Optimal Control of Moving Sets

Motivated by the control of invasive biological populations, we consider a class of optimization problems for moving sets $t\mapsto Ω(t)\subset\mathbb{R}^2$. Given an initial set $Ω_0$, the goal is to minimize the area of the contaminated set $Ω(t)$ over time, plus a cost related to the control effort. Here the control function is the inward normal speed along the boundary $\partial Ω(t)$. 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 $Ω(t)$ cannot have certain types of outward or inward corners. Finally, some explicit solutions are presented.

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