Optimal control of microalgae culture accounting for photoinhibition and light attenuation

Abstract

In a photobioreactor, due to the gradient of light, microalgae are successively exposed to conditions of low and high (inhibiting) light. This phenomenon can be captured by the Han model, which is a common mechanistic model of photoinhibition. Based on Han's description, we introduce a dynamic system of microalgae growth involving two control variables: the light intensity and the dilution rate of the reactor. This model is derived from slow/fast dynamic considerations in a chemostat system accounting for the light gradient due to absorption and scattering following the Lambert-Beer's law. Then, we formulate and study an optimal control problem in order to fully-characterize the optimal light supply and dilution strategies that maximize the harvested biomass. Our study, mainly based on Pontryagin's maximum principle (PMP), shows that singular arcs and turnpike-like behaviors appear in the optimal solution. In particular, we prove that the optimal strategy maintains the biomass at a constant level along singular arcs, and we determine its static value. The theoretical results are illustrated throughout this paper using a direct optimization method.