Optimally Controlled Moving Sets with Geographical Constraints

Abstract

The paper is concerned with a family of geometric evolution problems, modeling the spatial control of an invasive population within a region $V\subset {\mathbb{R}}^2$ bounded by geographical barriers. If no control is applied, the contaminated set $\hat{I}\text{c◯}(t)\subset V$ expands with unit speed in all directions. By implementing a control, a region of area $M$ can be cleared up per unit time. Given an initial set $\hat{I}\text{c◯}(0)=\hat{I}{\text{c◯}}_0\subseteq V$, three main problems are studied: (1) Existence of an admissible strategy $t\mapsto \hat{I}\text{c◯}(t)$ which eradicates the contamination in finite time, so that $\hat{I}\text{c◯}(T)=\varnothing$ for some $T>0$. (2) Optimal strategies that achieve eradication in minimum time. (3) Strategies that minimize the average area of the contaminated set on a given time interval $[0,T]$. For these optimization problems, a sufficient condition for optimality is proved, together with several necessary conditions. Based on these conditions, optimal set-valued motions $t\mapsto \hat{I}\text{c◯}(t)$ are explicitly constructed in a number of cases. \end{abstract}