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## ABSTRACT

This thesis is a contribution to Control Theory of some Partial Functional

Integrodifferential 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 integrodifferential 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 integrodifferential 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 tr 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 noncompactness.

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 Noncompactness . . . . . . . . . . . . . . . . . . . 32

xii

Abstract xiii

2.4 The Mönch Fixed Point Theorem and Balder’s Theorem . . . 35

I Controllability of some Partial Functional Integrodifferential

Equations in Banach Spaces 37

3 Controllability for some Partial Functional Integrodifferential

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 Integrodifferential

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 Integrodifferential

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 Integrodifferential

Equations in Banach Spaces 76

6 Solvability and Optimal Control for some Partial Functional

Integrodifferential 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

Integrodifferential 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

Integrodifferential 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 integrodifferential 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)

1

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 integrodifferential 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 Integrodifferential

equations in Banach spaces. It is made up of two parts:

• Part I: Controllability results for some partial functional integrodifferential

equations in Banach spaces.

• Part II: Optimal controls of some partial functional integrodifferential

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 integrodifferential 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 integrodifferential 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.

1.1.2 Delay Differential Equations

In the mathematical description of a great number of physical phenomena,

one usually suppose that the evolution of the system depends only on its

current state. However, there are situations where the evolution of a process

depends not only on its current state, but also on past states of the system.

Such phenomena arise in many areas, in particular, in population dynamics.

Amongst the mathematical models that can describe such situations are delay

differential equations whose delays can be of neutral type. One then has

equations whose temporal terms are, in a general way, nonlocal terms, involving

values of the state at past, discrete or distributed times. These nonlocal

terms are either of order 0 (delay equations) or of order 1 (neutral equations).

In genera1, the delays appear because of the necessary time for the system

to respond to certain evolution, or because a certain threshold (limit value)

must be attained before the system can be activated. Delay differential equations

lie at the interface between ordinary and partial differential equations.

The difference with ordinary differential equations is that the initial data are

themselves functions. This requires more elaborate mathematical study than

for ordinary differential equations, the nature of the delay (discrete, continuous,

infinite, state dependent, ) potentially complicating it.

Example.([5]) Let N(t) denote the density of adults at time t, in a biological

population composed of adult and juvenile individuals. Assume that the length

of the juvenile period is exactly r units of time for each individual. Assume

that adults produce offspring at a per capita rate and that their probability

per unit of time of dying is . Assume that a newborn survives the juvenile

period with probability and put t = . Then the dynamics of N can be

described by the differential equation

dN

dt

(t) = N(t) + kN(t r)

Introduction 4

which involves a nonlocal term, kN(t r) meaning that newborns become

adults with some delay. So the time variation of the population density N

involves the current as well as the past values of N. Such equations are called

delay equations.

In finite dimension, a complete theory has been developed for delay differential

equations, while the theory in infinite dimension is far from being

complete. Such equations in infinite dimension are represented by differential

equations which are nonlocal in time: knowing the solution at a given

time requires knowing it on a time interval whose lenght is equal to the delay.

These types of equations have lots of applications in physics, chemistry,

biology, population dynamics, .

Equations with finite delay genera1ly take the following abstract form:

8<

:

x0(t) = Ax(t) + F(t; xt) for t 0

x0 = ‘ 2 C = C([r; 0];X);

(1.1.1)

where F : R+ C ! X is a function satisfying some conditions; A : D(A)

X ! X is the infinitesimal generator of a C0-semigroup

T(t)

t0 on a

Banach space X; for t 0; C([r; 0];X) (the phase space), denotes the

Banach space of continuous functions x : [r; 0] ! X with supremum norm

kxk1 = supt2I kx(t)kX, 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. When r = 1, we say that the delay is infinite, and this

class of equations englobes equations known in mechanics, namely Volterra

equations. When A = 0, and X = Rn, we have delay differential equations in

finite dimension. For this type of equations, the solution operator is compact,

and this property has made it possible to develop a complete theory for delay

differential equations in finite dimension. In infinite dimension, this property

is no longer satisfied, and this lack of regularity requires the development of

new functional analysis tools to tackle problems related to the qualitative

aspect of the solutions.

In the literature devoted to equations with finite delay, the phase space is

the space of continuous functions on [r; 0], for some r > 0, endowed with

the uniform norm topology. But when the delay is unbounded (i.e., infinite),

the phase space denoted by B is a linear space of functions mapping ]1; 0]

into X satisfying some axioms. The selection of the phase space B plays

an important role in both qualitative and quantitative theories. A usual

choice is a normed space satisfying the following suitable axioms, which was

introduced by Hale and Kato [82]:

Introduction 5

(B; k kB) will be a normed linear space of functions mapping ] 1; 0] into

X and satisfying the following axioms:

(A1) There exist a positive constant H and functions K : R+ ! R+ continuous

and M : R+ ! R+ locally bounded, such that for a > 0, if

x : ] 1; a] ! X is continuous on [0; a] and x0 2 B, then for every

t 2 [0; a], the following conditions hold:

(i) xt 2 B,

(ii) kx(t)k HkxtkB, which is equivalent to k'(0)k Hk’kB for every

‘ 2 B,

(iii) kxtkB K(t) sup

0st

kx(s)k +M(t)kx0kB.

