Analysis of optimal control problems for the process of wastewater biological treatment

Abstract

We consider a three-dimensional deterministic control model of the process of aerobic wastewater biotreatment. For this model, we formulate and solve two optimal control problems, each of which has a corresponding minimizing functional. For the first problem, the functional is a weighted sum of the pollutant concentration at the end of a fixed time interval and the cumulative biomass con- centration over the interval. For the second problem, the functional is a weighted sum of the pollutant concentration at the end of the time interval and the cumulative oxygen and biomass concentra- tions over the interval. In order to solve these problems, we apply the Pontryagin Maximum Principle. The switching functions are analytically investigated and uniquely determine the type of the op- timal controls for the considered problems. Their properties allow the simplification of the optimal control problems to that of finite- dimensional constrained minimization. Numerical solutions of the optimal control problems are also provided.