Discrete-time hybrid control in Borel spaces: Average cost optimality criterion

Abstract

This paper addresses an optimal hybrid control problem in discrete-time with Borel state and action spaces. By hybrid we mean that the evolution of the state of the system may undergo deep changes according to structural modifications of the dynamic. Such modifications occur either by the position of the state or by means of the controller's actions. The optimality criterion is of a long-run ratio-average (or ratio-ergodic) type. We provide the existence of optimal average policies for this hybrid control problem by analyzing an associated dynamic programming equation. We also show that this problem can be translated into a standard (or non-hybrid) optimal control problem with cost constraints. Besides, we show that our model includes some special and important families of control problems, such as those with an impulsive or switching mode. Finally, to illustrate our results, we provide an example on a pollution-accumulation problem.