The Apics Team was created in January 2004 as a follow-up to the Miaou Team. On November 4th, the ``Comité des Projets'' of INRIA-Sophia recommended to the head of INRIA upgrading APICS to a Project Team.

The Team develops constructive methods for modeling, identification and control of dynamical systems.

Meromorphic approximation in the complex domain, with application to frequency identification and design of transfer functions, as well as singularity detection for the 2-D Laplace operator. Development of software for filter identification and the synthesis of microwave devices.

Inverse potential problems in 3-D and analysis of harmonic fields with applications to source detection and electro-encephalography.

Control and structure analysis of non-linear systems: continuous stabilization, linearization, and near optimal control with applications to orbit transfer of satellites.

Industrial collaborations with Alcatel Space, Alcatel-R&I, CNES, IRCOM.

Exchanges with UST (Villeneuve d'Asq), CMI-Université de Provence (Marseille), CWI (the Netherlands), CNR (Italy), SISSA (Italy), the Universities of Illinois (Urbana-Champaign), of South Florida (Tampa), of California (San Diego, Santa Barbara), of Alabama (Mobile), of Minnesota (Minneapolis), of Vanderbilt (Nashville), of Padova (Italy), of Beer Sheva (Israel), of Leeds (GB), of Maastricht and of Amsterdam (The Netherlands), of TU-Wien (Austria), of TFH-Berlin (Germany), of Kingston (Canada), of Szegëd (Hungary), of CINVESTAV (Mexico), ENIT (Tunis), VUB (Belgium).

The project is involved in a NATO Collaborative Linkage Grant (with Vanderbilt University and ENIT-LAMSIN), in a EMS21-RTG NSF program (with Vanderbilt University), in the ACI ``Obs-Cerv'' (with the Teams Caiman and Odyssée from Inria-Sophia Antipolis, among others), in a STIC Convention between INRIA and Tunisian Universities, in an EPSRC Grant with Leeds University (UK), in the ERCIM ``Working Group Control and Systems Theory'', in the ERNSI and TMR-NCN European research networks, and in a Marie-Curie EIF European program.

Let us first introduce the subject of Identification in some generality.

Abstracting in the form
of mathematical equations the behavior of a phenomenon is
a step called *modeling*. It typically serves two purposes: the first
is to describe the phenomenon with minimal complexity for some specific
purpose,
the second is to *predict* its outcome. This is general practice in
most applied sciences, be it for design, control or prediction,
although it is generally thought of as another optimization problem yet.

As a general rule, the user imposes the model to fit a parameterized form that reflects one's own prejudice, knowledge of the underlying physical system, and the algorithmic effort consented. Looking for such a trade-off usually raises the question of approximating the experimental data by the prediction of the model when the latter is subject to external excitations assumed to be the cause of the phenomenon under study. The ability to solve this approximation problem, which is often non-trivial and ill-posed, often conditions the practical usefulness of a given method.

It is when the predictive potential of a model is to be assessed that one is
led to *postulate* the existence of a *true* functional
correspondence between data and observations, thereby entering
the field of
*identification* itself. The predictive power of a model can be
expressed in various manners all of which are attempts to measure the
difference between the
true model and the observations. The necessity of taking into account the
difference between the observed behavior and the computed behavior induces
naturally the notion of *noise* as a corrupting factor
of the identification
process. This noise incorporates into the model, and can be handled in
a deterministic mode, where the quality of an identification algorithm is its
robustness to small errors. This notion is that of well-posedness in
numerical analysis or stability of motion in mechanics. The noise however is
often considered to be random, and then the true model is estimated by
averaging the data.
This notion allows approximate but reasonably simple descriptions of complex
systems whose mechanisms are not well known but plausibly antagonistic.
Note that,
in any case, some *assumptions* on the noise are required in order to
justify the approach (it has to be small in the deterministic case, and must
satisfy some independence and ergodicity properties in the stochastic
case). These assumptions can hardly be checked in practice, so that the
satisfaction of the end-user is the final criterion.

Hypothesizing an exact model also results in the
possibility of choosing the data in a manner suited for identifying a specific
phenomenon. This often interacts in a complex manner with the
*local* character of the model with respect to the data (for instance a
linear model is only valid in a neighborhood of a point).

We now turn to the activity of the team proper to identification.
Although the subject, on the academic level, has been the realm of
the stochastic paradigm for more than twenty years, it is in a
deterministic approach to identification of linear dynamical systems
(i.e. 1-D convolution processes) based on approximation in the complex domain,
that the Team made perhaps its most original contributions. Naturally, the
deep links stressed by the spectral theorem between time and frequency
domains induce well-known parallels between function theory and
probability, and the work of the Apics Team

The data are considered without postulating an exact model, but we simply look for a convenient approximation to the data in a range of frequency representing the working conditions of the underlying system. A prototypical example that illustrates our approach is the harmonic identification of dynamical systems which is widely used in the engineering practice, where the data are the responses of the system to periodic excitations in its band-width. We look for a stable linear model that describes correctly the behavior in this band-width, although the model can be inaccurate at high frequencies (that can seldom be measured). In most cases, we also want this model to be rational of suitable degree, either because this is imposed by the physical significance of the parameters or because complexity must remain reasonably low to allow the efficient use of the model for control, estimation or simulation. Other structural constraints, arising from the physics of the phenomenon to be modeled, often superimpose on the model. Note that, in this approach, no statistics are used for the errors, which can originate from corrupted measurements or from the limited validity of the linear hypothesis.

We distinguish between an identification step (called non-parametric in a certain terminology) that is provided with an infinite dimensional model, and an approximation step in which the order is reduced subject to certain specific constraints on the considered system. The first step typically consists, mathematically speaking, in reconstructing a function, analytic in the right half-plane, knowing its pointwise values on a portion of the imaginary axis, in other terms, to make the principle of analytic continuation effective on the boundary of the analyticity domain. This is a classical ill-posed issue (the inverse Cauchy problem for the Laplace equation) that we embed into a family of well-posed extremal problems, that may be viewed as a Tikhonov-like regularization scheme related to the spectral theory of analytic operators. This first step could in fact be made in higher dimensions, where analytic functions are replaced by harmonic fields. The second step is typically a rational or meromorphic approximation procedure (although other approximating families may arise as well) in some class of analytic functions in a simply connected domain, say the right half-plane in the case of harmonic identification. To make the best possible use of the allowable number of parameters, or to privilege some specific physical parameters of the system, it is generally important, in the second step, to compute optimal or nearly optimal approximants. Rational approximation in the complex plane is a classical and difficult problem, for which only few effective methods exist. In relation to system theory, mainly two difficulties arise: the necessity of controlling the poles of the approximants (to ensure the stability of the model), and the need to handle matrix-valued functions in the case where the system has several inputs and outputs.

Rational approximation in the
L^{p} sense to a transfer function on the imaginary axis (i.e the boundary
of the right half-plane) acquires a particular significance in this context
for p = 2 and p = .
If p = 2, it corresponds to parametric identification of minimum variance
when the system is fed with white noise input
(the case of colored noise corresponds to
weighted approximation), and it also corresponds
to the minimization of the error in operator norm in the
time domain. If p = , the approximation consists in minimizing the
power transfer L^{2}L^{2} of the error
(both in the time and frequency domains since
the Fourier transform is an isometry). These problems contribute
a generalization
(both rational and matrix-valued) of Szegö theory on orthogonal
polynomials, that seems the most natural framework for setting out
many optimization problems related to linear system identification.
Concerning this second step, it is worth pointing out that the analogs to
rational functions in higher dimensions are the gradients of
Newtonian potentials of discrete measures. Very little is known at present
on the approximation-theoretic properties of such objects.

We shall explain in more detail the above two steps in the sub-paragraphs
to come. For convenience, we shall approach them on the circle rather than
the line, which is the framework for discrete-time rather than continuous-time
systems. The two frameworks are mathematically equivalent *via* a Möbius
transform.

The title refers to the construction of a convolution model of
infinite dimension from frequency data in some bandwidth
and some reference gauge outside . The class of
models consists of stable transfer functions (*i.e.* analytic in the domain
of stability, be it the half-plane, the disk, etc),
and possibly also transfer functions with finitely many poles in the
domain of stability
*i.e,* convolution
operators corresponding to linear differential or difference equations
with finitely many unstable
modes. This issue arises in particular
for the design and identification of linear dynamical systems,
and in certain inverse problems for the Laplacian in dimension two.

Since the question under study may occur on the boundary of planar domains of
various shapes when it comes to inverse problems, it is common practice to normalize this boundary once and for
all, and to apply in each particular case a conformal transformation to
bring back to the normalized situation.
The normalized contour chosen here is
the unit circle. We denote by
D the unit disk, by H^{p} the Hardy space of exponent p, R_{N}
is the set of all rational functions having at most N poles in D,
and C(X) is the set of continuous functions on X. We are looking for a
function in H^{p} + R_{N}, taking on an arc K of the unit circle values that
are
close to some experimental data, and satisfying on some gauge
constraints, so that a prototypical Problem is:

*( P) Let p1, N0, K be an arc of the unit circle T,
fL^{p}(K), and M>0;
find a function gH^{p} + R_{N} such that
and such that g-f
is of minimal norm in L^{p}(K) under this constraint.*

In order to impose pointwise constraints in the frequency domain (for instance if the considered models are transfer functions of loss-less systems, see section ), one may wish to express the gauge constraint on in a more subtle manner, depending on the frequency:

*() Let p1, N0, K be an arc of the unit circle
T, fL^{p}(K), and
;
find a function
gH^{p} + R_{N} such that |g-|M a.e. on
and such that g-f is of minimal norm in
L^{p}(K) under this constraint.*

Problem (P) is an extension to the meromorphic case, and to incomplete data,
of classical analytic extremal problems (obtained
by setting K = T and N = 0), that generically go under the name
*bounded extremal problems*. These have been introduced and
intensively studied by the Team, distinguishing the case
p = from the cases 1p<, among which the
case p = 2 presents an unexpected link with the Carleman reconstruction
formulas .

