2025Activity​​​‌ reportProject-TeamSPHINX

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 PDEs 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 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 and​​​‌ almost all mathematical results​ on such FSIS have​‌ been obtained since 2000.​​

The most studied FSIS​​​‌ is the problem modeling​ a rigid body moving​‌ in a viscous incompressible​​ fluid. 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.​‌ Without approximations, the only​​ known results 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. 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.

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 66 or​​ Kaltenbacher, Neubauer, and Scherzer​​​‌ 67). 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 sensitivity, 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 reconstruction 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:​​​‌

• 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 54, 63​​​‌, 65, 74‌ for the design of‌​‌ feedback controllers), an input​​ (for instance source inverse​​​‌ problems 53, 58‌, 62) 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 61, specific‌​‌ one-dimensional techniques (like in​​ 55) or observer-based​​​‌ methods as in 70‌.

  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  69, 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 72 or 75‌​‌). Using observers, we​​ have proposed in 73​​​‌, 64 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  57‌​‌, 56.

• 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 $\Omega$ containing an​​​‌ (unknown) local heterogeneity $\mathrm{\omega ‌}$, we consider the‌​‌ boundary value problem of​​ the form

  $$L\mathrm{u‌}=0(\mathrm{\Omega ‌}\setminus \omega ),‌‌u=f(\partial \Omega ),‌Bu=\mathrm{0‌}(\partial \omega )‌‌$$

  where $L$ is a​​ given partial differential operator​​​‌ describing the physical phenomenon‌ under consideration (typically a‌​‌ second order differential operator),​​ $B$ the (possibly unknown)​​​‌ operator describing the behaviour‌ at the boundaries of‌​‌ the heterogeneity and $\mathrm{f}$ the exterior source used​​​‌ to probe the medium.‌ The question is then‌​‌ to recover the shape​​ of $\omega$ and/or the​​​‌ boundary operator $B$ from‌ some measurements on the‌​‌ outer boundary $\partial \mathrm{\Omega }$. This setting includes​​​‌ in particular inverse scattering‌ problems in acoustics and‌​‌ electromagnetics (in this case​​ $\Omega$ 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.,​ measurements 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. We​ plan to continue our​‌ efforts in this direction.​​ Our main objective is​​​‌ to improve our numerical​ codes in order to​‌ improve their efficiency. At​​ the moment,these codes are​​​‌ developed mainly for the​ scientific community and essentially​‌ to solve academic problems.​​ Below, we explain in​​​‌ detail the corresponding scientific​ program.

In order to​‌ simulate fluid-structure systems, it​​ is necessary to account​​​‌ for the moving fluid​ domain and the strong​‌ coupling between the structures.​​ 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​‌ 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.

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​​ consider only 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​‌ 59, 60,​​ 71, 68).​​​‌ Among all the important‌ issues, we aim to‌​‌ consider the following ones:​​

• Solve the control problem​​​‌ by limiting the set‌ of admissible deformations.
• 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 Modelization of​​ the lung

Various models​​​‌ exist for the human‌ lung and its functioning.‌​‌ A common approach involves​​ structuring the respiratory system​​​‌ into multiple levels (5‌ levels), each modeled using‌​‌ complex partial differential equations​​ (PDEs). An alternative approach​​​‌ is to develop simpler‌ models using ordinary differential‌​‌ equations (ODEs) that account​​ for the entire bronchial​​​‌ tree.

Our goal is‌ to study and enhance‌​‌ these models through optimal​​ control methods. The idea​​​‌ is to consider specific‌ aspects of the system‌​‌ (e.g., diaphragm force, position,​​ or velocity) and to​​​‌ optimize a cost function,‌ such as maximizing oxygen‌​‌ exchange in the bronchial​​ cells (alveoli) in order​​​‌ to improve the models.‌

5.1 Awards

• Yannick​​ Privat was awarded the​​​‌ "Perseverance in Action" prize‌ by the Fondation Mines‌​‌ Nancy, in recognition of​​ his commitment to students​​​‌ of the GIMA Department.‌
• During 2025, Yannick Privat‌​‌ was a Junior Member​​ of the Institut Universitaire​​​‌ de France (IUF).

6.1 Latest​​ software developments

7.1​​ Modeling and analysis

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Participants: Alessandro Duca,‌ Julien Lequeurre, Hugo‌​‌ Parada, Yannick Privat​​, Karim Ramdani,​​​‌ Jean-François Scheid, Takéo‌ Takahashi.

Negative materials‌​‌

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 10, we study‌​‌ composite assemblages of dielectrics​​ and metamaterials with respectively​​​‌ positive and negative material‌ parameters. In the continuum‌​‌ case, for a scalar​​ equation, such media may​​​‌ exhibit so-called plasmonic resonances‌ for certain values of‌​‌ the (negative) conductivity in​​​‌ the metamaterial. This work​ investigates such resonances, and​‌ the associated eigenfunctions, in​​ the case of composite​​​‌ conducting networks. Unlike the​ continuous media, we show​‌ a surprising specific dependence​​ on the geometry of​​​‌ the network of the​ resonant values. We also​‌ study how the problem​​ is affected by the​​​‌ choice of boundary conditions​ on the external nodes​‌ of the structure.

Fluid​​ and fluid-structure interaction systems​​​‌

In 49, we​ study the motion of​‌ a rigid body in​​ a viscous, compressible fluid​​​‌ filling the exterior of​ the domain of the​‌ solid. The rigid body​​ follows the Newton laws​​​‌ and the fluid is​ modeled by the Navier-Stokes​‌ system. Using a suitable​​ $L_p-{\mathrm{L‌}}_q$ approach, we analyze​ the corresponding coupled system​‌ and establish the existence​​ and uniqueness of strong​​​‌ solutions for small initial​ data, along with their​‌ decay rates. In particular,​​ we show that the​​​‌ position of the rigid​ body converges to a​‌ fixed position as $\mathrm{t}\to \infty$. To​​​‌ prove our results, we​ first apply a modified​‌ Lagrangian change of variables​​ to rewrite the system​​​‌ in a fixed spatial​ domain. Then, we analyze​‌ the corresponding linear system​​ and prove in particular​​​‌ some $L_p-L_q$ decay estimates​‌ by adopting a methodology​​ due to Kobayashi and​​​‌ Shibata. Finally, following the​ recent work of Shibata,​‌ we combine these decay​​ estimates with the maximal​​​‌ $L_p-$regularity​ of the linear system​‌ and with a fixed​​ point argument to obtain​​​‌ both the existence and​ the decay estimates of​‌ solutions for the nonlinear​​ system.

In 41,​​​‌ we consider a dissipative​ quantum fluid on the​‌ whole space $R^{\mathrm{d}}$ ($d\ge \mathrm{1‌}$) confined by an​ external harmonic potential. The​‌ dynamics of the quantum​​ fluid is described by​​​‌ the Quantum Navier-Stokes (QNS)​ system which is a​‌ particular case of the​​ Navier-Stokes-Korteweg systems. The goal​​​‌ of this paper is​ to prove the existence​‌ of global weak solutions​​ to the QNS system.​​​‌ To this end, we​ write the evolution equations​‌ with respect to a​​ Gaussian reference measure and​​​‌ follow the general strategy​ of previous works. Nevertheless,​‌ several substantial modifications have​​ to be done due​​​‌ to our choice of​ the reference measure.

