On the stability of a family of nonlinear time-varying systems

In the present paper we investigate several types of Lyapunov stability of an equilibrium x/sub e/ of a family of finite dimensional dynamical systems determined by ordinary differential (difference) equations. By utilizing the extreme systems of the family of systems, we establish sufficient conditions, as well as necessary conditions (converse theorems) for several robust stability types. Our results enable us to realize a significant reduction in the computational complexity of the algorithm of Brayton and Tong (1979) in the construction of computer generated Lyapunov functions. Furthermore, we demonstrate the applicability of the present results by analyzing robust stability properties of equilibria for Hopfield neural networks and by analyzing the Hurwitz and Schur stability of interval matrices.