**RESEARCH SCHOOL / ECOLE DE RECHERCHE **

CIRM-SMF PROGRAM

CIRM-SMF PROGRAM

#### Spectral Theory, Control and Inverse problems

*Théorie Spectrale, Contrôle et Problèmes Inverses*

#### 21 – 25 November 2022

#### Scientific Committee

Comité scientifique

Karine Beauchard (ENS Rennes)

Jean-Marc Bouclet (Institut de Mathématiques de Toulouse)

Enrique Fernández-Cara (University of Sevilla)

Jean-Pierre Raymond (Institut de Mathématiques de Toulouse)

Nicolas Raymond (Université d’Angers)

#### Organizing Committee

Comité d’organisation

Lucie Baudouin (CNRS – LAAS)

Franck Boyer (Institut de Mathématiques de Toulouse)

Jérémi Dardé (Institut de Mathématiques de Toulouse)

Sylvain Ervedoza (CNRS, Institut de Mathématiques de Bordeaux)

Julien Royer (Institut de Mathématiques de Toulouse)

contact: smf-stcip@listes.math.cnrs.fr

**This project aims to present a current view on these very active topics, through courses that will be given by internationally recognized experts, and will be addressed primarily to master, PhD and post-doctoral students, and to young researchers interested in these issues and willing to understand their connections.**

**The issues of spectral theory, control and inverse problems for partial differential equations, although usually studied separately, are in fact intimately related. Thus, many control results for partial differential equations can be reduced to the study of certain spectral properties : among the most spectacular ones, one can mention the spectral inequality of Lebeau – Robbiano – Jerison used in a key way to establish the controllability at zero of the heat equation, or the equivalence between the controllability of conservative systems and appropriate resolvent estimates (Burq – Zworski). Also, many techniques are common to inverse and control problems, such as Carleman estimates, also closely related to Agmon estimates in spectral theory.**

**Ce projet a pour objectif de présenter un point de vue actuel sur ces questions de recherche très actives, à travers des cours qui seront dispensés par des experts internationalement reconnus, et s’adressant avant tout aux étudiants en fin de master, en thèse et en post-doctorat, et aux jeunes chercheurs s’intéressant à certaines de ces questions et désireux de comprendre leurs connexions.**

**Les questions de théorie spectrale, contrôle et problèmes inverses pour des équations aux dérivées partielles, bien qu’étudiées généralement de façon distinctes, sont en fait intimement liées. Ainsi, beaucoup de résultat de contrôle pour des équations aux dérivées partielles peuvent se ramener à l’étude de certaines propriétés spectrales : parmi les plus spectaculaires, on peut mentionner l’inégalité spectrale de Lebeau – Robbiano – Jerison utilisée de façon clé pour établir la contrôlabilité à zéro de l’équation de la chaleur, ou encore l’équivalence entre la contrôlabilité de systèmes conservatifs et des estimées de résolvantes appropriées (Burq – Zworski). De même, de nombreuses techniques sont communes aux problèmes inverses et au contrôle, telles que les inégalités de Carleman, également en lien avec les estimées d’Agmon utilisées en théorie spectrale.**

#### LECTURES

After the introduction of the SSF and the statement of general properties, we will study it for perturbations of electrical and magnetic quantum Hamiltonian. We will be particularly interested in asymptotic properties and connection with resonances.

Vincent Bruneau (Institut de Mathématiques de Bordeaux)

Abstract: The propagation of waves in periodic media has known a regain of interest for many important applications, particularly in optics for micro and nano-technology. Indeed, in some frequency ranges, periodic structures behave as insulators or filters: the corresponding monochromatic waves, also called Floquet modes, cannot propagate in the bulk. The study of these modes is, from a mathematical point of view, related to the spectrum of the underlying operator that presents so-called band structures: the spectrum may contain some forbidden frequency intervals, called band gaps. Using the Floquet theory, we will prove that the spectrum of periodic operators (i.e. operators with periodic coefficients) has this band structure. In 1D, it is well-known that a periodic operator has gaps unless it is constant. By contrast, in 2d and 3d, a periodic operator might or might not have gaps. Even if necessary conditions for the existence of band gaps are not known, in lots of papers sufficient conditions are proposed. We will see that playing with the (high) contrast of the coefficients or the shape of the boundary of the medium, gaps can be created.

