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 consist of finding a solution with respect to the parameters of the problem, for instance, the propagation of waves with respect to the knowledge of the 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 Systems 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 105, 100, 79, 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 61, 58, 99, 69, 73, 102, 104, 89, 71.
Many other FSIS have been studied as well. Let us mention 91, 76, 72, 63, 49, 68, 50, 67 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: 54, 49, 82, 62, 52).
Without approximations, the only known results 59, 60 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 96. 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 77, 51, 93.

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 78 or Kaltenbacher, Neubauer, and Scherzer 80). 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 44, 70, 75, 101 for the design of feedback controllers), an input (for instance source inverse problems 41, 53, 64) 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 57, specific one-dimensional techniques (like in 45) or observer-based methods as in 85.

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 84, 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 90 or 103). Using observers, we have proposed in 92, 74 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 47, 46.

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 55, 56, 86, 83). 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 simulating the 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 contact 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.

Analysis

In 38, we
extend the theory of single and double layer potentials (well documented for functions with

In 11, 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 25, we study the weak uniqueness and the regularity of the weak solutions of a fluid-structure interaction system. More precisely, we consider the motion of a rigid ball in a viscous incompressible fluid and we assume that the fluid-rigid body system fills the entire space

In 34, we consider a fluid-structure interaction system coupling a viscous fluid governed by the compressible Navier-Stokes equations and a rigid body immersed in the fluid and modeled by Newton's law. In this work, we consider the Navier slip boundary conditions. Our aim is to show the local existence and uniqueness of the strong solution to the corresponding problem. The main step of this work is that we use a Lagrangian change of variables in order to handle the transport equation and to reduce the problem in the initial domain. However, the specificity here is that the Lagrangian coordinates do not coincide with the Eulerian coordinates at the boundaries since we consider slip boundary conditions. Therefore, it brings some extra nonlinear terms in the boundary conditions. The strategy is based on the study of the linearized system with nonhomogeneous boundary conditions and on the Banach fixed-point theorem.

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 this kind of problems in the context of fluid-structure interaction systems. More precisely, we obtained the following results.

In 12, we consider the controllability of an abstract parabolic system by using switching controls. More precisely, we show that under general hypotheses, if a parabolic system is null controllable for any positive time with

In 14, we prove the null-controllability of a fluid-structure system coupling the Navier-Stokes equation for the fluid and a plate equation at the boundary. The control acts on arbitrarily 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 nonlinear system. The proof relies on microlocal argument to handle the pressure terms.

In 32, we consider the controllability of a fluid-structure interaction system, where the fluid is modeled by the Navier-Stokes system and where the structure is a damped beam located on a part of its boundary. The motion of the fluid is bi-dimensional whereas the deformation of the structure is one-dimensional and we use periodic boundary conditions in the horizontal direction. Our result is the local null-controllability of this free-boundary system by using only one scalar control acting on an arbitrary small part of the fluid domain. This improves a previous result obtained by the authors where three scalar controls were needed to achieve the local null-controllability. In order to show the result, we prove the final-state observability of a linear Stokes-beam interaction system in a cylindrical domain. This is done by using a Fourier decomposition, proving Carleman inequalities for the corresponding system for the low-frequency solutions and in the case where the observation domain is a horizontal strip. Then we conclude this observability result by using a Lebeau-Robbiano strategy for the heat equation and a uniform exponential decay for the high-frequency solutions. Then, the result on the nonlinear system can be obtained by a change of variables and a fixed-point argument.

In 17, we study the local null-controllability of a modified Navier-Stokes system which includes 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 we consider a control with one vanishing component. The proof of the result is based on a Carleman estimate where the main difficulty consists of 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 18, 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 with the problem. Our method is based on the flatness approach that consists of 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 19, we study a one-dimensional cross diffusion system with a free boundary modeling physical vapor deposition. Using the flatness approach, we show several results of boundary controllability for this system in spaces of Gevrey class functions. One of the main difficulties consists in the physical constraints on the state and the control. More precisely, the state corresponds to volume fractions of the

In 33, we study the boundary controllability of

In 28, we deal with the controllability properties of a system of

In 27, we consider the controllability of a class of systems of

In 40, we consider a Stackelberg control strategy applied to the Boussinesq system. More precisely, we act on this system with a hierarchy of two controls. The aim of the "leader" control is the null-controllability property whereas the objective of "follower" control is to keep the state close to a given trajectory. By solving first the optimal control problem associated with the follower control, we are led to show the null-controllability property of a system coupling a forward with a backward Boussinesq type systems. Our main result states that for an adequate weighted functional for the optimal control problem, this coupled system is locally null controllable. To show this result, we first study the adjoint system of the linearized system and obtain a weighted observability estimate by combining several Carleman estimates and an adequate decomposition for the heat and the Stokes system.

In 31, we ensure an observability inequality, also known as spectral inequality, within spaces spanned by the first eigenfunctions for a family of one-dimensional degenerate operators

Finally, in 39, we 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 PDEs 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.

In 35, we consider the stabilization of a class of linear evolution systems

In 26, 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 with some numerical simulations that illustrate the stability results and the influence of the delay on the decay rate.

