CORIDA is a team labeled by INRIA, by CNRS and by University Henri Poincaré, via the Institut Élie Cartan of Nancy (UMR 7502 CNRS-INRIA-UHP-INPL-University of Nancy 2). The main focus of our research is the robust control of systems governed by partial differential equations (called pde's in the sequel). A special attention is devoted to systems with a hybrid dynamics such as the fluid-structure interactions. The equations modeling these systems couple either partial differential equations of different types or finite dimensional systems and infinite dimensional systems. We mainly consider inputs acting on the boundary or which are localized in a subset of the domain.

Infinite dimensional systems theory is motivated by the fact that a large number of mathematical models in applied sciences are given by evolution partial differential equations. Typical examples are the transport, heat or wave equations, which are used as mathematical models in a large number of problems in physics, chemistry, biology or finance. In all these cases the corresponding state space is infinite dimensional. The understanding of these systems from the point of view of control theory is an important scientific issue which has received a considerable attention during the last decades. Let us mention here that a basic question like the study of the controllability of infinite dimensional linear systems requires sophisticated techniques such as non harmonic analysis (cf. Russell ), multiplier methods (cf. Lions ) or micro-local analysis techniques (cf. Bardos–Lebeau–Rauch ). Like in the case of finite dimensional systems, the study of controllability should be only the starting point of the study of important and more practical issues like feedback optimal control or robust control. It turns out that most of these questions are open in the case of infinite dimensional systems. Consequently, our aim is to develop tools for the robust control of infinite dimensional systems. More precisely, given an infinite dimensional system one should be able to answer two basic questions:

Study the existence of a feedback operator with robustness properties.

Find an algorithm allowing the approximate computation of this feedback operator.

The answer to question 1 above requires the study of infinite dimensional Riccati operators and it is a difficult theoretical question. The answer to question 2 depends on the sense of the word “approximate”. In our meaning “approximate” means “convergence”, i.e., that we look for approximate feedback operators converging to the exact one when the discretization step tends to zero. From the practical point of view this means that our control laws should give good results if we use a large number of state variables. This fact is no longer a practical limitation of such an approach, at least in some important applications where powerful computers are now available. We intend to develop a methodology applicable to a large class of applications.

The problems we consider are modeled by the Navier-Stokes, Euler or Korteweg de Vries equations (for the fluid) coupled to the equations governing the motion of the solids. One of the main
difficulties of this problem comes from the fact that the domain occupied by the fluid is one of the unknowns of the problem. We have thus to tackle a
*free boundary problem*.

The control of fluid flows is a major challenge in many applications: aeronautics, pollution issues, regulation of irrigation channels or of the flow in pipelines, etc. All these problems cannot be easily reduced to finite dimensional models so a methodology of analysis and control based on pde's is an essential issue. In a first approximation the motion of fluid and of the solids can be decoupled. The most used models for an incompressible fluid are given by the Navier-Stokes or by the Euler equations.

The optimal open loop control approach of these models has been developed from both the theoretical and numerical points of view. Controllability issues for the equations modeling the fluid motion are by now well understood (see, for instance, Imanuvilov and the references therein). The feedback control of fluid motion has also been recently investigated by several research teams (see, for instance Barbu and references therein) but this field still contains an important number of open problems (in particular those concerning observers and implementation issues). One of our aims is to develop efficient tools for computing feedback laws for the control of fluid systems.

In real applications the fluid is often surrounded by or it surrounds an elastic structure. In the above situation one has to study fluid-structure interactions. This subject has been intensively studied during the last years, in particular for its applications in noise reduction problems, in lubrication issues or in aeronautics. In this kind of problems, a pde's system modeling the fluid in a cavity (Laplace equation, wave equation, Stokes, Navier-Stokes or Euler systems) is coupled to the equations modeling the motion of a part of the boundary. The difficulties of this problem are due to several reasons such as the strong nonlinear coupling and the existence of a free boundary. This partially explains the fact that applied mathematicians have only recently tackled these problems from either the numerical or theoretical point of view. One of the main results obtained in our project concerns the global existence of weak solutions in the case of a two-dimensional Navier–Stokes fluid (see ). Another important result gives the existence and the uniqueness of strong solutions for two or three-dimensional Navier–Stokes fluid (see ). In that case, the solution exists as long as there is no contact between rigid bodies, and for small data in the three-dimensional case.

The numerical methods used for computing the solutions of fluid or fluid structure problems in a direct setting (i.e., with given inputs) considerably progressed during the last years. In our project, we have proposed in an original numerical scheme to discretize the equations of motion for the system composed by a viscous incompressible fluid and several rigid bodies. One important characteristic of this scheme is that we use only a fixed mesh for the whole system, and therefore we do not need to remesh at some steps like for instance in the case of ALE methods. We have also developed two codes (in Matlab and in Scilab) based on our numerical scheme.

Another topic of great interest is the control of the interface of two fluids (typically water and air) by using as input the velocity of a moving wall which is a part of the boundary. One of the most popular models for this problem is given by the shallow water equations (Saint Venant equations) which neglect the dispersive effects. The controllability of several important systems governed by this type of equations has received a considerable attention during the last decade. Let us mention here the important work by Coron . If dispersive effects are considered, the relevant model is given by the Korteweg de Vries equation. The first work on the control of this equation goes back to Russell and Zhang (see ). An important advance in the study of this problem has been achieved in the work of Rosier where, for the first time, the influence of the length of the channel has been precisely investigated.