(A2) For the function x in (A1), t ! xt is a B-valued continuous function

for t 2 [0; a].

(A3) The space B is complete.

Example [83] Let the spaces

BC the space of bounded continuous functions defined from (1; 0] to X;

BUC the space of bounded uniformly continuous functions defined from

(1; 0] to X;

C1 :=

n

2 BC : lim!1 () exists

o

;

C0 :=

n

2 BC : lim!1 () = 0

o

, be endowed with the uniform norm

kk = sup

0

k()k:

We have that the spaces BUC; C1 and C0 satisfy conditions (A1) (A3).

Equations with infinite delay genera1ly take the following abstract form:

8<

:

x0(t) = Ax(t) + F(t; xt) for t 0

x0 = ‘ 2 B;

(1.1.2)

where F : R+B ! X is a function satisfying some conditions; A : D(A)

X ! X is the infinitesimal generator of a C0-semigroup

T(t)

t0 on a

Banach space X; for t 0; and the phase space B is a linear space of

functions mapping ]1; 0] into X satisfying axioms (A1)(A3), for every

t 0, xt denotes the history function of B defined by

xt() = x(t + ) for 1 0;

Introduction 6

1.2 Partial Functional Integrodifferential Equations

In many areas of applications such as Engineering, Electronics, Fluid Dynamics,

Physical Sciences, etc…, integrodifferential equations appear and

have received considerable attention during the last decades. Consider the

following system:

8>><

>>:

x0(t) = Ax(t) +

Z t

0

B(t s)x(s)ds + f(t) for t 2 I = [0; b]

x(0) = x0;

(1.2.1)

which is a linear Volterra integrodifferential equation that arises in the analysis

of heat conduction in materials with memory and viscoelasticity. Equation

(1.2.1) has been studied by many authors under various hypotheses concerning

the operators A and B, see for instance the works by Chen and Grimmer

[17], Hannsgen [32, 33], Miller [58, 59], Miller and Wheeler [60, 61], and the

references contained in them.

It was not until 1982 that Grimmer [85] proved the existence and uniqueness

of resolvent operators for this integrodifferential equation that give the variation

of parameter formula for the solutions.

In recent years, much work on the existence and regularity of solutions of the

nonlinear Volterra integrodifferential equations with delays and with nonlocal

conditions have been done by many authors by applying the resolvent

operator theory giving by Grimmer in [85], see e.g., [43, 106] and the references

therein. The objective of this thesis is to study the controllability

and the existence of optimal controls for some class of partial functional integrodifferential

equations in Banach spaces.

We motivate the study by giving the occurence of these partial functional

integrodifferential equations in different areas of applications in science.

1.2.1 A Model in Viscoelasticity

Integrodifferential equations have applications in many problems arising in

physical systems. The following one-dimensional model in viscoelasticity is

Introduction 7

one of the applications of that theory

8>>>>>>>>>>>><

>>>>>>>>>>>>:

@2!

@t2 (t; ) +

@!

@t

(t; ) =

@’

@

(t; ) + h(t; );

@!

@

(t; ) +

Z t

0

a(t s)

@!

@

(s; )ds = ‘(t; ); (t; ) 2 R+ [0; 1];

!(t; 0) = !(t; 1) = 0; t 2 R+;

!(0; ) = !0(); 2 [0; 1];

where, ! is the displacement, ‘ is the stress, h is the external force, ; > 0

and are constants. In this model, the first equation describes the linear

momentum while the second equation describes the constitutive relation between

stress and strain. Setting = 1; v = @!

@t , and u = @!

@ , the above

equations can be rewritten as follows

u0(t)

v0(t)

=

0 @

@

0

u(t)

v(t)

+

Z t

0

a(t s) 0

0 0

u(s)

v(s)

ds

+

0 0

0

u(t)

v(t)

+

0

h(t)

; t 0:

Setting

x(t) =

u(t)

v(t)

; A =

0 @

@

0

; B(t) =

a(t s) 0

0 0

K =

0 0

0

; and p(t) =

0

h(t)

;

we can rewrite the above equation into the following abstract form:

8>><

>>:

x0(t) = A

h

x(t) +

Z t

0

B(t s)x(s)

i

ds + Kx(t) + p(t) for t 0

x(0) = x0:

The operator A here is unbounded, while K and B(t) are bounded operators

for t 0 on a Banach space X. When AB(t) = B(t)A, we obtain the

Introduction 8

following equation:

8>><

>>:

x0(t) = Ax(t) +

Z t

0

B(t s)Ax(s)ds + Kx(t) + p(t) for t 0

x(0) = x0:

(1.2.2)

which has been studied in [23]. We note that in general, the equality AB(t) =

B(t)A does not hold.

