Two-Scale Optimal Control Problems

Abstract

In this chapter we study the limiting behavior of the optimal value of a cost functional for controlled two-scale stochastic systems with a small parameter tending to zero. In Section 5.1 we consider the Bolza problem where the cost functional contains an integral part (“running cost”) and a part depending only on terminal values of the phase variables of both types: slow and fast. The result is proved for the case where the model is “semilinear” in the sense that the coefficients depend linearly on the fast variable. This structure is essential: even in the deterministic setting the general nonlinear models are hardly tractable. It is assumed also that the diffusion coefficient of the fast variable is βε ^1/2 with β = o(|ln ε|^−1/2) which is our usual hypothesis. The admissible controls are open loop, i.e., adapted to the driving Wiener process. Such a setting seems to be the most developed: it allows to consider SDEs in the strong sense and use techniques similar to that of the classical theory of optimal control of ordinary differential equations. In the next sections we discuss more delicate subjects, namely, the behavior of the attainability sets for SDEs, aiming to prove a stochastic version of the Dontchev–Veliov theorem. In Section 5.2 we consider rather general models with open loop and closed loop (feedback) controls and compare the structure of attainability sets in these two settings. The main theorem of Section 5.2 clarifies the difference between these two concepts and shows why the model with closed loop controls (where solutions of SDEs are understood in the weak sense) suits more to the question we address later.