Optimally Controlled Moving Sets with Geographical Constraints

TL;DR

This paper studies geometric control problems for eradicating an invasive population within geographical constraints, focusing on existence, optimality, and explicit construction of control strategies to minimize contamination.

Contribution

It introduces a framework for controlling expanding sets with geographical barriers, providing existence, optimality conditions, and explicit solutions for eradication strategies.

Findings

Existence of strategies for finite-time eradication.

Necessary and sufficient conditions for optimal control.

Explicit construction of optimal set evolutions.

Abstract

The paper is concerned with a family of geometric evolution problems, modeling the spatial control of an invasive population within a region $V\subset \R^2$ bounded by geographical barriers. If no control is applied, the contaminated set $\Omega(t)\subset V$ 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 $\Omega(0)=\Omega_0\subseteq V$, three main problems are studied: (1) Existence of an admissible strategy $t\mapsto\Omega(t)$ which eradicates the contamination in finite time, so that $\Omega(T)=\emptyset$ 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 $t\mapsto \Omega(t)$ are explicitly constructed in a number of cases. \end{abstract}