We consider a controlled reaction–diffusion equation, motivated by a pest eradication problem. Our goal is to derive a simpler model, describing the controlled evolution of a contaminated set. In this direction, the first part of the paper studies the optimal control of 1-dimensional traveling wave profiles. Using Stokes’ formula, explicit solutions are obtained, which in some cases require measure-valued optimal controls. In the last section, we introduce a family of optimization problems for a moving set. We show how these can be derived from the original parabolic problems, by taking a sharp interface limit.

中文翻译：

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关于传播前沿的最优控制

我们考虑一个受控的反应扩散方程，由害虫根除问题驱动。我们的目标是推导出一个更简单的模型，描述受污染集的受控演化。在这个方向上，论文的第一部分研究了一维行波剖面的最优控制。使用斯托克斯公式，获得显式解决方案，在某些情况下需要测量值最优控制。在最后一节中，我们介绍了移动集的一系列优化问题。我们通过采用尖锐的界面限制来展示如何从原始抛物线问题中推导出这些问题。