In this project, we investigate theoretical and numerical mathematical issues concerning heterogeneous physical systems. The heterogeneities we consider result from the fact that the studied systems involve subsystems of different physical nature. In this wide class of problems, we study two types of systems: fluid-structure interaction systems (FSIS) and complex wave systems (CWS). In both situations, one has to develop specific methods to take the coupling between the subsystems into account.

(FSIS) Fluid-structure interaction systems appear in many applications: medicine (motion of the blood in veins and arteries), biology (animal locomotion in a fluid, such as swimming fishes or flapping birds but also locomotion of microorganisms, such as amoebas), civil engineering (design of bridges or any structure exposed to the wind or the flow of a river), naval architecture (design of boats and submarines, researching into new propulsion systems for underwater vehicles by imitating the locomotion of aquatic animals).
FSIS can be studied by modeling their motions through Partial Differential Equations (PDE) and/or Ordinary Differential Equations (ODE), as is classical in fluid mechanics or in solid mechanics.
This leads to the study of difficult nonlinear free boundary problems which have constituted a rich and active domain of research over the last decades.

(CWS) Complex wave systems are involved in a large number of applications in several areas of science and engineering: medicine (breast cancer detection, kidney stone destruction, osteoporosis diagnosis, etc.), telecommunications (in urban or submarine environments, optical fibers, etc.), aeronautics (target detection, aircraft noise reduction, etc.) and, in the longer term, quantum supercomputers.
Direct problems, that is finding a solution with respect to parameters of the problem, for instance the propagation of waves with respect to the knowledge of speed of propagation of the medium, most theoretical issues are now widely understood. However, substantial efforts remain to be undertaken concerning the simulation of wave propagation in complex media. Such situations include heterogeneous media with strong local variations of the physical properties (high frequency scattering, multiple scattering media) or quantum fluids (Bose-Einstein condensates). In the first case for instance, the numerical simulation of such direct problems is a hard task, as it generally requires solving ill-conditioned possibly indefinite large size problems, following from space or space-time discretizations of linear or nonlinear evolution PDE set on unbounded domains. Inverse problems are the converse problem of the direct problems, as they aim to find properties of the direct problem, for instance the speed of propagation in a medium, with respect to the solution or a partial observation of the solution. These problems are often ill-posed and many questions are open at both the theoretical (identifiability, stability and robustness, etc.) and practical (reconstruction methods, approximation and convergence analysis, numerical algorithms, etc.) levels.

Fluid-Structure Interaction System are present in many physical problems and applications. Their study involves solving several challenging mathematical problems:

In order to control such FSIS, one has first to analyze the corresponding system of PDE. The oldest works on FSIS go back to the pioneering contributions of Thomson, Tait and Kirchhoff in the 19th century and Lamb in the 20th century, who considered simplified models (potential fluid or Stokes system). The first mathematical studies in the case of a viscous incompressible fluid modeled by the Navier-Stokes system and a rigid body whose dynamics is modeled by Newton's laws appeared much later 114, 109, 86, and almost all mathematical results on such FSIS have been obtained in the last twenty years.

The most studied FSIS is the problem modeling a rigid body moving in a viscous incompressible fluid
( 67, 64, 107, 75, 80, 111, 113, 97, 77).
Many other FSIS have been studied as well. Let us mention 99, 83, 79, 69, 54, 74, 55, 73 for different fluids.
The case of deformable structures has also been considered, either for a fluid inside a moving structure (e.g. blood motion in arteries)
or for a moving deformable structure immersed in a fluid (e.g. fish locomotion).
The obtained coupled FSIS is a complex system and its study raises several difficulties. The main one comes from the fact that we gather two systems of different nature. Some studies have been performed for approximations of this system: 60, 54, 89, 68, 57).
Without approximations, the only known results 65, 66 were obtained with very strong assumptions on the regularity of the initial data.
Such assumptions are not satisfactory but seem inherent to this coupling between two systems of different natures. In order to study self-propelled motions of structures in a fluid, like fish locomotion, one can assume that the deformation of the structure is prescribed and known, whereas its displacement remains unknown (104). This permits to start the mathematical study of a challenging problem: understanding the locomotion mechanism of aquatic animals.
This is related to control or stabilization problems for FSIS. Some first results in this direction were obtained in 84, 56, 101.

