Reduction theorems for stability of compact sets in time-varying systems

Abstract

Reduction theorems provide a framework for stability analysis that consists in breaking down a complex problem into a hierarchical list of subproblems that are simpler to address. This paper investigates the following reduction problem for time-varying ordinary differential equations on Rn. Let Γ1 be a compact set and Γ2 be a closed set, both positively invariant and such that Γ1⊂Γ2⊂Rn. Suppose that Γ1 is uniformly asymptotically stable relative to Γ2. Find conditions under which Γ1 is uniformly asymptotically stable. We present a reduction theorem for uniform asymptotic stability that completely addresses the local and global version of this problem, as well as two reduction theorems for uniform stability and either local or global uniform attractivity. These theorems generalize well-known equilibrium stability results for cascade-connected systems as well as previous reduction theorems for time-invariant systems. We also present Lyapunov characterizations of the stability properties required in the reduction theorems that to date have not been investigated in the stability theory literature.