Probabilistic Reachability Analysis of Stochastic Control Systems

We address the reachability problem for continuous-time stochastic dynamic systems. Our objective is to present a unified framework that characterizes the reachable set of a dynamic system in the presence of both stochastic disturbances and deterministic disturbances. To achieve this, we devise a strategy that decouples the effects of deterministic disturbances and stochastic disturbances on the reachable sets of the system. For the deterministic part, we demonstrate that existing reachability methods effectively capture the impact of deterministic disturbances. As for the stochastic disturbances, we introduce a novel technique that probabilistically bounds the difference between a stochastic trajectory and its deterministic counterpart. The key to our approach is introducing a novel energy function termed the Averaged Moment Generating Function that yields a high probability bound for this difference. This bound is tight and exact for linear stochastic dynamics and applicable to a large class of nonlinear stochastic systems. By combining our innovative technique with existing methods for deterministic reachability analysis, we can compute estimations of reachable sets that surpass those obtained with current approaches for stochastic reachability analysis. We validate the effectiveness of our framework through various numerical experiments. Beyond its immediate applications in reachability analysis, our methodology is poised to have profound implications in the broader analysis and control of stochastic systems. It opens avenues for enhanced understanding and manipulation of complex stochastic dynamics, presenting opportunities for advancements in related fields.