ANALYSIS OF A TOY MODEL FOR OPTIMAL CROP PROTECTION

In this paper, we investigate an optimal control problem involving a toy model for the protection of a crop field. Precisely, we consider the protection of a crop field, and we want to place intervention zones represented by a control in order to maximize the protection of the field during a given period. We prove that there exists a unique control which maximizes the protection, and, moreover, it must be a bang-bang control. Furthermore, with additional assumptions on the crop field geometry, some results on the shape of the optimal intervention are proved using comparison results for elliptic equations via Schwarz and Steiner symmetrizations. Finally, some numerical simulations are performed in order to illustrate those results.