Section: New Results

Frequency domain methods for the analysis and control of systems governed by PDE's

With a numerical viscosity terms in the approximation scheme of second order evolution equations, we show in [11] the exponential or polynomial decay of the discrete scheme when the continuous problem has such a decay and when the spectrum of the spatial operator associated with the undamped problem satisfies the generalized gap condition. We further show the convergence of the discrete solution to the continuous one.

In [19] , we propose a strategy to determine the Dirichlet-to-Neumann (DtN) operator for infinite, lossy and locally perturbed hexagonal periodic media, using a factorization of this operator involving two non local operators. The first one is a DtN type operator and corresponds to a half-space problem, while the second one is a Dirichlet-to-Dirichlet (DtD) type operator related to the symmetry properties of the problem.

In [22] , we generalize to the case of acoustic penetrable scatterers the results derived by Hazard and Ramdani [54] for sound hard scatterers. In particular, we provide a justification of the DORT method in this case and we show that each small inhomogeneity gives rise to $3d+1$ eigenvalues of the time reversal operator. The selective focusing of the corresponding eigenfunctions is also proved.

In [17] , we consider the inverse problem of determining the potential in the dynamical Schrödinger equation on the interval by the measurement on the boundary. We use the Boundary Control Method to recover the spectrum of the problem from the observation at either left or right end points. Using the specificity of the one-dimensional situation we recover the spectral function, reducing the problem to the classical one which could be treated by known methods. We also consider the case where only a finite number ($N$) of eigenvalues are available and we prove the convergence of the reconstruction method as $N$ tends to infinity.

We give some spectral and condition number estimates of the acoustic single-layer operator for low-frequency multiple scattering in dense [15] and dilute [16] media.