Setting f(t; x(t)) = Kx(t) + p(t), equation (1.2.2) becomes

8>><

>>:

x0(t) = Ax(t) +

Z t

0

B(t s)Ax(s)ds + f(t; x(t)) for t 0

x(0) = x0:

(1.2.3)

which to the best of our knowledge has not been investigated for controllability

problem. This problem is addressed in the first part of this thesis, in

the case where equation (1.2.3) admits a nonlocal condition.

1.2.2 A Model in Heat Conduction in Materials with

Memory

Consider a heat flow in a rigid body

of a material with memory. Let

w(t; ); e(t; ); q(t; ) and s(t; ) denote, respectively, the temperature, the

internal energy, the heat flux, and the external heat supply at time t and

position . The balance law for the heat transfer is given by: (see e.g., [53]

et(t; ) + div q(t; ) = s(t; ) (1.2.4)

and the physical properties of the body suggest the dependence of e and q

on w and rw, respectively. For instance, assuming the Fourier Law, i.e.,

e(t; ) = c1w(t; ) (1.2.5)

q(t; ) = c2rw(t; ); (1.2.6)

where c1; c2 are positive constants, one deduces from (1.2.4) the classical

heat equation

wt(t; ) = cw(t; ) + f(t; ) (1.2.7)

with c = c1

1 c2 and f(t; ) = c1

1 s(t; ).

In many materials, the assumptions (1.2.5) and (1.2.6) are not justified because

they take no account of the memory effects: several models have been

Introduction 9

proposed to overcome this difficulty (see e.g. [100, 101, 102]): one of them

consists in substituting (1.2.6) with

q(t; ) = c2rw(t; )

Z t

1

h(t s)rw(s; )ds: (1.2.8)

Taking for simplicity c1 = c2 = 1, we get from (1.2.4), (1.2.5) and (1.2.8)

that

wt(t; ) = w(t; ) +

Z t

1

h(t s)w(s; )ds + s(t; ): (1.2.9)

If we assume that the thermal history w of the body

is known up to t = 0

and the temperature of the boundary of

is constant (=0) for all t, we

are led to the following system:

8>>>>>><

>>>>>>:

wt(t; ) = w(t; ) +

Z t

0

h(t s)w(s; )ds + f(t; ); (t; ) 2 [0; b]

w(0; ) = w0(); 2

w(t; ) = 0; (t; ) 2 [0; b] ;

(1.2.10)

where b > 0 is arbitrarily fixed. If we prescribe h (in addition to f) then

(1.2.10) is a Cauchy-Dirichlet problem for an integrodifferential equation in

the unknown w, which has been studied by several authors in the last decades

(see e.g., [103, 104, 105] and references therein).

Now, if we consider that the thermal history of the body

is known from

the time t r (for some r > 0) up to the present time t, the temperature of

the boundary @

of

is constant (= 0) for all t, and the external heat flux

depends on the thermal history of the body, then, system (1.2.10) becomes

the following integrodifferential equation with finite delay:

8>>>>>>>><

>>>>>>>>:

wt(t; ) = w(t; ) +

Z t

0

h(t s)w(s; )ds + f(t;w(t r; ))

for (t; ) 2 [0; b]

w(t; ) = (t; ) for t 2 [r; 0] and x 2

(1.2.11)

where is a given initial function and r is a positive number. In the one

dimensional setting, Travis and Webb [75] were the first to consider equation

Introduction 10

(1.2.11) with h = 0, and studied its existence and stability properties.

Now define

x(t)() = w(t; )

Ax = x

‘()() = (; ) for 2 [r; 0] and 2

:

f(t; ‘)() = f(t; ‘(r)()) for t 2 [0; b] and 2

(B(t)x)() = h(t)x(t)() for t 2 [0; b] and 2

:

Then, equation (1.2.11) can be transformed into the following abstract form:

8>><

>>:

x0(t) = Ax(t) +

Z t

0

B(t s)x(s)ds + f(t; xt) for t 2 I = [0; b];

x0 = ‘ 2 C([r; 0];X);

(1.2.12)

where X is a Banach space. Equation (1.2.12) has been studied by many

authors (see e.g., [83] and the references contained in it). But to the best

of our knowledge, this equation has never been considered for controllability

and existence of optimal controls and this motivates the work in this thesis.

An example of a material with memory is Shape-memory polymers (SMPs),

which are polymeric smart materials that have the ability to return from a deformed

state (temporary shape) to their original (permanent) shape induced

by an external stimulus (trigger), such as temperature change.

1.3 Controllability of Dynamical Systems

A dynamical system is a system that evolves in time through the iterated

application of an underlying dynamical rule. It is a mathematical model

that one usually constructs in order to investigate some physical phenomenon

that evolves in time. This model usually involves mainly ordinary differential

equations, partial differential equations or functional differential equations,

which describe the evolution of the process under study in mathematical

terms.

