In this paper we investigate POD discretizations of abstract linear-quadratic optimal control problems with control constraints. We apply the discrete technique developed by Hinze (Comput. Optim. Appl. 30:45-61, 2005) and prove error estimates for the corresponding discrete controls, where we combine error estimates for the state and the adjoint system from Kunisch and Volkwein (Numer. Math. 90:117-148, 2001; SIAM J. Numer. Anal. 40:492-515, 2002). Finally, we present numerical examples that illustrate the theoretical results.