Optimal impulse control of a diffusion process with both fixed and proportional costs of control

This paper concerns the optimal control of a system where the state is modeled by a homogeneous diffusion process in R1. Each time the system is controlled a fixed cost is incurred as well as a cost which is proportional to the magnitude of the control applied. In addition to the cost of control, there are holding or carrying costs incurred which are a function of the state of the system. Sufficient conditions are found to determine the optimal control for an infinite horizon problem. The optimal policy is one of "impulse" control originally introduced by Bensoussan and Lions [2] where the system is controlled only a finite number of times in any bounded time interval and the control requires an instantaneous finite change in the state variable. The issue of the existence of such controls is not addressed.