On the uniform asymptotic stability of certain linear nonautonomous differential equations

In this paper we give a simple characterization of the uniform asymptotic stability of equations $\dot x = - P(t)x$ where $P(t)$ is a bounded piecewise continuous symmetric positive semi-definite matrix. In the course of developing this characterization, a new and general sufficient condition is given for uniform asymptotic stability in terms of Lyapunov functions. The stability of this type of equation has come up in various control theory contexts (identification, optimization and filtering).