Deeply linked with Problem *( P)*, and meaningful for assessing
the validity of the linear approximation in the considered pass-band, is the
following completion Problem:

*() Let
p1, N0, K an arc of the unit circle T,
fL^{p}(K), and
M>0; find a function
such that
, and such that the distance to
H^{p} + R_{N} of the concatenated function fh
is minimal in L^{p}(T) under this constraint.*

A version of this problem where the constraint depends on the frequency is:

*() Let p1,
N0, K an arc the unit circle T, fL^{p}(K),
and
; find a function such that
|h-|M a.e. on , and such that the distance to
H^{p} + R_{N} of the concatenated function fh
is minimal in L^{p}(T) under this constraint.*

Let us mention that Problem *()* reduces to Problem
*( P)* that in turn reduces, although implicitly,
to an extremal Problem without
constraint, (i.e. a Problem of type

The solution to *( P_{0})* is classical if

The study of Problem *()* has
been carried out in the case where p = 2, = 0, and the function
M is in
of and bounded from below almost everywhere by a
strictly positive constant. Together with the existence and uniqueness of the
solution, we have proved that the constraint is saturated pointwise,
that is |g| = M a.e. on , this being perhaps
counter-intuitive. We obtained fixed point equations that characterize the
solution, involving the resolvant of a Toeplitz operator, but
with a multiplier that is here a function . The study of
the convergence of an iterative scheme is under
examination, the goal
being its software implementation. Note that if we approach
the multiplier by a step function, we get a string of spectral equations
similar to these used for solving Problem *( P)*.

An algorithm that consists in discretizing the modulus constraint and using
Lagrange duality-based optimization techniques has been implemented
and performs satisfactorily. It has interesting connections with
affine Riemann-Hilbert problems *cf.*.

We emphasize that *( P)* has many analogs, equally
interesting, that occur in different contexts
connected to conjugate functions.
For instance one may consider the following extremal Problem,
where the constraint on the approximant is expressed in terms of
its real and imaginary parts while the criterion takes only its real part
into account:

*Let p1, K be an arc of the unit circle T,
fL^{p}(K), , and , , M>0;
find a function gH^{p} such that
and
such that is of minimal norm in L^{p}(K)
under this constraint.*

This is a natural formulation for issues concerning Dirichlet-Neumann problem for the Laplace operator, see sections and , where data and physical prior information concern real (or imaginary) parts of analytic functions.

For p = 2, existence and uniqueness of the solution have been established in as well as a constructive solution procedure
which, in addition to the Toeplitz operator that characterizes the
solution of *( P)* in the case

Situations with other values of p will be considered, as well as a
suitable general weighted formulation of constrained extremal problems
on T.

In the non-Hilbertian case, where p2, , but still N = 0, the
solution of *( P)* can be deduced from that of

If p< and N>0, there is up to now no algorithmic solution to
Problem *( P_{0})* which is proved convergent. However, the
progress that were made allow us to conceive a coherent picture of
the main issues and to develop rather efficient
numerical schemes whose global convergence has been proved for prototypical
classes of functions in Approximation theory.
The essential features of the approach are summarized below.

First of all, in the case p = 2 and N>0 which is of particular importance,
Problem *( P_{0})* can be reduced to that of rational approximation
which is described in more details in
section . Here, the
link with classical interpolation theory, orthogonal polynomials, and
logarithmic potentials is strong and fruitful. Second, a general AAK theory
in

The case where 1p<2 remains largely open,
especially from the constructive
point of view, because if the approximation error can still be interpreted in
terms of singular values, the Hankel operator takes an abstract form
not permitting for a functional identification of its singular vectors.
Considering these values for p is not
simply an academic exercise:
the L^{1} criterion induces the operator norm
in the frequency domain, which is interesting for damping perturbations. It is
possible that some appropriate duality links the case p<2 to the case 2<p,
but this has not been established yet.

A valuable endeavor would be also to carry over to higher dimensions (in particular in 3-D) the above analysis, where harmonic fields replace analytic functions. On the ball or the half-space, it seems many of the necessary ingredients are available after the progress undergone by harmonic analysis in recent years, with the notable exception of multiplicative techniques. Pushing through some of them would be tantamount to making significant progress in harmonic identification.

Rational approximation is the second step mentioned in section and we first approach it in the scalar case, for complex-valued functions (as opposed to matrix-valued ones). The Problem can be stated as:

*Let 1 p, fH^{p} and n an integer;
find a rational function without poles in the unit disk, and of
degree at most n that is nearest possible to f in H^{p}.*

The most important values of p, as indicated in the introduction,
are p = and p = 2. In the latter case, the orthogonality between Hardy
spaces of the disk and of the complement of the disk (the last one being
restricted to functions that vanish at infinity to exclude
the constants) makes
rational approximation equivalent to meromorphic approximation, i.e. we
are back to Problem *( P)* of section with

It is only fair to say that the design of a numerically efficient
algorithm whose
convergence to the best approximant would be proved is the most
important problem from a practical perspective.
However, the algorithms developed by the team seem rather
effective and although their global
convergence has not been established.
*A contrario*, it is possible to consider an elimination algorithm
when the function to approximate is rational, in order to find all critical
points, since the problem is algebraic in this case. This method is surely
convergent, since it is exhaustive, but one has to compute the roots of an
algebraic system with n variables of degree N, where N is the degree of
the function to approximate and there can be as many as N^{n} solutions among
which it is necessary to distinguish those that are coefficients of polynomials
having all their roots in the unit disk; the latter indeed are
the only ones that generate critical
points. Despite the increase of computing capacity,
such a procedure is still unrealistic granted that
realistic values of n and N would be like a ten and a couple of hundreds
(cf. section ).

To prove or disprove the convergence of the above-described algorithms, and
to check them against practical situations, the team has undergone a
long-haul study of the number and nature of critical points, depending on the
class of functions to be approximated, in which
tools from differential topology and
operator theory team up with classical approximation theory.
The study of transfer functions of relaxation systems (*i.e.*
Markov functions) was initiated in and more or
less completed in , as well as the
case of e^{z} (the prototype of an entire function with convex
Taylor coefficients) and the case of meromorphic functions
(*à la* Montessus de Ballore) . After these studies, a
general principle has emerged that links the nature of the critical points in
rational approximation to the regularity of the decrease of the interpolation
errors with the degree, and a methodology to analyze
the uniqueness issue in the case where the
function to be approximated is a Cauchy integral on an open arc
(roughly speaking these functions
cover the case of singularities of dimension one that
are sufficiently regular, *cf.* section )
has been developed. This
methodology relies on the localization of the singularities *via*
the analysis of
families of non-Hermitian orthogonal polynomials, to obtain
strong estimates of the error that allow one to evaluate its relative decay.
Note
in this context an analogue of the Gonchar conjecture, that uniqueness ought
to hold at least for infinitely many values of the degree, corresponding to
a subsequence generating the *liminf* of the errors. This conjecture
actually suggests that uniqueness should be linked to the ratio of the
to-be-approximated function and its derivative on the circle.
When this ratio is pointwise
greater than 1 (*i.e.* the logarithmic variation is small), the Schwartz
lemma implies uniqueness in degree 1, and the generalization of this elementary
fact is an interesting open question.

Another uniqueness criterion has been obtained for rational functions, inspired from the spectral techniques of AAK theory. This result is interesting in that it is not asymptotic and does not require pointwise estimates of the error; however, it assumes a rapid decrease of the errors and the current formulation calls for further investigation.

The introduction of a weight in the optimization criterion is an interesting issue induced by the necessity to balance the information one has at the various frequencies. For instance in the stochastic theory, minimum variance identification leads to weight the error by the inverse of the spectral density of the noise. It is worth noting that most approaches to frequency identification in the engineering practice consists of posing a least-square minimization problem, and to weigh the terms so as to obtain a suitable result using a generic optimization toolbox. In this way we are led to consider minimizing a criterion of the form:

where, by definition,

and is a positive finite measure on T, p_{m} is a polynomial of
degree less or equal to m and q_{n} a monic polynomial of degree less or
equal to n. Such a problem is nicely put when is
absolutely continuous with respect to the Lebesgue measure and has invertible
derivative in . For instance when is the squared
modulus of an invertible analytic function,
introducing -orthogonal polynomials instead of the Fourier basis
makes the situation similar to the non-weighted case, at least if
mn-1. The corresponding algorithm was implemented
in the hyperion software. The analysis of the critical
points equations in the weighted case gives various counter-examples to
unimodality in maximum likelihood identification .

It is worth pointing out that *meromorphic* approximation is better
behaved (*i.e.* essentially invariant) with respect to the
introduction of a weight, see
.
Another kind of rational approximation, that arises in
several design problems where only constraints on the modulus are seeked,
consists in approximating the module of a function by the module of a rational
function, that is, solving for

This problem is strongly related to the previous ones;
in fact, it can be reduced to a convergent
series of standard rational approximation
problems. Note also that if p = , and if moduli are squared,
*i.e.* if the feasibility of

is required, one can use the Féjèr-Riesz characterization of positive trigonometric polynomials on the unit as squared moduli of algebraic polynomials to approach this issue as a convex problem in infinite dimension. This constitutes another fundamental direction for dealing with rational approximation in modulus that arises naturally in filter design problems.

We want here to study the behavior of poles of optimal
meromorphic approximants in L^{p} norm on a closed contour,
to functions defined by
Cauchy integrals of measures whose support lies inside the contour. If one
normalizes the contour to be the unit circle,
which is no restriction in principle thanks
to conformal mapping but raises of course difficult questions from
the constructive point of view for domains whose shape is not standard
(*i.e.* polygonal or
elliptic), we find ourselves again in the framework of
sections and
. The research so far has focused
on functions that are
analytic on and outside the contour, and
have singularities on an open arc inside the contour.

Generally speaking, the behavior of poles is particularly important in meromorphic approximation for the analysis of the error decrease with the degree and for most constructive aspects like uniqueness, so that everything here could take place in section . However, it is the original motivation of APICS to consider this issue in connection with the approximation of the solution to a Dirichlet-Neumann problem, so as to extract information on the singularities. This way to tackle a free boundary problem, classical in every respect but still widely open, illustrates the approach of the team to certain inverse problems, and gives rise to an active direction of research at the crossroads of function theory, potential theory and orthogonal polynomials.

