On Optimal Control for a Class of Partially-Observed Systems

Abstract

This note is intended to popularize a class of partially-observed control systems which, like the LQG-problem (cf. [5]), can be solved explicitly. The problem is to steer a linear system to a hyperplane in fixed time using bounded controls and having only partial information about the state available. While state- and observation-process evolve exactly as in the LQG-problem it differs from that one in that (1) the performance index is a quite different one and (2), more important, in that ‘hard constraints1 are put on the controls. In 1980 BeneS and Karatzas [l] have analyzed the one-dimensional problem and they have shown that the optimal control using partial observations is just u(t) = -sign(s(t)••x(t)) where x(t) is the conditional mean of the state given the observations up to time t and s(t) is a deterministic function depending on the given data in a prescribed way (cf. also [8]). Interestingly, although this is a non-linear problem it exhibits the ‘certainty-equivalence’ principle — the optimal control is obtained by estimating the state and then using this estimate as though it were the true state. Recently Christopeit and Helmes [3] derived the analogous result for the multidimensional problem. A cornerstone of their analysis is an existence result on weak solutions to certain stochastic differential equations with degenerate diffusions which has been derived by Christopeit [4] using Skorokhod’s imbedding technique.