  In this article, we focus on a periodic resource allocation problem applied
to a dynamical system which comes from a biological system. More precisely, we
consider a system with $N$ resources and $N$ activities, each activity use the
allocated resource to evolve up to a given time $T > 0$ where a control
(represented by a given permutation) will be applied on the system to
reallocate the resources. The goal is to find the optimal control strategies
which optimize the cost or the benefit of the system. This problem can be
illustrated by an industrial biological application, namely, the optimization
of a mixing strategy to enhance the growth rate in a microalgal 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. We prove that if the dynamics of the system is periodic, then the
period corresponds to one reallocation whatever the order of the involved
permutation matrix is. A nonlinear optimization problem for one reallocation
process is then introduced. Since $N!$ permutations need to be tested in the
general case, it can be numerically solved only for a limited number of $N$. To
overcome this difficulty, we introduce a second optimization problem which
provides a suboptimal solution of the initial problem, but whose solution 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.
