Asymptotic solutions of an optimal servo problem

This paper discusses a class of optimal servo problems with random input and with bounded control. The problem is to find a control
u
which minimizes the mean-square error and to calculate this minimum. Such problems can often be reduced to solving a non-linear partial differential equation. Since this equation is not amenable to an exact solution, a method of successive approximation based on singular perturbation is used to obtain asymptotic solutions for a simple system. The method is applicable for the case of small input, small disturbance, and a relatively large bound on the control
u
. Computational results for the mean-square error are shown, and the difficulty of this method is discussed.