Partially Observable Total-Cost Markov Decision Processes with Weakly Continuous Transition Probabilities

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.