Forward and Converse Lyapunov Theorems for Discrete Dynamical Systems

This technical note addresses the stability analysis of nonlinear dynamic systems. Three main contributions are made. First, we show that the standard assumption of a continuous Lyapunov function can be (and in some cases must be) relaxed. We introduce the concept of the `weak' Lyapunov function, which requires that an annulus condition be satisfied. We believe that this annulus condition is a more natural construct, because it is precisely what is needed to make the forward Lyapunov theorem true. Second, we provide an example of a nonlinear system with stable equilibrium point that cannot be shown to be stable with a continuous Lyapunov function. Finally, we demonstrate a simpler and less restrictive proof of the converse Lyapunov theorem.