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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 the liveness probability for a given safe set and set of initial states. This probability signifies the likelihood of the system remaining within the safe set indefinitely, starting from the set of initial states. To address this problem, we propose optimizations for verifying safe probabilistic invariance in discrete-time and continuous-time stochastic dynamical systems. These optimizations adapt classical stochastic barrier certificates, which are based on Doob's non-negative supermartingale inequality, and the equations described in 29,31, which can precisely define 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.
Wu et al. (Sat,) studied this question.