Reduction Theorems For Stability of Compact Sets in Time-Varying Systems

Submitted for evaluation and publication in AutomaticaReduction 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 R n . Let Γ 1 be a compact set and Γ 2 be a closed set, both positively invariant and such that Γ 1 ⊂ Γ 2 ⊂ R n . Suppose that Γ 1 is uniformly asymptotically stable relative to Γ 2 . Find conditions under which Γ 1 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 two reduction theorems for either local or global uniform stability and uniform attractivity. These theorems generalize well-known equilibrium stability results for cascade-connected systems as well 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