The​‌ paper 32 investigates a​​ boundary-value problem for the​​​‌ Korteweg-de Vries (KdV) equation​ on a star-graph structure.​‌ We develop a unified​​ framework introducing the notion​​​‌ of $s-$compatibility,​ which generalizes classical compatibility​‌ conditions to star-shaped and​​ more complex graph configurations,​​​‌ inspired by the works​ of Bona, Sun, and​‌ Zhang. By combining analytical​​ techniques with a fixed-point​​​‌ argument, we establish sharp​ global well-posedness for both​‌ the linear and nonlinear​​ problems at the ${\mathrm{H‌}}^s$ level. In this​ setting, our results extend​‌ the classical analysis for​​ a single KdV equation​​​‌ to star-shaped graphs composed​ of $N$ equations. These​‌ results provide the first​​ comprehensive well-posedness theory for​​​‌ KdV equations with coupled​ boundary conditions on graphs.​‌ Although control issues are​​ not treated in this​​ article, the analytic results​​​‌ obtained here address several‌ open problems, which will‌​‌ be addressed in the​​ future.

In 27,​​​‌ we consider the system‌ modeling the motion of‌​‌ a rigid body into​​ a viscous incompressible fluid.​​​‌ Such a system couples‌ the Navier-Stokes system for‌​‌ the fluid with the​​ Newton laws for the​​​‌ rigid body and has‌ a free boundary due‌​‌ to the motion of​​ the rigid body. We​​​‌ work here in the‌ case where the fluid‌​‌ domain and the structure​​ domain are confined into​​​‌ a bounded domain. We‌ show in this article‌​‌ that a weak solution​​ for this system such​​​‌ that the fluid velocity‌ satisfies a Prodi-Serrin condition‌​‌ is smooth in time​​ and space. As for​​​‌ the proof in the‌ case of the standard‌​‌ Navier-Stokes system, here we​​ consider a particular linearization​​​‌ of our system around‌ our weak solution. The‌​‌ corresponding linear system written​​ in a moving spatial​​​‌ domain is then studied‌ with the help of‌​‌ the Prodi-Serrin condition and​​ to show some uniqueness​​​‌ result that allows us‌ to identify the solutions‌​‌ of the linear system​​ and of the nonlinear​​​‌ system, we also study‌ the adjoint of this‌​‌ system.

Magnetic systems

In​​ 24, we investigate​​​‌ a real 3D stationary‌ flow characterized by chaotic‌​‌ advection generated by a​​ magnetic field created by​​​‌ permanent magnets acting on‌ a weakly conductive fluid‌​‌ subjected to a weak​​ constant current. The model​​​‌ under consideration involves the‌ Stokes equations for viscous‌​‌ incompressible fluid at low​​ Reynolds number in which​​​‌ the density forces correspond‌ to the Lorentz force‌​‌ generated by the magnetic​​ field of the magnets​​​‌ and the electric current‌ through the fluid. An‌​‌ innovative numerical approach based​​ on a mixed finite​​​‌ element method has been‌ developed and implemented for‌​‌ computing the flow velocity​​ fields with the electromagnetic​​​‌ force. This ensures highly‌ accurate numerical results, allowing‌​‌ a detailed analysis of​​ the chaotic behavior of​​​‌ fluid trajectories through the‌ computations of associated Poincaré‌​‌ sections and Lyapunov exponents.​​ Subsequently, an examination of​​​‌ mixing efficiency is conducted,‌ employing computations of contamination‌​‌ and homogeneity rates, as​​ well as mixing time.​​​‌ The obtained results underscore‌ the relevance of the‌​‌ modeling and computational tools​​ employed, as well as​​​‌ the design of the‌ magnetohydrodynamic device used.

In‌​‌ 39, we study​​ from a mathematical point​​​‌ of view the nanoparticle‌ model of a magnetic‌​‌ colloid, presented by G.​​ Klughertz. Our objective is​​​‌ to obtain properties of‌ stable stationary structures that‌​‌ arise in the long-time​​ limit for the magnetic​​​‌ nanoparticles dynamics following this‌ model. In this article,‌​‌ we present a detailed​​ study of two specific​​​‌ structures using techniques from‌ the calculus of variations.‌​‌ The first, called the​​ spear, consists of a​​​‌ chain of aligned particles‌ interacting via a Lennard-Jones‌​‌ potential. We establish existence​​ and uniqueness results, derive​​​‌ bounds on the distances‌ between neighboring particles, and‌​‌ provide a sharp asymptotic​​ description as the number​​​‌ of particles tends to‌ infinity. The second structure,‌​‌ the ring, features particles​​​‌ uniformly distributed along a​ circle. We prove its​‌ existence and uniqueness and​​ derive an explicit formula​​​‌ for its radius.

In​ 16, we investigate​‌ a simple model of​​ notched ferromagnetic nanowires using​​​‌ tools from calculus of​ variations and critical point​‌ theory. Specifically, we focus​​ on the case of​​​‌ a single unimodal notch​ and establish the existence​‌ and uniqueness of the​​ critical point of the​​​‌ energy. This is achieved​ through a lifting argument,​‌ which reduces the problem​​ to a generalized Sturm-Liouville​​​‌ equation. Uniqueness is demonstrated​ via a Mountain-Pass argument,​‌ where the assumption of​​ two distinct critical points​​​‌ leads to a contradiction.​ Additionally, we show that​‌ the solution corresponds to​​ a system of magnetic​​​‌ spins characterized by a​ single domain wall localized​‌ in the vicinity of​​ the notch. We further​​​‌ analyze the asymptotic decay​ of the solution at​‌ infinity and explore the​​ symmetric case using rearrangement​​​‌ techniques.

7.2 Control and​ stabilization

Participants: Rémi Buffe​‌, Alessandro Duca,​​ Hugo Parada, Yannick​​​‌ Privat, Karim Ramdani​, Takéo Takahashi,​‌ Julie Valein, Christophe​​ Zhang.

Controllability

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.

In 15​, 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-frequencies​ solutions and in the​‌ case where the observation​​ domain is an 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-frequencies​‌ solutions. Then, the result​​ on the nonlinear system​​​‌ can be obtained by​ a change of variables​‌ and a fixed-point argument.​​

In 37, we​​​‌ consider the local exact​ controllability to trajectories of​‌ the Navier-Stokes system with​​ distributed controls. Such a​​​‌ property was already obtained​ in previous works but​‌ here we improve the​​ "cost" of the control​​​‌ by refining the observability​ inequality associated with the​‌ Oseen system. More precisely,​​ our main result corresponds​​​‌ to a Carleman inequality​ for the Oseen system​‌ with weight functions similar​​ to the ones for​​ the heat equation. In​​​‌ our proof, we need‌ in particular to estimate‌​‌ precisely a pressure term​​ on the boundary and​​​‌ this is done by‌ treating differently the low-frequency‌​‌ part and the high-frequency​​ part of the solutions.​​​‌ For the low-frequency part‌ of the solution, we‌​‌ can obtain a Carleman​​ estimate without any boundary​​​‌ condition by using standard‌ methods of micro-local analysis‌​‌ whereas the high-frequency part​​ of the solution can​​​‌ be estimated by using‌ energy inequality of the‌​‌ Stokes system.

