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One of the most fundamental problems in Markov decision processes is analysisand control synthesis for safety and reachability specifications. We considerthe stochastic reach-avoid problem, in which the objective is to synthesize acontrol policy to maximize the probability of reaching a target set at a giventime, while staying in a safe set at all prior times. We characterize thesolution to this problem through an infinite dimensional linear program. Wethen develop a tractable approximation to the infinite dimensional linearprogram through finite dimensional approximations of the decision space andconstraints. For a large class of Markov decision processes modeled by Gaussianmixtures kernels we show that through a proper selection of the finitedimensional space, one can further reduce the computational complexity of theresulting linear program. We validate the proposed method and analyze itspotential with a series of numerical case studies.

The Linear Programming Approach to Reach-Avoid Problems for Markov Decision Processes