arXiv subjects

Idriss Mazari

Publications and source records attributed to Idriss Mazari.

18 recordsLinked to original sources

arXiv2026

Uniqueness and non-uniquess for the mean field control of fisheries

We study a Mean Field Control system arising in the management of fisheries with a special emphasis on non-uniqueness issues. Namely, we focus on a situation where a group of players coordinate in order to harvest a fishery in the most efficient way possible. A major challenge in such modelling is the coupling between the dynamics of fish population, which we model through a reaction-diffusion equation, and that of the players, which is seen through the lens of Mean Field Control. The resulting evolution system consists of four coupled equations. A central issue, both in the analysis and from the modelling perspective, is the uniqueness of solutions of this system. By focusing on the ergodic (or static) counterpart of the evolution equation, we show that one should in general expect the emergence of multiple solutions. Our approach relies on the theory of bifurcation, and the bifurcation parameter we take is the (biologically relevant) total amount of food available to the population. We also give refined uniqueness criteria that allow to bypass several limitations of previous works on this type of system [39]. This fits within two growing research lines: one on the optimal harvesting of fisheries [40, 39], one on questions of non-uniqueness in Mean Field Games and Mean Field Control [3, 27, 36].

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arXiv2024

Generalized principal eigenvalues of space-time periodic, weakly coupled, cooperative, parabolic systems

This paper is concerned with generalizations of the notion of principal eigenvalue in the context of space-time periodic cooperative systems. When the spatial domain is the whole space, the Krein-Rutman theorem cannot be applied and this leads to more sophisticated constructions and to the notion of generalized principal eigenvalues. These are not unique in general and we focus on a one-parameter family corresponding to principal eigenfunctions that are space-time periodic multiplicative perturbations of exponentials of the space variable. Besides existence and uniqueness properties of such principal eigenpairs, we also prove various dependence and optimization results illustrating how known results in the scalar setting can, or cannot, be extended to the vector setting. We especially prove an optimization property on minimizers and maximizers among mutation operators valued in the set of bistochastic matrices that is, to the best of our knowledge, new.

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arXiv2024

Analysis of a combined Filtered/phase-field approach to topology optimization in elasticit

We advance a combined filtered/phase-field approach to topology optimization in the setting of linearized elasticity. Existence of minimizers is proved and rigorous parameter asymptotics are discussed by means of variational convergence techniques. Moreover, we investigate an abstract space discretization in the spirit of conformal finite elements. Eventually, stationarity is equivalently reformulated in terms of a Lagrangian.

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arXiv2023

The bang-bang property in some parabolic bilinear optimal control problems \emph{via} two-scale asymptotic expansions

We investigate the bang-bang property for fairly general classes of $L^\infty-L^1$ constrained bilinear optimal control problems in two cases: that of the one-dimensional torus, in which case we consider parabolic equations, and that of general $d$ dimensional domains for time-discrete parabolic models. Such a study is motivated by several applications in applied mathematics, most importantly in the study of reaction-diffusion models. The main equation in the one-dimensional case writes $\partial_t u_m-Δu_m=mu_m+f(t,x,u_m)$, where $m=m(x)$ is the control, which must satisfy some $L^\infty$ bounds ($0\leq m\leq 1$ a.e.) and an $L^1$ constraint ($\int m=m_0$ is fixed), and where $f$ is a non-linearity that must only satisfy that any solution of this equation is positive at any given time. The time-discrete models are simply time-discretisations of such equations. The functionals we seek to optimise are rather general; in the case of the torus, they write $\mathcal J(m)=\iint_{(0,T)\times \mathbb T} j_1(t,x,u_m)+\int_{\mathbb T} j_2(x,u_m(T,\cdot))$. Roughly speaking we prove in this article that, if $j_1$ and $j_2$ are increasing, then any maximiser $m^*$ of $\mathcal J$ is bang-bang in the sense that it writes $m^*=1_E$ for some subset $E$ of the torus. It should be noted that such a result rewrites as an existence property for a shape optimisation problem. Our proofs rely on second order optimality conditions, combined with a fine study of two-scale asymptotic expansions. In the conclusion of this article, we offer several possible generalisations of our results to more involved situations (for instance for controls of the form $mφ(u_m)$), and we discuss the limits of our methods by explaining which difficulties may arise in other contexts.

