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 concentration 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 concentrations 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 optimal controls for the considered problems. Their properties allow the simplification of the optimal control problems to that of finitedimensional constrained minimization. Numerical solutions of the optimal control problems are also provided.