Study of a controlled Kolmogorov periodic equation connected to the optimization of periodic bioprocess

This paper is devoted to the study of a one-dimensional optimal control problem of Lagrange type under periodic constraints. The system is governed by a Kolmogorov equation including a control term. We show that the set of initial conditions for which there exists a control such that the associated trajectory is periodic is an interval. Thanks to Pontryagin Maximum Principle, we characterize optimal controls whenever the dynamics has two autonomous phases. Finally, we provide numerical simulation of the problem for a system describing a periodic photobioreactor.