This paper introduces different Model Predictive Control (MPC) strategies aimed at optimizing the energy production of a Pendulum Wave Energy Converter (PeWEC). Due to MPC ability of dealing with system constraints and considering future behaviors in optimal control computation, the first proposed MPC is applied to PeWEC as a classical problem of set-point tracking using a linear model to propagate the system behaviour. Moreover, since the MPC application in wave energy conversion deviates significantly from traditional MPC ones, an economic version is also explored. The objective function of the MPC thus realized directly considers a measure of the absorbed power. However, this formulation, together with the use of a nonlinear model in predicting the system evolution, leads to an optimization problem to be solved that is neither fully quadratic nor can be guaranteed to be convex. This paper shows that this second approach brings better performances, demonstrating that it is potentially more suitable for wave energy applications. On the other side, such approach has a computational drawback from both a real-time implementation and offline design perspectives. To avoid the potentially prohibitive computational costs that an online optimization would require, this work introduces a novel control based on a Deep Neural Network (DNN) able to mimic the nonlinear economic MPC. Results arising from simulations applying the proposed strategy demonstrate the effectiveness of the presented approach.