arXiv subjects

Grégoire Nadin

Publications and source records attributed to Grégoire Nadin.

At least 19 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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arXiv2026

Gaussian estimates for general parabolic operators in dimension 1

We derive in this paper Gaussian estimates for a general parabolic equation $u_{t}-\big(a(x)u_{x}\big)_x= r(x)u$ over $\mathbb{R}$. Here $a$ and $r$ are only assumed to be bounded, measurable and $\mathrm{essinf}_{\mathbb{R}} a>0$. We first consider a canonical equation $ν(x) \partial_{t}p - \partial_{x }\big( ν(x)a(x)\partial_{x}p\big)+W\partial_{x}p=0$, with $W\in \mathbb{R}$, $ν$ bounded and $\mathrm{essinf}_{\mathbb{R}} ν>0$, for which we derive Gaussian estimates for the fundamental solution: $$\forall t>0, x,y\in \mathbb{R}, \quad \displaystyle\frac{1}{Ct^{1/2}}e^{-C|T(x)-T(y)-Wt|^{2}/t} \leq P(t,x,y)\leq \frac{C}{t^{1/2}}e^{-|T(x)-T(y)-Wt|^{2}/Ct}.$$ Here, the function $T$ is a corrector, for which we are able to derive appropriate properties using one-dimensional arguments. We then show that any solution $u$ of the original equation could be divided by some generalized principal eigenfunction $ϕ_γ$ so that $p:=u/ϕ_γ$ satisfies a canonical equation. As a byproduct of our proof, we derive Nash type estimates, that is, Holder continuity in $x$, for the solutions of the canonical equation.

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arXiv2026

Optimization of the L1 norm of the solution of a Fisher-KPP equations in the small diffusivity regime

We investigate in the present paper the maximization problem for the L1 norm of the unique positive solution of an heterogeneous Fisher-KPP equation with respect to the growth rate. It is already known that the BV norms of maximizers of this functional blow up when the diffusivity tends to zero. Here, we first show that the maximizers are always BV. Next, we completely characterize the limit of the maximas of this functional as the diffusivity tends to zero, and we show that one can construct a quasi-maximizer which is periodic, in a sense, and with a BV norm behaving like the inverse of the square root of the diffusivity. Lastly, we prove that along a subsequence of diffusivities, any maximizer is periodic, in a sense.

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arXiv2026

Planar pulsating traveling wave solutions of non-cooperative Fisher--KPP systems in space-time periodic media

Non-cooperative Fisher-KPP systems with space-time periodic coefficients are motivated for instance by models for structured populations evolving in periodic environments. This paper is concerned with entire solutions describing the invasion of open space by a persistent population at constant speed. These solutions are important in the understanding of long-time behaviors for the Cauchy problem. Adapting methods developed for scalar equations satisfying the comparison principle as well as methods developed for systems with homogeneous coefficients, we prove, in each spatial direction, the existence of a critical speed such that: there exists no almost planar generalized transition waves with a smaller speed; if the direction is rational, each rational speed not smaller than the critical speed is the speed of a planar pulsating traveling wave with time and transverse space periodicity; if the coefficients are homogeneous in space, each speed not smaller than the critical speed is the speed of a planar pulsating traveling wave with time periodicity.

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arXiv2026

Optimal Dirac controls for time-periodic bistable ODEs, application to population replacement

This work addresses an optimal control problem on a dynamics governed by a nonlinear differential equation with a bistable time-periodic nonlinearity. This problem, relevant in population dynamics, models the strategy of replacing a population of A-type individuals by a population of B-type individuals in a time-varying environment, focusing on the evolution of the proportion of B-type individuals among the whole population. The control term accounts for the instant release of B-type individuals. Our main goal, after noting some interesting properties on the differential equation, is to determine the optimal time at which this release should be operated to ensure population replacement while minimizing the release effort. The results establish that the optimal release time appears to be the minimizer of a function involving the carrying capacity of the environment and the threshold periodic solution of the dynamics; they also describe the convergence of the whole optimal release strategy. An application to the biocontrol of mosquito populations using Wolbachia-infected individuals illustrates the relevance of the theoretical results. Wolbachia is a bacterium that helps preventing the transmission of some viruses from mosquitoes to humans, making the optimization of Wolbachia propagation in a mosquito population a crucial issue.