In 23, we show the stabilization by a finite number of controllers of a fluid-structure interaction system where the fluid is modeled by the Navier-Stokes system into a periodical canal and where the structure is an elastic wall localized on top of the fluid domain. The elastic deformation of the structure follows a damped beam equation. We also assume that the fluid can slip on its boundaries and we model this by using the Navier slip boundary conditions. Our result states the local exponential stabilization around a stationary state of strong solutions by using dynamical controllers in order to handle the compatibility conditions at initial time. The proof is based on a change of variables to write the fluid-structure interaction system in a fixed domain and on the stabilization of the linearization of the corresponding system around the stationary state. One of the main difficulties consists in handling the nonlinear terms coming from the change of variables in the boundary conditions.

In 36, 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 exhibits 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.

Optimization

In 16, 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 of 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 the reference domain of the elastic structure is proved under shape perturbations with diffeomorphisms. The shape-derivative is then calculated with the use of the velocity method. This derivative involves the material derivatives of the solution of this fluid–structure interaction problem. The adjoint method is then used to obtain a simplified expression for the shape derivative. The main difficulty for studying the shape sensitivity of this FSI problem lies in the coupling between the Stokes problem written in a Eulerian frame and the linear elasticity problem written in a Lagrangian form.

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 15, in collaboration with C. Timofte, K. Ramdani and R. Bunoiu investigate the homogenization of a diffusion-type problem, for sign-changing conductivities in the case of imperfect interface conditions is considered, by allowing flux jumps across their oscillating interface. The main difficulties of this study are due to the sign-changing coefficients and the appearance of an unsigned surface integral term in the variational formulation. A proof by contradiction (nonstandard in this context) and

Inverse problems

Supervised by A. Munnier and K. 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 88 and 87. 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 87 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. These results are detailed in 5.

In 29, we are interested in an inverse problem set on a tree shaped network where each edge behaves according to the wave equation with potential, external nodes have Dirichlet boundary conditions and internal nodes follow the Kirchoff law. The main goal is the reconstruction of the potential everywhere on the network, from the Neumann boundary measurements at all but one external vertices. Leveraging from the Lipschitz stability of this inverse problem, we aim to provide an efficient reconstruction algorithm based on the use of a specific global Carleman estimate. The proof of the main tool and of the convergence of the algorithm are provided; along with a detailed description of the numerical illustrations given at the end of the article.

In 37, we address the classical inverse problem of recovering the position and shape of obstacles immersed in a planar Stokes flow using boundary measurements. We prove that this problem can be transformed into a shape-from-moments problem to which ad hoc reconstruction methods can be applied. The effectiveness of this approach is confirmed by numerical tests that show significant improvements over those available in the literature to date.

The work in 13 is devoted to the modeling and numerical simulations of a one-dimensional model for localized corrosion phenomena. Localized corrosion involves the dissolution of metal in an aqueous solution of a number of chemical species together with their mass transport by diffusion and migration, and their reactions in solution. From a mathematical point of view, this problem can be identified as a Stefan problem involving a convection-reaction-diffusion system of PDEs with a moving boundary between the aqueous solution and the metal. The unknowns of this system are the concentrations of the chemical species, the electric potential and the position of the free boundary. The dissolution law steering the evolution of the free boundary is given by the nonlinear Butler-Volmer formula. In this work, the mathematical procedure for solving this strongly coupled differential equations system and the numerical development for simulations are presented. A finite-difference ALE (Abritrary Lagrangian Eulerian) scheme is used for the numerical computation of the solutions of this free boundary problem, leading to a nonlinear discrete system which is then solved using a Newton procedure. The numerical simulations obtained are in good agreement with experimental results.

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.

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 exists few tools available to obtain controllability results of these equations in arbitrarily small time 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.

Alessandro Duca was a reviewer for the 10th International Congress on Industrial and Applied Mathematics, ICIAM 2023 Tokyo.

Yannick Privat is a member of the editorial boards of the following journals and book series: AIMS Applied Math. books, Computational and Applied Mathematics, Evolution Equations and Control Theory, Journal of Optimization Theory and Applications, Mathematical Control and Related Fields (MCRF) and Numerical Algebra, Control and Optimization.

SPHINX members were reviewers for several scientific journals on control theory and PDEs. We mention for instance: Annales de l’Institut Henri Poincaré, Analyse non linéaire; Automatica; Communications in Mathematical Sciences; Computational and Applied Mathematics; ESAIM: Control, Optimisation and Calculus of Variations; Evolution Equations and Control Theory; Inverse Problems; Inverse Problems and Imaging; Journal de Mathématiques Pures et Appliquées; Journal of Dynamical and Control Systems; Journal of Mathematical Physics; Journal of Optimization Theory and Applications; Mathematical Control and Related Fields (MCRF); Mathematical Methods in the Applied Sciences; SIAM Journal on Applied Mathematics; SIAM Journal on Control and Optimization; etc.

Except for the researchers of the team (A. Duca, 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 levels (Licence, Master, Engineering school). Many of them also have pedagogical responsibilities. In 2023, several members of the team taught in the Mathematics Master 2 (second year) of Université de Lorraine: Rémi Buffe and Ludovick Gagnon a course on Control theory, David Dos Santos Ferreira a course on spectral theory, Alexandre Munnier a course on integral equations and potential theory, Yannick Privat and Christophe Zhang a course on Optimitazion, and Benjamin Florentin a course on Complex Analysis.

Karim Ramdani is a member (since October 2018) of the Working Group “Publications” of the “Committee for Open Science” of the French Ministry of Higher Education, Research and Innovation.

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.