  This paper presents a new safety specification method that is robust against
errors in the probability distribution of disturbances. Our proposed
distributionally robust safe policy maximizes the probability of a system
remaining in a desired set for all times, subject to the worst possible
disturbance distribution in an ambiguity set. We propose a dynamic game
formulation of constructing such policies and identify conditions under which a
non-randomized Markov policy is optimal. Based on this existence result, we
develop a practical design approach to safety-oriented stochastic controllers
with limited information about disturbance distributions. This control method
can be used to minimize another cost function while ensuring safety in a
probabilistic way. However, an associated Bellman equation involves
infinite-dimensional minimax optimization problems since the disturbance
distribution may have a continuous density. To resolve computational issues, we
propose a duality-based reformulation method that converts the
infinite-dimensional minimax problem into a semi-infinite program that can be
solved using existing convergent algorithms. We prove that there is no duality
gap, and that this approach thus preserves optimality. The results of numerical
tests confirm that the proposed method is robust against distributional errors
in disturbances, while a standard stochastic safety specification tool is not.
