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 73 or Kaltenbacher, Neubauer, and Scherzer 74). 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 61, 70, 72, 83 for the design of feedback controllers), an input (for instance source inverse problems 60, 65, 69) 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 68, specific one-dimensional techniques (like in 62) or observer-based methods as in 77.

  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  76, 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 81 or 84). Using observers, we have proposed in 82, 71 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  64, 63.

• 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 $\omega$, we consider the boundary value problem of the form

  $$Lu=0(\Omega \setminus \omega ),u=f\left(\partial \Omega \right),Bu=0\left(\partial \omega \right)$$

  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 $f$ 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 $\partial \Omega$. 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 66, 67, 78, 75). 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.

6.1 New software

7.1 Modeling and analysis

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Participants: Alessandro Duca, Julien Lequeurre, 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 18, we study the asymptotic behavior of a sign-changing transmission problem, stated in a symmetric oscillating domain obtained by gluing together a positive and a negative material, separated by an imperfect and rapidly oscillating interface. The interface separating the two heterogeneous materials has a periodic microstructure and is a small perturbation of a flat interface. The solution of the transmission problem is continuous and its flux has a jump on the oscillating interface. Under certain conditions on the properties of the two materials, we derive the limit problem and we prove the convergence result. The T-coercivity method is used to handle the lack of coercivity for both the microscopic and the macroscopic limit problems.

In 42, 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 31, 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 ${ℝ}^3$. 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.

A similar problem is considered in 55, where we work in the case where the fluid domain and the structure domain are confined into a bounded domain. We show that a weak solution 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. We also analyze the adjoint of this system to establish a uniqueness result, which allows us to identify the solutions of both the linear and nonlinear systems.

In 50, 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.

The work in 15 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 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.

Ferromagnetic nanowire

In 45, 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.

Schrödinger-Gross–Pitaevskii equations

In 12, we are interested in gradient flow type methods for computing the ground state of nonlinear Schrödinger-Gross–Pitaevskii equations. Fractional Normalized Gradient Flow methods, involving fractional derivatives and generalizing the celebrated Normalized Gradient Flow method are derived and analyzed. Several experiments are proposed to illustrate some convergence properties of the developed algorithms.

7.2 Control and stabilization

Participants: Rémi Buffe, Alessandro Duca, Julien Lequeurre, Alexandre Munnier, Yannick Privat, Karim Ramdani, Anthony Gerber-Roth, Jean-François Scheid, 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. More precisely, we obtained the following results.

In 17, 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 19, 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 21, we study the boundary controllability of $2\times 2$ 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 $T>T_0$ where $T_0\in [0,\infty ]$. Here we recover this result in the case that $T_0\in [0,\infty )$ 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 37, 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 38, 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 16, we ensure an observability inequality, also known as spectral inequality, within spaces spanned by the first eigenfunctions for a family of one-dimensional degenerate operators ${\partial }_x{x}^{\alpha }{\partial }_x$. We give a bound 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 $\alpha \ge 2$. we prove a bound for the blow-up of the constants of the spectral inequality when $\alpha \to 2^{-}$. The proof relies on a combination of the moment method and Carleman estimates.

In 30, we consider a nonlinear system of two parabolic equations, with a distributed control in the first equation and an odd coupling term in the second one. We prove that the nonlinear system is small-time locally null-controllable. The main difficulty is that the linearized system is not null-controllable. To overcome this obstacle, we construct odd controls for a nonlinear heat equation. The proof relies on three main steps. First, we obtain from the classical $L^2$ parabolic Carleman estimate, conjugated with maximal regularity results, a weighted $L^p$ observability inequality for the nonhomogeneous heat equation. Secondly, we perform a duality argument, close to the well-known Hilbert Uniqueness Method in a reflexive Banach setting, to prove that the heat equation perturbed by a source term is null-controllable thanks to odd controls. Finally, the nonlinearity is handled with a Schauder fixed-point argument.

Finally, in 35, 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.

In 53, we study the issue of the controllability assisted by computer. Given that there is little research on this topic, we consider a very simple model, a finite-dimensional autonomous controlled system subject to constraints on both the control and the state. Typically, the goal is to ensure that a finite-dimensional controlled linear system avoids certain regions. The method relies on reformulating this problem by introducing a functional associated with support hyperplanes, ensuring that the area to be avoided is not reached. To implement a computer-assisted approach, we introduce a temporal discretization of the problem and control rounding errors via interval arithmetic.

In 27, 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 20, we study the Schrödinger equation $i{\partial }_t\psi =-\Delta \psi +\eta \left(t\right){\sum }_{j=1}^J{\delta }_{x=a_j\left(t\right)}\psi$ on $L^2((0,1),ℂ)$ where $\eta :[0,T]\to {ℝ}^{+}$ and $a_j:[0,T]\to (0,1)$ with $j=1,...,J$. We show how to permute the energy associated to different eigenmodes of the Schrödinger equation via suitable choice of the functions $\eta$ and $a_j$. To the purpose, we introduce suitable control processes. We also propose a Galerkin approximation that we prove to be convergent and illustrate the control process with some numerical simulations.

