Optimal control of cell mass and maturity in a model of follicular ovulation

In this paper, we study optimal control problems associated with a scalar hyperbolic conservation law modeling the development of ovarian follicles. Changes in the age and maturity of follicular cells are described by a 2D conservation law, where the control terms act on the velocities. The control problem consists in optimizing the follicular cell resources so that the follicular maturity reaches a maximal value in fixed time. Formulating the optimal control problem within a hybrid framework, we prove necessary optimality conditions in the form of Hybrid Maximum Principle. Then we derive the optimal strategy and show that there exists at least one optimal bang-bang control with one single switching time.