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

4 TONUS - TOkamaks and NUmerical Simulations
IRMA - Institut de Recherche Mathématique Avancée, Inria Nancy - Grand Est
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 $\Omega$ 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.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03230860
Contributor : Yannick Privat Connect in order to contact the contributor
Submitted on : Thursday, May 20, 2021 - 11:41:08 AM
Last modification on : Saturday, December 4, 2021 - 4:09:31 AM

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• HAL Id : hal-03230860, version 1

### Citation

Idriss Mazari, Grégoire Nadin, Yannick Privat. Optimisation of the total population size for logistic diffusive equations: bang-bang property and fragmentation rate. 2021. ⟨hal-03230860⟩

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