In 28, we address the small-time controllability problem for a nonlinear Schrödinger equation (NLS) on ${ℝ}^N$ in the presence of magnetic and electric external fields. We choose a particular framework where the equation becomes $i{\partial }_t\psi =[-\Delta +u_0\left(t\right)h_{\overrightarrow{0}}+\langle u\left(t\right),P\rangle +{\kappa \left|\psi \right|}^{2p}]\psi$. Here, the control operators are defined by the zeroth Hermite function $h_{\overrightarrow{0}}\left(x\right)$ and the momentum operator $P=i\nabla$. 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 $u$. 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 the first author and Nersesyan, and on $S^2$ by Chambrion and the second author. 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.

The exact controllability of heat type equations in the presence of bilinear controls have been successfully studied in the recent works motivated by the numerous application to engineering, neurobiology, chemistry, and life science. Nevertheless, the result has been only achieved for one-dimensional domains due to limit of the existing techniques. In 44, we develop a new strategy to ensure the so-called exact controllability to the eigensolutions of heat-type equations via bilinear control on the two-dimensional domains. 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 w.r.t. 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 w.r.t 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. Afterwards, we apply our procedure to the bilinear heat equation on rectangular domains, and we ensure its exact controllability to the eigensolutions. Finally, we study the controllability issue on the square and we use perturbation theory techniques to deal with the presence of multiple eigenvalues for the spectrum of the Dirichlet Laplacian.

In 46, 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.

In the thesis 41, we demonstrate some controllability results for coupled heat equation systems. These results are obtained by exploiting the flatness property of these systems. The flatness of a system refers to the ability to parametrize its state and control by a function called a flat output, which can be freely chosen. In our work, we explicitly construct solutions in the form of series in space, where the coefficients are functions of time. By choosing a finite number of these coefficients, we can describe the system's trajectory. This thesis is written as a series of articles, and we successively consider three different systems. First, we show controllability results for a cross-diffusion system with a free boundary, which models a solar panel manufacturing process. We aim to respect sign constraints on the state and controls, as imposed by the modeling. Next, we examine a Stefan problem with two phases, a system of two heat equations with a free boundary that models the solid-liquid phase transition of a pure substance. The position of the interface between the two phases is given by a differential equation involving the solutions of the two heat equations. We establish controllability results for the temperature in each phase, as well as the position of the interface. Here too, we aim to respect sign constraints. Finally, we prove controllability results for a system of two heat equations in cascade. We improve upon known results for this system by explicitly constructing very regular controls.

In 23, we extend the concept of ensemble controllability to a class of linear partial differential equation. More precisely, we consider some abstract parabolic equation, where the system depends on some unknown parameter which is assumed to belong to a compact interval. We investigate the possibility of approximatively reaching (in L²-norm) any target state from any initial state, with an open loop control. Here the initial and target states might depend on the unknown parameter, but the control is assumed to be parameter independent.

In 24, we consider the linear time-invariant system $\dot{x}(t,\theta )=A\left(\theta \right)x(t,\theta )+B\left(\theta \right)u\left(t\right)$ with output $y(t,\theta )=C\left(\theta \right)x(t,\theta )$ where $A$, $B$ and $C$ are continuous matrices with respect to the constant parameter $\theta$, which belongs to some compact set P. Given any continuous initial state datum $\theta \to x_0\left(\theta \right)$ and any continuous output function $\theta \to y_1\left(\theta \right)$, we investigate the existence of a $\theta -$independent open loop control $u$ such that $x_0$ is steered, in finite time, arbitrarily closed to $y_1$ with the uniform norm. When $C\left(\theta \right)$ is the identity matrix, this notion is usually termed to be uniform ensemble controllability. In this paper, we extend some results on uniform ensemble controllability to the case where $C\left(\theta \right)$ is not the identity matrix. We also give some obstructions to ensemble controllability when the parameter set admits an interior point.

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 25, we consider the stabilization of a class of linear evolution systems $z^{\text{'}}=Az+Bv$ under the observation $y=Cz$ 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 than 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 diffusion system.

In 26, we show the stabilization by a finite number of controllers of a fluid-structure interaction system where the fluid is modeled by the Navier-Stokes system into a periodical canal and where the structure is an elastic wall localized on top of the fluid domain. The elastic deformation of the structure follows a damped beam equation. We also assume that the fluid can slip on its boundaries and we model this by using the Navier slip boundary conditions. Our result states the local exponential stabilization around a stationary state of strong solutions by using dynamical controllers in order to handle the compatibility conditions at initial time. The proof is based on a change of variables to write the fluid-structure interaction system in a fixed domain and on the stabilization of the linearization of the corresponding system around the stationary state. One of the main difficulties consists in handling the nonlinear terms coming from the change of variables in the boundary conditions.

