## TABLE OF CONTENTS

Acknowledgement ii

1 Introduction 1

1.1 Finite dimensional linear systems theory . . . . . . . . . . . . . . . . . . . . 2

1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Organization of the work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Spectral Theory and Semigroup Properties 7

2.1 Spectral Theory of Linear operators . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Semigroups of Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Controllability and Stablizability in Hilbert Spaces 26

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Controllability in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.1 A General Framework for Linear Control Systems . . . . . . . . . . . 31

3.2.2 Various Concept of Controllability . . . . . . . . . . . . . . . . . . . . 33

3.3 Stability and Stablizability in Hilbert Spaces . . . . . . . . . . . . . . . . . . 41

3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Bibliography 53

## CHAPTER ONE

Introduction

Questions about controllability and stability arise in almost every dynamical system problem.

As a result, controllability and stability are one of the most extensively studied subjects in

system theory. A departure point of control theory is the dierential equation

x_ = f(x; u); x(0) = x0 2 Rn; (1.0.1)

with the right hand side depending on a parameter u from a set U Rm. The set U is

called the set of control parameters. Controls are of two types: open and closed loops. An

open loop control can be basically an arbitrary function u() : [0;+1) ! U; for which the

equation

x_ (t) = f(x(t); u(t)); t 0; x(0) = x0 2 Rn; (1.0.2)

has a well dened solution.

A closed loop control can be identied with a mapping k : Rn ! U, which may depend

on t 0, such that the equation

x_ (t) = f(x(t); k(x(t))); t 0; x(0) = x0 2 Rn; (1.0.3)

has a well dened solution. The mapping k() is called feedback. Controls are called also

strategies or inputs, and the corresponding solutions of (1.0.2) or (1.0.3) are outputs of the

system.

We do not consider all the system theory concepts here, we will concentrate mainly here

on controllability and stability of linear system. To motivate our approach we present a brief

survey of nite dimensional theory concepts and results which we will generalize later.

Linear System Theory A linear system is described by a (linear) dierential equation

x_ (t) = Ax(t) + Bu(t); t 0; x(0) = x0; (1.0.4)

where u is the control, x is the state. These functions take their values in linear spaces U

and X, respectively. Furthermore, A and B are linear mappings between appropriate spaces.

If the spaces U and X are nite dimensional, then the system is called nite dimensional.

Otherwise, we have an innite dimensional linear system.

1

1.1 Finite dimensional linear systems theory

Consider the linear nite dimensional control system:

x_ (t) = Ax(t) + Bu(t); t 0; x(0) = x0; (1.1.1)

where x0; x(t) 2 Rn, u(t) 2 Rm, the linear transformations A : Rn ! Rn and B : Rm !

Rn will be identied with the representing matrices. System (1.1.1) is a dierential equation

on the state space Rn and the unique solution

x(t; x0; u) = eAtx0 +

Z t

0

eA(ts)Bu(s) ds: (1.1.2)

The following concepts of controllability, stability and stablizability are now standard.

Controllability One says that a state x0 2 Rn is reachable (or attainable) from x1 in

time T; if there exists an open loop control u() such that, for the output x(), one has

x(0) = x0; x(T) = x1: If an arbitrary state x0 is reachable from an arbitrary state x1 in a

time T; then the system (1.1.1) is said to be controllable.

The proposition below gives a formula for a control transferring x0 to x1. In this formula

the matrix QT , called the controllability matrix or controllability Gramian, appears:

QT =

Z t

0

eA(Ts)B(s)eA(Ts)B(s) ds; T > 0;

where A is the transpose matrix of A. Then QT is symmetric and nonnegative denite.

Proposition 1.1.1. Assume that for some T > 0 the matrix QT is nonsingular. Then for

arbitrary x0; x1 2 Rn the control

^u(s) = BeA(Ts)Q1

T (eA(Ts)x1 x0); s 2 [0; T];

transfers x0 to x1 at time T:

We now formulate algebraic conditions equivalent to controllability of (1.1.1).

Theorem 1.1.2. The folowing conditions are equivalent:

(i) An arbitrary state x1 2 Rn is attainable from 0.

(ii) System (1.1.1) is controllable.

(iii) System (1.1.1) is controllable at a given time T > 0:

(iv) Matrix QT is nonsingular for some T > 0.

(v) Matrix QT s nonsingular for an arbitrary T > 0.

(vi) rank

B : BA : BA2 : : BAn1

= n:

Condition (vi) is called the Kalman rank condition or the rank condition for short.

2

Stability An important problem in the control of linear systems is the study of its sability.

Consider the stability problem for the linear system

x_ (t) = Ax(t); t 0; x(0) = x0 2 Rn and A : Rn ! Rn: (1.1.3)

The system (1.1.3) is said to be assymptotically stable or strongly stable if for arbitrary

x0 2 Rn

lim

t!+1

kx(t; x0)k = 0;

and exponentially stable or uniformly stable if there exist positive constants M and such

that

kx(t; x0)k Metkx0k; t 0:

Instead of saying that (1.1.3) we will often say that the matrix A is stable. We have the

following well known result for nite dimensional linear system such as (1.1.3).

Theorem 1.1.3. The following condition are equivalent:

(i) System (1.1.3) is assymptotically stable.

