Abstract
We consider a control system which is described by a special but important class of timehomogeneous stochastic differential equations. The drift coefficients in the equations are considered controlled and the control policy is a measurable functioi of the present state of the system. A long-term average cost is associated with the system, and our problem is to choose a control which minimizes this cost. In this paper an extension of Ito's differentation rule to functions belonging to a Sobolev space is given, and this result is used to derive a sufficient condition for optimal controls which is analogous to that given by Wonham. A theorem on the existence of optimal stochastic controls is also given under some sufficiently strong assumption
