RESEARCH CENTRE

Inria Centre
at Université de Lorraine

IN PARTNERSHIP WITH:

CNRS, Université de Lorraine

2023
ACTIVITY REPORT

Project-Team

SPHINX

Heterogeneous Systems: Inverse Problems,
Control and Stabilization, Simulation

IN COLLABORATION WITH: Institut Elie Cartan de Lorraine (IECL)

DOMAIN

Applied Mathematics, Computation and
Simulation

THEME

Optimization and control of dynamic
systems

Contents

Project-Team SPHINX

1 Team members, visitors, external collaborators

2 Overall objectives

3 Research program

3.1 Analysis, control, stabilization and optimization of heterogeneous systems . . . . . . . . . .
3.2 Inverse problems for heterogeneous systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Numerical analysis and simulation of heterogeneous systems . . . . . . . . . . . . . . . . . .

4 Application domains

4.1 Robotic swimmers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Aeronautics

5 Highlights of the year

5.1 Awards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

3

3
3
4
5

6
6
6

7
7

6 New results

7
6.1 Analysis, control, stabilization and optimization of heterogeneous systems . . . . . . . . . .
7
6.2 Direct and inverse problems for heterogeneous systems . . . . . . . . . . . . . . . . . . . . . 10
6.3 Numerical analysis and simulation of heterogeneous systems . . . . . . . . . . . . . . . . . . 11

7 Partnerships and cooperations

11
7.1 International initiatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
7.1.1 Inria associate team not involved in an IIL or an international program . . . . . . . . 11
. . . . . . . . . . . . . . . . . . . . . . . . . . . 12
7.1.2 STIC/MATH/CLIMAT AmSud projects
7.2 National initiatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

8 Dissemination

13
8.1 Promoting scientific activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
8.1.1 Scientific events: organisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
8.1.2 Scientific events: selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
8.1.3 Journal
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
8.1.4 Invited talks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
8.1.5 Research administration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
8.2 Teaching - Supervision - Juries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
8.2.1 Teaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
8.2.2 Supervision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
8.2.3 Juries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
8.3 Popularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
8.3.1 Internal or external Inria responsibilities . . . . . . . . . . . . . . . . . . . . . . . . . . 16
8.3.2 Interventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

9 Scientific production

16
9.1 Major publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
9.2 Publications of the year
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
9.3 Cited publications

Project SPHINX

1

Project-Team SPHINX

Creation of the Project-Team: 2016 May 01

Keywords

Computer sciences and digital sciences

A6.1. – Methods in mathematical modeling

A6.1.1. – Continuous Modeling (PDE, ODE)

A6.2.1. – Numerical analysis of PDE and ODE

A6.2.6. – Optimization

A6.3.1. – Inverse problems

A6.3.2. – Data assimilation

A6.4. – Automatic control

A6.4.1. – Deterministic control

A6.4.3. – Observability and Controlability

A6.4.4. – Stability and Stabilization

A6.5. – Mathematical modeling for physical sciences

A6.5.1. – Solid mechanics

A6.5.2. – Fluid mechanics

A6.5.4. – Waves

A6.5.5. – Chemistry

Other research topics and application domains

B2. – Health

B2.6. – Biological and medical imaging

B5.3. – Nanotechnology

B5.11. – Quantum systems

B9. – Society and Knowledge

B9.5. – Sciences

B9.5.2. – Mathematics

B9.5.3. – Physics

B9.5.4. – Chemistry

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Inria Annual Report 2023

1 Team members, visitors, external collaborators

Research Scientists

• Karim Ramdani [Team leader, INRIA, Senior Researcher, HDR]

• Alessandro Duca [INRIA, ISFP]

• Ludovick Gagnon [INRIA, Researcher, until Nov 2023]

• Takéo Takahashi [INRIA, Senior Researcher, HDR]

• Jean-Claude Vivalda [INRIA, Senior Researcher, until Apr 2023, HDR]

Faculty Members

• Rémi Buffe [UL, Associate Professor]

• David Dos Santos Ferreira [UL, Associate Professor, HDR]

• Julien Lequeurre [UL, Associate Professor]

• Alexandre Munnier [UL, Associate Professor]

• Yannick Privat [UL, Professor, from Sep 2023, HDR]

• Jean-François Scheid [UL, Associate Professor, HDR]

• Julie Valein [UL, Associate Professor]

Post-Doctoral Fellows

• Baparou Danhane [INRIA, Post-Doctoral Fellow, until Jun 2023]

• Imene Aicha Djebour [INRIA, Post-Doctoral Fellow, until Aug 2023]

• Christophe Zhang [INRIA, from Sep 2023]

• Christophe Zhang [INRIA, until Aug 2023]

PhD Students

• Mabrouk Ben Jaba [INRIA, from Oct 2023]

• Blaise Colle [INRIA]

• Benjamin Florentin [INRIA]

• Anthony Gerber-Roth [UL]

Administrative Assistant

• Angélique Marc [INRIA, from Sep 2023]

Project SPHINX

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2 Overall objectives

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 heterogen-
eous 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 nu-
merical 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.

