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 ${\mathbb{R}}^n$. Let ${\Gamma }_1$ be a compact set and ${\Gamma }_2$ be a closed set, both positively invariant and such that ${\Gamma }_1\subset {\Gamma }_2\subset {\mathbb{R}}^n$. Suppose that ${\Gamma }_1$ is uniformly asymptotically stable relative to ${\Gamma }_2$. Find conditions under which ${\Gamma }_1$ 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 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.

归约定理为稳定性分析提供了一个框架，该框架包括将一个复杂的问题分解为一个由更容易解决的子问题组成的层次列表。本文研究了以下关于时变常微分方程的归约问题${\mathbb{R}}^n$. 让${\Gamma }_{\mathrm{1个}}$是紧集并且${\Gamma }_{\mathrm{2个}}$是一个闭集，既积极不变又使得${\Gamma }_{\mathrm{1个}}\subset {\Gamma }_{\mathrm{2个}}\subset {\mathbb{R}}^n$. 假设${\Gamma }_{\mathrm{1个}}$相对于${\Gamma }_{\mathrm{2个}}$. 找出条件${\Gamma }_{\mathrm{1个}}$一致渐近稳定。我们提出了一个统一渐近稳定性的缩减定理，它完全解决了这个问题的局部和全局版本，以及两个统一稳定性和局部或全局均匀吸引力的缩减定理。这些定理概括了级联系统的众所周知的平衡稳定性结果以及以前的时不变系统的约简定理。我们还介绍了迄今为止尚未在稳定性理论文献中研究过的归约定理中所需的稳定性特性的 Lyapunov 表征。