We use frequency tools to analyze different types of problems. The first one concerns the control, the optimal control and the stabilization of systems governed by pde's, and their numerical approximations. The second one concerns time-reversal phenomena, while the last one deals with numerical approximation of high-frequency scattering problems.

The first area concerns theoretical and numerical aspects in the control of a class of PDE's. More precisely, in a semigroup setting, the systems we consider have a skew-adjoint generator.
Classical examples are the wave, the Bernoulli-Euler or the Schrödinger equations. Our approach is based on an original characterization of exact controllability of second order conservative
systems proposed by K. Liu
. This characterization can be related to the Hautus criterion in the theory of finite dimensional systems (cf.
). It provides for time-dependent problems exact controllability criteria
**that do not depend on time, but depend on the frequency variable**conjugated to time. Studying the controllability of a given system amounts then to establishing uniform (with respect to
frequency) estimates. In other words, the problem of exact controllability for the wave equation, for instance, comes down to a high-frequency analysis for the Helmholtz operator. This
frequency approach has been proposed first by K. Liu for bounded control operators (corresponding to internal control problems), and has been recently extended to the case of unbounded
control operators (and thus including boundary control problems) by L. Miller
. Using the result of Miller, K. Ramdani, T. Takahashi, M. Tucsnak have obtained in
a new spectral formulation of the criterion of Liu
, which is valid for boundary control problems. This frequency test can be seen as an observability condition
for packets of eigenvectors of the operator. This frequency test has been successfully applied in
to study the exact controllability of the Schrödinger equation, the plate equation and the wave equation in a
square. Let us emphasize here that one further important advantage of this frequency approach lies in the fact that it can also be used for the analysis of space semi-discretized control
problems (by finite element or finite differences). The estimates to be proved must then be uniform with respect to
**both the frequency and the mesh size**.

In the case of finite dimensional systems one of the main applications of frequency domain methods consists in designing robust controllers, in particular of type. Obtaining the similar tools for systems governed by PDE's is one of the major challenges in the theory of infinite dimensional systems. The first difficulty which has to be tackled is that, even for very simple PDE systems, no method giving the parametrisation of all stabilizing controllers is available. One of the possible remedies consists in considering known families of stabilizing feedback laws depending on several parameters and in optimizing the norm of an appropriate transfer function with respect to this parameters. Such families of feedback laws yielding computationally tractable optimization problems are now available for systems governed by PDE's in one space dimension.

The second area in which we make use of frequency tools is the analysis of time-reversal for harmonic acoustic waves. This phenomenon described in Fink
is a direct consequence of the reversibility of the wave equation in a non dissipative medium. It can be used to
**focus an acoustic wave**on a target through a complex and/or unknown medium. To achieve this, the procedure followed is quite simple. First, time-reversal mirrors are used to generate an
incident wave that propagates through the medium. Then, the mirrors measure the acoustic field diffracted by the targets, time-reverse it and back-propagate it in the medium. Iterating the
scheme, we observe that the incident wave emitted by the mirrors focuses on the scatterers. An alternative and more original focusing technique is based on the so-called D.O.R.T. method
. According to this experimental method, the eigenelements of the time-reversal operator contain important
information on the propagation medium and on the scatterers contained in it. More precisely, the number of nonzero eigenvalues is exactly the number of scatterers, while each eigenvector
corresponds to an incident wave that selectively focuses on each scatterer.

Time-reversal has many applications covering a wide range of fields, among which we can cite
**medicine**(kidney stones destruction or medical imaging),
**sub-marine communication**and
**non destructive testing**. Let us emphasize that in the case of time-harmonic acoustic waves, time-reversal is equivalent to phase conjugation and involves the Helmholtz operator.

In , we proposed the first far field model of time reversal in the time-harmonic case.

This subject deals mainly with the numerical solution of the Helmholtz or Maxwell equations for open region scattering problems. This kind of situation can be met e.g. in radar systems in electromagnetism or in acoustics for the detection of underwater objects like submarines.

Two particular difficulties are considered in this situation

the wavelength of the incident signal is small compared to the characteristic size of the scatterer,

the problem is set in an unbounded domain.

These two problematics limit the application range of most common numerical techniques. The aim of this part is to develop new numerical simulation techniques based on microlocal analysis for modeling the propagation of rays. The importance of microlocal techniques in this situation is that it makes possible a local analysis both in the spatial and frequency domain. Therefore, it can be seen as a kind of asymptotic theory of rays which can be combined with numerical approximation techniques like boundary element methods. The resulting method is called the On-Surface Radiation Condition method.