3 Research program

3.1 Analysis, control, stabilization and optimization of heterogeneous systems

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

• Nonlinearity: One has to deal with a system of nonlinear PDE such as the Navier-Stokes or the

Euler systems;

• Coupling: The corresponding equations couple two systems of different types and the methods
associated with each system need to be suitably combined to successfully solve the full problem;

• Coordinates: The equations for the structure are classically written with Lagrangian coordinates

whereas the equations for the fluid are written with Eulerian coordinates;

• Free boundary: The fluid domain is moving and its motion depends on the motion of the structure.
The fluid domain is thus an unknown of the problem and one has to solve a free boundary problem.

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

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Inria Annual Report 2023

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, 68, 73, 102, 104, 89, 71]. Many other FSIS have been studied as well. Let us mention [91,
76, 72, 62, 50, 67, 49, 69] 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, 50, 82, 63, 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].

3.2 Inverse problems for heterogeneous systems

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:

• Uniqueness. The question here is to know whether the measurements uniquely determine the
unknown quantity to be recovered. This theoretical issue is a preliminary step in the study of any
inverse problem and can be a hard task.

• Stability. When uniqueness is ensured, the question of stability, which is closely related to sens-
itivity, deserves special attention. Stability estimates provide an upper bound for the parameter
error given some uncertainty on the data. This issue is closely related to the so-called observability
inequality in systems theory.

• Reconstruction. Inverse problems are usually ill-posed, one needs to develop specific reconstruc-
tion algorithms which are robust to noise, disturbances and discretization. A wide class of methods
is based on optimization techniques.

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

1. 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, 74, 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 regulariza-
tion 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].

Project SPHINX

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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, 75] 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].

2. 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 Ω containing an (unknown) local heterogeneity ω, we consider the boundary value
problem of the form






Lu = 0,
u = f ,
Bu = 0,

(Ω \ ω)
(∂Ω)
(∂ω)

where L is a given partial differential operator describing the physical phenomenon under consider-
ation (typically a second order differential operator), B the (possibly unknown) operator describing
the boundary condition on the boundary of the heterogeneity and f the exterior source used to
probe the medium. The question is then to recover the shape of ω and/or the boundary operator
B from some measurement on the outer boundary ∂Ω. This setting includes in particular inverse
scattering problems in acoustics and electromagnetics (in this case Ω is the whole space and the
data are far field measurements) and the inverse problem of detecting solids moving in a fluid. It
also includes, with slight modifications, more general situations of incomplete data (i.e., measure-
ments on part of the outer boundary) or penetrable inhomogeneities. Our approach to tackle this
type of problems is based on the derivation of a series expansion of the input-to-output map of the
problem (typically the Dirichlet-to-Neumann map of the problem for the Calderón problem) in
terms of the size of the obstacle.

3.3 Numerical analysis and simulation of heterogeneous systems

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.

• In the case of FSIS, our main objective is to provide computational tools for the scientific com-

munity, essentially to solve academic problems.

• In the case of CWS, our main objective is to build tools general enough to handle industrial
problems. Our strong collaboration with Christophe Geuzaine’s team in Liège (Belgium) makes
this objective credible, through the combination of DDM (Domain Decomposition Methods) and
parallel computing.

Below, we explain in detail the corresponding scientific program.

• Simulation of FSIS: In order to simulate fluid-structure systems, one has to deal with the fact that
the fluid domain is moving and that the two systems for the fluid and for the structure are strongly
coupled. To overcome this free boundary problem, three main families of methods are usually
applied to numerically compute in an efficient way the solutions of the fluid-structure interaction
systems. The first method consists in suitably displacing the mesh of the fluid domain in order to
follow the displacement and the deformation of the structure. A classical method based on this idea
is the A.L.E. (Arbitrary Lagrangian Eulerian) method: with such a procedure, it is possible to keep a
good precision at the interface between the fluid and the structure. However, such methods are

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Inria Annual Report 2023

difficult to apply for large displacements (typically the motion of rigid bodies). The second family
of methods consists in using a fixed mesh for both the fluid and the structure and to simultaneously
compute the velocity field of the fluid with the displacement velocity of the structure. The presence
of the structure is taken into account through the numerical scheme. Finally, the third class of
methods consists in transforming the set of PDEs governing the flow into a system of integral
equations set on the boundary of the immersed structure. The members of SPHINX have already
worked on these three families of numerical methods for FSIS systems with rigid bodies (see e.g.,
[97], [81], [98], [94], [95], [86]).