In 51​​, we study the​​​‌ controllability of the Navier-Stokes‌ system with distributed controls.‌​‌ These controls have a​​ vanishing component and are​​​‌ odd powers of smooth‌ functions. We prove that‌​‌ the corresponding system is​​ locally null-controllable for any​​​‌ positive time. In order‌ to prove this result,‌​‌ we first prove it​​ for the Stokes system,​​​‌ extending a previous similar‌ result obtained for the‌​‌ linear heat equation. The​​ method of proof combines​​​‌ a Carleman estimate with‌ maximal regularity results in‌​‌ $L_p$ for the​​ Stokes system and with​​​‌ a duality argument. The‌ local null-controllability for the‌​‌ Navier-Stokes system is then​​ obtained by using a​​​‌ Schauder fixed-point argument. We‌ then use this result‌​‌ to prove the local​​ null-controllability for a two-dimensional​​​‌ Boussinesq type system where‌ the coupling between the‌​‌ fluid system and the​​ heat equation is nonlinear​​​‌ and where the control‌ acts only on the‌​‌ heat equation.

The aim​​ of the work 40​​​‌ is to study the‌ controllability of the viscous‌​‌ Burgers equation in the​​ case of bilinear controls.​​​‌ We consider the problem‌ on the one-dimensional flat‌​‌ torus and on bounded​​ intervals equipped with Dirichlet​​​‌ or Neumann boundary conditions.‌ The controls depend solely‌​‌ on time and act​​ through a given family​​​‌ of spatial functions. We‌ first prove the small-time‌​‌ global approximate controllability of​​ the equation between states​​​‌ of the same sign.‌ This result is ensured‌​‌ by a saturating geometric​​ control approach with at​​​‌ least three controls that‌ are localized in frequency.‌​‌ Afterward, we show the​​ small-time global exact controllability​​​‌ to the non-zero constant‌ states of the equation‌​‌ via at least four​​ controls in the case​​​‌ of the flat torus‌ and Neumann boundary conditions.‌​‌ For this second result,​​ we proceed by studying​​​‌ the null-controllability of a‌ suitable linearized system. Then,‌​‌ we infer the controllability​​ for the initial bilinear​​​‌ Burgers equation via fixed-point‌ arguments. Explicit examples of‌​‌ bilinear controls verifying our​​ results are provided in​​​‌ the work.

In 22‌, we analyse the‌​‌ small-time reachability properties of​​ a nonlinear parabolic equation,​​​‌ by means of a‌ bilinear control, posed on‌​‌ a torus of arbitrary​​ dimension $d$. Under​​​‌ a saturation hypothesis on‌ the control operators, we‌​‌ show the small-time approximate​​ controllability between states sharing​​​‌ the same sign. Moreover,‌ in the one-dimensional case‌​‌ $d=1$,​​ we combine this property​​​‌ with a local exact‌ controllability result, and prove‌​‌ the small-time exact controllability​​ of any positive states​​​‌ towards the ground state‌ of the evolution operator.‌​‌

The exact controllability of​​​‌ heat-type equations in the​ presence of bilinear controls​‌ has been successfully studied​​ in recent works, motivated​​​‌ by numerous applications to​ engineering, neurobiology, chemistry, and​‌ life sciences. Nevertheless, the​​ result has only been​​​‌ achieved for 1-dimensional domains​ due to the limitations​‌ of the existing techniques.​​ In 36, we​​​‌ consider a fractional heat-type​ equation as ${\partial }_{\mathrm{t‌}}\psi +{(-\Delta )}^s\mathrm{\psi ‌}+\langle v(t),\mathrm{Q‌}\rangle \psi \left(\mathrm{t}\right)=0$ with​​​‌ $s>0$ and​ on a domain $\mathrm{\Omega ‌}\subset {ℝ}^N$ for​​ $N\in {\mathbb{N}}^{*‌}.$ We study the​ so-called exact controllability to​‌ the eigensolutions of the​​ equations when $s>‌max(\frac{4\mathrm{N}}{5},N-‌1)$. The​​ result is implied by​​​‌ the null controllability of​ a suitable linearized equation,​‌ and the main novelty​​ of the work is​​​‌ the strategy of its​ proof. First, the null​‌ controllability in a finite-dimensional​​ subspace has to be​​​‌ ensured via the solvability​ of a suitable moment​‌ problem. Explicit bounds on​​ the control cost with​​​‌ respect to the dimension​ of the controlled space​‌ are also required. Second,​​ the controllability can be​​​‌ extended to the whole​ Hilbert space, thanks to​‌ the Lebeau-Robbiano-Miller method, when​​ the control cost does​​​‌ not grow too fast​ with respect to the​‌ dimension of the finite-dimensional​​ subspace. We firstly develop​​​‌ our techniques in the​ general case when suitable​‌ hypotheses on the problem​​ are verified. Secondly, we​​​‌ apply our procedure to​ the bilinear heat equation​‌ on rectangular domains, and​​ we ensure its exact​​​‌ controllability to the eigensolutions.​

Analysing reachability associated to​‌ a control system is​​ a subtle issue, especially​​​‌ for infinite-dimensional dynamics, and​ when controls are subject​‌ to bounded constraints. In​​ 46, we develop​​​‌ a computer-assisted framework for​ establishing non-reachability in linear​‌ parabolic PDEs governed by​​ strongly elliptic operators, extending​​​‌ recent finite-dimensional techniques introduced​ in to the PDE​‌ setting. The non-reachability of​​ a given target is​​​‌ shown to be equivalent​ to proving that a​‌ properly defined dual functional​​ takes negative values. Our​​​‌ approach combines rigorous numerics​ with explicit convergence estimates​‌ for discretisations of the​​ adjoint equation, ensuring mathematically​​​‌ certified results with tight​ error bounds. We demonstrate​‌ the wide applicability of​​ our framework on Laplacian-driven​​​‌ control systems, showcasing its​ accuracy and reliability under​‌ various types of control​​ constraints.

The Sterile Insect​​​‌ Technique (SIT) is a​ biological control method used​‌ to reduce or eliminate​​ pest populations or disease​​​‌ vectors. This technique involves​ releasing sterilized insects that,​‌ upon mating with the​​ wild population, produce no​​​‌ offspring, leading to a​ decline or eventual eradication​‌ of the target species.​​ In the work 23​​​‌, we incorporate a​ spatial dimension by modeling​‌ the pest/vector population as​​ being distributed across multiple​​​‌ patches, with both wild​ and released sterile insects​‌ migrating between these patches​​ at predetermined rates. We​​​‌ mainly focus on a​ two-patch system. This study​‌ has two primary objectives:​​ first, to derive sufficient​​ conditions for achieving the​​​‌ elimination of the wild‌ population through SIT, whether‌​‌ releases occur in one​​ patch or in both​​​‌ patches. In particular, we‌ provide an estimate of‌​‌ the minimal release rates​​ to reach elimination thanks​​​‌ to the diffusion rates‌ between patches. This is‌​‌ the first time that​​ such a result is​​​‌ given in a general‌ manner. Second, we study‌​‌ an optimal SIT control​​ strategy, where we minimize​​​‌ the total amount of‌ sterile insects to release,‌​‌ and show that, within​​ one patch, it can​​​‌ successfully reduce the wild‌ population in that patch‌​‌ to a desired level​​ within a finite time​​​‌ frame, provided that the‌ migration rates between patches‌​‌ are sufficiently low. Numerical​​ simulations are employed to​​​‌ illustrate these results and‌ further analyze the outcomes.‌​‌

The work 26 studies​​ a basic safety question​​​‌ in control: can we‌ prove that a constrained‌​‌ linear system will never​​ reach a given unsafe​​​‌ region at a prescribed‌ final time? Rather than‌​‌ computing the whole reachable​​ set, the authors introduce​​​‌ a geometric criterion based‌ on supporting hyperplanes. This‌​‌ transforms the original control​​ problem into the search​​​‌ for a simple mathematical‌ certificate showing that the‌​‌ unsafe set lies beyond​​ everything the system can​​​‌ reach. The main contribution‌ of the paper is‌​‌ to make this idea​​ fully rigorous in practice,​​​‌ even when exact formulas‌ are not available.