In the presence of a boundary, an interface or more generally a lineic perturbation in a periodic medium, energy localization can be created. This is due to some monochromatic waves that propagate along the perturbation and are localized transversely to the perturbation, if the frequency is inside the spectral gap of the periodic medium. Such phenomena can be exploited in quantum, electronic or photonic device design. In the mathematical literature, sufficient conditions on the periodic media and the perturbations have been proposed in order to ensure the existence of such localized and guided waves. We will study various methods used to derive such conditions.

In general, the theoretical results will be illustrated by numerical results. We will describe the associated numerical methods if time permits.

Sonia Fliss (ENSTA PARISTECH)

Abstract: This course is an introduction to the question of unique continuation for differential operators, and some of its applications to control theory. The general question is whether a solution to P u=0 in a domain such that u vanishes on a subdomain has to vanish everywhere. The focus will be put on the wave operator.

During the course, we shall first review the classical local Hörmander theorem which holds under a pseudoconvexity condition. We shall discuss its advantages and weaknesses.

We shall then specialize our analysis to the case of a wave operator with time-independent coefficients. In this setting we shall prove that local unique continuation holds across any non-characteristic surface (as proved by Tataru, Robbiano-Zuily and Hörmander). This local result implies a global unique continuation statement which can be interpreted as a converse to finite propagation speed.

We shall finally discuss applications such as approximate controllability.

Matthieu Léautaud (Université Paris-Sud)

Abstract: We consider coefficient determination problems for hyperbolic partial differential equations. A typical problem of this type is to determine a spatially varying speed of sound in an acoustic wave equation given the Neumann-to-Dirichlet map associated to the equation. These problems are motivated by applications, for example, in geophysical imaging. There are two traditional approaches to solve coefficient determination problems for wave equations: the Boundary Control method originating from [1] and a method using geometric optics. The former applies only to equations with time-independent coefficients, but gives very strong results in this case, whereas the latter allows for time-dependent coefficients [2], but requires certain convexity from the geometric setup. We give an introduction to both the approaches in simple settings and then discuss recent developments of coefficient determination problems.

References

[1] Belishev, M. I. (1987). An approach to multidimensional inverse problems for the wave equation. Doklady Akademii Nauk SSSR, 297(3), 524–527.

[2] Stefanov, P. D. (1989). Uniqueness of the multi-dimensional inverse scattering problem for time dependent potentials. Math Z, 201(4), 541–559.

Lauri Oksanen (University of Helsinki)

**A poster session**will also be organized to allow the participants, especially the youngest ones, to present their work and their research topics. The posters will also be presented during oral sessions, to stimulate discussions and exchanges between the participants. The deadline to participate to the poster session is September 30, 2022.

Registration is mandatory to take part to the school. If accepted, your participation is free of charge. Your accommodation will be at the facilities of the CIRM. If you can take in charge your accommodation at the CIRM facilities (see https://www.fr-cirm-math.fr/tarifs.html), please indicate it by sending an email to the organizers. This will allow to fund other participants to the school.

Please note that the number of participants is limited, and participants taking part to the poster session will be confirmed primarily. In particular, if you have registered to the school and have to cancel your participation, we kindly ask you to inform us as soon as possible to allow other participants to join.

Funding request: Some partial fundings can be given for the travel expenses: if so, please register before September 30, 2022. Each applicant should give a short curriculum-vitae with the application. Priority will be given to young participants, in particular those proposing a poster.

This school is based on the conference https://summer-walk.sciencesconf.org/, originally planned on June 2020 and cancelled due to the COVID crisis.