  The potential of industrial applications for microalgae has motivated their
recent fast development. Their growth dynamics depends on different factors
that must be optimized. Since they get their energy from photosynthesis, light
is a key factor that strongly influences their productivity. Light is absorbed
and scattered in the liquid medium, and irradiance exponentially decreases
towards the darkest part of the photobioreactor at a rate nonlinearly depending
on the biomass concentration. Maximizing productivity is then a tricky problem,
especially when the growth rate is inhibited by an excess of light.
Productivity optimization turns out to be highly dependent on how light is
distributed along the reactor, and is therefore related to the extinction rate
and the background turbidity. The concept of optical depth productivity is
introduced for systems where background turbidity must be accounted for and a
global optimum maximizing productivity is proposed, extending the concept of
the compensation condition. This optimal condition consists in compensating the
algal growth rate at the bottom of the reactor by the respiration. This
condition can drive the optimization of the surface biomass productivity
depending on the minimum reachable depth. We develop a nonlinear controller and
prove the global asymptotic stability of the biomass concentration towards the
desired optimal value.
