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Stability and safety are crucial in safety-critical control of dynamicalsystems. The reach-avoid-stay objectives for deterministic dynamical systemscan be effectively handled by formal methods as well as Lyapunov methods withsoundness and approximate completeness guarantees. However, for continuous-timestochastic dynamical systems, probabilistic reach-avoid-stay problems areviewed as challenging tasks. Motivated by the recent surge of applications incharacterizing safety-critical properties using Lyapunov-barrier functions, weaim to provide a stochastic version for the probabilistic reach-avoid-stayproblems in consideration of robustness. To this end, we first establish aconnection between stochastic stability with safety constraints andreach-avoid-stay specifications. We then prove that stochastic Lyapunov-barrierfunctions provide sufficient conditions for the target objectives. We applyLyapunovbarrier conditions in control synthesis for reach-avoid-stayspecifications, and show its effectiveness in a case study.

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