A novel approach for the design of delayed output-feedback controllers that optimize a quadratic cost function is presented. The proposal is based on the formulation of the optimization of the quadratic performance criterion as a minimization problem whose objective function is given in terms of the so-called delay Lyapunov matrix. The computation of the delay Lyapunov matrix sensitivity with respect to the controller gains allows obtaining the gradient of the objective function and solving the minimization problem within a gradient-based optimization framework. The potential of the approach is shown by some examples of practical relevance. (C) 2019 Elsevier Ltd. All rights reserved.