  This paper studies an optimal control problem related to membrane filtration
processes. A simple 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
production of a membrane filtration system (i.e. the filtrate) over a finite
time horizon, where control variable is the sequence of filtration/backwashing
cycles over the operation time of process. Based on the Pontryagin Maximum
Principle, we characterize the optimal control strategy and show that it
presents 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 two different strategies are both
optimal.
