D(A) for each .n ∈ N and .fn (x) −→ f (x) for each .x ∈ X. Thus, the assumption .(ai) is readily satisfied. Then, all the hypotheses of Theorem 2.1.5 hold and therefore Eq. (2.7) with f having polynomial growth has a generalized solution. Example E2 Here we present a real physical example containing polynomial nonlinearity as seen in Example E1. This is described by a second order partial differential equation, .

vtt (t, ξ ) − v(t, ξ ) + m2 v(t, ξ ) = −β|v(t, ξ )|p−1 v(t, ξ ), t > 0, ξ ∈ , v(0, ξ ) = v0 (ξ ), vt (0, ξ ) = v1 (ξ ), ξ ∈ ,

(2.13)

where . is either an open bounded domain in .R d or a domain with finite volume. The initial data is given by .{v0 , v1 } and .β is generally a positive constant and .p ≥ 1. Defining .w1 (t) ≡ v(t, ·) and .w2 (t) ≡ vt (t, ·), Eq. (2.13) can be written as an abstract differential on a suitable Hilbert space as follows w˙ = Aw + f (w), t ≥ 0, w(0) = w0 ,

.

(2.14)

where the operator A is given by A≡

.

0 I −B 2 0

with B denoting the positive square root of the positive self-adjoint operator .(− + m2 ) realized on the Hilbert space .L2 (). The natural state space for the evolution equation is then given by the Hilbert space H ≡ D(B) × L2 ()

.

2.1 Evolution Equations with Continuous Vector Fields

25

with the scalar product (w, w) ˜ H ≡ (Bw1 , B w˜ 1 )L2 + (w2 , w˜ 2 )L2 .

.

Since B is strictly positive and closed, .D(B) is a Hilbert space with respect to the scalar product .(x, y)D(B) ≡ (Bx, By)L2 , .x, y ∈ D(B). The domain of A is given by .D(A) ≡ D(B 2 ) × D(B). In fact, both the domain and range of A are in H and .(iA)∗ = iA. Therefore, by Stone’s theorem (see [5, Theorem 3.1.4, p. 71]), A generates a unitary group .{U (t), t ∈ R} of bounded linear operators in H . The nonlinear operator f is given by f (w)(ξ ) ≡

.

0 , ξ ∈ , w ∈ H. −β|w1 (ξ )|p−1 w1 (ξ )

In fact, Eq. (2.14) can be considered as a general evolution equation containing Klein-Gordon equation and many other nonlinear equations of quantum mechanics on the Hilbert space H . Using variation of constants formula one can formulate Eq. (2.14) as an integral equation in H , 

t

w(t) = U (t)w0 +

.

U (t − s)f (w(s))ds, t ∈ I.

(2.15)

0

In nonlinear wave mechanics it is generally assumed that f is locally Lipschitz in H in the sense that there exists an increasing function .c : [0, ∞) × [0, ∞) −→ [0, ∞] vanishing at zero so that .

 f (w) − f (w) ˜ H ≤ c( w ,  w˜ )  w − w˜ H .

If c is continuous and bounded on bounded subsets of .[0, ∞) × [0, ∞) then the nonlinear operator f is locally Lipschitz and bounded on bounded subsets of H . Hence, using the integral equation (2.15) one can prove existence of mild solutions locally in time. In contrast, under the given assumptions one has measure-valued solutions globally. This follows from the fact that the sequence .{fn } defined by fn (w) ≡ nR(n, A)f (Qn w)

.

satisfies the assumptions .(ai) and .(aii) of Theorem 2.1.5, where .Qr denotes the retraction of the ball .Br ⊂ H of radius .r > 0 centered at the origin. In particular, for .d = 3, it follows from Sobolev’s embedding theorem [1] that the inclusion .H 1 () → L6 () is continuous and thus for .p = 3, the nonlinear operator f satisfies the local Lipschitz property in H . For .β > 0 one has apriori bounds and hence one can prove existence of strong or weak solutions. On the other hand, for existence of measure-valued solutions, the sign of .β is immaterial and .p ≥ 1 is any finite number.

26

2 Measure Solutions for Deterministic Evolution Equations

2.2 Evolution Equations Under Relaxed Hypothesis In the previous section we have considered the vector field f to be a continuous map from a Banach space E to itself (see Theorem 2.1.5). Here we admit stronger nonlinearities. This can be achieved by considering analytic semigroups. In this section we extend the results of Sect. 2.1 to a more general class of vector fields .{f } under the additional assumption that .−A is the generator of an analytic semigroup. We consider the system x˙ = −Ax + f (t, x), x(0) = x0 ∈ E.

.

(2.16)

Without loss of generality, we may assume that .0 ∈ ρ(−A). Let .α ∈ (0, 1) and Eα ≡ [D(Aα )] denote the Banach space with respect to the topology induced by the graph norm

.

|x|α ≡ |Aα x|E .

.

Again, we denote by .Eα+ the compactification of the topological space .Eα . Let us recall that a function .φ : Eα → R is said to be cylindrical if there exist an .n ∈ N , .ei∗ ∈ E ∗ , i = 1, 2, · · · , n, and .g ∈ C(R n ) such that .φ(ξ ) = g(e1∗ , ξ , e2∗ , ξ , · · · , en∗ , ξ ) for .ξ ∈ Eα . Let .ℱc denote the class of continuous real valued functions defined on .Eα having bounded supports and continuous Fréchet derivatives with values in .E ∗ . We choose this as the class of test functions. To consider measure-valued solution we need the operator .𝒜 given by   D(𝒜) ≡ φ ∈ ℱc : 𝒜φ ∈ BC(Eα+ ) ,

.

where (𝒜φ)(ξ ) = −A∗ Dφ(ξ ), ξ E ∗ ,E + Dφ(ξ ), f (t, ξ )E ∗ ,E for φ ∈ D(𝒜).

.

Theorem 2.2.1 Let .−A be the infinitesimal generator of an analytic semigroup in E and .f : I × Eα → E is strongly measurable and Bochner integrable in t on I, for each .x ∈ Eα , and for almost all .t ∈ I , it is locally Lipschitz and bounded in x on bounded subsets of .Eα . Then, for every .x0 ∈ Eα , the evolution equation (2.16) has + at least one (generalized or measure-valued) solution .μ ∈ Lw ∞ (I, Mrba (Eα )) in the w + ∗ sense of Definition 2.1.4. Further, .μ ∈ M∞ (I, rba (Eα )) and it is .w continuous. Proof Since .D(A) is dense in E and .x0 ∈ Eα ⊂ E, there exists a sequence .{x0,n } ⊂ s D(A) such that .x0,n −→ x0 . For each .r > 0, let .Qr denote the retract of the ball .Br ⊂ Eα , of radius r around the origin, that is,  Qr (x) ≡

x,

for x ∈ Br ,

(r/|x|α )x,

otherwise.

.

2.2 Evolution Equations Under Relaxed Hypothesis

27

Define fn (t, x) ≡ nR(n, −A)f (t, Qn x), n ∈ N ∩ ρ(−A)

.

and consider the Cauchy problem x˙ = −Ax + fn (t, x), x(0) = x0,n .

.

(2.17)

Clearly, by virtue of our assumption, for each .r > 0, there exists a .βr ∈ L+ 1 (I ), possibly .|βr |L1 (I ) → ∞ as .r → ∞, such that |f (t, x)|E ≤ βr (t)(1 + |x|α ) ≤ (1 + r)βr (t), x ∈ Br , .

|f (t, x) − f (t, y)|E ≤ βr (t)|x − y|α , x, y ∈ Br .

By virtue of these local estimates, for each .n ∈ N, .fn satisfies the following inequalities: |fn (t, x)| ≤ γn (t)(1 + |x|α ), x ∈ Eα , .

|fn (t, x) − fn (t, y)| ≤ γn (t)|x − y|α , x, y ∈ Eα ,

where .γn (t) ≡ |nR(n, −A)|βn (t). Further, for almost all .t ∈ I , .fn (t, x) ∈ D(A) for each .n ∈ N. Hence, for each .n ∈ N, Eq. (2.17) has a unique strong solution .xn = {xn (t), t ∈ I } and for almost all .t ∈ I , .xn (t) ∈ D(A) and it is differentiable. Now let .φ ∈ D(𝒜) with .φ having bounded support in .Eα . Then, for each .t ∈ I ,  φ(xn (t)) = φ(x0,n ) +

.

t

Dφ(xn (s)), −Axn (s) + fn (s, xn (s))E ∗ ,E ds.

0

Letting .δe (dξ ) denote the Dirac measure concentrated at the point .e ∈ E, and defining .λnt (dξ ) ≡ δxn (t) (dξ ), .t ∈ I , we can rewrite the above expression as  n .λt (φ)

= φ(x0,n ) +

t

0

λns (𝒜n (s)φ)ds, t ∈ I,

(2.18)

where (𝒜n (t)φ)(ξ ) = −A∗ Dφ(ξ ), ξ E ∗ ,E + Dφ(ξ ), fn (t, ξ )E ∗ ,E

.

w (I,  + for all .φ ∈ D(𝒜). Clearly, for each integer n, .λn ∈ M∞ rba (Eα )) and hence the functional .n , given by

 n (ψ) ≡

.

I ×Eα+

ψ(t, ξ )λnt (dξ )dt

28

2 Measure Solutions for Deterministic Evolution Equations

is well-defined for each .ψ ∈ L1 (I, BC(Eα+ )), and |n (ψ)| ≤ ψ L1 (I,BC(Eα+ )) , ∀ n ∈ N.

.

Thus, the sequence .{n } is contained in a bounded subset of .(L1 (I, BC(Eα+ )))∗ and hence the corresponding sequence of measures .{λn } is confined in the w (I,  + w + closed bounded set .M∞ rba (Eα )) of the vector space .L∞ (I, Mrba (Eα )). By Alaoglu’s theorem, there exists a generalized subsequence of the sequence .{λn }, w (I,  + w + relabeled as .{λn }, and a .λo ∈ M∞ rba (Eα )) ⊂ L∞ (I, Mrba (Eα )), such that w∗

λn −→ λo . Defining

.

(Fn (t)φ)(ξ ) ≡ Dφ(ξ ), fn (t, ξ ) − f (t, ξ )E ∗ ,E ,

.

we can rewrite Eq. (2.18) as  λnt (φ) = φ(x0,n ) +

.

0

t

 λns (𝒜(s)φ)ds +

t 0

λns (Fn (s)φ)ds, t ∈ I.

(2.19)

s

Since .x0,n −→ x0 and .φ is continuous and bounded, we have .φ(x0,n ) → φ(x0 ). For .φ ∈ D(𝒜), we have .𝒜(·)φ ∈ L1 (I, BC(Eα+ )) and hence by .w ∗ convergence of n o .λ to .λ we have  t  t . λns (𝒜(s)φ)ds → λos (𝒜(s)φ)ds, t ∈ I. 0

0

For convenience, letting J denote the identity operator in E and .Jn the operator nR(n, −A), we can write

.

(Fn (t)φ)(ξ ) = Dφ(ξ ), (Jn − J )f (t, Qn ξ )E ∗ ,E

.

+Dφ(ξ ), f (t, Qn ξ ) − f (t, ξ )E ∗ ,E ≡ I1 (n) + I2 (n). It is known that the Yosida approximation .Jn of the identity operator J converges to J on E in the strong operator topology of .L(E). Thus, its adjoint converges in the weak operator topology to the adjoint of the identity operator pointwise on the dual space .E ∗ . By our assumption .f (t, Qn ξ ) −→ f (t, ξ ) for almost every .t ∈ I uniformly on bounded subsets of .Eα . Since each .φ ∈ ℱc has bounded support, it follows from the above facts that I1 (n) ≡ (Jn − J )∗ Dφ(ξ ), f (t, Qn ξ )E ∗ ,E −→ 0,

.

2.2 Evolution Equations Under Relaxed Hypothesis

29

uniformly in .ξ on .Eα , for almost all .t ∈ I . Considering the second term, it is evident that I2 (n) = Dφ(ξ ), f (t, Qn ξ ) − f (t, ξ )E ∗ ,E −→ 0

.

uniformly in .ξ on .Eα as well. Therefore, .Fn (t)φ → 0 in the topology of .BC(Eα ) for almost all .t ∈ I , and by dominated convergence theorem, .Fn φ → 0 strongly in w∗

+ L1 (I, BC(Eα )). Using this fact and the fact that .λn −→ λo in .Lw ∞ (I, Mrba (Eα )), we conclude that  t . λns (Fn (s)φ)ds −→ 0 for each t ∈ I

.

0

as .n → ∞. Thus, taking the limits along a generalized subsequence, the right-hand t expression of Eq. (2.19) converges to .φ(x0 ) + 0 λos (𝒜(s)φ)ds, t ∈ I, for each .φ ∈ D(𝒜) having bounded support. Hence, for each .t ∈ I , the left-hand expression of (2.19) is well-defined and converges to .λot (φ). This shows that  o .λt (φ)

t

= φ(x0 ) + 0

λos (𝒜(s)φ)ds, t ∈ I,

(2.20)

for all .φ ∈ D(𝒜) having bounded supports. Hence, .λo is a generalized solution of Eq. (2.16) in the sense of Definition 2.1.4. The proof of the last assertion of the theorem follows from the fact that the approximating sequence is a sequence of Dirac measures and clearly are all positive and that positivity is preserved under w (I,  + w + weak star limit. Thus, .λo ∈ M∞ rba (Eα )) ⊂ L∞ (I, Mrba (Eα )). The weak o star continuity .t → λt follows easily from the expression (2.20). This completes 

the proof. As a corollary of Theorem 2.2.1 we have the following result. Corollary 2.2.2 Let .−A be the infinitesimal generator of an analytic semigroup in E and .f : I × Eα → E is strongly measurable in t on I for each .x ∈ Eα and for almost all .t ∈ I , it is locally Lipschitz in x on .Eα . Further, there exists a .K ∈ L+ 1 (I ) and an integer m such that |f (t, x)|E ≤ K(t)(1 + |x|m α ), t ∈ I, x ∈ Eα .

.

Then, for every .x0 ∈ Eα , the evolution equation (2.16) has at least one generalized + w + ∗ solution .μ ∈ Lw ∞ (I, Mrba (Eα )). Further, .μ ∈ M∞ (I, rba (Eα )) and it is .w continuous. Proof The assumptions of the corollary imply those of Theorem 2.2.1 and hence the statement of the corollary follows. 

30

2 Measure Solutions for Deterministic Evolution Equations

2.2.1 Competing Notions of Solutions In the case of infinite dimensional systems, it is well-known that there are several notions of solutions such as classical, strong, mild, and weak solutions [5]. It is interesting to note that every classical, strong, mild and weak solution is a generalized (or measure-valued) solution but the converse is false. Since classical and strong solutions are special cases of mild or weak solutions, we show that the mild and weak solutions are measure solutions. For illustration, let us consider the system given by Eq. (2.3). Its mild solution is given by the solution of the integral equation (2.4) on the Banach space X; and its measure solution is given by the functional equation (2.5) on the topological space w (I,  + w + like .M∞ rba (X )) which is a subset of .L∞ (I, Mrba (X )). First we show that any mild solution is a measure solution. The mild solution of the following equation y˙ = Ay + f (t, y), y(0) = y0 , t ∈ I ≡ [0, T ],

.

(2.21)

is given by the solution of the associated integral equation 

t

y(t) = S(t)y0 +

.

S(t − θ )f (θ, y)dθ, t ∈ I,

(2.22)

0

where .S(t), .t ≥ 0, is the semigroup corresponding to the generator A. We know that if f has linear growth, there exists a closed ball .Br ⊂ X of radius .r < ∞ such that the solutions are confined in .Br . Scalar multiplying on either side of the above expression by any .x ∗ ∈ X∗ we obtain x ∗ , y(t) = S ∗ (t)x ∗ , y0  +



t

.

S ∗ (t − θ )x ∗ , f (θ, y)dθ, t ∈ I.

(2.23)

0

Introducing the measure-valued function .μt (·) ≡ δy(t) (·) along the path y, we arrive at the following expression, 

x. ∗ , ξ μt (dξ ) = S ∗ (t)x ∗ , y0 

Br

+

 t 0

S ∗ (t − θ )x ∗ , f (θ, ξ )μθ (dξ )dθ, t ∈ I.

(2.24)

Br

Thus, .μ satisfies Definition 2.1.1 and hence it is a measure-valued solution of Eq. (2.21). This shows that every mild solution is a measure solution. Next we show that every weak solution is a measure solution. A function y is a weak solution of Eq. (2.21), if for every .η ∈ C 1 (I ) and .x ∗ ∈ D(A∗ ) ⊂ X∗ , it

2.2 Evolution Equations Under Relaxed Hypothesis

31

satisfies the following equation  .

−

x ∗ , y(t)ηdt ˙ = x ∗ , y0 η(0) +

I





 A∗ x ∗ , y(t) + x ∗ , f (t, y(t)) η(t)dt.

I

(2.25) Since the weak solution .y ∈ L1 (I, X), it is clear that .φ(y(t)) ≡ x ∗ , y(t) is a Lebesgue integrable scalar valued function for every .x ∗ ∈ X∗ . Thus, choosing ∗ .φ(ξ ) = x , ξ  and .λt = δy(t) , the Dirac measure concentrated along the path traced by y in the Banach space X, Eq. (2.25) can be written as  .



−

λt (φ)ηdt ˙ = λ0 (φ)η(0) + I

λt (𝒜φ)η(t)dt I

for every .η ∈ C 1 (I ) satisfying .η(T ) = 0. Integrating by parts, it follows from the above expression that  .

I

d λt (φ) − λt (𝒜φ) η(t)dt = 0, dt

(2.26)

for every .η ∈ C(I ) satisfying the terminal condition. Since this holds for every d such .η, we conclude that the identity . dt λt (φ) − λt (𝒜φ) = 0 holds in the sense of distribution. This shows that  t .λt (φ) = λ0 (φ) + λs (𝒜φ)ds, t ∈ I, (2.27) 0

and hence .λ is a measure-valued solution and thus we conclude that every weak solution is a measure-valued solution. Uniqueness Unfortunately, at this stage, we do not have a satisfactory result on the question of uniqueness of measure-valued solutions. We present here only a partial result. We will revisit this question in the sequel. For simplicity, consider the system (2.21) and suppose f is independent of time. If the operator .𝒜 has a countable set .{γi , φi } of eigenvalues and eigenvectors and .D(𝒜) is dense in + .BC(X ) then the measure solution is unique. This is justified as follows. Suppose + the system (2.21) has two measure-valued solutions .λ, .μ ∈ Lw ∞ (I, rba (X )). w + Then, it follows from Eq. (2.27) that .β ≡ (λ − μ) ∈ L∞ (I, Mrba (X )) satisfies the equation  βt (φ) =

.

0

t

βs (𝒜φ)ds for all φ ∈ D(𝒜) and t ∈ I.

32

2 Measure Solutions for Deterministic Evolution Equations

Thus, for each .i ∈ N, 

t

βt (φi ) = γi

.

βs (φi )ds for all t ∈ I,

0

and hence it follows from Grönwall’s inequality that for each .i ∈ N, .βt (φi ) ≡ 0 for all .t ∈ I . Since .D(𝒜) is dense in .BC(X+ ) we have .βt (φ) ≡ 0 for all .φ ∈ BC(X+ ) + and consequently .β = λ − μ = 0 ∈ Lw ∞ (I, Mrba (X )), justifying uniqueness. Thus, combining the existence theorem with the uniqueness result we conclude that the evolution equation .

(d/dt)μt = 𝒜 ∗ μt , t ∈ I, μ0 = δy0 ,

is well-posed in the Banach space .Mrba (Eα+ ) in the sense that .𝒜 ∗ is the infinitesimal generator of a .w ∗ continuous semigroup .𝒮(t), .t ≥ 0, of bounded linear operators on .Mrba (X+ ), and that .μt = 𝒮(t)μ0 , .t ≥ 0, is its mild solution. Further, .μ satisfies the above equation also in the weak sense, that is,  μt (φ) = μ0 (φ) +

.

t

μs (𝒜φ), t ≥ 0, for each φ ∈ D(𝒜).

0

2.2.2 Quasilinear Problems So far we have considered semilinear problems. Naturally, it would be interesting to consider and extend these results to quasilinear problems given by x˙ + A(x)x = f (x), t ∈ I ≡ (0, T ], x(0) = x0 .

.

(2.28)

Here we present a result with reference to quasilinear parabolic problems. The semilinear version of this problem has been studied extensively in the literature [39, 72, 74, 100] proving existence of pathwise solutions based on standard techniques. Let H be a Hilbert space and V a reflexive Banach space, a subspace of H , with .V ∗ being the dual of the space V giving the so-called Gelfand triple .{V , H, V ∗ }. We assume that the embeddings .V → H → V ∗ are continuous and dense. Suppose the operator valued function .A ∈ C(H, ℒ(V , V ∗ )) and it satisfies the following conditions:   (i) .A : H → ℒ(V , V ∗ ), sup |A(ξ )|ℒ(V ,V ∗ ) , ξ ∈ H < ∞, (ii) .A(ξ )v, vV ∗ ,V ≥ δ|v|2V , ∀ v ∈ V , ∀ ξ ∈ H for some constant .δ > 0, (iii) .f : H → V ∗ , is continuous and bounded on bounded sets. Let .ℱc ≡ {φ ∈ BC(V ∗ ) : Dφ ∈ BC(V ∗ , V )} denote the class of cylindrical derivatives continuous functions on .V ∗ having first Fréchet   and bounded. Define the operator .𝒜 by .D(𝒜) ≡ φ ∈ ℱc : 𝒜φ ∈ BC(H + ) , with .H + denoting the

2.2 Evolution Equations Under Relaxed Hypothesis

33

ˆ compact Hausdorff space obtained by Stone-Cech compactification of H . Set (𝒜φ)(ξ ) ≡ −A∗ (ξ )Dφ(ξ ), ξ V ∗ ,V + Dφ(ξ ), f (ξ )V ,V ∗ , for φ ∈ D(𝒜).

.

We prove the following result. Theorem 2.2.3 Consider the quasilinear system (2.28) and suppose the operator A satisfies the hypotheses (i) and (ii), and f satisfies the assumption (iii), and that the embeddings .V → H → V ∗ are compact. Then, for each .x0 ∈ H or + ∗ .λ0 ∈ rba (H ), the system (2.28) has at least one .w continuous measure-valued w + solution .λ ∈ M∞ (I, rba (H )) in the sense that  λt (φ) = λ0 (φ) +

.

t

λs (𝒜φ)ds, t ∈ I

0

for all .φ ∈ D(𝒜) with .φ having bounded supports in H . Proof For .0 < γ < ∞, define .fγ (ξ ) ≡ f (Qγ (ξ )), where .Qγ is the retract of the closed ball .Bγ ⊂ H . Then, consider the truncated system given by x˙ + A(x)x = fγ (x), t ∈ I, x(0) = x0 .

.

(2.29)

First we show that, for each .x0 ∈ H , this equation has at least one weak solution .xγ ∈ C(I, H ) ∩ L2 (I, V ). This is proved by using Galerkin approach. We construct an approximating sequence .{xγn , n ∈ N} which solves a sequence of finite dimensional (n-dimensional) ordinary differential equations representing finite dimensional projections of the infinite dimensional problem to a suitable increasing family of finite dimensional subspaces .Xn ⊂ V ⊂ H ⊂ V ∗ , with n n .xγ (0) = x (0) being the projection of .x0 to .Xn . Using the assumptions (i), (ii) and (iii), one can show that this sequence satisfies the following estimates:  .

|xγn (t)|2H 

t 0

+δ 0

t

|xγn (s)|2V ds ≤ |x0 |2H + (b2 (γ )/δ) < ∞,

|x˙γn (s)|V ∗ ds ≤ C 2 (γ ) < ∞, ∀ n ∈ N,

where .

  b(γ ) ≡ sup |fγ (ξ )|V ∗ , ξ ∈ H , C(γ ) ≡ 2b(γ )[(a 2 /δ 2 ) + T ]1/2 ,   a ≡ sup |A(ξ )|ℒ(V ,V ∗ ) , ξ ∈ H .

34

2 Measure Solutions for Deterministic Evolution Equations

Define .W ≡ {z : z ∈ L2 (I, V ), z˙ ∈ L2 (I, V ∗ )} with the norm topology given by 1/2  |z|W ≡ |z|2L2 (I,V ) + |˙z|2L2 (I,V ∗ ) .

.

It is a normed space and its completion with respect to the norm topology, denoted by .W again, is a Banach space. Since V is assumed to be a reflexive Banach space, .V ∗ is reflexive, and hence .W is also a reflexive Banach space. It follows from the above estimates that, for any fixed .γ ∈ (0, ∞), the set .{xγn , n ∈ N} is a bounded subset of .W. Hence, there exists a subsequence of the sequence .{xγn }n≥1 w

and an element .xγ ∈ W such that, along this subsequence, .xγn → xγ in .W. It is well-known that the embedding .W → C(I, H ) is continuous and hence .xγ ∈ C(I, H ) also. It remains to verify that .xγ is a weak solution of Eq. (2.29). Under the assumption of compact embeddings .V → H → V ∗ , it follows from a result due to Lions [127, p. 450] that the space .W is compactly embedded in n .L2 (I, H ). Thus, the sequence .{xγ , n ∈ N} not only converges weakly in .W but also strongly in .L2 (I, H ). Since .ξ → A(ξ ) is continuous and bounded from H to ∗ .ℒ(V , V ), we conclude that along a subsequence, if necessary, A∗ (xγn (t)) −→ A∗ (xγ (t)), for almost every t ∈ I

.

in .ℒ(V , V ∗ ) with respect to the strong operator topology. In other words, for each .v ∈ V , s

A∗ (xγn (t))v −→ A∗ (xγ (t))v, in V ∗ for almost every t ∈ I.

.

Since .ξ −→ A(ξ ), as an element of .L(V , V ∗ ), is uniformly norm bounded on H and I is a finite interval, it follows from dominated convergence theorem that ∗ n ∗ ∗ .A (xγ (·))v converges to .A (xγ (·))v (strongly) in .L2 (I, V ). Combining this with n the fact that .xγ converges weakly to .xγ in .L2 (I, V ), we obtain  .

lim

n→∞ I

∗

A

 (xγn (t))v, xγn (t)V ∗ ,V dt

A∗ (xγ (t))v, xγ (t)V ∗ ,V dt. (2.30)

= I

Using a similar argument it is easy to verify that 

 .

lim

n→∞ I

fγ (xγn (t)), vV ∗ ,V dt

=

fγ (xγ (t)), vV ∗ ,V .

(2.31)

I

According to weak formulation, one can verify that .xγn is a weak solution of the Cauchy problem x˙γn + A(xγn )xγn = f (xγn ), xγn (0) = x0n , t ∈ I.

.

2.2 Evolution Equations Under Relaxed Hypothesis

35

Scalar multiplying on either side of the above equation with the product .η(t)v, for any .η ∈ C 1 (I ) and .v ∈ Xn ⊂ V , and integrating by parts we obtain the following expression  −x n (0), vη(0) −

.

I

 = I

 xγn (s), vη(s)ds ˙ +

I

xγn (s), A∗ (xγn (s))vη(s)ds

f (xγn (s)), vη(s)ds.

(2.32)

Thus, letting .n → ∞ in the above expression and recalling that .xγn (0) strongly converges to .x0 in H , it follows from (2.30) and (2.31) that  −x0 , vη(0) −

.

 xγ (s), vη(s)ds ˙ +

I



xγ (s), A∗ (xγ (s))vη(s)ds

I

f (xγ (s)), vη(s)ds, ∀ v ∈ V .

=

(2.33)

I

This shows that the limit .xγ ∈ W is the unique weak solution of Eq. (2.29). It is easy to verify from the above expression that .xγ is differentiable in the sense of .V ∗ valued distribution and the identity (2.29) holds in this sense. As seen above, if .

sup { f (ξ ) V ∗ , ξ ∈ H } < ∞,

Eq. (2.28) has a weak solution. If this assumption does not hold, we look for a measure-valued solution. We have seen that for each finite .γ > 0, the truncated γ system (2.29) has the weak solution .xγ ∈ L2 (I, V ) ∩ L2 (I, H ). Defining .λt (·) = + δxγ (t) (·), we note that for every .ψ ∈ L1 (I, BC(H )) we have .

     I

H

   γ ψ(t, ξ )λt (dξ ) ≤ |ψ(t, ·)|BC(H + ) dt < ∞. + I

This shows that the set .{λγ , γ ∈ R0 }, where .R0 = {r ∈ R : r > 0}, is a bounded w (I,  + w + subset of the set .M∞ rba (H )) ⊂ L∞ (I, Mrba (H )). By virtue of Alaoglu’s theorem, weak star closure of any bounded set in the dual of a Banach space is weak star compact. Define .α : R0 −→ N by setting .α(γ ) = γ , the smallest integer equal to or greater than .γ . Clearly, .{λα(γ ) } is a subnet of the net .{λγ }. Thus, w (I,  + it follows from Alaoglu’s theorem that there exists a .λ ∈ M∞ rba (H )) such that, along a subnet if necessary, w∗

w + λα(γ ) → λ ∈ M∞ (I, rba (H + )) ⊂ Lw ∞ (I, Mrba (H )) as γ → ∞.

.

For any integer n, let .h ∈ C 1 (R n ) with compact support and define the cylinder function .φc (ξ ) = h((v1 , ξ ), (v2 , ξ ), · · · , (vn , ξ )) where .{vi } is any complete set of linearly independent elements orthogonal in V and .V ∗ and orthonormal in H .

36

2 Measure Solutions for Deterministic Evolution Equations

Letting .∂i h denote  the partial derivative of h with respect to its i-th coordinate, we have .Dφc = n1 (∂i h)vi . Using the cylinder function .φc in place of .v, ξ , and the measure .λα(γ ) in the expression (2.33) and integrating by parts one can verify that .

   α(γ ) α(γ ) (d/dt)λt (φc ) − λt (𝒜α(γ ) φc ) η(t)dt = 0, ∀ η ∈ C 1 (I ),

(2.34)

I

where (𝒜α(γ ) φc )(ξ ) = −A∗ (ξ )Dφc (ξ ), ξ  + fα(γ ) (ξ ), Dφc (ξ ).

.

Since this holds for every .η ∈ C 1 (I ), we have .

α(γ ) λt (φc ) α(γ )

= λ0

=

α(γ ) λ0 (φc ) +

 (φc ) +

t

α(γ )

λs



t

α(γ )

λs 0

(𝒜α(γ ) φc )ds



t

(𝒜φc )ds +

0

α(γ )

λs

(Fα(γ ) φc )ds, t ∈ I,

(2.35)

0

where (Fα(γ ) φc )(ξ ) = fα(γ ) (ξ ) − f (ξ ), Dφc (ξ ).

.

Since .φc is continuously Fréchet differentiable having compact support, .Dφc is continuous having compact support. Thus, .Fα(γ ) φc converges to zero uniformly s on H as .γ → ∞. Since I is a finite interval, it is clear that .Fα(γ ) φc −→ 0 ∈ L1 (I, B(E)) as .γ −→ ∞. Hence, along the subnet .α(γ ), letting .γ → ∞ in the expression (2.35) we obtain  λt (φc ) = λ0 (φc ) +

.

t

λs (𝒜φc )ds, t ∈ I.

(2.36)

0

It is clear from the proof that the above equation holds for all Fréchet differentiable functions (not only for cylinder functions) belonging to the domain of .𝒜 and having bounded supports. This completes the proof. 

2.3 Evolution Equations with Measurable Vector Fields In the previous sections we have dealt with continuous vector fields and regular bounded finitely additive measure-valued functions as solutions. In this section we consider measurable vector fields f possibly admitting discontinuities. Here the solution space is the space of finitely additive bounded measure-valued functions.

2.3 Evolution Equations with Measurable Vector Fields

37

2.3.1 Introduction Let Z denote any topological space and . a field of subsets of the set Z. Let .B(Z) ≡ B(Z, ) denote the class of . measurable bounded scalar valued functions. An element f of this space is said to be . measurable, if for every Borel set . in the range space, the set f −1 () = {z ∈ Z : f (z) ∈ } ∈ .

.

The space .B(Z) is furnished with the sup norm topology, |f | ≡ sup {|f (z)|, z ∈ Z} .

.

With respect to the norm topology, it is a Banach space. Let .Mba (Z) denote the class of all scalar valued finitely additive measures defined on the algebra .. Furnished with the total variation norm, .Mba (Z) is a Banach space. We have already seen in Chap. 1, that the dual of the Banach space .B(Z) is isometrically isomorphic to the space .Mba (Z), which is expressed symbolically as ∗ .B (Z) ∼ = Mba (Z). In other words, every continuous linear functional on .B(Z) has the representation  (f ) =

f (z)μ(dz), f ∈ B(Z)

.

Z

with a unique measure .μ ∈ Mba (Z), and conversely every such measure determines a continuous linear functional on .B(Z). We note that .Mba (Z) ⊃ Mrba (Z) ∼ = (BC(Z))∗ . This generalization allows measurable vector fields in the evolution equations. We consider the Cauchy problem in a Banach space E, x˙ = Ax + f (t, x), t ∈ I ≡ [0, T ], x(0) = x0 ∈ E,

.

(2.37)

where A is the generator of a .C0 semigroup .S(t), .t ≥ 0, in E, and f is a map from I × E to E. Let .B denote the sigma algebra of Borel subsets of the interval I and . a field or algebra of subsets of the set E, generated by closed subsets of E. Our general assumption is that f is a .B ×  measurable map with values in E. Let .φ ∈ BC(E) with .Dφ denoting its Fréchet derivative. We introduce the class of test functions .F given by .

  F ≡ φ ∈ BC(E) : Dφ exists, Dφ ∈ B(E, E ∗ ) .

.

Define the operator .A with its domain given by D(A) ≡ {φ ∈ F : Aφ ∈ L1 (I, B(E))}

.

38

2 Measure Solutions for Deterministic Evolution Equations

where (Aφ)(t, ξ ) = A∗ Dφ(ξ ), ξ E ∗ ,E + Dφ(ξ ), f (t, ξ )E ∗ ,E , for φ ∈ D(A).

.

Note that .D(A) = ∅, for example, for .ψ ∈ F, the function .φ given by .φ(x) ≡ ψ(λR(λ, A)x), belongs to .D(A) for each .λ ∈ ρ(A), the resolvent set of A. Conw (I,  (E)) sider the system (2.37) with A and f as defined above. Recall that .M∞ ba w is a subset of the Banach space .L∞ (I, Mba (E)). As in the previous sections, again ˆ we use the Stone-Cech compactification of E giving a compact Hausdorff space .E + . w + An element .μ ∈ M∞ (I, ba (E + )) ⊂ Lw ∞ (I, Mba (E )) is said to be a measurevalued (or generalized) solution of Eq. (2.37) if, for every .φ ∈ D(A) with .Dφ having bounded supports, the following identity holds,

.

 μt (φ) = φ(x0 ) +

.

t

μs (Aφ)ds, t ∈ I,

(2.38)

0

where, for .ψ ∈ B(E),  μt (ψ) ≡

.

E+

ψ(ξ )μt (dξ ), t ∈ I.

For simplicity of notation we shall use .D(A) to denote the common domain of the operators .A(t) for .t ∈ I .

2.3.2 Existence of Measure Solutions In the following theorem we prove existence of measure solutions for the evolution equation (2.37) under the assumption that f is a bounded .B ×  measurable map on .I × E with values in E. By .B ×  measurability we mean that for each . ∈ , the set .{(t, x) ∈ I × E : f (t, x) ∈ } ∈ B × . Theorem 2.3.1 Consider the system (2.37) and suppose E is a separable Banach space. Let A be the generator of a .C0 semigroup in E and .f : I × E −→ E be a bounded .B ×  measurable map satisfying the following approximation properties: (ai): there exists a sequence .{fn } such that .fn (t, x) ∈ D(A) for .x ∈ E and almost all .t ∈ I ; further, for almost all .t ∈ I , fn (t, x) −→ f (t, x)

.

uniformly on compact subsets of E;

2.3 Evolution Equations with Measurable Vector Fields

39

(aii): for any .r > 0, there exists a sequence .{αr,n } ∈ L+ 1 (I ), possibly .|αr,n |L1 (I ) → ∞ as .n → ∞, such that |fn (t, x) − fn (t, y)|E ≤ αr,n (t)|x − y|E , x, y ∈ Br ,

.

where .Br ⊂ E is a ball of radius r around the origin. Then, for every .x0 ∈ E, the evolution equation (2.37) has at least one + generalized solution .μ ∈ Lw ∞ (I, Mba (E )) satisfying Eq. (2.38). Further, .μ ∈ w + ∗ M∞ (I, ba (E )) and .t −→ μt is .w continuous. Proof Let .ρ(A) denote the resolvent set of the operator A and .R(λ, A) the corresponding resolvent operator for .λ ∈ ρ(A). Since A is the infinitesimal generator of a .C0 semigroup there exists a nonnegative number .ω such that .(ω, ∞) ⊂ ρ(A). Let .An ≡ nAR(n, A) denote the Yosida approximation of A defined for all .n ∈ ρ(A). Now consider the sequence of evolution equations x˙ = An x + fn (t, x), t ∈ I, x(0) = x0,n ≡ nR(n, A)x0 .

.

(2.39)

By assumption, f is bounded measurable, and the sequence .fn converges to f in the sense of assumption .(ai). Thus, .{fn } is also a bounded sequence. Let the common bound be denoted by .bf , that is, .

sup {|f (t, ξ )|E , |fn (t, ξ )|E , n ∈ N ∩ ρ(A), (t, ξ ) ∈ I × E} ≤ bf .

Since the sequence .{fn } is contained in .D(A) and, by assumption .(aii), they are locally Lipschitz, and the initial data .x0,n ∈ D(A), it follows from semigroup theory [5, p. 156] that for each .n ∈ ρ(A), this equation has a unique strong solution .xn with values .xn (t) ∈ D(A) and .x ˙n ∈ L1 (I, E) satisfying the first identity in Eq. (2.39), for almost all .t ∈ I . Since every strong solution is a mild solution, .xn must also satisfy the following integral equation 

t

xn (t) = Sn (t)x0,n +

.

Sn (t − s)fn (s, xn (s))ds, t ∈ I,

(2.40)

0

where .Sn (t), .t ≥ 0, is the semigroup generated by .An . In fact, these are uniformly continuous semigroups because their generators are bounded operators. Since .S(t), .t ≥ 0, is a .C0 semigroup (of bounded linear operators) and I is a finite interval, there exists a finite positive number M such that .

  sup  S(t) L(E) , t ∈ I ≤ M

and Sn (t) −→ S(t), t ∈ I,

.

40

2 Measure Solutions for Deterministic Evolution Equations

in the strong operator topology in .L(E) uniformly on the interval I . It follows from uniform boundedness principle that there exists a finite positive number .M˜ ≥ M such that .

  ˜ sup  Sn (t) L(E) , t ∈ I ≤ M.

Hence, it follows from the integral equation (2.40) that there exists a finite positive number .r˜ such that .

 sup {|xn (t)|E , t ∈ I } ≤ M˜ |xo |E + bf T ≡ r˜ , ∀ n ∈ N.

Thus, for any .r ≥ r˜ , we have .xn (t) ∈ Br ⊂ E for all .t ∈ I and all .n ∈ N. Since {xn } is a sequence of strong solutions of Eq. (2.39), it is clear that, for every .φ ∈ F,

.

 .

φ(xn (t)) = φ(x0,n ) +

t

Dφ(xn (s)), An xn (s) + fn (s, xn (s))E ∗ ,E ds, t ∈ I.

0

(2.41) Let .δe (·) denote the Dirac measure on E concentrated at the point .e ∈ E and define λnt (dξ ) ≡ δxn (t) (dξ ) and λn0 (dξ ) = δx0,n (dξ ).

.

Using this notation we can rewrite Eq. (2.41) in the form  λnt (φ) = λn0 (φ) +

t

.

0

λns (An (s)φ)ds, t ∈ I,

(2.42)

where the operator .An is given by (An φ)(t, ξ ) = A∗n Dφ(ξ ), ξ E ∗ ,E + Dφ(ξ ), fn (t, ξ )E ∗ ,E

.

w (I,  (E)) ⊂ for .φ ∈ D(A). Clearly, for each integer .n ∈ ρ(A), .λn ∈ M∞ ba w + .M∞ (I, ba (E )) and it follows from our preceding analysis that for any .r ≥ r ˜, .

supp(λnt ) ⊂ Br , ∀ t ∈ I, n ∈ N.

Our concern now is to show that the sequence .{λn } has a limit and that the limit is a measure solution of our original problem. To this end, we consider the sequence of linear functionals .{n } on the Banach space .L1 (I, B(E)) given by  n (ϕ) ≡

.

I ×E

ϕ(t, ξ )λnt (dξ )dt.

2.3 Evolution Equations with Measurable Vector Fields

41

Clearly, this is well-defined for each .ϕ ∈ L1 (I, B(E)), and we have |n (ϕ)| ≤ |ϕ|L1 (I,B(E)) , ∀ n ∈ ρ(A).

.

In other words, .{n } is a sequence of bounded linear functionals contained in a bounded subset of .(L1 (I, B(E)))∗ , the dual of .L1 (I, B(E)). Thus, it follows from the characterization of the dual space of the Banach space .L1 (I, B(E)), that the sequence .{λn } is confined in a bounded subset of .Lw ∞ (I, Mba (E)). Note that a bounded set B in a dual space is relatively .w ∗ (weak star) compact in the sense that its weak star closure is .w ∗ compact. Hence, by Alaoglu’s theorem there exists a subsequence (or subnet) of the sequence (or net) .{λn }, relabeled as .{λn }, and a o w w + .λ ∈ L∞ (I, Mba (E)) ⊂ L∞ (I, Mba (E )) such that w∗

λn −→ λo , in Lw ∞ (I, Mba (E)).

.

(2.43)

We show that .λo is a measure (generalized) solution of the evolution equation (2.37) in the sense of the functional equation (2.38). Let .φ ∈ D(A) with both .φ and .Dφ being continuous and bounded having compact supports (which may be different for different .φ). Define .

(Bn φ)(ξ ) ≡ (A∗n − A∗ )Dφ(ξ ), ξ E ∗ ,E , .

(2.44)

(Cn (t)φ)(ξ ) ≡ Dφ(ξ ), fn (t, ξ ) − f (t, ξ )E ∗ ,E , (t, ξ ) ∈ I × E.

(2.45)

Using these expressions, Eq. (2.42) can be rewritten as  n .λt (φ)

=

t

+

λn0 (φ)



0 t

+ 0

λns (A(s)φ)ds  λns (Bn φ)ds

+ 0

t

λns (Cn (s)φ)ds, t ∈ I.

(2.46)

Consider the expression (2.44) and recall that .An −→ A on .D(A) ⊂ E in the strong operator topology and, for .φ ∈ D(A) we have .Dφ(ξ ) ∈ D(A∗ ), and both .φ and s .Dφ are continuous having compact supports. Thus, .Bn φ ∈ B(E) and .(Bn φ) −→ 0 in B(E). Since .Bn φ is independent of t and I is a finite interval, it is evident that s

(Bn φ) −→ 0 in L1 (I, B(E))

.

(2.47)

as .n → ∞. Consider the expression (2.45). By use of similar arguments and the assumption .(ai), along with Lebesgue dominated convergence theorem, one can easily verify that s

(Cn φ) −→ 0 in L1 (I, B(E))

.

(2.48)

42

2 Measure Solutions for Deterministic Evolution Equations

as .n → ∞. Thus, letting .n → ∞ in the expression (2.46), it follows from (2.43), (2.47) and (2.48) that  λot (φ) = λ0 (φ) +

t

.

0

λos (A(s)φ)ds, t ∈ I,

where .λ0 (φ) = δx0 (φ) = φ(x0 ). Since .λo ∈ Lw ∞ (I, Mba (E)) and it is the (unique) w (I,  (E)) and that this set is weak star weak star limit of a sequence .{λn } ⊂ M∞ ba w (I,  (E)). This proves that .λo is a measureclosed, we conclude that .λo ∈ M∞ ba valued solution of the evolution equation (2.37). The last part of the theorem, asserting .w ∗ continuity, follows readily from the above integral expression. This completes the proof. 

The result given above depends on the existence of a sequence .{fn } approximating the nonlinear measurable vector field (map) f . The existence of such an approximating sequence follows from the following result (Proposition 2.3.2). Thus, these assumptions are natural and do not in any way limit the results. Proposition 2.3.2 Suppose E is a separable Banach space, A is a linear (generally unbounded) operator with domain and range in E having nonempty resolvent set .ρ(A) with the resolvent denoted by .R(λ, A) for .λ ∈ ρ(A). Then, for every bounded .B ×  measurable map .f = f (t, x) which is Lebesgue-Bochner integrable in t on I uniformly with respect to x in bounded subsets of E, there exists a sequence .{fn } satisfying the hypotheses (ai) and (aii) of Theorem 2.3.1. Proof By virtue of separability, the Banach space E has a Schauder basis .{ei }. Corresponding to this basis, let .{En } ⊂ E be an increasing family of n-dimensional subspaces of E and .{Qn } the corresponding family of projections of E onto .En . Let .n : E −→ R n denote the linear map taking each element x of E into its first n Fourier coefficients. That is, .n x = col {(i (x)), i = 1, 2, 3, · · · , n}, where .{i } is a sequence of linearly independent continuous linear functionals on E with .|i |E ∗ = 1 associated with the Schauder basis .{ei } of E. These functionals may be identified with a corresponding sequence .{ei∗ } ∈ B1 (E ∗ ) (closed unit ball). We use .C ∞ molifiers to construct a smooth family .{fn } approximating the given f . Let ∞ n ∞ functions on .R n with compact supports .n ∈ N and .ρn ∈ C (R ) be a family of .C 0 satisfying   ρn (ξ ) ≥ 0, ρn (ξ ) = ρn (−ξ ), supp(ρn ) ⊆ ξ ∈ R n : |ξ |R n ≤ (1/n)

.

and  .

Rn

ρn (ξ )dξ = 1, n ∈ N.

2.3 Evolution Equations with Measurable Vector Fields

43

Let .Jn ≡ nR(n, A) for .n ∈ ρ(A) ∩ N and recall that .Jn converges in the strong operator topology to the identity operator in E and that .Jn (E) ⊂ D(A). Define 

 fn (t, x) ≡

.

Rn

t, n x −

Jn f

n 

 ξi ei ρn (ξ )dξ.

i=1

By a simple change of variables this can be written as 

 fn (t, x) ≡

.

Rn

Jn f

t,

n 

 ηi ei ρn (n x − η)dη.

i=1

Since f is a bounded .B ×  measurable map on .I × E and .ρn has compact support, the integral is well-defined and hence the sequence .{fn } is well-defined satisfying .fn (t, x) ∈ D(A) for all .(t, x) ∈ I × E. Clearly it follows from the above expression that for each .(t, x) ∈ I × E, .fn (t, x) −→ f (t, x) in E. Pointwise convergence implies uniform convergence on compact sets. Thus, the hypothesis .(ai) of Theorem 2.3.1 is satisfied. For the local Lipschitz property, note that 

 fn (t, y) − fn (t, x) ≡

.

Rn

Jn f

t,

n 

 ηi ei (ρn (n y − η) − ρn (n x − η)) dη.

i=1

Using Lagrange formula applied to the molifier, we have 

1

ρn (η) = ρn (ξ ) +

.

(Dρn (ξ + θ (η − ξ )), η − ξ )dθ, η, ξ ∈ R n ,

0

where D denotes the first Fréchet derivative. Taking any ball .Br ⊂ E, .r > 0, and using this formula in the above equation, one can verify that there exists an element + .αr,n ∈ L (I ) such that 1 |fn (t, y) − fn (t, x)|E ≤ αr,n (t)|y − x|E , ∀ t ∈ I

.

for all .x, y ∈ Br . Indeed, by simple computation one can discover that the function αr,n can be chosen as

.

  n        .αr,n (t) ≡ γ Jn f t, ηi ei  gr,n (η)dη,   n R 

i=1

E

where the function .gr,n is given by gr,n (η) ≡

.

sup 0≤θ≤1;x,y∈Br

|Dρn (n x − η + θ (n y − n x))|R n ,

44

2 Measure Solutions for Deterministic Evolution Equations

and .γ is the smallest positive number for which .|n x|R n ≤ γ |x|E for all .x ∈ E and for all .n ∈ N . Since .Dρn is also a .C ∞ function having compact support, it is clear that .gr,n vanishes outside of a bounded subset of .R n . Thus, for the given f which is a bounded .B ×  measurable map taking values from E, the integral defining the function .αr,n is finite almost everywhere. Hence, .{αr,n } is a well-defined sequence of finite measurable functions. Since by our assumption, .t −→ f (t, x) is Bochner integrable, uniformly with respect to x on bounded subsets of E, it follows from the expression for .αr,n and Fubini’s theorem that .αr,n ∈ L+ 1 (I ). Thus, the hypothesis .(aii) of Theorem 2.3.1 holds. This completes the proof. 

Remark 2.3.3 Note that, for the proof of the previous proposition, we have only used the .B ×  measurability of f and its uniform boundedness on E. In other words, it is not necessary that f be a bounded .B ×  measurable map on .I × E. Now we are prepared to prove existence results with unbounded measurable vector fields .{f } in place of bounded vector fields thereby generalizing Theorem 2.3.1. This is presented in the following theorem. Theorem 2.3.4 Let A be the infinitesimal generator of a .C0 semigroup in the Banach space E and .f : I × E → E is .B ×  measurable, integrable in t on I uniformly with respect to x on bounded subsets of E; and for almost all .t ∈ I , it is bounded on bounded subsets of E. Then, for each .x0 ∈ E, the evolution w (I,  (E + )). equation (2.37) has at least one measure-valued solution .λ ∈ M∞ ba ∗ Further, .t → λt is .w continuous on I . Proof The basic technique is similar to that of [19, Theorem 3.3] and [13, Theorem 3.2]. Here we present a brief outline. For each finite number .γ > 0, define the truncated family .{fγ } as follows fγ (t, x) ≡ f (t, Rγ (x)),

.

where .Rγ is the retraction of the ball .Bγ ⊂ E, that is,  Rγ (ξ ) ≡

.

ξ,

if ξ ∈ Bγ ,

(γ /|ξ |)ξ,

otherwise.

Clearly, .fγ is .B ×  measurable, and, for each .ξ ∈ E, .t −→ fγ (t, ξ ) is integrable while for almost all .t ∈ I , .ξ −→ fγ (t, ξ ) is uniformly bounded on all of E. Thus, for each .γ < ∞, it follows from Theorem 2.3.1 that the evolution equation x˙ = Ax + fγ (t, x), t ∈ I, x(0) = x0

.

w (I,  (E)). In other words, .λγ satisfies has at least one measure solution .λγ ∈ M∞ ba the following functional equation γ

γ

λt (φ) = λ0 (φ) +

.



t 0

γ



t

λs (Aγ φ)ds = φ(x0 ) + 0

γ

λs (Aγ (s)φ)ds, t ∈ I

2.3 Evolution Equations with Measurable Vector Fields

45

for each .φ ∈ D(Aγ ) having bounded support, where the operator .Aγ is given by (Aγ (t)φ)(ξ ) ≡ A∗ Dφ(ξ ), ξ  + Dφ(ξ ), fγ (t, ξ ), t ∈ I.

.

Clearly, for .φ ∈ D(A), Aγ (t)φ = A(t)φ + Bγ (t)φ,

.

where Bγ (t)φ(ξ ) ≡ Dφ(ξ ), fγ (t, ξ ) − f (t, ξ )E ∗ ,E ,

.

and hence γ .λt (φ)



t

= φ(x0 ) +

γ λs (A(s)φ)ds

0



t

+

γ

λs (Bγ φ)ds, t ∈ I.

0

For each .γ > 0, the functional .γ given by  γ (ψ) ≡

.

I

γ

λt (ψ)dt

is well-defined on .L1 (I, B(E)) satisfying |γ (ψ)| ≤ |ψ|L1 (I,B(E))

.

for all .ψ ∈ L1 (I, B(E)). Since in the limit the support of the limit may escape w (I,  (E + )), E, we use the space .E + . Thus, the set .{λγ } is contained in .M∞ ba + )). Therefore, again by Alaoglu’s a .w ∗ closed bounded subset of .Lw (I, M (E ba ∞ theorem, there exists a subnet or a generalized subsequence .{λk ≡ λγk } and a o w + .λ ∈ M∞ (I, ba (E )) such that .

w∗

λk −→ λo as k → ∞

.

(2.49)

+ in .Lw ∞ (I, Mba (E )). Define .Bk ≡ Bγk . Since, for almost all .t ∈ I ,

fk (t, x) ≡ fγk (t, x) −→ f (t, x) as k → ∞

.

uniformly on compact subsets of E, and .fk is Bochner integrable on I uniformly with respect to x in bounded subsets of E, we conclude that, for each .φ ∈ D(A) having compact support, .Bk φ ∈ L1 (I, B(E)) and .Bk φ −→ 0 in .B(E) for almost all .t ∈ I . Thus, it follows from dominated convergence theorem that s

Bk φ −→ 0 in L1 (I, B(E)).

.

(2.50)

46

2 Measure Solutions for Deterministic Evolution Equations

Therefore, it follows from Eqs. (2.49) and (2.50) that for each .t ∈ I ,  .

0

t

λks (Bk φ)ds −→ 0.

Hence, it follows from similar arguments as in Theorem 2.3.1 that in the limit .λo satisfies the following equation  λot (φ) = λ0 (φ) +

t

.

0

λos (A(s)φ)ds, t ∈ I.

w (I,  (E + )) ⊂ Lw (I, M (E + )) is a measure solution of the Thus .λo ∈ M∞ ba ba ∞ evolution equation (2.37). The last part of the statement asserting .w ∗ continuity, follows trivially from the above functional equation. This completes the proof. 

Remark 2.3.5 Note that the evolution equation (2.37) has measure solution satisfying Eq. (2.38) not only for initial data given by a Dirac measure but also for any initial data described by a measure .π0 ∈ ba (E). Interested reader may see [19].

2.3.3 Differential Equations on the Space of Measures In view of our basic definition of measure solution (see Definition 2.1.4) and the preceding results, we can reformulate our original Cauchy problem defined on the Banach space E, as a Cauchy problem on the Banach space of finitely additive measures .Mba (E) ⊂ Mba (E + ) as follows: (d/dt)μt = A∗ (t)μt , t ≥ 0, μ0 = π0 .

.

(2.51)

This of course covers the original Cauchy problem as a special case. According to our existence results, we have seen that this equation has solution in the weak sense as implied by our definition. Hence, it follows from these results given by Theorems 2.3.1 and 2.3.4, that for each initial data .π0 ∈ ba (E) ⊂ Mba (E + ), the w (I,  (E + )) ⊂ evolution equation (2.51) has at least one weak solution .μ ∈ M∞ ba w + ∗ L∞ (I, Mba (E )) which is .w continuous in .t ∈ I . Consequently, there exists a ∗ ∗ .w continuous transition operator .U (t, s), .0 ≤ s ≤ t < ∞, which is a family of bounded linear operators on the Banach space .Mba (E + ), defining the evolution of the measure solution μt = U ∗ (t, 0)π0 , t ∈ I.

.

For any pair .(s, t) ∈ I satisfying .0 ≤ s < t, .U ∗ (t, s) is a bounded linear operator on the Banach space .Mba (E + ). In particular, for .s = 0, let .π0 ∈ ba (E) and let .μ denote the corresponding solution of Eq. (2.51). Then, for all .t ∈ I , there exists a

2.4 Bibliographical Notes

47

positive constant C, independent of .π0 , such that |μt |Mba (E + ) = |U ∗ (t, 0)π0 |Mba (E) ≤ C|π0 |Mba (E) .

.

Since we have only a partial result on uniqueness, the operator valued function U may not satisfy the desired evolution property U ∗ (t, r)U ∗ (r, s) = U ∗ (t, s), 0 ≤ s ≤ r ≤ t < ∞.

.

In the sequel we will discuss this question once again. For more discussion on the question of uniqueness see [19]. Remark 2.3.6 Recall that .E + is a compact Hausdorff space. With this extension, the solutions are countably additive measure-valued functions on the sigma algebra generated by open (or closed) sets in the compact Hausdorff space .E + . Remark 2.3.7 It is interesting to note that the assumption .(ai) of Theorem 2.3.1 can be relaxed to .(ai) as stated below, ∗ ∗ .(ai) : For each .e ∈ E , e∗ , fn (t, x)E ∗ ,E −→ e∗ , f (t, x)E ∗ ,E , for a.e. t ∈ I

.

uniformly on compact subsets of E.

2.4 Bibliographical Notes In the literature, the notion of generalized solutions (measure-valued solutions) was already introduced by DiPerna [68] in the study of conservation laws. These solutions, also called Young measures, introduced by DiPerna are suitable for systems governed by a specific class of partial differential equations. Later, Fattorini [73] and Ahmed [9, 11, 13, 15, 16, 19] formulated the notion of measure solutions in a much more general and abstract setting. These are applicable to a very broad class of abstract differential equations on Banach spaces, covering semilinear and many quasilinear partial differential equations, delay differential equations, stochastic differential equations, and many more, see [9, 11, 13, 15, 16, 73] and the references therein. In the abstract setting the measures can be either countably additive or finitely additive thus admitting substantial generality. Fattorini introduced the notion of relaxed solutions (measure solutions) for semilinear evolution equations on Banach spaces [72]. Theorem 2.1.2 is a slightly revised version of Fattorini’s original result. Fattorini assumed the existence of an apriori bound. This assumption was relaxed in the subsequent work of one of the authors of this monograph and an alternate definition of measure solutions was introduced as seen in Definition 2.1.4 which is used to prove Theorem 2.1.5 without any assumption on apriori bounds. For reference, see [9, 11, 19] and the references therein. We have also considered

48

2 Measure Solutions for Deterministic Evolution Equations

measure solutions for semilinear and quasilinear evolution equations with nonlinear terms which are merely continuous and bounded on bounded sets. Theorem 2.2.1 relaxes the assumption on the vector field f by use of analytic semigroup generated by the unbounded operator .−A. This result is extended to quasilinear problems in Theorem 2.2.3, where the unbounded operator A is also a nonlinear operator [14]. These results are extended later in the chapter to admit systems with measurable vector fields. Theorem 2.3.1 admits uniformly bounded measurable vector fields, and its extension to unbounded vector fields is given in Theorem 2.3.4. Many of these results were originally presented in several papers of one of the authors [9, 11, 13, 15, 16].

