Verifying Continuous-time Stochastic Hybrid Systems via Mori-Zwanzig model reduction

In this work, we develop a method for verifying Continuous-time Stochastic Hybrid Systems (CTSHSs) using the Mori-Zwanzig model reduction method, whose behaviors are specified by Metric Interval Temporal Logic (MITL) formulas. By partitioning the state space of the CTSHS and computing the optimal transition rates between partitions, we provide a procedure to both reduce a CTSHS to a Continuous-Time Markov Chain (CTMC), and the associated MITL formulas defined on the CTSHS to MITL specifications on the CTMC. We prove that an MITL formula on the CTSHS is true (or false) if the corresponding MITL formula on the CTMC is robustly true (or false) under certain perturbations. In addition, we propose a stochastic algorithm to complete the verification. Finally, as an example, we implement the method in a Billiard Problem.