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

In this paper we characterize (in Theorem 1) the uniform asymptotic stability of equations of the form \[[ {{array}{*{20}c} { x} \\ { y} \\ {array} } ] = [ {{array}{*{20}c} {A(t)} & & { - B(t)} \\ {B(t)} & & 0 \\ {array} } ][ {{array}{*{20}c} x \\ y \\ {array} } ]\] (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.