Hydrogen production systems supplied by photovoltaic solar energy have nonlinear dynamics and discontinuities which must be taken into account when a control system is applied. The main purpose of the control system is to maintain the electrolyzer current at the desired operating point and, at the same time, to optimize the grid energy consumption despite the solar energy variability. Classic controllers, like PID ones, are not able to obtain good performance over the whole operation range of these kinds of plants because of the aforementioned characteristics. To overcome these limitations, an optimal control strategy and a linear hybrid model predictive controller (HMPC) are applied to a hydrogen production system in this work. Regarding the optimal control design, a systematic framework is presented in order to obtain the optimal (in the sense of minimal grid energy consumption) trajectory of the states by converting the control problem into a boundary value problem by means of the Pontryagin's Maximum Principle. Interestingly, the resulting control law is explicit and piecewise continuous. Regarding the linear HMPC strategy, a mixed logical dynamical description of the linearized equations of the system is considered in order to obtain the control law by solving an optimization problem in the form of a mixed integer quadratic programming. For this control strategy three cost functions associating the grid energy consumption and the electrolyzer efficiency are presented. The proposed controllers are tested through numerical simulations for both the nominal and uncertain cases and different performance indexes are considered. Finally, a discussion of the main advantages and disadvantages of each controller in real-life applications is presented. (C) 2020 Elsevier Ltd. All rights reserved.