We discuss issues concerning the numerical approximation of optimal control problems where the dynamics are governed by partial differential equations. We focus on the linear quadratic case with infinite horizon. In this case, there is an approximation theory which can be used to compute a convergent solution for the regulation problem and to obtain optimal error estimates; this estimates will depend on the control configuration of the system, and we review the distributed/point control cases. In a more practical perspective, we also present a technique for a dimensional reduction of the discretized system which improves the computational efficiency of the algorithm.