Chapter 3

Measure Solutions for Impulsive Systems

3.1 Introduction In this chapter, we consider evolution equations subject to impulsive inputs or forces, and study the questions of existence and uniqueness of measure-valued solutions and regularity properties thereof. In applications, we find systems like the communication satellite which is intermittently hit by micrometeorites causing abrupt changes in the attitude dynamics and affecting communication signals. In medicine, laser power is used to burn tumor, where the system is the heat equation describing the temperature distribution in the tumor and the input is the impulsive (laser) heat source. There are many other practical examples involving impulsive forces. Aircraft flight dynamics can change abruptly as it suddenly enters dense clouds producing turbulence. In construction, foundations of tall and large buildings are reinforced by piles driven into ground by impulsive forces imparted by a freely falling hammer. Atmospheric electric storm can abruptly hit power grid causing blackout and cutting off communication signals. Tornadoes and earthquakes can hit geographical regions flattening buildings and bridges in a very short span of time. In the area of defence, missiles impart impulsive forces disintegrating targets instantly. Plague, epidemic, war can change the demography in a very short span of time. In fact one can find many examples in all fields of sciences and engineering, where the systems are subject to impulsive forces [40–42]. Let us consider the evolution equation dx = Axdt + f (x)dt + g(x)ν(dt), t ≥ 0, x(0) = ξ

.

in a Banach space E, where A is the infinitesimal generator of a .C0 semigroup .S(t), t ≥ 0, on E and .f, g : E → E are Borel measurable maps to be clarified shortly and .ν is a scalar valued signed measure. Even for the special case, .g = 0, we have seen that if E is an infinite dimensional space, mere continuity of f in x does

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. U. Ahmed, S. Wang, Measure-Valued Solutions for Nonlinear Evolution Equations on Banach Spaces and Their Optimal Control, https://doi.org/10.1007/978-3-031-37260-5_3

49

50

3 Measure Solutions for Impulsive Systems

not guarantee the existence of a mild or even a weak solution. In Chap. 2, we have seen a counter example due to Dieudonne [67]. There are other counter examples constructed by Godunov [80] demonstrating nonexistence of solutions in case f is merely continuous. In Chap. 2, we have considered measure solutions for semilinear and quasilinear evolution equations with nonlinear terms which are merely continuous and bounded on bounded sets and later in the same chapter, we have extended these results admitting systems with measurable vector fields. Many of these results were originally presented in several papers of one of the authors [9, 11, 13, 15, 16]. In this chapter we consider impulsive systems admitting vector fields f and g which are merely Borel measurable (not necessarily continuous) and bounded on bounded sets. In the literature, the notion of measure-valued solutions was already introduced by DiPerna [68] in the study of conservation laws. These solutions, also called Young measures, introduced by DiPerna are suitable for systems governed by a certain class of partial differential equations. Later, Fattorini [73] and Ahmed [9, 11, 13, 15, 16, 19] formulated the notion of measure solutions in a much more general and abstract setting. These are applicable to a very broad class of abstract differential equations on Banach spaces, covering semilinear and many quasilinear partial differential equations, delay differential equations, stochastic differential equations, and many more, see [9, 11, 13, 15, 16, 73] and references therein. In the abstract setting the measures can be either countably additive or finitely additive thus admitting substantial generality. We note that the impulsive component generated by scalar valued signed measures can be readily extended to vector measures with g denoting an operator valued function.

3.2 Spaces of Measure-Valued Functions Let E be a real separable Banach space and .I ≡ [0, T ] a finite interval and .L1 (I, E) the Banach space of Lebesgue-Bochner integrable functions with values in E. For the formulation of measure solutions, we have seen that the dual of the Banach space .L1 (I, E) plays a fundamental role. We have already noted that if the dual .E ∗ of the Banach space E satisfies the Radon-Nikodym property (RNP), then the dual of .L1 (I, E) is given by .L∞ (I, E ∗ ). In general it follows from the theory of lifting giving Theorem 1.2.7 [121, ∗ Theorem 7 and its Corollary, p.94] that the dual of .L1 (I, E) is given by .Lw ∞ (I, E ) ∗ ∗ which consists of .w measurable .E valued functions. Let . ≡ E denote the field (algebra) of subsets of the set E and .B(E) the space of bounded . measurable scalar valued functions defined on E and furnished with the topology of sup norm given by |f | ≡ sup {|f (e)|, e ∈ E} .

.

3.3 Measure-Valued Solutions

51

This is a Banach space. An element f of this space is said to be . measurable if for every Borel set F from the real line (the range space), the set f −1 (F ) ≡ {e ∈ E : f (e) ∈ F } ∈ .

.

Let .Mba (E) ≡ Mba (E, ) denote the class of all scalar valued finitely additive measures (set functions) defined on the algebra .. Furnished with the total variation norm, .Mba (E) is a Banach space. We have seen in Chap. 1 that the topological dual ∗ .B (E) of the Banach space .B(E) is the space of finitely additive measures .Mba (E) and that .B ∗ (E) ∼ = Mba (E). Let .ba (E) ⊂ Mba (E) denote the class of finitely additive probability measures furnished with the relative topology. The Banach space .B(E) and its dual .Mba (E) do not satisfy the RNP. Therefore, it follows from the characterization results discussed above that the dual of .L1 (I, B(E)) is w given by .Lw ∞ (I, Mba (E)). The space .L∞ (I, Mba (E)) is furnished with the weak star topology. In view of this duality, it is clear that for every continuous linear functional . on .L1 (I, B(E)), there exists .λ ∈ Lw ∞ (I, Mba (E)) such that  (ψ) =

 λt (ψ)dt =

.

I ×E

I

ψ(t, ξ )λt (dξ )dt,

and conversely every such .λ determines a continuous linear functional on L1 (I, B(E)). Similar results hold true for any countably additive bounded positive measure .β on I having bounded total variation. That is, the topological dual of w .L1 (β, B(E)) is given by .L∞ (β, Mba (E)). Thus, any continuous linear functional on .L1 (β, B(E)) is characterized by .

 (ψ) =

 λt (ψ)β(dt) =

.

I

I ×E

ψ(t, ξ )λt (dξ )β(dt).

Hence, considering the Banach space .L1 (I, B(E)) ⊕ L1 (β, B(E)), it follows from w the above results that its dual is given by .Lw ∞ (I, Mba (E)) ∩ L∞ (β, Mba (E)).

3.3 Measure-Valued Solutions We consider the following measure driven Cauchy problem in the Banach space E, dx(t) = Ax(t)dt + f (x(t))dt

.

+g(x(t−))ν(dt), t ∈ I ≡ [0, T ], x(0) = x0 ∈ E,

(3.1)

where A is the generator of a .C0 semigroup, .S(t), .t ≥ 0, in E, and .f, g are maps from E to E and .ν is a signed measure on the sigma algebra of subsets of the set I .

52

3 Measure Solutions for Impulsive Systems

Let .B(I, E) denote the vector space of bounded Borel measurable functions on I with values in the Banach space E. Since E is a Banach space, the vector space .B(I, E), furnished with the topology induced by the norm |z| ≡ sup {|z(t)|E , t ∈ I } ,

.

is a Banach space. In addition to the space .B(I, E), we shall occasionally need the class of bounded piecewise continuous functions on I with values in E which we denote by .P W C(I, E). Equipped with the same norm topology, this is a linear subspace of .B(I, E). Definition 3.3.1 An element .x ∈ B(I, E) is said to be a mild solution of the evolution equation (3.1) if it satisfies the following integral equation 

t

x(t) = S(t)x0 +

.



t

S(t − s)f (x(s))ds +

0

S(t − s)g(x(s−))v(ds), t ∈ I.

0

(3.2) Remark 3.3.2 Uniqueness of solutions of measure driven systems like (3.1) is guaranteed if one chooses the explicit scheme in the sense that at the atoms .{t} of the measure .ν, the jump is given by x(t) − x(t−) = g(x(t−))v({t}).

.

An alternate possibility is the choice x(t) − x(t−) = g(x(t))v({t}),

.

which is an implicit scheme and requires that, for any given .ξ ∈ E, the equation z = ξ + v({t})g(z)

.

has a solution. Even if g is a contraction on E this problem may not have a solution since .v({t}) may assume arbitrary values and so fixed point theorems based on contraction principle do not apply. On the other hand, in the absence of other regularities such as those possessed by compact maps (completely continuous), Schauder fixed point theorem does not apply. Thus, in the absence of concrete regularities, it is not possible to determine if the fixed point problem has at all any solution or has many solutions. In view of this we prefer the explicit scheme, since it eliminates this ambiguity and also satisfies the natural causality (nonanticipative) property of physical systems. Let .B denote the sigma algebra of Borel subsets of the interval I and . ≡ E a field or algebra of subsets of the set E generated by closed subsets of E. Our general assumption is that f and g are . measurable maps in E in the sense that for every

3.3 Measure-Valued Solutions

53

Borel set . in the range space E, h−1 ( ) ≡ {ξ ∈ E : h(ξ ) ∈ } ∈ , for h = f, g.

.

Let .M (E, E) denote the class of all . measurable maps from E to E in the sense defined above and .B (E, E) ⊂ M (E, E) the class of (uniformly) bounded . measurable maps. That is, for every .h ∈ B (E, E) there exists a finite positive number .bh such that .

sup {|h(e)|E , e ∈ E} ≤ bh .

For non-impulsive systems, the general notion of measure solutions was introduced in [9, 11, 13, 15, 16], where regular bounded finitely additive measures .Mrba (E) were used instead of .Mba (E) ⊃ Mrba ∼ = (BC(E))∗ . In [19], where measurable ( possibly discontinuous) vector fields were admitted, the most appropriate choice of the space of measures was found to be .Mba (E), and this is what was used. Due to the presence of the measure .ν, which may have atoms, we must slightly modify the definition. This generalization allows impulsive inputs as well as measurable vector fields describing the evolution equation. Let .Dφ denote the Fréchet derivative of .φ ∈ B(E) whenever it exists and introduce the class of test functions .F given by   F ≡ φ ∈ B(E) : Dφ exists, Dφ ∈ B(E, E ∗ ) .

.

For given .{A, f, g, ν}, we introduce the operators .A and .C as follows. Define the operator .A with its domain given by D(A) ≡ {φ ∈ F : Aφ ∈ B(E)} ,

.

where (Aφ)(ξ ) = A∗ Dφ(ξ ), ξ E ∗ ,E + Dφ(ξ ), f (ξ )E ∗ ,E

.

for .φ ∈ D(A). We know that .D(A) = ∅, since for .ψ ∈ F, the function .φ given by φ(x) ≡ ψ(λR(λ, A)x) belongs to .D(A) for each .λ ∈ ρ(A), the resolvent set of A. The operator .C is given by

.



1

(Cφ)(t, ξ ) ≡

.

Dφ(ξ + θg(ξ )ν({t})), g(ξ )E ∗ ,E dθ,

(3.3)

0

for all .t ∈ I at which .ν has an atom. If t is not an atom, the operator reduces to (Cφ)(t, ξ ) = Dφ(ξ ), g(ξ )E ∗ ,E .

.

54

3 Measure Solutions for Impulsive Systems

Formally, the operator .C is obtained by use of Lagrange formula, 

1

φ(y) = φ(x) +

.

  dθ Dφ(x + θ (y − x)), y − xE ∗ ,E ,

0

applied to the functional .φ(x(t)), where x is any pathwise solution of Eq. (3.1) whenever it exists. Since the measure .ν is not assumed to be non-atomic, the solutions are expected to have discontinuities at all its atoms. In case .{t} is an atom, the jump in x at time t is given by x(t) − x(t−) = g(x(t−))ν({t}).

.

Clearly, if t is not an atom, .x(t) = x(t−) which implies continuity at time t. In that case the operator .A remains unchanged and, as indicated above, .C reduces to (Cφ)(t, ξ ) = (C(t)φ)(ξ ) ≡ Dφ(ξ ), g(ξ ).

.

Note that the operator .A is time-invariant while the operator .C given by Eq. (3.3) is time-dependent. These are the appropriate operators arising in the study of measurevalued solutions for impulsive systems. Since, under the relaxed assumptions, we do not expect the Cauchy problem (3.1) to possess any pathwise E valued solution .x(t), .t ∈ I , the above arguments justifying the form of the operators .A and .C are formal. A rigorous justification follows later in the sequel. Consider the system (3.1) with A, f , g and .ν as defined above. Here we assume that .{A, f, g} are time-invariant. In order to cover time-dependent cases we shall write (A(t)φ)(ξ ) ≡ (Aφ)(t, ξ ),

.

(C(t)φ)(ξ ) ≡ (Cφ)(t, ξ ).

Because of the presence of the signed measure .ν in the evolution equation (3.1), we must modify the definition of measure solutions which was appropriate for systems not driven by measures. Let .|ν|(·) denote the positive measure induced by the variation of the signed measure .ν defined on the sigma algebra .B. Definition 3.3.3 A measure-valued function w w λ ∈ M∞ (I, ba (E)) ∩ M∞ (|v|, ba (E)) ⊂ Lw ∞ (I, Mba (E))

.

∩ Lw ∞ (|v|, Mba (E))

(3.4)

is said to be a measure solution (or generalized solution) of Eq. (3.1) if, for every φ ∈ D(A) having bounded supports, the following identity holds:

.

 λt (φ) = φ(x0 ) +

.

0

t



t

λs (Aφ)ds + 0

λs− (C(s)φ)ν(ds), t ∈ I,

3.3 Measure-Valued Solutions

55

satisfying .limt↓0 λt (φ) = δx0 (φ), where for any measure-valued function .μ, we defined its action on .ψ ∈ B(E) by  μt (ψ) ≡

ψ(ξ )μt (dξ ), t ∈ I.

.

E

3.3.1 Existence of Measure Solutions For proof of existence of (measure) solutions we make use of the following assumptions: (A1) E is a separable Banach space and A is the infinitesimal generator of a .C0 semigroup of operators .S(t), .t ≥ 0, in E and .ν is a countably additive bounded signed measure containing no atom at .t = 0. This is not a restriction since any atom of .ν at .t = 0 can be absorbed in the initial state by redefining the initial state as .x˜0 ≡ x0 + g(x0 )ν({0}). (A2) The function .h ∈ B (E, E) (h = f, g) satisfies the following approximation properties: (i) there exists a sequence of maps .{hn } ∈ B (E, E) uniformly bounded with respect to .n ∈ N, and for each .n ∈ N it is locally Lipschitz, .

 hn (x) − hn (y) E ≤ αn,r  x − y E , ∀ x, y ∈ Br (E) ≡ {e ∈ E : e ≤ r} ,

with Lipschitz constants .αn,r which are finite positive numbers for .r ≥ 0 and .n ∈ N; (ii) .hn (x) → h(x) uniformly on compact subsets of E. Remark 3.3.4 Note that, under the assumption (A2), for every .0 < r < ∞, we may have .limn→∞ αn,r → ∞. Sufficient conditions guaranteeing the existence of such an approximating sequence are given in Proposition 2.3.2 of Chap. 2. See also [19, Proposition 3.2]. Let .R(λ, A) denote the resolvent of the operator A for .λ ∈ ρ(A), the resolvent set of A, and .An ≡ nAR(n, A) the Yosida approximation of A. We shall need the following intermediate result corresponding to the sequence of approximating systems given by dx(t) = An x(t)dt + fn (x(t))dt + gn (x(t−))v(dt), x(0) = x0 , t ∈ I. (3.5)

.

Lemma 3.3.5 Consider the Cauchy problem (3.5) and suppose that the data {A, f, g, ν} satisfy the assumptions (A1) and (A2) with .{An , fn , gn } denoting the approximating sequence. Then, for each .x0 ∈ E, the evolution equation (3.5) has

.

56

3 Measure Solutions for Impulsive Systems

a unique mild solution .xn ∈ B(I, E) ∩ P W C(I, E) and that there exists a finite positive number .γ such that sup {sup { xn (t) E , t ∈ I }} ≤ γ < ∞.

.

n∈N

Proof Let .Sn (t), .t ≥ 0, denote the semigroup generated by the Yosida approximation .An of the generator A. Since .{An } is a sequence of bounded linear operators in E, the corresponding semigroups .Sn are continuous in the uniform operator topology. By definition, the question of existence of a mild solution for Eq. (3.5) is the same as the question of existence of a solution of the following integral equation  x(t) = Sn (t)x0 +

.

0

t

 Sn (t − s)fn (x(s))ds +

t

Sn (t − s)gn (x(s−))v(ds), t ∈ I.

0

(3.6) First we prove that any solution of this equation (if one exists) must satisfy the uniform bound as stated in the lemma. Since the functions f and g are in .B (E, E) (uniformly bounded . measurable maps) and by the assumption (A2) there exists a sequence .{fn , gn } ⊂ B (E, E) uniformly bounded in .n ∈ N that converges to .{f, g}, there exists a finite positive number c such that .

sup { f (ξ ) E ,  g(ξ ) E ,  fn (ξ ) E ,  gn (ξ ) E , ξ ∈ E} ≤ c.

Let .xn denote the solution satisfying the integral equation (3.6). Since I is a finite interval and .{Sn , S} are .C0 semigroups, there exists a finite positive number M such that .

  sup  S(t) L(E) ,  Sn (t) L(E) , t ∈ I ≤ M.

Clearly, for .x0 ∈ E, there exists a finite positive number .c0 such that . x0 ≤ c0 . Then, it follows from the expression (3.6) by direct computation using triangle inequalities that there exists a finite positive number .r0 such that .

sup { xn (t) E , t ∈ I } ≤ M(c0 + cT + c|ν|) ≡ r0 < ∞,

(3.7)

where .|ν| ≡ |ν|(I ) denotes the total variation norm. Since .An → A in the strong operator topology on .D(A), .Sn (t) → S(t) in the strong operator topology of .L(E) s uniformly on I . That is, for every .e ∈ E, .Sn (t)e → S(t)e uniformly on I . For detailed justification see the proof of Hille-Yosida theorem [5, Theorem 2.2.8, p. 27]. Since .{fn , gn } ⊂ B (E, E) are also uniformly bounded with respect to .n ∈ N, the estimate (3.7) holds true for all .n ∈ N. Next we prove existence and uniqueness of solutions. We show that, for an arbitrary but fixed .n ∈ N , Eq. (3.6) has a unique solution in .B(I, E) ∩ P W C(I, E). Let .Br (E) denote the closed ball in E of radius

3.3 Measure-Valued Solutions

57

r around the origin. For a finite .r ≥ r0 , define the set D ≡ {x ∈ B(I, E) : x(0) = x0 , x(t) ∈ Br (E), ∀ t ∈ I } .

.

Define the operator G on .D by 

t

(Gx)(t) ≡ Sn (t)x0 +

.



t

Sn (t −s)fn (x(s))ds +

0

Sn (t −s)gn (x(s−))ν(ds), t ∈ I.

0

Since by hypothesis .ν has no atom at 0, and it has bounded variation on I , it is easy to verify that G maps .D into itself. Define dt (x, y) ≡ sup {|x(s) − y(s)|E , 0 ≤ s ≤ t} , t ∈ I,

.

and set .d(x, y) ≡ dT (x, y). Since .D is a closed subset of the Banach space .B(I, E), furnished with the metric topology d it is a complete metric space. We show that G has a unique fixed point in .D. By virtue of Banach fixed point principle, it suffices to show that for some .k ∈ N , the k-th iterate (or power) .Gk of the operator G is a contraction on .D. Using the expression for G given above, it is easy to verify that for x, .y ∈ D, we have 

t

 (Gx)(t) . − (Gy)(t) ≤

Mαn,r  x(s) − y(s)  ds

0



t

+

Mαn,r  x(s−) − y(s−)  |ν|(ds), t ∈ I.

(3.8)

0

Since .r (≥ r0 ) and .n ∈ N are fixed, we may drop these subscripts and define 



β(σ ) ≡

αn,r ds +

.

σ

αn,r |v|(ds), σ ∈ B.

(3.9)

σ

Recalling that .αn,r is a finite positive number and .ν is a countably additive bounded signed measure having bounded variation on I , we conclude that the measure .β is also a countably additive bounded positive measure having bounded total variation on I . Define the function  β((0, t]), for t > 0, .θ (t) ≡ 0, for t = 0. This is a nonnegative non-decreasing right continuous function of bounded total variation on I . Hence, it is a bounded function. Any function of bounded variation is differentiable a.e. and the derivative is integrable. Thus, one can choose a nonnegative, integrable function .η that dominates the derivative of .θ a.e., while t its integral w dominates .θ itself everywhere on I satisfying .w(t) ≡ 0 η(s)ds ≥

58

3 Measure Solutions for Impulsive Systems

θ (t+), .t ∈ I . Without any loss of generality one can choose this w in place of .θ. Note that .ds− (x, y) ≤ ds (x, y) for x, .y ∈ D, .s ∈ I . In view of this, it follows from Eqs. (3.8) and (3.9) that  dt (Gx, Gy) ≤ M

.

t

 ds (x, y)dθ (s) ≤ M

0

t

ds (x, y)dw.

(3.10)

0

After k iterations, it follows from (3.10) that   dt (Gk x, Gk y) ≤ (M)k (w(t))k /k! dt (x, y), t ∈ I.

.

Since .t → w(t) and .t → dt are nonnegative and non-decreasing functions, it follows from the above expression that   d(Gk x, Gk y) ≤ (M)k (w(T ))k /k! d(x, y).

.

Thus, for k sufficiently large, .Gk is a contraction on the complete metric space .D, and hence by Banach fixed point theorem .Gk and consequently G, has a unique fixed point in .D. We denote this fixed point by .xn . Thus, we have proved the existence of a unique solution of the integral equation (3.6) in .B(I, E), and hence the existence of a unique mild solution of the evolution equation (3.5). Since .ν is a signed measure having bounded total variation, it has at most a countable set of atoms in I and hence .xn ∈ B(I, E) ∩ P W C(I, E). This completes the proof.   Using the above result we prove the existence of measure-valued solution for the original problem (3.1). In view of Lemma 3.3.5, it is clear that the solutions of the approximating sequence of evolution equation (3.5), denoted by .xn , are piecewise continuous and bounded uniformly with respect to .n ∈ N. Since .ν is a signed measure having bounded total variation on bounded sets, it has at most a countable set of atoms in I which we denote by .a(ν). Considering .t ∈ I \a(ν), and .t > 0 sufficiently small, such that .(t + t) ∈ I \a(ν), the value of .xn (t + t) can be approximated as follows xn (t +t) . ≈ xn (t)+An xn (t)t +fn (xn (t))t +gn (xn (t−))ν([t, t +t]). (3.11) Now let .φ ∈ BC(E) having bounded and continuous Fréchet derivative. By Lagrange formula we have φ(x. n (t + t)) ≈ φ(xn (t))  1   dθ Dφ(xn (t) + θ (xn (t + t) − xn (t))), xn (t + t) − xn (t)E ∗ ,E . + 0

(3.12)

3.3 Measure-Valued Solutions

59

Substituting (3.11) into (3.12), it follows from simple limiting arguments .t → dt, that dφ(xn (t)) = (An φ)(xn (t))dt + (Cn (t)φ)(xn (t−))v(dt),

.

(3.13)

where the operators .An and .Cn are given by .

(An φ)(ξ ) ≡ A∗n Dφ(ξ ), ξ E ∗ ,E + Dφ(ξ ), fn (ξ )E ∗ ,E , .

(3.14)

(Cn (t)φ)(ξ ) ≡ Dφ(ξ ), gn (ξ )E ∗ ,E .

(3.15)

Clearly, these expressions are time-invariant. In case .t ∈ I ∩ a(ν), observing that the first term on the right-hand side of (3.13) only involves Lebesgue measure and the Lebesgue measure of .a(ν) is zero, the operator .An remains unchanged. But the operator .Cn is no more independent of time. Since the last terms of (3.11) and (3.13) contain the measure .ν which may be atomic, again it follows from (3.12) by the same limiting argument that the operator .Cn is now given by 

1

(Cn (t)φ)(ξ ) ≡

.

Dφ(ξ + θgn (ξ )ν({t})), gn (ξ )E ∗ ,E dθ.

(3.16)

0

It is clear from the expression (3.16), that if t is not an atom, it reduces to the expression (3.15). Thus, in general, the operators .An and .Cn of Eq. (3.13) are given by the expressions (3.14) and (3.16), respectively. These are precisely the operators introduced earlier with f and g replaced by .fn and .gn , respectively. With the above preparation, we can now prove the existence of a measure-valued solution for the evolution equation (3.1). This is presented in the following theorem. Theorem 3.3.6 Consider the Cauchy problem (3.1) and suppose the data {A, f, g, ν} satisfy the assumptions (A1) and (A2). Then, for every .x0 ∈ E, the w (I,  (E)) ∩ evolution equation (3.1) has at least one measure solution .λ ∈ M∞ ba w M∞ (|v|, ba (E)) in the sense of Definition 3.3.3. Further, .t → λt is piecewise ∗ continuous on I except at the atoms of the measure .ν, and the support .w .S(λt ) ⊂ Br (E), .r ≥ r0 , for all .t ∈ I . .

Proof Let .An denote the Yosida approximation of A as mentioned above. By the assumption (A2), there exists a pair of sequences .{fn , gn } corresponding to the pair .{f, g} satisfying the approximation properties (i) and (ii) as stated in the assumption (A2). Then, it follows from Lemma 3.3.5 that the approximating sequence of Cauchy problems (3.5) has a sequence of solutions .{xn } ∈ B(I, E) ∩ P W C(I, E). Since these are E valued trajectories (paths), we can define a sequence of measure-valued functions .{λn }, each having all its mass concentrated along these paths, by setting λnt ( ) ≡ δxn (t) ( ), λ0 ( ) ≡ δx0 ( ), ∈ .

.

60

3 Measure Solutions for Impulsive Systems

Using this notation we can rewrite Eq. (3.13) in the form dλnt (φ) = λnt (An φ)dt + λnt− (Cn (t)φ)v(dt), .

λn0 (φ) = δxo (φ) = φ(x0 ) ≡ λ0 (φ).

Integrating this we obtain the following equivalent functional equation  λnt (φ) = λ0 (φ) +

.

0

t

 λns (An φ)ds +

t

0

λns− (Cn (s)φ)v(ds), t ∈ I.

(3.17)

w (I,  (E)) ∩ M w (|v|,  (E)) which Clearly, for each integer .n ∈ N, .λn ∈ M∞ ba ba ∞ w is a closed subset of .L∞ (I, Mba (E)) ∩ Lw ∞ (|v|, Mba (E)), and it follows from our preceding analysis that for any .r ≥ r0 , .

supp(λnt ) ⊂ Br , ∀ t ∈ I, n ∈ N.

Note that for .φ ∈ D(A), .An φ, .Cn φ ∈ B(E). Thus, for each .s ∈ I , the integrands in Eq. (3.17) are well-defined as the duality products related to the dual pair of Banach spaces .(B(E), Mba (E)). Our objective is to show that the sequence .{λn } has a convergent subsequence (subnet) and that the limit is a measure-valued solution (generalized solution) of our original problem. Towards this goal, consider the sequence of linear functionals .{n } given by  n (ϕ) ≡

 ϕ1 (t, ξ )λnt (dξ )dt +

.

I ×E

I ×E

ϕ2 (t, ξ )λnt− (dξ )v(dt).

Clearly, this is well-defined for each .ϕ = (ϕ1 , ϕ2 ) ∈ L1 (I, B(E)) ⊕ L1 (|v|, B(E)) and .|n (ϕ)| ≤ |ϕ|L1 (I,B(E))⊕L1 (|v|,B(E)) , .∀ n ∈ N, where .|ϕ|L1 (I,B(E))⊕L1 (|v|,B(E)) ≡ |ϕ1 |L1 (I,B(E)) + |ϕ2 |L1 (|v|,B(E)) . In other words, .{n } is a sequence of bounded linear functionals contained in a bounded subset of the dual .

(L1 (I, B(E)) ⊕ L1 (|v|, B(E)))∗ .

Thus, it follows from the characterization of the dual space of the Banach space L1 (I, B(E)) ⊕ L1 (|v|, B(E)), that the sequence .{λn } is contained in a bounded w subset of .Lw ∞ (I, Mba (E)) ∩ L∞ (|v|, Mba (E)). More precisely, they are con∗ w (I,  (E)) ∩ M w (|v|,  (E)). tained in the .w closed and bounded set .M∞ ba ba ∞ Hence, by Alaoglu’s theorem there exists a generalized subsequence (or subnet) w (I,  (E)) ∩ of the sequence (net) .{λn }, relabeled as .{λn }, and .λo ∈ M∞ ba w M∞ (|v|, ba (E)) such that

.

w∗

w w λn −→ λo , in M∞ (I, ba (E)) ∩ M∞ (|v|, ba (E)).

.

3.3 Measure-Valued Solutions

61

We show that .λo is a measure (generalized) solution of the evolution equation (3.1) in the sense of Definition 3.3.3. Let .φ ∈ D(A) with both .φ and .Dφ being continuous and bounded having compact supports (which may be different for different .φ). Clearly, .Dφ(ξ ) ∈ D(A∗ ) for all .ξ ∈ E. Define the operators A ≡ A1 + A2 ,

.

where (A1 φ)(ξ ) ≡ A∗ Dφ(ξ ), ξ E ∗ ,E , (A2 φ)(ξ ) ≡ Dφ(ξ ), f (ξ )E ∗ ,E

.

and An ≡ A1,n + A2,n

.

for (A1,n φ)(ξ ) ≡ A∗n Dφ(ξ ), ξ E ∗ ,E , (A2,n φ)(ξ ) ≡ Dφ(ξ ), fn (ξ )E ∗ ,E .

.

Clearly, for .φ ∈ D(A) we have An φ = Aφ + (A∗n − A∗ )Dφ, ξ  + Dφ, fn − f 

.

= Aφ + (A1,n − A1 )φ + (A2,n − A2 )φ.

(3.18)

Consider the second term of Eq. (3.18). Since .An → A in the strong operator topology on .D(A) and .D(A) is dense in E, for every .y ∈ D(A∗ ) ⊂ E ∗ , we have w∗

A∗n y −→ A∗ y in E ∗ .

.

For .φ ∈ D(A) we have .Dφ ∈ D(A∗ ). Since both .φ and .Dφ are continuous having compact supports, it follows from this that .

  sup |((A1,n − A1 )φ)(ξ )|, ξ ∈ E → 0,

(3.19)

as .n → ∞. Consider the third term of Eq. (3.18). Since .Dφ is continuous having compact support, and f , .fn are uniformly bounded and .fn → f uniformly on compact subsets of E, we have .

  sup |((A2,n − A2 )φ)(ξ )|, ξ ∈ E → 0,

(3.20)

as .n → ∞. Now consider the operator .C given by  (C(t)φ)(ξ ) =

.

0

1

Dφ(ξ + θg(ξ )ν({t})), g(ξ )E ∗ ,E dθ.

(3.21)

62

3 Measure Solutions for Impulsive Systems

Clearly, Cn = C + (Cn − C),

.

where .

((Cn (t) − C(t))φ)(ξ )  1   = dθ Dφ(ξ + θgn (ξ )ν({t})), gn (ξ )E ∗ ,E − Dφ(ξ + θg(ξ )ν({t})), g(ξ ) 

0 1

=

dθ {Dφ(ξ + θgn (ξ )ν({t})) − Dφ(ξ + θg(ξ )ν({t})), g(ξ )}

0



1

+

dθ {Dφ(ξ + θgn (ξ )ν({t})), gn (ξ ) − g(ξ )} .

(3.22)

0

Again, as .gn converges to g uniformly on compact sets and .Dφ is continuous with compact support, it follows from similar arguments as used for the operator .𝒜, that for each .t ∈ I , .

sup {|((Cn (t) − C(t))φ)(ξ )|, ξ ∈ E} → 0,

(3.23)

as .n → ∞. Clearly, we can express Eq. (3.17) in the form  λnt (φ) . = λ0 (φ) + 

t

− 0

t

0

 λns (Aφ)ds + 

λns (Aφ − An φ)ds −

t

0 t

0

λns− (C(s)φ)ν(ds)

λns− (C(s)φ − Cn (s)φ)ν(ds), t ∈ I.

(3.24)

It follows from the expressions (3.19), (3.20) and (3.23) and dominated convergence theorem that s

.

(Aφ − An φ) −→ 0 in L1 (I, B(E)), . s

(Cφ − Cn φ) −→ 0 in L1 (|v|, B(E)),

(3.25) (3.26)

where, we recall that

 L1 (|v|, B(E)) ≡ ψ(·) : ψ(t) ∈ B(E) and |ψ(t)|B(E) |v|(dt) < ∞ .

.

I s

w∗

Now we use the fact that if .zn → z in a Banach space X and .zn∗ −→ z∗ w∗ in the dual space .X∗ , then .zn∗ (zn ) → z∗ (z). We have seen above that .λn → w λo in .Lw ∞ (I, Mba (E)) ∩ L∞ (|v|, Mba (E)). It follows from these facts and the

3.3 Measure-Valued Solutions

63

expressions (3.25) and (3.26) that the last two components on the right-hand side of Eq. (3.24) vanish as .n → ∞. Thus, letting .n → ∞ in the expression (3.24), we arrive at the following expression  λot (φ) = λ0 (φ) +

.

0

t

 λos (Aφ)ds +

t 0

λos− (C(s)φ)ν(ds), t ∈ I.

(3.27)

Note that the first term on the right-hand side of Eq. (3.27) is bounded for any φ ∈ BC(E), and the second and third terms are bounded for any .φ ∈ D(A) ⊂ D(C). Thus, Eq. (3.27) holds for all .φ ∈ D(A), and not just for only those having compact supports. This proves that .λo is a measure-valued solution of the evolution equation (3.1) in the sense of Definition 3.3.3. For the last part of the theorem, first note that if .ν is absolutely continuous with respect to the Lebesgue measure, it follows directly from the expression (3.27) that .t → λot is weak star continuous on I . Jumps are expected whenever .ν has atoms. Since by our assumption, the measure .ν has bounded total variation, the set of atoms, .a(ν) ∩ I , is countable. Between any two atoms, .λo is weak star continuous having jumps only at the atoms. Hence, follows the assertion on piece-wise .w ∗ continuity and the identity .limI \a(ν)t↓0 λt (φ) = λ0 (φ). Recall that the supports of the sequence of measurevalued functions .λnt are contained in the closed ball .Br (E). Since .λo is the .w ∗ limit of this sequence, it is easy to verify that .λot (ϕ) = 0 for every .ϕ ∈ B(E) having support outside .Br (E). Thus, .Supp(λot ) ≡ S(λot ) ⊂ Br (E), t ∈ I . This completes the proof.   .

Remark 3.3.7 It follows from the proof of Theorem 3.3.6 that the set of mild solutions of Eq. (3.5) (approximating Eq. (3.1)) is dense (with respect to the .w ∗ topology) in the class of measure solutions of Eq. (3.1) in the sense of the embedding   w w x ∈ B(I, E) → δx(t) , t ∈ I ⊂ M∞ (I, ba (E)) ∩ M∞ (|v|, ba (E)).

.

Now we are prepared to prove existence of measure solutions for Eq. (3.1) with measurable vector fields which are merely bounded on bounded sets. We prove this result under the assumption that f and g are . measurable and that they are bounded on bounded subsets of E. In this case we use the compact Hausdorff space .E + in place of E to capture the supports of the measure solutions. Define for each (finite) real number .γ > 0, fγ (x) ≡ f (Rγ (x)), gγ (x) ≡ g(Rγ (x)), γ > 0,

.

where .Rγ is the retraction of the ball .Bγ ⊂ E, that is,  Rγ (ξ ) ≡

.

ξ,

if ξ ∈ Bγ ,

(γ /|ξ |E )ξ,

otherwise.

64

3 Measure Solutions for Impulsive Systems

Clearly, for each nonnegative .γ < ∞, .fγ and .gγ are bounded . measurable maps, that is, .fγ , .gγ ∈ B (E, E). Thus, for each finite positive number .γ , .

  sup |fγ (ξ )|E , |gγ (ξ )|E , ξ ∈ E < ∞.

Theorem 3.3.8 Let A be the infinitesimal generator of a .C0 semigroup in the Banach space E and .f, g : E → E, are . measurable maps bounded on bounded subsets of E and that .fγ , .gγ possess the approximation property stated in the assumption (A2) of Theorem 3.3.6. Then, for each .x0 ∈ E, the evolution w (I,  (E + )) ∩ equation (3.1) has at least one measure-valued solution .M∞ ba w + ∗ M∞ (|v|, ba (E )). Further, .t → λt is piecewise .w continuous. Proof The basic technique is similar to that of Theorem 3.3.6 (see also [13, Theorem 3.2, p. 1341]). Here we give a brief outline. Clearly, the truncated maps .{fγ , gγ , γ > 0}, are . measurable maps from E to E and bounded on all of E. Since for each .γ > 0, the maps .{fγ , gγ } possess the approximation property (A2), it follows from Theorem 3.3.6 that the evolution equation dx(t) = Ax(t)dt + fγ (x(t))dt + gγ (x(t−))v(dt), t ∈ I, x(0) = x0

(3.28)

.

w (I,  (E)) ∩ M w (|v|,  (E)). In has at least one measure solution .λγ ∈ M∞ ba ba ∞ γ other words, .λ is a measure solution of the evolution equation (3.28) in the sense of Definition 3.3.3 satisfying γ .λt (φ)

=

γ λ0 (φ) +



t

γ λs (Aγ φ)ds

0

 + 0

t

γ

(3.29)

λs− (Cγ (s)φ)v(ds), γ

for each .φ ∈ D(Aγ ) ∩ D(Cγ ) with .Dφ having bounded support, where .λ0 = δx0 , the Dirac measure supported at .{x0 }. The operators .Aγ and .Cγ are given by (Aγ φ)(ξ ) ≡ A∗ Dφ(ξ ), ξ  + Dφ(ξ ), fγ (ξ )  1 .   Dφ(ξ + θgγ (ξ )v({t})), gγ (ξ ) dθ. (Cγ (t)φ)(ξ ) ≡ 0

Again, following the same procedure using dominated convergence theorem as in Theorem 3.3.6, one can verify that, for each .φ ∈ D(A) with .Dφ(ξ ) ∈ D(A∗ ) and .Dφ continuous having compact support, s

Aγ φ −→ Aφ in L1 (I, B(E)), .

s

Cγ φ −→ Cφ in L1 (|v|, B(E)). This is an immediate consequence of the facts that, as .γ → ∞, .{fγ , gγ } converge to {f, g} uniformly on bounded subsets of E, and that, for every given .φ ∈ D(A) with

.

3.3 Measure-Valued Solutions

65

Dφ having compact support, the functions .{Aγ φ, Aφ, Cγ φ, Cφ} vanish outside this support. Hence, .Aγ φ, .Aφ ∈ L1 (I, B(E)) and .Cγ φ, .Cφ ∈ L1 (|v|, B(E)). Now for each .γ > 0, the functional .γ given by

.





γ

γ (ψ) ≡

.

i

λt (ψ1 )dt +

  ≡ i

E

γ

i

λt− (ψ2 )v(dt)

γ ψ1 (t, ξ )λt (dξ )dt

 

γ

+ i

E

ψ2 (t, ξ )λt− (dξ )v(dt)

is well-defined on .L1 (I, B(E)) ⊕ L1 (|v|, B(E)), and we have |γ (ψ)| ≤ |ψ|L1 (I,B(E))⊕L1 (|v|,B(E))

.

for all .ψ ∈ (L1 (I, B(E)) ⊕ L1 (|v|, B(E))). Thus, .{γ , γ > 0} is contained in a bounded subset of the dual .(L1 (I, B(E + )) ⊕ L1 (|ν|, B(E + )))∗ . Hence, by duality the associated set of measure-valued functions, .{λγ , γ > 0}, is contained in a + w + ∗ bounded subset of .Lw ∞ (I, Mba (E )) ∩ L∞ (|v|, Mba (E )). The .w closure of a ∗ bounded set in a dual space is .w compact. Therefore, again by Alaoglu’s theorem, there exists a subnet or a generalized subsequence .{λk ≡ λγk }, with .γk → ∞ as o w + w + .k → ∞, and a .λ ∈ L∞ (I, Mba (E )) ∩ L∞ (|v|, Mba (E )) such that w∗

λk −→ λo as k → ∞.

.

Thus, letting .γ → ∞ in the expression (3.29), along a subnet if necessary, we obtain  λot (φ) = λ0 (φ) +

t

.

0

 λos (A(s)φ)ds +

0

t

λos− (C(s)φ)v(ds), t ∈ I.

(3.30)

Following similar arguments as in Theorem 3.3.6, we conclude that w w λ o ∈ M∞ (I, ba (E + )) ∩ M∞ (|v|, ba (E + )).

.

Hence, .λo is a measure solution of the evolution equation (3.1) with . measurable vector fields .{f, g} which are merely bounded on bounded sets. The last part of the statement asserting piecewise .w ∗ continuity follows from identical arguments as   given in the proof of Theorem 3.3.6. This completes the proof. Remark 3.3.9 In case of unbounded vector fields .{f, g} as admitted in the above theorem, it is quite likely that the content of measure-valued solutions may leak ˆ out of the space E. In order to capture the support, we have used the Stone-Cech + compactification of E giving the compact Hausdorff space .E = βE (see [16]) and hence a larger space w w λ o ∈ M∞ (I, ba (E + )) ∩ M∞ (|v|, ba (E + ))

.

66

3 Measure Solutions for Impulsive Systems

as seen in the proof of Theorem 3.3.8. This process gives us countably additive measure-valued solutions taking values in .ca (E + ) ⊂ ba (E + ). Countably additive measures are obviously finitely additive and so we have retained the notation .ba (E + ) to indicate that they were generated from finitely additive counterparts. In fact, the evolution equation (3.1) has measure solutions not only for initial data given by a Dirac measure but also for any initial data described by a measure .π0 ∈ ba (E). For details see [19, Corollary 3.4]. The results presented above also hold for systems with time-dependent operators .{A(t), f (t, x), g(t, x), t ∈ I, x ∈ E} under the assumption that A generates a two parameter evolution operator .U (t, s), .0 ≤ s ≤ t < ∞ on the Banach space E. The proof needs only minor changes in the details of the analysis.

3.3.2 Measure Solutions vs. Pathwise Solutions It is interesting to enquire the possibility of existence of pathwise solutions, w (I,  (E + )) ∩ given that the system has measure-valued solutions .λ ∈ M∞ ba w + M∞ (|ν|, ba (E )). The answer is generally no, unless the vector fields .{f, g} are sufficiently regular. However, the question of existence of a mean trajectory may be more realistic. For example, if the vector fields .{f, g} are uniformly bounded on the Banach space .E, then the measure solution .λ has bounded support in E for finite intervals of time I , and hence one may expect the existence of an E valued function m such that  ∗ .e , m(t) ≡ e∗ , ξ  λt (dξ ), t ∈ I, E

for every .e∗ ∈ E ∗ . In the general case, for any .β > 0, one may consider the functional   2 ∗ (e∗ , ξ )e−β|ξ |E λt (dξ ), t ∈ I. .mβ (t, e ) ≡ E

Since the integrand enclosed by the parenthesis is in .B(E), this defines a continuous linear functional on .E ∗ . In other words, for each .t ∈ I and .β > 0, .e∗ → mβ (t, e∗ ) is a continuous linear functional on .E ∗ . Hence, in the sense of Dunford integral [66, Theorem 8, Corollary 9, p. 55–56], there exists an .E ∗∗ valued function denoted by .m ˜ β (t) such that mβ (t, e∗ ) = m ˜ β (t), e∗ E ∗∗ ,E ∗ , ∀ e∗ ∈ E ∗ .

.

In case .m ˜ β (t) is E valued, we have the Pettis integral [69] and only in this case we can hope for the existence of a mean process as there exists an E valued function

3.4 Differential Equations on the Space of Measures

67

m(t) such that

.

w

m ˜ β (t) → m(t) in E

.

as .β ↓ 0 for all .t ∈ I \a(ν). If the above integral exists only in the sense of Dunford, we have .m ˜ β (t) an .E ∗∗ valued function and so we do not have an E valued mean process. Similarly, we can define a correlation operator as follows. For simplicity, consider E to be a reflexive Banach space and define the operator valued function .Qβ by   2 (e∗ , ξ )2 e−β|ξ |E λt (dξ ), t ∈ I. .(Qβ (t)e , e ) ≡ ∗

∗

E

Again, it is clear that for .β > 0 the integrand is in .B(E), the space of bounded Borel measurable real valued functions on E. So, the operator .Qβ (t), .t ∈ I , is well-defined as elements of .L(E ∗ , E) and it is symmetric and positive. Thus, the measure solution has a correlation operator provided there exists an operator .Q(t) ∈ L(E ∗ , E) such that τwo

Qβ (t) −→ Q(t), for each t ∈ I,

.

in the weak operator topology as .β ↓ 0. If for any sequence of normalized family {ei∗ } ∈ E ∗ , we have

.

.

Tr(Qβ (t)) ≡

∞

Qβ (t)ei∗ , ei∗  < ∞

i=1

only for .β > 0, the measure solution does not possess a correlation operator. In other words, for the existence of a correlation operator corresponding to the measure solution .λt , .t ∈ I , it is necessary and sufficient that the limit .Q(t), .t ∈ I, is a nuclear operator having .Tr(Q(t)) < ∞ for each .t ∈ I .

3.4 Differential Equations on the Space of Measures In view of the preceding results on measure-valued solutions and the general definition of measure solutions as seen above, we can reformulate the Cauchy problems on the Banach space of finitely additive measures .Mba (E + ) as follows dμt = A∗ μt dt + C∗ (t)μt− ν(dt), t ≥ s, μs = π,

.

(3.31)

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3 Measure Solutions for Impulsive Systems

where .A∗ and .C∗ denote the adjoint operators corresponding to the operators given by (Aφ)(ξ ) ≡ A∗ Dφ(ξ ), ξ  + Dφ(ξ ), f (ξ ),  1 . {Dφ(ξ + θg(ξ )ν({t})), g(ξ )} dθ. (C(t)φ)(ξ ) ≡ 0

This is the differential version of the functional equation (3.30), with the latter being the weak form of the former. This of course covers the original Cauchy problem as a special case. According to our existence results, we have seen that this equation has solution in the weak sense. Hence, it follows from these results that for each initial state .π ∈ ba (E) ⊂ Mba (E), the evolution equation (3.31) has at least one w (I,  (E + )) ∩ M w (|v|,  (E + )) ⊂ Lw (I, M (E + )) ∩ weak solution .μ ∈ M∞ ba ba ba ∞ ∞ w + L∞ (|v|, Mba (E )) which is piecewise .w ∗ continuous. Consequently, there exists a piecewise .w ∗ continuous transition operator .U ∗ (t, s), .t ≥ s ≥ 0, which is a family of bounded linear operators on the Banach space .Mba (E + ) defining the flow of the measure solution μt = U ∗ (t, s)π, t ≥ s, s ≥ 0.

.

It is easy to verify that for each .t ∈ [0, ∞), .|μt (ϕ)| ≤ |ϕ|B(E) for all .ϕ ∈ B(E). Thus, the operator .U ∗ (t, s), .0 ≤ s ≤ t < ∞, is a contraction (more precisely, non-expansive), that is, for any arbitrary .ν ∈ Mba (E + ), we have |U ∗ (t, s)ν|Mba (E + ) ≤ |ν|Mba (E + ) , 0 ≤ s ≤ t ≤ T .

.

So far we have not discussed the question of uniqueness of solutions. This is of course equivalent to the question of uniqueness of the transition operator .U ∗ (t, s). In case of non-impulsive evolution equations [19], we had only partial answers based on spectral theory. For more on uniqueness the reader is referred to [23]. Here we present a result on uniqueness without proof. For proof the interested reader is referred to [23]. Theorem 3.4.1 Suppose the assumptions of Theorem 3.3.8 hold and let .{A∗ , C∗ (t)} denote the formal adjoints of the generating operators with .D(A) being dense in ∗ .B(E). Then, the corresponding evolution operator .U (t, s), .0 ≤ s ≤ t < ∞, is unique. Remark 3.4.2 In the preceding results we have considered .ν to be a signed measure. It is not necessary to restrict to signed measures. In fact, the results presented above can be easily extended to vector measure .ν with values in a Banach space F . In that case g is an operator valued function on E taking values .L(F, E) and the measure .ν is an F valued vector measure having bounded variation on I .

3.5 Differential Inclusions

69

3.5 Differential Inclusions In the study of systems and control theory, one often has to deal with systems governed by differential inclusions. One major source of such systems is imperfect knowledge of the associated vector fields like .{f, g}. Dynamic models are developed by scientists on the basis of the fundamental laws of physics. In the process, one has to include some basic parameters, the crucial building blocks of dynamic systems. For example, in the case of Navier-Stokes equations, one must know the coefficient of viscosity or the Reynolds number and strain rate etc; in the case of Maxwell’s equation the parameters are permittivity, permeability and speed of light; in the case of diffusion (in particular heat) equations one must know parameters like thermal diffusivity, heat conductivity and the material density; in the case of Schrödinger equation, the Planck’s constant and potentials. All these fundamental parameters are determined experimentally and therefore are subject to experimental errors and inaccuracies. Experimentalists often present the data with some standard deviation. Hence, these dynamic systems may be described by differential inclusions. Another situation that gives rise to differential inclusions is the dynamic models governed by variational inequalities determined by subdifferentials of proper convex functions. For more on physical interpretation see [17] and broad literature in general.

3.5.1 Classical Model Let us consider the following system, .

x(t) ˙ ∈ Ax(t) + F (x(t)), t ∈ I \D, x(0) = x0 , x(ti ) ∈ Gi (x(ti )), ti ∈ D ≡ {ti , i = 1, 2, · · · , n}, tn = T ,

(3.32)

where A is the infinitesimal generator of a .C0 semigroup in the Banach space E and the vector fields .F, Gi : E → 2E \∅, i = 1, 2, · · · , n, are multi functions. The symbol . denotes the jump operator given by .x(ti ) = x(ti + 0) − x(ti ) and .Gi denotes the set of admissible jump sizes. Here the first part of the system is governed by a differential inclusion which determines continuous evolution over the time interval .I \ D, and the second part of the system forces a change of the state through impulsive actions at a set of discrete points of time D, thereby changing the course of evolution. Let .SF and .SGi denote the set of all continuous selections of the multi functions F and .Gi , .i = 1, 2, · · · , n, respectively and suppose they are nonempty. For each .f ∈ SF and .gi ∈ SGi , we define the operators .

(𝒜f φ)(ξ ) ≡ A∗ Dφ(ξ ), ξ E ∗ ,E + Dφ(ξ ), f (ξ )E ∗ ,E ,  1 Dφ(ξ + θgi (ξ )), gi (ξ )E ∗ ,E dθ. (𝒞gi φ)(ξ ) = 0

70

3 Measure Solutions for Impulsive Systems

We show that for each pair of continuous selections of the multi functions F and Gi the system (3.32) has a solution. Clearly, this defines a set or a bundle of w (I,  + measure-valued trajectories in .M∞ rba (E )), which is called the solution of the evolution inclusions (3.32).

.

w (I,  + Definition 3.5.1 An element .λ ∈ M∞ rba (E )) is said to be a measure solution of the evolution inclusions (3.32) corresponding to the initial state .x0 ∈ E, if for each .f ∈ SF and .gi ∈ SGi , the following identities hold:

 .

λt (φ) = λti +0 (φ) +

t

λs (𝒜f φ)ds, t ∈ (ti , ti+1 ], i = 0, 1, 2, · · · , n − 1,

ti

λti +0 (φ) = λti (𝒞gi φ), i = 1, 2, 3, · · · , n − 1,

(3.33)

for each .φ ∈ D(𝒜f ) having bounded supports. Under some regularity assumptions on the multi functions .{F, Gi , i = 1, 2, · · · }, we can prove the existence of solutions of the evolution inclusions (3.32). Let .cbc(E) denote the class of nonempty closed bounded convex subsets of E. Theorem 3.5.2 Suppose (A1): A is the infinitesimal generator of a .C0 semigroup in E, (A2): .{F, Gi } : E → cbc(E + ) are lower semicontinuous multi functions mapping bounded sets into bounded sets. Then, for each initial state .x0 ∈ E, or more generally .L(x0 ) = λ0 ∈ rba (E + ), the system (3.32) has a nonempty set of w (I,  + measure-valued solutions in .M∞ rba (E )). Proof Under the assumptions of convexity and lower semicontinuity of the multi valued maps .{F, Gi }, it follows from Michael’s selection theorem (see Chap. 1) that the sets .SF and .SGi are nonempty. Hence, there exist .f ∈ SF and .gi ∈ SGi for all .{i = 1, 2, · · · , n − 1}. Since .{F, Gi } map bounded sets into bounded sets, the selections .{f, gi } also map bounded sets into bounded sets. Thus, in summary .{f, gi } are continuous maps from E to E and also bounded on bounded sets. It follows from Proposition 2.3.2 that the functions .{f, gi , i = 1, 2, 3, · · · , n−1} have the required approximation properties satisfying the assumptions of Theorem 2.1.5. Thus, it follows from this theorem that, on each interval .Ii = (ti , ti+1 ], given .λti and hence .λti +0 determined by the jump operator, there exists a measure solution .λ ∈ w (I ,  + M∞ i rba (E )) satisfying the system of Eqs. (3.33). To continue this process for the next interval .Ii+1 , the initial state is given by the expression λti+1 +0 (φ) = λti+1 (𝒞gi+1 φ),

.

which is a result of operation by the jump operator on the terminal state attained at the end of the previous interval. The process is continued until the last interval is covered. Thus, the system (3.32) has a nonempty set of measure-valued solutions. This completes the proof.   Remark 3.5.3 The system governed by the pair of differential and algebraic inclusions (3.32) can be used to model uncertain dynamic systems, where the multi

3.5 Differential Inclusions

71

function F may represent parametric or model uncertainty in the system equations. The multi functions .{Gi } can play several roles. They can be considered as intermittent impulsive disturbances experienced by the system. Also, they can be used to model controls such as Gi (x) = g(x, Ui ) = {η ∈ E : η = g(x, ζ ), ζ ∈ Ui }

.

with .Ui being any compact Polish space.

3.5.2 General Model Impulsive systems considered in the preceding subsection are special cases of more general systems driven by signed measures (or vector measures). There, the multi functions are assumed to be lower semicontinuous; here we relax this assumption by admitting measurable multi functions. For simplicity, here we consider systems governed by differential inclusions driven by signed measures. This is described as follows dx ∈ Axdt + F (t, x)dt + G(t, x)ν(dt), L(x0 ) = π0 ∈ ba (E),

.

(3.34)

where F and G are suitable multi functions. For technical reasons only, here we restrict ourselves to Borel measurable vector fields and multi functions. Let .BI ×E denote the Borel algebra of subsets of the set .I × E and .BM(I × E, E) denote the space of bounded .BI ×E measurable functions from .I × E to E in the sense that the inverse image (with respect to .f ∈ BM(I × E, E)) of any Borel set in the range space E is an element of .BI ×E . Since E is a Banach space, furnished with the sup norm topology .

sup {|f (t, x)|E , (t, x) ∈ I × E} ,

BM(I × E, E) is also a Banach space. Let .MU(I × E, 2E \∅) denote the class of nonempty .BI ×E measurable multi functions .{F } in the sense that for every open set E .O ⊂ E, and .F ∈ MU(I × E, 2 \∅), the set .

F − (O) ≡ {(t, ξ ) ∈ I × E : F (t, ξ ) ∩ O = ∅} ∈ BI ×E

.

and that |F (t, ξ )|o ≡ sup {|e|E : e ∈ F (t, ξ )} < ∞, ∀ (t, ξ ) ∈ I × E.

.

We are particularly interested in the following two classes of multi functions, MU(I × E, wkc(E)) and .MU(I × E, kc(E)), where .wkc(E)(kc(E)) denotes the class of nonempty weakly compact convex subsets of E. We continue to denote

.

72

3 Measure Solutions for Impulsive Systems

the class of Lebesgue-Bochner integrable functions on I with values in .B(E) by L1 (, B(E)) (. Lebesgue measure on I ) and those that are integrable with respect to the measure .|ν|(·) by .L1 (|ν|, B(E)). As seen in the preceding chapters, measure-valued solutions are defined in the weak sense. This requires test functions. Here, we introduce the following class of test functions,

.

.