The area of inverse problems covers a large class of theoretical and practical issues which are important in many applications (see for instance the books of Isakov 85 or Kaltenbacher, Neubauer, and Scherzer 87). Roughly speaking, an inverse problem is a problem where one attempts to recover an unknown property of a given system from its response to an external probing signal. For systems described by evolution PDE, one can be interested in the reconstruction from partial measurements of the state (initial, final or current), the inputs (a source term, for instance) or the parameters of the model (a physical coefficient for example). For stationary or periodic problems (i.e. problems where the time dependency is given), one can be interested in determining from boundary data a local heterogeneity (shape of an obstacle, value of a physical coefficient describing the medium, etc.). Such inverse problems are known to be generally ill-posed and their study raises the following questions:

We can split our research in inverse problems into two classes which both appear in FSIS and CWS:

Identification for evolution PDE.

Driven by applications, the identification problem for systems of infinite dimension described by evolution PDE has seen in the last three decades a fast and significant growth. The unknown to be recovered can be the (initial/final) state (e.g. state estimation problems 49, 76, 82, 110 for the design of feedback controllers), an input (for instance source inverse problems 46, 58, 70) or a parameter of the system. These problems are generally ill-posed and many regularization approaches have been developed. Among the different methods used for identification, let us mention optimization techniques ( 63), specific one-dimensional techniques (like in 50) or observer-based methods as in 93.

In the last few years, we have developed observers to solve initial data inverse problems for a class of linear systems of infinite dimension. Let us recall that observers, or Luenberger observers 91, have been introduced in automatic control theory to estimate the state of a dynamical system of finite dimension from the knowledge of an output (for more references, see for instance 98 or 112). Using observers, we have proposed in 100, 81 an iterative algorithm to reconstruct initial data from partial measurements for some evolution equations. We are deepening our activities in this direction by considering more general operators or more general sources and the reconstruction of coefficients for the wave equation. In connection with this problem, we study the stability in the determination of these coefficients. To achieve this, we use geometrical optics, which is a classical albeit powerful tool to obtain quantitative stability estimates on some inverse problems with a geometrical background, see for instance 52, 51.

Geometric inverse problems.

We investigate some geometric inverse problems that appear naturally in many applications, like medical imaging and non destructive testing. A typical problem we have in mind is the following: given a domain

where

Within the team, we have developed in the last few years numerical codes for the simulation of FSIS and CWS. We plan to continue our efforts in this direction.

Below, we explain in detail the corresponding scientific program.

Some companies aim at building biomimetic robots that can swim in an aquarium, as toys but also for medical purposes. An objective of SPHINX is to model and to analyze several models of these robotic swimmers. For the moment, we focus on the motion of a nanorobot. In that case, the size of the swimmers leads us to neglect the inertia forces and to only consider the viscosity effects. Such nanorobots could be used for medical purposes to deliver some medicine or perform small surgical operations. In order to get a better understanding of such robotic swimmers, we have obtained control results via shape changes and we have developed simulation tools (see 61, 62, 94, 90). Among all the important issues, we aim to consider the following ones:

The main tools for this investigation are the 3D codes that we have developed for simulation of fish in a viscous incompressible fluid (SUSHI3D) or in an inviscid incompressible fluid (SOLEIL).

