Uniqueness and stability results for an inverse spectral problem in a periodic waveguide

Let $\Omega =\omega\times\mathbb R$ where $\omega\subset \mathbb R^2$ be a bounded domain, and $V : \Omega \to\mathbb R$ a bounded potential which is $2\pi$-periodic in the variable $x_{3}\in \mathbb R$. We study the inverse problem consisting in the determination of $V$, through the boundary spectral data of the operator $u\mapsto Au := -\Delta u + Vu$, acting on $L^2(\omega\times(0,2\pi))$, with quasi-periodic and Dirichlet boundary conditions. More precisely we show that if for two potentials $V_{1}$ and $V_{2}$ we denote by $(\lambda_{1,k})_{k}$ and $(\lambda_{2,k})_{k}$ the eigenvalues associated to the operators $A_{1}$ and $A_{2}$ (that is the operator $A$ with $V := V_{1}$ or $V:=V_{2}$), then if $\lambda_{1,k} - \lambda_{2,k} \to 0$ as $k \to \infty$ we have that $V_{1} \equiv V_{2}$, provided one knows also that $\sum_{k\geq 1}\|\psi_{1,k} - \psi_{2,k}\|_{L^2(\partial\omega\times[0,2\pi])}^2 < \infty$, where $\psi_{m,k} := \partial\phi_{m,k}/\partial{\bf n}$. We establish also an optimal Lipschitz stability estimate. The arguments developed here may be applied to other spectral inverse problems, and similar results can be obtained

Imaging by modification: numerical reconstruction of local conductivities from corresponding power density measurements

International audienceWe discuss the reconstruction of the impedance from the local power density. This study is motivated by a new imaging principle which allows to recover interior measurements of the energy density by a non invasive method. We discuss the theoretical feasibility in two dimensions, and propose numerical algorithms to recover the conductivity in two and three dimension. The efficiency of this approach is documented by several numerical simulation

Polynomial stability of an abstract system with local damping

International audienceIn this paper, we consider the polynomial stability for an abstract system of the type utt+Lu+But=0, where L is a self-adjoint operator on a Hilbert space and operator B represents the local damping. By establishing precise estimates on the resolvent, we prove polynomial decay of the corresponding semigroup. The results reveal that the rate of decay depends strongly on the concentration of eigenvalues of operator L and non-degeneration of operator B. Finally, several examples are given as an application of our abstract results

Polynomial Stabilization of Solutions to a Class of Damped Wave Equations

We consider a class of wave equations of the type ∂ tt u + Lu + B∂ t u = 0, with a self-adjoint operator L, and various types of local damping represented by B. By establishing appropriate and raher precise estimates on the resolvent of an associated operator A on the imaginary axis of C, we prove polynomial decay of the semigroup exp(−tA) generated by that operator. We point out that the rate of decay depends strongly on the concentration of eigenvalues and that of the eigenfunctions of the operator L. We give several examples of application of our abstract result, showing in particular that for a rectangle Ω := (0, L 1) × (0, L 2) the decay rate of the energy is different depending on whether the ratio L 2 1 /L 2 2 is rational, or irrational but algebraic