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

Mazari, Idriss; Nadin, G.; Privat, Y. (2022), Optimisation of the total population size for logistic diffusive equations: bang-bang property and fragmentation rate., Communications in Partial Differential Equations, p. 1-32. 10.1080/03605302.2021.2007533

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Type
Article accepté pour publication ou publié
Date
2022
Journal name
Communications in Partial Differential Equations
Publisher
Taylor & Francis
Pages
1-32
Publication identifier
10.1080/03605302.2021.2007533
Metadata
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Author(s)
Mazari, Idriss
Nadin, G.
Privat, Y.
Abstract (EN)
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 Ω is an orthotope and if mμ is an optimal control, then ∥mμ∥BV≳μ−−√. The proof of this results relies on a very fine energy argument.
Subjects / Keywords
Population Size.; Total Population; Total Population Size; Fragmentation Rate; diffusive logistic equation; shape optimization; optimal control; bilinear optimal control; calculus of variations

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