Abstract We consider the fundamental problem of Linear Quadratic Gaussian Control on an infinite horizon with costly information acquisition. Specifically, we consider a two-dimensional coupled system, where one of the two states is observable, and the other is not. Additionally, to inference from the observable state, costly information is available via an additional, controlled observation process. Mathematically, the Kalman-Bucy filter is used to Markovianize the problem. Using an ansatz, the problem is then reduced to one of the control-dependent, conditional variance for which we show regularity of the value function. Using this regularity for the reduced problem together with the ansatz to solve the problem by dynamic programming and verification and construct the unique optimal control. We analyze the optimal control, the optimally controlled state and the value function and compare various properties to the literature of problems with costly information acquisition. Further, we show existence and uniqueness of an equilibrium for the controlled, conditional variance, and study sensitivity of the control problem at the equilibrium. At last, we compare the problem to the case of no costly information acquisition and fully observable states. Joint work with Christoph Knochenhauer and Yufei Zhang (Imperial College London).