As a general rule, critical point equations for these problems express that the
polynomial whose roots are the poles of the approximant
is a non-Hermitian orthogonal polynomial with
respect to some complex measure on the singular set
of the function to be approximated.
New results were obtained in recent years
concerning the location of such zeroes,
and the approach to inverse problem for the Laplacian that we outline
in this section appears to be attractive when the singularities
are one-dimensional, for instance in
the case of a cracked domain (see section
). In case the crack is
sufficiently smooth, the approach in question
is in fact equivalent to meromorphic
approximation of a function with two 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 .
Moreover the asymptotic density
of the poles is nothing but the equilibrium distribution on the geodesic arc
of the Green potential and it charges the end points, that are
*de facto* well localized if one is able to compute sufficiently many
zeros (this is where the method is not fully constructive).
It is interesting to note that these results
apply also, and even more easily, to the detection of monopolar and dipolar
sources, a case where poles as well as
logarithmic singularities exist. The case of more general cracks (for instance
formed by a finite union of analytic arcs) requires the analysis of the
situation where the number of branch points is finite but arbitrary. It is
conjectured that the poles tend to the contour
that links the end points
of these analytic arcs while minimizing the capacity of the condenser
, where T is the exterior boundary of the
domain (see section
). The conjecture is
confirmed numerically
and has been actually proved in the case where the locus of minimal capacity is
*connected*; this covers a large number of interesting cases, including
the case of general polynomial cracks, or of cracks consisting of
sufficiently smooth arcs.
This breakthrough, we hope, will constitute a
substantial progress towards a proof of the general case.
It would of course be very interesting to know what happens when
the crack is ``absolutely non analytic'', a limiting case
that can be interpreted as that of an infinite number of branch points,
and on which very little is known, although there are grounds to conjecture
that the endpoints at least are still accumulation points of the poles. This is
an outstanding open question for applications to inverse problems
.
Concerning the problem of a general
singularity, that may be two dimensional,
one can formulate the following
conjecture: if f is analytic outside and on the exterior
boundary of a domain D and if K is the minimal compact set
included in D that minimizes the capacity of the condenser
(T, K) under the constraint that f is analytic and single-valued outside
K (it exists, it is unique, and we assume it is of positive
capacity in order to
avoid degenerated cases), then every limit point (in the weak star sense)
of the sequence _{n}
of probability measures having equal mass at each pole of an optimal
meromorphic approximant (with at most n poles) of f in L^{p}(T)*has its support in K and sweeps out to the boundary of K
as the equilibrium measure on K of the condenser (T, K)*.
Yet this conjecture is far from being solved.

We conclude by mentioning 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 no
longer actual, and this makes for a long-term research project
with a wide range of applications.
It is interesting to mention that
the case of sources in dimension three in a spherical geometry, can
be attacked with the above 2D techniques as applied to planar
sections (see section
).

Matrix-valued approximation is necessary for handling systems with several
inputs and outputs, and 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 System-Theoretic sense) generalizes the degree. Hence the problem reads:
*Let 1 p, and n an
integer; find a rational matrix of size m×l without
poles in the unit disk and of McMillan degree at most n nearest possible
to in (H^{p})^{m×l}.*
To fix ideas, we may define
the

The main interest of the Apics Team lies in the case p = 2.
Then, the approximation algorithm
designed in the scalar case generalizes to
the matrix-valued situation . The
first difficulty consists here in the parametrization
of transfer matrices of given
McMillan degree n, and the inner matrices (*i.e.* matrix-valued functions
that are analytic in the unit disk and unitary on the circle) of degree n
enter the picture in an essential manner: they play the role of the denominator
in a fractional representation of transfer matrices using the so-called
Douglas-Shapiro-Shields 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 (parameterizations
valid in a neighborhood of a point), and he must handle changes of chart
in the course of the algorithm. The tangential Schur algorithm provides us with such a parameterization and allowed
the team to develop two rational approximation codes. The first one is
integrated in the hyperion software dealing with transfer
matrices while the other, which is developed under the matlab interpreter,
goes by the name of RARL2 and works with realizations.
Both have been experimented on
measurements by the CNES (branch of Toulouse),
IRCOM, and Alcatel Space, and
they give high quality results in all cases encountered so far.
These codes are now of daily use by
Alcatel Space and IRCOM, coupled with simulation software like EMXD to design
physical coupling parameters for the synthesis of hyperfrequency
filters made of resonant cavities, see .

In the above application, obtaining physical couplings requires the computation of realizations, also called internal representation in system theory. Among the parameterizations obtained via the Schur algorithm, some have a particular interest from this viewpoint . They lead to a simple and robust computation of balanced realizations and form the basis of the RARL2 algorithm.

Problems relative to multiple local minima naturally arise in
the matrix-valued case as well, but deriving criteria that
guarantee uniqueness is much
more difficult than in the scalar case. The case of rational functions
of the right degree already uses rather heavy machinery
, and that of matrix-valued Markov functions, that are
the first example beyond rational function
has made progress only very recently (*cf.*
section ).

In practice, a method similar to the one used in the scalar case has
been developed to generate local minima of a given order from those at
lower orders. In short, one sets out a matrix of degree n by
perturbation of a matrix of degree n-1 where the drop in degree is due to a
pole-zero cancellation. There is an important difference between
polynomial representations of transfer matrices and their
realizations: the former lead to an embedding in a ambient space of
rational matrices that allows a differentiable extension of the criterion on a
neighborhood of the initial manifold, but not the latter (the
boundary is strongly singular). Generating initial conditions in a recursive
manner is more delicate in terms of realizations, and some
basic questions on the boundary behavior of the gradient vector field
are still open.

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.

The asymptotic study of likelihood estimators is a natural companion
to the research on rational approximation described above. The context
is ultra-classical. Given a discrete process y(t) with values in
R^{p}, and another process with values in R^{m},
we check for an explanation of y in terms of u as
a finite order linear model:

where e is a white noise with p components, uncorrelated to u,
assumed to represent the uncertainty in y(t), and where the
transfer matrix [LH] that links (eu)^{t} to is rational and
stable of McMillan degree n, the matrix L being also of stable inverse
(among all noises with same covariance, and given innovation, we chose those
whose spectral factor has minimum phase). The number n is, by definition,
the order of the model. If we only suppose that [HL] belongs to the Hardy
space H^{2} and that L is outer (this means stably invertible in some
sense), such a representation is in fact general for *regular*
(i.e. purely non-deterministic) stationary processes.
Identification in this context appears then as a rational approximation
problem for
which the classical theory makes a trade-off between two antagonistic factors,
namely the bias error on the one hand that decreases when n increases
and the variance error on the other hand that increases with n
since the dispersion is
amplified with the number of parameters. This is the stochastic version of the
complexity versus precision alternative which is all-pervasive
in modeling.

If one introduces now as a new variable the rational matrix R defined by

and if T stands for the first block-row,
normalizing the variance of the noise to be the identity matrix,
the maximum likelihood
estimator is asymptotically equivalent, when the sample size increases,
to the minimization of

where is the spectral measure of the process (yu)^{t}
(which positive and matrix-valued)
and where Tr indicates the trace.
If we further restrict the class of
models by assuming that we deal with white noise, that is if L = I_{m},
one obtains a
weighted rational approximation problem corresponding to the minimization of the
variance on the output error. If moreover u itself is (observed)
white noise, the
situation becomes that of .

The consistency problem arises from the fact that the measure is
not available, so that one has to estimate () from time averages
of the observed samples, assuming that the process is ergodic. The question is
then to decide whether the argument of the minimum of the estimated
functional tends
to that of () when the sample size increases, and what is the
speed of convergence.
The most significant result here is perhaps
the one asserting that if there exists a functional model linking u to y
(*i.e. *u is indeed the cause of the phenomenon), and without
assuming compactness of the class of models ,
then consistency holds
under weak ergodicity conditions and persistent
excitation assumptions.
An analogous of the law of large numbers indicates, in this
context, that convergence is in the order of ,
where N is the sample size.

In the preceding result, consistency holds in the sense of pointwise
convergence of the estimates on the manifold of transfer functions
of given size
and order. One contribution of the former Miaou project has been to show that
the result holds
even if we do not postulate a causal dependency between inputs and outputs, the
measure being simply defined as the weak limit of the covariances.
A second contribution is that this convergence holds uniformly with all its
derivatives on each compact subset of the manifold of models,
thereby drawing a path
between the algorithmic behavior of the rational approximation problem (number
and nature of critical points, decrease of error, behavior of the poles) and
that of the minimization of empirical means. This allows one to
translate in terms
of asymptotic behavior of the estimators virtually all properties
that are uniform with
respect to the order of the approximants, and without having to assume that the
``true'' system belongs to the class of models. Let us mention for instance
that uniqueness of a critical point in H^{2} rational approximation, in the
case where the system to approximate is nearly rational of degree n,
implies uniqueness of a local minimizer for the output error when
the input is a white noise, asymptotically almost surely on every compact,
when the density of y with respect to u is nearly rational of degree n.
In the case of relaxation systems, with one input-output, that is, if the
transfer function is a Markov function, we obtain, in the light of
the results exposed in module , the same
conclusion when the order of approximation is large enough.
This is the first known case of unimodularity where the ``true'' system does
not belong to the class of models. An application to the
localization of the poles of rational estimates of the output error of
a long memory system was derived from this .
Here, we are faced again with the
question, already mentioned in the introduction,
of how to expand functions in bases that are
adapted to the singularities of the spectral density of long memory processes.
We believe this research direction would be worth exploring.

In order to control a system, one generally relies on a model, obtained from
*a priori* knowledge like physical laws or experimental observations. In many
applications, one is satisfied with a linear approximation around a nominal
point or trajectory. It is however very important to study non-linear
systems (or models) for the following reasons. First, some
systems have, near interesting working points, a linear approximation that is
non-controllable so that linearization is ineffective, even
locally. Secondly, even if the linearized model is controllable, one may wish to
extend the working domain beyond the validity domain of the linear
approximation. Work described in module dwells on
such issues. Finally, certain control problems, such as path planning,
are not of a local nature and cannot be answered *via* a linear
approximation. The structural study described in module
aims at exhibiting invariants that can be used,
either to reduce the study to simpler systems or to make grounds
for a non-linear identification theory, that would give informations
on model classes to be used in case there is no *a priori* reliable
information and still the black-box linear identification is not
satisfactory. The
success of the linear model, in control or in identification, is due to
the deep understanding one has of it; in the same fashion, a refined
knowledge of invariants of non-linear systems under basic transformations is a
prerequisite for a theory of non-linear identification and control. In
what follows, all non-linear systems are supposed to have a state space of
finite dimension.

