The Team aims at designing and developing constructive methods in modeling, identification and control of dynamical, resonant and diffusive systems.
Function theory and approximation theory in the complex domain, with applications to frequency identification of linear systems and inverse boundary problems for the Laplace and Beltrami operators:
System and circuit theory with applications to the modeling of analog microwave devices. Development of dedicated software for the synthesis of such devices.
Inverse potential problems in 2D and 3D and harmonic analysis with applications to nondestructive control (from magneto/electroencephalography in medical engineering or plasma confinement in tokamaks for nuclear fusion).
Control and structure analysis of nonlinear systems with applications to orbit transfer of satellites.
Collaboration under contract with Thales Alenia Space (Toulouse, Cannes, and Paris), CNES (Toulouse), XLim (Limoges), CEAIRFM (Cadarache).
Exchanges with UST (Villeneuve d'Asq), University BordeauxI (Talence), University of Orléans (MAPMO), University of Pau (EPI Inria Magique3D), University MarseilleI (CMI), CWI (the Netherlands), SISSA (Italy), the Universities of Illinois (UrbanaChampaign USA), California at San Diego and Santa Barbara (USA), Michigan at EastLansing (USA), Vanderbilt University (Nashville USA), Texas A&M (College Station USA), ISIB (CNR Padova, Italy), Beer Sheva (Israel), RMC (Kingston, Canada), University of Erlangen (Germany), Leeds (UK), Maastricht University (The Netherlands), Cork University (Ireland), Vrije Universiteit Brussel (Belgium), TUWien (Austria), TFHBerlin (Germany), CINVESTAV (Mexico), ENIT (Tunis), KTH (Stockholm).
The project is involved in the ANR projects AHPI (Math., coordinator) and Filipix (Telecom.), in a EMS21RTG NSF program (with Vanderbilt University, Nashville, USA), in an NSF Grant with Vanderbilt University and the MIT, in an EPSRC Grant with Leeds University (UK), in a InriaTunisian Universities program (STIC, with LAMSINENIT, Tunis).
Identification typically consists in approximating experimental data by the prediction of a model belonging to some model class. It consists therefore of two steps, namely the choice of a suitable model class and the determination of a model in the class that fits best with the data. The ability to solve this approximation problem, often nontrivial and illposed, impinges on the effectiveness of a method.
Particular attention is payed within the team to the class of stable linear timeinvariant systems, in particular resonant ones, and in isotropically diffusive systems, with techniques that dwell on functional and harmonic analysis. In fact one often restricts to a smaller class — e.g.rational models of suitable degree (resonant systems, see section ) or other structural constraints— and this leads us to split the identification problem in two consecutive steps:
Seek a stable but infinite (numerically: high) dimensional model to fit the data. Mathematically speaking, this step consists in reconstructing a function analytic in the right halfplane or in the unit disk (the transfer function), from its values on an interval of the imaginary axis or of the unit circle (the bandwidth). We will embed this classical illposed issue ( i.e.the inverse Cauchy problem for the Laplace equation) into a family of wellposed extremal problems, that may be viewed as a regularization scheme of Tikhonovtype. These problems are infinitedimensional but convex (see section ).
Approximate the above model by a lower order one reflecting further known properties of the physical system. This step aims at reducing the complexity while bringing physical significance to the design parameters. It typically consists of a rational or meromorphic approximation procedure with prescribed number of poles in certain classes of analytic functions. Rational approximation in the complex domain is a classical but difficult nonconvex problem, for which few effective methods exist. In relation to system theory, two specific difficulties superimpose on the classical situation, namely one must control the region where the poles of the approximants lie in order to ensure the stability of the model, and one has to handle matrixvalued functions when the system has several inputs and outputs, in which case the number of poles must be replaced by the McMillan degree (see section ).
When identifying elliptic (Laplace, Beltrami) partial differential equations from boundary data, point 1. above can be recast as an inverse boundaryvalue problem with (overdetermined DirichletNeumann) data on part of the boundary of a plane domain (recover a function, analytic in a domain, from incomplete boundary data). As such, it arises naturally in higher dimensions when analytic functions get replaced by gradients of harmonic functions (see section ). Motivated by free boundary problems in plasma control and questions of source recovery arising in magneto/electroencephalography, we aim at generalizing this approach to the real Beltrami equation in dimension 2 (section ) and to the Laplace equation in dimension 3 (section ).
Step 2. above, i.e., meromorphic approximation with prescribed number of poles—is used to approach other inverse problems beyond harmonic identification. In fact, the way the singularities of the approximant ( i.e.its poles) relate to the singularities of the approximated function is an allpervasive theme in approximation theory: for appropriate classes of functions, the location of the poles of the approximant can be used as an estimator of the singularities of the approximated function (see section ).
We provide further details on the two steps mentioned above in the subparagraphs to come.
Given a planar domain
D, the problem is to recover an analytic function from its values on a subset of the boundary of
D. It is convenient to normalize
Dand apply in each particular case a conformal transformation to meet a “normalized” domain. In the simply connected case, which is that of the halfplane, we fix
Dto be the unit disk, so that its boundary is the unit circle
T. We denote by
H^{p}the Hardy space of exponent
pwhich is the closure of polynomials in the
L^{p}norm on the circle if
1
p<
and the space of bounded holomorphic functions in
Dif
p=
. Functions in
H^{p}have welldefined boundary values in
L^{p}(
T), which make it possible to speak of (traces of) analytic functions on the boundary.
A standard extremal problem on the disk is :
(
P_{0}) Let
1
p
and
fL^{p}(
T); find a function
gH^{p}such that
g
fis of minimal norm in
L^{p}(
T).
When seeking an analytic function in
Dwhich approximately matches some measured values
fon a subarc
Kof
T, the following generalization of (
P_{0}) naturally arises:
(
P) Let
1
p
,
Ka subarc of
T,
fL^{p}(
K),
and
M>0; find a function
gH^{p}such that
and
g
fis of minimal norm in
L^{p}(
K)under this constraint.
Here
is a reference behavior capsulizing the expected behavior of the model off
K, while
Mis the admissible error with respect to this expectation. The value of
preflects the type of stability which is sought and how much one wants to smoothen the data.
To fix terminology we generically refer to (
P) as a
bounded extremal problem. The solution to this convex infinitedimensional optimization problem can be obtained upon iteratively solving spectral equations for appropriate Hankel and
Toeplitz operators, that involve a Lagrange parameter, and whose right handside is given by the solution to (
P_{0}) for some weighted concatenation of
fand
. Constructive aspects are described in
,
,
, for
p= 2,
p=
, and
1<
p<
, while the situation
p= 1is essentially open.
Various modifications of
(
P)have been studied in order to meet specific needs. For instance when dealing with lossless transfer functions (see section
), one may want to express the constraint on
in a pointwise manner:

g

Ma.e. on
, see
,
for
p= 2and
= 0.
The abovementioned problems can be stated on an annular geometry rather than a disk. For
p= 2the solution proceeds much along the same lines
. When
Kis the outer boundary, (
P) regularizes a classical inverse problem occurring in nondestructive control, namely to recover a harmonic function on the inner boundary from overdetermined DirichletNeumann data on
the outer boundary (see sections
and
). Interestingly perhaps, it becomes a tool to approach Bernoulli type problems
for the Laplacian, where overdetermined observations are made on the outer boundary and we
seek the inner boundaryknowing it is a level curve of the flux (see section
). Here, the Lagrange parameter indicates which deformation should be applied
on the inner contour in order to improve the fit to the data.
Continuing effort is currently payed by the team to carry over bounded extremal problems and their solution to more general settings.
Such generalizations are twofold: on the one hand Apics considers 2D diffusion equations with variable conductivity, on the other hand it investigates the ordinary Laplacian in . The targeted applications are the determination of free boundaries in plasma control and source detection in electro/magnetoencephalography (EEG/MEG, see section ).
An isotropic diffusion equation in dimension 2 can be recast as a socalled real Beltrami equation
. This way analytic functions get replaced by “generalized” ones in
problems (
P_{0}) and (
P). Hardy spaces of solutions, which are more general than Sobolev ones and allow one to handle
L^{p}boundary conditions, have been introduced when
1<
p<
. The expansions of solutions needed to constructively handle such
problems have been preliminary studied in
,
. The goal is to solve the analog of (
P) in this context to approach Bernoullitype problems (see section
).
At present, bounded extremal problems for the
nD Laplacian are considered on halfspaces or balls. Following
, Hardy spaces are defined as gradients of harmonic functions
satisfying
L^{p}growth conditions on inner hyperplanes or spheres. From the constructive viewpoint, when
p= 2, spherical harmonics offer a reasonable substitute to Fourier expansions
. Only very recently were we able to define operators of Hankel type
whose singular values connect to the solution of (
P_{0}) in BMO norms. The
L^{p}problem also makes contact with some nonlinear PDE's, namely to the
pLaplacian. The goal is here to solve the analog of (
P) on spherical shells to approach inverse diffusion problems across a conductor layer.
g
Let as before
Ddesignate the unit disk,
Tthe unit circle. We further put
R_{N}for the set of rational functions with at most
Npoles in
D, which allows us to define the meromorphic functions in
L^{p}(
T)as the traces of functions in
H^{p}+
R_{N}.
A natural generalization of problem (
P_{0}) is
(
P_{N}) Let
1
p
,
N0an integer, and
fL^{p}(
T); find a function
g_{N}H^{p}+
R_{N}such that
g_{N}
fis of minimal norm in
L^{p}(
T).
Problem (
P_{N}) aims, on the one hand, at solving inverse potential problems from overdetermined DirichletNeumann data, namely to recover approximate solutions of the inhomogeneous Laplace equation
u=
, with
some (unknown) distribution, which will be discretized by the process as a linear combination of
NDirac masses. On the other hand, it is used to perform the second step of the identification scheme described in section
, namely rational approximation with a prescribed number of poles to a function
analytic in the right halfplane, when one maps the latter conformally to the complement of
Dand solve (
P_{N}) for the transformed function on
T.
Only for
p=
and continuous
fis it known how to solve (
P_{N}) in closed form. The unique solution is given by the AAK theory, that allows one to express
g_{N}in terms of the singular vectors of the Hankel operator with symbol
f. The continuity of
g_{N}as a function of
fonly holds for stronger norms than uniform,
.
