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

Abstract

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

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