Stabilization by continuous state feedback — or output feedback, that is,
the partial information case — consists in designing a control law which is a
smooth (at least continuous) function of the state making a given
point (or trajectory) asymptotically stable for the closed system. One
can consider this as a weak version of the optimal control problem: to
compute a control that optimizes a given criterion (for instance to reach
a prescribed state in minimal time) leads in general to a very irregular
dependence on this state;
stabilization is a *qualitative* objective (*i.e.*
to reach that state
asymptotically) which is more flexible and allows one
to impose a lot more regularity.

Lyapunov functions are a well-known tool to study stability of
non-control dynamic systems. For a control system, a
*Control Lyapunov Function* is a Lyapunov function for the closed-loop
system where the feedback is chosen appropriately.
It can be expressed by a differential inequality
called the ``Artstein (in)equation '', that looks like
the Hamilton-Jacobi-Bellmann equation but is largely under-determined. One
can easily deduce from the knowledge of a control Lyapunov function
a continuous stabilizing
feedback.

The team is engaged in obtaining control Lyapunov functions for certain classes of systems. This should be the first step in synthesizing a stabilizing control, but even when such a control is known beforehand, obtaining a control Lyapunov function can still be very useful to study the robustness of the stabilization, or to modify the initial control law into a more robust one. 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 are exploited in the joint collaborative research conducted with Alcatel Space (see module ), 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.

A *static feedback* transformation of a dynamical control system is
a (non-singular) reparametrization of control, depending on the state, and
possibly, a change of coordinates in the state space. A *dynamic
feedback* transformation of a dynamic control system consists of a
dynamic extension (adding new states, and assigning then a new dynamics)
followed by a state feedback on the augmented system.

From the point of view of control, the interest of these transformations is that a command satisfying specific objectives on the transformed system can be used to control the original system including the possibly extended dynamics in the controller. Of course the favorable case is when the transformed system has a structure that can easily be exploited, for instance when it is a linear controllable system.

From the point of view of identification and modeling, in the non-linear case, the interest is either to derive qualitative invariants to support the choice of a non-linear model given the observations, or to contribute to a classification of non-linear systems which is missing sorely today.

These two aspects will now be developed.

The problem of dynamic linearization, still unsolved, is that of finding explicit conditions on a system for the existence of a dynamical feedback that would make it linear.

These last years , the following property of
control systems has been emphasized: for some systems (in particular
linear ones), there exists a finite number of functions of the state and of
the derivatives of the control up to a certain order,
that are differentially independent (*i.e.* coupled by no
differential equation) and
do ``parameterize all the trajectories''. This property and its importance in
control, has been brought in light in , where
it is called *differential flatness*, the above mentioned functions being
called *flat* or *linearizing functions*, and it was shown, roughly
speaking, that a system is differentially flat if, and only if, it can be
converted to a linear system by dynamic feedback. On one hand, this property
of the set of trajectories has in itself an interest at least as important for
control than the equivalence to a linear system, and on the other hand it
gives a handle for tackling the problem of dynamic linearization, namely to
find linearizing functions.

An important question remains open: how can one algorithmically decide
that a given system has this property or not, *i.e.* is dynamically
linearizable or not? This problem is both difficult and important for
non-linear control. For systems with four states and two controls, whose
dynamics is affine in the control (these are the lowest dimensions for which the
problem is really non-trivial), necessary and sufficient
conditions for the existence of linearizing functions
depending on the state and the control (but not the derivatives of the
control) can be given explicitly, but they do point to the complexity of the
issue.

From the algebraic-differential point of view, the module of differentials of
a controllable system is free and of finite dimension over the ring of
differential polynomials in d/dt with coefficients in the space of
functions of the system, and for which a basis can be explicitly
constructed . The question is to find out if it
has a basis made of closed forms, that is, locally exact forms.
Expressed in this
way, it is an extension of the classical integrability theorem of Frobenius
to the case where coefficients are differential operators. Together with
stability by exterior differentiation (the classical condition), further
conditions are required here to ascertain the degree of the
solutions is finite, the mid-term goal is to obtain a formal and implementable
algorithm, able to decide whether or not a given system is flat
around a regular point.
One can also consider sub-problems having their own interest, like deciding
flatness with a given pre-compensator, or characterizing ``formal'' flatness
that would correspond to a weak interpretation of the differential equation.
Such questions can be localized in the neighborhood of an equilibrium point.

In what precedes, we have not taken into account the degree of
*smoothness* of the transformations under consideration.

In the case of dynamical systems without control, it is well known
that, away from degenerate (non hyperbolic) points,
if one requires the transformations to be merely continuous,
every system is *locally* equivalent to a
linear system in a neighborhood of an equilibrium
(the Hartman-Grobman theorem).
It is thus tempting when
classifying *control* systems, to look for such
equivalence modulo non-differentiable transformations and to hope bring about some
robust ``qualitative'' invariants and perhaps stable normal forms.
A Hartman-Grobman theorem for control systems would say for instance, that
outside a ``meager'' class of models (for instance, those whose linear
approximation is non-controllable), and locally around nominal
values of the state
and the control, no qualitative phenomenon can distinguish a non-linear system
from a linear one, all non-linear phenomena being thus either of global
nature or singularities. Such a statement is wrong: if a system is locally
equivalent to a controllable linear system via a bi-continuous
transformation—a local homeomorphism in the state-control space—it is
*also* equivalent to this same controllable linear system via a
transformation that is as smooth as the system itself, at least in the neighborhood of a
regular point (in the sense that the rank of the control system is locally
constant), see for details; *a contrario*, under weak
regularity conditions, linearization can be done by non-causal
transformations (see the same report) whose structure remains unclear,
but acquire
a concrete meaning when the entries are themselves
generated by a finite dimensional dynamics.

The above considerations call for the following question, which is important for modeling control systems: are there local ``qualitative'' differences between the behavior of a non-linear system and its linear approximation when the latter is controllable?

The botton line of the team's activity is twofold, namely optimization in the frequency domain on the one hand, and the control of 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 or optical systems. For applications of the first type, approximation techniques as described in module 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 or optical regenerators. We describe all these below in more detail.

Localizing cracks, pointwise sources or occlusions in a two-dimensional material, using thermal, electrical, or magnetic measurements on its boundary is a classical inverse problem. It arises when studying fatigue of structures, behavior of conductors, or else magneto-encephalography as well as the detection of buried objects (mines, etc). However, no really efficient algorithm has emerged so far if no initial information on the location or on the geometry is known, because numerical integration of the inverse problem is very unstable. The presence of cracks in a plane conductor, for instance, or of sources in a cortex (modulo a reduction from 3D to 2D, see later on) can be expressed as a lack of analyticity of the (complexified) solution of the associated Dirichlet-Neumann problem that may in principle be approached using techniques of best rational or meromorphic approximation on the boundary of the object (see sections to and ,). In this connection, the realistic case where data are available on part of the boundary only is a typical opportunity to apply the analytic and meromorphic extension techniques developed earlier.

The 2D approach proposed here consists in constructing, from
measured data on a subset K of the boundary of a plane
domain D,
the trace on of a function F which is analytic in D except
for a possible singularity across some subset
(typically: a crack). One can then use
the approximation techniques described above in order to:

extend F to all if the data are incomplete
(it may happen that K) if the boundary is not
fully accessible to measurements),
for instance to identify an unknown Robin
coefficient, see where stability
properties of the procedure are established;

detect the presence of a defect in a computationally efficient manner, ;

Thus, inverse problems of geometric type that consist in finding an unknown
boundary from incomplete data can be approached this way
, usually in combination with other techniques . Preliminary numerical experiments have yielded
excellent results and it is now important to process
real experimental data, that the team
is currently busy analysing.
In particular, contacts with the Odyssée Team of Inria Sophia
Antipolis (within the ACI ``Obs-Cerv'') has provided us with
3D magneto-encephalographic data from which 2D information was extracted,
see section
. The team is also in contact with
other laboratories (*e.g.* Vanderbilt Univ. Physics Dept.)
in order to work out 2D or 3D data from physical experiments.

In the longer term, we envisage applying this type of methods to problems with variable conductivity, or to the Helmholtz equation. Using convergence properties of approximation algorithms in order to establish stability results for some of these inverse problems is also an appealing direction for future research.

One of the best training ground for the research of the team in function theory is the identification and design of physical systems for which the linearity assumption is well-satisfied in the working range of frequency, and whose specifications are made in frequency domain. Resonant systems, acoustic or electromagnetic, are prototypical examples of common use in telecommunications. We shall be more specific on two examples below.

Surface acoustic waves filters are largely used in modern telecommunications
especially for cellular phones. This is mainly due to their small
size and low cost. Unidirectional filters, formed of *Single Phase
UniDirectional Transducers* (in short: SPUDT)
that contain inner reflectors
(cf. Figure ), are increasingly used in
this technological area. The design of such filters is more
complex than traditional
ones.

We are interested here in a filter formed of two SPUDT transducers (Figure ). Each transducer is composed of cells of the same length each of which contains a reflector and all but the last one contain a source (Figure ). These sources are all connected to an electrical circuit, and cause electro-acoustic interactions inside the piezo-electric medium. In the transducer SPUDT2 represented on Figure , the reflectors are positioned with respect to the sources in such a way that near the central frequency, almost no wave can emanate from the transducer to the left (), this being called unidirectionality. In the right transducer SPUDT1, reflectors are positioned in a symmetric fashion so as to obtain unidirectionality to the left.

Specifications are given in the frequency domain on the amplitude and phase of the electrical transfer function. This function expresses the power transfer and can be written as

where Y is the admittance of the coupling:

The design problem consists in finding the reflection coefficients r
and the source efficiency in both transducers so as to meet
the specifications.

