The paper deals with the LPV stabilizability problem for linear plants whose parameters vary with time in a compact set. First, necessary and sufficient conditions for the existence of a linear gain-scheduled stabilizing compensator are given. Next, it is shown that, if these conditions are satisfied, any compensator transfer function depending on the plant parameters and internally stabilizing the closed-loop control system when the plant parameters are constant, can be realized in such a way that the closed-loop asymptotic stability is guaranteed under arbitrary parameter variations. To find one such realization, a reasonably simple and general algorithm based on Lyapunov equations and Cholesky factorization is provided. The result can profitably be used to achieve both pointwise optimality (or pole placement) and LPV stability. Some potential applications in adaptive control and online tuning are pointed out.