We consider the control of linear time-invariant (LTI) systems over erasure channels. We propose a class of controllers where, besides memoryless dependency on the plant to controller channel state, all processing is affine (and possibly time-varying). For such controller class, we show that optimal designs separate into an estimation problem and a linear quadratic regulator problem. The structure of the optimal controller is such that its affine part converges, as the horizon length tends to infinity, to an LTI filter under the same conditions which guarantee mean-square stability in the now well-known LQG control problem over erasure channels. Our infinite horizon proposal is computationally inexpensive and its steady-state behavior is easily characterized.