Myopically Verifiable Probabilistic Certificate for Long-term Safety

In this paper, we consider using barrier function-based approaches for the safe control problem in stochastic systems. In the presence of stochastic uncertainties, a myopic controller that ensures safe probability in infinitesimal time intervals may suffer from the accumulation of unsafe probability over time and result in a small long-term safe probability. Meanwhile, increasing the outlook time horizon may lead to significant computation burdens and delayed reactions, which also compromises safety. To tackle this challenge, we define a new notion of forward invariance on ‘probability space’ as opposed to the safe regions on state space. This new notion allows the long-term safe probability to be framed into a forward invariance condition, which can be efficiently evaluated. We use this safety condition to propose a controller that evaluates infinitesimal outlook horizon and guarantees long-term safe probability or fast recovery probability. The proposed controller ensures the safe probability does not decrease over time or informs the exposed levels of risks (unsafe probability) when it becomes infeasible. The performance of the proposed controller is evaluated in numerical simulations. Finally, we show that this framework can also be adapted to characterize the speed and probability of forward convergent behaviors, which can be of use to finite-time Lyapunov analysis in stochastic systems.