Stochastic Stability via Robustness of Linear Systems

Robust stability and stochastic stability have separately seen intense study in control theory since its inception. In this work we establish relations between these properties for discrete-time systems. Specifically, we examine a robustness framework which models the inherent uncertainty and variation in the system dynamics which arise in model-based learning control methods such as adaptive control and reinforcement learning. We provide results which guarantee mean-square stability margins in terms of multiplicative noises which affect the nominal dynamics, as well as connections to prior work which together imply that robust stability and mean-square stability are, in a certain sense, equivalent.