Section: New Results
Inverse problems for heterogeneous systems
In [7] , David Dos Santos Ferreira et al. obtain
global stability estimates for a potential in a Schrödinger equation on an open bounded set in dimension
In [20] , David Dos Santos Ferreira et al. proved uniform
In [10] , Munnier and Ramdani considered the 2D inverse problem of recovering the positions and the velocities of slowly moving small rigid disks in a bounded cavity filled with a perfect fluid. Using an integral formulation, they first derive an asymptotic expansion of the DtN map of the problem as the diameters of the disks tend to zero. Then, combining a suitable choice of exponential type data and the DORT method (french acronym for Diagonalization of the Time Reversal Operator), a reconstruction method for the unknown positions and velocities is proposed. Let us emphasize here that this reconstruction method uses in the context of fluid-structure interaction problems a method which is usually used for waves inverse scattering (the DORT method).
In [24] , Munnier and Ramdani proposed a new method to tackle a geometric inverse problem related to Calderón's inverse problem. More precisely, they proposed an explicit reconstruction formula for the cavity inverse problem using conformal mapping. This formula is derived by combining two ingredients: a new factorization result of the DtN map and the so-called generalized Polia-Szegö tensors of the cavity.
In [11] , Ramdani, Tucsnak and Valein tackled a state estimation problem for an infinite dimensional system arising in population dynamics (a linear model for age-structured populations with spatial diffusion). Assume the initial state to be unknown, the considered inverse problem is to estimate asymptotically on time the state of the system from a locally distributed observation in both age and space. This is done by designing a Luenberger observer for the system, taking advantage of the particular spectral structure of the problem (the system has a finite number of unstable eigenvalues).
In [12] , San Martin, Schwindt and Takahashi consider the geometrical inverse problem consisting in recovering an unknown obstacle in a viscous incompressible fluid by measurements of the Cauchy force on the exterior boundary. They deal with the case where the fluid equations are the non stationary Stokes system and using the enclosure method, they can recover the convex hull of the obstacle and the distance from a point to the obstacle. With the same method, they can obtain the same result in the case of a linear fluid–structure system composed by a rigid body and a viscous incompressible fluid. They also tackle the corresponding nonlinear systems: the Navier–Stokes system and a fluid–structure system with free boundary. Using complex spherical waves, they obtain some partial information on the distance from a point to the obstacle.