Probabilistic Reachability of Stochastic Control Systems: A Contraction-Based Approach

In this paper, we propose a framework for studying the probabilistic reachability of stochastic control systems. Given a stochastic system, we introduce a separation strategy for reachability analysis that decouples the effect of deterministic input/disturbance and stochastic uncertainty. A remarkable feature of this separation strategy is its ability to leverage any deterministic reachability framework to capture the effect of deterministic input/disturbance. Furthermore, this separation strategy encodes the impact of stochastic uncertainty on reachability analysis by measuring the distance between the trajectories of the stochastic system and its associated deterministic system. Using contraction theory, we provide probabilistic bounds on this trajectory distance and estimate the propagation of stochastic uncertainties through the system. By combining this probabilistic bound on trajectory distance with two computationally efficient deterministic reachability methods, we provide estimates for probabilistic reachable sets of the stochastic system. We demonstrate the efficacy of our framework through numerical experiments on a feedback-stabilized inverted pendulum.