Abstracts

Chapter I : Pseudo-differential operators and wave front set ( 2 hours )

1- Wave front set - Examples

2- Pseudo-differential operators

3- Symbolic calculus and operance

4- Wave front and PDO’s : pseudo-local property and microlocal elliptic regularity

Chapter II : Propagation of singularities ( 2 hours )

1-Geometric preliminaries

2- A micolocal ODE

3- The propagation theorem

4- Applications : a) Unique continuation for a semilinear wave equation. 

                             b) Egorov Theorem.

Chapter III : Control of wave equation ( 2 hours )

1- Microlocal defect measures - Examples and properties

2- Geometrical facts near the boundary

3- The Geometric control condition of Bardos-Lebeau-Rauch.

4- Observation of waves

5- The HUM control operator ( following Dehman-Lebeau )

Abstract:

This mini-course is based on the celebrated article of Guillemin and Kazhdan [GK] addressing the spectral inverse problem of recovering a Riemannian metric from the spectrum of the Laplace-Beltrami operator in a closed compact Riemannian manifold. 

I will explain the importance of the Duistermaat and Guillemin trace formula [DG] leading to some important geometric tomography problems.

A part of the mini course will be concerned with the exposition of these tomography problems in the setting of Anosov manifolds. 

References:

Abstract: 

In this talk, we will study the stability aspects of recovering the perturbations that appear in a higher-order Schrödinger operator from knowledge of two types of boundary measurements. 

First, we prove that the knowledge of Dirichlet-to-Neumann map for a second order perturbation of the polyharmonic operator $\LL_{A,B,q}(x,D)= (-\Delta)^m+A(x)D\cdot D+B(x)\cdot D+q(x)$,  with $m\geq2$, stably determines the symmetric tensor $A$, the vector field $B$ and the potential $q$, in a bounded domain of $\R^n$, here $n \geq 3$.  

Second, we show that it is possible to stably determine the first order perturbations $(B, q)$ of the perturbed bi-harmonic operator $\mathcal{H}_{B,q} = (-\Delta)^2 -2iB\cdot\nabla-i\dive B+q $, from the boundary spectral data. The spectral data consist of some asymptotic knowledge of a subset of the Dirichlet eigenvalues and Neumann traces of the associated eigenfunctions of the bi-harmonic operator on the boundary of a bounded domain of $\R^n$, here $n \geq 2$.

This is a work based on [ref1], [ref2] and [ref3].  

References:

Abstract :

The presentation explores inverse problems for non-self-adjoint equations, focusing on the stable recovery of coefficients from Neumann boundary data in dimensions greater than two. The methodology involves reducing these problems to auxiliary inverse problems and employing Carleman estimates.

Abstract: 

In this work we will study inverse problems associated with different PDE's: the wave equation and the magnetic Schrödinger equation.

For this, we focus on the stable determination of time-dependent coefficients appearing in such equations, from only considering arbitrary boundary measurements.

This is a work based on [ref1] and [ref2].  

References:

Abstract: 

Data-driven prediction in quantum mechanics consists in providing an approximative description of the motion of any particles at any

given time, from data that have been previously collected for a certain number of particles under the influence of the same Hamiltonian. The difficulty of this problem comes from the ignorance of the exact Hamiltonian ruling the dynamic. In order to address this problem, Alberto Ruiz and I have formulated an inverse problem consisting in determining the Hamiltonian of a quantum system from the knowledge of the state at some fixed finite time for each initial state. We focus on the simplest case where the Hamiltonian is given by 

$-\Delta + V$, where the electric potential $V$ is non-compactly supported. During the talk I will present several uniqueness results for time-dependent potentials $V = V(\mathrm{t}, \mathrm{x})$ and stationary potentials $V = V(\mathrm{x})$, and the difference between them. Roughly speaking,  these results are uniqueness theorems, that explain why the Hamiltonians ruling the dynamics of all quantum particles are determined by the corresponding initial and final states of all these particles. As a consequence, one expects to be able to solve the data-driven prediction problem in quantum mechanics.

The theorems I will discuss are the results of collaborations with Alberto Ruiz; and Manuel Cañizares, Ioannis Parissis and Athanasios Zacharopoulos.