In‌​‌ 19, we consider​​ the nonlinear Schrödinger equation​​​‌ (NLS) on a torus‌ of arbitrary dimension. The‌​‌ equation is studied in​​ presence of an external​​​‌ potential field whose time-dependent‌ amplitude is taken as‌​‌ control. Assuming that the​​ potential satisfies a saturation​​​‌ property, we show that‌ the NLS equation is‌​‌ approximately controllable between any​​ pair of eigenstates in​​​‌ arbitrarily small time. The‌ proof is obtained by‌​‌ developing a multiplicative version​​ of a geometric control​​​‌ approach introduced by Agrachev‌ and Sarychev. We give‌​‌ an application of this​​ result to the study​​​‌ of the large time‌ behavior of the NLS‌​‌ equation with random potential.​​ More precisely, we assume​​​‌ that the amplitude of‌ the potential is a‌​‌ random process whose law​​ is 1-periodic in time​​​‌ and non-degenerate. Combining the‌ controllability with a stopping‌​‌ time argument and the​​ Markov property, we show​​​‌ that the trajectories of‌ the random equation are‌​‌ almost surely unbounded in​​ regular Sobolev spaces.

In​​​‌ 21, we address‌ the small-time controllability problem‌​‌ for a nonlinear Schrödinger​​ equation (NLS) on ${\mathrm{ℝ‌}}^N$ in the presence‌ of magnetic and electric‌​‌ external fields. We choose​​ a particular framework where​​​‌ the equation becomes $\mathrm{i‌}{\partial }_t\psi =‌‌[-\Delta +u_0\left(\mathrm{t‌}\right)h_{\stackrel{\to ‌}{0}}+\langle u(‌‌t),\mathrm{P}\rangle +{\kappa |‌\psi |}^{2\mathrm{p‌}}]\psi$. Here,‌​‌ the control operators are​​ defined by the zeroth​​​‌ Hermite function $h_{\overrightarrow{\mathrm{0‌}}}(x)‌‌$ and the momentum operator​​ $P=i\mathrm{\nabla ‌}$. In detail, we‌ study when it is‌​‌ possible to control the​​​‌ dynamics of (NLS) as​ fast as desired via​‌ sufficiently large control signals​​ $u_0$ and $\mathrm{u‌}$. We first show​ the existence of a​‌ family of quantum states​​ for which this property​​​‌ is verified: this extends​ to ${ℝ}^N$ the​‌ validity of a small-time​​ control property recently shown​​​‌ on ${𝕋}^d$ by​ Duca and Nersesyan, and​‌ on $S^2$ by​​ Chambrion and Pozzoli. Secondly,​​​‌ by considering some specific​ states belonging to this​‌ family, as a physical​​ consequence we show the​​​‌ capability of controlling arbitrary​ changes of energy in​‌ bounded regions of the​​ quantum system, in time​​​‌ zero. Our results are​ proved by exploiting the​‌ idea that the nonlinear​​ term in (NLS) is​​​‌ only a perturbation of​ the linear problem when​‌ the time is as​​ small as desired. The​​​‌ core of the proof,​ then, is the controllability​‌ of the bilinear equation​​ which is tackled by​​​‌ using specific non-commutativity properties​ of infinite-dimensional propagators.

In​‌ 20, we consider​​ the 1D nonlinear Schrödinger​​​‌ equation with bilinear control.​ In the case of​‌ Neumann boundary conditions, local​​ exact controllability of this​​​‌ equation near the ground​ state has been proved​‌ by Beauchard and Laurent.​​ In this paper, we​​​‌ study the case of​ Dirichlet boundary conditions. To​‌ establish the controllability of​​ the linearised equation, we​​​‌ use a bilinear control​ acting through four directions:​‌ three Fourier modes and​​ one generic direction. The​​​‌ Fourier modes are appropriately​ chosen so that they​‌ satisfy a saturation property.​​ These modes allow to​​​‌ control approximately the linearised​ Schrödinger equation. We show​‌ that the reachable set​​ for the linearised equation​​​‌ is closed. This is​ achieved by representing the​‌ resolving operator as a​​ sum of two linear​​​‌ continuous mappings: one is​ surjective (here the control​‌ in generic direction is​​ used) and the other​​​‌ is compact. A mapping​ with dense and closed​‌ image is surjective, so​​ the linearised Schrödinger equation​​​‌ is exactly controllable. Then​ local exact controllability of​‌ the nonlinear equation is​​ derived using the inverse​​​‌ mapping theorem.

The local​ exact controllability of the​‌ one-dimensional bilinear Schrödinger equation​​ with Dirichlet boundary conditions​​​‌ has been extensively studied​ in subspaces of ${\mathrm{H‌}}^3$ since the seminal​​ work of K. Beauchard.​​​‌ In the work 35​, our first objective​‌ is to revisit this​​ result and establish the​​​‌ controllability in $H_{\mathrm{0}}^1$ for suitable discontinuous​‌ control potentials. In the​​ second part, we consider​​​‌ the equation in the​ presence of periodic boundary​‌ conditions and a constant​​ magnetic field. We prove​​​‌ the local exact controllability​ of periodic $H^{\mathrm{1‌}}-$states, thanks to​​ a Zeeman-type effect induced​​​‌ by the magnetic field​ which decouples the resonant​‌ spectrum. Finally, we discuss​​ open problems and partial​​​‌ results for the Neumann​ case and the harmonic​‌ oscillator.

The work 38​​ addresses the controllability of​​​‌ the energy of quantum​ bounded states through domain​‌ deformations in the two-dimensional​​ framework. We approach the​​​‌ controllability question from a​ practical point of view,​‌ with the primary goal​​ of providing simple and​​ implementable control processes involving​​​‌ moving rectangles. We numerically‌ validate the feasibility of‌​‌ our controls and analyze​​ the energy transitions that​​​‌ occur in quantum states‌ confined within specific deformations.‌​‌ This work presents two​​ possible approaches for numerically​​​‌ implementing domain deformations. We‌ show that one can‌​‌ either apply a suitable​​ change of variables to​​​‌ fix the domain, or‌ simulate the deformation using‌​‌ high-intensity confining potentials.

