Transmutation techniques and observability for time-discrete approximation schemes of conservative systems

International audienceIn this article, we consider abstract linear conservative systems and their time-discrete counterparts. Our main result is a representation formula expressing solutions of the continuous model through the solution of the corresponding time-discrete one. As an application, we show how observability properties for the time continuous model yield uniform (with respect to the time-step) observability results for its time-discrete approximation counterparts, provided the initial data are suitably filtered. The main output of this approach is the estimate on the time under which we can guarantee uniform observability for the time-discrete models. Besides, using a reverse representation formula, we also prove that this estimate on the time of uniform observability for the time-discrete models is sharp. We then conclude with some general comments and open problems

Convergence of an inverse problem for discrete wave equations

International audienceIt is by now well-known that one can recover a potential in the wave equation from the knowledge of the initial waves, the boundary data and the flux on a part of the boundary satisfying the Gamma-conditions of J.-L. Lions. We are interested in proving that trying to fit the discrete fluxes, given by discrete approximations of the wave equation, with the continuous one, one recovers, at the limit, the potential of the continuous model. In order to do that, we shall develop a Lax-type argument, usually used for convergence results of numerical schemes, which states that consistency and uniform stability imply convergence. In our case, the most difficult part of the analysis is the one corresponding to the uniform stability, that we shall prove using new uniform discrete Carleman estimates, where uniform means with respect to the discretization parameter. We shall then deduce a convergence result for the discrete inverse problems. Our analysis will be restricted to the 1-d case for space semi-discrete wave equations discretized on a uniform mesh using a finite differences approach

On the reachable set for the one-dimensional heat equation

International audienceThe goal of this article is to provide a description of the reachable set of the one-dimensional heat equation, set on the spatial domain x ∈ (−L, L) with Dirichlet boundary controls acting at both boundaries. Namely, in that case, we shall prove that for any L0 > L any function which can be extended analytically on the square {x + iy, |x| + |y| ≤ L0} belongs to the reachable set. This result is nearly sharp as one can prove that any function which belongs to the reachable set can be extended analytically on the square {x + iy, |x| + |y| < L}. Our method is based on a Carleman type estimate and on Cauchy's formula for holomorphic functions

Local controllability to trajectories for non-homogeneous incompressible Navier-Stokes equations

International audienceThe goal of this article is to show a local exact controllability to smooth (C2) trajectories for the 2-d density dependent incompressible Navier-Stokes equations. Our controllability result requires some geometric condition on the ow of the target trajectory, which is remanent from the transport equation satisfied by the density. The proof of this result uses a fixed point argument in suitable spaces adapted to a Carleman weight function that follows the ow of the target trajectory. Our result requires the proof of new Carleman estimates for heat and Stokes equations

Local exact controllability for the 2 and 3-d compressible Navier-Stokes equations

International audienceThe goal of this article is to present a local exact controllability result for the 2 and 3-dimensional compressible Navier-Stokes equations on a constant target trajectory when the controls act on the whole boundary. Our study is then based on the observability of the adjoint system of some linearized version of the system, which is analyzed thanks to a subsystem for which the coupling terms are somewhat weaker. In this step, we strongly use Carleman estimates in negative Sobolev spaces

Convergent algorithm based on Carleman estimates for the recovery of a potential in the wave equation.

International audienceThis article develops the numerical and theoretical study of the reconstruction algorithm of a potential in a wave equation from boundary measurements, using a cost functional built on weighted energy terms coming from a Carleman estimate. More precisely, this inverse problem for the wave equation consists in the determination of an unknown time-independent potential from a single measurement of the Neumann derivative of the solution on a part of the boundary. While its uniqueness and stability properties are already well known and studied, a constructive and globally convergent algorithm based on Carleman estimates for the wave operator was recently proposed in [BdBE13]. However, the numerical implementation of this strategy still presents several challenges, that we propose to address here

Global Carleman estimates for waves and applications

In this article, we extensively develop Carleman estimates for the wave equation and give some applications. We focus on the case of an observation of the flux on a part of the boundary satisfying the Gamma conditions of Lions. We will then consider two applications. The first one deals with the exact controllability problem for the wave equation with potential. Following the duality method proposed by Fursikov and Imanuvilov in the context of parabolic equations, we propose a constructive method to derive controls that weakly depend on the potentials. The second application concerns an inverse problem for the waves that consists in recovering an unknown time-independent potential from a single measurement of the flux. In that context, our approach does not yield any new stability result, but proposes a constructive algorithm to rebuild the potential. In both cases, the main idea is to introduce weighted functionals that contain the Carleman weights and then to take advantage of the freedom on the Carleman parameters to limit the influences of the potentials.Comment: 31 page