This report develops a framework for analyzing probabilistic reachability and safety problems for discrete time hybrid systems in a stochastic game setting. In particular, we formulate these problems as zero-sum stochastic games between the control, whose objective is to reach a desired target set or remain within a given safe set, and a rational adversary, whose objective is opposed to that of the control. It will be shown that the maximal probability of achieving the reachability and safety objectives subject to the worst-case adversary behavior can be computed through a suitable dynamic programming algorithm. Furthermore, there always exists an optimal control policy which achieves this worst-case probability, regardless of the choice of disturbance strategy, and sufficient conditions for optimality of the policy can be derived in terms of the dynamic programming recursion. We provide several application examples from the domains of air traffic management and robust motion planning to demonstrate our modeling framework and solution approach.




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