  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 the probabilistic reach-avoid-stay
problems in consideration of robustness. To this end, we first establish a
connection between stochastic 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
Lyapunovbarrier conditions in control synthesis for reach-avoid-stay
specifications, and show its effectiveness in a case study.
