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), as well as discretization issues of the gravitational potential in geophysics (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
fit is 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. 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. section ).
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
section
). 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.
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 dualmode 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. 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 functions, 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 closedloop 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 much 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.
A class of systems of interest to us has been the one of systems with a conservative drift and a small control (whose effect is small in magnitude compared to the drift). A prototype is the control of a satellite with low thrust propellers: the conservative drift is the classical Kepler problem and the control is small compared to earth attraction. We developed, starting with Alex Bombrun's PhD , original averaging methods, that differ from classical methods in that the average is a control system, i.e. the averaging process does not depend on the control strategy. A reference paper is still under preparation .
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 ).
Optimal transport is the problem of finding the cheapest transformation that moves a given initial measure to a given final one, where the cost of the transformation is obtained by integrating against the measure a pointtopoint cost that may be a squared Euclidean distance or a Riemannian distance on a manifold or more exotic ones where some directions are privileged that naturally lean towards optimal control.
The problem has a long history which goes back to the pioneering works ( , ), and was more recently revised and revitalized by and . At the same time, applications to many domains ranging from image processing to shape reconstruction or urban planning were developed, see a survey in .
We are interested in 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 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. 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 arrival of L. Rifford and the PhD of A. Hindawi, whose goal is to tackle the problem of systems with drift. See new results in section .
The motivations for a detailed study of equivalence classes and invariance of models of control systems under various classes of transformations 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.
The interested reader can find a richer overview (in french) in the first chapter of .
A static feedbacktransformation is a (nonsingular) reparametrization of the control depending on the state, together with a change of coordinates in the state space. Static equivalence has motivated a very wide literature; in the differentiable case, classification is performed in relatively low dimensions; it gives insight on models and also points out that this equivalence is “too fine”, i.e. very few systems are equivalent and normal forms are far from stable. This motivates the search for a rougher equivalence that would account for more qualitative phenomena. The HartmanGrobman theorem states that every ordinary differential equation (i.e. dynamical system without control) is locally equivalent, in a neighborhood of a nondegenerate equilibrium, to a linear system via a transformation that is solely bicontinuous, whereas smoothness requires many more invariants. This was a motivation to study topological(non necessarily smooth) equivalence. A “Hartman Grobman Theorem for control systems” is stated in under weak regularity conditions; it is too abstract to be relevant to the above considerations on qualitative phenomena: linearization is performed by functional noncausal transformations rather than feedback transformations stricto sensu; it however acquires a concrete meaning when the inputs are themselves generated by finitedimensional dynamics. A stronger Hartman Grobman Theorem for control systems (where transformations are homeomorphisms in the statecontrol space) in fact cannot hold : almost all topologically linearizable control systems are differentiably (in the same class of regularity as the system itself) linearizable. In general (equivalence between nonlinear systems), topological invariants are still a subject of interest to us.
A dynamic feedbacktransformation consists of a dynamic extension (adding new states, and assigning them new dynamics) followed by a state feedback on the augmented system; dynamic equivalence is another attempt to enlarge classes of equivalence. It is indeed strictly more general than static equivalence: it is known that many systems are dynamic equivalent but not static equivalent to a linear controllable system. The classes containing a linear controllable system are the ones of differentially flat systems; it turns out (see ) that many practical systems are in this class and that being “flat” also means that all the solutions to the systems are given by a (Monge) parametrizationthat describes the solutions without any integration.
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 differential operator giving the parametrization. Within the team, results on low dimensional systems have been obtained ; the above mentioned difficulty is not solved for these systems but results are given with a prioriprescribed bounds on this order.
For general dynamic equivalence as well as flatness, very few invariants are known. In particular, the fact that the size of the extra dynamics contained in the dynamic transformation (or the order of the above mentioned differential operator, for flatness) is not a priori bounded makes it very difficult to prove that two systems are notdynamic feedback equivalent, or that a system is notflat. Many simple systems pointed out in are conjectured not to be flat but no proof is available. The only known general necessary condition for flatness is the socalled ruled surface criterion; it was generalised by the team to dynamic equivalence between arbitrary nonlinear systems in and .
Another attempt towards conditions for flatness used the
differential algebraic point of view: the module of
differentials of a controllable system is, generically,
free and finitely generated over the ring of differential
polynomials in
d/
dtwith coefficients in the ring of
functions on the system's trajectories; flatness amounts to
existence of 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. Some non
classical conditions have to be added to the classical
stability by exterior differentiation, and the problem is
open. In
, a partial answer was given,
but in a framework where infinitely many variables are
allowed and a finiteness criterion is still missing.
