Mini-courses

• Yavar Kian, Aix Marseille Université, France
  2h30 mini-course : Introduction au problème de Calderón Abstract
  L'objectif de ce cours sera de présenter le problème de Calderón qui consiste à déterminer une conductivité apparaissant dans une équation elliptique à partir d'excitation et de mesures sur le bord du domaine. Nous étudierons ce problème en nous basant sur l'approche initiée par Sylvester et Uhlmann basée sur les solutions optique géométrique complexes.
• Éric Soccorsi, Aix Marseille Université, France
  2h30 mini-course: Introduction to Borg-Levinson Inverse Spectral Theory Abstract
• Louis Tébou, Florida International University, Miami, USA.
  2h30 mini-course: Simultaneous and indirect control of waves: some recent developments and open problems Abstract
  The control of a single wave equation has known some tremendous amount of activity since the pioneering works of Russell in the sixties and seventies, and especially after the publication of Lions' monographs on controllability. However, the control of multi-component systems of wave equations can arguably be said to be at an early stage. In the coming lectures, I will discuss some of my recent works in this area, focusing on the notion of simultaneous control (using the same control scheme in all components of the system at hand), and indirect control (using a single control for a system consisting of two components). I will also discuss some problems that I deem important to investigate.

Invited talks

• Moncef Aouadi, Université de Carthage, Tunisia
  Exact controllability for nonlinear thermoviscoelastic plate problem Abstract
  In this talk, we prove that after certain threshold time moment the nonlinear damped thermoviscoelastic plate is exactly controllable. We present a global well-posedness result for the nonlinear problem when the nonlinear functions are continuous and globally Lipschitz. Then we establish conditions on the physical constants of the plate to guarantee that the roots of the characteristic equation of the linear operator are simple. Subsequently, we find the necessary and sufficient conditions to guarantee the exponential decay of solutions. Then by a powerful spectral approach, we determine the optimal rate decay by the physical constants. The exact controllability of the linear and nonlinear problems is shown by proving that the corresponding controllability mappings are surjective.
• Ahmed Bchatnia, Université de Tunis El Manar, Tunisia
  Observability and stabilizability of the wave equation with moving boundary Abstract
  We deal with the one dimensional wave equation in a bounded interval with assigned moving interior point a(t) and Dirichlet boundary conditions are specified. Here a(t) is assumed to move slower than the light and periodically. Moreover a is continuous, piecewise linear with two independent parameters. Our major concern will be an observation problem which is based measuring, at each t>0 of the transverse velocity at a(t).
• Manuel González-Burgos, Universidad de Sevilla, Spain
  Sharp estimates of the one-dimensional boundary control cost for parabolic systems Abstract
  In this talk we will present new results on the cost of the boundary controllability of parabolic systems at time $T > 0$. In particular, we will study sharp estimates of the control cost at time T (T small enough) when the eigenvalues of the generator of the $C_0$ semigroup accumulate and do not satisfy a gap condition. The main ingredient we will use is the moment method.
• Makram Hamouda, Université de Tunis El Manar, Tunisia
  Theoretical and numerical aspects related to the energy of some Timoshenko systems Abstract
  In this talk, we consider a Timoshenko system subject to different types of dissipation (undamped, linear damping and nonlinear damping). First, we recall some theoretical asymptotic results related to the associated energy. Then, we design a discretization scheme, based on a combination of the finite element method and the finite difference one. The obtained scheme reaches to describe the discrete energy showing the positivity, the energy conservation property and the types of the decay rates in the case of a damped system. We numerically confirm the theoretical results established on the decay rates of the energy associated with each type of dissipation.
• Chaker Jammazi, Université de Carthage, Tunisia
  On the finite-time stabilization of some hyperbolic control systems by boundary feedback laws: Lyapunov approach Abstract
  This talk discusses the finite-time stabilization of some linear hyperbolic partial differential equations by means of persistently exciting boundary feedbacks. The construction of stabilizing feedbacks is based on Lyapunov approach, and the finite-time stability of the closed loop system is obtained by the persistent excitation condition of the boundary. A particular attention is paid to transport and wave control systems. The obtained results are used to solve two problems: the first one is finite-time stability of the SMB chromatography while the second application is the finite-time controllers of two strings connected by point mass.
• Souleymane Kadri Harouna, Université de La Rochelle, France
  A Stochastic Sub-grid Viscosity Model and Wavelet based Method for Images Assimilation Abstract
  The talk will concern the derivation of a stochastic model of the Navier-Stokes equations. We use a decomposition of the velocity field into a deterministic drift component and a random uncertainty term. The evolution equations of the deterministic component of the flow are derived from a stochastic version of the Reynolds Transport Theorem. These equations include in their general form an anisotropic diffusion term that represents the effect of unresolved turbulent scales modeled by stochastic uncertainty term. This is similar to the LES or RANS methods, in order to model the effects of the coarse component of a turbulent flow. Without forcing term, an analysis of the drift component equations allows to obtain physical properties such as the global energy dissipation similar to the classical Navier-Stokes equations. Then, some results on the existence of weak solutions can be obtained. Finally, we will present a wavelet based numerical discretization of the model and its application to variational image data assimilation.