• Simulation of CWS: Solving acoustic or electromagnetic scattering problems can become a tre-
mendously hard task in some specific situations. In the high frequency regime (i.e., for small
wavelength), acoustic (Helmholtz’s equation) or electromagnetic (Maxwell’s equations) scattering
problems are known to be difficult to solve while being crucial for industrial applications (e.g., in
aeronautics and aerospace engineering). Our particularity is to develop new numerical methods
based on the hybridization of standard numerical techniques (like algebraic preconditioners, etc.)
with approaches borrowed from asymptotic microlocal analysis. Most particularly, we contribute
to building hybrid algebraic/analytical preconditioners and quasi-optimal Domain Decomposition
Methods (DDM) [48, 65], [66] for highly indefinite linear systems. Corresponding three-dimensional
solvers (like for example GetDDM) will be developed and tested on realistic configurations (e.g.,
submarines, complete or parts of an aircraft, etc.) provided by industrial partners (Thales, Airbus).
Another situation where scattering problems can be hard to solve is the one of dense multiple
(acoustic, electromagnetic or elastic) scattering media. Computing waves in such media requires us
to take into account not only the interactions between the incident wave and the scatterers, but also
the effects of the interactions between the scatterers themselves. When the number of scatterers is
very large (and possibly at high frequency [42, 43]), specific deterministic or stochastic numerical
methods and algorithms are needed. We introduce new optimized numerical methods for solving
such complex configurations. Many applications are related to this problem, such as osteoporosis
diagnosis where quantitative ultrasound is a recent and promising technique to detect a risk of
fracture. Therefore, numerical simulation of wave propagation in multiple scattering elastic media
in the high frequency regime is a very useful tool for this purpose.

4 Application domains

4.1 Robotic swimmers

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:

1. Solve the control problem by limiting the set of admissible deformations.

2. Find the “best” location of the actuators, in the sense of being the closest to the exact optimal

control.

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).

4.2 Aeronautics

We will develop robust and efficient solvers for problems arising in aeronautics (or aerospace) like electro-
magnetic compatibility and acoustic problems related to noise reduction in an aircraft. Our interest for

Project SPHINX

7

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.

5 Highlights of the year

5.1 Awards

• Yannick Privat was awarded the Blaise Pascal Prize of the French Academy of Sciences in 2023.

6 New results

6.1 Analysis, control, stabilization and optimization of heterogeneous systems

Participants: Rémi Buffe, Imene Djebour, Alessandro Duca, Ludovick Gagnon, Ju-
lien Lequeurre, Karim Ramdani, Jean-François Scheid, Takéo Taka-
hashi, Julie Valein, Christophe Zhang.

Analysis

In [38], we extend the theory of single and double layer potentials (well documented for functions with
H 1
l oc regularity) to locally square integrable functions. Having in mind numerical simulations for which
functions are usually defined on a polygonal mesh, we wish this theory to cover the cases of non-smooth
domains (i.e., with Lipschitz continuous or polygonal boundaries).

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 R3. We prove that the corresponding
weak solutions that additionally satisfy a classical Prodi-Serrin condition, including a critical one, are
unique. We also show that the weak solutions are regular under the Prodi-Serrin conditions, with a
smallness condition in the critical case.

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.

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Inria Annual Report 2023

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 N controls, then it is also null controllable with the property that at each time, only one
of these controls is active. The main difference with previous results in the literature is that we can handle
the case where the main operator of the system is not self-adjoint. We give several examples to illustrate
our result: coupled heat equations with terms of orders 0 and 1, the Oseen system or the Boussinesq
system.

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
n + 1 chemical species and to the thickness of the film produced in the process, whereas the controls are
the fluxes of the chemical species. We obtain the local controllability in the case where we apply n + 1
nonnegative controls and a controllability result for large time in the case where we apply n controls
without any sign constraints. We also show in this last case that the controllability may fail for small times.
We illustrate these results with some numerical simulations.

In [33], we study the boundary controllability of 2 × 2 system of heat equations by using a flatness
approach. According to the relation between the diffusion coefficients of the heat equation, it is known
that the system can be not null controllable or null controllable for any T > T0 where T0 ∈ [0, ∞]. Here
we recover this result in the case that T0 ∈ [0, ∞) by using the flatness method, and we obtain an explicit
formula for the control and for the corresponding solutions. In particular, the state and the control have
Gevrey regularity in time and in space.

In [27], we deal with the controllability properties of a system of m coupled Stokes systems or m

Project SPHINX

9

coupled Navier-Stokes systems. We show the null-controllability of such systems in the case where the
coupling is in a cascade form and when the control acts only on one of the systems. Moreover, we impose
that this control has a vanishing component so that we control an m × N state (corresponding to the
velocities of the fluids) by N − 1 distributed scalar controls. The proof of the controllability of the coupled
Stokes system is based on a Carleman estimate for the adjoint system. The local null-controllability of the
coupled Navier-Stokes systems is then obtained by means of the source term method and a Banach fixed
point.

In [28], we consider the controllability of a class of systems of n Stokes equations, coupled through
terms of order zero and controlled by m distributed controls. Our main result states that such a system is
null controllable if and only if a Kalman type condition is satisfied. This generalizes the case of finite-
dimensional systems and the case of systems of coupled linear heat equations. The proof of the main
result relies on the use of the Kalman operator introduced and on a Carleman estimate for a cascade type
system of Stokes equations. Using a fixed-point argument, we also obtain that if the Kalman condition is
verified, then the corresponding system of Navier-Stokes equations is locally null controllable.