The case
p= 2is of special importance. In particular when
, the Hardy space of exponent 2 of the
complementof
Din the complex plane (by definition,
h(
z)belongs to
if, and only if
h(1/
z)belongs to
H^{p}), then (
P_{N}) reduces to rational approximation. Moreover, it turns out that the associated solution
g_{N}R_{N}has no pole outside
D, hence it is a
stablerational approximant to
f. However, in contrast with the situation when
p=
, this approximant may
notbe unique.
The former Miaou project (predecessor of Apics) has designed an adapted steepestdescent algorithm for the case
p= 2whose convergence to a
local minimumis guaranteed; it seems today the only procedure meeting this property. Roughly speaking, it is a gradient algorithm that proceeds recursively with respect to the order
Nof the approximant, in a compact region of the parameter space
. Although it has proved rather effective in all applications
carried out so far (see sections
,
), it is not known whether the absolute
minimumcan always be obtained by choosing initial conditions corresponding to
critical pointsof lower degree (as done by the Endymion software section
and RARL2 software, section
).
In order to establish convergence results of the algorithm to the global minimum, Apics has undergone a longhaul study of the number and nature of critical points, in which tools from
differential topology and operator theory team up with classical approximation theory. The main discovery is that the nature of the critical points (
e.g.
local minima, saddles...) depends on the decrease of the interpolation error to
fas
Nincreases
. Based on this, sufficient conditions have been developed for a
local
minimumto be unique. This technique requires strong error estimates that are often difficult to obtain, and most of the time only hold for
Nlarge. Examples where uniqueness or asymptotic uniqueness has been proved this way include transfer functions of relaxation systems (i.e., Markov functions)
, the exponential function, and meromorphic functions
. The case where
fis the Cauchy integral on a hyperbolic geodesic arc of a Dinicontinuous function which does not vanish “too much” has been recently answered in the positive, see section
. An analog to AAK theory has been carried out for
2
p<
. Although not computationally as powerful, it has better continuity
properties and stresses a continuous link between rational approximation in
H^{2}and meromorphic approximation in the uniform norm, allowing one to use, in either context, techniques available from the other
p<2, problem (
P_{N}) is still fairly open.
A common feature to all these problems is that critical point equations express nonHermitian orthogonality relations for the denominator of the approximant. This is used in an essential manner to assess the behavior of the poles of the approximants to functions with branched singularities which is of particular interest for inverse source problems ( cf. sections , ).
In higher dimensions, the analog of problem (
P_{N}) is the approximation of a vector field with gradients of potentials generated by
Npoint masses instead of meromorphic functions. The issue is by no means understood at present, and is a major endeavor of future research problems.
Certain constrained rational approximation problems, of special interest in identification and design of passive systems, arise when putting additional requirements on the approximant, for
instance that it should be smaller than 1 in modulus. Such questions have become over years an increasingly significant part of the team's activity (see sections
,
,
, and
). When translated over to the circle, a prototypical formulation consists in
approximating the modulus of a given function by the modulus of a rational function of degree
n. When
p= 2this problem can be reduced to a series of standard rational approximation problems, but usually one needs to solve it for
p=
. The case where

fis a piecewise constant function with values 0 and 1 can also be approached via classical Zolotarev problems
, that can be solved more or less explicitly when the passband
consists of a single arc. A constructive solution in the case where

fis a piecewise constant function with values 0 and 1 on several arcs (multiband filters) is one recent achievement of the team. Though the modulus of the response
is the first concern in filter design, the variation of the phase must nevertheless remain under control to avoid unacceptable distortion of the signal. This is an important issue, currently
under investigation within the team under contract with the CNES, see section
.
From the point of view of design, rational approximants are indeed useful only if they can be translated into physical parameter values for the device to be built. This is where system theory enters the scene, as the correspondence between the frequency response (i.e., the transferfunction) and the linear differential equations that generate this response (i.e., the statespace representation), which is the object of the socalled realizationprocess. Since filters have to be considered as dual modes cavities, the realization issue must indeed be tackled in a 2×2matrixvalued context that adds to the complexity. A fair share of the team's research in this direction is concerned with finding realizations meeting certain constraints (imposed by the technology in use) for a transferfunction that was obtained with the abovedescribed techniques (see section ).
We refer here to the behavior of the poles of best meromorphic approximants, in the
L^{p}sense on a closed curve, to functions
fdefined as Cauchy integrals of complex measures whose support lies inside the curve. If one normalizes the contour to be the unit circle
T, we are back to the framework of section
and to problem (
P_{N}); the invariance of the problem under conformal mapping was established in
. The research so far has focused on functions whose singular set
inside the contour is zero or onedimensional.
Generally speaking, the behavior of poles is particularly important in meromorphic approximation to obtain error rates as the degree goes large and also to tackle constructive issues like uniqueness. However, the original motivation of Apics is to consider this issue in connection with the approximation of the solution to a DirichletNeumann problem, so as to extract information on the singularities. The general theme is thus how do the singularities of the approximant reflect those of the approximated function?The approach to inverse problem for the 2D Laplacian that we outline here is attractive when the singularities are zero or onedimensional (see section ). It can be used as a computationally cheap preliminary step to obtain the initial guess of a more precise but heavier numerical optimization.
For sufficiently smooth cracks, or pointwise sources recovery, the approach in question is in fact equivalent to the meromorphic approximation of a function with branch points, and we were able to prove , that the poles of the approximants accumulate in a neighborhood of the geodesic hyperbolic arc that links the endpoints of the crack, or the sources . Moreover the asymptotic density of the poles turns out to be the equilibrium distribution on the geodesic arc of the Green potential and it charges the end points, that are thus well localized if one is able to compute sufficiently many zeros (this is where the method could fail). The case of more general cracks, as well as situations with three or more sources, requires the analysis of the situation where the number of branch points is finite but arbitrary, see section ). This are outstanding open questions for applications to inverse problems (see section ), as also the problem of a general singularity, that may be two dimensional.
Results of this type open new perspectives in nondestructive control, in that they link issues of current interest in approximation theory (the behavior of zeroes of nonHermitian orthogonal polynomials) to some classical inverse problems for which a dual approach is thereby proposed: to approximate the boundary conditions by true solutions of the equations, rather than the equation itself (by discretization).
Let us point out that the problem of approximating, by a rational or meromorphic function, in the
L^{p}sense on the boundary of a domain, the Cauchy transform of a real measure, localized inside the domain, can be viewed as an optimal discretization problem for a logarithmic potential
according to a criterion involving a Sobolev norm. This formulation can be generalized to higher dimensions, even if the computational power of complex analysis is then no longer available,
and this makes for a longterm research project with a wide range of applications. It is interesting to mention that the case of sources in dimension three in a spherical or ellipsoidal
geometry, can be attacked with the above 2D techniques as applied to planar sections (see section
).
Matrixvalued approximation is necessary for handling systems with several inputs and outputs, and it generates substantial additional difficulties with respect to scalar approximation, theoretically as well as algorithmically. In the matrix case, the McMillan degree (i.e., the degree of a minimal realization in the SystemTheoretic sense) generalizes the degree.
The problem we want to consider reads:
Let
and
nan integer; find a rational matrix of size
m×
lwithout poles in the unit disk and of McMillan degree at most
nwhich is nearest possible to
in
(
H
^{2})
^{m×
l}.Here the
L^{2}norm of a matrix is the square root of the sum of the squares of the norms of its entries.
The approximation algorithm designed in the scalar case generalizes to the matrixvalued situation
. The first difficulty consists here in the parametrization of
transfer matrices of given McMillan degree
n, and the inner matrices (i.e., matrixvalued functions that are analytic in the unit disk and unitary on the circle) of degree
nenter the picture in an essential manner: they play the role of the denominator in a fractional representation of transfer matrices (using the socalled DouglasShapiroShields
factorization).
The set of inner matrices of given degree has the structure of a smooth manifold that allows one to use differential tools as in the scalar case. In practice, one has to produce an atlas of charts (parametrization valid in a neighborhood of a point), and we must handle changes of charts in the course of the algorithm. Such parametrization can be obtained from interpolation theory and Schur type algorithms, the parameters being interpolation vectors or matrices , , . Some of these parametrizations have a particular interest for computation of realizations , , involved in the estimation of physical quantities for the synthesis of resonant filters. Two rational approximation codes (see sections and ) have been developed in the team.
Problems relative to multiple local minima naturally arise in the matrixvalued case as well, but deriving criteria that guarantee uniqueness is even more difficult than in the scalar case. The already investigated case of rational functions of the sought degree (the consistency problem) was solved using rather heavy machinery , and that of matrixvalued Markov functions, that are the first example beyond rational function has made progress only recently .
Let us stress that the algorithms mentioned above are first to handle rational approximation in the matrix case in a way that converges to local minima, while meeting stability constraints on the approximant.
State feedback stabilization consists in designing a control law which is a function of the state and makes a given point (or trajectory) asymptotically stable for the closed–loop system. That function of the state must bear some regularity, at least enough to allow the closedloop system to make sense; continuous or smooth feedback would be ideal, but one may also be content with discontinuous feedback if robustness properties are not defeated. One can consider this as a weak version of the optimal control problem which is to find a control that minimizes a given criterion (for instance the time to reach a prescribed state). Optimal control generally leads to a rather irregular dependence on the initial state; in contrast, stabilization is a qualitativeobjective (i.e., to reach a given state asymptotically) which is more flexible and allows one to impose a lot more regularity.
Lyapunov functions are a wellknown tool to study the stability of noncontrolled dynamic systems. For a control system, a Control Lyapunov Functionis a Lyapunov function for the closedloop system where the feedback is chosen appropriately. It can be expressed by a differential inequality called the “Artstein (in)equation” , reminiscent of the HamiltonJacobiBellmann equation but largely underdetermined. One can easily deduce a continuous stabilizing feedback control from the knowledge of a control Lyapunov function; also, even when such a control is known beforehand, obtaining a control Lyapunov function can still be very useful to deal with robustness issues. Moreover, if one has to deal with a problem where it is important to optimize a criterion, and if the optimal solution is hard to compute, one can look for a control Lyapunov function which comes “close” (in the sense of the criterion) to the solution of the optimization problem but leads to a control which is easier to work with.
These constructions were exploited in a joint collaborative research conducted with Thales Alenia Space (Cannes), where minimizing a certain cost is very important (fuel consumption / transfer time) while at the same time a feedback law is preferred because of robustness and ease of implementation (see section ).
