Published Paper
Inserted: 17 dec 2021
Last Updated: 4 jan 2024
Journal: Journal of Differential Equations
Year: 2023
Abstract:
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.
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