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
α∂x . We give a bound
by the first eigenfunctions for a family of one-dimensional degenerate operators ∂x x
for the blow-up of the constant when the frequency cut goes to infinity, which is known to be optimal
for the Laplace operator. A new application to observability for degenerate parabolic equations is given.
Finally, the associated degenerate parabolic equation is known to be not null controllable when α ≥ 2. we
prove a bound for the blow-up of the constants of the spectral inequality when α → 2
. The proof relies
on a combination of the moment method and Carleman estimates.

−

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 z

′ = Az + B v under the
observation y = C z by means of a finite dimensional control v. The control is based on the design of a
Luenberger observer which can be infinite or finite dimensional (of dimension large enough). In the
infinite dimensional case, the operator A is supposed to generate an analytical semigroup with compact
resolvent and the operators B and C are unbounded operators whereas, in the finite dimensional case, A
is assumed to be a self-adjoint operator with compact resolvent, B and C are supposed to be bounded
operators. In both cases, we show that if (A, B ) and (A,C ) verify the Fattorini-Hautus Criterion, then we
can construct an observer-based control v of finite dimension (greater or equal to the largest geometric
multiplicity of the unstable eigenvalues of A) such that the evolution problem is exponentially stable.
As an application, we study the stabilization of the N -dimensional convection-diffusion system with
Dirichlet boundary control and an internal observation.

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

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Inria Annual Report 2023

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 com-
patibility 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 bidi-
mensional 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.

6.2 Direct and inverse problems for heterogeneous systems

Participants:

Alessandro Duca, Anthony Gerber-Roth, Alexandre Munnier,
Karim Ramdani, Jean-François Scheid.

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 T -coercivity
techniques are used in order to cope with these difficulties.

Project SPHINX

Inverse problems

11

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.

6.3 Numerical analysis and simulation of heterogeneous systems

Participants:

Yannick Privat, Jean-François Scheid.

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.

7 Partnerships and cooperations

7.1 International initiatives

7.1.1 Inria associate team not involved in an IIL or an international program

MOUSTIQ

Participants:

Rémi Buffe, Ludovick Gagnon, Takéo Takahashi, Julie Valein.

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Inria Annual Report 2023

Title: Modelization and control of infectious diseases, wave propagation in heterogeneous media and

nonlinear dispersive equations

Duration: 2020 - 2024

Coordinator: Ludovick Gagnon

Partners: Universidade Federale da Paraiba (Brésil)

Summary: 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.

7.1.2 STIC/MATH/CLIMAT AmSud projects

SCIPInPDES

Participants:

Rémi Buffe, Ludovick Gagnon, Takéo Takahashi, Julie Valein.

Title: Stabilization, Control and Inverse Problems in PDEs

Program: MATH-AmSud

Duration: January 1, 2023 – December 31, 2024

Local supervisor: Ludovick Gagnon

Partners:

• Universidad Técnica Federico Santa María (Chili)

• Universidade Federal da Paraíba (Brazil)

Inria contact: Ludovick Gagnon

Summary: This project is devoted to the analysis of several physical processes and phenomena modeled
by Partial Differential Equations. In particular, the problems to be addressed are of great interest in
biology, engineering, fluid mechanics, aeronautics, chemistry and medicine. From a mathematical
point of view, we will study several qualitative properties for these problems, such as controllability,
observability, unique continuation, asymptotic behavior, uniqueness, stability, etc.

Project SPHINX

13

7.2 National initiatives
ANR TRECOS

Participants:

Ludovick Gagnon, Yannick Privat, Takéo Takahashi, Julie Valein.

Title: New Trends in Control and Stabilization: Constraints and non-local terms

Duration: 2021 – 2024

Coordinator: Sylvain Ervedoza

Partners:

• University of Bordeaux

• Sorbonne University of Paris

• University of Toulouse

Summary: The goal of this project is to address new directions of research in control theory for partial
differential equations, triggered by models from ecology and biology. In particular, our projet will
deal with the development of new methods which will be applicable in many applications, from
the treatment of cancer cells to the analysis of the thermic efficiency of buildings, and from control
issues for the biological control of pests to cardiovascular fluid flows. To achieve these objectives,
we will have to solve several theoretical issues in order to design efficient control methods.

ANR ODISSE

Participants:

Ludovick Gagnon, Takéo Takahashi, Karim Ramdani, Julie Valein, Jean-
Claude Vivalda.

Title: Observer Design for Infinite-dimensional Systems

Duration: 2019 – 2024

Coordinator: Vincent Andrieu

Partners:

• University Claude Bernard of Lyon

• University of Toulouse

• Inria Saclay - Ile de France

Summary: Methodologically, this project is at the crossroads of inverse problems and observers theory.
These two disciplines have a long and rich history of interactions and their overlap is becoming
more and more obvious. The project proposes fundamental/theoretical contributions in observer
design to reconstruct missing parameters in infinite dimensional systems. The methodological
contributions expected of this fundamental project will be illustrated on different test-beds.

