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 \[ array{*{20c} x \\ y \\ array } = array{*{20c} A (t) & & - B (t) \\ B (t) & & 0 \\ array } array{*{20c} x \\ y \\ array } \] (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.