 ℱ ≡ φ ∈ BC(E) : Dφ ∈ BC(E, E ∗ ), Dφ(ξ ) ∈ D(A∗ ),

 𝒜f (t)φ ∈ B(E), 𝒞g (t)φ ∈ B(E) for a.e. t ∈ I ,

for each pair f , .g ∈ BM(I × E, E). Theorem 3.5.4 Suppose the following assumptions hold. (a1): E is a separable Banach space and A is the infinitesimal generator of a .C0 semigroup on E, (a2): .F ∈ MU(I × E, wkc(E)) and .G ∈ MU(I × E, kc(E)), (a3): .ν is a signed measure having bounded variation on bounded sets. Then, for every .π0 ∈ ba (E), the system (3.34) has a nonempty set of measure-valued solutions denoted by w + w + .(F, G, π0 ) ⊂ M∞ (, ba (E )) ∩ M∞ (|ν|, ba (E )). Proof Under the assumption (a2), the multi functions F and G take values from c(E), the class of nonempty closed subsets of E, and that they are measurable with respect to the Borel field of sets .BI ×E . Since .(I × E, BI ×E ) is a measurable space and E a separable Banach space (hence a Polish space), it follows from the well-known Kuratowski-Ryll Nardzewski selection theorem (Theorem 1.3.3, see also [90, Theorem 2.1, p. 154]) that both F and G have .BI ×E measurable selections. We choose any such pair of measurable selections .{f, g} of the multi functions .{F, G} respectively and consider the following evolution equation, .

dx = Axdt + f (t, x(t))dt + g(t, x(t−))ν(dt), x(0) = x0 .

.

(3.35)

Then, it follows from Theorem 3.3.8 that for any fixed initial state, .L(x0 ) = π0 ∈ w (,  (E + )) ∩ ba (E), Eq. (3.35) has a measure solution .λ ≡ λ(f, g) ∈ M∞ ba w + M∞ (|ν|, ba (E )) in the sense of Definition 3.3.3. In other words, the evolution equation dλt = A∗f (t)λt dt + C∗g (t)λt− ν(dt), λ0 = π0

.

(3.36)

w (,  (E + )) ∩ M w (|ν|,  (E + )), has a measure solution .λ = λ(f, g) ∈ M∞ ba ba ∞ where the operators .{Af , Cg } are given by .

(𝒜f (t)φ)(ξ ) ≡ A∗ Dφ(ξ ), ξ E ∗ ,E + Dφ(ξ ), f (t, ξ )E ∗ ,E ,  1 dθ Dφ(ξ + θg(t, ξ )ν({t})), g(t, ξ )E ∗ ,E . (𝒞g (t)φ)(ξ ) = 0

3.5 Differential Inclusions

73

In fact, it follows from Theorem 3.3.8 that every pair .{f, g} of measurable selections of the multi functions .{F, G}, determines a measure-valued solution of the evolution equation (3.36) in the sense of Definition 3.3.3. Let .SF and .SG denote the set of all such .BI ×E measurable selections of the multi functions F and G, respectively. We conclude from this that the solution set denoted by (F, G, π0 ) ≡ {λ(f, g), f ∈ SF , g ∈ SG }

.

w (,  (E + )) ∩ M w (|ν|,  (E + )). Since the set is a nonempty subset of .M∞ ba ba ∞ w w M∞ (, ba (E + )) ∩ M∞ (|ν|, ba (E + ))

.

is bounded and .w ∗ closed, we conclude that it is a .w ∗ compact subset of w + w + ∗ .L∞ (, Mba (E )) ∩ L∞ (|ν|, Mba (E )). Thus the set .(F, G, π0 ) is relatively .w ∗ ∗ ∗ compact, that is, its .w closure is .w compact. Thus, for .w sequential compactness, it suffices to prove that .(F, G, π0 ) is .w ∗ sequentially closed. Let .{λn } be any sequence from the set .(F, G, π0 ) and suppose it converges in the .w ∗ topology w (,  (E + )) ∩ M w (|ν|,  (E + )). We must show that to an element .λo ∈ M∞ ba ba ∞ o .λ ∈ (F, G, π0 ). Since .λn ∈ (F, G, π0 ), there exists a sequence .{fn , gn } of n .BI ×E measurable selections of the multi functions .{F, G} so that .λ = λ(fn , gn ). Again we use the assumption (a2), in particular the properties that F takes values from .wkc(E) and G takes values from .kc(E). Based on these properties, one can show that there exists a subsequence of the sequence .{fn , gn }, relabeled as the original sequence, and a pair of .BI ×E measurable selections .{fo , go } of the multi functions F and G, respectively, such that w

fn (t, e) −→ fo (t, e) in E, .

s

gn (t, e) −→ go (t, e) in E, point-wise and hence on any compact subset of .I × E. Then, using the Definition 3.3.3, we write Eq. (3.36) in its weak form using the pair of vector fields .{fn , gn } and .{f0 , g0 } respectively for any .ϕ ∈ F having compact support. Taking the difference of the corresponding expressions and, using the convergence properties stated above, and dominated convergence theorem, one can verify that .λo coincides with the solution .λ(fo , go ) of Eq. (3.36) corresponding to .f = fo and .g = go in the weak sense. Thus, we have .λo ∈ (F, G, π0 ), thereby proving that the set ∗ .(F, G, π0 ) is sequentially .w closed and hence, being a subset of a weak star compact set, it is sequentially .w ∗ compact. This completes the proof.   Remark 3.5.5 Since the .w ∗ (weak star) topology is Hausdorff, we may consider w (,  (E + )) ∩ M w (|ν|,  (E + )) to be a compact Hausdorff space the set .M∞ ba ba ∞ + contained in the linear space .Lw (, M (E + )) ∩ Lw ba ∞ ∞ (|ν|, Mba (E )).

74

3 Measure Solutions for Impulsive Systems

3.6 Bibliographical Notes In this chapter we have considered impulsive systems driven by signed measures and presented several results on measure-valued solutions. Theorem 3.3.6 proves measure-valued solutions under the assumption that the operator A is the infinitesimal generator of .C0 -semigroup and the vector fields f and g are merely Borel measurable uniformly bounded maps (not necessarily continuous). Theorem 3.3.8 relaxes the assumption and covers vector fields .{f, g} which are Borel measurable and bounded on bounded sets. Using Dunford and Pettis integrals, the relationship between measure solutions and pathwise solutions are discussed. Theorem 3.5.2 presents measure-valued solutions for classical impulsive differential inclusions and Theorem 3.5.4 considers general impulsive differential inclusions. These results were proved by one of the authors [15, 23]. The authors are not familiar with any work in the literature considering measure-valued solutions for impulsive differential equations and inclusions on infinite dimensional spaces.

Chapter 4

Measure Solutions for Stochastic Systems

4.1 Introduction In this chapter we consider stochastic differential equations (SDE) on Hilbert spaces given by dx = Axdt + F (x)dt + σ (x)dW, t ≥ 0, x(0) = x0 ,

.

(4.1)

where A is the infinitesimal generator of a .C0 semigroup, .S(t), .t ≥ 0 on H , and F : H −→ H and .σ : H −→ L(E, H ) are continuous maps. The process .W = {W (t), t ≥ 0} is an E valued Brownian motion also called Wiener process, where E is a separable Hilbert space. It is well-known [64] that if F and .σ satisfy Lipschitz properties having at most linear growth and the initial state .x0 has finite second moment, the system (4.1) has a unique mild solution given by the solution of the integral equation

.



t

x(t) = S(t)x0 +

.



S(t − s)F (x(s))ds

0 t

+

S(t − s)σ (x(s))dW (s), t ∈ I = [0, T ], T < ∞.

(4.2)

0

Under the given assumptions, the solution .x = {x(t), t ∈ I } has a finite second moment in the sense that the expected value of the norm square is finite. Further, the solutions are sufficiently regular having continuous versions. In other words, the solutions are P-a.s continuous trajectories in H . The first integral is defined in the Lebesgue-Bochner sense and hence continuous. The second (stochastic) integral is defined in the sense of Itô. It is known that this stochastic integral has a continuous version. This is proved using the factorization technique developed in [64].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. U. Ahmed, S. Wang, Measure-Valued Solutions for Nonlinear Evolution Equations on Banach Spaces and Their Optimal Control, https://doi.org/10.1007/978-3-031-37260-5_4

75

76

4 Measure Solutions for Stochastic Systems

If the drift and the diffusion operators .{F, σ } are merely locally Lipschitz, by use of stopping time arguments one can prove the existence of mild solutions blowing up in finite time. Using the notion of measure solution these assumptions can be relaxed as we have seen in the case of deterministic systems. If the drift operator F is merely continuous and satisfies even polynomial growth, such as  p |F (x)|H ≤ K 1 + |x|H , p > 1,

.

standard results on stochastic differential equations in infinite dimensional Hilbert spaces do not hold [57, 64] unless some additional assumption such as dissipativity is imposed. The usual notions of mild and martingale solutions do not apply. However, the notion of measure or generalized solution does apply as demonstrated in this chapter. There is an interesting similarity between the notions of martingale [78, 79] and generalized solutions to be discussed later in this chapter. For simplicity of presentation, we have considered both F and .σ to be independent of time. However, the results given here can be easily extended to cover timedependent cases. Let Z denote any normal topological space and .BC(Z) the space of bounded continuous functions on Z with the topology of sup norm, and let .Mrba (Z) denote the space of regular bounded finitely additive set functions defined on the algebra generated by the closed subsets of Z. With respect to these topologies, these are Banach spaces and the dual of .BC(Z) is given by .Mrba (Z) as seen in Chap. 1, Theorem 1.1.8 (see [69, Theorem 2, p. 262]). Let .rba (Z) ⊂ Mrba (Z) denote the class of regular finitely additive probability measures furnished with the relative topology. It follows from the characterization results discussed in Chap. 1 that the dual of .L1 (I, BC(Z)) is given by .Lw ∞ (I, Mrba (Z)) which is furnished with the weak star topology. Let H , E be two separable Hilbert spaces and .(, F, Ft ↑, P ) a complete filtered probability space, .W (t), .t ∈ I , is an E valued .Ft -adapted cylindrical Wiener process and .σ : H → L(E, H ) the space of all bounded linear operators from E to H . Here W we consider .Ft ≡ FW t ∨ F0 (x0 ), where .Ft is the smallest sigma algebra generated by the Wiener Process W up to time .t, t ≥ 0, and .F0 (x0 ) is the smallest sigma algebra with respect to which .x0 is measurable. Let .I ×  be furnished with the predictable sigma field with reference to the filtration .Ft , .t ∈ I . Let .L1 (I × , BC(H )) denote the Banach space of integrable .BC(H ) valued random processes. By virtue of the theory of lifting we know that its topological dual is the space of .w ∗ measurable .Mrba (H ) valued .Ft -adapted random processes denoted by .Lw ∞ (I × , Mrba (H )).

4.2 Existence of Measure Solutions

77

4.2 Existence of Measure Solutions In the preceding chapters we have considered deterministic systems and proved existence of generalized or measure-valued solutions which consist of regular finitely additive measure-valued functions. Our objective here is to prove similar results for stochastic systems of the form (4.1). Since we do not use the standard assumptions such as Lipschitz and linear growth properties for the drift and the diffusion operators, we may expect the solutions to escape the original state space H at some point in time. Thus, as mentioned in the previous chapters, to capture the supports of our measure solutions, we may extend ˆ our state space through Stone-Cech compactification giving a compact Hausdorff space .H + = βH (see [16]). There are several techniques of compactification (e.g. Alexandrov one point compactification, Stone-Céch compactification, and Wallman compactification). For details see [123]. Unless the original space is a locally compact Hausdorff space, Alexandrov compactification does not produce a compact Hausdorff space. It is well-known that for any Tychonoff space .G, its Stone-Céch compactification denoted by .βG ≡ G+ is a compact Hausdorff space. In fact, for the purpose of this chapter, .G = H where H is a Hilbert space and hence a metric space with respect to its usual norm topology. Since every metric space is a Tychonoff space, H is a Tychonoff space. Hence, .H + is a compact Hausdorff space. Consequently, bounded continuous functions on H can be extended to continuous functions on .H + . So the appropriate choice of Banach spaces for the problems considered in + this section are .L1 (I × , BC(H + )) and .Lw ∞ (I × , Mrba (H )). Throughout this chapter, without any additional identifier, it is assumed that these are .Ft adapted processes. In particular, for measure-valued solutions, here we use the w (I × ,  + w + set .M∞ rba (H )) = L∞ (I × , rba (H )) consisting of probability + measure-valued processes, a subset of the vector space .Lw ∞ (I × , Mrba (H )). + + + Note that .Mrba (H ) = Mrca (H ) since .H is a compact Hausdorff space. In view of the fact that the measure solutions (of stochastic evolution equations) restricted to H are only finitely additive, we prefer to use the notation .Mrba (H + ) to emphasize this fact though they are countably additive on .H + . Throughout this chapter we use .Dφ and .D 2 φ to denote the first and second Fréchet derivatives of the function .φ. We denote by . the class of test functions as described below .

  ≡ φ ∈ BC(H ) : Dφ, D 2 φ exist and are continuous    having bounded supports in H and sup |D 2 φ(x)|L1 (H ) < ∞ , x∈H

where .L1 (H ) denotes the space of nuclear operators in H . Define the operators .A and .B with domain given by   D(A) ≡ φ ∈  : Aφ ∈ BC(H + ) ,

.

78

4 Measure Solutions for Stochastic Systems

where (Aφ)(ξ ) = (1/2) Tr(σ ∗ (D 2 φ)σ )(ξ ) + A∗ Dφ(ξ ), ξ  + F (ξ ), Dφ(ξ ),

.

(Bφ)(ξ ) ≡ (σ ∗ Dφ)(ξ ) ∈ E, for φ ∈ D(A). Note that .D(A) = ∅. This follows from the fact that for .ψ ∈ , the function .φ given by .φ(x) ≡ ψ(λR(λ, A)x) belongs to .D(A) for each .λ ∈ ρ(A), the resolvent set of A. We consider the stochastic system (4.1) and introduce the appropriate definition of measure-valued solutions for this system as follows. Definition 4.2.1 A measure-valued random process w + μ ∈ M∞ (I × , rba (H + )) ⊂ Lw ∞ (I × , Mrba (H ))

.

is said to be a generalized or measure solution of Eq. (4.1) if for every .φ ∈ D(A), the following equality holds  μt (φ) = φ(x0 ) +

.

t

 μs (Aφ)ds +

0

t

μs (Bφ), dW (s)E , P -a.s.,

(4.3)

0

for each .t ∈ I , where  μt (ψ) ≡

.

H+

ψ(ξ )μt (dξ ), t ∈ I.

In the following theorem we present a result on existence of measure-valued solutions and their regularity properties for the system (4.1). Theorem 4.2.2 Let A be the infinitesimal generator of a .C0 semigroup in H and F : H → H is continuous, and bounded in x on bounded subsets of H , and .σ : H → L(E, H ) is continuous and bounded on bounded subsets of H satisfying the following approximation properties: .

(ai): there exists a sequence .{Fn , σn } such that .Fn (x) ∈ D(A), .σn (x) ∈ L(E, D(A)) for each .x ∈ H , and Fn (x) −→ F (x) in H uniformly on compact subsets of H, .

σn∗ (x) −→ σ ∗ (x) strongly in L(H, E) uniformly on compact subsets of H.

(aii): there exists a pair of sequences .{αn , βn > 0} possibly .αn , .βn → ∞ as .n → ∞, such that |Fn (x) − Fn (y)|H ≤ αn |x − y|H ; |Fn (x)| ≤ αn (1 + |x|H ) .

|σn (x) − σn (y)|L2 (E,H ) ≤ βn |x − y|H ; |σn (x)|L2 (E,H ) ≤ βn (1 + |x|H )

4.2 Existence of Measure Solutions

79

for all x, .y ∈ H , where .L2 (E, H ) denotes the Hilbert space of Hilbert-Schmidt operators from E to H . Then, for every .x0 for which .P {ω ∈  : |x0 |H < ∞} = 1, the evolution + equation (4.1) has at least one generalized solution .λo ∈ Lw ∞ (I ×, Mrba (H )) in o w + the sense of Definition 4.3.1. Further, .λ ∈ M∞ (I × , rba (H )) and it is P -a.s ∗ .w continuous on I . Proof Since .D(A) is dense in H and .x0 ∈ H , a.s. (almost surely), there exists a s sequence .{x0,n } ∈ D(A) such that .x0,n −→ x0 a.s. We consider the sequence of Cauchy problems: dx = An xdt + Fn (x)dt + σn (x)dW (t), x(0) = x0,n ,

.

(4.4)

where .An = nAR(n, A), .n ∈ ρ(A), is the Yosida approximation of A. Since for each .n ∈ N and .x ∈ H , .Fn (x) ∈ D(A) and .σn (x) : E → D(A), it follows from the assumption (aii) that Eq. (4.4) has a unique .Ft − -adapted strong solution .xn = {xn (t), t ∈ I } which is P -a.s. continuous satisfying .

  sup E|xn (t)|2H , t ∈ I < ∞, for each n ∈ N,

and, for almost all .t ∈ I , .xn (t) ∈ D(A). Now let .φ ∈ D(A) with .Dφ and .D 2 φ having compact supports in H . Since .xn is a strong solution it follows from Itô’s formula that for each .t ∈ I ,  φ(xn (t)) = φ(x0,n ) +

t

(An φ)(xn (s))ds

.

0

 +

t

(Bn φ)(xn (s)), dW (s), t ∈ I,

(4.5)

0

where the angle brackets in the stochastic integral denote the scalar product in E (as indicated in the expression (4.3)) and .

(An φ)(ξ ) = (1/2) Tr((σn∗ (D 2 φ)σn )(ξ )) + A∗n Dφ(ξ ), ξ  + Fn (ξ ), Dφ(ξ ), (Bn φ)(ξ ) ≡ (σn∗ Dφ)(ξ ) ∈ E, for φ ∈ D(A).

Letting .δe (dξ ) denote the Dirac measure concentrated at the point .e ∈ H , and defining .λnt (dξ ) ≡ δxn (t) (dξ ), t ∈ I , with .λn0 (dξ ) ≡ δx0,n (dξ ), and using the notation of Definition 4.2.1, we can rewrite Eq. (4.5) as follows  λnt (φ) = λn0 (φ) +

.

0

t

 λns (An φ)ds +

t 0

λns (Bn φ), dW (s), t ∈ I.

(4.6)

80

4 Measure Solutions for Stochastic Systems

w (I × ,  n For each integer n, .λn ∈ M∞ rba (H )) and hence the set .{λ } is clearly contained in w + M∞ (I × , rba (H + )) ⊂ Lw ∞ (I × , Mrba (H )).

.

Consider the linear functional .n on .L1 (I × , BC(H + )) given by  n (ψ) ≡ E

.

 ψ(t, ξ )λnt (dξ )dt ≡

I ×H +

I ××H +

ψ(t, ω, ξ )λnt,ω (dξ )dP dt.

For any such .ψ ∈ L1 (I × , BC(H + )) one can readily verify that |n (ψ)| ≤ |ψ|L1 (I ×,BC(H + )) , for all n ∈ N.

.

Thus, the functional .n is well-defined on .L1 (I × , BC(H + )) and also it follows from the above inequality that the sequence .{n } is contained in a bounded subset of the dual .(L1 (I × , BC(H + )))∗ , the space of continuous linear functionals. Hence, it follows from the characterization of the dual space that the corresponding sequence of measures .{λn } is confined in a bounded subset of w + ∗ ∗ .L∞ (I ×, Mrba (H )). The .w closure of a bounded set is .w compact. Hence, by Alaoglu’s theorem, there exists a generalized subsequence (subnet) of the sequence + (net) .{λn }, relabeled as .{λn }, and a .λo ∈ Lw ∞ (I × , Mrba (H )), such that w∗

λn −→ λo . We show that .λo is a measure-valued solution of Eq. (4.1) in the sense of Definition 4.2.1. Define

.

ψ1,n (ξ ) ≡ (1/2) Tr(σn∗ (D 2 φ)σn )(ξ ), .

ψ1 (ξ ) ≡ (1/2) Tr(σ ∗ (D 2 φ)σ )(ξ ).

Since .σn∗ (x) −→ σ ∗ (x) strongly in .L(H, E) uniformly on compact subsets of H and .D 2 φ has compact support, and for each .φ ∈ D(A), .sup{|D 2 φ(ξ )|L1 (H ) , ξ ∈ H } < ∞, we have .ψ1,n , .ψ1 ∈ BC(H ) and .ψ1,n −→ ψ1 uniformly on H . Note that these functions are not stochastic processes, and hence independent of .(t, ω) ∈ I × . Since I is a finite interval and P is a probability measure, they can be treated as elements of .L1 (I × , BC(H + )). Hence, it follows from .w ∗ convergence of .λn to .λo that, for any .z ∈ L2 (, F, P ) = L2 () and .t ∈ I , we have 

 .

×[0,t]

z λns (ψ1,n )dsdP −→

×[0,t]

z λos (ψ1 )dsdP .

Defining ψ2,n (ξ ) ≡ A∗n (Dφ)(ξ ), ξ  and ψ2 (ξ ) ≡ A∗ (Dφ)(ξ ), ξ ,

.

(4.7)

4.2 Existence of Measure Solutions

81

and recalling that .An −→ A on .D(A) in the strong operator topology and, for φ ∈ D(A), .Dφ ∈ D(A∗ ), and both .φ and .Dφ are continuous having compact supports, we can deduce that .ψ2,n −→ ψ2 uniformly on H . Hence, again we have

.



 .

×[0,t]

z λns (ψ2,n )dsdP −→

z λo (ψ2 )dsdP .

(4.8)

×[0,t]

Similarly, defining ψ3,n (ξ ) ≡ Fn (ξ ), Dφ(ξ ) and ψ3 (ξ ) ≡ F (ξ ), Dφ(ξ ),

.

and recalling that both .φ(∈ D(A)) and .Dφ have compact supports and, by our assumption .Fn −→ F uniformly on compact subsets of H , it is clear that .ψ3,n −→ ψ3 in .BC(H ). Thus, for .z ∈ L2 (), we have   n . z λs (ψ3,n )dsdP −→ z λos (ψ3 )dsdP , t ∈ I. (4.9) ×[0,t]

×[0,t]

Using the expressions (4.7), (4.8) and (4.9) we conclude that for every .z ∈ L2 () and .φ ∈ D(A) with .Dφ, D 2 φ having compact supports, 

 .

×[0,t]

z λns (An φ)dsdP −→

×[0,t]

z λos (Aφ)dsdP , t ∈ I.

(4.10)

s

Since .x0,n −→ x0 a.s. in H and .φ ∈ BC(H + ), we have .φ(x0,n ) −→ φ(x0 ) a.s. Then, by Lebesgue dominated convergence theorem, for every .z ∈ L2 (), we have 

 z φ(x . 0,n )dP = 



z λn0 (φ)dP 

 z λ0 (φ)dP =

−→ 

z φ(x0 )dP ,

(4.11)



where .λ0 (φ) ≡ H φ(ξ )δx0 (dξ ). Considering the stochastic integral in the Eq. (4.6), we note that, since .Dφ is continuous having compact support, .Bn φ ∈ BC(H, E) and hence  |(Bn φ)(xn (s))|2E ds < ∞ .E I

uniformly with respect to .n ∈ N. Thus, the stochastic integral in the expression (4.6) is well-defined, and for any .z ∈ L2 () it follows from the properties of conditional expectation and the martingale theory that

 t 

 t  E z λns (Bn φ), dW (s) = E zt

λns (Bn φ), dW (s) ,

.

0

0

(4.12)

82

4 Measure Solutions for Stochastic Systems

where .zt ≡ E{z | Ft } is a square integrable .Ft martingale. Hence, there exists an Ft adapted E valued random process .η(t), .t ∈ I , belonging to .L2 (I × , E) and a square integrable .F0 measurable random variable .z0 independent of the Brownian increments such that  t .zt = z0 +

η(s), dW (s), t ∈ I,

.

0

and  E

.

I

|η(t)|2E dt < ∞.

Hence,

 t   t E z λns (Bn φ), dW (s) = E η(s), λns (Bn φ)ds.

.

0

0

Since .Dφ has compact support, it follows from the assumption (ai) that .Bn φ −→ Bφ in the topology of .BC(H, E) and hence s

η, Bn φ −→ η, Bφ

.

w∗

in .L1 (I × , BC(H )). It follows from this and the fact that .λn −→ λo in .Lw ∞ (I × , Mrba (H + )) which is the dual of .L1 (I × , BC(H + )), that, for each .t ∈ I ,

 E 0

t



η(s), . λns (Bn φ)ds



t

−→ E 0



η(s), λos (Bφ)ds



 t = E z λos (Bφ), dW (s) , t ∈ I.

(4.13)

0

Thus, multiplying both sides of Eq. (4.6) by an arbitrary .z ∈ L2 () and taking the limit of the expected values, it follows from (4.10), (4.11) and (4.13) that o .E(zλt (φ))

 t  o = E(zλ0 (φ)) + E z λs (Aφ)ds 0



 t o + E z λs (Bφ), dW (s) , t ∈ I.

(4.14)

0

Since this identity holds for arbitrary .z ∈ L2 () and .t ∈ I , we conclude that  λot (φ) = λ0 (φ) +

.

0

t

 λos (Aφ)ds +

t 0

λos (Bφ), dW (s), t ∈ I, P -a.s.

(4.15)

4.2 Existence of Measure Solutions

83

+ Due to the fact that .λo ∈ Lw ∞ (I × , Mrba (H )), it is evident that for each o o .φ ∈ D(A), .λt (Aφ) and .λt (Bφ) are well-defined .Ft -adapted square integrable random processes. Hence, Eq. (4.15) holds for all .φ ∈ D(A) and not just those having compact supports. Thus, .λo is a measure (generalized) solution of the stochastic evolution equation (4.1) in the sense of Definition 4.2.1. The proof of the last statement of the theorem follows from the fact that the approximating sequence is a sequence of Dirac measures and clearly are positive and that positivity w (I × ,  + is preserved under weak star limit. Thus, .λo ∈ M∞ rba (H )). The almost ∗ o sure .w continuity, .t −→ λt , follows trivially from the expression (4.15). This completes the proof.  

Remark 4.2.3 (1): In view of the above results, we observe that for existence of measure-valued solutions of the stochastic evolution equation (4.1), it suffices if the drift and diffusion operators .{F, σ } are merely continuous and bounded on bounded sets. This admits also .{F, σ } having polynomial growth. In contrast, for standard mild solutions, the usual assumptions are Lipschitz property and linear growth along with .σ being Hilbert-Schmidt in case W is a cylindrical Wiener process (see [64, Chapter 7]). This of course guarantees uniqueness and continuity of the path process. Our results sacrifice the path process and provide a stochastic finitely additive regular measure-valued process as the solution. (2): In the course of the proof, we have also observed that one can readily assume that the drift and diffusion operators are also functions of time and, under some additional conditions, stochastic. From the above theorem we have the following corollary. Corollary 4.2.4 Consider the Kolmogorov equation (d/dt)μt = A∗ μt , t ∈ I, μ0 = ν ∈ rba (H ),

.

(4.16)

where .A∗ is the formal dual of the operator .A with F and .σ satisfying the assumptions of Theorem 4.2.2. Then, for every .ν ∈ rba (H ) Eq. (4.16) has at least w (I,  + w + one weak solution .μ ∈ M∞ rba (H )) ⊂ L∞ (I, Mrba (H )) in the sense that for each .φ ∈ D(A) the following identity holds 

t

μt (φ) = ν(φ) +

.

μs (Aφ)ds, t ∈ I.

0

Proof For detailed proof see [12]. In fact a heuristic proof is straightforward. By applying the expectation operation on both sides of the expression (4.15) one finds that  t o .Eλt (φ) = Eλ0 (φ) + E λos (Aφ)ds t ∈ I, (4.17) 0

84

4 Measure Solutions for Stochastic Systems

for every .φ ∈ D(𝒜). Note that for any .ψ ∈ L1 (I, BC(H + )) the functional  ψ −→ E

λos (ψ)ds

.

I

is a well-defined continuous linear functional on .L1 (I, BC(H + )). Hence, by duality + there exists a .μo ∈ Lw ∞ (I, Mrba (H )) such that 



E

.

I

λos (ψ)ds =

I

μot (ψ)ds.

w (I × ,  + Since by virtue of Theorem 4.2.2 we have .λo ∈ M∞ rba (H )), it o w + is clear that .μ ∈ M∞ (I, rba (H )). Note that for any .φ ∈ D(𝒜), .𝒜φ ∈ L1 (I, BC(H + )). Thus,



t

E

.

0

 λos (𝒜φ)ds

t

=

μos (𝒜φ)ds, t ∈ I.

0

It follows from this argument that Eq. (4.17) is equivalent to the following equation,  o .μt (φ)

t

= μ0 (φ) + 0

μos (Aφ)ds, t ∈ I,

(4.18)

for every .φ ∈ D(𝒜) having compact support. Clearly, this is the weak form of the differential equation (4.16). This completes the proof.  

4.2.1 Martingale vs. Generalized Solutions As mentioned earlier, there is some similarity between the notion of martingale solution and the concept of measure-valued solution. In the formulation of martingale solution, a sample space (or a measurable space) is constructed using the standard canonical sample space, for example, . ≡ C(I, H ) equipped with the filtration .Ft ≡ σ {x(s), s ≤ t, x ∈ C(I, H )} and then looking for the existence of a probability measure P on the path space such that .(, Ft ↑, P ) is a filtered probability space and the functional .Ct (φ) given by  Ct (φ)(ω) ≡ φ(ω(t)) − φ(ω(0)) −

.

t

Aφ(ω(s))ds, t ∈ I, ω ∈ , (4.19)

0

is a P -.Ft martingale for each .φ in a suitable class of test functions ., such as, cylinder functions on H . For more details including interesting application to control theory see [78, 79].

4.2 Existence of Measure Solutions

85

On the other hand, according to the notion of measure-valued solutions (or generalized solution), we are looking for solutions .{μt , t ≥ 0}, which are .Ft − adapted .w ∗ continuous random processes with values in the space of regular bounded finitely additive probability measures .rba (H ) ⊂ rba (H + ) = rca (H + ) satisfying the identity in Definition 4.2.1. In the martingale formulation represented by Eq. (4.19), the structure and regularity of the path space is imposed on the problem requiring that the process evolves continuously in H . On the other hand, in the formulation of measure solution, the process evolves in the space .Mrba (H ) in place of H and the temporal regularity, such as .w ∗ continuity, is consequential not an imposition. This admits substantial relaxation on the drift and diffusion operators. For example, in the martingale formulation the operators F and .σ are assumed to have at most linear growth whereas in the formulation of measure-valued (or generalized) solutions polynomial growth (or even better) is allowed and further, the diffusion operator .σ is not required to be Hilbert-Schmidt valued.

4.2.2 Some Illustrative Examples For illustration of Theorem 4.2.2, we present here the following three examples. Example 4.2.5 First we provide a general characterization of the drift and dispersion operators which satisfy our basic assumptions and for which our results obviously hold. Let .H k ≡ H × H × H × · · · × H denote the k-fold Cartesian product of H and let .L(H k , H ) denote the class of bounded (multi) linear operators from .H k to H completed with respect to the norm topology induced by |Lk |L(H k ,H ) ≡ sup {|Lk (h1 , h2 , · · · , hk )|H , |hi |H = 1, i = 1, 2, · · · , k} ,

.

where .Lk ∈ L(H k , H ). For .k = 1, set .L(H 1 , H ) = L(H, H ) ≡ L(H ). Define Pk (x) ≡ Lk (x, x, · · · , x) and Pm (x) ≡



.

Pk (x), x ∈ H.

1≤k≤m

Then, we introduce the class   F ≡ Pm , m ∈ N, m < ∞

.

as the class of admissible drifts. We show that this class satisfies our basic assumptions. Let .Qr denote the retraction of the ball .Br ⊂ H of radius r centered at the origin. That is Qr (x) ≡

x,

.

(r/|x|)x,

for x ∈ Br , otherwise.

86

4 Measure Solutions for Stochastic Systems

For .F ∈ F, by definition there exists an integer .m ∈ N such that .F = Pm . Let .ρ(A) denote the resolvent set of A and .R(λ, A) the resolvent corresponding to .λ ∈ ρ(A). For any fixed .m ∈ N, define Fn (x) ≡ nR(n, A)F (Qn (x)) = nR(n, A)Pm (Qn (x)), n ∈ ρ(A).

.

Clearly, .{Fn } is a sequence of continuous and bounded maps in H and for each x ∈ H , .Fn (x) ∈ D(A) and .Fn (x) −→ F (x) point-wise in H and hence uniformly on compact subsets of H . It is straightforward to verify that for any fixed .m ∈ N, there exist constants .{αn = αn (m) > 0}, dependent on m, such that .limn αn = ∞ and for each n,

.

.

|Fn (x)| ≤ αn (1 + |x|H ), for all x ∈ H, |Fn (x) − Fn (y)| ≤ αn (|x − y|H ), for all x, y ∈ H.

For the diffusion operator we introduce the set .K, a subset of .C(H, L(E, H )), to denote the class of locally Lipschitz maps. Let .Pn be any increasing sequence of finite dimensional (possibly orthogonal) projections in the Hilbert space E converging strongly to the identity. For each .σ ∈ K we define σn (x) ≡ nR(n, A)σ (Qn x)Pn .

.

It is easy to verify that the sequence .{σn } satisfies our basic hypotheses and hence the class .K is covered by our result. In view of this characterization, for each .F ∈ F and .σ ∈ K, the system (4.1) reproduced below, dx = Axdt + F (x)dt + σ (x)dW, x(0) = x0 ,

.

has generalized (measure) solutions but not classical, weak, mild or even martingale solutions. Note that .σ is not required to be Hilbert-Schmidt. Example 4.2.6 For a more specific example, consider the system (4.1) with F p−1 given by .F (x) ≡ |x|H x, for any .p > 1. Clearly, F is locally Lipschitz but not dissipative. For .γ > 0, we define the family of operators on H as follows   p−1 p−1 Gγ (x) = |x|H /(1 + γ |x|H ) x, γ > 0.

.

It is easy to verify that .

|Gγ (x)|H ≤ (1/γ )|x|H |Gγ (x) − Gγ (y)|H ≤ (p/γ )|x − y|H , γ > 0.

4.2 Existence of Measure Solutions

87

Then, define Fn (x) = nR(n, A)G1/n (x), n ∈ N ∩ ρ(A).

.

One can easily check that, for .αn ≡ 2np|nR(n, A)|L(H ) , .

|Fn (x)|H ≤ αn (1 + |x|H ), for all x ∈ H, |Fn (x) − Fn (y)|H ≤ αn |x − y|H , for all x, y ∈ H.

Clearly, .αn −→ ∞ as .n −→ ∞. Note that .Fn (x) ∈ D(A) for each .n ∈ N and Fn −→ F uniformly on compact sets of H . Let .σ , given by

.

σ (x) = β(x),

.

denote the diffusion operator, where .β ∈ BC(H ) is Lipschitz, and . ∈ L(E, H ) (not necessarily Hilbert-Schmidt). Then, the sequence of operators .{σn }, given by σn (x) ≡ β(x)nR(n, A)Pn ,

.

satisfies all the hypothesis of Theorem 4.2.2. Indeed, one can easily verify the existence of a constant K such that √ |σn (x)|L2 (E,H ) ≤ K n|nR(n, A)|||L(E,H ) (1 + |x|H ) ≤ βn (1 + |x|H )

.

and |σn (x) − σn (y)|L2 (E,H ) ≤ βn (|x − y|H )

.

and that .βn −→ ∞. Further, .σn : H → L(E, D(A)) and .σn (x) −→ σ (x) strongly in .L(E, H ) uniformly on compact subsets of H . In other words, the sequence .{σn } is a Hilbert-Schmidt approximation of .σ in .L(E, H ). Thus, both F and .σ satisfy all the hypothesis of Theorem 4.2.2, and hence Eq. (4.1), with these F and .σ , has a generalized solution. Example 4.2.7 (Nonlinear Beams) Here we present a practical example arising from random vibration of mechanical structures. For moderately large vibrations, the following model has been considered to be appropriate for beam dynamics [38]:

.

ρ





2  ∂ y ∂2 ∂ 2y ∂ 2y + EI −N(y) + Kyt = q(t, x), t > 0, x ∈ (0, ), ∂t 2 ∂x 2 ∂x 2 ∂x 2

y(t, 0) = 0, Dy(t, 0) ≡ yx (t, 0) = 0, EI D 2 y(t, ) = u1 , D(EI (D 2 y(t, ))) − Nyx (t, ) = u2 ,

(4.20)

88

4 Measure Solutions for Stochastic Systems

where, in general, the nonlinear operator N is given by  N(y) ≡ a + b

 ∂y 2

.

∂x

0

dx.

Note that here D denotes the spatial derivative of .y(t, x), x ∈ (0, ). The nonlinear term represents membrane force. Here a and b are constants. If .a > 0 and .b = 0, it represents a linear extensible beam. The parameters .ρ, EI , and K denote the mass density (per unit length), flexural rigidity, and aerodynamic damping coefficient, respectively. Normally, feedback controls of the form .

u1 ≡ −δDyt (t, ), u2 ≡ γ yt (t, ) − N(y)Dy(t, )

(4.21)

are used to stabilize the system [11, 38]. The term q represents a stochastic load. We write the system of Eqs. (4.20) and (4.21) as an abstract (ordinary) differential equation on an appropriate Hilbert space. The most suitable space is the energy space given by H ≡ H02 × L2 (0, ),

.

where   ∂ϕ ≡ Dϕ ∈ L2 , D 2 ϕ ∈ L2 and ϕ(0) = 0, Dϕ(0) = 0 . H02 ≡ ϕ ∈ L2 : ∂x

.

Let B denote the formal beam operator given by Bψ ≡ (1/ρ)

.

∂2 ∂x 2

 ∂2 EI 2 ψ ≡ (1/ρ)D 2 (EI D 2 ψ). ∂x

Define the state as

z≡

.

z1 z2



≡

y yt

 .

Then, the system (4.20) and (4.21) can be written as (d/dt)z = Az + G(z) + q, ˜

.

(4.22)

where the operator A is given by the restriction of the formal differential operator

L≡

.

0 1 −B 0



4.2 Existence of Measure Solutions

89

to the domain .D(A) given by .

 D(A) ≡ z ∈ H : Lz ∈ H and EI D 2 z1 () + δDz2 () = 0,

 D(EI D 2 z1 ()) − γ z2 () = 0

and

 0 .q ˜≡ . q The operator A as defined above is dissipative and generates a contraction semigroup in H . The nonlinear operator .G is given by

G(z) ≡ −(1/ρ)

.

0 N(z1 )D 2 z1 + Kz2

 .

Since the modulus of rigidity EI and the mass density .ρ are strictly positive, in view of the given boundary conditions, the space H is a Hilbert space with respect to the scalar product

ϕ, ψ ≡ (EI D 2 ϕ1 , D 2 ψ1 ) + (ρϕ2 , ψ2 ),

.

where the first term corresponds to elastic potential energy and the second term corresponds to kinetic energy. For the stochastic load, we assume that .{ei , i = 1, 2, 3, · · · } is a complete orthonormal basis of .E ≡ L2 (0, ) and .{βi (t), t ≥ 0, i = 1, 2, 3, · · · } is a sequence of one dimensional standard independent Brownian motions. Define √ .W (t) ≡ λi βi (t)ei , t ≥ 0, i≥1

where .{λi } is a sequence of nonnegative numbers. This is an E valued Wiener process. It is easy to verify that E{(W (t), f )(W (s), g)} = (t ∧ s)



.

λi (f, ei )(g, ei ).

i≥1

For the random load q we choose the model q(t, ξ ) ≡

.

√ λi β˙i (t)ei (ξ ), t ≥ 0, ξ ∈ (0, ), i≥1

where .β˙i is the distributional derivative of the scalar Brownian motion .βi called white noise. If .{λi = 1, i = 1, 2, 3, · · · } then W is called a cylindrical Brownian

90

4 Measure Solutions for Stochastic Systems

motion in E. Using this model for the random load, Eq. (4.22) can be rigorously interpreted as the stochastic differential equation given by dz = (Az + G(z))dt + σ dW,

.

(4.23)

 0 ∈ L(E, H ) with I denoting the identity operator in E. It is easy I to verify that .G is locally Lipschitz and maps bounded sets of H into bounded sets of H . For the approximating sequence, we choose .σn ≡ nR(n, A)σ Pn and .Gn ≡ nR(n, A)G(Qn (·)) which satisfy the basic hypotheses of Theorem 4.2.2. Hence, for any given initial measure .μ0 ∈ rba (H ), the system (4.23) has generalized w (I × ,  + w + solutions in .M∞ rba (H )) ⊂ L∞ (I, Mrba (H )). where .σ ≡

4.3 Stochastic Systems Driven by Martingale Measures In the preceding section we have considered systems driven by Wiener processes. Here we consider more general martingales replacing Wiener process by stochastic vector measures which may contain both continuous and jump processes. This generalizes the results of the preceding section to systems driven by general martingales described as follows, dx(t) = Ax(t)dt + F (x(t))dt + G(x(t−))M(dt), t ≥ 0, x(0) = x0 ,

.

(4.24)

where A and F are as described in the previous section, and .G : H −→ L(E, H ) is a continuous map, and M is an E valued stochastic vector measure defined on the sigma algebra .B0 of Borel subsets of .R0 ≡ [0, ∞). For simplicity of presentation we have considered both F and G independent of time. However, the results presented here can be easily extended to the timedependent case without any difficulty.

4.3.1 Special Vector Spaces We introduce some special vector spaces used in this section. Let H , E be a pair of separable Hilbert spaces and .(, F, Ft ↑, t ≥ 0, P ) a complete filtered probability space, .M(J ), .J ∈ B0 , an E valued .Ft - adapted vector measure in the sense that for any .J ∈ B0 with .J ⊂ [0, t], .M(J ) or more precisely .e∗ (M(J )) is .Ft adapted for every .e∗ ∈ E ∗ = E. For the purpose of this section, we consider M M .Ft ≡ Ft ∨ F0 , where .Ft , t ≥ 0, and .F0 are the smallest sigma algebras with respect to which the measure M and the initial state .x0 , respectively, are measurable. The filtration denoted by .FM t , .t ≥ 0, is the smallest sigma algebra with respect to

4.3 Stochastic Systems Driven by Martingale Measures

91

s which the process . 0 M(ds), .s ≤ t, is progressively measurable. Let .I ×  be furnished with the predictable .σ -field with reference to the filtration .Ft , .t ∈ I . Let .Lw ∞ (I × , Mrba (H )) denote the vector space of .Mrba (H ) valued random processes .{λt , t ∈ I }, which are .Ft -adapted and .w ∗ measurable in the sense that, for each .φ ∈ BC(H ), .t −→ λt (φ) is an .Ft measurable random process possessing finite second moments. We furnish this space with the .w ∗ topology as before. We have seen before that .Lw ∞ (I × , Mrba (H )) is the dual of the Banach space L1 (I × , BC(H )),

.

where the latter space is furnished with the natural topology induced by the norm given by 

 (E {sup{|ϕ(t, ω, ξ )|, ξ ∈ H }}) dt =

|ϕ| ≡

.

I

E|ϕ|BC(H ) dt. I

Here we have chosen .X = BC(H ) and .X∗ = Mrba (H ). For .X = B(H ) (the space of Bounded Borel measurable functions on H ) and .X∗ = Mba (H ), it follows from the theory of lifting (see Chap. 1) that .Lw ∞ (I × , Mba (H )) is the dual of the Banach space .L1 (I × , B(H )). We will need both these spaces.

4.3.2 Some Basic Properties of the Martingale Measure M (M1): {M(J ), M(K), J ∩K = ∅, J, K ∈ B0 } are pairwise independent E valued random vectors satisfying E{(M(J ), ξ )} = 0, J ∈ B0 , ξ ∈ E, where E{z} ≡  zP (dω). (M2): There exists a countably additive positive measure π on B0 (π ∈ Mc (R0 )) having bounded variation on bounded sets, such that for every ξ , ζ ∈ E, E{ M(J ), ξ E M(K), ζ E } = (ξ, ζ )E π(J ∩ K).

.

Clearly, it follows from this last property that for any ξ ∈ E, E{ M(J ), ξ 2 } = |ξ |2E π(J ),

.

and that the process N, defined by  N(t) ≡

.

0

t

M(ds), t ≥ 0,

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4 Measure Solutions for Stochastic Systems

is a norm square integrable E valued Ft -martingale. A simple example is given by the stochastic Wiener integral  M(J ) ≡

f (t)dW (t), J ∈ B0 ,

.

J

where W is the cylindrical Brownian motion on R0 with values in the Hilbert space E and f is any locally square integrable scalar valued function. In this case π(J ) = J |f (t)|2 dt. If f is an Ft -adapted square integrable random process, the measure π is given by π(J ) = E J |f (t)|2 dt. If f ≡ 1, π is the Lebesgue measure. In the latter case, the system given by Eq. (4.24) reduces to the one driven by cylindrical Brownian motion [12].

4.3.3 Basic Formulation of the System Our objective here is to prove existence of measure-valued solutions for the stochastic system (4.24). This generalizes a similar result (Theorem 4.2.2) presented in the preceding section (see also [12]). Since the measure solutions may not be fully supported on the original state space H , again it is useful to extend the state space to a compact Hausdorff space containing H as a dense subspace. As ˆ seen before, this is done by using the Stone-Cech compactification of H denoted by .H + = βH . In view of this we shall often use .H + in place of H and + the spaces .L1 (I × , BC(H + )) with its dual .Lw ∞ (I × , Mrba (H )). We are particularly interested in the class of all finitely additive regular probability measurew (I × ,  + valued processes, .M∞ rba (H )), a subset of the linear vector space w + + is a compact Hausdorff space, .L∞ (I × , Mrba (H )). Recall that, since .H + + .Mrba (H ) = Mrca (H ). Again, in view of the fact that the measure solutions of stochastic evolution equations restricted to H are only finitely additive, we prefer to use the notation .Mrba (H + ) to emphasize this fact though they are countably additive on .H + . Without further notice, we use .Dφ and .D 2 φ to denote the first and second Fréchet derivatives of the function .φ whenever they exist. We denote by . the class of test functions as defined below,   2 . ≡ φ ∈ BC(H ) : Dφ, D φ exist, continuous and bounded on H . With a slight abuse of notations, we introduce the operators .A, .B, and .C with domains given by D(A) ≡ {φ ∈  : Aφ ∈ BC(H + )}, .

D(B) ≡ {φ ∈  : Dφ ∈ D(A∗ ) & Bφ ∈ BC(H + )},

4.3 Stochastic Systems Driven by Martingale Measures

93

where .

(Aφ)(ξ ) = (1/2) Tr(G∗ (D 2 φ)G)(ξ ) ≡ (1/2) Tr((D 2 φ)GG∗ )(ξ ), φ ∈ D(A), Bφ = (A∗ Dφ(ξ ), ξ ) + (F (ξ ), Dφ(ξ )) for φ ∈ D(B), Cφ(ξ ) ≡ G∗ (ξ )Dφ(ξ ).

(4.25)

Note that .D(A) ∩ D(B) ⊂ D(C). We consider the system given by the stochastic evolution equation (4.24) and introduce the following definition of measure-valued solutions. Definition 4.3.1 A measure-valued random process w + μ ∈ M∞ (I × , rba (H + )) ⊂ Lw ∞ (I × , Mrba (H ))

.

is said to be a measure (or generalized) solution of Eq. (4.24) if for every .φ ∈ D(A) ∩ D(B) and .t ∈ I , the following equality holds: 

t

μt (φ) = φ(x0 ) +

.





0

μs (Bφ)ds 0

t

+

t

μs (Aφ)π(ds) +

μs− (Cφ), M(ds)E P -a.s., t ∈ I,

(4.26)

0

where  μt (ψ) ≡

.

H+

ψ(ξ )μt (dξ ), t ∈ I.

Remark 4.3.2 Equation (4.26) can be written compactly in a differential form as follows dμt (φ) = μt (Aφ)π(dt) + μt (Bφ)dt + μt− (Cφ), M(dt), t ∈ I, μ0 (φ) = φ(x0 ),

.

for every .φ ∈ D(𝒜) ∩ D(ℬ). As seen in the previous chapters, this is the weak form of the stochastic evolution equation on the state space of measures .Mrba (H ) given by dμt = A∗ μt π(dt) + B∗ μt dt + C∗ μt− , M(dt)E , t ∈ I, μ0 = δx0 ,

.

where .{A∗ , B∗ , C∗ } are the duals of the operators .{A, B, C}, respectively.

94

4 Measure Solutions for Stochastic Systems

4.3.4 Existence of Measure Solutions To consider the question of existence of solutions we use the following assumptions: (A1): there exists a sequence .{Fn , Gn } with .Fn (x) ∈ D(A), .Gn (x) ∈ L(E, D(A)), for each .x ∈ H , and .Fn (x) −→ F (x) in H uniformly on compact subsets of H , .Gn (x) −→ G(x), in the strong operator topology in .L(E, H ), uniformly on compact subsets of H . (A2): there exists a pair of sequences of real numbers .{αn , βn > 0}, possibly .αn , βn → ∞ as .n → ∞, such that |Fn (x) − Fn (y)|H ≤ αn |x − y|H ; |Fn (x)|H ≤ αn (1 + |x|H ), .

|Gn (x) − Gn (y)|L2 (E,H ) ≤ βn |x − y|H ; |Gn (x)|L2 (E,H ) ≤ βn (1 + |x|H )

for all x, .y ∈ H , where .L2 (E, H ) denotes the Hilbert space of Hilbert-Schmidt operators from E to H . Now we are prepared to consider the question of existence of measure solutions for the stochastic evolution equation (4.24). This is presented as follows. Theorem 4.3.3 Suppose A is the infinitesimal generator of a .C0 semigroup in H and the maps .F : H −→ H and .G : H −→ L(E, H ) are continuous, and bounded on bounded subsets of H , satisfying the approximation properties (A1) and (A2); and M is a non-atomic stochastic vector measure satisfying the properties (M1) and (M2). Then, for every H valued random variable .x0 , for which .P {ω ∈  : |x0 |H < ∞} = 1, the evolution equation (4.24) has a measure-valued solution o w + .λ in the Banach space .L∞ (I × , Mrba (H )) in the sense of Definition 4.3.1. o w + w Further, .λ ∈ M∞ (I × , rba (H )) ⊂ L∞ (I × , rba (H + )). Proof Since .D(A) is dense in H and .x0 ∈ H , a.s., there exists a sequence of .F0 s measurable random vectors .{x0,n } ∈ D(A) such that .x0,n −→ x0 a.s. Consider the sequence of Cauchy problems: dx(t) = An x(t)dt + Fn (x(t))dt + Gn (x(t−))M(dt), x(0) = x0,n , (4.27)

.

where .An = nAR(n, A), .n ∈ ρ(A), is the Yosida approximation of A. Since for each .n ∈ N and .x ∈ H , .Fn (x) ∈ D(A), .Gn (x) : E → D(A), it follows from the assumption (A2) that Eq. (4.27) has a unique strong solution .xn = {xn (t), t ∈ I } which is .Ft -adapted, and for each .n ∈ N, .t ∈ I , .xn (t) ∈ D(A) and .

  sup E|xn (t)|2H , t ∈ I < ∞.

Since standard assumptions hold, this follows from classical results on existence of solutions and regularity properties thereof [64]. Now let .φ ∈ D(A) ∩ D(B) with 2 .Dφ and .D φ having compact supports in H . Since .xn is a strong solution, it follows

4.3 Stochastic Systems Driven by Martingale Measures

95

from Itô’s formula that, for each .t ∈ I ,  .

t

φ(xn (t)) = φ(x0,n ) +



t

(An φ)(xn (s))π(ds) +

0

(Bn φ)(xn (s))ds 0



t

+

(Cn φ)(xn (s−)), M(ds)E ,

(4.28)

0

where the operators .{An , Bn , Cn } are as those given by the expressions (4.25) with {An , Fn , Gn } replacing the operators .{A, F, G}. Letting .δζ (dξ ) denote the Dirac measure concentrated at the point .ζ ∈ H , and defining .λnt (dξ ) ≡ δxn (t) (dξ ), t ∈ I , .λn0 (dξ ) ≡ δx0,n (dξ ), and using the notation of Definition 4.3.1, we can rewrite Eq. (4.28) as

.

 λnt (φ)

.

=

λn0 (φ) +

t 0

 λns (An φ)π(ds) + 

t

+ 0

0

t

λns (Bn φ)ds

λns− (Cn φ), M(ds)E P -a.s., t ∈ I.

(4.29)

It is clear that, for each integer n, the functional .n given by  n (ψ) ≡ E

.

 I ×H +

ψ(t, ξ )λnt (dξ )dt

≡

I ××H +

ψ(t, ω, ξ )λnt,ω (dξ )dtdP

is a well-defined bounded linear functional on the Banach space .L1 (I × , BC(H + )) and that |n (ψ)| ≤ |ψ|L1 (I ×,BC(H + )) , for all n ∈ N.

.

Thus, the family of linear functionals .{n } is contained in a bounded subset of the dual of the space .L1 (I × , BC(H + )). In fact, the family .{λn } is contained in w (I × ,  + w + the set .M∞ rba (H )), a bounded subset of .L∞ (I × , Mrba (H )). ∗ ∗ The .w closure of a bounded set in any dual space is .w compact. Hence, by Alaoglu’s theorem, there exists a generalized subsequence (subnet) of the sequence + (net) .{λn }, relabeled as .{λn }, and a .λo ∈ Lw ∞ (I × , Mrba (H )), such that w∗

w (I × ,  + ∗ λn −→ λo . Since the set .M∞ rba (H )) is bounded and .w closed ∗ ∗ (so .w compact), and positivity is preserved under .w convergence, the limit o w + o .λ ∈ M∞ (I × , rba (H )). We show that .λ is a measure-valued solution of Eq. (4.24) in the sense of Definition 4.3.1. Define .

ψ1,n (ξ ) ≡ (1/2) Tr((G∗n (D 2 φ)Gn )(ξ )) = (1/2) Tr(((D 2 φ)Gn G∗n )(ξ )), .

ψ1 (ξ ) ≡ (1/2) Tr((G∗ (D 2 φ)G)(ξ )) = (1/2) Tr(((D 2 φ)GG∗ )(ξ )).

96

4 Measure Solutions for Stochastic Systems

It is clear that, if .Gn (x) −→ G(x) in the strong operator topology of .L(E, H ) uniformly on compact subsets of H , so does .G∗n (x) −→ G∗ (x) in .L(H, E) uniformly on compact subsets of H . Since .D 2 φ has compact support, and for each .φ ∈ D(A), we have .ψ1,n , .ψ1 ∈ BC(H ), it follows from the assumption (A1) that s + .ψ1,n −→ ψ1 uniformly on H , that is, .ψ1,n −→ ψ1 in .BC(H ). These functions are deterministic as they are independent of .(t, ω) ∈ I × . Since I is a finite time interval and P is a probability measure on ., this is a finite measure space. Thus, s it is evident that .ψ1,n −→ ψ1 in .L1 (I × , BC(H )). Combining this with the fact that the measure .π has bounded variation on bounded sets, it follows from the .w ∗ convergence of .λn to .λo that 

t

E

.

0

 λns (ψ1,n )π(ds) −→ E

t 0

λos (ψ1 )π(ds), t ∈ I.

(4.30)

Define ψ2,n (ξ ) ≡ (A∗n (Dφ)(ξ ), ξ )H and ψ2 (ξ ) ≡ (A∗ (Dφ)(ξ ), ξ )H .

.

Since .An −→ A on .D(A) in the strong operator topology and, for .φ ∈ D(A) ∩ D(B), .Dφ(ξ ) ∈ D(A∗ ), and further, by our choice, both .φ and .Dφ are continuous having compact support, we can deduce that .ψ2,n −→ ψ2 uniformly on H . These functions are deterministic and also time invariant. Thus, again by similar argument, s we conclude that .ψ2,n −→ ψ2 in .L1 (I × , BC(H + )). Hence, again by virtue of ∗ n o .w convergence of .λ to .λ , we have 

t

E

.

0

 λns (ψ2,n )ds −→ E

t 0

λos (ψ2 )ds, t ∈ I.

(4.31)

Similarly, define ψ3,n (ξ ) ≡ (Fn (ξ ), Dφ(ξ ))H and ψ3 (ξ ) ≡ (F (ξ ), Dφ(ξ ))H , .

ψ4,n (ξ ) ≡ G∗n (ξ )Dφ(ξ ) and ψ4 (ξ ) ≡ G∗ (ξ )Dφ(ξ ).

Since .φ ∈ D(A) ∩ D(B) and .Dφ has a compact support, and by our assumption s Fn −→ F uniformly on compact subsets of H , it follows that .ψ3,n −→ ψ3 in + the topology of .BC(H ) and, by similar arguments as above, strongly in .L1 (I × , BC(H + )). Thus, we have

.



t

E

.

0

 λns (ψ3,n )ds

t

−→ E 0

λos (ψ3 )ds, t ∈ I.

(4.32)

4.3 Stochastic Systems Driven by Martingale Measures

97

Using the expressions (4.30)–(4.32), we conclude that, for every .φ ∈ D(A) ∩ D(B) with .Dφ and .D 2 φ having compact supports, 

t

E

.

0

 λns (An φ)π(ds) −→ E

t 0

λos (Aφ)π(ds), t ∈ I,

(4.33)

λos (Bφ)ds, t ∈ I.

(4.34)

and 

t

E

.

0

 λns (Bn φ)ds −→ E

t 0

s

Since .x0,n −→ x0 a.s. and .φ ∈ BC(H + ), we have .φ(x0,n ) −→ φ(x0 ) a.s. Then, it follows from dominated convergence theorem that Eλn0 (φ) ≡ Eφ(x0,n ) −→ Eφ(x0 ) ≡ Eλ0 (φ).

.