Controllability plays an essential role in the development of modern mathematical

control systems. It has many important applications not only in

control theory and systems theory, but also in such areas as industrial and

chemical process control, reactor control, control of electric bulk power systems,

aerospace engineering and recently in quantum systems theory. It is

Introduction 11

one of the fundamental concepts in the mathematical control theory. This is

a qualitative property of dynamical control systems and is of particular importance

in control theory. A systematic study of controllability started at

the beginning of sixties in the last century, when the theory of controllability

based on the description in the form of state space for both time-invariant

and time-varying linear control systems was worked out [46]. Roughly speaking,

controllability generally means, that it is possible to steer a dynamical

control system from an arbitrary initial state to an arbitrary final state using

the set of admissible controls. The notion of controllability was identified by

Kalman [41], as one of the central properties determining system behavior.

The basic controllability problem in continuous time is formulated as follows:

one is interested in steering a control system, whose state x(t) (at time t)

defined on a fixed time interval 0 t b, is modelled by the solution of a

differential equation:

dx

dt

= F(t; x; u) on [0; b]; (1.3.1)

from an initial state x(0) = x0, to a desired state x1, using a control u from

the set of admissible controls, in time b.

For finite dimensional autonomous linear systems, when in (1.3.1), F(t; x; u) =

Ax + Bu, where A is an n n matrix and B is an n m matrix, the

Kalman rank condition [41] gives a necessary and sufficient condition for

controllability of the system. It says that system (1.3.1) with F(t; x; u) =

Ax + Bu is controllable if and only if the controllability matrix [AjB] =

[BjABjA2Bj jAn1B] of size nnm has rank n. This is a useful and simple

test, and much effort has been spent on trying to generalize it to nonlinear

systems in various forms. The systematic study of controllability questions

for continuous time nonlinear systems was begun in the early 70’s. At that

time, the papers [52], [74], and [47], building on previous work ([18], [34]) on

partial differential equations, gave a nonlinear analogue of the above Kalman

controllability rank condition. This analogue provides only a necessary test,

not sufficient. Also in infinite dimensional spaces, even for linear system,

one can hardly get a necessary and sufficient condition for controllability, for

more details see [37], and the references contained in it.

Consider the following infinite dimensional linear system:

x0(t) = Ax(t) + Bu(t) on [0; b];

x(0) = x0

(1.3.2)

where A generates a strongly continuous semigroup of bounded linear operators

T(t)

t0 on a Banach space X and B is a bounded linear operator from

a Banach space U into X.

Introduction 12

Now if x is a classical solution of (1.3.2), then x(t) 2 D(A), for all t 2 [0; b].

In the genera1 case, when A is unbounded, D(A) 6= X, which means that

the system cannot be steered to all of X. Therefore, only a mild solution of

(1.3.2) given by

x(t) = T(t)x0 +

Z t

0

T(t s)Bu(s)ds; (1.3.3)

will be considered with the following definition of controllability (exact controllability).

Definition 1.3.1 The system (1.3.2) is said to be controllable (exactly controllable)

on the interval I = [0; b] if for any two states x0; x1 2 X, there

exists a control u 2 L2(I;U) such that the mild solution x of (1.3.2) satisfies

x(b) = x1.

Solving the controllability problem in infinite dimension boils down to showing

the existence of mild solutions, using fixed point theorems, and finding

the appropriate control for which the mild solution satifies the equality condition

in the above definition. The appropriate control varies from system to

system; it is defined or constructed using the following assumption:

(H) The linear operator W from L2(I;U) into X, defined by

Wu =

Z b

0

T(b s)Bu(s) ds;

induces a bounded inverse operator fW1 defined on L2(I;U)=Ker(W).

Then by defining the control u by u(t) = fW1

x1 T(b)x0

(t), and using

it in equation (1.3.3), Hypothesis (H) yields x(b) = x1, showing that system

(1.3.2) is controllable on I = [0; b].

The construction of fW1 is outlined as follows ( see [7, 68]):

Let Y = L2(I;U)=Ker(W). Since Ker(W) is closed, Y is a Banach space

under the norm

k[u]kY = inf

u2[u]

kukL2(I;U) = inf

W^u=0

ku + ^uk

where [u] are the equivalence classes of u.

Define fW : Y ! X by

fW[u] = Wu; u 2 [u]:

Then, fW is one-to-one and kfW[u]k kWkk[u]kY .

Also, V = Range(W) is a Banach space with the norm kvkV = kfW1vkY .

Introduction 13

To see this, note that this norm is equivalent to the graph norm on D(fW1) =

Range(fW). fW is bounded, and since D(fW) = Y is closed, fW1 is closed.

So, the above norm makes Range(W) = V , a Banach space.

Moreover,

kWukV = kfW1WukY = kfW1fW[u]k = k[u]k = inf

u2[u]

kuk kuk;

So, W 2 B(L2(I;U);X).

Since, L2(I;U) is reflexive, and Ker(W) is weakly closed, the infimum is

actually achieved. Therefore, for any v 2 V , a control u 2 L2(I;U) can be

chosen so that u = fW1v.