In​​ 33, we introduce​​​‌ a novel concept called‌ the Graph Geometric Control‌​‌ Condition (GGCC). It turns​​ out to be a​​​‌ simple, geometric rewriting of‌ many of the frameworks‌​‌ in which the controllability​​ of PDEs on graphs​​​‌ has been studied. We‌ prove that (GGCC) is‌​‌ a necessary and sufficient​​ condition for the exact​​​‌ controllability of the wave‌ equation on metric graphs‌​‌ with internal controls and​​ Dirichlet boundary conditions. We​​​‌ then investigate the internal‌ exact controllability of the‌​‌ wave equation with mixed​​ boundary conditions and the​​​‌ one of the Schrödinger‌ equation, as well as‌​‌ the internal null-controllability of​​ the heat equation. We​​​‌ show that (GGCC) provides‌ a sufficient condition for‌​‌ the controllability of these​​ equations and we provide​​​‌ explicit examples proving that‌ (GGCC) is not necessary‌​‌ in these cases.

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 11, we​​​‌ analyze a system modeling‌ the evolution of an‌​‌ age and spatially structured​​ population (of Lotka-McKendrick type).​​​‌ We study it by‌ first writing it in‌​‌ an abstract form using​​ several operators. We show​​​‌ that the semigroup associated‌ with the corresponding system‌​‌ is differentiable. Using this​​ property, we show how​​​‌ to prove the exponential‌ stabilization with a finite-dimensional‌​‌ feedback control. We consider​​ two types of controls:​​​‌ one that acts directly‌ on the main equation‌​‌ of evolution and one​​ that acts on the​​​‌ birth equation. One of‌ the main difficulties in‌​‌ the analysis of this​​ system is that the​​​‌ operators involved in the‌ system can depend on‌​‌ the age variable. We​​ use in particular a​​​‌ parabolic evolution operator associated‌ with the main operator‌​‌ of the system. Our​​ stabilization result shows how​​​‌ to extend the framework‌ associated with parabolic system‌​‌ to the case of​​ differentiable semigroups.

In 44​​​‌, we analyze the‌ internal and boundary stabilization‌​‌ of the Cahn-Hilliard and​​ Kuramoto-Sivashinsky equations under saturated​​​‌ feedback control. We conduct‌ our study through the‌​‌ spectral analysis of the​​ associated linear operator. We​​​‌ identify a finite number‌ of eigenvalues related to‌​‌ the unstable part of​​ the system and then​​​‌ design a stabilization strategy‌ based on modal decomposition,‌​‌ linear matrix inequalities (LMIs),​​ and geometric conditions on​​​‌ the saturation function. Local‌ exponential stabilization in ${\mathrm{H‌‌}}^2$ is established.

The​​ paper 45 studies the​​​‌ rapid stabilization of a‌ multidimensional heat equation in‌​‌ the presence of an​​ unknown spatially localized disturbance.​​​‌ A novel multivalued feedback‌ control strategy is proposed,‌​‌ which synthesizes the frequency​​​‌ Lyapunov method with the​ sign multivalued operator. This​‌ methodology connects Lyapunov-based stability​​ analysis with spectral inequalities,​​​‌ while the inclusion of​ the sign operator ensures​‌ robustness against the disturbance.​​ The closed-loop system is​​​‌ governed by a differential​ inclusion, for which well-posedness​‌ is proved via the​​ theory of maximal monotone​​​‌ operators. This approach not​ only guarantees exponential stabilization​‌ but also circumvents the​​ need for explicit disturbance​​​‌ modeling or estimation.

In​ 43, we study​‌ the rapid stabilization of​​ an unstable wave equation,​​​‌ in which an unknown​ disturbance is located at​‌ the boundary condition. We​​ address two different boundary​​​‌ conditions: Dirichlet-Dirichlet and Dirichlet-Neumann.​ In both cases, we​‌ design a feedback law,​​ located at the same​​​‌ place as the unknown​ disturbance, that forces the​‌ exponential decay of the​​ energy for any desired​​​‌ decay rate while suppressing​ the effects of the​‌ unknown disturbance. For the​​ feedback design we employ​​​‌ the backstepping method, Lyapunov​ techniques and the sign​‌ multivalued operator. The well-posedness​​ of the closed-loop system,​​​‌ which is a differential​ inclusion, is shown with​‌ the maximal monotone operator​​ theory.

In 50,​​​‌ we show that the​ energy of classical solutions​‌ to the wave equation​​ with hyperbolic boundary condition​​​‌ (i.e., dynamic Wentzell boundary​ condition) and damping on​‌ the boundary decays like​​ $1/t$.​​​‌ In fact we allow​ mixed boundary conditions: a​‌ possibly empty, disjoint part​​ of the boundary may​​​‌ be kept at rest​ provided that the dynamic​‌ part satisfies the geometric​​ control condition. We also​​​‌ prove that this decay​ rate is sharp. Our​‌ results follow from resolvent​​ estimates, which we establish​​​‌ by studying high-frequency quasimodes.​

7.3 Optimal control and​‌ inverse problems

Participants: Benjamin​​ Florentin, Anthony Gerber-Roth​​​‌, Alexandre Munnier,​ Yannick Privat, Karim​‌ Ramdani, Jean-François Scheid​​, Takéo Takahashi,​​​‌ Julie Valein, Christophe​ Zhang.

Optimization problems​‌

The article 30 deals​​ with the existence of​​​‌ hypersurfaces minimizing general shape​ functionals under certain geometric​‌ constraints. We consider as​​ admissible shapes orientable hypersurfaces​​​‌ satisfying a so-called reach​ condition, also known as​‌ the uniform ball property,​​ which ensures $C^{\mathrm{1‌},1}$ regularity of​ the hypersurface. In this​‌ paper, we revisit and​​ generalise previous results on​​​‌ the subject. We provide​ a simpler framework and​‌ more concise proofs of​​ some of the results​​​‌ contained in these references​ and extend them to​‌ a new class of​​ problems involving PDEs. Indeed,​​​‌ by using the signed​ distance introduced by Delfour​‌ and Zolesio, we avoid​​ the intensive and technical​​​‌ use of local maps,​ as was the case​‌ in the above references.​​ Our approach, originally developed​​​‌ to solve an existence​ problem, can be easily​‌ extended to costs involving​​ different mathematical objects associated​​​‌ with the domain, such​ as solutions of elliptic​‌ equations on the hypersurface.​​

In 13, an​​​‌ approach based on the​ use of SympNet is​‌ employed to numerically solve​​ shape optimization problems. This​​​‌ is the first work​ on the subject, ultimately​‌ aiming to consider AI​​ methods for solving PDEs​​ without using mesh grids​​​‌ in order to address‌ problems that cannot be‌​‌ tackled with classical approaches,​​ such as shape optimization​​​‌ involving fluid dynamics systems‌ in turbulent regimes.

In‌​‌ 29, we investigate​​ the optimal shapes for​​​‌ the hydrodynamic resistance of‌ a rigid body set‌​‌ in motion in a​​ Stokes flow. At this​​​‌ low Reynolds number regime,‌ the hydrodynamic drag properties‌​‌ of an object are​​ encoded in a finite​​​‌ number of parameters contained‌ in the grand resistance‌​‌ tensor. Considering these parameters​​ as objective functions, we​​​‌ use calculus of variations‌ techniques to derive a‌​‌ general shape derivative formula,​​ allowing to specify how​​​‌ to deform the body‌ shape to improve the‌​‌ objective value of any​​ given resistance tensor entry.​​​‌ We then describe a‌ practical algorithm for numerically‌​‌ computing the optimized shapes​​ and apply it to​​​‌ several examples. Numerical results‌ reveal interesting new geometries‌​‌ for various criteria and​​ perspectives into optimal hydrodynamic​​​‌ profiles.

