  We address the question of optimization of the biomass long term productivity
in the framework of microalgal biomass production in photobioreactors under the
influence of day/night cycles. For that, we propose a simple bioreactor model
accounting for light attenuation in the reactor due to biomass density and
obtain the control law that optimizes productivity over a single day through
the application of Pontryagin's maximum principle, with the dilution rate being
the control. An important constraint on the obtained solution is that the
biomass in the reactor should be at the same level at the beginning and at the
end of the day so that the same control can be applied everyday and optimizes
the long term productivity. Several scenarios are possible depending on the
microalgae's strain parameters and the maximal admissible value of the dilution
rate: bang-bang or bang-singular-bang control or, if the growth rate of the
algae is very strong in the presence of light, constant maximal dilution. A
bifurcation diagram is presented to illustrate for which values of the
parameters these different behaviors occur.

Optimization and Control