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arXiv2025

Generalized principal eigenvalues for parabolic operators in bounded domains

We introduce here new generalized principal eigenvalues for linear parabolic operators with heterogeneous coefficients in space and time. We consider a bounded spatial domain and an unbounded time interval $I$ : $I=\mathbb{R},\ \mathbb{R}^+$ or $\mathbb{R}^-$, and operators with coefficients having a fairly general dependence on space and time. The notions we introduce rely on the parabolic maximum principle and extend some earlier definitions introduced for elliptic operators [BNV]. We first show that these eigenvalues hold the key to understanding the large time behavior and entire solutions of heterogeneous Fisher-KPP type equations. We then describe the relation of these quantities with principal Floquet bundles for parabolic operators which provides further characterizations of the principal eigenvalues. These allow us to derive monotonicity properties and comparisons between generalized principal eigenvalues, as well as perturbation results and further properties involving limit operators. We show that the sign of these eigenvalues encodes different versions of the maximum principle for parabolic operators. Lastly, we explicitly compute the generalized principal eigenvalues for several classes of operators such as spatial-independent, periodic, almost periodic, uniquely ergodic or random stationary ergodic coefficients.

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arXiv2025

Optimisation of space-time periodic eigenvalues

The goal of this paper is to provide a qualitative analysis of the optimisation of space-time periodic principal eigenvalues. Namely, considering a fixed time horizon $T$ and the $d$-dimensional torus $\mathbb{T}^d$, let, for any $m\in L^\infty((0,T)\times\mathbb{T}^d)$, $λ(m)$ be the principal eigenvalue of the operator $\partial_t-Δ-m$ endowed with (time-space) periodic boundary conditions. The main question we set out to answer is the following: how to choose $m$ so as to minimise $λ(m)$? This question stems from population dynamics. We prove that in several cases it is always beneficial to rearrange $m$ with respect to time in a symmetric way, which is the first comparison result for the rearrangement in time of parabolic equations. Furthermore, we investigate the validity (or lack thereof) of Talenti inequalities for the rearrangement in time of parabolic equations. The numerical simulations which illustrate our results were obtained by developing a framework within which it is possible to optimise criteria with respect to functions having a prescribed rearrangement (or distribution function).

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arXiv2023

On the instability of threshold solutions of reaction-diffusion equations, and applications to optimization problems

The first part of this paper is devoted to the derivation of a technical result, related to the stability of the solution of a reaction-diffusion equation $u_t-Δu = f(x,u)$ on $(0,\infty)\times \mathbb{R}^N$, where the initial datum $u(0,x)=u_0(x)$ is such that $\lim_{t\to +\infty} u(t,x)=W(x)$ for all $x$, with $W$ a steady state in $H^1(\mathbb{R}^N)$. We characterize the perturbations $h$ such that, if $u^h$ is the solution associated with the initial datum $u_0+h$, then, if $h$ is small enough in a sense, one has $u^h(t,x)>W(x)$ (resp. $u(t,x) 0$ for all $x$.

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arXiv2023

Large scale asymptotics of velocity-jump processes and non-local Hamilton-Jacobi equations

We investigate a simple velocity jump process in the regime of large deviation asymptotics. New velocities are taken randomly at a constant, large, rate from a Gaussian distribution with vanishing variance. The Kolmogorov forward equation associated with this process is the linear BGK kinetic transport equation. We derive a new type of Hamilton-Jacobi equation which is nonlocal with respect to the velocity variable. We introduce a suitable notion of viscosity solution, and we prove well-posedness in the viscosity sense. We also prove convergence of the logarithmic transformation towards this limit problem. Furthermore, we identify the variational formulation of the solution by means of an action functional supported on piecewise linear curves. As an application of this theory, we compute the exact rate of acceleration in a kinetic version of the celebrated Fisher-KPP equation in the one-dimensional case.

