An approximate linear programming solution to the probabilistic invariance problem for stochastic hybrid systems

We consider the problem of designing a feedback policy for a discrete time stochastic hybrid system that should be kept operating within some compact set A. To this purpose, we introduce an infinite-horizon discounted average reward function to be maximized, where a negative reward is associated to the transitions driving the system outside A and a positive reward to those leading it back to A. An approximate linear programming approach resting on randomization and function approximation is then proposed to solve the resulting dynamic programming problem. The performance of the obtained policy is assessed on a benchmark example and compared to standard solutions based on gridding.