# Séminaire EDPAN

Organisateurs : Nicolae Cindea
Les exposés ont lieu le jeudi à 11h15 en salle 2222 du bâtiment de mathématiques (consulter le plan d'accès au laboratoire).

### Mai 2022

• Jeudi 05 mai 2022 - Mohamed Boukraa

Fading regularization method for an inverse Cauchy problem in thin plate theory.

This work deals with the resolution of an inverse Cauchy type problem associated with the biharmonic equation such that the boundary conditions are known only on a part of the boundary of the domain. A problem often encountered in mechanics, especially in thin plate theory. Such a problem is called as ill-posed in the sense of Hadamard, i.e. the existence, uniqueness and stability of their solutions are not always guaranteed. Therefore, special methods should be designed to obtain a unique, stable and convergent solution to this inverse Cauchy problem. To address this issue, we choose to employ the fading regularization method. We investigate the numerical reconstruction of the missing boundary conditions on an inaccessible part of the boundary from the knowledge of over-prescribed exact or noisy data on the remaining and accessible part of the boundary. We present numerical implementations of this method using the method of fundamental solutions and the finite element method. We then propose to combine Discrete Kirchhoff plate elements with the fading regularization method to solve the Cauchy problem in thin plate theory.

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### Mars 2022

• Jeudi 17 mars 2022 - Hugo Martin

A kinetic equation modelling the collective dynamics of the rock-paper-scissors binary game: wellposedness in the Radon measure setting and subgeometric rate of convergence toward a limit measure that depends explicitely on the initial condition

The rock-paper-scissors game has been studied for many years from various points of view. Game theory established a long time ago that two rational players have a unique optimal strategy, which is playing one of the three moves at random, with probability 1/3. This very simple case was extended in various ways. In a recent paper, Pouradier Duteil and Salvarani considered a large amount of rational agents, that play a r-p-s game after encountering, and exchange a certain amount of money based on the outcome. This results in a PDE modeling a population structured in wealth, that can be interpreted as a discrete heat equation on the half-line with rate of diffusion depending on the amount of players that are rich enough (i.e. that would be able to pay their dept if they loose they next game), thus introducing a nonlinearity.

The goal of my work was to adapt a methodology in the measure framework that was successfully used on equations arising from biology. On the well-known renewal and growth-fragmentation equations, as well as equations steming for instance from neuroscience, convergences in total variation norm were obtained, sometimes supplemented by explicit exponential rates of convergence. The first step was to reformulate the equation in the space of signed measures, by mean of a duality approach. The dual problem in turn provided a explicit weak limit to the initial equation, which is new compared to the previous work.

The asymptotic behaviour is obtained using an unusual time rescaling. Like previously mentionned works, we obtain a decay in (weighted) total variation norm and an explicit limit, but unlike these works, the limit measure strongly depends on the initial measure and the decay rate is subgeometric.

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### Décembre 2021

• Jeudi 09 décembre 2021 - Kuntal BHANDARI

Boundary null-controllability of some 1-D coupled parabolic systems with Kirchhoff conditions

In this talk, we present the boundary null-controllability of some $2\times 2$ parabolic systems in 1-D that contains a linear interior coupling with real constant coefficient and a Kirchhoff-type condition through which the boundary coupling enters in the system. The control is exerted on a part of the boundary through a Dirichlet condition on either one of the two state components. We show that the controllability properties vary depending on which component the control is being applied; the choices of interior coupling coefficient and the Kirchhoff parameter play a crucial role to deduce positive or negative controllability results. To this end, we also discuss the controllability properties of some $3\times 3$ models with one or two boundary controls.

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