(4.35)

Recall that the vector  measure M induces a square integrable E valued .Ft  t martingale denoted by . N(t) ≡ 0 M(ds), t ≥ 0 with the quadratic variation t given by the measure . 0 π(ds) = π([0, t]). Consider the stochastic integral in Eq. (4.29). Since .Dφ is continuous having compact support and .Gn is continuous and bounded on bounded sets, we have .Cn φ ∈ BC(H + , E). Recall that .π is a positive measure having bounded total variation. Thus, it follows from this that  E

.

i

|(Cn φ)(xn (s−))|2E π(ds) < ∞

for all .n ∈ N. Hence, the last term in Eq. (4.29) containing the stochastic integral is a well-defined square integrable .Ft -martingale. Our objective is to show that for any .z ∈ L2 () we have     t   t n→∞ E z . λns− (Cn φ), M(ds)E −→ E z λos− (Cφ), M(ds)E , t ∈ I. 0

0

(4.36) This can be proved using well-known properties of iterated conditional expectations following similar arguments as in Theorem 4.2.2 (see also [12]). Consider the expression on the left of Eq. (4.36). For .z ∈ L2 (), it follows from the properties of conditional expectation and the martingale theory that    t    t n n .E z

λs− (Cn φ), dN(s)E = E zt

λs− (Cn φ), dN(s)E , 0

0

98

4 Measure Solutions for Stochastic Systems

where .zt ≡ E{z | Ft } is a square integrable .Ft martingale as introduced above. Hence, there exists an .Ft -adapted process .η(t), .t ≥ 0, with values in E, and an .F0 measurable random variable .z0 ∈ L2 () such that  .E |η(t)|2E π(dt) < ∞ I

and that 

t

z t = z0 +

.

η(s), dN(s)E .

0

Thus,   t   t  n n .E zt

λs− (Cn φ), dN(s)E = E

η(s), λs− (Cn φ)E π(ds) . 0

0

Since .Dφ has a compact support and .Cn φ −→ Cφ in the topology of .BC(H + , E), we have s

η(t), (Cn φ)(t, ξ )E −→ η(t), (Cφ)(t, ξ )E in BC(H + ) π × P -a.e.

.

In fact, due to norm square integrability of .η with respect to the measure .π × P on the predictable sigma field and boundedness of the sequence .{Cn (φ)}, it follows from dominated convergence theorem that s

η, Cn φE −→ η, CφE in L1 (I × , BC(H + ))

.

w∗

as .n → ∞. Using this result (strong convergence) and the fact that .λn −→ λo in w + .L∞ (I × , Mrba (H )), we conclude from the duality pairing of the two spaces, and the martingale argument that for each .t ∈ I , we have  E

t .

0

   t 

η(s), λns− (Cn φ)E π(ds) −→ E z λos− (Cφ), M(ds) .

(4.37)

0

Now multiplying both sides of Eq. (4.29) by an arbitrary .z ∈ L2 () and taking the limit of the expected values, it follows from the expressions (4.33)–(4.37) that E .



zλot (φ)



=E



zλo0 (φ)



  t  o +E z λs (Aφ)π(ds) 0

    t   t o o λs (Bφ)ds + E z λs− (Cφ), M(ds)E , t ∈ I. +E z 0

0

4.3 Stochastic Systems Driven by Martingale Measures

99

Since this holds for any .z ∈ L2 (), we conclude that for each .t ∈ I ,  λot (φ). = λo0 (φ) +

t 0

 λos (Aφ)π(ds) + 

t

+ 0

t 0

λos (Bφ)ds

λos− (Cφ), M(ds)E P -a.s.

(4.38)

w (I ×,  + w + We have already seen that .λo ∈ M∞ rba (H )) ⊂ L∞ (I ×, Mrba (H )) o o and that for each .φ ∈ D(𝒜) ∩ D(ℬ), .λ (Aφ) ∈ L2 (, L1 (π )), .λ (Bφ) ∈ L2 (, L1 (I )) and .λo (Cφ) ∈ L2 (, L1 (I )). Thus, Eq. (4.38) holds for all .φ ∈ D(A) ∩ D(B) and not only for those having first and second Fréchet differentials with compact supports. Hence, .λo is a measure-valued solution of Eq. (4.24) in the sense of Definition 4.3.1. This completes the proof.  

Remark 4.3.4 For existence of measure-valued solutions, we have assumed in the above theorem that F and G are continuous and bounded on bounded sets. Thus, these maps may have polynomial growth [12]. This result can be further extended to admit measurable vector fields using Proposition 2.3.2 as seen in Theorem 3.3.8. We will consider this briefly in the last section of this chapter. The following corollary is an immediate consequence of Theorem 4.3.3. Corollary 4.3.5 Consider the forward Kolmogorov equation dϑt = A∗ ϑt π(dt) + B∗ ϑt dt, ϑ(0) = ν0 ,

.

(4.39)

with .A∗ and .B∗ denoting the duals of the operators .A and .B, respectively. Suppose .{A, F, G, M, π } satisfy the assumptions of Theorem 4.3.3. Then, for each .ν0 ∈ .rba (H ), Eq. (4.39) has at least one weak solution w + ν ∈ M∞ (I, rba (H + )) ⊂ Lw ∞ (I, Mrba (H ))

.

in the sense that, for each .φ ∈ D(A) ∩ D(B), the following equality holds 

t

νt (φ) = ν0 (φ) +

.

0



t

νs (Aφ)π(ds) +

νs (Bφ)ds, t ∈ I.

0

Proof The proof is similar to that of Corollary 4.2.4. For detailed proof one can also follow similar arguments as in [12, Corollary 3.4, p. 85]. This completes the brief outline of our proof.   It is interesting to note that Corollary 4.3.5 proves existence of (measure) solutions for Kolmogorov equation with unbounded coefficients. This generalizes similar results of Cerrai [57] for parabolic and elliptic equations on finite dimensional spaces.

100

4 Measure Solutions for Stochastic Systems

Let us now consider the question of uniqueness of solutions. The uniqueness of measure solution was proved using spectral properties of the operator .A [19]. A direct proof based on a semigroup approach was given in [23]. Using similar techniques as in [23], we can prove the uniqueness of solution of Eq. (4.39) as presented below. Corollary 4.3.6 (Uniqueness) Suppose the assumptions of Corollary 4.3.5 hold and that .D(A) ∩ D(B) is dense in .BC(H ). Then, the solution (in the weak sense) of the evolution equation (4.39) is unique. Proof We prove uniqueness of (weak) solution using similar techniques as in the general semigroup theory (see also [23]). Accordingly, it suffices to demonstrate that for a given .ν0 ∈ rba (H ), the pair .{A∗ , B∗ } determines a unique evolution operator ∗ .{U (t, s), 0 ≤ s ≤ t < ∞} on .Mrba (H ). This will guarantee the uniqueness of weak solution of Eq. (4.39) (whenever it exists) having the representation μt = U ∗ (t, 0)ν0 , t ≥ 0.

.

By virtue of Corollary 4.3.5, for each given initial measure, Eq. (4.39) has at least one weak solution. For the given quadratic variation measure .π ∈ Mca (I ), corresponding to the martingale measure M, suppose the pair .{A∗i , B∗i }, .i = 1, 2, generates the evolution operator .Ui∗ (t, s), .0 ≤ s ≤ t < ∞, .i = 1, 2. For ∗ ∗ .ϕ ∈ D(Ai ) ∩ D(Bi ) and .ν ∈ D(A ) ∩ D(B ), we define the function i i h(r) ≡ U2∗ (t, r)U1∗ (r, s)ν, ϕ = ν(U1 (r, s)U2 (t, r)ϕ), r ∈ [s, t].

.

It is a well-known fact that on their domain, the infinitesimal generators commute with their corresponding evolution operators. Using this fact, it is easy to verify that, for .A1 = A2 and .B1 = B2 , the variation of h on .[s, t] is zero. Hence, h is constant on .[s, t] and so .h(t) = h(s), implying .ν(U1 (t, s)ϕ) = ν(U2 (t, s)ϕ). This holds for all .ϕ ∈ D(Ai ) ∩ D(Bi ) and .ν ∈ D(A∗i ) ∩ D(B∗i ). Since .D(Ai ) ∩ D(Bi ) is dense in .BC(H ), and .ν is arbitrary, we have .U1∗ (t, s) = U2∗ (t, s) for .0 ≤ s ≤ t < ∞, proving uniqueness.   Using the unique evolution operator corresponding to the pair .{A∗ , B∗ }, as stated in Corollary 4.3.6, and variation of constants formula, we can write the following stochastic evolution equation, dμt = A∗ μt π(dt) + B∗ μt dt + C∗ μt− , M(dt)E , t ∈ I, μ0 = δx0 (4.40)

.

as an integral equation on the Banach space .Mrba (H + ) ∗



μt = U (t, 0)μ0 +

.

0

t

U ∗ (t, s) C ∗ μs− , M(ds)E , t ∈ I.

4.4 Extension to Measurable Vector Fields

101

Remark 4.3.7 (Continuity of Measure Solution) If the martingale measure M is non-atomic, then the associated quadratic variation measure .π is also non-atomic and it is absolutely continuous with respect to the Lebesgue measure. Thus, it follows from the expression (4.38) that the measure solution .t −→ λot is weak star continuous on I . However, if the measure M has atomic component, then the measure .π also has atomic component. In this case, the measure solution .t −→ λot is only piecewise weak star continuous.

4.4 Extension to Measurable Vector Fields In Theorem 4.3.3 we have considered the system (4.24) with drift and diffusion operators .{F, G} assumed to be continuous. Here we consider them to be Borel measurable maps taking bounded sets into bounded sets. Hence, the following theorem is an extension of Theorem 4.3.3. In the present case, the measure solutions are no longer regular. Instead, they are bounded finitely additive measure-valued processes. Theorem 4.4.1 Consider the system (4.24). Suppose .{A, M} satisfy the assumptions of Theorem 4.3.3 and that .F : H → H and .G : H → L(E, H ) are Borel measurable maps and bounded on bounded sets. Then, for every .x0 for which .P {ω ∈  : |x0 |H < ∞} = 1, and independent of the martingale measure M, the evolution + equation (4.24) has a unique measure-valued solution .λo ∈ Lw ∞ (I ×, Mba (H )). o w + Further, .λ ∈ M∞ (I × , ba (H )). Proof We present a brief outline. First suppose that .{F, G} are bounded Borel measurable maps, i.e., uniformly bounded on H . Then, it follows from Proposition 2.3.2 (see also [19, Proposition 3.2]) that the pair .{F, G} has an approximating sequence .{Fn , Gn } satisfying (A1) and (A2) of Theorem 4.3.3. Given this fact, the proof is almost identical to that of Theorem 4.3.3 with .Mrba (H + ) replaced by + .Mba (H ). In other words, for uniformly bounded Borel measurable maps F and G, the system (4.24) has a unique measure-valued solution. In case the operators .{F, G} are only bounded on bounded sets and Borel measurable, we follow the following steps. Define the composition maps Fγ ≡ F ◦ Rγ , Gγ ≡ G ◦ Rγ , γ > 0,

.

where .Rγ is the retraction of the ball .Bγ ≡ {x ∈ H : |x|H ≤ γ } for .γ > 0. Clearly, these are uniformly bounded Borel measurable maps and it follows from the preceding results that the system (4.24) with .{F, G} replaced by .{Fγ , Gγ } has w (I × ,  (H + )). Let .γ be an a unique measure-valued solution .λγ ∈ M∞ ba n increasing sequence such that .γn → ∞ as .n → ∞. Then, following a similar limiting process as in Theorem 4.3.3 (see also [19]), one can prove that the sequence γ .{λ n }, along a generalized subsequence if necessary, converges in the weak star

102

4 Measure Solutions for Stochastic Systems

w (I ×,  (H + )) satisfying the following identity topology to an element .λo ∈ M∞ ba

 o .λt (φ)



t

= λ0 (φ) +

λos (𝒜φ)π(ds) +

0

0

t

λos (ℬφ)ds, t ∈ I,

for every .φ ∈ D(𝒜) ∩ D(ℬ). Thus .λo is a measure solution of Eq. (4.24). This completes the outline of our proof.   Remark 4.4.2 Under some additional assumptions on G, these results can be extended to include vector measure M with atoms and hence jump processes. In that case the operator .C is given by  (C(t)ϕ)(ξ ) ≡

.

1

G∗ (ξ )Dϕ(ξ + θ G(ξ )M({t}))dθ.

0

If t is not an atom, this operator reduces to the one given by (4.25).

4.5 Bibliographical Notes Stochastic differential equations on infinite dimensional spaces have been considered extensively in the literature using standard assumptions as seen in the following brief list of papers contributed by several well established authors [33, 34, 37, 59– 61, 63, 64, 70, 71, 83, 84, 104, 105, 110]. We have not considered set-valued stochastic differential equations [95], which are usually based on selection theorems for set-valued stochastic integrals [96]. We are not aware of any paper dealing with measure-valued solutions for infinite dimensional stochastic differential equations. In this chapter we have seen that if the drift and diffusion operators .{F, σ } are merely continuous and satisfy even polynomial growth, standard results on SDE in infinite dimensional Hilbert spaces do not hold [33, 57, 64, 82], unless some additional assumption such as dissipativity is imposed. Consequently, the usual notions of mild and martingale solutions do not apply. However, the notion of measure-valued or generalized solutions does apply as seen in this chapter. Here we have considered measure-valued solutions for stochastic systems in infinite dimensional spaces. In Theorem 4.2.2 we prove existence of measurevalued solutions for stochastic systems driven by Hilbert space valued Wiener process. Corollary 4.2.4 presents the associated Kolmogorov equation in infinite dimension proving existence of weak solutions. Theorem 4.3.3 proves existence of measure-valued solutions for stochastic differential equations driven by vectorvalued martingale measures generalizing Theorem 4.2.2. It is interesting to note that Corollary 4.3.5 proves existence of measure-valued solutions for the Kolmogorov equation with unbounded coefficients. This generalizes similar results of Cerrai [57] for parabolic and elliptic equations on finite dimensional spaces. Theorem 4.4.1 generalizes the above results to systems with measurable vector fields for the drift and diffusion operators.

Chapter 5

Measure Solutions for Neutral Evolution Equations

5.1 Introduction Neutral differential equations have found many applications in natural sciences and engineering, such as biological systems [103], neural networks of neutral type [2], distributed networks containing lossless transmission lines [87], neuromechanical systems [91], etc. We refer to [87] for more examples on applications in science and technology. In this chapter we consider neutral differential equations on infinite dimensional Banach spaces. Neutral systems include, among other examples, systems governed by parabolic partial differential equations with nonhomogeneous boundary data and control. For details the reader is referred to [32] and [4, Chapter 3], and the references therein. In case of neutral systems, temporal evolution is determined not only by the time derivative of its state but also that of a nonlinear function of the state. Here we consider systems governed by evolution equations of the form .

(d/dt)(x + g(t, x)) = Ax + f (t, x), x(0) = x0 ∈ E, t ∈ I ≡ [0, T ], T < ∞,

(5.1)

on a Banach space E, where A is the infinitesimal generator of a .C0 semigroup S(t), .t ≥ 0 in E, and the vector field .f : I × E −→ E is continuous and bounded on bounded sets and the map .g : I × E −→ E is also continuous and bounded on bounded sets. We consider the question of existence and uniqueness of measurevalued solutions for this class of systems. In Chap. 6 we consider related control problems.

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. U. Ahmed, S. Wang, Measure-Valued Solutions for Nonlinear Evolution Equations on Banach Spaces and Their Optimal Control, https://doi.org/10.1007/978-3-031-37260-5_5

103

104

5 Measure Solutions for Neutral Evolution Equations

5.2 Basic Background Materials Let X be any regular topological space and .BC(X) the Banach space of bounded real valued continuous functions endowed with sup norm topology. We have seen in the previous chapters that the topological dual of this space is given by .Mrba (X), the space of regular bounded finitely additive measures. Endowed with total variation norm it is a Banach space. We are interested in the space of regular bounded finitely additive probability measures on X which we have denoted by .rba (X). Clearly, this is a subset of .Mrba (X). Particularly, we need the space of measure-valued functions .μ : I ≡ [0, T ] −→ Mrba (X). We have already noted that the spaces .BC(X) and .Mrba (X) do not satisfy the RNP. Hence, the dual of .L1 (I, BC(X)) is not given by .L∞ (I, Mrba (X)). However, we know that by virtue of the theory of lifting, the (topological) dual of .L1 (I, BC(X)) is given by .Lw ∞ (I, Mrba (X)) which consists of weak star measurable .Mrba (X) valued functions. Any continuous linear functional . on .L1 (I, BC(X)) has the representation  (ϕ) =

.

I ×X

ϕ(t, x)μt (dx)dt

determined by a .μ ∈ Lw ∞ (I, Mrba (X)). ˆ We have seen in the previous chapters that, for .X+ = βX (Stone-Check comw + w + pactification X) the set .M∞ (I, rba (X )) ⊂ L∞ (I, Mrba (X )) is of particular + interest in the study of measure-valued solutions. The space .Lw ∞ (I, Mrba (X )) is endowed with the weak star topology. Then, it follows from Alaoglu’s theorem w (I,  + that the set .M∞ rba (X )) is weak star compact. The weak star topology is w Hausdorff and hence .M∞ (I, rba (X+ )) is a compact Hausdorff space.

5.3 Existence of Measure Solutions and Their Regularity Let us consider the evolution equation (5.1). For simplicity of presentation, we assume without any loss of generality that g is independent of time. This assumption is used only for the following result. Later in the sequel we use more relaxed assumptions. Theorem 5.3.1 Consider the system (5.1) and suppose that (i) A is the infinitesimal generator of a .C0 semigroup .S(t), .t ≥ 0, on E; (ii) .f : I × E −→ E is continuous and bounded on bounded sets satisfying the following approximation properties: (a) there exists a sequence .{fn } such that .fn (t, x) ∈ D(A) for .x ∈ E and ∗ ∗ .t ∈ I and further, for each .e ∈ E , as .n → ∞, e∗ , fn (t, x) E ∗ ,E −→ e∗ , f (t, x) E ∗ ,E , for each x ∈ E, t ∈ I ;

.

5.3 Existence of Measure Solutions and Their Regularity

105

(b) there exists a pair .{αn , βn } ∈ L+ 1 (I ), possibly .|αn |L1 , .|βn |L1 −→ ∞, such that |fn (t, x) − fn (t, y)|E ≤ αn (t)|x − y|E , and

.

|fn (t, x)|E ≤ βn (t)(1 + |x|E ), ∀ x, y ∈ E; (iii) .g : E −→ E is Lipschitz with Lipschitz constant .0 < K < 1. Then, for every initial state .x0 ∈ E (or initial measure .μ0 = ν ∈ rba (E)), the w (I,  + system (5.1) has at least one measure-valued solution .μ ∈ M∞ rba (E )). Proof Consider the algebraic equation .x + g(x) = y on the Banach space E. For any .y ∈ E, introduce the operator .Gy (x) ≡ y − g(x). It is easy to see that for each .y ∈ E, the operator .Gy is Lipschitz with Lipschitz constant .K(< 1) and hence by Banach fixed point theorem .Gy has a unique fixed point .x ∈ E. This defines the resolvent operator . : E −→ E given by .x = (y). The reader can easily verify that the operator . is also Lipschitz with Lipschitz constant .1/(1 − K). Thus, we can reformulate the original problem given by Eq. (5.1) in the following form: (d/dt)(y) = A(y) + f (t, (y)), t ∈ I ≡ [0, T ], y(0) = y0 ≡ x0 + g(x0 ) ∈ E.

.

(5.2) We prove that Eq. (5.2) has a measure-valued solution. For this we introduce the space of test functions,   ≡ ϕ ∈ BC(E) : Dϕ ∈ BC(E, E ∗ )

.

and the operator .A given by D(A) ≡ {ϕ ∈ : Aϕ ∈ BC(E)} ,

.

where (Aϕ)(t, ξ ) = A∗ Dϕ(ξ ), (ξ ) E ∗ ,E + Dϕ(ξ ), f (t, (ξ )) E ∗ ,E .

.

Since . is Lipschitz and .(t, x) −→ f (t, x) is continuous on .I × E, the composition map .f˜(t, y) ≡ f (t, (y)) is continuous and bounded on bounded sets. By use of Yosida approximation, we set .n (e) = nR(n, A)(e), .e ∈ E and .y0,n = nR(n, A)y0 for .n ∈ ρ(A), the resolvent set of A. Then, under the given assumptions all the conditions of Theorem 2.1.5 (see also [9, Theorem 3.2, p. 12]) are satisfied. Hence, it follows from this theorem that there exists at least one measure-valued w (I,  function .μ ∈ M∞ rba (E)) satisfying  μt (ϕ) = ϕ(y0 ) +

.

0

t

μs (Aϕ)ds, t ∈ I, ∀ ϕ ∈ D(A)

(5.3)

106

5 Measure Solutions for Neutral Evolution Equations

and therefore, by definition .μ is a measure-valued solution for the evolution equation (5.2). Let this solution be denoted by .μ(2) . We use this solution to determine the existence of a measure solution for the original problem (5.1), which we denote by .μ(1) . Clearly, it follows from the algebraic relation .y = x + g(x) that the measure solution .μ(2) is related to the measure solution .μ(1) as follows   (2) (1) . ϕ(η)μt (dη) = ϕ(ξ + g(ξ ))μt (dξ ) (5.4) E

E

for every .ϕ ∈ BC(E) and .t ∈ I . Clearly, the relation (5.4) prescribes and w (I,  characterizes the measure solution .μ(1) ∈ M∞ rba (E)) in terms of the (2) measure-valued solution .μ . An alternate relation, based on the transformation .x = (y), is given by  .

E



(1)

ϕ(ξ )μt (dξ ) =

E

(2)

ϕ((ξ ))μt (dξ )

for every .ϕ ∈ BC(E) and .t ∈ I . Thus, we have proved the existence of a measure-valued solution for the neutral evolution equation (5.1) given by .μ(1) ∈ w (I,  + M∞ 

rba (E )). This completes the proof. Remark 5.3.2 Let .μζ denote the solution of Eq. (5.3) for .y0 = ζ ∈ E. In case the measure induced by the initial state .y0 is given by .ν, Eq. (5.3) is replaced by  μˆ t (ϕ) = ν(ϕ) +

.

t

μˆ s (Aϕ)ds, t ∈ I,

(5.5)

0

where the solution .μˆ is now given by the convolution .μˆ t (ϕ) = .ϕ ∈ BC(E).



ζ

E

μt (ϕ)ν(dζ ) for

The method used above for proof of existence of measure-valued solution is very powerful. It applies to semilinear, quasilinear and nonlinear problems without requiring A to be the infinitesimal generator of a .C0 semigroup [9, 11– 14, 19, 20, 22, 23, 29]. However, it is required that the resolvent set .ρ(A) be nonempty. For neutral systems, the assumption that the map g be a contraction, is a limitation. So, we may use another technique which works very well for semilinear problems provided that A generates a .C0 semigroup. The technique that we are going to use now is known as the method of transposition. This is the technique we have used in Theorem 2.1.2 of Chap. 2 (see also [9, Theorem 2.2, p. 8]). Let .S(t), .t ≥ 0, denote the semigroup generated by A. Using the semigroup, we can formally use Duhamel’s formula and integration by parts to convert the differential

5.3 Existence of Measure Solutions and Their Regularity

107

equation (5.1) into the following integral equation  .

t

x(t) + g(t, x(t)) = S(t)(x0 + g(0, x0 )) +

S(t − r)f (r, x(r))dr

0



t

−

AS(t − r)g(r, x(r))dr, t ∈ I.

(5.6)

0

Since f and g are assumed to be merely continuous and bounded on bounded sets, this integral equation may have no mild solution. Thus, the identity is only formal. However, under fairly general assumptions, we can prove existence of measurevalued solutions. For this we need the following definition. For each .r ∈ [0, ∞), let .Br = Br (E) denote the closed ball in E of radius r centered at the origin. Definition 5.3.3 The evolution equation (5.1) is said to have a sign indefinite measure-valued solution if, for each .μ0 ∈ Mrba (E), there exists a .μ ∈ ∗ ∗ ∗ Lw ∞ (I, Mrba (E)) such that for each .e ∈ D(A ) ⊂ E and .r ∈ (0, ∞), .μ satisfies the following functional equation,  .

ξ + g(t, ξ ), e∗ E,E ∗ μt (dξ )

Br



S ∗ (t)e∗ , ξ + g(0, ξ ) μ0 (dξ )

= Br

+

 t 0

−

S ∗ (t − s)e∗ , f (s, ξ ) μs (dξ )ds

Br

 t 0

S ∗ (t − s)A∗ e∗ , g(s, ξ ) μs (dξ )ds, t ∈ I,

(5.7)

Br

in .Lw ∞ (I, Mrba (E)), and it is said to have a (probability) measure-valued solution w (I,  if .μ ∈ M∞ rba (E)) for .μ0 ∈ rba (E). Theorem 5.3.4 Consider the evolution equation (5.1) and suppose A generates a C0 semigroup .S(t), .t ≥ 0, in E and .f, g : I × E −→ E are continuous and bounded on bounded subsets of .I × E. Then, for any initial measure .μ0 = ν ∈ Mrba (E), the system (5.1) has at least one sign indefinite measure-valued solution w .μ ∈ L∞ (I, Mrba (E)) in the sense of Definition 5.3.3. .

Proof For any .r ∈ (0, ∞), let .Br denote closed ball of radius r in E centered at the origin. Then, for any given initial state .μ0 = ν ∈ Mrba (E) and .e∗ ∈ D(A∗ ) ⊂ E ∗ , we define the function .hr as follows,  .hr (t) ≡ S ∗ (t)e∗ , ξ + g(0, ξ ) E ∗ ,E ν(dξ ), t ∈ I. (5.8) Br

108

5 Measure Solutions for Neutral Evolution Equations

Since .S ∗ (t), .t ≥ 0, is continuous on .E ∗ with respect to its weak star topology, it follows from the assumption on g that the integrand is continuous and bounded on .I × Br (E) for every finite .r > 0. Hence, .hr ∈ L∞ (I ). Let us introduce the operator Lr ≡ Lr,1 + Lr,2 + Lr,3 ,

(5.9)

.

where  .

ξ + g(t, ξ ), e∗ E,E ∗ μt (dξ ), t ∈ I,

Lr,1 (μ)(t) ≡ Br

Lr,2 (μ)(t) ≡

 t 0

Lr,3 (μ)(t) ≡ −

S ∗ (t − s)A∗ e∗ , g(s, ξ ) E ∗ ,E μs (dξ )ds, t ∈ I,

Br

 t 0

S ∗ (t − s)e∗ , f (s, ξ ) E ∗ ,E μs (dξ )ds, t ∈ I.

Br

Note that the function .hr depends on the initial state (measure) .ν. To show this dependence explicitly, we introduce a linear operator .Hr on .Mrba (E) to .L∞ (I ) as follows:  .Hr (ν)(t) ≡ S ∗ (t)e∗ , ξ + g(0, ξ ) ν(dξ ) = hr (t), t ∈ I. Br

Clearly, Eq. (5.7) can be written in the following compact form Lr μ = Hr (ν) = hr .

.

(5.10)

Our problem is to prove existence of a measure-valued function .μ that satisfies Eq. (5.10). Under the assumptions on f and g, it is clear that the operators .{Lr,i , i = 1, 2, 3} map Lw ∞ (I, Mrba (Br (E))) to .L∞ (I ). Hence, for each .r ∈ (0, ∞), the map w .Lr is a bounded linear operator from .L∞ (I, Mrba (Br (E))) to .L∞ (I ). We use the method of transposition as seen in Theorem 2.1.2 of Chap. 2 (see also [9]). Using the adjoint of the operator .Lr we can construct an isomorphism as follows. First note that the adjoint operators corresponding to the operators .{Lr,i } are given by (L∗r,1 η)(t, ξ ) ≡ ξ + g(t, ξ ), e∗ η(t), (t, ξ ) ∈ I × Br (E),

.

(L∗r,2 η)(t, ξ ) ≡



T

.

∗ .(Lr,3 η)(t, ξ )

S ∗ (s − t)A∗ e∗ , g(t, ξ ) η(s)ds, (t, ξ ) ∈ I × Br (E),

t



T

≡− t

S ∗ (s − t)e∗ , f (t, ξ ) η(s)ds, (t, ξ ) ∈ I × Br (E),

5.3 Existence of Measure Solutions and Their Regularity

109

for .η ∈ L1 (I ). It is clear from the above expressions that for every .η ∈ L1 (I ), we have .(L∗r,i η) ∈ L1 (I, BC(Br (E))) for each finite .r > 0 and for each .i = 1, 2, 3. Hence, L∗r : L1 (I ) −→ L1 (I, BC(Br (E)))

.

for every finite .r > 0, where .L∗r = L∗r,1 + L∗r,2 + L∗r,3 . Now we introduce the space   Yr ≡ y ∈ L1 (I, BC(Br (E))) : y = L∗r η for some η ∈ L1 (I )

.

and furnish this with the norm topology given by   |y|Yr ≡ sup |η|L1 (I ) : η ∈ L1 (I ) and L∗r η = y .

.

The reader can easily verify that .Yr is a normed vector space and that, with respect to this norm topology, it is a Banach space. Clearly, it follows from the definition of the space .Yr , that .L∗r is surjective, that is, .L∗r (L1 (I )) = Yr . Further, it is noted that for every .η ∈ L1 (I ) we have the inequality |L∗r η|Yr ≥ |η|L1 (I ) , ∀ η ∈ L1 (I )

.

(5.11)

and so .L∗r is injective. Thus, .L∗r is a bijective map between the spaces .L1 (I ) and ∗ .Yr and hence .Lr ∈ Iso(L1 (I ), Yr ). For the given .hr ∈ L∞ (I ), it is clear that the  map .r (η) ≡ (hr , η)L∞ ,L1 = I hr (t)η(t)dt defines a continuous linear functional on the space .L1 (I ). Since .L∗r ∈ Iso(L1 , Yr ), this linear functional on .L1 (I ) is equivalent to the following linear functional ˜r (y) ≡ r ((L∗r )−1 (y)) ≡ (r o(L∗r )−1 )(y) on Yr .

.

Thus, it follows from the expression (5.11) that it is a continuous linear functional on .Yr . Hence, there exists a .yo∗ ∈ Yr∗ ⊂ Lw ∞ (I, Mrba (Br (E))) (dual of .Yr ) such that ˜r (y) = yo∗ , y Yr∗ ,Yr ≡ yo∗ (y)

.

for all .y ∈ Yr . It follows from the duality pairing as seen in Sect. 5.2 that .yo∗ is a measure-valued function .yo∗ ≡ λo , so that ˜r (y) = yo∗ (y) = .

 I ×Br

y(t, ξ )λot (dξ )dt ≡ y, λo Yr ,Yr∗

for all .y ∈ Yr . By transposing the isomorphism .L∗r , we obtain ˜r (L∗r η) = (Lr λo , η)L∞ (I ),L1 (I ) = r (η) = (hr , η)L∞ ,L1

.

110

5 Measure Solutions for Neutral Evolution Equations

for every .η ∈ L1 (I ). This implies the identity .Lr λo = hr , proving the existence of a solution of the functional equation (5.10). Clearly, for every .r (> 0) finite and ∗ ∈ E ∗ and .ν ∈ M .e rba (E) the function .hr ∈ L∞ (I ). Thus, we may conclude that, for any initial measure (state) .μ0 = ν ∈ Mrba (E), the corresponding solution o w .λ ∈ L∞ (I, Mrba (E)) and it satisfies the identity (5.10) for every finite .r > 0 and every .e∗ ∈ D(A∗ ) ⊂ E ∗ and .t ∈ I . Thus, Eq. (5.6) has a measure-valued solution 

in the sense of Definition 5.3.3. This completes the proof of existence. In the following corollary we prove uniqueness. Corollary 5.3.5 (Uniqueness and Well-Posedness) Under the assumptions of Theorem 5.3.4, the measure-valued solution of the functional equation (5.10) is unique and it is continuously dependent on the initial data. Proof Suppose the contrary and that there are two solutions .μ1 , .μ2 ∈ Lw ∞ (I, Mrba (E)) corresponding to the same initial state .μ0 ∈ Mrba (E). Then, it follows from Eq. (5.10) that, for every r .(> 0) finite, .Lr (μ1 − μ2 ) = 0 and so 0 = Lr (μ1 − μ2 ), η L∞ (I ),L1 (I ) = μ1 − μ2 , L∗r η Yr∗ ,Yr , ∀ η ∈ L1 (I ).

.

Since .L∗r ∈ Iso(L1 (I ), Yr ), this is equivalent to 0 = μ1 − μ2 , y Yr∗ ,Yr , ∀ y ∈ Yr .

.

Under the natural (canonical) embedding .Yr → Yr∗∗ , one can consider .Yr as a subspace of the space .Yr∗∗ , that is, .Yr ⊂ Yr∗∗ . By virtue of Goldstine’s theorem [69, Theorem 5, p. 424], the unit ball of .Yr is weak star dense in the unit ball of .Yr∗∗ , that is, B1 (Yr )

.

w∗

= B1 (Yr∗∗ ).

Thus, for every .y ∗∗ ∈ B1 (Yr∗∗ ), there exists a sequence .yn ∈ B1 (Yr ) converging to ∗∗ in the weak star topology. Clearly, . μ1 − μ2 , y ∗ .y n Yr ,Yr = 0 for all .n ∈ N, and we have 0 = lim μ1 − μ2 , yn Yr∗ ,Yr = lim μ1 − μ2 , yn Yr∗ ,Yr∗∗ = μ1 − μ2 , y ∗∗ Yr∗ ,Yr∗∗ .

.

n→∞

n→∞

Since .Yr∗∗ separates points of .Yr∗ , and .y ∗∗ is an arbitrary element of the unit ball ∗∗ 1 2 ∗ .B1 (Yr ), we have .μ = μ as elements of .Yr . Thus, the solution is unique. For n the proof of continuous dependence, let .{ν , ν o } ∈ Mrba (E) denote the initial states and .{μn , μo } ∈ Lw ∞ (I, Mrba (E)) the corresponding (sign indefinite) measure solutions of Eq. (5.10). For any finite .r > 0, recall the definition of the operator .ν −→ Hr (ν) from .Mrba (E) to .L∞ (I ) as shown on the right-hand side of the expression (5.10). Clearly, .Hr ∈ L(Mrba (E), L∞ (I )) and it is continuous with respect to the .w ∗ topologies on the space .Mrba (Br (E)) and the space .L∞ (I ), respectively. Then, it is easy to verify from the expression .Lr (μn ) − Lr (μo ) =

5.4 Stochastic Neutral Systems

111 w∗

w∗

n o Lr (μn −μo ) = Hr (ν n −ν o ) that .μn −→ μo in .Lw ∞ (I, Mrba (Br (E))) as .ν −→ ν in .Mrba (Br (E)) for every .r (> 0) finite. This completes the proof.



Remark 5.3.6 It follows from Theorem 5.3.4 that for any initial measure .μ0 ∈ Mrba (E), the functional equation (5.10) has a sign indefinite measure solution w .μ ∈ L∞ (I, Mrba (E)). This allows us to introduce a bounded linear evolution operator .V (t, s), .t ≥ s ≥ 0, on the Banach space .Mrba (E) so that μt = V (t, 0)μ0 , t ≥ 0.

.

It follows from the results on uniqueness, as proved in Corollary 5.3.5, that .V (t, s) satisfies the evolution (semigroup) property .V (t, s) = V (t, τ )V (τ, s), .0 ≤ s ≤ τ ≤ T < ∞. Remark 5.3.7 An important question is, if the initial state .μ0 = ν belongs to rba (E), does the corresponding measure solution belong to .Mw ∞ (I, rba (E)). The technique used in Theorem 5.3.1 and in [9, 11–14, 19, 20, 22, 23, 29] for proof of existence of measure solutions directly ensures this property. Unfortunately, the technique used here does not. One way to deal with the above problem using the present technique is to verify if the generalized Farkas-like theorem proved by Hernandez-Lerma and Lasserre in [89, Theorem 2.4, p. 152], and Craven and Koliha in [62, Theorem 1, Theorem 2, p. 987] hold. For details, interested readers are referred to [31].

.

Remark 5.3.8 In Theorem 5.3.4 we have seen that for every .r ∈ (0, ∞), the neutral evolution equation (5.1) has a measure-valued solution in .Lw ∞ (I, Mrba (Br (E))) ⊂ Lw ∞ (I, Mrba (E)). By virtue of Corollary 5.3.5 the solution is unique. For each w finite .r > 0, let .λr ∈ Lw ∞ (I, Mrba (Br (E))) ⊂ L∞ (I, Mrba (E)) denote the corresponding solution. It is left to the reader to find conditions on .{f, g} under which .{λr , r > 0} has a weak star limit in .Lw ∞ (I, Mrba (E)).

5.4 Stochastic Neutral Systems In this section we consider stochastic systems determined by neutral differential equations on Hilbert spaces. Let E and H be a pair of separable Hilbert spaces and consider the following system on E: d(x + g(t, x)) = Axdt + b(t, x)dt + σ (t, x)dW, x(0) = x0 , t ∈ I, (5.12)

.

where A is the infinitesimal generator of a .C0 semigroup .S(t), .t ≥ 0, in E, and the vector field .b : I × E −→ E is continuous and bounded on bounded sets and .σ : I × E −→ L(H, E) is continuous and bounded on bounded sets. The process .W = {W (t), t ∈ I } is an H valued cylindrical Brownian motion on a complete filtered probability space .(, F, Ft≥0 , P ), where .Ft≥0 ⊂ F is an increasing family of right

112

5 Measure Solutions for Neutral Evolution Equations

continuous complete sub-sigma algebras having left limits. We know that if .{b, σ } are merely continuous and bounded on bounded sets, Eq. (5.12) has no solution even in the mild sense. We have already mentioned, some counter examples due to Godunov [80] and Dieudonne [67]. As mentioned before, neutral systems include parabolic partial differential equations with nonhomogeneous boundary data. For details the reader is referred to [4, 32] and the references therein.

5.4.1 Basic Background Materials In this section, we are interested in stochastic measure-valued processes. These are Ft adapted .w ∗ measurable .Mrba (E) valued random processes. Consider the Banach space .L1 (I × , BC(E)) of random processes with the norm topology given by

.

 |ϕ| =

E sup{|ϕ(t, ω, ξ )|, ξ ∈ E}dt.

.

I

As seen before, it follows from the theory of lifting that the topological dual ∗ of this space is given by .Lw ∞ (I × , Mrba (E)) which consists of .w measurable .Ft -adapted .Mrba (E) valued random processes. In other words, .(L1 (I × ∗ , BC(E)))∗ ∼ = Lw ∞ (I × , Mrba (E)). Thus, for any . ∈ (L1 (I × , BC(E))) w there exists a .μ ∈ L∞ (I × , Mrba (E)) such that  (ϕ) =



.

I ×

μt,ω (ϕ)dtdP ≡

I ××E

ϕ(t, ω, x)μt,ω (dx)dtdP

+ for .φ ∈ L1 (I × , BC(E)). Here we need the set .Mw ∞ (I × , rba (E )) ⊂ w + L∞ (I × , Mrba (E )) denoting the class of regular probability measure-valued + random processes contained in the space .Lw ∞ (I ×, Mrba (E )) which is equipped ∗ with .w topology. Clearly, by virtue of Alaoglu’s theorem, the set .Mw ∞ (I × , rba (E + )) is .w ∗ compact. We consider the stochastic neutral system given by Eq. (5.12). As we have seen for deterministic systems, we transform this system into a regular (non-neutral) system.

Lemma 5.4.1 Consider the system given by Eq. (5.12). Suppose .g : I ×E −→ E is continuous with .g(·, 0) ∈ C(I, E) and further it is Lipschitz in the second argument |g(t, x) − g(t, z)|E ≤ η|x − z|E , ∀ t ∈ I,

.

5.4 Stochastic Neutral Systems

113

with .η ∈ (0, 1). Then, the system (5.12) can be written in the equivalent form as follows dy = A(t, y)dt + b(t, (t, y))dt + σ (t, (t, y))dW,

.

y(0) = x0 + g(0, x0 ) ∈ E,

(5.13)

with .t ∈ I , where . is described below. Proof Under the assumptions on g, it follows from Banach fixed point theorem that, for every .y ∈ E, the equation .x + g(t, x) = y has a unique solution .x = (t, y). It is easy to verify that . : I × E −→ E is continuous and further . is Lipschitz in the second argument with Lipschitz constant .γ ≡ 1/(1 − η). Thus, the system (5.12) can be written as Eq. (5.13). This completes the proof. 

5.4.2 Existence of Measure Solutions and Their Regularity For simplicity of presentation we introduce the following notations: ˜ y) = b(t, (t, y)), σ˜ (t, y) ≡ σ (t, (t, y)). b(t,

.

In passing we note that .b˜ and .σ˜ are composition of continuous maps and hence they are also continuous. Thus, the system (5.13) is equivalent to the following system, ˜ y)dt + σ˜ (t, y)dW, t ∈ I, y0 = x0 + g(0, x0 ). dy = A(t, y)dt + b(t,

.

(5.14) Because of the presence of the nonlinear operator . in front of the unbounded operator A, this is a nonlinear differential equation. Even if .b˜ and .σ˜ are Lipschitz in the state variable with . being Lipschitz, as seen above, we can not use the standard technique [64] to prove existence of even a mild solution. Further, we do not wish to impose any of the standard assumptions like local Lipschitz property and linear growth. We prove that under just continuity assumptions on b and .σ and some other properties, we have measure-valued solutions. Before we can present such a result, we need some preparatory results. Let A be the infinitesimal generator of a .C0 semigroup of operators .S(t), .t ≥ 0 on E with stability parameters .{M, ω} with .M ≥ 1 and .ω ≥ 0 so that .|S(t)|L(E) ≤ Meωt , .t ≥ 0. Clearly, the resolvent set .ρ(A) ⊃ (ω, ∞). Let .An ≡ nAR(n, A), a .n ∈ ρ(A), denote the Yosida approximation of A. Let .L (I, L2 (, E)) denote the 2 space of .Ft -adapted norm square integrable E valued random processes. We prove the following lemma. Lemma 5.4.2 Let .{An } denote the Yosida approximation of A and define .y0,n ≡ nR(n, A)y0 . Suppose .b˜ : I × E −→ E and .σ˜ : I × E −→ L2 (H, E) admit

114

5 Measure Solutions for Neutral Evolution Equations

the following approximation properties: there exists a sequence .{b˜n , σ˜ n } such that {b˜n (t, y) ∈ D(A), σ˜ n (t, y) ∈ L(H, D(A))} for all .(t, y) ∈ I × E satisfying the following properties:

.

˜ y) uniformly on compact subsets of .I × E; (A1): .b˜n (t, y) −→ b(t, (A2): .σ˜ n (t, y) −→ σ˜ (t, y) strongly in .L2 (H, E), uniformly on compact subsets of the set .I × E; (A3): There exists a sequence pair .{αn , βn } ∈ L+ 2 (I ) possibly satisfying .

lim |αn |L2 (I ) −→ ∞,

n→∞

lim |βn |L2 (I ) −→ ∞

n→∞

such that for all .t ∈ I and .y1 , y2 , y ∈ E, (i): .

|b˜n (t, y1 ) − b˜n (t, y2 )|E ≤ αn (t)|y1 − y2 |E , |b˜n (t, y)|E ≤ αn (t)(1 + |y|E ).

(ii): .

|σ˜ n (t, y1 ) − σ˜ n (t, y2 )|L2 (H,E) ≤ βn (t)|y1 − y2 |E , |σ˜ n (t, y)|L2 (H,E) ≤ βn (t)(1 + |y|E ).

Then, the approximating system dy = An (t, y)dt + b˜n (t, y)dt + σ˜ n (t, y)dW, t ∈ I, y(0) ≡ y0,n ,

.

(5.15) has a unique mild solution .yn ∈ La∞ (I, L2 (, E)) having continuous modification. Proof Since the proof is classical [64], we present an outline only. First let us note that the Yosida approximation of A given by .An ≡ nAR(n, A) is a bounded linear operator on E with bound .|An |L(E) ≤ an ≡ n(n(1 + M) − ω)/(n − ω) for all .n ∈ ρ(A) and hence for all .n > ω. Thus, the composition operator .An (t, y) is Lipschitz with Lipschitz constant .ηn = an /(1 − η) for all finite .n > ω. Further, one can easily verify that it satisfies the following growth property |An (t, y)|E ≤ (an /(1 − η))(1 ∨ |c|E )(1 + |y|E ), ∀ t ∈ I, y ∈ E,

.

where .c ≡ sup{|g(t, 0)|E , t ∈ I }. Thus, all the operators on the right-hand side of the evolution equation (5.15) are bounded, globally Lipschitz, and have at most linear growth. Hence, it follows from classical results on the existence and uniqueness of solutions of stochastic differential equations (SDE) on Hilbert spaces,

5.4 Stochastic Neutral Systems

115

that the system (5.15) has a unique mild solution .yn ∈ La∞ (I, L2 (, E)) admitting continuous modifications (see [64]). This completes the proof. 

Now we are prepared to consider the question of existence of measure-valued solution for system (5.14). Readers interested in more details may see [30]. Theorem 5.4.3 Suppose A is the generator of a .C0 semigroup .S(t), .t ≥ 0, with stability parameters .{M, ω} satisfying the properties as stated above. Let ˜ : I × E −→ E and .σ˜ : I × E −→ L2 (H, E) be continuous and bounded .b on bounded subsets of .I × E satisfying the assumptions (A1), (A2), and (A3) of Lemma 5.4.2. Then, for every .F0 measurable initial state .y0 with the law .μ0 ∈ rba (E), the evolution equation (5.14) has an .Ft -adapted measure solution o w o .μ ∈ M∞ (I × , rba (E)). Further, .t −→ μt is weak star continuous on I with probability one. Proof By virtue of Lemma 5.4.2, each of the approximating sequence of systems (5.15) has strong solution .{yn (t), t ∈ I } for .n ∈ N with .yn (0) = y0,n ∈ D(A) with probability one. Thus, Itô differential rule applies. Let .ϕ ∈ BC(E) having first and second Fréchet derivatives with bounded supports in E. Then, it follows from Itô formula that  t  t .ϕ(yn (t)) = ϕ(y0,n ) + (An ϕ)(s, yn (s))ds + (Bn ϕ)(s, yn (s))ds 0



0 t

+

(Cn ϕ)(s, yn (s)), dW (s) H , t ∈ I,

(5.16)

0

where the operators .{An , Bn , Cn } are given by (An ϕ)(t, ξ ) ≡ (1/2) Tr((D 2 ϕ)(σ˜ n σ˜ n∗ )(t, ξ )), (t, ξ ) ∈ I × E, .

(Bn ϕ)(t, ξ ) ≡ A∗n Dϕ(ξ ), (t, ξ ) E + Dϕ(ξ ), b˜n (t, ξ ) E , (t, ξ ) ∈ I × E, (Cn ϕ)(t, ξ ) = (σ˜ n∗ Dϕ)(t, ξ ), (t, ξ ) ∈ I × E.

As seen before, we introduce the sequence of Dirac measures .{δyn (t) (dξ )}n along the path process .{yn (t), t ∈ I }, and denote these by .{μnt (dξ )} defined on .B(E), the class of Borel sets in E. Using this notation we can rewrite the expression (5.16) in terms of the sequence of measure-valued random processes .{μn } as follows  μnt (ϕ) = μn0 (ϕ) +

t

.



0 t

+ 0

 μns (An ϕ)ds +

0

t

μns (Bn ϕ)ds

μns (Cn ϕ), dW (s) H , t ∈ I, P -a.s.,

(5.17)

where .μn0 ∈ rba (E) denotes the measure induced by the approximating sequence s

of initial states .{y0,n }. It is clear that if it were true that .yn (t) −→ y(t) in E, P -a.s.,

116

5 Measure Solutions for Neutral Evolution Equations

 then . E ϕ(ξ )μnt (dξ ) −→ ϕ(y(t)) P -a.s. Under the given (relaxed) assumptions on b and .σ and the initial condition, it is not expected that .yn converges to even a mild solution of Eq. (5.14). As seen in the preceding chapters, even in the deterministic case, if it were given that .yn (t) converges weakly to .y(t) for each .t ∈ I , .ϕ(yn (t))  ϕ(y(t)) for any .ϕ ∈ BC(E). However, there may exist a measure-valued .Ft -adapted process .νt , .t ∈ I , such that  .

lim ϕ(yn (t)) =

n→∞

ϕ(ξ )νt (dξ ) E

P -almost surely. We show that such a measure-valued process does exist. Considering the first term on the right-hand side of the expression (5.17), representing the initial condition, it is evident that for any .ϕ ∈ BC(E), .ϕ(y0,n ) −→ ϕ(y0 ) in probability. Thus, .μn0 (ϕ) −→ μ0 (ϕ). Returning to the sequence of Dirac measures .{μnt ≡ δyn (t) , t ∈ I }, we note that the sequence .{μn } is contained in w (I × ,  + w + the bounded set .M∞ rba (E )) ⊂ L∞ (I × , Mrba (E )). Recall that by virtue of the theory of lifting, the vector space of measure-valued processes + denoted by .Lw ∞ (I × , Mrba (E )) is the topological dual of the Banach space + w + .L1 (I × , BC(E )) and that .M∞ (I × , rba (E )) is a closed bounded convex w + subset of the dual .L∞ (I × , Mrba (E )). Thus, by Alaoglu’s theorem, the set w + .M∞ (I × , rba (E )) is compact in the weak star topology. Hence, there exists a generalized subsequence (subnet) of the sequence .{μn }, relabeled as the original w (I × ,  + sequence, and a .μo ∈ M∞ rba (E )) such that for any .ϕ ∈ L1 (I × + , BC(E)) ⊂ L1 (I × , BC(E )),  n . μ (ϕ) = ϕ(t, ω, ξ )μnt,ω (dξ )dtdP I ×E×

 −→

I ×E×

ϕ(t, ω, ξ )μot,ω (dξ )dtdP = μo (ϕ).

We show that .μo is a measure-valued solution of the stochastic differential equation (5.14) with .μ0 ∈ rba (E) being the measure induced by the initial state .y0 . By definition of measure-valued solutions, it suffices to verify that, for any o .ϕ ∈ D(A) ∩ D(B), .μ satisfies the following identity  o .μt (ϕ)

t

= μ0 (ϕ) + 

0 t

+ 0

 μos (Aϕ)ds

t

+ 0

μos (Bϕ)ds

μos (Cϕ), dW (s) H , t ∈ I, P -a.s.,

(5.18)

5.4 Stochastic Neutral Systems

117

where the operators .{A, B, C} are given by (Aϕ)(t, ξ ) ≡ (1/2) Tr((D 2 ϕ)(σ˜ σ˜ ∗ )(t, ξ )), (t, ξ ) ∈ I × E, .

˜ ξ ) E , (t, ξ ) ∈ I × E, (Bϕ)(t, ξ ) ≡ A∗ Dϕ(ξ ), (t, ξ ) E + Dϕ(ξ ), b(t, (Cϕ)(t, ξ ) = (σ˜ ∗ Dϕ)(t, ξ ), (t, ξ ) ∈ I × E.

From now on we follow a similar procedure as in Theorem 4.3.3 (see also [12, 20, 23]). For each .n ∈ ρ(A), and .ϕ ∈ D(A) ∩ D(B) having compact supports, we define ψ1,n (t, ξ ) ≡ (1/2) Tr((D 2 ϕ)(σ˜ n σ˜ n∗ ))(t, ξ ), (t, ξ ) ∈ I × E,

.

and ψ1 (t, ξ ) ≡ (1/2) Tr((D 2 ϕ)(σ˜ σ˜ ∗ ))(t, ξ ), (t, ξ ) ∈ I × E.

.

Clearly, under the assumption (A2), .σ˜ n (t, ξ ) −→ σ˜ (t, ξ ) strongly in .L(H, E) uniformly on compact subsets of .I × E. Since .D 2 ϕ has compact support and for each .ϕ ∈ D(A), we have .ψ1,n (t, ·), .ψ1 (t, ·) ∈ BC(E), it is clear that .ψ1,n (t, ·) → s ψ1 (t, ·) strongly in .BC(E) uniformly in .t ∈ I . In other words, .ψ1,n −→ ψ1 in the Banach space .BC(I × E) furnished with the sup norm topology. Thus, for any .F measurable random variable .ζ ∈ L2 () and any .t ∈ I , we have   t    t  E ζ μns (An ϕ)ds −→ E ζ μos (Aϕ)ds , t ∈ I.

.

0

(5.19)

0

Again, for any .ϕ ∈ D(A) ∩ D(B) having compact support, we define ψ2,n (t, ξ ) ≡ (Bn ϕ)(t, ξ ) ≡ A∗n Dϕ(ξ ), (t, ξ ) E + Dϕ(ξ ), b˜n (t, ξ ) E

.

and ˜ ξ ) E . ψ2 (t, ξ ) ≡ (Bϕ)(t, ξ ) ≡ A∗ Dϕ(ξ ), (t, ξ ) E + Dϕ(ξ ), b(t,

.

s

s

Since .An −→ A on .D(A) and E is a Hilbert space, .A∗n −→ A∗ on .D(A∗ ). For ∗ .ϕ ∈ D(B), it is clear that .Dϕ(ξ ) ∈ D(A ) for .ξ ∈ E, and it has compact support in E, and . is continuous on .I × E and bounded on bounded sets. Hence A∗n Dϕ(ξ ), (t, ξ ) E −→ A∗ Dϕ(ξ ), (t, ξ ) E

.

uniformly on .I × E. Again, since .Dϕ has compact support, by virtue of assumption (A1) we have ˜ ξ ) E Dϕ(ξ ), b˜n (t, ξ ) −→ Dϕ(ξ ), b(t,

.

118

5 Measure Solutions for Neutral Evolution Equations

uniformly on .I × E. Thus, .ψ2,n −→ ψ2 strongly in .BC(E) uniformly in .t ∈ I . It follows from these facts that for any .ζ ∈ L2 (),   t    t  E ζ μns (ψ2,n )ds −→ E ζ μos (ψ2 )ds , t ∈ I.

.

0

0

In other words, we have shown that for any .ϕ ∈ D(A) ∩ D(B) and .ζ ∈ L2 (),   t    t  E ζ μns (Bn ϕ)ds −→ E ζ μos (Bϕ)ds , t ∈ I,

.

0

(5.20)

0

as .n → ∞. Next, we consider the martingale term appearing in the expression on the right-hand side of Eq. (5.16). For each .n ∈ ρ(A), and .ϕ ∈ D(A) ∩ D(B) having compact support, we define ψ3,n (t, ξ ) ≡ (σ˜ n∗ Dϕ)(t, ξ ), (t, ξ ) ∈ I × E.

.

Clearly, this is a sequence of continuous functions on .I × E with values in the Hilbert space H and by virtue of assumption (A2), ψ3,n (t, ξ ) ≡ (σ˜ n∗ Dϕ)(t, ξ ) −→ (σ˜ ∗ Dϕ)(t, ξ ) ≡ ψ3 (t, ξ )

.

strongly in H uniformly on .I × E. One can easily verify that both .μn (ψ3,n ) and a a o .μ (ψ3 ) ∈ L (I, H ), where, we recall that .L (I, H ) denotes the space of H valued 2 2 norm square integrable .Ft -adapted random processes. For any .ζ ∈ .L2 (), we consider the integral    t n .E ζ μs (ψ3,n ), dW (s) H .

(5.21)

0

For .ζ ∈ L2 (), it is clear that .ζ (t) ≡ E{ζ | Ft } is a square integrable .Ft martingale. Thus, it follows from standard properties of conditional expectations that the expression (5.21) is equivalent to     t   t n n E ζ . μs (ψ3,n ), dW (s) H = E ζ (t) μs (ψ3,n ), dW (s) . 0

(5.22)

0

Now by virtue of the well-known martingale representation theorem, there exists an .F0 measurable random variable .ζ0 ∈ L2 () and an .Ft adapted random process a .z ∈ L (I, H ) (mutually stochastically independent) such that 2  ζ (t) = ζ0 +

.

0

t

z(s), dW (s) H , t ∈ I.

5.4 Stochastic Neutral Systems

119

By virtue of this representation used in (5.22), it follows from standard properties of stochastic integrals that    t  t E ζ (t) μns (ψ3,n ), dW (s) H = E z(s), μns (ψ3,n ) H ds, t ∈ I.

.

0

0

(5.23) w (I × ,  + Since .μn converges to .μo in .M∞ rba (E )) in the weak star topology, and a .ψ3,n converges strongly in H uniformly on .I × E, and .z ∈ L (I, H ), we conclude 2 that  t  t . lim E z(s), μns (ψ3,n ) H ds = E z(s), μos (ψ3 ) H ds, t ∈ I. n→∞

0

0

(5.24) By using the martingale representation once again in the reverse order, it is easy to verify that 

t

E

.

0

  t  z(s), μos (ψ3 ) H ds = E ζ μos (Cϕ), dW (s) , t ∈ I.

(5.25)

0

Thus, it follows from Eqs. (5.23)–(5.25) that   t    t  n o .E ζ μs (Cn ϕ), dW (s) → E ζ μs (Cϕ), dW (s) , t ∈ I. (5.26) 0

0

Now multiplying the expression (5.17) by .ζ on both sides and taking the expectation and letting .n → ∞ and using the expressions (5.19), (5.20), (5.26) and the measure .μ0 corresponding to the initial state, we obtain .

E



ζ μot (ϕ)



  t  t o = E ζ μ0 (ϕ) + ζ μs (Aϕ)ds + ζ μos (Bϕ)ds  +ζ 0

0

t

0

 o μs (Cϕ), dW (s) H , t ∈ I.

(5.27)

This identity holds for every .F-measurable random variable .ζ ∈ L2 (). Hence, the identity (5.18) holds P -a.s. for every .t ∈ I . Since all the integrals in the above expression are finite for any .ϕ ∈ D(A) ∩ D(B) having bounded supports, the identity (5.18) holds for all such test functions and not only those having w (I × ,  + compact supports. This proves that .μo ∈ M∞ rba (E )) is a measurevalued solution of the stochastic evolution equation (5.13). Hence, the evolution

120

5 Measure Solutions for Neutral Evolution Equations

w (I × ,  + equation (5.12) has a measure-valued solution .μ∗ ∈ M∞ rba (E )) given by

μ∗ = μo  −1 .

