Lasalle Invariance Principle for Ordinary Differential Equations and Applications
Chapter One
PREAMBLE OF THE STUDY
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,
xj(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,
xj(t) = (1 + t2)y2(t)
CHAPTER TWO
BASIC THEORY OF ORDINARY DIFFERENTIAL EQUATION
In this chapter we give a broad discussion of the existence and uniqueness of solutions of ordinary differential equations. We discuss equilibrium points, stability, fundamental matrix and variation of constants formula, and other key concepts of dynamical systems. We start this chapter with the following definitions;
Definitions and basic properties
Definition 2.1.1 Let I be an interval containing t0, let f : I × Rn → Rn be continuous and Lips- chitzian with respect to second variable, let x : I → Rn be continuous, then x is a solution of the following ordinary differential equation
on I, if
xis a C1 – function on
xj(t) = f (t, x(t)), t ∈ I x(t0) = x0, t0 ∈ I
(2.1.1)
xsatisfies the above ODE, for all t ∈
Theorem 2.1.2 [9](Peano’s Theorem) Let f : R × Rn → Rn be continuous in a neighbourhood of (t0, x0) then there exists a > 0 such that the initial value problem
xj = f (t, x), t ∈ R
x(t0) = x0 ∈ Rn.
has at least one solution on the interval I = [t0 − a, t0 + a] ⊆ R.
Proof . Define the set
(2.1.2)
E = C([t0 − a, t0 + a], Rn)
then E is a Banach space provided with the ” sup ” norm. Let
M = max ǁf (t, x)ǁ for Q = {(t, x) : −a ≤ t − t0 ≤ a, ǁx − x0ǁ ≤ b}
and define the set A ⊂ E by
A := {x ∈ E : sup ǁx(t) − x0ǁ ≤ b} = B(x0, b)C(I,Rn ⊆ E.
Then, A is a closed subset of E, as xn ∈ A implies that
lim
n→∞
xn = x ∈ A
(this follows from the uniform convergence in E). Also, A is convex (every ball is convex). Thus, by the Ascoli-Arzela theorem, A is compact, and A is complete as a closed subset of a complete metric space with the sup norm.
Also, let T : A → E be defined by
(Tx)(t) := x0 +
ˆ t
f (s, x(s))ds
t0
Let, (xn)n≥1 ⊆ A such that xn → x ∈ A with the ” sup ” norm, then,
xn(t) → x(t) implies sup ǁxn(s) − x(s)ǁ → 0, as n → ∞.
Therefore,
ˆ t
ǁTxn(t) − Tx(t)ǁ ≤ ǁf (s, xn(s)) − f (s, x(s))ǁds
ˆ t0+a
≤ t0−a
ǁf (s, xn(s)) − f (s, x(s))ǁd
≤ 2a sup ǁf (s, xn(s)) − f (s, x(s))ǁ.
f is continuous on I × B(x0, b) implies that f is uniformly continuous on I × B(x0, b). So, as
n → ∞, ǁTxn(t) − Tx(t)ǁ → 0.
Thus, T is continuous. Let Tz ∈ T (A)
ˆ t
ǁTz(t) − x0ǁ = ǁ
ˆ t
f (s, z(s))dsǁ
≤ t0
ˆ t
≤ t0
ǁf (s, z(s))ǁds
Mds
ˆ t0+a
≤ Mds = 2aM ≤ b
t0−a
for a small enough. Hence, T (A) ⊆ A.
We look for a fixed point of T , that is, we want to find
x ∈ E such that Tx = x.
A fixed point of T solves the IVP(2.1.2), and T has a fixed point as a consequence of the following Schauder – Tychonoff’s Theorem (If T : X → X is continuous and if A ⊂ X is a convex compact subset of the normed linear space X and T (A) ⊂ A, then T has a fixed point in A).
Example 2.1.3 Consider
Here, f (y) = √|y(t)|.
yj = |y(t)|, t ≥ 0, y(0) = 0.
Solving the given IVP, we have that,
t2
y(t) = − 4 , if y(t) < 0,
y(t) = 0, if y(t) = 0, t2
y(t) =
, if y(t) > 0.
4
So, the ODE does not have a unique solution. This is because f is not Lipschitzian.
CHAPTER THREE
CONCLUSION
The aim of this thesis is to study a long time behaviour solution using 3 approaches .
The firste is the linearization principle: If it works that fine, but most of cases it does not work well, like we have seen in many examples.
The second one is the Lyapunov functions, it is the best way to study the asymptotic behaviour of solutions, but the construction of the Lyapunov functions depends on the nature of the ODE. The third one is based on LaSalle invariance principle, it is an interesting working tool in dynam- ical systems and control theory.
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