This paper studies an optimal control problem related to membrane-filtration processes. A generic mathematical model of membrane fouling is used to capture the dynamic behavior of the filtration process, which consists in the attachment of matter onto the membrane during the filtration period, and the detachment of matter during the cleaning period. We consider the maximization of the net water production of a membrane filtration system (i.e., the filtrate) over a finite time horizon, where the control variable is the sequence of filtration/backwashing cycles over the operation time of process. Based on the Pontryagin's maximum principle, we characterize the optimal control strategy and show that it exhibits a singular arc. Moreover, we prove the existence of an additional switching curve before reaching the terminal state, and also the possibility of having a dispersal curve as a locus where both the different strategies are optimal.