.

Thus, for any .ϕ ∈ L1 (I × , BC(E)), the following identity holds 

ϕ(t, ω, ξ )μ∗t,ω (dξ )dtdP =

.

I ××E

 I ××E

ϕ(t, ω, (t, ξ ))μot,ω (dξ )dtdP .

The last statement asserting weak star continuity follows directly from the expression (5.18). This completes the proof. 

Remark 5.4.4 If the drift and the diffusion operators are Lipschitz having at most linear growth and the initial state has a finite second moment, then the measure solution .μo degenerates into a Dirac measure along the path process .y o (t), t ∈ I , which is a mild solution of the stochastic evolution equation (5.12). Note that the integral equation (5.18) is the weak form of the abstract stochastic evolution equation on .rba (E + ) ⊂ Mrba (E + ) given by dμt = A∗ μt dt + B∗ μt dt + C∗ μt , dW , t ∈ I,

.

(5.28)

with the initial state .μ0 ∈ rba (E). The operators .{A∗ , B∗ , C∗ } are the formal adjoints of the operators .{A, B, C}. Here we may look at .Mrba (E + ) as the state space. It is well-known that, with respect to the total variation norm, this is a Banach space. We look for solutions which are .Ft -adapted stochastic processes taking values in the .w ∗ closed convex subset .rba (E + ) ⊂ Mrba (E + ). Let us first consider the deterministic Kolmogorov equation associated with the stochastic system (5.14). This is given by dmt = A∗ mt dt + B∗ mt dt, t ∈ I, mo = π0 ∈ rba (E).

.

(5.29)

Since .b˜ and .σ˜ are only continuous and bounded on bounded sets and E is a Hilbert space, this equation has unbounded coefficients. So classical theory of Kolmogorov equations on Hilbert spaces does not hold. Here we use Theorem 5.4.3 to give a simple proof. Corollary 5.4.5 Consider the Kolmogorov evolution equation (5.29) on the Banach space .Mrba (E) with the initial state .π0 ∈ rba (E). Suppose the assumptions of Theorem 5.4.3 hold. Then, Eq. (5.29) has a measure-valued solution .m ∈ w (I,  + w + M∞ rba (E )) ⊂ L∞ (I, Mrba (E )) in the weak sense, that is, for any .ϕ ∈

5.4 Stochastic Neutral Systems

121

D(A) ∩ D(B) the following equation holds:  mt (ϕ) = π0 (ϕ) +

t

.

 ms (Aϕ)ds +

0

t

ms (Bϕ)ds, t ∈ I,

(5.30)

0

and there exists a .w ∗ continuous evolution operator .{U ∗ (t, s), 0 ≤ s ≤ T < ∞} on .Mrba (E + ) such that .mt = U ∗ (t, 0)π0 , .t ≥ 0. Proof The proof is very similar to that of Corollary 4.3.5. A detailed proof can be found in [20, Corollary 3.5]. An informal proof is very simple and can be obtained just by applying the expectation operator on either side of Eq. (5.18) and identifying the measure-valued function m through the relation 



E

.

E

ϕ(ξ )μot (dξ ) =

ϕ(ξ )mt (dξ ) E

for any .ϕ ∈ D(A) ∩ D(B). In other words, .mt (ϕ) ≡ Eμot (ϕ). This completes our proof. 

In view of this result, as we have seen before, we expect the existence of an evolution operator .U ∗ (t, s), .0 ≤ s ≤ T < ∞, on the Banach space .Mrba (E + ), such that mt = U ∗ (t, 0)π0 ,

.

and using the variation of constants formula, one can also write the stochastic evolution equation (5.28) as the following integral equation μot = U ∗ (t, 0)μ0 +



t

.

0

U ∗ (t, s) C∗ μos , dW (s) , t ∈ I.

These comments are rather informal. In order to justify these comments we need uniqueness of solution of Eq. (5.18). We have already seen that the stochastic evolution equation (5.28), as reproduced below dμt = A∗ μt dt + B∗ μt dt + C∗ μt , dW , t ∈ I,

.

is defined on the state space .Mrba (E + ). Given any initial state .μ0 = π ∈ rba (E), by Theorem 5.4.3 this equation has a solution .{μt , t ≥ 0}, a stochastic process possibly with values in .rba (E + ). In fact, one can verify that this is a Markov process generating a transition operator on .Mrba (E + ). For more on this the reader is referred to [30].

122

5 Measure Solutions for Neutral Evolution Equations

5.5 Second Order Neutral Differential Equations 5.5.1 Introduction As seen in the preceding sections, we have so far considered first order neutral differential equations. For application to hyperbolic neutral systems, in this section we consider second order neutral evolution equations. In previous studies [51, 52, 86], first order neutral evolution equations have been considered using standard assumptions and covering the questions of classical existence theory. First order equations are known to arise naturally from nonlinear boundary feedback control problems [4, 25, 26]. Here we consider second order neutral systems driven by vector measures covering impulsive systems as special cases [18, 27]. Moreover, we consider stochastic versions of the system driven by Brownian motion and vector measures.

5.5.2 Some Basic Notations Let X be any real Banach space and let .I ≡ [0, T ], with T finite, denote a closed bounded interval and let .T denote the sigma algebra of subsets of the set I . In compliance with our previous notations, let .B∞ (I, X) denote the Banach space of bounded .T measurable functions defined on I and taking values from the Banach space X. Let .Mba (T , X) denote the space of finitely additive X valued vector measures having bounded variations. For any .μ ∈ Mba (T , X) we recall the variation norm given by |μ|v ≡ sup



.

| μ(σ )|X ,

 σ ∈

where the summation is taken over ., consisting of any finite family of disjoint T measurable partitions of the interval .I, and the supremum is taken over all such partitions. Furnished with the total variation norm, this is a Banach space. Let .Mca (T , X) denote the class of X valued countably additive vector measures having bounded variations. This is a closed linear subspace of .Mba (T , X) and hence a Banach space. For details on vector measure the reader is referred to [66] and multi measure [90]. Let Y be another Banach space and denote by .L(X, Y ) the space of bounded linear operators from X to Y . This is a Banach space with respect to the uniform operator topology. We denote by .K(X, Y ) ⊂ L(X, Y ) the class of compact operators from X to Y .

.

5.5 Second Order Neutral Differential Equations

123

5.5.3 System Models We consider two different models. (M1): The first model is a control system described by dt (x˙ + B1 (t, x) + C1 (t, x)u) = Axdt + B2 (t, x)dt + C2 (t, x)μ2 (dt),

.

t ∈ I ≡ [0, T ], x(0+) = x0 , x(0) ˙ = x1 ,

(5.31)

where .{u, μ2 } will play the role of controls in Chap. 6. The control .u ∈ L1 (I, U ) where U is a Banach space and the control .μ2 ∈ Mca (T , V ) is a vector measure defined on the sigma algebra .T of subsets of the set I taking values in V , another Banach space. Formally we can write this as a system. Define y ≡ x˙ + B1 (t, x) + C1 (t, x)u

.

and .z ≡ (x, y) ≡ (z1 , z2 ) . Using this identity, Eq. (5.31) can be written formally as a first order evolution equation, dz = 𝒜zdt + ℱ(t, z)dt + 𝒢(t, z)μ(dt), z(0) = z0 ,

.

(5.32)

 where .μ ≡ (μ1 , μ2 ) , .μ1 (σ ) ≡ σ u(s)ds, σ ∈ T . Clearly, .μ ∈ Mca (T , W), where .W = U × V is a Banach space. The operators .{𝒜, ℱ(t, z), 𝒢(t, z)} are given by

−B1 (t, z1 ) .𝒜 ≡ , , ℱ(t, z) ≡ B2 (t, z1 )

0 −C1 (t, z1 ) . 𝒢(t, z) ≡ 0 C2 (t, z1 ) 0 I A 0





(M2): Another interesting model is given by .

dt (x˙ + B1 (t, x)) = Axdt + B2 (t, x, x)dt ˙ + C2 (t, x, x)μ ˙ 2 (dt), t ∈ I, x(0) = x0 , x(0) ˙ = x1 ,

(5.33)

where the vector measure .μ2 may be used as controls. Again, this can be written as a system as follows. Define y ≡ x˙ + B1 (t, x)

.

124

5 Measure Solutions for Neutral Evolution Equations

and .z ≡ (x, y) ≡ (z1 , z2 ) . Using this identity we obtain the evolution equation dz = 𝒜zdt + ℱ(t, z)dt + 𝒢(t, z)μ(dt), t ∈ I, z(0) = z0 , (5.34)

.

where .μ ≡ μ2 , and the operators .{𝒜, ℱ(t, z), 𝒢(t, z)} are given by 𝒜≡

.

0 I A 0



, ℱ(t, z) ≡



0 −B1 (t, z1 ) , 𝒢(t, z) ≡ ˜ , B˜ 2 (t, z1 , z2 ) C2 (t, z1 , z2 )

with B˜ 2 (t, z1 , z2 ) ≡ B2 (t, z1 , z2 − B1 (t, z1 )),

.

and C˜ 2 (t, z1 , z2 ) ≡ C2 (t, z1 , z2 − B1 (t, z1 )).

.

The appropriate √ state space for the above systems is given by the product space .Z ≡ D( A) × X as justified below.

5.5.4 System Models Generating C0 -Group The formal conversion of the second order system into a first order one is justified by the following well-known result. Lemma 5.5.1 ([81]) Let X be a real Banach space and suppose the following assumptions hold: (1): The operator A with domain and range, .D(A), R(A) ⊂ X, is the infinitesimal generator of a strongly continuous Cosine function (operator) .{C(t), t ∈ R} of bounded linear operators in X; (2): There exists an operator B satisfying .B 2 = A, with .0 ∈ ρ(B), and it commutes with any bounded linear operator that commutes with A; (3): The operator B generates a .C0 -group. Then, the operator .𝒜 with .D(𝒜) = D(A) × D(B) generates a .C0 -group .{U (t), t ∈ R} on the Banach space .Z ≡ [D(B)] × X, where .[D(B)] is the Banach space with respect to the topology induced by the graph norm, .|ξ |B ≡ |Bξ |X + |ξ |X . In view of the above result, the second order evolution equation, x¨ = Ax, x(0) = x1 , x(0) ˙ = x2 ,

.

5.5 Second Order Neutral Differential Equations

125

on the Banach space X can be formulated as an equivalent first order evolution equation on the Banach space .Z ≡ [D(B)] × X as follows z˙ = 𝒜z, z(0) = z0 , z0 ≡ (x1 , x2 ) .

.

It has a unique mild solution .z ∈ C(I, Z) given by .z(t) = U (t)z0 , t ≥ 0, where U (t), t ≥ 0, is the .C0 -group [5] of bounded linear operators with values in .L(Z) corresponding to .𝒜. This is given by

.

U (t) ≡

.

C(t) S(t) . B 2 S(t) C(t)

This family of operators .{U (t), t ∈ R} satisfies the following properties: (p1) (p2) (p3) (p4)

U (0) = IZ ; U (t + s) = U (t)U (s), t, s ∈ R; ˙ = C(t), C(t) ˙ .S(t) = B 2 S(t) = AS(t); ˙ .U (t)ξ = AU (t)ξ , for .ξ ∈ D(A). . .

2

d The reader can easily verify that .limt↓0 dt 2 (C(t)η) = Aη, for .η ∈ D(A). Using variation of constants formula, both the systems (5.32) and (5.34) can be written as an integral equation on the Banach space .Z = [D(B)] × X as follows,



t

z(t). = U (t)z0 +

U (t − s)ℱ(s, z(s))ds

0



t

+

U (t − s)𝒢(s, z(s))μ(ds), t ≥ 0,

(5.35)

0

where .μ ∈ Mca (T , W). As we have noted in Chap. 3, since .μ has bounded variation it may possess at most countably many atoms. Thus, any solution z of the integral equation (5.35) may have at most a countable set of discontinuities.

5.5.5 Existence and Regularity of Solutions We need the basic assumptions on the operators .ℱ and .𝒢 as stated below. First, we prove the existence and uniqueness of mild solutions under these assumptions. (A1): The map .ℱ : I × Z −→ Z is measurable in the first argument and Lipschitz in the second and there exists a .K ∈ L+ 1 (I ) such that the following conditions hold |ℱ(t, z)|Z ≤ K(t)(1 + |z|Z ), |ℱ(t, z1 ) − ℱ(t, z2 )|Z ≤ K(t)(|z1 − z2 |Z )

.

for .z, z1 , z2 ∈ Z.

126

5 Measure Solutions for Neutral Evolution Equations

(A2): The map .𝒢 : I × Z −→ L(W, Z) is measurable in the first argument and + Lipschitz in the second and there exists an .L ∈ L+ 1 (I, |μ|) = L1 (|μ|) such that the following conditions hold |𝒢(t, z)|L(W,Z) ≤ L(t)(1 + |z|Z ), |𝒢(t, z1 ) − 𝒢(t, z2 )|L(W,Z)

.

≤ L(t)(|z1 − z2 |Z ). for .z, z1 , z2 ∈ Z. Theorem 5.5.2 (Existence of Mild Solution) Consider the integral equation (5.35). Let .B∞ (I, Z) denote the Banach space of bounded measurable functions on I with values in Z endowed with the sup norm topology. Suppose the assumptions (A1), (A2) and those of Lemma 5.5.1 hold and that .μ has no atom at .{0}. Then, for every .z0 ∈ Z and .μ ∈ Mca (T , W), the integral equation has a unique solution .z ∈ B∞ (I, Z). Hence, the evolution equations (5.32) and (5.34) have each unique mild solutions. Proof For detailed proof see [27, Lemma 4.1]. Here we present only a brief outline. The question of existence of solution for the integral equation (5.35) is equivalent to the question of existence of a fixed point for the operator .Gμ on .B∞ (I, Z) given by  .

t

Gμ (z)(t) ≡ U (t)z0 +

U (t − s)ℱ(s, z(s))ds

0



t

+

U (t − s)𝒢(s, z(s))μ(ds), t ∈ I. (5.36)

0

The .C0 -group .U (t), t ∈ I , is a family of bounded linear operators with values in L(Z). Thus, there exists a finite positive number M such that .sup{|U (t)|L(Z) , t ∈ I } ≤ M. Using this fact and the assumptions (A1) and (A2) one can easily verify an a priori bound that follows from the integral inequality given by

.



t

ϕ(t) ≤ ζ (t) +

.

ϕ(s)m(ds), t ∈ I,

0

where .ϕ(t) = |z(t)|Z and m is a countably additive bounded positive measure given by  m(D) =

 K(s)ds +

.

D

L(s)|μ|(ds), D ∈ T , D

and the function .ζ is given by

 t  t ζ (t) = M |z0 |Z + K(s)ds + L(s)|μ|(ds) , t ∈ I.

.

0

0

5.5 Second Order Neutral Differential Equations

127

Hence, it follows from generalized Grönwall’s inequality that 



t

ϕ(t) ≤ ζ (t) +

t

exp

.

0

 m(dr) ζ (s)m(ds), t ∈ I.

s

Since .ζ is a nonnegative bounded measurable function and m is a bounded positive measure, it follows from the above inequality that .z ∈ B∞ (I, Z). Thus, .Gμ maps .B∞ (I, Z) onto itself. Using the Lipschitz property of the operators .ℱ and .𝒢, one can easily verify that  ρt (Gμ z1 , Gμ z2 ) ≤

.

t

ρs (z1 , z2 )m(ds), t ∈ I,

0

where .ρt (z1 , z2 ) ≡ sup{|z1 (s) − z2 (s)|Z , 0 ≤ s ≤ t}. Using the above inequality and following similar arguments as given in Lemma 3.3.5 one can verify that for a sufficiently large integer n, the n-th iterate .Gnμ is a contraction and hence, by Banach fixed point theorem, .Gnμ and therefore .Gμ itself has a unique fixed point, proving existence of mild solutions for the evolution equations (5.32) and (5.34). This completes a brief outline of our proof. 

In the rest of this section, we consider the question of existence of measurevalued solutions for the evolution equations like (5.32) and (5.34). In fact the results are similar to those already seen in Chap. 3. For the proof of existence of (measure) solutions we make use of the following assumptions: (A1): Z is a separable Banach space and .A is the infinitesimal generator of a .C0 -group of operators .{U (t), t ∈ R} in Z and .μ ∈ Mca (T , W) having bounded variations on bounded sets containing no atom at .t = 0. (A2): The functions .ℱ ∈ B(I × Z, Z) and .𝒢 ∈ B(I × Z, L(W, Z)) are Borel measurable maps and bounded on bounded subsets of Z. Theorem 5.5.3 Consider the system (5.32) and suppose that the assumptions (A1) and (A2) hold. Then, for any initial state .m0 ∈ ba (Z) the evolution equation (5.32) w (I,  (Z + )) ∩ M w (|μ|,  (Z + )). has a measure-valued solution .m ∈ M∞ ba ba ∞ ∗ Further, .t −→ mt is piecewise .w continuous on I with values in .ba (Z + ) ⊂ Mba (Z + ). Proof The proof is very similar to that of Theorem 3.3.8. We present a brief outline. First, we introduce the generating operators .𝒜 and .ℬ corresponding to the differential equation (5.32) as follows: .

𝒜(t)φ(ξ ) = A∗ Dφ(ξ ), ξ Z ∗ ,Z + Dφ(ξ ), ℱ(t, ξ ) Z ∗ ,Z , ℬ(t)φ(ξ ) = 𝒢 ∗ (t, ξ )Dφ(ξ ), ξ ∈ Z,

for all .φ ∈ BC(Z) having compact supports such that .𝒜(·)φ ∈ L1 (I, BC(Z)) and ℬφ(·) ∈ L1 (I, BC(Z, W∗ )), where .W∗ is the dual of the Banach space .W. Recall

.

128

5 Measure Solutions for Neutral Evolution Equations

that in the weak sense, the evolution equation on the space of measures is given by 

t

mt (φ) = m0 (φ) +

ms (𝒜(s)φ)ds

.

0



t

+ 0

ms− (ℬ(s)φ), μ(ds) W∗ ,W , t ∈ I,

(5.37)

for every .φ in .D(𝒜) ∩ D(ℬ) having compact supports. The objective is to w (I,  (Z + )) ∩ prove that there exists a measure-valued function .m ∈ M∞ ba w + w + w + M∞ (|μ|, ba (Z )) ⊂ L∞ (I, Mba (Z )) ∩ L∞ (|μ|, Mba (Z )) that satisfies the above integral relation. From here on we follow similar steps as seen in Theorem 3.3.8. For more details see [27]. This completes the brief outline of our proof. 

Remark 5.5.4 (1): We note that if the Banach space .W satisfies the Radon-Nikodym property, and .μ is a .W valued vector measure continuous with respect to Lebesgue measure, then there exists a Bochner integrable function .ϑ ∈ L1 (I, W) such that .dμ = ϑdt. In this case Eq. (5.37) reduces to the following equation, 

t

mt (φ) = m0 (φ) +

ms (𝒜(s)φ)ds

.

0



t

+ 0

ms− (ℬ(s)φ), ϑ(s) W∗ ,W ds, t ∈ I, (5.38)

and more importantly, the measure solution m is weak star continuous on I . (2): It is possible that, under certain stronger conditions, the supports of the measure-valued solutions may be contained in the basic space Z rather than the associated larger compact Hausdorff space .Z + .

5.6 Stochastic Second Order Neutral Systems In the preceding sections we have considered deterministic second order neutral systems. Here we consider stochastic versions of these neutral systems and present briefly some results on existence of measure solutions. These apply to stochastic hyperbolic neutral partial differential equations. In the context of stochastic systems it is convenient but not essential to assume that all the abstract spaces involved are Hilbert spaces. However, the results presented here can be extended to UMD Banach spaces, that is, Banach spaces having the UMD property (Banach space X in which the martingale difference

5.6 Stochastic Second Order Neutral Systems

129

sequences are unconditional). For simplicity we do not cover such generalizations. So we consider the vector spaces Z, .W, and H to be separable Hilbert spaces and the measure .μ ∈ Mca (T , W) is any .W valued countably additive vector measure defined on the sigma algebra .T of subsets of the interval .I ≡ [0, T ] having bounded variation on I . Let .(, F, Ft≥0 , P ) be a complete filtered probability space and W an H valued .Ft -adapted Brownian motion. The system we consider is given by dz = Azdt + ℱ(t, z)dt + 𝒢(t, z)μ(dt) + D(t)dW, z(0) = z0 , t ∈ I, (5.39)

.

where .A is the infinitesimal generator of a .C0 -group on Z, the functions .ℱ(t, z) and .𝒢(t, z) are Borel measurable maps and .W = {W (t), t ∈ I } is an H valued Brownian motion and .μ ∈ Mca (T , W). a (I, L (, Z)) denote the Banach space of .F -adapted Z valued norm Let .B∞ 2 t square integrable random processes furnished with the norm .

sup

  1/2 E|z(t)|2Z , t ∈ I < ∞.

We introduce the following basic assumptions. (B1): The process W is any H valued .Ft -adapted Brownian motion with incremental covariance operator .Q ∈ L+ 1 (H ) (positive nuclear), and the operator .D : I −→ L(H, Z) is strongly measurable satisfying .sup{|D(t)|L(H,Z) , t ∈ I } ≤ b < ∞. (B2): The operators .ℱ and .𝒢 satisfy the following properties: there exist .{K, α} ∈ L2 (I ), .{L, β} ∈ L2 (I, |μ|) = L2 (|μ|) such that (i): .

|ℱ(t, z1 ) − ℱ(t, z2 )|2Z ≤ K 2 (t)|z1 − z2 |2Z , z1 , z2 ∈ Z |ℱ(t, z)|2Z ≤ α 2 (t) + K 2 (t)|z|2Z , z ∈ Z.

(ii): .

|𝒢(t, z1 ) − 𝒢(t, z2 )|2L(W,Z) ≤ L2 (t)|z1 − z2 |2Z , z1 , z2 ∈ Z |𝒢(t, z)|2L(W,Z) ≤ β 2 (t) + L2 (t)|z|2Z , z ∈ Z.

First we present the following existence and regularity results. Theorem 5.6.1 Suppose the assumptions (B1), (B2) and Lemma 5.5.1 hold with A being the infinitesimal generator of the .C0 -group .U (t), t ∈ R. Then, for every initial state .z0 having finite second moment and .μ ∈ Mca (T , W) having

.

130

5 Measure Solutions for Neutral Evolution Equations

bounded variation, the stochastic system (5.39) has a unique mild solution .z ∈ a (I, L (, Z)). B∞ 2 Proof First let us consider the Ornstein-Uhlenbeck process given by the linear stochastic differential equation (SDE) dy = Aydt + D(t)dW, y(0) = 0.

.

This equation has a unique mild solution y given by 

t

y(t) =

.

U (t − s)D(s)dW, t ∈ I.

0

This is a Z valued Gaussian process with mean zero and covariance operator given by  Qy (t) =

.

t

U (t − s)D(s)QD∗ (s)U ∗ (t − s)ds, t ∈ I.

0

Since Q is nuclear and the operators .U (t), .D(t), .t ∈ I are bounded, and the latter strongly measurable, we conclude that the integrand is strongly measurable with values in .L+ 1 (Z) (Banach space of positive nuclear operators on Z) and Bochner integrable. Thus, for all .t ∈ I , .Qy (t) ∈ L+ 1 (Z) and there exists a finite positive number c such that .sup{Tr(Qy (t)), t ∈ I } ≤ c < ∞. This implies that a .y ∈ B∞ (I, L2 (, Z)). Using this process, the SDE (5.39) can be reduced to a standard evolution equation driven by a vector measure as follows dx = Axdt + ℱ(t, x + y)dt + G(t, x + y)μ(dt), t ∈ I, x(0) = z0 . (5.40)

.

In order to prove existence of a solution of this equation, we first derive an a priori bound. For given y, consider the following integral equation on the Hilbert space Z,  x(t) .= U (t)z0 +

t

U (t − s)ℱ(s, x(s) + y(s))ds

0

 +

t

U (t − s)𝒢(s, x(s) + y(s))μ(ds) ≡ (x)(t), t ∈ I. (5.41)

0

By definition, x is a mild solution of the system (5.40), if it satisfies the integral equation (5.41). We prove that the operator ., given by the expression on the righthand side of the above integral equation, has a unique fixed point in the Banach a (I, L (, Z)). First, we prove . maps .B a (I, L (, Z)) onto itself. For space .B∞ 2 2 ∞ this we need an a priori bound. Using Cauchy-Schwartz inequality and triangle

5.6 Stochastic Second Order Neutral Systems

131

inequality one can verify that  E|x(t)|2Z ≤ h2 (t) + C

t

.

E|x(s)|2 γ (ds), t ∈ I,

(5.42)

0

where .C = 32 and   t  t h2 (t). = 32M 2 E|x0 |2Z + α 2 (s)ds + β 2 (s)|μ|(ds) 0



t

+ 0

0



K 2 (s)E|y(s)|2Z ds +

t 0

 L2 (s)E|y(s)|2Z |μ|(ds) , t ∈ I,

(5.43)

L2 (s)|μ|(ds), σ ∈ T .

(5.44)

and the measure .γ is given by  γ (σ ) =

 K (s)ds + 2

.

σ

σ

Since .μ is a countably additive vector measure having bounded variation, the measure .|μ| induced by the variation is a countably additive bounded positive measure on .T . Thus, it follows from the above expression and the assumption (A2) that the measure .γ is a countably additive positive measure having bounded variation. Then, it follows from generalized Grönwall’s inequality applied to the expression (5.42) that  E|x(t)|2Z ≤ h2 (t) + C

t

.

  t  exp C γ (dθ ) h2 (s)γ (ds), t ∈ I.

0

(5.45)

s

Since the function .h2 (·) is an increasing function of t, it follows from the above expression that .

 sup E|x(t)|2Z ≤ h2 (T ) {1 + Cγ (I ) exp{Cγ (I )}} < ∞.

(5.46)

t∈I

Hence, we conclude that the operator ., as defined by the expression (5.41), maps a (I, L (, Z)) into .B a (I, L (, Z)). Next we verify that . has a unique fixed B∞ 2 2 ∞ a (I, L (, Z)). Again, using the assumption (A2) one can easily verify point in .B∞ 2 that  t E|(x. 1 )(t) − (x2 )(t)|2Z ≤ 2M 2 K 2 (s)E|x1 (s) − x2 (s)|2Z ds

.

 +2M 2 |μ| 0

0

t

L2 (s)E|x1 (s) − x2 (s)|2Z |μ|(ds), t ∈ I,

(5.47)

132

5 Measure Solutions for Neutral Evolution Equations

where .|μ| ≡ |μ|(I ) denotes the variation norm of the vector measure .μ. Defining C1 ≡ 2M 2 (1 + |μ|) and recalling the definition of the positive measure .γ , it follows from the above inequality that

.

 E|(x1 )(t) − (x2 )(t)|2Z ≤ C1

t

.

0

E|x1 (s) − x2 (s)|2Z γ (ds), t ∈ I.

(5.48)

Define

 ρt (x1 , x2 ) ≡ sup E|x1 (s) − x2 (s)|2Z , 0 ≤ s ≤ t , t ∈ I.

.

Using this .ρ we can rewrite the inequality (5.48) as follows 

t

ρt (x1 , x2 ) ≤ C1

.

ρs (x1 , x2 ) γ (ds), t ∈ I.

(5.49)

0

t Define .v(t) ≡ 0 γ (ds), t ∈ I . We have seen before that this integral is a monotone increasing function of bounded variation on I . So, v is differentiable a.e. on I and the differential is Lebesgue integrable. Hence, there exists a function .η ∈ L+ 1 (I ) t such that .η(t) > v(t) ˙ a.e. and .vo (t) ≡ 0 η(s)ds ≥ v(t+) for all .t ∈ I . Consequently, the inequality (5.49) can be replaced by 

t

ρt (x1 , x2 ) ≤ C1

.

ρs (x1 , x2 ) dvo (s), t ∈ I.

(5.50)

0

Using the above inequality one can easily verify that the second iterate satisfies ρt ( 2 x1 ,  2 x2 ) ≤ (1/2)(C1 vo (t))2 ρt (x1 , x2 ), t ∈ I.

.

(5.51)

Repeating this iterative process n times we find that ρt ( n x1 ,  n x2 ) ≤ (1/n!)(C1 vo (t))n ρt (x1 , x2 ), t ∈ I.

.

(5.52)

Since both .ρ and .vo are increasing functions of .t ∈ I , it follows from the above inequality that a (I,L (,Z)) ≤   n x. 1 −  n x2 B∞ 2

 a (I,L (,Z)) .(5.53) (C1 vo (T )/n!)  x1 − x2 B∞ 2

Thus, for n sufficiently large, the n-th iterate . n of . is a contraction on the Banach a (I, L (, Z)). Hence, both . n and . have one and the same fixed point. space .B∞ 2 a (I, L (, Z)) and This proves that Eq. (5.40) has a unique mild solution .x ∈ B∞ 2 a hence Eq. (5.39) has mild solution .z = x + y ∈ B∞ (I, L2 (, Z)). This completes the proof. 

5.6 Stochastic Second Order Neutral Systems

133

Now we are prepared to relax the assumptions on the operators .ℱ, .𝒢, and consider the question of existence of measure-valued solutions for the system (5.39) transformed into (5.40). In view of Theorem 5.6.1, it is equivalent to consider the transformed model. Define the family of operators .𝒜(t), .ℬ(t), .t ∈ I as follows .

𝒜(t)φ(ξ ) = A∗ Dφ(ξ ), ξ Z ∗ ,Z + Dφ(ξ ), ℱ(t, y(t) + ξ ) Z ∗ ,Z , ℬ(t)φ(ξ ) = 𝒢 ∗ (t, y(t) + ξ )Dφ(ξ ), ξ ∈ Z,

and note that these are operator valued .Ft -adapted stochastic processes. For measure-valued solution .λ, we consider the weak form given by  λt (ϕ) = λ0 (ϕ) +

.

0

t

 λs (𝒜(s)ϕ)ds + 0

t

λs (ℬ(s)ϕ), μ(ds) W∗ ,W , t ∈ I. (5.54)

Theorem 5.6.2 Suppose the operators .ℱ and .𝒢 satisfy the following properties: (a): .ℱ maps .I ×  × Z into Z, Borel measurable in all the variables, continuous and bounded on bounded subsets of Z in the third argument, and integrable on .I ×  uniformly with respect to z in bounded subsets of Z. (b): .𝒢 maps .I ×  × Z into .L(W, Z) Borel measurable in all the variables and continuous and bounded on bounded subsets of Z and integrable on .I ×  uniformly on bounded subsets of Z. Then, for each initial state .λ0 ∈ rba (Z), and vector measure .μ ∈ Mca (T , W) having bounded variation, Eq. (5.54) has a unique measure-valued w (I × ,  + solution .λ ∈ M∞ rba (Z )). Proof The proof follows from similar steps as in Theorems 3.3.6 and 4.3.3. Interested reader may carry out the details. 

Remark 5.6.3 Based on the approximation theory given by Proposition 2.3.2, the results presented above also hold for measurable vector fields .{ℱ, 𝒢} given that they are bounded on bounded sets of the state space. Remark 5.6.4 To avoid misunderstanding, we mention that it is not at all necessary to transform Eq. (5.39) to Eq. (5.40) to prove existence of measure-valued solutions. However, the transformed equation shows that the vector fields of any stochastic evolution equation can be stochastic processes by themselves and that as long as they are progressively measurable with respect to the filtration .Ft , .t ≥ 0, one can prove existence of measure-valued solutions.

134

5 Measure Solutions for Neutral Evolution Equations

5.7 Bibliographical Notes In this chapter we have considered both deterministic and stochastic neutral differential equations in infinite dimensional Banach spaces. These systems include parabolic and hyperbolic partial differential equations with nonhomogeneous boundary data [4, 32]. For neutral systems, temporal evolution is determined not only by the time derivative of its state but also by certain nonlinear functionals of both the state and its time derivative. In prior studies [51, 52, 86, 87], first order neutral evolution equations have been considered using standard assumptions, covering the questions of classical existence theory. First order equations are known to arise also naturally from nonlinear boundary feedback control problems for parabolic and hyperbolic equations [4, 25, 26]. A series of examples on engineering application of neutral differential equations is presented in [87]. Here we consider measure-valued solutions for abstract neutral differential equations on Banach spaces. The authors of this monograph are not aware of any studies in the literature concerning measure-valued solutions for neutral differential equations. In Theorems 5.3.1 and 5.3.4, we presented existence of measure solutions for deterministic neutral differential equations. We have also considered stochastic systems as presented in Theorem 5.4.3 and the associated Kolmogorov equation in Corollary 5.4.5. In Theorem 5.3.4, Fattorini’s technique [72, 73] based on the principle transposition, is used proving existence of measure solutions. In Theorem 5.3.1, we have used a more general technique as seen in [9, 11– 14, 19, 20, 22, 23, 29]. Further, we have also considered second order neutral evolution equations driven by vector measures covering impulsive systems as special cases. Interested reader is referred to [18, 27] and the references therein for more details.

Chapter 6

Optimal Control of Evolution Equations

In this chapter we consider optimal control problems for deterministic systems, impulsive systems, stochastic systems, and neutral systems. These systems are governed by evolution equations on the Banach space of regular bounded finitely additive measures. We focus predominantly on the questions of existence of optimal controls, including optimal feedback controls. We also present some necessary conditions of optimality. For non-convex control problems, we consider existence of optimal relaxed controls for differential inclusions [21] and other functional differential inclusions, see also [7, 8, 58, 72, 73] and the references therein. We will present several examples from physical sciences and engineering in Chap. 7.

6.1 Optimal Control of Deterministic Systems Let U be a compact subset of a Polish space .Z and .rba (U ) the class of regular finitely additive probability measures on the Borel subsets of U . For admissible w (I,  controls we choose the set .Uad ≡ M∞ rba (U )), a subset of the Banach space w .L∞ (I, Mrba (Z)) of weak star measurable functions on I with values in the space of regular finitely additive Borel measures .Mrba (Z). The space .Lw ∞ (I, Mrba (Z)) is given the weak star .(w ∗ ) topology and the set of admissible controls is endowed with the same topology. In short, the set of admissible controls w Uad ≡ M∞ (I, rba (U )) ⊂ Lw ∞ (I, Mrba (Z))

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. U. Ahmed, S. Wang, Measure-Valued Solutions for Nonlinear Evolution Equations on Banach Spaces and Their Optimal Control, https://doi.org/10.1007/978-3-031-37260-5_6

135

136

6 Optimal Control of Evolution Equations

consists of probability measure-valued functions called relaxed controls. The control system is given by the following differential equation on a Banach space E, .

x˙ = Ax + f (t, x, ut ), t ∈ I, x(0) = x0 ∈ E,  f (t, x, ζ )ut (dζ ), for u ∈ Uad , where f (t, x, ut ) ≡

(6.1)

U

with the cost functional given by  J (u) =

.

T

f0 (t, x(t), ut )dt + (x(T )).

(6.2)

0

The cost integrand .f0 (t, ξ, ζ ) and .(ξ ) are continuous nonnegative functions defined on .I × E × U and E, respectively. The problem is to find a control that minimizes the functional (6.2). This is how a typical control problem would be stated under standard assumptions on .{f, f0 , }. Under relaxed assumptions for the vector fields, such as mere continuity (or Borel measurability) and boundedness on bounded sets in the state variables, we only have measure-valued solutions. In this case the control problem is stated as follows. For each .u ∈ Uad , we define the operator .Au by the following expression (Au φ)(t, ξ ) = A∗ Dφ, ξ E ∗ ,E + f (t, ξ, ut ), Dφ(ξ )E,E ∗

.

for each .φ ∈ D(Au ), where its domain .D(Au ) is constant, that is, it does not vary with .(t, u) ∈ I × Uad and it is given by   D(Au ) ≡ φ ∈ F : Au φ ∈ L1 (I, BC(E))

.

with   F ≡ φ ∈ BC(E) : Dφ ∈ BC(E, E ∗ ), Dφ(ξ ) ∈ D(A∗ ) ⊂ E ∗ for ξ ∈ E .

.

The evolution equation on the space of measures is then given by (d/dt)λt = (Au )∗ λt , λ0 = ν, t ∈ I,

.

(6.3)

which can be written in the weak form as follows  t .λt (φ) = ν(φ) + λs (Au φ)ds, t ∈ I, λ0 (φ) = ν(φ), ∀ φ ∈ D(Au ), (6.4) 0

6.1 Optimal Control of Deterministic Systems

137

and the cost functional is given by  J (u) =



.

I ×E×U

f0 (t, ξ, ζ )ut (dζ )λt (dξ )dt +

(ξ )λT (dξ ),

(6.5)

E

where, corresponding to control u, .λ is the measure-valued solution of Eq. (6.3) in the weak sense given by the integral relation (6.4). The problem is to find a control that minimizes the functional (6.5). For this we need the following result. Without ˆ further notice, we use .E + to denote the Stone-Cech compactification of E. Corollary 6.1.1 Let A be the generator of a .C0 semigroup in E, and .f : I × E × U −→ E is Bochner (strongly) integrable in .t ∈ I , and continuous and bounded on bounded sets of .E × U, and for each .u ∈ Uad , .f u (t, x) ≡ f (t, x, ut ) admits the approximation property as stated in Theorem 2.1.5. Then, for each control .u ∈ w (I,  + Uad , there exists a .λ ∈ M∞ rba (E )) that satisfies Eq. (6.4) for every .φ ∈ u D(A ). Hence, the system (6.3), or equivalently (6.1), has at least one measurevalued solution. Further, the solution is .w ∗ continuous in .t ∈ I . Proof For each .u ∈ Uad , the function .f u satisfies the hypothesis of Theorem 2.1.5, and hence the conclusion follows from there.  Let . (u) denote the collection of all weak solutions of Eq. (6.3) or equivalently measure-valued solutions of Eq. (6.1) corresponding to the control .u ∈ Uad . In case of uniqueness . (u) = {λ(u)} is a singleton. Uniqueness holds under some additional assumptions. In the absence of uniqueness, we use the multi valued map . (u). It is easy to verify that . (u) is a closed convex multi function. For proof of existence of optimal controls, sequential (more generally net) closure of the graph   w w

≡ (u, λ) ∈ M∞ (I, rba (U )) × M∞ (I, rba (E + )) : λ ∈ (u)

.

plays a major role. We consider this problem below. Theorem 6.1.2 Under the assumptions of Corollary 6.1.1, the graph . of the multi function . , representing solutions of Eq. (6.3), or equivalently the system (6.1), is sequentially closed. Proof Let .{un , λn } ∈ , where by definition .λn ∈ (un ) is a measure solution of Eq. (6.1) corresponding to the relaxed control .un . By virtue of Alaoglu’s theorem, there exists a generalized subsequence (subnet), relabeled as the original sequence, and a pair .{uo , λo } such that w∗

.

w un −→ uo in M∞ (I, rba (U )), . w∗

w (I, rba (E + )). λn −→ λo in M∞

(6.6) (6.7)

138

6 Optimal Control of Evolution Equations

The problem is to show that .(uo , λo ) ∈ . By our definition of solution, for each .n ∈ N we have  t n .λt (φ) = ν(φ) + λns (An (s)φ)ds, t ∈ I, φ ∈ D(A), (6.8) 0

where (An (t)φ)(ξ ) = A∗ Dφ(ξ ), ξ  + Dφ(ξ ), f (t, ξ, unt ),  . Dφ(ξ ), f (t, ξ, unt ) = Dφ(ξ ), f (t, ξ, ζ )unt (dζ ). U

Note that the integral in the expression (6.8) is a triple integral with respect to the product measure .λns (dξ )×uns (dζ )×ds. Since U is bounded and f is continuous and bounded on bounded sets of .E ×U , for each .φ ∈ F the sequence .{An φ} is contained in a bounded subset of .L1 (I, BC(E)) and therefore this integral is well-defined for each .n ∈ N and further   t   n   . sup  λs (An (s)φ)ds  < ∞, t ∈ I. n∈N

0

Note that .μnt ≡ λnt × unt is the product measure on the algebra generated by the Cartesian product of closed subsets of the topological spaces .E + and U , respectively. Defining g(t, ξ, ζ ) ≡ A∗ Dφ(ξ ), ξ  + Dφ(ξ ), f (t, ξ, ζ ),

.

we can rewrite Eq. (6.8) as  n .λt (φ)

=

μnt (φ)

= ν(φ) +

t

0

μns (g)ds, t ∈ I.

(6.9)

It is clear that the sequence .{μn } is contained in w w w M∞ (I, rba (E + )) × M∞ (I, rba (U )) ⊂ M∞ (I, rba (E + × U )),

.

+ and hence it is contained in the closed bounded subset of .Lw ∞ (I, Mrba (E × U )). Thus, again by Alaoglu’s theorem, there exists a generalized subsequence (subnet) that converges to the product measure .μo ≡ λo ×uo . It follows from our assumptions on f , that for each .φ ∈ F, .g(·, ·, ·) ∈ L1 (I, BC(E + × U )) and hence, by virtue of

6.1 Optimal Control of Deterministic Systems

139

the weak star convergence, we have  .

0

t

 λns (An (s)φ)ds = =

 t 0



t

μns (g)ds −→

0

E + ×U

t 0

μos (g)ds

g(s, ξ, ζ )μos (dξ × dζ )ds for each t ∈ I.

Thus, for every .φ ∈ F, the left-hand member of Eq. (6.8), or equivalently Eq. (6.9), is also well-defined for each .t ∈ I and has a limit. Upon letting .n → ∞ in Eq. (6.8) and using these facts including (6.6), (6.7) and Fubini’s theorem, we obtain  λot (φ) = μot (φ) = ν(φ) + = ν(φ) +

.

t

0

μos (g)ds

 t  0

 = ν(φ) +

0

 E+

t

U

g(s, ξ, v)uos (dv)

λos (dξ )

ds,

λos (Ao (s)φ)ds, t ∈ I,

where .(Ao (t)φ)(ξ ) ≡ A∗ Dφ(ξ ), ξ  + Dφ(ξ ), f (t, ξ, uot ). Hence, .λo ∈ (uo ), proving sequential closure of the graph . . This completes the proof.  Next, we consider some typical control problems and focus on the question of existence of optimal controls. Our first control problem is to find a control in .Uad that minimizes the cost functional given by Eq. (6.5) subject to the dynamics (6.4). Theorem 6.1.3 Suppose the assumptions of Corollary 6.1.1 hold and that .f0 : I × E × U −→ R0 = [0, ∞) ∪ {∞} is measurable in the first variable and continuous in the rest, and . : E −→ R0 is continuous. Then, there exists an optimal control minimizing the cost functional (6.5) subject to the dynamics (6.4). Proof The cost functional (6.5) can be rewritten in the following equivalent form  C(u, λ) =

.

 I ×E + ×U

f0 (t, ξ, ζ )ut (dζ )λt (dξ )dt +

E+

(ξ )λT (dξ ), (6.10)

with .(u, λ) ∈ . Since the solution .λ is weak star continuous on I , the second integral involving .λT is well-defined and it belongs to .rba (E + ) and hence, .C(u, λ) is well-defined on . giving the cost functional (6.5). If .C(u, λ) = ∞ for all .(u, λ) ∈

, there is nothing to prove. So, let .(un , λn ) ∈ be a minimizing (generalized) sequence such that .limn→∞ C(un , λn ) = m0 for some .m0 ∈ [0, ∞). Clearly, the functional C is linear in its arguments and weak star continuous. By Theorem 6.1.2, w (I,  + the graph . is a closed subset of .M∞ rba (E × U )) which is a weak star w + compact subset of .L∞ (I, Mrba (E × U )). Thus, there exists a pair .(uo , λo ) ∈

such that, along a generalized subsequence if necessary, the pair .(un , λn ) converges to the pair .(uo , λo ) in the .w ∗ topology and .C(un , λn ) −→ C(uo , λo ). Since the pair

140

6 Optimal Control of Evolution Equations

(uo , λo ) ∈ , it is clear that

.

m0 ≤ C(uo , λo ) = lim C(un , λn ) = m0 .

.

n→∞

Hence, C attains its minimum on . . Therefore, there exists a control that minimizes the cost functional J . This completes the proof.  A closely related control problem is stated as follows. The basic objective is to find an admissible control corresponding to which the measure-valued solutions have supports close to a given target set. Let .cb(E) denote the class of nonempty closed bounded subsets of E furnished with the metric topology determined by the Hausdorff metric .dH . Since E is a Banach space, .(cb(E), dH ) is a complete metric space. Let .χG denote the characteristic function of any set .G ∈ cb(E) and .Spt (μ) denote the support of any measure .μ. Let .K ∈ cb(E) be the target set. The problem is to find a control .u ∈ Uad that imparts a minimum to the functional 



T

J (u) ≡

.

U ×E +

0

f0 (t, ξ, ζ )ut (dζ )λut (dξ )dt + dH (Spt(λuT ), K), (6.11)

where .Spt(λuT ) denotes the closure of the support of the measure .λuT . Theorem 6.1.4 Consider the system given by Eq. (6.3) with the cost functional (6.11) and suppose .f0 satisfies the assumptions stated in Theorem 6.1.3. Then, there exists an optimal control. Proof Define the functional C on the graph . of the multi function . denoting the control to solution map w

: Uad −→ 2L∞ (I,rba (E

.

+ ))

\∅

as follows  C(u, λ) ≡

T



.

0

U ×E+

f0 (t, v, ξ )ut (dv)λt (dξ )dt + dH (Spt(λT ), K).

Clearly, this is the cost functional .J (u) corresponding to .λ ∈ (u). Define m0 ≡ inf{C(u, λ), .(u, λ) ∈ }. Since .f0 is nonnegative it is clear that .m0 ≥ 0. Let .{un , λn } ∈ be a minimizing sequence so that

.

.

lim C(un , λn ) = m0 .

n→∞

Since . is .w ∗ compact, there exists a generalized subsequence of the sequence n n ∗ topology to .(uo , λo ) ∈ .{(u , λ )}, relabeled as such, that converges in the .w

. Clearly, the first term of the cost functional is .w ∗ continuous as seen in Theorem 6.1.3. The second term is .w ∗ lower semicontinuous. This follows from

6.1 Optimal Control of Deterministic Systems

141

the facts that the solutions are .w ∗ continuous in .t ∈ I and hence their point-wise w∗

values on I are well-defined and we have .λnT −→ λoT in .rba (E + ). This implies that .dH (Spt(λnT ), Spt(λoT )) −→ 0 and hence dH (Spt(λoT ), K) ≤ lim inf dH (Spt(λnT ), K).

.

n→∞

Thus, the functional C is .w ∗ lower semicontinuous giving, C(uo , λo ) ≤ lim inf C(un , λn ).

.

n→∞

Since .(uo , λo ) ∈ , we have m0 ≤ C(uo , λo ) ≤ lim inf C(un , λn ) ≤ lim C(un , λn ) = m0 .

.

n→∞

n→∞

This proves the existence of an optimal control.



Time Optimal Control Let .K ∈ cb(E) denote the target set and   w ϒ ≡ λ ∈ M∞ (I, rba (E + )) : λ ∈ (u), for some u ∈ Uad

.

the set of all (attainable) measure-valued solutions of the control system (6.1), or equivalently (6.3), with the initial state .ν = L(x0 ) having support .Spt (ν) satisfying .Spt(ν) ∩ K = ∅. Time optimal control problems can be formulated in several ways [9]. Let .0 < α < 1 as small as desired and define τ (λ) ≡ inf {t ∈ I : λt (K) ≥ α} .

.

The statement, .λt (K) ≥ α, means that at time t, .Spt(λt ) has nonempty intersection with the target set .K having measure .α > 0. If the underlying set is empty, we set .τ (λ) = T +. And if it is so for all .λ ∈ ϒ, it signifies that the target is not reachable. On the other hand, if the target is reachable, the problem is to find a .λo ∈ ϒ such that τ (λo ) ≤ τ (λ) for all λ ∈ ϒ.

.

In other words, the cost functional is given by J (u) = τ (λ), λ ∈ (u), for u ∈ Uad ,

.

(6.12)

and the objective is to find a control that minimizes this functional. It is clear that if λt (K) < α for all .t ≥ 0, and for all .λ ∈ ϒ, there is nothing to prove. However, if we have a choice, we may relax (reduce) .α so that the reachable set is nonempty and

.

142

6 Optimal Control of Evolution Equations

we have a viable problem. Another possible formulation is the contact time, that is,   τ1 (λ) ≡ inf t ≥ 0 : Spt(λt ) ∩ K = ∅ .

.

For other formulations see [9]. We consider the first problem related to the cost functional (6.12). Theorem 6.1.5 Consider the system (6.3) with the cost functional (6.12) and suppose the basic assumptions of Corollary 6.1.1 hold. Then, there exists an optimal control solving the time optimal control problem. Proof Let .{λn } ∈ ϒ be a minimizing sequence and .τn ≡ τ (λn ) the corresponding (decreasing) sequence of hitting times. Since .λn ∈ ϒ, there exists a sequence n n n n n .u ∈ Uad such that .λ ∈ (u ) and, by definition, .(u , λ ) ∈ (the graph of the ∗ multifunction . ). Since . is .w compact there exists a (generalized) subsequence of the given sequence, relabeled as the original sequence, and a pair .(uo , λo ) ∈ such w∗

w∗

w (I,  n o w + that .un −→ uo in .M∞ rba (U )) and .λ −→ λ in .M∞ (I, IIrba (E )) and .τn ↓ ∗ n n + τ > 0. Note that .λτn (K) ≥ α for all .n ∈ N and .λτn ∈ rba (E ) ⊂ Mrba (E + ). Since, by Alaoglu’s theorem, .rba (E + ) is a .w ∗ compact subset of .Mrba (E + ), there exists a generalized subsequence of this sequence, again relabeled as the original w∗ sequence, and a .μo ∈ rba (E + ) such that .λnτn −→ μo . We show that .μo = λoτ ∗ . u Since the domain .D(A ) is invariant with respect to .u ∈ Uad , for any .ψ ∈ D(Au ) we can write

λoτ ∗ − μo , ψ = λoτ ∗ − λnτ∗ , ψ + λnτ∗ − λnτn , ψ + λnτn − μo , ψ. (6.13)

.

The first and the third term on the right-hand side of (6.13) can be made as small as desired by choosing n sufficiently large. Thus, for any .ε > 0, there exists a number .n0 (possibly dependent on .ε and .ψ) such that |λoτ ∗ − μo , ψ| ≤ (2ε)/3 + |λnτn − λnτ∗ , ψ| for all n ≥ n0 .

.

Writing the remaining term explicitly we have  n .λτ n

− λnτ∗ , ψ

=

τn

τ∗

n

λns (Au ψ)ds.

Since .ψ ∈ D(Au ), both .ψ and its Fréchet derivative have bounded supports in E. Further, by our basic assumption, f is continuous and bounded on bounded sets of + n n .E × U and the fact that .(u , λ ) ∈ for all n, there exists a function .h ∈ L (I ) 1 such that n

|λnt (Au ψ)| ≤ h(t) for almost all t ∈ I

.

6.1 Optimal Control of Deterministic Systems

143

for all .n ∈ N. Thus, we can choose an integer .n1 ≥ n0 sufficiently large such that |λnτn − λnτ∗ , ψ| < (ε/3) for all n > n1 .

.

This means that |λnτ∗ − μo , ψ| ≤ ε.

.

for all .n > n1 . Since .ψ ∈ D(Au ) and .ε > 0 are arbitrary, we conclude that o o .μ = λ ∗ . Further, since these are regular measures and .K is a closed set, we have τ o o o ∈ ϒ, and hence there exists a control .μ (K) = λ ∗ (K) ≥ α. The process .λ τ ∗ o ∗ .u ∈ Uad such that .λ ∈ (u ), and this control is optimal. This completes the proof.  Remark 6.1.6 This result also holds for time-varying targets. It is only required that for each .t ≥ 0, .K(t) ∈ cb(E) and that it is continuous in the Hausdorff metric. Example: Fluid Dynamics Recently, Brezina and Feireis [54] used parametrized Young measures to construct measure-valued solution for a complete Euler system of equations describing the flow dynamics of inviscid compressible fluid. Here we consider measure-valued solutions for Navier-Stokes equations with nonhomogeneous dynamic boundary conditions driven by controls. The concept of measure-valued solutions may be an appropriate tool for quantifying turbulence in hydrodynamic problems. Taking Navier-Stokes equation (NSE) [55] with boundary controls, for example, we develop an abstract semilinear controlled evolution equation on a Banach space. In fact, the results of Chap. 2 apply to a wider class of flow problems including magnetohydrodynamic (MHD) equations. Using Navier-Stokes equation as a reference, one can develop an abstract model which includes a broad class of fluid dynamic problems [119]. Thus, the results of Chap. 2 also apply to this broader class of problems including Navier-Stokes and MHD equations. For incompressible fluid with nonslip boundary conditions, the Navier-Stokes equation is given by .

∂v/∂t + (v · ∇)v − γ v + ∇p = g, ˜ t > 0, ξ ∈  ⊂ R d , div(v) = 0,

(6.14)

v(0, ξ ) = v0 (ξ ), ξ ∈ ; v|∂ = 0, where . ⊂ R d (d = 1, 2, 3) is an open bounded domain with smooth boundary, v denotes the velocity distribution, p the pressure, .g˜ the volume force, and .γ the kinematic viscosity. Other more general equations of fluid dynamics can be found in [119].

144

6 Optimal Control of Evolution Equations

This equation can be reformulated as an abstract evolution equation like (6.1). Define H 1 (,R n ) 

 v ∈ C0∞ (, R n ) : div(v) = 0 , .  L (,R n )  v ∈ C0∞ (, R n ) : div(v) = 0 . H ≡ closure 2 V ≡ closure

Clearly, .V ⊂ H01 and that V can be continuously embedded in H . Identifying H with its dual .H ∗ , one has the standard Gelfand triple .V → H → V ∗ , where .V ∗ is the dual of V . In fact, the injections are compact. In the study of weak solutions, the Gelfand triple is widely used [39, 100]. Here we are interested in the formulation of NSE as an evolution equation in the Hilbert space H . Let P denote the orthogonal projection of .L2 (, R n ) onto H . Define Av ≡ −P (v), b(v, w) ≡ P [(v · ∇)w], .

B(v) ≡ b(v, v), f˜ = P g, ˜

for v, .w ∈ V ∩ H 2 . The operator A is known as the Stokes operator. By application of the projection operator P on either side of Eq. (6.14) and noting that .P (∇p) = 0, we obtain an abstract differential equation in the Hilbert space H , dv/dt + γ Av + B(v) = f˜, t ∈ I, v(0) = v0 .

.

(6.15)

For nonhomogeneous Dirichlet boundary value problems, the homogeneous boundary condition in Eq. (6.14) is replaced by 𝒯(v) ≡ v|∂ = (1/γ )G(v, u),

.

(6.16)

with .𝒯 denoting the trace operator. The operator G contains state feedback as well as open loop controls. For technical reasons, G is assumed to be a continuous bounded map from .V × L∞ (∂, R n ) to .H 1/2 (∂, R n ) transforming the control forces into boundary data. Solving the following Dirichlet problem, φ = 0, in ,

.

𝒯(φ) = (1/γ )G(v, u), in ∂, we have .φ = (1/γ )RG(v, u), where the operator .R ≡ (𝒯|Ker )−1 is the Dirichlet map. Using this map, one can verify that the NSE with nonhomogeneous boundary

6.1 Optimal Control of Deterministic Systems

145

data is governed by the evolution equation dv/dt + γ Av = f (t, v, u), t > 0, v(0) = v0 ,

.

(6.17)

in H , where .f (t, x, ζ ) ≡ −B(x) + ARG(x, ζ ) + f˜(t). In fact, an abstract formulation of MHD equations as well as many other fluid dynamic equations have the same representation involving similar Gelfand triples [119]. Let .D0 ⊂ ∂ be an open set and .β > 0 and introduce the set   U ≡ w ∈ L∞ (∂, R n ) : w(ζ ) = 0 for ζ ∈ / D0 , and |w|L∞ (∂,R n ) ≤ β .

.

Clearly, this is a .w ∗ compact subset of .L∞ (∂, R n ). Since .L1 (∂, R n ) is separable, the .w ∗ topology on U is metrizable with a metric that turns U into a compact (separable) metric space. For the class of admissible controls, we choose w w .Uad ≡ M∞ (I, rba (U )) = M∞ (I, rca (U )). For dynamic equations on the space of measures, we consider .E ⊆ H to be a Banach space with continuous embedding. The generating operator .𝒜, corresponding to the controlled NSE, is given by (𝒜 u φ)(ξ ) = −A(γ )∗ Dφ(ξ ), ξ E ∗ ,E + Dφ(ξ ), f u (t, ξ )E ∗ ,E

.

for .φ ∈ D(𝒜 u ) ≡ D(𝒜), a constant domain invariant with respect to .u ∈ Uad , where .

A(γ ) ≡ γ A, and  f (t, x, ς )ut (dς ), u ∈ Uad . f u (t, x) ≡ U

Theorem 6.1.7 Let .α ∈ (3/4, 1) and .Eα ⊂ E denote the Hilbert space induced by the scalar product .(x, y)Eα ≡ (Aα x, Aα y)H . The function f is Borel measurable on .I × Eα × U with values in E and Bochner integrable in .t ∈ I . Further, it is a continuous map taking bounded subsets of .Eα ×U onto bounded subsets of E. Then, for each control .u ∈ Uad and .v0 ∈ Eα or .ℒ(v0 ) ≡ λ0 ∈ rba (Eα+ ), Eq. (6.17) has w (I,  + a .w ∗ continuous measure-valued solution .λu ∈ M∞ rba (Eα )) satisfying  λut (ϕ) = λ0 (ϕ) +

.

