Optimal Solutions for a Class of Set-Valued Evolution Problems

Abstract

The paper is concerned with a class of optimization problems for moving sets $t\mapsto \Omega (t)\subset {\mathbb{R}}^2$, motivated by the control of invasive biological populations. Assuming that the initial contaminated set ${\Omega }_0$ 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 $\partial \Omega (t)$ where the curvature is maximal. In particular, this implies that $\Omega (t)$ 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.

41 pages, 18 figures