We address the question of optimization of the microalgal biomass long term
productivity in the framework of 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 main 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
some form of 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. Finally, a simple sub-optimal
bang-bang strategy is proposed that numerically achieves productivity levels
that almost match those of the optimal strategy