Several authors have studied the controllability of linear and nonlinear

systems with various initial conditions, and linear and nonlinear delay

systems in infinite dimensional Banach spaces (see [7] and the references

therein).

The controllability problem of nonlinear systems described by functional integrodifferential

equations with nonlocal conditions in infinite dimensional

Banach spaces, has been studied extensively by many authors, see for instance

[8]-[16], [40, 55, 56, 69, 70, 83, 71], [77], and the references therein.

For example in [77], the authors proved the controllability of an integrodifferential

system with nonlocal conditions basing on the measure of noncompactness

and the Sadovskii fixed-point theorem, and in [69], R. Atmania and

S. Mazouzi have proved the controllability of a semilinear integrodifferential

system using Schaefer fixed-point theorem and requiring the compactness of

the semigroup.

In [69], the authors assumed the compactness of the operator semigroup and

in [9], the authors assumed the compactness of the resolvent operator for

integral equations, whereas in [16, 40, 77], the authors managed to drop this

condition, in the same way as J. Wang, Z. Fan and Y. Zhou [40] have done for

the nonlocal controllability of some semilinear dynamic systems with fractional

derivative.

Many authors have also studied the controllability problem of nonlinear systems

with delay in inifinite dimensional Banach spaces: see for instance [8],

[55], [70], [83], [71], etc and the references therein. For example in [83], the authors

proved the controllability of semilinear functional evolution equations

with infinite delay using the nonlinear alternative of Leray-Schauder type.

In [56], Meili Li, Miansen Wang and Fengqin Zhang proved the controllability

of an impulsive functional differential system with finite delay using

the Schaefer fixed-point. In [71], S. Selvi and M. Mallika Arjunan proved

the controllability for impulsive differential systems with finite delay using

Introduction 14

the Mönch fixed-point theorem, and in [8], K. Balachandran and R. Sakthivel

studied the controllability of functional semilinear integrodifferential

systems in Banach spaces using Schaefer fixed point theorem and assuming

the compactness of the operator semigroup.

In this part of the thesis, motivated by the above works, we give sufficient

conditions that guarantee the controllability of the following dynamical systems

described by the following partial functional integrodifferential models:

• Partial functional integrodifferential 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);

(1.3.4)

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 integrodifferential equation with finite delay in a Banach

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);

(1.3.5)

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 tr up to the present

time t, and is defined by xt() = x(t + ) for r 0.

Introduction 15

• 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;

(1.3.6)

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 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.

To the best of our knowledge, up to now no work has reported neither on the

controllability of partial functional integrodifferential equation with nonlocal

initial condition (equation (1.3.4)), with finite delay (equation (1.3.5)) and

with inifinite delay (equation (1.3.6)) in Banach spaces, using the resolvent

operator approach. It has been an untreated topic in the literature, and this

fact is the main aim and motivation of the present work.

Our approach constists in transforming the problem into a fixed-point problem

of an appropriate operator and to apply the Mönch fixed-point theorem,

making use of the Hausdorff measure of noncompactness, and without assuming

the compactness of the resolvent operator for the associated linear

integral part. The results obtained in this part improve, extend and complement

many other important results in the litterature. They are summarized

in chapters 3, 4 and 5.

Equations (1.3.4), (1.3.5) and (1.3.6) are models that arise in the analysis

of heat conduction in materials with memory [85], and viscoelastici

Zty. The control term Cu(t) is the heating intensity and the integral part t

0

B(t s)x(s)ds is the memory of the system. Materials with memory are

interesting because they act adaptively to their environment. They can be

shaped easily into different forms at low temperature, but return to their

Introduction 16

original forms on heating. Steering such systems from an initial state (initial

condition) to a desired terminal one (boundary condition), by an appropriate

choice of a control (which could be the heating intensity), is of interest to

many scientists and engineers. The questions, whether one can heat the material

in such a way that the initial state is transferred onto a desired state in

time b and under which constraints on the control parameter u, are of interest.

1.4 Optimal Control of Dynamical Systems

In studying dynamical systems in order to improve the system behavior,

one problem that usually surfaces is that of optimal control. In the simplest

form, there is a given dynamical system (linear or nonlinear, discrete or

continuous), described by an ordinary differential equation, a partial differential

equation or a functional differential equation, for which input functions

(controls or commands) can be specified. There is also an objective or a cost

function whose value is determined by system behavior, and is in some sense

a measure of the quality of that behavior. The optimal control problem is

that of finding the input function ( the control or the command) so as to

optimize (maximize or minimize) the objective function. The problem seeks

to optimize the objective function subject to the constraints construed by

the model describing the evolution of the underlying system.

Before even precising mathematically this vocabulary, it is important to note

that we all do (more or less) optimal control without even paying attention

to it: to go from one place to another as fast as possible, or using the shortest

path, to maximize the income of investments or shares or minimize debts are

examples of problems of this type.

