# LaSalle Invariance Principle for Ordinary Differential Equations and Applications

**ABSTRACT**

The most popular method for studying **stability of nonlinear systems **is introduced by a Russian Mathematician named **Alexander Mikhailovich Lyapunov**. His work ”The General Problem of Motion Stability ” published in 1892 includes two methods: Linearization Method, and Direct Method. His work was then introduced by other scientists like Poincare and LaSalle .

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

In chapter two of this work, we study the existence and uniqueness of solutions of ordinary dif- ferential equations. Also, relevant theorems and concepts in ordinary differential equations was discussed in the chapter.

In chapter three, we study the stability of an equilibrium point and linearization principle. Also, relevant theorems and concepts in stability of an equilibrium point and linearization principle was discussed in the chapter

In chapter four, we study the various tools for determining stability of equilibrium points.

In chapter five, we discussed various applications of Lyapunov theorem, and LaSalle’s invariance principle.

**TABLE OF CONTENTS**

Acknowledgment i

Certi cation ii

Approval iii

Introduction v

Dedication vi

1 Preliminaries 2

1.1 De nitions and basic Theorems . . .. . 2

1.2 Exponential of matrices . . . . . . . 4

2 Basic Theory of Ordinary Dierential Equations 7

2.1 De nitions 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 De nitions and basic results . . . .. . . 21

4 Lyapunov functions and LaSalle’s invariance principle 26

4.1 De nitions 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

**INTRODUCTION **** **

**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 theroems in ordinary differential equations.

**Definition 1.1.1 **

*An equation containing only ordinary derivatives of one or more dependent vari- ables 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, **x*^{j}(*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, **x*^{j}(*t*) = (1 + *t*^{2})*y*^{2}(*t*)

**Definition 1.1.4 **

*f *: R*n *→ R*n **is said to be locally Lipschitz, if for all **r > *0 *there **exists **k*(*r*) *> *0 *such that *ǁ*f *(*x*) − *f *(*y*)ǁ ≤ *k*(*r*)ǁ*x *− *y*ǁ*, for all x, y *∈ *B*(0*,* *r*)*. **f *: R*n *→ R*n **is said to be Lipschitz, if there exists **k > *0 *such that*

ǁ*f *(*x*) − *f *(*y*)ǁ ≤ *k*ǁ*x *− *y*ǁ*, for all x, y *∈ R *. *

**Definition 1.1.5 ***(**Initial value problem (IVP) *

*Let **I **be an interval containing **x*_{0}*, the follow- ing problem **x*^{(}^{n}^{)}(*t*) = *f *(*t, x*(*t*)*, …, x*^{(}^{n}^{−}^{1)}(*t*)) *x*(*t*_{0}) = *x*_{0}*,* *x*^{j}(*t*_{0}) = *x*_{1}*,* *…,* *x*^{(}^{n}^{−}^{1)}(*t*_{0}) = *x*_{n}_{−}_{1 }*is called an initial value problem (IVP). **x*(*t*_{0}) = *x*_{0}*,* *x*^{j}(*t*_{0}) = *x*_{1}*,* *…,* *x*^{(}^{n}^{−}^{1)}(*t*_{0}) = *x*_{n}_{−}_{1}(1.1.2) *are called initial condition.*

BIBLIOGRAPHY

Chidume, Linear Functional Analysis. Ibadan University Press, 2014.

Cristian, C. Vidal, The Chetaev Theorem for Ordinary Difference Equations. Vol. 31 of Proyecciones Journal of Mathematics, 2012, 391-402.

Rowell, Computing the Matrix Exponential, The Cayley Hamilton Method. Massachusetts Institute of Technology, Department of Mechanical engineering, 2004, web.mit.edu/2.151/www/Handouts/CayleyHamilton.pdf

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CSN Team