Stochastic Lyapunov-Barrier Functions for Robust Probabilistic Reach-Avoid-Stay Specifications

Stability and safety are crucial in safety-critical control of dynamical systems. The reach-avoid-stay objectives for deterministic dynamical systems can be effectively handled by formal methods as well as Lyapunov methods with soundness and approximate completeness guarantees. However, for continuous-time stochastic dynamical systems, probabilistic reach-avoid-stay problems are viewed as challenging tasks. Motivated by the recent surge of applications in characterizing safety-critical properties using Lyapunov-barrier functions, we aim to provide a stochastic version for probabilistic reach-avoid-stay problems in consideration of robustness. To this end, based on the weak topology, we first establish a connection between probabilistic stability with safety constraints and reach-avoid-stay specifications. We, then, prove that stochastic Lyapunov-barrier functions provide sufficient conditions for the target objectives. We apply Lyapunov-barrier conditions in control synthesis for reach-avoid-stay specifications, and show its effectiveness in a case study.