For example, the dynamical system might be a space vehicule with inputs

corresponding to rocket thrust. The objective might then be to reach the

moon with minimum expenditure of fuel. As another example, the system

might be the nation’s economy, with controls corresponding to governement

monetary and fiscal policy. The objective might be to maximize the aggregate

deviations of unemployment and interest rates from fixed target values.

Finally, as a third example, which is of interest in this thesis, the system

might represent the dynamics of heat flow in materials with memory, with

controls corresponding to the heating intensities. The objective might be to

minimize or maximize the heating intensity so as to obtain a particular form

for the material.

The basic optimal control problem in continuous time is formulated as follows:

one is interested in a system (a space vehicule, a nation’s economy,

Introduction 17

heat flow in materials with memory, etc. . . ) whose state x(t) (at time t)

defined on a fixed time interval 0 t T, is modelled by the solution of a

differential equation:

dx

dt

= x_ = F(t; x; u) on [0; T]; x(0) = x0; (1.4.1)

where x0 2 Rn, and T 2 R are fixed with 0 < T; F is a given function

from R Rn U with values in Rn. As we can see, one of the arguments

of F is a function u defined on the interval [0; T] with values in a given set

U, which is the set of admissible controls. This function u (the control, or

the command) translates mathematically the actions (or decisions) that one

can exercise on the evolution of the system; the set U corresponds to the

restrictions or the constraints that must be respected by the controls ( for

example: limited ressources, bounds on the acceleration or the speed for the

driving of a vehicule or the heat for the heat flow in a material, etc. . . ).

To formulate an optimal control problem, is to define the state of the system

x(t) and the differential system that describes its evolution, the class of

admissible controls u(t) and finally an evolution criterion or a cost function;

most often, it is a criterion cumulated in time added to a final cost, whose

typical form is:

J(u) =

Z T

0

g(t; x(t); u(t)) dt + h(x(T)) (1.4.2)

where g and h are given functions, defined respectively in RRnU and Rn

with values in R. The problem to solve is then to determine the optimal cost

and an optimal control, that is to solve the following optimization problem:

8>>><

>>>:

inf

u(t)

Z T

0

g(t; x(t); u(t))dt + h(x(T))

dx

dt

= F(t; x; u) on [0; T]; x(0) = x0;

(1.4.3)

Problem (1.4.3) is called Bolza Problem, and when h 0, we have a

Lagrange Problem, which we are interested in in this thesis. The two popular

solution techniques of an optimal control problem are Pontryagin’s maximum

principle and the Hamilton- Jacobi-Bellman equation [50]. In infinite

dimension, methods and techniques of convex optimization are used. Optimal

control has become a highly established research front in recent years with

numerous contributions to the theory, in both deterministic and stochastic

contexts. Its application to diverse fields such as biology, economics, ecology,

Introduction 18

engineering, finance, management, and medicine cannot be overlooked (see,

e.g., [49] and the references contained in it). The associated mathematical

models are formulated, for example, as systems of ordinary, partial, delay, or

stochastic differential and integrodifferential equations or discrete dynamical

systems, for both scalar and multicriteria decision-making contexts.

Problems of existence of optimal controls for nonlinear differential systems

have been studied extensively by many authors under various hypotheses see

e.g.,[96], [97], [98], [87], [99], [95],[94], [88], [90], [89], and the references contained

in them.

In [96], Wang et al: studied the existence and continuous dependence of mild

solutions and the optimal controls of a Lagrange problem for some fractional

integrodifferential equation with infinite delay in Banach spaces using the

using the techniques of a priori estimation and extension of step by steps.

Wand and Zhou [97] discussed the optimal controls of a Lagrange problem

for fractional evolution equations. In [98], Wei et al: studied the optimal

controls for nonlinear impulsive integrodifferential equations of mixed type

on Banach spaces. In [99], the authors studied the existence of mild solutions

and the optimal controls of a Lagrange problem for some impulsive fractional

semilinear differential equations, using the techniques of a priori estimation.

In this part of the thesis, motivated by the above works, we establish the solvability

and existence of optimal controls of the following dynamical systems

described by the following partial functional integrodifferential models:

• Partial functional integrodifferential equation with finite delay in a Banach

space (X; k k) :

8>>>><

>>>>:

x0(t) = Ax(t) +

Z t

0

B(t s)x(s)ds + f(t; xt) + C(t)u(t);

for t 2 I = [0; T];

x0 = ‘ 2 C = C([r; 0];X);

(1.4.4)

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 a separable reflexive Banach space X; for t 0, B(t) is a closed

linear operator with domain D(B(t)) D(A). The control u takes

values from another separable reflexive Banach space U. The operator

C(t) belongs to B(U;X) which is the Banach space of bounded

linear operators from U into X, and C([r; 0];X) denotes the Banach

space of continuous functions x : [r; 0] ! X with supremum norm

kxk1 = supt2I kx(t)kX, 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

Introduction 19

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) + C(t)u(t);

for t 2 I = [0; b]

x0 = ‘ 2 B;

(1.4.5)

where A : D(A) ! X is the infinitesimal generator of a C0-semigroup

T(t)

t0 on a separable reflexive 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 separable reflexive 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.

