## Section: New Results

### Other topics

#### Spectral analysis of elliptic operators and application to control

Participants : Laurent Bourgeois, Jérôme Le Rousseau, Matthieu Léautaud.

The techniques used in control of evolution PDEs depend very often on some spectral properties of the generator if one uses the semigroup point of view. Such spectral properties rely on powerful functional inequalities such as Carleman estimates. The controlled systems are of parabolic type here.

We are interested in several subjects: (i) the case of non smooth coefficients, (ii) systems of coupled equations, and (iii) discretization issues.

(i) Non smooth coefficients

Carleman estimates have been known for a long time in the case of smooth coefficients in the principal part of the operator. In the case of a regularity as low as Hölder there are some counterexamples. Here, we are interested in coefficients that exhibit jumps across interfaces, for elliptic and parabolic operators. Classical techniques to prove such estimates fail to work in this situation. We manage to prove such Carleman estimates by relying on tools from semi-classical microlocal analysis (e.g. calderéron projectors).

We are also interested in models where diffusion phenomena take place in co-dimension one at the interface. This is joint work with Luc Robbiano (Université de Versailles).

(ii) systems of coupled equations

We address here the controllability of several coupled parabolic equations. This type of problem arises in particular in biology, considering for example coupled reaction and diffusion processes. The difficulty is that we want to control two different evolution problems with only one control force. We hence have to prove spectral estimates (by means of Carleman estimates) with only one observation. Moreover, these phenomena can be non-symmetric. In this situation the generator operator is no more diagonalizable. To treat this Problem, we have to introduce a different functional setting, relying on resolvent estimates for perturbed selfadjoint operators.

We are also interested in the stabilization of coupled hyperbolic equations (such as wave equations). This is joint work with Fatiha Alabau-Boussouira (Université de Metz).

(iii) Discretization issues

Discretization and control properties do not “commute” well in general. Exactly controllable equations can become uncontrollable (even for approximate controllability) after discretization. This happens for the wave equation and also for parabolic equations. For the parabolic case, we prove discrete Carleman estimates that yield uniform controllability results for the lower part of the spectrum in the case of a semi-discretization in space. Uniformity is then with respect to the discretization parameters. The analysis does not rely on filtering out high frequencies. In fact we prove that the high-frequency content of the controlled solution tends to zero exponentially as we refine the discretization. We also address the numerical analysis of the fully discretized problem. This is joint work in collaboration with Florence Hubert and Franck Boyer (Aix-Marseille universities), in the framework of the ANR project CoNum led by Jérôme Le Rousseau

#### Reactive transport in porous media

Participant : Adrien Semin.

This is a joint work with Jean-Baptiste Apoung from University of Paris-Sud, Pascal Havé from IFP, Jean-Gabriel Houot from University of Paris 5 and Michel Kern from team Estime (INRIA Rocquencourt).

We started a work last year about reactive transport in porous media. In this work, we developped a numerical method for coupling transport with chemistry in porous media. Our method is based on a fixed-point algorithm that enables us to coupled different transport and chemistry modules. This work finished at the beginning of this year with a publication in ESAIM Proceedings.

#### Linear elasticity

Participant : Patrick Ciarlet.

A collaboration with Philippe Ciarlet (City Univ. of Hong Kong).

Following a number of theoretical studies (carried out with the help of G. Geymonat and F. Krasucki (CRAS, 2007), C. Amrouche (Analysis and Applications, to appear)), we are considering the numerical approximation of linearized elasticity problems, via the St-Venant approach. Within this framework, one builds conforming finite element subspaces of the stress tensors. In this sense, this approach is different from the mixed approaches, whose aim is to compute approximations of both the displacement and stress.

This problem has been solved for the planar case. The stress tensors are approximated with the help of discrete, piecewise constant, symmetric tensor fields, defined on triangles, and one solves a discrete optimization problem. A paper on this topic has been published in M3AS (2009).

Higher degree approximations and extensions to quadrilaterals and to 3D configurations are currently investigated with Stefan Sauter (Zürich Univ.) and Blandine Vicard (ENSTA).