Nonlinear Systems Stability Analysis

3	Stability Analysis of Nonautonomous Systems

3.1 PRELIMINARIES

Consider the following nonlinear time-varying system with the equilibrium state at the origin:

$\dot{x}=f\left(t,x\right),x\in R^n,f\left(t,0\right)=0,\forall t\ge t_0.$

(3.1)

A function ϕ : R₊ → R₊ is of class K and is denoted by ϕ ∈ K if ϕ is continuous, strictly increasing, and ϕ(0) = 0. It is of class K infinity and is denoted by ϕ ∈ K_∞ if ϕ(p) → +∞ as p → +∞.

A function β : R₊ × R₊ → R₊ is of class KL and is denoted by β ∈ L if β(p,t) is continuous, strictly increasing with respect to p, strictly decreasing with respect to t, β(0,t) = 0, and β(p,t) → 0 as t → +∞.

Lemma 3.1 [k1]:

The zero equilibrium state of ẋ = f (t, x):

(i) Is uniformly stable (US) iff there exists an α ∈ K and a scalar c > 0 independent ...