Near Optimal Control in the Presence of Small Stochastic Perturbations

Abstract

We consider the stochastic system x˙(t) = f(x(t), u(t)) + ξ(t) where ξ(t) is a noise term, with loss criterion E∫0Tg(x,u)dt. A method of computing a correction to the optimal deterministic control, when the effects of ξ(t) are small, is presented. The method is based on some recent works in the stochastic calculus of variations which prove the applicability of a form of the Lagrange multiplier rule and the Hamiltonian formulation to stochastic extremum problems. The method is quite general and is capable of expansion to a greater degree of control correction as the noise effects increase.