A class of infinite horizon optimal control problems involving L p-type cost functionals with 0 < p ≤ 1 is discussed. The existence of optimal controls is studied for both the convex case with p = 1 and the nonconvex case with 0 < p < 1, and the sparsity structure of the optimal controls promoted by the L p-type penalties is analyzed. A dynamic programming approach is proposed to numerically approximate the corresponding sparse optimal controllers.