In this paper we present the approximation of an infinite horizon optimal control problem for evolutive advection-diffusion equations. The method is based on a model reduction technique, using a Proper Orthogonal Decomposition (POD) approximation, coupled with a Hamilton-Jacobi-Bellman (HJB) equation which characterizes the value function of the corresponding control problem for the reduced system. We show that it is possible to improve the surrogate model by means of a Model Predictive Control (MPC) solver. Finally, we present numerical tests to illustrate our approach and to show the effectiveness of the method in comparison to existing approaches.