Controllability and observability have been set at the center of control theory by the work of R. Kalman in the 1960's and soon they have been generalized to the infinite-dimensional context. The main early contributors have been D.L. Russell, H. Fattorini, T. Seidman, R. Triggiani, W. Littman and J.-L. Lions. The latter gave the field an enormous impact with his book , which is still a main source of inspiration for many researchers. Unlike in classical control theory, for infinite-dimensional systems there are many different (and not equivalent) concepts of controllability and observability. The strongest concepts are called exact controllability and exact observability, respectively. In the case of linear systems exact controllability is important because it guarantees stabilizability and the existence of a linear quadratic optimal control. Dually, exact observability guarantees the existence of an exponentially converging state estimator and the existence of a linear quadratic optimal filter. An important feature of infinite dimensional systems is that, unlike in the finite dimensional case, the conditions for exact observability are no longer independent of time. More precisely, for simple systems like a string equation, we have exact observability only for times which are large enough. For systems governed by other PDE's (like dispersive equations) the exact observability in arbitrarily small time has been only recently established by using new frequency domain techniques. A natural question is to estimate the energy required to drive a system in the desired final state when the control time goes to zero. This is a challenging theoretical issue which is critical for perturbation and approximation problems. In the finite dimensional case this issue has been first investigated in Seidman . In the case of systems governed by linear PDE's some similar estimates have been obtained only very recently (see, for instance Miller ). One of the open problems of this field is to give sharp estimates of the observability constants when the control time goes to zero.

Even in the finite-dimensional case, despite the fact that the linear theory is well established, many challenging questions are still open, concerning in particular nonlinear control systems.

In some cases it is appropriate to regard external perturbations as unknown inputs; for these systems the synthesis of observers is a challenging issue, since one cannot take into account the term containing the unknown input into the equations of the observer. While the theory of observability for linear systems with unknown inputs is well established, this is far from being the case in the nonlinear case. A related active field of research is the uniform stabilization of systems with time-varying parameters. The goal in this case is to stabilize a control system with a control strategy independent of some signals appearing in the dynamics, i.e., to stabilize simultaneously a family of time-dependent control systems and to characterize families of control systems that can be simultaneously stabilized.

One of the basic questions in finite- and infinite-dimensional control theory is that of motion planning, i.e., the explicit design of a control law capable of driving a system from an initial state to a prescribed final one. Several techniques, whose suitability depends strongly on the application which is considered, have been and are being developed to tackle such a problem, as for instance the continuation method, flatness, tracking or optimal control. Preliminary to any question regarding motion planning or optimal control is the issue of controllability, which is not, in the general nonlinear case, solved by the verification of a simple algebraic criterion. A further motivation to study nonlinear controllability criteria is given by the fact that techniques developed in the domain of (finite-dimensional) geometric control theory have been recently applied successfully to study the controllability of infinite-dimensional control systems, namely the Navier–Stokes equations (see Agrachev and Sarychev ).

This is a transverse research axis since all the research directions presented above have to be validated by giving control algorithms which are aimed to be implemented in real control systems. We stress below some of the main points which are common (from the implementation point of view) to the application of the different methods described in the previous sections.

For many infinite dimensional systems the use of co-located actuators and sensors and of simple proportional feed-back laws gives satisfying results. However, for a large class of systems of interest it is not clear that these feedbacks are efficient, or the use of co-located actuators and sensors is not possible. This is why a more general approach for the design of the feedbacks has to be considered. Among the techniques in finite dimensional systems theory those based on the solutions of infinite dimensional Riccati equation seem the most appropriate for a generalization to infinite dimensional systems. The classical approach is to approximate an LQR problem for a given infinite dimensional system by finite dimensional LQR problems. As it has been already pointed out in the literature this approach should be carefully analyzed since, even for some very simple examples, the sequence of feedbacks operators solving the finite dimensional LQR is not convergent. Roughly speaking this means that by refining the mesh we obtain a closed loop system which is not exponentially stable (even if the corresponding infinite dimensional system is theoretically stabilized). In order to overcome this difficulty, several methods have been proposed in the literature : filtering of high frequencies, multigrid methods or the introduction of a numerical viscosity term. We intend to first apply the numerical viscosity method introduced in Tcheougoué Tebou – Zuazua , for optimal and robust control problems.

As we already stressed in the previous sections the robust control of infinite dimensional systems is an emerging theory. Our aim is to develop tools applicable to a large class of problems which will be tested on models of increasing complexity. We describe below only the applications in which the members of our team have recently obtained important achievements.

One of the application domains of our work concerns the reduction of the noise due to the plane's engines during the take-off. This problem is addressed in the framework of the PhD thesis of Stefan Duprey at the Research Center of EADS (CIFRE contract). Antoine Henrot was his advisor, but his work was also supervised in Nancy by Karim Ramdani. In EADS, at Suresnes, his work was supervised by Isabelle Terrasse and François Dubois.

The main steps of this study of noise reduction are the following :

We write a code to compute the flow. Starting from the Euler equations in the potential and isentropic case, we are lead to solve a well–known non-linear elliptic problem, studied for example, in classical books like Glowinski or Nečas. To solve numerically this non linear problem, we use a fixed-point algorithm which turns out to be convergent.

We assume the acoustic perturbation to be potential and decoupled from the flow. By linearization of Euler equations, we get a linear problem satisfied by the acoustic potential. The coefficients of this equation involve the potential flow computed at step 1. The boundary conditions are either of Neumann or impedance type.

We have to write a code to compute the solution of step 2. The fact that the flow can be considered constant at infinity simplifies the equation outside a large domain containing the plane. This leads to two possible ways to solve this problem: using globally a Lorentz transform or using a domain decomposition method.

When the two previous codes work satisfactory, we can imagine optimization procedures by acting either on the shape of the engine or on its coating.

During his thesis (defended in October 2006), Stefan Duprey realized efficiently points 1 to 3. Moreover, he was also able to handle the theoretical questions of existence and uniqueness of a solution. At least three papers should follow from the whole work.