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arXiv2022

Spatial ecology, optimal control and game theoretical fishing problems

Of paramount importance in both ecological systems and economic policies are the problems of harvesting of natural resources. A paradigmatic situation where this question is raised is that of fishing strategies. Indeed, overfishing is a well-known problem in the management of live-stocks, as being too greedy may lead to an overall dramatic depletion of the population we are harvesting. A closely related topic is that of Nash equilibria in the context of fishing policies. Namely, two players being in competition for the same pool of resources, is it possible for them to find an equilibrium situation? The goal of this paper is to provide a detailed analysis of these two queries (\emph{i.e} optimal fishing strategies for single-player models and study of Nash equilibria for multiple players games) by using a basic yet instructive mathematical model, the logistic-diffusive equation. In this framework, the underlying model simply reads $-μΔθ=θ(K(x)-α(x)-θ)$ where $K$ accounts for natural resources, $θ$ for the density of the population that is being harvested and $α=α(x)$ encodes either the single player fishing strategy or, when dealing with Nash equilibria, a combination of the fishing strategies of both players. This article consists of two main parts. The first one gives a very fine characterisation of the optimisers for the single-player game. In the case where two players are involved, we aim at finding a Nash equilibrium. We prove the existence of Nash equilibria in several different regimes \textcolor{black}{and investigate several related qualitative queries}.Our study is completed by a variety of numerical simulations that illustrate our results and allow us to formulate open questions and conjectures.

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arXiv2021

Constrained control of gene-flow models

In ecology and population dynamics, gene-flow refers to the transfer of a trait from one population to another. This phenomenon appears in studying the evolution of social features, such as languages. From the mathematical point of view, gene-flow is modelled using bistable reaction-diffusion equations. The unknown is the proportion $p$ of the population possessing a certain trait, within a population $N$. Gene-flow is taken into account by assuming that the population density $N$ depends either on $p$ or on the location $x$. Recent applications stemming from mosquito-borne disease control problems or from the study of bilingualism have called for the investigation of the controllability properties of these models. At the mathematical level, this corresponds to boundary control problems and, since we are working with proportions, the control $u$ has to satisfy the constraints $0\leq u \leq 1$. In this article, we provide a thorough analysis of the influence of the gene-flow effect on boundary controllability properties. We prove that, when the population density $N$ only depends on the trait proportion $p$, the geometry of the domain is the only criterion that has to be considered. We then tackle the case of population densities $N$ varying in $x$. We first prove that, when $N$ varies slowly in $x$ and when the domain is narrow enough, controllability always holds. We then consider the case of sharp fluctuations in $N$: we give examples that prove that controllability may fail. Conversely, we give examples of $N$ such that controllability will always be guaranteed. All negative controllability results are proved by showing the existence of non-trivial stationary states, which act as barriers. The existence of such solutions and the methods of proof are of independent interest. Our article is completed by several numerical experiments that confirm our analysis.

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arXiv2021

Shape optimization of a weighted two-phase Dirichlet eigenvalue

Let $m$ be a bounded function and $α$ a nonnegative parameter. This article is concerned with the first eigenvalue $λ\_α(m)$ of the drifted Laplacian type operator $\mathcal L\_m$ given by $\mathcal L\_m(u)= -\operatorname{div} \left((1+αm)\nabla u\right)-mu$ on a smooth bounded domain, with Dirichlet boundary conditions. Assuming uniform pointwise and integral bounds on $m$, we investigate the issue of minimizing $λ\_α(m)$ with respect to $m$. Such a problem is related to the so-called "two phase extremal eigenvalue problem" and arises naturally, for instance in population dynamics where it is related to the survival ability of a species in a domain. We prove that unless the domain is a ball, this problem has no "regular" solution. We then provide a careful analysis in the case of a ball by: (1) characterizing the solution among all radially symmetric resources distributions, with the help of a new method involving a homogenized version of the problem; (2) proving in a more general setting, a stability result for the centered distribution of resources with the help of a monotonicity principle for second order shape derivatives which significantly simplifies the analysis.

