The Linear Programming Approach to Reach-Avoid Problems for Markov Decision Processes
This work provides a new theoretical characterization and practical approximation for safety and reachability control in stochastic systems, which is important for robotics and autonomous systems, though the results are incremental as they extend existing LP-based methods.
The paper addresses the stochastic reach-avoid problem for Markov decision processes, aiming to maximize the probability of reaching a target set by a given time while staying safe. It characterizes the solution via an infinite-dimensional linear program and develops a tractable finite-dimensional approximation, achieving reduced computational complexity for Gaussian mixture kernels.
One of the most fundamental problems in Markov decision processes is analysis and control synthesis for safety and reachability specifications. We consider the stochastic reach-avoid problem, in which the objective is to synthesize a control policy to maximize the probability of reaching a target set at a given time, while staying in a safe set at all prior times. We characterize the solution to this problem through an infinite dimensional linear program. We then develop a tractable approximation to the infinite dimensional linear program through finite dimensional approximations of the decision space and constraints. For a large class of Markov decision processes modeled by Gaussian mixtures kernels we show that through a proper selection of the finite dimensional space, one can further reduce the computational complexity of the resulting linear program. We validate the proposed method and analyze its potential with a series of numerical case studies.