The article 14‌ presents geometric optimal control‌​‌ techniques for analyzing geodesics​​ in time-optimal Zermelo navigation​​​‌ problems on $2-‌$spheres of revolution. We‌​‌ classify the problem by​​ analyzing the pair $(‌F_0,\mathrm{g‌})$, which represents‌​‌ the current (or wind)​​ and the Riemannian metric.​​​‌ Using the maximum principle,‌ the dynamics of geodesics‌​‌ are described by a​​ Hamiltonian vector field on​​​‌ the cotangent bundle $\mathrm{T‌}*S^2$.‌​‌ Our primary motivation is​​ the application to micromagnetism,​​​‌ specifically spin magnetization reversal‌ in ferromagnetic ellipsoidal samples.‌​‌ This model depends on​​ four parameters and the​​​‌ amplitude of the applied‌ magnetic field. The problem‌​‌ is formulated as a​​ Zermelo navigation on the​​​‌ $2-$sphere, where‌ geodesics are classified as‌​‌ elliptic, hyperbolic, or abnormal.​​ We demonstrate that the​​​‌ transition set $|{\mathrm{F‌}}_0{|}_g=‌‌1$, which separates​​ weak and strong current​​​‌ domains, is critical for‌ understanding optimality. A key‌​‌ result shows that abnormal​​ geodesics intersect this set​​​‌ with semi-cubical cusp singularities,‌ a phenomenon we term‌​‌ the Landau–Lifshitz billiard. The​​ analysis of the transition​​​‌ set's connected components is‌ complex and complemented by‌​‌ algebraic geometry and symbolic​​ computations. We further reveal​​​‌ that hyperbolic geodesics lose‌ optimality at their second‌​‌ intersection with the abnormal​​ arc. Our numerical simulations​​​‌ complement this analysis by‌ computing conjugate and cut‌​‌ loci, wavefronts, and accessibility​​ sets, providing new insights​​​‌ into optimal magnetization switching‌ under bounded control.

In‌​‌ 25, we consider​​ an optimal control problem​​​‌ for the Navier-Stokes system‌ with Tresca boundary conditions.‌​‌ With such boundary conditions,​​ the weak formulation of​​​‌ the system is a‌ variational inequality. We approximate‌​‌ this system and the​​ optimal control problem by​​​‌ regularizing the boundary conditions‌ leading to a variational‌​‌ equality. We show that​​ for the approximate system,​​​‌ there exists an optimal‌ control and we derive‌​‌ the first optimality condition​​ by using an adjoint​​​‌ system. We also prove‌ that the approximate optimal‌​‌ controls converge towards an​​ optimal control for the​​​‌ Navier-Stokes system with Tresca‌ boundary conditions. Finally we‌​‌ show that as the​​​‌ threshold of the Tresca​ law goes to infinity,​‌ the corresponding optimal controls​​ converge towards an optimal​​​‌ control for the Navier-Stokes​ system with the Dirichlet​‌ boundary condition.

The article​​ 34 explores recent geometric​​​‌ optimal control techniques for​ the analysis of geodesics​‌ in Zermelo navigation problems​​ on 2-spheres of revolution,​​​‌ focusing on accessibility and​ optimality properties. These techniques​‌ involve the classification of​​ pairs $(F_{\mathrm{0‌}},g)$,​ where $F_0$ represents​‌ the current and $\mathrm{g}$ is the Riemannian metric​​​‌ of revolution on the​ 2-sphere. By applying the​‌ maximum principle, the geodesic​​ dynamics are described by​​​‌ a Hamiltonian vector field​ on the cotangent bundle​‌ $T^{*}{S}^{\mathrm{2}}$, which remains invariant​​​‌ under positive homothety in​ the fiber. The primary​‌ motivation of this study​​ is to investigate the​​​‌ application of these techniques​ to micromagnetism, particularly in​‌ the context of spin​​ magnetization reversal. The underlying​​​‌ model is complex, depending​ on four parameters as​‌ well as the control​​ amplitude of the applied​​​‌ magnetic field. The analysis​ is further supported by​‌ algebraic geometry and numerical​​ simulations.

Coherent control protocols​​​‌ enabling fast and accurate​ implementations of logical gates​‌ is a key issue​​ in quantum computing. The​​​‌ work 48 deals with​ the optimal implementation of​‌ quantum gates in a​​ meansquared sense subject to​​​‌ experimental constraints on the​ multi-chromatic electromagnetic control pulses.​‌ Our assumptions encapsulate open​​ qudits of arbitrary finite​​​‌ dimension subject to decoherence​ models in accordance with​‌ the markovian GSK-Linblad formalism.​​ Given a unitary gate​​​‌ we show the existence​ of a time-minimal protocol​‌ minimizing the error. We​​ derive universal and easily​​​‌ computable a priori lower​ and upper bounds on​‌ both the minimal time​​ and minimal error. These​​​‌ bounds are not sharp​ but depend only on​‌ the dimension of the​​ qudit, the experimental constraints,​​​‌ the decoherence time and​ strength of the coupling​‌ between the qudit and​​ its environment. The wide​​​‌ applicability of these estimates​ helps in quantifying a​‌ posteriori the distance to​​ optimality of numerically calculated​​​‌ control protocols.

Controlling a​ damped oscillator is crucial​‌ in various technological and​​ scientific fields, such as​​​‌ structural engineering, aerospace, and​ noise reduction device design.​‌ The article 47 considers​​ a classical underdamped harmonic​​​‌ oscillator, focusing on its​ minimal-time control by modulating​‌ its time-dependent frequency. The​​ goal is to connect​​​‌ in minimal time two​ states with zero kinetic​‌ energy but different displacements.​​ We provide a comprehensive​​​‌ theoretical analysis of this​ problem, characterizing the set​‌ of reachable states and​​ detailing the structure of​​​‌ the optimal trajectories, which​ we precisely describe in​‌ phase space. These trajectories​​ correspond to bang-bang controls,​​​‌ and the number of​ commutations depends on the​‌ ratio between the final​​ and initial states. Our​​​‌ approach is based on​ combining the Pontryagin principle​‌ with a suitable choice​​ of "energy-angle" variables and​​​‌ a Bellman-like optimality principle.​

Inverse problems

The article​‌ 31 was written as​​ part of the thesis​​​‌ of Tom Sprunck (Inria,​ Macaron team), which was​‌ defended in Strasbourg in​​ December 2024. We present​​ an algorithm capable of​​​‌ fully inverting the "shoebox‌ image source method" (ISM),‌​‌ a room impulse response​​ (RIR) simulator for rectangular​​​‌ rooms. This algorithm reliably‌ retrieves the 18 input‌​‌ parameters, including the 3D​​ position of the source,​​​‌ the room dimensions, the‌ translations and orientations of‌​‌ the room, as well​​ as the absorption coefficients​​​‌ of the walls. It‌ is based on a‌​‌ recent gridless source localization​​ technique, combined with procedures​​​‌ to identify the room‌ axes and first-order reflections.‌​‌ Simulations show near-exact retrieval​​ of the parameters with​​​‌ a spherical array of‌ 32 microphones and a‌​‌ sampling frequency of 16​​ kHz.

