This paper describes sufficient conditions for the existence of optimal policies for Partially Observable Markov Decision Processes (POMDPs) with Borel state, observation, and action sets and with the expected total costs. Action sets may not be compact and one-step cost functions may be unbounded. The introduced conditions are also sufficient for the validity of optimality equations, semi-continuity of value functions, and convergence of value iterations to optimal values. Since POMDPs can be reduced to Completely Observable Markov Decision Processes (COMDPs), whose states are posterior state distributions, this paper focuses on the validity of the above mentioned optimality properties for COMDPs. The central question is whether transition probabilities for a COMDP are weakly continuous. We introduce sufficient conditions for this and show that the transition probabilities for a COMDP are weakly continuous, if transition probabilities of the underlying Markov Decision Process are weakly continuous and observation probabilities for the POMDP are continuous in the total variation. Moreover, the continuity in the total variation of the observation probabilities cannot be weakened to setwise continuity. The results are illustrated with counterexamples and examples.
Geometry and Determinism of Optimal Stationary Control in Partially Observable Markov Decision Processes
Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学研究所）；Faculty of Mathematics and Computer Science（数学与计算机科学学院）；Leipzig University（莱比锡大学）；Santa Fe Institute（圣塔菲研究所）
