Technical note-On the optimality of reflection control

The goal of this paper is to illustrate the optimality of reflection control in three different settings, to bring out their connections and to contrast their distinctions. First, we study the control of a Brownian motion with a negative drift, so as to minimize a long-run average cost objective. We prove the optimality of the reflection control, which prevents the Brownian motion from dropping below a certain level by cancelling out from time to time part of the negative drift; and we show that the optimal reflection level can be derived as the fixed point that equates the long-run average cost to the holding cost. Second, we establish the asymptotic optimality of the reflection control when it is applied to a discrete production-inventory system driven by (delayed) renewal processes; and we do so via identifying the limiting regime of the system under diffusion scaling. Third, in the case of controlling a birth-death model, we establish the optimality of the reflection control directly via a linear programming-based approach. In all three cases, we allow an exponentially bounded holding cost function, which appears to be more general than what's allowed in prior studies. This general cost function reveals some previously unknown technical fine points on the optimality of the reflection control, and extends significantly its domain of applications.