In the context​​​‌ of a network of‌ vibrating strings, modelled by‌​‌ interconnected linear partial differential​​ equations, in the work​​​‌ 12, we are‌ interested in the reconstruction‌​‌ of a zeroth order​​ term of each one-dimensional​​​‌ wave equation involved, using‌ some appropriate external boundary‌​‌ measurements. More precisely, 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 at‌ providing 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 the‌ paper 17, we‌​‌ study the heat equation​​ on a tree-shaped network​​​‌ with a piecewise regular‌ diffusion coefficient. By developing‌​‌ new Carleman estimates, we​​ establish stability results for​​​‌ the identification of the‌ diffusion coefficient. These stability‌​‌ estimates are derived using​​ either internal measurements or​​​‌ boundary observations, offering robust‌ insights into the inverse‌​‌ problem for this class​​ of equations.

The article​​​‌ 18 explores a variant‌ of Kac's famous problem,‌​‌ ‘‘Can one hear the​​ shape of a drum?”,​​​‌ by addressing a geometric‌ inverse problem in acoustics.‌​‌ Our objective is to​​ reconstruct the shape of​​​‌ a cuboid room using‌ acoustic signals measured by‌​‌ microphones placed within the​​ room. By examining this​​​‌ straightforward configuration, we aim‌ to understand the relationship‌​‌ between the acoustic signals​​ propagating in a room​​​‌ and its geometry. This‌ geometric problem can be‌​‌ reduced to locating a​​ finite set of acoustic​​​‌ point sources, known as‌ image sources. We model‌​‌ this issue as a​​ finite-dimensional optimization problem and​​​‌ propose a solution algorithm‌ inspired by super-resolution techniques.‌​‌ This involves a convex​​ relaxation of the finite-dimensional​​​‌ problem to an infinite-dimensional‌ subspace of Radon measures.‌​‌ We provide analytical insights​​ into this problem and​​​‌ demonstrate the efficiency of‌ the algorithm through multiple‌​‌ numerical examples.

The Steklov​​ spectrum of a smooth​​​‌ compact Riemannian manifold $(‌M,g)‌‌$ with boundary is the​​​‌ set of eigenvalues counted​ with multiplicities of its​‌ Dirichlet-to-Neumann map (DN map).​​ The work 42 is​​​‌ devoted to the Steklov​ spectral inverse problem of​‌ recovering the metric $\mathrm{g}$, up to natural​​​‌ gauge invariance, from its​ Steklov spectrum. Positive results​‌ are established in dimension​​ $n\ge 3$ for​​​‌ conformal metrics under the​ assumption that the geodesic​‌ flow on the boundary​​ is Anosov with simple​​​‌ length spectrum. The paper​ combines wave trace formula​‌ techniques with the injectivity​​ of the geodesic $\mathrm{X‌}-$ray transform for​ functions on closed Anosov​‌ manifolds. It is shown​​ that knowledge of the​​​‌ Steklov spectrum determines the​ jet at the boundary​‌ of the underlying Riemannian​​ metric within its conformal​​​‌ class. In this particular​ context, this parallels the​‌ well-known results of the​​ Calderon problem, where we​​​‌ are given the entire​ Dirichlet-to-Neumann map instead. As​‌ a simple corollary, assuming​​ real-analyticity of the conformal​​​‌ factor, Steklov isospectral metrics​ must coincide. Using similar​‌ arguments, we are also​​ able to prove under​​​‌ the same assumption of​ hyperbolicity of the geodesic​‌ flow on the boundary,​​ that generically any smooth​​​‌ potential $q$ can be​ recovered from the Steklov​‌ spectrum, in the sense​​ that its jet at​​​‌ the boundary is determined​ by the spectrum of​‌ the DN map for​​ the Schrödinger operator with​​​‌ potential $q$. Consequently,​ in this case, two​‌ analytic Steklov isospectral potentials​​ must be equal.

8.1 Bilateral​‌ contracts with industry

Participants:​​ Yannick Privat.

• 2024-2025​​​‌ : Yannick Privat had​ a scientific collaboration with​‌ W. Khettaf and the​​ start-up Flex-Horizon, in the​​​‌ framework of an industrial​ project at Mines Nancy.​‌

9.1 International research visitors​​​‌

9.2 National initiatives​‌

9.3 Regional​‌ initiatives

10.1​​​‌ Promoting scientific activities

10.2 Teaching - Supervision​​​‌ - Juries - Educational​ and pedagogical outreach

10.3​‌ Popularization

11.1 Major publications​​​‌

• 1 articleL.Loredana​ Bălilescu, J.Jorge​‌ San Martín and T.​​Takéo Takahashi. Fluid-structure​​​‌ interaction system with Coulomb's​ law.SIAM Journal​‌ on Mathematical Analysis2017​​HAL
• 2 articleR.​​​‌Renata Bunoiu, L.​Lucas Chesnel, K.​‌Karim Ramdani and M.​​Mahran Rihani. Homogenization​​​‌ of Maxwell's equations and​ related scalar problems with​‌ sign-changing coefficients.Annales​​ de la Faculté des​​​‌ Sciences de Toulouse. Mathématiques.​2020HAL
• 3 article​‌N.Nicolas Burq,​​ D.David Dos Santos​​​‌ Ferreira and K.Katya​ Krupchyk. From semiclassical​‌ Strichartz estimates to uniform​​ $L^p$ resolvent estimates​​​‌ on compact manifolds.​Int. Math. Res. Not.​‌ IMRN162018,​​ 5178--5218URL: https://doi.org/10.1093/imrn/rnx042DOI​​​‌
• 4 articleA.Anthony​ Gerber-Roth, A.Alexandre​‌ Munnier and K.Karim​​ Ramdani. A reconstruction​​​‌ method for the inverse​ gravimetric problem.SMAI​‌ Journal of Computational Mathematics​​92023, 197-225​​​‌HAL
• 5 articleO.​Olivier Glass, A.​‌Alexandre Munnier and F.​​Franck Sueur. Point​​​‌ vortex dynamics as zero-radius​ limit of the motion​‌ of a rigid body​​ in an irrotational fluid​​​‌.Inventiones Mathematicae214​12018, 171-287​‌HALDOI
• 6 article​​C.Céline Grandmont,​​​‌ M.Matthieu Hillairet and​ J.Julien Lequeurre.​‌ Existence of local strong​​ solutions to fluid-beam and​​​‌ fluid-rod interaction systems.​Annales de l'Institut Henri​‌ Poincaré (C) Non Linear​​ Analysis364July​​​‌ 2019, 1105-1149HAL​DOI
• 7 articleA.​‌Alexandre Munnier and K.​​Karim Ramdani. Calderón​​ cavities inverse problem as​​​‌ a shape-from-moments problem.‌Quarterly of Applied Mathematics‌​‌762018, 407-435​​HAL
• 8 articleK.​​​‌Karim Ramdani, J.‌Julie Valein and J.-C.‌​‌Jean-Claude Vivalda. Adaptive​​ observer for age-structured population​​​‌ with spatial diffusion.‌North-Western European Journal of‌​‌ Mathematics42018,​​ 39-58HAL
• 9 article​​​‌J.-F.Jean-François Scheid and‌ J.Jan Sokolowski.‌​‌ Shape optimization for a​​ fluid-elasticity system.Pure​​​‌ Appl. Funct. Anal.3‌12018, 193--217‌​‌