8 Dissemination

Participants: Mabrouk Ben Jaba, Rémi Buffe, Blaise Colle, Baparou Danhane, Imene
Aicha Djebour, David Dos Santos Ferreira, Alessandro Duca, Ben-
jamin Florentin, Ludovick Gagnon, Anthony Gerber-Roth, Julien Le-
queurre, Alexandre Munnier, Yannick Privat, Karim Ramdani, Jean-
François Scheid, Takéo Takahashi, Julie Valein, Jean-Claude Vivalda,
Christophe Zhang.

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Inria Annual Report 2023

8.1 Promoting scientific activities

8.1.1 Scientific events: organisation

Member of the organizing committees

• Rémi Buffe (since September 2023), Julien Lequeurre and Alexandre Munnier are co-organizers of

the annual workshop Journées EDP de l’IECL.

• David Dos Santos Ferreira is a member of the scientific committee of the GDR Analyse des EDP,

which organizes the "Journées des jeunes EDPistes".

• Rémi Buffe is the organizer of the IECL’s working group on PDEs.

• Ludovick Gagnon is a co-organizer of the IECL’s weekly seminar on PDEs in Nancy.

• Julien Lequeurre is a co-organizer of the IECL’s weekly seminar on PDEs in Metz

8.1.2 Scientific events: selection

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

8.1.3 Journal

Member of the editorial boards 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.

Reviewer - reviewing activities 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 Mathem-
atics; 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.

8.1.4 Invited talks

• Alessandro Duca was invited to give a talk at: 10th International Congress on Industrial and Applied
Mathematics, ICIAM 2023, in Tokyo. He presented his work at the workshops: “Journées QUACO”
(Laboratory Jacques-Louis Lions, Paris) and “Journée scientifique Matière, systèmes et dispositifs
quantique” (Université de Lorraine, Nancy). He was also invited to give a seminar at the New York
University of Shanghai.

• Julie Valein was invited to give a talk at the workshops: NANT and WG3 meeting “Graph struc-
ture generalizations in dynamical system modelling” (B˛edlewo, Pologne); “Contrôle : approches
mathématique et automatique ” (Fédération Charles Hermite, Nancy); workshop “Commande et
observation des systèmes” (LAAS, Toulouse); “Control of Partial Differential Equations” (Polytech-
nical University Hauts-de-France). She was also invited to give a seminar at the “Laboratoire de
Mathématiques de Besançon”.

• David Dos Santos Ferreira was invited to give a talk at the Ghent Methusalem colloquium (Belgium)
and Spectral and Resonance Problems for Imaging, Seismology and Materials Science (Reims).

• Christophe Zhang gave a seminar at the Laboratoire des Signaux et Systèmes (CNRS, Gif-sur-Yvette).

Project SPHINX

15

8.1.5 Research administration

• Karim Ramdani is the head of the PDE team of IECL laboratory (the Mathematics laboratory of

Université de Lorraine) since June 2021.

• Julie Valein is an elected member of the scientific pole AM2I of Université de Lorraine since 2022.
She was member of the CNU 26 in 2023 (elected member after the promotion of another member
from MCF to Professor during the mandate 2019-2023). She was also elected as a substitute member
for the CNU 26 for the mandate 2023-2027.

8.2 Teaching - Supervision - Juries

8.2.1 Teaching

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.

8.2.2 Supervision

• PhD in progress: Benjamin Florentin; “Déformations isospectrales pour l’opérateur de Steklov”;

Oct 2022; David Dos Santos Ferreira.

• PhD in progress: Anthony Gerber Roth; “Autour de quelques problèmes inverses géométriques”;

Set 2020; Alexandre Munnier and Karim Ramdani.

• PhD in progress: Mabrouk Ben Jaba; “Modélisation du transfert de dioxygène dans le poumon
humain, une approche ’contrôle optimal’”; Oct 2023; Yannick Privat and Jean-François Scheid.

• PhD in progress: Killian Lutz; “Méthodes hybrides couplant apprentissage par renforcement et
méthodes de contrôle optimal des EDP pour l’informatique quantique”; Sep 2023; Emmanuel
Franck and Yannick Privat.

• PhD in progress: Ivan Hasenohr; “Contrôlabilité sous contraintes géométriques et topologiques”;

Oct 2022; Sébastien Martin, Camille Pouchol, Yannick Privat and Christophe Zhang.

• PhD in progress: Blaise Colle; “Stabilisation et contrôlabilité du problème de Stefan”; Oct. 2021;

Jérôme Lohéac and Takéo Takahashi.

• PhD in progress: Yingying Wu-Zhang; “Algunos problemas de control en mecánica de fluidos y

algunas ecuaciones diferenciales parciales”; Sept. 2020; Luz de Teresa and Takéo Takahashi.

