  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 $t\geq 0$. The proof relies on the analysis of a
one-step constrained optimization problem, obtained by a time discretization.
