Scalable Underapproximative Verification of Stochastic LTI Systems using Convexity and Compactness

We present a scalable algorithm to construct a polytopic underapproximation of the terminal hitting time stochastic reach-avoid set, for the verification of high-dimensional stochastic LTI systems with arbitrary stochastic disturbance. We prove the existence of a polytopic underapproximation by characterizing the sufficient conditions under which the stochastic reach-avoid set and the proposed open-loop underapproximation are compact and convex. We construct the polytopic underapproximation by formulating and solving a series of convex optimization problems. These set-theoretic properties also characterize circumstances under which the stochastic reach-avoid problem admits a bang-bang optimal Markov policy. We demonstrate the scalability of our algorithm on a 40D chain of integrators, the highest dimensional example demonstrated to date for stochastic reach-avoid problems, and compare its performance with existing approaches on a spacecraft rendezvous and docking problem.