8.2.3 Juries

• Yannick Privat was the president of the jury of the Ph.D. thesis of M. Lajili (Université Haute Alsace,
September 2023). He was a reviewer of the Ph.D. theses of P. Ascensio (INRIA Sophia Antipolis,
September 2023) and E. Martinet (Université Savoie Mont Blanc, October 2023). He was also a
member of the juries of the Ph.D. theses of T. Delaunay (École Polytechnique, December 2023) and
V. Mons (Université Nice Côte d’Azur, December 2023).

• Karim Ramdani was president of the Ph.D. committee of Luis Alejandro Rosas Martinez (École

Polytechnique, November 2023).

• Takéo Takahashi was a reviewer of the Ph.D. thesis of Tiphaine Delaunay (École Polytechnique,

December 2023).

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Inria Annual Report 2023

• David Dos Santos Ferreira was a reviewer of the Ph.D. thesis of Spyridon Fillipas (Laboratoire de

Mathématiques d’Orsay, December 2023).

• Julie Valein was a reviewer of the Ph.D. thesis of F. Koudohode (Université Toulouse III - Paul
Sabatier, October 2023). She was also member of the juries of the Ph.D. theses of Z. Mohamad Ali
(Université de Lorraine - Metz, December 2023) and H. Parada (Laboratory Jean Kunztmann of
Grenoble, July 2023).

8.3 Popularization

8.3.1 Internal or external Inria responsibilities

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.

8.3.2 Interventions

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.

9 Scientific production

9.1 Major publications

[1] L. B˘alilescu, J. San Martín and T. Takahashi. ‘Fluid-structure interaction system with Coulomb’s
law’. In: SIAM Journal on Mathematical Analysis (2017). URL: https://hal.archives-ouverte
s.fr/hal-01386574.

[2] R. Bunoiu, L. Chesnel, K. Ramdani and M. Rihani. ‘Homogenization of Maxwell’s equations and
related scalar problems with sign-changing coefficients’. In: Annales de la Faculté des Sciences de
Toulouse. Mathématiques. (2020). URL: https://hal.inria.fr/hal-02421312.

[3] N. Burq, D. Dos Santos Ferreira and K. Krupchyk. ‘From semiclassical Strichartz estimates to
uniform Lp resolvent estimates on compact manifolds’. In: Int. Math. Res. Not. IMRN 16 (2018),
pp. 5178–5218. DOI: 10.1093/imrn/rnx042. URL: https://doi.org/10.1093/imrn/rnx042.

[4] L. Gagnon. ‘Lagrangian controllability of the 1-dimensional Korteweg–de Vries equation’. In: SIAM
J. Control Optim. 54.6 (2016), pp. 3152–3173. DOI: 10.1137/140964783. URL: https://doi.org
/10.1137/140964783.

[5] A. Gerber-Roth, A. Munnier and K. Ramdani. ‘A reconstruction method for the inverse gravimetric
problem’. In: SMAI Journal of Computational Mathematics 9 (2023), pp. 197–225. URL: https://i
nria.hal.science/hal-04157036.

[6] O. Glass, A. Munnier and F. Sueur. ‘Point vortex dynamics as zero-radius limit of the motion of a
rigid body in an irrotational fluid’. In: Inventiones Mathematicae 214.1 (2018), pp. 171–287. DOI:
10.1007/s00222-018-0802-4. URL: https://hal.archives-ouvertes.fr/hal-00950544.

[7] C. Grandmont, M. Hillairet and J. Lequeurre. ‘Existence of local strong solutions to fluid-beam
and fluid-rod interaction systems’. In: Annales de l’Institut Henri Poincaré (C) Non Linear Analysis
36.4 (July 2019), pp. 1105–1149. DOI: 10.1016/j.anihpc.2018.10.006. URL: https://hal.in
ria.fr/hal-01567661.

[8] A. Munnier and K. Ramdani. ‘Calderón cavities inverse problem as a shape-from-moments
problem’. In: Quarterly of Applied Mathematics 76 (2018), pp. 407–435. URL: https://hal.inri
a.fr/hal-01503425.

[9] K. Ramdani, J. Valein and J.-C. Vivalda. ‘Adaptive observer for age-structured population with
spatial diffusion’. In: North-Western European Journal of Mathematics 4 (2018), pp. 39–58. URL:
https://hal.inria.fr/hal-01469488.

[10]

J.-F. Scheid and J. Sokolowski. ‘Shape optimization for a fluid-elasticity system’. In: Pure Appl.
Funct. Anal. 3.1 (2018), pp. 193–217.

Project SPHINX

17

9.2 Publications of the year

International journals

[11] M. Badra and T. Takahashi. ‘Analyticity of the semigroup corresponding to a strongly damped
wave equation with a Ventcel boundary condition’. In: Dynamics of Partial Differential Equations
(2023). URL: https://hal.science/hal-03784384.