The study of optimal mass transport problems in the Euclidean or Riemannian setting has a long history which goes back to the pioneering works , , and was more recently revised and revitalized by , . It is the problem of finding the cheapest transformation that moves a given initial measure to a given final one, where the cost between points is a (squared) Euclidean or Riemannian distance.
There has been, quite newly, a lot of interest in the same transportation problems with a cost coming from optimal control, i.e. from minimizing an integral quadratic cost, among trajectories that are subject to differential constraints coming from a control system. The case of controllable affine control systems without drift (in which case the cost is the subRiemannian distance) is studied in , and .
This is a new topic in the team, starting with the PhD of A. Hindawi, whose goal is to tackle the problem of systems with drift. The optimal transport problem in this setting borrows methods from control and at the same time helps understanding optimal control because it is a more regular problem.
Here we study certain transformations of models of control systems, or more accurately of equivalence classes modulo such transformations. The interested reader can find a richer overview in the first chapter of the HDR Thesis . The motivations are twofold:
From the point of view of control, a command satisfying specific objectives on the transformed system can be used to control the original system including the transformation in the controller.
From the point of view of identification and modeling, the interest is either to derive qualitative invariants to support the choice of a nonlinear model given the observations, or to contribute to a classification of nonlinear models which is missing sorely today. This is a prerequisite for a general theory of nonlinear identification; indeed, the success of the linear model in control and identification is due to the deep understanding one has of it.
A static feedbacktransformation is a (nonsingular) reparametrization of the control depending on the state, together with a change of coordinates in the state space. A dynamic feedbacktransformation consists of a dynamic extension (adding new states, and assigning them a new dynamics) followed by a state feedback on the augmented system. Dynamic equivalence is obviously more general than static equivalence. Let us now stress two specific problems that we are tackling.
Dynamic Equivalence.Very few invariants are known. Any insight on this problem is relevant to the above questions. Some results are accounted for in section .
A special equivalence class is the one containing linear controllable systems. It turns out that a system is in this class — i.e.is dynamic linearizable— if and only if there is a formula that gives the general solution by applying a nonlinear differential operator to a certain number of arbitrary functions of time; such a formula is often called a (Monge) parametrizationand the order of the differential operator the order of the parametrization. Existence of such a parametrization has been emphasized over the last years as very important and useful in control, see ; this property (with additional requirements on the parametrization) is also called flatness.
An important question remains open: how can one algorithmically decide whether a given system has this property or not, i.e., is dynamic linearizable or not? The mathematical difficulty is that no a priori bound is known on the order of the above mentioned differential operator giving the parametrization. Within the team, results on low dimensional systems have been obtained , see also ; the above mentioned difficulty is not solved for these systems but results are given with prioriprescribed bounds on this order.
From the differential algebraic point of view, the module of differentials of a controllable system is free and finitely generated over the ring of differential polynomials in
d/
dtwith coefficients in the ring of functions on the system's trajectories; the above question is the one of finding out whether there exists a basis consisting of
closed differential forms. Expressed in this way, it looks like an extension of the classical Frobenius integrability theorem to the case where coefficients are differential operators. Of
course, some non classical conditions have to be added to the classical stability by exterior differentiation, and the problem is open. In
, a partial answer to this problem was given, but in a framework
where infinitely many variables are allowed and a finiteness criterion is still missing. The goal is to obtain a formal and implementable algorithm to decide whether or not a given system is
flat around a regular point.
Topological Equivalence.Compared to static equivalence, dynamic equivalence is more general, hence might offer some more robust “qualitative” invariants; another way to enlarge equivalence classes is to look for equivalence modulo possibly nondifferentiable transformations.
In the case of dynamical systems without control, the HartmanGrobman theorem states that every system is locally equivalent via a transformation that is solely bicontinuous, to a linear system in a neighborhood of a nondegenerate equilibrium. A similar result control systems would say, typically, that outside a “meager” class of models (for instance, those whose linear approximation is noncontrollable), and locally around nominal values of the state and the control, no qualitative phenomenon can distinguish a nonlinear system from a linear one, all nonlinear phenomena being thus either of global nature or singularities.
A stronger Hartman Grobman Theorem for control systems —where transformations are homeomorphisms in the statecontrol space— cannot hold, in fact; this is proved in , commented in section : almost all topologically linearizable control systems are differentiably linearizable. In general (equivalence between nonlinear systems), topological invariants are still to be investigated.
The bottom line of the team's activity is twofold, namely function theory and optimization in the frequency domain on the one hand, and the control of certain systems governed by differential equations on the other hand. Therefore one can distinguish between two main families of applications: one dealing with the design and identification of diffusive and resonant systems (these are inverse problems), and one dealing with the control of certain mechanical systems. For applications of the first type, approximation techniques as described in section allow one to deconvolve linear equations, analyticity being the result of either the use of Fourier transforms or the harmonic character of the equation itself. Applications of the second type mostly concern the control of systems that are “poorly” controllable, for instance low thrust satellites. We describe all these below in more detail.
We are mainly concerned with classical inverse problems like the one of localizing defaults (as cracks, pointwise sources or occlusions) in a two or three dimensional domain from boundary data (which may correspond to thermal, electrical, or magnetic measurements), of a solution to Laplace or to some conductivity equation in the domain. These defaults can be expressed as a lack of analyticity of the solution of the associated DirichletNeumann problem that may be approached, in balls, using techniques of best rational or meromorphic approximation on the boundary of the object (see section ).
Indeed, it turns out that traces of the boundary data on 2D cross sections (disks) coincide with analytic functions in the slicing plane, that has branched singularities inside the disk . These singularities are related to the actual location of the sources (namely, they reach in turn a maximum in modulus when the plane contains one of the sources). Hence, we are back to the 2D framework where approximately recovering these singularities can be performed using best rational approximation.
In this connection, the realistic case where data are available on part of the boundary only offers a typical opportunity to apply the analytic extension techniques (see section ) to Cauchy type issues, a somewhat different kind of inverse problems in which the team is strongly interested.
The approach proposed here consists in recovering, from measured data on a subset
Kof the boundary
Dof a domain
Dof
R^{2}or
R^{3}, say the values
F_{K}on
Kof some function
F, the subset
of its singularities (typically, a crack or a discrete set of pointwise sources), provided that
Fis an analytic function in
.
The analytic approximation techniques (section
) first allow us to extend
Ffrom the given data
F_{K}to all of
D, if
KD, which is a Cauchy type issue for which our algorithms provide robust solutions, in plane domains (see
for 2D annular domains, and
for 3D spherical situations, also discussed in section
). Note that identification schemes for an unknown Robin coefficient together
with stability properties have been obtained in the same way
.
From these extended data on the whole boundary, one can obtain information on the presence and the location of , using rational or meromorphic approximation on the boundary (section ). This may be viewed as a discretization of by the poles of the approximants .
This is the case in dimension 2, using classical links between analyticity and harmonicity , but also in dimension 3, at least in spherical or ellipsoidal domains, working on 2D plane sections, , .
The two above steps are shown in to provide a robust way of locating sources from incomplete boundary data in a 2D situation with several annular layers. Numerical experiments have already yielded excellent results in 3D situations and we are now on the way to process real experimental magnetoencephalographic data, see also sections , . The PhD theses of A.M. Nicu and M. Zghal are concerned with these applications, in collaboration with the Odyssée team of Inria Sophia Antipolis, and with neuroscience teams in partnerhospitals (hosp. Timone, Marseille).
Such methods are currently being generalized to problems with variable conductivity governed by a 2D Beltrami equation, see , , . The application we have in mind is to plasma confinement for thermonuclear fusion in a Tokamak, more precisely with the extrapolation of magnetic data on the boundary of the chamber from the outer boundary of the plasma, which is a level curve for the poloidal flux solving the original divgrad equation. Solving this inverse problem of Bernoulli type is of importance to determine the appropriate boundary conditions to be applied to the chamber in order to shape the plasma . These issues are the topics of the PhD theses of S. Chaabi and Y. Fischer, and of a joint collaboration with the CEAIRFM (Cadarache), the Laboratoire J.A. Dieudonné at the Univ. of NiceSA, and the CMILATP at the Univ. of Marseille I (see section ), see , .
Inverse potential problems are also naturally encountered in magnetization issues that arise in nondestructive control. A particular application, which the object of a joint NSFsupported
project with Vanderbilt University and MIT, is to geophysics where the remanent magnetization a rock is to be analyzed using a squidmagnetometer in order to analyze the history of the object;
specifically, the analysis of Martian rocks is conducted at MIT, for instance to understand if inversions of the magnetic field took place there. Mathematically speaking, the problem is to
recover the (3D valued) magnetization
mfrom measurements of the vector potential:
outside the volume of the object.
One of the best training grounds for the research of the team in function theory is the identification and design of physical systems for which the linearity assumption works well in the considered range of frequency, and whose specifications are made in the frequency domain. Resonant systems, either acoustic or electromagnetic based, are prototypical devices of common use in telecommunications.
In the domain of space telecommunications (satellite transmissions), constraints specific to onboard technology lead to the use of filters with resonant cavities in the microwave range. These filters serve multiplexing purposes (before or after amplification), and consist of a sequence of cylindrical hollow bodies, magnetically coupled by irises (orthogonal double slits). The electromagnetic wave that traverses the cavities satisfies the Maxwell equations, forcing the tangent electrical field along the body of the cavity to be zero. A deeper study (of the Helmholtz equation) states that essentially only a discrete set of wave vectors is selected. In the considered range of frequency, the electrical field in each cavity can be seen as being decomposed along two orthogonal modes, perpendicular to the axis of the cavity (other modes are far off in the frequency domain, and their influence can be neglected).
Near the resonance frequency, a good approximation of the Maxwell equations is given by the solution of a second order differential equation. One obtains thus an electrical model for our filter as a sequence of electricallycoupled resonant circuits, and each circuit will be modeled by two resonators, one per mode, whose resonance frequency represents the frequency of a mode, and whose resistance represent the electric losses (current on the surface).
