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 ô€€€ 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
ô€€€
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 ô€€€ 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
ô€€€
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 tô€€€r up to the present
time t, and is defined by xt() = x(t + ) for ô€€€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 ô€€€ s)x(s)ds + f(t; xt) + Cu(t);
for t 2 I = [0; b]
x0 = ‘ 2 B;
(0.0.3)
ô€€€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 ]ô€€€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 ô€€€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.
Â
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) = ô€€€N(t) + kN(t ô€€€ r)
Introduction 4
which involves a nonlocal term, kN(t ô€€€ 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
ô€€€
T(t)
t0 on a
Banach space X; for t 0; C([ô€€€r; 0];X) (the phase space), denotes the
Banach space of continuous functions x : [ô€€€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 ô€€€ r up to the present time t, and is defined by xt() = x(t + )
for ô€€€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 [ô€€€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 ]ô€€€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 ] ô€€€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 : ] ô€€€ 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 (ô€€€1; 0] to X;
BUC the space of bounded uniformly continuous functions defined from
(ô€€€1; 0] to X;
C1 :=
n
2 BC : lim!ô€€€1 () exists
o
;
C0 :=
n
2 BC : lim!ô€€€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) ô€€€ (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
ô€€€
T(t)
t0 on a
Banach space X; for t 0; and the phase space B is a linear space of
functions mapping ]ô€€€1; 0] into X satisfying axioms (A1)ô€€€(A3), for every
t 0, xt denotes the history function of B defined by
xt() = x(t + ) for ô€€€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 ô€€€ 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 ô€€€ 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 ô€€€ s) 0
0 0
u(s)
v(s)
ds
+
0 0
0 ô€€€
u(t)
v(t)
+
0
h(t)
; t 0:
Setting
x(t) =
u(t)
v(t)
; A =
0 @
@
0
; B(t) =
a(t ô€€€ s) 0
0 0
K =
0 0
0 ô€€€
; 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 ô€€€ 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 ô€€€ 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 ô€€€ 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; ) = ô€€€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 = cô€€€1
1 c2 and f(t; ) = cô€€€1
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; ) = ô€€€c2rw(t; ) ô€€€
Z t
ô€€€1
h(t ô€€€ 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
ô€€€1
h(t ô€€€ 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 ô€€€ of
is constant (=0) for all t, we
are led to the following system:
8>>>>>><
>>>>>>:
wt(t; ) = w(t; ) +
Z t
0
h(t ô€€€ s)w(s; )ds + f(t; ); (t; ) 2 [0; b]
w(0; ) = w0(); 2
w(t; ) = 0; (t; ) 2 [0; b] ô€€€;
(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 ô€€€ 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 ô€€€ s)w(s; )ds + f(t;w(t ô€€€ r; ))
for (t; ) 2 [0; b]
w(t; ) = (t; ) for t 2 [ô€€€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 ô€€€ 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 jAnô€€€1B] 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
ô€€€
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 ô€€€ 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 ô€€€ s)Bu(s) ds;
induces a bounded inverse operator fWô€€€1 defined on L2(I;U)=Ker(W).
Then by defining the control u by u(t) = fWô€€€1
x1 ô€€€ 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 fWô€€€1 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 = kfWô€€€1vkY .
Introduction 13
To see this, note that this norm is equivalent to the graph norm on D(fWô€€€1) =
Range(fW). fW is bounded, and since D(fW) = Y is closed, fWô€€€1 is closed.
So, the above norm makes Range(W) = V , a Banach space.
Moreover,
kWukV = kfWô€€€1WukY = kfWô€€€1fW[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 = fWô€€€1v.
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 ô€€€ 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
ô€€€
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 ô€€€ 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
ô€€€
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 tô€€€r up to the present
time t, and is defined by xt() = x(t + ) for ô€€€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 ô€€€ s)x(s)ds + f(t; xt) + Cu(t);
for t 2 I = [0; b]
x0 = ‘ 2 B;
(1.3.6)
ô€€€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 ] ô€€€ 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 ô€€€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 ô€€€ 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 ô€€€ 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
ô€€€
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([ô€€€r; 0];X) denotes the Banach
space of continuous functions x : [ô€€€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 ô€€€ r up to the present time t, and is defined by
Introduction 19
xt() = x(t + ) for ô€€€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 ô€€€ s)x(s)ds + f(t; xt) + C(t)u(t);
for t 2 I = [0; b]
x0 = ‘ 2 B;
(1.4.5)
ô€€€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 ] ô€€€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 ô€€€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 ô€€€ 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
ô€€€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
ô€€€
T(t)
t0 generated by A and the resolvent
operator
ô€€€
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.
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