  Motivated by the control of invasive biological populations, we consider a
class of optimization problems for moving sets $t\mapsto
\Omega(t)\subset\mathbb{R}^2$. Given an initial set $\Omega_0$, the goal is to
minimize the area of the contaminated set $\Omega(t)$ over time, plus a cost
related to the control effort. Here the control function is the inward normal
speed along the boundary $\partial \Omega(t)$. We prove the existence of
optimal solutions, within a class of sets with finite perimeter. Necessary
conditions for optimality are then derived, in the form of a Pontryagin maximum
principle. Additional optimality conditions show that the sets $\Omega(t)$
cannot have certain types of outward or inward corners. Finally, some explicit
solutions are presented.