We will develop robust and efficient solvers for problems arising in aeronautics (or aerospace) like electromagnetic compatibility and acoustic problems related to noise reduction in an aircraft. Our interest for these issues is motivated by our close contacts with companies like Airbus or “Thales Systèmes Aéroportés”. We will propose new applications needed by these partners and assist them in integrating these new scientific developments in their home-made solvers. In particular, in collaboration with C. Geuzaine (Université de Liège), we are building a freely available parallel solver based on Domain Decomposition Methods that can handle complex engineering simulations, in terms of geometry, discretization methods as well as physics problems, see here.

One of the members of the assessment panel that evaluated our team in 2021 wrote an email to the team leader wondering how it is possible that almost a year after he sent his report,he had still not received his honorarium. The problems due to the deployment of EKSAE (the new Inria Information System for finance and human resources) are detrimental to the people outside the institute who have agreed to collaborate with us. They also put us in a situation that is more than embarrassing, and damages the credibility of our institute. Finally, they are also detrimental to the staff of the institute by making them work in unacceptable conditions.

Analysis of fluid mechanics

In 18, we study a bi-dimensional viscous incompressible fluid in interaction with a beam located at its boundary. We show the existence of strong solutions for this fluid-structure interaction system, extending a previous result where they supposed that the initial deformation of the beam was small. The main point of the proof consists in the study of the linearized system and in particular in proving that the corresponding semigroup is of Gevrey class.

In 17, we consider a viscous incompressible fluid interacting with an elastic structure located on a part of its boundary. The fluid motion is modeled by the bi-dimensional Navier-Stokes system and the structure follows the linear wave equation in dimension 1 in space. The aim of the article is to study the linearized system coupling the Stokes system with a wave equation and to show that the corresponding semigroup is analytic. In particular the linear system satisfies a maximal regularity property that allows us to deduce the existence and uniqueness of strong solutions for the nonlinear system. This result can be compared to the case where the elastic structure is a beam equation (18) for which the corresponding semigroup is only of Gevrey class.

Control

Controlling coupled systems is a complex issue depending on the coupling conditions and the equations themselves. Our team has a strong expertise to tackle these kind of problems in the context of fluid-structure interaction systems. More precisely, we obtained the following results.

In 20, we prove an inequality of Hölder type traducing the unique continuation property at one time for the heat equation with a potential and Neumann boundary condition. The main feature of the proof is to overcome the propagation of smallness by a global approach using a refined parabolic frequency function method. It relies on a Carleman commutator estimate to obtain the logarithmic convexity property of the frequency function.

In 21, we are interested in the controllability of a fluid-structure interaction system where the fluid is viscous and incompressible and where the structure is elastic and located on a part of the boundary of the fluid's domain. In this article, we simplify this system by considering a linearization and by replacing the wave/plate equation for the structure by a heat equation. We show that the corresponding system coupling the Stokes equations with a heat equation at its boundary is null-controllable. The proof is based on Carleman estimates and interpolation inequalities. One of the Carleman estimates corresponds to the case of Ventcel boundary conditions. This work can be seen as a first step to handle the real system where the structure is modeled by the wave or the plate equation.

In 33, we prove the null-controllability of the non-simplified fluid-structure system (as opposed to 21), that is, a system coupling the Navier-Stokes equation for the fluid and a plate equation at the boundary. The control acts on arbitrary small subsets of the fluid domain and in a small subset of the vibrating boundary. By proving a proper observability inequality, we obtain the local controllability for the non-linear system. The proof relies on microlocal argument to handle the pressure terms.

In 25, an optimal control problem for the Navier–Stokes system with Navier slip boundary conditions is considered.
Denoting by

In 35, we study the local null controllability of a modified Navier-Stokes system where they include nonlocal spatial terms. We generalize a previous work where the nonlocal spatial term is given by the linearization of a Ladyzhenskaya model for a viscous incompressible fluid. Here the nonlocal spatial term is more complicated and they consider a control with one vanishing component. The proof of the result is based on a Carleman estimate where the main difficulty consists in handling the nonlocal spatial terms. One of the key points is a particular decomposition of the solution of the adjoint system that allows us to overcome regularity issues. With a similar approach, we also show the existence of insensitizing controls for the same system.

