Non-Convex Optimal Control for Wastewater Treatment Problem: Hamilton-Jacobi-Bellman Approach

Optimal control in wastewater treatment problem

Abstract

This paper deals with a non-convex optimal control problem, that models a wastewater treatment process, which aims at the degradation of pollutants by bacteria using dissolved oxygen. The object is to build an optimal control strategy, involving dilution rate, recycle rate, and aeration rate as control variables, to minimize the final value of the substrate biomass, running dissolved oxygen concentration and bacteria biomass, together with the cost of recycling and aeration. As results, we investigate the invariance and dissipation properties and determine the existence and uniqueness of a solution for the controlled dynamical system. Furthermore, because the dynamics are non-convex with respect to the controls, we use the Hamilton-Jacobi-Bellman equation and its viscosity solutions to show that the value function is the unique viscosity solution to this equation, which provides insight into the existence of an optimal control strategy achieving our goal. We finally, present some numerical simulations to support the theoretical outcomes.