In 51, 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.

In 43, 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.

7.3 Optimal control and inverse problems

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

Optimization problems

In 36, an existence result was established for optimal surfaces in problems involving surface quantities (curvature, perimeter, solution to a PDE on a manifold). Qualitative properties of solutions to shape optimization problems involving vector systems were provided in 32, 29, and 54. In particular, in the spirit of the famous Faber-Krahn inequality in spectral geometry, the optimality of the ball under measure constraint was studied.

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.

The article 14 is part of Mickael Bestard's PhD thesis and aims to determine traffic regulation measures to efficiently clear a road axis in case of an accident. It considers a hyperbolic model controlled on a graph, with controls positioned at certain nodes of the graph. An optimal control algorithm combining a gradient approach with a specific fixed-point method has been introduced to bypass the existence of a large number of local minima. The source code associated with this publication is available for free access.

The articles 10 and 48 address the control of a population of harmful insects using a laboratory-reared insect population. In both cases, the goal is to establish an optimal strategy for releasing the laboratory-reared insect population. From a mathematical perspective, this led to the development of an asymptotic analysis method for a perturbed optimal control dynamic system to account for the presence of insect migration. The underlying "Gamma-convergence" property is established by taking the limit in the optimality system, which is not standard.

In 11, we still consider an optimal control problem for a population of wild insects (pests) using a population of insects created in laboratories. It is assumed that the wild insects move spatially following a progressive wave-type dynamics. We are interested in finding an optimal strategy to block such a solution by taking action to eliminate the population in a prescribed subdomain (for example, modeling the effect of mechanical action or an insecticide applied in a specific region to reduce the number of individuals in the population). We develop an approach based on comparison principles to reduce the search for the optimal strategy.

The article 56 aims to develop tools to analyze the optimization of the observability constant associated with the heat equation with respect to the choice of the observation domain. This work constitutes a first (modest) contribution to this challenging problem. The developed approach allows for obtaining qualitative properties (existence of a limiting domain, estimation of convergence speed) of the optimal domains as the observation time tends to $+\infty$

Motivated by applications requiring sparse or nonnegative controls, we investigate in 57 reachability properties of linear infinite-dimensional control problems under conic constraints. Relaxing the problem to convex constraints if the initial cone is not already convex, we provide a constructive approach based on minimising a properly defined dual functional, which covers both the approximate and exact reachability problems. Our main results heavily rely on convex analysis, Fenchel duality and the Fenchel-Rockafellar theorem. As a byproduct, we uncover new sufficient conditions for approximate and exact reachability under convex conic constraints. We also prove that these conditions are in fact necessary. When the constraints are nonconvex, our method leads to sufficient conditions ensuring that the constructed controls fulfill the original constraints, which is in the flavour of bang-bang type properties. We show that our approach encompasses and generalises several works, and we obtain new results for different types of conic constraints and control systems

In 52, we show the existence of Nash equilibria for the Navier-Stokes system and for the Oseen system. We consider the cases of the Dirichlet boundary conditions and of the Navier slip boundary conditions. Then, we study the asymptotic behavior of the Nash equilibria as the friction coefficient goes to $\infty$ in the Navier slip boundary condition. More precisely, we show that the Nash equilibria for the Navier slip boundary condition converge towards a Nash equilibrium for the Dirichlet boundary condition. We also show the convergence of the corresponding direct and adjoint states.

The book chapter 39 was written in tribute to Ivan Kupka, who greatly influenced the field of optimal control. We revisit the famous Zermelo problem. Under certain symmetry assumptions on the system, we classify the extremal trajectories using a Morse-Reeb classification of geodesics.

In the book chapter 40, we study an optimization problem for the functional determinant of elliptic differential operators on the circle - over essentially bounded functions. In the one dimensional case, existence and uniqueness of maximisers and minimisers is proved.

Inverse problems

Supervised by A. Munnier and K. Ramdani, the PhD of Anthony Gerber-Roth is devoted to the investigation of some geometric inverse problems, and can be seen as a continuation of the work initiated by the two supervisors in 80 and 79. 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 79 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 4.

While the single-layer operator for the Laplacian is well understood, questions remain concerning the single-layer operator for the Bilaplacian, particularly with regard to invertibility issues linked with degenerate scales. In 33, we provide simple sufficient conditions ensuring this invertibility for a wide range of problems.

In 34, we address the classical inverse problem of recovering the position and shape of obstacles immersed in a planar Stokes flow using boundary measurements. We prove that this problem can be transformed into a shape-from-moments problem to which ad hoc reconstruction methods can be applied. The effectiveness of this approach is confirmed by numerical tests that show significant improvements over those available in the literature to date.

