Sufficient and Necessary Smooth Barrier-like Conditions for Continuous-Time Stochastic Reach-Avoid Verification

Abstract

In this paper, we study infinite-horizon reach-avoid verification for continuous-time stochastic systems modeled by stochastic differential equations (SDEs). We formulate this problem within a barrier-function-based framework, which transforms the verification problem into an existence problem for barrier functions satisfying barrier-like conditions expressed as functional inequalities. We provide sufficient and necessary barrier-like conditions in terms of polynomial barrier functions for infinite-horizon reach-avoid verification under suitable regularity and uniform ellipticity assumptions. We first construct a discounted value function that characterizes lower bounds on the reach-avoid probability. We then show that it is the unique classical solution of an associated elliptic Dirichlet problem. Based on this characterization, we further show that the barrier-like condition proposed in our previous work on finite-horizon reach-avoid verification is not only sufficient for infinite-horizon reach-avoid verification but also necessary whenever the reach-avoid probability is strictly larger than the specified threshold. In particular, whenever the reach-avoid probability is strictly larger than the specified threshold fro every state in the initial set, polynomial barrier functions satisfying this barrier-like condition exist. Furthermore, when the system dynamics are polynomial, we formulate the problem of finding polynomial barrier functions satisfying this barrier-like condition as sum-of-squares (SOS) programs, which are shown to be sound and complete. Finally, we demonstrate the theoretical results on two numerical examples.