Singular perturbation for the discounted continuous control of piecewise deterministic Markov processes

This paper deals with the expected discounted continuous control of piecewise deterministic Markov processes (PDMP´s) using a singular perturbation approach for dealing with rapidly oscillating parameters. The main goal is to show that the value function of the control problem for the system driven by the perturbed Markov chain converges to the value function of an averaged model in which the regimes within the same class are aggregated through a quasi-stationary distribution. This convergence is obtained by, roughly speaking, showing that the infimum and supremum limits of the value functions satisfy two optimality inequalities. This enables us to show the result by invoking a uniqueness argument, without needing any kind of Lipschitz continuity condition.