Fixed Time Stability of Discrete-Time Stochastic Dynamical Systems

In this paper, we address fixed time stability in probability of discrete-time stochastic dynamical systems. Unlike finite time stability in probability, wherein the finite time almost sure convergence behavior of the dynamical system depends on the system initial conditions, fixed time stability in probability involves finite time stability in probability for which the stochastic settling-time is guaranteed to be independent of the system initial conditions. More specifically, we develop Lyapunov theorems for fixed time stability in probability for Itô –type stationary nonlinear stochastic difference equations including a Lyapunov theorem that involves a Lyapunov difference satisfying an exponential inequality of the Lyapunov function that gives rise to a minimum bound on the average stochastic settling-time characterized by the primary and secondary branches of the Lambert W function.