Thesis Abstract

Abstract
The LaSalle invariance principle is a fundamental tool in the analysis of stability for systems described by ordinary differential equations. This principle provides a powerful method for proving the asymptotic stability of equilibrium points or limit cycles. The essence of the LaSalle invariance principle lies in constructing a Lyapunov function that decreases along solutions of the differential equation, eventually leading to the convergence of trajectories to a region called the largest invariant set. This region typically corresponds to the set of points where the derivative of the Lyapunov function is zero. The LaSalle invariance principle has a wide range of applications in various fields, including control theory, dynamical systems, and mathematical biology. In this research project, we delve into the theoretical foundations of the LaSalle invariance principle and explore its applications in the context of ordinary differential equations. We study the conditions under which the LaSalle invariance principle can be applied to guarantee stability properties of solutions to differential equations. By analyzing the behavior of trajectories near equilibrium points or limit cycles, we aim to establish the asymptotic stability of these critical points using Lyapunov functions constructed based on the LaSalle invariance principle. Furthermore, we investigate the robustness of the LaSalle invariance principle with respect to perturbations in the system dynamics. Understanding how small changes in the system parameters affect the stability analysis is crucial for practical applications of the principle. We explore techniques for incorporating uncertainties into the stability analysis framework and develop methods for quantifying the impact of perturbations on the stability properties of the system. Moreover, we consider practical examples to illustrate the effectiveness of the LaSalle invariance principle in analyzing stability properties of real-world systems. By applying the principle to specific differential equations modeling physical, biological, or engineering systems, we demonstrate how it can be used to predict the long-term behavior of trajectories and assess the stability of the system under different operating conditions. Overall, this research project contributes to the understanding and application of the LaSalle invariance principle for ordinary differential equations, shedding light on its significance in stability analysis and its relevance to a wide range of scientific and engineering disciplines.

Thesis Overview

PRELIMINARIES
1.1 Denitions and basic Theorems
In this chapter, we focussed on the basic concepts of the ordinary dierential equations. Also, we
emphasized on relevant theroems in ordinary dierential equations.
Denition 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 dierential 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(nô€€€1)) (1.1.1)
Denition 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))
Denition 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)
Denition 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) ô€€€ f(y)k k(r)kx ô€€€ 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) ô€€€ f(y)k kkx ô€€€ yk; for all x; y 2 Rn:
Denition 1.1.5 (Initial value problem (IVP) Let I be an interval containing x0, the follow-
ing problem (
x(n)(t) = f(t; x(t); :::; x(nô€€€1)(t))
x(t0) = x0; x0(t0) = x1; :::; x(nô€€€1)(t0) = xnô€€€1
(1.1.2)
is called an initial value problem (IVP).
x(t0) = x0; x0(t0) = x1; :::; x(nô€€€1)(t0) = xnô€€€1
are called initial condition.
2
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
u(x)v(x)
+

x
a
u(s)v(s)ds
ds

x
a
v(x)ds:
So, taking exponential of both side we get
u(x) +

x
a
u(s)v(s)ds

x
a
u(s)v(s)ds:
Thus,
u(x) e

x
a
v(s)ds
; x 2 [a; b]:
Corollary 1.1.7 Let u; v : [a; b] ! R+ be continuous such that
u(x)

x
a
u(s)v(s)ds; for all x 2 [a; b]:
Then, u = 0 on [a; b].
Proof . Now,
u(x)

