In this talk, I will present controllability results for some linear and non-linear wave equations. The linear equations will be vector-valued and at different levels of regularity. I will give the main ideas of the proof of a change of regularity result. For the non-linear equations, I will consider the case of the focusing cubic Klein-Gordon equation. I will state a local controllability result around a regular solution, and a null-controllability result for scattering solutions. In the presence of damping, I will give both a positive and a negative stabilization result. I will also provide some ideas of proofs.

La méthode Galerkin Discontinu permet d’approcher numériquement de façon très précise les solutions régulières des équations hyperboliques. Il est par contre plus délicat d’approcher des solutions discontinues ou des solutions perturbations autour de solutions stationnaires (pour des équations avec termes sources).

En effet, dans le premier cas, les oscillations de Gibbs générées aux discontinuités peuvent déstabiliser le schéma, tandis que dans le deuxième cas, l’erreur produite sur la solution stationnaire rend difficile l’étude des dynamiques perturbatives. Nous verrons dans cet exposé comment les réseaux de neurones peuvent être utilisés pour construire des viscosités artificielles qui stabilisent les schémas numériques et comment elles permettent de construire des bases Galerkin Discontinu adaptées aux solutions stationnaires du problème.

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Initialement considéré comme un lemme clé dans les structures de régularité, le théorème de reconstruction s’est avéré être un outil analytique très flexible pour étudier l’intégration à la fois stochastique et déterministe en dimension supérieure. Dans cet exposé, nous discuterons d’une extension particulière du théorème de reconstruction dans un contexte stochastique où la famille de distributions sous-jacente satisfait certaines conditions naturelles impliquant des incréments rectangulaires. Cela nous permet de prouver l’existence et l’unicité d’une nouvelle classe d’équations aux dérivées partielles stochastiques de type hyperbolique qui combine l’intégration stochastique standard à la Walsh et les produits de Young.

Travail en collaboration avec Hannes Kern (TU Berlin).

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We investigate the energy decay of hyperbolic system of wave-wave with generalized acoustic boundary conditions in N-dimensional space, with the equations being coupled through boundary connection. First, by spectrum approach combining with a general criteria of Arendt-Batty, we prove that our model is strongly stable. Then, after proving that this system lacks the exponential stability, we establish different type of polynomial energy decay rates provided that the coefficients of the acoustic boundary conditions satisfy some assumptions. Further, we present some appropriate examples and show that our assumptions have been set correctly. Finally, we prove that the obtained energy decay rate is optimal in particular case.

Abstract

Dans ce groupe de travail, je vais donner un aperçu général des différents résultats d’existence globale en temps d’une classe de systèmes de réaction-diffusion qui proviennent de la modélisation de l’écologie (Systèmes de Lotka-Volterra), , la chimie (réactions chimiques réversibles) et de nombreux autres domaines scientifiques.

This is a joint work with Dan Tiba (Institute of Mathematics, Romanian Academy, dan.tiba@imar.ro).
We study the penalized steady Navier-Stokes with Neumann boundary conditions system in a holdall domain, its approximation properties (with error estimates) and the uniqueness of the solution that is obtained in a non standard manner. Numerical tests are presented.

As part of control theory, stabilization consists in finding a way to make stable a trajectory of a system on which one has some means of action. In this talk, we will discuss recent advances in stabilization of PDEs, starting with one of the most natural approaches for nonlinear systems, quadratic Lyapunov functions, to more complex approaches such as Fredholm backstepping. Backstepping consists in finding a control operator such that the PDE system can be invertibly mapped to a simpler PDE system for which stability is known. Surprisingly powerful, this approach offers the possibility to deal with very general classes of systems. We will review the origin of the method and present new results that resolve a question opened in 2017 and illustrate it on the rapid stabilization of the linearized water-wave equations. Finally, if time allows we will talk about a completely different subject: teaching mathematics to an AI and we will consider two questions, can we train an AI to predict the solution of a mathematical problem? can we train an AI to prove a statement?

Dans ce groupe de travail, je vais donner un aperçu général des différents résultats d’existence globale en temps d’une classe de systèmes de réaction-diffusion qui proviennent de la modélisation de l’écologie (Systèmes de Lotka-Volterra), , la chimie (réactions chimiques réversibles) et de nombreux autres domaines scientifiques.

We consider the wave propagation in a periodic structure made of a honeycomb arrangement of thin tubes. We prove the presence of Dirac points (in the dispersion surfaces of the associated operator). In addition, cutting the structure in the Zig Zag direction creates guides modes. Those results are proved by asymtptotic analysis: as the width of the tubes goes to zeros the domain tends to a graph (where explicite computations can be done). This is a joint work with Sonia Fliss (POEMS, Inria-ENSTA-CNRS).

I will present some recent work in collaboration with Elisa Davoli (TU Wien) and Katerina Nik (University of Vienna) on a three-dimensional quasistatic morpholelastic model. The mechanical response of the body and its growth are modeled by the interplay of hyperelastic energy minimization and growth dynamics. An existence result is obtained by regularization and time-discretization, also taking advantage of an exponential-update scheme. Then, we allow the growth dynamics to depend on an additional scalar field modeling nutrient concentration, and formulate an optimal control problem. Eventually, we tackle the existence of coupled morphoelastic and nutrient solutions, when the latter is allowed to diffuse and interact with the growing body.