In this way, the filter can be seen as a quadripole, with two ports, when plugged on a resistor at one end and fed with some potential at the other end. We are then interested in the power
which is transmitted and reflected. This leads to defining a scattering matrix
S, that can be considered as the transfer function of a stable causal linear dynamical system, with two inputs and two outputs. Its diagonal terms
S_{1, 1},
S_{2, 2}correspond to reflections at each port, while
S_{1, 2},
S_{2, 1}correspond to transmission. These functions can be measured at certain frequencies (on the imaginary axis). The filter is rational of order 4 times the number of cavities (that is 16 in
the example), and the key step consists in expressing the components of the equivalent electrical circuit as a function of the
S_{ij}(since there are no formulas expressing the lengths of the screws in terms of parameters of this electrical model). This representation is also useful to analyze the numerical
simulations of the Maxwell equations, and to check the design, particularly the absence of higher resonant modes.
In fact, resonance is not studied via the electrical model, but via a lowpass equivalent circuit obtained upon linearizing near the central frequency, which is no longer conjugate symmetric (i.e., the underlying system may not have real coefficients) but whose degree is divided by 2 (8 in the example).
In short, the identification strategy is as follows:
measuring the scattering matrix of the filter near the optimal frequency over twice the pass band (which is 80Mhz in the example).
solving bounded extremal problems for the transmission and the reflection (the modulus of he response being respectively close to 0 and 1 outside the interval measurement, cf. section ). This provides us with a scattering matrix of order roughly 1/4 of the number of data points.
Approximating this scattering matrix by a rational transferfunction of fixed degree (8 in this example) via the Endymion or RARL2 software (cf. section ).
A realization of the transfer function is thus obtained, and some additional symmetry constraints are imposed.
Finally one builds a realization of the approximant and looks for a change of variables that eliminates nonphysical couplings. This is obtained by using algebraicsolvers and continuation algorithms on the group of orthogonal complex matrices (symmetry forces this type of transformation).
The final approximation is of high quality. This can be interpreted as a validation of the linearity hypothesis for the system: the relative
L^{2}error is less than
10
^{3}. This is illustrated by a reflection diagram (Figure
). Nonphysical couplings are less than
10
^{2}.
The above considerations are valid for a large class of filters. These developments have also been used for the design of nonsymmetric filters, useful for the synthesis of repeating devices.
The team investigates today the design of output multiplexors (OMUX) where several filters of the previous type are coupled on a common guide. In fact, it has undergone a rather general analysis of the question “How does an OMUX work?” With the help of numerical simulations and Schur analysis, general principles are being worked out to take into account:
the coupling between each channel and the “Tee” that connects it to the manifold,
the coupling between two consecutive channels.
The model is obtained upon chaining the corresponding scattering matrices, and mixes up rational elements and complex exponentials (because of the delays) hence constitutes an extension of the previous framework. Its study is being conducted under contract with Thales Alenia Space (Toulouse) (see sections ).
Generally speaking, aerospace engineering requires sophisticated control techniques for which optimization is often crucial, due to the extreme functioning conditions. The use of satellites in telecommunication networks motivates a lot of research in the area of signal and image processing; see for instance section for an illustration. Of course, this requires that satellites be adequately controlled, both in position and orientation (attitude). This problem and similar ones continue to motivate research in control. The team has been working for six years on control problems in orbital transfer with lowthrust engines, including four years under contract with Thales Alenia Space (formerly Alcatel Space) in Cannes.
Technically, the reason for using these (ionic) low thrust engines, rather than chemical engines that deliver a much higher thrust, is that they require much less “fuel”; this is decisive because the total mass is limited by the capacity of the launchers: less fuel means more payload, while fuel represents today an impressive part of the total mass.
From the control point of view, the low thrust makes the transfer problem delicate. In principle of course, the control law leading to the right orbit in minimum time exists, but it is quite
heavy to obtain numerically and the computation is nonrobust against many unmodelled phenomena. Considerable progress on the approximation of such a law by a feedback has been carried out
using
ad hocLyapunov functions.These approximate surprisingly well timeoptimal trajectories. The easy implementation of such control laws makes them attractive as compared to genuine optimal
control. Here the
n1first integrals are an easy means to build control Lyapunov functions since any function of these first integrals can be made monotone decreasing by a suitable
control. See
and the references therein.
The development of the
RARL2 (Réalisation interne et Approximation Rationnelle L2) is a software for rational approximation (see section
)
http://
This software takes as input a stable transfer function of a discrete time system represented by
either its internal realization,
or its first
NFourier coefficients,
or discretized values on the circle.
It computes a local best approximant which is
stable, of prescribed McMillan degree, in the
L^{2}norm.
It is akin to the arl2 function of Endymion (see section ) from which it differs mainly in the way systems are represented: a polynomial representation is used in Endymion, while RARL2 uses realizations, this being very interesting in certain cases. It is implemented in Matlab. This software handles multivariablesystems (with several inputs and several outputs), and uses a parametrization that has the following advantages
it incorporates the stability requirement in a builtin manner,
it allows the use of differential tools,
it is wellconditioned, and computationally cheap.
An iterative research strategy on the degree of the local minima, similar in principle to that of arl2, increases the chance of obtaining the absolute minimum (see section ) by generating, in a structured manner, several initial conditions.
RARL2 performs the rational approximation step in our applications to filter identification (section ) as well as sources or cracks recovery (section ). It was released to the universities of Delft, Maastricht, Cork and Brussels. The parametrization embodied in RARL2 was recently used for a multiobjective control synthesis problem provided by ESTECESA, The Netherlands (section ). An extension of the software to the case of triple poles approximants is now available. It gives nice results in the source recovery problem (section ). It is used by FindSources3D (see ).
The identification of filters modeled by an electrical circuit that was developed by the team (see section
) led us to compute the electrical parameters of the underlying filter. This
means finding a particular realization
(
A,
B,
C,
D)of the model given by the rational approximation step. This 4tuple must satisfy constraints that come from the geometry of the equivalent electrical network and
translate into some of the coefficients in
(
A,
B,
C,
D)being zero. Among the different geometries of coupling, there is one called “the arrow form”
which is of particular interest since it is unique for a given
transfer function and also easily computed. The computation of this realization is the first step of RGC. Subsequently, if the target realization is not in arrow form, one can nevertheless show
that it can be deduced from the arrowform by a complex orthogonal change of basis. In this case, RGC starts a local optimization procedure that reduces the distance between the arrow form and
the target, using successive orthogonal transformations. This optimization problem on the group of orthogonal matrices is nonconvex and has a lot of local and global minima. In fact, there is
not always uniqueness of the realization of the filter in the given geometry. Moreover, it is often interesting to know all the solutions of the problem, because the designer cannot be sure, in
many cases, which one is being handled, and also because the assumptions on the reciprocal influence of the resonant modes may not be equally well satisfied for all such solutions, hence some
of them should be preferred for the design. Today, apart from the particular case where the arrow form is the desired form (this happens frequently up to degree 6) the RGC software gives no
guarantee to obtain a single realization that satisfies the prescribed constraints. The software DedaleHF (see
), which is the successor of RGC, solves in a guaranteed manner this constraint
realization problem.
PRESTOHF: a toolbox dedicated to lowpass parameter identification for microwave filters http://wwwsop.inria.fr/apics/personnel/Fabien.Seyfert/Presto_web_page/presto_pres.html. In order to allow the industrial transfer of our methods, a Matlabbased toolbox has been developed, dedicated to the problem of identification of lowpass microwave filter parameters. It allows one to run the following algorithmic steps, either individually or in a single shot:
determination of delay components, that are caused by the access devices (automatic reference plane adjustment),
automatic determination of an analytic completion, bounded in modulus for each channel,
rational approximation of fixed McMillan degree,
determination of a constrained realization.
For the matrixvalued rational approximation step, PrestoHF relies either on hyperion (Unix or Linux only) or RARL2 (platform independent), two rational approximation engines developed within the team. Constrained realizations are computed by the RGC software. As a toolbox, PrestoHF has a modular structure, which allows one for example to include some building blocks in an already existing software.
The delay compensation algorithm is based on the following strong assumption: far off the passband, one can reasonably expect a good approximation of the rational components of
S_{11}and
S_{22}by the first few terms of their Taylor expansion at infinity, a small degree polynomial in
1/
s. Using this idea, a sequence of quadratic convex optimization problems are solved, in order to obtain appropriate compensations. In order to check the previous
assumption, one has to measure the filter on a larger band, typically three times the pass band.
This toolbox is currently used by Thales Alenia Space in Toulouse and a license agreement has been recently negotiated with Thales airborne systems. XLim (University of Limoges) is a heavy user of PrestoHF among the academic filtering community and some free license agreements are currently being considered with the microwave department of the University of Erlangen (Germany) and the Royal Military College (Kingston, Canada).
The development of
Endymion,
http://wwwsop.inria.fr/apics/endymion/index.htmlhas been stabilized. It is a software licensed under the CeCILL license version two, see
http://www.cecill.info. It has been registered under the
number IDDN.FR.001.310002.000.S.P.2009.000.10000 at the the APP It was developed on Linux, but works as well on MacOS. The core of the system is formed by a library that handles numbers (short
integers, arbitrary size rational numbers, floating point numbers, quadruple and octuple precision floating point numbers, arbitrary precision real numbers, complex numbers), polynomials,
matrices, etc. Specific data structures for the rational approximation algorithm
arl2and the bounded extremal problem
bepare also available. One can mention for instance splines, Fourier series, Schur matrices, etc. These data structures are manipulated by dedicated algorithms (matrix inversion, roots
of polynomials, a gradientbased algorithm for minimizing
, Newton method for finding a critical point of
, etc), and inputoutput functions that allow one to save data on disk, restore them, plot them, etc. The software is interactive: there is a symbolic interpreter based upon a Lisp
interpreter. For instance the coefficient of
z^{2}in
Pcan be obtained via Lisp syntax
(getcoef P 2)or modified via the symbolic syntax
P[2]++.
DedaleHF is a software meant to solve exhaustively the coupling matrix synthesis problem in reasonable time for the users of the filtering community. For a given coupling topology the coupling matrix synthesis problem (C.M. problem for short) consists in finding all possible electromagnetic coupling values between resonators that yield a realization of a given filter characteristics (see section ). Solving the latter problem is crucial during the design step of a filter in order to derive its physical dimensions as well as during the tuning process where coupling values need to be extracted from frequency measurements (see Figure ).
