Optimal feedback solutions to the infinite horizon LQR problem with state and input constraints based on receding horizon real-time quadratic programming are well known. In this paper we develop an explicit solution to the same problem, eliminating the need for real-time optimization. It is shown that the resulting feedback controller is piecewise linear. This explicit functional structure is exploited for efficient real-time implementation. A suboptimal strategy, based on a suboptimal choice of a finite horizon and imposing additional limitations on the allowed switching between active constraint sets on the horizon, is suggested in order to address the computer memory and processing capacity requirements of the explicit solution.