Short talks (20mn)

• Mama Abdelli, Université de Mascara, Algeria
  The universal bound property for a class of second order ODEs Abstract
• Abbes Benaissa, Université de Sidi Bel Abbes, Algeria
  A general decay result of a wave equation with a dynamic boundary control of diffusive type Abstract
  We study a wave equation with a dynamic boundary control of diffusive type. We establish optimal and explicit energy decay formula by using resolvent estimates. Our new result generalizes and improve the earlier related results in the literature.
• Alessandro Duca, Université de Grenoble, France
  Permutating quantum eigenmodes by using the quasi-adiabatic motion of a wall Abstract
• Elbahi Hadidi, Université de Sidi Bel Abbes, Algeria
  Un cycle limite d'un système planaire perturbé contrôlé Abstract
  Dans ce travail, nous étudions l'existence d'un cycle limite stable pour un type de système planaire perturbé contrôlé. Ensuite, nous avons consolidé cette étude par des exemples et des simulations avec Maple.
• Abdelhak Hafdallah, Université de Tebessa, Algeria
  On the optimal control of systems with missing data Abstract
  The aim of this talk is to present some basics and generalities about the optimal control of systems with incomplete data, the concept of no-regret control introduced by J.L.Lions to treat this kind of optimal control problems. The main idea of this method is to transfer an optimal control problem with missing data to a standard optimal control problem by using the concept of low-regret control. The no-regret control will be characterized by an optimality system independently of the missing data.
• Yamina Hamzaoui, Université de Oran, Algeria
  Existence of solutions to Neumann problem involving multiple critical exponents Abstract
  We consider the solvability of the Neumann problem for an elliptic system of two equations with weights involving two critical Sobolev exponents on a bounded domain in RN . By using variational methods, we investigate the e¤ect of the shape of the graph of the weight functionsand the geometry of the boundary on the existence of solutions.
• Ammar Khellaf, Université de Guelma, Algeria
  The Resolution of the Spectral Pollution the Generalized Spectrum method Abstract
  It is well known that the spectral pollution is the weakness of projection methods applied on unbounded operator. This phenomenon is considered as a serious problem in several areas in the field of applied mathematics. Thus, our technic is an alternative method based on disrupting an unbounded operator by a bounded one until that the spectral properties transform to controlled case. The generalized spectrum takes its place as the desired case. We show that every unbounded operator contains a decomposition of two bounded operators which carry all the spectral properties. Through this decomposition and basing upon its numerical approximations, the phenomenon of spectral pollution will be resolved. This phenomenon is considered as a serious problem in several areas in the field of applied mathematics that has been studied. The concept of generalized spectrum was constructed under the generalized spectrum of the matrices then extended for bounded operators case.
• Mohamed Zerguine, Université de Batna 2, Algeria
  Global persistence of geometric structures for stratfied Euler system with fractional dissipation Abstract