In 36, we show the boundary controllability to stationary states of the Stefan problem with two phases and in one dimension in the space variable. For an initial condition that is a stationary state and for a time of control large enough, we also obtain the controllability to stationary states together with the sign constraints associated to the problem. Our method is based on the flatness approach that consists in writing the solution and the controls through two outputs and their derivatives. We construct these outputs as Gevrey functions of order σ so that their solution and controls are also in a Gevrey class.

In 39, we consider a nonlinear system of two parabolic equations, with a distributed control in the first equation and an odd coupling term in the second one. We prove that the nonlinear system is small time locally null-controllable. The main difficulty is that the linearized system is not null-controllable. To overcome this obstacle, we extend in a nonlinear setting the strategy introduced in a previous article that consists in constructing odd controls for the linear heat equation. The proof relies on three main steps. First, we obtain from the classical

Finally, in 43, C. Zhang and co-authors consider the internal control of linear parabolic equations through on-off shape controls with a prescribed maximal measure. They establish small-time approximate controllability towards all possible final states allowed by the comparison principle with non negative controls and manage to build controls with constant amplitude.

Stabilization

Stabilization of infinite dimensional systems governed by PDE is a challenging problem. In our team, we have investigated this issue for different kinds of systems (fluid systems and wave systems) using different techniques.

The work 24 is devoted to the stabilization of parabolic systems with a finite-dimensional control subjected to a constant delay. Our main result shows that the Fattorini-Hautus criterion yields the existence of such a feedback control, as in the case of stabilization without delay. The proof consists in splitting the system into a finite dimensional unstable part and a stable infinite-dimensional part and to apply the Artstein transformation on the finite-dimensional system to remove the delay in the control. Using our abstract result, we can prove new results for the stabilization of parabolic systems with constant delay.

The aim of 31 is to study the asymptotic stability of the nonlinear Korteweg-de Vries equation in the presence of a delayed term. We first consider the case where the weight of the term with delay is smaller than the weight of the term without delay and we prove a semiglobal stability result for any lengths. Secondly we study the case where the support of the term without delay is not included in the support of the term with delay. In that case, we give a local exponential stability result if the weight of the delayed term is small enough. We illustrate these results by some numerical simulations.

In 42, we consider the Korteweg-de Vries equation with time-dependent delay on the boundary or internal feedbacks. Under some assumptions on the time- dependent delay, on the weights of the feedbacks and on the length of the spatial domain, we prove the exponential stability results, using appropriate Lyapunov functionals. We finish by some numerical simulations that illustrate the stability results and the influence of the delay on the decay rate.

In 32, we consider a wave equation with a structural damping coupled with an undamped wave equation located at its boundary. We prove that, due to the coupling, the full system is parabolic. In order to show that the underlying operator generates an analytical semigroup, we study in particular the effect of the damping of the "interior" wave equation on the "boundary" wave equation and show that it generates a structural damping.

In 38, we prove the rapid stabilization of the linearized water waves equation with the Fredholm backstepping method. This result is achieved by overcoming an important theoretical threshold imposed by the classical methodology, namely, the quadratically close criterion. Indeed, the spatial operator of the linearized water waves exhibit an insufficient growth of the eigenvalues and the quadratically close criterion is not true in this case. We introduce the duality compactness method for general skew-adjoint operators to circumvent this difficulty. In turn, we prove the existence of a Fredholm backstepping transformation for a wide range of equations, opening the path to an abstract framework for this widely used method.

In 37, I. Djebour investigates the stabilization of a fluid-structure interaction system composed by a three-dimensional viscous incompressible fluid and an elastic plate located on the upper part of the fluid boundary. The main result of this paper is the feedback stabilization of the strong solutions of the corresponding system around a stationary state for any exponential decay rate by means of a time delayed control localized on the fixed fluid boundary.

