Project Materials

Lasalle Invariance Principle for Ordinary Differential Equations and Applications

TABLE OF CONTENTS

Acknowledgment i
Certification ii
Approval iii
Introduction v
Dedication vi
1 Preliminaries 2
1.1 Definitions and basic Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Exponential of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Basic Theory of Ordinary Differential Equations 7
2.1 Definitions and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Continuous dependence with respect to the initial conditions . . . . . . . . . . . . 11
2.3 Local existence and blowing up phenomena for ODEs . . . . . . . . . . . . . . . . 12
2.4 Variation of constants formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Stability via linearization principle 21
3.1 Definitions and basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Lyapunov functions and LaSalle’s invariance principle 26
4.1 Denitions and basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Instability Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 How to search for a Lyapunov function (variable gradient method) . . . . . . . . . 30
4.4 LaSalle’s invariance principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.5 Barbashin and Krasorskii Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.6 Linear systems and linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5 More applications 40
5.1 Control design based on lyapunov’s direct method . . . . . . . . . . . . . . . . . . 41
Conclusion 48
Bibliography 48

CHAPTER ONE

PRELIMINARIES

1.1 Definitions and basic Theorems

In this chapter, we focussed on the basic concepts of the ordinary differential equations. Also, we emphasized on relevant theorems in ordinary differential equations.

Definition 1.1.1

An equation containing only ordinary derivatives of one or more dependent variables with respect to a single independent variable is called an ordinary differential equation ODE.

The order of an ODE is the order of the highest derivative in the equation. In symbol, we can
express an n-th order ODE by the form
x(n) = f(t; x; :::; x(n􀀀1)) (1.1.1)

Definition 1.1.2 (Autonomous ODE ) When f is time-independent, then (1.1.1) is said to be an autonomous ODE. For example, x0(t) = sin(x(t))

Definition 1.1.3 (Non-autonomous ODE ) When f is time-dependent, then (1.1.1) is said to be a non autonomous ODE. For example, x0(t) = (1 + t2)y2(t)

Definition 1.1.4 f : Rn ! Rn is said to be locally Lipschitz, if for all r > 0 there exists k(r) > 0
such that

kf(x) 􀀀 f(y)k k(r)kx 􀀀 yk; for all x; y 2 B(0; r):
f : Rn ! Rn is said to be Lipschitz, if there exists k > 0 such that kf(x) 􀀀 f(y)k kkx 􀀀 yk; for all x; y 2 Rn:

Definition 1.1.5 (Initial value problem (IVP) Let I be an interval containing x0, the following problem (x(n)(t) = f(t; x(t); :::; x(n􀀀1)(t))

x(t0) = x0; x0(t0) = x1; :::; x(n􀀀1)(t0) = xn􀀀1

(1.1.2) is called an initial value problem (IVP).

x(t0) = x0; x0(t0) = x1; :::; x(n􀀀1)(t0) = xn􀀀1 are called initial condition.

Lemma 1.1.6 [9](Gronwall’s Lemma) Let u; v : [a; b] ! R+ be continuous such that there exists
> 0 such that
u(x) +

x
a
u(s)v(s)ds; for all x 2 [a; b]:
Then,
u(x) e

x
a
v(s)ds
; for all x 2 [a; b]:
Proof .
u(x) +

x
a
u(s)v(s)ds
implies that
u(x)
+

x
a
u(s)v(s)ds
v(x):
So,
u(x)v(x)
+

x
a
u(s)v(s)ds
v(x);
which implies that
x
a