Equivalent Characterizations of Input-to-State Stability for Stochastic Discrete-Time Systems

Input-to-state stability (ISS) for stochastic difference inclusions is studied. First, ISS in probability relative to a compact set is defined. Subsequently, several equivalent characterizations are given. For example, ISS in probability is shown to be equivalent to global asymptotic stability in probability when the disturbance takes values in a ball whose radius is determined by a sufficiently small, but unbounded, function of the distance of the state to the compact set. In turn, a recent converse Lyapunov theorem for global asymptotic stability in probability provides an equivalent Lyapunov characterization. Finally, robust ISS in probability is defined and is shown to give another equivalent characterization.