Optimization

We have also considered optimization issues for fluid-structure interaction systems.

J.F. Scheid, V. Calisti and I. Lucardesi study an optimal shape problem for an elastic structure immersed in a viscous incompressible fluid. They aim to establish the existence of an optimal elastic domain associated with an energy-type functional for a Stokes-Elasticity system. They want to find an optimal reference domain (the domain before deformation) for the elasticity problem that minimizes an energy-type functional. This problem is concerned with 2D geometry and is an extension of 108 for a 1D problem. The optimal domain is searched for in a class of admissible open sets defined with a diffeomorphism of a given domain. The main difficulty lies in the coupling between the Stokes problem written in a eulerian frame and the linear elasticity problem written in a lagrangian form. The shape derivative of an energy-type functional has been formally obtained. This will allow us to numerically determine an optimal elastic domain which minimizes the energy-type functional under consideration. The rigorous proof of the derivability of the energy-type functional with respect to the domain is still in progress.

The article 92 is devoted to the mathematical analysis of a fluid-structure interaction system where the fluid is compressible and heat conducting and where the structure is deformable and located on a part of the boundary of the fluid domain. The fluid motion is modeled by the compressible Navier-Stokes-Fourier system and the structure displacement is described by a structurally damped plate equation. Our main results are the existence of strong solutions in an

In 17, we consider a viscous incompressible fluid interacting with an elastic structure located on a part of its boundary. The fluid motion is modeled by the bi-dimensional Navier-Stokes system and the structure follows the linear wave equation in dimension 1 in space. Our aim is to study the linearized system coupling the Stokes system with a wave equation and to show that the corresponding semigroup is analytic. In particular the linear system satisfies a maximal regularity property that allows us to deduce the existence and uniqueness of strong solutions for the nonlinear system. This result can be compared to the case where the elastic structure is a beam equation for which the corresponding semigroup is only of Gevrey class.

Direct problems

Negative materials are artificially structured composite materials (also known as metamaterials), whose dielectric permittivity and magnetic permeability are simultaneously negative in some frequency ranges. K. Ramdani continued his collaboration with R. Bunoiu on the homogenization of composite materials involving both positive and negative materials. Due to the sign-changing coefficients in the equations, classical homogenization theory fails, since it is based on uniform energy estimates which are known only for positive (more precisely constant sign) coefficients.

In 23, in collaboration with C. Timofte, the authors investigate the homogenization of a diffusion-type problem, for sign-changing conductivities with extreme contrasts (of order

Inverse problems

Supervised by Alexandre Munnier and Karim Ramdani, the PhD of Anthony Gerber-Roth is devoted to the investigation of some geometric inverse problems, and can be seen as a continuation of the work initiated by the two supervisors in 96 and 95. In these papers, the authors addressed a particular case of Calderòn's inverse problem in dimension two, namely the case of a homogeneous background containing a finite number of cavities (i.e. heterogeneities of infinitely high conductivities). The first contribution of Anthony Gerber-Roth was to apply the method proposed in 95 to tackle a two-dimensional inverse gravimetric problem. The strong connection with the important notion of quadrature domains in this context has been highlighted. An efficient reconstruction algorithm has been proposed (and rigorously justified in some cases) for this geometric inverse problem. This work, which is still in progress, has been presented to the conference WAVES 2022, the 15th International Conference on Mathematical and Numerical Aspects of Wave Propagation.

In 34, an optimal shape problem for a general functional depending on the solution of a bidimensional Fluid-Structure Interaction problem (FSI) is studied. The system is composed by a coupling stationary Stokes-Elasticity sub-system for modeling the deformation of an elastic structure immersed in a viscous fluid. The differentiability with respect to reference elastic domain variations is proved under shape perturbations with diffeomorphisms. The shape-derivative is then calculated. The main difficulty for studying the shape sensitivity, lies in the coupling between the Stokes problem written in a Eulerian frame and the linear elasticity problem written in a Lagrangian form.

