Computational techniques based on alternating projections are proposed to solve control design problems described by linear matrix inequalities (LMIs). In particular, we concentrate on the stabilization and the suboptimal H-proportional to output feedback control design problems. These problems can be described by a pair of LMIs and an additional coupling condition. This coupling condition is convex for the full-order control design problem, but convexity is lost for the control problem of order strictly less than the plant order. Wt : formulate these problems as feasibility problems with matrix constraint sets of simple geometry, and we utilize this geometry to obtain analytical expressions for the orthogonal projection operators onto these sets. Full-order and low-order controllers are designed using alternating projection methods. For the full-order controller case, global convergence of the alternating projection methods to a feasible solution is guaranteed. However, for the Low-order control case, only local convergence is guaranteed. An example is provided to illustrate the use of these methods for the full-order and the low-order controller design.