This domain is mostly connected to the techniques described in section .
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 3D spherical situations,
also discussed in section
).
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 were shown 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 Athena 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 ).
Inverse potential problems are also naturally encountered
in magnetization issues that arise in nondestructive control.
A particular application, which is 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.
In turns out that discretization issues in geophysics can also be undertaken by these approximation techniques. Indeed, in geodesy or for GPS computations, one may need to get a best discrete approximation of the gravitational potential on the Earth's surface, from partial data there. This is also the topic of the PhD theses of A.M. Nicu, and of a beginning collaboration with a physicist colleague (IGN, LAREG, geodesy). Related geometrical issues (finding out the geoid, level surface of the gravitational potential) should be considered either.
This domain is mostly connected to the techniques described in section .
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 ).
This domain is mostly connected to the techniques described in section .
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 low thrust 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 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. An extension of the software to the case of triple poles approximants is now available. It gives nice results in the source recovery problem. 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 (see ) (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 core of the Endymionsystem (a followup to hyperion) 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.
The development of Endymion, http://wwwsop.inria.fr/apics/endymion/index.htmlhas come to an end. The software is still maintained and sources are available on the ftp server.
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 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 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 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 software is described in
,
,
,
. Other applications of Tralics
consist in putting scientific papers on the Web; for instance
Cedram (
http://
This new research theme started in 2009. 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, thanks to a bunch of axioms designed by Carlos Simpson (CNRS, Nice). 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. The implementation is described in the technical report and led to a publication in the Journal of Formalized Reasoning .
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 . The software has been rewritten, using the ssreflect library.
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 generated 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 quite interesting but
considerably more difficult. A collaborative work is going
on, in the framework of the ANR project AHPI, aiming mainly
at the case
p=
.
We investigate the connections between the BMO
L^{2}harmonic gradients and valued in its orthogonal
space embedded in
L^{2}vector fields on the sphere whose tangent component
is a gradient. This issue is also considered in
L^{p},
1<
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 .
In 3D, functional or clinical active 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 Athena.
Further, it appears that in the rational approximation step of these schemes, multiplepoles possess a nice behaviour with respect to the branched singularities (see figure ). 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 so far 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 also appears to be interesting for geophysical issues, concerning the discretization of the gravitational potential by means of pointwise masses. This is another 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, September) and with IGN (Paris, LAREG, geodesy).
Magnetic sources localization from observations of the field away from the support of the magnetization is a topic under investigation in a joint effort with the Math. department of Vanderbilt University and the Earth Sciences department of MIT. The goal is to recover the magnetic properties of rock samples (meteorites) from fine measurements very close to the sample that can nowadays be obtained using SQUIDs (supraconducting coil device).
We completed the analysis of the kernel of the magnetization operator for 3D samples, which is the Riesz potential of the divergence of the magnetization assumed to be of bounded variation. It can be described in terms of measures whose balayage on the boundary of the sample vanishes, but this is not so effective, computationally.
The case of a thin slab (the magnetization is then modelled as a vector field on a portion of the plane) has proved more amenable. In the uni and bidirectional cases, using Hodge decomposition, one can show that the kernel contains only divergence free tangential vector field that are constant on lines, showing in particular that it reduces to zero for compactly supported magnetizations (which is always the case). A paper is being written on these results. Meanwhile, the severe illposedness of the reconstruction challenges discrete Fourier methods, one of the main problems being the truncation of the observations outside the range of the SQUID measurements. The next step will be to develop the extrapolation techniques initiated by the project team, using bounded extremal problems, in an attempt to overcome this issue.
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 Lipschitz 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
.
This year we generalized the construction to finitely
connected Dinismooth domains and
W^{1,
q}smooth conductivities, with
q2. The case of an annular geometry is
the relevant one for the application to plasma shaping
mentioned below. An article is currently being written on
these topics.
The application that initially motivated this work came from free boundary problems in plasma confinement (in tokamaks) for thermonuclear fusion. This work was initiated 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, the conductivity is
1/
xand the domain is an annulus
embedded in the right halfplane. We obtained a basis of
solutions (exponentials times Legendre functions) upon
separating variables in toroidal coordinates. This may be
viewed as a generalization to the annulus of the Bessel
type expansions derived in
,
for simply connected
geometries. This provides a computational setting to solve
the extremal problems mentioned before, and is the topic of
the PhD thesis of Y. Fischer.
On the halfplane, the conductivity
1/
xis severely unbounded but the
analysis of this test case is quite important for the
convergence of extrapolation algorithms to recover magnetic
quantities on the chamber. Additive decompositions into
Hardy solutions inside the outer boundary and outside the
inner boundary, with controlled vanishing on the imaginary
axis, have been obtained as part of the PhD work of S.
