Optimal Control of Moving Sets

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.