论文标题
移动集的最佳控制
Optimal Control of Moving Sets
论文作者
论文摘要
通过控制侵入性生物种群的控制，我们考虑了一类用于移动集的优化问题$ t \ mapstoω（t）\ subset \ mathbb {r}^2 $。鉴于初始集合$ω_0$，目标是随着时间的推移将受污染的集合$ω（t）$的面积最小化，再加上与控制工作相关的成本。在这里，控制功能是沿边界$ \ partialω（t）$的内向正常速度。我们在具有有限周围的一类集合中证明了最佳解决方案的存在。然后以最大原理的形式得出最佳的必要条件。其他最佳条件表明，集合$ω（t）$不能具有某些类型的外向或内向角。最后，提出了一些明确的解决方案。
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.