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arXiv2023

Final size and convergence rate for an epidemic in heterogeneous population

We formulate a general SEIR epidemic model in a heterogenous population characterized by some trait in a discrete or continuous subset of a space R d. The incubation and recovery rates governing the evolution of each homogenous subpopulation depend upon this trait, and no restriction is assumed on the contact matrix that defines the probability for an individual of a given trait to be infected by an individual with another trait. Our goal is to derive and study the final size equation fulfilled by the limit distribution of the population. We show that this limit exists and satisfies the final size equation. The main contribution is to prove the uniqueness of this solution among the distributions smaller than the initial condition. We also establish that the dominant eigenvalue of the next-generation operator (whose initial value is equal to the basic reproduction number) decreases along every trajectory until a limit smaller than 1. The results are shown to remain valid in presence of diffusion term. They generalize previous works corresponding to finite number of traits (including metapopulation models) or to rank 1 contact matrix (modeling e.g. susceptibility or infectivity presenting heterogeneity independently of one another).

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arXiv2022

Localising optimality conditions for the linear optimal control of semilinear equations \emph{via} concentration results for oscillating solutions of linear parabolic equations

We propose a fine analysis of second order optimality conditions for the optimal control of semi-linear parabolic equations with respect to the initial condition. More precisely, we investigate the following problem: maximise with respect to $y\in L^\infty({(0;T)\times Ω})$ the cost functional $J(y)=\iint_{(0;T)\times Ω}j_1(t,x,u)+\int_Ωj_2(x,u(T,\cdot))$ where $\partial_t u-Δu=f(t,x,u)+y\,, u(0,\cdot)=u_0$ with some classical boundary conditions, under constraints of the form $-κ_0\leq y\leq κ_1\text{ a.e.}\,, \int_Ωy(t,\cdot)=V_0$. This class of problems arises in several application fields. A challenging feature of these problems is the study of the so-called abnormal set $ \{-κ_0<y^*<κ_1\}$ where $y^*$ is an optimiser. This set is in general non-empty and it is important (for instance for numerical applications) to understand the behaviour of $y^*$ in this set: which values can $ y^*$ take? In this paper, we introduce a Laplace-type method to provide some answers to this question. This Laplace type method is of independent interest.

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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

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

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

Diameters of the level sets for reaction-diffusion equations in nonperiodic slowly varying media *

We consider in this article reaction-diffusion equations of the Fisher-KPP type with a nonlinearity depending on the space variable x, oscillating slowly and non-periodically. We are interested in the width of the interface between the unstable steady state 0 and the stable steady state 1 of the solutions of the Cauchy problem. We prove that, if the heterogeneity has large enough oscillations, then the width of this interface, that is, the diameter of some level sets, diverges linearly as t $\rightarrow$ +$\infty$ along some sequences of times, while it is sublinear along other sequences. As a corollary, we show that under these conditions generalized transition fronts do not exist for this equation.

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arXiv2020

A Turing mechanism in order to explain the patchy nature of Crohn's disease

Crohn's disease is an inflammatory bowel disease (IBD) that is not well understood. In particular, unlike other IBDs, the inflamed parts of the intestine compromise deep layers of the tissue and are not continuous but separated and distributed through the whole gastrointestinal tract, displaying a patchy inflammatory pattern. In the present paper, we introduce a toy-model which might explain the appearance of such patterns. We consider a reaction-diffusion system involving bacteria and phagocyte and prove that, under certain conditions, this system might reproduce an activator-inhibitor dynamic leading to the occurrence of Turing-type instabilities. In other words, we prove the existence of stable stationary solutions that are spatially periodic and do not vanish in time. We also propose a set of parameters for which the system exhibits such phenomena and compare it with realistic parameters found in the literature. This is the first time, as far as we know, that a Turing pattern is investigated in inflammatory models.

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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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