# Optimal control of membrane filtration systems

## TL;DR

This paper formulates and analyzes an optimal control problem for membrane filtration systems, using a mathematical model to determine the best filtration and cleaning cycles for maximizing net production.

## Contribution

It introduces a novel optimal control framework for membrane filtration, characterizing the optimal strategies and revealing complex control structures like singular and switching curves.

## Key findings

- Optimal control strategy includes singular arcs.
- Existence of a switching curve before the terminal state.
- Possibility of a dispersal curve with multiple optimal strategies.

## Abstract

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.