We address a discrete-time LQG control problem over a fixed performance window and apply a receding-horizon type control strategy, resulting in an exact solution to the problem in terms of semidefinite programming. The systems considered take parameters from a finite set, and switch between them according to an automaton. The controller has a finite preview of future parameters, beyond which only the set of parameters is known. We provide necessary and sufficient convex conditions for the existence of a controller which guarantees both exponential stability and finite-horizon performance levels for the system; the performance levels may differ according to the particular parameter sequence within the performance window. A simple, physics-based example is provided to illustrate the main results.