In this work, we study the infinite-time linear quadratic optimal control problem for systems with stochastic disturbances and constrained inputs. A number of stochastic problem formulations under the full state information (FSI) structure are considered with a particular focus on the subject of feedback structure and its impact on certainty equivalence. In particular, we clarify results concerning the open-loop hard constrained, closed-loop statistically constrained, and closed-loop hard constrained cases. Extension to the infinite-time framework provides a vehicle for interpreting these controllers and indicates that the last of the three is of most interest to regulation type applications. Additionally, the partial state information problem is considered, and conditions are given for which a separated configuration consisting of the optimal estimator cascaded with the FSI optimal controller remains optimal. (c) 2004 Elsevier Ltd. All rights reserved.