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arXiv2021

Qualitative analysis of optimisation problems with respect to non-constant Robin coefficients

Following recent interest in the qualitative analysis of some optimal control and shape optimisation problems, we provide in this article a detailed study of the optimisation of Robin boundary conditions in PDE constrained calculus of variations. Our main model consists of an elliptic PDE of the form $-Δu_β=f(x,u_β)$ endowed with the Robin boundary conditions $\partial_νu_β+β(x)u_β=0$. The optimisation variable is the function $β$, which is assumed to take values between 0 and 1 and to have a fixed integral. Two types of criteria are under consideration: the first one is non-energetic criteria. In other words, we aim at optimising functionals of the form $\mathcal J(β)=\int_{Ω\text{ or }\partial Ω}j(u_β)$. We prove that, depending on the monotonicity of the function $j$, the optimisers may be of \emph{bang-bang} type (in other words, the optimisers write $1_Γ$ for some measurable subset $Γ$ of $\partial \ Omega $) or, on the contrary, that they may only take values strictly between 0 and 1. This has consequence for a related shape optimisation problem, in which one tries to find where on the boundary Neumann ($\partial_νu=0$ ) and constant Robin conditions ($\partial_νu+u=0$) should be placed in order to optimise criteria. The proofs for this first case rely on new fine oscillatory techniques, used in combination with optimality conditions. We then investigate the case of compliance-type functionals. For such energetic functionals, we give an in-depth analysis and even some explicit characterisation of optimal $β^*$.

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arXiv2021

Optimisation of the total population size with respect to the initial condition for semilinear parabolic equations: Two-scale expansions and symmetrisations

In this article, we propose in-depth analysis and characterisation of the optimisers of the following optimisation problem: how to choose the initial condition $u_0$ in order to maximise the spatial integral at a given time of the solution of the semilinear equation $u_t-Δu=f(u)$, under $L^\infty$ and $L^1$ constraints on $u_0$? Our contribution in the present paper is to give a characterisation of the behaviour of the optimiser $\overline{u}_0$ when it does not saturate the $L^\infty$ constraints, which is a key step in implementing efficient numerical algorithms. We give such a characterisation under mild regularity assumptions by proving that in that case $\overline{u}_0$ can only take values in the "zone of concavity" of $f$. This is done using two-scale asymptotic expansions. We then show how well-known isoperimetric inequalities yield a full characterisation of maximisers when $f$ is convex. Finally, we provide several numerical simulations in one and two dimensions that illustrate and exemplify the fact that such characterisations significantly improves the computational time. All our theoretical results are in the one-dimensional case and we offer several comments about possible generalisations to other contexts, or obstructions that may prohibit doing so.

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arXiv2021

Spectral optimization of inhomogeneous plates

This article is devoted to the study of spectral optimisation for inhomogeneous plates. In particular, we optimise the first eigenvalue of a vibrating plate with respect to its thickness and/or density. Our result is threefold. First, we prove existence of an optimal thickness, using fine tools hinging on topological properties of rearrangement classes. Second, in the case of a circular plate, we provide a characterisation of this optimal thickness by means of Talenti inequalities. Finally, we prove a stability result when assuming that the thickness and the density of the plate are linearly related. This proof relies on H-convergence tools applied to biharmonic operators.

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arXiv2021

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

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 $Ø$ is an orthotope and if $m_μ$ is an optimal control, then $\Vert m_μ\Vert_{BV}\gtrsim \sqrtμ$. The proof of this results relies on a very fine energy argument.

