Robust optimal policies for Markov decision processes with safety-threshold constraints

We study the synthesis of robust optimal control policies for Markov decision processes with transition uncertainty (UMDPs) and subject to two types of constraints: (i) constraints on the worst-case, maximal total cost and (ii) safety-threshold constraints that bound the worst-case probability of visiting a set of error states. For maximal total cost constraints, we propose a state-augmentation method and a two-step synthesis algorithm to generate deterministic, memoryless optimal policies given the reward to be maximized. For safety threshold constraints, we introduce a new cost function and provide an approximately optimal solution by a reduction to an uncertain Markov decision process under a maximal total cost constraint. The safety-threshold constraints require memory and randomization for optimality. We discuss the use and the limitations of the proposed solution.