DedaleHF consists in two parts: a database of coupling topologies as well as a dedicated predictorcorrector code. Roughly speaking each reference file of the database contains, for a given coupling topology, the complete solution to the C.M. problem associated to a particular filtering characteristics. The latter is then used as a starting point for a predictorcorrector integration method that computes the solution to the C.M. problem of the user, i.e., the one corresponding to a userspecified filter characteristics. The reference files are computed off line using Groebner basis techniques or numerical techniques based on the exploration of a monodromy group. The use of such a continuation technique combined with an efficient implementation of the integrator produces a drastic reduction of the computational time, say, by a factor of 20.
Access to the database and integrator code is done via the web on http://wwwsop.inria.fr/apics/Dedale/WebPages. The software is free of charge for academic research purposes: a registration is however needed in order to access full functionality. Up to now 90 users have registered among the world (mainly: Europe, U.S.A, Canada and China) and 4000 reference files have been downloaded.
As mentioned in an extension of this software that handles symmetrical networks is under construction.
FindSources3D is a software dedicated to source recovery for the inverse EEG problem, in 3layer spherical settings, from pointwise data (see
http://
The major use of Tralics remains the production of the RaWeb (Scientific Annex to the Annual Activity Report of Inria),
. The input is a
Other applications of Tralics consist in putting scientific papers on the Web; for instance Cedram (
http://
The main philosophy of Tralics is to have the same parser as
A minor versions have been released this year, namely 2.15.6 in November. The documentation consists in some technical reports , , , they are regularly updated, especially the HTML version (produced by Tralics). Some new packages were added to the system (graphicx, xkeyval, color), and for efficiency reasons, part of the code is implemented in the C++ kernel. The referencing system was completely rewritten, so that for instance the XML document contains the same equation numbers as the PostScript version (in the RaWeb case, equation numbers are computed by the XMLtoHTML style sheet). The RaWeb preprocessor was removed: all commands specific to the Activity Report are now defined in package files, the kernel containing some primitives that can check the validity of some arguments versus a keyword list defined in the configuration file. One can add or removes entry types and fields in the bibliography.
This is a new research theme. Our objective is to use the proof assistant Coq in order to formally prove a great number of theorems in Algebra. We started with the first book (Theory of
sets,
) of the series “Elements of Mathematics”. The first chapter describes
Formal Mathematics, and we have shown that it is possible to interpret it in the Coq language. Note that Bourbaki expresses
in terms of
, which is not possible in Coq, and states that if
x,
Ris false, then
x, ¬
R. This is a nonconstructive statement. Moreover, this implies a general version of the axiom of choice (if for all
xthere is an
ysatisfying
P(
x,
y), then there is a mapping
fsuch that
P(
x,
f(
x))holds for all
x). We use some ideas of Carlos Simpson (University of Nice), and decide that a set is a type, and that
XYis true if and only if there is a representative of
Xof type
Y(this is nonconstructive, since “representative” is only defined through axioms).
The second chapter of Bourbaki covers the theory of sets proper. It defines ordered pairs, correspondences, union, intersection and product of a family of sets, as well as equivalence relations. Its implementation in Coq corresponds to 300 definitions and 1300 lemmas or theorems. It is described in . The third chapter of Bourbaki covers the theory of ordered sets, wellordered sets, equipotent sets, cardinals, natural integers, and infinite sets; its implementation in Coq is described in in . All results of the book been proved in Coq (230 definitions and 1200 lemmas), except inverse limits, direct limits and structures, which will be considered later; moreover there are more than one hundred exercises, most of them are nontrivial, and solving them will take some time.
Finite cardinals satisfy an induction principle (this is a special case of transfinite induction); This is the same induction principle as that of natural integers in Coq, so that these two
notions are isomorphic. This means that every theorem of the Coq library about natural integers translates directly into a theorem about finite cardinals. This allows us to prove theorems like:
The number of increasing (resp. strictly increasing) mappings of a set with
pelements into a set with
nelements is the number of subsets of
pelements of a set with
p+
n(resp.
n) elements.
We use the following 4 axioms. Let's denote by
Ethe type of sets. We assume existence of a function R, of type
x:
E,
xE, such that, if
x:
E, then for all
a:
xand
b:
x,
Ra=
Rbimplies
a=
b. The relation
abis defined by
c:
b,
Rc=
a. The first axioms says that if
aand
bare sets, then
implies
a=
b. The empty set
is inductively defined as a type without constructor. We assume existence of a function
C, of type
t:
E, (
tP)
Ntt(the first argument is a property
p, and the second is a proof
qthat the type
tis nonempty). The axiom of choice says that, if there exists
xsuch that
p(
x), then
C(
p,
q)satisfies
p(This corresponds to Bourbaki's axioms scheme S5 that says that
_{x}(
p)satisfies
pin such a case). We assume existence of a function
Iof type
x:
E, (
xE)
E. This means that, if
fis a function such that
f(
x)is a set for all
x, then
I(
f)is a set. The third axiom says
yI(
f)if and only if there exists
a:
xsuch that
f(
a) =
y. It implies existence of union of sets, but this Scheme of Substitution is slightly more general than Bourbaki's Scheme of Selection and Union, since it implies in
particular the axioms of the set of two elements. The final axiom says that for any property
P, if
Pis not false then it is true.
Solving overdetermined Cauchy problems for the Laplace equation on a spherical layer (in 3D) in order to treat incomplete experimental data is a necessary ingredient of the team's approach to inverse source problems, in particular for applications to EEG since the latter involves propagating the initial conditions from the boundary to the center of the domain where the singularities (i.e., the sources) are sought. Here, the domain is typically made of several homogeneous layers of different conductivities.
Such problems offer an opportunity to state and solve extremal problems for harmonic fields for which an analog of the Toeplitz operator approach to bounded extremal problems
has been obtained. Still, a best approximation on the subset of a
general vector field by a harmonic gradient under a
L^{2}norm constraint on the complementary subset can be computed by an inverse spectral equation for some Toeplitz operator. Constructive and numerical aspects of the procedure (harmonic
3D projection, Kelvin and Riesz transformation, spherical harmonics) and encouraging results have been obtained on numerically simulated data
. Issues of robust interpolation on the sphere from incomplete
pointwise data are also under study (splines, spherical harmonics, spherical wavelets, spherical Laplace operator, ...), in order to improve numerical accuracy of our reconstruction
schemes.
The analogous problem in
L^{p},
p2, is considerably more difficult. A collaborative work is going on, in the
framework of the ANR project AHPI, aiming mainly at the case
p=
. It was obtained that the BMO distance between a bounded vector field on the
sphere and a bounded harmonic gradient is within a constant of the norm of a Hankellike operator, acting on
L^{2}divergencefree vector fields with values in
L^{2}gradients. Estimating the constant requires solving further extremal problems in
L^{1}on the best approximation of a gradient by a divergence free vector field. This issue is currently being studied in
L^{p}where it leads to analyze particular solutions to the the
pLaplacian on the sphere.
The problem of sources recovery can be handled in 3D balls by using best rational approximation on 2D cross sections (disks) from traces of the boundary data on the corresponding circles (see section ).
The team started to consider more realistic geometries for the 3D domain under consideration. A possibility is to parametrize it in such a way that its planar crosssections are quadrature domains or Rdomains. In this framework, best rational approximation can still be performed in order to recover the singularities of solutions to Laplace equations, but complexity issues are delicate. The preliminary case of an ellipsoid, which requires the preliminary computation of an explicit basis of ellipsoidal harmonics, has been studied in and is one of the topics of the PhD thesis of M. Zghal.
In 3D, epileptic regions in the cortex are often represented by pointwise sources that have to be localized from measurements on the scalp of a potential satisfying a Laplace equation (EEG, electroencephalography). A breakthrough was made which makes it possible now to proceed via best rational approximation on a sequence of 2D disks along the inner sphere .
A dedicated numerical software “FindSources3D” (see section ) has been developed, in collaboration with the team Odyssée.
Further, it appears that in the rational approximation step of these schemes, multiplepoles possess a nice behaviour with respect to the branched singularities (see figures , ). This is due to the very basic physical assumptions on the model (for EEG data, one should consider triplepoles). Though numerically observed, there is no mathematical justification why these multiple poles have such strong accumulation properties, which remains an intriguing observation. This is the topic of .
Also, magnetic data from MEG (magnetoencephalography) will soon become available, which should enhance sources recovery.
This approach should also become interesting for geophysical issues, concerning the discretization of the gravitational potential by means of pointwise masses. This is another recent topic of A.M. Nicu's PhD thesis and of our present collaboration with LAMSINENIT, hence the reason why she also made a long working stay there (Univ. El Manar, Tunis, Tunisia, AprilJune).
Magnetic sources localization by analytic and rational approximation on plane sections is currently analyzed from experimental SQUID data, from Vanderbilt University Physics Dept. We have
started analyzing the kernel of the magnetization operator, which is the Riesz potential of the divergence. The natural assumptions to handle magnetizations that are piecewise constant
(allowing for characteristic elements embedded in the slab to be analyzed) has led us to study Riesz transforms and Hodge decompositions of functions of bounded variation, namely functions
whose distributional derivatives are signed measures. The kernel, it has been found, can be described in terms of measures whose balayage on the boundary of the object vanishes. The
constructive characterization of those is a rather difficult problem, but the restriction to more specific classes, like piecewise constant or unidirectional magnetizations that are of common
use in the field, seems better suited to the purpose of algorithmically recovering
m, up to a divergencefree term. The role of the extrapolation techniques initiated by the project team, using bounded extremal problems, should be important in this connection. This
research sheds light on the connections between inverse current problems (aiming at the inversion of the BiotSavart operator) and inverse magnetization problems (aiming at the inversion of
the potential of a divergence).
In collaboration with the CMILATP (University Marseille I) and in the framework of the ANR AHPI, the team considers 2D diffusion processes with variable conductivity. In particular its complexified version, the socalled real Beltrami equation, was investigated. In the case of a smooth domain, and for a smooth conductivity, we analyzed the Dirichlet problem for solutions in Sobolev and then in Hardy classes .
Their traces merely lie in
L^{p}(
1<
p<
) of the boundary, a space which is suitable for identification from pointwise
measurements. Again these traces turn out to be dense on strict subsets of the boundary. This allows us to state Cauchy problems as bounded extremal issues in
L^{p}classes of generalized analytic functions, in a reminiscent manner of what was done for analytic functions as discussed in section
. Recently, dual formulations were obtained and some multiplicative (fibered)
structure for the solution was obtained based on old work by Bers and Nirenberg on pseudoanalytic functions. An article is being written on these topics.