We began this year to study a new class of applications of observability theory. The investigated issues concern inverse problems in Magnetic Resonance Imaging (MRI) of moving bodies with emphasis on cardiac MRI. The main difficulty we tackle is due to the fact that MRI is, comparatively to other cardiac imaging modalities, a slow acquisition technique, implying that the object to be imaged has to be still. This is not the case for the heart where physiological motions, such as heart beat or breathing, are of the same order of magnitude as the acquisition time of an MRI image. Therefore, the assumption of sample stability, commonly used in MRI acquisition, is not respected. The violation of this assumption generally results in flow or motion artifacts. Motion remains a limiting factor in many MRI applications, despite different approaches suggested to reduce or compensate for its effects Welch et al. . Mathematically, the problem can be stated as follows: can we reconstruct a moving image by measuring at each time step a line of its Fourier transform? From a control theoretic point of view this means that we want to identify the state of a dynamical system by using an output which is a small part of its Fourier transform (this part may change during the measurement).

There are several strategies to overcome these difficulties but most of them are based on respiratory motion suppression with breath-hold. Usually MRI uses ECG information to acquire an image over multiple cardiac cycles by collecting segments of Fourier space data at the same delay in the cycle Lanzer et al. , assuming that cardiac position over several ECG cycles is reproducible. Unfortunately, in clinical situations many subjects are unable to hold their breath or maintain stable apnea. Therefore breath-holding acquisition techniques are limited in some clinical situations. Another approach, so called real-time, uses fast, but low resolution sequences to be faster than heart motion. But these sequences are limited in resolution and improper for diagnostic situations, which require small structure depiction as for coronary arteries.

The observation of nature and of the "perfection" of most of its mechanisms of living beings drives us to search
**a principle of optimality**which governs those mechanisms. If a mathematical model exists for describing a biological phenomenon or component of a living being, there is a temptation to
quantify the optimality by finding a functional which leads to the optimality principle. The confrontation between the computed optimum and the real one leads us to validate or invalidate the
model and/or the choice of the functional. This inverse modeling method consists in finding the mathematical model starting from observations and theirs consequences. If the optimal shape
which is issued from the mathematical model is close to the real shape, we have reasons to believe that the full model (equation and functional) is good. If not, one has to reject it and find
another one, or improve it.

The mathematical study of these questions strongly uses tools of "shape variation and optimization" as developed in the book .

We applied some modern techniques of automatic control to the command of the cooling of the fuel cells stack; these techniques result in a significant enhancement of the functioning of the cooling circuit. This problem is studied in the framework of the PhD thesis of Fehd Ben Aïcha (co-supervised with J.C. Vivalda and with M. Sorine, head of the SOSSO2 project).

Our aim is to develop Scilab tools for the numerical approximation of pde's. This task requires powerful sparse matrix primitives, which are not currently available in Scilab. We have thus developed the SCISPT Scilab toolbox, which interfaces the sparse solvers UMFPACK v4.3 of Tim Davis and TAUCS SNMF of Sivan Toledo. It also provides various utilities to deal with sparse matrices (estimate of the condition number, sparse pattern visualization, etc.). We intend to extend this work in the framework of collaborations with the Scilab consortium recently created.

A part of our activity in this field has been devoted to the study of well-posedness and to the numerical analysis of the equations modeling the motion of rigid bodies in an incompressible fluid. An important part of this work is done in collaboration with a group from the University of Chile through the associated team ANCIF. Many of the works we are quoting in what follows have been done with a member of this group (Jorge San Martín, Jaime Ortega, Carlos Conca, Patricio Cumsille).

Concerning the well-posedness results, the last main achievements are reported in the papers
,
. In reference
we gave an existence and uniqueness result in the case of a viscous fluid filling the exterior of an infinite
cylinder. The generalization of this result to more general geometries is studied in reference
. In this recent work, Cumsille and Takahashi have obtained the well-posedness in the case of a rigid body of
arbitrary shape moving in a viscous incompressible fluid. In the paper
, the authors prove the global in time existence of a
*classical*solution for the equations modeling the motion of a rigid body of arbitrary shape moving in a perfect incompressible fluid. The regularity of the solutions is an important
issue, as most of the control results for Euler systems are based of Coron's return method and so they involve smooth solutions. Notice also that a similar result for Navier-Stokes system is
not yet available.

In the last year, we have started to study the motion of aquatic organisms. More precisely, in , Takahashi and Tucsnak have considered the control of the motion of the rigid bodies moving in a fluid by using a velocity or a torque control acting on the rigid part only. They have obtained several reachability results at low Reynolds number. These results give a new insight of the very interesting propulsion mechanisms used by ocean micro-organisms (like ciliata). In , Scheid, Takahashi and Tucsnak have given and analyzed a model for the motion of a fish. The model consists in a solid undergoing an undulatory deformation, which is immersed in a viscous incompressible fluid. The displacement of the “creature” is decomposed into a rigid part and a deformation (undulatory) part. One of the particularities of the work is that no particular constitutive laws for the solid are used but instead the non–rigid part of the deformation is imposed. The advantages of this approach consist in the global character (up to possible contacts) of the obtained strong solutions and in the possibility of using numerical methods inspired by those developed in the rigid-fluid case. In fact, by considering a similar method to the one developed in the article , numerical simulations have been performed for the motion of a fish into a viscous incompressible fluid. Recall that the non-rigid part of the deformation is only imposed and the trajectory of the fish is not prescribed. We observe numerically that the fish manages to go ahead and thus a self-propelled motion is reached by the fish. A numerical code has been developed for the 2D case in Scilab with the use of the SCISPT toolbox for sparse solver (see Section 5).

