The Vanishing Approach for the Average Continuous Control of Piecewise Deterministic Markov Processes

TL;DR

This paper establishes conditions for optimal control strategies in long-term average cost problems for piecewise deterministic Markov processes, using a vanishing discount approach and integro-differential inequalities.

Contribution

It introduces a novel application of the vanishing discount method to PDMPs with general Borel state spaces and compact action sets, providing new sufficient conditions for optimality.

Findings

Derived sufficient conditions for optimal control existence.

Applied vanishing discount approach to PDMPs.

Connected growth conditions with control optimality.

Abstract

The main goal of this paper is to derive sufficient conditions for the existence of an optimal control strategy for 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. In order to do that we apply the so-called vanishing discount approach to obtain a solution to an average cost optimality inequality associated to the long run average cost problem. Our main assumptions are written in terms of some integro-differential inequalities related to the so-called expected growth condition, and geometric convergence of the post-jump location kernel associated to the PDMP.