Risk-Sensitive LQG Discounted Control Problems and Their Asymptotic Behavior

Abstract

In this paper we describe the certainty-equivalent of the infinite horizon with discounted cost for the linear-quadratic-Gaussian model. Existence of optimal solution is investigated as well as explicit formulas for the value function. This problem is related with the ergodic (average cost per unit time) criterion, analyzing the asymptotic behavior when the discount factor vanishes. It is shown that under a suitable log transformation, a functional of the risk-sensitive discounted cost converges to the risk-sensitive average cost. This convergence is studied through the solutions of the associated Bellman equations. In addition, we obtain explicitly the set of values for the risk parameter for which no breakdown exists for the value functions.