Chaabi.
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 related to the efficiency of the energy transfer. This issue is still untouched, but should be the topic of future studies, once the present approach will have been validated numerically.
Lossless systems and their transfer functions play a central role in system theory mainly, but not only, due to the DouglasShapiroShields factorization. Lossless matrixvalued functions generalize, in some sense, the notion of denominator to the matrix case. As such they are involved in many representations of stable systems. They are also present in many applications, as the scattering matrix of a resonant filter or the polyphase matrix of a filter bank, for example. Their parametrization is an important issue for many purposes going from optimization, model reduction to physical parameters recovery.
In , by making appropriate choices in the Schur algorithm, construction of balanced canonical forms is achieved in a recursive way with unitary matrix multiplications. Each step of the recursion involves an interpolation condition at a point located within the analyticity domain. Thereby, we get parametrizations which combine the computational interest of canonical forms and the conceptual interest of Schur analysis. However, for several reasons, it may prove helpful to allow for interpolation points on the boundary of the analyticity domain (the circle in discretetime and the imaginary axis in continuoustime). In , the tridiagonal canonical form of Ober in continuoustime was obtained from a recursion involving interpolation conditions at infinity. Vanishing moments, diagonal Markov parameters, can be interpreted in term of boundary interpolation conditions.
This year, we extended the work of
on
discretetime lossless systemsto allow for
interpolation conditions
on the unit circle. We investigate the possibility to
parametrize orthogonal wavelets with vanishing moments using
these results. A vanishing moment condition can be expressed
as a boundary interpolation condition for the lossless
polyphase filter. 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
.
The theory of orthogonal polynomials on the unit circle is
a most classical piece of analysis which is still the object
of intensive studies. The asymptotic behaviour of orthogonal
polynomials is of special interest for many issues pertaining
to approximation theory and the spectral theory of
differential operators. Its connection with prediction theory
of stationary stochastic processes has long been known
. Namely, the
nth orthonormal polynomial with respect to the
spectral measure of the process yields the optimal regression
coefficients of a linear onestep ahead predictor from the
n1st last values, in the sense of
minimum variance of the error. Likewise, the (inverse of) the
dominant coefficient of the polynomial gives the prediction
error. In particular, asymptotics for the dominant
coefficient determine the asymptically optimal prediction
error from the past as time goes large.
As compared to orthogonal polynomials, orthogonal rational functions have not been much considered up to now. They were apparently introduced by Dzrbasjan but the first systematic exposition seems to be the monograph by Bultheel et al. where the emphasis is more on the algebraic side of the theory. In fact, the asymptotic analysis of orthogonal rational functions is still in its infancy.
Having demonstrated last year, under mild smoothness assumptions, the possibility of convergent rational interpolation to Cauchy integrals of complex measures on analytic Jordan arcs and their strong asymptotics , we started investigating the case of Cauchy integrals on socalled symmetric contours for the logarithmic potential. These correspond to functions with more than two branched singularities, like those arising in the slicing method for source recovery in a sphere when there is more than one source (see section ). Recently we obtained weak asymptotics in this case, through an existence result and a characterization of compact of minimum capacity outside of which the function is singlevalued, teaming up with results from . A manuscript has just been completed on the subject.
To study strong asymptotics, at present we limit ourselves to a threefold geometry, and to the case of Padé approximants (interpolation at a single point with high order). The result is that uniform convergence can only take place if the weights of the branches of the threefold with respect to the equilibrium distribution are rationally dependent. If they are algebraically dependent, a spurious pole clusters to certain curves within the domain of analyticity, and if they are algebraically independent, exactly one pole exhibits chaotic behaviour in the complex plane. This generalizes results of Suetin on Cauchy integrals on disconnected pieces of a smooth contour. This investigation will continue.
We continued our work on the circuit realisations of filters' responses with mixed type (inductive or capacitive) coupling elements and constrained topologies. Our first paper describing the algebraic theory governing these equation was published . We focused on some practical application of it in where compact filter using zero generating irises were studied and designed. We now focus on the use of resonating couplings in the design of asymmetric filter's characteristics without the use crosscoupling in order to simplify their practical implementation. In parallel efforts are being spent to improve the synthesis method to higher order filters, having in mind some application to diplexer's filter with high number of symmetrically placed transmission zeros.
In another area of circuit realisation, we also started some work on lossy filter synthesis .