• Partial functional integrodifferential equation subject to Cauchy 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)) + C(t)u(t);

for t 2 I = [0; b]

x(0) = x0 2 X

(1.4.6)

where x0 2 X and f : I X ! X is a function satisfying some condit

ions; A : D(A) ! X is the infinitesimal generator of a C0-semigroup

T(t)

t0 on a separable reflexive Banach space X; for t 0, B(t) is

a closed linear operator with domain D(B(t)) D(A). The control u

takes values from another separable reflexive 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.

Introduction 20

Little is known and done about the existence of optimal controls for integrodifferential

equations with delay using the resolvent operator for integral

equations approach, especially the problem of existence of optimal controls

of a Lagrange problem for equations (1.4.4), (1.4.5) and (1.4.6) have been an

untreated topic in the literature, and this motivates the present work. We

tackle this problem by using the techniques of a priori estimation of mild

solutions, giving by the resolvent operator for the associated linear integral

part, and techniques of convex optimization. The results obtained in this

part complement many other important results in the literature. They are

summarized in chapters 6, 7 and 8.

1.5 Methods

In [68], Quinn and Carmichael proved that a controllability problem can be

converted into a fixed point problem. So our approach constists in transforming

the problems (1.3.4), (1.3.5) and (1.3.6) into fixed-point problems of

appropriate operators and to apply the Hausdorff measure of noncompactness

and the Mönch fixed-point theorem, without requiring the compactness

of the operator semigroup, in the same way as J. Wang, Z. Fan and Y. Zhou

[40] have done for the nonlocal controllability of some semilinear dynamic

systems with fractional derivative. This method enables us overcome the

resolvent operator case considered in this thesis. In contrary to the evolution

semigroup case considered in [55, 71], here the semigroup property can

not be used because resolvent operators in general do not form semigroups.

As for the existence of optimal controls of the associated Lagrange problems

to equations (1.4.4), (1.4.5) and (1.4.6), we apply techniques of convex

optimization together with Balder’s Theorem to obtain our results.

1.5.1 Measures of Noncompactness

Measures of noncompactness are very useful tools in nonlinear analysis. For

instance, in metric fixed point theory and in the theory of operator equations

in Banach spaces. They are also used in the studies of functional equations,

ordinary and partial differential equations, fractional partial differential equations,

integral and integrodifferential equations, optimal control theory, and

in the characterizations of compact operators between Banach spaces (see

e.g., [6, 11, 12, 67]). The concept of measure of noncompactness was first

defined and studied by Kuratowski [48] in 1930. In 1955, G. Darbo [21] used

it to prove his fixed point theorem. We have the following definition:

Introduction 21

Definition 1.5.1 Let E+ be the positive cone of an order Banach space

(E;). That is

E+ =

x 2 E : 0 x

:

And let X be an arbitrary Banach space. A function defined on the set of all

bounded subsets of X with values in E+ is called a measure of noncompactness

(MNC) on X if (co(D)) = (D) for every bounded subset D X, where

co(D) stands for the closed convex hull of D.

Example 1.5.2 ( [6], Example 1, p. 19]) Let X be an arbitrary metric

space and MX denote the set of all bounded subsets of X. Then the map

1 :MX ! [0;1) defined by

1(D) =

8<

:

0; if D is relatively compact

1; otherwise:

is a measure of noncompactness, the so-called discrete measure of noncompactness.

Two measures of noncompactness which are frequently used in many branches

of nonlinear analysis and its applications, are Kuratowski measure of noncompactness

and the Hausdorff measure of noncompactness, which we shall

use in this thesis. The frequent use of this latter one is due to the fact that

it is defined in a natural way and has several useful properties.

Definition 1.5.3 Let D be a bounded subset of a normed space Z. The

Kuratowski measure of noncompactness of D (shortly MNC) is defined by

(D) = inf

n

> 0 : D has a finite cover by sets of diameter less than

o

;

= inf

n

> 0 : D

[n

k=1

Sk; Sk Z; diam(Sk) < (k = 1; 2; ; n 2 N)

o

:

Definition 1.5.4 Let D be a bounded subset of a normed space Z. The

Hausdorff measure of noncompactness of D (shortly MNC) is defined by

(D) = inf

n

> 0 : D has a finite cover by balls of radius less than

o

= inf

n

> 0 : D

[n

k=1

B(xk; rk); xk 2 Z; rk < (k = 1; 2; ; n 2 N)

o

:

Introduction 22

The compactness conditions described by means of measures of noncompactness

are useful in showing the existence of solutions for differential and

integral equations in Banach spaces. Measures of noncompactness have applications

in many fields where loss of compactness arises. For example,

integral equations with strongly singular kernels, differential equations over

unbounded domains, functional differential equations of neutral type or with

deviating argument, linear differential operators with nonempty essential

spectrum, nonlinear superposition operators between various function spaces,

initial value problems in Banach spaces etc, see e.g.,[1, 11, 4, 25] and the references

therein.

