Average optimality for continuous-time Markov decision processes in polish spaces

This paper is devoted to studying the average optimality in continuous-time Markov decision processes with fairly general state and action spaces. The criterion to be maximized is expected average rewards. The transition rates of underlying continuous-time jump Markov processes are allowed to be unbounded, and the reward rates may have neither upper nor lower bounds. We first provide two optimality inequalities with opposed directions, and also give suitable conditions under which the existence of solutions to the two optimality inequalities is ensured. Then, from the two optimality inequalities we prove the existence of optimal (deterministic) stationary policies by using the Dynkin formula. Moreover, we present a ``semimartingale characterization'' of an optimal stationary policy. Finally, we use a generalized Potlach process with control to illustrate the difference between our conditions and those in the previous literature, and then further apply our results to average optimal control problems of generalized birth--death systems, upwardly skip-free processes and two queueing systems. The approach developed in this paper is slightly different from the ``optimality inequality approach'' widely used in the previous literature.Comment: Published at http://dx.doi.org/10.1214/105051606000000105 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Average Continuous Control of Piecewise Deterministic Markov Processes

This paper deals with the long run average continuous control problem of piecewise deterministic Markov processes (PDMP's) taking values in a general Borel space and with compact action space depending on the state variable. The control variable acts on the jump rate and transition measure of the PDMP, and the running and boundary costs are assumed to be positive but not necessarily bounded. Our first main result is to obtain an optimality equation for the long run average cost in terms of a discrete-time optimality equation related to the embedded Markov chain given by the post-jump location of the PDMP. Our second main result guarantees the existence of a feedback measurable selector for the discrete-time optimality equation by establishing a connection between this equation and an integro-differential equation. Our final main result is to obtain some sufficient conditions for the existence of a solution for a discrete-time optimality inequality and an ordinary optimal feedback control for the long run average cost using the so-called vanishing discount approach.Comment: 34 page

Average optimality for continuous-time Markov decision processes under weak continuity conditions

This article considers the average optimality for a continuous-time Markov decision process with Borel state and action spaces and an arbitrarily unbounded nonnegative cost rate. The existence of a deterministic stationary optimal policy is proved under a different and general set of conditions as compared to the previous literature; the controlled process can be explosive, the transition rates can be arbitrarily unbounded and are weakly continuous, the multifunction defining the admissible action spaces can be neither compact-valued nor upper semi-continuous, and the cost rate is not necessarily inf-compact