  This work is concerned with the safety controller synthesis of stochastic
hybrid systems, in which continuous evolutions are described by stochastic
differential equations with both Brownian motions and Poisson processes, and
instantaneous jumps are governed by stochastic difference equations with
additive noises. Our proposed framework leverages the notion of control barrier
certificates (CBC), as a discretization-free approach, to synthesize safety
controllers for stochastic hybrid systems while providing safety guarantees in
finite time horizons. In our proposed scheme, we first provide an augmented
framework to characterize each stochastic hybrid system containing continuous
evolutions and instantaneous jumps with a unified system covering both
scenarios. We then introduce an augmented control barrier certificate (ACBC)
for augmented systems and propose sufficient conditions to construct an ACBC
based on CBC of original hybrid systems. By utilizing the constructed ACBC, we
quantify upper bounds on the probability that the stochastic hybrid system
reaches certain unsafe regions in a finite time horizon. The proposed approach
is verified over a nonlinear case study.