Since September 2006, an INRIA associated engineer has been recruited to improve the current numerical code, to rewrite it in the Matlab software and finally to develop a 3D software for the
fish-like swimming. The improvement of the 2D code is in progress and deals with the Navier-Stokes solver together with the optimization of the Matlab implementation of the numerical code. The
Navier-Stokes solver has to allow to tackle realistic situations where the Reynolds number lies between
10
^{3}and
10
^{6}, depending on the nature of the fish-like swimming we consider (trout, salmon, eel, ...). The development of the 3D software has not started yet.

In , Houot and Munnier study the well-posedness of the equations modeling the motion of 3 dimensional rigid bodies immersed in a potential fluid flow. Although the existence and uniqueness of smooth solutions is well known in the particular case of a single solid in an infinitely extended fluid, the problem gets more involved in the presence of several bodies or in a bounded fluid cavity. By using shape sensitivity analysis, Houot and Munnier prove that even in this case there exists analytic solutions up to the collision between two solids or between a solid with the fluid boundary. Concerning this problem of collisions, they show that an infinite cylinder, filling with a potential fluid an half space, can collide with the boundary with non zero relative velocity. This result heavily contrasts with what happen in the case of a viscous fluid in which collisions are not possible. They compute the velocity damping of the cylinder when approaching the wall, proving that d'Alembert paradox does not hold in this case. In , Munnier extends the results of existence of analytic solutions to the case of deformable bodies.

Another control problem has been obtained by Takahashi (in collaboration with Imanuvilov) for a fluid–structure system ( ). More precisely, it is proved that we can control (locally) the motion of both the rigid bodies and of the fluid by using an input given by the velocity field of the fluid on a part of the exterior boundary of the domain. This question has a more theoretical motivation: we want to show that the presence of rigid bodies does not change in an essential way the controllability properties of the system.

The same problem may be tackled with a perfect fluid (not necessarily potential), the normal component of the fluid velocity being controlled on a small part of the boundary. For a ball in 2D, it has been shown by Rosier that we may control the position of the ball, the ball being at rest at the beginning and at the end of the control process. The extension to a solid with one axial symmetry (boat) is under investigation with Glass (Paris 6).

Another situation which has been treated is that of a rigid body moving in an inviscid fluid. More precisely, Chambrion and Sigalotti studied the controllability properties of a neutrally buoyant underwater vehicle immersed in an infinite volume of an inviscid incompressible fluid in irrotational motion (see ). Due to the potential nature of the flow, the state of the system is fully determined by a finite set of real variables which parameterize the set of configurations (position and attitude) of the moving body and its linear and angular momenta. The dynamics of such momenta, seen from the perspective of the solid, are described by the classical Kirchhoff equations whose control properties have been thoroughly studied (see for instance Leonardo et al., Astolfi et al. , ). In we tackle the less studied problem of controlling and stabilizing the full 12-dimensional nonlinear system describing the evolution of configuration and momenta.

Another quite different problem in fluid-structure interaction comes from the industrial PhD thesis of S. Baillet. At the end of the exploitation of an oil well, surface pumps are no longer
efficient enough to extract the oil remaining in the reservoir. Instead of closing the well, oil companies generally use well–pumps, introduced deep into the ground in order to maintain the
production. Such pumps are composed of a succession of identical stages (typically 15–20 stages, but it could go up to 100 stages) arranged in series. The optimization process needs numerical
simulation, but representing the whole pump numerically is impossible (for obvious calculation costs). Thus it is necessary to simplify the model, by representing only one stage. Therefore, a
very natural question which arises is the following: does the flow in one standard stage of the pump looks like the one we would obtain by considering periodic boundary conditions at the
entrance and the exit of the stage. Of course, one cannot hope that this kind of result holds for the first and the last stages which are still influenced by the boundary conditions at the top
and the bottom of the pump. Baillet, Henrot and Takahashi
have obtained an exponential convergence with respect to the number of stages when the fluid motion is modeled by
the Stokes equations. The main topic of this thesis was to look for an improvement of the shape of the pump, thanks to a classical
*shape optimization*approach. The objective we wanted to improve was the pressure gap between the top and the bottom of the pump. Numerical simulations were done using the commercial
software Fluent. The results of this approach appeared in
and is summed up in
.

It is well–known that the solution of LQR optimal control problems is given through a feedback operator involving a Riccati operator
P. This operator solves a Riccati equation in infinite dimension. Of course, in practice, one can only determine an approximate solution
P_{h}of this equation, and the natural question that arises is the following : does the approximate solution obtained using this operator
P_{h}(instead of
P) converge to the solution of the continuous problem? This question has been so far studied by many authors (see for instance
,
,
). In all these papers, one of the main assumptions is that the discretized systems should be
**uniformly stabilizable**with respect to the discretization parameter
h. Unfortunately, most of the standard numerical methods (finite element or finite differences) do not fulfill this condition. Using the frequency characterization of stabilizability
proposed by Liu and Zheng in
, we give in
a general technique ensuring the uniform stabilizability of classical numerical methods (finite element or
finite differences). This technique consists in adding to the standard numerical schemes a suitable numerical viscosity. Compared to the results of
, the main novelties brought in by our results lie in their generality, since they hold for the finite element
space semi-discretization of a wide class of second order evolution equations.

