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# Conferences

## Neumann-Poincaré-type operators, invisibility, and super-resolution

- to give a mathematical justification of cloaking due to anomalous localized resonance;
- to provide an original method for enhancing near cloaking;
- to achieve resolved inclusion imaging.

## Inverse acoustic and electromagnetic scattering for obstacles and cracks

In this talk, we will present some non-iterative methods to identify impenetrable obstacles or cracks with impedance boundary conditions from the scattering data for the 2D and 3D acoustic and electromagnetic problems. This is a joint work with Y. Boukari, K. Erhard, H. Haddar, O. Ivanyshyn, J. J. Liu, R. Potthast and M. Sini.

Let us consider a quantum particle in a potential and a uniform (in space) time-dependent electric field. It is a control system in which the state is the wave function of the particle and the control is the electric field. It is represented by a Schrödinger PDE, where the control acts bilinearly on the state. In this talk, we will present several recent results concerning the control of this equation : exact/approximate controllability, explicit feedback stabilization.

## Regularity properties for general free discontinuity problems

We give a general monotonicity formula for local minimizers of free discontinuity problems which have a critical deviation from minimality, of order d-1. This result allows to prove partial regularity results (i.e. closeness and density estimates for the jump set) for a large class of free discontinuity problems involving general energies associated to the jump set, as for example free boundary problems with Robin conditions. In particular, we give a short proof to the De Giorgi-Carriero-Leaci result for the Mumford-Shah functional. This is a joint work with Stephan Luckhaus.

## Transmission Eigenvalues in Inverse Scattering Theory

The transmission eigenvalue problem is a new class of eigenvalue problems that has recently appeared in inverse scattering theory for inhomogeneous media. Such eigenvalues provide information about material properties of the scattering object and can be determined from scattering data, hence can play an important role in a variety of problems in target identification. The transmission eigenvalue problem is non-selfadjoint and nonlinear which make its mathematical investigation very interesting.

In this lecture we will describe how the transmission eigenvalue problem arises in scattering theory, how transmission eigenvalues can be computed from scattering data and what is known mathematically about these eigenvalues. The investigation of transmission eigenvalue problem for anisotropic media will be discussed and Faber-Krahn type inequalities for the first real transmission eigenvalue will be presented. We conclude our presentation with some recent preliminary results on transmission eigenvalues for absorbing and dispersive media, i.e. with complex valued index of refraction, as well as for anisotropic media with contrast that changes sign.

Our presentation contains a collection of results obtained with several collaborators, in particular with David Colton, Drossos Gintides, Houssem Haddar and Andreas Kirsch.

## Control and inverse problems for degenerate parabolic operators

Degeneracy is a `natural' phenomenon that occurs in applied and theoretical sciences where parabolic operators are used to study diffusions: from climatology models to stochastic viability, from mathematical finance to subriemannian manifolds.

While the general theory of degenerate parabolic equations has been developed for almost a century, interest in specific issues, such as control and inverse problems, is by far more recent. It turns out that, from the point of view of control theory, parabolic operators that degenerate on subsets of the space domain may exhibit a wider range of behaviors than uniformly parabolic ones. For instance, null controllability in arbitrary time may be true or false according to the degree of degeneracy, and there are also examples where a finite time is needed to ensure such a property. Similar considerations could be made for inverse problems.

This talk will survey most of the theory that has been established, so far, for operators with boundary degeneracy, and discuss recent results for operators of Grushin type which degenerate in the interior.

## Regularity estimate in high conductivity homogenization and application

In collaboration with Eric Bonnetier and Habib Ammari, we studied the effect of a periodic microstructure near an inhomogeneity near a in a time reversal experiment. The purpose of this work was to provide a mathematical explanation for the "resolution beyond the diffraction limit" observed experimentally by Lerosey, de Rosny, Tourin and Fink. From an homogenization theory perspective, this is a quantification of the effect of defects on effective properties of periodic microstructure.

In a recent work with Marc Briane and Luc Nguyen, we considered the case of a periodic micro-structure with highly conducting fibres (i.e. metal rods). Fenchenko and Khruslov showed 30 years ago that for a particular scaling range, the effective problem includes a non-local term. This model has received a renewed interest recently, after Bouchitté and Felbacq showed that such structures could be used to construct composite meta-materials with unusual properties. A key issue in this model is the regularity of the solution.

We will see that the energy estimate is the only uniform bound: even locally, any higher Sobolev norm blows up. On the other hand, a uniform holder estimate for the gradient holds on a (moving) sub-domain. This sub-domain fills in the whole domain at given rate when the period tends to zero. This allows us in turn to quantify the effect of impurities in the substrate.

We will discuss the mechanical bases of cellular motility by swimming and crawling. Starting from observations of biological self-propulsion, we will analyze the geometric structure underlying motility at small scales, the swimming strategies available to microscopic swimmers and recipes to optimize their strokes. The talk is based on joint work with F. Alouges.