The case of a conductivity that is merely in
, which is important for inverse conductivity problems, is under examination (PhD thesis of S. Chaabi). There, it is still unknown whether solutions exist for all
p.
The application that initially motivated this work comes from free boundary problems in plasma confinement (in tokamaks) for thermonuclear fusion. This work was started in collaboration with the Laboratoire J. Dieudonné (University of Nice) and is now the topic of a collaboration with two teams of physicists from the CEAIRFM (Cadarache).
In the transversal section of a tokamak (which is a disk if the vessel is idealized into a torus), the socalled poloidal flux is subject to some conductivity outside the plasma volume for some simple explicit smooth conductivity function, while the boundary of the plasma (in the Tore Supra Tokamak) is a level line of this flux . Related magnetic measurements are available on the chamber, which furnish incomplete boundary data from which one wants to recover the inner (plasma) boundary. This free boundary problem (of Bernoulli type) can be handled through the solutions of a family of bounded extremal problems in generalized Hardy classes of solutions to real Beltrami equations, in the annular framework. Such approximation problems also allow us to approach a somewhat dual extrapolation issue, raised by colleagues from the CEA for the purpose of numerical simulation. It consists in recovering magnetic quantities on the outer boundary (the chamber) from an initial guess of what the inner boundary (plasma) is.
In the particular case at hand, it is possible to explicitly compute a basis of solutions (Bessel functions) that help the computations, see , . However, many other choices are possible, which are under study. This is the topic of the PhD thesis of Y. Fischer.
In the most recent tokamaks, like Jet or ITER, an interesting feature of the level curves of the poloidal flux is the occurrence of a cusp (a saddle point of the poloidal flux, called an X point), and it is desirable to shape the plasma according to a level line passing through this X point for physical reasons relating to the efficiency of the energy transfer. This will be the topic of future studies.
Our work of the past ten years on balanced realizations of lossless systems, Schur parameters, canonical forms and applications were the topic of a semiplenary session, given by R. Peeters at the conference Sysid09 . Our last results on subdiagonal pivot structure for inputnormal pairs and associated canonical forms were also presented at this conference . These forms generalise to the MIMO case the wellknown Hessenberg form in discretetime and SchwarzOber form in continuoustime. Their use for model reduction purposes seems to be relevant and is currently under investigation.
For the class of lossless discretetime systems, subdiagonal forms can be computed from a specific backward recursive Schur algorithm. In continuoustime, the relevant recursive algorithm in connection with these forms involves a boundary interpolation problem. We got a parametrization of the (subdiagonal) OberSchwarz canonical form (SISO) in terms of boundary interpolation values (angular derivatives). These results were presented at the ERNSI meeting (poster). Boundary interpolation of matrix lossless functions and its applications to the parametrization of filter banks leading to orthogonal wavelets is under study (see section ).
An application of our rational approximation methods to orthogonal wavelets has been investigated. The problem is to implement wavelets in analog circuits in view of medical signal
processing applications. A dedicated method has been developed
based on an
L^{2}approximation of the wavelet by the impulse response of a stable causal low order filter. However, this method fails to find an accurate and sufficiently small order approximation in
some difficult cases (Daubechies db7 and db3). The idea was to use the software RARL2 to perform a model reduction on an accurate high order (100200) approximation. However, an admissibility
condition for wavelets is that the integral of a wavelet equals zero, which means that it has one vanishing moment. The low order approximation is still required to have an integral zero,
otherwise undesired bias will show up when the wavelet is used in an application. We thus had to adapt a version of the RARL2 software to address this constraint. Since we are dealing with a
quadratic optimization problem under a linear constraint, this can be solved analytically. We could thus reformulate the problem of
L^{2}approximation subject to this constraint into an optimization problem over the class of lossless systems. This could be handled by the software with only minor changes and we were able
to perform an accurate approximation of order 8 for db7 (Figure
). This way to address a linear or a convex constraint could be used for other
purposes, for example to impose passivity.
In close connection, we investigate the possibility to parametrize wavelets with (more) vanishing moments using interpolation theory at the boundary. A very useful and concise description of
the class of filter banks leading to orthogonal wavelets is by means of its associated
losslesspolyphase filter
. A vanishing moment condition can be expressed as a boundary
interpolation condition for the lossless polyphase filter. We thus exploited our previous works on the parametrization of lossless matrix functions with interpolation conditions. We got
explicit parametrizations of
2×2polyphase matrices of arbitrary order
nwith (up to) 3 vanishing moments built in, in terms of angular derivative (positive) parameters. However, the conditions were cleverly handled in an unusual recursive fashion that we
still do not completely understand. These results have been presented at the ANRAHPI meeting.
Passive devices play an important role in many application areas: telecommunication, chemical process control, economy, biomedical processes. Network simulation software packages (as ADS or
SPICE) require passive models for their components. However, identifying a passive model from band limited frequency data is still an open and challenging problem. Schur rational approximation
is a new way to approach this problem and has been the subject of
. In this work, a parametrization of all strictly Schur rational
functions of degree n is constructed from a multipoint Schur algorithm, the parameters being both the interpolation values and interpolation points. Examples are computed by an
L^{2}norm optimization process and the results are validated by comparison with the unconstrained
L^{2}rational approximation. Over the last two years, the results of
on the hyperbolic convergence of the classical Schur algorithm were
generalized to the case of the multipoint Schur algorithm, which is a more delicate situation because we allow for the interpolation points to approach the boundary circle. Orthogonal rational
functions and a recent generalization of Geronimus theorem were used
, combined with Hilbertian techniques from reproducing kernel spaces
and new quantitative versions of the Beurling theorem, to obtain an analog of the Szegö theorem where the interpolation points tend to the boundary, provided the approximated function is
continuous and less than 1 in a neighborhood of the accumulation set of the interpolation points. This generalized the results in
and was of novel type since the nth orthogonal rational function
inverts the Szegö function modulo the Poisson kernel. This yields a rather unexpected theorem on the behaviour of certain orthogonal polynomials with varying weight. This year we obtained new
bounds for orthogonal rational functions, based on
estimates for the squared modulus of the Szegö function, that yield new results even in the case of polynomials since they yield information even in cases where the measure has
vanishing density provided it is Sobolev
W^{11/
p,
p}smooth on the circle for some
p>1. An article is being written that summarizes these results.
The research has been pursued on a major open issue, namely how to choose the interpolation points with respect to the approximated Schur function so as to yield the best convergence possible. We also started analyzing the consequences of our work for the representation of certain nonstationary stochastic processes. In this connection the case of vanishing densities and singular components is under investigation.
The results of
and
were extensively used over the last years to prove the convergence in
capacity of
L^{p}best meromorphic approximants on the circle (
i.e.solutions to problem (
P_{N}) of section
) when
p2, for those functions
fthat can be written as Cauchy transforms of complex measures supported on a hyperbolic geodesic arc
,
,
,
. A rational function can also be added to
fwithout modifying the results, which is useful for applications to inverse sources problems. Some mild conditions (bounded variation of the argument and powerthickness of the total
variation) were required on the measure. Here, we recall that convergence in capacity means that the (logarithmic) capacity of the set where the error is greater than
goes to 0 for each fixed
>0. This convergence can be quantified, namely it is geometric with
pointwise rate
exp{1/
C
G}where
Cis the capacity of the condensor
and
Gthe Green potential of the equilibrium measure. The results can be adapted to somewhat general interpolation schemes
,
. From this work it follows that the counting measures of the poles of
the approximants converge, in the weak* sense, to the Green equilibrium distribution on
. In particular the poles cluster to the endpoints of the arc, which is of fundamental use in the team's approach to source detection (see section
).
This year the weak* convergence of the poles of best
H^{2}rational approximants to Cauchy integrals over general symmetric contours for the Green Potential, and not merely geodesic arcs, were established using the reflected symmetry of the
poles and the interpolation nodes of such approximants across the circle, and analyzing the location of continua of minima weighted through a discretization of the weight and a limiting
process. This warrants the use of rational approximation to functions with arbitrarily many branchpoints in source detection. A paper is currently being written on this topic.
The technique we just described only yields convergence in capacity and nth root asymptotics. To obtain strong asymptotics, additional assumptions must be made on the approximated function. Last year, we proved strong asymptotics of multipoint Padé interpolants, in appropriate interpolation nodes, to Cauchy integrals over arbitrary analytic arcs, when the density of the measure with respect to a positive power of the equilibrium distribution on the arc is Dinismooth. In addition, the density may in fact vanish in finitely many points like a small fractional power of the distance to these point . Moreover, the polar singularities of the function, if any, are asymptotically reproduced by the approximants with their multiplicities. This is important for inverse problem of mixed type, like those mentioned in section , where monopolar and dipolar sources are handled simultaneously.
This year we proved under appropriate smoothness assumptions that the result still holds without restrictions on the density, that is, the power of the equilibrium distribution with respect
to which we compute its derivative needs no longer be positive (in the language of orthogonal polynomials, this means we can handle arbitrary Jacobi weights). The lower the power the smoother
the density should be. Typically, if the power is zero (so that we only consider the density with respect to arclength on the arc), a fraction of a derivative is sufficient (i.e. the density
should belong to a
W^{11/
p,
p}class on the arc for some
p>2. When the power gets negative,
classes of Höldersmoothness for the
kth derivative are required, where
kis related to the integer part of the Jacobi exponents and
to their fractional part. This time however, the density is not allowed to vanish.
This result more or less settles the issue of convergence of multipoint Padé approximants to Cauchy integrals over arcs, because it asserts that uniform convergence holds, under mild assumptions on the density, when the interpolation points are chosen in some appropriate manner (symmetric with respect to a weighted equilibrium potential adapted to the contour), and because we also proved that whenever a convergent interpolation scheme exists to a Cauchy integral with smooth density on an arc, with interpolation keeping off the arc, then the arc must be analytic.
The technique of proof uses a generalization, over varying contours, of the RiemannHilbert approach to the asymptotics of orthogonal polynomials as adapted to the segment in This provides us with precise (PlancherelRotach type) asymptotics for the nonHermitian orthogonal polynomials which is the denominator of the approximant. Asymptotics for the latter are even obtained on the arc where the measure of orthogonality is supported. In the case of nonpositive Jacobi powers, estimates and Muckenhoupt weights are also needed. A paper has been written and submitted to report on this research .
