  This paper focuses on mixing strategies to enhance the growth rate in an
algal raceway system. A mixing device, such as a paddle wheel, is considered to
control the rearrangement of the depth of the algae cultures hence the light
perceived at each lap. The dynamics of the photosystems after a rearrangement
is accounted for by the Han model. Our approach consists in considering
permanent regimes where the strategy is parametrized by a permutation matrix
which modifies the order of the layers at the beginning of each lap. It is
proven that the dynamics of the photosystems is then periodic, with a period
corresponding to one lap of the raceway whatever the order of the considered
permutation matrix is. An objective function related to the average growth rate
over one lap is then introduced. Since N ! permutations (N being the number of
considered layers) need to be tested in the general case, it can be numerically
solved only for a limited number of layers. Consequently, we propose a second
optimization problem associated with a suboptimal solution of the initial
problem, which can be determined explicitly. A sufficient condition to
characterize cases where the two problems have the same solution is given. Some
numerical experiments are performed to assess the benefit of optimal strategies
in various settings.