The transducers are described by analytic transfer functions called mixed matrices, that link input waves and currents to output waves and potentials. Physical properties of reciprocity and energy conservation endow these matrices with a rich mathematical structure that allows one to use approximation techniques in the complex domain (see module ) according to the following steps:

describe the set of electrical transfer functions obtainable from the model,

set out the design problem as a *rational approximation problem*
in a normed space of analytic functions:

where D is the desired electrical transfer,

use a rational approximation software (see module ) to identify the design parameters.

The first item, is the subject of ongoing research. It connects the geometry of the zeroes of a rational matrix to the existence of an inner symmetric extension without increase of the degree (reciprocal Darlington synthesis), see . Let us mention that the interest of the team for such filters was triggered by a collaboration with Thomson Microsonics.

In the domain of space telecommunications (satellite transmissions), constraints specific to onboard technology lead to the use of filters with resonant cavities in the hyperfrequency 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 away, 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 electrically-coupled 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
plug on a resistor at one end and fed with some potential at the other.
We are
then interested in the power 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 for expressing the length of the screws in terms of parameters of this
electrical model). On the other hand, this is also useful for the design of
the filter, for analyzing numerical simulations of the Maxwell equations, and
for checking the design, particularly the
absence of higher resonant modes.

In reality, resonance is not studied via the electrical model,
but via a low pass
equivalent 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, in H^{2} norm for the transmission
and in Sobolev norm for the reflection (the module of he response
being respectively
close to 0 and 1 outside the interval measurement) cf. module .
This gives a scattering matrix of order roughly 1/4 of the number of
data points.

Then one rationally approximate with fixed degree (8 in this example) via the hyperion software cf. module .

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 non-physical couplings. This is obtained by using algebraic-solvers and continuation algorithms on the group of orthogonal complex matrices (symmetry forces this kind of change of basis).

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 ). Non-physical coupling 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 unsymmetric filters, useful for the synthesis of repeating devices.

The team extends today its investigations, to the design of output multiplexors (OMUX) that couple several filters of the previous type on a manifold. The objective is to establish a global model for the behavior that would take into account:

within each channel the coupling between the filter and the Tee that connects it to the manifold,

the coupling between two consecutive channels.

The model is obtained upon chaining the transfer matrices associated to the scattering matrices. It mixes rational elements and complex exponentials (because of the delays) and constitutes an extension of the previous framework. Under contract with the CNES (see ), the team has started a study of the design with gauge constraints, based on function theoretical tools.

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 located and positioned (correct orientation). This problem and other similar ones continue to motivate research in control from the part of the team. Aerospace engineering in general is a domain that requires sophisticated control techniques, and where optimization is often crucial, due to the extreme conditions.

The team has been working for two years on control problems in orbital transfer with low-thrust engines, under contract with Alcatel Space Cannes, see module . 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, and fuel represents 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 non-robust against many unmodelled phenomena.

The increased capacity of numerical channels in information technology is a major industrial challenge. The most performing means nowadays for transporting signals from a server to the user and backwards is via optical fibers. The use of this medium at the limit of its capacity of response causes new control problems in order to maintain a safe signal, both in the fibers and in the routing and regeneration devices.

In a recent past, the team has worked in collaboration with Alcatel R&I (Marcoussis) on the control of ``all-optic'' regenerators. Although no collaboration is presently active, we consider this a potentially rich domain of applications

The works presented in module lie upstream with respect to applications. However, beyond the fact that deciding whether a given system is linear modulo an adequate compensator is clearly conceptually important, it is fair to say that ``flat outputs'' are of considerable interest for path planning . Moreover, as indicated in section , a better understanding of the invariants of non-linear systems under feedback would result in significant progress in identification.

The development of a

RARL2 (Réalisation interne et Approximation Rationnelle L2) is a software for rational approximation (see module ). Its web page is http://www-sop.inria.fr/miaou/RARL2/rarl2.html. This software takes as input a stable transfer function of a discrete time system represented by

either its internal realization

or its first N Fourier 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 germane to the arl2 function of hyperion from which it
differs mainly in the way
systems are represented: a polynomial representation is used in hyperion, while
RARL2 uses realizations, this being very interesting in certain cases. It is
implemented in MATLAB.
This software handles *multi-variable* systems (with several inputs and
several outputs), and uses a parameterization that has the following
advantages

it incorporates the stability requirement in a buit-in manner,

it allows the use of differential tools,

it is well-conditioned, 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 module ) by generating, in a structured manner, several initial conditions. Contrary to the polynomial case, we are in a singular geometry on the boundary of the manifold on which minimization takes place, which forbids the extension of the criterion to the ambient space. We have thus to take into account a singularity on the boundary of the approximation domain, and it is not possible to compute a descent direction as being the gradient of a function defined on a larger domain, although the initial conditions obtained from minima of lower order are on this boundary. Thus, determining a descent direction is nowadays, to a large extent, a heuristic step. This step works well in the cases handled up to now, but research is under way in order to make this step truly algorithmic.

The identification of filters modeled by an electrical
circuit that was developed inside the team (see module )
has led to compute the electrical parameters of the
filter. This means finding a particular realization (A, B, C, D) of the model
given by the rational approximation step. This 4-tuple 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. However if
the desired realization is not in arrow form, one can show that it can be
deduced by an orthogonal change of basis (in general complex). 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 non-convex
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. Work is in progress,
see section .

PRESTO-HF: a toolbox dedicated to lowpass parameter identification for hyperfrequency filters http://www-sop.inria.fr/miaou/Fabien.Seyfert/Presto_web_page/presto_pres.html

In order to allow the industrial transfer of our methods, a Matlab-based toolbox has been developed, dedicated to the problem of identification of low-pass hyperfrequency filter parameters. It allows to run the following algorithmic steps, one after the other, or all together in a single sweep:

determination of delay components, that are caused by the access devices (automatic reference plane adjustment);

automatic determination of an analytic completion, bounded in module for each channel, (see module );

rational approximation, of fixed McMillan degree;

determination of a constrained realization.

For the matrix-valued rational approximation stage Presto-HF relies either on hyperion (Unix or Linux only) or RARL2 (platform independent), both rational approximation engines were developed within the team. Constrained realizations are computed by the RGC software. As a toolbox, Presto-HF 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 Alcatel Space in Toulouse.

The great novelty in the RAWEB2002 (Scientific Annex to the Annual Activity
Report of Inria), was the use of XML as intermediate language, and the
possibility of bypassing

The construction of the raweb is explained schematically on figure
. The input is either a *Tralics* software,
described in module ; it was originally a Perl script,
nowadays it is a C++ executable. An XSLT processor
(for instance `xsltproc`, from the Gnome tools) is used to convert
the XML either into HTML, or into an XSL-FO document, by adding some
formatting instructions (in this phase, we explain for instance that the
text font should be Times). This file is formatted by `xmltex` package that teaches

In the original version, one could instruct *Tralics* to produce the XML
output, or to convert it also to HTML or Pdf. One could also ask for a
direct PostScript version (by-passing the XML phase). This is now governed
by a Perl script, called `rahandler.pl`. One can modify this script
(for instance, change the name or the pathname of the XSLT processor,
or the location of the SGML catalog file); this is now the recommended
procedure (of course, it is still possible to specify in the *Tralics*
configuration file these names, which are transmitted to the script).
The raweb package uses a Makefile to call *Tralics* without options,
and then all other tools, (in this case `rahandler.pl` is unused).

As a byproduct, all bibliographical references of years 2000 to 2003 have been translated to XML, sorted by authors, type, year, and put on the web (currently the internal server http://www.inria.fr/interne/disc/).

One important issue was the choice of the DTD (*document type definition*).
On one hand, it should follow the pseudo-DTD as defined for the RAWEB
six years ago (the Activity Report is a set of modules, with contributors,
key-words, etc), and on the other hand, it must be as close as possible to
standard DTDs. We have decided to use a variant of the TEI
(*text encoding initiative*,
see http://www.tei-c.org/) for the
text, MathML for the mathematics, and an ad-hoc DTD for the bibliography.
This DTD was modified in 2004, independently of *Tralics*.
In other words, on Figure , a new arrow has to be added:
it goes from the old DTD to the new one.

The main difficulty comes from the mathematics: consider a formula like
. This is translated
by *Tralics* into a formula that contains a script X,
coded as `<mi>𝒳</mi>`. After conversion to the new DTD, entities
are replaced by Unicode characters, so that the X becomes
`<mi>𝒳</mi>`. This character seems to be unknown by browsers like
Amaya or Mozilla, and is rendered by a question mark or a little box
containing the Unicode value (here 01D4B3). This is one of the reasons why
math formulas are still replaced by images; in the case `$x+\alpha$`,
only the is converted; this has the advantage to reduce the number
of images, but in some cases is not very elegant.

Conversion is done by a dedicated Perl script that extracts from the XML
file all formulas, and converts them to a set of pages in a dvi file
(we use here the same algorithm for converting the XML to PostScript).
Each page is converted to an image via pstoimg, which is a Perl code,
part of latex2html.
We try to associate each image an Alt field that describes the formula, but
this is difficult: for the example we get
`${\#119987 _y=lim_{x\#8594 0}sin^2{(x)}}$`.

The *Tralics* software is a C++ written

A second application is the following:
when researchers wish to publish an Inria Research Report, they send their
PostScript or Pdf document, together with the start of the *Tralics*, using a special
configuration file, that extracts only the title page information
(author names, abstract, etc). A perl script removes useless pieces, and
produces an HTML notice (see for instance
http://www.inria.fr/rrrt/rr-5316.html).
As you can see, math formulas like `$(2^n+1)$`, `$n\times n$`
are output more or less verbatim, by changing the `\catcode` of some
characters, and by redefining all Greek letters and symbols like `\times`.

The main philosophy of *Tralics* is to have the same parser as `\chardef`,
`\catcode`, `\ifx`, `\expandafter`, `\csname`, etc.,
that are not described in the `\documentclass`. All element names (except p) can be changed by the user.

