We consider a controlled reaction-diffusion equation, modeling the spreading of an invasive population, and derive a simpler model describing the controlled evolution of a contaminated set. We first analyze the optimal control of 1-dimensional traveling wave profiles. Using Stokes’ formula, explicit solutions are obtained, which in some cases require measure-valued optimal controls. Then we introduce a family of optimization problems for a moving set and show how these can be derived from the original parabolic problems, by taking a sharp interface limit. Assuming that the initial contaminated set is convex, we prove that an eradication strategy is optimal if and only if at each given time the control is active along the portion of the boundary where the curvature is maximal.

This is a joint work with Stefano Bianchini, Alberto Bressan, and Najmeh Salehi.