The above results on time reversal and photonic crystals, added to other contributions on control of PDE's constitute the material of the “Habilitation à Diriger des Recherches” of Karim Ramdani (see ), defended on october 19th, 2007.

In , a new On-Surface Radiation Condition is introduced. The construction takes care of the different kinds of contributions which compose the scattered ray. Mathematically, the condition is given through a square-root operator. A paraxialization process of this operator based on Padé approximants in combination with an iterative finite element scheme leads to an efficient and accurate numerical method. A complete review of the On-Surface Radiation Condition method is provided in the Chapter Book . In a collaboration with Y. Boubendir from the University of Minneapolis, the development of integral operator based preconditioners for solving the scattering problem by a penetrable body is presented in .

In we obtain new blow-up rates for fast controls of one dimensional Schrödinger and heat equations, by combining techniques of co analysis and non harmonic Fourier analysis. The obtained results significantly improve the best existing estimates which have been published in and .

In
we consider the two dimensional Schrödinger and Euler-Bernoulli plate equations. By generalizing results from
Ramdani, Takahashi, Tenenbaum and Tucsnak
, we prove that these systems are exactly observable
*in arbitrarily small time*. Moreover, we show that the above results hold even if the observation regions have
*arbitrarily small measures*. More precisely, we prove that in the case of homogeneous Neumann boundary conditions with Dirichlet boundary observation, the exact observability property
holds for every observation region with non empty interior. In the case of homogeneous Dirichlet boundary conditions with Neumann boundary observation, we show that the exact observability
property holds if and only if the observation region has an open intersection with an edge of each direction. Moreover, we give
*explicit estimates for the blow-up rate*of the observability constants as the time and/or the size of the observation region tend to zero. The main ingredients of the proofs are an
effective version of a theorem of Beurling and Kahane on non harmonic Fourier series and an estimate for the number of lattice points in the neighborhood of an ellipse.

The results have been essentially used in to get sharp conditions for the stabilizability of the Schrödinger and plate equations. Our method combines results from with some estimates of an appropriate transfer function and with the functional analytic results from .

We have also obtained some results on the stabilization for visco-elastic materials. In many mechanical applications, the damping effects of materials are non-local and of memory type. It is therefore important to obtain asymptotic properties of solutions as time grows, to know for instance for applications, if undesired vibrations of solutions will asymptotically decrease for large time. For such materials, the solutions are described by hyperbolic PDE's, for which the feedback law is of memory type, which means that the feedback operator is a convolution operator in time, therefore nonlocal (in time). One of the difficulty is that for such problems, the invariance by translation in time is lost, as well as the Lasalle invariance principle. Also, dissipation properties and multiplier method is more involved. This leads to many challenging open questions. In the paper , Alabau studies visco-elastic materials for which the feedback law is of memory type. Alabau assumes that the kernel of the convolution operator is exponentially or polynomially decaying at infinity. Alabau proves stabilization properties with exponential or polynomial decay of the energy for general abstract hyperbolic equations with applications to various models : waves, elasticity, Petrowsky equations. In collaboration with Cannarsa, Alabau is studying the case of different growing rates than exponential or polynomial of the kernel. This case is much more involved and requires new mathematical techniques. We have new results in this direction.

Damping and stabilization properties of materials are an important field in control theory. In many applications, such damping phenomenas are highly nonlinear and the way of decay of the feedback is not necessarily known. In , Alabau shows her method can be applied with success to coupled systems such as Timoshenko beams, subjected to first order coupled and a single control force. Alabau proves general decay estimates (of polynomial, logarithmic ..., type) depending on the growth properties of the feedback at the origin in case of equal speeds of propagation for the equation of the transverse displacement of the beam and of the rotation angle. In case of different speeds of propagation, Alabau proves polynomial decay for smooth solutions in case of globally Lipschitz nonlinear feedbacks, with linear growth.

Henrot and Cox were interested in a musical problem which strongly use the eigenfrequency of the damped wave equation. It is a question relating to a model for harmonics on stringed
instruments which could be set as: is it possible to achieve “Correct Touch” in the pointwise damping of a fixed string? By correct touch, we consider the following. When we place a finger
lightly at one of the nodes of the low frequency harmonics, it forces the string to play a note that sounds like a superposition of those normal modes with nodes at the location of the finger.
Now, the question is to determine what should be the pressure of the finger in order to best damps the remaining modes. From a mathematical point of view, we consider the wave equation with a
damping as
b_{a}u_{t}where
u_{t}is the speed,
athe location of the pressure,
_{a}is the Dirac distribution at
aand
bthe intensity of the pressure. We want to determine, for each
a, the
bwhich minimizes the spectral abscissa of the modes not vanishing at
a. This involves a precise qualitative analysis of the spectrum of the damped operator in the complex plane. This work will appear in
.

In Sigalotti together with Boscain and Agrachev considered a generalization of Riemannian geometry which naturally arises in quantum control theory (see, for instance, Boscain et al. ) and in the study of hypoelliptic operators (Franchi et al. ). We obtain in this framework a generalization of the Gauss-Bonnet formula.

