Optimal solutions for a class of set-valued evolution problems

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