A Framework for Safe Probabilistic Invariance Verification of Stochastic Dynamical Systems

Abstract

Ensuring safety through set invariance has proven to be a valuable method in various robotics and control applications. This paper introduces a comprehensive framework for the safe probabilistic invariance verification of both discrete- and continuous-time stochastic dynamical systems over an infinite time horizon. The objective is to ascertain the lower and upper bounds of liveness probabilities for a given safe set and set of initial states. The liveness probability signifies the likelihood of the system remaining within the safe set indefinitely, starting from a state in the initial set. To address this problem, we propose optimizations for verifying safe probabilistic invariance in discrete-time and continuous-time stochastic dynamical systems. These optimizations are constructed via either using the Doob’s nonnegative supermartingale inequality-based method or relaxing the equations described in \cite{xue2021reach,xue2023reach}, which can precisely characterize the probability of reaching a target set while avoiding unsafe states. Finally, we demonstrate the effectiveness of these optimizations through several examples using semi-definite programming tools.

Keywords: Stochastic Systems, Safe Probabilistic Invariance Verification, Liveness Probabilities, Lower and Upper Bounds