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arXiv · 2201.01723

Optimal Control of Moving Sets

Alberto Bressan·Maria Teresa Chiri·Najmeh Salehi
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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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Alberto Bressan, Maria Teresa Chiri, Najmeh Salehi. 2022-01-05. Optimal Control of Moving Sets. https://doi.org/10.13140/rg.2.2.13589.65766

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