Optimal Solutions for a Class of Set-Valued Evolution Problems

The paper is concerned with a class of optimization problems for moving sets $t\mapsto \mathrm{\Omega }(t)\subset {\mathbb{R}}^2$, motivated by the control of invasive biological populations. Assuming that the initial contaminated set ${\mathrm{\Omega }}_0$ is convex, we prove that a strategy is optimal if an only if at each given time $t\in [0,T]$ the control is active along the portion of the boundary $\mathrm{\partial }\mathrm{\Omega }(t)$ where the curvature is maximal. In particular, this implies that $\mathrm{\Omega }(t)$ is convex for all $t\ge 0$. The proof relies on the analysis of a one-step constrained optimization problem, obtained by a time discretization.