Controlling the degradation of a protection effect

Overview & Synopsis

We study the optimal placement of limited intervention zones that reduce the degradation of a protection effect in a crop field. To this end, we formulate an optimal control problem for a protection model described by a parabolic equation. The control represents the intervention zones, and the objective is to maximize the protection over the field throughout the season. We prove the existence of an optimal control using a relaxation method, and show that it is necessarily a bang-bang control. Then, in some specific cases, symmetrization methods allow us to derive geometric properties of the optimal intervention based on the geometry of the field. Numerical simulations are included to support and illustrate the theoretical results.