The objective of our work is the derivation of efficient algorithms for the synthesis of microwave multiplexers. Although the functioning of such devices relies primary on Maxwell equations, classical modal analysis techniques yield equivalent reciprocal electrical circuits that exhibit good approximation properties in a restricted frequency range. These models present an interesting trade off between flexibility and accuracy that justifies their central position among the techniques used for the design of microwave devices. In particular, the design of microwave filters relies heavily on the polynomial structure of 2×2reciprocal lossless squattering matrices. This structure allows to cast the design, under modulus constraints, of the response of a filter to a quasiconvex optimisation Zolatariov problem.
The polynomial structure of
3×3reciprocal lossless squattering
matrices turn out to be more involved and mainly unknown. An
important step was performed this year in that direction
using the results obtained in
on reciprocal Darlington
synthesis. We derived a method
to count and compute all possible
3×3reciprocal lossless extensions
of a rational Schur function
p/
q(a reflection of the scattering
matrix). From a practical synthesis point of view, the
extension process that starts with the numerators
p_{1}and
p_{2}of the transmission elements is more relevant.
Divisibility conditions have been derived which can be
interpreted as interpolation conditions connecting
p/
qon one hand and
p_{1}and
p_{2}on the other hand. The Shur function
p/
qcan thus be computed from
p_{1}and
p_{2}running a Schur type algorithm. 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. Of
particular interest is the case where the underlying circuit
is composed of two filters connected at a common port. Based
on some of the characteristics the rational response posseses
in this case, the design and realization of a compact
diplexer was performed in the context of the ANR Filipix
.
Some progress was made on the deembedding of filter
responses when starting from the external measurements of a
diplexer. The problem states as followed. 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. It was shown that unless some additional hypothesis
are made on the responses
Aand
Bthe problem has no unique solution. More precisely, at
every frequency point the solution set is a complex algebraic
variety of dimension one which, for example, can be
parametrised by the transmission term of one the two
filtering devices. Under some losslessness assumption on all
system components the dimension of the solution set drops to
a single real one and the phase of one transmission term (of
Aor
B) can be taken as a free parameter. Eventually the
problem is being studied under some hypothesis about the
rationality of the device's response or when junctions with
more than one possible state are used (resulting in several
measurements of the diplexer). This work is pursued in
collaboration with Thales Alenia Space and the Polytecnico di
Milano as well as with the support of the ANR Filipix.
There has been a recurrent collaboration with X. Litrico on simple nonlinear models obtained from a family of linear systems with delay. This comes from situations where a PDE is linearized in the neighborhood of a steadystate behavior and then approached by a linear system with delay. In the paper , we develop this method, aiming at control or forecast of river flow.
Publications and are related to optimal transport in that they study geometric conditions in Riemannian geometry that are either sufficient or necessary for continuity of transport map. is a deep study of the MaTrudingerWang Tensor while studies convexity of the injectivity domain; both are restricted to surfaces.
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.
Contract (reference CNES: RS10/TG0001019) involving CNES, University of Bilbao and Inria whose objective is to set up a methodology for testing the stability of amplifying devices. The work at Inria concerns the design of frequency optimization techniques to identify the linearized response and analyze the linear periodic components.
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.
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).
APICS is part of the European Research Network on System Identification (ERNSI)
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).
Cyprus NF grant“Orthogonal polynomials in the complex plane: distribution of zeros, strong asymptotics and shape reconstruction.”
A program InriaTunisian Universities(STIC) links Apics to the LAMSINENIT (Tunis).
 Elodie Pozzi (Univ. Lyon I).
 Yannick Privat (CNRS, ENS Cachan, antenne Bretagne).
 Doug Hardin (Vanderbilt University).
 Vladimir Peller (Michigan State University).
 Nikos Stylianopoulos (University of Cyprus).
 Maxim Yattselev (University of Oregon at Eugene).
 Andrea Gombani (University of Padova).
 Smain Amari (Royal Military College of Canada)
L. Baratchart is Inria's representative at the « conseil scientifique » of the Univ. Provence (AixMarseille). He was a member of the “Comité de sélection” of the Univ. of Bordeaux I (section 25).
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.
L. Baratchart was a semiplenary speaker at the Conference “Mathematical theory of Networks and Systems”, Budapest (Hungary). He was an invited speaker at the conference “Finite and infinitedimensional complex analysis and applications”, Macau (China). He gave a communication at the Workshop “Boundary value problems and related questions” Beijing (China). He was a “colloquium speaker” at the Morningside Institute of the Chinese Academy of Sciences (Beijing). He was an invited participant to the Workshop “New perspectives on univariate and multivariate orthogonal polynomials”, BIRS, Banff (Canada). He gave a seminar at the Technische Universität (Berlin).
