Stochastic Reachability of Discrete-Time Stochastic Systems via Probability Measures

Abstract: We develop a framework for stochastic reachability for discrete-time systems in terms of probability measures. We reframe the stochastic reachability problem in terms of reaching and avoiding known probability measures, and seek to characterize a) the initial set of distributions which satisfy the reachability specifications, and b) a Markov based feedback control that ensures these properties for a given initial distribution. That is, we seek to find a Markov feedback policy which ensures that the state probability measure reaches a target probability measure while avoiding an undesirable probability measure over a finite time horizon. We establish a backward recursion for propagation of state probability measures, as well as sufficient conditions for the existence of a Markov feedback policy via the 1-Wasserstein distance. Lastly, we characterize the set of initial distributions which satisfy the reachability specifications. We demonstrate our approach on an analytical example.