Abstract:

This presentation focus on the null-controllability problem for the \emph{generalized Baouendi-Grushin equation} $(\partial_t - \partial_x^2 - q(x)^2\partial_y^2)f = \mathbf{1}_\omega u$ on a rectangular domain. Sharp controllability results already exist when the control domain $\omega$ is a vertical strip, or when $q(x) = x$. In a recent work with Armand Koenig and Julien Royer, we provide upper and lower bounds for the minimal time of null-controllability for general $q$ and non-rectangular control region $\omega$. In some geometries for $\omega$, the upper bound and the lower bound are equal, in which case, we know the exact value of the minimal time of null-controllability.

I will present this new results and some key points of their proofs: known results when $\omega$ is a vertical strip and cutoff arguments for the upper bound of the minimal time of null-controllability; spectral analysis of the Schrödinger operator $-\partial_x^2 + \nu^2 q(x)^2$ when $\Re(\nu)>0$, pseudo-differential-type operators on polynomials and Runge's theorem for the lower bound. 

Abstract:

Carleman estimates are weighted a priori estimates for differential operators (in our case the Laplace-Beltrami operator in a compact Riemannian  manifold with boundary). One usually refers to these estimates as global when they involve boundary terms. Originally used to prove unique continuation results outside the analytic framework, they became a very useful tool in control theory and in the resolution of inverse problems. The $L^2$ theory based on integration by parts and commutator estimates is fairly well understood and advanced. The $L^p$ theory initiated by Jerison and Kenig usually requires parametrices constructions and the analysis of boundedness estimates for oscillatory integral operators. The first global $L^p$ Carleman estimates were proved by Dehman, Ervedoza and Thabouti for the Euclidean Laplacean. We aim to initiate an analysis involving boundary terms in the case of variable coefficients elliptic operators, and intend to apply it to the stabilization of the damped wave equation involving unbounded potentials. This is a joint work with Rémi Buffe.

Abstract: 

In this presentation, we will give sufficient conditions for the observability of wave equations on intersecting manifolds with a transmission condition at the intersection. A jump condition of the metric is assumed at the interface, so that the propagation of the bicharacteristic rays are given by the Snell's law. The bicharacteristic rays become a graph, and tracking the observability become very intricate. Using microlocal defect measure to characterize observability, we will give a sufficient condition for the observability of rays at the interface in term of a rank condition. We will illustrate our results with explicit examples, including 1-D network of wave equations. 

Abstract: 

We know that the strong dissipation of the heat equation of the heat equation implies its null-controllability, thanks to the so-called Lebeau-Robbiano’s method [4]. In fact, if we consider the fractional heat equation $(\partial_t + (-\Delta)^\alpha)f = \mathbf 1_\omega u$, this Lebeau-Robbiano's method works as long as $\alpha > 1/2$. On the other hand, null-controllability does not hold when $\alpha < 1/2$ [1,3] or when $\alpha=1/2$ in dimension $1$ [2].

In the case $\alpha=1/2$ and dimension $1$, the proof uses a natural connection between solutions of the half-heat equation and harmonic functions on the unit disk. In a joint work with Andreas Hartman, we precise this result and study the space of null-controllable initial states. Leveraging tools from complex and harmonic analysis (Hardy and Bergman spaces, separation of singularities, Carleson measures, etc.), we prove

• the space of null-controllable initial states does not depend on time;

• it is dense with dense complement in every $W^{s,2}$;

• it is a subset of the projection on positive frequencies of functions in $L^2(\omega)$.

References:

Abstract: 

An acoustic medium occupying a compact domain with non-constant sound speed, is probed by an impulsive plane wave, and the far-field response is measured in all directions for all frequencies. A longstanding open problem, called the fixed angle scattering inverse problem, is the recovery of the sound speed from this far-field response. In some situations, the acoustic properties of the medium are modeled by a Lorentzian metric and then the goal is the recovery of this metric from the far field measurements corresponding to a finite number of incoming plane waves.

We consider a time domain, near field version of this problem and show that natural fixed angle measurements distinguish between a constant velocity (the Minkowski metric) medium and a non-constant velocity (a general Lorentzian metric) medium. The talk is based on a joint work with Rakesh (Delaware) and Mikko Salo (Jyväskylä).

Abstract: 

In this talk, I will first present recent results on global Lp Carleman estimates for the Laplace operator in dimensions d>2 and its application to quantifying unique continuation with respect to lower-order terms. I will start with the motivation and then briefly outline the proof. This is joint work with Belhassen Dehman and Sylvain Ervedoza. If time permits, I will also discuss how to achieve sharper results for first-order terms using Thomas Wolff’s osculation argument. This is a work in progress with Pedro Caro and Sylvain Ervedoza.