The adaptive LQG problem--Part I

This paper is concerned with a rigorous study of the "dual control" problem of Fel\´dbaum, i.e., the LQG optimal control problem in the presence of Bayesian parameter uncertainty. The solution of this problem involves two parts, one relating to filtering and one to control. Although we establish our filtering result in complete generality in the last section, most of the paper concentrates on the finite parameter case to ease the exposition. The control result that we establish is incomplete in the sense that the smoothness of the optimal cost function is assumed rather than proved. Nevertheless, our results and methods are such that we arrive at a new proof of the classical separation theorem showing that the well-known LQG feedback law is optimal within the widest possible class of admissible controls. As this new proof avoids all talk of "dependence of the sigma algebra on the control," "weak solutions," "measure transformation techniques," etc., we feel that this result will help to clarify what is involved in the classical separation theorem.