0

t

λus (Au ϕ)ds, t ∈ I, for each ϕ ∈ D(𝒜).

(6.18)

Proof The proof is entirely similar to that of Theorem 2.2.1. See also [10, Theorem  2A]. Remark 6.1.8 In many practical applications of NSE and MHD equation, the initial state (velocity) is either unknown or difficult to estimate. In this situation it may be chosen as a suitable measure possibly with bounded support.

146

6 Optimal Control of Evolution Equations

We can apply these results to control problems related to hydrodynamics and magnetohydrodynamics. The admissible controls .Uad may be taken as stated above. Similar control problems, as considered above the section, Example: Fluid Dynamics, can be solved by the same technique requiring only changes in the state space and admissible controls as specified in this section for NSE and MHD problems [14]. We will return to NSE once again as we consider their stochastic versions. We end this section by presenting the necessary conditions of optimality for the boundary control problem (as described above) with the cost functional  J (u) =



.

I ×Eα

(t, ξ, ut )λut (dξ )dt +

Eα

(ξ )λuT (dξ ),

(6.19)

where .(t, ξ, ς ) ≥ 0 is Borel measurable in all the variables, and continuous and bounded on bounded sets of .Eα × U , and . ≥ 0 is continuous and bounded on bounded sets of .Eα . Given the system (6.18), the problem is to find a control that minimizes the functional (6.19). For simplicity we use the notations  .

 =  (t, ξ ) = (t, ξ, ut ) = u

(t, ξ, ς )ut (dς ), (t, ξ ) ∈ I × Eα ,

u

U



F u (ψ) = F (t, ξ, ut ), Dψ(ξ )Eα ,Eα∗ =

 U

 F (t, ξ, ς ), Dψ(ξ )Eα ,Eα∗ ut (dς ),

where .Dψ always denotes the first Frèchet derivative of .ψ. Theorem 6.1.9 Consider the system (6.18) with the cost functional (6.19) satisfying the assumptions as stated above, and suppose the assumptions of Theorem 6.1.7 w (I,  + hold. Then, for any control .uo ∈ Uad , with .λo ∈ M∞ rba (Eα )) being the corresponding solution of Eq. (6.18), to be optimal, it is necessary that there exists a + .ψ ∈ L1 (I, BC(Eα )) satisfying the following inequality and the evolution equations  .

I ×Eα

u−uo  o  + F u−u (ψ) λot (dξ )dt ≥ 0, ∀ u ∈ Uad , . o

o

−dψ = Au ψdt + u dt, t ∈ I, ψ(T ) = , .

o ∗ dλot = Au λot dt, t ∈ I, λo0 = λ0 .

(6.20) (6.21) (6.22)

Proof We present a brief outline. Let .uo ∈ Uad be the optimal control and .uε = uo + ε(u − uo ) for .u ∈ Uad and .ε ∈ [0, 1]. Since .Uad is a closed convex subset of w ε .L∞ (I, Mrba (U )), we have .u ∈ Uad . Computing (1/ε)(J (uε ) − J (uo ))

.

6.1 Optimal Control of Deterministic Systems

147

and letting .ε ↓ 0, one can verify, with a slight abuse of notation, that  dJ (u0 , u − uo ) =

.

 where L(m) =

o

I ×Eα

u−u (t, ξ )λot (dξ )dt + L(m) ≥ 0, ∀ u ∈ Uad , 

uo

I ×Eα

 (t, ξ )mt (dξ )dt +

(ξ )mT (dξ ) and, for t ∈ I, Eα

mt (φ) ≡ lim (1/ε)(λεt (φ) − λot (φ)) for each φ ∈ BC(Eα ).

(6.23)

ε→0

Carrying out some straightforward computations one can verify that the (signed) measure-valued function m satisfies the following evolution equation in the weak sense, o

o

dmt (φ) = mt (Au φ)dt + λot (F u−u φ)dt, t ∈ I, m0 (φ) = 0, ∀ φ ∈ BC(Eα ).

.

(6.24) It is clear from the expression above Eq. (6.23) that the map .m −→ L(m) is a + continuous linear functional on .Lw ∞ (I, Mrba (Eα )) and it follows from Eq. (6.24) o ∗ o u−u + that .(F ) λ −→ m is a continuous linear operator from .Lw ∞ (I, Mrba (Eα )) to itself. Thus, for every .u ∈ Uad , the composition map u−u ∗ o ˜ (F u−u )∗ λo −→ m −→ L(m) ≡ L((F ) λ ) o

o

(6.25)

.

+ is a continuous linear functional on .Lw ∞ (I, Mrba (Eα )). Hence, there exists a .ψ in w + ∗ its topological dual .(L∞ (I, Mrba (Eα ))) such that o u−uo ∗ o ˜ .L((F ) λ ) = F u−u (ψ), λo  =



o

I ×Eα

F u−u (ψ) λot (dξ )dt.

Under the canonical embedding of any Banach space in its bidual, we have + ∗ L1 (I, BC(Eα+ )) ⊂ (Lw ∞ (I, Mrba (Eα ))) . Thus, the above expression is also well+ defined for any .ψ ∈ L1 (I, BC(Eα )). In fact, it turns out, as seen below, that + .ψ ∈ L1 (I, BC(Eα )). Substituting this expression in the first inequality appearing in Eq. (6.23), we obtain the necessary condition given by the expression (6.20). To prove that .ψ satisfies Eq. (6.21), we use the variational Eq. (6.24) with .ψ in place of .φ and substitute in the above expression and integrate by parts to obtain .

 .

I ×Eα

u−uo F ψ)λot (dξ )dt = mT (ψ(T , ·)) −



T 0

o

mt (dψ + Au ψ)dt.

(6.26)

148

6 Optimal Control of Evolution Equations o

o

Thus, by setting .dψ + Au ψ = −u and .ψ(T , ξ ) = (ξ ), it follows from the above expression that 



.

I ×Eα



o

F u−u ψ)λot (dξ )dt = mT () +

T

o

mt (u )dt.

(6.27)

0

Clearly, the above identity coincides with the equality (6.25) as it must. This proves the necessary condition given by Eq. (6.21). It follows from [65, Theorem 1.1, p. 171] that Eq. (6.21) has a unique solution in .C(I, BC(Eα+ )). Equation (6.22) is the original system Eq. (6.18) written in differential form corresponding to the optimal control and so nothing to prove. This completes the proof.  For proof of existence of optimal controls and several interesting applications, interested reader may see [10]. We do not present them here to avoid duplication with similar results appearing in other chapters. If f and .{, } satisfy standard regularity conditions, then we have pathwise solutions. In that case, the measure solution has Dirac structure along the trajectory, that is, .λot (dξ ) = δx o (t) (dξ ), for all .t ∈ I for some .x o ∈ C(I, Eα ). In the following corollary, we show that the necessary conditions of optimality given by Theorem 6.1.9 reduce to classical necessary conditions of optimality from which one can easily derive the classical minimum principle similar to that of Pontryagin. Corollary 6.1.10 Consider the system (6.17) with .−A generating an analytic semigroup on H and suppose that both f and . are continuously Gâteaux differentiable in the state variable in the direction of the dense subspace .D(A) ⊂ Eα . Let o ∈ U o .u ad be the optimal control and .x ∈ C(I, Eα ) the corresponding (mild) solution of Eq. (6.17) starting from the state .x0 ∈ Eα , and .ψ(t, ξ ), .(t, ξ ) ∈ I × Eα , is the solution of Eq. (6.21) corresponding to the optimal control .uo . Then, the necessary conditions of optimality given by Theorem 6.1.9 reduce to the following set of inequality and evolution equations,  .

I



 u (t, x o (t)) + F u (t, x o (t)), ϕ(t)Eα ,Eα∗ dt ≥

 I

o o u (t, x o (t)) + F u (t, x o (t)), ϕ(t)Eα ,Eα∗ dt, ∀ u ∈ Uad , (6.28)

− (d/dt)ϕ = −νA∗ ϕ + (Fxu (t, x o ))∗ ϕ + ux (t, x o ), t ∈ I, o

.

o

ϕ(T ) = x (x o (T )),

(6.29) o

(d/dt)x o = −νAx o + F u (t, x o ), t ∈ I, x o (0) = x0 ,

.

(6.30)

6.1 Optimal Control of Deterministic Systems

149

where 

o

.

F u (t, x o (t)) = f (t, x o (t), uot ) = uo

Fx (t, x (t)) = Dx f (t, x o

o

(t), uot )

U

f (t, x o (t), ς )uot (dς ), 

= U



o

ux (t, x o (t)) =

U

Dx f (t, x o (t), ς )uot (dς ),

Dx (t, x o (t), ς )uot (dς ).

Proof We present a brief outline. Existence of pathwise solution .x o means that the measure solution is a Dirac measure concentrated along the path, .{x o (t), t ∈ I }, giving .λot (dξ ) = δx o (t) (dξ ), t ∈ I , with initial state given by .λ0 = δx0 (dξ ). Using this in Eq. (6.22) one can readily obtain the state Eq. (6.30). For Eq. (6.29), we make use of Eq. (6.21). Computing the total time derivative of .ψ(t, x o (t) + εξ ) and .ψ(t, x o (t)) for any .ξ ∈ D(A) and .ε > 0, and subtracting the latter from the former and dividing by .ε > 0, and letting .ε ↓ 0, we obtain the adjoint Eq. (6.29). In the process one finds that .ϕ(t) = Dx ψ(t, x o (t)), t ∈ I and .ϕ(T ) = x (x o (T )). Using the measure-valued function .λot = δx o (t) , t ∈ I and .ϕ as given above, in the expression (6.20), we obtain the necessary condition (6.28). It is interesting to note that the optimal cost is given by the familiar expression  ψ(0, x0 ) = (x o (T )) +

.

T

o

u (t, x o (t))dt.

0

This completes the outline of our proof.



For more details on application of Navier-Stokes equations to artificial heart based on classical theory, the reader may see [6, 10]. Application to Artificial Heart Pump The boundary control problem considered here can be used to operate artificial hearts. Consider an inflatable-contractible device (like a balloon) housed in the interior of an artificial heart chamber and attached to the boundary port open to a hydraulic pump. The chamber is designed with a pair of inlet and outlet ports. By inflating the flexible device, blood is pumped out through the outlet port transporting oxygen-rich blood to the entire body through the network of arteries. By contracting the balloon, blood is returned through veins to the lungs and back to the heart chamber through the inlet port, thereby completing one cycle. Given the geometric shape of the heart chamber, it is the control strategy that determines the efficacy of the system. The necessary conditions of optimality given above can be used to find optimal control strategy. For numerical results see [118].

150

6 Optimal Control of Evolution Equations

6.2 Optimal Control of Impulsive Systems Let E and . be a pair of real Banach spaces, and .f : I × E ×  −→ E be a Borel measurable map. Throughout the rest of this section, the vector field f is assumed to satisfy the following regularity condition: (a): The vector field .f : I × E ×  −→ E is Borel measurable in the first two arguments and continuous in the third. Here we consider several control problems including state feedback controls. Consider the control system dx(t) = Ax(t)dt + f (t, x(t), u)dt + g(t, x(t−))ν(dt), x(0) = x0 , t ∈ I,

.

(6.31) with .ν being a signed measure and the control .u ∈ Uad , where .Uad is the class of admissible controls as described below. Admissible Feedback Controls Let . be a separable Banach space and .BM(I × E, ) the vector space of bounded Borel measurable functions defined on .I × E and taking values from the Banach space .. Furnished with the sup norm topology, this is a Banach space. We give this space a weaker topology. Let .τwub denote the topology of weak convergence in ., uniformly on bounded subsets of .I × E. That is, a sequence .{un } ⊂ BM(I × E, ) is .τwub convergent to .uo ∈ BM(I × E, ), τwu written as .un −→ uo , if and only if, for every .η ∈ BM(I × E, ∗ ) having bounded support in .I × E, .

lim (η(t, ξ ), un (t, ξ ) − uo (t, ξ ))∗ , = 0

n→∞

uniformly on .I × E. Let .U be a weakly compact subset of . and .U : I × E −→ cc(U), a Borel measurable multi function with nonempty closed convex values. For admissible controls we choose the set .Uad given by the family of Borel measurable selections of the multi function U , Uad ≡ {u ∈ BM(I × E, ) : u(t, x) ∈ U (t, x), ∀ (t, x) ∈ I × E} .

.

Note that by virtue of Kuratowski-Ryll Nardzewski selection theorem (Theorem 1.3.3 and [90, Theorem 2.1, p. 154]), this is a nonempty set. We consider .Uad endowed with the relative .τwub topology. Let .λu denote the measure solution of Eq. (6.31) corresponding to the control .u ∈ Uad . The first problem we consider is stated as follows. Problem (P1) The problem is to find a control from the set .Uad that maximizes the probability of closely following a moving target, . (t), t ∈ I . In other words, the

6.2 Optimal Control of Impulsive Systems

151

objective is to find .u ∈ Uad that maximizes the functional  J (u) =

.

I

(λut ( (t)))dt,

(6.32)

where . is any nonnegative non-decreasing real valued function and .λu is a measure-valued solution of the evolution Eq. (6.31) corresponding to control .u ∈ Uad . Define the multi function F (t, x) ≡ f (t, x, U (t, x)), (t, x) ∈ I × E.

.

Since f satisfies the assumption (a) and U is a measurable multi function, the composition map F is a measurable multi function [47, Theorem 8.2.8, p. 314]. Clearly, the system (6.31) can be considered as a particular realization of the differential inclusion dx ∈ Axdt + F (t, x)dt + g(t, x−)ν(dt), x(0) = x0 , t ∈ I,

.

(6.33)

through a measurable selection f of the multi function F . Thus, the corresponding family of evolution equations on the space of measures .Mba (E) (the Banach space of finitely additive measures on the field of subsets of the set E) can be written as follows dλt = A∗f (t)λt dt + C∗g (t)λt− ν(dt), π0 = δx0 , t ∈ I, f ∈ SF ,

.

(6.34)

where .SF denotes the set of all measurable selections of the multi function F , and the generating operators .Af and .Cg are given by (Af φ)(t, ξ ) = A∗ Dφ(ξ ), ξ  + Dφ(ξ ), f (t, ξ )E ∗ ,E ,

.

and 

1

(Cg φ)(t, ξ ) =

.

Dφ(ξ + θg(t, ξ )ν({t})), g(t, ξ )E ∗ ,E dθ.

0

We are now prepared to consider the control Problem (P1). Theorem 6.2.1 Consider the system (6.34), related to the differential inclusion (6.33), and suppose .{A, F, g, ν} satisfy the assumptions of Theorem 3.5.4 and let .L(x0 ) = π0 ∈ ba (E) denote the measure corresponding to the initial state. Suppose the multi function .t → (t) with values in .cb(E) is continuous in the Hausdorff metric, and . is an upper semicontinuous real valued nonnegative, monotone increasing function on R. Then, there exists an optimal control maximizing the functional (6.32) solving the Problem (P1).

152

6 Optimal Control of Evolution Equations

Proof First note that the functional (6.32) defined on .Uad is equivalent to the following functional J˜(λ) ≡

 (6.35)

(λt ( (t)))dt

.

I w (I,  (E + )) ∩ M w (|ν|, defined on the attainable set . (F, g, π0 ) ⊂ M∞ ba ∞ + )) ∩ Lw (|ν|, M (E + )) representing the set of ba (E + )) ⊂ Lw (I, M (E ba ba ∞ ∞ solutions of the differential inclusion (6.34). Let .{λn } ⊂ (F, g, π0 ) be a maximizing sequence. Since the set . (F, g, π0 ) is .w ∗ sequentially compact, there exists a subsequence relabeled as the original sequence, and a .λo ∈ (F, g, π0 ) w∗

such that .λn −→ λo . By our assumption, the measure .ν has bounded variation on I , and hence it may have at most a countable set of atoms. Thus, both .λn and .λo are .w ∗ continuous a.e. on I (except at a countable set of points in I ) and since . (t) ∈ cb(E), t ∈ I (closed set), .

lim λnt ( (t)) ≤ λot ( (t)) a.e. on I.

Thus, by monotone increasing property and upper semicontinuity of the function ., we have .

lim (λnt ( (t))) ≤ (λot ( (t))) a.e. t ∈ I.

n→∞

Integrating this over the interval I , it follows from Fatou’s lemma that 

 .

lim

n→∞ I

(λnt ( (t)))dt ≤

 lim (λnt ( (t)))dt ≤

I n→∞

I

(λot ( (t)))dt.

Thus, the functional .J˜ is .w ∗ upper semicontinuous on the set . (F, g, π0 ). This set is ∗ o .w compact and hence .J˜ attains its maximum on . (F, g, π0 ). Let .λ ∈ (F, g, π0 ) denote the maximiser. Letting .SF denote the Borel measurable selections of the multi function F , we conclude that there exists an .f o ∈ SF so that .λo = λ(f o , g). Clearly, the multi function .H given by   H(t, x) ≡ v ∈ U (t, x) : f o (t, x) = f (t, x, v), (t, x) ∈ I × E ,

.

is nonempty and it is also measurable since .f : I × E ×  −→ E satisfies the assumption (a) and the multi function U is assumed to be measurable with values in .cc(U). Thus, the graph of the multi function .H given by   Gr (H) ≡ (t, ξ, v) ∈ Gr (U ) : f o (t, ξ ) = f (t, ξ, v)

.

is an element of .BI ×E × B . Then, it follows from Yankov-Von Neumann-Aumann selection theorem (Theorem 1.3.4 and [90, Theorem 2.14, p. 158]) that there exists

6.2 Optimal Control of Impulsive Systems

153

a measurable selection .uo of .H such that .uo (t, x) ∈ H(t, x) for all .(t, x) ∈ I × E. Hence, .f o (t, x) = f (t, x, uo (t, x)) for all .(t, x) ∈ I × E. Thus, we have proved that .J (uo ) = J˜(λo ) and hence .uo is the optimal feedback control. This completes  the proof. Another interesting problem, similar to Problem (P1), arises in applications requiring obstacle avoidance strategies. Problem (P2) Let .b(E) denote the class of nonempty bounded subsets of E and D(t), t ∈ I , a measurable multi function with values in .b(E). This is considered a danger zone changing with time. The concern is to stay away from this region, as far as possible. The problem can be formulated as follows. Let .d(ξ, D(t)) denote the distance of .ξ from the set .D(t) given by

.

d(ξ, D(t)) ≡ inf |ξ − γ |E .

.

γ ∈D(t)

For any positive number r, and .t ∈ I , define the multi function Qr (t) ≡ {ξ ∈ E : d(ξ, D(t)) ≥ r} ,

.

and, for any Borel measurable set .K ⊂ E, let .χK denote the characteristic function of the set K. The problem is to find a control that maximizes the functional  J (u) ≡



.

I ×E

χQr (t) (ξ )λut (dξ )dt =

I

λut (Qr (t))dt.

(6.36)

Maximizing this functional is equivalent to reducing the risk of hitting the danger zone. In this regard we have the following result. Theorem 6.2.2 Suppose .{A, F, g, ν} satisfy the assumptions of Theorem 6.2.1. Let t −→ D(t) be a nonempty open bounded measurable set-valued function. Then, there exists an optimal control .uo ∈ Uad that maximizes the functional (6.36).

.

Proof Measurability of the multi function D implies measurability of the function t −→ d(ξ, D(t)). This in turn implies that .Qr is also a measurable multi function. Thus, the function .ψr (t, ξ ) ≡ χQr (t) (ξ ) is a Borel measurable function and it belongs to .L1 (I, B(E + )). Hence, the functional C, defined by

.

 C(λ) ≡

.

I ×E +

ψr (t, ξ )λt (dξ )dt,

w (I,  (E + )) ⊂ is a .w ∗ upper semicontinuous bounded linear functional on .M∞ ba + )). This is precisely the cost functional given by the expresLw (I, M (E ba ∞ w (I,  (E + )). Since sion (6.36) with .λu replaced by .λ ∈ (F, g, π0 ) ⊂ M∞ ba ∗ compact subset of .M w (I,  (E + )), the functional C as . (F, g, π0 ) is a .w ba ∞ defined above attains its maximum at some point .λo ∈ (F, g, π0 ). The rest of

154

6 Optimal Control of Evolution Equations

the proof is based on measurable selection theorem similar to that in Theorem 6.2.1. This completes the proof.  Next we consider the Bolza problem (containing both running cost and terminal cost). Problem (P3) Consider the following system with state feedback control u dx = Axdt + f (t, x)dt + K(t, x)u(t, x)dt + g(t, x)ν(dt), t ∈ I, x(0) = x0 ,

.

(6.37) with the cost functional given by 

T

J (u) =

.

{0 (t, x(t)) + ρ(t, x(t))|u(t, x(t))| } dt + (x(T )),

(6.38)

0

where .{A, f, g, ν} are as stated before, .K ∈ BM(I × E, L(, E)) and the control .u ∈ BM(I × E, ). The objective is to find a control u that minimizes the functional (6.38). Clearly, this is how the original problem would have been stated under standard assumptions on the vector fields .{f, g}. However, since we admit measurable vector fields, Eq. (6.37) may not possess any E valued pathwise solutions. So we consider, instead, the corresponding system on the space of measures. This is given by dλt = A∗ (t)λt dt + B∗u (t)λt dt + C∗ (t)λt− ν(dt), λ0 = π0 , t ∈ I,

.

(6.39)

corresponding to control u. Here the operators .A, .Bu and .C are given by (Aφ)(t, ξ ) = A∗ Dφ(ξ ), ξ  + Dφ(ξ ), f (t, ξ )E ∗ ,E ,

.

(Bu ϕ)(t, ξ ) ≡ K ∗ (t, ξ )Dϕ(ξ ), u(t, ξ )∗ , ,

.



1

(Cφ)(t, ξ ) =

.

Dφ(ξ + θg(t, ξ )ν({t})), g(t, ξ )E ∗ ,E dθ, for φ ∈ D(A),

0

respectively. We choose the following weighted cost functional in terms of measurevalued solutions, Thus, again, the appropriate formulation of this problem is to find an admissible control that minimizes the functional  .J (u) ≡ wε (ξ ) {0 (t, ξ ) + ρ(t, ξ )|u(t, ξ )| } λut (dξ )dt + λuT (wε ), I ×E

(6.40)

6.2 Optimal Control of Impulsive Systems

155

  where .wε (ξ ) ≡ exp −ε|ξ |2E , for .ε > 0, and .λu is the weak solution of the evolution Eq. (6.39) corresponding to control .u ∈ Uad , endowed with the .τwub topology as described above. We prove the following result. Theorem 6.2.3 Consider the system (6.39) with the cost functional (6.40) and suppose that the admissible set of controls .Uad is a .τwub compact subset of ∗ .BM(I × E, ), and the dual . is a uniformly convex Banach space. Suppose .{A, f, g, ν} satisfy the assumptions of Theorem 3.5.4 with .T ∈ / a(ν) and .K : I × E −→ L(, E) is a bounded Borel measurable map. The cost integrand .0 is a nonnegative Borel measurable function on .I × E, integrable on I for each .ξ ∈ E having at most polynomial growth on E. The function ., determining the terminal cost, is a nonnegative Borel measurable function having at most polynomial growth on E. The function .ρ is a nonnegative function on E having bounded support. Then, there exists an optimal control for the Problem (P3). Proof Consider the cost functional .J (u) given by (6.40). We prove that J is .τwub τwub lower semicontinuous on .Uad . Let .un −→ uo . Then, by virtue of the assumption on K, it can be shown that the measure-valued solution .{λn }, corresponding to the sequence of controls .{un } of the Cauchy problem (6.39), has a .w ∗ convergent w (I,  (E + )) which is the unique weak subsequence having the limit .λo ∈ M∞ ba (or measure-valued) solution of (6.39) corresponding to the control .u◦ . Clearly, by definition, the corresponding value of the cost functional is given by  J (uo ) ≡

.

I ×E

  wε 0 (t, ξ ) + ρ(t, ξ )|uo (t, ξ )| λot (dξ )dt + λoT (wε ). (6.41)

Let .β :  −→ ∗ denote the normalized duality map, that is, for each .z ∈ ,   β(z) ≡ z∗ ∈ ∗ : (z∗ , z) = |z| .

.

By Hahn-Banach theorem this set is nonempty. Since .∗ is uniformly convex, the duality map is single valued and uniformly continuous. Hence, we can write |uo (t, ξ )| = (β(uo (t, ξ )), uo (t, ξ )) ≡ (ηo (t, ξ ), uo (t, ξ )),

.

where we have defined .ηo ≡ β(uo ). Since .β is continuous and .uo is a Borel measurable . valued function, .ηo (t, ξ ) is a bounded .∗ valued Borel measurable function on .I × E satisfying .|ηo (t, ξ )|∗ = 1 for all .(t, ξ ) ∈ I × E. Using the above expression in Eq. (6.41) and carrying out some simple algebraic operations, it is easy to verify that for all .n ∈ N, J (uo ) ≤ |I1,n | + |I2,n | + J (un ),

.

(6.42)

156

6 Optimal Control of Evolution Equations

where  .

I1,n ≡

I ×E

wε (ξ )(0 (t, ξ ) + ρ(t, ξ )|uo (t, ξ )| )(λot (dξ ) − λnt (dξ ))dt 

+ λoT (wε ) − λnT (wε ) ,

 I2,n ≡

I ×E

wε (ξ ){ρ(t, ξ )ηo (t, ξ ), (uo (t, ξ ) − un (t, ξ ))}λnt (dξ )dt.

w (I,  (E + )) ∩ M w (|ν|,  (E + )) in the .w ∗ Since .λn converges to .λo ∈ M∞ ba ba ∞ topology of + w + Lw ∞ (, Mba (E )) ∩ L∞ (|ν|, Mba (E )),

.

and the integrand, .wε (o + ρ|uo |) ∈ L1 (I, B(E + )) ⊕ L1 (|ν|, B(E + )), the first component of .I1,n converges to zero as .n −→ ∞. By assumption .T ∈ / a(ν) (atoms w∗ o n n o + of .ν), and thus .{λT , λT } ∈ ba (E ) and .λT −→ λT . Since .wε  ∈ B(E + ) we conclude that the last term of .I1,n also converges to zero as .n → ∞. Since .wε ρ is uniformly bounded on E for almost all .t ∈ I and the set of admissible controls .Uad is also uniformly bounded, it is clear that the integrand of the term n τwub o .I2,n is contained in .L1 (I, B(E)) ⊕ L1 (|ν|, B(E)) and as .u −→ u , it follows from Lebesgue dominated convergence theorem that the integrand converges strongly in n o ∗ .L1 (I, B(E)) ⊕ L1 (|ν|, B(E)) while .λ converges to .λ in the .w topology of its dual. Hence, .limn→∞ I2,n = 0. Thus, it follows from the inequality (6.42) that J (uo ) ≤ lim inf J (un ),

.

n→∞

proving that J is .τwub lower semicontinuous on .Uad . Since .Uad is .τwub compact, J attains its minimum on .Uad . This proves the existence of an optimal control.  Remark 6.2.4 For the proof of existence of optimal controls it is not essential to assume uniform convexity of the Banach space .∗ . All that is required is the existence of a Borel measurable selection .ηo (t, ξ ) ∈ β(uo (t, ξ )) for .(t, ξ ) ∈ I × E. ∗ It is easy to verify that the multi function .β :  −→ 2B1 ( ) \∅ is monotone with values from the class of nonempty .w ∗ compact convex subsets of the unit ball .B1 (∗ ). If . is separable then the unit ball .B1 (∗ ) is metrizable and hence, with respect to the metric topology it is a Polish space. If E is separable then .I × E is a separable Borel measurable space. Hence, by virtue of KuratowskiRyll Nardzewski selection theorem (Theorem 1.3.3), the multi function .I × E  (t, ξ ) −→ β(uo (t, ξ )) has measurable selections. Control of Uncertain Systems Consider a system that is intermittently hit by impulsive forces of uncertain intensities. The upper bound of the intensities is known and can be characterized by a multi function G. Let us consider the control

6.2 Optimal Control of Impulsive Systems

157

system governed by the following differential inclusion, dx ∈ Axdt + f (t, x, u)dt + G(t, x(t−))ν(dt), x(0) = x0 ,

.

(6.43)

where G is assumed to satisfy the following property. The multi function .G : I × E −→ kc(E), with .kc(E) denoting the class of nonempty compact convex subsets of E. Our problem is to control this system in the presence of uncertainty. Adopting the pessimistic point of view one tries to minimize the maximum risk (or loss). Thus, the optimal control problem may be stated as follows. Problem (P4) Let .L : I × E ×  −→ R and . : E −→ R be Borel measurable maps satisfying certain properties stated later. The problem is to find a control .uo that minimizes the maximum loss determined by the functional 

 L(t, x(t), u)dt + (x(T )) : x ∈ 𝒳(u) ,

J0 (u) = sup

.

(6.44)

I

where .𝒳(u) denotes the family of solutions of the differential inclusion (6.43) for a given control policy u and initial state .x0 . The supremum is taken over the set .𝒳(u). This is how the problem would be stated under standard assumptions on the vector fields. Due to the presence of measurable vector fields (nonstandard assumptions), the corresponding formulation on the space of measures is given as follows. Suppose the distribution of the initial state is known, given by .L(x0 ) = π0 . For the contingent function F we take F (t, ξ ) ≡ f (t, ξ, U (t, ξ )), (t, ξ ) ∈ I × E.

.

Accordingly, we reformulate the control problem as follows. The system is governed by the evolution inclusion .

dλt ∈ A∗F (t)λt dt + C∗G (t)λt− ν(dt), λ0 = π0 , t ∈ I,

(6.45)

on the Banach space .Mba (E), where .AF ≡ {Af , f ∈ SF }, .CG ≡ {Cg , g ∈ SG } with .SF and .SG denoting the class of Borel measurable selections of the multi functions F and G, respectively. The objective functional (6.44) is replaced by the following functional  .

J0 (u) = sup

I ×E

 L(t, ξ, u(t, ξ ))λt (dξ )dt +

(ξ )λT (dξ ) ≡ C(u, λ), E

 λ ∈ (f u , G, π0 ) ,

(6.46)

where . (f u , G, π0 ) denotes the set of measure solutions of the system (6.45) corresponding to an admissible control u, giving .f u ≡ f (·, ·, u(·, ·)) a measurable selection of F . Our objective is to prove the existence of an optimal control minimizing the functional .J0 (u).

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6 Optimal Control of Evolution Equations

Theorem 6.2.5 Suppose .{A, F, G, π0 } satisfy the assumptions of Theorem 3.5.4, the cost integrand .L : I × E ×  −→ R is Borel measurable and bounded on bounded sets, and the map v −→ (L(t, ξ, v), f (t, ξ, v))

.

is continuous from . to .R × E for every .(t, ξ ) ∈ I × E, and that the set-valued function Q(t, ξ ) ≡ {(z, η) ∈ R × E : z ≥ L(t, ξ, v), η = f (t, ξ, v), v ∈ U (t, ξ )}

.

defined on .I × E is closed and convex valued. Further, suppose there exists an h ∈ L1 (I, B(E)) such that .L(t, ξ, v) ≥ h(t, ξ ) for all .v ∈ U (t, ξ ) and .(ξ ) ≥ 0, .ξ ∈ E and .T ∈ / a(ν). Then, there exists a control .uo ∈ Uad such that .J0 (uo ) ≤ J0 (u), .u ∈ Uad . .

Proof By Theorem 3.5.4, for every .u ∈ Uad , the set . (f u , G, π0 ) ⊂ w (I,  ∗ ∗ M∞ rba (E)) is .w compact and .λ −→ C(u, λ) is .w continuous. Hence, for a fixed but arbitrary .u ∈ Uad , the functional .λ −→ C(u, λ), given by the expression within the parenthesis of Eq. (6.46), attains its supremum on . (f u , G, π0 ). Thus, the functional .J0 (u) is well-defined possibly taking values from the extended real line. Since .h ∈ L1 (I, B(E)), and . is nonnegative, we have .J0 (u) > −∞ for all .u ∈ Uad . If .J0 (u) ≡ +∞ for all .u ∈ Uad , there is nothing to prove. So we may assume the contrary. In this case there exists an .m ∈ (−∞, +∞) such that .

inf {J0 (u), u ∈ Uad } = m.

We prove the existence of at least one control .uo ∈ Uad at which .J0 (uo ) = m. Let n .{u } ⊂ Uad be a minimizing sequence such that .

lim J0 (un ) = m.

n→∞

Corresponding to the sequence .{un } ∈ Uad , define the sequence .{n , f n } by n n n n .{ (t, ξ ) ≡ L(t, ξ, u (t, ξ ))} and .{f (t, ξ ) ≡ f (t, ξ, u (t, ξ ))}. Since the set n ∗ n ∗ . (f , G, π0 ) is .w compact, and .λ −→ C(u , λ) is .w continuous, there exists a .λn ∈ (f n , G, π0 ) at which the supremum is attained for every .n ∈ N. Thus, by construction we have   n n n .J0 (u ) =  (t, ξ )λt (dξ )dt + (ξ )λnT (dξ ) I ×E

E

and (n (t, ξ ), f n (t, ξ )) ∈ Q(t, ξ ), (t, ξ ) ∈ I × E,

.

6.2 Optimal Control of Impulsive Systems

159

ˆ ξ) ≡ for all .n ∈ N. Since L is bounded on bounded sets, the function .(t, sup{L(t, ξ, v), v ∈ U (t, ξ )} is a well-defined Borel measurable function on .I × E. ˆ given Thus, without any change of the original problem we can substitute Q by .Q by ˆ ξ ) ≥ z ≥ L(t, ξ, v), ˆ ξ ) ≡ (z, η) ∈ R × E : (t, Q(t,

η = f (t, ξ, v), v ∈ U (t, ξ ) .

.

By our assumption the multi function F is .wkc(E) valued and hence it follows ˆ is also .wkc(R × E) valued. from the above expression that the multi function .Q ∗ By Theorem 3.5.4, . (F, G, π0 ) is .w sequentially compact. Thus, there exists a generalized subsequence of the sequence .{λn } and a corresponding subsequence of the sequence .{n , f n }, relabeled as the original sequence, and .λo ∈ (F, G, π0 ) and Borel measurable functions .{o , f o } defined on .I × E taking values from .R × E such that w∗

w λn −→ λo in M∞ (I, ba (E))

.

and w

(n (t, ξ ), f n (t, ξ )) −→ (o (t, ξ ), f o (t, ξ )) in R × E.

.

In fact, this convergence is also uniform on compact subsets of .I × E. Since .Q(t, ξ ) is closed convex valued we have (◦ (t, ξ ), f o (t, ξ )) ∈ Q(t, ξ ) for all (t, ξ ) ∈ I × E.

.

Here we have used the well-known Mazur’s theorem which states that a convex set in a locally convex topological vector space is weakly closed if, and only if, it is strongly closed. Define the set-valued function B on .I × E with values   B(t, ξ ) ≡ v ∈ U (t, ξ ) : o (t, ξ ) ≥ L(t, ξ, v), f o (t, ξ ) = f (t, ξ, v) .

.

By virtue of measurability of the defining functions, B is a nonempty measurable multi function. Further, for any fixed .(t, ξ ) ∈ I × E, it follows from continuity of the map .v −→ (L(t, ξ, v), f (t, ξ, v)) from . to .R × E, that .B(t, ξ ) has closed values in .U ⊂ . Thus, again by Kuratowski-Ryll Nardzewski selection theorem (Theorem 1.3.3) there exists a measurable selection .uo of B such that o (t, ξ ) ≥ L(t, ξ, uo (t, ξ )), f o (t, ξ ) = f (t, ξ, uo (t, ξ )), ∀ (t, ξ ) ∈ I × E. (6.47)

.

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6 Optimal Control of Evolution Equations

Since .n (t, ξ ) −→ o (t, ξ ) uniformly on compact subsets of .I × E, for any nonnegative bounded continuous function .ρ defined on .I × E having compact s support, we have .ρn −→ ρo in .L1 (I, B(E)). Thus, for every such .ρ it follows ∗ from .w (weak star) convergence of .λn to .λo that 

 .

I ×E

(n (t, ξ )ρ(t, ξ ))λnt (dξ )dt −→

I ×E

(o (t, ξ )ρ(t, ξ ))λot (dξ )dt.

Since this holds for every continuous function .ρ(≥ 0) having compact support, and |m| < ∞ and .T ∈ / a(ν), we have   .m ≡ lim n (t, ξ )λnt (dξ )dt + (ξ )λnT (dξ )

.

n→∞ I ×E



=

I ×E

E



o (t, ξ )λot (dξ )dt +

E

(ξ )λoT (dξ ).

(6.48)

From this and the first inequality of (6.47) we conclude that  m=

.

I ×E

 ≥

I ×E

 o (t, ξ )λot (dξ )dt +

E

(ξ )λoT (dξ )

L(t, ξ, uo (t, ξ ))λot (dξ )dt +

 E

(ξ )λoT (dξ ) ≡ J0 (uo ). (6.49)

Since m is the infimum of .J0 (·) on .Uad and .uo ∈ Uad , we have .J0 (uo ) ≥ m. Thus, it follows from the above expression that .J0 (uo ) = m, proving the existence of an optimal control.  Some Comments on Application So far we have presented some applications of measure-valued solutions. We have also considered optimal control of fluid dynamics involving Navier-Stokes equation with boundary controls. Existence of optimal controls including necessary conditions of optimality has been developed along the line presented in [10]. In the sequel we consider stochastic Navier-Stokes equation and its control. In fact, it is believed that the notion of measure solution may have a significant impact on the study and control of hydro-dynamic turbulence. Concept of measure solution has also found application in the study of nonlinear conservation laws [68]. We believe that many of the nonlinear wave equations and Schrödinger equations with potentials having polynomial or even exponential growth, which do not admit any (strong, mild, or weak) solution, possess measure solutions globally.

6.3 Optimal Control of Stochastic Systems

161

6.3 Optimal Control of Stochastic Systems In this section we consider optimal control problems for stochastic systems on Hilbert spaces. In particular we are interested in optimal feedback controls. The system is governed by the following stochastic differential equation on a Hilbert space H as seen in Sects. 4.2 and 4.3, .

dx(t) = Ax(t)dt + F (x(t))dt + (x(t))u(t, x(t))dt + G(x(t−))M(dt), x(0) = x0 , t ∈ I = [0, T ],

(6.50)

where . : H −→ L(, H ) is a bounded Borel measurable map with . being another separable Hilbert space and .u : I × H −→  is any bounded Borel measurable function representing state feedback (Markovian) control. Let .BM(I ×H, ) denote the class of bounded Borel measurable functions from .I × H to .. Furnished with the uniform norm topology, |u| ≡ sup {|u(t, x)| , (t, x) ∈ I × H } ,

.

it is a Banach space. Again, for admissible controls, we choose a weaker topology and introduce the following class of functions. Let U be a closed bounded (possibly convex) subset of . and U ≡ {u ∈ BM(I × H, ) : u(t, x) ∈ U, ∀ (t, x) ∈ I × H } .

.

On .BM(I ×H, ), we introduced the topology of weak convergence in . uniformly on bounded subsets of .I × H and denoted this topology by .τwub . In other words, a sequence .{un } ⊂ BM(I × H, ) is said to converge to .u0 ∈ BM(I × H, ) in the .τwub topology if, for each .v ∈ , (un (t, x), v) −→ (u0 (t, x), v)

.

uniformly on bounded subsets of the set .I × H . We assume that .U has been furnished with the relative .τwub topology. Let .Uad ⊂ U be any .τwub compact set and choose this set for admissible controls. Later in the sequel we will discuss admissible topologies in more detail. We consider the following Bolza problem. Problem (S-P1) The problem is to find .uo ∈ Uad that minimizes the cost functional  T  .J (u) ≡ E (t, x(t))dt + (x(T )) , (6.51) 0

162

6 Optimal Control of Evolution Equations

with . and . being any real valued Borel measurable functions on .I × H and H , respectively. Under the general assumptions of Sect. 4.3 (see Theorem 4.3.3), the control system given by Eq. (6.50) may not have pathwise solution. Hence, it is necessary to reformulate the problem in terms of measure solutions. For this purpose we introduce the operator .Bu associated with the control u as follows. For each .u ∈ Uad , we define the operator .Bu as (Bu φ)(t, ξ ) ≡ (u(t, ξ ), ∗ (ξ )Dφ(ξ )) , (t, ξ ) ∈ I × H,

.

(6.52)

for every .φ ∈ D(A)∩D(B), where for each .ξ ∈ H , . ∗ (ξ ) ∈ L(H, ) is the adjoint of the operator . (ξ ). Then, the correct formulation of the Bolza problem is to find o .u ∈ Uad that minimizes the functional  J (u) ≡ E

T

0

 ≡E

0

H T







.

(t, ξ )λut (dξ )dt + 

H

(ξ )λuT (dξ )

ˆ λut )dt + (λ ˆ uT ) , (t,

(6.53)

w (I ×!,  (H + )) ⊂ Lw (I ×!, M (H + )) is the (weak) solution where .λu ∈ M∞ ba ba ∞ of equation

dλt =. A∗ λt π(dt) + B∗ λt dt + B∗u λt dt + C∗ λt− , M(dt)E , λ0 = δx0 . (6.54) We use the following result on continuous dependence of solutions on control. Lemma 6.3.1 Consider the system given by Eq. (6.54) with the set of admissible controls .Uad as defined above. Suppose the assumptions of Theorem 4.3.3 hold and that . : H −→ L(, H ) is a bounded Borel measurable map. Then, for every .u ∈ w (I × !,  (H + )) and Uad the system (6.54) has a unique weak solution .λu ∈ M∞ ba w (I × !,  (H + )) further, the control to solution map, .u −→ λu , from .Uad to .M∞ ba is continuous with respect to the topologies .τwub on .Uad and weak star topology on w + .M∞ (I × !, Mba (H )). Proof Existence and uniqueness of solution follows from Theorem 4.3.3 with the operator .B replaced by the sum .(B + Bu ). We prove continuity. Let .{un , uo } ⊂ τwub w (I × !,  (H + )) denote the Uad and .un −→ uo and suppose .{λn , λo } ⊂ M∞ ba associated weak solutions of the system (6.54). Then, for each .ϕ ∈ D(A) ∩ D(B), + the difference .μn ≡ λn − λo ∈ Lw ∞ (I × !, Mba (H )) is a sequence satisfying P -a.s. the following identity  μnt (ϕ). =

t

0



t

+ 0

 μns (Aϕ)π(ds) +

t 0

μns ((B + Buo )ϕ)ds

μns− (Cϕ), M(ds) +



t 0

λns ((Bun − Buo )ϕ)ds, t ∈ I,

(6.55)

6.3 Optimal Control of Stochastic Systems

163

w (I × !, . (H + )), with .μn0 = μn0− = 0. Since the set .{λn , λo } is contained in .M∞ ba n n the difference sequence .{μ } with the initial condition .μ0 = 0 is contained in a + ∗ bounded subset of .Lw ∞ (I × !, Mba (H )). Thus, there exists a .w closed ball containing the bounded set referred to above. Hence, by virtue of Alaoglu’s theorem, both the sequences have .w ∗ convergent generalized subsequences or subnets which we relabel as the original sequence. Let .λ∗ denote the .w ∗ limit of .λn and .μ∗ the ∗ n n o .w limit of .μ . Since .u converges to .u in .τwub topology, and .Dϕ has a compact support and . is a uniformly bounded Borel measurable (operator valued) function, it follows from the dominated convergence theorem that, for any .z ∈ L2 (!), s

z((Bun − Buo )ϕ) ≡ z(un − uo , ∗ Dϕ) −→ 0 in L1 (I × !, B(H + )).

.

w∗

w (I × !,  (H + )), we obtain Using this and the fact that .λn −→ λ∗ in .M∞ ba

  t lim . E z λns ((Bun − Buo )ϕ)ds

n→∞

0



t

= lim E n→∞

0

λns (z((Bun

− Buo )ϕ))ds

= 0, t ∈ I.

Then, multiplying on either side of Eq. (6.55) by any .z ∈ L2 (!) and taking the expectation, and letting .n → ∞ and then recalling that .z ∈ L2 (!) is arbitrary, one can verify that ∗ .μt (ϕ)

 = 0

t

μ∗s (Aϕ)π(ds)

 +

t

0

 +

0

t

μ∗s ((B + Buo )ϕ)ds μ∗s− (Cϕ), M(ds)ds, t ∈ I, P -a.s.,

for all .ϕ ∈ D(A) ∩ D(B). This is a linear homogeneous Volterra type functional equation for .μ∗ , and hence, following the same procedure as in [23, Theorem 4.1] we find that .μ∗ = 0. In other words, the weak star limit .λ∗ coincides with .λo , the weak solution of Eq. (6.54) corresponding to the control .uo . This proves the  continuity of control to solution map as stated. Now we consider the control of the Bolza Problem (S-P1). Theorem 6.3.2 Consider the system given by Eq. (6.54) and the Bolza problem (6.53) with admissible controls .Uad . Suppose the assumptions of Lemma 6.3.1 hold, and that . and . are real valued nonnegative Borel measurable functions on .I × H and H respectively and bounded on bounded sets and .{T } is not an atom of the martingale measure M. Then, there exists an optimal control solving the Problem (S-P1).

164

6 Optimal Control of Evolution Equations

Proof Since both . and . are nonnegative, .J (u) ≥ 0. Clearly, if .J (u) = +∞ for all .u ∈ Uad , there is nothing to prove. So, suppose there are controls for which .J (u) is finite. Define .inf{J (u), u ∈ Uad } = m, and let .{un } ⊂ Uad be a minimizing sequence. Since .Uad is .τwub compact, there exists a generalized subsequence (subnet), relabeled as the original sequence, and a control .uo ∈ Uad τwub such that .un −→ uo . Then, by virtue of Lemma 6.3.1, along a further subnet if n

w∗

o

necessary, we have .λu −→ λu . Note that the functional (6.53) is linear in .λu and bounded (since .{un } is a minimizing sequence). Thus, by virtue of continuity, n o o .limn→∞ J (u ) = J (u ) = m and hence .u is the optimal control.  Next we consider a control problem (S-P2) (Lagrange problem) that includes the cost of control, with the objective functional given by  J (u) ≡ E

.

I ×H

{(t, ξ ) + ρ(ξ )|u(t, ξ )| } λut (dξ )dt −→ inf,

(6.56)

where .ρ is a positive bounded Borel measurable function on H having bounded support and .λu is the weak solution of the stochastic evolution Eq. (6.54) corresponding to the control .u ∈ Uad . Theorem 6.3.3 Consider the system (6.54) with admissible controls .Uad and the objective functional given by (6.56). Suppose . is a real valued nonnegative Borel measurable function on .I × H and .ρ is any real valued nonnegative bounded Borel measurable function on H having bounded support. Then, there exists an optimal control for the problem (S-P2). Proof Again, by virtue of the assumption on ., we have .J (u) ≥ 0. If .J (u) ≡ +∞ for all .u ∈ Uad , there is nothing to prove. So, we may assume the contrary. Let .{un } be a minimizing sequence so that .

˜ lim J (un ) = inf {J (u), u ∈ Uad } ≡ m.

n→∞

We show that the second term of the objective functional (6.56), denoted by .J2 , is .τwub lower semicontinuous on .Uad . Since .Uad is .τwub compact, the sequence n .{u } contains a generalized subsequence, relabeled as the original sequence, which o converges in .τwub topology to an element .uo ∈ Uad . Let .λo = λu denote the solution of Eq. (6.54) corresponding to the control .uo . Consider the value of .J2 at o .u ,  o o .J2 (u ) ≡ E ρ(ξ )|uo (t, ξ )| λut (dξ )dt. I ×H

Since .uo (t, ξ ) is a . valued bounded (Borel) measurable function, by Riesz theorem there exists a .B1 () valued (Borel) measurable function .ηo on .I × H such that |uo (t, ξ )| = (uo (t, ξ ), ηo (t, ξ )) , ∀ (t, ξ ) ∈ I × H.

.

6.3 Optimal Control of Stochastic Systems

165

In fact, one can take .ηo (t, ξ ) = uo (t, ξ )/|uo (t, ξ )| with the convention that .0/0 ≡ 0. Hence, .J2 (uo ) can be written as  o o .J2 (u ) ≡ E ρ(ξ )(uo (t, ξ ), ηo (t, ξ )) λut (dξ )dt  =E

I ×H

o

I ×H

n

ρ(ξ )(uo (t, ξ ), ηo (t, ξ )) (λut − λut )(dξ )dt

+E



n

I ×H

ρ(ξ )(uo (t, ξ ) − un (t, ξ ), ηo (t, ξ )) λut (dξ )dt



+E

n

I ×H

ρ(ξ )(un (t, ξ ), ηo (t, ξ )) λut (dξ )dt.

(6.57)

It follows from Lemma 6.3.1 that the first term on the right-hand side of τwub Eq. (6.57) converges to zero as .n → ∞. Since .ρ has bounded support and .un −→ o o n o u , it is clear that .ρ(u − u , η ) −→ 0 uniformly on .I × H , hence strongly n o in .L1 (I × !, BC(H )), while .λu converges to .λu in the weak star topology of w + .L∞ (I × !, Mba (H )). Thus, the second term of (6.57) also converges to zero as .n → ∞. Clearly, it follows from positivity of both .ρ and the measure solutions, and the fact that .|ηo (t, ξ )| ≡ 1, that the third term is majorized by .J2 (un ). From these facts we have J2 (uo ) ≤ lim inf J2 (un ).

.

n→∞

Thus, .J2 is .τwub lower semicontinuous. It was already seen in Theorem 6.3.2 that the first term of the cost functional J is continuous. Thus, .u −→ J (u) is .τwub lower semicontinuous. Since .{un } is a minimizing sequence with the limit .uo ∈ Uad , it follows from .τwub lower semicontinuity that ˜ m ˜ ≤ J (uo ) ≤ lim inf J (un ) ≤ lim J (un ) = inf {J (u), u ∈ Uad } ≡ m.

.

Hence, .uo is the optimal control, proving existence.



Remark 6.3.4 The assumption that .ρ has bounded support seems to be theoretically restrictive. In many practical applications, however, the controls are bounded and allowed to use only limited state information such as a closed bounded subset of a finite dimensional subspace .H0 ⊂ H . Thus, for practical applications, the assumption on .ρ having bounded support is not so restrictive. However, this limitation can be relaxed by use of the family .{ρr , r > 0} where each .ρr has support .Br (H ), the closed ball in H of radius r centered at the origin and then taking the limit .r → ∞ provided the supports of the measure solutions decay sufficiently as .r → ∞. There are many other interesting control problems involving the system (6.54). We present here a few simple examples.

166

6 Optimal Control of Evolution Equations

Problem (S-P3) Let K be a closed subset of H and suppose we wish to find a control that maximizes the expected value of the mass of the corresponding measure solution on K at the terminal time T . This can be interpreted as maximizing the expected value of hitting the target. The corresponding objective functional is given by  J (u) = E

.

H

  χK (ξ )λuT (dξ ) = E λuT (K) ,

(6.58)

where .χK is the indicator function of the set K. Since K is a closed set, the indicator function is upper semicontinuous. Thus, if .{T } is not an atom of the measure .M, the functional .u −→ J (u) is .w ∗ upper semicontinuous and hence it attains its maximum on .Uad , guaranteeing the existence of an optimal control. Problem (S-P4) Let .D(t) ⊂ H , .t ∈ I , be a nonempty open set-valued function considered as a forbidden zone. The problem is to find a control to evade this danger zone during the time period I . The objective functional can be chosen as follows  



J (u) = E

.

I

H

χD(t) (ξ )λut (dξ )dt

=E I

λut (D(t))dt.

(6.59)

The problem is to find a control that minimizes the above functional. In Theorem 4.3.3, we have assumed that the martingale measure M, and hence its quadratic variation measure .π , is non-atomic. Thus, the measure solution .λu corresponding to any control .u ∈ Uad is almost surely .w ∗ continuous in .t ∈ I . We prove existence of an optimal control in the following theorem. Theorem 6.3.5 Consider the system given by the evolution Eq. (6.54) with admissible controls .Uad and the cost functional given by the expression (6.59). Suppose the assumptions of Lemma 6.3.1 hold and, for each .t ∈ I , the set .D(t) is a nonempty open subset of H and its closure is continuous on I in the Hausdorff metric. Then, there exists an optimal control. Proof By assumption, for each .t ∈ I , the set .D(t) is open. Thus, the characteristic τwub function .χD(t) (ξ ) is lower semicontinuous on H . Let .un −→ uo in .Uad . Then, it w∗

w (I × !,  (H + )). Hence, we follows from Lemma 6.3.1 that .λu −→ λu in .M∞ ba have n

o

n

o

λut (D(t)) ≤ lim λut (D(t)) for a.e. t ∈ I, P -a.s.

.

Then, it follows from Fatou’s lemma that   n uo .E λt (D(t))dt ≤ lim E λut (D(t))dt, I

I

(6.60)

6.3 Optimal Control of Stochastic Systems

167

and therefore .J (uo ) ≤ lim J (un ). Thus, the cost functional J , given by the expression (6.59), is lower semicontinuous on .Uad in the .τwub topology. Since .Uad is .τwub compact, J attains its infimum on .u ∈ Uad , proving existence of an optimal  control. This completes the proof. Another interesting problem, identified as (S-P5), can be stated as follows. Let  ∈ B(H ), the space of real valued bounded Borel measurable functions on the Hilbert space H , and .g ∈ Cb (R) (bounded continuous real valued functions defined on R) be given. The problem is to find a control that minimizes (maximizes) the functional

.

  J (u) ≡ E g(λuT ()) .

.

(6.61)

Theorem 6.3.6 Consider the system given by the evolution Eq. (6.54) with admissible controls .Uad and the objective functional given by (6.61). Suppose the assumptions of Theorem 4.3.3 and Lemma 6.3.1 hold. Further suppose . ∈ B(H ) and .g ∈ Cb (R) and that T is not an atom of the martingale measure M. Then, the problem (S-P5) has a solution. Proof Since T is not an atom of the martingale measure M, it is clear that for  ∈ B(H ), .{λuT (), u ∈ Uad } is a family of well-defined real random variables. Without loss of generality, we may assume that .||B(H ) ≤ 1. Clearly, it follows from this that

.

|λuT ()| ≤ 1, P -a.s., for all u ∈ Uad .

.

For each .u ∈ Uad , define the probability measure .μu on .B(R), the Borel sets of R, by setting   μu (S) = P λuT () ∈ S

.

for .S ∈ B(R). Then, the functional (6.61) is equivalent to  J (u) =

g(ζ )μu (dζ ) ≡ L(μu ).

.

R

Since .g ∈ Cb (R), it is clear that L is a weak star continuous linear functional on the space of probability measures .M1 (R). Note that the family of probability measures,   M0 ≡ μu , u ∈ Uad

.

is contained in .M1 and has compact support given by the closed unit interval [−1, +1]. Therefore, .M0 is uniformly tight and hence a relatively weakly compact subset of .M1 . Using the assumption of .τwub compactness of the set .Uad , one can verify that .M0 is also weakly closed. Thus, .M0 is a weakly compact subset of .M1 (R). Therefore, the linear functional L, which is weak star continuous, attains .

168

6 Optimal Control of Evolution Equations

its minimum (maximum) on the set .M0 . Consequently, the cost functional J given by Eq. (6.61) attains its minimum (maximum) on .Uad . This proves the existence of an optimal control for the problem (S-P5).  Remark 6.3.7 (1): We have assumed the martingale measure M and the related quadratic variation measure .π to be non-atomic. It is not difficult to admit martingale measures having atomic components. This will allow inclusion of continuous as well as jump processes. (2): For similar results on differential inclusions interested readers may see [15]. (3): For many more examples of this kind interested readers may see [20].

6.4 Optimal Control of Neutral Systems 6.4.1 Deterministic Neutral Systems (DNS) In this section we consider optimal control problems of neutral systems as presented in Chap. 5. Let F be any real Banach space and .U ⊂ F a weakly compact convex set. Consider the following control system (a controlled version of Eq. (5.1)) on the Banach space E as in Chap. 5, (d/dt)(x . + g(t, x)) = Ax + f (t, x) + B(t)u(t, x), t ∈ I, x(0) = x0 ∈ E, (6.62) where B is a bounded strongly measurable (that is measurable in the strong operator topology) operator valued function with values .B(t) ∈ L(F, E) for .t ∈ I , and u is the state feedback control defined on .I × E taking values in U . Let .Uad denote the class of admissible feedback controls consisting of functions defined on .I × E and taking values in U so that for each .v ∈ F ∗ , (t, ξ ) −→ u(t, ξ ), vF,F ∗

.

is continuous on every bounded subset of .I × E, for example, .I × Br (E), where Br (E) is a closed ball in E of finite radius .r (> 0) centered at the origin. The set .Uad is endowed with the topology of weak convergence in F uniformly on bounded subsets of .I × E. Recall that we have denoted this topology by .τwub . Let D denote a directed set .(D, ≤) and .{uα , α ∈ D} a net in .Uad . The net .uα is said to converge τwub in this topology to .uo ∈ Uad , denoted by .uα −→ uo , if for every .v ∈ F ∗ , .

v, uα (t, ξ )F ∗ ,F −→ v, uo (t, ξ )F ∗ ,F uniformly on I × Br (E)

.

for every finite .r > 0. We assume that .Uad is compact with respect to the topology τwub . In fact, by use of Hahn-Banach separation theorem and the assumption that U is a weakly compact convex subset of the Banach space F , one can easily verify that .Uad is .τwub compact. Let .Cwub (I × E, F ) denote the class of functions defined

.