The work in 19 is devoted to the long time behaviour of the solution of a one dimensional Stefan problem arising from corrosion theory. It is rigorously proved that under rather general hypotheses on the initial data, the solution of this free boundary problem converges to a self-similar profile as the time

The paper 13 is devoted to the numerical computation of fractional linear systems. The proposed approach is based on an efficient computation of Cauchy integrals allowing to estimate the real power of a (sparse) matrix A. A first preconditioner M is used to reduce the length of the Cauchy integral contour enclosing the spectrum of M A, hence allowing for a large reduction of the number of quadrature nodes along the integral contour. Next, ILU-factorizations are used to efficiently solve the linear systems involved in the computation of approximate Cauchy integrals. Numerical examples related to stationary (deterministic or stochastic) fractional Poisson-like equations are finally proposed to illustrate the methodology.

Several contributions have been devoted to the numerical approximation of problems set in unbounded domains, appearing in acoustics, electromagnetics, quantum field theory, fluid mechanics and continuum mechanics. More precisely, absorbing boundary conditions (ABC) have been used to solve acoustic scattering problems 28, the linearized Green-Naghdi system in fluid dynamics 29 and a mechanical problem from peridynamics 30. Perfectly matched players (PML) have been proposed for the numerical solution of nonlinear Klein-Gordon equations 15. In electromagnetics, coupling between high-order finite elements and boundary elements has been used to tackle time-harmonic scattering by inhomogeneous objects 16. In acoustics, other methods have been also proposed: integral equations methods for 3D high-frequency acoustic scattering problems 27 and on-surface radiation conditions (OSRC) combined with isogeometric (IGA) finite elements 11. Finally, the acoustic scattering problem by small-amplitude boundary deformations has been studied in 26 using a multi-harmonic finite element method.

In collaboration with Emmanuel Lorin, Xavier Antoine investigated numerical methods to tackle fractional equations, either in the PDE case 12, 13 or for algebraic linear systems 14.

The three industrial PhD theses of I. Badia, D. Gasperini and P. Marchner have been defended in 2022.

This project is divided into three research axes, all in the field of control theory and within the field of expertise of the Sphinx project team. Although covering several fields of applications, the problems studied here can be handled with similar mathematical techniques.

The first axis consists in improving a network transport model of virus spread by mosquitoes such as Zika, Dengue or Chikungunya. The objective is to introduce time-delay terms into the model to take into account delays such as incubation time or reaction time of health authorities. The study of the controllability of the model will then be carried out in order to optimize the reaction time as well as the coverage of the population in the event of an outbreak.

The second axis concerns the controllability of waves in a heterogeneous environment. These media are characterized by discontinuous propagation speed at the interface between two media, leading to refraction phenomena according to Snell's law. Only a few controllability results are known in restricted geometric settings, the last result being due to the Inria principal investigator. Examples of applications of the controllability of these models range from seismic exploration to the clearance of anti-personnel mines.

Finally, the last axis aims to study the controllability of nonlinear dispersive equations. These equations are distinguished by a decrease of the solutions due to the different propagation speed of each frequency. There only exist few tools available to obtain arbitrarily small time controllability results of these equations and many important questions remain open. These equations can be used to model, for example, the propagation of waves in shallow waters as well as the propagation of signals in an optical fiber.

Except L. Gagnon, K. Ramdani, T. Takahashi and J.-C. Vivalda, SPHINX members have teaching obligations at “Université de Lorraine” and are teaching at least 192 hours each year. They teach mathematics at different level (Licence, Master, Engineering school). Many of them have pedagogical responsibilities.

Karim Ramdani gave several talks to review the most recent changes in scientific publishing, especially concerning the emergence of the dangerous author-pays model of open science.