## On the inverse conductivity problem with complex coefficient.

In this talk I will address some issues related to the inverse conductivity problem with a complex valued coefficient. In particular I will take into consideration the possibility of recovering from boundary data a limited number of features of the conductivity. (in collaboration with E. Beretta and S. Vessella)

A body of work has emerged concerning "metamaterials." The unexpected behavior of these materials or structures is typically due to microstructural resonance. I will discuss some examples, including current work with Jens Jorgensen, Jianfeng Lu, and Michael Weinstein on "membrane-type acoustic metamaterials" (a class of structured membranes with unexpected transmission properties, introduced by Z. Yang et al in 2008).

## Stabilization of discrete-time quantum systems subject to non-demolition measurements with imperfections and delays

This talk develops the mathematical methods underlying a quantum feedback experiment stabilizing photon-number states that has been realized recently in the Laboratoire Kastler-Brossel at Ecole Normale Supérieure. It considers a controlled system those quantum state, a finite-dimensional density operator, is governed by a discrete-time non linear Markov process. In open-loop, the measurements are assumed to be quantum non-demolition (QND) measurements. This Markov process admits thus a set of stationary pure states associated to an ortho-normal basis of the underlying Hilbert space. These stationary states provide martingales that are crucial to characterizing the open-loop stability: under simple assumptions, almost all trajectories converge to one of these stationary states; the probability to converge to a stationary state is given by its overlap with the initial quantum state. From these open-loop martingales, we construct a super-martingale whose parameters are given by inverting a Laplacian matrix characterizing the impact of the control input on the Kraus operators defining the Markov process. This super-martingale measures the distance between the current quantum state with the goal state chosen from one of the stationary pure states. At each step, the control input minimizes the conditional expectation of this super-martingale. It is proven that the resulting feedback scheme stabilizes almost surely towards the goal stationary state whatever the initial quantum state. This state feedback takes into account a known constant delay of arbitrary length in the control loop. This control strategy is proved to remain also convergent when the state is replaced by its estimate based on a quantum filter relying on measurements that are corrupted by random errors with conditional probabilities described by a known left stochastic matrix. Closed-loop simulations corroborated by experimental data illustrate the interest of such nonlinear feedback scheme.

This mathematical analysis was done in collaboration with H. Amini, R. Somaraju, I. Dotsenko, C. Sayrin and M. Mirrahimi.## Differentiating discretized metrics and applications

Let us consider a positive function *g* on a domain Ω and use it as a conformal metric,
so as to define a distance *d* through

*
d(x,y)=inf
{
∫
0
1
g(σ(t))|σ'(t)|dt : σ(0)=x,σ(1)=y
}.
*

What happens if one perturbates *g* ? which is the first variation of *d* ? In the "continuous case'' it is easy to check that pertubating *g* into *g+εh* the first variation of *d(x,y)* is given by the integral of *h* along the geodesics connecting *x* to *y*.
Yet, for computational purposes one needs to discretize the computation of distances, and one of the most efficient way is to use the so-called Fast Marching Algorithm which gives discretizations of the viscosity solutions of the Eikonal Equation
*|∇ u|=g*.
We will see a corresponding algorithm, that we called Subgradient Marching Algorithm, computing in the same loop the distance
as well as its gradient with respect to metrics perturbations in *O(N^2 ln(N))*. This discretization being consistent with convexity
properties and we will use it so as to attack some variational problems such as the traffic
congestion equilibria, the optimal obstacles to slow down the enemies and some inverse problems based on travel-time tomography.

## Optimal design problems for conservative equations

We consider a conservative evolution equation on a given domain Omega of R^n. The purpose of this talk is to investigate some natural shape optimization problem arising in the context of mathematics, physics or engineering. Given an initial state, one may observe on a measurable subset omega of Omega with given measure the solution of the equation, or control it (by HUM) or stabilize it (by a linear damping) to rest, with a control supported on omega. In the three cases, we focus on the question to know how to determine the best possible domain omega over all subsets of Omega of fixed measure (say L|Omega| with 0<L<1) ensuring either the best observation, or the smallest possible norm of control, or the best rate of convergence for the stabilization. These questions are first investigated with fixed initial data. We then provide relevant criterions that do not depend on the initial conditions and analyze the related shape optimization problems. In particular, we comment on the regularity of the optimal domain, which can be regular or of fractal type according to the problem under consideration. One of these problems consists of the optimization (with respect to the domain omega) of observability constants. Finally, we provide approximation procedures in order to compute numerically the best domain. In particular in dimension one efficient algorithms can be developed by using an interpretation of the problem in terms of optimal control. This is a work in collaboration with Y. Privat (ENS Rennes, France) and E. Zuazua (BCAM Bilbao, Spain).