The article 58 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 22, we address the problem of organ registration in augmented surgery, where the deformation of the patient’s organ is reconstructed in real-time from a partial observation of its surface. Physics- based registration methods rely on adding artificial forces to drive the registration, which may result in implausible displacement fields. In this paper, we look at this inverse problem through the lens of optimal control, in an attempt to reconstruct a physically consistent surface load. The resulting optimization problem features an elastic model, a least-squares data attachment term based on orthogonal projections, and an admissible set of surface loads defined prior to reconstruction in the mechanical model. After a discussion about the existence of solutions, we analyze the necessary optimality conditions and use them to derive a suitable optimization algorithm. We implement an adjoint method and we test our approach on multiple examples, including the so-called Sparse Data Challenge. We obtain very promising results, that illustrate the feasibility of our approach with linear and nonlinear models.

8.1 Bilateral contracts with industry

Participants: Yannick Privat.

• 2024-2025 : Yannick Privat started 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

10.1 Promoting scientific activities

10.2 Teaching - Supervision - Juries

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 Analysis2017HAL
• 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 articleN.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 Mathematics92023, 197-225HALback to text
• 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 Mathematicae21412018, 171-287HALDOI
• 6 articleC.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, 1105-1149HALDOI
• 7 articleA.Alexandre Munnier and K.Karim Ramdani. Calderón cavities inverse problem as a shape-from-moments problem.Quarterly of Applied Mathematics762018, 407-435HAL
• 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 articleJ.-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

• 60 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
• 61 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
• 62 articleM. 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
• 63 articleM.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.2011.5.745DOIback to text
• 64 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 Problems26122010, 125010, 30URL: http://dx.doi.org/10.1088/0266-5611/26/12/125010DOIback to text
• 65 articleG.Gottfried Bruckner and M.Masahiro Yamamoto. Determination of point wave sources by pointwise observations: stability and reconstruction.Inverse Problems1632000, 723--748back to text
• 66 articleT.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
• 67 articleT.Thomas Chambrion and A.Alexandre Munnier. Locomotion and control of a self-propelled shape-changing body in a fluid.J. Nonlinear Sci.2132011, 325--385URL: http://dx.doi.org/10.1007/s00332-010-9084-8DOIback to text
• 68 articleC.Cheok Choi, G.Gen Nakamura and K.Kenji Shirota. Variational approach for identifying a coefficient of the wave equation.Cubo922007, 81--101back to text
• 69 articleA.A. El Badia and T.T. Ha-Duong. Determination of point wave sources by boundary measurements.Inverse Problems1742001, 1127--1139back to text
• 70 articleE.Emilia Fridman. Observers and initial state recovering for a class of hyperbolic systems via Lyapunov method.Automatica4972013, 2250 - 2260back to text
• 71 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
• 72 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
• 73 bookV.Victor Isakov. Inverse problems for partial differential equations.127Applied Mathematical SciencesNew YorkSpringer2006back to text
• 74 bookB.Barbara Kaltenbacher, A.Andreas Neubauer and O.Otmar Scherzer. Iterative regularization methods for nonlinear ill-posed problems.6Radon Series on Computational and Applied MathematicsWalter de Gruyter GmbH & Co. KG, Berlin2008back to text
• 75 articleJ.J. Lohéac and A.A. Munnier. Controllability of 3D Low Reynolds Swimmers.ESAIM:COCV2013back to text
• 76 articleD.D.G. Luenberger. Observing the state of a linear system.IEEE Trans. Mil. Electron.MIL-81964, 74-80back to text
• 77 articleP.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
• 78 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/S0218202510004829DOIback to text
• 79 articleA.Alexandre Munnier and K.Karim Ramdani. Calderón cavities inverse problem as a shape-from-moments problem.Quarterly of Applied Mathematics762018, 407-435HALDOIback to textback to text
• 80 articleA.Alexandre Munnier and K.Karim Ramdani. Conformal mapping for cavity inverse problem: an explicit reconstruction formula.Applicable Analysis2016HALDOIback to text
• 81 bookJ.John O'Reilly. Observers for linear systems.170Mathematics in Science and EngineeringOrlando, FLAcademic Press Inc.1983back to text
• 82 articleK.K. Ramdani, M.M. Tucsnak and G.G. Weiss. Recovering the initial state of an infinite-dimensional system using observers.Automatica46102010, 1616-1625back to text
• 83 articleP.Plamen Stefanov and G.Gunther Uhlmann. Thermoacoustic tomography with variable sound speed.Inverse Problems2570750112009, 16back to text
• 84 bookH.Hieu Trinh and T.Tyrone Fernando. Functional observers for dynamical systems.420Lecture Notes in Control and Information SciencesBerlinSpringer2012back to text