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arXiv2021

Quantitative estimates for parabolic optimal control problems under $L^\infty$ and $L^1$ constraints in the ball:Quantifying parabolic isoperimetric inequalities

In this article, we present two different approaches for obtaining quantitative inequalities in the context of parabolic optimal control problems. Our model consists of a linearly controlled heat equation with Dirichlet boundary condition $(u_f)_t-Δu_f=f$, $f$ being the control. We seek to maximise the functional $\mathcal J_T(f):=\frac12\int_{(0;T)\times Ω} u_f^2$ or, for some $ε>0$, $\mathcal J_T^ε(f):=\frac12\int_{(0;T)\times Ω} u_f^2+ε\int_Ωu_f^2(T,\cdot)$ and to obtain quantitative estimates for these maximisation problems. We offer two approaches in the case where the domain $Ω$ is a ball. In that case, if $f$ satisfies $L^1$ and $L^\infty$ constraints and does not depend on time, we propose a shape derivative approach that shows that, for any competitor $f=f(x)$ satisfying the same constraints, we have $\mathcal J_T(f^*)-\mathcal J_T(f)\gtrsim \Vert f-f^*\Vert_{L^1(Ω)}^2$, $f^*$ being the maximiser. Through our proof of this time-independent case, we also show how to obtain coercivity norms for shape hessians in such parabolic optimisation problems. We also consider the case where $f=f(t,x)$ satisfies a global $L^\infty$ constraint and, for every $t\in (0;T)$, an $L^1$ constraint. In this case, assuming $ε>0$, we prove an estimate of the form $\mathcal J_T^ε(f^*)-\mathcal J_T^ε(f)\gtrsim\int_0^T a_ε(t) \Vert f(t,\cdot)-f^*(t,\cdot)\Vert_{L^1(Ω)}^2$ where $a_ε(t)>0$ for any $t\in (0;T)$. The proof of this result relies on a uniform bathtub principle.

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arXiv2020

A fragmentation phenomenon for a non-energetic optimal control problem: optimisation of the total population size in logistic diffusive models

Following some recent works, we investigate the problem of optimising the total population size for logistic diffusive models with respect to resources distributions. Using the spatially heterogeneous Fisher-KPP equation, we obtain a surprising fragmentation phenomenon: depending on the scale of diffusivity (i.e the dispersal rate), it is better to either concentrate or fragment resources. Our main result is that, the smaller the dispersal rate of the species in the domain, the more optimal resources distributions tend to oscillate. This is in sharp contrast with other criteria in population dynamics, such as the classical problem of optimising the survival ability of a species, where concentrating resources is always favourable, regardless of the diffusivity. Our study is completed by numerous numerical simulations that confirm our results.

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arXiv2020

Quantitative stability for eigenvalues of Schrödinger operator, Quantitative bathtub principle \& Application to the turnpike property for a bilinear optimal control problem

This work is concerned with two optimisation problems that we tackle from a qualitative perspective. The first one deals with quantitative inequalities for spectral optimisation problems for Schrödinger operators in general domains, the second one deals with the turnpike property for optimal bilinear control problems. In the first part of this article, we prove, under mild technical assumptions, quantitative inequalities for the optimisation of the first eigenvalue of $-Δ-V$ with Dirichlet boundary conditions with respect to the potential $V$, under $L^\infty$ and $L^1$ constraints. This is done using a new method of proof which relies on in a crucial way on a quantitative bathtub principle. We believe our approach susceptible of being generalised to other steady elliptic optimisation problems. In the second part of this paper, we use this inequality to tackle a turnpike problem. Namely, considering a bilinear control system of the form $u_t-Δu=\mathcal V u$, $\mathcal V=\mathcal V(t,x)$ being the control, can we give qualitative information, under $L^\infty$ and $L^1$ constraints on $\mathcal V$, on the solutions of the optimisation problem $\sup \int_Ωu(T,x)dx$? We prove that the quantitative inequality for eigenvalues implies an integral turnpike property: defining $\mathcal I^*$ as the set of optimal potentials for the eigenvalue optimisation problem and $\mathcal V_T^*$ as a solution of the bilinear optimal control problem, the quantity $\int_0^T \operatorname{dist}_{L^1}(\mathcal V_T^*(t,\cdot)\,, \mathcal I^*)^2$ is bounded uniformly in $T$.

