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# Control of Singularly Perturbed Hybrid Stochastic Systems

## Jerzy A. Filar, V Gaitsgory, and Alain Haurie

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In this paper we study a class of optimal stochastic control problems involving two different time scales. The fast mode of the system is represented by deterministic state equations whereas the slow mode of the system corresponds to a jump disturbance process. Under a fundamental "ergodicity" property for a class of "infinitesimal control systems" associated with the fast mode, we show that there exists a limit problem which provides a good approximation to the optimal control of the perturbed system. Both the finite and infinite discounted horizon cases are considered. We show how an approximate optimal control law can be constructed from the solution of the limit control problem. In the particular case where the infinitesimal control systems possess the so-called turnpike property, i.e. are characterized by the existence of global attractors, the limit control problem can be given an interpretation related to a decomposition approach.

, 31 pages