The observability of discrete time systems has been studied by Vivalda in . In , we deal with the notion of practical observers for affine systems with unknown inputs. These systems can be written as

We show that a system which is observable with respect to the unknown input can be put into triangular form. Moreover, for this kind of systems, we design a practical observer, that is to say a family of auxiliary dynamical system written as

such that

with and .

A relatively recent activity of our group consists in proposing new methods for the reconstruction of moving objects via Magnetic Resonance Imaging (MRI). This is the main topic of the Phd
thesis of Cîndea, which is directed jointly by Tucsnak and Vuissoz from the MRI laboratory of the University Hospital of Nancy. One ideas of this works is to use the theory of observers in
order to reconstruct the state from the measurements in the Fourier space. This work is based on solving preliminary modeling and identification problems which require computations which
cannot, for the moment, be performed in real time. These problems are tackled by our group by two methods. The first one, presented in a paper submitted to
*MRM*reduces the problem to an integral equation which is solved using a regularization procedure. This method is based on the fact that the motion of the moving body has been partially
detected by appropriate sensors.

The second method uses weaker information on the motion of the body, namely that it is periodic. The aim of our work is to provide a method for cardiac imaging reconstruction at each cardiac or respiratory phase. We propose a method inspired by Zwaan . In his method, heart motion during breath-hold was studied with the assumption that each heartbeat is a rescaled copy of a standard heartbeat. Here, the same assumption is extended to respiratory motion. Similarly to the algorithm presented by Zwaan , our method is based on the properties of the Reproducing Kernel Hilbert Spaces (RKHS) , which provide a general and rigorous framework for handling interpolation problems, and have been widely used in signal and image processing. This framework is of particular relevance here, as retrospective gating can be reformulated as a scattered data interpolation in a 1D or 2D space.

is the dichotomic structure of the lung tree and the dimension of each branch optimal?

is the cylindrical shape of a given branch optimal?

A CIFRE contract has been signed with IFP (French Center of Research and Industrial Development for oil and automotive industries, based in Rueil-Malmaison). Indeed, Baillet began a PhD
program (advisor Henrot), funded by a CIFRE grant, in June 2004. The aim of this work consists in improving the efficiency of the pump sucking the crude oil from the subsoil. The control
variable is the geometry of the pump and the output of the system is the gap of pressure. The system is governed by the three-dimensional Navier–Stokes equations. Instead of computing the exact
gradient of the criterion (which seems too difficult or too costly), we intend to compute an incomplete gradient, possibly coupled with
*one-shot*type methods. Then, a classical gradient-type or Quasi-Newton optimization algorithm is performed.

The work in the field of automotive industry described in subsection 4.4 is formalized by a CIFRE contract with Renault (the PhD Student is employed by that firm since June 2005).

At INRIA: Tucsnak is member of the Executive Team and of the Project Committee of the INRIA Nancy-Grand Est Research Centre .

In the Universities and in CNRS committees:

Henrot is the head of the Institut Élie Cartan de Nancy (IECN).

Tucsnak is member of the Scientific Council of UHP.

Alabau is member of CNU, section 26 and member of the Scientific Council of University Paul Verlaine-Metz.

Our team is part of the GDR entitled “Fluid-Structure Interactions”.

CPER (“Contrat Plan Etat Région”):

Mason, Sigalotti (leader) and Vivalda are in “Stabilité et Commande des Systèmes à Commutations”. This is project in the AOC theme, in collaboration with the Automatic Control team at CRAN, is devoted to the stabilization of hybrid systems arising in the domain of DC-DC converters.

Pinçon, Scheid, Takahashi (leader), Tavernier and Tucsnak are in the project “Se propulser dans un fluide, analyse, contrôle et visualisation” (AOC theme), in collaboration with the INRIA team, ALICE.

ANR (“Agence Nationale de la recherche”):

Pinçon, Scheid, Takahashi and Tucsnak are in the ANR project “MOSICOB” (“ANR Blanche”) for three years in collaboration with the University of Paris Sud and with the University of Grenoble.

Rosier and Takahashi are in the ANR project “Contrôle d'équations aux dérivées partielles en mécanique des fluides” (in collaboration with the University of Paris 6, the University of Versailles and the University of Nanterre).

Our team is part of the GDR entitled “Fluid-Structure Interactions”.

The CORIDA project is linked with a group of the University of Chile through the associated team ANCIF.

A IFCPAR grant with the Tata Institute (Bengalooru, India).

A “Partenariat Hubert Curien” (PHC) with the IST (Lisbon, Spain).

A “Partenariat Hubert Curien” (PHC) with the Mathematical Institute of the Academy of Sciences of the Czech Republic (Prague).

A INRIA–DGRSRT project (with the Faculty of Monastir, Tunisia).

Evans Harrell (Georgia Tech, Atlanta), Gérard Philippin (U. Laval, Québec), George Weiss (Imperial College, London, England), Ana Leonor Silvestre and Carlos Alves (IST Lisbon, Portugal), Kais Ammari (Monastir, Tunisie).

Alabau:

ICIAM 2007, Zurich, co-organization with Cannarsa (Univ. Rome II) and Vancostenoble (Univ. Toulouse III) of a mini-symposium entitled: Control and stabilization of PDE's of interest to the applied sciences.

6th European-Maghreb Conference on Evolution Equations, CIRM Luminy, co-organized with Ouhabaz (Univ. Bordeaux I) and Monniaux (Univ. Aix Marseille 3).