We have pursued this line of research for functions defined as Cauchy integrals over union of (possibly intersecting) arcs, and obtained convergence results over regular threefolds. The current goal is to understand which systems of arcs can be construed as critical configurations for weighted potential problems, and whether the above analysis can be extended beyond arcs to 2D singular sets.
In another collection, the results of
have been carried over for analytic approximation to the matrix case
in
. The surprising fact was that not every matrix valued function
generates a vectorial Hankel operator meeting the AAK theorem when
p<
. This led us to the generalization of the latter based on Hankel operators with
matrix argument.
Groomed by industrial users like Thales AleniaSpace, we made some progress in the analysis of the realizations of 2x2 lossless scattering systems whose scattering response
(
S
_{i,
j})satisfies the socalled
autoreciprocalcondition
S_{1, 1}=
S_{2, 2}. It was shown that autoreciprocal inner responses admit a canonical circuit realisation of the form of Fig.
. The length difference (
m
l) of the two antennas of Fig.
is equal to the Cauchy index on the imaginary axes of the filter function to be
realised. Surprisingly enough this form appears to be central in the new modal framework S.Amari is currently developing on dual mode filters (
). It was shown that the classical folded form can be advantageously
replaced by the latter yielding a design procedure with nearly no tuning required (all the physical dimensions of the filter can be computed exactly from the circuit parameters): a paper has
been published on this topic
. In future work, we will focus on the practical implementation of
this analysis within the software DedaleHF
.
We also made some progress on the problem of circuit realisations with mixed type (inductive or capacitive) coupling elements. An algebraic formulation of the synthesis problem of circuits with mixed type elements has been obtained which relies on a set of two matricial equations. As opposed to the classical low pass case with frequency independent couplings the unknown is no longer a similarity transform but a general nonsingular matrix acting on two coupling matrices: the capacitive and the inductive one. First results were obtained in this field allowing the exact synthesis of filters with resonating coupling elements, see . Applications of this technique to synthesise extremely compact filters, with sharp responses, is being studied in collaboration with the Royal Military College (Canada). Note however, that the filter orders for which this synthesis is computationally tractable, for the moment, is modest (no more than 5 or 6). Further developments, focusing in particular on an efficient algebraic formulation of the problem, are needed in order to convince engineers of its relevancy when compared to generic local optimisation techniques. The state of our work was presented at Rome (European Microwave Conference) and Toulouse (CNESESA filter workshop) while a publication is currently being reviewed.
The objective of our work in the ANR Filipix is the derivation of efficient algorithms for the synthesis of microwave multiplexers. In our opinion, an efficient frequency design process
calls for the understanding of the structure of
n×
nlossless reciprocal rational functions for
n>2. Whereas the case
n= 2is completely understood and a keystone of filter synthesis, very little seems to be known about the polynomial structure of such matrices when they involve more
than 2 ports.
We therefore started with the analysis of the 3×3case typically of practical use in the manufacturing of diplexers. Based on recent results obtained on minimal degree reciprocal Darlington synthesis we derived a polynomial model for 3×3reciprocal inner rational matrices with given MacMillan degree. The latter writes as follow:
where we define
rr^{*}=
p_{1}p_{1}^{*}+
p_{2}p_{2}^{*}
and the following divisibility conditions must hold
If the polynomials
pand
rof degree
kn1are given together with a condition at infinity on
S, then one can show that there exist
2
^{k}(
3×3) inner extensions (of MacMillan degree
n) of
S_{1, 1}. Their computation involves mainly linear algebra. During his internship, Amine Rouini designed and programmed a procedure which makes effective this extension process. From a practical
synthesis point of view the extension process that starts with the polynomials
p_{1}and
p_{2}is more relevant but remains technically problematic: some issues concerning the stability of the derived polynomial
qare still unsolved for the moment. The study of particular forms of the polynomial model in connexion with some special circuit topologies used for the implementation of the diplexer are
also currently under investigation.
The theoretical developments took place over the last two years, while deepenings of the numerical aspects were carried out in 2007. This study was conducted under contract with the CNES and ThalesAleniaSpace (Toulouse), and was part of V. Lunot's doctoral work . The problem goes as follows. On introducing the ratio of the transmission and reflexion entries of a scattering matrix, the design of a multiband filter response (see section ) reduces to the following optimization problem of Zolotarev type :
where
(resp.
) is a finite union of compact intervals
I_{i}of the real line corresponding to the passbands (resp. stopbands), and
P_{m}(
K)stands for the set of polynomials of degree less than
mwith coefficients in the field
K. Depending on the physical symmetry of the filter, it is interesting to solve problem (
) either for
(“real” problem) or
(“mixed” problem), or else
(“complex” problem). The “real” Zolotarev problem can be decomposed into a sequence of concave maximization problems, whose solution we were able to characterize in terms of an
alternation property. Based on this, a Remezlike algorithm has been derived in the polynomial case (i.e., when the denominator
qof the scattering matrix is fixed), which allows for the computation of a dualband response (see Figure
) according to the frequency specifications (see Figure
for an example from the spacecraft SPOT5 (CNES)). We have designed an algorithm
in the rational case which, unlike linear programming, avoids sampling over all frequencies. This raises the issue of the “generic normality” (
i.e.the maximum degree) of the approximant with respect to the geometry of the intervals. This question remains open. The design of efficient procedures to tackle the “mixed” and
“complex” cases remains a challenge. The software
easyFFwas registered at the APP under the number IDDN.FR.001.150004.000.S.P.2009.000.3150. A license agreement is beeing worked for a permanent distribution of this software to our
academic partners: Xlim and the Royal Military College of Canada. Applications of the Remez algorithm to filter synthesis are described in
,
. An article on the general approach based on linear programming has
been published
.
Some important results have been obtained in order to handle tuning and synthesis of broad band filters. One of the major problems when dealing with wide band filters is the break down of
the classical low pass model which relies on a narrow band assumption. We showed however that there exists a unifying “low pass formalism” which is valid in the narrow band as well as in the
wide band situations. The latter relies on the following remark. Let
Sbe any inner, real, symmetric (
S^{t}=
S), rational matrix, which is identity at infinity and has MacMillan degree
n. Then the rational matrix
S_{r}defined by:
is again an inner, complex, symmetric matrix, which is identity at infinity, and has MacMillan degree
n. It can be shown that
Sis entirely characterised by the knowledge of its reduced “complex” version
S_{r}. Measurements of
Son two conjugate frequency bands are mapped to measurements of
S_{r}on a single band, which up to the use of a linear frequency transformation can be cast to the normalized band
[1, 1]. Usual techniques used to recover rational models from low pass responses measured on a single frequency interval can therefore be used to recover high pass
responses via the use of the generalized reduced response
S_{r}. Implementation attempts of the latter in the PrestoHf software were started and encouraging results where obtained for the tuning of an ultrawide band filter realized with suspended
strip lines. Figure
shows data and their rational approximation of this 10
^{th}order filter (reduced order 5) with a bandwidth ratio of approximately
10%(in collaboration with the university of Ulm, Germany).
Concerning the synthesis of the response of such filters we had already shown that the latter amounts to a Zolotarev problem with a nonpolynomial weight (with a square root singularity).
For fixed transmission zeros we were however able to derive explicit formulas for the optimal (in the Chebychev sense) filter function
F_{n}:
where
is a suitable parabolic frequency transformation and the
z_{k}^{'}sare prescribed transmission zeros. Recurrence formulas for the practical computation of
F_{N}have been derived and implemented as part of the DedaleHF software package. As for the realization of such responses, first results were obtained with resonant coupling elements
In collaboration with the RMC and possibly with XLim and STMicroelectronics (Tours) our goal is now to test the validity of our unified approach on real examples. Data collection campaigns obtained during tuning phases are scheduled. Joint publications about the topic are also in progress.
An OMUX (Output MUltipleXor) can be modeled in the frequency domain through scattering matrices of filters, like those described in section , connected in parallel onto a common guide. The problem of designing an OMUX with specified performance in a given frequency range naturally translates into a set of constraints on the values of the scattering matrices and of the phase shift introduced by the guide in the considered bandwidth.
An OMUX simulator on a Matlab platform was designed last year and checked against a number of designs proposed by Thales Alenia Space (a.k.a. TAS). Under the terms of a contract with TAS (2007), it has been used to design a dedicated software to optimize OMUXes whose second release to TAS has taken place this year.
The software proceeds by adding channels recursively, applying to the new channel the above shortcircuit and reflectioninthe bandwidth rules. This yields an initial guess for the global “optimizer” which seems to regularly outperform those currently used by TAS. More extensive tests are being conducted. A natural sequel should consist of the study of the socalled “manifoldpeaks” that may impede a design based on ideal assumptions of losslesness.
A new problem was brought in by Damien Pacaud (TAS) concerning a deembedding problem one encounters while tuning Tjunction diplexers. Let
Sbe the measured scattering matrix of a diplexer composed of a junction with response
Tand two filtering devices with response
Aand
Bas plotted on figure
. The deembedding question is the following: given
Sand
T, is it possible to derive
Aand
B? Although the question may appear classical very little seems to be known about it in the literature and among measurements specialists.
Using algebraic elimination techniques we derived the following interesting relation:
S_{1, 1}
S_{1, 2}*
S_{1, 3}/
S_{2, 3}=
T_{1, 1}
T_{1, 2}*
T_{1, 3}/
T_{2, 3}
which shows that the reflexion term
S_{1, 1}can be deduced, from the remaining measurements (independently from
Aand
B). As a consequence of this redundancy we showed that deembedding problem, in its current statement, is ill posed. Even if additional lossless conditions are made, the following
statement holds: for every phase parameter of the transmission
A_{1, 2}there exists a unique set of measurements
Aand
Bthat are compatible with
Sand
T. Moreover closed form expressions exists for the derivation
Aand
B.
In order to overcome the ill posed character of the deembedding problem we currently study approaches where several measuring campaigns are made, for example while varying the short circuit
position of the junction
T. Generalisation of our preliminary results to general multiplexers are also of great interest.
This negative answer to the question raised in the last paragraph of section calls for the following question, which is important for modeling control systems: are there local “qualitative” differences between the behavior of a nonlinear system and that of its linear approximation when the latter is controllable? It would also be interesting to know whether, for equivalence between arbitrary systems (not assuming that one of them is linear controllable), the gap between topological and smooth equivalence is still negligible.