1.5.2 Fixed Point Theory

The fixed point technique is one of the useful methods mainly applied in the

existence and uniqueness of solutions of differential equations and the controllability

of dynamical systems. One of the main branches of fixed point

theory deals with the topological properties of the operators involved. With

respect to the topological aspect, the two main theorems are Brouwer’s theorem

and its infinite dimensional version, Schauder’s fixed point theorem.

In both theorems, compactness plays an essential role. In 1955, Darbo [21]

extended Schauder’s theorem to the setting of noncompact operators, introducing

the notion of k-set contraction which is closely related to the idea

of measures of noncompactness. In 1967, Sadovskii [26] gave a fixed point

result more general than Darbo’s theorem using the concept of condensing

map. Thus, the fixed point theory for condensing mappings allows us to

obtain a relationship between the two theories. In 1980, Mönch [62] gave

a fixed point theorem for maps between Banach spaces, which extends the

Schauder and more generally, Sadovskii fixed point theorems.

In this thesis, we use the Mönch fixed point theorem and the Hausdorff measure

of noncompactness to prove the controllability results for partial functional

integrodifferential equations. The advantage of using this fixed point

theorem is to weaken the compactness assumption of the operator semigroup

and the resolvent operator for integral equations.

1.5.3 Semigroup Theory

The theory of semigroups of bounded linear operators is part of functional

analysis. It is an extensive mathematical subject with substantial applications

to many fields of analysis, and has developed quite rapidly since the

discovery of the generation theorem by Hille and Yosida in 1948. Semigroups

are a powerful tool in solving evolution equations. They give the variation of

Introduction 23

parameter formula for mild solutions. In [66], Pazy discussed the existence

and uniqueness of mild, strong and classical solutions of evolution equations

using semigroup theory and fixed point theorems.

1.5.4 Resolvent Operator for Integral Equations

This concept was first introduced in 1982 by Ronald Grimmer [85]. He

proved the existence and uniqueness of resolvent operators for integrodifferential

equations that give the variation of parameter formula for the solution.

Resolvent operators in general do not form semigroup, and are a better tool

in solving integrodifferential equations than semigroups. In [24], W. Desch,

R. Grimmer and W. Schappacher proved the equivalence of the compactness

of the resolvent operator for integral equations and that of the operator semigroup.

In this thesis, we show a similar result on the equivalence between

the operator-norm continuity of the resolvent operator for integral equations

and the C0-semigroup. This result is important because it allows to drop the

compactness assumption on the resolvent operator and assume its operatornorm

continuity, in proving our controllability results. In [69], the authors

assumed the compactness of the operator semigroup and in [9], the authors

assumed the compactness of the resolvent operator whereas in [16, 40, 77],

the authors managed to drop this condition which motivates our current

work. Our contributions in this direction are summerized in Chapters 3 and

4.

1.6 Organization of the Thesis

The present study in this thesis deals firstly with the fixed point approach

via measures of noncompactness for proving controllability results for partial

functional integrodifferential systems in Banach spaces.

We note that the partial functional integrodifferential equations (1.3.4), (1.3.5)

and (1.3.6) have not been investigated (to the best of our knowledge) for controllability.

The second part of the thesis deals with the existence of optimal controls of

the associated Lagrange problems to equations (1.4.4), (1.4.5) and (1.4.6),

we apply techniques of convex optimization together with Balder’s Theorem.

In chapter 2, we give preliminary results that will be used in proving our

main results. In particular, we establish the equivalence between operatornorm

continuity of the semigroup

T(t)

t0 generated by A and the resolvent

operator

R(t)

t0 corresponding to the associated linear equation.

Introduction 24

In chapter 3 , we prove the controllability result for some partial functional

integrodifferential equation with nonlocal initial conditions in Banach spaces

(equation (1.3.4)), by using resolvent operators for integral equations and the

Mönch fixed point theorem.

In chapter 4, we establish a controllability result for some partial functional

integrodifferential equation with finite delay in Banach spaces (equation

(1.3.5)), by using resolvent operators for integral equations and the Mönch

fixed point theorem.

In chapter 5, we establish a controllability result for some partial functional

integrodifferential equation with infinite delay in Banach spaces (equation

(1.3.6)), by using resolvent operators for integral equations and the Mönch

fixed point theorem.

In chapter 6, we prove the solvability and the existence of optimal controls

for some partial functional integrodifferential equation with finite delay in

Banach spaces (equation (1.4.4)), by using the techniques of convex optimization,

a-priori estimation and contraction mapping principle.

In Chapter 7 , we prove the solvability and the existence of optimal controls

for some partial functional integrodifferential equation with infinite delay in

Banach spaces (equation (1.4.5)), by using the techniques of convex optimization,

a-priori estimation and contraction mapping principle.

In Chapter 8, we prove the solvability and the existence of optimal controls

for some partial functional integrodifferential equation with classical initial

conditions in Banach spaces (equation (1.4.6)), by using the techniques of

convex optimization, a-priori estimation and contraction mapping principle.