Recursive Lyapunov Functions

The paper presents the concept of recursive Lyapunov function. The concept is applied to investigation of asymptotic stability problem of autonomous systems. The sequence of functions {U(alpha)((i))} and corresponding performance measures lambda((i)) are introduced. It is proven that lambda((i+1)) <= lambda((i)) and in most cases the inequality is a strong one. This fact leads to a concept of a recursive Lyapunov function. For the very important applications case of exponential stability the procedure is effective under very weak conditions imposed on the function V = U((0)). The procedure may be particularly applicable for the systems dependent on parameters, when the Lyapunov function determined from one set of parameters may be employed at the first step of the procedure.