Y. Fischer gave a presentation at the meeting of the ANR project AHPI, Bordeaux (Jan.), and at the seminar of the team APICS (Oct.). He presented a poster at the conference PICOF 2010, “Inverse Problems, Control and Shape Optimization”, Carthagène (Spain, Apr.).
J. Leblond was an invited speaker at the Workshop in “Operator Theory and Complex Analysis”, Lyon (Nov.), She was invited to give a talk at the working group of the team Défi, INRIASaclay and CMAPPoytechnique (June) and at the seminar of the team Analyse et Géométrie, LATPCMI, Univ AixMarseille I (Dec.), and gave a communication at the meeting of the ANR project AHPI, INRIASophia (Nov.). She was a member of the “Conseil Scientifique” of the X ^{th}Forum for Young Mathematicians, CIRM, Luminy (Nov.).
J. Grimm gave a talk at the DML 2010 conference. He presented .
A.M. Nicu gave a communication at the meeting of the ANR project AHPI, INRIASophia (Nov.) and at the X ^{th}Forum for Young Mathematicians, CIRM, Luminy (Nov.). She presented a poster at the conference PICOF 2010, Carthagène (Spain, Apr.).
M. Olivi coorganized with A. Gombani two sessions on "Interpolation and approximation" and gave a communication at the Conference “Mathematical theory of Networks and Systems”, Budapest (Hungary). She gave a talk at the 2010 ERNSI Meeting in Cambridge (UK).
F. Seyfert coorganized with S.Bila (Xlim) a workshop on “Advanced topics in design and realization of microwave filters" at the European Microwave Conference in Paris (Sept.).
A meeting of the ANR AHPI was organized at INRIA (1719 Nov.).
L. Baratchart is a member of the editorial board of Computational Methods and Function Theoryand Complex Analysis and Operator Theory.
The following scientists gave a talk at the team's seminar:
Jana Nemcova, Rational Systems in Control and System Theory.
Pierre Vacher(ONERA, Toulouse), Recherche de paramétrisations adaptées à l'identification de systèmes linéaires MIMO.
Nicolas Brisebarre(CNRS, LIP, Arénaire, ENS Lyon) and Guillaume Hanrot(ENS Lyon, LIP, Arénaire), Approximants polynomiaux efficaces en machine.
Sylvain Chevillard(INRIA Nancy  Équipeprojet Caramel), Outils pour l'évaluation efficace de fonctions numériques.
Nataliya Shcherbakova(ENSEEIHT, Toulouse), Optimal control of 2level dissipative quantum control systems.
Laurent Baratchart, Problèmes inverses de magnétisation et décomposition de Hodge pour les champs à variation bornée.
Lydiya Yuschenko(Centre de Physique Théorique,
Luminy), Approximants de Padé à
Npoints complexes pour les fonctions de
Stieltjes.
Bernard Bonnard(Institut de Mathématiques de Bourgogne), Applications du contrôle des spins 1/2 dissipatifs en imagerie médicale.
Elodie Pozzi(Institut Camille Jordan, Lyon), Universalité du shift et des opérateurs de composition.
Andrea Gombani(ISIBInformation, Padova, Italy), On the partial realization problem.
Guillaume Charpiat(EPI PULSAR), Estimation de métriques convenant à une variété empirique de formes.
Maxym Derevyagin(Technische Universität Berlin), On the convergence of Pade approximants to rational perturbations of Markov functions.
Vladimir Peller(Michigan State University), Perturbations d'opérateurs normaux.
Yannick Privat(ENS Cachan), Comment déterminer le domaine de contrôle optimal d'une membrane en vue de sa stabilisation ?
Yannick Fischer, Problèmes extrémaux bornés pour une équation elliptique et applications à la résolution d'un problème inverse pour le tokamak Tore Supra.
Martine. Olivi (with Maureen. Clerc), Mathématiques pour l'ingénieur (Fourier analysis and integration), section Mathématiques Appliquées et Modélisation, 3rd year, École Polytechnique Univ. NiceSophia Antipolis (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 in MEEG and in geophysics », Univ. NiceSophia Antipolis.
Meriem Zghal, , coadvised, Univ. Tunis El Manar (Tunisia), until October.
Martine Olivi, «Parametrization of rational lossless matrices with application to linear system theory», defended in october .
JeanBaptiste Pomet was a member of the PhD defense committees of Li Shunjie (Université de Rouen), Gauthier Picot and Gabriel Janin (Université de Bourgogne, Dijon).