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arXiv2020

Shape optimization of a Dirichlet type energy for semilinear elliptic partial differential equations

Minimizing the so-called "Dirichlet energy" with respect to the domain under a volume constraint is a standard problem in shape optimization which is now well understood. This article is devoted to a prototypal non-linear version of the problem, where one aims at minimizing a Dirichlet-type energy involving the solution to a semilinear elliptic PDE with respect to the domain, under a volume constraint. One of the main differences with the standard version of this problem rests upon the fact that the criterion to minimize does not write as the minimum of an energy, and thus most of the usual tools to analyze this problem cannot be used. By using a relaxed version of this problem, we first prove the existence of optimal shapes under several assumptions on the problem parameters. We then analyze the stability of the ball, expected to be a good candidate for solving the shape optimization problem, when the coefficients of the involved PDE are radially symmetric.

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arXiv2020

Quantitative inequality for the eigenvalue of a Schrödinger operator in the ball

The aim of this article is to prove a quantitative inequality for the first eigenvalue of a Schrödinger operator in the ball. More precisely, we optimize the first eigenvalue $λ(V)$ of the operator $\mathcal L_v:=-Δ-V$ with Dirichlet boundary conditions with respect to the potential $V$, under $L^1$ and $L^\infty$ constraints on $V$. The solution has been known to be the characteristic function of a centered ball, but this article aims at proving a sharp growth rate of the following form: if $V^*$ is a minimizer, then $λ(V)-λ(V^*)\geq C ||V-V^*||_{L^1(Ω)}^2$ for some $C>0$. The proof relies on two notions of derivatives for shape optimization: parametric derivatives and shape derivatives. We use parametric derivatives to handle radial competitors, and shape derivatives to deal with normal deformation of the ball. A dichotomy is then established to extend the result to all other potentials. We develop a new method to handle radial distributions and a comparison principle to handle second order shape derivatives at the ball. Finally, we add some remarks regarding the coercivity norm of the second order shape derivative in this context.

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arXiv2019

Optimal location of resources maximizing the total population size in logistic models

In this article, we consider a species whose population density solves the steady diffusive logistic equation in a heterogeneous environment modeled with the help of a spatially non constant coefficient standing for a resources distribution. We address the issue of maximizing the total population size with respect to the resources distribution, considering some uniform pointwise bounds as well as prescribing the total amount of resources. By assuming the diffusion rate of the species large enough, we prove that any optimal configuration is bang-bang (in other words an extreme point of the admissible set) meaning that this problem can be recast as a shape optimization problem, the unknown domain standing for the resources location. In the one-dimensional case, this problem is deeply analyzed, and for large diffusion rates, all optimal configurations are exhibited. This study is completed by several numerical simulations in the one dimensional case.

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arXiv2018

Optimal control of resources for species survival

Consider a species whose population density solves the steady diffusive logistic equation in a heterogeneous environment modeled with the help of a spatially non constant coefficient standing for a resources distribution in a given box. We look at maximizing the total population size with respect to resources distribution, under some biologically relevant constraints. Assuming that the diffusion rate of the species is large enough, we prove that any optimal configuration is the characteristic function of a domain standing for the resources location. Moreover, we highlight that optimal configurations look {\it concentrated} whenever the diffusion rate is large enough. In the one-dimensional case, this problem is deeply analyzed, and for large diffusion rates, all optimal configurations are exhibited.

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