Henrot:
*Mini-workshop: Shape Analysis for Eigenvalues*, Oberwolfach, Germany, April 8–13 2007, co-organized with D. Bucur and G. Buttazzo.

Tucsnak: co-organizer of the International Workshop on Analysis and Control of pde's, Pont-à-Mousson, France, June 25–29, 2007

Alabau:

September–October 2008, Workshop on Direct, Inverse and Control Problems for PDE's DICOP, Cortona, Italie, Plenary conference.

May 2008 7th AIMS Conference, Arlington, USA, 18-21 may, invited to two special sessions : “Nonlocal equations and diffusion equations” and “Qualitative behavior of solutions to evolutionary PDE's”

April 2008 One week-invitation at Univ. of Halle, Germany.

December 2007 Invitation to “Journées de Théorie Spectrale et EDP”, Kairouan, Tunisia, December 17–19.

October 2007 Invitation to Scuola Normale di Pisa, Italy, October 28–31.

September 2007 Workshop on “Contrôle et les Problèmes Inverses”, Besançon.

August 2007 6th Worskhop on PDE and Applications, Rio de Janeiro, Brazil, August 28–31. Plenary conference.

September 2007 II symposium on Partial Differential Equations, Maringa, September 3–6. Plenary conference.

July 2007 23rd IFIP TC 7 Conference on System Modeling and Optimization, July 23–27, 2007, Cracovia (Poland).

July 2007 ICIAM Conference, Zurich.

June 2007 Workshop on Direct, Inverse and Control Problems for PDE's DICOP, Roma, Italy, June 25–28 2007. Plenary conference.

April 2007 Two weeks invitation at INDAM, Roma, April 1–15 2007.

Antoine:

Invited talk at the Workshop "Computational Electromagnetism and Acoustics", Oberwolfach, Germany, February 2007.

Invited talk at the Workshop "High Frequency Scattering and Propagation", 24-26 July, 2007, Reading University, England, Waves 2007 (satellite workshop of the program "Highly Oscillatory Problems"), organized by the Isaac Newton Institute for Mathematical Sciences, Cambridge, England.

Invitation to the program "Bose-Einstein Condensation and Quantized Vortices in Superfluidity and Superconductivity", Institute for Mathematical Sciences (IMS), the National University of Singapore, November 2007.

Invited talk to the Workshop: "Méthodes pour les problèmes direct et inverse de diffraction: progrès récents", Pau, France, December 2007.

Invited talk to the Workshop "Analysis of Boundary Element Methods", Oberwolfach, Germany, April 2008.

Chambrion:

*Control of the Schrödinger equation in infinite dimensions*, joint work with Sigalotti and Mason, ESF conference “Control, Constraints and Quanta” held in Bedlewo, Poland, 10–16
October 2007.

Henrot:

*Paris*: Journées dédiées à la mémoire de Thomas Lachand-Robert, January 2007.

*Neuchatel (Suisse)*: Semaine de Théorie Spectrale et Géométrie, February 2007.

*Oberwolfach (Germany)*: Mini-workshop “Shape analysis for eigenvalues”, April 2007.

*Banff (Canada)*: Workshop “Geometric Inequalities”, June 2007.

*Cracovie (Pologne)*: 23rd IFIP TC 7 Conference on System Modeling and Optimization, July 2007.

*Stockholm (Sweden)*: Conference on potential analysis dedicated to Björn Gustafsson for his 60th birthday, October 2007.

Sigalotti

Trieste (Italy): Workshop on “Control, optimization and stability of non-linear systems:geometric and analytic methods”, May 2007.

Montpellier: Workshop “Contrôle et Optimisation”, September 2007.

Takahashi

International Workshop on Analysis and Control of pde's, Pont-à-Mousson, France, June 25–29, 2007

Equadiff'07 (invited minisymposium “Recent progress in mathematical fluid mechanics”), Vienna, Austria, August 5–11, 2007.

Workshop “Contrôle et les Problèmes Inverses”, Besançon, September 26–27, 2007.

Workshop French–Indian, Toulouse and Bangalore (Video-conferencing), October 25–26, 2007.

Tucsnak

SIMUMAT, Suumer School on Fluid Structure Interaction, Castro Urdiales, Spain, July 2007

IFAC Workshop on Control of Distributed Parameter Systems, University of Namur (FUNDP), Namur, Belgium, July 23-27, 2007

International Conference of Theoretical and Numerical Fluid Mechanics III, Vancouver, Canada, August 13-17, 2007

CEDYA'07 (Spanish Conference on Differential Equations), Seville, Spain, September 2007.

Workshop French–Indian, Toulouse and Bangalore (Video-conferencing), October 25–26, 2007.

Antoine

City University of Hong Kong, February 2008.

Scheid

CMM, Santiago de Chile, January 2007

Takahashi

Portugal, November 2007

Tucsnak

Portugal, October 2007

Tucsnak is associated editor of “SIAM Journal on Control” and of “ESAIM COCV”.

Most of the project members are professors or assistant professors so they have an important teaching activity. We mention here only the graduate courses.

Non linear analysis of PDE's and applications (Alabau);

Scientific Computing (Henrot);

Numerical analysis of the Navier-Stokes equations (Scheid);

Introduction to the Control Theory (Sigalotti and Takahashi);

The Navier–Stokes Equations (Tucsnak).