Discrete-time stochastic control systems: Examples of robustness to strictly causal perturbations

Robust stability for discrete-time stochastic systems employing possibly discontinuous control laws is the focus of this paper. Through novel examples, we illustrate the fact that the existence of a continuous stochastic Lyapunov function implies robustness of stability to sufficiently small, state-dependent, strictly causal, worst-case perturbations, irrespective to the continuity of the stabilizing control law. We emphasize the role of strict causality through an example for which a continuous stochastic Lyapunov function is not sufficient for robustness to arbitrarily small worst-case perturbations, which are not strictly causal. Finally we illustrate our main result for a stochastic control system which admits a continuous Lyapunov function, but associated with a necessarily discontinuous control law.