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

x
a
u(s)v(s)ds
1
n
+

x
a
u(s)v(s)ds; for all n 1:
So, by Gronwall’s lemma,
u(x)
1
n
e

x
a u(s)v(s)ds;
so as
n ! 1; u(x) ! 0:
Thus, u(x) = 0, since u(x) 0. Hence, u = 0 on [a; b].
3
1.2 Exponential of matrices
Denition 1.2.1 Let A 2 Mnn(R), then eA is an n n matrix given by the power series
eA =
1X
k=0
Ak
k!
The series above converges absolutely for all A 2 Mnn(R)
Proof . The n-th partial sum is
Sn =
Xn
k=0
Ak
k!
So, let n > m Then,
Sn ô€€€ Sm =
Xn
k=m+1
Ak
k!
:
So,
kSn ô€€€ Smk
Xn
k=m+1
kAkk
k!
:
So as
m ! 1; kSn ô€€€ Smk ! 0
So, (Sn)n is Cauchy. Thus, converges.
Theorem 1.2.2 [3](Cayley Hamilton Theorem)
Let A 2 Mnn(R) and () = det(I ô€€€ A) its characteristic polynomial then
(A) = 0:
Proof . Let A 2 Mnn(R);
() = det(I ô€€€ A) = c0 + c1 + c22 + ::: + cnn:
adj(A ô€€€ I) = B0 + B1 + B22 + ::: + Bnô€€€2nô€€€2 + Bnô€€€1nô€€€1;
where Bi 2 Mnn(R) for i = 0; 1; 2; :::; n; but, from linear algebra we have that
Aô€€€1 =
adj(A)
det(A)
;
where adj(A) denotes the adjugate or classical adjoint of A. So,
det(I ô€€€ tA)I = (I ô€€€ tA)adj(I ô€€€ tA):
(A ô€€€ I)(B0 + B1 + B22 + ::: + Bnô€€€2nô€€€2 + Bnô€€€1nô€€€1) = (c0 + c1 + c22 + ::: + cnn)I:
Observe that the entries in adj(I ô€€€tA) are polynomials in of degree at most nô€€€1. So, Bi is the
zero matrix for i = n. Equating the coecients of n on both sides gives
c0I + c1A + c2A2 + ::: + cnAn = 0:
Thus,
(A) = 0:
4
Example 1.2.3 (Application of Cayley Hamilton Theorem)
Find etA for A =

0 1
ô€€€1 0
!
Solution:
The characteristic equation is s2 + 1 = 0, and the eigenvalues are 1 = i, and 2 = ô€€€i. So, by
Theorem 1.2.2 we have that,
etA = 0I + 1A;
where we are to nd the value of 0, and 1. So,
eti = cos t + i sin t = 0 + 1i
eô€€€ti = cos t ô€€€ i sin t = 0 ô€€€ 1i
which implies that 0 = cos t, and 1 = sin t. So,
etA = cos(t)I + sin(t)A =

cos t sin t
ô€€€sin t cos t
!
Theorem 1.2.4 [11] Let A;B 2 Mnn(R). Then,
(1) If 0 denotes the zero matrix, then e0 = I, the identity matrix.
(2) If A is invertible, then eABAô€€€1
= AeBAô€€€1.
Proof . Recall that, for all integers s 0, we have (ABAô€€€1)s = ABsAô€€€1. Now,
eABAô€€€1
= I + ABAô€€€1 +
(ABAô€€€1)2
2!
+ :::
= I + ABAô€€€1 +
AB2Aô€€€1
2!
+ :::
= A(I + B +
B2
2!
+ :::)Aô€€€1
= AeBAô€€€1:
(3) If A is symmetric such that A = AT , then
e(AT ) = (eA)T :
Proof .
eA =
1X
k=0
Ak
k!
:
Then
eAT
=
1X
k=0
(AT )k
k!
=
1X
k=0
(Ak)T
k!
= (
1X
k=0
Ak
k!
)T = (eA)T :
(4) If AB = BA, then
eA+B = eAeB:
Proof .
eAeB = (I + A +
A2
2!
+
A3
3!
+ :::)(I + B +
B2
2!
+
B3
3!
+ :::)
= (
1X
k=0
Ak
k!
)(
1X
j=0
Bj
j!
)
=
1X
k=0
1X
j=0
(A + B)k+j
j!k!
5
Put m = j + k, then j = m ô€€€ k then from the binomial theorem that
eAeB =
1X
m=0
1X
k=0
AmBmô€€€k
(m ô€€€ k)!k!
=
1X
m=0
Am
m!
1X
k=0
m!
(m ô€€€ k)!
Bmô€€€k
k!
=
1X
m=0
(A + B)m
m!
= eA+B:
Theorem 1.2.5 [9]
detA
dt
= AetA = etAA; for t 2 R:
Proof . x(t; x0) = etAx0. Then,
dx(t; x0)
dt
= etAx0A =
1X
k=0
tkAk
k!
x0A = ( lim
n!1
Xn
k=0
tkAk
k!
)x0A
= lim
n!1
Xn
k=0
tkAk+1
k!
x0 = lim
n!1
Xn
k=0
AtkAk
k!
x0 = A
1X
k=0
tkAk
k!
x0 = AetAx0
So,
detA
dt
= AetA = etAA:
Proposition 1.2.6 The solution x(:; x0) of the following linear space
(
x0(t) = Ax(t); t 2 R
x(0) = x0 2 Rn
where A 2 Mnn(R), is given by
x(t; x0) = etAx0:
6