6.4 Optimal Control of Neutral Systems

169

on .I × E taking values in F which are weakly continuous uniformly on bounded subsets of .I × E. This is a locally convex topological space and .Uad is a compact subset of .Cwub (I × E, F ). Let .Lr (u) denote the operator .Lr , as seen in Eq. (5.9), corresponding to the evolution Eq. (6.62) with the control .u ∈ Uad . As seen in Theorem 5.3.4, these operators are contained in the space of bounded linear operators, w .L(L∞ (I, Mrba (E)), L∞ (I )). Under the assumption on the operator valued function B and the admissible set of controls .Uad , it is easy to verify that Theorem 5.3.4 also holds for the control system given by Eq. (6.62). Thus, by Theorem 5.3.4, given the initial (state) measure .μ0 ∈ rba (E) and a control .u ∈ Uad , the system (6.62) has a unique measure-valued solution u ∈ M w (I,  + w + .μ rba (E )) ⊂ M∞ (I, Mrba (E )). A general control problem is ∞ o to find a control law .u ∈ Uad that extremizes (minimizes or maximizes) a given objective functional J (u) ≡ G(u, μu ),

.

w (I,  + ¯ (the extended real number system). To where .G : Uad × M∞ rba (E )) −→ R consider such problems we require continuous dependence of solutions with respect to control. This is proved in the following theorem.

Theorem 6.4.1 Suppose the assumptions of Theorem 5.3.4 hold. Let B be a strongly measurable uniformly bounded operator valued function on I with values in .L(F, E) and .Uad the class of admissible feedback controls endowed with the .τwub topology. Then, the measure solution of Eq. (6.62) is continuously dependent τwub

w∗

on control in the sense that whenever .uα −→ uo in .Uad , .μα −→ μo in w + α o .M∞ (I, rba (E )), where .{μ , μ }, α ∈ D, are the measure-valued solutions of Eq. (6.62) corresponding to the controls .{uα , uo }, respectively. Proof Recall the functional Eq. (5.7) for the uncontrolled system (5.34). For the control system (6.62), the functional Eq. (5.7) must be modified by an additional term associated with the control as follows   ∗ . ξ + g(t, ξ ), e E,E ∗ μt (dξ ) = S ∗ (t)e∗ , ξ + g(0, ξ )μ0 (dξ ) Br

+

Br

 t 0

S ∗ (t − s)e∗ , f (s, ξ ) + B(s)u(s, ξ )μs (dξ )ds Br

−

 t 0

S ∗ (t − s)A∗ e∗ , g(s, ξ )μs (dξ )ds, t ∈ I.

(6.63)

Br

Under the assumptions on the operator valued function B and the admissible control Uad , one can verify that .fˆ ≡ f + Bu satisfies the same properties as f as seen in Theorem 5.3.4. Thus, it follows from Theorem 5.3.4 that, for each fixed initial state .μo and control .u ∈ Uad , the functional equation has a unique solution

.

170

6 Optimal Control of Evolution Equations

w (I,  + μu ∈ M∞ rba (E )). For a fixed initial state (measure) .μ0 ∈ rba (E), α o w let .{μ , μ } ∈ M∞ (I, rba (E + )) denote the solutions of the above functional equation corresponding to the controls .{uα , uo } ∈ Uad , respectively. Introduce the + operator .Lr,4 mapping .Lw ∞ (I, Mrba (E )) onto .L∞ (I ) as follows

.

(Lr,4 μ)(t) ≡

 t

.

0

S ∗ (t − s)e∗ , B(s)uo (s, ξ )μs (dξ )ds, t ∈ I,

Br

and define .Lˆ r = Lr + Lr,4 . Then, it follows from simple algebraic manipulations, involving the expression (6.63) corresponding to the controls .{uα , uo } and the associated solutions .{μα , μo } respectively, that .

Lˆ r (μα − μo )(t)  t B ∗ (s)S ∗ (t − s)e∗ , uα (s, ξ ) − uo (s, ξ )F ∗ ,F μαs (dξ )ds, t ∈ I. = 0

Br

(6.64) Since U is a weakly compact subset of F , it is bounded. By our assumption, B is a uniformly bounded operator valued function with values in .L(F, E). Thus, it follows from these properties that, for any .e∗ ∈ E ∗ and any finite .r > 0, the righthand expression of (6.64) is in .L∞ (I ) for all .α ∈ D. Scalar multiplying the above expression by any .η ∈ L1 (I ) and using Fubini’s theorem, we arrive at the following identity ˆ r (μα − μo ), ηL∞ ,L1 = μα − μo , (Lˆ r )∗ ηY ∗ ,Yr = .L r

 I ×Br

ψα (s, ξ )μαs (dξ )ds, (6.65)

where, for .(s, ξ ) ∈ I × Br ⊂ I × E,  ψα (s, ξ ) ≡

.

T

dt η(t)B ∗ (s)S ∗ (t − s)e∗ , uα (s, ξ ) − uo (s, ξ )F ∗ ,F , s ∈ I.

s τwub

Since .uα −→ uo , it follows from the above expression and Lebesgue bounded convergence theorem that .ψα → 0 strongly in .L1 (I, BC(Br (E))) for every finite α α .r > 0. Combining this with the fact that .0 ≤ μs (Br ) ≤ μs (E) ≤ 1 for all .s ∈ I , and .α ∈ D, we conclude that the expression on the right-hand side of Eq. (6.65) converges to zero (along a subnet if necessary). Then, it follows w∗ from the fact that .(Lˆ r )∗ ∈ Iso(L1 , Yr ) (see Theorem 5.3.4) that .μα −→ μo in ∗ w + .Yr ⊂ M∞ (I, rba (E )). This completes the proof.  Now we are prepared to consider some optimal control problems.

6.4 Optimal Control of Neutral Systems

171

Problem (DNS-1) (Following a Moving Target) Let .cbc(E) denote the class of nonempty closed bounded convex subsets of the Banach space E and let .K : I −→ cbc(E) be a multi function continuous in the Hausdorff metric. The problem is to find a control law such that the distance of the support of the measure solution from the set-valued function K is minimized. This can be formulated as the problem of minimizing the functional  J1 (u) ≡

.

I ×E

d(ξ, K(t))μut (dξ )dt,

(6.66)

where .d(ξ, K(t)) denotes the distance of the point .ξ from the set .K(t). Theorem 6.4.2 Consider the control Problem (DNS-1) subject to the dynamic system (6.62) and suppose the assumptions of Theorem 6.4.1 hold and that .J1 (u) is not identically .+∞. Then, there exists an optimal control .u◦ ∈ Uad at which .J1 attains its minimum. Proof Since the function .ξ −→ d(ξ, K(t)) is continuous and bounded on bounded sets, but may become unbounded as .|ξ |E → ∞, there may be controls for which the integral will diverge to .+∞. If for all .u ∈ Uad , .J1 (u) = +∞, there is nothing to prove. Under the given assumption there are controls for which the functional is finite. We want to prove that there is one that minimizes the functional .J1 . Consider the weight function .wρ (ξ ) ≡ exp {−ρ|ξ |E } for .ρ > 0. Clearly, this is a continuous, nonnegative, symmetric real valued function on E and, since the multi function K takes values from .cbc(E), the product .wρ (·)d(·, K(·)) ∈ L1 (I, BC(E)). Thus, the regularized functional  J1,ρ (u) ≡

.

I ×E

wρ (ξ )d(ξ, K(t))μut (dξ )dt

(6.67)

is well-defined for all .u ∈ Uad . Hence, it follows from Theorem 6.4.1 that .J1,ρ is continuous on .Uad with respect to the .τwub topology. Since the admissible set .Uad is .τwub compact and .J1,ρ is continuous in this topology, it attains its minimum on .Uad . Hence, for each .ρ > 0, there exists an optimal control ρ ∈ U .u ad for the regularized problem with the corresponding measure solution ρ w .μ ∈ M∞ (I, rba (E)). Since the set of controls for which .J1 is finite is nonempty, we can choose a decreasing sequence .ρn ↓ 0 such that .J1,ρn (uρn ) is monotone decreasing and it is a minimizing sequence. By virtue of .τwub compactness of the set .Uad , there exists a generalized subsequence of the sequence .{ρn }, relabeled as τwub the original (generalized) sequence .{ρn }, and a .uo ∈ Uad such that .uρn −→ uo . Let ρ o .{μ n , μ } denote the corresponding measure solutions of the evolution Eq. (6.63). w∗

Then, by Theorem 6.4.1, along a further subsequence if necessary, .μρn −→ μo . Thus, taking the limit of the functional (6.67) along this generalized subsequence,

172

6 Optimal Control of Evolution Equations

we obtain  J1 (uo ) =

.

I ×E

d(ξ, K(t))μot (dξ )dt. 

This proves the existence.

Problem (DNS-2) (Concentration Control of Probability Mass) Let C be a closed bounded subset of E and suppose it is required to find a control policy that maximizes the concentration of (probability) mass on C over the time period I . We consider a slightly more general problem. Let .λ ∈ L+ 1 (I ), a nonnegative integrable function, and consider the weighted objective functional  J2 (u) ≡

.

I

λ(t)μut (C)dt.

(6.68)

Our problem is to find a control .u ∈ Uad that maximizes this functional. Theorem 6.4.3 Consider the Problem (DNS-2) with the objective functional (6.68) subject to the dynamic system (6.62). Suppose the assumptions of Theorem 6.4.1 hold. Then, there exists an optimal control .uo ∈ Uad maximizing the functional (6.68). Proof We regularize the problem as follows. Let .ε > 0 (as small as desired) and let Cε ≡ {ξ ∈ E : d(ξ, C) < ε} be an open neighborhood of C and .Cε its complement. Clearly, .infξ ∈C,ζ ∈Cε d(ξ, ζ ) = ε > 0. Then, it follows from [113, Theorem 1.6, p. 4] that there exists a uniformly continuous function .ψε ∈ BU C(E) ⊂ BC(E) such that .ψε (ξ ) = 1 for .ξ ∈ C and equal to zero for .ξ ∈ Cε , and otherwise .0 ≤ ψε (ξ ) ≤ 1. We consider the regularized problem, .

 J2,ε (u) =

.

I ×E

λ(t)ψε (ξ )μut (dξ )dt,

(6.69)

and find .u ∈ Uad that maximizes the functional. Note that the function .ϕε given by .ϕε (t, ξ ) ≡ λ(t)ψε (ξ ) is an element of .L1 (I, BC(E)) and hence the functional (6.69) is in the form of the standard duality pairing between the dual pair of spaces .{L1 (I, BC(E)), Lw ∞ (I, Mrba (E))}. It follows from Theorem 6.4.1 that the functional .J2,ε (·) is continuous in the .τwub topology. Since .Uad is .τwub compact, .J2,ε (·) attains its maximum on .Uad . Let .uε ∈ Uad denote any of the w (I,  maximizing net and .με ∈ M∞ rba (E)) the corresponding net of measure solutions of Eq. (6.62) so that .J2,ε (uε ) ≥ J2,ε (u) for all .u ∈ Uad . Again, by virtue of compactness of the set .Uad , there exists an .uo ∈ Uad such that, τwub along a generalized subnet if necessary, .uε −→ uo in .Uad as .ε ↓ 0, and by w∗

w (I,  + o Theorem 6.4.1, .με −→ μo in .M∞ rba (E )), where .μ is the measure solution o of the system (6.62) corresponding to the control .u . Since .ψε ∈ BU C(E) and it converges uniformly to the characteristic function of the set C and the measures

6.4 Optimal Control of Neutral Systems

173 w∗

{με , μo } are regular and bounded and .με −→ μo , we can let .ε ↓ 0 and obtain  ε ε o o .limε↓0 J2,ε (u ) = limε↓0 I ×E λ(t)ψε (ξ )μt (dξ )dt = I λ(t)μt (C)dt = J2 (u ). o This proves that the functional .J2 attains its maximum at .u ∈ Uad and hence an optimal control exists.  .

Problem (DNS-3) (Exit Time Problem) Let .C0 be a closed subset of E denoting the support of the initial measure .μ0 and suppose it is contained in the interior of a closed ball .Bγ ⊂ E of radius .γ such that .∂Bγ ∩ C0 = ∅. Here, we define the exit time to be the first time, the support of the measure solution has nonempty intersection with the set .Bγ having positive measure. The problem is to find a control that maximizes the first exit time. This can be formulated as follows. Since w (I,  the measure solution of Eq. (6.62) is an element of the space .M∞ rba (E)) ⊂ w + L∞ (I, Mrba (E )) it may not be set-wise continuous on I . However, the function  t −→ νtu (·) ≡ (1/t)

.

0

t

μus (·)ds, t ∈ (0, T ],

(6.70)

is a weak star continuous (probability) measure-valued function. Thus, the functional   J3 (u) ≡ τγ (u) ≡ inf t ∈ I = [0, T ] : νtu (Bγ ) < 1

.

(6.71)

is well-defined. If the underlying set is empty, we set .τγ (u) = T +. This is the situation when the support of the measure solution .μut remains confined in the closed ball .Bγ for all .t ≥ 0. On the other hand, if the set is nonempty, it makes sense to consider and prove the following result. Theorem 6.4.4 Consider the exit time Problem (DNS-3) and suppose the assumptions of Theorem 6.4.1 hold. Then, there exists a control .uo that maximizes the functional .J3 (·) (and hence the exit time). Proof We prove that .J3 is upper semicontinuous in the .τwub topology. Let {un } ∈ Uad , and .{μn } be the corresponding measure solutions of the evolution Eq. (6.62), where .n ∈ N with N denoting the set of positive integers. Since o ∈ U .Uad is .τwub compact, there exists an element .u ad such that, along a n generalized subsequence if necessary, .u converges to .uo in .τwub topology. Let o ∈ M w (I,  .μ rba (E)) denote the measure solution of Eq. (6.62) corresponding ∞ to .uo . Then, by Theorem 6.4.1, along a subsequence, relabeled as the original .

w∗

w (I,  + sequence, .μn −→ μo in .M∞ rba (E )). Since .Bγ is a closed set it follows from a well-known result [113, Theorem 6.1, p. 40] that .

lim νtn (Bγ ) ≤ νto (Bγ ), t ∈ I.

(6.72)

174

6 Optimal Control of Evolution Equations

By the definition of limsup, for every .ε ∈ (0, 1), there exists an integer .nε ∈ N such that νtn (Bγ ) ≤ lim νtn (Bγ ) + ε, ∀ n > nε , t ∈ I.

.

Hence, .

    inf t ≥ 0 : νtn (Bγ ) < 1 ≤ inf t ≥ 0 : lim νtn (Bγ ) + ε < 1 , ∀ n > nε .

Using the inequality (6.72), one can readily verify that .

    inf t ≥ 0 : lim νtn (Bγ ) + ε < 1 ≤ inf t ≥ 0 : νto (Bγ ) + ε < 1 .

Thus, it follows from the above inequalities that .

    inf t ≥ 0 : νtn (Bγ ) < 1 ≤ inf t ≥ 0 : νto (Bγ ) + ε < 1 , ∀ n > nε .

Hence, it is clear from the above expression that .

     lim inf t ≥ 0 : νtn (Bγ ) < 1 ≤ inf t ≥ 0 : νto (Bγ ) + ε < 1 .

Since this holds for any arbitrary .ε ∈ (0, 1), it follows from the definition of .τγ (u) and the above inequality that .lim τγ (un ) ≤ τγ (uo ). Thus, we have .

lim J3 (un ) ≤ J3 (uo ),

proving upper semicontinuity of .J3 . Hence, .J3 attains its maximum on the set .Uad , proving the existence of an optimal policy.  Problem (DNS-4) (Lagrange Problem) Next we consider the following Lagrange problem, which is to find a feedback control .u ∈ Uad that minimizes the cost functional  .J4 (u) = (t, ξ, u(t, ξ ))μut (dξ )dt, (6.73) I ×E

where .μu is the measure solution of Eq. (6.62) corresponding to the control .u ∈ Uad . Theorem 6.4.5 Consider the system (6.62) with admissible controls .Uad and the cost functional (6.73). Suppose the assumptions of Theorem 6.4.1 hold and the following conditions are satisfied: (A1) The function . is nonnegative and lower semicontinuous on .Uad in the τwub sense that whenever the net .uα −→ uo , .α ∈ (D, ≤) (directed set),

6.4 Optimal Control of Neutral Systems

175

limα (t, ξ, uα (t, ξ )) ≥ (t, ξ, uo (t, ξ )) uniformly on bounded subsets of .I × E.  (A2) The integrals . I ×E (t, ξ, u(t, ξ ))μut (dξ )dt are well-defined for all .u ∈ Uad with values in .R + ≡ [0, ∞) ∪ {+∞}. .

Then, there exists an admissible control at which J attains its minimum. Proof We leave the proof for the readers. Hint: The proof follows from lower semicontinuity of the functional.  Remark 6.4.6 In this subsection we have presented existence of optimal controls for several control problems. We leave the questions of necessary conditions of optimality for the readers.

6.4.2 Stochastic Neutral Systems (SNS) In this section we consider optimal control problems for stochastic neutral systems governed by the following evolution equation on a Hilbert space E, d(x + g(t, x)) = Axdt + f (t, x)dt + B(t)u(t, x)dt + σ (t, x)dW,

.

x(0) = x0 ∈ E, t ∈ I,

(6.74)

where .{A, f, σ, g} are as in the previous subsection. In physical and social sciences, it is very important to have feedback controls, that is, controls based of state information. Let U be a real Hilbert space endowed with weak topology and let B be a continuous (in the strong operator topology) operator valued function defined on I and taking values in .L(U, E). The control .u : I × E −→ U is assumed to be continuous and bounded in norm on bounded subsets of .I × E. Under the assumption that .x −→ g(t, x) is a contraction, using similar transformation as in Sect. 5.4 of Chap. 5, we obtain the following system .

dy = Ah(t, y)dt + f (t, h(t, y))dt + B(t)u(t, h(t, y))dt + σ (t, h(t, y))dW, y(0) = y0 ≡ x0 + g(0, x0 ), t ∈ I.

(6.75)

where .x = h(t, y) solving the equation .x + g(t, x) = y. Again, for economy of notation, we define .v(t, y) ≡ u(t, h(t, y)). In this case we must add to the generating family .{A, B, C} an additional operator .Dv associated with the control as follows (Dv ϕ)(t, ξ ) ≡ B ∗ (t)Dϕ(ξ ), v(t, ξ )U , for any ϕ ∈ BC(E),

.

176

6 Optimal Control of Evolution Equations

giving the complete family of generating operators .{A, B, C, Dv }. Using this we modify the uncontrolled system (5.28) to obtain the control system given by dμt = A∗ μt dt + B∗ μt dt + D∗v μt dt + C∗ μt , dW , t ∈ I,

.

(6.76)

with initial condition .μt |t=0 = μ0 ∈ rba (E). This equation is understood in its weak form given by dμt (ϕ) . = μt (Aϕ)dt + μt (Bϕ)dt + μt (Dv ϕ)dt + μt (Cϕ), dW , t ∈ I, (6.77) for .ϕ ∈ D(A) ∩ D(B) with initial condition .μ0 (ϕ). Now we are prepared to present a result on existence of measure solutions for the control system. Corollary 6.4.7 Consider the system (6.76) and suppose the assumptions of Theorem 5.4.3 hold. Further, assume the control component .Bv satisfies similar approximation properties. Then, the controlled evolution Eq. (6.75) has a unique w (I × !,  + w + measure-valued solution .μv ∈ M∞ rba (E )) ⊂ L∞ (I × !, Mrba (E )) in the sense that  t  t μvt (ϕ). = μ0 (ϕ) + μvs (Aϕ)ds + μvs (Bϕ)ds 0



t

+ 0



μvs (Dv ϕ)ds +

0

0

t

μvs (Cϕ), dW (s)H , t ∈ I, P -a.s., (6.78)

for every .ϕ ∈ D(A) ∩ D(B). Proof The proof is identical to that of Theorem 5.4.3.



In view of the above result we can state that the evolution Eq. (6.76) on the w (I × !,  + Banach space .Mrba (E + ) has a solution .μv ∈ M∞ rba (E )) in the weak sense. Before we proceed further, we introduce the set of admissible controls for the system (6.76), or equivalently (6.77). For many applications, it is preferable to use feedback controls. Let .U0 be a weakly compact (possibly convex) subset of the Hilbert space U . Consider the class of functions mapping .I × E into .U0 which are continuous in the weak topology of U on compact subsets of .I ×E. In this topology, a sequence .v n converges to .v o , if and only if, for each .z∗ ∈ U , (v n (t, ξ ), z∗ )U −→ (v o (t, ξ ), z∗ )U

.

uniformly on compacts of .I × E. We denote this topology by .τwuc and reiterate that this is the topology of weak convergence in U uniformly on compact subsets of .I × E. Let .Uad denote the class of admissible controls which is .τwuc compact. To study control problems for the system (6.77), we use the following result on continuity of the control to solution map .v −→ μv .

6.4 Optimal Control of Neutral Systems

177

Theorem 6.4.8 Consider the system (6.77) with the admissible controls .Uad endowed with the .τwuc topology. Suppose the assumptions of Corollary 6.4.7 hold. Then, the control to solution map, .v −→ μv , is continuous with respect to .τwuc w (I × !,  + topology on .Uad and .w ∗ topology on .M∞ rba (E )). Proof For any control .v ∈ Uad and any initial state .μ0 ∈ rba (E), Eq. (6.77) is written in its weak form as follows  t  t v .μt (ϕ) = μ0 (ϕ) + μvs (Aϕ)ds + μvs (Bϕ)ds 0

 +

0

0

t

 μvs (Dv ϕ)ds +

t

0

μvs (Cϕ), dW (s)H , t ∈ I, (6.79)

for each .ϕ ∈ D(A) ∩ D(B) having compact support. Let .{v n , v o } be a sequence τwuc w (I × !,  + in .Uad and suppose .v n −→ v o . Let .{μn , μo } ∈ M∞ rba (E )) denote the corresponding solutions of Eq. (6.79). By Alaoglu’s theorem, there exists a generalized subsequence (subnet), relabeled as the original sequence (net) and a w∗

w (I × !,  + n o o λ o ∈ M∞ rba (E )) such that .μ −→ λ with .λ (0) = μ0 . We show o o n n o that .λ = μ . Define .ϑ = μ − μ . Note that this is a signed measure and hence + an element of .Lw ∞ (I × !, Mrba (E )). Now computing the difference of the two identities similar to (6.79) corresponding to the controls .v n and .v o respectively, we obtain  t  t  t n n n ϑt (ϕ) = ϑs (Aϕ)ds + ϑs (Bϕ)ds + ϑsn (Cϕ), dW (s)

.

0

.

0



t

+ 0

ϑsn (Dv o ϕ)ds +

0

 t 0

E

(B ∗ (s)Dϕ, v n − v o )U μns (dξ )ds, t ∈ I.

Here the last term on the right-hand side of the above expression is crucial. Since τwuc v n −→ v o , and .B ∗ (t)Dϕ(ξ ) has compact support on .I × E, the integrand in the last term converges to zero strongly in .L1 (I × !, BC(E)). On the other hand, w∗ n −→ w (I × !,  + w + .μ λo in .M∞ rba (E )) ⊂ L∞ (I × !, Mrba (E )). Thus, in the limit the last term converges to zero for each .t ∈ I . Then, letting .n → ∞ and recalling that the initial state is given by .ϑ0o = 0, it follows from the above equation that .ϑ o ≡ λo − μo satisfies the following linear homogeneous stochastic functional equation .

 ϑto (ϕ) =

.

0

t

 ϑso (Aϕ)ds +

t 0

 ϑso (Bϕ)ds + 

t

+ 0

t 0

ϑso (Dv o ϕ)ds

ϑso (Cϕ), dW (s), t ∈ I.

178

6 Optimal Control of Evolution Equations

This equation holds P -a.s. for all .t ∈ I and for all .ϕ ∈ D(A) ∩ D(B) having compact supports. Thus, it can have only the trivial solution .ϑ o = 0, and hence o o w + n w∗ o .λ = μ as elements of .M∞ (I × !, rba (E )), proving that .μ −→ μ . For a different proof based on semigroup arguments, see Corollary 4.3.6 and interested  readers may also see [23]. This proves the continuity as stated. Now we are prepared to consider control problems. Control Problem (SNS-P1) Suppose C is a closed possibly bounded subset of E and .t ∈ I . It is required to find a control law that maximizes the probability of hitting it at time t. For simplicity, we consider the problem of finding a control law that maximizes the expected value of the mass of the stochastic measure on the target set C at time .t ∈ I . This can be formulated as follows   J1 (v) = E μvt (C) .

.

(6.80)

Let .$ denote the sigma algebra generated by weak star closed subsets of the set rba (E + ) and consider the measurable space .(rba (E + ), $). The set .rba (E + ), endowed with the .w ∗ topology, is a compact Hausdorff space denoted by .
. Then, we introduce the space of probability measures .M1 (
) on the space .
. For each control .v ∈ Uad and .t ∈ I , the measure solution .μv of Eq. (6.76) defines a probability measure on .
as follows

.

Qvt ( ) ≡ (P (μvt )−1 )( ) = Prob .{μvt ∈ }, ∈ $, ⊂
.

.

Clearly, .Qvt ∈ M1 (
). Then, for each .t ∈ I , we can define the attainable set   R(t) ≡ Qvt , v ∈ Uad ⊂ M1 (
).

.

In view of this it is clear that the objective functional (6.80) is equivalent to the following functional  J1 (v) =

.




μ(C)Qvt (dμ).

(6.81)

Our objective is to find a .v ∈ Uad at which .J1 (v) attains its maximum. In particular, one is often interested in the terminal control. The corresponding objective functional can be written in terms of the attainable set .R(T ), giving .Jˆ(Q) =

 μ(C)Q(dμ), Q ∈ R(T ).

(6.82)




Corollary 6.4.9 Consider the control system given by the stochastic evolution Eq. (6.76). Suppose the assumptions of Theorem 6.4.8 hold. Then, there exists an optimal control maximizing the functional (6.82).

6.4 Optimal Control of Neutral Systems

179

Proof Since .
is a compact Hausdorff space and .M1 (
) is the space of probability measures on it, .M1 (
) itself is compact with respect to weak star topology (on the space of probability measures). It is well-known that, for any fixed closed set C, the map .μ −→ μ(C) is upper semicontinuous and hence the functional .Q −→ J˜1 (Q) is upper semicontinuous on .M1 (
). Since .M1 (
) is compact, .J˜ attains its maximum on it. Hence, the problem (SNS-P1) has a solution, proving existence of an optimal control. This completes the proof.  There are many such interesting control problems. Interested readers may refer to [24, 30, 31]. We conclude this section after considering an exit time problem. Control Problem (SNS-P2) Here we consider an exit time problem. Consider the system (6.75), or equivalently (6.76), with initial condition .μ0 = % ∈ rba (E). Let .C ⊂ E be a nonempty closed set with .supp(%) ⊂ O ⊂ C, where .O is an open set containing the support of .% while it is contained in C. Let .δ ∈ (0, 1) be w (I × chosen as small as desired. For any admissible control .v ∈ Uad , let .μv ∈ M∞ + !, rba (E )) denote the solution of the evolution Eq. (6.76). Define the exit time as the first time there is leakage of probability mass as follows   τ (v) ≡ inf t ∈ I : μvt,ω (C) < 1 − δ .

.

Clearly, at time .t = 0, .μv0,ω (C) = ν(C) = 1 P -a.s. So, the above condition certainly signifies leakage of mass from the closed set C. Clearly, for any .t ∈ I , the set .{τ (v) < t} is .Ft measurable and so it is a well-defined stopping time. Our problem is to find a control that maximizes the expected value of the exit time .τ (v). Hence, the objective functional is given by J2 (v) ≡ E {τ (v)} , v ∈ Uad .

.

we prove the following result. Corollary 6.4.10 Consider the system given by (6.76) with the initial condition μ0 = % and control .v ∈ Uad and the objective functional .J2 (v) ≡ Eτ (v) with C being a closed subset of E satisfying the properties stated in the problem description above. Suppose the assumptions of Theorem 6.4.8 hold. Then, there exists a control ∗ ∗ .v ∈ Uad such that .J2 (v ) ≥ sup{J2 (v), v ∈ Uad }. .

Proof Since .Uad is .τwuc compact, it suffices to prove that .J2 (v) is upper semicontinuous in this topology. We present a brief outline here. For detailed proof see [30, τwuc Theorem 5.6]. Suppose .vn −→ v ∗ and consider the random variables   n τ (vn )(ω) ≡ inf t ∈ I : μvt,ω (C) < 1 − δ ,

. ∗ τ (v ∗ )(ω) ≡ inf t ∈ I : μvt,ω (C) < 1 − δ .

180

6 Optimal Control of Evolution Equations

w (I × !,  + v Recall that for .v ∈ Uad , .μv ∈ M∞ rba (E )) and hence clearly .μt,ω ∈ + + ∗ rba (E ) for a.e. .t ∈ I and P -a.s. Since .rba (E ) is a .w compact subset of + .Mrba (E ), and C is a closed set in .E, it follows from a well-known result [113, Theorem 6.1, p. 40] that there exists a generalized subsequence, relabeled as the original sequence, such that ∗

.

n lim μvt,ω (C) ≤ μvt,ω (C), for a.e. t ∈ I, P -a.s.

Clearly, it follows from this inequality that .



  ∗ n t ∈ I : μvt,ω (C) < 1 − δ ⊂ t ∈ I : limμvt,ω (C) < 1 − δ ,

and hence we have .

  n inf t ∈ I : lim μvt,ω (C) < 1 − δ ≤ τ (v ∗ )(ω), P -a.s.

From this inequality one can deduce that .

lim τ (v n )(ω) ≤ τ (v ∗ )(ω), P -a.s.

Using the above inequality and the definition of the functional .J2 , we obtain .

lim J2 (v n ) = lim Eτ (v n ) ≤ Eτ (v ∗ ) = J2 (v ∗ ).

This shows that .J2 is upper semicontinuous on .Uad . Since .Uad is .τwuc compact, it is clear that .J2 attains its maximum on it, proving existence of an optimal control. 

6.5 Bibliographical Notes Optimal control theory for infinite dimensional deterministic systems is very well developed over the last five decades as reflected in the pioneering works of Lions, Fattorini, Cesari, Warga, Ahmed, Li, and Teo, among many others, as seen in the brief list of references [4–8, 25–27, 29, 39, 43, 44, 74, 99, 100, 102, 107, 112, 122, 124]. For deterministic impulsive systems, see [17, 18] and the references therein. Control theory for stochastic systems on infinite dimensional spaces is not as well developed as that for deterministic systems though substantial work has been done and currently in progress as seen in the brief list [33–37, 39, 78, 79, 83, 84, 104, 110] and the references therein. All these work is based on classical pathwise solutions and their optimal control. As seen throughout this monograph, we have considered measure-valued solutions under very relaxed assumptions on the vector fields. In this chapter, under the

6.5 Bibliographical Notes

181

relaxed assumptions, we have considered optimal control problems for deterministic systems, impulsive systems, stochastic systems, and neutral systems governed by evolution equations on the Banach space of regular bounded finitely additive measures, with a primary focus on the questions of existence of optimal controls [10]. Other general equations of fluid dynamics based on the concept of classical pathwise solutions [119] including the earth’s climate system modeling [94] are well-known. Taking Navier-Stokes equation (NSE) [55, 94, 119] with boundary controls as an example, we have developed an abstract semilinear controlled evolution equation on a Banach space proving existence of measure-valued solutions and optimal controls. The concept of measure solutions has also found application in the study of nonlinear conservation laws [68]. For non-convex control problems, we consider existence of optimal relaxed controls for differential inclusions and other functional differential inclusions, see also [7, 8, 58, 72, 73] and the references therein. Interested reader may refer to [24, 30, 31] for many other interesting control problems.

Chapter 7

Examples From Physical Sciences

In this chapter we present few examples from physical sciences with applications to physics and engineering. Our first example comes from nonlinear optics, the second example is from hydrodynamics, and the third one is related to biomedical science.

7.1 Nonlinear Schrödinger Equation It is well-known that the Schrödinger equation plays a fundamental role in quantum physics and physical chemistry. In the area of optics, the optical field is generally governed by a class of nonlinear Schrödinger equations. For such equations there is an extensive literature [56, 66, 92, 93, 106, 108, 109, 111, 117, 120] devoted to the basic questions of existence and uniqueness of solutions and their regularity properties. Further, some of these studies also present sufficient conditions for the existence of solutions that blow up in finite time. In addition, nonlinear Schrödinger equation has also gained special importance in the field of optics and its engineering applications [46, 49, 50, 97, 111, 115]. Development of fiber optics and its applications in the fields of communications and quantum computing are the driving forces for extensive research in nonlinear optics and its controls [46, 49, 50, 98, 115]. Most of the work on control of Schrödinger equations is focused on deterministic linear problems with quadratic cost functionals, such as [46, 49, 50, 97, 111]. These studies employ standard assumptions on the vector fields and construct pathwise solutions. Using classical Pontryagin minimum principle they develop necessary conditions of optimality. In a recent paper [36] we have studied control problems of nonlinear Schrödinger equation, where we proved existence of optimal feedback controls and developed necessary conditions of optimality using standard assumptions on the vector fields.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. U. Ahmed, S. Wang, Measure-Valued Solutions for Nonlinear Evolution Equations on Banach Spaces and Their Optimal Control, https://doi.org/10.1007/978-3-031-37260-5_7

183

184

7 Examples From Physical Sciences

Here we relax the assumptions significantly. Under these general assumptions Schrödinger equation does not have pathwise solutions. So, we prove existence of measure-valued solutions and their regularity properties. Using these results we prove existence of optimal feedback control laws, which is highly desirable in engineering fields.

7.1.1 Basic Formulation of the System Model The system is governed by a nonlinear stochastic Schrödinger equation with homogeneous Dirichlet boundary conditions defined on a domain .D ⊂ R d with smooth boundary .∂D. This is given by the following equation, .

i∂t ψ = (− + V0 (ξ ))ψ + Vc (t, ξ )ψ + λf (|ψ|)ψ + σ (t, ξ )N(t, ξ ),

(7.1)

ψ(0, ξ ) = ψ0 (ξ ), ξ ∈ D ⊂ R and ψ(t, ξ ) = 0, ξ ∈ ∂D, t ∈ I ≡ (0, T ], d

where .∂t denotes the partial with respect to time and the operator . stands for the standard Laplacian in dimension .d (d = 1, 2, 3). The function .V0 is the ground state potential independent of time and .Vc (t, ·) is the control potential induced by the interaction of a laser beam with the nonlinear medium. We assume that these potentials are real valued measurable functions on D. The equation is written in the atomic unit. The function f is a nonnegative real valued function, .ψ is the wave function, .λ is a real parameter, .N = N1 + iN2 is the space-time complex white noise, and .σ is a given function of time and space determining the range of noise affecting the system. The initial state .ψ0 is given. The wave function .ψ takes values in the field of complex numbers and it is given by .ψ = ψ1 + iψ2 . We formulate this as a pair of coupled partial differential equations for .ψ1 and .ψ2 . Define the operator .A0 ≡ (− + V0 ) satisfying the homogeneous Dirichlet boundary conditions as indicated in Eq. (7.1) and introduce the matrix of operators as follows       0 Vc (t) 0 σ (t) 0 A0 , B(t) ≡ , (t) ≡ .A ≡ , −A0 0 −Vc (t) 0 −σ (t) 0 where .Vc (t) ≡ Vc (t, ·) and .σ (t) ≡ σ (t, ·) are functions with values in suitable Banach spaces to be introduced shortly. Using the notation .x = (x1 , x2 ) ≡ (ψ1 , ψ2 ) and the nonlinear function f , we introduce the Nemytski operator F as follows F (x) = (λf (|x|)x2 , −λf (|x|)x1 ) ,

.

where f is a nonnegative real valued function. Using the above notations the Schrödinger equation (7.1) can be written as a system of two coupled partial

7.1 Nonlinear Schrödinger Equation

185

differential equations presented in the abstract form as follows (d/dt)x = Ax + B(t)x + F (x) + (t)W˙ (t), x(0) = x0 ,

.

(7.2)

where .W˙ (t) now stands for the white Gaussian random field with the time derivative understood in the sense of distribution. The original equation is defined on a complex Hilbert space. With the transformation introduced, we can consider the equivalent system (7.2) defined on a real Hilbert space given by the Cartesian product of two real Hilbert spaces. We denote this space by .H ≡ L2 (D) × L2 (D) and consider this as the state space of the system. Thus, the system (7.2) is considered as an abstract differential equation on the Hilbert space H . The system as written is formal, which can be rigorously formulated as an Itô stochastic differential equation on H as follows dx = Axdt + B(t)xdt + F (x)dt + (t)dW (t), x(0) = x0 ,

.

(7.3)

where W stands for a Gaussian random process with values in an infinite dimensional Hilbert space E, where E may be an infinite dimensional subspace of the state space H such as .L2 (D0 , R 2 ), with .D0 being any nonempty closed bounded subset of D. For consistency, the operator valued function .(t) is assumed to take values from .L(E, H ). Let .( , F, P ) denote a complete probability space with a family of non-decreasing right continuous filtration .Ft≥0 ⊂ F having left limits, on which the E valued Brownian motion .{W (t), t ∈ I } is supported.

7.1.2 Existence and Uniqueness of Solutions Before considering measure-valued solutions, let us take a look at classical results on the question of existence and uniqueness of solutions revealing some physical properties of the system. Without loss of generality, we consider .A0 ≡ (−0 + V0 ) to be a positive self adjoint operator on the Hilbert space .L2 (D) ≡ L2 (D, R) with domain .D(A0 ) = H 2 (D) ∩ H01 (D) ⊂ L2 (D). Next we consider the operator A and note that its domain is given by .D(A) = D(A0 ) × D(A0 ) ⊂ H and range .R(A) ⊂ H , and that .D(A) is dense in H . It is not difficult to verify that the operator A is skew adjoint, that is, .(iA)∗ = iA. Thus, it follows from Stone’s theorem [5, Theorem 3.1.4, p. 71] that A generates a unitary group of bounded linear operators .{U (t), t ∈ R} on H with .U (0) = IH , the identity operator in H . For simplicity we assume that the potential .Vc ∈ B∞ (I × D) ⊂ L∞ (I × D), where .B∞ (I × D) denotes the Banach space of bounded measurable functions on .I ×D with respect to the standard sup norm topology. Under this assumption, the operator .B(t) ∈ L(H ) for all .t ∈ I, and one may assume that the operator valued function .B = B(·) is measurable in the strong operator topology. Using the unitary group .U (t), t ∈ I , we

186

7 Examples From Physical Sciences

can transform the evolution equation (7.3) into an integral equation on the space H as follows  t x(t) = . U (t)x0 + U (t − s)B(s)x(s)ds  +

0 t



t

U (t − s)F (x(s))ds +

0

U (t − s)(s)dW (s), t ∈ I. (7.4)

0

a (I, L ( , H )) which consists of .F Recall the definition of the Banach space .B∞ 2 t adapted random processes defined on I with values in .L2 ( , H ) and furnished with the norm topology

|x| ≡ sup

.



E|x(t)|2H

1/2

 , t ∈I .

Under standard assumptions on the operators .{A, B, F, }, using Banach fixed point theorem one can prove the existence and uniqueness of (pathwise) solutions of the integral equation as stated below. Hence, the stochastic differential equation (7.3) has a unique mild solution. Theorem 7.1.1 Consider the stochastic integral equation (7.4) on the Hilbert space H and suppose the following assumptions hold: (A1) B is a strongly (strong operator topology) measurable operator valued function defined on I with values in .L(H ) satisfying .sup{|B(t)|L(H ) , t ∈ I } ≤ β < ∞; (A2) .F : H −→ H is continuous and there exits a .K > 0 such that |F (x)|2H ≤ K(1 + |x|2H ), and |F (x) − F (y)|2H ≤ K|x − y|2H , ∀ x, y ∈ H ;

.

(A3) . is strongly measurable on I with values in the space of Hilbert-Schmidt operators .L2 (E, H ) and Bochner integrable, that is, . ∈ L1 (I, L2 (E, H )), and W is a cylindrical Brownian motion with values in E. Then, for any .F0 measurable initial state .x0 ∈ L2 ( , H ), the integral equaa (I, L ( , H )) having tion (7.4) has a unique .Ft -adapted mild solution .x ∈ B∞ 2 continuous (version) sample paths. Proof The proof is based on Banach fixed point theorem. For detailed proof see [36]. 

It is interesting to note that the nonlinear Schrödinger equation as presented above satisfies conservation of mass. Indeed, by scalar multiplying on either side

7.1 Nonlinear Schrödinger Equation

187

of the evolution equation (7.3) by .x(t) in H , we obtain the following expression  2 2 E|x(t)| . H = E|x0 |H + 2E



t

+2E

t



0

(B(s)x(s), x(s))ds 0



t

(F (x(s)), x(s))ds + 2E

0

t

(Ax(s), x(s))ds + 2E

 ∗ (s)x(s), dW (s)E ds, t ∈ I. (7.5)

0

It is easy to verify that  E

.

I

| ∗ (s)x(s)|2E ds ≤ |x|2B∞ a (I,L ( ,H )) 2

 0

T

|(s)|2L2 (H,E) ds < ∞

and hence 

t

E

.

 ∗ (s)x(s), dW (s)E = 0, t ∈ I.

0

Since the scalar products .(Ax, x)H = (Bx, x)H = (F (x), x)H = 0, and the stochastic integral vanishes, it follows from Eq. (7.5) that E|x(t)|2H = E|x0 |2H , t ∈ I.

.

(7.6)

This shows that the stochastic Schrödinger equation also preserves the law of conservation of mass. In [36] we made a mistake in regard to conservation of mass and reported in the negative and presented an incorrect formula. Except for this, all the results of [36] are correct. Remark 7.1.2 If the function f , defining the nonlinear operator F , is uniformly bounded then the assumption (A2) holds. If f is a polynomial of finite degree, then the operator F is only locally Lipschitz and bounded on bounded sets. In this case a solution exists locally and may blow up in finite time. However, if f is merely continuous and bounded on bounded sets, we do not expect any pathwise solution. But we may have measure-valued solutions. Before proceeding further, we introduce the class of admissible feedback control operators. Let .L(H ) denote the space of bounded linear operators in the Hilbert space H and suppose it is endowed with the strong operator topology denoted by .τso with the corresponding topological space denoted by .Lso (H ). It is known that this is a locally convex sequentially complete Hausdorff topological vector space [97, p. 156]. Let .B∞ (I, Lso (H )) denote the class of functions defined on I and taking values in .Lso (H ) which are bounded and measurable in the strong operator topology. For compatibility with the structure of Schrödinger equation, a

188

7 Examples From Physical Sciences

set . ⊂ Lso (H ) is admissible if each element . ∈ has the form 

≡

.

0 V −V 0

 ,

where .V ∈ L(L2 (D)). We assume that the set . is compatible and bounded in the sense that there exists a finite positive number .β such that .



sup | |L(H ) , ∈ ≤ β < ∞,

and that it is compact in the strong operator topology. For the class of admissible control policies we choose a set .Kad ⊂ B∞ (I, ) ⊂ B∞ (I, Lso (H )). The function space .B∞ (I, Lso (H )) is given the Tychonoff product topology which is equivalent to the topology of point-wise convergence on the interval I with respect to the strong operator topology .τso . The set of admissible controls (feedback operators) .Kad is furnished with the relative Tychonoff product topology denoted by .τπ . We present the following theorem for existence of measurevalued solutions for the Schrödinger equation (7.3). For study of measure-valued solutions we need the generating operators corresponding to this equation. For each control .B ∈ Kad , the infinitesimal generators are given by .

(Aφ)(t, ξ ) = (1/2) Tr((t)∗ (D 2 φ)(t)(ξ )) + (A∗ Dφ(ξ ), ξ ) + (F (ξ ), Dφ(ξ )), (Bφ)(t, ξ ) ≡ ((t)∗ Dφ)(ξ ) ∈ E, (CB φ)(t, ξ ) ≡ Dφ(ξ ), B(t)ξ , for φ ∈ D(A).

In general, on the space of measures .Mrba (H + ), the corresponding evolution equation in its weak form is given by  .

t

μt (φ) = μ0 (φ) + 0

 μs (Aφ)ds +

t

μs (CB φ)ds 0



t

+

μs (Bφ), dW (s)E , ∀ t ∈ I, P -a.s.

(7.7)

0

for each .φ ∈ D(A). Theorem 7.1.3 Let A be the generator of a unitary group of operators in H and .F : H → H is continuous and bounded on bounded sets of H satisfying the approximation properties as in Theorem 4.2.2, and . : I → L2 (E, H ) is Borel measurable and W is a cylindrical Brownian motion with values in a Hilbert space E. Then, for each initial state .μ0 ∈ rba (H ) and control (feedback operator) .B ∈ Kad , the system (7.7) has a unique measure-valued solution .μ ∈ + w + ∗ Mw ∞ (I × , rba (H )) ⊂ L∞ (I × , Mrba (H )). Further, .t −→ μt is .w continuous.

7.1 Nonlinear Schrödinger Equation

189

Proof Define .F˜ (t, x) = F (x) + B(t)x, .t ∈ I , .x ∈ H , and note that B takes values in . , a compact set in .Lso (H ). So, under the given assumptions on the operators .{A, F, B, }, the domain .D(A) of the operator .A is time-invariant. Thus, following 

the same steps as in Theorem 4.2.2, one can obtain the proof. For existence of optimal controls we need the following corollary, an immediate consequence of the above theorem. Corollary 7.1.4 Suppose the assumptions of Theorem 7.1.3 hold. Then, the control to solution map, .B −→ μ, is continuous with respect to Tychonoff product topology w ∗ + .τπ on .Kad and the .w topology on .M∞ (I × , rba (H )). Proof The proof is similar to that of Theorem 6.4.8. So, we present a brief outline. In Theorem 6.4.8, the control was a general function of time and space denoted by .v(t, x), .(t, x) ∈ I × H . Here .v(t, x) = B(t)x and it is the operator valued function B that is chosen as the feedback control induced by the laser beam. The set of admissible controls is a subset .Kad ⊂ B∞ (I, Lso (H )) which is compact in the Tychonoff product topology .τπ . Using this topology and the .w ∗ topology on the + target space .Lw ∞ (I × , Mrba (H )), we arrive at the conclusion as stated. This completes our brief outline of the proof. 

Using the above results we can prove existence of optimal controls for both the problems (DNS-1) and (DNS-2) as seen in Sect. 6.4. We consider one problem. Let .t −→ K(t) be a multi function with values in .cb(H ), the class of nonempty closed bounded subsets of H , and let .γ be a nonnegative countably additive bounded measure on I . The problem is to find a control .B ∈ Kad that maximizes the functional 

T

J (B) = E

.

0

μB t (K(t))γ (dt).

(7.8)

Corollary 7.1.5 Consider the system (7.7) with the set of admissible controls .Kad and suppose the assumptions of Corollary 7.1.4 hold and the multi function K, with values in .cb(H ), is continuous in the Hausdorff metric. Then, the problem (7.8) has a solution. Proof We prove that J is upper semicontinuous on .Kad . Let .Bn ∈ Kad and suppose it converges to .Bo in the Tychonoff product topology .τπ . Since .Kad is .τπ compact, + the limit .Bo ∈ Kad . Let .{μn , μo } ∈ Mw ∞ (I × , rba (H )) denote the solutions of Eq. (7.7) corresponding to the controls .{Bn , Bo } ∈ Kad , respectively. It follows w∗

w (I × ,  + from Corollary 7.1.4 that .μn −→ μo in .M∞ rba (H )) and, as a function + ∗ of .t ∈ I with values in .rba (H ), it is .w continuous P -a.s. Since for each .t ∈ I , ∗ .K(t) is a closed set, it follows from .w continuity that .

lim μnt (K(t)) ≤ μot (K(t)), ∀ t ∈ I, P -a.s.

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7 Examples From Physical Sciences

Now integrating the above expression with respect to the measure .γ , one can verify that   . lim E μnt (K(t))γ (dt) ≤ E μot (K(t))γ (dt). I

I

Clearly, this means .

lim J (Bn ) ≤ J (Bo ).

Thus, J is an upper semicontinuous functional on the set .Kad with respect to the product topology .τπ . Since .Kad is compact in this topology, J attains its maximum on it, proving existence of an optimal (feedback) control. 

There are several control problems of practical interest in nonlinear optics. One interesting problem is the Bolza problem as stated below. Let .G0 be a measurable multi function with nonempty closed values, with .d(ξ, G0 (t)) denoting the usual distance of .ξ from the set .G0 (t) and let .G1 ⊂ H be a nonempty closed set. The objective is to find a control policy .B ∈ Kad that maximizes the concentration of mass of the measure-valued solution along the prescribed running target .G0 (·) and the terminal target .G1 . This is equivalent to minimizing the following weighted cost functional   .J (B) = E (t, ξ )μt (dξ )dt + E (ξ )μT (dξ ), (7.9) I ×H +

H+

where .(t, ξ ) = ρ(ξ )d(ξ, G0 (t)) and .(ξ ) = (ξ )d(ξ, G1 ) and .μ is the solution of Eq. (7.7) corresponding to the control .B ∈ Kad . Both the running and the terminal costs are weighted by continuous nonnegative real valued functions .ρ and ., respectively. These functions are assumed to be uniformly bounded on H . Our objective here is to develop necessary conditions of optimality. Before we proceed with this, let us recall and review that an optimal control .Bo ∈ Kad exists. τπ Let .Bn , Bo ∈ Kad and suppose .Bn −→ Bo . Then, by virtue of Corollary 7.1.4, w∗

along a subsequence if necessary, relabeled as the original sequence, .μn −→ μo . Hence, it is clear from the expression (7.9) that J is continuous in the .τπ topology on every bounded subset of H and so, by virtue of compactness of the set .Kad , there exists a minimizing control .Bo . Next, we present necessary conditions of optimality, whereby one can construct the optimal controls. Theorem 7.1.6 Consider the system given by Eq. (7.7) and the cost functional (7.9) with nonnegative weights .ρ,  uniformly bounded on H . Suppose the set of admissible controls .Kad is convex, and compact in the Tychonoff product topology .τπ . Then, for a control .Bo ∈ Kad to be optimal, with the corresponding solution .μo ∈ w (I × ,  + + M∞ rba (H )), it is necessary that there exists a .ψ ∈ L1 (I × , BC(H ))

7.1 Nonlinear Schrödinger Equation

191

such that the triple .{Bo , μo , ψ} satisfy the following evolution equations dμot = (A∗ μot + C∗Bo μot )dt + B∗ μot , dW E , t ∈ I, μo0 = μ0 , . (7.10)

.

−dψ = (Aψ + CBo ψ)dt + dt, t ∈ I, ψ(T , ·) = ,

(7.11)

and the following inequality, 

T

dJ (Bo , B − Bo ) = E

.

0

μot (CB−Bo ψ)dt ≥ 0, ∀ B ∈ Kad .

(7.12)

Proof Let .Bo ∈ Kad denote the optimal control and .B ∈ Kad any other. Define Bε = Bo + ε(B − Bo ) for .ε ∈ [0, 1]. Since the set .Kad is convex and compact, o ε .Bε ∈ Kad for all .ε ∈ [0, 1] and .B ∈ Kad . Let .μ , .μ denote the solutions of Eq. (7.7) corresponding to the controls .Bo and .Bε , respectively. Then, for every .φ ∈ D(A) we have .

 ε .μt (φ)

= μ0 (φ) +

t

0

 +

t

0

 μεs (Aφ)ds

t

+ 0

μεs (CBε φ)ds

μεs (Bφ), dW (s)E , ∀ t ∈ I, P -a.s.

(7.13)

and  o .μt (φ)

= μ0 (φ) +

t

0

 +

0

t

 μos (Aφ)ds

t

+ 0

μos (CBo φ)ds

μos (Bφ), dW (s)E , ∀ t ∈ I, P -a.s.

(7.14)

Defining mo = lim(1/ε)(με − μo )

.

and subtracting Eq. (7.14) from Eq. (7.13) term by term and dividing by .ε and letting ε → 0, we arrive at the following expression

.

 .

mot (φ)

= 0

t

 mos (Aφ)ds 

t

+ 0

+ 0

t

 mos (CBo φ)ds

+ 0

t

mos (Bφ), dW (s)

μos (CB−Bo φ)ds, t ∈ I, for each φ ∈ D(A).

(7.15)

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7 Examples From Physical Sciences

Clearly, this is equivalent to the following differential equation in weak form, .

dmot (φ) = mot (Aφ)dt + mot (CBo φ)dt + mot (Bφ), dW (t) + μot (CB−Bo φ)dt, t ∈ I, mo0 (φ) = 0 for each φ ∈ D(A).

(7.16)

Note that Eq. (7.16) is nonhomogeneous; it is driven by the difference .B − Bo . The map C∗B−Bo μo −→ mo

.

(7.17)

w (I × , M + is a continuous linear map from .M∞ rba (H )) to itself. Since by assumption .Bo is optimal, it is clear that .J (Bε ) ≥ J (Bo ) for all .ε ∈ [0, 1] and all .B ∈ Kad . Considering the difference of the two functionals .{J (Bε ), J (Bo )}, and dividing it by .ε and letting .ε ↓ 0, one can readily verify that the Gâteaux differential of the functional J at .Bo in the direction .(B − Bo ) satisfies the following inequality

 dJ (B. o , B − Bo ) = E

I ×H +

(t, ξ )mot (dξ )dt



+E H

(ξ )moT (dξ ) ≡ L(mo ) ≥ 0, ∀ B ∈ Kad .

(7.18)

+ Clearly, .mo −→L(mo ) is a continuous linear functional on .Lw ∞ (I × , Mrba (H )). Thus, it follows from (7.17) and (7.18) that the composition map

˜ ∗B−B μo ) C∗B−Bo μo −→ mo −→ L(mo ) ≡ L(C o

.

(7.19)

+ is a continuous linear functional on .Lw ∞ (I × , Mrba (H )). Hence, there exists a w + ∗ .ψ ∈ (L∞ (I × , Mrba (H ))) (the dual of the former) such that

˜ ∗B−B μo ) = E L(C o



.



I



I

=E =E I

C∗B−Bo μo , ψdt CB−Bo ψ, μo dt μo (CB−Bo ψ)dt.

(7.20)

It turns out that, under canonical embedding of any Banach space X into its bidual X∗∗ , there exists a .ψ ∈ LI (I × , BC(H + )) such that the above expression holds. Let us now consider the temporal variation of the scalar product .mo , ψ giving

.

dmot , ψ(t, ·) = dmot , ψ + mot , dψ + dmot , dψ

.

= dmot (ψ) + mot , dψ + dmot , dψ,

(7.21)

7.1 Nonlinear Schrödinger Equation

193

where the last term denotes the quadratic variation. Using the variational equation (7.16) for the first term with .φ replaced by .ψ, we obtain o o o o dm . t , ψ(t, ·) = mt (Aψ)dt + mt (CBo ψ)dt + mt (Bψ), dW (t)

+ μot (CB−Bo ψ)dt + mot , dψ + dmot , dψ, t ∈ I. (7.22) Integrating the above identity we have  .

T

E 0

 dmot , ψ(t, ·) = E

T 0

 mot (Aψdt + CBo ψdt + dψ) + E 

T

+E 0

 μot (CB−Bo ψ)dt + E

T

I

mot (Bψ), dW 

dmo , dψ.

0

Using stopping time argument one can verify that the second term on the right-hand side of the above equation, giving the stochastic integral, vanishes leaving 



T

E

dmot , ψ(t, ·)

.

0

T

=E 0



T

+E 0

mot (Aψdt + CBo ψdt + dψ) 

μot (CB−Bo ψ)dt

T

+E

dmo , dψ.

(7.23)

0

Since the first term on the right-hand side of the above equation does not contain any stochastic integral with respect to the Brownian motion W , the last term giving the quadratic variation also vanishes. Thus, we are left with 

T

E

.

0

 dmot , ψ(t, ·)

T

=E 0

mot (Aψdt + CBo ψdt + dψ) 

T

+E 0

μot (CB−Bo ψ)dt.

(7.24)

Setting .

− dψ = Aψdt + CBo ψdt + dt, t ∈ I,

(7.25)

in the above equation and completing the integration and recalling that .mo0 = 0, we obtain the following identity, E

.



moT (ψ(T , ·))







+E I

mot ()dt

=E I

μot (CB−Bo ψ)dt.

(7.26)

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7 Examples From Physical Sciences

Choosing .ψ(T , ·) = (·), we arrive at the adjoint system .

− dψ = Aψdt + CBo ψdt + dt, t ∈ I, ψ(T , ·) = ,

(7.27)

and Eq. (7.26) reduces to

E moT () + E





.

I

mot ()dt = E

I

μot (CB−Bo ψ)dt,

(7.28)

where the left-hand side coincides with .L(mo ) as defined in the expression (7.18). And the identity satisfies the expression (7.19) as required. Note that Eq. (7.27) coincides with Eq. (7.11), and Eqs. (7.28) and (7.18) give the necessary condition (7.12). Equation (7.10) is the system equation corresponding to the optimal control .Bo , so nothing to prove. This completes the proof of all the necessary conditions of optimality as stated. 

For some problems in quantum optics it may be necessary to avoid certain states, which can be formulated as a control problem. Let .O ⊂ H be an open bounded set to be avoided. In other words, the problem is to find a control .Bo ∈ Kad so that the support of the corresponding measure solution has minimal intersection with the forbidden zone .O. It may not be possible to have the intersection empty. So, we formulate this as 

T

J (B) = E

.

