Rational Inattention in scalar LQG control

Motivated in part by the "rational inattention" framework of information-constrained decision-making by economic agents, we have recently introduced a general model for average-cost optimal control of Markov processes subject to mutual information constraints [1]. The optimal informationconstrained control problem reduces to an infinite-dimensional convex program and admits a decomposition based on the Bellman error, which is the object of study in approximate dynamic programming. In this paper, we apply our general theory to an informationconstrained variant of the scalar linear-quadratic-Gaussian (LQG) control problem. We give an upper bound on the optimal steady-state value of the quadratic performance objective and present explicit constructions of controllers that achieve this bound. We show that the obvious certainty-equivalent control policy is suboptimal when the information constraints are very severe, and exhibit another policy that performs better in this low-information regime. In the two extreme cases of no information (open-loop) and perfect information, these two policies coincide with the optimum.