Laboratoire de Recherche Analyse et Contrôle des Equations aux Dérivées Partielles

JLAB’24 - Journée du Labo

26 Juin 2024

Hotel Monastir Center

 

9h : Accueil des participants

9h20 – 10h: Mohamed Ali Hamza (Imam Abdulrahman Bin Faisal University)

Blow-up and lifespan estimate of the solution of the damped wave equation in the scale-invariant case

 

10h-10h40: Stéphane Gerbi (Savoie Mont Blanc University)

A finite volume method for a 1-d wave equation with non smooth wave speed and localized Kelvin-Voigt damping

 

 

10h40-11h10: Pause-café 

11h10-11h50: Makram Hamouda(Imam Abdulrahman Bin Faisal University)

 Machine Learning in solving PDEs: A case study on the Timoshenko system.

 

11h50-12h10:  Ammar Moulahi(University of Monastir)

Stabilité d'un système de transmission onde/chaleur dégénéré 

 

12h10-12h30: Wafa Ahmedi (University of Monastir)

Stabilization of a locally transmission problems of two strongly-weakly coupled  wave systems

12h30-14h: Pause déjeuner

14h-14h40: Carlos Castro (Polytechnical University of Madrid)

Numerical approximation of some inverse source problems associated to the wave equation.

14h40-15h20: Jamel Ferchichi (University of Sousse)

Singular and shape design optimal control tasks relayed to porous media

 

15h20-15h40: Habib Ayadi (University of Sfax)

Finite and fixed time stability, Examples and counter examples

 

15h40-16h: Clôture et pause-café

 

 

 

 

 

Mohamed Ali Hamza

 

Blow-up and lifespan estimate of the solution of the damped wave equation in the scale-invariant case

 

      In this talk we discuss the damped wave equation in the scale-invariant case with time derivative nonlinearity. In the first part, we study the influence of the damping term  μ/(1+t) u_t  in the global dynamics of the solution, under the assumption of small initial data. In fact, we obtain a new interval for µ that we conjecture to be closer to optimality, or probably optimal, and, thus, characterizes the threshold between the blow-up and the global existence regions. The blow-up region is given by p ∈ (1, pG(n + µ)) where pG(n) is the Glassey exponent. The second part will be devoted for the same problem in the case whith combined nonlinearities.

 

Stéphane Gerbi

 

A finite volume method for a 1-d wave equation with non smooth wave speed and localized Kelvin-Voigt damping

 

       In this talk, we study the numerical solution of an elastic/viscoelastic wave equation with non smooth wave speed and localized distributed Kelvin-Voigt damping.

Our method is based on the Finite Volume Method (FVM) and we are interested in deriving the stability estimates and the convergence of the numerical solution to the continuous one.

Numerical experiments are performed to confirm the theoretical study on the decay rate of the solution to the null one when a localized damping acts.

This is a joint work with Rayan Nasser and Ali Wehbe from the Lebanese University in Beirut.

 

Makram Hamouda

 

Machine Learning in solving PDEs: A case study on the Timoshenko system.

     This talk delves into the application of ML techniques in addressing the Timoshenko system—a set of PDEs governing the behavior of beam structures accounting for shear deformation and rotational inertia effects. Traditional methods for solving these equations, while robust, often face challenges in computational cost and complexity, especially for high-dimensional problems or complex geometries.

Our approach leverages deep learning frameworks, specifically physics-informed neural networks (PINNs), which integrate the underlying physical laws directly into the training process. This enables the network to not only approximate solutions but also respect the intrinsic properties of the Timoshenko system. We will explore the architecture and training methodologies of these networks, highlighting how they are designed to capture the physical properties of beams.

The talk will present comparative analyses demonstrating the accuracy and computational advantages of our ML-based methods against classical numerical solvers. Additionally, we will discuss the potential for generalization to other PDE systems and the implications for future research in computational mechanics and structural engineering.

 

Ammar Moulahi 

 

Stabilité d'un système de transmission onde/chaleur dégénéré 

 

     Dans cet exposé, nous considérons un système couplé sur une surface cylindrique où la composante parabolique est dégénérée.   Cela implique une équation d'onde et une équation de chaleur avec transmissions naturelles. 

Nous discutons des problèmes de stabilité de ce système sous les conditions aux limites de Dirichlet au niveau des cercles de bord. Nous montrons que lorsque la dégénérescence est faible, le semi-groupe associé est polynomialement stable, ce qui fournit des taux de stabilité élevés. En particulier, notre résultat de stabilité généralise et améliore dans un certain sens ceux existants dans le contexte unidimensionnel et bidimensionnel.

 

Wafa Ahmedi

 

Stabilization of a locally transmission problems of two strongly-weakly coupled  wave systems

 

    In this talk,  we embark on a captivating exploration of the stabilization of locally transmitted problems within the realm of two interconnected wave systems.

 

To begin, we wield the formidable Arendt-Batty criteria to affirm the resolute stability of our system. Then, with an artful fusion of a frequency domain approach and the multiplier method, we unveil the exquisite phenomenon of exponential stability, a phenomenon that manifests when the waves of the second system synchronize their propagation speeds.

 

 In cases where these speeds diverge, our investigation reveals a graceful decay of our system's energy, elegantly characterized by a polynomial decline at a rate of  t^(-1) .

 

Carlos Castro

Numerical approximation of some inverse source problems associated to the wave equation.

 

      We analyze the use of two different numerical methods to approximate the solutions of inverse source problems associated to the wave equation. The inverse problem consists in recovering the source term from known boundary information of a single solution for sufficiently large time. It is well-known that a suitable boundary observability inequatity gives the stability of such inverse problems.

 However, natural discretizations of these inverse problems (as finite elements or finite differences) may not inherit the observability inequality, uniformly with respect to the discretization parameter. In this case, the solution of the discrete inverse problem cannot be used to approximate the solution of the continuous one. In this talk we present two numerical methods that overcome this difficulty. The first one is based on a mixed finite element formulation and can be extended to variable coefficients in one dimension. This extension is  delicate and it is based on nonstandard uniform spectral properties of the associated discrete operator. The second is based on a polynomial spectral collocation method and can be extended to the elasticity system. The first part of this work is done in collaboration with S. Micu and the second one with S. Boumimez. 

 

Jamel Ferchichi

 

Singular and shape design optimal control tasks relayed to porous media

We undertake rigorous analysis of the geometric inverse problem concerning the detection and identification of an elusive obstacle denoted as ω, submerged within a bounded fluid flow domain Ω, governed by the time-dependent Brinkman model. Our methodological approach involves transforming it into an optimization task, employing a least squares functional. We prove the well-posedness of an optimal solution to the optimization problem. Subsequently, we perform topological asymptotic expansion of the cost function using a simple penalty method combined with a prior estimate of the state solution. An important advantage of this method is that it avoids the truncation method used in the literature.

 

 

Habib Ayadi

 

Finite and fixed time stability

 Examples and counter examples

 

     Finite Time Stability (FTS) is a critical concept in control theory and dynamic systems, focusing on the behavior of systems within a finite time interval. Unlike traditional stability notions that concern long-term behavior, FTS ensures that a system’s state reaches a desired condition within a specified finite period. This mini-course will delve into the theoretical foundations, practical implications, and methods for achieving finite time stability in various systems, including both linear and nonlinear dynamics and distributed systems. 

 

The notion of finite-time stability has been proposed in 60s by Emilo Roxin (Roxin, 1966) and it has been developed in many works later, where a particular attention is paid to the time of convergence to a steady state. The fixed-time stability concept is quite recent (Polyakov,2012). Some examples and countre examples are done to illustrate the theory concept.