Stability of Retarded Dynamical Systems: A Lyapunov Function Approach1

Four new stability theorems for general classes of retarded dynamical systems are established by using the Lyapunov function approach in this paper. A new technique is proposed for estimating the derivative of the Lyapunov function along the solution of a system at some specific instant. It is remarked that the stability conditions obtained by the established stability theorems together with the proposed technique are less conservative than those obtained by the existing Razumikhin- type theorems and techniques in the literature. Four illustrative examples and two corollaries for systems with a single time-varying delay or multiple time-varying delays are given to show the applications of the established stability theorems and the proposed technique.