0

μB t (O)dt,

(7.29)

where .μB is the solution of the system given by Eq. (7.7) corresponding to the control .B ∈ Kad . The problem is to find a control that minimizes this functional. For existence of optimal controls, we note that B −→ μB t (O)

.

is lower semicontinuous with respect to the Tychonoff product topology .τπ on Kad . Using this fact and Fatou’s lemma one can verify that the functional J is lower semicontinuous. Hence, the existence of an optimal control follows from compactness of the set .Kad . For necessary conditions of optimality, we can use Theorem 7.1.6. Consider the cost functional given by (7.9) and set .(t, ξ ) = χO (ξ ), the characteristic function of the set .O, and . = 0. This gives us the desired cost functional (7.29). Thus, Theorem 7.1.6 with the above changes gives the necessary conditions of optimality. Using the above results we can develop a numerical algorithm, whereby one can construct the optimal policy. Here we present a convergence theorem that guarantees the existence of a sequence of control operators in .Kad along which the cost functional decreases monotonically to at least a local minimum.

.

7.1 Nonlinear Schrödinger Equation

195

Proposition 7.1.7 Consider the system (7.7) with the cost functional (7.9) and suppose the assumptions of Theorem 7.1.6 hold. Then, there exists a sequence of controls .{Bn } in .Kad along which the sequence of cost functionals .{J (Bn )} monotonically converges to (possibly) a local minimum. Proof (Step 1): Choose any .B1 ∈ Kad and let .μ1 denote the corresponding solution of Eq. (7.10) with .Bo replaced by .B1 . (Step 2): Let .ψ 1 denote the solution of the adjoint system (7.11) with .Bo replaced by .B1 . (Step 3): Use the pair .{B1 , ψ 1 } in place of the pair .{Bo , ψ} in the functional given by the expression (7.12). At this stage we have  dJ (B1 , B − B1 ) = E

.

I

μ1t (CB−B1 ψ 1 )dt.

(7.30)

If .dJ (B1 , B − B1 ) ≥ 0 holds for all .B ∈ Kad , then .B1 is optimal. This is a rare event and may be ignored. (Step 4): At this step, we must choose .B2 ∈ Kad so that the value of the cost functional .J (B2 ) ≤ J (B1 ). We do so by use of the duality map .D mapping .L1 (H ) (the space of nuclear operators or trace class operators) onto its dual ∗ .L(H ) = (L1 (H )) . Let .L ∈ L1 (H ) and define the duality map as follows  D(L) = B ∈ L(H ) : T r(L∗ B) = T r(B ∗ L) = B 2L(H ) = L 2L1 (H ) .

.

Before using this duality map we rewrite the functional (7.30) in more detail as follows. Using Fubin’s theorem, we have  dJ (B1 , B − B1 ) = E

.

I

 =

E 

I



I

= = I

μ1t (CB−B1 ψ 1 )dt,    Dψ 1 (t, ξ ), (B(t) − B1 (t))ξ H μ1t (dξ )dt H

   Dψ 1 (t, ξ ) ⊗ ξ, B(t) − B1 (t)L1 (H ),L(H ) μ1t (dξ )dt E H

F1 (t), B(t) − B1 (t)L1 (H ),L(H ) dt, B ∈ Kad ,

(7.31)

where .F1 is given by the following integral of the tensor product (.h⊗g for .h, g ∈ H ) F1 (t) = E

 

.

H

 Dψ 1 (t, ξ ) ⊗ ξ μ1t (dξ ), t ∈ I.

(Step 5): Construct B2 (t) ≡ B1 (t) − εL1 (t), for an L1 (t) ∈ D(F1 (t)), t ∈ I.

.

196

7 Examples From Physical Sciences

Choosing .B2 for B in the expression (7.31) we obtain the Gâteaux differential of J at .B1 in the direction .−εL1 (t), .t ∈ I ,  .dJ (B1 , −εL1 ) = −ε F1 (t), L1 (t)L1 (H ),L(H ) dt I

 = −ε

I

 F1 (t) 2L1 (H ) dt

 = −ε

I

 L1 (t) 2L(H ) dt.

(7.32)

Thus, for .ε > 0 sufficiently small, the value of the cost functional J at .B2 is given by J (B2 ) = J (B1 ) + dJ (B1 , B2 − B1 ) + o( (B2 − B1 ) )

.

= J (B1 ) + dJ (B1 , −εL1 ) + o(ε)  = J (B1 ) − ε  L1 (t) 2L(H ) dt + o(ε).

(7.33)

I

(Step 6): Choose .B2 as given at the end of Step 5, and return to (Step 1) to continue the process for .{Bn }, .n ≥ 2. It is clear from this process that we obtain a monotone decreasing (nonincreasing) sequence .J (B1 ) > J (B2 ) > J (B3 ) > · · · > J (Bn ) > J (Bn+1 ) > · · · . Since both the cost integrands are nonnegative, .J (B) ≥ 0 for all .B ∈ Kad . Thus, it is clear that there exists a finite number .c0 ≥ 0 such that .limn→∞ J (Bn ) = c0 (possibly a local minimum). This completes the proof. 

7.2 Stochastic Navier-Stokes Equation In Sect. 6.1 we have considered boundary control problems for deterministic Navier-Stokes equation (NSE). Here we consider the stochastic NSE with locally distributed controls. Using projection operator P to divergence free vector fields, stochastic NSE with nonslip boundary conditions can be written as an abstract SDE on a Hilbert space [10, 28, 75]. This is given by .

dv + (γ Av + B(v))dt = F0 (t, u)dt + F (t)dw, t ≥ 0, v(0) = v0 , F0 (t, u) ≡ f0 (t) + u,

(7.34)

where .γ > 0 is the coefficient of kinematic viscosity, .Av ≡ P (−v), .P (p) = 0, and .Bv = P (b(v, v)). We have already seen that the nonlinear operator B arises

7.2 Stochastic Navier-Stokes Equation

197

from the convective term, where the bilinear form b is given by  b(u, v) =

(u(z) · )v(z)dz

.



with . ⊂ R d , d = 1, 2, 3, being a bounded open domain with smooth boundary .∂. The operator A, known as the Stokes operator, is an unbounded positive self adjoint operator in H with strictly positive eigenvalues. Hence, A has continuous inverse .A−1 , which is a compact operator in H . The most important component in the dynamics is the convective term represented by the nonlinear operator B. The property of this operator is studied through the trilinear form given by  L(x, y, z) ≡

(x(ξ ) ·

.





)y(ξ ), z(ξ )R d dξ.

This is a well-defined function on .V × V × V and the reader can easily verify that on the divergence free vector fields, .L(x, y, y) = 0 for all x, .y ∈ V . Further, it follows from this fact that the operator .B(·) maps V into .V ∗ . For further details on the properties of this trilinear form see [75, 119]. In the expression, .F0 = f0 + u, .f0 is the natural volume force with values in H and u is the volume force generated by the feedback controller defined below, w is the standard .R m valued Brownian motion, and F is an operator valued function with values .F (t) ∈ L(R m , H ). We need the divergence free Hilbert space H = P (L2 (, R d )) ≡ Lσ2 (, R d ),

.

and the NSE-related Gelfand triple: .V → H → V ∗ , where .V ⊂ H01 () as mentioned above. The corresponding infinitesimal generators for evolution equations on the space of measure-valued processes are given by A(t)ϕ(ξ ) ≡ (1/2) Tr(D 2 ϕ(ξ )(F (t)F ∗ (t))),

.

B(ϕ)(ξ ) ≡ −γ Aξ − B(ξ ), Dϕ(ξ )V ∗ ,V ,

(7.35)

C1 (u)(t)ϕ(ξ ) ≡ F0 (t, u), Dϕ(ξ )H , C2 (t)ϕ(ξ ) ≡ F ∗ (t)Dϕ(ξ ), ξ ∈ H. Then, the controlled stochastic Navier-Stokes equation (SNSE) on the space of measures is defined in the weak form as follows .

dμt (ϕ) = μt (Aϕ)dt + μt (Bϕ)dt + μt (C1 (u)ϕ)dt + μt (C2 ϕ), dw, μ0 (ϕ) = ν0 (ϕ), for t ∈ I = [0, T ], ϕ ∈ D(A) ∩ D(B).

(7.36)

Here we wish to use feedback controls based on partial state information. Let .H0 be a closed linear subspace of H and .P0 the projection of H onto .H0 = P0 (H ). Let .ϒ be a continuous bounded possibly nonlinear operator (called observer) mapping

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7 Examples From Physical Sciences

H0 to Y , where Y is another Hilbert space (called observation space). The output of the observer is given by .y = ϒ(P0 x). This data is used by a linear controller (operator) . ∈ L(Y, H ), providing the control .u = ϒ(P0 x). With this feedback control,

.

F0 (t, u) = f0 + u = f0 + ϒ(P0 x),

.

and hence, with a slight abuse of notation, the operator .C1 (u) can be rewritten as C1 (u)ϕ ≡ C1 ( )ϕ = f0 + ϒ(P0 (ξ )), Dϕ(ξ ).

.

Let .FCad ≡ B1 (L(Y, H )) (closed unit ball in .L(Y, H )) denote the class of admissible controls (control laws). The objective is to find a feedback control law . ∈ FCad that minimizes a certain cost functional to be introduced shortly. It is well-known that the closed unit ball .B1 (L(Y, H )) is compact in the weak operator topology .τwo [69]. In fact, any positive multiple of the closed unit ball is compact in the weak operator topology. We assume that the admissible set .FCad has been endowed with this topology. Note that one can replace the closed unit ball by any closed ball of finite radius. We consider control problems after presenting some essential results concerning existence and regularity of measure-valued solutions of the evolution equation (7.36). In the study of NSE, the most appropriate spaces are the Gelfand triple .{V , H, V ∗ } as introduced above. Here, for the study of stochastic NSE, we use probability measures on such spaces. Let .C(H ) denote the class of all real valued continuous (not necessarily bounded) functions on H and define .k(x) = (1+|x|2H ). Let us introduce the class of functions   C2 (H ) ≡ ϕ ∈ C(H ) : sup {|ϕ(x)|/k(x)} < ∞ .

.

x∈H

With respect to the norm topology, .|ϕ|C2 (H ) = supx∈H {|ϕ(x)|/k(x)}, .C2 (H ) is a Banach space. The topological dual of this space, denoted by .M2 (H ), is the space of regular signed Borel measures on H satisfying  k(x)|μ|(dx) < ∞,

.

H

where .|μ| is the positive measure on H induced by the variation of the measure .μ. Thus, a continuous linear functional . on .C2 (H ) has the form  .(ϕ) = ϕ(x)μ(dx). H

7.2 Stochastic Navier-Stokes Equation

199

Clearly,





|(ϕ)| =

ϕ(x)μ(dx)

≤ (|ϕ(x)|/k(x)) k(x)|μ|(dx)

.

H

H

≤ |ϕ|C2 (H ) |μ|M2 (H ) < ∞. Thus, .C2∗ (H ) ∼ = M2 (H ) is an isometric isomorphism. Let X be any Hilbert space and .M+ (X) ⊂ M2 (X) denote the space of Borel probability measures on X having 2 finite second moments. That is, any .μ ∈ M+ 2 (X) has the property that 

 k(ξ )μ(dξ ) =

.

H

H

(1 + |ξ |2X )μ(dξ ) < ∞.

We consider measure-valued stochastic processes. In particular, we use the following spaces. Let .La∞ (I, L1 ( , M+ 2 (X))) denote the class of .Ft - adapted essentially bounded measurable stochastic processes defined on I and taking values in + + a .L1 ( , M (X)). Similarly, we have the space .L (I, L1 ( , M (X))). Below we 2 2 1 take .X = {H, V }, the energy spaces. First, we consider the question of existence and regularity properties of measure solutions of the system (7.36) before considering control problems. Theorem 7.2.1 Consider the system (7.36) with admissible controls from .FCad , F ∈ L2 (I, L2 (R m , H )), .f0 ∈ La2 (I, L2 ( , H )) and suppose .ϒ has at most linear growth satisfying .ϒ(0) = 0. Then, for any initial state .ν0 satisfying .E|ν0 |M + (H )
0. Letting .ic denote the embedding constant .V → H , and defining δc ≡ max{1, ic2 } and using it in the expression (7.42) we obtain

.

 t

 t  t

2

.2 E μs (f0 , ξ )ds ≤ (1/ε)E |f0 |H ds + (δc ε)E |μs |M+ (V ) ds, t ∈ I.

2 0

0

0

(7.43) Under our assumption, the observer .ϒ has at most linear growth satisfying .ϒ(0) = 0, and the feedback control law . ∈ B1 (L(Y, H )). Thus, there exists a finite positive number .β such that .| ϒ(P0 ξ )|H ≤ β|ξ |H . Hence, the second term in the expression (7.41) satisfies the following inequality



 t



μs ( ϒ(P0 ξ ), ξ H )ds ≤ 2β .2 E |μs |M+ (H ) ds, t ∈ I.

2

(7.44)

0

Thus, it follows from the expression (7.41) and the inequalities (7.43) and (7.44) that

 t

 t  t

2

≤ (1/ε)E E . μ (C ( )ϕ)ds |f | ds + (δ ε)E |μs |M+ (V ) ds s 1 0 c H

0

0

0



t

+2βE 0

2

|μs |M+ (H ) ds, t ∈ I. (7.45) 2

Considering the fifth term (Itô integral) on the right-hand side of Eq. (7.37), and noting that .F (·) is a Hilbert-Schmidt (deterministic) operator valued function and .μt is .Ft -adapted (in the weak star sense), it is transparent that the integrand .t −→ μt (C2 ϕ) is .Ft -adapted. Thus, it follows from stopping time argument that the expected value of the stochastic integral vanishes. Since .ε > 0 is otherwise arbitrary, by choosing .ε = (γ /δc ) in the above expression, and using

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7 Examples From Physical Sciences

the inequalities (7.38), (7.39), (7.40) and (7.45) we obtain  E|μ . t| + M (H ) + γ E 2

t 0

 |μs |M+ (V ) ds ≤ E|ν0 |M+ (H ) + E 2



t

+(δc /γ )E 0

2

 |f0 |2H ds + 2βE

t 0

t

T r(F F ∗ )ds

0

|μs |M+ (H ) ds, t ∈ I. 2

(7.46)

By virtue of Grönwall’s inequality, it follows from the above expression that .

E|μt |M+ (H ) 2

  t ≤ C E|ν0 |M+ (H ) + E T r(F F ∗ )ds 2

0



t

+(δc /γ )E 0

 |f0 |2H ds , t ∈ I,

(7.47)

where .C = exp{2βT } and hence  E I

  + |μ . s| + ds ≤ (4βT C/γ ) E|ν | + E 0 M (H ) M (V ) 2

2

T

T r(F F ∗ )ds

0



T

+(δc /γ )E 0

 |f0 |2H ds

.

(7.48)

F is assumed to be deterministic, the expectation operation on the integral Since t T r(F (s)F ∗ (s))ds is irrelevant. In view of the inequalities (7.47) and (7.48), it is 0 clear that

.

+ a μ ∈ La∞ (I, L1 ( , M+ 2 (H ))) ∩ L1 (I, L1 ( , M2 (V ))),

.



proving the regularities as stated.

Remark 7.2.2 In the proof we have used stopping time argument to justify that the expected value of the Itô integral vanishes. A direct approach is as follows. It is easy to verify that the stochastic integral satisfies the following inequality,

 t

 t



∗

. E μs (C2 ϕ), dw = 2 E μs (F ξ ), dw

0 0  t

≤2 0

|F (s)|2H.S E|μs |M+ (H ) ds. 2

Adding this term on the right-hand side of the inequality (7.46) and using Winter’s inequality [28, Theorem 7.8.1, p. 191], [39, Lemma 5.2.1, p. 338], one can verify that .E|μs |M+ (H ) < ∞, ∀ s ∈ I . Hence, .t −→ μt (C2 ϕ) is square integrable. 2 Thus, the stopping time argument is justified to conclude that the stochastic integral vanishes. For detailed proof see [28].

7.2 Stochastic Navier-Stokes Equation

203

Remark 7.2.3 Note that the inequalities (7.47) and (7.48) hold for all . ∈ FCad . Let .μ denote the solution of Eq. (7.36) corresponding to the control law . ∈ FCad . Then, it follows from the above theorem that the solution set

S ≡ μ , ∈ FCad

(7.49)

.

is a bounded subset of .M+ N . According to the norm topology used in the above theorem, these are probability measure-valued stochastic processes having finite w (I × ,  + second moments. Clearly, the solution set .S ⊂ M∞ rba (H )). In any practical feedback control problem, the sensor (or observer) has limited access to data surrounding the system and hence it can provide only limited information for feedback. This is assumed in the following theorem. In order to consider control problems, we need the following result on the continuity of control to solution map. Theorem 7.2.4 Consider the system (7.36) with the set of admissible (feedback) controls .FCad and suppose the assumptions of Theorem 7.2.1 hold and the observer .ϒ maps bounded subsets of .P0 (H ) into weakly sequentially compact subsets of Y . Then, the control to solution map . −→ μ is continuous with respect to the weak w (I × ,  + operator topology on .FCad and the .w ∗ topology on .M∞ rba (H )). Proof Let .{ n } be any generalized sequence in .FCad and suppose it converges in the weak operator topology .(τwo ) to . o . Since .FCad is .τwo compact, . o ∈ FCad . Let .μn and .μo denote the solutions of Eq. (7.36) corresponding to controls (feedback laws) . n and . o , respectively. Clearly, it follows from Eq. (7.36), written in the weak form, that the difference .(μo − μn ) satisfies the following identity,  .

E(μot − μnt )(ϕ) = E 

t

+E 0

t 0

(μos

 (μos − μns )(Aϕ)ds + E

t 0

 − μns )(C1 ( o )ϕ)ds 

t

+E 0

(μos − μns )(Bϕ)ds t

+E 0

μns (C1 ( o − n )ϕ)ds

(μos − μns )(C2 ϕ), dw(s), t ∈ I,

(7.50)

for every .ϕ ∈ D(A) ∩ D(B) having compact support. Again, one can justify that the last term, representing the stochastic integral, vanishes. Thus, we have  o n E(μ . t − μt )(ϕ) = E

t 0

 (μos − μns )(Aϕ)ds + E 

t

+E 0

t 0

(μos − μns )(Bϕ)ds

(μos − μns )(C1 ( o )ϕ)ds 

t

+E 0

μns (C1 ( o − n )ϕ)ds, t ∈ I.

(7.51)

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7 Examples From Physical Sciences

w (I × ,  + Since the sequence .{μn } is contained in a bounded set .S ⊂ M∞ rba (H )) ∗ w + and the later set is a .w closed bounded subset of the space .L∞ (I × , Mrba (H )), it follows from Alaoglu’s theorem that there exists a generalized subsequence, w (I × ,  + relabeled as the original sequence, and a .λo ∈ M∞ rba (H )) such that n o .μ → λ in the weak star topology. Recall that the last term in the above expression is given by

 E

t

.

0

μns (C1 ( o − n )ϕ)ds =E

 t 0

H

(( o − n )ϒ(P0 ξ ), Dϕ(ξ )H )μns (dξ )ds, t ∈ I. (7.52)

Since . n converges to . o in the weak operator topology on .B1 (L(Y, H )), and the observer .ϒ is a weakly compact map from .H0 = P0 (H ) to Y and .Dϕ has compact support, it is clear that the integrand converges to zero in .BC(H + ). Note that the integrand is deterministic and also independent of time, and I is a finite interval. Hence, one can consider the convergence of the integrand to zero in the space .L1 (I × , BC(H + )). On the other hand, the measure .μn converges in the .w ∗ w (I × ,  + w + topology to .λo in .M∞ rba (H )) ⊂ L∞ (I × , Mrba (H )). Thus, the above expression, being a duality pairing of the spaces .L1 (I × , BC(H + )) and w + .L∞ (I × , Mrba (H )), converges to zero as .n → ∞. Based on this result and letting .n → ∞, it follows from Eq. (7.51) that  .

E(μot − λot )(ϕ) = E

t 0

 (μos − λos )(Aϕ)ds + E 

t

+E 0

t 0

(μos − λos )(Bϕ)ds

(μos − λos )(C1 ( o )ϕ)ds, t ∈ I,

(7.53)

for every .ϕ ∈ D(A) ∩ D(B) having compact support. It follows from linearity and homogeneity of this equation that .μo = λo . This proves .w ∗ convergence of .μn to o .μ . Thus, the control to solution map is continuous with respect to the topologies as stated. This completes the proof. 

Remark 7.2.5 In the above theorem the test function .ϕ was assumed to have its first Fréchet derivative .Dϕ with compact support. This can be dispensed with if the nonlinear (observer) operator .ϒ is completely continuous having at most linear growth. In what follows, we consider few control problems arising from practical applications. (A-P1): (Turbulence Control) Let .ξd ∈ H denote the desired flow velocity and let .Br (ξd ) ⊂ H denote the closed ball of radius r around .ξd and .d(ξ, Br (ξd )) the distance of the point .ξ from the ball .Br (ξd ) with .r > 0 chosen as small as desired. Define .ψε = ψε (ξ ) = d(ξ, Br (ξd )) exp{−(ε|ξ |2H )} for .ε > 0 as small as desired,

7.2 Stochastic Navier-Stokes Equation

205

and .w1 ∈ L+ 1 (I ). The objective functional is given by the following cost functional 

 

J ( ) ≡ E

.

I

w1 (t)μ t (ψε )dt

=E I

H+

w1 (t)ψε (ξ )μ t,ω (dξ )dt.

(7.54)

The problem is to find a feedback control . o ∈ FCad that minimizes the functional .J ( ). Minimizing this functional is equivalent to minimizing the level of fluctuation of kinetic energy of the fluid beyond the prescribed sphere. Corollary 7.2.6 Consider the system (7.36) with the set of admissible controls FCad and the cost functional given by (7.54). Suppose the assumptions of Theorem 7.2.4 hold. Then, there exists an optimal feedback control law.

.

Proof By Theorem 7.2.4, the control to solution map is continuous in the weak w (I × ,  + operator topology on .FCad and the weak star topology on .M∞ rba (H )). w + w Recall that the predual of .L∞ (I × , Mrba (H )) (containing the set .M∞ (I × , rba (H + ))), is given by .L1 (I × , BC(H + )) and the integrand .w1 ψε is clearly an element of the latter space. In other words, the integral is a natural duality pairing. Thus, it follows from Theorem 7.2.4 that the functional . −→ J ( ) is continuous in the weak operator topology. Since .FCad is compact in this topology, there exists a o . ∈ FCad at which J attains its minimum, proving existence of optimal feedback controls. 

(A-P2): Another important application is found in artificial heart [6, 118]. In a normal cardiovascular system, blood flow is essentially laminar except in the arteries. It is known to cardiologists that abnormal and turbulent blood flow appears to be present in many cardiovascular diseases and may even contribute to their initiation and progression. Thus, according to cardiologists, the blood flow in any artificial heart should be (smooth) laminar. An artificial heart is a small hydraulic device that pumps blood through the lung out into the arteries carrying oxygenated blood to the entire body and then draws back into the pump through the veins to complete one cardiac cycle. This mechanically induced flow can cause hemolysis (rupture of red blood cells releasing hemoglobin into the blood stream), resulting in inefficient heart function. The objective of control is to minimize blood clots and hemolysis. The appropriate cost functional for this is given by the integral of the sum of the norms of the curl vector and the gradient matrix of the velocity vector. These are denoted by .K(v) = curl(v) and .G(v) =  · v, respectively. It is known that curl may induce blood clots and shear stress may cause hemolysis. Note that the operators .K : V −→ H and .G : V −→ H are linear and continuous. The cost integrand can be chosen as    ε (t, ξ ) ≡ α(t)|K(ξ )|2H + β(t)|G(ξ )|2H exp −ε|ξ |2V

.

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7 Examples From Physical Sciences

for .ε > 0 as small as desired, and .{α, β} ∈ L+ ∞ (I ) are nonnegative measurable functions assigning weights to the two components of the cost integrand. The cost functional is then taken as   .J ( ) = E ε (t, ξ )μ t,ω (dξ )dt. (7.55) I

V

Corollary 7.2.7 Consider the system (7.36) with the set of admissible controls FCad and the cost functional given by (7.55). Suppose the assumptions of Theorem 7.2.4 hold. Then, there exists an optimal feedback control law.

.



Proof The proof is similar to that of Corollary 7.2.6.

The problem becomes difficult if the spatial weight (spatial discount) is removed and the cost functional is given by   J ( ) = E

.

I

V

0 (t, ξ )μ t,ω (dξ )dt,

(7.56)

where the integrand is unbounded on V . However, as seen above, Navier-Stokes equation has measure-valued solutions with stronger regularities. They are not only probability measure-valued processes, but also have finite second moments in V as well as in H . Using this additional property we can solve the above control problem. Theorem 7.2.8 Consider the system (7.36) with the set of admissible controls .FCad and the cost functional given by (7.56). Suppose the assumptions of Theorem 7.2.4 hold and that the operator .ϒ is completely continuous (mapping bounded sets onto relatively compact sets) satisfying .ϒ(0) = 0. Then, the control to solution map, . −→ μ , is continuous with respect to the weak operator topology on .FCad and the norm topology on .La∞ (I, L1 ( , M2+ (H ))) ∩ La1 (I, L1 ( , M2+ (V ))). Proof Let . n ∈ FCad be a generalized sequence (net) and . o its limit in the weak operator topology. Let .{μn , μo } denote the corresponding solutions of Eq. (7.37). Thus, we have  Eμ. nt (ϕ)



t

= Eν0 (ϕ) + E

μns (Aϕ)ds

0



t

+E

 Eμ. ot (ϕ) = Eν0 (ϕ) + E

0

t 0

t

+E

0

μns (Bϕ)ds 

μns (C1 ( n )ϕ)ds + E

0



t

+E

 μos (Aϕ)ds + E

0 t

0

μos (C1 ( )ϕ)ds + E

t

μns (C2 ϕ), dw(s).

(7.57)

μos (Bϕ)ds



t 0

μos (C2 ϕ), dw(s).

(7.58)

7.2 Stochastic Navier-Stokes Equation

207

Subtracting Eq. (7.57) from Eq. (7.58) term by term, we obtain  o n E(μ . t − μt )(ϕ) = E

t 0



t

+E 0



t

+E 0

μns (C1 ( o

 (μos − μns )(Aϕ)ds + E

t 0

(μos − μns )(Bϕ)ds

(μos − μns )(C1 ( o )ϕ)ds 

t

− )ϕ)ds + E n

0

μos (C2 ϕ), dw(s), t ∈ I.

(7.59)

First, we consider the second term on the right-hand side of the above expression. For convenience, this term is split into two parts as follows  E

t

.

0

(μos − μns )(Bϕ)ds = −γ E +E

 t 0

V

 t 0

V

(Aξ, Dϕ(ξ ))(μos − μns )(dξ )ds

(B(ξ ), Dϕ(ξ ))(μos − μns )(dξ )ds, t ∈ I, ϕ ∈ D(B). (7.60)

Choosing .ϕ(ξ ) = (1 + |ξ |2V ), an element of the Banach space .C2 (V ), in the above equation and recalling that .Aξ, ξ  = |ξ |2V , we obtain 

t

E . 0

(μos

− μns )(Bϕ)ds

+2E

 t 0

V

= −2γ E

 t 0

V

|ξ |2V (μos − μns )(dξ )ds

B(ξ ), ξ )(μos − μns )(dξ )ds, t ∈ I, ϕ ∈ D(B). (7.61)

It follows from the property of the trilinear form, as seen in the introduction of this section, that the convective term vanishes, leaving 

t

E

.

0

(μos − μns )(Bϕ)ds = −2γ E = −2γ E

 t 0

V

0

V

 t 

t

= −2γ E 0

|ξ |2V (μos − μns )(dξ )ds (1 + |ξ |2V )(μos − μns )(dξ )ds

(μos − μns )(ϕ)ds.

(7.62)

The latter identity follows from the fact that both .{μo , μn } are probability measure-valued processes. Note that .ϕ ∈ ∂B1 (C2 (V )) given by .ϕ(ξ ) = (1 + |ξ |2V ).

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7 Examples From Physical Sciences

Using the above expression in Eq. (7.59) we obtain  E(μot. − μnt )(ϕ) + 2γ E 

t

=E 0

t 0

(μos − μns )(ϕ)ds 

(μos − μns )(Aϕ)ds + E 

t

+E 0

μns (C1 ( o

t 0

(μos − μns )(C1 ( o )ϕ)ds 

t

− )ϕ)ds + E n

0

μos (C2 ϕ), dw(s).

(7.63)

+ a By Theorem 7.2.1, .{μn , μo } ∈ La∞ (I, L1 ( , M+ 2 (H ))) ∩ L1 (I, L1 ( , M2 (V ))). We use this property to complete the proof. Considering the integrand of the last term (Itô integral) in the above equation, it is easy to verify that

E|μos (C2 ϕ)|2R m ≤ |F ∗ (s)|2L(H,R m ) E|μos |M+ (H ) , s ∈ I.

.

2

Since .μo ∈ La∞ (I, L1 ( , M+ 2 (H ))), it is clear that there exists a finite positive number .η such that ess-.sup{E|μos |M+ (H ) , s ∈ I } ≤ η. On the other hand, .F ∈ 2  L2 (I, L(R m , H )) and hence . I E|μos (C2 ϕ)|2R m ds < ∞. Thus, the Itô integral vanishes, turning Eq. (7.63) into the following equation  E(μot.

− μnt )(ϕ) + 2γ E 

t

+E 0

t 0

 (μos

− μns )(ϕ)ds 

(μos − μns )(C1 ( o )ϕ)ds + E

t 0

t

=E 0

(μos − μns )(Aϕ)ds

μns (C1 ( o − n )ϕ)ds, t ∈ I. (7.64)

Considering the first term on the right-hand side of the above equation for the same ϕ giving .Aϕ = (1/2)T r(D 2 ϕF F ∗ ) = T r(F F ∗ ), one can verify that

.

 t

 t

o n

≤E . E (μ − μ )(Aϕ)ds |F (s)|2L(R m ,H ) |μos − μns |M+ (H ) ds, t ∈ I. s s

2 0

0

(7.65) Next, we consider the second term on the right. Recalling the observation operator ϒ having at most linear growth (with growth rate .β), one can readily verify that

.

.

 t

 t

E (μo − μn )(C1 ( o )ϕ)ds ≤ 2β E|μos − μns |M+ (H ) ds, t ∈ I. (7.66) s s

2 0

0

7.2 Stochastic Navier-Stokes Equation

209

Considering the third term on the right and taking the same .ϕ, we have 

t

en (t) ≡ E

.

0

= 2E

μns (C1 ( o − n )ϕ)ds

 t  0

H

 ( o − n )ϒ(P0 ξ ), ξ H μns (dξ ) ds, t ∈ I.

(7.67)

Since, by assumption, the observer .ϒ is a compact map, and convergence in the weak operator topology is uniform on compacts, and solutions are contained in a + a bounded set .S ⊂ La∞ (I, L1 ( , M+ 2 (H ))) ∩ L1 (I, L1 ( , M2 (V ))), it is clear that .

lim en (t) = 0, ∀ t ∈ I.

(7.68)

n→∞

Thus, letting .n → ∞, it follows from the expression (7.64) that  Emot (ϕ) + 2γ E

t

.

0

 mos (ϕ)ds = E

t 0

 mos (Aϕ)ds + E

t 0

mos (C1 ( o )ϕ)ds, t ∈ I, (7.69)

where .mo is a signed measure being the limit of the difference of two probability measure-valued processes .{μo − μn }n≥1 . It follows from linearity and homogeneity of this equation that .mo = 0, implying convergence of .μn to .μo , where .μo ∈ + a La∞ (I, L1 ( , M+ 2 (H ))) ∩ L1 (I, L1 ( , M2 (V ))). Under the stronger assumptions used here, we have shown continuity of the control to solution map, . −→ μ , with respect to the weak operator topology on .FCad and the standard (norm) topology on + + a a .L∞ (I, L1 ( , M (H ))) ∩ L (I, L1 ( , M (V ))). This completes the proof. 

2 2 1 Now we are prepared to consider the control problem (7.56). Corollary 7.2.9 Consider the system (7.36) with admissible controls .FCad and the cost functional given by (7.56). Suppose the assumptions of Theorem 7.2.8 hold and that the operator .ϒ is completely continuous (mapping bounded sets onto relatively compact sets) satisfying .ϒ(0) = 0. Then, there exists an optimal feedback control in .FCad . Proof Considering the cost functional (7.56), it is noted that .0 has quadratic growth on V and hence unbounded on V . However, .0 ∈ L∞ (I, C2 (V )) (deterministic) and .μ ∈ La1 (I, L1 ( , M2 (V ))). Hence, we have    |J ( )| ≤ |0 |L∞ (I,C2 (V )) |μ |La (I,L1 ( ,M + (V ))) < ∞.

.

1

2

Thus, it follows from boundedness of the solution set .S (see (7.49)) that .

sup {J ( ), ∈ FCad } < ∞.

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7 Examples From Physical Sciences

It follows from continuity of the cost functional . −→ J ( ) in the weak operator topology and compactness of the admissible set .FCad in the same topology, that J attains its minimum on .FCad , proving existence of an optimal feedback control 

(operator). Remark 7.2.10 (1): In the above theorem, we have considered the set of admissible feedback controls to be constant linear operators contained in the closed unit ball .B1 ⊂ L(Y, H ). Endowed with the weak operator topology .τwo , this set is compact. This remains true for any positive multiple .rB1 , r > 0. (2): We can also extend this to time-dependent operators. Again, assume that .Br = rB1 is  furnished with the weak operator topology. Consider the product space .Fc = t∈I Ct , where .Ct = rB1 for all .t ∈ I . Endowed with the Tychonoff product topology [123], this set is also compact. The results presented for constant feedback controls can be extended to this space .Fc of time-dependent feedback operators without any extra efforts. We wish to present a simpler and direct proof of Theorem 7.2.8. Proposition 7.2.11 Suppose all the assumptions of Theorem 7.2.8 hold. Then, the control to solution map, . −→ μ , is continuous with respect to the weak operator topology on .FCad and the norm topology on .L∞ (I, L1 ( , M+ 2 (H ))) ∩ (V ))). L1 (I, L1 ( , M+ 2 Proof We start with the expression (7.59) and note that it holds for all .ϕ ∈ D(A) ∩ D(B). We have also proved that the Itô integral vanishes. Now in the limit, .n → ∞, the control term also vanishes, as seen in the preceding theorem, and the difference o n .(μ − μ ) converges to some signed measure, say .ν, giving the expression 

t

Eνt (ϕ) = E

.



t

νs (Aϕ)ds + E

0



t

νs (Bϕ)ds + E

0

νs (C1 ( o )ϕ)ds, t ∈ I.

0

(7.70) Clearly, it follows from triangle inequality that 

t

|Eνt (ϕ)| . ≤E

|νs (Aϕ)|ds

0



t

+E 0



t

|νs (Bϕ)|ds + E

|νs (C1 ( o )ϕ)|ds, t ∈ I.

(7.71)

0

Consider the Banach space .C2 (V ) and let .B1 (C2 (V )) denote the closed unit ball in the Banach space .C2 (V ). Let .|ν| denote the positive measure induced by the variation of the signed measure .ν. Using the dual pair .{C2 (V ), M2 (V )} and Hölder

7.3 Reaction Diffusion Equation (Biomedical Application)

211

inequality in the above expression, one can easily verify that 



t

|Eν. t (ϕ)| ≤

E|νs |M+ (V ) |(Aϕ)|C2 (V ) ds + 2

0



t

+

0

t

E|νs |M+ (V ) |(Bϕ)|C2 (V ) ds 2

E|νs |M+ (V ) |(C1 ( o )ϕ)|C2 (V ) ds, t ∈ I. 2

0

(7.72)

This inequality holds for every .ϕ ∈ B1 (C2 (V )) that also belongs to the domains of A and .B. Since the embedding .V → H is continuous with embedding constant .ic , one can easily verify that .C2 (H ) → C2 (V ), with embedding constant .δc = max{1, ic2 }. Hence, .|Aϕ|C2 (V ) ≤ δc |Aϕ|C2 (H ) . Recall that .F ∈ L2 (I, L(R m , H )) and therefore .F F ∗ ∈ L1 (I, L1 (H )), where .L1 (H ) denotes the space of nuclear operators in H . Thus, there exists a constant .a > 0 such that .

|Aϕ|C2 (V ) ≤ aδc |F (s)F ∗ (s)|L1 (H ) ≡ g(s), s ∈ I,

.

where .g ∈ L+ 1 (I ). Similarly, one can verify that there exist two constants .b > 0, o .c > 0 such that for .ϕ ∈ B1 (C2 (V )), we have .|Bϕ|C2 (V ) ≤ b and .|C1 ( )ϕ)|C2 (V ) ≤ c. Hence, for .f (t) ≡ g(t) + b + c, we have 

t

|Eνt (ϕ)| ≤

.

0

f (s)E|νs |M+ (V ) ds, t ∈ I, 2

(7.73)

which holds for all .ϕ ∈ B1 (C2 (V )) that belong to the domains of the operators .A, B and .C1 ( o ). Hence, it follows from the above inequality that

.

 E|νt |M+ (V ) ≤

t

.

2

0

f (s)E|νs |M+ (V ) ds, t ∈ I, 2

(7.74)

and consequently, by Grönwall’s inequality, .E|νt |M+ (V ) = 0, .t ∈ I . Again, one can 2 verify that the embedding .M2 (V ) → M2 (H ) is continuous with the embedding constant .δc . Thus, E|νt |M+ (H ) ≤ δc E|νt |M+ (V ) = 0, t ∈ I.

.

2

2

This shows that .ν = 0, proving strong convergence of .μn to .μo .



7.3 Reaction Diffusion Equation (Biomedical Application) We consider deterministic reaction diffusion equation with application to immunology, particularly immunotherapy, to fight cancer [35]. There are two competing populations, namely antigens (pathogens) and antibodies. Antibodies are naturally

212

7 Examples From Physical Sciences

produced by the body’s immune system in response to invading (pathogens) cancer causing antigens. In case the immune system is compromised, it is necessary to use appropriate drugs to boost up the immune system, leading to the questions of optimal choice of drugs and their doses and hence optimal controls. A complete description, starting from partial differential equations leading to an abstract model, is given in [35]. For lack of space we do not go into details. Let .Q ⊂ R 3 be an open bounded set with smooth boundary .∂Q and .C(Q, R 2n ) ≡ C(Q, R n )×C(Q, R n ) the Banach space of continuous functions on .Q taking values in .R 2n with 2n denoting the n pairs of antigens and antigen-specific antibodies. Let .Q0 ⊂ Q be a closed bounded set and .U = C(Q0 , R m ), where m denotes the number of distinct choices of drugs used to produce antibodies or empower the natural antibodies to bind onto cancer cells and destroy them. In the abstract form, the system is governed by a differential equation on the Banach space .E0 = C(Q, R 2n ), dy/dt = Ay + F (t, y, u), y(0) = y0 ∈ E0 , t ∈ I,

.

(7.75)

where .y = (y 1 , y 2 ) denotes the pair of spatial population densities of antibodies and antigens respectively as a function of time. The operator A is derived from .D subject to homogeneous Neumann boundary conditions with D denoting a diagonal .2n × 2n matrix with positive entries representing the diffusion coefficients. It is known [35] that A generates a compact .C0 semigroup .S(t), .t > 0, on .E0 and that .E0 is invariant under .S(t), .t ≥ 0. The function .F : I × E0 × U −→ E0 is Borel measurable in all the variables, continuous in the last two arguments having polynomial growth (of finite order) in the state variable .y ∈ E0 . Under these very general assumptions, it is shown that Eq. (7.75) has a unique mild solution [35, Theorem 2.6]. For measure-valued solutions it suffices if F is merely continuous and bounded on bounded sets. Before we consider control problems, we must have complete characterization of the class of admissible controls .Uad . To include nonconvex control problems, we consider relaxed controls. Recall that U is a separable Banach space. Let .Ub ⊂ U be a weakly compact set with elements having norm equal to or less than b where .b > 0. Since U is separable, its dual .U ∗ is weak star separable and hence .Ub is metrizable [69, Theorem V.6.3] with respect to which it is a compact Polish space. Let .M1 (Ub ) denote the space of probability measures on the Borel subsets of the set .Ub and .BC(Ub ) the space of bounded continuous functions on .Ub with the usual sup norm topology. Since .Ub is a compact Polish space, .M1 (Ub ) is also a compact Polish space. w (I, M (U )) ⊂ Lw (I, M(U )) denote the class of weak star measurable Let .M∞ 1 b b ∞ functions defined on I and taking values from the space .M1 (Ub ). By Alaoglu’s w (I, M (U )) is a weak star compact subset of .Lw (I, M(U )), theorem, the set .M∞ 1 b b ∞ where .M(Ub ) is the space of regular Borel measures on the Polish space .Ub . For w (I, M (U )). It is clear that the admissible controls, we choose the set .Uad = M∞ 1 b w embedding .L∞ (I, Ub ) → M∞ (I, M1 (Ub )) is continuous. In preparation, we present the following theorem on continuous dependence of solutions on control.

7.3 Reaction Diffusion Equation (Biomedical Application)

213

Theorem 7.3.1 Consider the system (7.75) with the admissible controls .Uad . Suppose A is the infinitesimal generator of a .C0 semigroup on .E0 and .F : I × E0 × Ub −→ E0 is Borel measurable in all the variables, integrable on I and continuous and bounded on bounded subsets of .E0 ×Ub . Then, (i): for each .y0 ∈ E0 or .μ0 ∈ M1 (E0 ) and .u ∈ Uad , Eq. (7.75) has a unique measure-valued solution w u .μ ∈ M∞ (I, M1 (E0 )); (ii): the control to solution map, .u −→ μ , is continuous w with respect to weak star topologies on .Uad and .M∞ (I, M1 (E0 )), respectively. Proof The proof of existence is similar to those in Chap. 3 and the proof of continuous dependence of solution with respect to controls is similar to that of Theorem 6.1.1. 

Now we are prepared to consider control problems. For distinction, we write E0 = EB × EG , where .EB = EG = C(Q, R n ) ≡ E. Elements of .EB and .EG are denoted by .ζ and .η, representing the spatial density of antibodies and antigens, respectively. For the elements of .E0 , we use the pair .ξ = (ζ, η) ∈ EB × EG . w (I, M (E )) denote the measure-valued For any control .u ∈ Uad , let .μu ∈ M∞ 1 0 solution of Eq. (7.75) in the sense that for every .ϕ ∈ D(A) ∩ D(B(u)), .μu satisfies the following equation

.

 μut (ϕ) = μ0 (ϕ) +

.

0

t

 μus (Aϕ)ds +

0

t

μus (B(u)(s)ϕ)ds, t ∈ I,

(7.76)

where the operators .A and .B(u) are given by .(Aϕ)(ξ ) = A∗ Dϕ(ξ ), ξ E0∗ ,E0 and .(B(u)ϕ)(t, ξ ) = Dϕ(ξ ), F (t, ξ, u)E ∗ ,E . 0 0 The objective is to find a control that minimizes the antigen population and any possible disparity between a desired (prescribed) level and the actual level of antibody population. With this objective, the cost functional can be chosen as  J (u) =

.

I ×EB ×EG

  w(ζ, η) α(t)|ζ − ζd |2EB + β(t)|η|2EG μut (dζ × dη)dt,

 ≡

I ×E0

(t, ξ )μut (dξ )dt ≡ , μu ,

(7.77)

where .ζd ∈ EB is the desired level of antibody population and .(ζ, η) ∈ EB × EG is the actual population of antibodies and antigens respectively. The functions .α, β ∈ L+ 1 (I ) are temporal weights and .w ∈ BC(E0 ) is a positive spatial weight. Corollary 7.3.2 Suppose the assumptions of Theorem 7.3.1 hold. Consider the system given by Eq. (7.76) with the set of admissible controls .Uad and the cost functional given by (7.77), with .w ∈ BC(E0 ) a positive weight having bounded support and .α, .β ∈ L+ 1 (I ). Then, there exists an optimal control minimizing the cost functional J .

214

7 Examples From Physical Sciences

Proof Note that, under the given assumption on w, the cost functional is a natural duality product in the sense that  J (u) =

.

I ×E0

(t, ξ )μut (dξ )dt ≡ , μu L1 (I,BC(E0 )),Lw∞ (I,M(E0 )) ,

(7.78)

where .μu is the solution of Eq. (7.76) corresponding to the control u. Consider a generalized sequence (net) .{un } ∈ Uad converging to .uo ∈ Uad in the relative w (I, M (U )). Let .{μn , μo } ∈ M w (I, M (E )) denote weak star topology on .M∞ 1 b 1 0 ∞ w∗

the corresponding solutions of Eq. (7.76). It follows from Theorem 7.3.1 that .μn → w (I, M (E )) (in the weak star topology). Under the assumptions, . ∈ μo in .M∞ 1 0 L1 (I, BC(E0 )) and hence, .

J (un ) = , μn L1 (I,BC(E0 )),Lw∞ (I,M(E0 )) −→ , μo L1 (I,BC(E0 )),Lw∞ (I,M(E0 )) = J (uo )

(7.79)

as .n → ∞. This shows that J is continuous on .Uad in the .w ∗ topology. Since .Uad is weak star compact, it is clear that J attains its minimum on .Uad . This proves the existence of an optimal control. 

The fundamental objective is to administer optimal doses of drugs so as to produce sufficient population of effective antibodies which can recognize and bind onto invading antigens to destroy them. So, an alternate approach can be formulated as follows. Recall the state space .E0 = EB × EG with .EG denoting the state space of antigens. Let .K ⊂ EG be a closed bounded set given by

K ≡ η ∈ EG : |η|EG ≤ ε ,

.

where .ε > 0 may be considered as the harmless level of antigens present in the patient’s body at the end of the treatment period .[0, T ]. Consider the cost functional  J (u) ≡

.

EB

μuT (dζ × K ) = μuT (EB × K ) ≡ muT (K ),

(7.80)

where .K = EG \K is the complement of the set .K. The objective is to find a control that minimizes .muT (K ). Corollary 7.3.3 Suppose the assumptions of Theorem 7.3.1 hold. Consider the system given by Eq. (7.76) with the set of admissible controls .Uad and the cost functional given by (7.80). Then, there exists an optimal control minimizing the cost functional J . Proof By Corollary 6.1.1, the function .t −→ μut is continuous on I in the weak star sense. So, .μuT is well-defined. Note that the marginal .μuT (EB × (·)) is a probability measure on the Borel sets of .EG . Hence, .muT (·) ≡ μuT (EB × (·)) is a Borel

7.3 Reaction Diffusion Equation (Biomedical Application)

215 w∗

w∗

probability measure on .EG . We know that, as .un → uo in .Uad , .μn → μo in  w .M∞ (I, M1 (E0 )). Since .K is an open set, it follows from this that moT (K ) ≤ lim mnT (K ).

.

Hence, .J (uo ) ≤ lim J (un ). Thus, J is weak star lower semicontinuous on the weak star compact set .Uad . Hence, it attains its minimum on .Uad , proving existence of an optimal control. 

We present below necessary conditions of optimality for a Bolza problem. Theorem 7.3.4 Consider the system given by Eq. (7.76) with the set of admissible controls .Uad and the cost functional given by  J (u). =

I ×E0

 (t, ξ, ut )μut (dξ )dt +

E0

(ξ )μuT (dξ ) = u , μu  + , μuT , (7.81)

for .u ∈ Uad with .u being a bounded Borel measurable real valued function on .I × E0 ×Ub , continuous on .E0 ×Ub for almost all .t ∈ I ; and . is a real valued function continuous and bounded on .E0 . Then, for a control .uo ∈ Uad to be optimal with .μo being the corresponding solution of Eq. (7.76), it is necessary that there exists a .ψ o , continuous and bounded on bounded sets, such that the triple .{uo , μo , ψ o } satisfies the following inequality and evolution equations,  .

dJ (uo ; u − uo ) = 0

T

o

o

[u−u + Bu−u (ψ o )], μot C(E0 ),M(E0 ) dt ≥ 0, ∀ u ∈ Uad , (7.82)

where .ψ o is a mild solution of the following evolution equation on .Eo , o

.

− dψ o = Aψ o dt + Bu ψ o dt + (t, ·, uot )dt, t ∈ I, ψ o (T , ·) = (·),

(7.83)

w (I, M (E )) is the measure solution of the state equation correand .μo ∈ M∞ 1 0 sponding to the control .uo ,

dμot = A∗ μot dt + (Bu )∗ μot dt, t ∈ I, μo0 = μ0 . o

.

(7.84)

Proof Following similar steps as seen in the proof of Theorem 7.1.6, one can derive the necessary conditions as stated. Interested readers may try for detailed proof.  Optimal Strategy for Immunotherapy For cancer treatment based on immunotherapy, we can also formulate the control problem as follows. Consider the state space .E0 = EB × EG , where .EB and .EG denote the state space of antibodies and antigens, respectively. Let .I ≡ [0, T ] denote the treatment period. The objective

216

7 Examples From Physical Sciences

is to administer optimal doses of antibody producing drugs for treatment. This is achieved by increasing the antibody production to the same level as that of antigen population in the patient’s body. For any prescribed .ε > 0 (as small as desired) and any desirable level .r > 0 (finite but as large as required), define the target set as

 

Kε,r ≡ ξ = (ζ, η) ∈ EB × EG

|ζ − η|E ≤ ε and |ζ |EB ≤ r, |η|EG ≤ r .

.

This set reflects the requirement that the antibody-antigen population difference remains equal or less than .ε, and that their individual population sizes do not exceed r. Here we can choose . = 0 and . as the indicator function of the set .Kε,r , giving .(ξ ) = χKε,r (ξ ) and the objective functional as  J (u) =

.

E0

(ξ )μuT (dξ ) = μuT (Kε,r ).

(7.85)

Note that maximizing the above objective functional is equivalent to maximizing the concentration of probability mass on the target set .Kε,r . Corollary 7.3.5 Consider the system (7.76) with the set of admissible controls Uad and the objective functional given by (7.85). Suppose the assumptions of Theorem 7.3.1 hold. Then, there exists an optimal control maximizing the objective functional (7.85). The necessary conditions of optimality are given by those of Theorem 7.3.4, with . ≡ 0, the terminal cost replaced by (7.85), and the inequality (7.82) reversed.

.

Proof The target set .Kε,r is a closed bounded subset of .E0 = EB × EG , and, by Theorem 7.3.1, the control to solution map, .u −→ μu , is continuous with respect w (I, M (E )), respectively. Also, we know to weak star topologies on .Uad and .M∞ 1 0 that .t −→ μt is weak star continuous on I . Hence, .μT is well-defined and whenever w∗

un −→ uo we have

.

n

.

o

lim μuT (Kε,r ) ≤ μuT (Kε,r ).

Thus, .lim J (un ) ≤ L(uo ) and hence J is upper semicontinuous on .Uad . Since .Uad is weak star compact and J is upper semicontinuous on it with respect to the same topology, J attains its maximum on .Uad . Thus, an optimal control exists. The necessary conditions of optimality for this problem are given by the necessary conditions as stated in Theorem 7.3.4 by setting . ≡ 0 in Eqs. (7.82) and (7.83) and choosing the terminal cost as given by Eq. (7.85) and reversing the inequality in the expression (7.82). This completes the proof. 

Numerical Challenges Currently we are not aware of any existing computational technique that can be directly used or adapted and expanded for numerical solution of differential equations on the space of measure-valued functions on Banach spaces. This remains an interesting open problem. We expect that in the future,

7.4 Bibliographical Notes

217

with the advance of quantum computers, the computational challenges will be effectively overcome. Once such development takes place, many problems based on nonstandard differential equations will be solvable leading to inspiring and possibly unforeseen advancement in applied sciences.

7.4 Bibliographical Notes In this chapter we have presented three examples from physical sciences with applications to physics and engineering, including nonlinear optics, hydrodynamics, and biomedical science. In the area of optics, the optical field is generally governed by a class of nonlinear Schrödinger equations. For such equations there is an extensive literature [56, 66, 92, 93, 106, 108, 109, 111, 117, 120] devoted to the basic questions of existence and uniqueness of pathwise solutions and their regularity properties. Nonlinear Schrödinger equations have also gained special importance in the field of optics and its engineering applications [46, 49, 50, 97, 111, 115]. However, most of the work on control of Schrödinger equations focuses on deterministic linear problems with quadratic cost functionals, such as [46, 49, 50, 97, 111]. These studies employ standard assumptions on the vector fields and construct standard pathwise solutions. In contrast, we have used significantly relaxed assumptions as seen in the theoretical development presented in the preceding chapters. Under our general assumptions Schrödinger equation does not have pathwise solutions. Thus, we have proved existence of measure-valued solutions and their regularity properties. Using these results we prove existence of optimal feedback control laws, which is very useful for engineering applications. For hydrodynamics described by Navier-Stokes equation (NSE) [55], we have considered its stochastic version with locally distributed controls. Using projection to divergence free vector fields, stochastic NSE with nonslip boundary conditions can be written as an abstract SDE on a Hilbert space [10, 28, 75]. We consider the question of existence and regularity properties of measure solutions in Theorem 7.2.1, with applications to optimal control of turbulence [102] and artificial heart [6, 118]. Finally, we have considered reaction diffusion equations with application to immunology, particularly immunotherapy, to fight cancer. A complete description, starting from partial differential equations leading to an abstract model, is given in [35].

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Index

A Alaoglu’s theorem, 9

B Banach space, 1 Bartle-Dunford-Schwartz theorem, 7 Bochner integrals, 5 Boolean isomorphism, 8 Bounded variation, 4 Brooks-Dinculeanu theorem, 8

C Cauchy problem, 16 Continuous selection, 10 Countably additive measure, 4 Cylindrical, 26

D Duhamel’s formula, 106 Dunford-Pettis theorem, 6

E Evolution equations under relaxed hypothesis, 26 competing notions of solutions, 30 existence of solutions, 26 quasilinear problems, 32 existence of measure-valued solutions, 33

Evolution equations with continuous vector fields, 15 existence of measure-valued solutions, 17 generalized solutions, 17 Evolution equations with measurable vector fields, 36 existence of measure solutions, 38 Examples from physical sciences, 183 nonlinear Schrödinger equation, 183 existence and uniqueness of solutions, 185 system model, 184 reaction diffusion equation, 211 cancer immunotherapy, 212 stochastic Navier-Stokes equation, 196 artificial heart (A-P2), 205 turbulence control (A-P1), 204

F Finitely additive measure, 4 Fréchet derivative, 20 Function spaces, 1

G Galerkin approach, 33 Gelfand triple, 144 Goldstine’s theorem, 110

H Hölder inequality, 211 Hahn-Banach extension theorem, 19

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. U. Ahmed, S. Wang, Measure-Valued Solutions for Nonlinear Evolution Equations on Banach Spaces and Their Optimal Control, https://doi.org/10.1007/978-3-031-37260-5

225

226 Hahn-Banach separation theorem, 168 Hausdorff space, 4 Hille-Yosida theorem, 56

I Impulsive systems, 49 classical model, 69 different inclusions, 69 general model, 71 measured valued solutions, 51 Cauchy problem, 51 existence, 55 mild solutions, 52 pathwise solutions, 66 uniqueness of solutions, 68 measure-valued functions, 50

K Kuratowski and Ryll-Nardzewski selection theorem, 11

L Lower semicontinuous, 10

M Martingale representation theorem, 118 Mazur’s theorem, 159 Measurable selection, 10 Method of transposition, 106 Michael’s selection theorem, 10 Multi valued functions, 9

N Neutral evolution equations existence of measure solutions, 104 uniqueness and well-posedness of solutions, 110

O Optimal control of evolution equations, 135 deterministic neutral systems (DNS), 168 Problem (DNS-P1): following a moving target, 171 Problem (DNS-P2): concentration control of probability mass, 172 Problem (DNS-P3): exit time problem, 173

Index Problem (DNS-P4): Lagrange problem, 174 deterministic systems, 135 time optimal control, 141 impulsive systems, 150 Problem (P1): target following problem, 150 Problem (P2): avoiding a danger zone, 153 Problem (P3): Bolza problem, 154 Problem (P4): Bolza problem with uncertainty, 157 stochastic neutral systems (SNS), 175 Problem (SNS-P1): hitting a target, 178 Problem (SNS-P2): exit time problem, 179 stochastic systems, 161 Problem (S-P1): Bolza problem, 161 Problem (S-P2): Lagrange problem, 164 Problem (S-P3): hitting a target set, 166 Problem (S-P4): evasion problem, 166 Ornstein-Uhlenbeck process, 130

P Paracompact space, 10 Polish space, 10

R Radon-Nikodym derivative (RND), 5 Radon-Nikodym property (RNP), 6 Riesz representation theorem, 4

S Second order neutral equations, 122 existence of mild solutions, 126 regularity of solutions, 127 Selection theorem, 10 Kuratowski and Ryll-Nardzewski selection theorem, 11 Michael’s selection theorem, 10 Yankov-von Neumann-Aumann selection theorem, 11 Semi-variation, 5 Set-valued maps, 9 Souslin space, 10 Space of set functions, 2 duality, 2 Stochastic neutral equations, 128 existence of solutions, 129

Index Stochastic neutral systems, 111 existence of measure solutions, 113 regularity of measure solutions, 115 Stochastic systems, 75 existence of measure solutions, 78 martingale measures, 90 basic properties, 91 existence of solutions, 94 measurable vector fields, 101 uniqueness of solutions, 100 measure solutions, 78 Stone’s theorem, 185 T Test functions, 20 Theory of lifting, 8 Topological space Hausdorff, 3 normal, 3 regular, 3

227 U Uniform boundedness principle, 40 Upper semicontinuous, 9

V Vector-valued measures, 4

W Weak Radon-Nikodym property (RNP), 6 Weak star (.w ∗ ) compact, 9 Weak star (.w ∗ ) sequentially compact, 9 Winter’s inequality, 202

Y Yankov-von Neumann-Aumann selection theorem, 11