(ii) System (1.1.3) is exponentially stable.

(iii) supfRe; 2 (A)g < 0:

(vi)

R +1

0 kx(t; x0)k2 dt < +1:

(v) The matrix equation

AQ + AQ = I (1.1.4)

has a nonnegative symmetric solution Q.

Equation (1.1.4) is called Lyapunov equation.

Stabilizability The central theme within the system theory is to design a control u such

that the corresponding state has desired behaviour. A typical example is stabilization:

Design an control u such that x(t) ! 0 as t ! +1: A control designed by feedback achieves

this goal. Feedback means that we do not calculate u only based on x0; A; and B but also

on the current state. This results in a control of the form u(t) = Dx(t); and the question of

stabilization turns into a question of nding D.

One says that the system (1.1.1) is stablizable if there exists an mn marix D such that

A + BD is stable.

1.2 Motivation

From nite to innite dimensional system To get an appreciation of the technical

problems associated with the control of innite dimensional systems, consider the distributed

control of wave motion. Let y(x; t) denote the transverse displacement at timt t 0 of

3

a vibrating medium in an ndimensional bounded open region

with smooth boundary

@

Rn: Let us assume that

y(x; t) = 0; for x 2 @

and t 0

and let the initial data at time t = 0 be

y(x; 0) = y0 and

@

@t

y(x; 0) = y1;

for some suciently smooth functions y0 and y1 dened on

. Consider the partial dierential

equation

@2y

@t2 (t) + Ay(t) = Bu(t) (1.2.1)

or the equivalent rst order system

8>>><

>>>:

@y

@t

(t) z(t) = 0

@z

@t

(t) + Ay(t) = Bu(t);

(1.2.2)

where the operator

A :=

Xn

i;j=1

@

@xi

aij(x)

@

@xj

+ a0(x); a0(x) > 0

is uniformly elliptic operator with C1coecients in

, that is aij 2 C1(

). Let the

controller u() 2 L2(0;1;Rn) and let

B(x) :=

B1(x) Bm(x)

be an n m matrix with each column in C1(

). We consiedr the state space

w :=

y

z

2 W := H1

0(

)

L2(

);

where H1

0(

) denotes the usual Sobolev space of L2functions, with derivatives, in the

distribution sense, belong to L2(

) and vanishing at the boundary. Then (1.2.2) can be

formally written as

dw

dt

(t) + ~ Aw(t) = ~B u(t); (1.2.3)

where

A~ =

0 I

A 0

and ~B =

0

B

:

Here ~B2 L(Rm;W) and the operator ~ A is an unbounded opertor with a dense domain

D(A~) = [H1

0(

) \ H2(

)]

H1

0(

) W:

4

Because of the presence of the unbounded operator A~; it is clear that the concept of a

solution for (1.2.3) is not immediate. Intuitively, if we want (1.2.3) to have classical solution,

then we would need

w0 =

y0

z0

2 D(A~)

and u() 2 C1(

): On the other hand, A~ generates a strongly continuous semigroup

(S(t))t0 on W, i.e., S(t) is a bounded linear opearator on W satisfying

(i) S(0) = I.

(ii) S(t1 + t2) = S(t1)S(t2); 0 < t1; t2 < +1:

(iii) t 7! S(t)w : [0;+1) ! W is continuous for each w 2 W:

Then we say that w(t) is a solution of (1.2.3) if it satises

w(t) = S(t)w0 +

Z t

0

S(t s)~B u(s) ds; (1.2.4)

where the integral is interpreted in the Bochner sense.

Now, given a strongly continuous semigroup S(t) on X (say a Banach space), it has a

unique innitesimal generator A, a closed unbounded operator that is dened on some dense

domain D(A). The converse question, namely, when does A generate a strongly continuous

semigroup, is a far more dicult issue and is the content of the Hille-Yosida theorem. Furthermore,

there are dierent kinds of semigroups, and each plays an important role in the

study of controllability and stability of innite dimensional systems.

1.3 Organization of the work

The project is divided into two parts. Firstly, a general framework for the concepts of

controllability and stability of linear systems in a Hilbert spaces is developed. Secondly, the

concepts of controllability and stability of linear system in a Banach space is discussed.

Chapter 2

In chapter 2 we review some basic concepts of spectral theory and semigroup of linear

operators, all the results discussed in this chapter are preliminary and can be found in any

materials on the subject. The materials follows from Khalil Ezzinbi [3]; Klaus-Jochen Engel

and Rainer Nagel [5]; R. F. Curtain and H. J. Zwart [6]; and Zheng-Hua Luo, Zhu Guo and

Ömer Mörgul [8].

Chapter 3

Chapter 3 is dealed with the controllability and stabilizability of linear systems in Hilbert

spaces. A general framework of concept of controllability of linear systems with bounded

5

control is discussed and then the various concepts of controllability and sabilizability of

linear systems and the the relationships between them is stated. almost all the results in

this chapter are from A. J. Pritchard and T. Zabczyk [1]; Alain Bensoussen, Guiseppe Da

Prato, Michel C. Delphour, Sanjoy K. Mitter [2]; Jerzy Zabczyk [4] and Ruth F. Curtain

and Anthony J. Pritchard [7].

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