This year we added constructions like `\newcolumntype` (in fact,
we re-implemented completely the array package, although it is still
impossible to put arrays into arrays in the Raweb),
`\newtheorem` (with all the bells and whistles), and a lot of Unicode
characters (there are 1700 commands defined nowadays). We also changed
the output of font changing commands: instead of an element named `hi`
with an attribute `rend` that could be bold, italic, etc, one can ask
*Tralics* to output an element named, for instance, `bold`.
We added commands to manipulate MathML objects. Given the following piece
of code

` \providecommand\operatorname[1]{\mathmo{#1}`

` \mathattribute{form}{prefix}\mathattribute{movablelimits}{true}}`

` \def\Dmin{\operatorname{dmin}}`

you can use `\Dmin` like `\min`, for instance min_{x}f(x)>dmin_{x}f(x).

There are some unsolved problems: for instance, a figure environment should contain only graphics together with a single caption, commands defined by the picture environment are translated (but refused by the style sheet), non-math material in a math formula is rejected (unless it is formed of characters only).

For more information, see the Tralics web page. It contains a description of each command. A reference manual is in preparation.

The magnetic field produced by a magnetic dipole located at a point is

The problem is to identify the location ,
and the momentum of a sequence of N
dipoles indexed by k = 1, ..., N,
given measurements from
a SQUID (superconducting quantum interference device). The
assumption that z_{k} is independent of k (*i.e.* all dipoles lie in a
plane) is made, and we assume also that is parallel to the
z-axis for all k. In this case the previous formula simplifies to

The effect of the pick-up coil needed by the SQUID can be modeled
by averaging over a small disk, of radius a. Thus we measure a quantity of the
form

These parametrization issues have been studied for several years in the
project. The main motivation was to find good optimization parameters for
our approximation problems .
Atlases of charts have been derived from a matrix Schur algorithm associated
with Nevanlinna-Pick interpolation data. In a chart, a lossless function
can be represented by a balanced realization computed as a product of
unitary matrices from the Schur parameters . Moreover, an
adapted chart for a given lossless
function can be built from a realization in Schur form.
Such a parametrization presents a lot of advantages in view of the
approximation problems we have in mind: it ensures
identifiability, takes into account the stability constraint, preserves the
order and presents a good numerical behavior.
This parametrization has been used in the software RARL2 which
deals with rational approximation in L^{2} norm.

This year, we studied atlases built from a more general interpolation problem, the contour integral interpolation problem of Nudelman. These atlases were introduced last year to deal with real-valued systems. We paid a particular attention to an atlas which uses a favorable mutual encoding property of lossless functions . It works for both real and complex systems and has been implemented in a new version of the software RARL2 .

Following a different approach based on so-called *nice selections*,
balanced canonical forms were constructed which have the property
that the corresponding controllability matrix is positive upper-triangular,
up to a column permutation. Such canonical forms
present, in particular, good truncation properties. It was proved that
this atlas can also be obtained from a tangential Schur algorithm by choosing
the interpolation points at zero and the directions in a well-specified
manner among standard basis vectors . This atlas is
minimal in the sense that no chart can be left out without losing the
property that the atlas cover the manifold.

The objective of these studies is to have at one's disposal a panel of parametrizations that could be used as optimization parameters and also that could take into account some particular property coming from the physics. This could be symmetry or some other constraint on the realization matrix like for example the structure imposed by the couplings of an hyperfrequency filter .

In view to enlarge the field of application of these parametrizations we started a collaboration with the Delft Center for Systems and Control. The idea was to implement our parametrizations in the software of stochastic identification that they developed so far, and to test the advantages and disadvantages of our different parametrizations on their problems. Another possible field of application is multiobjective control. An approach developed by the robust control community, consists in transforming the multiobjective control problem into a tractable LMI problem whenever the lossless factor of the Youla parameter is known. Then, the search of this lossless factor is limited to a particular form which makes the optimization easier. We think that this approach could be significantly improved using our parametrizations to optimize within the set of all lossless factors of fixed degree.

Surface Acoustic Waves (in short: SAW) filters consist in a series of transducers which
transmit electrical power by
means of surface acoustic waves propagating on a piezoelectric medium. They
are usually described by a mixed scattering matrix which
relates acoustic waves, currents and voltages. By reciprocity and energy
conservation, these transfers must be either lossless, contractive or
positive real, and symmetric.
In the design of SAW filters, the desired electrical power transmission is
specified. An important issue is to characterize the functions that can
actually be realized for a given type of filter.
In any case, these functions are Schur and can be completed into a
conservative matrix with an increase of at most 2 of the McMillan degree,
this matrix describing the global behavior of the filter.
Such a completion problem is known as Darlington synthesis
and has always a solution for any higher McMillan degree
in the rational case if the
symmetry condition is of no concern. However in our case,
additional constraints arise from the geometry of the filter as the
symmetry and certain interpolation condition. In ,
a complete mathematical description of such devices is given,
including realizations
for the relevant tranfer-functions, as well as a necessary and sufficient
condition for symmetric Darlington synthesis preserving the McMillan degree.
More generally, we characterized in the existence
of a symmetric Darlington synthesis with specified increase of
the McMillan degree: a symmetric extension of a symmetric contractive matrix
S of degree
n exists in degree n + k if, and only if,
I-SS^{*} has at most k zeros with odd multiplicity. This results tells
something about the minimal number of gyrators to be used in
circuit synthesis; an article is currently being written to report on these
results.

There are natural links
between meromorphic approximation of Markov functions and the
n-widths of the unit ball of H^{p} in L^{q}(), because
the extremal functions are essentially the singular vectors
of the Hankel operator associated with the approximation problem
with exponent s such that 1/s + 2/p = 1. Previous
work when
pq generalizes the asymptotics obtained by
Fisher and Stessin for the n-width, and we have taken up this year the
study of the case where the support of the measure is a general regular
compact set of the disk in an attempt to carry over the
relation between embedding operators and rational or meromorphic approximation
to 2-D singular sets. This is important for applications to the detection of
occlusions in conducting media, see section
. We have shown the accumulation
points of the zeros of singular vectors of such operators are carried by the
support of the measure, which is an important step to carry out the
asymptotics of the singular values. These were obtained when the singular
set is a disk, and more general cases are currently under study.

The study of matrix-valued rational approximation to
matrix-Markov functions (*i.e.* Cauchy transforms of a
positive matrix valued measure) has been pursued, with the aim of proving that
every critical point is Markov (former work of the team showed that the best
approximant is Markov). There are no new results in this direction so far.
We were able, however, to generalize the error rates known in the scalar case
under Szegö-type conditions to the matrix case.

It is known after that the denominators of best rational
of meromorphic approximants in the L^{p} norm on a closed curve (say the
unit circle T to fix ideas) satisfy for p2 a
non-Hermitian orthogonality relation for
functions described as Cauchy transforms of complex measures on a
curve (locus of singularities) contained in the unit disk
D. This has been used to assess the asymptotic behavior of the poles of
such an approximant when is a hyperbolic geometric arc,
that is, under weak conditions on the measure, the counting measure of these
poles converges weak-star to the equilibrium distribution of the
condenser (T, ) where T is the unit circle.
Non asymptotic bounds have been obtained for the sum of the complement to
of the hyperbolic angles under which the poles ``see'' :
the sum of these complements over all the poles (they are n in total if
the approximant has degree n) is bounded by the aperture of
plus twice
the variation of the argument of the measure (which is independent of n).
This produces ``hard'' testable inequalities for the location of the poles,
that should prove particularly valuable in inverse source problem
(because they are not asymptotic in nature), see
module . This research has been the object of an
article .

The more general situation where is a so-called
``minimal contour'' for the Green
potential (of which a geodesic arc is an example) has been essentially settled
with the same conclusion concerning the convergence of the counting measure of
the poles. The writing up of this (rather technical) result is underway, and
of particular significance with respect to the determination of 2-D sources
or piecewise analytic cracks from overdetermined boundary data, see
module and .

Another issue which especially interesting in connection with crack detection is the behavior of the poles in the case of a more general (not necessarily piecewise analytic) crack. We have shown that, using conformal maps of a ringed domain, one is led to a question similar to that on a geodesic arc but with a less regular measure, and we conjecture that the poles at least accumulate towards the endpoints of the crack. The proof, however, is still far from complete and will require further efforts. A pending issue is also the behavior of poles for 2D singular sets. No results in this direction were obtained so far.

The fact that 2D harmonic functions are real parts of analytic functions allows one to tackle issues in singularity detection and geometric reconstruction from boundary data of solutions to Laplace equations using the meromorphic and rational approximation tools developed by the team. Some electrical conductivity defaults can be modeled by pointwise sources inside the considered domain. In dimension 2, the question made significant progress in recent years: the singularities of the function (of the complex variable) which is to be reconstructed from boundary measures are poles (case of dipolar sources) or logarithmic singularities (case of monopolar sources). Hence, the behavior of the poles of the rational or meromorphic approximants, described in modules to , allows one to efficiently locate their position. This, together with corresponding software implementation, is part of the subject of the Ph.D. thesis of F. Ben Hassen and of an article to appear , where the related situation of small inhomogeneities connected to mine detection is also considered.

In 3D, epileptic regions in the cortex are often represented by pointwise sources that have to be localized from potential measures on the scalp of a potential difference, that is the solution of a Laplace equation (EEG, electoencephalography). Note that the head is here modeled as a sequence of spherical layers. This inverse EEG problem is the object of a collaboration between the Apics and Odyssée Teams through the ACI ``Obs-Cerv''. An interesting breakthrough was made last year which makes it possible now to proceed via best rational approximation on a sequence of 2D disks along the inner sphere , . The point here is that, up to an additive function harmonic in the 3D ball, the trace of the potential on each boundary circle coincides with a function having branched singularities in the corresponding disk. The behavior along the family of disks of the poles of their best rational approximants on each circle is strongly linked to the location of the sources, using properties discussed in sections and . (in the particular case of a unique source, we end up with a rational function); this is under study as well as a number of important related issues.