If two control systems on manifolds of the same dimension are dynamic equivalent (see section ), we prove in that either they are static equivalent – i.e.equivalent via a classical diffeomorphism– or they are both ruled; for systems of different dimensions, the one of higher dimension must be ruled. A ruled system is one whose equations define at each point in the state manifold, a ruled submanifold of the tangent space. It was already known that a differentially flat system must be ruled; this is a particular case of the present result, in which one of the systems is “trivial” (i.e., linear controllable).
This is an important contribution because it is difficult in general to prove that two systems are notdynamic equivalent. No general general necessary condition (or obstruction) was known; that condition is also the only general obstruction known for flatness.
In the terminology of
,
, a
Kepler control systemis a system in dimension
nwhose drift has
n1first integral and compact trajectories and where the control is “small” in the sense that we are interested in asymptotic properties as the bound on the control
tends to zero. It is the case in low thrust orbital transfer, see section
, for negative energy,
i.e.in the socalled elliptic domain.
For this class of systems, a notion of average control systemis introduced in , . Using averaging techniques in this context is rather natural, since the free system produces a fast periodic motion and the smallcontrol a slow one; averaging is a widespread tool in perturbations of integrable Hamiltonian systems, and the small control is in some sense a “perturbation”. In some recent literature, one proceeds as follows: the control is preassigned, for instance to time optimal control via Pontryagin's Maximum Principle or else to some feedback designed beforehand. Then, averaging is performed on the resulting ordinary differential equation, whose limit behavior is analyzed when the control magnitude tends to zero.
The novelty of (see also ) is to average beforeassigning the control, hence getting a control systemthat describes the limit behavior better. For that reason, the average control system is a convenient tool when comparing different control strategies.
It allowed us to answer an open question stated in on the minimum transfertime between two elliptic orbit when the thrust magnitude tends to zero, see .
Under some controllability conditions that are trivially satisfied in the case at hand, we proved that the average system is one where the velocity set has nonempty interior, i.e. all velocity directions are allowed at any point, and the constraint is convex; mathematically this yields a Finsler structure (in the same way as a controllable system without drift with a quadratic constraint on the control yields a subRiemannian structure). An article is in progress, reporting on these results .
Contract (reference Inria: 2470, CNES: 60465/00) involving CNES, XLim and Inria, whose objective is to work out a software package for identification and design of microwave devices. The work at Inria concerns the design of multiband filters with constraints on the group delay. The problem is to control the logarithmic derivative of the modulus of a rational function, while meeting specifications on its modulus.
L. Baratchart is a member of the editorial board of Computational Methods and Function Theoryand Complex Analysis and Operator Theory.
AHPI (Analyse Harmonique et Problèmes Inverses), is a “Projet blanc” in Mathematics involving InriaSophia (L. Baratchart coordinator), the Université de Provence (LATP, AixMarseille), the Université Bordeaux I (LATN), the Université d'Orléans (MAPMO), InriaBordeaux and the Université de Pau (Magique 3D). It aims at developing Harmonic Analysis techniques to approach inverse problems in seismology, Electroencephalography, tomography and nondestructive control.
Filipix (FILtering for Innovative Payload with Improved fleXibility) is a “Projet Thématique en Télécommunications”, involving InriaSophia (Apics), XLim, Thales Alenia Space (Centre de Toulouse, coordinator).
NSF EMS21RTG is a students exchange program with Vanderbilt University (Nashville, USA).
NSF CMGcollaborative research grant DMS/0934630, “Imaging magnetization distributions in geological samples”, with Vanderbilt University and the MIT (USA).
EPSRCresearch grant EP/F020341/1 (Operator theory in function spaces on finitelyconnected domains), with Leeds University (UK) and the University Lyon I, 20072009.
A program InriaTunisian Universities(STIC) links Apics to the LAMSINENIT (Tunis).
Apics is linked with the CEAIRFM (Cadarache), through a grant with the Région PACA, for the thesis of Y. Fischer.
Apics is part of the regional working group SBPI (Signal, Noise, Inverse Problems), with teams from Observatoire de la Côte d'Azur and Géoazur (CNRS) http://wwwsop.inria.fr/apics/sbpi.
The following scientists gave a talk at the team's seminar:
Slim Chabaane, Univ. Sfax (Tunisie), Quelques estimations logarithmiques de type optimal dans les espaces de Hardy Sobolev .
Camilla Colombo, Univ. Glasgow, Méthodes d'optimisation et de contrôle pour des missions d'interception d'astéroïdes.
Karine Dadourian, Ecole Centrale Marseille, Approximation nonlinéaire multiéchelles. Application à la compression d'images.
Blaise Faugeras, LJADUniv. Nice SA (Nice), Identification de l'équilibre du plasma dans un Tokamak en temps réel.
Moncef Mahjoub, LAMSINENIT (Tunis), Complétion de données dans un domaine annulaire. Application à la résolution de quelques problèmes inverses.
Jean Baptiste Pomet, Équivalence et linéarisation des systèmes de contrôle.
Laurent Praly, CAS, Mines ParisTech, Nonlinear Observer Design with an Appropriate Riemannian Metric.
Pierre Rouchon, CAS, Mines ParisTech, Feedback generation of quantum Fock states by discrete QND measures.
Ed Saff, Axissupported External Fields on the Sphere.
Meriem Zghal, Problème inverse d'identification de sources en EEG: Résolution et étude de la stabilité.
L. Baratchart, Mathematics, Vanderbilt University (Nahville, TN, USA), since August.
Y. Fischer, Mathématiques pour l'ingénieur, section Mathématiques Appliquées et Modélisation, 3rd year, École Polytechnique Univ. NiceSophia Antipolis (EPU).
J. Leblond, Centre Montessori, collège, MouansSartoux, until June.
A.M. Nicu, Mathématiques, section Génie des Eaux, 3rd year, EPU, since September.
M. Olivi, Mathématiques pour l'ingénieur (Fourier analysis and integration), section Mathématiques Appliquées et Modélisation, 3rd year, EPU.
M. Zghal, Mathématiques, sections Mathématiques Appliquées et Modélisation, Génie des Eaux, 3rd year, EPU.
Slah Chaabi, « Problèmes extrémaux pour l´équation de Beltrami réelle 2dimensionnelle et application à la détermination de frontières libres », coadvised, Univ. AixMarseille I.
Yannick Fischer, « Problèmes inverses pour l'équation de Beltrami et extrapolation de quantités magnétiques dans un Tokamak », Univ. NiceSophia Antipolis.
Ahed Hindawi « Transport optimal en contrôle », Univ. NiceSophia Antipolis.
AnaMaria Nicu, « Inverse potential problems for MEG/EEG », Univ. NiceSophia Antipolis.
Meriem Zghal, « Constructive aspects of some inverse problems (Cauchy, sources) for Laplace equation in ellipsoidal domains », coadvised, Univ. Tunis El Manar (Tunisia).
JeanBaptiste Pomet, « Equivalence et linéarisation des systèmes de contrôle », defended in october .
J. Leblond was a member of the PhD defense committees of R. Mdimegh, Univ. Tunis El Manar (Tunisie), and of H. Meftahi, Univ. Lille I.
L. Baratchart is Inria's representative at the « conseil scientifique » of the Univ. Provence (AixMarseille). He was a member of the “Commissions de spécialistes” of the Univ. of Lille and Bordeaux.
J. Grimm is a representative at the « comité de centre » (Research Center INRIASophia).
J. Leblond is a member of the « Commission d'Évaluation » (CE) of INRIA
M. Olivi is a member of the CSD (Comité de Suivi Doctoral) of the Research Center.
J.B. Pomet is a representative at the « comité technique paritaire » (CTP) of INRIA.
F. Seyfert is a member of the CDL (Comité de Développement Logiciel) of the Research Center.
L. Baratchart was a plenary speaker at the Conference “Time series analysis and system identification”, Vienna (Austria). He gave a communication at the Conference “Computational Methods in Function Theory” (CMFT, Ankara), and was a “colloquium speaker” at Univ. of Michigan (East Lansing, USA), and at Univ. of Mississipi (Oxford, USA). He participated to the Colloque “Aspects géométriques des EDP”, CIRM, Luminy (Mars), together with Y. Fischer.
Y. Fischer gave a presentation at the First FrancoTunisian Conference on Mathematics, CFTM1, Djerba (Tunisia, Mars), and at the Congress SMAI 2009, La Colle sur Loup (France, May). He was also invited to give a talk at the meeting of the ANR project AHPI, SophiaAntipolis (April), at a working group of Labo. J.A. Dieudonné, Univ. Nice SophiaAntipolis (Nice, May), and at the “Séminaires Croisés”, INRIASophia, session “Simulations numériques pour ITER et la Fusion”, October. He participated to the Summer School of the Large Scale Initiative "Fusion", IRMA, Strasbourg (September).
J. Leblond was an invited speaker at the Colloque “Aspects géométriques des EDP”, CIRM, Luminy (Mars). She gave a plenary talk at the Conference on Distributed Parameter Systems (CDPS), IFAC, Toulouse (July), and at the Workshop “Operator Theory and Applications”, ICMS, Edinburgh (Scotland, Sept.). She was invited to give a talk at the Seminars of the team Algorithms, INRIARocquencourt (January) and of the team Analyse et Géométrie, LATPCMI, Univ AixMarseille I (April). She organized at INRIASophiaAntipolis the meeting of the ANR project AHPI (together with E. Russ and S. Sorres, April), and the session “Simulations numériques pour ITER et la Fusion” of the “Séminaires Croisés” (October).
A.M. Nicu participated to the Summer School “Traitement du signal et des images”, Peyresq (France, July) and to the School CEAEDFINRIA, “Eléctrophysiologie cardiaque et cérébrale”, INRIARocquencourt (November).
M. Olivi attented the Sysid Meeting in StMalo (France). She gave a presentation at the Groupe de Travail Identification of the GDR MACS (Paris, France) and at the Department of Knowledge Engineering, University of Maastricht. She presented a poster at the 2009 ERNSI Meeting in Stift Vorau (Austria).
F. Seyfert was invited to give a talk on “Broad band filter synthesis” at the European Microwave Conference in Roma (Italy). He also gave a talk at the CNESESA International Workshop on Microwave Filters in Toulouse (France).
M. Zghal presented a poster at the Congress SMAI 2009, La Colle sur Loup (France, May), for which she got a price of the best conference posters.