Notions of exponential robust stochastic stability, ISS and their Lyapunov characterizations

Concepts of exponential global robust stability for stochastic control systems are analysed in terms of Lyapunov functions. The main result of the paper constitutes a generalization of a converse stability theorem due to Khasminskii for stochastic differential equations and establishes that, under certain hypotheses, the origin is robustly exponentially stable in the r th mean, if and only if the system admits a Lyapunov function which is smooth except possibly at the origin. The main result concerning robust asymptotic stability enable us to derive a Lyapunov‐like characterization for the concept of stochastic input‐to‐state stability (ISS).