Controlled Traveling Profiles for Models of Invasive Biological Species

TL;DR

This paper investigates optimal control strategies to shape traveling profiles of invasive species spread modeled by reaction-diffusion equations, establishing existence, conditions for optimality, and limitations in controllability.

Contribution

It proves the existence of optimal controls for various nonlinear models and derives necessary conditions, extending understanding of controlling invasive species spread.

Findings

Existence of optimal controls for different nonlinear models.

Necessary conditions for optimality of controls.

Identification of cases where wave speed cannot be controlled with finite cost.

Abstract

We consider a family of controlled reaction-diffusion equations, describing the spatial spreading of an invasive biological species. For a given propagation speed $c\in{I\!\!R}$, we seek a control with minimum cost, which achieves a traveling profile with speed $c$. For various nonlinear models, the existence of a (possibly measure valued) optimal control is proved, together with necessary conditions for optimality. In the last section we study a case where the wave speed cannot be modified by any control with finite cost. The present analysis is motivated by the recent results in arXiv:2201.01723 and arXiv:2108.09321, showing how a control problem for a reaction-diffusion equation can be approximated by a simpler problem of optimal control of a moving set.