A Decompositional Proof Scheme for Automated Convergence Proofs of Stochastic Hybrid Systems

In this paper, we describe a decompositional approach to convergence proofs for stochastic hybrid systems given as probabilistic hybrid automata. We focus on a concept called “stability in probability,” which implies convergence of almost all trajectories of the stochastic hybrid system to a designated equilibrium point. By adapting classical Lyapunov function results to the stochastic hybrid case, we show how automatic stability proofs for such systems can be obtained with the help of numerical tools. To ease the load on the numerical solvers and to permit incremental construction of stable systems, we then propose an automatable Lyapunov-based decompositional framework for stochastic stability proofs. This framework allows conducting sub-proofs separately for different parts of the automaton, such that they still yield a proof for the entire system. Finally, we give an outline on how these decomposition results can be applied to conduct quantitative probabilistic convergence analysis, i.e., determining convergence probabilities below 1.