An LQ strong stabilization problem is proposed. To determine when a controller with periodic gains is locally superior to a linear time invariant compensator for this problem, a Pi test is presented. For systems with strictly proper transfer functions, it is proven that the frequency range where stable periodic controllers based on weak variations about the LTI case can give better performance than stable LTI compensators is finite. In the development, a means to evaluate the second partials of functions with respect to matrix-valued parameters is introduced. For those systems where periodic control is warranted, techniques for designing optimal periodic strongly stabilizing controllers are presented. Two examples detailing the application of the Pi test are provided, as well as an optimal periodic controller design example. (C) 2003 Elsevier Ltd, All rights reserved.