Optimisation of the total population size for logistic diffusive equations: bang-bang property and fragmentation rate

TL;DR

This paper analyzes how to optimize total population size in a logistic-diffusive model, proving the bang-bang property and exploring the fragmentation of maximizers using novel spectral methods.

Contribution

It establishes the bang-bang property for optimizers in general settings and introduces a spectral method applicable to various bilinear control problems.

Findings

Proves bang-bang property under general conditions.

Shows a blow-up rate of the BV-norm of maximizers as diffusivity decreases.

Provides a new spectral approach for control optimization problems.

Abstract

In this article, we give an in-depth analysis of the problem of optimising the total population size for a standard logistic-diffusive model. This optimisation problem stems from the study of spatial ecology and amounts to the following question: assuming a species evolves in a domain, what is the best way to spread resources in order to ensure a maximal population size at equilibrium? {In recent years, many authors contributed to this topic.} We settle here the proof of two fundamental properties of optimisers: the bang-bang one which had so far only been proved under several strong assumptions, and the other one is the fragmentation of maximisers. Here, we prove the bang-bang property in all generality using a new spectral method. The technique introduced to demonstrate the bang-bang character of optimizers can be adapted and generalized to many optimization problems with other classes of bilinear optimal control problems where the state equation is semilinear and elliptic. We comment on it in a conclusion section.Regarding the geometry of maximisers, we exhibit a blow-up rate for the $BV$-norm of maximisers as the diffusivity gets smaller: if $\O$ is an orthotope and if $m_\mu$ is an optimal control, then $\Vert m_\mu\Vert_{BV}\gtrsim \sqrt{\mu}$. The proof of this results relies on a very fine energy argument.