First, solving Cauchy problems on an annulus or on a spherical layer in order to treat incomplete experimental data is also a necessary ingredient of the methodology, since it is involved in the propagation of initial conditions from the boundary to the center of the domain, where singularities are seeked, when this domain is formed of several homogeneous layers of different conductivities. On a spherical layer, this is the aim of the post-doctoral trainee of B. Atfeh (it has also been preliminary handled by J. Chetboun and C. Aziadjonou, together with uniqueness issues for constant conductivities). Constructive and numerical aspects of the expected procedures (harmonic 3D projection, Kelvin and Riesz transformation, spherical harmonics) are under study and encouraging results are already available on numerically computed data. This offers an opportunity to state and solve extremal problems for harmonic fields.

We also 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 cross-sections are quadrature domains or R-domains. In this framework, best rational approximation can still be performed in order to recover the singularities of solutions to Laplace equations (F.-O. Helme, summer trainee) but complexity issues have to be examined carefully.

Finally, we begin to consider actual 3D approximation for such inverse problems. Quaternionic analysis seems to be an appropriate tool, but much of the theory (in particular the multiplicative side) remains to be developed.

In the 2D case again, with incomplete data, the geometric problem
of finding, in a stable and constructive way, an unknown (insulating) part of
the boundary of a domain is considered in the Ph.D. thesis of
I. Fellah. Approximation and analytic extension techniques
described in section together with numerical conformal
transformations of the disk
provide here also interesting algorithms for the inverse problem under
consideration. A related result that was obtained this year is an L^{p}
existence
and uniqueness result for the Neumann problem on a piecewise
domain with inward pointing cusps (note that
the endpoints of a crack are such cusps) when 1<p<2. Although it is
reminiscent of classical L^{p} theorem on Lipschitz domains ,
it seems to be a new result (observe that Lipschitz domains cannot have cusps)
exploiting weighted norm inequalities. Moreover, a Cauchy-type representation
for the solution was obtained using Smirnov classes representation properties,
and the technique generalizes to mixed boundary conditions that occur when the
crack is no longer assumed to be a perfect insulator. An article
is being written on these aspects.

Finally, solving Cauchy problems on an annulus is the main theme of the PhD thesis of M. Mahjoub. This arises when identifying a crack in a tube or a Robin coefficient on its inner skull. It can be formulated as a best approximation problem on part of the boundary of a doubly connected domain, which allowed both numerical algorithms and stability results to be obtained in this framework , thereby generalizing the simply connected situation , .

To carry out identification and design of filters under passivity constraints
(such constraints are common since passive devices are ubiquitous,
including in particular hyperfrequency filters), it is natural to consider the
mixed bounded extremal problem *()*
stated in section . An algorithm to asymptotically
solve this problem in nested spaces of polynomials has been obtained, and its
connection to affine Rieman-Hilbert problems has also been carried out. This
connection provides a handle to analyze regularity properties of the solution,
and gives us an alternative process based on dichotomy.
It should be valuable to estimate delays in waveguides, and could complement
the existing procedures dealing with this issue in PRESTO-HF.

We studied in some generality the case of parameterized linear systems characterized by the following classical state space equation,

where is a finite set of r parameters and
(A(p), B(p), C(p)) are matrices whose entries are polynomials (over the field
) of the variables
. For a parameterized system and we call
the transfer function of the system (p). Some
important questions in filter synthesis concern the determination
of the following parameterized sets

General results were obtained about these sets, in particular a
necessary and sufficient condition ensuring their
cardinals are finite. In the special case of coupled-resonators an
efficient algebraic formulation has been derived which allowed
us to compute for nearly all common filter geometries.
However for a new class of high order filters first presented in
the latter procedure breaks down because of the computational complexity of
the Gröbner basis computation. This led us to consider homotopic methods
based on continuation techniques in order to solve the algebraic system
defining . The usual framework of these methods that is based
on the Bezout bound or on mixed volume computations appeared to be
intractable in our case mainly because of the degeneracy of our algebraic
systems: for example for a 10^{th} order filter, the Bezout bound is about
10^{44} whereas the number of solutions over the ground field is
known to be only 384. To overcome this difficulty we are currently developing
a continuation method which consists of the exploration of the monodromy
group of an algebraic variety by following a family of paths that separate
the branch points. This method is still under study but preliminary numerical
results that yielded the exhaustive computation of in the
latter 10^{th} order case are very encouraging. Using this method, we
envisage to build up a precomputed filter database that would
allow a fast computation of high order filters for every specific filtering characteristic.

Results were also obtained about the existence of a ``real solution'' in the set in the case of loss-less characteristics. For the 5^{th} order coupling topology of figure it was shown that one can find an open set U of for which for all pU the set contains no ``real'' element. Conversely it was shown, by an argument based on the Borsuk-Ulam antipodal theorem, that for lossless characteristics and the 6^{th} order coupling topology of figure there generically exists at least one ``real'' element in .

An OMUX (Output MUltipleXor) can be modeled in the frequency domain by chaining
of scattering matrices of filters as those described in
section , connected in parallel to a common access *via* a
wave guide, see figure . The problem of designing
the OMUX so as to satisfy
gauge constraints is then naturally translated into a set of constraints on the
values of the scattering matrices and phase shift introduced by the guides
in the considered bandwidth.

An OMUX simulator on a matlab platform was designed last year, and we began this year a study of dedicated optimization procedures. The direct approach, as used by the manufacturer, is of course to couple this simulator with an optimizer, in order to reduce transmission and reflection wherever they are too large. This yields unsatisfactory results in cases of high degree and narrow bandwidth, in particular because peaks arise with the dilation of the cavities caused by an increase of the temperature (when the satellite gets exposed to the sun).

The thesis of V. Lunot has just started on the problem of
approximation by Schur rational functions (*i.e.* rational functions of
modulus bounded by 1 in a half-plane), for this issue arises naturally in
connection with the hyperbolic distance induced by chain scattering matrices.
We hope that solving iteratively approximations problems of this type
with all but one channel being fixed can lead to a tuning of
each filter in a diagonal manner. This is an interesting question, both for
applications and in itself.

As a result, we expect to be able to produce a multi-phased tuning procedure, first relaxed, channel after channel, then global, using a quasi-Newton method. Note the discretizations in frequency of the integral criterion and the near periodicity of the exponentials (that express the delays) interact in a complex manner, and generate numerous local minima, which is one reason why the optimization problem should be analyzed in depth.

We focus on what we call the controlled Keplerian problem. A satellite is equipped with a low thrust engine and we want to perform an orbital transfer, i.e. to drive the satellite from an initial orbit to a final one. This problem was raised by Alcatel Space. As the old ballistic commands do not work for the low thrust we have to explore new techniques and chose to pursue our ideas on time optimal GTO-GEO transfer.

One of the achievements this year, in the course of the PhD work of A. Bombrun, was to build a family of control Liapunov functions based on the first five integrals of the classical two body problem (this is not new), in such a way that a given transfer (say time-optimal) can be approached very closely by fitting correctly the parameters in the family (this is the novelty). In numerical simulations we show such a Liapunov feedback not only is close to the given time optimal trajectory, but also gives very satisfactory trajectories for different initial conditions. We are trying to show that we can approach any orbital transfer within this family. Of course, this control law enjoys the natural robustness properties of feedback control.

We also proposed a strategy to achieve the full rendez-vous (transfer plus longitude assignment). This is not a full closed-loop strategy, but performs well.

We designed tools for analyzing some over-determined
systems of PDEs for which neither the number of independent variables nor
the order is *a priori* fixed.
It is based on a valuation adapted to the control system .

The equations arising when characterizing flatness involve a number of
variables which is finite, but not known *a priori*... so it is tempting to take as solutions
formal power series in infinitely many variables. The tools above allow us
to give a meaning to solutions in such formal power series.
A notion of ``very'' formal integrability was introduced, meaning existence of
solutions in this class.

We are not yet able to use this tool for all ``equations'' of flatness, so
that the equations we write in are only
*necessary conditions* for flatness.
However, if, for some systems, these equations
were not very formally integrable, then these systems would not be flat.
Unfortunately we did prove that these equations are *always* very
formally integrable.
Putting a full characterization of flatness in this form is still under course.

This is the topic of David Avanessoff's PhD.

Here we studied the smallest nontrivial dimensions, i.e. three states and two
controls.
One can prove that this is the same as studying systems with four states and two controls, whose
dynamics are affine in the control.
In , a sufficient condition for such systems to be flat was
given, and this condition was also proved to be necessary for
``(x, u)-flatness'' (in the language of the above paragraph, a version of
flatness where the number of variables to consider is decided in advance).
It is conjectured that it is in fact necessary for
flatness itself, and even that systems that do not satisfy this condition do
not admit a parameterization.

The proof in was very intricate and used computer algebra
at one point in the argument.
Using a different approach we are able to prove, in a much simpler way, that
systems that do not satisfy this condition do not admit a parameterization of
order less than 4 (this implies that they are not ``(x, u)-flat''; the
systems proved to be flat in admit a parameterization of
order 1), and in fact we hope to conclude, by this method, that these systems indeed do not
admit a parameterization (it was out of question with the
method in ).
This is also the topic of David Avanessoff's PhD.

Controllability results for drifted systems are usually obtained by a combination of local and global properties of the system under study. Local controllability properties basically follow from the knowledge of the Lie bracket configuration of the system, while global ones require particular symmetries or some sort of ergodicity. A typical example is the one of a left-invariant control system on a Lie group. Classically, the homogeneity of the manifold given by the group structure is used to obtain global properties out of local ones.

The aim of this research line is to obtain controllability/non-controllability results for special but inhomogeneous drifted systems.

The main object of our research is given by Dubbins-like systems
on Riemannian surfaces.
The goal is to answer, using control techniques, the following natural
question, arising from the works of Dubbins:
given a complete, connected, two-dimensional Riemannian
manifold M, and (p_{1}, v_{1}), (p_{2}, v_{2}) in TM, does there exist a curve
in M, with arbitrary small geodesic curvature, such that
connects p_{1} to p_{2} and, for i = 1, 2, is
equal to v_{i} at p_{i}?
The answer clearly depends on the geometric properties of M, and
gives a meaning to such properties from a control viewpoint. In we proved that the small-curvature-connectness introduced above
holds for compact surfaces, for unbounded surfaces whose Gaussian curvature tends to zero at infinity, and for surfaces which are non-negatively curved outside a compact set.

The case of non-positively curved surfaces was addressed in , where necessary and sufficient conditions ensuring such connectness have been established.

