Analysis of an Optimal Control Problem Related to the Anaerobic Digestion Process

Abstract

Our aim in this work is to synthesize optimal feeding strategies maximizing over a time period the biogas production in a continuously filled bioreactor controlled by its dilution rate. Following [5], such an anaerobic process is described by the so-called AM2 model which is a four-dimensional dynamical system. Instead of modeling the optimization of the biogas production as an optimal control problem of Lagrange type, we propose in this paper a slightly different optimal control approach : we study the minimal time control problem to reach a target point which is chosen in such a way that it maximizes the biogas production in the AM2 model at steady state. Thanks to the Pontryagin Maximum Principle and to geometric control theory, we provide an optimal feedback control for the minimal time control problem when initial conditions are taken on the invariant and attractive manifold of the system. The optimal synthesis exhibits turnpike and anti-turnpike singular arcs and a cut-locus.