[12] M. Badra and T. Takahashi. ‘Switching controls for parabolic systems’. In: Palestine Journal of

Mathematics (2023). URL: https://hal.science/hal-03417266.

[13] M. Bouguezzi, J.-F. Scheid, D. Hilhorst, H. Matano, C. Bataillon, F. Lequien and F. Rouillard. ‘Anodic
dissolution model with diffusion-migration transport for simulating localized corrosion’. In:
Electrochimica Acta (2024). URL: https://inria.hal.science/hal-04160934.

[14] R. Buffe and T. Takahashi. ‘Controllability of a fluid-structure interaction system coupling the
Navier-Stokes system and a damped beam equation’. In: Comptes Rendus. Mathématique (2023).
DOI: 10.5802/crmath.509. URL: https://hal.science/hal-03878011.

[15] R. Bunoiu, K. Ramdani and C. Timofte. ‘Homogenization of a transmission problem with sign-
changing coefficients and interfacial flux jump’. In: Communications in Mathematical Sciences
21.7 (2023), pp. 2029–2049. DOI: 10.4310/CMS.2023.v21.n7.a13. URL: https://inria.hal
.science/hal-03712444.

[16] V. Calisti, I. Lucardesi and J.-F. Scheid. ‘Shape sensitivity analysis of a 2D Fluid-Structure Inter-
action problem’. In: Journal of Optimization Theory and Applications 199 (2023), pp. 36–79. DOI:
10.1007/s10957-023-02213-4. URL: https://hal.science/hal-03810334.

[17] N. Carreño and T. Takahashi. ‘Control problems for the Navier-Stokes system with nonlocal spatial
terms’. In: Journal of Optimization Theory and Applications (2023). URL: https://hal.science
/hal-03819799.

[18] B. Colle, J. Lohéac and T. Takahashi. ‘Controllability of the Stefan problem by the flatness ap-
proach’. In: Systems and Control Letters 174 (Apr. 2023), p. 105480. DOI: 10.1016/j.sysconle.2
023.105480. URL: https://hal.science/hal-03721544.

[19] B. Colle, J. Lohéac and T. Takahashi. ‘Controllability results for a cross diffusion system with a free
boundary by a flatness approach’. In: Acta Applicandae Mathematicae 187 (30th Jan. 2023), p. 14.
DOI: 10.1007/s10440-023-00607-0. URL: https://hal.science/hal-03969875.

[20] S. Cotin, G. Mestdagh and Y. Privat. ‘Organ registration from partial surface data in augmented
surgery from an optimal control perspective’. In: Proceedings of the Royal Society A: Mathematical,
Physical and Engineering Sciences 480 (22nd Mar. 2023), p. 20230197. DOI: 10.1098/rspa.2023
.0197. URL: https://hal.science/hal-04043695.

[21] B. Danhane and J. Lohéac. ‘Ensemble controllability of parabolic type equations’. In: Systems
and Control Letters 183 (Jan. 2024), p. 105683. DOI: 10.1016/j.sysconle.2023.105683. URL:
https://hal.science/hal-04174446.

[22] B. Danhane, J. Lohéac and M. Jungers. ‘Conditions for uniform ensemble output controllability,
and obstruction to uniform ensemble controllability’. In: Mathematical Control and Related Fields
(2024). DOI: 10.3934/mcrf.2023036. URL: https://hal.science/hal-03824645.

[23]

I. A. Djebour and T. Takahashi. ‘Feedback boundary stabilization for a viscous incompressible
fluid with Navier slip boundary conditions in interaction with a damped beam’. In: Nonlinear
Analysis: Real World Applications (2024). DOI: 10.1016/j.nonrwa.2023.104022. URL: https:
//hal.science/hal-04012901.

[24] A. Gerber-Roth, A. Munnier and K. Ramdani. ‘A reconstruction method for the inverse gravimetric
problem’. In: SMAI Journal of Computational Mathematics 9 (2023), pp. 197–225. URL: https://i
nria.hal.science/hal-04157036.

[25] D. Maity and T. Takahashi. ‘Uniqueness and regularity of weak solutions of a fluid-rigid body
interaction system under the Prodi-Serrin condition’. In: Discrete and Continuous Dynamical
Systems - Series A 44.3 (2024), pp. 718–742. DOI: 10.3934/dcds.2023123. URL: https://hal.sc
ience/hal-04075090.

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[26] H. Parada, C. Timimoun and J. Valein. ‘Stability results for the KdV equation with time-varying
delay’. In: Systems and Control Letters 177 (July 2023), p. 105547. DOI: 10.1016/j.sysconle.202
3.105547. URL: https://hal.science/hal-03819356.

[27] T. Takahashi, L. de Teresa and Y. Wu-Zhang. ‘Controllability results for cascade systems of m
coupled N -dimensional Stokes and Navier-Stokes systems by N =1 scalar controls’. In: ESAIM:
Control, Optimisation and Calculus of Variations (2023). DOI: 10.48550/arXiv.2108.09856.
URL: https://hal.science/hal-03325287.

