Results and open questions on the boundary control of moving sets

TL;DR

This paper surveys recent advances and open challenges in boundary control problems for moving sets, focusing on minimizing invasive species spread through boundary speed control with constraints.

Contribution

It provides a comprehensive overview of recent results and open questions in boundary control of moving sets, especially in ecological invasion scenarios.

Findings

Existence of eradication strategies in minimum time.

Optimal boundary control for minimizing contaminated area.

Impact of geographical constraints on control strategies.

Abstract

These notes provide a survey of recent results and open problems on the boundary control of moving sets. Motivated by the control of an invasive biological species, we consider a class of optimization problems for a moving set $t\mapsto \Omega(t)$, where 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, assigned along the boundary $\partial \Omega(t)$. We also consider problems with geographical constraints, where the invasive population is restricted within an island. Existence and structure of eradication strategies, which entirely remove the invasive population in minimum time, is also discussed.