Reachability analysis for controlled discrete time stochastic hybrid systems

A model for discrete time stochastic hybrid systems whose evolution can be influenced by some control input is proposed in this paper. With reference to the introduced class of systems, a methodology for probabilistic reachability analysis is developed that is relevant to safety verification. This methodology is based on the interpretation of the safety verification problem as an optimal control problem for a certain controlled Markov process. In particular, this allows to characterize through some optimal cost function the set of initial conditions for the system such that safety is guaranteed with sufficiently high probability. The proposed methodology is applied to the problem of regulating the average temperature in a room by a thermostat controlling a heater.