On Optimal Control of Singularly Perturbed Stochastic Differential Equations

Abstract

An ordinary differential equation of the first order is referred to as singularly perturbed if it contains a small parameter at the left-hand side, e.g. see [2]. Such equations are used for the description of fast variables in two-scale systems. Assume that we have a two-scale system where on the observed time-interval the fast variables almost attain their steady states. In the limit, when the parameter is assumed to be zero, singularly perturbed differential equations have to be replaced by algebraic equations, and the order of the system diminishes. Quite often the limiting model is much more simple than the original one. For the optimal control problem the change in the structure can imply the discontinuity of the extremal value of the cost functional and the limit behaviour of the extremal value is the question of interest.