Optimal control of moving sets

Motivated by the control of invasive biological populations, we consider a class of optimization problems for moving sets t↦Ω(t)⊂IR². Given an initial set Ω₀, 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 ∂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.