Contributions to the Control Theory of Some Partial Functional Integro-differential Equations in Banach Spaces

Contributions to the Control Theory of Some Partial Functional Integro-differential Equations in Banach Spaces

ABSTRACT

This thesis is a contribution to Control Theory of some Partial Functional Integro-differential Equations in Banach spaces. It is made up of two parts: controllability and existence of optimal controls. In the first part, we consider the dynamical control systems given by the following models that arise in the analysis of heat conduction in materials with memory, and viscoelasticity, and take the form of a:

Partial functional integro-differential equation subject to a nonlocal initial condition in a Banach space (X; k k) :
8>>>>< >>>>:
x0(t) = Ax(t) +
Z t
0
B(t ? s)x(s)ds + f(t; x(t)) + Cu(t)
for t 2 I = [0; b];
x(0) = x0 + g(x);
(0.0.1)
where x0 2 X; g : C(I;X) ! X and f : I X ! X are functions satisfying some conditions; A : D(A) ! X is the infinitesimal generator of a C0-semigroup
?
T(t)
t0 on X; for t 0, B(t) is a closed
linear operator with domain D(B(t)) D(A). The control u belongs to L2(I;U) which is a Banach space of admissible controls, where U is a Banach space.

Partial functional integro-differential equation with finite delay in a Bavii

Abstract viii
nach space (X; k k) :
8>>>>< >>>>:
x0(t) = Ax(t) +
Z t
0
B(t ? s)x(s)ds + f(t; xt) + Cu(t) ;
for t 2 I = [0; b];
x0 = ‘ 2 C = C([?r; 0];X);
(0.0.2)
where f : I C ! X is a function satisfying some conditions; A :
D(A) ! X is the infinitesimal generator of a C0-semigroup
?
T(t)

t0
on X; for t 0, B(t) is a closed linear operator with domain D(B(t))
D(A). The control u belongs to L2(I;U) which is a Banach space of admissible controls, where U is a Banach space, and xt denotes the history function of C of the state from the time t?r up to the present time t, and is defined by xt() = x(t + ) for ?r 0.

Partial functional integrodifferential equation with infinite delay in a Banach space (X; k k) :
8>>>>< >>>>:
x0(t) = Ax(t) +
Z t
0
(t ? s)x(s)ds + f(t; xt) + Cu(t);
for t 2 I = [0; b]
x0 = ‘ 2 B;
(0.0.3)
?where A : D(A) ! X is the infinitesimal generator of a C0-semigroup
T(t)

t0 on a Banach space X; for t 0, (t) is a closed linear operator with domain D( (t)) D(A). The control u takes values from another Banach space U. The operator C(t) belongs to L(U;X) which is the Banach space of bounded linear operators from U into X, and the phase space B is a linear space of functions mapping ]?1; 0] into X satisfying axioms which will be described later, for every t 0, xt denotes the history function of B defined by xt() = x(t+) for ?1 0; f : I B ! X is a continuous function satisfying some conditions.

We give sufficient conditions that ensure the controllability of the systems without assuming the compactness of the semigroup, by supposing that their linear homogeneous and undelayed parts admit a resolvent operator in the sense of Grimmer and by making use of the Hausdorff measure of non-compactness.

In the second part, we consider equations (0.0.1), (0.0.2) and (0.0.3), in the

Abstract ix

case where the operator C = C(t) (time dependent), the function g = 0, the Banach spaces X and U are separable and reflexive. Using techniques of convex optimization, a priori estimation, and applying Balder’s Theorem, we establish the existence of optimal controls for the following Lagrange optimal control problem associated to each of the equations:

(LP)
8< : Find a control u0 2 Uad such that J (u0) J (u) for all u 2 Uad; where J (u) := Z T 0 L t; xut ; xu(t); u(t) dt; L is some functional, xu denotes the mild solution corresponding to the control u 2 Uad, and Uad denotes the set of admissible controls. TABLE OF CONTENTS