A different field of application of the analysis of controllability of inhomogeneous drifted systems is given by non-linear switched systems.
More precisely, given a switched system of the type , u[-1, 1], qR^{2}, where X + Y and X-Y are globally asymptotically stable, we study its stability properties (global uniform asymptotic stability, uniform stability, boundedness, ...) in terms of the topology of the set where X and Y are parallel. All such stability properties can be re-interpreted in terms of the behavior of attainable sets.

Contracts n^{o}1 03 E 1034

In the framework of a contract that links CNES, IRCOM and Inria, whose objective is to realize a software package for identification and design of hyperfrequency devices, the work of Inria included:

modeling of delay, see module ,

exhaustive coupling determination on case studies ),

OMUX simulator with exact computation of derivatives,

This contract has been renewed for 16 months starting November 2004, in order to develop a generic code for coupling determination and to carry out the optimization of OMUX.

Contract n^{o}1 01 E 0726.

This contract started in 2001, for three years. The topic is the design of control laws for satellites with low-thrust engines,

It finances Alex Bombrun's PhD.

L. Baratchart is member of the editorial board of *Computational
Methods in Function Theory*.

Together with project-teams Caiman and Odyssée
(INRIA-Sophia Antipolis, ENPC), the University of Nice (J.A. Dieudonné lab.),
CEA, CNRS-LENA (Paris), and a few French hospitals, we are part of the
national action **ACI Masse de données « OBS-CERV »**, 2003-2006 (inverse
problems, EEG).
C. Aziadjonou and J. Chetboun were supported by this ACI.

The **region PACA** (Provence Alpes Côte d'Azur) has been partially
supporting
the post-doctoral stay of Per Enquist until May, 2004. We also obtained a (modest) grant from
the region for exchanges with SISSA Trieste (Italy), 2003-2004.

The post-doctoral training of B. Atfeh is funded by **INRIA**.

The team enjoys a **Marie Curie EIF** (Intra European Fellowship)
FP6-2002-Mobility-5-502062, for 24 months (2003-2005). This finances Mario
Sigalotti's post-doc.

The Team is a member of the **Marie Curie multi-partner training site***Control Training Site*, number HPMT-CT-2001-00278, 2001-2005. See
http://www.supelec.fr/lss/CTS/.

The project is member of Working Group Control and System Theory
of the **ERCIM** consortium, see
http://www.ladseb.pd.cnr.it/control/ercim/control.html.

**NATO CLG** (Collaborative Linkage Grant), PST.CLG.979703,
« Constructive approximation and inverse diffusion problems », with
Vanderbilt Univ. (Nashville, USA) and LAMSIN-ENIT (Tunis, Tu.), 2003-2005.

**EPSRC** grant (EP/C004418) «Constrained approximation in
function spaces, with applications», with Leeds Univ. (UK) and
Univ. Lyon I.

**STIC-INRIA** grant with LAMSIN-ENIT (Tunis, Tu.), « Problèmes
inverses du Laplacien et approximation constructive des fonctions ».

The following scientists visited us and gave a seminar:

Amina Amassad, UNSA, Nice, et équipe APICS,
*Contribution à l'analyse et au contrôle des inclusions différentielles en mécanique du contact*.

Alexander Aptekarev, Keldysh Institute of Applied Mathematics, Moscow,
*Family of equilibrium measures and variational principle for Burger's
equation*.

Bilal Atfeh, équipe APICS,
*Méthode des lignes de courant appliquée à la modélisation
des bassins*.

Ugo Boscain, SISSA, Trieste, Italy,
*Stability of planar switched systems for arbitrary switchings*.

Imen Fellah, Lamsin-ENIT, Tunisie,
*Complétion de données dans les espaces de Hardy et problèmes inverses
pour le Laplacien en 2D*.

Stanislas Kupin, Université de Provence,
*Comportement asymptotique de polynômes orthogonaux sur le cercle
d'après la régularité des coefficients*.

Moncef Mahjoub, Lamsin-ENIT, Tunisie,
*Complétion de données dans une couronne
et ses applications à quelques problèmes inverses*.

Sylvain Neut, LAIL, Univ de Lille,
*Implantation et nouvelles applications de la
méthode d'équivalence de Cartan*.

Laurent Niederman, Département de Mathématiques,
Université Paris-Sud, Orsay,
*Stabilité Hamiltonienne et théorie de Morse-Sard*.

Laurent Praly, Centre Automatique et Systèmes, Ecole des mines de Paris,
*Diverses stratégies de synthèse de commande stabilisante pour
le transfert d'orbite*.

Witold Respondek, Laboratoire de Mathématiques INSA de Rouen,
*Bifurcations des systèmes non linéaires de contrôle sur le plan*.

Pierre Rouchon, Centre Automatique et Systèmes, Ecole des Mines de Paris,
*Invariant observer for mechanical systems*.

Edward B. Saff, Dept. of Mathematics, Vanderbilt University,
*Discretizing manifolds via minimum energy points*.

Mario Sigalotti, équipe APICS,
*Dubbins' problem on surfaces of nonpositive curvature*.

Nikos Stylianopoulos,
Department of Mathematics and Statistics, University of Cyprus,
*Conformal mapping of elongated domains with applications
to the solution of Laplacian problems*.

Jan H. Van Schuppen, CWI/VU,
*Control and realization of piecewise-affine hybrid systems*.

Igor Zelenko, SISSA, Trieste, Italy,
*Variational approach to problem of equivalence of
rank 2 vector distributions*.

L. Baratchart, DEA Géométrie et Analyse, LATP-CMI, Univ. de Provence (Marseille).

Jonathan Chetboun and Christelle Aziadjonou,
*Problème inverse en Electroencéphalographie*

Frank-Olivier Helme (Mémoire de DEA de Mathématiques pures, Université de Provence, Aix-Marseille I.)
*Résolution de problèmes inverses de source dans des domaines
paramétrés en dimension 3 par approximation méromorphe*

Jean-Michel Guieu (Mémoire de DEA de Mathématiques pures, Université de Provence, Aix-Marseille I.)
*Comportement asymptotique des pôles dans l'approximation méromorphe
des fonctions analytiques en dehors d'un compact du disque*

David Avanessoff, « Linéarisation dynamique des systèmes non linéares et paramétrage de l'ensemble des solutions » (dynamic linearization of non linear control systems, and parameterization of all trajectories).

Alex Bombrun, « Commande optimale, feedback, et tranfert orbital de satellites » (optimal control, feedback, and orbital transfert for low thrust satellite orbit transfer)

Imen Fellah, ``Data completion in Hardy classes and applications to inverse problems'', co-tutelle with Lamsin-ENIT (Tunis).

Vincent Lunot, « Optimisation et synthèse d'OMUX »,

Moncef Mahjoub, ``Complétion de données et ses application à la détermination de défauts géométriques.'' co-tutelle with Lamsin-ENIT (Tunis).

Fehmi Ben Hassen, «Recovery and identification of pointwise sources and small size inclusions», Lamsin-ENIT, Univ. Tunis II (Tunisie), December 11th 2004.

M. Sigalotti is in charge of organizing a seminar on control and identification.

L. Baratchart is a member of the ``bureau'' of the CP (Comité des Projets) of INRIA-Sophia Antipolis.

J. Grimm is a member of the CUMI (Comité des utilisateurs des moyens informatiques) of the Research Unit of Sophia Antipolis (dissolved in September 2004).

J.-B. Pomet is a representative at the ``comité de centre'' (until October 2004).

J. Grimm is a representative at the ``comité de centre'' (Starting October 2004).

J. Leblond is co-directing - together with J.-D. Fournier (CNRS, Obs. Nice) - and J. Grimm is participating in the edition of the proceedings (to appear in 2005) of the CNRS-INRIA summer school ``Harmonic analysis and rational approximation: their rôles in signals, control and dynamical systems theory'' (Porquerolles, 2003) http://www-sop.inria.fr/apics/anap03/index.en.html.

M. Olivi is a member of the CSD (Comité de Suivi Doctoral) of the Research Unit of Sophia Antipolis.

F. Seyfert is a member of the CDL (Comité de développement logiciel) of the Research Unit of Sophia Antipolis.

J.-B. Pomet was an invited researcher at Banach center in Warsaw, for one month in January.

J.-B. Pomet was an invited speaker at the Colloquium on Dynamical Systems, Control and Applications, organized by Univ. Autonoma de Mexico, December, Mexico City.

F. Seyfert was an invited speaker at the ``International Workshop on Microwave Filters'', co-organized by the CNES and ESA, September, Toulouse.

David Avanessoff and Mario Sigalotti gave talks at the ``2nd Junior European Meeting Control Theory and Stabilization'', Torino, It.

Mario Sigalotti gave a talk at the ``First CTS Workshop'', University of Coimbra, Portugal, July 1-3.

Juliette Leblond was invited to give a plenary talk at the Forum des Jeunes Mathématiciennes, IHP, Paris (January), gave a communication at Constructive Functions Tech. 04, Georgia Tech., Atlanta, Georgia, USA (November), and at the annual workshop of the ACI ``Obs-Cerv'', ENPC, Champs sur Marne (September).

M. Olivi was invited to give a talk at the Delft Center for systems and control of the Delft University of Technology.

M. Olivi gave two talks at the SSSC 2004, Oaxaca, Mexico, 8-10 December.

A. Bombrun gave a talk at a meeting on spatial mechanics at CNES, Toulouse, in September.

J.-B. Pomet gave a talk at a meeting on geometrical methods and PDEs at CIRM, Luminy, in November.

F. Seyfert gave a talk at the ``Journées nationales du calcul formel 2003'' about the use of computer algebra based methods for the exhaustive computation of couplings parameters, at ``Advances in constructive approximation (Nashville)'' about the mixed bounded extremal problem and at the ``IMS 2003 (Philadelphia)'' about the determination of a rational stable model from measured scattering data.

L. Baratchart was an invited speaker at the ``Funktionentheorie'' meeting, Oberwolfach, feb. 2004, at the "Journee Approximation", Lille, march 2004, and at the ``Tech '04 Function Theory'' Conference, November 2004, Atlanta (Georgia). He was a regular speaker at the MTNS, Leuwen, July 2004.