On the Stability of Nonautonomous Differential Equations $\dot x = [A + B(t)]x$, with Skew Symmetric Matrix $B(t)$

Abstract

In this paper we characterize (in Theorem 1) the uniform asymptotic stability of equations of the form \[\left[ {\begin{array}{*{20}c} {\dot x} \\ {\dot y} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {A(t)} &\vline & { - B(t)} \\\hline {B(t)} &\vline & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} x \\ y \\ \end{array} } \right]\] (where $A(t) + A(t)^T $ is negative definite) in terms of the “richness” of $B(t)$. The equation is uniformly asymptotically stable if and only if $B(t)$ is sufficiently rich. We actually obtain stability results for a much broader class of systems (Theorems 2 and 3) whose behavior is similar to the one above. Such systems have come up recently in some adaptive control problems.