11.2 Publications of the​​ year

11.3 Cited publications​​

• 53 articleC.Carlos​​​‌ Alves, A. L.​Ana Leonor Silvestre,​‌ T.T. Takahashi and​​ M.Marius Tucsnak.​​​‌ Solving inverse source problems​ using observability. Applications to​‌ the Euler-Bernoulli plate equation​​.SIAM J. Control​​​‌ Optim.4832009​, 1632-1659back to​‌ text
• 54 articleD.​​D. Auroux and J.​​​‌J. Blum. A​ nudging-based data assimilation method​‌ : the Back and​​ Forth Nudging (BFN) algorithm​​​‌.Nonlin. Proc. Geophys.​15305-3192008back​‌ to text
• 55 article​​M. I.M. I.​​​‌ Belishev and S. A.​S. A. Ivanov.​‌ Reconstruction of the parameters​​ of a system of​​​‌ connected beams from dynamic​ boundary measurements.Zap.​‌ Nauchn. Sem. S.-Peterburg. Otdel.​​ Mat. Inst. Steklov. (POMI)​​​‌324Mat. Vopr. Teor.​ Rasprostr. Voln. 342005​‌, 20--42, 262back​​ to text
• 56 article​​​‌M.Mourad Bellassoued and​ D.David Dos Santos​‌ Ferreira. Stability estimates​​ for the anisotropic wave​​​‌ equation from the Dirichlet-to-Neumann​ map.Inverse Probl.​‌ Imaging542011​​, 745--773URL: http://dx.doi.org/10.3934/ipi.2011.5.745​​​‌DOIback to text​
• 57 articleM.Mourad​‌ Bellassoued and D. D.​​David Dos Santos Ferreira​​​‌. Stable determination of​ coefficients in the dynamical​‌ anisotropic Schrödinger equation from​​ the Dirichlet-to-Neumann map.​​​‌Inverse Problems2612​2010, 125010, 30​‌URL: http://dx.doi.org/10.1088/0266-5611/26/12/125010DOIback​​ to text
• 58 article​​​‌G.Gottfried Bruckner and​ M.Masahiro Yamamoto.​‌ Determination of point wave​​ sources by pointwise observations:​​ stability and reconstruction.​​​‌Inverse Problems163‌2000, 723--748back‌​‌ to text
• 59 article​​T.Thomas Chambrion and​​​‌ A.Alexandre Munnier.‌ Generic controllability of 3D‌​‌ swimmers in a perfect​​ fluid.SIAM J.​​​‌ Control Optim.505‌2012, 2814--2835URL:‌​‌ http://dx.doi.org/10.1137/110828654DOIback to​​ text
• 60 articleT.​​​‌Thomas Chambrion and A.‌Alexandre Munnier. Locomotion‌​‌ and control of a​​ self-propelled shape-changing body in​​​‌ a fluid.J.‌ Nonlinear Sci.213‌​‌2011, 325--385URL:​​ http://dx.doi.org/10.1007/s00332-010-9084-8DOIback to​​​‌ text
• 61 articleC.‌Cheok Choi, G.‌​‌Gen Nakamura and K.​​Kenji Shirota. Variational​​​‌ approach for identifying a‌ coefficient of the wave‌​‌ equation.Cubo9​​22007, 81--101​​​‌back to text
• 62‌ articleA.A. El‌​‌ Badia and T.T.​​ Ha-Duong. Determination of​​​‌ point wave sources by‌ boundary measurements.Inverse‌​‌ Problems1742001​​, 1127--1139back to​​​‌ text
• 63 articleE.‌Emilia Fridman. Observers‌​‌ and initial state recovering​​ for a class of​​​‌ hyperbolic systems via Lyapunov‌ method.Automatica49‌​‌72013, 2250​​ - 2260back to​​​‌ text
• 64 articleG.‌Ghislain Haine and K.‌​‌Karim Ramdani. Reconstructing​​ initial data using observers:​​​‌ error analysis of the‌ semi-discrete and fully discrete‌​‌ approximations.Numer. Math.​​12022012,​​​‌ 307-343back to text‌
• 65 articleG.Ghislain‌​‌ Haine. Recovering the​​ observable part of the​​​‌ initial data of an‌ infinite-dimensional linear system with‌​‌ skew-adjoint generator.Mathematics​​ of Control, Signals, and​​​‌ Systems2632014‌, 435-462back to‌​‌ text
• 66 bookV.​​Victor Isakov. Inverse​​​‌ problems for partial differential‌ equations.127Applied‌​‌ Mathematical SciencesNew York​​Springer2006back to​​​‌ text
• 67 bookB.‌Barbara Kaltenbacher, A.‌​‌Andreas Neubauer and O.​​Otmar Scherzer. Iterative​​​‌ regularization methods for nonlinear‌ ill-posed problems.6‌​‌Radon Series on Computational​​ and Applied MathematicsWalter​​​‌ de Gruyter GmbH &‌ Co. KG, Berlin2008‌​‌back to text
• 68​​ articleJ.J. Lohéac​​​‌ and A.A. Munnier‌. Controllability of 3D‌​‌ Low Reynolds Swimmers.​​ESAIM:COCV2013back to​​​‌ text
• 69 articleD.‌D.G. Luenberger. Observing‌​‌ the state of a​​ linear system.IEEE​​​‌ Trans. Mil. Electron.MIL-8‌1964, 74-80back‌​‌ to text
• 70 article​​P.P. Moireau,​​​‌ D.D. Chapelle and‌ P.P. Le Tallec‌​‌. Joint state and​​ parameter estimation for distributed​​​‌ mechanical systems.Computer‌ Methods in Applied Mechanics‌​‌ and Engineering1972008​​, 659--677back to​​​‌ text
• 71 articleA.‌Alexandre Munnier and B.‌​‌Bruno Pinçon. Locomotion​​ of articulated bodies in​​​‌ an ideal fluid: 2D‌ model with buoyancy, circulation‌​‌ and collisions.Math.​​ Models Methods Appl. Sci.​​​‌20102010,‌ 1899--1940URL: http://dx.doi.org/10.1142/S0218202510004829DOI‌​‌back to text
• 72​​ bookJ.John O'Reilly​​​‌. Observers for linear‌ systems.170Mathematics‌​‌ in Science and Engineering​​Orlando, FLAcademic Press​​​‌ Inc.1983back to‌ text
• 73 articleK.‌​‌K. Ramdani, M.​​​‌M. Tucsnak and G.​G. Weiss. Recovering​‌ the initial state of​​ an infinite-dimensional system using​​​‌ observers.Automatica46​102010, 1616-1625​‌back to text
• 74​​ articleP.Plamen Stefanov​​​‌ and G.Gunther Uhlmann​. Thermoacoustic tomography with​‌ variable sound speed.​​Inverse Problems257​​​‌0750112009, 16​back to text
• 75​‌ bookH.Hieu Trinh​​ and T.Tyrone Fernando​​​‌. Functional observers for​ dynamical systems.420​‌Lecture Notes in Control​​ and Information SciencesBerlin​​​‌Springer2012back to​ text