Acknowledgements iv
Abstract vii
1 Introduction 1
1.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Nonlocal Differential Equations . . . . . . . . . . . . . 2
1.1.2 Delay Differential Equations . . . . . . . . . . . . . . . 3
1.2 Partial Functional Integrodifferential Equations . . . . . . . . 6
1.2.1 A Model in Viscoelasticity . . . . . . . . . . . . . . . . 6
1.2.2 A Model in Heat Conduction in Materials with Memory 8
1.3 Controllability of Dynamical Systems . . . . . . . . . . . . . . 10
1.4 Optimal Control of Dynamical Systems . . . . . . . . . . . . . 16
1.5 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5.1 Measures of Noncompactness . . . . . . . . . . . . . . 20
1.5.2 Fixed Point Theory . . . . . . . . . . . . . . . . . . . . 22
1.5.3 Semigroup Theory . . . . . . . . . . . . . . . . . . . . 22
1.5.4 Resolvent Operator for Integral Equations . . . . . . . 23
1.6 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . 23
2 Preliminaries 25
2.1 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Resolvent Operator for Integral Equations . . . . . . . . . . . 27
2.3 Measure of Non-compactness . . . . . . . . . . . . . . . . . . . 32
Abstract xiii
2.4 The Mönch Fixed Point Theorem and Balder’s Theorem . . . 35
I Controllability of some Partial Functional
Integro-differential Equations in Banach Spaces 37
3 Controllability for some Partial Functional
Integro-differential Equations with Nonlocal Conditions in Banach Spaces 38
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 Example of Application . . . . . . . . . . . . . . . . . . . . . . 47
4 Controllability for some Partial Functional
Integro-differential Equations with Finite Delay in Banach Spaces 50
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Controllability result . . . . . . . . . . . . . . . . . . . . . . . 53
4.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5 Controllability Results for some Partial Functional
Integro-differential Equations with Infinite Delay in Banach Spaces 62
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . 64
5.3 Controllability result . . . . . . . . . . . . . . . . . . . . . . . 66
5.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
II Optimal Controls of some Partial Functional
Integro-differential Equations in Banach Spaces 76
6 Solvability and Optimal Control for some Partial Functional
Integro-differential Equations with Finite Delay 77
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2 Existence of mild solutions for equation (6.1.1) . . . . . . . . . 79
6.3 Continuous Dependence . . . . . . . . . . . . . . . . . . . . . 82
6.4 Existence of the Optimal Controls . . . . . . . . . . . . . . . . 86
6.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7 On the Solvability and Optimal Control of some Partial Functional
Integro-differential Equations with Infinite Delay in Banach Spaces 92
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Table of Contents xiv
7.2 Existence of mild solutions for equation (7.1.1) . . . . . . . . . 94
7.3 Continuous Dependence and Existence of the Optimal Control 99
7.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8 Solvability and Optimal Controls for some Partial Functional
Integro-differential Equations with Classical Initial Conditions
in Banach Spaces 109
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.2 Existence of mild solutions for equation (8.1.1) . . . . . . . . . 111
8.3 Continuous Dependence and Existence of the Optimal Control 114
8.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Conclusion and Perspectives 123
Bibliography 124

CHAPTER ONE

Introduction

1.1 General Introduction

In various fields of science and engineering such as Electronics, Fluid Dynamics, Physical Sciences, etc…, many problems that are related to linear viscoelasticity, nonlinear elasticity and Newtonian or non- Newtonian fluid mechanics have mathematical models which are described by differential or integral equations or integro-differential equations which have received considerable attention during the last decades. Control Theory arises in many
modern applications in engineering and environmental sciences [2]. It is one of the most interdisciplinary research areas [22][63] and its empirical concept for technology goes back to antiquity with the works of Archimede, Philon, etc…, [73]. A control system is a dynamical system on which one can act by the use of suitable parameters (i.e., the controls) in order to achieve a desired behavior or state of the system. Control systems are usually modeled by mathematical formalism involving mainly ordinary differential equations, partial differential equations or functional differential equations. In condensed expression, they often take the form of differential equation :
x0(t) = F(t; x(t); u(t)) for t 0; where x is the state and u is the control. While studying a control system, two most common problems that appear are the controllability and the optimal controls problems. The controllability problem consists in checking the possibility of steering the control system from an initial state (initial condition)

Introduction 2

to a desired terminal one (boundary condition), by an appropriate choice of the control u, while the optimal control problem consists in finding the input function (the control or the command) so as to optimize (maximize or minimize)
the objective function. Control Theory of integro-differential equations with classical initial conditions and with delays, have received considerable attention by researchers during the last decades.

This thesis is a contribution to Control Theory of Partial Functional Integro-differential equations in Banach spaces. It is made up of two parts:

• Part I: Controllability results for some partial functional integro-differential equations in Banach spaces.

• Part II: Optimal controls of some partial functional integro-differential equations in Banach spaces.

It lies at the interface between Nonlinear Functional Analysis, Optimization Theory and Dynamical Systems. In the first part, we establish the controllability for some partial functional integro-differential equations, with nonlocal initial conditions, with finite delay and then with infinite delay. The second part deals with the solvability and the existence of optimal controls for these partial functional integro-differential equations, with Cauchy initial conditions, with finite delay and then with infinite delay. We use fixed point techniques to solve the controllability problem and convex optimization techniques to solve the optimal control problems.

1.1.1 Nonlocal Differential Equations

Many problems arising in engineering and life sciences are modeled mathematically by differential equations. Differential equations are one of the most powerful and frequently used tools in mathematical modeling. Depending on the nature of the problem, these equations may take various forms like ordinary differential equations, partial differential equations or functional differential equations. In condensed expression, they often take the following
form:

x0(t) = F(t; x(t)) for t 0;

where x is the state. Most often, these problems are subject to some initial conditions. The classical initial condition is that referred to as the Cauchy initial condition, given by x(0) = x0, where x0 is some initial state of the system at time t = 0. However, in many real world contexts such as Engineering, Environmental sciences, Demography, etc…, nonlocal constraints (such as isoperimetric or energy condition, multipoint boundary condition

Introduction 3

and flux boundary condition) appear and have received considerable attention during the last decades, cf. [14] and [15]. They usually take the following form: x(0) = x0+g(x), where g is a function satisfying some conditions, and
x is the state or a solution of the differential equation in question. Observe that the initial condition in this case depends on the solution of the system.

So, the concept of nonlocal initial condition not only extends that of Cauchy initial condition, but also turns out to have better effects in applications as it may take into account future measurements over a certain period after the initial time t equals 0.