[28] T. Takahashi, L. de Teresa and Y. Wu-Zhang. ‘A Kalman condition for the controllability of a
coupled system of Stokes equations’. In: Journal of Evolution Equations (2024). URL: https://ha
l.science/hal-03936869.

Reports & preprints

[29] L. Baudouin, M. de Buhan, E. Crépeau and J. Valein. Carleman-Based Reconstruction Algorithm

on a wave Network. 1st Dec. 2023. URL: https://hal.science/hal-04361363.

[30] M. Bestard, E. Franck, L. Navoret and Y. Privat. Optimal scenario for road evacuation in an urban

environment. 20th Oct. 2023. URL: https://hal.science/hal-04253010.

[31] R. Buffe, K. D. Phung and A. Slimani. An optimal spectral inequality for degenerate operators.

26th Sept. 2023. URL: https://hal.science/hal-04219477.

[32] R. Buffe and T. Takahashi. Controllability with one scalar control of a system of interaction between
the Navier-Stokes system and a damped beam equation. 24th June 2023. URL: https://hal.scie
nce/hal-04140088.

[33] B. Colle, J. Lohéac and T. Takahashi. Flatness approach for the boundary controllability of a system

of heat equations. 6th June 2023. URL: https://hal.science/hal-04119834.

[34]

[35]

I. A. Djebour. Existence of strong solutions for a compressible fluid-solid interaction system with
Navier slip boundary conditions. 13th Apr. 2023. URL: https://hal.science/hal-04068293.

I. A. Djebour, K. Ramdani and J. Valein. Observer-based feedback-control for the stabilization of a
class of parabolic systems. 7th July 2023. URL: https://hal.science/hal-04155214.

[36] L. Gagnon, A. Hayat, S. Xiang and C. Zhang. Fredholm backstepping for critical operators and
application to rapid stabilization for the linearized water waves. 12th Jan. 2024. URL: https://ha
l.science/hal-03892656.

[37] A. Munnier. Reconstruction of obstacles in a Stokes flow as a shape-from-moments problem.

26th Sept. 2023. URL: https://hal.science/hal-04218495.

[38] A. Munnier. Square integrable surface potentials on non-smooth domains and application to the

Laplace equation in L2. 25th May 2023. URL: https://hal.science/hal-03942972.

[39] C. Pouchol, E. Trélat and C. Zhang. Approximate control of parabolic equations with on-off shape
controls by Fenchel duality. 11th Dec. 2023. URL: https://hal.science/hal-03889865.

[40] T. Takahashi, L. de Teresa and Y. Wu-Zhang. Stackelberg exact controllability for the Boussinesq

system. 3rd Oct. 2023. URL: https://hal.science/hal-04228391.

9.3 Cited publications

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[42] X. Antoine, C. Geuzaine and K. Ramdani. ‘Computational Methods for Multiple Scattering at
High Frequency with Applications to Periodic Structures Calculations’. In: Wave Propagation in
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Scattering Problems by Disks’. In: Journal of Algorithms & Computational Technologies 6.2 (2012),
pp. 241–259.

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<scratchpad>
- Main topic: Research Centre at Université de Lorraine, focusing on heterogeneous systems, fluid-structure interaction, and control theory.
- Key points:
  - Project-Team SPHINX: investigates theoretical and numerical mathematical issues concerning heterogeneous physical systems.
  - Fluid-structure interaction systems (FSIS) and complex wave systems (CWS): two types of systems studied.
  - Analysis, control, stabilization, and optimization of heterogeneous systems.
  - Applications in robotics, aeronautics, and other fields.
  - Research program includes direct and inverse problems, numerical analysis, and simulation.
  - Partnerships and cooperations with international and national organizations.
- Structure: The document is divided into sections, including Project-Team SPHINX, Activity Report, Contents, Project-Team SPHINX, and so on.
- Potential artifacts to ignore: web-related elements, unnecessary formatting, and links.
</scratchpad>

<final_summary>
The document presents the Research Centre at Université de Lorraine, focusing on heterogeneous systems, fluid-structure interaction, and control theory. The Project-Team SPHINX investigates theoretical and numerical mathematical issues concerning heterogeneous physical systems, including fluid-structure interaction systems and complex wave systems. The research program encompasses direct and inverse problems, numerical analysis, and simulation, with applications in robotics, aeronautics, and other fields. The team collaborates with international and national organizations, aiming to advance the understanding and control of heterogeneous systems.
The document presents the Research Centre at Université de Lorraine, focusing on heterogeneous systems, fluid-structure interaction, and control theory. The Project-Team SPHINX investigates theoretical and numerical mathematical issues concerning heterogeneous physical systems, including fluid-structure interaction systems and complex wave systems. The research program encompasses direct and inverse problems, numerical analysis, and simulation, with applications in robotics, aeronautics, and other fields. The team collaborates with international and national organizations, aiming to advance the understanding and control of heterogeneous systems.